Thermodynamics of bcc metals in phase-field-crystal models

Jaatinen, A.; Achim, C. V.; Elder, K. R.; Ala-Nissilä, Tapio

doi:10.1103/physreve.80.031602

Cited by 183 publications

(300 citation statements)

References 22 publications

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“…Consider Eq. (20) in the moving reference frame with the origin at the front invading unstable phase. Then, in Eq.…”

Section: Selection Of the Lattice Parametermentioning

confidence: 99%

“…It can be related to other continuum fields theories such as classical density-functional theory 8,9 and the atomic density function theory 10 . The PFC-model may also be considered as a conserved version of the Swift-Hohenberg equation and provides an efficient method for simulating liquid-solid transitions 11,12 , colloidal solidification 13 , dislocation motion and plasticity 14,15 , glass formation 16 , epitaxial growth 6,17 , grain boundary premelting 18 , surface reconstructions 19 , and grain boundary energies 20 .…”

Section: Introductionmentioning

confidence: 99%

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Marginal stability analysis of the phase field crystal model in one spatial dimension

Galenko

Elder

2011

Phys. Rev. B

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The problem of wave-length k f and velocity V selection for a solid front invading an unstable homogeneous phase is considered. A marginal stability analysis is used to predict k f and V for the parabolic and hyperbolic (or modified) phase-field crystal models in one-dimension. It is shown that the marginally selected wave-number of the periodic crystal monotonically increases with increasing undercooling and relaxation times. At high undercooling and relaxation times it is found that the system can select a k f that is unstable to an Eckhaus instability in the bulk phase. This may imply a transition to highly defected or glassy states in higher dimensions.

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“…Consider Eq. (20) in the moving reference frame with the origin at the front invading unstable phase. Then, in Eq.…”

Section: Selection Of the Lattice Parametermentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Marginal stability analysis of the phase field crystal model in one spatial dimension

Galenko

Elder

2011

Phys. Rev. B

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“…With suitable choices of parameters for Fe (see Ref. 44), this model has been shown to predict reasonable values of solid-liquid interfacial energy 44,54 and GB energies 46 , despite the uncontrolled truncation of many other sets of reciprocal lattice vectors. It has also proven capable of reproducing a subtle dislocation-pairing GB structural transition at high homologous temperatures 23 , previously evidenced in a 2D PFC study of GB premelting 29 , and also observed in MD simulations 23 .…”

Section: Introductionmentioning

confidence: 99%

“…In this paper, we investigate GB premelting and shearing and their relationship using the PFC method [40][41][42][43][44][45][46] . We use the simplest PFC model 40,41 with the same freeenergy function as the Swift-Hohenberg model of pattern formation 47 , which favors hexagonal and bcc ordering in two and three dimensions (2D and 3D), respectively.…”

Section: Introductionmentioning

confidence: 99%

Phase-field-crystal study of grain boundary premelting and shearing in bcc iron

et al. 2013

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We use the phase-field-crystal (PFC) method to investigate the equilibrium premelting and nonequilibrium shearing behaviors of [001] symmetric tilt grain boundaries (GBs) at high homologous temperature over the complete range of misorientation 0 < θ < 90 • in classical models of bcc Fe. We characterize the dependence of the premelted layer width W as a function of temperature and misorientation. In addition, we compute the thermodynamic disjoining potential whose derivative with respect to W represents the structural force between crystal-melt interfaces due to the spatial overlap of density waves. The disjoining potential is also computed by molecular dynamics (MD) simulations, for quantitative comparison with PFC simulations, and coarse-grained amplitude equations (AE) derived from PFC that provide additional analytical insights. We find that, for GBs over an intermediate range of misorientation (θmin < θ < θmax), W diverges as the melting temperature is approached from below, corresponding to a purely repulsive disjoining potential, while for GBs outside this range (θ < θmin or θmax < θ < 90 • ), W remains finite at the melting point. In the latter case, W corresponds to a shallow attractive minimum of the disjoining potential. The misorientation range where W diverges predicted by PFC simulations is much smaller than the range predicted by MD simulations when the small dimensionless parameter ǫ of the PFC model is matched to liquid structure factor properties. However it agrees well with MD simulations with a lower ǫ value chosen to match the ratio of bulk modulus and solid-liquid interfacial free-energy, consistent with the amplitude-equation prediction that θmin and 90 • − θmax scale as ∼ ǫ 1/2 . The incorporation of thermal fluctuations in PFC is found to have a negligible effect on this range. In response to a shear stress parallel to the GB plane, GBs in PFC simulations exhibit coupled motion normal to this plane or sliding. Furthermore, the coupling factor exhibits a discontinuous change as a function of θ that reflects a transition between two coupling modes. Sliding is only observed over a range of misorientation that is a strongly increasing function of temperature for T /TM ≥ 0.8 and matches roughly the range where W diverges at the melting point. The coupling factor for the two coupling modes is in excellent quantitative agreement with previous theoretical predictions [J. W. Cahn, Y. Mishin, and A. Suzuki, Acta Mater. 54, 4953 (2006)].

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“…Analyses are mainly limited to visual comparisons and relative energetics. Often, it is stated that the PFC-computed energies are not quantitatively comparable to molecular methods but that they follow similar trends as a function of some structural parameter, for instance grain boundary orientation 26,27 . However, to capture the structure, properties, and stress-driven thermally-activated phenomena that control kinetics requires accurate absolute energies.…”

Section: Introductionmentioning

confidence: 99%

Assessment of phase-field-crystal concepts using long-time molecular dynamics

Baker

2015

Phys. Rev. B

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The ability of the phase-field-crystal (PFC) model to quantitatively predict atomistic defect structures in crystalline solids is addressed. First, general aspects of the PFC model are discussed within the context of obtaining quantitative results in solid materials. Then a specific example is used to illustrate major points. Specifically, accelerated molecular dynamics is used to compute the oneparticle probability density, ρ (1) (r), in a complex atomistic defect consisting of a Lomer dislocation with an equilibrium distribution of vacancies in the core, and the results are considered within the general framework of the PFC model. As expected, ρ (1) (r) shows numerous spatially localized peaks with integrated densities smaller than unity, as would arise in a PFC computation. However, the ρ (1) (r) actually corresponds to a time-averaged superposition of a few well-defined atomic configurations each having a well-defined energy. The deconvolution of ρ (1) (r) to obtain the actual distinct atomic configurations is not feasible. Using a potential energy functional that accurately computes the energies of distinct configurations, the potential energy computed using ρ (1) (r) differs from the actual average atomistic energy by ∼50 eV divided among approximately 46 atoms in the core of the defect. Attempts to rectify this deviation by introducing correlations can not significantly reduce this error. The simulations show that energy barriers between distinct configurations varying by up to 0.5 eV, indicating that the simple kinetic evolution law used in PFC cannot accurately capture the true time evolution in this problem. Overall, these results demonstrate, in one non-trivial case, that the PFC model is probably unable to predict atomistic defect structures, energies, or kinetic barriers, at the quantitative levels needed for application to problems in materials science.

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Thermodynamics of bcc metals in phase-field-crystal models

Cited by 183 publications

References 22 publications

Marginal stability analysis of the phase field crystal model in one spatial dimension

Marginal stability analysis of the phase field crystal model in one spatial dimension

Phase-field-crystal study of grain boundary premelting and shearing in bcc iron

Assessment of phase-field-crystal concepts using long-time molecular dynamics

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