Rigid Dumbbell Model Research Articles

Stress Tensor of Single Rigid Dumbbell by Virtual Work Method

We derive the stress tensor of a rigid dumbbell by using the virtual work method. In the virtual work method, we virtually apply a small deformation to the system, and relate the change of the energy to the work done by the stress tensor. A rigid dumbbell consists of two particles connected by a rigid bond of which length is constant (the rigid constraint). The energy of the rigid dumbbell consists only on the kinetic energy. Also, only the deformations which do not violate the rigid constraint are allowed. Thus we need the dynamic equations which are consistent with the rigid constraint to apply the virtual deformation. We rewrite the dynamic equations for the underdamped SLLOD-type dynamic equations into the forms which are consistent with the rigid constraint. Then we apply the virtual deformation to a rigid dumbbell based on the obtained dynamic equations. We derive the stress tensor for the rigid dumbbell model from the change of the kinetic energy. Finally, we take the overdamped limit and derive the stress tensor and the dynamic equation for the overdamped rigid dumbbell. We show that the Green-Kubo type linear response formula can be reproduced by combining the stress tensor and the dynamic equation at the overdamped limit.

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.

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.

Multi-scale molding and numerical simulation of the flow-induced crystallization of polymer

A numerical simulation for the isothermal flow-induced crystallization (FIC) of polymer melt in a simple shear flow is presented. The model is based on a two-phase suspension model established for crystallization and has the advantage of taking into account the polymer melt rheological behavior through the first normal stress difference calculation. A FENE dumbbell model and a rigid dumbbell model are used to describe the amorphous phase and the semi-crystalline phase, respectively. Crystallization is described as a spherulitical nucleation and growth process. The results show that the short-term shear has a large effect on the crystallization dynamics of polymer.

Simulations of concentrated suspensions of rigid fibers: Relationship between short-time diffusivities and the long-time rotational diffusion

Brownian dynamics simulations of the behavior of suspensions of fibers demonstrate that the scaling of the rotational diffusivity with respect to the number density (nL3) is a sensitive function of the thickness and the parameter L2D(R0)/D(T0), where D(R0) is the rotational diffusivity at infinite dilution, D(T0) is the average center-of-mass diffusivity at infinite dilution, and L is the fiber length. Existing theories for the long-time rotational diffusivities of rigid fibers in the semidilute and concentrated regimes fail to accurately account for the relationship with the dilute values of the rotational and translational diffusivities of the various physical models used to simulate the fibers. The concentration regime studied in this work ranges from a number density of nL3 approximately 0-150, which is below the transition from an isotropic to nematic state. The effect of the fiber thickness was studied by performing simulations of rods with aspect ratios (fiber length over diameter) of 25, 50, and 500, as well as performing projections for infinitely thin fibers. The excluded volume of the rods was enforced through the use of short-range potentials. For a rod with an aspect ratio of 50 with a parameter of L2D(R0)/D(T0)=9, which corresponds to a slender-body model of the individual fibers, the rotational diffusivity (D(R)) scales as D(R)/D(R0) approximately (nL3)(-1.9) in the concentration regime of 70 < or = nL3 < or = 150. Similarly with a parameter of L2D(R0)/D(T0)=4, corresponding to a rigid-dumbbell model, the rotational diffusivity scales as D(R)/D(R0) approximately (nL3)(-1.1) over the same range of concentrations. For rods with aspect ratios of 25, it is observed that a difference in the scaling is seen for L2D(R0)/D(T0) approximately < 8, with higher values of this ratio exhibiting essentially the same scaling. Additional values of the ratio L2D(R0)/D(T0) were investigated to determine the overall behavior of the suspension dynamics with respect to this parameter. These findings resolve discrepancies between simulation results for rotational diffusivities reported by previous investigators and provide new insights for the development of an accurate theory for the diffusivity of rigid rods suspended in solution.

A model for post-flow induced crystallization: General equations and predictions

The aim of this work is to model the flow-enhanced crystallization and the flow-induced morphological changes of semi-crystalline materials during and after shearing flow. A FENE-P dumbbell model and a rigid dumbbell model are used to describe the molecular chain conformation and the orientation evolution for the amorphous phase and the semi-crystalline phase, respectively. The effect of flow on crystallization is considered by relating excess free energy and flow-induced orientation to crystallization kinetics. The crystallization of the material couples back to influence the solidification rheology of the crystallizing system. An isotactic polypropylene is used as an example to illustrate model predictions. We predict a pronounced effect of short-term shear treatments in accelerating nucleation and changing rheological behavior. Results are compared with available experimental data.

Formation of the shear-induced state in dilute cationic surfactant solutions

In cationic surfactant solutions a change of state occurs due to mechanical stresses. In the dilute regime of rodlike micelles the formation of a so-called Shear-Induced State (SIS) occurs above a critical shear rate. In this context ‘dilute’ means that there is no sterical interaction between rodlike micelles, the solution is below the overlap concentration. Employing a mathematical model, it is shown that aggregation forces are weak compared to hydrodynamic forces. The mathematical formulation is based on a model of Israelachvili which describes the chemical potential of micelles. Hydrodynamic forces are calculated with a rigid-dumbbell model. SIS formation can be explained by the destruction of rodlike micelles.

Dependence of the Stress Tensor on the Intramolecular Viscosity

A family of rheological models that take into account intramolecular friction, various types of intra‐ and intermolecular interactions, and various types of flexibility mechanisms are developed on the levels of configuration space kinetic theory and of conformation tensor theory. Particular attention is paid to formulae for the extra stress tensor. Predictions of rheological properties are calculated for one selected model corresponding to stiff macromolecular chains and large intramolecular friction. The predictions are found to be very similar to the predictions implied by the rigid dumbbell model. The main difference is in the mathematical complexity of the governing equations. While all solutions of the governing equations that are presented herein are found analytically in closed form, solutions of the governing equations of the rigid dumbbell model require tedious numerical calculations. Since mathematical simplicity of rheological models is particularly useful in numerical calculations of flows of ...

Dilute solutions of macromolecules in a rectilinear Poiseuille flow

Rheological results for a dilute solution of elastic and rigid dumbbells in a rectilinear Poiseuille flow are obtained within the framework of the kinetic theory of polymeric liquids. The values of the stresses and viscometric functions resulting from the two models of a macromolecule are calculated and compared. In both cases some of the stress tensor components show contributions due to the coupling of Brownian motion to convection in the nonhomogeneous velocity field of the solvent. The coupling parameter ε∼λγ̄L/B is assumed small and used to develop a perturbation scheme. Here, γ̄ is the shear rate at the wall, B is the half-width of the channel, and L and λ are the characteristic length and relaxation time of the macromolecule. Unlike the elastic dumbbell model, the rigid dumbbell model predicts a viscosity increase with increasing channel width and a negative value for the second normal stress difference.

Behavior of Solutions of Linear Macromolecules in Steady Shear Flow with Superposed Oscillations

The rotational diffusion equation has been solved for a suspension of rigid dumbbells in a velocity field which is the superposition of steady shearing flow (large deformation) and an infinitesimal oscillatory motion (very small deformation). An expression is obtained for the associated shear stresses and the results are expressed in terms of a complex viscosity, appropriately modified for the superposed steady shearing flow. The rigid dumbbell model gives a maximum in the real part of the complex viscosity as well as negative values for the imaginary part at low frequencies. Such behavior has been observed experimentally. The elastic dumbbell model (two beads connected by a Hookean spring) does not give results which are qualitatively similar to the experimental data.

Stress Relaxation in Solutions of Linear Macromolecules

The rigid dumbbell model has been used to derive expressions for the relaxation of shear and normal stresses after the cessation of steady shearing flow. It is found that the shear stresses relax more rapidly than the normal stresses, and that all stresses relax more rapidly as the velocity gradient during the prior steady state shearing flow increases; this behavior is in qualitative agreement with experimental data. The results further show that the integral under the shear-stress relaxation curve is simply related to the normal stress in steady flow. The elastic dumbbell model, with a Gaussian spring, gives results inconsistent with experiment.

Rotational Specific Heat and Half Quantum Numbers

Application of half quantum numbers to the theory of the rotational specific heat of hydrogen.—It is shown from a consideration of infra-red rotation-oscillation spectra, that the lowest possible azimuthal quantum number for a non-oscillating rotating molecule of the rigid dumb-bell model can have only the values zero, one, or one-half. An elementary theory of quantization in space for the new case of half quantum numbers is then developed which shows that the a priori probabilities for successive levels of rotational energy stand in the ratios of 1, 2, 3,.... The specific heat curve for diatomic hydrogen to 300°K is then calculated on the basis of the energy levels and of a priori probabilities corresponding to half quantum numbers, and is compared with the experimental points and with the curves calculated by Reiche using zero and one as the lowest possible azimuthal quantum number. At low temperatures the new curve agrees with the experimental data as well as any curve of Reiche's. At the higher temperatures, none of the curves agree with all the experimental points. The moment of inertia for the hydrogen molecule corresponding to the new curve is J=1.387×10^-41 gm cm2, about two-thirds the values assumed by Reiche, 2.095 to 2.293×10^-41, and agrees better with the conclusion of Sommerfeld from the separation of lines in the many lined spectrum of hydrogen, that the moment of inertia of an excited hydrogen molecule is 1.9×10^-41 gm cm2, which should be greater than that of the unexcited molecules involved in specific heats. Hence the possibility of half quantum numbers seems worthy of consideration.