Non-Gaussian chain statistics and finite extensibility in liquid crystal elastomers

In this work we will derive an anisotropic generalisation of the finitely extensible chain model, due to Kuhn and Gr\"un, which is well known in rubber elasticity. This provides a chain energy that couples elastic behaviour to a probability distribution describing the orientations of liquid crystal monomers within a main chain elastomer. The key point is to invoke a maximum relative entropy assumption on the distribution of bond angles in an observed chain. The chain energy's fourth order Taylor expansion is also given, which couples to the second and fourth moments of the nematic distribution function only

Maximum entropy methods as the bridge between macroscopic and microscopic theory

This paper investigates a function of macroscopic variables known as the singular potential, building on previous work by Ball and Majumdar. The singular potential is a function of the admissible statistical averages of probability distributions on a state space, defined so that it corresponds to the maximum possible entropy given known observed statistical averages, although non-classical entropy-like objective functions will also be considered. First the set of admissible moments must be established, and under the conditions presented in this work the set is open, bounded and convex allowing a description in terms of supporting hyperplanes, which provides estimates on the development of singularities for related probability distributions. Under appropriate conditions it is shown that the singular potential is strictly convex, as differentiable as the microscopic entropy and blows up uniformly as the macroscopic variable tends to the boundary of the set of admissible moments. Applications of the singular potential are then discussed, and particular consideration will be given to certain free-energy functionals typical in mean-field theory, demonstrating an equivalence between certain microscopic and macroscopic free-energy functionals. This allows statements about L^1-local minimisers of Onsager's free energy to be obtained which cannot be given by two-sided variations, and overcomes the need to ensure local minimisers are bounded away from zero and infinity before taking bounded variations. The analysis also permits the definition of a dual order parameter for which Onsager's free energy allows an explicit representation. Also the difficulties in approximating the singular potential by everywhere defined functions, in particular by polynomials, are addressed with examples demonstrating the failure of the Taylor approximation to preserve shape properties of the singular potential

Convex Integration Arising in the Modelling of Shape-Memory Alloys: Some Remarks on Rigidity, Flexibility and Some Numerical Implementations

We study convex integration solutions in the context of the modelling of shape-memory alloys. The purpose of the article is two-fold, treating both rigidity and flexibility properties: Firstly, we relate the maximal regularity of convex integration solutions to the presence of lower bounds in variational models with surface energy. Hence, variational models with surface energy could be viewed as a selection mechanism allowing for or excluding convex integration solutions. Secondly, we present the first numerical implementations of convex integration schemes for the model problem of the geometrically linearised two-dimensional hexagonal-to-rhombic phase transformation. We discuss and compare the two algorithms from [RZZ16] and [RZZ17].Comment: 35 pages, 14 figure

Contributions of Repulsive and Attractive Interactions to Nematic Order

Both repulsive and attractive molecular interactions can be used to explain the onset of nematic order. The object of this paper is to combine these two nematogenic molecular interactions in a unified theory. This attempt is not unprecedented; what is perhaps new is the focus on the understanding of nematics in the high density limit. There, the orientational probability distribution is shown to exhibit a unique feature: it has compact support on configuration space. As attractive interactions are turned on, the behavior changes, and at a critical attractive interaction strength, thermotropic behavior of the Maier-Saupe type is attained.Comment: 14 pages, 4 figure

The clinical application of PET/CT: a contemporary review

The combination of positron emission tomography (PET) scanners and x-ray computed tomography (CT) scanners into a single PET/CT scanner has resulted in vast improvements in the diagnosis of disease, particularly in the field of oncology. A decade on from the publication of the details of the first PET/CT scanner, we review the technology and applications of the modality. We examine the design aspects of combining two different imaging types into a single scanner, and the artefacts produced such as attenuation correction, motion and CT truncation artefacts. The article also provides a discussion and literature review of the applications of PET/CT to date, covering detection of tumours, radiotherapy treatment planning, patient management, and applications external to the field of oncology

Density functional theory for dense nematics with steric interactions

The celebrated work of Onsager on hard particle systems, based on the truncated second order virial expansion, is valid at relatively low volume fractions for large aspect ratio particles. While it predicts the isotropic-nematic phase transition, it fails to provide a realistic equation of state in that the pressure remains finite for arbitrarily high densities. In this work, we derive a mean field density functional form of the Helmholtz free energy for nematics with hard core repulsion. In addition to predicting the isotropic-nematic transition, the model provides a more realistic equation of state. The energy landscape is much richer, and the orientational probability distribution function in the nematic phase possesses a unique feature: it vanishes on a nonzero measure set in orientational space

On a probabilistic model for martensitic avalanches incorporating mechanical compatibility

Building on the work in [BCH15, CH18, TIVP17], in this article we propose and study a simple, geometrically constrained, probabilistic algorithm geared towards capturing some aspects of the nucleation in shape-memory alloys. As a main novelty with respect to the algorithms in [BCH15, CH18, TIVP17] we include mechanical compatibility. The mechanical compatibility here is guaranteed by using convex integration building blocks in the nucleation steps. We analytically investigate the algorithm’s convergence and the solutions’ regularity, viewing the latter as a measure for the fractality of the resulting microstructure. We complement our analysis with a numerical implemenation of the scheme and compare it to the numerical results in [BCH15, CH18, TIVP17]