Coexistence of menisci and the influence of neighboring pores on capillary displacement curvatures in sphere packings

Coexistence of Menisci and the Influence of Neighboring Pores on Capillary Displacement Curvatures in Sphere Packings G E O F F R E Y MASON ~ AND N O R M A N R. M O R R O W New Mexico Petroleum Recovery Research Center, New Mexico Institute of Mining and Technology, Socorro, New Mexico 87801 Received November 8, 1983; accepted March 31, 1984 The methods of calculating meniscus curvatures given by Mayer and Stowe and also independently by Princen are essentially the same. The method is exact for pores defined by rods. From comparison with experimental results, the method provides, for zero contact angle at least, a close approximation for pores defined by spheres. The application of the method to model pores defined by rods and spheres is discussed with particular attention being paid to the effects of neighboring pores. The merits of defining the neighbors of a particular pore as mirror images are discussed together with the effect of neighboring pores on the determination of pore sizes from capillary displacement curvatures. Meniscus curvatures of a family of pore shapes defined by three equal rods and mirror image neighbors are tabulated. A simple correlation was found between these values and estimates of the curvature given by the Haines incircle approximation. INTRODUCTION

The curvatures of the menisci in the pores of a porous material are central to the analysis of processes such as capillary condensation, adsorption, mercury porosimetry, fluid displacement, and tertiary oil recovery by blob mobilization. Frequently, the porous material is modeled by a structure such as a packing of spheres. Pore size, fluid densities, and interfacial tensions are usually such that the change in meniscus curvature caused by gravity over the region of a few pores can be neglected. Thus, on the scale of a few pores, at least, the meniscus curvature is constant. The analysis of the displacement of a wetring phase from sphere packings involves two main mechanistic features, one concerning the meniscus curvatures in the pores and the other arising from the behavior of large assemblies of pores. The first involves the passage of menisci through constrictions between spheres Dept. Chemical Engineering, Loughborough University of Technology, Loughborough, Leicestershire, England.

and, in particular, the m a x i m u m curvature required for meniscus movement. The menisci have complex shapes determined by the pore boundary conditions and these boundary conditions may or may not involve the presence of neighboring pores. The exact computation of these configurations by finite difference methods is possible in theory but tedious and awkward in practice. In the past, various approximations to the meniscus curvature have been used to obtain estimates of this maximum curvature, the most widely applied being the Haines approximation which takes the meniscus to be a portion of the spherical surface which just fits into the pore. So this facet of the displacement problem requires, in essence, models of both the geometry of the pores in the sphere packing and also of the meniscus behavior in a particular pore geometry. The second main mechanistic feature involves the way that the emptying of one pore affects the accessibility of menisci to neighboring pores. Bulk immiscible invading phase usually breaks into a porous material at a fairly well-defined displacement curvature. This is to be expected if all of the pores are

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Journal of ColloMand InterfaceScience, Vol. 100,No. 2, August1984

0021-9797/84 $3.00 Copyright© 1984by AcademicPress,Inc. All fightsof reproductionin any formreserved.

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MASON AND MORROW

the same size and shape, but it also occurs when the pores are of different sizes and shapes. This facet of the displacement problem involves a cascade process whereby one emptying pore allows access to neighbors which can also be emptied. In this paper we try to draw together and extend some previous analyses of the first mechanistic feature with particular emphasis on the influence of immediately neighboring pores on the values of m a x i m u m meniscus curvature of a given pore. EXACT SOLUTION FOR SPHERES The general equation for the shape of a capillary surface of constant curvature without cylindrical symmetry is a nonlinear, secondorder, partial differential equation which, provided the boundary conditions can be handled, can be solved numerically (1). For most practical purposes the procedure is still too time consuming and difficult to be run on a routine basis. The rigorous analysis of the stability conditions that define the m a x i m u m curvature would be particularly difficult to apply. As a result, there has been a search for more readily obtainable approximate solutions to this problem. HAINES INCIRCLE APPROXIMATION After observing the meniscus displacements in various pore shapes that arose in regular sphere packings, Haines (2) proposed that the curvature o f a meniscus that just passed through the central constriction of a pore could be approximated to that of the sphere which just touches the spheres defining the pore. There is a lot to be said for the Haines approximation in terms of utility. It is certainly the easiest to apply. But it is not a very precise approximation as is shown in Fig. 1 where the Haines approximation (2) is compared with the experimental results of Hackett and Strettan (3). MAYER

AND STOWE'S ANALYSIS FOR SPHERES

Mayer and Stowe (4) proposed a solution for the sphere problem based upon virtual disJournal of Colloid and InterfaceScience, Vol. 100, No. 2, August 1984

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FIG. 1. Comparison of the experimental results of Hackett and Strettan with various theoretical approximations. The model pore consists of four contacting spheres with cell angle varying such that the pore varies from triangular to square. The Haines approximation assumes the meniscus is spherical. Hwang obtained curvatures which are inverselyproportional to the hydraulic radius. The Mayer and Stowe analysisis actually for rods and not spheres as they claim. The "point of separation" is wherethe meniscuscurvaturebecomeslowerfor a single pore defined by four rods than for a pair of pores each defined by three rods. If there is no mechanistic pathway for "separation" to occur because, for example, the meniscus wedges (see text) where the rods are close are bounded, then the change over point is shifted to the rupture of the back-to-backwedges. Note the goodness of fit of the Mayer and Stowe approximation for rods to the experimental data from spheres.

placement of a meniscus in a pore, the cross section of which was defined by four circles. The pore shapes varied between the dosed rhombus and the closed square array. The underlying theory presented by Mayer and Stowe has been questioned by Melrose (5) and later by Haynes (6) because an exact solution to the sphere problem must take into account the nonuniform cross section of a pore formed by spheres.

521

MENISCUS CURVATURES PRINCEN'S ANALYSIS FOR RODS

LINK BETWEEN RODS AND SPHERES

The true nature of the theoretical results presented by Mayer and Stowe is revealed by examination of the formative equations used by Mayer and Stowe with respect to pore geometry, and also by results presented later by Princen (8). Princen, in a series of papers (79), presented approximate solutions for the capillary rise in pore geometries of uniform cross section created by rods rather than spheres. Princen's analysis is exact for surfaces of constant curvature in rods, and involves precisely the same computational procedure as that of Mayer and Stowe. This is confirmed by the identical numerical values obtained by Mayer and Stowe and Princen for equivalent configurations, namely three rods in mutual contact and four contacting rods in a square array. For zero contact angle, 11.32 was obtained as the dimensionless curvature of the meniscus in three contacting rods, and 4.49 for the curvature in the square configuration of four contacting rods.

Mayer and Stowe did not test their results against experiment, but it is seen from comparison with results reported by Hackett and Strettan (see Fig. 1), that, for zero contact angle at least, the Mayer and Stowe theory gives a remarkably good fit over the full range of packing angles. This observation establishes an important link between systems formed by rods and systems formed by spheres. For rods, displacement curvatures can be calculated exactly, whereas the mathematics of calculating displacement curvatures in sphere packings is close to intractable. Princen's studies were related to behavior of liquids in bundles of fibers. However, results for sphere packings are of more general interest because sphere packings are the most commonly used models of porous media. The experimental link between the two types of systems suggests that the theoretical method, which we now refer to as the MS-P method, is not only of value with respect to computing meniscus curvatures in capillaries of uniform cross section, but also provides a useful approximation for displacement curvatures in sphere packing arrangements and so can be applied to pore configurations beyond those discussed so far. In the present work we will employ the MS-P method to examine the effect of boundary conditions and in particular the presence of neighboring pores on theoretical values of displacement curvatures. Even though displacement curvatures for systems of parallel rods are relatively easy to calculate, the problem is still not trivial. The MS-P method requires that the basic arrangement of the liquid in the pore be known before the analysis can be applied. For discussion of possible arrangements, it is helpful to have a terminology that covers the main features of meniscus behavior in systems of parallel rods.

DODDS' APPROXIMATION

Dodds (10) noted that for triangular pores given by close packing (hereafter referred to as closed-triangular pores), there was excellent agreement between Princen's results calculated for rods and Hackett and Strettan's experimental results for spheres. Dodds went on to calculate displacement curvatures for pores formed by three rods with size and spacing varied. The configurations were taken as models for pores found in sphere packings. HWANG'S HYDRAULIC RADIUS M E T H O D

Hwang (11) presented a derivation of the Gauss equation which relates change in interracial areas to volumetric displacements. In application of the method, Hwang implicitly assumed that the capillary interface spanned the total cross section of a given tube. This assumption led to calculated curvatures being inversely proportional to hydraulic radius (ratio of pore perimeter to pore area).

TERMINOLOGY

For regions of a capillary formed by rods where relative amounts of each phase are essentially constant per unit length of the capJournal of Colloid and Interface Science, Vol. 100, No. 2, August 1984

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MASON AND MORROW

illary, menisci, if present, will be referred to generally as wedges. The term wedge derives from wedge shaped menisci as shown in section and profile in Fig. 2a. Wedges at line contacts, such as those in Fig. 2a, will be referred to as corner wedges. A drop of wetting liquid placed in a corner will form a corner wedge and spread indefinitely to give a surface of infinite curvature. Thus the existence of a c o m e r wedge of finite curvature requires that the wedge be bounded. In practice this merely requires that the system be of finite length with suitable bounds. When rods are in near contact a double wedge can form, such as shown in Fig. 2b, which is essentially two back-to-back wedges. If a drop of wetting liquid is placed between two rods in near contact, it will form a double wedge with the drop spreading along the rods for only a limited distance. The spreading is limited by the curvature set by double-wedge terminal menisci which wilt bound each end of the elongated drop. Curvatures of double wedge menisci were reported by Princen for various distances of rod separation (7). For a tube which has a meniscus with no

iI

wedges, such as a cylinder, the meniscus occupies the region over which there is essentially a total change in phase along the tube. Analogous regions can also be observed for tubes with wedges, but then the phase change with distance is partial. Menisci associated with either this total or partial phase change will be referred to as main terminal menisci. The curvature of the main terminal meniscus is set by the geometry of the given tube together with any existing wedges. The wetting phase does not necessarily occupy all of the tube cross section on the wetting phase side of the terminal meniscus. Depending on pore geometry and precise wetting conditions, the nonwetting phase will sometimes propagate a wedge on the wetting phase side of the main terminal meniscus (see Fig. 2c). Such structures, while they m a y scarcely look like wedges, will be counted as such for general discussion purposes and will be called

nonwetting phase wedges. In considering the behavior of constant curvature systems, an important principle to bear in mind is that when a main terminal meniscus exists it must set the curvature. Any two distinct kinds of terminal menisci such as a main terminal meniscus and a doublewedge terminal meniscus cannot coexist in a stable system (9). The mere existence of the main terminal meniscus demands that all wedge menisci be bounded in such a way that their curvature can adjust to that set by the main terminal meniscus. Wedges of this kind will be referred to as bounded wedge menisci. PRINCIPLE AND APPLICATION OF MS-P METHOD

C

Fl~. 2. Examples of some meniscus wedge configurations. (a) shows a pair of wedge menisci between two contacting rods. We will callthese particularwedgescomer wedges. (b) showstwo separated rods containing a double wedge and double wedge terminal meniscus. (c) shows three rods with two contacts and one gap. It contains a main terminal meniscus, a pair of wetting phase comer wedges, and a single nonwetting phase wedge. All the wedgemenisciin this configurationare bounded, otherwise the main terminal meniscus cannot exist. Journal of Colloid and Interface Science, V o L 100, No. 2, A u g u s t 1984

We will explain the principle of the MS-P method by a worked example in which the basic fluid arrangement is obvious. Consider the meniscus in the kite-shaped section pore of Fig. 3. For a perfectly wetting liquid, there is always a comer-wedge in the angled comer. The main terminal meniscus spans the main circular body of the tube. Let the area spanned by the main terminal meniscus be A. The perimeter of this area is

MENISCUS CURVATURES

523

A

In the diagram of the pore cross section both A and P can be seen to be simple geometric functions of r, and hence depend on the position of the corner wedge meniscus. The solution to the problem is thus to find the particular meniscus position which satisfies Eq. E [7]. In principle this is straightforward, alFIG. 3. This figure shows a tube of uniform kite-shaped though in practice it can involve the iterative cross section containing a comer wedge meniscus (BCD) solution of quite lengthy equations. and a main terminalmeniscus(ABDE).Thisconfiguration Notice how, in this simple case, the particis used as a simple example of the MS-P method. ular values of A and P as functions o f r depend upon there being a single wedge meniscus in the only corner. Now, if the pore had two made up of two parts, the corner wedge BD comers, then it is possible that, depending on (length PL), and the solid perimeter (AB + DE the contact angle, a wedge meniscus might + EA) (length Ps). If the capillary pressure is only exist in one corner and not the other. Pc, and a is the interfacial tension and C the This changes the basic meniscus configuration meniscus curvature, then and hence the functions which determine A Pc = aC. [1] and P. There is also the possibility of there Let the contact angle with which the liquid being more than one solution. However, the meets the solid be/9. (There will be a range, required solution is usually obvious. In many up to a particular value of O, over which the instances, if an unrealistic basic meniscus wedge will exist.) A virtual work calculation configuration is taken, the solutions may give based on a small displacement of the meniscus physically unreal roots, or wedges that overlap, and these are clearly mathematical rather than gives e c A d x = (Ps cos 0 + eL)dX. [2] physical. In the absence of wedges, the meniscus curvature is simply given by the hyEliminating Pc between 1 and 2 gives draulic radius. CA = Ps cos 0 -4- PL[3] PHYSICAL SIGNIFICANCE OF HAINES AND H W A N G APPROXIMATIONS

If we define an effective perimeter, P, as P = Ps cos 0 + PL

[4]

C = P/A.

[5]

then Thus the curvature equals the effective perimeter divided by the area of projection of the main terminal meniscus. Now at a sufficient distance (in practice quite close) from the main terminal meniscus, the cross section of the comer wedge becomes a circular arc, radius r. This is consistent with the corner wedge having zero curvature in the direction parallel to the tube axis at sufficiently large distance from the main terminal meniscus. Thus, C = 1/r

[6]

r = A/P.

[7]

SO

Now that the principle of the MS-P method has been described, it is instructive to consider the physical significance of the other methods o f determining curvatures which were mentioned previously. The main deficiencies of these approximations becomes apparent from consideration of the wedges. The Haines incircle approximation implies that the wedges are at the point of overlap and are at their maximum possible size. Use of the Haines approximation implies a discontinuity in curvature. The curvature of the spherical main terminal meniscus is always twice that of the cylindrical wedge meniscus which has the same radius. For example, for a closed-triangular pore the curvature of the m a i n terJournal of Colloid and Interface Science, Vol. 100, No. 2, August 1984

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MASON AND MORROW

minal meniscus is 12.9, but the curvature of the wedges will be only half of this value, 6.45. If, starting with the Haines approximation, the size of the wedge meniscus is decreased, the curvature of the wedge will increase, while that of the main terminal meniscus will decrease. The MS-P method finds, in essence, the position at which the main terminal meniscus and the wedges have equal curvature, so that the condition of constant curvature is satisfied. Hwang's solution, which involves use of the hydraulic radius, leads to a main terminal meniscus curvature of 19.7 for the triangular pore, a much higher curvature than the incircle, implying that this meniscus does not touch the pore. Furthermore, the absence of wedges implies that the wedge curvatures are infinite. Both implications are physically unrealistic. With respect to the wedge size, the Haines approximation assumes a maximum wedge size while the Hwang approximation assumes that it is zero. The calculated values of displacement curvatures for these extremes of wedge size are both higher than the curvature calculated by the MS-P method, which is consistent with the MS-P approach of finding the minimum curvature for a given basic meniscus configuration. DISPLACEMENT CURVATURES IN SINGLE PORES AND PORES WITH NEIGHBORS Although the methods used by Mayer and Stowe and Princen are basically the same, there is, apart from the previously mentioned results for hexagonal and cubic packings, surprisingly little duplication between their presented tables of results. In the main, they are complementary. Princen investigated rod configurations for different separations of the rods, whereas Mayer and Stowe treated the rhomboidal pore, with shape ranging from closed-triangular to square. Both configurations provide useful illustrations of the effect of neighboring pores on displacement curvature. The rhomboidal pores considered by Mayer and Stowe were always bounded by four contacting rods. The meJournal of Colloid and Interface Science, Vol. 100, No. 2, August 1984

niscus properties of such bounded pores are not affected by external changes such as the introduction or movement of a neighboring external rod. The meniscus properties do, however, show a change in the meniscus configuration as the cell half-angle is varied. For the hexagonal arrangement of the four rods there are two distinct pores each defined by three contacting rods. If the rhombus halfangle is increased, a small gap appears between two of the initially contacting rods. This gap is, of course, internal to the boundary of the four rods defining the pore. It contains a bounded double wedge. For small separations, the system still behaves as two three-sided pores which are independent of each other and two main terminal menisci will exist, one in each pore. If the gap is increased further, there comes a critical gap size at which the back-to-back menisci of the bounded double wedge touch and the wedge ruptures. This point corresponds to termination of the line in the theoretical dimensionless curvature versus cell angle relationship shown in Fig. 1. A single terminal meniscus now forms which contacts all four rods. Nevertheless, the pore can still be viewed as a pair of three-sided pores but they are a mirror image pair with two of the rods of each pore being shared between the two pores. The menisci in either three-sided pore are no longer independent of each other. Each is fused to a mirror image meniscus provided by the neighboring pore. Thus, we have an example of a pore with a neighbor, with the behavior of one half of the pore being affected by changes in the neighboring half pore. The effect of the presence of the neighboring pore can be appreciated by considering, for example, the single pore formed by three rods. Let gaps of equal size be introduced between adjacent pairs of rods. Tabulated solutions from the analysis of this case are given by Princen (8). At small separations the system has three bounded double wedges, one between each rod pair, and these isolate the main terminal meniscus behavior from external influences. As the gaps are increased in size there comes a point, as in the

MENISCUS C U R V A T U R E S

rhomboidal pore, where the back-to-back menisci of the bounded double wedges much. The double wedges disappear and three nonwetting phase wedges form on the opposite side of the main terminal meniscus. Now, after rupture, the terminal meniscus between the three rods is no longer isolated from external surfaces and its behavior becomes determined to some extent by the neighboring interfaces. Further example of the influence of neighboring pores is provided by comparing Princen's results for three equispaced rods with those for equispaced rods in infinite array. Results for the latter can be obtained from analytic expressions also given by Princen (8). When the gaps are small, all of the pores behave independently and the presence, or not, of the neighboring pores makes no difference to the curvature of the main terminal meniscus in a unit pore. Beyond the critical separation, even though the basic pore shape is the same for a given rod spacing, meniscus curvatures differ for the two systems as will be shown later. C U R V A T U R E S IN SINGLE PORES

Analysis of a pore given by three rods with a single gap was performed by Dodds (10). At small separations the single gap contains a double wedge and the curvatures are identical to those in the rhomboidal pore investigated by Mayer and Stowe. However, for the Dodds pore, as the gap widens beyond the critical separation, a nonwetting phase wedge forms. The perimeter force of the nonwetting phase wedge is of opposite sign to the wetting phase wedges, and so the meniscus curvature decreases dramatically as the gap is increased further. This is shown in Fig. 4 where the tabulated values of Dodds (three rods) are plotted together with the tabulated values given by Mayer and Stowe for four rods. The important difference between these results, even though the basic geometry is the same, is that the four rod system consists of a pair of mirror image neighbors, whereas the three rod pore is alone. Also shown are the two

525

tabulated values obtained by Princen for the two limiting configurations. They coincide exactly with two Mayer and Stowe points. Dodds noted that the displacement curvature for three contacting rods calculated by the Princen analysis was in excellent agreement with the displacement curvatures measured by Hackett and Strettan for the closed-triangular pore. We have already shown (Fig. 1) that there is agreement across the complete range between the numerical results of Mayer and Stowe and the experimental results presented by Hackett and Strettan. The conclusion to be drawn from this agreement is that interface curvatures calculated for rods can, under the right circumstances, provide a useful approximation for curvatures in sphere packs. Dodds made this conclusion and went on, using the approach described above, to calculate curvatures for single pores formed by rods of various size and spacing. One use made of the results was to test the validity of the Haines incircle approximation. Dodds showed that the Haines approximation was valid at small separations (cell angles) provided a correction factor of 0.8755 was used. As the gap was increased beyond the point at which the double wedge meniscus disappeared, the deviation of results for three rods from the incircle approximation becomes increasingly severe. Dodds ascribed this difference to errors in the Haines approximation. We will show later that the severe deviation is a consequence of assuming that neighboring pores do not exist. SEPARATED R O D S - M I R R O R IMA G E NEIGHBORS

For purposes of analysis, the pore space in a random packing of equal spheres can be reduced to an assembly of individual tetrahedral subunits (12). The behavior of the whole connected assembly depends on a combination of the behavior of each tetrahedral pore (which also depends upon its neighboring tetrahedra) and also the way in which emptying one tetrahedron permits its neighbors to empty (13). For desaturation, the capillary Journal of Colloid and Interface Science, Vol. 100, No. 2, August 1984

526

MASON AND MORROW 13

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FIG. 4. Comparison of the normalized curvature values calculated by Mayer and Stowe (4), Princen (8), and Dodds (10). Up to the point of separation, the pore is effectivelyclosedby the double wedge meniscus and all the theories agree. Afterthe point of separation, the Dodds analysistakes a nonwettingphase wedge as existing, whereas the Mayer and Stowe analysis, regardingthe pore as being defined by four rods, takes the triangular pore to have a reflecting neighbor. properties of the triangular pores given by the faces of these tetrahedral subunits are important because they correspond t o the constrictions in the pore space that determine displacement curvatures for drainage. For random sphere packings, Mason (12) has given the population distributions of various properties of these tetrahedra including the size distribution o f the Haines inspheres of the triangular pores. He used these, together with the Haines approximation, to calculate displacement curvatures of the individual faces. We now examine the degree to which such a model, using the Haines insphere, corresponds to the displacement curvatures calculated by the MS-P method. Journal of Colloid and Interface Science, Vol. 100, No. 2, August 1984

The configurations considered by Mayer and Stowe involved four spheres in a plane. Generally, a pore consists o f a pair of triangular pores with two contacts per triangular pore. There is also the special case of three contacts per pore for closed-triangular pores. In random packings of equal spheres, the triangular pores can be classified according to whether there are 0, 1, 2, or 3 contacts per triangle. In general, each triangular pore shape can have a range of displacement curvatures which depend on the size and shape of neighboring pores. The problem now arises as to what is the most satisfactory representation of neighboring pores when calculating the capillary be-

MENISCUS CURVATURES havior of individual pores in, for example, a random packing of equal spheres. O f all possible ways of modeling neighbors, the assumption of mirror image neighboring pores has distinct advantages. Analysis of such arrangements of rods gives unique solutions for particular geometries. With respect to neighboring pores in sphere packs, the mirror image model is physically realistic in that where neighboring menisci exist in porous media, all menisci in the region have the same curvature. For rod systems, if neighboring pores were not mirror images, adjacent menisci would have unequal curvatures and could not coexist. Further consideration of the appropriateness of mirror image pores as boundary conditions will be given later. We now consider a rod system with pores that have mirror image neighbors, and solidliquid contact angle of zero. The MS-P method will be applied to triangular pores, with rods of unit diameter at each vertex, and sides representing all combinations of side lengths, 1, 1.1, 1.2, 1.3, and 1.4 diameters. The section through the geometry of a general triangular pore is shown in Fig. 5. There may or may not be double wedge menisci between rod pairs. The analysis starts by calculating the meniscus curvature on the assumption that no wedges exist. Now, starting with the shortest side first it finds whether the existence of a wedge meniscus will reduce the computed meniscus curvature. If it does, then the area and perimeter force are modified accordingly and the next separation is tested. Should a separation be tested such that incorporating a meniscus increases the curvature, then that gap and all larger gaps are taken to have no wedge menisci. This analysis assumes that the meniscus can always adopt the configuration which gives the lowest curvature. With reference to Fig. 1, this means that the lowest curvature follows the lower branch starting at the point of separation rather than the upper branch which terminates at the point where back-to-back wedges rupture. The MS-P approach is to calculate the meniscus curvature from the area of liquid sup-

527

FIG. 5. The section through a general three-sided arrangement of three rods. The side lengths used in the analysis vary from one rod diameter, a contact, to 1.4 rod diameters, the maximum likely to occur in random packings of spheres. ported by surface tension forces at the perimeter. If there were no wedge menisci present, then the meniscus curvature is the area ABCDEF in Fig. 5, Atot, divided by the wetted perimeter (BC + DE + FA), Ptot. So, atot - -

rinit = Ptot

°

[8]

The curvature (1/rinit) represents the maxim u m curvature that the system might possibly produce. If wedge menisci exist, then they must reduce this curvature. Figure 6 shows a wedge meniscus for a general side length 2L, the wedge meniscus having a radius r, and the rods having a radius R. The contact angle is zero. If this wedge meniscus is incorporated, then the area is reduced by ABCD and the perimeter undergoes two changes. The section of perimeter equal to 2AB is no longer included but the section equal to BC is added. Let the area ABCD equal AA, then AA is given by AA=(R+r)

2sinacosa - [R2o~ + r2(90 - a)]Ir/180.

[9]

Journal of Colloid and Interface Science, Vol. 100, No. 2, August 1984

528

MASON AND M O R R O W

I I

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for rl. If the new radius rl was larger than the initial meniscus radius, rinit, then the wedge meniscus was taken to exist and rl became the current best estimate of the meniscus in the configuration. The whole procedure was then repeated on the assumption that there were wedge menisci in sides 1 and 2, and that both had a radius, r:. AAI was obtained from Eqs. [9] and [10] in terms of r2, and API similarly obtained. AA2 and APE were arrived at similarly, r2 from Eq. [ 11 ] being different for the side length Lz. r2 = (L2 - R c o s a 2 ) / c o s a 2.

If the change in perimeter is Ap, then A P = [R~ - r(90 - a)]Ir/90.

[10]

Simple geometry gives r = (L - R cos a)/cos a.

[11]

Initially, if there are no wedge menisci present, the area, Atot, can be obtained from geometry. Let S = (Ll + L2 + L3)/2 [12] then Atot = 4[S(S - L I ) ( S - L2) × ( S - L3)] 1/2 - IrR2/2.

[13]

The initial perimeter force is given by Ptot = a-R.

[14]

We can obtain the initial meniscus radius ri,it, from r i n i t = Atot/Ptot [ 15] which is the key MS-P equation. Now starting with the shortest side length 2Ll, a wedge meniscus can be assumed to exist and A l and P1 and the meniscus radius, rl, evaluated by solving Eqs. [9], [ 10], and [ 11 ] and using rl -

Atot Ptot

-

AAl API

[16]

where AAl and Ap~ are the changes in area and perimeter force brought about by the existence of the wedge meniscus in side 1. These equations were solved numerically Journal of Colloid and Interface Science, Vol. 100, No. 2, August 1984

[17]

Again, the numerical solution for r2 was found. If the new radius rE was larger than the current best estimate, rt, then this wedge meniscus was taken to exist and rE became the current best estimate. The whole procedure was repeated to test for the existence of the wedge meniscus in the third side. Of course, if wedges are found to be absent for the first or second rod pair, then wedges will be absent in all sides of greater length. This procedure can be summarized as the systematic calculation of all possible meniscus configurations and the adoption of the configuration which gave the largest meniscus radius, r. The normalized curvature, C, is given by C = R/r.

[ 18]

Iterative solutions were obtained for all triangular pores with combinations of side lengths of 1, 1.1, 1.2, 1.3, and 1.4 diameters. The results are given in Table I. Results obtained for equally spaced rods in infinite array and for three rods alone are shown in Fig. 7 as examples of pores with and without neighbors respectively. At small distances of separation, results for both examples are identical. The high curvatures predicted by Hwang result from not taking wedge menisci into account. At larger separations the results obtained for equally spaced rods in infinite array agree with analytic expressions for hydraulic radius given by Princen (8) and also by Hwang (11). Com-

MENISCUS CURVATURES TABLEI Parameters of Pore Geometry and Meniscus Curvatures for Triangular Configurations of Spaced Equal Rods and Zero Contact Angle Liquid Side lengths in rod diameters LI

Lz

L3

Curvature Cm.p MS-P method

1

1

1 1.1 1.2 1.3 1.4 1.1 1.2 1.3 1.4 1.2 1.3 1.4 1.3 1.4 1.4

11.320 8.051 5.801 4.829 4.498 6.715 5.232 4.361 3.925 4.122 3.508 3.159 2.959 2.629 2.303

1.1

1.2 1.3 1.4 1.1

1.1

1.1

1.2

5.984a 4.831 4.034 3.583 3.928 3.366 3.019 2.874 2.558 2.257

7.403 6.401 5.502 4.695 5.712 5.056 4.440 4.601 4.146 3.829

1.4

1.2 1.3 1.4 1.3 1.4 1.4 1.3 1.4 1.4

3.402a 2.987a 2.712a 2.625a 2.377a 2.144a 2.316a 2.099~ 1.900a

5.186 4.678 4.189 4.297 3.920 3.640 3.991 3.685 3.444

1.4

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3.244

1.4 1.2 1.3 1.3

1.4

12.928 10.133 8.000 6.331 4.997 8.512 7.117 5.919 4.888 6.249 5.427 4.663 4.898 4.360 4.009

1.1 1.2 1.3 1.4 1.2 1.3 1.4 1.3 1.4 1.4

1.3

1.2

Curvature Cn Haines incircle approximation

1.4 1.3

a Indicates no wedge menisci.

parison o f these results with those reported for three rods alone (8, 11) shows the extent to which meniscus curvatures are changed by the presence o f neighboring pores. F o o t n o t e a in Table I indicates that no wedge menisci are stable in the configuration

529

and consequently the total normalized meniscus curvature is simply R P t o t / A t o t , the reciprocal o f hydraulic radius. Also listed in Table I is the curvature (CH) obtained f r o m the Haines incircle approximation (2). It is instructive to c o m p a r e the incircle approximation with the curvature CMS.P obtained with the MS-P method. This is d o n e graphically on Fig. 8 with the M S - P curvatures being plotted against the incircle curvatures. The points are scattered but for m o s t practical purposes, a line can be fitted (weighted by passing through the point for the 1, 1, 1 arrangement which is the one occurring m o s t frequently in sphere packings) which gives CMS-P = CH -- 1.5.

[19]

This is a particularly simple approximation and it enables curvatures obtained from the simple incircle approximation to be corrected for some o f their grosser deficiencies. The MSP curvature is obtained simply by subtracting 1.5 from the normalized curvature given by the Haines insphere. A n examination o f Table I reveals that, n o t surprisingly, as the triangle edge lengths increase, the meniscus curvatures decrease. Figure 9 gives the meniscus curvature plotted against the s u m o f the gaps in the triangle sides (normalized by division by a rod diameter) for the tabulated results. The general trend o f falling curvature with increasing gap lengths can be seen. In fact, all o f the points from the table fall within the b a n d shown. This even simpler correlation enables the meniscus curvature to be estimated from the sum o f the three gaps in the sides o f the triangle defining the pore. While it is not exact, it is a convenience over the MS-P m e t h o d o f calculation. ASSEMBLIES OF PORES IN SPHERE PACKINGS The justification for the application o f the MS-P analysis to pores in sphere packings is that the curvatures measured for spheres are in close agreement with the theoretical curvatures calculated for rods. This link is only Journal of Colloid and InterfaceScience, Vol. 100, No. 2, August 1984

530

MASON AND MORROW 20

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established for closed-rhombus configurations of the type investigated by Hackett and Strettan. In extending application of the MS-P analysis to sphere packings, it is assumed that it applies to the additional configurations which arise in random packings of spheres. The extension of the analysis from pores with closed boundaries raised the question o f the adequate representation of neighboring pores. This was resolved by the assumption of mirror image neighbors. The influence of neighboring pores on displacement curvatures will now be reexamined together with the appropriateness Journal of Colloid and Interface Science, Vol. 100, No. 2, August 1984

of application of the rods analysis to pores in sphere packs.

Constant Curvature and Coexistence The liquid interface in a sphere pack will have a constant curvature, at least over the range of a few pores. Pores formed from assemblies of spheres can, unlike rods, support a range of meniscus curvatures. For any particular pore (hereafter referred to as the central pore), the most important curvature is the value at which it empties. If a meniscus exists in a neighboring pore, then its varying cross

MENISCUS CURVATURES i

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dron edges. The probability of a cell edge representing a sphere-sphere contact was approximately 1/2 and the probability of the d edge representing a gap was also about 1/2. Wz 10 ~ So, classifying triangular pores as S (for the 3: 9 shortest cell length, given by contacting spheres) and L (for any edge length that. is greater than 2R), the probability of a triangular og G pore being a closed-triangular pore is (1/2) 3 ~ G or 1/8. Similarly, triangular pores of the type SSL (two short and one long side) have a ~ 3 probability of 3/8, the SLL of 3/8 and the g z 2 L L L of 1/8. The percolation threshold of a i four-connected Bethe tree network, such as 0 I I I I I I I { I I I might logically be chosen to model the tet0 I 2 5 4 5 6 7 8 9 IO II 12 13 rahedrally structured pore space of a sphere NORMALIZED CURVATURE,MS-P, CMS_P packing has a threshold value of 1/3. At this FiG. 8. Graphicalcomparisonof the curvaturespredicted stage of drainage, the menisci will have mainly by the Haines incircle and the MS-P method for varying passed through pores in the L L L and LLS size and shape sectioned pores. Note that a linear approximation can be used to simply relate the values given categories. (This is based on the observation that pore size of the categories will tend to by the two methods. decrease in the sequence LLL, LLS, LSS, SSS.) Now consider the neighbors of particular sides of the central triangular pore. If the parsection will permit the coexistence of a meticular side is a contact (S), then the neighbors niscus which will have the same curvature as that in the central pore. Thus, at the emptying of the central pore, all the neighboring menisci 12 , , , , , i , i i , i i will have the same curvature and will thus II mimic reflective neighbor boundary condio_ ~ ° tions. For-a central pore defined by an ar~9 rangement of rods with neighboring pores also defined by rods, then, if the curvatures of such ~7 pores are different, it is not possible for main %6 terminal menisci to coexist in all pores. In ~ s order to model the constant curvature conN ~4 dition for the neighboring pores that occurs 13

12

in sphere packings, the meniscus curvatures of all of the rod pores (central and neighbors) have to be made the same. This can be achieved by making the neighboring pores mirror images of the central pore.

Probabilities of Pore Types Mason (12) has shown that the tetrahedral cells comprising a sphere packing can in turn be made up by random assembly of tetrahe-

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FIG. 9. Correlation of the total gaps in the pore sides against normalized Curvature. This correlation gives a simple way of estimating the meniscus curvature of a particular triangular pore with mirror image neighboring pores. Journal of Colloid and Interface Science,

Vol.

100, No.

2, August

1984

532

MASON AND

have no effect. If the side is long (L), then the other two sides of the neighbors may be SS (probability 1/4), SL (probability 1/2), or I,L (probability 1/4). The most likely neighbor of the LLS central pore thus has SL sides, and this is true for both of the triangular pores which share the L side. Because these types are in the same category (although not necessarily the same size), this again provides support (additional to the constant curvature argument) for the contention that the mirror image neighbor assumption is reasonable.

MORROW

Ls equivalent to the point of separation for rhombohedral pores, each pore will behave independently and there is no difference in drainage curvatures for single pores and pores with reflective neighbors (see Fig. 4). If a common edge length exceeds the value of Ls for the given central pore and neighbor configuration, the effect of neighbors can become significant. Even though the pores are not identical, the meniscus has constant curvature and the larger pore is expected from foregoing arguments to drain at a curvature very close to that given if the neighbor were reflective. Once the central pore drains, the Influence of Neighboring Pores on potential boundary condition for the neighbor Displacement Curvatures at the common edge is that given by an empty The importance of taking the presence of pore. Two possibilities now exist for the neighbors into account in the calculation of neighboring pores. If the pore remains filled, displacement curvatures can be gauged by the the central pore, in effect, is no longer a neighcomparison shown in Fig. 4 between isolated bor and boundary conditions more akin to pores defined by three rods and pores having the single pore shown in Fig. 4 now apply. mirror image neighbors. Dodds suggested that With further increase in curvature, this pore the computed solutions for three rods would will drain, but at a curvature corresponding be representative of equivalent triangular pores to a single pore. This will generally be distinctly in sphere packings. However, pores in sphere lower than that given by the same pore having packs clearly have neighbors. The curvatures a reflective neighbor. The other possibility is for mirror image pores calculated by Mayer that once the central pore drains, the limiting and Stowe (and they were in good agreement curvature for the neighboring pore after losing with experiment) are seen to be generally the central pore as a neighbor is exceeded. As much higher than those calculated by Dodds a result, the central pore and its neighbor drain for single pores. The behavior of a particular simultaneously. In either case, the curvature individual pore does not therefore necessarily for drainage of the neighboring pore is lower simply depend on its own geometry but is than the value that would be given assuming also, to some extent, dependent on the size a reflective neighbor. In summary, several possibilities exist for and behavior of neighboring pores. It follows that a given triangular pore geometry will not the drainage behavior of an accessible central pore with the following sequence of occuralways drain at a given meniscus curvature. Effects on meniscus displacement curva- rences being the more likely with increase in tures by neighbors can be classified by con- curvature (decrease in pore size). sidering the behavior of a particular pore (1) The central pore drains at a curvature throat (the central pore) with respect to one of its neighbors. Both pores are considered to corresponding to its mirror image boundary have gained accessibility in the percolation condition. The neighbor may or may not drain sense and therefore share a common meniscus. simultaneously. This behavior is more likely The central pore and the neighbors in question for the larger pores which tend to empty in share a tetrahedral edge. If this edge corre- the initial stages of desaturation. (2) If the central pore has a larger neighbor, sponds to a contact or is no greater than length Journal of Colloid and Interface Science, Vol. 100, No. 2, August 1984

MENISCUS CURVATURES

then it may drain at a curvature corresponding to that for drainage of the larger neighboring pore with mirror image boundary conditions. (3) The central pore has remained filled after the larger neighbor has drained at a curvature set by boundary conditions akin to the Dodds model of pore emptying established after drainage of one or possibly more neighbors. (4) The central pore is so small that its neighbors have no effect on its drainage curvature, the line of c o m m o n contact being less than the critical distance. (5) Menisci may not pass through any o f the pores of a tetrahedral subunit and so it may not drain at all. The still-full pore becomes hydraulically isolated through drainage of neighboring pores and associated funicular structures. Liquid contained in the undrained pore contributes to the irreducible saturation retained by the sphere pack.

Neighbors Not in the Same Plane When the MS-P analysis is applied to an arrangement of rods, then the rods have to be parallel and therefore the central pore and its neighbors are all in the same plane. In a random packing o f spheres it is most unlikely that the triangular pores will be in the same plane as their neighbors. However, the random packing does have some structure and this makes it possible to estimate these angles. For one pair of spheres which give a tetrahedron edge, there are approximately five tetrahedra sharing the edge. The angle between adjacent triangular faces will thus be about 360°/5 (about 72°). Neighboring faces can therefore be at up to an angle of about 72 ° to one another, rather than the 180 ° used by the rod analysis. This largest possible change in angle will clearly make a difference to the calculated curvatures and will tend to raise them more toward the Haa'nes insphere values. But because our values for the rods are always reasonably close to those given by the Haines insphere, the curvature change arising from

533

the adjacent faces not being in the same plane will be modest (an absolute maximum normalized curvature difference of 1.5). The gist of arguments based on curvatures calculated for rod systems should therefore still hold for sphere packings.

Pore Space Connectivity If the pore space of a random sphere packing is divided into tetrahedral subunits, then there are four triangular pores per subunit. From a percolation standpoint, such a system can be modelled conveniently as a Bethe tree network with four bonds (triangular pores) per node (tetrahedral subunit). As we have seen, there is a possibility that a large triangular pore can act in conjunction with a neighbor so that both pass a meniscus through the gap between two adjacent, but separated, spheres. This means that the meniscus leaving a tetrahedral subunit gains entrance not only to the two directly adjacent subunits, which is consistent with the Bethe tree network, but also to two other tetrahedra which share the large side length in c o m m o n - - a possibility that increases the apparent connectivity (bonds per node) from four to a larger value. However, it may be that the use of a connectivity of four for the modeling of tetrahedral unit connections in a sphere packing is still about right because the increased connections created by menisci passing through large gaps between spheres roughly balances the dosed loops of subunits which exist in the real network but not in the Bethe tree.

Pore Size and Capillary Displacement Curvatures It has been shown that properties of neighboring pores affect the capillary behavior of a central pore. Thus, there is no unique relation between the size and shape of a particular pore and its capillary displacement curvature but rather a range of displacement curvatures decided by the neighboring pores. If a group of pores has a geometric pore size distribution, Journal of Colloid and Interface Science, Vol. 100; No. 2, August 1984

534

MASON

AND

then an associated capillary curvature distribution cannot be evaluated simply from the size and shape of each pore combined with some topological model of connectivity. If the effects of neighbors is to be taken into account, the range of neighboring pores and their probabilities have to be allowed for. The capillary properties of pores of a particular size will depend on the distribution of probabilities of the neighbors and the likelihood of them being larger or smaller than the particular pores. In practice, the situation occurs in reverse when one seeks to obtain a geometric pore size distribution from measured capillary pressure curveS, such as might be obtained by mercury porosimetry. The pore size associated with a particular capillary pressure ought to allow for the effects of neighbors. From previous discussion, it was shown that the effects of neighboring pores will tend to make pore sizes measured from capillary pressure displacement curvatures more nearly alike. This occurs either through simultaneous drainage of neighboring pores of different size or change in boundary conditions, of the type illustrated in Fig. 4, which lower the displacement curvature for a given pore. The cascade mechanism, mentioned in the introduction of this paper, is associated with pores draining at a higher capillary pressure than might be computed from their size. In this respect, the effects of pore blocking are opposite to those of neighboring pores. However, both effects will contribute to making the pore size distribution obtained from capillary pressure curves much narrower than the actual geometric distribution. SUMMARY

The method presented by Mayer and Stowe (4) of calculating meniscus curvatures in pores formed by spheres, in fact, applies to parallel rods and is the same in principle as that presented later by Princen (7-9) for rods. A comparison of theoretical results for rods with the experimental results of Hackett and Strettan Journal of Colloid and Interface Science, Vol. 100, No. 2, August 1984

MORROW

(3) for sphere configurations indicated that the MS-P method, which is exact for rods, gave a good approximation, for zero contact angle, of meniscus behavior for pores defined by spheres. This observation provides an important bridge between theory for rods and its approximation to curvatures for pores formed by spheres. The mathematics for the latter are generally intractable. Application of the MS-P method is not trivial and requires that the basic configuration of the wedge menisci in pores of uniform cross section be known. The consequences o f not taking wedge menisci into account are demonstrated by the results of Hwang. In situations where wedges are present, Hwang's results show wide deviations for systems which form wedges. This is demonstrated by both experimental results and all other theoretical approximations. Estimation of displacement curvatures for assemblies of pores requires that the boundary conditions which are affected by the presence of neighboring pores be realistically defined. A convenient and plausible (because of the constant curvature condition) method of allowing for the presence of undrained neighbors is to assume that neighboring pores are mirror images of the pore in question. The assumption of mirror image neighbors, together with the assumption that curvatures for pores defined by spheres can be approximated by pores formed from rods, permits displacement curvatures to be calculated for a wide range of both symmetric and asymmetric configurations. Comparison of the complete range of results with the Haines incircle approximation reveals, to a first approximation, that the MS-P method gives dimensionless curvatures which were about 1.5 less than those given by the incircle. This result permits ready estimation of meniscus curvature in model pores. It involves only a simple correction to the easily determined values given by the incircle. If the change in boundary conditions caused by drainage of one pore affects its neighbor,

MENISCUS CURVATURES the result m a y be either s i m u l t a n e o u s drainage o f b o t h pores, o r i f the n e i g h b o r i n g p o r e rem a i n s filled, its d r a i n a g e c u r v a t u r e will b e reduced because the mirror image neighbor b o u n d a r y c o n d i t i o n n o longer applies. B o t h o f these m e c h a n i s m s will t e n d to cause p o r e sizes e s t i m a t e d f r o m c a p i l l a r y pressure relat i o n s h i p s to be n a r r o w e r t h a n t h o s e given b y the g e o m e t r i c a l structure o f the packing. ACKNOWLEDGMENTS Funding for this work was provided by the U. S. Department of Energy, Contract No. DE-AS 19-80BC10310 and the New Mexico Energy Research and Development Institute, Project No. 2-69-3309. G. Mason was on sabbatical leave from the Department of Chemical Engineering, Loughborough University of Technology, England.

535 REFERENCES

1. Orr, F. M., Scriven, L. E., and Rivas, A. P., J. Colloid Interface Sci. 52, 602 (1975). 2. Haines, W. B., J. Agric. Sci. 17, 264 (1927). 3. Hackett, F. E., and Strettan, J. S., J. Agric. Sci. 18, 671 (1928). 4. Mayer, R. P., and Stowe, R. A., J. Colloid Interface Sci. 20, 893 (1965). 5. Melrose, J. C., J. Colloid Interface Sci. 20, 911 (1965). 6. Haynes, J. M., Colloid Sci. (Chemical Society, Special Periodical Reports) 2, 101 (1975). 7. Princen, H. M., J. ColloidlnterfaceSci. 30, 69 (1969). 8. Princen, H. M., J. Colloid Interface Sci. 30, 359 ( 1969). 9. Princen, H. M., J. Colloid Interface Sci. 34, 171 (1970). 10. Dodds, J. A., Powder Technol. 20, 61 (1978). 11. Hwang, S. K., Z. Physik. Chernie Neue Folge 105, 225 (1977). 12. Mason, G., J. Colloid Interface Sci. 35, 279 (1971). 13. Mason, G., J. Colloid Interface Sci. 41, 208 (1972).

Journal of Colloid and Interface Science, Vol. 100, No. 2, August 1984