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 fundaments 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.Keywords: phase-field-crystal (PFC) models, static and dynamical density functional theory (DFT and DDFT), condensed matter dynamics of liquid crystals and polymers, nucleation and pattern formation, simulations in materials science, colloidal crystal growth and growth anisotropy * Corresponding authors. Emails: heike.emmerich@uni-bayreuth. de, hlowen@thphy.uni-duesseldorf.de, and grana@szfki.hu

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Vesicle membranes with two coexisting fluid phases can exhibit a variety of domain patterns. These patterns and the associated vesicle shapes are studied theoretically by minimization of energy functionals that depend on the membrane composition and on the material parameters of the membrane. The latter parameters are (i) the bending rigidities and (ii) the Gaussian curvature moduli of the two types of membrane domains as well as (iii) the line tension of the domain boundaries. It is shown that the interplay between these different parameters leads to stable multi-domain patterns with more than two domains for a wide range of membrane composition and material parameters. As the membrane composition or the material parameters are varied, the vesicles can undergo transitions between different patterns of membrane domains. For fixed vesicle volume, an additional domain-induced transition is observed, in which the vesicle shape changes even though the domain pattern does not. The different multi-domain patterns and vesicle shapes are summarized in terms of morphology diagrams.

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The microscopic structure of a molecularly thin liquid-crystal film confined between two plane parallel surfaces ͑i.e., walls͒ composed of rigidly fixed atoms is investigated in grand canonical ensemble Monte Carlo simulations in which the temperature T, the chemical potential , and the wall separation s z are the relevant thermodynamic state variables. These conditions correspond to those encountered in related experiments employing the surface forces apparatus ͑SFA͒. Wall atoms are distributed according to the ͑100͒ configuration of a face-centered cubic ͑fcc͒ lattice. Film molecules interact with each other via the Gay-Berne potential which may be viewed as a Lennard-Jones ͑12,6͒ potential modified to account for the anisotropy of the interaction between two ellipsoidal film molecules. Parameters governing the film-wall interaction are chosen such that molecules tend to arrange their symmetry axes parallel with the plane of a wall ͑i.e., the x-y plane͒. The thermodynamic state of a bulk phase in equilibrium with the confined film pertains to the isotropic phase of the Gay-Berne fluid, so that preferred orientations in the film are unambiguously ascribed to confinement ͑i.e., to the presence of the walls͒. In general, film structure is characterized by stratification, that is, the tendency of film molecules to arrange their centers of mass in individual strata parallel with the walls. The strata are more diffuse than in films composed of ''simple'' molecules without rotational degrees of freedom due to a larger geometric incompatibility between film and wall structure and to orientability of film molecules in the present model. As s z is increased at fixed T and , molecularly thin liquid-crystal films undergo complex structural changes resulting from a competition between wall-induced orientation and lack of space. These effects are analyzed in depth by density-alignment histograms and correlated with variations of the normal stress T zz exerted by the film on the walls. The normal stress, which is in principle accessible in SFA experiments, depends strongly on s z even in rather thick films, indicating the importance of cooperative wall-induced phenomena for materials properties of confined liquid-crystal films. ͓S1063-651X͑97͒04702-8͔