3.1. Derjaguin-Landau-Verwey-Overbeek (DLVO) theory

In the DLVO theory, van der Waals attraction force and electrostatic repulsion force are suggested as the dominant interactions between two charged hydrophobic or lyophobic particles/surfaces in electrolyte solution. Moreover, the total interaction between particles and solid surfaces in a liquid is assumed as the sum of the two interactions. This is the first theory enabling to explain and predict the experimental observations of particle deposition and aggregation in a quantitative way. The van der Waals interaction arises from the electromagnetic effects of the molecules composing the particles while the electric double layer interaction is caused by the overlapping of the electric double layers of two particles/surfaces in an aqueous medium. Normally, the former is attractive and the latter is repulsive, which could be changed depending on the material properties in some specific cases [4].

### 3.1.1. van der Waals force

The van der Walls force, also known as London-van der Waals force, originates from a fluctuating electromagnetic field in particles and between particle and solid surface which is induced by the spontaneous magnetic and electrical polarisation. The van der Waals force can be either attractive or repulsive depending on the material property and is always attractive between identical materials. A number of methods have been proposed to calculate the van der Waals interaction energy [7–11]. Basically, there are two computation methods: the microscopic and the macroscopic.

For the microscopic methods, perturbation theory was initially adopted to solve the Schrӧdinger equation for the interactions between two hydrogen atoms at a large separation distance by Wang [12] and London [13], and they considered the interactions between the protons and electrons of the two atoms in the calculation. Their study provides a basis of quantum-mechanical analysis of the interaction between two non-polar molecules. Margenau [14] improved the analysis with consideration of higher moments. The retardation effect for the interactions was further investigated by Casimir and Polder [15] when the separation distance was shorter than the characteristic wavelength of radiation. Subsequently, Hamaker [11] proposed a simplified microscopic approximation in which the interaction between two solids is pair-wise additive. In another words, the total interaction force can be obtained by simply summing up the forces over all pairs of atoms in both solids. It is worth mentioning that Hamker's microscopic method neglects the retardation effect and many-body interactions. However, the influence of neighbouring atoms cannot be ignored, especially for condensed medium such as liquid. As a result, the pair additivity is difficult to be implemented for interacting objects in aqueous medium.

<sup>A</sup><sup>132</sup> <sup>≈</sup> <sup>3</sup> 4 kBT <sup>ε</sup><sup>1</sup> � <sup>ε</sup><sup>3</sup> ε<sup>1</sup> þ ε<sup>3</sup>

100 nm for most materials.

3.1.2. Electrostatic double layer force

Figure 1. Interaction between the sphere and the plate.

� � <sup>ε</sup><sup>2</sup> � <sup>ε</sup><sup>3</sup>

ε<sup>2</sup> þ ε<sup>3</sup> � �

temperature and non-polar media have values in the range 2–3.

Fvdw <sup>¼</sup> �A<sup>132</sup> 6kT

þ 3hν<sup>e</sup> 8 ffiffiffi 2 p

n2 <sup>1</sup> <sup>þ</sup> <sup>n</sup><sup>2</sup> 3 � �<sup>0</sup>:<sup>5</sup>

where the refractive index ni and zero frequency term ε0,i are the temperature-dependent factors for van der Waals interaction, kB is the Boltzmann constant, and T is the absolute temperature of the system. The zero frequency term for most of the aqueous colloids is about equal to 3kT/4 or around 3 � <sup>10</sup>�<sup>21</sup> J. Water has a dielectric constant of about 80 at room

In consideration of the retardation effect, Suzuki and Higuchi [22] proposed an approximated expression for the van der Waals interaction potential between the sphere and the plate as

where a is the radius of the sphere, h is the minimum surface-to-surface separation between the sphere and the plate, and H (h/a) and λ (λ/a) are the separation distance between the sphere and plate surface and the dimensionless characteristic wavelength, respectively, as illustrated in Figure 1. λ is the characteristic wavelength of the interaction which has a value of about

Because of the electric double layer (EDL) force, particles can be well dispersed in liquids other than forming aggregation. The EDL force originates from the repulsion between the charged surfaces of the particles and solid surfaces immersed in liquids of high dielectric constants. The charges form the so-called electric double layer in the vicinity of the particles and the solid surfaces. The charging mechanism of a solid surface in a liquid medium can be categorised into two: (1) ionisation or dissociation of surface groups on the solid surface and (2) adsorption or binding of ions from electrolyte solutions onto a surface with oppositely charged sites or an

n2 <sup>1</sup> � <sup>n</sup><sup>2</sup> 3 � � n<sup>2</sup>

n2 <sup>2</sup> <sup>þ</sup> <sup>n</sup><sup>2</sup> 3 � �<sup>0</sup>:<sup>5</sup>

<sup>λ</sup> <sup>λ</sup> <sup>þ</sup> <sup>22</sup>:232<sup>H</sup> � �

<sup>2</sup> � <sup>n</sup><sup>2</sup> 3 � �

� �<sup>0</sup>:<sup>5</sup> h i

http://dx.doi.org/10.5772/intechopen.78240

<sup>2</sup> <sup>þ</sup> <sup>n</sup><sup>2</sup> 3

(3)

107

n2 <sup>1</sup> <sup>þ</sup> <sup>n</sup><sup>2</sup> 3 � �<sup>0</sup>:<sup>5</sup> <sup>þ</sup> <sup>n</sup><sup>2</sup>

Particle Deposition in Microfluidic Devices at Elevated Temperatures

<sup>H</sup><sup>2</sup> <sup>λ</sup> <sup>þ</sup> <sup>11</sup>:116<sup>H</sup> � �<sup>2</sup> (4)

A more rigorous approach, macroscopic theory, was proposed in order to account for the aforementioned challenges. Dzyaloshinskii et al. [9] developed a new theory to avoid the problem of additivity encountered in the microscopic methods, known as Lifshitz theory. In this theory, large subjects are treated as a continuous medium without considering the atomic structure. The interaction forces between the subjects are calculated based on the bulk material properties including dielectric constants and refractive indices. The retardation effect is implicitly considered in the full Lifshitz treatment, but it is readily take account of the effect via modifying the Hamaker constant. In terms of calculation of Hamaker constant, various approaches have been developed and details can be found in the literatures [16–19].

Hamaker constants are most accurately calculated by Lifshitz theory, which determines the magnitude of the interaction through the frequency dependent dielectric properties of the intervening media [9, 18]. The Hamaker constant is estimated from the frequency dependent dielectric properties of the individual materials comprising the system as

$$A\_{132} = \frac{3}{2} kT \begin{pmatrix} \frac{1}{2} \left( \frac{\varepsilon\_1(\text{iv}\_0) - \varepsilon\_3(\text{iv}\_0)}{\varepsilon\_1(\text{iv}\_0) + \varepsilon\_3(\text{iv}\_0)} \right) \left( \frac{\varepsilon\_2(\text{iv}\_0) - \varepsilon\_1(\text{iv}\_0)}{\varepsilon\_2(\text{iv}\_0) + \varepsilon\_1(\text{iv}\_0)} \right) \\ + \sum\_{j=1}^{m} \left( \frac{\varepsilon\_1(\text{iv}\_j) - \varepsilon\_3(\text{iv}\_j)}{\varepsilon\_1(\text{iv}\_j) + \varepsilon\_3(\text{iv}\_j)} \right) \left( \frac{\varepsilon\_2(\text{iv}\_j) - \varepsilon\_1(\text{iv}\_j)}{\varepsilon\_2(\text{iv}\_j) + \varepsilon\_1(\text{iv}\_j)} \right) \end{pmatrix} \tag{1}$$

where A<sup>132</sup> is the Hamaker constant between particles '1' and the plate '3' in medium '2' and is a measure of the magnitude of the interaction between two objects. Israelachvili [20] proposed a simplified expression for the function ε ivj � � based on the refractive index and the absorption frequency of materials.

$$
\varepsilon(i\upsilon\_{\rangle}) = 1 + \frac{n^2 - 1}{\upsilon^2} \tag{2}
$$

In practice, it is difficult to obtain all the parameters in Eq. (2), mainly the absorption frequency. Assuming the absorption frequencies of the three media are the same, the Tabor-Winterton (TW) expression was developed to overcome the difficulties, shown as [21]

$$A\_{132} \approx \frac{3}{4} k\_B T \left(\frac{\varepsilon\_1 - \varepsilon\_3}{\varepsilon\_1 + \varepsilon\_3}\right) \left(\frac{\varepsilon\_2 - \varepsilon\_3}{\varepsilon\_2 + \varepsilon\_3}\right) + \frac{3h\nu\_\varepsilon}{8\sqrt{2}} \frac{\left(n\_1^2 - n\_3^2\right) \left(n\_2^2 - n\_3^2\right)}{\left(n\_1^2 + n\_3^2\right)^{0.5} \left(n\_2^2 + n\_3^2\right)^{0.5} \left[\left(n\_1^2 + n\_3^2\right)^{0.5} + \left(n\_2^2 + n\_3^2\right)^{0.5}\right]}\right) \tag{3}$$

where the refractive index ni and zero frequency term ε0,i are the temperature-dependent factors for van der Waals interaction, kB is the Boltzmann constant, and T is the absolute temperature of the system. The zero frequency term for most of the aqueous colloids is about equal to 3kT/4 or around 3 � <sup>10</sup>�<sup>21</sup> J. Water has a dielectric constant of about 80 at room temperature and non-polar media have values in the range 2–3.

In consideration of the retardation effect, Suzuki and Higuchi [22] proposed an approximated expression for the van der Waals interaction potential between the sphere and the plate as

$$F\_{vdw} = \frac{-A\_{132}}{6kT} \frac{\overline{\lambda} (\overline{\lambda} + 22.232H)}{H^2 (\overline{\lambda} + 11.116H)^2} \tag{4}$$

where a is the radius of the sphere, h is the minimum surface-to-surface separation between the sphere and the plate, and H (h/a) and λ (λ/a) are the separation distance between the sphere and plate surface and the dimensionless characteristic wavelength, respectively, as illustrated in Figure 1. λ is the characteristic wavelength of the interaction which has a value of about 100 nm for most materials.

#### 3.1.2. Electrostatic double layer force

For the microscopic methods, perturbation theory was initially adopted to solve the Schrӧdinger equation for the interactions between two hydrogen atoms at a large separation distance by Wang [12] and London [13], and they considered the interactions between the protons and electrons of the two atoms in the calculation. Their study provides a basis of quantum-mechanical analysis of the interaction between two non-polar molecules. Margenau [14] improved the analysis with consideration of higher moments. The retardation effect for the interactions was further investigated by Casimir and Polder [15] when the separation distance was shorter than the characteristic wavelength of radiation. Subsequently, Hamaker [11] proposed a simplified microscopic approximation in which the interaction between two solids is pair-wise additive. In another words, the total interaction force can be obtained by simply summing up the forces over all pairs of atoms in both solids. It is worth mentioning that Hamker's microscopic method neglects the retardation effect and many-body interactions. However, the influence of neighbouring atoms cannot be ignored, especially for condensed medium such as liquid. As a result, the pair additivity is

A more rigorous approach, macroscopic theory, was proposed in order to account for the aforementioned challenges. Dzyaloshinskii et al. [9] developed a new theory to avoid the problem of additivity encountered in the microscopic methods, known as Lifshitz theory. In this theory, large subjects are treated as a continuous medium without considering the atomic structure. The interaction forces between the subjects are calculated based on the bulk material properties including dielectric constants and refractive indices. The retardation effect is implicitly considered in the full Lifshitz treatment, but it is readily take account of the effect via modifying the Hamaker constant. In terms of calculation of Hamaker constant, various

Hamaker constants are most accurately calculated by Lifshitz theory, which determines the magnitude of the interaction through the frequency dependent dielectric properties of the intervening media [9, 18]. The Hamaker constant is estimated from the frequency dependent

� � � <sup>ε</sup><sup>3</sup> ivj

� � <sup>þ</sup> <sup>ε</sup><sup>3</sup> ivj

where A<sup>132</sup> is the Hamaker constant between particles '1' and the plate '3' in medium '2' and is a measure of the magnitude of the interaction between two objects. Israelachvili [20] proposed

In practice, it is difficult to obtain all the parameters in Eq. (2), mainly the absorption frequency. Assuming the absorption frequencies of the three media are the same, the Tabor-

Winterton (TW) expression was developed to overcome the difficulties, shown as [21]

� � <sup>ε</sup>2ð Þ� iv<sup>0</sup> <sup>ε</sup>1ð Þ iv<sup>0</sup>

� �

� � ! <sup>ε</sup><sup>2</sup> ivj

<sup>n</sup><sup>2</sup> � <sup>1</sup>

ε2ð Þþ iv<sup>0</sup> ε1ð Þ iv<sup>0</sup> � �

ε<sup>2</sup> ivj

� � � <sup>ε</sup><sup>1</sup> ivj

!

� � <sup>þ</sup> <sup>ε</sup><sup>1</sup> ivj

� � based on the refractive index and the absorption

� �

1

CCCCA

(1)

� �

<sup>v</sup><sup>2</sup> (2)

approaches have been developed and details can be found in the literatures [16–19].

ε1ð Þ� iv<sup>0</sup> ε3ð Þ iv<sup>0</sup> ε1ð Þþ iv<sup>0</sup> ε3ð Þ iv<sup>0</sup>

ε<sup>1</sup> ivj

ε<sup>1</sup> ivj

ε ivj � � <sup>¼</sup> <sup>1</sup> <sup>þ</sup>

dielectric properties of the individual materials comprising the system as

1 2

0

BBBB@

þ X∞ j¼1

<sup>A</sup><sup>132</sup> <sup>¼</sup> <sup>3</sup> 2 kT

a simplified expression for the function ε ivj

frequency of materials.

106 Microfluidics and Nanofluidics

difficult to be implemented for interacting objects in aqueous medium.

Because of the electric double layer (EDL) force, particles can be well dispersed in liquids other than forming aggregation. The EDL force originates from the repulsion between the charged surfaces of the particles and solid surfaces immersed in liquids of high dielectric constants. The charges form the so-called electric double layer in the vicinity of the particles and the solid surfaces. The charging mechanism of a solid surface in a liquid medium can be categorised into two: (1) ionisation or dissociation of surface groups on the solid surface and (2) adsorption or binding of ions from electrolyte solutions onto a surface with oppositely charged sites or an

Figure 1. Interaction between the sphere and the plate.

originally uncharged solid surface. For a single particle suspended in a liquid medium, the particle is covered by the electric double layer (Figure 2). With consideration of the finite size of ions, Stern [23] developed an electric double layer model in which one immobilised layer of ions is absorbed onto the particle surface and the other layer is filled with diffusive space charges from the liquid medium (Figure 2). The former layer is termed as Stern layer and the latter layer is called as diffuse or Gouy layer.

Due to the nonuniform distribution of charges around the charged surface, electric potential reduces gradually with the separation distance from the solid surface to the bulk liquid phase. In the electric double layer model, several potentials are defined including surface potential on the solid surface, Stern potential at the Stern layer and zeta potential (ζ) at slipping plane. Assuming ions of identical property and average surface charge over the whole solid surface, the electric potential (ψ) and the average charge distribution in the diffuse layer of the electric double layer can be computed based on the non-linear Poisson-Boltzmann equation (PBE) as [24–26]

$$\nabla^2 \psi = \frac{-1}{\varepsilon\_0 \varepsilon} \sum\_i n\_i^0 z\_i e \exp\left(\frac{-z\_i e \psi}{k\_B T}\right) \tag{5}$$

made: interactions with constant surface potential and constant surface charge density. For the constant surface potential cases, surface-chemical equilibrium is maintained while two particles/ surfaces are approaching in a very short time. This may not be realistic for some practical cases [27]. For the constant surface charge cases, two particles/surfaces have fixed surface charge densities in the approaching process. These two assumptions are applied to the potential and charge on the particle/solid surfaces, whereas the interaction between electrical double layers is determined by the potential at the Stern plane. The charges at the Stern layer may behave differently from those on the particle/solid surface during the approach process. Recently, Barisik et al. [28] and Zhao et al. [29] have applied a complex charge regulation as boundary conditions

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Generally, the EDL interaction energy can be computed based on two methods. One method is to directly solve the Poisson-Boltzmann equation for systems of particle/solid surfaces. Normally, it is difficult to obtain simple analytical solutions by this method. The other method is to construct the formula based on known expressions for each of the surfaces involved without consideration of influences of the other surfaces. The approximations of EDL interaction energy obtained in this way are often more attractive for practical applications which require

In 1934, Derjaguin [31] developed an integration method to calculate the electric double layer interactions between two spheres in a dilute suspension. It has become a widely adopted method in colloidal chemistry since then. The EDL interaction energy between two spheres

where h denotes the minimum separation distance between two sphere surfaces, and a<sup>1</sup> and a<sup>2</sup> are the radii of two spheres. The EDL interaction force can be obtained by differentiating the

It should be noted that the above expressions are only applicable for cases in which κap > 5 and h ≪ a<sup>p</sup> are valid. By allowing one of the radii to approach infinity, the sphere-plate interactions can be derived from the sphere-sphere interactions. Assuming constant surface potential, a theoretical expression of EDL interaction energy between a sphere and a plane was

> ln <sup>1</sup> <sup>þ</sup> exp ð Þ �κ<sup>h</sup> 1 � exp ð Þ �κh � �

<sup>þ</sup> ln 1 � exp ð Þ �2κ<sup>h</sup> � � " #: (8)

ð ∞

υEdh (6)

υEð Þh (7)

h

Vedl <sup>¼</sup> <sup>2</sup>πa1a<sup>2</sup> a<sup>1</sup> þ a<sup>2</sup>

Fedl <sup>¼</sup> <sup>2</sup>πa1a<sup>2</sup> a<sup>1</sup> þ a<sup>2</sup>

to calculate the EDL interactions in nanoscale.

fairy accuracy and simplicity [30].

developed by Hogg et al. [32] as

Vedl <sup>¼</sup> πε0ε<sup>a</sup> <sup>ξ</sup><sup>2</sup>

3.1.3. Sphere-plate double layer interactions

with overlapping electric double layers can be calculated as

interaction energy, Vedl, with the separation distance, h, as

<sup>p</sup> <sup>þ</sup> <sup>ξ</sup><sup>2</sup> w � � <sup>2</sup>ξ<sup>2</sup>

pξ2 w

ξ2 <sup>p</sup> <sup>þ</sup> <sup>ξ</sup><sup>2</sup> w

where n<sup>0</sup> is the number density of ions in bulk, i represents the component i, z and e are the valence and the elementary electric charge, ɛ<sup>0</sup> is the permittivity of vacuum, and ɛ is the static dielectric constant.

As illustrated in Figure 2, a particle approaches a solid surface in an electrolyte solution or two charged particles approach each other, and their diffuse layers would overlap with each other. EDL force is repulsive for two surfaces with charges of same sign, while it becomes attractive for two particles with charges of opposite sign. The accuracy of calculating the EDL interaction is influenced by various factors. To simplify the calculation, two important assumptions are

Figure 2. Schematic of a diffuse double layer of a charged particle in the vicinity of a charged solid/wall surface.

made: interactions with constant surface potential and constant surface charge density. For the constant surface potential cases, surface-chemical equilibrium is maintained while two particles/ surfaces are approaching in a very short time. This may not be realistic for some practical cases [27]. For the constant surface charge cases, two particles/surfaces have fixed surface charge densities in the approaching process. These two assumptions are applied to the potential and charge on the particle/solid surfaces, whereas the interaction between electrical double layers is determined by the potential at the Stern plane. The charges at the Stern layer may behave differently from those on the particle/solid surface during the approach process. Recently, Barisik et al. [28] and Zhao et al. [29] have applied a complex charge regulation as boundary conditions to calculate the EDL interactions in nanoscale.

Generally, the EDL interaction energy can be computed based on two methods. One method is to directly solve the Poisson-Boltzmann equation for systems of particle/solid surfaces. Normally, it is difficult to obtain simple analytical solutions by this method. The other method is to construct the formula based on known expressions for each of the surfaces involved without consideration of influences of the other surfaces. The approximations of EDL interaction energy obtained in this way are often more attractive for practical applications which require fairy accuracy and simplicity [30].

#### 3.1.3. Sphere-plate double layer interactions

originally uncharged solid surface. For a single particle suspended in a liquid medium, the particle is covered by the electric double layer (Figure 2). With consideration of the finite size of ions, Stern [23] developed an electric double layer model in which one immobilised layer of ions is absorbed onto the particle surface and the other layer is filled with diffusive space charges from the liquid medium (Figure 2). The former layer is termed as Stern layer and the

Due to the nonuniform distribution of charges around the charged surface, electric potential reduces gradually with the separation distance from the solid surface to the bulk liquid phase. In the electric double layer model, several potentials are defined including surface potential on the solid surface, Stern potential at the Stern layer and zeta potential (ζ) at slipping plane. Assuming ions of identical property and average surface charge over the whole solid surface, the electric potential (ψ) and the average charge distribution in the diffuse layer of the electric double layer

where n<sup>0</sup> is the number density of ions in bulk, i represents the component i, z and e are the valence and the elementary electric charge, ɛ<sup>0</sup> is the permittivity of vacuum, and ɛ is the static

As illustrated in Figure 2, a particle approaches a solid surface in an electrolyte solution or two charged particles approach each other, and their diffuse layers would overlap with each other. EDL force is repulsive for two surfaces with charges of same sign, while it becomes attractive for two particles with charges of opposite sign. The accuracy of calculating the EDL interaction is influenced by various factors. To simplify the calculation, two important assumptions are

Figure 2. Schematic of a diffuse double layer of a charged particle in the vicinity of a charged solid/wall surface.

<sup>i</sup> zie exp �zie<sup>ψ</sup>

kBT � �

(5)

can be computed based on the non-linear Poisson-Boltzmann equation (PBE) as [24–26]

∇2

<sup>ψ</sup> <sup>¼</sup> -1 ε0ε X i n0

latter layer is called as diffuse or Gouy layer.

dielectric constant.

108 Microfluidics and Nanofluidics

In 1934, Derjaguin [31] developed an integration method to calculate the electric double layer interactions between two spheres in a dilute suspension. It has become a widely adopted method in colloidal chemistry since then. The EDL interaction energy between two spheres with overlapping electric double layers can be calculated as

$$V\_{cell} = \frac{2\pi a\_1 a\_2}{a\_1 + a\_2} \Big|\_{h}^{\approx} \nu\_E dh \tag{6}$$

where h denotes the minimum separation distance between two sphere surfaces, and a<sup>1</sup> and a<sup>2</sup> are the radii of two spheres. The EDL interaction force can be obtained by differentiating the interaction energy, Vedl, with the separation distance, h, as

$$F\_{cell} = \frac{2\pi a\_1 a\_2}{a\_1 + a\_2} \nu\_E(h) \tag{7}$$

It should be noted that the above expressions are only applicable for cases in which κap > 5 and h ≪ a<sup>p</sup> are valid. By allowing one of the radii to approach infinity, the sphere-plate interactions can be derived from the sphere-sphere interactions. Assuming constant surface potential, a theoretical expression of EDL interaction energy between a sphere and a plane was developed by Hogg et al. [32] as

$$V\_{\rm eff} = \pi \varepsilon\_0 \varepsilon a \left(\xi\_p^2 + \xi\_w^2\right) \left[\frac{2\xi\_p^2 \xi\_w^2}{\xi\_p^2 + \xi\_w^2} \ln\left[\frac{1 + \exp\left(-\kappa h\right)}{1 - \exp\left(-\kappa h\right)}\right] + \ln\left[1 - \exp\left(-2\kappa h\right)\right]\right].\tag{8}$$

The electrical double layer (EDL) interaction force can be obtained by differentiating electrical double layer interaction energy with separation between two surfaces.

$$F\_{\rm eff} = -\frac{\partial V\_{\rm eff}}{\partial h} = 2\pi\varepsilon\_0 \varepsilon a\kappa \left(\xi\_p^2 + \xi\_w^2\right) \left[\frac{2\xi\_p^2 \xi\_w^2}{\xi\_p^2 + \xi\_w^2} \frac{\exp\left(-\kappa h\right)}{1 - \exp\left(-2\kappa h\right)} - \frac{\exp\left(-2\kappa h\right)}{1 - \exp\left(-2\kappa h\right)}\right] \tag{9}$$

where ζ<sup>p</sup> and ζ<sup>w</sup> denote the zeta potentials of colloid particle and channel wall, respectively. ɛ<sup>0</sup> and ε<sup>r</sup> represent the permittivity of vacuum and relative permittivity, respectively. The EDL thickness, also known as Debye length, κ�<sup>1</sup> is defined as

$$\kappa^{-1} = \left(\frac{2e^2 z^2 n\_{\circ}}{\varepsilon k\_{\text{B}} T}\right)^{-0.5} \tag{10}$$

V ¼ Vvdw þ Vedl (11)

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111

Eq. (11) gives both a theoretical framework to predict and compare experimentally measured colloidal interactions, and the knowledge of how surface interactions can be controlled.

In the classical DLVO theory, van der Waals and electrical double layer interactions play vital roles in colloidal particle interactions. This theory has been successfully utilised to explain many experimental observations. Whereas, there are situations in which theoretical predications based on interactions of electrical double layer and van der Waals force cannot provide reasonable agreement with experimental results [6, 37]. For instance, the classic DLVO theory fails to explain the interactions with ultra-short separation distance (i.e., shorter than a few nanometres). The continuum theories are not valid in such short distances, and bulk material properties (e.g., refractive index, density, and dielectric constant) cannot be used to describe such interactions. For these cases, some additional non-DLVO forces can be introduced into the DLVO theory, such as Born repulsion [38], polymer bridging [39], hydration forces [40], hydrophobic interaction [41], and steric interaction [42]. In this section, a brief introduction for non-DLVO forces is provided and more details can be found in a comprehensive review by

Polymer bridging theory applies to polymer flocculation. It is postulated that polymer bridges are built between neighbouring solid particles in a suspension in order to form a loose porous 3D network of solid particles (i.e., floc). When the detailed spatial variation of the short-range forces are not crucial, Elimelech et al. [30] reported that the microscopically averaged Born repulsion could be a convenient approach to consider effects of non-DLVO interactions. As particles interact with adsorbed fluid layers, solvation or hydration forces begin to take effects. Grabbe and Horn [43] suggested that the repulsive hydration force plays a dominant role for two interacting silica surfaces in a short range immersed in an electrolyte solution (NaCl). Unlike the electric double layer force, the hydration force was found to be independent on the electrolyte concentration over the range in their experiments. However, the physical mechanism of the hydration force is still unclear. The anomalous polarisation of water near the interfaces could generate the hydration force. It also could originate from the entropic repulsion of thermally activated molecular groups from protrusions on the surfaces [44–46]. Water molecules between two hydrophobic surfaces tend to migrate from the narrow gap to the bulk liquid at extremely short separation distance. It is because that the opportunities for hydrogen bonding are unlimited in the bulk liquid and free energy is lower than in the gap. As a result, an attractive force, hydrophobic force, would be generated between the two surfaces. This attractive force works in much greater range (up to 80 nm) than the van der Waals force and is one to two orders of magnitude stronger [41, 47, 48]. When two polymer-covered particles approach to each other, steric or osmotic forces would be developed between the particles. The steric force is related to the repulsive entropic force caused by the entropy of confining these chains for overlapping polymer molecules. So far, theories of steric forces are not wellestablished. Many components can affect the magnitude of the steric forces, such as bonding

3.2. Non-DLVO forces

Liang [4].

where e represents the electron charge, z is the valance of ions and n<sup>∞</sup> denotes the bulk number density of ions. When the conditions of h ≪ a<sup>p</sup> and κap ≫ 1 are satisfied, the expressions above can work well with cases of small potential. Alternatively, the EDL interaction energy can be computed with either the linear superposition method or the complete numerical solution of the nonlinear Poisson-Boltzmann equation [33]. Considering ion-ion interactions, a complex statistical mechanical model was developed for calculating the EDL interaction based on the thermodynamic entropy and Helmholtz free energy approach. Different boundary conditions have been studied, such as charge regulation and constant surface charge density [28, 34, 35].

The total interaction energy, V, in the DLVO theory is obtained by the summation of the electrostatic and van der Waals contributions as illustrated in Figure 3. With the electronic double layer potential and van der Waals potential described under previous mentioned assumptions, the total interaction is calculated as

Figure 3. Example diagram of potential energy vs. separation distance [36].

$$V = V\_{vdw} + V\_{ell} \tag{11}$$

Eq. (11) gives both a theoretical framework to predict and compare experimentally measured colloidal interactions, and the knowledge of how surface interactions can be controlled.

#### 3.2. Non-DLVO forces

The electrical double layer (EDL) interaction force can be obtained by differentiating electrical

pξ2 w

where ζ<sup>p</sup> and ζ<sup>w</sup> denote the zeta potentials of colloid particle and channel wall, respectively. ɛ<sup>0</sup> and ε<sup>r</sup> represent the permittivity of vacuum and relative permittivity, respectively. The EDL

where e represents the electron charge, z is the valance of ions and n<sup>∞</sup> denotes the bulk number density of ions. When the conditions of h ≪ a<sup>p</sup> and κap ≫ 1 are satisfied, the expressions above can work well with cases of small potential. Alternatively, the EDL interaction energy can be computed with either the linear superposition method or the complete numerical solution of the nonlinear Poisson-Boltzmann equation [33]. Considering ion-ion interactions, a complex statistical mechanical model was developed for calculating the EDL interaction based on the thermodynamic entropy and Helmholtz free energy approach. Different boundary conditions have been studied, such as charge regulation and constant

The total interaction energy, V, in the DLVO theory is obtained by the summation of the electrostatic and van der Waals contributions as illustrated in Figure 3. With the electronic double layer potential and van der Waals potential described under previous mentioned

exp ð Þ �κh

<sup>1</sup> � exp ð Þ �2κ<sup>h</sup> � exp ð Þ �2κ<sup>h</sup>

" #

1 � exp ð Þ �2κh

(9)

(10)

ξ2 <sup>p</sup> <sup>þ</sup> <sup>ξ</sup><sup>2</sup> w

<sup>κ</sup>�<sup>1</sup> <sup>¼</sup> <sup>2</sup>e<sup>2</sup>z<sup>2</sup>n<sup>∞</sup> εkBT � ��0:<sup>5</sup>

double layer interaction energy with separation between two surfaces.

<sup>p</sup> <sup>þ</sup> <sup>ξ</sup><sup>2</sup> w � � <sup>2</sup>ξ<sup>2</sup>

<sup>∂</sup><sup>h</sup> <sup>¼</sup> <sup>2</sup>πε0εaκ ξ<sup>2</sup>

thickness, also known as Debye length, κ�<sup>1</sup> is defined as

Fedl ¼ � <sup>∂</sup>Vedl

110 Microfluidics and Nanofluidics

surface charge density [28, 34, 35].

assumptions, the total interaction is calculated as

Figure 3. Example diagram of potential energy vs. separation distance [36].

In the classical DLVO theory, van der Waals and electrical double layer interactions play vital roles in colloidal particle interactions. This theory has been successfully utilised to explain many experimental observations. Whereas, there are situations in which theoretical predications based on interactions of electrical double layer and van der Waals force cannot provide reasonable agreement with experimental results [6, 37]. For instance, the classic DLVO theory fails to explain the interactions with ultra-short separation distance (i.e., shorter than a few nanometres). The continuum theories are not valid in such short distances, and bulk material properties (e.g., refractive index, density, and dielectric constant) cannot be used to describe such interactions. For these cases, some additional non-DLVO forces can be introduced into the DLVO theory, such as Born repulsion [38], polymer bridging [39], hydration forces [40], hydrophobic interaction [41], and steric interaction [42]. In this section, a brief introduction for non-DLVO forces is provided and more details can be found in a comprehensive review by Liang [4].

Polymer bridging theory applies to polymer flocculation. It is postulated that polymer bridges are built between neighbouring solid particles in a suspension in order to form a loose porous 3D network of solid particles (i.e., floc). When the detailed spatial variation of the short-range forces are not crucial, Elimelech et al. [30] reported that the microscopically averaged Born repulsion could be a convenient approach to consider effects of non-DLVO interactions. As particles interact with adsorbed fluid layers, solvation or hydration forces begin to take effects. Grabbe and Horn [43] suggested that the repulsive hydration force plays a dominant role for two interacting silica surfaces in a short range immersed in an electrolyte solution (NaCl). Unlike the electric double layer force, the hydration force was found to be independent on the electrolyte concentration over the range in their experiments. However, the physical mechanism of the hydration force is still unclear. The anomalous polarisation of water near the interfaces could generate the hydration force. It also could originate from the entropic repulsion of thermally activated molecular groups from protrusions on the surfaces [44–46]. Water molecules between two hydrophobic surfaces tend to migrate from the narrow gap to the bulk liquid at extremely short separation distance. It is because that the opportunities for hydrogen bonding are unlimited in the bulk liquid and free energy is lower than in the gap. As a result, an attractive force, hydrophobic force, would be generated between the two surfaces. This attractive force works in much greater range (up to 80 nm) than the van der Waals force and is one to two orders of magnitude stronger [41, 47, 48]. When two polymer-covered particles approach to each other, steric or osmotic forces would be developed between the particles. The steric force is related to the repulsive entropic force caused by the entropy of confining these chains for overlapping polymer molecules. So far, theories of steric forces are not wellestablished. Many components can affect the magnitude of the steric forces, such as bonding stress between the polymer molecules, the quantity or coverage of polymer molecules on each solid surface and solid surfaces (i.e., reversible process or not) [49–52].
