Extending phase field models of solidification to polycrystalline materials

Warren, James A.; Kobayashi, Ryo; Lobkovsky, Alexander E.; Carter, W. Craig

doi:10.1016/s1359-6454(03)00388-4

Cited by 309 publications

(283 citation statements)

References 33 publications

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“…The simulation was started with the liquid of average density ρ 0 = 0.285 and dimensionless temperature r = −0.25; other parameters were set to (∆x, ∆t, α, β) = (π/8, 0.001, 15, 0.9). The measured grain boundary energies per unit length are consistent with the usual Read-Shockley form [4,7] To demonstrate the presence of elastic relaxation modes in the MPFC model, we performed simulations of an effectively one dimensional single-crystal specimen under uniaxial tension. The system was prepared in the coexistence region as given by the phase diagram, and the solid sample was surrounded by liquid.…”

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confidence: 62%

Phase-Field Crystals with Elastic Interactions

2006

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We report on a novel extension of the recent phase-field crystal (PFC) method introduced in [Elder et al., Phys. Rev. Lett., 88, 245701:1-4 (2002)], which incorporates elastic interactions as well as crystal plasticity and diffusive dynamics. In our model, elastic interactions are mediated through wave modes that propagate on time scales many orders of magnitude slower than atomic vibrations but still much faster than diffusive times scales. This allows us to preserve the quintessential advantage of the PFC model: the ability to simulate atomic-scale interactions and dynamics on time scales many orders of magnitude longer than characteristic vibrational time scales. We demonstrate the two different modes of propagation in our model and show that simulations of grain growth and elasto-plastic deformation are consistent with the microstructural properties of nanocrystals.PACS numbers: 61.82.Rx, 62.25.+g, 62.30.+d, 63.22.+m The deformation of a solid triggers processes which operate across several length and time scales. On long length and time scales its behavior can be described by a set of hydrodynamic equations [1, 2], which describe, e.g., elastic deformation dynamics of the body. On atomic length (∼ 10 −10 m) and time (∼ 10 −13 s) scales, on the other hand, the dynamics can be captured by direct molecular dynamics (MD) simulations, which incorporate local bonding information either through direct quantum-mechanical calculations or semi-empirical many-body potentials. While innovations in computing methods have greatly improved the efficiency of MD simulations, standard atomistic computer simulations are still limited to fairly small system sizes (∼ 10 9 atoms) and short times (∼ 10 −8 s). This limitation is most severe when developing simulation models to study the physics and mechanics of nanostructured materials, where the relevant length scales are atomic and time scales are mesoscopic. In this regime, the available numerical tools are rare.Progress towards alleviating this limitation has recently been made by the introduction of a new modeling paradigm known as the phase-field crystal (PFC) method [3]. This method introduces a local atomic mass density field ρ(r) in which atomic vibrations have been integrated out up to diffusive time scales. Dissipative dynamics are then constructed to govern the temporal evolution of ρ. Unfortunately, the original PFC model evolves mass density only on diffusive time scales. In particular, it does not contain a mechanism for simulating elastic interactions, an important aspect for studying, for example, the deformation properties of nanocrystalline solids.In this Letter, we introduce a modified phase-field crystal (MPFC) model that includes both diffusive dynamics and elastic interactions. This is achieved by exploiting the separation of time scales that exists between diffusive and elastic relaxation processes in solids. In particular, the MPFC model is constructed to transmit long wavelength density fluctuations with wave modes that propagate up to a time scale t...

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confidence: 62%

Phase-Field Crystals with Elastic Interactions

2006

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“…As s introduces a misorientation dependence to the surface energy [51], it is possible to introduce favored misorientations through this coefficient. However, it is also possible to introduce misorientation dependencies via a coupling to gradients in θ.…”

Section: Phase Field Theory With Crystallographic Branchingmentioning

confidence: 99%

Growth and form of spherulites

et al. 2005

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Many structural materials (metal alloys, polymers, minerals, etc.) are formed by quenching liquids into crystalline solids. This highly non-equilibrium process often leads to polycrystalline growth patterns that are broadly termed 'spherulites' because of their large-scale average spherical shape. Despite the prevalence and practical importance of spherulite formation, only rather qualitative concepts of this phenomenon exist. The present work explains the growth and form of these fundamental condensed matter structures on the basis of a unified field theoretic approach. Our phase field model is the first to incorporate the essential ingredients for this type crystal growth: anisotropies in both the surface energy and interface mobilities that are responsible for needle-like growth, trapping of local orientational order due to either static heterogeneities (impurities) or dynamic heterogeneities in highly supercooled liquids, and a preferred relative grain orientation induced by a misorientation-dependent grain boundary energy. Our calculations indicate that the diversity of spherulite growth forms arises from a competition between the ordering effect of discrete local crystallographic symmetries and the randomization of the local crystallographic orientation that accompanies crystal grain nucleation at the growth front (growth front nucleation or GFN). The large-scale isotropy of spherulitic growth arises from the predominance of GFN.

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“…The measured grain boundary energies per unit length for such a polycrystalline sample are consistent with the usual Read-Shockley form. 38,41,42,50 A comparison of the grain boundary energy from the PFC model, the Read-Shockley equation, and several experiments are shown in Figure 2.…”

Section: The Phase-field-crystal Methodsmentioning

confidence: 99%

“…[33][34][35][36][37] These approaches have proven quite useful in various applications such as polycrystalline solidification. 28,[33][34][35][36][37][38][39][40] Nevertheless, it has proven quite challenging to incorporate elastoplasticity, diffusive phase transformation kinetics, and anisotropic surface energy effects into a single, thermodynamically consistent model.…”

Section: Phase-field Modelsmentioning

confidence: 99%