J. Phys. Chem. 1986, 90, 125-127
125
Hydrostatic Approach to Caplllarlty L. Boruvka, Y. Rotenberg, and A. W. Neumann* Department of Mechanical Engineering, University of Toronto, Toronto, Ontario, Canada M5S 1 A4 (Received: April 15, 1985; In Final Form: September 6, 1985)
A hydrostatic treatment of capillary systems containing a single interface is presented. This nonthermodynamic theory is based on the concept of virtual work and the concept of parallel dividing surfaces. The hydrostatic results are shown to support and supplement the generalized thermodynamic theory of capillarity.
Introduction When Gibbs' developed the classical theory of capillarity more than 100 years ago, he suggested some possibilities for refining and generalizing his theory. Subsequent developments in differential geometry and calculus of variations provided sufficient mathematical tools to pursue Gibbs' suggestions. The first attempt at a generalized theory of capillarity was carried out by Buff et aLZ4 in the 1950s. A second, improved version of the generalized theory was presented by two of the present authors5 in 1977. The basis for a thermodynamic treatment of interfaces is a fundamental equation for a dividing surface. Gibbs postulated a form of this equation which included two curvature terms but subsequently eliminated both of them. Buff retained the first curvature term as the cornerstone of his generalized theory. Our own 1977 fundamental equation for surfaces not only retains both curvature terms but also is of a different form. It eliminates problems with mixing intensive and extensive thermodynamic parameters and also avoids the use of noninvariant surface geometric properties. However, it still is a postulated equation and as such remains in need of verification and interpretation. Also, in retaining the second curvature term, we ignored the question of whether this term is or is not negligible for all practical purposes. This is where the hydrostatic approach to capillarity makes its contribution. Buff2 has shown that this nonthermodynamic approach to capillarity, which is based on the recognition of an interfacial stress tensor field, is capable of an independent verification of some of the results of his generalized thermodynamic theory. In Buffs hydrostatic theory the excess hydrostatic equation is integrated across the interface. This procedure leads to the Laplace equation, Le., one of the equilibrium conditions. However, our motivation was not to derive merely an equilibrium condition but to corroborate the form of the proper fundamental equation for curved interfaces. The necessary connection between the hydrostatic approach and thermodynamics can be attained by considering virtual work. The principle of virtual work is equivalent to the minimum principle for the thermodynamic free energy. Finally, the free energy can be linked to the internal energy formulation through the conventional Legendre transformations. For this reason the present hydrostatic approach is based on a virtual work principle rather than the hydrostatic equation. Hydrostatic Equation The basic equation of hydrostatics can be written as
?.a where
balances the internal forces in a fluid body represented by 2 with the external body forces represented by the potential, $. Equation 1 can also be viewed as a momentum balance equation for a stationary fluid, and it can be obtained by reducing the momentum conservation equation of fluid dynamics. The hydrostatic treatment of capillarity starts by assuming that eq 1 holds everywhere inside the fluid with 8 and p varying quickly but smoothly across the interface. This means, from the molecular point of view, that a time averaging has been applied and that any further molecular considerations are eliminated in favor of the usual continuum treatment of fluids. Inside each bulk phase, away from the influence of an interface, the stress tensor becomes isotropic (diagonal) 5 = -pi
(2)
where P is pressure and i is a unit tensor. So, inside a bulk phase the hydrostatic equation reduces to
dP = -pa+
Next, a dividing surface is placed within an interface and the bulk properties, P and p , are extrapolated such that they satisfy eq 3 right up to the dividing surface:
ape= -ped$
(1) J. W. Gibbs, "The Collected Works of J. Willard Gibbs", Vol. I, Longmans, Green and Co., New York, 1928. (2) F. P. Buff, J . Chem. Phys., 25, 146 (1956). (3) F. P. Buff and H. Saltsburg, J . Chem. Phys., 26, 23 (1957). (4) F. P. Buff and H. Saltsburg, J. Chem. Phys., 26, 1526 (1957). (5) L. Boruvka and A. W. Neumann, J. Chem. Phys., 66, 5464 (1977).
0022-3654/86/2090-0125$01 S O / O
(4)
The extrapolated bulk properties, denoted by subscript e, exhibit finite jumps across the dividing surface, in contrast to the smooth actual properties. The interfacial excess quantities
will satisfy the excess hydrostatic equation
d'aE = pEd#
(7)
which follows by adding eq 1 and 4. Both sides of the excess hydrostatic equation have finite jumps across the dividing surface and are quickly vanishing away from it.
Stress Tensor Field The following parametrization of the position vector, 2, in the space occupied by the interface has been considered by Buff2 2 = d(u,v,A) = F ( U , V ) + An'(u,v) (8) where 7 is the position vector of a selected dividing surface and n' is the unit normal to the surface
= pa4
a is the stress tensor and p is the density. The equation
(3)
(9) I
Y
"I
By definition, the surfaces, d (A = constant), represent a oneparameter family of parallel surfaces, with the dividing surface at A = 0. Using the above parametrization, Buff assumed the following form of the stress tensor a= + aNn'n' (10) where i, is the unit surface tensor 0 1986 American Chemical Society
(i, = i - En').
Equation 10
126 The Journal of Physical Chemistry, Vol. 90, No. 1 , 1986
shows that at any point the normal, n', is an axis of symmetry of the assumed stress tensor. The corresponding excess stress tensor will be of the same form + ?E = UTE12 UNEn'n' (1 1)
+
as can be seen from eq 6 and 10. The assumed stress tensor field is exact only for a spherical interface. In this case the normal and tangential excess stress components (oNEand uTE)will be some functions of X alone. Otherwise, in the cases with nonuniform curvatures, eq 11 is expected to be a good approximation of the actual form of the interfacial excess stress tensor. Buffs approximation of the interfacial excess stress tensor, eq 11, appears indispensable in making the hydrostatic approach to capillarity tractable. Buff also provided a first indication that the use of eq 11 does lead to meaningful results.
Virtual Work in a Hydrostatic System The equation of hydrostatics, eq 1, is a necessary condition for equilibrium of a static fluid body. As such, eq 1 may be replaced by another necessary condition 6W= 0
(12)
the condition that the total virtual work in a fluid body be zero at equilibrium. The appropriate expression for 6W is SW = 6Wi SW, + 6 W o (13)
+
The first part of 6W
is the virtual work of (done by) the internal forces. Both the internal forces and the virtual displa,ceFents of 6Wl are represented by tensors: 5, the stress tensor; VSR, the virtual strain tensor. The second part of 6W
Boruvka et al. two-phase fluid body with an interface is considered, and the usual treatment is applied: the Gibbs's dividing surface is placed within the interface, the bulk properties are extrapolated, and the interfacial excess quantities are introduced. From eq 6 the virtual work of the internal forces, eq 14, separates into 6wi = 6Wie 6wiE (21)
+
where (VI is the virtual work in the extrapolated bulk phases and 6WiE
=
-SS Si;E:(?6d)d V
(23)
(V
is the interfacial excess part of 6Wi.
Hydrostatic Properties of a Dividing Surface When the actual bulk phases in a hydrostatic system are approximated by the extrapolated bulk phases, the interfacial excesses of physical properties have to be assigned to the dividing surface. Otherwise, the system totals of physical properties would not be preserved. This is the main idea of the dividing surface concept. The resulting approximation is expected to be very good because only very small displacements are required when making the transition from an excess property distributed in the thin interface to the corresponding superficial property distributed along the dividing surface. The simplest way of assigning the interfacial excess properties to the dividing surface is, again, by making use of the parallel surfaces. A volume element of an interfacial region spanned by the parallel surfaces, eq 8, can be expressed as (see ref 2, eq 14 with C, + ~2 = -J, C I C ~ = K )
+ X 2 K ) dX dA
d V = (1 - XJ
(24)
where J, K , and dA are the first (mean) curvature, the second (Gaussian) curvature, and an element of area, respectively, of the dividing surface at X = 0. The quantity (1 - XJ X2K) dA is an element of area of the A-parallel surface, as determined by the normals erected along the boundary of dA on the dividing surface. The interfacial excess mass in a hydrostatic system containing an interface is
+
is the virtual work of the-external body force (-p?$) due to the virtual displacements, 6R. The third part of 6 W
6Wo=
Sn'.i+,dl? dA
(16)
(4)
is the virtual work of thezxternal surface forces (Z.5,) due to the virtual displacements, 6R,at the boundary, A,, of the system; Z is the outward normal. If the virtual work expression is set up correctly, the hydrostatic equation should follow from 6 W = 0. This will be shown next. If one uses the identity ?.(&6k) = (?.5).6i? 5:(?6i?) (17)
+
(v)
Using eq 24, we assign this mass to the dividing surface as M E
(4
p(A) = S p E ( 1 -
(VI (4) = &Si?, the virtual work given by eq 13-16 becomes d V - JJ(A.5
- Z*iio)-6ddA (19)
(4)
(v)
Now, in order to satisfy 6W = 0 for an arbitrary Si?, the hydrostatic equation must hold everywhere inside the volume. In addition, a surface force balance Z.5 =
z.ijo
S S P (dA~ )
where
(product rule) and the Gauss divergence theorem
with F 6W= J J J ( a - 5 - p?$).Si?
= MA)=
(20)
has to be satisfied across the boundary. If the boundary is taken inside the fluid body, eq 20 will be satisfied through B = a,, the continuity of ii. Otherwise, eq 20 may hold with ii # ii0 (fluid in a stressed container). With eq 13 formulated explicitly, the hydrostatic approach to capillarity can now be based on the virtual work concept. A
XJ
X2K) dh
(25)
is the superficial (per unit area) mass density. Note that, in general, p(A) will vary along the dividing surface [pE = pE(u,u,X), J = J(u,u), K = K(u,u), and therefore p ( A ) = p ( A ) ( u , u ) ] . Also, note that the integration limits on X in eq 25 are implied by the vanishing of pE and that all centers of curvature of the dividing surface are assumed to fall outside these limits. The virtual work of the excess internal forces is the last and the most important attribute of the dividing surface to be obtained. First, by use of eq 1 1 and 24 in eq 23, this term becomes SW,, = S(rTEi2 CTN&):[( 1 - XJ X2K)a6d]dX dA (26)
-11
+
+
An approximation for 6 2 is obtained by taking the variation of eq 8 and simplified by setting 6A = 0:
62 = 6 7 +
X6Z
(27)
This approximation is in line with the assumptions already made. It means that the variation is such that the parallel surfaces
The Journal of Physical Chemistry, Vol. 90,No. 1, 1986 127
Hydrostatic Approach to Capillarity spanning the interface remain parallel and at the original distances from each other. It might be of interest to note that the term Z6X,if retained in eq 27, does not contribute to the SW,,expression in the spherical and cylindrical cases. In these situations, the contribution of Z6X vanishes due to the hydrostatic relations between uT and uN. The key step in further treatment of eq 26 is the following decomposition of the space gradient: (1 - XJ
+ X2K)?
= $2 - X f i 2 *
+ (1 - XJ + X 2 K ) an ' ~ (28)
A derivation of eq 28 follows from eq 8 and 9 a_"d fro? the definitions of the surface differential operators, V2 and V2*.6 When eq 27 and 28 are used, the double dot product in eq 26 is carried out term by term and simplified by using the following identities: (ZZ):(iisZ) = Z.6Z = 0
-l*:(ZZ) = 0
=0 -i2:(v2z) - . =(ZZ):($2*a) e2.a, i2:(e2*&) = V2*.2
(29)
The resulting expression for 6WiEcan be written as
-s1
6w,E
6Wi(A)=
+
[X062.67 Xl(92.6Z- f i 2 * - 6 i . ) - X2fi2*.6Z] dA
(4
(30) where Xk
= JX'C~TE
dX, k
0, 1, 2
(31)
It can be seen that the virtual work, 6WiE,of the excess interfacial stress tensor, a,, has been approximated by the virtual work, 6w(A), of the internal surface forces, X1,X 2 , and X,, in the selected dividing surface. In eq 30 the internal surface forces are multiplied by the corresponding virtual displacements, which may be put in a simpler form. Using the following expressions for variations of dA, J, and K (see ref 6, Vol. 2, sections 103-105) 6dA = 92-67 dA 6J = f i 2 * * 6 7 - ?y6Z - f l y 6 7
(32)
6K = - f i 2 * * 6 Z - f l y 6 7
(34)
and introducing the total curvatures, 6 and
(33)
X,by
dd = JdA, d X = KdA (35) we obtain the final expression for the virtual work of the internal forces in a dividing surface 6W$A) =
-s
$(Xo6 dA - X,6 d d
+ X26 d X )
(36)
(4
(6) C. E. Weatherburn, 'Differential Geometry of Three Dimensions", Vols. I, 11, Cambridge University Press, Cambridge, 1930.
Comparison with the General Thermodynamic Theory It has been shown that the general thermodynamic theory of capillarity5 yields, via a free energy analysis,' the following expression for the virtual work of internal forces in a dividing surface 6W,(A) = - s l ( y 6 dA
+ C16 d d + C26 d 2 )
(37)
(4
where y is the surface tension and C1and C2 are the first and the second bending moments, respectively. Equation 37 is matched term by term to eq 36 together with eq 31 by setting y = Xo = s u T E dX
(38)
C1 = -XI = - S A U T E dh
(39)
These equations show that the thermodynamic quantities y, CI, and C2 correspond to the first three moments of the tangential excess stress component, UTE, about the dividing surface at X = 0. As far as the treatment of interfaces is concerned, the exact match between eq 36 and 37 makes the hydrostatic approach to capillarity equivalent to the mechanical part of the general thermodynamic theory. As the hydrostatic approach naturally merges into the wider and more general thermodynamic theory, it becomes clear that eq 36-40 are the main results of consequence. First, it should be emphasized that eq 36 and 37 actually prove the correctness of the general fundamental equation for surfaces, as it was set up in ref 5, eq 13-22. This confirms that A, 6, and X do indeed constitute the proper set of extensive geometric properties one must consider when generalizing the classical theory of capillarity.
Conclusions A hydrostatic treatment of capillary systems with a single interface has been presented. This alternate approach to capillarity supports the general thermodynamic theory5 by proving the correctness of the thermodynamic fundamental equations used therein. The resulting hydrostatic expressions for the bending moments, C1and C,,provide a justification and a physical interpretation of these previously postulated terms. Even with no knowledge of the interfacial stress distribution, eq 38-40 will be useful in examining the aspects of selection and shifting of the dividing surface, curvature dependence of surface tension, etc. A hydrostatic treatment of capillary systems with a three-phase contact line has not been carried out. A fully general approach will have to overcome the limitations of the parallel dividing surface concept. (7) L. Boruvka, Y. Rotenberg, and A. W. Neumann, Langmuir, 1, 40 (1985).
