Curvature effects on the phase rule

Fluid Phase Equilibria,

98 (1994) 13-34

Curvature effects on the phase rule Dongqing

Li

Department of Mechanical Engineering, University of Alberta, Edmonton, Alta. TM; 2G8, Canada (Received October 4, 1993; accepted in final form January 14, 1994)

Abstract

Li, D., 1994. Curvature effects on the phase rule. Fluid Phase Equilibria, 98: 13-34. Counting the number of degrees of freedom for capillary systems is often not simple because of complications with the mechanical equilibrium constraints associated with curved liquid-fluid interfaces and curved three-phase contact lines. In this paper, the inapplicability of the Gibbs phase rule to capillary systems containing curved interfaces is discussed. For a general capillary system, the presence of curved interfaces results in fewer mechanical equilibrium constraints and hence more degrees of freedom, in comparison with composite bulk phase systems. A recently derived phase rule for moderately curved capillary systems is reviewed in this paper. Some experimental confirmation of the number of degrees of freedom predicted by such a phase rule is discussed. Furthermore, by counting properly the total number of variables required to describe the equilibrium state and the total number of equilibrium constraints, a phase rule for capillary systems containing highly curved surfaces and three-phase lines is derived. The comparisons show that there is not only a difference between the phase rule for capillary systems, but also a difference between the phase rule for moderately curved capillary systems and the phase rule for highly curved capillary systems. Such differences in the number of degrees of freedom result from the curvature effects of the interfaces. Keywords: Theory; Phase rule for capillary systems; Interfaces; Curvature

1. Introduction For bulk phase systems, the Gibbs phase rule is a well-known, fundamental component of classical thermodynamics. Such a phase rule predicts the number of thermodynamic degrees of freedom for a multicomponent, multi-bulk-phase system. The number of degrees of freedom indicates the number of necessary, independent, intensive variables required to describe the equilibrium state of a composite system. The phase rule, therefore, plays an important role, both theoretically and experimentally, in thermodynamics. A less well-known side of the phase rule is the phase rule for capillary systems in which the curved interfaces (i.e. bulk phase boundaries) and the curved intersection lines of bulk phases 0378-3812/94/%07.00 0 1994 - Elsevier Science B.V. All rights reserved SSDZ 0378-3812(94)02482-G

14

D. Li 1 Fluid Phase Equilibria 98 (1994) 13-34

are important. Past investigations by Crisp (1947, 1948), Defay (1934) and Defay and Prigogine (1966) have considered the effect of both planar and curved surfaces between heterogeneous bulk phases. A recent work by Li et al. ( 1989) derives a phase rule considering bulk phases, moderately curved surface phases, and moderately curved line-phases simultaneously. Generally, all the past studies were restricted to moderately curved interfaces or did not consider the distinction between a “moderately curved” capillary system and a “highly curved” capillary system. However, as the difference between planar, moderately curved, and highly curved interfaces determines the number of variables required to characterize an interface, and the forms of mechanical equilibrium conditions (and hence the number of mechanical equilibrium constraints), such a difference will affect the number of degrees of freedom of capillary systems. It is, therefore, the purpose of this paper to study not only the difference between the phase rule for bulk phase systems and the phase rule for capillary systems containing curved interfaces and curved three-phase contact lines, but also the high curvature effects on the number of degrees of freedom of capillary systems.

2. Variables required to characterize the equilibrium state of a simple, single-phase system Generally, the phase rule or the number of degrees of freedom varies from system to system. However, a basic rule to evaluate the number of degrees of freedom or variance of any composite system is to subtract the number of equilibrium constraint equations from the number of variables used to describe the composite system., It is worth while, therefore, to review variables required to characterize the equilibrium state of a simple, single-phase system or a subsystem in a composite system. Within the framework of Gibbsian thermodynamics, the basic postulate (Callen, 1960) is that the, equilibrium states of a given system can be completely characterized, at the macroscopic level, by an appropriate set of n independent extensive variables ~,,~2,..t,X

(1)

and that the fundamental equations, as functions of the IE independent extensive variables, contain all the thermodynamic information of the system. Usually, the energetic form of the fundamental equation gives the internal energy U of the system as a function of the IZ independent extensive variables, i.e. U=U(X,,X2,...,X,) The particular number n and the “appropriate set” (X, , X2, . . . , X,) for a system will depend on what the’system is and on “at what level” or “under what approximation” the system is going to be studied. _ Generally, a simple system is considered as g homogeneous, isotropic, chemically inert and uncharged system with no interactions with any external field. For a simple bulk phase system with r components, the equilibrium state is completely characterized by (r + 2) independent extensive variables, given by S,V,N,,N,,...,N,

D. Li 1 Fluid Phase Equilibria 98 (1994) 13-34

15

where S and V are the entropy and volume of the system, respectively, and Ni , N2, . . . , IV, are the mole numbers of the independent chemical components in this system. For this case, n is equal to (r + 2). The fundamental equation will be: U=U(S,V,N,,N,

,...,

N,)

(4) When the free energies associated with interfaces plays a significant role compared with the bulk-phase energy, such as in wetting and adhesion phenomena, the boundaries between bulk phases have to be taken into account. Physically, the boundary between two bulk phases is a heterogeneous region with a finite thickness. For most systems, the thickness of such a heterogeneous interfacial region is only about a few molecular diameters. Compared with its lateral dimensions, the thickness of the interfacial region is usually neglected. According to the Gibbs surface model (Gibbs, 1961), in the absence of external fields, two contiguous bulk phases which form an interface are considered to remain uniform right up to a mathematical surface, called the dividing surface. Excess quantities, which are differences between the model system and the real system with a non-homogeneous interfacial region, are assigned to the surface so as to make these two systems equivalent. Thus, while a bulk phase is three-dimensional, a surface phase is considered as a two-dimensional phase. The classical theory of capillarity as formulated by Gibbs (Gibbs, 1961) is concerned with the description of moderately curved capillary systems. Usually, for moderately curved surfaces, the radius of curvature ranges from micrometres to millimetres. Such a moderately curved simple surface phase is considered as a homogeneous, isotropic, chemically inert and uncharged two-dimensional phase with no significant curvature effects. Its equilibrium state can be completely characterized by the following (r + 2) independent extensive variables: SA, A, Nf, N$, . . . , N? The fundamental

(5)

equation for such a surface phase is

UA = UA(SA, A, Nf, . . . , Np?)

(6) where UA is the internal energy of the surface phase; SA and A are the entropy and surface area of the surface, respectively, Nf and Np are the mole numbers of the independent chemical components of the surface phase. Again, the number II for this case is (I + 2). When the moderate curvature approximation is not valid, i.e. when the thickness of the non-homogeneous interfacial region between two bulk phases is no longer negligible in comparison with the radii of curvature of the interface, it is necessary to consider an interface as a highly curved surface. For a highly curved surface phase, not only is the energy associated with the surface area important, but also the energies contributed by the curvature effects are significant. Therefore, additional new variables to characterize the curvature effects have to be introduced (Boruvka and Neumann, 1977). The two new variables, additional to the set of variables shown in Eq. (5), are J and K, the mean curvature and gaussian curvature, respectively. However, J and K generally may not be constant along the surface. In order to be consistent with our foregoing thermodynamic formalism, there have to be two corresponding properties representing the entire surface phase, these being extensive quantities as are SA, A and N?. Two such variables can be defined as J=

JdA

and

R=

KdA

16

D. Li 1 Fluid Phase Equilibria 98 (1994) 13-34

where the integrations are carried out over the whole surface area. The fundamental for this highly curved, simple surface phase is then UA = UA(SA, A, IA, RA, Nf, Nf, . . . , Nf)

equation (8)

Here the number of fundamental extensive variables, n, is (Y+ 4). By analogy with a two-dimensional surface phase, the three-bulk-phase contact line or the intersecting line of three interfaces is considered as a linear phase. A rigorous treatment of line phases within the framework of Gibbsian thermodynamics is given by_ a generalized theory of capillarity (Boruvka and Neumann, 1977). For a simple, moderately curved line phase, similar to the fundamental equations for bulk phases and moderately curved surface phases, the fundamental equation can be written as uL = UL(SL, L, Nf, N$, . . . , N;)

(9)

where UL and SL are the internal energy and the entropy of the line phase, L is the length of theline,andNF(i=1,2 ,..., r) is the mole number of the ith component in the line phase. For such a simple, moderately curved linear phase, the number of fundamental extensive variables is (r + 2). However, like highly curved surface phases, a highly curved line phase requires more new variables in its fundamental equation. Discussions of the highly curved line phases can be found elsewhere (Boruvka and Neumann, 1977). Generally, the intensive properties of a system are defined as the derivatives of the fundamental equation, as Eq. (2), with respect to the fundamental variables, as in Eq. (l), i.e.

au

yi=

(ax,>q+i

Temperature

(10)

i=l,2,...,n

and chemical potentials are defined as -- 8UU pi - aN;

i=l,2,...,r

(11)

where a = B (bulk phase), A (surface phase) and L (line phase). For a simple bulk phase, the pressure is defined as i3UB

(12)

p=-w

By analogy with the bulk phase, the surface tension of a simple surface phase is defined as follows:

au-4 I)= aA

(13)

The surface tension is a mechanical property for a surface phase as is the pressure for a bulk phase. For a highly curved surface phase, there are two additional intensive properties corresponding to the curvature variables, C

=au_A 1aJ

_ auA C'=aa

(14)

D. Li 1 Fluid Phase Equilibria 98 (1994) 13-34

17

c, and C2 are the first surface curvature potential and the second surface curvature potential, respectively. For a moderately curved linear phase, a mechanical property analogous to surface tension for a two-dimensional surface phase or pressure for a three-dimensional bulk phase is called line tension, defined as WL O=aL

(1%

The thermodynamic equilibrium conditions of a system, including thermal equilibrium conditions, chemical equilibrium conditions and mechanical equilibrium conditions, generally consist of all the intensive variables Yi (i = 1,2, . . . , n). As long as all the intensive variables Yi in the equilibrium conditions are specified, the equilibrium conditions, hence the equilibrium state, will be determined. From this one might conclude that describing the equilibrium state of a simple, single-phase system requires n intensive variables Yi (i = 1,2, . . . , n). However, these n intensive variables are not all independent. Instead, they are related by the Gibbs-Duhem equation: i

xdY,=O

i=l

(16)

Equation (16) implies that one of the Yi (i = 1,2, . . . , n) will be a function of the others. The equilibrium state, therefore, can be determined by (n - 1) independent intensive variables. However, to describe the equilibrium states, one is not restricted to these intensive variables n). The specific extensive variables are also intensive variables because their Yi(i=1,2,..., values no longer depend on the mass, the size or the “extent” of the system. Such (n - 1) specific extensive variables (17) will be sufficient to determine the equilibrium states. Thus, because the thermodynamic equilibrium conditions are completely determined by intensive variables, any combinations of (n - 1) independent intensive variables chosen from the set of Y1, Y2, . . . , Y, and the set of states. Here “independent” implies that x1,x2,. * * 9 x, can be used to describe the equilibrium one cannot have both Yi and xi in the same set of (n - 1) intensive variables, because Yi and Xi are related by the definition of Yi.

3. Inapplicability of Gibbs phase rule to capillary systems The Gibbs phase rule is a well-known relation that is employed to determine the number of thermodynamic degrees of freedom, or the number of independent intensive variables for a composite bulk phase system. Let us consider a multicomponent, multiphase system in equilibrium. If there are a total of n bulk or volume phases and each phase has r independent chemical components, then in order to describe the state of a particular phase o! for this heterogeneous system, one would require (r + 1) independent variables, T”, Pa, where x; for (i = 1,2, . . . , r - 1) is the bulk mole fraction of the ith xf, x;, . . . , x;_1,

18

D. Li 1 Fluid Phase Equilibria 98 (1994) 13-34

component in phase a. Consequently, if one enumerates all n phases the required number of intensive variables will be n(r + 1). Furthermore, since these co-existing phases are all in equilibrium the intensive variables are constrained to satisfy the following conditions: Thermal equilibrium conditions: T* = 7-B= . . . = T”

(n -

1) equations

(18)

Mechanical equilibrium conditions: p”=PS=.

. . =p”

(n -

1) equations

(19)

Chemical equilibrium conditions: p; = j$ = * . . = py where i=l,2

,...,

r(n -

1) equations

(20)

r.

These equilibrium conditions, given in Eqs. (18)-(20), give a total of (r + 2)(n - 1) constraint amongst the n(r + 1) intensive variables; therefore, the number of independent intensive variables or the number of degrees of freedom, f, is f=(r+l)n-((r+2)(n-l)=r+2-n

(21)

Relation (21) represents the standard Gibbs phase rule. It simply states that in a system with r independent chemical components and n co-existing phases, the maximum number of independent intensive variables that may be used to describe the state of the system is J To avoid the possibility of any misunderstanding as to when Eq. (21) is applicable, the conditions that the system must satisfy for Eq. (21) to hold are summarized below (Kirkwood and Oppenheim, 1961; Miinster, 1970): (1) Boundary surface effects are neglected and all boundaries between bulk phases are thermally conducting, deformable and permeable to all components. (2) Volume is the only work coordinate, i.e. P dv work is the only mode of work. (3) No chemical reactions. Therefore, as capillary systems do not satisfy all of these conditions the classical Gibbs phase rule expressed by Eq. (21) is not applicable. This fact is readily illustrated by considering a one-component capillary system consisting of a liquid droplet surrounded by its vapour (Fig. l(a)). In this case, there are two bulk phases and one liquid-vapour interface, but only one component. That is, n = 3 while r = 1. According to Eq. (21) the number of degrees of freedom fwould be zero. This is obviously unreasonable since it is well known from experimental work that both temperature T and pressure P can be chosen as independent variables in open systems; thus the number of degrees of freedom should be two. As another example, one may consider a sessile drop system (see Fig. l(b)). In this case the capillary system contains three bulk phases (i.e. solid, liquid and the liquid’s vapour), and three surface phases (i.e. solid-liquid, solid-vapour and liquid-vapour), so that n = 6; there are two components (i.e. solid and liquid), so r = 2. Using Eq. (21) the number of degrees of freedom for this system would be -2. This is unacceptable as it is well known that the number of degrees of freedom cannot be negative.

D. Li / Fluid Phase Equilibria 98 (1994) 13-34

19 Fluid 1

SOW

(a)

@)

.

w

6.9

Fig. 1. Examples of common capillary systems.

One may understand the reason why Eq. (21) is not generally applicable to capillary systems by considering the first example in more detail. For this arrangement the number of degrees of freedom was calculated to be zero. This is obviously incorrect as seen by comparing the number of intensive variables required to describe the state of this liquid-vapour system with the number of equilibrium constraint equations. The complete set of intensive variables consists of the temperatures, pressures and surface tension of the interface which are denoted by T’, P’, T”, P’, T” and y I”, where the superscript “1” represents the liquid phase and the superscript “v” represents the vapour phase. The corresponding equilibrium condition are Thermal equilibrium:

T’ = T” = T”

Chemical equilibrium: p’ = p’ = #’ Mechanical equilibrium: P’ - P’ = y”J”

CW

where J” is the mean curvature of the liquid-vapour interface and Eq. (22a) is the classical Laplace equation of capillarity. The key point is that Eq. (22a) does not function as a constraint relation among the variables listed above since J” is not a member of the set of intensive variables required for the moderately curved surface considered above. Alternatively, it is obvious that P’, P’ and y” may all be freely assigned; J” will then acquire a value such that Eq. (22a) is satisfied. Therefore, the above equations provide only four constraint equations among our six intensive variables. Therefore, there are two degrees of freedom as stated above. Equation (21) does not provide the correct result since it fails to account properly for the reduction in the number of mechanical equilibrium constraint equation(s). According to Eq. ( 19), which was used to derive the phase rule given by relation (21), this one-component, three-phase system should exhibit (n - 1) = 3 - 1 = 2 mechanical constraints. Physically, these constraint equations demand that the pressures in all three phases are equal. However, for a curved surface the pressures cannot be equal in all phases. The appropriate mechanical equilibrium conditions are given by relations of the type shown in Eq. (22a) and not by those shown in Eq. (19). Equation (22a) does not provide a constraint on the intensive variables (i.e. the pressures and surface tension). Thus, a reduction in the number of constraint conditions is necessary whenever a curved surface separates two bulk phases. This is the case for our one-component liquid droplet in equilibrium with its vapour.

D. Li 1 Fluid Phase Equilibria 98 (1994) 13-34

20

It is worth while to emphasize that, whenever using the Laplace equation, Eq. (22a), an additional degree of freedom is operative. For example, for pure water at 1 atm, the curvature or Laplace pressure (i.e. the pressure difference given by the Laplace equation) causes the boiling temperature of a bubble of one micrometre in radius to be 121°C (Lupis, 1983). In comparison with the boiling temperature in the case of a flat surface (i.e. lOO’C), this is a 21% increase in the boiling temperature of water, due to the interfacial curvature effects. As another example, consider several capillary tubes of identical radii but different materials vertically dipping into a bulk liquid. Since the contact angles of the liquid on the surfaces of these tubes are different, the curvatures of the three menisci will be different; hence the Laplace pressures are different so that there can be large differences in capillary rise between these tubes. Such a phenomenon can be observed for the cases where the radii of the capillary tubes are of the order of millimetres. This illustrates the significance of the additional degrees of freedom for systems containing moderately curved interfaces with radius of curvature of the order of millimetres. Further, one may consider a capillary of radius infinity, i.e. a meniscus at a planar vertical wall. For water at room temperature the capillary rise at the wall will be 3.85 mm for zero contact angle, 2.08 mm for a 45” contact angle, and zero for a contact angle of 90”. For contact angles above 90”, there are corresponding capillary depressions. The cause for three different states is simply different Laplace pressures. The additional degree of freedom due to Laplace pressure, therefore, plays a significant role in determination of the equilibrium state of contact angle or wetting systems. For a radius of curvature in the milimetre range, the shape of drops and other menisci is determined by a balance between hydrostatic pressure and the Laplace pressure. As the effect of the hydrostatic pressure is well known, the experimental menisci and shapes of drops provide a means for the determination of interfacial tensions and contact angles (Adamson, 1982; Cheng et al., 1990; Li et al., 1992). 4. Phase rule for moderately curved surface systems For moderately curved capillary systems, the governing boundary conditions or mechanical equilibrium conditions are the Laplace equation and Young’s equation; the Laplace equation may be written (in a more general notation than Eq. (22a)) as p* _ pfl=

r”8J”8

( 22b)

where y@ is the surface tension of the surface phase that is sandwiched between bulk phases a and j?, and Jab is the mean curvature of that surface. Along the edge of the surface phases, where three surfaces meet to form region or contact line, the boundary condition is given by the classical Neumann triangle relation if all phases are deformable: 2py*3

cos 8 = (y12)2 - (y=)*

- (y13)2

(23)

where the phases are defined as in Fig. l(d) and 0 is the angle within phase 3 between the l-3 and 2-3 interfaces. The mechanical equilibrium condition of the three-phase contact line is given by the classical Young equation, if one phase is a rigid solid: JPcose

=yS”-ysl

(24)

21

D. Li 1 Fluid Phase Equilibria 98 (1994) 13-34

where y” and ys’ represent the surface free energies of the solid-vapour interface and solid-liquid interface, respectively, and 0 is the contact angle between the liquid-vapour and the solid-liquid interfaces. Recently, a phase rule for capillary systems containing moderately curved interfaces has been derived (Li et al., 1989). The basic approach is summarized as follows. Consider a system of r independent chemical components and n phases, of which n(*) are contiguous, single-phase surface phases. For these types of composite system the independent, intensive parameters of the system are given completely by

T',...,T";P’, . . . , Pn-“‘A’; xi, . . . , x::fA’; y’, . . . , Yn(A).

9

](A)

x1

,...,x,-1

,,(A)

(25)

where T” represents the temperature of the nth phase, PnmncA’represents the pressure for the (n - n(*))th bulk phase, xl -n(A)(i = 1, . . . , r - 1) represents the bulk phase mole fractions, yncA) represents the surface tension of the n(*)th surface phase, and ~1~~)(i = 1, . . . , r - 1) represents the surface phase mole fractions of the n (*)th surface phase. There are n intensive parameters that characterize the temperatures, (n - n(*)) parameters for the pressures, (r - l)(n - n(*)) parameters for the bulk phase mole fractions, n(*) parameters for the surface tensions, and (r - l)n(*) parameters for the surface phase mole fractions. Thus, this composite system has n(r + 1) intensive variables. The set of intensive parameters given above is not unique and other equally valid choices are available. For example, Defay (Defay, 1934) used T’, . . . , T”; c;, . . . , c;-“(~); T;(A), . . . r:(*)

Wa)

where T” is as defined above, cl -n(A) is the molar concentration of the ith component in the (n - n(*))th bulk phase and Ii(A) 1s * the equivalent molar concentration of the ith component in the n(*)th surface phase. Once again, the total number of intensive variables is equal to n(r + l), as expected. This set was arrived at by considering initially a set of variables which contained, in addition to the variables in Eq. (25a), the mean curvature JncA)as an independent parameter. After employing the classical Laplace equation (22b) as a constraint condition they concluded that the set of variables shown in Eq. (25a) was sufficient to characterize completely any composite capillary system with surfaces, and that the mean curvature was not independent of the other parameters listed in Eq. (25a). Generally, for such a moderately curved capillary systems (a number of common systems are illustrated in Fig. 1) the total number of intensive variables is still n(r + l), as shown in Eq. (25). The thermal and chemical equilibrium conditions given by Eqs. (18) and (20) above yield the same number of constraint relations as previously used to derive Eq. (21). However, for moderately curved capillary systems the mechanical equilibrium conditions ( 19) no longer apply between all phases. In place of pressure equality relations of the form shown in Eq. (19) one must insert the appropriate mechanical equilibrium conditions, as given by the Laplace equation (22b), the classical Neumann triangle relation (23) and the classical Young equation (24) for each curved surface in the capillary system. The Laplace equation (22b) represents a relation between the independent intensive variables (see Eq. (25)), i.e. Pa, Ps and y”p, on the one hand, and a new variable, J”@, on the other. Through Eq. (22), the latter quantity is completely determined by the former three, or in other words, Eq. (22b) can always be satisfied by adjusting the value of Jab while the values of the pressures and the surface tension in Eq. (22b) can be

22

D. Li / Fluid Phase Equilibria 98 (1994) 13-34

assigned freely. Therefore, it can be concluded that Eq. (22b) does not represent a constraint among P”, PB and y “fl. Similarly, the classical Young equation and the classical Neumann triangle relation also contain a new variable, the contact angle 8, in addition to the surface tensions which are members of the original set of intensive variables given in Eq. (25). Thus, neither the classical Young equation nor the classical Neumann triangle relation will provide any restrictions or constraints on the surface tensions. Therefore, among the mechanical equilibrium conditions for the moderately curved surface systems, it is only relations of the form Pa = PB that provide constraints. Consequently, for capillary systems which possess non-planar, moderately curved surfaces the number of mechanical equilibrium constraint equations is less than (n - I), the number which arises from the equality of the pressures as shown in Eq. ( 19). Now, defining N as the total number of distinct P” = Ps type relations among the mechanical equilibrium conditions, then for a capillary system with r chemical components and n phases, the total number of constraint equations in equilibrium is given by (n - 1) Thermal equilibrium

+

r(n - 1) Chemical equilibrium

+

N Mechanical equilibrium

Remembering that the number of intensive variables in the system is n(r + l), one obtains the number of degrees of freedom f as f=n(r+l)-[(n-l)+r(n-l)+N]=r+l-N

(26) where Y is the number of independent chemical components or species in this system, and N is the number of P” = Ps pressure equality relations among the mechanical equilibrium conditions. In this fashion one sees that the number of mechanical equilibrium conditions, of the type shown in Eq. ( 19), is reduced to N from a total number of (n - n (*) - 1). Consequently, the number of degrees of freedom is increased by (n(*) - N). The above analysis was also extended to consider the three-phase contact line phase. By analogy with two-dimensional surface phases, the three-phase contact line is considered as a one-dimensional linear phase formed by the intersection of three surfaces, with line tension CJas the mechanical property corresponding to surface tension in the two-dimensional surface phase. By analogy with both the bulk and surface phases let us define the corresponding intensive variables for a linear phase as T"bV

&V

9x1

@V

7x2

@V

,.*.,x,-1

@Y

(27)

whe’re aapy represents the line tension at the three-phase contact line boundary between the CI,/_I and y phases, and x ;Byrepresent the mole fractions in the line phase. Consider a system of three bulk phases (a, p and y), three moderately curved surface phases (a - p, a - y and /I - y) and one moderately curved linear phase (a - p - y). For this system there are a total of 7(r + 1) intensive variables used to defme all bulk, surface and line phases. If these seven co-existing phases are in equilibrium, then the corresponding thermal and chemical equilibrium conditions for the system will be given by ~=Tfl=T'=~!=-V=TflV='=~Y

and

D. Li 1 Fluid Phase Equilibria 98 (1994) 13-34

where i = 1,2, . . . , r. Coupled equilibrium conditions:

with these conditions

23

will be several kinds of mechanical

(1) For planar surfaces p”=pS

(2) For curved surfaces p” _ PB = y”flJ”b (3) For a liquid on a solid substrate, the corrected Young equation (Boruvka and Neumann, 1977) yaficos 8 + fJ%$, = yW- yfiy

(28)

where lcBsis the geodesic curvature of the three-phase contact line in the plane of the solid or y phase. (4) For a liquid lens between two bulk fluid phases, the corrected Neumann triangle relation (Boruvka and Neumann, 1977)

where mj is a unit vector tangent to the adjacent curved surface and normal to the three-phase line, and K is the curvature vector of the three-phase line. Like the Laplace equation (22b), and the classical Young equation (24), the corrected Young equation (28) also does not present a constraint, since, in addition to the contact angle 19,there is a second quantity, the geodesic curvature Key, which is also not a member of the set of intensive parameters in Eq. (27). Thus, (28) provides even greater flexibility and hence is not a constraint equation for the surface tensions and the line tension. Similar arguments apply to the corrected Neumann triangle relation (29). Again, only relations of the form Pa = Ps provide restrictions or constraints on the number of degrees of freedom since the pressures in adjacent bulk phases lose their ability to be freely assigned. Therefore, the above mentioned thermal, chemical and mechanical equilibrium conditions give 6(r + 1) + N constraint relations, where N is the number of P” = Ps type relations. Thus, comparing the number of variables and the number of constraint equations, the number of degrees of freedom for the above system is f=7(r+l)-6(r+l)-N=r+l-N It is interesting to note that this equation is exactly the same as Eq. (26), the, phase rule for moderately curved capillary systems without consideration of the linear phase. This result becomes plausable when one realizes that the presence of the linear phase does not increase or decrease the number of degrees of freedom since the intensive thermal and chemical variables of the line phase introduced in Eq. (27) are constrained by the thermal and chemical equality relations; and the mechanical equilibrium conditions involving the line tension do not increase or decrease the number of mechanical equilibrium constraints.

24

D. Li 1 Fluid Phase Equilibria 98 (1994) 13-34

5. Application of the phase rule for moderately curved surface systems

The phase rule for capillary systems is of particular interest in the studies of interfacial phenomena such as wetting and adhesion. For example, knowledge of the number of degrees of freedom as given by Eq. (26) provides a theoretical justification for functional relationships between interfacial free energies or interfacial tensions. Of course, the precise form of the relationship only becomes apparent with suitable experimental measurements. It will also become apparent later that, in this respect, there is a considerable difference between solid-liquid-vapour and liquid-liquid-vapour systems. 5.1. Surface systems with curved fluid interfaces Consider a capillary system consisting of three bulk fluid phases of different densities as shown in Fig. Id. Along with the three bulk phases there are also three surface phases (i.e. those between fluid l-fluid 3, fluid 1-fluid 2 and fluid 2-fluid 3). If all the surfaces are curved the surface mechanical equilibrium conditions will be P3 _ P’ = $1 J31

~3 _ ~2 = ?32~32

p2 _ p’

= y’2J’2

There are no pressure equality relations of the type P” = PB since all the interfaces are curved. As a consequence, N = 0, and the phase rule stated in Eq. (26) yields f=r+l

(30)

A special case of the liquid lens system illustrated above occurs when the interface between fluids 1 and 2 is planar (see Fig. Id). When this occurs J l2 = 0 and the mechanical equilibrium condition given above is replaced by the pressure equality relation P2 = P’. For this case, iV = 1. Applying the phase rule (26) gives f=r

(31)

For a two-component system, with r = 2, Eq. (30) will give the number of degrees of freedom as f = 3 while Eq. (31) will yield the result f = 2. These results are the same as those derived by Defay and Prigogine (1966) using a different approach. 5.2. Surface systems with a planar solid substrate Consider a sessile drop system (a two-component, three-phase system) as shown in Fig. l(b) which has three bulk phases (i.e. solid, liquid and the liquid’s vapour) and three surface phases (i.e. solid-vapour, solid-liquid and liquid-vapour). Under the simplifying assumption that the solid phase is isotropic such that uniaxial normal stresses and tangential stresses are zero one may characterize its bulk mechanical state by a single hydrostatic pressure P” (Miinster, 1970; McLellan, 1980). This simplifying approximation will prove suitable for our present purpose of demonstrating that the number of degrees of freedom of a system with an ideal solid as outlined above is not the same, in general, as the number of degrees of freedom of a liquid-liquid lens-fluid system.

D. Li / Fluid Phase Equilibria 98 (1994) 13-34

In order number N equilibrium conditions p"=p'

for us to apply the phase rule for capillary systems, Eq. (26), one must determine the of pressure equality relations of the form P” = Ps which exist among the mechanical conditions. For the capillary system shown in Fig. l(b) the mechanical equilibrium are given by (Li et al., 1989) p'-p'=

yl' J'v

yl"cos fg= yS"-yS~

Therefore only one pressure equality relation exists, so N = 1. Accordingly, gives the number of degrees of freedom of this system as

f=r

25

the phase rule (26)

(32)

Realizing that this system has just two components (i.e. solid and liquid) one can readily see that the number of degrees of freedom is just two. That is, for a two-component, solid-liquid-vapour capillary system as illustrated in Fig. l(b) or Fig. l(c), there are only two degrees of freedom. Consequently, this result means that any two of the original set of intensive variables used to describe this system are sufficient to characterize the system completely. Alternatively, any two intensive variables may be chosen as the independent variables and the remaining intensive variables can be expressed as functions thereof. Thus, if one chooses y”’ and y” as the two independent variables then one may express y’i, the remaining surface free energy, as a function of the other two surface free energies in the following functional form: (33) Equation (33) is referred to as an equation of state for interfacial tensions (Neumann et al., 1974; Li and Neumann, 1992, a, b). Explicit forms of such an equation of state can be found elsewhere (Neumann, et al., 1974; Li and Neumann, 1992a, b). Comparing the situation of the solid-liquid-vapour system with that of a liquid-liquid lens-fluid system one sees that while the number of degrees of freedom of a two-component solid-liquid-vapour system is two, the number of degrees of freedom for a two-component liquid-liquid lens-fluid system is three. This result implies that a relationship of the form y23 = ~~~(y’~,y13) generally does not exist for liquid-liquid lens-fluid systems. At this point, let us consider experimental data. Figure 2 reproduces the results of contact angle measurements performed with three very smooth and homogeneous solid surfaces and a range of pure liquids (Li and Neumann, 1992b). The contact angle measurements were performed by means of axisymmetric drop shape analysis (Cheng et al., 1990; Li et al., 1992); the accuracy of such measurement is of the order of 0.1”. The key point to note is that for each of the three solids, the data points for the different liquids fall on a smooth curve. From Fig. 2, one can see that, as one goes from the most hydrophobic surface, the FC721, to the most hydrophilic one, the PET surface, one shifts the curves. This shift is consistent with the idea of different surface free energies, y sv,of the three surfaces. Changing the liquid surface tension continuously by changing liquids from relatively low surface tension to higher surface tension causes the data points to move along the three smooth curves. It is quite clear a priori and certainly born out by Fig. 2 that one can change the contact angle and, because of Young’s equation. Eq. (24), the solid-liquid interfacial free energy by changing the liquid-

D. Li /Fluid Phase Equilibria 98 (1994) 13-34

26

a-FC721

40-

-20 -

-40 20

I 30

I 40

’

.

I.,

50

60

, 70

1 80

Y’” (tin& Fig. 2. ylv cos 0 vs. y’” curves for FC721, FEP and PET surfaces.

vapour and the solid-vapor interfacial free energies through the simple mechanism of changing liquid and solid. If there was only one degree, this most obvious procedure would not be possible. Further, as seen from Fig. 2, each of these curves may be represented by a correlation y” cos 8 =f(y’“, r”“) where y”’ is constant for a given curve (i.e. a given solid surface), as there is no appreciable vapour adsorption on the hydrophobic solid surfaces. Combining this equation with the Young equation, Eq. (24), gives yS”- ys’ =f(y’“, y”) Rearranging

this equation, one has

y” = ysv _-f(y’“, y”) = ySl(yi”,7”) That is, the experimental capillary systems.

data indeed confirm Eq. (33), the prediction

of the phase rule for

6. Phase rule for highly curved surface systems In the present section, high curvature effects on the number of degrees of freedom will be analysed. Without loss of generality, let us consider a capillary system consisting of three bulk phases, three surface or interface phases and a three-phase line phase. The curvature effects are significant for some of the surface phases and the line phase. To count the number of degrees of freedom for such an r-component system with highly curved surfaces and three-phase lines,

27

D. Li 1 Fluid Phase Equilibria 98 (1994) 13-34

one first needs to find the necessary number of intensive variables to describe the equilibrium states of each of the phases in this system. As mentioned before, for highly curved surfaces, in order to consider the high curvature effects, two new variables have to be introduced. The two additional variables reflecting the high curvature effects on the surface phase are the mean curvature

and the gaussian curvature

where RI and R2 are the principal radii of curvature. A fundamental equation describing the entire highly curved surface is given in the previous section section (see Eqs. (7) and (8)). However, J and K generally are not constant along the surface. For highly curved surfaces, the dependence of surface properties on local curvature needs to be considered. This is a different feature of highly curved surface systems in comparison with moderately curved surface systems. Therefore, in order to consider this fact, a more general and appropriate approach to characterize a highly curved surface phase is to model a local surface area element dA. Consequently, such a hghly curved surface phase can be described in terms of a density form of fundamental equation with respect to a surface area element dA. The fundamental equation for such a surface area element dA of a highly curved surface phase, a/I, in the form of internal energy density (Boruvka and Neumann, 1977; Gaydos and Neumann, 1992) can be written as .‘fl = uafl(sa~,J@, K”, ~78, . . . , p;q Where ,S@ is the entropy density concentration of ith component in the chemical potentials, pi (i = 1,2, (or bending moments) are defined 1992)

(34) of the surface a@, and pyb (i = 1,2, . . . , r) is the molar the surface phase a/3. In addition to the temperature T and . . . , r), the first and the second surface curvature potentials as (Boruvka and Neumann, 1977; Gaydos and Neumann,

(35) The surface tension is defined by (Boruvka and Neumann,

1977; Gaydos and Neumann,

1992)

The above gives a total (r + 4) intensive variables. Since they are related by the Gibbs-Duhem equation (37)

D. Li / Fluid Phase Equilibria 98 (1994) 13-34

28

only (r + 3) of them are independent. One can choose the (r + 3) independent intensive variables to describe the equilibrium state of this highly curved surface phase as T, yaB,Cyp, C& pfS, . . . , /I;!_,

(38)

By analogy with the highly curved surface phase, the fundamental equation for a highly curved three-phase line phase ~$7 in the density form is given by (Boruvka and Neumann, 1977; Gaydos and Neumann, 1992) U@Y = u=~y(s@y, {@}, {e,>, {Q, ‘cgi,zgi>,

(39)

where s@yand pyPyare the entropy density and the molar concentration of ith component of the linear phase C&J. The contact angles {Q> (ij = c$, fir, ~a) and the curvatures {Knj,K~, zgi} (j = IX,/I, y) are the two additional sets of geometric variables required to describe the equilibrium states of a highly curved three-phase line phase. 8, is the angle formed by the ith and jth interfaces at the point at which they intersect to form the line. Kaj, ugi and rgi are the normal curvature, the geodesic curvature and the geodesic torsion of the three-phase line at each point along the length of the line, relative to the jth interface. In addition to the temperature T and the chemical potentials pi (i = 1,2, . . . , r), the linear curvature potentials are defined as

au @Y

co*,= ae.

, ij =

cnj =

HAPY,w

I/

cgi

auJ = @Y

ale

a9 B,Y

nJ

2fg ,j =a, B,Y gr

c,

(9

2g ,.i=a, P,Y gr

The line tension is defined by (Boruvka and Neumann,

1977; Gaydos and Neumann,

1992)

However, they are not all independent. The above yields (r + 14) intensive ‘variables. First, the fundamental variables 0, and Knj,lcgi are related, respectively, by the geometric constraints 1 BiJ= 2n ii =@J,~~Y,Y~ rc&

+

u&

=

u$

(42) +

u$

=

rcEy

+

u&

=

(43)

u2

Obviously, the relations (42) and (43) impose four constraints on Co,, and C’nj,C,, respectively. Secondly, the Gibbs-Duhem equation s”’ dT + do + C pgsy dpi + C 8, dC,,. + C [unj dC,j +

UH

dCa + rgi dCrj] = 0

(44)

will give another constraint among the intensive variables. Therefore, only (r + 9) intensive variables are independent. One may choose these (r + 9) independent intensive variables at T, c, CBoIB, CBsy,C,,, C,,, C,, C,,, C+, Cy, PY? . . . , ~9%

(45)

D. Li 1 Fluid Phase Equilibria 98 (1994) 13-34

29

Since a planar or moderately curved surface does not require the two curvature potentials in Eq. (38), the number of intensive variables required for a planar or moderately curved surface phase is (r + l), the same as that for a bulk phase. Therefore, for convenience, in the following, let us define S,, as the number of planar and*moderately curved surface phases, and define S as the numbers of highly curved surface phases. Thus, for the system consisting of three bulk phases, S, planar or moderately curved interface phase(s), S highly curved interface phase(s) and one highly curved line phase, the total number of required intensive variables for this system will be (3 + &-Jr + 1) + S(r + 3) + (Y+ 9) When the multiphase system under our consideration is in a thermodynamic equilibrium state, the equilibrium conditions will bring constraints on these variables so that they will not all be independent of each other. Generally, the thermal and chemical equilibrium conditions, given by Eqs. (18) and (20), yield (n - 1) and r(n - 1) constraint equations, respectively. The mechanical equilibrium conditions usually include the following types: For two bulk phases separated by a planar interface, the mechanical equilibrium condition is the pressure equality. For two bulk phases separated by a moderately curved surface, the mechanical equilibrium condition is the classical Laplace equation. For two bulk phases separated by a highly curved interface a/I with constant yap, CyP and C$p, the mechanical equilibrium condition is the generalized Laplace equation (Boruvka and Neumann, 1977; Gaydos and Neumann, 1992). J+"fl

+ 2K”bC;fl

= p” _ pfl

(46)

For a highly curved liquid-liquid-fluid three-phase line, the mechanical equilibrium conditions is the Neumann triangle relation. The most general Neumann triangle relation is extremely complex and can be seen elsewhere (Boruvka and Neumann, 1977; Gaydos and Neumann, 1992). If the line tension d and these linear mechanical potentials Ceil, Cnj, C,, C, are constant along the three-phase line, then, the generalized Neumann triangle relation reduces to (Boruvka and Neumann, 1977; Gaydos and Neumann, 1992)

[

1

d + C Gije, IZ= C [Yj+ CcOi,, - G,~JKnjl~j

(47)

Where R is the curvature vector of the three-phase line and &j is a unit vector tangent to the adjacent curved surface and normal to the three-phase line. For a solid-liquid-fluid three-phase line, the mechanical equilibrium condition is the generalized Young equation (Boruvka and Neumann, 1977; Gaydos and Neumann, 1992)

(48) However, the above Laplace equations. Young equations and the Neumann triangle equation generally do not provide any constraints on the intensive variables in these equations, unless the curvatures and the contact angles in these equations are pre-fixed. Only P" = Pp type relations

D. Li 1 Fluid Phase Equilibria 98 (1994) 13-34

30

in the mechanical equilibrium conditions act as the constraints on pressures between bulk phases. Therefore, if one assumes N is the number of P’ = Pp type relations in the mechanical equilibrium~conditions, the total number of the constraint equations for such a system with n (here n = 3 + S,, + S + 1) phases will be [(3+S,+S+l)-1}+r[(3+&+S+l)-l]+N Thermal Chemical equilibrium equilibrium conditions conditions

mechanical equilibrium conditions

Then, it follows that the number of degrees of freedom of the above system containing highly curved surfaces and a highly curved three-phase line will be

f = (3 + &)(r + 1) + S(r + 3) + (r + 9) -([3+S,,+s+

l] - 1) -r([3+S,+S+

13 - 1) -N=2S+8+r

+ 1 -N

(49)

where s is the number of highly curved interface phases, r is the number of chemical components in this system and N is the number of P = Pfi type relations among the mechanical equilibrium conditions. If one compares the phase rule for highly curved capillary systems, Eq. (49), with the phase rule for moderately curved capillary systems, Eq. (26), ,it will be noted that the phase rule (49) has two additional terms, 2s and 8. In fact, the two numbers, 2 and 8, are just the additional independent variables required to count the high curvature effects for a surface phase and a line phase, respectively. If one considers that the system does not contain any highly curved surfaces and a highly curved three-phase line, and has only moderately curved surfaces, then it can be expected that 2s and 8 will drop out of Eq. (49); hence, the phase rule (49) is reduced to the phase rule for moderately curved capillary systems, Eq. (26). Similarly, in the case of no highly curved line phase, the number 8 should not appear, and the number of degrees of freedom is f=2S+r+l-N

(50)

As an example, consider a droplet in equilibrium with its vapour phase. If the drop-vapour interface is moderately curved (e.g. a drop with a radius of one millimetre), the phase rule for moderately curved systems,applies. Since, for the case, the mechanical equilibrium condition is the classical Laplace equation of capillarity, N = 0. For such a one-component system, r = 1. Equation (26) predicts that the number of degrees of freedom of this system is two. The two independent ‘intensive variables to characterize such a system may be the temperature and the vapour phase pressure. Now, let us consider a very small droplet in equilibrium with its vapour phase. For this sytem, there are only two bulk phases and one highly curved interface phase, and no three-phase line, so S = 1. Again, there is no pressure equality relation between the two bulk phases and N = 0. The phase rule for highly curved capillary systems, Eq. (50) in this case predicts f=2+r+l=r+3

(51)

D. Li / Fluid Phase Equilibria 98 (1994) 13-34

31

Since for this one-component drop-vapour system, r = 1, Eq. (51) gives f = 4, i.e. the number of degrees of freedom is four. Comparing this result with that of the moderate curvature case, the additional two degrees of freedom reflect the high curvature effects on thermodynamic equilibrium. The physical or experimental implications of such a difference in the number of degrees of freedom require further study. However, it should be realized that applicability of the phase rule to high-curvature surface systems is limited by the validity of the thermodynamic approach, i.e. the validity of macroscopic concepts of density, surface tension, radius of curvature, etc. Overall, it should be noted that, as elaborated in this paper, the curvature effects require more variables on the one hand, and do not provide any new constraint relations on the other. This is the cause of the differences in the number of degrees of freedom between the moderately curved surface systems and the bulk phase systems, and between the highly curved surface systems and the moderately curved surface systems. For example, if specifying the temperature, composition, interfacial tension and curvature potentials could determine the mean and the gaussian curvatures, then a high-curvature surface would not result in any more independent variables and hence more degrees of freedom than a low curvature surface.

7. Summary The classical Gibbs phase rule for bulk phase systems in the form off = r + 2 - m is not applicable to capillary systems. This is because the above form of the phase rule does not consider the effects of curved interfaces. Owing to the presence of curved interfaces, the number of mechanical equilibrium constraints is smaller in a capillary system, and hence the number of degrees of freedom is larger than that of a bulk phase system. Generally, capillary systems can be divided into two categories: highly curved capillary systems and moderately curved capillary systems. Because of the difference in curvature, the fundamental equations and variables as well as the mechanical equilibrium conditions to characterize these two types of systems are different. Consequently, there is not only a difference between the phase rule of bulk phase systems and that of capillary systems, but also a difference between the phase rule for moderately curved capillary systems and that for highly curved capillary systems. A recently derived phase rule for capillary systems containing moderately curved interfaces and contact lines of bulk phases is reviewed. Implications of such a phase rule on a functional relationship of interfacial free energies and some experimental confirmation are also discussed in this paper. By counting properly the total number of independent, intensive variables required to describe the equilibrium state and the total number of equilibrium constraint equations, a phase rule for highly curved capillary systems is derived in this paper. Comparison with the phase rule for moderately curved capillary systems shows that the high-curvature capillary systems have more degrees of freedom than the moderately curved capillary systems, owing to the high curvature of surfaces and three-phase line which require more variables.

32 8.

D. Li 1 Fluid Phase Equilibria 98 (1994) 13-34

Acknowledgements This study was supported

by a Central Research Fund of the University of Alberta.

9. List of symbols

Co,, Cnj, Ca and C, f J

J K

R

z L

mj, Gj

T U

u V xi

xi K

surface area first curvature potential corresponding to the mean curvature J second curvature potential corresponding to the gaussian curvature K average first curvature potential corresponding to the average mean curvature J average 2nd curvature potential corresponding to the average gaussian curvature R linear curvature potentials number of degrees of freedom mean curvature average mean curvature defined by Eq. (7) gaussian curvature average gaussian curvature defined by Eq. (7) curvature vector of the three-phase line length of a three-phase contact line unit vector tangent to the adjacent curved surface and normal to the three-phase line number of co-existing phases number of pressure equality relationships in the mechanical equilibrium conditions molar number of the ith component pressure number of independent chemical components in a system principal radii of curvature entropy density entropy number of planar or moderately curved interface phases number of highly curved surface phases temperature internal energy density internal energy volume mole fraction of the ith component ith extensive variable intensive variable corresponding to Xi

D. Li 1 Fluid Phase Equilibria 98 (1994) 13-34

33

9.1. Greek letters Y

e 0,

K

% % %j Pi Pi d

%

surface tension of surface free energy contact angle angle formed by the ith and the jth interfaces at the three-phase line curvature vector of the three-phase line local geodesic curvature of the three-phase line geodesic curvature of the three-phase contact in the plane of the solid surface local normal curvature of the three-phase line chemical potential of the ith component molar density of the ith component line tension or linear free energy local geodesic torsion of the three-phase line

9.2. Subscripts and superscripts A L

a,BPY aB, BY,aY @Y

surface phase line phase bulk phases a, p, y, respectively interfaces of bulk phases a, /I, y, respectively line phase or the three-phase line

References Adamson, A.W. 1982. Physical Chemistry of Surfaces, 4th edn. Wiley, New York. Boruvka, L. and Neumann, A.W., 1977. Generalization of the classical theory of capillarity. J. Chem. Phys., 66: 5464-5476. Crisp, D.J., 1947. Surface Chemistry: Proceedings of joint meeting of the Faraday Society of So&t& de Chimie Physique, Bordeaux. Crisp, D.J., 1948. Discuss. Faraday Sot., 3: 98-9. Callen, H.B., 1960. Thermodynamics. Wiley, New York. Cheng, P., Li, D., Boruvka, L., Rotenberg, Y. and Neumann, A.W., 1990. Automation of axisymmetric drop shape analysis for measurements of interfacial tensions and contact angles. Colloids Surfaces, 43: 151167. Defay, R., 1934. Etude Thermodynamique de la Tension Superficielle. Gauthier-Villars, Paris. Defay, R. and Prigogine, I., 1966. Surface Tension and Adsorption. (Translation of collaboration between A. Bellemans and D.H. Everett). Longmans, Green & Co., London. Gibbs, J.W., 1961. The Scientific Papers of J. Willard Gibbs, Vol. 1. Dover, New York. Gaydos, J. and Neumann, A.W., 1992. Generalized theory of capillarity for axisymmetric men&i. Adv. Colloid Interface Sci., 38: 143-227. Rirkwood, J.G. and Oppenheim, I., 1961. Chemical Thermodynamics. McGraw-Hill, Toronto, Ont. Li, D., Gaydos, J., and Neumann, A.W., 1989. The phase rule for systems containing surfaces and lines, I. Moderate curvature. Langmuir, 5: 1133- 1140. Li, D., Cheng, P. and Neumann, A.W., 1992. Contact angle measurement by axisymmetric drop shape analysis (ADSA). Adv Colloid Interface Sci., 39: 347-382.

D. Li 1 Fluid Phase Equilibria 98 (1994) 13-34 Li, D. and Neumann, A.W., 1992a. Equation of state for interfacial tensions of solid-liquid systems. Adv. Colloid Interface Sci., 39: 229-345. Li, D. and Neumann, A.W., 1992b. Contact angles on hydrophobic solid surfaces and their interpretation. J. Colloid Interface Sci., 148: 190-200. Lupis, C.H.P., 1983. Chemical Thermodynamics of Materials. North-Holland, New York. Mtinster, A., 1970. Classical Thermodynamics. (ES. Halberstadt, trans.), Wiley, Toronto, Ont. McLellan, A.G., 1980. The Classical Thermodynamics of Deformable- Materials. Cambridge University Press, New York. Neumann, A.W., Good, R.J., Hope, C.J. and Sejpal, M., 1974. An equation-of-state approach to determine surface tensions of low-energy solids from contact angles. J. Colloid Interface Sci., 49: 291-304.