Frenkel-Kontorova model with nonconvex-interparticle interactions and strain gradients

We study the statics and dynamics of a chain of atoms moving in a periodic potential with non-linear, nonconvex-interparticle interactions, and with strain gradients which we model by including next-nearest-neighbor interactions through the discrete Hamiltonian $H={\Sigma }_n\frac{{\dot{u}}_n^2}{2}+\alpha {(u_{n+1\mathrm{}}-u_n)}^4-\beta {(u_{n+1\mathrm{}}-u_n)}^2+\frac{(\gamma }{2)}{(u_{n+1\mathrm{}}-2\mathrm{}{u}_n+u_{n-1\mathrm{}})}^2-{cosu}_n$. We obtain the phase diagram within an ansatz of periodically modulated configurations. These generalize the homogeneous (for $\beta <\frac{1}{8}$) and dimerized (for $\beta <\frac{1}{8}$) configurations reported previously for $\gamma =0\mathrm{}$, and are given by $u_n=n{a}_1+b_1\mathrm{for} n=1,.\mathrm{}.\mathrm{}.,N$ and $u_n=n{a}_2+b_2\mathrm{for} n=N+1,.\mathrm{}.\mathrm{}.,M$. The dynamics of transitions between different configurations when the parameters are varied is also investigated and we show that these are dominated by nucleation processes, which occur on short time scales compared with the subsequent slow growth. The possible relevance of the model to the dynamics of twin boundaries recently observed in the copper-oxide high-temperature superconductors is discussed.