In this paper the relationship between the classical density functional theory of freezing and phase-field modeling is examined. More specifically a connection is made between the correlation functions that enter density functional theory and the free energy functionals used in phase-field crystal modeling and standard models of binary alloys ͑i.e., regular solution model͒. To demonstrate the properties of the phase-field crystal formalism a simple model of binary alloy crystallization is derived and shown to simultaneously model solidification, phase segregation, grain growth, elastic and plastic deformations in anisotropic systems with multiple crystal orientations on diffusive time scales.

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We report on a novel extension of the recent phase-field crystal (PFC) method introduced in [Elder et al., Phys. Rev. Lett., 88, 245701:1-4 (2002)], which incorporates elastic interactions as well as crystal plasticity and diffusive dynamics. In our model, elastic interactions are mediated through wave modes that propagate on time scales many orders of magnitude slower than atomic vibrations but still much faster than diffusive times scales. This allows us to preserve the quintessential advantage of the PFC model: the ability to simulate atomic-scale interactions and dynamics on time scales many orders of magnitude longer than characteristic vibrational time scales. We demonstrate the two different modes of propagation in our model and show that simulations of grain growth and elasto-plastic deformation are consistent with the microstructural properties of nanocrystals.PACS numbers: 61.82.Rx, 62.25.+g, 62.30.+d, 63.22.+m The deformation of a solid triggers processes which operate across several length and time scales. On long length and time scales its behavior can be described by a set of hydrodynamic equations [1, 2], which describe, e.g., elastic deformation dynamics of the body. On atomic length (∼ 10 −10 m) and time (∼ 10 −13 s) scales, on the other hand, the dynamics can be captured by direct molecular dynamics (MD) simulations, which incorporate local bonding information either through direct quantum-mechanical calculations or semi-empirical many-body potentials. While innovations in computing methods have greatly improved the efficiency of MD simulations, standard atomistic computer simulations are still limited to fairly small system sizes (∼ 10 9 atoms) and short times (∼ 10 −8 s). This limitation is most severe when developing simulation models to study the physics and mechanics of nanostructured materials, where the relevant length scales are atomic and time scales are mesoscopic. In this regime, the available numerical tools are rare.Progress towards alleviating this limitation has recently been made by the introduction of a new modeling paradigm known as the phase-field crystal (PFC) method [3]. This method introduces a local atomic mass density field ρ(r) in which atomic vibrations have been integrated out up to diffusive time scales. Dissipative dynamics are then constructed to govern the temporal evolution of ρ. Unfortunately, the original PFC model evolves mass density only on diffusive time scales. In particular, it does not contain a mechanism for simulating elastic interactions, an important aspect for studying, for example, the deformation properties of nanocrystalline solids.In this Letter, we introduce a modified phase-field crystal (MPFC) model that includes both diffusive dynamics and elastic interactions. This is achieved by exploiting the separation of time scales that exists between diffusive and elastic relaxation processes in solids. In particular, the MPFC model is constructed to transmit long wavelength density fluctuations with wave modes that propagate up to a time scale t...

We study dendritic microstructure evolution using an adaptive grid, finite element method applied to a phase-field model. The computational complexity of our algorithm, per unit time, scales linearly with system size, rather than the quadratic variation given by standard uniform mesh schemes. Time-dependent calculations in two dimensions are in good agreement with the predictions of solvability theory, and can be extended to three dimensions and small undercoolings. 05.70.Ln, 81.30.Fb, 64.70.Dv, 81.10.Aj Dendrites are the primary component of solidification microstructures in metals. The formation, shape, speed and size of dendritic microstructures has been a topic of intense study in the past 10-15 years. Experiments [1,2] by Glicksman and coworkers on succinonitrile (SCN) and other transparent analogues of metals have been accurate enough to provide tests of theories of dendritic growth, and have stimulated considerable theoretical progress [3][4][5]. The experiments have clearly demonstrated that naturally growing dendrites possess a unique steady state tip, characterized by its velocity, radius of curvature and shape, which leads the to time-dependent sidebranched dendrite as it propagates.The earliest theories of dendritic growth solved for the diffusion field around a self-similar body of revolution propagating at constant speed [6,7]. In these studies the diffusion field determines the product of the dendrite velocity and tip radius, but neither quantity by itself. Adding capillarity effects to the theory predicts a unique maximum growth speed, [8] but experiments showed that this point does not represent the operating state for real dendrites.The goal of contemporary research has been to predict steady state features of dendritic growth and to compute time-dependent microstructures from numerical solutions of the equations of motion. The purpose of this letter is to present a computationally efficient method for time-dependent calculations, and to verify that the steady state properties are in excellent agreement with those predicted by analysis of the steady state problem.Insight into the steady state dendrite problem was first obtained from local models [3,4,[9][10][11][12][13][14][15][16][17][18] describing the evolution of the interface, and incorporating the features of the bulk phases into the governing equation of motion for the interface. These models were the first [10] to show that a nonzero dendrite velocity is obtained only if a source of anisotropy -for example, anisotropic interface energy -is present in the description of dendritic evolution. Subsequently, it was shown that the spectrum of allowed steady state velocities is discrete, rather than continuous, and the role of anisotropy was understood theoretically, both in the local models and the full moving boundary problem [4,5,14,19]. Moreover, only the fastest of a spectrum of steady state velocities is stable, thus forming the operating state of the dendrite. It is widely believed that sidebranching is generated by thermal or other ...

Amplitude representations of a binary phase field crystal model are developed for a two dimensional triangular lattice and three dimensional BCC and FCC crystal structures. The relationship between these amplitude equations and the standard phase field models for binary alloy solidification with elasticity are derived, providing an explicit connection between phase field crystal and phase field models. Sample simulations of solute migration at grain boundaries, eutectic solidification and quantum dot formation on nano-membranes are also presented.

The phase field crystal (PFC) method has emerged as a promising technique for modeling materials with atomistic resolution on mesoscopic time scales. The approach is numerically much more efficient than classical density functional theory (CDFT), but its single mode free energy functional only leads to lattices with triangular (2D) or BCC (3D) symmetries. By returning to a closer approximation of the CDFT free energy functional, we develop a systematic construction of two-particle direct correlation functions that allow the study of a broad class of crystalline structures. This construction examines planar spacings, lattice symmetries, planar atomic densities and the atomic vibrational amplitude in the unit cell of the lattice and also provides control parameters for temperature and anisotropic surface energies. Solid-state transformations form the basis of many problems in materials science and physical metallurgy [1]. They involve complex structural changes between parent and daughter phases and couple atomic-scale elastic and plastic effects with diffusional processes. These phenomena are presently impossible to compute at experimentally relevant time scales using molecular dynamics simulations. On the other hand, meso-scale continuum models wash out most of the relevant atomic scale physics that leads to elasticity, plasticity, defect interactions and grain boundary nucleation and migration. Phase field studies of precipitate and ledge growth [2-4] must thus re-introduce these effects phenomenologically. There is presently no continuum method to efficiently simulate solid-state transformations on diffusional time scales that self-consistently computes elastic and plastic effects at the atomic scale.Classical density function theory (CDFT) provides a formalism that can accurately describe the emergence of crystalline order from a liquid or solid phase through a coarsegrained density field [5]. Unfortunately, this approach requires very high spatial resolution and is highly inefficient for dynamical calculations [6]. A recent model, coined the phase field crystal (PFC) model, has been gaining widespread recognition as a hybrid method between CDFT and traditional phase field methods. PFC models capture most of the essential physics of CDFT without having to resolve atomically sharp atomic density peaks [7][8][9][10][11]. In spite of their successes, however, only PFC free energy functionals minimized by triangular (2D) or BCC (3D) lattices have been seriously studied. While there have been attempts to extend the PFC model to describe other crystal symmetries such as square [12,13] and FCC [6,14,15] lattices, these have been somewhat ad-hod and not self-consistently connected to material properties. PFC modeling is presently lacking a generalized free energy formulation that allows the study of important phase transformations between different crystalline states.This Letter proposes a new CDFT/PFC model based on two-point direct correlation function that is systematically constructed to be minimized by arbitrary...

We introduce and characterize free-energy functionals for modeling of solids with different crystallographic symmetries within the phase-field-crystal methodology. The excess free energy responsible for the emergence of periodic phases is inspired by classical density-functional theory, but uses only a minimal description for the modes of the direct correlation function to preserve computational efficiency. We provide a detailed prescription for controlling the crystal structure and introduce parameters for changing temperature and surface energies, so that phase transformations between body-centered-cubic (bcc), face-centered-cubic (fcc), hexagonal-close-packed (hcp), and simple-cubic (sc) lattices can be studied. To illustrate the versatility of our free-energy functional, we compute the phase diagram for fcc-bcc-liquid coexistence in the temperature-density plane. We also demonstrate that our model can be extended to include hcp symmetry by dynamically simulating hcp-liquid coexistence from a seeded crystal nucleus. We further quantify the dependence of the elastic constants on the model control parameters in two and three dimensions, showing how the degree of elastic anisotropy can be tuned from the shape of the direct correlation functions.

The use of continuum phase-field models to describe the motion of well-defined interfaces is discussed for a class of phenomena that includes order-disorder transitions, spinodal decomposition and Ostwald ripening, dendritic growth, and the solidification of eutectic alloys. The projection operator method is used to extract the "sharp-interface limit" from phase-field models which have interfaces that are diffuse on a length scale xi. In particular, phase-field equations are mapped onto sharp-interface equations in the limits xi(kappa)<<1 and xi(v)/D<<1, where kappa and v are, respectively, the interface curvature and velocity and D is the diffusion constant in the bulk. The calculations provide one general set of sharp-interface equations that incorporate the Gibbs-Thomson condition, the Allen-Cahn equation, and the Kardar-Parisi-Zhang equation.