Disclinations in a homogenously deformed nematic elastomer

We consider a nematic elastomer which has been cross-linked in an isotropic state. As an application, we consider the time-independent, isochoric, homogeneous deformation of a right circular cylinder wherein each circular cross section of the specimen is deformed into an ellipse. We explore the possibility of the existence of disclinations when the molecular conformation is uniaxial. Numerical solution of the governing boundary-value problem indicates the presence of an isotropic core (the disclination core) of material surrounding the cylinder axis in which the polymeric chains are shaped as spherical coils. The isotropic core is bounded by a narrow transition layer across which the molecular conformation changes from spherical to uniaxial. The material thereby becomes anisotropic away from the disclination. The anisotropic regions show a markedly different distribution for the free energy. This distribution of energy determines the radius of the core to be on the order of 10⁻² μm. This work is a first step toward gaining an understanding of the existence of defective states in homogenous deformed nematic elastomers.     

Quasiconvex envelopes of energies for nematic elastomers in the small strain regime and applications

We provide some explicit formulas for the quasiconvex envelope of energy densities for nematic elastomers in the small strain regime and plane strain conditions. We then demonstrate their use as a powerful tool for the interpretation of mechanical experiments. Analytical formulas characterizing the stress-strain response in pure shear are derived, providing an easily testable benchmark for future numerical and experimental investigations on the mechanics of nematic elastomers.     

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.     

Rheo-dielectric behavior of low molecular weight liquid crystal 2. Behavior of 8CB in nematic and smectic states

Rheo-dielectric behavior was examined for 4−4^′−n-octyl-cyanobiphenyl (8CB) having large dipoles parallel to its principal axis (in the direction of the C≡N bond). In the quiescent state at all temperatures (T) examined, orientational fluctuation of the 8CB molecules was observed as dielectric dispersions at characteristic frequencies ω_c>10⁶ s⁻¹. In the isotropic state at high T, no detectable changes of the complex dielectric constant ɛ^*(ω) were found under slow flow at shear rates ˙γ≫ω_c. In the nematic state at intermediate T, the terminal relaxation intensity of ɛ^*(ω) was decreased under such slow flow. In the smectic state at lower T, the flow effect became much less significant. These results were related to the flow-induced changes of the liquid crystalline textures in the nematic and smectic states, and the differences of the rheo-dielectric behavior in these states are discussed in relation to a difference of the symmetry of molecular arrangements in the nematic and smectic textures. Received: 1 October 1998 Accepted: 13 January 1999     

Liquid–crystalline nematic polymers revisited

A new model for nematic polymers is proposed, based on the probability ψ(u,u,t) for a macromolecule to be oriented along direction u while embedded in a u environment created by its neighbours. The potential of the internal forces is written Φ(u,u) accordingly. The free energy contains a contribution ν Φ + k_BT ln ψ where the brackets mean an average over the probability distribution, while ν is the (uniform) polymer number density. An equation is derived for the time-evolution of the order parameter S = uu − I/3, together with an expression for the stress tensor. These two results offer a generalization of the Doi Model in so far as they include a distortional energy, analogue to the Frank elastic energy for low molecular mass nematics. Extending the Maier–Saupe variational procedure, we specify the way that the internal potential Φ(u,u) must be written for it to favour non-zero values of the order parameter, while giving a penalty to situations with gradients of the order parameter. The result is quite different from the potential proposed a decade ago by Marrucci and Greco (their Φ depends on u only), while it has a clear connection with the so-called Landau-de Gennes (LdG) tensor models, which are based on a free-energy depending on the order parameter and its gradients.     

Transitions in Poiseuille flow of nematic liquid crystal

Recent experiments by Sengupta et al. (Phys. Rev. Lett. 2013) [9] revealed interesting transitions that can occur in flow of nematic liquid crystal under carefully controlled conditions within a long microfluidic channel of width much larger than height, and homeotropic anchoring at the walls. At low flow rates the director field of the nematic adopts a configuration that is dominated by the surface anchoring, being nearly parallel to the channel height direction over most of the cross-section; but at high flow rates there is a transition to a flow-dominated state, where the director configuration at the channel centerline is aligned with the flow (perpendicular to the channel height direction). We analyze simple channel-flow solutions to the Leslie–Ericksen model for nematics. We demonstrate that two solutions exist, at all flow rates, but that there is a transition between the elastic free energies of these solutions: the anchoring-dominated solution has the lowest energy at low flow rates, and the flow-dominated solution has lowest energy at high flow rates.     

Mullins effect in elastomers filled with particles aligned by a magnetic field

Magnetorheological elastomers (MRE) are particulate composite materials, whose fillers are structured by a magnetic field during curing. These particles are brought in quasi-contact by the field, in a chain-like unidirectional structure. Due to this organization the local stresses between the particles is high and debonding between particles and elastomer occur at low strain. We have experimentally studied and analytically modeled the progressive breaking of the polymer-to-particle bonds. The two cases of strong and weak bonds between elastomers and particles have been studied. The analytical model correctly reproduces the stress strain curve in the presence of a debonding process although overestimating the size of the debonding cavity which is obtained by comparison between experiments and FEM simulations. The extension of the model to a chain of spheres allows to well explain the Mullins effect on MRE. Furthermore it is shown that the quality of the bonds between the particles and the elastomer does not influence the change of stiffness brought by the application of a magnetic field.     

Stress relaxation of large amplitudes and long timescales in soft thermoplastic and filled elastomers

We propose a model for describing mesoscale relaxation mechanisms in soft thermoplastic elastomers and also in the high-temperature regime of filled rubbers. The model consists of hard spheres embedded in an elastic matrix. It is solved by dissipative particle dynamics. We study the response of the model to deformations of various amplitudes. We show that it displays slow relaxation processes of large amplitudes that are related to irreversible reorganizations at a mesoscopic scale. We characterize these reorganizations as buckling of instabilities that change the local environment of the hard inclusions. Paper presented at the 3rd Annual Rheology Conference, AERC 2006, April 27–29, 2006, Crete, Greece.     

Likelihood and expected-time statistics of monodomain attractors in sheared discotic and rod-like nematic polymers

Employing a mesoscopic Doi tensor model, we develop transient statistical properties of sheared nematic polymer monodomains consistent with typical experimental protocols. Our goal is to convey to the experimentalist a list of expected outcomes, based not only on properties of the nematic liquid and imposed flow rate, but also on the timescale of the experiment and variability in the initial conditions. Step 1 is deterministic: we solve the model equations completely, then compile the flow-phase diagram of all monodomain attractors and phase transitions versus nematic concentration and Peclet number (shear rate normalized by molecular relaxation rate). Step 2 is to overlay on the phase diagram a statistical diagnostic of the expected time, t_A, to reach a small neighborhood of every attractor A. The statistics are taken over the arbitrary quiescent director angle on the sphere, modeling experiments that begin from rest. Step 3 is to explore parameter regimes with multiple attractors, where we statistically determine the likelihood of convergence to each attractor. These statistical properties are critical for any application of theoretical models to the interpretation of experimental data. If t_A is longer than the timescale of the experiment, attractor A is never fully resonated and the relevant stress and scattering predictions are those of the transients, not the attractor. In bi-stable and tri-stable parameter regimes, which are typical of nematic polymers, a distribution of monodomains of each type will populate the sample, so experimental data must be compared with weighted averages based on the likelihood of each attractor (see Grosso et al (2003) Phys Rev Lett 90:098304). The final step is to give statistics of shear stress and normal stress differences during the approach to each attractor type, as well as typical paths of the major director that are contrasted with the results of Van Horn et al (Rheol Acta (2003) 42(6):585–589) with Leslie-Ericksen theory.     

Magnetoactive elastomers with periodic and random microstructures

We investigate the behavior of magnetoactive elastomers (MAEs) with periodic and random distributions of circular and elliptical fibers. For the MAEs with periodic microstructures, we develop finite element models and determine the local fields as well as the effective properties of MAEs with rectangular and quasi-hexagonal unit cells. For the MAEs with random microstructures, we derive a closed-form expression for the effective response making use of a recently developed theory (Ponte Castañeda and Galipeau, 2011). In particular, we determine the responses to pure shear loading in the presence of a magnetic field, both of which are aligned with the geometric axes of the fibers, and examine the roles of the deformation, concentration, particle shape, and distribution on the magnetostriction, actuation stress, and the magnetically induced stiffness of the composite. We show that the coupling effects are of second order in the concentration. This is consistent with the fact that these effects are primarily the result of the interaction between inclusions. We also demonstrate explicitly that the magnetomechanical coupling of these MAEs, when subjected to aligned loading conditions, depends not only on the magnetic susceptibility, but also, crucially, on its derivative with respect to the deformation. As a consequence, we find that the magnetoelastic effects may be quite different, even for composites with similar effective susceptibilities.     

Dynamics of shear aligning of nematic liquid crystal monodomains

The equations of linear and angular momentum for nematic liquid crystals have been described with Ericksen's transversely isotropic fluid [TIF] model and solved for start-up of shear flow at constant rate and varying initial alignment conditions. An analytical solution for the rotation provides predictions of the nematic director which closely agree with experimental results of Boudreau et al. (1999), supporting the validity of Ericksen's TIF model. The solution is limited to flows where the effects of director gradients are negligible. Received: 13 September 1999/Accepted: 24 January 2000     

Director reorientation and order reconstruction: competing mechanisms in a nematic cell

We propose a model to explore the competition between two mechanisms possibly at work in a nematic liquid crystal confined within a flat cell with strong uniaxial planar conditions on the bounding plates and subject to an external field. To obtain an electric field perpendicular to the plates, a voltage is imposed across the cell; no further assumption is made on the electric potential within the cell, which is therefore calculated together with the nematic texture. The Landau-de Gennes theory of liquid crystals is used to derive the equilibrium nematic order tensor Q. When the voltage applied is low enough, the equilibrium texture is nearly homogeneous. Above a critical voltage, there exist two different possibilities for adjusting the order tensor to the applied field within the cell: plain director reorientation, i.e., the classical Freedericksz transition, and order reconstruction. The former mechanism entails the rotation of the eigenvectors of Q and can be described essentially by the orientation of the ordinary uniaxial nematic director, whilst the latter mechanism implies a significant variation of the eigenvalues of Q within the cell, virtually without any rotation of its eigenvectors, but with the intervention of a wealth of biaxial states. Either mechanism can actually occur, which yields different nematic textures, depending on material parameters, temperature, cell thickness and the applied potential. The equilibrium phase diagram illustrating the prevailing mechanism is constructed for a significant set of parameters.      

Soft elasticity and microstructure in smectic C elastomers

Smectic C elastomers are layered materials exhibiting a solid-like elastic response normal to the layers and a rubbery response in the plane. The set of strains K _C minimizing the elastic energy contains a one-parameter family of simple stretches associated with an internal degree of freedom, coming from the in-plane component of the director. We investigate soft elasticity and the corresponding microstructure by determining the quasiconvex hull of the set K _C , and use this to propose experimental tests that should make the predicted soft response observable.      

Landau theory for biaxial nematic liquid crystals with two order parameter tensors

We present a phenomenological theory for the homogeneous phases of nematic liquid crystals constituted by biaxial molecules. We propose a general polynomial potential in two macroscopic order parameter tensors that reproduces the mean-field phase diagram confirmed by Monte Carlo simulations [De Matteis et al. in Phys Rev E 72:041706 (2005)] and recently recognized to be universal [Bisi et al. in Phys Rev E 73:051709 (2006)] for dispersion force molecular pair-potentials enjoying the D _2h symmetry. The requirement that the phenomenological theory comply uniquely with this phase diagram reduces considerably the admissible phenomenological coefficients, both in their number and in the ranges where they can vary.      

The weak shear kinetic phase diagram for nematic polymers

We study the shear problem for nematic polymers as modeled by the molecular kinetic theory of Doi (1981), focusing on the anomalous slow flow regime. We provide the kinetic phase diagram of monodomain (MD) attractors and phase transitions vs normalized nematic concentration (N) and weak normalized shear rate (Peclet number, Pe). We then overlay all rheological features typically reported in experiments: alignment properties, normal stress differences and shear stress. These features play a critical role in the synthesis between theory and experiment for nematic polymers (Larson 1999; Doi and Edwards 1986). MD type is routinely used for rheological shear characterization: cf., flow-aligning 5CB (Mather et al. 1996a), tumbling PBT (Srinivasarao and Berry 1991), and 8CB (Mather et al. 1996b), evidence for a wagging regime (Mewis et al. 1997), out-of-plane kayaking modes (Larson and Ottinger 1991), and evidence for chaotic major director dynamics (Bandyopadhyay et al. 2000). MD transitions correlate with sign changes in normal stresses (Larson and Ottinger 1991; Magda et al. 1991; Kiss and Porter 1978, 1980). Furthermore, structure formation in shear devices appears to be correlated with monodomain precursor dynamics (Tan and Berry 2003; Forest et al. 2002a). In this paper we combine seminal kinetic theory results (Kuzuu and Doi 1983, 1984; Larson 1990; Larson and Ottinger 1991; Faraoni et al. 1999; Grosso et al. 2001), symmetry observations (Forest et al. 2002b), and mesoscopic results on the fate of orientational degeneracy in weak shear (Forest and Wang 2003; Forest et al. 2003a), together with our resolved numerical simulations, to provide the kinetic flow-phase diagram of Doi theory in the weak shear regime, 0<Pe<1, for infinitely thin rods. We report the "birth" of key rheological features at the onset of flow: sign changes and local maxima and minima in normal stress differences (N₁ and N₂) associated with MD transitions. These results serve as the basis for continuation of the kinetic phase diagram to Pe>1 ; as the definitive benchmark for any mesoscopic or continuum model; and experimental data can be compared in order to determine accuracy and limitations of the Doi theory in weak shear.     

Effects of strong anchoring on the dynamic moduli of heterogeneous nematic polymers

We focus on the linear viscoelastic response of heterogeneous nematic polymers to small amplitude oscillatory shear, paying special attention to the macroscopic influence of strong plate anchoring conditions. The model consists of the Stokes hydrodynamic equations with viscous and nematic stresses, coupled to orientational dynamics and structure driven by the flow gradient, an excluded-volume potential, and a two-constant distortional elasticity potential. We show that the dynamical response simplifies when plate anchoring is either tangential or homeotropic, recovering explicitly solvable Leslie–Ericksen–Frank behavior together with weakly varying order parameters across the plate gap. With these plate conditions, we establish “model consistency” so that all experimental driving conditions (plate-controlled velocity [strain] or shear stress, imposed oscillatory pressure) yield identical dynamic moduli for the same material parameters and anchoring conditions, eliminating the culpability of device influence in scaling behavior. Two physical predictions emerge that imply significant macroscopic elastic and viscous effects controlled by plate anchoring relative to flow geometry: (1) The storage modulus is enhanced by two to three orders of magnitude for homeotropic relative to parallel anchoring, across all frequencies. (2) The loss modulus exhibits enhancement of a factor of two to three for homeotropic over tangential anchoring, restricted to low frequencies. We further deduce a scaling law for the dynamic moduli versus anisotropy of the distortional elasticity potential.

基于辛本征值方法的受约束介电弹性体中褶皱不稳定性的分析

辛弹性力学已广泛应用于弹性学中各种边值问题的精确解、计算表面波模式以及预测多层超弹性薄膜中的表面褶皱。本文展示了辛分析框架还可应用于受约束介电弹性体中的表面褶皱。机械和电位移向量是两个基本变量来描述介电弹性体中机械变形与电场紧密耦合。褶皱的临界电压可以通过引入基本变量的对偶变量来从辛本征值问题中解决。本文采用扩展的W-W（Wittrick-Williams）算法和精确的积分方法,准确而高效地解决制定的辛本征值问题的本征值。通过将褶皱电压和波数与有无表面能的褶皱基准结果进行比较,验证了辛分析的有效性。辛分析框架简洁且适用于其他不稳定问题,如分层电介质弹性体、磁弹性不稳定性以及层压复合结构的微观和宏观不稳定性。     

Spinodal decomposition and nematic coarsening in a rigid-rod solution

The dynamics of a solution of rodlike liquid-crystalline molecules are simulated for the related problems of isotropic–nematic spinodal decomposition and the coarsening of misaligned nematic grains. The Doi diffusion equation for the rod distribution function is coupled with the full Onsager intermolecular potential, discretized by the finite-element method, and integrated forward in time by using a parallel, semi-implicit scheme. The Onsager potential models rod interaction on the scale of a single rod length in order to resolve accurately defects and interfaces in structure. Simulation results show the effects of rotational and translational diffusivity ratios on the mechanisms for alignment and phase separation in spinodal decomposition. As rotational motion is restricted, individual grains become more aligned prior to coalescence events. When rods are restricted to diffusive motion along their axis, the spinodal decomposition process is arrested, and the system will reach a pseudo-steady state featuring misaligned nematic grains. These results mark the first dynamic computation of the Doi diffusion equation for spinodal decomposition in nonhomogeneous rigid-rod systems. Nematic coarsening simulations show the effects of misalignment between neighboring ordered domains on the coarsening time and director field around structured interfaces. Results show that the coarsening time is dependent not only on misalignment between grain directors, but also on the tilt angle of the directors into the interface.     

磁流变弹性体若干物理量的数值分析

应用有限元方法,考虑了颗粒的磁化饱和过程与非线性磁化过程,计算得到了磁流变弹性体中的磁场分布,进而研究了在不同磁场大小、不同颗粒体积比浓度下磁流变弹性体在成链方向的相对磁导率,计算结果和实验结果取得了一致。利用Maxwell应力张量,计算了磁场引起的磁流变弹性体的附加剪切模量。分析了颗粒体积比浓度、外加磁场对磁流变弹性体的磁致剪切模量的影响。研究了颗粒为旋转椭球形状时,颗粒的放置方式与其长短轴之比对磁流变弹性体的磁致剪切模量的影响。计算结果表明,磁流变弹性体的相对磁导率随颗粒体积比浓度的增大而增大,随磁场强度的增大而减小,颗粒的形状和放置方式对磁流变效应有很大的影响。     

Superposition principles for small amplitude oscillatory shearing of nematic mesophases

The linear viscoelasticty of Leslie-Ericksen monodomain liquid crystals subjected to a bend distortion through a small amplitude oscillatory shear flow driven by harmonic wall stress is analyzed, using numerical and asymptotic methods. The viscoelastic material functions were derived using a new scaling approach that extracts the material parameters that control superposition. Small and high frequency superposition schemes for linear viscoleasticity were derived. The schemes were successfully applied to collapse the predicted loss and storage linear viscoelastic moduli of seven experimental data sets. Comparisons between different shear flows (simple shear and capillary Poiseuille) and different director distortion modes (splay and bend) shows that the superposition schemes are applicable to shear flows in any single director distortion mode.