10414
J. Phys. Chem. 1996, 100, 10414-10422
Statistical Thermodynamics of the Fluid-Solid Interface D. J. Diestler Department of Agronomy, UniVersity of NebraskasLincoln, Lincoln, Nebraska 68583-0915 ReceiVed: January 31, 1996X
Although the fluid-fluid interface has been fully described within the framework of classical statistical thermodynamics, a similarly complete treatment of the fluid-solid interface is lacking. Fluid-solid interfacial tension γf, which cannot be directly measured, is here defined operationally by a gedanken experiment (analogous to the classic experiment measuring fluid-fluid interfacial tension) in which the fluid is allowed to spread uniformly over the solid surface while the local intensive properties of the two phases are kept fixed. A general statistical-mechanical expression for γf is derived, which for the first time accounts for dynamic coupling between molecular motions in the solid and fluid phases. In the limit of decoupling (the molecules in the solid become fixed, the effective temperature of the solid approaches T ) 0 K) this expression reduces to a previous formula for the interfacial tension based on a rigid solid (T ) 0 K). A new contribution to the interfacial tension in the rigid-solid limit, due to the reaction of the fluid to the “stretching” of the solid, is found. By means of thermodynamic perturbation theory an explicit expression for γf is derived from a simple modelsa dilute vapor in contact with an Einstein solidsand used to demonstrate the substantial influence on interfacial tension of thermal coupling between the phases.
1. Introduction The behavior of the interface between fluid phases in thermodynamic equilibrium can be described unambiguously, with a few weak assumptions, by both thermodynamics1-6 and statistical mechanics.4-6 Perhaps the simplest example, the single-component liquid-vapor system involving a planar interface, is schematized in Figure 1. The standard thermodynamic treatment3 leads to the Gibbs fundamental relation
dF ) -S dT + µ dN - p dV + γlv dA
(1.1)
where the symbols assume their customary meanings (F ≡ Helmholtz energy, S ≡ entropy, T ≡ absolute temperature, V ≡ total volume of system, p ≡ pressure of homogeneous phases, N ≡ total number of molecules, µ ≡ chemical potential, A ≡ interfacial area, and γlv ≡ interfacial (surface) tension). From eq 1.1 it follows that the increase in the Helmholtz energy accompanying an increase in the interfacial area at constant T, N, and V (depicted in Figure 1) is given by
∆F ) γlv∆A
(1.2)
where the constancy of γlv under the conditions of the described transformation has been exploited. The increase in the Helmholtz energy, ∆F, is just the reversible isothermal work that must be done on the system to effect a concerted movement of the horizontal piston in the normal (z) direction and the vertical piston in the transverse (either x or y) direction. The net result of this movement is that the interfacial area is increased by ∆A; the local densities, as well as other intensive properties of the two phases, remain constant. From a microscopic perspective, the molecules of both phases reorganize themselves as the pistons are moved to increase the interfacial area. Molecules from the homogeneous interiors of the two contiguous phases are brought to the interface. The work done on the system is just that needed to move molecules from the interior of the liquid, where they are surrounded completely by other molecules and thus reside in a X
Abstract published in AdVance ACS Abstracts, May 15, 1996.
S0022-3654(96)00328-0 CCC: $12.00
Figure 1. Liquid-vapor system. Arrows indicate movements of pistons necessary to increase the interfacial area at constant volume of the whole system.
region of relatively low potential, to the interface, where the density is much reduced from that of the homogeneous liquid and the molecules find themselves in a region of higher potential. One purpose of this article is to present an extension of the above thermodynamic analysis of the fluid-fluid system to the situation in which one of the phases is solid. In particular, a fluid-solid interfacial tension (γf) analogous to the surface tension γlv is defined operationally. The chief conceptual difficulty encountered in the attempt to formulate such a definition is that the solid is elastic. When one proceeds to increase the interfacial area using the apparatus depicted in Figure 1, the solid is stretched transversely, its molecules tending on average to move farther apart in the transverse dimensions. That is, the molecules of the solid do not simply rearrange themselves so as to leave the local intensive properties at the interface unaltered. Nevertheless, in principle the goal is to be able to change the fluid-solid interfacial area (i.e., to wet the solid) under the latter constraint. There seems to exist no experimental means of measuring directly the work required to wet a solid surface under these conditions.5,7,8 In section 2 a gedanken apparatus for the measurement of γf is described. The essential idea is to permit transverse expansions of the fluid in the presence of the solid maintained under fixed transVerse strains (or their conjugate stresses). This is accomplished by confining the fluid and solid phases in transverse dimensions by separate, independently movable © 1996 American Chemical Society
Statistical Thermodynamics of the Fluid-Solid Interface
Figure 2. Fluid-solid system. Arrows suggest concerted movements of the pistons necessary to increase interfacial area while the intensive properties of the system are kept fixed.
imaginary pistons. Section 2 also presents a thermodynamic treatment of the system and develops formulas for the interfacial tension in terms of the components of the stress tensor. Section 3 is given to the statistical-thermodynamic description of the gedanken experiment. Molecular statistical expressions for the interfacial tension are derived. Navascue´s and Berry9,10 have previously reported a statistical-mechanical treatment of the fluid-solid interface, but the linkage to thermodynamics is murky and, moreover, the solid is assumed to be rigid (that is, the solid is at T ) 0 K; the molecular motions of solid and fluid are uncoupled). Their expression for the interfacial tension is equivalent to those employed in the context of computer simulations of fluid confined between planar rigid solid surfaces (walls).11-13 The results of recent Monte Carlo simulations14 indicate, however, that thermal motions in the walls can have a substantial impact on the interfacial tension. In section 4 the theory is applied to the following simple model: a dilute gas in contact with an Einstein solid. An approximate canonical partition function is derived by means of thermodynamic perturbation theory. An explicit formula for γf is obtained for the special case of a smooth solid surface (i.e., one in which the molecular structure of the solid has been so averaged as to produce a mean fluid-solid potential energy that depends only on the distance of the fluid atom from the surface). The effects of dynamic coupling between the solid and fluid phases on the interfacial tension are examined. Section 6 closes the article with a discussion of the principal findings. 2. Thermodynamics of the Fluid-Solid Interface A. The Gedanken Experiment. Figure 2 depicts the system, which comprises fluid and solid phases separated by a planar interface. The system is assumed to be contained within the rectangular prism, whose dimensions are large compared with the width of the interface (i.e., the range of the normal (z) direction over which the local intensive properties differ substantially from those of either bulk phase). It is assumed that the solid and fluid are entirely insoluble in each other. Further, the solid is taken to be Hookean (i.e., to be fully describable within the harmonic approximation for the potential energy). To define and compute a fluid-solid interfacial tension analogous to ordinary surface tension (i.e., liquid-vapor interfacial tension), one needs to be able to alter the area of the interface between the fluid and solid phases while maintaining the local intensive properties (e.g., the local densities) fixed. This can be accomplished by the following device: the system is confined in transverse (x and y) directions by separate and
J. Phys. Chem., Vol. 100, No. 24, 1996 10415 independently movable pistons (labeled 1 and 2 for the x direction in Figure 2), which are part of the surroundings. That is, the fluid and solid are respectively constrained by pistons 1 and 2, whose faces lie in the planes x ) Lf and x ) Ls, and by pistons 1′ and 2′, with faces lying in the respective planes y ) Wf and y ) Ws. The fluid is confined in the normal direction by piston 3 having its face in the plane z ) H. The system is also restricted to the octant x > 0, y > 0, and z > 0 by stationary walls contained in the planes x ) 0, y ) 0, and z ) 0. Since shearing is neglected, the gedanken system has five independent mechanical degrees of freedom (i.e., strains): Ls, Ws, Lf, Wf, and H. Note that the “experimental” setup does not allow the normal dimensions of the separate phases (Hf and Hs) to be controlled independently, but they obey the relation H ) Hf + Hs. The redundant degrees of freedom associated with the transverse dimensions of the system are introduced merely for mathematical convenience. They make possible the definition of the interfacial tension, which involves an additive combination of the stresses conjugate to Lf and Wf. These stresses are analogous to the (negative) partial pressures of a gaseous mixture. To be specific, piston 1 (1′) is acted upon by only the fluid, and piston 2 (2′) by only the solid. The conjugate stresses (defined quantitatively below) Txx,f (Tyy,f) and Txx,s (Tyy,s) that must be separately applied to hold pistons 1 (1′) and 2 (2′) in place, and thereby to maintain the system in a state of thermodynamic equilibrium under specified conditions, are just the negatives of the partial pressures exerted by fluid and solid, respectively. Since it is implicitly assumed that the system is to retain its rectangular prismatic shape, the faces of the pistons 1 and 2 and 1′ and 2′ must coincide for all states of the system. That is, the stresses are to be computed only for the situation where Lf ) Ls ) L and Wf ) Ws ) W. They are therefore functions of only the three strains L, W, and H, as well as thermal (S or T) and chemical (N or µ) variables. B. Fundamental Relations. The following fundamental relation is assumed to govern reversible transformations of the hypothetical system described in part A:
dF ) -S dT + µf dNf + µs dNs + Txx,fWfHf dLf + Tyy,fLfHf dWf + Txx,sWsHs dLs + Tyy,sLsHs dWs + TzzLW dH (2.1) In eq 2.1 Nf and Ns stand for the respective amounts (numbers of molecules) of fluid and solid, and µf and µs for the chemical potentials. The interpretation of the work terms in eq 2.1 merits some care. For example, the xx component of the stress applied to piston 1 is given formally by
Txx,fWfHf ) (∂F/∂Lf)T,Nf,Ns,Wf,Ls,Ws,H|Lf)Ls)L;Wf)Ws)W
(2.2)
where the vertical bar signifies that the partial derivative is to be evaluated by setting the transverse dimensions of the fluid phase equal to those of the solid phase. The symbol Txx,f represents the mean stress that must be applied perpendicularly to the effective area WfHf of the plane segment (i.e., the face of piston 1), which is by convention directed outwardly. Hence, the quantity Txx,fWfHf dLf is the work done on the system by the surroundings during the reversible displacement of piston 1. It is perhaps worth reiterating that the stresses conjugate to the strains are functions of just the three strains L, W, and H in addition to T, Nf, and Ns. The treatment can be further simplified by assuming the solid phase to be homogeneous and isotropic in transVerse dimensions. Then Txx,f ) Tyy,f ) T|,f and Txx,s ) Tyy,s ) T|,s, and eq
10416 J. Phys. Chem., Vol. 100, No. 24, 1996
Diestler
2.1 can be rewritten as
dF ) -S dT + µf dNf + µs dNs + γf′ dAf + γs′ dAs + TzzA dH (2.3) where Af ) LfWf, As ) LsWs, and
γf′ ) (∂F/∂Af)T,Nf,Ns,As,H|Af)As)A ) T|,fHf
(2.4a)
γs′ ) (∂F/∂As)T,Nf,Ns,Af,H|As)Af)A ) T|,sHs
(2.4b)
TzzA ) (∂F/∂H)T,Nf,Ns,Af)As)A
(2.4c)
C. Homogeneity of the Helmholtz Energy. Now consider a process in which the interfacial area is changed while the intensive variables (i.e., T, µf, µs and the stresses and tensions) are held constant. Let dAf ) dAs ) dA and eliminate H in favor of the volume of the whole system via the relation V ) AH. Then eq 2.3 becomes
dF ) -S dT + µf dNf + µs dNs + (γf + γs)dA + Tzz dV (2.5) where the (partial) interfacial tensions γf and γs are defined by
γf ) γf′ - TzzHf
(2.6a)
γs ) γs′ - TzzHs
(2.6b)
Integration of eq 2.5 under the specified conditions yields
F ) µfNf + µsNs + (γf + γs)A + TzzV ) µfNf + µsNs + (γf′ + γs′)A
(2.7)
where the zero of the Helmholtz energy is taken when Nf ) Ns ) A ) V ) 0. Equation 2.7 can be rationalized by imagining the system to be formed by moving the pair of (coincident) pistons 1 and 2 (see Figure 2) from x ) 0 to x ) L or the pair 1′ and 2′ from y ) 0 to y ) W, respectively. It is interesting that eq 2.5 in principle applies to the liquidvapor system. In this case N ) Nf + Ns and µ ) µf ) µs, since each phase consists of one and the same chemical component and the two phases are in chemical equilibrium. Moreover, because no distinction is made between the transverse stresses applied separately to the liquid and vapor phases, γlv ) γf + γs. Hence, substituting these relations into eq 2.5, along with the identity -p ) Tzz, one gets eq 1.1. D. Wetting the Surface As It Is Stretched. Consider a transformation in which the interfacial area is increased at constant T, Nf, Ns, and V, which process can be effected by downward movement of piston 3 and simultaneous outward movements of (coincident) pair 1 and 1′, or 2 and 2′. From eq 2.5 one has
dF ) (γf + γs) dA
(2.8)
For a finite increase in A, the isothermal work done on the system is given by
∆F ) ∫A (γf + γs) dA A2
(2.9)
1
Equation 2.9 is the analogue of eq 1.2. As the solid is stretched transversely, its molecules cannot rearrange so as to keep the intensive properties at the interface constant, as can the
molecules of the liquid in the liquid-vapor system. Hence, γf and γs both depend on A. E. Wetting the Solid under Constant Tension. Now suppose the interfacial area is increased while T, Nf, Vf ) AHf, the tension γs′, and the solid density ns ) Ns/(AsHs) are held fixed. Under these constraints Tzz, µf, µs, and γf are also constant. The contemplated process can be carried out by concerted movements of the pistons similar to those described just above in section D. The only difference is that piston 2 (or 2′), which is moved coincidently with 1 (or 1′), is under no applied stress. It merely demarcates the solid phase from the surroundings, admitting additional solid to the system as it moves outward. Note that Hs is also fixed under the conditions of the transformation. Then one has dH ) dHf and A dHf ) dVf - Hf dA and, from eq 2.3,
dF ) -S dT + µf dNf + µs dNs + (γf + γs′) dA + Tzz dVf (2.10) The change in the Helmholtz energy for the constrained wetting process is therefore
∆F ) γf∆A + (γs′∆A + µs∆Ns)
(2.11)
where the first term of the right member of eq 2.11 is the reversible isothermal work done on the system in deforming the fluid phase so as to increase the interfacial area and thereby to wet the surface of the solid, with the latter maintained under constant stress so that its intensive properties stay constant. The remaining terms constitute the free energy imported into the system with the entry of additional solid. This interpretation is bolstered by an independent computation of the work done on the system in the wetting process. For an infinitesimal increase of A in which the fluid volume Vf ) AHf is kept fixed, one has
dW ) TzzA dHf + γf′ dA ) (γf′ - TzzHf) dA ) γf dA
(2.12)
Since γf is constant under the conditions of the wetting, eq 2.12 is easily integrated to yield
W ) γf∆A
(2.13)
which is the total work expended to wet the solid. Equation 2.11 is the analogue of eq 1.2 pertaining to the liquid-vapor system. The difference between eqs 1.2 and 2.11 is the additional free energy possessed by the admitted solid on account of its state of stress (γs′∆A) and chemical potential (µs∆Ns). The admission of solid is of course occasioned by the requirement that the local intensive properties of the two phases be maintained constant during the wetting. In the foregoing discussion it is implicit that the area of interfaces between the fluid and the stationary walls and piston faces is negligible or that the gedanken experiment is rigged so that the net change in these “extraneous” interfacial areas is zero. 3. Statistical-Mechanical Treatment of the Fluid-Solid Interface A. Molecular Description of the System. In the interest of notational simplification the internal degrees of freedom of all molecules are neglected (that is, both fluid and solid are
Statistical Thermodynamics of the Fluid-Solid Interface
J. Phys. Chem., Vol. 100, No. 24, 1996 10417
assumed monatomic). According to the standard theory,15 the Helmholtz energy is given by
F ) -kBT ln Q
(3.1)
derive a statistical formula for the interfacial tension γf. It is convenient to focus on just Txx,f, since Tyy,f and Tzz,f may be written by analogy. Equations 2.2, 3.1, and 3.2 can be combined to give
In the classical limit the canonical partition function Q can be written
WHfTxx,f ) -kBTZ-1(∂Z/∂Lf)|Lf)L
Q ) Z/(Nf! Λf3NfΛs3Ns)
(3.2)
The partial derivative in eq 3.7 can be evaluated by the more or less standard procedure used to derive the so-called “pressure equation” for homogeneous phases.16 Introducing the scaled coordinates
(3.3)
x′i ) xi/Lf
where the configuration integral is defined by
Z ) ∫drs∫drf exp(-U/kBT)
Here rs and rf collectively denote the solid-atom and fluid-atom positions, which range over the volumes available to the respective phases. The thermal de Broglie wavelengths are defined by Λf,s ≡ (h2/2πmf,skBT)1/2, mf,s being the mass of the fluid or solid atom, h Planck’s constant, and kB Boltzmann’s constant. The total potential energy is taken to be expressible as a sum of solid-solid, fluid-fluid, and solid-fluid contributions:
U ) Uss(rs) + Uff(rf) + Ufs(rs,rf)
(3.4)
To facilitate formal manipulations to be carried out presently, the various contributions to the potential energy are assumed to consist of sums of pairwise interactions of the following forms:
Nf
Z)
∫0 dx′i∫0 ∏ i)1
LfNf
1
dyi∫0
Wf
dzi∫drs exp(-U/kBT)
H-Hs
(3.9)
where the origin of the z-coordinate has been shifted to Hs. The arguments of the (implicitly) transformed potential energy function U contain Lf through the interatomic distances
[rijff]2 ) Lf2(x′i - x′j)2 + (yif - yjf)2 + (zif - zjf)2 (3.10a) [rijfs]2 ) (Lfx′i - xjs)2 + (yif - yjs)2 + (zif - zjs)2
(3.10b)
(3.5b)
∂Z/∂Lf ) NfZ/Lf - (kBT)-1∫drs∫drf exp(-U/kBT) ×
Nf Nf
i)1 j*i
Nf Ns
Ufs ) ∑∑φfs(rij)
one can recast the configuration integral in eq 3.3 as
(3.5a)
i)1 j*i
Uff ) 1/2∑∑φff(rij)
(3.8)
that appear in the fluid-fluid (Uff) and fluid-solid (Ufs) contributions, respectively. Superscripts ff and fs have been attached to rij temporarily for emphasis. Differentiation of eq 3.9 yields
Ns Ns
Uss ) 1/2∑∑φss(rij)
(3.7)
(3.5c)
Nf Nf
Nf Ns
i)1 j*i
i)1 j)1
[1/2∑∑φff′ ∂rijff/∂Lf + ∑∑φfs′ ∂rijfs/∂Lf]
(3.11)
i)1 j)1
Here φss, φff, and φfs are interatomic potentials, and rij is the distance between a pair of atoms. The stationary walls and the piston faces that confine the fluid phase can be treated as hard walls. Therefore, the integrations over the hypervolumes available to the solid and fluid can be represented by Ns
∫drs ) ∏∫0 dxj∫0 dyj∫0 dzj Ls
Ws
Hs
i)1
Wf
H
(3.12a)
∂rijfs/∂Lf ) xifxijfs/(Lfrijfs)
(3.12b)
Combining eqs 3.7, 3.11, and 3.12, one obtains
Nf
Lf
∂rijff/∂Lf ) xijff2/(Lfrijff)
(3.6a)
j)1
∫drf ) ∏∫0 dxi∫0 dyi∫H dzi
The expressions in eqs 3.5 for Uff and Ufs have been used here to construct the partial derivatives of U. The prime denotes the derivative with respect to the argument (e.g., φff′ ) dφff/drijff). From eqs 3.10 follow the relations
Nf Nf
(3.6b)
s
where the (laboratory) origin of the coordinates is shown in Figure 2. In “practical” computer simulations one would take the system to consist of a slab of fluid (or solid) sandwiched between slabs of solid (or fluid) and impose periodic boundary conditions to avoid complications due to the interfaces with pistons and walls. B. Statistical Formulas for the Interfacial Tension. Using the above statistical-mechanical expressions and the thermodynamic ones derived in section 2, one can now proceed to
Txx,f ) -NfkBT/(AHf) + (AHf)-1[1/2∑∑〈φff′xij2/rij〉 + i)1 j*i Nf Ns
∑ ∑〈φfs′xixij/rij〉] i)1 j)1
(3.13)
The angular brackets in eq 3.13 stand for the canonical ensemble (configurational) average, which is defined in general by
〈X〉 ) ∫drs∫drf P(Ns,Nf)(rs,rf) X(rs,rf)
(3.14)
where X is a dynamical function of the configuration (rs,rf) and
10418 J. Phys. Chem., Vol. 100, No. 24, 1996
Diestler
P(Ns,Nf) is the canonical specific configurational probability distribution (i.e., the probability that the configuration lies in the hypervolume element drs drf about (rs, rf)), which is given by
the normal modes as 3Ns
Z ) exp[-Uss(0)/kBT]∫drf∏∫-∞dqj × ∞
j)1
(rs,rf) ) exp(-U/kBT)/Z
(Ns,Nf)
P
(3.15)
3Ns
k)1 Nf Ns
In a similar fashion analogous expressions for Tyy,f and Tzz can be derived. Combining these with that (eq 3.13) for Txx,f according to eq 2.6a, one arrives finally at
γf ) (2A) { /2∑∑〈φff′[xij + yij - 2zij ]/rij〉 + 2
2
2
i)1 j*i Nf Ns
∑ ∑〈φfs′[xixij + yiyij - 2zizij]/rij〉} i)1 j)1
i)1 j*i
φfs(|rif - (rj(0) + ∆rj)|)]} ∑ ∑ i)1 j)i
(3.21)
From this relation, along with those given in eqs 2.4b, 3.1, and 3.2, one obtains
Nf Nf
-1 1
Nf Nf
exp{-(kBT)-1[1/2∑ms ωk2qk2 + 1/2∑∑φff(rijff) +
γs′ ) -kBTZ-1 ∂Z/∂As (3.16)
3Ns
) ∂Uss(0)/∂As + ∑ms〈qj2〉ωj ∂ωj/∂As j)1
Nf Ns
By a sequence of manipulations paralleling that leading to eq 3.16 one can deduce the following expression for γs: Ns Ns
γs ) (2A)-1{1/2∑∑〈φss′[xij2 + yij2 - 2zij2]/rij〉 i)1 j*i Nf Ns
∑ ∑〈φfs′[xjxij + yjyij - 2zjzij]/rij〉} i)1 j)1
(3.17)
Adding eqs 3.16 and 3.17 yields the total interfacial tension
γf + γs ) (4A)-1∑∑〈φ′[xij2 + yij2 - 2zij2]/rij〉
(3.18)
i j*i
where the double sum enumerates all possible pairs without regard to order. The form of the right member of eq 3.18 is identical with that derived previously for γlv,17 which corroborates the thermodynamic demonstration of the equivalence between γf + γs and γlv given at the end of section 2C. C. Alternative Statistical Expression for the Tension γs′. Another form for the tension γs′ associated with the solid phase can be derived within the framework of the harmonic approximation for the solid. In terms of normal modes the contribution Uss of the isolated solid to the potential energy can be written 3Ns
Uss ) Uss(0) + 1/2∑msωj2qj2
(3.19)
j)1
where qj is a normal coordinate and ωj is its fundamental frequency. In eq 3.19 Uss(0), the potential energy of the solid with the atoms fixed in the equilibrium configuration {rj(0), j ) 1, 2, 3, ... Ns}, is given by Ns Ns
Uss(0) ) /2∑∑φss(rij(0)) 1
(3.20)
i)1 j*i
where rij(0) ) |ri(0) - rj(0)|. The normal coordinates, their associated frequencies, and equilibrium potential energy Uss(0) all depend on the transverse strains (Ls, Ws), or As, and the solid density ns. The partition function can be expressed in terms of
〈φfs′(rij)rij-1rij‚∂[rj(0) + ∆rj]/∂As〉 ∑ ∑ i)1 j)1
(3.22)
Although the variation of the tension γs′ with the transverse stretching of the solid phase is not of primary interest in the context of this article, it is nevertheless clear from eq 3.22 that γs′ depends on the state of the fluid through the ensemble average. It is likewise clear from eq 3.16 that the interfacial tension γf depends in a similar way on the state of the solid. In other words, the two phases react dynamically to each other at the interface. D. Expressions for γf in Terms of Pair Distribution Functions. Since the quantities in brackets in eq 3.16 depend on the positions of only pairs of atoms, the integrations over all other coordinates can be carried out formally to give Nf Nf
γf ) (2A) { /2∑∑∫dri∫drj Pff(2)(ri,rj) φff′(rij) × [xij2 + -1 1
i)1 j*i
Nf Ns
yij2
-
2zij2]/rij
+ ∑∑∫dri∫drj Pfs(2)(ri,rj) φfs′(rij) × i)1 j)1
[xixij + yiyij - 2zizij]/rij} (3.23) where the configurational distribution functions for specific pairs of atoms are given by Nf
Ns
k*i,j
l)1
Nf
Ns
k*i
l*j
Pff(2)(rif,rjf) ) ∏∫drfk∏∫drls P(Ns,Nf)(rs,rf) Pfs(2)(rif,rjs) ) ∏∫drfk∏∫drls P(Ns,Nf)(rs,rf) (3.24) Because the fluid atoms are equivalent, every pair involving two fluid atoms contributes the same to γf, and every pair involving any fluid atom and the same solid atom contributes the same. One can therefore rewrite eq 3.23 as
γf ) (2A)-1{1/2∫dr1∫dr2 Fff(2)(r1,r2) φff′(r12) × [x122 + Ns
y122 - 2z122]/r12 + Ns-1∑∫dr1∫drj Ffs(2)(r1,rj) φfs′(r1j) × j)1
[x1x1j + y1y1j - 2z1z1j]/r1j} (3.25)
Statistical Thermodynamics of the Fluid-Solid Interface
J. Phys. Chem., Vol. 100, No. 24, 1996 10419 In the rigid-solid limit, qj ) ∆xi ) 0 for all i and j, and eq 3.22 reduces to
where
Fff(2) ) Nf(Nf - 1)Pff(2) Ffs(2) ) NfNsPfs(2)
Nf Ns
(3.26)
are the generic pair distribution functions. If the solid atoms could be grouped into sets of equivalent (by symmetry) atoms, then the sum over j in eq 3.25 would reduce to a sum over such sets. In case all atoms were equivalent (e.g., if only the atoms in the top layer of a simple cubic lattice were included in Ufs), then the sum on j would collapse completely and the expression in eq 3.25 would simplify to
γf ) (2A)-1∫dr1∫dr2 Fff(2)(r1,r2) φff′(r12)[x122 - z122]/r12 +
A-1∫dr1∫dr2 Ffs(2)(r1,r2) φfs′(r12)[x1x12 - z1z12]/r12 (3.27)
where the equivalence of the x and y directions has also been accounted for. It is stressed that even though the expression for γf is greatly simplified under the presumed idealized circumstances, a formidable complexity engendered by the dynamic coupling between fluid and solid phases nevertheless inheres in the pair distribution functions. E. The Limit of a Rigid Solid. If the solid is assumed to be rigid, then the integrations over the solid-atom positions rj in eq 3.23 can be formally carried out [Pfs(2)(ri,rj) ) Pf(1)(ri) δ(rj - rj(0))] to yield Nf Nf
γf ) (2A)-1{1/2∑∑∫dri∫drj Pff(2)(ri,rj) φff′(rij) × [xij2 + i)1 j*i Nf
Nf
yij2 - 2zij2]/rij + ∑∫dri Pf(1)(ri)∑φfs′(rij(0)) × [xixij(0) + i)1
γs′ ) ∂Uss(0)/∂As - ∑∑〈φfs′(rij(0))rij(0)-1rij(0)‚∂rj(0)/∂As〉 i)1 j)1
(3.33)
which shows that γs′ depends not only upon the behavior of the isolated solid (first term on the right side of eq 3.33) but also upon the reaction of the fluid to the solid (second term). 4. Perturbation Theory of the Fluid-Solid Dynamic Coupling The effects of dynamic coupling between the fluid and solid phases can be delineated by a simple model system describable to a good approximation by thermodynamic perturbation theory. According to the latter,18 the Helmholtz energy can be expressed as
F = -kBT ln Q(0) + 〈U(1)〉0
(4.1)
where Q(0) is the partition function of the unperturbed (zeroorder) system and 〈U(1)〉0 is the (assumed) small correction due to the perturbation. The subscript 0 on the angular brackets signifies the ensemble average over the unperturbed configurational probability distribution function. The unperturbed system is taken to be the isolated solid plus the fluid in the field of the rigid solid. The perturbation thus arises from the small (thermal) displacements of the solid atoms from their equilibrium positions and is contained in the fluid-solid interaction Ufs. The zero-order potential energy and the perturbation can be identified by expanding Ufs in powers of the Cartesian displacements. One gets
j)1
yiyij(0) - 2zizij(0)]/rij(0)} (3.28)
3Ns
Ufs(rs,rf) ) Ufs(0,rf) + ∑(∂Ufs/∂∆xj)0∆xj + j)1 3Ns 3Ns
Defining the single-atom potential ψ(ri) by
/2∑∑(∂2Ufs/∂∆xj ∂∆xk)0∆xj∆xk (4.2)
1
Ns
ψ(ri) ) ∑φfs(rij ) (0)
j)1 k)1
(3.29)
j)1
Nf Nf
where powers higher than the second are neglected and the argument (or subscript) 0 implies evaluation at the equilibrium configuration of the isolated solid. Combining eqs 3.4 and 4.2 yields
i)1 j*i
U(0) ) Uss(rs) + Uff(rf) + Ufs(0,rf)
one can rewrite eq 3.28 as
γf ) (2A)-1{1/2∑∑∫dri∫drj Pff(2)(ri,rj) φff′(rij) × [xij2 + Nf
) Us(0)(rs) + Uf(0)(rf)
yij2 - 2zij2]/rij + 2∑∫dri Pf(1)(ri) ∇riψ(ri)‚r* i} (3.30) i)1
(4.3a)
3Ns
U(1) ) ∑(∂Ufs/∂∆xj)0∆xj +
where
j)1
r*i ≡ /2(xiex + yiey) - ziez 1
(3.31)
Again invoking the equivalence of fluid atoms, one has finally
3Ns 3Ns
1
/2∑∑(∂2Ufs/∂∆xj ∂∆xk)0∆xj∆xk (4.3b) j)1 k)1
γf ) (2A)-1∫dr1∫dr2 Fff(2)(r1,r2) φff′(r12)[x122 - z122]/r12 +
The second line of eq 4.3a indicates that the unperturbed potential energy separates into contributions from the solid and fluid. Therefore, the whole zero-order partition function factors as
where Ff(1) and Fff(2) are the local density and pair distribution function of the fluid, now in the field of the rigid solid. Equation 3.32 agrees with the principal result (for γSF given in eq 27 of refs 9 and 10) of Navascue´s and Berry.5,9,10
Q(0) ) Qs(0)Qf(0)
A-1∫dr1 Ff(1)(r1) ∇r1ψ(r1)‚r* 1 (3.32)
(4.4)
A. The Unperturbed Solid Phase. For the sake of simplicity Einstein’s model is adopted for the solid. Then from
10420 J. Phys. Chem., Vol. 100, No. 24, 1996
Diestler where Z1 is the single-atom configurational integral (i.e., free volume),
eqs 3.19 and 4.3a follows Ns
Us (rs) ) Uss(0) + /2∑msω2(∆xi2 + ∆yi2 + ∆zi2) (4.5) (0)
1
Z1 ) ∫0 dx∫0 dy∫H dz exp[-ψ(r)/kBT] Lf
i)1
Wf
H
(4.16)
s
where ω is the fundamental vibrational frequency of every atom, which moves in the mean potential field of its neighbors. The canonical partition function for the unperturbed solid is given by
Qs(0) ) Zs(0)/Λs3Ns
(4.6)
In general, ψ(r) is periodic in x and y. However, one may smooth the interface by averaging ψ(r) over the positions of the solid atoms. The solid atoms are smeared with equal weight over the region z e Hs occupied by the solid phase. If the resulting smooth-wall potential ψ(z) is substituted into eq 4.16, the integrals on x and y can be done to give
where the zero-order configurational integral is 3Ns
Zs(0) ) exp[-Uss(0)/kBT]∏∫-∞d∆xi exp(-msω2∆xi2/2kBT) ∞
i)1
Z1 ) Afη(H - Hs)
(4.17)
η(H - Hs) ≡ ∫H dz exp[-ψ(z)/kBT]
(4.18)
where
(4.7)
H s
Evaluating the integrals in eq 4.7 and plugging the results back into eq 4.6 yield
Qs(0) ) exp[-Uss(0)/kBT](2πkBT/hω)3Ns
(4.8)
The mean square displacement of the ith solid atom, 〈∆xi2〉0, is given formally by
〈∆xi2〉0 ) ∫d∆rs P0(Ns)(∆r1, ∆r2, ∆r3, ... ∆rNs)∆xi2
(4.9)
defines the one-dimensional “normal” configuration integral. The above-described smearing yields a potential that can in general be represented in the form ψ(z) ) cr (z - Hs)-n - ca(z - Hs)-m, n > m > 0, where cr and ca are constants. Hence, in this approximation a sharp boundary (z ) Hs, where ψ(z) becomes infinite) exists between the fluid and solid phases. By analogy with eq 4.10 the zero-order configurational probability distribution for the unperturbed fluid is expressible as
where the configurational probability distribution function for the unperturbed solid, P0(Ns), is defined by
P0(Ns) ) exp[-Us(0)(rs)/kBT]/Zs(0)
Nf
) ∏{exp[-ψ(ri)/kBT]/Z1}
(4.10)
Substituting the explicit expressions in eqs 4.5, 4.7, and 4.10 for Us(0), Zs(0), and P0(Ns), respectively, into eq 4.9 and simplifying the result, one obtains
〈∆xi2〉0 ) kBT/msω2
P0(Nf) ) exp[-Uf(0)(rf)/kBT]/Zf(0)
(4.11)
(4.19)
i)1
where the second line of eq 4.19 follows from eqs 4.13 and 4.15. C. Correction to the Free Energy. From eq 4.3b one has 3Ns
〈U(1)〉0 ) ∑〈(∂Ufs/∂∆xj)0〉0,f〈∆xj〉0 +
Similarly it is easy to show that
〈∆xi〉0 ) 0 〈∆xi∆xj〉0 ) 0, i * j
j)1
(4.12a)
3Ns 3Ns
1
(4.12b)
B. The Unperturbed Fluid Phase. Again in the interest of simplicity, let us take the fluid to be a dilute vapor, so that fluid-fluid interactions are negligible (i.e., Uff ) 0). Then from eq 4.3a one has
/2∑∑〈(∂2Ufs/∂∆xj ∂∆xk)0〉0,f〈∆xj∆xk〉0 (4.20) j)1 k)1
Note that the complete zero-order ensemble average factors into separate averages over unperturbed fluid (denoted by the subscripts 0,f) and solid phases. Use of the relations in eqs 4.11 and 4.12 simplifies eq 4.20 to
Nf
Uf(0)(rf) ) ∑ψ(ri)
(4.13)
i)1
Ns
〈U 〉0 ) (kBT/2msω )∑〈[(∂2Ufs/∂∆xj2)0 + (∂2Ufs/∂∆yj2)0 + (1)
2
j)1
where eq 3.29 defines the potential energy ψ(ri) of a single fluid atom in the field of the rigid solid. The partition function for the unperturbed fluid is therefore (0)
Qf
) Zf
(0)
/Nf! Λf3Nf
(4.14)
The zero-order configurational integral Zf(0) can be written as
Zf(0) ) ZN1 f
(4.15)
(∂2Ufs/∂∆zj2)0]〉0,f (4.21) From eq 3.5c one deduces Nf
∂2Ufs/∂∆xj2 ) ∑[φfs′′xij2/rij2 + φfs′/rij - φfs′xij2/rij3]
(4.22)
i)1
as well as analogous expressions for ∂2Ufs/∂∆yj2 and ∂2Ufs/∂∆zj2.
Statistical Thermodynamics of the Fluid-Solid Interface
J. Phys. Chem., Vol. 100, No. 24, 1996 10421
Addition of these and substitution of the sum into eq 4.21 yield Nf Ns
〈U(1)〉0 ) (kBT/2msω2)∑∑〈[φfs′′(rij) + 2rij-1φfs′(rij)]0〉0,f i)1 j)1
(4.23)
Since the fluid atoms are equivalent, eq 4.23 can be rewritten as
〈U 〉0 ) (NfkBT/2msω )〈ψ (r)〉0,f (1)
2
(1)
(4.24)
Ns
ψ (r) ) ∑[φfs′′(|r - rj(0)|) + 2φfs′(|r - rj(0)|)|r - rj(0)|-1] (1)
(4.25)
The ensemble average value of
ψ(1)
H
where the second line follows from eq 4.19. Note that in the smooth-surface approximation described above both ψ and ψ(1) depend only on z, and the formula in eq 4.26 reduces to
〈ψ(1)〉0,f ) η-1∫H dz exp[-ψ(z)/kBT] ψ(1)(z) H
(4.27)
s
Combining eqs 4.1, 4.4, 4.8, 4.14, 4.15, and 4.24, one finally reaches
) Uss(0) - 3NskBT ln(2πkBT/hω) NfkBT ln Z1 + 3NfkBT ln Λf NfkBT(ln Nf - 1) + NfkBT〈ψ(1)〉0,f/(2msω2) (4.28) Equation 4.28, the primary result of this section, is the perturbation-theory expression for the Helmholtz energy of a dilute gas in thermodynamic equilibrium with an isotropic (Einstein) solid. 5. Influence of Thermal Coupling on Interfacial Tension The effects of thermal coupling between the solid and fluid phases on the interfacial tension can be conveniently examined within the confines of the smooth-wall version of the model system treated in section 4. Assume that ψ(z) and ψ(1)(z) both vanish when the separation between the fluid atom and the interface is greater than a cutoff value ∆H. Then from eqs 4.17 and 4.18 one obtains
(5.1)
and from eq 4.27
〈ψ(1)〉0,f ) c1(c0 + H - Hs - ∆H)-1, H g Hs + ∆H
(5.2)
where
c0 ≡ η(∆H) c1 ≡ ∫H
Hs+∆H s
dz exp[-ψ(z)/kBT]ψ(1)(z)
where eqs 5.1 and 5.2 have been used to compute the derivatives. Similarly eq 2.4c and 4.28 yield
AfTzz ) ∂F/∂H
) -NfkBT(c0 + H - Hs - ∆H)-1 NfkBTc1(c0 + H - Hs - ∆H)-2/(2msω2)
(5.5)
Combining expressions in eqs 5.4 and 5.5 according to eq 2.6a, one gets
γf ) -NfkBTAf-1{1 - (H - Hs)[c0 - ∆H + (H - Hs)]-1 c1(2msω2)-1(H - Hs)[c0 - ∆H + (H - Hs)]-2} ) -NfkBTVf-1{(c0 - ∆H)[1 + (c0 - ∆H)/Hf]-1 c1(2msω2)-1[1 + (c0 - ∆H)/Hf]-2} (5.6)
F ) -kBT{ln Qs(0) + ln Qf(0)} + 〈U(1)〉0
Z1 ) Af(c0 + H - Hs - ∆H), H g Hs + ∆H
(5.4)
NfkBT[∂〈ψ(1)〉0,f/∂H]/(2msω2)
) Z1-1∫0 dx∫0 dy∫H dz exp[-ψ(r)/kBT] ψ(1)(r) s (4.26) Wf
) -NfkBTZ1-1 ∂Z1/∂Af + NfkBT[∂〈ψ(1)〉0,f/∂Af]/(2msω2)
) -NfkBTZ1-1 ∂Z1/∂H +
is given by
〈ψ(1)(ri)〉0,f ) ∫drf P0(Nf)(rf) ψ(1)(ri) Lf
γf′ ) ∂F/∂Af ) -NfkBT/Af
where ψ(1)(r) is defined in analogy to ψ (eq 3.29):
j)1
Since the range of interaction (∆H) is on the order of the thickness of the interface and the dimensions of the system are immensely greater than the thickness, the inequality H - Hs ) Hf . ∆H obtains. The interfacial tension is calculated by the formula in eq 2.6a. From eqs 2.4a and 4.28 follow the expressions
(5.3)
where the second line of eq 5.6 exploits the relations H - Hs ) Hf and AfHf ) Vf. Because the dimensions of the fluid phase are much greater than the width of the interface (i.e., the range of the intermolecular forces), one has the inequality Hf . (c0 - ∆H), and eq 5.6 thus reduces to
γf ) -p[(c0 - ∆H) - c1/2msω2]
(5.7)
where p ) NfkBT/Vf is the pressure of the bulk vapor. The first term in square brackets on the right side of eq 5.7 arises solely from the unperturbed (rigid-solid) system and can be explicitly rewritten in terms of the zero-order interaction as
γf(0) ) -p(c0 - ∆H) ) -p∫H dz{exp[-ψ(z)/kBT] - 1} s (5.8) ∞
where the upper limit has been extended to infinity, since the integrand decays rapidly with increasing distance z of the fluid atom from the interface. The sign of γf(0) can be either positive or negative, depending upon the nature of ψ(z). If ψ(z) is on balance attractiVe (i.e., if ψ(z) < 0 on average over ∆H), then γf(0) is negative and the system does work on the surroundings with an increase in the interfacial area. Molecules in the vapor phase lower their mean potential energy in coming to the attractive interface. On the other hand, if ψ(z) is on balance repulsive, then the creation of interfacial area requires that work be done on the system to bring molecules from the relatively lower potential of the bulk vapor to the interface.
10422 J. Phys. Chem., Vol. 100, No. 24, 1996 The second term in eq 5.7, the correction due to thermal motion in the solid phase, can be cast explicitly as
γf(1) ) p(2msω2)-1∫H dz exp[-ψ(z)/kBT] ψ(1)(z) (5.9) ∞ s
where the upper limit on the integral in c1 (see eq 5.3) is replaced by ∞, since ψ(1)(z) falls off very rapidly with increasing z. The magnitude of the correction is governed by both c1 and the force constant (msω2) of the effective harmonic potential binding the solid atoms. In the limit of the rigid solid (msω2 f ∞), the correction γf(1) vanishes, as it should. The sign of γf(1), which is determined by the factor c1, results from an interplay between the Boltzmann factor (determined by both ψ(z) and T) and ψ(1)(z). 6. Conclusion and Discussion The central problem addressed in this article is to define operationally a fluid-solid interfacial tension analogous to ordinary surface tension (i.e., liquid-vapor interfacial tension). To render the analogue accurate, one needs a means to change the interfacial area (wet the solid) while keeping the local intensive properties of both phases constant. This process is easy to accomplish in the liquid-vapor system, for the molecules of both phases can instantaneously reorganize themselves (the fluids assuming the shape of their container) as the interfacial area is altered at constant T, N, and V (where V is the volume of the whole system). However, the molecules of an elastic solid cannot rearrange themselves under an imposed strain. The virtual device of confining the fluid and solid phases in transVerse dimensions by distinct and independently movable pistons allows one to spread the fluid uniformly over the solid surface while the local properties (in particular, the densities) of both phases are held constant at the interface (indeed throughout each phase). This process necessitates the system’s being open with respect to the solid phase. The total change in the Helmholtz free energy is therefore not simply the work of creating an interface, as it is in the liquid-vapor system. It includes the additional free energy possessed by the admitted solid by virtue of its state of stress (γs′∆A) and its chemical potential (µs∆Ns). The statistical-mechanical expression for γf (eq 3.16) is based on the fundamental thermodynamic treatment of the gedanken system and takes into account the dynamic coupling of molecular motions in the solid and fluid phases. It is noteworthy that the new general formula in eq 3.23 reduces to that in eq 3.32 for the rigid solid, which is equivalent to the expression derived earlier by Navascue´s and Berry,5,9,10 whose treatment is predicated on a rigid (T ) 0 K) solid. It is also worthy of remark that the new rigid-wall limiting expression for γs′ (eq 3.33) comprises contributions from both the “isolated” solid and the “reaction” of the fluid to the stretching of the solid. The latter contribution was neglected by Navascue´s and Berry,9,10 who were mainly concerned with computing the work of adhesion between liquid and solid. The work of adhesion involves the difference between γl + γs,l′ and γv + γs,v′ for the
Diestler liquid-solid and vapor-solid interfaces, where the subscripts l and V on γs′ distinguish between the tensions of solid in contact with liquid and vapor, which are in general different because of the “reaction” term. If one includes only the “bare” solid contribution, then γs,l′ exactly cancels γs,v′. How good this approximation is remains to be investigated. An investigation of the importance of dynamic coupling between the solid and fluid phases is carried out by application of thermodynamic perturbation theory to a highly idealized model: dilute vapor in contact with an Einstein solid. The simple model obviously neglects instantaneous correlations in the motions of both fluid and solid atoms. Nevertheless, it is adequate to underscore the importance of thermal coupling between the phases. One would expect such coupling to be just as significant for more sophisticated models. The model also neglects molecular structure of the solid surface; that is, it takes the surface to be smooth on an infinitesimal scale, the potential energy of the fluid atom depending only on its distance from the surface. The virtue of these simplifications is that they lead to a physically transparent expression for the interfacial tension, analysis of which demonstrates clearly the interplay of factors determining the sign and magnitude of γf. If accurate values for the interfacial tension for realistic systems are required, these can be obtained by use of the exact formula (eq 3.16) within the framework of computer simulation. Acknowledgment. The author is grateful for financial support of the U.S. Department of Energy under Grant No. DEFG02/85ERGO310 and of the U.S. Army Research Office under Grant No. OAAL03-90-G-0074. References and Notes (1) The Scientific Papers of J. Willard Gibbs; Dover: New York, 1961; Vol. I, p 219. (2) Guggenheim, E. A. Thermodynamics; North-Holland: Amsterdam, 1967; p 45. (3) Hill, T. L. Thermodynamics for Chemists and Biologists; Addison-Wesley: Reading, MA, 1968; Chapter 1. (4) Defay, R.; Prigogine, I. Surface Tension and Adsorption; Wiley: New York, 1966. (5) Navascue´s, G. Rep. Prog. Phys. 1979, 42, 1132. (6) Rowlinson, J. S.; Widom, B. Molecular Theory of Capillarity; Clarendon: Oxford, U.K., 1984. (7) Adamson, A. W. Physical Chemistry of Surfaces; Wiley: New York, 1976; Chapter VII. (8) Castellan, G. W. Physical Chemistry; Addison-Wesley: Reading, MA, 1983; p 418. (9) Navascue´s, G.; Berry, M. V. Mol. Phys. 1977, 34, 649. (10) Navascue´s, G.; Berry, M. V. Wetting, Spreading and Adhesion; Padday, J. F., Ed.; Academic: London, 1978; p 83. (11) Toxvaerd, S. J. Chem. Phys. 1981, 74, 1998. (12) Magda, J. J.; Tirrell, M.; Davis, H. T. J. Chem. Phys. 1985, 83, 1888. (13) Curry, J. E.; Cushman, J. H.; Schoen, M.; Diestler, D. J. Mol. Phys. 1994, 81, 1059. (14) Diestler, D. J.; Schoen, M. J. Chem. Phys., in press. (15) McQuarrie, D. A. Statistical Mechanics; Harper and Row: New York, 1976; Chapter 2. (16) Reference 15, p 261. (17) Reference 6, p 90. (18) Reference 15, Chapter 14.
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