Energy Localization and Transport in Binary Isotopically Disordered Fermi-Pasta-Ulam Chains

Abstract

Energy transport in binary isotopically disordered (BID) nonlinear Fermi-Pasta-Ulam (FPU) chains is a competition between localization and mode transitions. Starting from an arbitrary localized pulse, energy will dissipate ballistically until either Anderson localization (a disorder effect) or phonon scattering (a nonlinearity effect) slow the rate of dissipation. To reduce computational effort, we propose starting from a localized energy eigenstate so that in the absence of anharmonicity the energy is stationary and there is no transport. The second moment of the site energies is used to characterize an effective thermal conductivity as a function of impurity concentration and nonlinearity strength. Calculating the properties of harmonic BID chains at arbitrary impurity concentration is complicated by the pure-disordered-pure transition that occurs as the impurity concentration varies from zero to one. The localization length of dilute impurity harmonic BID chains is calculated exactly using scaling laws and the scattering cross section of a single impurity, which is calculated for discrete systems, differs from the continuum result. For arbitrary impurity concentration, the localization length is estimated by assuming independent contributions from the two limiting cases of pure material. Information entropy was used to show that the number of modes excited by phonon scattering decreased with increasing impurity concentration, a fact that consistent with density of states calculations. At all impurity concentrations, the second moment of the site energies increases linearly in time, a fact that is corroborated by the number of masses participating in energy transport, as calculated from the localization parameter. The dilute concentration dependence of the effective thermal conductivity was consistent with kinetic theory. At the highest concentrations the thermal conductivity was proportional to the original localization length because mode suppression and dense impurities meant that the same length scale remained dominant over a long period of time.