Long-range Elasticity Research Articles

A Nonhomogeneous Kinetic Model of Liquid Crystal Polymers and Its Thermodynamic Closure Approximation

Ageneral nonhomogeneous extension of the Doi’s kinetic theory with translational diffusion and nonlocal potential is proposed to describe the microstructures and defect dynamics of Liquid Crystal Polymer (LCP) solutions. The long-range elasticity of polymer molecules is depicted by a kernel type potential, from which one can derive the well-known Marrucci-Greco potential with weak spatial distortion assumption. Applying quasi-equilibrium closure approximation, we get a second-order moment model for isotropic long-range elasticity, and this reduced moment model maintains the energy dissipation. Implemented by the invariant-based fitting method, the moment model is a decent tool for numerical simulations of defect dynamics and texture evolution in LCP solutions. The numerical results of in-plane rotational case show that the reduced second-order moment model qualitatively predicts complicated nonhomogeneous director dynamics under moderate nematic potential strength, and the translational diffusion plays an important role in defect dynamics. PACS: 61.30.Cz, 61.30.Dk, 61.30.Vx

Forces in nematic liquid crystals: from nanoscale interfacial forces to long-range forces in nematic colloids

In this paper we give an overview of experiments that provided an insight into the nature of forces between surfaces and objects in a nematic liquid crystal. These forces, also called ‘structural forces’, are the consequence of the long-range orientational order and orientational elasticity of nematic liquid crystals. Owing to their fundamental as well as technological importance, forces between objects in liquid crystals have been a subject of growing interest during the last decade. Experimental observations and studies of structural forces are described from nanoscale interfacial forces, measured by an atomic force microscope, to the micro-scale forces between colloidal particles in nematics, studied by laser tweezers and optical video microscopy.

Bond Scrambling and Network Elasticity

Dynamic networks (DNs) recently reported in the literature are based on cross-linked supramolecular chains or on covalent chains with reversible bonds. As originally pointed out by Lehn, these networks should be regarded as dynamic materials exhibiting adaptive features due to continuous scrambling of their bonds and sequences. Results in the recent literature reveal that these networks undergo reversible long-range deformation resembling that of rubber networks. The present analysis of this process in terms of the theory of composite networks is based on the expectation that the scrambling process should allow rupture of bonds in the undeformed state and their reformation in the stretched state. Accordingly, a permanent set of the resting length of DNs should generally be expected and set materials should retain long-range elasticity relative to the set state. However, only a limited set is shown by DNs, implying that a strong memory of the initial network topology assists elastic recovery of the original dimensions. The analysis of reported experimental data further reveals that the stress-strain dependence of dynamic networks accurately follows the classical rubber elasticity theory. In this respect, DNs show better rubber behavior than typical covalent networks. Consistently with theoretical predictions, this surprising finding suggests that bond scrambling relieves local strain constraints on the fluctuations of networks junctions and favors the recovery of the initial network topology. Scrambling therefore allows compliance under stress and enhanced recovery when stress is released. Unprecedented applications of these advanced materials thus become foreseeable.

Numerical Simulation of Emergence of Textures in Flows of Liquid Crystalline Polymers using a Constitutive Model with Long-Range Elasticity

Simple shear and Poiseuille flows of liquid crystalline polymers between parallel plates were numerically simulated using the Marrucci-Greco model, which is a constitutive equation that includes long-range elasticity effects. Homogeneous, inhomogeneous decoupled, and inhomogeneous full coupled models [Kupferman et al., J Non-Newtonian Fluid Mech, 91, 255 (2000)] were considered in the present simulation. In the simulation of Poiseuille flows, the relation between the long-range elasticity and the emergence of textures caused by the distribution of orientation angle of directors was investigated. The molecular orientation on the channel plate restricts rotation of directors near the plate through the long-range elasticity, and irregular rotation motions of directors were observed in the inhomogeneous full coupled simulation. Full coupled computations of the velocity field and the director motion predicted the emergence of textures during flows. In addition, the simulation results suggested that a director rotation and the molecular interaction due to the long-range elasticity are necessary for the emergence of textures.

Thermal depinning and transverse-field tilting transitions in a planar vortex array pinned by a columnar defect

We study the thermal and transverse-magnetic-field response of a vortex line array confined to a plane with a single columnar pinning defect. By integrating out ``bulk'' degrees of freedom away from the columnar defect we reduce this two-dimensional problem to a one-dimensional one, localized on the defect and exhibiting a long-range elasticity along the defect. We show that as a function of temperature, for a magnetic field aligned with the defect this system exhibits a one-dimensional analog of a roughening transition, with a low-temperature ``smooth'' phase corresponding to a vortex array pinned by the defect, and a high-temperature ``rough'' phase in which at long scales thermal fluctuations effectively average away pinning by the defect. We also find that in the low-temperature pinned phase, the vortex lattice tilt response to a transverse magnetic field proceeds via a soliton proliferation ``transition,'' governed by an integrable sine-Hilbert equation and analogous to the well-known commensurate-incommensurate transition in sine-Gordon systems. The distinguishing feature here is the long-range nature of the one-dimensional elasticity, leading to a logarithmic soliton energy and interaction. We predict the transverse-field--temperature phase diagram and discuss extension of our results to a bulk vortex array in the presence of a dilute concentration of columnar defects.

Kinetic Structure Simulations of Nematic Polymers in Plane Couette Cells. II: In-Plane Structure Transitions

Nematic, or liquid crystalline, polymer (LCP) composites are composed of large aspect ratio rod-like or platelet macromolecules. This class of nanocomposites exhibits tremendous potential for high performance material applications, ranging across mechanical, electrical, piezoelectric, thermal, and barrier properties. Fibers made from nematic polymers have set synthetic materials performance standards for decades. The current target is to engineer multifunctional films and molded parts, for which processing flows are shear-dominated. Nematic polymer films inherit anisotropy from collective orientational distributions of the molecular constituents and develop heterogeneity on length scales that are, as yet, not well understood and thereby uncontrollable. Rigid LCPs in viscous solvents have a theoretical and computational foundation from which one can model parallel plate Couette cell experiments and explore anisotropic structure generation arising from nonequilibrium interactions between hydrodynamics, molecular short- and long-range elasticity, and confinement effects. Recent progress on the longwave limit of homogeneous nematic polymers in imposed simple shear and linear planar flows [R. G. Larson and H. Ottinger, Macromolecules, 24 (1991), pp. 6270--6282], [V. Faraoni, M. Grosso, S. Crescitelli, and P. L. Maffettone, J. Rheol., 43 (1999), pp. 829--843], [M. Grosso, R. Keunings, S. Crescitelli, and P. L. Maffettone, Phys. Rev. Lett., 86 (2001), pp. 3184--3187], [M. G. Forest, Q. Wang, and R. Zhou, Rheol. Acta, 43 (2004), pp. 17--37], [M. G. Forest, Q. Wang, and R. Zhou, Rheol. Acta, 44 (2004), pp. 80--93], [M. G. Forest, Q. Wang, R. Zhou, and E. Choate, J. Non-Newtonian Fluid Mech., 118 (2004), pp. 17--31], [M. G. Forest, R. Zhou, and Q. Wang, Phys. Rev. Lett., 93 (2004), 088301] provides resolved kinetic simulations of the molecular orientational distribution. These results characterize anisotropy and dynamic attractors of sheared bulk domains and underscore limitations of mesoscopic models for orientation of the rigid rod or platelet ensembles. In this paper, we apply our resolved kinetic structure code [R. Zhou, M. G. Forest, and Q. Wang, Multiscale Model. Simul., 3 (2005), pp. 853--870] to model onset and saturation of heterogeneity in the orientational distribution by coupling a distortional elasticity potential (with distinct elasticity constants) and anchoring conditions in a plane Couette cell. For this initial study, the flow field is imposed and the orientational distribution is confined to the shear deformation plane, which affords comparison with seminal [T. Tsuji and A. D. Rey, Phys. Rev. E (3), 62 (2000), pp. 8141--8151] as well as our own mesoscopic model simulations [H. Zhou, M. G. Forest, and Q. Wang, J. Non-Newtonian Fluid Mech., submitted], [H. Zhou and M. G. Forest, Discrete Contin. Dyn. Syst. Ser. B}, to appear].

Regional differences in degree of resilin cross-linking in the desert locust, Schistocerca gregaria

Various cuticular regions from the desert locust, Schistocerca gregaria, were quantitatively analyzed for two cross-linking amino acids, dityrosine and trityrosine, characteristic constituents of the rubberlike cuticular protein, resilin. These amino acids were found in all regions of cuticle investigated, but in widely varying amounts. In fully mature adult locusts the largest amounts of di- and trityrosine were obtained from the prealar arms and wing-hinges, structures possessing long-range elasticity and being involved in energy storage in the flight system. In structures where deformations tend to occur more slowly, such as the clypeo-labral springs and tracheae, di- and trityrosine are less abundant. In sclerotized cuticle from femur and tibia, as well as in cornea and in the highly stretchable intersegmental membranes of mature females, they are only found in trace amounts and are probably unrelated to elasticity. The trityrosine-to-dityrosine ratio in the variuos cuticular regions vary from nearly equal amounts of the two amino acids to about ten times more dityrosine than trityrosine, indicating that the regions differ in degree of cross-linking; the tracheal wall is the material with the highest trityrosine-to-dityrosine ratio. In some cuticular regions the ratio increases during maturation from newly moulted (teneral) adults to reproductively active locusts; the most pronounced increase was observed for the wing-hinges, and only a small increase was observed for the abdominal tergal plates. In most cuticular regions in fifth instar locust nymphs the contents of di- and trityrosine corresponded to the contents measured for the adult cuticular regions, but only trace amounts of the two amino acids were obtained from the region of the nymphal wing base which corresponds to the wing-hinge containing cuticular region in adult locusts.

The Mechanical Properties of Hydrated Intermediate Filaments: Insights from Hagfish Slime Threads

Intermediate filaments (IFs) impart mechanical integrity to cells, yet IF mechanics are poorly understood. It is assumed that IFs in cells are as stiff as hard α-keratin, F-actin, and microtubules, but the high bending flexibility of IFs and the low stiffness of soft α-keratins suggest that hydrated IFs may be quite soft. To test this hypothesis, we measured the tensile mechanics of the keratin-like threads from hagfish slime, which are an ideal model for exploring the mechanics of IF bundles and IFs because they consist of tightly packed and aligned IFs. Tensile tests suggest that hydrated IF bundles possess low initial stiffness (Ei=6.4MPa) and remarkable elasticity (up to strains of 0.34), which we attribute to soft elastomeric IF protein terminal domains in series with stiffer coiled coils. The high tensile strength (180MPa) and toughness (130 MJ/m3) of IF bundles support the notion that IFs lend mechanical integrity to cells. Their long-range elasticity suggests that IFs may also allow cells to recover from large deformations. X-ray diffraction and congo-red staining indicate that post-yield deformation leads to an irreversible α→β conformational transition in IFs, which leads to plastic deformation, and may be used by cells as a mechanosensory cue.

Squeezing molecular thin alkane lubrication films between curved solid surfaces with long-range elasticity: Layering transitions and wear

The properties of alkane lubricants confined between two approaching solids are investigated by a model that accounts for the curvature and the elastic properties of the solid surfaces. We consider linear alkane molecules of different chain lengths, C3H8, C4H10, C8H18, C9H20, C10H22, C12H26, and C14H30 confined between smooth gold surfaces. In most cases we observe well defined molecular layers develop in the lubricant film when the width of the film is of the order of a few atomic diameters. An external squeezing-pressure induces discontinuous, thermally activated changes in the number n of lubricant layers. We find that with increasing alkane chain length, the transition from n to n−1 layers occurs at higher pressure, as expected based on the increasing wettability (or spreading pressure) with increasing chain length. Thus, the longer alkanes are better boundary lubricants than the shorter ones, and this should result in less wear. We obtain good correlation between our theoretical results and wear experiments.

Monte Carlo simulations and dynamic field theory for suspended particles in liquid crystalline systems

Monte Carlo simulations and dynamic field theory are used to study spherical particles suspended in a nematic liquid crystal. Within these two approaches, we investigate the binding of the defects to the particles, the adsorption of a particle at a solid surface, and two particles interacting with each other. Quantitative comparisons indicate good agreement between the two approaches. A Monte Carlo method based on the combination of canonical expanded ensemble simulations with a density-of-state formalism is used to determine the potential of mean force between one particle and a hard wall. On the other hand, the potential of mean force is evaluated using a dynamic field theory, where the time-dependent evolution of the second rank tensor includes two major aspects of liquid crystalline materials, namely the excluded volume and the long-range order elasticity. The results indicate an effective repulsive force that acts between the particle and the wall. Layer formation at the surface of the hard wall gives rise to local minima in the potential of mean force. The director profile for a particle at contact with a solid surface is characterized by a disclination line distorted and attracted towards the wall. The structure of the nematic for two particles at short distances is also investigated. Our results indicate a structure where the two particles are separated by a circular disclination line. The potential of mean force associated with this configuration indicates an effective attractive interaction between the two particles.

Dynamic interaction between suspended particles and defects in a nematic liquid crystal.

Insertion of spherical particles into a uniform nematic liquid crystal gives rise to the formation of topological defects. In the present work, we investigate how a spherical particle accompanied by its topological defects interacts with neighboring disclination lines. We perform two- and three-dimensional dynamic simulations to analyze the effect of a particle on the annihilation process of two disclination lines. The dynamics of the liquid crystal is described by a time-dependent evolution equation on the symmetric traceless order parameter that includes some of the salient features of liquid crystalline materials: excluded volume effects, or equivalently, short-range order elasticity and long-range order elasticity. At the surface of the particle, the liquid crystal is assumed to exhibit strong homeotropic anchoring. The particle is located between two disclination lines of topological charges +1/2 and -1/2. Two-dimensional simulations indicate that the topological defects bound to the particle mediate an interaction between the two disclination lines which increases the attraction between them. This result is confirmed by three-dimensional simulations that provide a complete description of the director field and of the order parameter around the particle. These simulations indicate that a spherical particle between two disclination lines can be surrounded by a Saturn ring, and suggest that the dynamic behavior of disclination lines could be used to report the structure of a defect around the particle.

Theoretical and Computational Rheology for Discotic Nematic Liquid Crystals

This paper presents an analysis of the role of orientation on the rheology of discotic nematic liquid crystals. The shear rheological properties exhibited by flow-aligning discotic mesophases are calculated by using a complete generalized nonlinear second-order tensor Landau-de Gennes model that takes into account short-range order elasticity, long-range elasticity, and viscous effects. A unified expression for the extra stress tensor is given. The second-order tensor Landau-de Gennes model was reduced to the uniaxial Leslie-Ericksen to obtain limiting rheological material functions valid at low and high shear rates. Analytical results are able to predict the material functions computed by the full Landau-de Gennes model for nonhomogeneous flow-aligning discotic nematic liquid crystals. Experimentally reported changes in the sign of the first normal stress differences with shear rate are captured by the model. A new Carreau-Yasuda liquid crystal model has been used to characterize the shear rheology for characteristic boundary conditions, and a viscosity power law exponent of 0.5 and a normal stress coefficient power law exponent of 0.44 have been obtained.

Two-loop functional renormalization group theory of the depinning transition

We construct the field theory which describes the universal properties of the quasi-static isotropic depinning transition for interfaces and elastic periodic systems at zero temperature, taking properly into account the non-analytic form of the dynamical action. This cures the inability of the 1-loop flow-equations to distinguish between statics and quasi-static depinning, and thus to account for the irreversibility of the latter. We prove two-loop renormalizability, obtain the 2-loop beta-function and show the generation of "irreversible" anomalous terms, originating from the non-analytic nature of the theory, which cause the statics and driven dynamics to differ at 2-loop order. We obtain the roughness exponent zeta and dynamical exponent z to order epsilon^2. This allows to test several previous conjectures made on the basis of the 1-loop result. First it demonstrates that random-field disorder does indeed attract all disorder of shorter range. It also shows that the conjecture zeta=epsilon/3 is incorrect, and allows to compute the violations, as zeta=epsilon/3 (1 + 0.14331 epsilon), epsilon=4-d. This solves a longstanding discrepancy with simulations. For long-range elasticity it yields zeta=epsilon/3 (1 + 0.39735 epsilon), epsilon=2-d (vs. the standard prediction zeta=1/3 for d=1), in reasonable agreement with the most recent simulations. The high value of zeta approximately 0.5 found in experiments both on the contact line depinning of liquid Helium and on slow crack fronts is discussed.

Phenomenology of squeezing and sliding of molecularly thin Xe, CH4 and C16H34 lubrication films between smooth and rough curved solid surfaces with long-range elasticity

The properties of Xe, CH4 and C16H34 lubricant confined between two approaching solids are investigated by a model that accounts for the curvature and elastic properties of the solid surfaces. We consider both smooth surfaces, and surfaces with short-scale roughness. In most cases we observe well defined molecular layers develop in the lubricant film when the width of the film is of the order of a few atomic diameters, but in some cases atomic scale roughness inhibit the formation of these layers, and the lubricant exhibit liquid-like properties. An external squeezing-pressure induces discontinuous, thermally activated changes in the number n of lubricant layers. We observe that the layering transition tends to nucleate in disordered or imperfect regions in the lubrication film. We also present and discuss results of sliding dynamics for Xe and C16H34 lubrication films.

Kinetic theory for flows of nonhomogeneous rodlike liquid crystalline polymers with a nonlocal intermolecular potential.

The Doi kinetic theory for flows of homogeneous, rodlike liquid crystalline polymers (LCPs) is extended to model flows of nonhomogeneous, rodlike LCPs through a nonlocal (long-range) intermolecular potential. The theory features (i) a nonlocal, anisotropic, effective intermolecular potential in an integral form that is consistent with the chemical potential, (ii) short-range elasticity as well as long-range isotropic and anisotropic elasticity, (iii) a closed-form stress expression accounting for the nonlocal molecular interaction, and (iv) an extra elastic body force exclusively associated with the integral form of the intermolecular potential. With the effective intermolecular potential, the theory is proven to be well posed in that it warrants a positive entropy production and thereby the second law of thermodynamics. Approximate theories are obtained by gradient expansions of the number density function in the free energy density.

Fibrillin: from microfibril assembly to biomechanical function.

Fibrillins form the structural framework of a unique and essential class of extracellular microfibrils that endow dynamic connective tissues with long-range elasticity. Their biological importance is emphasized by the linkage of fibrillin mutations to Marfan syndrome and related connective tissue disorders, which are associated with severe cardiovascular, ocular and skeletal defects. These microfibrils have a complex ultrastructure and it has proved a major challenge both to define their structural organization and to relate it to their biological function. However, new approaches have at last begun to reveal important insights into their molecular assembly, structural organization and biomechanical properties. This paper describes the current understanding of the molecular assembly of fibrillin molecules, the alignment of fibrillin molecules within microfibrils and the unique elastomeric properties of microfibrils.

Roughness at the depinning threshold for a long-range elastic string.

In this paper, we compute to high precision the roughness exponent zeta of a long-range elastic string, at the depinning threshold, in a random medium. Our numerical method exploits the analytic structure of the problem ("no-passing" theorem), but avoids direct simulation of the evolution equations. The roughness exponent has recently been studied by simulations, functional renormalization-group calculations, and by experiments (fracture of solids, liquid meniscus in 4He). Our result zeta=0.388 +/- 0.002 is significantly larger than what was stated in previous simulations, which were consistent with a one-loop renormalization-group calculation. Furthermore, the data are incompatible with the experimental results for crack propagation in solids and for a 4He contact line on a rough substrate. This implies that the experiments cannot be described by pure harmonic long-range elasticity in the quasistatic limit.

Defect Nucleation and Annihilation in Sheared Polymeric Liquid Crystals

ABSTRACTShear–induced nucleation and annihilation of topological defects due to hydrodynamic instability in nematic liquid crystals is a phenomenon of both scientific interest and practical importance. We use a complete generalized non-linear second order tensor Landau-de Gennes model that takes into account short range order elasticity, long-range elasticity and viscous effects, to simulate the nucleation and annihilation of twist inversion walls in flow-aligning nematic polymers subjected to shear flow. Shearing a homogeneous nematic sample perpendicular to the director results in an linear instability that maybe symmetric at low shear rates, and antisymmetric at higher shear rates. At even higher shear rates the onset of nonlinearities results in the nucleation of a parallel array of twist inversion walls, such that asymmetry prevails. By increasing the shear rate the following director symmetry transition cascade is observed: symmetric → antisymmetric → asymmetric → symmetric. The nucleation of the parallel array of twist inversion walls in the asymmetric mode is due to the degeneracy in reorientation towards the shear plane. The annihilation of twist walls is mediated by twist waves along the velocity gradient direction. Twist walls annihilate by three mechanisms: wall-wall annihilations, wall-wall coalescence, and wall-bounding surface coalescence. The annihilation rate increases with increasing shear rate and at sufficiently high rates the layered structure is replaced by a homogeneously aligned system. The role of short range and long range elasticity on defect nucleation and annihilation is characterized in terms of the Deborah and Ericksen numbers. Close form solutions to approximated equations are used to explain the numerical results of the full Landau-de Gennes equations of nematodynamics.

Fibrillin-rich microfibrils: elastic biopolymers of the extracellular matrix.

Fibrillin-rich microfibrils are evolutionarily ancient macromolecular assemblies of the extracellular matrix. They have unique extensible properties that endow vascular and other tissues with long-range elasticity. Microfibril extensibility supports the low pressure closed circulations of lower organisms such as crustaceans. In higher vertebrates, microfibrils act as a template for elastin deposition and are components of mature elastic fibres. In man, the importance of microfibrils is highlighted by the linkage of mutations in their principal structural component, fibrillin-1, to the heritable disease Marfan syndrome which is characterised by severe cardiovascular, skeletal and ocular defects. When isolated from tissues, fibrillin-rich microfibrils have a complex ultrastructural organisation with a characteristic 'beads-on-a-strong' appearance. X-ray fibre diffraction studies and biomechanical testing have shown that microfibrils are reversibly extensible at tissue extensions of 100%. Ultrastructural analysis and 3D reconstructions of isolated microfibrils using automated electron tomography have revealed new details of how fibrillin molecules are aligned within microfibrils in untensioned and extended states, and delineated the role of calcium in regulating microfibril beaded periodicity, rest length and molecular organisation. The molecular basis of how fibrillin molecules assemble into microfibrils, the central role of cells in regulating this process, and the identity of other molecules that may coassemble into microfibrils are now being elucidated. This information will enhance our understanding of the elastic mechanism of these unique extracellular matrix polymers, and may lead to new microfibril-based strategies for repairing elastic tissues in ageing and disease.

Development of wavy texture in startup flows of liquid crystalline polymer solution through a slit cell

The development of wavy texture in startup flows of liquid crystalline polymers through a slit cell is examined to clarify the mechanism of this phenomenon, using a liquid crystalline solution of 50 w % hydroxypropylcellulose. The startup flow at a constant flow rate is generated by a piston moving at a constant speed. It takes a comparatively long time until the wavy texture appears, and the induction time decreases with increasing apparent shear rate. However, it is found that the apparent shear strain required for the onset of wavy texture is independent of the apparent shear rate except at low apparent shear rates. For the slit with 0.52 mm thickness the wide wavy texture is observed at the initial stage of the emergence of wavy texture, and after that the wavelength of the wide wavy texture approximately halves regardless of the apparent shear rate. Furthermore, it is suggested that a twisted structure of molecular orientation exists in the induction period. The results obtained in the present experiments imply that the long-range order elasticity contributes to the emergence of wavy texture.