Fault migration vs. two-dimensional ostwald ripening as a mechanism for coarsening of rod eutectics

Scripta

METALLURGICA

V o l . 9, Printed

pp. 1 2 1 9 - 1 2 2 3 , in t h e U n i t e d

FAULT MIGRATION VS. TW0-DIMENSIONAL

1975 States

Pergamon

Press~

Inc

OSTWALD RIPENING

AS A MECHANISM FOR COARSENING OF ROD EUTECTICS

T. H. Courtney Department of Metallurgical Engineering Michigan Technological University, Houghton, Michigan (Received

There apparently

still exist

eutectics at elevated temperatures sional 0stwald ripening. dominate

coarsening

increased

August

differences

is caused by termination

subsequently

(fault) migration or by two-dimen-

(i) predicted that fault migration would

at longer times, when the standard deviation in fiber radius

due to fault migration and/or concurrent

dimensional

1975)

of opinion as to whether coarsening of rod-like

Cline's original analysis

initially;

25,

two-dimensional

0stwald ripening, two-

Ostwald ripening would then dominate the coarsening process. extended by Weatherly and Nakagawa

(4) more recently,

reanalyzed the kinetics

(2) and these workers

coarsening mechanism.

ever, Smartt (6,7) et al have concluded that the initial coarsening Ostwald ripening.

sibly be due to a difference et al and by Bayles et al.

in the initial

How-

in this system is con-

Jones pointed out that this dichotomy could posfault density of the materials

Indeed, examination

et al (6) showed very few faults present

(3), as well as Jones

of coarsening in AI-AI3Ni , as reported by Bayles

et al (5), and concluded that fault migration was the controlling

trolled by two-dimensional

Cline's analysis was

utilized by Smartt

of deep etched longitudinal

initially.

There were, however,

sections by Smartt

no such examinations

made by Bayles et al so Jones assumption must still be classified as a surmise. The present communication

is designed to clarify those conditions

gration can dominate coarsening

in rod-like eutectics.

coarsening is dominated by fault migration This will be done in two ways. migration analysis

only when the as-grown

First, the basic equations

and due to two-dimensional

under which fault mi-

It will be shown that the original faulty density is very large.

describing mass flux due to fault

Ostwald ripening will be compared.

Second, the original

of Cline will be utilized to estimate what the expected change in rod density due to

fault migration sequently.

is at that point in time where Ostwald ripening is expected to dominate sub-

Finally, the results of these analyses will be discussed in light of more recent

work in which the microstructural

evolution during coarsening was closely correlated to the

kinetics thereof. TQ estimate the relative mass flux due to Ostwald ripening and termination migration we will first consider a termination a fiber of larger radius R.

on a fiber of radius r which is migrating via dissolution to

Two-dimensional

time, i.e. away from the termination,

Ostwald ripening is assumed occurring at the same

the smaller fiber is also being dissolved and the

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larger one is growing.

OF

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Vol.

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Assuming local equilibrium, the concentration driving force for fault

migration is either

~c~

2__1 Ac~2 ~ r R °r r-['

when ~ is the average fiber radius.

(i)

The first term will be more appropriate if the diffusion

fields of the fibers do not overlap; the second term will be so if they do.

For most systems,

the volume fraction of the dispersed phase is sufficiently large that the second expression is more correct. the latter.

Either expression, however, will suffice for our purposes so we will use

The corresponding concentration driving force for two-dimensional 0stwald ripen-

ing is

ACoR ~ l_ _ Ri " r

(2)

The factors 2/r and i/r in Eqs. (i) and (2) come from the Thomson-Freundich equation, i.e. the rod termination has 2 radii of curvature of magnitude r; the fiber has one radius of curvature of magnitude r and one of infinity.

The mass transfer ratio for the two processes is

obtained by multiplying the ratio of the concentration

differences by the diffusion area

ratio, AF/AoR E NLr , where N L

(3)

is the number of terminations per unit fiber length.

Thus, the mass flux ratio is

D

JF/JoR = 2 NLr

{ ~ } ,

and this ratio can be determined if NLr and r/R are known.

(4) For example, if r/R = 0.98 (an

apparently conservative estimate on the spread in fiber radii, even in as-grown eutectics which are characterized by narrow distributions in this parameter), the termination flux will be equal to the Ostwald ripening flux if there is one termination every fifty radii along the length of a fiber.

For the termination flux to be the dominant coarsening mechanism,

say JF/JoR = 5, there would have to be a termination every ten fiber radii along the fiber length.

As the spread in fiber radius becomes greater, for termination migration to dominate

the mass transfer process requires an even higher termination density.

For example, if r/R =

0.95, JF/JoR = 5 when a fiber termination exists every four fiber radii along its length. For such a case, the morphology is obviously not representative of that amenable to analysis using either two-dimensional Ostwald ripening or termination migration kinetics.

Table I

lists the fault densities necessary, corresponding to different ratios of the mass fluxes, as a function of r/~.

It is apparent that fault migration will dominate the mass transfer

process only for very high fault densities; it will equal the Ostwald ripening flux at mod.erate fault densities only when the distribution in fiber radii is very narrow.

Thus, while

the above analysis is overly simplified, it nonetheless shows that fault migration dominates fiber coarsening only in those systems having a very large initial fault density and/or characterized by a narrow distribution fiber radii.

Since the distribution in fiber radii will

increase w i t h increasing time due to fault migration and concurrent two-dimensional Ostwald ripening, even in those systems for which it is initially the dominant mechanism, it will give way to classical two-dimensional Ostwald ripening after a period of time as noted by Cline.

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0.98

0.9>

0.90

0.80

i

1/51

1121

i/ii

1/6

5

lll0.2

m14.2

112.2(!)

111.2(!)

TABLE I

~r if J--~-~R :

for

rl[ :

=

To estimate what changes in fiber density correspond to the fault dominated period, we shall reconsider the original analysis of Cline.

He deduced that the rod density N is

related to the coarsening time, t, and the original rod density, N , as o

: NO

1 i + l-- (Bm + B B) nO%

,

(5)

2

where n

= number of faults per unit volume, o B T = ratio of termination velocity to rod density, B B = ratio of branch velocity to rod density.

(If the rod density does not change along the length of the specimen, and number of terminations a hexagonal

must be equal.)

the number of branches

Cline deduced an explicit expression

for B T for

array of rods;

BT = _34 (.~) 2 where ~ is the average

Ao, in -1 R'/}~ ~./'R"

(6)

fiber radius and I the interrod spacing; thus the parameter I/~ is

determined by the volume

fraction of rods; A is essentially a kinetic term in Cline's analo Because of the complex geometry associated with a branch, Cline did not deduce an ex-

ysis.

pression Eq.

for BB, but it certainly should be of the order of B T.

(6) can be written

Assuming B B and B T are equal,

as

~/~o =

i

(7)

i + BTnot Cline predicted that termination migration would dominate the initial stages of coarsening since there will be a transient

time, tc, before fibers will begin to dissolve by Ostwald

ripening effects and this time was related to the standard deviation in fiber sizes. mated that two-dimensional sizes approaches

0.2.

effects would become important when the standard deviation in fiber

Thus t

e

was estimated as tc = ~3/Ao i n ( 0 " 2 / G o )

where a at t

c

o

is the original standard deviation in fiber sizes.

(8)

If we take

o to be 0.02, N/N o

is given by N

~o

He esti-

(tc) =

i

l+

3// / ~ \ a

~

)

in X/~ n ~ l n lO 1-~/x o

(9)

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The parameter n o is simply N N L where N L is the number of terminations per rod length and N the number of rods per unit area.

For a hexagonal array, k 2 / ~ 2 =

~/V F cos 30 ° and N ~ 2 = VF/w.

Taking V F = 0.10 (appropriate to the A1-A13Ni system for which most studies have been made);

N_ (tc) No

1

(10)

i + 3.71 NLR

If we consider a 10% change in rod density (barely within most experimental measurements of rod density), this would correspond to a rod termination density of approximately one every 17 rod diameters.

If the termination density was greater, termination migration would dominate the

coarsening process, as measured in terms of rod density, to greater than a 10% change; if less, it would dominate for even smaller changes in rod density.

For example in order for the ter-

mination migration to dominate changes in rod density to a decrease in 20% of such, would require ca one termination every 7 fiber diameters -- an unusually high fault density.

(Inci-

dentally, Eq. (4) predicts that when r/~ = 0.8, there would have to be terminations every 3 rod diameters (cf. Table I) in order for termination migration to be the dominant mass transfer mechanism.)

It can be concluded, therefore, that analyses utilizing termination migration

models when applied to systems undergoing large changes in fiber density such as those observed by Bayles et al are inappropriate and apparent kinetic agreement can be only fortuitous. On an analytical basis, Cline's Eq. 45 predicted as such when both termination migration and Ostwald ripening occur simultaneously.

Furthermore, the fault densities necessary for termi-

nation migration to dominate the process are much greater than those normally observed in rodlike eutectics. In view of the above conclusion, it is worthwhile to discuss some recent results on the AI-A13Ni (8) system and in the Cu-Cr system (9).

In the material studied here, the termina-

tion density was sufficiently small in the as-grown material that two-dimensional Ostwald ripening (allowing for an incubation period) dominates even the initial stages of coarsening. Subsequent to this initial stage, microscopic observations indicated that terminations are eventually produced by coarsening.

The origin of these is not specified, but they can come

about via two-dimensional Ostwald ripening operating on a fiber with a diametrical perturbation or via a Raleigh instability (i).

Irrespective of their origin, however, once present in

sufficient numbers, the coarsening rate increases and appears to be due to both two-dimensional 0stwald ripening and to termination migration. systems (i0).

Similar observations have been made in other

The interesting conclusion, therefore, is that if a eutectic does not possess

a very large number of as-grown terminations, two-dimensional 0stwald ripening dominates coarsening originally with termination migration coming into play subsequently.

This is the

opposite conclusion to ClOne's original analysis and results from his assumption that the termination density will actually decrease with time whereas in fact it increases with time, at least in the systems discussed above. Finally, it should be remarked that the above discussion is restricted to a fibrous geometry.

With the intrinsically more stable lamellar geometry, termination migration appears

to be the dominant mechanism in most systems studied to date (9,11). This work was supported by the National Science Foundation.

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References i.

H. E. Cline, Acta Met., 3-9, 481 (1971).

2.

G. C. Weatherly and Y. G. Nakagawa, Scripta Met., 5, 777 (1971).

3.

Y. G. Nakagawa and G. C. Weatherly, Acta Met., 20, 345 (1972).

4.

H. Jones, Scripta Met., 8, i011 (1974).

5.

B. J. Bayles, J. A. Ford and M. J. Sackind, Trans. TMS-AIME, 239, 844 (1967).

6.

H. B. Smartt, L. K. Tu and T. H. Courtney, Met. Trans., 2, 2717 (1971).

7.

H. B. Smartt and T. H. Courtney, Met. Trans., 4, 217 (1973).

8.

H. B. Smartt and T. H. Courtney, accepted for publication in Met. Trans.

9.

L. Y. Lin, Ph.D. Thesis, University of Texas at Austin, Austin, Tex., August, 1975.

i0.

J. L. Walter and H. E. Cline, Met. Trans., 4, 33 (1973).

/i.

L. D. Graham and R. W. Kraft, Trans. TMS-AIME, 236, 94 (1966).

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