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Capillary penetration of liquids between periodically corrugated plates
Capillary Penetration of Liquids between Periodically Corrugated Plates A L l B O R H A N , *'~ K A M A L K. R U N G T A , *
AND
ABRAHAM MARMURt
* Department of Chemical Engineering, Pennsylvania State University, University Park, Pennsylvania 16802; and ~fDepartment of Chemical Engineering, Technion--Israel Institute of Technology, 32000 Haifa, Israel
Received December 17, 1990; accepted February 21, 1991 The capillary rise of a liquid between two sinusoidally corrugated plates in a gravitational field is considered in order to study the effect of geometrical variations perpendicular to the direction of penetration. The height of capillary rise at equilibrium is calculated over the cross section of the capillary using the method of local variations, and the effect of geometrical parameters is elucidated. It is found that the spatial variations in the rise height at the centerline are negligible while those at the capillary wall follow the variations in the capillary cross section. The centefline rise height is found to increase with the corrugation amplitude, particularly at small wavelengths. The relative enhancement in the centerline rise height compared with a parallel plate capillary is shown to be independent of the contact angle over the range of contact angles studied. Based on these theoretical predictions, it is concluded that the experimentally observed saturation front phenomenon in porous structures is probably of kinetic rather than thermodynamic origin. © 1991AcademicPress,Inc. I. INTRODUCTION The capillary penetration o f liquids into p o r o u s structures has been the subject o f num e r o u s theoretical and experimental studies due to its significant role in a variety o f technological fields. Wetting and drying o f powders and fabrics, wicking in textile and paper, water m o v e m e n t in soil or rocks, and liquid transfer in a microgravity e n v i r o n m e n t are just a few well-known examples o f capillary phenomena. The complexity o f real porous media and the lack o f m e a n s for defining local structural parameters in these systems have resulted in the use of simplified models in studies of the capillary penetration process. In the simplest model, the pore space is visualized as an array o f isolated cylindrical capillaries. However, such a m o d e l ignores local details and is, therefore, incapable of describing essential features such as hysteresis a n d spatial saturation gradients which have been observed experimentally ( 1 - 5 ) . A n u m b e r o f m o r e sophisticated approaches have been devised to a c c o u n t for the effect o f l To whom correspondence should be addressed.
local variations in the structure o f porous media on capillary penetration. Models o f capillaries with n o n u n i f o r m cross sections have been formulated (2, 5 - 8 ) to a c c o u n t for the variations in the direction o f penetration, while treatments o f penetration into ordered packings o f spheres have attempted ( l, 5-7, 9) to define the geometry o f the porous m e d i u m in m o r e detail. Both o f these models are capable o f predicting hysteresis phenomena. However, n o n e o f the existing models can explain, even qualitatively, the observed saturation gradients at equilibrium. A different a p p r o a c h has recently been formulated by M a r m u r (10), which uses periodic functions as a first-order a p p r o x i m a t i o n o f the local variations in the structural properties o f a porous m e d i u m . This procedure is suitable for a n y periodic function which is believed to represent the structure o f the porous m e d i u m m o r e realistically. This model seems to predict m a n y o f the experimentally observed equilibrium p h e n o m e n a without attempting to define the geometry o f the p o r o u s m e d i u m . Hence, it seems feasible that this line o f development could be extended to model structural varia-
425 0021-9797/91 $3.00 Journal of Colloid and lnteoCace Science, Vol. 146, No. 2, October 15, 1991
Copyright © 1991 by Academic Press, Inc. All rights of reproduction in any form reserved.
426
BORHAN,
RUNGTA,
tions perpendicular to the direction of penetration and their effect on the spatial saturation gradients at equilibrium. In this work, we use periodic functions as a first-order approximation of the local structural variations perpendicular to the direction of penetration and consider the capillary rise of a liquid between two periodically corrugated plates in a gravitational field. The height of capillary rise at equilibrium will be calculated using the method of local variations, and the effect of structural parameters (e.g., amplitude and wavelength of corrugation) will be elucidated. II. T H E V A R I A T I O N A L
PROBLEM
Consider the capillary space between two sinusoidally corrugated plates containing two different homogenous fluids with densities p l and p2 in a constant gravitational field, g, as shown in Fig. 1. Both fluids are connected to infinite reservoirs. Let ai be the interfacial tension on the boundary Si of the vessel walls with the ith fluid (i = 1, 2), and let a be that on the interface between the two fluids. In the absence of surface tension, the fluid interface will be a plane in the equilibrium state. This plane is taken as the xy-plane of the rectangular Cartesian coordinate system considered here, with the z-axis directed opposite to the gravitational field. In dimensionless form, the cross section of the corrugated plates in the
AND MARMUR
xy-plane can be described by an equation of the form y=+)7(x)=+_
. [2~rx'~] l+Asln[---~-)j,
where A and X are the amplitude and wavelength of the sinusoidal corrugation, respectively, made dimensionless by w, half the gap width of the plates at x = 0. The walls of the capillary are assumed to be smooth, undeformable, and unbounded in the x and z directions. The objective is to determine the equilibrium height of capillary rise between the plates over the entire cross section. The equilibrium conditions include the Young-Laplace equation on the interface between the two fluids 2H~ = 2xP,
[ 2a]
where H is the local mean curvature and &P is the pressure drop across the interface, and the Duprr-Young condition on the contact line cos 02 = al - ~2,
h 2dxdy + ff-L~SI + az S2 + S, O"
Journal of Colloid and Interface Science, Vol. !46, No. 2, October 15, 1991
[2b]
where 02 is the equilibrium contact angle of fluid 2 with the smooth solid surface. It is well known that these equations can be derived from the variational principle of stationary potential energy ( 1 1 ). In this work, we use the variational approach to the equilibrium problem of determining the stationary state rather than solving the two-dimensional nonlinear elliptic PDE given by [2a]. The potential energy for the mechanical system under consideration (fluids + corrugated plates) can be written as U = ~ Bo
FiG. 1 A. sketch of the periodically corrugated capillary.
[1]
O"
[3]
with the dimensionless Bond number defined as Bo = (t)2 - 01)g602/o'. Here, h ( x , y) is the elevation of the interface above the xy-plane and D is the projection of the surface S on this
PENETRATION BETWEEN CORRUGATED PLATES plane. Using [2b], functional [3] may be written, with accuracy to within a constant term, as
V(h))=ffo[Vl × dxdy - cos 02 fr hdl, [4] where F is the boundary of region D. Thus, the problem reduces to finding the continuously differentiable function h(x, y) defined in region D which minimizes the functional U(h) in [4 ]. We shall use the method of local variations, developed by Petrov and Chernougko (12), in conjunction with finite differences to achieve the minimization of the functional [4] for Bo > 0. The capillary geometry has one plane of reflective symmetry (xz-plane) and is periodic in the x direction. Hence, the computational domain is taken as one-half the capillary cross section bounded by I'per, I~sym, and Psolid denoting the periodic, symmetry, and solid wall boundaries for one wavelength, respectively (Fig. 2). Along I'solid the meniscus meets the capillary wall at the prescribed contact angle, 02, such that
-hx__fix+hy 1/1 + hx2 + hy2 ~ J
=[
N.n
]= cos 02,
-2
[Sa]
427
where N is the unit outward vector everywhere normal to the interface and n is the inward directed unit normal to the capillary wall. This condition will be automatically satisfied by the function h which minimizes [4]. On Pper, the function h(x, y) satisfies the periodic boundary conditions h(0, y) = h(X, y)
[5bl
hx(O, y) = hx(X, y),
[5c]
and
while on Fsy~, it satisfies the symmetry condition
hy(x, 0) = 0.
[5d]
IlL NUMERICAL PROCEDURE The variational difference solution of the problem is greatly facilitated if the difficulty associated with the existence of a sinusoidal boundary is removed by introducing a transformation of the form = x n
[6a]
Y y(x)
[6b]
This transformation maps the physical domain, D, into the rectangular region D' in the ~7/-plane given by 0 ~ ~ ~< X, 0 ~ ~/~ 1. The transformed variational problem then becomes the minimization of
U(h) 1
h,)2+
Fsolid
rpor
J
rpor
................ i
Boh2"l
+ -~ ];d~d~7-cos02 fr, hV1 + fi~d~
1
[7a] subject to
FIG.2. Crosssectionof the sinusoidallycorrugatedcapillary in the xy-plane.
h(0, n) = h(X, 7)
[7b]
hal0, 7) = h~(X, 7)
[7c1
h,((, 0) = 0
[7d]
Journal of Colloid and Interface Science, Vol. 146, No. 2, October 15, 1991
BORHAN, RUNGTA, AND MARMUR
428
and to condition [5a] on the contact line which takes the form
+(1 +372 + h~) 1/2cot 02].
[7e]
The computational domain, D', is divided into cells by a collection of points, ( ~ , nj), whose coordinates are given by ~=iA~;
A~=--; m
nj=jAT/;
An=-;
1
n
i=0 .....
m
j=0 .....
n.
Expression [ 7a] is then approximated as
U(h) ~- E Io + Z X~, Lj
[8a]
i
where I a and K~, respectively, represent the contributions to the integrals over D' and its boundary I" in [7a] and are given by
h~,j] Bo
\ 37~/
-2]
+ 2 - h;j 37,~A. (i=0,...
,m- 1; j=0,.
. .,n-
1)
[8b] IV. R E S U L T S A N D D I S C U S S I O N
Ki = - cos 02hi; Vl + 37~i/x~ (j=n;i=O,
1. . . . .
Starting with an initial approximation h ~.0) ( i = 0..... m; j = 0 . . . . , n) and a small variation step, 6, the values of h a at iteration k are varied one point at a time according to hO (k+l) =.-,J h!k) + f i o r , h!k+D -J = . ,h, j!k) - 6, such that the RHS of [8a] is reduced. If neither variation decreases the RHS of [ 8a], htr/j!k+l) is set equal to h ~). After each variation, the values of h ~k) are updated according to [ 8e ] over the entire domain. Clearly, the functional U monotonically decreases after each iteration. This process is repeated until the relative change in the value of the potential, I Au(K)/ u(k) l, falls below a specified tolerance value. The value of 6 is then reduced and the calculations are repeated to achieve the desired degree of accuracy. Finally, the values of m and n in our calculations are increased until a further change in the mesh size does not alter the values of h 0. In most of our calculations, an initial approximation of ,*h lJ!9) = 0 was used, although several calculations with other initial approximations were performed to check for proper convergence of the algorithm. The minimum variational step, 6, was 10 -8 and the largest mesh size in the worst case corresponded to m = n = 70. A typical tolerance of 10 -5 for the value of I Au(k)/u(K)] was used in the convergence criterion.
m-l).
[8c]
In expressions [ 8b ] and [ 8c], the quantities with carets represent the average values over the four vertices of each cell, the spatial derivatives of h are determined using central differencing, and h i j = l[~li, j -]- h i + l , j ]
[8d]
hij = ~[hi, j+l + hi.j-1 + hi+~j + h i - j
[8e]
hij = h ( ~ , nj)
[8f]
y i = 1 + A sin ( ~ - ~ Z ) .
[8g]
Journal of CoUoid and Interface Science, Vol. 146, No. 2, October 15, 1991
Equilibrium rise heights and meniscus shapes for the sinusoidally corrugated plates have been calculated over a wide range of parameters 02, X, and A, using the algorithm described in the previous section. To check the accuracy of our numerical procedure, the results for the limiting case of A = 0, corresponding to a parallel-plate capillary, were also determined through a numerical solution of the Young-Laplace equation (Eq. [2a] ). Figure 3 shows a comparison of these results with those obtained from the method of local variations for contact angles of 45 ° and 60 ° . Clearly, the results are virtually indistinguishable over the entire cross section of the cap-
P E N E T R A T I O N BETWEEN C O R R U G A T E D PLATES
429
4), particularly for large amplitudes of corrugation. On the other hand, the spatial variations in the liquid height at the capillary wall seem to follow the capillary geometry; that is, the variations in h near the capillary wall are 6 sinusoidal in nature, with the largest rise height occurring near the widest region of the capillary (around ~ = k/4) and with the variations increasing with increasing amplitude of corrugation. Evidently, the oscillations in the height of the liquid are averaged at relatively 0.0 O. 1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 short distances from the wall. Y A comparison of the centerline rise heights FIG. 3. The equilibrium meniscus profile forA = 0 and in Figs. 4-6 clearly indicates a major increase Bo = 0.1. ([5) Variational method (02 = 45°); (A) variin this quantity with amplitude of corrugation. ational method (02 = 60 ° ). The solid curve represents the This is not surprising since an increase in the solution of Eq. [23] for the corresponding equilibrium amplitude results in larger solid-liquid and contact angle. solid-vapor areas while leaving the cross-sectional area of the capillary unchanged. Conillary and our numerical procedure has con- sequently, the difference between the solidvapor and the solid-liquid interfacial energies, verged satisfactorily. The meniscus profiles, h (~, n), for the case which is the driving force for penetration, is of ~, = 2 are shown in Figs. 4-6 for various increased while the liquid-vapor interfacial amplitudes. It is interesting to note that the energy remains unchanged. This leads to a spatial variations in the centefline (~7 = 0) rise higher rise of the liquid since the decrease in height are negligible in contrast to the intuitive free energy gained by lowering the solid-vapor expectation of larger rise heights in the narrow area outweighs the gravitational energy inregions of the capillary (i.e., around ~ = 3k/ crease due to the higher rise of the liquid. Fig-
5.33
5.13
503
2.00
~
~
~133 0.00
FIG. 4. Meniscus profile, h(~, 7), for ~, = 2, A = 0.1 and Bo = 0.1, 02
=
60 °.
Journal of Colloid and Interface Science, Vol. 146, No. 2, October 15, 1991
430
BORHAN,
RUNGTA,
AND
MARMUR
I'. 6.18
6.09 ]
5.8,9~
0.67 "
T/
0.33
0 O0 0.00
FIG. 5. M e n i s c u s profile, h(~, 7 ) , f o r X = 2, A = 0.3 a n d Bo = 0.1, 02 = 6 0 °.
ure 7 demonstrates the effect of amplitude on the centerline rise height for different values of corrugation wavelength. Apparently, the enhancement in the rise height with increasing amplitude is most pronounced for smaller wavelengths, as expected based on the above reasoning. For X = 5, there is almost no increase in the centerline rise height with amplitude, and the rise height remains close to that in a parallel-plate capillary (A = 0). Spatial variations in the geometry of the capillary seem to have little effect on the equilibrium
rise height for wavelengths of corrugation greater than about 5. The effect of the equilibrium contact angle on the observed trends can be realized from a comparison of Figs. 7-9. The increase in the centerline rise height with the amplitude of corrugation is more pronounced at lower contact angles due to the larger driving force at the contact line. However, the relative enhancement in the centefline rise height compared with the parallel-plate capillary seems to be about the same for the range of contact
7.52
7.42
2.00
0.33
0 O0 0.00
FIG. 6. M e n i s c u s profile, h(~, 7 ) , for X = 2, A = 0.5 a n d B o = 0,1, 02 = 6 0 °. Journal of Colloid andlnterface Science, Vol. 146, No. 2, October 15, 1991
PENETRATION
BETWEEN
CORRUGATED
PLATES
431
12-
•
1o
#J c~
9-
L, q) ..~
8-
w
o
7-
6
rn
c
......... 0.0
J ......... 0,1 Amplitude
i ......... 0.2 of
z ......... 0.3
Corrugation,
i ......... 0.4.
0.5
A
FIG. 7. Effect o f a m p l i t u d e o n the centerline rise height for 02 = 60 ° a n d Bo = 0.1. ( D ) ~, = 1; ( A ) ~ = 2; (*) X = 3; ( © ) X = 5.
.~
19-
i9 -r"
18-
• Is
17-
16-
--
15
tO
~) 1 1 "
9-
B0.0
0.1 Amplitude
0.2 of
0.3 Corrugation,
0.4
0.5
A
FIG. 8. Effect o f a m p l i t u d e on the centerline rise height for 02 = 30 ° a n d Bo = 0.1. ([]))~ = 1; (/~) X = 2; (*) X = 3; ( 0 ) X = 5. Journal of Colloid and Interface Science, Vol. 146, No. 2, October 15, 1991
432
BORHAN, R U N G T A , A N D M A R M U R ~_ 2 2 ~ •~
21
-i- 2O
Q:~ 18 .c 17-
c 15e cJ t 6 M
12-
c
0,0
0.1
0.2
Amplitude
0,3
of
0.4
Corrugotlon,
FIG. 9. Effect of amplitude on the centerline rise height for
02 =
0.5
A
15 ° and Bo = 0.1. ([3) k = 1; (/x) ~ = 2;
( * ) X = 3 ; ( < ? ) X = 5.
3'
O v
5 e-
2,
1, 1 . . . . . . . . .
0.0
z . . . . . . . . .
0.1
i . . . . . . . . .
0.2
Amplitude
| . . . . . . . . .
0.3
of
~ . . . . . . . . .
0.4
i . . . . . . . . .
0.5
Corrugation,
i
0.8
......... 0.7
A
FIG, 10. Relative enhancement in the centerline rise height with amplitude for different equilibrium contact angles. ( × ) 02 = 60°; (ZX) 02 = 30°; (F1) 02 = 15 °. Journal of Colloid and Interface Science, Vol. 146, No. 2, Ocfober 15, 1991
PENETRATION BETWEEN CORRUGATED PLATES
angles considered, as shown in Fig. 10. It should be noted that a contact angle of 10 ° was the lowest value used in our calculations due to numerical difficulties associated with contact angles near zero. The results of our calculations do not predict the existence of a saturation gradient for contact angles greater than 10 °, nor do they indicate a transition to such behavior as the contact angle is reduced. Based on these results, it seems likely that the reason for the experimentally observed saturation gradients is of a kinetic origin rather than a thermodynamic one. The wetting of the solid surfaces, for example, may play an important role in the development of these saturation gradients by allowing the formation of dispersed vapor pockets. This would explain the absence of the saturation gradients for nonzero contact angles and is a subject for future study. ACKNOWLEDGMENTS The authors thank Dr. A. H. Sporer for valuable dis-
433
cussions and the IBM Almaden Research Center for partial support of this work.
REFERENCES 1. Melrose, J. C., Soc. Pet. Eng. J. 5, 259 (1965). 2. Morrow, N. R., l n d Eng. Chem. 62, 32 (1970). 3. Van Brakel, J., and Heertjes, P. M., Nature 254, 585 (1975). 4. Marmur, A., in "Modern Approaches to Wettability" (M. E. Schrader and G. Loeb, Eds.). Plenum, New York, in press. 5. Van Brakel, J., Powder Technol. 11, 205 (1975). 6. Smith, W. O., Foote, P. D., and Busang, P. F., Physics 1, 18 (1931). 7. Smith, W. O., Physics 4, 184 (1933). 8. Levine, S., Lowndes, J., and Reed, P., J. Colloid Interface Sci. 77, 253 (1980). 9. Ban, S., Wolfram, E., and Rohrsetzer, S., Colloids Surf 22, 301 (1987). 10. Marmur, A., J. Colloidlnterface Sci. 129, 161 (1989). 11. Myshkis, A. D., Babskii, V. G., Kopachevskii, N. D., Slobozhanin, L. A., and Tyuptsov, A. D. "Low Gravity Fluid Mechanics." Springer-Verlag, New York/Berlin, 1987. 12. Petrov, V. M., and Chernou~ko, F. L., Izv. Akad. Nauk SSR, Mekh. Zhidk. Gaza 5, 152 (1966).
Journalof Colloidand Interface Science, Vol. 146,No. 2, October15, 1991
AND
ABRAHAM MARMURt
* Department of Chemical Engineering, Pennsylvania State University, University Park, Pennsylvania 16802; and ~fDepartment of Chemical Engineering, Technion--Israel Institute of Technology, 32000 Haifa, Israel
Received December 17, 1990; accepted February 21, 1991 The capillary rise of a liquid between two sinusoidally corrugated plates in a gravitational field is considered in order to study the effect of geometrical variations perpendicular to the direction of penetration. The height of capillary rise at equilibrium is calculated over the cross section of the capillary using the method of local variations, and the effect of geometrical parameters is elucidated. It is found that the spatial variations in the rise height at the centerline are negligible while those at the capillary wall follow the variations in the capillary cross section. The centefline rise height is found to increase with the corrugation amplitude, particularly at small wavelengths. The relative enhancement in the centerline rise height compared with a parallel plate capillary is shown to be independent of the contact angle over the range of contact angles studied. Based on these theoretical predictions, it is concluded that the experimentally observed saturation front phenomenon in porous structures is probably of kinetic rather than thermodynamic origin. © 1991AcademicPress,Inc. I. INTRODUCTION The capillary penetration o f liquids into p o r o u s structures has been the subject o f num e r o u s theoretical and experimental studies due to its significant role in a variety o f technological fields. Wetting and drying o f powders and fabrics, wicking in textile and paper, water m o v e m e n t in soil or rocks, and liquid transfer in a microgravity e n v i r o n m e n t are just a few well-known examples o f capillary phenomena. The complexity o f real porous media and the lack o f m e a n s for defining local structural parameters in these systems have resulted in the use of simplified models in studies of the capillary penetration process. In the simplest model, the pore space is visualized as an array o f isolated cylindrical capillaries. However, such a m o d e l ignores local details and is, therefore, incapable of describing essential features such as hysteresis a n d spatial saturation gradients which have been observed experimentally ( 1 - 5 ) . A n u m b e r o f m o r e sophisticated approaches have been devised to a c c o u n t for the effect o f l To whom correspondence should be addressed.
local variations in the structure o f porous media on capillary penetration. Models o f capillaries with n o n u n i f o r m cross sections have been formulated (2, 5 - 8 ) to a c c o u n t for the variations in the direction o f penetration, while treatments o f penetration into ordered packings o f spheres have attempted ( l, 5-7, 9) to define the geometry o f the porous m e d i u m in m o r e detail. Both o f these models are capable o f predicting hysteresis phenomena. However, n o n e o f the existing models can explain, even qualitatively, the observed saturation gradients at equilibrium. A different a p p r o a c h has recently been formulated by M a r m u r (10), which uses periodic functions as a first-order a p p r o x i m a t i o n o f the local variations in the structural properties o f a porous m e d i u m . This procedure is suitable for a n y periodic function which is believed to represent the structure o f the porous m e d i u m m o r e realistically. This model seems to predict m a n y o f the experimentally observed equilibrium p h e n o m e n a without attempting to define the geometry o f the p o r o u s m e d i u m . Hence, it seems feasible that this line o f development could be extended to model structural varia-
425 0021-9797/91 $3.00 Journal of Colloid and lnteoCace Science, Vol. 146, No. 2, October 15, 1991
Copyright © 1991 by Academic Press, Inc. All rights of reproduction in any form reserved.
426
BORHAN,
RUNGTA,
tions perpendicular to the direction of penetration and their effect on the spatial saturation gradients at equilibrium. In this work, we use periodic functions as a first-order approximation of the local structural variations perpendicular to the direction of penetration and consider the capillary rise of a liquid between two periodically corrugated plates in a gravitational field. The height of capillary rise at equilibrium will be calculated using the method of local variations, and the effect of structural parameters (e.g., amplitude and wavelength of corrugation) will be elucidated. II. T H E V A R I A T I O N A L
PROBLEM
Consider the capillary space between two sinusoidally corrugated plates containing two different homogenous fluids with densities p l and p2 in a constant gravitational field, g, as shown in Fig. 1. Both fluids are connected to infinite reservoirs. Let ai be the interfacial tension on the boundary Si of the vessel walls with the ith fluid (i = 1, 2), and let a be that on the interface between the two fluids. In the absence of surface tension, the fluid interface will be a plane in the equilibrium state. This plane is taken as the xy-plane of the rectangular Cartesian coordinate system considered here, with the z-axis directed opposite to the gravitational field. In dimensionless form, the cross section of the corrugated plates in the
AND MARMUR
xy-plane can be described by an equation of the form y=+)7(x)=+_
. [2~rx'~] l+Asln[---~-)j,
where A and X are the amplitude and wavelength of the sinusoidal corrugation, respectively, made dimensionless by w, half the gap width of the plates at x = 0. The walls of the capillary are assumed to be smooth, undeformable, and unbounded in the x and z directions. The objective is to determine the equilibrium height of capillary rise between the plates over the entire cross section. The equilibrium conditions include the Young-Laplace equation on the interface between the two fluids 2H~ = 2xP,
[ 2a]
where H is the local mean curvature and &P is the pressure drop across the interface, and the Duprr-Young condition on the contact line cos 02 = al - ~2,
h 2dxdy + ff-L~SI + az S2 + S, O"
Journal of Colloid and Interface Science, Vol. !46, No. 2, October 15, 1991
[2b]
where 02 is the equilibrium contact angle of fluid 2 with the smooth solid surface. It is well known that these equations can be derived from the variational principle of stationary potential energy ( 1 1 ). In this work, we use the variational approach to the equilibrium problem of determining the stationary state rather than solving the two-dimensional nonlinear elliptic PDE given by [2a]. The potential energy for the mechanical system under consideration (fluids + corrugated plates) can be written as U = ~ Bo
FiG. 1 A. sketch of the periodically corrugated capillary.
[1]
O"
[3]
with the dimensionless Bond number defined as Bo = (t)2 - 01)g602/o'. Here, h ( x , y) is the elevation of the interface above the xy-plane and D is the projection of the surface S on this
PENETRATION BETWEEN CORRUGATED PLATES plane. Using [2b], functional [3] may be written, with accuracy to within a constant term, as
V(h))=ffo[Vl × dxdy - cos 02 fr hdl, [4] where F is the boundary of region D. Thus, the problem reduces to finding the continuously differentiable function h(x, y) defined in region D which minimizes the functional U(h) in [4 ]. We shall use the method of local variations, developed by Petrov and Chernougko (12), in conjunction with finite differences to achieve the minimization of the functional [4] for Bo > 0. The capillary geometry has one plane of reflective symmetry (xz-plane) and is periodic in the x direction. Hence, the computational domain is taken as one-half the capillary cross section bounded by I'per, I~sym, and Psolid denoting the periodic, symmetry, and solid wall boundaries for one wavelength, respectively (Fig. 2). Along I'solid the meniscus meets the capillary wall at the prescribed contact angle, 02, such that
-hx__fix+hy 1/1 + hx2 + hy2 ~ J
=[
N.n
]= cos 02,
-2
[Sa]
427
where N is the unit outward vector everywhere normal to the interface and n is the inward directed unit normal to the capillary wall. This condition will be automatically satisfied by the function h which minimizes [4]. On Pper, the function h(x, y) satisfies the periodic boundary conditions h(0, y) = h(X, y)
[5bl
hx(O, y) = hx(X, y),
[5c]
and
while on Fsy~, it satisfies the symmetry condition
hy(x, 0) = 0.
[5d]
IlL NUMERICAL PROCEDURE The variational difference solution of the problem is greatly facilitated if the difficulty associated with the existence of a sinusoidal boundary is removed by introducing a transformation of the form = x n
[6a]
Y y(x)
[6b]
This transformation maps the physical domain, D, into the rectangular region D' in the ~7/-plane given by 0 ~ ~ ~< X, 0 ~ ~/~ 1. The transformed variational problem then becomes the minimization of
U(h) 1
h,)2+
Fsolid
rpor
J
rpor
................ i
Boh2"l
+ -~ ];d~d~7-cos02 fr, hV1 + fi~d~
1
[7a] subject to
FIG.2. Crosssectionof the sinusoidallycorrugatedcapillary in the xy-plane.
h(0, n) = h(X, 7)
[7b]
hal0, 7) = h~(X, 7)
[7c1
h,((, 0) = 0
[7d]
Journal of Colloid and Interface Science, Vol. 146, No. 2, October 15, 1991
BORHAN, RUNGTA, AND MARMUR
428
and to condition [5a] on the contact line which takes the form
+(1 +372 + h~) 1/2cot 02].
[7e]
The computational domain, D', is divided into cells by a collection of points, ( ~ , nj), whose coordinates are given by ~=iA~;
A~=--; m
nj=jAT/;
An=-;
1
n
i=0 .....
m
j=0 .....
n.
Expression [ 7a] is then approximated as
U(h) ~- E Io + Z X~, Lj
[8a]
i
where I a and K~, respectively, represent the contributions to the integrals over D' and its boundary I" in [7a] and are given by
h~,j] Bo
\ 37~/
-2]
+ 2 - h;j 37,~A. (i=0,...
,m- 1; j=0,.
. .,n-
1)
[8b] IV. R E S U L T S A N D D I S C U S S I O N
Ki = - cos 02hi; Vl + 37~i/x~ (j=n;i=O,
1. . . . .
Starting with an initial approximation h ~.0) ( i = 0..... m; j = 0 . . . . , n) and a small variation step, 6, the values of h a at iteration k are varied one point at a time according to hO (k+l) =.-,J h!k) + f i o r , h!k+D -J = . ,h, j!k) - 6, such that the RHS of [8a] is reduced. If neither variation decreases the RHS of [ 8a], htr/j!k+l) is set equal to h ~). After each variation, the values of h ~k) are updated according to [ 8e ] over the entire domain. Clearly, the functional U monotonically decreases after each iteration. This process is repeated until the relative change in the value of the potential, I Au(K)/ u(k) l, falls below a specified tolerance value. The value of 6 is then reduced and the calculations are repeated to achieve the desired degree of accuracy. Finally, the values of m and n in our calculations are increased until a further change in the mesh size does not alter the values of h 0. In most of our calculations, an initial approximation of ,*h lJ!9) = 0 was used, although several calculations with other initial approximations were performed to check for proper convergence of the algorithm. The minimum variational step, 6, was 10 -8 and the largest mesh size in the worst case corresponded to m = n = 70. A typical tolerance of 10 -5 for the value of I Au(k)/u(K)] was used in the convergence criterion.
m-l).
[8c]
In expressions [ 8b ] and [ 8c], the quantities with carets represent the average values over the four vertices of each cell, the spatial derivatives of h are determined using central differencing, and h i j = l[~li, j -]- h i + l , j ]
[8d]
hij = ~[hi, j+l + hi.j-1 + hi+~j + h i - j
[8e]
hij = h ( ~ , nj)
[8f]
y i = 1 + A sin ( ~ - ~ Z ) .
[8g]
Journal of CoUoid and Interface Science, Vol. 146, No. 2, October 15, 1991
Equilibrium rise heights and meniscus shapes for the sinusoidally corrugated plates have been calculated over a wide range of parameters 02, X, and A, using the algorithm described in the previous section. To check the accuracy of our numerical procedure, the results for the limiting case of A = 0, corresponding to a parallel-plate capillary, were also determined through a numerical solution of the Young-Laplace equation (Eq. [2a] ). Figure 3 shows a comparison of these results with those obtained from the method of local variations for contact angles of 45 ° and 60 ° . Clearly, the results are virtually indistinguishable over the entire cross section of the cap-
P E N E T R A T I O N BETWEEN C O R R U G A T E D PLATES
429
4), particularly for large amplitudes of corrugation. On the other hand, the spatial variations in the liquid height at the capillary wall seem to follow the capillary geometry; that is, the variations in h near the capillary wall are 6 sinusoidal in nature, with the largest rise height occurring near the widest region of the capillary (around ~ = k/4) and with the variations increasing with increasing amplitude of corrugation. Evidently, the oscillations in the height of the liquid are averaged at relatively 0.0 O. 1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 short distances from the wall. Y A comparison of the centerline rise heights FIG. 3. The equilibrium meniscus profile forA = 0 and in Figs. 4-6 clearly indicates a major increase Bo = 0.1. ([5) Variational method (02 = 45°); (A) variin this quantity with amplitude of corrugation. ational method (02 = 60 ° ). The solid curve represents the This is not surprising since an increase in the solution of Eq. [23] for the corresponding equilibrium amplitude results in larger solid-liquid and contact angle. solid-vapor areas while leaving the cross-sectional area of the capillary unchanged. Conillary and our numerical procedure has con- sequently, the difference between the solidvapor and the solid-liquid interfacial energies, verged satisfactorily. The meniscus profiles, h (~, n), for the case which is the driving force for penetration, is of ~, = 2 are shown in Figs. 4-6 for various increased while the liquid-vapor interfacial amplitudes. It is interesting to note that the energy remains unchanged. This leads to a spatial variations in the centefline (~7 = 0) rise higher rise of the liquid since the decrease in height are negligible in contrast to the intuitive free energy gained by lowering the solid-vapor expectation of larger rise heights in the narrow area outweighs the gravitational energy inregions of the capillary (i.e., around ~ = 3k/ crease due to the higher rise of the liquid. Fig-
5.33
5.13
503
2.00
~
~
~133 0.00
FIG. 4. Meniscus profile, h(~, 7), for ~, = 2, A = 0.1 and Bo = 0.1, 02
=
60 °.
Journal of Colloid and Interface Science, Vol. 146, No. 2, October 15, 1991
430
BORHAN,
RUNGTA,
AND
MARMUR
I'. 6.18
6.09 ]
5.8,9~
0.67 "
T/
0.33
0 O0 0.00
FIG. 5. M e n i s c u s profile, h(~, 7 ) , f o r X = 2, A = 0.3 a n d Bo = 0.1, 02 = 6 0 °.
ure 7 demonstrates the effect of amplitude on the centerline rise height for different values of corrugation wavelength. Apparently, the enhancement in the rise height with increasing amplitude is most pronounced for smaller wavelengths, as expected based on the above reasoning. For X = 5, there is almost no increase in the centerline rise height with amplitude, and the rise height remains close to that in a parallel-plate capillary (A = 0). Spatial variations in the geometry of the capillary seem to have little effect on the equilibrium
rise height for wavelengths of corrugation greater than about 5. The effect of the equilibrium contact angle on the observed trends can be realized from a comparison of Figs. 7-9. The increase in the centerline rise height with the amplitude of corrugation is more pronounced at lower contact angles due to the larger driving force at the contact line. However, the relative enhancement in the centefline rise height compared with the parallel-plate capillary seems to be about the same for the range of contact
7.52
7.42
2.00
0.33
0 O0 0.00
FIG. 6. M e n i s c u s profile, h(~, 7 ) , for X = 2, A = 0.5 a n d B o = 0,1, 02 = 6 0 °. Journal of Colloid andlnterface Science, Vol. 146, No. 2, October 15, 1991
PENETRATION
BETWEEN
CORRUGATED
PLATES
431
12-
•
1o
#J c~
9-
L, q) ..~
8-
w
o
7-
6
rn
c
......... 0.0
J ......... 0,1 Amplitude
i ......... 0.2 of
z ......... 0.3
Corrugation,
i ......... 0.4.
0.5
A
FIG. 7. Effect o f a m p l i t u d e o n the centerline rise height for 02 = 60 ° a n d Bo = 0.1. ( D ) ~, = 1; ( A ) ~ = 2; (*) X = 3; ( © ) X = 5.
.~
19-
i9 -r"
18-
• Is
17-
16-
--
15
tO
~) 1 1 "
9-
B0.0
0.1 Amplitude
0.2 of
0.3 Corrugation,
0.4
0.5
A
FIG. 8. Effect o f a m p l i t u d e on the centerline rise height for 02 = 30 ° a n d Bo = 0.1. ([]))~ = 1; (/~) X = 2; (*) X = 3; ( 0 ) X = 5. Journal of Colloid and Interface Science, Vol. 146, No. 2, October 15, 1991
432
BORHAN, R U N G T A , A N D M A R M U R ~_ 2 2 ~ •~
21
-i- 2O
Q:~ 18 .c 17-
c 15e cJ t 6 M
12-
c
0,0
0.1
0.2
Amplitude
0,3
of
0.4
Corrugotlon,
FIG. 9. Effect of amplitude on the centerline rise height for
02 =
0.5
A
15 ° and Bo = 0.1. ([3) k = 1; (/x) ~ = 2;
( * ) X = 3 ; ( < ? ) X = 5.
3'
O v
5 e-
2,
1, 1 . . . . . . . . .
0.0
z . . . . . . . . .
0.1
i . . . . . . . . .
0.2
Amplitude
| . . . . . . . . .
0.3
of
~ . . . . . . . . .
0.4
i . . . . . . . . .
0.5
Corrugation,
i
0.8
......... 0.7
A
FIG, 10. Relative enhancement in the centerline rise height with amplitude for different equilibrium contact angles. ( × ) 02 = 60°; (ZX) 02 = 30°; (F1) 02 = 15 °. Journal of Colloid and Interface Science, Vol. 146, No. 2, Ocfober 15, 1991
PENETRATION BETWEEN CORRUGATED PLATES
angles considered, as shown in Fig. 10. It should be noted that a contact angle of 10 ° was the lowest value used in our calculations due to numerical difficulties associated with contact angles near zero. The results of our calculations do not predict the existence of a saturation gradient for contact angles greater than 10 °, nor do they indicate a transition to such behavior as the contact angle is reduced. Based on these results, it seems likely that the reason for the experimentally observed saturation gradients is of a kinetic origin rather than a thermodynamic one. The wetting of the solid surfaces, for example, may play an important role in the development of these saturation gradients by allowing the formation of dispersed vapor pockets. This would explain the absence of the saturation gradients for nonzero contact angles and is a subject for future study. ACKNOWLEDGMENTS The authors thank Dr. A. H. Sporer for valuable dis-
433
cussions and the IBM Almaden Research Center for partial support of this work.
REFERENCES 1. Melrose, J. C., Soc. Pet. Eng. J. 5, 259 (1965). 2. Morrow, N. R., l n d Eng. Chem. 62, 32 (1970). 3. Van Brakel, J., and Heertjes, P. M., Nature 254, 585 (1975). 4. Marmur, A., in "Modern Approaches to Wettability" (M. E. Schrader and G. Loeb, Eds.). Plenum, New York, in press. 5. Van Brakel, J., Powder Technol. 11, 205 (1975). 6. Smith, W. O., Foote, P. D., and Busang, P. F., Physics 1, 18 (1931). 7. Smith, W. O., Physics 4, 184 (1933). 8. Levine, S., Lowndes, J., and Reed, P., J. Colloid Interface Sci. 77, 253 (1980). 9. Ban, S., Wolfram, E., and Rohrsetzer, S., Colloids Surf 22, 301 (1987). 10. Marmur, A., J. Colloidlnterface Sci. 129, 161 (1989). 11. Myshkis, A. D., Babskii, V. G., Kopachevskii, N. D., Slobozhanin, L. A., and Tyuptsov, A. D. "Low Gravity Fluid Mechanics." Springer-Verlag, New York/Berlin, 1987. 12. Petrov, V. M., and Chernou~ko, F. L., Izv. Akad. Nauk SSR, Mekh. Zhidk. Gaza 5, 152 (1966).
Journalof Colloidand Interface Science, Vol. 146,No. 2, October15, 1991