**3. Basic thermodynamics of interfaces**

For an open system of variable surface area, the Gibbs free energy must depend on composition, temperature, *T*, pressure, *p*, and the total surface area, *A*:

$$\mathbf{G} = \mathbf{G}(T, p, A, n\_1, n\_2, \dots, n\_k) \tag{33}$$

From this function it follows that:

$$d\mathbf{G} = \left(\frac{\partial \mathbf{G}}{\partial T}\right)\_{r, \mathbf{n}\_i} dT + \left(\frac{\partial \mathbf{G}}{\partial p}\right)\_{r, \mathbf{n}\_j} dp + \left(\frac{\partial \mathbf{G}}{\partial A}\right)\_{r, \mathbf{n}\_i} dA + \sum\_{i=1}^{i=1} \left(\frac{\partial \mathbf{G}}{\partial n\_i}\right)\_{T, p, \mathbf{n}\_j} dn\_i \tag{34}$$

The first two partial differentials refer to constant composition, so we my use the general definitions:

$$G = H - TS = \mathcal{U} + PV - TS \tag{35}$$

To obtain

$$S = -\left(\frac{\partial G}{\partial T}\right)\_{p,nj} \tag{36}$$

and

$$V = -\left(\frac{\partial G}{\partial p}\right)\_{p, \text{eq}}\tag{37}$$

Insertion of these relations into (35) gives us the fundamental result

$$\text{s-dG} = -\text{SdT} + \text{Vdp} + \text{\textchi} \, \text{dA} + \sum\_{i=1}^{i=k} \mu\_i dn\_i \tag{38}$$

where the chemical potential µi is defined as:

$$
\mu\_i = \left(\frac{\partial G}{\partial \mathbf{n}\_i}\right)\_{\mathbf{r}\_{,\mathbf{p},\mathbf{n}\_j}} \tag{39}
$$

and the surface energy γ as:

$$\mathcal{V} = \left(\frac{\partial G}{\partial A}\right)\_{\mathbb{T}\_{\mathcal{V}, \text{adj}}} \tag{40}$$

The chemical potential is defined as the increase in free energy of a system on adding an infinitesimal amount of a component (per unit number of molecules of that component added) when *T*, *p* and the composition of all other components are held constant. Clearly, from this definition, if a component '*i*' in phase A has a higher chemical potential than in phase B (that is, *A B i i* ) then the total free energy will be lowered if molecules are transferred from phase A to B and this will occur in a spontaneous process until the

For an open system of variable surface area, the Gibbs free energy must depend on

, , , , , ,

*GGG G dG dT dp dA dn T pA n*

The first two partial differentials refer to constant composition, so we my use the general

*<sup>G</sup> <sup>S</sup> T* 

*<sup>G</sup> <sup>V</sup>*

Insertion of these relations into (35) gives us the fundamental result

where the chemical potential µi is defined as:

and the surface energy γ as:

phase B (that is, *A B*

*i i* 

*dG SdT Vdp*

*i*

*p ni i p nj T pn <sup>i</sup> Tpnj*

*p* ,*nj*

*p* ,*ni*

1 *i k*

 *dA dn* 

) then the total free energy will be lowered if molecules are

*i*

*i i*

(38)

*p* 

, ,

*Tpn* , , *j*

*<sup>i</sup> <sup>T</sup> <sup>p</sup> nj*

*G n*

*G A*

The chemical potential is defined as the increase in free energy of a system on adding an infinitesimal amount of a component (per unit number of molecules of that component added) when *T*, *p* and the composition of all other components are held constant. Clearly, from this definition, if a component '*i*' in phase A has a higher chemical potential than in

transferred from phase A to B and this will occur in a spontaneous process until the

 

(34)

1 2 ( , , , , ,... ) *G GT p An n n <sup>k</sup>* (33)

1

*G H TS U PV TS* (35)

*i i*

(36)

(37)

(39)

(40)

*i k*

 

composition, temperature, *T*, pressure, *p*, and the total surface area, *A*:

**3. Basic thermodynamics of interfaces** 

From this function it follows that:

definitions:

To obtain

and

chemical potentials equalize, at equilibrium. It is easy to see from this why the chemical potential is so useful in mixtures and solutions in matter transfer (open) processes (Norde, W., 2003). This is especially clear when it is understood that m*i* is a simple function of concentration, that is:

$$
\mu\_i = \mu\_i^0 + kT \ln \mathbf{C}\_i \tag{41}
$$

for dilute mixtures, where m*i* o is the standard chemical potential of component '*i*', usually 1 M for solutes and 1 atm for gas mixtures. This equation is based on the entropy associated with a component in a mixture and is at the heart of why we generally plot measurable changes in any particular solution property against the log of the solute concentration, rather than using a linear scale. Generally, only substantial changes in concentration or pressure produce significant changes in the properties of the mixture. (For example, consider the use of the pH scale.) (Koopal L.K., and et al. 1994).

### **3.1 Thermodynamics for closed systems**

The First Law of Thermodynamics is the law of conservation of energy; it simply requires that the total quantity of energy be the same both before and after the conversion. In other words, the total energy of any system and its surroundings is conserved. It does not place any restriction on the conversion of energy from one form to another. The interchange of heat and work is also considered in this first law. In principle, the internal energy of any system can be changed, by heating or doing work on the system. The First Law of Thermodynamics requires that for a closed (but not isolated) system, the energy changes of the system be exactly compensated by energy changes in the surroundings. Energy can be exchanged between such a system and its surroundings in two forms: heat and work. Heat and work have the same units (joule, J) and they are ways of transferring energy from one entity to another. A quantity of heat, Q, represents an amount of energy in transit between a system and its surroundings, and is not a property of the system. Heat flows from higher to lower temperature systems. Work, W, is the energy in transit between a system and its surroundings, resulting from the displacement of external force acting on the system. Like heat, a quantity of work represents an amount of energy and is not a property of the system. Temperature is a property of a system while heat and work refer to a process. It is important to realize the difference between temperature, heat capacity and heat: temperature, T, is a property which is equal when heat is no longer conducted between bodies in thermal contact and can be determined with suitable instruments (thermometers) having a reference system depending on a material property (for example, mercury thermometers show the density differences of liquid mercury metal with temperature in a capillary column in order to visualize and measure the change of temperature). Suppose any closed system (thus having a constant mass) undergoes a process by which it passes from an initial state to a final state. If the only interaction with its surroundings is in the form of transfers of heat, Q, and work, W, then only the internal energy, U, can be changed, and the First Law of Thermodynamics is expressed mathematically as (Lyklema, J. ;2005 & Keller J.U.;2005)

$$
\Delta \mathcal{L} I = \mathcal{U}\_{final} - \mathcal{U}\_{initial} = \mathcal{Q} + \mathcal{W} \tag{42}
$$

where Q and W are quantities inclusive of sign so that when the heat transfers from the system or work is done by the system, we use negative values in Equation (11). Processes

Fig. 1. Diagram of the variation in solute concentration at an interface between two phases.

Fig. 2. Diagrammatic illustration of the change in surface energy caused by the addition of a

where A is the interfacial area (note that Γi may be either positive or negative). Let us now examine the effect of adsorption on the interfacial energy (γ). If a solute 'i' is positively adsorbed with a surface density of Γi, we would expect the surface energy to decrease on increasing the bulk concentration of this component (and vice versa). This situation is illustrated in Figure 10.2, where the total free energy of the system GT and mi are both increased by addition of component i but because this component is favourably adsorbed at the surface (only relative to the solvent, since both have a higher energy state at the surface),

solute.

*i i n A* 

(44)

where heat should be given to the system (or absorbed by the system) (Q > 0) are called endothermic and processes where heat is taken from the system (or released from the system) (Q < 0) are called exothermic. The total work performed on the system is W. There are many different ways that energy can be stored in a body by doing work on it: volumetrically by compressing it; elastically by straining it; electrostatically by charging it; by polarizing it in an electric field E; by magnetizing it in a magnetic field H; and chemically, by changing its composition to increase its chemical potential. In interface science, the formation of a new surface area is also another form of doing work. Each example is a different type of work – they all have the form that the (differential) work performed is the change in some extensive variable of the system multiplied by an intensive variable. In thermodynamics, the most studied work type is pressure–volume work, *W*PV, on gases performed by compressing or expanding the gas confined in a cylinder under a piston. All other work types can be categorized by a single term, *non-pressure–volume work*, *W*non-PV. Then, *W* is expressed as the sum of the pressure–volume work, *W*PV, and the non pressure– volume work, *W*non-PV, when many types of work are operative in a process (Miladinovic N., Weatherley L.R. 2008).

Equation (11) states that the internal energy, Δ*U* depends only on the initial and final states and in no way on the path followed between them. In this form, heat can be defined as *the work-free transfer of internal energy from one system to another*. Equation (11) applies both to *reversible* and *irreversible* processes. A *reversible process* is an infinitely slow process during which departure from equilibrium is always infinitesimally small. In addition, such processes can be reversed at any moment by infinitesimal changes in the surroundings (in external conditions) causing it to retrace the initial path in the opposite direction. A reversible process proceeds so that the system is never displaced more than differentially from an equilibrium state. An *irreversible process* is a process where the departure from equilibrium cannot be reversed by changes in the surroundings. For a differential change, Equation (11) is often used in the differential form (Scatchard, G. 1976), (Zeldowitsch J., 1934):

$$d\mathcal{L}I = \partial \mathcal{W} + \partial \mathcal{Q} \tag{43}$$

for reversible processes involving infinitesimal changes only. The internal energy, *U* is a function of the measurable quantities of the system such as temperature, volume, and pressure, which are all state functions like internal energy itself. The differential d*U is* an exact differential similar to d*T*, d*V*, and d*P*; so we can always integrate 2 *<sup>f</sup>* ( ) *U dU* expression.

#### **3.2 Derivation of the gibbs adsorption isotherm**

Let us consider the interface between two phases, say between a liquid and a vapor, where a solute (i) is dissolved in the liquid phase. The real concentration gradient of solute near the interface may look like Figure 10.1. When the solute increases in concentration near the surface (e.g. a surfactant) there must be a surface excess of solute *ni* , compared with the bulk value continued right up to the interface. We can define a surface excess concentration (in units of moles per unit area) as:

1

where heat should be given to the system (or absorbed by the system) (Q > 0) are called endothermic and processes where heat is taken from the system (or released from the system) (Q < 0) are called exothermic. The total work performed on the system is W. There are many different ways that energy can be stored in a body by doing work on it: volumetrically by compressing it; elastically by straining it; electrostatically by charging it; by polarizing it in an electric field E; by magnetizing it in a magnetic field H; and chemically, by changing its composition to increase its chemical potential. In interface science, the formation of a new surface area is also another form of doing work. Each example is a different type of work – they all have the form that the (differential) work performed is the change in some extensive variable of the system multiplied by an intensive variable. In thermodynamics, the most studied work type is pressure–volume work, *W*PV, on gases performed by compressing or expanding the gas confined in a cylinder under a piston. All other work types can be categorized by a single term, *non-pressure–volume work*, *W*non-PV. Then, *W* is expressed as the sum of the pressure–volume work, *W*PV, and the non pressure– volume work, *W*non-PV, when many types of work are operative in a process (Miladinovic N.,

Equation (11) states that the internal energy, Δ*U* depends only on the initial and final states and in no way on the path followed between them. In this form, heat can be defined as *the work-free transfer of internal energy from one system to another*. Equation (11) applies both to *reversible* and *irreversible* processes. A *reversible process* is an infinitely slow process during which departure from equilibrium is always infinitesimally small. In addition, such processes can be reversed at any moment by infinitesimal changes in the surroundings (in external conditions) causing it to retrace the initial path in the opposite direction. A reversible process proceeds so that the system is never displaced more than differentially from an equilibrium state. An *irreversible process* is a process where the departure from equilibrium cannot be reversed by changes in the surroundings. For a differential change, Equation (11) is often used in the differential form (Scatchard, G. 1976), (Zeldowitsch J.,

for reversible processes involving infinitesimal changes only. The internal energy, *U* is a function of the measurable quantities of the system such as temperature, volume, and pressure, which are all state functions like internal energy itself. The differential d*U is* an exact differential similar to d*T*, d*V*, and d*P*; so we can always integrate

Let us consider the interface between two phases, say between a liquid and a vapor, where a solute (i) is dissolved in the liquid phase. The real concentration gradient of solute near the interface may look like Figure 10.1. When the solute increases in concentration near the

bulk value continued right up to the interface. We can define a surface excess concentration

*dU W Q* (43)

, compared with the

Weatherley L.R. 2008).

1934):

2

*<sup>f</sup>* ( ) *U dU* 

expression.

(in units of moles per unit area) as:

**3.2 Derivation of the gibbs adsorption isotherm** 

surface (e.g. a surfactant) there must be a surface excess of solute *ni*

1

Fig. 1. Diagram of the variation in solute concentration at an interface between two phases.

Fig. 2. Diagrammatic illustration of the change in surface energy caused by the addition of a solute.

where A is the interfacial area (note that Γi may be either positive or negative). Let us now examine the effect of adsorption on the interfacial energy (γ). If a solute 'i' is positively adsorbed with a surface density of Γi, we would expect the surface energy to decrease on increasing the bulk concentration of this component (and vice versa). This situation is illustrated in Figure 10.2, where the total free energy of the system GT and mi are both increased by addition of component i but because this component is favourably adsorbed at the surface (only relative to the solvent, since both have a higher energy state at the surface),

the surface. The validity of this fundamental equation of adsorption has been proven by comparison with direct adsorption measurements. The method is best applied to liquid/vapor and liquid/liquid interfaces, where surface energies can easily be measured. However, care must be taken to allow equilibrium adsorption of the solute (which may be

Finally, it should be noted that (51) was derived for the case of a single adsorbing solute (e.g. a non-ionic surfactant). However, for ionic surfactants such as CTAB, two species (CTA+ and Br-) adsorb at the interface. In this case the equation becomes(Murrell, J.N. and Jenkins, A.D.

1

because the bulk chemical potentials of both ions change with concentration of the

Adsorption equilibria information is the most important piece of information in understanding an adsorption process. No matter how many components are present in the system, the adsorption equilibria of pure components are the essential ingredient for the understanding of how many those components can be accommodated by a solid adsorbent. With this information, it can be used in the study of adsorption kinetics of a single component, adsorption equilibria of multicomponent systems, and then adsorption kinetics of multicomponent systems. In this section, we present the fundamentals of pure component equilibria. Various fundamental equations are shown, and to start with the proceeding we will present the most basic theory in adsorption: the Langmuir theory (1918). This theory allows us to understand the monolayer surface adsorption on an ideal surface. By an ideal surface here, we mean that the energy fluctuation on this surface is periodic and the magnitude of this fluctuation is larger than the thermal energy of a molecule (kT), and hence the troughs of the energy fluctuation are acting as the adsorption sites. If the distance between the two neighboring troughs is much larger than the diameter of the adsorbate molecule, the adsorption process is called localised and each adsorbate molecule will occupy one site. Also, the depth of all troughs of the ideal surface are the same, that is the adsorption heat released upon adsorption on each site is the same no matter what the loading is. After the Langmuir theory, we will present the Gibbs thermodynamics approach. This approach treats the adsorbed phase as a single entity, and Gibbs adapted the classical thermodynamics of the bulk phase and applied it to the adsorbed phase. In doing this the concept of volume in the bulk phase is replaced by the area, and the pressure is replaced by the so-called spreading pressure. By assuming some forms of thermal equation of state relating the number of mole of adsorbate, the area and the spreading pressure (analogue of equations of state in the gas phase) and using them in the Gibbs equation, a number of fundamental equations can be derived, such as the linear isotherm, etc (Mohan D., Pittman Jr

Following the Gibbs approach, we will show the vacancy solution theory developed by Suwanayuen and Danner in 1980. Basically in this approach the system is assumed to consist of two solutions. One is the gas phase and the other is the adsorbed phase. The

1

(52)

2 ln *TT RT c* 

1

**4. Fundamentals of pure component adsorption equilibrium** 

slow) during measurement.

surfactant.

C.U. 2006).

1994), (Ng J.C.Y., and et al. 2002):

the work required to create new surface (i.e. γ) is reduced. Thus, although the total free energy of the system increases with the creation of new surface, this process is made easier as the chemical potential of the selectively adsorbed component increases (i.e. with concentration). This reduction in surface energy must be directly related to the change in chemical potential of the solute and to the amount adsorbed and is therefore given by the simple relationship (Zeldowitsch J., 1934):

$$d\chi = -\Gamma\_i d\mu\_i \tag{45}$$

or, for the case of several components,

$$d\mathcal{Y} = -\sum\_{i} \Gamma\_{i} d\mu\_{i} \tag{46}$$

The change in mi is caused by the change in bulk solute concentration. This is the Gibbs surface tension equation. Basically, these equations describe the fact that increasing the chemical potential of the adsorbing species reduces the energy required to produce new surface (i.e. γ). This, of course, is the principal action of surfactants, which will be discussed in more detail in a later section. Using this result let us now consider a solution of two components

$$d\eta = -\Gamma\_\text{i} d\mu\_\text{i} - \Gamma\_\text{i} d\mu\_\text{i} \tag{47}$$

and hence the adsorption excess for one of the components is given by

$$
\Gamma\_1 = -\left(\frac{\partial \mathcal{Y}}{\partial \mu\_1}\right)\_{\Gamma, \mu\_2} \tag{48}
$$

Thus, in principle, we could determine the adsorption excess of one of the components from surface tension measurements, if we could vary m1 independently of µ2. But the latter appears not to be possible, because the chemical potentials are dependent on the concentration of each component. However, for dilute solutions the change in µ for the solvent is negligible compared with that of the solute. Hence, the change for the solvent can be ignored and we obtain the simple result that

$$d\gamma = -\Gamma\_\text{\tiny\kern-1.2ex/c\, 1em/c} \tag{49}$$

Now, since µ1 = µ2+RTlnc1, differentiation with respect to c1 gives

$$
\left(\frac{\partial \mu\_{\rm i}}{\partial \mathbf{c}\_{\rm i}}\right)\_{\rm T} = RT \left(\frac{\partial \ln c\_{\rm i}}{\partial \mathbf{c}\_{\rm i}}\right)\_{\rm T} = \frac{RT}{c\_{\rm i}}\tag{50}
$$

Then substitution in (49) leads to the result:

$$\Gamma\_{\rm 1} = -\frac{1}{RT} \left( \frac{\partial \mathcal{Y}}{\partial \ln \mathcal{C}\_{\rm 1}} \right)\_{\rm T} = \frac{\mathcal{C}\_{\rm 1}}{RT} \left( \frac{\partial \mathcal{Y}}{\partial \mathcal{C}\_{\rm 1}} \right)\_{\rm T} \tag{51}$$

This is the important Gibbs adsorption isotherm. (Note that for concentrated solutions the activity should be used in this equation.) An experimental measurement of γ over a range of concentrations allows us to plot γ against lnc1 and hence obtain Γ1, the adsorption density at

the work required to create new surface (i.e. γ) is reduced. Thus, although the total free energy of the system increases with the creation of new surface, this process is made easier as the chemical potential of the selectively adsorbed component increases (i.e. with concentration). This reduction in surface energy must be directly related to the change in chemical potential of the solute and to the amount adsorbed and is therefore given by the

> *i i d d*

*ii i d d*

The change in mi is caused by the change in bulk solute concentration. This is the Gibbs surface tension equation. Basically, these equations describe the fact that increasing the chemical potential of the adsorbing species reduces the energy required to produce new surface (i.e. γ). This, of course, is the principal action of surfactants, which will be discussed in more detail in a later section. Using this result let us now consider a solution of two

11 22 *d dd*

 

 

1

and hence the adsorption excess for one of the components is given by

Now, since µ1 = µ2+RTlnc1, differentiation with respect to c1 gives

1

1

 

1 *T* ,2

Thus, in principle, we could determine the adsorption excess of one of the components from surface tension measurements, if we could vary m1 independently of µ2. But the latter appears not to be possible, because the chemical potentials are dependent on the concentration of each component. However, for dilute solutions the change in µ for the solvent is negligible compared with that of the solute. Hence, the change for the solvent can

> 1 1 *d d*

1 1

This is the important Gibbs adsorption isotherm. (Note that for concentrated solutions the activity should be used in this equation.) An experimental measurement of γ over a range of concentrations allows us to plot γ against lnc1 and hence obtain Γ1, the adsorption density at

1 11 ln

*RT c RT c* 

*T T*

*c RT RT c cc*

1

1 1

ln *T T c*

(45)

(46)

(47)

(48)

(49)

(50)

(51)

simple relationship (Zeldowitsch J., 1934):

or, for the case of several components,

be ignored and we obtain the simple result that

Then substitution in (49) leads to the result:

components

the surface. The validity of this fundamental equation of adsorption has been proven by comparison with direct adsorption measurements. The method is best applied to liquid/vapor and liquid/liquid interfaces, where surface energies can easily be measured. However, care must be taken to allow equilibrium adsorption of the solute (which may be slow) during measurement.

Finally, it should be noted that (51) was derived for the case of a single adsorbing solute (e.g. a non-ionic surfactant). However, for ionic surfactants such as CTAB, two species (CTA+ and Br- ) adsorb at the interface. In this case the equation becomes(Murrell, J.N. and Jenkins, A.D. 1994), (Ng J.C.Y., and et al. 2002):

$$\Gamma\_{\rm i} = -\frac{1}{2RT} \left( \frac{\partial \mathcal{Y}}{\partial \ln c\_{\rm i}} \right)\_{TT} \tag{52}$$

because the bulk chemical potentials of both ions change with concentration of the surfactant.
