Coalescence in the 1D Cahn–Hilliard model

Abstract

We present an approximate analytical solution of the Cahn-Hilliard equation describing the coalescence during a first order phase transition. We have identified all the intermediate profiles, stationary solutions of the noiseless Cahn-Hilliard equation. Using properties of the soliton lattices, periodic solutions of the Ginzburg-Landau equation, we have construct a family of ansatz describing continuously the process of destabilization and period doubling predicted in Langer's self similar scenario [1].

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

“…Therefore we can describe the coalescence by a transformation at constant segregation parameter k, while the degree of freedom φ, associated with the relative phase between the two profiles, evolves in time from 0 to 1 (or −1) according to the C-H dynamics [22].…”

“…Even if this operator doesn't have simple (algebraic) exact eigenfunction of period 2λ C−H [21], Ψ L (x, k, φ), for φ = 0 and k = k s n+1 , happens nevertheless to be a good approximation for the eigenfunction of lowest eigenvalue [22]. Due to the concavity of F GL (φ) around φ = 0, (see Figure 2), this eigenvalue will be negative, triggering a linear destabilization of the pattern Ψ n and an exponential amplification of the perturbation, i.e.…”

Ballooning, bulging and necking: an exact solution for longitudinal phase separation in elastic systems near a critical point

“…In such a region, the interfaces are self-consistently diffuse, and expansion in the vicinity of the critical point leads to a simple and universal energy for near-critical 1D phase separation. The resultant energy is similar to near-critical van der Waals theory [57,65] as studied in the Cahn-Hillard equation [66][67][68], but with the addition of a linear term from the elastic tension; equivalently it is similar to the Ginzburg-Landau magnetic energy [69] but in the presence of an external field.…”

Well-posedness of the classical solution for the Kuramto–Sivashinsky equation with anisotropy effects

“…From a mathematical point of view, in [2], the existence of some extremely slowly evolving solutions for (1.5) is proven, considering a bounded domain, while, in [6,22], the problem of a global attractor is studied. Instead, in [27,65], numerical schemes for (1.5) are analyzed, while, in [60], an approximate analytical solution is studied.…”