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On the driving force for the growth and dissolution of grain boundary allotriomorphs by the “collector plates” mechanism
Scripta
METALLURGICA
Vol. 8, pp. 5 5 9 - 5 6 2 , P r i n t e d in the U n i t e d
1974 States
Pergamon
Press,
Inc.
ON THE DRIVING FORCE FOR THE GROWTH AND DISSOLUTION OF GRAIN BOUNDARY ALLOTRIOMORPHS BY THE "COLLECTOR PLATE" MECHANISM
K. C. Russell Department of Metallurgy and Materials Science and Center for Materials Science and Engineering Massachusetts Institute of Technology Cambridge, Mass. 02139 H. I. Aaronson Department of Metallurgical Engineering Michigan Technological University Houghton, Mich. 49931
(Received
March
21,
1974)
Introduction Grain boundary allotriomorphs are precipitate crystals which nucleate at grain boundaries in the matrix phase and grow preferentially and more or less smoothly along them (1,2).
At
temperatures above about 0.9 of the absolute solidus temperature (Tm), allotriomorphs both grow (3,4) and dissolve (5-8) largely or entirely by volume diffusion in substitutional alloys with an f.c.c, matrix.
When growth of @ allotriomorphs in A1-4% Cu (4,9) and of ~ allotrio-
morphs in Ag-5.6% A1 (3,4) takes place at lower temperatures, the "collector plate" mechanism is found to play an increasingly important role in the growth process.
This mechanism consists
of volume diffusion to the grain boundaries (which serve as "collector plates"), diffusion along the grain boundaries to the growing allotriomorphs and then diffusion along the predominantly disordered broad faces of the allotriomorphs and deposition on these faces (9). The driving force for diffusion along the matrix:allotriomorph (m:a) boundaries was suggested to be the capillarity-induced differences in solute concentration between the sharply curved edges of allotriomorphs and the relatively flat centers of their broad faces (9). Similarly, the dissolution of grain boundary allotriomorphs has been shown to proceed to an increasing extent by the reverse of the collector plate mechanism as the aging temperature is decreased below ca. 0.9 T
m
in various AI-Cu alloys and also in an AI-15.8% Ag alloy (7,8).
question of the driving force for solute diffusion along the broad faces of allotriomorphs during dissolution was not treated in the latter investigation.
559
Clearly, however the
The
560
GROWTH
AND
DISSOLUTION
OF
GRAIN
BOUNDARY
ALLOTRIOMORPHS
Vol.
8, No.
5
capillarity approach so far employed for allotriomorph growth by the collector plate mechanism is incomplete with respect to dissolution by this mechanism operating in reverse since this approach yields a concentration gradient of solute in the m:a boundaries which is directed toward the centers of the allotriomorph faces under all circumstances.
In the present Letter,
the capillarity approach is further developed to the point where it can encompass both growth and dissolution by the collector plate mechanism.
These considerations are then applied to
ascertain the relative kinetics of mass transport along ~:~ and ~:@ boundaries during the growth of 8 allotriomorphs in AI-4% Cu.
For convenience in the following discussions, the
matrix phase will be designated as ~ and the precipitate phase as e, and the precipitate will be assumed solute-rich. ~x@ O e--~<<
~x D
D--F-
The flux along a boundary is proportional to the product of the boundary diffusivity and the concentration gradient along the boundary.
Let Des and D @ be the diffusivities in
the e:a and the ~:@ boundaries, and ~xc~/~i and ~x 0/~£ be the maximum possible concentration gradients along the ~:~ and e:@, respectively.
We first consider the above situation in which
mass transport along ~:8 boundaries is appreciably more difficult than along ~:~ boundaries. During Dissolution During the initial stages of the dissolution process, the edges of the allotriomorphs recede preferentially, resulting in the development of "humps" of high curvature (corresponding to a roughly circular ridge on each broad face in three dimensions).
Visualizing the
allotriomorphs as joined caps of spheres of equilibrium curvature just prior to dissolution leads to the conclusions that the regions between the allotriomorph edges and the humps will have a lower chemical potential than those as yet unaffected by dissolution, while the humps themselves will have the highest chemical potential of all.
Hence solute will diffuse from
the humps to the edges along the m:a interfaces, causing dissolution of the allotriomorphs.* As dissolution proceeds, the humps will move toward the centers of the broad faces, eventually becoming a central peak in each (Figure i).
Diffusion toward the centers of the allotriomorphs will also occur, but this will only delay somewhat the completion of the dissolution process.
Vol.
8, No.
q
° .
5
GROWTH AND D I S S O L U T I O N OF GRAIN BOUNDARY A L L O T R I O M O R P H S
561
°
Growth
FIG. i Equilibrium, growth and dissolution shapes of a grain boundary allotriomorph. The equilibrium shape is composed of two spherical caps. The dissolution morphology has a larger dihedral angle, but the curvature increases continuously from the three grain junctions to the centers of the broad faces. The growth shape has a smaller dihedral angle, and its curvature decreases continuously from the three grain junctions to the centers of the broad faces.
During Growth As concluded by Brailsford and Aaron (I0), the allotriomorph will be highly oblate relative to the equilibrium shape (Figure i).
The rapid ingestion of solute at the edges will
not be sufficiently well shared with the broad faces to enable these faces to thicken at proportionate rates.
Departure of the allotriomorph edges from the equilibrium dihedral
angle -- in this case yielding a smaller angle, but a higher chemical potential of solute -and the relative flatness of the broad faces will combine to provide the driving force for such diffusion of solute from edges to centers as is able to take place.
~xa~ ~xa@ Da~--~--<< D~e ~ Under this condition,
departures from equilibrium shape will be small during growth (i0)
and also during dissolution, since the high relative rate of mass transport along the m:a boundaries will tend strongly to minimize the extent to which non-equilibrium shapes can evolve.
Nonetheless, during growth by the collector plate mechanism there must be a sharper
curvature at the edges of the allotriomorphs than at the centers of their broad faces, while during dissolution the reverse must obtain.
The directions in which the necessary departures
from equilibrium shape should occur are not clear; the essential point is that at best they will be very difficult to detect experimentally.
562
GROWTH
AND
DISSOLUTION
OF
GRAIN
BOUNDARY
ALLOTRIOMORPHS
Vol.
8, No.
5
Comparison with Experiment Measurements of the shapes of 8 allotriomorphs precipitated from Ai-4% Cu during growth and after equilibrium have shown that the average dihedral angle during growth is 94 I/2 ° as compared with an average equilibrium angle of 113 @ (ii).
On the foregoing considerations this
result indicates that mass transport is significantly more rapid along e:~ than along ~:8 boundaries in this system.
However, as Brailsford and Aaron (i0) have pointed out, the Cu
concentration in ~:8 boundaries is not known, so the value of D 8 cannot be established with certainty.
Pasparakis and Brown (8) have recently obtained data on the dissolution kinetics
of 8 AI-Cu allotriomorphs.
Information on the dihedral angle at the edges of these allo-
triomorphs during dissolution would be of considerable interest in the present context. Acknowledgements The contribution of K.C. Russell to this investigation was supported by N.S.F. Grant GH-37103 (for nucleation studies); that of H.I. Aaronson was supported both by this Grant and by ARO(D) Grant No. DA-ARO-D-31-124-73-GI44
(for growth studies).
This support is gratefully
acknowledged. References i.
C. A. DubS, H. I. Aaronson and R. F. Mehl, Rev. de Met., 5_#_5,201 (1958).
2.
H. I. Aaronson, Decomposition of Austenlte by Diffuslonal Processes, p. 387, Intersclence Publishers, New York (1962).
3.
E. B. Hawbolt and L. C. Brown, Trans. AIME, 239, 1916 (1967).
4.
J. Goldman, H. I. Aaronson and H. B. Aaron, Met. Trans., l, 1805 (1970).
5.
M. G. Hall and C. W. Haworth, The Mechanics of Phase Transformations in Crystalline Solids, p. 117, Inst. of Metals, London (1969).
6.
M. G. Hall and C. W. Haworth, Acta Met., 18, 331 (1970).
7.
A. Pasparakis, D. E. Coates and L. C. Brown, Acta Met., 21, 991 (1973).
8.
A. Pasparakls and L. C. Brown, Acta Met., in press.
9.
H. B. Aaron and H. I. Aaronson, Acta Met., 16, 789 (1968).
I0.
A. D. Brailsford and H. B. Aaron, Jnl. App. Phys., 4_O0, 1702 (1969).
ii.
H. B. Aaron and H. I. Aaronson, Acta Met., 18, 699 (1970).
METALLURGICA
Vol. 8, pp. 5 5 9 - 5 6 2 , P r i n t e d in the U n i t e d
1974 States
Pergamon
Press,
Inc.
ON THE DRIVING FORCE FOR THE GROWTH AND DISSOLUTION OF GRAIN BOUNDARY ALLOTRIOMORPHS BY THE "COLLECTOR PLATE" MECHANISM
K. C. Russell Department of Metallurgy and Materials Science and Center for Materials Science and Engineering Massachusetts Institute of Technology Cambridge, Mass. 02139 H. I. Aaronson Department of Metallurgical Engineering Michigan Technological University Houghton, Mich. 49931
(Received
March
21,
1974)
Introduction Grain boundary allotriomorphs are precipitate crystals which nucleate at grain boundaries in the matrix phase and grow preferentially and more or less smoothly along them (1,2).
At
temperatures above about 0.9 of the absolute solidus temperature (Tm), allotriomorphs both grow (3,4) and dissolve (5-8) largely or entirely by volume diffusion in substitutional alloys with an f.c.c, matrix.
When growth of @ allotriomorphs in A1-4% Cu (4,9) and of ~ allotrio-
morphs in Ag-5.6% A1 (3,4) takes place at lower temperatures, the "collector plate" mechanism is found to play an increasingly important role in the growth process.
This mechanism consists
of volume diffusion to the grain boundaries (which serve as "collector plates"), diffusion along the grain boundaries to the growing allotriomorphs and then diffusion along the predominantly disordered broad faces of the allotriomorphs and deposition on these faces (9). The driving force for diffusion along the matrix:allotriomorph (m:a) boundaries was suggested to be the capillarity-induced differences in solute concentration between the sharply curved edges of allotriomorphs and the relatively flat centers of their broad faces (9). Similarly, the dissolution of grain boundary allotriomorphs has been shown to proceed to an increasing extent by the reverse of the collector plate mechanism as the aging temperature is decreased below ca. 0.9 T
m
in various AI-Cu alloys and also in an AI-15.8% Ag alloy (7,8).
question of the driving force for solute diffusion along the broad faces of allotriomorphs during dissolution was not treated in the latter investigation.
559
Clearly, however the
The
560
GROWTH
AND
DISSOLUTION
OF
GRAIN
BOUNDARY
ALLOTRIOMORPHS
Vol.
8, No.
5
capillarity approach so far employed for allotriomorph growth by the collector plate mechanism is incomplete with respect to dissolution by this mechanism operating in reverse since this approach yields a concentration gradient of solute in the m:a boundaries which is directed toward the centers of the allotriomorph faces under all circumstances.
In the present Letter,
the capillarity approach is further developed to the point where it can encompass both growth and dissolution by the collector plate mechanism.
These considerations are then applied to
ascertain the relative kinetics of mass transport along ~:~ and ~:@ boundaries during the growth of 8 allotriomorphs in AI-4% Cu.
For convenience in the following discussions, the
matrix phase will be designated as ~ and the precipitate phase as e, and the precipitate will be assumed solute-rich. ~x@ O e--~<<
~x D
D--F-
The flux along a boundary is proportional to the product of the boundary diffusivity and the concentration gradient along the boundary.
Let Des and D @ be the diffusivities in
the e:a and the ~:@ boundaries, and ~xc~/~i and ~x 0/~£ be the maximum possible concentration gradients along the ~:~ and e:@, respectively.
We first consider the above situation in which
mass transport along ~:8 boundaries is appreciably more difficult than along ~:~ boundaries. During Dissolution During the initial stages of the dissolution process, the edges of the allotriomorphs recede preferentially, resulting in the development of "humps" of high curvature (corresponding to a roughly circular ridge on each broad face in three dimensions).
Visualizing the
allotriomorphs as joined caps of spheres of equilibrium curvature just prior to dissolution leads to the conclusions that the regions between the allotriomorph edges and the humps will have a lower chemical potential than those as yet unaffected by dissolution, while the humps themselves will have the highest chemical potential of all.
Hence solute will diffuse from
the humps to the edges along the m:a interfaces, causing dissolution of the allotriomorphs.* As dissolution proceeds, the humps will move toward the centers of the broad faces, eventually becoming a central peak in each (Figure i).
Diffusion toward the centers of the allotriomorphs will also occur, but this will only delay somewhat the completion of the dissolution process.
Vol.
8, No.
q
° .
5
GROWTH AND D I S S O L U T I O N OF GRAIN BOUNDARY A L L O T R I O M O R P H S
561
°
Growth
FIG. i Equilibrium, growth and dissolution shapes of a grain boundary allotriomorph. The equilibrium shape is composed of two spherical caps. The dissolution morphology has a larger dihedral angle, but the curvature increases continuously from the three grain junctions to the centers of the broad faces. The growth shape has a smaller dihedral angle, and its curvature decreases continuously from the three grain junctions to the centers of the broad faces.
During Growth As concluded by Brailsford and Aaron (I0), the allotriomorph will be highly oblate relative to the equilibrium shape (Figure i).
The rapid ingestion of solute at the edges will
not be sufficiently well shared with the broad faces to enable these faces to thicken at proportionate rates.
Departure of the allotriomorph edges from the equilibrium dihedral
angle -- in this case yielding a smaller angle, but a higher chemical potential of solute -and the relative flatness of the broad faces will combine to provide the driving force for such diffusion of solute from edges to centers as is able to take place.
~xa~ ~xa@ Da~--~--<< D~e ~ Under this condition,
departures from equilibrium shape will be small during growth (i0)
and also during dissolution, since the high relative rate of mass transport along the m:a boundaries will tend strongly to minimize the extent to which non-equilibrium shapes can evolve.
Nonetheless, during growth by the collector plate mechanism there must be a sharper
curvature at the edges of the allotriomorphs than at the centers of their broad faces, while during dissolution the reverse must obtain.
The directions in which the necessary departures
from equilibrium shape should occur are not clear; the essential point is that at best they will be very difficult to detect experimentally.
562
GROWTH
AND
DISSOLUTION
OF
GRAIN
BOUNDARY
ALLOTRIOMORPHS
Vol.
8, No.
5
Comparison with Experiment Measurements of the shapes of 8 allotriomorphs precipitated from Ai-4% Cu during growth and after equilibrium have shown that the average dihedral angle during growth is 94 I/2 ° as compared with an average equilibrium angle of 113 @ (ii).
On the foregoing considerations this
result indicates that mass transport is significantly more rapid along e:~ than along ~:8 boundaries in this system.
However, as Brailsford and Aaron (i0) have pointed out, the Cu
concentration in ~:8 boundaries is not known, so the value of D 8 cannot be established with certainty.
Pasparakis and Brown (8) have recently obtained data on the dissolution kinetics
of 8 AI-Cu allotriomorphs.
Information on the dihedral angle at the edges of these allo-
triomorphs during dissolution would be of considerable interest in the present context. Acknowledgements The contribution of K.C. Russell to this investigation was supported by N.S.F. Grant GH-37103 (for nucleation studies); that of H.I. Aaronson was supported both by this Grant and by ARO(D) Grant No. DA-ARO-D-31-124-73-GI44
(for growth studies).
This support is gratefully
acknowledged. References i.
C. A. DubS, H. I. Aaronson and R. F. Mehl, Rev. de Met., 5_#_5,201 (1958).
2.
H. I. Aaronson, Decomposition of Austenlte by Diffuslonal Processes, p. 387, Intersclence Publishers, New York (1962).
3.
E. B. Hawbolt and L. C. Brown, Trans. AIME, 239, 1916 (1967).
4.
J. Goldman, H. I. Aaronson and H. B. Aaron, Met. Trans., l, 1805 (1970).
5.
M. G. Hall and C. W. Haworth, The Mechanics of Phase Transformations in Crystalline Solids, p. 117, Inst. of Metals, London (1969).
6.
M. G. Hall and C. W. Haworth, Acta Met., 18, 331 (1970).
7.
A. Pasparakis, D. E. Coates and L. C. Brown, Acta Met., 21, 991 (1973).
8.
A. Pasparakls and L. C. Brown, Acta Met., in press.
9.
H. B. Aaron and H. I. Aaronson, Acta Met., 16, 789 (1968).
I0.
A. D. Brailsford and H. B. Aaron, Jnl. App. Phys., 4_O0, 1702 (1969).
ii.
H. B. Aaron and H. I. Aaronson, Acta Met., 18, 699 (1970).