Doi Theory Research Articles

Orientational dynamics of nematic liquid crystals under shear flow

The orientational dynamics of low molecular weight and polymeric nematic liquid crystals in a flow field is investigated, based on a nonlinear relaxation equation for the second rank alignment tensor. Various approximations are discussed: Assuming uniaxial alignment with a constant order parameter, the results of the Ericksen–Leslie theory are recovered. The detailed analysis to be presented here for plane Couette flow concerns (i) uniaxial alignment with a variable degree of order and (ii) the tensorial analysis involving the three symmetry-adapted components of the five components of the alignment tensor. The transitions between tumbling, wagging and aligning behavior observed in polymeric liquid crystals and described by the Doi theory of rod-like nematic polymers are recovered. Consequences for the rheological behavior are indicated.

Rotational and Translational Dynamics of Rodlike Polymers: A Combined Transient Electric Birefringence and Dynamic Light Scattering Study

Dynamic light scattering (DLS) and transient electric birefringence decay measurements are performed on poly(γ-benzyl-α-l-glutamate) solutions over a wide concentration range. We interpret the primary dynamic decay rate always observed from dynamic light scattering experiments by extending the expression for the decay rate beyond the original limits of the mean field (DSO theory) theory of Doi, Shimada and Okano, (J. Chem. Phys. 1988, 88, 4070) to more general time and length scales. Our result, expressed as a power series in the scattering vector magnitude q, shows that, as expected, the primary decay rate from dynamic light scattering experiments at small q directly provides diffusion coefficients for mutual translation, whereas that at larger q contains contributions from both rotation and translation, including terms containing the translational anisotropy. The rotational relaxation times are obtained independently from the transient electric birefringence experiments and are used in our formula to extract concentration dependencies of the self-diffusion coefficients from the DLS time correlation functions at large scattering vector q. Finally, from the data at large q, we report the transverse and longitudinal translational self-diffusion coefficients as functions of concentration.

Smoothed particle hydrodynamics techniques for the solution of kinetic theory problems: Part 2. The effect of flow perturbations on the simple shear behavior of LCPs

The effect of slight perturbations to simple shear flow of liquid-crystalline polymers (LCPs) is explored by using the SPH technique to solve the unapproximated orientation distribution function equation arising from the Doi Theory. First, the case of simple shear flow is outlined, and it is shown that skewed distributions play an important role in the transition from periodic to steady behavior as the shear rate is increased. Next, we consider perturbations to flows that are slightly more extensional than simple shear, parametrized by the flow type parameter α. They are shown to eliminate all periodic director behavior (tumbling and wagging), even when the relative increment in flow type is small. At lower shear rates (or more properly, lower Peclet number Pe based upon the rotational diffusivity), the elimination occurs through a homoclinic bifurcation, the transition being rather abrupt as the flow type is changed. At higher Pe, periodic behavior is suppressed more gradually through a Hopf bifurcation, with tumbling being replaced by wagging and negative θ flow-aligning, where θ is the angle of the director in the shear plane. The effect of these perturbations on rheological behavior is also explored. As the flow is made slightly more extensional, the zero-shear rate limiting value of the generalized viscosity η decreases dramatically, due to the slowing down of tumbling as the system approaches a homoclinic orbit; as Pe is increased, the viscosity rises again before falling, due to the induction of wagging behavior where tumbling would normally prevail in simple shear. Finally, it is found when the flow type is changed sufficiently, the interesting, non-monotonic behavior of rheological functions seen in simple shear of LCPs is replaced by monotonic behavior, even though the flow is still relatively close to simple shear.

The effect of temperature and concentration on N1 and tumbling in a liquid crystal polymer

The shear rate γ̇min at the relative minimum in the N1 flow curve is studied as a function of temperature and concentration for liquid crystalline (hydroxypropyl)cellulose (HPC). For lyotropes, at least, γ̇min is the shear rate necessary to halt director “tumbling” and align the molecules. HPC is a convenient polymer for studying the relationship between lyotropic and thermotropic liquid crystalline polymers because it exhibits a pure thermotropic phase at elevated temperatures, and room temperature lyotropic phases at moderate concentrations in m-cresol. At the highest possible polymer concentration at which reliable rheology data can be obtained (around 70 wt % polymer), indirect evidence for director tumbling is observed, in that N1 retains a local minimum versus shear rate. For the highest concentrations this minimum N1 value is positive, rather than negative, as is the case at lower concentrations and as is predicted by the Doi theory. Empirical time–temperature and time–concentration shifting can be used to estimate γ̇min from measured values of the shear viscosity.

Introductory Lecture: Inertia, coarsening and fluid motion in binary mixtures

Symmetric binary fluids, quenched into a regime of immiscibility, undergo phase separation by spinodal decomposition. In the late stages, the fluids are separated by sharply defined, but curved, interfaces: the resulting Laplace pressure drives fluid flow. Scaling ideas (of Siggia and of Furukawa) predict that, ultimately, this flow should become turbulent as inertial effects dominate over viscous ones. The physics here is complex: mesoscale simulation methods (such as lattice Boltzmann and dissipative particle dynamics) can play an essential role in its elucidation, as we describe. Likewise, it is a matter of experience that immiscible fluids will mix, on some lengthscale at least, if stirred vigorously enough. A scaling theory (of Doi and Ohta) predicts the dependence of a steady state domain size on shear rate, but assumes low Reynolds number (inertia is neglected). Our preliminary simulation results (three-dimensional, so far only on small systems) show little sign of the kind of steady state envisaged by Doi and Ohta; they raise instead the possibility of an oriented domain texture which can continue to coarsen until either inertial effects, or (in our simulations) finite size effects, come into play.

Some mathematical issues in the modeling of flow phenomena of polymeric liquid crystals

We consider shear flow of polymeric liquid crystals for large and small values of the Ericksen number ℰ. The model consists of governing equations for the velocity field v, the pressure p, the director n, and the order parameter s. The constitutive functions for the Leslie coefficients, αi, derived from the molecular theory of Doi play a crucial role in the modeling. One of the goals of the analysis is to examine the role of s in describing singularities as well as in obtaining new regimes which cannot be predicted by the previous Leslie–Ericksen model. In particular, solutions are obtained that correspond to domain structures parallel to the flow. The domains are separated by singular lines across which the director experiences jumps of ±45°. Such flows correspond to regimes with large values of ℰ. Another type of configuration analyzed in the article corresponds to periodic stripes parallel to the flow and such that the optic fields vary along the direction perpendicular to the plane of shear. Such configurations occur in the limit of small ℰ.

Closure approximations for the Doi theory: Which to use in simulating complex flows of liquid-crystalline polymers?

The goal of this article is to determine which closure model should be used in simulating complex flows of liquid-crystalline polymers (LCPs). We examine the performance of six closure models: the quadratic closure, a quadratic closure with finite molecular aspect ratio, the two Hinch–Leal closures, a hybrid between the quadratic and the first Hinch–Leal closures and a recently proposed Bingham closure. The first part of the article studies the predictions of the models in homogeneous flows. We generate their bifurcation diagrams in the (U,Pe) plane, where U is the nematic strength and Pe is the Peclet number, and place special emphasis on the effects of the flow type. These solutions are then compared with the “exact solutions” of the unapproximated Doi theory. Results show the Bingham closure to give the best approximation to the Doi theory in terms of reproducing transitions between the director aligning, wagging and tumbling regimes at the correct values of U and Pe and predicting the arrest of period...

Coupling effects between stress and concentration changes in polymers

Abstract The flow-induced concentration changes in polymers are analysed using the two-fluid theory of Doi and Milner, which predicts migration in flows with either curvilinear trajectories or non-uniform shear rates. Here we show that, even in simple shear flows, migration can be relevant since it can make the stationary solution with uniform concentration linearly unstable in fast flows.

Evaluation of the Entanglement Molecular Weights of Maize Starches from Solution Rheological Measurements

ABSTRACTThe entanglement molecular weights of waxy maize (WM) and normal maize (NM) starches were calculated from solution rheological data. The viscoelastic behavior of both WM and NM starches were measured at several different concentrations and then shifted to produce a master curve for each of the materials. The theory of Doi and Edwards was used to calculate the plateau moduli from which values for the entanglement molecular weights for WM and NM starches were calculated. The entanglement molecular weights were 100 ± 15 kg/mol for WM starch and 96 ± 8 kg/mol for NM starch. These two values were within experimental error of one another and represent the entanglement molecular weight of amylopectin, the major component of WM and NM starches. The entanglement degrees of polymerization for WM and NM starch, using a value of 162 g/mol for the monomer molecular weight of amylopectin, were 617 ± 92 and 592 ± 49, respectively. The values for the entanglement molecular weight and the entanglement degrees of polymerization for WM and NM starch were markedly higher than those quoted for many commercial polymers. This finding indicates that molecular weights of >1 million are required to produce starch‐based materials with consistent physical properties.

Effect of long range order on sheared liquid crystalline materials Part 1: compatibility between tumbling behavior and fixed anchoring

A viscoelastic constitutive equation for liquid crystalline flows is presented in terms of tensor order parameter dynamics. The presented equation takes into account short range order, long range order (Frank elasticity), and flow effects, and captures the spatio-temporal evolution of the tensor order parameter. The dimensionless parameters arising in the theory are the Deborah (De) number and Ericksen (Er) number, and their ratio is a new dimensionless number R which represent the ratio of short range elasticity to long range elasticity. At Er → ∞ the model converges to the Doi equation, and for De → 0 to the Leslie-Ericksen director equation; the consistency with the Doi theory is shown by the classical shear flow solutions: tumbling, oscillating, and steady state. The model is used to analyze rectilinear simple shear flow subjected to strong anchoring conditions. For the range of governing parameters studied in this paper it is found that for small Er, a spatially nonhomogenous steady state of the tensor order parameters exists. For De corresponding to the tumbling solutions of the Doi equation, the present theory predicts the presence of a composite three layer structure: (a) a center bulk region characterized by director tumbling; and (b) two director oscillating regions next to the bounding surfaces. The amplitude of the director oscillations decrease from a maximum at the boundaries between tumbling-oscillating modes to zero at the bounding surfaces. The boundary between the tumbling-oscillating mode is characterized by the time periodic emergence of an abnormal nematic state, which is characterized by a tensor order parameter with two equal eigenvalues on the shear plane. The emergence of the abnormal nematic state provides for a compatibility mechanism between oscillations and rotations. Thus, we have found through the use of computational modeling a new mechanism that explains the compatibility of Doi's tumbling solutions and fixed surface anchoring conditions. The new flow mechanism is independent of the type of strong anchoring conditions, and arises with planar, homeotropic, and any arbitrary director anchoring.

Simulating complex flows of liquid-crystalline polymers using the Doi theory

We simulate the startup flow of lyotropic liquid-crystalline polymers (LCPs) in an eccentric cylinder geometry. The objectives are to explore the mechanisms for the generation of disclinations in a nonhomogeneous flow and to study the coupling between the flow and the polymer configuration. The Doi theory, generalized to spatially varying flows and approximated by the quadratic closure, is used to model the evolution of LCP configurations. This, along with the equations of motion for the fluid, is solved by a finite-element method. The flow modification by the polymer stress is mild for the parameters used, but the LCP exhibits complex orientational behavior in different regions of the flow domain. For relatively weak nematic strength, a steady state is reached in which the director is oriented either along or transverse to the streamline, depending upon local flow conditions and the deformation history. A pair of disclinations, with strength ±1/2, are identified in the steady state, and the LCP configuration at the disclinations confirms the model of a structured defect core proposed by Greco and Marrucci (1992). For strong nematic strength, director tumbling occurs in the more rotational regions of the flow field, giving rise to a polydomain structure. The boundary of the tumbling domain consists of two disclinations of ±1/2 strength, a structure very similar to previous experimental observations of LCP domains.

The electrorheology of rigid rod poly(n-hexyl isocyanate) solutions

The electroviscosities of solutions of a lyotropic liquid crystalline polymer, poly(n-hexyl isocyanate), were investigated by a slit viscometer. The morphologies of the solutions being studied include the isotropic, the biphasic, and the fully liquid crystalline. All three morphologies exhibit viscosity enhancements with imposition of an electric field. However, the electrorheological behavior of the isotropic solution is different from those of other morphologies. The isotropic solution starts with a higher field free viscosity and its electroviscosity increases gradually with the increasing electric field strength. In contrast, the anisotropic solutions begin with lower zero field viscosities and the electroviscosities increase sluggishly until a critical field strength is reached; the viscosities then increase rapidly and finally exceed that of the isotropic solution. For the morphologies of the biphasic and the fully liquid crystalline, the dependence of the viscosity enhancements on field strength and shear rate can be described by a single variable. The variable scales with the square of the electric field strength and the reciprocal of the shear rate. By introducing the effect of the molecular permanent dipole moments into Doi's theory, the electrorheological effects of PHIC solutions can be satisfactorily interpreted.

Smoothed particle hydrodynamics techniques for the solution of kinetic theory problems Part 1: Method

The smoothed particle hydrodynamics (SPH) technique has been applied to a problem in kinetic theory, namely, the dynamics of liquid crystalline polymers (LCPs). It is a Lagrangian solution method developed for fluid flow calculations; its adaption to kinetic theory is outlined. The Lagrangian formulation of the Doi theory for LCPs is first described, and the problem is presented in the general framework of nonparametric density estimation. The implementation of the SPH technique in this specific problem is given, highlighting particular aspects of our implementation of SPH, including the form of the kernel function and use of an adaptive kernel. We then present results which demonstrate convergence and other details of the solution method, and also make comparisons with other solution techniques and discuss other potential applications.

On Poiseuille flow of liquid crystals

We consider Poiseuille flow of polymeric liquid crystals corresponding to large values of the velocity gradient. The model employed [Ericksen, 1991] proposes governing equations for the velocity field, v, the pressure p, the director n, and the order parameter s. The constitutive functions for the Leslie coefficients alphai derived from the molecular theory of Doi [1981] play a crucial role in the modelling. In addition to the Ericksen number, E, the present model exhibits a new non-dimensional parameter I, that represents the contribution of the elastic free energy of non-gradient type with respect to Frank-Oseen's elasticity. One of the goals of the analysis was to examine the role of s in describing singularities as well as in obtaining regimes which are not predicted by the previous Leslie-Ericksen model. In particular, solutions are obtained that correspond to domain structures parallel to the flow. Such domains are separated by singular lines across which the director experiences jumps of, approximately, 45 degrees with respect to the flow direction. A condition on the size of I is required in order to support such layered structures. The contribution of the energy associated with I turns out to play the role of an elastic surface energy which is, otherwise, neglected in the present model.

Flow birefringence of poly(n-hexyl isocyanate) solutions

A correction for optical quantities measured by a slit rheometer is proposed in order to obtain the corresponding optical properties at wall. This correction together with the Rabinowitch correction enables studies of the relation between the optical and the mechanical properties. A simple optical flow cell of a slit geometry was designed for the simultaneous measurements of the mechanical and the optical properties of flowing birefringent materials. The setup and the correction were applied to investigate the flow birefringence of the isotropic solutions of a liquid crystalline polymer, poly(n-hexyl isocyanate). The apparent birefringence defined as the refractive index difference between the ordinary light and the extraordinary light was found to increase with the increasing shear rates and was higher for solutions of high concentrations at a given shear rate. The stress optical law was checked. The ratio of the apparent birefringence to the shear stress was found to be independent of the shear rate at concentration of 10 wt%. For solutions of 15 and 20 wt%, the ratios increase with the shear rate when the shear rate is small, and they reach a maximum then decrease with increasing shear rate. The optical-stress behavior of PHIC solutions qualitatively follows the theoretical prediction of the Doi theory.

Two‐dimensional modeling of the electrorheological behavior of liquid crystalline polymer solutions

The molecular Doi theory of rigid rod polymer rheology is used to describe the experimentally observed electrorheological effect of liquid crystalline polymer solutions. Two‐dimensional Doi theory has been modified by including the electric potential energy of a dipole in the energy term of the diffusion equation and in the free energy expression. Based on the same model, a linear theory of viscoelasticity has been developed to predict the storage and loss moduli under the influence of an applied electric field. Calculations indicate that molecular orientation induces viscosity enhancement, while the model attributes nonzero storage and loss moduli to the distortion of the orientation distribution function. Molecular parameters in the model can be estimated from either theory or experiment. Using a constant rotational diffusivity, model predictions of low shear electroviscosity and low‐frequency storage and loss moduli agree well with experimental data for poly(n‐hexyl isocyanate) solutions.

Untersuchungen zur irreversibilität von Netzwerkentschlaufungen beim Fließen von Polymerschmelzen

AbstractExperimental and theoretical investigations of stress relaxation after “double‐step” shear experiments lead to a better understanding of the irreversibility assumption of polymer melts. Here we compare measurements of shear stress relaxation for a branched polyethylene and a (linear) polyisobutene melt to theoretical predictions of Wagner's first and second equation and to a molecular theory of Doi. None of these theories is able to describe all the results of the measurements. These investigations demonstrate, however, what types of “double‐step” shear experiments are most sensitive to the irreversible disentanglement of polymer melts.

A rheo‐optical study of near‐critical polymer solutions under oscillatory shear flow

The flow‐induced structure and dynamics of concentration fluctuations in a near‐critical polymer solution have been investigated using the technique of polarimetry. Specifically, this article presents scattering dichroism of a single‐phased, semidilute solution of polystyrene in dioctyl phthalate (PS/DOP) subjected to small amplitude oscillatory shear flow. The measured frequency response is compared to results obtained using the Onuki–Doi theory, which relates dichroism to structure factor, and the Helfand–Fredrickson model, from which the structure factor is calculated. The predictions are in good qualitative agreement with the experimental data when stress relaxation is taken into account in the constitutive equation relating stress to strain in the system. Our results demonstrate that the strength of the coupling between viscoelastic stress and flow governs the behavior of concentration fluctuations in semidilute polymer solutions under oscillatory shear flow.

A comparison of closure approximations for the Doi theory of LCPs

The behavior of the Doi model for liquid crystalline polymers, using a closure approximation due to Hinch and Leal (HL1), is studied and compared to the behavior using the simple quadratic closure. It is found that the HL1 closure gives some aphysical results for simple shear flow, but describes a much richer solution structure for this flow than the quadratic closure, including tumbling and wagging phenomena that are found both experimentally, and in solutions of the Doi theory without closure approximations. However, there is a very sharp transition in the predicted behavior as the flow is changed very slightly to one in which the strain rate exceeds the vorticity: the orientation distribution becomes flow aligning for all values of the nematic potential, and the phenomena of tumbling and wagging are suppressed completely. Although the quadratic closure model does not exhibit tumbling in simple shear flow, and thus has been sharply criticized as an approximation of the Doi model, it does exhibit tumbling for flows that are only very slightly more rotational than simple shear flow, and the sharp transition to flow aligning solutions with added strain rate occurs just prior to simple shear, instead of just after as in the HL1 model. Finally, we find that the transition from flow‐aligning to periodic solutions occurs via two distinct mechanisms, one occurring for Pe≳20 where there is a strong coupling between the director orientation and the degree of alignment and the other for smaller Pe where the degree of orientation is unchanged by the flow.

Dynamics of a 2311 Base Pair Superhelical DNA in Dilute and Semidilute Solutions

Dynamic light scattering-photon correlation spectroscopy (PCS) has been used to study the dynamics of a 2311 base pair (bp) superhelical plasmid DNA in dilute and semidilute solutions. In dilute solution, translational and rotational/internal modes could be resolved. Assuming that the DNA is a rigid rod, the translational diffusion coefficient determined at infinite dilution gives cylinder dimensions consistent with a calculated length L of 230 nm and a diameter of 13.4 nm. If the rotational internal mode is interpreted as rotation of a rigid rod, the derived rotational diffusion coefficient is consistent with a shorter rod of a smaller length to diameter ratio. The dilute and semidilute regimes are clearly distinguished by a break in the concentration dependence of the mutual diffusion coefficient. The transition concentration, using the L given above, was found to lie at cL 3 between 3 and 5. In the semidilute regime, a slow mode was detected in addition to the main diffusion mode. The relative amplitude of the slow mode increases with increasing concentration until it is the dominant mode in the DLS frequency distribution. The slow mode frequency decreases with increasing concentration. The characteristics of the slow mode are not in accord with those of the slow mode predicted by the spinodal decomposition theory of Doi, Shimada, and Okano (DSO) for stiff rodlike polymers. The observed mode is consistent with the formation of aggregates in the semidilute regime. The mutual diffusion coefficient in the semidilute regime shows a linear concentration dependence, which is in qualitative agreement with the predictions of the DSO theory. The slope of this linear concentration dependence, however, differs markedly from the theoretical prediction assuming a concentration-independent self-diffusion coefficient. In addition, no deviation from quadratic scattering vector length dependence of the mutual diffusion mode, as predicted by the DSO theory, is observed. The rotational diffusion/internal relaxation frequencies appear to decrease with increasing concentration. However, the rather large scatter in the data (due to the relatively small contribution of this mode to the spectrum in the presence of the slow mode) prevented a quantitative comparison with existing theories