Nonlinear growth of periodic patterns

Abstract

We study the growth of a periodic pattern in one dimension for a model of spinodal decomposition, the Cahn-Hilliard equation. We particularly focus on the intermediate region, where the nonlinearity cannot be neglected anymore, and before the coalescence dominates. The dynamics is captured through the standard technique of a solubility condition performed over a particular family of quasistatic solutions. The main result is that the dynamics along this particular class of solutions can be expressed in terms of a simple ordinary differential equation. The density profile of the stationary regime found at the end of the nonlinear growth is also well characterized. Numerical simulations correspond satisfactorily to the analytical results through three different methods and asymptotic dynamics are well recovered, even far from the region where the approximations hold.

Thermodiffusion in binary liquids: the role of irreversibility

“…Much less is known, however, on thermal diffusion in molecular mixtures. In spite of the many experimental [16][17][18][19][20][21][22][23] and molecular dynamics data [24][25][26] on the composition, temperature, and mass dependencies, available models diverge on the drivinng mechanism [27][28][29][30][31][32][33][34]. Although the observed isotope effect [18] indicates irreversible behavior, there is at present no general agreement whether the Soret effect can be described by equilibrium thermodynamics, or on the contrary reflects the irreversible character of the underlying diffusion process.…”

1D Cahn–Hilliard dynamics: Ostwald ripening and application to modulated phase systems

“…describe very well both the linear growth and the saturation of the spinodal decomposition [17]. Thus, the dynamics is now reduced to the time evolution of the single free parameter:…”

“…Using a solubility condition, it is possible to transform this equation into a differential equation for k(t) and to compute the full non linear part of this dynamics (i.e. the saturation of the spinodal decomposition), which leads the system in a well defined stationary state [17].…”

Do thermal diffusion and Dufour coefficients satisfy Onsager’s reciprocity relation?

“…For mixtures such as CCl 4 -CBr 4 the B k have been expressed through the activation free energy [6,24]. The isotope effect observed upon deuteration in benzenecyclohexane mixtures, has been related to the molecular collision rates [25]. [30] (squares) and Blanco et al [31] (circles).…”