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Precipitate Dissolution
complicated and difficult to model in a satisfactory manner.
Precipitate Dissolution Precipitates form as a result of processing, often nucleating and\or growing as the temperature of the material falls. In some alloys they are a desirable feature, and their stability against further change is important. In other alloys their presence is undesirable, and homogenization is the term used to describe a heat treatment (usually at a high temperature such as Td in Fig. 1) to dissolve precipitates and produce a single-phase solid solution of effectively uniform composition. It is generally possible to dissolve precipitates if the solubility in the matrix for the element enriched in the precipitate is greater than its concentration in the alloy, as illustrated at temperature Td in Fig. 1 where Cαβ Co. The rate at which precipitates dissolve is generally thought to be controlled by the solubility factor and by long-range diffusion (see Diffusion in Solids) of the enriched element(s) in the surrounding matrix, although there have been cases where an interface-controlled process, depending on the rate of transfer of atoms across the interface between precipitate and matrix, has been proposed as the rate-limiting process. Theoretical treatments are available to describe the detailed kinetics of precipitate dissolution and various approximations have been used to describe the diffusion process for isolated precipitates, for cases where the diffusion fields impinge, making various assumptions and for different geometries. Direct experimental tests of whether it is diffusion or an interface reaction that controls the process have been made but are rare. Applications to typical industrial homogenization practice exist, even where the alloys are relatively
Td
Tg
1. Mathematical Models for Isolated Precipitates The general treatment of precipitate dissolution was well formulated by Aaron and Kotler (1971) who made a detailed theoretical study of the subject, building on the work of others (in particular Whelan (1969) and Nolfi et al. (1969)). This showed (see Fig. 2), for a precipitate of initial radius Ro, which shrinks to a radius R l R(t) at time t, that in the region r Ro the matrix concentration is increased above its initial value, whilst in the region r Ro the matrix concentration at first decreases with time and then increases (at r just less than Ro). This is a different and more complex behavior than the corresponding profile for a growing precipitate (see Precipitate Particle Coarsening), and is the reason why it has not been possible to find a closed analytic solution for the volume diffusion-controlled rate of dissolution of a single isolated spherical precipitate in an infinite matrix. This requires the solution of the field equation D]#C l bC \bt where D, the volume interdiffusion coefficient, is assumed to be independent of concentration and C l C(r,t) is the concentration for r R, with appropriate boundary conditions which include in particular R l Ro at t l 0 and C(r l R,t) l Cαβ, i.e., the composition in the matrix adjacent to the precipitate is fixed at its equilibrium value at all t, and usually Cβα l Cβ, i.e., the composition of the precipitate does not vary with temperature. The various solutions were reviewed by Aaron et al., who concluded that, provided CβαkCαβ CαβkCα, i.e., the precipitate is relatively rich in solute, then R is a slowly varying function of time and the ‘‘ stationary interface approximation, ’’ where the field equation is solved for the assumption dR\dt l 0, is the most accurate approximation. (The assumption here is that the diffusion field outside the precipitate is not affected by the movement of the interface.) It follows that, for small values of k, where k l 2(CαβkCα)\(CβαkCαβ), an expression can be obtained for the interface velocity: dR\dt lkk(D\Rj(D\πt)"/#)
Cα
Cαβ
Cβα Cβ
Cο
Figure 1 Equilibrium diagram showing growth temperature Tg and dissolution temperature Td, and other compositions (see Fig. 2).
This can be integrated (following Whelan 1969) to give an expression for R(t). Platelike precipitates can be treated in a similar way; Readey and Cooper (1966) have discussed the kinetics of dissolution from a more physical point of view. It should be noted that, for such ‘‘diffusion control,’’ the concentration gradient in the matrix adjacent to the interface is initially infinite, implying an infinite 1
Precipitate Dissolution Cβ Cβα
Cαβ Cο Cα R2 R1 R0
r
Figure 2 Initial particle of radius Ro. Profiles shown for two dissolution times t (R ) and t (R ), where t t . " " # # # "
rate of dissolution, and it is certain in these circumstances, if not more generally, that an interface reaction will be dominant. The gradient then reduces and diffusion control may become dominant; for a planar precipitate the dissolution rate will become progressively slower with time while for a spherical particle the rate will increase as R 4 0 because of geometrical effects. The possible influence of curvature (Gibbs–Thompson) effects were considered and found only to be of possible significance when R 4 0 and then only if CαβkCα became very small. If an interface process controls dissolution and causes the dissolution rate to be slower than volume diffusion control would predict, then the matrix concentration at the interface CαI will be less than the equilibrium diagram value Cαβ. It is convenient to define ∆CK as the driving force for the reaction, where ∆CK l CαβkCαI, the value by which the interface concentration is less than the equilibrium value. One of two assumptions is then commonly made, either following Nolfi et al. (1969) that the flux from the interface is proportional to ∆CK, i.e., J lkµo∆CK or (Aaron and Kotler 1971) that the velocity of the interface is taken as proportional to ∆CK, i.e., V l kKo∆CK. The function V (or J ) can be varied to accommodate different mechanisms at the interface, the linear form given above corresponding to uniform atomic detachment from the interface and, depending on the form of V (or J ), the concentration CαI can be calculated and the subsequent diffusion-controlled dissolution rate computed numerically. There have been further refinements such as introducing the effect of change of Cβα with temperature, and consequential diffusion inside the precipitate and change of the precipitate composition (Tanzilli and Heckel 1969), and different particle size distributions (Tundal and Ryum 1992). 2. Experimental Studies of Interface Control Most studies have involved measuring volume fraction of precipitate, alloy content of matrix, or some such parameter, and using the results to justify some model 2
of the process and a corresponding solution of the diffusion problem. The relatively few definitive studies to measure ∆CK have involved microprobe measurements of composition near precipitate particles. Hall and Haworth (1970) studied the dissolution of large Widmansta$ tten plates of Θ phase in Al–Cu alloys by partially dissolving them and analyzing the diffusion field in their neighborhood to obtain a value for ∆CK. They concluded that ∆CK l 0, i.e., diffusion controlled the dissolution, for all sensible dissolution times. Similar work by Abbott and Haworth (1972) on plates of γ phase in Al–Ag alloys found nonzero values of ∆CK, clear evidence for an interface-controlled process, for dissolution times of up to and over three hours. A similar study by Pabi (1980) of the dissolution of silicon in an aluminum-rich matrix reached the same conclusion but also found evidence for variation of the solution kinetics from one particle to another, which was possibly due to variation in structure of the interphase boundaries.
3. Precipitate Arrays In many practical situations the precipitates vary in size, are not formed in a matrix of constant composition, and are not effectively isolated, and Aaron and Kotler (1971) did include consideration of an array of precipitates and of different geometries. If the diffusion fields overlap (termed ‘‘ soft impingement ’’) then the rate of dissolution is slower than predicted by the theory given above. Tundal and Ryum (1992) considered a size distribution of particles and included the phenomenon of coarsening during the dissolution process but required a computer-based solution of the problem. They studied dissolution of silicon particles in an Al–Si alloy and concluded that such a treatment was necessary if an accurate prediction of dissolution kinetics in practical alloys was to be achieved.
4. Applications and Multicomponent Systems In commercial alloys there are usually more than two components and there is a size distribution of precipitates. A method to deal with this situation which is typical of one approach is due to Goodrich (1993) where a simple Avrami kinetic model is used to fit some variable such as the fraction of manganese precipitated (measured in this case using electrical resistivity to determine the manganese content of the matrix in a type 3104 aluminum can alloy) and measurements made over a range of temperatures and times to cover possible commercial practice. This leads to a relatively easy measurement to examine correlation with some desirable or undesirable property, or to study production variables. However, this approach does not give any better understanding of
Precipitate Dissolution the phenomenon and should not be used to extrapolate outside the temperature\time regime that has been studied. An alternative for binary alloys follows the method of Tundall and Ryum (1992) who studied the dissolution of silicon particles in an Al–Si alloy. They found by both computer simulation and observation that the particle size distribution was an important factor and soft impingement had to be allowed for in the calculation. AH gren (1990, 1992) studied the dissolution of carbides in steels containing another metal M as well as iron. Although progress can be made and a relatively simple approximate analysis can be produced by making certain simplifying assumptions, such as no partitioning of M between matrix and carbide, partitioning can be brought about by time at temperature and have a profound effect on the subsequent dissolution rate. A multicomponent analysis is more complicated and the local interface equilibrium has to be calculated from the corresponding multicomponent multidimensional phase diagram for which AH gren used the THERMOCALC software. However, the appropriate tieline is not defined by this calculation and a flux balance at the interface for each species has to be introduced to remove this degeneracy. Calculation of these fluxes depends on the availability of multicomponent diffusion data. AH gren used the DICTRA software to perform these calculations, and it is not easy to specify the required accuracy of the thermodynamic and diffusion data. However, for both aluminum and iron alloys good agreement is found between theory and experiment if sufficiently sophisticated methods of computation are used.
Bibliography Aaron H B, Kotler G R 1971 Second phase dissolution. Metall. Trans. 2, 393–408 Abbott K, Haworth C W 1972 Dissolution of γ phase in Al–Ag alloys. Acta Metall. 21, 951–60 AH gren J 1990 Kinetics of carbide dissolution. Scand. J. Metall. 19, 2–8 AH gren J 1992 Computer simulations of diffusional reactions in complex steels. ISIJ Int. 32 (3), 291–6 Goodrich 1993 A model for the precipitation\dissolution of Mn during commercial homogenization of aluminium alloy 3104. In: Morris J G et al. (eds.) Aluminium Alloys for Packaging. The Minerals, Metals and Materials Society, Warrendale, PA, pp. 47–60 Hall M G, Haworth C W 1970 Dissolution of Θ phase in Al–5% Cu. Acta Metall. 18, 331–7 Nolfi F V Jr, Shewman P G, Foster J S 1969 The dissolution and growth kinetics of spherical precipitates. Trans. Metall. Soc. AIME 245, 1427–33 Pabi S K 1980 On the dissolution kinetics of silicon in an aluminium-rich matrix. Mater. Sci. Eng. 43, 151–8 Ready D W, Cooper A R 1966 Molecular diffusion with a moving boundary and spherical symmetry. Chem. Eng. Sci. 21, 917–22 Tanzilli R A, Heckel R W 1969 A normalized treatment of solution of second phase particles. Trans. Metall. Soc. AIME 245, 1363–6 Tundal U H, Ryum N 1992 Dissolution of particles in binary alloys. Metall. Trans. A. 23A, 433–49 Whelan M J 1969 On the kinetics of precipitate dissolution. Met. Sci. J. 3, 95–7
C. W. Haworth
Copyright ' 2001 Elsevier Science Ltd. All rights reserved. No part of this publication may be reproduced, stored in any retrieval system or transmitted in any form or by any means : electronic, electrostatic, magnetic tape, mechanical, photocopying, recording or otherwise, without permission in writing from the publishers. Encyclopedia of Materials : Science and Technology ISBN: 0-08-0431526 pp. 7818–7820 3
Precipitate Dissolution Precipitates form as a result of processing, often nucleating and\or growing as the temperature of the material falls. In some alloys they are a desirable feature, and their stability against further change is important. In other alloys their presence is undesirable, and homogenization is the term used to describe a heat treatment (usually at a high temperature such as Td in Fig. 1) to dissolve precipitates and produce a single-phase solid solution of effectively uniform composition. It is generally possible to dissolve precipitates if the solubility in the matrix for the element enriched in the precipitate is greater than its concentration in the alloy, as illustrated at temperature Td in Fig. 1 where Cαβ Co. The rate at which precipitates dissolve is generally thought to be controlled by the solubility factor and by long-range diffusion (see Diffusion in Solids) of the enriched element(s) in the surrounding matrix, although there have been cases where an interface-controlled process, depending on the rate of transfer of atoms across the interface between precipitate and matrix, has been proposed as the rate-limiting process. Theoretical treatments are available to describe the detailed kinetics of precipitate dissolution and various approximations have been used to describe the diffusion process for isolated precipitates, for cases where the diffusion fields impinge, making various assumptions and for different geometries. Direct experimental tests of whether it is diffusion or an interface reaction that controls the process have been made but are rare. Applications to typical industrial homogenization practice exist, even where the alloys are relatively
Td
Tg
1. Mathematical Models for Isolated Precipitates The general treatment of precipitate dissolution was well formulated by Aaron and Kotler (1971) who made a detailed theoretical study of the subject, building on the work of others (in particular Whelan (1969) and Nolfi et al. (1969)). This showed (see Fig. 2), for a precipitate of initial radius Ro, which shrinks to a radius R l R(t) at time t, that in the region r Ro the matrix concentration is increased above its initial value, whilst in the region r Ro the matrix concentration at first decreases with time and then increases (at r just less than Ro). This is a different and more complex behavior than the corresponding profile for a growing precipitate (see Precipitate Particle Coarsening), and is the reason why it has not been possible to find a closed analytic solution for the volume diffusion-controlled rate of dissolution of a single isolated spherical precipitate in an infinite matrix. This requires the solution of the field equation D]#C l bC \bt where D, the volume interdiffusion coefficient, is assumed to be independent of concentration and C l C(r,t) is the concentration for r R, with appropriate boundary conditions which include in particular R l Ro at t l 0 and C(r l R,t) l Cαβ, i.e., the composition in the matrix adjacent to the precipitate is fixed at its equilibrium value at all t, and usually Cβα l Cβ, i.e., the composition of the precipitate does not vary with temperature. The various solutions were reviewed by Aaron et al., who concluded that, provided CβαkCαβ CαβkCα, i.e., the precipitate is relatively rich in solute, then R is a slowly varying function of time and the ‘‘ stationary interface approximation, ’’ where the field equation is solved for the assumption dR\dt l 0, is the most accurate approximation. (The assumption here is that the diffusion field outside the precipitate is not affected by the movement of the interface.) It follows that, for small values of k, where k l 2(CαβkCα)\(CβαkCαβ), an expression can be obtained for the interface velocity: dR\dt lkk(D\Rj(D\πt)"/#)
Cα
Cαβ
Cβα Cβ
Cο
Figure 1 Equilibrium diagram showing growth temperature Tg and dissolution temperature Td, and other compositions (see Fig. 2).
This can be integrated (following Whelan 1969) to give an expression for R(t). Platelike precipitates can be treated in a similar way; Readey and Cooper (1966) have discussed the kinetics of dissolution from a more physical point of view. It should be noted that, for such ‘‘diffusion control,’’ the concentration gradient in the matrix adjacent to the interface is initially infinite, implying an infinite 1
Precipitate Dissolution Cβ Cβα
Cαβ Cο Cα R2 R1 R0
r
Figure 2 Initial particle of radius Ro. Profiles shown for two dissolution times t (R ) and t (R ), where t t . " " # # # "
rate of dissolution, and it is certain in these circumstances, if not more generally, that an interface reaction will be dominant. The gradient then reduces and diffusion control may become dominant; for a planar precipitate the dissolution rate will become progressively slower with time while for a spherical particle the rate will increase as R 4 0 because of geometrical effects. The possible influence of curvature (Gibbs–Thompson) effects were considered and found only to be of possible significance when R 4 0 and then only if CαβkCα became very small. If an interface process controls dissolution and causes the dissolution rate to be slower than volume diffusion control would predict, then the matrix concentration at the interface CαI will be less than the equilibrium diagram value Cαβ. It is convenient to define ∆CK as the driving force for the reaction, where ∆CK l CαβkCαI, the value by which the interface concentration is less than the equilibrium value. One of two assumptions is then commonly made, either following Nolfi et al. (1969) that the flux from the interface is proportional to ∆CK, i.e., J lkµo∆CK or (Aaron and Kotler 1971) that the velocity of the interface is taken as proportional to ∆CK, i.e., V l kKo∆CK. The function V (or J ) can be varied to accommodate different mechanisms at the interface, the linear form given above corresponding to uniform atomic detachment from the interface and, depending on the form of V (or J ), the concentration CαI can be calculated and the subsequent diffusion-controlled dissolution rate computed numerically. There have been further refinements such as introducing the effect of change of Cβα with temperature, and consequential diffusion inside the precipitate and change of the precipitate composition (Tanzilli and Heckel 1969), and different particle size distributions (Tundal and Ryum 1992). 2. Experimental Studies of Interface Control Most studies have involved measuring volume fraction of precipitate, alloy content of matrix, or some such parameter, and using the results to justify some model 2
of the process and a corresponding solution of the diffusion problem. The relatively few definitive studies to measure ∆CK have involved microprobe measurements of composition near precipitate particles. Hall and Haworth (1970) studied the dissolution of large Widmansta$ tten plates of Θ phase in Al–Cu alloys by partially dissolving them and analyzing the diffusion field in their neighborhood to obtain a value for ∆CK. They concluded that ∆CK l 0, i.e., diffusion controlled the dissolution, for all sensible dissolution times. Similar work by Abbott and Haworth (1972) on plates of γ phase in Al–Ag alloys found nonzero values of ∆CK, clear evidence for an interface-controlled process, for dissolution times of up to and over three hours. A similar study by Pabi (1980) of the dissolution of silicon in an aluminum-rich matrix reached the same conclusion but also found evidence for variation of the solution kinetics from one particle to another, which was possibly due to variation in structure of the interphase boundaries.
3. Precipitate Arrays In many practical situations the precipitates vary in size, are not formed in a matrix of constant composition, and are not effectively isolated, and Aaron and Kotler (1971) did include consideration of an array of precipitates and of different geometries. If the diffusion fields overlap (termed ‘‘ soft impingement ’’) then the rate of dissolution is slower than predicted by the theory given above. Tundal and Ryum (1992) considered a size distribution of particles and included the phenomenon of coarsening during the dissolution process but required a computer-based solution of the problem. They studied dissolution of silicon particles in an Al–Si alloy and concluded that such a treatment was necessary if an accurate prediction of dissolution kinetics in practical alloys was to be achieved.
4. Applications and Multicomponent Systems In commercial alloys there are usually more than two components and there is a size distribution of precipitates. A method to deal with this situation which is typical of one approach is due to Goodrich (1993) where a simple Avrami kinetic model is used to fit some variable such as the fraction of manganese precipitated (measured in this case using electrical resistivity to determine the manganese content of the matrix in a type 3104 aluminum can alloy) and measurements made over a range of temperatures and times to cover possible commercial practice. This leads to a relatively easy measurement to examine correlation with some desirable or undesirable property, or to study production variables. However, this approach does not give any better understanding of
Precipitate Dissolution the phenomenon and should not be used to extrapolate outside the temperature\time regime that has been studied. An alternative for binary alloys follows the method of Tundall and Ryum (1992) who studied the dissolution of silicon particles in an Al–Si alloy. They found by both computer simulation and observation that the particle size distribution was an important factor and soft impingement had to be allowed for in the calculation. AH gren (1990, 1992) studied the dissolution of carbides in steels containing another metal M as well as iron. Although progress can be made and a relatively simple approximate analysis can be produced by making certain simplifying assumptions, such as no partitioning of M between matrix and carbide, partitioning can be brought about by time at temperature and have a profound effect on the subsequent dissolution rate. A multicomponent analysis is more complicated and the local interface equilibrium has to be calculated from the corresponding multicomponent multidimensional phase diagram for which AH gren used the THERMOCALC software. However, the appropriate tieline is not defined by this calculation and a flux balance at the interface for each species has to be introduced to remove this degeneracy. Calculation of these fluxes depends on the availability of multicomponent diffusion data. AH gren used the DICTRA software to perform these calculations, and it is not easy to specify the required accuracy of the thermodynamic and diffusion data. However, for both aluminum and iron alloys good agreement is found between theory and experiment if sufficiently sophisticated methods of computation are used.
Bibliography Aaron H B, Kotler G R 1971 Second phase dissolution. Metall. Trans. 2, 393–408 Abbott K, Haworth C W 1972 Dissolution of γ phase in Al–Ag alloys. Acta Metall. 21, 951–60 AH gren J 1990 Kinetics of carbide dissolution. Scand. J. Metall. 19, 2–8 AH gren J 1992 Computer simulations of diffusional reactions in complex steels. ISIJ Int. 32 (3), 291–6 Goodrich 1993 A model for the precipitation\dissolution of Mn during commercial homogenization of aluminium alloy 3104. In: Morris J G et al. (eds.) Aluminium Alloys for Packaging. The Minerals, Metals and Materials Society, Warrendale, PA, pp. 47–60 Hall M G, Haworth C W 1970 Dissolution of Θ phase in Al–5% Cu. Acta Metall. 18, 331–7 Nolfi F V Jr, Shewman P G, Foster J S 1969 The dissolution and growth kinetics of spherical precipitates. Trans. Metall. Soc. AIME 245, 1427–33 Pabi S K 1980 On the dissolution kinetics of silicon in an aluminium-rich matrix. Mater. Sci. Eng. 43, 151–8 Ready D W, Cooper A R 1966 Molecular diffusion with a moving boundary and spherical symmetry. Chem. Eng. Sci. 21, 917–22 Tanzilli R A, Heckel R W 1969 A normalized treatment of solution of second phase particles. Trans. Metall. Soc. AIME 245, 1363–6 Tundal U H, Ryum N 1992 Dissolution of particles in binary alloys. Metall. Trans. A. 23A, 433–49 Whelan M J 1969 On the kinetics of precipitate dissolution. Met. Sci. J. 3, 95–7
C. W. Haworth
Copyright ' 2001 Elsevier Science Ltd. All rights reserved. No part of this publication may be reproduced, stored in any retrieval system or transmitted in any form or by any means : electronic, electrostatic, magnetic tape, mechanical, photocopying, recording or otherwise, without permission in writing from the publishers. Encyclopedia of Materials : Science and Technology ISBN: 0-08-0431526 pp. 7818–7820 3