NUMERICAL STUDY OF THE ELECTRICAL DOUBLE-LAYER REPULSION BETWEEN NON-PARALLEL CLAY PARTICLES OF FINITE LENGTH

Transcription

1 NTERNATONAL JOURNAL FOR NUMERCAL AND ANALYnCAL METHODS N GEOMECHANCS, VOL. 15, (1991) NUMERCAL STUDY OF THE ELECTRCAL DOUBLE-LAYER REPULSON BETWEEN NON-PARALLEL CLAY PARTCLES OF FNTE LENGTH A. ANANDARAJAH AND NlNG LU Department of Civil Engineering, The Johns Hopkins University, Baltimore. MD. V.S.A. SUMMARY The study presented herein concerns the double-layer repulsive force that exists between clay particles in a clay-water-clectrolyte system. The objective is to examine the nature of the double-layer repulsive force between two negatively charged clay particles that are inclined to each other and have finite length. The finite element numerical technique is used first to determine the spatial distribution of the electric potential. The net repulsive force and its location are then evaluated from this known distribution of the potential on the basis of the theory of electrostatics. A systematic parametric study is performed by varying certain dimensionless system variables, particularly to investigate the effect of orientation, surface potential and particle length. t is found that for non-parallel orientation, as the value of inclination and surface potential increase, the repulsive force decreases and its location shifts towards the particle ends that are closer to each other. NTRODUCTON The theory of the electrical double layer surrounding particles in colloidal systems has been treated extensively in several text books (e.g. References 1 and 2). These classical theories are based on several idealizing assumptions. For example, the following assumptions are made in the Gouy-Chapman one-dimensional theory of a diffuse double layer surrounding a single clay (1) neglecting their size, the ions in the double layer may be considered as point charges; (2) the surface charge is uniformly distributed; (3) the clay particle is an infinitely long plate; and (4) the electric potential is reasonably small. With a further assumption that the solution contains ions of only one valence, complete analytical solutions were derived for the variation of electric potential, ion concentration and charge density with distance from the particle surface. Along the same lines, equations were derived for the double layer around a spherical particle (e.g. Debye-Huckel theory'). Stern developed an improved theory that takes into account the finite dimensions of the ions.6 Considering a system consisting of two infinitely long, parallel clay particles and making the aforementioned simplifying assumptions, analytical equations were subsequently derived for interacting double layers.' A quantity of considerable importance is the repulsive force that develops between the clay particles when the double layers interact with each other. An expression for the repulsive force may be derived in several different ways, using, for example, an equilibrium condition,' the osmotic pressure con~ept,~ the theory of electrostatics,* etc. All of these approaches lead to exactly the same result. By making use of the theoretical equations of an interacting double layer and considering a simple clay-water4ectrolyte system, Bolt developed a theory for the compressibility of saturated /9 lpo683-2 ls by John Wiley & Sons, Ltd. Received 19 February 199 Revised 6 June 199

2 684 A. ANANDARAJAH AND N. LU clays.9 t was assumed that the clay particles are arranged parallel to each other and that the pressure applied externally to the system is completely counterbalanced by the net double-layer repulsive force. By relating the average distance between the clay particles to the void ratio of the system, he predicted a relationship between the external pressure and the void ratio in terms of system variables such as the concentration and valence of ions, the specific surface of the clay particle, etc. To verify the validity of the theory, Bolt compared the theoretical predictions with experimental data obtained using specimens of clays such as montmorillonite and illite. The agreement was found to be encouraging in that the predicted variation in the behaviour with changes in system variables such as the electrolyte type and concentration was qualitatively in accord with experimental data. Quantitatively, however, there was significant discrepancy in some cases. Among other factors, non-parallel arrangement and non-uniform sizes of particles were considered to have contributed to the failure of the theory to predict the behaviour quantitatively. The objective of the study presented in this paper is to develop an understanding of the nature of interaction between double layers surrounding clay particles that are of finite size and have arbitrary orientation with respect to one another. Particular emphasis is given to the net repulsive force between the particles. Assumptions, such as that the ions in the system are point charges, the surface charge is uniformly distributed and the value of the electric potential is reasonably small, are retained in the present study as well. Since the governing partial differential equation in the electric potential cannot be solved analytically in the present case, it is solved numerically using the finite element technique. Once the spatial distribution of the electric potential is determined, the net repulsive force acting on the clay particles and its point of action are then determined. On the basis of the theory of electrostatics, a procedure is developed for this purpose. A systematic parametric study is carried out by varying the particle size, surface potential and angle between particles. For simplicity, a two-dimensional problem is considered. One application of the results of this study concerns the numerical simulation of the stress-strain behaviour of an assembly of arbitrarily oriented clay particles. n addition to double-layer repulsive forces, interparticle forces in real saturated clays include mechanical forces'o and other type of physicwhemical forces such as the Van der Waal's forces.' Such numerical studies carried out on an assembly of granular particles (where the interparticle forces are purely mechanical) have proven to be invaluable in understanding the microstructural aspects of the stress-strain behaviour." An example of a numerical simulation of an assembly of platy particles is given by Scott and Craig," who have not, however, considered the physicwhemical forces. Results presented in this paper should facilitate consideration of the net repulsive forces in such studies. FUNDAMENTAL DFFERENTAL EQUATON GOVERNNG THE VARATON OF ELECTRC POTENTAL n regard to the notation used in this paper, boldface quantities denote vectors or matrices. For detailed treatment of electrical double layers in colloidal systems, the reader is referred to standard texts (e.g. References 1 and 2). A summary is presented here for convenience and for later discussion. The present study concerns a colloidal system in which the solid phase consists of clay particles that are negatively charged on the surface. Here we assume that the intensity of the surface charge is characterized by a specified surface electric potential Jlo. The liquid phase consists of an electrolyte-water solution. We consider a particular simple system where the solution contains

3 ELECTRC DOUBLE-LAYER REPULSON N CLAY 685 ions of one valence. Let n denote the concentration (in ions cm-j) in the equilibrium solution (i.e. at a point far away from the surface) and v the valence of ions, both cations and anions. The ions are assumed to be point charges that are free to move around in the liquid phase. Concentrations of positive and negative charges, which are spatially uniform in the absence of clay particles, are redistributed and become spatially non-uniform when clay particles are added to the system in order to neutralize the negative charges on the surface of clay particles. This gives rise to the diffuse double layer around the particles. n such a system the electric potential $ also varies from point to point. Considering a two-dimensional domain R bounded within the boundary r, the spatial variation of the electric potential is governed by the Poisson equation as where p is the charge density, x and y are Cartesian co-ordinates and E is the dielectric constant of the medium. For reasonably small values of $, the cation and anion concentrations at any point, n- and n,, may be related to $ with the aid of Boltzmann s theorem as n- = nerp( *) ve* n+ = nexp( -*T) ye* (3) where e, k and Tare the electronic unit charge, Boltzmann s constant and absolute temperature respectively. The charge density may then be related to the electric potential by p = ve(n+ -n-) = -2nvesinh (4) (%) Substitution of (4) into (1) yields the following partial differential equation governing the spatial variation of electric potential in terms of the system variables: To complete the definition of the above boundary value problem, a set of boundary conditions must be specified on the boundary r, which are $ =& on r* (6) -=& a* on r,, as (7) + r,, 6 is the prescribed value of the potential on r*, b is the prescribed value of where l- = the gradient of $ with respect to s and s is the unit vector normal to rq. t is convenient to present the above boundary value problem in terms of dimensionless variables as follows. Let us introduce dimensionless variables 4, [ and q defined as

4 686 A. ANANDARAJAH AND N. LU where t may be noted that the reciprocal of K (i.e. 1 / K) given by (1 1) is the standard double-layer thickness of a single double layer surrounding an infinitely long clay particle placed in a monovalent electrolyte solution. The co-ordinates x and y are linearly scaled by the constant K in (9) and (1) to obtain ( and q. Thus the orientation of a unit vector in R will remain unchanged in the transformed domain (say fi), but the dimensions of fi will be linearly scaled by K. The boundary value problem is then transformed in terms of these dimensionless variables as where - ve$ Q=kT f = -, + is the boundary defining the transformed domain fi. SOLUTON BY THE FNTE ELEMENT METHOD The boundary value problem defined by (12)-(14) may be solved analytically only in certain special cases. The problem of interest in the present study, which involves arbitrarily oriented clay particles, requires a suitable numerical procedure. The finite element method is chosen herein because of its versatility in accommodating complicated boundary conditions. n the finite element method the boundary value problem is first transformed into a minimization problem with the aid of variational principles. The principles may be found in any standard text on the finite element method (e.g. Reference 13) and thus are not repeated herein; only the key steps are summarized. t may be shown that among all admissible fields of 4 within a the one that minimizes the quadratic functional will be the solution to the problem defined by (12)-(14). An admissible field in this case is one that is continuous and piecewise smooth in fi and satisfies the boundary condition given by (13). This variational statement forms the basis for deriving the finite element matrix equation. The domain fi is divided into a number of finite elements interconnected at a finite number of nodes. The spatial variation of 4 within each finite element is then approximated using smooth shape functions defined in terms of the values of Q at the nodes surrounding the element, called the nodal unknown degrees of freedom. The functional of (1 7) is then minimized with respect to

5 ELECTRC DOUBLE-LAYER REPULSON N CLAY 687 the unknown nodal degrees of freedom to arrive at a matrix equation of the form KU = F (18) where K is a banded, symmetric, positive definite matrix, u is a vector containing the unknown nodal degrees of freedom to be evaluated and F is a vector depending on the distribution of Q, on TQ and the spatial variation of 4 within a. The global matrix K can be formed by assembling element matrices k,. After imposing the boundary condition given by (13), the above matrix equation is solved to obtain u. n the present analysis the standard four-noded isoparametric element is employed. The element matrix k, in this case is identical to that involved in the solution of Laplace's equation and the reader is referred to any standard finite element analysis text. The special nature of F requires further comments. F depends on contributions from the second (say F,) and third (say F,) integrals in (17). Note that F = F, + F,. F, is again identical to that involved in the solution of the Laplace equation. F,, however, depends on the solution u itself (note that the third integral in (17) is a function of 4). thus making (18) non-linear. F2 can be formed by assembling the element vectors f;, which are of the form f; = j5* N sinh( 4) dae (19) where a, is the volume of the finite element e and N is the shape matrix containing the isoparametric shape functions. These shape functions relate the spatial variation of 4 within a given finite element e to the nodal unknowns, say $,, 4, 43 and 6,, as 4 = N14, + N242 + N343 + N444 (2) where N,, N,, N3 and N4 are the elements of N. As is normally done in the isoparametric formulation, the integral in (19) can be evaluated numerically as i=4 f; = 1 w'n'sinh(4')lj' i= 1 where a superscript i over a variable designates its value at the ith Gauss quadrature point. w is an appropriate weight and J is the absolute value of the Jacobian involved in the co-ordinate transformation from (c, q) (equations (9) and (1)) to the intrinsic co-ordinates used in the isoparametric formulation. 4i may be evaluated from the unknown degrees of freedom surrounding the element c$j, j = 1-4, as (equation (2)) Owing to the dependence of F on u, an iterative procedure is required to solve (18). At a given iteration, (18) may be written as Kd+l =F,+ Fi+, 1 (23) where u'+l is the solution at the ( + 1)th iteration and FF1 is the approximation to F, at the (+ 1)th iteration. Note that F, is independent of u. n the present analysis, FF' is evaluated using a weighted mean of the solutions obtained during the previous two iterations as $+ = (1 --)u'+ou'-' (24) where w in the above equation is a relaxation factor. This procedure was found to be stable and

6 688 A. ANANDARAJAH AND N. LU convergent. A value of o =.1 was found to work very well in all cases analysed. Typically, convergence was achieved within.1 YO accuracy in about 5 iterations. The validity of the numerical results was verified for two simple cases for which analytical solutions are available. The numerical results were compared with the analytical solutions. The case of a one-dimensional single particle is shown in Figure 1. An infinitely long clay particle is placed in an electrolyte system. A demensionless surface potential 4,-, is prescribed on the surface. Since the problem is symmetric about a plane passing through the centreline of the particle (i.e. i; = ), it suffices to consider one half of the domain. Also, the problem is one-dimensional and thus a unit length is considered in the q-direction. The boundary conditions are shown in the figure. n the finite element analysis the electrolyte solution is assumed to be at equilibrium (i.e. 4 = ) at [ = 3., which corresponds to a physical distance (x) of three times the double-layer thickness (equation (9)). The problem was analysed by the finite element method and the variation of 4 (equation (8)) with i; (equation (9)) is compared with the analytical solution. The analytical solution to 4 as a function of i; is given by' ew (ew2-1 )e-( e/2 = ew (e+o/z- 1)e-C (25) The results obtained for two cases, 4o = 5 and 1, are presented in Figures 2(a) and 2(b) respectively. The numerical results agree very well with the analytical results. Numerical results corresponding to a range of prescribed surface potential values compared well with analytical results. For the sake of brevity, these comparisons are not presented. t may be of interest to know that, using the values k = x erg K-', e = x 1- lo esu, T = 29" K and v = 2, the surface potential is evaluated from (15) to be 125 and 25 mv for b = 5 and 1 respectively. d/dv = a 1 a fluld n t e Figure 1. Schematic diagram of a non-interacting onedimensional double-layer problem

7 ELECTRC DOUBLE-LAYER REPULSON N CLAY 689 The numerical procedure was then used to analyse the case of one-dimensional interacting double layers. n this case, two infinitely long clay particles are placed parallel to each other at a certain separation distance. Equal potential values 4o are prescribed on the surface of the two clay particles. The analytical solution to the mid-plane potential 4d is evaluated from the following elliptic integral: [2cosh(y)-2cosh(~o)]- 2dy = -Co (26) where c is half the dimensionless distance between particles. The above integral can be evaluated using standard tables. The domain confined between the particles and having a unit length in the q-direction (Figure 3) was analysed using the finite element method. The variation of 4d for different values of Numcr i ca Nuncr i ca 1 --_. - Ana y t ical Analytical a c s (a) (b) Figure 2. Electrical potential distribution around a single clay particle clay particles - J fluid Figure 3. Schematic diagram of an interacting one-dimensional double-layer problem

8 69 A. ANANDARAJAH AND N. LU to is compared with the corresponding analytical values of 4,, in Figures 4(a) and 4(b) for #o = 5 and 1 respectively. t can be observed that the results compare very well. The case of inclined particles of finite lengths was then numerically analysed and the repulsive forces determined. The details are described in the following sections. NET REPULSVE FORCE BETWEEN TWO NCLNED CLAY PARTCLES A schematic diagram of the problem of interest is presented in Figure 5. Two clay particles A and B of equal length, are placed in an electrolyte solution as shown. The angle between the particles is 8. The potential around the surface of both particles is assumed to be a constant Jl, and that far away from the particles is assumed to be zero. The problem is symmetric about a plane midway between the particles (mid-plane). The objective is to evaluate the magnitude and location of the net electrical repulsive force between the two particles as a function of,, 8 and Jlo. A number of different methods have been employed (e.g. References 1, 7 and 9) in the past to evaluate the double-layer repulsion between two infinitely long, parallel clay particles. n the Analytical Numcr i ca Anal y t i csl Numer i ca lo' CO CO (a) (b) Figure 4. Mid-plane potential as a function of particle separation distance = ' >' * - *o \ Particle A! Length - &, Figure 5. Schematic diagram of the two-particle system to bc analyscd

9 ELECTRC DOUBLE-LAYER' REPULSON N CLAY 69 1 present analysis we evaluate the repulsive force using the theory of electricity. This method is most convenient and straightforward when evaluating the repulsive force from the spatial distribution of electric potential, which in the present study is evaluated by the finite element procedure described in the preceding section. Owing to the non-uniform distribution of charges in the domain of interest (Figure 5), there will be a net electrostatic force at every point in the domain. This net force may be evaluated through the concept of force density. Crowley gives the force on a unit volume of the medium containing the charges as14 wheref, and& are the force densities in the x- and ydirections and i and j are the unit basis vectors in the x- and y-directions. The net force over a finite domain R can be obtained by integrating the force densities as n the system shown in Figure 5, the net force over an appropriate portion of the domain will be felt directly by each of the clay particles. This electrostatic force is the double-layer repulsive force of interest. Since the problem is symmetric about the mid-plane, a$/ay = on this plane, where the y-axis is chosen to be normal to the mid-plane. t follows from this that the net force over the domain R, will act on particle A and that over R, will act on particle B, where R, is one half of the domain R cut along the mid-plane and R, is the other half. Because of symmetry, the forces felt by particles A and B are equal to each other. t therefore suffices to consider either R, or R, in the analysis. t is convenient to cast the problem in a non-dimensional form. Defining dimensionless net forces R, and R, as R, Fz = - 2nk T R, = - FY 2nkT and substituting for p in terms of $ from (4). the normalized net forces may be written in terms of the dimensionless quantities defined as where fi is the transformed domain. The finite element analysis procedure requires a finite domain. A rectangular domain fi, with respect to the transformed axes C and q, shown in Figure 6, is chosen herein, where the boundaries r,, T3 and r4 are to be placed sufficiently far away from the particle so that 4 on these boundaries is approximately zero. The lower boundary rl represents the mid-plane that divides the domain shown in Figure 5 into two equal halves. Evaluation of the net force over can be achieved using the integral expression (31). However, this expression requires, in addition to the spatial variation of 4, the spatial variation of its gradients (note that 8 cosh( 4)/aq = sinh( $)&$/dq, etc.). n the finite element method of solving

10 692 A. ANANDARAJAH AND N. LU Figure 6. Rectangular domain a, with respect to the transformed co-ordinates C and q (12)-( 14), 4 is treated as the primary variable and its gradients are treated as secondary variables, i.e. nodal values of 4 are first solved for and the gradients are then evaluated from these nodal values in accordance with the approximate shape function employed in the analysis. This results in lower accuracy for the gradients compared to that for the nodal values of 4 itself. n view of the fact that in the present analysis the spatial variation of gradients is highly non-linear near the particle surface and that higher-order shape functions may be required to capture such gradients, it would be more accurate to evaluate the net forces using an expression that involves just 4 and not its gradients. Such an expression can be derived from (31) using the well-known gradient theorem. Consider the closed boundary r shown in Figure 6, where r = rl + r2 + r3 + r4 + Ts + r6 + r7 + re. encloses the fluid in a, but excludes the clay particle. The volume integral over fil in (31) is transformed with the aid of the gradient theorem into a surface integral over as R = R,i + R,,j = - (s,i + s,j)cosh(#)dt where sc and s, are the (- and q-components of the unit vectors normal to the surface and i and j are the unit basis vectors in the (- and q-directions. This integral may further be simplified by splitting it into integrals over rl through re and noting the following. The unit vector normal to Ts is equal and opposite to the unit vector normal to r6, but the distribution of 4 along r5 is identical to that over r6. Therefore the sum of integrals over rs and r6 is zero. By the same argument, the sum of integrals over r7 and Ts is zero and that of integrals over r2 and is zero. Observing that rl and are normal to the q-axis and are of equal length, (32) simplifies to R, = (33) c c c R, = - J s,cosh(+)dr - J s,cosh(4)dr =J [cash($)- l]dt rl r3 rl t follows from (34) that in the case of infinitely long, parallel particles, where 4 is constant along rl, the repulsive force per unit length of the particle is ra = cosh(4) - 1 which is identical to the analytical solution.' According to (33) and (34) the net repulsive force is (34) (35)

11 ELECTRC DOUBLE-LAYER REPULSON N CLAY 693 normal to the mid-plane. The repulsive force and its location on rl are respectively the area and centroid of the distribution of cosh(4) - 1 along the mid-plane. NUMERCAL RESULTS Using the procedure described in the preceding sections, a systematic parametric study was carried out in order to investigate the influence of the parameters L (the dimensionless length of particles), 8 (the angle between particles) and do (the dimensionless surface potential) on the nature of the interparticle electric repulsive force between two inclined clay particles of finite length. Note that L =,K (equation (9)) and 4o = ve#,/kt(equation (15)). The finite element mesh employed in predicting the spatial variation of the dimensionless potential 4 is shown in Figure 7. n evaluating 4 as accurately as possible, a very fine mesh as shown was used. Neglecting the thickness, the particle was approximated by a straight line placed between points C and D shown in Figure 7. The dimensions of this rectangular domain, shown in Figure 7, were found to be satisfactory to simulate the boundary conditions on r2, T3 and r4 (Figure 6) properly. The analyses were repeated by varying the values of L, 8 and do. The results of all cases analysed are presented numerically in Tables -V. Results of selected cases are presented graphically in Figures A discussion of the results is presented in the following paragraphs. Recalling that the repulsive force is a function of the distribution of cosh( 4) - 1 along the midplane (equation (34)), it will be illustrative first to examine its variation for a few cases. Results are presented only for the cases in which L = 2. Figure 8 presents the variation of cosh( 4) - 1 along the mid-plane from C = to 1 for the case with 4o = 4. The particle is placed such that point C (Figure 7) is at { = 4 and q =. Different curves in the figure correspond to different &values. Similar results for +o = 6, 8 and 1 are presented in Figures For the sake of brevity, results obtained for particles of other lengths considered are not presented, but the pattern of variation was found to be similar in these cases. L. 5-1 e -1~1~ - -- / - 4. L CoSitmt 4.OO rl a4/ar, = o Figure 7. Mesh used for finite element computation of the electrical potential field

12 694 A. ANANDARAJAH AND N. LU Table 1. Normalized net repulsive force as a function of angle between particles () and normalized particle length (L) for normalized surface potential qjo = 4. L= 1 L=2 L=4 L= ~oooo 1 ~oooo 1 ~oooo 1 ~oooo ~oooo Table 11. Normalized net repulsive force as a function of angle between particles () and normalized particle length (L) for normalized surface potential qjo = Q

13 ~ ~~ ELECTRC DOUBLE-LAYER REPULSON N CLAY 695 Table 111. Normalized net repulsive force as a function of angle between particles () and normalized particle length (L) for normalized surface potential t$o = , Table V. Normalized net repulsive force as a function of angle between particles () and normalized particle length (L) for normalized surface potential 4o = L= 1 L=2 L=4 L=8 ~

14 ~ 696 A. ANANDARAJAH AND N. LU Table V. Normalized location of net repulsive force as a function of angle between particles () and normalized particle length (L) for normalized surface potential 4o = 4. - (deg) L= 1 L=2 L=4 1= Table V. Normalized location of net repulsive force as a function of angle between particles () and normalized particle length (L) for normalized surface potential 4o = 6. 2rj~c~(8/2) (deg) L=l L=2 L=4 1= O~oooo oooo o m oooo

15 ~~~~~ ELECTRC DOUBLE-LAYER REPULSON N CLAY 697 Table V. Normalized location of net repulsive force as a function of angle between particles (9) and normalized particle length (L) for normalized surface potential 4o = 8. 2r/ L cos( 12) (deg) L=l L=2 L=4 L= o m O.oo O.oo o m o m o m Table V. Normalized location of net repulsive force as a function of angle between particles (9) and normalized particle length (L) for normalized surface potential do = VL cos( /2) (deg) L= 1 L=2 L=4 L= , Coo O-oooO oooo oooo O.oo O@KHl o ~ m m.m QOOOO Several interesting observations can be made from the results presented in Figures When the particle is placed on the mid-plane (i.e. 8 = Oo), the values of 4 over the length of the particle (i.e. from t; = 4 to 6) correspond to the prescribed values. For 4o = 4,6,8 and 1, cosh( 4) - 1 is 263, 2.7, and respectively. t may be pointed out that cosh(4) is quite sensitive to the value of 4. n all four cases presented in Figures 8-1 1, it may be noted that as 8 increases, the shape of the cosh( 4) - 1 diagram changes. The value of cosh( 4) - 1 decreases for t; > 4 from the prescribed value at t; = 4. There is, however, some increase for t; 2 6. There are two reasons for this general behaviour. Firstly, since the particle length is kept constant, the far ends of the particles (point D in Figure 7) move towards [ = 4 as 8 increases. n other words, Lcos(8/2) decreases. Thus the effective length along the mid-plane over which the double layers interact with each other

16 698 A. ANANDARAJAH AND N. LU a - 2 L e - eoo e ) e 9 1 c Figure 8. Potential distribution along the mid-plane for &o = 4M and L = a L u 1 5, decreases. Secondly, the vertical distance between the far ends of the particles increases as increases. This again decreases the interaction for C > 4. As the level of interaction decreases, the rate of decay of q5 away from the particle increases, giving rise to a lower mid-plane potential. t follows from (32) that when the total areaof the cosh(q5) - 1 versus 6 diagram decreases, the net repulsive force should decrease. Also, the centroid of this diagram shifts towards C = 4 from a

17 ELECTRC DOUBLE-LAYER REPULSON N CLAY c Figure 1. Potential distribution along the mid-plane for & = 8. and L = Figure 11. Potential distribution along the mid-plane for &, = 1 and L = 2. value of ( = 5 for 8 = "; i.e. the location of the repulsive force moves closer to ( = 4. The influence of the surface potential may be seen by comparing the results presented in Figures with one another. The decrease in the value of cosh( 4) - 1 with 8 near the far end increases with &. This is due to the fact that the rate of decay of 4 away from the particle surface increases with an increase in the value of the surface potential. n other words, for a given angle 8

18 7 A. ANANDARAJAH AND N. LU the ratio between the mid-plane potential and the surface potential near the far end becomes smaller and smaller as the surface potential increases. This is clearer in the one-dimensional results presented in Figures qa) and qb). Note that when C = 1, the ratio between the mid-plane potential and the surface potential drops from a value of about 4 to.2 as 4. increases from 5 to 1. From the numerical results, the net repulsive force and its line of action for the various cases considered were determined by computing the area of the cosh(4) - 1 diagram and its centroid respectively (equation (34)). The results are presented in Tables -V. For the purpose of visual examination and discussion, the variation of the repulsive force with the variables involved (8, 4. and L) is presented for a few cases in Figures 12 and 14 and the variation of its location for the same cases is presented in Figures 13 and 15. The repulsive forces presented in these figures were normalized with respect to the electrostatic force (R,) acting over a length L on one side of an infinitely long, single particle placed in the respective reservoir. n other words, if an infinitely long particle was placed in the same electrolyte system, there would be an equal and opposite pressure (R,/L) acting on both sides of this particle and, of course, the net force would be zero. t follows from (31) that =[cosh(4)- l]l (37) The net force R, for the two-particle system evaluated by (34) is normalized with respect to R,. The location of R, was computed as a distance ( along the c-axis from point C (Figure 7) and normalized with respect to Lcos(4/2)/2, the projection of one-half the particle length on the mid-plane. Note that for the cases with 8 =, the distribution of cosh(4) - 1 along the mid-plane is symmetric about the centre of the particle and thus 2r/Lcos(8/2) = 1... go - 4,-6 go = 8 v go = 1. B D O O D a o o s o.1 l o eo ioo ieo 8 (degrees) Figure 12. Variation of normalized repulsive force with angle between particles for different surface potential values (L = 1.)

19 ELECTRC DOUBLE-LAYER REPULSON N CLAY 7 1 Figure 13. (degrees) Variation of normalized location of repulsive force with angle between particles for ditterent surface values (L = 1.) potential l2 1.o 8- a L-1 L = 2 L - 4 L = 8 Y' \*O 6 x a Y." Figure 14. Variation of normalized repulsive force with normalized surface potential for different values of normalized particle length ( = 45") The influence of 8 on the value of RJR, is presented in Figure 12 and that on the value of 2r/Lcos(8/2) is presented in Figure 13. For all four cases of &, presented, R,/R, is very near 1. at 8 =, indicating that the particle length does not have a noticeable effect on the repulsive pressure (force/unit length) in these parallel cases. This was found to be true for all L-values considered (i.e. L 2.5). As expected, 2vL cos( 8/2) in these cases is computed to be unity (i.e. the net force passes through the centre of the particle).

20 72 A. ANANDARAJAH AND N. LU J L - L = 2 A L - 4. v L = a ooo _ Figure 15. Variation of normalized location of repulsive force with normalized surface potential for different values of normalized particle length ( = As 8 increases, R,/R, decreases and the rate of decrease increases with increasing value of 4,. Figure 13 shows that the location shifts towards the left and the rate at which it shifts increases with increasing 4,. As indicated earlier, this is due to the way the distribution of cosh(4) - 1 along the mid-plane changes with these variables (Figures 8-1 2). The results presented in Figures 14 and 15 for a particular value of = 45" show that the values of R,/R, and 2VLcos(8/2) for a given value of 4, decrease with increasing value of L. This is due to the fact that because the particles are inclined, as L increases, the far ends of the particles move farther and farther away in the q-direction. This reduces the overall interaction between the particles, reducing the value of R,/R,. The interaction becomes centred around the left ends of the particles and therefore 21/L cos( 8/2) decreases. n summary, the numerical results indicate that an increase in the values of the three variables considered (, 4, and L) has the same effect on the repulsive force and its location, i.e. decreases the value of R, with respect to R, and shifts the location of R, towards the left ends of the particle. One must, however, be careful in evaluating the absolute value of the repulsive force, i.e. F, (equations (28) and (3)). For example, even though R, decreases relative to R, as 4, increases, Fy may or may not decrease because R, increases with 4, (equation(37)). Denoting R,/R, by U, F, may be computed using (3), (37), (S), (15) and (1 1) in terms of the system variables as F, = ZnkTR, (38) = 2nkTUR, (39) = 2nkTUL[cosh(4,)- 13 =4nl,euJ(7)[cosh($)-] 2nnkT

21 ELECTRC DOUBLE-LAYER REPULSON N CLAY 73 For the benefit of those readers who may be interested in estimating the value of the repulsive force and its location as a function of the system variables, the results of all cases considered in this study are presented numerically in Tables -V. t may be pointed out that because of its non-linear nature, the finite element analyses involved in this study are computationally complex and costly. The analyses were carried out on a Micro Vax 1 computer and a typical run takes about 6 h of CPU time. The results presented in Tables -V were obtained from 32 such runs, after several other preliminary runs. CONCLUSONS The nature of the double-layer interaction between two clay particles that are of finite length and arbitrarily oriented has not been investigated in the past owing to the difficulty that the problem does not lend itself to analytical treatment. n the study presented in this paper, this is achieved numerically using the finite element technique. The objective is to investigate the nature of the magnitude, direction and location of the net double-layer repulsive force between two clay particles as a function of particle inclination, surface potential and particle length. The spatial distribution of the electric potential is determined using the finite element technique and the net force is computed from this on the basis of the theory of electricity. Extensive parametric study is carried out. The results indicate that the overall double-layer interaction decreases as the angle between particles, surface potential and particle length increase. This has an effect of decreasing the value of the net repulsive force and shifting its location towards the particle ends that are closer to each other. The results are presented in terms of dimensionless variables and, given the values of the system variables, these results may be used to estimate the value of the net repulsive force and its location. ACKNOWLEDGEMENT The computer facilities required for this study and partial financial support to Ning Lu, the coauthor of this paper, were provided by the Department of Civil Engineering of The Johns Hopkins University. REFERENCES 1. E. J. W. Verwey and J. Th. G. Overbeek, Theory ofthe Stability of Lyophbic Colloids, Elsevier, Amsterdam, H. van Olphen, An ntroduction to Cloy Colloid Chemistry, Wiley, New York, G. Gouy, 'Sur la constitution de la charge electrique a la surface d'un electrolyte', Ann. Phys. (Paris), Ser. 4, 9, (191). 4. D. L. Chapman, 'A contribution to the theory of electrocapillarity', Phil. Mug. 25, (1913). 5. P. Debye and E. Huckel, Phys. Z., 24, (1923); 25, (1924). 6.. Stem, 'Zur Theorie der Elektrolytixhen Doppelschriht', Z. Elektrochem. 3, (1924) Langmuir, 'The role of attractive and repulsive forces in the formation of tactoids, thixotropic gels, protein crystals and coacervates', J. Chem. Phys. 6, (1938). 8. B. Derjaguin, 'On the repulsive forces between charged colloid particles and on the theory of slow coagulation and stability of lyophobe sols', Trans. Farduy Soc., 36, (194). 9. G. H. Bolt. 'Physic-hemical analysis of the compressibility of pure clays', Ceotechnique, 6, 8693 (1956). 1. J. K. Mitchell, Fundamentals of Soil Behavior, Wiley, New York, P. A. Cundall, A. Drexher and. D. L. Strack, 'Numerical experiments on granular assemblies; measurements and observations', Proc. UTAM Symp. on Deformution and Failure of Granular Materials, Delft, 31 August-3 September 1982, pp. 35S R. F. Scott and M. J. K. Craig, 'Computer modeling of clay structure and mechanics' 1. Ceotech. Eng.,ASCE, 16, (GTl), (198) C. Zienkiewia, The Finite Elemenr Method, McGraw-Hill, New York, J. M. Crowley, Fundamentals of Applied Electrostatics. Wilcy, New York, 1986.