EMPIRICAL ESTIMATION OF DOUBLE-LAYER REPULSIVE FORCE BETWEEN TWO INCLINED CLAY PARTICLES OF FINITE LENGTH

Transcription

1 EMPIRICAL ESTIMATION OF DOUBLE-LAYER REPULSIVE FORCE BETWEEN TWO INCLINED CLAY PARTICLES OF FINITE LENGTH INTRODUCTION By Ning Lu 1 and A. Anandarajah 2 The electrical double layer surrounding particles in a clay-water-electrolyte system has analytically been treated in the past for a limited number of simple cases; for example, the case of interacting double layers surrounding parallel particles of infinite length (van Olphen 1977; Verwey and Overbeek 1948). These mathematical equations have served both qualitatively and quantitatively to explain the behavior of colloidal systems. For example, the existence of various types of microstructures such as flocculated and dispersive structures in saturated clays can be explained qualitatively on the basis of the existing physico-chemical theories. Using these theories, Bolt (1956) attempted to predict the compressibility characteristics of saturated clays. The agreement between the theory and experiment was found to be encouraging in that the predicted variation in the behavior with changes in system variables was qualitatively in accord with the experimental data. Quantitatively, however, there was significant discrepancy in some cases. A number of factors may be considered to have attributed to the failure of the theory to predict the behavior quantitatively; for example, cross linking and nonparallel particle arrangement (Mitchell 1976), presence of "dead" volume of liquid resulting from terraced particle surfaces (Bolt 1956), existence of strong attractive forces (Sposito 1984), and influence of the structure of interlayer water (Low and Margheim 1979; Low 1980, 1981). Furthermore, several other idealizing assumptions, e.g., ions are point charges, were made in deriving the double-layer equations (van Olphen 1977; Verwey and Overbeek 1948; Mitchell 1976). Recently, Anandarajah and Lu (1990) studied the influence of nonparallel particle orientation and the particle length on the double-layer repulsion between clay particles. The attractive forces were not considered in this study and the other assumptions made in the derivation of classical equations (van Olphen 1977; Verwey and Overbeek 1948) were retained. The spatial distribution of the electrical potential was first determined by employing an iterative finite element technique to solve the nonlinear governing equation. The magnitude and location of the net repulsive force were then evaluated from this known distribution of the potential based on the theory of electrostatics. By employing this procedure, a systematic numerical study of the dependence of the normalized net repulsive force on the system variables including the particle length, surface potential, and particle inclination was 'Grad. Student, Dept. of Civ. Engrg., The Johns Hopkins Univ., Baltimore, MD Asst. Prof., Dept. of Civ. Engrg., The Johns Hopkins Univ., Baltimore, MD Note. Discussion open until September 1, To extend the closing date one month, a written request must be filed with the ASCE Manager of Journals. The manuscript for this paper was submitted for review and possible publication on September 24, This paper is part of the Journal of Geotechnical Engineering, Vol. 118, No. 4, April, ASCE, ISSN /92/ /$ $.15 per page. Paper No

2 conducted. were presented in both graphical and tabular forms for a wide range of system variables. These numerical results would be most useful if they could be cast into some form of empirical relationship so that they could easily be implemented in an analysis of interest. For example, such formulas would allow one to efficiently, but approximately calculate the electrical double-layer repulsive force in such analysis as the numerical simulation of the behavior of an assembly of clay particles subjected to an external load. Such previous studies on discs, spheres, and plates that considered mechanical effects have provided significant insight into their micromechanics (Cundall et al. 1982; Scott and Craig 1980). The objective of this note is to present empirical relationships describing the relationship between both the magnitude and location of the net repulsive force and the system variables. NET REPULSIVE FORCE BETWEEN TWO CLAY PARTICLES The boundary value problem involved is stated and solved by Anandarajah and Lu (1990). For convenience of the reader, the pertinent details are summarized herein. Considering a clay-water-electrolyte system shown in Fig. 1 and making some variable transformations, the boundary value problem associated with the spatial distribution of the electrical potential v i can be cast in a dimensionless space as (Fig. 2): 4> = ds where + - d <f> = 0 d 2 <\> = sinh(4>) in A (1) on f* (2) on T Q (3) n - n,+n 2 0=0 FIG. 1. Schematic of Two-Particle System to Be Analyzed 629

3 0=0 0=0 Q. =0 Mid-plane FIG, 2. Rectangular Domain a, with Respect to Transformed Coordinates and T) "3 <«> =.Kx (5) n = «y (6) K= Sjl (7) and * " # ' < " > e- = # m where T - T^ + f e is the boundary defining the transformed domain ft. In the aforementioned equations, x and y are Cartesian coordinates; e = >e 0,e 0 = x 10 ~ 12 c/vm;d is the dielectric constant of the medium; e = X 10~ 19 ; C is the elementary charge; n is the concentration of ions; v is the valence of ions; k and T are the Boltzmann's constant and absolute temperature, respectively; i i is the electric potential at any point; and4> and q s are the prescribed values of the electric potential and its gradient, respectively. Also note that UK is the double-layer thickness (or Debye length) in the case of a noninteracting, single particle of infinite length. Applying a variational principle and the finite element method and using the principles of electrostatics, the boundary value problem [(l)-(3)] was solved numerically in order to determine the net double-layer repulsive force and its location for a wide range of system variables. The results were 630 r 4

4 presented in tabular and graphical forms in terms of a repulsive force index If and a location index /, defined as: '/= (10) z = ' L^2e where R^ = a normalized repulsive force in the ^-direction (Fig. 2); R x = the value of R^ when 9 is zero [note:_i? = (cosh 4> - 1)L]; L = the normalized length of the particle; and / = the location measured from the left end of the particle as shown in Fig. 2. It was shown that i? = 0. R^ is related to the corresponding force in the x - y plane as: *--dfe (11) (12 > The validity of the numerical procedure was verified considering two simple problems for which analytical solutions are available; i.e., the case of a noninteracting single particle of infinite length and the case of two interacting parallel particles of infinite length. Excellent agreements were found in both cases. For details of the experimental verification of the onedimensional equations, the reader is referred to Mitchell (1976). CHOSEN EMPIRICAL FORMULAS The numerical analyses were carried out for the following values of the normalized system variables: <j> 0 = 4, 6, 8, and 10, L = 1, 2, 4, and 8, and 8 starting from 5 in increment of 5 until I f becomes zero. The empirical equations to be chosen should be able to closely approximate the numerical data for the range of values of the system variables considered so that they can be used to quantify the net repulsive force and its location for a given set of system variables. The following expressions have been found to mathematically represent the numerical data: 7 1_ ' = r ra,* ^ 1 > = TT^< where (13 > (14 > Of = 0.005L 2-5 4)i- - 61n ( L ) +31 ] (15) b f = <t> ln(l) d> o (16) a, = O.Offl&L x 10-6 <K (17) b, = 0.001( (18) The coefficients a f, b f, a,, and b t were determined from the numerical data by a least-square matching procedure. In order to illustrate how well 631

5 ' 1 ' Eq " 4.00, Numeri cal 00 ' 6 >00- Numeri cal 0 O - e.oo. Numeri cal , Numerical 0.0 _L (degrees ) FIG. 3. Comparison of Empirical and Numerical Relationships for L = ' 6 ~ 0.4 n_ " \\ 4 \ V 1 ' 1 ' 1 \o <>V 0 0 A D 1 ' 1 ' 1 -Eq " 4.00, Numer 00-6,00, Numer , Numer 00 =10.00, Numer ^i Hac 0.0!, 1, i l. l. l 1, 1, (degrees ) 1 eel eel cal eel,, FIG. 4. Comparison of Empirical and Numerical Relationships for L = 4 these equations represent the numerical data, the variations of the repulsive force index l f and the location index /, with the system variables given by these empirical formulas are compared with the numerical results for two typical cases (L = 4 and L = 8) in Figs The agreement is very good in some cases (e.g., see Fig. 3) and somewhat less in others (e.g., see Fig. 5). For the sake of brevity, comparisons for other cases are not presented herein, but the error was found to be in the same order as seen in these figures. In general, the discrepancy decreases when the particle length L increases and increases when surface potential increases. Further results and discussion can be found in Lu (1990), where it is shown that the variations rep

6 i i i r T T Eq Q a 4.00, Numerical D 0 O , Numerical o 0 O , Numerical 4 0 O , Numerical 0.0 _. L _L (degrees ) FIG. 5. Comparison of Empirical and Numerical Relationships for L = 8 "T T~ Numerical Numerical Numerical 0 O , Numerical I degrees ) FIG. 6. Comparison of Empirical and Numerical Relationships for L = 8 resented by the empirical equations and the numerical data has correlation coefficient varying between 95% and 99%. CONCLUSIONS The double-layer repulsive force that exists between two inclined clay particles of finite length, immersed in an electrolyte solution, has recently been quantified using a numerical procedure. Its magnitude and location have been calculated for a wide range of system variables, the surface potential, angle between particles, and length of particles. These numerical data are cast in the form of empirical equations in this note. It is shown that the relationships between the magnitude of the repulsive force and its 633

7 location and the system variables given by these equations agree reasonably well with the numerical results. These equations may, for example, be used in such studies as the numerical simulation of the stress-strain behavior of an assembly of clay particles subjected to an external load, where it is necessary to evaluate the repulsive forces as a function of the system variables. APPENDIX. REFERENCES Anandarajah, A., and Lu, N. (1990). "Numerical study of the electrical double-layer repulsion between non-parallel clay particles of finite length," Int. J. Numerical and Analytical Methods in Geomechanics (in Press). Bolt, G. H. (1956). "Physico-chemical analysis of the compressibility of pure clays," Geotechnique, London, England, 6(2), Cundall, P. A., Drescher, A., and Strack, O. D. L. (1982). "Numerical experiments on granular assemblies; measurements and observations." Proc. of the IUTAM Symp. on Deformation and Failure of Granular Materials, Aug. 31-Sept. 3, Low, P. F. (1980). "The swelling of clay: II. Montmorillonites." Soil Sci. Soc. Am. J., 44, Low, P. F. (1981). "The swelling of clay: III. Dissociation of exchangeable cations." Soil Sci. Soc. Am. J., 45, Low. P. F., and Margheim, J. F. (1979). "The swelling of clay: I. Basic concepts and empirical equations." Soil Sci. Soc. Am. J., 43, Lu, N. (1990). "Numerical study of the electrical double-layer repulsion between nonparallel clay particles of finite length," thesis presented to The Johns Hopkins University, at Baltimore, Maryland, in partial fulfillment of the requirement for the degree of Doctor of Philosophy. Mitchel, J. K. (1976). Fundamentals of soil behavior. John Wiley & Sons, New York, N.Y. Scott, R. F. and Craig, M. J. K. "Computer Modeling of Clay Structure and Mechanics," Journal of Geotechnical Engineering, ASCE, Vol. 106, No. GT1, pp , Jan Sposito, G. (1984). Surface chemistry of soils. Oxford Univ. Press, van Olphen, H. (1979). An introduction to clay colloid chemistry. John Wiley & Sons, New York, N.Y. Verwey, E. J. W., and Overbeek, J. Th. G. (1948). Theory of the stability oflyophobic colloids. Elsevier Publishing Company, Inc. 634