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Grain Shape, Grain Boundary Mobility,
and the Herring Relation
Cooperative Research Team on
Fundamentals of Dirty Interfaces: From Atoms to Alloy Microstructures
A fundamental understanding of grain boundary motion is of paramount importance for mesoscale simulations of polycrystalline microstructures. This motion can generally be driven both by the local "curvature" of a grain boundary to reduce interfacial area or by a thermodynamic driving force originating from spatial variations of free-energy in the bulk of the material, such as an inhomogeneous distribution of strain or an external magnetic field. A puzzling observation, in both recent experiments [Gottstein and Shvindlerman, 1998; Molodov, 2004; Huang and Humphreys, 2001; Taheri et al., 2004] and simulations of Ising-like representations of grain boundaries, is that curvature-driven motion leads to round isotropic grains, whereas bulk-driven motion can produce highly anisotropic grains. This has raised the question: is grain boundary mobility unique, or does it depend on the nature of the driving force? New results from theory, simulations, and experiments, obtained through a unique collaborative effort have shed light on this long-standing question and have led to the development of a new molecular dynamics- based method for the prediction of fundamentally important grain boundary properties. The basic conclusion is that, although mobility may well be consistent across driving forces, interfacial stiffness plays a more important role in determining boundary migration than has been appreciated hitherto.
This answer to the question has been developed by application of the
computational simulation of boundary migration using the Ising model,
coupled with theoretical development. An important realization is that the
dramatic change of grain shape with driving force is only paradoxical if
interpreted using the law V = M*κ + MH which has been widely
used in mesoscale
simulations of polycrystalline microstructure to describe the velocity V of boundaries driven by the local curvature κ and an
external field H.
Implicit in this law is the assumption that the reduced mobility for
curvature driven motion, M* is proportional to the interface
energy γ.
If, instead, one uses the thermodynamically correct expression due to
Herring [1951] for the driving force for curvature-driven motion,
(γ + γ'') κ,
where γ'' is the second derivative of γ with
respect to inclination of the
boundary normal, the law
V = M[(γ + γ'')κ + H]
describes both curvature- and bulk-driven motion
with a unique "bare mobility" M. Solving the original puzzle
is now reduced to understanding why a curvature and a bulk driving
force can yield isotropic and very anisotropic grain shapes,
respectively, within this thermodynamically consistent law of interface
motion.
In the Ising model, the answer is subtle [Lobkovsky et al. 2003]. The key insight is that the entropic part of γ'' can be highly anisotropic, such that the reduced mobility (i.e., the product of interface stiffness, γ + γ'', and mobility) can be nearly isotropic even though the mobility itself is highly anisotropic. The cancellation of these two anisotropies associated with stiffness and mobility reflects the fact that the number of geometrically necessary kinks on the boundary, and hence its configurational entropy, varies rapidly with inclination near low-energy/low-mobility interfaces, but slowly near high-energy/high-mobility interfaces, where the kink density is high. That is: the stiffness is high where the mobility is low and vice versa. Consequently, the grain shape can appear isotropic or highly anisotropic depending on whether its motion is curvature-driven or bulk-driven, respectively, but the mobility itself is independent of driving force.
Figure. (left) Diagram of the molecular dynamics simulation cell with a pair of crystals forming a boundary. (right) A snapshot of the atoms colored by the local coordination such that red indicates full fcc coordination whereas atoms in the boundary region have lower atomic coordination and are colored accordingly.
Many clues that support the conclusion are evident in the computational work performed in this CMSN project. The main tool used here is a molecular dynamics simulation of boundary migration in bicrystal systems, using a standard interatomic potential (for aluminum, by reference to the experimental system with the greatest amount of literature). The standard method consists of measuring the retraction velocity of a U-shaped grain [Upmanyu, 1999] using curvature as a driving force. The limitation of this approach is that it only yields an average of the reduced mobility over the spectrum of boundary normals present in the U-shaped bicrystal configuration. Thus this limits the value of the mobility information because it is not possible to extract the dependence on normal. The results do show cusps and peaks in both boundary energy and (apparent) mobility so the potential for interfacial stiffness to influence migration rates is there. Clearly then, the extension of these concepts to real grain boundaries in polycrystalline materials requires a computational method to calculate the bare mobility with its normal dependence. A new method to compute this fundamental quantity has recently been developed which consists of driving a flat grain boundary with a tensile or compressive strain as illustrated in the Figure [Zhang et al., 2004]. This method exploits the difference of elastic energy between the two strained crystals to drive the boundary between them. The results are in good agreement with the previous results based on a curvature driving force with similar crystallographic anisotropy and activation energies. A wider range of boundary types will have to be investigated, however, before it will be possible to form a judgment about the effect of interfacial stiffness on apparent mobilities described here.
The experimental evidence supports the conclusion in a limited but intriguing fashion. It is known, for example, that certain lattice misorientations at a boundary are associated with high mobility, e.g., 40°<111> in fcc metals [Gottstein and Shvindlerman, 1998]. When volumetric driving forces, such as the stored energy of cold work, are used to drive boundaries, the mobility is clearly highly dependent on the boundary normal [Huang and Humphreys, 2001]. When the boundary plane comprises close-packed planes (i.e., a pure twist <111> boundary), the mobility is low, as expected, but for the same misorientation, pure tilt configurations exhibit maximum mobility. For this driving force, only the boundary energy and mobility enter the velocity expression. When curvature is used to drive boundaries in fcc aluminum, however, [Molodov, 2004] the normal dependence is not present even though the variation in mobility with crystallographic type is very similar to that observed for stored energy. Thus the mobility dependence on boundary normal may be cancelled out by the effect of the interface stiffness near the energy cusp associated with <111> twist boundaries [Rohrer et al., 2004]. This tentative view will have to be reinforced by further experimental investigation.
The outlook in this area is both challenging and stimulating. Although we now have a substantially improved understanding that now needs to be properly validated in experiments, much remains to be done. The activation energies for boundary migration as extracted from molecular dynamics simulations are rather small in magnitude and much less than the activation energies that are typical of self-diffusion. Although typical activation energies for boundary migration under volumetric driving forces (e.g., stored energy of cold work) are similar to self-diffusion energies, those observed in curvature-driven migration are highly variable. For certain boundary types, the activation energies are observed to be substantially larger even than some of the slow-diffusing solutes present in the experimental materials. These discrepancies point to a lack of understanding of the mechanism(s) of boundary migration. The textbook account of individual atom-hopping enshrined in the Burke and Turnbull [1952] theory cannot be correct! The challenge, then, is to develop a viable theory for boundary migration in the solid state that can explain the known facts. Given the complexity of the problem, it seems self-evident that computational materials science has a vital role to play.
Consequences for Theory and Computation
In general terms, the impact of this work on Theory and Computation in DOE-BES is that (a) adopting a computational perspective on interfacial properties has stimulated a new understanding of the (equilibrium and) kinetic properties of boundaries; (b) the new theoretical insight related to interfacial stiffness is, in turn, stimulating new computational (and experimental) effort to validate the concepts; and (c) that, notwithstanding the advances reported here, the current theoretical understanding of interfacial properties is inadequate and needs further development.
References
Burke, J. and D. Turnbull (1952): Progress in Metal Physics 3, 220.
Herring, C. (1951), "Surface tension as a motivation for sintering," in The Physics of Powder Metallurgy, W.E. Kingston, ed., McGraw-Hill Book Co., New York, pp. 143-179.
Huang, Y. and F. J. Humphreys (2001). The effect of solute elements on grain boundary mobility during recrystallization of single-phase aluminium alloys. Proc. 1st Intl. Conf. on Recrystallization & Grain Growth, Aachen, Germany, Springer, p. 409-414.
Lobkovsky, A., A. Karma, M. I. Mendelev, M. Haataja, and D. J. Srolovitz (2004): "Grain shape, grain boundary mobility, and the Herring relation," Acta Materialia 52, 285-292.
Molodov, D. (2004): Unpublished Research.
Saylor, D. M., B. S. El-Dasher, A. D. Rollett and G. S. Rohrer (2004). "Distribution of Grain Boundaries in Aluminum as a Function of Five Macroscopic Parameters." Acta materialia: accepted for publication (2004).
Upmanyu, M., D. Srolovitz, L. Shvindlerman, and G. Gottstein (1999): "Misorientation dependence of intrinsic grain boundary mobility: Simulation and experiment," Acta materialia 47, p. 3901.
Zhang, H., M.I. Mendelev, and D.J. Srolovitz (2004): "Computer simulation of the elastically driven migration of a flat grain boundary," Acta materialia, submitted.
Contact: Alain Karma, Northeastern University
