Labyrinth-like Patterns Research Articles

Ferroelectric incommensurate spin crystals.

Ferroics, especially ferromagnets, can form complex topological spin structures such as vortices1 and skyrmions2,3 when subjected to particular electrical and mechanical boundary conditions. Simple vortex-like, electric-dipole-based topological structures have been observed in dedicated ferroelectric systems, especially ferroelectric-insulator superlattices such as PbTiO3/SrTiO3, which was later shown to be a model system owing to its high depolarizing field4-8. To date, the electric dipole equivalent of ordered magnetic spin lattices driven by the Dzyaloshinskii-Moriya interaction (DMi)9,10 has not been experimentally observed. Here we examine a domain structure in a single PbTiO3 epitaxial layer sandwiched between SrRuO3 electrodes. We observe periodic clockwise and anticlockwise ferroelectric vortices that are modulated by a second ordering along their toroidal core. The resulting topology, supported by calculations, is a labyrinth-like pattern with two orthogonal periodic modulations that form an incommensurate polar crystal that provides a ferroelectric analogue to the recently discovered incommensurate spin crystals in ferromagnetic materials11-13. These findings further blur the border between emergent ferromagnetic and ferroelectric topologies, clearing the way for experimental realization of further electric counterparts of magnetic DMi-driven phases.

Repulsive inter-layer coupling induces anti-phase synchronization

We present numerical results for the synchronization phenomena in a bilayer network of repulsively coupled 2D lattices of van der Pol oscillators. We consider the cases when the network layers have either different or the same types of intra-layer coupling topology. When the layers are uncoupled, the lattice of van der Pol oscillators with a repulsive interaction typically demonstrates a labyrinth-like pattern, while the lattice with attractively coupled van der Pol oscillators shows a regular spiral wave structure. We reveal for the first time that repulsive inter-layer coupling leads to anti-phase synchronization of spatiotemporal structures for all considered combinations of intra-layer coupling. As a synchronization measure, we use the correlation coefficient between the symmetrical pairs of network nodes, which is always close to -1 in the case of anti-phase synchronization. We also study how the form of synchronous structures depends on the intra-layer coupling strengths when the repulsive inter-layer coupling is varied.

Patterning in Mussel Beds Explained by the Interplay of Multi-Level Selection and Spatial Self-Organization

Cooperation, ubiquitous in nature, is difficult to explain from an evolutionary perspective. Many modelling studies strive to resolve this challenge, but their simplifying assumptions on population and interaction structure are rarely met in ecological settings. Here we use a modelling approach that includes more ecological detail to investigate evolution of cooperation in spatially self-organized mussel beds. Mussels cooperate with each other through aggregative movement and attachment using byssal threads. These cooperative behaviours shape the spatial structure of the mussel bed, which can range from scattered distributions to labyrinth-like patterns and dense mussel clumps. The spatial pattern in turn impacts an individual’s fitness at two levels: (i) proper attachment to neighbouring individuals decreases predation risk, and (ii) attachment to a sufficiently large group prevents dislodgement by wave stress. Without this second level of selection, our simulations do typically not result in evolutionary attractors that lead to the labyrinth-like spatial patterns that are characteristic for natural mussel beds. Yet, when group-level selection is included, labyrinth-like patterns emerge under a wide range of conditions. Our model demonstrates that multiple selection factors working at different spatial scales – predation of individuals and dislodgement of entire mussel clumps – combinedly determine evolution of cooperative traits in mussels and thereby result in emergence of the labyrinth-like spatial patterns that we observe in natural mussel beds.

Surface resonance of the (2×1) reconstructed lanthanum hexaboride (001)-cleavage plane: A combined STM and DFT study

We performed a combined study of the (001)-cleavage plane of lanthanum hexaboride (LaB$_\text{6}$) using scanning tunneling microscopy (STM) and density functional theory (DFT). Experimentally, we found a (2$\times$1) reconstructed surface on a local scale. The reconstruction is only short-range ordered and tends to order perpendicularly to step edges. At larger distances from surface steps, the reconstruction evolves to a labyrinth-like pattern. These findings are supported by low-energy electron diffraction (LEED) experiments. Slab calculations within the framework of DFT shows that the atomic structure consists of parallel lanthanum chains on top of boron octahedra. Scanning tunneling spectroscopy (STS) shows a prominent spectral feature at -0.6 eV. Using DFT, we identify this structure as a surface resonance of the (2$\times$1) reconstructed LaB$_\text{6}$ (100)-surface which is dominated by boron dangling bond-states and lanthanum d-states.

Random scalar fields and hyperuniformity

Disordered many-particle hyperuniform systems are exotic amorphous states of matter that lie between crystals and liquids. Hyperuniform systems have attracted recent attention because they are endowed with novel transport and optical properties. Recently, the hyperuniformity concept has been generalized to characterize two-phase media, scalar fields, and random vector fields. In this paper, we devise methods to explicitly construct hyperuniform scalar fields. Specifically, we analyze spatial patterns generated from Gaussian random fields, which have been used to model the microwave background radiation and heterogeneous materials, the Cahn-Hilliard equation for spinodal decomposition, and Swift-Hohenberg equations that have been used to model emergent pattern formation, including Rayleigh-Bénard convection. We show that the Gaussian random scalar fields can be constructed to be hyperuniform. We also numerically study the time evolution of spinodal decomposition patterns and demonstrate that they are hyperuniform in the scaling regime. Moreover, we find that labyrinth-like patterns generated by the Swift-Hohenberg equation are effectively hyperuniform. We show that thresholding (level-cutting) a hyperuniform Gaussian random field to produce a two-phase random medium tends to destroy the hyperuniformity of the progenitor scalar field. We then propose guidelines to achieve effectively hyperuniform two-phase media derived from thresholded non-Gaussian fields. Our investigation paves the way for new research directions to characterize the large-structure spatial patterns that arise in physics, chemistry, biology, and ecology. Moreover, our theoretical results are expected to guide experimentalists to synthesize new classes of hyperuniform materials with novel physical properties via coarsening processes and using state-of-the-art techniques, such as stereolithography and 3D printing.

Structure preserving integration and model order reduction of skew-gradient reaction–diffusion systems

Activator-inhibitor FitzHugh-Nagumo (FHN) equation is an example for reaction-diffusion equations with skew-gradient structure. We discretize the FHN equation using symmetric interior penalty discontinuous Galerkin (SIPG) method in space and average vector field (AVF) method in time. The AVF method is a geometric integrator, i.e. it preserves the energy of the Hamiltonian systems and energy dissipation of the gradient systems. In this work, we show that the fully discrete energy of the FHN equation satisfies the mini-maximizer property of the continuous energy for the skew-gradient systems. We present numerical results with traveling fronts and pulses for one dimensional, two coupled FHN equations and three coupled FHN equations with one activator and two inhibitors in skew-gradient form. Turing patterns are computed for fully discretized two dimensional FHN equation in the form of spots and labyrinths. Because the computation of the Turing patterns is time consuming for different parameters, we applied model order reduction with the proper orthogonal decomposition (POD). The nonlinear term in the reduced equations is computed using the discrete empirical interpolation (DEIM) with SIPG discretization. Due to the local nature of the discontinuous Galerkin (DG) method, the nonlinear terms can be computed more efficiently than for the continuous finite elements. The reduced solutions are very close to the fully discretized ones. The efficiency and accuracy of the POD and POD-DEIM reduced solutions are shown for the labyrinth-like patterns.

Mucinous ovarian neoplasms. Prognostically mostly excellent, infrequently a wolf in sheep's clothing

Mucinous ovarian neoplasms represent the second largest group of epithelial ovarian tumors after serous neoplasms, of which benign cystadenomas constitute more than 80 %. Mucinous cystadenomas and carcinomas cannot be distinguished by the clinical features or the mean age of onset of the disease. They typically occur unilaterally, are confined to the adnexae (FIGO stageI) and clinically present with non-specific abdominal symptoms or are diagnosed by chance. The mean age of disease onset is around 50years old. The prognosis is excellent. Implants, peritoneal metastases and bilateral occurrence of ovarian mucinous neoplasms should lead to the suspicion of metastasis particularly from a gastrointestinal tumor. Neither microinvasion defined as a maximum extent of invasion of 5mm, nor intraepithelial carcinoma characterized by high grade atypia without invasion, affect the prognosis of mucinous borderline tumors. Mucinous carcinomas typically show confluent glandular, expansile growth that leads to a labyrinth-like pattern. A destructive infiltrative or nodular growth pattern, however, should lead to the consideration of metastasis. Mural nodules that may reveal a spindle cell sarcomatous or anaplastic carcinomatous pattern occur infrequently in mucinous and do not affect the prognosis. Pax8 positivity is indicative of a primary ovarian neoplasm. In this case, however, mucinous tumors associated with teratomas may show the colonic immunoreaction pattern (CK7-/CK20+/CDX2+). The rare mucinous tumors with endocervical differentiation are now designated as seromucinous tumors and consist of two or more distinct cell types, are frequently associated with endometriosis and seem to show a molecular genetic relationship to endometrioid neoplasms.

Lymphatic differentiation in renal angiomyolipomas

Renal angiomyolipomas are mesenchymal neoplasms with varying proportions of smooth muscle, adipose tissue, and abnormal blood vessels. Although the presence of lymphangiomatous-like foci is frequently noted in large series of angiomyolipoma, lymphatic differentiation has not been previously studied. Twelve angiomyolipomas from 10 patients were identified. All tumors expressed a melanocytic marker, HMB-45 or Melan-A. Twenty-eight paraffin blocks (1-4 per tumor) were stained for lymphatic endothelial cell markers, podoplanin, and D2-40, and the presence and distribution of lymphatic differentiation were recorded. The angiomyolipomas ranged from typical triphasic tumors to leiomyoma-like and lipoma-like tumors. All 12 tumors showed positive staining with podoplanin, and all 6 tumors stained for D2-40 were also positive, indicative of lymphatic differentiation. Lymphatic differentiation was variably observed throughout the tumors. It was most prevalent in myoid areas of the triphasic angiomyolipomas and in the leiomyoma-like variant, but infrequent and widely scattered within the adipose regions of triphasic angiomyolipoma and in the lipoma-like variant. The lymphatics were usually small, often irregularly shaped, and isolated vessels in fat, whereas in myoid regions lymphatics were clustered and in some areas formed a sinusoidal or labyrinth-like pattern. Lymphatics were commonly adjacent to abnormal arteries. However, unlike the lymphatics in the normal renal cortex, a consistent adventitial association was not observed and the clustering around arteries is regarded as reflecting the myoid regions that typically exist in these areas. In conclusion, lymphatic differentiation is common in angiomyolipomas, preferentially located in myoid regions. These data expand the mesenchymal pluripotential profile of renal angiomyolipomas.

Moiré Patterns in Superimposed Nanoporous Thin Films Derived from Block-Copolymer Assemblies

Nanoscale rotation moire patterns are observed in double-layer nanoporous thin films produced with use of block-copolymer self-assembly. Periodic hexagonal moire superstructures appear when the films possessing long-range order are superimposed at small misorientation angles. Overlapping films misoriented by angles close to 30° generate aperiodic quasi-crystal-like superstructures with 5-fold symmetries. The stacking of the disordered nanoporous films produces labyrinth-like patterns. Block-copolymer moire superstructures are promising as nanolithography masks with controllable morphology and periodicity.