J. Phys. Chem. 1989, 93, 4813-4811 (see Figure 10) and new particles are nucleated and grow in the same way as for nonpercolating systems. It must be noted that a structure similar to that proposed by Ninham et al.I7 would lead to a very similar mechanism; in this case, the flexible polymer chain would be confined inside a flexible cylinder instead of a necklace of connected beads. Conclusion
The present study was performed to furnish insight into the mechanism of polymerization in inverse microemulsions and, in particular, to provide a better understanding of the process of particle nucleation and the effect of the initial microemulsion composition upon particle nucleation during the course of the polymerization. Conductivity measurements have shown that under certain conditions addition of AM to toluene/AOT/H20 microemulsions was sufficient to convert a nonpercolating system into a percolating one. The percolation threshold depends not only upon the volume fraction of the dispersed phase but also upon the quantitative relationships among the various components of the disperse phase. Nevertheless, percolation in these cases was
4873
shown to obey previously known scaling laws. The study of the evolution of the particle size with the degree of polymerization has revealed the role played by the percolation process. For nonpercolating systems, a small decrease in particle size is observed from the beginning of polymerization to the end, which can be accounted for by a selective swelling of the polymer particles. For percolating systems, an intermediate stage, not yet fully understood, occurs in which extended particles, or perhaps aggregates, are formed. It should be noted that, at full conversion, particles from both types of systems exhibit the same structure and size (D = 400 A) with a rather small index of polydispersity. Finally, a particularly important result is the confirmation by TEM that a continuous particle nucleation occurs throughout the course of the polymerization.
Acknowledgment. We thank Drs. R. Zana, B. Ninham, T. Zemb, and S. Candau for their very stimulating and helpful discussions and Dr. C. Holtzscherer and C. Straupe for their help with some of the light scattering and microscopy experiments. Registry No. Aerosol OT, 577-1 1-7; acrylamide, 79-06-1.
Capillary Rise in Thin Porous Media Abraham Marmur Department of Chemical Engineering, Technion-Israel (Received: August 29, 1988)
Institute of Technology, 32000 Haifa, Israel
A theory for the equilibrium capillary rise inside a thin porous medium is presented. It is shown that the height of rise depends on the thickness of the medium and is lower than the corresponding value for an infinitely thick slab. The extent of deviation is shown to depend on all the parameters which characterize the system. For high true contact angles, the capillary rise in a thin porous medium may be only a very small fraction of the value for a thick one. This phenomenon is explained in terms of the reexposure effect, which stems from the exposure of the liquid rising inside the porous medium to the outside fluid, through the pores in the side surfaces. A symmetry with respect to the axis of 90° for the true contact angle is identified, so that the results for capillary rise can be also applied to capillary depression.
Introduction Capillary penetration of liquids into porous media is of great interest for a variety of scientific and technological purposes. Oil recovery, coating of wood, and printing on paper are some of the well-known applications, for which capillary penetration is of crucial importance. The thermodynamics of penetration into capillaries and porous media has been studied for a long time, theoretically as well as However, as has been recently pointed out,13,14most of the studies have dealt with penetration from infinite liquid reservoirs into infinite capillaries or porous media. In many systems, the porous medium as well as the liquid reservoir are of limited size. Examples of such systems are liquid ( I ) Lord Rayleigh, Proc. R. SOC.London, Ser. A 1915, 92, 184. (2) Smith, W. 0.; Foote, P. D.; Busang, P. F. Physics 1931, I , 18. (3) Adam, N. K. Discuss. Faraday SOC.1948, No. 3 , 5. (4) Melrose, J. C. SOC.Pet. Eng. J . 1965, 5 , 259. (5) Morrow, N. R. Ind. Eng. Chem. 1970, 62, 32. (6) van Brakel, J. Powder Technol. 1975, 11, 205. (7) Everett, D. H.; Haynes, J. M. Z . Phys. Chem. 1975, 97, 301. ( 8 ) Everett, D. H. J . Colloid Interface Sci. 1975, 52, 189. (9) Van Brakel, J.; Heertjes, P. M. Nafure (London) 1975, 254, 585. (IO) Boucher, E. A. Rep. Prog. Phys. 1980, 43, 32. ( I I ) White, L. R. J . Colloid Interface Sci. 1982, 90, 536. (12) Yang, Y.-W.;Zografi, G.; Milier, E. E. J . Colloid Interface Sei. 1988, 122. 24.
0022-3654 I 8 9 12093-4873$01 .SO ,IO I
,
drops on textile yarns or on paper. The small size of a drop enhances capillary penetration, because of the added pressure due to the curvature of the d r ~ p . ’ ~ -Thinness ’~ of the porous medium has a more subtle influence, which has been termed “the reexposure effect”.14 When a liquid penetrates into a thin porous medium, it is reexposed to the outside fluid (vapor or liquid) through the pores in the side surfaces of the medium. This is similar to penetration into a capillary with perforated walls. When the porous medium is infinite, the reexposure effect is negligible, because the influence of the sides is small compared with the contribution of the bulk. However, when the porous medium is thin, the reexposure effect may be the dominant factor, as has been shown for the case of penetration of a drop.I4 It is therefore of interest to study the effect of a limited thickness of a porous medium on the extent of capillary rise in it. Such information can be especially useful for characterization of paper, textiles, etc., using tensiometric measurements. One of the main problems in modeling penetration into porous media is their complex structure, which leads to hysteresis p h e n ~ m e n a . ~ * ~ J ~ J ~ However, as a first approximation, average properties of the porous (13) Marmur, A. J . Colloid (14) Marmur, A. J . Colloid (1 5) Marmur, A. J . Colloid (16) Marmur, A. J . Colloid (17) Marmur, A. J . Colloid
Inferface Sci. 1988, 122, 209. Interface Sci. 1988, 123, 161. Interface Sci. 1988, 124, 30 1. Interface Sci., in press. Interface Sci., in press.
0 1989 American Chemical Society
4874
The Journal of Physical Chemistry, Vol. 93, No. 12, 1989
Marmur where Ap p - pr, g is the gravitational acceleration, Alf is the surface area of the liquid-fluid interface outside the porous medium, on one side of it (lfl in Figure la), and f is the local height above the horizontal of this interface, at a distance x from the slab surface. The liquid-fluid interfacial free energy inside the slab is described by
Glf = Yldaes, + 2I.Y - k)
b
C
Figure 1. The model system: a porous slab, initially saturated with the fluid, is dipped into the liquid: (a) equilibrium withy < I; (b) equilibrium with y > I; (c) equilibrium with y = 1.
medium may be used to define the system, so that the effect of the thinness can be studied independently of other complicating factors. Thus, the purposes of the present paper is to theoretically study the thermodynamics of capillary rise in a thin porous medium, which can be characterized by a small number of average parameters.
Gsl
Theory The system to be analyzed consists of a porous slab which is initially saturaed with a fluid, and then dipped into a liquid which is at equilibrium with the fluid, as shown in Figure la. The fluid may be a gas or a liquid. The thickness of the slab is a, and its bottom is located at a depth z (the numerical value of which is negative) with respect to the horizontal surface of the liquid. The slab may be assumed infinite in its other dimensions. The liquid outside the slab rises to a height y above its horizontal level, and makes an apparent contact angle 8, with the slab surface. The liquid inside the slab penetrates to a height I above the horizontal level of the liquid. Both y and I may assume negative values under appropriate conditions. The porous medium is assumed to be characterized only by its bulk porosity, t, surface porosity, ts, and specific area (area of solid per unit volume of solid), s. Such a porous medium may be termed “microscopically uniform”, since no structural details are recognized. Also, this assumption implicitly implies that the pores are small in comparison with the thickness of the slab. The surface porosity may be different from the bulk porosity, because of geometrical reasons, surface treatment, or coating. The liquid is characterized by its density, p, and its interfacial tensions with the fluid, Ylf, and the solid, ysl. The density of the fluid is pf, and its interfacial tension with the solid is Ysf. The equilibrium positions of the system can be found by minimization of the free energy, which is given by
- Gsf = (Ysl - Ysf)[(l
- €)(I-
z)as
+
- d0,- z + (a/2))1 (4)
The first term in the brackets accounts for the interfacial contact between the liquid and the solid inside the porous medium (sll in Figure la). The second term stands for the area of contact between the liquid and the sides of the porous slab, including the bottom side (s12 in Figure la). Equation 4 can be rewritten as
GI - Gsf =
-ylf cos 0 [(I - t ) ( l - z)as
+ 2(1 - t,)O, - z + (a/2))]
(2)
(5)
where 0 is the true contact angle in the solid-liquid-fluid system that would have been measured had the solid been perfectly flat, nonporous, and homogeneous. This is given by the well-known Young equation1’ Ysr
-
Ysl
cos % = Ylf
The gravitational energy of the liquid inside the porous medium is given by Ut
G, = Apg-(I2 - z2) 2
(7)
-
When the slab is very thick ( a a),all the terms which are not multiplied by the thickness, a, become negligible, and the expression for the free energy is G YIta
- = ts, - cos 0 [(l - c ) ( l - z)s
where Glfmis the free energy of the liquid-fluid interface of the meniscus outside the porous slab, G,, is the gravitational energy of this meniscus, and GIf, Gsl, Gsf, and G, are the free energy components due to the liquid-fluid interface, solid-liquid interface, solid-fluid interface, and gravitation, respectively, inside the slab. All free energy values are defined per unit width (in the direction perpendicular to the plane of the figure), because of the twodimensional nature of the problem. The detailed equations for these free energy components are given in the following. The free energy of the liquid-fluid system outside the porous slab, on its two sides, is given by
(3)
The first term on the right-hand side of eq 3 accounts for the upper liquid-fluid interface (lf2 in Figure la). The factor s1 accounts for the fact that the liquid-fluid interfaces inside the pores are not flat, and is defined by the ratio of the liquid-fluid interfacial area to the cross-sectional area of the pore. This term is constant and therefore does not affect the equilibrium states of the system. Thus, there is no need to elaborate on the calculation of sl. The last term in eq 3 stands for the liquid-fluid interface at the sides of the porous slab (lf3 in Figure la). The absolute magnitude of 0,- I ) is taken, since the liquid in the porous slab may be either higher or lower than the liquid outside (compare Figure 1, a and b). Also, the factor SIis not taken into account, since the liquid is prevented from coming out by forming a flat liquid-fluid interface. This is different from the case of the upper interface inside the porous medium, where equilibrium is achieved by a balance between gravity and interfacial forces. The solid-liquid and solid-fluid free energies are always expressed together, as is well-known, because the liquid displaces the fluid. Thus
+ (1 -
APgt
t,)]
+ -(I2 2Ylf
2
-z ) (8)
When this expression is differentiated with respect to I , the well-known result” is recovered, which can be written in a dimensionless form as
cl, = (l cos 0 where I , refers to the height of rise at a Y1r
-
(9) a,
S
p
SIC, and (10)
(18) Adamson, A. W. Physical Chemistry of Surfaces, 4th ed.; Wiley: New York. 1982.
The Journal of Physical Chemistry, Vol. 93, No. 12, 1989 4875
Capillary Rise in Thin Porous Media In this case there is only one independent variable, 1. When the porous slab is thin, there are two independent variables in the system, e.g., I and y , so the minimization of the free energy has to be done by partial differentiation with respect to each of them. In addition, the term with the absolute magnitude sign in eq 3 may cause a discontinuity in the derivative of the free energy. Therefore, the minimum has to be separately searched for in the regions where y C I , where y > 1, and on the line y = 1. These three possible situations are shown schematically in Figure 1, a-c. It is well-known (e.g., ref 19), that a minimum in the free energy of a system containing liquid-fluid interfaces and solid-liquid-fluid contact lines can, in general, be assured by the simultaneous fulfillment of two equations: (a) the Young-Laplace equation, which determines the shape of the interface, and (b) the Young equation for the true contact angle, eq 6, which serves as a boundary condition for the former. It is very difficult to solve the present problem by directly applying these two equations, because the solid surface is nonuniform, and the properties of the porous medium are averaged ("microscopic uniformity"), so that the true contact angle cannot be identified in the model system. Therefore, eq 1, 2, 3, 5, and 7, which are formulated in terms of average properties wherever necessary, have to be differentiated to formulate the necessary conditions of equilibrium. The conditions of equilibrium must specify the equilibrium height of liquid rise in the porous slab, I,, the equilibrium height of liquid rise outside the porous slab, at its surface, ye, and the shape of the liquid-fluid interface outside the slab. The latter can be determined independently of the former two by minimization of eq 2. The result is the well-known solution for a cylindrical interface*O cr = 2 sin (4/2) (11) where 4 is the angle between the normal to the interface and the {direction. This solution assures that the contributions of the liquid-fluid interface and gravity to the differential of eq 2 cancel each other, except for the following term, which stands for the differential change in the interfacial area at the solid surface (e.g., ref 19): (when eq 11 holds) (12) dGlrm= 2ylf cos Oa dy The apparent contact angle, which is related to 4 by Os = [(*/2) - do],where $o 1 $(x=O), is the boundary condition necessary for determining the height of the liquid in contact with the outside surface of the slab, y = {(x=O). Its equilibrium value for the present system, e,, is calculated by partially differentiating the equation for the free energy with respect t o y and requiring the derivative to be zero, as a necessary equilibrium condition. For y < I , this differentiation, followed by a slight rearrangement leads to cos O,, = (1 - e,) cos 0 e, for ye < I, (13)
+
Similarly, for y > I cos Oae = (1
- e,) cos 0 - e,
for ye > I,
- €)as + 2(1 - e,)]
+ c2aeye = 0
(14) For y = I , there is only one independent variable, and the differentiation leads to 2 cos
o,,
- cos 0 [(l
for ye = I, (15) Here, in contrast to the previous two equations, ye appears in the equation for Oa. Using eq 11 for ye, we can arrange eq 15 to yield 2 cos Oae - cos 0 [(l - €)as 2(1 - e,)]
+
(2cae) sin
(a
+
-
$) = 0
for y , = I, (16)
In addition to the above equilibrium conditions, which determine the height of the liquid outside the porous slab, the height of capillary rise inside the porous medium is required. This condition (19) Marmur, A. Ado. Colloid Interface Sci. 1983, 19, 15. (20) Princen, H. M. In Surface and Colloid Science; Matigivic, E., Ed.; Wiley: New York, 1969; p 1.
10
L
-'
i
"rl
t 10
-3;.
'
'
,
' ' ' ' '1 a ',' ' '' ' ' ! ' ' ' ' ' ' ' ' ' "' 20 40 60 80 TRUE CONTACT ANGLE, deg. I '
I
'
" "
"
Figure 2. Dimensionless heights of rise inside the slab, L, and outside
it, Y,vs the true contact angle, which characterizes the surface energies of the system. S = 100, E = tl = 0.4, A = 0.06. is formulated by partially differentiating the expression for the free energy with respect to I and requiring the derivative to be zero. Again, the equilibrium condition has to be separately formulatzd for each of the cases shown in Figure 1, because of the absolute magnitude sign in eq 3. For y C I, the equilibrium condition turns out, after some rearrangement, to be L = l -
28' A(l - e)S cos 8
for y e C I,
(17)
where L = l e / l mis dimensionless height of rise and A 2 ac is the dimensionless slab thickness. For y > I , the minimization leads to
For ye = I,, the equilibrium condition is already given by eq 16 in terms of O,,, from which the equilibrium height should be calculated by using eq 11.
Results and Discussion The first observation that should be made regarding the equilibrium conditions, eq 13, 14, 16, 17, and 18, is that they are characterized by a symmetry with respect to the axis of 0 = 90'. Thus, the following discussion will concentrate on the case of capillary rise only (0 C 90°), as in Figure la, with the understanding that the results and conclusions apply also to situations of capillary depression (0 > 90°, Figure lb), taking the above symmetry into account. An additional minor observation is that, as can be expected, the depth, z, has no influence on the equilibrium conditions. Equation 17 presents one of the main results of the present analysis. It shows that the dimensionless height of rise for a thin slab is lower than for an infinite porous medium. The extent of the deviation depends on the numerical values of all the parameters which define the system, as will be shown with the aid of the figures. The other main point of interest is the transition between the region of ye C I, and the region of equal heights, y e = I,. It is interesting to notice that by substituting cos e, from eq 13 (which is valid for ye C le) into eq 15 (which is valid for y , = Z,) a value of y e results, which equals I, as calculated from eq 17. This means that the region of ye = I, starts just at the point where the condition ye C I, cannot be fulfilled anymore. This is not an obvious result, since the free energy could have had, in principle, more than one local minimum with an overlap of the various regions. Figure 2 presents a sample calculation for the case of S = 100, e = es = 0.4, and A = 0.06. This figure shows the dimensionless heights of capillary rise inside the porous slab, L, and outside it,
4876 The Journal of Physical Chemistry, Vol. 93, No. 12, 1989
Marmur E
0.05 '\
* 0 4
0.04
1 E
0.03
i
0.02
/
0.01
20 40 60 80 T R U E CONTACT ANGLE, deg
Figure 3. Dimensionless heights of rise inside the porous slab, L, vs the true contact angle, for various thicknesses. The dimensionless thickness, A, is indicated by the numbers in the figure. S = 100, c = cg = 0.4.
Y = ye/lm,vs the true contact angle (for an infinite porous medium, L = 1, but Y would still depend on the surface properties and the contact angle). It is important to realize that the true contact angle serves here only as a convenient measure of the surface energies which are involved, via eq 6, and cannot be geometrically identified in the present model system. The upper curve in Figure 2 shows the calculated values of L from eq 17. The lower curve shows the calculated values of V from eq 13 and 11 (a logarithmic scale was chosen so that the small changes in Y could be observed). It is clearly seen that the value of L is appreciably smaller than unity even for 0 = 0, and it further decreases when the true contact angle increases. For true contact angles above 78' in this case, the region of equal heights starts, as calculated from eq 16 and 1 1. In this region, the capillary rise is only a very small fraction of the corresponding value for an infinite slab. Therefore, from a practical point of view, capillary rise is arrested for values of 0, which are appreciably less than 90". The value of Y does not change much with 0, but a shallow minimum can be observed, followed by a rise toward the value of 8 for which the equal heights region begins. The explanation of this behavior is as follows. The driving force for penetration into a capillary or a porous medium is the gain in the free energy due to the creation of a solid-liquid interface instead of the liquid-fluid and solid-fluid interfaces. This is, of course, true also for a thin porous medium. However, as a liquid penetrates into a thin porous slab, it gets reexposed to the outside fluid through the pores in the side surfaces (If3 in Figure la). This reexposure reduces the gain in free energy and leads to lower heights of capillary rise, as shown by eq 17. Obviously, when the porous medium is sufficiently thick, the effect of the sides is expected to become marginal. As the true contact angle, 0, increases, the driving force for penetration decreases. Therefore, for high values of 8, the reexposure effect becomes more meaningful. This is demonstrated in Figure 2 by the dramatic decrease in L, which leads to the region of ye = I,. This decrease results from the tendency of the system to minimize the area for reexpure, tv - /I, by lowering I, to values close to ye. Ultimately, the two heights become equal, and the reexposure effect is prevented by eliminating the liquid-fluid interface at the sides of the slab (lf3 in Figure la, which does not exist in the situation shown in Figure IC). Thus, for a thin porous medium, for which the inside liquid interacts with the outside fluid, the free energy is minimized at the "cost" of a low capillary rise. A similar effect was suggested for the penetration of a drop into a thin porous media.14 Figures 3 and 4 show the effect of the dimensionless thickness of the slab, A , on the heights of rise inside and outside the porous medium, respectively. The numbers in the figures indicate the values of A . As can be expected, the values of L are higher and
0.00
0
" ' ~ " " ' ' " " " '
" ' ~ ' ' " " '
'
" " "
23
I '
20 40 60 80 T R U E CONTACT ANGLE, deg
4 ' ' ?
Figure 4. Dimensionless heights of rise outside the porous slab, Y,which correspond to the values of L in Figure 3.
-
0.0
1
0.1
0.2
0.3
DIMENSIONLESS SLAB THICKNESS
Figure 5. Effect of the bulk and surface porosity on the transition true contact angle, which separates the region of y e < I, from the region of ye = I,. S = 100; c = cs and their values are indicated by the numbers in the figure.
closer to unity as the thickness is higher. Also, the value of '6 at which the inside and outside heights begin to equal, as shown by the breaks in the curves, becomes closer to 90' when the thickness increases. The Y values follow a base curve, which is common to all thicknesses, and then depart from this curve close to the point which marks the beginning of the equal heights region. It is worthwhile at this point to get a physical feeling for the dimensionless thickness, A , and the dimensionless specific area, S. If the porous medium is composed of fibers, for example, it can be assumed that the real specific area, s, is inversely proportional to the average radius of the fiber, s = k / r , where k is a dimensionless constant and r is the radius of the fiber. For cylindrical fibers, k = 2. The value of c for water displacing air, for example, is about 3.7 cm-I. So, a value of S = 100 implies s = 370 cm-I and a fiber radius of about 50 wm. Regarding the slab thickness, if the crude model of a cylindrical fiber is followed, then the ratio of the thickness to the fiber radius is A S / k . Thus, for example, a value of 0.06 in conjunction with S = 100 represents a thickness of 3 fiber radii, while a value of 0.3 for the same S corresponds to 15 fiber radii. Of course, it should be kept in mind that various porous media may largely differ in structure from the fiber model. However, the values chosen for S and A in the present study seem to be representative. Figure 5 demonstrates the effect of the porosity on capillary penetration, for S = 100 and various porosity values, as indicated by the numbers in the figure, assuming c = cs. Each curve in the
The Journal of Physical Chemistry, Vol. 93, No. 12, 1989 4877
Capillary Rise in Thin Porous Media
nkl! 0.0
I
,
,
,
,
0.1
,
, , , ,
,
, ,
,
, , , , , , , ,
0.2
,i 0.3
DIMENSIONLESS SLAB THICKNESS
0.0
0.1 0.2 DIMENSIONLESS S U B THICKNESS
0.3
Figure 6. Effect of the surface porosity on the transition true contact angle, which separates the region of y, < I, from the region of ye = 1,. S = 100, c = 0.5; tois indicated by the numbers in the figure.
Figure 7. Effect of the dimensionless specific area, S,on the transition true contact angle, which separates the region of ye < I, from the region of ye = I,. S is indicated by the numbers in the figure, t = c, = 0.4.
figure divides the 0-A space into two regions: below the curve, equilibrium for the given porosity is attained for y e < le, and eq 13 and 17 prevail; above the curve, equilibrium requires ye = I,, and eq 16 should be used. In other words, each curve in the figure shows the value of 0 corresponding to a given thickness, at which the transition to ye = le occurs. Figure 5 shows that, as the porosity increases, the reexposure effect becomes more prominent. This is indicated by the lower values of the transition 0. Figure 6 demonstrates the effect of varying only the surface porosity, e,, the value of which is indicated in the figure, for S = 100 and a bulk porosity of e = 0.5. The reexposure effect is seen to be more prominent as the surface porosity increases; however, the change becomes smaller as cs is increased. This is so because, when the surface porosity is higher than the bulk porosity, a further increase does not add much to the reexposure effect. When the surface porosity is made small, the interaction between the inside and outside becomes more difficult, and the value of L approaches unity, as indicated by the fact that the transition 0 approaches 90'. It should be remembered, however, that the model cannot be carried too far with respect to lowering the value of c,, since for c, = 0 the inside and outside are independent by definition, and for very small values of e,, the implicit assumption of a continuous slab surface with average properties does not hold. Figure 7 presents the effect of varying S from 50 to 400, as indicated by the numbers in the figure. It shows that, as the value
of S is increased for a given thickness, the transition value of 0 approaches 90'. A high S value means a small characteristic size of the particles which form the porous medium. Thus, the same actual thickness for a higher S means a higher number of particles within the width of the slab, which makes the reexposure effect less pronounced.
Summary and Conclusions A theory was developed for the capillary rise of a liquid into a thin porous slab. The results of the calculations show, in general, that the height of rise is sensitive to the thickness of the porous medium. The specific results can be summarized as follows: 1. The equations are symmetrical with respect to 0 = 90°,and therefore all the following conclusions which apply to the case of 0 < 90' can be properly adapted for 0 > 90'. 2. Capillary rise in a thin porous slab is lower than in an infinite porous medium, due to the reexposure effect. 3. Two thermodynamic regions can be identified: (a) capillary rise inside the slab is higher than outside it, and (b) the two heights of rise are equal. The latter prevails at high values of the true contact angle and is characterized by a very low height of rise. Thus, capillary rise is practically arrested at values of 0 which are lower than 90'. 4. Increasing the porosities and decreasing the specific area of the solid enhance the reexposure effect and thus decrease the capillary rise for a given thickness.
