1. technology and computing

Discreteness-of-Charge Adsorption Micropotentials
III. Dielectric-Conductive Imaging
J. Ross Macdonald and C. A. Barlow, Jr.
Texas Instruments Incorporated, DaZlas, Texas
ABSTRACT
A c o n v e n i e n t general method for calculating potentials and fields arising
from p l a n a r arrays of discrete adions u n d e r a variety of i m a g i n g conditions
is described and illustrated. Adions are perfectly imaged by one conducting
plane (single imaging) and are also imaged by a dielectric constant discont i n u i t y at a plane on their other side. The method employs only solutions
of the single imaging problem, is readily applied w i t h o u t a computer, a n d
is p e r t i n e n t to the usual electrolyte compact layer adjoining either a n electrode or a dielectric material, which m a y be air. The single image solutions
used in calculating more complex i m a g i n g results may be exact values obtained from a previous rather complicated approach, or for ease in calculation, m a y f r e q u e n t l y be approximate b u t quite accurate values calculated
by a simple method described herein. Using the exact approach, one can
calculate, for the full range of the dielectric reflection parameter, fields and
potentials along a n y line p e r p e n d i c u l a r to the conducting plane. Here we
are p r i m a r i l y concerned with potentials and fields along the line through a
removed adion, a n d the approximate single imaging solution is especially
useful. Although we apply the method to r e g u l a r hexagonal arrays, in the
.
.
.
.
" 1 ly
latter case it
is equally
apphcable
to
arrays described by. G r a h a m e ' s partla
smeared, cut-off model for single imaging. Some comparison with the results
of this model is presented. In addition to calculating and illustrating the v a r i ation of field and potential w i t h i n the compact layer and the adjoining dielectric medium, we have e x a m i n e d in detail the difference between the
micropotential and the macropotential for m a n y different imaging situations.
The p r e s e n t study includes the previously treated single conductive plane
imaging and also the (infinite) conductive-conductive i m a g i n g situations as
special cases. It is found that special care is needed to describe the latter
situation by the present model. Finally, the effect of possible conductive
imaging by the electrolyte diffuse layer is considered qualitatively.
The situation is illustrated in Fig. 1. The circle centers denote the positions of adion charge centroids.
The diagram is not to scale since /9 is r o u g h l y equal
to an ionic radius, a n d the m i n i m u m distance b e t w e e n
spherical adions i n the plane is thus approximately
The system we shall consider in this paper is the
electrolyte compact double layer (1). We shall assume it consists of a m o n o l a y e r of ions (effective
charge zve and average surface charge density qz)
b o u n d e d on one side by a p l a n e interface which we
shall call the electrode-surface p l a n e (ESP), g e n e r ally (but not always) associated with an adsorbing
conductor, and on the other side by an i m a g i n a r y plane
m a r k i n g the points of closest approach of the charge
centroids of ions in the electrolyte, or diffuse layer.
The p l a n e of closest approach is k n o w n as the outer
Helmholtz plane (OHP), and the plane passing through
the charge centroids of the adions i n the monolayer
is the i n n e r Helmholtz plane (IHP). We shall define
to be the distance b e t w e e n the OHP and the I H P
and ~ to be the I H P - E S P separation; d --- /9 + "r is
therefore the total thickness of the compact layer.
2;~. T h e present treatment of course applies for adsorption on a curved electrode surface such as a
m e r c u r y drop provided, as is the case in practice,
that the radius of curvature of the surface is m u c h
greater than the characteristic microscopic distances
involved in the situation such as d and the effective
D e b y e length in the diffuse layer.
T h e possibility immediately arises that other material besides the ions, such as adsorbed water molecules, m a y reside in the compact layer; to represent
this possibility approximately as well as to take into
account to the same degree of a p p r o x i m a t i o n the finite
polarizability of the adions themselves, we regard
these ions as point charges l y i n g on the I H P and take
the dielectric constant of the compact layer i n which
they reside to be el. We have discussed the i n t r o d u c tion of such an ~1 in some detail elsewhere (1-4).
The present paper is concerned p r i m a r i l y with the
d e t e r m i n a t i o n of the electrical potential w i t h i n the
compact layer; this is because the potential's behavior is centrally related to v i r t u a l l y every m e a s u r a b l e electrochemical property of the system (5).
As it t u r n s out, the potential w i t h i n the system is
sensitively d e p e n d e n t on certain s t r u c t u r a l aspects of
the system; not only is it necessary to take the discreteness of the elements into account (6), b u t correspondingly one m a y only do so if reasonable models
a r e used for the spatial distribution of the discrete
adions w i t h i n their p l a n e (7). W h a t is "reasonable"
depends, it seems, on the temperature, the surface
density of the monolayer, the quantities zv, P, and %
and other, more subtle, characteristics of the system.
I n the present work we shall usually assume that
the ions form a perfect hexagonal a r r a y with lattice
REGION I
-((I)
ESP IHP
9 ..
9 -.
9 ..
0
0
0
0
0
0
O
0
0
REGION "iT ((~)
0
0
0
0
0
0
"--t- -v~-VA CANCY
0
0
0
9 ..
0
0
0
0
0
"
O
0
0
Z-AXIS
ooJoo.
9 .o
I
0
0
0
ORIGIN~
9 ..oo
'1'
OHP
0
0
0
oo...
0
0
0
0
..-
0
0
0
0
.o.
T I
1
IMAGES
IMAGES
IMAGES
IMAGES
(-~Zve)
(-Zve)
(-~Zve)
(-~2Zve)
IMAGES
(~Zve)
CHARGES
(Zve)
...o.
"'"
IMAGES
(,,vzv e)
IMAGES
(m2Zv e)
Fig. 1. Cross-sectional diagram of the double layer region showing charges, and some of their images applicable for calculating
potential in region I, and the distances #, % and z.
978
Vol. 113, No. 10
DISCRETENESS-OF-CHARGE
constant rl, although our general analysis is applicable
even without this specific restriction. This is a popular model (3, 4, 7a, 8); and even though it is by no
means applicable u n d e r all conditions (7), it seems
to be a model which is approximately valid for m a n y
conditions of interest. As might be suspected, the
hexagonal model begins to become inapplicable at
low surface densities a n d / o r high temperatures; f u r ther discussion of the applicability of this and other
models appears elsewhere (7).
There is one other m a j o r consideration involved
before one can d e t e r m i n e the potential in the compact
layer: What is the effect of those charges outside the
compact layer on the potential? If one of the b o u n d i n g
planes is the surface of a conductor, the effect of
those charges on and beyond this plane is straightforward; the conductor acts as an electrical imaging
plane, a n d charges on the .conductive surface in excess over those (proportional to ql) involved in the
imaging process set up an additional u n i f o r m electric
field in the compact layer. If one or both of the
b o u n d i n g surfaces is that of a n e u t r a l dielectric i n sulator, t h e n again a type of imaging is involved. A
more difficult question, one which is responsible for
m u c h of the variety i n the various theories of double
layer structure and whose solution has significant i m pact on the expected properties of such systems, is
the effect of the diffuse layer on the potential in the
compact layer. One possible approximation is simply
to neglect a n y variation in charge density or polarization w i t h i n the diffuse layer along directions parallel
to the OHP; that is, to assume that these quantities
vary only i n the direction perpendicular to the OHP.
T h a t this is not strictly true is a result of the discreteness-of-charge in the compact layer; that this
assumption might still be a good approximation rests
on the t h e r m a l motion of the ions i n the diffuse layer,
motion which tends to disrupt a n y "shadowing"
within this layer of the charge density variations on
the IHP through pairing of adions with counterions
in the diffuse layer. If one makes this approximation,
the explicit effect of the diffuse layer goes to nought,
and the potential w i t h i n the compact layer is determ i n e d completely b y the m o n o l a y e r ions and the
b o u n d a r y conditions at the other interface. Several
more or less correct, exact or approximate, t r e a t m e n t s
of this model have appeared i n p r i n t (4, 8c-f, 9, 10).
The above situation, with a metal electrode, has been
termed by the present authors the "single-imaging"
case i n a t r e a t m e n t (4) hereafter r e f e r r e d to as II.
At the extreme opposite end of things, another
model of the diffuse layer's effect has been proposed
and treated (3,7b, 8b-f, 11). This might be termed
the "Ershler model" after its first proponent; the
present authors ha,ve dealt with this model in some
detail and have termed it the "infinite-imaging" case
i n a t r e a t m e n t (3) hereafter referred to as I. As the
latter terminology implies to some of us, this model
assumes that mobile diffuse-layer ions are capable of
a r r a n g i n g themselves so as to m a k e the OHP a n equipotential surface. Effectively then, the OHP becomes
a second c o n d u c t i v e - i m a g i n g plane (exactly as though
it were a m e t a l surface) and the outcome of this is
a n infinite set of images, as in a hall of mirrors.
U n f o r t u n a t e l y it has not been possible to establish
unequivocally, either e x p e r i m e n t a l l y or theoretically,
which of these two models more n e a r l y represents
the actual behavior of the diffuse-layer ions. A n adequate t r e a t m e n t of the diffuse layer i n the vicinity of
the OHP, where the influence of the discrete compactlayer ions is greatest and the usual approximations
most hazardous, m a y be as yet an unfinished assignment. The most promising approaches have been those
of Stillinger and K i r k w o o d (12) and some of the
Russian workers (13); however, even these t r e a t m e n t s
invoke expansions i n v o l v i n g the ratio of electrical
energy to t h e r m a l e n e r g y i n the diffuse layer, a n d
for concentrations of interest the validity of such
MICROPOTENTIALS
979
theories is uncertain. The e x p e r i m e n t a l problems are
almost as sticky: The quantities actually measured
are only related to local potentials after an " i n t e r p r e tive process" which is somewhat questionable; for
example, isothermal m e a s u r e m e n t s leading to adsorption coverages (ql) only yield adsorption potentials
vs. coverage if the form of the isotherm is known.
F u r t h e r m o r e , it is possible that the state of ionization
of the m o n o l a y e r elements m a y change over certain
ranges of surface coverages for some systems (2, 14,
15).
R e t u r n i n g to the refuge of our idealized situation,
however, there is one diffuse-layer effect which should
exist i n d e p e n d e n t of the behavior of the diffuse-layer
ions: I n most situations there is a fairly a b r u p t
change in the dielectric constant in going across the
OHP. If we denote the dielectric constant of the b u l k
electrolyte, containing no excess ions, by ~2, the difference e 2 - E1 will induce dielectric imaging at the
OHP. [We should note that the m a j o r dielectric constant change m a y not occur precisely at the OHP, so
that the imaging p l a n e m a y lie somewhat w i t h i n the
compact layer (16). This possibility will be neglected
here.] W h e n this imaging is accompanied by metallic
i m a g i n g at the ESP, resulting in an infinite set of
imperfect images, as though i n a hall of imperfect
mirrors, we have described this dielectric-conductive
situation as the " p a r t i a l - i m a g i n g " case. This case has
the following features of p a r t i c u l a r interest. First,
this type of imaging should be present in most systems, whatever additional effect m a y be present due
to the diffuse-layer ions. The dielectric discontinuity,
or a n approximation thereto, would be expected in
most cases, f o r m i n g a "background effect" for any additional action b y the diffuse-layer ions. I n view of
this effect, it is i n fact difficult to construct a n electrolytic system which displays "single imaging," and
we now regard this case to be of interest only in reference to nonelectrolyte systems or as a l e a s t - i m a g i n g
limiting case for electrolytes.
Since we p l a n to compare m a n y of the theoretical
single, partial, and infinite imaging discreteness-ofcharge treatments m e n t i o n e d above in a forthcoming
review (17), we shall omit much of such cross-comparison from the present paper. It is perhaps worth
m e n t i o n i n g that the present work contains i n a certain sense all the results of the limiting cases I a n d II
and also yields a continuous bridge b e t w e e n them.
Although the p r e s e n t paper does not include a t r e a t m e n t of dielectric-dielectric imaging, a case treated
to some extent by Levine et al. (7b), our present,
easily applied methods require, as we shall show
elsewhere, only simple modifications to apply to this
situation as well.
General Analysis
I n the r e m a i n d e r of this paper we shall be p r i m a r i l y
concerned with t h e potential which results w h e n a
single vacancy is present in an otherwise perfect
hexagonal a r r a y of adsorbed ions. As has been discussed in I and II, this leads to the interesting part
of the micropotential; the effects of omitted selfimages .are i n d e p e n d e n t of q and ql and m a y be s u b sumed into the "chemical" part of the adsorption potential. The potential ~ in our present work consists
of two contributions: a discrete-charge c o n t r i b u t i o n
~a which is always present w h a t e v e r the charge on
the E S P and a u n i f o r m - d i s p l a c e m e n t c o m p o n e n t Ce
which results from an additive constant D field n o r m a l to the E S P and v a n i s h i n g w h e n the ESP is
g r o u n d e d so its charge q becomes equal to - - q l for
dielectric-conductive imaging. We first d e t e r m i n e the
discrete-charge c o n t r i b u t i o n ~a. The basic method is
to replace the ESP and OHP interfaces w i t h a system
of fictitious "image charges;" the particular system
depends on w h e t h e r we seek the potential on the
m e t a l side or the solution side of the OHP. Thus,
r e f e r r i n g to Fig. 1, the potential b e t w e e n the m e t a l
a n d the OHP (region I) will derive from a n infinite
980
JOURNAL
ESP IHP
D
D
9
D
I
9
0
9
Q
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
OF THE
ELECTROCHEMICAL
OHP
~
6
VIRTUAL
IMAGES
(-~Zve)
O c t o b e r 1966
w h e r e 1]el/e2 = 1 - - ~. Thus, w e h a v e expressed t he
p a r t i a l - i m a g e p o t en t i al in t e r m s of a single series
(instead of the double or triple series used by others)
i n v o l v i n g the accurately k n o w n (4) s i n g l e - i m a g e potentials. L e v i n e et at. (7b, 11) h a v e e m p l o y e d a similar approach (for the infinite i m ag i n g case), in which
t h e y used G r a h a m e ' s cut-off m o d el (8c) for the single
i m ag e potential tao(0,0,z). Similarly, defining ~a as
the electric field associated w i t h %a in the pr e s e nt
r e g i m e and gao as the field associated with the singlei m ag i n g potential tao, one obtains in regions I and II,
respectively
Z-AXIS
Q
SOCIETY
VIRTUAL
IMAGES
~a ( X , y , Z ) ~- ~ a o ( X , y , Z )
(-~z v e)
VIRTUAl.
IMAGES
VIRTUAL
CHARGES
(',/(~Zve)
('~ Zve)
+ ~
Fig. 2. Cross-sectional diagram of the double layer region shawing effective charges and some of their virtual images applicable
for calculating potential in region IL
~n{gao(X,Y,Z -t- 2nd) -F ~ao(X,y,2nd-- z)} [3]
~a(X,y,z) = (~lez/e2) ~
=nSao(X,Y,Z + 2rid)
[4]
n~O
set of images (for each charge on the I H P ) as though
the whole space w e r e of dielectric constant ~l; r e f e r r i n g to Fig. 2, t h e p o t e n ti a l on the solution side of the
O H P (region II) will likewise i n v o l v e an infinite set
of images, and the w h o l e space of region II is to be
r e g a r d e d as h a v i n g dielectric constant e2. T h e sets of
images for each charge on the I H P a r e chosen so that
the resulting potentials satisfy the a p p r o p r i a t e conditions on the E S P and OHP. In particular, the r e s u l t ing potential should be zero on the ESP, continuous
across the OHP, and r e s u l t in a continuous v a r i a t i o n
in the n o r m a l component of electric displacement
across the OHP. It m a y be verified by inspection that
the set of images indicated in Fig. 1 and 2 lead to
potentials satisfying these conditions.
We shall set up a coordinate system as follows: The
positive z-axis is ta k e n n o r m a l to the ESP, e x t e n d i n g
o u t w a r d s towards the OHP. The x and y axes lie in
the ESP, and the origin of coordinates is on the E S P
and the n o r m a l line passing t h r o u g h the vacancy.
A l t h o u g h the methods of the p r e s e n t paper m a y be
r e a d i l y e x t e n d e d to give potentials for positions o t h er
than x = y = 0, in the following w e shall only be
concerned with actually calculating the potential (and
field) along this line.
Let us now define ~ - (~2
~1)/(~2 -F ~,), a n d ~1
--- 1 -t- ~ = 2e2/(~1 -F ~2). R e f e r r i n g to Fig. 1, w e see
that the a r r a y of adions and their images giving rise
to Wa in region I f o r m sheets of nonideal dipoles
whose centers lie on the planes z = z , ~ 2rid w h e r e n
is an integer. The separation b e t w e e n t h e positive
and n e g a t i v e charge of each nonideal dipole is 2/~,
and the strength of t h e dipole sheets drops off as ~1-1.
Similarly, r e f e r r i n g to Fig. 2, it m a y be seen t h a t t h e
potential Wa in region II derives f r o m a set of n o n ideal dipoles similar to those p e r t a i n i n g to r e g i o n I.
T h e differences b e t w e e n the sets for the two regions
a re that the p o s i t i v e - n sheets are missing for the r e gion II calculation a n d the dipole strengths ar e all
m u l t i p l i e d by the factor 0.
If w e n o w define @ao(X,y,z) to be the potential p r o duced at (x,y,z) by our a r r a y of adions in t h e singleimage r e g i m e (the q u a n t i t y k n o w n as @a in II) w e
m a y take the foregoing into account and w r i t e
-
~ (x,yz) = r
+ ~
-
(x,y,z)
~" {@ao(X,y,z "t- 2nd) -- Wao(X,y, 2rid-- z) }
[1]
These s i n g l e - s u m m a t i o n expressions for ~ba and ~ a
c o n v e r g e quite r a p i d l y ; e v e n for ~ = 1, t h e e x p r e s sion for potential in region I converges well. As ~ approaches unity, the v al u e of Wa in region II is bounded
n~ 0
w h e r e the upper bound, ~ , is defined as in II; thus
~ =- 4n~ql/E1. In c a r r y i n g out the sums for Wa(0,0,z)
- Wa(z) it was usually found to be m o r e c o n v e n i e n t to
use an a p p r o x i m a t e but highly accurate analytic
expression for ~ao(0,0,z) -- ~ao(Z) than to use the exact
results obtainable f r o m II. The a p p r o x i m a t i o n used and
its accuracy is discussed f u r t h e r in ref. (18) an d in the
Appendix. Calculation of ~ o ( z ) by this a p p r o x i m a t i o n
is far simpler than by the l en g t h y expressions given in
II, y et yields a d e q u a t e accuracy. The accuracy of the
results in the p r e s e n t p a p e r have, in fact, been
checked by calculating m a n y of t h e m w i t h both exact
and a p p r o x i m a t e values for Wao(Z).
To complete the potential, w e must add the u n i f o r m
D field part, ~e. The b o u n d a r y conditions at the E S P
r e q u i r e that, if the total surface charge density on the
E S P is q, then t h e u n i f o r m field produced is simply
4n(q + ql). T a k i n g the zero of We to lie on the ESP,
we obtain We = --(4~/~!) (q -F qDz for region I and
te = --4~(q -F ql) [(d/~l) -F { ( z - d)/~2}] for region
II. We note, incidentally, that if Waic and ~aic (see II)
are substituted for Wao and 8ao in the foregoing e q u a tions, the r esu l t i n g potentials and fields are the ones
p e r t a i n i n g to a complete adion array.
B e f o r e p r e s e n t i n g detailed results of our calculation of potentials, fields, and other quantities of i n terest, w e point out an i n t e r e s t i n g f e a t u r e p e r t a i n i n g
to the limit ~ --> 1. I n t u i t i v e l y one m i g h t e x p e c t ~a (in
region I) to approach the infinite-image ( c o n d u c t i v e conductive) ~a as ~ approaches unity. As it turns out,
however, this preconceived notion is e n t i r e l y wrong:
Th e limiting b e h a v i o r of the p r esen t ~a is distinctly
different from the ~a d e t e r m i n e d in I, w h i c h we shall
here denote as Wacc. Thus one cannot simply take
= 1 herein and directly obtain ~acc. What does occur
for g i v en q and ql is that the limiting v a l u e of the
p r esen t W = Wa "~ ~e is identical to the v a l u e of
r162
We obtained by redefining We (in I) so as to
place its zero on the E S P (as has been done w i t h our
present p a r t i a l - i m a g i n g We). In other words, the actual
limiting b e h a v i o r of the p r esen t Wa is given by lira
~-~1
Wa = tacc ~ (4aqlz~/eld). In particular, at the O H P
w h e r e @ace is zero, lim @a = W| This b e h a v i o r r e p r e -
in region I, and in region II
~)-->1
Sa(X,y,z) = (~lel/e2) ~ ~" ~ao(X,Y,Z + 2rid)
[2]
sents a real physical effect and not just a t r i v i a l definition of the zero of potential. Th e physical o r i g i n of
the effect lies in the f u n d a m e n t a l difference b e t w e e n
V o l . 113, N o . 10
DISCRETENESS-OF-CHARGE
a dielectric, no matter how "strong," and a conductor.
O n l y t h e c o n d u c t o r a l l o w s c h a r g e to b u i l d u p o n its
s u r f a c e so as to r e n d e r t h e field i n s i d e e x a c t l y zero.
Normalized Equations
F o r p r e s e n t a t i o n a n d c o m p a r i s o n p u r p o s e s , it is
c o n v e n i e n t to d e a l w i t h n o r m a l i z e d e q u a t i o n s w h e n ever practical. We shall normalize potentials with the
V| i n t r o d u c e d a b o v e a n d s h a l l d e n o t e s u c h q u a n t i t i e s
as Vao/V| a n d Va/V| a s VaoN a n d VaN, r e s p e c t i v e l y . N o t e
t h a t s i n c e ~l o c c u r s i n V~, a p o t e n t i a l s u c h as V i n
r e g i o n II, w h e r e e ---- E2, is s t i l l n o r m a l i z e d b y a q u a n t i t y i n v o l v i n g ~1, n o t ~2, i n t h i s c o n v e n t i o n . I t w i l l
p r o v e c o n v e n i e n t to n o r m a l i z e fields i n a n e q u i v a l e n t
w a y , u s i n g ~| -- - - 4nql/el. F i n a l l y , w e w i l l n o r m a l i z e
e l e c t r i c d i s p l a c e m e n t s w i t h ~)~ -- ~ 4 ~ q l .
F r o m n o w o n l e t u s t a k e x = y ---- 0 a n d d e f i n e
Z -- z/~ a n d Ri - rl/~. F u r t h e r , l e t ~ / ~ ~- r ,
Zo - 1 + r - d/~, a n d p -- 2Zo. T h e n t h e I H P o c c u r s
a t Z = 1 a n d t h e O H P a t Z ---- Zo. F o r s o m e p u r p o s e s ,
as w e s h a l l see, i t is u s e f u l also to u s e t h e n o r m a l i z e d
v a r i a b l e } -- Z/R1 -- z/rl.
W e m a y n o w o m i t r e f e r e n c e to x a n d y a n d w r i t e i n
p l a c e of [1] a n d [2]
981
MICROPOTENTIALS
O H P a n d t h e b u l k of t h e s o l u t i o n . I t is a c t u a l l y
V1 + V2 w h i c h a p p e a r s i n a n a d s o r p t i o n i s o t h e r m ,
b u t w e s h a l l n e g l e c t V2 h e r e a f t e r s i n c e i t d e p e n d s o n
q a n d ql a n d c a n b e a d d e d i n w h e n e v e r p e r t i n e n t .
R e m a r k , h o w e v e r , t h a t u n l e s s t h e O H P is t a k e n as a n
eq~ipotential surface, which implies conductive imaging there, the usual one dimensional diffuse-layer
t h e o r y is i n c o n s i s t e n t w i t h t h e r e q u i r e m e n t t h a t t h e
potential be continuous across the OHP. On the other
h a n d , e v e n w h e n t h e O H P is n o t a n e q u i p o t e n t i a l , t h e
v a l u e of V2, n o w a f u n c t i o n of x a n d y, w i l l p r o b a b l y
b e q u i t e close to t h a t p r e d i c t e d b y t h e u s u a l o n e - d i mensional theory.
F r o m t h e a b o v e d i s c u s s i o n , w e h a v e Vi -- V(~)
- - V ( d ) . W e w i s h to c o m p a r e t h i s p. d. to t h e a v e r a g e
p. d., VI, a c r o s s t h e i n n e r r e g i o n . T h e a v e r a g e p o t e n t i a l itself, V ( z ) , m a y b e w r i t t e n i n n o r m a l i z e d f o r m
as (3)
VN(Z) =[Z(q/ql) + (Z--1) ]
[15]
f o r Z --~ 1. I t is t a k e n z e r o a t Z = 0. V i s is t h e n
- - V N ( Z o ) . I t w i l l b e c o n v e n i e n t to i n t r o d u c e t h e n o r m a l i z e d q u a n t i t y k -- ~/d = r / ( 1 + r ) = r Z o -1. F o r
V1N w e r e a d i l y o b t a i n
V1N = vaN(I) - - V a N ( g o )
VaN~- VaN(Z;R1) = VaoN(Z;R1)
Jr r{1 + ( q / q t ) }
A s i n II, l e t u s f o r m t h e r a t i o ( a p p l i c a b l e f o r Z o -
eo
-~- ~-~ (on{VaoN (rip ~- Z ; R 1 ) - - VaoN (rip - - Z;nl) )
n=l
( r e g i o n I, O --~ Z ~ Zo)
[5]
A ~- vI/Vl
= --vIN/VN(Zo)
r { 1 + ( q / q l ) } ~- AVaN
1~ -[- ( i -b F)
~nVaoN(np+Z;R1)
[6]
n= 0
( r e g i o n II, Z ~ Zo).
E q u a t i o n s f o l l o w i n g f r o m [3] a n d [4] m a y b e s i m i l a r l y
w r i t t e n ; i n s t e a d , s i n c e i t p r o v e s m o r e c o n v e n i e n t to
p l o t d i s p l a c e m e n t s t h a n fields, w e g i v e e q u a t i o n s f o r
~a s ~ ~a/~,
e q u a l to el~a/el~| i n r e g i o n I a n d to
e2~a/el~ i n r e g i o n II. T o o b t a i n 8 a N i n r e g i o n II, w e
n e e d o n l y m u l t i p l y ~)aN i n t h i s r e g i o n b y (ei/~2) ---(1 - - ~ ) / ( 1 + r
The results for ~a N are
D , ~ = ~,o N ( Z ; R D
1)
~ ~,(1 -~ A)
oo
VaN~-~-~ ( 1 - - c o )
[16]
(q/ql)
[17]
where A is defined in [17] and AVaN - VaN(1)
--VaN(Zo). The quantity A, which measures the deviation from linear proportionality of the micropotential to Vi, follows from [17] and may be written
1 +
=
r + (I
)~--IAVaN
+ r ) (q/ql)
[18]
When q = 0, we define A - Ao, where
Ao = r-ill + ;~,-IAVaN]
[19]
Discussionof Results
Conducting E S P
"~- ~
~on{~aoN(np -~-Z;R1) -~-
~aoN(np--Z;R1)}
[7]
(region I)
D~N = (1 + ,~) ~
~ ' g ~ o S ( n p + Z;R1)
[8]
( r e g i o n 1i)
For use in comparing with experimental results,
i t is f r e q u e n t l y u s e f u l to h a v e a v a i l a b l e c o m p l e t e p o t e n t i a l s a n d fields w h i c h i n c l u d e t h e u n i f o r m D field
contributions. We may write
VN = VaN -~- VeN
[9]
6 N = ~ a N -~- ~e N
[10]
w h e r e eaN = ~)aN a n d { ( 1 - - ~ ) / ( 1
I a n d II, r e s p e c t i v e l y , a n d
Ve N
=
--
{1 -]-
+ ~)}~)a s i n r e g i o n s
(q/ql)}Z "l
[II]
O~_Z ~_Zo
~e N = - -
VeN=--{I+
{1 + ( q / q l ) }
(q/ql)}[Z~
+ ( ( 1 - - , ~ ) / (1 + ~) } ( Z - - Zo)
~eN = - - { I -b (q/q,)}[(l - - ~ ) / ( I + ~)
I}
[12]
Z~Zo
[13]
[14]
There is, of course, no average D field contribution
when q ~ - - ql and the ESP is grounded.
The mieropotential Vi is related to the energy r e quired to move an ion at the OHP to its adsorbed
position at the IHP. This definition does not include
the small p. d. V2 in t h e diffuse layer between the
In this section, we shall consider the usual electrol y t e s i t u a t i o n s h o w n i n Fig. 1 of a c o n d u c t i n g e l e c trode, a compact inner layer, and a diffuse layer ext e n d i n g b e y o n d t h e O H P . W e shall, f o r t h e t i m e b e i n g ,
i g n o r e effects a r i s i n g f r o m m o v e a b l e i o n s i n t h e d i f f u s e l a y e r a n d s h a l l o n l y b e c o n c e r n e d w i t h t h e effect
of t h e r - - ~ d i e l e c t r i c d i s c o n t i n u i t y a t t h e O H P .
S u g g e s t e d v a l u e s of ~1 i n t h e c o m p a c t l a y e r h a v e
r a n g e d f r o m 6 to a b o u t 15 [see r e f ( 1 ) ] . T h e v a l u e of
~2 f o r a n a q u e o u s e l e c t r o l y t e c a n b e n o g r e a t e r t h a n
81 a t r o o m t e m p e r a t u r e a n d w i l l l i k e l y b e a p p r e c i a b l y
r e d u c e d i n t h e d i f f u s e l a y e r b e c a u s e of d i e l e c t r i c s a t u r a t i o n a r i s i n g f r o m t h e a v e r a g e field t h e r e a n d t h e
effects of c l o s e l y n e i g h b o r i n g i o n s (1, 19, 20). A r e a s o n a b l e a p p r o x i m a t e l o w e r v a l u e f o r e2 m i g h t t h e r e f o r e b e 50. T h o s e c o m b i n a t i o n s of t h e a b o v e v a l u e s
of el a n d e2 w h i c h g i v e t h e s m a l l e s t a n d l a r g e s t ~'s
l e a d to ~mln - - 0.54 a n d r
~ 0.86. T h e s m a l l e r is ~,
t h e less is d i e l e c t r i c i m a g i n g a l o n e l i k e l y to a p p r o x i m a t e w e l l to t h e a c t u a l e l e c t r o l y t e s i t u a t i o n i n c l u d i n g
diffuse layer conductive imaging. In the succeeding
figures, h o w e v e r , w e h a v e s h o w n r e s u l t s f o r a v a r i e t y
of v a l u e s of ~ b e t w e e n -El a n d --1, f o r c o m p l e t e n e s s ,
f o r c o m p a r i s o n w i t h t h e r e s u l t s of I a n d II, a n d b e c a u s e s o m e of t h e n e g a t i v e v a l u e s a r e p e r t i n e n t to
the situation discussed in the next section.
F i g u r e s 3 a n d 4 s h o w t h e d e p e n d e n c e of n o r m a l i z e d
potential and displacement on normalized distance
f r o m t h e E S P , ~, f o r a n u m b e r of ~, v a l u e s f o r t h e
c h o i c e R1 ---- 5 a n d s e v e r a l v a l u e s of r . T h e s e v a l u e s
of r p r o b a b l y c o v e r t h e e x p e r i m e n t a l l y l i k e l y p o s s i b i l i t i e s f o r s u c h e l e c t r o l y t e s as KI. W e h a v e u s e d
982
JOURNAL OF THE ELECTROCHEMICAL SOCIETY
l
r
OHPI
I
e I
I
1
I
. . . .
, / ' . - - i - I o ~ ~
r
I
0.8
0.~ ,'4 /-_o,
o
'
I
o4
._0~-_~
0.8~.
{~)o
1.2
~
.
i
~
October 1966
T
T
:',t
~o,.~
,~-~--.-~'--~"
~ ~------'-r~~::.~"/q~
i.6
1(52
r
I
0.4
0
~" 0.8
1.2
(a)
ot
N 1.__
~
nil~~ I /
///~
~176
~
1-._._~
-1
~
~
:
0.2
I
0
0.4
0.8
L2
1.6
(b)
f
i
I
i.o t__
i
[
OH~ "78
08
I
L
IHPi
1620
I
l
'
~
,
r =u
! ~ o . 8 ~ -o.6
~
\i
~
~
(
I
"~
0.8
1.2
;o
,
,o,,
:=:,
.El
0.4
~
(b)
,,,
o
/
-0.6
0.4
4
-I
0.8
1.2
r=2
-1
1.6
Fig. 3. Normalized discrete potential V/N = V/aN vs. normalized
distance from ESP, ~ = Z/R1, for R~ = 5 and r _= ,~/fl equal
to 1~, I, and 2. The parameter is ~ ~ (e2 - - el)/(e2 Jr- el).
1(52
0
0.4
0.8
1.2
(c)
the parameter ~ here instead of Z itself because we
have found elsewhere
(4, 18) t h a t f o r s i n g l e i m a g i n g (~, = 0) V/aN = V/aoN is a f u n c t i o n p r i m a r i l y of
a n d d e p e n d s o n l y s l i g h t l y o n Rz s e p a r a t e l y . T h e p r e s ent curves may thus be compared directly with others
p l o t t e d v s . ~. U n f o r t u n a t e l y , w h e n ~ r 0, s u c h v i r t u a l
i n d e p e n d e n c e of RI is less m a r k e d , as w e s h a l l d e m o n strate elsewhere.
For greatest clarity and distinction between curves,
w e h a v e t a k e n q = - - q ; i n Fig. 3 a n d 4. T h i s c h o i c e
h a s t h e effect of m a k i n g ~ s _~ V/aN a n d Z)N = Z)aN.
Although q seldom equals --ql in electrolyte situat i o n s of t h e k i n d d i s c u s s e d i n t h e p r e s e n t s e c t i o n ,
V/N a n d f)N f o r q r - - q ; m a y b e r e a d i l y d e r i v e d f r o m
t h e c u r v e s g i v e n t h r o u g h t h e u s e of Eq. [9] a n d [10],
[11], a n d [12], a n d t h e k n o w n r e l a t i o n s b e t w e e n ~N
a n d Z)N. W e h a v e e l e c t e d to p l o t Z)N i n Fig. 4 r a t h e r
Fig. 4. Normalized discrete displacement ~ N = ~ a N vs. ~ for
R; = $, F = l/z, 1, 2, and a variety of , values.
t h a n ~N to a v o i d t h e d i s c o n t i n u i t i e s w h i c h w o u l d
o t h e r w i s e a p p e a r a t t h e O H P . I t is i n t e r e s t i n g to n o t e
t h a t d i s c o n t i n u i t i e s s t i l l a p p e a r i n t h e first d e r i v a t i v e
of 9 e x c e p t w h e n • ---- 0. B e c a u s e of t h e i n f i n i t e d i s c o n t i n u i t i e s w h i c h o c c u r i n s o m e q u a n t i t i e s a t ~ ---- 1
a n d --1, o u r p r e s e n t r e s u l t s do n o t s e r v e to y i e l d
~a N f o r Zo ~-- 1 w h e n ~ ---- --1, b u t t h e y do s h o w t h a t
~a N ~ 0 f o r Zo ~ 1 w h e n ~ ---- 1. T h e m i s s i n g ~ a N p o r t i o n f o r ~ = --1 c a n b e r e a d i l y c a l c u l a t e d d i r e c t l y
f r o m Eq. [8], h o w e v e r , b y first m u l t i p l y i n g i t b y
(1 - - ~ ) / ( 1 + ~) a n d t h e n l e t t i n g ~ = ~ 1 .
A l t h o u g h t h e p r e s e n t m e t h o d of c a l c u l a t i n g V/N a n d
~ N f o r a n y p e r t i n e n t v a l u e s of ~, Z, a n d Rz is suffi-
V o l . 113, N o . 10
DISCRETENESS-OF-CHARGE
ciently simple that no c o m p u t e r need be used to
achieve 1% or better accuracy, w e have r e q u i r e d
for this paper so m a n y values of these and o t h er
r e lat ed quantities that w e have, in fact, used a comp u t e r for their calculation. We h a v e m a d e a careful
comparison b e t w e e n the results of t h e a p p r o x i m a t e
formulas for ~aoN and ~ao N given in the A p p e n d i x and
v e r y accurate values of these quantities calculated as
in II. This comparison shows, for example, that R1 ---5 the use of the (3/3) a p p r o x i m a n t for P ( D g i v en as
(a) in Table I of the A p p e n d i x yields values of ~a~
for an y reasonable Z, r, and ~ choices w h ic h g e n e r a l l y
differ f r o m the accurate ones by only a small a m o u n t
in the fifth significant figure. T h e results for ~ a N a r e
thus at least as accurate as those for P ( O itself and
are usually m o r e accurate. This approximant, or an
e v e n simpler one g i v e n in ref. (18), can be used in
calculating results such as those shown in Fig. 3 and
4. A n a p p r o x i m a n t such as the (2/3) one g iv en for
F ( D in the A p p e n d i x is also r e q u i r e d in calculating
~)a N by the modified cut-off method. W h e n an app r o x i m a n t for p ( O d e r i v e d for R1 = 5 is used to calculate values for ~aoN o r ~ a o N w h e n R1 ~ 5, almost as
high accuracy as w i t h Rz = 5 can be e x p e c t e d w h e n
Rz > 5 since, as stated, ~aoN depends v e r y little on
Rz alone but p r i m a r i l y on ~. In particular, R1 = 5
values are v e r y close to R1 ---- oo values of ~ a N for a
gi v en ~ (18). W h e n R~ is < 5, t h e r e is s o m e w h a t m o r e
d e p e n d e n c e on R~, a l t h o u g h the difference is still sm al l
e v e n for R~ ---- 2, the smallest v a l u e of R1 that usually
need be considered (18). For example, using the R~
= 5 (3/3) P (D a p p r o x i m a n t to calculate values of
~ N for Rz ---- 4 leads to deviations b e t w e e n accurate
and a p p r o x i m a t e values of ~a~ w h i c h usually occur
in the f o u r t h significant figure and r a r e l y in the third.
As discussed and listed in ref. (18), even simpler
a p p r o x i m a n t s than those g i v e n h e r e in the A p p e n d i x
m a y o r d i n a r i l y be used to obtain a d e q u a t e accuracy
in calculated fields and potentials.
In o rd er to achieve high accuracy in the present
c o m p u t e r results, we s u m m e d such series as those of
Eq. [5] to [8] to f a i r l y high o r d e r a n d / o r used the
epsilon al g o r i t h m (21) to e x t r a p o l a t e to accurate
final values w h e n necessary. The series are all v e r y
r a p i d l y c o n v e r g e n t w h e n ~ < < 1. T h e y are g e n e r a l l y
most slowly c o n v e r g e n t w h e n ~, = 1 so w e e x a m i n e d
this limit. W h e n o~ ---- 1, the series of Eq. [5] is only
conditionally convergent. E v e n so, it g e n e r a l l y converges quite rapidly. Thus for R~ ---- 5, r ---- 1, and Z
= 1 w e find that ~a~ is w i t h i n 1.7% of its final v a l u e
w h e n only t h r e e (accurate) t e r m s of the series are
used. With five terms the p e r c e n t a g e is 0.71, and if
the epsilon al g o r it h m is used on these five terms, the
percentage drops to 0.28.
Since [5] is only conditionally c o n v e r g e n t w h e n
---- 1, r e a r r a n g e m e n t of its te r m s should change its
value. We h a v e found that this circumstance can be
put to good use in calculating h ~
~baN(1)
~aN (Zo) in the infinite im a g i n g case w h e r e ~ = 1 and
c o n v e r g e n c e is slowest. O r d i n a r i l y this q u a n t i t y w o u l d
h a v e to be calculated by tw o separate evaluations
of ~as using [5]. A f t e r a slight r e a r r a n g e m e n t of t h e
te r m s of [5], we find the surprising result, h o w e v e r ,
that
983
MICROPOTENTIALS
q. There, G r a h a m e ' s (22) values for qz as a function
of q, d e t e r m i n e d f r o m m e a s u r e m e n t s on a 1N K I
electrolyte at 25~
w e r e used in calculating g. The
above result suggested to us that it w o u l d be w o r t h w h i l e to e x a m i n e the field at the IHP, ~ (~), for se ve r a l
r and ~ values. Results are shown in Fig. 5. We h a v e
chosen to look at the field at Z = 1 since its v a l u e at
the site of a r e m o v e d ion is p e r t i n e n t to ionic p o l a r ization, dielectric saturation, and ionic compressibility
--~'
I I I'
"~--
I'
I'
1 I I'
I I Ir = 112
""-.'"-
Zv.-,
_
-1
- - -
_~I
_
I I I I I ~ I i I I I J I J I I 1%
16
12
8
4
0
-4
-8
-12
-16
q (FC/Cm ~)
(o)
~
I
i: I : i : 1 : I : I .I
o I--
I.
i. I
I
I
m I
. . . . . . ~os'-=...
-~ ~
z,=-,
"o.7.---'.-..:..
~"~,~<~,:-._
D= 2~
---:'-.
1 I I
16
ORA,A,E K1
"-%.
-
_21
l
,,l,)o,,,
I ~ I ~ I I I i I I l I b-\l
12
8
4
0
-4
-8
-12
-16
q (/~c/crn 2)
(b)
o|
~.-J_l
F
-
V
I
---o
v I
I
I
i
I
I
I
.... :: ....
....
""
~.-Ib.~
I
I
I
I
r. 2
-
" "
-
~RA,A,E KI
__
A~a N = ~ao~ (1; Rt) -- ~aoN(p -- I; R1)
DO
q- ~-~ {~aoN(~o q- 1; Rz) -- ~aoX[(n -F 1),o-- 1; R1]} [20]
n=l
(~ = 1 only)
a series whose c o n v e r g e n c e is still g e n e r a l l y good.
The above is a lu c k y result; e v e n it is unnecessary,
however, if w e note that w h e n ~, = 1, ~o~(Zo) - 1.
Thus, w h e n ,o = 1, h~a N can still be calculated using
[5] only once to first obtain +aN(l).
In II, we noted the e x t r a o r d i n a r y constancy of ~ (d)
for /~ = 3A, ~ ---- 0, and r = 2/3 over a wide r a n g e of
_21
t
16
t
I
12
f
I
8
f
I
4
r
I
0
f
I r
-4
I
-8
I
I
-12
f 1%,
-16
q (/zc/cm2)
(c)
Fig. 5. Full electric field at the IHP, 8(~), v s . average electrode
charge density, q, using Grahame's KI ql(q) data for 1N and
0.025N concentrations; r = V2, 1, 2; ~ = 2~; and several values
of Cal.
984
October 1966
JOURNAL OF THE ELECTROCHEMICAL SOCIETY
effects (1). W e s h o w r e s u l t s i n Fig. 5 u s i n g G r a h a m e ' s
ql(q) d a t a f o r b o t h 1N a n d 0.025N c o n c e n t r a t i o n s f o r
t h e c h o i c e 8 = 2A, w h i c h w e n o w b e l i e v e is a m o r e
l i k e l y v a l u e t h a n t h e 3 A u s e d i n II. T h e r e s u l t s i n
Fig. 5 w e r e c a l c u l a t e d u s i n g t h e ( b ) a n d (c) a p p r o x i m a n t s of T a b l e I i n t h e A p p e n d i x . T h e ( b ) a p p r o x i m a n t f o r P ( D is p a r t i c u l a r l y a p p r o p r i a t e h e r e s i n c e
Z is fixed a t o n e f o r t h e s e c a l c u l a t i o n s a n d R1 v a r i e s .
T h e r e s u l t i n g a p p r o x i m a n t is t h e r e f o r e s o m e w h a t s u p e r i o r to t h a t i n ( a ) w h e r e R1 is fixed a t 5 a n d Z
v a r i e s to p r o d u c e c h a n g e s i n 4. S i n c e F ( O is a s m a l l
c o r r e c t i o n t e r m , its R1 = 5 a p p r o x i m a n t is q u i t e a d e q u a t e i n t h e p r e s e n t case.
I t w i l l b e s e e n f r o m Fig. 5 t h a t ~ (8) r e m a i n s q u i t e
c o n s t a n t w h e n ~ = 0 a n d r e m a i n s so also e v e n w h e n
~, ~ 1 f o r r - - 2. F o r r = 1, 8 ( ~ ) f o r t h e 0.025N c o n c e n t r a t i o n goes t h r o u g h z e r o i n t h e e x p e r i m e n t a l
r a n g e of q w h e n ~ is a b o u t 0.4 o r g r e a t e r . F i n a l l y ,
w h e n r = 89 a l l of t h e c u r v e s s h o w n c h a n g e s i g n
i n t h e e x p e r i m e n t a l r a n g e e x c e p t t h o s e f o r ~ = 0.
If w e a s s u m e t h a t t h e r a p p r o p r i a t e to t h e a c t u a l e x p e r i m e n t a l s i t u a t i o n is 1 or less a n d t h a t 0.7 < ~ ~ 1,
t h e n i t is e v i d e n t t h a t ~ (8) w i l l v a r y s u f f i c i e n t l y o v e r
t h e q r a n g e , f o r e i t h e r 1N o r 0.025N c o n c e n t r a t i o n s ,
t h a t i t w i l l n o t b e a g o o d a s s u m p t i o n to t a k e el c o m p l e t e l y s a t u r a t e d a n d e q u a l to 6 o v e r t h e e n t i r e q
r a n g e . F o r r = 1, ~ = 0.9, a n d t h i s c o n s t a n t v a l u e of
~1, t h e a p p r o p r i a t e c u r v e s h o w s t h a t ~ ( 8 ) v a r i e s f r o m
a b o u t 3.5 x 107 v / c m to 4 x 106 v / c m as q goes f r o m
--18 to + 1 8 ~C/cm~. W i t h e, b e c o m i n g less s a t u r a t e d
a n d ~ u s l a r g e r as ~ ( 8 ) d e c r e a s e s ( 2 0 ) , ~ ( ~ ) a t q =
18 ~ C / c m 2 m i g h t b e as s m a l l as 2 x 106 v / c m . W e
r e m a r k a g a i n , h o w e v e r , t h a t t h e i n t r o d u c t i o n of a n ,~
a t a l l i n t h e p r e s e n t s i t u a t i o n is a c o n s i d e r a b l e a p p r o x i m a t i o n , m a k i n g a n y c o n c l u s i o n s a b o u t its v a r i a t i o n
uncertain.
F i g u r e 6 s h o w s t h e n o n l i n e a r i t y p a r a m e t e r 4o f o r
t h e u s u a l v a l u e s of r a n d m a n y p o s i t i v e v a l u e s of ~.
These results form a bridge between the ~ = 1 res u l t s of I a n d t h e ~ = 0 r e s u l t s of II. H e r e a n d e l s e where in this paper all w = 1 and ~ = 0 results agree
e x c e l l e n t l y w i t h t h o s e p r e v i o u s l y g i v e n i n I a n d II.
We can now follow in detail the continuous change
f r o m o n e l i m i t i n g c a s e to t h e o t h e r , h o w e v e r . I t s h o u l d
be noted that in the present paper the w values 0 and 1
represent only limiting situations and are not exami n e d i n d e t a i l ; m o r e r e s u l t s f o r t h e s e specific v a l u e s
t h u s a p p e a r i n I a n d II t h a n a r e g i v e n h e r e .
I t w i l l b e n o t e d f r o m Fig. 6 t h a t , e x c e p t f o r
v a l u e s e q u a l to or v e r y n e a r u n i t y , ~o is b y n o m e a n s
n e g l i g i b l e o v e r m o s t of t h e r a n g e of Rz s h o w n . A t
t h e t o p of Fig. 6a is s h o w n a q~ scaI'e f o l l o w i n g f r o m
t h e R~ s c a l e w h e n 8 = 2A, p r o b a b l y a r e a s o n a b l e
v a l u e f o r t h e K I s y s t e m . T h e R , = 2 v a l u e , w h i c h is
t h e s m a l l e s t p o s s i b l e R~ f o r c l o s e - p a c k e d s p h e r i c a l
a d i o n s , c o r r e s p o n d s to a q~ of 115.6 ~ C / c m ~ f o r t h i s
c h o i c e of 8. O n t h e o t h e r h a n d , t h e l a r g e s t v a l u e d e r i v e d f r o m G r a h a m e ' s (22) e x p e r i m e n t s is a b o u t 43
~ C / c m 2 ( c o r r e s p o n d i n g to R~ ~ 3.3 f o r 8 = 2 A ) , i n d i c a t i n g a m a x i m u m a d i o n s u r f a c e c o v e r a g e of a b o u t
37% for these experiments.
T h e d a s h e d a n d d o t t e d l i n e s of Fig. 6b a r e c a l c u l a t e d f o r t h e u s u a l c u t - o f f m o d e l of t h e i n n e r r e g i o n
(8c, 18), n o t f o r a h e x a g o n a l a r r a y . T h e y u s e v a l u e s
of ~aos o b t a i n e d f r o m Eq. [ A - l ] w i t h c o n s t a n t p v a l ues, a n d [ A - l ] is a p p l i e d f o r all Z a n d R~ c o m b i n a t i o n s s i n c e [ A - 4 ] is o n l y a p p l i c a b l e to t h e h e x a g o n a l
a r r a y . T h e d o t t e d - c u r v e v a l u e p = (~/3--/2~)~/2
0.5250376 is G r a h a m e ' s o r i g i n a l figure. I t is t h e v a l u e
a p p r o p r i a t e f o r a s m e a r e d a r r a y of c h a r g e s a n d is
t h u s also t h e v a l u e f o r t h e fixed h e x a g o n a l a r r a y
w h e n ~ --> r
I t is t h e s m a l l e s t v a l u e of p p o s s i b l e
for such an array. The dashed-curve value p = 4~/
~ / 3 ~ 0.65752059, w h e r e ~ ~ 11.034175, is t h e v a l u e
a p p r o p r i a t e f o r a n a r r a y of i d e a l d i p o l e s w h e n ~ ->
0 [see ref. ( 1 8 ) ] . T h i s p is t h u s t h e l a r g e s t v a l u e
p o s s i b l e w i t h a h e x a g o n a l a r r a y . I t is i n t e r e s t i n g t h a t
o v e r m u c h of t h e R1 scale of i ~ t e r e s t , t h e a c c u r a t e
462.5 115.6 50
I
I
I
~
,621
30
I
20
I
-
-
12
I
I
~
5
3
I
2.06
0.4 ~
\,"+--'T"~I
I
4
I
I
_
, ~
7
t
9
II
I
I
13
15
Ri
(o)
I
-
I
I
(
I
'
I
i
I
i
0
L
id2
'
I
i
I
' ~
0.98~
I
\
3
I
~
i
7
5
9
[
"~
,
II
I
i
13
15
Ri
(b)
I ~
I
I
I
I
I
I
I
I
I
I
I
~ ~
F
iiI
q
,62
,
0
0.97 - - - ' ~
I
3
i
I \~
5
~ ~
7
9
II
t3
15
Ri
(c)
Fig. 6. The nonlinearity parameter 4o vs. Rz for r ~ 1/~, 1, 2
and positive values of m. Dotted lines were calculated using
Grahame's cut-off model involving the constant value of p, p| ~ 0.52.50376. Dashed lines were calculated similarly with p again
constant and equal to po ~ 0.65752059. The nonlinear ql scale at
the top of (a) is applicable for the choice fl -~-- 2,~, only.
h e x a g o n a l - a r r a y r e s u l t s lie b e t w e e n t h o s e o b t a i n e d
w i t h t h e a b o v e t w o l i m i t i n g v a l u e s of p. O n t h e o t h e r
h a n d , i n t h e r e g i o n of a p p r e c i a b l e I H P c h a r g e d e n s i t y ,
s a y 2 ~ R1 < 6, w h e r e t h e f i x e d h e x a g o n a l a r r a y
m o d e l is m o s t a p p r o p r i a t e a t r o o m t e m p e r a t u r e , w e
n o t e r e g i o n s of v e r y a p p r e c i a b l e d e v i a t i o n o f t h e
VoI. 113, No. 10
li ~
20
0!I
n,~l
DISCRETENESS-OF-CHARGE MICROPOTENTIALS
::::
0
ii ~176
/i
A
/
I0.6
i
/ /\ \
1//
II
o./,//i,
,//I
oA
/i
//
T-//,~
o
...r lX,/i
-44
I
-40
_
~ I
-36
-
i
-32
I
I
-28
I
i
-24
I
i
-20
I
i
-16
I
i
I
-12
i
-8
I
I
-4
ql (/~C/cm2)
(o)
2.0
'
I
'
I
i
I
~ I
;
I
'
I
i
I
i
-\
I
~ I
i
I
f
ro,
--
ZV= - I
L6 -i
~
#=2~
A '~ X
_\
\
_\
\
0.8
\ ~
0.4
I
-44
GRAHAME 1N
KI qI(q}DATA
0.4
O6
~_
I = 1 ~ I [ I ] I J I i I M I ~ I = I
-40 -36 -32 -28
-24 -20
-16 -12
-8
-4
q~ ( Fc/cm
2}
(b)
I.O
_,
I
~
I
~
I
i
I
i
I
i
I
I
I
i
I
irl
2i
I
i
ZV: -I
~004~
~_
~
00.8
GRAHAME 1N
.
8
KI qllq) DATA
06
,
-44
[ i I = I z I i ] , 1 I I J I * I ~ I
-40 -36 -32 -28
-24 -20
-16 -12
-8
-4
ql (vC/cm2)
0
(c)
Fig. 7. Quantity
A ---- 42/V2
vs.
= 2.~, :F = ]/2, ], 2, and several
lines denote negative
values.
Grohome's
derived
q2(q) for
positive values of (o. Dashed
fixed-p results f r o m the h e x a g o n a l a r r a y results. In
such a region, the h e x a g o n a l m o d e l is definitely p r e f e r ab l e to a constant-p cut-off model. T h e e x a c t
charge density w h e r e t h e l a t t e r m i g h t be m o r e p r e ferable is difficult to d e t e r m i n e [see (7, 17)], h o w e v e r .
F i g u r e 7 shows t h e ratio of the m i c r o p o t e n t i a l to
the macropotential, A -- 42/Vz, vs. q2 for r = 89 1,
and 2, and s ev era l values of ~. H e r e again G r a h a m e ' s
(22) q2(q) values for 1N KI w e r e e m p l o y e d in calcu-
985
lating A f r o m Eq. [17]. Fo r a g i v en q, the corresponding qz was used to calculate Rz, using # = 2A, then
A4aN was calculated and was finally used in Eq. [17].
This calculation of A4aN, and also most of those for
Ao in Fig. 6, used the Rz = 5 (3/3) a p p r o x i m a n t for
P ( D discussed in the A p p e n d i x together w i t h Eq.
[ A - l ] and [A-4]. We do not show A curves for the
0.025N concentration, but calculations for this case
showed t h e m to be quite similar to the 1N curves.
W e h a v e elected to show A curves h e r e instead of t he
corresponding ~ curves g i v en in I and II because
turns out to be v e r y appreciable for most of the
curves (nonlinear distance d e p e n d e n c e of 4 N) and the
ratio A is itself t h e n of most direct interest.
E q u a t i o n [17] shows t h a t w h e n 5 = 0, A = L This
is, of course, the case at the r i g h t of t h e A curves in
Fig. 7 w h e r e qz ~ 0. Dashed lines are used in Fig.
7a to indicate n e g a t i v e values of A. These arise be cause Vt m a y change sign, leading to a pole in A
near qz = --35 ~,C/cm ~. A l t h o u g h both 41 and V2 are
continuous, their ratio need n o t be. Th e peculiar be h a v i o r of the ~ = 1 cu r v e in Fig. 7a is e v i d e n t l y p r o duced by a zero in 4z near b u t not at that of Vz.
Th e q u a n t i t y of most interest for adsorption isot h e r m s is 42 itself. It can, of course, be r ead i ly obtained f r o m the A values g i v en here by m u l t i p l y i n g
by calculated v a l u e s of V1. In this paper, w e shall not
be directly concerned with adsorption isotherms and
w i t h the 41 w h i c h enters them, r e s e r v i n g such discussion for an o t h er place (17). It is w o r t h w h i l e to
point out, h o w e v e r , that w h e n A is neglected (often
a good a p p r o x i m a t i o n for infinite imaging) and 41
is t h e n t a k e n as kV1, G r a h a m e (22) and G r a h a m e and
Parsons (23) h a v e found that the 42 d e r i v e d by using
e x p e r i m e n t a l results in a simple adsorption isotherm
leads to a v a r i a b l e L A l t h o u g h this specific approach
can p r o b a b l y be improved, it does lead to ~ v ar i a t i on
w i t h q or q2 of m u c h the same f o r m as that of A in
Fig. 7b and c. In particular, ~. is f o u n d to increase
continuously for K I as q increases f r o m --18 ~ C / c m 2
to 18 ~ C / c m 2. Reasonable values of r and ~ can even
be selected that lead to A v ar i at i o n q u a n t i t a t i v e l y
v e r y similar to that found for k, but w e do not wish
to stress this a g r e e m e n t e v e n though the G r a h a m e Parsons k ( = 4 2 / V 2 ) d e t e r m i n e d as ab o v e is f o r m a l l y
f u l l y e q u i v a l e n t to our A and t h e y both equal the
constant ~ (a ratio of distances) of the p r esen t paper
w h e n q2 = 0.
Dielectric E S P
W h i l e in t h e foregoing w o r k metallic i m agi ng is
assumed to occur at a conducting electrode and dielectric i m a g i n g at the OHP, our m a t h e m a t i c a l mode l
also pertains a p p r o x i m a t e l y to an e n t i r e l y different
system: A t an e l e c t r o l y t e - d i e l e c t r i c interface, w e
m a y also h a v e a surface phase in w h i c h ions are
h e x a g o n a l l y a r r a y e d on an " I H P . " S e p a r a t i n g this
phase f r o m the bulk electrolyte w o u l d be an " O H P ; "
separating it f r o m the b u l k dielectric w o u l d be an
"ESP." W h i l e e x t e r n a l charged electrodes m i g h t be
present near the surface layer, producing a u n i form field 4e, these electrodes need only be a m i c r o scopically l ar g e distance r e m o v e d f r o m the l ay e r (a
probable situation here) for their effect upon 4a to
be negligible.
Our m o d el in the p r esen t situation is that the O H P
a p p r o x i m a t e s to a metallic i m ag i n g plane and the
E S P forms a dielectric i m ag i n g plane. Clearly, the
O H P will a p p r o x i m a t e better and b e t t e r to a cond u ct i v e i m ag i n g plane the higher the concentration
of ions in the diffuse layer. To m a i n t a i n our p r e vious equations w i t h m i n i m u m change r e q u i r e s that
we now define # as the I H P - O H P separation, 7 as
the I H P - E S P separation, q as the total surface
charge density on the OHP, that w e m e a s u r e z f r o m
the O H P (thus z = # + 7 at the E S P ) , and that w e
define e] and e2 to be the effective dielectric constants
of the surface phase and bulk dielectric, r e s p e c t i v e l y
(see Fig. 8).
October 1966
J O U R N A L OF THE ELECTROCHEMICAL SOCIETY
986
BULK
ELECTROLYTE
SURFACE REGION
(El)
:L
,I.
OHP
iHP
BULK
DIELECTRIC {E))--~
just the d i s c r e t e - c h a r g e contribution to t h e n o r m a l i z e d
potential at the IHP. The choice q = - - q , means
that t h e charge on the I H P is e n t i r e l y balanced by
that in t h e diffuse layer, h e r e t a k e n to be on t h e (conducting) OHP. One w o u l d only expect q ~ --qz in
the present situation w h e n t h e i n n er l ay er was e x posed to an e x t e r n a l l y applied field.
W e h a v e plotted curves for A vs. R1 in Fig. 9. F or
completeness and because some authors h a v e t a ke n
ESP
9
o
•
.
.
.
.
.
o-
iiAi:
!i!!iiiiiiiiiiii
]iiiiiiiiiiiiiii
o
SURFACE- LAYER tONS
Fig. 8. Cross-sectional diagram showing appropriate situation
and distances when the OHP is taken as a conducting plane.
T h e r e still r e m a i n certain differences b e t w e e n our
system an d the original compact l a y e r for w h i c h our
equations w e r e derived. Most i m p o r t a n t of these is
that the adion density on the I H P w i l l now depend
on the potential difference b e t w e e n the I H P and the
conductive plane; thus, 4, -- 4(fi) - - 4(0) = 4(fi),
w h e r e "/Y' h e r e is p r o b a b l y n u m e r i c a l l y equal to the
"~" of our original system. The reason for the present i n v o l v e m e n t of t h e conductive plane potential,
w h i ch is of course zero, is that in this system th e ions
are p r e s u m e d to originate (again neglecting V2)
f r o m the conductive (outer Helmholtz) plane w h e r e a s
earlier they originated f r o m the dielectric i m ag i n g
plane, and t h e r e the finite potential at the dielectric
discontinuity had to be incorporated in the m i c r o p o tential. Finally, apart f r o m the n u m e r i c a l differences
expected b e t w e e n n e w and old /~'s, 7's, q's, and so
forth, w e m a y now obtain w i t h g r e a t e r likelihood
n e g a t i v e v a l u e s for the p a r a m e t e r ~. L e v i n e et al.
(7b, 11) h a v e discussed, to some extent, a situation
w h e r e ~2 is t ak en as 15, p e r t a i n i n g to silver chloride,
a n d el taken as 10 or 15. In these cases, ~ = 0.2 and 0,
respectively. On the other hand, the i m p o r t a n t a i r electrolyte interface is a m o r e common situation. F o r
e, = 10 and 5, ~ is a b o u t - - 0 . 8 2 and --0.67, respectively,
for such a boundary.
We h a v e al r ead y plotted in Fig. 3 and 4 n o r m a l i z e d
discrete potentials and displacements for q = - - q ,
and negative values of w. To m a k e use of these curves
in the p r es en t situation, however, w e must i n t e r p r e t
the plane m a r k e d O H P in those figures as being the
ESP, and the plane ~ = 0, in the e a r l i e r case the ESP,
as the OHP. Th e I H P is the same in both situations.
The different i n t e r p r e t a t i o n s now g i v e n to /~ and -/
result in probable changes in t h e n u m e r i c a l v a l u e s
a p p r o p r i a t e to these quantities; correspondingly, the
n u m e r i c a l v a l u e of 4| is likely to be different. Note
that if t h e n u m e r i c a l values of 7 and p are simply
i n t e r c h a n g e d in going f r o m a conductive E S P to a
dielectric E S P situation, t h e n r in the dielectric
situation has a m a g n i t u d e w h ic h is just the reciprocal
of that in the usual situation. A l t h o u g h w e h a v e only
shown curves of potential and field applying w h e n
q -- ---ql, t h e extension to the m o r e g e n e r a l case is
straightforward. T h e expression for 4~ in the p r esen t
situation is identical to t h a t g i v e n for the case of a
conducting electrode.
Finally, w e wish to define and calculate a q u a n tity A analogous to that defined for our original system. In the p res e n t situation, we shall again define
i ~ 411Vl
=
10-1 -
-04
-
~
IO2
3
I -0.9
5
7
9
II
13
15
Rt
(o)
I_
~
~
I
I
[
I
I
I
I
J
I
~
- - - . ~ ~
\~--~...~~~.~------
I
i-
q =-ql
I
-
I
-
o.8
o.6
idl _-_
-0.2
1(5: ~
5
5
7
9
II
15
15
R=
(b)
I
-
I
f
'
i
I
I
I
I
I
I
I
I
I
q=-q/
~
~
~
~
o.2 ~
o --.__.~
d/(fi)/Vl = 4N(1)/VN(Zo)
=
[1 + (q/q,)] - - ~ N ( 1 )
r + (1 + F) (q/q,)
5
[21]
This e q u a t i o n shows t h a t w h e n q, ~ - - q A --> (1 +
r ) - * = Zo -1 = r - l k as R, -> Do for fixed q. S p e c i a l izing now to t h e case most in te r e s t in g and p e r t i n e n t
for the present situation, q = - - q , , w e find A = 4aN(l),
5
7
9
II
13
15
RI
(c)
Fig. 9. Quantity A = 41/V1 appropriate when the OHP is taken
conducting v s . R1 for F = ~, 1, 2 for a full range of m values.
Here q = --ql and A = 4a(fl)/4|
pertinent to the conducting
ESP case as well.
V o k 113, No. 10
DISCRETENESS-OF-CHARGE
the OHP to be a metallic i m a g in g plane in situations
w h e r e the m a t e r i a l beyond th e E S P m a y lead to a
positive v a l u e for ~, w e show curves for positive
values as w e l l as n e g a t i v e ones. It will be noted that
as R1 increases all the ~ = 1 curves approach the
limiting v a l u e (1 + r ) - 1 : /~/(/~ q_ -x). In this situation, ~1 ---- ~a(/~) is just the proper linear proportion,
~/(/~ -k 7), of V~ ----- ~| and
h e r e defined t h r o u g h
A _= r - l ~ ( 1 + ~), w o u l d be zero as it should be.
Again, w e h a v e p r e s e n t e d curves of A r a t h e r t h a n
since w e b e l i e v e A to be m o r e significant in the
pres en t case.
The curves of Fig. 9 are p a r t i c u l a r l y v a l u a b l e since
w h e n q = - - q l , A = ~a(/~)/~|
and the curves show
i m m e d i a t e l y how large the d i s c r e t e - c h a r g e potential
at t h e I H P is c o m p a r e d to the total potential (at
la r g e distances) set up by t h e array, r174 These r e suits for ~a N ( 1 ) are, of course, j u s t as a p p r o p r i at e for
the situation of a conducting E S P discussed in the last
section since t h e y i m m e d i a t e l y give the r e l a t i v e ~a
at the I H P in this case as well. Note f u r t h e r that
since the a v e r a g e potential at the IHP, V(~), is also
~ for q
the curves show in addition h o w ~a(~)
differs f r o m the a v e r a g e potential at the IHP.
In the present paper, w e h a v e m a d e calculations
based on a m o d e l in w h i c h t h e r e is one conductive
i m ag i n g plane and one dielectric i m a g in g plane (and
h a v e also included results a p p r o p r i a t e for two conducting planes). These calculations w e r e r e f e r r e d to
two physical situations. In the first situation treated,
the e l e c t r o l y t e was assumed to b e h a v e like a simple
dielectric, mobile ions in the diffuse layer w e r e not
explicitly t ak en into account, and the O H P t h e r e f o r e
was considered a dielectric i m a g i n g plane. In the
second situation, the O H P was considered effectively
to be a conductive i m a g in g plane, and the dielectric
i m ag i n g occurred elsewhere, at the surface of a t r u e
dielectric. H o w v a li d is our m o d e l for these two situations?
The theoretical difficulties associated w i t h the first
system h a v e al r e a d y been discussed s o m e w h a t and
preclude a precise a n s w e r to this question for this
case. H o w e v e r , w e m a y m a k e some reasonable guesses
as follows. I n a s m u c h as t h e ions in the diffuse l a y e r
w i n cause the O H P to b e c o m e to some e x t e n t a conductive imaging plane, we anticipate that the present
model will not accurately p o r t r a y th e first system
except possibly at ionic concentrations so low that
the effective Deb y e lengths are l a r g e r than other
characteristic dimensions of the system (e.g., ~ and
r~), concentrations w h i c h are often l o w e r than those
of most interest. Note that the D e b y e length for 0.025N
at 25~ is about 20A. If /~ ---- 2A, this corresponds to
R1 ----- 10, and complete neglect of any diffuse layer
conductive i m a g i n g would probably then only be
justified for R1 values a p p r e c i a b ly s m a l le r than 10
for this b u l k concentration. A l t h o u g h one m i g h t be
t e m p t e d to apply the present m o d e l by using an eff e c t i v e ~, larger than that calculated on the basis of
dielectric constants alone, which h o p e f u ll y w o u l d ap p r o x i m a t e l y take into account the mobile ions, we
do not h e r e advocate this p r o c e d u r e as an e n t i r e l y
satisfactory solution to the p r o b l e m for two reasons:
O u r o w n estimates of th e i m p o r t a n c e of c o n d u c t i v e
im ag i n g at the O H P for ionic concentrations such as
1N wo u l d i m p l y effective ~ values e x c e e d i n g ly close
to unity, for wh i c h the E r s h l e r m o d e l is appropriate.
F u r t h e r m o r e , any actual small difference b e t w e e n the
predictions of t h e E r s h l e r m o d e l and those of the
present m o d el w i t h ~, ~ 1 w o u l d p r o b a b l y be no
l a r g e r than the errors introduced into t h e p r e s e n t
t r e a t m e n t by the a t t e m p t to subsume the action of
the diffuse-layer ions into an effective ~. N e v e r t h e less, the present approach a d e q u a t e l y illustrates for
the first t i m e the effects of the e v e r - p r e s e n t u n d e r l y ing dielectric imaging.
Concerning the second system considered, w h e r e t h e
O H P is t a k e n to be a conductive i m a g i n g plane, our
A,
=--ql,
987
MICROPOTENTIALS
e v a l u a t i o n of the m o d el validity is accordingly considerably higher. The f o r eg o i n g a r g u m e n t s w h i c h favor taking the O H P as just such an i m ag i n g plane a r e
as encouraging in this situation as they w e r e discouraging in the f o r m e r one. We t h e r e f o r e feel that
the p r esen t t r e a t m e n t pertains to a dielectric E S P
(for the usual appreciable ionic concentrations) fairly
well. If an accurate f u t u r e t h e o r y should d e m o n strate that our p r esen t opinions are o v er est i m a t e s of
conductive i m ag i n g at the OHP, then w e w o u l d h a v e
to i n t e r c h a n g e our evaluations of the applicability of
the present m o d e l to the t w o situations.
Acknowledgment
Th e authors g r eat l y appreciate the help of M a r i l y n
W h i t e and Charles Ratliff in w r i t i n g the several comp u t e r p r o g r a m s used in this work.
APPENDIX
In an o t h er paper (18), w e h a v e discussed in some
detail how G r a h a m e ' s (8c) cut-off a p p r o x i m a t i o n for
single i m ag i n g d i s c r e t e n e s s - o f - c h a r g e potential calculations can be modified in a simple w a y to yi e l d
highly accurate results for h e x a g o n a l p l a n a r arrays
of ideal or n o n - i d e a l dipoles. Here, w e thus shall only
q u o t e results used in the p r e s e n t work.
Fo r ~baoN and 8ao N we m a y w r i t e
~baoN ----- 1/2 { [ ( p R 1 ) 2 -[- ( Z -[- 1 ) 2 ] 1/2
- -
[(pR1) 2 + ( Z - - 1)2] 1/2} [ A - l ]
Z + 1 - - [(pR1)2/Z]F
~a~ ---- 1/2
[ (pR1) 2 q- (Z q- 1)2] 1/2
Z--l--
[ (pR1)2/Z]F
"l
[(PR1)2-b ( Z - - l ) 2 ] 1/2 f
[A-2]
In these equations, p and F (which ar e constant
and zero, r e s p e c t i v e l y in G r a h a m e ' s w o r k ) are p r i m a r i l y functions of } -- Z/R1 b u t depend slightly on
R1 as w e l l (18). H e r e F =_ --dlnp/dln}. Using C h e b y chev rational f u n c t i o n a p p r o x i m a t i o n methods, w e obtained in ref. (18) v e r y accurate, yet simple, a p p r o x imations of t h e p and F functions w h i ch m a k e [ A - l ]
and [A-2] exact for h e x a g o n a l arrays. This m e t h o d
of approach was used because it was found that p
changed only over a l i m i t ed r a n g e for any fixed R1
w h e n } v a r i e d f r o m 0 to oo and F was small over most
of the range, r e a c h i n g a m a x i m u m of about 0.14 near
}-----1.
The rational function a p p r o x i m a t i o n s for p and F
are all of t h e t y p e
f(,)
:
~
ai, i /
i=0
/
~
b~'
[A-3]
i=0
and m a y be t e r m e d ( n / m ) approximants. Note that
bm - 1. In Table I we give the coefficients of (3/3)
and (0/2) a p p r o x i m a n t s for p(~) and of a (2/3) a pp r o x i m a n t for F(~). Coefficients of other simpler a pp r o x i m a n t s are g i v en in ref. (18). Here, w e h a v e
used m o r e complicated a p p r o x i m a n t s than w o u l d n o r m a l l y be necessary in order to ensure accuracy of at
least sev er al decimal places in all the results calculated. In Table I, (a) values ar e for a fit w h i ch rainTable I. Rational function coefficients for (a) p(~) (RI = 5 fit),
(b) p(~) (Z = 1 fit) and (c) F(/~) (R1 = 5 fit)
(a) #.~ = 1.355 • 10-~
(c)
t
O
1
a~
3
b~
(a)
(b)
-- 4 1 . 6 0 7 0 3 5
6.2168220
(c)
(a)
O
27.059236
(c)
(a)
(b)
-- 0.21067153
--33.039613
6.4039198
--69.285095
(c)
-0.058259894
1.0
- 5.2036756
(a)
0.75227306
(b)
2
(b) , ~ = 1.261 • 10-8
• 10 -~
#~ = 3.965
- -
(b)
--
(e)
- -
-- 6 3 . 8 5 7 6 8 0
9.4332222
-- 3.3093430
41.149777
0.72151658
1.0
- 1.0
988
J O U R N A L OF THE ELECTROCHEMICAL SOCIETY
imized the absolute value of the relative deviations,
6R, while (b) and (c) values were for fits which m i n imized the absolute values of the absolute deviations,
6A, between accurate values and approximate p r e dictions. Values of the p e r t i n e n t 8R and 5A'S are
shown in the table and illustrate the high accuracy
of the approximants.
We have used the above expressions for ~aoN and
~ a o N for a hexagonal a r r a y for Z < 1 + 3R1. W h e n
Z --~ 1 + 3R1, however, the results of II reduce to
and
fao N --~ 1 - - (~/3/4n) JR12/(Z 2 - 1) ]
[A-4]
~aoN --~ (~/3/2;~) [ZR12/(Z 2 -- 1) 2]
[A-5]
accurate to about eight figures w h e n Z ---- 1 -{- 3R1 and
becoming even more accurate as Z/R1 increases f u r ther. Therefore, these simple expressions were e m ployed in place of [ A - l ] or [A-2] w h e n Z --~ 1 + 3R1.
e
ql
q
zv
~
~2
-~
d
rl
GLOSSARY
Basic Parameters of System
Charge of proton
Average surface charge density on IHP
Total surface charge density on conductive
plane
Effective valence of surface-layer ions
Effective dielectric constant of surface region
Effective dielectric constant of dielectric region
Distance from IHP to conductive plane
Distance from IHP to dielectric imaging plane
~ + 7 = thickness of surface region
Lattice spacing of surface-layer ions
W
(e 2 - - e l ) / (e 2 -~- e l )
,]
1 -}- w : 2e2/(~2 na el)
r
p
(x,y)
z
Z
R1
Zo
~,I~
2dl~
Coordinates in a plane parallel to IHP, generally set to zero here
Coordinate of position n o r m a l to E S P
zl~
rl/~
1 + r = d/3
ZIR1 = z/rl
Unnormalized Potentials and Fields
~a + .% = actual potential at a point (x,y,z)
~e
Portion of potential arising from a u n i f o r m electric displacement
r
Portion of potential arising from discrete system of charges and images
fao Discrete potential in single image regime
~acc Discrete potential in infinite image regime
Actual electric field, associated with
~a
Field associated with ~a
@ao Field associated with r
Z)a Displacement associated with ~,
Potential Di~erences, Average Quantities, and
Normalizing Facfors
r
Micropotential -- ~ ( a t IHP) - - r
OHP)
V1
Average potential drop across surface layer
V2
Potential drop across diffuse layer
V (z) Average potential in a plane z = constant
r174
( 4 ~ q f f e O = average potential drop arising from
discrete a r r a y of charges and images; used as a
normalizing factor
~|
Normalizing q u a n t i t y for fields - - - ~ / ~
~
Normalizing q u a n t i t y for displacements -- elg|
Normalized Potentials and Fields
~.oN
~]eN
~N
VI N
VN(Z)
a~ N
~N
~.o/~,~
~,J~,|
~1/~,~
Vll~|
V(z)I~|
~a~ (i) - - ~.~ (Zo)
~ao N
~aol~|
~/~
D . I D|
~N
D. N
~I~|
O c t o b e r 1966
Other Quantities
~l/V1 - a q u a n t i t y m e a s u r i n g ratio of micropotential to full surface layer average p.d.
A q u a n t i t y m e a s u r i n g d e p a r t u r e from a strict
proportionality b e t w e e n 41 and V1. W h e n the
ESP is conductive, A -- k(1 + 4 ) ; w h e n the ESP
is dielectric, A = r - l ~ ( 1 + 4)
4o
Value of 4 for q = 0
p,F Quantities used i n our modified cut-off a p p r o x imation for ~a and ~a
REFERENCES
1. J. R. Macdonald and C. A. Barlow, Jr., Proc. First
A u s t r a l i a n Conf. on Electrochemistry, Sydney,
Australia, 15 F e b r u a r y 1963, pp. 199-247, A.
F r i e n d and F. G u t m a n n , Editors, P e r g a m o n
Press, Oxford (1964).
2. J. R. Macdonald and C. A. Barlow, Jr., J. Chem.
Phys., 39, 412 (1963); 40, 237 (1964). A n i m p o r t a n t correction to this paper is discussed i n
ref. (14) below.
3. C. A. Barlow, Jr., and J. R. Macdonald, ibid., 4@,
1535 (1964). I n Eq. [23 ]of this paper, the term
in the square bracket should be in the d e n o m i nator, not the n u m e r a t o r , of the equation. The
term h - h which appears twice on p. 1543 should
be h-h~.
4. C. A. Barlow, Jr., and J. R. Macdonald, ibid., 43,
2575 (1965). The colon i n ref. (2) of this paper
should be a semicolon; ~ in the Glossary should
be ~a; there should be a h y p h e n b e t w e e n "average" and "discrete" at the bottom of p. 2586;
and Mignolet (28) on p. 2590 should be Mignolet
(14).
5. For discussion of some of these relations see: (a)
P. Delahay, "Double Layer and Electrode K i netics," Interscience, New York (1965); (b) ref.
(3) and (4) above; ( c ) A . N. F r u m k i n , This
Journal, 107, 461 (1960); a n d (d) A. A r a m a t a
and P. Delahay, J. Phys. Chem., 68, 880 (1964).
6, See ref. (5a) and (5b) and references therein.
Also, J. R. Macdonald and C. A. Barlow, Jr., J.
Phys. Chem., 68, 2737 (1964).
7. (a) J. R. Macdonald and C. A. Barlow, Jr., Can.
J. Chem., 43, 2985 (1965); (b) S. Levine. J.
Mingins, and G. M. Bell, ibid., 43, 2834 (1965).
8. (a) O. Esin and V. Shikov, Zhur. Fiz. Khim., 17,
236 (1943); (b) B. V. Ershler, ibid., 20, 679 (1946);
(c) D. C. Grahame, Z. Elektrochem., 62, 264
(1958); (d) V. G. Levich, V. A. K i r ' y a n o v , and
V. S. Krylov, Dokl. Akad. Nauk SSSR, 135, 1193
(1960); (e) V.S. Krylov, ibid., 144, 356 (1962);
(f) V. S. Krylov, Electrochim. Acta, 9, 1247
(1964).
9. N. F. Mott and R. J. Watts-Tobin, Electrochim.
Acta, 4, 79 (1961). P r i n t i n g and other errors in
this and some of the other papers referenced in
the present work are listed i n the references of
the present ref. (1)
10. J. O'M. Bockris, M. A. D e v a n a t h a n , and K. Miiller,
Proc. Roy. Soc. (London), A274, 55 (1963). See
also ref. (1), pp. 832-863.
11. S. Levine, G. M. Bell, and D. Calvert, Can. J.
Chem., 40, 518 (1962).
12. F. H. Stillinger a n d J. G. Kirkwood, J. Chem.
Phys., 33, 1282 (1960).
13. (a) V. S. K r y l o v and V. G. Levich, Zhur. Fiz.
Khim., 37, 106 (1963); (b) ibid., 37, 2273 (1963).
14. (a) J. R. Macdonald a n d C. A. Barlow, Jr., J.
Chem. Phys., 44, 202 (1966); (b) ibid., J. Appl.
Phys., To be published.
15. L. Schmidt and R. Gomer, J. Chem. Phys., 42, 3573
(1965).
16. N. F. Mott, R. Parsons, and R. J. W a t t s - T o b i n ,
Phil. Mag., 7, 483 (1962).
17. C. A. Barlow, Jr., and J. R. Macdonald, " A d vances in Electrochemistry a n d Electrochemical
Engineering," P. Delahay, Editor. To appear.
18. J. R. Macdonald and C. A. Barlow, Jr., Surface
Science, 4, 381 (1966).
19. J. R. Macdonald, J. Chem. Phys., 22, 1857 (1954).
20. J. R. Macdonald a n d C. A. Barlow, Jr., ibid., 36,
3062 (1962).
21. J. R. Macdonald, J. Appl. Phys., 35, 3034 (1964).
22. D. C. Grahame, J. Am. Chem. So c., 80, 4201 (1958).
23. D. C. G r a h a m e and R. Parsons, ibid., 83, 1291
(1961).
A
Discussion
H. D. Hurwitz: It m u s t be stressed that the v a l u e of
the micropotential is not sufficient to account for the
total coulombic i n te r a c ti o n among adsorbed particles.
This is shown easily by deriving, as indicated next,
the total coulombic e n e r g y of i n t e r a c t io n for a set of
charges of species a adsorbed at the inner H e l m h o l t z
plane ( I H P ) , assuming as usual that: 1, the i n t e r a c tion a - s o l v e n t is implicitly stated in the definition of
the local dielectric constant eo and of the pair c o r r e l a tion function gl,2 introduced below; 2, the interaction
a-diffuse l ay er charge is described by a simple a v e r age electrostatic effect in w h i c h the ionic distribution
in the diffuse layer is s m e a r e d out o v e r parallel planes
to t h e electrode; 3, the dielectric discontinuity near
the outer H e l m h o l t z plane (OHP) leads to partial or
total reflection of the charge e~ of a.
U n d e r these conditions w e assume that th e surface
density of ~ is p~. H e n c e p a l gives the probability to
find a at one given position, 1, of th e IHP. The p r o b a bility to en co u n t e r a neighbor to a at position 2 is
then by definition gl,2 p2 w h e r e gl,2 is the radial
distribution function. T h e coulombic interaction b e t ween these two particles is (e2J,o)71,2 w h e r e 71,2
represents t h e reduced coulombic potential at 2 corresponding to an isolated charge located at the I H P
at 1. The q u a n t i t y 71,2 includes the effects of p ar t i al
or total infinite reflections into the dielectric discontinuities of the model.
The p r o b a b l e pair interaction for positions 1 and 2 is
e2a
71,2 gl,2 p a l pa2
[1]
eO
Le t us sum over all particles on the I H P considered
to interact with ~ at 1 and over all c e n t r a l positions, 1, in order to get finally the total i n t e r act i o n
energy
e2~
Z 71,2 gl,2 pal pa2
2Co 1 2
[2]
(Because of s y m m e t r y of ~1,2 gI,2 t h e quadratic sum
is divided by 2.) Instead of s u m m i n g over all 1 and 2
w e m a y i n t e g r a t e o v e r - a l l distances s12 b e t w e e n 1
and 2 and o v e r - a l l positions 1 in o r d e r to gee finally
e2a
f gl,2 71,2 p 2 dSl2
9
[3]
2go
w h e r e II is the total area of the IHP.
The change of in te r a c ti o n energy on introduction of
one particle a is obtained t h r o u g h differentiation of
[3]. T h e r e f o r e
e2a
0
2co S7(s12)
Op~ (g(si2) p2~)ds12
[4]
It is clear that [4] is the contribution of coulombic
interactions among specifically adsorbed charges to
the change of potential e n e r g y of adsorption.~A m o r e
convenient w a y to w r i t e [4] is to decompose this e x pression in the f o l lo w i n g m a n n e r
The second t e r m is then a correction w h i ch results
f r o m d i s c r e t e n e s s - o f - c h a r g e effects [g(s12) is different
f r o m unity] and the first and second t e r m together
r e p r e s e n t t h e so-called micropotential.
A t this point w e h a v e to realize that w h e n e v e r we
introduce in our system at position 1 an ion a w e
do not alter only the local density Pa, but w e modify
simultaneously t h e radial distribution of particles.
This in t u r n gives rise to some interaction energy.
The last t e r m of [5] reflects such an effect of radial
redistribution. The contribution of this effect to the
free energy of adsorption has y et been considered only
f r o m a q u a l i t a t i v e v i e w p o i n t 1,2 and is g en er al ly n e glected. H o w e v e r , a careful evaluation based on a
simple m o d el for gl,2 has led 3 to the conclusion that
such w o r k of r e d i s t r i b u t i o n is not negligible at all and
m a y even be of the order of magnitude of the discreteness-of-charge correction as given by the second
term in expression [5]. In the crude hexagonal lattice
approximation of Ershler, in which no allowance is
m a d e for thermal motion, the gl,2 functions are represented by delta Dirac functions.
C. A. Barlow, Jr.: Three points seem to have been
raised here. First, "that the value of the micropotential
is not sufficient to account for the total coulom.bic interaction a m o n g adsorbed particles;" second, "that the
w o r k of redistribution is not negligible at all and m a y
even be of the order of magnitude of the dicretenessof-charge correction;" and third, that the hexagonal
lattice approximation is crude, the use of exact twoparticle correlation functions being preferable.
W e disagree with the first point; however, the disagreement possibly stems merely from a different use
of words: W e assert4 that the total coulombic energy
U(N) of a system of N charges is given by a s u m
over all particles of the local potential times onehalf the charge; in other words, w e agree entirely
with Dr. Hurwitz's Eq. [2] within the approximation
of the model. N o w except for a contribution, V2, from
the diffuse layer, the "local potential" occurring above
is just the micropotential. To equate the two involves
the neglect of V2 in the micropotential, to be sure,
but this is discussed in our present paper just after
Eq. [14], and this doesn't seem to be Dr. Hurwitz's
objection. His objection, as seen f r o m the second point
raised, is that w e are so m eh o w neglecting r e a r r a n g e m e n t energy. This is not true; in the first place, in the
present paper we did not need to calculate the e n e r g y
of adsorbing one additional ion, hU - U ( N + 1) - U ( N ) . We h a v e shown explicitly 4 that, h a v i n g found
the micropotential, we h a v e a means of calculating
the r el at ed q u a n t i t y U ( N ) in accordance with the
foregoing. In the second place, our approach does lead
to the redistribution energy in essentially the same
w a y that it arises in Dr. H u r w i t z ' s discussion w h e n
one finally gets down to the business of calculating ~U,
the e n e r g y of adsorption. T h e only difference in approach is that w e initially calculate U ( N ) and save
the differencing with respect to N until the v e r y last
(as m a y be seen f r o m our papers 5,6 w h e r e w e d e t e r mine the total e n e r g y for adsorbing an additional
z D. C. G r a h a m e , Z. Elektrochem., 62, 264 (1958).
ea
P~ ~eo f ~(sl2)dSl2 + P= ea f T(Sl2)[g(sl2)-e2a
~- Y~ P ~ a - - f T ( s l 2 )
9o
C. A . B a r l o w , J r . a n d J . R. M a c d o n a l d , J. Chem. Phys., 43, 2575
(1965).
0
'
g(sze)dsle
0 ph
a I-I. D . H u r w i t z , Z. physik. Chem., T o b e p u b l i s h e d .
[5]
C. A . B a r l o w ,
2575 (1965).
In v i e w of the fact that the first t e r m in [5] c o r r e sponds to the effect at [1] of charges u n i f o r m l y dist r i b u t e d o v e r the IHP, it is r e a d i l y i n f e r r e d that by
definition this t e r m pertains to th e a v e r a g e e l e c t r o static potential (macroscopic p o te n t ia l) .
Jr.
and
J . It. M a c d o n a l d ,
J. Chem. Phys., 43,
s J'. R . M a c d o n a l d a n d C. A . B a r l o w , J r . , " T h e P e n e t r a t i o n P a r a m e t e r f o r a n A d s o r b e d L a y e r o f P o l a r i z a b l e I o n s , " J. Appl.
Phys., T o b e p u b l i s h e d i n 1966.
6 J . R. M a c d o n a l d a n d C. A , B a r l o w , J r . , Can. J. Chem., 43, 2985
(1965).
989
990
JOURNAL
OF THE
ELECTROCHEMICAL
ion). We ag r ee with Dr. H u r w i tz ' s s t a t e m e n t about
orders of m a g n i t u d e ; w e were, in fact, the first to
m a k e a correct q u a n t i t a t i v e study of redistribution
e n e r g y in the present system. Indeed, although this
question was considered q u a n t i t a t i v e l y by us in the
w o r k cited by Dr. H u r w i t z 4 and a c u r v e plotted to
show the n u m e r i c a l m a g n i t u d e of the effect for a
wide ran g e of surface coverages, in his discussion
Dr. H u r w i t z only references his own unpublished
w o r k to support "the conclusion that such w o r k of
redistribution is not negligible at all and m a y even
be of the order of m a g n i t u d e of the discreteness-ofcharge contribution." It is of interest to note that in
our p r e v i o u s l y published w o r k 4 w e explicitly state
that the w o r k of r e d i s t r i b u t i o n is of t h e same order
of m a g n i t u d e as the full adsorption e n e r g y and is
not at all negligible. Dr. H u r w i t z seems to h a v e missed
noticing the significance of the q u a n t i t y n in our
paper, 4 since he refers to this p a p e r as considering
the question "only f r o m a q u a l i t a t i v e viewpoint."
F u r t h e r q u a n t i t a t i v e t r e a t m e n t of this m a t t e r will
soon appear.~
Finally, that the h e x a g o n a l model m a y be " c r u d e "
cannot be denied. We h a v e already p r e s e n te d a discussion of just h o w c r u d e it m a y be u n d e r some conditions. 6 If the proper correlation functions w e r e
known, they would c e r t a i n l y be p r e f e r a b l e to the
g-function lattice model. They are not known, h o w ever, so our approach has been to try to keep the
following s t at em e n t (concerning statistical physics)
of Richard F e y n m a n 7 in mind:
" A n y o n e wh o wants to analyze the properties of
m a t t e r in a real p r o b l e m m i g h t w a n t to start by
w r i t i n g down the f u n d a m e n t a l equations and t h e n
try to solve them m a th e m a ti c a ll y . A l t h o u g h t h e r e
are people who tr y to use such an approach, these
people are t h e failures in this field; the r e a l successes
come to those who start f r o m a physical point of view,
people who h a v e a rough idea w h e r e t h e y are going
and then begin by m a k i n g the r ig h t kind of a p p r o x imations, k n o w i n g w h a t is big and w h a t is small in
a g i v en complicated situation. These problems are so
complicated that e v e n an e l e m e n t a r y understanding,
a l t h o u g h inaccurate and incomplete, is w o r t h w h i l e
having . . . . "
We believe the h e x a g o n a l a r r a y approach to be the
" r i g h t kind of a p p r o x i m a t i o n " for a ql r a n g e of p r actical importance.6
Richard Payne: According to the results in Fig. 7,
the ratio of ~1/V1 is u n i f o r m l y larger for the single
i m ag i n g l i m i t than for the infinite imaging limit. This
does not seem consistent w i t h the Esin and S h i k o v
and G r a h a m e calculations for the single im a g i ng case
which o v e r e x p l a i n e d the Esin and M a r k o v effect (i.e.,
r
too s m a l l ) ; w h e r e a s Ershler's t r e a t m e n t of the
infinite i m ag i n g case gave l a r g e r values of ~1/V1. It
seems, therefore, that the tr e n d in Fig. 7 as ~ varies
is inverted.
J. Ross Macdonald and C. A. Barlow, Jr.: The a p p a r e n t
discrepancy pointed o u t by Dr. P a y n e is not significant for several reasons. First, the s i n g l e - i m a g i n g
t r e a t m e n t s of Esin and S h i k o v and G r a h a m e not only
tre at a physically v e r y different situation than the
present s i n g l e - i m a g i n g one b u t the situation tr eat ed is
much less physically appropriate than is ours. Second,
we p r o p e r l y include t h e u n i f o r m D field contribution
to the s i n g l e - i m a g i n g micropotential, omitted by other
authors. The E s i n - S h i k o v and G r a h a m e t r e a t m e n t s are
not r e a l l y s i n g l e - i m a g e ones at all since they consider
fixed arrays of adsorbed ions each with a rigidly paired
counterion in the diffuse layer. T h e counterions are
not t a k e n as images of th e adions but as r e a l associated ions a fixed p e r p e n d i c u l a r distance from the
R. F e y n m a n , R. B. Leighton, a n d M. Sands, " T h e F e y n m a n Lect u r e s on Physics," Vol. I, p. 39-2, Addison-Wesley, Reading, Mass.
(1963).
SOCIETY
O c t o b e r 1966
adions. Er sh l er has modified this approach by considering single i m a g i n g of the adions in t h e diffuse
layer, a p r e f e r a b l e assumption. This is indeed a singleimaging treatment, but it necessarily still disagrees
with our ~, = 0 results in Fig. 7 since there w e take
i m ag i n g in the electrode instead of the solution, take
q variable, and include the D field contribution.
The significant point r e g a r d i n g Fig. 7 is that for
reasonable r values the ~ = 1 curves g e n e r a l l y yield
micropotential values w h i ch are smaller t h a n they
should be to explain the E s i n - M a r k o v effect (A too
small) while the ~ = 0 results yield micropotentials
w h i ch are too large to explain it. F o r an ap p r o pr i a t e
v a l u e of r, it appears possible to pick a ~ v a l u e near
but less than unity which will lead to results w hi c h
will ex p l ai n the E s i n - M a r k o v effect (within the l i m itations of the p r e s e n t t r e a t m e n t ) better than will
results w i t h either ~ = 1 or 0.
Richard Payne: It is not clear w h y g r o u n d i n g of the
electrode should affect the charge on the m e t a l since
the w h o l e system is electrically neutral. If the effect
of grounding the electrode w e r e to e l i m i n a t e the u n i f o r m displacement component ~e, then it w o u l d be
possible to d e t e r m i n e the point of zero charge simply
by m e a s u r i n g the potential of the grounded electrode
w i t h respect to a r e f e r e n c e electrode.
J. Ross Macdonald and C. A. Barlow, Jr.: A c t u a l l y one
could not m e a s u r e the ecm potential in this w a y w i t h out additional i n f o r m at i o n about the system: The
condition that the electrode is grounded is not the
same as the condition of zero charge. The charge on
the electrode which produces the ground condition depends on the i m a g i n g situation w h i c h applies, as well
as the adsorbed surface charge density. Specifically,
for single and for dielectric i m ag i n g the condition is
that q = --ql, w h e r e a s for i n f i n i t e- co n d u ct i v e imaging
,y
we have q -- ql at the ground condition. H o w ever, if one k n e w the physical (imaging) situation and
m e a s u r e d ql w i t h the electrode grounded, one mi ght
be able to i n f er from the a v e r a g e potential drop
across the compact l ay er w h a t the ecru potential is.
Richard Payne: I t h i n k it is i m p o r t a n t to point out
that ~ in the G r a h a m e - P a r s o n s analysis is n o t fully
e q u i v a l e n t to ~l/V1 in this t r e a t m e n t as stated by the
author for the following reason. In the G r a h a m e - P a r sons analysis k is obtained f r o m the e x p e r i m e n t a l r e sults by investigating the concentration dependence
of the adsorption e n e r g y at constant charge in the
electrode. ~ is t h e r e f o r e an a v e r a g e v a l u e over a
r an g e of concentrations and amounts adsorbed. In
other words, the imaging conditions in the diffuse
l a y e r are not k e p t constant during the calculation
w h e r e a s the calculations g i v e n h e r e r e f e r to definite
i m ag i n g conditions in the solution.
J. Ross Macdonald: Our A - ~l/V1 is f o r m a l l y e q u i v alent, as stated, to G r a h a m e and Parson's (~i _ ~o)/~u,
w h i c h they set eq u al to ~ on the basis of an a p p r o x imate infinite-imaging treatment. E x p e r i m e n t a l r e sults seem to indicate that the ratio ( ~ - - ~ o ) / ~ de pends only slightly, if at all, on the concentration.
It t h e r e f o r e seems reasonable to co m p ar e our t h e oretical results, w h i ch assume a definite d e g r e e of
dielectric i m ag i n g at the OHP, with e x p e r i m e n t a l l y
d e r i v e d results obtained using various concentrations.
We do not expect m u c h change of ~ w i t h c o n c e n t r a tion, and the change in the a v e r a g e c h a r g e density
in the diffuse l a y e r a p p a r e n t l y does not affect the
ratio w i t h w h i ch w e compare appreciably. The comparison thus appears useful even though t h e r e are
approximations i n v o l v e d w h i c h preclude perfect
a g r e e m e n t b e t w e e n t h e o r y and experiment.
V o l . 113, N o . I 0
DISCRETENESS-OF-CHARGE
Richard Payne: T h e authors h a v e pointed out that
the v a l u e of }, (and h e n c e 41) obtained in the G r a h a m e - P a r s o n s analysis is d e p e n d e n t on an assumed
f o r m of the isotherm w h i c h I think is an i m p o r t a n t
point. Incidentally, I would also like to point out
that ~. also depends on w h e t h e r the anion activity or
the salt activity is used in t h e isotherm.
In this context, it seems to me that the F r u m k i n
isotherm or b e t t e r the F l o r y - H u g g i n s modification
of the F r u m k i n isotherm introduced by Parsons offers
an e x c e l l e n t starting point for the e x p e r i m e n t a l study
of discreteness of charge effects for the following r e a sons. If we w r i t e down the S t e r n f o r m of the isotherm,
8
a___ exp [(41 -5 r
-----1--0
[1]
and compare this w i t h the F r u m k i n isotherm in the
form
1 -- e
=
-
rs
exp
----+f(q)--A0
RT
[2]
it seems reasonable to equate the specific adsorption
potential ~ w i t h the standard free e n e r g y of adsorption at zero c o v e r a g e --AG ~, i.e.
Fr =
In rs
[3]
= 1(q) --A0
[4]
--AG o-
and to w r i t e 41 as
RT
where f(q) is the form of the charge dependence of
the standard free energy of adsorption at zero coverage and A is the lateral interaction coefficient of the
adsorbed anions. I think it is clear from [4] that the
Frumkin isotherm with a constant value of A represents the infinite imaging limit to a close approximation since the adsorption energy is linearly dependent
on the amount adsorbed. Deviation from the linear e
dependence of the la te r a l interactions, e.g., a ea/2
d e p e n d e n c e for single imaging should show up as
v a r i a t i o n in A w h e n the F r u m k i n isotherm is applied.
It is interesting to note that a p p a r e n t v a r i a t i o n of A
is often found for adsorption of anions f r o m solutions
of a single salt.
A f u r t h e r point I w o u l d like to m a k e h e r e is that
the infinite imaging case also leads to linear charge
dependence of --AG ~ p r o v id in g the effect of r e p l a c e m e n t of oriented solvent dipoles is neglected and
p r o v i d i n g the dielectric constant is assumed constant:
4 1 = ~ +"rv
V1
=
m7
-
-
~ +~v
q+
_49
_~ [
4~
. .
q+
.
~2
.
~ +"rv
.ql
]
~q_,,/
ql
Deviations f r o m the linear charge dependence of
--AG ~ can arise (i) f r o m r e p l a c e m e n t of oriented dipoles wh i ch w o u l d contribute a t e r m 4~n~/e to VI
w h e r e n is the n u m b e r of dipoles replaced by each
ion and ~ i s the m e a n n o r m a l c o m p o n e n t of the dipole
m o m e n t and (ii) th r o u g h deviations of 41/V1 f r o m
constancy r es u l t i n g from i m p e r f e c t imaging in the
solution.
J. Ross Macdonald and C. A. Barlow, Jr.: We do not
,y
a gr ee that 41 is g i v e n by - -
V1 only that the u n i -
f o r m field part is g iv e n by this q u a n t i t y in the infiniteconductive i m ag i n g situation. The linearity of 41 w i t h
charge on the electrode, q, is i n d e p e n d e n t of this, h o w ever, and in fact does not depend on the type of i m a g ing present. We only r e f e r to the explicit dependence
of ~1 on q. Since e v i d e n t l y as q changes ql will generally change as well, we do not include all q - d e p e n d -
991
MICROPOTENTIALS
ence in the present discussion. We are in fact r e f e r r i n g
to the l i n ear i t y of 4i w i t h q u n d er h y p o t h e t i c a l c i r cumstances w h e r e the constitution of the compact
l ay er is held fixed and only q is allowed to change. A c cordingly, w e agree w i t h the st at em en t that 41 w i l l
be explicitly linear w i t h q, but disagree with the
reasoning w h i ch led to this conclusion and w i t h the
comments concerning causes for "deviations from the
linear c h a r g e dependence o f - - A G o . '' On the other
hand, the r e m a r k s about "deviations . . . (etc.)" are
interesting and p e r t i n e n t if by "ch ar g e d ep en de nc e "
is m e a n t " d e p e n d e n c e on ql." Note, h o w e v e r , that if
one is concerned w i t h the dependence of 41 on ql, he
must include the contribution f r o m the discrete ions
and their i m a g e s (r
and that such inclusion will
itself destroy the l i n e a r i t y w i t h o u t the additional
help of the two effects cited by Dr. Payne.
Roger Parso~,~: I t h i n k it is fair to say that the g e n eral idea of the discreteness of charge effect is w i d e l y
accepted and is supported by s e v e r a l pieces of e x p e r i m e n t a l evidence. H o w e v e r , it is m u c h m o r e difficult to
d e m o n s t r a t e e x p e r i m e n t a l l y the fine details of the
models proposed, e.g., w h e t h e r i m a g i n g is p ar ti a l or
infinite, w h e t h e r the h e x a g o n a l lattice or the diso r d er ed models p r o v i d e the closest a p p r o x i m a t i o n to
reality. I should emphasize the i m p o r t a n c e of att e m p t i n g to v e r i f y these models by e x p e r i m e n t a l test.
We h a v e t r i ed to e x a m i n e i m a g i n g in t h e diffuse l a y e r
r e c e n t l y 8 and suggest that it is not complete. H o w ever, this w o r k does not test t h e efficiency of i m a g ing in the dielective jump.
In connection w i t h the p r o b l e m of t h e r m a l motion,
is it correct to say that some of t h e difference be t w e e n you and L e v i n e et al. is due to the fact that
you assume a l o w er dielective constant in the i nne r
layer? A v al u e of 6 or 7 seems m o r e reasonable
than 15.
I w o u l d like to point out that the v a r i a b l e ~. obtained by G r a h a m e and Parsons was f u r t h e r analyzed
by P a r r y and Parsons 9 and an i m p r o v e d model suggested.
Finally, I t h i n k it is i n t er est i n g to note that the
elegant m e t h o d of s u m m a t i o n proposed in this paper
has some r e l a t i o n to the i n t u i t i v e w a y in which D a vi d
G r a h a m e discussed the potential drop in the i nne r
l a y e r 10.
J. Ross Macdonald and C. A. Barlow: We believe this
difference of opinion as to t h e p r o p e r v a l u e of the
dielectric constant to use in these circumstances is
the only significant d i s a g r e e m e n t b e t w e e n L e v i n e and
his c o - w o r k e r s and ourselves although t h e r e a r e
small p e r t u r b a t i o n s on the final n u m b e r s for lattice
stability coming from slight differences in stability
criteria, etc. In a r e c e n t p ap er by Bell, Mingins, and
L e v i n e 11 the i n n e r - l a y e r dielectric constant is t a ke n
to be 10. As y o u h a v e pointed out w e h a v e g e n e r a l l y
taken 5 or 6 to be m o r e typical of the i n n e r - l a y e r
dielectric constant, to the e x t e n t that one can define
such an object. This c h o i c e was m o t i v a t e d by our
prior w o r k on differential capacitance in the electrical
double layer. However, in l at er t h eo r et i cal studies on
electrode w o r k function change, we calculate that the
dielectric constant in some systems m a y be c o m p l e t e l y
different f r o m b u l k values insofar as w o r k function
effects are concerned. The significant phrase is "insof ar as w o r k function effects are concerned." It turns
out that t h e use of a single dielectric constant to
characterize dielectric effects in the i n n er l a y e r is
incorrect, since this region is so c o m p l e t e l y different
f r o m t h r e e - d i m e n s i o n a l b u l k matter. W e h a v e r e S E. D u t k i e w i c z
(1966).
a n d R. P a r s o n s ,
J. Eleetroanal. Chem., 11, 100
D5. M. P a r r y a n d R, P a r s o n s , Trans. Faraday Soe., 59, 241 (1963).
"to F i g . 5 i n Z. Elektroehem., 6"$, 264 (195B).
Bell, M i n g i n s , a n d L e v i n e , Trans. Faraday Soc., 62, 949 (1966).
992
October 1966
JOURNAL OF THE ELECTROCHEMICAL SOCIETY
cently estimated the dielectric constant effective in
reducing the lateral interaction b e t w e e n adions and
find that it is no g r e a t e r than 2, and m o r e likely to
be closer to unity. So you see the question of the proper
dielectric constant to use is a r a t h e r confused one.
It seems to us that all previous estimates of lattice
stability in the i n n e r - l a y e r (including our own) h a v e
e m p l o y e d a dielectric constant w h i c h is at least 3
times too large, and possibly m u c h worse than that.
It is completely correct that our i m a g e s u mm at i o n
method stems f r o m that originally used (for ~ -= 1)
by Grahame. We have generalized the approach to
include the I~1 < 1 case. In this connection, we think
it is interesting to note a certain difficulty not f u l l y
appreciated by G r a h a m e w h i c h t r o u b l e d us for a
while: Whereas the s u m m a t i o n t ech n i q u e used here
is absolutely c o n v e r g e n t for all I~l < 1, the series is
only conditionally c o n v e r g e n t for ~ = 1. As it happens, the particular a r r a n g e m e n t of terms e m p l o y e d
by G r a h a m e and ourselves in the ~, = 1 case leads to
the potential appropriate for q = --ql. Since this does
not coincide w i t h the condition q ~ - ~/ql/(~ + ~)
obtaining w h e n both i m ag i n g planes are at zero potential, this accounts for G r a h a m e ' s conclusion that
the infinite set of images and the adions contribute
to the total p.d. across the inner layer, an incorrect
conclusion for the model w i t h conductive i m ag ing at
the OHP.
The Adsorption of Aromatic Sulfonates at a Mercury Electrode
II. Sodium p-Toluenesulfonate--An
Example of Two-Position Adsorption
J. M. Parry1 and R. Parsons
D e p a r S m e n t o~ P h y s i c a l Che~nistry, T h e U n i v e r s i t y , Bristol Er~gland
ABSTRACT
The adsorption of the p - t o l u e n e s u l f o n a t e ion on a m e r c u r y surface f r o m
aqueous solution has been studied by m e a s u r i n g the i n t er f aci al tension and
the capacity of the electrode as a function of concentration. A n i n t e r p r e t a t i o n
of the results is proposed in terms of two orientations of the adsorbed ion.
A simple model of this type of system is proposed, at first neglecting i n t e r action b e t w e e n adsorbed ions other than due to their space-filling properties.
A v e r y a p p r o x i m a t e allowance for interaction is introduced, and it is concluded that consideration of interaction is essential for a f u l l description of
the system.
In the previous paper of this series (1), w e d escribed the adsorption of the benzene m - d i s u l f o n a t e
(BMDS) ion wh ic h m a y be t r e a t e d by methods
closely analogous to those used for simple ions such
as the halides. This is essentially because adsorption
of this ion occurs with the plane of the benzene ring
oriented p a r a l l e l to the plane of the m e r c u r y solution
interface over t h e w h o l e of the e x p e r i m e n t a l l y accessible range. As we m e n t i o n e d previously, the b eh a v i o r of the p - t o l u e n e s u l f o n a t e (PTS) ion is m o r e
complicated and, as we shall show in the present p aper, it cannot be analyzed by using the surface pressure in a simple way. This is due to the existence
of the adsorbed species in m o r e t h a n one orientation,
and we b e l i e v e that this system provides a c l e a r - c u t
e x a m p l e of two position adsorption. This pr o b l em
has been discussed in a q u a l i t a t i v e w a y by D a m a s k i n
et al. (2); here a m o r e q u a n t i t a t i v e description is at tempted.
Experimental
M e a s u r e m e n t s of differential capacity and i n t e r facial tension w e r e carried out as described previously
(1). Sodium p - t o l u e n e s u l f o n a t e was recrystallized
thr ee times f r o m e q u i l i b r i u m water; it crystallizes as
the h e m i h y d r a t e from concentrated solutions. T h e
w a t e r content wa s d e t e r m i n e d by heating and w e i g h ing and also v o l u m e t r i c a l l y after e x c h a n g i n g the Na +
for H + on an ion e x c h a n g e resin.
Results
The concentration dependence of the capacity at
constant t e m p e r a t u r e is shown in Fig. 1. It is ev i d en t
f r o m these data that the b e h a v i o r of P T S is q u a l i t a t i v e l y different f r o m that of BMDS as shown in Fig.
1 of ref. (1). A t the l o w e r concentrations up to 0.1M
the capacity curves for both ions are similar in
showing a m a r k e d peak at a potential of about --0.6v
(SCE). H o w e v e r , as the concentration is increased
above 0.1M this peak is lowered, e v e n t u a l l y being
1 P r e s e n t a d d r e s s : Tyeo L a b o r a t o r i e s , Inc., W a l t h a m , M a s s a c h u setts.
!
l
i
l
C//uFcm-'
-30 I
h
g
e
I
,o
-0;5
-1;o
-1;5
Fig. !. Differential capacitance per unit area of a mercury
electrode in aqueous solutions of sodium p-toluenesulfonate at
30~ plotted as a function of potential with respect to a saturated
calomel electrode: a, 0.0113M; b, 0.0227M; c, 0.0567M; d, 0.113M;
e, 0.227M; f, 0.567M; g, 1.134M; h, 2.268M.
replaced by a m i n i m u m , w h i l e a higher and n a r r o w e r
peak develops at m o r e n e g a t i v e potentials (up to
--1.4v vs. SCE). These results suggest that at low b u l k
concentrations the P T S ion is lying flat on the m e r cury surface like the BMDS ion w h i l e at h i g h e r concentrations r e - o r i e n t a t i o n occurs to allow closer pa c king of the P TS ion.
Th e capacity curves w e r e i n t e g r a t e d n u m e r i c a l l y
using a digital c o m p u t e r (Elliott 803) as described
previously (3). Th e i n t e g r a t i o n constants w e r e obtained f r o m the electrocapillary curves and ar e given
in Table I. Th e potential of zero charge was found
from the electrocapillary c u r v e by e x t r a p o l a t i n g the