Article pubs.acs.org/Langmuir
Surface Tension and Adsorption without a Dividing Surface Abraham Marmur* Department of Chemical Engineering, Technion − Israel Institute of Technology, Haifa 32000, Israel ABSTRACT: The ingenious concept of a dividing surface of zero thickness that was introduced by Gibbs is the basis of the theory of surface tension and adsorption. However, some fundamental questions, mainly those related to the location of the dividing surface and the proper definition of relative adsorption, have remained open over the years. To avoid these questions, the present paper proposes to analyze an interfacial phase by defining a thermodynamic system of constant, but nonzero thickness. The interfacial phase is analyzed as it really is, namely a nonuniform three-dimensional entity. The current analysis redevelops the equation for calculating surface tension, though with different assumptions. However, the main point in the proposed model is that the thermodynamic interfacial system, due to its fixed thickness, conforms to the requirement of first-order homogeneity of the internal energy. This property is the key that allows using the Gibbs adsorption isotherm. It is also characteristic of the Gibbs dividing surface model, but has not always been discussed with regard to subsequent models. The resulting equation leads to a simple, “natural” expression for the relative adsorption. This expression may be compared with simulations and sophisticated surface concentration measurements, and from which the dependence of interfacial tension on the solution composition can be derived. Finally, it is important to point out that in order to calculate the interfacial tension as well as the relative adsorption from data on the properties of the interfacial phase, there is no need to know its exact thickness, as long as it is bigger than the actual thickness but sufficiently small.
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INTRODUCTION Interfacial phenomena have been attracting more and more attention during recent decades. This is mainly due to the steady trend in science and technology of going down in systems size to the micrometer and nanometer ranges. Consequently, it is becoming more and more important to understand the challenging fundamentals of interfacial science. As is very well-known, there exists a very thin region between two phases at equilibrium, within which the thermodynamic properties change gradually from those of one phase to those of the other (assuming the system not to be too close to the critical point). This region is referred to as the “surface phase,” when the system includes only one dense phase in contact with vapor or gas, or “interfacial phase” when the system contains more than one dense phase. For the sake of brevity, the term “surface” will be used here to also represent the term “interface” or “interfacial.” Interestingly, there are still open questions surrounding the definition of the most fundamental concepts in the field: surface tension and adsorption. The nonuniformity of properties in the surface phase requires thermodynamic considerations that are somewhat different from those used for bulk phases (that are isotropic and uniform at equilibrium). To avoid dealing with nonuniform properties, Gibbs1 developed the ingenious concept of a dividing mathematical surface. This zero-thickness surface separates two model bulk phases that are supposedly uniform in properties all the way to the dividing surface. In order to account for the error involved in this assumed model, the dividing surface is assigned appropriate position and thermodynamic properties, which ensure that the model system © 2015 American Chemical Society
as a whole has the same total energy and overall composition as the real one. The correction required for the internal energy, U, is attributed to the dividing surface, and is defined as σA, where A is the surface area and σ is the surface tension. In addition, in order to account for the true mass of each species in the system, the proper amounts are attributed to the dividing surface per unit area. This was described by Gibbs as “adsorption” to the surface. The question of the effect of the position of the dividing surface within the surface phase on the values of surface tension and adsorption has been debated for many years,2−5 although not intensely. The existing literature has been recently thoroughly reviewed by Radke.5 This question is important, since without a clear definition of this position it is difficult, if not impossible, to directly compare experimental data or molecular simulations with calculations based on the dividing surface concept. Such a comparison is not only essential for fundamental understanding, but also for the ability to predict surface properties of novel soft matter systems. The relevance of this question will keep increasing as the means of experimental testing and simulations of surfaces keep improving. Of special importance are multicomponent systems, for which the effects of adsorption may be complicated. In order to avoid the uncertainty regarding the effect of positioning of the dividing surface on surface tension, Bakker suggested an approach that assumed the surface phase to be a Received: September 30, 2015 Revised: November 2, 2015 Published: November 2, 2015 12653
DOI: 10.1021/acs.langmuir.5b03647 Langmuir 2015, 31, 12653−12657
Article
Langmuir three-dimensional, though very thin region.3 The properties of the system were assumed to vary continuously across this molecularly thin surface phase, from those of one bulk phase to those of the other. In particular, the stress tensor was assumed to vary continuously within the surface phase. For a plane surface, the stress tensor component, τ, that is parallel to the surface was assumed to vary in the direction perpendicular to the surface, z, while the component that is perpendicular to the surface was considered constant and equal to the pressure in the bulk phases, P∞. The mechanical balance led to the following definition of surface tension:3
In other words, if all the extensive parameters of the system (entropy, Ŝ, volume, V̂ , and all values of the number of moles of species i, ni) are multiplied by an arbitrary factor, t, then Û increases by the same factor t. Starting with this fundamental property, the Gibbs−Duhem equation for bulk phases is derived as3 m
∑ nidμi = 0
where μi is the chemical potential of component i, and m is the number of components. The Gibbs dividing surface methodology considers the surface phase to be characterized by its surface area instead of its volume. Therefore, the internal energy of the surface phase has to obey the condition
∞
σ≡
∫−∞ [P∞ − τ(z)] dz
(3)
i=1
(1)
It should be noted that the quantity in square brackets in this equation becomes zero outside the surface region, where all components of the stress tensor are equal and their value is the pressure, P∞. Therefore, there is no need to know the actual thickness of the surface region. This equation is the working tool for calculating the value of surface tension by molecular simulations. The Bakker approach for calculating surface tension indeed enables consideration of the nonuniform surface phase as a whole, without using the dividing surface concept. While this approach satisfactorily covers the case of pure liquids, it does not deal with the issue of surface adsorption in multicomponent systems. Several other attempts have been made to solve this problem;4,5 however, it seems that the thermodynamic requirement for the validity of the Gibbs− Duhem equation for multicomponent systems (which for surface phases is known as the Gibbs adsorption equation) has not always been considered. Thus, the purpose of the present paper is to suggest a coherent approach that enables the calculation of both surface tension and adsorption in multicomponent systems, without using the dividing surface concept and in full agreement with the Gibbs−Duhem equation. To achieve this goal, a novel, though simple analysis of surface phases is presented that treats the surface phase as it really is, namely, as a nonuniform phase. Following the background on the Gibbs−Duhem equation and the definition of the thermodynamic system, two results will be presented. First, a general definition of surface tension that turns out to be identical to that of Bakker, despite the difference in the system definitions. This identity is taken as a confirmation of the correctness of the present approach. Then, the main result is a “natural”, straightforward definition of relative adsorption. Both definitions may be applied to data (e.g., from molecular simulations) without the need to exactly know the thickness of the surface thermodynamic surface phase.
s s s s Û (tS ̂ , tAs , tn1s , tn2s , tn3s , tn4s ...) = tÛ (S ̂ , As , n1s , n2s , n3s , n4s ...)
(4)
where the superscript s indicates the surface phase. So, the surface area is assumed to play the same role as that of the volume in bulk systems. Indeed, if t identical surface areas (subsystems) of zero thickness, with the same number of adsorbed molecules and having the same entropy are considered together as a system, the internal energy of the system is t times that of each subsystem. Thus, the Gibbs dividing surface model is compatible with the condition underlying the Gibbs−Duhem equation. The latter, for a surface phase, under constant pressure and temperature of the environment becomes m m m ⎛ ns ⎞ dσ = − ∑ ⎜ i ⎟ dμi s ≡ − ∑ Γ isdμi = − RT ∑ Γ is d ln(γixi) ⎝A⎠ i=1 i=1 i=1
(5)
Γsi
where is the number of moles of component i in the surface phase per unit area, xi is the mole fraction of this component, and γi is its activity coefficient. It should also be noted that the chemical potential of each component must be uniform throughout the whole system. Thus, the bulk values of the chemical potentials, mole fractions, and activity coefficients may be plugged into eq 5, instead of those of the surface phase. This form of the equation is called the Gibbs adsorption isotherm. It correlates the surface tension of a solution with its composition. Nonetheless, the uncertainty mentioned above regarding the effect of the position of the dividing surface remains: different positions may yield different adsorption values. Therefore, it is highly desirable to find a way to consider adsorption with regard to a more realistic, three-dimensional entity, similarly to the Bakker approach to the mechanical aspects of the surface phase. However, if the surface phase is considered three-dimensional, namely, with a nonzero thickness, hs, the condition of first order homogeneity is not necessarily fulfilled. The volume of the phase, V̂ s = hsAs, can be determined by many (in principle, infinite) combinations of hs and As. Thus, multiplying the volume by a given factor t does not guarantee that the energy would do the same. The behavior of the system will be different for each of these combinations, since the proportion between surface area and thickness will change. Thus, a way is needed to solve this dilemma. The challenge is to construct a model that will be realistic (three-dimensional), yet compatible with the Gibbs−Duhem equation.
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THE GIBBS−DUHEM AND GIBBS ADSORPTION EQUATIONS In order to study and understand surface tension and adsorption in multicomponent systems, it is necessary to consider the Gibbs−Duhem equation.3 The main underlying assumption in the development of this equation for any phase is the fundamental thermodynamic requirement that the internal energy, Û , must be a first-order homogeneous function, namely, Û (tS ̂, tV̂ , tn1 , tn2 , tn3 , tn4...) = tÛ (S ̂, V̂ , n1 , n2 , n3 , n4...) (2) 12654
DOI: 10.1021/acs.langmuir.5b03647 Langmuir 2015, 31, 12653−12657
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Langmuir The Suggested Thermodynamic System. As is wellknown, there may be a few possible ways to define a thermodynamic system that represents a real system with its boundaries and constraints. It is usually advantageous to choose the thermodynamic system in a way that makes the analysis as easy as possible. Here, we define the thermodynamic system as comprising two bulk fluid subsystems and a surface subsystem. The latter consists of the actual surface phase with the addition of very small parts of the other bulk phases. Thus, the borders of the surface subsystem, which are parallel to the surface, are inside the bulk phases, very close to the surface phase (Figure 1). This definition of the borders of the surface system should
results. Then, the main result of the paperthe treatment of adsorptionwill be developed and presented.
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SURFACE TENSION DEFINITION As is well-known, in the absence of external fields (which is assumed to be the present case), the internal energy may change by three processes: heat transfer, mass transfer, and mechanical work. At equilibrium, heat and mass transfer lead to uniformity of temperature and chemical potentials, respectively. According to a foot note by Gibbs1 that was later proven in detail,8 uniformity of temperature and chemical potentials is sufficient to define equilibrium states. However, in order to analyze and understand phenomena related to surface tension, we need to take a close look at the mechanical part of the internal energy. For the sake of brevity, the uniformity in temperature and chemical potentials throughout the whole system is a priori assumed here. Thus, only the mechanical part of the equilibrium criteria needs to be discussed. The terms representing this part in the differential expression of the internal energy (that add up to zero at equilibrium) are given by c
d
s
− P c dV ̂ − P d dV ̂ − P ̅ s dV ̂ = 0 Figure 1. Thermodynamic isolated system and its subsystems (not to scale).
(6)
where P is pressure, the superscripts c and d denote the two bulk phases (c is the bulk phase that encloses the other phases when a situation like that exists). The first two terms are the usual work terms related to volume changes of the bulk phases. The third term represents the contribution of the volume change of the surface phase (that in the present model is strictly proportional to the area change) to the mechanical part of the internal energy. P̅s is an “effective pressure” that expresses the average of the stress parallel to the surface
not be associated with the treatment of Russanov6,7 that used two Gibss dividing surfaces for the two sides of a thin film. The borders in the current model are inside the bulk phases; therefore, they do not play the same role as the Gibbs dividing surface. They are simply “regular” borders assigned to a thermodynamic system as it is, and do not define a substitute system that is different from the real one. The system is considered to be isolated. The equilibrium state of an isolated system is defined by a minimum in the internal energy, as done below. Obviously, the same final results would be obtained if the system were kept under different conditions, as long as the proper thermodynamic function was chosen for minimization in each case. The crux of the matter is the definition of the boundaries of the surface phase as being at a fixed, constant distance from each other, and parallel. By this fixation of the thickness, the thermodynamic surface system is made compatible with the Gibbs−Duhem equation. This is so, since the volume and area are uniquely determined by each other, and eqs 2 and 4 become equivalent. Both approaches, the Gibbs approach and the current one, are compatible with the Gibbs−Duhem equation because of the constancy of the thickness of the studied systems (zero in the Gibbs approach, and nonzero, but constant in the present approach). However, this similarity is only formalit does not represent a physical similarity. In the Gibbs approach, a simplified model that assumes two uniform bulk phases is analyzed. The zero-thickness, dividing surface is a tool used to correct the deficiencies of the simplified model. The present approach considers the system as it really is. The fixed distance between the borders is an arbitrary, though very useful means, for simplifying the thermodynamic analysis, not the system itself. The next step is deriving the mechanical definition of surface tension, using the present approach. It will be shown to be identical with eq 1, despite the fact that in the Bakker approach the thickness was not assumed constant. This will serve as an indication that the present approach indeed yields correct
P̅ s ≡
1 hs
∫0
hs
τ (z ) d z
(7)
Since the thickness of the surface phase is constant, there is no work associated with the stress component normal to the surface. Taking into account the constraint that the whole system, being isolated, has a constant volume, namely, dV̂ c + dV̂ d + dV̂ s = 0, eq 6 turns into d
− (P d − P c ) d V ̂ − (P ̅ s − P c ) d V ̂
s
d
= −(P d − P c) dV̂ − (P ̅ s − P c)hs dAs (8)
=0
̂s
s
s
where the relationship dV = h dA is adopted, remembering that hs is constant. Surface tension is defined as the work per unit area needed to increase the surface area. According to eq 8, surface tension is then given by σ ≡ (P c − P ̅ s )h s hs
=
∫0
=
∫−∞ [P c − τ(z)] dz
[P c − τ(z)] dz
∞
(9)
It is important to notice that the surface tension is independent of our choice of the value of hs, the arbitrary thickness assigned to the surface phase, as long as it is bigger than the actual thickness of this phase. The reason is that, outside the surface region, the integrand in eq 9 is identically zero. This equation is 12655
DOI: 10.1021/acs.langmuir.5b03647 Langmuir 2015, 31, 12653−12657
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Langmuir
surface to be in a location that leads to Γ1 = 0. Then, the Gibbs adsorption isotherm for a binary solution, Γ2(x2), can be empirically developed from the experimental dependence of surface tension on the bulk concentration, σ(x2). However, this approach does not work for a larger number of components, since we cannot elucidate the individual contribution of each component. Therefore, in general, models for surface adsorption have to address a “relative adsorption,” as deduced from eq 12 to be
the same as eq 1 that was developed by Bakker without assuming the thickness to be constant.3 The identity between the two results lends support to the present approach, in the sense that, despite the difference in assumptions, the equation for calculating the surface tension comes out to be identical. The physical interpretation is that surface tension, namely, the work per unit area required to change the surface area, is the difference between volume work had it been a bulk phase and the actual volume work done by the surface phase. It is also of interest to note that if the thickness of the surface phase is considered sufficiently small relatively to the radius of curvature, eq 8 may be rewritten as ⎛1 dA 1 ⎞ (P d − P c ) = σ d = σ ⎜ + ⎟ R2 ⎠ ⎝ R1 dV ̂
⎛ Γ s ⎞ ⎛ Γ s dγi Γ s dγ ⎞ Γ(si ,rel) ≡ ⎜Γ is − xi 1 ⎟ + ⎜⎜ i − χi 1 1 ⎟⎟ x1 ⎠ ⎝ γi d ln xi γ1 dx1 ⎠ ⎝
s
This expression is simplified for ideal solutions to be (10)
Γ is,rel ≡ Γ is − xi
This is, of course, the Young−Laplace equation, from which experimental values of surface tension can be deduced. This equation may not hold for systems that, for example, either include very small (nano) drops or are close to the critical point. In these cases, the surface tension may depend on the drop size.
ADSORPTION Next, we discuss the Gibbs adsorption isotherm, eq 5, as it evolves from the present model. This relationship is extremely important, since it determines the dependence of surface tension of a solution on its composition. The present approach assures the required first-order homogeneity of the internal energy of the surface phase, thus fulfilling the basic condition underlying the Gibbs adsorption isotherm. As mentioned above, the chemical potential of each component must be uniform throughout the whole system. Thus, the bulk values of the chemical potentials may be plugged into eq 5, instead of those of the surface phase. This is more practical, since the concentrations of the components can be more readily measured in the bulk. Equation 5 can be rewritten as
(11)
Γsi
It should be remembered that relates to the surface phase, but xi and γi are values in the bulk phases. If we use the fact that Σmi=1xi = 1, we can write dx1 = −Σmi=2 dxi. Then, eq 11 can be rewritten as
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⎛ Γ s dγi Γs Γ s dγ ⎞ dσ = − RT ∑ ⎜⎜Γ is − xi 1 + i − xi 1 1 ⎟⎟ d ln xi x1 γi d ln xi γ1 dx1 ⎠ i=2 ⎝ m
(12)
Suppose, for example, that component 1 is the solvent, then eq 12 describes the effect of the solutes on the surface tension of the solution. To simply demonstrate the point, the example of an ideal binary solution can be used. In this case, eq 12 becomes ⎛ x Γ⎞ dσ = −RT ⎜Γ2 − 2 1 ⎟ dx 2 x1 ⎠ ⎝
Γ1s x1
(15)
Thus, the relative adsorption may, for example, express the difference between the adsorption of the solute and that of the solvent, the latter being “normalized” to the mole fraction of the solute. This is the parameter that has to be calculated by molecular simulations, or measured by sophisticated techniques, in order to understand surface adsorption and its dependence on each component. To calculate the adsorption from the results of simulations or measurements, simple s integration can be used: Γi = ∫ h0 Ci dz, where Ci is the local molar concentration. Various other definitions of relative adsorption have been suggested in the past,4,5 but they originated from models that did not necessarily consider the surface phase as a whole, nonuniform entity. Equation 15 itself was also suggested; however, it was considered to be only an approximation that is relevant when one fluid phase is gaseous.4 An important advantage of the present definition of relative adsorption is that, beyond the actual limits of surface phase, the same sum of terms that is used in eq 11 is identically zero. This conclusion is derived from the Gibbs−Duhem equation for bulk phases, eq 3. Thus, as in the case of surface tension definition, we do not need to know the exact thickness hs, since the value of the sum will remain constant once we go beyond the actual limits of the surface phase. We just need to ascertain that the borders of the surface phase in the model are slightly beyond the “real” borders.
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m ⎛ s Γ Γ s dγ ⎞ dσ = −RT ∑ ⎜⎜ i + i i ⎟⎟ dxi γi dxi ⎠ x i=1 ⎝ i
(14)
(13)
Clearly, as is well-known, it is impossible to relate Γ2 directly to the bulk concentration, x2 (or its equivalents), without assuming some physical model for the adsorption process. A common way of overcoming this problem, when using the Gibbs dividing surface approach, is to assume the dividing 12656
SUMMARY AND CONCLUSIONS • The present paper proposes to define a surface phase as having a constant, but nonzero thickness. This definition appears to eliminate altogether the difficulties related to the uncertainty in the location of the Gibbs dividing surface. • The current model leads to an equation for calculating surface tension that is identical to that of Bakker, despite the difference in the assumptions underlying the two models. This identity supports the validity of the present approach. • The present model conforms to the requirement of firstorder homogeneity of the internal energy function, thus allowing the use of the Gibbs−Duhem equation. From this equation, a “natural” definition of relative adsorption is deduced. In addition to being a direct outcome of the Gibbs−Duhem equation, an important advantage of the present definition is that the relative adsorption becomes DOI: 10.1021/acs.langmuir.5b03647 Langmuir 2015, 31, 12653−12657
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Langmuir identically zero when calculated outside the actual surface phase. The present conclusions will hopefully lead to further research on surface adsorption and the dependence of surface tension on concentration, either by molecular simulations or by sophisticated techniques of chemical analysis of surfaces.
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Notes
The authors declare no competing financial interest.
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REFERENCES
(1) Gibbs, J. W. The collected works of J. Willard Gibbs, V. I; Yale University Press: New Haven, CT, 1948. (2) Tolman, R. C. J. Chem. Phys. 1948, 16, 758−774. (3) Guggenheim, E. A. Thermodynamics: An Advanced Treatment for Chemists and Physicists, 4th ed.; North-Holland: Amsterdam, 1959. (4) Deafay, R.; Priggogine, I. Surface Tension and Adsorption; Longmans: London, 1966. (5) Radke, C. J. Adv. Colloid Interface Sci. 2015, 222, 600−614. (6) Rusanov, A. I. Colloids J. USSR 1966, 28, 583−588. (7) Amirfazli, A. J. Adhes. 2004, 80, 1003−1016. (8) Marmur, A. Thermodynamic Equilibrium of Nonhomogeneous and Nonisotropic Chemically Reactive Systems. Chem. Eng. Commun. 1985, 39, 381−388.
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DOI: 10.1021/acs.langmuir.5b03647 Langmuir 2015, 31, 12653−12657
