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Progress in Surface Science, Vol. 23(1), pp. 3-154, 1986 Printed in the U.S.A. All rights reserved.

0079-6816/86 $0.00 + .50 Copyright © 1987 Pergamon Journals Ltd.

SURFACE ELECTRON SCREENING THEORY AND ITS APPLICATIONS TO METAL-ELECTROLYTE INTERFACES V.J. FELDMAN and M.B. PARTENSKII Ural Timber Technology Institute Siberian Tract, 37, Sverdlovsk 620032, U.S.S.R.

M.M. VOROB'EV Institute of Electrochemistry, Ural Science Centre of AS of the U,S.S,R. S. Kovalevskaya, 20, Sverdlovsk 620219 U.S.S,R. Abstract The modern theory of the static electron response of a metal surface is reviewed. The basic motive of the survey, originated from the analysis of the contradictions between the widely accepted sharp boundary ucdels and experiment, is self-consistency. Applications of i:he various versions of the density functional fomnalism to the surface response calculations are considered. in particular, the screening of the uniform electrostatic field is discussed on the basis of the local (Thomas-Fermi type) and nonlocal statistical models, within the liohn-Sham scheme (in the linear response aoproximatJon and beyond it) and using sum ~.ules. The rrsults of the self-consistent analysis of a number of phenomena at the metal-vacuum interphase (e.g. electron and ion field emission or ionization and polarization of a minute metal particle) are briefly described. The main attention is given to the effect of a metal on the electrical properties ( the bilayer capacity first of all) of the metal-electro]yte interphaseso The results obtained in this field aggravsted the question about the possibility of the negative capacity values, sharply arised in connection with the "Cooper-i~arrison catastrophe". This question and the associated problem of the bilayer instability are discussed in the survey applying the results to the model microscopical calculations and the "gedanken experiments" with the electro-mechanical "catastrophe machir~es".

V.J. Feldman, M.B. Partenskii and M.M. Vorob'ev Contents I.

In~roduc tion

6

2.

Basic Theory

10

A. The key formulas of the density-i'unctional formalism B. Sum rules C. Some double layer characteristics in the one-dimensional approximation 3.

.

~etal S~rface in Electrostatic

I0 17 20

Field

25

A. The requirement of self-consistency and the boundary conditions B. Statistical mlectron theory of the metal surface response C. ~uantum-mecnanical approach D. Some phenomena at a charged metal surface

28 37 47

Effect o f Surface Electron Resoonse on the Equilibrium Electrical Properties of a Metal- Electrolyte Solution Interphase

54

25

A. ~ain results and cortradictions of the traditional mogels of the compact layer B. The self-consistent electron response and associated effects in the theory of the compact layer at the metal - surface-inactive electrolyte solution ~nterphase .

56

61

Effect of Surface Electron Response on the Capacity of a Bilayer at a ~etal - Solid Electrolyte Interphase

83

A. The problem of the abno~i;ally iligh values of the bilayer capacity B. The self-consistent electron response in the theory of the 1~etal - silver ion conductor solid electrolyte in~erphase .

85 87

Critical peculiarities of the Double Layer. The ~qegative Capacity Problem A. The negative capacity values in the double layer theory B. Are negative capacit 5 values possible in nature C. Critical peculiarities of the double layer under the potential control

96

?

96 107

110

Surface Electron Response Theory With Applications Acknowledgments

120

Appendices

121

A. Some useful relations and estimates B. Local statistical models in the case of the nonamenable potentials and negative charges C. The dipole lattice in a capacitor D. Capacitors with a gap relaxing upon chargi.ng E. On the physical interpretation of t h e - ~ term in Eq.(6.15)

122 124 137 138 141

References Abbreviations CL

Compact layer

IBA LR

Infinite B a r r i ~ Approximation ~inear Response

~FA

~ean Field Approximation

~SA EC

A~ean Spherical Approximation Negative 0apacity

RC

Relaxing Capacitor

HPA

Random Phase Approximation

SIES pzc

Surface-Inactive Electrolyte Solution point (potential) of zero charge atomic units

a.u.

121

Other abbrmviations are those of names o f authors as explained in the text.

6

V.J. Feldman, M.B. Partenskii and M.M. Vorob'ev I. Introduction One of the most distinctive

properties

of a metal surface

the strong electron screening of the electrostatic limits metal

the influence

of external

ed in the concept

This phenomenon is reflect-

of the ideal conductor with indefenite material accepted

is that details

in classical

of the field

electrostatics.

the nature of the conductor manifests ning charge is located

exactly

itself,

field drops

drop inside

in which

are invisible

surface of the

there abruptly

to zero

the perfect conductor.

It is clear that the ideal conductor model is justified the cases with the atomic

the actual distance S from the surface

size.

The opposite situation when

more detailed microscopic

consideration.

nature of the conductor becomes induced charge distribution, into the metal, tribution stitute

the potential

potential

of importance.

drop inside

in calculations

determining

means

of the external

field.

to the use of fixed model

on the distribution

density or conductivity)

naturally

roscopic

describing

electrodyn~ics

artificial

is rest-

in the unperturbed

sta-

and the extra potenti-

the various

of a conductor

Self-consisten-

Self-consistency

of electrons

self-consistently),

is in

surface and for that

barriers,

te(even if the induced charge distribution

~The characteristics

that the

the electron distribution,

by the electron distribution.

al are obtained

that con-

of the screened field is

both for the undisturbed

rictions

the

the metal and its con-

It is these effects

its turn determined in the presence

the

It influences

cy must be achieved alternative

requires

In the latter case

The demand of self-consistency

barrier,

exceeding A

of the present survey.

The key consideration self-consistency.

S ~10

only in

the depth of the field penetration

to the contact capacity.

the subject

after

theory ~. Thus tile s c r e e

~t the geometrical

the external

and there is no potential

The matter

attenuation at the surface,

the averaging inherent in the macroscopic ideal conductor,

field, which

charges upon the vol~1~e of the

to the first few atomic lajers.

characteristics,

is

approximations

(for instance,

the electron

appear in the equations the metal surface

because the depth oi' the skin layer can exceed ing length considerably i.

of

of mac-

screening,

the static

screen-

Surface Electron Response Theory With Applications the dielectric

function of the metal.

ted approaches

turn out to be useful in the description

tal surface responce.

In some cases the enumera-

This concerns primarily

the field attenuation into the metal.

But we shall also demonst-

to apply the non-self-consistent

in the problems,

the outer space adjacent

is of importance,

of a me-

the character of

rate that the attempts where

7

models

to a metal surface

may often lead to delusions sometimes

even gre-

ater then those of the ideal conductor model. It ~ust be mentioned

that "self-consistency"

with broadly refered "first-principles". ses a rigorous

(exact)

description

Self-consistency

suppo-

of a model system irrespective

of the roughness of the IHodel itself. functional

is not identical

In terms of the density-

formalism it means that one may choose a functional,

only distantly related to real electrons

(for example,

Fermi model, widely used in calculations

of the electron respon-

se), and a model distribution sembling real cores

(for ex~iple,

a basis of practically electron response). we must determine

of the positive

the Thomas-

charge vaguely re-

the jellium model,

assumed

all the theoretical works devoted

as

to the

But after having chosen these approximations

the exact ground state of the model electron gas

moving in the field of the model external charge.

Then it is pos-

sible to judge the adequacy of the used theoretical model comparing its predictions with the reliable the case of non-self-consistent to understand periment.

the origin of the discrepancy

First-principle

stency, make extra demands

calculations,

(for exa~z~ple, to the pseudopotential

data, while in

it is hardly possible between theory and ex-

in addition to self-consi-

to the exactness

free electrons with the lattice 2 gy functional 3

experimental

calculations

of the model itself

describing the interaction of and (or)

to the electron ener-

). But the exact sense of these requirements

is

not dete~nined in practice and they apply to the desired situation rather than to the real one. Many findings of the self-consistent

theory of the electron res-

ponse of a metal surface have been recognized as classical, instance,

the results of the works where

the screening of the fi-

eld of the external point charges and various applications chemisorption

theory are discussed

for to the

(for a review see Ref. 4). To

refrain from repetitions we mention the corresponding results only in rare necessary cases.

8

V.J. Feldman, M.B. Partenskii and M.M. Vorob'ev Our attention

ble electrical properties

is concentrated layer.

of the free uniformly

ties of the double phases.

layer

The influence

of the double sidere~

discussion

of the metal

There

I. The a c c u m u l a t i o n

problem

dulation

optical

by different

layer

theory

in the theoretical

differential siderable

interest,

In the m a j o r i t y

after

and

far beyond

efi'ects

surface. 5,6 relaxation

picture.

and negative

electrochemical discussed

this model the cores are smeared into lly) with the positive cha~"ge density - nj(x)

eff-

',Ve bear in values

of the

problem°

of' these questions

model 7 was adopted

n+(x)

of the mo-

of charge

the bilayer instability

the discussion

of the works

unifo1~n background)

the electron

of the infinite

capacity

in elec-

used for the

.im~erpretatlon~ " of

of the metal

ects had been included

ding to our opinion,

physics"°

there is a nta~ber of problems

sharp

the p o s s i b i l i t y

of the

obtained

the analysis

which beesu~e extremely mind

t~eo-

in the descrip-

methods

for the correct

data via the properties

the

the bilayer

the solution

is easily

Thus

is con-

of the s e l f - c o n s i s t e n t

for "electrochemical

surfaces

is important

3. In the double

the p r o p e r t i e s

interphases

between

Thus,

density

spectroscopy.

with other

upon

That is why they are widely

of metal

in these contacts the optical

charge

contacts.

and the proper-

electron response layer.

is necessary

2. The higi~ surface investigation

surface

both the

interest.

the importance

electrical

in the dou-

three reasons m a k i n g

of contradictions

of the metal

tion of the double

trochemical

surface

of particular

showed up

effects includes

of metals

electrode

are at least

of this p r o b l e m

ry and experiment

mentioned

chargeo

at the contacts

layer at the metal-electrolyte

in detail.

inclusion

on the charge

This range of problems

Accor-

is of con-

applications°

the "jellituJi" (planar

for the metal title uniform

= ~.~(-x)

lattice°

background

In (je-

(Ioi)

where @(x) is a H e a v i s i d e step-function. The jellium model works well for simple metals 7, and ii'J the calculations of the sui'face electron

response

it is als~

succesfully

used for the noble me-

tals (for an example see l~ef.S), in the description of problems considered in this review, the outer screening space plats a most important

role.

Then

the relative

weight

of the s- and p-electrons in

Surface Electron Response Theory With Applications the density cability portion

of states

criterion

near

the Fermi level ~ay serve

as arL appli--

for the free electron description,

of highly-energetic

electrons

increases

because

the

with the distance

from the metal 9. For further discussion

it is inlportant to consider

the position

of the jellimn

edge relative

to the centers

Let us ex~z~ine

the "genetic"

way of the jellium background

ruction, charges

that is, firstly,

along the atomic planes,

yer I at Fig.l), of each layer dently, from

the sequantal

and then the charge

the edge of the Obtained

the first lattice

distance

plane

between nearest

ions. const-

averag.ing o£ the ion

parallel

till its contact with

of the surface

to the surface

(area)

conserving

the neighbouring

(the la-

expansion

layers.

jelly lies at the distance

of the ionic

lattice

planes

array,

where

d

in the direction

Evid7/2

is the normal

to a surface.

I

FT.

Fig.1.

The "genetic"

foz,m positively arc given

,~

w a y of the construction

charged b a c k g r o u n d

In the calculations

of the surface

discreetness

the help of one-dimensienal tentJ.al correction

( the jellium).

umi-

Explanations

In the description

used.

are presented

SI and atomic units given in Appendix

A.

barriers

with

response

the effects

r:~odels; the ex~u.~ples are the pseudopoaveraging

over the

of the type suggested

oi" ~he electron

(in accordance

nal formalism)are

electron

are usuall 2 taken into account with

to the jelliu.,a potential,

ato~lic layers 7, or model

results

o£ the plana~

in the text.

of the lattice

units

i

screening

the tradition

in the S1 units.

theor 3 the atolr~ic

of the density

But in electrochelnical

functio-

applications

The relations

for some of tile re]eavant

in Ref.10

between

quantaties

the the

are also

10

V.J. Feldman, M.B. Partenskii and M.M. Vorob'ev

2. A.

Basic Theor~

The key formulas of the density-functional formalism

(i~ The variational principle and the condition o f the electron gas equilibrium.

The density functional formalism is based on

the fundamental Hohenberg-hohn theorem 11~o The theorem leads to the variational principle for an energy functional

E In]

of a

system of electrons in its ground state, moving in a static field of external charges (

n(r) is the electron density). Because of

the long range of the Coulomb interaction it is often convinient 11 to extract classical electrostatic energy

Ees[n]

from the ener-

gy functional : S[n] = Here

G[n]

Ees[n ] +

is an unique functional of the electron density, which

includes all the contributions to one

(2.1)

Gin]

E[~

except the electrostatic

( id est kinetic and so called exchange-correlation contri-

butions). Its universality

( independence from an external fi-

eld) is guaranteed by the Hohenberg-~ohn theorem. Let us introduce potential ~(E)

of an electron. It distinguishes

from the electrostatic potential by the sign only (in atomic units) and satisfies the Poisson equation, which we shall write as follows:

(2.2)

where pm(#)=~j(~)-~(r)

is the metal charge density

(for the

sake of simplicity we restrict ourselves by the jellium model), pi(9) and ~ ( 9 ) are the free (ionic) and the bound external charge distributions correspondingly. The ground state energy ~ n~nber of electrons ]n(~) d~

of the system with the fixed total

= ~

(2.3)

~The modern discussion of the Hohenberg-hohn theorem and its extensions is contained in Refs. 3,12.

11

Surface Electron Response Theory With Applications coincides with the low boundar,y

infE[n]

of the energy f u n c t i o -

nal for the set of trial functions n(r) satisfying Eq.(2.3),

and

the corresponding electron density is the ground state density of the system. Providing the functional E In] obey certain restrictions (convex,etc.) infE[n] = rain E[n I , and we arrive to the traditional variational problem

E]---

(2.4)

o

together with usually omitted stability conditions. Here ff Lagrange multiplier; chemical potential

is a

for a large N it is equal 7 to the electro-

~i/~N.

The substitution of Eq.(2.1) into

(2.4) gives the Euler-Lagrange

2G

+~(~)

=/q"

(2.5)

(2.5)

gives in principle a possibili-

ty to an exact quantum-mechanical

description of the system in its

The

equilibrium equation

J!,q.

equation

ground state via macroscopic quantaties such as density. It is useful to strengthen the outlined analogy with the classical description introducing a new characteristic quantity, n ~ e l y electron pressure pin(r)]. Bearing this idea in mind we may try to transform Eq.(2.5)

to the equation vp

= -nV~

(2.6)

which has a form of the equilibrium condition for a liquid or a gas placed in the external field. Here the electron pressure p(r) is determined by the equation ~p

~G

n(~) V(T~)

~_

(2.7)

It is easy to show that Eqs.(2.6) and (2.7) are correct at least in two cases: (a) if n(r) depends only upon one coordinate and (b) if Gin] is given in the Local Density Approximation in the form

that is

Gin] = J~'o(n)d~

(2.8)

~o(n)

(2.9)

where ~

f(n)n

12


is a f u n c t i o n is

(not ~ f u n c t i o n a l )

the n o n - e l e c t r o s t a t i c

Density

Approximation

energy

of' the e l e c t r o n per one electron.

the p r e s u r e p(n)

density,

has

f(n)

In the L o c a l

a particularly

simple

form

: n 2 ~f (2.10)

In

the a b o v e - p o i n t e d

tor f i e l d ponds

n(r) w x °

and

(2.22), we

for all of the n o n e l e c t r o s t a t i c

to the surface energy

of the jellium:

Surface Electron Response Theory With Applications

w k = 3 PI

55/3

Z~RI

( ~I

R2

wx = 3 P2 ~4/3 ( T

W

C

15

xo

+ 0"2"~2- + T ) xo

~° + T ) + O. 2 5/~2

R3 nR 4 + x ° ) = 3 P3 ~ ( ~ T - + ~ 3R 5

ww = P4 ~'[ ~I (t3o - 31nto - I )

(2.25)

+ ~ 2 t3]

Here PI = 0"3~3S~2)2/3; P4 = 1/72;

P2 = - 0 " 7 5 ( 3 ~ ) I / 3 ;

a R i = Ri(t o) - Ri(1);

P3 = -0.056

t o = (I + A) I/3

and R I = 0.2t 5 + 0.5t 2 - 0.51nlt2+t+11 R 2 = 0.25t 4 + t - 0.51nlt2+t+11

•2t+I + 3-1/2arct3~-/~

- 3-I/2arc t

R3 = t3/3 + 0.5pt 2 + p2 t _ p6 lnl1+pl p3+I _

p(p+1)

-

_

2t+I

P(P-1)inlt2÷t+11

_

2(p3+I)

2t+I

31/2(p3+1)arc~~ R 4 = t3/3 - 0.5pt 2 + p2t _ p31nlp+tl R 5 = p +I ,

p = 0 . 0 7 9 / E I/3

In a number of works the higher order corrections of gradient expansion (2.13) have been taken into account in the calculations of the electronic characteristics of a metal surface, but their expedience has not been proved yat.

16

V.J. Feldman, M.B, Partenskii and M.M. Vorob'ev

(i i i) The quantum-mechanical

(extended H a r t r e e ) a p p r o a c h .

kohn and Sham 20 suggested to extract the kinetic energy of n o n i n t e r a c t i n g electrons having the same density teracting electron gas, from the nonelectrostatic

tion functional.

functional:

(2.26)

is by definition

The decision of ~qs.

lent to the selfconsistent

s

n(r) as in-

G In] = ~-~Ln] + ~xc in] Here the functional Exc[nJ

T {n]

the exchange-correla-

(2.2) and (2.4) is equiva-

solution of the system of the one-

particle Schr~dinger equations (2.27) where =

2

i=1

(2.28)

and the effective potential v(~)

~Exc

:~(~)

+ - -

n (2.29) is given by ~ohn in Ref.3 .

The deduction o f gqs.(2.26)-(2.29)

The K o h n - S h a m scheme is more exact than the previously described extended Thomas-Fermi method,

since the kinetic functional

Ts[n ]

is calculated exactly for the chosen approximation of [,.xc ' ' hi, without using the gradient expansion. In the case of the semiinfinite jellium metal with the flat surface foll

the eigenfunctions

of Eo.(2.27)

can be factorized 7 as

OVCS

exp (ikl,r ) ~ k(X) -

~, , k (~)

where

?k(x)

satisfies

(2.30)

the equation

(2.31) and boundary conditions

~ k (x) = {sin(!,,_-:< + ~ (I~)), x 0

(the latter is correct only in acting field). functions

,

~

-~

x -~ +co

(2.32)

~he absence of the external extr-

The electron density is obtained from the eigen-

n(x)

_

~L~' rk2 j'g2 ] dk , , F 1

O

_ k 2 )i ~ k ( X ) l

2

(2.33)

Surface Electron Response Theory With Applications B.

17

Sum ~ules The self-consistent analysis of a surface electron structure

requires the cumbersome computer calculations even in the most simple one-dimensional models. Therefore exact results obtained from some general principles avoiding calculations are of interest. Among them one finds the so called "sum rules" for the potential. Bearing in mind that sum rules play an important role in the surface theory and, in particularly, in the surface response theory, while their use is sometimes accompanied by uncertanties, we shall discuss them in sufficient detail. The first of sum rules was derived by Theophilou 21 from KohnSham equations, and after that for the second time by Budd and Vanninlennus 22 with the help of the Pauli-Hellman-Feynman theorem. It is as follows: ~ [ C~(O) - % ] where

~

and

= p([)

(2.34)

p(H) are the values of the potential and the elec-

tron pressure in the bulk of the macroscopic metal. This equation permits to determine the "internal" barrier contribution for a metal-vacuum interface without the calculation of the function ~ ( x ) . Later on analogous sum rules were obtained for more complex systems23-31o The works where sum rules for a bimetal contact 26 and for a metal sphere 27 were derived from the equilibrium condition (2.6) should be pointed out. Their derivation is simpler and more obvious than the use of the Pauli-Hellman-Feynman theorem. We shall profit by the technique 26-27 to obtain sum rules for layered surface systems 32 (see also the review33). So, let us consider layered systems with a geometry of a plane, a cylinder and a sphere (in all of these cases equation (2.6) remains correct), in which the layer number i consists from the uniform background of the d e n s i t y ~ , mittivity 6~.

I ~ the case ~ =

a vacuum gap, in the case jellium, in the case ~i=O,

having the dielectric per-

1,Hi=O the layer i coincides with

6~=I, ~i~0 we get uniform unpolarized ~{>1

we arrive to a dielectric con-

tinuum. Following I0 we shall take into account the contribution to the potential due to the lattice discreetness in the one-dimensional model barrier ~(x)° Then the potential ~(x) in Eq.(2.6) should be substituted with


18

v(x) where and

~(x) u(x)

Eq.(2.6)

:

~(x)

u(x)

+

(2.35)

is detezunined, as before, by Poisson equation (2.2), is the given one-dimensional

barrier. Integrating the

transformed in this way over layer i and using Eq.(2.2),

one obtaines the following reccurent expression

p(xi)

- p(xi-1)

= s~j

(xi)

-

(xi-1

"

where

)] %[T(x (2.36)

x{

×i-I

D(x) is dielectric displacement.

×i-I

The parameter i

equals

0, I and 2 for the plane, cylinder and sphere geometry correspondingly. Taking into account the continuity of D(x) and p(x) and summarizing Eq.(2.36) over all layers, we obtain a stnn rule for any layered system of the considered a metal-adsorbate-dielectric

type 32'53. For example, for

system in an external electrostatic

field D o we have

(2.37) g-1

D~(a)

I

D2

Here the charge density of an adsorbate is described in the Lang model 34 ]]a d ( x ) : %

~ (x) I 8 (a-x)

(2°38)

and, following I0, it is assumed that

u(x) It is appropriate

= cm @ ( l - x )

(2.39)

to mention here that instead of Eq.(2.39)

any

other one-dimensional barrier can be considered. For instance, the linear barrier ~=~' x ~(-x) results i n - ~ } e D d , instead of the last te~n in Eq.(2.37), Harrison potential 35 V0 ~ ( X _ X l ) ,

while the~ one-dimensional

which is sometimes used for the

description of the interaction between metal electrons and inner electron shells of a d a t o m ~ g i v e s

the contribution V ° E(Xl). In

Surface E1ectron Response Theory With Appl i cat ions

19

different particular cases Eq.(2.37) gives sum rules, which have been previously derived by various technique in works] 0'21-23'27-30 For the future applications it is necessary to discuss carefully

account of an external field in sum rules. In order to di-

sengage ourselves from extraneous details let us write sum rule (2.37) for a metal-vacuum interface in an external electrostatic field F 0 :

~ [.C~(o) - ~

= pC~.) + 8 ~ F2

(2.40)

This sum rule was firstly obtained in Ref. 21.

It

is important

to mention that strictly speaking an electron system has no stationary ground state for orem

Fox~O.

Therefore, Hohenberg-Kohn the-

(because the ground state does not exist)

and the equili-

briunl condition (2.37), as well as sum rules (.2.3.7),(2.40) could not be justified in this case. That is why sum rule (2.40) is supplemented 21 by the condition Fox>O. For Fox0,

where

(see the previous subsection) the former are defined.

metals with a great enough background density

For

( n ~ 8n ° ) the ag-

reement is even quantatively good (see Fig.2~and Table 1 ). In work 73,

the field p e n e t r a t i o n into the metal was cal-


Xe,l

35

\

LO

-0.5

0

~0,

40,

60Z,~s~

2.0 @-(( ~P

1

4.0

0 0

o+ o

I

-OA

0

~

I

I

o..i-

I

0.4 0.6 0.8 ~,~'rruP-

Figo2. The centroid of the charge distribution

imduced by an

external field F 0 = # 3 ~ . a. Local statistical models. I: Na ( r s = 3.99 ); 2: C u ( r s = 2 . 6 3 ) ; 3 , 4 : A 1 (rs=2.07). I-3: TFD model, 4: TF model. For A1 t h e r e sults 72 of the variational calculation adopting the trial function (2.24) with three independent parameters are represented for comparison ( the dashed line). b. Nonlocal statistical and quantum - mechanical calculations for Au ( rs=3): statistical caculation of Xim ( ~ ) 7 3 ( curvel); semi-self-consistent calculation 68 (c~rve 2), statistical calculation 72 with trial f~nction (2.2@) ( o ), quantum- mechanical calculation 92 (+)of x e ( ~ ) .

36


culated using the oscillating

trial functions

f 1 - A exp(~x)cos(rx+ ~)

n(x)

l

Beside

the normalization

n'(x)

and

Bexp(-/~) condition

and the ~equirements

n(x) should be continuous

from sum rule 23 was introduced,

that

one extra demand following

so that only two parameters

re-

main free. It should be mentioned with a regret that a mistake was made in this procedure. As it is repeated in some other works, we shall discuss it in detail. Let us write sm~ rule (2.40) using Eq.(2.45):

~,. [ CID(O) - ~:;~.]

= p(7,_) + 23"C ,~.~2

With the help of the Poisson Eq.(3.8)

equation

(..3.8)

(2.2) we may rewrite

as follows:

Z 2 - n ( x ) ] .xd~ - ~'(~)n + 2 ~ n-

- 4 ~ l : (nxj ) , , Denoting

SnE(x)=n (x)-no(X)

we get 23 from Eq.(3.9)

>-2

o

~nz(~)~a~ = Using Eq.(3.10)

(3.9)

2~

(3.1v)

the expression

may be t r a n s f o r m e d

Xe(~)

(2.47)

for the charge centroid

x

e

to I

=--~

~

~n~(x)xdx

(3.11a)

O

Analogously,

using Eq.(2.51) Xim(~ ) =

In Ref.

73

Eq.(3.9)

~

we have d

S ~n~(x)xdx

(3.11b)

was used with a wrong sign for

The same error was made in the calculation this is the reason why (see Fig.2B)

Xim(~)

of

2~gT_.2/n

Xim(~).

Probably,

in Ref. 73

increa-

ses so slowly with the deepening into the anodic range. The correct use of sum rule (3.8) leads to asymptotically linear deepen-

Surface Electron Response Theory With Applications ing of ----I -n

Xe(~)and

xi~into

a metal with slopes

37

-(2n) -I and

correspondingly. This is confir~ned by calculations within lo-

cal statistical models, where the exact variational problem is solved and sum rule is kept automatically (see subsection 3.B.(i)). The same ~esult follows from the quantum-mechanical consideration (see next section 3.C.) and from nonlocal statistical calculations • In the variational approach the existence of the stationary ground state of an electron system is assumed. But it is clear that for ~ < 0

this assumption is only an approximation (see the text

following Eq.(2.40)

), which validity depends on the value of the

negative charge. The strong electron tunneling at large enough I~I

leads to the failure of this approximation. In the variatio-

nal calculations this failure manifests i t s e l ~ n cal "jump"

the catastrophi-

of the induced charge centroid away from a metal,

which happens (with the ~

decreasing) the later, the greater is

. For instance 72, for A1 (rs=2) the curve

Xe(}-,) sharply be-

comes very steep (approximately vertical) when >-.' approaches O. I C/m 2. In the case of finite systems the restrictions on the range of the permissible charges~, where the ground state exists, are not so severe . The matter is that the field disappears far away from the surface of these systems. In the interesting (for us) case of the metal-electrolyte systems there are some factors, which suppress the electron tunneling from a metal

(see Section 4), thus

also expanding the range of the applicability of the variational approach. C.

quantum-mechanical approach.

(i) Linear Response Approximation.

The majority of the quantum-

mechanical calculations of a metal surface electron response are based on Linear Response Approximation (LRA). Therefore before we shall go to nonlinear calculations it is relevant to discuss briefly foundations of LRA and those its results which demonstrate For example, for a negatively charged molecule the solution of the variational problem within the extended TF-model with the first gradient correction in the Weizsacker form exists 58 for some charges O > q > -

2 z , where z is the total charge of the nucleus (however, the exact value of the lower bound of the permissible charges is not established yet).

38


the important role of self-consistency. Let us transform the total effective potential V = V

0

+

into the for~

AV

(3.12)

where

~V = Here

Z~v +

~v

~

(r)

#xc (n) n

j Ir-r'l

z~n +

dr'

is an external p e r t u r b i n g potential.

The charge-

induced correction to the electron density may be expressed as

n(r)

=~

~o(~,~,) ~V(r') d~'

=

(3.~4)

2 ~2((~,~') av(~')d~' where

X.(r,r')

and

X_°(f,f ')

"non-self-consistent"

response

(3.15)

are the "self-consistent" functions correspondingly.

and The

difference

between these equations is that after substitution of

Eq.(3.13)~

Eq.(3.14)

~n(r) tain

transforms into an integral equation for

while Eq.(3.15) z~n(r)

gives in principle

is known, so that all dificulties nation of ~ ( r , r ' ) . To express ntial

X- via

~ ~(~')=

~(~-~')

~

~Q(,~,~')=2C°(~,~')+~

= ~

+

~(r),

are transfered

if

to ob2C(r,r')

to the determi-

o , we shall consider a perturbing p o t e , where

~(r)

l t i p l y i n g both sides of Eq.(3.13) by over ~' and using Eqs.(3.14)-(3.15)

V(rl'r2)

a possibility

by direct i n t e g r a t i o n for any

is the Dirac function, i~u 26 (r'r') Z integrating one gets 74'Y5

arldr22C_°{r,~2)V(~l,~2)2(~(~2,~

.~

n "

, ~(~1_~2 )

')

(3o16)

(3.17)

In the important for us case of a p e r t u r b a t i o n conserving the one-dimensional character of problem, simplified according to the rule

Eqs.(3.16)-(3.17)

can be


r , r , , r l , r 2 --~-x,x',x1,x 2 ; !r1-r2!-1 --~--2~rrlxl-x2! The function

39

(3.18)

2~°(~,r2 ) is easily expressed through eigenfunc-

tions of the unperturbed Hamiltonian.

To obtain this expression

we shall take the linear correction to eigenfunction in the form of

' o

where

GE(r,r')

(_ ~ v 2 +

vo

is a Green function:

(9) _ s ) c s ( ~ , 9 , ) = E ( 9 _ ~ , )

S u b s t i t u t i n g Eq.(3.19) tion, linear in ~ V

E

_

and extracting a contribu-

(~)0~(~,~,)~ o (~,; + c.c. E

d~'~

o

(3.21)

Z

Eq.(3.21) with Eq.(3.14)

~o(~,~,) = Using t h e

into Eq.(2.28)

(3.20)

, we get

~n(~) -- -# d~'g ~ o Comparing

(3.19)

(~)GE(~,~,)~

we find

~ (~,)

+

(3.22)

CoO

expression

o, (.9)~ ;s,(9 o ,)

%(~,~,) =~-~~

(3.23)

E'

E - E'

for Green function, we finally obtain 2

X°(~,9')=

dg'

o

o

Z

(~,)

,. E

-

c.c.

E'

In the one-dimensional case it may be more convinient another representation for the Green function,

to use

expressing it

via the so-called "right" and "left" solutions of the S c h r ~ d i n -

40


ger equation, whJICh excludes ferent energies:

the suI~nnation over states with dif-

e(x-x,>

~k(X) O(x'-x)

+

o

Gk(X,X') I o ~,~k(X>) Here

x~ (x) --o ¢~k(X0:

curve 2.

46


In the case rs=3 considered the potential

barrier

ectrons.

92

, the distance

and the edge of the background

to be 4.8 A . A c c o r d i n g influence

in Ref.

to the authors,

of an electrolyte

in m a t a l - e l e c t r o l y ~

strong electrostatic

field created

by a charged

lized within

of the atomic

size,

cause

a space

of the strong interfacial

fore, the electron cathodic region,

emission

and to restrict

ground

state

Equations

dic charges.

For

similar

to Eq.(2.31)

Therefore

with

strong

near ~- 1

the authors

It seems

effects field

charge

restricts level

barrier

possible

ate

of electrons calculations

catho-

=~1~-0.08 leaves

a me-

~

bar-

by the charges

-0.08

C/m 2 is a

of the influence

3b).

charges

at negative

and the electron

conclusions

of the variational

charges

The "vacuum electrons

of elec-

statistical

are stronger

away from

"inhabit"

the

On the the vacu-

in vacuum more

agreement with

(see Section

for positive for

them with

gently.

the results

calculations 92 showed

The results 92 of the X e ( ~ ) calculations compared

the inver-

steeper.

calculations 66-71

the q u a n t u m - m e c h a n i c a l

2b . The authors

move

becomes

decreases

demonstrated

with

tail" is of most

extra electrons

density

are in a qualitative

oscillations ones.

are clearly

distribution

and an electron profile

contrary,

ted at Fig.

the infinite

well near the infinite

description

(Fig.

um region

the Friedell for negative

a number

in the response

At positive

all this

of a

of negative

electrons.

sion of external region,

range

the

out that a top of the effective

restricted

in the screening

sensitivity.

by a problem

that the strong r e s t r i c t i o n ~

on metal

The nonlinear

with

to exclude

solved 92 s e l f - c o n s i s t e n t l y

the Fermi

in the potential

of a too simplified

by changes

were

The used model

coincides ~

themselves

potential,

rs=3 it turned

"hump"

tal and is located

Beside

even for rather

even in the "dangerous"

spectrum.

C/m 2. At charges

These

There-

well

potential

"tail"

4).

be-

the potential

for a discrete

trolyte

is loca-

Gies and Cerharts 92 used

to the effective

result

can be neglected

a

a surface,

(see Section

the el-

of the interracial

charges.

~>)-~1"

contacts

surface

adjacent

screening

on metal

the structure

stationary

rier.

is assumed

Not d e t a i l i n g

electron

added

tunneling

charging.

between

this barrier imitates

in contact with a metal

As it is well.known,

a

rs=3

charges

3.B). that

than

are represen-

the results 73 and


47

discovered strong discrepancies for large ~ ~. As it was previously mentioned, because of the incorrect use of the su~J rule the results 73 are erroneous just for large ~-.. For the aim of comparison we display also the dependence

X e ( ~ ) obtained 72 by the

variational procedure using three-parametric

trial function

(2.24)° The agreement with the results 92 is good for all considered ~ , while the agreement with the semi-self-consistent ations due to Theophilou and ~odinos 88 is worse°

calcul-

The "spread" ~ of a charge-induced electron density profile, defined 92 as the full width at half maximum of the induced charge density, weakly depends on ~_~, varying from ~1.5 ~

for rs=2 till

~2.5 A for rs=5. This conclusion is important for a discussion of optical measurements (see, for example, Refs. 6,93). Gies and Gerhardt have also shown that the induced electron density profile differs substantially from that obtained by a rigid shift of the neutral profile by s=- ~'/~. The same follows from the re~ sults of the statistical calculations (Section 3.B.). D.

Some phenomena a t a charged metal surface

Now we shall discuss a number of experimental situations, where microscopic details of the electron density distribution at a homogeneously charged metal surface and the nonlinearity a metal surface response are of importance. It is well known that within a broad range of the field strength F 0 the field emission current j obeys the Fowler-~ordheim law (see, for example, Ref. 94): lg(j/F~) = c - d/F 0 where c and d are independent from F 0. But in the case of strong fields (e.g. F 0 ~ 5 109 V/m for tungsten) j deviates from the Fowler-Nordheim lawtowards lower values. ~Gies and Gerhardts 92 illegaly compared their dependences X e ( ~ ) with the results 73 for x. (not xA). However, it can be shown, i~I ~ 73 that the reconstructed (from results ) dependence X e ( ~ ) also substantially branches off the curve 92 at large ~- . In particulars6, Xe ( ~ ) 7 3 should increase in the asymptotic region while X e ( 7 ) 9 2 decreases as - ~ / 2 ~ .

48


We shall discuss the attempts to interprete this phenomenon from the point of view of the nonlinear surface electron response. Some of these attempts were based on the extended image force model, in which the effective potential (refered to an electrochemical p o t e n t i a l ~

(x)

T

) well outside the surface is as follows:

=WI~ -

~4(x - Xim) I - I

_ FoCX_Xe)

Then the nonlinear electron response results in the following contribution to the barrier height (work function)

~@

= FO(Xe-Xim)

which should b e summarized together with well-known image correction

-F01/2

If one supposes

xim=Xe(F0=O)=const,

field-induced displacement of X e ( ~ ) to the increasing of ~ fields (when F0

xe

force then the

away from a metal leads 7

and decreasing of

j ~. But in strong

depends on F 0) the displacement of

xim

with

should be also taken into account. To estimate this effect we

use expression (2.52). As follows from Eq.(2.52)

that

d Xe/d~

~ 0

Xim~ xe

(see Section 3) then it

for ~-~ ~ 0 , id est the

image plane moves away from a surface faster than the centroid of the induced charge. The corresponding contribution to the work function $ @

=-F02 I d Xe/d F0t

is negative. Thus, the more con-

sistent account of the nonlinear static electron response decreases the barrier height compared with that of the classical image force model and increases the extra tunneling current with increasing.

F0

This tendency is opposite to the experimental one.

However, it can not be concluded that it is impossible in principle to explain the experiment by the nonlinear electron response effects. For example, it seems necessary to go beyond the image force model for the surface barrier and to solve self-consistently the stationary (not static) quantum-mechanical problem of field emission.

The semi-self-consistent calculations 88 results in the

deviations from Fowler-Nordheim law towards the needed side but too small in value. But the nonlinear effect in

X e ( ~ ) is also

The discussion of these effects within a non-self-consistent

picture of the surface response see, for exmrlple, in works by I~Ks 97 and Sidyakin 46.


49

substantially lower in this calculation in comparison with the self-consistent results 92 (Fig. 2b). Therefore, it is still not clear whether self-consistency alters the dependence jCF O) towards the needed side increasing the deviations from the Fowler~ordheim law, or it will result in the opposite tendency similar to the image force model. Beside this, dynamical aspects of a surface screening may be of importance. Dynamical effects result from the inertia of a metal plasma response upon the field of a moving (emitted) charge. It was shown 95 within the sharp boundary approximation that these effects removes the image plane towards a metal with F 0 increasing~ uprising the barrier height. ~ay this effect compensate the barrier decreasing contribution obtained previously in the "static limit"? ~ h e answer is not found yet. If the dynamic effects are substantial, we must take them into account while constructing the effective potential within the Kohn-Sham approach. In addition, the space charge effects also can not be ignored~ 4'96 The nonlinearity of the metal electron screening may play even a greater role in the field ion emission, the field evaporation and the field ionization of surface atoms, because the fields typical for these experiments are of an o r d e ~ a g n i t u d e

greater than

those of the field electron emission. But in these cases it turned out also impossible to extract the corresponding contributions in the pure form. Beside the necessity of taking into account the real electmon structure of an emitted ion, its polarizability and the kinetics of the charge transfer during the act of emission 97'98, the situation is complicated also because of field induced lattice relaxation 90'91. To describe the latter effect we must go beyond the jelli~n model and simultaneously achieve self-consistency for both electronic and ionic variables. The corresponding theory is in a conceiving state yet. The effect of the "field penetration into a metal" upon the capacity of a thin dielectric g a ~ b e t w e e n ~Of course, the mentioned effect

two metal conductors was

is considerable only if the

width of a dielectric gap is comparable with X e N ~ . clear that

it

has

Thus it is

no relation with the size effect in the ca-

pacity of thin dielectric gaps 102 - 103 ~

wide, investigated in

typical experiments. I00 T~e origin of the size effect obtained in these experiments is still vague. 101,102

50


first discussed 51'52'99 within the non-self-consistent models of the Rice's type.

~he sign independent penetration of an elect-

rostatic field into a metal resulted 51'51'99 in the increasing o~ the effective gap in comparison with its geometrical value. In Ref. 39

the Xe(O) value self-consistently obtained for a me-

tal -vacuum interface have been accepted for the discussion of the capacity of thin gaps. Positive Xe(O) values lead to the decreasing of the effective gap. In Ref. 33

the dependence of the

capacity on the magnitude and sign of the applied voltage have been analysed. In all these works the influence of the dielectric on the dependences X e ( ~ )

had not been taken into account.

The simplest way

to consider this effect is to describle the dielectric by the local dielectric permittivity ~ , In the density-functional description of the metal-dielectric contacts this model was introduced by Hyrabayashi. I02 H o w e v e ~ i n the density functional E~n] I02 the polarisation of the dielectric had been taken into account only in the classical electrostatic contribution, id est via introducing ~ in the Poisson equation. The result is that the electron "tail" stretches outwards a metal! 02-I04 But this approximation is adequate only in the TF model (the polarization effect within the TF model has been calculated in Ref.

104 ) or in the Hartree

model within the quantum-mechanical approach, which are rather crude. If the many-particle effects are included in the theoretical picture,

then the renormalization of the Coulomb interact-

ion should be taken into account in the exchange-correlation energy as well. 105-I06 It has been shown by variational calculations, 107 that the resulting changes in the surface barrier are comparable with those, obtained in Refs.

102-104. This conclusion I07 is of

importance for the theory of the metal-electrolyte interphases, since the Hyrabayashi model I02 is widely used nowa,~ays for the description of the "background permittivity" of these contacts (see Sections 4 and 5). A comparatively new and perspective field of application of the surface electron screening theory is the self-consistent calculations of the widely and for a long time investigated electron properties of a metal microstructures,

first of all their ioniza-

tion potential and electron affinity. The ionization (electron) potential I

and the electron affinity S

are defined as the modu-

lus of the differences between energies of a neutral particle and a particle with subtracted (I) or added (S)

electron. For extre-

Surface Electron Response Theory With Applications mely minute electron

particles

leads

the subtraction

to high charge densities,

the electron response

are of importance

The strong I and S dependences rticles

have

reasons:

or by the contribution

rosphere,

so n o n l i n e a r

classical

of the self-energy

in

electrostatics of image

into I and S, except

the self-energy

by

forces

of an extra charge.

are the same for a macrometal

and n e g l e c t i n g

effects

in this case°

by the size dependence

suming all of the contributions

of even one

on the size of a metal m i c r o p a -

been explained within

two alternative

image forces,

(or addition)

51

As-

that of the

and for a metal mic-

of the extra charge

ari-

sing due to subtruction or addition of an electron, we obtain the "3/8 law ''110, which for ionization potential, for ex~aple, has a form

I (R) where

WF

is the work

dius of a metal ergies

: WF + 3/(8R) function

sphere°

of a charged

,

of a macrometal

Assuming

~id neutral

that the difference particles

self-energy of the extra charge, ssion due to Kubo 111

and

reduces

we obtain

I (R) = WF + I/(2R)

R

is a ra-

between

only

en-

to the

the electrostatic

expre-

.

The experimental curves 112 lie ~ between the size dependence due to kubo 111 and that of Smith 110 , and do not provide a criterion to choose

one of them.

ary to clarify self-consistent in the

~We m e n t i o n asured

Thus microscopic

the origin

of the considered

jellium model

size dependence.

size dependences

(see also the review I09 and references probably,

I(R),

therein).

from island

me-

of a charge

with a substrate and, perhaos, be taken into account.

films

This contradic-

fro,~ a fact that in the d i s c u s s i o n

latter data the relaxation

The

of a metal vapour 112, are in disagree-

ment with the data 111 on electron photoemission tion arises,

are necess-

calculations 114-118 have resulted

that the ahnost monotonous

by p h o t o i o n i z a t i o n

calculations

on a particle

of the

in contact

with other microparticles,

should

52


dependences I(R)

:

WF + (2R) -1

The m a i n p o r t i o n i~ -I

of the numerical

in accordance

I(R)

( 2 R ) -1

+ O(R - 2 )

in the chief

term

and S(R) 114-118 is deterhypothesis

by the self-

R

a small metal

have been also discussed

in capacity

has rather

and

However,

the a n

conceptial

than

significance.

Among other

equilibrium

electron p r o p e r t i e s

ticles we w i s h

to m e n t i o n

dipole moment.

The inherent

sphere,

because

arisen

stributions

only

the centroids

able degree predicted

an anomalous the response

field ,as large consistent

of the electron in Ref.

by Gor'kov

increasing

120

as two-three

. An inte-

in consider-

and E l i a s h b e r g 121, who

of the susceptibility,

of the system upon external orders

c a l c u l a t i o n 122 gives

and ion di-

to a nonspherical

of metal m i c r o p a r t i c l e s

had been stimulated

and own

of small metal hemi-

with each other due

have been calculated

to the p o l a r i z a b i l i t y

of metal micropar-

their p o l a r i z a b i l i t y

dipole moment

do not coincide

form of a particle,

terizing

of the capacity~of

value

R(R) have been calculated.

of the size effect

practical

rest

coefficient

with Kubo's

the deviations

/rom its classical

the size dependencees alysis

: WF -

of the extra charge.

In works 70'118 sphere

S(R)

of the size dependences

mined 78'118 energy

+ O(R-2);

charac-

(not total 122)

of magnitude.

The non-self-

the p o l a r i z a b i l i t y

smaller

than the classical value ~ d = R 3 . For small external fields s e l f - c o n s i s t e n t calculations results 70'118'119 in the value °~- = ~sff~16,117which is a little shown / ' charge

that

densities

microsphere

on a particle

behaves

bit greater

Reff=R(~=O)=R+Xe(O

itself

than

R 3 . It can be

) , so that for small

and in a w e a k

as a classical

the

external

particle

field a

with

the radi-

us Ref f. In the K o h n - S h a m problem

results

monotonous rse,

in c~early

size dependence

a nonspherical

quantum-mechanical effects

calculations 114 the spherical seen shell ~ ( Y R ) -I

effects . Real

form and the m a j o r i t y calculations

as unphysical

ones.

symmetry

of the

superimposed

clusters

have,

of the authors

of

join Kubo I09, who considers

on the of couthe shell

Surface Electron Response Theory With Applications In conclusion we wish to mention that in all of the considered situations promote

the calculation

the cardinal

of a metal surface response did not

solution of the problems raised by experi-

ment. We shall show in the further sections

that the application

of the metal surface

electron screening theory

of metal-electrolyte

interphases

to

a description

turned out to be more fruitful.

53

54


4. Effect of Surface Electron Response on the Equilibrium Electrical Properties of a Metal Electrolyte Solution Interphase We consider only "ideal polarized" or "blocked" interphases 40-42. When a potential difference is applied to such contacts,

the rever-

sible charge separation occurs, while a current through the interphase is negligible. Theoretical and experimental investigations of these systems resulted in an important conclusion that for the high enough electrode charge density and (or) the ion concentration the interphasial bilayer has a width of atomic size. Then the contact capacity may be very high (up to F/m 2) so that potential values M S V, typical for electrochemical experiment, lead to high surface charge densities (tenths or units of C/m2). To obtain a more clear image we mention that

~=

0 . 1 C / m 2 corresponds

to the field strength P O = 4 ~ ~ I V/~. This is an extremely high field strength of the same order of magnitude as in the field ion emission. So, what electrode equilibrium can be spoken about in this case? In answering this question a considerable distinction between electrochemical and vacuum experiments must be taken into account. The matter is that the emission current is determined by a form and a height of the surface potential barrier and not by the F 0 value itself. For example, in the field electron emission the barrier width at Fermi level is of importance. In electrolyte the field is screened within a short distance from a metal and therefore either the second intersection of V(r) w i t h ~

is absent

or the barrier width at Fermi level is great enough, so that the electron current is negligible. O

Because of a small effective bilayer gap ~ I A the diffuseness of the electron distribution should be taken into account self-consistently while describing bilayer electrochemical properties. As high bilayer field strength is achieved in these contacts, nonlinear effects in the electron response should be important. The main information about the interfacial properties of ideal polarized interphases has been obtained from differential capacity data. Measure@ capacity dependences are usually interpreted according to Grahame 123, who first reveals that the dependence of a bilayer differential capacity C on an electrode charge ~ and an


55

electrolyte concentration c is accurately enough approximated by the formula

C-I(c,T.. ) = 0~1(~)

+ Cd(C,~')

(4.1)

where Cd is calculated from gouy-Chapman formula (2.59), and ~ ( ~ ) is independent from c. It has been proved mostly by Parson-Zobel plots 124, that Grahame's phenomenological parametrlzation adequately describes capacity data for a great number of electrochemical contacts 40"42' 123-125, though there are some uncertainties 126"129 and exceptlons 126'130-131. The microscopic interpretation of Eq. (4.1) proposed by Graha-

me123 was based on the Gouy-Chapma~-Stern model 132 . Thus a metal

electrode is imitated by the ideal conductor while point electrolyre ions, distributed in the homogenous dielectric medium with permittivity ~ V coinciding with the bulk permlttivlty of a solvent, are considered be an ideal gas placed in the mean field created by a metal and by ions themselves. It was assumed also that electrolyte ions could not approach a metal closer than the so-called "distance of the closest approach" xH ~ A, introduced to take into account a finite ion size. In the case of surface-lnactive electrolyre ions adsorb on a metal electrode without breaking a "solvation sheath" consisting from solvent molecules associated with an ion. Then in this case x H should be chosen equal to a radius of a solvated ion Rsi ~ R i ÷ Ds, where R i is anion radius and D s a diameter of a solvent molecule. Resuming this discussion we mention that the concentration independent contribution CH is interpreted in the Grahame model as a capacity of the "compact layer" (CL) filling the region 0 < x xH). A reasonable agreement between the theory and values ( 4 0 ~ CH~I~0. I-0.5 A, extracted from the experiment assuming the validity of Eq. (4.1), is achieved by introducing a model parameter x H / ~ H " So the compact layer is finally represented by a homogenous dielectrlc slab 0 dl

+ d2

(4.5)

+ d2

where ~ v is the bulk permittivity of a solvent and ~ H is the "compact layer dielectric permittivity" ( E H ~ ~ v ). The "ortogonalization" repulsion of electron shells of solvent molecules from free electrons of a metal due to Pauli principle following Duke and Alferrief361s approximated by the repulsive part of the Harrison's pseudopotential W(r i - r) = ~ f~(r i - r). The corresponding contribution to the surface energy (2.11) is thus as fol-

64


lows Wfs =

In

(x =

d I + d2 ) 2

(4.6)

Here ~ = Ns, ~ fs ~O'I-I; N s is the concentration of solvent molecules in the interracial layer, sok~0.4 for H s = I015cm -2 = 0.027 a.u. and ~ f s = 15 a.u. The ion screening charge distribution was approximated by the expression ni(x) = -~-" ~ ( x -

d I - d 2)

(4.7)

which is reasonable, as the details of this distribution weakly influences the compact layer capacity. The statistical density ftmctional approach was used and the nonlocal correction (2.22) to the kinetic energy was adopted. The surface energy including repulsive ferm (4.6) was minimized by the Raleigh-Ritz procedure using trial functions (2.24) restricted by the condition of the n'~x) continuity. The C H ( ~ ) dependences156 monotonously fall down with ~ increasing. This conclusion is in fact predetermined by the vacuum calculations (Section 3) together with the results of the investigation of the ~n~luence of the dielectric permittivity on the surface potential barrier 102-104, 107. In all cases, considered in the above-mentioned self-consistent calculations, the electron plate x e approaches the metal with ~ increasing, that is the metal capacity, referred to the fixed point outside a metal, decreases. The account of a metal interaction with molecular orbitals influences these dependences quantatively, but does not alter their qualitative form, which in its turn is rather naturally. The discouraging fact is, that the obtained monotonous CH( ~ ) dependences~ natural from the viewpoint of the above-represented picture of the electron response of a pure metal surface, contradicts the form of the experimental c a p a c i t y c u r v e s . The a g r e e ment w i t h t h e e x p r e i m e n t , p o i n t e d o u t by BBG156 f o r t h e c h a n g e s o f t h e OH( 7 = O) v a l u e s when g o i n g f r o m Hg t o Ga i s a n a r t i f a c t o f t h e i r c h o i c e o f t h e J e l l i u m edge p o s i t i o n and s h o u l d n o t be considered seriously. According to our point of view the main result of the investig a t i o n 156 i s t h a t ~ m ~ t a t i o n o f t h e s o l v e n t by c o n t i n u o u s media with the f~xed boundary, results in the behavior of a cont a c t i n a g r e e m e n t w i t h vacuum c a l c u l a t i o n s and a r e u n a b l e t o d e s cribe the electrochemical experiment.


65

~ii~ Extended molecular models: the metal surface electron response and orientational relaxation of solvent molecules. Now we shall consider the attempts to construct a theory uniting the account of the electron relaxation wlth the orientational relaxation of solvent molecules. These attempts are a natural development of molecular models. The superposition of the vacuum-like dependences CHI(~)156 with the susceptibility ~ (~_) of the twodimensional lattice of point dipoles (see section 4.A) predicts the typical for molecular models hump near ~'= O, and a small negative slope of the capacity curve at great I~l • The results of model calculations 157' 159 confirm this forecast. It both papers 157,158 a metal was considered in Jellium model, and screening counterions were approximated by a charged plane - ~ ( x xH), but different models were used for description of the solvent. Schmickler 157 Imltated solvent molecules adjacent to a metal surface by a two-dimensional lattice of point dipoles, using the popular "spin I model" or " ~ f i n i t e spin model ''157. To avoid treedimensional calculations he proposed to describe the electron-dipole interaction in such a way that the electron part of a problem became one-dimensional and the dipole part reduced to the problem of the dipole lattice in the uniform external field. Thus in the calculation 157 of the equilibrium electron density the energy of the dipole-metal interaction assumed to be

Wdi p C n] =

x [n(x)-nj(x)] • Fdi

,

(4.8)

o

where (4.9) Pdlp = < > " Px/ , and n(x) is approximated by trial function (3.6). At t h e same time was determined from the analysis of the relaxation of a two-dimensional lattice of point dipoles upon the uniform external field. The results of the self-consistent calculation of the electron-dipole system (the algoritm is described in Ref. 157) for model parameters values ~ = 6.12.10 -20 C-m, R = 3~, N s = 1019 m -2, x H = 6.4 ~ lead to dependences C H ( 7 ), represented at Pig. 6. The temperature dependence of the capacity maximum ( see Fig. 6) is weaker than i% is predicted by the well-known result 0-I(0) = const + const/(kT) of the molecular models. This is consequence of the appearance of large temperature indepen-

66


dent contributions in the surface energy of the contact. The ellium background density H strongly ~nfluences C(O). With H increasing the height of the capacity hump increases too, because x(O) is greater for metals with greater H (in full analogy with the results for a vacuum-metal interphase disclused in Section 3). Kornyshev et a1158 restricted themselves by a rather crude "capacitor " or "slab" approximations while describing the charge distribution in the monolayer of the water dipole molecules adjacent the surface, but in opposite to Schmickler 157, they provide a more consistent analysis of the model. In the first of the proposed 158 models ("c-model") the solvent was approximated by the uniformly charged slabs with a width L/2 and a surface charge density ~ s = O~ qNs(1-2 ~ ). Here L is a water molecule diameter (3.3 A) , ~ is a portion of water molecules orientated with oxygen end towards a metal, N s = 1019 m -2 is, as previously, the surface density of water molecule, and q characterizes the charge separation in the water molecule (qL = ~ ). In the second o~ models 158 ("d-model") the sol vent was imitated by two uniformly charged planes with the equal charge density ~:s = qNs ( I - 2 ~ ) and with a gap d = 1.07 a . u . . The charge dependence of the distance between the centre of this capacitor and the background edge, resulted from the water molecule asymmetry, was also taken into account. The "ortogonalizational" repulsion energy Wfc was approximated similar to Ref. 155 (see Eq. (4.6)) but in the case of c-model the pseudopotential had been smeared over the slab. The pseudopotential parameters were considered as the model ones and their influence on the capacity was discussed. The same is true for the contribution, describing the metal pseudopotential in the model (2.39) with i = O. In both models 158 the "background dielectric permittivity" ~ H was 158 assigned to the solvent. The model of the solvent in the compact layer, supplemented by the ideal conductor model for an electrode, leads to the restangular hump with a width 2qNs/3 (c-model) or 2qN s (d-model). The electron response uprises the capacity values (the increasing is stronger the greater is H), transforms the restangle to the trapezoid and causes a slight negative slope of the capacity curve at its table-land regions (the "plateau" and the "valleys") 158. * The geometrical distance were determined in 158 from the wellknown Shuster's model of a water molecule.

Surface Electron ResponseTheory With Applications

67

The p o s i t i o n and t h e w i d t h o f t h e r e s u l t i n g t r a p e z o i d a l hump a r e v e r y s e n s i t i v e t o t h e v a l u e s o f t h e model p a r a m e t e r s and t o t h e s o l v e n t model e m p l o y e d . I n p a r t i c u l a r , t h e hump i s much w i d e r i n d-model , t h a n i n c - m o d e l , so t h a t f o r c h o s e n v a l u e s o f q, Ns , e t ~ o n l y t h e u p p e r p l a t e a u o f t h e hump was s e e n 158 i n d - m o d e l w i t h i n t h e e l e c t r o c h e m i c a l r a n g e of c h a r g e s . The r e s u l t s 157' 158 show t h a t s e l f - c o n s i s t e n t caclulations of t h e c h a r g e i n d u c e d e l e c t r o n r e l a x a t i o n r e t a i n t h e c a p a c i t y hump t y p i c a l f o r m o l e c u l a r m o d e l s . However, t h o s e new f e a t u r e s i n t h e c a p a c i t y d e p e n d e n c e s , which a p p e a r i f e l e c t r o n r e s p o n s e i s s e l f c o n s i s t e n t l y t a k e n i n t o a c c o u n t , do n o t h e l p t o u n d e r s t a n d t h e peculearities of the experimental capacity curves, discussed in S e a t i o n 4.A, and t o overcome t h e w e l l - k n o w n d i f f i c u l t i e s 4 0 ' 1 3 0 ' 1~1, 142, 145 o f t h e m o l e c u l a r m o d e l s * . I n t h i s c o n n e c t i o n u n s u c c e s f u l a t t e m p t s 159' 160 to reproduce t h e m e a s u r e d a n o d i c p o s i t i o n o f t h e c a p a c i t y hump b y t h e e l e c tron relaxation effect are significant. The m e t a l and d i p o l e con@ t r i b u t i o n i n t o i n v e r s e c a p a c i t y were s,--m-~ized i n R e f s . 159, 160 as follows

0-1 = ~dZ.- nT-'(x)zdx + L - Ns ~-~ < px'~ ,

(4.10)

where T. is the diate~oe between the Jellium edge and the plane of co1~terions, O a l c u l a t ~ the f i r s t ¢antribtttion the authors of t h e works 159' 160 made am e r r o r w h i l e u s i n g t h e sum r u l e ( 3 . 1 0 ) (compare t h e i r n e g l i g e n c e w i t h t h a t o f R e f . 73 and i t s d i s c u s s i o n i n S e c t i o n 3 . B ) . As a r e s u l t t h e l i n e a r c o n t r i b u t i o n - ~ / i (ins t e a d o f Y / n ) a r o s e i n C-1 , d i s p l a c i n g t h e c a p a c i t y hump t o w a r d s t h e n e e d e d d i r e c t i o n . The c o n t r a d i c t i o n w i t h t h e r e s u l t s o f v a r i a t i o n a l calculations (see, for example, Refs. 66, 67, 71, 156), leading to the opposite displacement of the hump, was explained 160 by the error due to the use of trial function within the RaleighRitz procedure. We have proved (Section 3.B) that predictions of the variational calculations and those of sum rules are the same. Consequens

F o r example, t h e a n o d i c ( ~ > O ) p o s i t i o n Of t h e c a p a c i t y hump as i t i s i n t e r p r e t e d i n m o l e c u l a r m o d e l s seems t o be i n d i s agreement w i t h t h e c a t h o d i c p o s i t i o n o f t h e mAX4mum o f t h e ~nner l a y e r e n t r o p y o f f o r m a t i o n , w i t h d a t a on a d s o r p t i o n o f s m a l l o r g a n i c m o l e c u l e s and t e m p e r a t u r e d e p e n d e n c e s o f t h e p o t e n t i a l s o f zero

Ohawge.

68


fly, they both contradict the experiment, if the symplified models of the above-considered type are adopted. Thus the necessity to improve the existing compact layer models seems even more evident. (iii) The metal s u r f a c e electron response and the displacementtype relaxation of electrol~te s~ecies in the compact la~er. How can we make the BRG model 15b more natural and, simultaneously, instead of the monotonous uprising of the capacity with ~ decreasing obtain the capacity curve falling down near pzc, but uprzzing at large negative ~ as it happens in the experiment (see Refs. 40, 135, 140-143) ?. Such capacity behaviour will follow, if the effective bilayer gap i (~-) = IZH-Xel initially becomes larger with >- increasing, but begins to contract for great enough ~ I. FolioBRG, let us freeze the rotation of solvent molecules. Then the needed bahaviour of the effective gap will follow, if a monolayer of the solvent molecules adjacent the surface, and the Helmholz plane bonded with this monolayer will move upon charging, initially going away from the surface (faster than Xe), but breaking and turning backwards for I ~ I further uprizing. It is easy to understand that such a behaviour is natural~ if the "adhesion well" for a metal-SIES system is smooth enough, so that solvent molecules are weakly bounded with a metal*. Then the displacement of x H towards the vacuum for small ~ naturally results from the extra "ortogonizational" repulsion of solvent The smooth bottom of the adhesion well usually means a small value of the adhesion energy Wadh E(a =oo) - E (ae~) = win+ ws + Wm/ES ) where w m, wES and Wm/ES are the surface energies (according to Gibb's definition 167.) of a metal, an electrolyte solution and of a contact metal-elect3olyte correspondin@ solution (the contact area A s assumed equal unity). Neglecting the effect of electrolyte ions, let us estimate wad h using the measured values of the closely connected with of wetting energy Wwe%t and the heat of adsorption Earls. The w e t t ~ energy is determined by Dupre's for~ , u l a 141 ,

the Por ges for

162

Wwe~t = win/a + Ws/a + win/s, whwre win/a and ws/a are surface energies of the metal-air and solvent-air contacts. instance, the energy of we~ting mercury by pure liquids chA~from 113.6 erg/cm 2 for metanol till 169.7 erg/cm 2 for anilin| water Wwett = 130.6 erg/cm 2 = 1.8 kCal/mole.


69

molecules from a m~tal, a ~ o c i a t ~ d with the increasing of the metal electron d~nsity near a surzace. The charge-induced motion o~ x H which is faster than that o~ x e is not unr~aAistlc. TO accept this ices one may assume, for simplicity, that the induced electron density distribution has a bell shape, and this "bell" blows upon charging without the displacement of its centre ( in the spirit of LR). Them the "ortogonalizational"

repulsion will

increase x H, while x e is motionless. The "contraction" of the gap 1 (~) at large J~lis naturally explained by the electrostatic attraction, which increases proportionally ~ ,

so that at large

enough IZlit finally dominates the repulsion (compare with the Mackdonald's 136 "electrostriction"

of the compact layer).

The attempts to take this effect into account were made for metal-SIES interphase on the phenomenological basis 166 and for a model metal-solid electrolyte interphase within a density functional approach 167. The subsequent calculations of the microscopic models for the metal-SIES interphase have shown, that the situation in the compact layer may be much richer that in the phenomenological pictures q66 or in a simple bilayer model~67Inparti cular,

in works q66'167 the capacity anomalies and problems asso-

ciated with them (see,e.g.,Section6),

have not been noticed. The-

refore, we shall restrict our discussion only to "microscopic " calculations70,168, q69. The bilayer models, accepted in these calculations, are shown at Fig.7. A metal was described in the jellium model. The discreetness of the ionic lattice was takan into account only in Ref.169, where the Ashcroft pseudcpotential was adopted within the first order parturbation theory. The solvent was imitated by a layered medium with a sharp edge at x = a. The adsorption energy ~ a = ,

characterizing the adsorption of a

single molecule, can be related" " with Wad h by the Born-Haber cycle. In the very interesting case of water adsorption on mercury the experimental data are rather controversial in this connection (see the review163); the recommended 151 estimate equals several kCal/mole, which is much less than th~ well-known result by Kemball q6@ Ead s = 17.6 kCal/mole. As the energy of hydrogen bonding in water is approximately 5 kCal/mole, then the estimate Wadh ~ w w e t t ~ - w a ~ O . q eV/molecule leads to the picture of solvent molecules comparatively weakly bonded with mercury.

70


f l i I t l i

cv

1 0

a

a+d

x

FigoT. The one-dimensional models 168'169 for the metal- SIES interphase. er region,

In the model 168 the space x ~ a + d is the diffuse laythe dielectric slab with the permittivity ~ H

and the

electron - repulsive potential imitates the solvent molecules and electrolyte ions adjacent to an electrode surface (d is the radius of the solvated ion and the Helmholz plane positiom is a+d)o The region O ~ x < a is the vacuum gap (E =I), the space x •0 is occupied by the jelli~m with the backgroumd density (and background permittivity ~ m )" In the HJPS model 169 the gap width a equals the distance of the closest approach of the electrolyte ions to a metal, ~H = I (while calculating the compact layer capacity ) and the distance d, to which the "electron contribution" in the compact layer capacity (coinciding in the model 169 with the latter) should be referred, is determined by the condition n ( x + d ) = O.

Surface Electron Response Theory With Applications The main difference between the models 168 sad 169 is the different physical interpretation of the compact layer. In Ref.169 the "distance of the closest approach" of ions (the outer Helmholz plane) x H coincides with the solvent edge a; it means that the finite size of ion amd solvent molecules, forming its solvation sheath, is neglected. In the works 168'70 the compact layer model was chosen in accordance with the traditional picture assigned to Stern and Grahame, so that its width was assumed to be equal to the vacuum gap a plus the radius of the solvated ion Rsi=Ri+D s , where R i is the ion radius and D s is the diameter of the solvent molecule. Beside this, the solvent polarizability was taken into account in Refs.70,168 within layer dielectric medium approximation 156. Because of the "polarization" contributiom into the surface energy, the dielectric slab, imitating a solvent, desires to favour the region of the highest field strength, i.e. the space near the jellium edge. This ~ffect was fully neglected by Halley, Johnson, Price and Schwalm (HJPS). 169 The repulsion of free (f) and core (c) electrons of a metal from the orbitals of the solvent (s) molecule, resulting from the Pauli principle, was imitated analogously Refs. 156,158 :

Wfs = V o ~

n(x)dx

(~.11 a)

Wcs = V o ~

nc(x)dx

(4.11 b)

where V o is the intensity of the repulsion sad n(x), nc(X ) are the densities of the free,core electrons of a metal correspondingly (mc(X) is frozen). The energy Wcs (4.11 b) of the cs-repulsion was approximated by either the infinite potential wall, forbidding the solvemt to penetrmte into the jellium168,or by the expressions of the Born-~eyer's type 70, or by formula (4.11 b) with n~(x) calculated with the Hartree-Fock wave functions for free a t o ~ 169. A very important question is that about the origin of the electrostatic attraction between a solvent and a metal at pzc( at ~ @0 the electrostatic attraction force ~ ~ Z a r i s e s , but at ~ = 0 the attraction disappears in the one-dimensional models 168,169. This attraction should struggle with the " ortoganalizational "

71

72


repulsion, resulting in the equilibrium gap a(>-= 0 )~'1 ~. The attempts to obtain the attraction from the microscopic consideram tion were unsuccesful so model expressions C(a) 70'q68 and Elong rang

169 were introduced

U(a) = - . ~

:

(a+d)

Elong range (a)= - AY(a+x d - Xim(a,Z)

(4o12 a)

)P

(~.la b)

Here A is the intensity of attraction, which is, in fact, a model pal~am~ter; d approximately equals the solvent molecule diameter tit was put = 8 a.u.168); x d is the radius of solvent molecule tit was put q.5 a.u-q69); xim is the position of th~ "image" plane, determined by expression ( 5. 11 b) 73. It was shown 70'q69 that the form of the attz'actlon "potential" is not important because the relative changes of the CL width, following changes of the gap a, are small. Uniting contributions into the surface energy, we get finally:

w In;a] = wj [n;~ + wfsIn;a j + Wcf(a)+U(a)

(~.13)

The first contribution wj is the surface energy of a jellium metal. The electrostatic contribution into wj includes the energy of the electrostatic attraction between the metal and an electrolyte solution; the nonelectrostatic contribution was calculated either within local density approximation for exchange and correlation 169 or according ~qs. (2.25)70'168o The equilibrium gap a and the equilibrium electron density n(x) for each given ~ had been determined in these works by minimization of the surface energy (4.13). The electron distribution was obtained either by the solution of the Kohn-Sham equations (2.27) -(2.33) 169, or by Raleigh-Ritz procedure,using trial functions (3.7) 70'168, The first method describes the electron distribution better,but, as w e have already mentioned (see Section3),in the calculation of the surface electron response both these approaches " For example,ta~ing into account the interaction of the molecular dipoles and solvated ions with their images,}~JPSobtained the estimate for A from Eq.(@.12b),which t~rned out to be for an order of magnitude lower,than the value,chosen in Ref.169 to fit the theory to experiment.


~0

+

too

•. o -

!tt

I

I

-0.005

~1

0

0.005 Z,a.~..

Fig.8. The theoretical dependences OH-1 ( ~ )169 for the model contact of a monocrystalline silver electrode with a SIES.

H9 0

!

l

-0.009 0

l

0.002 Z , ~ . .

Fig.9. The experimental dependences CH -q ( ~ ) for the contacts of Hg,ln, Cd, Pb, with a SIES, extracted 169 from the data f l 2 3 ~ I ~ 9 adopting the Grahame-Parsons-Zobel technique. CwI a.

9.0

J9

f5

---

~4o 400

--'--

ll4

~0 5 - ~005

|

|

0

&005

Z,a.~

FiglO. The e x p e r i m e n t a l d e p e n d e n c e s CH-1 ( ~ . ) f o r t h r e e f a c e s of a silver monocrystal in contact with a SIES, extracted 169 from the data fl53 adopting the Grahame-Parsons-Zobel technique°

73

74


give similar results. Summarizing the description of the models 168,169 we underline that in its essential part the models 168

differ from the model 156 by the physical interpreta-

tion of the vacuum gap a and by the way of its determination;the electrolyte part of the model q69, which analyses the metal electron distribution more carefully, than in Eels.q68,70, cial case of the models 168 at S u =q, a=xu o Now we discuss those of the results, 70'q~8'q69=

is the spe-

which, accor-

ding to our opinion, are of main interest. The inverse differential capac~ity of a compact layer for Ag/SIES interphase,obtained by HJPS q69 is represented at Fig.8 (the capacity dependences for Au and Cu, also obtained in this paper, are qualitatively the same). If the cs-repulsion is neglec ted, then the capacity curve 169 in the range of the investigated has a foz~n of convexed downwards parabola - like curve which is in qualitative agreement with the experimental capacity curves for interphases with Hg,Pb,In,Cd (Figo9) and Ga electrodes, but in a contradiction with the experimental data for the case of noble metals (Fig.t0). The inclusion of the repulsion of the solvent from the metal cores slows down the decreasing of the gap a (~)

at small ~ . In one-dimensional models "velocity"Ida/d~i

decreases faster if the distance between the jellium edge and the centres of the cores is smaller, that is if the face is less closely packed° For the face AS(It0), which is the rarest packed face among the three considered in the paper 169 the Ida/d~| decreasing results in the increasing of CH-I (see ~q.(2.51) ), and the hump appears on the curve C H ( ~ ) (Fig.8). The experimental capacity curve C H ( ~ ) for Ag~SI~S interphase has the maximum approximately at the s a m e ~ , but,in opposite to the theory, the hump is experimentally observed for all of three mostly closepacked faces of silver, and its overall shape is rather far from the spike on the theoretical curves 1 6 9 ~ o m p a r e Figs.8,9). The typical dependences C H (~-) for Hg/SILS, In/~InS, Ga/~I~S interpbases, obtained in Ref.q68, are shov~ ( in a slightly corrected 70 form) at Fig.11. As it is seen from Fig.11, for reasonable choice of the papametres the parabolic branch of the theoretical capacity curve closely approximates the "experimental" dependences C H ( 5- ) for gallium, indium and, at ~ 0 , mercury electrodes.

75


0.4

O.5 0.2

~li\

1

\

"'--~'

\ \ "~.

-0.9.

I

Gti

i

0

-8

-t6

Z, ~0-~ ~.~

! L, I 0.3

~

\

0.t

i

i

168~ Fig.11. The con~arison of the theoratical dependences u H ( 7. ) ( ) for Hg/SIES (a), In/SIES (b), Ga/SI~S (c) model interphase with the o ~ r i m e n t a l data ( - - ) for Hg/H20 - NaF inter" and In/SIES and Ga/SI~S in~erpna~es • _ , ~ 148 phase 123 . The measured dependences C ( ~ ) are represented by the curves C H ( Y ~ ) , calculated by Grahame 123 and Trassatti 127 adopting Eq.(4.1); at the insertion of Fig.qqe the original data C ( ~ ) due to Frumkin,Grigor'ev and BaGotskaya 148 for Ga/~20 - In NaClO@ ( ~ ) and G a / ~ O -In Na2SO~( ) ar~ also given.

76


0.3

3~\I h'\.

'1 -0.3

I

I

;

-8

-,6

I

|

I

~, ~0-~ el.~

Fig.12. Thc compact layer capacity CH( ~ ) for various values of the radius Rcs of the cs-repulsion. I-3: the theoretical c~rves 7° CH( ~ ) for the model H ~ S I E S interphase for Rcs=0.5, 0.48, 0.3 a H correspondingly. The experimental data by Grahame q23 for Hg/H20 - NaF interphase are also represented.

g,e~ ~.iO "~ 5.0

215 ~.4 +

~lO

1.0

1.5

a, a.~.

Fig.q3. The adhesion energy ~ ( a ) = w(a) - w( °o ) of the model system 168 for some ~ near the bifurcation point ~cr"


77

~oble metal electrodes were not considered in Ref.168, but it is reasonable to expect,that the change of the background density and the pseudopotential correction should not change the qualitative character of the capacity dependences,so model 168 should contradict the experimental data for noble metals as well as model 169. It was stated in Ref.168,tbat the capacity singularity (asymptote), incorrectly interpreted by the authors as the finite jump of the capacity curve due to the stop of the solvent at the infinite potential wall, may lead to the capacity hump a more reasonable approximation for the cs-repulsion will be used. But a more rigorous analysis 70 has shown the existence of other possibilities. One of them is illustrated by Fig.q2, where the results of the calculation 70, in which cs-repulsion was approximated by the Born~ayer type expression

Wcs = Bcs, exp(- a/ ~cs) w i t h B c s ~ 2 - 3 • 10 " ~ a . u . , Rcs ~ 0 . 5 , We s e e t h a t t h e s m o o t h c s - r e p u l s i o n raction

radius

~cs results

away t h e e l e c t r o c h e m i c a l

(4.1~) are represented. w i t h a l a r g e enough i n t e -

in the displacement r~ge

of charges,

of the asymptote

but the reasonably

small Rcs leads to the capacity singularity as previously, and not to the a hump. This distinction of the model q68'70 from the model q69 follows from the catastrophical jump of the vacuum gap, which is absent in the model ~69. The s ~ f a c e

energy per unit area

(a) as a function of the gap a for s e v e r a l ~

, obtained by the

minimization of the s~rface energy over electron variables,

is

represented at Fig.q3. It is clear from the picture that at some ~=~the distant minimum disappears, the energy ~ (a) becomes the one-modal function and the gap suddenly collapses. The existence of the nearest minimum resulting in the ~ (a) bimodality and in follows 70 from the above-mentioned

the cusp catstrophy q71'172,

pulling of the dielectric medium into the region of the highest total (i.e. including the zero-charge contribution)

electrostatic

field strength. Of course,the displacement type relaxation may lead q68 to the hump at the capacity curve. But its existence should be a result of a very fine interplay between the attraction and the repulsion, both smooth and continuous 70.

78


The models 108'169 have at least three common shortcomings associated with a primitive description of the solvent . The first is a very rough description of the solvent p o l a r i z a tion. The majority of the authors, HJPS 169 among them, believe that the saturation of bhe solvent molecule polarization in the compact layer considerably contributes into the C H ( ~ ) ,

espe-

cially to its temperature dependence. However, in model q69 the solvent polarization in the compact layer does not influence the CL capacity directly, while its contribution into the surface energy is taken into account very crudely, if it is taken into

"

It is also appropriate to mention some difficulties follow-

ing from the definition of the Helmholz plane position, proposed inq69 1. The requirement are point-like,

a = x H , implying the solvent molecules

is in a certain contradiction with the microsco-

pic character of the description of the contact. Beside this,it means, that at the stage of the diffuse layer capacity calculation the region x > x H = a

is assumed to have the bulk permitti-

vity of the solvent, while at the stage of the compact layer capacity calculation the same region x > a zable.

assumed to be unpolari-

2. HJPS stated that the difference between

x H and a, absent

in their model, can be taken into account post faotum by adding the correction 4 o r ~

to Eg. ( 2.51

) for CH-q , where ~ = x H- a.

The negative ~ value adopted in Ref. 169 from hard sphere results 173 ( see also @1'174-175 ) makes the agreement with the experiment better 169. Howaver, the microscopic picture of the interphase ~compare with model 168 ) requires that x H the medium, imitating the solvent,

should be in the interior of at the distance approxima-

tely equal to the radius of the solvated ion Asi " Then ~ , 0 and the agreement of the results q69 with experiment becomes WOrSe.

3. Since

xH = a

is too small to provide reasonable C L

capacity values , HJPS wer~ forced to introduce, beside new model parameter to which the capacity was referred.

xH, a


79

account at all (it was claimed 169, that the polarization contribution constitute a part of the "stabilizing" term Elong range )" In model 168 only the influence of the solvent polarization on the classical electrostatic energy was taken into account, and even this within the local dielectric permittivity concept, which imitated the effect v~ry primitively and might lead 157 to substantial errors. The dependence of the exchange-correlation energy of metal electrons on the solvent polarization (see the prelimenary discussion of this effect in Ref.qo4) was fully neglected in Refs.7o,168. The second defect concerns the artificial imtroduction of the non-self-consistent " stabilizing contribution " U (a)(or Elong range) of unclear origin. With the existing estimates for U (a) and the results of model calculations 7°'168'q69 may preElong range tend only to show the principal possibility of the explanation of some peculiarities o~ the compact layer capacity behavio~r via the displacement (alternative to the 0rlentational) relaxation of the compact layez. The third defect of models 168'q69 is in close connection with the second one. We mean the one-dimensional character of the models, which neglects the individuality of the s o l v e n t m o l e c u l e s and electrolyte ions and the bonding between them, thus omitting a number of associated effects (perharps, a significant temperatu~e dependence of the CL capacity among them). In conclusion we wish to mention one important and inspirating result of our ~ a l y s i s . We have already seen that models 168'~69 are very different, when comparing the separate contributions in th~ total energy, the physical interpretation of the models and the assumed approximations.

However, they have a very important

feature in common: a new mechanism involved, which manifests itself only when the charge-induced relaxation of metal electrons and electrolyte species is considered simultaneously in a selfconsistent manner. Therefore, if model parameters are fitted to @

obtain the reasonable gap value a ( ~ =

0)~A

and, consequent-

ly, C H ( ~ = O ) ~ 0.1 F/m 2, then similiar parabola - like capacity behaviour at ~ - g 0 inevitably follows in each of the discussed models. The same capacity dependences are observed in the experiment. Thus it seems reasonable to assume, that the physical origin of such capacity behaviour is " caught " by the mo-

80


dels168,169 (iv)

I

The metal surface electron response and the re!socation of

hard sphere ion-dipole mixtures. In all of the previously considered models for the metal-SIES interphases bilayer was assumed to consist of the compact and the diffuse parts. The properties of the solvent within the diffuse layer were assumed to be the same as in the bulk, while its bebaviour in the compact layer was aproximated by different and primitive models, suco as the lattice of point dipoles or the homogeneous dielectric slab, displacing upon charging. The ion distribution im the diffuse layer has been always described within the Go~y-Cbapman theory developed for point -like ions. The hard sphere model 172 for an electrolyte solution,which approximates the electro±yte ion amd the solvent molecule by a hard sphere with the embedded point charge or point dipole in its centre, gives 1'?~-176'~q in principle, a possibility to consider the ions equally with the molecular dipoles,

taking into account their

finite size and without the division of the bilayer into the compact and diffuse parts with the special solvent properties in them. The interphase between the jellium metal and the hard sphere ion and dipole mixture has been considered by Schmickler with Henderson 176 and by Badiali et al. q77. In the work qW7,~ as in earlier works 155 by Badiali et.al.,

the interpretation of the v a c u u m

gap between the jellium edge and the solvent has been used,which contradicts the tradition and common sense of the jelli~m model (see Section 4 ~ ) .

Beca~s~ of this circumstance the value of the

bilayer capacity remains,in fact, indeterminate. Basiaes,the mode1177 is less general than that by Schmickler and H e _~~r ~_ uo ~_ _ 1 7 6 • Therefore we shall dwell on the work q76. The problem was solved in 176 by a not rigorous reauction to the problems for a system of ~lectrons, described in the Badiali 156like model and for s system of hard sphere ions and dipoles,bounded by a hard charge plane wall. The approximate solution of the latter problem had been derived by Blum with Henderson q75 and by Carnie with Chen q74 within the ~ean Spherical Approximation ( ~ A ) ~1'172 justii'ied only for small Z on th~ wall ~md for relatively low ion concentrations with

ae~ R s (the hard sphere region)as well as from the region 0 < x < Rs ( Ri)" The results of C H calculations 76 for a number of metals are given at

Surface Electron Response Theory With Applications Fig.14. They demonstrate that bably, caught

a

83

qualitative tendency is, pro-

by this model. We also mention that the extrapola-

ted C H curve becomes negative for ~ ~ 0.22 a.u. The NBA gives the bilayer capacity only at the pzc. Recently, the heurustic extension of the MSA results to higher electrode char. ges have been developed and the capacity vs charge dependences have been calculated 178'179 • The hard sphere model is of importance for the bilayer theory, since it is a step towards the self- consistent description of the metal elecbrons, solvent molecules and electrolyte ions all together at the metal-electrolyte interface. Besides, already in the field linear ~ A it predicted the possibility of the concentration independent contribution into the bilayer capacity, which follows from the solvent polarization within the region available for ions. Nevertheless, the hard sphere model can not pretend on the realisof tic description of the interaction ~lectrolyte species with metal, which may 168'169 substantially influence the electrical properties of the bilayer.

5. Effect of Surface Electron Response on the Capacit2 of a Bilayer at a Metal-Solid Electrolyte Interphase It has been shown in the preceeding section, that the uprising of the bilayer capacity with the electrode charge decreasing, which is natural from the view~point of the electron response of a pure metal surface, contradicts the capacity behaviour measured for a metal-electrolyte solution interphase. To explain the experiment the charge-induced relaxation of the equilibrium position of solvent molecules and electrolyte ions turned oat to be of importance. However, this effect should be small, if the interaction between electrolyte species and a metal is strong. For example, mobile ions of a solid electrolyte, having no solvation shells, can approach a metal closely and interact strongly with it. Therefore,it is expected that the equilibrium positions of the mobile ions, neighbouring a metal, change only slightly upon charging, id est the charge-imduced displacement of the Helmholz plane may be neglected. Then the capacity of the "compact" layer of a molecular size will be determined by the charge-induced displacement of t~e electrQD "plate" alone. The corresponding vacuum tenden.

84


c,-~ (,.:o1.

n~~

.\ "Sn.

0.2 0.0

,

,

40

~5

~0

~.,~0 a . u .

Fig.qz~. The compact laye~ capacity C H ( ~

=0) in the pzc for the

various jellium metals in contact with a SIESo The theoretical res,~lts 156 (

) are obtained within the hard sphere model for a

SIES. The experimental data (references are given in the paper 176) are ~epresent~d by dots.

I0_, I.~em ~'

~00 50 {

4.o

-02

-~0

-~0 Zo

-40

0 Z~ tO-~ e/'m~

Fig.15. The dependezaces of the bilayer capacity C, the chargeinduced potential drop ~ = ~ (~) -- ~ = O) and the induced charge centroid x vs the electrode cha~g~ ~ for the model AujAgnX interphas e180-188 ( ~ = 8.84 10 -3 a.~.,d=2o52 A,&m=~H='I) in the ca~e of the submonolayer electroacLsorption (k=fl). 1: C(~.), I,: -cCZ); a : ~ z ( ~ ) ; 3: x e ( ~ ) .


85

cy in the total bilayer capacity behaviour may be strengthened by a counter displacement of the ionic "plate". This situation seems to take place in a number of metal-solid electrolyte interphases. A.

The problem of the abnormally high values of th e bilayer capacity.

The deficiences of the non-self-consistent approach manifested themselves extremely sharp in the attempts to interprete the abnormally high capacity values, observed in contacts of some metals (Au,Pt) with solid electrolytes, which are silver ion conductors. To clear up the contradiction arisen, we would remind, that in the traditional theoretical picture the compact layer capacity C H is assumed to be in series with other contributions into the total inverse capacity. It is also always supposed that C H is positive, as well as othercontributions.

Then the total bilayer capacity

achieves its maximum when the electrode charge is totaly screened within the compact layer, that is

c ~ CH

(5.1)

The estimates for C H are usually obtained from a simple formula @5 (w~ change only the notations)

CH = @~(XH~Xe(O)

)

(5.2)

where, as previously~ ~ H is the compact layer permittivity and x H is the distance be~leen the centres of the ions adjacent to a metal and an electrode;

in the Stern model, usually used in these

cases, X H ~ R i = 1.26 ~ (for Ag+). Putting Xe(O) = 0 (the ideal conductor model) and ~ H = 3 @ 5 we get from Eqs. (5.1) and (5.2) the estimate C ~ 0.2 F/m 2 . Taking into account the field penetration into a metal in the spirit of the Rice model 50, that is putting x e = - 0.5 ~, we arrive @5 to the inequality C 4 0 . 1 6 F#m 2. The obtained estimates significantly (more than for an order of magnitude ) differ from the measured capacity values approach• This inequality is invalid, if either CH, or the contribution in series with it is negative. This possibility have not been considered in the traditional models of a bilayer at a metalsolid electrolyte interphase.

86


ing hundreds of microfarads per centimeter in square (£or a review of experimental results see Re£s.18o-q82). To overcome the pointed out contradiction Raleigh q83 suggested to modify the description

of the ionic distribution,

r~serving

the ideal conductor model for a metal. In his moael the ionic screening charge density was chosen in the two-layered form nl(x) = ~1"

~(x)

.

~(d-x) + n 2 ~(x-d), @(2d-x)

where d is the width of each layer and nq d ~ f l

= - k~

(5.~) is

the surface charge density in the first layer. Due to the requirement of the screening of the electrode charge within the compact la~er the surface density of charge in the second ionic layer equals ~ 2 ~ n2"d = - ~ - ~ 1 ionic screening charge lies at x i : d (-k + 3/2

=(k-1)~-~

o The centroid of the

),

(5.~)

so that for k=3/2, assumed by Raleigh, C = o o for e v e r y ~ (in this case the position of the "ionic plate" of the bilayer coincides with the boundary of an ideal conductor sod consequently with the position of the "electron plate"). Any high value of the capacity can be obtained by a slight variation of k near i~aloigh's value. The "overscreening effect" ( k 7 1 ) ,

well-known

in electro-

chemistry as "superequivalent adsorption" is a rather wide-spread phenomenon, so essence of the Raleigh model is verisimilar. Perhaps, this model catches the oscillations of the io~ screeni~g charge d~nsity, which appear if ionic correlations are taken into account 184'185 '172. Then Eq.(5.5) may be regarded as a rough approximation

of the first two oscillation peaks of this distrib~-

tiom. Using sum rules (2.~o) and (3.qo) it is easy to verify that similar oscillations of the screening charge distribution in a metal result in the negligible contribution of the region x O into the inverse inberfacial capacity, in evident analogy with Raleigh's hypothesis. However,it seems to us the fixed val,~e of 3/2183 for k has no reliable foundation. Though the Raleigh's model~f~ore reasonable,than the l~acdonald's ~5 "compact layer shorting" due to "electron tunneling from the electrode to an Ag + ion"j the choice of the fixed k and the use of the ideal conductor model makes the former somewhat artificial. Beside this, it is hardly possible for the model 183 to explain the very important peculiarity of the


87

measure~ capacity curves, that is the sharp uprising el C (U) (U is the el~ctrod~ overvoltage) when approaching to a low ("cathodic") boundary of the "Oioc~ing r ~ g i o n

[for platium the same

capacity uprising is observed also for large U, but the Giscussion of this peculiarity of platiIlum we reserve for better ~lmes). A more natural explanation of the abnormally high capacity values together with the ~bove-pointed peculiarity of 5he capacity curves is provided by the s~lf-consistent accoumt of the metal surface electron r~sponse.

B.

The sei£-consistent electrom respons~ in the ~h~ory or the metal-silver iom conductor - solid electrolyte interphase.

(i) The extended Raleigh model for the ion screening charge distribution. We would remind,

that in opposite to the Rice model 5o

the self-consistent calculations

(see Sections 3 and 4) give

x e (0)> O, increasing CH(O) ( see Eq.(5.2~. The dependence of x e on ~ is even more important for the explanation of the abnormally high capacity values and the sharp capacity uprising at some . The matter is that the charge-induced displacement of x e out of a metal results in any high capacity values, up to the divergency of C (Fig.15). The origin of this phenomenon is analysed qualitively in Appendix D, taking a capacitor with a gap, relaxing upon charging as an example. In the first calculations 186-188 of the effect of the electron response on the bilayer capacity for a metal-solid electrolyte interphase the Raleigh model 183 was adopted for the ion screening charge ~istrlbution.

~owever,the fixed value of 3/2 was not assi-

gned to the "overscreening coefficient" k, instead of this the effect of overscreening was considered; the particular case k = 1 of the submomolayer electroadsorption (the Lang model (2.38)) was also investigated 186. The background dielectric permittivity was chosen in the three-layer form:

&Cx) =£mr(-x) + ~

~(x) . 8(2d-x) + 6vS(X-2d).

(5.5) O__

The values of model parameters were put as follows: ~=O.06 A 3(Au), o

d = 2.52 A tAg+), ~ m = 1, g H = 3(Ref.@5), 6 v =10 ( the latter parameter practically does not influence the results). The ekctron distribution was determined by minimization of the surface energy ( 2.11 ) with the non-electrostatic contribution

88


(2.25), r~sulted from the substitution of the trial functions (2.2~) into the functional G In] as given by Eqns.(2.19) (2.1o), (2.22). The centroid x e of the extra electron distribution,the potential d~op qb in the bilayer and the bilayer capacity C = C~ were obtained from ~qns.(2.4$) -(2.5o), (2.55). The obtainedfl87, q88 dependence are given at Fig.16. For every k value the C ( Z

) dependence is identical to that,obtained for

the relaxing capacitor (Appendix D ): there is a divergency point >-~o' to the right of which the capacity can achieve any high value,while to the left C ( ~ ) becomes negative. The increasing of the "overscreening parameter" removes the divergency point towards the more positive ~

,so that for k > 1 . 3

the divergency

point enter the anodic range. This can be easily understood using Eq.(5o4), as with k increasing from I to 3/2, x i moves from d/2 to On As it is seen from Appendix D and the preceeding caculations to obtain the divergency the gap should compress upon charging,that is the electron plate should approach the ion plate. Therefore, for those k, for which x i < x e ( 0 ) ,

to get C = o ~

the

electrone plate x e should move towards a metal s u r f a c ~ o t away from it] and this is achieved by positive charging of a metal. The negative capacity values repzesented at Figo16 trill be specially discussed ~] Section b. The positive capacity branch, transformed into the curve C ( ~ ) , has been used 18G-18~ for interpretation of the measured capacity d~pendencos. The tohal potential drop in the bilayer was top-resented as follows: @ =

@eq +

U

whet ~ eq is the equilibrium potential drop for a dead voltage a source and U is the overvoltage,supplied by sou_rce and marked on the x-axis of the experimental graphs Cex p (U). According to the theoretical C ( ~ ) curve (Fig.17), the closer is ~ e q to ~ o, the greater is the C(U=O) valu~ and the steeper is the capacity decreasing with U increasing. Analogous regularities are observed in the experiments p~rformed i or AuYAgC1 and Au#Ag4Rb J5 interphases. The increasing of the capacity value C(C=O) at the low boundary of the blocking regime, obtained by the variation of the temp~rat~_ce and (or) the concentratio~] of the dopping impurities,follows in th~ increasing of the steepness of the C ~xp (U) curves in their beginning at low polarizations (see Figs.4, 5,9,q0 in the review q82). It was also sho~n 187'188 that the dependences C

exp

(U) measu-


89

c,F/.~~ I

I I I I

0

I

I

i

I

°

-o.~ -0.3 - ~

-O.t

0

o.~

Z, ~

Fig.q6. The effect of the o v a r s c r e e n ~ g on the equilibrium electrical properties of the model interphase A ~ A g n X 1 8 7 - 1 8 8 ( ~ = ~ , the other parameters are the same as in Fig.15). 1: ~ 1 ; 2: k=1.3; 3: k=1.5.

4

0

I

I

I

1.6

4.4

4.~

~,eV

Fig.17. The theoretical dependence C ( ~ n ) for the model conSact Au/AgmX in the case of the submonolayer electrodsorption

(~I).

90


red for contacts Au/Ag 4 RbJ 5 and Au/AgCI may be reproduced on the basis of extended Raleigh model with the simplest choice k=l introducing the single fitting paramete~ qbeq and choosing it in such a way that C ( ~ eq ) coincides with the measured C value exp in the beginning of the curve. The main difference between the experimental capacity curves for contacts Au/Ag~ RbJ 5 and Au/AgC1 is that in the first contact the capacity practically monotonously decreases to small values ( o.1 - o.3 F/m 2 ) within the range 0.1 -0.5 V, while in the second one Ce~p (C) rather steeply achieves the approximately constant value, C, which is compa~atevely high ( ~ 1 . 5 F/m 2 for t = ~ 6 ° C 18~,182 J. The Cex p (U) dependence for the Au/Ag 4 RbJ 5 interphase (Fig. 18) and the contribution Cexp(U) -C for the Au/AgC1 interphase, steeply increasing with U decreasing, is described by the results of the self-consistent calculations rather w~ll. However in the case of Au#AgC1 contact to explain the high capacity values in the wide range of charges th~ more exact description of the ionic subsystem is necessary.

This remark applies

equally to the contacts of the considered solid electrolytes with platinum fo~ which the strongly nonmonotonous behaviou~ of the Cex p in the intermediate range of overvoltag,~s and the steep capacity uprising with the further U increasing should be explained tOO° (ii) The metal s~rface ~lectron response supplemented by the di~fuse lager ion relaxation. In the previously discussed caculations 186-188 the position of the ionic plate was frozen, so the ion relaxation efl ects were neglected.

Thus the inclusion

of these

effects in the theoretical picture of the interphase is of interest. The simplest model of the ion relaxation,that

is the Gouy-

Chapman model of the diffuse layer (see Section2),

leads to the

capacity uprising, parabolic at small I~I and linear at large (see Eq°(2.61) ). The "lattice saturation" slows the C uprising~ the braking being stronger at greater i~l ,and at gr~at enough I~I it even decreases the capacity. This effect is a stmaitforward consequence of the finitn~ss of the number of ion positions in the compact layer,which makes harder the accamulation of ions near an electrode when their numbe2 increases, so that the ionic plate approach towards the electrode is firstly braked, and then it is changed by the plate's movement in the opposite direction. These effects in the capacity have been discussed,

for example,in


0.6 5 0.4 31

O.Z

0

O.i

O.E

0.3

~ voE£s

Fig.18. The fitting of the th3ory to the experimental dependence C(U) (U is the overvoltage supplied by a potential source) for AuJAg~RbI 5 interphase. 1-5: the experiment ( t=22, 6o, 100,145, • • 190 O C correspondingly ); 1'-5': the theory187'q88

91

92


Refs.~O-45,189. Assuming the existence of the Ste~n gap between a~ electrode and the centres of the ions adjacent to its surface but

reser-

ving the ideal conductor model for a metal, we decrease the capacity. However,

the qualitative form of the capacity curve does

not change. Thus, in the Gouy-Chapmao-St~rn model and its extensions the bilayer capacity is fimite ior any finite

~-

and

~b

, in agre-

ement with a more general result (2.61) for any local statistical

f(~

)- model; What will happen,i i" the charge induced

elec-

tron relaxation will be included in these models? ~will capacity peculiarities (divergency, negativity) so naturally arising in the previously considered models with the frozen ionic " p l a t e " , s u r vive ? In order to answer these questions the theory of the GouyCbapman-~tern-Grahame ~ype was supplemented 190 by the analysis of the diffuse metal electron distribution. It was shown that lot any positive and not very strong negative charging the electron response effect can be accurately taken into account, if the ~t~rn gap width xH

is to be replaced by x H - x e ( ~ ) .

In other words,

the relaxing ~lectron plate plays a role of a metal su~lace. The q x e ( ~ ) dependence was approximated 90 by the following expantion: xe ( ~ )

The parameters x e (O), ~

= xe

and

(o) + ~ + ~

~ 2

were considered as model para-

meters, varying in the ramge of values d~termined ~rom the calculations for two limiting cases oi the screening charge localization: far away from a metal 53'68'7~A'92 and in the layer adjacent to a surface 186-188. The real ionic distribution is intermediate b~tween these opposite limits. The difference between the limitimg cases concerns mostly the p a r a m ~ o r

#

; for example, for

Au in both limits x e (0) lies in the range u.7 @ u.8 ~, and ~ ~ O, so that x e moves away from a metal, as ~ becomes more negative, but in the first limit it accelerates ( ~ > O), while in the second cas~ it moves with decc~leration ( # K 0). The ion contribution to the bilay~r capacity was calculated using expressions(2.61)-(2.64). The capacity dependence


93

"surface disorder"" have been analyzed qg°, assuming that the energies of the defect formation differ from those in the bulk only in the int~rfacial layer baying width of atomic size. (The Frenkel defects were considered). The obtained 19o dependences C ( ~ ) t u r n ed out to be strongly asymmetric functions,steeply uprising with the deepening into the cathodic region. In the whole considered range of the model p a r a m e t e r s ~

,~

, Xe(0) , characteristics of

the "surface disorder" and temperatures T, the capacity has s divergency point I 0 lying b e t w e e n - O . q

a n d - 0 . 3 C i m 2. For z ~ z _ ~ 0 the

capacity is negative. Perhaps,it seems strange that the capacity assymptote lies at smalll~l,

since in the case of the diffuse ion distribution x i

may greatly exceed the ion radius R i (in the extended Raleigh model186 -q88 x i ~ R i , so that x i ~ x e and the ,position of the asymptote within the electrochemical range of charging is quite natural). However, one should bear in mind, that in the diffuse distribution case the rapid coumter motion of ion "plate " u p o n cathodic charging favours the steep increasing and divergency of the bilayer capacity,while in works 186-188 the charge induced displacement of the ion "plate" was neglected. If the "lattice saturation" is taken into accoumt,then C ( ~ quickly decreases at large anodic charges, because there the electron plate displacement acts in alliance with the lattice satuma" The "surface disorder" is the result of the difference between the energies of the defect formation (U i for defects of the type i) in the bulk and at the surface of the ionic crystal q91-qr6. It means that U i is a function of the distance from the surface. The influence of the surface disorder upon the bilayer capacity was first discussed by Chebotin and Solov'eva 197. But instead of the self-consistent solution of the Poisson equation for ionic charge distribution they express the "surface charge" via the surface potential with a help of some heu~-istic procedure. In Ref.190 a more simple model of the surface disorder ( a step function for Ui(x) ) was adopted, but the potential drop was determined more rigorously by a computer solution of the Poisson's equation.

94


tion effect. The decreasing of the capacity from the infinite values at ~ = ~ o

to zero a t ~ - ~

+o~

may be not monotonous:

for so-

me values of the model parameters and characteristics of the "su_~face disorder" a minimum and a subsequent maximum may appear near >- = O. Thus a form of the capacity curve near pzc is very sensitive to the values of parameters. In particular ,the capacity minimum shifts towards the anodic (cathodic) region due to the "surface disorder effects",

if the interracial layer is enriched

by positive (nagative) defects. The typical dependence C ( Z ), obtained 19° for oC = -180, ~ =-2800167 , and T =~50°C in absence of any "surface disorder" is given at Fig.q9 for A g C ~ the influence of the teu~erature on the defect bulk concentration and the dielectric permittivity of a solid electrolyte were taken from the data198). The possibility of the existence of the second singular point J

~ o ~ > 2 o and the second positive branch of the capacity curve in ! the cathodic r a n g e ~ -. becoming more negatiw~, and the ion plat~ does the same (because,for example,of the "lattice saturation" effect), then ~ decreasing w i t h > - increasing(C ~ 0!) should brake and then be replaced by ~ uprising. The qualitative character of the curve ~ ( T ) in this case is displayed at Fig.20. The extremal of this kind should result 19° in the hysteresis of the capacity dependences C(U) ( see S e c t i o n 6 and Fig.20). The results of this section show, that the electron response lead to the divergency and negativity of the capacity in all considered models of the ionic screening charge. At the same time, the existence of the second asymptote and the correspond ng positive branch of the capacity strongly depends on the parameters of the models; in particular ,in the limiting case66-71,when an external field is applied to a metal,they do not appear at all.Since expected values of~-~ are not small (it was o b t a i n e d q 9 ° ~ o' -0.7 C/m2), their reliable determination demands a rigorous enough description of ion subsystem as well. The matter is that the ion concentration near a surface may be so high for largel~l, that a consistent account of the short-range repulsion and a more presise descriotion of the coulomb effects become of particular importance°

Surface Electron Response Theory With Applications I

'

,

I I I I zol I'-~0

95

F

~o 20' Z,em

,

I I

- t.00-

Fig.19. The theoretical dependences 19° C ( ~ ) in the Stern-GouyChapman - like model for the diffuse layer, supplemented by the charge-induced relaxation of the electron "plate" position x e.

~F

-Z~

f~

400

Ij|l

I I

~ll

I I

I

I/-°

-0.~

I

I

0

0.5

. (:~,V

Fig.20. The illustration of the hystersis of the ~ ( ~ ) ( at the insertion qualitatively ) and C ( ~ ) depeodences for the model metal-solid electrolyte interphase. The C ( ~ ) curve has been calculated 19° adopting the X e ( ~ ) dependence 186-188 ( for k =1)and including the "lattice relaxation" effect.

96

V.J. Feldman, M.B. Partenskii and M.M. Vorob'ev 6. Critical

i~eculiarities of the Double Layer.

The N e g a t i v e

Capacity 2roblem.

A. ~i;he ne~4ative capacity values in the double la,yer tmeor.y It seems

to us that in the tileory of the double l a y e r at the me-

tal-electrolyte psychological

interphase

factor,

which we shall c o n v e n t i o n a l l y

i d i n g of the n e g a t i v e negative

a certain role was and is p l a y e d

capacity

(NC) values.

c a p a c i t y in the c a l c u l a t i o n s have

sufficient

ground

ximations,

i n v o l v e d in its solution,

to declare

the I~C values were

the avo-

The a p p e a r e n c e

of the

been c o n s i d e r e d

a theoretical model

and

as a

(or) appro-

non adequate ~. In other words,

assuJ~ed to be 2 h y s i c a l l y

this v i e w p o i n t ~as no, consistently

call

by a

impossible,

though

argued in the c o r r e s p o n d i n g

works. In this s e c t i o n we shall negative intend

capacity values

to show

try to prove

that in co,non case

are not p r o h i b i t e d ~ .

In particular,

of the usual

electrochemical

experiment.

firstly we shall discuss

some exsu~ples of the a p p e a r a n c e

negative

in the double layer

capacity values treatment

by Lott

just the

a v o i d i n g of the NC values is the origin of the a r t i f i c i a l of the inverse

zero.

can be found in the n o n e l e c t r o c h e m i c a l

stated, case".

For instance,

in ~ e

that "the dili erential It should

attention mulated

be m e n t i o n e d

capacity curves

the d e v e l o p m e n t should like

analysis

before

inter-

they achieve literature

textbook 200 on r a d i o t e c h n i q u e capacity can not be n e g a t i v e that in some cases

to I~C values had played a positive

the critical

of the

in the cri-

et al. 165. To our mind,

r u p t i o n 170'176

as well.

But

theor2.

of NC Values may be found already

ticism 139 of the model

Examples

we

that the compact layer c a p a c i t y may be negative

in the c o n d i t i o n s

This

the

the h e i g h t e n e d

role,

of the existed m o d e l s

of the double layer

theory.

it is in any

since it stiand favours

In this c o n n e c t i o n we

the violent debate 2 0 1 - 2 0 4 ' 1 4 6 ' 1 8 8 initia.~ ted by C o o p e r and H a r r l o o n 2O5 , and an analysis by Blum et al. 186,

who

showed

to m e n t i o n

that

the potential

f u n c t i o n of an electrode

drop in the b i l a y e r is m o n o t o n o u s

charge,

if certain c o n d i t i o n s

are fulfil-

ed. The systems of the n e g a t i v e sidered,

in the e q u i l i b r i u m capacities

for ex~nple,

are discussed.

in the excited dynamic

in Ref.

207

.

The p o s s i b i l i t y systems

is con-


97

(i) The Cooper-Harrison catastrophe. It follows from ~q.(4.3) that the capacity of the model system becomes negative for those electrode charges~-~ , for which the susceptibility of the dipole lattice exceeds d / ( 4 ~ ) . In the double layer theory this situation was first considered by Cooper and Harrison (CH) 205, who imitated solvent molecules in the bilayer by the two-dimensional Ising lattice of point dipoles, placed in the external field F O = 4 ~

.

While calculating the capacity they approximated the compact layer by a flat thin circular disk-like capacitor with the gap filled by a dielectric, having such a uniform normal polarization per unit volume,

that multiplied by 4

d

it equals the potential

difference 4 ~ N s < p x > created by a dipole lattice. The "susceptibility" of the dipole lattice X =Nsd ~ p ~ / d ~ calculated within the 1~ean Field Approximation (~FA), turned out 205 to be greater than d/4, with chosen d=3 ~. Then Eq.(4.3)

"predicts a negative value

of CTn at (px > =0 ,,205 . If < p ~ =0 at ~ = 0 , then, as Parsons mentioned 201 , integral capacity K H also becomes negative in the vicinity o f ~ =0. The statement 205 that negative capacity values are inevitable in the "molecular" Ising models of the compact layer, was called the "Cooper-Harrison catastrophe "201 and, later~ the " p 0 1 a r i z a t i ° 2 on catastrophe" 03, probably, by analogy with the " 4 ~ / 3 - c a t a s t rophe ''208 (or "Lorenz catastrophe ''209, or "~osotti catastrophe ''210) which dea~s with the phase transition from para- to ferroelectric state. This "CH-catastrophe" has been widely discussed in the literature 201-204'146'188. In this connection we wish to mention the following. I. A rigorous solution of the discussed problem is absent in Ref. 205. This is not strange, since a rigorous solution of the twodimensional Ising model for a lattice in the nonzero external field have not been obtained yet 211. CH used the Weiss's mean field approximation 212. But the estimates, made by CH themselves, show that for reasonable values of the model parameters ( ~ = 2 D and a=3.6 ~) the energy of interaction between the nearest neighbours in the lattice equals Jo=~2/a3~7-- 10 -21 J, which is twice as large as

kT~4

10 - 2 1 J

at T=300 K, so that in the considered

dipole model the water should be in the antiferroelectric

state

already a~ t=lO0°C. In other words, in the temperature range t . 0

RC at fixed

where i(,~-.) is determined by Eqs.(6.2). C is given by the expression (4~[

values,

In

d i

~2~2=m d2E > 91 -d-l~ ° The potential

capacity

(RC) with

temn is the electrostatic

_

ere v e r i s i m i -

of the problem.

solved models,

capacitor"

equilibrium~ of an isolated

91

in the

There-

"by hands"

the former is the n o n e l e c t r o s t a t i c

the stable

considered

values

decision

by art-

curves.

in their nature.

of the negative

may be c o n s t r u c t e d

Let us consider

The results

capacity

the final

now a nmnber

in principle,

Here

models.

"excluded"

capacity

are approximate

they show that the negative but can not provide

order

while

theoretical

The electromecnanical

two previous lar,

of the

see Secti-

in the similar model 169 and

in the hard sphere model 176 the i~C values were ificial

of char-

the form


( 4~TC

)-I =

1 (~)

+

2 d E o . {k---~o2 d2E -I d 1 dl ~

It follows from the equilibrium ble equilibrium

103

conditions

(6.5)

(6.2) that in the sta-

state of the relaxing capacitor

the second term

in Eq.(6.5) is negative. Then C ( ~ ) < 0 for those F , where the modulus of this term exceeds the first term. In particular, it follows from Eq.(6.5) stable, near ~

that, if

1--O at ~

and the system is

then the capacity is necessarily negative in some range ( l i l l L ( 4 ~ C ( ~ ) ) -I = - 4 X _ ~ 2 / ( d 2 E o / d 1 2 ) < 0 ).

Now we conslaer an example. Let the following approximation be reasonable in the actual range 0 < 1 < io: Eo ( 1 ) :

( 1 - 1o

+

( 1 - 1 o )3 ,

(6.6)

where k 0 , while p is ambiguous. For p=O Eq. (6.6) gives the energy of the "elastic capacitor", while for k=O it gives the energy of the lineary relaxing capacitor,

which were intro-

duced in Ref. 187 as the electromechanical models of the bilayer (see Appendix D). Let us introduce for convinience the undimensional variables ° 2)

W=

E / ( k l

T2

= ~ 2/( 4 ~ k

Then Eqs.(6.1)

W

=

;

Wo :

Eo/

( k 1 2 )o

Io3 ); d 2 : 4 ~ 2 / (

k 1 o );

and (6.6) are transformed

Wo + ½ d 2 z

,

W° =

( z - I )2+

;

z=

i/i

o

;

(6.7) s = p 1° / k .

into

s

3

( z - I )

9

(6.8)

From the stable equilibrium conditions d Wo/ d z + we obtain

~2/

2 = 0 ,

d2Wo/dZ2> O

(6.9)

104

V.J. Feldman, M.B. Partenskii and M.M. Vorob'ev z

(d)

=

I -

(2s)

-I-

(2s)

-I

(I - 2 s ~ 2 ) I/~- (6.10)

Calculating C -I from Eq.(6.4) withl/10=z(6 ) from Eq.(6.10)

it is

easy to verify that for any s a range of d exists, where C-1(d )< 0 . In the particular case s =0 of the "elastic capacitor" (see Appendix D)

C(d ) diverges at I d o I =(2/3) I/2, where

the gap width Zo=2/3 and the potential difference

~ o = Z o 2/3. At

I~I>16o [ C(6 ) < 0 . If s > 0 . 5 , then a new peculiarity in the behaviour of the relaxing capacitor appears: RC looses stability at 16crL =(2s) -I/2. Thus at d = 6cr the gap between plates collapses. A "stopper" (hard wall) performed at a distance Zst < 0.5 the relaxing capacitor becomes a Zeeman-like "catastrophe machine" (see Refs.170,171 and references therein). Its energy W(z,d) has a canonical form of the "fold catastrophe with a restriction ''170'171. The RC capacity for

s=±1 is represented at Fig. 21 . As for 0~her

s values, it diverges at some ~ . F o r 6>~cr' C-1=Zst=C°nst ( Z s t ~ Z ~ r ) ) Lt is interesting to mention that the capacity becomes negative ~arlier, then the capacitor looses stability (Fig. 22):l~cr|> l~ot. This inequality is a general one (see Section 6.C.). The similar behaviour is displayed by the relaxing capacitor 222 with energy E (1) = E ° (1) + 2 ~ Z

2

1

12

, Eo(1)=~(l-lo)2+ 7-1n 1

(6.11)

The second term in Eo(1) may be interpreted as the energy of the Ampere interaction between antiparallel currents I frozen in the plates of the capacitor. The dependences C(~[~), even more strange than those discussed above, may arise if a relaxing capacitor with the nonelectrostatic energy Eo(Z)=sgn ~. z -~ could be realised. In this case, at any ~- , C ( ~ ) = o o forl~l=1, and C ( ~ ) < 0 for i~l < I . The relaxing capacitors with energies (6.6) and (6.11) clearly illustrate the origin of the negative capacity values. At the same time, since in these models the divergency point go¢O , the integral capacity K is always positive. We may propose also a model with the negative integral capacity.


I

C,(4~o) -~ 4 -5

J II

0

,/ I

!

0.5

I

i

I I

I I les~

I

I I

I

Figo21. The capacity of the anharmonic relaxing capacitor221for two values of the anharmonisity parameter s=l (curve I) and s= -I (curve 2). Tb~ stopper position Zst = 0.25.

0 o.5

~

S

Fig.22. The state diagram for the anharmonic relaxing capacitor 221. 1: l~-,crI ; 2: I ~ •

105

106


It is a "bicapacitor", which has two outer plates fixed at a distance a from each other, and two internal plates with fixed charges (per unit area) ~ p and -~-~p on them, bonded by a spring. Let the gap between internal plates be

I . The energy of this

system is

E = 2X ~ 2 (

a-1 ) + 2 / [ : ( ~ p

-,~_.)21 + k( 1_1o )2

(6.12)

The differential and integral capacities of this "bicapacitor" are equal: ( 4 ~ C ) -I= (4JgK) -1= a-~-pl Z = a

4YU ~ 2 k

,

(6.13)

because 2~ I

:

1o

-

k

2

47 +

k

Zp'Z

is a l i n e a r function of the charge ~ . It follows from Eq.(6.13) that if a g 4 ~ - p 2 / k =K-l< 0 . It is useful to study a special case

, then C-I= lo=2~ Zp2/k

,

when the gap between internal plates is negligible at Z =0 . The charging of the external plates results in the appearance of two competitive contributions in the total potential drop:

@i=4~.a

and q b 2 = - 4 ~ . ~ l . The second contribution @ 2 arises due to the attraction between internal and external plates and resembles the dipole contribution in the molecular models of the compact layer. For a < 4 ~ ~-.p2/k the induced contribution dominates over @ I' id est the integral capacity is negative.

@2

So, we have proved that the divergency and the negative capacity values naturally arises if the plates of the capacitor move to each other upon charging. This result is of extra interest, because the relaxing capacitor elearly demonstrates the origin of the negative capacities in the microscopical models of the bilayer, where the translational relaxation upon charging is taken into account, while the "bicapacitor" illustrates molecular models.

the situation in the


107

B. Are negative capacit,y values possible in nature? To answer this question ~ let us turn to the fundamental monograph

by Landau and Lifshitz I. The statements

that the capacity

should not be negative are met in two chapters of this book. Are there any contradictions tical predictions

between these state~ents

of the negative

I. The conclusion

capacity of any two conductors

(see Ref.1,

the energy of the electrostatic ntaty,

capacity values discussed above?

that the mutual

should be positive was derived

§2) from the fact that

field is a positevily defined qus-

assuming the linear dependence

ges of the conductors.

and the theore-

However,

between potentials

and char-

in the previously discussed micro-

scopical models

(Sections 4 and 5) and in the relaxing capacitor

this dependence

is substantially

"bicapacitor" near,

the dependence

but the capacity

electrostatic

energy,

C=K does not determine

termined.

the sign of the

C < 0 . discussed calculations

the electrode

and only after that the potential" ~b was de-

For purity let us consider

We fix a charge ~

and charges is li-

and the total energy of the electrostatic

2. In all of the previously Y'~ was fixed,

In the case of the

between potential

field remains positive even if charge

nonlinear.

the relaxing capacitor

(6.1),

at one of its plates, which is assumed to be i

isolated before charging and after it ("~-.-control").

The second

grounded plate of the capacitor gets the charge of the opposite sign.

The increased electrostatic

wards each other,

force removes

so a new equilibrium gap

tential difference

~

(~)

are established.

wn, starting from some ~ o the capacity ample,

for "elastic capacitor

klo/(6~)

ding the region with

C

that everywhere at extrema inclu-

, the relaxing capacitor state is sta-

v a r i a t i a n

Ell ~

of

o

1 at

f i x e d

(6.14)

sign of the static di-

E (k) of matter is considered

Dolgov et al.223and references this fundamental

l()-) and a new po-

becomes negative.

The similar question about the admissible electric function

to-

As we have already sho-

(see Appendix D)

. It is easy to verify

the plates

ther~[u. However,

in the review by the solution of

problem has a little concern to our discussion.

108


We

see thatforY:control

does not break with respect But usually

the appearance

the stability

of the electrically

to the variation

we deal with a " ~ - c o n t r o l "

ence between electrodes

source

is fixed.

but it should be determined

equilibrium "energy"

of the "global"

of the "global"

,

Here E( )-. ) is the energy

to consider

23a)

the investigated differ-

no more a free vari-

from the condition

of the stable

(including a potentiostat)

dp

) = s(Z.,

system. I The

) -7.,@

of the previous

extrema l( Z ), id est E( ~- ) = E( ~ with the i n t e r e s t i n g d i s c u s s i o n 224,225 fu~

experiment:

so that the potential

Then ~ i s

system

(See Fig.

system is

Z

~(

capacity

isolated

of the state variable.

cell is joined to a potential able,

of the negative

the physical

origin

(6o~5)

isolated

system

at the

, l ( ~ )). In c o n n e c t i o n it is, probably, use-

of the second

contribution

in Eq.(6.15); this is done in Appendix E. The stable cont~tions for the "global" system are expressed as

equilibrium

dZ and ---d2E = C - I ( z

)>

0

(6 17)

dZ2 It follows ntrol"

from the condition

the stable

Eq.(6.16)

states

correspond

with

We see that negative ntrol", only

"isolated"

capacities

(~),

given by

values

only.

The condi-

Thus,

system p r e d i c t e d

C>O,

considered

because

the i n t e r d i c t i o n

-control",

capacity

~

the requirement

by this r e q u i r e m e n t ,

while

to " 4

that in the case of " d p - c o -

on the extrema curve

to positive

tion (6.17) coincides Ref. 1, § 25.

forbidden

(6.17)

previously,

they appeared

on the negative

the negativity by a theoretical

In connection to achieve

with Eq.(6.17) the states

in the " ~ - - c o n t r o l " a capacitor

we must

with

experiment

about

it is interesting ~-I >

}~'01

refers of an

does not conthe shortcom-

to discuss,

, f o r which

The matter

link it to a potential

for " ~- -co-

capacity

model,

in

are not

of the capacity

tradict physical principles and says nothing ings of the theoretical approach.

way

obtained

the

C < O,

is that to charge

source,

and in this


109

moment an isolated system transforms into a global one. Thus one may think that a charge corresponding to a negative capacity value can not be transferred to a capacitor. However, this opinion is wrong. For example, if a source supplies a potential i~l~l~ol then a nonequilibrium process of the gap contraction will start accompanied by the i n c r e a ~ o f

15- I

. Interrupting this process

in the suitable moment (switching a source off) we can obtain any desired ~

value, including

I~'1 ~1>"ol

. After isolation the

capacitor will achieve its equilibrium state (for a given ~

a

system has only one equilibrium state). This is one way to observe the negative branch of the capacity. The other possibility is to fix the plates positions thus preventing a gap relaxation, and to supply a potential ~

, required to get a desired

~-=~/(4~d).

This situation can be realized without any external interference, if a characteristic time interval of the mechanical relaxation of the capacitor (associated with inertia, friction etc.) greatly exceeds the charging time, so that during the charging process one deals, practically, with a non-relaxing capacitor. An important question is can we observe negative capacity in the potential-controlled experiment? The answer is yes, if a RClike system is in series with a capacitor having CoY O. Then always a range of charges exists , where the total inverse capacity

C-1 = Crc -1 though

+

Col

is positive and the state of the system is stable,

CrcK O. It follows that the negative values of the com-

pact layer capacity are not forbidden, if the total capacity of the compact and diffuse layers connected in series is positive. H Thus the negative CH values deemed pathological evolve naturally from very concrete strictly solvable models and should in a sense be a rule rather than exception.

To verify this conclusion

~According to Grahame's factorization ~ o r m u l a

(4.1~

the concen-

tration independent contribution C H should be equal to the total capacity C in the hypothetical limit of the infinitely high concentration c o Then the negative C H value should mean that in the corresponding range of charges the bilayer is unstable at I

I However, CH should depend on

c -~o

c at least at high enough con-

centrations, and this effect should be taken into account in the investigation of the correspondence between the CH value and the stability of the bilayer.

110


the extra experimental efforts are necessary ~. But one should bear in mind that the interpretation of the experiment in the vicinity o f ~

should be extremely carefull, since in this range o -I of charges the contribution C H is small and may be of compara-

ble size with errors involved in the Gouy-Chapman theory (or its extensions) and (or) in the experimental data. Besides,

to extract

the reliable information near pzc (the problem, arising if

~o

is

close to zero) the roughness of an electrode surface should be 126 properly taken into account C. Critical properties of the double layer under the potential control. Equations (6.16) for the extremal @ ( Z ) are identical for " Z - " and " @ - c o n t r o l ''I~7'188 while ~-~ is not too close to the critical point 7"~ =~-o " In tile vicinity of ~-~o the fluctuations in the "global"system become of importance, but for the present we shall pay no attention to the fluctuation contribution into the "energy" E and corresponding changes in the extremal,

thus acting in spirit

of the Landau theory of phase transitions. So we can use the traditional density functional approach with the normalization condition (2.3) to analyze not only an isolated system but a "global" system, in the vicinity of ~ o

as well. At the s ~ e

time se-

rious distinctions exist between these systems. We have already mentioned one of them: the negative capacity branch is attainable only at ~ - c o n t r o l ,

while at @ -control the corresponding ~-. va-

lues are the points of the local maximums of the energy curves E(~-,~=const)

and are "unattainable" in the equilibriums. ~ow we

shall discuss some additional peculiarities, inherent to a "global" system with the relaxing subsysten~ of the type considered above. (i) The degenerate critical point and the canonical form of the double layer energy.

In correspondence with Eqs.(6.15)-(6.17)

the divergency point ~ o

' where the capacity has its vertical

asymptote, ( C-I(~-o)=0 ) is a degenerate critical point of the energy E~'(5-,@) of the "global" system: The Wittgenstein's "...a question (can exist) only where an answer exists" and " if a question can be framed at all, it is also possible to answer it ''226 supports our hope that these efforts will not be useless.


111

W 0

I

I

•, q87 Fig.2}. The "enorg~' surface of the elastic capacz~or . The extremal ~ ( ~ ) is represented by . the dashed c u r. v~.eW~, ~ ,,X ~ s -~. -t/=j~ ~_T,_Y~ . . -L,~"IJ -- ~ ~a,-~,. a. The "e~er~' W (d ,~ ) of the isolated elastlc capacl~or~--~ b. The"energy" ~ (g ,~ ) = W ( g ) - d ~ of the elastic capacitor joined to a potentiostat.

112

V.O. Feldman, M.B. Partenskii and M.M. Vorob'ev

(6.18)

=0 , for q~ = @ o

£ Expanding

- >- = )-o = ~£2 E(Y7)

~- = ~-o

in the Taylor series in the point %- o up to the

third order, one can easily obtain

~(Z,cP)

where

= ~- (Z:_£::o)3

2~ =(dC-I/d~)'

a = ~-I/2(qDo-qD)

_>-:(@_~o

) +s(2_o,~o)

(6.19)

>-= ~-o" The "diffeomorphism"

x= o(1/3()--)-o) ,

after omitting the "varied constant ''170

~(~o,@o)-~-~o(~-~o ) transforms the energy ~(~ ,@) canonical form of the "fold" catastrophy 170'171

into the

E ( x, a) : ~ x3 + a x

In the "typical ''170'171 case o( = 0

(6.20)

and the qualitative

our of the system (strictly speaking, degenerate

critical

behavi-

in a some domain near the

point) is determined by the characteristic

form (6.20) of the energy of the system. In other words,

the high-

er order terms can be omitted in the typical case, if there is no any special

(for exs~aple, synm~etrical) restrictions.

This nontri-

vial statement follows i rom the Thom theorem (for details see books 170, 171 and references therein). The Thom theorem gives also a guarantee

that the canonical form

of the energy of a system in the vicinity of the degenerate critical point is a catastrophy.

Typically

the energy has a fold ca-

tastrophy canonical form, if there is one control variable our case). If the number

(@

in

p of the control variables increases

(a role of a control variable may be played by the c o n c e n t r a t i o n and other par~neters of an electrolyte, parameters of a metal, temperature etc.)

then the elementary catastrophes

order (from the Thorn's list) will

of the higher

typically appear.

if p=2 and there is onlyl one state v a r i a b l e ( ~ ) , catastrophy E(x,a,b)= @ x 4 @ a x 2 + b x will arise.

For example,

then the "cusp"

Surface Electron Response Theory With Applications 113 The Them theorem also states that typically the catastrophy form is structurely stable. Thus we can be sure that small "deformations" of the energy family (like that given by Eq.(6.20) ), produced, for example, by alteration of the external parameters of the model, will not change the qualitative "catastrophical" form of the energy and will not lead to the disappearence of the degenerate critical point. (ii) The catastrophy fla~s. If the energy of a system has a degenerate critical point the system gets a number of critical peculiarities ("catastrophy flags") 170'171. The four among them: "unattainability", instability, divergency of the linear response function and critical softening of a mode were cohsidered in Refs. 187,188 with an elastic capacitor as an example. The energy of this capacitor after switching to a potential source is (compare with Eq.(6.15)

~(d ,~)

):

= w(d)

- d.,:p

5 2

, W ( d ) = '--~--(1- d 2 / 4 )

(6.21)

The energy surface W ( ~ , ~ ) is represented at Fig.23h. It clearl5 illustrates the geometry of the fold catastrophy, the existence el unattainable stationary states (with C < O ) , the presence of a bifurcation point ~ o and of an unstable r e g i o n j ~ | ~ I~o ~ , where W(~)Imonotonously, decreases and the gap collapses. The divergency of the capacity C =(I-3 d 2 / 2 ) - I ~ ( 6o_6)-I at ~ = Go=(2/3) 1/2 have been already discussed. Here we only mention that in the vicinity of the degenerate critical p o i n t ~ l ~ - d o ~ I ~ o ( ~ o - ~ ) I I/2, I~l< ~oI' id est the dependence ~ - ~ o form with the critical exponent ~=2,

on ~ o - ~ so that

has a power law C ( ~ ) = (31TJ) -1/2

. .1~o_ ~.1-1/2~ t T o _ T I ( 1 - ~ ) 1 ~ . The dynamics of the relaxing capacitor near the degenerate critical point is also of interest. The equation of the plate's motion in the simplest ease s=O at fixed ~ is:

dYz ~ ~ = z - 1 + ~2/(2z2) m

(6.22)

114 where

V.J. Feldman, M.B. Partenskii and M.M. Vorob'ev m is the mass of a plate (per unit area). It follows from

Eq.(6.22)

that the own frequency ~ o of the mechanical oscillati-

ons of the plate decreases proportionally ( ~ o - ~ ) 1/4, when ~ approaches

~o'

id est it depends on ~

according to a power law with

the critical exponent I/4. Some effects of the critical softening of the oscillation mode of the charge upon the measured impedance of the electrochemical cell is discussed in Her. 188 ~. The "fold" catastrophy has only one minimum, so after loosing stability the system enters a dynamic regime. Another situation, when instabil~ty transfers system into a new equilibrium state, is also rather typical. In this case other catastrophe "flags" also manifest themselves. We mean the "multimodality" (a system may have two (and more) equilibrium states)

, the equilibrium:

state "branching off" (small changes of the initial values of the control variables may result ~n great changes of the final states of the state variables) and the hysteresis (the process is not reversible, when the change of the control variables is inversed). These flags are clearly demonstrated by Figs. 9.1;9.3;9.4 from Ref. 171 . The bimodality of the energy, taking the catastrophically relaxing capacitors with s stop in the gap as an example, have been discussed in Refs. 70,187-188,221-222 . It was also shown 70'221' 222 that in these models the dependence ~ ( @ ) maY, have a hysteresis. A role of a stop at a metal-electrolyte interphase may be played by any constraint on the electrolyte ion concentration and (or) on the bilayer gap value (associated, for exs~iple, with the finiteness of the number of positions allowed for ions in the compact layer) 188. This possibility is confirmed by a model calculation 190 of the metal-solid electrolyte system (see discussion in Section 5 and Fig. 20). To study this problem in detail it is useful to solve Eq.(6.22) together with the Kirchgoff equations,

thus taking into account the

potential drop in other elements of a current. In connection with the a.c. measurements the case of the ha~lonic change of the electromotion force is of main interest.

Considering, for example,

the

relaxing capacitor in series with an active resistor, one must supplement Eq.(6.22) by equation q=(~/z)~

zq/A + R d q / d t = ~ o ( 1 +

bc o s ~ t ) ,

where

is the charge of the capacitor (A is the plate's area),

R is the resistence (both in the corresponding units), and the


115

(iii) The problem of critical exponents. As we have already seen, the divergency of the bilaye~" capacity testifies to the instabilitj of the bilayer in the " ~ - c o n t r o l " experime~t. Our analysis (see, particularly, Section 5) has sho~vn, that the appearance of this peculiarity in the elctrochemical range of charging is real, and the comparison o:' t~le obtained C((~) dependences ~ith the ex' periment confirmed this suspicion. At the same time the analysis of Sections 4-6 could not pretend to a reliable description of the bilayer in a close vicinity O f ~ o and, in particular, to the prediction of the correct values of critical exponents. This happens not only due to the simpleness of the microscopical models of the bila~er (for instance, their onedimensionality),but, and this is, perhaps, more important, due to the accepted "quasi-equilibrium"

manner of their analysis. While

calculating the ene±~gy of the system we adopt the extrema values of the microscopical variables (n(x), ni(x), the gap a, etc.), but neglect their large-scale critical fluctuationso A result is that similar to the Landau 228'1 or mean-field theories the quasi-equllibrium analysis gives the values of the critical expJnents, which often disagree with experiment.~ In application to the bilayer critical properties this general conclusion may be illustrated by the results 232 of the phenomenological analysis of the "temperature jump" investigations 232'233 of the bilayer (within the fluctuation theory 234 of phase transitions). (iv) The connection_.between the bilayer critical properties for " ~ - " and for " ~ control~ We have considered the bilayer instability in the "(~-control" experiment. Its main symptom is the divergency of the bilayer capacity, while its main feature is the bilayer instability u.ith respect to the process of charge transfer from a potential source to an electrode. V/e have also considered the instabilities

(phase transitions) in

right-hand side of the equation is the electromotive force. The nontrivial dynamic properties of this circuit are discussed in Ref. 227. Here we only n~ention that in the vicinity of the degenerate critical point even the extraction of the capacity from the impedance measurements become a problem. ~ o r e o v e r , ~he Thom theorem predicts170,229 the universality of the critical exponents witnin the Landau-like or mean-field theory. This fact superfluously confirms that nonadequacy of the classical critical exponents follows from the attempts to describe the critical behaviour of the system involving only macroscopical variables, id est it can be traced to the failure of a finite-dimensional approximation to t h ~ 2 ~ a ~ i t i o n function on the infinite-dimensional space of all states. ~ - ~ "

116


the ">--control"

gedanken experiments,

arising in some model sys-

tems having no charge exchange with a potential source. Examples of this kind are the "displacement-type" transition in the mo168 del of the compact layer at the metal-electrolyte solution interphase (Section 4oB.) and its analogs in the "anharmonic" relaxing capacitors 221'222 (Section 6.A.). Another example is the "order-disorder" transition in the dipole lattice of the "molecular capacitor", if it exists (see Appendix C ). Let us analyze how the instabilities and phase transitios, if they are inherent to the compact layer, artificially isolated in some gedanken experiment, will manifest themselves in the real "qb-control" experiment. For simplicity, we shall assume, that in any equilibrium state t i l l the degenerate critical point ~ o ' the bilayer capacity can be adequately represented by the compact layer capacity C H in series with the diffuse layer capacity C d . As we have already mentioned in Section 6.B. , to analyze the system stability under " R - c o n t r o l " one may consider the charge dependence of the capacity of the isolated system. In the " ~ - c o n t r o l " experiment the role of the response function is played by the inverse capacity CH-S = d ~ / d ~ since in this case the state variable is ~ and the control variable i s ~ , In the bifurcation point ~-cr the inverse capacity of the isolated system approaches infinity. ~ioreover, the analysis of the model systems discussed shows that H

CH-I~

-Oo

for

I~I-~

~cr~

- 0

(6.23)

Then the total bilayer capacity approaches minus infinity at For example, the inverse capacity of the anharmonic capacitors goes to minus infinity due to the divergency of the derivative d 1 / d ~ at ~- = ~ c r (see Section 6.A.). The same is true for the model metal-SIES interphase70, 169 where the "velocity" d a/d of the gap compression upon charging also approaches infinity at the bifurcation point (see Eg.(2.53) ). The inverse capacity of the "molecular capacitor" C-1 ~ -co , if the dipole lattice susceptibility ~--~ oo dix C ).

at I~.l-~

'I~rL -0 (see Eq.(4.3)

and Appen-


117

~=~cr' if O d ( ~ c r ) is finite. It follows, if C(~-=0) > 0, that the total inverse capacity should change a sign at some intermediateS- = ~ o , id est the critical charge ~ o of the "global" system is smaller in modulus, than the bifurcation charge~'~cr of the isolated s y s t e m : I T ol J ~ c r l . This means that in the system, joined to a potential source, the bilayer instability with respect to a charge transfer from a potentiostat arises at smaller J~-I, than the instability (in particular, phase transition) in the compact layer. It also i follows, that near the bifurcation point ~ o' where C-1=0, C H ( ~ ) is inevitably finite and negative, since CH-I= -Cd-l~ 0 . These general results are confirmed by the analysis of the "anharmonic" capacitor (see Section 6.A.) . So, when ~ = ~ o , the "global" system looses the equilibrium. This instability may transfer the system into a new equilibrium state with a bilayer charge Z i ( ~'l > 1~-~ ). If l~'l > I ~

,

then a phase transition in the compact layer may occur ate- ~ c r during the nonequilibrium process of the bilayer charging from a source. For example, in the case of the surface inactive electrolytes the compact layer may exhibit a charge-induced phase transition 232 of the type, typical for antiferroelectrics (see also Appendix C), or the cooperative

"displacement-type"

reconstruc-

tion, as discussed in Refs. 70,168-169 , while in the case of the specifically adsorbing electrolytes the compact layer instability may be additionally associated,

for example, with the charge-

induced cooperative destruction of the solvation shells of the electrolyte ions. The discussed instabilities, if appear, will "interfere" with each other, thus manifesting themselves in the experiment in some complex way. This possibility should be taken into account, for example, when the peaks of the relaxation time of the bilayer are interpreted. If the instability of the bilayer under "~-control', is "driven" by the compact layer instability ( i.g., the "orderdisorder" phase transition) then near ~ the relaxation time o associated with the charging of the diffuse layer, and not with the reorientation of the solvent molecules in the compact layer, goes to infinity. At the same time, when during the non-equilibrium process of the instability growth the bilayer charge approaches ~-~cr' the relaxation time of the reorientation of the solvent molecules should sharply increase. Of course, the dynamical analysis is necessary to obtain the

118


reliable information about the development of the instability, and our "quasi-equilibrium" consideration is only a prelude to it. In conclusion we wish to remind that the instability in the "global" system is not inevitably associated with the instability of an "isolated" bilayer or its compact part. This immediately follows from the analysis 187-188 of the model metal-solid electrolyte interphase (Section 5) and from the investigations 167 of the relaxing capacitors (Appendix D)

. Perhaps,

the "phase tran-

sition" in the bilayer at the metal-electrolyte solution interphase, discussed in work 235 also belongs to this class of instabilities. (v~ Some peculiarities in the experimental behaviour of the metalelectrolyte interphases.

Now we shall consider those peculiari-

ties in the properties of the double layer at metal-electrolyte interphase, which can be interpreted as the catastrophe "flags" or critical phenomena, associated with potential-induced phase transitions. We are aware of the possibility of other explanations of these peculiarities. However, we wish to underline that this approach gives a simple and rather universal explanation of the various existing experimental facts and predicts a number of new ones. Abnormally high values of the bilayer capacity have been measured at interphases between metals (Au and Pt) and solid electrolytes, which are silver ion conductors (imperfect ionic crystals AgBr, AgC1; ionic superconductors ~-Agl, Ag4RbI 5 180-183). In these systems the differential capacity achieves tenth and hundreds of F/cm 2 (for instance, in the case of Pt/AgC1, C(~-) ~ 2 . 5 F/m 2 in the temperature range t=300-400°C 216). In the contacts of Ga, In, Au, Ag and some other metal electrodes with highly concentrated solutions of the surface-inactive electrolytes the bilayer capacity near pzc is of the same order of magnitude ( I. F/m 2 )125-129,147-154. ~oreover, in these systems the differential capacity usually increases very steeply, as if it approaches asymptote. Thus, the measured capacities in a whole number of metalelectrolyte systems behave themselves so, as one may expect for the susceptibility near the degenerate critical point. This is confirmed also by a fact,

that a sharp capacity uprising usually occu-

res near the boundary of the stable ("unpolarized" or "blocking") range of the electrode potentials. For example, as Frumkin, Grlgor'


119

ev and Bagotskaya 148 have pointed out, in a contact of Ga with a highly concentrated (c ~In) SIES after steep uprising of the capacity up to the values of 0.3 F/m 2 ( EaClO 4) and 1.5 F/m 2 (Na2S04, KC1, KBr, KI) the bilayer capacity sharply falls down accompanied by the "creation of the film of the oxide phase". In other words, immediately after the steep uprising of the susceptibility the instability arises, which may be interpreted as a manifestation of the catastrophyo The gedanken experiment with the bimodal electromechanical models has shown, that the transition of the system from one stable equilibrium state to another, accompanied by the strong charging induced by a small change of the potential near the bifurcation point may be erroneously interpreted as the Jump up of the capacity at corresponding~ . The attempt to display this discontinuous behaviour by a continuous curve will result in the sharp peak on the capacity dependence. Thus, sharp peaks on the capacity curves are also suspicious as the "fingerprints" of the catastrophe. The "critical softening of the mode" means that the time ~ of the relaxation of the charge in the bilayer approachs the infinity at the degenerate critical point@ The critical behaviour of the relaxation time is characterized by a large critical exponent. To confirm this for metal-electrolyte interphase Benderskii et alo 232 used the results of the relaxation measurements obtained by the "temperature jump" technique 233. As it has been mentioned earlier, the "critical softening" of the charge relaxation mode should be accompanied by a sharp increasing of the frequency dispersion of the capacity, measured in the potential range near the bifurcation point. Indeed, the low frequency dispersion of the capacity was discovered long ago both for metal-electrolyte solution 40,42,236 and for metal-solid electrolyte interphases 180. In many cases the dispersion turned out to by associated with the surface roughness 237 ,238 or some artefacts of the experiment 236'239, but in other cases the opposite was established 180,240-242 ~. The comparison of the dispersion curves 240-242 with dependences 180,240-242 C ( ~ ) for the same ~Some new ideas in the analysis of the influence of the surface roughness upon the dispersion of the impedance,

associated with

the application of the fractal geometry 243 to description of the metal-electrolyte interphases, may be found in works 244 and references therein.

120


contacts where

shows

C(~P)

red capacity ponds near

steeply

the boundary

at those potentials,

At Ga/SiES

at potentials,

region,

of the frequency may be interpreted point

~ o' where

One sees a lot of problems this means

that

the m e a s u -

which

corres-

(see Fig. 14), id est

thus revealing

to the case of the n~etal-solid

at some @

interphase

of the capacity

of the stable

to the bifurcation opinion,

increases

on frequency

decrease"

the i n c r e a s i n g

pacity

increases.

"depends

to the sharo

our similar So,

that d i s p e r s i o n

electrolyte

dispersion

the behavi interphase

of the bilayer

as a result

of the a p p r o a c h

C ( @ ) has an asymptote.

in this field

of research.

this area is a prominent

To our

one.

Acknowled~nents

We are grateful

to i.V.Dobrovolskaya,

L.S.Feld~an,

makova, V . J . K r i c h e v s k a y a a~id M . V J ~ o r o b , e v a tance at all s t ~ e s of this work.

ca-

L.I{.Eol-

for helpful

assis-

Surface Electron Response Theory With Applications 121 Appendices A. Some useful relations and estimates (i I Atomic units (a.u.).lel= ~ = m e I . The length unit is Bohr (a H) and the energy unit is Hartree (Har): aH= h2/(me e2) = 0.529177 ~, H a r = 27.21160 eV. ~ i ) The dielectric permittivity. The dielectric constant of vacuum g o = 8.85419 10 -12 F/m = 8 . 8 5 4 1 9 ~ F / c m 2 ~ . The bulk dielectric permittivity ~V of water equels ~78; for silver conducting solid electrolytes, discussed in this paper, ~¥ ~I0.20. The "compact layer permittivity" 10 for various systems. (iii) The charge density

~

an~ield

~ H lies between 3 and

F 0 = 4yK~.

In electroche-

mistry the charge density is usually measured in /~C/cm 2 . I ~ C / c m 2 = 6.24146 10 -4 e/A 2 = 1.74779 10 -4 e/aH2 . The field FO= I V/A = 1010V/m corresponds to the charge density ~ = &oFo = 8 . 8 5 4 1 9 ~ C / c m 2 = 1.54752 10 -3 e/a~, , so that the electrochemical range of charges I~I ~ 20 /~C/cm = 3.5.10 -3 e/aH 2 corresponds to the field F 0 ~ 2 V/A. (iv I The capacity C and the effective bilayer gap i* = ( $ ~ C ~ -I. In electrochemistry C is usually measured in ~ F / c m 2 = 0.01 F/m2. The capacity value I F/m 2 correspond~ 1 ~ = 0.0885 ~ = 0.167 a-, so that typical electrochemical capacity values I0$I00 /~F/cm ~ corresponds to l~= 1,0.1 A. (v~ The dipole moment _~. The dipole moment of molecules is usually measured in Debae (D). ID = 10 -18 electrostatic units = 3.336 10 -30 C.m = 0.2082 e A = 0.3935 e-a H. The d~pole moment of a water molecule is usually taken as 1.835 D = 6.12 10 -30 C.m = o 0.382 e A = 0.722 e.a H. (vi) Size of water molecules and their surface density N S in the interfacial (compact) layer. The "diameter" of a water molecule is usually choosen as 3 A . Assuming that the solvent molecules in the "compact" layer constitute the two-dimensional hexagonal lattice, one obtains N~= 2 / ( a 2 ~ ) , where a is the lattice parameter. For water NS= I01~m -2 = 0.1 ~-2 = 0.0280 -2 is usually accepted. V#ith this value for NS, ~ N S = 2.02 10-2e/aH. (vii) Some energy values. The ionization potential I and the affinity S for water molecule: I = 12.60 eV, S = 0.9 eV. The energy (heat) of adsorption and similar quantaties chemists usually measure in kCal/mole. I eV/molecule = 23.045 kCal/mole, since the International Calory is defined by equality I Cal = 4.1868 J (Ther-

122


mochemical on energy

Calory

and similar

1 eV/A 2 =1.029 monolayer B.

4.1840

J ). The surface

adhesi-

quantaties

are usually

calculated

J/m 2 =16020

erg/cm 2 . For water

statistical

models

and negative

The potential

energy,

10 -2 e/aH 2 =16.02

I eV/A 2 a p p r o x i m a t e l y

Local

tentials

equals

equals

in eV/A 2.

10 eV/molecule.

po-

in the case of the nonamenable

charges

produced

by the uniforla background

is amenable

if

~n where

no

is the greatest

Eq.(2.18)).

If inequality

ous electron case

distribution

the lower bound

derived

root of the equation (B.I)

f(n)

is defined

in Eq.(2o9).

varistional

possible

approach.

distributions

leading

We start with a layered Fig.

slabS).

of the electron

(B.2),

if the local

energy will

equal

division nitely

density

transverse

form of the curve, to Peano's

of the type,

electron equals

by the

suptra-

one of the

densities

represented no,

at

separated

is assumed.

bound The

if the electrostatic

can be achieved on the

The resulting fills

total

contribu-

by the infinite

"cells" w i t h infidistribution

the total

area

has a (similar

curves).

be a greatest

~n p(n)

root

= 0

by

the total n o n e l e c t r o s t a t i c the lower energy

distribution size.

volmue,

to Eq.(B.2).

which u n i f o r m l y

plane-filling

can be

density,

be calculated

approximation

The latter

of the electmon

small

Let n'o

sysZem

metal

Let us try to guess

the same value,

tion is negligible.

system

the electron

cannot

For this d i s t r i b u t i o n

energy

In this

(Bo2)

in the considered

structure

24 (layers with constant

jellium

then the h o m o g e n e unstable.

of the electron

However,

lower bound

ditional

(see

= ~ f(n o)

of electrons

the obtained

p(n)=O

of the "j-model". 58 The r e s u l t is

inf S Z n ]

plying

is disturbed,

in the bulk becomes

of the energy

from the analysis

here N is a n ~ b e r

(B.I)

o

of the equation

(B.3)


o

123

;ss

Fig.24. The model electron distribution in the b~lk of a metal with the background density ~ 4 n o at the intermediate step of the construction of the electron density distribution, for which inf E In] is achieved. The transverse size of the neutral "elementary cell" is denoted by So

I1.

./

. ,,,,.

rL(xl

//1.~ / X)

w'D a,I

0 x

0

X

Fig°25. The profile of the electron density near a surface for the metastable state of the electron system of a jellium metal with the background density n ot

~ n ~ n

0 .

124


which

coincides

pressibility. bution

with

In the range

is still

electron

ing the lower

bound

(A2.2).

(near a metal

represented

the electron ces,

state

only

discuss

functional

leave

"Thomas-Fermi

the charged

Thomas-Fermi-Dirac so does not exist

Dirac

Q = ~ < S x >

(c.2)

=~th~

where th x =(eX-e-X)/(eX+e -x) is the hypertangent, sionless energy

:

(Fo+~x) : . ~ o kT kT

JS o + _ _ kT

and the dimen-

(0.3)

Our goal is to determine or ~ , and then to obtain the susceptibility ~ ( T , F o) of the dipole lattice from the expression (4.4):

d (T'F°) = ~ N s d

Fo

~2N s (c.4)

- kT/coth~ -JS °

The requirement of self-consistency of wing state equation

0 . 170~171

ity of any critical point the potential is transformed by a homomorphism) is equivalent

In other words,

, ISx~ ~z 1 kT

Then it follows from the Thom theorem

tly, Eq.(C.5)

+

kT

has no degenerate

te temperatures,

~F o o O

~

127

is negative,

the dipole lattice is always in

the paraelectric state, and ~ is finite. Taking into account the terms of the higher orders in the expansion of ~ with respect to

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