Theory for Atomic Diffusion on Fixed and Deformable Crystal Lattices (original) (raw)

Transport and diffusion on crystalline surfaces under external forces

New Journal of Physics, 2005

We present a numerical study of classical particles obeying a Langevin equation and moving on a solid crystalline surface under an external force that may either be constant or modulated by periodic oscillations. We focus on the particle drift velocity and diffusion. The roles of friction and equilibrium thermal fluctuations are studied for two nonlinear dynamical regimes corresponding to low and to high but finite friction. We identify a number of resonances and antiresonances, and provide phenomenological interpretations of the observed behaviour.

Coupled diffusion and phase transition: Phase fields, constraints, and the Cahn–Hilliard equation

Meccanica, 2021

We develop a constrained theory for constituent migration in bodies with microstructure described by a scalar phase field. The distinguishing features of the theory stem from a systematic treatment and characterization of the reactions needed to maintain the internal constraint given by the coincidence of the mass fraction and the phase field. We also develop boundary conditions for situations in which the interface between the body and its environment is structureless and cannot support constituent transport. In addition to yielding a new derivation of the Cahn–Hilliard equation, the theory affords an interpretation of that equation as a limiting variant of an Allen–Cahn type diffusion system arising from the unconstrained theory obtained by considering the mass fraction and the phase field as independent quantities. We corroborate that interpretation with three-dimensional numerical simulations of a recently proposed benchmark problem.

Continuous displacement of “lattice” atoms

Physica A: Statistical Mechanics and its Applications, 1992

In the existing CVM (cluster variation method) formulations, atoms are placed on lattice points. A new formulation is proposed in which atoms can be displaced from a lattice point. The displaced position is written by a vector r, which varies continuously. This model is treated in the CVM framework by regarding an atom at r as a species r. The probability of finding an atom displaced at r in dr is written as f(r) dr, and the corresponding pair probability is written as g(r,, r2) dr, dr,. We formulate using the pair approximation of the CVM in the present paper. The interatomic potential is assumed given, for example as the Lennard-Jones form. The entropy is written in terms of f(r) and g(r,, r2) using the CVM formula. The special feature of the present formulation, which is different from the prevailing no-displacement cases of the CVM, is that rotational symmetry of the lattice is to be satisfied by the f(r) and g(r,, rz) functions. After the general equations are written in the continuum vector form and in the integral equation formulation, examples of a single-component system are solved by changing integrals into summations over finite intervals. Further we construct simulations of displacement patterns in such a way that the pattern satisfies the pair probability distribution which has been calculated as the output of the CVM analysis. The simulated pattern shows the wavy behavior of phonons.

Atomistic study of diffusion-mediated plasticity and creep using phase field crystal methods

Physical Review B, 2015

The nonequilibrium dynamics of diffusion-mediated plasticity and creep in materials subjected to constant load at high homologous temperatures is studied atomistically using Phase Field Crystal (PFC) methods. Creep stress and grain size exponents obtained for nanopolycrystalline systems, m 1.02 and p 1.98, respectively, closely match those expected for idealized diffusional Nabarro-Herring creep. These exponents are observed in the presence of significant stress-assisted diffusive grain boundary migration, indicating that Nabarro-Herring creep and stress-assisted boundary migration contribute in the same manner to the macroscopic constitutive relation. When plastic response is dislocation-mediated, power law stress exponents inferred from dislocation climb rates are found to increase monotonically from m 3, as expected for generic climb-mediated natural creep, to m 5.8 as the dislocation density ρ d is increased beyond typical experimental values. Stress exponents m 3 directly measured from simulations that include dislocation nucleation, climb, glide, and annihilation are attributed primarily to these large ρ d effects. Extrapolation to lower ρ d suggests that m 4 − 4.5 should be obtained from our PFC description at typical experimental ρ d values, which is consistent with expectations for power law creep via mixed climb and glide. The anomalously large stress exponents observed in our atomistic simulations at large ρ d may nonetheless be relevant to systems in which comparable densities are obtained locally within heterogeneous defect domains such as dislocation cell walls or tangles.

Nucleation and phase propagation in a multistable lattice with weak nonlocal interactions

Continuum Mechanics and Thermodynamics, 2007

We study the overdamped gradient flow dynamics of a chain of massless points connected by bistable nearest-neighbor (NN) interactions and harmonic next-nearest-neighbor (NNN) interactions under quasistatic loads of assigned displacements. The model reproduces experimental observations on the phase transition of shape-memory wires with the possibility of different microstructure evolution strategies: internal or boundary nucleations and one or two coherently propagating phase fronts. The presence or absence of a stress peak is also obtained by considering nonlocal interaction effects with the loading device. Similar results are also obtained under the hypothesis of global energy minimization. The system also retains the described properties in the continuum limit. Some rate effects are numerically analyzed.

Dynamical theory of migration of an adsorbed atom on solid surfaces

The Journal of Chemical Physics, 1976

A dynamical theory of an adsorbed atom on solid surfaces is developed starting from a Hamiltonian which includes: (I) motion of the adsorbed atom parallel to the surface; (2) the vibration of the adsorbed atom perpendicular to the surface; (3) the lattice vibrations (phonons) of the solid; (4) the coupling between the vibration of the adsorbed atom and its parallel motion; (5) the coupling between the lattice vibrations and the parallel motion of the adsorbed atom; (6) the concerted interaction among the phonons, the vibration, and the parallel motion. Using a canonical transformation, we describe the motion of the adsorbed atom as one of a pseudomolecule formed from the adsorbed atom and the distorted lattice. An equation for the probability of finding the adsorbed atom at a lattice site is derived and the mean square displacement of the motion of the adsorbed atom is calculated. At low temperature, the migration of the adsorbed atom is coherent and the mean square displacement is proportional to the square of time. The adsorbed atom behaves like a free particle with a certain velocity. At high temperature, the migration has diffusional character and the mean square displacement is proportional to time. This is ~ue to the interaction between the parallel motion of the adsorbed atom and phonons as well as to the one between the vibration of the adsorbed atom and phonons. At intermediate temperature, after a certain time, which becomes shorter as the temperature increases, the migration becomes diffusional. The resulting diffusion coetfl.Cient consists of two parts: one represents the diffusional character of the migration, and the other its coherent nature. The temperature dependence of the diffusional part is of the Arrhenius type, while that of the coherent part increases as temperature decreases. The coherent part is small when the interaction between the adsorbed atom and the lattice atoms is very strong. Thus the theory gives reasonable trends with respect to the variation of parameters taken into account. Comparison with existing experiments on tungsten and rhodium is briefly made to estimate the order of magnitude of the lattice distortion due to the strong interaction between the adsorbed atom and the lattice atoms. Another criticism may be directed against the assumption of the diffusional character of the migration, Although it is to be expected that at long times the mean square displacement is proportional to the time and a proper diffusion coefficient is defined by Eq. (1. 1), re

Lattice continuum and diffusional creep

Diffusional creep is characterized by growth/ disappearance of lattice planes at the crystal boundaries that serve as sources/sinks of vacancies, and by diffusion of vacancies. The lattice continuum theory developed here represents a natural and intuitive framework for the analysis of diffusion in crystals and lattice growth/loss at the boundaries. The formulation includes the definition of the Lagrangian reference configuration for the newly created lattice, the transport theorem and the definition of the creep rate tensor for a polycrystal as a piecewise uniform, discontinuous field. The values associated with each crystalline grain are related to the normal diffusional flux at grain boundaries. The governing equations for Nabarro– Herring creep are derived with coupled diffusion and elasticity with compositional eigenstrain. Both, bulk diffusional dissipation and boundary dissipation accompanying vacancy nucleation and absorption, are considered, but the latter is found to be negligible. For periodic arrangements of grains, diffusion formally decouples from elasticity but at the cost of a complicated boundary condition. The equilibrium of deviatorically stressed polycrystals is impossible without inclusion of interface energies. The secondary creep rate estimates correspond to the standard Nabarro–Herring model, and the volumetric creep is small. The initial (primary) creep rate is estimated to be much larger than the secondary creep rate.