Quantitative criterion for recrystallization nucleation in single-phase alloys: Prediction of critical strains and incubation times

Acta Materialia 54 (2006) 3983–3990 www.actamat-journals.com

Quantitative criterion for recrystallization nucleation in single-phase alloys: Prediction of critical strains and incubation times H.S. Zurob b

a,*

, Y. Bre´chet b, J. Dunlop

b

a Department of Materials Science and Engineering, McMaster University, 1280 Main Street West, Hamilton, Ont., Canada L8S 4L7 Laboratoire de Thermodynamique et Physico-Chimie Me´tallurgiques, Domaine Universitaire, BP 75, 38402, St. Martin d’He`res Cedex, France

Received 23 September 2005; received in revised form 24 April 2006; accepted 24 April 2006 Available online 7 July 2006

Abstract A simple physically based model is presented to describe the nucleation step of recrystallization in single-phase materials. The model is based on a nucleation criterion which captures the effects of subgrain growth and recovery on the competition between the capillary forces, which oppose the nucleation of recrystallization, and the stored energy, which drives recrystallization. The key quantities predicted by the model are the critical strain, the critical temperature, the incubation time and the nucleation rate for recrystallization. The key inputs needed in the model are the recovery kinetics and the effective subgrain boundary mobility. The model is successfully applied to describe the nucleation of recrystallization in oxygen-free Cu and Al–1% Mg. These two materials are taken to as representative of low and high stacking fault energy systems.  2006 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Nucleation of recrystallization; Recovery; Subgrain growth; Incubation time; Modeling

1. Introduction Recrystallization is used industrially to manipulate the microstructure of alloys through grain refinement, texture control and relief of deformation stresses. Extensive investigations have been carried out to describe the recrystallization kinetics in a variety of industrial and model alloys. At present, powerful semiempirical models are available to describe and control industrial processes involving recrystallization [1,2]. These models predict the recrystallization time, critical temperature, critical strain and final grain size. Due to their empirical nature, the models lack the power to predict recrystallization kinetics outside the range over which they were developed. This has stimulated continued fundamental interest in recrystallization modeling. Analytical models have been developed to predict recrystallization kinetics and textures (e.g., Refs. [3–10]); however, none of these models addresses directly the questions of *

Corresponding author. Tel.: +1 905 525 9140; fax: +1 905 528 9295. E-mail address: [email protected] (H.S. Zurob).

the recrystallization incubation time and critical strain. The purpose of the work reported here was to develop a model that can predict the quantities mentioned above for single-phase materials. 2. The model The current contribution presents a predictive physically based model for the nucleation of recrystallization. The inputs of the model are well-defined and measurable quantities such as the recovery kinetics, grain boundary mobility and initial subgrain size. The model outputs include the incubation time, critical strain and critical temperature for recrystallization as well as the nucleation rate. Ultimately, a complete description of the evolution of the recrystallized fraction with time could be obtained by coupling the present model to a growth model. For simplicity, the following discussion is limited to the process of straininduced boundary migration (SIBM) which was first described by Bailey and Hirsch [11]. This process is expected to be dominant in single-phase materials at small

1359-6454/$30.00  2006 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2006.04.028

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and medium strains [12]. Other nucleation processes such as particle-stimulated nucleation (PSN) and nucleation in shear bands, could, in principle, be described using the framework of the present model. Starting with the classic work of Bailey and Hirsch [11], SIBM takes place when the driving force due to the stored energy of dislocations is sufficient to overcome the boundary curvature, 2c/r, where c is the boundary energy and r is the radius of the subgrain. This leads to a simple expression for the critical size of the recrystallization nucleus: rc ðtÞ ¼

2c GðtÞ

ð1Þ

where rc is the radius that a subgrain needs to reach in order to become a viable nucleus. The nucleation step is visualized in Fig. 1. Starting with Fig. 1(a), which is typical of a low stacking fault energy (SFE) material, the dislocations do not organize into subgrains and the nucleation site is then visualized in terms of a relatively dislocation-free region/cell in the vicinity of the grain boundary. In high SFE materials (Fig. 1(c)), the subgrain structure is well defined and contains most of the dislocations present in the deformed state. The nucleation site is simply a subgrain which has a lower stored energy than its surroundings. In both of the above examples, the capillary term, 2c/r(t), gradually decreases as subgrain growth progresses. When the capillary term drops below the instantaneous value of the stored energy, G(t), the subgrain will bulge into the

(a)

neighboring grain, as shown in Fig. 1(b) and (d) and a nucleus is said to be formed. This nucleus will preferentially grow into the neighboring grain because it benefits from a very mobile high-angle boundary. The effect of recovery on recrystallization is captured through a time-dependent stored energy [6]. The size of the critical nucleus, 2c/G(t), will increase with time as G(t) decreases due to recovery. The occurrence of nucleation is then dependent on the competition between the rate at which the critical subgrain radius increases and the rate at which a given subgrain (of mobility M) can grow to reach the critical size. In what follows, we start by describing this competition in detail and we demonstrate how this naturally leads to the concepts of the critical strain and critical temperature for recrystallization (Section 2.1). In Section 2.2, we describe the kinetics of subgrain growth in terms of the driving force and subgrain mobility. The estimation of the nucleation rate and the number of nuclei is discussed in Section 2.3. Finally, the distinguishing features of the model are highlighted in Section 2.4. 2.1. Nucleation criterion The critical subgrain size, rc(t), above which a subgrain overcomes the capillary forces and starts to grow rapidly, is given by Eq. (1). This equation captures the competition between the increase in the critical subgrain size, 2c/G(t), as a result of the recovery of the global driving force,

(c)

Grain II Grain I

Grain II Grain I

cell

SG

(d)

(b)

Grain II Grain I

nucleus

Grain II Grain I

nucleus

Fig. 1. Possible examples of recrystallization nucleation sites. In (a) and (c), the cell/subgrain initially grows within Grain I. When the cell/subgrain reaches the critical size which allows it to overcome the capillary force, it bulges into Grain II and a nucleus is formed by SIBM as shown in (b) and (d).

H.S. Zurob et al. / Acta Materialia 54 (2006) 3983–3990

and the increase in actual subgrain size, r(t), due to the growth of a particular subgrain in response to its local environment. This concept is illustrated in Fig. 2, where both rc and r are plotted as a function of time. The dotted lines indicate the average subgrain size, Ær(t)æ, and the shaded area reflects the fact that a range of subgrain sizes would exist in any given sample. In Fig. 2(a), none of the subgrains present in the sample reach the critical size and as such no recrystallization nuclei are formed. This would be the case, for example, when the initial strain, e, is less than the critical strain, ec, or when the annealing temperature, T, is less than the recrystallization temperature, Tc. In contrast, nucleation takes place for the cases illustrated in Fig. 2(b) which could be interpreted as corresponding to e > ec or T > Tc. The first nucleus is formed when the shaded area first touches the curve for rc. This corresponds to the largest subgrain in the population reaching the critical size. With time, increasingly smaller grains will attain the critical size and more nuclei will be formed. Progressively, subgrains, which were initially too small, will reach the critical size. The nucleation rate may then be calculated if the subgrain size distribution and the number of nucleation sites are known (Section 2.3).

ε<ε c

r

c

Sub-Grain Size

The detailed mechanisms of dislocation motion leading to cell and subgrain growth have been reviewed by a number of authors [6,13]. It is usually assumed that the rate of subgrain growth can be described using a simple law of the type [6,13,14] vðtÞ ¼ MGðtÞ

ð2Þ

where v is the rate of subgrain growth and M is the subboundary mobility. The driving force used here, G(t), is a global quantity which comprises both the dislocations in the subgrain boundaries and those in the subgrain interior. For a subgrain of initial size r0, the size at time t is given by Z t MGðtÞdt ð3Þ rðtÞ ¼ r0 þ 0

In the present treatment, it is assumed that the subgrain size distribution remains self-similar during subgrain growth, which means that the distribution of the normalized subgrain size, r(t)/Ær(t)æ, is invariant. As such, it is not necessary to follow the evolution of individual subgrains. Rather it is sufficient to monitor the evolution of the average subgrain size, Ær(t)æ: Z t hrðtÞi ¼ hr0 i þ MGðtÞdt ð4Þ For a subgrain of size r(t) = vÆr(t)æ, the nucleation criterion may be expressed as a critical value of the normalized subgrain size, v: v > vc ¼



Time

(b)

2.2. Subgrain growth

0

Sub-Grain Size

(a)

3985

ε>ε c

r

c



2c   Rt GðtÞ hr0 i þ 0 MGðtÞdt

ð5Þ

The time evolution of vc is illustrated in Fig. 3, which also shows the definition of the recrystallization incubation time. The incubation time corresponds to the time necessary for vc(t) to fall below the value of v corresponding to the largest subgrain in the distribution, vmax = rmax/Æræ. In a log-normal distribution the largest subgrain (for practical purposes) would have a radius which is about 3 times the average, Æræ. In a Rayleigh distribution, the largest subgrain is only about 2.5 times the average [6]. In the example shown in Fig. 3, there is an initial increase in the value of vc due to the occurrence of rapid recovery at the onset of annealing. However, as recovery slows down, the evolution of vc becomes dominated by subgrain growth and the value of vc decreases, meaning that increasingly smaller subgrains would be able to develop into nuclei. 2.3. Nucleation rate and number of nuclei

Time

Fig. 2. Recrystallization nucleation takes place when the subgrain size, r, reaches the critical radius, rc. (a) When e < ec or when T < Tc, r will be less than rc and recrystallization does not take place during the time considered. (b) When e > ec or when T > Tc, r eventually exceeds rc and recrystallization takes place.

When the stored energy is sufficiently large, the value of vc will eventually drop to within the range of subgrain sizes present in the sample and nucleation will take place. The nucleation rate and the number of nuclei can be derived from Fig. 2(b) if the subgrain size distribution and the number of nucleation sites are known. Let P(v) be the

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(a)

1 - Nucleated Fraction

0.8

χ (t1) c

P(χ χ)

0.6

0.4

0.2

0

Normalised Sub-grain Size χ (b)

1

t2 > t1

- Nucleated Fraction

0.8

density function of the subgrain size distribution; the fraction of subgrains that would develop into recrystallization nuclei during an annealing treatment of duration, t, is then given by Z 1 f ðtÞ ¼ P ðvÞdv ð6Þ vc ðtÞ

As vc(t) decreases (due to subgrain growth) the fraction of subgrains which are able to form recrystallization nuclei increases as shown schematically in Fig. 4. This fraction can be related to the actual number of nuclei if the type of nucleation site is known. For example, when nucleation takes place on grain boundaries, it is reasonable to assume that the number of potential nucleation sites per unit volume is proportional to 1/(DÆræ2), where D is the grain size. This is similar to the approach used by Vatne et al. [7]. The key difference between the two approaches is that Ref. [7] uses the assumption of site saturation meaning that the critical radius, as defined by the Bailey–Hirsch criterion [11], is constant. In our case, the critical radius evolves due to recovery and this leads to a time-dependent nucleation rate. 2.4. Summary of the main features of the model We have described the nucleation of recrystallization in terms of the unstable growth of a cell/subgrain situated in the vicinity of a high-angle boundary. Humphreys and colleagues [8–10] presented a similar model with the key difference being the treatment of the recovery kinetics. Although

χ (t2) c

0.6

P(χ χ)

Fig. 3. Recrystallization takes place when the critical normalized subgrain size, vc, becomes smaller than the normalized size of the largest subgrain in the sample (i.e., vc < vmax). The initial increase in vc is due to recovery, which was assumed to follow logarithmic kinetics in the present illustration.

0.4

0.2

0

Normalised Sub-grain Size χ Fig. 4. Fraction of subgrains which gives rise to viable recrystallization nuclei consists of subgrains whose size is greater than vc. As the value of vc decreases with increasing annealing time (due to subgrain growth), more subgrains are able to nucleate.

both subgrain growth and individual dislocation recovery are expected to contribute to the softening and to the decrease of stored energy, the classic model proposed by Humphreys only accounts for subgrain growth. We took the more general view, in which the energy released by recovery is not necessarily equal to the energy released by subgrain or cell growth. Physically, this implies that recovery is not limited to subgrain growth, but may also involve the ‘‘cleaning’’ of the subgrain and cell interiors. We are thus able to model medium and low SFE materials in which a non-negligible density of dislocations remains inside the cells/subgrains. The simplicity of the present model conceals two subtle points which are of great importance. The first of these concerns the value of the sub-boundary mobility in Eq. (5) and the role of orientation gradients in the development of recrystallization nuclei. The second point concerns the grain size dependence of recrystallization. Both issues are discussed below.

H.S. Zurob et al. / Acta Materialia 54 (2006) 3983–3990

2.4.1. Sub-boundary mobility The relevant mobility in Eqs. (2)–(5) is the average mobility of the subgrain population in contact with the grain boundary. This value is calculated as Z hHAG hMi ¼ UðhÞMðhÞdh ð7Þ 0

where U(h) is the distribution of sub-boundary misorientation in the vicinity of the grain boundary, M(h) is the subboundary mobility as a function of misorientation and hHAG is the cut-off misorientation at which the sub-boundary is considered to act as a high-angle boundary (15). Following Humphreys, we approximate M(h) by the function [8–10] "  4 # MðhÞ h ¼ 1  exp 5 ð8Þ M HAG hHAG Based on the evidence in Pantleon and Hansen [15], the sub-boundary distribution is well described by a Rayleigh distribution: "    2 # p h p h UðhÞ ¼ exp  ð9Þ 2 havg 4 havg where havg is the average subgrain misorientation in the vicinity of the boundary. Taking this quantity to be in the range 3–5 [15], we obtain an average mobility of 0.02MHAG–0.2MHAG, by applying Eqs. (7)–(9). Given the fact that accurate estimates of MHAG are seldom available, the effective sub-boundary mobility is best treated as an adjustable parameter whose value is in the range of 0.01MHAG–1MHAG. A second important point, which follows from the above calculations, is that nucleation is faster for larger misorientations (i.e., larger havg). This is consistent with existing evidence on the important role of misorientation gradients in the nucleation of recrystallization [6]. 2.4.2. Effect of grain size If recrystallization is dominated by SIBM, the rate of recrystallization would be expected to increase as the initial grain size (D0) decreases [6]. At first sight, this may seem at odds with Eq. (5) which shows no dependence on grain size. The key to resolving this apparent contradiction is that Eq. (5) is strictly applicable if there is no lack of nucleation sites (infinite sample) and if the detection method (e.g., metallography) is capable of detecting all of the nuclei formed. Under these conditions, the occurrence or absence of recrystallization does not depend on the grain size. The origin of the experimental dependence on grain size is a statistical one. In a finite sample and for a given nucleus detection efficiency, the chance of observing a nucleus after a given annealing time increases as the number of nucleation sites (grain boundaries) increases. To quantify the above effect, we argue that the recrystallized fraction needs to exceed a critical value of Xd in order

3987

for the experimentalist to have a reasonable chance of detecting nucleation. If Ær*æ is the average size of the recrystallization nuclei (i.e., the average subgrain size for r > rc), then Xd can be expressed as " # 1=D0 f ðtÞhr i  3 i  ¼ ð10Þ X d / ½f ðtÞ ½hr 2 D0 hr i The first term in brackets is the fraction of subgrains capable of forming recrystallization nuclei as defined by Eq. (6), the second term is proportional to the number of grain boundary nucleation sites per unit volume and the third term is proportional to the size of a critical nucleus. For a given value of Xd, Eq. (10) predicts that recrystallization will be detected earlier (i.e., at smaller f(t)) when the initial grain size, D0, is smaller. This is nicely illustrated in terms of Fig. 4(a) which could be taken to represent the point at which recrystallization becomes detectable in the smallgrain sample, while Fig. 4(b) could be taken to correspond to the detection of nucleation in the large-grain sample. The model is thus capable of capturing the effect of grain size on recrystallization kinetics. In Section 3.1.2, the model is used to calculate the effect of grain size on the critical strain of recrystallization in Cu. 3. Application of the model In this section, the model is used to describe quantitatively the nucleation of recrystallization in two classes of materials. The first class consists of small and medium SFE materials in which negligible recovery of the coldrolled structure is observed prior to recrystallization [6,16]. The second class consists of high SFE metals and alloys. Extensive recovery is expected to take place prior to recrystallization in these materials [6,16–18]. Consequently, the nucleation of recrystallization will be sensitive to the competition between the recovery and mobility terms in Eq. (5). 3.1. Nucleation in the absence of recovery Cu and its alloys have a low to medium SFE. It is generally observed that very little recovery takes place prior to recrystallization in cold-worked Cu. In this section, we predict the incubation time and critical strain for the recrystallization of oxygen-free Cu. 3.1.1. Incubation time Huang and Form [19] measured the recrystallization incubation times for Cu bars which had been cold-drawn 52% and annealed at 270, 290, 310 and 330 C. In the absence of recovery, our model can be simplified to derive an analytical expression for the recrystallization incubation time. Starting with Eq. (5) and assuming that the stored energy does not change with time, we obtain for the appearance of the first nucleus

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Dt ¼

H.S. Zurob et al. / Acta Materialia 54 (2006) 3983–3990

2c r0  2 MG vmax MG

ð11Þ

In the following calculations we used c = 0.625 J/m2 [19], vmax = 2.5 and r0 = 0.25 lm [19]. The value of M was set equal to the high-angle boundary mobility measured by Viswanathan and Bauer [20]: M ¼ M HAB ¼ 0:0002 expð126000=RT Þ

Grain size (lm)

Temperature (C)

Measured critical strain

Calculated critical strain

100

550 600 650

4 3.5 2.6

4.6 3.5 2.8

250

550 600 650

5.5 4.7 3.6

5.8 4.4 3.5

600

550 600 650

7.2 6.1 5.2

7.5 5.6 4.4

ð12Þ

As for the stored energy, it was estimated from the flow stress, r, using the relation [21] G¼

Table 2 Comparison of the predicted and measured critical strains for the recrystallization of pure Cu for various temperatures and grain sizes [25]

1 ðr  ryield Þ2 2 M 2Taylor a2 l

ð13Þ

with the Taylor factor, MTaylor, being 3.1 [22], the constant a being 0.5 [22] and the shear modulus, l, being 42.1(1  0.54(T  300)/1356) GPa [23]. The change in flow stress due to cold work (r  ryield) was estimated from the stress–strain curves of Gourdin and Lassila [24] who studied a Cu alloy with essentially the same composition and grain size as the one being discussed. The measured and calculated incubation times are compared in Table 1. The agreement between the measured and calculated values is very good, given the uncertainties associated with the experimental values for the mobility and incubation times. The key point here is that the predicted incubation times have both the right magnitude and the right temperature dependence. This lends support to the idea that nucleation, in low/medium SFE materials, is controlled by the rate of subgrain/cell growth. 3.1.2. Critical strain In this section, we model the data of Cigdem [25] who reported the critical strain for recrystallization as a function of the annealing temperature and the initial grain size. The experimental data is conveniently summarized in Table 2. As before, we assume that no recovery takes place and, as such, Eq. (11) can be used. Once again, c and vmax were set equal to 0.625 J/m2 and 2.5, respectively. The annealing time, Dt, is 1 h as reported by Cigdem [25]. Eq. (13) is again used to relate the stored energy to the flow stress, which is, itself, related to the strain, e, by the approximate relation [22] r  ryield ¼ 0:139e GPa for 0 < e < 0:1

ð14Þ

Table 1 Comparison of the measured [19] and predicted recrystallization incubation times in oxygen-free Cu cold drawn to a reduction of 52% and annealed between 270 and 330 C Temperature (C)

Measured incubation time (min)

Predicted incubation time (min)

270 290 310 330

33 9 5.4 2.4

39 15 6.0 2.6

The critical strain can now be calculated by combining Eqs. (11), (13) and (14). We use the critical strain for the 100 lm sample which was annealed at 600 C as a reference point, meaning that the mobility is adjusted to fit this data point. An exact fit of this experimental point is obtained using a mobility of 0.3MHAB. To calculate the critical strains for the 100 lm sample at 550 and 650 C it suffices to substitute the appropriate temperature into Eq. (11). The calculation of the critical strain for the other grain sizes is slightly more involved. Starting with Eq. (10) and assigning the subscript S to the 100 lm sample and subscript B to the 250 lm sample, we can write fB ðtÞhr i fS ðtÞhr i ¼ DB DS

ð15Þ

The functions f(t) and Ær*æ are then expressed in terms of vc such that Z 1 f ðtÞ ¼ P ðvÞdv ð16aÞ vc ! Z 1

hr i ¼

vP ðvÞdv =f ðtÞ

ð16bÞ

vc

Substitution of Eqs. (16a) and (16b) into Eq. (15) results in an expression which only contains vcB and vcS. In our earlier calculations for the 100 lm sample we already used vcS = 2.5, which leaves vcB as the only unknown in Eq. (15). This equation can now be solved to obtain vcB. Once this quantity is obtained, Eq. (11) can be used to calculate the stored energy, which is then related to the critical strain through Eqs. (13) and (14). The results of the calculations are listed in Table 2 along with the experimental data. Overall very good agreement is observed between the predicted and measured critical strains. It is worth emphasizing that in the above calculations one value of the mobility (i.e., M = 0.3MHAB) was used to fit all 9 experimental points, thus demonstrating the efficiency of the current approach. 3.2. Nucleation in the presence of significant recovery When cold-worked Al–Mg alloys are annealed, significant recovery is observed prior to the onset of recrystalliza-

H.S. Zurob et al. / Acta Materialia 54 (2006) 3983–3990

(a) Barioz et al. [26] and Legresy and Guyot [27] measured the recovery kinetics of Al–1% Mg in terms of the evolution of the yield stress with time. The measurements spanned cold deformation strains from 0.1 to 3 and annealing temperatures from 160 to 190 C. We used the recovery model of Verdier et al. [21] to fit the above data and extrapolate them to the higher temperatures at which recrystallization takes place. (b) The experimental constants needed to relate the flow stress to the stored energy in Al–Mg alloys were taken from the investigation of Guyot and Raynaud [28]. (c) The average initial subgrain size was calculated as a function of stress using the empirical relation suggested in Ref. [29]. (d) An estimate of the high-angle grain boundary mobility (MHAB) is available from the work of Huang and Humphreys [30] on Al–0.9% Mg. In the following calculations we used an effective sub-boundary mobility of 0.032MHAB. 3.2.1. Incubation time The experimental nucleation data which is used to test the model comes from the investigation by Koizumi et al. [31]. In this work, a cold rolling reduction of 95% was used and the annealing temperature was varied between 225 and 300 C. Fig. 5 is a reproduction of the recrystallization curves reported in Ref. [31]. The incubation times predicted by the model for the above conditions are shown in Table 3. These values were calculated for vmax = 3 (lower limit) and vmax = 2.5 (upper limit). In each case, the predicted

Recrystallized Fraction (%)

60 o

50

300 C

40

o

275 C

30

Temperature (C)

Predicted incubation time (s)

225 250 275 300

50,000–75,000 4600–6700 500–750 60–100

1.2 Experimental data of Perryman Model Predictions

1 0.8 0.6 0.4 0.2 0 0

20

40

60

80

100

120

140

Annealing time (s)

Fig. 6. Comparison of the measured and predicted nucleation rate evolution for an Al–1% Mg alloy cold rolled to a reduction of 60% and annealed at 350 C. The data were normalized by the largest nucleation rate (source: Perryman [17]).

incubation times correspond exactly to the times at which the experimentally measured recrystallized fractions start to increase. 3.2.2. Nucleation rate We next turn our attention to the experimental data of Perryman [17] who estimated the recrystallization nucleation rate from the number of recrystallized grains observed by optical metallography. Fig. 6 compares the measured and predicted nucleation rates for an Al–1% Mg alloy, cold-rolled to a reduction of 60% and annealed at 350 C. Good agreement is observed between the experimental data and the model predictions for t > 30 s. The disagreement at shorter times is likely due to nucleation within highly deformed regions of the microstructure [6]. This effect is not captured in our model, which assumes a uniform stored energy throughout the microstructure. 4. Summary and discussion

20 o

250 C

10

o

0 10

Table 3 Predicted recrystallization incubation times for Al–1% Mg cold rolled to a reduction of 95% and annealed between 225 and 300 C

Normalized Nucleation Rate

tion [17,21]. As such, the nucleation behavior of these alloys will be influenced by recovery. In this section, our model is used to describe the nucleation of recrystallization in Al–1% Mg. We start by summarizing the input data that are used in all of the subsequent calculations.

3989

225 C 100

1000

10000

100000

Annealing time (s) Fig. 5. Recrystallization kinetics for Al–1% Mg cold deformed to a reduction of 95% and annealed at various temperatures (source: Koizumi et al. [31]).

We have presented a simple physically based model for the nucleation of recrystallization in single-phase materials. The model can predict the incubation time, critical strain, critical temperature and nucleation rate of recrystallization. All of the quantities that enter into the model are physically meaningful and measurable. These quantities are: the recovery kinetics, the grain boundary mobility and the initial subgrain size and size distribution. The last two parameters are relatively easy to estimate; extensive

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tabulations of the subgrain size as a function of deformation exist in the literature [29] and, for most materials, the subgrain size distribution can be approximated by a log-normal or a Rayleigh distribution [32]. Concerning the recovery kinetics, it is essential to obtain accurate recovery data when trying to model the nucleation behavior of high SFE materials. Recovery becomes less important, however, when the SFE is low (e.g., Section 3.1). Fortunately, reliable recovery data appear to be available for many pure metals and alloys. In the absence of existing data, quick estimates of the recovery kinetics could frequently be obtained from simple hardness or yield stress experiments [6,21]. This leaves the boundary mobility as the one input parameter, which is hardest to measure reliably. In some well-characterized systems, various estimates of the mobility are available. In the absence of such estimates, a starting value based on the Turnbull estimate [33] could be used. The key point is that while one may not be able to estimate the boundary mobility exactly, one could at least estimate its temperature dependence which is approximately related to the grain boundary self-diffusion activation energy [33]. An adjustable parameter could then be introduced into the pre-exponential of the mobility and this could be used to fit the data as described in the examples of Sections 3.1 and 3.2. The present model then provides a one-parameter framework for fitting and rationalizing existing recrystallization data. Once this parameter is established, the model could be used in a predictive manner as long as the elementary nucleation mechanism remains the same. The present treatment is amenable to coupling to a growth model which would allow the prediction of the complete recrystallization kinetics. The concepts of incubation time, critical strain and recrystallization temperature would all enter naturally into the combined model and there would be no need to introduce them empirically. In addition, the present framework could easily be adapted to describe non-isothermal heating and cooling cycles. Work is currently under way to incorporate these generalizations into the model. Acknowledgements The authors acknowledge many stimulating discussions with Professors D. Embury and G. Purdy (McMaster University). H.S.Z. gratefully acknowledges the financial sup-

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