THE MOLECULAR HYDROSTATIC ANALYSIS OP GIBBS’ THEORY QF CAlpIEEmY BY FRANK P. BUFF* Institute for Theoretical Physics of the University, Utrecht, Netherlands Received 27th July, 1960 Gibbs’ theory of surface phenomena is based on the postulate that the intrinsic energy depends on the area and principal curvatures of the dividing surface located in the non- spherical transition zone. This assumption leads to first-order correction terms to the classical formulas which may be shown to provide asymptotic corrections to the free energy and thus to provide criteria for the breakdown of thermodynamic concepts. It is the purpose of this paper to examine the Gibbs postulate from the point of view of molecular hydrostatics. With use of the equation of hydrostatics and a stress tensor with unequal tangential components, the equilibrium conditions and work elements are computed.This general stress tensor implies that the work and the surface analogue of the equation of hy- drostatics contain two tensions rather than the earlier single surface free energy. The normal component of the two-dimensional hydrostatic equation leads to a generalized Laplace equation and the tangential components determine the spatial dependence of the tensions. The Gibbs postulate is found correct only when to first-order terms the two tensions are taken to be equal. This equality is shown valid for representative surfaces. I The classical investigations of the phenomenological description of surface phenomena culminated in Gibbs’ thermodynamic theory of capillarity.1 The basis of his detailed treatment of curved fluid interfaces is as follows. First, the geometric parameters of the interface are made precise by the introduction of a family of parallel dividing surfaces.They are located in the transition zone separating the bulk phases in such a manner as to be “similarly situated with respect to the condition of adjacent matter ”. Secondly, a small region is con- sidered which terminates within the adjacent bulk phases and encloses the inhomo- geneous film. A specific fundamental equation, summarizing the first and second laws of thermodynamics, is then postulated for reversible changes in state of this region. The analytical consequences of these concepts follow with the use of known mathematical techniques. Since some features of Gibbs’ theory have long been controversial, it is proper to ask whether his fundamental equation is valid and what its physical interpretation might be.The most familiar answers are associated with the investigations of Guggenheim 2 and Tolman.3 In Guggenheim’s view, the detailed aspects of Gibbs’ theory of curved interfaces are vague and irrelevant since, intuitively, thermodynamic concepts break down for systems comparable in size with the thickness of the inhomogeneous layer. On the other hand, Tolman, in his sub- sequent recapitulation of Gibbs’ theory of spherical drops, accepted the funda- mental equation and, with Gibbs, interpreted the results to be valid for micro- scopic droplets. The actual state of affairs 4 is nearer Guggenheim’s position.A careful interpretation of Gibbs’ theory leads to a quantitative criterion which confirms the conclusion that macroscopic concepts cannot be extrapolated into microscopic domains. This arises from the observation that, upon accep tiiig the * permanent address : Department of Chemistry, University of Rochester, Rochester, New York. 52F. P . BUFF 53 correctness of the fundamental equation, the retention of the detailed terms leads to first-order corrections which enter into the asymptotic expansion of the free energy with respect to the geometrical parameters of external force. Although the physical interpretation of Gibbs' detailed theory is now quali- tatively clear, the type of surface to which the fundamental equation applies rigorously is not established.On the basis of the statistical-mechanical theory of the grand ensemble, the fundamental equation has been confirmed for spherical interfaces.5 However, for non-spherical surfaces, encountered in the presence of the gravitational field, only that underlying model 4 is available which is sufficient for the derivation of Gibbs' formalism. Furthermore, important consequences of this theory are only obtained via the thermodynamic route. It is the purpose of the present discussion to explore the general case in greater detail with use of molecular hydrostatics. For comparison with the later analysis, it will now be instructive to review briefly the basic equations in the extended thermodynamic formulation. A region is considered, small on the macroscopic scale, but large compared with the range of intermolecular forces, which encloses the transition zone.The region terminates at points within the fluid phases a and p where bulk properties subsist. The transi- tion zone is spanned by an initially arbitrary set of parallel dividing surfaces which separate the total volume into volumes ucc and vp. For the dividing surface of area s and principal curvatures c1 and c2, the intrinsic energy E is assumed to depend on the entropy S, composition (Nil and the geometrical parameters of the systems : E = E(S,(Ni),u~,vp,s,cl,cz). (1) With this set of molar variables, the following hndamental equation is then postulated for reversible changes in state : T is the temperature, pi is the chemical potential of component i, pa and pp are the respective bulk pressures extrapolated up to the dividing surface and y is identified with the surface tension. The thermodynamic curvature terms Cf and C2 are recognized equal to within first-order terms, i.e., C1 = C2 = C.The fundamental relation, eqn. (2), pertaining to the actual system, is next contrasted with its analogue pertaining to a hypothetical system consisting of bulk properties within volumes v, and up. Subtraction from eqn. (2) of the respective extrapolated fundamental equations which hold for the bulk phases a and p and division by the area s leads to the following relations for the excess intensive thermodynamic functions : n E, = TS, + c piri+ 7, (3) i = 1 and n dE, = TdS, + 1 pidFi + (C/s)d(c, + cJ. i = 1 (4) Es(Ss) is the excess, per unit area, of the actual energy (entropy) diminished by the energy (entropy) within phases a and p if bulk properties were to subsist up to the dividing surface.Ti is the corresponding excess, per unit area, of the composition of component i. The free-energy relation (3) arises from the assump- tion that y possesses the excess free-energy property. The criterion for thermo- dynamic equilibrium, in conjunction with eqn. (3) and (4) for the excess intrinsic energy, leads to the familiar conditions of constancy of T and partial potential pi. It is recalled that - pi = pi+Ii'fi$54 GIBBS’ THEORY OF CAPILLARITY where Mi is the moIecuIar weight of component i andq is the external potential. The remaining equilibrium condition is found to be the extended Kelvin (Laplace) equation, r is the superficial density of mass and Ngp is the unit normal to the arbitrary dividing surface, directed from phase a to phase p.Additional implications follow from the differentiation of eqn. (3) and equating the result to eqn. (4). It leads to the generalized Gibbs adsorption equation Pu - Pp = Y(%+ c,) - (4 + c22)(cls) + w$w. (5) n dy + S,dT + C ridji, = (C/s)d(c, + c,) + Td@. (6) i= I Under the equilibrium conditions of constancy for T and @$, eqn. (6) can be immediately integrated to within the desired first-order terms The subscript 00 designates evaluation at a planar interface and the superscript ‘’ designates the selection of the I?” = 0 dividing surface. The explicit representation of the free energy of the system is the final and most important inference that follows from this approach.It is most con- veniently represented by the T, p work function a, which appears in the grand partition function and is equal to the difference between the Helmholtz and Gibbs free energies. For a two-phase system,6 with use of eqn. (3), the asymptotic repre- sentation of GI with respect to the geometrical parameters of external force is given by r-Y,+(Cjs)’;(c~+c,)+ . . (7) The integrations are extended over the whole system and y is given by eqn. (7). Since it can be shown 4 that (CiSK = -&oY, and that 6 is comparable with the range of intermolecular forces, it is clear that the usefulness of eqn. (8) breaks down for systems comparable in dimension with the range of intermolecular forces. Thus, the retention of the correction terms in the formulation provides the criterion which heralds the breakdown of thermo- dynamic concepts.Once this analysis has been carried out, it is, of course, un- important to retain these terms in applications to macroscopic systems. In conclusion, it should be recognized that, although in the detailed treatment the thermodynamic functions depend on the reference surface selected, those combinations which are operationally meaningful must be invariant to this arbitrary choice. For example, since IR is an invariant, it leads to the familiar variational principle of capillarity, With its use, eqn. (5) follows from eqn. (8). I1 The preceding treatment of curved interfaces has been based on the validity of Gibbs’ fundamental equation and on the assumption that y possesses the free- energy property.In order to analyze these assumptions, it is first convenient to examine the spatial variation of the concentration throughout the system. On the basis of statistical-mechanical considerations,4 it is found that, in the interior of bulk phases, the concentration is constant apart from negligible terms con- taining the local mass fluctuations. However, as the transition zone is approached,F. P. BUFF 55 the concentration variation is more complex. It varies rapidly between its limiting bulk values in a direction normal to the dividing surfaces, but is sensibly constant over a small lateral extension on the various dividing surfaces. The detailed variation of the concentration is fortunately not required for the assessment of the phenomenological description.It is sufficient to examine gross properties of that member of the chain of Born and Green equations 7 which summarizes the balance of external and intermolecular forces. Since this equation is equivalent to the equation of hydrostatics, the subsequent discussion will be formulated in molecular hydrostatic terms. In this approach, the properties of the stress tensor a are of primary importance. Although Q is a known 8 function of the molecular variables, this explicit representation may be postponed to the end of the analysis. The required basic relation may thus be exhibited in the familiar form v . a = pV$, wherep is the mass density. Its application to the theory of capillarity requires further specification of p and a.It is first convenient to divide the whole system into small macroscopic cells which are either located in the interior of the bulk phases or which enclose the transition zone. In the first case, apart from very small corrections, p is constant. Also, for those regions located in the interior of bulk phases CI andp, it is sufficient to assume that the stress tensor is isotropic : Q;. = - p j l ; j = a$. 1 is the three-dimensional unit tensor and p is the pressure at the point considered. In the second case, the region is considered to terminate within the two bulk phases and its transition zone is spanned by a family of parallel surfaces. When generalized co-ordinates u, 0 are introduced, the current point r of a given surface s may be represented parametrically by r = r(u, v), while the current point W of the parallel surface S, located at a constant distance R along the normal from surface s, is related to the corresponding point 1: by r x r , R = r(u,u)+AN(u,v); N = --!-- 1 % X T , I' N is the common unit normal to the surfaces s and S.Our assumption concerning the spatial dependence of the density, within a given small region, may be expressed in the form p = p(A). The stress tensor within the transition zone will be taken to be of the general form a = oT,elel + oT2e2e2 + a,NN. N is the unit normal and el and e2 are the unit vectors along the lines of curvature of the dividing surface under consideration. This stress tensor serves to remove simplifications in our earlier 4 model calculations.Although it will lead to more involved final expressions, it provides greater insight into the problem considered. The phenomenological results of chief interest are the analytic form of the thermodynamic work element and the equation summarizing the equilibrium conditions. Their derivation on the basis of eqn. (11) and (12) is carried out with the use of known9 techniques involving the mathematical theory of parallel surfaces. Accordingly, we shall only examine the main concepts that are en- countered. In both cases, eqn. (12) is extended over the complete small macro- scopic region under consideration. For the work element it is required on its boundaries, while for the equilibrium condition it enters through the hydrostatic condition. It is then recognized that, although the pressure in the interior of bulk phase varies slowly, the components of stress undergo drastic change as the two- phase transition zone is traversed.This complication is circumvented by the respective extrapolation of the bulk stresses for phases a and p, eqn. (lQ), up to the dividing surface. The subtractive procedure finally yields relations which56 GIBBS' THEORY OF CAPILLARITY contain convergent integrals whose integrands involve the difference between the true state of affairs and the extrapolated properties. These integrals represent the thermodynamic functions required for the phenomenological description of surface phenomena. Thus the thermodynamic parameters finally make their appearance in the form of averages of the interfacial variation of the stress and of the density. It should again be recognized that, although the individual functions depend on the reference surface selected, those Combinations which are operationally meaningful must be invariant to this choice.The stress averages that appear are as follows : Y i = CJS = where 0 R<O i o o . A(A) = The distances la and ;ls are located in regions where bulk properties subsist. The physical interpretation of yj and Ci follows from the context in which they appear in the final relations. We shall find that Y j is the generalized excess surface free- energy and Ci is the thermodynamic curvature term. These considerations lead to the following phenomenological description. The dividing surface s, bounded by the closed curve C, separates the total volume into the volume va of phase a and the volume up of phase p.The stress tensor, eqn. (12), then implies that the work done by the system corresponding to an actual (not virtual) increase in extent of the region is given by &,x is the boundary displacement, 4 is the unit tangent along C', t x Nap is drawn outward and yi is given by eqn. (13). For the representation of the equilibrium condition, it is convenient to introduce the two-parametric gradient a v "I v2 =- [r,xWz+Nxru- a . I ru r, I With its use, the equilibrium condition is found to be expressible as a two-dimen- sional analogue of the equation of hydrostatics, v2 . a2 3- (Pa - Pp)Norp - r w = 0, (17) (18) where the phenomenological surface stress ~2 is given by 2 i = 1 a2 = C (yi - ciCi/s)eiei.yj is identical with that appearing in the work corresponding to an actual dis- placement and C; is given by eqn. (14). In order to interpret eqn. (17), it is first necessary to decompose it into its normal and tangential components. With use of the relation 10 e i . V2ei = -ciN-ejV2 . e j ; i # j , (19)F . P. BUFF 57 the respective components are found to be 2 and ei V2(yi - ciCi/s) + (yi - y j - C ~ C ~ / S + cjcj/s)v2 , ei = IT+ . ei; i # j . (21) The condition under which these results reduce to those of the extended Gibbs theory is that the two tensions y1 and y 2 are equal, This can be seen in the following manner. For y1 = y2 = Y, the actual work done by the system reduces to Thus y is the contribution to the excess free energy. Furthermore, this substitution reduces eqn.(20) to the earlier Kelvin (Laplace) eqn. (5). Finally, with use of the Mainardi-Codazzi relation,ll ei . V,cj = (ci-cj)V2 . ei, V2(Y - (c, + CZ)(C/S), - L+) = 0. (23) (24) Y "YCo+(Cl +c,>(Cls)l&. (25) the tangential component reduces to Its integral is identical with the curvature dependence of Y found from the extended Gibbs adsorption eqn. (6), The parameters ya, and (Cis),, evaluated at a planar interface, are by eqn. (13) and (14) given by 03 Yco = 1 (%--%W, (26) -03 (C/s), = 1 z(o,-oN)dz. -03 z is directed along the space-fixed axis. We observe that (i) only at this point of the analysis is it fruitful to represent the stresses explicitly in terms of molecular variables and (ii) that, within the framework of classical statistical mechanics, only the intermolecular force contribution to the stress tensor is required in eqn.(26). Since the condition of validity of the detailed Gibbs formalism has been shown to be y1 = y2, we now examine this equality with use of eqn. (21. Upon ap- plication of the Mainardi-Codazzi relation, eqn. (21) may be transformed to We henceforth restrict ourselves to surfaces of revolution which constitute the main example encountered in practice. Here eqn. (27) may be readily integrated to first-order terms with the result, yi N 700 + (c1+ cZ)( CIS): + ac j i Zj. (28) The integration constant a may be determined by this argument. If it is assumed that the first-order principal curvature expansion of yi applies generally, then by comparison with the rigorous result for a sphere 5 Y N Y m +2c(c/s>:, (29) it follows that a = 0.Under this condition y1 is indeed equal to y2.58 GIBBS' THEORY OF CAPILLARITY The conclusion of this paper is as follows. The detailed Gibbs theory of capil- larity is a device for calculating the higher terms in the asymptotic expansion of the free energy with respect to the geometrical parameters characterizing the system. This expansion provides an explicit criterion which shows the breakdown of macro- scopic thermodynamic concepts for systems comparable with the range of inter- molecular forces. An analysis of the phenomenological equations of capillarity on the basis of molecular hydrostatics shows that the Gibbs theory is valid only when the two tensions that appear in the general formalism are equal. For representative surfaces this condition appears to be satisfied. I gratefully acknowledge the award of a National Science Foundation Senior Fellowship for 1959-60 during which period I had a stimulating discussion with Prof. L. Van Move on the subject of this paper. 1 Gibbs, Collected Works (Yale University Press, New Haven, 1948), vol. I, pp. 55-353. 2 Guggenheim, Trans. Faraday Soc., 1940,36, 397. 3 Tolman, J. Chern. Physics, 1949,17, 333. 4 Buff, J. Chein. Physics, 1956, 25, 146. SBufl', J. Chem. Physics, 1955, 23, 419. 6 At the confluence of three phases, D must be supplemented by the inclusion of a thermodynamic length parameter, Buff and Saltsburg, J. Chern. Physics, 1957,26,1526. 7 Born and Green, Proc. Roy. Sac. A, 1946,188, 10. 8 Irving and Kirkwood, J. Chem. Physics, 1950, 18, 817. 9 Buff and Saltsburg, 9; Chern. Physics, 1957, 26, 23. 10 Weatherburn, Diferential Geometry (Cambridge University Press, Cambridge, 1930), 11 ref. (lo), p. 52. vol. lT, p. 19.