Dynamical scaling behavior of the Swift-Hohenberg equation following a quench to the modulated state

Abstract

We study the kinetics of phase transitions in a Rayleigh-Benard system after onset of convection using 2D Swift-Hohenberg equation. An initially uniform state evolves to one whose ground state is spatially periodic. We confirmed previous results which showed that dynamical scaling occurs at medium quench (ǫ = 0.25) with scaling exponents 1/5 and 1/4 under zero noise and finite noise respectively. We find logarithmic scaling behavior for a deep quench (ǫ = 0.75) at zero noise. A simple method is devised to measure the proxy of domain wall length. We find that the energy and domain wall length exhibit scaling behavior with the same exponent. For ǫ = 0.25, the scaling exponents are 1/4 and 0.3 at zero and finite noise respectively.

Grain boundary pinning and glassy dynamics in stripe phases

“…Such a description, valid for distances much larger than the defect core, typically involves time dependent Ginzburg-Landau equations or their generalizations. A few cases have been studied extensively, including domain coarsening in O(N) models [3,4], in nematics [5][6][7][8], and in smectic phases as effectively encountered in models of Rayleigh-Bénard convection or lamellar phases of block copolymers [9][10][11][12][13][14][15]. In the case of modulated phases, the motion of a single defect has been widely studied within the well known amplitude equation formalism.…”

“…The results of Sections II and III provide a possible interpretation of conflicting results in the literature. Previous studies of this problem [10][11][12][13][14][15] addressed the existence of self-similarity during domain coarsening and attempted to quantify the time dependence of the linear scale of the coarsening structure. The statistical self-similarity hypothesis asserts that after a possible transient, consecutive configurations of the coarsening structure are geometrically similar in a statistical sense.…”

Hexagonal phase ordering in strongly segregated copolymer films

“…In this case, a wide variety of scaling exponents may be observed [21] which is dependent on a parameter describing the depth of the quench which leads to crystallization. Dynamic scaling of pattern domains has also been studied in the context of convection patterns [24][25][26][27]. Scaling exponents in these studies range from 1/5 to 1/3, depending on how far the system was from onset of instability and whether noise was included.…”

Defect formation in the Swift-Hohenberg equation

“…A canonical measure for this length is the width of the structure factor [7,8,9,10,11,12]: in a defect-free domain the order parameter field takes the form φ(x, t) ≃ A(x, t) cos(q(x, t) · x)+higher harmonics (7) where A(x, t) and q(x, t) vary slowly within the domain. Thus, in the absence of any defects one expects the structure factor S(q, t) = |φ q (t)| 2 of an infinite system to be a delta function centered around q = q 0 .…”

“…Similar to other models, in which this coarsening process is self-similar [6], any linear scale of the structure is expected to grow as a power law in time ξ ∼ t 1/z . However, simulations of sudden quenches of the SH equation [7,8,9,10,11,12] have revealed that the observed exponent z is sensitive to non-universal model features such as the quench depth or noise strength. Moreover, different definitions of the length scale have led to different exponents with values reported in the interval 2 ≤ z ≤ 5.…”