Design of Parametrically Forced Patterns and Quasipatterns

Rucklidge, Alastair M.; Silber, Mary

doi:10.1137/080719066

Cited by 40 publications

(78 citation statements)

References 71 publications

(210 reference statements)

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“…Second, strong reinforcing (i.e., resonant) nonlinear interactions between these two characteristic density waves are required [17,19,20]. Earlier work on quasipatterns observed in Faraday wave experiments reveals similar requirements [19,[21][22][23]. We demonstrate here, following Mermin and Troian [24], that these same requirements suffice to stabilize icosahedral QCs in 3D.…”

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confidence: 51%

Three-Dimensional Icosahedral Phase Field Quasicrystal

et al. 2016

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We investigate the formation and stability of icosahedral quasicrystalline structures using a dynamic phase field crystal model. Nonlinear interactions between density waves at two length scales stabilize threedimensional quasicrystals. We determine the phase diagram and parameter values required for the quasicrystal to be the global minimum free energy state. We demonstrate that traits that promote the formation of two-dimensional quasicrystals are extant in three dimensions, and highlight the characteristics required for three-dimensional soft matter quasicrystal formation. DOI: 10.1103/PhysRevLett.117.075501 Periodic crystals form ordered arrangements of atoms or molecules with rotation and translation symmetries, and possess discrete x-ray diffraction patterns, or equivalently, discrete spatial Fourier spectra. In contrast, quasicrystals (QCs) lack the translational symmetries of periodic crystals, yet also display discrete spatial Fourier spectra. QCs made from metal alloys were discovered in 1982 [1] and attracted the Nobel prize for chemistry in 2011. QCs can be quasiperiodic in all three dimensions (e.g., with icosahedral symmetry), or can be quasiperiodic in two (or one) directions while being periodic in one (or two). The vast majority of the QCs discovered so far are metallic alloys (e.g., Al/Mn or Cd/Ca). However, QCs have recently been found in nanoparticles [2], mesoporous silica [3], and soft matter [4] systems. The latter include micellar melts [5,6] formed, e.g., from linear, dendrimer or star block copolymers. Recently, three-dimensional (3D) icosahedral QCs have been found in molecular dynamics simulations of particles interacting via a three-well pair potential [7].In recent years, model systems in two dimensions (2D) have been studied in order to understand soft matter QC formation and stability [8][9][10][11][12]. Phase field crystal models have been employed to simulate the growth of 2D QCs [13] and the adsorption properties on a quasicrystalline substrate [14]. The ingredients for 2D quasipattern formation are, first, a propensity towards periodic density modulations with two characteristic wave numbers k 1 and k 2 [15-18]. The ratio k 2 =k 1 must be close to certain special values; e.g., for dodecagonal QCs the value is 2 cosðπ=12Þ. Second, strong reinforcing (i.e., resonant) nonlinear interactions between these two characteristic density waves are required [17,19,20]. Earlier work on quasipatterns observed in Faraday wave experiments reveals similar requirements [19,[21][22][23]. We demonstrate here, following Mermin and Troian [24], that these same requirements suffice to stabilize icosahedral QCs in 3D. In contrast, nonlinear resonant interactions between density waves at a single wavelength are important in stabilizing simple crystal structures, such as body-centered cubic (bcc) crystals [25] although, with the right coupling, QCs can also be stabilized [26].We consider a 3D phase field crystal (PFC) model, appropriate for soft matter systems, that generates modulations with two l...

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Three-Dimensional Icosahedral Phase Field Quasicrystal

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“…• apart and whose lengths differ by 0.05% [17]. Starting from an initial condition with these 12 modes set to a small amplitude, we observe that the system easily forms a perfect example of a QC (Fig.…”

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confidence: 94%

“…This system exhibits quasipatterns (the fluid dynamical analog of quasicrystals), as discovered in the early 1990s [15,16], and two different mechanisms for their formation have been identified (see [17] for a more detailed discussion). Briefly, patterns with Q-fold symmetry are expressed as sums of modes with Q wave vectors spaced at equal angles and weakly nonlinear theory is used to compute how waves with different orientations affect each other.…”

Section: Introductionmentioning

confidence: 99%

“…Briefly, patterns with Q-fold symmetry are expressed as sums of modes with Q wave vectors spaced at equal angles and weakly nonlinear theory is used to compute how waves with different orientations affect each other. One mechanism relies on strong self-coupling to downplay the effect of waves with different orientations [18][19][20], so permitting 8-, 10-, 12-, 14-, 16-, 18-, 20-fold, or higher quasipatterns [17]. The second mechanism invokes nonlinear coupling between the primary waves with secondary weakly damped (or weakly excited) waves such that primary waves with wave vectors separated by a certain angle determined by the ratio of the primary to secondary wave number are favored [16,[21][22][23][24][25][26][27].…”

Section: Introductionmentioning

confidence: 99%

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Soft-core particles freezing to form a quasicrystal and a crystal-liquid phase

2015

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Systems of soft-core particles interacting via a two-scale potential are studied. The potential is responsible for peaks in the structure factor of the liquid state at two different but comparable length scales and a similar bimodal structure is evident in the dispersion relation. Dynamical density functional theory in two dimensions is used to identify two unusual states of this system: a crystal-liquid state, in which the majority of the particles are located on lattice sites but a minority remains free and so behaves like a liquid, and a 12-fold quasicrystalline state. Both are present even for deeply quenched liquids and are found in a regime in which the liquid is unstable with respect to modulations on the smaller scale only. As a result, the system initially evolves towards a small-scale crystal state; this state is not a minimum of the free energy, however, and so the system subsequently attempts to reorganize to generate the lower-energy larger-scale crystals. This dynamical process generates a disordered state with quasicrystalline domains and takes place even when this large scale is linearly stable, i.e., it is a nonlinear process. With controlled initial conditions, a perfect quasicrystal can form. The results are corroborated using Brownian dynamics simulations.

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“…We substitute the solution u 1 = Bp 1 and v 1 = Bq 1 into (20) and use ∂ ∂t − μ p 1 = −ωq 1 , so the equation can be then written as We find coefficients of the above equation by computing the inner products numerically. Therefore, the PDE is reduced to the Allen-Cahn equation as (22) B T = 1.5687λB + 11.1591B XX + 9.4717B 3 ,…”

Section: Reduction Of the Pde To The Allen-cahn Equationmentioning

confidence: 99%

Localized Patterns in Periodically Forced Systems

Alnahdi¹,

Niesen²,

Rucklidge³

et al. 2014

SIAM J. Appl. Dyn. Syst.

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Spatially localized, time-periodic structures are common in pattern-forming systems, appearing in fluid mechanics, chemical reactions, and granular media. We examine the existence of oscillatory localized states in a PDE model with single frequency time dependent forcing, introduced in [A. M. Rucklidge and M. Silber, SIAM J. Appl. Math., 8 (2009), pp. 298-347] as a phenomenological model of the Faraday wave experiment. In this study, we reduce the PDE model to the forced complex Ginzburg-Landau equation in the limit of weak forcing and weak damping. This allows us to use the known localized solutions found in [J. Burke, A. Yochelis, and E. Knobloch, SIAM J. Appl. Dyn. Syst., 7 (2008), pp. 651-711]. We reduce the forced complex Ginzburg-Landau equation to the Allen-Cahn equation near onset, obtaining an asymptotically exact expression for localized solutions. We also extend this analysis to the strong forcing case, recovering the Allen-Cahn equation directly without the intermediate step. We find excellent agreement between numerical localized solutions of the PDE, localized solutions of the forced complex Ginzburg-Landau equation, and analytic (hyperbolic) solutions of the Allen-Cahn equation. This is the first time that a PDE with time dependent forcing has been reduced to the Allen-Cahn equation and its localized oscillatory solutions quantitatively studied. Introduction.Localized patterns arise in a wide range of interesting pattern-forming problems. Much progress has been made on steady problems, where bistability between a steady pattern and the zero state leads to localized patterns bounded by stationary fronts between these two states [6,12]. In contrast, oscillons, which are oscillating localized structures in a stationary background, are relatively less well understood [18,25]. Fluid [3], chemical reaction [21], and granular media [25] problems have been studied experimentally. When the surface of the excited system becomes unstable (the Faraday instability), standing waves are found on the surface of the medium. Oscillons have been found where this primary bifurcation is subcritical [11], and these take the form of alternating conical peaks and craters against a stationary background.Previous studies have averaged over the fast timescale of the oscillation and have focused on PDE models where the localized solution is effectively steady [2,7,11]. Here we will seek localized oscillatory states in a PDE with time dependent parametric forcing. We find

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Design of Parametrically Forced Patterns and Quasipatterns

Cited by 40 publications

References 71 publications

Three-Dimensional Icosahedral Phase Field Quasicrystal

Three-Dimensional Icosahedral Phase Field Quasicrystal

Soft-core particles freezing to form a quasicrystal and a crystal-liquid phase

Localized Patterns in Periodically Forced Systems

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