As the compression flow proceeds, particles are pushed towards each other by the imposed forcing, thus driving the system out of equilibrium with particles being closer to each other on average than they would be at equilibrium. This shows up as an isotropic accumulation of particles at contact (r =2) in the pair-distribution functiong(r), which is a function of the scalar particle separation only because of the isotropic forcing. Brownian motion of the particles causes them to diffuse against the concentration gradient that is built up near contact and makesg(r) decay with increasingr. For small rates of compression the perturbation to the microstructure decays as 1/r. At higher compres- sion rates a boundary layer ofO
|Pe|−1
forms near contact in which Brownian diffusion balances the compression flow and the microstructural perturbation decays to zero inside the boundary layer.
The magnitude of the microstructural perturbation inside the boundary layer isO(|Pe|) from dilute theory analysis. Figure 6.1 shows a planar cross-section of the three-dimensional pair distribution function computed from Brownian Dynamics compression simulations with 1000 particles. The figure illustrates the isotropic nature of the forcing in compression and the stronger accumulation
and faster decay of particle concentration near contact with increasingPe. At higher volume frac- tions the formation of an additional ring of higher concentration can be seen as the caging effect of particles becomes more prominent, but the behavior ofg(r) with varyingPeis still the same.
In all the simulations we start with an equilibrium particle configuration, so the microstructure must evolve as part of the simulation. The time it takes to reach a steady state would be determined by the diffusive time-scale of the particles and the distance over which they need to diffuse in order to balance the compression flow. The simplest estimate for the diffusive time isb2/2Ds∞(φ), where D∞s is the long-time self-diffusivity which accounts for the hindrance encountered by a particle as it makes its way through the surrounding particle structure. In regions of accumulation of particles the time scale of diffusion would be even slower than that given byDs∞. The long-time self-diffusivity is a decreasing function of particle concentration so we expect that the approach to steady state would be slower with increasingφ. However it is possible that the gradients in the microstructure reach a steady state before reaching high values ofφso that only the particle density is increasing while g(r) remains unchanged.
The imposed compression forcing is balanced by Brownian motion only in the boundary layer so particles need to diffuse over a distance ofO
|Pe|−1
for the microstructure to reach a steady state.
Figure 6.2 and Figure 6.3 show the radial pair-distribution function from BD and ASD simulations respectively at Pe = −1 and Pe = −10 for φ = 0.2. The data from both simulations is almost identical except for the value at contact, where lubrication forces in ASD simulations reduce the particle mobility significantly, causing a stronger accumulation of particles. Clearlyg(r) decays over a much longer distance atPe=−1 than atPe=−10, because particles need to diffuse over a shorter distance at larger values ofPein order to balance the imposed forcing with Brownian diffusion. At the same time the volume fraction also changes at a faster rate for largerPe. Specifically the volume fraction as a function of time is given by
φ(τ)=φ(τ)e−3Peτ. (6.2)
The change in volume fraction is exponential so that it changes very slowly at the start of our simulations whenφ ∼ 0.01 but changes very quickly at higher volume fractions which are in the regime of interest. From the analysis in Appendix A in Chapter 1, the microstructural disturbance
also approaches its long-time limit exponentially ase−3Peτin the dilute limit for large Pe. Hence even in the dilute limit we would not expect the microstructure to be able to keep up with the change in volume fraction in the highPelimit. Essentially the particles need to move a distance ofO(1) on average to be pushed into the boundary layer and this would not happen fast enough compared to the rate of change of the volume fraction, so the boundary layer formation remains incomplete. For small Pe the microstructure would change even more slowly (algebraically in the linear response limit) as shown in Chapter 3 §3.4, and therefore would not achieve a steady state at any volume fraction.
In the above discussion we assumed that the initial configuration of particles is spatially homo- geneous. While this is true in an average sense because we start from equilibrium configurations, any actual realization would have spatial inhomogeneities due to the random placement of particles.
Thus there would be some regions where particles are already very close to each other and some other regions where particles are not close to any neighboring particle. At smallPeBrownian mo- tion of the particles would cause these variations to dissipate and allow the particles to sample many random configurations, so that relative to the rate of the imposed compression flow the particle dis- tribution is mostly homogeneous. In this case there is actual competition between the compression flow pushing the particles closer and Brownian motion dissipating the accumulation of particles.
Particles would need to move anO(1) distance in order to be pushed into the boundary layer. There- fore at smallPeof compression we don’t expect the microstructural disturbance to reach a steady state.
At largePehowever, Brownian motion is negligible so groups of particle that were already close to each other are pushed even closer without any competition from Brownian diffusion, thereby cre- ating clusters of particles scattered throughout the space where the boundary layer is formed very quickly. This is evident from Figure 6.4 which shows a histogram of the instantaneous hydrody- namic pressure moment of particles at rates ofPe = −1 andPe = −1000 compiled over 200 runs of ASDB-nf compression simulations. At smallPethe pressure moment values are all very close to the average indicating that the particle distribution has been homogenized by Brownian motion. At highPehowever the distribution of pressure moments is more spread out and there is a significant number of particles that are quite a bit far from the mean, and these must be the particles that are close to other particles and thereby increase the overall stress due to lubrication forces. A look at
Figure 6.5, which shows two such examples of particles with pressure moment greater than a cutoff value (hSHi/Pe>10) deduced from the histogram withPe=−1000, reveals that this is indeed the case. Clusters and chains of particles distributed throughout space are clearly visible in the figures and these are the regions of high particle pressures. There is a balance between the imposed flow and the boundary condition at contact that is set up in these regions, due to hard sphere collisions in the absence of hydrodynamics and lubrication forces near contact when hydrodynamics are im- portant. The particle phase pressure in these regions would be much higher than in the rest of the suspension, resulting in a total stress that is much closer to the steady state value than at smallerPe.
Therefore at largePe even though the suspension on an average does not reach steady state there will be particle clusters throughout space depending on the initial configuration where the steady state boundary layer is formed and these clusters would grow in size as the compression of particles proceeds. The total particle-phase stress would be dominated by the stress in these clusters.
This does not undermine the results of the current topic because the main purpose of this study is to determine the bulk viscosity at high Pevia simulation. Whether the microstructure reaches the correct steady state or not may also depend on the number of particles used in the periodic simulation cell, which determines the cell size for a givenφ. If the size of the boundary layer is comparable to or greater than the cell length, the decay ofg(r) may not be captured correctly as the periodic boundary conditions impose an additional constraint ong(r) at the cell boundaries.