Three-dimensional nucleation with diffusion controlled growth

Three-dimensional nucleation with diffusion controlled growth

J. Efectroanal. Chem., 177 (1984) 13-23 Elsevier Sequoia S.A., Lausanne - Printed THREE-DIMENSIONAL GROWTH PART I. NUMBER 13 in The Netherlands NUC...

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J. Efectroanal. Chem., 177 (1984) 13-23 Elsevier Sequoia S.A., Lausanne - Printed

THREE-DIMENSIONAL GROWTH PART I. NUMBER

13 in The Netherlands

NUCLEATION

DENSITY

WITH DIFFUSION

CONTROLLED

OF ACTIVE SITES AND NUCLEATION

RATES

PER SITE

B.R. SCHARIFKER Departamento (Received

and J. MOSTANY

de Quimica, Universidad Simbn Bolioar, Apartado 80659, Caracas 1080

3rd March

-A (Venezuela)

1984)

ABSTRACT

The current transient for three-dimensional nucleation on a finite number of active sites, followed by diffusion controlled growth, has been analysed. Both the number density of active sites and the true nucleation rate per site can be obtained from the current maximum of single-step potentiostatic experiments. Instantaneous and progressive nucleation are shown to be special cases of the more general situation described.

LIST OF SYMBOLS A z c c(r, t) c(z, t) D E F I IIn

J,

JP k M N

N, N 0.1 P R

nucleation rate per active site (s-t) = zFD’/~~/~‘/’ (A s’12 cm-*) = &rkD (s-l) bulk concentration (mol cme3) instantaneous concentration at a radial distance r from hemispherical centre (mol cm-‘) instantaneous concentration at a distance z from the planar surface (mol cm-3) diffusion coefficient (cm* s-t) expectation number the Faraday constant (C mol-‘) current density (A cm-*) maximum of current (A cmm2) flux density at the surface of hemispherical centre (mol cmm2 s-‘) flux density at the surface of planar electrode (mol cmm2 s-t) = (In&/p) ‘I2 dimensionless constant affecting growth rate of diffusion zones molar mass of the deposit (g mol-‘) number density of growing centres (cmm2) number density of active sites (cme2) number density of active sites not converted into growth centres at time t (cmm2) probability radial distance from a representative point (cm)

0022-0728/84/$03.00

0 1984 Elsevier Sequoia

S.A.

14

r

‘d

t u W x ,? a z

radial distance from hemispherical centre (cm) radius of diffusion zone (cm) time (s) age of diffusion zone (s) Jacobian matrix = bt, dimensionless time of the maximum distance normal to the plane (cm) = b/A dimensionless parameter density of the deposit (g cmm3) fractional area covered by diffusion zones

INTRODUCTION

The experimental study of electrochemical three-dimensional nucleation processes usually relies upon the determination of the number of crystallites on an electrode surface, either from the direct microscopic observation of the surface or by correlating the measured electrical variable, most commonly the current, to the number of crystallites. The result has invariably been the product of the nucleation rate per active site on the surface, A, and the number density of active sites for nucleation, NO [1,2]. Both quantities, however, vary with overpotential and in order to establish the exact relationship between the overpotential and the kinetics of nucleation, it is necessary to determine separately NOand A. In particular, the value deduced for the free energy change of nucleation from experimental data on the variation of the product AN, with overpotential would be in gross error if N,, itself varied significantly with overpotential. The fact that the number of active sites for nucleation may not be constant throughout the overpotential range has been recognised already [3] and even a distribution of activities of the sites has been proposed [4] but, to the knowledge of the present authors, the number of active sites has been assumed to be constant in all experimental investigations of three-dimensional nucleation to date, and thus the variation of the apparent nucleation rate, AN,, with overpotential, has been equated to the variation of true nucleation rate A with overpotential, notwithstanding the fact that it is the latter quantity which is needed for a knowledge of the energetics of nucleation. Thus it is important to be able to determine NO and A separately if meaningful comparisons are to be made with the different theories of nucleation

[5,61. In this paper, we will develop the method of calculating current transients for three-dimensional nucleation with diffusion controlled growth described in previous communications [7,8], in order to determine NO and A simultaneously from single step potentiostatic current transients. The second paper of this series will present the basic results of the experimental investigation of the number density of active sites and the nucleation rates per site for the nucleation of lead on vitreous carbon. Further aspects, such as the modification of the number density of sites and of the true nucleation rates by adsorption of ions from solution, will be reported in later communications.

15

Projection

of hemispherical

diffusional fields onto the plane of the electrodes

In order to calculate the current transient, we will consider the equivalent area of plane surface towards which diffuses, by linear diffusion, the same amount of material that would be transferred, through spherical diffusion, to a hemispherical growing centre [7]. The total amount of material that diffuses to a hemispherical electrode of radius rO is 2ar,2J,(t)

= -2 sr,2D(ac(r,

t)/ar).=.,

(1)

where J,(t) is the flux at the surface of the hemisphere. Under conditions of semi-infinite diffusion and with zero surface concentration of electroactive species, the mass transport equation (1) results in the well known expression [9] (ac( r, t)/ar).=,,

= c/( 7rDt)1’2 + c/r0

(2)

but, because of the small size of the nuclei in the early stages of their growth, the second term in eqn. (2) dominates at all times and, to a good approximation, mass transport may be considered to proceed in a steady state, i.e. (a+,

t)/a+ro

= c/r0

(3)

Hence, 2rrtJF(t)

= -2ar,,Dc

(4

The equivalent area of plane surface, rr:, towards which the same amount of of material diffuses as that given by eqn. (1) can be defined through eqn. (5) ‘rrrd2Jp(t) = -nriD(ac(z,

t)/&),_,

(5)

where Jr< t) is the flux at the surface of the planar electrode and z is the direction normal to the plane. For semi-infinite linear diffusion and complete diffusion control eqn. (5) results in 7rrd2Jp(t) = -~ar~~D’/~c/~~/~t’/~

(6)

Now, since we have defined r, in such a way as to equate the amount of material (i.e. flux x area) diffusing either to the hemisphere or to the segment of the plane of area mr2, then 2rriJ,(

t) = mriJ,,( t)

(7)

and from eqns. (4), (6) and (7) it follows that 2mr0 Dc = vr’/2riD1/2c/t’/2 form which the radius of the diffusion zone is given by rd = (2rO)1’2( nDt)1’4

(8)

This result applies to a stationary hemisphere sitting on the plane. For a growing

16

hemispherical

nucleus,

its radius can be expressed

as [2,10]

‘a ( t ) = (2DcMt/p)“*

(9)

and eqn. (8) becomes Id(t)

= (kDt)“2

(10)

where

k = (8m~M/‘p)~‘” The earlier justified

(11)

conjecture

[7] that the diffusion

zones

radii grew with t’/2

is thus

by eqns. (10) and (11).

Calculation

of coverage probabilities

and of the current transients

Having reduced the mass transport from spherical coordinates around each growing centre to linear, the overlap of hemispherical diffusion fields of individual nuclei is also reduced from overlap in “2:” dimensions [11,12] to overlap in two dimensions, projection

for which the calculation of the diffusional

that nucleation the surface, growing

nuclei

available i.e.

occurs

initially

randomly

available

is uniform

on a limited

for nucleation.

with time,

due to their conversion

- dN,,,/dt

of the true coverage

of the electrode

fields onto the plane is exact [13-161.

then

number

density

If their probability the decay

N, of active sites on of conversion

rate of the number

into nuclei is proportional

by the

We will consider

to the number

of sites,

= AN,,,

(12)

where N, f is the number density of active sites remaining available for nucleation time t. Integrating eqn. (12) and given the initial condition that N,,a = N,, then N,,, = N, exp( -At) thus the rate of formation dN/dt This

only takes

at

(13) of nuclei on the electrode

becomes

= AN,,, = AN, exp( -At)

equation

into

of, sites

into

(14) account

the reduction

of the rate of formation

of

growing nuclei due to the decay of the number of active sites because of their conversion into growing nuclei. The further decay of the number of active sites due to their ingestion either by the growing nuclei themselves or by exclusion zones for nucleation associated with the diffusion zones discussed above is taken into account by the statistical treatment of collision and overlap of diffusion zones that follows. Since the growth rate of diffusion zones is always larger than the growth rate of the nuclei that originate them, cf. eqns. (9) and (lo), then ingestion by diffusion zones will necessarily precede ingestion by growing nuclei, and thus the latter need not be considered. Furthermore, it is not even necessary to identify diffusion zones with exclusion zones, for any nucleus actually growing inside a diffusion zone will not

17

contribute to the observed nucleus [14].

current

Given the rate of formation

and will therefore

behave just

as a “phantom”

of nuclei by eqn. (14) and the growth rate of diffusion

zones by eqn. (lo), the true coverage of the electrode by them can be calculated by a number of methods, all leading to the same result. We will follow here the method of Evans [15-171

but we wish to point out that that of Avrami

[14] produced

the same

final expression for the true coverage. The probability that a representative point, chosen at random at the surface of the electrode, shall be crossed by exactly m diffusion

zones can be found from the Poisson

equation

P,=E”exp(-E)/m!

(15)

where E is the expectation number. the representative point is then

The probability

that no diffusion

field will cover

Po=exp(-E)

(16)

P, is the fractional area of the electrode that remains uncovered and, since the expectation number varies with time, is also a function of t. The radius of the projection onto the plane of the diffusion field of a growing nucleus, the diffusion zone, is given by eqn. (lo), and it follows that any diffusion zone of age u > R*/kD developed within an annulus of width dR and radial distance R from the representative point

will pass it and that the longest

originated

at time t can reach the representative

[17] that the expectation point is E = (W’0”2

(’

Jo

JR2/kD

=N,nkD[f+(l and hence

number

distance

of diffusion

2mRAN,,_ expl __ -A(t- .

from

which a diffusion

point is (kDt)‘/*.

zone

It then follows

zones that can cover a representative

u)]dudR (17)

-e-“‘)/A]

that the fractional

area covered

by diffusion

zones, 8 = 1 - exp( - E), is

B=l-exp{-NomkD[t-(l-e-A’)]}

(18)

The radial flux density of electroactive material through the boundaries of the diffusion fields will be given by the equivalent planar diffusive flux to an electrode of fractional 1=

area 0 [7]. Thus the current

( ,SD1’*c/7r1’*t’/*)(1

density

to the whole electrode

- exp{ - NomkD [ t - (1 - ebA’)/A]

This expression can be presented in non-dimensional t/t,, Fig. 1, for different values of the dimensionless Parameters

of the current

surface is

})

form [8] by plotting Z’/Zi parameter (Y = N,rkD/A.

(19) vs.

maximum

The current described by eqn. (19) passes through a maximum and therefore the current Z, and the time t, corresponding to the maximum can be evaluated by

18

equating the first derivative of eqn. (19) to zero. t, is thus given by ln(1 + 2bt, - 2btmeeA’m) - bt, + (b/A)(l

- eCA’m) = 0

(20)

where b = NgrkD

(21)

Making the substitutions x = bt m

(22)

a=b/A

(23)

we obtain from eqn. (20)

ln(1 + 2x - 2xeCxla)

-x

+ a(1 - eeX’,)

= 0

(24)

Equation (19) can now be written in terms of x and a for the current maximum, i.e. Im=(a/t~2){1-exp[-x+a(l-e-“/“)]}

(25)

where a = ~FD’/~c/,rr’/~

(26)

and thus knowledge of I,,, and t, from experimental current transients allows the simultaneous determination of x and a by solving the system of transcendental (non-linear) equations ln( 1 - I,ty’/a)

+ x - a(1 - eCxlrr) = 0

ln[l + 2x(1 - eCXILI)] -x

+ a(1 - e-X’u) = 0

(27.1) (27.2)

from which N, and A are obtained, simultaneously, from single-step potentiostatic experiments. In order to solve the system of equations (27), we first note that eqn. (27.2) can be solved numerically to obtain x as an function of a. A plot of values of x that satisfy eqn. (27.2) for different values of a, found using Newton’s method, is shown in Fig. 2. On the other hand, adding eqn. (27.1) to eqn. (27.2) one finds that -ln(l

- I,ty’/a)

= ln[l + 2x(1

- e-x/a)]

(28)

+ 2x(1 - e-x/a)]

(29)

from which I,tz’/a

= 2x(1 - e- "'")/[l

We can then use the appropriate values of x and a found above, cf. Fig. 2, to construct a plot of Z,,,t~‘/a vs. log a from eqn. (29), which is shown in Fig. 3. From these two plots and through the definitions of b, x, a and a given in eqns. (21)-(23) and (26), it is possible to find both N, and A from the experimental values of Z, and t In’ We will now examine the two limiting cases of small a, or “instantaneous”

19

IO’1

/Aem-*

i

( I/Im)’

I.C

0.5

t /tm I

0

1

2

3

4

Fig. 1. (a) Calculated mol-‘,

p=11.3

Non-dimensional

current transients for D = 1 X10-s cm’ s-l, c = 1 X10-s mol cmm3, M = 207.2 g s-‘. (1) a = 0.16, (2) LY= 0.50, (3) (Y = 2.0. (b) g cmm3, z = 2 and AN, =109 cm-’ plots of the transients in (a) together with the corresponding

plots for instantaneous

nucleation (upper continuous curve) and progressive nucleation (lower continuous curve).

20

Fig. 2. Solution of eqn. (27.2). (0) Numerical solution using Newton’s method; (-- -) asymptotic ) asymptotic solution for large values of a, eqn. (35). solution for low values of a, eqn. (32); (-

0.95 -

l/2

Imtm a 0.90 -

to9

0.70 -

-3

-2

Fig. 3. Plot of I,,,f~,/‘/u solution of eqn. (27.2).

-I

0

I

2

a 3

vs. log a, eqn. (29), using values of x corresponding

to OLfound

by the numerical

21

nucleation, and large (Y,or “progressive” nucleation. They correspond to the extreme situations of fast nucleation on a small number of active sites and slow nucleation on a large number of active sites, respectively, and have been discussed in previous communications [7,8]. For instantaneous nucleation (LX-+ 0) it has been shown that [7] 1= (~~/t’/‘)[l

-exp(-bt)]

and that t, = 1.2564/b. obtains I,,,t’,‘/a

(30) Introducing

this value into eqn. (30) and rearranging

= 1 - exp( - 1.2564) = 0.7153

one

(31)

which is the limiting value for a + 0 of I,ty’/a obtained from the solution of eqn. (27), shown in Fig. 3. Furthermore, the asymptotic solution of eqn. (27.2) for small values of (Yis x = 1.2564

lx-+0

(32)

and is shown as a broken line in Fig. 2. For progressive nucleation (CY+ cc), similarly I = (a/~‘/~)[

1 - exp( -Abt2/2)]

and t, = (4.6733/Ab)‘j2. I,t~‘/u

[7], (33)

Thus

= 1 - exp( - 2.3367) = 0.9034

(34)

which is the limiting value for cy -+ cc found from the solution of eqn. (27), Fig. 3. nucleation can be considered as special, Then “instantaneous” and “progressive” extreme cases of a more general phenomenon of heterogeneous nucleation on a finite number of active sites on the surface, as recently pointed out by Abyaneh and Fleischmann [17]. Now, from eqns. (22) and (23), Abti/2 in the exponential term in eqn. (33) can be equated to x2/2~ and then lim ( x2/2cr) a-a,

= 2.3367

thus x = (4.6733~~)~‘~ = 2.1618~~“~

(r-cc

(35)

gives the asymptotic solution of eqn. (27.2) for large values of (Ywhich is shown by the continuous line in Fig. 2. The set of equations (27) can be solved graphically, by the procedure outlined above, using the plots presented ‘in Figs. 2 and 3 as working curves. It is a simple matter, however, to find its solution by means of numerical analysis [18]. Given a set of continuous and differentiable equations in the vicinity of the roots x,(Y: (36) the Jacobian

matrix

of the set can be constructed

as the derivatives

off,

and f2 with

22

respect to their variables,

w(x, a) =

afox [ afox

af,m afdaa

1

(37)

Having an approximation to the roots of the set, xp and (Ye, a better approximation can be found from the expression ap+1

Xptl7

(38) The solution of the set of equations (27) is thus found iteratively by successive approximations of x and (Yuntil the residuals fi( x, a) and f2(x, a) are smaller than a preestablished convergence criterion. The experimental determinations of N, and A for the nucleation of lead on vitreous carbon described in Part II of this series were accomplished with this numerical method. SUMMARY

Three-dimensional nucleation on a finite number of active sites followed by diffusion controlled growth of nuclei has been analysed, and an expression for the current transient was obtained by consideration of the two-dimensional projections of three-dimensional diffusional fields of individual nuclei onto the plane of the electrode. The current maximum is shown to provide the information needed for the simultaneous evaluation of the number density of active sites on the surface and of the rate of nucleation per active site. A procedure is described whereby working curves allow the determination of both quantities from single-step potentiostatic experiments. Finally, instantaneous and progressive nucleation are shown to be special cases of the more general situation described here, and they were recovered through the asymptotic solutions for small and large values of the dimensionless parameter (Y, respectively. ACKNOWLEDGEMENT

One of us (B.S.) gratefully acknowledges the Consejo National de Investigaciones Cientificas y Tecnologicas (CONICIT) of Venezuela for financial support through grant number Sl-1227. REFERENCES 1 2 3 4 5 6

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