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Conditions for stability against precipitate coarsening
Scripta METALLURGICA
Vol. 8, pp. 91-92, 1974 Printed in the United States
Pergamon Press, Inc.
CONDITIONS FOR STABILITY AGAINST PRECIPITATE COARSENING
J. E. Morral* and N. P. Louat+ *Department of Metallurgy tDepartment of Physics University of Connecticut Storrs, Connecticut 06268
(Received October 4, 1973;
Revised December 6, 1973)
A precipitate model proposed by Aubauer (1,2) has recently been criticized by Hopper and Uhlmann (3) and by deFontaine (4).
The significant feature reported about this model system
was that coarsening would cease in i t once a c r i t i c a l particle size was reached. The ensuing debate between the persons mentioned above (3-6) has been concerned primarily with details of the proposed model. However, another deficiency exists in Aubauer treatment:
this deficiency
is in the s t a b i l i t y c r i t e r i a he used to prove that a cessation of coarsening would occur. The necessary and sufficient conditions for s t a b i l i t y are that the system free energy be minimum with respect to precipitate composition, size, and shape. Although Aubauer minimized his system free energy with respect to the f i r s t two of these variables, he did not consider changes in shape; instead, he assumed the precipitates were always spherical.
This is a ques-
tionable assumption, because the term proportional to area in his free energy equation is always negative at a stable particle size.
This can be seen in equation (7) of reference (2),
viz, AG = V(AGv + 3(o/D) (d) f(d) -I + kE g(d) f(d) - l )
(1)
Here d is D/R, the ratio of precipitate interface thickness to precipitate radius; V is the precipitate volume which remains constant during coarsening; ~ is chemical surface energy; AGv is the volume free energy and KE is the elastic strain energy coefficient. involving d can be expressed in the range 0 < d ~ l as g(d) f(d) - l ~ l - .6d f(d) -I ~ l + 1.5 (d+d2)
The functions (2) (3)
with these substitutions, the resulting equation is AG = V[AGv + KE) + (3o/D - .6kc)d + (l.5o/D)d 2 + (l.5o/D)d 3] (4) In equation (4), (AGv + kE) is the coefficient of volume while (3o/D - .6k E) is the coefficient of VD/R which is proportional to surface area. By inspection, one finds that equation (4) contains a minimum with respect to R only in cases when the coefficient of surface area is negative. A negative coefficient for surface area strongly suggests that a perturbation in pre-
91
92
CONDITIONS
POR STABILITY A G A I N S T PRECIPITATE
CDARSENING
Vol.
8, No.
cipitate shape could further lower the system free energy and, therefore, the system would not be stable.
What Aubauer thought was a minimum might actually be a saddle point in free
energy that allows for continuous changes in both size and shape. We conclude that insufficient proof has been given that Aubauer's model can achieve a stable state. Sufficient proof would require formulating the system free energy in terms of both shape and size and then finding extrema with respect to these two variables. References I.
H. -P. Aubauer, Acta Met. 20, 165 (1972).
2. 3. 4. 5. 6.
H. R. D. H. R.
-P. Aubauer, Acta Met. 20, 173 (1972). W. Hopper and D. R. Uhlmann, Scripta Met. 6, 327 (1972). deFontaine, Scripta Met. 7, 463 (1973). -P. Aubauer, Scripta Met. 6, 1061 (1972). W. Hopper and D. R. Uhlmann, Scripta Met. 4, 467 (1973).
2
Vol. 8, pp. 91-92, 1974 Printed in the United States
Pergamon Press, Inc.
CONDITIONS FOR STABILITY AGAINST PRECIPITATE COARSENING
J. E. Morral* and N. P. Louat+ *Department of Metallurgy tDepartment of Physics University of Connecticut Storrs, Connecticut 06268
(Received October 4, 1973;
Revised December 6, 1973)
A precipitate model proposed by Aubauer (1,2) has recently been criticized by Hopper and Uhlmann (3) and by deFontaine (4).
The significant feature reported about this model system
was that coarsening would cease in i t once a c r i t i c a l particle size was reached. The ensuing debate between the persons mentioned above (3-6) has been concerned primarily with details of the proposed model. However, another deficiency exists in Aubauer treatment:
this deficiency
is in the s t a b i l i t y c r i t e r i a he used to prove that a cessation of coarsening would occur. The necessary and sufficient conditions for s t a b i l i t y are that the system free energy be minimum with respect to precipitate composition, size, and shape. Although Aubauer minimized his system free energy with respect to the f i r s t two of these variables, he did not consider changes in shape; instead, he assumed the precipitates were always spherical.
This is a ques-
tionable assumption, because the term proportional to area in his free energy equation is always negative at a stable particle size.
This can be seen in equation (7) of reference (2),
viz, AG = V(AGv + 3(o/D) (d) f(d) -I + kE g(d) f(d) - l )
(1)
Here d is D/R, the ratio of precipitate interface thickness to precipitate radius; V is the precipitate volume which remains constant during coarsening; ~ is chemical surface energy; AGv is the volume free energy and KE is the elastic strain energy coefficient. involving d can be expressed in the range 0 < d ~ l as g(d) f(d) - l ~ l - .6d f(d) -I ~ l + 1.5 (d+d2)
The functions (2) (3)
with these substitutions, the resulting equation is AG = V[AGv + KE) + (3o/D - .6kc)d + (l.5o/D)d 2 + (l.5o/D)d 3] (4) In equation (4), (AGv + kE) is the coefficient of volume while (3o/D - .6k E) is the coefficient of VD/R which is proportional to surface area. By inspection, one finds that equation (4) contains a minimum with respect to R only in cases when the coefficient of surface area is negative. A negative coefficient for surface area strongly suggests that a perturbation in pre-
91
92
CONDITIONS
POR STABILITY A G A I N S T PRECIPITATE
CDARSENING
Vol.
8, No.
cipitate shape could further lower the system free energy and, therefore, the system would not be stable.
What Aubauer thought was a minimum might actually be a saddle point in free
energy that allows for continuous changes in both size and shape. We conclude that insufficient proof has been given that Aubauer's model can achieve a stable state. Sufficient proof would require formulating the system free energy in terms of both shape and size and then finding extrema with respect to these two variables. References I.
H. -P. Aubauer, Acta Met. 20, 165 (1972).
2. 3. 4. 5. 6.
H. R. D. H. R.
-P. Aubauer, Acta Met. 20, 173 (1972). W. Hopper and D. R. Uhlmann, Scripta Met. 6, 327 (1972). deFontaine, Scripta Met. 7, 463 (1973). -P. Aubauer, Scripta Met. 6, 1061 (1972). W. Hopper and D. R. Uhlmann, Scripta Met. 4, 467 (1973).
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