K. R. Elder

Diffusive Atomistic Dynamics of Edge Dislocations in Two Dimensions

The fundamental dislocation processes of glide, climb, and annihilation are studied on diffusive time scales within the framework of a continuum field theory, the phase field crystal model. Glide and climb are examined for single edge dislocations subjected to shear and compressive strain, respectively, in a two-dimensional hexagonal lattice. It is shown that the natural features of these processes are reproduced without any explicit consideration of elasticity theory or ad hoc construction of microscopic Peierls potentials. Particular attention is paid to the Peierls barrier for dislocation glide or climb and the ensuing dynamic behavior as functions of strain rate, temperature, and dislocation density. It is shown that the dynamics are accurately described by simple viscous motion equations for an overdamped point mass, where the dislocation mobility is the only adjustable parameter. The critical distance for the annihilation of two edge dislocations as a function of separation angle is also presented.

Flame propagation in random media.

We introduce a phase-field model to describe the dynamics of a self-sustaining propagating combustion front within a medium of randomly distributed reactants. Numerical simulations of this model show that a flame front exists for reactant concentration {ital c}{gt}{ital c}{sup *}{gt}0, while its vanishing at {ital c}{sup *} is consistent with mean-field percolation theory. For {ital c}{gt}{ital c}{sup *}, we find that the interface associated with the diffuse combustion zone exhibits kinetic roughening characteristic of the Kardar-Parisi-Zhang equation.

Phase-field-crystal model for magnetocrystalline interactions in isotropic ferromagnetic solids.

An isotropic magnetoelastic phase-field-crystal model to study the relation between morphological structure and magnetic properties of pure ferromagnetic solids is introduced. Analytic calculations in two dimensions were used to determine the phase diagram and obtain the relationship between elastic strains and magnetization. Time-dependent numerical simulations in two dimensions were used to demonstrate the effect of grain boundaries on the formation of magnetic domains. It was shown that the grain boundaries act as nucleating sites for domains of reverse magnetization. Finally, we derive a relation for coercivity versus grain misorientation in the isotropic limit.

Influence of modulated structures on ordering dynamics in CuAu

Using a microscopic model, we have studied the evolution of microstructure in a model metallic alloy. The Hamiltonian was derived from the effective medium theory of cohesion in metals (EMT): an approximation scheme for integrating out the electronic degrees of freedom and constructing an effective classical Hamiltonian. The alloy chosen for this study was CuAu which exhibits a sequence of first-order phase transitions: disordered → modulated → ordered. To describe the dynamics of ordering, a free energy functional was constructed from the EMT Hamiltonian and used in a Langevin equation. This study demonstrates the feasibility of predicting microstructure in alloys starting from a description based on the electronic structure of alloys. The simulations exhibit interesting features in late-stage growth which are attributed to the presence of the modulated phase as a metastable phase in the ordered regime.

Novel mechanical properties in lamellar phases of liquid-crystalline diblock copolymers

Abstract.Structural properties of flexible nematic diblock copolymers in the lamellar phase are investigated using a mean-field model. We address two complementary questions on the mechanics of the system: 1) How does the nematic order affect the elasticity of the one-dimensional solid? 2) What effect does the block copolymer microstructure has on the orientation of the nematic director? In the limit when the microstructure does not influence the nematic director orientation we predict a soft lamellar compression mode. When the microstructure does influence the nematic director orientation, small compressions lead to conventional elasticity, until a critical strain is reached, where there is then a transition to a softer response. On the other hand, we show that an identifiable lamellar symmetry provides a direction along which the nematic director prefers to align. Our model provides avenues to explore nonlinear properties of flexible diblock copolymers in which the monomers on both sides have mesogenic side groups.

Comment on "Pipe Network Model for Scaling of Dynamic Interfaces in Porous Media"

We argue that a proposed exponent identity [Phys. Rev. Lett 85, 1238 (2000)] for interface roughening in spontaneous imbibition is wrong. It rests on the assumption that the fluctuations are controlled by a single time scale, but liquid conservation imposes two distinct time scales.

Modelling Pattern Formation on Primate Visual Cortex

We begin this chapter by giving an overview of the types of theoretical model used to describe pattern formation related to ocular dominance on the visual cortex. We then present a new neural network model for the formation of ocular dominance stripes on primate visual cortex and examine the generic phase behavior and dynamics of the model. The dynamical equation of ocular dominance development can be identified with a class of Langevin equations with a non-conserved order parameter. We find that the phase diagram for our model comprises three phases: a striped phase, a hexagonal “bubble” phase and a uniform paramagnetic phase. We then examine the dynamics of the striped phase by solving the Langevin equation both numerically and by singular perturbation theory. Finally, we compare the results of the model with physiological data. The typical striped structure of the ocular dominance columns corresponds to the zero-field configurations of the model. We also use our model to simulate mononuclear deprivation.