Suspension stress and the bulk viscosity

Chapter 5

Equilibrium Properties via Simulation

5.1 Introduction

Several transport properties of particle systems in the regime of small deviations from equilibrium can be deduced from knowledge of the particle microstructure and fluctuations in the microstructure at equilibrium. At high particle concentrations the simple closures used for the dilute theory are no longer applicable and one must resort to numerical simulations to account for interactions between multiple particles. In this chapter we describe the Brownian Dynamics, Stokesian Dynamics and Accelerated Stokesian Dynamics simulations performed for Brownian particles at equilibrium and the calculation of the suspension bulk viscosity from simulations.

form

DΣp

E=−nkTI+n[hS^Ei+hS^Bi+hS^Pi], (5.2) where −nkTI is simply the osmotic pressure due to the thermal kinetic energy of the Brownian particles andhS^Ei,hS^BiandhS^Piare the average hydrodynamic, Brownian and interparticle-force particle stresslets (symmetric first moment of the force distribution integrated over the particle sur- face). The number density of particles is given byn, andk andT are the Boltzmann constant and fluid temperature, respectively. The particle stresslets include the stress due to the many body in- teractions among all the particles, and this is the part that we compute via particle simulations. At any given time the particle stresslets are a function only of the instantaneous particle configuration.

Finally, as defined in this work the effective bulk viscosity of the suspension is given by κe f f =

κ+ ⁴₃ηφ 1

1−φ +κ^E +κ^B+κ^P, (5.3)

where the last three terms are the hydrodynamic, Brownian and interparticle-force contributions to the bulk viscosity coming from many-body interactions and are obtained directly from the corre- sponding average particle stresslets. The first term gives the single particle correction to the bulk viscosity in a uniformly expanding fluid.

The direct hydrodynamic contribution to the stress due to the disturbance flows caused by the finite size of the particles in an imposed flow is given bynhS^Ei, and it is directly proportional to the applied rate of deformation. The Péclet number for an expansion flow defined as

Pe=

1 3heib²

2D ,

is the rate of expansionheinondimensionalized by the diffusive time-scale of the particleb²/2D.

HereD = kT/6πηb is the Stokes-Einstein-Sutherland diffusivity of an isolated particle, and each particle acts as a hard sphere of radiusb. The corresponding hydrodynamic contribution to the bulk viscosity is obtained from the hydrodynamic stresslet as

κ^E = nhS^Ei

3hei , (5.4)

and therefore to first order inPe it can be computed directly from the equilibrium microstructure because the scaling with the rate of deformationhei cancels out. HereS denotes the trace of the corresponding stresslet, i.e., hSi = ¹₃hSiI for an isotropic stresslet. In practice the hydrodynamic contribution to the bulk viscosity for small deviations from equilibrium (|Pe| 1) is computed by averaging the hydrodynamic stress over an ensemble of equilibrium particle configurations.

The Brownian and interparticle-force contributions to the stress have a finite average value at equilibrium independent of the rate of deformation because they originate from the thermal mo- tion of the particles. Deviations from the equilibrium value occur only when the microstructure is perturbed from equilibrium. The corresponding contributions to the bulk viscosity are given by

κ^B= nhS^Bi −nhS^Bi^eq

3hei and κ^P= nhS^Pi −nhS^Pi^eq

3hei , (5.5)

where the superscripteqdenotes the equilibrium value. The κ^B andκ^P contributions can be eval- uated numerically from dynamic simulation by generating deviations from equilibrium in the mi- crostructure and computing the resulting change in the stress. However, for small deviations from equilibrium it becomes difficult to isolate the excess stress caused by the imposed flow from thermal fluctuations in the stress and increasingly longer simulations would be required to get good aver- aging and obtain an accurate value for the bulk viscosity contributions. We avail ourselves of an alternate approach to evaluating the transport properties at equilibrium, using Green-Kubo relations to get them from stress autocorrelation functions [Green 1952]. The stress autocorrelation function characterizes the nature and rate of relaxation of fluctuations in the stress due to Brownian motion.

For the shear viscosity one would use the shear stress autocorrelation function at equilibrium, and for the bulk viscosity the pressure autocorrelation is used as [Nagèle and Bergenholtz 1998]

κ^B= V kT

Z ∞ 0

hδΠ(0)δΠ(t)idt, (5.6)

where the angle brackets denote an ensemble average,Vis the volume of the system over which the averaging is done,tis the time, andδΠ(t) is the instantaneous deviation of the osmotic pressure of the suspension from its equilibrium value. Nondimensionalizing the pressure fluctuations withnkT

and time with the relative diffusive time of the particles asτ=t(2D/b²) we obtain the relation κ^B = 9

4NφZ ∞ 0

hδΠ(0)δΠ(τ)idτ, (5.7)

whereN is the total number of particles in the volumeV,φis the volume-fraction of the particles andκ^B = (κ^B+κ^P)/ηis the total Brownian contribution to the bulk viscosity nondimensionalized with the fluid viscosity. We define the nondimensional pressure autocorrelation function as

C^P(τ)=hδΠ(0)δΠ(τ)i.

In the absence of hydrodynamic interactions the direct Brownian contribution to the stressnhS^Bi arising from interactions between the particles as they undergo Brownian motion becomes zero and only the interparticle-force contributionnhS^Piremains. This work is restricted to hard spheres so that the interparticle-force stress comprises only the force due to hard sphere collisionsF^P= ¹₂nδ(r−

2b), where nis the normal vector along the line joining the centers of two touching spheres andδ is the delta function at the surface of contact. The interparticle force produces the total Brownian stress in the case of hard spheres undergoing Brownian motion. Conversely, in the presence of hydrodynamic interactions the particle surfaces never touch due to the strong lubrication forces near contact and accordinglynhS^Pi = 0, whilenhS^Biaccounts for all of the Brownian stress. The hard sphere nature of the particles is preserved by the no-slip hydrodynamic boundary condition on the particle surface. In the following sections we describe the determination of the bulk viscosity via Brownian Dynamics in the absence of hydrodynamics, and using Stokesian Dynamics [Brady and Bossis 1988] and Accelerated Stokesian Dynamics — near field [Banchio and Brady 2003]

simulations to account for the presence of hydrodynamic interactions.