**Capillary Bridges — A Tool for Three-Phase Contact Investigation**

Boryan P. Radoev, Plamen V. Petkov and Ivan T. Ivanov

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/60684

#### **Abstract**

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Subject of investigation are capillary bridges (CB) between two parallel solid plates normally oriented to the gravity field. Presented are results of study of CB with negligible gravity effects and CB undergoing observable gravitational deforma‐ tions. Among the discussed problems some new aspects of the CB behavior are for‐ mulated. One of them is the so-called stretching thickness limit, i.e. the maximal thickness above which a CB of given volume and contact angles cannot exist. It is shown that the stretching thickness limit of a concave CB substantially differs from that of a convex one. Analysis of the forces acting on CB plates is presented. It clear‐ ly demonstrates that the gravity part of the forces, relative to the part of capillary forces, increases with stretching. Most of the observed effects are interpreted on the basis of the two CB radii of curvature analysis, thus avoiding the ponderous proce‐ dures of obtaining (integrating) the CB generatrix profile. The success of this ap‐ proach lies in its combination with image analysis of CB profile. Discussed are the contact angle hysteresis effects at CB stretching and pressing.

**Keywords:** capillary bridge, image analysis, contact angle hysteresis

## **1. Introduction**

As every classical subject, the capillary bridge (CB) has its centuries-old history and continuous reincarnation in the science and praxis. A characteristic feature of the classical subject is the

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richness of problems they produce. CB related human activity goes back to the ancient times [1, 2]. Nowadays CB provide well-known effects in atomic force microscopy [3, 4] and lithography [5-8]. CB also play important role in soil-water interactions [9-11].

The scientific interest of CB originates from the mathematical problem of construction of figures of minimal area, defined by the French astronomer and mathematician C. Delaunay [12]. He found a new class of axially symmetric surfaces of constant mean curvature. Much later Kenmotsu [13] solved the complex nonlinear equations, describing this class of surfaces. However, the solution that he had proposed has little practical importance because the representation there has no geometrical interpretation.

An important step in the analysis of minimal surfaces is the understanding of the role of their curvatures. Today it is well known that a surface of minimal area has a constant mean curvature and vice versa [12], but the formulation and the proof of this theorem have their long story. It begins with the remarkable figure proposed by Euler [14], named later catenoid [15]. In the mathematical studies, the catenoid is defined as a figure of revolution with zero mean curvature but for the rest of the natural sciences it is popular as an object, formed by a soap film stretched between two parallel rings. The fundamental question here is what physical feature of soap films determines their minimal areas. The answer of this question is the surface tension, nowadays given at school, but it took almost a whole (18th) century for that answer. Many scientists have contributed to the elucidation of the problem, two of them: Pierre Simon Laplace (1749-1827) and Thomas Young (1773-1829) [16] are of the greatest merit. To these two scientists and especially to one of them, Laplace, we owe the introduction of the notion of capillary tension, which opened the way of modern investigation of capillary phenomena. Moreover, Laplace has practically formulated the currently used condition for mechanical equilibrium between two fluids divided by a capillary surface, *p*<sup>σ</sup> =∆*p*. Here *p<sup>σ</sup>* =*σ*(1 / *R*<sup>1</sup> + 1 / *R*2) is the capillary pressure with surface tension *σ*, radii of curvature are *R*1,2 and the pressure difference between the adjacent bulk phases is ∆*p* (isotropic fluids). The sum (1 / *R*<sup>1</sup> + 1 / *R*2) is twice the so-called mean curvature. Note that the cited above Laplace formula has its gener‐ alized form for anisotropic fluids (e.g. liquid crystals) [17]. Among all parameters of the Laplace formula, most interesting are the radii of the curvature R1,2. They are the tool for investigation and analysis of the variety of capillary shapes. The analytical interpretation of the mean curvature is based on the differential geometry but its combination with the direct geometrical meaning of the radii proves to be very fruitful. For surfaces of revolution, this geometrical meaning becomes very transparent. Along with that, the analytical expressions of their radii R1,2 take on specific form facilitating their further analysis. Expressive examples in this respect with wide application in the academic and applied research are CBs [18, 19]. As a geometric subject, surfaces of revolution are formed by rotation of a curve (called generatrix) around a given axis. From here it follows that each of their cross-sections with a plane perpendicular to the axis of revolution is a circle. In the reality, two solid plates play the role of the parallel planes. This simple geometric figure imposes strict requirements regarding the solid CB plates. More about the solid plates, used in our experiments, can be found in Section 4. They must be: (1) macroscopically smooth (without roughness) and (2) homogeneous (with constant surface energy). An axial symmetrical CB can be formed between many combinations of solid surfaces (e.g. between two co-axial spheres, cone and plane, etc.) but the prevailing part of the re‐ searches are focused exclusively on bridges between parallel flat plates. As we will discuss the role of gravity, the axis of revolution must be parallel to the gravity field. A general survey of a CB in a gravitational field at arbitrary orientation and inertia fields can be found in [20]. The smoothness and homogeneity of solid surfaces are genetically related to the so-called threephase contact (TPC) angle hysteresis. In more details, this topic will be discussed in Sections 3-4. Here we will make only a short remark. A widely spread opinion is that hysteresis is a result of surface roughness and heterogeneity, i.e. on macroscopic smooth and homogeneous surfaces hysteresis should not be observed [21]. As an illustration of this opinion usually the undisputed lack of hysteresis of oil droplet on water (immiscible liquids) is considered. Liquid surfaces are naturally smooth and homogeneous but in the same time they (actually the Newtonian liquids) have a very specific rheological property, they do not bear shear stress at equilibrium in contrast to the solids. This difference is crucial for the dynamics of the systems. A droplet on liquid will be set in motion by any nonzero (tangential to the three phase contact) force, while a droplet on solid could stay at rest, as the solid substrate can react to the opposed force by static shear stress [22, 23]. Here it is important to remind the experimentally proven nonslip model in the fluid dynamics. The nonslip model is a special case of the more general concept of continuum (e.g. [24]). According to this model, within fluids as well as their boundaries, the velocity field is continuous. This continuity model is based on the intermo‐ lecular (van der Waals) forces' existence. More precisely, on the presumption that the part of an external force per intermolecular region is negligible with respect to the van der Waals forces [25]. Hydrodynamic problems connected with the TPC line motion have made some authors to give up the nonslip presumption [26, 27, 28, 29]. Most popular are two alternatives: the hypothesis of local slippage in the TPC vicinity [30] and the model based on Eyring theory [31, 32]. A weakness of these approaches is neglecting of the so-called surface forces [33, 34]. These forces result directly from intermolecular forces and become significant in thin gaps (thickness < 10 nm) between two surfaces, i.e. just in the TPC zones, where the hydrodynamic singularities arise. The interested reader can find more about the role of surface forces for the wetting in [35]. In our experiments we have observed hysteresis in a wide range: from practically pinned contacts (strong hysteresis) to contacts with nearly constant contact angles and we have tried to interpret it in the framework of nonslip convention.

richness of problems they produce. CB related human activity goes back to the ancient times [1, 2]. Nowadays CB provide well-known effects in atomic force microscopy [3, 4] and

The scientific interest of CB originates from the mathematical problem of construction of figures of minimal area, defined by the French astronomer and mathematician C. Delaunay [12]. He found a new class of axially symmetric surfaces of constant mean curvature. Much later Kenmotsu [13] solved the complex nonlinear equations, describing this class of surfaces. However, the solution that he had proposed has little practical importance because the

An important step in the analysis of minimal surfaces is the understanding of the role of their curvatures. Today it is well known that a surface of minimal area has a constant mean curvature and vice versa [12], but the formulation and the proof of this theorem have their long story. It begins with the remarkable figure proposed by Euler [14], named later catenoid [15]. In the mathematical studies, the catenoid is defined as a figure of revolution with zero mean curvature but for the rest of the natural sciences it is popular as an object, formed by a soap film stretched between two parallel rings. The fundamental question here is what physical feature of soap films determines their minimal areas. The answer of this question is the surface tension, nowadays given at school, but it took almost a whole (18th) century for that answer. Many scientists have contributed to the elucidation of the problem, two of them: Pierre Simon Laplace (1749-1827) and Thomas Young (1773-1829) [16] are of the greatest merit. To these two scientists and especially to one of them, Laplace, we owe the introduction of the notion of capillary tension, which opened the way of modern investigation of capillary phenomena. Moreover, Laplace has practically formulated the currently used condition for mechanical equilibrium between two fluids divided by a capillary surface, *p*<sup>σ</sup> =∆*p*. Here *p<sup>σ</sup>* =*σ*(1 / *R*<sup>1</sup> + 1 / *R*2) is the capillary pressure with surface tension *σ*, radii of curvature are *R*1,2 and the pressure difference between the adjacent bulk phases is ∆*p* (isotropic fluids). The sum (1 / *R*<sup>1</sup> + 1 / *R*2) is twice the so-called mean curvature. Note that the cited above Laplace formula has its gener‐ alized form for anisotropic fluids (e.g. liquid crystals) [17]. Among all parameters of the Laplace formula, most interesting are the radii of the curvature R1,2. They are the tool for investigation and analysis of the variety of capillary shapes. The analytical interpretation of the mean curvature is based on the differential geometry but its combination with the direct geometrical meaning of the radii proves to be very fruitful. For surfaces of revolution, this geometrical meaning becomes very transparent. Along with that, the analytical expressions of their radii R1,2 take on specific form facilitating their further analysis. Expressive examples in this respect with wide application in the academic and applied research are CBs [18, 19]. As a geometric subject, surfaces of revolution are formed by rotation of a curve (called generatrix) around a given axis. From here it follows that each of their cross-sections with a plane perpendicular to the axis of revolution is a circle. In the reality, two solid plates play the role of the parallel planes. This simple geometric figure imposes strict requirements regarding the solid CB plates. More about the solid plates, used in our experiments, can be found in Section 4. They must be: (1) macroscopically smooth (without roughness) and (2) homogeneous (with constant surface energy). An axial symmetrical CB can be formed between many combinations of solid surfaces

lithography [5-8]. CB also play important role in soil-water interactions [9-11].

representation there has no geometrical interpretation.

24 Surface Energy

Most of the problems discussed in this chapter are well known. They concern not only CB between two flat parallel solid surfaces but also other specific CB types (Appendix C). Some nontrivial results obtained here are related to the study of the CB upper stretching limit. Usually this problem is discussed from the stability viewpoint but it should be treated as a critical point. The critical point is defined as a boundary equilibrium state, i.e. a system cannot exist outside the critical point. Of course, this state could be unachievable, if it is preceded by instable states. As known from the theory [36], the stability concerns the reaction of the system at perturbation of a given equilibrium state. If the reaction is a tendency to return the system in the equilibrium, this state is stable, otherwise the state is unstable. A typical example, illustrating the difference between critical and instable states, is a CB with cylindrical form, i.e. a CB with a 90° contact angle. From an equilibrium viewpoint, a cylindrical CB can be stretched without limitations, i.e. *H*cr → ∞ (*H* – CB thickness). On the other hand, cylindrical CB have stability limit *Н*stab=π*R* (*R −* cylinder radius), the so-called Rayleigh instability, above which a cylindrical CB becomes unstable [37, 38]. A more detailed analysis of CB critical points is provided in Section 2. Actually, the stability problems are beyond the scope of this study.

## **2. Definition of the subject**

Subject of investigation, as already mentioned, is a capillary bridge between two parallel solid plates, normally oriented to the gravity field (Figure 1). Only axisymmetric bridges are considered, i.e. CBs with circle three phase contacts. Although the analysis is performed for equal contact angles on the upper/lower plate (θ+=θ–;*R*+=*R*–), it is shown that most of the theoretical results are also applicable for different contact angles/radii − Section 2.

**Figure 1.** *Sketch of capillary bridge in gravity field*. *Left*: CB geometric parameters; *Right*: CB dynamic parameters, *G –* CB weight, *F*<sup>±</sup> *–* external forces, *G* <sup>→</sup> <sup>+</sup> *<sup>F</sup>*<sup>+</sup> <sup>→</sup> <sup>+</sup> *<sup>F</sup>*<sup>−</sup> <sup>→</sup> =0 (about the notations see in the glossary and text)

The processes of evaporation, condensation and the related potential temperature effects [39, 40, 41] won't be discussed. Water bridges have shown observable evaporation, which was neglected, because the evaporation rates were low enough, thus ensuring quasi-static states of the CB, Section 4. Moreover, the scaled forms of the theoretical results are invariant to the bridge volume, Eq. (8). Concerning the RTIL bridges, they are practically non-volatile.

As known, the mechanical equilibrium of a capillary system obeys the pressure balance, *p<sup>σ</sup>* =*Δp*, where *p*<sup>σ</sup> is the capillary pressure and ∆*p* is the pressure difference of both sides of the capillary surface. Because of the axial symmetry, *p<sup>σ</sup>* <sup>=</sup>*σ<sup>r</sup>* <sup>−</sup><sup>1</sup> *d*(*r*sin*φ* / *dr*), where *r* and *φ* are current coordinates. In this symmetry, the separate curvatures (1/*R*1,2) are defined as follows: 1 / *R*<sup>1</sup> =sin*φ* /*r* (the so-called azimuthal curvature); 1 / *R*<sup>2</sup> =*d*sin*φ* / *dr* (meridional curvature). The angle *φ* is defined between the normal vector at a given point of the generatrix and the vertical axis *z*, Figure 1 left. In this study the CB pressure difference is defined as *Δp* = *pi* − *pe* where *p*i/e are internal/external pressures, Figure 1 right. The external pressure is practically constant (*pe* = *const*), while the inside pressure varies with height (hydrostatic pressure), *pi* = *p*<sup>0</sup> −*ρgz*. The precise expression of the gravitational term is *Δρgz*, where *Δρ* =*ρliquid* −*ρair* ≈*ρliquid* ≡*ρ* (the air density is negligible compared to liquid density). We use traditional notations as: *ρ* − density and *g* – gravity of earth. The inside pressure *p*0 at level *z*=0 deserves special attention. In its turn, the zero vertical position is a question of choice, Sections 2 and 3. Inserting all briefly explained quantities in the pressure balance, one obtains [18, 42]:

CB have stability limit *Н*stab=π*R* (*R −* cylinder radius), the so-called Rayleigh instability, above which a cylindrical CB becomes unstable [37, 38]. A more detailed analysis of CB critical points is provided in Section 2. Actually, the stability problems are beyond the scope of this study.

Subject of investigation, as already mentioned, is a capillary bridge between two parallel solid plates, normally oriented to the gravity field (Figure 1). Only axisymmetric bridges are considered, i.e. CBs with circle three phase contacts. Although the analysis is performed for equal contact angles on the upper/lower plate (θ+=θ–;*R*+=*R*–), it is shown that most of the

**Figure 1.** *Sketch of capillary bridge in gravity field*. *Left*: CB geometric parameters; *Right*: CB dynamic parameters, *G –* CB

The processes of evaporation, condensation and the related potential temperature effects [39, 40, 41] won't be discussed. Water bridges have shown observable evaporation, which was neglected, because the evaporation rates were low enough, thus ensuring quasi-static states of the CB, Section 4. Moreover, the scaled forms of the theoretical results are invariant to the bridge volume, Eq. (8). Concerning the RTIL bridges, they are practically non-volatile.

As known, the mechanical equilibrium of a capillary system obeys the pressure balance, *p<sup>σ</sup>* =*Δp*, where *p*<sup>σ</sup> is the capillary pressure and ∆*p* is the pressure difference of both sides of the

coordinates. In this symmetry, the separate curvatures (1/*R*1,2) are defined as follows: 1 / *R*<sup>1</sup> =sin*φ* /*r* (the so-called azimuthal curvature); 1 / *R*<sup>2</sup> =*d*sin*φ* / *dr* (meridional curvature). The angle *φ* is defined between the normal vector at a given point of the generatrix and the vertical axis *z*, Figure 1 left. In this study the CB pressure difference is defined as *Δp* = *pi* − *pe* where *p*i/e

<sup>→</sup> =0 (about the notations see in the glossary and text)

*d*(*r*sin*φ* / *dr*), where *r* and *φ* are current

theoretical results are also applicable for different contact angles/radii − Section 2.

**2. Definition of the subject**

26 Surface Energy

weight, *F*<sup>±</sup> *–* external forces, *G*

<sup>→</sup> <sup>+</sup> *<sup>F</sup>*<sup>+</sup> <sup>→</sup> <sup>+</sup> *<sup>F</sup>*<sup>−</sup>

capillary surface. Because of the axial symmetry, *p<sup>σ</sup>* <sup>=</sup>*σ<sup>r</sup>* <sup>−</sup><sup>1</sup>

$$
\sigma \frac{1}{r} \frac{d}{dr} (r \sin \phi) = \Delta p\_0 - \rho \mathbf{g} \mathbf{z},
\tag{1}
$$

with *Δ p*<sup>0</sup> = *p*<sup>0</sup> − *pe*. The main mathematical complication in Eq. (1) comes from the non-linear relation between sin*φ* and the generatrix equation *z*(*r*), sin*φ* =(*dz* / *dr*) / 1 + (*dz* / *dr*) 2 . At the same time, the use of sin*φ* as variable show advantages which are repeatedly demonstrated in the present study.

A preparatory and very important step for the successful solution and analysis of any differential equation is its scaling. Particularly the scaled form of Eq. (1) reads:

$$\frac{1}{\infty} \frac{d}{d\mathbf{x}} (\mathbf{x} \sin \varphi) = P\_0 - Bo \,\mathbf{y},\tag{2}$$

where *x* ≡*r* / *L <sup>r</sup>* ; *y* ≡ *z* / *L <sup>z</sup>* ; *P*<sup>0</sup> ≡(*Δ p*0)*L <sup>r</sup>* / *σ* ; *Bo* ≡*ρgL <sup>r</sup>L <sup>z</sup>* / *σ*. Generally scaling lengths (here, *Lr*, *Lz*) are derived from the dimensions of the system and are chosen so that the order of the scaled (dimensionless) variable is of the order of unity. So, for instance, in the case of a CB, the vertical scaling length obviously is the bridge height, i.e. *Lz*=*H*. To know more about the concrete scaling procedures, see in Sections 2 and 3. The dimensionless parameter *Bo*, known as Bond number [43, 20] is decisive for the role of gravity in the capillary systems. Most often the criterion for neglecting the effects of gravity is written as *Bo*<<1 but as it can be seen from Eq.(2), the correct condition is *Bo*<<*P*0. More generally, the scaled form Eq. (2) is the stepping stone for solving the generatrix equation *z*(*r*) in *Bo* powers series, *z*(*x*)= *z*0(*x*) + *z*1(*x*)*Bo* + ... for arbitrary values of *Bo*<*P*0. Another often noticed misleading usage of the Bond number concerns the scaling lengths *Lk*. Usually they are reduced to only one length, which in the case of a CB is most often the contact radius, i.e. *Bo* ≡*ρgR* <sup>2</sup> / *σ* [44, 45]. Eq. (1) also allows an interesting comparative analysis of the two curvatures 1 / *R*1,2, Section 3. As an illustration of such analysis, let's consider the curvatures in the CB waist/haunch point. The azimuthal curvature reduces to 1 /*rm* (*r*m, waist/haunch radius, Figure 1), while the meridional curvature preserves its form (*d*sin*φ* / *dr*). Now let us imagine a flattening of CB at constant contact angles. It is obvious that the azimuthal curvature will diminish (*r*m grows), while the meridional curvature *d*sin*φ* / *dr* increases. Moreover, in Section 2 it is shown that *d*sin*φ* / *dr* →*∞* at *H/R* → 0, Appendix A. In conclusion, it follows that the balance, Eq. (1) allows different type of approximations, e.g. a weightless CB (*Bo*=0, Section 2), a heavy CB (*Bo*≠0, Section 3); 2*D*/3*D* bridges (Section 3).

## **3. Capillary bridge in the absence of gravity**

This section is devoted to a CB for which the role of gravity is negligible compared with the capillary pressure. According to Eq. (2), the pressure balance in this case takes the form [42]:

$$\frac{1}{\chi} \frac{d}{d\chi} (\chi \sin \varphi) = P\_0,\tag{3}$$

with a criterion for its validity *Bo* < < *P*0. The CB Bond number is expressed as *Bo* =*ρgL <sup>r</sup>H* / *σ*, where *H* is the bridge thickness and *Lr* is the radial scaling length (Section 1). As shown at Figure 1, there are two variants for choice of *Lr*: the contact radius *R* or the haunch/neck radius *r*m and below it will be shown that *r*m is the preferable one. Before that, let's briefly consider the factors determining the *Bo* magnitude. The parameters in *Bo* can be grouped in two parts, a physical part (ρ*g*/σ) and a geometric (*LrH*) one. The physical complex (ρ*g*/σ), often related to the so-called capillary length, *L <sup>σ</sup>* ≡ *σ* / *ρg* is a constant for a given liquid. For instance the capillary length of water is of the order of millimeters (σ≈70 mN/m; ρ≈10<sup>3</sup> Kg/m3 ; *g*≈10 m/s<sup>2</sup> ), *L <sup>σ</sup>* ≈1 *mm*. The variable parameters in our experiments are the geometric lengths, *Lr* and *H*. They are not independent but coupled, roughly speaking, as *L <sup>r</sup>* <sup>2</sup> ~1 / *H* (constant volume, *V* ~ *L <sup>r</sup>* 2 *H* ). From here it follows that *Bo* ~ *H* , i.e. at CB flattening (thinning) the role of gravity decreases.

The first integral of the pressure balance Eq. (3) causes no difficulty and using the boundary condition *φ*(*r* =*r*m) =90° , one obtains,

$$\alpha \sin \varphi = C \left(\mathbf{x}^2 - \mathbf{l}\right) + \mathbf{l},\tag{4}$$

Note that here, in accordance with the above-mentioned boundary condition, the scaling length is the waist/haunch radius *r*m and the dimensionless parameters are *x* ≡*r* /*rm* and *C* ≡*Δ p*0*rm* / 2*σ* respectively. The notation *C* of the scaled capillary pressure instead of *P*0 is to emphasize the difference between ∆*p*0 of heavy and weightless CB. In presence of gravity (heavy CB), ∆*p*<sup>0</sup> is referred to the CB bottom while in absence of gravity (weightless CB) ∆*p*0 is a global characteristic. The scaled capillary pressure *C* plays central role in the entire CB analysis below. Very indicative, for instance is its relation with the TPC radii *R* and angles*θ*. Applying Eq. (4) to the upper/lower contact (*x*=*X*±*)* one obtains:

$$C = \frac{X\_{\pm} \sin \theta\_{\pm} - 1}{X\_{\pm}^{\prime 2} - 1},\tag{5}$$

where, *X*<sup>±</sup> ≡*R*<sup>±</sup> /*rm*. Actually Eq. (4) holds for any cross section of CB with a plane (normal oriented to the axis of symmetry) relating a (*X*,θ) couple at the respective level *y*. This general consideration allows us for the sake of simplicity to use Eq. (5) in form *C*(*X*,θ), Figure 2. Relation (5) clearly shows the algebraic character of *C*, i.e. the algebraic character of the capillary pressure *P*γ. It can be positive, zero or negative. At convex generatrix (*X*<1), in the entire interval of angles (*π* / 2<*θ* ≤*π*), the capillary pressure is positive (*C* > 0). The intercept points of the dashed line *C*=1 with the isogons (Figure 2) correspond to spherical CB forms, i.e. circular arc generatrix curves. As seen from Eq. (5), the spherical CB parameters satisfy the condition *X* =sin*θ*. An exception is the point *C*(*X*=0)=1, which is a peculiar end point of all convex CB (*θ*>π/2) isogons. As it will be discussed below in this section, approaching (by stretching) *X* → 0, i.e. at *r*<sup>m</sup> → 0, the convex isogon generatrix *y*(*x*) pass through inflexion point (d2 y/dx2 =0, Eq. (10)). Another noteworthy region of *C* vs. *X* diagrams is the line *X*=1. It is the asymptote of concave/convex isogons (C → ±∞) and the discontinuity point {C=1/2, *X*=1} lies also on it. The regions C → ±∞ concern the so-called thin CB, Appendix A, while the point {C=1/2, *X*=1} acquires cylindrical shape (contact angle *θ*=π/2). Cylindrical CB is an attractive capillary subject because of its simple form, making the stability problems very transparent [46].

**3. Capillary bridge in the absence of gravity**

This section is devoted to a CB for which the role of gravity is negligible compared with the capillary pressure. According to Eq. (2), the pressure balance in this case takes the form [42]:

with a criterion for its validity *Bo* < < *P*0. The CB Bond number is expressed as *Bo* =*ρgL <sup>r</sup>H* / *σ*, where *H* is the bridge thickness and *Lr* is the radial scaling length (Section 1). As shown at Figure 1, there are two variants for choice of *Lr*: the contact radius *R* or the haunch/neck radius *r*m and below it will be shown that *r*m is the preferable one. Before that, let's briefly consider the factors determining the *Bo* magnitude. The parameters in *Bo* can be grouped in two parts, a physical part (ρ*g*/σ) and a geometric (*LrH*) one. The physical complex (ρ*g*/σ), often related to the so-called capillary length, *L <sup>σ</sup>* ≡ *σ* / *ρg* is a constant for a given liquid. For instance the

*L <sup>σ</sup>* ≈1 *mm*. The variable parameters in our experiments are the geometric lengths, *Lr* and *H*.

The first integral of the pressure balance Eq. (3) causes no difficulty and using the boundary

Note that here, in accordance with the above-mentioned boundary condition, the scaling length is the waist/haunch radius *r*m and the dimensionless parameters are *x* ≡*r* /*rm* and *C* ≡*Δ p*0*rm* / 2*σ* respectively. The notation *C* of the scaled capillary pressure instead of *P*0 is to emphasize the difference between ∆*p*0 of heavy and weightless CB. In presence of gravity (heavy CB), ∆*p*<sup>0</sup> is referred to the CB bottom while in absence of gravity (weightless CB) ∆*p*0 is a global characteristic. The scaled capillary pressure *C* plays central role in the entire CB analysis below. Very indicative, for instance is its relation with the TPC radii *R* and angles*θ*.

> 2 sin 1, <sup>1</sup> ± ± q

where, *X*<sup>±</sup> ≡*R*<sup>±</sup> /*rm*. Actually Eq. (4) holds for any cross section of CB with a plane (normal oriented to the axis of symmetry) relating a (*X*,θ) couple at the respective level *y*. This general consideration allows us for the sake of simplicity to use Eq. (5) in form *C*(*X*,θ), Figure 2. Relation

± - <sup>=</sup> -

*<sup>X</sup> <sup>C</sup>*

( ) <sup>2</sup> *x Cx* sin 1 1,

j

Applying Eq. (4) to the upper/lower contact (*x*=*X*±*)* one obtains:

*H* ). From here it follows that *Bo* ~ *H* , i.e. at CB flattening (thinning) the role of gravity

<sup>1</sup> ( sin ) , j<sup>=</sup> *<sup>d</sup> x P*

capillary length of water is of the order of millimeters (σ≈70 mN/m; ρ≈10<sup>3</sup>

They are not independent but coupled, roughly speaking, as *L <sup>r</sup>*

*V* ~ *L <sup>r</sup>* 2

28 Surface Energy

decreases.

condition *φ*(*r* =*r*m) =90° , one obtains,

0

*x dx* (3)

= -+ (4)

*<sup>X</sup>* (5)

Kg/m3

<sup>2</sup> ~1 / *H* (constant volume,

; *g*≈10 m/s<sup>2</sup>

),

**Figure 2.** *Left* – Capillary pressure, *C vs X,* calculated from Eq. (5); *Right* – arbitrary cross section illustrating a convex CB with constant *C* but different contact angles θ, θ<sup>i</sup> .

Zero capillary pressure (*C*=0) defining a catenoid state is one of the most popular capillary figures. We have already mentioned about it in the Introduction and here will add only the condition for its realization, *X* sin*θ* =1, Eq. (5), illustrated by the cross point of the dashed horizontal line *C*=0 with the isogons *θ* <*π* / 2 (Figure 2 left). The region *X* >1 corresponds to stretching of concave CB. Below in the text it will be shown that there are critical values of (*X, θ)* above which concave CB do not exist.

The external forces, *F*<sup>±</sup> supporting a CB are another very important characteristic, which can be obtained via elementary tools. At negligible gravity, *F*± are equal in absolute value but oppositely directed, *F*<sup>+</sup> <sup>→</sup> <sup>+</sup> *<sup>F</sup>*<sup>−</sup> <sup>→</sup> =0 (Figure 1). As derived in Appendix B, due to the gravity, the relation between *F*± gets more complicated. In contrast to liquid|gas surface where the mechanical equilibrium is described by a local balance, Eq. (3), the forced balance on solid| liquid; solid|gas can be derived only globally. Due to the force acting on the entire solid area and on the TPC contact contour, one obtains [42].

$$F\_{\pm} = \pm (2\pi R\_{\pm} \sigma \sin \theta\_{\pm} - \pi R\_{\pm}^2 \Delta p\_0) \tag{6}$$

Because of the trivial symmetry, we will further analyse only the balance on the upper plate, thus omitting the subscript (+). The term 2*πRσ*sin*θ* is the normal (toward the contact plane) component of the TPC force and the term *πR* <sup>2</sup> *Δ p*0 is the capillary pressure force. Upon expressing ∆*p*0 through *С* from Eq. (5), we finally obtain *F*:

$$F = 2\pi\sigma R \frac{X - \sin\theta}{X^2 - 1}.\tag{7}$$

Like the capillary pressure, Eq. (5), the force *F* can alter its sign or become zero. At concave CB, since *X* ≥1, *F* is positive for all angles in the interval 0≤*θ* ≤*π* / 2 and acts as a stretching force. At convex CB (*X*<1) the external force *F* is positive at (sin*θ* − *X* )>0, zero (*F*=0) in the point sin*θ* = *X* , (spherical CB, Eq. (4) at *C*=1) and negative pressing force (*F*<0) at *X*>sin *θ*.

The complete analysis of CB behavior requires integration of Eq. (4). In essence, this integration yields the generatrix equation *y*(*x*) which in our notations can be presented as [47]:

$$\text{If } y(\mathbf{x}, C) = \frac{RI\_0(\mathbf{x}, C)}{X}, \text{ with } I\_0(\mathbf{x}, C) = \pm \int\_1^\mathbf{x} \frac{1 + C\left(\boldsymbol{\xi}^2 - 1\right)}{\sqrt{\boldsymbol{\xi}^2 - \left[1 + C\left(\boldsymbol{\xi}^2 - 1\right)\right]^2}} d\boldsymbol{\xi}, \tag{8}$$

The integral *I*0(*x*,*C*) describes the upper part of the generatix curve, i.e. at 0 ≤*y*≤ *H*/2*r*m (above the neck, Figure 1 left). Note that here, for the sake of convenience, the co-ordinate system is established on the CB neck/haunch with (scaled) radial coordinate defined in the interval 1≤*x*≤*X* (=*R*/*r*m) for concave CB and *X*≤*x*≤1 for convex CB respectively. The sign '±' accounts whether the CB is concave (positive sign, *X* >1, 0≤*θ* <*π* / 2), or convex (negative sign, *X* <1, *π* / 2<*θ* ≤*π*). Further the signs of *I*0 will be omitted, given the correct sign in every particular situation. Traditionally *I*0 is presented by (Legendre's) elliptic integrals first and second kind *F*, *E* (e.g. [18]), but for its evaluation we apply another calculation scheme (see further).

As seen *I*0 is integrable, but singular (in the lower limit *ξ*=1), which gives rise to significant instability of the numerical results. We have settled the issue by dividing the integrals into singular and regular parts. The singular part allows direct integration, while the regular part is estimated numerically and as a result, for *x*=*X* one obtains:

$$I\_0\left(X,C\right) = \frac{\pi}{4C} - \frac{1}{2C} \arcsin\left[\frac{\left(1 - 2C\right) - 2C^2\left(X^2 - 1\right)}{\left(1 - 2C\right)}\right] - \int\_1^\cdot \sqrt{\frac{\xi - \left[C\left(\xi^2 - 1\right) + 1\right]}{\xi + \left[C\left(\xi^2 - 1\right) + 1\right]}} d\xi,\tag{9}$$

In practice, the evaluation of *I*0, according to Eq. (9), is performed by assigning a series of values of *X* at a fixed contact angle *θ*. The computation procedure is split into two subintervals *X* >1, 0≤*θ* <*π* / 2 (concave CB) and *X* <1, *π* / 2<*θ* ≤*π* (convex CB). We have used for *X* step of ∆*X* = 0.05; the angles subject to computation were: 15°, 30°, 45°, 60°, 89°; 91°, 100°, 120°, 179° (Figure 3, Figure 4).

Most of the evaluations here concern experimentally measured quantities. As all measure‐ ments are optical, it is preferable the experimental data to be presented in a form invariant with respect to the optical magnification. An appropriate presentation is H/*R* vs *X*, very convenient for interpretation by the theoretical relation following from Eq. (8) at *x* = *X* , *H* / *R* =2*I*0(*X* , *C*) / *X* . Formally the ratio (*H*/*R*) is function of *X* and *C*, but the parameter *C* is function of *X* and contact angles *θ*± Eq. (5) which transforms (*H*/*R*) equivalently in a function of *X* and *θ*, *H* / *R* = *f* (*X* , *θ*). It's important to note that the additional condition of constant volume *V* =2*πrm* 3 *∫* 1 *X ξ* 2 *d I*0(*ξ*, *C*)=*const* is necessary for the determination of the radii *R*

#### or *r*m separately [47].

8

2

) *F R Rp* ± ± ±± =± - D (6)

*Δ p*0 is the capillary pressure force. Upon

*<sup>X</sup>* (7)

( )

( ) ( )

x

 x

 x x

(8)

2

x

*C*

+ -

1 1

é ù -+ - ê ú ë û

 x

<sup>0</sup> <sup>2</sup> 2 2 <sup>1</sup>

x

<sup>0</sup> (2 sin

Because of the trivial symmetry, we will further analyse only the balance on the upper plate, thus omitting the subscript (+). The term 2*πRσ*sin*θ* is the normal (toward the contact plane)

> 2 sin 2 .

Like the capillary pressure, Eq. (5), the force *F* can alter its sign or become zero. At concave CB, since *X* ≥1, *F* is positive for all angles in the interval 0≤*θ* ≤*π* / 2 and acts as a stretching force. At convex CB (*X*<1) the external force *F* is positive at (sin*θ* − *X* )>0, zero (*F*=0) in the point

The complete analysis of CB behavior requires integration of Eq. (4). In essence, this integration

1 1 , , , with , ,

ò

The integral *I*0(*x*,*C*) describes the upper part of the generatix curve, i.e. at 0 ≤*y*≤ *H*/2*r*m (above the neck, Figure 1 left). Note that here, for the sake of convenience, the co-ordinate system is established on the CB neck/haunch with (scaled) radial coordinate defined in the interval 1≤*x*≤*X* (=*R*/*r*m) for concave CB and *X*≤*x*≤1 for convex CB respectively. The sign '±' accounts whether the CB is concave (positive sign, *X* >1, 0≤*θ* <*π* / 2), or convex (negative sign, *X* <1, *π* / 2<*θ* ≤*π*). Further the signs of *I*0 will be omitted, given the correct sign in every particular situation. Traditionally *I*0 is presented by (Legendre's) elliptic integrals first and second kind *F*, *E* (e.g. [18]), but for its evaluation we apply another calculation scheme (see

As seen *I*0 is integrable, but singular (in the lower limit *ξ*=1), which gives rise to significant instability of the numerical results. We have settled the issue by dividing the integrals into singular and regular parts. The singular part allows direct integration, while the regular part

2 2 2

é ù -- - - -+ é ù

<sup>ò</sup> *C CX <sup>X</sup> <sup>C</sup>*

x

x

1

*C C <sup>C</sup> <sup>C</sup>* (9)

( )

*I XC d*

12 2 1 1 1 <sup>1</sup> , arcsin , 4 2 1 2 1 1

ë û <sup>=</sup> - - ê ú - + -+ é ù ë û ë û

In practice, the evaluation of *I*0, according to Eq. (9), is performed by assigning a series of values of *X* at a fixed contact angle *θ*. The computation procedure is split into two subintervals *X* >1, 0≤*θ* <*π* / 2 (concave CB) and *X* <1, *π* / 2<*θ* ≤*π* (convex CB). We have used for *X* step of ∆*X* = 0.05; the angles subject to computation were: 15°, 30°, 45°, 60°, 89°; 91°, 100°, 120°, 179°

0 2

1 q

 q p

ps

ps - <sup>=</sup> - *<sup>X</sup> F R*

sin*θ* = *X* , (spherical CB, Eq. (4) at *C*=1) and negative pressing force (*F*<0) at *X*>sin *θ*.

yields the generatrix equation *y*(*x*) which in our notations can be presented as [47]:

( ) ( ) ( ) ( )

*<sup>x</sup> RI x C <sup>C</sup> y xC I xC d*

= = ±

is estimated numerically and as a result, for *x*=*X* one obtains:

( ) ( ) ( )

p

(Figure 3, Figure 4).

component of the TPC force and the term *πR* <sup>2</sup>

0

further).

30 Surface Energy

*X*

expressing ∆*p*0 through *С* from Eq. (5), we finally obtain *F*:

Figure 3 and Figure 4 present series of curves (*H* / *R*)vs.*X* <sup>−</sup><sup>1</sup> , respectively vs. *X* calculated via Eqs. (8)-(9) for a set of contact angles. The coordinate *Х*-1 in Figure 3 has been favored for its more compact presentation (0≤ *X* <sup>−</sup><sup>1</sup> ≤1), compared to *Х* (1≤ *X* <*∞*). The limit *X* → 1 named here thin CB will be discussed in more details in Appendix A. Figure 3 and Figure 4 present series of curves <sup>1</sup> ( / ) vs. *HR X* , respectively vs. *X* calculated via Eqs. (8)-(9) for a set of contact angles. The coordinate *Х*-1 in Figure 3 has been favored for its more compact presentation ( <sup>1</sup> 0 1 *X* ), compared to *Х* (1 *X* ). The limit *X*1 named here thin CB will be discussed in more details in Appendix A.

measured contact angles indicated apparent hysteresis. The photo series above illustrate the real CB shape deformation at stretching. **Figure 3.** Data from several experiments (triangles, hollow circles and squares) of stretching concave CB. The meas‐ ured contact angles indicated apparent hysteresis. The photo series above illustrate the real CB shape deformation at stretching.

**Figure 3.** Data from several experiments (triangles, hollow circles and squares) of stretching concave CB. The

The most substantial difference between the graphics of Figure 3 and Figure 4 is the presence of maximum only in one of them. On the one hand, (*H*/*R*)max is a critical value, i.e. a (concave) CB can't be stretched more than this degree. On the other hand, the existence of extremum puts the question which of the two 1/ *X*m at a given (*H*/*R*) is the real one? This well-known problem in the literature is solved via thermodynamic considerations. The right branch of an concave isogone [(1/ *X*m)< (1/ *X*)<1] is the so called thermodynamic stable branch, while the left branch is thermodynamic unstable one. These two branches define equilibrium states but with different liquid|gas area. The CBs on the left branch are with bigger area compared with the CBs on the right branch and this is maybe the most transparent elucidation of the question [46]. Now the question is why the convex CB (>/2) show no extremum? A short (and correct) but not so transparent answer is because the two curvatures of convex CB have same sign in contrast to concave CB, where the curvatures have opposite signs. The The most substantial difference between the graphics of Figure 3 and Figure 4 is the presence of maximum only in one of them. On the one hand, (*H*/*R*)max is a critical value, i.e. a (concave) CB can't be stretched more than this degree. On the other hand, the existence of extremum puts the question which of the two 1/ *X*m at a given (*H*/*R*) is the real one? This well-known problem in the literature is solved via thermodynamic considerations. The right branch of an concave isogone [(1/ *X*m)< (1/ *X*)<1] is the so called thermodynamic stable branch, while the left branch is thermodynamic unstable one. These two branches define equilibrium states but with different liquid|gas area. The CBs on the left branch are with bigger area compared with the CBs on the right branch and this is maybe the most transparent elucidation of the question [46].

upper limit of stretching of convex CB is at *X*=0, i.e. at *R*=0. It is the same limit as in the case of *C*(*X*=0)=1 (Figure 2), where all convex isogons end at one point. It is interesting to note that in the state *R*=0 the parameter *C* acquires the value coinciding with the value of *C* for a sphere (*C*=1). The

while for all other angles the asymptote *R*0 calls for additional analysis. The solution of the problem is associated with the appearance of an inflexion point *x*i in the generatrix, i.e. with the

, (10)

<sup>2</sup> 1 sin 1 sin *<sup>i</sup> C X x X C X*

=180º,

problem here is that the complete (closed) sphere is only congruent with a contact angle

appearance of a root in 2 2 ( /)0 *<sup>i</sup> <sup>x</sup> d y dx* . From Eq. (8) one obtains:

9

its own analysis.

Now the question is why the convex CB (θ>π/2) show no extremum? A short (and correct) but not so transparent answer is because the two curvatures of convex CB have same sign in contrast to concave CB, where the curvatures have opposite signs. The upper limit of stretching of convex CB is at *X*=0, i.e. at *R*=0. It is the same limit as in the case of *C*(*X*=0)=1 (Figure 2), where all convex isogons end at one point. It is interesting to note that in the state *R*=0 the parameter *C* acquires the value coinciding with the value of *C* for a sphere (*C*=1). The problem here is that the complete (closed) sphere is only congruent with a contact angle *θ* =180°, while for all other angles the asymptote *R* → 0 calls for additional analysis. The solution of the problem is associated with the appearance of an inflexion point *x*<sup>i</sup> in the generatrix, i.e. with the appearance of a root in (*d* <sup>2</sup>*y* / *d x* <sup>2</sup> )*xi* =0. From Eq. (8) one obtains:

$$\alpha\_i^2 = \frac{1-C}{C} = X \frac{\sin \theta - X}{1 - X \sin \theta},\tag{10}$$

It ensues from Eq. (10) that the inflexion emerges in the range 0< *X* <sin*θ*, i.e. beyond the "sphere" state *X* =sin*θ*. More details about the inflexion point can be found in [42, 47]. Figure 5 illustrates the appearance of inflexion point for two angles (95°, 120°) at stretching. It is clearly seen that at *θ* =180° any signs of inflexion are absent.

On Figure 3 and Figure 4 experimentally measured points of CB stretching are drawn. The direct measurements of contact angles show good coincidence with the theoretical isogons angle, which is a positive test of the method. The experimental data show also some interesting features of the TPC hysteresis. As seen, concave CB exhibits expressed hysteresis (contact angles *θ* change 10°-15°) while convex CB show now detective one (all experimental points lie close to θ≈100° isogone). Another noticeable difference in the TPC behavior between concave and convex CB is the reaction of the lower/upper contacts at stretching. Convex CBs show observable difference between the upper/lower contact radii (*R*±) and angles (*θ*±), while the same parameters of concave CBs remain practically equal. upper/lower contact radii (*R*) and angles (), while the same parameters of concave CBs remain practically equal.

(*X*=0.990.91) is practically constant. The deviation from the isogone at stronger stretching is due to gravity. The photo series above illustrate the real CB shape deformation at stretching. **Figure 4.** Data from stretching convex CB experiments (triangles and squares). The contact angle at weak stretching (*X*=0.99÷0.91) is practically constant. The deviation from the isogone at stronger stretching is due to gravity. The photo series above illustrate the real CB shape deformation at stretching.

The domain of rupture of the investigated convex CB turned to be at 0.84 0.80 *X* , significantly earlier than the definition limit *Х* = 0. The reason for the premature rupture could be the rise of *Rayleigh* instability [37], combined with the gravitational deformation. Yet, this phenomenon requires

**Figure 4.** Data from stretching convex CB experiments (triangles and squares). The contact angle at weak stretching

Now the question is why the convex CB (θ>π/2) show no extremum? A short (and correct) but not so transparent answer is because the two curvatures of convex CB have same sign in contrast to concave CB, where the curvatures have opposite signs. The upper limit of stretching of convex CB is at *X*=0, i.e. at *R*=0. It is the same limit as in the case of *C*(*X*=0)=1 (Figure 2), where all convex isogons end at one point. It is interesting to note that in the state *R*=0 the parameter *C* acquires the value coinciding with the value of *C* for a sphere (*C*=1). The problem here is that the complete (closed) sphere is only congruent with a contact angle *θ* =180°, while for all other angles the asymptote *R* → 0 calls for additional analysis. The solution of the problem is associated with the appearance of an inflexion point *x*<sup>i</sup> in the generatrix, i.e. with

=0. From Eq. (8) one obtains:

q

*C X* (10)

), while the same parameters of concave CBs remain

q

It ensues from Eq. (10) that the inflexion emerges in the range 0< *X* <sin*θ*, i.e. beyond the "sphere" state *X* =sin*θ*. More details about the inflexion point can be found in [42, 47]. Figure 5 illustrates the appearance of inflexion point for two angles (95°, 120°) at stretching. It is clearly

On Figure 3 and Figure 4 experimentally measured points of CB stretching are drawn. The direct measurements of contact angles show good coincidence with the theoretical isogons angle, which is a positive test of the method. The experimental data show also some interesting features of the TPC hysteresis. As seen, concave CB exhibits expressed hysteresis (contact angles *θ* change 10°-15°) while convex CB show now detective one (all experimental points lie close to θ≈100° isogone). Another noticeable difference in the TPC behavior between concave and convex CB is the reaction of the lower/upper contacts at stretching. Convex CBs show observable difference between the upper/lower contact radii (*R*±) and angles (*θ*±), while the

**Figure 4.** Data from stretching convex CB experiments (triangles and squares). The contact angle at weak stretching (*X*=0.990.91) is practically constant. The deviation from the isogone at stronger stretching is due to gravity. The photo series

**Figure 4.** Data from stretching convex CB experiments (triangles and squares). The contact angle at weak stretching (*X*=0.99÷0.91) is practically constant. The deviation from the isogone at stronger stretching is due to gravity. The photo

Stretching Stretching

The domain of rupture of the investigated convex CB turned to be at 0.84 0.80 *X* , significantly earlier than the definition limit *Х* = 0. The reason for the premature rupture could be the rise of *Rayleigh* instability [37], combined with the gravitational deformation. Yet, this phenomenon requires

)*xi*

<sup>2</sup> 1 sin , 1 sin


the appearance of a root in (*d* <sup>2</sup>*y* / *d x* <sup>2</sup>

32 Surface Energy

9

seen that at *θ* =180° any signs of inflexion are absent.

same parameters of concave CBs remain practically equal.

upper/lower contact radii (*R*) and angles (

above illustrate the real CB shape deformation at stretching.

series above illustrate the real CB shape deformation at stretching.

practically equal.

its own analysis.

**Figure 5.** Capillary bridge profiles computed for three different states (*Х*= 0.01, 0.1, 0.5) of three isogones in dimensionless coordinates *y = y*(*x*): (a) = 95; (b) = 120; (c) = 180. The appearance of an inflexion is distinctly perceptible at increased stretching (*Х* = 0.1, 0.5) at ≠ 180. **Figure 5.** Capillary bridge profiles computed for three different states (*Х*= 0.01, 0.1, 0.5) of three isogones in dimension‐ less coordinates *y = y*(*x*): (a) *θ* = 95°; (b) *θ* = 120°; (c) *θ* = 180°. The appearance of an inflexion is distinctly perceptible at increased stretching (*Х* = 0.1, 0.5) at *θ* ≠ 180°.

10

The domain of rupture of the investigated convex CB turned to be at *X* =0.84÷0.80, significantly earlier than the definition limit *Х* = 0. The reason for the premature rupture could be the rise of *Rayleigh* instability [37], combined with the gravitational deformation. Yet, this phenomenon requires its own analysis.

## **4. Capillary bridge in a gravity field**

In this section we will look into an aspect that is always present but often neglected. It is the role of gravity field on capillary forms. The work of Bashforth & Adams [48] from over a century ago is among the first reports exploring the effects of gravity on capillary shapes applying numerical methods for calculation of their shape. Latter many authors, e.g. [48, 49, 50, 51, 52, 20], proposed in the manner of Bashforth & Adams' work, new parameterization of the Laplace equation and thus defining the shape parameters of both bound and unbound axisymmetric menisci. Although these works have basically resolved the issue, there are still blank spots which need to be filled. We propose a slightly different variant of the classical approach, consisting of relatively simple instruments which allow us to obtain many interest‐ ing results, some of which are used for the interpretation of the experimental data.

A basis for the further analysis is the capillary gravitational balance in its dimensional Eq. (1) and scaled Eq. (2) forms. In contrast to CB, in the absence of gravity where the first integral of the pressure balance causes no difficulties, Eq.(4), the appearance of the gravitational term makes only its global integration transparent. In Appendix B one can find detailed derivation of all interesting relations, while here are given only the most substantial results. Among them are the external forces *F*± supporting CB in mechanical equilibrium (Figure 1 right). The emphasis *mechanical* is because the system could be thermodynamically non-equilibrium (e.g. to evaporate) which does not disturb the mechanical equilibrium.

As derived in Appendix B, Eq.(28), the upper/lower external forces F± , get the form:

$$F\_{+} = \pi R\_{+} \frac{2\sigma r\_{m} (R\_{+} - r\_{m} \sin \theta\_{+}) + R\_{+} (G\_{+} / \pi - \rho \text{gr}\_{m}^{2} h\_{+})}{R\_{+}^{2} - r\_{m}^{2}}$$

$$F\_{-} = -\pi R\_{-} \frac{2\sigma r\_{m} (R\_{-} - r\_{m} \sin \theta\_{-}) - R\_{-} (G\_{-} / \pi - \rho \text{gr}\_{m}^{2} h\_{-})}{R\_{-}^{2} - r\_{m}^{2}},$$

The two parameters *R*±, *θ*± depend differently on gravity. In the case of identical substrates and ideal contacts (i.e. without hysteresis), the contact angles must be equal (*θ*<sup>+</sup> =*θ*<sup>−</sup> =*θ*) regardless of gravity, as far as they reflects situations of TPC governed by the van der Waals and electrostatic forces [33, 35]. Generally *θ*<sup>+</sup> ≠*θ*− as far TPC hysteresis is always present at solid surfaces. Compared to contact angles, contact radii *R*<sup>±</sup> are functions of gravity and differ one from another (*R*<sup>+</sup> ≠*R*−) even at equal angles *θ*. This can be proved easily by putting *θ*<sup>+</sup> =*θ*−; *R*<sup>+</sup> =*R*<sup>−</sup> in Eq. (11) from where it follows *G*=0. From Eq. (11) follows that only *F*– may nullify (*F*<sup>−</sup> =0), while *F*<sup>+</sup> is non-zero, except at the trivial case when *R*<sup>+</sup> =0. Note that in the case *R*<sup>+</sup> =0(*R*<sup>−</sup> ≠0) we don't have bridge but a sessile droplet.

The domain of rupture of the investigated convex CB turned to be at *X* =0.84÷0.80, significantly earlier than the definition limit *Х* = 0. The reason for the premature rupture could be the rise of *Rayleigh* instability [37], combined with the gravitational deformation. Yet, this phenomenon

In this section we will look into an aspect that is always present but often neglected. It is the role of gravity field on capillary forms. The work of Bashforth & Adams [48] from over a century ago is among the first reports exploring the effects of gravity on capillary shapes applying numerical methods for calculation of their shape. Latter many authors, e.g. [48, 49, 50, 51, 52, 20], proposed in the manner of Bashforth & Adams' work, new parameterization of the Laplace equation and thus defining the shape parameters of both bound and unbound axisymmetric menisci. Although these works have basically resolved the issue, there are still blank spots which need to be filled. We propose a slightly different variant of the classical approach, consisting of relatively simple instruments which allow us to obtain many interest‐

ing results, some of which are used for the interpretation of the experimental data.

As derived in Appendix B, Eq.(28), the upper/lower external forces F± , get the form:

 q

+ + ++ + + + +

*r R r R G gr h F R*



 q


*r R r R G gr h F R*

2 2

2 ( sin ) ( / )

*m m m m m m m m*

*R r*

2 2

2 ( sin ) ( / ),

*R r*

The two parameters *R*±, *θ*± depend differently on gravity. In the case of identical substrates and ideal contacts (i.e. without hysteresis), the contact angles must be equal (*θ*<sup>+</sup> =*θ*<sup>−</sup> =*θ*) regardless of gravity, as far as they reflects situations of TPC governed by the van der Waals and electrostatic forces [33, 35]. Generally *θ*<sup>+</sup> ≠*θ*− as far TPC hysteresis is always present at solid surfaces. Compared to contact angles, contact radii *R*<sup>±</sup> are functions of gravity and differ one

2

 pr

> pr

2

(11)

to evaporate) which does not disturb the mechanical equilibrium.

s

s

p

p

A basis for the further analysis is the capillary gravitational balance in its dimensional Eq. (1) and scaled Eq. (2) forms. In contrast to CB, in the absence of gravity where the first integral of the pressure balance causes no difficulties, Eq.(4), the appearance of the gravitational term makes only its global integration transparent. In Appendix B one can find detailed derivation of all interesting relations, while here are given only the most substantial results. Among them are the external forces *F*± supporting CB in mechanical equilibrium (Figure 1 right). The emphasis *mechanical* is because the system could be thermodynamically non-equilibrium (e.g.

requires its own analysis.

34 Surface Energy

**4. Capillary bridge in a gravity field**

The situation (*F*<sup>−</sup> =0 ; *F*<sup>+</sup> ≠0 ;*R*<sup>±</sup> ≠0) is equivalent to a pendant CB, similar to pendant droplet, where the whole weight is balanced by the external force at the upper plate, *F*<sup>+</sup> =*G*. Of course for the sake of simplicity here the weight of CB plates is neglected.

As mentioned many times, the solution for heavy CB profile, *z*(*r*) is a matter of profound numerical calculus, which actually is a combination of different approximations with semiempirical numerical methods. Over the past decade, the development of this methodology has seemingly been completed. The research now is focused mainly on applications related to practical problems, [53, 54]. Below we demonstrate a known but not often used approach [55], marked by its geometrical transparency revealing some new insides of CB profiles.

To begin with, let's remind of the comparative analysis of the main curvatures 1 / *R*1,2 made in Section 1.The radii of curvature *R*1,2 depends quite differently on the CB dimensions, height *H* and contact radius *R.* On flattering the azimuthal radius *R*<sup>1</sup> =*r* / sin*φ* increases while the meridional *R*<sup>2</sup> =*dr* / *d*sin*φ* always decreases. This behavior derives directly from CB volume conservation condition. In addition, the meridional radius *R*2 depends on the contact angle*θ*. So for instance, at *θ*=90° (cylindrical bridge), *R*2=0 and the capillary pressure solely remains functional on *R*1, (*p<sup>σ</sup>* =*σ* / *R*1). A detailed analysis of this and many other aspects can be found in Section 2. What is important here is the existence of CB heights interval, where 1 / *R*<sup>1</sup> <1 / *R*<sup>2</sup> so that the left hand side of Eq. (1) to be simplified as *σd*(sin*ϕ*) / *dr* =*Δ p*<sup>0</sup> −*ρgz*. From geometric viewpoint, neglecting the azimuthal curvature reduces the problem from 3*D* to 2*D*, and thus facilitating its analysis. The advantage comes from the trigonometric relation, *d*sin*φ* / *dx* = −*d*cos*φ* / *dz* so that we can rewrite the equation of balance as

$$(\sigma d(\cos \phi) / \, dz = -\Delta p\_0 + \rho \text{gz}, \tag{12}$$

Now Eq. (12) can be integrated directly but before that let's look at the validity of this approx‐ imation from experimental viewpoint.

In Figure 6, experimental results about curvature ratio (1 / *R*2) / (1 / *R*1) in the waist (*R*1=*r*m) of a concave CB as a function of its thickness ratio *H*/*H*cr are given. The meridional curvature is notated as *K* and the thickness is scaled by the CB rupture thickness *Hcr*. The negative sign of the ordinate accentuates on the different curvatures' orientations (a concave CB), while for a convex CB the same ordinate would be positive. As seen from the diagram for a thickness up to *H* / *Hcr* ≤0.6, the meridional curvature is bigger (in absolute values) than the azimuthal curvature, i.e. in this range the 2*D* approximation holds. Series of experimental photos corresponding to the diagram data are shown above the diagram.

the balanced equation as

<sup>2</sup> *R dr d* / sin

<sup>1</sup> ( /) *p R* 

To begin with, let's remind of the comparative analysis of the main curvatures 1,2 1 / *R* made in Section 1.The radii of curvature *R*1,2 depends quite differently on the CB dimensions, height *H* and

=90 (cylindrical bridge), *R*2=0 and the capillary pressure solely remains functional on *R*<sup>1</sup> ,

important here is the existence of CB heights interval, where 1 2 1/ 1/ *R R* so that the left hand side

azimuthal curvature reduces the problem from 3*D* to 2*D*, and thus facilitating its analysis. The

 

*d dz p gz* (cos ) /

Now Eq. (12) can be integrated directly but before that let's look at the validity of this approximation

0

*d dr p gz* (sin ) /

. A detailed analysis of this and many other aspects can be found in Section 2. What is

always decreases. This behavior derives directly from CB volume conservation

increases while the meridional

. So for instance, at

so that we can rewrite

, (12)

. From geometric viewpoint, neglecting the

contact radius *R.* On flattering the azimuthal radius 1*R r* / sin

of Eq. (1) to be simplified as 0

condition. In addition, the meridional radius *R*2 depends on the contact angle

advantage comes from the trigonometric relation, *d dx d dz* sin / cos /

of CB rupture; green – IL\_4; blue – IL\_5; red –IL\_6; the transition 2*D*3*D* is marked on the photo series. **Figure 6.** Experimental results for the curvature ratio (*K*m*r*m) vs. the thickness ratio *H*/*H*cr; *H*cr is the thickness of CB rupture; green – IL\_4; blue – IL\_5; red –IL\_6; the transition 2*D* → 3*D* is marked on the photo series.

In Figure 6, experimental results about curvature ratio 2 1 (1 / ) / (1 / ) *R R* in the waist (*R*1=*r*m) of a

*K* and the thickness is scaled by the CB rupture thickness *Hcr*. The negative sign of the ordinate

**Figure 6.** Experimental results for the curvature ratio (*K*m *r*m) vs. the thickness ratio *H*/*H*cr; *H*cr is the thickness

The first integral of Eq. (12) in scaled version reads, concave CB as a function of its thickness ratio *H*/*H*cr are given. The meridional curvature is notated as

$$\cos\varphi = B\imath\jmath\gamma^2 - C\jmath + A,\tag{13}$$

with *Bo* <sup>=</sup>*ρgH* <sup>2</sup> / <sup>2</sup>*σ*;*<sup>C</sup>* <sup>≡</sup>(*<sup>Δ</sup> <sup>p</sup>*0)*<sup>H</sup>* / *<sup>σ</sup>*; *<sup>y</sup>* <sup>≡</sup> *<sup>z</sup>* / *<sup>H</sup>* . In 2*D*, there is only one scaling length − *H* as far as only one length variable *z* figures in the balance Eq. (12). The two constants of Eq. (13) *A* and *C* are determined by the boundary conditions at the lower/upper plate: 12

$$\cos\varphi \mathbf{o}\_- = -\cos\theta\_- = A \text{ at } \mathbf{y} = 0 \text{ and } \cos\varphi\_+ = \cos\theta\_+ = Bo - C + A \text{ at } \mathbf{y} = \mathbf{l},\tag{14}$$

The signs of cos*φ*± depend on the orientation of the normal vector towards the axis of symmetry, Figure 7.

The system of Eq. (13), (14) is still not the complete solution of the problem (12) but it contains abundant information about the characteristic 2D bridge profile points. For example the extremum points (waist/haunch) for which accounting that the current angle *φ* equals to 90°, from Eqs. (13) and (14) one obtains a quadratic equation with roots,

$$2\,\mathrm{y}\_e = \mathrm{l} - \frac{\cos\theta\_+ + \cos\theta\_-}{Bo} \pm \sqrt{(\frac{\cos\theta\_+ + \cos\theta\_-}{Bo})^2 - 2\frac{\cos\theta\_+ - \cos\theta\_-}{Bo} + 1},\tag{15}$$

From Eq. (15) one can reveal many details connected with 2D bridge profile but before that it must be defined the area of *θ*± and *Bo* values inside which a 2*D* bridge exists. The solution of this problem follows from the consideration that |cos*φ* | ≤1, i.e. | *Boy* <sup>2</sup> −*Cy* + *A*| ≤1. As far the left-hand side is a function of *y*, the inequality concerns its extremum point, i.e. | −*C* <sup>2</sup> / 4*Bo* + *A*| ≤1 and after substituting the already determined constant *A* from (14) one obtains,

$$Bo \le 2(\mathbf{l} + \cos \theta\_-) [\mathbf{l} + \frac{\cos \theta\_+ + \cos \theta\_-}{\mathbf{l} + \cos \theta\_-} + \sqrt{\mathbf{l} + 2 \frac{\cos \theta\_+ + \cos \theta\_-}{\mathbf{l} + \cos \theta\_-}}],\tag{16}$$

From Eq. (16) it follows that CBs with θ–<θ+ allow higher *Bo* values in comparison with θ–>θ+. For instance, at θ+ =180°; θ–=0°, *Bo*≤8, while in the reverse case, θ+ =0; θ–=180°, *Bo*=0. Moreover, at θ–=180° any heavy bridge cannot exist.

**Figure 7.** Sketch illustrating the *relations between the angles θ±and φ±.*

The first integral of Eq. (12) in scaled version reads,

cos cos j

Figure 7.

12

 q <sup>2</sup> cosj

rupture; green – IL\_4; blue – IL\_5; red –IL\_6; the transition 2*D* → 3*D* is marked on the photo series.

*C* are determined by the boundary conditions at the lower/upper plate:

from Eqs. (13) and (14) one obtains a quadratic equation with roots,

qq

at os c 0 and c

with *Bo* <sup>=</sup>*ρgH* <sup>2</sup> / <sup>2</sup>*σ*;*<sup>C</sup>* <sup>≡</sup>(*<sup>Δ</sup> <sup>p</sup>*0)*<sup>H</sup>* / *<sup>σ</sup>*; *<sup>y</sup>* <sup>≡</sup> *<sup>z</sup>* / *<sup>H</sup>* . In 2*D*, there is only one scaling length − *H* as far as only one length variable *z* figures in the balance Eq. (12). The two constants of Eq. (13) *A* and

**Figure 6.** Experimental results for the curvature ratio (*K*m *r*m) vs. the thickness ratio *H*/*H*cr; *H*cr is the thickness of CB rupture; green – IL\_4; blue – IL\_5; red –IL\_6; the transition 2*D*3*D* is marked on the photo series.

In Figure 6, experimental results about curvature ratio 2 1 (1 / ) / (1 / ) *R R* in the waist (*R*1=*r*m) of a concave CB as a function of its thickness ratio *H*/*H*cr are given. The meridional curvature is notated as *K* and the thickness is scaled by the CB rupture thickness *Hcr*. The negative sign of the ordinate accentuates on the different curvatures' orientations (a concave CB), while for a convex CB the same

**Figure 6.** Experimental results for the curvature ratio (*K*m*r*m) vs. the thickness ratio *H*/*H*cr; *H*cr is the thickness of CB

To begin with, let's remind of the comparative analysis of the main curvatures 1,2 1 / *R* made in Section 1.The radii of curvature *R*1,2 depends quite differently on the CB dimensions, height *H* and

=90 (cylindrical bridge), *R*2=0 and the capillary pressure solely remains functional on *R*<sup>1</sup> ,

important here is the existence of CB heights interval, where 1 2 1/ 1/ *R R* so that the left hand side

azimuthal curvature reduces the problem from 3*D* to 2*D*, and thus facilitating its analysis. The

 

*d dz p gz* (cos ) /

Now Eq. (12) can be integrated directly but before that let's look at the validity of this approximation

0

*d dr p gz* (sin ) /

. A detailed analysis of this and many other aspects can be found in Section 2. What is

always decreases. This behavior derives directly from CB volume conservation

increases while the meridional

. So for instance, at

so that we can rewrite

, (12)

. From geometric viewpoint, neglecting the

contact radius *R.* On flattering the azimuthal radius 1*R r* / sin

of Eq. (1) to be simplified as 0

condition. In addition, the meridional radius *R*2 depends on the contact angle

advantage comes from the trigonometric relation, *d dx d dz* sin / cos /

<sup>2</sup> *R dr d* / sin

<sup>1</sup> ( /) *p R* 

36 Surface Energy

the balanced equation as

from experimental viewpoint.

j

The signs of cos*φ*± depend on the orientation of the normal vector towards the axis of symmetry,

The system of Eq. (13), (14) is still not the complete solution of the problem (12) but it contains abundant information about the characteristic 2D bridge profile points. For example the extremum points (waist/haunch) for which accounting that the current angle *φ* equals to 90°,

+- +- +- ++ -

cos cos cos cos cos cos <sup>2</sup> 2 1 ( ) 2 1,

*ye* = -± - +

 qq

 qos at 1, - - =- = *A* = + + = = *Bo* - = *C* + *Ay y* (14)

> qq

*Bo Bo Bo* (15)

= -+ *Boy Cy A*, (13)

Other indicative points are the TPC co-ordinates *X*±, where *X*– =*x*(*y*=0), *X*+ =*x*(*y*=1). These coordinates are related with the parameters, *θ*± and *Bo* via the integral of Eq. (13), whose concise form reads,

$$
\Delta X = -\oint\_0 \cot \varphi \, d\varphi,\tag{17}
$$

It should be noted that here only the difference *X*+ – *X*– = ∆*X* is reasonable, because of the translation invariance of 2*D* problem.

According to Eqs. (13), (14), and (17), ∆*X* depends on three parameters *ΔX* (*θ*±, *Bo*). As men‐ tioned many times, the value and behavior of *θ*± depend on the particular TPC rheology. So for instance, without hysteresis *θ*±= *const*, while in the general case (in presence of hysteresis) they are function of Bo, *θ*±(Bo). Figure 8 represents experimental data of a typical contact angles dependence on bridge thickness *H*. With stretching the two contact angles show increasing difference, more accurate the upper angle *θ*+ changes while the lower angle *θ*<sup>−</sup> remains practically constant.

**Figure 8.** *Experimental results illustrating the ratio of the upper/lower contact angle θ±as a function of the thickness ratio H/Hcr* of IL concave CBs; green– IL\_4; blue – IL\_5; red–IL\_6;

Figure 9 presents theoretical curves (isogons) calculated on the basis of Eq. (17) compared with experimental data. The isogons are calculated at constant parameter *θ*-– = 0 (complete wetting) and increasing *θ*+ from 0° to 45° with step of 5°. Note that due to the relation *Bo* ~ *H*<sup>2</sup> , the isogons actually describe the dependence ∆*X*(*H*). The *θ* -values are chosen to be close to the experi‐ mental ones.

**Figure 9.** *Theoretical curves* (isogones) ∆*X vs. Bond number*, calculated by Eq. (17). The points represent experimental da‐ ta from concave CBs (hydrophilic plates): green – IL\_4; blue – IL\_5; red –IL\_6.

## **5. Experimental section**

they are function of Bo, *θ*±(Bo). Figure 8 represents experimental data of a typical contact angles dependence on bridge thickness *H*. With stretching the two contact angles show increasing difference, more accurate the upper angle *θ*+ changes while the lower angle *θ*<sup>−</sup> remains

**Figure 8.** *Experimental results illustrating the ratio of the upper/lower contact angle θ±as a function of the thickness ratio H/Hcr*

Figure 9 presents theoretical curves (isogons) calculated on the basis of Eq. (17) compared with experimental data. The isogons are calculated at constant parameter *θ*-– = 0 (complete wetting)

actually describe the dependence ∆*X*(*H*). The *θ* -values are chosen to be close to the experi‐

**Figure 9.** *Theoretical curves* (isogones) ∆*X vs. Bond number*, calculated by Eq. (17). The points represent experimental da‐

ta from concave CBs (hydrophilic plates): green – IL\_4; blue – IL\_5; red –IL\_6.

, the isogons

and increasing *θ*+ from 0° to 45° with step of 5°. Note that due to the relation *Bo* ~ *H*<sup>2</sup>

practically constant.

38 Surface Energy

mental ones.

of IL concave CBs; green– IL\_4; blue – IL\_5; red–IL\_6;

There are number of experimental methodologies concerning capillary bridges. Most of them are developed in order to utilize CBs as convenient tool for TPC properties, adhesive and capillary forces investigation. Over the past decade, there are some data in the literature which covers mainly the force aspect of capillary bridges. Most authors as Wei et al. [56], Yang et al. [57], Bradley et al., [58] etc. prefer to form capillary bridge at the tip of an AFM where they can measure the force directly. Others like Lee et al. [59] and Lipowsky et al. [60] offer various setups, which focus on directly capturing the CB profile. In this section, we will present similar setup with original image and statistical analysis

## **5.1. Experimental setup**

Our experimental setup consists of a micrometer, onto the measuring arms of which two square (20x20x2 mm) stainless steel supporting plates were fixed, parallel to one another.

Two 22x22 mm microscope cover glasses (ISOLAB) of soda lime silica composition were selected as working surfaces. They were glued to the supporting plates for static measure‐ ments. Images were recorded by using a high speed camera, MotionXtra N3, which was mounted onto a horizontal optical tube with appropriate magnification, Figure 10.

**Figure 10.** A picture and a schematic representation of the experimental setup

The light system was designed for the bundle of light to be directed perpendicularly to the electronic sensor of a high speed camera and at the same time the waist of the photographed CB to appear exactly at the middle position in the light bundle.

## **5.2. Solid surface preparation**

*Hydrophilic glass* surfaces were pre-cleaned with 99.9% C2H5OH and washed with deionized (Millipore) water before being glued to the supporting plates.

All experiments were carried out with deionized (Milipore) water or ionic liquids (IL).

Hydrophobized glass cover slides have been used for the convex CB. The preliminary hydrophobization was done with PDMS (Rhodia Silicones, 47V1000), following the procedure described in [61]. Before gluing the slides, they were washed with 99.9% C2H5OH.

The experimental protocol is the same for all surfaces and samples.

#### **5.3. Water capillary bridges**

A small droplet of ≈ 1 mm3 volume was placed in the middle of the lower glass slide. The upper glass slide was moved toward the droplet until a capillary bridge was formed. Further, several equilibrium states were recorded; pressing the shape until thin film was formed. Afterwards stretching took place until breakage occurred (some selected sequential pictures of the experimental part are presented in Figure 3, Figure 4, and Figure 6. The experiment was repeated several times with varying initial droplet volume. Concerning the effects due to evaporation, the direct volume decrease played no role, since the theoretical relations are in scaled (volume invariant) form (Section 2, Eq. (8)). Other effects related to the evaporation (e.g. thermo-effects) were not observed.

## **6. Ionic liquid capillary bridges**

Room-temperature ionic liquids (RTIL) show very promising properties in studies of liquid CBs. They are salts in liquid states and usually exhibit very low vapor pressure (10-10 Pa at 25˚C), i.e. no volume changes occurred during the experiment. Three RTIL were used for CB formation between hydrophilic glass surfaces. Summary of their physical properties is presented in Table 1 and their ion structural formulae are given in Figure 11 [62].


**Table 1.** Physical properties of IL

From the presented data in Table 1, it is seen that ratio ρ/σ determining the Bond number, Eqs. (2) remains constant for all RTILs.

From the presented data in Table 1, it is seen that ratio ρ/σ determining the Bond number, Eqs. (2) remains constant for Capillary Bridges — A Tool for Three-Phase Contact Investigation http://dx.doi.org/10.5772/60684 41

properties is presented in Table 1 and their ion structural formulae are given in Figure 11 [62].

**Ionic Liquid Surface Tension,**

**σ [mN/m]** 

IL 4 EMIM BTA 33.6 1548 0.046 IL 5 DiEMIM BTA 31.6 1450 0.046 IL 6 Et3Pic BTA 32.9 1513 0.046

**[s2/m3 Cation Anion ]** 

no role, since the theoretical relations are in scaled (volume invariant) form (Section 2, Eq. (8)). Other effects related to

Room-temperature ionic liquids (RTIL) show very promising properties in studies of liquid CBs. They are salts in liquid states and usually exhibit very low vapor pressure (10-10 Pa at 25C), i.e. no volume changes occurred during the experiment. Three RTIL were used for CB formation between hydrophilic glass surfaces. Summary of their physical

> **Density, ρ [kg/m3]**

**ρ/σ,**

the evaporation (e.g. thermo-effects) were not observed.

**6. Ionic liquid capillary bridges**

**Sample number** 

Table 1. Physical properties of IL

**6.1. Image analysis**

all RTILs.

**Figure 11.** Structural formulaе of ionic liquid ions Figure 11. Structural formulaе of ionic liquid ions

#### **6.1. Image analysis**

All experiments were carried out with deionized (Milipore) water or ionic liquids (IL).

described in [61]. Before gluing the slides, they were washed with 99.9% C2H5OH.

The experimental protocol is the same for all surfaces and samples.

**5.3. Water capillary bridges**

40 Surface Energy

A small droplet of ≈ 1 mm3

thermo-effects) were not observed.

**Table 1.** Physical properties of IL

(2) remains constant for all RTILs.

**6. Ionic liquid capillary bridges**

Hydrophobized glass cover slides have been used for the convex CB. The preliminary hydrophobization was done with PDMS (Rhodia Silicones, 47V1000), following the procedure

glass slide was moved toward the droplet until a capillary bridge was formed. Further, several equilibrium states were recorded; pressing the shape until thin film was formed. Afterwards stretching took place until breakage occurred (some selected sequential pictures of the experimental part are presented in Figure 3, Figure 4, and Figure 6. The experiment was repeated several times with varying initial droplet volume. Concerning the effects due to evaporation, the direct volume decrease played no role, since the theoretical relations are in scaled (volume invariant) form (Section 2, Eq. (8)). Other effects related to the evaporation (e.g.

Room-temperature ionic liquids (RTIL) show very promising properties in studies of liquid CBs. They are salts in liquid states and usually exhibit very low vapor pressure (10-10 Pa at 25˚C), i.e. no volume changes occurred during the experiment. Three RTIL were used for CB formation between hydrophilic glass surfaces. Summary of their physical properties is

**[mN/m]**

From the presented data in Table 1, it is seen that ratio ρ/σ determining the Bond number, Eqs.

**] Cation Anion**

presented in Table 1 and their ion structural formulae are given in Figure 11 [62].

IL 4 EMIM BTA 33.6 1548 0.046

IL 5 DiEMIM BTA 31.6 1450 0.046

IL 6 Et3Pic BTA 32.9 1513 0.046

**Sample number Ionic Liquid Surface Tension, σ**

volume was placed in the middle of the lower glass slide. The upper

**Density, ρ [kg/m3 ]**

**ρ/σ, [s2 /m3** The image analysis consist of three essential steps: capturing the CB image, detection of CB profile edge and statistical approximation of the determined profile [63]. Below we've presented schematically each step with short comments. The image analysis consist of three essential steps: capturing the CB image, detection of CB profile edge and statistical approximation of the determined profile [63]. Below we've presented schematically each step with short comments.

*Capturing the CB image Capturing the CB image* (Figure 12)

Figure 12. *A typical CB image.* It's important to stress out that although the TPC region looks sharp, the contrast there isn't good enough. That's one of the reasons we choose to restrict us on the waist region. **Figure 12.** *A typical CB image.* It's important to stress out that although the TPC region looks sharp, the contrast there isn't good enough. That's one of the reasons we choose to restrict us on the waist region.

#### **6.1.1. Determining the CB profile** *6.1.1. Determining the CB profile*

Determining the precise CB profile is the most vital part of the whole image analysis. This is why we pay special attention to it. It is performed in two steps: first we perform a rough edge detection (filtering the image) to get rid of the Determining the precise CB profile is the most vital part of the whole image analysis. This is why we pay special attention to it. It is performed in two steps: first we perform a rough edge detection (filtering the image) to get rid of the unnecessary information (background, light effects, etc.) and then scanning the filtered image to determine the precise profile.

**•** Finding edges with Sobel operator based filter [64].

**Figure 13.** Resulting image after applying the Sobel Filter on the image from Figure 12.

**•** Scanning the filtered image (normal to the obtained contours)

**Figure 14.** For each point, *y*<sup>i</sup> over a vertical stripe of the filtered picture (the white dashed line), the intensity value *I*<sup>i</sup> is mapped, Figure 15.

**•** Determining the dynamic threshold

The average intensity of the signal *I* ¯ can be determined on every stripe of the image (Figure 15). Assuming that the obtained useful information is a small part of the entire signal, one can use the so-called "three-sigma rule" (webpage [65]) which is a measure for the noise exclusion. In the current case we use 3 D for the coefficient *k*, which multiplies the standard deviation.

$$t = \overline{I} \pm k\sqrt{\mathbf{D}} \qquad \overline{I} = \frac{\sum\_{\substack{Strip\text{e} \\ n}} I\_l}{n} \qquad \mathbf{D} = \frac{\sum\_{\substack{Strip\text{e} \\ n-1}} \left(I\_l - \overline{I}\right)^2}{n-1}$$

Here *t* is the calculated dynamic threshold; *Ii* is the intensity of the *i*–th point; *I* ¯ is the average intensity over a given stripe; *k* is a coefficient which controls the selection filter quality; D is the mean square deviation of the intensities; *n* is the number of *y*<sup>i</sup> points.

**Figure 15.** Pixel intensity over a stripe of the image. The exact positions *yi* see below.

Averaging is done by a standard numerical approximation

**•** Scanning the filtered image (normal to the obtained contours)

mapped, Figure 15.

42 Surface Energy

**•** Determining the dynamic threshold

The average intensity of the signal *I*

**Figure 14.** For each point, *y*<sup>i</sup> over a vertical stripe of the filtered picture (the white dashed line), the intensity value *I*<sup>i</sup>

15). Assuming that the obtained useful information is a small part of the entire signal, one can use the so-called "three-sigma rule" (webpage [65]) which is a measure for the noise exclusion. In the current case we use 3 D for the coefficient *k*, which multiplies the standard deviation.

> å å *i i Stripe Stripe*

*I II*

D D

*tIk I n n*

Here *t* is the calculated dynamic threshold; *Ii* is the intensity of the *i*–th point; *I*

the mean square deviation of the intensities; *n* is the number of *y*<sup>i</sup>

**Figure 15.** Pixel intensity over a stripe of the image. The exact positions *yi*

=± = = -

intensity over a given stripe; *k* is a coefficient which controls the selection filter quality; D is

¯ can be determined on every stripe of the image (Figure

( ) 2


1

see below.

points.

is

¯ is the average

$$\mathbf{y}\_i = \frac{\sum\_{I\_j \ge 1} I\_j \mathbf{y}\_j}{\sum\_{I\_j \ge 1} I\_j}$$

Here *yj* is the coordinate of the *i*-th point and *yi* is the averaged pixel coordinate (Figure 15). For all obtained points, *x* coordinates is with 1px precision (based on the scanning frequency), while *y* coordinates have sub-pixel precision compensating the Cartesian sampling of the sensor (Figure 16).

**Figure 16.** Resultant curve of established co-ordinates of the points in x,y [pixel] space.

#### *6.1.2. Processing the obtained coordinates*


$$\left(\frac{1}{K}\right)^2 = \left(\mathbf{x} - \mathbf{x}\_0\right)^2 + \left(\boldsymbol{\wp} - \boldsymbol{\wp}\_0\right)^2$$


This simple procedure gives us the opportunity to evaluate the curvature *K*-1 directly from the real CB profile and thus to make the validation of 2*D* → 3*D* transition possible (Figure 6).

## **7. Summary**

Presented are results of study of two types of CB systems. The first one covered investigations of CB in the absence of gravity. This is a demonstration of a geometrical approach combined with the analytical description of the system, showing very good correlation with the experi‐ mental data. The second aspect is dedicated to CB behavior in gravity field. Here again the benefit of the combination of geometric and analytical methods is employed. The transparency of the obtained results holds out hope, the same approach to be successfully applied to other CB problems, e.g. CB stability.

## **Appendix A**

In contrast to the variety of CB parameters near the critical height, their flattening (thinning) toward zero thickness, is of much more universal character. This universality starts to show itself at a thickness *Н*, smaller than the radius of the contact *R* (*R*>>*H*). Taking into account the self-evident fact that *R* →*rm*→*∞* and*X* (≡*R* /*rm*)→1at*H* →0 makes suitable the substitution *Х* = 1+*∆*, with *∆* → 0. Thus, the parameter *С* in the thin CB region tends to (Eq. (5)):

$$C(X = \mathbf{l} - \boldsymbol{\Delta}) \approx -\frac{\mathbf{l} - \sin \theta}{2\Delta} + \frac{\mathbf{l} + \sin \theta}{4} \tag{18}$$

The two terms on the right-hand side represent the two (dimensionless) curvatures: the meridional curvature (first term) and the azimuthal curvature (second term). For example, at *θ* =90° we obtain a cylinder; the generatrix turns into a straight line of zero curvature, which leaves only the second (azimuthal) curvature equal to 1/2 and *С* = 1/2. The meridional curva‐ ture's change of sign as a function of the contact angle is allowed for by the sign of Δ.

It is worth commenting on the fact that the thin CB generatrices converge to equations of circle. It follows by inserting Eq. (18) in Eq. (4), which after integration yields,

$$\left(\mathbf{y}^{\,2} + (\mathbf{x} - \mathbf{l} + X\_{\mathcal{C}})^{\,2} = X\_{\mathcal{C}}\right)^{\,2} \tag{19}$$

Note that within the framework of the thin CB approximation, the correct range of integration is 1≤ *x* ≤1 + *Δ*. The dimensionless circular radius *Xc* =1 / 2*C* (coinciding with the CB generatrix radii of curvature) is related to the CB thickness, via *H* / 2*rm* = *Xc* |cos*θ* |. The use of modulus sign is meant to eliminate the sign alteration when the angle θ passes through *π*/2 (see above). The capillary pressure, *C* is represented by the first term on the right-hand side of Eq. (18), *Xc* =1 / 2*C*. Equation (19) can be generalized as:

$$\left(\mathbf{y}^{2} + \left(\mathbf{x} - \mathbf{l} \pm X\_{\mathbf{c}}\right)^{2} = X\_{\mathbf{c}}^{2}\right.\tag{20}$$

Where, the positive sign is for *θ* < *π*/2 and negative for *θ* > *π*/2.

Estimation of the external force *F* acting upon thin CB can be obtained from Eq. (7) at *Δ* →0 as follows:

$$F(X = \mathrm{l} + \Delta) \approx \pi \gamma R(\mathrm{l} + \frac{\mathrm{l} - \sin \theta}{\Delta}) = \pi \gamma R \left(\mathrm{l} + 2R \frac{\cos \theta}{H}\right) \tag{21}$$

By analogy to the capillary pressure, again for the case of *θ* ≠90°, the second term in the righthand side is of interest, which eventually (upon sufficient thinning) becomes dominant *F* (*H* →0)=2*πγR* <sup>2</sup> cos*θ* / *H* . Allowing for the volume constancy (*πR* <sup>2</sup> *H* =*const*), makes the force *F* in the asymptotic dependence inversely proportional to the thickness of a square: *F* (*H* →0)~1 / *H* <sup>2</sup> . In the estimate of the volume, we have assumed that it equals to its cylindrical part, disregarding the menisci − an entirely correct approximation in case of sufficiently thin bridges.

## **Appendix B**

with the analytical description of the system, showing very good correlation with the experi‐ mental data. The second aspect is dedicated to CB behavior in gravity field. Here again the benefit of the combination of geometric and analytical methods is employed. The transparency of the obtained results holds out hope, the same approach to be successfully applied to other

In contrast to the variety of CB parameters near the critical height, their flattening (thinning) toward zero thickness, is of much more universal character. This universality starts to show itself at a thickness *Н*, smaller than the radius of the contact *R* (*R*>>*H*). Taking into account the self-evident fact that *R* →*rm*→*∞* and*X* (≡*R* /*rm*)→1at*H* →0 makes suitable the substitution *Х* =

1+*∆*, with *∆* → 0. Thus, the parameter *С* in the thin CB region tends to (Eq. (5)):

1 sin 1 sin ( 1) 2 4

ture's change of sign as a function of the contact angle is allowed for by the sign of Δ.

It follows by inserting Eq. (18) in Eq. (4), which after integration yields,

*Xc* =1 / 2*C*. Equation (19) can be generalized as:

Where, the positive sign is for *θ* < *π*/2 and negative for *θ* > *π*/2.

 = -D »- + D

The two terms on the right-hand side represent the two (dimensionless) curvatures: the meridional curvature (first term) and the azimuthal curvature (second term). For example, at *θ* =90° we obtain a cylinder; the generatrix turns into a straight line of zero curvature, which leaves only the second (azimuthal) curvature equal to 1/2 and *С* = 1/2. The meridional curva‐

It is worth commenting on the fact that the thin CB generatrices converge to equations of circle.

Note that within the framework of the thin CB approximation, the correct range of integration is 1≤ *x* ≤1 + *Δ*. The dimensionless circular radius *Xc* =1 / 2*C* (coinciding with the CB generatrix radii of curvature) is related to the CB thickness, via *H* / 2*rm* = *Xc* |cos*θ* |. The use of modulus sign is meant to eliminate the sign alteration when the angle θ passes through *π*/2 (see above). The capillary pressure, *C* is represented by the first term on the right-hand side of Eq. (18),


 q

*C X* (18)

2 22 + -+ = (1 ) *c c yx X X* (19)

( ) 2 2 <sup>2</sup> + -± = <sup>1</sup> *c c yx X X* (20)

CB problems, e.g. CB stability.

**Appendix A**

44 Surface Energy

A brief derivation and a mathematical analysis of the supporting forces (*F*±) follows. Because of the double-meaning of the *z*(*r*) curve, the integration is carried out separately for the two parts of CB volume (above and below the point *r*m, Figure 1). And so, for the two parts we obtain:

$$\begin{aligned} \sigma \, 2\pi (r\_m - R\_- \sin \theta\_-) &= \pi (r\_m^2 - R\_-^2) \Delta p\_0 - \rho \, \text{g}\, \pi \int\_{R\_-}^{r\_m} z\_- dr^2 \quad - \text{ for the lower part} \\ &\qquad \begin{aligned} \sigma \, 2\pi (R\_+ \sin \theta\_+ - r\_m) &= \pi (R\_+^2 - r\_m^2) \Delta p\_0 - \rho \, \text{g}\, \pi \int\_{r\_m}^{r\_m} z\_+ dr^2 \quad - \text{ for the upper part}, \end{aligned} \end{aligned} \tag{22}$$

To clarify the matter, we present the integrals in (22) as follows:

$$\begin{aligned} \int\_{-R\_{-}}^{r\_{m}} z\_{-} dr^{2} &= r\_{m}^{2} h\_{-} - \int\_{0}^{R\_{-}} r^{2} dz \to r\_{m}^{2} h\_{-} - V\_{-} / \,\pi; \\\ R\_{+} & \int\_{0}^{R\_{+}} z\_{+} dr^{2} = R\_{+}^{2} H - r\_{m}^{2} h\_{-} - \int\_{h\_{-}}^{H} r^{2} dz \to R\_{+}^{2} H - r\_{m}^{2} h\_{-} - V\_{+} / \,\pi, \end{aligned} \tag{23}$$

and when substitute (23) in (22), one obtains:

$$\begin{split} \sigma 2\pi (r\_m - R\_- \sin \theta\_-) &= \pi (r\_m^2 - R\_-^2) \Delta p\_0 - \rho \text{g}\pi h\_- r\_m^2 + G\_- \\ \sigma 2\pi (R\_+ \sin \theta\_+ - r\_m) &= \pi (R\_+^2 - r\_m^2) \Delta p\_0 - \rho \text{g}\pi (R\_+^2 H - r\_m^2 h\_-) + G\_+, \end{split} \tag{24}$$

where *G*± = ρg*V*<sup>±</sup> is the respected weights of the upper/lower CB parts; *V*± − their volume parts. Actually Eqs. (24) define the pressure difference ∆*p*<sup>0</sup>

$$\begin{split} \Delta p\_0 &= -\frac{2\sigma (r\_m - R\_- \sin \theta\_-) + \rho \text{g} \, h\_- r\_m^2 - G\_- / \pi}{(R\_-^2 - r\_m^2)} = \\ &= \frac{2\sigma (R\_+ \sin \theta\_+ - r\_m) + \rho \text{g} \, (H R\_+^2 - h\_- r\_m^2) - G\_+ / \pi}{(R\_+^2 - r\_m^2)}, \end{split} \tag{25}$$

As discussed in Section 3, resultant forces *F*± at CB plates can be expressed as follows,

$$
\pm F\_{\pm} = (p\_{\pm} - p\_e) \pi R\_{\pm}^2 - 2\pi \sigma R\_{\pm} \sin \theta\_{\pm}, \tag{26}
$$

where *p*± are the internal pressures on the upper/lower plate, i.e. *p*<sup>−</sup> = *p*0; *p*<sup>+</sup> = *p*<sup>0</sup> −*ρgH* . Making use of the notations introduced in Section 2, the above equation can be rewritten as (compare with Eq.(6)),

$$\begin{aligned} F\_{+} &= \pi R\_{+}^{2} (\Delta p\_{0} - \rho \text{g}H) - 2\pi \sigma R\_{+} \sin \theta\_{+}, \\ -F\_{-} &= \pi R\_{-}^{2} \Delta p\_{0} - 2\pi \sigma R\_{-} \sin \theta\_{-}, \end{aligned} \tag{27}$$

After substituting ∆*p*0 from (25) in (27), we come to the final expression for the force

$$\begin{aligned} \text{a)} &F\_{+} = 2\pi\sigma R\_{+} \sin\theta\_{+} - \pi R\_{+}^{2} (\Delta P - \rho gH) = \\ &= \pi R\_{+} \frac{2\sigma r\_{m} (R\_{+} - r\_{m} \sin\theta\_{+}) + R\_{+} (G\_{+} / \pi - \rho gr\_{m}^{2} h\_{+})}{R\_{+}^{2} - r\_{m}^{2}}; \\\\ \text{b)} &F\_{-} = \pi R\_{-}^{2} \Delta P - 2\pi\sigma R\_{-} \sin\theta\_{-} = \\ &= -\pi R\_{-} \frac{2\sigma r\_{m} (R\_{-} - r\_{m} \sin\theta\_{-}) - R\_{-} (G\_{-} / \pi - \rho gr\_{m}^{2} h\_{-})}{R\_{-}^{2} - r\_{m}^{2}}. \end{aligned} \tag{28}$$

As at any mechanical equilibrium the following ballance is valid *F*<sup>+</sup> + *F*<sup>−</sup> + *G* =0


*m*

## **Appendix C**

(24)

(25)

(27)

(28)

2 2 2 0 2 2 2 2 0


*mm m*

where *G*± = ρg*V*<sup>±</sup> is the respected weights of the upper/lower CB parts; *V*± − their volume parts.


*m m m m*

r p

r p

2

 p

p

(26)

2 2

2 ( sin ) ( ) ( ) ,

2 2

2 ( sin ) ( ) / , ( )

As discussed in Section 3, resultant forces *F*± at CB plates can be expressed as follows,

<sup>2</sup> ±= - - *F p pR R* ± ± ± ±± ( ) 2 sin , *<sup>e</sup>* p

 ps

where *p*± are the internal pressures on the upper/lower plate, i.e. *p*<sup>−</sup> = *p*0; *p*<sup>+</sup> = *p*<sup>0</sup> −*ρgH* . Making use of the notations introduced in Section 2, the above equation can be rewritten as (compare

> ( ) 2 sin , 2 sin ,

 ps

> q

 r

> pr

> > pr

 r

= D- -

*F Rp R*

a) 2 sin ( )

*F R R P gH*



As at any mechanical equilibrium the following ballance is valid *F*<sup>+</sup> + *F*<sup>−</sup> + *G* =0

 q p

+ ++ +

 ps

= D- =


*F R p gH R*


 ps

After substituting ∆*p*0 from (25) in (27), we come to the final expression for the force

2

 q

= - D- =

2 2

2 ( sin ) ( / );

*m m m m*

+ + ++ + <sup>+</sup> +

*r R r R G gr h <sup>R</sup> R r*

2 2

2 ( sin ) ( / ).

*m m m m*


 q

*r R r R G gr h <sup>R</sup> R r*

 q

+ + + +

 q

> q

> > 2

2

2 ( sin ) / ( )

*R r R r g HR h r G R r*


 qr

*r R gh r G <sup>p</sup>*


*m m*

 r


*R r R r p g RH rh G*



*r R r R p g hr G*

*mm m*

2 ( sin ) ( )

s

sq

 p

 p

0 2 2

p

2

s

p

ps

s

p

p

b) 2 sin

*F RP R*

p


 q

 q

Actually Eqs. (24) define the pressure difference ∆*p*<sup>0</sup>

sp

46 Surface Energy

sp

with Eq.(6)),



## **Glossary**


*θ*± − upper/lower contact angle ρliquid|air − liquid|air density

σ − surface tension

**References Topics**

R. Aveyard, P. Cooper, P.D.I. Fletcher, C.E. Rutherford, Langmuir, 9 (1993) 604 R. Aveyard, B.P. Binks, P.D.I. Fletcher, T.G. Peck, C.E. Rutherford, Adv.

P.R. Garrett, in: P.R. Garrett (Ed.), Defoaming: Theory and Industrial

R. Aveyard, J.H. Clint, J. Chem. Soc. Faraday Trans., 93 (1997) 1397 N.D. Denkov, P. Cooper, J.-Y. Martin, Langmuir, 15 (1999) 8514

Applications, M. Dekker, New York, 1993; Chapter 1

ColloidInterface Sci., 48 (1994) 93

**Glossary**

48 Surface Energy

*Bo* − bond number

*F*±*–* external forces

*H* − bridge thickness

*G –* CB weight

*K* − curvature

*CB* − capillary bridge

N.D. Denkov, Langmuir, 15 (1999) 8530

*C* − dimensionless capillary pressure

*Lr* / *Lz* – radial/axial scaling length *R* − equilibrium dimension radius

*R*± − upper/lower dimension radius

TPC − Three-Phase Contact

*g* – gravity of earth

*p*σ − capillary pressure

∆*p* − pressure difference

*rm* − dimension waist radius

*X* − dimensionless radius *R*/*rm*

*p*i/e − internal/external pressures

RTIL − Room-Temperature Ionic Liquid

*R*1,2 − meridional/azimuthal curvature radiuses

*x* ≡*r* /*rm*; *y* ≡ *z* / *H* − current dimensionless coordinates

## **Acknowledgements**

The authors are thankful for the financial support of the project No. 159/2015 financed by the Scientific Research Foundation at University of Sofia "St. Kliment Ohridski".

## **Author details**

Boryan P. Radoev1\*, Plamen V. Petkov2 and Ivan T. Ivanov1

\*Address all correspondence to: fhbr@lchem.uni-sofia.bg

1 Sofia University "St. Kliment Ohridski", Faculty of Chemistry and Pharmacy, Department of Physical Chemistry, Sofia, Bulgaria

2 Sofia University "St. Kliment Ohridski", Faculty of Chemistry and Pharmacy, Department of Pharmaceutical and Chemical Engineering, Sofia, Bulgaria

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## **Chapter 3**

## **Solid-Liquid-Solid Interfaces**

## Jeffrey L. Streator

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/61572

#### **Abstract**

Interfaces comprised of a liquid interposed between two solids in close proximity are common in small-scale devices. In many cases, the liquid induces large and undesired ad‐ hesive forces. It is of interest, therefore, to model the way in which forces are developed in such an interface. The following chapter presents several models of liquid-mediated adhesion, considering the roles of surface geometry, liquid surface tension, elastic defor‐ mation, surface roughness, and surface motion on the development of interfacial forces.

**Keywords:** Capillary film, liquid-mediated adhesion, liquid bridge

## **1. Introduction**

Phenomena related to the wetting of solid–solid interfaces are of technological importance. When two surfaces are in close proximity, the presence of a liquid film may cause the surfaces to stick together. Such liquid-mediated adhesion can negatively affect the operation of micro/ nanoscale systems [1–7]. The interfacial liquid film, which may be present due to condensation, contamination, or lubrication, may experience large concave curvatures at the liquid-vapor interface and large negative pressures. These negative pressures give rise to large adhesive forces, which can have a potentially deleterious effect on the performance of small-scale devices.

In this chapter, we will discuss the behavior of an interface comprised of a liquid interposed between two solids. Throughout this chapter, we are concerned with the role of liquid films in regimes where gravitational effects are negligible, which generally implies that the vertical length scale is small. As an illustration, it can be easily shown that the change in pressure due to gravity within a near-hemispherical water droplet (resting on a horizontal surface) from just within the top of the free surface to the bottom of the droplet is given by (Δ*p*) *gravity* =*ρgR*, where *ρ* is the water mass density, *g* is the gravitational acceleration, and *R* is the approximate radius

© 2015 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

of the droplet. By comparison, the change in pressure across the free surface of the droplet is given by (Δ*p*) *surface tension* =2*γ* / *R*, where *γ* is the liquid surface tension. Thus, the ratio of gravi‐ tational effects to surface tension effects is equal to *ρgR* <sup>2</sup> / 2*γ*. For water at room temperature (and ambient pressure), one has *ρ* = 1000 kg/m3 and *γ* = 0.0727 N/m, so that for a radius of 1.0 mm, we have a ratio of about 0.07, meaning that the change in pressure due to gravity is only about 7% of that due to surface tension. Moreover, it is seen that the relative effects of gravity decrease in proportion to the square of the droplet radius. In general, the smaller the vertical scale, the less important are the effects of gravity in comparison to those of surface tension.

Of particular interest in this chapter is the topic of liquid-mediation adhesion, a mechanism by which the liquid film pulls inward on the solid surfaces. We consider the effects of liquid surface tension, liquid viscosity, surface geometry, surface roughness, surface elasticity, and surface motion on the development of adhesive forces in the interface. Our approach to discussing the recent literature on the topic of liquid-mediated adhesion is to organize things according to several basic characteristics: gross interface geometry (flat or curved), surface topography (smooth or rough), structural properties (rigid or deforming), meniscus type (constant-volume or constant-pressure) and separating process (quasi-static or dynamic). In this context, Table 1 categorizes recent research that is particularly relevant to the subject of this chapter. It is noted that an entry of "volume" under the "film constant" heading means that the volume of the liquid bridge is held fixed during the separation process, while an entry of "pressure" indicates that the liquid is assumed to remain in thermodynamic equilibrium with its vapor during the separation process.



**Table 1.** Recent research on the topic of the liquid-mediated adhesion

## **2. Models of solid surfaces bridged by a liquid**

#### **2.1. Liquid film between smooth, rigid, parallel flats**

#### *2.1.1. Static and quasi-static conditions*

of the droplet. By comparison, the change in pressure across the free surface of the droplet is

tational effects to surface tension effects is equal to *ρgR* <sup>2</sup> / 2*γ*. For water at room temperature

mm, we have a ratio of about 0.07, meaning that the change in pressure due to gravity is only about 7% of that due to surface tension. Moreover, it is seen that the relative effects of gravity decrease in proportion to the square of the droplet radius. In general, the smaller the vertical scale, the less important are the effects of gravity in comparison to those of surface tension.

Of particular interest in this chapter is the topic of liquid-mediation adhesion, a mechanism by which the liquid film pulls inward on the solid surfaces. We consider the effects of liquid surface tension, liquid viscosity, surface geometry, surface roughness, surface elasticity, and surface motion on the development of adhesive forces in the interface. Our approach to discussing the recent literature on the topic of liquid-mediated adhesion is to organize things according to several basic characteristics: gross interface geometry (flat or curved), surface topography (smooth or rough), structural properties (rigid or deforming), meniscus type (constant-volume or constant-pressure) and separating process (quasi-static or dynamic). In this context, Table 1 categorizes recent research that is particularly relevant to the subject of this chapter. It is noted that an entry of "volume" under the "film constant" heading means that the volume of the liquid bridge is held fixed during the separation process, while an entry of "pressure" indicates that the liquid is assumed to remain in thermodynamic equilibrium

> **Loading Process**

flat on flat rough elastic-plastic quasi-static pressure Del Rio et al. 2008 19

flat on flat rough elastic quasi-static pressure Peng et al. 2009 21

flat on flat rough elastically hard quasi-static pressure de Boer 2007 17 flat on flat rough elastic quasi-static pressure Persson 2008 20

flat on flat rough elastic quasi-static volume Streator 2009 33

flat on flat smooth elastic quasi-static volume Zheng and

flat on flat rough elastic quasi-static pressure Wang and

flat on flat rough elastic quasi-static volume Streator and

flat on flat rough elastic quasi-static volume Rostami and

**Film Constant**

volume

*surface tension* =2*γ* / *R*, where *γ* is the liquid surface tension. Thus, the ratio of gravi‐

and *γ* = 0.0727 N/m, so that for a radius of 1.0

**Author(s) Year**

Streator

Regnier

Jackson

Streator

de Boer and de Boer

**Publ.**

2004 28

2015 37

2007 18

2009 34

2015 35

**Ref. No.**

given by (Δ*p*)

56 Surface Energy

**Gross Interface Geometry**

(and ambient pressure), one has *ρ* = 1000 kg/m3

with its vapor during the separation process.

**Deform. Behavior**

flat on flat rough rigid quasi-static pressure or

**Surface Type**

> Consider the problem of a continuous liquid film that is at static equilibrium between two rigid, parallel flats in close proximity as shown in Figure 1. In this idealized case, the liquid forms an axisymmetric configuration, so that any horizontal cross section is circular. Because the liquid is in static equilibrium, the entire film must be at a single pressure. Per the Young-Laplace equation [8], the pressure drop Δ*p* across the free surface is given by

$$
\Delta p = p\_a - p = \gamma \left(\frac{1}{R\_1} + \frac{1}{R\_2}\right). \tag{1}
$$

Where *pa* is the ambient pressure, *p* is the film pressure, and *R*1*,*2 are the principal radii of normal curvature of the free surface at any given point on the free surface. Since we are dealing with small vertical spacing, it is reasonable to assume the radius of curvature (*R*2) that exists in the plane of the figure at each free surface point is much smaller than the other principal radius of curvature (*R*1), which lies in a plane that is perpendicular to the plane of the figure as well as perpendicular to the tangent plane to the free surface at the point in question. In Figure 1, we have chosen to illustrate the value of *R*1 that exists in the plane of minimum horizontal diameter. Assuming *R*<sup>1</sup> is sufficiently larger than *R2* that 1/*R*<sup>1</sup> may be neglected, the pressure drop in Eq. (1) becomes

**Figure 1.** Profile of an axisymmetric liquid film between rigid, parallel plates. (a) The liquid wets both surfaces leading to a concave film shape. (b) The liquid wets neither surface, leading to a convex film shape. (c) The liquid wets one surface but not the other, with the wet surface closer to complete wetting than the non-wet surface is to complete nonwetting, leading to a convex film. (d) The liquid wets one surface, but not the other, with the non-wet surface closer to complete non-wetting than the wet surface is to complete wetting, leading to a convex film.

Moreover, owing to the fact that the liquid film, being continuous and in static equilibrium, must experience a uniform pressure, one may conclude that the radius of curvature *R*2 is the same at every point of the free surface. Thus, the free surface profile is in the shape of a circle. Using this result leads to the geometrical relationship depicted in Figure 2, by which one concludes that

$$h = R\_2(\cos \theta\_1 + \cos \theta\_2) \tag{3}$$

so that

$$
\Delta p = \frac{\gamma (\cos \theta\_1 + \cos \theta\_2)}{h} \tag{4}
$$

where *θ*1,2 are the liquid–solid contact angles. A liquid is considered to "wet" a given surface if its contact angle (measured from solid–liquid interface to the solid-vapor interface) is less than 90 degrees, and a liquid is considered to be non-wetting if its contact angle is greater than 90 degrees. Complete wetting is associated with a contact angle of 0 degrees and complete nonwetting corresponds to a contact angle of 180 degrees. While Figure 2 shows a case for which the liquid wets both upper and lower surfaces, Eq. (3) holds for each of the configurations depicted in Figure 1. Depending on both signs and relative magnitudes of the cosine terms in Eq. (4), the film pressure may be greater than, equal to, or less than atmospheric pressure. For the case of Figure 1a, where both surfaces are wet by the liquid, both cosine terms are positive and the film pressure is sub-ambient. In contrast, for the case of Figure 1b, where neither surface is wet by the liquid, both cosine terms are negative and the film pressure exceeds the ambient pressure. When one of the surfaces is wet by the film while the other is not (Figures 1c and 1d), the sign of the film pressure depends on the relative magnitudes of the two cosine terms. When the contact angle associated with the surface that is wet by the liquid is closer to zero than the other contact angle is to 180 degrees (Figure 1c), then the film shape is concave and the film pressure is sub-ambient. On the other hand, if the contact angle associated with the surface that is wet by the liquid departs from zero degrees more than the opposing surface departs from 180 degrees (Figure 1d), then the film shape is convex and the film pressure is greater than ambient.

as perpendicular to the tangent plane to the free surface at the point in question. In Figure 1, we have chosen to illustrate the value of *R*1 that exists in the plane of minimum horizontal diameter. Assuming *R*<sup>1</sup> is sufficiently larger than *R2* that 1/*R*<sup>1</sup> may be neglected, the pressure

> 2 <sup>Δ</sup>*<sup>p</sup> <sup>R</sup>* g

**Figure 1.** Profile of an axisymmetric liquid film between rigid, parallel plates. (a) The liquid wets both surfaces leading to a concave film shape. (b) The liquid wets neither surface, leading to a convex film shape. (c) The liquid wets one surface but not the other, with the wet surface closer to complete wetting than the non-wet surface is to complete nonwetting, leading to a convex film. (d) The liquid wets one surface, but not the other, with the non-wet surface closer to

Moreover, owing to the fact that the liquid film, being continuous and in static equilibrium, must experience a uniform pressure, one may conclude that the radius of curvature *R*2 is the same at every point of the free surface. Thus, the free surface profile is in the shape of a circle. Using this result leads to the geometrical relationship depicted in Figure 2, by which one

> 21 2 *h R*= + (cos cos ) q

1 2 (cos cos ) <sup>Δ</sup>*<sup>p</sup> <sup>h</sup>* gq

where *θ*1,2 are the liquid–solid contact angles. A liquid is considered to "wet" a given surface if its contact angle (measured from solid–liquid interface to the solid-vapor interface) is less than 90 degrees, and a liquid is considered to be non-wetting if its contact angle is greater than 90 degrees. Complete wetting is associated with a contact angle of 0 degrees and complete nonwetting corresponds to a contact angle of 180 degrees. While Figure 2 shows a case for which the liquid wets both upper and lower surfaces, Eq. (3) holds for each of the configurations depicted in Figure 1. Depending on both signs and relative magnitudes of the cosine terms in

 q

 q

complete non-wetting than the wet surface is to complete wetting, leading to a convex film.

= (2)

(3)

<sup>+</sup> <sup>=</sup> (4)

drop in Eq. (1) becomes

58 Surface Energy

concludes that

so that

The value of the contact angle for a particular case is determined by a local thermodynamic equilibrium among the three relevant interfaces, which can be expressed in the Young-Dupree equation [8]

$$
\gamma\_{SV} = \gamma\_{SL} + \gamma\_{LV} \cos \theta \tag{5}
$$

where *γSV* , *γSL* , and *γLV* are the surface energies per unit area of the solid–vapor, solid–liquid, and liquid–vapor interfaces, respectively. Note that *γLV* is the same as the surface tension of the liquid *γ*.

**Figure 2.** Geometrical relationship between plate spacing and radius of curvature of free surface for an assumed circu‐ lar profile. Without loss of generality, the liquid is shown here as wetting both surfaces.

For a concave film shape (Figures 1a and 1c) the sum on the right-hand side of Eq. (4) is positive, yielding a positive pressure drop relative to atmospheric pressure. Thus, in terms of gauge pressure, the pressure within the film is negative. One important consequence is that the liquid exerts a force that pulls inward on the two plates so that the force exerted on either of the plates may be considered the force of adhesion due to the presence of the film. With reference to Figure 3, this adhesive force (*Fad*) can be expressed as

$$F\_{ad} = \left(-p\right)\pi R\_1^2 + 2\pi R\_1 \chi = \frac{\chi(\cos\theta\_1 + \cos\theta\_2)}{h} \left(\pi R\_1^2 + 2\pi R\_1 \chi\right) \tag{6}$$

The first term on the right-hand side is the contribution to the adhesive force arising from the pressure drop across the free surface, while the second term is the adhesive force exerted by the free surface itself. Note that the total force exerted on the bottom of this upper section of the liquid film is simply transmitted to the upper plate, so the force given by Eq. (6) is indeed the adhesive force. Now under the assumption that *R*1 is much greater than *R*2, it can be shown that the force contribution to the pressure drop dominates the force contribution due to the free surface. Let the first term on the far right-hand side be denoted by *F*Δ*<sup>p</sup>* and the second term be denoted by *Fγ*. Then *F<sup>γ</sup>* / *F*Δ*<sup>p</sup>* =2 *R*<sup>2</sup> / *R*1. Thus, to the extent that 1/*R*<sup>1</sup> can be neglected in Eq. (1), which leads to Eq. (2), the force contributed by the free surface 2*πR*1*γ* may be neglected in Eq. (6).

**Figure 3.** Sources of force exerted on the upper section of the fluid (dividing line chosen at the plane of minimum di‐ ameter).

Suppose now that the liquid film has a fixed volume *Vo*. To a good approximation, this volume may be expressed as *Vo* =*πR*<sup>1</sup> 2 *h* , where the deviation from a cylindrical geometry has been ignored. Then the adhesive force may be written as:

$$F\_{ad} = \frac{\gamma (\cos \theta\_1 + \cos \theta\_2)}{h^2} \ V\_o \tag{7}$$

This equation shows that under the conditions of fixed liquid volume the adhesive force is inversely proportional to the square of the film thickness.

When a quantity of a pure liquid of given chemical species is at thermodynamic equilibrium, the partial pressure of the vapor phase of the species is equal to the vapor pressure of the liquid phase for the given temperature. For a curved free surface, there is a small deviation in the vapor pressure from that corresponding to a planar free surface. This deviation is accounted for by the well-known Kelvin equation [8]

$$\ln \frac{p\_s}{p\_v} = \gamma \left(\frac{1}{R\_1} + \frac{1}{R\_2}\right) \frac{V\_m}{RT} = \frac{\gamma}{R\_K} \frac{V\_m}{RT} \tag{8}$$

where *ps* is the saturation pressure at the given temperature, *pv* is the pressure of the vapor just outside of the liquid film, *Vm* is the molar volume, *RK* is the Kelvin radius, *R* is the universal gas constant, and *T* is the absolute temperature. Assuming, as before, *R*1≫*R*2, and then isolating *R*2, one obtains

$$R\_2 = R\_K = -\frac{\mathcal{V}V\_m}{RTl\eta \frac{p\_v}{p\_s}}\tag{9}$$

Using this result in Eq. (2) gives

The first term on the right-hand side is the contribution to the adhesive force arising from the pressure drop across the free surface, while the second term is the adhesive force exerted by the free surface itself. Note that the total force exerted on the bottom of this upper section of the liquid film is simply transmitted to the upper plate, so the force given by Eq. (6) is indeed the adhesive force. Now under the assumption that *R*1 is much greater than *R*2, it can be shown that the force contribution to the pressure drop dominates the force contribution due to the free surface. Let the first term on the far right-hand side be denoted by *F*Δ*<sup>p</sup>* and the second term be denoted by *Fγ*. Then *F<sup>γ</sup>* / *F*Δ*<sup>p</sup>* =2 *R*<sup>2</sup> / *R*1. Thus, to the extent that 1/*R*<sup>1</sup> can be neglected in Eq. (1), which leads to Eq. (2), the force contributed by the free surface 2*πR*1*γ* may be

**Figure 3.** Sources of force exerted on the upper section of the fluid (dividing line chosen at the plane of minimum di‐

Suppose now that the liquid film has a fixed volume *Vo*. To a good approximation, this volume

1 2 2 (cos cos ) *ad <sup>o</sup> F V h*

This equation shows that under the conditions of fixed liquid volume the adhesive force is

When a quantity of a pure liquid of given chemical species is at thermodynamic equilibrium, the partial pressure of the vapor phase of the species is equal to the vapor pressure of the liquid phase for the given temperature. For a curved free surface, there is a small deviation in the vapor pressure from that corresponding to a planar free surface. This deviation is accounted

1 2

æ ö =+ = ç ÷ è ø

g

1 1 *<sup>s</sup> m m v K p VV ln p R R RT R RT*

where *ps* is the saturation pressure at the given temperature, *pv* is the pressure of the vapor just outside of the liquid film, *Vm* is the molar volume, *RK* is the Kelvin radius, *R* is the universal

g

 q

gq

*h* , where the deviation from a cylindrical geometry has been

<sup>+</sup> <sup>=</sup> (7)

(8)

neglected in Eq. (6).

60 Surface Energy

ameter).

may be expressed as *Vo* =*πR*<sup>1</sup>

2

inversely proportional to the square of the film thickness.

for by the well-known Kelvin equation [8]

ignored. Then the adhesive force may be written as:

$$p\_a - p = -\frac{RT\ln\frac{p\_v}{p\_s}}{V\_m} \tag{10}$$

Now, suppose the chemical species in question is water, so that the ratio *pv* / *ps* represents the relative humidity. Then Eq. (9) states that, at thermodynamic equilibrium, the radius of curvature of any free surface of the liquid film is determined by the relative humidity. For example, taking properties of water at room temperature and assuming a relative humidity of 95%, we have

$$R\_2 = -\frac{\left(0.0727 \frac{\text{N}}{\text{m}}\right) \left(0.018 \text{ kg} / \text{mol}\right) / \left(0.0100 \text{m}^3 / \text{kg}\right)}{8.314 \frac{\text{J}}{\text{mol} - \text{K}} (293 \text{K}) \ln(0.95)} = 10.5 \,\text{nm} \tag{11}$$

so that, from Eq. (10),

$$p\_a - p = -\frac{RTl n \frac{p\_v}{p\_s}}{V\_m} = \frac{\left(0.0727 \frac{\text{N}}{\text{m}}\right)}{10.5 \text{ nm}} = 6.9 \text{ MPa} \tag{12}$$

If we take the contact angles to be zero, then, from Eq. (4) and Eq. (11), *h* =2*R*<sup>2</sup> =21 nm. Now, for this mathematically idealized case of perfectly parallel plates, 21 nm would be the only spacing for which a liquid film could exist at thermodynamic equilibrium at 95% relative humidity. On the other hand, if the surfaces were curved, even slightly, then there would be a range of humidity values for which a film could be sustained at thermodynamic equilibrium. It should be noted that, in practical situations, the establishment of thermodynamic equilibri‐ um may require a considerable amount of time, such as hours, or even days. In the interim, the adhesive forces will be dictated by the current amount of liquid within the interface.

#### *2.1.2. Dynamic separation*

The foregoing analysis is applicable to conditions of static (or quasi-static) equilibrium. Additional effects may arise from viscous interactions. Consider now a situation where the upper plate is pulled upward at a prescribed rate, while the lower plate is held fixed. One approach to analyzing such a situation [9] is to assume that the liquid flow is governed by the Reynolds equation of lubrication [10].

$$\frac{1}{r}\frac{\partial}{\partial r}\left(rh^3\frac{\partial p}{\partial r}\right) = 12\,\mu\frac{\partial h}{\partial t} \tag{13}$$

where *r* is the radial coordinate measured from the center of the axisymmetric film crosssection. Assuming that the gap, *h* , is uniform (i.e., independent of *r*), the above equation can be integrated twice to give

$$p\left(r,t\right) = \Im \mu \frac{\dot{\hbar}}{h^3} + c\_1\left(t\right) \ln\left(r\right) + c\_o\left(t\right) \tag{14}$$

where *h*˙ <sup>≡</sup> <sup>∂</sup> *<sup>h</sup>* <sup>∂</sup> *<sup>t</sup>* and *c*<sup>1</sup> (*t*) and *co* (*t*) are constants of integration (i.e., independent of *r*).

To obtain the constants of integration, we assume that (1) the pressure just inside the free surface is that corresponding to the static case (see Eq. 4), and (2) the pressure is finite at *r =* 0. Then letting *R*1, as before, denote the inner radius of the droplet, we obtain (in terms of gauge pressure):

$$p\left(r,t\right) = -\frac{\gamma(\cos\theta\_1 + \cos\theta\_2)}{h} + 3\mu \frac{\dot{h}}{h^3} \left(r^2 - R\_1^2\right) \tag{15}$$

Now, the adhesive force is just given by

$$F\_{ad} = \int\_0^b -p\left(r, t\right) 2\pi r dr = \frac{\gamma(\cos\theta\_1 + \cos\theta\_2)}{h} \pi R\_1^2 + \frac{3}{2} \pi \mu \frac{\dot{h}}{h^3} R\_1^4 \tag{16}$$

For a fixed liquid volume (*Vo*), which is approximated by *Vo* =*πR*<sup>1</sup> 2 *h* , we arrive at

$$F\_{ad} = \int\_0^b -p\left(r, t\right)2\pi r dr = \frac{\gamma(\cos\theta\_1 + \cos\theta\_2)}{h^2}V\_o + \frac{3}{2}\mu\frac{\dot{h}}{h^5}\;V\_o^2\tag{17}$$

The above equation shows that adhesive force grows in proportion to the rate *h*˙ at which the plates are being pulled apart and is quite sensitive to the value of separation. Small separations require a much larger separating force than what is required at larger separations for the given separating rate. One caveat, however, is that there is a practical limit as to the magnitude of negative pressure that can be sustained during the separation of surfaces. Whereas thermo‐ dynamic equilibrium suggests that the liquid will cavitate once its absolute pressure ap‐ proaches zero (e.g., [11]), it has also been found that, in certain cases, the liquid film may achieve absolute pressures that are negative [7, 12–15]. For example, an analytical model was developed [15] for the dynamic vertical separation of opposing plates and, based on fitting with experimental data, a tensile strength of 35 kPa was found for a mineral oil. In any case, if one denotes *p*cav as the cavitation pressure (relative to atmospheric pressure), then the maximum possible adhesive force can be written as:

$$F\_{ad} = -p\_{cav} \pi R\_1^2 \tag{18}$$

#### **2.2. Liquid film between rigid, inclined surfaces**

*2.1.2. Dynamic separation*

62 Surface Energy

be integrated twice to give

<sup>∂</sup> *<sup>t</sup>* and *c*<sup>1</sup>

(*t*) and *co*

Now, the adhesive force is just given by

0

0

*b*

*ad*

*b*

where *h*˙ <sup>≡</sup> <sup>∂</sup> *<sup>h</sup>*

pressure):

Reynolds equation of lubrication [10].

The foregoing analysis is applicable to conditions of static (or quasi-static) equilibrium. Additional effects may arise from viscous interactions. Consider now a situation where the upper plate is pulled upward at a prescribed rate, while the lower plate is held fixed. One approach to analyzing such a situation [9] is to assume that the liquid flow is governed by the

> <sup>1</sup> <sup>3</sup> <sup>12</sup>*<sup>p</sup> <sup>h</sup> rh rr r t*

( ) ( ) ( ) ( ) <sup>3</sup> <sup>1</sup> , 3 ln *<sup>o</sup> <sup>h</sup> prt c t r c t <sup>h</sup>* =+ +

To obtain the constants of integration, we assume that (1) the pressure just inside the free surface is that corresponding to the static case (see Eq. 4), and (2) the pressure is finite at *r =* 0. Then letting *R*1, as before, denote the inner radius of the droplet, we obtain (in terms of gauge

( ) ( ) 1 2 2 2

 q

( ) 1 2 2 4

( ) 1 2 <sup>2</sup>

+

(cos cos ) <sup>3</sup> , 2 <sup>2</sup>

*ad o o <sup>h</sup> F p r t rdr V V*

gq

=- = + <sup>ò</sup> &

+ =- = + <sup>ò</sup> &

 q

(cos cos ) <sup>3</sup> , 2 <sup>2</sup>

*<sup>h</sup> F p r t rdr R R*

gq

(cos cos ) , 3 *<sup>h</sup> prt r R h h*

+

gq

p

For a fixed liquid volume (*Vo*), which is approximated by *Vo* =*πR*<sup>1</sup>

p

m

m

(*t*) are constants of integration (i.e., independent of *r*).

3 1

*h h*

2 5

*h h*

 q p

=- + - & (15)

1 1 3

2

*h* , we arrive at

(16)

(17)

 pm

 m

m

where *r* is the radial coordinate measured from the center of the axisymmetric film crosssection. Assuming that the gap, *h* , is uniform (i.e., independent of *r*), the above equation can

¶ ¶ æ ö ¶ ç ÷ <sup>=</sup> ¶¶ ¶ è ø (13)

& (14)

Consider the situation depicted in Figure 4, where there is a liquid film between two flat surfaces whose planes intersect. The configuration of Figure 4a is a non-equilibrium state owing to the greater free-surface curvature on the right than on the left, and the associated lower pressure (i.e., greater reduction in pressure compared to ambient). Thus, the fluid will flow from left to right, all the way up to the edge (Figure 4b) until achieving a configuration with equal free-surface curvature at left and right ends, thereby yielding the same pressure drop. The two-dimensional depiction of Figure 4, of course, obscures the required re-config‐ uration that happens in three dimensions. In fact, the entire free surface must attain the same curvature, which means that liquid would find its way to both the front and back edges as well as the right edge.

**Figure 4.** Liquid film between inclined surfaces: (a) non-equilibrium configuration and (b) equilibrium configuration.

#### **2.3. Liquid film between a smooth, rigid sphere and a rigid flat**

The sphere-flat configuration is of interest in its own right and as an important part of a rough surface contact model, in which contributions from various asperity-asperity liquid bridges are summed by viewing each pair as reflecting the interaction between a pair of spheres having the asperity curvatures.

#### *2.3.1. Static and quasi-static conditions*

The interaction between a sphere and flat bridged by a liquid film, as illustrated in Figure 5, has been analyzed in [16]. When the radial width of the liquid film (*b*) is sufficiently small compared to the radius of the sphere, the slope of the sphere at the location of the free surface may be taken as horizontal. In this case, the pressure within the film will be given by Eq. (4) with *h* replaced by *h* (*b*), the film thickness at the radial location of the free surface. Additionally, the sphere contour can be approximated well by that of a paraboloid, so that

$$h(b) = D + \frac{b^2}{2R} \tag{19}$$

This gives

$$p\_a - p = \frac{\gamma(\cos \theta\_1 + \cos \theta\_2)}{h(b)} = \frac{\gamma(\cos \theta\_1 + \cos \theta\_2)}{D + \frac{b^2}{2R}} \tag{20}$$

The force of adhesion is obtained by multiplying this pressure difference by the cross-section area of the liquid bridge (*πb* <sup>2</sup> ), giving (after re-arrangement)

$$F\_{ad} = 2\pi R\gamma(\cos\theta\_1 + \cos\theta\_2) \left(1 - \frac{D}{h(b)}\right) = \frac{2\pi R\gamma(\cos\theta\_1 + \cos\theta\_2)}{1 + \frac{2RD}{b^2}}\tag{21}$$

Several studies have considered the role of relative humidity on the adhesion between a sphere and a flat (or sphere on sphere) [9, 17–23], where, at thermodynamic equilibrium, the radius of the curvature of the free surface of the meniscus would be equal to the Kelvin radius, per Eq. (9). Such analysis is most appropriate for volatile liquids [24]. In this case the value of *h* (*b*) appearing in Eq. (21) would be determined directly by the relative humidity, via Eqs. (3) and (9).

**Figure 5.** Liquid film between a rigid sphere and a rigid flat.

#### *2.3.2. Dynamic separation*

*2.3.1. Static and quasi-static conditions*

This gives

64 Surface Energy

(9).

area of the liquid bridge (*πb* <sup>2</sup>

*ad*

pg

**Figure 5.** Liquid film between a rigid sphere and a rigid flat.

The interaction between a sphere and flat bridged by a liquid film, as illustrated in Figure 5, has been analyzed in [16]. When the radial width of the liquid film (*b*) is sufficiently small compared to the radius of the sphere, the slope of the sphere at the location of the free surface may be taken as horizontal. In this case, the pressure within the film will be given by Eq. (4) with *h* replaced by *h* (*b*), the film thickness at the radial location of the free surface. Additionally,

> 2 2

*R*

12 12

qgq

(cos cos ) (cos cos )

The force of adhesion is obtained by multiplying this pressure difference by the cross-section

( )

Several studies have considered the role of relative humidity on the adhesion between a sphere and a flat (or sphere on sphere) [9, 17–23], where, at thermodynamic equilibrium, the radius of the curvature of the free surface of the meniscus would be equal to the Kelvin radius, per Eq. (9). Such analysis is most appropriate for volatile liquids [24]. In this case the value of *h* (*b*) appearing in Eq. (21) would be determined directly by the relative humidity, via Eqs. (3) and

2 (cos cos ) 2 (cos cos ) 1 <sup>2</sup> <sup>1</sup>

ç ÷

), giving (after re-arrangement)

2

 q

*R*

2

+

*h b RD*

pg

æ ö +

è ø +

= + (19)

1 2

 q

2

*b*

 q (20)

(21)

the sphere contour can be approximated well by that of a paraboloid, so that

( )

( )

gq

1 2

= + -= ç ÷

 q

*<sup>D</sup> <sup>R</sup> F R*

 q

*<sup>a</sup> p p h b <sup>b</sup> <sup>D</sup>*

+ + - = <sup>=</sup>

*<sup>b</sup> hb D*

Now we consider the forces that arise when a sphere of mass *m* and radius *R* is separated from the flat in a dynamic fashion, so that the minimum spacing *D* is a function of time. Denoting the instantaneous vertical spacing between the sphere surface and the flat as *h* (*r*, *t*), we have *D* =*h* (0, *t*). When the wetted radius *b* is much smaller than the sphere radius, one gets

$$h(r,t) = D\left(t\right) + \frac{r^2}{2R} \tag{22}$$

When a net external force *F* (i.e., an applied force less the sphere weight) acts on the sphere (positive upward), the governing equation becomes [25]

$$m\frac{d^2D}{dt^2} = F - F\_m - F\_v \tag{23}$$

where *Fm* is the "meniscus" force, which accounts for the effect of the pressure drop across the curved free surface of the liquid meniscus and *Fv* is the "viscous" force, which arises from the deformation of the liquid bridge. It is assumed that any buoyancy forces are negligible. Following [26], the pressure field, as derived from the solution of the Reynolds equation (e.g., [10]), can be written as

$$p\left(r,t\right) = p\left(b,t\right) - 3\,\mu R \left(\frac{1}{h\left(r,t\right)^2} - \frac{1}{h\left(b,t\right)^2}\right) \frac{1}{D} \frac{dD}{dt} \tag{24}$$

where *μ* is the liquid viscosity. Note that the wetted radius *b* is itself a function of time. Assuming that the free surface of the liquid has the same shape as when the film is quasi-static, the pressure just inside the meniscus, *p*(*b*, *t*) is given by Eq. (4). Thus, integrating over the meniscus area gives [25]

$$F\_{ad} = F\_m + F\_v = 2\pi R\gamma (\cos\theta\_1 + \cos\theta\_2) \left(1 - \frac{D}{h\left(b, t\right)}\right) + 6\pi\mu R^2 \left(1 - \frac{D}{h\left(b, t\right)}\right)^2 \frac{1}{D} \frac{dD}{dt} \tag{25}$$

where *Fad* is the adhesive force. Direct integration of the film thickness profile (19) provides the liquid volume:

$$V = \pi b^2 D + \frac{\pi b^4}{4R} = \pi R \left[ h\left(b, t\right)^2 - D^2 \right] \tag{26}$$

Assuming the meniscus volume is fixed, we set *V* =*Vo* and express the film thickness at the free surface as

$$h(b, t) = \sqrt{D^2 + \frac{V\_o}{\pi R}}\tag{27}$$

Using this result in Eq. (25) allows the force exerted by the liquid to be expressed in terms of the separation *D*

$$F\_{ad} = F\_m + F\_v = 2\pi R\gamma(\cos\theta\_1 + \cos\theta\_2) \left(1 - \frac{D}{\sqrt{D^2 + \frac{V\_o}{\pi R}}}\right) + 6\pi\mu R^2 \left(1 - \frac{D}{\sqrt{D^2 + \frac{V\_o}{\pi R}}}\right)^2 \frac{1}{D} \frac{dD}{dt} \tag{28}$$

In cases where the inertial term of Eq. (23) is negligible, the net applied load *F* is equated with the sum of the capillary and viscous forces *Fad*. Moreover, in cases where the variation in the capillary force is small compared to the variation in the viscous force, Eq. (28) can be integrated to give [9, 27]

$$\int\_{0}^{t} \left( F - F\_{m} \right) dt = \int\_{0}^{D} F\_{v} dt = \int\_{0}^{D} D\_{v} \frac{dt}{dD} dD = 6 \pi \mu R^{2} \ln \left| \frac{D\_{s} \left( D\_{o} + \sqrt{D\_{o}^{2} + \frac{V\_{o}}{\pi R}} \right)^{2} \sqrt{D\_{s}^{2} + \frac{V\_{o}}{\pi R}}}{D\_{o} \left( D\_{s} + \sqrt{D\_{s}^{2} + \frac{V\_{o}}{\pi R}} \right)^{2} \sqrt{D\_{o}^{2} + \frac{V\_{o}}{\pi R}}} \right| \tag{29}$$

where *ts* is the time to completely separate the surfaces, *Ds* is the distance at which the sphere and flat are considered completely separated, and *Do* is the sphere-flat spacing at the beginning of the separation process. It was argued in [27] that even in the case of initial solid-solid contact, the value of *Do* should not be zero, but should be selected in accordance with the interface roughness. Eq. (29) suggests that the separation process for a viscous-dominated interaction is governed by the time integral of the viscous force, known as the viscous impulse, the value of which is a function only of the input parameters of the problem [27]. Now the distance of complete separation *Ds* is taken as infinite in [27], and finite in [9]. For *Ds* →*∞*, Eq. (29) can be simplified [27], giving

$$I\_{v} = 6\pi\mu R^{2} \ln \left[ \frac{\left(D\_{o} + \sqrt{D\_{o}^{2} + \frac{V\_{o}}{\pi R}}\right)^{2}}{4D\_{o}\sqrt{D\_{o}^{2} + \frac{V\_{o}}{\pi R}}} \right] = 6\pi\mu R^{2} \ln \left[ \frac{\left(D\_{o} + \sqrt{D\_{o}^{2} + \frac{V\_{o}}{\pi R}}\right)^{2}}{4D\_{o}\sqrt{D\_{o}^{2} + \frac{V\_{o}}{\pi R}}} \right] \tag{30}$$

where *Iv* is the viscous impulse. Applying Eq. (30) to the case of a sphere interacting with a surface that is initially covered with a thin film of uniform thickness *ho*, the above is approxi‐ mated by [27]

$$I\_v = 6\pi\mu R^2 \ln\left(h\_o / 2D\_o\right) \tag{31}$$

One important result of the above relationship is that the rate of applied loading determines the peak adhesive load developed during separation, which we label here the "pull-off force" (*F*pull−off).

For example, when the externally applied force increases at a constant rate (*F*˙ ), the pull-off force takes the form:

$$F\_{\text{pull-off}} = F\_m + \sqrt{2\dot{F}I\_v} \tag{32}$$

A modified approach is needed to analyze the "fully-flooded" case, where the sphere interacts with a sufficiently thick lubricant film that further increases to the film thickness have negligible impact on the adhesive force. In this case, Eq. (32) still holds, but the viscous impulse becomes [25]

$$I\_v = 6\pi\mu R^2 \ln\left| 0.1 \frac{\left(6\pi\mu R^2\right)^{\frac{3}{2}}}{mD\_o\sqrt{2\bar{F}}} \right| \tag{33}$$

where *m* is the mass of the sphere.

Assuming the meniscus volume is fixed, we set *V* =*Vo* and express the film thickness at the free

Using this result in Eq. (25) allows the force exerted by the liquid to be expressed in terms of

p

In cases where the inertial term of Eq. (23) is negligible, the net applied load *F* is equated with the sum of the capillary and viscous forces *Fad*. Moreover, in cases where the variation in the capillary force is small compared to the variation in the viscous force, Eq. (28) can be integrated

2

where *ts* is the time to completely separate the surfaces, *Ds* is the distance at which the sphere and flat are considered completely separated, and *Do* is the sphere-flat spacing at the beginning of the separation process. It was argued in [27] that even in the case of initial solid-solid contact, the value of *Do* should not be zero, but should be selected in accordance with the interface roughness. Eq. (29) suggests that the separation process for a viscous-dominated interaction is governed by the time integral of the viscous force, known as the viscous impulse, the value of which is a function only of the input parameters of the problem [27]. Now the distance of complete separation *Ds* is taken as infinite in [27], and finite in [9]. For *Ds* →*∞*, Eq. (29) can be

ò òò (29)

0 00 2 2

2 2

p

p

4 4

è øè ø = =

6 6

*I R ln R ln*

pm

*t DD so o s*

è ø -== =

6

*F F dt F dt F dD R ln*

p*R*

= + (27)

2

*o o*

*R R*

*V V D dt*

 p

2 2 2

*o o*

 p

 p

*R R*

2

 p

 p

*o o os s <sup>o</sup>*

p

p

*V V DD D D*

æ ö ç ÷ ++ + ë û è ø

2 2

*R R*

*R R*

2 2

*o o o o o o*

é ùé ù æ öæ ö ê úê ú ç ÷ç ÷ + + + +

*V V D D D D*

2 2

*V V D D D D*

pm

*o o o o o o*

+ + ë ûë û

é ù æ ö ê ú ç ÷ ++ +

2 2

+ + è øè ø

pm

æ öæ ö ç ÷ç ÷

*D D*

*dt R R*

*dD V V DD D D*

2

(28)

(30)

( ) <sup>2</sup> , *Vo h bt D*

1 2 (cos cos ) 1 6 1 *ad m v*

*<sup>D</sup> D dD F FF R <sup>R</sup>*

1 2

 q

=+= + - + -

 q

pg

surface as

66 Surface Energy

the separation *D*

to give [9, 27]

( )

simplified [27], giving

*v*

pm

*s s s*

*m vv*

It is emphasized here that Eqs. (28)-(33) presume the liquid film is not experiencing any cavitation. As discussed previously (see Eq. (18)), the potential development of a fully cavitated film would provide an upper bound for the adhesive force.

#### **2.4. Liquid film between smooth, elastic flats**

Figure 6 depicts a scenario when a liquid film interacts with two semi-infinite elastic bodies, where *E*1,2 and *ν*1,2 are the elastic moduli and Poisson ratios of bodies 1 and 2, respectively and *H* is the uniform gap between the surfaces that exists in the absence of deformation. For this situation, the pressure within the liquid film causes elastic deformation of the half-spaces. Here we focus our attention on the case were the liquid film wets both surfaces such that they each experience a contact angle less than 90 degrees. This problem has been analyzed previously [28] and that work is summarized here. Letting Δ*p* represent the pressure drop across the free surface of the liquid (from outside to inside), the film pressure (gauge) can be expressed as

$$p(r) = \begin{cases} -\Delta p & r \le b \\ 0 & r > b \end{cases} \tag{34}$$

This pressure field causes an associated deformation field [29]

$$u(r) = u\_1(r) + u\_2(r) = \begin{cases} \frac{4\Delta p b}{\pi E'} \int\_0^{\pi/2} \sqrt{1 - \frac{r^2}{b^2} \sin^2 \varphi} \, d\varphi & r \le b \\\\ \frac{4\Delta p b}{\pi E'} \int\_0^{\pi/2} \frac{b^2 \cos^2 \varphi}{\sqrt{1 - \frac{r^2}{b^2} \sin^2 \varphi}} \, d\varphi & r > b \end{cases} \tag{35}$$

**Figure 6.** A liquid film bridging two elastic half-spaces.

In the above equation, *u*<sup>1</sup> (*r*) and *u*<sup>2</sup> (*r*) are the normal surface displacements of bodies 1 and 2, respectively (each positive toward the opposing body), *u*(*r*) is the total displacement, and the reduced modulus *E* ′ is given by 1 / *E* ′ ≡(1−*ν*<sup>1</sup> 2 ) / *E*<sup>1</sup> + (1−*ν*<sup>2</sup> 2 )/ *E*2. It is assumed here that the bulk positions of the bodies are fixed, so that opposing surface points remote to the interface are maintained at a spacing of *H.* At any radial position within the wetted film, the film thickness can be expressed as

$$h(r) = H - \mu(r) \tag{36}$$

Using Eq. (35), the volume of the liquid bridge (*Vo*), which is assumed to be fixed, is given by

$$V\_o = \bigcup\_{o}^{b} 2\pi r h(r) dr = \pi Hb^2 - \frac{16}{3E} \Delta p b^3 \tag{37}$$

The equilibrium configuration can be determined by considering the minimization of the free energy, which is comprised of elastic strain energy (*US*) and surface energy *UE.* The elastic strain energy is simply given by the work done in creating the deformation field

$$
\hbar \mathcal{U}\_E = \frac{1}{2} \sum\_{o}^{b} \Delta p u \left( r \right) 2\pi r dr \tag{38}
$$

Using Eq. (35) and carrying out the integration gives

( ) <sup>0</sup> *prb p r r b* ìï-D £ <sup>=</sup> <sup>í</sup>

/2 2 2 1 2

p

ï ï ¢ <sup>ï</sup>

î

p

/2 2

<sup>4</sup> 1 sin

0

ò

p

*ur u r u r pb <sup>b</sup> d rb*

¢

p

4 cos

*E r*

respectively (each positive toward the opposing body), *u*(*r*) is the total displacement, and the

positions of the bodies are fixed, so that opposing surface points remote to the interface are maintained at a spacing of *H.* At any radial position within the wetted film, the film thickness

Using Eq. (35), the volume of the liquid bridge (*Vo*), which is assumed to be fixed, is given by

( ) 2 3 <sup>16</sup> <sup>2</sup>

*V rh r dr Hb pb <sup>E</sup>*

 p

3

) / *E*<sup>1</sup> + (1−*ν*<sup>2</sup>

2

≡(1−*ν*<sup>1</sup> 2

*E b*

2

<sup>ì</sup> <sup>D</sup> <sup>ï</sup> - £ <sup>ï</sup>

2 0 2 2

1 sin

*b*


2

y y y

y

y

(*r*) are the normal surface displacements of bodies 1 and 2,

*hr H ur* ( ) = - ( ) (36)

¢ = = -D ò (37)

)/ *E*2. It is assumed here that the bulk

>

<sup>ò</sup> (35)

*pb <sup>r</sup> d rb*

This pressure field causes an associated deformation field [29]

<sup>ï</sup> =+= <sup>í</sup> <sup>D</sup>

( ) ( ) ( )

**Figure 6.** A liquid film bridging two elastic half-spaces.

reduced modulus *E* ′ is given by 1 / *E* ′

(*r*) and *u*<sup>2</sup>

*b*

p

*o o*

In the above equation, *u*<sup>1</sup>

68 Surface Energy

can be expressed as

<sup>ï</sup> <sup>&</sup>gt; <sup>î</sup> (34)

$$
\Delta U\_E = \frac{8}{3E'} \Delta p^2 b^3 \tag{39}
$$

Now the surface energy consists for energy contributions from the solid-vapor, solid-liquid, and liquid-vapor interfaces, so that

$$\mathcal{U}L\_S = \pi \left(R\_o^2 - b^2\right) \left(\mathcal{V}\_{SV\_1} + \mathcal{V}\_{SV\_2}\right) + \pi b^2 \left(\mathcal{V}\_{SL\_1} + \mathcal{V}\_{SL\_2}\right) + A\_{LV}\mathcal{V} \tag{40}$$

where subscripts 1 and 2 refer to the upper and lower surfaces, respectively and *Ro* is the radius of an arbitrary control region that encloses the interface. It is reasonable to assume here that the film thickness at the free surface is sufficiently small compared to the meniscus radius that the energy contributions from the liquid-vapor interface are negligible. Then the total free energy is given by

$$\mathcal{U}I\_T = \mathcal{U}\_S + \mathcal{U}\_E = \pi \left(R\_o^2 - b^2\right) \left(\gamma\_{SV\_1} + \gamma\_{SV\_2}\right) + \pi b^2 \left(\gamma\_{SL\_1} + \gamma\_{SL\_2}\right) + \frac{8}{3E} \Delta p^2 b^3 \tag{41}$$

Applying Eq. (5) to each surface and recalling that *γ* ≡*γLV* one obtains

$$\mathcal{U}L\_T = \pi R\_o^2 \left(\mathcal{Y}\_{SV\_1} + \mathcal{Y}\_{SV\_2}\right) - \pi b^2 \gamma \left(\cos\theta\_1 + \cos\theta\_2\right) + \frac{8}{3E} \Delta p^2 b^3 \tag{42}$$

A stable equilibrium corresponds to the minimization of the free energy *UT* under the constraint of constant liquid volume *Vo*. Let us now introduce dimensionless quantities:

$$
\eta \equiv \frac{2\Delta p b}{E'H} \tag{43}
$$

$$\mathcal{U}\_T^\* \equiv \frac{\mathcal{U}\_T - \pi \mathcal{R}\_o^2 \left(\mathcal{V}\_{SV\_1} + \mathcal{V}\_{SV\_2}\right)}{\sqrt{\pi} E' H^{3/2} V\_o^{1/2}} \tag{44}$$

$$\Gamma \equiv \frac{\gamma^2 \left(\cos\theta\_1 + \cos\theta\_2\right)^2 V\_o}{4E^2 H^5} \tag{45}$$

With these definitions, the dimensionless free energy can be expressed as

$$
\Delta U\_T = \frac{2\eta^2}{3\pi\sqrt{1 - \left(8/3\pi\right)\eta}} - 2\sqrt{\frac{\Gamma}{\pi}} \frac{1}{1 - \left(8/3\pi\right)\eta} \tag{46}
$$

Note also that from Eqs. (35), (36) and (43), the minimum film thickness is given by

$$\dot{m}\_{\rm min} = H - \mu \left( 0 \right) = H - \frac{2\Delta p b}{E'} = H \left( 1 - \eta \right) \tag{47}$$

So that *η* provides a dimensionless measure as to the degree of surface approach. When *η* = 0, the surfaces are at the original separation throughout, and when *η* = 1 the surfaces come into point contact. For a stable equilibrium, the dimensionless energy achieves a local minimum with respect to *η*. Thus, a necessary (but not sufficient) condition for stable equilibrium is that *dUT* \* / *dη* =0, which yields the following requirement:

$$
\sigma^2 \left( 1 - \frac{2}{\pi} \eta \right)^2 \left( 1 - \frac{8}{3\pi} \eta \right) = \frac{16\Gamma}{\pi} \tag{48}
$$

The solution space of Eq. (48) is shown in Figure 7. An investigation of *d* <sup>2</sup> *UT* \* / *dη* <sup>2</sup> at the various equilibrium values of *η* reveals that values to the left of the peak (*η* <0.5577) correspond to stable equilibrium configurations, whereas those to the right of the peak, correspond to unstable equilibrium configurations. From the graph, we see that no equilibrium configurations exist for values of Γ greater than 0.0134. This result implies that solid-solid contact must occur whenever Γ>0.0134. Further, once solid-solid contact occurs, the contact region grows without bound (i.e., the free energy, which now includes a solid-solid contribution, decreases monot‐ onically with increasing contact radius).

**Figure 7.** Equilibrium configurations for two half-spaces bridged by a liquid film.

Using Figure 7, one can determine the adhesive force. Letting the subscript "eq" identify values corresponding to a stable equilibrium configuration, it can be shown using Eqs. (35)–(37), (43), and (48), that

$$b\_{eq} = \sqrt{\frac{V\_o}{\pi H \left(1 - \frac{8}{3\pi} \eta\_{eq}\right)}}\tag{49}$$

$$h\_{eq}(b\_{eq}) = H\left(1 - \frac{2}{\pi}\eta\_{eq}\right) \tag{50}$$

$$
\Delta p\_{eq} = \frac{\gamma (\cos \theta\_1 + \cos \theta\_2)}{h\_{eq}(b\_{eq})} \tag{51}
$$

Then, the adhesive force is given by

2 *pb E H*

2


( ) 2 2 1 2 '2 5 cos cos

*E H*

4

<sup>2</sup> \* 2 1 <sup>2</sup> 3 1 8/3 1 8/3

Note also that from Eqs. (35), (36) and (43), the minimum film thickness is given by

min ( ) ( ) <sup>2</sup> 0 1 *pb h Hu H H*

2 <sup>2</sup> 2 8 16 1 1 3

hh

p

The solution space of Eq. (48) is shown in Figure 7. An investigation of *d* <sup>2</sup>

 ph

<sup>G</sup> º -

pg

*EH V*

( ) 1 2

 g

*Vo*

*o*

3/2 1/2 *T o SV SV*

> q

( ) ( )

*E*

 h

 pp

equilibrium values of *η* reveals that values to the left of the peak (*η* <0.5577) correspond to stable equilibrium configurations, whereas those to the right of the peak, correspond to unstable equilibrium configurations. From the graph, we see that no equilibrium configurations exist for values of Γ greater than 0.0134. This result implies that solid-solid contact must occur whenever Γ>0.0134. Further, once solid-solid contact occurs, the contact region grows without bound (i.e., the free energy, which now includes a solid-solid contribution, decreases monot‐

æ öæ ö <sup>G</sup> ç ÷ç ÷ - -= è øè ø

So that *η* provides a dimensionless measure as to the degree of surface approach. When *η* = 0, the surfaces are at the original separation throughout, and when *η* = 1 the surfaces come into point contact. For a stable equilibrium, the dimensionless energy achieves a local minimum with respect to *η*. Thus, a necessary (but not sufficient) condition for stable equilibrium is that

p

<sup>D</sup> <sup>º</sup> ¢ (43)

º (44)

<sup>+</sup> <sup>º</sup> (45)

p h

h

<sup>D</sup> =- =- = - ¢ (47)

*UT* \* / *dη* <sup>2</sup> (48)

at the various


h

*U R*

gq

With these definitions, the dimensionless free energy can be expressed as

h

p¢

\*

*U*

*T*

Γ

p

\* / *dη* =0, which yields the following requirement:

onically with increasing contact radius).

*UT*

*dUT*

70 Surface Energy

$$F\_{ad} = \pi b\_{eq}^2 \Delta p\_{eq} = \frac{\gamma (\cos \theta\_1 + \cos \theta\_2) V\_o}{H^2 \left(1 - \frac{8}{3\pi} \eta\_{eq}\right) \left(1 - \frac{2}{\pi} \eta\_{eq}\right)}\tag{52}$$

#### **2.5. Liquid film between smooth, elastic spheres**

When a liquid bridges two elastic spheres [30], as illustrated in Figure 8, the situation is similar to the case of two elastic half-spaces (discussed above), but with an added feature due the surface curvature. The displacement profile is still given by Eq. (35), but the film thickness profile is now given by

$$
\mu h(r) = H + \frac{r^2}{2R} - \mu(r) \tag{53}
$$

where *R* is the composite radius of curvature, defined by 1 / *R* =1 / *R*<sup>1</sup> + 1 / *R*2. Thus, upon integrating, the liquid volume takes the form

$$V\_o = \pi Hb^2 - \frac{16}{3E'} \Delta p b^3 + \frac{\pi b^4}{4R} \tag{54}$$

Note that the expressions for the elastic strain energy and surface energy are the same as those for the two half-spaces, so that the total free energy is still given by Eq. (41). In addition to nondimensional parameters *η* and Γ (Eqs. 43 and 45), we introduce a dimensionless volume according to

$$
\Psi \equiv \frac{V\_o}{RH^2} \tag{55}
$$

and use a different form for the dimensionless free energy

$$\mathcal{U}\_T^\* \equiv \frac{\mathcal{U}\_T - \pi \mathcal{R}\_o^2 \left(\mathcal{V}\_{SV\_1} + \mathcal{V}\_{SV\_2}\right)}{\pi \mathbb{E}' H^{5/2} \mathcal{R}^{1/2}} \tag{56}$$

This results in

$$\mathbf{M}\_{T}^{\*} = 4\sqrt{\frac{\Gamma}{\pi}} \left[ 1 - \frac{8}{3\pi}\eta - \sqrt{\left(1 - \frac{8}{3\pi}\eta\right)^{2} + \frac{\Psi}{\pi}} \right] + \frac{2}{3\pi}\eta^{2}\sqrt{-2\left(1 - \frac{8}{3\pi}\eta\right) + \sqrt{\left(1 - \frac{8}{3\pi}\eta\right)^{2} + \frac{\Psi}{\pi}}} \tag{57}$$

Setting *dUT* \* / *dη* =0 leads to the following necessary condition on *η* for a stable equilibrium to be achieved

$$\frac{1}{2}\eta^2 \left[\frac{2}{3\pi}\eta + \sqrt{\left(1 - \frac{8}{3\pi}\eta\right)^2 + \frac{\Psi}{\pi}}\right]^2 \left[1 - \frac{8}{3\pi}\eta + \sqrt{\left(1 - \frac{8}{3\pi}\eta\right)^2 + \frac{\Psi}{\pi}}\right] = \frac{32\Gamma}{\pi} \tag{58}$$

It can readily be shown that for Ψ=0, Eq. (58) reduces to Eq. (48), which is applicable to two half-spaces. This result is expected because for a finite liquid volume and a finite, non-zero value of surface spacing, an infinite value of sphere radius (which corresponds to a flat-flat interface) causes Ψ to vanish as per Eq. (55).

The solution space for Eq. (58) is plotted in Figure 9 for several values of dimensionless volume Ψ. As observed, the smallest chosen value of Ψ yields a curve that is quite close to the halfspace solution (Figure 7). It turns out that stable equilibrium configurations exist without solidsolid contact for a range of Γ values that depends on the value of Ψ. This relationship is summarized in Figure 10, which reveals regions with and without solid-solid contact within Ψ−Γ space. The boundary curve is given by Γc=0.03007Ψ1.3955 <sup>+</sup> 0.01336 . If, for a given Ψ, Γ<Γ<sup>c</sup> then there is no solid-solid contact, whereas for Γ≥Γc the spheres must experience contact.

For equilibrium configurations that do not involve solid-solid contact, the pressure drop is given by Eq. (51), but with the gap at the free-surface given by

$$h\_{eq}(b\_{eq}) = H \left[ \frac{2}{3\pi} \eta\_{eq} + \sqrt{\left( 1 - \frac{8}{3\pi} \eta\_{eq} \right)^2 + \frac{\Psi}{\pi}} \right] \tag{59}$$

and the wetted radius given by (via solution of Eq. 54)

$$b\_{eq} = \left[\frac{2V\_o}{\pi H \left[1 - \frac{8}{3\pi}\eta\_{eq} + \sqrt{\left(1 - \frac{8}{3\pi}\eta\_{eq}\right)^2 + \frac{V\_o}{\pi RH^2}}\right]}\right]^{\frac{1}{2}}\tag{60}$$

Thus, the adhesive force then becomes

.

surface curvature. The displacement profile is still given by Eq. (35), but the film thickness

where *R* is the composite radius of curvature, defined by 1 / *R* =1 / *R*<sup>1</sup> + 1 / *R*2. Thus, upon

2 3 16 3 4 *<sup>o</sup> <sup>b</sup> V Hb pb E R*

Note that the expressions for the elastic strain energy and surface energy are the same as those for the two half-spaces, so that the total free energy is still given by Eq. (41). In addition to nondimensional parameters *η* and Γ (Eqs. 43 and 45), we introduce a dimensionless volume

> 2 *Vo RH*

2


*EH R* pg

33 3 3 3

 h

2 2

 h

 p

\* / *dη* =0 leads to the following necessary condition on *η* for a stable equilibrium to

é ù G Y ê ú æ ö æ öæ ö <sup>Y</sup> = - - - + + -- + - + ç ÷ ç ÷ç ÷ è ø è øè ø ë û

*U R*

p

\* 2 88 2 8 8 41 1 21 1

 pp

2

33 33

 p

<sup>2</sup> 2 8 8 8 <sup>32</sup> 1 11

<sup>é</sup> ù é <sup>ù</sup> æ ö <sup>Y</sup> æ ö Y G <sup>ê</sup> ú ê <sup>ú</sup> +- + - +- += ç ÷ ç ÷ <sup>ê</sup> ú ê <sup>ú</sup> è ø è ø êë ú ê û ë úû

( ) 1 2

 g

2 2

 h

 p

> h

> > pp

 p

 h

> p

(57)

(58)

 p

5/2 1/2 *T o SV SV*

p

and use a different form for the dimensionless free energy

\*

*U*

hh

 p

 h

 p *T*

4

p

=+ - (53)

¢ = - D+ (54)

Y º (55)

¢ <sup>º</sup> (56)

( ) ( ) 2 2 *<sup>r</sup> hr H ur R*

profile is now given by

72 Surface Energy

according to

This results in

p

hh

p

 p

*UT*

Setting *dUT*

be achieved

integrating, the liquid volume takes the form

$$F\_{at} = \pi b\_{eq}^2 \Delta p\_{eq} = \frac{2\gamma(\cos\theta\_1 + \cos\theta\_2)V\_o}{H^2 \left[1 - \frac{8}{3\pi}\eta\_{eq} + \sqrt{\left(1 - \frac{8}{3\pi}\eta\_{eq}\right)^2 + \frac{V\_o}{\pi RH^2}}\right] \left[\frac{2}{3\pi}\eta\_{eq} + \sqrt{\left(1 - \frac{8}{3\pi}\eta\_{eq}\right)^2 + \frac{\Psi}{\pi}}\right]}\tag{61}$$

The above force represents the external, separating force (over and above the weight of the sphere) required to maintain the spheres at the given configuration (i.e., with undeformed separation, *H*).

In cases where Γ>Γc, the solids come into contact over some contact radius *a*, with the contact region surrounded by an annulus of liquid. The presence of contact modifies the form of the free energy, which becomes [31]

$$\mathcal{U}L\_T = \pi \left(R\_o^2 - b^2\right) \left(\gamma\_{SV\_1} + \gamma\_{SV\_2}\right) - \pi \left(b^2 - a^2\right) \left(\gamma\_{SL\_1} + \gamma\_{SL\_2}\right) + \pi a^2 \gamma\_{S\_{12}} \tag{62}$$

where *γS*12 is the surface tension associated with the solid-solid interface. The dimensionless formulation involves two additional ratios [31]:

$$am \equiv \frac{b}{a} \tag{63}$$

$$\Phi \equiv \frac{\Delta \mathcal{Y}}{\gamma \left(\cos \theta\_1 + \cos \theta\_2\right)} - 1 \tag{64}$$

where Δ*γ* is the well-known work of adhesion and is given by [8]

$$
\Delta \chi = \chi\_{SV\_1} + \chi\_{SV\_2} - \chi\_{S\_{12}} \tag{65}
$$

The equilibrium solution, for given values of Γ and Ψ, is now expressed in terms of both *η* and *m*. In the case of solid-solid contact, no analytical expressions exist for the elastic strain energy, owing to the unknown contact solid-solid pressure distribution. Therefore, the equilibrium configurations and associated adhesive forces must be acquired through a numerical process [31].

It can be shown [31] that the advent of solid-solid contact introduces hysteresis, just as in the case of the JKR contact model [32], which applies to dry contact. Thus, the set of configurations that the interface would pass through when breaking the contact, such as during a controlled separation process, would be different from those experienced upon its formation. For example, the value of *H* at which the solid-solid contact is lost during a separation process is different from the value of *H* that corresponds to the formation of solid-solid contact during an approach process. Put another way, there is a jump-on instability at a certain *H* upon approach, where the interface goes suddenly from no contact to contact, as well as a jump-off instability upon separation (at a larger *H*), where the interface proceeds suddenly from having a contact radius *a* to having no solid-solid contact. One convenient experimental measure of the strength of an adhesive contact is the pull-off force, which can take on different values depending upon how the pull-off process is conducted. When the separation *H* (which is defined by the minimum gap between the undeformed sphere contours) is specified and increased quasi-statically, the interface will reach a configuration that is unstable and then abruptly lose contact. The magnitude of external, separating force required to reach this point of instability during separation is defined as the pull-off force during a controlled separation process.

In cases where Γ>Γc, the solids come into contact over some contact radius *a*, with the contact region surrounded by an annulus of liquid. The presence of contact modifies the form of the

> ( )( ) ( )( ) 1 2 1 2 <sup>12</sup> 2 2 2 2 <sup>2</sup> *U Rb T o SV SV SL SL <sup>S</sup>* = - +-- ++

> > *b m*

( ) 1 2 <sup>Δ</sup> Φ 1 cos cos g

 q

1 2 12

The equilibrium solution, for given values of Γ and Ψ, is now expressed in terms of both *η* and *m*. In the case of solid-solid contact, no analytical expressions exist for the elastic strain energy, owing to the unknown contact solid-solid pressure distribution. Therefore, the equilibrium configurations and associated adhesive forces must be acquired through a

It can be shown [31] that the advent of solid-solid contact introduces hysteresis, just as in the case of the JKR contact model [32], which applies to dry contact. Thus, the set of configurations that the interface would pass through when breaking the contact, such as during a controlled separation process, would be different from those experienced upon its formation. For example, the value of *H* at which the solid-solid contact is lost during a separation process is different from the value of *H* that corresponds to the formation of solid-solid contact during an approach process. Put another way, there is a jump-on instability at a certain *H* upon approach, where the interface goes suddenly from no contact to contact, as well as a jump-off instability upon separation (at a larger *H*), where the interface proceeds suddenly from having a contact radius *a* to having no solid-solid contact. One convenient experimental measure of the strength of an adhesive contact is the pull-off force, which can take on different values

 g

Δ *SV SV S*

 g

gq

 g

 g

 pg*b a a* (62)

*<sup>a</sup>* <sup>º</sup> (63)

º - <sup>+</sup> (64)

=+- (65)

 p

free energy, which becomes [31]

74 Surface Energy

numerical process [31].

p

 g  g

where *γS*12 is the surface tension associated with the solid-solid interface.

The dimensionless formulation involves two additional ratios [31]:

where Δ*γ* is the well-known work of adhesion and is given by [8]

gg

**Figure 8.** A liquid film bridging two smooth, elastic half-spaces with no solid-solid contact.

**Figure 9.** Equilibrium configurations for two spheres bridged by a liquid film without solid-solid contact.

**Figure 10.** Regions in Γ-Ψ space showing regions with and without solid-solid contact.

#### **2.6. Liquid film between contacting rough, elastic surfaces**

Adhesive forces arising due to the presence of a liquid film between rough, elastic (or elasticplastic) surfaces have been the subject of several recent works [17, 19–21, 33–37]. Figure 11 depicts a situation where two rough, elastic surfaces are in contact in the presence of an intervening liquid film. Taking into consideration a three-dimensional geometry, the assump‐ tion here is that the liquid film is continuous, so that there are no regions of liquid completely encased within a zone of solid-solid contact. Now in the case where the liquid wets the surfaces (i.e., the contact angles are less than 90°), the free surface of the liquid is concave and the film pressure is sub-ambient. Assuming that the lateral dimensions are much greater than the liquid film thickness, the pressure drop across the free surface is given by

$$\Delta p = \ p\_a - p = \frac{\gamma(\cos \theta\_1 + \cos \theta\_2)}{h\_{fs}} \tag{66}$$

where *h fs* is the film thickness at the location of the free surface. For a continuous liquid film in static equilibrium, the pressure throughout the film must be the same, so Eq. (66) suggests that the periphery of the liquid film is at a constant height. The overall equilibrium shape of the film will depend upon the details of the gap distribution within the surfaces, which itself will be modulated by surface deformation due to the tensile stresses exerted by the liquid film. In general, the equilibrium configuration of the liquid film will not be axisymmetric. However, for surfaces that are nearly flat aside from a small-scale roughness, we can expect that the equilibrium film shape will be nearly axisymmetric. Assuming an axisymmetric liquid film, one can describe the establishment of equilibrium as follows: First, suppose a quantity of liquid is placed upon a surface that will serve as the lower surface of the contacting pair. Then, the other surface is placed in contact with the lower surface (and the liquid film) under some external load *P*. As solid-solid contact is first formed at the mutual asperity peaks, the liquid film will be quickly squeezed out to a radius that is determined by the given liquid volume and the average gap between the surfaces within the wetted region. However, as capillary forces take effect, the elastic surfaces will further deform, thereby reducing the mean gap between the surfaces and causing an increase in the wetted radius. An increase in the wetted radius in conjunction with a decreasing interfacial gap will cause a greater tensile force *Ft* and greater surface deformation. Hence, there is the possibility that the rate of increase of the capillary force (with expanding wetted radius) will exceed the rate of increase of the com‐ pressive force coming from the contacting asperities. In such a situation, the interface is expected to collapse, whereby the solid surfaces come into complete or nearly complete contact.

One numerical model of such an interface appears in [35]. Here it is assumed that the liquid film is axisymmetric and that deformation of the asperities is modeled according to the multiscale contact model of [38]. Thus, the surface topography is characterized by its spectral content and algebraic formulas are applied to compute the effects of external and capillary forces on the average spacing within the interface. Another important assumption is that the mean spacing *h* ¯ within the wetted region is a good approximation to the spacing at the free surface *h fs* so that pressure drop within the film is approximated by

$$\Delta p = \frac{\gamma (\cos \theta\_1 + \cos \theta\_2)}{\bar{h}} \tag{67}$$

**Figure 11.** A liquid film bridging two rough, elastic surfaces.

Thus, the tensile force (*Ft*) becomes

**Figure 10.** Regions in Γ-Ψ space showing regions with and without solid-solid contact.

film thickness, the pressure drop across the free surface is given by

Adhesive forces arising due to the presence of a liquid film between rough, elastic (or elasticplastic) surfaces have been the subject of several recent works [17, 19–21, 33–37]. Figure 11 depicts a situation where two rough, elastic surfaces are in contact in the presence of an intervening liquid film. Taking into consideration a three-dimensional geometry, the assump‐ tion here is that the liquid film is continuous, so that there are no regions of liquid completely encased within a zone of solid-solid contact. Now in the case where the liquid wets the surfaces (i.e., the contact angles are less than 90°), the free surface of the liquid is concave and the film pressure is sub-ambient. Assuming that the lateral dimensions are much greater than the liquid

1 2 (cos cos ) *<sup>a</sup>*

where *h fs* is the film thickness at the location of the free surface. For a continuous liquid film in static equilibrium, the pressure throughout the film must be the same, so Eq. (66) suggests that the periphery of the liquid film is at a constant height. The overall equilibrium shape of the film will depend upon the details of the gap distribution within the surfaces, which itself will be modulated by surface deformation due to the tensile stresses exerted by the liquid film. In general, the equilibrium configuration of the liquid film will not be axisymmetric. However, for surfaces that are nearly flat aside from a small-scale roughness, we can expect that the equilibrium film shape will be nearly axisymmetric. Assuming an axisymmetric liquid film, one can describe the establishment of equilibrium as follows: First, suppose a quantity of liquid is placed upon a surface that will serve as the lower surface of the contacting pair. Then, the other surface is placed in contact with the lower surface (and the liquid film) under some external load *P*. As solid-solid contact is first formed at the mutual asperity peaks, the liquid

gq

*ppp <sup>h</sup>*

*fs*

 q+

D= - = (66)

**2.6. Liquid film between contacting rough, elastic surfaces**

76 Surface Energy

$$F\_t = \pi b^2 \frac{\chi(\cos \theta\_1 + \cos \theta\_2)}{\tilde{h}} \tag{68}$$

where *b* is the radius of the wetted region. If one defines the adhesive force as the tensile contribution to the net force exerted on either solid body, then *Ft* is the just the adhesive force (*Fad* ). An alternative definition for the adhesive force would be the value of the tensile external load required to maintain static equilibrium, or required to achieve a certain separation and separation rate. The latter definition views the adhesive force as the interfacial tensile force less the interfacial compressive force.

Sample results of the analysis are displayed in Figure 12, for the following input parameters: *Vo* = 0.1 mm3 , γ = 72.7 mN/m, σ = 0.4 μm, and *An* = 4 cm<sup>2</sup> , where σ is the r.m.s. surface roughness of a 3D isotropic surface with a Gaussian height distribution, and *An* is the nominal contact area of the interface (i.e., the projected area of the interface in Figure 11). Figure 12 shows the influence of external load on several contact parameters, including tensile force and contact area (Fig. 12a) as well as average gap and wetted radius (Fig. 12b). The tensile force is seen to grow steadily with increasing external load until approaching a critical load, where the rate of increase of tensile force with load approaches infinity. The attainment of a near vertical slope in the curve suggests that the interface is unstable: no equilibrium configurations could be found for values of external load beyond the critical value. Analogous results are found for the average gap, tensile radius and solid-solid contact area. Such behavior suggests interface collapse, whereby beyond the critical point, the surfaces come into complete and near complete contact [33–36]. By introducing certain dimensionless parameters, the results can be general‐ ized. Let an adhesion parameter Γ be defined according to

$$\Gamma \equiv \frac{\gamma \left(\cos \theta\_1 + \cos \theta\_2\right) V\_o}{\pi \sqrt{2} A\_n^{1/2} E' \sigma^3} \tag{69}$$

and let the dimensionless versions of external load, tensile force, and liquid volume be defined respectively as

$$P^\* \equiv \frac{P}{\gamma V\_o \left(\cos \theta\_1 + \cos \theta\_2\right) / \sigma^2} \tag{70}$$

$$F\_t^\* = \frac{F\_t}{\gamma V\_o \left(\cos \theta\_1 + \cos \theta\_2\right) / \sigma^2} \tag{71}$$

$$V\_o^\* \equiv \frac{V\_t}{A\_n \sigma} \tag{72}$$

The results for dimensionless tensile force versus the adhesion parameter are depicted in Figure 13 at several values of dimensionless volume. This figure reveals that, for each dimen‐ sionless volume considered, there is a critical value of the adhesion parameter whereby the force curve becomes vertical, suggesting the onset of surface collapse.

separation rate. The latter definition views the adhesive force as the interfacial tensile force

Sample results of the analysis are displayed in Figure 12, for the following input parameters:

of a 3D isotropic surface with a Gaussian height distribution, and *An* is the nominal contact area of the interface (i.e., the projected area of the interface in Figure 11). Figure 12 shows the influence of external load on several contact parameters, including tensile force and contact area (Fig. 12a) as well as average gap and wetted radius (Fig. 12b). The tensile force is seen to grow steadily with increasing external load until approaching a critical load, where the rate of increase of tensile force with load approaches infinity. The attainment of a near vertical slope in the curve suggests that the interface is unstable: no equilibrium configurations could be found for values of external load beyond the critical value. Analogous results are found for the average gap, tensile radius and solid-solid contact area. Such behavior suggests interface collapse, whereby beyond the critical point, the surfaces come into complete and near complete contact [33–36]. By introducing certain dimensionless parameters, the results can be general‐

> ( ) 1 2 1/2 3

> > *n*

*A E*

and let the dimensionless versions of external load, tensile force, and liquid volume be defined

 q

 s *o*

2

2

<sup>+</sup> <sup>º</sup> ¢ (69)

*<sup>V</sup>* <sup>º</sup> <sup>+</sup> (70)

*<sup>V</sup>* <sup>º</sup> <sup>+</sup> (71)

<sup>º</sup> (72)

*V*

cos cos

2

( ) \*

 q

( ) \*

 q

\* *t o*

*n V*

The results for dimensionless tensile force versus the adhesion parameter are depicted in Figure 13 at several values of dimensionless volume. This figure reveals that, for each dimen‐ sionless volume considered, there is a critical value of the adhesion parameter whereby the

*A* s

*<sup>P</sup> <sup>P</sup>* g

*o*

*V*

g

force curve becomes vertical, suggesting the onset of surface collapse.

1 2 cos cos / *<sup>o</sup>*

1 2 cos cos / *t*

*F*

 qs

 qs

gq

p

, where σ is the r.m.s. surface roughness

, γ = 72.7 mN/m, σ = 0.4 μm, and *An* = 4 cm<sup>2</sup>

ized. Let an adhesion parameter Γ be defined according to

Γ

*t*

*F*

less the interfacial compressive force.

*Vo* = 0.1 mm3

78 Surface Energy

respectively as

**Figure 12.** The effect of external load: (a) tensile force and contact area; (b) average gap and wetted radius.

**Figure 13.** Dimensionless tensile force as a function of adhesion parameter for several dimensionless liquid volumes.

## **Acknowledgements**

The author would like to thank the National Science Foundation (US) for support of this work and Amir Rostami, a graduate research assistant, for performing some calculations used herein.

## **Author details**

Jeffrey L. Streator

Address all correspondence to: jeffrey.streator@me.gatech.edu

G.W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA, USA

## **References**


**Acknowledgements**

herein.

80 Surface Energy

**Author details**

Jeffrey L. Streator

GA, USA

**References**

2003;17(4):563-582.

1967.

The author would like to thank the National Science Foundation (US) for support of this work and Amir Rostami, a graduate research assistant, for performing some calculations used

G.W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta,

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[3] Van Spengen WM, Puers R, De Wolf I. On the physics of stiction and its impact on the reliability of microstructures. Journal of Adhesion Science and Technology.

[4] Komvopoulos K. Adhesion and friction forces in microelectromechanical systems: Mechanisms, measurement, surface modification techniques, and adhesion theory.

[5] Liu H, Bhushan B. Adhesion and friction studies of microelectromechanical systems/ nanoelectromechanical systems materials using a novel microtriboapparatus. Journal

[6] Lee S-C, Polycarpou AA. Adhesion forces for sub-10 nm flying-height magnetic stor‐

[7] Yang SH, Nosonovsky M, Zhang H, Chung K-H. Nanoscale water capillary bridges under deeply negative pressure. Chemical Physics Letters. 2008;451(1):88-92.

[8] Adamson AW, Gast AP. Physical chemistry of surfaces: JWiley and Sons. New York;

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**Section 2**

**Surface Properties**

[24] Rabinovich YI, Esayanur MS, Moudgil BM. Capillary forces between two spheres with a fixed volume liquid bridge: Theory and experiment. Langmuir.

[25] Streator JL. Analytical instability model for the separation of a sphere from a flat in the presence of a liquid. Proceedings of the ASME/STLE International Joint Tribology

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[28] Zheng J, Streator J. A liquid bridge between two elastic half-spaces: A theoretical

[30] Zheng J, Streator JL. A micro-scale liquid bridge between two elastic spheres: Defor‐

[31] Zheng J, Streator J. A generalized formulation for the contact between elastic spheres: Applicability to both wet and dry conditions. Journal of Tribology. 2007;129:274. [32] Johnson KL, Kendall, K, Roberts, AD. Surface energy and the contact of elastic solids. Proceedings of the Royal Society (London), Series A. 1971;324(1558):301-313.

[33] Streator JL. A model of liquid-mediated adhesion with a 2D rough surface. Tribology

[34] Streator JL, Jackson RL. A model for the liquid-mediated collapse of 2-D rough surfa‐

[35] Rostami A, Streator JL. Study of Liquid-mediated adhesion betweeen 3D rough sur‐

[36] Rostami A, Streator JL. A deterministic approach to studying liquid-mediated adhe‐ sion between rough surfaces. Tribology Letters. 2015;84:36-47. DOI: 10.1007/

[37] Wang L, Regnier S. A more general capillary adhesion model including shape index: Single-asperity and multi-asperity cases. Tribology Transacations. 2015;58:106-112.

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## **Adhesive Properties of Metals and Metal Alloys**

## Anna Rudawska

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/60599

#### **Abstract**

The paper presents the effect of some surface treatment on the bonded joints strength of selected construction materials, adhesive properties of adherends after surface treatment and surface roughness. The aluminium alloys sheets, the titanium sheets and the stainless steel sheets were tested. In the experiments the following surface treatments were investigated: degreasing (chemical cleaning), mechanical treatment, mechanical treatment and degreasing, etching, anodising and chromate treatment. Adhesive joints were formed with a two component epoxy adhesive, Loctite 3430. Adhesive joint tensile-shear strength tests were performed in accordance with EN DIN 1465 standard on Zwick/Roell Z100 and Zwick/Roell Z150 testing machines. Adhesive properties were determined by surface free energy and surface free energy was determined by the Owens-Wendt method. The roughness of specimens was qualified by the method for measuring contact roughness, using an M2 profilometer manufactured by Mahr. The surface view was obtained by used NanoFocus uscan AF2. Results obtained from adhesive joint strength tests of materials evidence that surface treatment plays an important role in increasing strength of analysed joints. Tests indicate that in numerous instances this is mechanical treatment only or mechanical treatment followed by chemical cleaning which translate to the highest joint strength. The surface treatment method which introduces extensive changes in the analysed materials surface geometry is mechanical treatment. The results of surface roughness parameters measurement carried out on test samples subjected to anodising indicate that anodising has an impact on the height of surface irregularities. The application of various surface treatments in different structural materials allows modification of their adhesive properties, determined by the surface free energy. It was noted that different surface treatments contribute not only to the surface free energy changes but to the SFE components share in the total value. In the majority of variants of EN AW-2024PLT3 aluminium alloy sheet surface treatment the dispersive

© 2015 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

component amounted to the 93-99% of the total surface free energy. The assumption then should be that in order for the determination of a particular surface for adhesive processes to be comprehensive it should account for the adherends surface geometry as well as its adhesion properties. The geometry of surface can influence the mechan‐ ical adhesion and the surface free energy is connected with both mechanical adhesion and the other constituent of adhesion – proper adhesion.

**Keywords:** adhesive properties, surface free energy, contact angle, metals, metals alloys, surface treatment

## **1. Introduction**

Adhesive bonding provides an invaluable alternative to other modern methods of joining structural materials [1-3]. At present, bonding technology offers a range of applications for a number of various branches of industry, including in building, automotive, aircraft, machinebuilding, packaging manufacturing, and marine applications [4]. Bonded joints find applica‐ tions in various structures and constitute an extensively used form of adhesive joint [5]. Adhesive bonding technology offers numerous advantages over other methods, and joining materials of dissimilar physical or chemical properties is a prominent mark of superiority. This feature frequently determines that adhesive bonding is the only applicable method, particu‐ larly in the case of adherends of different chemical composition and physical properties, which could pose a considerable problem if, e.g., a welded joint were to be applied.

The increasing popularity of adhesive bonding as a method for joining metals is a result of several factors [1-10], e.g., high joint strength and lack of stresses within the joint, along with low cost per unit resulting predominantly from the amount of adhesive used to form a single joint. Further advantages of adhesive joints are as follows: vibration damping, forming the joint without machine tools, expensive equipment or materials (nevertheless in certain cases the cost of technological instrumentation may prove to be high), lack of electrochemical phenomena usually accompanying other methods of joining metals, and the joining of dissimilar structural materials, frequently of substantial disproportions in geometric dimen‐ sions [2, 5]. Currently, adhesive joining is frequently applied in bonding polymer composite and metal substrates. Structural adhesive bonding is an indispensable method of joining thinwalled elements of sandwich construction, whose advantages are lightness and rigidity, which are essential properties in aircraft constructions [1, 3].

One requirement of structural bonded joints is proper strength [2, 3, 5]. This is important due to the fact that one of the basic requirements to be fulfilled by an adhesive joint is obtaining the desired static strength. The strength of adhesive joints is determined by several major factors: technological, structural, material and environmental [6, 7, 11-19].

Adhesive bonding technology comprises several consecutive technological operations: surface treatment, preparation and application of adhesive, joining substrates, cure conditioning, finishing and joint quality control [2, 3, 5]. Individual operations may consist of a number of stages of specific technological parameters; they might also require various items of equipment and instrumentation. Detailed conditions of adhesive bonding operation are selected based on, inter alia, the type of substrate, the geometry of elements and structures, joint formation conditions, production type, etc. [1-4, 7, 14, 18, 20, 21].

A summary of the previous studies of some issues in adhesive joints and bonding technology is presented in Table 1.


**Table 1.** Summary of some issues in adhesive joints and bonding technology

The subject of the test and analysis were the issues of surface treatment, particularly the influence of surface treatment on surface free energy and strength of adhesive joints, and also the influence on the geometric structure of adherends.

## **2. Adherend surface treatment for adhesive bonding**

## **2.1. The aim of surface treatment**

component amounted to the 93-99% of the total surface free energy. The assumption then should be that in order for the determination of a particular surface for adhesive processes to be comprehensive it should account for the adherends surface geometry as well as its adhesion properties. The geometry of surface can influence the mechan‐ ical adhesion and the surface free energy is connected with both mechanical adhesion

**Keywords:** adhesive properties, surface free energy, contact angle, metals, metals

Adhesive bonding provides an invaluable alternative to other modern methods of joining structural materials [1-3]. At present, bonding technology offers a range of applications for a number of various branches of industry, including in building, automotive, aircraft, machinebuilding, packaging manufacturing, and marine applications [4]. Bonded joints find applica‐ tions in various structures and constitute an extensively used form of adhesive joint [5]. Adhesive bonding technology offers numerous advantages over other methods, and joining materials of dissimilar physical or chemical properties is a prominent mark of superiority. This feature frequently determines that adhesive bonding is the only applicable method, particu‐ larly in the case of adherends of different chemical composition and physical properties, which

The increasing popularity of adhesive bonding as a method for joining metals is a result of several factors [1-10], e.g., high joint strength and lack of stresses within the joint, along with low cost per unit resulting predominantly from the amount of adhesive used to form a single joint. Further advantages of adhesive joints are as follows: vibration damping, forming the joint without machine tools, expensive equipment or materials (nevertheless in certain cases the cost of technological instrumentation may prove to be high), lack of electrochemical phenomena usually accompanying other methods of joining metals, and the joining of dissimilar structural materials, frequently of substantial disproportions in geometric dimen‐ sions [2, 5]. Currently, adhesive joining is frequently applied in bonding polymer composite and metal substrates. Structural adhesive bonding is an indispensable method of joining thinwalled elements of sandwich construction, whose advantages are lightness and rigidity, which

One requirement of structural bonded joints is proper strength [2, 3, 5]. This is important due to the fact that one of the basic requirements to be fulfilled by an adhesive joint is obtaining the desired static strength. The strength of adhesive joints is determined by several major

Adhesive bonding technology comprises several consecutive technological operations: surface treatment, preparation and application of adhesive, joining substrates, cure conditioning,

factors: technological, structural, material and environmental [6, 7, 11-19].

could pose a considerable problem if, e.g., a welded joint were to be applied.

are essential properties in aircraft constructions [1, 3].

and the other constituent of adhesion – proper adhesion.

alloys, surface treatment

**1. Introduction**

86 Surface Energy

Surface pretreatment is one of the first and most important technological stages in the adhesive bonding process. It is preceded by the analysis of properties, type and geometrical structure of a material surface for adhesive bonding, as the choice of an appropriate surface pretreatment method depends on these data [4-8]. In adhesive bonding, the surface of joined elements is defined as the part of the material where interactions with an adhesive occur [2]. This is connected both with the area and depth of interaction. In order to produce strong adhesive joints, surface pretreatments for adhesive bonding should ensure the following [3, 5, 13, 20]:


Surface treatment includes, inter alia, the following operations [5, 7, 23-26]:


Critchlow et al. [27] underline that a particular pretreatment for structural bonding will ideally produce a surface which is free from contamination, wettable by the adhesive, highly macroor micro-rough, mechanically stable, and hydrolytically stable.

Spadaro, Dispenza and Sunseri [28] systematized surface treatment operations when present‐ ing test results of the impact of surface treatment operations on adhesively bonded joints of aluminium alloys.

The selection of surface preparation method, including the choice of proper technological operations aimed at developing a desired structure and energetic properties, is dependent on multiple factors, predominantly the type of materials to be joined with an adhesive bond.

In the test of adhesive properties and adhesive strength the following materials were used:


The following surface treatments were tested:


## **2.2. Characteristics of some surface treatment methods**

## *2.2.1. Degreasing*

**•** correct surface development,

of, for instance, adhesive or paint,

which is of significance to the wetting process.

or micro-rough, mechanically stable, and hydrolytically stable.

**•** aluminium alloy sheets: EN AW-2024PLT3 and EN AW-7075PLT0,

**•** titanium sheets: CP1 (Grade 1) and CP2 (Grade 2),

**•** degreasing (chemical clearing) with degreasing agents

**•** a combination of selected aforementioned methods.

The following surface treatments were tested:

**•** mechanical treatment with abrasive tools ,

surface layer,

88 Surface Energy

aluminium alloys.

**•** X5CrNi181 stainless steel.

**•** anodizing,

**•** etching,

**•** chromate treatment,

**•** good activation of surfaces of elements being bonded.

Surface treatment includes, inter alia, the following operations [5, 7, 23-26]:

**•** removal of surface contaminants, residues of technological processes (e.g., stamping, rolling, forging or machining) or of a protective layer providing, e.g., corrosion protection in storage and transport, and in the case of polymers, removal of additives migrating to the

**•** changes in the geometric structure of the surface, through, e.g., increasing the surface roughness, which results in the development of a "true" wetting surface, the contact surface

**•** reduction of intermolecular forces' range, leading to the increase of the surface free energy,

Critchlow et al. [27] underline that a particular pretreatment for structural bonding will ideally produce a surface which is free from contamination, wettable by the adhesive, highly macro-

Spadaro, Dispenza and Sunseri [28] systematized surface treatment operations when present‐ ing test results of the impact of surface treatment operations on adhesively bonded joints of

The selection of surface preparation method, including the choice of proper technological operations aimed at developing a desired structure and energetic properties, is dependent on multiple factors, predominantly the type of materials to be joined with an adhesive bond.

In the test of adhesive properties and adhesive strength the following materials were used:

Chemical cleaning of a surface, frequently referred to as degreasing, constitutes, in the majority of instances, the first surface treatment operation in the preparation of adherends for adhesive joining [5]. The surface of adherends is covered with a layer of grease, dust, various machining residues, or organic and non-organic substances. That is why chemical degreasing is applied to remove from the surfaces of adherends all contaminants which could decrease the strength of the adhesive bond in the adherends' contact area [20]. The selection of chemical cleaning method and degreasing agent is contingent on numerous factors, such as: the efficiency of chemical cleaning, the dimensions of adherends, the type of material, technological equipment available, etc. [2]. The degreasing process can be conducted with different degreasing agents and instrumentation [2, 3, 5].

The following cleaning methods are used in industrial manufacture [5]:


The most frequently applied degreasing agents include: acetone, petrol, benzene, ethanol, trichloroethane, tetrachloroethane, toluene, methyl ethyl ketone (MEK), to name but a few [2, 5]. A variety of cleansing agents such as organic solvents and water-miscible detergents are commonly available. The chemical cleaning procedure is carried out in immersion washers, treatment chambers or with a soaked cloth [5].

## *2.2.2. Mechanical treatment*

Mechanical methods find applications in both the removal of contaminants and the changing of surface geometry [29]. The structure of the surface layer is a feature heavily dependent on the surface roughness. Although mechanical treatment acts towards changing the surface structure, and therefore towards its development, it fails to activate it for bonding.

Methods of mechanical treatment, employed as preparation of the adherend surface, include the following: abrasive tool treatment, abrasive blasting, grit blasting, peening, brushing, scraping and sanding [5, 11, 30, 31].

When employing these treatment methods, particular attention should be given to surface roughness geometry [32] so as not to generate excessive internal stresses, which could contribute to lowering the adhesive joint's strength. L.F.M. da Silva et al. [33] highlight that considerable surface roughness might result in increased stress concentration and consequent‐ ly lower joint strength. Extreme surface roughness of adherends can lead to lowering of the adhesively bonded joint's strength when the adhesive fails to penetrate and wet the surface irregularities. Analysis of surface topography with regard to the wetting angle proves that irregularities in the shape of elongated ridges, characterized by low apex angle value and high accumulation, are the most advantageous. It has been shown in the literature [21] that the highest joint strength properties are obtained when the surface roughness parameter of adherends (Maximum Profile Valley Depth) Rm = 7 ÷ 25 μm.

Some researchers argue that mechanical methods applied in surface treatment generate nonaxial shear stresses, which lead to structural micro-cracks, e.g., micro-stresses, dislocations, sharp edges, crevices, etc. It was, furthermore, concluded that mechanical treatment can cause compressive stresses in the surface layer of adherends and, consequently, their plastic deformation. This, in turn, is a factor introducing stresses into the adhesive layer, which could reduce adhesive joint strength by 10-50 % [21].

## *2.2.3. Chemical treatment*

Surface preparation with chemical treatment methods is employed in the case of adherends of a substantial amount of surface contamination. Chemical methods, based on liquid chemical compounds, enable surface and surface layer development. Their chemical constitution ensures high physicochemical activity of the surface with a bonding agent (e.g., adhesive), applied or medium substance [2, 5].

Aluminium alloy etching [21] showed that the constitution of etching bath has a great impact on the adhesive joint's strength; for instance, shear strength of aluminium alloys subjected to etching in 4 % NaOH is 40 % higher than in the case of 20 % HNO3.

## *2.2.4. Electrochemical treatment*

Anodizing is a widely used surface treatment operation applied in metals, consisting in electrolytic formation of oxide film. Anodizing finds applications predominantly in alumini‐ um and its alloys; however, it may be used in certain types of steel, titanium and magnesium alloys. Eloxal process (electrolytic oxidation of aluminium) is a term frequently encountered in reference to aluminium anodizing [34].

Anodic oxidation of aluminium, known as anodizing, is a process during which on the surface of metal a thicker oxide layer is formed, providing superior corrosion protection to the natural passivation layer [35-37].

The anodizing process consists in the aluminium surface being transformed into aluminium oxide when exposed to electrolyte solution yielding OH- hydroxide ions [38]:

$$2\text{Al} + 6\text{OH}^{\cdot} \rightarrow \text{Al}\_2\text{O}\_3 + 3\text{H}\_2\text{O} + 6\text{e}^{\cdot} \tag{1}$$

The resulting oxide layer thickness grows with anodizing time. It may amount to several dozen mm for protective and decorative layers or even exceed 100 mm for durable engineering application aluminium layers. The final structure of layers depends on the alumina-solubility of the electrolyte solution [39].

adhesively bonded joint's strength when the adhesive fails to penetrate and wet the surface irregularities. Analysis of surface topography with regard to the wetting angle proves that irregularities in the shape of elongated ridges, characterized by low apex angle value and high accumulation, are the most advantageous. It has been shown in the literature [21] that the highest joint strength properties are obtained when the surface roughness parameter of

Some researchers argue that mechanical methods applied in surface treatment generate nonaxial shear stresses, which lead to structural micro-cracks, e.g., micro-stresses, dislocations, sharp edges, crevices, etc. It was, furthermore, concluded that mechanical treatment can cause compressive stresses in the surface layer of adherends and, consequently, their plastic deformation. This, in turn, is a factor introducing stresses into the adhesive layer, which could

Surface preparation with chemical treatment methods is employed in the case of adherends of a substantial amount of surface contamination. Chemical methods, based on liquid chemical compounds, enable surface and surface layer development. Their chemical constitution ensures high physicochemical activity of the surface with a bonding agent (e.g., adhesive),

Aluminium alloy etching [21] showed that the constitution of etching bath has a great impact on the adhesive joint's strength; for instance, shear strength of aluminium alloys subjected to

Anodizing is a widely used surface treatment operation applied in metals, consisting in electrolytic formation of oxide film. Anodizing finds applications predominantly in alumini‐ um and its alloys; however, it may be used in certain types of steel, titanium and magnesium alloys. Eloxal process (electrolytic oxidation of aluminium) is a term frequently encountered

Anodic oxidation of aluminium, known as anodizing, is a process during which on the surface of metal a thicker oxide layer is formed, providing superior corrosion protection to the natural

The anodizing process consists in the aluminium surface being transformed into aluminium


The resulting oxide layer thickness grows with anodizing time. It may amount to several dozen mm for protective and decorative layers or even exceed 100 mm for durable engineering

23 2 2Al + 6OH Al O + 3H O + 6e ® (1)

oxide when exposed to electrolyte solution yielding OH- hydroxide ions [38]:

etching in 4 % NaOH is 40 % higher than in the case of 20 % HNO3.

adherends (Maximum Profile Valley Depth) Rm = 7 ÷ 25 μm.

reduce adhesive joint strength by 10-50 % [21].

*2.2.3. Chemical treatment*

90 Surface Energy

applied or medium substance [2, 5].

*2.2.4. Electrochemical treatment*

passivation layer [35-37].

in reference to aluminium anodizing [34].

In the initial stages of aluminium oxide film formation, a dense thin layer (0.01 – 0.1 μ*m*) of Al2O3 is formed (the so-called barrier or blocking layer), which subsequently changes into a porous layer as a result of barrier layer reformation at the oxide-electrolyte interface. The barrier layer is generated owing to Al3+ ions' migration in the electric field and their reaction with O2- or OH ions producing anhydrous Al2O3. The final stage consists in the porous layer's expansion in thickness (up to 100 μ*m*) [36].

Oxide films obtained from poorly soluble solutions are characterized by a specific porous structure. Regularly distributed pores almost throughout the oxide layer and perpendicular to the surface have a diameter ranging from several to several tens of nanometres, depending on the conditions of anodic oxidation. The barrier layer is a thin non-porous layer situated from the inside. Due to the fact that the pores are densely distributed and small in diameter, the surface of such aluminium oxide is well-developed and is characterized by high adsorp‐ tivity, which is used in, e.g., introducing dyes.

For protective and decorative applications, it is the sulphuric acid bath which is the most commonly used. The layer thickness is adjusted according to norms depending on the operating conditions of the product [40].

The characteristic porous structure of oxide layers formed in the process of anodizing in sulphuric acid facilitates subsequent chemical dyeing operations. Organic dyes are frequently applied, most of which easily adsorb on the developed aluminium oxide surface. The process is usually performed in diluted dye solution at elevated temperature and in adjusted dye solution pH. The process, however, has one disadvantage, namely the poor light-fastness of the resulting surface colouring, particularly when exposed to direct sunlight. That is why only a small number of organic dyes are permitted for dyeing aluminium architectural elements [38, 41].

Chromate coatings are recognized as conversion coatings, i.e., coatings generated as a result of the chemical or electrochemical reaction of a metal surface layer with certain chemical compounds in which the produced salt is practically insoluble in the medium where the reaction is conducted [42, 43].

The previous study [44] provides results of the application of surface treatment operations on aluminium alloy sheets: yellow and black anodizing in sulphuric acid, self-colour anodizing and chromate conversion coating lead to certain changes in the surface free energy values. However, different variants of anodizing in sulphuric acid resulted in changes in percentage distribution of the dispersive and the polar component of the surface free energy. The results of the surface free energy and its component calculations indicate that the highest surface free energy value was obtained in the case of chromate coating (69.8mJ m-2) and self-colour anodizing in sulphuric acid (68.5mJ m-2). Simultaneously, the lowest value of the surface free energy was observed for black anodization.

A summary of the previous studies of selected surface treatment of various adherends are presented in Table 2.


**Table 2.** The examples of type of surface treatment and adherends

Numerous studies [2, 5, 7, 23-26] stress the importance of the effect of surface pretreatments for adhesive bonding and their effect on adhesive joint strength and the quality of adhesively bonded joints.

#### **2.3. Surface preparation methods and adhesive joint strength**

#### *2.3.1. Adhesive joints: Types and dimensions*

The research subject was a shear-loaded single lap adhesive joint of selected structural materials. Sample dimensions are presented in Figure 1: length of adherends l = 100±0.4 mm, width b = 20±0.3 mm, bond-line thickness: gk = 0.1±0.02 mm [46].

**Figure 1.** Single lap adhesive joint: b – adherends width, g – adherends thickness, gk – bond-line thickness, l – adher‐ ends length, lz – overlap length, P – force representing load type and direction

Thickness of adherends *g* and overlap length *lz*, are presented in Table 3 [46].


**Table 3.** Dimensions of analysed adhesive joints

#### *2.3.2. Adhesive joints forming*

**No. Type of surface treatment Type of adherend**

3 Chemical treatment polymers [23], titanium [4]

coating aluminium [24, 27, 39, 41-44]

2 Mechanical treatment aluminium [6, 30], steel [2, 32], titanium [4]

Numerous studies [2, 5, 7, 23-26] stress the importance of the effect of surface pretreatments for adhesive bonding and their effect on adhesive joint strength and the quality of adhesively

The research subject was a shear-loaded single lap adhesive joint of selected structural materials. Sample dimensions are presented in Figure 1: length of adherends l = 100±0.4 mm,

**Figure 1.** Single lap adhesive joint: b – adherends width, g – adherends thickness, gk – bond-line thickness, l – adher‐

1 Degreasing aluminium [12]

5 Others: plasma treatment aluminium [47]

**2.3. Surface preparation methods and adhesive joint strength**

width b = 20±0.3 mm, bond-line thickness: gk = 0.1±0.02 mm [46].

ends length, lz – overlap length, P – force representing load type and direction

Thickness of adherends *g* and overlap length *lz*, are presented in Table 3 [46].

<sup>4</sup> Electrochemical treatment: anodizing, chromate

**Table 2.** The examples of type of surface treatment and adherends

*2.3.1. Adhesive joints: Types and dimensions*

bonded joints.

92 Surface Energy

Adhesive joints were formed with a two-component epoxy adhesive, Loctite 3430 [47], suitable for the analysed adherends, with a short cure time at room temperature. The adhesive was prepared with a static mixer and applied on one of the adherends, in accordance with the manufacturer's recommendations. Even bond-line thickness was ensured in preliminary research by, inter alia, selecting a suitable amount of adhesive and pressure while curing.

Proper surface treatments were selected according to the given structural material:


Degreasing in experimental tests was carried out with Loctite 7036 degreasing agent, mostly containing aliphatic hydrocarbons. Chemical cleaning with Loctite 7063 was a three-stage process consisting in spraying the surface of adherends, removing the agent with a cloth, and after the final application of the degreaser leaving the sample to dry. Chemical cleaning took place at an ambient temperature of 20±2 °C, and in relative humidity of 32-40 % [47].

Since chemical cleaning is rarely sufficient for providing good adhesion, as previously mentioned, the sample preparation included an abrasive mechanical treatment stage, per‐ formed with abrasive paper, which is the most convenient material for mechanical treatment of adherends. This approach is selected because of its high efficiency and accessibility com‐ bined with low cost, and its uncomplicated and versatile application in various conditions. Another important advantage of this method is that it requires little effort to ensure the machined surface shows marks in no direction.

Mechanical treatment in experimental tests was carried out with P320 abrasive paper.

The last stage of this surface treatment operation, having treated the surface with abrasive paper, was to remove the remaining contaminants from the surface with Loctite 7036 degreas‐ ing agent.

Sulphuric acid anodizing was another surface treatment used in tests. Adherends were immersed in 180-200 g/l solution of sulphuric acid at the temperature of 10÷15 °C for 35 minutes. Afterwards, the samples were dyed yellow in a 40÷55 g/l solution of K2Cr2O7, pH 4.5÷6.2, at the temperature of 90÷95 °C for 35 minutes.

Another batch of samples was subjected to chromate treatment, consisting of a 30 second immersion in a 5.5 g/l solution of Na2Cr2O7; a 4 g/l solution of Na2SO4; 4.5-5.5 g/l solution of H3BO3 and a 1.5 ml/l solution of HNO3, pH 1.4-1.6.

Finally, the last surface treatment applied in tests for comparison was etching in a 40-60 g/l aqueous solution of sodium hydroxide. The adherends were immersed for 3-4 minutes at 45-55 °C, subsequently rinsed with warm running water and finally left to dry.

For the sake of control, a part of the samples remained untreated. This allowed the determi‐ nation of the actual impact of surface treatments on the surface free energy values of adherends.

Joint forming conditions were as follows:


## *2.3.3. Strength tests*

Adhesive joint tensile-shear strength tests were performed in accordance with EN DIN 1465 standard on Zwick/Roell Z100 and Zwick/Roell Z150 testing machines. Testing speed was equal to 5 mm/min.

#### *2.3.4. Test results*

The results of shear strength tests on EN AW-2024PLT3 aluminium alloy sheet adhesive joints are presented in Figure 2. The results presented in figures are mean values of 8-2 measurements performed for each surface treatment variant.

Application of anodizing and chromate operations as an EN AW-2024 aluminium alloy sheet surface treatment method produced adhesive joints of maximum shear strength of 11.09 MPa and 12.39 MPa, respectively. Similarly good results in promoting joint strength were observed when the surface of EN AW-2024PLT3 aluminium alloy sheets was subjected to exclusively mechanical treatment; the results in that case amounted to 8.43 MPa and 8.66 MPa; therefore, the joint strength was six times higher than in the case where no surface treatment was applied. In the failure of EN AW-2024PLT3 aluminium alloy sheet adhesive joints, a characteristic and

The last stage of this surface treatment operation, having treated the surface with abrasive paper, was to remove the remaining contaminants from the surface with Loctite 7036 degreas‐

Sulphuric acid anodizing was another surface treatment used in tests. Adherends were immersed in 180-200 g/l solution of sulphuric acid at the temperature of 10÷15 °C for 35 minutes. Afterwards, the samples were dyed yellow in a 40÷55 g/l solution of K2Cr2O7, pH

Another batch of samples was subjected to chromate treatment, consisting of a 30 second immersion in a 5.5 g/l solution of Na2Cr2O7; a 4 g/l solution of Na2SO4; 4.5-5.5 g/l solution of

Finally, the last surface treatment applied in tests for comparison was etching in a 40-60 g/l aqueous solution of sodium hydroxide. The adherends were immersed for 3-4 minutes at 45-55

For the sake of control, a part of the samples remained untreated. This allowed the determi‐ nation of the actual impact of surface treatments on the surface free energy values of adherends.

Adhesive joint tensile-shear strength tests were performed in accordance with EN DIN 1465 standard on Zwick/Roell Z100 and Zwick/Roell Z150 testing machines. Testing speed was

The results of shear strength tests on EN AW-2024PLT3 aluminium alloy sheet adhesive joints are presented in Figure 2. The results presented in figures are mean values of 8-2 measurements

Application of anodizing and chromate operations as an EN AW-2024 aluminium alloy sheet surface treatment method produced adhesive joints of maximum shear strength of 11.09 MPa and 12.39 MPa, respectively. Similarly good results in promoting joint strength were observed when the surface of EN AW-2024PLT3 aluminium alloy sheets was subjected to exclusively mechanical treatment; the results in that case amounted to 8.43 MPa and 8.66 MPa; therefore, the joint strength was six times higher than in the case where no surface treatment was applied. In the failure of EN AW-2024PLT3 aluminium alloy sheet adhesive joints, a characteristic and

°C, subsequently rinsed with warm running water and finally left to dry.

4.5÷6.2, at the temperature of 90÷95 °C for 35 minutes.

H3BO3 and a 1.5 ml/l solution of HNO3, pH 1.4-1.6.

Joint forming conditions were as follows:

**•** seasoned for 48 h at ambient temperature of 20±2 °C.

performed for each surface treatment variant.

**•** cure temperature 20±2 °C,

**•** relative humidity 32-40 %,

*2.3.3. Strength tests*

equal to 5 mm/min.

*2.3.4. Test results*

**•** pressure when curing 0.02 MPa;

ing agent.

94 Surface Energy

**Figure 2.** Shear strength tests of EN AW-2024PLT3 aluminium alloy sheet adhesive joints after different surface treat‐ ment methods: U - untreated, D - degreasing (chemical cleaning), M - mechanical treatment, MD - mechanical treat‐ ment and degreasing, E - etching A – anodizing, Ch - chromate treatment

repeatable shape of failed elements can be noticed. An observed plastic deformation of adherends results from shear and bending stresses.

Test results for homogenous EN AW-2024PLT3 aluminium alloy sheet adhesive joints' strength after certain surface treatment procedures are presented in Figure 3.

Adhesive joints were formed on 0.80 mm thick sheet samples. The highest sheet adhesive joint strength was observed after mechanical treatment (5.75 MPa), while the lowest was produced in the case of untreated adherends (4.30 MPa). In the latter case (variant M), the increase in joint strength was equal to 25 % as compared with variant U (untreated surface). The analysis of chemical treatment leads to the conclusion that the application of this particular operation generates conditions promoting joint strength. In addition, a positive impact of degreasing on joint strength can be observed in relation to the variant with no chemical treatment (approx. 9 % higher). It was observed that in each case when preparation of adherends' surface for adhesive joining was performed, higher joint strength is produced in comparison with variant U (untreated surface).

Adhesive joint strength tests were carried out on two types of titanium adherend: CP1 and CP3. CP1 titanium sheet adhesive joint strength after the analysed surface treatments is presented in Figure 4.

This shows that the highest strength was demonstrated by samples subjected to degreasing, and mechanical and degreasing. The shear strength of these joints was nearly four times higher than in the case of an untreated surface, whereas exclusively mechanical treatment only proved

**Figure 3.** Shear strength tests of EN AW-7075PLT0 aluminium alloy sheet adhesive joints after different surface treat‐ ment methods: U - untreated, D - degreasing (chemical cleaning), M - mechanical treatment, MD - mechanical treat‐ ment and degreasing

**Figure 4.** Shear strength tests of CP1 titanium sheet adhesive joints after different surface treatment methods: U - un‐ treated, D - degreasing (chemical cleaning), M - mechanical treatment, MD - mechanical treatment and degreasing

slightly less effective. The shear strength developed here was notably lower by 7 MPa as compared to variant D (degreasing).

**Figure 5.** Shear strength tests of CP3 titanium sheet adhesive joints after different surface treatment methods: U - un‐ treated, D - degreasing (degreasing), M - degreasing, MD - mechanical treatment and degreasing

Strength test results for CP3 titanium sheet adhesive joints after the analysed surface treatment are shown in Figure 5.

The application of mechanical and chemical treatment as surface treatment operations for CP3 titanium sheets was translated into the highest joint strength (14.20 MPa). The shear strength of such joints was seven times greater than in the case of an untreated surface. When treated mechanically, with no chemical cleaning, or in the case where only mechanical cleaning was applied, the resulting joint shear strength was lower. Homogeneous CP3 titanium sheet adhesive joint shear strength after degreasing constituted 68 % of strength value obtained following variant MD of surface preparation and 76 % of variant M (mechanical treatment).

Stainless steel adherend samples were treated in an identical manner to the case of aluminium and titanium sheet adherends. Joint strength test results after the analysed surface treatment are presented in Figure 6.

It was observed that the highest values of joint strength were obtained after mechanical and chemical treatment (14.84 MPa), but it was observed that there were large differences in the obtained test results. Standard deviation is significantly greater than for other types of surface treatment. The difference in the values of the strength after the application of degreasing and after machining and degreasing is about 10 %. Based on the results of the statistical analysis, it can be seen that the use of both degreasing and mechanical treatment makes it possible to obtain the same strength, with a higher reproducibility for the degreasing. Comparing the results of the bond strength after the surface treatment and without treatment by test materials

slightly less effective. The shear strength developed here was notably lower by 7 MPa as

**Figure 4.** Shear strength tests of CP1 titanium sheet adhesive joints after different surface treatment methods: U - un‐ treated, D - degreasing (chemical cleaning), M - mechanical treatment, MD - mechanical treatment and degreasing

**Figure 3.** Shear strength tests of EN AW-7075PLT0 aluminium alloy sheet adhesive joints after different surface treat‐ ment methods: U - untreated, D - degreasing (chemical cleaning), M - mechanical treatment, MD - mechanical treat‐

compared to variant D (degreasing).

ment and degreasing

96 Surface Energy

**Figure 6.** Shear strength tests of stainless steel adhesive joints after different surface treatment methods: U - untreated, D - degreasing (chemical cleaning), M - mechanical treatment, MD - mechanical treatment and degreasing

to prepare the surfaces for bonding, it can be noted that each of the analysed types of surface treatment allows for a greater strength to be obtained than the variant of untreated surface.

In numerous instances, when degreasing follows mechanical treatment, the resulting joint strength is higher as compared to joint strength of materials subjected exclusively to mechan‐ ical treatment. These results are evident in the case of CP1 and CP3 titanium sheet adhesive joints, while in EN AW-2024PLT3 and EN AW-7075PLTO aluminium alloy sheets the increase was insignificant. It ought to be mentioned, however, that chemical cleaning produced the highest scatter of joint strength value results as compared with other surface treatment operations. It could be indicative of higher non-uniformity of adhesive properties obtained in the case in question, which would in turn result in considerable differences in adhesive joint strength values obtained in tests. This observation is, furthermore, confirmed by the SFE value analysis carried out after degreasing of, e.g., CP1 (Figure 4) and CP3 (Figure 5) titanium sheets.

## **3. Adhesive properties, wettability and surface free energy**

#### **3.1. The characteristics of surface properties**

Adhesive properties of the surface layer of structural materials determine the adequacy of the constituted surface layer for the processes where adhesion plays an essential role [48, 49].

Adhesive properties can be described with different physical quantities: the contact angle Θ and related wetting phenomenon, the work of adhesion Wa and the surface free energy. The contact angle Θ is an indicator of wettability – good wettability is marked by a small contact angle (Θ < 90°), whereas poor wettability co-occurs with a high contact angle (Θ > 90°) [50-52].

The small contact angle of water is presented in Figure 7 and the contact angle of diidomethane is presented in Figure 8. Figure 9 presents the high contact angle of water (Θ > 90°). Direct measurement of the contact angle of a liquid drop on the analysed surface is presented in Figures 7-9.

Many researchers argue that adhesive properties can be determined with surface free energy (SFE). This thermodynamic quantity describes the surface energetic state and is characteristic of particular solids or liquids. There are a number of SFE calculation methods, e.g., the Fowkes method, the Zisman method, the Owens-Wendt method and the van Oss-Chaudhury-Good method, the Neumann method and the method of Wu [49, 43-56].

**Figure 7.** The small contact angle of water (Θ < 90°)

to prepare the surfaces for bonding, it can be noted that each of the analysed types of surface treatment allows for a greater strength to be obtained than the variant of untreated surface. In numerous instances, when degreasing follows mechanical treatment, the resulting joint strength is higher as compared to joint strength of materials subjected exclusively to mechan‐ ical treatment. These results are evident in the case of CP1 and CP3 titanium sheet adhesive joints, while in EN AW-2024PLT3 and EN AW-7075PLTO aluminium alloy sheets the increase was insignificant. It ought to be mentioned, however, that chemical cleaning produced the highest scatter of joint strength value results as compared with other surface treatment operations. It could be indicative of higher non-uniformity of adhesive properties obtained in the case in question, which would in turn result in considerable differences in adhesive joint strength values obtained in tests. This observation is, furthermore, confirmed by the SFE value analysis carried out after degreasing of, e.g., CP1 (Figure 4) and CP3 (Figure 5) titanium sheets.

**Figure 6.** Shear strength tests of stainless steel adhesive joints after different surface treatment methods: U - untreated,

D - degreasing (chemical cleaning), M - mechanical treatment, MD - mechanical treatment and degreasing

Adhesive properties of the surface layer of structural materials determine the adequacy of the constituted surface layer for the processes where adhesion plays an essential role [48, 49].

Adhesive properties can be described with different physical quantities: the contact angle Θ and related wetting phenomenon, the work of adhesion Wa and the surface free energy. The

**3. Adhesive properties, wettability and surface free energy**

**3.1. The characteristics of surface properties**

98 Surface Energy

Numerous methods for direct measurement of the surface free energy are applied in liquids; however, in solids only indirect methods for determining the SFE can be applied.

## **3.2. Surface free energy after various surface treatments**

#### *3.2.1. Characteristics of tested materials and surface treatment*

Tests were conducted on the following types of material:

**•** EN AW-2024PLT3 aluminium alloy sheets,

**Figure 8.** The small contact angle of diidomethane (Θ < 90°)

**Figure 9.** The high contact angle (Θ > 90°)

**•** EN AW-7075PLTO aluminium alloy sheets,


Surface treatment operations were selected according to a given structural material [47]:


**•** EN AW-7075PLTO aluminium alloy sheets,

**Figure 9.** The high contact angle (Θ > 90°)

**Figure 8.** The small contact angle of diidomethane (Θ < 90°)

100 Surface Energy

**6.** a combination of selected methods listed above.

#### *3.2.2. Method for determining free surface energy*

Adhesive properties were determined by free surface energy. Free surface energy, in turn, was determined by the Owens-Wendt method. The method assumes that free surface energy (γS) is a sum of two components: polar (γ<sup>S</sup> p) and dispersive (γ<sup>S</sup> d), and that they all are characterized by the following dependence [5, 12, 22]:

$$
\gamma\_S = \gamma\_S^{\;d} + \gamma\_S^{\;p} \tag{2}
$$

The polar component is defined as a sum of components generated by intermolecular forces, including polar, hydrogen, inductive, acidic and basic, excluding dispersive forces. Dispersive forces, on the other hand, are components of free surface energy. To deter‐ mine polar and dispersive components of free surface energy, it is necessary to measure wetting angles of the surfaces of the materials being tested using two measuring liquids. Measuring liquids used to this end are liquids whose free surface energy and its polar and dispersive components are known. One of the liquids is apolar, while the other is bipo‐ lar. Distilled water was used as the bipolar liquid and diiodomethane was used as the apolar liquid. The components γ<sup>S</sup> <sup>d</sup> and γ<sup>S</sup> p of the tested materials can be determined using the relevant formulas given in the studies [12, 22].

The essential values of the applied measuring liquids' surface free energy γ*<sup>S</sup>* and its compo‐ nents are listed in Table 4.

To calculate free surface energy, the wetting angle Θ of the surfaces of the tested materials was measured. The measurements were made using the method for direct measurement of the angle formed by a measuring liquid drop and the surface being examined. The measurements of the wetting angle were performed at a temperature of 26±2 °C and air humidity of 30±2 %. The volume of drops of the measuring liquids ranged from 0.8 to 1.5 μl. To every sample surface (there were 10 samples for each material), five drops of the measuring liquids were applied.


**Table 4.** Values of free surface energy *γL* of the measuring liquids applied and its components [22]

After that, 5-10 measurements were made and the mean for each sample batch was calculated. The wetting angle was measured immediately following the application of a drop of the measuring liquid (after a few seconds). The measurements were made using a PGX goniometer manufactured by Fibro System (Sweden) and a PG programme for computer image analysis.

#### *3.2.3. Results of surface free energy*

The preparation of adherends for bonding was material-dependent, i.e., certain treatments, such as anodizing, chromate treatment or etching, were only applicable for 2024 aluminium alloy sheet adherends. The remaining samples were subjected to standard operations, mainly degreasing, mechanical treatment and the combination of the two. Figure 10 and Figure 11 present the values of the surface free energy *γS* and its components for the analysed aluminium alloys after particular surface treatments.

**Figure 10.** Surface free energy of EN AW-2024PLT3 aluminium alloy sheets: U – untreated, D – degreasing (chemical cleaning), M – mechanical treatment, MD – mechanical treatment and degreasing, A – anodizing, Ch – chromate treat‐ ment, E – etching

Figure 12 and Figure 13 present the values of the surface free energy *γSp* and its components for titanium sheets after particular surface treatments.

Figure 14 presents the values of the surface free energy *γS* and its components for stainless steel after particular surface treatments.

After analysis of the test results presented in Figures 10-14, it becomes apparent that there is a direct correlation between different surface treatments and the surface free energy of adherends. The final values of *γS*, however, depend on the type of material rather than the type of surface treatment. Furthermore, the values of polar *γ<sup>S</sup> p* and dispersive *γ<sup>S</sup> d* components of the surface free energy exhibit the tendency to vary not only within the same type of material but within the same type of surface treatment as well. In each of the analysed instances (excluding anodizing, Table 2.15) the *γ<sup>S</sup> d* component significantly dominates over the *γ<sup>S</sup> <sup>p</sup>* in total *γS*. It can be nevertheless noted that although anodizing produces a higher polar compo‐ nent of the surface free energy, *γS* component values are comparable.

After that, 5-10 measurements were made and the mean for each sample batch was calculated. The wetting angle was measured immediately following the application of a drop of the measuring liquid (after a few seconds). The measurements were made using a PGX goniometer manufactured by Fibro System (Sweden) and a PG programme for computer image analysis.

**[mJ/m2 ]**

1 Distilled water 72.8 21.8 51.0 2 Diiodomethane 50.8 48.5 2, 3

**Table 4.** Values of free surface energy *γL* of the measuring liquids applied and its components [22]

*γ<sup>L</sup> <sup>d</sup>* **[mJ/m2 ]**

*γ<sup>L</sup> <sup>p</sup>* **[mJ/m2 ]**

The preparation of adherends for bonding was material-dependent, i.e., certain treatments, such as anodizing, chromate treatment or etching, were only applicable for 2024 aluminium alloy sheet adherends. The remaining samples were subjected to standard operations, mainly degreasing, mechanical treatment and the combination of the two. Figure 10 and Figure 11 present the values of the surface free energy *γS* and its components for the analysed aluminium

**Figure 10.** Surface free energy of EN AW-2024PLT3 aluminium alloy sheets: U – untreated, D – degreasing (chemical cleaning), M – mechanical treatment, MD – mechanical treatment and degreasing, A – anodizing, Ch – chromate treat‐

*3.2.3. Results of surface free energy*

102 Surface Energy

**No. Measuring liquid** *<sup>γ</sup><sup>L</sup>*

alloys after particular surface treatments.

ment, E – etching

**Figure 11.** Surface free energy of EN AW-7075PLTO aluminium alloy sheets: U – untreated, D – degreasing (chemical cleaning), M – mechanical treatment, MD – mechanical treatment and degreasing

**Figure 12.** Surface free energy of CP1 titanium sheets: U – untreated, D – degreasing (chemical cleaning), M – mechani‐ cal treatment, MD – mechanical treatment and degreasing

**Figure 13.** Surface free energy of CP3 titanium sheets: U – untreated, D – degreasing (chemical cleaning), M – mechani‐ cal treatment, MD – mechanical treatment and degreasing

**Figure 14.** Surface free energy of stainless steel: U – untreated, D – degreasing (chemical cleaning), M – mechanical treatment, MD – mechanical treatment and degreasing

Bearing in mind the frequently negligible differences between the values of the surface free energy of different adherends after different surface treatments, the results were subjected to statistical analysis [46].

## **4. Geometric structure of adherends**

**Figure 12.** Surface free energy of CP1 titanium sheets: U – untreated, D – degreasing (chemical cleaning), M – mechani‐

**Figure 13.** Surface free energy of CP3 titanium sheets: U – untreated, D – degreasing (chemical cleaning), M – mechani‐

cal treatment, MD – mechanical treatment and degreasing

104 Surface Energy

cal treatment, MD – mechanical treatment and degreasing

## **4.1. The characteristics of geometric structure**

The geometric structure of substrate surfaces is of considerable significance from the perspec‐ tive of adhesive bonding. According to the mechanical theory of adhesion, penetration of micropores in adhered elements is an essential condition to be fulfilled in order for the mechanical interlocking to bear loads. The mechanical theory of adhesion recognizes different factors contributing to increasing adhesive joint strength [2, 3, 5, 21]. One of these dependencies is that with increased surface roughness of a given material, the number of irregularities which can be penetrated by the adhesive grows. This leads to the conclusion that the strength of adhesive joints formed on porous substrates is significantly higher than in the case of smoothsurface adherends, resulting from a considerably larger contact surface [3].

Excessive numbers of narrow micropores may hinder adhesive penetration, particularly in the case of high viscosity adhesive or that of high surface tension, where the adhesive may stop at the peaks of irregularities. In such a situation, micropores tend to trap air bubbles to form an additional weak boundary layer, acting to the detriment of adhesion. Therefore, there exists a degree of surface roughness the exceeding of which produces disadvantageous conditions for the intermolecular bonds between adhesive and substrate.

The literature [2, 3, 23] presents dependencies used to describe the penetration of adhesive into micropores and irregularities of the surface in adhesive bonding. Depth of penetration is contingent on several factors, e.g., diameter of pores, viscosity, the surface free energy of adhesive or the wetting angle.

Different studies have analysed phenomena relevant to the relationship between surface roughness of adherends, their energetic state and wettability [32, 57, 58].

Mechanical treatment is one of the methods applied for the purpose of preparing the surface of adherends for bonding. Mechanical treatment consists in the removal of various surface contaminants, e.g., corrosion layers, and in addition, it enables surface development by constituting the geometric structure of adhered surface.

Of the numerous mechanical pretreatment methods, some prominent examples that could be mentioned are: sand- or grit-blasting, grinding or using coated abrasives [32, 58, 59]. Studies of the correlation between surface preparation and adhesive joint strength rarely offer detailed analysis of the impact of surface roughness parameters on adhesively bonded joint strength [59]. Moreover, certain researchers [2, 3] argue that surface roughness profile parameters alone fail to produce a satisfactory description of the degree of surface development.

#### **4.2. The characteristic of geometric structure tests**

The surface tests presented here indicate diversification of the geometric structure of adher‐ ends with respect to mechanical adhesion. Geometric structure often determines penetration of the adhesive into the surface irregularities and might promote mechanical adhesion, which is of great importance to adhesive bond strength.

Surface profiles, obtained from, inter alia, profilometer measurements, reveal that chemical cleaning fails to modify the surface geometry of analysed materials. On the other hand, mechanical, chemical and electrochemical treatments do generate considerable changes which consequently promote adhesion. It is for that reason that surface images and profiles were used in the characterization of the surface after chemical cleaning and mechanical treatment. In the case of EN AW-2024 aluminium alloy sheet, profiles and images were obtained following all applied surface treatments, i.e., anodizing, chromate treatment and etching, whereas for CP3 titanium sheets they were obtained after etching. The tables collate the results of measurements of surface roughness parameters, which represent mean values of 10-12 repetitions for each parameter.

The roughness of specimens was qualified by the method for measuring contact roughness, using an M2 profilometer manufactured by Mahr. The surface view was obtained using NanoFocus μscan AF2.

## **4.3. Results of geometric structure – Aluminium alloy sheets**

Two types of aluminium alloy sheets were tested in this study: EN AW-2024 (according to EN AW-2024-AlCu4Mg1 [60]) and EN AW-7075 (according to EN AW-7075-ALZn5.5MgCu [60]). The tests were performed on samples of different thicknesses and tempers [61-63]. Thickness and temper characteristics of analysed materials are presented in Table 5 and Table 6.


Designations [350]:

O - thermally treated to produce stable tempers to develop mechanical properties, as after annealing,

T3 – heat treated, subjected to cold working and natural ageing until reaching a stable condition,

PL – plated.

a degree of surface roughness the exceeding of which produces disadvantageous conditions

The literature [2, 3, 23] presents dependencies used to describe the penetration of adhesive into micropores and irregularities of the surface in adhesive bonding. Depth of penetration is contingent on several factors, e.g., diameter of pores, viscosity, the surface free energy of

Different studies have analysed phenomena relevant to the relationship between surface

Mechanical treatment is one of the methods applied for the purpose of preparing the surface of adherends for bonding. Mechanical treatment consists in the removal of various surface contaminants, e.g., corrosion layers, and in addition, it enables surface development by

Of the numerous mechanical pretreatment methods, some prominent examples that could be mentioned are: sand- or grit-blasting, grinding or using coated abrasives [32, 58, 59]. Studies of the correlation between surface preparation and adhesive joint strength rarely offer detailed analysis of the impact of surface roughness parameters on adhesively bonded joint strength [59]. Moreover, certain researchers [2, 3] argue that surface roughness profile parameters alone

The surface tests presented here indicate diversification of the geometric structure of adher‐ ends with respect to mechanical adhesion. Geometric structure often determines penetration of the adhesive into the surface irregularities and might promote mechanical adhesion, which

Surface profiles, obtained from, inter alia, profilometer measurements, reveal that chemical cleaning fails to modify the surface geometry of analysed materials. On the other hand, mechanical, chemical and electrochemical treatments do generate considerable changes which consequently promote adhesion. It is for that reason that surface images and profiles were used in the characterization of the surface after chemical cleaning and mechanical treatment. In the case of EN AW-2024 aluminium alloy sheet, profiles and images were obtained following all applied surface treatments, i.e., anodizing, chromate treatment and etching, whereas for CP3 titanium sheets they were obtained after etching. The tables collate the results of measurements of surface roughness parameters, which represent mean values of 10-12 repetitions for each

The roughness of specimens was qualified by the method for measuring contact roughness, using an M2 profilometer manufactured by Mahr. The surface view was obtained using

fail to produce a satisfactory description of the degree of surface development.

for the intermolecular bonds between adhesive and substrate.

constituting the geometric structure of adhered surface.

**4.2. The characteristic of geometric structure tests**

is of great importance to adhesive bond strength.

parameter.

NanoFocus μscan AF2.

roughness of adherends, their energetic state and wettability [32, 57, 58].

adhesive or the wetting angle.

106 Surface Energy

**Table 5.** Characteristics of EN AW-2024 aluminium alloy sheets


Designations [350]:

O - thermally treated to produce stable tempers to develop mechanical properties, as after annealing,

PL – plated.

**Table 6.** Characteristics of EN AW-7075 aluminium alloy sheets

Table 7 and Table 8 present characteristic surface roughness profile parameters (average from 10-12 values) of the analysed aluminium alloys subjected only to degreasing (chemical cleaning).


**Table 7.** Surface roughness parameters of EN AW-2024 aluminium alloy sheets


**Table 8.** Surface roughness parameters of EN AW-7075PLTO aluminium alloy sheets

It was noted that it could be as a result of rolling that particular surface roughness values are obtained. In EN AW-2024 aluminium alloy sheets' temper O there was a marked increase in surface roughness as compared with EN AW-7075 aluminium alloy sheets' temper O.

The surface roughness parameter results of the analysed aluminium alloy sheets (after degreasing) are furthermore reflected by surface images and profiles from a 3D profiler. Representative sheet specimens of different tempers are presented in Figures 15-18 [44].

**Figure 15.** Surface of EN AW-2024PLT3 aluminium alloy sheet; degreasing, 3D profiler

**Figure 16.** Representative profile of EN AW-2024PLT3 aluminium alloy sheets; degreasing

**Figure 17.** Surface of EN AW-7075PLTO aluminium alloy sheets; degreasing, 3D profiler

**No. Type of aluminium alloy Mean values of surface roughness parameters, μm**

**Table 8.** Surface roughness parameters of EN AW-7075PLTO aluminium alloy sheets

108 Surface Energy

**Figure 15.** Surface of EN AW-2024PLT3 aluminium alloy sheet; degreasing, 3D profiler

**Figure 16.** Representative profile of EN AW-2024PLT3 aluminium alloy sheets; degreasing

1 EN AW-7075PLTO 0.15 1.12 2.06 203 0.54 0.22 0.41

It was noted that it could be as a result of rolling that particular surface roughness values are obtained. In EN AW-2024 aluminium alloy sheets' temper O there was a marked increase in surface roughness as compared with EN AW-7075 aluminium alloy sheets' temper O.

The surface roughness parameter results of the analysed aluminium alloy sheets (after degreasing) are furthermore reflected by surface images and profiles from a 3D profiler. Representative sheet specimens of different tempers are presented in Figures 15-18 [44].

Ra Rz Rz max Sm Rp Rpk Rvk

**Figure 18.** Representative profile of EN AW-7075PLTO aluminium alloy sheets; degreasing

In comparing images and profiles of the analysed surfaces, it emerges that it was probably the differences in the rolling process of the analysed sheets that produced differences in the geometric structure of the surface.

Surface roughness measurements of EN AW-2024PLT3 and EN AW-7075PLTO aluminium alloy sheets after different surface treatments are presented in Table 9 and Table 10.

As opposed to chemical cleaning, mechanical treatment did modify the geometric structure of aluminium alloy sheet surface. Surface roughness parameters of EN AW-7075PLTO following mechanical treatment and degreasing are decreased in comparison with samples subject to mechanical treatment only, which could be a consequence of removing mechanical treatment residue from the surface. Model images and roughness parameters (average from 10-12 values) after this surface treatment method are presented in Figures 19-22.


**Table 9.** Surface roughness parameters of EN AW-2024PLT3 after different surface treatments


**Table 10.** Surface roughness parameters of EN AW-7075PLTO after different surface treatments

**Figure 19.** Surface of EN AW-2024PLT3 aluminium alloy sheets; mechanical treatment, 3D profiler

**Figure 20.** Representative profile of EN AW-2024PLT3 aluminium alloy sheets; mechanical treatment

**No. Surface treatment Mean values of surface roughness parameters, μm**

**Table 9.** Surface roughness parameters of EN AW-2024PLT3 after different surface treatments

**Table 10.** Surface roughness parameters of EN AW-7075PLTO after different surface treatments

**Figure 19.** Surface of EN AW-2024PLT3 aluminium alloy sheets; mechanical treatment, 3D profiler

3

110 Surface Energy

3

Mechanical treatment and chemical cleaning

**No. Surface treatment**

Mechanical treatment and chemical cleaning

1 Chemical cleaning 0.20 1.36 2.59 270 0.65 0.28 0.64 2 Mechanical treatment 1.31 9.83 14.24 137 4.87 2.17 2.52

4 Anodizing 0.41 3.78 5.79 122 1.49 0.52 1.02 5 Chromate treatment 0.37 3.48 5.70 108 1.52 0.73 0.77 6 Etching 0.33 2.72 4.63 145 1.04 0.48 0.87

1 Chemical cleaning 0.31 2.31 4.64 290 1.13 0.64 0.72 2 Mechanical treatment 1.58 12.07 16.96 142 6.17 2.68 3.14

Ra Rz Rz max Sm Rp Rpk Rvk

1.40 9.58 12.46 125 4.44 2.00 2.73

**Mean values of surface roughness parameters, μm** Ra Rz Rz max Sm Rp Rpk Rvk

1.28 9.16 12.12 113 4.41 1.79 2.53

**Figure 21.** Surface of EN AW-7075PLTO aluminium alloy sheets; mechanical treatment, 3D profiler

**Figure 22.** Representative profile of EN AW-7075PLTO aluminium alloy sheets; mechanical treatment

The surface of EN AW-2024PLT3 and EN AW-7075PLTO aluminium alloys after mechanical treatment manifests a considerably higher number of micro-irregularities in the uniform distribution on the surface (Figures 19 and 21). Simultaneously, an increase in surface rough‐ ness resulting from mechanical treatment (4-6 times) (Table 9 and Table 10), compared with most of the other surface treatments, should be noted.

**Figure 23.** Surface of EN AW-7075PLTO aluminium alloy sheets; anodizing, 3D profiler

The surface roughness parameter results for the analysed aluminium alloy sheets EN AW-2024PLT3 after anodizing, chromate treatment and etching are reflected by surface images and profiles from a 3D profiler. Representative sheet specimens are presented in Figures 23-28.

**Figure 24.** Representative profile of EN AW-2024PLT3 aluminium alloy sheets; anodizing

**Figure 22.** Representative profile of EN AW-7075PLTO aluminium alloy sheets; mechanical treatment

most of the other surface treatments, should be noted.

112 Surface Energy

**Figure 23.** Surface of EN AW-7075PLTO aluminium alloy sheets; anodizing, 3D profiler

The surface of EN AW-2024PLT3 and EN AW-7075PLTO aluminium alloys after mechanical treatment manifests a considerably higher number of micro-irregularities in the uniform distribution on the surface (Figures 19 and 21). Simultaneously, an increase in surface rough‐ ness resulting from mechanical treatment (4-6 times) (Table 9 and Table 10), compared with

**Figure 25.** Surface of EN AW-2024PLT3 aluminium alloy sheets; chromate treatment, 3D profiler

**Figure 26.** Representative profile of EN AW-2024PLT3 aluminium alloy sheets; chromate treatment [44]

**Figure 27.** Surface of EN AW-2024PLT3 aluminium alloy sheets; etching, 3D profiler

**Figure 28.** Representative profile of EN AW-2024PLT3 aluminium alloy sheets; etching

The results of surface roughness parameter measurement carried out on test samples subjected to anodizing indicate that anodizing has an impact on the height of surface irregularities.

Comparing analysed samples subjected to surface treatment technologies, it was chromate conversion coating which proved to have the greatest impact on both the height and the structure of surface irregularities, as shown in Figure 25 and Figure 26. In the case of this method, the decrease in the values of irregularities' height parameters ranged between -6.2 % for the *Rk* and -42.6 % for the *Rvk*, while there was practically no difference in the values of the two parameters *Rp* and *Rpk*. As a result, noticeable smoothing of the test sample surface was achieved. Reduced mean spacing of profile peaks' *Sm* by 18 % and the local *S* by 12.5 % together represent an increased number of irregularities, as confirmed by the representation of the roughness profile in Figure 26 and of the structure in Figure 25 [44].

## **5. Conclusions and summary**

0.0 1600.0 3200.0 4800.0 6400.0 8000.0

**Figure 26.** Representative profile of EN AW-2024PLT3 aluminium alloy sheets; chromate treatment [44]

**Figure 27.** Surface of EN AW-2024PLT3 aluminium alloy sheets; etching, 3D profiler




0.00

1.60

3.20 [µm]

114 Surface Energy

TU-Dresden

[µm]

0.80 µm

Rauheit

Results obtained from adhesive joint strength testing of the materials show that surface treatment plays an important role in increasing the strength of analysed joints. Moreover, the application of the same surface treatment in the context of different structural materials produces different strength values.

Tests indicate that in numerous instances it is mechanical treatment only or mechanical treatment followed by degreasing which translate into the highest joint strength. Statistical analysis of adhesive joint strength test results demonstrated that in certain cases no significant differences between different surface treatments effects can be detected. Moreover, degreasing conducted after mechanical treatment is not reflected in a significant increase in joint strength values.

In certain joints, e.g., CP1 titanium sheets, degreasing is sufficient to provide relatively high joint strength. Furthermore, in certain applications, the differences between the effects of degreasing and mechanical treatment with degreasing showed no statistical significance. It was also noted that mechanical treatment can produce joints of substantial strength (e.g., EN AW-7075PLTO aluminium sheet adhesive joints).

The applied surface treatment methods, such as mechanical, chemical and electrochemical methods, modify geometry and roughness parameters. Chemical cleaning, however, has little impact on surface geometry, in the case of both untreated and mechanically treated surfaces. Nevertheless, in certain applications, change in surface parameters can be observed.

The surface treatment method that introduces extensive changes in the analysed materials' surface geometry is mechanical treatment. In some cases it has, moreover, played a role in increasing surface roughness parameters, e.g., in aluminium alloys and titanium sheets.

The application of various surface treatments in different structural materials allows for modification of their adhesive properties, determined by the surface free energy. Statistical analysis proved that in the majority of cases the surface free energy values responded consid‐ erably to the surface treatment operations. Although in some of the analysed variants the differences in γ<sup>S</sup> values were negligible, they were nevertheless statistically relevant (level of significance of 0.05).

It was noted that different surface treatments contribute not only to the surface free energy changes but to the SFE components' share in the total value. In the majority of variants of EN AW-2024PLT3 aluminium alloy sheet surface treatment, the dispersive component amounted to the 93-99 % of the total surface free energy. Electrochemical treatment (anodizing and chromate treatment) produced very different results. The polar component γSp after chromate treatment was greater than the dispersive component γSd, and after anodizing the polar component amounted to 41 % of total SFE. It appears, then, that chemical treatment methods manifest capabilities to increase the surface free energy and to balance the dispersive-to-polar component ratio. This ratio is typical of particular technologies of surface treatment; for instance in CP1 and CP3 titanium sheets, the dispersive component of the SFE ranges between 73 % and 87 % of the total surface free energy, whereas for stainless steel it is 78 %-90 %.

With regard to the surface free energy values in EN AW-2024PLT3 aluminium alloy sheets, it is electrochemical treatment (chromate treatment and anodizing) which is the most advanta‐ geous. In the case of other treatments, the second-best mechanical treatment is additionally beneficial in EN AW-7075PLTO aluminium alloy sheets and CP1 titanium sheets.

Development of surface geometry and roughness parameters of the analysed adherends does not always correlate with the increase in the values of the surface free energy of these materials. It is a non-linear dependence, in which the increase of surface roughness parameters is on several occasions not reflected in an identical increase in the surface free energy.

The assumption, then, should be that in order for the determination of a particular surface for adhesive processes to be comprehensive, it should account for the adherend's surface geom‐ etry as well as its adhesion properties. The geometry of the surface can influence the mechanical adhesion, and the surface free energy is connected with both mechanical adhesion and the other constituent of adhesion: proper adhesion.

## **Author details**

conducted after mechanical treatment is not reflected in a significant increase in joint strength

In certain joints, e.g., CP1 titanium sheets, degreasing is sufficient to provide relatively high joint strength. Furthermore, in certain applications, the differences between the effects of degreasing and mechanical treatment with degreasing showed no statistical significance. It was also noted that mechanical treatment can produce joints of substantial strength (e.g., EN

The applied surface treatment methods, such as mechanical, chemical and electrochemical methods, modify geometry and roughness parameters. Chemical cleaning, however, has little impact on surface geometry, in the case of both untreated and mechanically treated surfaces.

The surface treatment method that introduces extensive changes in the analysed materials' surface geometry is mechanical treatment. In some cases it has, moreover, played a role in increasing surface roughness parameters, e.g., in aluminium alloys and titanium sheets.

The application of various surface treatments in different structural materials allows for modification of their adhesive properties, determined by the surface free energy. Statistical analysis proved that in the majority of cases the surface free energy values responded consid‐ erably to the surface treatment operations. Although in some of the analysed variants the differences in γ<sup>S</sup> values were negligible, they were nevertheless statistically relevant (level of

It was noted that different surface treatments contribute not only to the surface free energy changes but to the SFE components' share in the total value. In the majority of variants of EN AW-2024PLT3 aluminium alloy sheet surface treatment, the dispersive component amounted to the 93-99 % of the total surface free energy. Electrochemical treatment (anodizing and chromate treatment) produced very different results. The polar component γSp after chromate treatment was greater than the dispersive component γSd, and after anodizing the polar component amounted to 41 % of total SFE. It appears, then, that chemical treatment methods manifest capabilities to increase the surface free energy and to balance the dispersive-to-polar component ratio. This ratio is typical of particular technologies of surface treatment; for instance in CP1 and CP3 titanium sheets, the dispersive component of the SFE ranges between 73 % and 87 % of the total surface free energy, whereas for stainless steel it is 78 %-90 %.

With regard to the surface free energy values in EN AW-2024PLT3 aluminium alloy sheets, it is electrochemical treatment (chromate treatment and anodizing) which is the most advanta‐ geous. In the case of other treatments, the second-best mechanical treatment is additionally

Development of surface geometry and roughness parameters of the analysed adherends does not always correlate with the increase in the values of the surface free energy of these materials. It is a non-linear dependence, in which the increase of surface roughness parameters is on

beneficial in EN AW-7075PLTO aluminium alloy sheets and CP1 titanium sheets.

several occasions not reflected in an identical increase in the surface free energy.

Nevertheless, in certain applications, change in surface parameters can be observed.

AW-7075PLTO aluminium sheet adhesive joints).

values.

116 Surface Energy

significance of 0.05).

Anna Rudawska\*

Address all correspondence to: a.rudawska@pollub.pl

Faculty of Mechanical Engineering, Lublin University of Technology, Lublin, Poland

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## **Chapter 5**
