5.3 Brownian Dynamics simulations
5.3.1 Simulation results
The equilibrium pressure obtained from the simulations is shown in Figure 5.1 and serves as a primary check for the validity of the simulation method. The analytical value of the equilibrium osmotic pressure given by
Π0=1+4φg0(2;φ) (5.11)
is also shown, whereg0(2;φ) is the equilibrium pair-distribution function at contact evaluated using the well known Carnahan-Starling (CS) equation of state [Carnahan and Starling 1969] forφ≤0.55:
g0(2;φ)= 1− 12φ
(1−φ)3. (5.12)
For higher volume fractions we have used the very precise values forg0(2;φ) determined by Rintoul and Torquato [1996], which also match the CS equation of state at its limiting value ofφ=0.5. The osmotic pressure obtained from Brownian Dynamics is seen to slightly underestimate the theoretical value. We attribute this discrepancy to the error associated with the determination of the interparticle force from particle collisions, which is larger for higher volume fractions as the number of collisions to be resolved increases.
The pressure autocorrelation function was evaluated as an ensemble average over all the particle configurations over the course of the simulation. Specifically, for any given simulation run
hδΠ(0)δΠ(τ)i= 1 M−τ/∆τ
M−τ/∆τ
X
m=0
δΠ(m∆τ)δΠ(m∆τ+τ), (5.13)
where M is the total number of time steps from the run included in the calculation, andδΠ(t) =
Π(t)−D ΠE
. Further averaging was done over all the runs performed for a givenφ. Since the pressure was calculated as an average of the diagonal components of the stress, the cross-correlations among the diagonal stress components are automatically included in the above expression for the pressure autocorrelation function.
Figure 5.2 and 5.3 show the pressure autocorrelation function from Brownian Dynamics simu- lations for variousφwithN = 100 and 1000 particles, respectively. In the short-time limit the data exhibits the expectedt−1/2 scaling [Cichocki and Felderhof 1991; Brady 1993b] . At longer times the decay is very slow due to the isotropic nature of the dissipation of pressure fluctuations. Figure 5.4 shows the simulation data scaled to match the theoretical prediction for the pressure autocorre- lation function forφ=0.2 to 0.35. The analytical expression for the scaled pressure autocorrelation function using the time-scale for simulationsτ=t(2D/b2) is given by
CPNH(τ)= 4 3
√2
πτ −eτ/4Erfc rτ
4
!!
, (5.14)
whereCNHP (τ)=CNHP (τ)N/(φg0(2;φ)) as described in the next section. At short times the pressure autocorrelation function is predicted to scale ast−1/2with the asymptoteCPNH ∼ 4/3
2/√ πτ−1
, and ast→ ∞it decays asCPNH ∼(16/3√
π)τ−3/2. The pressure autocorrelation data is in excellent agreement with the theoretical prediction in thet→0 limit. At longer times the data for the smallest volume fractionφ=0.05 is in good agreement with the theory but there are variations in the rate of decay at higher volume fractions. Both theN=100 andN =1000 systems exhibit similar behavior so the system size apparently does not have any effect on the pressure autocorrelation data when periodic boundary conditions are imposed. For 1000 particles atφ >0.52 the particles were prone to crystallization resulting in a pressure autocorrelation function that is much higher and does not decay at all. The crystallization took place starting atφ >0.5 with 100 particles.
For all the data thet−3/2 rate of decay of the pressure autocorrelation function is so slow that the data gets reduced to just noise before being able to capture the long-time behavior for both the N = 100 andN = 1000 systems. Therefore it is important to know from theory how the long- time tails decay so that the pressure autocorrelation function can be integrated correctly to find the bulk viscosity. The long-time tails were obtained by fitting the analytical curve from dilute theory to the region shortly before the data becomes too noisy. The noise in the simulation results could
be reduced by using a smaller time-step thereby capturing more of the long-time behavior, but that would also increase the computational cost significantly. Having analytical knowledge of the pressure autocorrelation curve alleviates the need for more simulation data for the long-time tails, particularly because it is apparent from the figures that the simulation results follow the analytical curve in the regions of low noise. We simply fit the dilute theory curve to the simulation data in the region before the data becomes too noisy in order to obtain the long-time tails.
The Green-Kubo bulk viscosity in the absence of hydrodynamic interactions is shown in Figure 5.9. The Molecular Dynamics simulation results of Sigurgeirsson and Heyes [2003] for the bulk viscosity of hard sphere fluids scaled with the zero-density bulk viscosity are also shown for com- parison. Both the data are in reasonable agreement at small φand have the same scaling at large φbut there are significant differences in the intermediateφregime. Although both the hard sphere systems must have the same equilibrium properties, the mechanism of stress dissipation is differ- ent in MD and BD simulations, hence the transport properties need not be identical. The particle motion in BD is heavily damped because of the drag force exerted by the surrounding fluid, there- fore particle momentum is not conserved. Additionally, at each time-step the particles experience a random Brownian kick which completely changes the spatial distribution of particle momentum.
On the other hand, in MD simulations the particles start with an initial set of positions and mo- menta which may be random but as the simulation proceeds the particle collisions are such that the total momentum is conserved. Thus the distribution of momentum in time and space is different in BD and MD. Since the dynamic properties such as the shear and bulk viscosity depend on the rate of change of momentum in addition to the spatial distribution of particles, we do not expect the transport properties obtained from BD and MD to be identical.
The shear stress autocorrelation function was also determined from theN = 1000 particle sys- tems for validation of the simulations, and is shown in Figure 5.10. It decays much faster ast−7/2 and consequently the long-time tail is captured correctly in the simulation results. The Brownian shear viscosityη0P was calculated by numerically integrating the shear stress autocorrelation func- tion without any fitting because the noise in the data is adequately low. The Green-Kubo expression for shear viscosity in nondimensional form given by [Nagèle and Bergenholtz 1998]
ηB = 9
4NφZ ∞ 0
hΣxy(0)Σxy(τ)idτ, (5.15)
was used to calculateηB, which is the Brownian contribution to the shear viscosity scaled with the fluid viscosity. Since there is no preferred direction at equilibrium theyzandxzcomponents of the stress were also autocorrelated and included in the ensemble average in the Green-Kubo formula.
Figure 5.12 shows the Green-Kubo shear viscosity in the absence of hydrodynamic interactions scaled with the fluid viscosity along with the simulation results of Foss and Brady [2000], which are noticeably lower than our results at higher volume fractions. The difference is due to the higher accuracy of the results from the present work because the time-step used is smaller (Foss and Brady used a time-step of∆τ= 2.5×10−4), and also because we fitted the analytical dilute theory curve to the simulation data in order to obtain the long-time tails while Foss and Brady simply fitted the tails with t−7/2, thereby losing some area under the stress autocorrelation curve. The total shear viscosity for Brownian particles in a suspension would also includes the high frequency dynamic viscosity, given byη0∞=η
1+ 52φ
in the absence of inter-particle hydrodynamic interactions. The high frequency dynamic viscosity represents the direct viscous contribution to the suspension stress.