Iain W. Stewart

Density conservation for a coagulation equation

A Theorem is presented which proves that solutions to the coagulation equation are density conserving whenever certain growth assumptions are made on the coagulation kernel.

On the coagulation-fragmentation equation

Two Theorems are presented on the subject of non-density conserving solutions to the coagulationfragmentation equation. The first deals with pure coagulation processes and the second with pure fragmentation. A third Theorem provides an asymptotic property for general solutions of the pure fragmentation equation. Some examples of exact solutions are given.

Dynamics of jumpwise temperature pitch variations in planar cholesteric layers for a finite strength of surface anchoring

The dynamics of pitch jumps in cholesteric layers with a finite surface anchoring strength under variations in temperature is investigated theoretically. General expressions are presented that connect the dynamics of pitch jumps with the parameters that determine the process, such as the viscosity, the specific form of the anchoring potential, and the dimensionless parameter Sd = K22/Wd, where W is the depth of the anchoring potential, K22 is the twist elastic modulus, and d is the layer thickness. It is found that the shape of the anchoring potential significantly influences the temporal behavior of the cholesteric helix in the process of a pitch jump. To illustrate this revealed dependence of the pitch jump dynamics on the shape and strength of the anchoring potential, the problem was investigated for two different models of the surface anchoring potential for a jump mechanism in connection with the director at the surface slipping over the barrier of the anchoring potential. Calculations for the unwinding (winding) of the helix in the process of the jump were performed to investigate the case of infinitely strong anchoring on one surface and finite anchoring on the other, which is important in applications. The results show that an experimental investigation of the dynamics of the pitch jumps will make it possible to distinguish different shapes of the finite strength anchoring potential and, in particular, it will provide a means for determining whether the well-known Rapini-Papoular anchoring potential is the best suited potential relevant to the dynamics of pitch jumps in cholesteric layers with a finite surface anchoring strength. The optimal conditions for experimental observation of these phenomena are briefly considered.

A Novel Method for Measuring Compression Constants in Smectics

This article considers a refined analysis of the Helfrich–Hurault effect in smectic A when the layer normal a and the director n need not coincide. We demonstrate that the Helfrich-Hurault transition threshold decreases as a and n decouple. The dependence of this transition upon the elastic coupling constant B 1 is discussed and its qualitative universal influence upon the reduced threshold is depicted. The implications for the measurement of compression and coupling constants is discussed.

Nonsingular walls in plane cholesteric layers

The structure of a straight interface (wall) between regions with differing values of the pitch in planar cholesteric layers with finite strength of the surface anchoring is investigated theoretically. It is found that the shape and strength of the anchoring potential influences essentially the structure of the wall and a motionless wall between thermodynamically stable regions without a singularity in the director distribution in the layer can exist for sufficiently weak anchoring only. More specifically, for the existence of such a wall the dimensionless parameter Sd = K22/Wd (where W is the depth of the anchoring potential, K22 is the elastic twist modulus and d is the layer thickness) should exceed its critical value, which is dependent on the shape of the anchoring potential. General equations describing the director distribution in the wall are presented. Detailed analysis of these equations is carried out for the case of infinitely strong anchoring at one surface and finite anchoring strength at the second layer surface. It is shown that the wall width L is directly dependent upon the shape and strength of the anchoring potential and that its estimate ranges from d to (dLp)1/2 (where Lp = K22/W is the penetration length), corresponding to different anchoring strengths and shape potentials. The dependence of the director distribution in the wall upon all three Frank elastic moduli is analytically found for some specific limiting cases of the model anchoring potentials. Motion of the wall is briefly investigated and the corresponding calculations performed under the assumption that the shape of a moving wall is the same as a motionless one. It is noted that experimental investigation of the walls in planar cholesteric layers can be used for the determination of the actual shape of surface anchoring potentials.

Layer distortions induced by a magnetic field in planar samples of smectic C liquid crystals

This article derives theoretical results for the onset of the Helfrich-Hurault transition in smectic C liquid crystals induced by a magnetic field applied parallel to the smectic layers. A suitable quadratic energy in terms of the smectic layer displacement u is derived from the nonlinear version of the smectic C energy. This energy is minimized via averaging to enable the calculation of a critical field strength H c for the onset of layer distortions. Comparisons are made with known results for the corresponding geometry in the smectic A case. An estimate for the value of the smectic C elastic constant A 12 can also be made by considering characteristic length scales.

Director orientation in nematic liquid crystals using crossed electric and magnetic fields

A theoretical investigation is made of static and dynamic effects when a nematic liquid crystal is subjected to crossed electric and magnetic fields. In the static problem a twist-wall solution is discussed for a semi-infinite sample of nematic; a control parameter, q, describes the relationship between the fields and their crossed angle and is used to characterize the solution. For an infinite sample of nematic this parameter also turns out to characterize the types of solution (travelling waves) which are available for the nonlinear dynamic equation when certain approximations are made. The type of solution which occurs is shown to depend crucially on the boundary conditions, the relative magnitudes of the electric and magnetic fields and their crossed angle.

Energetics of lipid bilayers with applications to deformations induced by inclusions

A new energy for the description of large deformations of lipid bilayers is formulated with mathematical rigor. This energy is derived by considering the smectic A liquid crystalline nature of lipid bilayers and the coupling between the deformations of the layers and their constituent lipid molecules. Analogies between smectic A liquid crystals, with an infinite number of layers, and lipid bilayers, with a finite number of layers, are further discussed. The novelty of the energy density is demonstrated by studying the large deformations of planar lipid bilayers induced by cylindrical inclusions. The results of this study are directly compared with the results obtained using Mays theoretical framework [May, Eur. Biophys. J., 2000, 29, 17–28] in which small deformations are assumed. As expected, the proposed energy density predicts larger distortions of the lipid molecules and deformations of the lipid bilayers close to an inclusion.

Pressure effects on the equilibrium configurations of bilayer lipid membranes

Novel circulation-active cyclohexenecarboxylates of the formula (I) in which R1 is an organic radical, R2 is an optionally substituted aryl radical, R3 is an optionally substituted alkyl or aryl radical, X and Y are -CN, -COR4, -SO2R4, -COOR5, -CONR6R7 or -SO2NR6R7, and R4, R5, R6 and R7 are various organic radicals.

Flow past finite obstacles in smectic liquid crystals: permeative flow induced SA to SC phase transition

Abstract The hydrodynamic equations for flow past a finite barrier within a ‘bookshelf’ aligned smectic A liquid crystal are solved. Solutions are found for the flow components within (v x) and normal (v z) to the smectic layers. The behaviour of the v z component is confirmed by experimental observation of both the buckling instability and a permeative flow induced SA to Sc phase transition.