Effect of particle-size dynamics on properties of dense spongy-particle systems: Approach towards equilibrium

Open-porous deformable particles, often envisaged as sponges, are ubiquitous in biological and industrial systems (e.g., casein micelles in dairy products and microgels in cosmetics). The rich behavior of these suspensions is owing to the elasticity of the supporting network of the particle, and the viscosity of permeating solvent. Therefore, the rate-dependent size change of these particles depends on their structure, i.e., the permeability. This work aims at investigating the effect of the particle-size dynamics and the underlying particle structure, i.e., the particle permeability, on the transient and long-time behavior of suspensions of spongy particles in the absence of applied deformation, using the dynamic two-scale model developed by Hütter et al. [Farad. Discuss. 158, 407 (2012)]. In the high-density limit, the transient behavior is found to be accelerated by the particle-size dynamics, even at average size changes as small as $1\%$. The accelerated dynamics is evidenced by (i) the higher short-time diffusion coefficient as compared to elastic-particle systems and (ii) the accelerated formation of the stable fcc crystal structure. Furthermore, after long times, the particle-size dynamics of spongy particles is shown to result in lower stationary values of the energy and normal stresses as compared to elastic-particle systems. This dependence of the long-time behavior of these systems on the permeability, that essentially is a transport coefficient and hence must not affect the equilibrium properties, confirms that full equilibration has not been reached.

Figure 1

Evolution of energy per particle for a system with particle with ${\zeta }^{*}=0.005$ and $S^{*}=2.5\times {10}^{-3}$. The volume fraction of the system is ${\phi }_{\text{free}}=0.766$ ($n=1.17\times {10}^{19}{\text{m}}^{-3}$).

Figure 2

Particle configuration at (a) $t/{\tau }_Q^{\text{Br}}=0$ and (b) $t/{\tau }_Q^{\text{Br}}=384.6$. Particle coordinates are scaled by the simulation box length and the particle size is reduced for better visualization.

Figure 3

Energy per particle for systems with different ${\zeta }^{*}$ and ${\phi }_{\text{free}}$ averaged over (a) five different initial configurations and certain realization of the noise, (b) five different realizations of the noise and a fixed initial configuration. The standard deviation of the data is shown by the error bars.

Figure 4

Transition time ${\tau }_{\text{t}}$ for systems with different ${\zeta }^{*}$ and ${\phi }_{\text{free}}$ averaged over (a) five different initial configurations and a certain realization of the noise, (b) five different realizations of the noise and a fixed initial configuration. The spread in data based on the standard error is shown by dashed lines.

Figure 5

Components of the energy per particle for ${\zeta }^{*}=0$ (closed symbols) and ${\zeta }^{*}=0.005$ (open symbols), and $S^{*}=2.5\times {10}^{-3}$. The error bars are based on the standard deviation obtained from running the simulation with five different realizations of the noise.

Figure 6

Radius distribution at $t/{\tau }_Q^{\text{Br}}=384.6$ for a system at ${\phi }_{\text{free}}=0.916$ ($n=1.4\times {10}^{19}{\text{m}}^{-3}$) with (a) ${\zeta }^{*}=0$ with $\langle R\rangle /R_{\text{eq}}=1$ and ${\sigma }_R/R_{\text{eq}}=0.00637$, and (b) ${\zeta }^{*}=0.005$ with $\langle R\rangle /R_{\text{eq}}=0.9934$ and ${\sigma }_R/R_{\text{eq}}=0.00643$. The red line represents the Boltzmann distribution for nonoverlapping particles as given in Eq. (25).

Figure 7

Normal stress for systems with ${\zeta }^{*}=0$ (closed symbols) and ${\zeta }^{*}=0.005$ (open symbols), and $S^{*}=2.5\times {10}^{-3}$. The error bars are based on the standard deviation obtained from running the simulation with five different realizations of the noise.

Figure 8

Scaled short-time diffusion coefficient $D^{\text{S}}$ for porous systems with ${\zeta }^{*}=0.005$ (open symbols) and nonporous systems with ${\zeta }^{*}=0$ (closed symbols), and $S^{*}=2.5\times {10}^{-3}$ at different number densities. $D_0$ denotes the single-particle diffusion coefficient in the dilute limit calculated based on the equilibrium size. The standard error of the data based on simulations run with five different realizations of the noise is smaller than the symbol size.

Figure 9

Scaled long-time diffusion coefficient $D^{\text{L}}$ for porous systems with ${\zeta }^{*}=0.005$ (open symbols) and nonporous systems with ${\zeta }^{*}=0$ (closed symbols), and $S^{*}=2.5\times {10}^{-3}$ at different number densities. $D_0$ denotes the single-particle diffusion coefficient in the dilute limit calculated based on the equilibrium size. The error bars are the standard error of simulations run with five different realizations of the noise.

Figure 10

Average structural composition $f$ of bcc, hcp, and fcc structures for systems with $S^{*}=2.5\times {10}^{-3}$ and ${\zeta }^{*}=0$ (closed symbols) and ${\zeta }^{*}=0.005$ (open symbols). Error bars are the standard error based on simulations run at five different realizations of the noise.

Figure 11

Order parameter $({\overline{q}}_4,{\overline{q}}_6)$-density maps at $t=0.5{\tau }_{\text{t}}$ (top row), at $t={\tau }_{\text{t}}$ (middle row), and at $t=2{\tau }_{\text{t}}$ (bottom row) for systems with ${\phi }_{\text{free}}=0.916$ ($n=1.4\times {10}^{19}{\text{m}}^{-3}$) and ${\zeta }^{*}=0$ (left column), and ${\zeta }^{*}=0.005$ (right column), respectively. The contour lines correspond to $5\%$ of the maximum density, for better visualization.

Figure 12

Evolution of fraction of crystalline phase for the system in Fig. 11 at a density of ${\phi }_{\text{free}}=0.916$ ($n=1.4\times {10}^{19}{\text{m}}^{-3}$) with $S^{*}=2.5\times {10}^{-3}$ and ${\zeta }^{*}=0$ (black) and ${\zeta }^{*}=0.005$ (red), respectively.

Figure 13

Evolution of number of clusters of ordered particles for the system in Fig. 11 at a density of ${\phi }_{\text{free}}=0.916$ ($n=1.4\times {10}^{19}{\text{m}}^{-3}$) with $S^{*}=2.5\times {10}^{-3}$ and ${\zeta }^{*}=0$ (black) and ${\zeta }^{*}=0.005$ (red), respectively.

Figure 14

The first moment of cluster-size distribution $M_n$ of ordered particles for systems with different realizations of the noise at a density of ${\phi }_{\text{free}}=0.916$ ($n=1.4\times {10}^{19}{\text{m}}^{-3}$) with $S^{*}=2.5\times {10}^{-3}$ and ${\zeta }^{*}=0$ (black) and ${\zeta }^{*}=0.005$ (red).

Figure 15

The polydispersity in cluster-size distribution $M_w/M_n$ of ordered particles for systems with different realizations of the noise at a density of ${\phi }_{\text{free}}=0.916$ ($n=1.4\times {10}^{19}{\text{m}}^{-3}$) with $S^{*}=2.5\times {10}^{-3}$ and ${\zeta }^{*}=0$ (black) and ${\zeta }^{*}=0.005$ (red).