Thermomechanical modeling of the thermo-order-mechanical coupling behaviors in liquid crystal elastomers

Liquid crystal elastomer is a kind of anisotropic polymeric material, with complicated micro-structures and thermo-order-mechanical coupling behaviors. In this paper, we propose a method to systematically model these coupling behaviors. We derive the constitutive model in full tensor structure according to the Clausius-Duhem inequality. Two of the constitutive equations represent the mechanical equilibrium and the other two represent the phase equilibrium. Choosing the total free energy as the combination of the neo-classical free energy and the Landau-de Gennes nematic free energy, we obtain the Cauchy stress-deformation gradient relation and the order-mechanical coupling equations. We find the analytical homogeneous solutions of the deformation for the typical mechanical loadings, such as uniaxial stretch, and simple shear in any directions. We also compare the compression behavior of prolate liquid crystal elastomers with the stretch behavior of oblate liquid crystal elastomers. As a result, the stress, strain, temperature, order parameter, biaxiality and the direction of the director of liquid crystal elastomers couple with each other. When the prolate liquid crystal elastomer sample is stretched in the direction parallel to its director, the deviatoric stress makes the mesogens more order and increase the transition temperature. When the sample is sheared or stretched in the direction non-parallel to the director, the director of the liquid crystal elastomer will rotate, and the biaxiality will be induced. Because of the order-mechanical coupling, under infinitesimal deformation, liquid crystal elastomer has anisotropic Young’s modulus and zero shear modulus in the direction parallel or perpendicular to the director. While for the oblate liquid crystal elastomers, the stretch parallel to the director will cause the rotation of the director and induce the biaxiality.     

Continuum Electromechanical Theory for Nematic Continua with Application to Freedericksz Instability

Of interest in this work are nematic continua that exhibit electromechanical coupling. The first part of this paper presents a novel variational formulation with a potential energy depending on four independent variables (the displacement, director, specific polarization and electric displacement perturbation). Variations of the potential energy with respect to each one of these variables lead to the governing mechanical equilibrium and constitutive relations plus Maxwell’s equations.The proposed variational formulation is next applied to the study of bifurcation of an infinite layer of a nematic liquid crystal confined between two parallel plates and subjected to a uniform electric field perpendicular to these plates under full anchoring boundary conditions. As the electric field exceeds a critical value, the nematic directors which are initially parallel to the plates, rotate and tend to align with the electric field orientation. This phenomenon, termed in the literature as Freedericksz transition, is treated here as a bifurcation problem using a fully 2D formulation. It is shown that the solution corresponding to the lowest applied electric field, also termed the critical load, is uniform in the direction parallel to the plates and that the corresponding bifurcated path is stable near this critical load. This result holds for arbitrary positive constants of the Frank-Oseen energy (and values of electric susceptibility constants that allow bifurcation) and justifies the 1D treatment of the Freedericksz transition in 2D settings that is widely adopted in the liquid crystal literature. An asymptotic analysis of the supercritical, stable bifurcated equilibrium path about the critical load is also presented and compared with the exact bifurcated solution.     

Relaxation of Multiwell Energies in Linearized Elasticity and Applications to Nematic Elastomers

The relaxation of a free-energy functional which describes the order–strain interaction in nematic elastomers is obtained explicitly. We work in the regime of small strains (linearized kinematics). Adopting the uniaxial order tensor theory (Frank model) to describe the liquid crystal order, we prove that the minima of the relaxed functional exhibit an effective biaxial nematic texture, as in the de Gennes order tensor model. In particular, this implies that, at a sufficiently macroscopic scale, the response of the material is soft even if the order of the system is assumed to be fixed at the microscopic scale. The relaxed energy density satisfies a solenoidal quasiconvexification formula.     

On the Derivative of Some Tensor-Valued Functions

An explicit expression of the derivative of the square root of a tensor is provided, by using the expressions of the derivatives of the eigenvalues and eigenvectors of a symmetric tensor. Starting from this result, the derivatives of the right and left stretch tensor U, V and of the rotation R with respect to the deformation gradient F, are calculated. Expressions for the material time derivatives of U, V and R are also given. This revised version was published online in August 2006 with corrections to the Cover Date.     

Magnetic field-induced changes in molecular order in nematic liquid crystals

d there are two inherent characteristic lengths, the nematic correlation length and the magnetic coherence length . As the magnetic field increases the magnetic coherence length decreases and the relative ordering of the three length scales determines the director and scalar order parameter configuration through the cell. We use asymptotic expansions in regions defined by these length scales to analytically determine the molecular configuration in terms of these variables. Specifically, we investigate the boundary layer between the cell substrate and the bulk nematic material when strong anchoring forces the nematic director in a different direction to that of the applied field. We find that at low field strengths the classical picture of liquid crystal/magnetic field interaction occurs, that is, the director orientation is governed by the surface alignment until a transition occurs as the magnetic coherence length becomes comparable to the cell thickness and the director changes orientation so as to align with the magnetic field. At high field strengths, we find that a field-induced reduction of the molecular order occurs in a region close to the cell boundary. We are able to analytically determine the director and scalar order parameter configurations for the majority of field strengths and where analytical solutions are not found a numerical solution is presented. It is hoped that further work will extend this basis of analytical solutions to include a solution for all field strengths and for different cell configurations. Received July 31, 2001 / Published online May 21, 2002     

The shear flow behavior of LCPs based on a generalized Doi model with distortional elasticity

We consider the shear flow behavior of nematic LCPs, modeled via an extension of the Doi theory that incorporates the mean-field nematic potential due to Marrucci and Greco to account for distortional elasticity. Based upon the constitutive model that derives from this starting point, we utilize finite-element methods to investigate the LCP behavior in a planar shear flow. We assume that the LCP is pinned at the walls and is initially in its equilibrium configuration. The goal of our simulations is to explore the evolution of the LCP structure and the flow. Our results show that in-plane tumbling instabilities lead to a non-uniform orientation field, which, in turn, arrests tumbling. The resulting quasi-steady-state texture is characterized by a length scale that seems to be consistent with a Marrucci-like scaling. When we allow for out-of-plane tipping of the director, we predict an out-of-plane director instability, which is qualitatively consistent with what has been observed in experiments.     

Basic equations for the microscopic theory of plasticity of polycrystalline materials with given texture

An expression for the yield stress of anisotropic materials is applied to the anisotropic strength of hard rolled copper foils whose crystallographic texture is known. We assume that this crystallographic texture is the only cause of the anisotropic plastic behaviour of the material. The model used for the yield stress is also used to deduce:

1. Stress-strain relations for isotropic polycrystalline materials;
2. A formula for the fully plastic strain tensor, applied to anisotropic hard rolled copper foils.

For the anisotropic copper foils considered the calculated curves of the yield stress and of the strain tensor as a function of the angle x between rolling and tensile direction agree qualitatively with the measured values. However, the theory is not complete, since the yield stress and the plastic strain tensor are both a function of a parameter Q, the fraction of the number of available crystallographic slip planes on which the maximum shear stress has reached the critical value τ_a. We assume that for “fully” plastic deformation a certain critical fraction Q _e of the total number of slip planes has to be active. The fraction Q _e is called the critical active quantity. With the parameter Q _e we adjust the calculated curves to the measured ones. The dependence of Q _e on the properties of the material (e.g. the crystallographic texture) is discussed in Appendix I.     

Parallel transport and defects on nematic shells

Nematic shells are thin films of nematic liquid crystal deposited on the boundary of colloidal particles, where liquid crystal molecules may freely glide, while remaining tangent to the surface substrate. The surface nematic order is described here by an appropriate tensor field Q, which vanishes wherever a defect occurs in the molecular order. We show how the classical concept of parallel transport on a manifold introduced by Levi-Civita can be adapted to this setting to define the topological charge m of a defect. We arrive at a simple formula to compute m from a generic representation of Q. In a number of separate appendices, we revisit in a unified language several, apparently disparate applications of Levi-Civita’s parallel transport.     

Rotation of liquid crystalline macromolecules in shear flow and shear-induced periodic orientation patterns

We have measured the shear-induced rotation of the nematic director in a liquid crystalline polymer using poly benzyl glutamate (PBG) as model system. PBG is a well characterized synthetic poly ( amino acid) with rigid chain architecture and well defined conformations. For the experiments it is important to start out with a sample in which the molecules are highly aligned with a uniform director. This so-called monodomain morphology is obtained by use of strong magnetic fields and surface modifications of the sample holders. When shearing the monodomain at a constant rate, the macromolecules rotate initially homogeneously until a periodic director pattern develops. These spatially periodic structures emerge in a narrow range of shear strain and, as shearing continues, disintegrate into a chaotic texture. By varying the initial monodomain director with respect to the flow direction (but within the shear planes) we could show that the periodic patterns do not depend on the shear direction; they are governed by the director of the initial monodomain. We observe conoscopically that at high shear rates the texture becomes uniformly aligned. The molecules are aligned preferentially with an angle of about 4° to the shear direction (against vorticity direction). Interestingly, this agrees very well with predictions made by Larson (1990).Dedicated to Prof. Dr. J. Meissner on the occasion of his retirement from the chair of Polymer Physics at the Eidgenössische Technische Hochschule (ETH) Zürich, Switzerland     

The role of divergent terms in the Frank energy of nematic liquid crystals

The effect of divergent terms in the Frank orientation energy of nematic liquid crystals on the equilibrium state of the director field is studied. Such terms have no effect on the equations of motion or on the equilibrium of the medium under consideration; however, they should be taken into account in the derivation of boundary conditions. It is shown that, in the case of boundary perturbations or in the case of polar orientation angle perturbations, the divergent terms can be considered as a surface energy for the azimuth angle (this energy is similar to the Rapini-Papoular energy). In addition, these terms may cause a deviation of the director in the plane parallel to the boundary. The equilibrium problem for a nematic liquid crystal is considered as an example in the case of small periodic boundary perturbations.     

Electrorheology of nematic liquid crystals in uniform shear flow

A hydrodynamic model for the electrorheological effect in a polymeric nematic confined in a rectangular cell is studied. The competition between a constant electric and a uniform shear flow is explicitly considered. For the final stationary state where the induced reorientation of the director has already occurred, we show that the averaged viscosity is enhanced. For this same state several rheological properties such as the first normal stress difference and the force between the cell plates are also analytically calculated as a function of position, the applied field, and Reynolds' number. These results are compared with those obtained previously for a pressure driven flow. The scope and limitations of the model and methods employed are discussed. Received: 23 November 1999/Accepted: 13 June 2000     

The effects of resolution and noise on kinematic features of fine-scale turbulence

The effect of spatial resolution and experimental noise on the kinematic fine-scale features in shear flow turbulence is investigated by means of comparing numerical and experimental data. A direct numerical simulation (DNS) of a nominally two-dimensional planar mixing layer is mean filtered onto a uniform Cartesian grid at four different, progressively coarser, spatial resolutions. Spatial gradients are then calculated using a simple second-order scheme that is commonly used in experimental studies in order to make direct comparisons between the numerical and previously obtained experimental data. As expected, consistent with other studies, it is found that reduction of spatial resolution greatly reduces the frequency of high magnitude velocity gradients and thereby reduces the intermittency of the scalar analogues to strain (dissipation) and rotation (enstrophy). There is also an increase in the distances over which dissipation and enstrophy are spatially coherent in physical space as the resolution is coarsened, although these distances remain a constant number of grid points, suggesting that the data follow the applied filter. This reduction of intermittency is also observed in the eigenvalues of the strain-rate tensor as spatial resolution is reduced. The quantity with which these eigenvalues is normalised is shown to be extremely important as fine-scale quantities, such as the Kolmogorov length scale, are showed to change with different spatial resolution. This leads to a slight change in the modal values for these eigenvalues when normalised by the local Kolmogorov scale, which is not observed when they are normalised by large-scale, resolution-independent quantities. The interaction between strain and rotation is examined by means of the joint probability density function (pdf) between the second and third invariants of the characteristic equation of the velocity gradient tensor, Q and R respectively and by the alignments between the eigenvectors of the strain-rate tensor and the vorticity vector. Gaussian noise is shown to increase the divergence error of a dataset and subsequently affect both the Q–R joint pdf and the magnitude of the alignment cosines. The experimental datasets are showed to behave qualitatively similarly to the numerical datasets to which Gaussian noise has been added, confirming the importance of understanding the limitations of coarsely resolved, noisy experimental data.     

Derivation of a dual-mixed <Emphasis Type="Italic">hp</Emphasis>-finite element model for axisymmetrically loaded cylindrical shells

The two-field dual-mixed Fraeijs de Veubeke variational formulation of three-dimensional elasticity serves as the starting point of the derivation of a dimensionally reduced shell model presented in this paper. The fundamental variables of this complementary energy-based variational principle are the not a priori symmetric stress tensor and the skew-symmetric rotation tensor. The tensor of first-order stress functions is applied to satisfy translational equilibrium, while the rotation tensor plays the role of a Lagrange multiplier to ensure rotational equilibrium. The volumetric locking-free shell model uses unmodified three-dimensional constitutive equations, and no classical kinematical hypotheses are employed during the derivation. The numerical performance of the related low-order h-, and higher-order p-version finite elements developed for axisymmetrically loaded cylindrical shells is investigated by two representative model problems. It is numerically proven that no negative effect can be experienced when the thickness is small and tends to zero.     

Ordering effects in electric splay Freedericksz transitions

We analyze the interaction between a nematic liquid crystal and an electric field, in a cell in which splay Freedericksz geometry is enforced. Equilibrium configurations are explored both close to the Freedericksz threshold and in the limit of strong applied voltages. We frame within de Gennes’ order-tensor theory, which allows us to detect the effects of a variable degree of orientation on critical fields and bifurcation shapes. The applied voltage induces nontrivial effects on the degree of orientation as well. Up to the Freedericksz transition, the degree of orientation decreases, whereas ordering is recovered when the applied voltage drop increases. We also stress the role played by the dielectric anisotropy. In particular, the limit in which the dielectric anisotropy approaches the dielectric permittivities deserves attention, since the order-tensor theory regularizes some of the critical phenomena exhibited by classical Frank solutions.      

Mechanical model for fiber-laden membranes

An integrated mechanical model for fiber-laden membranes is presented and representative predictions of relevance to cellulose ordering and orientation in the plant cell wall are presented. The model describes nematic liquid crystalline self-assembly of rigid fibers on an arbitrarily curved fluid membrane. The mechanics of the fluid membrane is described by the Helfrich bending-torsion model, the fiber self-assembly is described by the 2D Landau-de Gennes quadrupolar Q-tensor order parameter model, and the fiber-membrane interactions (inspired by an extension of the 2D Maier-Saupe model to curved surfaces) include competing curvo-philic (curvature-seeking) and curvo-phobic (curvature-avoiding) effects. Analysis of the free energy reveals three fiber orientation regimes: (a) along the major curvature, (b) along the minor curvature, (c) away from the principal curvatures, according to the competing curvo-philic and curvo-phobic interactions. The derived shape equation (normal stress balance) now includes curvature-nematic ordering contributions, with both bending and torsion renormalizations. Integration of the shape and nematic order equations gives a complete model whose solution describes the coupled membrane shape/fiber order state. Applications to cylindrical membranes, relevant to the plant cell wall, shows how growth decreases the fiber order parameter and moves the fibers’ director from the axial direction towards the azimuthal orientation, eventually leading to a state of stress predicted by pure membranes. The ubiquitous 54.7° cellulose fibril orientation with respect to the long axis in a cylindrical plant cell wall is shown to be predicted by the preset model when the ratio of curvo-phobic and curvo-philic interactions is in the range of the cylinder radius.     

Blow Up Criterion for Compressible Nematic Liquid Crystal Flows in Dimension Three

In this paper, we consider the short time strong solution to a simplified hydrodynamic flow modeling compressible, nematic liquid crystal materials in dimension three. We establish a criterion for possible breakdown of such solutions at a finite time in terms of the temporal integral of both the maximum norm of the deformation tensor of the velocity gradient and the square of the maximum norm of the gradient of a liquid crystal director field.     

Landau–De Gennes Theory of Nematic Liquid Crystals: the Oseen–Frank Limit and Beyond

We study global minimizers of a continuum Landau–De Gennes energy functional for nematic liquid crystals, in three-dimensional domains, subject to uniaxial boundary conditions. We analyze the physically relevant limit of small elastic constant and show that global minimizers converge strongly, in W ^1,2, to a global minimizer predicted by the Oseen–Frank theory for uniaxial nematic liquid crystals with constant order parameter. Moreover, the convergence is uniform in the interior of the domain, away from the singularities of the limiting Oseen–Frank global minimizer. We obtain results on the rate of convergence of the eigenvalues and the regularity of the eigenvectors of the Landau–De Gennes global minimizer. We also study the interplay between biaxiality and uniaxiality in Landau–De Gennes global energy minimizers and obtain estimates for various related quantities such as the biaxiality parameter and the size of admissible strongly biaxial regions.     

Analysis of Nematic Liquid Crystals with Disclination Lines

We investigate the structure of nematic liquid crystal thin films described by the Landau–de Gennes tensor-valued order parameter model with Dirichlet boundary conditions on the sides of nonzero degree. We prove that as the elasticity constant goes to zero in the energy, a limiting uniaxial nematic texture forms with a finite number of defects, all of degree or all of degree , corresponding to vertical disclination lines at those locations. We also state a result on the limiting behavior of minimizers of the Chern–Simons–Higgs model without magnetic field that follows from a similar proof.     

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