Phase-field-crystal models for condensed matter dynamics on atomic length and diffusive time scales: an overview

Abstract

Here, we review the basic concepts and applications of the phase-field-crystal (PFC) method, which is one of the latest simulation methodologies in materials science for problems, where atomic- and microscales are tightly coupled. The PFC method operates on atomic length and diffusive time scales, and thus constitutes a computationally efficient alternative to molecular simulation methods. Its intense development in materials science started fairly recently following the work by Elder et al. [Phys. Rev. Lett. 88 (2002), p. 245701]. Since these initial studies, dynamical density functional theory and thermodynamic concepts have been linked to the PFC approach to serve as further theoretical fundamentals for the latter. In this review, we summarize these methodological development steps as well as the most important applications of the PFC method with a special focus on the interaction of development steps taken in hard and soft matter physics, respectively. Doing so, we hope to present today's state of the art in PFC modelling as well as the potential, which might still arise from this method in physics and materials science in the nearby future.

PACS :

• 64.70.D- Solid–liquid transitions
• 81.10.Aj Theory and models of crystal growth
• physics and chemistry of crystal growth
• crystal morphology
• and orientation
• 68.08.-p Liquid–solid interfaces
• 61.30.-v Liquid crystals

Keywords:

• phase-field-crystal models
• static and dynamical density functional theory
• condensed matter dynamics of liquid crystals
• nucleation and pattern formation
• simulations in materials science
• colloidal crystal growth and growth anisotropy

Acknowledgements

This work has been supported by the EU FP7 Projects “ENSEMBLE” (contract no. NMP4-SL-2008-213669) and “EXOMET” (contract no. NMP-LA-2012-280421, co-funded by ESA), by the ESA MAP/PECS project “MAGNEPHAS III”, and by the German Research Foundation (DFG) in the context of the DFG Priority Program 1296.

Notes

It is interesting to note that there are also mesoscopic particle systems with Newtonian dynamics, which are virtually undamped. These are realized in the so-called complex plasmas 75 76, where dust particles are dispersed and levitated in a plasma.

This follows from the representation (9) under consideration of the translational and rotational symmetries of the isotropic bulk fluid that also apply to the direct correlation function

More refined approaches include also the third-order term 115 with an approximate triplet direct correlation function 116 117.

However, one should also note that density fluctuations, which are, for example, embodied in the liquid structure factor, are not reproduced by Equation (33), since the one-particle density is the only variable here.

Note that the order-parameter field introduced here is not identical with the field in Equation (1), although both fields are dimensionless.

This Taylor approximation of the logarithm has the serious consequence that the non-negative-density constraint gets lost in the PFC model.

A recent comparison between DFT and PFC models for the structure of the hard-sphere crystal-fluid interface was performed in reference 134.

To keep the notation simple, we ignore and write ℱ instead of throughout this article.

Turnbull's coefficient C _T is a reduced liquid-solid interfacial free energy defined via the relationship , where , , N _A, and v _m are the total liquid-solid interfacial free energy, the molar heat of fusion, the Avogadro number, and the molar volume, respectively. C _T is expected to depend only on the crystal structure. Recent results indicate that besides structure the interaction potential also has influence on its magnitude.