Turab Lookman

The climate attractor over short timescales

Recent work has highlighted the possibilities of using certain ideas from the theory of dynamical systems for the study of global climate. These ideas include the geometrical notion of correlation or scaling dimension, first used to analyse attractors arising in mathematical and laboratory systems1–4 and later applied in a geophysical context5,6. The conclusion of the original geophysical work5 has been criticized in the light of a reanalysis7 which questions the existence of a low-dimensional climate attractor. Here results of a similar analysis conducted on daily meteorological observations from 1946 to 1982, are announced. This new work overcomes limitations of previous analyses, and supports the existence of such an attractor.

Phase separation and shape deformation of two-phase membranes

Within a coupled-field Ginzburg-Landau model we study analytically phase separation and accompanying shape deformation on a two-phase elastic membrane in simple geometries such as cylinders, spheres, and tori. Using an exact periodic domain wall solution we solve for the shape and phase separating field, and estimate the degree of deformation of the membrane. The results are pertinent to preferential phase separation in regions of differing curvature on a variety of vesicles.

Statistical error in a chord estimator of correlation dimension: The rule of five''

The statistical precision of a chord method for estimating dimension from a correlation integral is derived. The optimal chord length is determined, and a comparison is made with other estimators. The simple chord estimator is only 25% less precise than the optimal estimator which uses the full resolution and full range of the correlation integral. The analytic calculations are based on the hypothesis that all pairwise distances between the points in the embedding space are statistically independent. The adequacy of this approximation is assessed numerically, and a surprising result is observed in which dimension estimators can be anomalously precise for sets with reasonably uniform (nonfractal) distributions.

Hierarchical pattern formation in elastic materials

Abstract We study hierarchical structures such as branched twins in elastic materials based on a model of martensitic materials in which hierarchical twinning near the habit plane (austenite-martensite interface) is a new and crucial ingredient. The model includes (1) a triple-well potential ( θ 6 model) in local strain, (2) strain gradient terms up to second order in strain and fourth-order in gradient, and (3) all symmetry-allowed compositional fluctuation-induced strain gradient terms which favor hierarchical structures and enable communication between macroscopic (cm) and microscopic (A˚) regions essential for shape memory. Below the transition temperature ( T 0 ) we obtain the conditions under which branching of twins is energetically favorable. This hierarchy of length scales provides a related hierarchy of time scales and thus the possibility of non-exponential decay. Results based on 2D simulations of the time-dependent Ginzburg-Landau (TDGL) equation are shown for twins, tweed and hierarchy formation. We also apply stability analysis to study the formation of modulated structures at early time and obtain an approximate phase diagram for the model.

Computer simulation of martensitic textures

We consider a Ginzburg-Landau model free energy F(ϵ, e1, e2) for a (2D) martensitic transition, that provides a unified understanding of varied twin/tweed textures. Here F is a triple well potential in the rectangular strain (ϵ) order parameter and quadratic e12, e22 in the compressional and shear strains, respectively. Random compositional fluctuations η(r) (e.g. in an alloy) are gradient-coupled to ϵ, ~ − ∑rϵ(r)[(Δx2 − Δy2)η(r)] in a “local-stress” model. We find that the compatibility condition (linking tensor components ϵ(r) and e1(r), e2(r)), together with local variations such as interfaces or η(r) fluctuations, can drive the formation of global elastic textures, through long-range and anisotropic effective ϵ-ϵ interactions. We have carried out extensive relaxational computer simulations using the time-dependent Ginzburg-Landau (TDGL) equation that supports our analytic work and shows the spontaneous formation of parallel twins, and chequer-board tweed. The observed microstructure in NiAl and FexPd1 − x alloys can be explained on the basis of our analysis and simulations.

Deformable curved magnetic surfaces

We study curved magnetic surfaces in the context of soft condensed matter. Specifically, we consider classical Heisenberg spins on elastically deformable curved geometries in orthogonal curvilinear coordinates such as a cylinder, a torus, etc. We find that a mismatch of length scales (geometrical frustration) in the presence of magnetic solitons leads to an elastic soliton (deformation) on the magnetic surfaces. We illustrate the results on (i) a circular cylinder with either spin anisotropy or external magnetic field or multiple solitons, (ii) an elliptic cylinder and (iii) a torus section. Our results are applicable to microtubules and vesicles (spheroidal or toroidal) comprised of magnetic organic materials such as magnetic polymers.

Late stage cluster growth: spatial correlations and ordering on surfaces

Abstract The widely accepted models for late stage clustering, the Lifshitz-Slyozov-Wagner theory for Ostwald ripening and the self-similar Monte Carlo simulations by Family and Meakin for coalescence growth, are consistent with random spatial distributions of the clustered phase. Several detailed investigations have revealed that cluster-cluster interactions have to be further considered. Only one of these studies lead to predictions of spatial ordering when long-range repulsive forces were included. In this paper we investigate nearest neighbor cluster distance distributions for both late stage growth processes (coalescence for Ga on GaAs(0 0 1) and Ostwald ripening for Sn and In on Si(1 1 1)) and correlate the results with experimental and theoretical results for the cluster-cluster interaction in these systems. Non-random spatial distributions are found for ripening even at rather low areal cluster densities, indicating that (i) cluster-cluster interactions play a major role in the morphological evolution of ripening structures at much smaller areal fractions than previously assumed and (ii) the Gibbs-Thomson effect, in combination with a diffusion controlled exchange of matter between clusters, is sufficient as a driving force to obtain partially ordered structures.

Spinodal decomposition in binary fluids under shear flow

Abstract We discuss the effects of shear rates on the structure, rheology and kinetics of phase separation in binary fluids through numerical Langevin simulations in both two- and three-dimensions. Our major findings are as follows: (1) shear flow distorts the isotropy properties in normal spinodal decomposition. Under stronger shear, a string phase appears; (2) domains grow differently in directions parallel and perpendicular to the flow direction; (3) excess shear viscosities due to shear flow have a characteristic peak as a function of shear rates. We compare our findings with the experimental data and simulation results which do not incorporate hydrodynamics.

Critical exponents for simple non-uniform polymer networks

The authors study a number of non-uniform specified topologies and show rigorously that for certain topologies with cut edges, the critical exponent gamma t is in agreement with a conjecture given by Gaunt et al. (1984) and that the exponent nu t= nu , the exponent for self-avoiding walks. The authors also find that the scaling relations gamma t- gamma t1 and gamma t- gamma t11 are the same as for self-avoiding walks, previously conjectured only for uniform networks. By assigning an interaction energy to a nearest neighbour contact, they prove that the collapse transition for these topologies is the same as that for self-avoiding walks.

On the convergence of the series expansion analysis for self-avoiding walks attached to a surface

Extensions of exact enumeration data for self-avoiding walks attached to the surfaces of face-centered cubic and triangular lattices are reported. The convergence of the estimates for the critical points pc and exponent gamma 1 obtained by the Baker-Hunter (1973) confluent singularity analysis is discussed and comparison made with estimates of pc from the corresponding bulk series.