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Comments on “reply to discussion on ‘the growth kinetics of plate-shaped precipitates’”
Scripta METALLURGICA
Vol, ii, pp, 741-744, 1977 Printed in the United States
Pergamon Press,
Inc.
COMMENTS ON "REPLY TO DISCUSSION ON 'THE GROWTH KINETICS OF PLATE-SHAPED PRECIPITATES'" H. I. Aaronson Department of Metallurgical Engineering Michigan Technological University Houghton, Michigan 49931 U.S.A. (Received June 15, 1977)
The Reply by Doherty, Ferrante and Chen (i) makes clear that the basic difference between us lles in divergent views of what constitutes volume diffusion-controlled growth. Doherty et al take an essentially phenomenological tack: if growth kinetics (as measured by observations made at relatively long time intervals) are the same as those calculated on the assumption of volume dlffuslon-control, then the growth mechanism is accepted as such. This writer prefers an atomistic approach, based upon the definition of F. S. Ham (2,3), in which volume dlffusion-control is interpreted as an effectively uniform probability of atomic attachment or detachment at all sites along an Interphase boundary. The transfer of atoms across the boundary is taken to be uninhibited to the point where the mole fraction of solute in the matrix at the boundary is essentially that of the solvus at the reaction temperature, corrected as necessary for curvature of the boundary. Only a disordered Interphase boundary can meet these specifications. (It should be noted, t h o u g ~ that even a disordered boundary will fall to do so when the boundary serves as a conduit for mass transport (4,5); or when an interstitial and a substitutional solute are present, only the interstltial solute diffuses during growth, and the substitutional solute is adsorbed at the boundary, thereby modifying a boundary condition for the volume diffusion of interstitial solute (6,7).) On the basis of experimental evidence, some of which is defended below, it is accepted that partially or fully coherent boundaries between crystals differing in structure (7) or markedly in degree of long-range order (8) can migrate only by the ledge mechanism. The broad faces of plates, which are the primary subject of this exchange, have such structures (7,8). Although the risers of the ledges can migrate (but do not always do so (9,10)) with volume dlffuslon-controlled kinetics, the broad faces of the ledges, which constitute much the largest proportion of the interphase boundary area, are immobile and hence the growth kinetics of the boundary as a whole cannot be described as being volume diffuslon-controlled. The marked difference in the equations for the growth rates of partially or fully coherent boundaries by the ledge mechanism and of disordered boundaries by volume diffuslon-control emphasizes the distinction made between their migrational processes. Consider a planar boundary. This is a good morphological representation of a broad face of a plate of low aspect (thickness/length) ratio. The growth rate, G, for a planar, disordered interphase boundary migrating with volume diffusion-controlled kinetics may be written (17): Sa ) KD1/2(x8 - xB a
=
_
_
741
(1)
742
COMMENTS ON REPLY TO DISCUSSION
where K = a supersaturation-dependent
Vol.
Ii, No. 9
constant varying between 1.13 and^l.41,
~ i ~ i i ~ i ! i ~ ! ~ i i ~mi ! ~ i ~ i i ~ i i ! ~ i ! ~ ! t i $ i ~ ! i ~ i ~~i ~ i i ! ~ i ~ !v~ i ! ~ joP x ) the 6 matrix prior to transformation. For a ledged interphase boundary whose broad faces are immobile and whose risers migrate with volume diffusioncontrolled kinetics Jones and Trivedi (12) write: O =
D(x~ ~
-
x B)
d
(2)
ab(X~ ~ - X ~8) where ~ = a mathematical constant complexly dependent upon supersaturation (and physically equivalent to an effective diffusion distance) and b = spacing between ledges. Atkinson et al (13) have pointed out that these growth rates are equal when: b =
2(Dt)i/2(x6 -
x~6)I/2
(3)
6~ x~6)I/2 K~(x 8 Since b at the broad faces of plates tends to diminish at later reaction times (9,10,14), it is feasible for plate thickening by the ledge mechanism to occur with simulated volume dlffuslon-controlled kinetics. Unless the technique used to measure thickening kinetics provides sufficient resolution in both space and time to detect the discontinuities in thickness along a fiducial line normal to the broad faces caused by the passage of ledges, the error in interpretation made possible by such a simulation will not be detected. Chen and Doherty (15) appear to have encountered just this difficulty in their studies of the thickening kinetics of 8' Al-Cu plates. Although Sankaran and Laird (14) used the same experimental technique to study the thickening of 8' plates, they also measured b by means of TEM observations, back-calculated D from Eqn. (2) and obtained values thereof in very good agreement with those extrapolated from conventional interdiffusion measurements made at higher temperatures. Hence this is a case in which, to paraphrase Doherty et al, an experimental result is not a (valid) experimental result as a consequence of an error of analysis. One source of the phenomenological approach developed by Doherty et al appears to be a fundamental misunderstanding of two aspects of diffusional growth theory. They envision such kinetics as a two step sequence: diffusion by random walk across the solute-depleted region round the precipitate, and transfer across the interphase boundary. If the reaction is dlffuslon-llmlted, they conclude, the mechanism of atomic attachment will not affect growth kinetics Provided that "random walk" is replaced with "gradient" diffusion and that atomic transfer across the boundary at areas where this is feasible is uninhibited in the sense previously described, this view Is correct insofar as it is extended. However, these considerations fall to consider the influence upon growth of: (a) the availability of only a small proportion of the Interphase boundary area as atomic attachment sites when growth takes place by the ledge mechanism, and (b) the drastically different diffusion field round the riser of a ledge vis-avis that at a planar, disordered boundary. In respect of (a), the proportion of the interfaclal sites at which atomic transfer can be achieved is given by the ratio of the ledge height to the inter-ledge spacing. This is normally appreciably less than 0.i (9,10,14). Concerning (b), Jones and Trivedi (12) have shown that the concentration gradients about the riser of a ledge are time-lndependent, complex, steep and strikingly different from those in front of a planar, disordered interphase boundary. As Atkinson et al (13) have remarked, it is thus possible for a ledged interphase boundary to migrate more rapidly than a planar, disordered one under the same conditions of temperature and composition, provided that the ledges are sufficiently closely spaced and the maximum diffusion distance in front of a ledge is comparable to or less than the Inter-ledge spacing. (Such accelerated growth kinetics of the broad
Vol.
ii, No. 9
COMMENTS ON REPLY TO DISCUSSION
743
faces of precipitate plates have been observed in several systems (9,10,14), but they occur during only a brief portion of the growth process.) The high-temperature TEM studies which Laird and Aaronson (LA) (9) reported on the growth kinetics of y A1-Ag plates have provided data essential to the validation of our general theory of precipitate morphology (17,7,8). Doherty et al (1) have made the following criticisms and reinterpretation of this data. (a) As a consequence of the complex heat treatment cycle which had to be used in order to observe ledge growth within the time limitations of the experimental technique, "the solute content close to the precipitate will be non-unlform and the growth rate on a diffusion limited model would be expected to fall as the effective supersaturation fell". (This statement was evidently offered to rebut the significance of effectively zero thickening rates measured at times when thickening should have been proceeding quite rapidly.) (b) The experimental results were compromised by the use of thin foils, since the diffusion distance, estimated as (Dt)l/2. is much greater than the foil thickness. (c) Re-examlning the experimental data presented on y plate thickening, Doherty et al concluded that, at least during the initial stages, there is no evidence for inhibition of diffuslon-limited growth. Replies are offered to these comments in the order presented. (a) LA plotted volume diffusion-controlled thickness vs. time for the maximum and the minimum mole fractions of Ag in the a matrix which resulted from their complex solution annealing cycle. Both plots exhibited appreciable slopes at growth times when the experimental thickening rate was essentially zero. (b) The estimate of the diffusion distance given by Doherty et al, 800 nm, may be suitable for a planar, disordered boundary. In the case of y A1-Ag plates, it is appropriate to use the diffusion distance associated with the riser of a ledge. From the JonesTrivedi (12) treatment, this is found to be ca. 30 nm, significantly less than the 100 nm thickness of the usual thin foil. (c) Further consideration of Figure 7 in LA (in which experimental data on y plate thickness vs. time are compared with values calculated assuming volume diffusion-control) strengthens the case made by LA for interface-controlled kinetics. The hot stage TEM observations which LA reported on the broad faces of plates, viewed flat-on, make clear that the quite steep "risers" in the experimental plot of Figure 7 of plate thickness measured edge-on must have resulted from the passage of many 0.46 nm high ledges in quick succession; the initial "riser" was thus produced by nearly 50 individual ledges. Had the resolution of the microscopy technique employed been better, each of these large "risers" would evidently have been resolved into a sequence of 0.46 nm high "risers" of infinite slope, separated from each other by a short "flat" of zero slope. Neither such "risers" and "flats", nor the much longer "flats" between clusters of ledges actually resolved in Figure 7 resemble in any way volume dlffusion-controlled growth. The conclusion is thus reached that the predictions of a general theory of precipitate morphology (17,7,8,) as to the thickening mechanism and kinetics of precipitate plates are fully supported by the available experimental evidence Doherty et al support the suggestion of Nicholson (18) that precipitate crystals maintain their equilibrium shape continuously during growt h . A separate response will shortly be made to this view upon the basis of recently obtained experimental data (19). Acknowledgements Support by the Army Research Office is gratefully acknowledged.
744
COMMENTS ON REPLY TO DISCUSSION
Vol. ii, No. 9
References Io
2. 3. 4. 5. 6. . .
9. I0. ii. 12 13 14 15 16 17 18. 19.
R. D. Doherty, M. Ferrante and Y. H. Chen, Scripta Met., ~, 733, this issue. F. S. Ham, J. Phys. Chem. Solids, 6, 335 (1958). F. S. Ham, Quart. Appl. Math., 17, 137 (1959). H. B. Aaron and H. I. Aaronson, Acta Met., 16, 789 (1968). J. Goldman, H. I. Aaronson and H. B. Aaron, Met. Trans., ~, 1485 (1970). P. G. Boswell, K. R. Kinsman and H. I. Aaronson, unpublished research, Ford Motor Co., Dearborn, Michigan (1968). H. I. Aaronson, C. Laird and K. R. Kinsman, Phase Transformations, p. 313. ASM, Metals Park, Ohio (1970). H. I. Aaronson, Jnl. of Microscopy, 102, 275 (1974). C. Laird and H. I. Aaronson, Acta Met., 17~ 505 (1969). K. R. Kinsman, E. Eichen and H. I. Aaronso---n, Met. Trans., 6A, 303 (1975). C. Zener, Jnl. App. Phys., 2__00,950 (1949). G. J. Jones and R. K. Trivedi, ibid, 4__22,4299 (1971). C. Atkinson, K. R. Kinsman and H. I. Aaronson, Scripta Met., ~, 1105 (1973) R. Sankaran and C. Laird, Acta Met., 22, 957 (1974). Y. H. Chen and R. D. Doherty, Scripta-Met., p. 725, this issue. K. R. Kinsman, H. I. Aaronson and E. Eichen, Met. Trans., ~, 1041 (1971). H. I. Aaronson, Decomposition of Austenite by Diffusional Processes, p. 387 Interscience, NY (1962). R. B. Nicholson, Interfaces Conference--Melbourne, p. 139. Butterworths, Sydney (1969). S. J. Murray, H. I. Aaronson and M. R. Plichta, to be submitted to Scripta Met.
Vol, ii, pp, 741-744, 1977 Printed in the United States
Pergamon Press,
Inc.
COMMENTS ON "REPLY TO DISCUSSION ON 'THE GROWTH KINETICS OF PLATE-SHAPED PRECIPITATES'" H. I. Aaronson Department of Metallurgical Engineering Michigan Technological University Houghton, Michigan 49931 U.S.A. (Received June 15, 1977)
The Reply by Doherty, Ferrante and Chen (i) makes clear that the basic difference between us lles in divergent views of what constitutes volume diffusion-controlled growth. Doherty et al take an essentially phenomenological tack: if growth kinetics (as measured by observations made at relatively long time intervals) are the same as those calculated on the assumption of volume dlffuslon-control, then the growth mechanism is accepted as such. This writer prefers an atomistic approach, based upon the definition of F. S. Ham (2,3), in which volume dlffusion-control is interpreted as an effectively uniform probability of atomic attachment or detachment at all sites along an Interphase boundary. The transfer of atoms across the boundary is taken to be uninhibited to the point where the mole fraction of solute in the matrix at the boundary is essentially that of the solvus at the reaction temperature, corrected as necessary for curvature of the boundary. Only a disordered Interphase boundary can meet these specifications. (It should be noted, t h o u g ~ that even a disordered boundary will fall to do so when the boundary serves as a conduit for mass transport (4,5); or when an interstitial and a substitutional solute are present, only the interstltial solute diffuses during growth, and the substitutional solute is adsorbed at the boundary, thereby modifying a boundary condition for the volume diffusion of interstitial solute (6,7).) On the basis of experimental evidence, some of which is defended below, it is accepted that partially or fully coherent boundaries between crystals differing in structure (7) or markedly in degree of long-range order (8) can migrate only by the ledge mechanism. The broad faces of plates, which are the primary subject of this exchange, have such structures (7,8). Although the risers of the ledges can migrate (but do not always do so (9,10)) with volume dlffuslon-controlled kinetics, the broad faces of the ledges, which constitute much the largest proportion of the interphase boundary area, are immobile and hence the growth kinetics of the boundary as a whole cannot be described as being volume diffuslon-controlled. The marked difference in the equations for the growth rates of partially or fully coherent boundaries by the ledge mechanism and of disordered boundaries by volume diffuslon-control emphasizes the distinction made between their migrational processes. Consider a planar boundary. This is a good morphological representation of a broad face of a plate of low aspect (thickness/length) ratio. The growth rate, G, for a planar, disordered interphase boundary migrating with volume diffusion-controlled kinetics may be written (17): Sa ) KD1/2(x8 - xB a
=
_
_
741
(1)
742
COMMENTS ON REPLY TO DISCUSSION
where K = a supersaturation-dependent
Vol.
Ii, No. 9
constant varying between 1.13 and^l.41,
~ i ~ i i ~ i ! i ~ ! ~ i i ~mi ! ~ i ~ i i ~ i i ! ~ i ! ~ ! t i $ i ~ ! i ~ i ~~i ~ i i ! ~ i ~ !v~ i ! ~ joP x ) the 6 matrix prior to transformation. For a ledged interphase boundary whose broad faces are immobile and whose risers migrate with volume diffusioncontrolled kinetics Jones and Trivedi (12) write: O =
D(x~ ~
-
x B)
d
(2)
ab(X~ ~ - X ~8) where ~ = a mathematical constant complexly dependent upon supersaturation (and physically equivalent to an effective diffusion distance) and b = spacing between ledges. Atkinson et al (13) have pointed out that these growth rates are equal when: b =
2(Dt)i/2(x6 -
x~6)I/2
(3)
6~ x~6)I/2 K~(x 8 Since b at the broad faces of plates tends to diminish at later reaction times (9,10,14), it is feasible for plate thickening by the ledge mechanism to occur with simulated volume dlffuslon-controlled kinetics. Unless the technique used to measure thickening kinetics provides sufficient resolution in both space and time to detect the discontinuities in thickness along a fiducial line normal to the broad faces caused by the passage of ledges, the error in interpretation made possible by such a simulation will not be detected. Chen and Doherty (15) appear to have encountered just this difficulty in their studies of the thickening kinetics of 8' Al-Cu plates. Although Sankaran and Laird (14) used the same experimental technique to study the thickening of 8' plates, they also measured b by means of TEM observations, back-calculated D from Eqn. (2) and obtained values thereof in very good agreement with those extrapolated from conventional interdiffusion measurements made at higher temperatures. Hence this is a case in which, to paraphrase Doherty et al, an experimental result is not a (valid) experimental result as a consequence of an error of analysis. One source of the phenomenological approach developed by Doherty et al appears to be a fundamental misunderstanding of two aspects of diffusional growth theory. They envision such kinetics as a two step sequence: diffusion by random walk across the solute-depleted region round the precipitate, and transfer across the interphase boundary. If the reaction is dlffuslon-llmlted, they conclude, the mechanism of atomic attachment will not affect growth kinetics Provided that "random walk" is replaced with "gradient" diffusion and that atomic transfer across the boundary at areas where this is feasible is uninhibited in the sense previously described, this view Is correct insofar as it is extended. However, these considerations fall to consider the influence upon growth of: (a) the availability of only a small proportion of the Interphase boundary area as atomic attachment sites when growth takes place by the ledge mechanism, and (b) the drastically different diffusion field round the riser of a ledge vis-avis that at a planar, disordered boundary. In respect of (a), the proportion of the interfaclal sites at which atomic transfer can be achieved is given by the ratio of the ledge height to the inter-ledge spacing. This is normally appreciably less than 0.i (9,10,14). Concerning (b), Jones and Trivedi (12) have shown that the concentration gradients about the riser of a ledge are time-lndependent, complex, steep and strikingly different from those in front of a planar, disordered interphase boundary. As Atkinson et al (13) have remarked, it is thus possible for a ledged interphase boundary to migrate more rapidly than a planar, disordered one under the same conditions of temperature and composition, provided that the ledges are sufficiently closely spaced and the maximum diffusion distance in front of a ledge is comparable to or less than the Inter-ledge spacing. (Such accelerated growth kinetics of the broad
Vol.
ii, No. 9
COMMENTS ON REPLY TO DISCUSSION
743
faces of precipitate plates have been observed in several systems (9,10,14), but they occur during only a brief portion of the growth process.) The high-temperature TEM studies which Laird and Aaronson (LA) (9) reported on the growth kinetics of y A1-Ag plates have provided data essential to the validation of our general theory of precipitate morphology (17,7,8). Doherty et al (1) have made the following criticisms and reinterpretation of this data. (a) As a consequence of the complex heat treatment cycle which had to be used in order to observe ledge growth within the time limitations of the experimental technique, "the solute content close to the precipitate will be non-unlform and the growth rate on a diffusion limited model would be expected to fall as the effective supersaturation fell". (This statement was evidently offered to rebut the significance of effectively zero thickening rates measured at times when thickening should have been proceeding quite rapidly.) (b) The experimental results were compromised by the use of thin foils, since the diffusion distance, estimated as (Dt)l/2. is much greater than the foil thickness. (c) Re-examlning the experimental data presented on y plate thickening, Doherty et al concluded that, at least during the initial stages, there is no evidence for inhibition of diffuslon-limited growth. Replies are offered to these comments in the order presented. (a) LA plotted volume diffusion-controlled thickness vs. time for the maximum and the minimum mole fractions of Ag in the a matrix which resulted from their complex solution annealing cycle. Both plots exhibited appreciable slopes at growth times when the experimental thickening rate was essentially zero. (b) The estimate of the diffusion distance given by Doherty et al, 800 nm, may be suitable for a planar, disordered boundary. In the case of y A1-Ag plates, it is appropriate to use the diffusion distance associated with the riser of a ledge. From the JonesTrivedi (12) treatment, this is found to be ca. 30 nm, significantly less than the 100 nm thickness of the usual thin foil. (c) Further consideration of Figure 7 in LA (in which experimental data on y plate thickness vs. time are compared with values calculated assuming volume diffusion-control) strengthens the case made by LA for interface-controlled kinetics. The hot stage TEM observations which LA reported on the broad faces of plates, viewed flat-on, make clear that the quite steep "risers" in the experimental plot of Figure 7 of plate thickness measured edge-on must have resulted from the passage of many 0.46 nm high ledges in quick succession; the initial "riser" was thus produced by nearly 50 individual ledges. Had the resolution of the microscopy technique employed been better, each of these large "risers" would evidently have been resolved into a sequence of 0.46 nm high "risers" of infinite slope, separated from each other by a short "flat" of zero slope. Neither such "risers" and "flats", nor the much longer "flats" between clusters of ledges actually resolved in Figure 7 resemble in any way volume dlffusion-controlled growth. The conclusion is thus reached that the predictions of a general theory of precipitate morphology (17,7,8,) as to the thickening mechanism and kinetics of precipitate plates are fully supported by the available experimental evidence Doherty et al support the suggestion of Nicholson (18) that precipitate crystals maintain their equilibrium shape continuously during growt h . A separate response will shortly be made to this view upon the basis of recently obtained experimental data (19). Acknowledgements Support by the Army Research Office is gratefully acknowledged.
744
COMMENTS ON REPLY TO DISCUSSION
Vol. ii, No. 9
References Io
2. 3. 4. 5. 6. . .
9. I0. ii. 12 13 14 15 16 17 18. 19.
R. D. Doherty, M. Ferrante and Y. H. Chen, Scripta Met., ~, 733, this issue. F. S. Ham, J. Phys. Chem. Solids, 6, 335 (1958). F. S. Ham, Quart. Appl. Math., 17, 137 (1959). H. B. Aaron and H. I. Aaronson, Acta Met., 16, 789 (1968). J. Goldman, H. I. Aaronson and H. B. Aaron, Met. Trans., ~, 1485 (1970). P. G. Boswell, K. R. Kinsman and H. I. Aaronson, unpublished research, Ford Motor Co., Dearborn, Michigan (1968). H. I. Aaronson, C. Laird and K. R. Kinsman, Phase Transformations, p. 313. ASM, Metals Park, Ohio (1970). H. I. Aaronson, Jnl. of Microscopy, 102, 275 (1974). C. Laird and H. I. Aaronson, Acta Met., 17~ 505 (1969). K. R. Kinsman, E. Eichen and H. I. Aaronso---n, Met. Trans., 6A, 303 (1975). C. Zener, Jnl. App. Phys., 2__00,950 (1949). G. J. Jones and R. K. Trivedi, ibid, 4__22,4299 (1971). C. Atkinson, K. R. Kinsman and H. I. Aaronson, Scripta Met., ~, 1105 (1973) R. Sankaran and C. Laird, Acta Met., 22, 957 (1974). Y. H. Chen and R. D. Doherty, Scripta-Met., p. 725, this issue. K. R. Kinsman, H. I. Aaronson and E. Eichen, Met. Trans., ~, 1041 (1971). H. I. Aaronson, Decomposition of Austenite by Diffusional Processes, p. 387 Interscience, NY (1962). R. B. Nicholson, Interfaces Conference--Melbourne, p. 139. Butterworths, Sydney (1969). S. J. Murray, H. I. Aaronson and M. R. Plichta, to be submitted to Scripta Met.