Phase Separation Dynamics and External Force Field

Kitahara, Kakuo; Oono, Yoshitsugu; Jasnow, David

doi:10.1142/s0217984988000461

Cited by 60 publications

(36 citation statements)

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“…Physically, it is clear that an external field of this form accelerates the phase separation, so λ must be φ-dependent. Indeed, phenomenological derivations [2,9] of λ yield precisely the form λ ∝ (1 − φ 2 ) alluded to above. Furthermore, the coarsening dynamics of this model has been studied by computer simulations, both with [5,6,8] and without [3] external driving forces.…”

mentioning

confidence: 81%

“…This interest has a physical origin. It has been noticed [2] that when one models the coupling to an external driving field E, such as gravity, through an additional term(where here the field E is in the z-direction), this extra term does not change (1) unless λ depends on φ. This is because δF 1 /δφ = −Ez, and ∇ 2 z = 0.…”

mentioning

confidence: 99%

“…independent of the order parameter φ. Recently, however, there has been considerable interest [2][3][4][5][6][7][8] in cases where λ depends explicitly on φ, notably through the dependence λ(φ) = λ 0 (1 − φ 2 ). This interest has a physical origin.…”

mentioning

confidence: 99%

“…This interest has a physical origin. It has been noticed [2] that when one models the coupling to an external driving field E, such as gravity, through an additional term…”

mentioning

confidence: 99%

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Lifshitz-Slyozov scaling for late-stage coarsening with an order-parameter-dependent mobility

Bray

Emmott

1995

Phys. Rev. B

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The coarsening dynamics of the Cahn-Hilliard equation with order-parameter dependent mobility, λ(φ) ∝ (1 − φ 2 ) α , is addressed at zero temperature in the Lifshitz-Slyozov limit where the minority phase occupies a vanishingly small volume fraction. Despite the absence of bulk diffusion for α > 0, the mean domain size is found to grow as R ∝ t 1/(3+α) , due to subdiffusive transport of the order parameter through the majority phase. The domain-size distribution is determined explicitly for the physically relevant case α = 1. 05.70.Ln, 64.60.Cn, 05.70.Jk The phenomenon of spinodal decomposition in, e.g. binary alloys, is usually modelled by the Cahn-Hilliardfor the order-parameter field φ. Eq. (1) takes the form of a continuity equation, ∂ t φ = −∇ · j, with current j = −λ∇µ, where λ is a transport coefficient ('mobility') and the chemical potential µ is the functional derivative, µ = δF/δφ, of a Ginzburg-Landau free energy functionalgiven byHere V (φ) is the usual double-well potential whose minima (taken here to be at φ = ±1) represent the equilibrium phases.In conventional treatments of (1), the mobility λ is taken to be a constant, i.e. independent of the order parameter φ. Recently, however, there has been considerable interest [2][3][4][5][6][7][8] in cases where λ depends explicitly on φ, notably through the dependence λ(φ) = λ 0 (1 − φ 2 ). This interest has a physical origin. It has been noticed [2] that when one models the coupling to an external driving field E, such as gravity, through an additional term(where here the field E is in the z-direction), this extra term does not change (1) unless λ depends on φ. This is because δF 1 /δφ = −Ez, and ∇ 2 z = 0. Physically, it is clear that an external field of this form accelerates the phase separation, so λ must be φ-dependent. Indeed, phenomenological derivations [2,9] of λ yield precisely the form λ ∝ (1 − φ 2 ) alluded to above. Furthermore, the coarsening dynamics of this model has been studied by computer simulations, both with [5,6,8] and without [3] external driving forces. It is therefore interesting to study this problem in its own right, even without external driving forces.In this Communication, we study a general class of systems described by Eq. (1) with(we absorb the constant λ 0 into the timescale). To make analytical progress, we specialize to the case where the minority phase occupies a vanishingly small volume fraction. For the conventional case (α = 0), this is the limit treated by the seminal work of Lifshitz and Slyozov (LS) [10], and by Wagner [11], which leads to the result R ∝ t 1/3 for the mean domain size, and gives an exact expression for the domain-size distribution. For general α ≥ 0 we find R ∝ t 1/(3+α) . We also determine explicitly the domain-size distribution for the physically relevant case α = 1. (The other physically relevant case, α = 0, has been treated by LS.) For small volume fractions, coarsening proceeds by nucleation and growth rather than by spinodal decomposition. For present purposes we limit discussion to ...

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mentioning

confidence: 81%

mentioning

confidence: 99%

mentioning

confidence: 99%

“…This interest has a physical origin. It has been noticed [2] that when one models the coupling to an external driving field E, such as gravity, through an additional term…”

mentioning

confidence: 99%

See 2 more Smart Citations

Lifshitz-Slyozov scaling for late-stage coarsening with an order-parameter-dependent mobility

Bray

Emmott

1995

Phys. Rev. B

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“…In order to capture the dynamics of a system in the presence of an external driving field, an order-parameter-dependent diffusion coefficient (or 'mobility') is required [2,5,6]. The resulting modification of the Cahn-Hilliard equation has been studied by several authors both analytically and numerically [2,[5][6][7][8][9].…”

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confidence: 99%

Coarsening dynamics of a one-dimensional driven Cahn-Hilliard system

1996

Phys. Rev. E

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We study the one-dimensional Cahn-Hilliard equation with an additional driving term representing, say, the effect of gravity. We find that the driving field E has an asymmetric effect on the solution for a single stationary domain wall (or 'kink'), the direction of the field determining whether the analytic solutions found by Leung [J. Stat. Phys. 61, 345 (1990)] are unique. The dynamics of a kink-antikink pair ('bubble') is then studied. The behaviour of a bubble is dependent on the relative sizes of a characteristic length scale E −1 , where E is the driving field, and the separation, L, of the interfaces. For EL ≫ 1 the velocities of the interfaces are negligible, while in the opposite limit a travelling-wave solution is found with a velocity v ∝ E/L. For this latter case (EL ≪ 1) a set of reduced equations, describing the evolution of the domain lengths, is obtained for a system with a large number of interfaces, and implies a characteristic length scale growing as (Et) 1/2 . Numerical results for the domain-size distribution and structure factor confirm this behaviour, and show that the system exhibits dynamical scaling from very early times.

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