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Langmuir 1990,6, 888-892
(zero contact angles) and the corresponding ysl values, as calculated from the equation of state, are plotted as a function of the liquid surface tension ylv. It is evident from this figure that the arguments given by Johnson and Dettre to disclaim the equation of state, on the basis of the assumption of ysl = 0 as the limiting value, are completely incorrect since the equation of state does not predict zero ys, values for situations where ylv # ysv.Of course, the equation of state is also not applicable for liquid-liquid systems, as explained above. We find it surprising that Johnson and Dettre question the validity of the assumption that the limiting value of yal.is zero. This condition is completely consistent with Fowkes’ interfacial tension equation, when both solid and liquid are completely dispersive, or with Good’s approach,” when the interaction parameter equals one. Lifshitz theory considerations21 indicate that the limiting value for the interaction of two identical slabs across a medium is also zero when the dielectric properties of the medium approach that of the slabs. Criticizing only the equation of state on this point when this assumption is widely used by other workers is misleading. The question of whether ysl = 0 is indeed the limiting value of ysl has been examined in detail,” using contact angle measurements, liquid-liquid interfacial tensions, and freezing front experiments. The conclusion was that there is overwhelming evidence that zero is the limiting value for ysl. (20) Good, R.J. J . Colloid Interface Sci. 1977, 59, 398. (21) Israelachili, J. N. Intermolecular and Surface Forces; Academic Press: New York, 1985. (22) Neumann, A. W.; Spelt, J. K.; Smith, R. P.; Francis, D. W.; Rotenberg, Y.; Absolom, D. R. J . Colloid Interface Sci. 1984, 102, 278.
Reply to “On the Existence of an Equation of State for Interfacial Free Energies” Capillary effects such as wetting, spreading, and adhesion, which are either directly or indirectly dependent upon the value of the specific free energy of the solidfluid interface, are less well quantified because the latter quantity is not directly amenable to experimental measurement. Past attempts to quantify these values have relied heavily upon models. These models have generally been grouped and treated as two “distinct” schools of thought. The approach adopted by Fowkes”’ and coworkers is based primarily upon the simple assumption that the specific free energy may be determined as a linear sum of individual intermolecular interactions across the interface in question. The second approach, known as the equation-of-state method, is exclusivelya phenomenal treatment based on established surface thermodynamic principles. In a recent paper, Morrison3 criticized the equation of state school of thought by claiming that three errors of “omission“,“mathematics”, and “thermodynamics” exist in the original derivation of their relation. These “errors” may be summarized as follows: (1) Fowkes, F. M. Ind. Eng. Chem. 1964,56,40. (2) Fowkes, F. M. J. Adhesion Sci. Technol. 1987, 1, 7. (3) Morrison, I. D.Langmuir 1989,5, 540.
0743-7463/90/2406-0888$02.50/0
The important test of any equation of state, or more generally, any interfacial tension equation, is whether such an equation can correctly estimate interfacial tensions involving a solid and whether the equation can correctly predict independent phenomena. The question whether the equation of state, in the present form, gives correct estimates of solid-liquid and solid-vapor interfacial tensions has been examined previou~ly.~~’ In ref 7-9, several equation of state type relations were used to interpret results from different experimental techniques for determining solid surface tensions. It was shown that only the equation of state passed the stringent requirements of all these experiments. Direct force measurements provide the most direct approach of estimating solid-liquid interfacial tensions. In ref 8, we compare solid-vapor and solid-liquid interfacial tensions, obtained from direct force meas u r e m e n t ~between ~~ two surfactant-coated mica sheets across water, to those from equation of state considerations obtained by using contact angle data from ref 23. There is excellent agreement between the values obtained by the two approaches indicating that the equation of state does indeed correctly interpret contact angle data.
D.Li, E. Moy, and A. W.Neumann* Department of Mechanical Engineering, University of Toronto, 5 King’s College Road, Toronto, Canada M5S 1 A4 Received June I , 1989. In Final Form: September 6, 1989 (23) Claesson, P.M.; Blom, C. E.; Herder, P. C.; Ninham, B. W. J . Colloid Interface Sci. 1980, 214, 234.
(1) The “equations (cf. eqs 13-15 below) for interfacial tensions show the interfacial energies to depend on the properties of only one of two components, the chemical potential of the fluid, pZ”, and thus these equations ”appear to be independent of the solid”. (2) Equations 13-15 of the present manuscript do not follow from the Gibbs adsorption equations 10-12 and are, in the words of Morrison, “true but incomplete”. (3) The number of degrees of freedom calculated from the bulk phase rule (see below) is only “one, most conveniently, temperature”. We shall demonstrate in what follows that these criticisms are incorrect. It does not readily appear to us that the two “schools of thought” are mutually exclusive; rather, they are simply limited. Ideally, in a perfect, fully cognizant world one would be able to evaluate the complete collection of intermolecular interactions and thus evaluate surface energies or tensions without recourse t o phenomenal approaches. Surface thermodynamics, like its volume thermodynamics cousin, would acquire the status of a useful phenomenal discipline that would enhance practical calculation. In reality, however, the status of the molecular approach does not lend itself to a level of practicality required in many applications. This state of affairs is ~ell-known.‘~We quite Croxton7 on the surface tension of water as an example: “The lower electropolar con0 1990 American Chemical Society
Langmuir, Vol. 6, No. 4, 1990 889
Comments
tributions to the surface tension, ...for liquid water at 4 "C ...account for approximately two thirds of the experimental value of 75 dyn cm-': contributipns arising from electrostatic image effects, higher electropolar and polarization effects, and the use of an anisotropic rather than the sphericalized X-ray radial distributions will all tend to increase the computed value of the surface tension. However, more elaborate assumptions are hardly justified a t our present level of understanding." and "whilst it is clear that long range dipolar contributions do make a substantial contribution to the final value, it is virtually impossible to disentangle and assess the effects of the various approximations...". In light of these limitations, even for a relatively simple molecule like H,O, it is not surprising that the quasi-molecular approach of Fowkes is somewhat misleading in its simplicity, and it is hopelessly limited. An alternative route is through the phenomenal methodology embodied by surface thermodynamics and the phase rule, which informs us that an equation of state type relation exists among suitably chosen surface variables. It does not imply that the functional relation is universal; rather, we discover through experimental work that one particular form of the functional relationeiBdoes a reasonable job of correlating the wetting behavior of liquids on many "low" energy solids. An analysis of the phase rule for moderately curved capillary systems" shows that a surface relation or function exists that is analogous to eq 4 below; however, it does not yield the explicit form of the relation. Thus, the existence of a relation is established by the phase rule but not its universality. One may like to consider this relation in the same light as the ideal gas law in that era when gas dynamics was in its infancy. Both relations share the common ground of representing a universal correlation which is materially nonspecific. Fallacious conclusions are drawn in Morrison's paper regarding this point of material dependence (Le., his first "error" of "omission") that require additional commentary. Morrison does not seem to realize that the p, chemical potentials used in the equations of Ward and Neumannll are interfacial chemical potentials defined by
where NI(lv) 0 by definition and not the bulk liquid (Le., component 2) chemical potential defined by
as claimed in his article. Thus, our first point is that a quantity like p2(lV)is dependent upon both the liquid and vapor phases. It is not so surprising that pZ(lv)has properties which reflect the explicit nature of its interface as similar effects occur with surface tension. For example, if we consider the surface tension y(lV)of an oil-water ~~~
(4) Crorton, C. A. Statistical Mechanics of the Liquid Surface; Wiley: Toronto, 1980. (5)Gray, C. G.; Gubbine, K. E. Theory of Molecular Fluids; Fundamentab; Clarendon: Oxford, 19W,Vol. l. (6) R o w l i o n , J. S.;Widom, B. Molecular Theory of Capillarity; Clarendon: Oxford, 1982. (7) Crorton, C. A. ref 4, pp 212-15. (8) Spelt, J. K.; Absolom, D. R.; Neumann, A. W. Langmuir 1986,2,
620.
(ow) interface and the y"") of a nitrogen-water (nw) interface, we do not expect to find that yow= ynW. Likewise, one would not expect that pZow = pZnW.This suggests that the value of pZ(lv)is a reflection of the material properties of the interface's adjacent bulk phases. The same statement cannot be made for p,('), which depends only upon bulk liquid properties. Morrison confuses these two quantities throughout his article with vague utterances about "material dependent quantities". The entire claim of the need for these equations to be functions of an unknown number of material dependent parameters is just as flawed as claiming that the functional dependence of the pressure P on the temperature T and specific molar volume u given by the van der Waals equation (3)
is not expressible as
P = P(T,u)
(4)
but should be written as P = P(T,u,a,b) (5) to include the particular van der Waals constants a and b explicitly. The second area of confusion in Morrison's paper, which is connected to this idea of material dependence, seems to arise from a lack of understanding about the difference between the "surface of tension" dividing surface and the dividing surface of "zero mass". For any planar interface (i.e., the solid-vapor or solid-liquid interfaces are prime examples), it is possible to shift these dividing surfaces to the same location. Consequently, in a relation like
or (7)
one discovers that both p,(") and p,(sl) are evaluated at a dividing surface position such that rl E 0 (i.e., zero mass for solid). It was also explicitly stated by Ward and Neumann" that "there is no dissolution of the solid nor is there any absorption of any of the components from the liquid or gaseous phase by the solid." Taken together, it should not be too surprising that the solid's properties do not appear explicitly in the final expressions for the interfacial energies. However, this is not the same as the statement made by Morrison' that "the interfacial energies appear to be independent of the solid." Even a casual glance at the original paper by Ward and Neumann'' reveals (cf. their eq 12) that the equation of state depends upon the presence of the solid. Thus, Morrison's first claim to an "error of omission" in the original equation of state derivation is false. If the system under consideration is sufficiently small so that capillary or surface effects are significant (i.e., in cases where a curved meniscus is present), then it is possible to determine the degrees of freedom or variance of the composite system by either counting the number of variables and subtracting the number of constraints" or via employing the appropriate Gibbs adsorption equations for the system." In the latter case, each adsorp-
(9) Spelt, J. K.; Neumann, A. W. Langmuir 1987,3,588.
(IO) Li,D.;Gaydos, J.; Neumann, A. W. Langmuir 1989,5,1133. (11)Ward, C.A.;Neumann, A. W. J . Colloid Interface Sci. 1974,49, 286.
(12) Ward, C. A.; Neumann, A. W. J . Colloid Interface Sci. 1974,40, 281.
890 Langmuir, Vol. 6, No. 4, 1990
Comments
tion equation (cf. eqs 10-12 below) may be written in the form of a total differential equation dY = E x i d x i i=l
where the quantities X i are functions of some or all of the independent variables. Expressions of this kind are called Pfaff differential^.'^ The second supposed error of Morrison in the equation of state school of thinking is essentially a mathematical question that queries when one may proceed from a differential relation such as eq 8 to an integrated relation such as
Y = f(3Cl,X2,..4") (9) The requirements necessary to proceed from eq 8 to eq 9 have been known since Euler's original work of 1770,14 and they are certainly satisfied here.'"17 For the particular case of a liquid drop resting on an "ideal" solid in equilibrium with its vapor, one may certainly proceed from the three appropriate Gibbs adsorption equations dy(sV)= -S(sv)dT(1) r(sv)dpz 2(1)
(10)
dy(lV)= -s(lV)dT(1) rilV)dp2
(12)
where I$\ is the specific adsorption of the second component (i.e., liquid) on the solid-vapor interface at the 0, T is the temdividing surface position where perature, and p z is the chemical potential of the second component, to the integrated relations p v )
= Y(sV)(T,p2)
(13)
y(sl)
= Y(sl)(T,p2)
(14)
p)= y('V)(T,p2) (15) From these relations, one may then conclude that y("l)
= f(y(sv),y(lv))
(16)
Apprehension with the conclusions presented above in eqs 13-16, which are analogousto those required to present the relation in eq 4,is unwarranted and may be partially related to confusion about the classical thermodynamic formalism for capillary systems. According to Morr i ~ o n"the , ~ functional form y = y ( T , p 2 )can be written, but (it) is also seen to be incomplete, as it neglects the dependence of interfacial energy on the possible energy levels of each interface." Fortunately, the question of whether or not these equations are complete depends upon the classical definition of what constitutes a fundamental equation for a capillary system and not upon vague notions about microscopic energy levels, intermolecular interactions, or radial distribution functions. These relations are not equivalent to a fundamental equation since they, like any specific equation of state for a bulk phase, are zeroth-order homogeneous in their independent variables. As a consequence, they do not contain the same (13) Schouten, J. A,; v.d. Kulk, W. Pfaffs Problem and Its Generalizations; Chelsea: New York, 1969. (14) Ince, E. L. Ordinary Differential Equations; Longmans, Green & Co.: Toronto, 1927; pp 52,53. (15) Mbster, A.Classical Thermodynamics (ES . Halberstadt,trans.); Wiley: Toronto, 1970; p 26. (16) Sneddon, In. N. Elements of Partial Differential Equations; McGraw-Hill: Toronto, 1957. (17) Defay, R.; Prigogine, I. Surface Tension and Adsorption (A. Bellemans, collab. with D. H. Everett, trans.); Longmans, Green & Co.: London, 1966.
complete level of information that is possessed by the fundamental equation.18 These considerations show quite explicitly that there is no "error of mathematics" and that eqs 13-16 above follow as a direct consequence of the classical fundamental equation for capillary systems. The final perceived difficulty presented by Morrison3 in his "error of thermodynamics" which is connected with his statement that "by the phase rule, fixing one thermodynamic variable (e.g., temperature) fixes the values of all the thermodynamic functions, including the interfacial energies, for any combination of a fluid and a solid, ...". Here, the lack of thermodynamic understanding is epitomized by the erroneous application of the bulk phase rule to capillary systems, apparently oblivious to the existing evidence to the ~ o n t r a r y . ' ~ ? All ~ " ~of~these authors have shown that the phase rule for capillary systems of moderate curvature is different from that for bulk systems. Recently, we have summarized the theory and extended its application." For capillary systems, we find that the number of degrees of freedom predicted by the classical bulk phase rule is not sufficient to determine the total state of the system. This arises because there is a reduction in the number of mechanical constraint equations for a composite system with curved surfaces. The reason for this reduction is that the mechanical equilibrium conditions which require pressure equality between all bulk phases no longer apply between adjacent phases separated via a curved (even a moderately curved) surface. The most commonly observed manifestation of this additional degree of freedom is the capillary rise of a liquid at a solid surface, which is a manifestation of the Laplace pressure. In place of pressure equality relations of the form P* = P@,where a and 0 represent adjacent bulk phases, one must insert the appropriate mechnical equilibrium conditions for each pair of curved surfaces in the capillary system. For moderately curved surfaces, the appropriate condition is the classical Laplace equation of capillarity p* - pB = yaflJafl (17) where yaB is the surface tension of the a-0 interface and Ja8 is the mean curvatuve.28 The mean curvature, Ja8, does not vary freely but is fixed by both the pressure difference across the surface and the surface tension, and as a result, eq 17 imposes no a priori constraint on the values of the system. If we define N as the total number of distinct P = Pe type relations among the mechanical equilibrium conditions, then for a capillary system with r chemical components and n phases, the number (18) Callen, H. B. Thermodynamics; Wiley: New York, 1960. (19) Crisp, D. J. Surface Chemistry: Proc. of Joint Meeting of Faraday Society and Societe de Chimie Physique, Bordeaux, 1947. See aleo: Discuss. Faraday SOC.1948,3,98. (20) Defay, R. Etude Thermodynamique de la Tension Superficielle; Gauthier-Villars:Paris, 1934. (21) Dufour, L.; Defay, R. Thermodynamics of Clouds (M. Smyth and A. Beer, trans.); Academic: New York, 1963. (22) Guggenheim, E. A. Thermodynamics, 6th ed.; North-Holland Amsterdam, 1977. (23) Hunter, R. J. Foundations of Colloid Science;Clarendon: Oxford, 1987; Vol. 1. (24) Kirkwood, J. G.;Oppenheim, I. Chemical Thermodynamics; McGraw-Hill:Toronto, 1961. (25) Lupis, C. H. P. Chemical Thermodynamics of Materials; NorthHolland: New York, 1983. (26) Modell, M.; Reid, R. C. Thermodynamics and Its Applications, 2nd ed.; Prentice-Hall: Englewood Cliffs, NJ, 1983. (27) Rusanov, A. I. The Centennial of Gibbs' Theory of Capillarity. In The Modern Theory of Capillarity; Goodrich, F. C., Rusanov, A. I., Eds.; Akademie-Verlag:East Berlin, 1981; pp 1-18. (28) Kreyszig, E.Introduction to Differential Geometry and Riemannian Geometry; University of Toronto Press: Toronto, 1968.
Langmuir, Vol. 6, No.4, 1990 891
Comments
of degrees of freedom f is" f=r+l-N
n-Hexauiacontnne
Thus, for a capillary system consisting of immiscible bulk fluid phases, one may easily see from eq 18 that the number of degrees of freedom will depend explicitly on the number of planar surfaces that are present. The phase rule does not in itself admit order of magnitude arguments, such as those suggested by Morrison, once one chooses to model a system as a capillary system which possess both bulk and surface regions. Consequently, we infer from this that for a two-component fluid-liquid lens-fluid system the number of degrees of freedom is three and that an equation of state dependent on just two variables is not possible. Only under the very special condition where the interface between fluids is planar would one expect that a relation of this type could exist. These results have been previously derived by Defay and Prigogine"" and discussed by Dufour and Defay.21 Obviously, for a sessile drop system (Le., a two-component, three-phase system) which has three bulk phases and three surface phases, there is only one pressure equality relation. Realizing that this system has just two components (Le., solid and liquid), we see that the number of degrees of freedom is likewise two and not one, as claimed by Morrison. Thus, with y('") and ?(Iv)selected as the two independent variables, one may express y("), the remaining surface tension, as a function of the other two surface tensions in the functional form given by eq 16. This provides the thermodynamic basis for an equation of state. Quite apart from the phase rule arguments above, one may use the Gibbs adsorption equation (see eq 20 below) to demonstrate that the conditions under which the surface tension would be dependent upon just one variable, as claimed by Morrison: are unreasonably restrictive and are not obeyed experimentally. Morrison suggested that the one variable on which the surface tension is dependent could be chosen "most conveniently" as temperature, e.g., y = y ( T ) , or equivalently, that ( 6 ~=)0~(i.e., variations of surface tension are zero at constant temperature). It has been recognized by other^^^^^^ that the surface tension will only be a function of temperature alone if
r$; = o
for all i = 1 , 2 , ...,r (19) where the subscript (1) refers to the dividing surface of zero mass for component number one. This conclusion is apparent directly from the Gibbs adsorption equation for either a planar or a moderately curved surface, which is given by
Physically,this would represent a rather unique or unusual situation that may be approximately satisfied in some cases but is not satisfied exactly as a general rule. For this relation to be satisfied, a capillary system would need to exhibit absolutely no pressure dependence of the surface tension, i.e., y # y ( P ) . However, the existence of this dependence is attested to by a number of pressuredependent measurements of surface tension of multicomponent system^^'-^^ which show that eq 19 is not satis(29) Defny, R.; Prigogine, I. ref 17, p 96. (30)Modell, M.; Reid, R. C. ref 26, p 420. (31) Motomurn, K. J . Colloid Interface Sei. 1978,64, 348. (32) Good,R. J. Pure Appl. Chem. 1976,48,427.
-\
(18) 0.4
-0.4
4 u)
1
30
40
SO
60
? (mud)
70
80
Figure 1. Plot of cos B versus ?(Iv) for a n-hexatriacontanesurface. The liquids used were n-dodecane, n-tetradecane, nhexadecane, ethylene glycol, thiodiglycol, glycerol, and water listed in order of ascending surface tension.
fied for all components. As a consequence, it is incorrect to claim that the surface tension is a function of the temperature alone. In connection with these arguments, it is also incorrect to claim, as done by Morrison: that "the curvature of an interface introduces a new degree of freedom only when the interface is less than a micron or SO." While it is true that the value of the surface tension of a liquidvapor interface does not change appreciably until one is dealing with very small drops, it is not correct to claim that capillarity effects are not noticeably until such small systems occur. Numerous examples are available of the effect of curved interfaces upon the mechanical equilibrium of capillary systems. The curvature of the surface produces easily observed effects like the meniscus rise of a liquid on a solid boundary while more subtle influences occur on the vapor pressure (e.g., Thomson's equation), boiling point, and solubility of a pure species as well as the chemical potential of a solute. Essentially, the effect is present whenever the pressure difference between two adjacent bulk phases is non-zero (see eq 17). In all these situations, the equations used to describe these phenomena pertain to moderately curved, rather than highly curved, capillary systems. Thus, in contrast to the claims of Morrison, we find that even moderately curved capillary systems exhibit more degrees of freedom than planar capillary systems. The fact that there may be even more degrees of freedom for highly curved capillary systems, for which eq 17 is modified to include bending is of no consequence for the present purpose. In conclusion, the phenomenal methodology embodied by surface thermodynamics and the phase rule for moderately curved capillary systems show that a functional relationship or an equation of state type relation will exist for suitably chosen surface variables. Universality, that is, an identical functional relationship for different solid-liquid systems, cannot, however, be proven from surface thermodynamics considerations. Nevertheless, if the equation of state is indeed universal, if the equilibrium spreading pressure is negligible, and if the measured contact angles represent the true Young contact angles, then the data points plotted as cos 0 versus y(lV)should fall on a single smooth curve for each solid-liquid system. From our experience, we have found that the better the quality of the surface and the more satisfactory the contact angle measurement procedure, the smoother the cos 0 versus y(lV)curves tend to be. In Figure 1, we reproduce a cos 0 versus y(l")plot for (33) DeFilippis, F. M.A.Sc. Thesis, University of Toronto, Toronto, 1989. (34) Boruvkn, L.; Neumnnn, A. W. J. Chem. Phys. 1977,66,5464.
892 Langmuir, Vol. 6, No. 4, 1990
n-he~atriacontane~' surfaces. The first three data points in Fi ure 1 for the n-alkanes are those of Fox and Zisman!6 Only those measurements were considered for which the contact angle 6 > 30°, in order to exclude systems where equilibrium spreading pressure might be pronounced. The data for the remaining four liquids were taken from Hellwig and N e ~ m a n n . ~The ' n-hexatriacontane surfaces were obtained by vapor deposition under vacuum and were so smooth that there was no contact angle hysteresis when checked with water. For the latter systems, the contact angles were measured dynamically, using the capillary rise at a vertical plate method, at very low advancing rates of the three-phase line. The smooth curve connecting the points is not a curve fit but was calculated b using the equation of state3* for a surface tension y ( 4 of 20 mJ/m2. Curves like that illustrated in Figure 1 suggest that the equation of state is indeed applicable for low-energy solids3' and that equilibrium spreading pressures are negligible, at least in this case. The fact that the equilibrium spreading pressure (35)Neumann, A. W.Adv. Colloid Interface Sci. 1974,4, 105. (36)Fox, H. W.;Zisman, W. A. J. Colloid Interface Sci. 1952, 7,428. (37)Hellwig, G.E. H.; Neumann, A. W. Int. Congr. Surface Activity, Barcelona, 1968, Section B, p 687. (38)Neumann, A. W.;Good,R. J.; Hope, C. J.; Sejpal, M. J. Colloid Interface Scr. 1974, 49, 291.
Comments is probably negligible for certain systems does not guarantee that this is universally true for all solid-liquid systems. If either the quality of the solid surfaces or the contact angle acquisition is not satisfactory, then a variety of patterns of experimental data, which are not consistent with a single smooth curve, are also possible. Thus, it is clear that scatter in experimental points alone does not disprove the universality of the equation of state. Clearly, in order to investigate the question of universality of the equation of state further, more high-quality contact angle data on high-quality surfaces and more information on equilibrium spreading pressure are required. J. Gaydos, E. Moy, and A. W. Neumann'
Department of Mechanical Engineering, University of Toronto, Toronto, Ontario, Canada M5S 1A4 Received July 24, 1989. In Final Form: November 6, 1989 (39)Li,D.;Moy,E.; Neumann, A. W. The Equation of State Approach for Interfacial Tensions: Comments to Johnson and Dettre. Langmuir, in press.
