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Modeling microstructural evolution during recrystallization
Scripta METALLURGICA et MATERIALIA
Vol. 27, pp. 1563-1568, 1992 Printed in the U.S.A.
Pergamon Press L t d
CONFERENCE SET No.
MODELING MICROSTRUCTURAL EVOLUTION DURING RECRYSTALLIZATION R. A. Vandermeer
Code 6320.1 Naval Research Laboratory Washington D. C. 20375-5000 USA
(Received September 16, 1992) General Persuective The modeling of nucleation and growth processes from a kinetics and global microstructural evolution point of view has been the subject of much interest for many years. The earliest attempts to model isothermal nucleation and growth processes were due to Fraenkel and Goez (1), Goler and Sachs (2), and Tammann (3). These attempts all had one serious drawback; they failed to take into account the impingements of the new, growing grains with one another. At the point of impingement, the boundaries of these grains stop migrating; therefore, the overall reaction kinetics in the later stages of the process slow down. Accounting for impingement, thus, is critical to the development of proper microstructural models of nucleation and growth processes. Austin and Rickett (4) suggested an empirical approach to aid in appraising kinetic data and comparing it at different temperatures but there was no apparent relationship between the methodology and the mechanism by which transformations were thought to proceed, i. e. the method was not deduced from a microstructural model. It was the work of Kolmogorov (5), Johnson and Mehl (6) and especially Avrami (7) that led to the first kinetic developments where the impingement factor was taken into account. The method was based on an abstract consideration of the nucleation and growth process; the new grains were imagined to grow unimpeded through one another and to continue to nucleate in already-transformed regions as well as in the untransformed volumes. The totality of the volume transformed in such an abstraction was called the extended volume transformed. In this fictitious view of the process some regions may be counted as transformed more than once. Therefore, the extended volume fraction transformed can reach values greater than one. The extended volume concept is important because, if the grains of the new phase are distributed randomly throughout the volume in a statistical sense, then there is a simple mathematical relationship between the real volume fraction transformed and the extended volume fraction transformed which in differential form is given by dVv dVvex = 1 - V v
(1)
where Vv is the actual volume fraction transformed and V~exis the extended volume fraction transformed*. Eq. 1 provides the fundamental starting point for the modeling of phase transformation kinetics by the method of Kolmogorov, Johnson and Mehl and Avrami. The modeling process is implemented by posing the nucleation and growth characteristics of the product phase in terms of the extended volume fraction transformed. When inserted into Eq. 1 and integrated, the posed model yields a relationship between the actual fraction transformed, a nucleation rate, a growth rate, a geometrical factor and reaction time. To understand how this is accomplished consider the following general case as an example: Let N(x) be the nucleation rate, i. e. the number of new grains appearing per unit volume of material during the time increment between t and t + dr. If v(t - x) is the volume of a new grain nucleated at x and growing unimpingedfor time t - x (t is the overall reaction time), then dVvex = N(%). v ( t - X). dx (2) This particular idealization assumes that the grains of the transformation product all have the same geometry * Previously, many, including the preseat author, have used Xv and Xvexfor the fraction transformed and the extended fraction transformed, respectively. When these quantities are in fact the true microstructural properties, the accepted conventions from quantitative metallography (8) should be adopted instead. Hence, Vv and V,ex, are used in this paper.
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and once nucleated each grows independently of the others. The shape of the new grains and the interface migration rate are contained in the v(t - x) term. Thus, assuming a spheroidal shape and the preservation of that shape during the growth process, the new grain volume can be expressed as 3 v ( t - x) = K v • a (3) where a is the grain "radius" (major semi-axis) at time t - ~ and Kv is a shape factor. Finally, a may be related to an interface migration rate, G, and the growth time, t', by the equation a---
G . dt'
(4) Eqs. 1 to 4 are combined and, after carrying out the integrations, yield the sought for relationship. The extended volume concept was discussed and applied most extensively by Avrami (7), who examined several different transformation models. At about the same time, Johnson and Mehl (6) were deriving a specific kinetic model applicable to reactions involving a constant nucleation rate and a constant growth or interface migration rate where the new grains have spherical shapes; this case was treated by Avrami also. Earlier Kolmogorov (5) had modeled the crystallization process statistically. He employed a methodology equivalent to the extended volume concept to derive kinetic equations identical to Johnson and Mehl and Avrami. Avrami (7) noted that the limiting cases for the idealized models he considered all had a particular form for the time dependence of the volume fraction. Therefore, he suggested the general expression for reaction kinetics Vv=l-exp[ -B'tk] (5) where t is the reaction time. For the cases Avrami considered, B and k were model dependent constants. Eq. 5 has become known as the Avrami equation, although recently it is sometimes referred to as the KJMA equation. Annlieation to Recrvstallizafion As with reconstructive phase transformations, the evolution of microstructure during recrystallization can be described phenomenologically as a nucleation and growth process. Identifiable, new strain-free grains emerge from the cold-worked microstructure ("nucleation") surrounded by high angle grain boundaries which migrate (growth) until the cold worked material is consumed. Since its inception, the Avrami equation has received an enormous amount of attention among researchers studying the kinetics of recrystallization. In the past, it has been used indiscriminately as an almost universal description of experimental data. This is due to the circumstance that when Eq. 5 is written in logarithmic form, ln{ln[l/(1-Vv)] } versus In t, the function plots as a straight line with slope k and intercept In B (at t = 1). In many recrystallization studies, attempts are made to plot data in this fashion to determine experimental values for k and B. The Avrami equation has come under scrutiny lately and criticized because of its failure to describe much observed recrystallization kinetic behavior adequately (9-13). Primarily two types of failure have been discussed. First, the exponent k is noted frequently to have low values (< 2) even though the re,crystallized grains are essentially equiaxed, whereas growth models such as those proposed by Johnson and Mehl and Avrami would predict 3 < k < 4 for this type of shape (9,14,15). Second, there are a number of clear examples where recrystallization data, when plotted as ln{ln[l/(1-Vv)]} versus In t (the Avrami equation format), do not fall on a straight line and prominent, negative deviations from straight line behavior are detected (16,17). These deviations are usually more severe in the latter stages of recrystaUization and at lower annealing temperatures. But, it should be borne in mind that Eq. 5 is strictly valid only under certain very limiting conditions. These failures, however, should caution about the indiscriminate use of the equation. Recently, Price (18), noting the failures of the Avrami equation, considered the impingement problem from another perspective -- one in which a growing recrystallized grain is simulated on a computer as a geometrical figure expanding within a Wigner-Seitz impingement cell. Growth of the grain beyond the cell itself was forbidden, i.e. the walls of the Wigner-Seitz cell are impenetrable. The simplified computer simulation (SCS) model of recrystallization developed by Price from this approach was based on the growth of equi-sized, regularly-spaced spheroidal grains all nucleated at the same instant of time. In this model the extended volume concept of Avrami and Johnson and Mehl is discarded and the shape of the recrystallized grains is dictated by the shape of the impingement cell. Orthorhombic impingement cells, designated by Price as "the axial symmetry constraint", were suggested to be the most relevant ones for recrystallization (18-19). The kinetics of re,crystallization can be deduced within the framework of the SCS model. When considered in the light of real microstructures, however, the Price model has two serious limitations: 1) time-dependent nucleation is not dealt with, and 2) the recrystaUized microstructure is represented as an ordered arrangement of growing grains (unlike
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the Avrami model where grains are randomly distributed) all starting growth at the same time. Another drawback to the model is the somewhat arbitrary choice of the impingement cell shape, although Price suggested (20) that the geometry and amount of deformation may dictate this choice. Rather than seek another method of dealing with the impingement problem as Price did, the viewpoint is taken here that the failures of the Avrami equation cited above may simply mean the micmstructural model used to derive the parameters, k and B, in Eq. 5 from Eqs. 1-4 does not conform to the reality of the physical, microstructural situation in these circumstances. Certainly, if the assumption of randomness, from which Eq. 1 is deduced, is violated, then failure should probably be expected. A situation where such may ¢¢cur is the case of grain boundary nucleated processes where the nuclei of the product form exclusively at the parent grain beandaries or grain edges in obvious clusters. Even under these circumstances, as Cahn (21) and Vandermeer and Masumura (22) have shown, use of the concept of extended space to develop microstructural models and deduce kinetic equations from them, was invaluable. Therefore, the failures noted above should not be presumed to negate the value of the extended volume concept to model impingement effects in phase transformations and recrystallization in particular. Instead, improvements in the microstructural models themselves would seem to be an appropriate step to take to understand the discrepancies to Eq. 5. To illustrate this point, consider a hypothetical microstructural model for recrystallization in which the new grains emerge from locations exclusively at the grain boundaries of the cold-worked structure. Let there be N ° pre-existing "nuclei" per unit area of grain boundary which all become active at t = 0. Let the interface migration rate, G, be a constant and let the new grains be spherically shaped. The total nucleating grain boundary area per unit volume is taken to be S. The kinetic equation for this case was derived by Vandermeer and Masumura (22) based on an approach by Cahn (21). Extended space concepts were employed in the development. This model is similar to one of the Avrami models in all respects but one -- the distribution of the nucleation sites. (It should be noted that with only slight modification this model might be applicable to recrystallization processes where the new recrystallized grains emerge from shear bands (or transition bands) which to a first approximation may be represented as planar nucleation sites. The S would then be the total nucleating shear band area per unit volume.) In Fig. 1, five examples, each representing a different cold-worked grain size (which is inversely proportional to the parameter, S ), are plotted in the standard Avrami log-log format for 0.005 < Vv < 0.995. To analyze these hypothetical cases in terms of the Avrami equation clearly is problematic. At best, Eq. 5 can only approximate the behavior for the highest and lowest values of S; also, the time exponent, k, in the Avrami equation for the example with the least S would have a low value -- near 1. The low experimental value of k results from the fact that the nuclei are clustered on "planes" rather than randomly distributed in the volume and significant impingement can set in at a very early stage in the overall re.crystallization process. In light of the earlier discussion, the Avrami equation should be viewed as a failure when applied to these cases, but not because extended volume concepts failed. Instead, the microstructural models for which the equation was developed are inappropriate. Other complicating factors, in addition to the just-discussed, early site-saturated nucleation of recrystallized grains at microstructural heterogeneities, have also been identified which can interfere with the recrystallization kinetics. These also result in an apparent failure of the Avrami equation by causing negative curvatures on an Avrami plot and/or low apparent Avrami exponents, k. They are : 1) the sometime-occurring competition between recovery processes and recrystallization to dissipate the stored energy of cold work which is the driving force for grain boundary migration (16); 2) an initially heterogeneous distribution of stored energy due to the complex deformation mechanisms at play during cold work (13, 23, 26); and 3) the dynamic interaction of moving grain boundaries with dissolved impurity atoms (24). Indeed there could be other factors which may yet need to be ascertained. The three factors noted above all influence re.crystallization kinetics through the grain boundary migration rate which is the product of a mobility term and a driving force. The first two of these factors cause the growth rate to decrease with annealing time because the driving force for migration changes with time/space as the grains grow larger. (This assumes in the second case that the new grains emerge from regions of highest stored energy.) The third factor, which relates to the impurity drag mechanism (25), would affect the migration rate through the grain boundary mobility.
ImprovedModels The above discussion suggests that by devising improved, albeit more complicated, microstructural models, the kinetics of recrystaUization may be better understood. The formulation of improved microstructural models need not be left merely to empiricism; methodologies exist for introducing these complicating factors into recrystallization models. For example, take the case of the competion between recovery and recrystalliza-
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tion which was recently discussed by Vandermeer and Rath (26) for cold worked iron single crystals. They developed a recovery model from existing experimental recovery data (27) and incorporated it into a re,crystallization model (23) which was based on observed microstructural property measurements. The important features of the microstruetural model included 1) the "nucleation" of recrystallized grains from pre-existing, randomlyplaced sites (subgrains) all emerging at times short enough to be considered zero (compared to the remainder of the process) and 2) the growth of the "nucleated" grains by interface migration to geometrical shapes approximated by spheres. The model for growth was represented by Eqs. 3 and 4 and incorporated an algorithm which took into account recovery of the cold worked matrix in regions just ahead of the migrating grain boundaries. The recovery process was modeled semi-empirically from measurements of the recovery of flow stress (27). The effect of competing recovery is to modify the driving force for boundary migration, and the magnitude depends on the relative rates of recovery processes versus the rate of boundary migration. Because the activation energies for recovery processes are in general lower than the activation energy for recrystallization, the extent of the competition depends on temperature. The influence of recovery is stronger at lower annealing temperatures, a characteristic that was known qualitatively for many years (16). Thus, at the highest temperatures in the iron (23), essentially no effect of recovery was detected because the grain boundaries were migrating too fast. At the lowest temperatures, on the other hand, recovery processes competed effectively and were noted to induce a retardation of recrystallization as evidenced by negative deviations in the Vv rate data in the Avrami plot (23). The experimental microstructural properties themselves can assist in the modeling process. It is now clear that while many recrystallization kinetics studies in the past focussed only on how Vv varied with time, greater insight can be gained if additional microstructural properties are measured as well as Vv. Two other measurements which have been shown to be very useful for devising improved models of recrystaUization are Sv, the instantaneous, global value of the interfacial area per unit volume separating recrystallized grains from the cold worked matrix and 7~ax, the length of the largest intercept-free distance (cord length) observed on the plane of polish in an unimpinged recrystallized grain. (A plot of Xtmxversus time can be used to estimate the maximum G during the early stages of recrystallization before impingement effects become important.) When experimental Sv data are available for analysis, a set of complementary equations for Sv, equivalent to Eqs. 1- 3 and 5 for Vv, is needed. These equations for the simple model detailed earlier are given by S v= {Sve~" {1- Vv}
(6)
dSvex = N(x). s ( t - x). d'¢ 2 s ( t - x) = Ks. a
(7) (8)
Sv = {1- Vv}. K.t m (9) where S~x is the extendedinteffacial area per unit volume, s(t - x) is the inteffacial area of a single recrystailized grain "nucleated" at~ and growing unimpinged for t - x, Ks is another shape factor and K and m are model dependent constants analogous to B and k in Eq. 5. With the additional microstructural property characterization, it may be possible to deduce, unequivocally, the global time dependent (or lack thereof) character of the "nucleation" process. Vandermeer and Rath (23), analyzed mierostructural property measurements using a Laplace transform methodology first proposed by Gokhale and DeHoff (30), and were able to conclude that on annealing, "nucleation" of recrystallized grains in a deformed single crystal of iron occurred so rapidly that all "nucleation" was accomplished in times short enough to be considered zero at all temperatures studied. If both Vv and Sv are measured as functions of time, then an interface-averaged boundary migration rate <(3> can be estimated by using the Cahn-Hagel formulation (31), 1
d Vv
(G) = S'--$" dt (10) When <(3> is compared to Gmax= (l/2).(d hrax / dt) an estimation of grain anisotropy may be deduced (32). Eqs. 1-9 can be used for recrystallization analysis if the new grains are randomly distributed even though N(x) and G are not necessarily constants. A class of sensible microstructural models can be developed if N(x) and G can be reasonably represented by power-law functions of time. Such was seen to be the case for recrystallizadon in cold rolled iron single crystals (23, 28, 29, 32). Additionally it was found to be possible to incorporate a further complexity in the analysis in two instances (28, 29). In these, the new grains were modeled as growing spheroids whose eccentricities varied with time. This was possible, as noted above, because Gtrax could be estimated experimentally. The changing eccentricity is yet another factor, with those mentioned earlier, that can lower the Avrami exponent, k, for recrystallization to a value below 3.
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Mierostruetural path The relationship between the various global microstructural properties that describes the microstructural evolution constituting the recrystallization process is called the microstructural path function. The nature of this relationship depends on both the geometrical and the nucleation characteristics of recrystallization. Vandermeer Masumura and Rath (33) have derived the path function for random, shape-preserved nucleation and growth transformations and obtained the following equation Sv = C. (1 - Vv). [ - In(1 - Vv)]q
(11)
where C and q are constants which depend on the specifics of the chosen model. (A discussion of the classes of models which demonstrate this behavior is presented in references 33 and 22 .) Eqs. 5, 9 and 11 form the basic set of equations upon which an appropriate microstructural model can be deduced from experiment. Only two of the three equations are independent, however. When recrystallization is characterized by both Vv and Sv, it may be easier to develop a proper microstructural model using the microstructural path function (Eq. 11) in conjunction with one kinetic function (Eqs. 5 and 9 are examples of kinetic functions) rather than use the two kinetic functions. The use of the microstructural path function is an advantage when time dependent complications such as competing recovery or non-homogeneous stored energy distributions are present. While these factors manifest themselves in the kinetic functions, they do not affect the path function (33) which reflects the geometrical and nucleation aspects of recrystallization. It should be remembered that Eq. 11 like Eq. 5 has certain limitations for its application to experiment and should be used with discretion. Path functions for more complicated microstructures are now becoming available (22). Summary
An attempt has been made to defend the use of the extended volume concept to develop microstructural models for recrystallization based on a nucleation and growth formalism. It is argued that failures of the Avrami equation to characterize recrystallization kinetics are more than likely due to an inappropriate choice of microstructural model rather than failure of the extended volume concept to correct for impingement. Improved models are becoming available now and use of the microstructural properties themselves can aid in the development of more realistic models. Acknowlfdgement This work was performed at the Naval Research Laboratory under the sponsorship of the Office of Naval Research of the US Department of Navy whose support is gratefully acknowledged. References
1) Fraenkel, W. and Gocz, W.: Zeits f anorg Chemie, 1925, 144. 45 2) v. Goler, F. and Sachs G.: Zeits f Physik, 1932, 77, 281 3) Tammann, G.: Zeits f anorg Chemie, 1933, 214, 407 4) Austin, I. B. and Rickett, R. L.: Trans.AIME, 1939, 135, 396 5) Kolmogorov, A.N.: Izv.Akad.Nauk.USSR-Ser.Matemat., 1937,1(3), 355 6) Johnson, W.A. and Mehl, R.F.: Trans.AIME, 1939, 135,416 7) Avrami, M.: l.Chem.Phys., 1939, 7, 1103; ibid., 1940, 8, 212 8) "Quantitative Microscopy" (eds R.T.DeHoff and F.N.Rhines), 1968, McGraw-Hill, New York NY, p.77 9) Loria, E.A., Detert, K. and Morris, LG.: Acta Met, 1965, 13,929 10) Price, C.W.: Scripta Met, 1985, 19, 669 11) Price, C.W.: Scripta Met, 1989, 23, 1273 12) Doherty, R.D., Rollett, A.R. and Srolovitz, D.L: "Annealing Processes - Recovery, Recrystallization and Grain Growth", (eds. N.Hansen, D.luul-Jensen, T.Lcffers and B.Ralph), 1986, Riso National Lab., Roskilde, Denmark, p.53 13) RoUett, A.D., Srolovitz, D.J., Doherty, R.D. and Anderson, M.P.: Acta Met, 1989, 37,627
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14) Perryman, E.C.W.: Trans.AIME, 1955, 203, 369 15) Leslie, W.C., Plecity, F.J. and Michalak, J.T.: Trans.TMS-AIME, 221, 1961, 691 16) Vandermeer, R.A. and Gordon, P.: "Recovery and Recrystallization of Metals" (ed. L.Himmel), 1963, Interscience Publ,, N'Y, p.221 17) Rosen, A., Burton, M.S. and Smith, G.V.: Trans.TMS-AIME, 2_30,1964, 205 18) Price, C.W.: Acta Met, 1987, 35, 1377; ibid., 1990, 38,727, ibid., 1991, 39, 1807 19) Price, C.W.: "Recrystallization '92", (ed. T. Chandra), 1990, TMS, Warrendale, PA, p. 199 20) Price, C.W.: Private communication 21) Cahn, J.W.: Acta Met, 1956, 4, 449 22) Vandermeer, R.A. and Masumura, R.A.: Acta Metall Mater, 40, 1992, 877 23) Vandermeer, R.A. and B.B.Rath:, Met Trans A, 20A, 1989, 391 24) Vandermeer, R.A.: Acta Met, 15, 1967, 447 25) Cahn, J.W.: Acta Met, 10, 1962, 789 26) Vandermeer, R.A. and Rath, B.B.: Met Trans A, 21A, 1990, 1143 27) Michalak, J.T. and Paxton, H.W.: Trans TMS-AIME, 221, 1961, 850 28) Vandermeer, R.A. and Rath, B.B.: Met Trans A, 20A, 1989, 1933 29) Vandermeer, R.A. and Rath, B.B.: "Recrystallization '92'%(ed. by T.Chandra), 1990, TMS, Warrendale, PA, p.49 30) Gokhale, A.M. and DeHoff, R.T.: Met Trans A, 16A, 1985, 559 31) Cahn, J.W. and Hagel, W.: "Decomposition of Austenite by Diffusional Processes", (ed. Z.D.Zackey and H.I.Aaronson), 1960, Interscience Publ., NY, p. 131 32) Vandermeer, R.A. and Rath, B.B.: "Materials Architecture", (ed by J.B.Bilde-Sorensen, N.Hansen, D.Juu!Jensen, T.Leffers, H.Lilholt and O.B.Pedersen), 1989, Riso National Laboratory, Roskilde, Denmark, p.589 33) Vandermeer, R.A., Masumura, R.A. and Rath, B.B.: Acta Met, 39, 1991, 383
1
o.1 _ffi
O.Ol
10.2
10 "~
10 ° ( Ns ° G 2 )~
10 ~
10 ~
10 3
"t
Figure 1 -- Effect of amount cold-worked grain boundary area per unit volume on the recrystallization kinetics for a hypothetical case of grain boundary nucleated, growth controlled recrystallization.
Vol. 27, pp. 1563-1568, 1992 Printed in the U.S.A.
Pergamon Press L t d
CONFERENCE SET No.
MODELING MICROSTRUCTURAL EVOLUTION DURING RECRYSTALLIZATION R. A. Vandermeer
Code 6320.1 Naval Research Laboratory Washington D. C. 20375-5000 USA
(Received September 16, 1992) General Persuective The modeling of nucleation and growth processes from a kinetics and global microstructural evolution point of view has been the subject of much interest for many years. The earliest attempts to model isothermal nucleation and growth processes were due to Fraenkel and Goez (1), Goler and Sachs (2), and Tammann (3). These attempts all had one serious drawback; they failed to take into account the impingements of the new, growing grains with one another. At the point of impingement, the boundaries of these grains stop migrating; therefore, the overall reaction kinetics in the later stages of the process slow down. Accounting for impingement, thus, is critical to the development of proper microstructural models of nucleation and growth processes. Austin and Rickett (4) suggested an empirical approach to aid in appraising kinetic data and comparing it at different temperatures but there was no apparent relationship between the methodology and the mechanism by which transformations were thought to proceed, i. e. the method was not deduced from a microstructural model. It was the work of Kolmogorov (5), Johnson and Mehl (6) and especially Avrami (7) that led to the first kinetic developments where the impingement factor was taken into account. The method was based on an abstract consideration of the nucleation and growth process; the new grains were imagined to grow unimpeded through one another and to continue to nucleate in already-transformed regions as well as in the untransformed volumes. The totality of the volume transformed in such an abstraction was called the extended volume transformed. In this fictitious view of the process some regions may be counted as transformed more than once. Therefore, the extended volume fraction transformed can reach values greater than one. The extended volume concept is important because, if the grains of the new phase are distributed randomly throughout the volume in a statistical sense, then there is a simple mathematical relationship between the real volume fraction transformed and the extended volume fraction transformed which in differential form is given by dVv dVvex = 1 - V v
(1)
where Vv is the actual volume fraction transformed and V~exis the extended volume fraction transformed*. Eq. 1 provides the fundamental starting point for the modeling of phase transformation kinetics by the method of Kolmogorov, Johnson and Mehl and Avrami. The modeling process is implemented by posing the nucleation and growth characteristics of the product phase in terms of the extended volume fraction transformed. When inserted into Eq. 1 and integrated, the posed model yields a relationship between the actual fraction transformed, a nucleation rate, a growth rate, a geometrical factor and reaction time. To understand how this is accomplished consider the following general case as an example: Let N(x) be the nucleation rate, i. e. the number of new grains appearing per unit volume of material during the time increment between t and t + dr. If v(t - x) is the volume of a new grain nucleated at x and growing unimpingedfor time t - x (t is the overall reaction time), then dVvex = N(%). v ( t - X). dx (2) This particular idealization assumes that the grains of the transformation product all have the same geometry * Previously, many, including the preseat author, have used Xv and Xvexfor the fraction transformed and the extended fraction transformed, respectively. When these quantities are in fact the true microstructural properties, the accepted conventions from quantitative metallography (8) should be adopted instead. Hence, Vv and V,ex, are used in this paper.
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and once nucleated each grows independently of the others. The shape of the new grains and the interface migration rate are contained in the v(t - x) term. Thus, assuming a spheroidal shape and the preservation of that shape during the growth process, the new grain volume can be expressed as 3 v ( t - x) = K v • a (3) where a is the grain "radius" (major semi-axis) at time t - ~ and Kv is a shape factor. Finally, a may be related to an interface migration rate, G, and the growth time, t', by the equation a---
G . dt'
(4) Eqs. 1 to 4 are combined and, after carrying out the integrations, yield the sought for relationship. The extended volume concept was discussed and applied most extensively by Avrami (7), who examined several different transformation models. At about the same time, Johnson and Mehl (6) were deriving a specific kinetic model applicable to reactions involving a constant nucleation rate and a constant growth or interface migration rate where the new grains have spherical shapes; this case was treated by Avrami also. Earlier Kolmogorov (5) had modeled the crystallization process statistically. He employed a methodology equivalent to the extended volume concept to derive kinetic equations identical to Johnson and Mehl and Avrami. Avrami (7) noted that the limiting cases for the idealized models he considered all had a particular form for the time dependence of the volume fraction. Therefore, he suggested the general expression for reaction kinetics Vv=l-exp[ -B'tk] (5) where t is the reaction time. For the cases Avrami considered, B and k were model dependent constants. Eq. 5 has become known as the Avrami equation, although recently it is sometimes referred to as the KJMA equation. Annlieation to Recrvstallizafion As with reconstructive phase transformations, the evolution of microstructure during recrystallization can be described phenomenologically as a nucleation and growth process. Identifiable, new strain-free grains emerge from the cold-worked microstructure ("nucleation") surrounded by high angle grain boundaries which migrate (growth) until the cold worked material is consumed. Since its inception, the Avrami equation has received an enormous amount of attention among researchers studying the kinetics of recrystallization. In the past, it has been used indiscriminately as an almost universal description of experimental data. This is due to the circumstance that when Eq. 5 is written in logarithmic form, ln{ln[l/(1-Vv)] } versus In t, the function plots as a straight line with slope k and intercept In B (at t = 1). In many recrystallization studies, attempts are made to plot data in this fashion to determine experimental values for k and B. The Avrami equation has come under scrutiny lately and criticized because of its failure to describe much observed recrystallization kinetic behavior adequately (9-13). Primarily two types of failure have been discussed. First, the exponent k is noted frequently to have low values (< 2) even though the re,crystallized grains are essentially equiaxed, whereas growth models such as those proposed by Johnson and Mehl and Avrami would predict 3 < k < 4 for this type of shape (9,14,15). Second, there are a number of clear examples where recrystallization data, when plotted as ln{ln[l/(1-Vv)]} versus In t (the Avrami equation format), do not fall on a straight line and prominent, negative deviations from straight line behavior are detected (16,17). These deviations are usually more severe in the latter stages of recrystaUization and at lower annealing temperatures. But, it should be borne in mind that Eq. 5 is strictly valid only under certain very limiting conditions. These failures, however, should caution about the indiscriminate use of the equation. Recently, Price (18), noting the failures of the Avrami equation, considered the impingement problem from another perspective -- one in which a growing recrystallized grain is simulated on a computer as a geometrical figure expanding within a Wigner-Seitz impingement cell. Growth of the grain beyond the cell itself was forbidden, i.e. the walls of the Wigner-Seitz cell are impenetrable. The simplified computer simulation (SCS) model of recrystallization developed by Price from this approach was based on the growth of equi-sized, regularly-spaced spheroidal grains all nucleated at the same instant of time. In this model the extended volume concept of Avrami and Johnson and Mehl is discarded and the shape of the recrystallized grains is dictated by the shape of the impingement cell. Orthorhombic impingement cells, designated by Price as "the axial symmetry constraint", were suggested to be the most relevant ones for recrystallization (18-19). The kinetics of re,crystallization can be deduced within the framework of the SCS model. When considered in the light of real microstructures, however, the Price model has two serious limitations: 1) time-dependent nucleation is not dealt with, and 2) the recrystaUized microstructure is represented as an ordered arrangement of growing grains (unlike
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the Avrami model where grains are randomly distributed) all starting growth at the same time. Another drawback to the model is the somewhat arbitrary choice of the impingement cell shape, although Price suggested (20) that the geometry and amount of deformation may dictate this choice. Rather than seek another method of dealing with the impingement problem as Price did, the viewpoint is taken here that the failures of the Avrami equation cited above may simply mean the micmstructural model used to derive the parameters, k and B, in Eq. 5 from Eqs. 1-4 does not conform to the reality of the physical, microstructural situation in these circumstances. Certainly, if the assumption of randomness, from which Eq. 1 is deduced, is violated, then failure should probably be expected. A situation where such may ¢¢cur is the case of grain boundary nucleated processes where the nuclei of the product form exclusively at the parent grain beandaries or grain edges in obvious clusters. Even under these circumstances, as Cahn (21) and Vandermeer and Masumura (22) have shown, use of the concept of extended space to develop microstructural models and deduce kinetic equations from them, was invaluable. Therefore, the failures noted above should not be presumed to negate the value of the extended volume concept to model impingement effects in phase transformations and recrystallization in particular. Instead, improvements in the microstructural models themselves would seem to be an appropriate step to take to understand the discrepancies to Eq. 5. To illustrate this point, consider a hypothetical microstructural model for recrystallization in which the new grains emerge from locations exclusively at the grain boundaries of the cold-worked structure. Let there be N ° pre-existing "nuclei" per unit area of grain boundary which all become active at t = 0. Let the interface migration rate, G, be a constant and let the new grains be spherically shaped. The total nucleating grain boundary area per unit volume is taken to be S. The kinetic equation for this case was derived by Vandermeer and Masumura (22) based on an approach by Cahn (21). Extended space concepts were employed in the development. This model is similar to one of the Avrami models in all respects but one -- the distribution of the nucleation sites. (It should be noted that with only slight modification this model might be applicable to recrystallization processes where the new recrystallized grains emerge from shear bands (or transition bands) which to a first approximation may be represented as planar nucleation sites. The S would then be the total nucleating shear band area per unit volume.) In Fig. 1, five examples, each representing a different cold-worked grain size (which is inversely proportional to the parameter, S ), are plotted in the standard Avrami log-log format for 0.005 < Vv < 0.995. To analyze these hypothetical cases in terms of the Avrami equation clearly is problematic. At best, Eq. 5 can only approximate the behavior for the highest and lowest values of S; also, the time exponent, k, in the Avrami equation for the example with the least S would have a low value -- near 1. The low experimental value of k results from the fact that the nuclei are clustered on "planes" rather than randomly distributed in the volume and significant impingement can set in at a very early stage in the overall re.crystallization process. In light of the earlier discussion, the Avrami equation should be viewed as a failure when applied to these cases, but not because extended volume concepts failed. Instead, the microstructural models for which the equation was developed are inappropriate. Other complicating factors, in addition to the just-discussed, early site-saturated nucleation of recrystallized grains at microstructural heterogeneities, have also been identified which can interfere with the recrystallization kinetics. These also result in an apparent failure of the Avrami equation by causing negative curvatures on an Avrami plot and/or low apparent Avrami exponents, k. They are : 1) the sometime-occurring competition between recovery processes and recrystallization to dissipate the stored energy of cold work which is the driving force for grain boundary migration (16); 2) an initially heterogeneous distribution of stored energy due to the complex deformation mechanisms at play during cold work (13, 23, 26); and 3) the dynamic interaction of moving grain boundaries with dissolved impurity atoms (24). Indeed there could be other factors which may yet need to be ascertained. The three factors noted above all influence re.crystallization kinetics through the grain boundary migration rate which is the product of a mobility term and a driving force. The first two of these factors cause the growth rate to decrease with annealing time because the driving force for migration changes with time/space as the grains grow larger. (This assumes in the second case that the new grains emerge from regions of highest stored energy.) The third factor, which relates to the impurity drag mechanism (25), would affect the migration rate through the grain boundary mobility.
ImprovedModels The above discussion suggests that by devising improved, albeit more complicated, microstructural models, the kinetics of recrystaUization may be better understood. The formulation of improved microstructural models need not be left merely to empiricism; methodologies exist for introducing these complicating factors into recrystallization models. For example, take the case of the competion between recovery and recrystalliza-
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tion which was recently discussed by Vandermeer and Rath (26) for cold worked iron single crystals. They developed a recovery model from existing experimental recovery data (27) and incorporated it into a re,crystallization model (23) which was based on observed microstructural property measurements. The important features of the microstruetural model included 1) the "nucleation" of recrystallized grains from pre-existing, randomlyplaced sites (subgrains) all emerging at times short enough to be considered zero (compared to the remainder of the process) and 2) the growth of the "nucleated" grains by interface migration to geometrical shapes approximated by spheres. The model for growth was represented by Eqs. 3 and 4 and incorporated an algorithm which took into account recovery of the cold worked matrix in regions just ahead of the migrating grain boundaries. The recovery process was modeled semi-empirically from measurements of the recovery of flow stress (27). The effect of competing recovery is to modify the driving force for boundary migration, and the magnitude depends on the relative rates of recovery processes versus the rate of boundary migration. Because the activation energies for recovery processes are in general lower than the activation energy for recrystallization, the extent of the competition depends on temperature. The influence of recovery is stronger at lower annealing temperatures, a characteristic that was known qualitatively for many years (16). Thus, at the highest temperatures in the iron (23), essentially no effect of recovery was detected because the grain boundaries were migrating too fast. At the lowest temperatures, on the other hand, recovery processes competed effectively and were noted to induce a retardation of recrystallization as evidenced by negative deviations in the Vv rate data in the Avrami plot (23). The experimental microstructural properties themselves can assist in the modeling process. It is now clear that while many recrystallization kinetics studies in the past focussed only on how Vv varied with time, greater insight can be gained if additional microstructural properties are measured as well as Vv. Two other measurements which have been shown to be very useful for devising improved models of recrystaUization are Sv, the instantaneous, global value of the interfacial area per unit volume separating recrystallized grains from the cold worked matrix and 7~ax, the length of the largest intercept-free distance (cord length) observed on the plane of polish in an unimpinged recrystallized grain. (A plot of Xtmxversus time can be used to estimate the maximum G during the early stages of recrystallization before impingement effects become important.) When experimental Sv data are available for analysis, a set of complementary equations for Sv, equivalent to Eqs. 1- 3 and 5 for Vv, is needed. These equations for the simple model detailed earlier are given by S v= {Sve~" {1- Vv}
(6)
dSvex = N(x). s ( t - x). d'¢ 2 s ( t - x) = Ks. a
(7) (8)
Sv = {1- Vv}. K.t m (9) where S~x is the extendedinteffacial area per unit volume, s(t - x) is the inteffacial area of a single recrystailized grain "nucleated" at~ and growing unimpinged for t - x, Ks is another shape factor and K and m are model dependent constants analogous to B and k in Eq. 5. With the additional microstructural property characterization, it may be possible to deduce, unequivocally, the global time dependent (or lack thereof) character of the "nucleation" process. Vandermeer and Rath (23), analyzed mierostructural property measurements using a Laplace transform methodology first proposed by Gokhale and DeHoff (30), and were able to conclude that on annealing, "nucleation" of recrystallized grains in a deformed single crystal of iron occurred so rapidly that all "nucleation" was accomplished in times short enough to be considered zero at all temperatures studied. If both Vv and Sv are measured as functions of time, then an interface-averaged boundary migration rate <(3> can be estimated by using the Cahn-Hagel formulation (31), 1
d Vv
(G) = S'--$" dt (10) When <(3> is compared to Gmax= (l/2).(d hrax / dt) an estimation of grain anisotropy may be deduced (32). Eqs. 1-9 can be used for recrystallization analysis if the new grains are randomly distributed even though N(x) and G are not necessarily constants. A class of sensible microstructural models can be developed if N(x) and G can be reasonably represented by power-law functions of time. Such was seen to be the case for recrystallizadon in cold rolled iron single crystals (23, 28, 29, 32). Additionally it was found to be possible to incorporate a further complexity in the analysis in two instances (28, 29). In these, the new grains were modeled as growing spheroids whose eccentricities varied with time. This was possible, as noted above, because Gtrax could be estimated experimentally. The changing eccentricity is yet another factor, with those mentioned earlier, that can lower the Avrami exponent, k, for recrystallization to a value below 3.
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Mierostruetural path The relationship between the various global microstructural properties that describes the microstructural evolution constituting the recrystallization process is called the microstructural path function. The nature of this relationship depends on both the geometrical and the nucleation characteristics of recrystallization. Vandermeer Masumura and Rath (33) have derived the path function for random, shape-preserved nucleation and growth transformations and obtained the following equation Sv = C. (1 - Vv). [ - In(1 - Vv)]q
(11)
where C and q are constants which depend on the specifics of the chosen model. (A discussion of the classes of models which demonstrate this behavior is presented in references 33 and 22 .) Eqs. 5, 9 and 11 form the basic set of equations upon which an appropriate microstructural model can be deduced from experiment. Only two of the three equations are independent, however. When recrystallization is characterized by both Vv and Sv, it may be easier to develop a proper microstructural model using the microstructural path function (Eq. 11) in conjunction with one kinetic function (Eqs. 5 and 9 are examples of kinetic functions) rather than use the two kinetic functions. The use of the microstructural path function is an advantage when time dependent complications such as competing recovery or non-homogeneous stored energy distributions are present. While these factors manifest themselves in the kinetic functions, they do not affect the path function (33) which reflects the geometrical and nucleation aspects of recrystallization. It should be remembered that Eq. 11 like Eq. 5 has certain limitations for its application to experiment and should be used with discretion. Path functions for more complicated microstructures are now becoming available (22). Summary
An attempt has been made to defend the use of the extended volume concept to develop microstructural models for recrystallization based on a nucleation and growth formalism. It is argued that failures of the Avrami equation to characterize recrystallization kinetics are more than likely due to an inappropriate choice of microstructural model rather than failure of the extended volume concept to correct for impingement. Improved models are becoming available now and use of the microstructural properties themselves can aid in the development of more realistic models. Acknowlfdgement This work was performed at the Naval Research Laboratory under the sponsorship of the Office of Naval Research of the US Department of Navy whose support is gratefully acknowledged. References
1) Fraenkel, W. and Gocz, W.: Zeits f anorg Chemie, 1925, 144. 45 2) v. Goler, F. and Sachs G.: Zeits f Physik, 1932, 77, 281 3) Tammann, G.: Zeits f anorg Chemie, 1933, 214, 407 4) Austin, I. B. and Rickett, R. L.: Trans.AIME, 1939, 135, 396 5) Kolmogorov, A.N.: Izv.Akad.Nauk.USSR-Ser.Matemat., 1937,1(3), 355 6) Johnson, W.A. and Mehl, R.F.: Trans.AIME, 1939, 135,416 7) Avrami, M.: l.Chem.Phys., 1939, 7, 1103; ibid., 1940, 8, 212 8) "Quantitative Microscopy" (eds R.T.DeHoff and F.N.Rhines), 1968, McGraw-Hill, New York NY, p.77 9) Loria, E.A., Detert, K. and Morris, LG.: Acta Met, 1965, 13,929 10) Price, C.W.: Scripta Met, 1985, 19, 669 11) Price, C.W.: Scripta Met, 1989, 23, 1273 12) Doherty, R.D., Rollett, A.R. and Srolovitz, D.L: "Annealing Processes - Recovery, Recrystallization and Grain Growth", (eds. N.Hansen, D.luul-Jensen, T.Lcffers and B.Ralph), 1986, Riso National Lab., Roskilde, Denmark, p.53 13) RoUett, A.D., Srolovitz, D.J., Doherty, R.D. and Anderson, M.P.: Acta Met, 1989, 37,627
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14) Perryman, E.C.W.: Trans.AIME, 1955, 203, 369 15) Leslie, W.C., Plecity, F.J. and Michalak, J.T.: Trans.TMS-AIME, 221, 1961, 691 16) Vandermeer, R.A. and Gordon, P.: "Recovery and Recrystallization of Metals" (ed. L.Himmel), 1963, Interscience Publ,, N'Y, p.221 17) Rosen, A., Burton, M.S. and Smith, G.V.: Trans.TMS-AIME, 2_30,1964, 205 18) Price, C.W.: Acta Met, 1987, 35, 1377; ibid., 1990, 38,727, ibid., 1991, 39, 1807 19) Price, C.W.: "Recrystallization '92", (ed. T. Chandra), 1990, TMS, Warrendale, PA, p. 199 20) Price, C.W.: Private communication 21) Cahn, J.W.: Acta Met, 1956, 4, 449 22) Vandermeer, R.A. and Masumura, R.A.: Acta Metall Mater, 40, 1992, 877 23) Vandermeer, R.A. and B.B.Rath:, Met Trans A, 20A, 1989, 391 24) Vandermeer, R.A.: Acta Met, 15, 1967, 447 25) Cahn, J.W.: Acta Met, 10, 1962, 789 26) Vandermeer, R.A. and Rath, B.B.: Met Trans A, 21A, 1990, 1143 27) Michalak, J.T. and Paxton, H.W.: Trans TMS-AIME, 221, 1961, 850 28) Vandermeer, R.A. and Rath, B.B.: Met Trans A, 20A, 1989, 1933 29) Vandermeer, R.A. and Rath, B.B.: "Recrystallization '92'%(ed. by T.Chandra), 1990, TMS, Warrendale, PA, p.49 30) Gokhale, A.M. and DeHoff, R.T.: Met Trans A, 16A, 1985, 559 31) Cahn, J.W. and Hagel, W.: "Decomposition of Austenite by Diffusional Processes", (ed. Z.D.Zackey and H.I.Aaronson), 1960, Interscience Publ., NY, p. 131 32) Vandermeer, R.A. and Rath, B.B.: "Materials Architecture", (ed by J.B.Bilde-Sorensen, N.Hansen, D.Juu!Jensen, T.Leffers, H.Lilholt and O.B.Pedersen), 1989, Riso National Laboratory, Roskilde, Denmark, p.589 33) Vandermeer, R.A., Masumura, R.A. and Rath, B.B.: Acta Met, 39, 1991, 383
1
o.1 _ffi
O.Ol
10.2
10 "~
10 ° ( Ns ° G 2 )~
10 ~
10 ~
10 3
"t
Figure 1 -- Effect of amount cold-worked grain boundary area per unit volume on the recrystallization kinetics for a hypothetical case of grain boundary nucleated, growth controlled recrystallization.