Clays and Clay Minerals, Vol. 44, No. 4, 479-484, 1996.
P O T E N T I A L - D I S T A N C E RELATIONSHIPS OF CLAY-WATER
S Y S T E M S C O N S I D E R I N G THE STERN THEORY
A. SRIDHARAN 1 AND P. V. SATYAMURTY2
Dept. of Civil Engineering, Indian Institute of Science, Bangalore 560 012, India
Abstract--This paper deals with the use of Stern theory as applied to a clay-water electrolyte system,
which is more realistic to understand the force system at micro level than the Gouy-Chapman theory.
The influence of the Stern layer on potential-distance relationship has been presented quantitatively for
certain specified clay-water systems and the results are compared with the Gouy-Chapman model. A
detailed parametric study concerning the number of adsorption spots on the clay platelet, the thickness
of the Stern layer, specific adsorption potential and the value of dielectric constant of the pore fluid in
the Stern layer, was carried out. This study investigates that the potential obtained at any distance using
the Stern theory is higher than that obtained by the Gouy-Chapman theory. The hydrated size of the ion
is found to have a significant influence on the potential- distance relationship for a given clay, pore fluid
characteristics and valence of the exchangeable ion.
Key Words--Cation Size, Clay Minerals, Diffuse Double Layer, Electric Potential, Stern Theory.
INTRODUCTION
Several investigators have studied the diffuse double layer theory (Verwey and O v e r b e e k 1948; Bolt
1956; L o w 1959; van Olphen 1963; Mitchell 1976;
Sridharan and Jayadeva 1982) to understand better the
behavior of a clay-water electrolyte system. M a n y of
these researchers have used the G o u y - C h a p m a n theory
to explain their experimental observations (Bolt 1956;
Olsen and Mesri 1970; Mitchell 1976; Sridharan and
Jayadeva 1982). However, the G o u y m o d e l of an electric double layer contains some unrealistic elements.
F o r example, the ions are treated as point charges and
any specific effects related to the ion size are neglected. However, in the vicinity o f particle surface, where
the concentration is high, the distance of the closest
approach to the surface is important and the size of
the ions must be considered to give an accurate description o f the status o f cations adsorbed on clay particles as in the S t e m model. A n attempt was m a d e here
to study the effect of various parameters on the potential-distance relationship assuming the S t e m model
and compared the effect to the G o u y - C h a p m a n model.
DOUBLE LAYER AS PER STERN THEORY
In the S t e m theory, the distance of the closest approach of counter-ions to the charged clay surface is
limited by the size of these ions. The counter-ions
charge is separated from the surface charge by a layer
o f thickness, ~ in which there is no charge (van Olphen
1963). A molecular condenser is formed by the surface
charge and the charge in the plane of the centers of
the closest-ions (Figure 1). B e y o n d the molecular condenser, the remainder o f the counter-ion charge is distributed as in a diffuse G o u y atmosphere. A low dielectric constant is assigned to the m e d i u m within the
Stern layer (van Olphen 1963). M a n y investigators
C o p y r i g h t 9 1996, The Clay Minerals Society
479
have discussed the dielectric constant o f the m e d i u m
next to the charged plate (Conway et al. 1951; C o w n i e
and Palmer 1952; Palmer 1952; Palmer et al. 1952).
Sridharan (1968) discussed the variation of a dielectric
constant with distance, closer to the clay platelet as
brought out by the above investigators and concluded
that there is a general agreement between different approaches and could be taken as 3 to 6.
Single Platelet (van Olphen 1963)
Clay particles carry net negative charges on their
surfaces (Grim 1962; L a m b e and W h i t m a n 1969;
Mitchell 1976). T h e s e negative charges are primarily
due to isomorphous substitution. These charges depend upon the type of the clay and the environmental
conditions in which they are formed. They can be
treated as constant charged platelets.
Or =
(It1 ~- O"2
[1]
where ~r = Total surface charge = B E C / S , ~rl =
Charge in the Stern layer, tr z = Charge in the G o u y
layer, where B E C = Base exchange capacity o f soil
and S = Specific surface area o f soil.
A c c o r d i n g to the Stern theory, ~rl and ~z are given as:
o', =
tr2
Nlve
1 + (NA/Mn)exp(-(vedP 8 + qr)/kT)
= (enkT~sinh/Ve~ ~
\ 2~r /
\2kT/
[2]
[3]
where N1 = No. of adsorption spots on 1 cm 2 area o f
the surface, N A = A v o g a d r o Number, M = M o l e c u l a r
weight of the solvent, n = Concentration o f ion electrolyte, v = Valence of cation, e = Unit electrostatic
charge, qba = Potential on the border o f Stern and G o u y
layers (Stern Potential), ~ = Specific adsorption po-
480
Sridharan and Satyamurty
-,fs
Clays and Clay Minerals
Table 1. Values of valence, hydrated radius and N1 for various ions.
~o---"
Stern layer
( molecular condenser)
Platelet
surface
( negatively
charged )
Ion
Valence
Hydrate~ radius
(A)
K
Na
Li
Ca
1
1
1
2
3.8
7.9
10.3
9.6
N1
(ions/cm 2)
1.731
4.006
2.357
2.713
•
•
•
•
10 la
1013
1013
l013
potential) at a distance, 8. B e y o n d the Stern layer, the
potential distribution, which follows the G o u y distribution, considering qb~ as the starting potential of the
G o u y layer, is given by:
%-I'
e z~/2 + 1 + (e z~/2 - 1)e
layer
ey/2 =
Distance from
Figure 1.
Stern Layer
Schematic figure of Stern and Gouy layers.
e z~/2 + 1 - (e z~/2 - 1)e-~
[6]
where y = ve(qb/kT) (Non-dimensional potential at a
distance x) % = (veqbJkT) and ~ = Kx, in which K
= ~/87rne2v2/ 9
Thus, for any given soil and pore
fluid, qb~ can be determined from Equation [5], tbo from
Equation [4], and potential, (b at any distance, x beyond the Stern layer using Equation [6].
Parametric Study
tential of the counter-ions at the surface, k = Boltzmann constant, T = Absolute temperature and 9 =
Dielectric constant of the pore medium.
The surface charge is related to the Stern potential and
surface potential as:
Er
tr = ~
(dO0 - ~ )
[4]
where 9 =- Dielectric constant of the m e d i u m within
the field of the molecular condensor (that is, within
the Stern layer), 8 = Thickness of the Stern layer, ~b0
= Potential at the surface of the clay platelet and +8
= Stern potential.
Substituting the Equations [2] and [3] into [1], one can
get the relation b e t w e e n a and qb~ as
O" z
NlVe
1 + ( N A / M n ) e x p ( - ( v e ~ ~ + 'tr)/kT)
(9
+ k 2~r /
/
k2kT]
[5]
By solving Equation [5], one can get the value of +~.
By knowing the value of qba, using Equation [4], surface potential, qb0 can be calculated.
POTENTIAL-DISTANCE RELATIONSHIP. Within the literature, the potential-distance relationship for clays using
the Stern theory has been dealt with qualitatively, but
not quantitatively. Therefore within this paper, an attempt was m a d e to give a quantitative picture. For the
Stern layer, the potential drops linearly with distance
f r o m a value o f qb0 at the surface to a value of +~ ( S t e m
The influence of various parameters, such as, the
number of adsorption spots on 1 c m 2 area of the plate,
N 1, the thickness of Sternlayer, 8, specific adsorption
potential, t~, the value o f dielectric constant of pore
fluid within the Stern layer, and 9 on potential-distance relationships have been studied. Also, the distribution of potential according to the Stern theory is
compared to the G o u y - C h a p m a n theory.
The thickness of the Stern layer was taken as the
radius of the hydrated ion. The number of adsorption
spots on 1 c m 2 area of the plate, N 1 depends upon the
hydrated radius of the ion. The value of N1 is determ i n e d as the number o f equivalent squares of size
equal to the diameter of the hydrated ion of those can
be a c c o m m o d a t e d on 1 c m 2 area o f the platelet. Hence,
the thickness of the Stern layer and the number o f
adsorption spots per c m 2 of the plate are interdependent.
Table 1 presents the type o f ion, valence, hydrated
radius (Mitchell 1976) and corresponding value o f N~.
W h e n the type of ion is specified, corresponding values of 8, N~, and v were used.
Table 2 gives the values o f B E C , Base E x c h a n g e
Capacity, and S, Specific Surface Area, for the three
types of clays, which were used in this study (Sridharan and Jayadeva 1982).
Table 3 gives the assumed values of dielectric constant of the pore fluid in the G o u y atmosphere, 9 dielectric constant of the fluid in the Stern layer, 9 and
the molecular weight of the solvent, M, which were
used in this study. The values were taken at a r o o m
temperature of 25 ~
Vol. 44, No. 4, 1996
Potential-Distance relationships
Table 2. Clay properties.
Clay type
BEC
(meq/100 gm)
S
(m2/gm)
Montmorillonite
Illite
Kaolinite
100
40
3
800
Table 3. Values of dielectric constants in Gouy and Stren
layers and molecular weights for various pore fluids
100
15
EFFECT OF N~. Figure 2 hypothetically shows the effect
of N~ on the potential-distance relationship, b e y o n d the
Stern layer, for a fixed value of 8 of 6 ,~. As N~ is
increased, the Stern potential is decreased at any distance. B e y o n d the value of N~ ~ 1016 ions/cm 2, the
potential at any distance by the Stern theory is less
than that obtained by the G o u y - C h a p m a n theory. Figure 3 shows the potential-distance relationship for Na
and Ca montmorillonites, with corresponding values
o f NI, 8 and v (Table 1). For both N a and Ca montmorillonites, it was seen that the potential at any point
that is obtained using the Stern m o d e l is higher than
that is calculated by using the G o u y - C h a p m a n m o d e l
throughout.
EFFECT OF e'. F r o m Equations [4] and [5], it can be
seen that the change in the value of e' does change the
value of qb~, but not qb0. The relation between 4)0 and
e' is given by Equation [6]. Figure 4 gives the varia-
Pore fluid
Dielectric
conslant in
Gouy layer
(e)
Dielectric
constant in
Stern layer
(e')
Molecular
weight of
pore fluid
(M)
Water
Ethanol
Carbon tetrachloride
78.54
24.3
2.28
6
3
2.28
18
46
154
tion in surface potential with e'. It can be o b s e r v e d
that the increase o f the value of e' decreases the surface potential, ~b0 significantly. Since ~b~ was not affected by e', b e y o n d the Stern layer, the potential distribution was the same for a given soil and pore fluid
characteristic and was independent of e'.
EFFECT OF 0. The relation between the specific adsorption potential, t~, thickness o f Stern layer, ~, and surface negative charge, (r, is given by Equation [5]. The
change within the value of t~ causes change within the
Stern potential. Figure 5 gives the potential-distance
relationship for Li and K montmorillonites for ~ = 0
and ~ = 0.1 eV. Corresponding values o f NI and 8
given in Table 1 were used. It is observed that the
increase of t~ reduces the Stern potential significantly
that is for K, especially w h e n N1 is large.
300
Montmorillonite
Water
v:l
,,,, ,,
. ,.is.
.
,,.,,1'41=1 u i o n s / c m
'/, I 015
> 2 0 0 "kk"\:'~
E
~'~'~
2
n:0.0001M
5_6A 0
I/#~-~0
_
. . . ."'%. . .........
. ~ i i l r n
0
481
Gouy-Chapman
Stern tl"
Theory
Theory
e ~
E
o
a..
100
10"
-...........
~
~
o
o
_
06 .......21611 ii , i ~I 6 ..... i1616~ ~ i....816 .... ~ li I106
I I i a i I i i12-.~6..... i14-6
Distance
Figure 2.
(A ~
Effect of N1 on potential-distance relationship.
482
Clays and Clay Minerals
Sridharan and Satyamurty
250 -
',,
',
200
Monfmorillonife
Water
n=O.O001M
",,
",,
!/,=0
Gouy-Chapman Theory
>
.__.. ...... Stern Theory
E 150
-6
loo
*5
r~
50
q 0'"'"'~'6'"'"'B'6"'"'"~'6""'"9'6'"'" i'i ~""' i'~b""
Distance (A ~
Figure 3.
>
v
E
Potential vs distance curves for Na and Ca montmorillonites.
4
Monfmorillonite
~,~..
3
..~Ca
"'-,~....
Water
n=O.O001 M
Stern Theory
o
m
0
O 0
.+..
oCl00091
"'--.
"
0
n
o
0
3
5
4
33
I
4
I
5
I
6
I
7
t
8
I
9
I
......
I
2
10
I
Figure 4.
Effect of e' on surface potential.
I
3
I,
I,~-
J
4
5
6
7
8
Vol. 44, No. 4, 1996
Potential-Distance relationships
483
Montmorillonite
Water
n=O.OOO1M
270
v:l
. %%
Gouy-Chapman
........ Stern Theory
"
",,..
>
E 22O
D, ~
,,,
.,~,,, \ .
0
Jl
Theory
=0
L,, t//=0.1eV
= 170
',-I,,,,-
0
13._
120
K,~
7q0
III
II
II
II
I III
Iii
50
Figure 5.
II
III
II
50
III
~
-~
E
11
I II
II
II
IIIII
90
(A ~
II
I tl
II
IIIII
110
II
II
II/''-
130
150
Effect of specific adsorption potential on potential-distance relationship.
_,'i
v
II
70
Distance
300
>
IIII
I=lllite
3 5 0 --,~
: ~
K=Kaolinife
- ,',/I
M=Montmorillonite
- .. ,..,/K
: -.2".
M
:
250 -
"-'-"~-_
- c --.\.
~,',, ~
2OO
12I
o~
_i., ""~'~
~~- ~~~. ~~ , , , . , .
,4,,,-
r'-
I 5q 0
15
2C
ID
0
IO0
L
I
Water
n=O.O001M
1#=0
Stern Theory (Lithium ion) .
......... Stern Theory (.Potassium ion)
K+
Gouy-Chapmen
Theory (for Li + &
ions)
cl6 .......
s'6 ......
's'6'"""Y6
.......
Distance
Figure 6.
.....
(A ~
Potential-distance curves for three types of clays.
484
Sridharan and Satyamurty
EFFECT OF CLAY TYPE. F i g u r e 6 s h o w s the potentialdistance r e l a t i o n s h i p for the three types o f clays. F o r
a g i v e n p o r e fluid, t h e d i f f e r e n c e s in potentials at any
d i s t a n c e for the t h r e e t y p e s o f clays is negligible. But,
the type o f e x c h a n g e a b l e ion h a s a significant effect
on the p o t e n t i a l - d i s t a n c e relationship for a g i v e n clay
a n d pore fluid, a l t h o u g h the v a l e n c e is the same.
CONCLUSIONS
A d e t a i l e d q u a n t i t a t i v e study o f the Stern theory,
w h i c h is m o r e realistic for use w i t h i n the c l a y - w a t e r
s y s t e m since it takes the size o f the ion into consideration, has b e e n carried out. A q u a n t i t a t i v e study h a s
b e e n c o m p l e t e d u s i n g the S t e m theory. T h e S t e m theo r y is c o m p a r e d w i t h the G o u y - C h a p m a n theory. T h i s
detailed p a r a m e t r i c study d e t e r m i n e d the influence o f
the various p a r a m e t e r s affecting the surface a n d S t e m
potentials. T h e potentials o b t a i n e d at any d i s t a n c e
f r o m the clay surface u s i n g the Stern t h e o r y are h i g h e r
t h a n t h o s e o b t a i n e d b y t h e G o u y - C h a p m a n theory. T h e
size o f ion is f o u n d to h a v e a significant influence o n
the p o t e n t i a l - d i s t a n c e r e l a t i o n s h i p for a g i v e n clay,
pore fluid characteristics a n d valence.
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(Received 26 January 1995; accepted 27 September 1995;
Ms. 2615)