Recurrent flow patterns as a basis for two-dimensional turbulence: Predicting statistics from structures.

A dynamical systems approach to turbulence envisions the flow as a trajectory through a high-dimensional state space [Hopf, Commun. Appl. Maths 1, 303 (1948)]. The chaotic dynamics are shaped by the unstable simple invariant solutions populating the inertial manifold. The hope has been to turn this picture into a predictive framework where the statistics of the flow follow from a weighted sum of the statistics of each simple invariant solution. Two outstanding obstacles have prevented this goal from being achieved: 1) paucity of known solutions and 2) the lack of a rational theory for predicting the required weights. Here, we describe a method to substantially solve these problems, and thereby provide compelling evidence that the probability density functions (PDFs) of a fully developed turbulent flow can be reconstructed with a set of unstable periodic orbits. Our method for finding solutions uses automatic differentiation, with high-quality guesses constructed by minimizing a trajectory-dependent loss function. We use this approach to find hundreds of solutions in turbulent, two-dimensional Kolmogorov flow. Robust statistical predictions are then computed by learning weights after converting a turbulent trajectory into a Markov chain for which the states are individual solutions, and the nearest solution to a given snapshot is determined using a deep convolutional autoencoder. In this study, the PDFs of a spatiotemporally chaotic system have been successfully reproduced with a set of simple invariant states, and we provide a fascinating connection between self-sustaining dynamical processes and the more well-known statistical properties of turbulence.

Exploring the free-energy landscape of a rotating superfluid.

The equilibrium state of a superfluid in a rotating cylindrical vessel is a vortex crystal-an array of vortex lines, which is stationary in the rotating frame. Experimental realizations of this behavior typically show a sequence of transient states before the free-energy-minimizing configuration is reached. Motivated by these observations, we construct a new method for a systematic exploration of the free-energy landscape via gradient-based optimization of a scalar loss function. Our approach is inspired by the pioneering numerical work of Campbell and Ziff [Phys. Rev. B. 20, 1886 (1979)] and makes use of automatic differentiation, which crucially allows us to include entire solution trajectories in the loss. We first use the method to converge thousands of low free-energy relative equilibria in the unbounded domain for vortex numbers in the range 10≤N≤30, which reveals an extremely dense set of mostly saddle-like solutions. As part of this search, we discover new continuous families of relative equilibria, which are often global minimizers of free energy. These continuous families all consist of crystals arranged in a double-ring configuration, and we assess which state from the family is most likely to be observed experimentally by computing energy-minimizing pathways from nearby local minima-identifying a common entry point into the family. The continuous families become discrete sets of equal-energy solutions when the wall is introduced in the problem. Finally, we develop an approach to compute homoclinic orbits and use it to examine the dynamics in the vicinity of the minimizing state by converging connections for low-energy saddles.

Exact Traveling Wave Solutions in Viscoelastic Channel Flow.

Elasto-inertial turbulence (EIT) is a new, two-dimensional chaotic flow state observed in polymer solutions with possible connections to inertialess elastic turbulence and drag-reduced Newtonian turbulence. In this Letter, we argue that the origins of EIT are fundamentally different from Newtonian turbulence by finding a dynamical connection between EIT and an elasto-inertial linear instability recently found at high Weissenberg numbers [Garg et al., Phys. Rev. Lett. 121, 024502 (2018)PRLTAO0031-900710.1103/PhysRevLett.121.024502]. This link is established by isolating the first known exact coherent structures in viscoelastic parallel flows-nonlinear elasto-inertial traveling waves (TWs)-borne at the linear instability and tracking them down to substantially lower Weissenberg numbers where EIT exists. These TWs have a distinctive "arrowhead" structure in the polymer stretch field and can be clearly recognized albeit transiently in EIT as well as being attractors for EIT dynamics if the Weissenberg number is sufficiently large. Our findings suggest that the dynamical systems picture in which Newtonian turbulence is built around the coexistence of many (unstable) simple invariant solutions populating phase space carries over to EIT, though these solutions rely on elasticity to exist.

Nanohybrid Membrane Synthesis with Phosphorene Nanoparticles: A Study of the Addition, Stability and Toxicity.

Phosphorene is a promising candidate as a membrane material additive because of its inherent photocatalytic properties and electrical conductance which can help reduce fouling and improve membrane properties. The main objective of this study was to characterize structural and morphologic changes arising from the addition of phosphorene to polymeric membranes. Here, phosphorene was physically incorporated into a blend of polysulfone (PSf) and sulfonated poly ether ether ketone (SPEEK) doping solution. Protein and dye rejection studies were carried out to determine the permeability and selectivity of the membranes. Since loss of material additives during filtration processes is a challenge, the stability of phosphorene nanoparticles in different environments was also examined. Furthermore, given that phosphorene is a new material, toxicity studies with a model nematode, Caenorhabditis elegans, were carried out to provide insight into the biocompatibility and safety of phosphorene. Results showed that membranes modified with phosphorene displayed a higher protein rejection, but lower flux values. Phosphorene also led to a 70% reduction in dye fouling after filtration. Additionally, data showed that phosphorene loss was negligible within the membrane matrix irrespective of the pH environment. Phosphorene caused toxicity to nematodes in a free form, while no toxicity was observed for membrane permeates.

CaMn0.5Ir0.5O2.5: An Anion-Deficient Perovskite Oxide Containing Ir3.

Reaction between CaMn0.5Ir0.5O3 and NaH, either through solid-solid contact or via a gas mediated reaction process, yields the topochemically reduced phase CaMn0.5Ir0.5O2.5 in which Mn3+ and Ir3+ cations are located within a partially anion-vacancy disordered lattice. Magnetization data from CaMn0.5Ir0.5O2.5 can be fit by the Curie-Weiss law to yield C = 1.586 cm3 K mol-1 and θ = -86.9 K, consistent with a combination of S = 2, Mn3+ and S = 0, Ir3+. On cooling below T ∼ 110 K, the system undergoes a transition to a spin-glass state, consistent with the observed Mn/Ir cation disorder and frustration between Mn-O-Mn and Mn-O-Ir-O-Mn magnetic couplings. The degree of reduction and the observed anion-vacancy disorder are discussed on the basis of the d-orbital filling of the transition-metal cations.

Structure and Magnetism of (La/Sr)2M0.5IrV0.5O4 and Topochemically Reduced (La/Sr)2M0.5IrII0.5O3 (M = Fe, Co) Complex Oxides.

Neutron powder diffraction data show that Sr2Fe0.5Ir0.5O4, Sr2Co0.5Ir0.5O4, and La0.5Sr1.5Co0.5Ir0.5O4 all adopt undistorted, n = 1 Ruddlesden-Popper structures in which the Ir5+ and Fe3+/Co3+/Co2+ cations are statistically disordered over all the octahedral coordination sites. Magnetization data indicate the two cobalt phases are spin glasses at low temperature, while Sr2Fe0.5Ir0.5O4 appears to adopt an antiferromagnetic state with very small magnetically ordered domains. Topochemical reduction with a Zr getter converts the tetragonal A2M0.5Ir0.5O4 phases to the corresponding orthorhombic A2M0.5Ir0.5O3 phases in which the Ir2+ and Fe2+/Co2+/Co1+ cations are located in approximately square-planar coordination sites. Magnetization data indicate Sr2Fe0.5Ir0.5O3 is a spin glass below TG ∼ 30 K, while Sr2Co0.5Ir0.5O3 appears to be antiferromagnetic below TN ∼ 25 K and La0.5Sr1.5Co0.5Ir0.5O3 shows no sign of magnetic order for T > 5 K. The magnetic behavior of both the A2M0.5Ir0.5O4 and A2M0.5Ir0.5O3 phases is discussed on the basis of metal d-electron count and structural features.

Sr2FeIrO4: Square-Planar Ir(II) in an Extended Oxide.

Topochemical reduction of the double-perovskite oxide Sr2FeIrO6 under dilute hydrogen leads to the formation of Sr2FeIrO4. This phase consists of ordered infinite sheets of apex-linked Fe2+O4 and Ir2+O4 squares stacked with Sr2+ cations and is the first report of Ir2+ in an extended oxide phase. Plane-wave density functional theory calculations indicate high-spin Fe2+ (d6, S = 2) and low-spin Ir2+ (d7, S = 1/2) configurations for the metals and confirm that both ions have a doubly occupied d z2 orbital, a configuration that is emerging as a consistent feature of all layered oxide phases of this type. The stability and double occupation of d z2 in the Ir2+ ions invites a somewhat unexpected analogy to the extensively studied Ir4+ ion as both ions share a common near-degenerate (d xy/ xz/ yz)5 valence configuration. On cooling below 115 K, Sr2FeIrO4 enters a magnetically ordered state in which the Ir and Fe sublattices adopt type II antiferromagnetically coupled networks which interpenetrate each other, leading to frustration in the nearest-neighbor Fe-O-Ir couplings, half of which are ferromagnetic and half antiferromagnetic. The spin frustration drives a symmetry-lowering structural distortion in which the four equivalent Ir-O and Fe-O distances of the tetragonal I4/ mmm lattice split into two mutually trans pairs in a lattice with monoclinic I112/ m symmetry. This strong magneto-lattice coupling arises from the novel local electronic configurations of the Fe2+ and Ir2+ cations and their cation-ordered arrangement in a distorted perovskite lattice.

Doped Sr2FeIrO6-Phase Separation and a Jeff ≠ 0 State for Ir5.

High-resolution synchrotron X-ray and neutron powder diffraction data demonstrate that, in contrast to recent reports, Sr2FeIrO6 adopts an I1̅ symmetry double perovskite structure with an a-b-c- tilting distortion. This distorted structure does not tolerate cation substitution, with low levels of A-site (Ca, Ba, La) or Fe-site (Ga) substitution leading to separation into two phases: a stoichiometric I1̅ phase and a cation-substituted, P21/ n symmetry, a-a-c+ distorted double perovskite phase. Magnetization, neutron diffraction, and 57Fe Mössbauer data show that, in common with Sr2FeIrO6, the cation substituted Sr2- xA xFe1- yGa yIrO6 phases undergo transitions to type-II antiferromagnetically ordered states at TN ∼ 120 K. However, in contrast to stoichiometric Sr2FeIrO6, cation substituted samples exhibit a further magnetic transition at TA ∼ 220 K, which corresponds to the ordering of Jeff ≠ 0 Ir5+ centers in the cation-substituted, P21/ n symmetry, double perovskite phases.