Snaking without subcriticality: grain boundaries as non-topological defects

Subramanian, Priya; Archer, Andrew J.; Knobloch, Edgar; Rucklidge, Alastair M.

doi:10.1093/imamat/hxab032

Cited by 6 publications

(4 citation statements)

References 36 publications

(41 reference statements)

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“…Although we do not observe any stable quasicrystals composed of particles with five-fold rotational symmetry, it seems that five-fold symmetry prominently occurs in defects along grain boundaries as shown in figure 11. The grain boundaries contain the well known pairs of dislocations with five and seven neighbors [38]. Yet the density peaks with five neighbors stand out, as the orientation field on the peaks is large, whereas it vanishes on the peaks with six or seven neighbors.…”

Section: Defects and Metastable Statesmentioning

confidence: 99%

Phase field crystal model for particles with n-fold rotational symmetry in two dimensions

Weigel

Schmiedeberg

2022

Modelling Simul. Mater. Sci. Eng.

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We introduce a Phase Field Crystal (PFC) model for particles with n-fold rotational symmetry in two dimensions. Our approach is based on a free energy functional that depends on the reduced one-particle density, the strength of the orientation, and the direction of the orientation, where all these order parameters depend on the position. The functional is constructed such that for particles with axial symmetry (i. e. n = 2) the PFC model for liquid crystals as introduced by H. Löwen [J. Phys.: Condens. Matter 22, 364105 (2010)] is recovered. We discuss the stability of the functional and explore phases that occur for 1 ≤ n ≤ 6. In addition to isotropic, nematic, stripe, and triangular order, we also observe cluster crystals with square, rhombic, honeycomb, and even quasicrystalline symmetry. The n-fold symmetry of the particles corresponds to the one that can be realized for colloids with symmetrically arranged patches. We explain how both, repulsive as well as attractive patches, are described in our model.

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Section: Defects and Metastable Statesmentioning

confidence: 99%

Phase field crystal model for particles with n-fold rotational symmetry in two dimensions

Weigel

Schmiedeberg

2022

Modelling Simul. Mater. Sci. Eng.

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“…There is experimental evidence that quasicrystals are associated with the presence of two length scales in the system , in a wide range of examples going beyond soft matter and materials science (for example, fluid dynamics − and nonlinear optics). The presence of different length scales is qualitatively clear in the structure of the components for several of the soft matter systems that form QCs, including micelles with a soft corona , and star copolymers with arms of different lengths. , The connection between having two length scales and the stability of QCs is supported by a large body of theoretical work, including from the fields of fluid dynamics and pattern formation, ,,− phase field crystals, − classical density functional theory of interacting particles, − molecular dynamics, , and self-assembly of hard particles , and hard particles with shoulder potentials. , At the most basic level, the theoretical work attributes the stability of QCs to the nonlinear three-wave interaction of waves of density fluctuations on the two length scales. For example, when the ratio of those length scales is 2 cos 15° ≈ 1.93, the nonlinear interactions between two waves of one length scale and one of the other favor density waves that are spaced 30° apart in Fourier space, − ,,, giving 12-fold symmetry.…”

Section: Introductionmentioning

confidence: 99%

Design of Linear Block Copolymers andABCStar Terpolymers That Produce Two Length Scales at Phase Separation

Joseph,

Read,

Rucklidge

2023

Macromolecules

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Quasicrystals (materials with long-range order but without the usual spatial periodicity of crystals) were discovered in several soft matter systems in the last 20 years. The stability of quasicrystals has been attributed to the presence of two prominent length scales in a specific ratio, which is 1.93 for the 12-fold quasicrystals most commonly found in soft matter. We propose design criteria for block copolymers such that quasicrystal-friendly length scales emerge at the point of phase separation from a melt, basing our calculations on the Random Phase Approximation. We consider two block copolymer families: linear chains containing two different monomer types in blocks of different lengths, and ABC star terpolymers. In all examples, we are able to identify parameter windows with the two length scales having a ratio of 1.93. The models that we consider that are simplest for polymer synthesis are, first, a monodisperse A L BA S B melt and, second, a model based on random reactions from a mixture of A L, A S, and B chains: both feature the length scale ratio of 1.93 and should be relatively easy to synthesize.

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“…These periodic patterns can be combined to form more complex patterns in two-dimensions with sharp interfaces at the transition between the patterns like so-called grain boundaries [18,25,31,39,52]. These are often seen in nature, for instance in Rayleigh-Benard convection [20,25], gannets nesting [41], graphene [24], and phyllotaxis [35].…”

Section: Introductionmentioning

confidence: 99%

Analysing transitions from a Turing instability to large periodic patterns in a reaction-diffusion system

Brown¹,

Derks²,

Heijster³

et al. 2023

Preprint

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Analytically tracking patterns emerging from a small amplitude Turing instability to large amplitude remains a challenge as no general theory exists. In this paper, we consider a three component reaction-diffusion system with one of its components singularly perturbed, this component is known as the fast variable. We develop an analytical theory describing the periodic patterns emerging from a Turing instability using geometric singular perturbation theory. We show analytically that after the initial Turing instability, spatially periodic patterns evolve into a small amplitude spike in the fast variable whose amplitude grows as one moves away from onset. This is followed by a secondary transition where the spike in the fast variable widens, its periodic pattern develops two sharp transitions between two flat states and the amplitudes of the other variables grow. The final type of transition we uncover analytically is where the flat states of the fast variable develop structure in the periodic pattern. The analysis is illustrated and motivated by a numerical investigation. We conclude with a preliminary numerical investigation where we uncover more complicated periodic patterns and snaking-like behaviour that are driven by the three transitions analysed in this paper. This paper provides a crucial step towards understanding how periodic patterns transition from a Turing instability to large amplitude.

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Snaking without subcriticality: grain boundaries as non-topological defects

Cited by 6 publications

References 36 publications

Phase field crystal model for particles with n-fold rotational symmetry in two dimensions

Phase field crystal model for particles with n-fold rotational symmetry in two dimensions

Design of Linear Block Copolymers andABCStar Terpolymers That Produce Two Length Scales at Phase Separation

Analysing transitions from a Turing instability to large periodic patterns in a reaction-diffusion system

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