Classical dynamical density functional theory (DDFT) is one of the cornerstones of modern statistical mechanics. It is an extension of the highly successful method of classical density functional theory (DFT) to nonequilibrium systems. Originally developed for the treatment of simple and complex fluids, DDFT is now applied in fields as diverse as hydrodynamics, materials science, chemistry, biology, and plasma physics. In this review, we give a broad overview over classical DDFT. We explain its theoretical foundations and the ways in which it can be derived. The relations between the different forms of deterministic and stochastic DDFT as well as between DDFT and related theories, such as quantum-mechanical time-dependent DFT, mode coupling theory, and phase field crystal models, are clarified. Moreover, we discuss the wide spectrum of extensions of DDFT, which covers methods with additional order parameters (like extended DDFT), exact approaches (like power functional theory), and systems with more complex dynamics (like active matter). Finally, the large variety of applications, ranging from fluid mechanics and polymer physics to solidification, pattern formation, biophysics, and electrochemistry, is presented.

Classical dynamical density functional theory: from fundamentals to applications

In this paper, a two-dimensional phase field crystal model of graphene and hexagonal boron nitride (hBN) is extended to include out-of-plane deformations in stacked multilayer systems. As proof of principle, the model is shown analytically to reduce to standard models of flexible sheets in the small deformation limit. Applications to strained sheets, dislocation dipoles, and grain boundaries are used to validate the behavior of a single flexible graphene layer. For multilayer systems, parameters are obtained to match existing theoretical density functional theory calculations for graphene/graphene, hBN/hBN, and graphene/hBN bilayers. More precisely, it is shown that the parameters can be chosen to closely match the stacking energies and layer spacing calculated by Zhou et al. [Phys. Rev. B 92, 155438 (2015)]. Further validation of the model is presented in a study of rotated graphene bilayers and stacking boundaries. The flexibility of the model is illustrated by simulations that highlight the impact of complex microstructures in one layer on the other layer in a graphene/graphene bilayer.

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Modeling buckling and topological defects in stacked two-dimensional layers of graphene and hexagonal boron nitride

We show that one can employ well-established numerical continuation methods to efficiently calculate the phase diagram for thermodynamic systems. In particular, this involves the determination of lines of phase coexistence related to first order phase transitions and the continuation of triple points. To illustrate the method we apply it to a binary Phase-Field-Crystal model for the crystallisation of a mixture of two types of particles. The resulting phase diagram is determined for one- and two-dimensional domains. In the former case it is compared to the diagram obtained from a one-mode approximation. The various observed liquid and crystalline phases and their stable and metastable coexistence are discussed as well as the temperature-dependence of the phase diagrams. This includes the (dis)appearance of critical points and triple points. We also relate bifurcation diagrams for finite-size systems to the thermodynamics of phase transitions in the infinite-size limit.

Efficient calculation of phase coexistence and phase diagrams: application to a binary phase-field crystal model

In the past few years, binary colloidal crystals (BCCs) composed of both large and small particles have attracted considerable attention from the scientific community as an exciting alternative to single colloidal crystals (SCCs). In particular, more complex structures with diverse nanotopographies and desirable optical properties of BCCs can be obtained by various colloidal assembly methods, as compared to SCCs. Furthermore, high accuracy in crystal growth with controllable stoichiometries allows for a great deal of promising applications in the fields of both interfacial and material sciences. The visible-light diffraction property of BCCs is more superior than that of SCCs, which makes them have more promising applications in the fabrication of photonic crystals with full band gaps. On the other hand, their spherical shapes and ease of removal property make them ideal templates for ordered porous material fabrication. Hence, this perspective outlined recent advances in assembly approaches of BCCs, with an emphasis on their promising applications for advanced photonics and multifunctional porous material fabrication. Eventually, some challenging yet important issues and some future perspectives are further discussed.

Recent Advances in Binary Colloidal Crystals for Photonics and Porous Material Fabrication.

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Grain rotation and coupled grain boundary motion in two-dimensional binary hexagonal materials

The formation and dynamics of a wide variety of binary two-dimensional ordered structures and superlattices are investigated through a phase field crystal model with sublattice ordering. Various types of binary ordered phases, the phase diagrams, and the grain growth dynamics and structural transformation processes, including the emergence of topological defects, are examined. The results are compared to the ordering and assembly of two-component colloidal systems. Two factors governing the binary phase ordering are identified, the coupling and competition between the length scales of two sublattices and the selection of average particle densities of two components. The control and variation of these two factors lead to the prediction of various complex binary ordered patterns, with different types of sublattice ordering for integer vs. noninteger ratios of sublattice length scales. These findings will enable further systematic studies of complex ordering and assembly processes of binary systems particularly binary colloidal crystals.

Phase ordering, transformation, and grain growth of two-dimensional binary colloidal crystals: A phase field crystal modeling

The formation and dynamics of a wide variety of binary two-dimensional ordered structures and superlattices are investigated through a phase field crystal model with sublattice ordering. Various types of binary ordered phases, the phase diagrams, and the grain growth dynamics and structural transformation processes, including the emergence of topological defects, are examined. The results are compared to the ordering and assembly of two-component colloidal systems. Two factors governing the binary phase ordering are identified, namely the coupling and competition between the length scales of two sublattices, and the selection of average particle densities of two components. The control and variation of these two factors lead to the prediction of various complex binary ordered patterns, with different types of sublattice ordering for integer versus noninteger ratios of sublattice length scales. These findings will enable further systematic studies of complex ordering and assembly processes of binary systems, particularly binary colloidal crystals.

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Phase ordering, transformation, and grain growth of two-dimensional binary colloidal crystals: A phase field crystal modeling

Phase ordering, transformation, and grain growth of two-dimensional binary colloidal crystals: A phase field crystal modeling