A kinetic–hydrodynamic simulation of microstructure of liquid crystal polymers in plane shear flow

We study the microstructure formation and defects dynamics arising in liquid crystalline polymers (LCPs) in plane shear flow by a kinetic–hydrodynamic coupled model. The kinetic model is an extension of the Doi theory with a non-local intermolecular potential, including translational diffusion and density variation. LCP molecules are ensured anchoring at the boundary by an additional boundary potential, meanwhile mass conservation of LCPs holds in the whole flow region. Plane Couette flow and Poiseuille flow are studied using the kinetic–hydrodynamic model and the molecular director is restricted in the shear plane. In plane Couette flow, the numerical results predict seven in-plane flow modes, including four in-plane modes reported by Rey and Tsuji [Macromol. Theory Simul. 7 (1998) 623–639] and three new complicated in-plane modes with inner defects. Furthermore, some significant scaling properties were verified, such as the thickness of the boundary layer is proportional to molecular length, the tumbling period is proportional to the inverse of shear rate. In plane Poiseuille flow, the micro-morph is quasi-periodic in time when flow viscosity and molecular elasticity are comparable. Different local states, such as flow-aligning, tumbling or wagging, arise in different flow region. The difference of the local states, or difference of the tumbling rates in near-by regions causes defects and form branch pattern in director spatial–temporal configuration figure.     

Molecular orientation of a commercial thermotropic liquid crystalline polymer in simple shear and complex flow

In-situ X-ray scattering methods have been used to measure the average degree of molecular orientation in the commercial thermotropic copolyesteramide, Vectra B. Experiments were conducted in both homogeneous shear flow and in extrusion-fed channel flows that provided mixed shear/extensional deformations. In the channel flows, extension has a dramatic effect on the average orientation state in the vicinity of stagnation points or expansions/contractions in cross-sectional area. Of particular note, a temporary increase and subsequent decay in orientation observed in a 4:1 slit-contraction flow provides additional indirect evidence supporting the hypothesis that Vectra B exhibits director tumbling. This is consistent with results from other fully aromatic copolyesters but contrasts with findings in model thermotropes incorporating flexible spacers. Thus, it seems that the stiffer backbone of commercial main chain LCPs is the main feature which, apparently, leads to tumbling. Measurements of average molecular orientation in transient shear flows show some connections with the corresponding mechanical response, but fail to show the distinctive characteristics that have previously been associated with either tumbling or aligning in LCPs using similar procedures. These experiments might be adversely affected by the comparatively slow rate of data acquisition, which leads to lengthy experiments in which the sample is more prone to degradation.     

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.     

Brownian dynamics simulation of rodlike polymers under shear flow

The highly nonlinear behaviors of rodlike polymers in nematic phase under shear flow are studied with Brownian dynamics simulation. The LebwohlLasher nematogen model is taken as the prototype of the simulation and the mean-field approximation is avoided. By considering the nearest-neighbor intermolecular interaction, the spatial orientational correlation is introduced and therefore the spatial inhomogeneity such as the multiple-domain effect can automatically be incorporated. The transient order parameters, birefringence axes, shear stresses and first normal stress differences are calculated. The important finding of this work is that the director wagging and damped oscillation share the same molecular origin as director tumbling. The only difference is that the system is split into micro-domains which tumble with different phase angles in the wagging and damped oscillation regimes. The tumbling of the director of the whole system is suppressed due to the spatial inhomogeneity of director fields and then the damped oscillation of macroscopic stresses becomes predominant. The negative first normal stress difference exists at moderate shear rates, where both elasticity and viscosity play important role. Our simulation results including some dimensionless scaling parameters find good agreement with experimental observations in literature.     

Periodic and Homoclinic Motions in Forced,Coupled Oscillators

We study periodic and homoclinic motions in periodically forced, weakly coupled oscillators with a form of perturbations of two independent planar Hamiltonian systems. First, we extend the subharmonic Melnikov method, and give existence, stability and bifurcation theorems for periodic orbits. Second, we directly apply or modify a version of the homoclinic Melnikov method for orbits homoclinic to two types of periodic orbits. The first type of periodic orbit results from persistence of the unperturbed hyperbolic periodic orbit, and the second type is born out of resonances in the unperturbed invariant manifolds. So we see that some different types of homoclinic motions occur. The relationship between the subharmonic and homoclinic Melnikov theories is also discussed. We apply these theories to the weakly coupled Duffing oscillators.     

Robustness of pulsating jet-like layers in sheared nano-rod dispersions

Nano-rod dispersions in steady shear exhibit persistent transient responses both in experiments and simulations. The rotational contribution from shear flow couples with orientational diffusion, excluded-volume interactions, and distortional elasticity to yield complex dynamics and gradient morphology of the rod ensemble. The classification of sheared responses has mostly focused on “nematodynamics” of the collective particle response known as tumbling, wagging and kayaking; in heterogeneous simulations, one monitors the variability in nematodynamics across the domain. In this paper, we focus on flow coupling and non-Newtonian feedback in transient heterogeneous simulations, and in particular on a remarkable effect: the formation of localized, pulsating jet layers in the shear gap. We solve the Navier–Stokes momentum equations coupled through an orientation-dependent stress to three different orientational models (a kinetic Smoluchowski equation and two tensor models, one from kinetic closure and another from irreversible thermodynamics). A similar spurt phenomenon was reported in 1D simulations of a model for planar nematic liquids by Kupferman et al. [R. Kupferman, M. Kawaguchi, M.M. Denn, Emergence of structure in models of liquid crystalline polymers with elasticity, J. Non-Newt. Fluid Mech. 91 (2000) 255–271], which we extend to full orientational configuration space. We show: the pulsating jet layers correlate, in space and time, with the formation of a non-topological “oblate defect phase” in which the principal axis of orientation spreads from a unique direction to a circle; the jet-defect layers form where the local nematodynamics transitions from finite oscillation (wagging) to continuous rotation (tumbling), and when neighboring directors lose phase coherence; and, a negative first normal stress difference develops in the pulsating jet-defect layers. Finally, we extend one model algorithm to two space dimensions and show numerical stability of the jet-defect phenomenon to 2D perturbations.     

Orientation dynamics in commercial thermotropic liquid crystalline polymers in transient shear flows

In situ X-ray scattering measurements of molecular orientation under shear are reported for two commercial thermotropic liquid crystalline polymers (TLCPs), Vectra A950® and Vectra B950®. Transient shear flow protocols (reversals, step changes, and flow cessation) are used to investigate the underlying director dynamics. Synchrotron X-ray scattering in conjunction with a high-speed area detector provides sufficient time resolution to limit the total time spent in the melt during testing, whereas a redesigned X-ray capable shear cell provides a more robust platform for working with TLCP melts at high temperatures. The transient orientation response upon flow inception or flow reversal does not provide definitive signatures of either tumbling or shear alignment. However, the observation of clear transient responses to step increases or step decreases in shear rate contrasts with expectations and experience with shear-aligning nematics and suggests that these polymers are of the tumbling class. Finally, these two polymers show opposite trends in orientation following flow cessation, which appears to correlate with the evolution of dynamic modulus during relaxation. Specifically, Vectra B shows an increase in orientation upon flow cessation, an observation that can only be rationalized by the assumption of tumbling dynamics in shear. Together with prior observations of commercial LCP melts in channel flows, these results suggest that this class of materials, as a rule, exhibits director tumbling.     

Polydomain model predictions of liquid crystalline polymer orientation in mixed shear/extensional channel flows

 The Larson-Doi (LD) polydomain model is used to simulate orientation development along the centerline of slit-expansion and slit-contraction flows of liquid crystalline polymers (LCPs). Orientation is computed using the LD structural evolution equations, subject to an imposed velocity field that accounts for the spatial variation of both shear and extension rates characteristic of this class of flows. Computed axial distributions of orientation averaged through the sample thickness are qualitatively similar to birefringence and X-ray scattering measurements of molecular orientation in similar flows of lyotropic and thermotropic LCPs. In slit-expansion flows, the simulations predict a 90^∘ flip in orientation direction near the midplane due to transverse stretching in the expansion region. Far away from the midplane where shear gradients dominate, orientation remains primarily along the flow direction. Within the LD model, tumbling and flow aligning materials respond in a qualitatively similar manner to mixed shear and extension, although tumbling materials are systematically more susceptible to the effects of extension. Received: 22 October 1999/Accepted: 13 January 2000     

Homoclinic bifurcation at resonant eigenvalues

We consider a bifurcation of homoclinic orbits, which is an analogue of period doubling in the limit of infinite period. This bifurcation can occur in generic two parameter vector fields when a homoclinic orbit is attached to a stationary point with resonant eigenvalues. The resonance condition requires the eigenvalues with positive/negative real part closest to zero to be real, simple, and equidistant to zero. Under an additional global twist condition, an exponentially flat bifurcation of double homoclinic orbits from the primary homoclinic branch is established rigorously. Moreover, associated period doublings of periodic orbits with almost infinite period are detected. If the global twist condition is violated, a resonant side switching occurs. This corresponds to an exponentially flat bifurcation of periodic saddle-node orbits from the homoclinic branch.Partially supported by DARPA and NSF.Partially supported by the Deutsche Forschungsgemeinschaft and by Konrad-Zuse-Zentrum für Informationstechnik Berlin.     

Extensional motions of spatially periodic lattices

The behavior of microrheological models for multiphase fluids that have spatially periodic structure depends on certain kinematic properties of the unit cell. Anomalous results associated with identical objects approaching too closely during the flow can be reduced if not eliminated by satisfying lattice compatibility conditions. This is straightforward for simple shearing flow but subtle for extensional flows. Using the connection between lattice compatibility and lattice reproducibility (periodic lattice behavior with the flow) we establish sufficient conditions for compatibility of arbitrary lattices in planar extensional flow. Detailed results for square and hexagonal unit cells include: initial orientations for periodic behavior; strain periods; and minimum lattice spacings D. We identify the orientation of a square unit cell that leads to periodic behavior (with the minimum period) and the largest D of any lattice in planar extensional flow. We show that no lattice exhibits periodic behavior in uniaxial extensional flow (or biaxial extensional flow) even though Adler & Brenner have established the existence of compatibility.     

Linear stability and dynamics of viscoelastic flows using time-dependent numerical simulations

Ultimately, numerical simulation of viscoelastic flows will prove most useful if the calculations can predict the details of steady-state processing conditions as well as the linear stability and non-linear dynamics of these states. We use finite element spatial discretization coupled with a semi-implicit θ-method for time integration to explore the linear and non-linear dynamics of two, two-dimensional viscoelastic flows: plane Couette flow and pressure-driven flow past a linear, periodic array of cylinders in a channel. For the upper convected Maxwell (UCM) fluid, the linear stability analysis for the plane Couette flow can be performed in closed form and the two most dangerous, although always stable, eigenvalues and eigenfunctions are known in closed form. The eigenfunctions are non-orthogonal in the usual inner product and hence, the linear dynamics are expected to exhibit non-normal (non-exponential) behavior at intermediate times. This is demonstrated by numerical integration and by the definition of a suitable growth function based on the eigenvalues and the eigenvectors. Transient growth of the disturbances at intermediate times is predicted by the analysis for the UCM fluid and is demonstrated in linear dynamical simulations for the Oldroyd-B model. Simulations for the fully non-linear equations show the amplification of this transient growth that is caused by non-linear coupling between the non-orthogonal eigenvectors. The finite element analysis of linear stability to two-dimensional disturbances is extended to the two-dimensional flow past a linear, periodic array of cylinders in a channel, where the steady-state motion itself is known only from numerical calculations. For a single cylinder or widely separated cylinders, the flow is stable for the range of Deborah number (De) accessible in the calculations. Moreover, the dependence of the most dangerous eigenvalue on De≡λV/R resembles its behavior in simple shear flow, as does the spatial structure of the associated eigenfunction. However, for closely spaced cylinders, an instability is predicted with the critical Deborah number De_c scaling linearly with the dimensionless separation distance L between the cylinders, that is, the critical Deborah number De_Lc≡λV/L is shown to be an O(1) constant. The unstable eigenfunction appears as a family of two-dimensional vortices close to the channel wall which travel downstream. This instability is possibly caused by the interaction between a shear mode which approaches neutral stability for De ≫ 1 and the periodic modulation caused by the presence of the cylinders. Nonlinear time-dependent simulations show that this secondary flow eventually evolves into a stable limit cycle, indicative of a supercritical Hopf bifurcation from the steady base state.     

Bifurcations and chaotic dynamics in a 4-dimensional competitive Lotka–Volterra system

In this paper, we study the Hopf bifurcation for the four-dimensional competitive Lotka–Volterra system, and give an example which can display chaotic dynamics apparent like Rössler’s folded band attractor. This demonstrates that Smale’s conclusions in (J. Math. Biol., 3:5–7, 1976) are true even for the simplest competitive Lotka–Volterra systems when the dimension n is four. We explore the mechanism of occurrence of chaotic behavior for the four-dimensional competitive Lotka–Volterra system. The numerical study indicates that a periodic solution by Hopf bifurcation can undergo successive period-doubling cascades, and a homoclinic orbit can undergo homoclinic bifurcation by Shil’nikov theorem.     

Homoclinic bifurcation and chaos attractor in elastic two-bar truss

In this paper, we present a dynamic bifurcation analysis of the non-linear Duffing's equation on a simple elastic structure. The structure is a two-bar elastic truss with a damper, and possesses geometrical non-linear stiffness. We consider the dynamic instability of its structure based on Duffing's oscillation, which shows bifurcation behavior of the homoclinic orbit. We could numerically forecast the trajectory near the invariant saddle point of homoclinic bifurcation on this model, and we found that it is possible to solve dynamic bifurcation and strange attractors (chaos) on this non-linear structure. On this truss, we could investigate the dynamic stability of the strange attractor using Lyapunov exponents under the frequency and/or the amplitude parameter of periodic load.     

An unsymmetric constitutive equation for anisotropic viscoelastic fluid

A continuum constitutive theory of corotational derivative type is developed for the anisotropic viscoelastic fluid–liquid crystalline (LC) polymers. A concept of anisotropic viscoelastic simple fluid is introduced. The stress tensor instead of the velocity gradient tensor D in the classic Leslie–Ericksen theory is described by the first Rivlin–Ericksen tensor A and a spin tensor W measured with respect to a co-rotational coordinate system. A model LCP-H on this theory is proposed and the characteristic unsymmetric behaviour of the shear stress is predicted for LC polymer liquids. Two shear stresses thereby in shear flow of LC polymer liquids lead to internal vortex flow and rotational flow. The conclusion could be of theoretical meaning for the modern liquid crystalline display technology. By using the equation, extrusion–extensional flows of the fluid are studied for fiber spinning of LC polymer melts, the elongational viscosity vs. extension rate with variation of shear rate is given in figures. A considerable increase of elongational viscosity and bifurcation behaviour are observed when the orientational motion of the director vector is considered. The contraction of extrudate of LC polymer melts is caused by the high elongational viscosity. For anisotropic viscoelastic fluids, an important advance has been made in the investigation on the constitutive equation on the basis of which a series of new anisotropic non-Newtonian fluid problems can be addressed. The project supported by the National Natural Science Foundation of China (10372100, 19832050) (Key project). The English text was polished by Yunming Chen.     

The stability of two-dimensional linear flows of an Oldroyd-type fluid

A linear stability analysis is made for an Oldroyd-type fluid undergoing steady two-dimensional flows in which the velocity field is a linear function of position throughout an unbounded region. This class of basic flows is characterized by a parameter λ which ranges from λ = 0 for simple shear flow to λ = 1 for pure extensional flow. The time derivatives in the constitutive equation can be varied continuously from co-rotational to co-deformational as a parameter β varies from 0 to 1. The linearized disturbance equations are analyzed to determine the asymptotic behavior as time t → ∞ of a spatially periodic initial disturbance. It is found that unbounded flows in the range 0 < λ ? 1 are unconditionally unstable with respect to periodic initial disturbances which have lines of constant phase parallel to the inlet streamline in the plane of the basic flow. When the Weissenberg number is sufficiently small, only disturbances with sufficiently small wavenumber α₃ in the direction normal to the basic flow plane are unstable. However, for certain values of β, critical Weissenberg numbers are found above which flows are unstable for all values of the wavenumber α₃.     

Ordering effects in shear flows of discotic polymers

The tensorial mechanical model of Farhoudi and Rey (1993) for uniaxial, rodlike, spatially homogeneous and monodomain nematics is modified to describe the microstructural response of discotic nematic network polymers in rectilinear simple shear flow. The particular topological features of the discotic phase are taken into account by a proper modification of the phenomenological parameters. Asymptotic and numerical solutions of the microstructural balance equations indicate the appearance of tumbling, oscillating, and stationary flow regimes as the strength of shear increases, as is the case for rod-like nematic polymers (Marrucci, 1991). The tumbling-oscillating transition is characterized by a diverging tumbling function , while the oscillating-stationary transition is characterized by a single steady value smaller than —1. The stable steady states of the stationary regime are shown to belong to the family of unstable isotropic solutions that exist at small shear rates, and are characterized by a director angle close to, but less than +90° to the flow direction.     

Rheological predictions of a transversely isotropic fluid model with extensible microstructure

A continuum extensible director theory was formulated to describe the isothermal, incompressible flow of uniaxial rodlike semiflexible liquid crystalline polymers. The model is strictly restricted to material that flow-align in shear, and that, in the absence of flow, are sufficiently far from the nematic-isotropic phase transition. The microstructure of the continuum is described by a variable length director, but the extensibility is finite. The model is an extension of the TIF (Transversely Isotropic Fluid) model of Ericksen (1960). The thermodynamic restrictions on the model parameters are found using the non-negative definiteness of the entropy production. The rheological material functions predicted by the model are calculated for steady simple shear and steady uniaxial extensional flows. In the rigid rod limit the model predictions agree with those of the TIF model, and for the finite extensibility case the model predictions are in agreement with those associated with flexible isotropic polymers: strong non-Newtonian shear viscosity, positive first normal stress differences, recoverable shear of order one, negative second normal stress differences, and a maximum in the steady uniaxial extensional viscosity.     

Numerical simulation of entry flow of the IUPAC-LDPE melt

Numerical simulations have been undertaken for the creeping entry flow of a well-characterized polymer melt (IUPAC-LDPE) in a 4:1 axisymmetric and a 14:1 planar contraction. The fluid has been modeled using an integral constitutive equation of the K-BKZ type with a spectrum of relaxation times (Papanastasiou–Scriven–Macosko or PSM model). Numerical values for the constants appearing in the equation have been obtained from fitting shear viscosity and normal stress data as measured in shear and elongational data from uniaxial elongation experiments. The numerical solutions show that in the axisymmetric contraction the vortex in the reservoir first increases with increasing flow rate (or apparent shear rate), goes through a maximum and then decreases following the behavior of the uniaxial elongational viscosity. For the planar contraction, the vortex diminishes monotonically with increasing flow rate following the planar extensional viscosity. This kinematic behavior is not in agreement with recent experiments. The PSM strain-memory function of the model is then modified to account for strain-hardening in planar extension. Then the vortex pattern shows an increase in both axisymmetric and planar flows. The results for planar flow are compared with recent experiments showing the correct trend.     

Twisted and nontwisted bifurcations induced by diffusion

We discuss a diffusively perturbed predator-prey system. Freedman and Wolkowicz showed that the corresponding ODE can have a periodic solution that bifurcates from a homoclinic loop. When the diffusion coefficients are large, this solution represents a stable, spatially homogeneous time-periodic solution of the PDE. We show that when the diffusion coefficients become small, the spatially homogeneous periodic solution becomes unstable and bifurcates into spatially nonhomogeneous periodic solutions. The nature of the bifurcation is determined by the twistedness of an equilibrium/homoclinic bifurcation that occurs as the diffusion coefficients decrease. In the nontwisted case two spatially nonhomogeneous simple periodic solutions of equal period are generated, while in the twisted case a unique spatially nonhomogeneous double periodic solution is generated through period-doubling.