An analytical model for stress-induced grain growth in the presence of both second-phase particles and solute segregation at grain boundaries
Introduction
Grain growth has been the topic of many investigations over the past few decades, and despite the vast amount of theoretical and experimental studies, many important questions remain unanswered [1], [2], [3], [4], [5], [6], [7], [8]. It is well established that the underlying mechanisms that govern grain growth primarily involve grain boundary (GB) migration [1], [2], [3], [4] and grain rotation (followed by coalescence) [5], [6], [7], [8]. The driving forces for GB migration and grain rotation stem from two sources: (i) internal structures, i.e., surface tension due to GB curvature and gradient in dislocation densities between neighboring grains that cause GB migration [1], [2] and net torque due to a reduction in GB energy that triggers grain rotation [5], [9], [10], [11], [12]; and (ii) externally applied stresses [3], [4], [6], [7], [8].
Internal structure driven GB migration (for reviews, see Refs. [1], [2]) and grain rotation [5], [9], [10], [11], [12] have been well documented. Recently, stress-induced GB migration and grain rotation have been extensively investigated using molecular dynamics (MD) simulations [13], [14], [15], [16], [17], [18], [19], [20], [21], and in situ [3], [4], [6], [7], [8], [22], [23], [24] and ex situ [25], [26], [27], [28], [29] transmission electron microscopy (TEM) observations have confirmed the occurrence of stress-induced GB migration and grain rotation. Furthermore, MD simulations [15], [17], [18], [19], [20], [21] and experimental studies via optical microscopy [30], [31] and TEM [23], [24] have revealed that, during stress-induced grain growth, there exists a coupling effect between grain translation (sliding) along a GB caused by shear stress along the GB and GB migration, the former giving rise to grain rotation in the case of a curved GB [19], [32]. The coupling phenomenon has also been studied by theoretical modeling [32], [33], [34].
Inspection of these and other published studies, however, reveals that theoretical [13], [14], [15], [16], [17], [18], [21] and experimental [3], [4], [6], [7], [8], [22], [23], [24], [25], [26], [27], [28], [29] efforts relevant to stress-induced GB migration and grain rotation involve mostly single-phase materials and consequently, material systems containing second-phase particles have received very limited attention [35]. As a consequence, an important question remains unanswered. That is, what is the interaction between second-phase particles at GBs and stress-induced GB migration and grain rotation? In addition, inspection of the published theoretical investigations [13], [14], [15], [16], [17], [18] indicates that only very limited studies [21] address the effect of solute atoms segregated at GBs on stress-induced grain growth. Interestingly, despite the fact that electrodeposited materials, which almost always contain a high level of impurities as solute atoms segregated at GBs [8], [25], [27], [36], [37], are widely used as model materials to study stress-induced GB migration and grain rotation, the interactions between GB segregation and stress-induced GB migration and grain rotation are seldom discussed [35], [38].
Most recently, Lin et al. experimentally studied the stress-induced grain growth phenomenon in the presence of both second-phase particles and solutes segregated at GBs during hot extrusion of an ultra-fine-grained (UFG) 5083 Al (an Al–Mg–Mn–Cr–Fe alloy) synthesized via the consolidation of mechanically milled powders [35]. The results show that second-phase particles and solute segregation at GBs generate such high resistance forces that grain growth was essentially inhibited during annealing of the UFG 5083 Al at 400 °C for a prolonged period of 5 h in the absence of an externally applied stress. In contrast, the average grain size increased by a factor of ∼2.7 during hot extrusion of the UFG 5083 Al at 400 °C, suggesting that the externally applied stresses originating from the state of stress imposed during extrusion overcame these resistance forces, enabling the operation of GB migration and grain rotation and thus the occurrence of grain growth.
In view of the above discussion, it is the objective of the present study to quantify the influence of second-phase particles and solute segregation at GBs on GB migration and grain rotation during stress-induced grain growth. To accomplish this goal, a theoretical framework is first formulated by incorporating the drag stresses generated by second-phase particles and by solute atoms segregated at GBs into the well-established Cahn–Taylor model [32] and then implemented to rationalize stress-induced grain growth phenomenon reported in Ref. [35]: (i) to analyze and discuss stress-induced grain growth during hot extrusion and (ii) to quantify the influence of second-phase particles and solute atoms segregated at GBs on stress-induced GB migration and grain rotation. In order to provide proper context to the present study, the experimental results reported in Ref. [35] are reviewed in Section 2.
Section snippets
Published experimental results on stress-induced grain growth phenomenon in UFG 5083 Al containing both second-phase particles and solute segregation at GBs [35]
An UFG 5083 Al (chemical composition: Mg 4.50, Mn 0.70, Cr 0.16, Fe 0.20, Ni 0.019, Si 0.13, Cu 0.014, C 0.019, O 0.45, N 0.004, Al balance, wt.%) used as the model material for the study of stress-induced grain growth was prepared by hot isostatic pressing (HIPing) of mechanically milled pre-alloyed 5083 powders. In order to study stress-induced grain growth, the UFG 5083 Al was extruded at 400 °C with area reduction ratio of 10. The extrusion process was estimated to last ∼36 s, with an
Theoretical model
It has been well established that the velocities of GB migration (vn) and/or of grain rotation (vt) provide indications as to whether GB migration and/or grain rotation occur, leading to grain growth or not [1], [20]. Based on this viewpoint, a theoretical model used to investigate grain growth must quantify vn and vt. In the experimental study reported in Ref. [35], the mechanisms underlying the stress-induced grain growth include both GB migration and grain rotation; moreover, the two
Results and discussion
Using the equations presented in Section 3.2, vn as well as vt and for the grains with average size () and average misorientation (45°) were calculated under the conditions of externally applied stress and temperature (400 °C) as used in Ref. [35], based on the relevant physical and mechanical properties (as shown in Section 3.4 and Table 1) and the published experimental data in Ref. [35] as reviewed in Section 2. By doing so, the stress-induced grain growth phenomenon during hot extrusion
Conclusions
By incorporating the drag stresses generated by second-phase particles and solute atoms segregated at GBs into the well-established Cahn–Taylor model, a new theoretical framework was formulated. Rationalization of the grain growth phenomenon during hot extrusion reveals that the externally applied stresses, together with driving forces provided by GB curvature and reduction in dislocation density, exceeded the resistance forces to both GB migration and grain rotation, enabling significant grain
Acknowledgements
This work was supported by the US Office of Naval Research N00014-09-1-0437. Y.J.L. acknowledges the financial support from the One Hundred Talents Project (Project no. 2010100005) and the Research Program of Department of Education (Project no. ZH2011111), Hebei, China.
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