^{1}, Run Jiang

^{1}, Roland G. Winkler

^{1}and Gerhard Gompper

^{1,a)}

### Abstract

In order to study the dynamics of colloidal suspensions with viscoelasticsolvents, a simple mesoscopic model of the solvent is required. We propose to extend the multiparticle collision dynamics (MPC) technique—a particle-based simulation method, which has been successfully applied to study the hydrodynamic behavior of many complex fluids with Newtonian solvent—to shear-thinningviscoelasticsolvents. Here, the normal MPC particles are replaced by dumbbells with finite-extensible nonlinear elastic (FENE) springs. We have studied the properties of FENE-dumbbell fluids under simple shear flow with shear rate. The stress tensor is calculated, and the viscosity η and the first normal-stress coefficient Ψ_{1} are obtained. Shear-thinning behavior is found for reduced shear rates, where τ is a characteristic dumbbell relaxation time. Here, both η and Ψ_{1} display power-law behavior in the shear-thinning regime. Thus, the FENE-dumbbell fluid with MPC collisions provides a good description of viscoelastic fluids. As a first application, we study the flow behavior of a colloid in a shear-thinningviscoelastic fluid in two dimensions. A slowing down of the colloid rotation in a viscoelastic fluid compared to a Newtonian fluid is obtained, in agreement with recent numerical calculations and experimental results.

We thank Jan Vermant (Leuven), Ingo O. Götze, and Chien-Cheng Huang (Jülich) for stimulating discussions. Financial support by the EU FP7 Collaborative Research Project “Nanodirect” (NMP4-SL-2008-213948) is gratefully acknowledged.

I. INTRODUCTION

II. MODEL AND METHOD

A. Multiparticle collision dynamics

B. FENE Dumbbells

C. Shear flow and stress tensor

D. Simulation parameters

III. FENE-DUMBBELL FLUID IN SIMPLE SHEAR FLOW

A. Dumbbell orientation, stretching, and tumbling

B. Rheology of FENE-dumbbell fluids

C. FENE-dumbbell fluids in three dimensions

IV. COLLOIDDYNAMICS UNDER SHEAR FLOW

V. SUMMARY AND CONCLUSIONS

### Key Topics

- Colloidal systems
- 42.0
- Viscoelasticity
- 37.0
- Shear flows
- 27.0
- Shear thinning
- 26.0
- Solvents
- 21.0

## Figures

Velocity (■) and density (▲) profiles in simple shear flow as a function of position *y* in the gradient direction. The shear rate is , the density of MPC particles is ρ = 10, and the collision time is *h* = 0.02. The spring constant of the FENE dumbbell is *k* = 0.2, and the FENE parameter is *b* = 8.

Velocity (■) and density (▲) profiles in simple shear flow as a function of position *y* in the gradient direction. The shear rate is , the density of MPC particles is ρ = 10, and the collision time is *h* = 0.02. The spring constant of the FENE dumbbell is *k* = 0.2, and the FENE parameter is *b* = 8.

Density distribution of dumbbell monomers relative to the dumbbell's center of mass (arb. units) for the indicated reduced shear rates . The other parameters are the same as in Fig. 1. The maximum extension is . The integrated probability is the same in all three plots.

Density distribution of dumbbell monomers relative to the dumbbell's center of mass (arb. units) for the indicated reduced shear rates . The other parameters are the same as in Fig. 1. The maximum extension is . The integrated probability is the same in all three plots.

Extension *R*/*R* _{0} (•) and inclination tan (2θ) (▼) as a function of the reduced shear rate Γ. The other parameters are the same as those in Fig. 1. The numbers indicate the exponents of power-law regimes.

Extension *R*/*R* _{0} (•) and inclination tan (2θ) (▼) as a function of the reduced shear rate Γ. The other parameters are the same as those in Fig. 1. The numbers indicate the exponents of power-law regimes.

Probability distribution of the dumbbell extension for various shear rates Γ = 0 (▲), 0.92 (+), 1.84 (▼), 4.6 (•), and 9.2 (✦). The solid line is obtained from the Boltzmann factor with the FENE potential. The other parameters are the same as in Fig. 1.

Probability distribution of the dumbbell extension for various shear rates Γ = 0 (▲), 0.92 (+), 1.84 (▼), 4.6 (•), and 9.2 (✦). The solid line is obtained from the Boltzmann factor with the FENE potential. The other parameters are the same as in Fig. 1.

Reduced rotation frequency of dumbbells as a function of reduced shear rate Γ. The other parameters are the same as those in Fig. 1. The green and black dashed lines indicate the asymptotic behavior for small and large reduced shear rates, respectively.

Reduced rotation frequency of dumbbells as a function of reduced shear rate Γ. The other parameters are the same as those in Fig. 1. The green and black dashed lines indicate the asymptotic behavior for small and large reduced shear rates, respectively.

The reduced viscosity η_{ r } as a function of the reduced shear rate Γ. The parameters are the same as Fig. 1. The solid line shows a fit to the Carreau-type function (18) with the parameters μ = 0.60, *q* = 1.38, and Γ_{0} = 3.24, the dashed line indicates the asymptotic scaling law for high Weissenberg numbers.

The reduced viscosity η_{ r } as a function of the reduced shear rate Γ. The parameters are the same as Fig. 1. The solid line shows a fit to the Carreau-type function (18) with the parameters μ = 0.60, *q* = 1.38, and Γ_{0} = 3.24, the dashed line indicates the asymptotic scaling law for high Weissenberg numbers.

The reduced viscosity η_{ r } as a function of the reduced shear rate Γ for dumbbells with (a) same FENE parameter *b* = 8 but different spring constant *k*; (b) same spring constant *k* = 0.2 but different FENE parameters. Dashed lines show fits to the Carreau-type expression (18) with the parameters μ = 0.60, *q* = 1.38, and Γ_{0} = 3.24 for *b* = 8, Γ_{0} = 4.40 for *b* = 18, Γ_{0} = 8.27 for *b* = 50, and Γ_{0} = 27.5 for *b* = 200.

The reduced viscosity η_{ r } as a function of the reduced shear rate Γ for dumbbells with (a) same FENE parameter *b* = 8 but different spring constant *k*; (b) same spring constant *k* = 0.2 but different FENE parameters. Dashed lines show fits to the Carreau-type expression (18) with the parameters μ = 0.60, *q* = 1.38, and Γ_{0} = 3.24 for *b* = 8, Γ_{0} = 4.40 for *b* = 18, Γ_{0} = 8.27 for *b* = 50, and Γ_{0} = 27.5 for *b* = 200.

The reduced coefficient of the first normal stress difference as a function of the reduced shear rate Γ. The other parameters are the same as those in Fig. 1. The solid line is a guide to the eye, the dashed line indicates the asymptotic scaling law.

The reduced coefficient of the first normal stress difference as a function of the reduced shear rate Γ. The other parameters are the same as those in Fig. 1. The solid line is a guide to the eye, the dashed line indicates the asymptotic scaling law.

The reduced viscosity η_{ r } as a function of the reduced shear rate Γ in three dimensions. The parameters are collision time *h* = 0.02, density ρ = 10, spring constant *k* = 0.2, and FENE parameter *b* = 8. The solid line shows a fit to the Carreau-type function (18) with the parameters μ = 0.60, *q* = 1.38, and Γ_{0} = 3.61, the dashed line indicates the asymptotic scaling law for high Weissenberg numbers.

The reduced viscosity η_{ r } as a function of the reduced shear rate Γ in three dimensions. The parameters are collision time *h* = 0.02, density ρ = 10, spring constant *k* = 0.2, and FENE parameter *b* = 8. The solid line shows a fit to the Carreau-type function (18) with the parameters μ = 0.60, *q* = 1.38, and Γ_{0} = 3.61, the dashed line indicates the asymptotic scaling law for high Weissenberg numbers.

Reduced rotation frequency of a colloid in a Newtonian fluid with indicated collision time *h* and particle density ρ. The size of the simulation box is *L* _{ x } = 120, *L* _{ y } = 80.

Reduced rotation frequency of a colloid in a Newtonian fluid with indicated collision time *h* and particle density ρ. The size of the simulation box is *L* _{ x } = 120, *L* _{ y } = 80.

Snapshot of dumbbell conformations in shear flow near a colloidal particle (blue circle). Only a fraction of 0.6% of dumbbells is shown. The reduced shear rate is Γ = 23.5 (). Note that the dumbbells in the interior of the colloid are not stretched. The parameters are collision time *h* = 0.01, density ρ = 50, spring constant *k* = 0.2, and FENE parameter *b* = 8. The system size is *L* _{ x } = 120, *L* _{ y } = 80.

Snapshot of dumbbell conformations in shear flow near a colloidal particle (blue circle). Only a fraction of 0.6% of dumbbells is shown. The reduced shear rate is Γ = 23.5 (). Note that the dumbbells in the interior of the colloid are not stretched. The parameters are collision time *h* = 0.01, density ρ = 50, spring constant *k* = 0.2, and FENE parameter *b* = 8. The system size is *L* _{ x } = 120, *L* _{ y } = 80.

Reduced rotation frequency of a colloid in a viscoelastic and Newtonian fluid, respectively. The parameters are the same as in Fig. 11.

Reduced rotation frequency of a colloid in a viscoelastic and Newtonian fluid, respectively. The parameters are the same as in Fig. 11.

Streamlines in Newtonian fluids with (a) , (b) , and in viscoelastic fluids with (c) Γ = 2.35 (), (d) Γ = 23.5 (). The parameters are the same as in Fig. 11.

Streamlines in Newtonian fluids with (a) , (b) , and in viscoelastic fluids with (c) Γ = 2.35 (), (d) Γ = 23.5 (). The parameters are the same as in Fig. 11.

Net velocity fields **v** − **v** ^{ ideal } and corresponding streamlines of (a) a Newtonian fluid with shear rate , (b) a viscoelastic fluid with Γ = 2.35 (), and (c) a viscoelastic fluid with Γ = 23.5 (). Parameters are ρ = 50, *h* = 0.01, and for the viscoelastic case *k* = 0.2, and *b* = 8. The blue circle indicates the position and size of the colloid.

Net velocity fields **v** − **v** ^{ ideal } and corresponding streamlines of (a) a Newtonian fluid with shear rate , (b) a viscoelastic fluid with Γ = 2.35 (), and (c) a viscoelastic fluid with Γ = 23.5 (). Parameters are ρ = 50, *h* = 0.01, and for the viscoelastic case *k* = 0.2, and *b* = 8. The blue circle indicates the position and size of the colloid.

Reduced rotation velocity of FENE dumbbells spatially resolved in the vicinity of a colloid in shear flow for the shear rates (a) Γ = 2.35 () and (b) Γ = 23.5 (). The other parameters are ρ = 50, *h* = 0.01, *k* = 0.2, and *b* = 8.

Reduced rotation velocity of FENE dumbbells spatially resolved in the vicinity of a colloid in shear flow for the shear rates (a) Γ = 2.35 () and (b) Γ = 23.5 (). The other parameters are ρ = 50, *h* = 0.01, *k* = 0.2, and *b* = 8.

Average extension *R* of FENE dumbbells spatially resolved in the vicinity of a sphere in shear flow for the shear rates (a) Γ = 2.35 () and (b) Γ = 23.5 (). The other parameters are ρ = 50, *h* = 0.01, *k* = 0.2, and *b* = 8.

Average extension *R* of FENE dumbbells spatially resolved in the vicinity of a sphere in shear flow for the shear rates (a) Γ = 2.35 () and (b) Γ = 23.5 (). The other parameters are ρ = 50, *h* = 0.01, *k* = 0.2, and *b* = 8.

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