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Brownian dynamics simulations of shear-thickening in dilute polymer solutions

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Abstract

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.

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Abbreviations

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

a :

Dimensionless anisotropy parameter

\(\dot y\) :

Shear rate

ζ i :

Friction coefficient of singlets (i = 1) and doublets (i = 2)

η i :

Intrinsic viscosity of singlets (i = 1) and doublets (i = 2)

η red :

Reduced viscosity of solution

η sp :

Specific viscosity

η :

Viscosity of the polymer solution of concentration c

η s :

Viscosity of the solvent

η(t) :

White noise

K T :

Velocity gradient tensor

λ Hi :

Time constant of a singlet (i = 1) and a doublet (i = 2)

σ1 :

Length scale of singlets (standard deviation of singlet lengths at equilibrium)

σ2 :

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)

ψ0 :

Equilibrium configuration distribution function of Q

ψ i :

Configuration distribution function of singlets (i = 1) and doublets (i = 2)

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Authors and Affiliations

  1. Department of Chemical Engineering, The University of British Columbia, V6T 1Z4, Vancouver, B.C., Canada

    Savvas G. Hatzikiriakos

  2. Foundation for Research & Technology — Hellas, Institute of Electronic Structure & Laser, Heraklion, 71110, Crete, Greece

    Dimitrios Vlassopoulos

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  1. Savvas G. Hatzikiriakos
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  2. Dimitrios Vlassopoulos
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Hatzikiriakos, S.G., Vlassopoulos, D. Brownian dynamics simulations of shear-thickening in dilute polymer solutions. Rheola Acta 35, 274–287 (1996). https://doi.org/10.1007/BF00366914

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