Rigid Dumbbell Research Articles

Order in polymeric liquids under oscillatory shear flow

We examine the second order orientation tensor for the simplest molecular model relevant to a polymeric liquid in large-amplitude oscillatory shear (LAOS) flow, the rigid dumbbell suspension. For this, we use an approximate solution to the diffusion equation for rigid dumbbells, an expansion for the orientation distribution function truncated after the fourth power of the shear rate amplitude. We then calculate the second order orientation tensor, and then use this to calculate the order parameter tensor. We next examine the invariants of both the second order orientation tensor and the order parameter tensor. From the second invariant of the order parameter tensor, we calculate the scalar, the nematic order, and examine its evolution for a polymeric liquid in LAOS. We find this nematic order, our main result, to be even. We use Lissajous figures to illustrate the roles of the Weissenberg and Deborah numbers on the evolving order in LAOS. We use the low frequency limit of our main result to arrive at an expression for the nematic order in steady shear flow. Our work gives a first glimpse into macromolecular order in LAOS. Our work also provides analytical benchmarks for numerical solutions to the diffusion equation for both oscillatory and steady shear flows.

Macromolecular tumbling and wobbling in large-amplitude oscillatory shear flow

For a suspension of rigid dumbbells, in any simple shear flow, we recently solved for the diffusion equation for the orientation distribution function by a power series expansion in the shear rate magnitude. In this paper, we focus specifically on large-amplitude oscillatory shear flow, for which we extend the orientation distribution function to the 6th power of the shear rate amplitude. We arrive at the Fourier solution for each harmonic contribution to the total orientation distribution function, separating each harmonic into its coefficients in and out-of-phase with cos nωt, ψn′ and ψn″, respectively. We plot, for the first time, the evolving normalized alternant macromolecular orientation. Moreover, to deepen our understanding of the macromolecular motions, we distinguish and study the two types of possible rotations, tumbling and wobbling.

Exact coefficients for rigid dumbbell suspensions for steady shear flow material function expansions

From kinetic molecular theory, we can attribute the elasticity of polymeric liquids to macromolecular orientation. For a suspension of rigid dumbbells, subject to a particular flow field, we must first solve the diffusion equation for the orientation distribution function. From this distribution, we then calculate physical properties such as the steady shear flow material functions. We thus arrive at power series expansions in the shear rate for both the orientation distribution function and for the steady shear flow material functions. Analytical work on many viscoelastic material functions must be checked for consistency, in their steady shear flow limits, against these power series. For instance, for large-amplitude oscillatory shear flow, we recover the coefficients of these expansions in the limits of low test frequency. The coefficients of the steady shear viscosity and the first normal stress coefficient functions are not known exactly beyond the fourth power. In this work, for both of these functions, we arrive at exact expressions for the first 20 coefficients. We close with five worked examples illustrating uses for our new coefficients.

Orientation Distribution Function Pattern for Rigid Dumbbell Suspensions in Any Simple Shear Flow

AbstractFor rigid dumbbells suspended in a Newtonian solvent, the viscoelastic response depends exclusively on the dynamics of dumbbell orientation. The orientation distribution function ψ(θ, φ, t) represents the probability of finding dumbbells within the range (θ, θ + dθ) and (φ, φ + dφ). This function is expressed in terms of a partial differential equation (the diffusion equation), which, for any simple shear flow, is solved by postulating a series expansion in the shear rate magnitude. Each order of this expansion yields a new partial differential equation, for which one must postulate a form for its solution. This paper finds a simple and direct pattern to these solutions. The use of this pattern reduces the amount of work required to determine the coefficients of the power series expansion of the orientation distribution function, ψ i . To demonstrate the usefulness of this new pattern, new expressions for these coefficients are derived up to and including the sixth power of the shear rate magnitude. This work also completes previous findings that ended at the fourth power of the shear rate magnitude.

Rheological consequences of wet and dry friction in a dumbbell model with hydrodynamic interactions and internal viscosity.

The effect of fluctuating internal viscosity and hydrodynamic interactions on a range of rheological properties of dilute polymer solutions is examined using a finitely extensible dumbbell model for a polymer. Brownian dynamics simulations are used to compute both transient and steady state viscometric functions in shear flow. The results enable a careful differentiation of the influence, on rheological properties, of solvent-mediated friction from that of a dissipative mechanism that is independent of solvent viscosity. In particular, hydrodynamic interactions have a significant influence on the magnitude of the stress jump at the inception of shear flow, and on the transient viscometric functions, but a negligible effect on the steady state viscometric functions at high shear rates. Zero-shear rate viscometric functions of free-draining dumbbells remain essentially independent of the internal viscosity parameter, as predicted by the Gaussian approximation, but the inclusion of hydrodynamic interactions induces a dependence on both the hydrodynamic interaction and the internal viscosity parameter. Large values of the internal viscosity parameter lead to linear viscoelastic predictions that mimic the behavior of rigid dumbbell solutions. On the other hand, steady-shear viscometric functions at high shear rates differ in general from those for rigid dumbbells, depending crucially on the finite extensibility of the dumbbell spring.

Molecular continua for polymeric liquids in large-amplitude oscillatory shear flow

In this paper, we connect a molecular description of the rheology of a polymeric liquid to a continuum description, and then test this connection for large-amplitude oscillatory shear (LAOS) flow. Specifically, for the continuum description, we use the 6-constant Oldroyd framework, and for the molecular, we use the simplest relevant molecular model, the suspension of rigid dumbbells. By relevant, we mean predicting at least higher harmonics in the shear stress response in LAOS. We call this connection a molecular continuum, and we examine two ways of arriving at this connection. The first goes through the retarded motion expansion, and the second expands each of a set of specific material functions (complex, steady shear, and steady uniaxial extensional viscosities). Both ways involve in comparing the coefficients of expansions and then solve for the six constants of the continuum framework in terms of the two constants of the rigid dumbbell suspension. The purpose of a molecular continuum is that many well-known results for rigid dumbbell suspensions in other flow fields can also be easily obtained, without having to firstly find the orientation distribution function. In this paper, we focus on the recent result for the rigid dumbbell suspension in LAOS. We compare the accuracies of the retarded motion molecular continuum (RMMC) with the material function molecular continuum (MFMC). We find the RMMC to be the most accurate for LAOS.

An exploration and further study of an enhanced Oldroyd model

The Oldroyd 6-constant constitutive model for polymeric liquids [J. G. Oldroyd, “On the formulation of rheological equations of state,” Proc. R. Soc. A 200, 523 (1950)] was shown in the work of Bird [“A modification of the Oldroyd model for rigid dumbbell suspensions with Brownian motion,” Z. Angew. Math. Phys. 23, 157 (1972)] to have insufficient flexibility even to describe all second-order time-dependent behaviors of the simple molecular model of rigid dumbbells in solution. Bird proposed an enhancement of the Oldroyd model that would remove this deficiency. The advantage of such a continuum constitutive model is that it is far easier to use in solving specific flow problems than the more physical, but far more cumbersome, molecular models. Thus, if this enhanced constitutive model is indeed able to replicate the results of the molecular models, it should be extremely useful over the applicable range. Here, we employ the enhanced constitutive model of Bird [Z. Angew. Math. Phys. 23, 157 (1972)] to solve four different flow problems, and we verify that our solutions coincide exactly with those of a molecular model: through third order for rectilinear steady shear flow and steady extensional flow, and for one steady non-rectilinear flow (eccentric disk rheometer flow); and through second order for one unsteady (i.e., oscillatory) shear flow, thus providing strong confirmation of the physical veracity and utility of this continuum constitutive model, at least for small and moderate shear or extension rates.

Polymer orientation contributions in large-amplitude oscillatory shear flow

Previously, we used a dilute suspension of rigid dumbbells as a model for the dynamics for polymeric liquids in large-amplitude oscillatory shear (LAOS) flow. We then use dumbbell orientation to explain fluid elasticity. We derived the expression for the polymer orientation distribution, and then we decomposed this function into its first five harmonics (the zeroth, first, second, third and fourth harmonics). We further separated the harmonics into their components, in-phase and out-of-phase with cos nωt. In this work, we deepen our understanding of the relationship between the orientation distribution function and the shear stress and normal stress differences. We also investigate the components of orientation that make no contribution at all to measured rheological responses. Further, the larger the γ˙0 of the oscillatory shear flow, the greater the fraction of polymer that escapes rheological measurement. Our analysis focuses on the nonlinear viscoelastic regime, and specifically, where both λω and λγ˙0 are unity. We learn that all orientation contributions to the normal stresses also contribute to the shear stress. The parts of the orientation distribution contributing to the shear stress take on the familiar peanut shapes of the total orientation distribution. The parts of the orientation distribution causing the normal stress differences are subsets of the part of the orientation distribution contributing to the shear stress.

Kinetics and Morphology of Flow Induced Polymer Crystallization in 3D Shear Flow Investigated by Monte Carlo Simulation

To explore the kinetics and morphology of flow induced crystallization of polymers, a nucleation-growth evolution model for spherulites and shish-kebabs is built based on Schneider rate model and Eder model. The model considers that the spherulites are thermally induced, growing like spheres, while the shish-kebabs are flow induced, growing like cylinders, with the first normal stress difference of crystallizing system being the driving force for the nucleation of shish-kebabs. A two-phase suspension model is introduced to describe the crystallizing system, which Finitely Extensible Non-linear Elastic-Peterlin (FENE-P) model and rigid dumbbell model are used to describe amorphous phase and semi-crystalline phase, respectively. Morphological Monte Carlo method is presented to simulate the polymer crystallization in 3D simple shear flow. Roles of shear rate, shear time and shear strain on the crystallization kinetics, morphology, and rheology are analyzed. Numerical results show that crystallization kinetics, morphology and rheology in shear flow are qualitatively in agreement with the theoretical, experimental and other numerical works which verifies the validity and effectiveness of our model and algorithm. To our knowledge, this is the first time that a model and an algorithm revealing the details of crystal morphology have been applied to the flow induced crystallization of polymers.

Macromolecular Origins of Fifth Shear Stress Harmonic in Large-Amplitude Oscillatory Shear Flow

This study examines the simplest relevant molecular model of a polymeric liquid in large-amplitude oscillatory shear (LAOS) flow: the suspension of rigid dumbbells in a Newtonian solvent. For such suspensions, the viscoelastic response of the polymeric liquid depends exclusively on the dynamics of dumbbell orientation. We have previously derived explicit analytical expressions for the shear rate amplitude and frequency dependences of the first and third harmonics of the alternating shear stress response in LAOS. Higher harmonics sculpt the shear stress, distorting it from its sinusoidal shape. In this work, we derive the polymer contribution to the shear stress response up to and including the next higher, fifth harmonic. For this, we use the fourth order term in the expansion of the orientation distribution to calculate the shear stress response. Our analysis employs the general method of Bird and Armstrong [J Chem Phys, 56, 3680 (1972)]. Our expression is the only one to have been derived from a molecular theory for a fifth harmonic. Our paper thus provides the first glimpse of the molecular origins of a shear stress harmonic higher than the third.

Existence and stability of periodic oscillations of a rigid dumbbell satellite around its center of mass

We study the existence and stability of periodic solutions of a differential equation that models the planar oscillations of a satellite in an elliptic orbit around its center of mass. The proof is based on a suitable version of Poincare-Birkhoff theorem and the third order approximation method.

Molecular origins of higher harmonics in large-amplitude oscillatory shear flow: Shear stress response

Recent work has focused on deepening our understanding of the molecular origins of the higher harmonics that arise in the shear stress response of polymeric liquids in large-amplitude oscillatory shear flow. For instance, these higher harmonics have been explained by just considering the orientation distribution of rigid dumbbells suspended in a Newtonian solvent. These dumbbells, when in dilute suspension, form the simplest relevant molecular model of polymer viscoelasticity, and this model specifically neglects interactions between the polymer molecules [R. B. Bird et al., “Dilute rigid dumbbell suspensions in large-amplitude oscillatory shear flow: Shear stress response,” J. Chem. Phys. 140, 074904 (2014)]. In this paper, we explore these interactions by examining the Curtiss-Bird model, a kinetic molecular theory designed specifically to account for the restricted motions that arise when polymer chains are concentrated, thus interacting and specifically, entangled. We begin our comparison using a heretofore ignored explicit analytical solution [X.-J. Fan and R. B. Bird, “A kinetic theory for polymer melts. VI. Calculation of additional material functions,” J. Non-Newtonian Fluid Mech. 15, 341 (1984)]. For concentrated systems, the chain motion transverse to the chain axis is more restricted than along the axis. This anisotropy is described by the link tension coefficient, ϵ, for which several special cases arise: ϵ = 0 corresponds to reptation, ϵ > 1/8 to rod-climbing, 1/5 ≤ ϵ ≤ 3/4 to reasonable predictions for shear-thinning in steady simple shear flow, and ϵ = 1 to the dilute solution without hydrodynamic interaction. In this paper, we examine the shapes of the shear stress versus shear rate loops for the special cases ϵ=0,1/8,3/8,1, and we compare these with those of rigid dumbbell and reptation model predictions.

Complex polymer orientation

Previously, we explored the dynamics of a dilute suspension of rigid dumbbells as a model for polymeric liquids in large-amplitude oscillatory shear (LAOS) flow. We derived the expression for the polymer orientation distribution, and then we decomposed this function into its first five harmonics (the zeroth, first, second, third and fourth harmonics). In this work, we separate each harmonic into its components in-phase and out-of-phase with cosnωt, and then use three-dimensional imaging to investigate complex polymer motion in LAOS. Our analysis focuses on the nonlinear viscoelastic regime, specifically, where both λω and λγ˙0 are unity. We learn that harmonic components, in-phase and out-of-phase with cosnωt, contribute to orientation differently, and that out-of-phase components tend to give more probable orientation away from the flow direction. We also find that some orientation contributions are caused by interferences between complex components, some constructive and some destructive.

Fourier decomposition of polymer orientation in large-amplitude oscillatory shear flow.

In our previous work, we explored the dynamics of a dilute suspension of rigid dumbbells as a model for polymeric liquids in large-amplitude oscillatory shear flow, a flow experiment that has gained a significant following in recent years. We chose rigid dumbbells since these are the simplest molecular model to give higher harmonics in the components of the stress response. We derived the expression for the dumbbell orientation distribution, and then we used this function to calculate the shear stress response, and normal stress difference responses in large-amplitude oscillatory shear flow. In this paper, we deepen our understanding of the polymer motion underlying large-amplitude oscillatory shear flow by decomposing the orientation distribution function into its first five Fourier components (the zeroth, first, second, third, and fourth harmonics). We use three-dimensional images to explore each harmonic of the polymer motion. Our analysis includes the three most important cases: (i) nonlinear steady shear flow (where the Deborah number is zero and the Weissenberg number is above unity), (ii) nonlinear viscoelasticity (where both and exceed unity), and (iii) linear viscoelasticity (where exceeds unity and where approaches zero). We learn that the polymer orientation distribution is spherical in the linear viscoelastic regime, and otherwise tilted and peanut-shaped. We find that the peanut-shaping is mainly caused by the zeroth harmonic, and the tilting, by the second. The first, third, and fourth harmonics of the orientation distribution make only slight contributions to the overall polymer motion.

Constitutive model fingerprints in medium-amplitude oscillatory shear

Rheologists have expectations for signatures of linear viscoelastic properties, such as shapes of G′(ω) and G″(ω). Medium amplitude (or asymptotically-nonlinear) oscillatory shear (MAOS) provides additional nonlinear rheological information with low dimensional, well-defined material functions [Ewoldt and Bharadwaj, Rheol. Acta 52, 201–209 (2013)]. Here, we develop expectations of signatures (or fingerprints) for the four asymptotically-nonlinear material functions associated with MAOS, [e1](ω), [e3](ω), [v1](ω), [v3](ω). Although the linear fingerprints may be identical for different models, the asymptotically-nonlinear fingerprints may be different in magnitude, frequency-scaling, curve shapes, and sign changes. To perform the analysis, we collect/translate a library of available analytical strain-controlled MAOS fingerprints for seven different constitutive models. Using this library, we identify general trends and highlight key differences of asymptotic-nonlinear viscoelasticity. Asymptotic nonlinearities for all models considered here obey the terminal regime inter-relations and frequency scaling predicted by Bharadwaj and Ewoldt [J. Rheol. 58, 891–910 (2014)]. Unlike the positive linear viscoelastic measures, at least one of the four asymptotic nonlinearities changes signs with Deborah number (De). Following sign interpretations of Ewoldt and Bharadwaj [Rheol. Acta 52, 201–209 (2013)], we show that nonlinearities tend to be driven by strain-rates at small De, and by strains at large De, a trend observed for nearly all the constitutive models studied here, the exception being the model for dilute rigid dumbbell suspensions of Bird et al. [J. Chem. Phys. 140, 074904 (2014)]. Some constitutive models exhibit multiple sign changes at intermediate De and there may be no universal behavior of asymptotically-nonlinear fingerprints in this regime. Therefore, frequency-dependent signatures can be material-specific. This will allow inverse problems to infer structure, select models, and fit model parameters using asymptotically-nonlinear signatures. To illustrate this aspect, we demonstrate a fingerprint matching exercise with experimental measurements on a transiently cross-linked hydrogel system. We find that currently available model fingerprints can match the qualitative magnitudes and frequency dependence, but not the signs of the experimental transient network response.

Dynamics of elastic dumbbells sedimenting in a viscous fluid: oscillations and hydrodynamic repulsion

AbstractHydrodynamic interactions between two identical elastic dumbbells settling under gravity in a viscous fluid at low Reynolds number are investigated using the point-particle model. The evolution of a benchmark initial configuration is studied, in which the dumbbells are vertical and their centres are aligned horizontally. Rigid dumbbells and pairs of separate beads starting from the same positions tumble periodically while settling. We find that elasticity (which breaks the time-reversal symmetry of the motion) significantly affects the system dynamics. This is remarkable when taking into account that elastic forces are always much smaller than gravity. We observe oscillating motion of the elastic dumbbells, which tumble and change their length non-periodically. Independently of the value of the spring constant, a horizontal hydrodynamic repulsion appears between the dumbbells: their centres of mass move apart from each other horizontally. This motion is fast for moderate values of the spring constant $k$, and slows down when $k$ tends to zero or to infinity; in these limiting cases we recover the periodic dynamics reported in the literature. For moderate values of the spring constant, and different initial configurations, we observe the existence of a universal time-dependent solution to which the system converges after an initial relaxation phase. The tumbling time and the width of the trajectories in the centre-of-mass frame increase with time. In addition to its fundamental significance, the benchmark solution presented here is important to understanding general features of systems with a larger number of elastic particles, in regular and random configurations.

Orientation in Large-Amplitude Oscillatory Shear

We examine the simplest relevant molecular model for large-amplitude oscillatory shear flow of a polymeric liquid: The dilute suspension of rigid dumbbells in a Newtonian solvent. We find explicit analytical expressions for the orientation distribution, and specifically for test conditions of frequency and shear rate amplitude that generate higher harmonics in the shear stress and normal stress difference responses. We solve the diffusion equation analytically to explore molecular orientation induced by oscillatory shear flow. We see that the orientation distribution is neither even nor odd. We find zeroth, first, second, third, and fourth harmonics of the orientation distribution, and we have derived explicit analytical expressions for these. We provide a clear visualization of the orientation distribution in large-amplitude oscillatory shear flow in spherical coordinates all the way around one full alternant cycle. The Newtonian distribution is nearly isotropic, the linearly viscoelastic, only slightly anisotropic, and the nonlinear viscoelastic, highly anisotropic.

Normal stress differences in large-amplitude oscillatory shear flow for dilute rigid dumbbell suspensions

We examine the simplest relevant molecular model for large-amplitude oscillatory shear (LAOS) flow of a polymeric liquid: the suspension of rigid dumbbells in a Newtonian solvent. We find explicit analytical expressions for the shear rate amplitude and frequency dependences of the zeroth, second and fourth harmonics of the first and second normal stress difference responses. We include a detailed comparison of these predictions with the corresponding results for the simplest relevant continuum model: the corotational Maxwell model. We find that the responses of both models are qualitatively alike. The rigid dumbbell model relies entirely on the dumbbell orientation to explain the viscoelastic response of the polymeric liquid, including the higher harmonics in large-amplitude oscillatory shear flow. Our analysis employs the general method of Bird and Armstrong (1972) for analyzing the behavior of the rigid dumbbell model in any unsteady shear flow.We derive the first three terms of the deviation of the orientational distribution function from the equilibrium state. Then, after getting the “paren functions,” we use these for evaluating the normal stress differences for large amplitude oscillatory shear flow. We find the shapes of the first normal stress difference versus shear rate loops predicted to be reasonable [see Fig. 1(a)]. We find that the second normal stress difference is not proportional to the first, and that its shape differs markedly from that of the first [cf. Fig. 1(a) and (b)]. We discover the same remarkable qualitative similarity between the predictions of the rigid dumbbell model and the corotational Maxwell model for the first normal stress difference. We find no qualitative similarities between the dumbbell and the continuum models for any of the predicted coefficients of the second normal stress differences in large-amplitude oscillatory shear flow.

Dilute rigid dumbbell suspensions in large-amplitude oscillatory shear flow: Shear stress response

We examine the simplest relevant molecular model for large-amplitude shear (LAOS) flow of a polymeric liquid: the suspension of rigid dumbbells in a Newtonian solvent. We find explicit analytical expressions for the shear rate amplitude and frequency dependences of the first and third harmonics of the alternating shear stress response. We include a detailed comparison of these predictions with the corresponding results for the simplest relevant continuum model: the corotational Maxwell model. We find that the responses of both models are qualitatively similar. The rigid dumbbell model relies entirely on the dumbbell orientation to explain the viscoelastic response of the polymeric liquid, including the higher harmonics in large-amplitude oscillatory shear flow. Our analysis employs the general method of Bird and Armstrong ["Time-dependent flows of dilute solutions of rodlike macromolecules," J. Chem. Phys. 56, 3680 (1972)] for analyzing the behavior of the rigid dumbbell model in any unsteady shear flow. We derive the first three terms of the deviation of the orientational distribution function from the equilibrium state. Then, after getting the "paren functions," we use these for evaluating the shear stress for LAOS flow. We find the shapes of the shear stress versus shear rate loops predicted to be reasonable.

On the periodic solutions of a rigid dumbbell satellite in a circular orbit

The aim of the present paper is to provide suficient conditions for the existence of periodic solutions of the perturbed attitude dynamics of a rigid dumbbell satellite in a circular orbit.