5.4 E ff ect of hydrodynamic interactions
5.4.2 Temporal scaling
mobility of a particle can be computed from an average over all configurations as
Ds0=kTD R−1FU,iiE
, (5.37)
where the subscriptiiindicates that only the diagonal terms are included, and for the translational diffusivity only the terms giving the translational velocity are included in the average. From (5.36) and (5.37) we can deduce thatD
FB(0)FB(t)E
∼ 1/D0s. Since the Brownian particle stress is simply the first moment of the force distribution it must have the same scaling with φ as the Brownian forces, thus giving 1/D0sscaling for the Brownian stress autocorrelation. While the forces are cor- related only instantaneously, the stress depends on the configuration of particles as well, hence the stress autocorrelation decays with time as the configuration changes. Equivalently, one could invoke the generalized Stokes-Einstein relation
η0∞(φ) 1
D0s(φ), (5.38)
for all volume fractions coupled with the observation that the resistance matrices must scale asη0∞ which is the effective solvent viscosity at high volume fractions, to get the above scaling for the Brownian forces.
The SD simulations exhibit a sudden break in the increase of the relaxation time scale forφ >
0.45, not seen in the ASDB-nf simulations. This can be attributed to the small size of the particle system used for SD simulations (N=27). The simulation cell size keeps decreasing asφincreases, and forφ≥0.45 it is the cell length that sets the time scale of diffusion and stress relaxation. Further, the long-time decay forφ ≥ 0.45 seems to be almost exponential as is characteristic of diffusion in a closed box, instead of the expected asymptote to t−3/2. If the particles were diffusing in a finite-sized box instead of an infinite suspension, the fluctuations in particle density would dissipate exponentially due to the finite boundary conditions. Therefore the stress autocorrelation functions would also have an exponential decay in the long-time limit. In an infinite suspension there are no spatial boundary conditions forcing an exponential decay so the decay is algebraic. However, we implement periodic boundary conditions to simulate an infinite suspension. When the simulation cell size is smaller than the length over which the particles need to diffuse for fluctuations to decay algebraically, the periodic boundary conditions begin to simulate the effect of being in a finite box thus causing an exponential decay of the stress autocorrelation functions. We will use only the ASDB-nf simulations for all the scaling analysis in order to minimize any finite-size effects. Even with the ASDB-nf simulations forφ > 0.52, CPH decays much more slowly because the particles are in the glassy regime so that the microstructure is unable to relax by diffusion. At long times the finite size of the simulation cell causes the pressure autocorrelation to decay exponentially as the periodic boundary conditions simulate diffusion of the particles in a finite box.
Figure 5.15 shows the scaled pressure autocorrelation functionCPH=CHPNDs0(φ)/(φg0(2;φ)) as a function of the diffusive timeτfor (a)φ≤0.35 and (b)φ≥0.35. The values ofDs0were taken from the ASD simulations of Sierou and Brady [2001]. The scaling works very well for collapsing all the curves in the short time limit but variations in the rate of decay at longer times prevent a complete collapse of the data over all time. The zero-time limit from scaled dilute theory given by (5.35), also shown on the plots, underestimates the simulation data by a small amount. The analytical curve for the pressure autocorrelation function without hydrodynamics is also shown, and it is clear that at long times the simulation data has the same asymptotic behavior as the analytical curve. The slight decrease in the time scale of relaxation for smaller volume fractions was not expected and needs further examination. First we need to eliminate cell-size effects so we compare the data for two different volume fractions but withNsuch that the simulation cell has the same size for both cases.
Figure 5.16(a) shows the data for φ = 0.05,N = 100 andφ = 0.25,N = 500, both having a cell length= 20.31b. Figure 5.16(b) shows the data forφ = 0.1,N = 1000 andφ = 0.25,N = 2500, both having a cell length=34.73b. The decrease in the relaxation time scale is evident in from both figures, hence we can conclude that finite-size effects are not responsible. To quantify the rate of decay we compute the relaxation timeτκas defined in (5.24), but since computing it over the entire range ofτwould require fitting the tails to a known curve we compute
τκ(1)= R1
0hτδΠ(0)δΠ(τ)idτ R1
0hδΠ(0)δΠ(τ)idτ (5.39)
instead, i.e., the integral is evaluated only up-toτ=1 where the data has not yet become too noisy.
This way we can get a more rigorous estimate of the change in time-scale obtained strictly from the simulation data. The values ofτκ(1) for allφare shown in Figure 5.17. Evidently the relaxation time decreases rapidly for very smallφand slowly for largerφup toφ = 0.35 after which it increases, indicating that there might be competing effects that determine the time scale at small and largeφ, respectively.
An explanation for the low-φtemporal scaling can be had from observing the nature of contri- butions to the Brownian stress and the process of structural relaxation. The Brownian stress can be separated into a contribution ΣB1 from particles in contact, which is of the same form as in hard sphere fluids, and a contributionΣB2from hydrodynamic interaction between all the particles:
nD SBE
=−n2kT a Z
r=2b
ˆ
rˆrg(r)dS +nkTD
RS U·R−1FU· ∇lnPNE
, (5.40)
wherePN(r,t) is the probability density for theN−particle configuration [Brady 1993a]. At small volume fractions bothΣB1 andΣB2contribute significantly to the Brownian stress, which implies that the total stress and its fluctuations are determined not only by the particles at contact but also by the configuration of surrounding particles over the distance in which the hydrodynamic interactions decay. The rate of decay of fluctuations inΣB2 is determined by the rate at which the surround- ing particles away from contact rearrange into a different configuration. As the volume fraction increases, the number of particles and therefore the number of different configurations that can be sampled in the region around a particle also increases, and consequently the stress gets de-correlated
faster as new configurations are achieved more easily. This explains the initial decline inτκ seen in Figure 5.17. On the other handg(r) at contact also increases with increasing φ, and due to the strong lubrication forces between particles at contact the ΣB1 contribution becomes more impor- tant thanΣB2, thereby diluting the effect of fluctuations inΣB2on the stress relaxation. The stress relaxation rate is eventually determined byΣB1only for large volume fractions. The reduced short- time self-diffusivity of the particles due to stronger hydrodynamic interactions with increasing φ also slows down the rate at which new configurations can be sampled but evidently it is not the rate-determining factor at smallφ. The long-time self-diffusivity does not govern the relaxation rate until ‘caging’ effects in the microstructure become important, typically forφ >0.35 as observed in our BD simulations.
Thus there are several different competing processes that influence the rate of stress relaxation for small volume fractions: the fluctuations inΣB2decreaseτκwith increasingφ, while the increas- ing dominance ofΣB1and the reduction inD0stend to increaseτκand eventually win out atφ=0.35, at which pointD∞s also begins to influence the relaxation rate. It’s not clear how the combination of these processes can be quantified, so we have approximated the relaxation time scale forφ≤ 0.35 usingτκ(1). For largeφone would expect the long-time self-diffusivity with hydrodynamic interac- tionsD∞,Hs (φ) to set the time scale of relaxation but we found that for 0.35≤φ≤0.5 the long-time self-diffusivity without hydrodynamicsDs∞,NH(φ) provides a better temporal scaling for the simula- tion data. This does not mean that hydrodynamics are not important in the temporal scaling, rather D∞,NHs (φ) being a weaker function of φthan Ds∞,H(φ) happens to yield a good approximation for the combined effect of the competing processes discussed above. Forφ > 0.5 the data scales well withD∞,Hs (φ). Taking all these considerations into account we define the characteristic diffusivity for stress relaxation with hydrodynamic interactions as
DˆH(φ)=
τf it(0.05)/τf it(φ), 0.05≤φ≤0.35 Ds∞,NH(φ)τf it(0.05)
Ds∞,NH(0.35)τf it(0.35), 0.35≤φ≤0.5 D0s(φ)D∞,NHs (φ)τf it(0.05)
Ds0(0.5)D∞,NHs (0.35)τf it(0.35), φ >0.5
(5.41)
such that ˆDH(φ) is a continuous function of φ. Here we have used the approximation Ds∞,H = D0sD∞,NHs [Brady 1994]. The pressure autocorrelation data with long-time tails fitted with the ana- lytical expression for no hydrodynamics is shown in Figure 5.19(a) as a function of the bare diffusive time and in Figure 5.19(b) as a function of the scaled time given by ˆτ = b2/DDˆH(φ). The analyt- ical curves used for fitting were computed as a function of the scaled time ˆτ. The simulation data collapses reasonably well onto a single curve with this scaling. The time-scale for stress relaxation computed using data between 0< τ≤200 from ASDB-nf simulations with fitted long-time tails is shown in Figure 5.18 and has the expected behavior.