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SHORT

COMMUNICATION

DIFFUSION

TO FRACTAL

L. NYIKOS* Central H-1525

and

SURFACES

T. PAJKOSSY

Research Institute for Physics Budapest, P.O.Box 49, Hungary

ABSTRACT Diffusion (Warburg) impedance is generalized for irregular electrode surfaces characterized by their fractal dimension exponent of the impedance is shown to be Df. The frequency by computer simulation. (Df-1)/2, a result verified INTRODUCTION Fractal

geometry

in destroying lytic

(1) is an efficient

tool

for characterizing

has recently been in very general terms. Much attention and reaction on fractal surfaces (2-9) and to the role

surfaces diffusion

metal

(10) and

producing

deposition

fll- 16)

experiments

fractal

(17-221

and

surfaces

colloid

irregular paid

to

of diffusion

including

electro-

aggregation

(23-25).

Diffusion

to fractal surfaces, however, has received little attention alit is touqht to play an important role in such diverse fields as ca-

though

fluorescence quenching talysis, enzyme kinetics, We have recently published a fractal model pedance

of blocking

showed that the

the

is, given

model

as

Z

the

!iw)-aB

=

as

electrode

experiment

nature

manifests

of the

itself

time-domain

equivalent

of

proportional

to teaB

rather

Df=2

with

The

is the

Df being

aB(D,) L

angular

the

relation

frequency

been

im-

arguments we element (CPE),

effective has

(26,271. the

and

fractal verified

diin a

interface,

described

in non-conventional

the CPE than

behaviour being

by the

electrode

corresponds

exponential

fractal

kinetics to a current

as in the

dimen-

since

the

response

conventional

case. Here,

of charge

we extend across

this

a fractal

impedance

is a CPE with

computer

simulation.

treatment interface,

aD = (Df-1)/2

DIFFUSION

tal

describes

Using scaling to be a constant plase

i = fl,w

l/(Df-1)

surface.

relaxation

that

junctions.

interface where

spin

(29).

irreqular

Df>2,

Z of

aB is given

of the

The sion

impedance

exponent

mension

metal/electrolyte

and (28)

to describe

diffusion-limilted

transfer

we show that this generalized diffusion and, finally, we verify the result by

IMPEDANCE

CALCULATION

To derive the diffusion impedance we consider a totally interface to which only diffusion carries the particles. 1347

absorbing fracAt time zero

1348 the

L. NYIKOS concentration

troughout case

the

of the

electroactive

system,

a potential and we are interested

(30)1,

T. PAJKOSSV

AND

species

is assumed

to be uniform

step is applied [similarly to the Cotrell in the time dependence of the Faradaic

current. Let us denote effective

by a factor both

of

Ai and

1.

vector

5,

every

= c(r,t')

this,

the

quire

t' = %*t.

plied

by the area

area

at the

ways

and

point onto

of the the

chosen

coordinate

with

the

surface. Since 2 a we obtain

as

flux

(31), and

2:

one

between

enlarge

thus

fractal system

area

scales

enlargement in the bulk

as g + Br, where For

the time

the first argument In case Q the total

EDf

the boundary

is obtained

for

case

(5) with (1).

a

or at the

in-

= is a position map

appropriately.

and

proportional

as

of

For

conditions

5 as the

re-

flux multi-

to the

concentration gra-1 this latter quantity is proportional to 8 and the conventional scaling law for the current as (1)

jati3,a2t) = aj(l,t) where

the

interface

the concentration

is scaled

equation

being

which the

In the conventional case interface scales simply

unchanged

system.

that

current

of the

original

enlarged

provided

total

cutoffs

(Fig.l).

the area

of the diffusion

The

outer

X0, respectively,

(2) hi is left

holds

structure

scales

the

of a: conventional, system.

be mapped

in a suitably

ca(B=,t)

dient

case

Illustration hypothetical

can

and

t3 in two different

fractal

In case terface

inner

by Ai and

Xo are magnified,

a2 . In the

Fig.

the

Df prevails,

refers to' system size. current is proportional

to the area

for

times

that

are short in relation to r0 = X~/D, where D is the diffusion coefficient of the electroactive species. In other words, as long as the root mean square displacement of the diffusing particles is small compared with the outer cut-

Diffusion to fractd surfaces

1349

dominate and the large-scale off, local properties effect on the diffusion current. Therefore, since

structure for case

has very 2 area

little

scales

as

BDf we obtain jb(B,t) The

key

gularities

point

is that

disappears

As a consequence,

Eqs.

D.f j(l,t)

= X:/D

for t'ri

and we have

from

= 6

(2)

the effect

the approximate 2 we get, after

1 and

of the

equality dropping

smallest

irre-

j,(B,t) = jbt6,t). the first argument,

1-Df jtt32t)/j(t) which

has

the

solution

j = t

--o D

with

The

power-law

behaviour

is valid

the exponent =

aD

(3)

= 6

(Df-1)/2

between

the

(4)

temporal

cutoffs

~~ and

~~ cor-

responding to the spatial ones Xi and ho. -0 i.e. Z = (ia,) D. In the frequency domain Eqs. 3 and 4 lead to CPE behaviour, interface eD = l/2 i.e. the Warburg limit is For a smooth, two-dimensional recovered. that

For

the

of the

bulk

The

decay

lines

result

differently Since

D f>2 the

initial

within

of Eq.

for

a given

data

at interfaces

we decided

are

than

to the

distance

case,

with

to perform

is greater owing

4 is identical

an analogous

no experimental

measured

exponent is faster

from

with

and

the

that

simulations

reflects

a greater

fractal with

(26),

that

determined

fractal

in Eq.

obtained

of Ref.

for the diffusion

to see

simply proportion

surface.

of de Gennes

as yet

an independently

this

that

is in conflict

available

computer

l/2:

fact

32.

impedance dimension,

4 is correct.

SIMULATION For

simulation

rectangular hitting

the wall.

quarter

(Fig.

a stepsize wall

quarter wall

several

it took line

required

thousands

shown

generated

to perform

smaller

the dotted

at the place

after

were

allowed

of steps

boundary

a computationally

Particles

considerably

crossing

fractal

allow

2) and

the number

ticles

the

segments

than was

with

the

inner

were

walk

cutoff.

2 was

chosen

detection

probability

its

inside lattice

If a particle

re-introduced

since

of a particle

on a square

and a new particle

2 were

by symmetry. of steps

uniform

a random

noted

in Fig.

in Fig.

efficient

hit

generated. to the

Finally,

particles

"killed"

and disregarded

one with

the Par-

same

not hitting since

the they

do not contribute to the short-time current. After generating s104 reacting particles, the decay curve was constructed from the statistics of the arrival times. The algorithm was tested by replacing non-fractal one: the diffusion current was indeed,

the fractal boundary by a straight, then found to decay as t -l/2 as ,

it should. For the fractal interface of Fig. 2 with Df = 2.5, Eq. 4 predicts aD = =3/4. Figure 3 shows that this is in fact the case: the simulated diffusion current decays as t -3/4 showing that Eq. 4 is the correct result in agreement

L. NYIKOS AND T. PAIKO~SY

1350

wi th

Ref.

Fig.

2.

26.

An approximant, obtained by iteration (cf. Ref. l), to the fractal border curve used in the computer simulation. The boundaries used in the actual calculations were characterized by "o/Xo > 256.

Kinetic haviour, terms.

and The

transfer

at

fractals

offer

tc at

irregular an

generalization

impedance,

considered occuring

processes

be

to

suitable

realistic,

surfaces

efficient

fractai

developed a

and

Fig.

non-idealized

often

way

to

surfaces

verified

starting

point

Simulated current plotted in double fashion. The line with the slope of that 5redicted by

3.

show

handle

of

decay logarithmic is drawn -314, i.e. Eq. 4.

non-conventional irregularity

diffusion-controlled

in

the

present

in

the

investigations

paper,

bein

general charge

is

therefore

of

processes

surfaces. REFERENCES

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SahAni,

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T.

33.. ~~fi?_eR&a?i~e%[email protected]&~~Solid .

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