On the minimum grain size obtainable by high-pressure torsion

Materials Science & Engineering A 558 (2012) 59–63

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On the minimum grain size obtainable by high-pressure torsion Farghalli A. Mohamed n, Shehreen S. Dheda Department of Chemical Engineering and Materials Science, University of California, Irvine, CA 92697-2575, USA

a r t i c l e i n f o

abstract

Article history: Received 10 March 2012 Received in revised form 15 July 2012 Accepted 16 July 2012 Available online 24 July 2012

Recently, a dislocation model that quantitatively relates the minimum grain size obtainable by ball milling, dmin, to several physical parameters, such as the activation energy for self-diffusion and the stacking fault energy, in a nanocrystalline (nc) material was developed. In this paper, it is shown that the predictions of the model are consistent with the characteristics of the minimum grain size, dmin, obtainable in FCC and BCC metals by high-pressure torsion. Such a consistency indicates that the dislocation model for ball milling is quantitatively applicable to the description of other severe plastic deformation (SPD) processes. & 2012 Elsevier B.V. All rights reserved.

Keywords: Activation energy Ball milling Bulk modulus Hardness High-pressure torsion Minimum grain size Nanocrystalline materials Recovery Stacking fault energy

1. Introduction An effective approach to enhance the strength of structural materials is grain refinement. In general, as the grain size, d, is refined, the strength increases as according to the Hall–Petch relation [1,2] that can be represented by pffiffiffi s ¼ s0 þc= d ð1Þ where s is the flow stress, so is a friction stress and c is a constant. Eq. (1) is also valid when the yield strength is replaced by the hardness, H (H¼3s). The mechanistic source of the Hall–Petch relation is not entirely clear, and more than one mechanism may pertain [3–5]. Severe plastic deformation (SPD) has been employed in recent years as a top-down method for grain refinement, which occurs as a result of structural decomposition. In the past two decades, considerable efforts have been devoted for exploring novel top-down processing techniques and for understanding the underlying mechanisms for structural decomposition. As a result of these efforts, a variety of severe plastic deformation (SPD) techniques have been developed including ball milling (BM) [6–9], high-pressure torsion (HPT) [10,11], and equal channel angular pressing (ECAP) [12,13]. The refinement of grain size during BM is governed by deforming individual particles using the energy generated by ball collision in a milling medium either at room temperature or at a n

Corresponding author. Tel.: þ1 949 824 5807; fax: þ1 949 824 2541. E-mail address: [email protected] (F.A. Mohamed).

0921-5093/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.msea.2012.07.066

low temperature. During milling, materials experience severe impact deformation, resulting in grain refinement. Over the past several years, the characteristics of crystal refinement and development of nanostructures during ball milling have been studied extensively. These studies have led to several important findings. First, Oleszak and Shingu [14] have observed that the crystallite size decreases with milling time and that continuous milling leads to a minimum grain size, which is a characteristic of each metal. Second, Eckert et al. [15] have suggested that the minimum average grain size, dmin, is obtained as a result of a balance between the formation of dislocation structure and its recovery by thermal processes. Third, Fecht [16] has proposed a phenomenological approach for grain refinement during milling that involves the following three stages: (a) the localization of a high-dislocation density in shear bands, (b) the annihilation and recombination of dislocations, forming cells and subgrains (recovery), and (c) the transformation of subboundaries into high-angle grain boundaries. By utilizing the aforementioned findings and suggestions, Mohamed [17] has recently developed a theoretical dislocation model, which quantitatively describes the dependence of the minimum grain size on several physical parameters in an nc-material. The model may be represented by [17] 2

dmin =b ¼ A3 expðbQ =4RTÞðDPO Gb =no kTÞ0:25 ðg=GbÞ0:5 ðG=sÞ1:25

ð2Þ

where b is the value of Burgers vector, A3 is a dimensionless constant, b is a constant (  0.04), Q is the self-diffusion activation energy, R is the gas constant, T is the absolute temperature, DPO is the frequency factor for pipe diffusion, G is the shear modulus, no

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F.A. Mohamed, S.S. Dheda / Materials Science & Engineering A 558 (2012) 59–63

is the initial dislocation velocity, k is Boltzmann’s constant, g is the stacking fault energy and H is the hardness ( ¼3s, where s is the normal stress). As shown by Eq. (2), the model predicts that the minimum grain size, dmin, depends on hardness, stacking fault energy, and an exponential function of the activation energy for recovery. The predictions of the above model were found to be in good agreement with the experimental data reported for nc-FCC and nc-BCC metals [17]. In the paper, the validity of the above model to the description of the minimum grain size obtainable as a result of processing via HPT is examined. Such an examination is motivated by two considerations. First, the refining mechanism in HPT, like that in BM, involves two competing processes: the introduction of dislocation structure by severe deformation and its recovery by thermal processes. Second, during all SPD processes, grain size decreases rapidly at the early stage of processing and then approaches a constant value, dmin, representing steady-state conditions, with further straining [18].

of a few GPa. Very large shear strain can be obtained after several revolutions without changing the sample dimensions. HPT is of scientific interest for various research activities. For example, a recent study has exploited the nc- and near nc-structures in a single HPT sample to investigate the grain size effect on the deformation mechanism in nanostructured copper [10]. Ball milling (BM) and high-pressure torsion (HPT) are characterized by three common features. First, in these two processes, coarse-grained materials are refined to nanostructured (nc) or ultrafine-grained materials by structural decomposition. Second, the final grain size obtainable via is determined by two competing processes: the introduction of dislocation structure by severe deformation and its recovery by thermal processes. Third, during these processes, the microstructural evolution appears to follow the same pattern [25]. For example, the microstructural evolution during processing of Al and its alloys by either ball milling [26] or high-pressure torsion [27,28] follows these steps: dislocation multiplication and accumulation, formation of subgrains, the increase of the misorientation angle and the transformation of subgrains into high-angle grains, and a balance between dislocation multiplication due to severe deformation and dislocation annihilation at boundaries. The preceding discussion leads to the implication that the model developed for ball milling should be equally applicable to HPT. This implication is quantitatively examined herein.

2. Analysis and discussion 2.1. The model In developing the model for the steady-state grain size obtainable by ball milling, an analysis was performed in detail to obtain two expressions that were related to: (a) the rate at which the grain size decreases during milling, (dd/dt)  , and (b) the rate of grain size increase, (dd/dt) þ . It was assumed that the former rate and the latter rate were proportional to the deformation energy and the rate of recovery, respectively. The details of the physical parameters that contributed to the deformation energy (for example, dislocation multiplication, strain produced by the movement of dislocations) and the rate of recovery (for example, dislocation climb velocity) were identified in the original article [17]. The minimum grain size was then given by the following condition: ðdd=dtÞ þ ¼ ðdd=dtÞ

2.3. Sources of the data related to HPT Data on the minimum grain size, dmin, are available for six FCC metals (Al, Ni, Cu, Pd, Ag, and Au), two FCC alloys (Ni-40% Co and Ni-65% Co) and seven BCC metals (Cr, Fe, Nb, W, V, Mo, Ta). The values of the minimum grain size, dmin, reported for nc-FCC and nc-BCC metals are given in Table 1. These values were taken from Refs. [29–42]. Also included in this table are physical parameters characterizing the metals listed in the table. These parameters are the Burgers vectors, b, the stacking fault energy,g, the activation energy for diffusion, Q, the melting temperature, Tm, the shear modulus, G, and the bulk modulus, B. The values of the parameters were previously documented in Ref. [17].

ð3Þ

2.4. Correlation between the model and explicit parameters 2.2. Basis of the applicability of the model to high-pressure torsion Consideration of Eq. 2 indicates the presence of three explicit parameters that can be correlated with available experimental data. These parameters are the activation energy, Q, hardness, H, and the normalized stacking fault energy, g=Gb.

In high-pressure torsion (HPT) processing [10,11,19–24], disk samples are torsionally deformed between two anvils rotating with respect to each other under a normal compression pressure Table 1 Materials’ parameters. Crystal structure

Metal

dmin (mm)

FCC

Al Ni Cu Pd Ag Au Ni-40Co Ni-65Co Cr Fe Nb W V Mo Ta

1.70 0.24 0.39 0.23 0.48 0.52 0.12 0.09 0.50 0.20 0.18 0.17 0.33 0.27 0.18

BCC

[29,30] [31] [31,32] [31,33] [34] [34] [35] [35] [36] [37] [38,39] [40] [41] [41,42] [39]

b (nm)

dmin/b

G (Gpa)

g (mJ/m2)

g/Gb

Q (kJ/mol)

Tm (K)

B (Gpa)

Bb3

Hv (Gpa) [43]

0.286 0.249 0.255 0.275 0.288 0.2884 0.25 0.25 0.249 0.248 0.286 0.274 0.2618 0.2725 0.2856

5944 964 1510 836 1667 1803 460 340 2008 806 617 620 1261 972 630

26.2 76 48 43 30.3 27.7 80 80 115 81 37 160 46.7 125.6 69

200 250 55 130 23 50 100 20 380 180 150 500 150 430 210

0.02669 0.01321 0.00449 0.01099 0.00264 0.00626 0.00500 0.00100 0.01327 0.00896 0.01418 0.01141 0.01227 0.01256 0.01066

143 289 197 264 189 177

933 1726 1356 1825 1235 1336

75 177 138 187 104 171

1.8 2.7 2.3 3.9 2.5 4.1

0.313 3.021 1.298 2.127 0.941 0.804

306 281 420 567 308 465 413

2130 1808 2742 3695 2175 2888 3253

160 180 170 311 158 261.2 196.3

2.5 2.7 4.0 6.4 2.8 5.3 4.6

4.756 3.020 2.354 8.830 2.354 6.669 4.132

Data for b, G, g, Q, Tm, B were taken from Ref. [17].

F.A. Mohamed, S.S. Dheda / Materials Science & Engineering A 558 (2012) 59–63

Al

61

Al Cr Au Ag Cu

Pd Fe

Cr

Ag Au

V

Pd

Mo

Ni Ta Nb

V Cu

W

Ni

Mo W Ta

Fig. 1. Normalized minimum grain size, dmin/b obtained by HPT versus the activation energy for the self-diffusion, Q.

2.4.1. Minimum grain size vs. the activation energy According to Eq. (2), when the logarithm of the normalized minimum grain size is plotted against the activation energy for lattice diffusion, a straight line having a slope of  b/RT results. The data on dmin for FCC metals and BCC metals that are given in Table 1 are plotted in Fig. 1. An examination of this figure reveals that the data of all metals scatter about a straight line, which represents the best fit, in agreement with the above prediction. The slope of this straight line was estimated as 0.0037, and the equation of the plot can be best described by the following equation: dmin =b ¼ 3750e0:0037Q

Fe

Nb

Fig. 2. Normalized minimum grain size, dmin/b, obtained by HPT as a function of the normalized hardness, H/G, (logarithmic scale).

Al

Ag

Cu

Au

Cr V Pd Fe

Mo Ni

Ta W Ni-40wt%Co

Nb

Ni-65wt%Co

ð4aÞ

Eq. (4a) is essentially similar to the following equation that was previously reported for the curve describing the behavior of nc-FCC and nc-BCC metals [17] under the condition of ball milling dmin =b ¼ 112e0:0037Q

ð4bÞ

The difference in the value of the pre-exponential term between Eqs. 4(a) and 4(b) reflects the fact that the grain sizes produced via HPT are in general larger than those obtained as a result of milling [18]. A comparison between Eq. (6) and Eq. (2) indicates that: b/4RT¼0.0037. By substituting T 300 K and R¼8.31 J/mol, b is estimated as 0.037. This value is close to that predicted from a previous analysis (b  0.04) [17]. 2.4.2. Minimum grain size vs. hardness Eckert et al. [15] have suggested the minimum grain size, dmin, for FCC metals (Al, Cu, Ni, Pd, Rh, and Ir) produced via milling scales with the hardness of the specific metal, as expressed in terms of the Vickers hardness of these metals, Hv (cold worked conventional polycrystalline materials). The validity of such a suggestion in nc-FCC and nc-BCC metals was quantitatively examined by Mohamed and Xun [44], who found that an increase in the normalized hardness, H/G, results in a corresponding decrease in the normalized minimum grain size for metals, dmin/b. Eq. (2) relates the normalized minimum grain size obtainable by milling, dmin/b, to s. When s is replaced by hardness, H, using the following relation: H¼3s, Eq. (2) can be written as 2

dmin =b ¼ A3 ðebQ =4RT ÞðDpo Gb =no kTÞ0:25 ðg=GbÞ0:5 ðG=HÞ1:25

ð5Þ

where A3 is a dimensionless constant. According to Eq. (5), a plot of the logarithm of the normalized minimum grain size, dmin/b, against hardness, H/G, results in a straight line having a slope of 1.25. This prediction was verified in the case of nc-FCC and ncBCC metals processes via ball milling [17]. In order to examine the validity of such prediction to metals when prepared via HPT, the above plot was constructed in Fig. 2 for metals listed in Table 1. As shown by the figure, all datum points fit a straight line having

Fig. 3. Normalized minimum grain size, dmin/b obtained by HPT as a function of the normalized stacking fault energy, g/Gb, (logarithmic scale).

a slope of about 1.22, a finding that is consistent with the above prediction.

2.4.3. Minimum grain size vs. stacking fault energy One of the important predictions of the model as represented by Eq. (2) is the dependence of dmin on the stacking fault energy, g. According to the equation, the normalized minimum grain size dmin/b depends on the square root of the normalized stacking fault energy, g/Gb. It was shown [17] that for nc-FCC and nc-BCC metals processed by milling, there was good agreement between experimental data and prediction. In Fig. 3, the data available on nc-metals in Table 1 are plotted as dmin/b vs. the normalized stacking fault energy, g/Gb, on a logarithmic scale. An examination of the plot of Fig. 3 reveals three main points. First, despite some scatter, the datum points can be fitted by a straight line. Second, metals such as Ta and W that have similar normalized stacking fault energies exhibit similar dmin/b. Second, the slope of the straight line in Fig. 3 is equal 0.4, which is in good agreement with the theoretically predicted value of 0.5; the equation of the straight line may be given by dmin =b ¼ Cðg=GbÞ0:4

ð6Þ

Two comments are in order regarding the relation between dmin/b and g/Gb. First, such a relation reflects that recovery plays an important role in determining dmin; without the occurrence of recovery the steady-state grain size would not be obtained. In general, recovery of dislocation structures during the plastic deformation occurs via dislocation climbs and/or cross slip, both of which are sensitive to the value of the stacking fault energy of the material. Second, Edalati and Horita [39] have recently reported that dmin/b obtainable by HPT is nearly insensitive to g/Gb as far as the data are evaluated at a given T/Tm. This report appears in contradiction not only to the previous results on BM

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[17] and the present results on HPT but also to the basic concept discussed above that the occurrence of recovery, which is sensitive to g, is necessary in terms of reaching a steady-state grain size during a severe plastic deformation process. In this context, Ito and Horita [27] have concluded that microstructure evolution during HPT involves dislocation absorption at subgrain boundaries and as a result of this process, the misorientation angle increases and subgrain boundaries transform into high-angle grain boundaries. As indicated elsewhere [17,45], dislocation absorption at boundaries occurs via dislocation climb, a process that is sensitive to the value of g.

plotted as normalized minimum grain size, dmin/b, on a logarithmic scale vs. the bulk modulus, B, the data fitted a straight line whose equation was described by the following expression:

2.5. Correlation between the model and implicit parameters

Q ¼ c1 a3 Eo

There are two implicit parameters, the melting point, Tm, and the bulk modulus, B, in Eq. (2). The correlation between the model and these two implicit parameters are examined herein. 2.5.1. Minimum grain size vs. melting temperature According to well-documented information, the self-diffusion activation energy, Q, scales with melting temperature, Tm, according to the following relation [46]: Q =RT m  l

ð7Þ

where l is about 17–18. By replacing Q in Eq. (2) by lRTm, one obtains 2

dmin =b ¼ A3 expððbl=T m ÞT=4ÞðDpo Gb =no kTÞ

0:25

0:5

ðg=GbÞ

ðG=sÞ

1:25

ð8Þ Eq. (8) predicts that a plot of the logarithm of the normalized minimum grain size against the melting temperature yields a straight line. The validity of such a prediction was demonstrated in case of nc-FCC and nc-BCC processed by ball milling [17]. Fig. 4 shows a plot of the normalized minimum grain size, dmin/b, against the melting temperature, Tm, of all metals that are listed in Table 1, and that were produced via HPT. In agreement with the prediction of Eq. (8), the data fit a straight line, which may be described by the following equation: dmin =b ¼ 3818e0:00056Tm

dmin =b ¼ 100e0:003B

ð10Þ

As discussed elsewhere [17], the origin of the above equation is not related to the presence of the shear modulus (shear modulus in Eq. (2) has no net effect) but most likely arises from the relation between the bulk modulus and the activation energy for self-diffusion, Q, which can be represented by the following expression [47]: ð11Þ

where c1 is the proportionality constant, Eo is an appropriate elastic modulus, and a is the lattice parameter. In order to examine whether the behavior of FCC and BCC metals prepared by HPT is consistent with the above trend, two steps were carried out. First, the data of all metals listed in Table 1 were plotted as Q against Bb3 on a logarithmic scale in Fig. 5(a); for the sake of consistency with previous analyses [17,18], data of b rather than a was used. As can be seen, the data can be fitted by a straight line whose equation is represented by 3

Q ¼ 85b B

ð12Þ

Second, the logarithm of dmin/b, for all metals was plotted vs. Bb3 in Fig. 5(b). As can be seen, the data of FCC and BCC metals scatter about a straight line that can be described by the following equation: dmin =b ¼ 3528e0:3b

3

B

ð13Þ

The agreement between Eq. (13) and Eq. (6) can be verified by showing that starting with Eq. (4a) and utilizing Eq. (12) yields Eq. (13). This procedure is carried out as 3

3750e0:0037Q ¼ 3750e0:00380b

B

3

¼ 3750e0:3145b

3

3

¼ 3750e0:0145b B e0:3b

B

B

ð14aÞ

ð9Þ

According to Eq. (8), the slope of the straight line should be equal to  bl/4T. Substituting b ¼0.037, l ¼17.5 and T¼300 K yields 0.00055 as the value of  bl/4T (the slope). This value agrees very well with the slope of the straight line of Fig. 4 as represented by Eq. (9). Also, the value pre-exponential term in Eq. (9) is in good agreement with that of pre-exponential term in Eq. (4a).

W Nb Cr

Ni V Fe

Mo

Pd

Cu Ag

2.5.2. Minimum grain size vs. bulk modulus The results of a previous analysis [17] that were based on Eq. (2) have led to the following finding: when dmin, reported for nc-FCC metals and nc-BCC metals processed by ball milling were

Ta

Au

Al

Al Al

Cr Au Au

Cu Ag Ni

Cr

Ag Cu Ni

Fe

Pd

V

Fe

Mo W Nb

V

Mo

Pd Nb

Ta

W

Ta

Fig. 4. Normalized minimum grain size, dmin/b obtained by HPT versus melting temperature, Tm.

Fig. 5. (a) The activation energy for the self-diffusion, Q, versus the bulk modulus multiplied by the cube of the Burgers vector, Bb3. (b) The logarithm of the normalized minimum grain size, dmin/b obtained by HPT versus the bulk modulus multiplied by the cube of the Burgers vector, Bb3.

F.A. Mohamed, S.S. Dheda / Materials Science & Engineering A 558 (2012) 59–63

By taking an average of four for b3B in the first exponential 3 term, e0:0145b B , one obtains 3

3750e0:0037Q ¼ 3528e0:3b

B

ð14bÞ

This finding not only demonstrates that there is good agreement between the prediction of Eq. (2) and the effect of the bulk modulus on the values of dmin obtainable by HPT but also indicates that the origin of the dependence of dmin/b, on the bulk modulus is the relation between the bulk modulus and the activation energy for self-diffusion.

3. Conclusions It is demonstrated that the data on the normalized minimum grain size obtainable as a result of processing via high–pressure torsion (HPT) agree reasonably well with the predictions of the dislocation model that was recently developed to account for the characteristics associated with the minimum grain size obtainable by ball milling (BM). The origin of the agreement is related to two observations. First, the refining mechanism in HPT, like that in BM, involves two competing processes: the introduction of dislocation structure by severe deformation and its recovery by thermal processes. Second, grain refinement during HPT, like that during (BM), involves dislocation multiplication and accumulation, formation of subgrains, the increase of the misorientation angle and the transformation of subgrains into high-angle grains, and a balance between dislocation multiplication due to severe deformation and dislocation annihilation at boundaries.

Acknowledgments This work was supported by US National Science Foundation (Grant no: DMR-0702978). References [1] [2] [3] [4] [5] [6]

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