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On the Methods To Determine Surface Energies Lasse Makkonen* Technical Research Centre of Finland, Box 18071, 02044 VTT, Finland Received June 24, 1999. In Final Form: May 30, 2000 There is a prevailing dispute on methods to determine surface energies of surfaces involving a solid. The most widely used methods are the surface component (STC) theory and the equation of state (EQS) approach. Several versions of these basic theories have been developed, and their predictions for various systems differ but are of the same order of magnitude. However, the solid/liquid surface energies determined by the methods based on the effect of interface curvature on the equilibrium phase change temperature, as described by the Gibbs-Thomson (G-T) equation, are 2 orders of magnitude higher than those determined by the STC and the EQS. This controversy is addressed here by critically analyzing both the thermodynamic and mechanical derivations of the G-T equation. It is concluded that none of the arguments recently presented against the use of the G-T equation, to explain the discrepancy, appear to be valid. Consequently, it appears that both the STC theory and the EQS approach may be incorrect.
1. Introduction A common approach to estimate the surface energy of solids is based on an interpretation of sessile drop contact angle data (Figure 1). In Figure 1 θ is the contact angle, γs the solid/vapor surface energy, γl the liquid/vapor surface energy, and γ the solid/liquid surface energy. By the analogy of the surface energy and surface tension of a liquid, γl can easely be determined separately, but the Young-Dupre´ equation for the equilibrium of Figure 1
γs ) γl cos θ + γ
(1)
includes two unknowns, the solid/vapor and solid/liquid surface energies, γs and γ. Therefore, models have been developed to relate the interfacial energies with each other. One line of these models is based on the assumption that the free energy can be calculated as a sum of components that represent specific types of intermolecular interactions. This surface component (STC) theory was pioneered by Fowkes1,2 and has later been advanced to the Lifshitzvan der Waals/acid-base approach and the van OssChaudhury-Good equation.3-7 Another model is the socalled equation of state (EQS) developed by Neumann et al.8-13 based on a thermodynamic derivation. In the EQS approach the solid/liquid interfacial energy is a function of the total solid and liquid surface energies only, i.e., γ ) f(γs,γl). * E-mail:
[email protected]. (1) Fowkes, F. M. J. Phys. Chem. 1962, 66, 382. (2) Fowkes, F. M. Ind. Eng. Chem. 1964, 56, 40. (3) van Oss, C. J.; Chaudhury, M. K.; Good, R. J. Adv. Colloid Interface Sci. 1987, 28, 35. (4) van Oss, C. J.; Good, R. J.; Chaudhury, M. K. Langmuir 1988, 4, 884. (5) van Oss, C. J.; Chaudhury, M. K.; Good, R. J. Chem. Rev. 1988, 88, 927. (6) Good, R. J.; van Oss, C. J. Wettability; Plenum Press: New York, 1992, Chapter 1. (7) Lee, L.-H. Langmuir 1996, 12, 1681. (8) Ward, C. A.; Neumann, A. W. J. Colloid Interface Sci. 1974, 49, 286. (9) Spelt, J. K.; Absolom, D. R.; Neumann, A. W. Langmuir 1986, 2, 620. (10) Spelt, J. K.; Neumann, A. W. Langmuir 1987, 3, 588. (11) Li, D.; Neumann, A. W. Adv. Colloid Interface Sci. 1992, 39, 299. (12) Kwok, D. Y.; Li, D.; Neumann, A. W. Langmuir 1994, 10, 1323. (13) Neumann, A. W.; Spelt, J. K. Applied Surface Thermodynamics; Marcel Dekker: New York, 1997; 239 pp.
Figure 1. Sessile drop on a solid surface. The arrows represent the surface energies when interpreted as surface tensional forces.
There has been a long controversy on the correctness of these models. A fundamental uncertainty of these theories is the application of a combining rule of some form to evaluate the parameters of unlike-pair molecular interactions in terms of the like-pair interactions. The combining rules, such as the commonly used Berthelot rule, are hypothesis only. It is beyond the scope of this paper to criticize these theories in detail, but it may be pointed out that a number of strong arguments have been presented previously against the acid-base approach12,13 while also EQS has been severely criticized.4,14-17 In fact, Morrison16 showed that the derivation of EQS is invalid for various reasons. Nevertheless, EQS is widely used and was again put forward in a recent text book.13 The solid/liquid surface energies resulting from the two abovementioned approaches do not generally quite agree but are of the same order of magnitude.13 An entirely different approach to determine surface energies, also widely used, involves the application of the Gibbs-Thomson equation (G-T). The G-T equation forms the basis of nucleation science and many other fields of microscopic phenomena,18,19 such as dendritic crystal growth, sedimentation, and freezing of porous materials. The G-T equation relates the equilibrium phase change temperature to the interface curvature and surface energy. The G-T equation can thus be used to estimate the solid/ liquid interfacial energy directly, e.g., from a nucleation (14) van de Ven, T. G. M.; Smith, P. G.; Cox, R. G.; Mason, S. G. J. Colloid Interface Sci. 1983, 91, 299. (15) van de Ven, T. G. M. J. Colloid Interface Sci. 1984, 102, 301. (16) Morrison, I. D. Langmuir 1989, 5, 540. (17) Wu, W.; Giese, R. F., Jr.; van Oss, C. J. Langmuir 1995, 11, 379. (18) Tiller, W. A. The Science of Crystallization; Cambridge University Press: 1991; 391 pp. (19) Kurz, W.; Fisher, D. J. Fundamentals of Solidification; Trans Tech Publ.: Enfield, NH, 1989; 305 pp.
10.1021/la990815p CCC: $19.00 © 2000 American Chemical Society Published on Web 08/31/2000
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Makkonen
ficiencies of the Gibbs-Thomson equation”, where this conclusion is prompted. Here the controversial situation described above is dicussed further, in particular as it relates to the arguments presented against the use of the G-T equation. To this aim, the derivation of the G-T equation is presented in the following, first in its classical form, then by using general thermodynamics, and finally by mechanical terms applying the interpretation of the interfacial energy as the surface tensional force. 3. The Gibbs-Thomson Equation
Figure 2. Principle of determining the solid/liquid surface energy γ by the Gibbs-Thomson equation based on the curvature of a grain boundary groove in a linear temperature gradient. Within the curved interface the equilibrium temperature T differs from that of a planar interface far away from the groove TM.
experiment20 or an experiment on a curvature of a grain boundary groove21,22 (see Figure 2). This then makes it possible to solve also the surface energy of the solid γs from the Young-Dupre´ equation.
3.1. Classical Derivation. In many text books the derivation of the G-T equation is presented as follows. Consider a drop in a gas or a bubble in a liquid having a radius of R, surface area A, and surface energy γ. The change in the total surface energy resulting from shrinking the drop or the bubble (e.g., by letting fluid out of it) is
∆Aγ ) 8πR ∆R γ
(2)
because the equilibrium shape here is a sphere with A ) 4πR2. Since shrinking decreases the total surface energy, the tendency to do so must, at equilibrium, be balanced by a pressure difference ∆p across the film. Upon a change of ∆R, the work against this pressure difference is
2. The Discrepancy In a recent effort to further justify the EQS approach, Neumann and Spelt13 compared the solid/liquid surface energies based on different methods including EQS and the Fowkes equation, the G-T equation, and the Lifshitz theory23 and the gradient theory.24 The results show that these methods give values mostly of the same magnitude except for the G-T equation, which results in solid/liquid surface energies that are 2 orders of magnitude larger. As an example, for the ice/water interface various studies based on G-T give25 a surface energy of around 30 mJ/m2 whereas EQS predicts 13 a value of 0.38 mJ/m2. Corresponding values for, e.g., naphthalene are 61 and 1.44 mJ/m2 and for biphenyl 50 and 0.67 mJ/m2. Little attention has been paid previously to this very alarming situation. The serious quantitative discrepancy between the G-T method and the indirect methods points out that there is something fundamentally wrong with either, or both, of the commonly used methods of the very advanced fields of science. If all other methods except the G-T were incorrect, a calamity would follow in those fields of science where these methods have generally been used. On the other hand, if the G-T were incorrect, the foundations of, e.g., the nucleation theory and dendritic growth modeling would collapse. The consequences would be enormous considering, e.g., that in the field of dendritic growth alone roughly 1000 papers have appeared in scientific journals during only the last 5 years.26 Almost all these studies utilize the G-T equation both directly, as it operates at the dendrite tip interface, and indirectly by using the solid/liquid surface energies based on the G-T equation. Neumann and Spelt13 concluded from the quantitative discrepancy discussed above that it is the G-T equation which is inapplicable to determining the solid/liquid interfacial energies. Their book includes a chapter “De(20) Jones, D. R. H. J. Mater. Sci. 1974, 9, 1. (21) Schaefer, R. J.; Glicksman, M. E.; Ayers, J. D. Philos. Mag. 1975, 32, 725. (22) Hardy, S. C. Philos. Mag. 1977, 35, 471. (23) Lifshitz, E. M. Zh. Eksp. Teor. Fiz. 1955, 29, 94. (24) Cahn, J. W.; Hilliard, J. E. J. Chem. Phys. 1958, 28, 258. (25) Hillig, W. B. J. Cryst. Growth 1998, 183, 463. (26) Caroli, B.; Muller-Krumbhaar, H. ISIJ Int. 1995, 35, 1541.
A ∆p ∆R ) 4πR2 ∆p ∆R
(3)
The change in the total surface energy due to shrinking equals the work done, so that equating the right-hand sides of eqs 2 and 3 results in the Young-Laplace equation for a sphere
∆p ) 2γ/R
(4)
In this situation, where the pressure changes on the other side of the interface only, the change in pressure is connected to a change in the equilibrium phase change temperature by the Gibbs-Duhem relation
∆p ) -∆TL/TM
(5)
where ∆T ) T - TM is the difference between the actual interface temperature T and the equilibrium phase change temperature TM of a planar interface and L is the volumetric latent heat of fusion. Combining eqs 4 and 5 yields the Gibbs-Thomson equation for a sphere
∆T ) -2γTM/(RL)
(6)
3.2. Thermodynamic Derivation. The total Gibbs free energy ∆G for a solid particle, with the volume V and surface area A, submerged in its own liquid is the sum of the volume and interfacial terms
∆G ) V ∆Gv + Aγ
(7)
where ∆Gv is the Gibbs free energy difference between the liquid and the solid per unit volume and γ is the solid/ liquid surface energy. At a thermodynamic equilibrium
d(∆G)/dV ) 0
(8)
i.e.
∆Gv +
dA γ)0 dV
(9)
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by the force γδl whose projection F normal to the surface is
F1 ) γδl sin φ1
(15)
where sin φ1 ) R/r1. Considering the two orthogonal sections with the principal radii r1 and r2, the combined normal force F ) F1 + F2 is
F ) 2γδlR(1/r1 + 1/r2)
(16)
To obtain the total normal force, F is integrated along a quarter of the circumference of the circular surface (with the radius of R) giving28 Figure 3. Infinitesimally small curved surface zone with a radius of R acted upon by surface tension γ. F is the normal component of the surface tensional force γδl acting on a contour. Adopted from ref 28.
The Gibbs free energy difference per unit volume ∆Gv can be described by18,19
Ftot ) πR2γK
(17)
The related normal pressure
∆p ) Ftot/πR2
(18)
exerted on a solid by the surface tension is thus
∆Gv ) -∆Sf ∆T
(10)
where ∆Sf is the entropy of fusion per unit volume. Since ∆Sf ) L/TM, eq 10 becomes
∆Gv ) -L ∆T/TM
(11)
and eq 9 yields
∆T ) -
γTM dA L dV
(12)
In the case of surfaces in three dimensions with the principal radii of curvature r1 and r2, the mean curvature K can be defined19 as
K ) (1/r1 + 1/r2) ) dA/dV
(13)
so that
∆T ) -
γTMK L
(14)
which is the Gibbs-Thomson equation in its general form. 3.3. Mechanical Derivation. For the mechanical derivation of the G-T equation at a solid/liquid interface, the surface energy γ needs to be interpreted as a surface tension acting parallel to the interface. This situation is analogous to a surface of a liquid, for which the surface tensional force is readily measurable. Such an analogy between the liquid, solid/liquid, and solid interfaces has been a subject of some criticism.27 However, the force interpretation of γ is quite appropriate also on an interface involving a solid because the adjustment of the sessile drop contact angle occurs by the horizontal motion along the solid/liquid and solid interfaces (Figure 1). According to the Newtonian definition, any motion may be interpreted as a consequence of a force. Thus, when a contour at a constant curvilinear distant R from a point on the solid/liquid interface is described (see Figure 3), any element δl of this contour is acted on (27) Bikerman, J. J. Top. Curr. Chem. 1978, 77, 1.
∆p ) γK
(19)
which is the Young-Laplace equation. Combining eq 19 with the Gibbs-Duhem equation (eq 5) results in the G-T equation (14). 4. Discussion The principal arguments presented against the use of the G-T equation in evaluating surface energies have been 1. The derivation of the G-T equation is for a fluid and is not valid for a solid surface.27 2. The existence of the G-T effect for solids requires that the stresses within the bulk are hydrostatic, so that a thermodynamic pressure used in its derivation can be defined for the solid.13 3. The conditions of the second argument result in the Gibbs-Wulff theorem,29 and therefore, the G-T equation is valid only for solid crystals whose shapes are “Wulffian”.13 Smoothly curved interfaces, such as a grain boundary groove, do not satisfy this requirement. In light of the classical derivation of the G-T equation (section 3.1), these arguments may appear appropriate. Clearly, the classical derivation assumes a spherical equilibrium shape and a fluid which is deformable without changing its surface structure. Also the pressure inside of a drop or a bubble assumed in the classical derivation is hydrostatic. However, when we address these same arguments in the light of the thermodynamic (section 3.2) and mechanical (section 3.3) derivations of the G-T equation, the situation is very different. The thermodynamic derivation points out that argument 1 against the G-T equation is unjustified, as no assumptions of fluidlike properties are required in this derivation. On the other hand, externally caused elastic strains may additionally be present on a solid surface, and it is obvious that the G-T equation should not be applied to surfaces under high elastic strain.30 There is no reason, however, to assume that grain boundary grooves or nucleating particles, which form freely by a phase change process, would involve such strains. On the contrary, crystal growth clearly represents (28) Dufour, L.; Defay, R. Thermodynamics of Clouds; Academic Press: New York, 1963; 255 pp. (29) Wulff, G. Z. Kristallogr. 1901, 34, 449 (30) Wu, C. H.; Chen, C.-H. Acta Mater. 1998, 46, 3755.
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the creation of a new surface and not the deformation of an existing surface. Argument 2 originates from the classical derivation of the Young-Laplace equation (eq 4) for a fluid, which applies the concept of pressure. The thermodynamic derivation given here, however, shows that it is not necessary to use pressure in order to obtain the G-T equation. Furthermore, the mechanical derivation points out an interpretation that the mechanical forces are exerted by an imaginary surface film and countered by the solid. This situation may be seen as a pressure at the capillary interface only and does not require an assumption of a hydrostatic pressure within the bulk of the solid. Note also that, unlike the classical derivation, the mechanical derivation implies no changes in the volume that would necessitate the assumption of a fluid. Argument 3 may now be accounted for in two ways. First, the thermodynamic derivation shows that the G-T equation is valid regardless of the shape. Second, the mechanical derivation shows that the G-T equation is valid for any infinitesimal surface area, that is, locally. Consequently, at equilibrium, the solid/liquid interface curvature K must everywhere be at the local equilibrium stated by the degree of undercooling ∆T according to eq 14. Thus, argument 3 claiming that the shape of a grain boundary groove in a temperature gradient cannot be described by the G-T equation is unjustified.
Makkonen
In summary, none of the arguments presented recently against the validity of the G-T equation on curved solid/ liquid interfaces appears to be valid. Therefore, the wide use of the G-T equation in studying capillary phenomena, as well as in determining the solid/liquid and solid/vapor surface energies, appears well justified. 5. Conclusions There is a severe discrepancy between the solid-liquid surface energies obtained by the Gibbs-Thomson equation and by the other relevant methods. The thermodynamic and mechanical derivations of the Gibbs-Thomson equation are considered in this paper. On the basis of them it is concluded that none of the recently presented arguments against the method of a boundary-groove curvature measurement or other experiments utilizing the GibbsThomson equation are valid. This, together with the abovementioned discrepancy, suggests that the surface energies based on the indirect methods, such as the Fowkes equation, the Lifshitz-van der Waals acid-base approach, the van Oss-Chaudhury-Good equation, and Neumann’s equation of state, are incorrect. LA990815P
