Brownian dynamics simulations of shear-thickening in dilute polymer solutions

A versatile model describing the shear thickening behaviour of dilute polymer solutions in high shear flows is presented. The polymer macromolecules are modelled as Hookean elastic dumbbells which deform affinely during flow. In addition, the dumbbells feel a retractive anisotropic hydrodynamic drag and an isotropic Brownian force. Furthermore, it is assumed that high shear rate increases the probability of molecules forming associations and this is described through expressions for the frequencies of association and dissociation, without explicitly accounting for finite extensibility, hydrodynamic interaction or excluded volume effects. Thus, a reversible kinetic process is incorporated into the model, which results in two diffusion equations for the associated and dissociated dumbbells. Numerical simulations predict shear thickening for specific range of parameters, which are physically meaningful and related to molecular characteristics of the polymer. A comparison against experimental data reported in the literature revealed very promising results, thus confirming the ability of this model to predict shear thickening under a wide range of conditions, for various polymer models.Nomenclature A A factor in the frequency of association - B Frequency of dissociation - B ₀ Reference frequency of dissociation - c Concentration of polymer solution - c _i Concentration of singlets (i = 1) and doublets (i = 2) in the solution - c ^* The overlap concentration - D _t Translation coefficient of molecule - F _i (Q) Spring force for a singlet (i = 1) and for a doublet (i = 2) - F Frequency of association - F ₀ Reference frequency of association - H _i Dumbbell spring constant for a singlet (i = 1) and for a doublet (i = 2) - k Boltzman's constant - k _H Huggins constant - MW Molecular weight - MW _c Critical molecular weight for formation of entanglements - n Number density of molecules in the polymer solution - n ₀ Number density of dumbbells at equilibrium - n _i Number density of singlets (i = 1) and doublets (i = 2) - Q Vector defining the size and orientation of a dumbbell - t Time - T Absolute temperature - x Degree of multimerization - W Interaction energy between the two components of a doublet Greek letters a Dimensionless anisotropy parameter - Shear rate - _i Friction coefficient of singlets (i = 1) and doublets (i = 2) - _i Intrinsic viscosity of singlets (i = 1) and doublets (i = 2) - _red Reduced viscosity of solution - _sp Specific viscosity - Viscosity of the polymer solution of concentration c - _s Viscosity of the solvent - (t) White noise - K ^T Velocity gradient tensor - _Hi Time constant of a singlet (i = 1) and a doublet (i = 2) - ₁ Length scale of singlets (standard deviation of singlet lengths at equilibrium) - ₂ Length scale of doublets - T _p Stress tensor - T _xy Shear Stress (xy element of T _p ) - T _pi Contributions to the stress tensor of singlets (i = 1) and doublets (i = 2) - ₀ Equilibrium configuration distribution function of Q - _i Configuration distribution function of singlets (i = 1) and doublets (i = 2)     

Threshold-type shear-thickening in polymeric solutions

Threshold-type shear-thickening has been reported in the literature for relatively complex mixtures and one of the most famous is a solution of poly(vinylalcohol) in water in which a substantial amount of sodium borate is added. As far as we know, relatively few polymer solutions have been shown to be shear-thickening, even less of the threshold type. Two cases of threshold-type shear-thickening are examined. One of them is reversible [HPAM Pusher 700 or partially hydrolyzed poly(acrylamide) aqueous solution], the other one (in vitro-synthesized native dextran aqueous solution) leads to the formation of an irreversible structure provided sufficient strain is applied. In both cases, the occurrence of shear-thickening above a critical value of shear rate is related with a change in macromolecular conformation. This conformational change (reversible) can be followed by the formation of intermolecular bonds (irreversible).Presented at the 57^th Annual Meeting of the Society of Rheology, Ann Arbor, Michigan, Oct. 1985     

Thickening behaviour of dilute polymer solutions in non-inertial elongational flows

A molecular interpretation is proposed to interpret the thickening behaviour of dilute solutions of high molecular weight flexible polymers in non-intertial flows having an elongational character. A set of new results has been gathered showing that the onset of very high end-pressure losses at high deformation rates in capillary flow can be explained by a flow-induced coil-stretch transition of macromolecules in solution. When a high degree of elongation is achieved a marked increase in viscous dissipation occurs in elongational flows.     

On the mechanism of drag reduction in dilute polymer solutions

Experimental evidence is given that drag reducing polymer molecules are preferentially collected by strained vortices. This can explain why extremely small amounts of additives can be so effective. They become concentrated in areas of a turbulent flow where they are most efficient.     

Influence of the formation of associations of macromolecules on the rheo-optical properties of their dilute solutions. I. Theory

Assuming the formation of doublets in the flow according to a mass action law, the shear rate and the concentration dependence of the extinction angle, of the birefringence, and of the average coil expansion are calculated for dilute solutions of flexible macromolecules. It is shown that this reversible association process has a strong influence on the measurable parameters in a flow birefringence experiment. c concentration (g/cm³) - h ² mean square end-to-end distance at shear rate - h ₀ ² mean-square end-to-end distance at zero-shear rate - n refractive index of the solution (not very different from the solvent for a very dilute solution) - E mean coil expansion - K ₀,K constant of the mass action law - M molecular weight - R _G gas constant - T absolute temperature - ₁ – ₂ optical anisotropy of the segment - ₀ Deborah number: - Deborah number: - shear rate - ₀, reduced concentration - _s viscosity of the solvent - [] ₀ intrinsic viscosity at zero-shear rate - [] intrinsic viscosity at shear rate - extinction angle - N _a Avodagro's number - _n magnitude of the birefringence     

Anomalous diffraction model of linear optical dichroism in shear-thickening polymer solutions

The anomalous diffraction approximation (ADA) has bees recently applied to interpret measurements of the linear optical dichroism induced by shear in shear-thickening polymer solutions. A conceptual problem in this application is discussed, and a minor modification to the interpretation is proposed which is concordant with earlier magneto-optic results, but retains the correct use of the ADA.     

Dynamics of dilute polymer solutions

Taking volume effects, hydrodynamic no-flow conditions, and internal viscosity into account in the molecular dynamics has made it possible to formulate an equation of flow for a dilute polymer solution which in the region of comparatively slowly varying motions is described by the available experimental facts [9].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 138–146, January–February, 1988.     

Extensional rheology of dilute polymer solutions in oscillatory cross-slot flow: the transient behaviour of birefringent strands

Birefringent strands are key to understanding polymeric non-Newtonian flows, especially in extension. Utilising microfluidic extensional flow oscillatory rheometry coupled with microvelocimetry (μ-PIV), we report experiments on the genesis, steady state and decay of such strands, together with rheological consequences. For closely monodisperse atactic polystyrene, we report massive effects of the polymer on flow even at low concentrations. The often observed startup “overshoot” in stress and birefringence is observed at unprecedented dilution and discussed in terms of the local strain rate. Strand decay shows pronounced hysteresis. These factors are most important in modelling real flows such as cyclic and capillary entrance flows. Even with the closely monodisperse and well-characterised samples used, residual polydispersity plays a vital role in flow behaviour.     

Effect of internal viscosity and elasticity of ellipsoidal macromolecules on the rheological behavior of dilute polymer solutions. rheological equations of state

The structural-continuum approach is used to obtain the Theological equations of state for dilute polymer solutions whose macromolecules can be modeled by an ellipsoid of rotation with internal viscosity and elasticity.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 94–98, May–June, 1976.     

Influence of the formation of associations of macromolecules on the rheo-optical properties of their dilute solutions. II. Comparison with experimental results

The results of the theory presented in part I are compared to already published flow birefringence data obtained with dilute up to semi-dilute solutions of polyethylene oxide which have also shown shear thickening effects. The agreement with the theory is quite good so that the reversible formation of associations in flow seems to be a process which has to be taken in account for high molecular weight flexible macromolecules solutions in relatively poor solvents.     

Bead-spring chain model for the dynamics of dilute polymer solutions : Part 2. Comparisons with experimental data

Comparisons are made between experimental rheological data and theoretical predictions obtained from a recently developed algorithm which incorporates three major molecular concepts in a theory for dilute polymer solutions (hydrodynamic interaction, excluded volume and nonlinear springs). These predictions include the radius expansion factor, the apparent chain expansion factor, the molecular weight dependence of the intrinsic viscosity, the frequency dependence of oscillatory flow birefringence, and the shear rate dependence of the intrinsic viscosity. This paper shows that a bead-spring chain model quantitatively predicts these quantities when the relevant molecular concepts are incorporated, suggesting that the rheological properties of dilute polymer solutions can be explained and predicted in terms of these molecular parameters.     

Thermal diffusion of dilute polymer solutions

In this paper diffusion of a dilute solution of elastic dumbbell model macromolecules under non-isothermal conditions is studied. Using the center of mass definition for the local polymer concentration, the diffusive flux contains a thermal diffusion dyadic d _T .  To get some idea of thermal diffusion d _T is evaluated for steady state isothermal conditions. Explicit results are presented for some homogeneous flows. It is shown that if the polymeric number density is defined via the beads (of the dumbbell) – termed n _b – then the diffusive flux j contains , where τ ^c is the intramolecular contribution to the bulk stress. Though the form of the diffusion equation for n _b thus differs from the corresponding one for n, it is shown that for essentially unbounded systems differences between n and n _b are small. Since the results involve the translational diffusion coefficient they can readily be taken over for Rouse coils. Received: 23 September 1997 Accepted: 5 June 1998     

Dissipative stresses in dilute polymer solutions

Planar extensional flows of a dilute polymer solution are investigated using a free-draining bead-rod model. For steady flows, an analytic expression for the probability density of the polymer configuration is available. It is found that part of the associated steady polymer stress is unambiguously viscous at all time scales, in the sense that on cessation of flow it disappears instantaneously, but, except at very high flow rates, the elastic component is larger.A Brownian dynamics simulation of the chain is constructed for start-up flows for which no analytic expression is known. A stress that is apparently viscous is found to develop alongside the elastic stress, having comparable magnitude at moderate flow rates. An interpretation of this result for a system having a wide spectrum of relaxation times is given. This feature is not captured by conventional FENE constitutive equations, and a novel model is developed. The consequences for calculations of complex flows are briefly discussed.     

Converging radial flow of dilute polymer solutions

A characteristic time of dilute polymer solutions is determined from a converging radial flow experiment. The influence of the intradisk separation and of the polymer concentration on this time is studied. Present results are compared to results obtained with a diaphragm. The main limitations of the experiments are pointed out.     

On the dynamics of coiled macromolecules

Summary A modified Rouse model of dilute polymer solution in flow was considered and applied to describe the rheological behaviour of this system. The modification of the model was carried out in the following direction. The restoring force acting between two neighbouring beads was calculated accounting the fact, that the deformation of the real chain was accompanied by change not only of entropy, but also of internal energy, since during the deformation a certain number of links can shift from one energetical state to another. The calculations were carried out in the rotational isomeric state approximation.

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Zusammenfassung Es wird ein modifiziertes Rouse-Modell einer verdünnten Polymer-Lösung betrachtet und zur Beschreibung des rheologischen Verhaltens solcher Systeme verwendet. Die Modifikation wurde in folgender Weise vorgenommen: Die zwischen zwei benachbarten Elementen (beads) wirkenden Rückstellkräfte wurden unter Berücksichtigung der Tatsache berechnet, daß die Deformation der wirklichen Kette nicht nur mit einer Änderung der Entropie,sondern auch der inneren Energie verknüpft ist, da während der Deformation eine gewisse Anzahl von Verbindungsgliedern von einem energetischen Zustand in einen andern übergehen können. Diese Rechnungen wurden mit der Approximation für rotationsisomere Zustände durchgeführt.

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Rheological behaviour of dilute suspensions of platelet particles

The rheological behaviour of Newtonian suspensions of platelets with an aspect ratio ranging from 0.006 to 0.2 is studied in this work. Using model particles to correlate the aspect ratios square-shaped particles with their sedimentation behaviour, the results obtained were used to estimate the average aspect ratio of mica particles. A comparison is made between the aspect ratio measured by sedimentation experiments and that determined from the rheological behaviour of dilute suspensions. Their good agreement shows that the average aspect ratio of a polydispersed suspension of platelets estimated using packing experiments correlates well with the apparent aspect ratio given by the intrinsic viscosity of the suspension.