Langmuir 1991, 7, 1035-1038
1035
Comments Reply to the Comment Is It Really Impermissible To
Shift the Gibbs Dividing Surface in the Classical Theory of Capillarity? In a recent communication Markin and Kozlovl criticized one of our earlier papers2 that dealt with the possibility of shifting dividing surfaces. In that paper: we had reiterated the reasons which led to our conclusion that the surface of tension dividing surface, as introduced by Gibbs? could not be shifted within the framework of the classical theory of capillarity. We then proceeded to demonstrate that shifting the dividing surface can be accomplished readily and satisfactorily within the framework of our generalized theory of capillarity. The paper by Markin and Kozlovl challengesnot only the correctness of our paper but also several basic aspects of our generalized theory of Capillarity,' such as our definition of surface tension. We shall demonstrate below that the criticisms of Markin and Kozlov are neither justified nor correct after a brief review of certain key concepts of Gibbsian thermodynamics. The first important point to stress is that Markin and Kozlov1ps have criticized our paper on the basis of definitions and conventions that are very different from those used by us.2*49mIn order to address the real issues, they would clearly need to argue on the basis of our definitions, which we believe to be quite standard. In order to avoid more confusion, we wish to state briefly the key definitions and provide explanations of what is meant by "Gibbsian thermodynamics", 'Gibbs' model" of a surface, the "classical theory of capillarity", and the "generalized theory of capillarity". Definitions 1. Gibbsian Thermodynamics. The preeminent concept of Gibbsian thermodynamics is the fundamental equation. According to Gibbs? the internal energy form of the fundamental equation for a bulk phase is given by
where Sv is the total entropy, V is the system's volume, and Miv represents the total mass of the ith component. The essential benefit of employing this standard textbook definition is that intensive parameters like the temperature, pressure, and the chemical potentials are defined quite simply as partial derivatives. For a capillary system not subjected to any external body forces, the classical surface fundamental equation, as suggested by Gibbs, is (1) Markin, V. S.; Kozlov, M. M. Langmuir 1989,5, 1130.
(2) Rotenberg, Y.; Boruvka, L.; Neumann,A. W.Langmuir 1986, 1, 533. (3) Gibbs, J. W. The Scientific Papers of J. Willard Gibbs; Dover: New York, 1981; Vol. 1, pp 55-371. (4) Boruvka, L.; Neumann, A. W. J. Chem. Phys. 1977,66,5464. (5) Markin, V. S.; Kozlov, M. M.; Leikin, S. L. J. Chem. SOC.,Faraday Trans. 2 1988,84, 1149. (6) Boruvka, L.; Rotenberg,Y.; Neumann, A. W.J . Phys. Chem. 1985, 89, 2714. (7) Boruvka, L.; Rotenberg,Y.; Neumann, A. W. Langmuir 1985,1,40. (8) Boruvka, L.; Rotenberg,Y.; Neumann, A. W. J. Phys. Chem. 1986, 90,125.
UA= UA(P,A,iUP) (2) where A is the surface area. According to Gibbsg this particular dependence applies for a moderately curved surface when the surface is in the surface of tension position. Equation 2 is completely analogous to the bulk expression given by eq 1. All quantities, like the surface tension y, are defined as partial derivatives of UAwith respect to the independent variables. 2. Gibbs' Model of a Surface. The Gibbsian model for a capillary system considers two bulk fluid phases separated by a mathematical dividingsurface. The surface is introduced in such a manner that certain quantities like the energy, entropy, and mass are conserved. These conservationrequirements are satisfied by assigning excess energy, entropy, or mass to the surface such that in a location that is near to the surface phase both bulk fluid phases would be considered as remaining homogeneous up to the dividing surface. Obviously, any arbitrary shift of this dividing surface position would normally result in some change to the extensive properties of each of the adjacent bulk phases. This in turn would necessitate a corresponding change in the surface excess quantities to ensure that all conservation requirements were still satisfied. 3. Classical Theory of Capillarity. The classical theory of capillarity is the theory of capillarity due to Gibbs. The key point is that the proper fundamental equation for surfaces that are planar or moderately curved is given by eq 2 provided one places the dividing surface in the so-called "surface of tension" ["surface of pure tension" would be a far better term] position. The use of the term 'Classical" in this context is well-defined and understood.lb13 4. Generalized Theory of Capillarity. Unlike the classical fundamental equation given above, generalized fundamental equations do contain curvature-dependent terms. Gibbs himself presented one form of a generalized fundamental equation for a surface
UA= UA(SA,A,MP,c1,c2) (3) where c1 and c2 are principal curvatures that are assumed to be uniform on the surface. The other expressions implied by eq 3 are dU = T d S + &dMi
+ y dA + Cl dcl + C2dc2
(4)
i
and
u = TS '+ &i i ~ i + y~
(5)
where eq 4 is integrated at constant principal curvatures to yield eq 5 since both principal curvaturea are intensive. Gibbs did not expand upon the physical meaning of the quantities C1 and CZ but proceeded immediately to (9) Gibbs, J. W. Reference 3, p 231. (10) Buff, F. P. J. Chem. Phys. 1961, 19, 1691; see p 1592. (11) Hill, T. L. J. Chem. Phys 1961,19,1203. (12) Melrose, J. C. Thermodynamics of Surface Phenomena. In Proceedings of the International Con erence on Thermodynamics; Cardiff, U.K. (1970);Landeberg, P. T., B u t t e m o h : London, 1970, pp 273-286; BW p 279. (13) Tolman, R. C. J. Chem. Phys. 1948,16,758; nee p 760.
d;
0743-7463/91/2407-1035$02.50/0 0 1991 American Chemical Society
1036 Langmuir, Vol. 7.,No. 5, 1991
Comments
eliminate any consideration of these terms by shifting the dividing surface to the surface of tension position defined by the condition C1+ C2 = 0.14 It is not surprising that he eliminated these dependencies on the curvature almost immediately since he was primarily interested in investigating the effects of capillarity for systems that are15 "composed of parts which are approximately plane" or for those common situations in whichls "our measurements are practically confined to cases in which the difference of the pressures in the homogeneous masses is small." Virtually all current research in the curvature dependence of surface tension and related topics is either based on eq 2, which is not applicable for high curvature situations, or on deficient modifica.tions of this equation. Although these deficiencies have been discussed elsewhere,4J*ea few words are necessary to explain the nature of this problem since an understanding of this feature is central to our main difficulty with the critique of Markin and Koz1ov.l For a system kept a t thermal and chemical equilibrium (via contact with a suitable reservoir), the remaining mechanical equilibrium conditions can be found by minimizing the overall grand canonical potential Q of the system. If eqs 3-5 are employed, then the surface contribution to this potential is
nA= UA- TSA- &MP i
or, by eq 5 QA
= yA
(7)
If we assume that the dividing surface is not in the surface of tension position, so that C1+ C2 # 0, then we can easily see from eq 7 that on a shifted dividing surface the mechanical behavior of the interface would still be represented by just a uniform surface tension y. However, on the shifted dividing surface, the values of both y and A would be different than the values of y and A, which occur for the dividing surface in the surface of tension position. Just such a conclusion follows from the Kondo equation. This, however, violates physical reality. It is well-known from mechanics that if there is pure tension along one surface, then a shifted surface must be described by a tension and one or two distributed bending moments; otherwise, the system described by the shifted surface will not be statically equivalent to the original, unshifted surface. What should be obvious from this argument is that eq 4 is incorrect. When this equation is employed to describe a system with a dividing surface not in the surface of tension position, one finds that it is not even able to reproduce the classical behavior of an interface, let alone any highcurvature refinements. Any comparison of our generalized theory of capillarity with the theory of Gibbs based on eq 3 must realize that this form of the fundamental equation is not suitable if one wishes to describe surface systems with arbitrarily curved surfaces. We believe that Gibbs was well aware of this deficiency but that he used eq 4 only as a procedural means to get to a formulation for a fundamental equation for moderate curvature without explicit curvature terms. Finally, our generalized theory of capillarity2*416+J is based on the fundamental equation (14) Gibbs, J. W. Reference 3, p 225. (15) Gibbs, J. W. Reference 3, p 228. (16) Gibbs, J. W. Reference 3, p 232.
UA(SA,A,MP,d,X) (8) where the totalmean and Gaussian curvatures17are defined by d = $$JdA
and X=$SKdA
(10)
with the mean and Gaussian curvatures defined by J = c1+ c2 and K = clc2. The definition of the surface tension y, which is completely analogous to the definition of the pressure, follows directly from the assumed form of eq 8 as
or
after a suitable Legendre transformation from UAto QA. If all surface properties are uniform or constant at each point on the surface, then one may replace the integral expressions in eqs 9 and 10 for d and 31 by J A and KA, respectively. These expressions for d and X are not as general as those considered in the generalized theory of Boruvka and N e ~ m a n nbut , ~ for the sake of comparison with the definitions of Markin and K o z ~ o vwe , ~ shall ~~ restrict the discussion to this lower level of generality. It should be obvious that this form of the fundamental equation, with these restrictions on d and X,greatly curtails its ability to describe the shape of capillary systems that one might wish to consider.'E When one uses the fundamental equation given by eq 2, one finds that the surface of tension position for the dividing surface is the only position that is an equilibrium position. The discussion of this restriction may seem aimless; however, one often encounters in the literature cases were the dividing surface is shifted to another position even when eq 2 is taken as the expression for the fundamental equation and the classical Laplace equation, y J=A I', is taken as the mechanical equilibrium condition across the surface. This restriction is not a severe drawback of the classical theory of capillarity since the theory was envisaged by Gibbs to deal primarily with surfaces that may be regarded as nearly planar, that is, when the radii of curvature are very large in proportion to the thickness of the nonhomogeneousinterface. However,as established by Rotenberg et the position of the dividing surface in this case is invariant if energy is to be conserved. Specific Comments to the Arguments of Markin and Kozlov In what follows we shall address the objections of Markin and Kozlov,l but in a sequence that is different from theirs. Tension versus the Position of the Dividing Surface and Adequacy of Thermodynamic Characteristics. Central to many of Markin and Kozlov's arguments is the definition of the surface tension.'^^ We are unable to agree that their definition of the quantity y~ is the proper Gibb(17)Weatherburn, C. E. Differential Geometry of Three Dimensions; Cambridge University Press: London, 1930; 2 vola. (18) Neogi, P.; Friberg, S.E. J. Colloid Interface Sci. 1989,127,492; see p 496.
Langmuir, Vol. 7, No. 5, 1991 1037
Comments
sian definition of the surface tension. First, the partial derivative used by them (see their eq 15) to define the quantity YG,which they call the Gibbsian surface tension
is in reality the average specific free energy density of the interface (that is sometimes referred to as the average specific grand canonical potential) or &and not the surface tension y.7 This conclusion follows easily since the variations in u* as given by r
dwA= -gA d T -
Cp: dpi + C, dJ + C, dK
(14)
i=l
are zero for a homogeneous portion of surface in which the temperature, T, all chemical potentials, pi, the mean, and Gaussian curvatures are all constant as required by the partial derivative in eq 14. Second,the followingwas made quite clear by Gibbs.lg “The value of y is therefore independent of the position of the dividing surface, when this surface is plane. But when we call this quantity the superficial tension, we must remember that it will not have its characteristic properties as a tension with reference to any arbitrary surface. Considered as a tension, its position is in the surface which we have called the surface of tension, and, strictly speaking, nowhere else.” In the current vernacular this means that & = y only at the surface of tension position where y is a pure tension. At any other position the equality between uAand y will not hold since the specific free energy of the surface will also contain energetic curvature contributions. Throughout his analysis Gibbs was very much aware of the constraints that he imposed on his formalism. Thus, when he considered, for instance, the surface tension y, he was very careful to distinguish between its value at the surface of tension and its value at any other dividing surface location. It would seem quite apparent that Gibbs had no intention of generalizing his analysis beyond capillary systems with moderate curvatures. The specific point here is that both definitions (i.e., Gibbs’ and ours) of the “superficial tension” or specific free energy yield a quantity that is a pure tension at the “surface of tension” dividing surface location. At any other position of the dividing surface, the quantity y remains as a pure tension so that there is a distinction between the surface tension y and the specific free energy wA.7 Finally, a small difficulty with the definitions employed by Markin and Kozlov is that there is a duplication of terminology involving the specific free energy wA. Markin and Kozlov refer to the specific free energy by using the symbol yo, while the pure tension quantity y, that was originally used by Gibbs, is without any designation. After realizing these differences, one may quickly see that their eq 14 is nothing more than the definition for the specific free energy, which is given in general by uA=y+CJ+CKK (15) In conclusion, the claim by Markin and Kozlov that their quantity represents the Gibbsian surface tension is not correct because it is defined as an average specific free energy density and not as a tension in the fashion suggested by Gibbs. Displacement of the Gibbs Dividing Surface. With this background we may now frame the question: What does it mean when one considers shifting adividing surface (19) Gibbs, J. W. Reference 3, p 234.
within the classical theory of capillarity? Three items in addition to the conservation requirements discussed above need to be satisfied to ensure that the above question has meaning within the framework of Gibbs’ classical theory of capillarity. They are (i) dividing surface is at the “surface of tension” location, (ii) all bending moments are zero, and (iii) the surface tension is an isotropic tension. Thus, we find in Rotenberg et the solution to this original problem for the spherical interface that Gibbs had previously considered.20 They showed that the dividing surface’s position under the condition that this surface remain as a surface of pure tension is invariant or that it is impossible to shift the surface of tension dividing surface while conserving energy within the classical framework unless the surface is planar. This conclusion applies to both the classical fundamental eq 2 and the more generalized form suggest by Gibbs (i.e., eq 3). Consequently, any relation, like the Kondo equation,21 which presupposes the possibility of a real shift of the dividing surface, is impossible. On a hypothetically shifted dividing surface the mechanical behavior of the interface would still be represented by a uniform surface tension in the classical theory, but the magnitudes of both y and the area A would need to be different from the classical values as a direct consequence of the shift. Obviously, thio situation violates physical reality. First, one cannot just impose a shift on the dividing surface as this would also permit one to impose any arbitrary magnitude on the surface tension contrary to the experimental evidence which suggests that y has a fixed value when one fixes both the temperature T and the chemical potential PI. Second,from the theory of elasticity22we understand that if there is a pure tension along one surface, then a shifted surface must be described by a given tension and corresponding bending moments so as to maintain the static equivalence of the shifted surface with the originalsurface. If these compensating bending moments are to be described a priori, they must be explicitly present in the expression for the specific surface free energy density, wA; that is, they must be present in the fundamental equation. In the classical theory, which is based on eq 2 or in the more general expression given by eq 3,they are not properly represented, whereas in the generalized theory, which is based on eq 8,they are present from the beginning. Finally, equations like the Kondo equation do not follow directly or indirectly from the classical fundamental equation for the surface (Le., they are not Gibbsian) and they do not pertain to a system in equilibrium as noted by Kondo himself. However, the relation that does follow from the classical fundamental equation is well-known as the classical Laplace equation, y J = AP. Thus, to claim, as was done by Markin and Kozlov, that one should use “a generalized Laplace equation ... at an arbitrary dividing surface” is incorrect. Their generalized Kondo equation also suffers from these three deficiencies. Generalized Kondo Equation. A relation like the Kondo equation, which is meant to yield the dependence of the surface tension on position of the dividing surface when shifts are performed in a uniform othogonal manner away from the original surface, is not a complete or proper generalization of the classical theory of capillarity as discussed above. Furthermore, it should be realized that one may not properly discuss any shift of the surface of tension dividing surface to any other arbitrary dividing surface position unless one is also able to describe suitably, (20) Gibbs, J. W. Reference 3, p 226. (21) Kondo, S. J . Chem. Phys. 1966,25,662. (22) Fung,Y.C. Foundations of Solid Mechanics;Prentice-Hall: Englewood Cliffs, NJ, 1965.
1038 Langmuir, Vol. 7, No. 5, 1991
through the use of a proper fundamental equation, the manner in which the free energy is affected by these changes. Relations like the Kondo equation given by Markin and Kozlov’s eq 13 are incorrect because they depend on a classical expression for the free energy (see their eq 51, which does not attempt to describe the proper dependence of the free energy on the surface’s bending moments. Finally, any properly generalized Kondo equation, even for a system that is free of any gravitational fields, would not be an isolated relation but would depend intimately for its existence on the presence of two additional relations which would yield the variation of the bending moments with dividing surface position.2 All these relations would need to be satisfied if conservation requirements were to hold at every point of the capillary system. For spherical systems the dependence of both the surface tension and the one remaining bending moment on the choice of dividing surface has been recognized elsewhere.ll The absence of these additional relations in the work of Markin and Kozlov is cause for concern. Several other points require specific elucidation and shall be addressed in a separate ~ a p e r . ~ 3 Conclusions. The generalized form of the fundamental equation for surfaces was postulated4 by recognizing that certain requirements must be satisfied. Specifically, the additional, nonclassical, independent variables of the free energy must have extensive geometric properties and the geometric variables must also be expressible in terms of simple, low-order scalar differential invariants of the surface.” The two obvious variables that satisfy these conditions are the first and second total curvatures of the surface defined above by eqs 9 and 10. The initial generalization of the classical theory of capillarity to systems with nonmoderately curved interfaces is due to BufPop24-26and latter Buff and Saltsburg27*=who supplemented the uniform tension term, which is the only term present in the classical theory, with an expression proportional to the mean curvature.a They considered this term as the dominant term in which one could account for corrections due to high curvature effects. A useful elucidation of these ideas is provided by Melrose.12Jo However, we disagree with Markin and Kozlov’s claim31 that the necessity of taking due account of all the terms which can be now defined as first and second bending moments in the generalized Laplace equation was clearly formulated by Melrose because he never introduces the second bending moment in either of his papers, uses the first bending moment as an extensive rather than an (23) Gaydos, J.; Boruvka, L.; Neumann, A. W. In preparation. (24) Buff, F. P. J . Chem. Phys. 1956,25, 146. (26) Buff, F. P. The Theory of Capillarity. In Encyclopedia ofPhysics; FIClgge, S., Ed.;Springer-Verlag: Berlin, 1960; Vol. X (Structure of Liquids), pp 281-304. 1960,30, 52. (26) Buff, F. P. Discuss. Farnday SOC. (27) Buff, F. P.; Saltsburg, H.J. Chem. Phye. 1957,26, 23. (28) Buff, F. P.; Saltsburg, H.J. Chem. Phys. 1957,26,1526. (29) Neogi, P.; Friberg, S. E. Reference 18, p 492. (30) Melrose, J. C. Industrial Eng. Chem. 1968, 60 (3), 53. (31) Markin, V. S.; Kozlov, M. M.; Leikin, S. J. Kolloidn. Zh. 1989,51,
888.
Comments
intensive quantity as done by later a ~ t h o r s , 1and * ~does ~~~~ not introduce any corrections to Buffs work. Next, it must be recognized that a fundamental equation for any particular potential is useless unless it can be associated with a stationary principle so that both the equilibrium states and the stability of the system may be determined.M-M The stationary problem involves an integration process that is similar to the procedure for obtaining the Euler relation for a bulk system. All intensive parameters of the particular thermodynamic potential have to be kept constant, or more generally, they have to be kept a t their predetermined equilibrium values and this predetermination must have a real physical basis. For example, the use of the Helmholtz potential in a stationary problem requires that the system be in diathermal contact with a temperature or heat reservoir of known temperature. However, this usual treatment of intensive parameters is impossible for the parameters c1 and c2 used in eq 3 since there does not exist a curvature reservoir entity. After all these comments, the fundamental point that is missed by Markin and Kozlov should be reiterated: Gibbsian thinking begins with the concept of a fundamental equation like eq 2 for a planar surface as well as a moderately curved surface with the dividing surface in the surface of tension position. Shifting the dividing surface “by force” destroys the validity of the fundamental equation, even if we blame the breakdown of the conservation of free energy on a putative dependence of the surface tension on the position of the dividing surface. Within the Gibbsian formalism (i.e., of establishing equilibrium and stability conditions), terms such as the formal derivative cannot rigorously arise from the fundamental equation proposed by Gibbs. To call a treatment “Gibbsian” which introduces such ad hoc terms in order to maintain conservation of energy is inappropriate and misguided. If one utilizes fundamental eq 3 rather than eq 2 to describe curved surfaces, one is still not able to properly describe a shifted dividing surface for the reasons discussed above. Finally, all these difficulties are avoided in the treatment of Boruvka and Neumann4since a specific choice of dividing surface is not necessary. J. Gaydos, L. Boruvka, and A. W. Neumann’
Department of Mechanical Engineering, University of Toronto, Toronto, Ontario, Canada M5S l A 4 Received December 8, 1989. In Final Form: October 19, 1990 (32) Scriven, L. E. Nature 1976,263, 123. (33) Scriven, L. E. Equilibrium Bicontinuoue Structures. In Surfactants in Solution; Mittal, K. L., Ed.; Plenum Press: New York, 1977;Vol. 2, pp 877-893.
(34)Courant, R.; Hilbert,D. Methods ofMathemtica1Physics; WileyInterscience Publishers: New York, 1989; 2 vole. (35) Forsyth,A.R. Calculusof Varioti0ns;CambridgeUniversityPreea: Cambridee. 1927. (36) F k , C. An Introduction to the Calculus of Variations; Dover: New York, 1963.
