The initiating rate of a heterogeneous reaction

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The initiating rate of a heterogeneous reaction

Yu. Buyevich, M. Tret’Yakov

To cite this version:

Yu. Buyevich, M. Tret’Yakov. The initiating rate of a heterogeneous reaction. Journal de Physique

II, EDP Sciences, 1994, 4 (9), pp.1605-1616. �10.1051/jp2:1994220�. �jpa-00248064�

Classification Physi<'s Abstra<.ts

82.20F 82.65J 05.40

The initiating rate of

_a

heterogeneous reaction

Yu. A.

Buyevich

(1. ^2,

*)

and M. V.

Tret'yakov (2)

(')

Laboratoire

d'Aerothermique

du CNRS, 92190 Meudon, France

(2) Department of Mathematical Physics, Urals State

University,

620 083 Ekaterinburg, Russia

(Rec'eived 15 July ^{1993, revised 24}

April

1994, ac'c'epted 19 May 1994)

Abstract. A stochastic model of nucleation and subsequent

growth

of _new

phase

islands _on solid surfaces is

proposed

and expressions ^for the _mean induction time and its variance _are obtained in _an

explicit

form. The intermediate stage ^of the _new surface

phase

evolution is studied under conditions of the supersaturation

gradually decreasing

_{as a} result of the

growth

of the islands. The island size distribution _as well _as the supersaturation and the fraction of the surface

area covered with the _new phase are found _as functions of time. It is proved ^{that both} the induction

period

and decrease in surface

metastability

must be taken into account when

trying

_to treat the surface reaction kinetics, and _some conclusions of conventional models with _no allowance for the

aforementioned factors have certainly to be revised.

I. Introduction.

It is the

development

of the thin film microstructure

during

the

origination

and

growth

^of a film of _{a new} surface

phase

that determines the distribution of both

grain

size and orientation in

continuous

polycrystalline

thin films. The formulation of _a

general physical

model of the _new

phase

formation and evolution _on solid substrates is of considerable interest since microstructu- ral

peculiarities

_are

key

^{factors which} are

responsible

for _a great ^{deal of}

properties

of various

thin films, such _as mechanical

strength, rigidity,

electrical

conductivity, magnetic susceptibi- lity,

diverse

optical properties,

etc.

[1, 2].

^{This is} the _reason

why

^the

subject

of _« kinetics of

growth

and

coarsening

of _new

phase

^islands on solid substrates _» has been

extensively

studied for several decades

by using

^different

approximate approaches [1-6].

An initial

period

of any interracial and

topochemical

reaction consists in

covering

_a reaction

surface with a reaction

product layer.

It involves two basic stages ^which determine the

induction

period

of the reaction and _so affect the overall reaction _rate. The first stage ^consists in stochastic

origination

of critical nuclei

(or seeds)

of _{a new} surface

phase

influenced

by

(*) Present address NASA/Ames Research Center. Moffett Field. CA 94035. U-S-A-

extemal random fluctuations of temperature, pressure, adatom concentration and heat and

mass transfer coefficients. This stage ^leads to formation of

stably growing

islands of the _new

phase

and may be

justly

referred _{to as an «} initial _» stage. ^{The second} stage ^{includes the} more or less

rapid growth

of these islands

accompanied by

the

continuing

_emergence of _new critical nuclei in the metastable

surroundings,

until either the whole interface is

completely

covered with _{a new}

phase layer

_or the adatom concentration diminishes to its

equilibrium value,

and the

island

growth

comes to an end. This is _{an «} intermediate _» stage ^{of the surface} coverage.

We shall

primarily

_pay attention in this paper to induction

phenomena

which may be

identified _as those

resulting

in _a

relatively long period during

^{which there} are no

macroscopic

stable islands on the surface and _so

providing

for _an observable tiine

lag

in formation of the first initial nucleus. Such _a

period

is

commonly ignored,

in

spite

of the fact that it _can well be

comparable

with the overall time of the surface

covering, especially

_so as far _as surfaces of

relatively

small _{area are} concerned.

In what

follows,

the familiar concept ^{of first} passage time

[7, 8]

will be used to find _out the

lag

time that

elapses

from the termination of the

adsorption

of _{atoms onto} the surface till the

beginning

of the

relatively

fast intermediate stage ^{of the surface}

covering. Early

_stages of the

origination

of critical nuclei will be accounted for in _terms of solutions _{to a}

simple

stochastic differential

equation [8, 9]

that governs the _new

phase

island

growth

_on the average.

After

that,

the surface

covering

is

investigated

with allowance for _a

progressive

decrease in the surface

metastability.

The

supersaturation

is

commonly

assumed to be constant

during

the entire _course of _a surface reaction

[2-4].

Such _an

assumption

_may be true,

however, only

at the very

beginning

of the reaction and in _{a case} of fast

chemosorption.

Otherwise, the island

growth usually

leads to a decrease in the _mean surface concentration of adatoms which is

capable

of

greatly affecting

both initiation of _new critical nuclei and the

growth

rate of stable

macroscopic

islands. It is

quite

clear that the

supersaturation

remains

high enough

^{for the} nucleation _to continue

during

_a

sufficiently long period,

so that both the

origination

of _new nuclei and _a fall in the

growth

rate of

macroscopic

islands caused

by

the

metastability

decrease have

simultaneously

to be taken into _{account at} the intermediate stage ^{of the}

covering

that

follows the initial stage.

Allowance for such _a nonlinear feedback between the kinetics of the stable island formation and

growth

and the _rate with which the

supersaturation

is

decreasing gives

rise _{to a severe} mathematical

problem

that has _not been _overcome until very

recently.

A method _to deal with the

problem

has been

developed

in

[10]

in connection with the batch

crystallization

from

supersaturated

solutions and

supercooled

melts. After

that,

the method has been

successfully generalized

to processes of the

phase separation

of colloids in

[I Ii

where _a

general

discussion

of the _{current state} of _{art can} be found. In the present paper, the method of

[10,

II is

somewhat modified and

adapted

to

study

the _new

phase

island evolution _on solid substrates.

The aforementioned feedback becomes irrelevant

merely

in _cases when the

adsorption

from _an ambient gaseous medium is fast

enough

to

provide

for _a

continuing supply

of _new adatoms _to the reaction surface.

2. Stochastic model.

Let _us consider _a surface coverage process when the radius

R(t)

of a nucleus presents a

solution of the stochastic differential

equation

dR/dt

= a

j

R~'

a~

R~'~

₊

_{f (t)}

,

(2,

I

which has the

meaning

of a

Langevin equation.

Here coefficients _aj and a~ are functions of temperature, pressure and other external conditions. If it is the adatom transport to the nucleus

or island

perimeter

that has _to be

regarded

as a

rate-controlling

step, ^these coefficients may be found from

solving

a

pertinent

diffusion

problem

_on the

surface,

and their form

depends

on the

relevant transport ^mechanisms

[12].

In such a case, k

= 0 if the islands _are of

cylindrical

form

[4, 13]

and k

=

if the islands _are

spherical [14].

The island

shape depends

on the

equilibrium

conditions at the curved surface of the island, also _on

peculiarities

of the process of attachment to or reaction at the island surface of _new adatoms.

According

to the

thermodynamic theory

^{of the} first-order

phase transitions,

the first _{term on} the

right-hand

side of

(2. corresponds

to

changes

in the

original phase

free energy whereas the second _term is

responsible

for

changes

ⁱⁿ energy of the _new

phase,

as the island and its

perimeter

are

growing.

There is _a clear

analogy

^{with the} nucleation in the bulk of _a metastable

medium

(formation

of bubbles in

superheated

and

crystals

in

supercooled liquids

and of

droplets

in

supersaturated

_vapours, also

origination

of _new

phase

elements in metallic

alloys

in consequence of either first _or second order

phase

transitions, _etc. ). Then the first and second

terms in

(2. I) play basically

the _same role _{as terms} associated with _a difference in the Gibbs

free energy inside and outside _a

growing

_new

phase

element and with the surface tension effect in the classical

theory

of nucleation,

respectively.

It is worth

noting right

_away, in order _{not to}

return to the

question afterwards,

that the

theory

to be

developed

for islands of _{a new} surface

phase

_can

easily

be

expanded

to familiar

problems

of the critical nucleus formation in the bulk, without any difficulties of

principal

nature.

The last _term f

(t)

of the

Langevin equation (2.

I

)

is

interpreted

_as

being

due _to either internal random fluctuations in the system or an effect of _a

randomly fluctuating

environment. The

most evident _reasons of this term

making

^its _appearance are connected with inevitable

fluctuations of the adatom concentration, _pressure and temperature as well as of the island

form and local orientation of the

crystallographic

_{symmetry axes} at its surface. Random

fluctuations of relevant transfer coefficients and reaction kinetic constants are also

capable

of

contributing

to this _term. On the

whole,

the fluctuations _are to be understood in the _{same sense}

as in books

[7-9]

and in _numerous papers on noise-induced

phase

transitions in

dynamic

systems. ^In a

general

_case,

f (t

_may also

depend

_on the _current island

radius,

which reflects _an influence of the island size _on the fluctuation

intensity.

However,

solely

those fluctuations that

pertain

to the reaction kinetics and to diffusional transport ^{in the closest}

vicinity

of the island

surface can be

dependent

on R, whereas all the environmental fluctuations have to be

insensitive to the island size. In order to avoid excessive mathematical

complications

and to make final conclusions _more transparent, we shall

neglect

the

possible dependence

of the

external noise _{term on} the island radius.

For the sake of

simplicity,

we shall also confine ourselves _to

studying

_an additive noise _case.

This _{seems to} be

fully appropriate

because _we have

apparently good

reasons not to expect

multiplicative

noise contributions to

change

the relevant induction time statistics to a

significant

extent. The latter would result in coefficients aj and a~

being necessarily

looked

upon as random functions _as well. An

analysis

based on

(2,I)

with random coefficients

requires

a

special

mathematical

problem

to be

posed

and solved which has _not much in

common with the

problem

considered below and merits _an

independent

treatment.

Also to

simplify

the matter, we shall

regard fluctuating

function

f(t)

_as a Gaussian _non- correlated random process with _{zero mean.} Then it is _to be described with the

help

of the

following

correlation function _:

(f(t) f(t'))

=

2 _~r 3(t t'),

(2.2)

with «

characterizing

the noise

intensity

and

angular

^brackets

denoting

the

averaging

over all

possible

realizations of

f(t).

The white noise is known _to

provide

for _a

reasonably good

approximation

when

modelling

random fluctuations with _a finite

sufficiently

small correlation

time

[9, 16].

Because of the above

assumption

of

$(t) being independent

of

R,

_« must be

regarded

as a constant

preassigned

_parameter.

3. Induction time statistics.

From _a deterministic

point

of view, there exists such _a critical value

R~,

of the island radius that islands with R

~ R~~ grow and islands with R

~ R~~ dissolve and

eventually

vanish. The critical value is to be found from _an evident relation

R~,

₌

a~laj

,

(3.1) again reminding

of the classical

problem

of nucleation in the bulk of _a metastable medium.

When either intemal noise due to the thermal fluctuations _or extemal noise caused

by

environmental fluctuations is present, ^it

helps

to drive _a nucleus

through

this critical size.

Obviously,

such _a

driving

_process is

essentially random,

and an observed induction time may

considerably

_vary from

experiment

to

experiment

under otherwise identical conditions, all the

more so, the

larger

the parameter «. To allow for such

variations,

_we have to consider _a nucleation process associated with the presence of _some

potential

barrier U

(R )

which _an island must pass over in order _to survive. The process resembles, ^of _course, that of nucleation of _new

phase

elements in metastable continua _save for the fact that the conventional

thermodynamic theory

^of fluctuations

extensively

used in the classical

theory

of nucleation appears to be

inapplicable

to the

problem

under

study,

with fluctuations of _an

arbitrary physical origin.

There is _a

large variety

of

problems

in

physics

and

chemistry

which _can be treated with the aid of the idealized

concept

^of a Brownian

panicle moving

_{over a} finite

potential

barrier

[7, 17].

What _{we are}

really

^interested in is the well-known first passage time

problem

that involves

calculation of the distribution of the time that it takes for _a Brownian

particle

to reach _a

preassigned position.

The

practical significance

of the

problem

lies in that the mean first passage time may be

thought

of _{as a} convenient _measure of the inversed reaction rate constant.

The

potential

function

U(R)

_can be introduced

by

means of

rewriting equation

(2.I) in the

following

form _:

dR

=

(dU/dR

dt ₊

(2 «)~'2 dW, (3.2)

with

R

^R

U(R)=-aj x~'dx+a~ x~'~'dx

~~

standing

for the wanted

potential

function with _a local maximum _at R

= R~~ and

W(t being

_a

Wiener process. The rate with which the random fluctuations drive the island radius _over the barrier determines the kinetics of the initial stage ^{of the} surface

covering

and the first passage time statistics.

A suitable

quantity

that may be called _to characterize the nucleus evolution kinetics is

presented precisely by

the first passage time

[17, 18].

The latter has _to be understood _as the time

T(R~~(Ro)

^{that is} necessary for stochastic process

R(t)

to reach the critical value

R~,

^for the first time,

given

that _an initial value R

(to)

=

Ro

^lies somewhere within the range

(0,

_R~~), that

is,

T(R~~(Ro)

₌ ^min

(t _to (R (t)

=

R~,,

^R

(to)

=

Ro) (3.3)

This time varies from realization to realization and, thus, represents a stochastic

quantity

moments of which _can be obtained from

equations [7, 8]

«

~~

T~(R~~

(Ro) $~ l~

T~(R~~

(Ro

₌

nT~ (R~, (Ro) (3.4)

dRo

_{R Ro o}

at the

following boundary

conditions _:

T~(R~, (R~,)

₌

0, dT~(R~~(0 )/dRo

₌

0,

_n

~ 0

(3.5)

These conditions have _to be

supplemented

^with an

identity To(R~,(Ro)

= I.

Expressions

for the _mean value

Tj (R~~(Ro)

^of

T(R~~(Ro) representing

^the mean first passage time and for the second _moment

T~(R~,(Ro)

of the first passage time distribution follow from

solution of the

corresponding problems

identified

by (3.4)

and

(3.5). They

take the form

l~l "

l~'dXl j~'

^{dX2 ~XP}

^[U(Xl U(X2)1'")

,

Rjj ⁰

(3.6)

T~ =

~

j~" ^dxj

^{~' dx~}

^~~'

^~ix.~

^{~~ dx~}

^exp

^{[U (xi}

^{U(x~ +}

^U(x3

^{) U}

^{(x4 )1'«}

«~

Rt, 0 Rt, ⁰

Results of numerical calculation of

Tj

and T~

regarded

as functions of _mare

presented

in

figure

I and

2, respectively,

for different values of R~~ and

Ro

=

0.

It is worth

noting

that all the

problems (3.4), (3.5) degenerate

and become

singular

_{as «}

approaches

_zero, since the senior derivatives vanish and it is

impossible

for _a solution of _a first- order differential

equation

to

satisfy

two

imposed boundary

conditions. It is _a

simple

matter to

get

corresponding asymptotics by making

use of the standard method of matched

asymptotic expansions.

However, this

hardly

_pays because the process of _an island

passing

_over the

x~=o.9

= o.7

£

o.5

lo 25 40

lla, oec/mm'

Fig. 1. Mean induction time for a, 0.04 mm/s, k

= 0 and different critical radii R~, (mm).

p f

o.7

©

£

o.5

6 26 46

lla, »ec/mm~

Fig.

2. Variance of induction

period

notation is the _{same as} in

figure1.

potential

barrier is known in advance _to

gradually

lose its stochastic features and to become

wholly

deterministic _as

« is

coming

to zero. It _means that

Tj

tends to

infinity

for any

Ro

_~ R~~.

It is

expedient

to

point

out that the first passage

time,

sometimes referred _{to as an}

«induction»

time,

bears upon the

anticipated

behaviour of one island. It should be

distinguished

from the induction time that has relevance _to the emergence of the first critical nucleus in _a

given sample

of finite size and is

usually

measured in

experiments.

The latter time has to decrease _as the

sample

size grows. It _can be related _to the former _one with the

help

of

solving

_a familiar

problem

of the

theory

of

probability,

what goes

beyond

the intended scope of this paper. However, the indicated times

approximately

coincide if _a

sufficiently

small

sample

is in

question,

so that the

generation

of _more than _one critical nucleus

during

the _same

period

of

observation of the

sample

_appears to be _a

highly improbable

_event.

4. Island

growth

under

decreasing metastability.

When many islands _are

growing

due to

consuming

adatoms from the surface

adsorption layer,

and _an external inflow of the adatoms from the ambient gaseous

phase

is either absent _or insufficient _to compensate ^the

consumption,

the surface

metastability

is bound _to decrease. In

such a case, it is necessary to take into _account that reaction parameters aj and

a~ as well _{as «} become functions of the relative

supersaturation

_~

(t)

which is _to be defined _as

~

(t )

=

(c

_c~

(co c~)~

~, c~, c-o and _c

= c

(t being

understood _as the

equilibrium,

initial and actual current interstitial surface concentrations of the adatoms,

respectively.

For definiteness,

we shall focus attention

only

on the _case of islands of

cylindrical shape (k

=

0)

with _no

appreciable supply

of the adatoms from the environment. Then

equation (2.

_can be rewritten in _{a more} convenient from

dR/dt

= a

(R R~,)

R~ _~ (t +

f (t) (4,1)

The island radius has

already

been mentioned to be _a random function of time. The evolution of _an

assemblage

of _new

phase

islands has therefore to be described in terms of an island size

distribution

density

function and of

ensuing

statistical moments of the distribution. The master Fokker-Planck

equation

for the island size distribution

density

_p

(R,

t) ^that

corresponds

to stochastic differential

equation (4.I)

is of the form

k

=

I (a

~

~~'

~p + «

£

p

,

p

(R, 0)

₌

0,

R _~

R~,, (4.2)

?t ?R R aR

where

p(R, t)

is assumed to be normalized _to the average number of the islands per unit

surface _area. We choose the

following boundary

condition _:

j R

R~,

"

aR ^{~ ~ ~} R ^'~~

R

R

~~'~'

_'~

' ~ ~

'

~~'~~

where

J(~, x)

is the nucleation _rate and _x

=

x(t)

stands for the

time-dependent

surface coverage, that is, the fraction of the surface _area that is covered with the _new

phase.

The _mass balance law

yields

^the

following

relations _:

x(ti

= I exp

i- wK~(tit

,

K~

₌

j~

^R~ p

(R,

t) dR,

(4.4)

o

y~

(i

₌ Ax (i

,

A

~ po(<.o c~

)-1 (4.5)

po

being

the _new

phase

surface

density.

The surface coverage.< is similar _{to a} volume fraction in bulk systems. ^{It is defined} as the total _area of all the islands per unit _area of the surface. If

possible overlapping

of

neighbouring

islands _were

entirely ignored,

this

quantity

would be

equal

to.i~ =

arK~

(t ). This

equality

holds

approximately

true and is

quite compatible

with

experimental

evidence

during

the initial stage of the surface reaction

[15].

Such _{a case} has

previously

been treated in

[19]. According

to the

now classical

theory

of

Mampel [20]

and its various modifications elaborated in

[15],

the allowance for the

overlapping

results in formula

(4.4).

Equation (4.5)

is written _out when

assuming

^{that there is} no external influx of the adatoms onto the surface. It _can

readily

be

modified,

however, to allow for such _an influx. The result

can be

put

^down as follows _:

1~(t i

=

P

j

_~

(t ),

_.;

(i )j

Ax (t

)

,

(4.6)

where the first _{term on} the

right-hand

side presents ^{the mentioned influx of the adatoms} to the surface,

admittedly

from _a

surrounding

_gaseous

phase.

In a

comparatively simple

case this

influx _may be taken

equal

to

B[1-

_~

(t)]

[I

x(t)],

which

corresponds

to the

Langmuir

kinetics of

adsorption. Having

in mind

purely

illustrative purposes, we are

going

to _use further

only expression (4.5).

All

general

results of this paper remain valid, however, ^when

equation (4.6)

is used.

We _assume, next, the noise

intensity

to be

proportional

to the

supersaturation

(« =

b~ (t ))

and take

R~,

=

0. The first

assumption places

some restrictions _on the types ^{of random}

fluctuations

being

considered, which _we do _not intend _to discuss here. The second

assumption

is

entirely justified

when _a characteristic island radius

considerably

exceeds that of critical nuclei. It is convenient _to introduce dimensionless variables and parameters ^{in the}

following

way :

Jo

= J(~,

x)(

~,

to

=

(a/Jo)~'~

to ₌

rota,

s ₌

R/f~,

T =

t/to,

1(~,x)=J/Jo, F=pi(, F=blare.

By making

use of these

quantities

we become able _to rewrite

equations (4.2)

^and

(4.3)

_as follows _:

I=-~i+e~~, _{F(s,0)=0, s~0, (4.7)}

?T ?s

s

j- e(atlas)

+ F j~ _o =

i (y~,

xi/y~

=

i,(y~, x). (4.8)

When

following

the

methofl

of

[10, III,

we introduce _{a new} time variable

o (T)

=

j~

^~

^(t)

^dt,

^(4.9)

and

equation (4.7)

reduces _to

aF aF

a2F

j

= + e j ,

F (s, 0

=

0

,

F j

(s,

o)

=

F (s, _T

(4. lo)

? ° ?s as

A relevant solution of

problem (4. lo)

can be obtained

by applying

the

Laplace

transform. It is

Fj (s,

^o ₌

I

j~ ^{I~(T) 2} j~

exp

(-

^~~

^~~

^~~~~ exp

~

_erfc

y))

dT 2F

~

ar(H-T) 4E(H-T)

_E

(4.ll)

y =

Is (0 T)~

^~~~ ₊

(0 T)~~~]/2

^F'~~

It is easy ^to get ^from

(4.

II _an

expression

^{for the} second moment

~i2(~ )

"

=

F2 j~ ^{I,(T ) (erfc (z)}

^{+ [8 z~ + 8 z2}

^1]

^{erfc (- z) +}

⁽⁴

^z/

^/)(2

^{z2 + 1) exp}

^{(- z2))}

^dT ,

o

z = (o

T)~'2/2,[. _(4.12)

Now, by using (4.4), (4.5)

and

(4.12)

_{we are} able _to find _out

explicit expressions

for the relative

supersaturation

and coverage

degree.

At small

times,

_we arrive at the

following asymptotics

_:

y~

(o)

=1

(213)A(gr~)"2 05'2, _,;(o)

=

(2/3)(gr~)1'2 05'2 (4,13)

It should be

emphasized

that the limit

expressions

which could be deduced from

(4.4)

and

(4.5)

when o and _F

simultaneously

tend _{to zero} appear to be

essentially dependent

_on the order

according

_to which these

partial

limits _are calculated. Formulae

(4,13)

_are derived _at o « ~, F

being

looked upon as a

fixed,

however small parameter. ^In

particular,

it _means that the limits of

(4.13)

at _{e -} 0 _are

physically meaningless.

A

significant

feature of the above model consists in that _not

only

the

growth

^of a fixed number of _new

phase

islands is taken into account, but also the nucleation of _new critical nuclei

leading

to formation of _new islands is allowed for. The latter item

happens

to be of great

importance

at the

beginning

of any Surface reaction. It is the _reason

why

the behaviour of the coverage

degree

^{is determined}

by ^05'2

and _{ceases to} be

quadratic

in

time,

as it

usually

results from conventional models

[3, 15].

In order _to

provide

for _a

representative example,

_{we are}

going

_to

study

a

particular

_case of the nucleation kinetics. While

using

the classical nucleation

theory

and

theory

of

topochemical

reactions,

_one arrives at the

following Simple

formulae _: J

=Joi(~, x), 1(~, x)

=

wi~(T)i(i -x), wj~ (T)j

_{= exp}

lpg(T ii, g(T

= ~ ^'

(4.14)

Here

W[~ (T )]

is the flux of critical nuclei evaluated in

conformity

with the Weber-Volmer- Frenkel-Zel'dovich kinetics

[2 II.

The

physical meaning

of parameter p

(sometimes

referred _to

as a Gibbs

number)

is that of _a dimensionless activation _energy of nucleation _;

usually

p ^» I, and it

happens

to be of

significant

consequence whenever the

general

method of lo, I

II

is

applied.

The critical nuclei _are well-known _to be

originated

at so-called _« active sites _», such _as

dislocations, vacancies, admixtures,

etc.

[15].

Therefore, the flux of critical nuclei has to

depend

on the uncovered _area fraction _x, and

Jo ought

to be

proportional

to the _mean

surface concentration of such sites.

When p»

I,

_{we can} derive _an

approximate

relation for the second moment

K~(o)

immediately

from

(4,12),

with the

help

of the

asymptotic

method

by Laplace.

The result is

K~(H )

=

(2/5) r(2/5)1(2/3)Ap (El w)~'~i~

^~'~

je~

erfc

(zi

+

(H~

₊ 4 FH 2

F~)

_x

x erfc (- _?

)/2

₊

(solar

)~'2 (o ₊ 2 _F exp (- z~

)],

_z

=

Hip

)"~/2 (4.15) F(x) being

the Eulerian

gamma-function. Corresponding expressions

for the

supersaturation

and

degree

of surface coverage follow from

equations (4.4), (4.5)

and

(4,15).

Time

dependences

^{of relative}

supersaturation

_~

(T

and

degree x(T

) ^of surface coverage are

plotted

in

figure

3. It should be remembered that dimensionless variable o has _no

explicit physical meaning

of time since it is _a

special

functional of the

supersaturation

identified in

X(T),7j(T)

i o.5

o-o

o 6 to T

Fig.

3. Time

dependence

of supersaturation _~ and

degree

of coverage x at A 1.02, _p

=

50 and e=0.I; solid lines correspond to actual decreasing

supersaturation,

dashed _{ones- to constant} supersaturation that coincides with its initial value, dotted line describes x(t) with no correction for islands

overlapping.

accord with

(4.9).

However, it is

proportional

to Tat small _T, when _~ is

approximately equal

to its initial value. In _a

general

case,

equations (4.5), (4.9)

and

(4,12)

describe the

dependence

^of

the

supersaturation

_on time in _a

parametric

form. Solid _curves

correspond

to numerical solutions of

(4.4)

and

(4.5)

into which the _exact formula

(4.12)

for the second moment

K~

^{and formula} (4,14) for the nucleation _{rate are} introduced. One may compare the

shape

^{of the} theoretical _curve of

x(T

in

figure

3 with those of

repeatedly

obtained

experimental

_curves to

be well convinced of

quite

_a

satisfactory qualitative

agreement ^between the

theory

^and

experiments.

Dashed _curves in

figure

3 result from model calculation of relative

supersaturation

and

coverage

degree

in _case of

metastability being presumed

_constant,

~(T)

=

~(0).

It is

absolutely

clear from

figure

3 that such _a

presumption,

which is

commonly

taken

advantage

of in

compliance

with known

models,

_may be _{a cause} of false results and wrong final

inferences. A dotted _curve

presents

^the coverage

degree

when calculated _at

neglect

of the effect of island

overlapping.

It has been determined earlier in

[19]. Figure

3 shows the last

assumption being approximately

true

merely

at the very

beginning

of _a surface reaction.

5. Discussion.

The first of main issues of the present paper lies in the

development

of _a

general

stochastic model of the _new

phase

evolution _{on a} solid substrate, which

permits

the statistics of the time needed for spontaneous ^nuclei to get over the critical

potential

barrier to become

stably growing

_new

phase islands,

also of the induction

period

of _a

topochemical

reaction to be

rigorously

studied. The latter

period

coincides with the former time if the substrate _area is

sufficiently

small, in which _case it _can be rather

long,

as it follows from

figures

and 2. It shows that

neglect

of the

lag period inevitably

leads _to wrong theoretical and

experimental

conclusions, and this is

likely

to be of great

practical

consequence. Moreover, there _are known

experiments

which prove a

large

variation in observed induction times to occur

[22].

An

unusually long

induction

period

can be

explained by

the presence of _a slow

rate-controlling

process, which is slow at least in

comparison

^{with the} characteristic time scale of

adsorption

of adatoms

by

the substrate. The

developed theory

_proves the kinetics of the substrate

covering

to be

possibly

affected

by

random fluctuations of different

physical origin

_up to an order of

magnitude,

_as

compared

with available conventional ideas and concepts.

Apart

from other

things,

the above inferences enable _us to offer _a natural

explanation

to the well-known

irreproducibility

from

experiment

to

experiment

that

frequently

_occurs in

practice

and somewhat blurs obtained results and conclusions. Researchers do _not report

usually

on the mentioned

irreproducibility

of their observations and hesitate to

regard

it _as an

objective

and

quite

natural

phenomenon, supposedly

because the conventional deterministic theories cannot

interpret

Such _a

phenomenon

that has _an outward appearance of _an artefact. As _a result, vital information about

physical

_essence of processes

being investigated

often

happens

to be lost.

We

hope

this paper will prove to be of _use for

experimentalists

in the _matter of

helping

them _to

gain

a more concise and informative

interpretation

of their observations.

We have

employed

parameter « in _our consideration as a sole _measure of noise

properties.

It is

apparently impossible

to infer this parameter ^from

experiments

alone in _a direct and

conclusive way. It _can be estimated, however, with the aid of derived formulae and

independent

measurements of the induction

period, degrees

^of

metastability

and surface coverage and island size distribution at the initial stage ^{of surface reactions.} Furthermore, the random fluctuations

influencing

the reaction have been modelled as a Gaussian white noise. In addition to the usual

problem

of

allowing

for _a finite correlation time that describes _a deviation of _an actual noise from _a Markovian random process, it remains _to be _seen

just

how successful such _a model

might

be in different circumstances,

especially

_so because the very type ^{of noise}

is known _to affect final results _even

qualitatively

and _to

quite

a considerable extent

(for

an

example,

see

[23]).

The other main issue of the paper concems the intermediate stage ^{of surface}

covering

affected

by

the

gradual

decrease of the interface

metastability

caused

by

the _reverse influence of

growing

new

phase

islands. This stage

during

which _most of _a reaction

product

is

being emerged

has been

investigated

under _a

requirement

of _an extemal inflow of adatoms

being

absent. The derived

expressions

for the island size distribution

density, supersaturation

and

degree

of surface coverage can

readily

be

modified, nevertheless,

_so as to

incorporate

a

nonzero adatom flux _to the reaction surface.

The aforementioned _monotonous decrease in

metastability

has been shown _to

dramatically

influence the reaction rate, so that it must

certainly

be allowed for in both theoretical _treatment and conclusive

interpretation

of

experimental findings. Unfortunately,

we are not in _a

position

to

precisely

_compare theoretical

predictions

with

experimental

data since, as a matter of

fact,

authors do not

commonly

indicate

important experimental conditions,

such _as the initial and

equilibrium

adatom concentrations, activation energy of

nucleation,

and others, which _are

sorely

needed _to evaluate all the dimensionless parameters involved in the model.

In the

circumstances, only

_a

qualitative correspondence

between

major

theoretical expec- tations and

experimental

evidence could be discussed. This fact

notwithstanding,

_we have

attempted

_to check the present

theory by comparing

the theoretical _curve of

x(T

with _a number of

corresponding

curves inferred

by

different researchers from their

experiments. Usually

the

agreement

happens

to be rather

good,

but the trouble is that _we have been

compelled

to

regard

A, _{F, p} and to as some

adjustable

parameters.

By

_way of

example,

in

figure

4 _we illustrate the

comparison

of the theoretical function

x(T )

with

experiments

obtained in

[13]

where

covering

of surfaces made of

spectroscopically

_pure nickel with its oxide has been studied in full detail.

However, pertinent

values of the above-indicated parameters ^needed to ensure

adequacy

of theoretical calculation _are not

wholly

clear from the available

description

of these

experiments.

Moreover, reliable

experimental

determination of these parameters ^{would involve} substantial

difficulties,

_even

though

the authors of

[13]

_were

fully

aware of urgent

necessity

to report on

X(T)

^~~

o.5

o-o 2.0 4.0 T

Fig.

4. Time dependence ^of degree of coverage curve

theory

at A

= 1.05, _p

= 10, _~ 0.085 jm).

(al, (Aj experimental data for

covering

of surfaces of pure Ni with its oxide _at 300 K and pressure

p 10~~ ₃

x lo- and 6 _x lo- ^~ tom to 40.5, 108.7 and 250 _s, respectively.

them. This is

why

^the values indicated in the

figure caption

have been chosen in _a somewhat

arbitrary

_manner, which

undoubtedly

makes the

comparison

less

convincing

than it should and could be. All the _Same, the very form of the J.(T curve as follows from the

theory

is rather

suggestive

and _{seems to} be well confirmed

by

the

experimental

evidence.

Certain

important

features of the process under

study

have been

wholly

overlooked in this paper. We have

investigated

both nucleation and the island

growth

but have not

paid

attention to coalescence

phenomena,

_or Ostwald

ripening,

that often take

place

in the end of _a first-order

phase

transition at low

supersaturation [1,

2

II,

after the termination of the intermediate stage of the transformation. It

might

be rather

interesting

and instructive to consider _an interrelation

between the mentioned stages ^of surface

covering. Also,

_we have not taken into _{account an} effect of

partial

_coverage of the surface with _{a new}

phase

_on effective values of relevant _mass

transfer coefficients that

happens

sometimes to be essential

[24].

In

conclusion,

_we have to

point

out that the

necessity

often arises to solve the Fokker-Planck

equation (4.2)

for another condition

imposed

_on the island size distribution

density,

rather than for

(4.3).

A sound

example

is

provided by

the condition _p

(0, R)

₌ 3

(R ),

^with _p

(t,

R

) being

normalized _to

unity.

In such _{a case,} the flux of critical nuclei is _to be found _{as a} solution _to the Fokker-Planck

equation.

The method

developed

in the present paper can also be

generalized

^and

successfully applied

to the

grain growth

in

polycrystalline

solids

[5],

_as well _as while

dealing

with _some processes in

catalytic reactions,

in

thermodesorption,

and so forth, not to mention familiar processes of the

origination

and

subsequent

evolution of _new

phase

elements in the bulk of metastable

media.

References

[1] Alwater H. A.,

Yang

C. M., J. Appl. Phys. 67 (1990) 6202.

[2] Atkinson H. V., Acta Meiall. Mater. 36 (1988) 469.

[3]

Holloway

P. H.. Outlaw R. A.. Surf. Sci. iii (1981) 300.

[4] Atkinson A., Rev. Mod. Phys. 57 (1985) 437.

[5] Pande C. S., Dantsker E., Acta Metall. Mater. 38 (1990) 945 and 39 (1991) 300.

[6] Lawless K. R.,

Rep.

Progr.

Phys.

37 (1974) 231.

[7] Van Kampen N. G.. Instabilities and Nonequilibrium Structures, E. Tirapequi and D. Villaroel Eds.

(Reidel pub(. Co., 1987) p. 247.

[8] Gardiner C. M., Handbook of Stochastic Methods,

Springer

Series in Sinergetics, vol.13

(Springer-Verlag,

Berlin, 1985).

[9] Horsthemke W., Lefever R., Noise-Induced Phase Transitions. Springer Series in

Sinergetics,

vol. 15

(Springer-Verlag,

Berlin, 1984).

[10]

Buyevich

Yu. A., Mansurov V. V., J. Crystal Grow,th 104 (1990) 861.

[I Ii

Buyevich

Yu. A., Ivanov A. O., Ph_vsica A 193 (1993) 221.

[12] Kuan D. Y., Davis H. T.. Aris R.. Chem.

Engng

Sci. 78 (1983) 719.

[13] Borman V. D., Gusev E. M., Devyatko Yu. N. et al., Poveikhnosi, _n 8 (1990) 22.

[14] Thomson C. V., Acia Meiall. Mater. 36 (1988) 2929.

[15] Barret P.,

Cindtique

Hdtdrogbne (Gauthier-Villars, Paris, 1973).

[16] Van Kampen N. G.. Stochastic Processes _in Physics and

Chemistry

(North-Holland Physics Publ..

19851.

[17] Fedotov S. P., Tret'yakov M. V., Combust. Sci. Technol. 78 (1991) 1.

[18] Nicolis G., Baras F., J. Stat. Phys. 48 (1987) 1071.

[19] Buyevich

Yu. A..

Tret'yakov

M. V.,

Doklady

AN SSSR 323 (1992) 306.

j20]

Mampel

K. L., Z. Phys. Chem. 187A (1940) 235.

j21] Lifshitz J. M., Slezov V. V., J. Phys. Chem. Solids 19 (1961) 35.

j22]

Nagypal

I..

Epstein

I. R., J.

Phys.

Chem. 90 j1986) 6285 J. Chem. Phys. 89 (1988) 6925.

j23]

Buyevich

Yu. A.. Fedotov S. P., Tret'yakov M. V., Ph_vsi<.a A 198 (1993) 354.

j24]

Buyevich

Yu. A., Komarinski S. L., J. Engng Phys. 53 (1987) 389.