Hydrodynamic influence on the aggregation of colloids with short-range depletion potential in dilute suspensions

This work studies the aggregation of colloids under depletion attraction focusing on the effect of hydrodynamic interactions on cluster kinetics and average structures. Our numerical study is based on a variant of the immersed boundary method (inertial coupling method) which solves the solvent fluctuating hydrodynamics and instantaneously couples the particle and the fluid momentum transfer. Comparison is made with Langevin dynamics and Monte Carlo methods, which do not include hydrodynamics. We consider a small system (20 and 40 particles) which reproduces the system considered by Whitmer and Luijten [J. Phys. Chem. B 115, 7294 (2011)] (analyzed by multiparticle rotation dynamics). In that work, the authors report substantial hydrodynamic effects altering the cluster sizes and shapes. By contrast, within statistical error, we have not found any hydrodynamic effect on the average clusters structure. We also analyze the time-evolution of the aggregation kinetics, revealing a slowing down of the process due to the reduction of the mobility close to the clusters. The time correlation of the cluster number, shape, and size reveals two characteristic times: a decay at short times faster than the single-particle diffusion time $t_D$, followed by a slower exponential decay at long times with a typical time of several $t_D$.

Figure 1

Depletion potentials used in this work: see Table 1 for details.

Figure 2

Time evolution of the total clusterized mass (particles in clusters of size $N\ge 2$ in simulations with and without hydrodynamics (ICM and LD, respectively). Error bars represent the standard error.

Figure 3

Probability $P\left(N\right)$ defined by Eq. (14). Results for systems with 20 (open symbols) and 40 particles (closed symbols) are presented for ICM (red), Langevin dynamics (blue), and Monte Carlo (pink) simulations. Dashed line corresponds to $P\left(N\right)\approx \frac{1}{\langle N\rangle }exp\left(-\frac{N}{\langle N\rangle }\right)$. The error bars represent the standard error.

Figure 4

Probability of the number of bonds per particle for a system with 20 particles interacting with the potential 1. Results are presented for ICM (red), Langevin dynamics (blue), and Monte Carlo (pink) simulations. The error bars represent the standard error. The average number of bonds in a cluster of size $N\ge 2$ is 1.55.

Figure 5

Probability density of the asphericity for clusters of size $N=5,10$. Results are presented for ICM (red), Langevin dynamics (blue), and Monte Carlo (pink) simulations. The error bars represent the standard error. Results correspond to potential 1 in Fig. 1.

Figure 6

Probability of cluster sizes with two different potentials (potential 1, open symbols, and potential 2, closed symbols) with the same second virial. Results are presented for ICM (red), Langevin dynamics (blue), and Monte Carlo (pink) simulations. The error bars represent the standard error.

Figure 7

Top: Probability density of the asphericity, obtained from ICM simulations, for clusters of size $N=5$ with two different potentials with the same second virial: potential 1 (red) and potential 2 (blue). Bottom: The same quantity for the potential 2 from ICM (red), LD (blue), and MC (pink). The error bars represent the standard error.

Figure 8

Comparison between 9 long ($\approx 1000t_D$) and 96 short simulations ($\approx 88t_D$) (like Ref. [9]) for $N_T=20$ and $N_T=40$ simulations. The error bars represent the standard error.

Figure 9

Probability of survival times for three different sizes of clusters. Results are presented for ICM (red) and LD (blue). The error bars represent the standard error. Results obtained for potential 1.

Figure 10

Results from ICM simulations. Top: time correlation function of the number of clusters of sizes 1,2,3,4, and 5. Bottom: the same function has been drawn with the axis $y$ in logarithmic scale. For clarity, just two different size of clusters have been depicted. Results obtained for potential 1.

Figure 11

Results from ICM simulations: time correlation function of the number of big ($N\ge 5$) and small ($N<5$) clusters. In the inset, the same graph have been drawn with the y axis in logarithmic scale. In the latter case, for clarity, just two of the curves have been depicted. Results obtained for potential 1.

Figure 12

Time correlation function of the total number of clusters of size $N$. The case $N=1$ indicates detachment/attachment events of single particles. Results with ICM and LD simulations are shown. In the inset, the same graph with the y axis in logarithmic scale. Results obtained for potential 1.

Figure 13

Results from ICM simulations. Time correlation function of the asphericity for clusters of size $N=3,5$. Results obtained for potential 1. The error bars represent the standard error.