Inverse Hall–Petch effect and grain boundary sliding controlled flow in nanocrystalline materials

Materials Science and Engineering A 452–453 (2007) 462–468

Inverse Hall–Petch effect and grain boundary sliding controlled flow in nanocrystalline materials K.A. Padmanabhan a,∗ , G.P. Dinda b , H. Hahn b , H. Gleiter b a

b

Anna University, Administration Building, Students’ Affairs Wing, Chennai 600 025, India Institute of Nanotechnology, Research Centre Karlsruhe, P.O. Box 3640, D-76021 Karlsruhe, Germany Received 12 February 2006; received in revised form 17 October 2006; accepted 18 October 2006

Abstract A simplified procedure for an order of magnitude experimental validation of a recently proposed model for the strain-rate dependent deformation of nanostructured materials is given. The grain size dependence of hardness predicted by the model in the range where the inverse Hall–Petch effect is observed is validated using reliable experimental results. The significance of some related observations is also discussed. © 2006 Elsevier B.V. All rights reserved. Keywords: Inverse Hall–Petch effect; Strain-rate dependent flow; Mesoscopic grain boundary sliding; Grain boundary migration; Rate controlling process/step

1. Introduction Ambient temperature mechanical softening with decreasing grain size in nanostructured (n-) materials – the so called “inverse Hall–Petch effect” – was first observed experimentally in 1989 [1]. This report generated a debate because the relative density of the compacts used was low and the change in grain size was achieved through annealing, which could have altered in addition the density. Inverse Hall–Petch effect was reported subsequently by others as well [2–4]. Koch and Narayan [5] have identified the problems associated with obtaining artefact-free samples and the accurate determination of grain size and its distribution. According to them, till 2001 only two sets of experimental results, viz., that of Erb et al. [6,7] and the unpublished results of Koch and Narayan (included in ref. [5]) truly demonstrate the inverse Hall–Petch effect. More recently, Wolf et al. [8] have expanded this list to include five more publications [9–13] as reporting genuine inverse Hall–Petch effect. However, results presented in refs. [5,9,12] are the same except that in ref. [9] the hardness values are converted into stress values by dividing the former set of values by 3 (a well-known relationship between hardness and flow stress in microcrystalline materials [14]). Ref. [11] was

∗

Corresponding author. Tel.: +91 44 22352270; fax: +91 44 22352270. E-mail addresses: [email protected], [email protected] (K.A. Padmanabhan). 0921-5093/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2006.10.084

cited erroneously in ref. [8]. Therefore, as on date there are four sets of results reporting inverse Hall–Petch effect, which are considered to be non-controversial [5,8]: (a) in laser ablated Zn [5,9,12], (b) in electrodeposited Ni [6,7], (c) in electrodeposited Ni tested by a nanoscratch technique [10] and (d) in pulsed laser deposited Cu [13]. Based on an approach rooted in micromechanics, the inverse Hall–Petch effect in nanocrystalline materials was predicted in 1995 [15,16] by extending and refining an earlier model for mesoscopic (cooperative) grain/interphase boundary sliding controlled flow in microcrystalline materials [17]. This effect was predicted again in 1998, using computer simulation [18]. In materials science as well as in mechanics the equivalence among an increase in temperature, T, a decrease in strain-rate, ε˙ , and a decrease in grain size, L, is well recognized. Therefore, it is not unexpected that in some systems with a grain size in the lower nm range, creep effects (in which flow stress/hardness increases with increasing grain size) are seen at ambient temperatures. A model to understand optimal structural superplasticity in micro- and sub-microcrystalline metallic and ceramic materials and strain-rate dependent deformation in nanocrystalline materials on a common basis has been presented [16,17,19–22]. It was realized at an early stage [19,20] that if rate controlling flow were confined to the grain boundary regions, an infinite continuum would be available for deformation. Of course, grain deformation is necessary to overcome the geometric constraints and to obtain coherent strain across the grains. But, if the assumption is that the rate controlling flow is confined

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to the grain boundary regions only, grain deformation can then be regarded as a faster accommodation process and it will, therefore, make no contribution to the rate equation. In this paper, this model [16,17,20–22] is: (a) further developed to consider hardness variation as a function of the experimental variables, (b) simplified to reduce the number of empirical constants required to facilitate easier order of magnitude validation of theoretical predictions, (c) invoked to verify its predictions concerning the inverse Hall–Petch effect using experimental data considered to be non-controversial and (d) shown to account for the mechanical response of a not-fully dense nanocrystalline palladium compact [23]. This simplification allows three temperature dependent constants to be given realistic mean values (see below) and so the solution of the transcendental equations resulting from the model becomes easier. In Appendix A, a brief description of the model is presented. 2. Further development of the model 2.1. Re-examination of Eqs. (A.2)–(A.7) Eqs. (A.2)–(A.7) are based on the premises (a) that grain boundary sliding, that develops to a mesoscopic scale, controls the rate of deformation, (b) that grain boundary migration allows boundary sliding to develop to a mesoscopic scale by the alignment of contiguous boundaries to form a plane interface, (c) that at coarser grain sizes boundary migration is brought about by the combined action of dislocation emission from the boundaries in a barrier-free manner or during a thermal event [34] and rate controlling boundary diffusion of the order of a fraction of unit interatomic distance on average and (d) at grain sizes in the lower range of the nm scale, boundary migration is caused entirely by diffusion of the order of unit interatomic distance within the grain boundary. This picture is consistent with what was suggested by Herring [35] for the migration of an “ideal” grain boundary [22]. Very recently, Cahn et al. [31] have proposed a diffusionless boundary migration mechanism based on the coupling action between a relative displacement between two grains parallel to their common boundary and a displacement perpendicular to the grain boundary. It is well known that a perpendicular displacement of the boundary (say, caused by diffusion) should be accompany a displacement parallel to the boundary, if continuity of the material is to be ensured and the density maintained constant (see, for example, refs. [36,37]). What is new in the proposal of ref. [31] is that it maintains that the perpendicular displacement of the boundary that will accompany a shear parallel to the boundary (the mechanism responsible for this latter process has not been identified) is sufficient to account for boundary migration in its entirety, without invoking diffusional flow. The present authors experience some difficulties with that approach. (a) Ref. [31] considers two solutions of the Frank–Bilby equation applicable to small angle boundaries in terms of dislocation concepts and suggests that these equations apply to high angle boundaries also. In our opinion, a good correlation with experimental results alone cannot be used as an argument to justify the extrapolation of theoretical results to regions outside the domain

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within which the solutions of the Frank–Bilby equation are valid. The work of Pond, Sutton, Wolf and others has clearly demonstrated that dislocation concepts are not useful in describing the disorder present at high-angle grain boundaries [28,29]. (b) It is yet to be shown that the velocity of migration predicted by the model is equal to the velocity of migration of high angle grain boundaries observed experimentally. Therefore, in our opinion, there is no need, at least at this stage, to revise the details proposed in ref. [22] and Eqs. (A.2)–(A.7) may be used as they are. 2.2. Hardness–flow stress correlation Hardness H, and the effective shear flow stress that causes rate controlling flow, τ eff , are directly related [14]. In the present model [16,17,22] (see Appendix A) a threshold shear stress of τ 0 is necessary for the onset of rate controlling mesoscopic boundary sliding. That is, τ eff = (τ − τ 0 ), where τ is the externally applied shear stress. Therefore, H = D1 (τ − τ0 )

(1a)

In the earlier papers [16,17,22], by treating the√material to be a von Mises solid the relationship (σ − σ0 ) = 3(τ − τ0 ) was written. But, according to ref. [33], it should be written as (σ − σ0 ) = 3.06(τ − τ0 ). In contrast, Hill [32] has argued that macroscopic mechanical response is usually measured at the √ surface, where a conversion factor of 3 is adequate. Unfortunately, the accuracy of measurements is never sufficient to decide in favour of one proposal or the other. Therefore, in this paper both these possibilities are considered. That is,   D1 H= √ (σ − σ0 ) (1b) 3 or   D1 H= (σ − σ0 ) (1c) 3.06 √ For materials of conventional grain sizes, (D1 / 3) (or (D1 /3.06)) is equal to 3 [14].If the constant of proportionality between H and (σ − σ 0 ) is taken to be unchanged for materials of grain size in the nm scale also, compared√ with that for materials of microcrystalline grain sizes, i.e., (D1 / 3) (or (D1 /3.06)) = 3, for example, as done in ref. [38], further analysis is possible. (The procedure will remain unchanged, even if the constant of proportionality were different.)Then, from Eqs. (A.7) and (1b),  4 2    W γ0 υH −F0 γ˙ sp = 0.4025 exp (2a) LkT kT From Eqs. (A.7) and (1c)  4 2    W γ0 υH −F0 γ˙ sp = 0.7111 exp LkT kT From Eqs. (A.3)–(A.5), (A.7) and (1a)     GυH 0.5 −F0 2 τ0 diff = 0.3288W γ0 exp Dgb aC 2kT

(2b)

(3)

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K.A. Padmanabhan et al. / Materials Science and Engineering A 452–453 (2007) 462–468

2.3. Simplifications For an order of magnitude calculation, it is possible to assign the values indicated below for all systems, except where experimental measurements result in different values. Apart from simplifying the calculations, this will also prevent them from being used as adjustable constants. From the computer simulations of Wolf [28,29] and bubble raft experiments [27] the mean shear strain associated with a sliding event in a basic unit of boundary sliding, γ 0 , is equal to 1 if the factor that converts the shear mode 0.1. Then, ε0 = 0.057, √ to the tensile is 3. This would be equal to 0.033, if the Taylor factor were used as the conversion factor, in accordance with the suggestion of ref. [33]. For many materials, for an order of magnitude calculation, p may be taken as equal to (1/3) [14]. There has been some controversy on the width of high angle/general grain boundaries in nanostructured materials. Van Swygenhoven et al. [39], after a careful examination of available literature and computer simulation, have concluded that there is no unambiguous evidence in support of the early view that grain boundaries in nanocrystals are highly disordered, amorphous or liquid-like interfaces. Their results suggest that the grain boundary structure in nanocrystalline materials is “similar to that found in larger grain polycrystals”. Haasen [40], based on a thorough scrutiny of field-ion micrographs and simulations, has presented a sketch (see Fig. 3.10 of ref. [40]), which reveals that the width of the grain boundary is equal to 2–3a, where a is the atomic diameter. Therefore, in the present calculations, the grain boundary width, W, is taken as equal to 2.5a. The vacancy concentration, C, at temperatures where contribution from sliding cannot be ignored, is taken as equal to 10−4 [30]. If the free energy of activation/activation energy for the rate controlling process is determined from the experimental results, the mean value of G for the range of the experiments can be calculated using Eq. (A.1). In this simplified picture, accurate knowledge of F0 and σ 0 (which can be determined from isostructural steady state logarithmic tensile stress–tensile strain rate plots for different temperatures) are enough to describe flow quantitatively (Eq. (A.7)). From Eq. (A.2), the value of r can be determined, if the specific grain boundary energy is known (usually 0.5–1.0 J m−2 [22]). 3. Experimental validation 3.1. Simplification of earlier procedure for experimental validation In the earlier paper [22], for experimentally validating the model by an iteration method, γ 0 , F0 and τ 0 in Eq. (A.7) were allowed to change. But, Eq. (A.7) is transcendental in nature. Therefore, a solution cannot be obtained straightaway. Initially, the values of γ 0 and τ 0 are chosen arbitrarily and the corre1 In refs. [16,17,22], ε was rounded off to 0.05 (Tresca), as the effect of this 0 approximation on the numerical values computed was negligible.

sponding F0 value is calculated. Then, the values of γ 0 and τ 0 are individually changed to re-determine the value of F0 and so on. The combination for which the sum of the squares of the difference between the experimental and the predicted values of the strain rate is the least is taken as the best fit. As can be seen readily, this is a trial and error procedure and the reliability as well as the time needed for obtaining a robust solution will be positively influenced by a decrease in the number of constants in the rate equation. In view of what was stated in the previous section, in the present simplified calculations ε0 in Table 1 of ref. [22] is taken as equal to 0.057 and F0 (temperature independent) and σ 0 (temperature dependent) are estimated by the method of iteration more easily. A comparison between the present results and those obtained in ref. [22] is made in Table 1. From the new F0 and σ 0 values and by taking p = (1/3) and the appropriate value of Γ B for each system, G and r values for the four systems could be computed. These values are also given in Table 1. A careful examination of the results reveals that a simplification that uses for all systems ε0 = 0.057, p = (1/3) and G = constant in the temperature interval of the experiments may be employed with success for order of magnitude calculations.2 It is evident that this simplified method leads to a significant reduction in the experimental and computational efforts needed to validate the model and it also predicts the strain rate within an order of magnitude.

3.2. Experimental validation of inverse Hall–Petch effect The data on inverse Hall–Petch relationship seen in n-Zn [5,9,12], electrodeposited n-Ni [6,7], n-Ni tested by a nanoscratch technique [10] and pulsed laser deposited Cu [13] were digitised and they are reproduced in Fig. 1. If boundary migration/plane interface formation takes place by a combination of dislocation emission and diffusion over a distance on average equal to a fraction of unit interatomic distance in the boundary region, from Eqs. (1a)–(1c) and (A.2), it follows that H = A1 −

B1 L

(4)

where A1 and B1 are the constants.3 If G, Γ B and r are known for these systems under the given experimental conditions, A1 and B1 can be calculated a priori using Eqs. (1a)–(1c) and (A.2). A least squares fit between H and L was attempted in terms of Eq. (4) to see if it is obeyed. When boundary migration is entirely by diffusion over a distance of the order of unit interatomic distance

2 Even when ε is taken as equal to 0.033 (Taylor factor conversion [33]), 0 the method will remain basically the same. For fixed values of F0 and τ 0 , the values of G and r will change by factors of 1.1108 (increase) and 0.9003 (decrease), respectively. If F0 and σ 0 are kept the same instead of F0 and τ 0 (same accuracy of prediction), r will increase by a factor of 1.5593, but G will increase only, as before, by a factor of 1.1108. √ 3 It should be noted that regardless of whether the factor of conversion is 3 (von Mises) or 3.06 (Taylor factor), Eq. (4) is valid.

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Table 1 Simplified calculations for Tables 1–4 of ref. [22] with ε0 = 0.057, p = (1/3) and G as Gmean Table number

Temperature (K)

Grain size (␮m)

7.6 7.6 7.6

ε0 (ref. [22]) Original F0 [22] (kJ/mol)

New F0 (kJ/mol)

0.067 0.067 0.068

148 148 148

141.5 141.5 141.5

30.74 30.74 30.74

0.47 0.48 0.49

0.50 0.70 0.90

0.064 0.126 0.209

1.6 1.4 1.8

1.2 1.3 1.7

0.066 0.066 0.066 0.064

140 140 140 140

135 135 135 135

29.34 29.34 29.34 29.34

0.198 0.200 0.202 0.204

0.30 0.32 0.34 0.36

0.136 0.155 0.175 0.196

2.2 2.5 3.6 3.7

2.3 2.1 2.9 3.7

2.90 2.92 2.94 2.96 2.98

0.0071 0.0022 0.0009 0.0009 0.0009

26.9 17.7 10 15.4 14.9

10.6 6.4 9.7 17.1 16.9

0.0011

4.7

1.3

Gmean (GPa)

Original σ 0 [22] (MPa)

New r (nm)

Maximum error factor between observed and present strain rate Ref. [22] New value

1

793 753 713

2

831 811 791 763

3

1723 1723 1723 1673 1623

1.2 0.66 0.41 0.41 0.41

0.044 0.044 0.044 0.044 0.044

390 390 390 390 390

393 393 393 393 393

129.2 129.2 129.2 129.2 129.2

1.015 1.845 2.970 3.026 3.082

4

1333

0.113

0.038

350

342

112.4

11.753

18 18 18 18

New σ 0 (MPa)

in the boundary region, from Eq. (A.4)   B2 (L − B3 )0.5 H = A2 − (5) L √ with A2 , B2 and B3 (=2 6W) as constants. The data [5–7,9,10,12,13] were subjected to a least squares fit in terms of Eq. (5) also. The results of both the analyses are included in Fig. 1. It is clear that both Eqs. (4) and (5) fit the data pertaining to three of the four systems (including with respect to the value of W in case of Eq. (5)) adequately. But, in case of n-Cu [13], the fit in terms of Eq. (5) is bad below a grain size of about 4.9 nm.

Fig. 1. Inverse Hall–Petch effect in four systems. Symbols represent the experimental points. Full curve is according to Eq. (4). Dashed curve is as per Eq. (5). () n-Zn [5,9,12]. Best fit values: A1 = 1.76, B1 = 7.50; A2 = 3.95, B2 = 10.70, B3 = 2.45 (i.e., W = 0.5 nm); maximum error in the fit as per Eq. (4) is 2.18%; as per Eq. (5) is 4.37%. (䊉) n-Ni [6,7]. Best fit values: A1 = 7.94, B1 = 13.72; A2 = 13.38, B2 = 24.50, B3 = 2.45 (i.e., W = 0.5 nm); maximum error in the fit as per Eq. (4) is 2.00%; as per Eq. (5) is 1.96%. () n-Ni by nanoscratch test [10]. Best fit values: A1 = 8.785, B1 = 33.500; A2 = 12.95, B2 = 27.00, B3 = 2.45 (i.e., W = 0.5 nm); maximum error in the fit as per Eq. (4) is 0.13%; as per Eq. (5) is 0.07%. (*) n-Cu [13]. Best fit values: A1 = 21.60, B1 = 50.00; maximum error in the fit as per Eq. (4) is 0.11%. Eq. (5) gave a poor fit for this set of data.

11.5

However, as the fit is good for both the equations in most cases, the question whether boundary migration involved dislocation emission from grain boundary obstacles and diffusion over a distance of the order of a fraction of unit interatomic distance in the boundary region or pure diffusion of the order of unit interatomic distance should be settled by transmission electron microscopy, as done in ref. [23]. 3.3. Estimation of γ˙ sp , Lmax , N at L = Lmax , τ 0diff Pd compacts of relative density 91.7% and average grain size 11–33 nm could be rolled at room temperature to a maximum true strain of about 0.65. The average strain rate was in the range of 0.06–0.138 s−1 . At the end of the deformation, the relative density had increased to 96.4% [23]. Such densification accompanying the deformation of partially dense compacts of conventionally grain sized materials has been reported in the powder metallurgy literature (see, for example, refs. [41–44]). Studies of this nature are essential in case of nanostructured compacts also because for catalytic and dental applications, to cite but two examples, one needs to make partially dense (porous) nanocompacts. Notwithstanding this, unlike in the powder metallurgy literature where the mechanical properties of partially dense compacts are systematically related to those of fully dense material, in the literature on the mechanical properties of nanocrystalline materials one finds a general reluctance to undertake studies of this kind. To the best of our knowledge, the first study of this nature on nanostructured metallic and composite compacts was reported recently [45]. It has been shown that the mechanical response of well prepared, partially dense nanostructured materials is similar to that of fully dense nanocrystalline compacts, with the difference that: (a) the elastic constants change with a change in density and (b) under identical experimental conditions, the strain rate of deformation decreases with an increase in density. Therefore, we believe that it is reasonable to understand the experimental results of Markmann et al. [23] using the present model.

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Rolling is a direct compression process [46] in which the deviatoric (shear) component, responsible for plastic deformation, is identical to that present in a uniaxial tensile test. The hydrostatic component is equal but has an opposite sign (compressive in rolling; tensile in a tension test). The compressive hydrostatic component present during rolling might have helped retard crack growth. Still, a room temperature true strain of 0.65 is a remarkable result because the melting point of Pd is 1827 K [47]. To understand this behavior, a Pd compact was produced by physical vapour deposition and consolidation (same process as the one used in ref. [23]). The average grain size was 15 nm and the average relative density was 89%. These values are very close to those used earlier [23] and so the mechanical response is expected to be similar. Microhardness was measured on a machine (Helmut Fischer make) in which the corresponding Young’s modulus, E, of the specimen also is revealed simultaneously. p is equal to 0.39 for Pd [47]. Using the relationship E = 2G(1 + p), the shear modulus, G, could be calculated. As an average of four readings (which were close), G was obtained as 15.76 GPa and H as 1.63 GPa. For Pd, a = 2.75 × 10−10 m [47], T = 300 K, ν = 1013 s−1 and γ 0 = 0.10 (see Section 2.3). Hence, from Eqs. (2a) and (A.1), a longitudinal strain rate of 0.312 s−1 is predicted.4 In calculations of the present kind, a prediction is considered to be good if it is within an order of magnitude of the measured value. It is noteworthy that the entire experimentally measured range of strain rate is within an order of magnitude of the predicted value, regardless √ of whether the Taylor factor of 3.06 or the Mises factor of 3 is used for converting the shear stress into its tensile equivalent. Plane interface formation entirely by diffusion of the order of a unit interatomic distance in a n-Pd compact of 91.7% relative density – which increased to 96.4% after deformation – and a grain size of 33 nm has been reported in ref. [23]. For n-Pd, Γ B = 1.0 J m−2 [48]. From Eq. (A.6) it is deduced that the following correlation between Dgb and Lmax will be present. Dgb (m2 s−1 ) Lmax (nm)

3 × 10−23 13.4

5 × 10−23 15.9

1 × 10−22 20

1 × 10−21 29.5

From Eq. (A.5) the corresponding N value at L = Lmax is obtained as: Dgb (m2 s−1 ) N at L = Lmax

3 × 10−23 221

5 × 10−23 342

1 × 10−22 608

1 × 10−21 14565

Results concerning grain boundary diffusivity in n-Pd are conflicting. After a careful review, W¨urschum et al. [49,50] have concluded that the most reliable values available at present for Dgb in n-Pd of near-full density lie in the range of

4

−1 ,

This will be equal √ to 0.5406 s of the von Mises 3.

if the Taylor factor of 3.06 is used, instead

10−21 to 10−19 m2 s−1 at a temperature interval of diffusion of 423–523 K. In view of this, the Dgb value required to obtain realistic values for Lmax and N at L = Lmax for the room temperature deformation of palladium, as estimated above, appears to be meaningful. The best experimental technique available for determining the threshold stress, particularly when a single mechanism is rate controlling, is the stress reduction test. Blum and co-workers have discussed the scope and limitations of this test [51,52]. Using this method, suitably adapted for steady state flow, τ 0disloc and τ 0diff should be measured and compared with the corresponding values estimated in this paper. That will provide further support for the model or suggest that it needs modification/refinement. Evidently, the numerical agreement between the predictions of the present model and the experimental results reported in the literature is satisfactory. But, in this analysis the inverse Hall–Petch effect and related phenomena are attributed to creep effects, in which grain boundary sliding, assisted by boundary migration, develops to a mesoscopic scale (plane interface formation) and controls the rate of deformation. No doubt, the extent of diffusion required in this model to produce the plane interfaces is over distances of the order of a unit interatomic distance or less and hence much less diffusion flux is needed than in either Coble or Herring-Nabarro creep. Still, grain boundary sliding controlled processes usually give rise to relatively high values for the strain-rate sensitivity index, m. For example, in high temperature creep m is in the range of 0.1–0.3, while during superplastic flow its value lies between 0.3 and 0.9 [53]. In contrast, both m and the strain-hardening index, N, have very low values during room temperature deformation of nanostructured (n-) materials—see Ma [54]. A careful examination of Fig. 1 of ref. [54] reveals that nanocrystalline copper of grain size ∼200 nm deforms with near-zero N value to a uniform elongation value of slightly more than 40%. N ∼ 0.0 is also reported during superplastic flow, when grain size is relatively stable [53]. In the absence of strain-rate sensitivity of flow, Considere’s criterion, i.e., uniform strain εu = N [46], would suggest that necking should start almost immediately after yielding, which is not observed. Therefore, there is some strain-rate sensitivity in this n-Cu specimen and experimentally a m value of 0.01–0.025 has been reported for n-Pd compacts, deformed at room temperature [45]. In a very early work, Woodford [55] collated elongation at fracture—m data for a number of systems and this was presented as Fig. 2.10 in ref [53]. It is clear from this plot that a value if m ∼0.05–0.06 is sufficient to give rise to an elongation of about 40%. As m ∼0.01–0.025 during the room temperature deformation of n-Pd (melting point—1827 K), a value of m ∼0.05–0.06 during the room temperature deformation of copper (melting point = 1356 K) can be expected realistically. This can account for the result presented in Fig. 1 of ref [54]. In fact, strain-rate sensitivity of flow is invoked to explain ductility in nanostructured materials by others also [54]. In the rate equation (Eq. (A.7)) the non-linearity in the relationship between stress and strain rate is introduced by the threshold stress, τ 0 , which is directly related to G0.5 , with G the shear modulus (Eqs. (A.2) and (A.3)). G decreases with

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increasing temperature and so τ 0 also will decrease with increasing temperature. In view of Eq. (A.7), m (inverse of the stress exponent) will increase with an increase in temperature and attain large values at high homologous temperatures. Thus, if stability of grain size is ensured, at higher homologous temperatures significant elongation at reduced temperatures, compared to materials of conventional grain sizes, may/can be achieved in nanostructured materials. 4. Conclusions 1. By assuming, as an approximation, fixed values for all systems in the narrow temperature and grain size ranges of the experiments with respect to the strain associated with the unit boundary sliding event, the Poisson ratio, the vacancy concentration at a high angle/high energy grain boundary and its width, it is possible to simplify significantly the procedure for an order of magnitude experimental validation of a recently proposed model [22]. 2. The grain size dependence of hardness predicted [16,22] for the range in which inverse Hall–Petch effect is present in nmaterials is verified to be accurate using data considered to be reliable [5–7,9,10,12,13]. 3. The rather high strain rate of deformation withstood by a n-Pd compact, that was not fully dense, during room temperature rolling [23] is accounted for quantitatively. Appendix A The model is described in detail elsewhere, see [16,17,20– 22]. Briefly, rate controlling flow is assumed to be confined to a three-dimensional, continuous network of grain- and interphaseboundaries that surround grains which do not deform except for what is required to ensure strain and geometric compatibility (see Fig. 1 of ref. [17]). Boundaries of high viscosity are bypassed by grain rotation [24,25]. Grain shape is assumed to be rhombic dodecahedron. Every grain/interphase high angle/high energy boundary, conducive to boundary sliding, is divided into a number of atomic scale ensembles that surround free volume sites present at discrete locations characteristic of the boundary. Each of these ensembles – assumed to be oblate spheroids (Eshelby representation [26]) located symmetrically about the plane of the boundary with one-half falling in each of the two grains that meet to form the boundary – constitutes a basic unit of sliding. The presence of free volume ensures that the shear resistance of these basic sliding units is less than that of the rest of the boundary. The ground area in the grain boundary plane of the oblate spheroid is assumed to be πW2 (diameter of this circle is ∼5 atomic diameters on the boundary plane) [27] and the radius on either side of the boundary to be (W/2), where W is the grain boundary width. This will ensure: (a) that the internal stresses developed by shearing along the boundary are of relatively short range and (b) that shear in a basic unit is independent of that present in other similar units. As the excess free volume at a general/random high angle boundary changes only slightly with misorientation [28,29], the individual shear transformations, γ i , and the transitory volume expansions, εi (arising

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due to the sliding unit being embedded inside a solid matrix), will not differ much for different basic units or for the different metastable states within a basic unit. Therefore, they can be treated in terms of their average values γ 0 and ε0 , respectively [17]. The work of Wolf [28,29] and bubble raft experiments – see, for example, ref. [27] – suggest that γ 0 ≈ 0.1. The elastic energy of the shear and volumetric distortions accompanying the unit shear, F0 , constitutes the activation energy for the boundary sliding process [17]. That is, F0 = 21 (β1 γ02 + β2 ε20 )GV0

(A.1)

where G is the shear modulus, and for an oblate spheroid, V0 = (2/3)πW3 (the volume of the basic sliding unit), β1 = 0.944((1.590 − p)/(1 − p)), β2 = (4/9)((1 + p)/(1 − p)) [26] and p is the Poisson’s ratio. Steric hindrance, e.g., a triple junction, will make this kind of sliding ineffective. It has been suggested [16,17,22] that when sliding at a boundary develops to a mesoscopic scale – defined to be of the order of a grain diameter or more – aided by boundary migration, one of the angles at the triple junction tends towards 180◦ . The driving force for plane interface formation is the minimisation of the total free energy of the deforming system [22], in addition to this process leading to maximum work being done in the direction of the applied stress—Taylor’s principle of maximum work [17]. Then, large scale sliding will result from two or more boundaries aligning themselves to form a plane interface. Plane interface formation requires energy expenditure and gives rise to a long range threshold stress. An analysis [22] has revealed that below a certain grain size, Lmax , in the lower end of the nm range, boundary migration takes place entirely by diffusion covering a distance of about unit interatomic distance. But above this grain size, it is caused by a combination of non-rate controlling dislocation emission from grain boundary obstacles and rate controlling boundary diffusion over an average distance less than unit interatomic distance. Of the many equations derived [16,17,22], the following are necessary for further analysis. If τ 0disloc is the threshold stress in the shear mode, when boundary migration takes place by a combination of dislocation emission and diffusion over a length scale less than unit interatomic distance on average and τ 0diff its value when boundary migration is caused entirely by diffusion over a length scale of unit interatomic distance, τ0 disloc =

C1 ; L

 C1 =

8GΓB r 30.25

0.5 ,

L > Lmax

(L − L0 )0.5 ; LN 0.25 0.5  21.5 GΓB , L ≤ Lmax C2 = 30.75

(A.2)

τ0 diff = C2

 τ0 diff ≈ C3

 (L − L0 )0.5 ; L

C3 = C2 N −0.25

(A.3)

(A.4)

468

K.A. Padmanabhan et al. / Materials Science and Engineering A 452–453 (2007) 462–468

if the grain size dependence of N is ignored in the narrow interval where the inverse Hall–Petch effect is observed.    Dgb aCΓB L − L0 0.5 ; L ≤ Lmax (A.5) N = 5.7325 kT γ˙ sp L3  Lmax =  γ˙ sp =

28.0835Dgb aCΓB r kT γ˙ sp

2.0944W 4 γ0 2 ν kTL

1/3



(A.6) 

(τ − τ0 ) exp

−F0 kT

 (A.7)

energy, r the boundwhere Γ B is the specific grain boundary √ ary misfit removed by diffusion, L0 = 2 6W, N the number of grain boundaries that align to form a plane interface when boundary migration takes place entirely by diffusion, Dgb the grain boundary diffusivity, a the atomic diameter, C the vacancy concentration at a high angle/high energy grain boundary at the temperature of deformation (≈10−4 [30]), ν the thermal vibration frequency, k the Boltzmann constant, τ 0 is either τ 0disloc or τ 0diff depending on grain size and γ˙ sp is the exter5 nal shear √ strain rate. If von Mises yield behaviour is assumed, γ˙ sp = 3˙εsp . The threshold stress in uniaxial tension mode, √ σ0 = 3τ0 [17,32].6 τ 0 is governed by Eq. (A.2) if L > Lmax and by Eq. (A.3) if L ≤ Lmax . If σ is the externally applied tensile stress, an effective stress (σ − σ 0 ) will be available to drive the atomic/microscopic boundary sliding process [16,17,22]. References [1] A.H. Chokshi, A. Rosen, J. Karch, H. Gleiter, Scr. Metall. Mater. 23 (1989) 1679–1683. [2] K. Kim, K. Okazaki, Mater. Sci. Forum 88–90 (1992) 553–560. [3] H. Chang, C.J. Alstetter, S.J. Kim, in: Nanophase Metals–Processing and Properties, Proc. 2nd Pacific Rim International Conference on Advanced Materials and Processing (PRICM-2), Kyongju, K.S. Shin, J.K. Yoon, S.J. Kim (Eds.), The Korean Institute of Metal and Materials, 1995, pp. 2107–2112. [4] G. Palumbo, U. Erb, K.T. Aust, Scr. Metall. Mater. 24 (1990) 2347–2350. [5] C.C. Koch, J. Narayan, in: D. Farkas, H. Kung, M. Mayo, H. Van Swygenhoven, J. Weertman (Eds.), Structure and Mechanical Properties of Nanophase Materials—Theory and Computer Simulation Versus Experiment, Symposium Proceedings, vol. 634, MRS, Warrendale, PA, USA, 2001, pp. B5.1.1–B5.1.11. [6] A.M. El-Sherik, U. Erb, G. Palumbo, K.T. Aust, Scr. Metall. Mater. 27 (1992) 1185–1188. [7] U. Erb, Nanostruct. Mater. 6 (1995) 533–538. [8] D. Wolf, V. Yamakov, S.R. Phillpot, A. Mukherjee, H. Gleiter, Acta Mater. 53 (2005) 1–40. [9] H. Conrad, J. Narayan, Acta Mater. 50 (2002) 5062–5078. [10] C.A. Schuh, T.G. Nieh, T. Yamasaki, Scr. Mater. 46 (2002) 735–740. [11] A.V. Sergueeva, V. Stolyarov, R.Z. Valiev, A.K. Mukherjee, Scr. Mater. 45 (2001) 747–752.

5

Very recently, a diffusionless model for grain boundary migration has been proposed [31]. Therefore, there is a need to consider that alternative point of view before using Eqs. (A.2)–(A.7) for further analysis. See Section 2.1. 6 In a paper published in 2000 [33], it was argued that to convert a shear stress/strain to a tensile stress/strain in case of polycrystalline (BCC, FCC) mate√ rials the former should be multiplied by 3.06 (the Taylor factor) and not 3. This view also needs to be examined. See Sections 2–3.

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