Phase field crystal

Phase-field crystal modelling of crystal nucleation, heteroepitaxy and patterning

László Gránásy1,2, György Tegze1, Gyula Tóth3, Tamás Pusztai1

1Institute for Solid State Physics and Optics, Wigner Research Centre for Physics, P.O. Box 49, Budapest H-1525, Hungary
2BCAST, Brunel University, Uxbridge, Middlesex, UB8 3PH, United Kingdom
3Department of Mathematical Sciences, Loughborough University, Loughborough, Leicestershire, LE11 3TU, U.K.

A simple dynamical density functional theory, the phase-field crystal (PFC) model, was used to describe homogeneous and heterogeneous crystal nucleation in two-dimensional (2D) monodisperse colloidal systems and crystal nucleation in highly compressed Fe liquid. External periodic potentials were used to approximate inert crystalline substrates in addressing heterogeneous nucleation. In agreement with experiments in 2D colloids, the PFC model predicts that in 2D supersaturated liquids, crystalline freezing starts with homogeneous crystal nucleation without the occurrence of the hexatic phase. At extreme supersaturations, crystal nucleation happens after the appearance of an amorphous precursor both in two and three dimensions. Contrary to expectations based on the classical nucleation theory, it is shown that corners are not necessarily favourable places for crystal nucleation. Finally, it is shown that by adding external potential terms to the free energy, the PFC theory can be used to model colloid patterning experiments.

Topics: Phase field crystal

Diffusion-controlled anisotropic growth of stable and metastable crystal polymorphs in the phase-field crystal model

György Tegze1, László Gránásy1,2, Gyula Tóth3, Frigyes Podmaniczky1, A Jaatinen4, T Ala-Nissila4, Tamás Pusztai1

1Institute for Solid State Physics and Optics, Wigner Research Centre for Physics, P.O. Box 49, Budapest H-1525, Hungary
2BCAST, Brunel University, Uxbridge, Middlesex, UB8 3PH, United Kingdom
3Department of Mathematical Sciences, Loughborough University, Loughborough, Leicestershire, LE11 3TU, U.K.
4Department of Applied Physics, Helsinki University of Technology, Post Office Box 1100, FI-02015 TKK, Finland

We use a simple density functional approach on a diffusional time scale, to address freezing to the body-centered cubic (bcc), hexagonal close-packed (hcp), and face-centered cubic (fcc) structures. We observe faceted equilibrium shapes and diffusion-controlled layerwise crystal growth consistent with two- dimensional nucleation. The predicted growth anisotropies are discussed in relation with results from experiment and atomistic simulations. We also demonstrate that varying the lattice constant of a simple cubic substrate, one can tune the epitaxially growing body-centered tetragonal structure between bcc and fcc, and observe a Mullins-Sekerka/Asaro-Tiller-Grinfeld-type instability.

Topics: Phase field crystal

Advanced operator-splitting-based semi-implicit spectral method to solve the binary phase-field crystal equation with variable coefficients

György Tegze1, Gurvinder Bansel2, Gyula Tóth3, Tamás Pusztai1, Zhongyun Fan2, László Gránásy1,2

1Institute for Solid State Physics and Optics, Wigner Research Centre for Physics, P.O. Box 49, Budapest H-1525, Hungary
2BCAST, Brunel University, Uxbridge, Middlesex, UB8 3PH, United Kingdom
3Department of Mathematical Sciences, Loughborough University, Loughborough, Leicestershire, LE11 3TU, U.K.

We present an efficient method to solve numerically the equations of dissipative dynamics of the binary phase-field crystal model proposed by Elder et al. [K.R. Elder, M. Katakowski, M. Haataja, M. Grant, Phys. Rev. B 75, 064107 (2007)] characterized by variable coefficients. Using the operator splitting method, the problem has been decomposed into sub-problems that can be solved more efficiently. A combination of non-trivial splitting with spectral semi-implicit solution leads to sets of algebraic equations of diagonal matrix form. Extensive testing of the method has been carried out to find the optimum balance among errors associated with time integration, spatial discretization, and splitting. We show that our method speeds up the computations by orders of magnitude relative to the conventional explicit finite difference scheme, while the costs of the pointwise implicit solution per timestep remains low. Also we show that due to its numerical dissipation, finite differencing can not compete with spectral differencing in terms of accuracy. In addition, we demonstrate that our method can efficiently be parallelized for distributed memory systems, where an excellent scalability with the number of CPUs is observed.

Topics: Phase field crystal

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