We preferred the shooting method which is realized in the next steps: 1. Using the comparation theorems (10), an interval of the initial values ~b0' (or. ~b,,) was ...
On the Electrical Double Layer Theory I. A Numerical Method for Solving a Generalized Poisson-Boltzmann Equation YIGAL GUR,* ISRAELA RAVINA,t AND A L E X A N D E R J. BABCHIN$ *Soils and Fertilizers Laboratory, Faculty of Agricultural Engineering, Technion, Haifa, Israel; tFaculty of Agricultural Engineering, Technion, Haifa, Israel; SDepartment of Chemistry, Technion, Haifa, Israel Received January 18, 1977; accepted October 3, 1977 Based on asymptotic behavior of the Poisson-Boltzmann equation, a numerical procedure for its solving is suggested. The method may be used both for single and for overlapped double layers and does not depend on the explicit form of the equation. The method is tested by comparing the numerical results with the analytical solution of the equation that was obtained for a 1-1 electrolyte, using the Gouy-Chapman theory. The accuracy of the method is found to be up to the 4th significant digit in a wide interval of the surface potentials. INTRODUCTION
PBE can be obtained only for binary symmetric electrolytes, in two cases: single interface (infinite problem or one plate case); overlapped interfaces (finite problem or two plates case) with small surface potentials - 2 5 mV at room temperature. The solution of the PBE can b e reduced to algebraic equation with elliptic functions in the two plates case and high surface potentials. This equation was obtained and tabulated by Verwey and Overbeek (4). Unfortunately, comparisons of the theoretical predictions and experimental results revealed some inconsistencies that led to attempts to revise the Gouy-Chapman theory by modifying the PBE. Without going into the details of these modifications we may note that they introduce intractable analytic problems as solutions of highly nonlinear differential equation. Therefore, numerical analysis methods and computers were applied to obtain the solutions of the PBE. One of the first attempts was made b y Bolt (5). In his study, the computer was used, actually, as an electronic calculator followed by graphical integration
The conception of a diffuse double layer introduced by Gouy (1) and Chapman (2) describes the equilibrium state at a solidliquid interface. This enables us to compute ions' concentrations and adsorptions and to predict interparticle forces in the Derjaguin-Landau (3), V e r w e y - O b e r b e e k (4) theory of the colloid stability (DLVO theory). The main idea of the G o u y Chapman theory is that the interface ions produce an electrical field and this field acts on the distribution of the ions. To obtain a mathematic description of such self-consistent problem the Poisson-Boltzmann equation (PBE) was suggested. The explicit expression of the PBE in the G o u y Chapman theory was obtained with the next simplifying assumptions: a. Ions are considered as point charges. b. The dielectric permeability of the liquid at the interface is constant. c, The energy of the ion is only the electrostatic energy of the self-consistent field-ion interaction. d. The interface is planar. Nevertheless, the analytic solution of the 326 0021-9797/78/0642-0326502.00/0 Copyright © 1978 by Academic Press, Inc. All rights of reproduction in any form reserved.
Journal of Colloid and Interface Science, Vol. 64, No. 2, April 1978
327
NUMERICAL METHOD FOR PBE
of a nonlinear PBE. Levine and Bell (6) have already applied the Runge-Kutta method for integration of a modified PBE in the case of a 1-1 electrolyte. Bresler (7) suggested a numerical procedure based also on the Runge-Kutta method for the case of a mixture of electrolytes in a system of overlapping double layers. The intensive numerical treatment of a modified PBE was submitted by Levine (8). The algorithms reported in (6-8) have some weak points and can not be applied for solution of the generalized PBE. Thus, the method suggested in (6) is valid only for sufficiently low surface potentials. Any attempts to use it in the wider boundary conditions region lead to its numeric instability. The methods (7, 8) assume a linear relationship between surface charge and surface electric field in interface. This relation is used to reduce the boundary problem to an initial problem for given PBE. The validity of this relationship is restricted with constant permittivity in an interface. Thus, it is impossible to apply these algorithms when corrections of the PBE include the general case of variable dielectric properties of interfaces. In this work we suggest a numerical method with high stability and accuracy for the solution of a general PBE in case of planar double layers as for single interface as for overlapped interfaces. The suggested algorithm is not dependent on any explicit form of the PBE, as it is based on the asymptotic behavior of the PBE. STATEMENT OF THE PROBLEM
The general PBE is given by:
nf
bulk concentration of the ith species of ions N total number of ion species in a solution z~ signed valency of the ith ion U~ energy of the ith ion k Boltzmann constant T absolute temperature. Introducing now the coordinate system for the planar interface where the x-axis is directed into the liquid phase, Eq. [1] attains its common implicit form: d24) dx 2
4zre
•+E--
N
x ~ ni°z~ e x p [ - UJkT], i=l
where • q5 e
dielectric permeability averaged electrical potential elementary electric charge
[1]
[2]
where E = -d4~/dx is the average self-consistent electrical field. In the classical Gouy-Chapman theory the following assumptions are made: e = •o = const,
[3]
U~ = ez~(o~.
[4]
and The usage of this expression enables us to obtain the explicit PBE for Gouy-Chapman theory: d24~ -dx 2
47re X ~ ni°zi exp[-ez~c~/kT]. Eo
[5]
~=1
A revision of the Gouy-Chapman theory needs more general assumptions than Eqs. [3] and [4], that leads to complicated analysis of the nonlinear second-order equation in general form: dx 2
N
dE
x ~ n?z~ e x p [ - U j k T ] , i=1
d~4~
div (• grad qS) = -4~re
dE
dq~ -- ~ X , 6 , ~ X ) •
[6]
In Eq. [6] the form of the f-function depends on the explicit expressions used for the dielectric permeability and the energy state of the ions. An unique solution of Eq. [2] or [6] can be obtained when appropriate boundary Journal o f Colloid and Interface Science, Vot. 64, No. 2, April 1978
328
GUR, RAVINA, AND BABCHIN
conditions are added. For a single interface, these conditions are usually formulated in the form of a constant surface potential 60 6(0) = ~b0; ~b(o~) = 0,
[7]
or in the form of a constant surface charge o-0 -
f0op ( x ) d x
= ~o,
=
60;
-ff2x
~=~o
or
-
p ( x ) d x = o'0.
f(x,~bAb') = f(xo,,h(xo),
6'(xo))
[8]
where p ( x ) = charge density distribution in the double layer. The equivalent boundary conditions for overlapping double layers are formulated as the next:
4,(0)
coincides identically with a solution of an asymptotic form of the PBE. To obtain this form of the PBE, we will expand the right side of Eq. [6] around some point x0:
[10]
/xof × [~b'(x) - ~b'(xo)] + . . . .
In this equation d~' = dch/dx. If Xo is far enough from the interface, the functionf(x,~b,(b') is proportional to the electric charge density. Thus, the next relations are valid: lim f ( X o , ¢ ( X o ) , ~b'(x0)) = O,
X0---->cc
In these equations x m is the half distance between the two plates. As is known, analytical methods for solving the general nonlinear boundary problem, Eqs. [6]-[8], or Eqs. [6]-[10], do not exist. Applications of n u m e r i c a l methods is limited because of the next two reasons: first, the presence of the infinite point in the boundary conditions Eqs. [7] and [8] for the one plate case; second, wide range of changes and rapid variation of the unknown potential function 6(x). The second reason is true in both: in single and overlapped double layers cases. Often, this leads to the failure of the numerical procedure because of the accumulation of random and tranctation errors even for small integration steps. The failure often occurs for high surface potential ~bo values and small overlapping (big Xm). TO circumvent these difficulties, we will consider in more detail the asymptotic behavior of the implicit PBE in its general form Eq. [2] or [6]. A S Y M P T O T I C SOLUTION OF THE IMPLICIT PBE
We assume that the asymptotic solution of the generalized implicit PBE, Eq. [6], Journal of Colloid and Interface Science, Vol. 64, No. 2, April 1978
[11]
[12]
The potential and the field intensity which are proportional to ~b(x) and qf'(x), respectively, must decrease with increasing of x. This leads to: lim 6(x0) = 0,
[14]
,2'O---->~
lira ~b'(x0) = 0.
[15]
Taking now in Eq. [11] the limit x0 --~ ~ and using Eqs. [12]-[15], we get for the asymptotic behavior of the function f ( x , ¢ , ¢ ' ) : f(x,~b,4)') = a 6 ( x ) + b~b'(x),
[16]
where we note a = lim O f ~--'~ 0 6 '
[17]
of b = lim ~ x-.~ 0¢'
[18]
Note that the limits [17] and [18] exist a priori in accordance with the Lipschitz conditions, which are necessary for a unique
329
NUMERICAL METHOD FOR PBE
solution of the boundary problem [6][8]. Certainly, the magnitude of the limits depend on the explicit form of the right side of the PBE and, in particular, on the dielectric state of the interface liquid and energy state of the ions in the interface, thus, being different for different double layer models. We obtain the asymptotic form of the PBE by substituting Eqs. [16] and [6]:
d26 - -
dx 2
the PBE boundary problem [6]-[8] to a finite one. The asymptotic behavior of the PBE Eq. [20] implies the following transformation of the independent variable x: X = exp[-Xx]. The derivatives spectively: d$
b
dx
dx -
a4~
=
0.
[19]
Because of Eq. [14], the general solution of Eq. [19] can be written as 6(x) = C exp[-Xx],
a = K z --
4~-e2 u
~ zi2ni °, e o k T ~=1
b = 0,
re-
XX,
[25]
dX
[26] "
Instead of [6]-[8] we can write:
dZ6 -~(X,+, d4~) dX 2
--d--2
[27] '
6(0) = 0, 4)(1) = 4~0,
[28]
or
dX -
o(x)
X
=
~°"
[291
The new function • is accordingly related to the f-function of Eq. [6] as follows:
h
1 d6
[30]
X dX"
[21] [22]
so that the eigenvalue of Eq. [19] in this case is X = K and corresponding asymptotic solution of the PBE in the Gouy-Chapman theory is: 4)(x) = C exp[-Kx].
--
d24~ - X z x z d2go + dx 2 dX 2 X dX
[20]
where h is the positive eigenvalue of Eq. [19] and C is an arbitrary constant. According to our assumption listed above, we identify expression [20] with the asymptotic solution of the generalized implicit PBE. We emphasize that the exponential asymptotic behavior of the potential function 6(x) is a consequence of Eqs. [12][15], otherwise, it results from the weakening of the boundary influence with increasing distance from the boundary. For illustration, we consider the PBE for the G o u y - C h a p m a n double layer theory, Eq. [5]. Differentiating of the right side of Eq. [5] according to Eqs. [17] and [18], gives the values of the coefficients a and b:
change,
d$ -
do5 -
of 6(x)
[24]
[23]
THE NUMERICAL ALGORITHM
Using the previous results we will attempt to reduce the semi-infinite interval [0,w) of
An important advantage of this transformation is that upon integration of the Eq. [26] on the interval [0,1] by any one of the numerical methods, there occurs an automatic variation of the step size. Actually, the equidistant integration of the Eqs. [26]-[28] corresponds in x-system to the steps, changing from large steps in the region of slow variations of the qS(x) (in the vicinity of x ~ 2), to small steps when the function 4~(x) varies rapidly (in the vicinity of x = 0). The increments 2~X can always be chosen so that the ratio of the derivatives 6 x ' ( 1 ) / 6 x ' ( O ) corresponds to Journal of Colloid and Interface Science, Vol. 64, No. 2, April 1978
330
GUR, R A V I N A , A N D B A B C H I N
the ratio of the step sizes (Ax)J(Ax)o, thus creating the necessary stability and accuracy of the integration procedure in the X-system. The advantage discussed can be usefully realized for the overlapping double layers also, particularly in case of high surface potentials and small overlapping (Xm > 1). The boundary conditions [9] and [10] are simply transformed by means of Eqs. [24] and [25] to a new set: (d6) -'~
-
=0;
4)(1)=60,
[31]
X=Xm
fx
p(X)
dX X
=
ho'o.
[32]
m
In these equations we noted
Xm = exp[-hxm].
[33]
Upon choosing a definite method of integration, preference is given to those methods which reduce the boundary problem to an equivalent initial problem. These methods are more accurate and better developed (10). For the initial problem of Eq. [27], the system of boundary conditions is: ~b(0) = 0;
OSx'(0) = ~b0',
[34]
for infinite case and
~(Xm) = f~m; ~)x'(Xm) =-0,
[35]
for overlapping double layers. We preferred the shooting method which is realized in the next steps: 1. Using the comparation theorems (10), an interval of the initial values ~b0' (or ~b,,) was determined in which the solution of Eq. [27] exists and is unique. 2. An integration procedure was begun as initial problem Eqs. [27]-[34] (or Eqs. [27]-[35] for overlapping case) with a random initial value ~b0' (or ~bm) from the interval of existence that has been determined previously. 3. The last value qS(1) was compared with the given qS0, Eqs. [28] and [30] or if the constant surface charge o-0 was given, the Journal of Colloid and Interface Science, Vol. 64, No. 2, April 1978
left side integrals in Eqs. [29] and [32] were computed and compared with the value ~O" 0.
4. If the difference exceeded a desirable value, a correction procedure was called to find a new initial value and the integration procedure was repeated. The shooting from X = 0 to X = 1 corresponds in the x-system to a " b a c k " shooting going from infinity towards the interface. Using the back shooting procedure in the X-system enables solution of the more general problem, namely to find the whole class of the PBE solutions for some range of surface potentials because every shoot gives us a real physical solution. It results from step 1 of the procedure. Thus, one can determine parameters of the double layer as functions of the surface potential without usage of step 4, that leads to the more efficient usage of computing time. In general, the expenditure of the computing job in case of given ~b0or o'0 depends on the desirable accuracy, type of the integration procedure and the correction procedure. We used the Hamming modified predictor-corrector method (11, 12) for numerical integration of Eq. [27]. The correction procedure was based on a continued fraction interpolation (12). We used Eq. [5] and binary symmetric electrolyte to test the back shooting method. The numerical solutions obtained by this method TABLE I Maximal Absolute Difference between Explicit Eq. [36] and Numerical Solution for Single Double Layers
~T4'o 10 12 14 16 18 20
Abs. error
5.34 3.46 1.05 2.68 7.58 2.12
× × x × × ×
10-4 10 -6 10 -5 10-5 10 -5 10-4
331
N U M E R I C A L METHOD FOR PBE
and
T A B L E II Maximal Absolute Difference between Explicit Eq. [39] and Numerical Solution for Different Overlapping ~ Kx,n
Abs. error x 10 -4
0.1 0.5 1.0 1.5 2.0 2.5 3.0 3.5 5.0 20.0 25.0
5.72 5.70 5.51 5.39 5.32 5.27 5.22 5.30 5.26 5.28 5.43
F(K,q2") =
(1 - K ~ sin 2 q01/~ '
is the elliptic integral of the first kind. The usage of the Jacobian elliptic function Sn(F,K) where F and K are the same as in Eq. [38], permits us to find: ez
kT 49(x) = U - 2 In Sn(Kx/2 e x p [ - U / 2 ] + f ( e x p [ - U], arc sin e x p [ - ( Z - U)/2]), e x p [ - U]).
Surface potential (ez/kT)d~o = 10.
were compared with appropriate analytical solutions of the nonmodified PBE Eq. [5]. Tables I - I V present the results of the tests. Table I presents the maximal absolute difference between numerical and explicit solution of the infinite double layer problem as a function of the surface potentials qS0. The next expression was used for computation of the explicit solution:
ez
(1+ A exp[-Kx]]
k-~ ~b(x) = 2 In
[38]
[39]
The back shooting method for overlapped double layers was tested in the wide range of the surface potentials: - 2 0 --
ez
~b0 - 20,
kT
and for different overlapping: 0.5 - KXm
