Langmuir 2002, 18, 1445-1448 The Gibbs-Thomson Equation and the Solid-Liquid Interface
Della Volpe et al.1 (DVSM in the following) discuss two recent papers, one of which is my article “On the Methods To Determine Surface Energies”.2 They present some criticism of my attempt to show that the Gibbs-Thomson equation (GTE) is valid in the context of determining interfacial surface energies. The comments by DVMS are welcome and expected considering the striking conclusion2 that, because GTE appears to be valid, the generally used indirect methods, such as the Fowkes equation, the Lifshitz-van der Waals acid-base approach, the van OssChaudhury-Good equation, and Neumann’s equations of state, provide solid-liquid surface energies that are in error by 2 orders of magnitude. The comments presented by DVSM are discussed below in the order that they appear in their paper.1 First, DVSM pay attention to the discrepancy between the ice-water interfacial energy obtained by using GTE and the reported spreading behavior of water on ice. This problem has been solved in ref 3 by experimentally showing that the contact angle of water on ice is 37° with the resulting ice surface energy slightly smaller than that of water. The problem of surface melting on ice, mentioned by DVSM, was also extensively discussed in ref 3. One of the consequences of these findings is that the free energy of an ice surface is not minimized by surface premelting, thus showing that the widely held theories4-7 on the formation of a liquidlike layer on ice are unfounded, as previously noted by Knight.8 The problem raised by DVMS on the ice-water system is thus obsolete. Furthermore, it existed in the first place only due to a misinterpretation on the spreading behavior of water on ice not due to the use of GTE. Next, DVMS point out that eq 10 in ref 2 for the Gibbs free energy difference ∆Gv
∆Gv ) -∆Sf ∆T
(1)
is not exact from a theoretical point of view. Here ∆Sf is the entropy of fusion and ∆T is the supercooling. Indeed, eq 1 is a linear approximation by Turnbull,9 which assumes that the difference ∆C in the specific heats of the liquid and the solid equals zero. Turnbull’s approximation is involved also in the Clapeyron equation (eq 5 in ref 2). DVMS do not discuss the numerical effect of this approximation, which has, in fact, been thoroughly analyzed before. One can use either the tabulated values of the specific heats of ice Cs and water Cl10,11 in numerical integration of the exact integral equation for ∆Gv (see, e.g., ref. 12, p 201) or one of the other well-known approximations13-16 that are more accurate than that by Turnbull. The results of these methods are quite similar (1) Della Volpe, C.; Siboni, S.; Morra, M. Langmuir 2001, 18, 0000. (2) Makkonen, L. Langmuir 2000, 16, 7666. (3) Makkonen, L. J. Phys. Chem. 1997, B101, 6196. (4) Lachmann, R.; Stranski, I. N. J. Cryst. Growth 1972, 13/14, 236. (5) Baker, M.; Dash, J. G. J. Cryst. Growth 1989, 97, 770. (6) Dash, J. G. Science 1989, 246, 1591. (7) Wettlaufer, J. S. Philos. Trans. R. Soc. London 1999, A357, 3403. (8) Knight, C. A. J. Geophys. Res., D 1996, 101, 12921. (9) Turnbull, D. J. Appl. Phys. 1950, 21, 1022. (10) Franks, F. Water: A Comprehensive Treatise; Plenum Press: New York, 1982; Vol. 7, 498 pp. (11) Leyndekkers, J. V.; Hunter, R. J. J. Chem. Phys. 1985, 82, 1440. (12) Kurz, W.; Fisher, D. J. Fundamentals of Solidification; Trans Tech Publ.: Enfield, NH, 1989; 305 pp.
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Table 1. Correction for the Conventional Gibbs-Thomson Equation Due to Taking into Account the Temperature Dependence of the Specific Heat Difference between Water and Ice supercooling ∆T (K)
correction (%)
-0.1 -0.5 -1 -5 -10
0.03 0.16 0.32 1.60 3.25
but that by Dubey and Ramachandrarao16 has the best theoretical justification. Their expression using the definitions here, is
∆Gv ) -∆Sf ∆T - [∆Cf(∆T)2/(2T)] [1 + ∆T/6T]
(2)
where ∆Hf is the latent heat of fusion per mole, Tf is the equilibrium melting temperature of a planar interface, and ∆Cf is Cl - Cs at Tf. As noted in ref 12, the right-hand correction term in eq 2 is negligibly small for metals. DVMS mention hydrocarbons and water. For those hydrocarbons, such as succinonitrile (ref 11), for which GTE has been widely used, ∆C is actually very close to zero, so that the situation is the same. Water, however, is indeed interesting here, because of its significance and the wide use of GTE for the water-ice interface as well as due to the fact that the value of Cl is about twice the value of Cs for this system. It is, therefore, necessary to consider the ratio of the two terms of eq 2 for the water-ice system at different temperatures T. Inserting into eq 2 ∆Hf ) 6002 J mol-1, ∆Cf ) 37.8 J mol-1, and Tf ) 273.15 K from ref 10 gives the results presented in Table 1 for the percentile correction provided by the right-hand-side term into eq 2. From the derivation of the GTE (eqs 9-14 in ref 2), it can be seen that this is the correction that should be applied to the measured surface energy γ (or to ∆T in eq 14 of ref 2). The modern measurements of the solid-liquid interfacial energy related to the use of GTE are done close to the equilibrium, i.e., at very small supercoolings. For example, the most recent measurements for the waterice system17 were made at ∆T of 0.147-0.412 K. As shown in Table 1, the inaccuracy, for which DVMS have concern, is approximately 0.1%. Noting that the general accuracy of determining the solid-liquid surface energies (see ref 18) is 10-20%, it is not easy to share the concern of DVMS. Certainly, the nondetectable effect of neglecting the temperature dependence of ∆C in deriving GTE has no relevance to the real issue here, which is the 2 orders of magnitude difference between the solid-liquid surface energies obtained by GTE and the other widely used methods. DVMS have further concern that the integration over the temperature and curvature regimes, to obtain the simple form of GTE, cannot be done because the thermodynamic quantities depend on temperature T and curvature K. DVMS proceed by presenting GTE in an (13) Hoffman, J. D. J. Chem. Phys. 1958, 29, 1192. (14) Jones, D. R. H.; Chadwick, G. A. Philos. Mag. 1971, 24, 995. (15) Thompson, C V.; Spaepen, F. Acta Metall. 1979, 27, 1855. (16) Dubey, K. S.; Ramachandrarao, P. Acta Metall. 1984, 32, 91. (17) Hillig, W. B. J. Cryst. Growth 1998, 183, 463. (18) Jones, D. R. H. J. Mater. Sci. 1974, 9, 1.
10.1021/la011290f CCC: $22.00 © 2002 American Chemical Society Published on Web 01/23/2002
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integral form, the difference from the conventional GTE being that the volumetric enthalpy of fusion ∆hV (L in ref 2) depends on T and K. This is incorrect as far as the dependence on K is concerned. The volumetric enthalpy of fusion cannot depend on the curvature K because K is a property of the interface only. In a thermodynamic context, the interface is “Gibbsian”, i.e., a discrete interface that has no volume. Therefore, in the thermodynamic framework discussed here, ∆hV, defined as a volumetric quantity, cannot depend on K. As to the dependence of ∆hV on temperature, the fusion enthalpy per mole ∆Hf is defined at Tf, so that the dependence of the volumetric fusion enthalpy ∆hV on T is due to thermal expansion only. Turning again to water as an example, the volumetric thermal expansion coefficient at ∆T of a few kelvin is of the order of 10-4 K-1, so that ∆hV can be considered to be independent of temperature in the narrow range of supercoolings where GTE is used to determine surface energies. The thermal expansion coefficients of other supercooled liquids are of the same small order of magnitude. Thus, the concern by DVSM regarding the effect of intergrating over a temperature regime in deriving GTE is unnecessary. Related to the mechanical derivation of GTE in ref 2, DVMS present, from ref 19, an equation
fij ) γδij + ∂γ/∂ij
(3)
where fij is the surface stress, δij is the Kronecker delta, and ij is the surface strain tensor. DVSM point out that eq 3 relates surface stress and surface tension through elastic deformations in the solid but later go on to argue that the surface tension is an invalid concept for solids. With this inconsistency aside, this author claims that eq 3 is simply wrong. DVSM note that “In a solid the elastic components for the formation of a new surface are predominant.” This implies that DVSM interpret the second term on the right side of eq 3 as a result of an elastic distortion, whereby a newly formed surface attains the final equilibrium. However, from both the practical (measurements) and theoretical (thermodynamics) point of view the surface tension γ and any surface stress that is related to it are defined at equilibrium. Because γ is the work of creating a new surface at equilibrium, it includes surface distortions of all kinds. Thus, internal stresses, such as those formed by elastic deformations of a newly generated surface, are already subsumed into the surface tension γ, and no such additional effect as the second term of eq 3 exists. On the other hand, the original derivation by Shuttleworth20 that results in eq 3 starts from assuming that the surface stress is balanced by external forces, not the forces that are related to the formation of a surface at equilibrium, as implied by DVSM. However, when a solid is stretched by external forces, the stress tensor fij is determined by the elastic modulus of the materialsa parameter that does not appear in eq 3 at all. Consequently, eq 3 is wrong whatever the assumption of the origin of the surface strain ij is. The incorrect nature of eq 3 deserves further attention here, because this equation appears in many textbooks. Equation 3 originates from Shuttleworth,20 who used a different terminology (“surface tension” for f and “surface energy” for γ). Using here the terms adopted by DVSM, (19) Neumann, A. W.; Spelt, J. K. Applied Surface Thermodynamics; Marcel Dekker: New York, 1997; 239 pp. (20) Shuttleworth, Proc. Phys. Soc. (London) 1950, A63, 444.
and simplifying into one-dimensional stretching of an area A, Shuttleworth’s starting equation reads
f dA ) d(Aγ)
(4)
where f is the component of the surface stress in the direction of stretching. According to the explanation in ref 20, the left side of eq 4 is the work done against the surface stress when stretching the surface by a small amount dA. When the deformation is reversible and occurs at constant temperature, this work must, according to ref 20, be equal to the increase in the total surface energy d(Aγ), represented by the right side of eq 4. By derivation it follows from eq 4 that
f ) d(Aγ)/dA ) γ + A(dγ/dA)
(5)
which is analogous to the more general form in eq 3. Equations 3 and 5 are supposed to show that the surface stress f and the surface tension (i.e., surface energy) γ on a solid surface are not equal. The reasoning behind eq 4 includes a number of errors, however. The first error is an unjustified assumption that γ depends on A, but f does not. When the concept of virtual work is applied, as is done here, the left side of eq 4 should include an integral over f dA. The obscure eq 5 is a direct result of this omission. The second error is a more fundamental one. The surface tension γ is caused by the molecular force imbalance perpendicular to the surface and manifests itself as a surface stress parallel to a liquid surface, because new molecules are lifted from the bulk into the surface when a surface is stretched. However, when a solid is stretched, the molecular density on the surface decreases, and no new molecules are brought onto the surface. Therefore, when a solid is stretched, no work is done against such a surface stress that has any relation to the concept of γ. Instead, work is done to increase the distance between the molecules parallel to the surface. Thus, the work of deformation of a solid should be related to the strain energy, not the surface energy γ, as done in eq 4. The third error is the inclusion of the term on the right side in eq 4. When a solid is stretched, the number of molecules in the surface remains constant and their separation decreases.1,19-21 Because of the latter, dγ/dA is not zero as noted in ref 20. However, because of the former, the change in the total surface energy d(Aγ) is identically zero. The surface energy originates from the lack of counterparts of the molecules at the surface and thus relates directly to their number. Thus, when a solid is stretched, γ is inversely proportional to A, and the righthand side of eq 4 does not exist. A conclusion of the errors pointed out above is that eq 4 is entirely meaningless and that its corollary eq 5 should thus be abandoned. This, of course, applies to the more general form in eq 3, as well. Accordingly, the surface energy γ and the surface tension (the stress related to γ) should be considered numerically equal also on solids, in contrast to eq 5. However, one must be careful not to confuse such an equilibrium surface stress with various types of externally caused stresses mentioned by DVSM. Their point here appears to be the one noted already in ref 2, i.e., that GTE may not be applicable on solids strained due to a high external stress. The question then is, are surfaces, such as grain boundary grooves used in determining the surface energies by GTE, under a high external stress? The view of the present (21) Bikerman, J. J. Top. Curr. Chem. 1978, 77, 1.
Comments
author, shared in ref 12, is that solidification of a liquid represents creation of a new solid surface and not stretching or deformation of an existing surface. In solidification the molecules of the liquid phase freely adhere to the solid surface, which then is free from external stresses. A further argument against the significance of stresses in the solid on γ arises from noting that the solid-liquid interface can also be viewed from the liquid’s side and thus be seen as the surface of the liquid. A liquid cannot maintain significant elastic deformations, so that this view suggests that a solid-liquid interface is free from all such three types of external stresses as described by DVMS that may exist in a bulk solid. The claim by DVMS that the concept of surface tension is invalid for a solid-liquid interface reflects a rather generally accepted view outlined in ref 21 and also deserves a detailed discussion here. DVMS write “In a crystalline solid to move an atom or a molecule from the bulk to the surface cannot be made in a quasistatic, reversible way, because, contrary to what happens in the liquid state, strong bonds exist. As a result the concept of surface tension for a solid is invalid.” Similarly, Bikerman21 states that solid surfaces have no tendency to contract. These views reflect a way of thinking where a lack of a process means that there is a lack of a property. This is not so, because there may be constraints (strong bonds between molecules in this case). Let us consider a simple analogy of a lamp bonded on a roof of a room. If the bond is removed the lamp falls down. Only then a process occurs and the weight of the lamp can be directly measured. However, both with and without the bond, the lamp is affected by gravity and its relevant property, the weight, can be defined. To this author, claiming that, the property, surface tension, cannot be defined for a solid is as obscure as to say that the weight of the lamp, bonded on a roof, cannot be defined because it does not fall down. Similarly, the claim that the surface of a solid has no tendency to contract is as obscure as to say that the lamp has no tendency to fall. The surface tension, i.e., the limitless tendency of a surface to contract, is a property, always related to an interface, whereas capillary dynamics is a process that only takes place when the constraints for minimizing the total energy are removed. The lamp, considered above, may fall down even if the bond remains, however. For example, the entire roof may give up. There is an analogy here with the contact angle measurement (Figure 1 in ref 2) in that the free energy of a solid-liquid-vapor system may be minimized also by a process occurring within the liquid (droplet spreading), not the solid. Nevertheless, the liquid forming the drop moves because of the properties related to the solid as well, i.e., the solid-liquid and solid-vapor surface tensions. Thus, even though no process occurs within the solid, the solid-liquid surface tension must be considered real, and it satisfies the Young-Dupre equation (eq 1 in ref 2), as in a balance of forces. A further argument is the Newtonian definition of a force. According to Newton’s second law any change in motion is a consequence of a force. In a capillary rise and in a contact angle experiment, the liquid moves along the well-defined solid-liquid surface, so that a force must exist that pulls the solid-liquid-vapor interaction line toward the solid (as shown in Figure 1 in ref 2). Thus, the claim,1,21 that the surface tension is an invalid concept for a solid, contradicts the fundamentals of classical physics. Because of the constraint that prevents molecular mobility, the surface tension on a solid without a liquid counterpart
Langmuir, Vol. 18, No. 4, 2002 1447
is unable to do mechanical work and no work is done against surface tension when a solid is stretched, as discussed above. In this meaning only, it is correct to say that the surface tension on a solid does not exist. It is noteworthy here that GTE is not used for solids as such, but only in the context of a phase equilibrium. The long controversy on the interpretation of a surface tension of a solid, continued by DVSM and here, reflects different approaches by thermodynamicists and physicists. The former prefer to explain phenomena in terms of energy principles and the latter in terms of mechanisms. It is emphasized here that there is no reason for a controversy between these two approaches, since the classical physics defines work, i.e., energy, as the product of a force and a distance. Therefore, in principle, any dynamic process that can be explained in terms of energy can also be explained by a force and vice versa. This is why the Young-Dupre equation and GTE can be derived2 both by the principle of minimizing the free energy and a balance of forces, the result being the same. The last point by DVSM is the concern about the size and roughness of the solid particles used when applying GTE. The problem of size was recently extensively discussed by Bogdan22 who, for example, showed that the ice-water surface energy starts to change in the order of 10% only when the ice crystal radius of curvature gets smaller than about 10-8 m. The relevant radius of curvature in the most recent determinations of the icewater surface energy17 was 10-8 to 3 × 10-7 m. In the measurements of γ using the grain boundary groove method the curvature is orders of magnitude higher than these values. The effect of the roughness of the solid surface in the contact angle measurements should be considered as a constraint only. This is because the energy proof of the Young-Dupre equation is derived for the average geometry and surface energies and does not concern the local structure of the interface. The possible local constraint caused by the roughness of the solid is usually dealt with by measuring both the advancing and the receding angles. Also, there is no reason to expect that a solid surface formed by solidification of a liquid at equilibrium (a grain boundary groove) is sufficiently rough to affect the contact angles. If it were that rough, it would not be an equilibrium surface. Finally, it should be noted that the curvature in the modern measurements of the solid-liquid surface energy (e.g., refs 23 and 24) is not based on a curvature at some local point, as indicated by DVSM. The measurement is based on comparing the measured overall geometry of a grain boundary groove to its theoretical shape in its entire length. Thus, the determination of K is done well above the optical detection limit and in a way directly applicable to utilizing the Young-Dupre equation and GTE. In conclusion, no unreasonable approximations, as claimed by DVSM, were made in the derivations of GTE in ref 2. To the extent that the aspects pointed out by DVSM are valid at all, their effect is nondetectable in practice. The comments by DVSM on the mechanical derivation of GTE relate to the fundamentals of capillarity, i.e., the definition and existence of surface tension on solids, as well as to the effect of deformations in a solid at its interface with a liquid. These comments reflect a rather common view in the literature. In this paper, a somewhat (22) Bogdan, A. J. Chem. Phys. 1997, 106, 1921. (23) Schaefer, R. J.; Glicksman, M. E.; Ayers, J. D. Philos. Mag. 1976, 32, 725. (24) Bayender, B.; Marasli, N.; Cadirli, E.; Sisman, H.; Gunduz, M. J. Cryst. Growth 1998, 194, 119.
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different nature of the solid-liquid interface is outlined. It is shown here that a generally used equation, quoted by DVSM, that relates the surface tension (i.e., stress) and surface energy, is incorrect. The conclusion of these developments is that the criticism by DVSM on the concept of surface tension on solids and the use of GTE is not valid. It appears, therefore, that the origin of the severe discrepancy between the solid-liquid surface energies provided by the use of GTE and the indirect methods, commonly used in physical chemistry, must be sought from the indirect methods.
Comments
Acknowledgment. Thanks are due to K. Kolari and C. A. Knight for fruitful discussions and the Academy of Finland and Jenny and Antti Wihuri Foundation for financial support. Lasse Makkonen
Technical Research Centre of Finland, Box 1805, 02044 VTT, Finland Received August 9, 2001 In Final Form: October 17, 2001 LA011290F
