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

Villain-Guillot, Simon

doi:10.1016/j.physleta.2008.10.051

Cited by 5 publications

(7 citation statements)

References 28 publications

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“…A similar result, using a different approach, was found by Villain-Guillot in Ref. [9]. If λ GS is not dynamically relevant, what is the relevant final state?…”

Section: Dynamicssupporting

confidence: 83%

Dynamics versus energetics in phase separation

Politi

Torcini

2015

J. Stat. Mech.

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Phase separation may be driven by the minimization of a suitable free energy F . This is the case, e.g., for diblock copolimer melts, where F is minimized by a steady periodic pattern whose wavelength λ GS depends on the segregation strength α −1 and it is know since long time that in one spatial dimension λ GS ≃ α −1/3 . Here we study in details the dynamics of the system in 1D for different initial conditions and by varying α by five orders of magnitude. We find that, depending on the initial state, the final configuration may have a wavelength λ D with λ min (α) < λ D < λ max (α), where λ min ≈ ln(1/α) and λ max ≈ α −1/2 . In particular, if the initial state is homogeneous, the system exhibits a logarithmic coarsening process which arrests whenever λ D ≈ λ min .

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“…A similar result, using a different approach, was found by Villain-Guillot in Ref. [9]. If λ GS is not dynamically relevant, what is the relevant final state?…”

Section: Dynamicssupporting

confidence: 83%

Dynamics versus energetics in phase separation

Politi

Torcini

2015

J. Stat. Mech.

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“…Nevertheless, if one looks carefully, the domains are no longer homogeneous but present a small concavity. The interface profile is no longer monotonous, leading to a breakdown of Politi and Misbah analysis which would have predicted a scaling of λ in ln(1/ β) [25]. Moreover, if we impose boundary conditions (i.e.…”

Section: Discussionmentioning

confidence: 99%

1D Cahn–Hilliard equation for modulated phase systems

Villain-Guillot¹

2010

J. Phys. A: Math. Theor.

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Formation of modulated phase patterns can be modelized by a modi…ed Cahn-Hilliard equation which includes a non local term preventing the formation of macroscopic domains. Using stationnary solutions of the original Cahn-Hilliard equation as analytical ansatzs, we compute the thermodynamically stable period of a 1D modulated phase pattern. We …nd that the period scales like the power-1/3 of the strength of the long range interaction

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“…Let us start with the simplest and well known models: the GL model (4a) and its conserved version, the CH model (5a). They admit analytical solutions based on the Jacobi elliptic function Sn(x; p) [24], allowing us to test our numerical procedure with a controlled result. This family of solution is parametrized by the elliptic modulus p ∈ [0, 1] (for more details see Appendix B) and take the form…”

Section: Resultsmentioning

confidence: 99%

“…where the functions N n = e nω k ∆t N [u k (t − n∆t)] k are computed during the time stepping of the algorithm by successive multiplications of the factor exp(ω k ∆t). As before, the corrector step is applied to the variable z k (t + ∆t) and is computed from the value of the predictor and the value of u k at three previous time steps The stationary states of the Ginzburg-Landau equation can be expressed analytically by means of the Jacobi elliptic function sine-amplitude Sn(x; p) [24]. This family of solutions is parametrized by the elliptic modulus p ∈ [0, 1] and take the form…”

Section: Appendix A: Pseudospectral Algorithmmentioning

confidence: 99%

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Coarsening dynamics in one dimension: The phase diffusion equation and its numerical implementation

Nicoli

Misbah²,

Politi³

2013

Phys. Rev. E

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Many nonlinear partial differential equations (PDEs) display a coarsening dynamics, i.e., an emerging pattern whose typical length scale L increases with time. The so-called coarsening exponent n characterizes the time dependence of the scale of the pattern, L(t)≈t(n), and coarsening dynamics can be described by a diffusion equation for the phase of the pattern. By means of a multiscale analysis we are able to find the analytical expression of such diffusion equations. Here, we propose a recipe to implement numerically the determination of D(λ), the phase diffusion coefficient, as a function of the wavelength λ of the base steady state u(0)(x). D carries all information about coarsening dynamics and, through the relation |D(L)|=/~L(2)/t, it allows us to determine the coarsening exponent. The main conceptual message is that the coarsening exponent is determined without solving a time-dependent equation, but only by inspecting the periodic steady-state solutions. This provides a much faster strategy than a orward time-dependent calculation. We discuss our method for several different PDEs, both conserved and not conserved.

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

Cited by 5 publications

References 28 publications

Dynamics versus energetics in phase separation

Dynamics versus energetics in phase separation

1D Cahn–Hilliard equation for modulated phase systems

Coarsening dynamics in one dimension: The phase diffusion equation and its numerical implementation

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