M. Warner

Elasticity of entangled networks

Entanglements are modelled by links which make a sliding contact between polymer networks. A formal solution to this problem is given using the replica formalism and the contribution of an entanglement to the free energy of shear is given by: 12kT∑iλi2(1+η)1 + ηλi2 + kT∑log(1+ηλi2 where λi are the Cartesian extension ratios and η is a measure of the freedom of a link to slide compared with the freedom of movement of a chain. The expression quoted gives correct limits in the (trivial) case of η zero, and η infinity, when it is merely a negative contribution to the osmotic pressure of the network.

Host-symbiont recombination versus natural selection in the response of coral-dinoflagellate symbioses to environmental disturbance.

Mutualisms between reef-building corals and endosymbiotic dinoflagellates are particularly sensitive to environmental stress, yet the ecosystems they construct have endured major oscillations in global climate. During the winter of 2008, an extreme cold-water event occurred in the Gulf of California that bleached corals in the genus Pocillopora harbouring a thermally ‘sensitive’ symbiont, designated Symbiodinium C1b-c, while colonies possessing Symbiodinium D1 were mostly unaffected. Certain bleached colonies recovered quickly while others suffered partial or complete mortality. In most colonies, no appreciable change was observed in the identity of the original symbiont, indicating that these partnerships are stable. During the initial phases of recovery, a third species of symbiont B1Aiptasia, genetically identical to that harboured by the invasive anemone, Aiptasia sp., grew opportunistically and was visible as light-yellow patches on the branch tips of several colonies. However, this symbiont did not persist and was displaced in all cases by C1b-c several months later. Colonies with D1 were abundant at inshore habitats along the continental eastern Pacific, where seasonal turbidity is high relative to offshore islands. Environmental conditions of the central and southern coasts of Mexico were not sufficient to explain the exclusivity of D1 Pocillopora in these regions. It is possible that mass mortalities associated with major thermal disturbances during the 1997–1998 El Niño Southern Oscillation eliminated C1b-c holobionts from these locations. The differential loss of Pocillopora holobionts in response to thermal stress suggests that natural selection on existing variation can cause rapid and significant shifts in the frequency of particular coral–algal partnerships. However, coral populations may take decades to recover following episodes of severe selection, thereby raising considerable uncertainty about the long-term viability of these communities.

Nematic elastomers : A new state of matter ?

Recent experimental and theoretical work shows that nematic elastomers have a highly mobile director, despite being nominally solid liquid crystals. We review this work and show that such a system represents an essentially new state of matter with non-symmetric elasticity, new nematic transitions, discontinuous stress-strain relations and an ability to behave like liquids for some special class of deformations. These phenomena have been seen in large monodomain samples made in nematic fields or by multistage crosslinking under stresses. Modelling of monodomain elastomers is straightforward and successfully explains the unusual mechanical behaviour observed. The theory is a direct extension of the classical theory of conventional rubbers; we call it the neo-classical theory of elastomers. Being a molecular theory, it describes the very large extensions at which the new transitions occur. We outline problems for the future, principally those of polydomains.

Theory of nematic networks

Using classical elasticity theory, the rise in free energy upon crosslinking nematogenic polymers into a network is calculated for the isotropic and nematic phases. Spontaneous strains are allowed for in the network. The consequence of network formation upon nematic–isotropic equilibria is calculated by adding these elastic contributions to a conventional Landau theory. Memory of the crosslinking conditions yields quartic and quadratic additions to the standard Landau theory. We find that crosslinking in the isotropic state lowers the nematic–isotropic phase transition temperature compared with the unlinked case and the application of suitable stress raises it again. Crosslinking in the nematic state raises the transition temperature. We recover the mechanical critical point proposed long ago by de Gennes. Our Gaussian theory encompasses both main‐ and side‐chain polymers. The hairpin limit for main chain networks yields a modulus varying exponentially with temperature. The Landau–de Gennes free energy for comb polymers is presented for the first time.

The phase equilibria in thermotropic liquid crystalline systems

The lattice theory of hard rods with anisotropic dispersion forces presented recently is extended to the case where the rods are dispersed in a diluent. Two systems are treated: (i) rods of axial ratio x diluted with solvent molecules having x=1, and (ii) rods of axial ratio xα mixed with homologs of axial ratio xβ. Treatment of the latter system can be regarded as an extension of the Flory and Abe theory for ’’athermal’’ polydisperse systems, in which dispersion forces are inconsequential. Solutions of sufficiently long (x≳20) rods subject to anisotropic dispersion forces are predicted to exhibit complex phase equilibria involving a re‐entrant nematic phase (with a critical point) and a triple point, as well as the usual nematic–isotropic coexistence. These deductions should be relevant to thermotropic solutions of rodlike synthetic polymers with phenylene groups or other groups exhibiting anisotropic interactions. Calculations for mixtures of short rods yield more prosaic phase diagrams. It is suggested...

Rod to coil transitions in nematic polymers

The order parameter and anisotropy (elongation) of the configurations of a nematic polymer in the nematic phase are calculated. At low temperatures exponentially rapid growth of chain dimensions as a function of inverse temperature is found. In the nematic direction the chain eventually adopts a rod-like state. The thermal activation of hairpins (abrupt reversals in chain directions) causes this behaviour. However, at even lower temperatures the deviation from rod-like alignment is governed by gentle meandering away from the mean field direction. A Maier-Saupe mean field theory of the nematic-isotropic transition is constructed calculating for long chains the transition temperature as a function of chain properties and predicting a universal jump in the order parameter at the transition, Delta Sni=1/3.

Gaussian curvature from flat elastica sheets

We discuss methods of reversibly inducing non-developable surfaces from flat sheets of material at the micro-scale all the way to macroscopic objects. We analyse the elastic ground states of a nematic glass in the membrane approximation as a function of temperature for disclination defects of topological charge +1. An aim is to show that by writing an appropriate director field into such a solid, one could create a surface with Gaussian curvature, dynamically switchable from flat sheets while avoiding stretch energy. In addition to the prospect of programmable structures, such surfaces offer actuation via stretch in thin systems since when illumination is subsequently removed, unavoidable stretches return.

Changing liquid crystal elastomer ordering with light – a route to opto-mechanically responsive materials

Long, flexible chains with rods and spacers give nematic polymer melts. Crosslinking yields highly extensible nematic elastomers. Molecular fluidity and orientational order are retained. Uniquely, these elastomers change shape drastically when orientational order is lost with temperature since chains lose their elongation. Reversible changes of 400% are easy to achieve. de Gennes envisaged this coupling between macroscopic strain and nematic (and other) liquid crystalline order. Dye molecular rods are excited by photons into bent conformations. Bent rods depress orientational order and large thermal shape changes will be duplicated by light. Equally, rods may rotate away from the optical electric vector to avoid bending, thus also leading to mechanical strain. Mechanical recovery follows decay back to the molecular ground state. We sketch how these effects arise, explain how even polydomain networks deform under polarised and unpolarised light, and present new results on how nuclear magnetic resonance can reveal details of polydomain optical response. We explore non-linear absorption (photo-bleaching) leading to mechanical response where, by Beers law, little light should theoretically penetrate. In actuation, light penetrates quickly, is easy to deliver remotely, both excites and stimulates decay, and offers polarisation control over mechanics. Non-uniform director fields also control response. We illustrate novel photo-mechanical effects.

Theory of nematic backbone polymer phases and conformations

The nematic-isotropic phase transition of long flexible polymers using mean-field theory with the Maier-Saupe expression of the van der Waals interaction are described. The order parameter of nematic polymers is calculated as a function of reduced temperature, kB/T(a epsilon )1/2, where a is the quadrupolar mean-field strength per unit length of the molecules and epsilon is the bend elasticity of the polymers. The reduction factor (a epsilon )1/2 is in contrast to the factor in the Maier-Saupe theory of conventional nematics, a, indicating via epsilon the polymer aspect to the problem. The transition temperature, critical-order parameter and latent heat are given. For very long chains the transition between the nematic and isotropic phases occurs when T=0.38775 (a epsilon )1/2/kB where the order parameter is equal to 0.35642. The authors predict the temperature dependence of the order parameter and transition behaviour for different lengths of the polymer chain. A detailed procedure is suggested for comparison with experiment involving order parameters, latent entropy and configurational properties as revealed by small-angle neutron and X-ray scattering.

Mathematical modelling of the elastic properties of retina: a determination of Young's modulus.

The retina can be regarded as an elastic membrane or sheet which stretches and deforms when a force is applied to it. Isolated bovine retina was taken and a graded traction force applied to determine retinal profile as a function of force. The resulting profile can be modelled mathematically and the model then used to determine a value for the elastic constant. The value of the elastic constant obtained by this method is approximately 2 N/m. This value of the elastic constant, combined with the observed retinal thickness, yields a value of Youngs modulus for retina of approximately 2 × 104 Pa, which is about 2 orders of magnitude weaker than typical rubber. This value can then be used in modelling retinal behaviour in vivo when forces are applied to detached retina.