Pattern dynamics of Rayleigh-Bénard convective rolls and weakly segregated diblock copolymers

Christensen, Jacob Højgaard; Bray, A. J.

doi:10.1103/physreve.58.5364

Cited by 63 publications

(96 citation statements)

References 13 publications

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“…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.…”

Section: Introductionmentioning

confidence: 99%

“…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.…”

Section: Introductionmentioning

confidence: 99%

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Grain boundary pinning and glassy dynamics in stripe phases

2002

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We study numerically and analytically the coarsening of stripe phases in two spatial dimensions, and show that transient configurations do not achieve long ranged orientational order but rather evolve into glassy configurations with very slow dynamics. In the absence of thermal fluctuations, defects such as grain boundaries become pinned in an effective periodic potential that is induced by the underlying periodicity of the stripe pattern itself. Pinning arises without quenched disorder from the non-adiabatic coupling between the slowly varying envelope of the order parameter around a defect, and its fast variation over the stripe wavelength. The characteristic size of ordered domains asymptotes to a finite value Rg ∼ λ0 ǫ −1/2 exp(|a|/ √ ǫ), where ǫ ≪ 1 is the dimensionless distance away from threshold, λ0 the stripe wavelength, and a a constant of order unity. Random fluctuations allow defect motion to resume until a new characteristic scale is reached, function of the intensity of the fluctuations. We finally discuss the relationship between defect pinning and the coarsening laws obtained in the intermediate time regime.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Grain boundary pinning and glassy dynamics in stripe phases

2002

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“…This force is given by F = qٌ͓A ϫ v͔, where the three variables q, A, and v denote the charge͑s͒ in the PZT precursor, the magnetic vector potential due to the applied field, and the thermodynamic diffusion velocity of the polymer. This driving force can be included with the Cahn-Hilliard equation describing the evolution of the lamellar order 25,26 the Lorentz term that drives the curvature. The linearized equation that can model the initial evolution of the observed 2D onion structure is then…”

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confidence: 99%

Self-organized two-dimensional onions

Ren

Briber

Wuttig

2009

Applied Physics Letters

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Spontaneously self-assembled onion-type nanostructures based on block copolymers as templating materials are reported. Polystyrene-poly͑ethylene oxide͒ diblock copolymer containing CoFe 2 O 4 and Pb 1.1 ͑Zr 0.53 Ti 0.47 ͒O 3 precursors segregated to the two microdomains forms well-ordered templated lamellar structures. Onion-type nanostructures have been induced by room temperature solvent annealing for 64 h in a magnetic field of 0.8 T oriented perpendicularly to the plane of film. The recorded images suggest that the Lorentz force acting on charges in the paraelectric precursor induces a circular component of the diffusion flux that leads to the onion formation. This templating process opens a route for nanometer-scale patterning of magnetic toroids. © 2009 American Institute of Physics. ͓DOI: 10.1063/1.3101373͔ Block copolymers have attracted much attention as prominent systems for nanotechnological applications because of their ability to spontaneously develop patterns on a nanoscale based on self-assembly. [1][2][3][4] Currently, a great deal of attention is being paid to the synthesis of complex nanostructured inorganic-organic hybrid materials. 5-10 A typical approach is the use of organic structures formed through self-assembly as structure directing agents. The final morphology is then determined by the cooperative organization of inorganic and organic molecular species into threedimensionally structured arrays. 11 In thin films, in addition to composition and molecular weight, the domain structure is also dependent on the surface energies of the blocks and on geometrical constraints introduced by the film. A recent example of inorganic-organic hybrid templating is the fabrication of ultrahigh density ferromagnetic nanocylinders that are embedded in a piezoelectric matrix by utilizing a precursorcontaining polystyrene-block-polyethylene oxide block copolymer. 12 The morphology and orientations of microdomains are the key factors controlling the properties of the patterned materials. The limited control over order and orientation of the microdomain in thin films poses restrictions on possible usages of patterning innovative material configurations via block copolymers. 13 Recent work demonstrates the alignment of block copolymers by electromagnetic fields. 14,15 Electric field induced alignment has been shown to be especially useful in thin film systems in which the two blocks have different dielectric constants. It does, however, require special sample geometries and surface preparation in the form of electrodes and patterned surfaces. Ultimately, dielectric breakdown limits the applicability of electric fields to alter block copolymeric patterns. [16][17][18] Magnetic fields should in principle also lead to spatial reorientations of copolymer blocks. They are attractive as no limitations to the field strength and sample configuration exist. Magnetic field induced alignment was reported in 1998; 19 a 2.4 T magnetic field was imposed on a diblock copolymer comprising both block chains of liquidcrystallin...

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“…These defects are reminiscent of the perforated lamellar phase described in reference [1]. One way to study this is to use a directional order parameter measure introduced by Bray et al [21] to study the 2D problem. This order parameter is similar to the order parameter for complex fluids and provides directional as well as density information.…”

Section: Future Workmentioning

confidence: 99%

Kinetics of ordering in fluctuation-driven first-order transitions: Simulation and theory

2000

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Many systems where interactions compete with each other or with constraints are well described by a model first introduced by Brazovskii. Such systems include block copolymers, alloys with modulated phases, Rayleigh-Benard Cells and type-I superconductors. The hallmark of this model is that the fluctuation spectrum is isotropic and has a minimum at a nonzero wave vector represented by the surface of a d-dimensional hyper-sphere. It was shown by Brazovskii that the fluctuations change the free energy structure from a φ 4 to a φ 6 form with the disordered state metastable for all quench depths.The transition from the disordered to the periodic, lamellar structure changes from second order to first order and suggests that the dynamics is governed by nucleation. Using numerical simulations we have confirmed that the equilibrium free energy function is indeed of a φ 6 form. A study of the dynamics, however, shows that, following a deep quench, the dynamics is described by unstable growth rather than nucleation. A dynamical calculation, based on a generalization of the Brazovskii calculations shows that the disordered state can remain unstable for a long time following the quench.

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Pattern dynamics of Rayleigh-Bénard convective rolls and weakly segregated diblock copolymers

Cited by 63 publications

References 13 publications

Grain boundary pinning and glassy dynamics in stripe phases

Grain boundary pinning and glassy dynamics in stripe phases

Self-organized two-dimensional onions

Kinetics of ordering in fluctuation-driven first-order transitions: Simulation and theory

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