Subsonic phase transition waves in bistable lattice models with small spinodal region

Phase transitions waves in atomic chains with double-well potential play a fundamental role in materials science, but very little is known about their mathematical properties. In particular, the only available results about waves with large amplitudes concern chains with piecewise-quadratic pair potential. In this paper we consider perturbations of a bi-quadratic potential and prove that the corresponding three-parameter family of waves persists as long as the perturbation is small and localised with respect to the strain variable. As a standard Lyapunov-Schmidt reduction cannot be used due to the presence of an essential spectrum, we characterise the perturbation of the wave as a fixed point of a nonlinear and nonlocal operator and show that this operator is contractive in a small ball in a suitable function space. Moreover, we derive a uniqueness result for phase transition waves with certain properties and discuss the kinetic relation.Comment: revised version with extended introduction, improved perturbation method, and novel uniqueness result; 20 pages, 5 figure

Phase Boundary Propagation in Mass Spring Chains and Long Molecules

Martensitic phase transitions in crystalline solids have been studied and utilized for many technological applications, including biomedical devices. These transitions typically proceed by the nucleation and propagation of interfaces, or phase boundaries. Over the last few decades, a continuum theory of phase transitions has emerged under the framework of thermoelasticity to study the propagation of these phase boundaries. It is now well-established that classical mechanical and thermodynamic principles are not sufficient to describe their motion within a continuum theory, and a kinetic relation must be supplied to complete the constitutive description. A few theoretical techniques that have been used to infer kinetic relations are phase-field models and viscosity-capillarity based methods, within both of which a phase boundary is sharp, but smeared over a short length. Here we use a different technique to infer a kinetic relation. We discrete a one-dimensional continuum into a chain of masses and springs with multi-well energy landscapes and numerically solve impact and Riemann problems in such systems. In our simulations we see propagating phase boundaries that satisfy all the jump conditions of continuum theories. By changing the boundary and initial conditions on the chains we can explore all possible phase boundary velocities and infer kinetic relations that when fed to the continuum theory give excellent agreement with our discrete mass-spring simulations. A physical system that shares many features with the mass-spring systems analyzed in this thesis is DNA in single molecule extension-rotation experiments. DNA is typically modeled as a one-dimensional continuum immersed in a heat bath. It is also known from fluorescence experiments that some of these transitions proceed by the motion of phase boundaries, just as in crystalline solids. Hence, we use a continuum theory to study these phase boundaries in DNA across which both the stretch and twist can jump. We show that experimental observations from many different labs on various DNA structural transitions can be quantitatively explained within our model

Shocks versus kinks in a discrete model of displacive phase transitions

We consider dynamics of phase boundaries in a bistable one-dimensional lattice with harmonic long-range interactions. Using Fourier transform and Wiener-Hopf technique, we construct traveling wave solutions that represent both subsonic phase boundaries (kinks) and intersonic ones (shocks). We derive the kinetic relation for kinks that provides a needed closure for the continuum theory. We show that the different structure of the roots of the dispersion relation in the case of shocks introduces an additional free parameter in these solutions, which thus do not require a kinetic relation on the macroscopic level. The case of ferromagnetic second-neighbor interactions is analyzed in detail. We show that the model parameters have a significant effect on the existence, structure and stability of the traveling waves, as well as their behavior near the sonic limit

Beyond Kinetic Relations

We introduce the concept of kinetic equations representing a natural extension of the more conventional notion of a kinetic relation. Algebraic kinetic relations, widely used to model dynamics of dislocations, cracks and phase boundaries, link the instantaneous value of the velocity of a defect with an instantaneous value of the driving force. The new approach generalizes kinetic relations by implying a relation between the velocity and the driving force which is nonlocal in time. To make this relations explicit one needs to integrate the system of kinetic equations. We illustrate the difference between kinetic relation and kinetic equations by working out in full detail a prototypical model of an overdamped defect in a one-dimensional discrete lattice. We show that the minimal nonlocal kinetic description containing now an internal time scale is furnished by a system of two ordinary differential equations coupling the spatial location of defect with another internal parameter that describes configuration of the core region.Comment: Revised version, 33 pages, 9 figure

Solitary waves for nonconvex FPU lattices

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Solitary waves for nonconvex FPU lattices

Solitary waves in a one-dimensional chain of atoms {q j} j∈Z are investigated. The potential energy is required to be monotone and grow super-quadratically. The existence of solitary waves with a prescribed asymptotic strain is shown under certain assumptions on the asymptotic strain and the wave speed. It is demonstrated the invariance of the equations allows one to transform a system with non-convex potential energy density to the situation under consideration