ON THE T H E O R Y OF THE E L E C T R O C H E M I C A L DIODE

(UDC 541.13)

B. M. G r ~ f o v

Institute of Electrochemistry, Academy of Sciences, USSR Translated from Izvestiya Akademii Nauk SSSR, Seriya Khimicheskaya, No. 5,

pp. 814-821, May, 1964 Original article submitted October 26, 1962

The electrochemical diode represents an electrochemical cell, the electrodes of which possess different areas.

Oxidation-reduction reactions can occur on the surface of the electrodes. In the solution, in addition to the oxidized and reduced ions, there are also ions of a third substance, which guarantee electrical neutrality of the volume of the

solution. We shall investigate the properties of the diode for the example of the reaction I 3" + 2e ~ 3I ' . Electrical �9 +

neutrality of the solution is guaranteedbyK ions. On account of the great difference in the surface of the electrodes,

we can l imit ourselves to a consideration of the processes only at the small electrode [1]. A model convenient for

theoretical investigations is the model of a spherical diode with diameter 2a.

1. Let us consider first of all the natural convection arising during electrolysis on account of the appearance

of an electrolyte density gradient, due to the ion concentration gradient. For simplicity, we shall consider that natu-

ral convection is determined by the gradient of the concentration of ions of one kind. Let us denote the concentra-

tion of this ion as c. Let us consider a steady-state system of l imit ing current. Let us direct the Z-.axis vert ically

downward. We shall count the coordinatex from the lower point of the sphere along the surface, the coordinate.y_.per- pendieular to the surface of the sphere. Let us denote as0 the angle between the Z- and Y-axes. The ascending motion of the electrolyte that arises will possess axial symmetry with respect to the Z-axis. The rate of motion of

the solution along the X- and Y-axes will be denoted as u and v, respectively.

The equations of the diffusion boundary layer, describing the approach of ions toward the electrode, can be

represented in the form [2, 3]

Ou Ou 02u u - - -{- v age 0 sin 0 (1) = o7

02 + , ~ = D (2) Ox Oy Off'-

where (p -- is a dimensionless concentration, equal to (p = (c o - - c ) / e0; c O -- is the ion concentration far from the electrode; v is the kinemat ic viscosity of the solution; D is the diffusion coefficient; a is a dimensionless coeffi-

cient determining the dependence of the solution density p on the concentration c

a - - Pc=c0 ~ c=co'

~.is the value of the acceleration of free fall.

In the equation of motion (1), we actually can neglect the inertial factors. On account of the small thickness

of the boundary layer, due to the large value of the Prandtl number, the velocity of the solution does not have t ime

to rise appreciably, Hence the inertial factors, which are squares of the velocity, prove to be small in size. To Eqs. (1), (2) we should add the equation of continuity of the liquid medium. The system of equations written should satis- fy the following conditions: the velocities u, v_ become 0 at the surface of the sphere, the velocity u becomes 0 at infinity, the concentration ~o is equal to I at the surface of the sphere and 0 in the volume of the solution. Let us introduce the function of the current ~0 according to the equations:

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t 0 r t 0r l Z - - _ _ V~---

R (z) Oy B (x) Ox

where R(x) is the distance from the point of the surface to the Z-axis. The equation of continuity will be automat- ically satisfied.

To obtain a solution of the system (1), (2), in which the inertial factors are omitted, let us use the method of similarity. Let us assume:

e %

(n) 0

where 7/ is a new independent variable, according to which self-simulation is accomplished

[ a g ~' /~ sin%0 o 1/, " Y

11 = \4--~-a/ i s sin%0 d0] 0

then Eqs. (I) and (2) are rewritten in the form

/ " ' + Cp = 0 q~" + 3Pr/tp" = 0 (8)

The prime denotes differentiation with respect to ~. The functions land ~0 should satisfy the conditions

/ I~:=o-- l ' Ir,=o = 0 ] ' I~=co = 0 ~P/a=o = 1 ep I~=co = 0 (4)

The system (8), (4) formally coincides with the system considered in a study of the natural convection around a plate [3], if in this system we omit the quadratic components with respect to the velocity, which actually are not considered in [3] on account of their smallness in the calculation. Hence we can immediately write

c = 0 ,7 .Pr 'hco.~l

From this, we find for the density of the diffusion flux:

0c = 0,7.Dc0Pr % \4--~-a] 1--=D ~ v=0 sin%0

i' sin%0 dO] '/` 5

(5)

In order to obtain the complete flux, we should integrate (5) with respect to the entire surface of the sphere. Then the dimensionless flux, determined by the Nusseh number, will be equal to

N U ~ I

Dc.._o . 4na~ a

- - 0,35 Pr'/,Gr'/, f o sin%0 dO

~ [ ! sin%0 dO] V"

where the Grasshof number Gr is defined by the function

G r = aga 3 / (4v 2)

Numerical integration gives a value of 2.0 for the integral determined, Hence we have:

5 u = 0,7 .Pr'/,. GrV, * (Sa)

It follows from expression (5) that the thinnest diffusion layer exists at the base of the sphere, where the angle 0 is equal to 0. The thickness of the diffusion layer at this point is equal to:

�9 In [5], an expression analogous to (5a) with numerical coefficient 0.664 ~ 0.05 was obtained by a more complex method, using integral functions of the type of the theorem of momentum in a hydrodynamic layer [2]. Thus, both methods lead to the same results.

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(5 (0) = 1/0,7- (3I. v D a / a g ) '/. C6)

The thickness of the diffusion layer around the sphere in a fixed medium is equal to the radius of the sphere a [4].

Hence, if we find that 5(0) >> a, then delivery of the substance to the electrode will be accomplished chiefly on

account of molecular transport. When the opposite inequality 5(0) << a is fulfilled, the influence of natural convec- tion must be taken into consideration. Using (6), we can obtain an expression for the critical dimensions of the sphere

act [aer = 5(0)]

acr = 1 / (0 ,7) ' / ' . (~1~ ~,D I (~g)'/~.

If the dimensions of the electrode are much smaller than the critical value, then the influence of natural convection

on the transport of matter can be neglected. Numerical estimates give the values acr(a=10 ~ = 0.9 �9 10 -2 cm ;

a c r ( a = 10 "-6) =0.9 �9 10 - a c m for the crit ical dimensions when Pr = 10 s, g = 10 a c m / sec ~, v = 10 "2 cm2/sec.

in a consideration of the natural convection around a spherical electrode, we did not consider the influence of

a l l types of ions on the electrolyte density, nor did we consider the presence of migration fluxes. However, we might

think that the order of magnitude of the cr i t ical dimensions would not change in a more correct formulation of the

problem.

2. We shall consider that the spherical electrode possesses sufficiently small diameter, so that we can be l im-

ited to a consideration of the phenomena in a fixed medium. In an electrochemical diode, the diffusion and migra-

tion fluxes of ions, generally speaking, are of the same order of magnitude. Hence, the steady-state flow of the cur-

rent will 'be described by the system of equations [1, 3]

it3) (a) . a_~ ~ = dn (3) F En(3) (7) D(a) r 2 dr R T

i ( - ) ( ~ _ ) . ._2_ ~ = _ dn(-) _ F En(- (s) D(-P r2 dr R T

0 = - - dn(+) F (9) dr -~ ~ En(+)

where j(~)(a) and j ( ' ) ( a ) - are the ion fluxes on the surface of the electrode, D(s), D(-)-- are the diffusion coefficients,

n ( s ) n(-) n(+)_ are the ion concentrations. The indices (9), (-) , (+) pertain to ions Is-, I - , and K +, respectively; r

is the distance from a given point to the center of the sphere; E is the electric field intensity; F is Faraday's number;

R is the gas constant; T is the absolute temperature.

To these three equations we should add the condition of electr ical neutrality of the solution

rt (+) = n(-) -k n (s) ( t0)

From (7)- (10Ne can obtain the fact that the concentration of nondischarging ions n (+) should satisfy the equa- tioll.

an(+) ( 1~ 3) (a) i (-) ( a ) ) .2/r~. dr = - - 1/2 k - b ~ + D(-)

From this, by direct integration we fend the distribution of the concentration of nondischarging ions around the elec- trode. Knowing the function n (+) =n (+)(r) from (9), we find the value of the electric field E = E(r). After this, the

linear Eqs. (7) and (8) are already integrated. The integration constants are determined from the conditions for infi-

nity

n ( + ) ( ~ ) = no(+) n(-) (oo) = n0~-) n(3)(c~) = n0 (a)

and for the conditions at the electrode

](3) (a) = I ] 8•a2F ](-) (a) = - - 3 1 / 8na2F

where ! is the electric current passing through the diode. The direction of the external normal of the spherical e lec-

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trode is selected as the positive direction. If the e lec t rochemica l react ion on the electrode can be considered reversi-

ble, then the following expression is obtained for the polar izat ion curve of the diode:

{ [ ')1} R T V = - - ~/, in I - - S--E D(-~ ,~ , , , ,~- '~ i 32:~F,,,,~o§ ~

where V is the voltage drop on the diode, considering both the potent ial drop between the electrode and the layer of e lect rolyte near the e lectrode, and the vol tage drop in the e lectrolyte . The equi l ibr ium value of the voltage drop on the diode is equal to 0, since it is assumed that the second (large) e lectrode is made out of the same meta l as the

small e lectrode.

An investigation of expression (11) shows that if the ini t ial concentrations of the ions I" and I[ are commen- surate, then the polar izat ion curve possesses a more or less symmetr ica l form. If there is much of one substance and l i t t le of the other, then the symmetry of the curve is disturbed, and the ion begins to manifest definite rect ifying properties. Hence, the presence of great ly differing areas in the electrodes is a necessary, but by no means sufficient condition for the appearance of rec t i f ica t ion effects. In addition to different areas, a different concentration of the

ions par t ic ipat ing in the oxidat ion-reduct ions on the electrodes is also required.

An exper imenta l study [6] of the e i ec t rochemica l diode shows that there is satisfactory quanti ta t ive agreement

between the exper imenta l polar izat ion curves and the curve (11).

3. From (11) for the case when there is l i t t le of the substance I.q" in solution, we have an approximate expres-

sion for the polar izat ion curve

I; / R T . V : lh In {1 + I / 8z~D(3)aFno( 3)} (12)

If there is l i t t le of the I- ions, on the other hand, then from (11) we have approximate ly

F / RT. V ~--- - - a/~ In {t - - 3 I / 8~D(-)aFno(-)} (13)

The polar izat ion curves (12) and (13) can be obtained on the assumption that the transport of ions present in

negl igible amounts is accomplished only by a diffusion mechanism, while the concentrations of the other types of ions (including the ions par t ic ipat ing in the e lect rode reaction) general ly do not change, in spite of the passage of current . This result agrees with the well-known premise that the migrat ion forces do not influence the transport of

ions in the presence of an excess of an indifferent e lec t ro ly te [1, 3],

The diode contains no ions of an extraneous e lect rolyte in the l i tera l sense. However, on account of the fact that the fluxes of discharging ions are of the same order of magnitude, while the ini t ia l ion concentrations differ greatly, prac t ica l ly no change occurs in the concentration of the ions present in large amounts, and they form the background together with K+ ions. Hence our further consideration of the diode will be performed considering only the concentrat ion changes of the ions present in smal l amounts, where these changes will satisfy the diffusion equa-

tion:

On I at = D . 1 / r . 0 2 1 Ote(rn) (14)

where D is the diffusion coeff ic ient , n is the concentration. In the case of a reversible react ion, the boundary condi-

tion on the e lect rode can be written in the form [1]

F V } (15) n ]r=a = no e x p q

V is the voltage drop on the diode, no is the initial ion concentration, q = 2, if n~)<<n0 (') and q = - 2/3, if nl ")

<< n (s). Another boundary condition is the condition at infinity:

n i r ~ = no (16)

The boundary value problem (14)-(16) is standard for the problems of ma themat i ca l physics, and its solution is written

in the form of quadratures in the case of an arbitrary dependence of the vol tage drop on the t ime.

The t ime character izing the re laxat ion of the diffusion process around a spherical e lectrode is equal to [1]

T = a 2 / z~D (17)

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We shall be interested in the case of a periodic process, when the voltage on the diode is a periodic continuation

with period T o of the function

t

-3 Wo(i ,i r o < t < r o - - 3- "-TT) when y

(18)

In this case it is expedient to consider an established periodic process, when the concentration and flux of the ions

are periodic functions of the period To. Then they permit resolution in a trigonometric Fourier series:

+co To n(r , t) = ~_j nk ( r ) e i~ n~(r) = l / T o l n (r, t) e -i~kt dt

~=--r 0 (19)

{ok = 2~k / To

From (14), (15), (16), (18), and (19) we can obtain an expression for the flux density Jr=a in the form

D no / - - T - ] ' r = a ( t ) = T I n [ r = a - - n o ] - i - D T ] -~o .F (t) (20)

where l +oo 8n V 2 ~ [sh qWo when k is even e2i=/~

F (t) = ~ 4q~W ~ + g2kZ [ch qWo when k is odd k ~ - - c o

(21)

The function F(t) possesses the following property, which is obtained directly from (21)

F ( t - lhT0) = - -F( t ) (22)

Expression (20) shows that if the period of the set voltage T O is much greater than the relaxation t ime r, i. e. T O >> a2/D, then the second component in (20)is small in comparison with the first. A quasisteady-state process will

exist in the diode, and, consequently, the diode will manifest rectifying properties. If the period of the set voltage

is reduced such that the inequali ty T O << aZ/D is fulfilled, then the flux on the electrode will be determined mainly

by the second component in (20). Hence, in view of the property (22) of the function F(t), we shall have j r=a ~t--�89 0) = - J r = a(t). This means that the amount of electr ici ty flowing in the forward and reverse directions will be prac-

t ica l iy the same if only T << a2/D.

4. Formulas (17) and (20) show that the diode can be forced to operate normally on sufficiently high frequen-

cies if the dimensions of the electrode are made sufficiently small. However, small dimensions of the electrode lead

to a great resistance and very small currents. Hence it is natural to attempt to construct a diode by taking many small electrodes, connected in parallel . Then the smallness of the electrode dimensions will guarantee operation at

high frequencies, while the large number of the electrodes will remove the diffusion difficulties.

Let us find the condition of independence of the work of the diodes from one another, As a model, let us con-

sider a plane on which hemispheric small electrodes, representing a single electrode, are applied in a definite order.

The distance between individual small electrodes significantly exceeds the dimensions of the electrodes themselves. Let us consider the steady-state case, when a total flux I passes on each of the small electrodes. We need to find the

solution of the Laplace equation satisfying the conditions: I. The flux on each of the small electrodes is equal to some constant value. II. The flux to the interelectrode surface is equal to zero.III. The concentration far from the

small electrodes takes a definite value n 0,

Let us introduce the following expression into the discussion:

n ~) -- ~ I/2nDr~ -[- no (23) k

where k is the number of the small electrode, r k is the distance from the point under consideration to the center of

the hemispherical k-electrode. Expression (23)is a harmonic function and satisfies condition III. Moreover, it can be

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determined by direct differentiation that expression (23) satisfies condition II. However, expression (23) does not satisfy condition I. Expression (28) can be rewritten in the form

n (r) = no q- I / 2nDr -4- ~, ' I /2nDrk (24) k

The prime after the sign of summation means that the electrode close to the surface of which the expression (23) is investigated was omitted in the summation. Here r is the distance from a given point to the center of the iso- lated electrode. The first two components in (24) coincid~ with the expression for the concentration close to the iso- lated hemispheric electrode. Hence expression (23) will approximately satisfy condition I if the value of the sum with the prime sign is small in comparison with the value of I/21r Da close to the surface of the isolated electrode, when r ~ a. Let the small electrodes be distributed with some density o , Then, summing with respect to all the electrodes falling in a circle of radius b, the center of which coincides with the center of the investigated hemisphere,

m

we obtain

(25) Y~'I/2z~Drk = I~ /D . ( V ' ~ + a ~ - - [ ' -~) ~ I~b/D

In obtaining (25), we considered the fact that the dimensions of the electrode a m are small in comparison with the in- terelectrode distance L and in comparison with the radius of the circle b, within which many electrodes were placed. Hence we could replace the summation by integration.

The value of E'I /2~Dr k depends on the arrangement of the investigated electrode in the system of small elec- trodes. The greatest value of this sum will occur for the electrode situated in the center of the system. If the system of electrodes is made in the form of a circle with radius bsyst ' then we obtain for the greatest value of the sum

E ' I = --D bsyst 2~Drk )max I~

Hence, in order for each small electrode of the system of electrodes to work independently from the other electrodes, we must require that the fonowing inequality be fulfilled:

a ~ t / (bsyst6) ~ L ~ / bsyst (26)

This result was obtained from a consideration of the steady-state case, The same inequality can be arrived at if we consider the process of establishment of constant current (relaxation process) or if we consider established peri-

odic processes.

The author would like to express his gratitude to V. G. Levieh for his supervision of the work, in addition to thanking P. D. Lukovtsev, I. V. Strizhevskii, L. A. Sokolov, and M. A, Novitskii for their discussion of the results.

S U M M A R Y

]. The condition of the absence of influence of convective motion on the transport of ions in an electrochemi- cal diode was elucidated.

2. An expression was obtained for the polarization curve of the diode, considering diffusion and migration

fluxes of the ions.

3, The rectification effect disappears if the period of variation of the applied voltage is significantly smaller than the relaxation time of the diffusion process close to the small electrode.

4. The condition of independence of the work of individual small electrodes, forming a single system of many

electrodes, was obtained.

L I T E R A T U R E C I T E D

1. A .N . Frumkin, V. S. Bagotskii, Z. A. Iofa, and B. N. Kabanov, Kinetics of Electrode Processes [in Russian],

Moscow University, 1952. 2. G. Schlichting, Theory of the Boundary Layer [in Russian], Foreign Literature Press, Moscow, 1956. 3. V.G. Levich, Physicoehemical Hydrodynamics [in Russian], State Press for Physical and Mathematical Literature,

Moscow, 1959.

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4. P, Delachey, New Instruments and Methods in Electrochemistry [Russian translation], Foreign Literature Press Moscow, 1957.

5, H, J. Merk and I. A. Prins, A ppl. Sci. Res., A_4, 207 (1984). 6. L. A, Sokolov and I. V. Strizhevskii, Transactions of the Academy of Communal Economy [in Russian], United

Scientific and Technical Press, 1962.

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