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EMPIRICAL ESTIMATION O F DOUBLE-LAYER
REPULSIVE F O R C E BETWEEN T W O INCLINED CLAY
PARTICLES O F FINITE L E N G T H
By Ning Lu1 and A. Anandarajah2
INTRODUCTION
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
21218-2699.
2
Asst. Prof., Dept. of Civ. Engrg., The Johns Hopkins Univ., Baltimore, MD
21218-2699.
Note. Discussion open until September 1, 1992. 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, 1990. This paper is part of the Journal of Geotechnical Engineering,
Vol. 118, No. 4, April, 1992. ©ASCE, ISSN 0733-9410/92/0004-0628/$1.00 + $.15
per page. Paper No. 570.
628
J. Geotech. Engrg. 1992.118:628-634.
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conducted. Results 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):
dd2<\>
+ d
4> =
ds
<f>
= 0
= sinh(4>)
in A
(1)
on f*
(2)
on TQ
(3)
where
n - n,+n2
0=0
FIG. 1. Schematic of Two-Particle System to Be Analyzed
629
J. Geotech. Engrg. 1992.118:628-634.
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0=0
r4
0=0
Mid-plane
Q. =0
FIG, 2. Rectangular Domain a , with Respect to Transformed Coordinates £
and T)
•"3
<«>
£ = .Kx
(5)
•n = «y
(6)
K= S j l
(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
= £>e0,e0 = 8.854 x 10 ~ 12 c/vm;D is the dielectric constant of the medium;
e = 1.602 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;
and 4> and qs 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
J. Geotech. Engrg. 1992.118:628-634.
presented in tabular and graphical forms in terms of a repulsive force index
If and a location index /, defined as:
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' / = §
z =
'
•
••
(10)
(11)
L^2e
where R^ = a normalized repulsive force in the ^-direction (Fig. 2); Rx =
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:
(12
*--dfe
>
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 If 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_
(13
' = r ra,*^
>
(14
1
> = TT^<
>
where
Of = 0.005L 2 - 5 4)i-°- 61n ( L ) +31 ]
(15)
85
bf = -0.725<t>-°- ln(L) + 2.3 - 0.18d>o
(16)
+ 0.7 x 10-6<K-7 - 0.3
(17)
a, = O.Offl&L
b, = 0.001(|£9 + 1 . 2
(18)
The coefficients af, bf, a,, and bt were determined from the numerical
data by a least-square matching procedure. In order to illustrate how well
631
J. Geotech. Engrg. 1992.118:628-634.
1.2
1
• 1 '
1 ' .1
- E q . 13
00 " 4 . 0 0 , Numeri c a l
00 ' 6 > 0 0 - Numeri c a l
0O - e.oo. Numeri c a l
- 1 0 . 0 0 , Numerical
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1.0
0.0
0
20
10
60
80
100
120
1
1
Results
Results
Results
Results
_L
140
160
180
8 (degrees )
FIG. 3.
Comparison of Empirical and Numerical Relationships for L = 4
1.2
1
'
1
'
1
1
'
1
'
- E q . 14
00 " 4 . 0 0 ,
• 00 - 6 , 0 0 ,
0
00 - 8 . 0 0 ,
A
00 = 1 0 . 0 0 ,
0
1 .0
0.8
8 °'
~
Numer
Numer
Numer
Numer
1
eel
eel
cal
eel
, | ,
Results
Results
Results
Results
.
-
V
6
- °\\
\o
0.4
4
n_
"
1
<>V
\
0.2
-
• D
^ i £ H a c
0.0
!
0
20
,
1
40
,
l . l . l
i
60
80
100
1
120
140
,
1
160
,
180
9 (degrees )
FIG. 4.
Comparison of Empirical and Numerical Relationships for L = 4
these equations represent the numerical data, the variations of the repulsive
force index lf 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. 3-6. 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 rep632
J. Geotech. Engrg. 1992.118:628-634.
1.2
-
0
D
o
4
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1.0
i
—
|
—
i
—
|
—
i
Eq. 13
0Q a 4.00,
0O - 6.00,
0O - 8.00,
0O -10.00,
—
r
T
Numerical
Numerical
Numerical
Numerical
T
Results
Results —
Results
Results
0.8
0.6
0.4
0.2
_. L
120 140
0.0
0
20
40
60
80
8
FIG. 5.
100
(degrees
180
)
Comparison of Empirical and Numerical Relationships for L = 8
"T
Numerical
Numerical
Numerical
0O -10.00, Numerical
0
20
40
60
80
9
FIG. 6.
_L
160
100
120
T~
Results
Results
Results
Results
140
160
180
I degrees )
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
J. Geotech. Engrg. 1992.118:628-634.
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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), 86-93.
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, 355370.
Low, P. F. (1980). "The swelling of clay: II. Montmorillonites." Soil Sci. Soc. Am.
J., 44, 667-676.
Low, P. F. (1981). "The swelling of clay: III. Dissociation of exchangeable cations."
Soil Sci. Soc. Am. J., 45, 1074-1078.
Low. P. F., and Margheim, J. F. (1979). "The swelling of clay: I. Basic concepts
and empirical equations." Soil Sci. Soc. Am. J., 43, 473-481.
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.
17-34, Jan. 1980.
Sposito, G. (1984). Surface chemistry of soils. Oxford Univ. Press, 205-227.
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.
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J. Geotech. Engrg. 1992.118:628-634.