Mesoscopic Lattice Boltzmann Modeling of Flowing Soft Systems

A mesoscopic multicomponent lattice Boltzmann model with short-range repulsion between different species and short (midranged) attractive (repulsive) interactions between like molecules is introduced. The interplay between these composite interactions gives rise to a rich configurational dynamics of the density field, exhibiting many features of disordered liquid dispersions (microemulsions) and soft-glassy materials, such as long-time relaxation due to caging effects, anomalous enhanced viscosity, aging effects under moderate shear and flow above a critical shear rate.

Figure 1

The two components $A$ and $B$ interact via a repulsive pseudopotential, which supports a surface tension ${\sigma }_{AB}$. Moreover, each component experiences an attractive interaction in the first Brillouin zone and a repulsive one acting on both Brillouin zones. Each of these interactions can be tuned through a separate coupling constant.

Figure 2

The nominal surface tension of the two-component fluid as a function of the reference density ${\rho }_0$ (left panel). The coupling parameters are $G_A^a=-12.55$, $G_A^r=11.70$, $G_B^a=-11.71$, $G_B^r=10.95$, $G_{AB}=0.58$. Above a critical value, ${\rho }_{\mathrm{crit}}\sim 0.72$, the nominal surface tension turns negative. Right panel: A typical snapshot of the density field ${\rho }_A$ at time $2\times {10}^6$ (lattice units) at resolution ${512}^2$. Once the separation between $A$ and $B$ phases has occurred, we obtain ${\rho }_A\approx 1.2$ in the region occupied by phase $A$, and ${\rho }_A\sim 0.1$ elsewhere (the same values, although complementary in space, occur for fluid $B$). The interface width is $w\sim 5$ lattice points wide. A cut of the density distribution across the interface shows that the profile is not monotonic, with small bumps on both high and low-density ends (bottom panel, dashed line, open symbols, LR). To investigate the nature of these bumps, we have performed additional simulations at higher resolution (black solid line, HR), by tuning the parameters $G$ and ${\rho }_0$ as outlined in [5] ($G_A^a=-8.2$, $G_A^r=7.38$, $G_B^a=-7.29$, $G_B^r=6.47$, $G_{AB}=0.49$, ${\rho }_0=0.83$). The bumps are indeed seen to attenuate at higher resolution, which points to a numerical rather than physical origin. Because of the small but positive value of $\sigma$ (for ${\rho }_0=0.7$, we find $\sigma =0.0525$), the system proves capable of supporting fairly complex metastable density configurations.

Figure 3

The response function $R(t)$ and the surface indicator $I_{AB}(t)$ as a function of time, (expressed in units of ${10}^3$ LB time steps). The forcing is $U_0=0.1$, the domain is ${128}^2$ and the other parameters are defined in the text. The sharp decrease of the response function in the initial stage indicates the structural arrest of the system, associated with an anomalous enhancement of the flow viscosity, about 4 orders of magnitude above the molecular value. In the left panel, cages are present, which manage to “trap” microstructures inside. In the right panel, the cages break down and the system is now able to flow again.

Figure 4

Aging of the system. Correlation function corresponding to different waiting times $t_w$ ($t_w=5\times {10}^4$, red, squares; $t_w=2\times {10}^5$, green, circles; and $t_w=3\times {10}^5$, blue, triangles) with shear stress $U_0=0.02$. In the inset, we show the correlation function for $t_w=3\times {10}^5$ and $U_0=0.03$: with increasing shear stress the structural arrest disappears.