THERMODYNAMIC ASPECTS OF CAPILLARITY
I
JAMES C. MELROSE
The thermodynamic treatment of a fluid interfacial region is sysfematically developed and, introducing hydrostatic principles and the properties of the stress tensor, extended The advantages which result from this approach include both those of rigor and those
of augmented physical insight
c
a p h t y was first recognized as an important area :. of scienhfic study at a time when the laws of thermodynamics were still unknown. Only much later, when physical chemistry had developed into a distinct discipline within chemical science, did capillarity become part of what is now referred to as colloid and surface chemistry. As a result of this shift in emphasii, those aspects of the subject most easily treated by a thermodynamic type of analysis have become dominant, with other aspects, not strictly thermodynamic in nature, becoming somewhat obscured. This course of development explains why, in spite of the relatively remote origins of the subject, some of the basic principles of capillarity have only recently been fully elucidated. This increased understanding should provide considerable stimulus to the further application of these principles to a wide variety of problems in colloid and surface chemistry. Many technological problems are
concerned with the statics and dynamics of multiphase fluids. These problems, too, require basic knowledge concerning the behavior of fluid/fluid and fluidlsolid interfaces for their solution. Thus, such technological areas as soil physics and the recovery of petroleum rely quite directly on the subject of capillarity for an understanding of many of their phenomena. The main features of the phenomenological theory of capillarity, so far as statics is concerned, were outlined by Laplace and Thomas Young, using concepts drawn from mechanics. The application of thermodynamics to capillarity was introduced by Kelvin and thoroughly explored by Gibbs (28). Unfortunately, certain limitations inherent in Kelvin’s approach have led to much confusion. Even at the present time, the concept of interfacial tension is occasionally regarded as equivalent to the interfacial Gibbs free energy. Also, many writers on thermodynamics, particularly among chemists, have failed to appreciate the mathematical motivation and significance of a concept introduced by Gibbs, the socalled “dividing surface.” This has led to a number of attempts to circumvent the Gibbs treatment and to provide alternative approaches. As justification for these attempts it is often stated or implied that the Gibbs formalism constitutcs a model in which the properties of each bulk region persist without change up to the mathematical surface known as the dividing surface, which is then assumed to poasw the V O L 6 0 NO. 3 M A R C H 1 9 6 8
53
energy, entropy, and mass of the ‘(true” interface. The disadvantages of such a model are, of course, obvious. This interpretation, however, does not reflect a correct assessment of the motivation underlying the formalism. That is to say, the model of a n interface which the formalism actually assumes is that of a region in which the variations of the energy, entropy, and mass densities are continuous. Also, the model explicitly recognizes the fact that the detailed nature of these variations is not subject to experimental observation and description. Thus, the formalism provides a method which is precisely suited to the problem of treating an interfacial region in phenomenological terms. I n recent years, a re-examination of the approach introduced by Gibbs has taken place. Discussions due to Tolman (56-58), Koenig (35, 36), Hill (33), and Kondo (37, 38) have clarified several aspects of the dividing surface concept, as applied in a strictly thermodynamic context. However, the contributions made by Buff (7-77) are of even more significance. As an example, by introducing appropriate mathematical techniques, but retaining the dividing surface concept, a rigorous Gibbs treatment of arbitrarily curved fluid/ fluid interfaces has been achieved. A second and more important advance, included in Buff’s analysis, is the rigorous treatment of the anisotropy of the stress tensor within fluid interfacial regions. This treatment entails an explicit consideration of the basic principle of hydrostatics, namely, that in the absence of external macroscopic fields, the divergence of the stress tensor must vanish. Since this principle was ignored in Gibbs’s strictly thermodynamic approach, it is clear that Buff’s treatment goes beyond that of Gibbs. I n fact, this development succeeds in unifying the hydrostatic and thermodynamic approaches to capillarity. Hence, the long-standing and unfortunate debate as to the “reality” of a tensile force existing in fluid interfaces is resolved. Also, the extended formalism provides the basis for developing an asymptotic expansion of the free energy with respect to the geometrical parameters of the system. This expansion then accounts quantitatively for the breakdown of thermodynamic concepts for very small systems. The present paper will be devoted to a review and elementary exposition of the principal features of the phenomenological treatment of static fluid interfaces which is now available. Significant aspects of this treatment appear to be either ignored or misunderstood at the present time, and it is hoped that the present review will aid in making clear the advantages to be gained from a rigorous treatment. For a comprehensive treatment of the basic principles of capillarity, but with emphasis laid on the strictly thermodynamic approach, the reader is referred to the 54
INDUSTRIAL A N D E N G I N E E R I N G CHEMISTRY
treatise of Defay, Prigogine, Bellemans, and Everett (20) and to the article by Ono and Kondo (48). A detailed exposition of the approach followed in the present paper has been given by Buff (73). A valuable survey of related work, both basic and applied in nature, will be found in the collection of papers presented at a previous symposium (57). This volume, edited by Ross, indicates the scope of the technological problems which involve capillarity and presents a suitable perspective for assessing the applicability and importance of the theoretical concepts surveyed in the present work. Before proceeding with the theoretical discussion, however, one particular area of application should be mentioned. This area, in which capillarity plays an all-important role, is that which is concerned with the static and dynamic behavior of multiphase fluids contained within the interstices or void space of fine-grained porous materials. At the level of the smallest pore sizes to which the concepts apply, the capillary properties of materials such as adsorbents and catalysts must be considered (6, 19). Here, in interpreting gas phase adsorption phenomena at moderate to high relative pressures, it is necessary to take into account the simultaneous occurrence of capillary condensation, as Wade (59) has shown. I n the case of porous solids characterized by larger pore sizes, the displacement of one fluid phase by another from the pore space is of importance in a variety of technological fields. Two of these fields, as mentioned above, are the recovery of petroleum and the transport of moisture in soils. A common feature of the capillary phenomena observed in such contexts should be pointed out. The pore sizes will usually be sufficiently small that the fluid/fluid interfaces can be described mathematically as having a constant mean curvature (50). Insight into the behavior of interfaces under this constraint can be gained from studies on model pore systems. Such a study has recently been carried out by Haynes (32). One of the most interesting aspects of the capillary action encountered in fluid displacement processes is that of hysteresis. The subject has been reviewed by Everett (24), with particular emphasis on the role of hysteresis in gas adsorption. I t should be noted that capillary pressure experiments (29, 53, 54) utilizing materials having much larger pore sizes exhibit hysteretic
C. Melrose i s a Senior Research Associate f o r the M o b i l Research and Development Corp., Field Research Laboratory, Dallas, Tex. AUTHOR J a m e s
P HEAT
I
n
surface, called by G i b b the "dividing surface." As will be seen, such a surface can be located withii the zone of interphase contact by any one of several conventions which are suggested by the formalism itself. Among other consequmes, employing such a convention divides the total volume of the system to be considered into two parts, v" and V? Except in the immediate vicinity of the reference surface-i.e., within the contact zone-the phases a and B are taken as homogeneous. The term, bulk fluid region, will be used to refer to these homogeneous parts. The properties of the bulk fluid regions will therefore be described by the standard Gibbs formalism for fluid phases. Thus, any subvolume, V, within either d the bulk fluid phases is characterized by a set of extensive macroscopic quantities. These quantities are the energy, entropy, and mole numbers, denoted respectively by U,S,and N 4 (i = 1, . . ., n), where n is the number of components. Consequently, for either bulk fluid region, a Gib& fundamental equation can be written :
.
behavior which is in many respects analogous to that which is observed in adsorption. A revealing study of hysteresis in the capillary p m u r e versus saturation relationship has recently been reported by Morrow and Harris (47). The importance of the technological processes which involve m u l t i p k fluids commingled within porous solids is evident. Better understanding and control of the capillary phenomena associated with these processes should result from further studies of the type to which attention has been drawn. It may be anticipated that these studies will require for their full interpretation an adequate formulation of the thermodynamic and hydrostatic principles which govern such phenomena. Specifkalion of Variables k r Copillary Systems
Description of thermodynamic system. The principal problems to be faced in developing a rigorous theory of capillarity involve the establishment of the conditions for mechanical equilibrium for curved fluid interfaces (70, 77) and for &-phase confluent zones (contact lines). Since the curved fluid interface represents a problem of less difficulty, the discussion will be devoted to this case. Techniques suitable for treating the more complex situation of a contact line have been developed by Buff and Saltsburg (76, 77). A detailed specification of the thermodynamic variables appropriate to the fluid interface situation is given by Tolman (56) and Koenig (35). Following these authors, we consider the system to be comof two fluid phases, a and 8, in contact. In Figure 1 an idealized system in the gravitational field is shown. To develop a rigorous thermodynamic treatment, it is necmary to use the concept of a mathematical reference
dU
TdS
- PdV +
I
i-1
pdNI
(11
Since U is a homogeneous function of order one in the extensive variables S, V, and Nu this equation,can be integrated by means of Euler's theorem (49), giving
U = TS - PV
+
" i-1
pcNr
These equations imply the Gibbs-Duhem form,
VdP
= SdT
+
I
EN& i-l
(3)
We also assume that the phases a and 8 are in thermal, material, and mechanical equilibrium. The assump tion of thermal and material equilibria implies that the partial derivatives of U with respect to the extensive quantities S and N , are the same in both phasen. This assumes, of course, that all components are present in both a and 8. These partial derivatives are homogeneous functions of order zero and hence correspond to n 1 intensive functions of state which are identical in the two phases. The additional complications which arise when some components are present in only one phase have been d s c m e d by Koenig (35). In writing Fiquations 1 to 3, it is ale0 assumed that we can neglect all external fields, such as the gravitational field and mamscopic electric and magnetic fields. As Figwe 1 implies, the experimental study of fluid interfaces usually relies on the gravitational field, in conjunction with appropriate boundary conditions and the density daercnce between the fluids, to provide the
+
V O L 6 0 NO. 3 M A R C H I968
55
Table I.
Variance and Geometrical Variables: Two Fluid Phases
Interface
pressure difference which distorts the interface. However, it will aid in clarity if the intrinsic contributions of the gravitational field to the thermodynamic state, both of the bulk fluid regions and of the interfacial zone, are neglected in the present discussion. The appropriate modifications in the formalism which are due to these contributions have been considered by Buff ( 7 7). Mechanical equilibrium assumption. Unlike the intensive variables T and p t , the pressures in the bulk phases will not in general be equal. I n connection with this unique behavior of the partial derivative of U with respect to V , it should be recalled that whereas U , S, and N , are always scalar quantities, the pressure in a fluid phase arises from a consideration of the forces or stress vectors acting on a fluid element ( 7 , 55), The resultant of these forces is expressed by the stress dyadic or stress tensor, a. In order to establish the conditions under which the existence of a thermodynamic pressure can be asserted, it is necessary to impose two restrictions on the stress tensor. The first of these is the condition for mechanical equilibrium, which we write as ( 7 , 55)
v.cr=o
(4)
This expression, it should be noted, follows from the equation of motion of a fluid which is assumed to be at rest and not subject to body forces. I t therefore can be regarded as expressing the principle of momentum balance for such a fluid. Under this restriction, all components of the stress tensor vanish except the diagonal components (normal stresses). The second restriction on the stress tensor which can be imposed in the case of a bulk fluid region follows from the assumption that the fluid is isotropic. Hence the diagonal (nonvanishing) components of the stress tensor can be taken as equal. This statement expresses Pascal’s law and permits us to treat the thermodynamic or hydrostatic pressure as a scalar quantity :
-P
= 011 = 6 2 2 = 6 3 3
(5)
If we now consider the nonhomogeneous region which corresponds to the interfacial zone, it is clear that Equation 4 must still be imposed. However, since such a region is not isotropic, we must expect that the normal stresses are no longer identical, and hence Equation 5 will no longer be true (30). This has the further consequence that, unless the interfacial region has a special configuration (e.g., planar), the pressures in the isotropic regions on either side of the interface will not be the same. At this stage is it useful to point out that the variance of the system which is predicted by the Gibbs phase rule is correct only if the pressures in the contiguous phases are equal. I n the general case of a curved interface, the actual variance is increased by unity, Le., for the 56
INDUSTRIAL A N D ENGINEERING CHEMISTRY
configuration
Extensive geom. var.
None, Pa = P P
V ( 0 = 0)
Planar, Pa = P P
v, 0
(J =
Curved, Pa # PP
VU, VP, 0
lo,I, K
Curved, Pa # P P
GJ
?.a,
a
n
Nonextens. geom. vat.
... K = 0)
hP
No. of geom. ~ a r . ~ l b
26 4c 3c
In fundamental equation, no. of nongeometrical variables =
+ 1.
b Variance = n. c Variance = n
+ 1.
+
system to be considered it is n 1. Extensive discussions of the phase rule, as applied to more complex capillary systems, have been given by Defay et al. (20) and by Rusanov (52). Geometrical variables. As a first step in extending the thermodynamic description to the interfacial region, a selection of appropriate geometrical variables is required. I n Table I several such sets are listed. These sets correspond to successively more complicated situations, beginning with the familiar case in which interfacial effects are totally neglected, while maintaining thermodynamic equilibrium. When this condition is relaxed but a planar interface assumed, it is clear that the area, Q, of the interface must be included as a variable in the fundamental Gibbs equation. I n this case no problem arises in connection with the location of the mathematical surface which serves to define the area of the interface. This follows from the fact that for a planar interface the portion of the interfacial region to be considered is bounded by a family of parallel normals. Hence, regardless of the position which is chosen for the location of a mathematical reference surface within the nonhomogeneous region, the area of such a surface is invariant. Also, we may note that, since the pressures in the two homogeneous regions are equal, only changes in the total volume of the system need be considered. I n the case of a curved interface, the family of normals is no longer parallel, and the area of any reference surface will depend on its location. Hence, the formalism must be capable of precisely specifying the position of the reference within the interfacial region. Also, as we have already noted, the pressures are now different in the bulk phase regions, and the variance of the system as a whole is increased to n 1. This has the further consequence, indicated in Table I, that the number of geometrical variables required in the fundamental equation is increased, That is, it is no longer possible to express the mechanical work associated with the expansion or contraction of the system volume by a single PdV term.
+
As will be shown later, it is possible to choose a set of three independent geometrical variables such that the problem of specifying the area of the reference surface is avoided. This set, given in the final row of Table I, is them sufficient to ddine the state of the system-i.e., to account for the mechanical work which the system can perform on, or receive from, its surroundings. However, if the more obvious set of extensive variables, P,p, and a, is chosen, a supplemental nonextensive variable is required, as is indicated in Table I. This variable serves to locate the reference surface and hence enables the extensive geometrical variables to be precisely defined. In discussing curved interfaces, we must also note the existence of intensive variables specifying the curvature of the reference surface. The mean curvature, J, is often regarded as the appropriate additional variable used in fixing the intensive state, corresponding to the increase in the variance. In addition to J, the Gaussian curvature, K, as will be seen, plays an important roie in developing a suitable description of the thermodynamic state of a curved fluid interface. To proeeed with such a description, it is necessary to review briefly some of the mathematical properties characterizing curved surfaces. This is done in the following section.
X
/
Malhemalical Description d Curved Budaces
Measure8 of Curvature. In Figure 2 the mathematical concepts leading to the definitions of two measures of curvature are illustrated. Thus, at any point of a curved surface, it is useful to consider a unit normal vector, F, and two orthogonal unit tangent vectors, g,, and fe. There exists a unique orientation of the tangent vectors, demoted by gl and fa, such that the normal curvatures, k. and k,, in the corresponding directions take on maximum and minium values (Euler's theorem). The sum and product of these principal curvatures then define the mean curvature, J , and the Gaussian curvature, K,
JIki+k,
(64
K = kika
(6P) These two measures of curvature pogPess several important mathematical properties. For example, if now we take the arc lengths in any two orthogonal directions as x. and xs, a surface gradient operator may be defined (4) as follows:
V a s C"
a a -+ e ax"
When this operator is applied to the unit normal vector, a symmetric dyadic, Ve, is formed. I t then turns out that VOL 6 0
NO. 3 M A R C H 1 9 6 0 57
VI.
= .k
+ k.
J
Hence, J is the first scalar invariant of the dyadic, V@, and consequently is to be regarded as a point function over the surface. Although the magnitude of the product &.k is not independent of the orientation of the orthogonal unit tangent vectors, the Gaussian curvature, K,does have this property and in fact is the second scalar invariant of Vg. It has a rather more complex representation (60),
ParalIel surfaces. It is essential in considering curved interfaces to be able to specify precisely how the two measures of curvature vary for different positions of the mathematical reference surface located within the interfacial region. This can be accomplished by utilizing the concept of parallel surfaces (21,39). I n Figure 3, a suitable coordinate system is indicated. Choosing an arbitrary normal passing through the interfacial region, we may denote the distance from some arbitrary point, measured along such a normal, by the parameter, X. This parameter then specifies the position of a given member of a family of surfaces, all members of which are to be considered parallel. Three particular surfaces are identified by the superscripts a,@, and The surfaces which are located at A = X" and X = Xp are within the bulk fluid regions. The surface whose position is specified by X = Xo is the reference surface lying within the transition zone (interfacial region). The fundamental mathematical property defining such a family of surfaces is that all of the members are characterized by the same set of normal directions. That is to say, displacing the set of unit normal vectors a given constant distance yields the set of unit normal vectors ddining another surface parallel to the first. This mathematical definition enables us to formulate properly, as first pointed out by Buff (7 I), the concept of two-dimensional homogeneity with respect to an interfacial region. We first recognize that we are unable to measure or observe either the actual thickness of an interface or the precise way in which the mass, energy, and entropy densities vary as the interface is traversed in a normal direction. Nevertheless, we can assume that fluid interfaces possess the following fundamental property. If a given mathematical surface within the nonhomogeneous region is characterized by the same value of the point density of matter at every point of the surface, then all surfaces which are parallel to the given surface will also be uniform with respect to mass density. I n other words, if the interfacial region is homogeneous in a two-dimensional sense, then the mass, energy, and
'.
58
INDUSTRIAL A N D E N G I N E E R I N G CHEMISTRY
entropy density profiles must be invariant over the set of normal directions. Since we can assume that the interfacial region is homogeneous in this two-dimensional sense, it wiIl be necessary, in much of the discussion, to establish only a local description of the interfacial thermodynamic properties. This will also prove to be convenient, because we need then to consider only a s m a l l area on any given reference surface within the interfacial region. Such an area should be sufficiently s m a l l that we can regard the two measures of curvature as uniform over the defined region. The values of the scalar point functions representing the curvature will, however, vary for different positions of the reference surface denoted by A". It is easily shown that for differential changes in the distance parameter, A,
dki -= dx From
- k,'
(i = 1, 2)
this it follows that
Integrating Equation 9, we obtain a useful expression for the ratio of the Gaussian curvature measured on the reference surface at Xo to that measured on a parallel surface at an arbitrary distance X:
K(XO)/K(X)= 1
+
J(XO)(
- A"} + K ( X O ) ( X -
XO)'
(11)
Spherical representation. The obvious extensive variable which the thermodynamic formalism requires is the area, fl, of the reference surface located at A". We have noted, however, that this parameter will vary as X" varies, even when the set of normals defining the external boundaries of that portion of the interface which we wish to consider is fixed. Hence, for displacements of this type, fl cannot be considered as a homogeneous first-order function with respect to the matter, energy, and entropy included within any given pair of parallel surfaces. This difficulty has, in fact, rarely been emphasized in the literature. Thus, the restrictions under which Euler's theorem on homogeneous first-order functions can be applied are usually not specified. However, in the approach developed by Hill (33) for the special case of a spherical interface, a device was introduced which successfully deals with this problem. By use of the concept of spherical mapping (22, 27, a), this device can easily be generalized to nonspherical interfaces, as will be indicated.
These expresions are applicable only at a point on the surface, that is, as the area 0 converges to a point. Nevertheless, it is just in this local sense that a rigorous geometrical treatment is required at this stage of the discussion. Transformation of variables. The desirable property of the parameter w as an independent geometrical variable for use in the thermodynamic description can now be demonstrated. Writing the total differential of the area, a, as
rr
L
-
UHII SfHElE
and using Equations 13a and 15, we find
a
Figure 4. Rcjnerenfafimt of a rphm'tal image
In Figure 4, a cross-section of a curved surface, S, is shown, together with the cross-section of its spherical image. The latter is the portion, 9, of the surface of the unit sphere which is formed by a set of normals parallel to the set defining the boundaries of the surface under consideration. Thus, corresponding to any spherical image is a solid angle, w, equal in magnitude to the area of the image, a*. The property of the spherical image which is to be employed is contained in a theorem which asserts that as the area of the surface vanishes, that is, for a given point on the surface, the ratio of the area of the image to the area of the surface (the normal vectors having the same sense) is equal to the Gaussian curvature, K ,
lim (Q*/hl)
=
K
(17)
& = - d ww + J R d x o
The integration of t h i s result by means of Euler's theorem makes it apparent that 0 is homogeneous of order one in the variable w but not in the variable A'. A similar situation arises in considering the independent geometrical variables upon which the volumes v" and Va depend. Noting that
we see that the total differentials of V" and p a r e written
as
(12)
0 4
If we now recognize that in the neighborhood of a given p i n t the area, 0, is a function of two independent variables, w and A", we see that the two partial derivatives are just =
K-l
-
n w
Thermodynamics of Curved Fluid Interfaces
From Equation 12 it follows that
(23- ii a
=
This expression, together with Equation lob, enables us towrite Equation 13b as
(E)=
JO
Equations 17 and 19 provide transformations from the set of variables given in the final row of Table I: w, A", and AB; to the set: Vu, VB, a, and A'. As a result of the homogeneousfirst-order property of the parameter, w, the set which includes w will prove to have a rather unique advantage. Associated with this advantage is the fact that this set, when used in the fundamental equation, possesses one less member than is required when the set including D is used.
(15)
Gibbs excess functions. The key features of the thermodynamic treatment of interfacial regions will involve the application of the mathematical concepts just discussed. Before this can be done, however, we must introduce some further definitions which describe the variation of the energy, entropy, and mass densities in the nonhomogeneous region. Thus, we can write as point functions of the distance, A, VOL 60
NO. 3
MARCH 1968
59
Q(x)
lim v-0
(u/v);,?(A)
lim V+O
(s/v);
= lim ( N J V ) ; (i = v40
p&)
1,
,
..
.)
n)
(20)
Within the bulk fluid regions, the densities of energy, entropy, and mass are, of course, independent of X. When we denote these values of the densities by the superscripts a and /3, the extrapolation of the bulk fluid densities to a reference surface located at the position A" can be formally represented, as Buff (70) has suggested, in the following way:
rr. Thus, although these quantities have been defined in terms of an arbitrarily selected normal direction through the interfacial region, they will actually be invariant over any particular reference surface which may be chosen. Such behavior may be regarded as an essential characteristic of properly defined surface excess quantities. Functions of state. We now define extensive properties for the total (nonhomogeneous) system by taking the integral over the functions, 0, 3, and pi alone, and then multiplying by the area of the reference surface, a. These properties are given by
and for each component,
Here, A(X) is a unit step function A@)
= O
for X
< Xo;
A(X) = 1 for
X2
Xo
(22)
The so-called Gibbs excess properties are then obtained by subtracting each of the extrapolated densities from the corresponding actual density and integrating the result over the entire range of the parameter, XLe., through the interfacial region and out to the external boundaries, X" and hP. This integration is carried out by introducing a factor accounting for the change in the area of the surface on which the density functions may be regarded as defined. From Equation 12 it follows that the area to be used for this purpose varies inversely as the Gaussian curvature. Hence, the factor required is simply the ratio of Gaussian curvatures given by Equation 11. Thus, the following integrals define the surface densities of energy, entropy, and mass, u =
s =
rt
=
l@{a(X)- e@(Xo) l@ -
) { K ( X " ) / K ( X ))dX
{,?(A)
LP{
PdX>
pp(Xo) ](K(X")/K(X))dX
-
Piap(x0>
(23a)
n
+,
The generalized work variable, is postulated to be homogeneous first order with respect to the functions of state and is therefore also a function of state. Integrating, we obtain @ =
U - TS
-
n
i=l
(23b)
)1NX0)/K(X)JdX (234
Definitions given by Equations 23 do not in any sense imply that the mathematical surface located at X" possesses mass or any other physical attribute. The surface excess quantities are, however, defined in terms of a specific location of a reference surface. Hence, their magnitudes are highly dependent on the position of the reference surface. In fact, this surface can be so located as to make an)' one of the surface densities vanish. As Buff (72) has pointed out, two such locations will then define a distance which is an excellent measure of the sharpness of density profiles represented by the point functions defined by Equation 20. The definitions of the surface excess quantities given by Equations 23 are such that the two-dimensional homogeneity associated with the energy, entropy, and mass density profiles can also be attributed to u, s, and 60
The quantities denoted here by superscripts a and /3 clearly will be subiect to the Gibbs formalism applicaable to bulk fluid-i.e., Equations 1 to 3. The fundamental equation and Gibbs-Duhem equation interrelating the extensive quantities, 77,S, and N t , will be more complex. I t should be emphasized that these properties are not defined in terms of a particular location of the reference surface. Hence, we may say that U, S, and N ( are functions of state, restricting this term to quantities which are not dependent on l o . A fundamental Gibbs equation for the functions of state may now be written. This equation is
I N D U S T R I A L A N D ENGINEERING C H E M I S T R Y
p&t
(26)
I n writing Equation 25 for the quantities, U, S, and AT,, it is, of course, assumed that the partial derivatives, T and p i , are identical with the corresponding functions characterizing the bulk phase regions. Such an assumption implies merely that the interfacial zone is in thermal and material equilibrium with the contiguous bulk phases. Analysis of mechanical work terms. It is now clearly necessary to relate the thermodynamic potential (free energy available for mechanical work at constant T , p i ) to the geometrical variables introduced in the preceding section. We choose as a set of independent geometrical variables the set given in the last row of Table I. This set is not defined in terms of a reference surface and hence is suitable for expressing changes in the potential @. We write, therefore,
+
d@ = $*dX"
+ $ @ d i p + $"@dm
(27)
The pressures in the bulk fluid regions become explicit by introducing the partial derivatives of the total volume
with respect to a change in the position of the outer boundary surfaces,
The final coefficient, by virtue of the first-order homogeneity of the solid angle, w, is simply (29)
$a@ = @ / w
Introducing the transformations represented by Equations 19, we can rewrite Equation 27 in the form d 9 = -P"dVa
- P@dVs+ { ( a + P"V" + P@V@)/wfdw + (P" - P@)QdX"
(30)
A further transformation can be effected by using Equation 17 to express the variation in w in terms of the corresponding variation in Q. This yields da
-PadVa
=
When we recall Equations 15 and 18, it is seen that
- PpdVB+ { (9+ P V " + P@V@)/Q)dQ +
{ (P"- P@)Q- (a + P"Va + PaVa)J]dAo
(31)
The form of Equation 31 suggests a thermodynamic definition of interfacial tension. Thus, writing y
formalism does not give rise to any inconsistencies precisely because of the presence in the fundamental equation of the term in dX". Hence, this term provides an essential feature of the formalism, without which it would not be self-consistent. Variation in reference surface location. To demonstrate the significance of the parameter .$ we first note that the definitions adopted for y and .$ are such that both quantities depend on the location of the reference surface, that is, on A", as well as on the state of the system. We can, in fact, differentiate y with respect to A", holding all functions of state constant. Denoting such a derivative by square brackets, we obtain
= (a + P V "
+ P@V@)/Q
[~P"-P@-yyJ
(32) (33)
(37) The nature of the parameter, f , can be still further elucidated by employing an additional geometrical transformation. For this, we use Equation 10a and change the independent variable denoting a relocation or shift of the reference surface to the mean curvature of this surface, J . This gives
we obtain Equation 31 in a more concise form,
+
d 9 = -PadVu - P@dV@ ydQ
+ EdX"
c
where
E
=
gJ-2
[$I
= ((2 K
- P)-1
(34)
(35)
Equation 34 shows that to express the change in the potential 9 in terms of the changes in the set of extensive variables, V", V@,and Q, the additional variable, A", must be included in the fundamental Gibbs equation. I t should be emphasized that this result is obtained simply by applying suitable geometrical transformations to a n equation in which only three independent variables appear. Furthermore, these variables are not defined in terms of a reference surface. Thus, the parameter X o appears in the fundamental equation in a special role, serving only to specify the location of the reference surface. A comparison of Equations 32 and 34 shows that in integrating the latter, the parameters V", V@,and Q are homogeneous of order one with respect to 9,while A" is not. Since a change in X o refers to a shift or relocation of the reference surface entering into the definitions of V", V@,0, and y, it is clear that such a change does not alter the thermodynamic state of the system. I n other words, the geometrical variables V", V', and Q, and the interfacial tension, y, are not pure state functions, in spite of the fact that the geometrical variables possess the desirable first-order homogeneity behavior. This unusual and rather subtle aspect of the
A second differentiation shows that the value of y obtained by choosing a reference surface such that c vanishes is a minimum. Following Gibbs (ZS), this particular reference surface is referred to as the surface of tension. We have not yet established that the particular placement of the reference surface which makes c vanish is, in fact, within the interfacial region. If we accept for the moment that this can be done, the parameter c is seen to possess some of the characteristics of a surface excess quantity, such as were defined in Equation 23. We can also rewrite Equations 33 and 34 as follows:
P" - Pa = y J + ~ ( K2 - J2) d@ = - PdV" - P@dV@ + ydQ + CdJ
(39)
c = CQ
(41)
(40)
with
These expressions provide the motivation for selecting the surface of tension as a primary reference surface, since then Equations 39 and 40 reduce to forms obtained through more familiar but less rigorous procedures. It should be noted that for the special case of a planar interface the final terms in Equations 39 and 40 vanish without requiring that the reference surface be the surface of tension. Thus, for this case, any choice of X" leads to VOL. 6 0
NO. 3
MARCH 1968
61
Equation 39 relates the thermodynamically defined parameters, y and c, to the pressures in the two bulk phases and to the two measures of curvature. As will be shown in a later section, this relationship constitutes an essential link with the hydrostatic treatment based on the necessary condition for mechanical equilibrium, Equation 4. The hydrostatic analysis also yields the desired intepretation of y and c as surface excess parameters. Hence these quantities, due to the two-dimensional homogeneity of the contact zone, will be uniform over a macroscopic portion of the interface. I t then follows from Equation 39 that the surface of tension must be a surface of constant mean curvature-Le., every point of the surface of tension will have the same mean curvature, J . If this curvature is not too high, then other reference surfaces will deviate only slightly from this condition. When this is so, we may conclude that J is an appropriate intensive parameter which characterizes the whole extent of the interface we wish to consider. We thus avoid the arbitrary aspect of the treatment inherent in the use of X o as a parameter specifying the location of the reference surface. This aspect follows from the fact that X o is defined in terms of an arbitrary selection of a normal direction. Consequently, we may now regard Equations 39 and 40 as applying to the whole interface, rather than being restricted to just a local validity, as has been implicit in the previous discussion. Formalism for surface excess quantities. At this point we are able to return to a further consideration of the surface densities defined previously and to interrelate them within a surface thermodynamic formalism. Combining Equations 25 and 26 with Equations 40 and 32, respectively, we obtain the following : dU = T d S
+
These two expressions yield a Gibbs-Duhem equation in the form V*dPa
Table II.
+ Qdy - CdJ
n
d r = -sdT
-
C F,dp, 2=1
+ cdJ
These expressions link the Gibbs excess quantities, rZ,with the intensive variables, identical in the two bulk phases, and with the interfacial tension, also an intensive variable. I t is useful, following Defay e t a l . (ZO),to regard these equations as defining the formalism for a nonautonomous system. This terminology emphasizes the fact that the system in question actual1)- has physical existence only insofar as two bulk fluid phases are in mutual contact. The resulting molecular interactions in the zone of contact are then accounted for within the structure of the nonautonomous thermodynamic formalism. An important feature of the Gibbs-Duhem relation, Equation 47, is the presence of the reference surface curvature, J , as a variable. Also, this expression includes one more independent variable than is required by the variance. This aspect of the formalism, far from being a disadvantage, provides just the self-consistency which the unavoidable features of the geometrical part of the treatment make necessary. Thus, setting any u, s, and
(43)
Surface Tension Dependence on Intensive State Variables U
r=o
r
1
-(") ar 0
INDUSTRIAL A N D ENGINEERING CHEMISTRY
(45)
(47)
Surface excess properly
Convention
62
iVZdp, =1
Siiriilarly, writing Equation 3 for both a and /3 and subtracting from Equation 45 yield
ptdS, -
I
+
If now Equation 2 is written for each of the homogeneous regions 01 and p, and both expressions are subtracted from Equation 44, we can then utilize Equations 24 to obtain
i=1
+ ydQ + CdJ
SdT
=
t
n
P*dVa - P P d V P
+ VPdPo
I
0
C
(2)r
r- o I)
/
P"
I I I
I
,
..
, .,?.,
' : .I
. . ,.I.., &
one of the excess quantities, c, u, r,, or J, equal to zero corresponds to a convention which locates the reference surface within the nonhomogeneous region of the system -i.e., the interfacial zone. This will clearly, except for the case of the u = 0 surface, eliminate a variable in Equation 47. Transforrningzhis equation by an appropriate substitution for the exentropy, J, gives
d(7/T) = - ( u / l W T
+ 5 rd(-,JT) + (c/T)dJ
pretations of the excess pmperties, it should be recalled that for each set of partial derivatives the surface tension and the curvature are defined according to a different convention. Hence, these parameters, as they occur in different sets of derivatives, should not be equated, except in the special case of vanishing curvature. We may also note at this p i n t a slight inconsistency in the formalism arising from the fact that only for the convention c = 0 is the reference surface characterized by a uniform value of the mean curvature, J . Except in the case of vesy high curvatures, however, this inconsistency may be ignored. Also, if attention is restricted to interface configurations of simple shape, such as can be represented by spherical and cyhdrical reference surfam, then the treatment can be extended to high curvatures. Under this restriction, as will be seen later, the flexibility inherent in Equation 47 with respect to the reference surface which may be chosen can be used to advantage. This flexibility, in fact, provides the key to the problem of developing an exact thermodynamic expression relating the curvature dep d e n c e to a measure of interfacial thickness. T o establish more concretely the significance of the various conventions suggested by the formalism, Figure 5 presents schematically the profiles of the m a s and energy densities for a single component system. The interface thus corresponds to the transition region between liquid and vapor. As shown, the p d e s refer to the situation encountered when the more dense phase is on the concave side of the interface-i.e., a liquid drop. I t is reasonable to expect, as is indicated in Figure 5, that the u = 0 reference surface will always lie on the liquid side of the reference surface corresponding to the r = 0 convention. For the converse situation encountered with a vapor bubble, the sense of X must, therefore, be reversed.
i-1
(48)
This expression shows that a similar d u c t i o n to the number of variables required by the variance can be accomplished with the convention, u = 0. It should be emphasized that the choice of a particular reference surface has no physical significance whatsoever. Such a choice is arbitrary and hence dictated only by convenience. To illustrate this fact, Table I1 provides a concise summary of the various interpretations of the dependence of the surface tension (assuming a single component system) on temperature, chemical potential, and on curvature. I t is seen that each different convention gives rise to a different set of partial derivatives corresponding to the set of surface excess properties. I n connection with these varying inter-
Hydrorlcllicr of C u d Fluid Inhrfclser
Di5uentkl eqnation for the stress tensor. The discussion in the pmious section was devoted exclusively to arguments of a thermodynamic nature. In fact, the discussion gave no indication as to the relationship between Equation 39 and the conditions necessary for establishing that mechanical equilibrium actually pubsists in the system described. Thus, in Equation 39 the pressures in the bulk phases were related to the measures of curvature by means of the interfacial tension, y, and the excess quantity denoted by c, but b t h of these quantities were defined in a purely thermodynamic way. Furthermore, it was only stated, and not proved, that the location ofthe reference surface which is specified by the c = 0 convention does lie within the interfacial region. V O L 6 0 NO. 3 M A R C H 1 9 6 8
63
The neglect of a fundamental equafion
of hydrostatics has, in the past led to the use of various intuitive arguments
Finally, it should be noted that while Equation 46 related the interfacial tension to a set of surface excess quantities, no insight has been provided as to the significance of the interfacial tension itself as a surface excess quantity. As the result of the investigations carried out by Buff (70, 7 7 ) , it is now known that the solutions to all of these problems lie outside of the strictly thermodynamic approach followed by Gibbs and most subsequent writers, The additional basic principle which must be invoked is simply the statement of the condition for mechanical equilibrium provided by Equation 4. The neglect of this fundamental equation of hydrostatics has, in the past, led to the use of various intuitive arguments. These approaches, of course, have been motivated by the lack of a suitable basis for relating Equation 39 to the mechanical equilibrium condition. Thus, interfacial models involving an infinitesimally thin stretched membrane have been proposed. Such naive models have inevitably evoked criticism from thermodynamicists. Unfortunately, this criticism has also failed to recognize the basic hydrostatic principle which is involved. I n the following, therefore, we briefly sketch the manner in which this principle is to be used in the present discussion. T o apply Equation 4 it is first of all necessary to recognize the two-dimensional character of the interfacial region. Thus, the components of the stress tensor are to be defined with respect to the tangential and normal directions for members of a family of parallel surfaces. If we furthermore assume that the interfacial region will be subject to a two-dimensional analog of Pascal's law, we see that the tangential components will obey a condition of isotropy. The stress tensor under these circumstances can be written as
I n this expression the tangential and normal components, uT and u N , are to be regarded as functions of the distance parameter, A. When Equation 49 is substituted in Equation 4 and the required manipulations are carried out, as Foster (26) has shown, the follc Jing differential equation is obtained : (50) 64
I N D U S T R I A L AND E N G I N E E R I N G CHEM S T R Y
Here, the mean curvature, J , is also a continuous function of A. The integration of Equation 50 can, of course, be carried out without introducing the concept of a reference surface. However, the resulting expression involves the values of the Gaussian curvature at the external boundaries represented by the surfaces at X = Xu and X = AB. This defect can be eliminated by again introducing an extrapolation procedure. Thus, employing the step function defined by Equation 22 and noting that -pa
=
-pp
=
uTa
= uNa
(5 11.
=
(51b)
we can write uup(Xo)
{ 1 - A ( X ) )u N a
$. A ( X ) U N
B
(52)
With the use of this expression the integration of Equation 50 can be written in the form
2 K(X")
lB
{uT(X)
- U~'(X"))
(X - X " ) d X
(53)
The curvature measures appearing in this relationship now correspond to the reference surface located at X = A".
Mechanical definition of interfacial tension. Equation 53 clearly represents the condition for mechanical equilibrium for the two-phase system and associated transition or interfacial region. This particular form, however, does not provide an immediately apparent link with the thermodynamic treatment considered in the previous section. Hence, a suitable transformation must be sought. Such a transformation, of course, will involve the introduction of a mechanical definition of the interfacial tension. The definition which turns out ( 7 7 ) to have the required form is the following : y =
lp
(uT (X)
-
u u p ( X o ) )[l
+ J(Xo){X
- A")
K(X"){X - X o j 2 ] d h
+ (54)
This expression may be differentiated with respect to the variable Xo, considering the function uT(X) as fixed.
sive quantity which, when integrated through the interfacial region, gives rise to the interfacial tension is simply the deviation of the tangential component of the stress tensor from its bulk phase value. That the resulting integral takes the form of a surface excess quantity may be seen by referring to Equation 11. When this relationship is substituted in Equation 54, we obtain an expression which has precisely the same form as the integrals defined in Equations 23. As a further consequence, we verify the assertion made in the previous section that the interfacial tension can be regarded as invariant over that macroscopic portion of the interface which is of interest. A third important result obtained from the hydrostatic analysis is that we can now verify the fact that the particular choice of the reference surface location which makes the parameter c vanish lies within the interfacial region. To demonstrate this, it is useful first to write Equations 54 and 55 for the special case of a planar interface. Indicating the properties for this case by the subscript 03 (infinite radii of curvature) and recalling Equation 50, we have Figure 6. Interface projles for tangential stress component
If the resulting expression is then transformed to the derivative with respect to J(Xo),we obtain c =
L@
{ a&)
J(XO)K(XO){
-
aQ@(XO))
2 K ( X O ) - J'(X0))- q x
-
XO)
IdX
(55)
Equation 39. With the appropriate expansions, Equation 53 is then recovered. For the sake of emphasis, we now rewrite Equation 39 as
- Pf+ =
y(X0)J(X0)
=
La
(aT@)
- V N ) (X - XO)dX
If we now set c = 0, these expressions yield
( A - A") [l -
To establish that Equation 54 does, in fact, have the proper form, we substitute Equations 54 and 55 in
PQ
c,
+ c(X0)(2 K ( X O ) - P(XO)] (56)
This result may be called, following Buff ( I I ) , the generalized Gibbs-Kelvin relation. When it is derived, as in the previous section, from a strictly thermodynamic analysis, it links the interfacial tension, y, to other thermodynamic functions, as in Equations 44 and 46. When, on the other hand, the fundamental equation of hydrostatics serves as the basis for the derivation of Equation 56, we see that the relationship expresses the condition for mechanical equilibrium between the two bulk fluids in contact. Furthermore, we obtain the interpretation of y as a surface excess function. As Equation 54 shows, the inten-
We see that the surface of tension is located a t a position within the interface which corresponds to a mean distance, as defined in terms of the distribution of the stress deviation from isotropy. I n Figure 6, two such distributions are schematically represented for the more general situation corresponding to a curved interface. Assuming a single component substance, the two cases illustrated refer to the vaporliquid interface configurations for a drop ( p a > pp) and a bubble (pa < pa). Owing to differences in the mass, energy, and entropy profiles as the interface is traversed, the distribution of stress anisotropy will also differ. Hence, the position within the interface, defined as the surface of tension, will also differ. As will be seen later, the interfacial tension itself will be slightly different in these two cases. A further consequence of the hydrostatic treatment briefly outlined above may be mentioned. O n any member of a family of parallel surfaces a two-dimensional unit tensor may be defined. Since such a surface is also a reference surface for computing surface excess quantities, we may associate particular values of the parameters y and c with the surface. As Buff and Saltsburg VOL 6 0
NO. 3 M A R C H 1 9 6 8
65
(76) have shown, these values can then be used to define a surface stress tensor. This tensor will be isotropic only when defined on the surface of tension. .4n interesting consequence of introducing such a tensor is that it is then possible to develop a surface or two-dimensional analog of the fundamental equation of hydrostatics ( 7 6 ) . The two-dimensional gradient operator defined by Equation 7 can be utilized to obtain the divergence of the surface stress tensor. I t is found that the pressure difference, Pa - PO, now appears as a body force. This formulation thus provides a suitable context for interpreting the much disputed vectorial nature of surface forces. The intuitive concept of a stretched membrane, often invoked, is given a precise meaning, and the relationship of this concept to welldefined thermodynamic parameters is elucidated.
Surface of tension. The experimental measurement of surface and interfacial tension involves, for all known methods except that of the Wilhelmy plate ( 2 ) ,the use of the mechanical equilibrium condition, Equation 39. The convention used with this expression is that corresponding to the surface of tension, c = 0. The justification for adopting this convention is provided by the discussion presented in the previous section, in which it was proved that this convention corresponds to choosing a reference surface which does in fact lie within the transition zone. Consequently, we can now turn our attention to the possible curvature dependence of the interfacial tension defined according to this convention. Since available experimental evidence does not indicate that such a dependence exists, we can conclude that for the range of curvatures employed in such measurements the dependence is negligible. I n nucleation phenomena (78, 25) and in other contexts (43), however, much larger interfacial curvatures are encountered. I t is desirable, therefore, to predict from thermodynamic arguments alone the range of curvatures in which curvature dependence may be expected to become appreciable in magnitude. For this purpose we now restrict the discussion to the case of a single component system with an interface of spherical shape separating the vapor and liquid phases. This restriction will not obscure the physical result which is of interest and at the same time will enable us to avoid developing a set of formulas of undue complexity. From the thermodynamic identities listed in Table 11, we note that for the c = 0 reference surface,
66
= -($)T
where now the curvature k is simply
42
k = '/z J =
(e)T-
Differentiating Equation 39, with c pa
-
pB=
2y
(611
=
0, then gives
2kr
When we use this result, Equation 60 yields
T o eliminate r, we recall Equation 23c. Defining AX as the distance between the c = 0 reference surface and the r = 0 reference surface,
Surface Tension Curvature Dependence
r
It follows that we can write for the curvature dependence,
(59)
INDUSTRIAL A N D ENGINEERING CHEMISTRY
AX =
xo(r =
0)
- x0(c
=
01
(64)
kz(AX)']
(65)
we find that = (pa
- ps)AX( 1 + kAX
+
'/3
Substituting in Equation 63 now yields = T
+
+
2 AX11 kAX [l 2 kAX{ 1 kAX
+
+
+
k2(AX)2) '/3 k2(AX)2f ]
(66) This result is known as the Gibbs-Tolman-Koenig (35, 58) equation. The expression has been integrated numerically by Tolman. Before examining these numerical results in detail, however, it will be useful to consider the curvature dependence for the surface tension defined according to an alternative convention. Surface of zero mass density. A second reference surface to which primary importance may be attached is that specified by the convention F = 0. As is indicated in Table 11, the usual interpretations of the quantities derived from the temperature dependence of the surface tension are based on this choice of a reference surface. We may also note that for this surface, c = - 1 (
2
)
2 bk
This relationship, to which attention has been called by Buff (9), shows that the curvature dependence arising from a physical change in state is closely linked with the dependence of the surface tension on the choice of the reference surface. Recalling Equation 38, we may regard Equation 39, if the thermodynamic state is held fixed, as a differential equation with the curvature of the
Figure 7 . Dependence of surface tension on reference
suiface
location
reference surface as the dependent variable. equation may be integrated, with the result that
The
I n Figure 7 , the dependence of the surface tension on the reference surface location, as expressed by Equation 68, is plotted. The quantities Y~~~ and kmin refer, of course, to the surface of tension, since this choice is specified by the condition, c = 0. The larger the curvature of the surface of tension the more the surface tension for the I' = 0 reference surface is enhanced. We may also note that for a spherical interface, Equation 64 can be written as
Figure 7 then indicates, if taken in conjunction with Equations 38 and 67, that the surface tension will be decreased or increased with increasing curvature according to whether the sign of AX is positive or negative. Equation 67 can be employed to carry out still another transformation of Equation 39. If the latter is written as
we find that
If we now introduce Equation 68, the following is obtained
Expanding this expression then gives a result which may be compared with Equation 66, since in the latter, k is in fact kmin. This comparison, as first noted by Buff (70), shows that
Figure 8. Dependence of surface tension on interface curvature
Numerical results. Turning now to the integration of Equation 66, we first observe that the quantity AX itself may be curvature-dependent. I n the numerical integration carried out by Tolman (58), it is assumed that any such dependence can be neglected. These results are plotted in Figure 8 for various values of AX. Tolman (57) and Kirkwood and Buff (34) have provided evidence showing that the c = 0 reference surface will always lie on the liquid or more dense side of the r = 0 surface. I t then follows that AX will be positive for the liquid drop configuration and negative for a vapor bubble. As pointed out previously and as indicated in Figure 8, the surface tension at high curvatures will be decreased or enhanced accordingly. In the following section a method will be described by means of which the location of the u = 0 surface, relative to the I' = 0 surface, can be determined. As will be seen, the distance between the u = 0 surface and the I? = 0 surface depends on the choice of the reference state for computing the bulk phase internal energies. Subject to an appropriate choice of the reference state, the relative positions of these surfaces which are indicated in Figure 5 can be confirmed. I t also follows, however, that the distance between the u = 0 and I ' = 0 surfaces does not provide a suitable estimate for the magnitude of AX. We may conclude, then, that the position of the c = 0 reference surface cannot be precisely located in the absence of a reliable statistical mechanical theory for the interfacial region. Nevertheless, it seems clear in the light of present knowledge that the magnitude of AX will be of the order of liquid phase intermolecular distances-Le., 2 to 4 X lo-* cm. Referring to Figure 8, we see that the effect of curvature on surface tension only becomes appreciable when the radius of curvature, k-l, is itself approaching molecular dimensions. Furthermore, as Buff (9, 70, 74) VOL. 6 0
NO. 3
MARCH 1968
67
A consistent and versatile thermodynamic formalism can be developed for the region of mutual contact between
fluid
phases
has emphasized, the integration of Equation 66 should properly be interpreted as yielding only a first-order or asymptotic correction to the surface tension of a planar interface. I n view of Equation 73, we then conclude that for all reference surfaces, 7 -7,(l
- 2AA,k
+ . . .)
(74)
Here, the quantities 7, and Ah, refer to the planar interface, k = 0. We also see that, as a consequence of Equation 67, we can write Equation 74 as ~ - ~ m + 2 ~ m k + ..
.
(75)
where c, refers to the r = 0 reference surface. It should be pointed out that Equation 74 provides a simple and straightforward means of assessing so-called "pore-size" measurements obtained by applying the Kelvin relation (5, 23). The latter is based on Equation 39, implicitly assuming the c = 0 convention. I t is also assumed that the surface tension is curvature-independent. According to Equation 74, then, the difference between the calculated pore size, RKelvin, and the radius of curvature corresponding to the surface of tension will not exceed 2jAAmI. However, a physically more appealing choice of the radius of curvature is that for r = 0. This will introduce a correction of -\AAm\, Thus, the maximum error introduced by employing the classical expression is given by the following,
R(r
= 0)
- RKelvin < IAAmI
(76)
In the range of curvatures in which the existence of such an error should be recognized, other important corrections to the usual form of the Kelvin equation are required (42). Thermodynamic measures of interfacial thickness. We have seen in the preceding discussion that a consistent and versatile thermodynamic formalism can be developed for the region of mutual contact between fluid phases. An essential feature of the formalism is the use of a reference surface, the position of which within the interfacial zone must be specified by convention. This device is introduced precisely because a detailed description of the structure of this zone cannot be given in terms of macroscopic variables. 68
INDUSTRIAL AND ENGINEERING CHEMISTRY
It was pointed out, however, that when two such reference surfaces are considered, the distance between them constitutes a well-defined and useful measure of the thickness of the interfacial zone. That is to say, this distance provides a measure of the steepness of the mass, energy, and entropy density profiles which characterize the interface. I t has only recently been pointed out by Buff (72) that such measures are in fact accessible in terms of the measured thermodynamic properties of an interface and its contiguous bulk phases. To illustrate this, we again consider a single component system and assume a planar interface. Two characteristic distances can then be defined as AX+ = A A - = {A"(,
- A"(U
o))~,~ = 0 ) - xo(r= o ) J J G 0 = 0)
=
(77a) (77b)
Employing appropriate thermodynamic transformations in conjunction with Equations 46 and 47 then yields (72)
s
Here, 0 and are the molar energy and entropy for the coexisting bulk phases. The values of these bulk phase properties are often defined with reference to arbitrarily chosen reference states. I t then follows that if AA* and AA- are computed using such '(standard state" values, these distances will also be dependent on the reference states chosen. I t is possible, however, to choose reference states for 0 and such that the distances AX+ and AA- will have physical significance as measures of the thickness of the interfacial region. Computations based on such reference states have been carried out by Buff and co-workers (75) for some simple fluids over a wide range of temperatures, approaching the critical temperature. In connection with these thermodynamic measures of interfacial thickness, it should be recalled that molecular theories of surface tension are frequently based on
s
rather simple assumptions concerning the orientation and packing of molecules in a single “surface layer.” A typical recent example of a theory based on such a viewpoint is the significant structure theory of surface tension (47). T o the extent that approaches of this kind are successful, we should expect that measures of the interfacial thickness which are physically significant will not differ greatly from the liquid phase intermolecular distances. Interface Configurations
The Laplace equation. The condition for the mechanical equilibrium of a portion of a fluid system, within which an interfacial region subsists, has been treated in detail in the preceding sections. Hydrostatic principles extend the thermodynamic analysis. By this means, the intuitive mechanical approach is shown to be equivalent to writing Equation 39 for a particular reference surface, the surface of tension. We may then, following traditional usage, refer to this expression as the Laplace equation of capillarity. The range of validity of the Laplace equation can also be assessed if the extended formalism introduced by Buff (10, 77) is employed. For this purpose we substitute Equation 39 in the expression for the free energy, @, which corresponds to the thermodynamic definition of the interfacial tension, Equation 32. If now phase a is taken to be a spherical liquid drop of radius R (defined on the I‘ = 0 surface), and if Equation 75 is employed, we obtain
This result is clearly to be interpreted as an asymptotic expansion for the free energy in terms of the geometrical parameters characterizing the system. Since the magnitude of the succeeding constant term in Equation 79 is unknown, the physical significance which may be attached to the term involving the parameter c,(r = 0) is restricted. That is, this term only provides an upper bound for the error which results when all terms beyond the ym term are omitted. Consequently, as Buff has emphasized (II), Equation 79 provides a means of quantitatively accounting for the breakdown of thermodynamic concepts for fluid phases of very small size. Turning now to a consideration of the application of the Laplace equation in broader physical contexts, we may briefly mention several well-known problems. These problems arise because in most physical applications, the interfaces of interest are not spherical in shape. However, in such cases the Laplace equation provides a differential equation by means of which the shapes or configurations of the fluid interfaces within the system can be determined. Thus, in the first instance, the boundary conditions for the Laplace equation must be specified. A second general problem then involves the mathematical solution of the differential equation itself. Finally, the stability requirements applicable
both to the boundary conditions and to the interfacial configurations must be taken into account. With respect to boundary conditions, the physical situation which must be treated is that corresponding to the mutual confluence of three interfacial regions. When these regions can be characterized as fluid interfaces, the required analysis has been carried out in detail by Buff and Saltsburg (76, 77). This analysis leads to the description of the confluent zone as a three-phase line of contact. Associated with this description is a onedimensional formulation of the thermodynamics and hydrostatics applicable to such a region. It is much more frequently the case, however, that only two fluid phases are present in the system. Hence, the fluid interfaces are terminated by contact with a solid phase. Under the assumption, introduced by Gibbs (28), that the thermodynamic state of the solid phase is not perturbed by the presence of the fluid phases, a suitable thermodynamic treatment of the three-phase line of contact can be formulated (44). This discussmn provides a thermodynamic context for the classical equation of Thomas Young, which relates the several interfacial tensions to the contact angle. The latter quantity then specifies, in conjunction with the configuration of the solid phase surface, the boundary condition‘ for the Laplace equation. The mathematical problems encountered in obtaining solutions to the Laplace equation can be minimized by ensuring that the contact angle is uniform over the line of contact and by choosing the configuration of the solid surface in such a way that the fluid interface (surface of tension) is a surface of revolution. The classical development of the subject of capillarity was largely concerned with such solutions, as is indicated by the standard treatises in the field (2, 3). In most cases, the axis of symmetry is the vertical axis, as defined by the gravitational field. The mean curvature then varies directly as the vertical distance. The numerical solutions appropriate in this situation are well known. If the physical problem of interest is such that relatively complex solid surface configurations are involved, the solution of the Laplace equation will present serious difficulties. As an example, we may consider a surface of revolution such that the Gaussian curvature is everywhere negative-i.e., an anticlastic surface. Even if the gravitational field can be ignored, so that the surface is characterized by a constant mean curvature, the solution will in this case require the use of incomplete elliptic integrals (42). I n the general case, we may represent such a surface in a Cartesian coordinate system by the function, z(x,y), as in Figure 2. The differential equation which describes such a surface is then given in vector notation by Equation 8a. When expressed in Cartesian form, this equation is written as r(1
+ q2) - 2&s + t ( 1 + F 2 )
= J(1 + p 2
+ 42}3’2
(80)
where
az VOL 6 0
NO. 3 M A R C H 1 9 6 8
69
Thus, the general case corresponds to a nonlinear, second-order partial differential equation of the elliptic type. Stability conditions. As has been stated, any solution to the Laplace equation which is intended to describe a physical situation will be subject to stability requirements. The first of these requirements relates to the thermodynamic interpretation of the boundary condition when the contact angle is zero. Such a situation is encountered if one of the fluid phases (the wetting liquid) spreads over the solid surface, at least to the extent that is permitted by the amount of the liquid which is present in the system, The question then arises as to the thermodynamic significance of a contact angle of zero magnitude. Following Gibbs (28), a stability condition can be invoked under such circumstances. This requirement is applicable to the thin film of the spreading liquid which is present at the interface between the solid and the nonwetting fluid (gas or immiscible liquid). Such a film, of course, is to be regarded as arising from an adsorption process and not from the spreading process. The stability requirement (28, 31, 45) then provides the appropriate thermodynamic interpretation of the observed contact angle. The second stability condition is concerned with the configuration or shape of the fluid interface described by the Laplace equation. Depending on the constraints imposed by the solid surface configuration, the fluid interface may be unstable to small displacements. Such behavior will in fact be possible when the interface is situated within commonly encountered “porespace” channels. Thus, the solid surface will often be such that the pore walls have a converging or diverging radial dimension. We may then consider the set of unit normal vectors (the sense being defined as in Figure 2) for a curved fluid interface bounded by walls of this type. Within this set there will be a vector which is oriented parallel to the pore axis. If this vector points in the direction of increasing pore diameter, the fluid interface will be unstable with respect to small perturbations of the pressures in the contiguous bulk phases. Conversely, if the interface curvature is in the opposite sense, stability will be maintained. This particular kind of stability requirement was also noted by Gibbs (28). I t has, however, received little attention by later authors, except in the field of soil physics. The pore configurations which are envisaged in this field are of the type just described. Thus, the hysteresis in the capillary pressure us. fluid saturation relationship which is encountered in fine-grained porous materials (29, 46) can be explained in terms of the stability limits for fluid interface configurations (44,53, 54). Acknowledgment
I should like to thank Professor F. P. Buff of the University of Rochester for comments and interpretations relating to several of the more important concepts discussed in this work. I n addition, I a m indebted to lectures given by Professor F. 0. Koenig of Stanford 70
INDUSTRIAL A N D E N G I N E E R I N G CHEMISTRY
University for insights concerning many aspects of the thermodynamic treatment and to discussions with Dr. W. R . Foster of Mobil Research and Development Corp. for essential features of the hydrostatic treatment. Appreciation is also expressed to the Mobil Research and Development Corp. for permission to publish this work. REFER ENCES (1) Aris, R., “Vectors, Tensors, and the Basic Equations of Fluid Mechanics,”
Chap. 5, Prentice-Hall, Englewood Cliffs, 1962. (2) Bakker G “Ka illaritat ‘und Oberflachenspannung,” in “Hondbuch der Expcrimedrolpjr;rik,”$01. VI, Akademische Verlagsgesellschaft, Leipzig, 1928. (3) Bouasse, H., “CapillaritC, phenomknes superficiels,” Delagrave, Paris, 1924. (4) Brand, L., “Vector and Tensor Analysis,” Chap. VIII, Wiley, N e w York, 1947. (5) Brewer, D.F.,Champeney, D. C., Proc. Phys.Soc. 79,885 (1962). (6) Brunauer, S., “The Adsorption of Gases and Vapors,” Chap. 5, Princeton Univ. Press, Princeton, 1943. (7) Buff, F. P., Adoan. Chem. Ser., No. 93,340 (1961). ( 8 ) Buff, F. P., Discusxioar Faraday Soc., No. 30, 52 (1960). (9) Buff, F. P., J . Cham. Phys. 19, 1591 (1951). (IO) Zbid., 23, 419 (1955). (11) Zbid., 25, 146 (1956). (12) Buff, F. P., “Range of Surface Interactions,” Paper No. 1, 38th National Colloid Symposium, Austin, Tex., June 11-13,1964. (13) Buff, F. P., “The Theorv of Capillarity,” in Handbuch der Physik, Vol. X,pp. 281-304, Springer-Verlag, Gerlin, 1960. (14) Buff, F. P., 2. Elektrochem. 56, 311 (1952). (15) Buff, F. P.,Lovett, R. A., Stewart, C. W., Vicieli, J., to be published. (16) Buff, F. P., Saltsburg, H., J . Chcm. Phys. 26, 23 (1957). (17) Ibid., 26, 1526 (1957). (18) Byers, H.R., Chary,S. K., 2eits.JiirAngew. Math. Phys.14,428 (1963). (19) Carman, P. C., J . Phys. Chem. 57, 56 (1953). (20) Defay, R.,Prigogine, I., Bellemans, A., Everett, D. H., “Surface Tension and Adsorption,” Wiley, New York, 1966. (21) Eisenhart, L. P., “Treatise on the Differential Geometr of Curves and Surfaces,” pp. 177-9, Ginn, New York, 1909; Dover reprint, d w York, 1960. (22) Eisenhart, L. P., Zhid., pp. 141-5. DeWitt,T. W., J. Am. Chem.Soc. 65,1253 (1943). (23) Emmett, P. H., (24) Everett, D. H., “Adsorption Hysteresis,” in “The Solid-Gas Interface,” Vol. 2, E. A. Flood, Ed., pp. 1055-1113, Dekker, New York, 1967. (25) Feder, J., Russell, K. C., Lothe, J., Pound, G. M., Aduan. Physics 15, 1 1 1 (1966). (26) Foster, W.R., Mobil Research and Development Corp., personal communication. (27) Gauss, K . F., “General Investigations of Curved Surfaces,” tr. by J. C. Morehead and A . M . Hiltebeitel, pp. 9-1 l, Princeton University Library, 1902. (28) Gibbs, J. W., “Scientific Papers,” Vol. 1, pp. 219-331, Longmans, London, 1906; Dover reprint, New York, 1961. (29) Haines, W. B., J . Agri. Sci. 20, 97 (1930). (30).Harasima, A,, “Advances in Chemical Physics,” Vol. 1, pp. 203-37, Interscience, New York, 1958. (31) Harkins, W. D.,Livingston, H. K., J. Cham. Phys. 10, 342 (1942). (32) Haynes, J. M.,“Capillary Properties of Some Model Pore Systems,” Ph.D. Thesis, University of Bristol, 1965. (33) Hil1,T. L., J . Cham. Phys. 19,1203 (1951); J . Phys. Chem. 56, 526 (1952). (34) Kirkwood, J. G., Buff, F. P., J. Chem. Phys. 17, 338 (1949). (35) Koenig, F. O.,Zbid., 18, 449 (1950). (36) Koenig, F. O.,2. Elektrochem. 57, 361 (1953). (37) Kondo, S.,f. Chem. Phys. 25, 662 (1956). (38) Kondo, S., J. Phys. SOC. Japnn 10, 381 (1955). (39) Kreysig, E,, “Differential Geometry,” pp. 277-9, University of Toronto Press, Toronto, 1959. (40) Kreysig, E.,Ibid., pp. 186-190. (41) Lu, W‘.-C., Jhon, M . S., Ree,T., Eyring, H., J . Chem. Phys. 46,1075 (1967). (42) Melrose, J. C., A.Z.Ch.E. Journal 12, 986 (1966). (43) Melrose, J. C., J . Colloid Sci. 20, 801 (1965). (44) Melrose, J. C., Soc. Petrol. Engrs. J . 5, 259 (1965). (45) Melrose, J. C.,“Symposium on Wetting,” S.C.I. Monograph No. 25,pp. 12343, London (1967). (46) Miller, E. E., Miller, R. D., 1. Apfil. Phys. 27, 324 (1956). (47) Morrow, N.R., Harris, C. C., Soc. Petrol. Engrr. J . 5, 15 (1965). (48) Ono, S., Kondo, S., “Molecular Theory of Surface Tension in Liquids,” in Hnndbach dtr Physik, Vol. X, pp. 134-280, Springer-Verlag, Berlin, 1960. (49) Osgood, W. F., “Advanced Calculus,” pp. 121-3, Macmillan, New York, 1925. (50) Plateau, J. A. F., “Statique expCrimentale et theorique des liquides,” Vol. 1, Gauthier-Villars, Paris, 1873. (5!) Ross, S.,ed., “Chemistry and Physics of Interfaces,” Am. Chem. SOC.,Washington, D. C., 1965. (52) Rusanov, A. I., Colloid J . (USSR) 27, 360 (1965). (53) Smith, W.O.,Physics 4, 184 (1933). (54) Smith, W. O.,Foote, P. D., Busang, P. F., Zbid., 1, 18 (1931). (55) Sokolnikoff,I. S.,“Tensor Analysis,” Chap. VI, Wiley, New York, 1951. (56) Tolman, R. C., f. Chem. Phys. 16, 758 (1948). (57) Zbid., 17, 118 (1949). (58) Ibid., p. 333. (59) Wade, W. H., J . Phys. Chem. 68, 1029 (1964). (60) Weatherburn, C. E., Quart. J. Moth. 50, 230 (1925).
