5.3 Brownian Dynamics simulations
5.3.3 Temporal scaling
The collapse of the scaled pressure autocorrelation data onto the analytical curve is perfect in the short-time limit but at longer times the rate of decay deviates from the analytical curve by varying amounts depending on the volume fraction. The pressure autocorrelation obtained from Brownian Dynamics simulations for volume fractions up-toφ = 0.35 are shown in Figure 5.4 forN = 1000 and N = 100. The agreement of the simulation data with the analytical curve is very good for φ=0.05, but asφincreases there is a perceptibly faster decay of the pressure autocorrelation up to φ=0.2, where it stops becoming faster. Forφ≥0.4 the rate of decay is slower with increasingφas shown in Figure 5.5, indicating an increase in the time scale for relaxation.
For additional guidance we also look at the shear stress autocorrelationCSNH = hΣxy(0)Σxy(τ)i, shown in Figure 5.10. The simulation data forCSNHhas much lesser noise and captures the long-time decay correctly. Since both the shear and the pressure autocorrelation are just different measure- ments from the same physical process we expect their magnitude and temporal behavior to have
similar scaling withφ. The scaled shear stress autocorrelation defined as
CSNH= N
φg0(2;φ)CSNH, (5.23)
is shown in Figure 5.11(a) and indeed for small volume fractions it coincides very well with the dilute-theory analytical curve 5.14. There is a slight decrease in the rate of decay with increasingφ at small values ofφbut it is very slight compared to the variations in the pressure autocorrelation function. The theoretical curve forCNHS was obtained from a numerical inverse Laplace transform [Hollenbeck 1998] of the frequency-dependent shear viscosity for hard spheres without hydrody- namics. It is quite clear from both figures that the time scale of relaxation is constant forφ ≤0.35 and increases monotonically withφ for higher volume fractions, similar toCPNH in Figure 5.8(a).
Therefore the characteristic diffusivity scaling for stress relaxation must be ˆD(φ) =1 forφ≤ 0.35 and a decreasing function ofφforφ >0.35.
The exact behavior of the characteristic diffusivity ˆD(φ) governing the relaxation of fluctuations is not clear from theory, and several quantities have been suggested earlier. Brady [1993b] suggested scaling time with the equilibrium short-time self-diffusivity as ˆD(φ) = D0s(φ)/D, as it accounts for the slowing down of the dynamics due to stronger hydrodynamic interactions between particles as the volume fraction increases. He successfully used this scaling to collapse the experimental data of van der Werffet al.[1989] for the frequency-dependent dynamic viscosity for the concentration range 0.46 ≤ φ ≤ 0.6. In the absence of hydrodynamic interactions the short-time self-diffusivity is simply the bare diffusivity for allφand therefore scaling withDs0(φ) would not account for the variations in temporal decay in our simulations.
A more suitable choice for the temporal scaling is the equilibrium long-time self-diffusivity D∞s(φ) because it incorporates the effects of particle interactions with and without hydrodynamics.
The long-time self diffusivity corresponds to the motion of a particle on times long compared to a2/Dso that the particle has wandered far compared to its size, and in doing so exchanged places with its neighbors and experienced many different configurations [Brady 1994]. The time-scale of stress relaxation would scale asD∞s if it was necessary for the particles to exchange places with a neighbor in order to achieve a significantly different particle configuration such that the fluctuations in particle-phase stress are no longer correlated. This is the case at high volume fractions when par-
ticles are likely to get trapped in a ‘cage’ of neighboring particles. At low volume fractions particles have more freedom to move around so that the particles can achieve significantly different config- urations via collisions with neighboring particles alone, without having to travel far or exchange places with a neighbor. This explains the uniform rate of stress relaxation at low concentrations and the slowing down of the stress relaxation at higher concentrations. At intermediate concentrations the particles would only need to distort the surrounding cage sufficiently without actually breaking out of it for the microstructure to relax to a different state. Hence the relaxation time-scale would be influenced byD∞s but not be completely determined by it.
Our simulations indicate that the temporal scaling with D∞s should start at approximatelyφ = 0.35. For smaller volume fractions the temporal scaling can be quantified by computing the relax- ation time directly from the stress autocorrelation functions. From Figure 5.4 it is evident that the change in relaxation rate does not depend on the number of particles, therefore it must be the result of some physical process. The relaxation time for the pressure autocorrelation is defined as
τκ = R∞
0 hτδΠ(0)δΠ(τ)idτ R∞
0 hδΠ(0)δΠ(τ)idτ. (5.24)
However, since computing the relaxation time over the entire range ofτwould require fitting the tails to a known curve (for which we don’t know the correct temporal scaling yet) we compute
τκ(0.7)= R0.7
0 hτδΠ(0)δΠ(τ)idτ R0.7
0 hδΠ(0)δΠ(τ)idτ (5.25)
instead, i.e., the integral is evaluated only up-to τ = 0.7 where the data has not yet become too noisy. Besides, the time scale integral computed withτ→ ∞in (5.25) will be unbounded because hτδΠ(0)δΠ(τ)i ∼t−1/2asτ→ ∞. The change in the relaxation rate is apparent even within this short time boundary, henceτκ(0.7) should give a reasonable estimate for the change in temporal scaling.
Theτκ(0.7) data is shown in Figure 5.6. There is a rapid decrease in the relaxation time-scale for φ ≤ 0.2 and after that the relaxation rate is almost constant, indicating that there are possibly two competing effects governing the time scale. At very small volume fractions, the addition of more particles in the system would cause an increase in the number of collisions taking place and thereby help in dissipating the stress and microstructural deformation faster. At small φ, an increase in
the particle density is not sufficient to hinder the motion of the particles. Therefore there is a net decrease in the time scale of relaxation as more collisions are taking place but the hindrance due to the additional collisions is not enough to perceptibly slow down the movement of the particles. Asφ increases the hindering effect of more particles eventually catches up and balances out the entropic effect that was helping to dissipate the stress, so that the relaxation time-scale stops changing after φ = 0.2. Since there is no clear scaling to account for the competing effects we simply use the polynomial fit given byτf it(φ) for the temporal scaling in this regime. For higher volume fractions the relaxation time-scale starts increasing due to ‘caging’ effects and thereforeD∞s is a better scaling forφ≥0.35.
Accordingly we define the characteristic diffusivity scaling as
DˆNH(φ)=
τf it(0.05)/τf it(φ), ifφ <0.35 D∞,NHs (φ)τf it(0.05)
D∞,NHs (0.35)τf it(0.35), ifφ≥0.35 (5.26)
so that ˆDNH(φ) is a continuous function ofφ, where the subscriptNH stand for no hydrodynamics.
HereD∞s(φ) is the long-time self-diffusivity in the absence of hydrodynamic interactions. The values for D∞s were calculated from interpolation of the simulation results of Foss [1999]. The pressure autocorrelation data plotted against time scaled as ˆτ=tb2/DDˆNH(φ) is shown in Figure 5.8(b), and the shear stress autocorrelation data with the same temporal scaling is shown in Figure 5.11(b). For both sets of data the long-time tails were obtained by fitting the corresponding analytical curves to the simulation data with the temporal scaling given by (5.26). The simulation results and the fitted tails collapse quite nicely onto the theoretical curve with this scaling. The time-scale of decay for larger times including the long time tails given byτκ(200) and shown in Figure 5.7 has roughly the same behavior asτκ(1). The scaled theoretical prediction for the bulk viscosity given by (5.20) is shown in Figure 5.9 along with the Brownian Dynamics simulation results. The MD simulation results of Sigurgeirsson and Heyes [2003] for the bulk viscosity of hard spheres scaled with the zero-density shear viscosity are also shown on the same plot. The MD data does not have the same scaling as our BD data forφ <0.35 because the entropic effect of more Brownian particles helping to dissipate the stress with increasingφis absent. For higherφthe MD data also scales as D∞s as
expected. The theoretical prediction is in good agreement with our BD results withN = 1000 and N =100 particles. The scaled theory for the shear viscosity is shown in Figure 5.12 and is in very good agreement with the BD results.