Effect of noise on ordering of hexagonal grains in a phase-field-crystal model

Ohnogi, Hiroshi; Shiwa, Yuh

doi:10.1103/physreve.84.051603

Cited by 12 publications

(17 citation statements)

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“…When grain boundaries (GBs) between domains of different crystal orientation are mobile, those patterns generally coarsen in time to reduce GB length or area by elimination of smaller grains. This coarsening behavior has been extensively studied because of its practical importance for engineering polycrystalline materials [7] and its fundamental relevance for our general understanding of nonequilibrium ordering phenomena.The ordering dynamics of modulated phases and NE patterns has been extensively studied theoretically [8][9][10][11][12][13][14][15][16] in the framework of model equations of the formwhere ψ is an order parameter appropriate to each system that can be globally conserved (n = 1) or non-conserved (n = 0), η is a noise uncorrelated in space and time with a variance determined by the fluctuation-dissipation re-, and F is a Lyapounov functional with a minimum in a lattice ordered state. Eq.(1) has also been proposed as a theoretical framework− the phase-field-crystal (PFC) model− to study polycrystalline materials on diffusive time scales with ψ representing the crystal density field [17,18].…”

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Unified Theoretical Framework for Polycrystalline Pattern Evolution

Adland

2013

Phys. Rev. Lett.

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The rate of curvature-driven grain growth in polycrystalline materials is well-known to be limited by interface dissipation. We show analytically and by simulations that, for systems forming modulated phases or non-equilibrium patterns with crystal ordering, growth is limited by bulk dissipation associated with lattice translation, which dramatically slows down grain coarsening. We also show that bulk dissipation is reduced by thermal noise so that those systems exhibit faster coarsening behavior dominated by interface dissipation for high Peierls barrier and high noise. Those results provide a unified theoretical framework for understanding and modeling polycrystalline pattern evolution in diverse systems over a broad range of length and time scales.PACS numbers: 61.72. Mm, 05.40.Ca, 61.72.Hh, 62.20.Hg Polycrystalline patterns are observed in very diverse systems including crystalline solids [1], colloidal systems [2,3], various spatially modulated phases of macromolecular systems such as diblock copolymers [4,5], and nonequilibrium (NE) dissipative structures [6]. When grain boundaries (GBs) between domains of different crystal orientation are mobile, those patterns generally coarsen in time to reduce GB length or area by elimination of smaller grains. This coarsening behavior has been extensively studied because of its practical importance for engineering polycrystalline materials [7] and its fundamental relevance for our general understanding of nonequilibrium ordering phenomena.The ordering dynamics of modulated phases and NE patterns has been extensively studied theoretically [8][9][10][11][12][13][14][15][16] in the framework of model equations of the formwhere ψ is an order parameter appropriate to each system that can be globally conserved (n = 1) or non-conserved (n = 0), η is a noise uncorrelated in space and time with a variance determined by the fluctuation-dissipation relation η( r, t)η( r , t) = 2αT (−∇ 2 ) n δ( r − r )δ(t − t ), and F is a Lyapounov functional with a minimum in a lattice ordered state. Eq. (1) has also been proposed as a theoretical framework− the phase-field-crystal (PFC) model− to study polycrystalline materials on diffusive time scales with ψ representing the crystal density field [17,18]. While Eq. (1) has been traditionally studied for purely relaxational (p = 0) dynamics [8][9][10][11][12][13][14][15][16][17], propagative (p = 1) wave-like dynamics has also been introduced in the PFC framework to mimic phonon-mediated relaxation of the strain field [18].Extensive computational studies of Eq. (1) have shown that the characteristic domain or grain size for both roll patterns [9][10][11][12] and hexagonal lattices [13][14][15][16] grows ∼ t q . The exponent q is typically much smaller than the q = 1/2 value expected for "normal grain growth" in polycrystalline materials [19], and depends on parameters and noise strength [15,16]. While there have been theoretical attempts to explain those exponents for roll patterns [9][10][11][12], the origin of this sluggish (low q) coarse...

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“…The ordering dynamics of modulated phases and NE patterns has been extensively studied theoretically [8][9][10][11][12][13][14][15][16] in the framework of model equations of the form…”

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confidence: 99%

Unified Theoretical Framework for Polycrystalline Pattern Evolution

Adland

2013

Phys. Rev. Lett.

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“…The grain size of the hexagonal lattice growths in time (t) scaling as t  , where  is the growth exponent. [11,12] The ordering process is relevant both fundamentally to understand the far from equilibrium coarsening phenomena and practically because the order affects the functional properties of the systems.…”

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confidence: 99%

Ordering kinetics in two-dimensional hexagonal pattern of cylinder-forming PS- b -PMMA block copolymer thin films: Dependence on the segregation strength

Seguini¹,

Zanenga

Laus

et al. 2018

Phys. Rev. Materials

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This paper reports the experimental determination of the growth exponents and activation enthalpies for the ordering process of standing cylinder-forming all-organic polystyrene-block-poly (methyl methacrylate) (PS-b-PMMA) block copolymer (BCP) thin films as a function of the BCP degree of polymerization (N). The maximum growth exponent of 1/3 is observed for the BCP with the lowest N at the border of the order disorder transition. Both the growth exponents and the activation enthalpies exponentially decrease with the BCP segregation strength (•N) following the same path of the diffusivity.

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“…If the symmetry φ → −φ is violated (due to an external field [117] or a cubic term in the free energy density [118]), a competition between stripes and hexagons takes place. As an example, let us mention ordering in patterns governed by a generalized Swift-Hohenberg equation…”

Section: Pattern Orderingmentioning

confidence: 99%

Coarsening versus pattern formation

Nepomnyashchy

2015

Comptes Rendus. Physique

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It is known that similar physical systems can reveal two quite different ways of behavior, either coarsening, which creates a uniform state or a large-scale structure, or formation of ordered or disordered patterns, which are never homogenized. We present a description of coarsening using simple basic models, the Allen-Cahn equation and the Cahn-Hilliard equation, and discuss the factors that may slow down and arrest the process of coarsening. Among them are pinning of domain walls on inhomogeneities, oscillatory tails of domain walls, nonlocal interactions, and others. Coarsening of pattern domains is also discussed.

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Effect of noise on ordering of hexagonal grains in a phase-field-crystal model

Cited by 12 publications

References 30 publications

Unified Theoretical Framework for Polycrystalline Pattern Evolution

Unified Theoretical Framework for Polycrystalline Pattern Evolution

Ordering kinetics in two-dimensional hexagonal pattern of cylinder-forming PS- b -PMMA block copolymer thin films: Dependence on the segregation strength

Coarsening versus pattern formation

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