Shear cell rupture of nematic liquid crystal droplets in viscous fluids

We model the hydrodynamics of a shear cell experiment with an immiscible nematic liquid crystal droplet in a viscous fluid using an energetic variational approach and phase-field methods [86]. The model includes the coupled system for the flow field for each phase, a phase-field function for the diffuse interface and the orientational director field of the liquid crystal phase. An efficient numerical scheme is implemented for the two-dimensional evolution of the shear cell experiment for this initial data. The same model reduces to an immiscible viscous droplet in a viscous fluid, which we simulate first to compare with other numerical and experimental behavior. Then we simulate drop deformation by varying capillary number (independent of liquid crystal physics), liquid crystal interfacial anchoring energy and Oseen–Frank distortional elastic energy. We show the number of eventual droplets (one to several) and “beads on a string” behavior are tunable with these three physical parameters. All stable droplets possess signature quadrupolar shear and normal stress distributions. The liquid crystal droplets always possess a global surface defect structure, called a boojum, when tangential surface anchoring is imposed. Boojums [79], [32] consist of degree +1/2 and ?1/2 surface defects within a bipolar global orientational structure.     

Drops of nematic liquid crystals

Dedicated to Bernard D. Coleman on the occasion of his sixtieth birthday     

Self-propelling,coalescing droplets

This work proposes and explores a new propulsion mechanism for sessile droplets which could be of interest for microfluidic applications. This mechanism relies on the Marangoni stresses resulting from the surface tension gradient arising when two droplets of different surface tensions coalesce. We argue that the tendency of the fluid to flow towards regions of higher surface tension is sufficient to displace the droplet. The coalescence of two miscible, partially wetting droplets with different surface tensions is investigated theoretically in this paper and modeled in the lubrication approximation framework. The problem is described by a set of three highly non-linear, coupled partial differential equations which is solved with a commercial finite element code. The analysis reveals two important dimensionless numbers which govern the flow characteristics, one related to the strength of the surface tension gradient and the other to the diffusion time scale. The numerical results confirm the occurrence of the self-propulsion behavior and a parametric study is performed to explore the role of the two dimensionless numbers on the propulsion speed and the total displacement. Unsurprisingly, self-propulsion is enhanced for larger values of the surface tension contrast between the two droplets and smaller values of the diffusion time scale which results in more time for the surface tension gradient to act.     

Disclinated states in nematic elastomers

We present a theory for uniaxial nematic elastomers with variable asphericity. As an application of the theory, we consider the time-independent, isochoric radial expansion of a right circular cylinder. Numerical solutions to the resulting differential equation are obtained for a range of radial expansions. For all expansions considered, there exists an isotropic core of material surrounding the cylinder axis where the asphericity vanishes and in which the polymeric chains are shaped as spherical coils. This region, corresponding to a disclination of strength +1 along the axis, is bounded by a narrow transition layer across which the asphericity increases rapidly and attains a non-trivial positive value. The material thereby becomes anisotropic away from the disclination so that the polymeric chains are shaped as ellipsoidal coils of revolution prolate about the cylinder radius. In accordance with the area of steeply changing asphericity between isotropic and anisotropic regimes, a marked drop in the free-energy density is observed. The boundary of the disclination core is associated with the location of this energy drop. For realistic choices of material parameters, this criterion yields a core on the order of 10⁻² μm, which coincides with observations in conventional liquid-crystal melts. Also occurring at the core boundary, and further confirming its location, are sharp transitions in the behavior of the constitutively determined contributions to the deformational stress and a change in the pressure. Furthermore, the constitutively determined contribution to the orientational stress is completely concentrated at the core boundary. The total energy shows a definitive preference for disclinated states.     

Jeffrey-Hamel flow of Leslie-Ericksen nematic liquids

A similarity solution of the Leslie-Ericksen equations for nematic liquid crystals is obtained for flow between converging and diverging planar walls (Jeffrey-Hamel flow). There are three regions in the flow: extensional or compressional flow near the centerline, shear near the wall, and a wall boundary layer in which elastic stresses control the transition from the wall-induced orientation to the bulk behavior. The boundary layer thickness is obtained in closed form; the scaling with the Ericksen number depends on whether or not the boundary layer extends into the region of extensional flow. Imposition of a magnetic field with an azimuthal component in a converging flow can result in a Freedericksz-like transition from radial to transverse orientation at the center line at a critical field strength. This transition provides a new means to measure the irrotational viscosity λ₂.     

Ogden-type energies for nematic elastomers

Ogden-type extensions of the free-energy densities currently used to model the mechanical behavior of nematic elastomers are proposed and analyzed. Based on a multiplicative decomposition of the deformation gradient into an elastic and a spontaneous or remanent part, they provide a suitable framework to study the stiffening response at high imposed stretches. Geometrically linear versions of the models (Taylor expansions at order two) are provided and discussed. These small strain theories provide a clear illustration of the geometric structure of the underlying energy landscape (the energy grows quadratically with the distance from a non-convex set of spontaneous strains or energy wells). The comparison between small strain and finite deformation theories may also be useful in the opposite direction, inspiring finite deformation generalizations of small strain theories currently used in the mechanics of active and phase-transforming materials. The energy well structure makes the free-energy densities non-convex. Explicit quasi-convex envelopes are provided, and applied to compute the stiffening response of a specimen tested in plane strain extension experiments (pure shear).     

Viscous forces on nematic defects

The effect of backflow in defect dynamics is assessed by computing the viscous force on point and line defects that move due to reorientation. It is found that defects with a positive winding number are accelerated while defects with a negative winding number are slowed down by the backflow. The results are in agreement with experimental and numerical results for defect annihilation.     

Theory of interfacial dynamics of nematic polymers

 The interfacial momentum and torque balance equations for deforming interfaces between nematic polymers and isotropic viscous fluids are derived and analyzed with respect to shape selection and interfacial nematic ordering. It is found that the interfacial momentum balance equation for nematic interfaces involves bending forces that act normal to the interface, and that interfacial pressure jumps may exist even for planar surfaces. In addition tangential forces on nematic interfaces arise in the presence of surface gradients of the tensor order parameter. The torque balance equation shows that couple stress jumps are balanced by the surface molecular field. The interfacial balance equations are shown to be coupled such that nematic ordering depends on shape and vice versa. The governing dimensionless numbers for deforming nematic polymer interfaces are identified and the limiting regimes are discussed in reference to related experimental data. It is found that the ratio of Frank elasticity to surface anchoring controls whether the surface tensor order parameter deviates from its preferred equilibrium value. Whether the shape is affected, depends on the relative magnitudes of the isotropic surface tension, Frank bulk elasticity, and anchoring energy, and capillary number. Received: 16 April 1999/Accepted: 19 August 1999     

Binary collision dynamics of fuel droplets

An experimental study has been carried out in which the collision dynamics of two n-haxane fuel droplets are studied. The experiments are performed on the collision of two burning droplets, as well as two nonburning droplets, to assess the influence of the high temperature combustion environment on the dynamics of the collision.The results indicate that as the Weber number is increased, the collision type moves toward higher energy collision, and for the same Weber number, different types of collisions, depending on the local value of the collision impact parameter, may occur. In the range of the Weber numbers studied, the results show that for the nonburning droplets, the collision type can be bouncing, grazing, temporary coalescence-satellite generating, or permanent coalescence, depending on the local value of the impact parameter. For the burning droplets in the same initial Weber number range, only temporary coalescence and permanent coalescence are observed.     

Deposition of droplets from turbulent stream

A model of particle deposition from a turbulent stream is presented. It is based on the modified stopping distance concept which allows for the difference between particle and eddy diffusivities, recognizing that the particle has equal probability to move toward the wall or back into the turbulent core. The theory is tested against the data which cover a wide range of droplet size and duct Reynolds number for different surface configuration. A quite satisfactory agreement has been found in all examined cases.     

Free-energy density functions for nematic elastomers

The recently proposed neo-classical theory for nematic elastomers generalizes standard molecular-statistical Gaussian network theory to allow for anisotropic distributions of polymer chains. The resulting free-energy density models several of the novel properties of nematic elastomers. In particular, it predicts the ability of nematic elastomers to undergo large deformations with exactly zero force and energy cost—so called soft elasticity. Although some nematic elastomers have been shown to undergo deformations with unusually small applied forces, not all do so, and none deform with zero force. Further, as a zero force corresponds to infinitely many possible deformations in the neo-classical theory, this non-uniqueness leads to serious indeterminacies in numerical schemes. Here we suggest that the neo-classical free-energy density is incomplete and propose an alternative derivation that resolves these difficulties. In our approach, we use the molecular-statistical theory to identify appropriate variables. This yields the choice for the microstructural degrees of freedom as well as two independent strain tensors (the overall macroscopic strain plus a relative strain that indicates how the deformation of the elastomeric microstructure deviates from the macroscopic deformation). We then propose expressions for the free-energy density as a function of the three quantities and show how the material parameters can be measured by two simple tests. The neo-classical free-energy density can be viewed as a special case of our expressions in which the free-energy density is independent of the overall macroscopic strain, thus supporting our view that the neo-classical theory is incomplete.     

Continuum theory for nematic liquid crystals

This paper presents a formulation of continuum theory for nematic liquid crystals based upon the balance laws for linear and angular momentum, that derives directly expressions for stress and couple stress in these transversely isotropic liquids. This approach therefore avoids the introduction of generalised forces or torques associated with the director describing the axis of transverse isotropy.     

Constrained stability for biaxial nematic phases

We study the invariance properties of the molecular Hamiltonian interaction put forward by Straley to describe biaxial nematic phases. We show that the reduction to two out of four scalar order parameters, which was accidently remarked upon in the literature, is indeed a rigorous consequence of the Hamiltonian invariance for specific values of the interaction parameters. The stability analysis of the mean-field free energy in the reduction classes for the order parameters reveals a sequence of Landau triple points.     

Surface waves in nematic liquid crystals

The problem of small-amplitude harmonic wave propagation along the surface of an incompressible nematic liquid crystal is considered when the evolution of the orientation vector is specified by viscous stresses and the orientational elasticity can be ignored. An analytic solution and a dispersion relation are obtained in the case of the planar and homeotropic initial orientation of this vector.     

Adhesion of viscoelastic bodies

The influence of surface energy on the contact between viscoelastic bodies is investigated. An equation is derived for its effect upon the contact zone after reaching the equilibrium position. As an example of an application, the adhesion of a viscoelastic standard solid, which may be considered as a model of the mechanical properties of living cells, is described.     

Elastic forces on nematic point defects

Received April 30, 2002 / Published online May 21, 2002 Communicated by Harald Ehrentraut, Darmstadt     

On the number of droplets in aerosols

The number of droplets which may be formed with a supersaturated vapor in presence of a gas cannot exceed a number proportional to (p_v−p_v0)⁴ where p_v and p_v0 denote at the same temperature the pressure of the supersaturated vapor–gas mixture and the pressure of the saturated vapor–gas mixture. The energy necessary to the droplet formation is also bounded by a number proportional to (p_v−p_v0)².     

Axisymmetric configurations of bipolar liquid crystal droplets

Received July 4, 2001 / Published online April 10, 2002