Model of the kinetics of polymorphous crystallization

Quantitative phase field simulations of polycrystalline solidification using a vector order parameter

A vector order parameter phase field model derived from a grand potential functional is presented as an alternative approach for modeling polycrystalline solidification of alloys. In this approach, the grand potential density is designed to contain a discrete set of finite wells, a feature that naturally allows for the growth and controlled interaction of multiple grains using a single vector field. We verify that dendritic solidification in binary alloys follows the well-established quantitative behavior in the thin interface limit. In addition, it is shown that grain boundary energy and solute back-diffusion are quantitatively consistent with earlier theoretical work, with grain boundary energy being controlled through a simple solid-solid interaction parameter. Moreover, when considering polycrystalline aggregates and their coarsening, we show that the kinetics follow the expected parabolic growth law. Finally, we demonstrate how this vector order parameter model can be used to describe nucleation in polycrystalline systems via thermal fluctuations of the vector order parameter, a process that cannot be treated consistently with multiphase or multi-order-parameter based phase field models. The presented vector order parameter model serves as a practical and efficient computational tool for simulating polycrystalline materials. We also discuss the extension of the order parameter to higher dimensions as a simple method for modeling multiple solid phases.

Crystallization in organic semiconductor thin films: a diffuse-interface approach

The crystallization of organic semiconductor thin films from an amorphous phase often results in a broad range of microstructures and molecular arrangements that in turn critically impact the electronic properties of the film. Here we present a diffuse-interface model of thin film crystallization that accounts for out-of-plane tilting of the kinetically favored crystalline orientation as well as the simultaneous appearance of multiple polymorphs. By adjusting the relative thermodynamic stability of grains oriented with the fast-growing axis either parallel or perpendicular to the substrate, crystallization can be made to occur in the form of either commonly observed spherulites or more complex morphologies such as sectors and centers. Furthermore, tuning the relative kinetic coefficients and free energies of multiple polymorphs can result in a spherulite of one crystal structure embedded within a spherulite of another crystal structure. A parametric study of the effects of anisotropy, densification, time-varying treatments, and substrate patterning reveals a wide variety of morphologies that are possible in these thin films, driven by a combination of kinetic and thermodynamics effects.

Classical density functional theory and the phase-field crystal method using a rational function to describe the two-body direct correlation function

We introduce a new approach to represent a two-body direct correlation function (DCF) in order to alleviate the computational demand of classical density functional theory (CDFT) and enhance the predictive capability of the phase-field crystal (PFC) method. The approach utilizes a rational function fit (RFF) to approximate the two-body DCF in Fourier space. We use the RFF to show that short-wavelength contributions of the two-body DCF play an important role in determining the thermodynamic properties of materials. We further show that using the RFF to empirically parametrize the two-body DCF allows us to obtain the thermodynamic properties of solids and liquids that agree with the results of CDFT simulations with the full two-body DCF without incurring significant computational costs. In addition, the RFF can also be used to improve the representation of the two-body DCF in the PFC method. Last, the RFF allows for a real-space reformulation of the CDFT and PFC method, which enables descriptions of nonperiodic systems and the use of nonuniform and adaptive grids.

Coarsening in polycrystalline material using quaternions

We develop a phase field model to study the phenomenon of recrystallization and grain coarsening in polycrystalline material. A unique feature of our model is that it can time-evolve the actual orientation field of a material, expressed in terms of quaternions, a four-dimensional non-conserved vector field. The quaternions evolve in time following a Langevin dynamics. The free energy that drives the evolution contains bulk energy for various preferred grain types and anisotropic grain boundary energy. As a proof of principle for the new formalism we show that the average grain size (L) follows the usual L ∼t(1/2) scaling law when the grain boundary energy is independent of the misorientation angle between neighboring grains, whereas the scaling exponent is less (∼0.42) when the grain boundary energy follows the misorientation-dependent, phenomenological Read-Shockley formula.

Elastic deformation of polycrystals

We propose a framework to model elastic properties of polycrystals by coupling crystal orientational degrees of freedom with elastic strains. Our model encodes crystal symmetries and takes into account explicitly the strain compatibility induced long-range interaction between grains. The coupling of crystal orientation and elastic interactions allows for the rotation of individual grains by an external load. We apply the model to simulate uniaxial tensile loading of a 2D polycrystal within linear elasticity and a system with elastic anharmonicities that describe structural phase transformations. We investigate the constitutive response of the polycrystal and compare it to that of single crystals with crystallographic orientations that form the polycrystal.

Modeling elasticity in crystal growth

A new model of crystal growth is presented that describes the phenomena on atomic length and diffusive time scales. The former incorporates elastic and plastic deformation in a natural manner, and the latter enables access to time scales much larger than conventional atomic methods. The model is shown to be consistent with the predictions of Read and Shockley for grain boundary energy, and Matthews and Blakeslee for misfit dislocations in epitaxial growth.

Sharp interface limit of a phase-field model of crystal grains

We analyze a two-dimensional phase field model designed to describe the dynamics of crystalline grains. The phenomenological free energy is a functional of two order parameters. The first one reflects the orientational order, while the second reflects the predominantally local orientation of the crystal. We consider the gradient flow of this free energy. Solutions can be interpreted as ensembles of grains (in which the orientation is constant in space) separated by grain boundaries. We study the dynamics of the boundaries as well as the rotation of the grains. In the limit of an infinitely sharp interface, the normal velocity of the boundary is proportional to both its curvature and its energy. We obtain explicit formulas for the interfacial energy and mobility, and study their behavior in the limit of a small misorientation. We calculate the rate of rotation of a grain in the sharp interface limit, and find that it depends sensitively on the choice of the model.