Comments on the stability of dispersed particles

Scripta METALLURGICA

Vol. 7, pp. 467-470, 1973 P r i n t e d in the U n i t e d States

Pergamon

Press,

C O M M E N T S ON THE S T A B I L I T Y OF D I S P E R S E D P A R T I C L E S

R. W. Hopper and D. R. U h l m a n n

D e p a r t m e n t of M e t a l l u r g y and M a t e r i a l s

Science

Center for M a t e r i a l s Science and E n g i n e e r i n g Massachusetts Cambridge,

I n s t i t u t e of T e c h n o l o g y Massachusetts

(Received March

Aubauer

02139

8, 1973)

(i, 2) has a n a l y s e d the s t a b i l i t y of "rimmed"

precipitates,

c o n c l u d i n g that elastic strain energies can sta-

bilize an array of such p r e c i p i t a t e s against O s t w a l d ripening. The p r e s e n t authors p o i n t e d out interfaces

(3) that the theory of diffuse

(4, 5) p r e d i c t s no such stable state.

"the origin of the c o n f l i c t i n g c o n c l u s i o n s

We noted that

is unclear," but

p o i n t e d out that some of the higher terms w h i c h are r e q u i r e d to r e p r e s e n t the c o m p o s i t i o n are sharply bent Aubauer

in regions where the contour surfaces

"might serve to stabilize the p r e c i p i t a t e s . "

(6) has r e c e n t l y s u g g e s t e d that the source of the con-

flicting conclusions

lies in an i n a p p r o p r i a t e a p p l i c a t i o n of

diffuse i n t e r f a c e theory.

Toward the end of the p r e s e n t dis-

cussion, we take e x c e p t i o n to this view. More recently, (i, 2, 3, 6).

de F o n t a i n e

He c o r r e c t l y draws a t t e n t i o n to an i n a d v e r t e n t

error in the paper

(3) by the p r e s e n t authQrs.

strain energy density in eqn. been B(c-c o)

2

(7) d i s c u s s e d all four articles

(i) of Ref.

and not B(C-Co) . 467

3

The elastic

should indeed have

We were not p r o p o s i n g a new

Inc

468

COMMENTS ON STABILITY OF DISPERSED PARTICLES

diffuse interface model with a linear dependence strain energy density on the e x p r e s s i o n

subsequent m a n i f e s t a t i o n s

attention he noted eqn.

in eqns.

apparent

to the problem,

(2) and

erroneous

to de Fontaine

our intent.

also predicts intermediate

did not affect our

of our paper.

(7) that the basic d i f f i c u l t y

that since the p e r f e c t l y instability,

between

This assertion justification--for

is an

cannot, reasons

sharp interface model

it is highly unlikely

"sharp"

and

De Fontaine

that cases

"diffuse" would be stable.

however,

be made without detailed

posed by the higher terms in the

free energy expansion which are r e q u i r e d to represent bent contour surfaces, apparently

as noted in our previous

sharp diffuse

(8), cited in Ref.

paper.

interfaces.

~, the present

contour

surfaces

In a separate

importance

of high curvature

of this and other higher terms on stability and until such an e v a l u a t i o n

seems premature of diffuse

paper

authors have d e m o n s t r a t e d in

(cf. Fig i,

~), tends to make the interface more diffuse.

considered;

De

the use of such terms only to

that a term in V4c, w h i c h will be of notable representing

sharply

confused our concern about sharply bent

surfaces and considers

treat r e l a t i v e l y

Ref.

As

treatment of the m a t r i x strain energy in Aubauer's

further argues

contour

interest

for calling our

(!), a suggestion w h i c h we shall not discuss.

Fontaine

3,

to anyone with a serious

([), this p r o b l e m of t r a n s c r i p t i o n

asserts

and its

(3) of Ref.

even if m i s c o n s t r u i n g

(4) or the conclusions

theory

This mis-transcription,

but we are grateful

De Fontaine

of elastic

(~), a reference which was cited

in this regard in our paper.

in the field;

7, No.

(C-Co), but simply m i s - t r a n s c r i b e d

from Hilliard

should be i m m e d i a t e l y

Vol.

The effects

have not yet been

is carried out,

it

to comment in any general way on the stability

interfaces.

S

Vol.

7, No.

5

COMMENTS ON STABILITY OF DISPERSED PARTICLES

In Aubauer's firstly,

"Response"

(6) to Ref. ~, it is suggested,

that the assumption of small concentration gradients,

implicit in diffuse interface theory, physical

469

situation.

is inappropriate

It seems unlikely,

however,

theoretical differences originate here: stability criteria predicts increasing

that the

For one of Aubauer's stability with increasing

rim thickness

(eqn. 13, Ref.

diffuse limit

(where diffuse interface theory should be at its

best)

i).

in this

Thus,

it is precisely in the

that the two theories conflict. We also do not concur with Aubauer's

second contention

(~)

that the free energy density should depend explicitly on the spatial coordinates, particle center, Aubauer

in general,

in particular.

or on the distance r from a In support of the latter,

(6) observes that because the matrix strain energy

density decreases with increasing r, the free energy density should depend on r.

This is true;

But it does so implicitly, atives.

it should;

through c(~)

and it does.

and its spatial deriv-

More cannot logically be demanded.

It is true, of course,

that the dependence of the elastic

energy density on r will be different for each model diffuse interface,

and sharp interface).

(rimmed,

Such quantitative

details might explain the differing stability predictions. Aubauer asserts

(6) that the strain energy density depends

"sensitively on the details of the particle rim."

Such a state-

ment would seem to support the use of diffuse interface theory, which makes no ad hoc assumptions about the form of the spatial dependence of the intensive parameters but determine this by variational

stability conditions.

The authors are grateful to the National Science Foundation who provided financial support for the work.

470

COMMENTS ON S T A B I L I T Y OF D I S P E R S E D P A R T I C L E S

Vol.

7, No.

REFERENCES i.

H. -P. Aubauer,

A c t a Met.

20

(1972)

165

2.

H. -P. Aubauer, A c t a Met.

20

(1972)

173

3.

R. W. H o p p e r and D. R. Uhlmann,

4.

J. W. C a h n and J. E. Hilliard,

5.

J. E. H i l l i a r d in P h a s e T r a n s f o r m a t i o n s

6.

H. -P. Aubauer,

S c r i p t a Met. 6

7.

D. de Fontaine,

S c r i p t a Met.,

8.

R. W. H o p p e r and D. R. uhlmann,

S c r i p t a Met.

6 (1972)

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4043

5