Frank Elasticity Research Articles

Dynamic texture scaling of sheared nematic polymers in the large Ericksen number limit

In steady shear experiments, nematic polymers and rigid Brownian rod dispersions develop unsteady fine structure (Larson and Mead, 1992 [33]; Larson and Mead, 1993 [34]; Tan and Berry, 2003 [42]). These features are representative of the dynamic morphologies generated during film and mold processes, which translate to the two banes of materials engineering: heterogeneous properties on unknown length scales, and dynamic fluctuations in processing outcomes. Our goal here is to quantify the length scales of these dynamic textures in the realistic parameter regime of extremely high Ericksen number, Er, the ratio of viscous to elastic stress. To avoid model-specific or numerical anomalies, we simulate three distinct models of flowing nematic polymers employing three different algorithms run on different computer systems, following our earlier studies at moderate Ericksen numbers (Forest et al., 2008 [11,12]). Our strategy recognizes the rich history on this problem as well as practical prohibitive constraints. First, the relevant experimental and processing conditions dictate E r ≈ O ( 1 0 5 ) and order unity Deborah number ( De, the bulk shear rate normalized by the rotational relaxation rate of the nematic phase). Second, theoretical estimates of the smallest length scales of morphology and their sources exist only for steady-state structures, originally from liquid crystal models (cf. Carlsson, 1984 [2]; Carlsson, 1976 [3]; Cladis and Torza, 1976 [4]; de Gennes, 1974 [21]; Manneville, 1981 [35]) and subsequently from nematic polymer orientation tensor and full kinetic models (cf. Cui, 2006 [5]; Forest and Wang, 2003 [13]; Forest et al., 2004 [15]; Forest et al., 2007 [19]; Marrucci, 1985 [36]; Marrucci, 1990 [37]; Marrucci, 1991 [38]; Marrucci and Greco, 1993 [39]; Zhou and Forest, 2006 [50]; Zhou et al., 2007 [52]). Third, numerical experiments for E r ≈ O ( 1 0 2 , 3 ) and D e ≈ O ( 1 ) confirm experimental observations of unsteady long-time attractors (Denn and Rey, 2002 [6]; Feng et al., 2001 [10]; Forest et al., 2008 [11,12]; Forest et al., 2005 [17]; Han and Rey, 1995 [23]; Klein et al., 2005 [30]; Kupferman et al., 2000 [32]; Sgalari et al., 2002 [40]; Tsuji and Rey, 1997 [43]; Yang et al., 2008 [46]; Yang et al., in press [47]). Fourth, the unsteady regime persists to E r ≈ O ( 1 0 5 ) (Sgalari et al., 2002 [40]), where single runs, much less two or three decades of Er, are prohibitively expensive in two or three space dimensions due to severe spatial and temporal resolution constraints. Here we propose and implement a new strategy to finesse these limitations. The key observation arises from comprehensive 2D simulations (Klein et al., 2005 [30]; Yang et al., 2008 [46]; Yang et al., in press [47]) which implicate transient defect cores (local isotropic or oblate disordered phases) as the source of the finest texture, or equivalently, the strongest gradients. In light of these results, we simulate 1D flow-orientational heterogeneity spanning the plate gap, which preserves the fine-scale transient defect cores and associated shear flow and stress features, while screening higher-dimensional, nonlocal defect topology and cellular flow patterns. We simulate spatio-temporal attractors for Er out to 1 0 5 for all three models. Each model reveals a transient staircase of tumbling-wagging layers spanning the shear gap, where the smallest length scales of morphology are associated with intermittent oblate defects between adjacent tumbling and wagging layers, which form precisely while neighboring layers become out of phase. With increasing Er, the number of tumbling, wagging and oblate defect layers grows, meanwhile layers propagate, collide, merge and reform ad infinitum. To quantify these dynamic textures, we calculate a time-averaged morphology length scale distribution function, defined by the reciprocal of the gradient of the oblate defect metric. The peak of this PDF characterizes a dominant morphology length scale which, for all three models, remarkably adheres to the Marrucci scaling law E r − ( 1 / 2 ) , consistent with earlier numerical studies (Sgalari et al., 2002 [40]) on yet another nematic flow-tensor model. This result yields an a priori estimate for the finest length scales in shear-dominated nematic polymer films and molds based on processing conditions and Frank elasticity constants, which confirms a prediction of Marrucci, 25 years ago (Marrucci, 1985 [36]), based on dimensional analysis in the dynamic tumbling regime.

Non-classical scaling for forced wetting of a nematic fluid on a polymeric fiber

We report on the forced wetting of a liquid crystal on a polymer fiber in the nematic and isotropic phase under partial wetting conditions. As the cylindrical fiber exits from the fluid reservoir, the fluid forced-coated on the fiber is immediately broken into droplets due to capillary-driven instability, even in the wetting conditions. For the nematic fluid, the initial film thickness, h, before breakup, scales almost linearly with the capillary number, Ca, as h ∼ Ca0.94, while h ∼ Ca2/3 for the isotropic fluid, as predicted for a Newtonian fluid. The amount of the fluid coated on a fiber is larger in the nematic phase than in the isotropic phase at a given velocity within the velocity range studied. Analysis using Ericksen-Leslie equations shows that Frank elasticity plays no role in increasing the coating thickness for the nematic fluid, while viscous anisotropy is the source of observed rescaling, h ∼ Ca. This non-classical scaling is attributed to the deformation-stress cross-coupling and the existence of extensional kinematics in the meniscus formation region.

Size Dependent Hardness of Polyamide/Imide

Indentation experiments with indentation depths in the nano- and micrometer ranges in the literature have shown that that an indentation size effect similar to metals is present in polymers. In this paper we present indentation ex- periments of a polyamide/imide polymer of indentation depths in the range of 1 to 30 microns. An increase in hardness of about 70% was observed at small indentation depths compared to hardness at large indentation depths. The indentation re- sults are discussed with respect to an indentation hardness rule based on Frank elasticity suggested in a previous paper by the authors. It is found that the hardness model agrees well with the experiments.

Dynamics of Stimulus Response of Swollen Nematic Elastomers

Nematic elastomers are rubbery polymer networks containing rigid mesogenic molecules. The nematic elastomers exhibit unique physical properties stemming from the strong coupling of rubber elasticity and liquid crystallinity.[1] We reported that the nematic elastomers swollen by solvents (nematic gels) showed the shape and volume variations upon temperature change (anisotropic swelling)[2], and that they exhibited the macroscopic deformation with a significant change in optical birefringence in fast response to electric fields (electro-opto-mechanical effect)[3]. In this talk, we present the dynamic aspects of these phenomena. 2. Dynamics of Temperature Response [4] The nematic gels surrounded by nematic solvents vary their shape and volume depending on the nematic order of each nematogen: The gels swell largely in the totally isotropic or nematic states, but in the latter state they are elongated along the director; In the temperature region where the states inside and outside the gels are nematic and isotropic, respectively, the gels are considerably shrunken with an elongation along the director. We have observed the processes of shape and volume variations after the temperature-jumps with several combinations of initial and destination temperatures. The nematic gels show the complicated kinetics of swelling and shrinking: The dimension along or normal to the director exhibit overshoot or undershoot depending on the locations of the initial and destination temperatures in the phase diagram. We have revealed that the swelling kinetics of nematic gels is composed of the volume and shape variation modes with markedly different characteristic times. 3. Dynamics of Electric-Field Response [5] We have evaluated the characteristic times for the optical and mechanical responses of nematic gels to the imposition and removal of electric fields. The gels exhibit the fast decay times (regarding the field removal) in the order of millisecond which are a few orders of magnitude smaller than that of the pure swelling nematic solvent (5CB) confined in a cell with the same thickness as the gel thickenss (ca. 20 ∝m). This indicates that the nematic solvents in the gels (solvent content: more than 70 vol%) move fast together with the nematic networks, and that the driving force for recovery is rubber elastic force instead of Frank elasticity. The rise times (regarding the field imposition) become smaller with increasing the field strength. We estimate the elastic and viscous parameters of the gels from the rise and decay times, and the threshold field-strengths for the onset of the responses.

Low-Frequency Dilational Elasticity of the Nematic 4‘-Pentyl-4-biphenylcarbonitrile (5CB)/Water Interface

Axisymmetric oscillating pendant drop shape analysis has been used to study the interfacial rheology of the liquid crystal 4'-pentyl-4-biphenylcarbonitrile (5CB) in water with homeotropic anchoring. Nearly spherical 5CB droplets were subjected to low frequency (1-5 mHz) volume oscillations, and the increase in tension with surface dilation was used to calculate the complex modulus. The droplet interface response is completely elastic, with no relaxations occurring on the experimental time scale. This surprising result is attributed to droplet storage of elastic energy in the form of distorted orientational distributions within the bulk (Frank elasticity) and on the surface (anchoring elasticity).

Indentation size effects in polymers and related rotation gradients

Similar to metals, the hardness of many polymers increases with decreasing indentation depths at depth ranges from several microns down to several nanometers. While for metals such phenomena are commonly attributed to geometrically necessary dislocation densities, such an explanation cannot be applied to polymers. To provide a micromechanically motivated model for the indentation size effect in polymers, here we propose an elasto-plastic extension of the higher order elasticity model recently developed by the authors. In this model, size effects in polymers (as well as in nematic liquid crystals) are related to Frank elasticity arising from bending distortions of the polymer chains and their interactions. On the basis of this theory, we derive a simple model for indentation size effects in polymers. Unlike other models, our model includes only elastic size effects due to rotational gradients. It is shown that the proposed model can explain the experimentally observed size effects in polymers. Together with the existing experimental data mentioned here, new experimental data for silicon rubber are also presented and discussed.

The role of Frank elasticity in cholesteric elastomers

The effect of Frank elasticity on deformations of cholesteric elastomers by mechanical stress applied perpendicular to the helix axis is studied by numerical minimization of the free energy. Above a critical strain, a solution with an only oscillating director is found to be stable in comparison to a distorted helix. At the critical strain, the contractions perpendicular to stress change discontinuously. The critical strain is found to increase with increasing Frank elasticity contribution to the free energy density, and to diverge when the conformation anisotropy of the polymer backbone vanishes. The results are compared with recent experiments which indicated that, in case of weak conformation anisotropies, the Frank elasticity contribution to the free energy cannot be neglected.

Unwinding of a strained cholesteric elastomer by disclination loop nucleation

The application of a sufficiently strong strain perpendicular to the pitch axis of a monodomain cholesteric elastomer unwinds the cholesteric helix. Previous theoretical analyses of this transition ignored the effects of Frank elasticity which we include here. We find that the strain needed to unwind the helix is reduced because of the Frank penalty and the cholesteric state becomes metastable above the transition. We consider in detail a previously proposed mechanism by which the topologically stable helical texture is removed in the metastable state: namely, by the nucleation of twist disclination loops in the plane perpendicular to the pitch axis. We present an approximate calculation of the barrier energy for this nucleation process which neglects possible spatial variation of the strain fields in the elastomer, as well as a more accurate calculation based on a finite-element modeling of the elastomer.

The Phase Transitions from Chiral Nematic Toward Smectic Liquid Crystals

A Chen-Lubensky energy is used to investigate phase transitions from chiral nematic to smectic C* and smectic A* liquid crystal phases. We consider a liquid crystalline material Ω confined between two parallel plates, where the dimensions of Ω are assumed to be large relative both to the width of a smectic layer and the material’s chiral pitch. We take boundary conditions so that the smectic phase melts at the plates’ surfaces and prove the existence of energy minimizers in an admissible set consisting of order parameters \({\Psi \in \mathcal{H}^{2}_{0}(\Omega)}\) and molecular directors \({\mathbf{n} \in \mathbf{W}^{1,2} (\Omega; \mathbb{S}^{2})}\) . Then under the physically observed assumption that the Frank elasticity constants become large near a phase transition, we establish estimates for the transition region separating phases. In particular we derive analytic estimates proving that chirality lowers the transition temperature regime above which minimizers are nematic and below which minimizers are in a smectic phase.

Microscopic Observations and Simulations of Bloch Walls in Nematic Thin Films

We study Bloch wall defects formed by quenching nematic thin films from planar anchoring to homeotropic anchoring through a temperature-driven anchoring transition. The director profiles of the walls are directly visualized using fluorescence confocal polarizing microscopy, and shown to agree well with the simulation based on the Frank elasticity theory. A pure twist wall exists if the ratio of sample thickness to surface extrapolation length p is smaller than or close to 1; while a diffuse Bloch wall is obtained if p is much greater than 1.

Isotropic-Nematic Interface and Wetting in Suspensions of Colloidal Platelets

We study interfacial phenomena in a colloidal dispersion of sterically stabilized gibbsite platelets, exhibiting coexisting isotropic and nematic phases separated by a sharp horizontal interface. The nematic phase wets a vertical glass wall and polarized light micrographs reveal homeotropic surface anchoring both at the free isotropic-nematic interface and at the wall. On the basis of complete wetting of the wall by the nematic phase, as found in our density functional calculations and computer simulations, we analyze the balance between Frank elasticity and surface anchoring near the contact line. Because of weak surface anchoring, the director field in the capillary rise region is uniform. From the measured rise (6 microm) of the meniscus at the wall we determine the isotropic-nematic surface tension to be 3 nN/m, in quantitative agreement with our theoretical and simulation results.

On the origin of size effects in small-strain elasticity of solid polymers

The micromechanical origin of size effects in elasticity of solid polymers is discussed in this article. It is shown that size effects related to rotational gradients can be interpreted in terms of Frank elasticity arising from the finite bending stiffness of the polymer chains and their interactions. We derive a relationship between the gradient of the nematic director field, related to the orientation of the polymer segments, and the curvature tensor associated with rotational gradients. In the resulting couple stress model, the total deformation energy of the material contains a Frank elasticity term. The energy contribution from rotational gradients is related to an effective Frank elastic constant K ∼ [N]. The corresponding characteristic length for size effects is obtained as l = K ∼ / 3 μ where μ is the shear modulus. Our estimations suggest that the characteristic length for size effects in polymers is larger than that obtained for small-molecules materials. Experimental data from the literature are also discussed.

A model of weak viscoelastic nematodynamics

The paper develops a continuum theory of weak viscoelastic nematodynamics of Maxwell type. It can describe the molecular elasticity effects in mono-domain flows of liquid crystalline polymers as well as the viscoelastic effects in suspensions of uniaxially symmetric particles in polymer fluids. Along with viscoelastic and nematic kinematics, the theory employs a general form of weakly elastic thermodynamic potential and the Leslie–Ericksen–Parodi type constitutive equations for viscous nematic liquids, while ignoring inertia effects and the Frank (orientation) elasticity in liquid crystal polymers. In general case, even the simplest Maxwell model has many basic parameters. Nevertheless, recently discovered algebraic properties of nematic operations reveal a general structure of the theory and present it in a simple form. It is shown that the evolution equation for director is also viscoelastic. An example of magnetization exemplifies the action of non-symmetric stresses. When the magnetic field is absent, the theory is reduced to the symmetric, fluid mechanical case with relaxation properties for both the stress and director. Our recent analyses of elastic and viscous soft deformation modes are also extended to the viscoelastic case. The occurrence of possible soft modes minimizes both the free energy and dissipation, and also significantly decreases the number of material parameters. In symmetric linear case, the theory is explicitly presented in terms of anisotropic linear memory functionals. Several analytical results demonstrate a rich behavior predicted by the developed model for steady and unsteady flows in simple shearing and simple elongation.

Exact dynamical reorientation in nematic liquid crystals: Green's function approach

The exact dynamical behavior of the nematic director field in a cell limited by two planar surfaces, having inhomogeneous distribution of easy axes, is theoretically investigated within the Green's function approach. The equilibrium configurations are determined for splay-bend deformations in the framework of Frank elasticity, in the one-constant approximation. Both, the strong anchoring (Dirichlet's problem) and the weak anchoring cases (mixed Dirichlet–Neumann problem) are solved in the presence of an external time dependent field. The general solutions for both problems are obtained for the more general case in which the boundary conditions are position and time dependent.

Direct Computation of the Twist Elastic Coefficient of a Nematic Liquid Crystal Via Monte Carlo Simulations

A direct method for the computation of the twist Frank elasticity coefficient of a nematic liquid crystal is presented. The method, suitable for numerical calculations, is tested and applied in Monte Carlo simulations for a model system. The dependence of the elasticity coefficient on the temperature and density and its relation to the nematic order parameter are analyzed and discussed.

Texture dependence of capillary instabilities in nematic liquid crystalline fibres

Static and dynamic linear analyses of axisymmetric capillary instabilities in textured nematic liquid crystalline fibres are performed using the equations of nemato-statics and inviscid nemato-dynamics. Three representative textures, viz. axial, onion, and radial, are analysed to show all possible effects of Frank gradient elasticity on the wavelength selection and growth rate of peristaltic modes driven by surface area reduction. It is found that Frank elasticity may tend to stabilize or destabilize the fibre, depending on the initial fibre texture. Axial textures tend to stabilize the fibre through the director splay–bend distortions driven by surface tilting. Onion textures are destabilized by decreasing azimuthal bend elastic energy caused by surface displacement. Radial textures exhibit a stabilizing tilt mechanism due to bend modes and a destabilizing displacement mechanism due to splay modes, but the former is predicted to be dominant. The static analysis provides good estimates of the instability thresholds while the transient energy balance provides information on the fastest growing modes. The static and dynamic results are compared and shown to be fully consistent. The couplings between splay and/or bend distortions, surface tilting, and surface displacement in nematic fibres are characterized and used to explain the deviations from the classical Rayleigh instability.

Thermodynamics of soft anisotropic interfaces

The Gibbs-Duhem equation for interfaces between nematic liquid crystals and isotropic fluids is formulated and shown to be a generic equation for soft anisotropic surfaces. The one-to-one correspondence between the nematic and crystalline surface Gibbs-Duhem equations is established. Consistency between the surface Gibbs-Duhem equation and the classical equations of interfacial nematostatics is shown. Using a phase space that takes into account thermodynamics, liquid crystalline order, and geometric variables, the generalized nematic surface Gibbs-Duhem equation reveals the presence of couplings between shape, adsorption, temperature, and average molecular orientation. Merging the thermodynamic analysis with nematostatics results in a model for morphactancy, that is, adsorption-induced interfacial shape selection. The specific roles of gradient bulk Frank elasticity, interfacial tension, and anchoring energy are elucidated by analyzing particular paths in the thermodynamic-geometric phase space.

Free Energy of Rigid Rods: Contribution of Rotational Entropy to Frank Elasticity

We apply the density functional formalism developed by the present authors [J. Fukuda and H. Yokoyama: J. Chem. Phys. 115 (2001) 4930] for the calculation of the free energy of rigid rods as a functional of the compositional scalar order parameter φ and the orientational order parameter of second-rank tensor Q i j We model rigid rods as semiflexible wormlike chains with infinite bending elasticity and the exact expression for the entropic part of the free energy up to the second order in the gradients can be obtained. Discussion is devoted to the contribution of rotational entropy to Frank elasticity. Although none of the previous molecular theories for the Frank elasticity of liquid crystals has taken rotational entropy into account, our results indicate that rotational entropy can give a non-negligible contribution to Frank elasticity.

Cholesteric elastomers: deformable photonic solids.

A mechanical strain applied to a monodomain cholesteric elastomer modulates and eventually unwinds the helical director distribution. There are similarities with the classical problem of an electric field applied to a cholesteric liquid crystal, but also differences. Frank elasticity is of minor importance unless the gel is very weak. The interplay is rather between the director being helically anchored to the rubber elastic matrix and the external mechanical field. Stretching perpendicular to the helix axis induces the uniform unwound state via the elimination of sharp, pinned twist walls above a critical strain. Below the critical strain the coarsening of the director distribution is not accompanied by an increase but rather by an affine decrease in the pitch. Unwinding through conical director states occurs when the elastomer is stretched along the helical axis. Finally we consider cholesteric elastomers in a classical device geometry with an electric field applied along the pitch axis and hence transverse to the director orientation.

Effect of the network on the director fluctuations in a nematic side-group elastomer analysed by static and dynamic light scattering

Static and dynamic light scattering experiments were performed on monodomains of a crosslinked nematic side-group polysiloxane. The depolarized scattering caused by fluctuations of the nematic director was analysed. Due to a superposed strong static scattering produced by heterogeneities the fluctuating part of the scattering intensity was found to be rather small. In contrast to low molar mass nematics, the relaxation rates and strength did not show a significant dependence on the scattering vector. The results can be explained by restoring torques on the director originating from the coupling to the network elasticity which dominate those originating from the Frank elasticity. Quantities describing the coupling as well as effective viscosities for the director reorientation were obtained.