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Domain growth of a BPI droplet in an isotropic environment. The droplet was equilibtated close to the isotropic-BPI boundary and then quenched to τ = 0 , κ = 2 , where BPII is stable. (See phase diagram in Fig.1.) The pictures show isosurfaces ( q = 0 . 13 ) of the scalar order parameter: during the equilibratio and at intermediate and late times (at t = 4 × 10 3 ; t = 1 . 2 × 10 5 ; t = 6 × 10 5 ) (a-c). In picture (b)
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The cubic blue phases of liquid crystals are fascinating and technologically promising examples of hierarchically structured soft materials, comprising ordered networks of defect lines (disclinations) within a liquid crystalline matrix. We present large-scale simulations of their domain growth, starting from a blue phase nucleus within a supercoole...
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Context 1
... were performed as described in the Methods sec- tion of the main text. Supporting Table S1 specifies the com- plete simulation parameters (all in LBU) for each of the runs presented in Figures 2-5 of the main text and Figures S1-S2. ...
Context 2
... address elsewhere (manuscript in preparation) the question of whether the chosen free energy density, Eq.A2, can indeed predict a stable BPIII phase in that regime. Of significance here is the fact that it does predict a metastable BPIII phase for τ = 0 and κ = 2, 3 (relevant to Figs.4,5, and S2). ...
Citations
... Similarly it was proposed by Marcus [5] that BPIII is an amorphous melt of DTCs. In agreement with the latter two predictions, recent mesoscopic numerical simulations of the BP structures [29,30] predict the amorphous structure of the disclination lines enmeshed with DTCs. While its properties have been studied in considerable detail, direct experimental observations of BPIII structure on the microscopic level are still scarce, performed mainly on the frozen [31] or polymerised [32] structures using electron microscopy. ...
... We therefore want to fully explain the observed structure of BPIII using Landau-de Gennes numerical simulations of the bulk structure. Similar to previous works by Henrich et al. [29,30] the unbounded stable BPIII structure appeared after numerical relaxation as an intertwined structure of regions with reduced scalar order parameter -disclinations -and elongated double-twisted regions -skyrmion filaments -shown in the left panel of Figure 6. The disclinations, shown in yellow, form an irregular mesh with all observed disclination junctions appearing 4-ended, similarly as in BPII. ...
Using high-resolution microscope observations, we show that both the static cubic (BPI) and the dynamic amorphous (BPIII) blue phases consist of fractional skyrmion filaments that transform into the same quasi two-dimensional structure of ‘baby’ half-skyrmions upon confinement to ultra-thin layers. We confirm this using numerical simulations of BPIII skyrmion structures in confinement combined with optical simulations, which reproduce experimental images, shown to originate at the glass-sample boundaries. Chiral nature of skyrmion filaments is demonstrated through experimental observations and optical simulations using circularly polarised light of opposite handedness producing images that vary strongly in comparative optical contrast. Direct observations and light scattering measurements of BPIII dynamics in bulk reveal that it is a highly dynamic phase, distinguished by two well-separated dynamic regimes, the slower of which can be observed even with a naked eye. We show that the slow dynamics corresponds to the thermally driven collective reshaping of the amorphous structure, while the fast mode is due to the director fluctuations within the filaments. Chirality of the relaxation modes in the bulk isotropic phase is also demonstrated and discussed. We show the dynamics in thinnest layers still show a two phase behaviour, with overall dynamics of half-skyrmions dramatically slowed down.
... For this, we followed the general strategy for parameterizing the LB simulations described in Ref. 30. We set the solvent viscosity to that of water, but, following standard practice with LB simulations of microscopic systems, we set the density to several orders of magnitude larger than the density of water 55,[70][71][72] . This allows increasing the timestep ∆t (thereby accelerating simulations) while ensuring that the system is still in the low Reynolds number regime. ...
We consider an immersed elastic body that is actively driven through a structured fluid by a motor or an external force. The behavior of such a system generally cannot be solved analytically, necessitating the use of numerical methods. However, current numerical methods omit important details of the microscopic structure and dynamics of the fluid, which can modulate the magnitudes and directions of viscoelastic restoring forces. To address this issue, we develop a simulation platform for modeling viscoelastic media with tensorial elasticity. We build on the lattice Boltzmann algorithm and incorporate viscoelastic forces, elastic immersed objects, a microscopic orientation field, and coupling between viscoelasticity and the orientation field. We demonstrate our method by characterizing how the viscoelastic restoring force on a driven immersed object depends on various key parameters as well as the tensorial character of the elastic response. We find that the restoring force depends non-monotonically on the rate of diffusion of the stress and the size of the object. We further show how the restoring force depends on the relative orientation of the microscopic structure and the pulling direction. These results imply that accounting for previously neglected physical features, such as stress diffusion and the microscopic orientation field, can improve the realism of viscoelastic simulations. We discuss possible applications and extensions to the method.
... Among others, Blue Phases (BP) are examples of CLC that occur in proximity of the isotropic-nematic transition and manifest themselves as a network of double twist cylinders that self-assemble into three-dimensional (3D) structures [1,[13][14][15], which appear in vivid shades of blue -to which they owe their name. In 3D BPs come in three different forms. ...
We study the phase behaviour of cholesteric liquid crystal shells with different geometries. We compare the cases of tangential and no anchoring at the surface, focussing on the former case, which leads to a competition between the intrinsic tendency of the cholesteric to twist and the anchoring free energy which suppresses it. We then characterise the topological phases arising close to the isotropic-cholesteric transition. These typically consist of quasi-crystalline or amorphous tessellations of the surface by half-skyrmions, which are stable at lower and larger shell size respectively. For ellipsoidal shells, defects in the tessellation couple to local curvature, and according to the shell size they either migrate to the poles or distribute uniformly on the surface. For toroidal shells, the variations in the local curvature of the surface stabilises heterogeneous phases where cholesteric or isotropic patterns coexist with hexagonal lattices of half-skyrmions.
... We show two main sets of parameters, corresponding to those used for the droplet pulling results (Tables I, III, IV, and V) and those used for the instability results (Tables II, IV, and VI). We note that, following standard practice with LB simulations, the density of water is set to several orders of magnitude larger than its actual value [58][59][60][61]. This allows increasing the time step (thereby speeding up simulations) while still ensuring that the system has a small Reynolds number. ...
Non-reciprocal interactions fueled by local energy consumption are found in biological and synthetic active matter, where both viscosity and elasticity are often important. Such systems can be described by "odd" viscoelasticity, which assumes fewer material symmetries than traditional theories. In odd viscoelastic systems there is an interplay between the energy-consuming odd elastic elements and the traditional stabilizing elements. This leads to rich dynamical behavior which, due to a lack of appropriate numerical methods, has remained relatively unexplored. Furthermore, the implications associated with the presence of such odd terms in actomyosin and other similar anisotropic systems has not been addressed. Here, we study odd viscoelasticity analytically and using hydrodynamic simulations based on the lattice Boltzmann algorithm. We first outline how odd effects may naturally emerge from a theory of polymeric elasticity which can describe anisotropic systems like actomyosin. Next, we report on two striking features of odd viscoelastic dynamics: a pattern-forming instability which produces an oscillating array of fluid vortices, and strong transverse and rotational forces during a simulated rheological experiment. These findings can guide efforts to detect or engineer odd dynamics in soft active matter systems.
... Icosahedral or bond orientational models of the BPIII phase structure are suggested by various theories [39][40][41][42] but not confirmed experimentally. Mesoscopic numerical simulations [43,44] predict the structure of BPIII as an amorphous network of disclination lines permeated by double-twisted regions, which is in agreement with the proposal by Marcus [26]. Although a range of evidence is consistent with BPIII as a liquid tangle of DTCs [45][46][47], the microscopic structure of BPIII remains unclear, due to the lack of convincing experimental evidence. ...
... To corroborate the experiments, we perform extensive Landau-de Gennes (LdG) numerical modeling of stable and metastable skyrmion structures in thin layers and bulk BPIII; details can be found in Appendix C. Similar to previous work by Henrich et al. [43,44], we reproduce the bulk BPIII of mutually enmeshed structures of singular defect lines and skyrmion filament-DTCs, as shown in Fig. 8 Fig. 8(e)]. DTCs in bulk BPIII can, therefore, be considered as quarter-skyrmions, although, more precisely, there is a distribution of maximum radial twist angles of DTCs (i.e., their actual size) in bulk BPIII, as discussed in the continuation. ...
Skyrmions are topologically protected, vortexlike formations of a field that cannot be removed by any smooth transformation and emerge in a range of fundamentally different, either quantum or classical systems, from spin textures to chiral ferromagnets and chiral complex fluids. Notably, they are generally observed in thin ordered or disordered quasi-2D layers, but little is known about their three-dimensional structuring and organization, including structural transitions from 2D to 3D. Here, we show experimentally and numerically that the blue phase (BP) III of a chiral liquid crystal is a 3D fluid of chiral skyrmion filaments of the nematic orientational field, entangled with a 3D network of topological defect lines. It is an effective 3D dynamic fluid determined by the thermal fluctuations of two distinct branches of excitations: rapid internal fluctuations of the skyrmion structure and a slow collective motion of the skyrmion filaments. When confined to less than an approximately 150-nm layer, the 3D bulk skyrmion fluid transforms into a different effectively 2D liquid of half-skyrmions, with the dynamics of the skyrmion liquid slowing down by an order of magnitude and with the individual skyrmions lingering, and even disappearing into, and reappearing from the homogeneous liquid crystal. The thickness-temperature phase diagram actually shows that both the BPIII and BPI phases are made of skyrmions, which when confined to less than approximately 150 nm cells transform equally into a 2D half-skyrmion liquid. The temperature range of this 2D half-skyrmion liquid is much broader than the temperature interval of BP phases, which makes BP materials interesting for broad-temperature-range skyrmionic applications. We envisage a soft matter skyrmionic device, in which skyrmions are created and detected by light.
... To date, the phase transformation of BPLCs has been characterized using polarized optical microscopy (POM) 17,[42][43][44][45][46][47][48][49] . The DTCs, which are analogous to the atoms of the atomic crystals, can be considered as structural units of BPIII 15,39,40 , BPII 8,10,31,[34][35][36]or BPI 8,10,34,35,37,38 , in which their microstructures have been observed using confocal laser scanning microscopy 41,50 , transmission electron microscopy (TEM) 38,51 , and corresponding simulations 32,52,53 . However, the phase-transition process has yet to be observed at the submicrometer scale owing to the poor stability of the transition states, and thus the transition mechanism remains unclear. ...
In a narrow temperature window in going from the isotropic to highly chiral orders, cholesteric liquid crystals exhibit so-called blue phases, consisting of different morphologies of long, space-filling double twisted cylinders. Those of cubic spatial symmetry have attracted considerable attention in recent years as templates for soft photonic materials. The latter often requires the creation of monodomains of predefined orientation and size, but their engineering is complicated by a lack of comprehensive understanding of how blue phases nucleate and transform into each other at a submicrometer length scale. In this work, we accomplish this by intercepting nucleation processes at intermediate stages with fast cross-linking of a stabilizing polymer matrix. We reveal using transmission electron microscopy, synchrotron small-angle X-ray diffraction, and angle-resolved microspectroscopy that the grid of double-twisted cylinders undergoes highly coordinated, diffusionless transformations. In light of our findings, the implementation of several applications is discussed, such as temperature-switchable QR codes, micro-area lasing, and fabrication of blue phase liquid crystals with large domain sizes.
... The texture in BPI in the cell with GS&CS is more uniform than that in cells made of only GSs or only CSs. Studies have shown that the formation of BPs could depend on homogeneous nucleation in the bulk 35 and heterogeneous nucleation on patterned substrates. 33 While homogenous nucleation is supposed to proceed through a series of metastable states, nucleation in confined systems has been shown to be promoted by the patterned substrates which act as nucleation sites. ...
Optimizing strategies used for improving the stability and properties of blue phase (BP) liquid crystals directly impact device performance. Various factors ranging from molecular structure to sample size and substrate conditions can influence selective reflection and electro-optics of BPs. More recently, the technique of incorporating colloidal nanoparticle (NP) assemblies has been used to enhance BP ranges. In cubic BPs, disclination networks can act as trapping centers for NPs, reducing the high elastic energy cost of these regions, favoring BP stability. Organization of NPs in the defect regions can sustain stable 3D colloidal structures, widening the scope and applicability of BPs as photonic materials. Physical and chemical properties, size, and shape of the NPs can also determine the utilization of BPs for advanced applications like lasers and high quality displays. In view of this, a mixture of two calamitic chiral compounds in which all three BPs, viz., BPI, BPII, and BPIII, were induced was combined with rod-shaped CdSe/CdS quantum rods (QRs) and spherical CdSe quantum dots (QDs), which were specifically chosen due to their exceptional optical properties. This also provided an opportunity to investigate the effect of the shape of the NPs on the preferential stabilization of the BPs and on the electro-optic Kerr effect. QRs were found to be more efficient in enhancing the overall BP range, with an almost twofold increase of ∼27 °C with ∼0.5 wt. %. On the other hand, with QDs, the BP range showed an initial increase of 20 °C for ∼0.3 wt. %, which, however, decreased with a further increase in QDs. Another major difference is that the Kerr effect was active only in the BPIII in the case of QDs but is measurable in both cubic BPI and BPII in the case of QRs. The results have been described in terms of the organization of the nanocrystals within the defect lines and the lattice orientations imposed by the substrates.
... Nevertheless, several fundamental aspects of these exotic phases are yet to be explored. For example, recently, several theoretical predictions were made on the defect dynamics and rheological properties of BP-II and BP-I [20][21][22][23][24], which are yet to be explored experimentally. ...
... Both BP-II and BP-I also exhibit the flow-induced nematic texture at a shear rate much higher than 200 s −1 . Recently, Henrich et al. studied the rheology of cubic blue phases by computer simulation [20,21,23]. They have identified various flow regimes with increasing shear rate in both BP-I and BP-II. ...
Topological defects are important in determining the properties of physical systems and are known varyingly depending on the broken symmetry. In superfluid helium, they are called vortices; in periodic crystals, one refers to dislocations; and in liquid crystals, they are disclinations. The defects and the inter-defect interaction in some highly chiral liquid crystals stabilize some intermediate complex phases such as Blue Phases (BPs) and Twist Grain Boundary-A (TGBA) phases. The defect dynamics of these phases contributes to the rheological properties. The temperature range of these intermediate phases usually are very small in pure liquid crystals; consequently, a detailed experiment has been difficult to achieve. However, the temperature range could be enhanced significantly in multicomponent systems. In this review article, we discuss some recent experimental progress made in understanding the rheological properties of the wide-temperature-range TGBA and BP liquid crystals.
... Specifically, these authors simulated BPI or BPII nuclei embedded in an isotropic or a cholesteric background (31), and found that BPs grow disorderly from the nuclei, leading to metastable states characterized by an amorphous defect network. In their work, the nuclei were quenched to temperatures where the BPI and BPII are thought to be stable; these authors then suggested that the ordering dynamics of BPs exhibits a hierarchical nature, which requires a secondary nucleation stage to control the nucleation and orientation of bulk BPs. ...
... It is instructive to highlight several differences in the formation of BPs that arise between homogeneous nucleation (in the bulk) and heterogeneous nucleation (on patterned surfaces). As mentioned earlier, Henrich et al. (31) have shown that, during homogeneous nucleation, the formation of BPI starting from Chol, and the formation of BPII starting from Iso appear to follow a sequence of metastable states. In our calculations and experiments, the system is confined and the patterned surfaces act as the preferred nucleation site. ...
Liquid-crystal blue phases (BPs) are highly ordered at two levels. Molecules exhibit orientational order at nanometer length scales, while chirality leads to ordered arrays of double-twisted cylinders
over micrometer scales. Past studies of polycrystalline BPs were challenged by the existence of grain boundaries between randomly oriented crystalline nanodomains. Here, the nucleation of BPs is
controlled with precision by relying on chemically nanopatterned surfaces, leading to macroscopic single-crystal BP specimens where the dynamics of mesocrystal formation can be directly observed.
Theory and experiments show that transitions between two BPs having a different network structure proceed through local reorganization of the crystalline array, without diffusion of the double-twisted cylinders. In solid crystals, martensitic transformations between crystal structures involve the concerted motion of a few
atoms, without diffusion. The transformation between BPs, where crystal features arise in the submicron regime, is found to be martensitic in nature when one considers the collective behavior of the double-twist cylinders. Single-crystal BPs are shown to offer fertile grounds for the study of directed crystal nucleation and the controlled growth of soft matter.
... While the director field can theoretically develop out-of-plane components, for active experiments and the parameter values used in this study, it does not move out of plane. Discrete space and time steps are chosen as unity and all quantities can be converted to physical units in a material-dependent manner 22,51,52 . Simulations are performed with the parameters A ¼ 0, B ¼ 0.3, C ¼ À 0.3, G ¼ 0.34, K ¼ 0.04, l ¼ 0.3, r ¼ 1 and m ¼ 2/3, in lattice Boltzmann units. ...
Meso-scale turbulence is an innate phenomenon, distinct from inertial turbulence, that spontaneously occurs at low Reynolds number in fluidized biological systems. This spatiotemporal disordered flow radically changes nutrient and molecular transport in living fluids and can strongly affect the collective behaviour in prominent biological processes, including biofilm formation, morphogenesis and cancer invasion. Despite its crucial role in such physiological processes, understanding meso-scale turbulence and any relation to classical inertial turbulence remains obscure. Here we show how the motion of active matter along a micro-channel transitions to meso-scale turbulence through the evolution of locally disordered patches (active puffs) from an ordered vortex-lattice flow state. We demonstrate that the stationary critical exponents of this transition to meso-scale turbulence in a channel coincide with the directed percolation universality class. This finding bridges our understanding of the onset of low-Reynolds-number meso-scale turbulence and traditional scale-invariant turbulence in confinement.
