Self similar size distributions in particle coarsening

Scripta METALLURGICA et MATERIALIA

Vol.

28, pp. 1471-1476, 1993 Printed in the U.S.A.

Pergamon Press Ltd. All rights reserved

SELF SIMILAR SIZE DISTRIBUTIONS IN P A R T I C L E COARSENING

S_D. Coughlan* and M.A_ Fortes Departamento de Engenharia de Materiais, Instituto Superior T6cnico. Lisboa, Portugal.

(Received December 28, 1992) (Revised March ii, 1993) Introduction The classical LSW description of particle coarsening due to Lifshitz and Slyozov [1] and to Wagner [2] was recently reexamined by Brown [3]. While the LSW theory predicts a unique steady state distribution of the reduced diameters of the particles (i.e. diameter, a, divided by average diameter, g), Brown concluded that there are infinite steady states of the a/2 distribution, forming a one-parameter family of curves. The LSW distribution is among the distributions in this family. Under such steady state conditions coarsening leads to a distribution of sizes which is simply changed by scaling (self-similarity). The paper by Brown has been the object of much discussion in the literature [-4-12], but his findings seem to be essentially correct. Indeed, Brown has subsequently confirmed [8] by a direct computational sinmlation of coarsening, that there are steady state distributions distinct from the LSW distribution. The purpose of this paper is to correct and improve Brown's original treatment, particularly an error in his equation for the rate of change of the total number of particles. This error has led Brown to disregard some of the solutions of "his" continuity equation on the grounds that they did not give an average value of a/g equal to unity. We also find that there are stationary solutions of a type not identified by Brown. These discrepancies may also help to clarify nmch of the published discussion of Brown's original paper. The starting point of the classical mean field description of particle coarsening is the equation for the rate of change of the diameter, a, of an individual particle. We call this the growth function, G: _da_ where k is a kinetic constant. The reduced diameter, will be denoted G*. It is given by G*

a/Y,

k

a

1)

(1)

will be denoted u, using Brown's notation. The rate of change of u

du 1 da = d~ = ~ d t -

a ~

dg 1 (G- udg dt -- ~ ~-)

(2)

It is also possible to write, following Browm G* where

3 d g f(u) =~3i-

(3)

b E,,c,,-,)-.'] and

v = k ( g 2 d~ ;( v, 1

= ak (9-~t---~a)-~--

(s)

The distribution of diameters and that of reduced diameters are described by functions ¢(a,t) and ¢*(u,t), respectively. For example 4~*(u,t) is defined such that. ¢*du is the fraction of particles with reduced diameter in the interval u, u + d u (for short: u,du) at time t. The two functions are not independent, since the fraction in u,du is the same az that, in a,da with a=gu. Hence ¢*(u)=~¢(~)

;

u

=

~a

* On leave from D e p a r t m e n t of Physics, Trinity College Dublin, Ireland.

1471 0956-716X/93 $6.00 + .00 Copyright (c) 1993 Pergamon Press Ltd.

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We also have f °Ma ¢ ( a ) d a = 1 ; f o Ma

a¢(a)da=~

(7)

where aM(t ) is the m a x i m u m value of a in tile distribution and $ouM ¢*(u)du = 1

(8a)

f : M u¢*(u)du = 1

(8b)

where uM(t ) is the maximum u in the distribution. The last equation expresses the fact that ~=1. The

continuity

equation

We shall now derive the continuity equation (or evolutionary equation) for the distribution of reduced diameters. If N(t) is the number of particles at time t, the number of particles in u,du at t is N(t)d~*(u,t)du. Due to coarsening there is a flux of particles: in the time interval dt the number of particles that "enter" the interval u,du is N(t)d,*(u,t) G*(u,t) dt and the number of those "leaving" the same interval u,du is N(t)¢*(u+du,t) G*(u+du,t)dt. The difference between these quantities -N(t) a¢*G* T du dt

(9)

is the change that occurs in dt in the number of particles in u,du. Integrating for all values of u we obtain the change in the total munber of particles, dN(t): u be*G* dN(t) = - N(t)dt f o M T du (10) The integral equals -(~b*G*)0, the value of -~*G* for u=0, because (~b*G*)uM= 0 since no large particles leave the system. Then i dN _ (¢*G*)0 = (¢ G)0 (11) N dt since G~ = Go/~ (eq. 2) and ~b*=~¢ (eq. 6). Using the appropriate forms of G and G* for coarsening (eqs.1 and 3, respectively) we have 1 dN v d~ ¢'* ¢ N dt - - Z ~" (~'~)0 = - k (_)0

(12)

This shows that for 1 ~t to be finite,¢* must tend to zero with u ~ aa u tends to zero (or ~ must tend to zero with a2)_ Tile number of particles in u,du at time t is N(t)¢*(u,t)du and changes to N(t+dt)¢*(u,t+dt)du at time t+dt. The net change in the number of particles in u,du in dt is then a¢*N du dt (13) 0t Equating (13) and (9) and using (ll) to eliminate N, we obtain the continuity equation, i_e.,the equation that governs the evolution of the distribution ¢*(u,t) of the reduced diameters: ~ t * = - (¢*G*)0 ¢ * -

0¢*G* 0u

(14)

A similar equation can be derived, along the same lines, for tile evolution of the distribution of sizes, ~b(a,t): 0~ _

0¢G

(15)

- - ( ¢ G)0 ¢- o.

In Appendix 1 we use eq. 15 to show that the form (1) of the growth function G (or the equivalent form of G*, eq. (3)), ensures that the total volume of particles is conserved. St&tion&ry

distributions

(scaling)

The steady state distributions of the reduced size are described by the solutions ~b* of (14) such that 0~b*/Ot=0. Therefore, the steady state distributions ~b*(u) are the solutions of - (¢*G*)0 ¢*

-

d ¢*G* du

(16)

The solutions must satisfy conditions (8). If eq_ (16) is multiplied by du and integrated between 0 and UM, it turns out that condition (8a) is "automatically" satisfiedby any solution. However (8b) has to be imposed. As shown in Appendix i, the quantity u, defined in (5), must be a constant in a steady state of u. Brown [3] assumed this without proof. Since v is a constant, a cubic growth law holds in every steady state (see eq. 5):

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da - 3 _ _ 3k_ dt - v

(17)

Using (3) for G* we may write the steady state equation (16) in the form -

(¢*1"}°¢*

d(C,*f) du

--

(18)

This equation differs from the equation obtained by Brown in ref. 3 in the factor -(¢*f)a, which in Brown's equation is unity. We integrate (18) in the same way as Brown did. We introduce a new function H(u) = - ¢*f in eq.(18). Integration is then straightforward and leads to

(19)

H(u) = Ho exp (Ho f ~ ~ )

(20)

Finally Haexp(H o fu ~ )

f

The constant H0=H(0 ) is, from (19) and (4),

(21)

¢* Ha = - (¢*f)0 = v~ (~'-~)0

(22)

Eq. 21 coincides with Brown's eq. 12 in reL 3 (in which an obvious minus sign was omitted) if and only if H0=l. We will show, however, that H 0 may not be unity. The solutions (21) for ¢*(u) depend ou two parameters, v and H 0. The solutions are normalized, as already noted, but the condition ~ = 1 (eq. 8b) has to be imposed and leads to a definite relation between H 0 and v. By solving numerically eq. (8b) we obtained It0(v ) as shown in Fig.1. The acceptable steady state solutions of coarsening therefore form a one-parameter family. It is convenieqt to choose v as the parameter. For v>~6.75 it is H o = I independent of v. For v<6.75, H 0 decreases as v decreases and tends to 1/3 as v tends to zero. It should be noted that H a varies very slowly with v for v~6.75. For example, for v=4, H0=0.98. This led Brown, who assumed H0=l for any u, to consider as admissible the solutions for u larger than 3 (in ref. 3) or 4 (in refs. 6 and 7), but to disregard the solutions for smaller v on the grounds that they did lint give ~=1. The integral in the exponential of (21), i.e., f0u du/f, can be calculated analytically. This is accomplished by finding the negative root u 0 of f. There is such a root for any v>0. The value of -u 0 increases with increasing u. For v=6.75 it is -u0=3. Writing u3 - v(u-l) = (u-uo) (u2 + uo u - ~-~o) (23) the integral is easily calculated with the result _fu

.~ = en

uS_..(nl)

[ . (u u0)3 ] 3 u0.

+ ~-B en L

u-~ u 3 - u ( u - 1 ) - - ~

J

i

(24}

where B = 2 uo~ - ~ ,

(25)

A = - ~ (3 + uo)

(26)

and I depends on the sign of

as follows v>6.75 ; ,~<0

v=6.75 ; A = 0

[4:~ ~ ~-(2--~+%) -(2U+Uo)

I = -~A~n I=-2[

4:S " -~-u

+U o~ d

~-~u% u o - +0~

(27a) (27b)

u<6.75 ; A > 0

For u>6.75 the interval of u is finite, the extreme value, u =ut, being the smallest positive root of f(u). This is UM=l.5 for v=6.75 and decreases, approaching UM=l, as u increases. T~lere is a second positive root, u2, of f(u). The steady state solutions for v>6.75 have a simple analytic form, which can be obtained from (21), (24) and (27a). The result is ¢*=3v(- _..v u a) -D . (u1~DC ,W2, . u ~. (u-u0) -~-2D . (u2_u)-2+D(1+C) . (ul-u) -2+D(1"C)

(26a)

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where

D-

-2~

v

c

_ 3Us

-~:Z

(285)

Depending on the sign of the exponent of (ui-u) in (28a) the solutions fall into two types. For 6.758.64, the exponent of (Ul-U) is negative and ¢* tends to infinite as u approaches u M. Examples of both types of stationary solutions are shown in Fig.2c,d. For ~,=6.75 the solution also has a simple form which can be obtained from (21), (24) and (27b): ¢* = (20.25x4 l / s ) u 2 (u+3) -7/3

(1.5-u) -ills

exp ( - 1.---~_u)

(29)

This is the LSW solution. It is plotted in Fig_2b. The slope at UM=l.5 is zeroFor v<6.75 the interval of u is (0,c~) and the analytic form of ¢*, which can be obtained from (21), (24) and (27c), is more complicated containing exponentials of the tan -1 function. As u tends to infinite, ¢* tends to zero with u -(l+3tt°) (not as u -4 as stated in ref. 6). Examples of stationary solutions for v<6,75 are shown in Fig. 2a. The

variation

in

the

number

of

particles

Combining (12) with (22) yields

~1 -d -N _ . 3 H ° ~1 d~

(30)

N ~ 3H0 -- coustant

(31)

which can be integrated to

Since N=constant (volume conservation) we conclude that for v>~6_75 (H0---1)the average volume of a particle(which is proportional to ) is proportional to ~3_ For u>~6.75 the average volume is finite,as shown in Appendix 2. W e also obtain, by eliminating d~/dt between (5) and (30): den N 1 dN 3Hok dt = ~ =( ~)-3

(32)

The rate of change of ~n N in a steady state of the reduced sizes is inversely proportional to ~s. For u>~6.75 the rate is therefore inversely proportional to the average volume of a particle. Conclusions The main conclusions of the present analysis of the continuum, mean field approach of coarsening are as follows: 1 - The growth function G - -da ~ for particle coarsening eq 1 implies that the total volume of particles is conserved d ~ ' ' 2 Steady states of the reduced size distribution imply that u (defined by eq_ 5) is a constant, or equivalently, in a steady state the rate of change of ~a is a constant (eq.17) and there are no steady states for which that rate is not a constant. 3 - The steady state distributions of the reduced dlaineter form a one-parameter family. The parameter, v, appears in the parabolic equation that relates ~ with time (eq. 17). There are solutions for all positive values of ~'. The stationary solutions can be written in analytic form for any u. We have written down the solutions for t,>~6_75 (eqs. 28 and 29). 4 - The stationary solutions fall into three types, depending on the value of v_ Essentially, for ~,<6.75 the solutions have a tail to infinite (UM=OO); for 6.75~8.24, the interval of u is finite but ~* tends to infinite at the upper limit of the interval of u and there is no maximum. 5 - In steady states, the rate of change of the number of particles (eqs_ 30 and 32) depends on a second parameter, H0, related to v. For v>~6.75 it is H0--1 and the average volume of a particle is proportional to -~3. For v<6.75 it is H0
Appendix

1 -

Volume

conservation

and

r&te

of

change

of

Consider a function X(a) of the partieh diameter. Its average value in the distribution defined by ¢(a,t) is = f : M X ~b da

A1

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where aM(t ) is the maximum a in the distribution. The rate of change of is obtained as follows d _ daM dt - ~ (Xo)a M + where

aM 0¢ da fo X

A2

daM/dt=G( aM).

Using the eontiuuity equation (15) for ~ - we obtain d aM 0¢G dt - (GX4,)a M - (¢G)o - fo X~ da

A3

The integral can be calculated by parts (with x'-dX-~-~) raM X 0¢G ~-da=[¢GX]0

aM - f0aM X ' ¢ G d a

d ~

A4

Inserting in (A3) finally gives d - (¢G)0 + - (¢GX)0 dt -

A5

We apply this general equation to two particular cases: X=a and X=a 3, with G given by eq.1. We have, from eqs_ (11), (12) and (5) ¢* (¢G)0 = - k7 ( ~ ) 0 A6 Introducing u in (1) and noting that < u > = f i = l we have = k ~

(<~>_ <3>)

AT

and = 0 Inserting

A6-A8 in

A8

A5 we finally obtain, for X=a, dt - ~ 2

and, for X=a a,

(~'-~)o+<1>" <

>

A9

d dt _ k.~ (~'~)0 < as>

A10

The total volume of particles is proportional to N and we have d (N d-~

) = N

~

+

= 0

~

All

in virtue of (AI0), (12) and (5). The total volume is therefore always conserved, even if the distribution of reduced diameters is not stationary. In a steady state distribution of u, the quantity in brackets in (A9) is a constant. Therefore both 3 2. d~/dt and v in eq. 5 are also constant in a stationary state_ Appendix

2

- The

average

volume

of

a

pa.r*cicle

The average volume of a particle is proportional to given by = f : U ~(a) as da = ~-s f : U ~*(u) u s du

A12

If v>~6.75, u M is finite and the last integral is finite even if if* tends to infinite at u M (for L,>8.64)_ If u<6.75 it is UM=C~ and, a.s previously noted, uZ¢*(u)c~ u 2-3H°, as u approaches infinity. However 2-31-10>-1 since H0
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References 1 - I.M. Lifshitz and V.V. Slyozov, J. Phys. Chem. Solids , USSR, 19, 315 (1961). 2 - C.Wagner, Z. Elektrochem. 65, 581 (1961). 3 - L.C. Brown, Acta metall. 37, 71 (1989). 4 - M. tIillert, O. Hunderi, N.Ryum and T.O.Saetre, Scripta metall. 23, 1979 (1989). 5 - J.J. Itoyt, Scripta metall, et materialia 24, 163 (1990). 6 - L.C. Brown, Scripta metall, et materialia, 24, 963 (1990). 7 - L.C. Brown, Scripta metall, et materialia, 04, 2231 (1990). 8 - L.C. Brown, Scripta metall, et materia]ia, ~ , 261 (1991). 9 - M.l-Iillert, O.Hunderi and N. Ryum, Scripta metall, et materialia, 26, 1933 (1992). 10 - L.C. Brown, Scripta metal], et materialia, 26, 1939 (1992). 11 - M.Hillert, O_Hunderi and N. Ryum, Scripta metall, et materialia, 26, 1943 (1992). 12 - L.C. Brown, Scripta metall, et materialia, 26, 1945 (1992).

Acknowledgement

FCl*/CT92/0777.

This work was carried out within the EEC Science Program, Contract

1.0 0.8 H 0 0.6 0.4 i l , I L l i l i l i l i l i l i l l J

0.2

1

0

2

3

4

5

6

7

8

9

10

v Fig-1 - Plot, of t,he paramet,er H o as a function of v. For v>~6.75 ir~ is Ho=I. 3

3 (~

Qe lu)

0.01

2

/ ,.%.o.i ii

I

~.,f.l'" 0.3

.."'..376

; 4-..'-.1,o .,\ - . ".-..

, 6.75

1

. I , . I , , --,', 08 0.6 1.2 1.8

--.--

~

O'(u)

0 1.8

.~.b ~ , i , I i I . ~ i , I 0.3 O.O 0.9 1.2 1.6 1.8

2.1

U

U

(a)

(b)

~,7.0

.........

~

Jq

"

lOOO

100

Oalu)

O'(ul

10

1 ~,6.7

0.1

• 0.3

0.0

i , i 0.6 1.2

I , I 1.6 1.8

1I

0.001 0

0.3

0.6

0.9

u

u

(c}

(d)

I , 1.2 1.5

I 1.8

Fig.2- Examples of stationary distribution functions (~*(u) of' the reduced sixes, u: a) for v<6.75 ; b) for v=6.75; c) For 6.758.64. The value of v is indicated for each curve. The ¢*(u) scale in d) is logarithmic.

12