Ising model for phase separation in alloys with anisotropic elastic interaction—I. Theory

Elastic forces drive nonequilibrium pattern formation in a model of nanocrystal ion exchange

Chemical transformations, such as ion exchange, are commonly employed to modify nanocrystal compositions. Yet the mechanisms of these transformations, which often operate far from equilibrium and entail mixing diverse chemical species, remain poorly understood. Here we explore an idealized model for ion exchange in which a chemical potential drives compositional defects to accumulate at a crystal's surface. These impurities subsequently diffuse inward. We find that the nature of interactions between sites in a compositionally impure crystal strongly impacts exchange trajectories. In particular, elastic deformations which accompany lattice-mismatched species promote spatially modulated patterns in the composition. These same patterns can be produced at equilibrium in core/shell nanocrystals, whose structure mimics transient motifs observed in nonequilibrium trajectories. Moreover, the core of such nanocrystals undergoes a phase transition-from modulated to unstructured-as the thickness or stiffness of the shell is decreased. Our results help explain the varied patterns observed in heterostructured nanocrystals produced by ion exchange and suggest principles for the rational design of compositionally patterned nanomaterials.

Consequences of Lattice Mismatch for Phase Equilibrium in Heterostructured Solids

Lattice mismatch can substantially impact the spatial organization of heterogeneous materials. We examine a simple model for lattice-mismatched solids over a broad range of temperature and composition, revealing both uniform and spatially modulated phases. Scenarios for coexistence among them are unconventional due to the extensive mechanical cost of segregation. Together with an adapted Maxwell construction for elastic phase separation, mean field theory predicts a phase diagram that captures key low-temperature features of Monte Carlo simulations.

Modeling the elastic energy of alloys: Potential pitfalls of continuum treatments

Some issues that arise when modeling elastic energy for binary alloys are discussed within the context of a Keating model and density-functional calculations. The Keating model is a simplified atomistic formulation based on modeling elastic interactions of a binary alloy with harmonic springs whose equilibrium length is species dependent. It is demonstrated that the continuum limit for the strain field are the usual equations of linear elasticity for alloys and that they correctly capture the coarse-grained behavior of the displacement field. In addition, it is established that Euler-Lagrange equation of the continuum limit of the elastic energy will yield the same strain field equation. This is the same energy functional that is often used to model elastic effects in binary alloys. However, a direct calculation of the elastic energy atomistic model reveals that the continuum expression for the elastic energy is both qualitatively and quantitatively incorrect. This is because it does not take atomistic scale compositional nonuniformity into account. Importantly, this result also shows that finely mixed alloys tend to have more elastic energy than segregated systems, which is the exact opposite of predictions made by some continuum theories. It is also shown that for strained thin films the traditionally used effective misfit for alloys systematically underestimate the strain energy. In some models, this drawback is handled by including an elastic contribution to the enthalpy of mixing, which is characterized in terms of the continuum concentration. The direct calculation of the atomistic model reveals that this approach suffers serious difficulties. It is demonstrated that elastic contribution to the enthalpy of mixing is nonisotropic and scale dependent. It is also shown that such effects are present in density-functional theory calculations for the Si-Ge system. This work demonstrates that it is critical to include the microscopic arrangements in any elastic model to achieve even qualitatively correct behavior.

Interface properties and microstructural stabilization in elastically inhomogeneous multiphase systems

Using a two-dimensional multiscale model based on the classical time-dependent density-functional theory for lattice systems we recently introduced, we numerically study the impact of elastic inhomogeneities on the growth of isolated inclusions in multiphase alloys. We demonstrate that the coupling between the overall interface structure (as determined by the shape of the inclusions) and the local, atomic-scale structure can be very large, and is able to significantly affect the behavior of inclusions during growth. Elasticity is shown to have a strong influence on the local energetics at interfaces, leading to shape modulations and kinetic faceting. Morphological and energetic changes induce oscillations in the chemical potential of the inclusions, opening the way to the stabilization of multi-inclusion microstructures, consistent with experiment. We demonstrate that the interface elastic misfit is a key parameter for controlling the strength of elastic effects.

Multiscale model for microstructure evolution in multiphase materials: Application to the growth of isolated inclusions in presence of elasticity

We present a multiscale model based on the classical lattice time-dependent density-functional theory to study microstructure evolution in multiphase systems. As a first test of the method, we study the static and dynamic properties of isolated inclusions. Three cases are explored: elastically homogeneous systems, elastically inhomogeneous systems with soft inclusions, and elastically inhomogeneous systems with hard inclusions. The equilibrium properties of inclusions are shown to be consistent with previous results: both homogeneous and hard inclusions adopt a circular shape independent of their size, whereas soft inclusions are circular below a critical radius and elliptic above. In all cases, the Gibbs-Thomson relation is obeyed, except for a change in the prefactor at the critical radius in soft inclusions. Under growth conditions, homogeneous inclusions exhibit a Mullins-Sekerka shape instability [W. Mullins and R. Sekerka, J. Appl. Phys. 34, 323 (1963)], whereas in inhomogeneous systems, the growth of perturbations follows the Leo-Sekerka model [P. Leo and R. Sekerka, Acta Metall. 37, 3139 (1989)]. For soft inclusions, the mode instability regime is gradually replaced by a tip-growing mechanism, which leads to stable, strongly out-of-equilibrium shapes even at very low supersaturation. This mechanism is shown to significantly affect the growth dynamics of soft inclusions, whereas dynamical corrections to the growth rates are negligible in homogeneous and hard inclusions. Finally, due to its microscopic formulation, the model is shown to automatically take into account phenomena caused by the presence of the underlying discrete lattice: anisotropy of the interfacial energy, anisotropy of the kinetics, and preferential excitation of shape perturbations commensurate with the rotational symmetry of the lattice.

Kawasaki-type dynamics: diffusion in the kinetic Gaussian model

In this Brief Report, we retain the basic idea and at the same time generalize Kawasaki's dynamics, the spin-pair exchange mechanism, to a spin-pair redistribution mechanism, and present a normalized redistribution probability. This serves to unite various order-parameter-conserved processes into a universal framework in microscopics and provides the basis for further treatment. As an example of the applications, we treated the kinetic Gaussian model and obtained the exact diffusion equation. We observed critical slowing down near the critical point and found that the critical dynamic exponent z = 1/v = 2 is independent of space dimensionality and the assumed mechanism, whether Glauber type or Kawasaki type.

Suppression of the order parameter correlation length by inhomogeneous strains

Very recently we presented puzzling results of diffuse neutron scattering experiments on KSCN and RbSCN. The data yield an increase of the diffuse intensity with increasing temperature below T(c), whereas the width remains constant. Using molecular dynamics and 3D Monte Carlo simulations, we have shown that below T(c) the width of the correlation functions can be stabilized by strain fields originating from the order parameter strain interactions. Here we construct a novel analytic model which predicts the existence of a second characteristic length scale and explains the suppression of the growth of precursor clusters by the influence of inhomogeneous strain fields.