Deformation Mechanisms in Nanocrystalline Materials

2.1.1 Activation volume for deformation

2.1.2 Activation energy

2.1.3 Magnitude of deformation rates

Any successful deformation mechanism is required not only to account for deformation characteristics, such as the activation volume and the activation energy for deformation, but also to predict the magnitudes of the deformation rates. The data given in Table I will permit examining whether a proposed deformation mechanism for nc materials meets this requirement.

2.1.4 Nanoscale softening

Experimental observations have indicated that when the grain sizes of nc Cu and nc Ni (and their alloys) fall below a critical value in the nanoscale range, the strength decreases, i.e., nanoscale softening (inverse Hall–Petch behavior) occurs. According to a very recent analysis,[28] the critical grain sizes for Cu and Ni are ~25 and 15 nm, respectively. The results of molecular dynamics (MD) simulations[29] have shown the presence of a maximum in the flow stress of Cu when its grain size becomes smaller than 15 nm.

2.2.1 Coble creep

First, the activation volume associated with Coble creep is b ³. This value is much smaller than those reported for nc materials (10 to 40 b ³).

Second, deformation data reported for nc Ni and nc Cu (Table II) are plotted in the form of normalized creep rate \( \left( {{{\mathop \gamma \limits^{ \bullet } {\text{k}}T} \mathord{/ {\vphantom {{\mathop \gamma \limits^{ \bullet } {\text{k}}T} {D_{gb} Gb}}} \kern-\nulldelimiterspace} {D_{gb} Gb}}} \right)\left( {{d \mathord{\left/ {\vphantom {d b}} \right. \kern-\nulldelimiterspace} b}} \right)^{3} \) as a function of normalized stress (τ/G) on a logarithmic scale in Figure 1(a). The shear moduli for Ni and Cu were taken from Reference 31. Included in Figure 1(a) is the prediction of Eq. [7] that is given as a solid line at 45 deg. In plotting the deformation data, the following parameters were used: (1) for Ni, Q _gb and D _gb0 are equal[28] to 156.6 kJ/mole and 1.92 cm²/s, respectively; (2) for Cu, Q _gb and D _gb0 are equal[28] to 120 kJ/mole and 1.6 cm²/s, respectively; and (3) the Burgers vectors of Ni and Cu are 0.249 and 0.256 nm, respectively. An inspection of Figure 1(a) shows that the agreement between the prediction Coble creep and the data for nc Cu and nc Ni is poor in terms of trend and magnitude. This disagreement remains even if different values of Q _gb and D _gb0 reported by Frost and Ashby were used.[32] Figure 1(b) illustrates this point under the following conditions: (1) for Ni, Q _gb and D _gb0 are equal[32] to 115 kJ/mole and 0.1 cm²/s, respectively; and (2) for Cu, Q _gb and D _gb0 are equal[32] to 104 kJ/mole and 7 × 10⁻² cm²/s, respectively.

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Third, the Hall–Petch relations for Cu and Ni as described by Eqs. [5] and [6], respectively, are depicted as solid straight lines (labeled as Hall–Petch Equation) in Figures 2 and 3, respectively, where applied shear stress τ in the case of Cu and hardness, and H in the case of Ni, are plotted against \( 1/\sqrt d . \) In plotting Eq. [7] representing Coble creep in Figures 2 and 3 as τ or H against \( 1/\sqrt d \) (labeled as Coble creep), the following information and conditions in addition to the aforementioned data on Q _gb and D _gb0 were used: (1) the activation volumes v for Cu and Ni were taken[28] to be 12 and 10 b ³, respectively; and (2) the shear strain rate = 10⁻³ s⁻¹[28] and T = 300 K. As shown by Figures 2 and 3, the curves representing Coble creep (Eq. [7]) for Cu and Ni intersect the straight lines representing the Hall–Petch behavior at 0.44 and 0.008 nm, respectively. It is clear that these values for the critical grain sizes are not realistic and that Coble creep cannot account for the transition from hardening (conventional Hall–Petch behavior) to softening (inverse Hall–Petch relation). The values of the critical grain size remain unrealistic even if different activation energies[32] were used. For example, the critical grain size for Cu becomes 1.6 nm when Q _gb  = 104 kJ/mole and D _gb0 = 0.1 cm²/s. In addition, the critical grain size for Ni becomes 0.3 nm when Q _gb  = 115 kJ/mole and D _gb0 = 7 × 10⁻² cm²/s.

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Finally, the results of investigations of nc materials have shown[33] that elongations to failure (ductility) are in the range 2 to 8 pct. These elongations to failure are too low to be attributed to the operation of Coble creep, the stress exponent of which is unity. As reported elsewhere,[34] ductility is a function of the reciprocal of the stress exponent. Accordingly, it is expected that elongations to failure arising from the Coble process (n = 1) will be much higher than those associated with the micrograin superplasticity process[35] (n = 2). For the latter process, elongations to failure are >500 pct.

2.2.2 Triple junction model

2.2.3 Atomic scale boundary sliding model

Hahn and Padmanabhan[38] developed a model for deformation in nc materials. The model is based on the concept that deformation in nc materials is controlled by atomic scale boundary sliding. The model, which is phenomenological in nature, involves many adjustable parameters, some of which cannot be evaluated with certainty. Accordingly, it is not possible to provide a comparison between the prediction of the model and present experimental data.

2.2.4 Modified Coble creep

Despite these merits, present attempts to provide a comparison between the prediction of Eq. [9a] and the experimental data given in Table II were not successful. The source for this problem was not initially clear. However, a careful consideration of the expression for the threshold stress in Eq. [9b] led to the identification of the most probable reason. It was found that for most of the experimental datum points given in Table II, the values of the normalized threshold stress were higher than those of the normalized applied stress. For example, the grain sizes used in investigations of nc Ni are in the range 25 to 100 nm. By using the value b for nc Ni (0.249 nm), the normalized threshold stress b/d in Eq. [9b] was estimated to be in the range 2.5 × 10⁻³ to 10⁻². By normalizing applied stresses in Table II using the reported value of the shear modulus[31] for Ni, it was determined that most of the values of the normalized stresses used in testing nc Ni are in the range 7 × 10⁻⁴ to 4 × 10⁻³. Similar results were obtained for nc Cu. While the values of the normalized threshold stress are in the range 5 × 10⁻³ to 3 × 10⁻², those of applied stress are in the range 2 × 10⁻³ to 4 × 10⁻³. These findings show that the expression of the threshold stress proposed by Masumara et al.[39] is not realistic, because it yields normalized threshold stresses higher than applied stresses.

In addition to this problem, which is related to the unrealistic nature of the expression for the threshold stress, another issue exists. The results of a very recent investigation of nc Ni have shown[5] that over several orders of magnitude of the strain rate, the stress exponent for creep in nc Ni at 393 K decreases from a value of 30 to a value of 4.5 with decreasing applied stress. This behavior cannot be explained by a modified Newtonian Coble process that incorporates a threshold stress τ ₀, because the presence of τ ₀ would lead to an increase in the stress exponent with decreasing applied stress, a trend that contrasts with the experimental behavior.

2.2.5 Grain boundary sliding with back stress

There are two problems with this modified model, as represented by Eq. [10b]. First, a true stress exponent of approximately 2 is a characteristic of the superplastic flow that is associated with extensive ductility. However, as mentioned earlier, the results of investigations of nc materials have shown that these materials exhibit poor ductility.[33] Second, the presence of a back stress in Eq. [10b] leads to an increase in the apparent stress exponent with decreasing applied stress. This prediction contrasts with recent experimental evidence,[5] which shows that the stress exponent for creep over several orders of magnitude of the strain rate (10⁻⁹ s⁻¹ to 10⁻³ s⁻¹) in nc Ni at 393 K decreases with decreasing applied stress.

2.2.6 Boundary sliding under the condition of nonlinear-viscous behavior

2.2.7 Thermally activated grain boundary shearing process

The model predicts the activation volume to be b ³. This value contrasts with the experimental values of 10 to 30 b ³ that were reported for the activation volume for deformation in nc materials (Table I).

In Figure 4(a), Eq. [11b] is depicted graphically by plotting \( {{\mathop \gamma \limits^{ \bullet } b^{2} } \mathord{/ {\vphantom {{\mathop \gamma \limits^{ \bullet } b^{2} } {D_{gb} }}} \kern-\nulldelimiterspace} {D_{gb} }} \) against \( {{b^{3} \upsilon_{D} } \mathord{\left/ {\vphantom {{b^{3} \upsilon_{D} } {dD_{gb0} }}} \right. \kern-\nulldelimiterspace} {dD_{gb0} }}\left[ {\sinh \left( {{{\tau \upsilon } \mathord{\left/ {\vphantom {{\tau \upsilon } {{\text{k}}T}}} \right. \kern-\nulldelimiterspace} {{\text{k}}T}}} \right)} \right] \) on a logarithmic scale. The solid line at 45 deg represents the prediction of the model. The experimental data given in Table II are plotted in the figure in terms of these two dimensionless parameters. It is clear that the experimental data, when normalized in terms of \( {{\mathop \gamma \limits^{ \bullet } b^{2} } \mathord{/ {\vphantom {{\mathop \gamma \limits^{ \bullet } b^{2} } {D_{gb} }}} \kern-\nulldelimiterspace} {D_{gb} }} \) and \( {{b^{3} \upsilon_{D} } \mathord{/ {\vphantom {{b^{3} \upsilon_{D} } {dD_{gb0} }}} \kern-\nulldelimiterspace} {dD_{gb0} }}\left[ {\sinh \left( {{{\tau \upsilon } \mathord{\left/ {\vphantom {{\tau \upsilon } {{\text{k}}T}}} \right. \kern-\nulldelimiterspace} {{\text{k}}T}}} \right)} \right], \) are not in agreement with the prediction of the model, as represented by the solid line at 45 deg. This disagreement remains even if different activation energies for the grain boundary[32] are used. Figure 4(b) demonstrates this point under the following conditions: (1) for Cu, Q _gb , and D _gb0 are equal[32] to 104 kJ/mole and 0.1 cm²/s, respectively; and (2) for Ni, Q _gb and D _gb0 are equal[32] to 115 kJ/mole and 7 × 10⁻² cm²/s, respectively.

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Equation [12a] is plotted in Figure 5 as τ against \( 1/\sqrt d \) for Cu. Also, included in the figure is a solid line representing conventional Hall–Petch behavior for that metal, as described by Eq. [5]. As shown by the figure, the curves representing the model (Eq. [12a]) for Cu intersects the straight line representing conventional Hall–Petch behavior at 0.44 nm, a value that is not realistic. The behavior for Ni is illustrated in Figure 6, in which hardness H is plotted against \( 1/\sqrt d . \) Similarly, the intersection of the curve representing the model with the solid line representing the conventional Hall–Petch behavior results in a critical grain size the value of which is not realistic (0.16 nm). The values of the critical grain sizes remain, even if different activation energies[32] were used. For example, the critical grain size for Cu becomes 1.8 nm when Q _gb  = 104 kJ/mole and D _gb0 = 0.1 cm²/s. Also, the critical grain size for Ni becomes 1.0 nm when Q _gb  = 115 kJ/mole and D _gb0 = 7 × 10⁻² cm²/s.

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The preceding discussion shows that although the model proposed by Conrad and Narayan has attractive features, it does not account for the deformation behavior of nc Cu and nc Ni in terms of the following: (1) the value of the activation volume, (2) the trend and position of the experimental data, and (3) the values of the critical grain sizes for a transition from hardening to softening (inverse Hall–Petch behavior) in nc Cu and nc Ni.

2.2.8 Composite model involving amorphous boundary layer

Fan et al.[45] applied the equation of the model along with the Hall–Petch relations to fit the experimental data on Cu and Ni. On the basis of the best fit, the values of the critical grain size for Cu and Ni were estimated as 25 and 7.7 nm, respectively. The value of 25 nm for Cu fully agrees with that inferred from the experimental data for this metal,[28] but the value of 7.7 nm for Ni is smaller than that inferred from the experimental data (12 to 15 nm).

It is now necessary to examine whether Eq. [13a] can explain the trends in the positions of the experimental data for Cu and Ni. For this purpose, Eq. [13a] is re-expressed in the following normalized form:

$$ {\frac{{\mathop \gamma \limits^{ \bullet } {\text{k}}T}}{{D_{gb}Gb}}} = 6\pi \left\{ {{\frac{r\tau }{bG}}\left. {\left[ {1 +g\left( {{\frac{d}{w}} - 1} \right)} \right]^{ - 1} }\right\}}\right. $$

(13b)

In Figure 7(a), Eq. [13b] is depicted graphically by plotting \( {{\mathop \gamma \limits^{ \bullet } {\text{k}}T} \mathord{/ {\vphantom {{\mathop \gamma \limits^{ \bullet } {\text{k}}T} {D_{gb} Gb}}} \kern-\nulldelimiterspace} {D_{gb} Gb}} \) against \( {{{\text{R}}\tau } \mathord{/ {\vphantom {{{\text{R}}\tau } {bG}}} \kern-\nulldelimiterspace} {bG}}\left[ {1 + g\left( {{d \mathord{\left/ {\vphantom {d w}} \right. \kern-\nulldelimiterspace} w} - 1} \right)} \right]^{ - 1} \) on a logarithmic scale. The solid line at 45 deg represents the prediction of the model. The experimental data given in Table II are plotted in the figure in terms of these two dimensionless parameters. An examination of the figure indicates that the model as represented by the solid line at 45 deg cannot account for the trends and positions of the experimental data for nc Cu and nc Ni. This finding does change if different values for Q _gb and D _gb0 are used. Figure 7(b) demonstrates this behavior under the following conditions: (1) for Cu, Q _gb and D _gb0 are equal[32] to 104 kJ/mole and 0.1 cm²/s, respectively; and (2) for Ni, Q _gb and D _gb0 are equal[32] to 115 kJ/mole and 7 × 10⁻² cm²/s, respectively.

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There are three additional problems regarding the prediction of the composite model just discussed.[45] The first problem is related to the grain size dependence of the strain rate. As indicated by Fan et al.,[45] the grain exponent for deformation as predicted by Eq. [13b] is less than 1. This value is very low when compared with the experimentally reported value[5] of approximately 3. The second problem concerns ductility. The composite model, as represented by Eq. [13a], is characterized by a linear dependence of the strain rate on the applied stress (Newtonian behavior). As a result, the model predicts extensive ductility. This prediction is not consistent with present experimental evidence, which shows that ductility in nc material is very limited. Finally, grains in a real structure are not isolated but are surrounded by other grains. Accordingly, deformation in the boundary layer of one grain needs to be accommodated to relieve stress concentrations.

2.2.9 Model of strongest grain size

The model of Argon and Yi[46] appears to be attractive, because it is based on the idea that the deformation behavior of nc materials arises from the operation of grain boundary shear and dislocation plasticity.

At a constant strain rate, Eq. [14] leads to a maximum flow stress at a critical normalized grain size d _c . For grain sizes <d _c , the process of grain boundary shear controls the deformation and for grain sizes >d _c , the process of dislocation plasticity controls the deformation. By selecting certain material parameters and by taking \( \dot{\gamma } = 10^{ 8} \,{\text{s}}^{ - 1} , \) Argon and Yip[46] plotted the variation in the normalized stress as a function of the normalized grain size. Such a plot exhibited a transition from hardening (conventional Hall–Petch behavior) to softening (inverse Hall–Petch behavior) at d _c  = 12.2 nm. Consideration of the analysis of Yip and Argon[46] indicates that the critical grain size is independent of the value of the shear strain rate. This indication is illustrated in Figure 8, in which Eq. [14] is plotted as applied shear stress τ against \( 1/\sqrt d , \) using a shear strain rate of 10⁻³ s⁻¹. In addition, included in the figure is the prediction for 10⁸ s⁻¹. As shown by Figure 8, the value of the critical grain size remains as 12.2 nm, but the value of the maximum stress decreases with decreasing the shear strain rate. The critical grain size of 12.2 nm predicted by the model for Cu is close to that obtained from the results of MD simulations[29] performed on Cu (d _c  = 15 nm), but is smaller than that inferred from experimental data (d _c  = 25 nm). In addition, the peak stress predicted by the model is smaller than that inferred from experimental results and the left branch of the curve that represents hardening (Hall–Petch conventional behavior) extends to the origin. This trend is different from the real Hall–Petch behavior that intercepts the y-axis at τ ₀ (Eq. [4]). Despite these differences, the model of Argon and Yip,[46] unlike other models discussed earlier,[30,44] predicts a transition from hardening to nanoscale softening at a realistic value for the grain size.

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The next step is to examine whether Eq. [14] can account for the trends and positions of the experimental data for Cu and Ni. For the purpose of this examination, Eq. [14] is plotted as \( {{\mathop \gamma \limits^{ \bullet } }/{\left( {v_{D} {\cdot} F(d/b)} \right)}} \) vs τ/G on a logarithmic scale in Figure 9. The function F(d/b) under the condition that m = 30 is given by

$$ F\left( {{\frac{d}{b}}} \right) = A\left( {{\frac{d}{b}}} \right)^{ - 1}\, +\, B\left( {{\frac{d}{b}}} \right)^{15}\, -\, 6B\left( {{\frac{d}{b}}} \right)^{14} $$

(16)

where \( A = {{6\gamma^{T} } \mathord{/ {\vphantom {{6\gamma^{T} } {\left( {\hat{\tau }_{is} /\mu } \right)}}} \kern-\nulldelimiterspace} {\left( {\hat{\tau }_{is} /\mu } \right)}}^{m} \) and \( B = {{\left( {{{v_{G} } \mathord{/ {\vphantom {{v_{G} } {v_{D} }}} \kern-\nulldelimiterspace} {v_{D} }}} \right)} \mathord{\left/ {\vphantom {{\left( {{{v_{G} } \mathord{\left/ {\vphantom {{v_{G} } {v_{D} }}} \right. \kern-\nulldelimiterspace} {v_{D} }}} \right)} {\alpha^{m} }}} \right. \kern-\nulldelimiterspace} {\alpha^{m} }}. \) Included in Figure 9 are the data on nc Cu and nc Ni. An examination of the figure shows that Eq. [14] is represented by a solid line the slope of which is 30 (m = 30), that the agreement between experimental datum points and the positions of the solid line is poor with the exception of at very high stresses, and that the disagreement in position and trend between the experimental data and the prediction of the model is clearly evident at intermediate and low stresses; the data for these stresses fit short segments of straight lines the slopes of which are in the range 5 to 7. The origin of this disagreement is most likely related to two factors. First, Argon and Yip[46] have assumed that grain boundary shear and dislocation plasticity operate close to their athermal threshold level. This is reflected in the observation that Eq. [14] does not include a term related to thermal activation. The absence of this term is not consistent with the experimental results obtained on nc materials,[3,5,16, 17, 18, 19, 20, 21, 22, 23, 24, 25] which clearly show that the deformation behavior of nc material is not athermal and that strain rates measured during the deformation depend on temperature according to the following expression:

$$ \mathop \gamma \limits^{ \bullet } \propto e^{{ - Q/{\text{R}}T}} $$

(17)

Second, in order to simplify the analysis, Argon and Yip[46] assumed that the stress exponents characterizing both grain boundary shear and dislocation plasticity are not only the same (n = 30) but are also independent of temperature. There is no definitive experimental information that lends support for such a simplification. In addition, as acknowledged by Argon and Yip,[46] if deformation occurred at different temperatures other than room temperatures, m would be a function of temperature.

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2.2.10 Dislocation-accommodated boundary sliding

Calculations[5] based on deformation characteristics in UFG ceramics and the value of the stress required to move a boundary dislocation into the interior of the grain have suggested that the value of M is the range 5 to 20. This range, in turn, results in an activation volume in the range 10 to 40 b ³, which is consistent with the reportedly experimental range; as mentioned earlier and shown by Table I, the activation volume measured during the deformation of nc materials is in the range 10 to 50 b ³.

The data on nc Ni and nc Cu that are given in Table II were plotted in Figure 10(a) in terms of the two normalized parameters: \( \mathop \gamma \limits^{ \bullet } \left( {{d \mathord{\left/ {\vphantom {d b}} \right. \kern-\nulldelimiterspace} b}} \right)^{3} \left( {{{b^{2} } \mathord{\left/ {\vphantom {{b^{2} } {D_{gb} }}} \right. \kern-\nulldelimiterspace} {D_{gb} }}} \right) \) and exp (τυ/kT) − 1. Also included in the figure is a solid straight line that represents the prediction of Eq. [18c]. It is clear that the data points cluster about the solid line, demonstrating the presence of good correspondence between the rate equation representing this deformation process and the experimental data for nc Ni and nc Cu, which were obtained at different conditions of temperature, grain size, and strain rates. If different values for diffusivities in Cu and Ni (for Cu, Q _gb  = 104 kJ/mole and D _gb0 = 0.1 cm²/s; for Ni, Q _gb  = 115 kJ/mole and D _gb0 = 7 × 10⁻² cm²/s) are used,[32] most of the datum points cluster about a broken line that is parallel to the solid line representing the prediction of Eq. [18c]; the agreement is good between the prediction and experimental data in terms of trend. Figure 10(b) demonstrates this behavior.

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When Eq. [18b] was combined with the equation representing conventional Hall–Petch behavior, two findings were reported.[28] First, the variation in stress and hardness for Cu and Ni with \( 1/\sqrt d \) exhibits a transition from hardening to softening. Second, the critical grain sizes at this transition occur for Cu and Ni are 25 and 13 nm, respectively. These values agree well with those estimated by a recent analysis[28] of the experimental data reported for both metals.

Furthermore, this model provides explanations for two experimental observations. The first observation is related to the apparent activation energy for deformation. As stated earlier, the activation energy for deformation in nc Ni decreases with increasing applied stress. Applying the definition of the activation energy (Eq. [2]) to Eq. [18b] leads to the following expression that describes the dependence of the apparent activation energy on applied stress:

$$ Q = Q_{gb} - {\frac{{{\text{R}}\upsilon \tau }}{\text{k}}}\,{\frac{1}{{\left[ {1 - \exp \left( { - {{\upsilon \tau } \mathord{\left/ {\vphantom {{\upsilon \tau } {{\text{k}}T}}} \right. \kern-\nulldelimiterspace} {{\text{k}}T}}} \right)} \right]^{ - 1} }}} $$

(19)

Equation [19] is depicted graphically in Figure 11 using T = 393 K and ν = 19 b ³. Included in the figure are the data on the variation of the activation energy for deformation in nc Ni (40 nm) with stress.[5] It is clear that there is good agreement between the prediction of Eq. [20] and the experimental data on nc Ni in terms of trend and magnitude.

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Finally, the values of stresses along with the values of the stress exponents estimated from Eq. [22] are combined with Eq. [20] for the purpose of plotting e _f pct against the tensile strain rate at 393 K. This plot is shown in Figure 12. Included in the plot are the ductility data obtained for nc Ni (40 nm) at 393 K.[25] It is clear that the agreement between prediction and experiment is good in terms of the trend. However, the values of the predicted ductility are higher than those experimentally reported. Two possible factors may account for this discrepancy. First, the prediction is based on the value of the stress exponent estimated from Eq. [21]. However, ductility can be influenced by other factors such as impurities and the size of the tensile sample. Second, during the deformation of nc Ni, the occurrence of boundary sliding leads to the formation of voids and cavities the growth and linking of which result in premature failure. The process of cavitation and the linking of cavities and voids assumes significance at low strain rates. In this regard, it is interesting to notice that the separation between the predicted values and the experimental values of ductility at 393 K increases with decreasing strain rate.

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