Timothy J. Atherton

Maximizing the hyperpolarizability poorly determines the potential

The dimensionless zero-frequency electronic first hyperpolarizability 3−1/4βE107/2m3/2(eℏ)−3 of an electron in one dimension was maximized by adjusting the shape of a piecewise linear potential. Careful maximizations converged quickly to 0.708951 with increasing numbers of parameters. The Hessian shows that β is strongly sensitive to only two parameters in the potential: sensitivity to additional parameters decreases rapidly. With more than two parameters, a wide range of potentials and an apparently narrower range of wavefunctions have nearly optimal hyperpolarizability. Modulations of the potential to which the unique maximum is insensitive were characterized. Prospects for concise description of the two important constraints on near-optimum potentials are discussed.

Observations of focal conic domains in smectic liquid crystals aligned on patterned self‐assembled monolayers

Patterned Self‐Assembled Monolayers (SAMs) promoting both homeotropic and planar degenerate alignment of nCBs in their smectic‐A phase were created using microcontact printing of functionalized organothiols on gold films. By patterning the surface with homeotropic and planar aligning SAMs, the location and formation of the focal conic domains (FCDs) can be controlled. Polarizing microscopy was used to study the formation of FCDs in circle, stripe and checkerboard pattern geometries. Fluorescent confocal microscopy (FCM) was used for the first time to measure the eccentricity of FCDs that form along a stripe pattern.

Neuronal alignment on asymmetric textured surfaces

Axonal growth and the formation of synaptic connections are key steps in the development of the nervous system. Here, we present experimental and theoretical results on axonal growth and interconnectivity in order to elucidate some of the basic rules that neuronal cells use for functional connections with one another. We demonstrate that a unidirectional nanotextured surface can bias axonal growth. We perform a systematic investigation of neuronal processes on asymmetric surfaces and quantify the role that biomechanical surface cues play in neuronal growth. These results represent an important step towards engineering directed axonal growth for neuro-regeneration studies.

Optimizing the second hyperpolarizability with minimally parametrized potentials

The dimensionless zero-frequency intrinsic second hyperpolarizability γint=γ/4E10−5m−2(eℏ)4 was optimized for a single electron in a 1D well by adjusting the shape of the potential. Optimized potentials were found to have hyperpolarizabilities in the range −0.15⪅γint⪅0.60; potentials optimizing gamma were arbitrarily close to the lower bound and were within ∼0.5% of the upper bound. All optimal potentials possess parity symmetry. Analysis of the Hessian of γint around the maximum reveals that effectively only a single parameter, one of those chosen in the piecewise linear representation adopted, is important to obtaining an extremum. Prospects for designing chromophores based on the design principle here elucidated are discussed.

Energy Minimization for Liquid Crystal Equilibrium with Electric and Flexoelectric Effects

This paper outlines an energy-minimization finite-element approach to the modeling of equilibrium configurations for nematic liquid crystals in the presence of internal and external electric fields. The method targets minimization of free energy based on the electrically and flexoelectrically augmented Frank--Oseen free energy models. The Hessian, resulting from the linearization of the first-order optimality conditions, is shown to be invertible for both models when discretized by a mixed finite-element method under certain assumptions. This implies that the intermediate discrete linearizations are well-posed. A coupled multigrid solver with Vanka-type relaxation is proposed and numerically vetted for approximation of the solution to the linear systems arising in the linearizations. Two electric model numerical experiments are performed with the proposed multigrid solver. The first compares the algorithms solution of a classical Freedericksz transition problem to the known analytical solution and demons...

An Energy-Minimization Finite-Element Approach for the Frank--Oseen Model of Nematic Liquid Crystals

This paper outlines an energy-minimization finite-element approach to the computational modeling of equilibrium configurations for nematic liquid crystals under free elastic effects. The method targets minimization of the system free energy based on the Frank-Oseen free-energy model. Solutions to the intermediate discretized free elastic linearizations are shown to exist generally and are unique under certain assumptions. This requires proving continuity, coercivity, and weak coercivity for the accompanying appropriate bilinear forms within a mixed finite-element framework. Error analysis demonstrates that the method constitutes a convergent scheme. Numerical experiments are performed for problems with a range of physical parameters as well as simple and patterned boundary conditions. The resulting algorithm accurately handles heterogeneous constant coefficients and effectively resolves configurations resulting from complicated boundary conditions relevant in ongoing research.

Thermal characterisation of thermotropic nematic liquid-crystalline elastomers

ABSTRACT Nematic liquid-crystalline elastomers (LCEs) are weakly cross-linked polymeric networks that exhibit rubber elasticity and liquid-crystalline orientational order due to the presence of mesogenic groups. Three end-on side-chain nematic LCEs were investigated using real-time synchrotron wide-angle X-ray scattering (WAXS), differential scanning calorimetry (DSC), and thermogravimetry (TG) to correlate the thermal behaviour with structural and chemical differences among them. The elastomers differed in cross-linking density and mesogen composition. Thermally reversible glass transition temperature, Tg, and nematic-to-isotropic transition temperature, Tni, were observed upon heating and cooling. By varying the heating rate, Tg0 and Tni0 were determined at zero heating rate. The temperature dependence of the orientational order parameter was determined from the anisotropic azimuthal angular distribution of equatorial reflections seen during real-time WAXS. Results show that the choice of cross-linking unit, its shape, density, and structure of co-monomers, all influence the temperature range over which the thermal transitions take place. Including multi-ring aromatic groups as cross-linkers increased the effective stiffness of the cross-linking, resulting in a higher glass transition temperature. The nematic-to-isotropic transition temperature increased in the presence of multi-ring aromatic structures, as either cross-linkers or mesogens, particularly when the multi-ring structures were larger than the low-molar-mass mesogen common to all three samples. Graphical Abstract

Shape minimisation problems in liquid crystals

ABSTRACT We consider a class of liquid crystal free-boundary problems for which both the equilibrium shape and internal configuration of a system must simultaneously be determined, for example in films with air–liquid or fluid–liquid crystal interfaces and elastomers. We develop a finite element algorithm to solve such problems with dynamic mesh control, achieved by supplementing the free energy with an auxiliary functional that promotes mesh quality and is minimised in the null space of the energy. We apply this algorithm to a flexible capacitor, as well as to determine the shape of liquid crystal tactoids as a function of the surface tension and elastic constants. These are compared with theoretical predictions and experimental observations of tactoids from the literature. GRAPHICAL ABSTRACT

Neuronal growth as diffusion in an effective potential.

Current understanding of neuronal growth is mostly qualitative, as the staggering number of physical and chemical guidance cues involved prohibit a fully quantitative description of axonal dynamics. We report on a general approach that describes axonal growth in vitro, on poly-D-lysine-coated glass substrates, as diffusion in an effective external potential, representing the collective contribution of all causal influences on the growth cone. We use this approach to obtain effective growth rules that reveal an emergent regulatory mechanism for axonal pathfinding on these substrates.

Anomalous uniform domain in a twisted nematic cell constructed from micropatterned surfaces

We have discovered an optically uniform type of domain that occurs in twisted nematic (TN) cells that are constructed from substrates chemically patterned with stripes via microcontact printing of self-assembled monolayers; such domains do not occur in TN cells constructed from uniform substrates. In such a cell, the azimuthal anchoring at the substrates is due to the elastic anisotropy of the liquid crystal rather than the conventional rubbing mechanism. A model is presented that predicts the relative stability of the twisted and anomalous states as a function of the material and design parameters.