Activated dynamics in dense fluids of attractive nonspherical particles. II. Elasticity, barriers, relaxation, fragility, and self-diffusion

In paper II of this series we apply the center-of-mass version of Nonlinear Langevin Equation theory to study how short-range attractive interactions influence the elastic shear modulus, transient localization length, activated dynamics, and kinetic arrest of a variety of nonspherical particle dense fluids (and the spherical analog) as a function of volume fraction and attraction strength. The activation barrier (roughly the natural logarithm of the dimensionless relaxation time) is predicted to be a rich function of particle shape, volume fraction, and attraction strength, and the dynamic fragility varies significantly with particle shape. At fixed volume fraction, the barrier grows in a parabolic manner with inverse temperature nondimensionalized by an onset value, analogous to what has been established for thermal glass-forming liquids. Kinetic arrest boundaries lie at significantly higher volume fractions and attraction strengths relative to their dynamic crossover analogs, but their particle shape dependence remains the same. A limited universality of barrier heights is found based on the concept of an effective mean-square confining force. The mean hopping time and self-diffusion constant in the attractive glass region of the nonequilibrium phase diagram is predicted to vary nonmonotonically with attraction strength or inverse temperature, qualitatively consistent with recent computer simulations and colloid experiments.

Figure 1

Schematic, but qualitatively accurate, synthesis of all the kinetic arrest diagrams inferred from the dynamic free-energy calculations in paper I [1]. Trajectories studied in this paper are indicated by the arrows. Solid lines denote ideal NMCT kinetic arrest boundaries, and dashed lines indicate “soft” crossover boundaries where sharp, but continuous, changes of the barrier location or localization length occur.

Figure 2

(a) Natural logarithm of the CM localization length in the glassy regime, i.e., below the glass-to-gel transition curve (direction b in Fig. 1), as a function of volume fraction for spherical particles at $\varepsilon =1.02$ (open black circles), eight-site rods at $\varepsilon =1.56$ (filled dark blue triangles), eight-site rods at $\varepsilon =0.51$ (open dark blue triangles), eight-site disks at $\varepsilon =1.62$ (filled red squares), eight-site disks at $\varepsilon =0.51$ (open red squares), snub disphenoids at $\varepsilon =1.72$ (filled green diamonds), and $\varepsilon =0.83$ (open green diamonds). (b) Analogous plot of the elastic shear modulus in units of $k_{B}T/{\sigma }^3$.

Figure 3

Natural logarithm of the elastic modulus as a function of the distance from the ideal gel or glass transition. The open symbols show the behavior at fixed $\varepsilon$ and increasing $\varphi$ (top axis). The filled symbols show the behavior at fixed $\varphi$ and increasing $\varepsilon$ (bottom axis). The shapes included are six-site rods at $\varphi =0.20$ (filled blue triangles), octagonal particles at $\varphi =0.25$ (pink circles), eight-site disks at $\varphi =0.25$ (red squares), octahedra at $\varphi =0.45$ (filled green diamonds), spheres at $\varepsilon =3.38$ (open black circles), eight-site disks at $\varepsilon =3.26$ (open red squares), and octahedral particles at $\varepsilon =3.84$ (open green diamonds).

Figure 4

Natural logarithm of the normalized hopping time as a function of the barrier height in the gel region, for the eight-site rod at $\varphi =0.20$ (filled dark blue triangles), eight-site disks at $\varphi =0.35$ (filled red squares), octahedra at $\varphi =0.45$ (filled green diamonds), spheres at $\varepsilon =3.38$ (filled black circles), and snub disphenoids at $\varepsilon =3.97$ (filled pink pentagons). Shown is the universal exponential dependence of the hopping time on the barrier height in the gel region. Inset: Analogous results in the repulsive glass region for the eight-site rod at $\varepsilon =0.5$ (open dark blue triangles), eight-site disks at $\varepsilon =1.0$ (open red squares), and snub disphenoids at $\varepsilon =1.7$ (open green diamonds).

Figure 5

Kinetic arrest diagram corresponding to an arrest criterion of ${\tau }_{\mathrm{hop}}/{\tau }_s=1000$ for several shapes (dotted lines and filled symbols): eight-site rods (filled dark blue triangles), spheres (filled black circles), eight-site disks (filled red squares), and snub disphenoids (filled green pentagons). The ideal NMCT dynamic crossover boundary (filled dark blue triangles and solid lines), glass-to-gel curves (open triangles and solid lines), and the kinetic arrest boundary (open triangles and dotted lines) at the ${\tau }_{\mathrm{hop}}/{\tau }_s=10$ criterion, are also shown for fluids of eight-site rods.

Figure 6

Dynamic free-energy barrier height as a function of the difference variable, $\varphi -{\varphi }_c$. The open symbols indicate results for fixed $\varepsilon =1$ (direction b in Fig. 1), whereas the filled symbols indicate results at a fixed $\varepsilon =3$ (direction c in Fig. 1). Spheres are indicated by black circles, octagons by pink triangles, eight-site disks by red squares, and snub disphenoids by green pentagons.

Figure 7

Dynamic free-energy barrier height as a function of the dimensionless and reduced variable $(\varepsilon /{\varepsilon }_c-1){}^2$ (where here ${\varepsilon }_0={\varepsilon }_c$ is the ideal NMCT transition value), for eight-site rods at $\varphi =0.20$ (filled dark blue triangles), six-site rods at $\varphi =0.25$ (filled brown circles), eight-site disks at $\varphi =0.25$ (filled red squares), and at $\varphi =0.35$ (filled pink diamonds), spheres at $\varphi =0.35$ (filled light blue circles), snub disphenoids at $\varphi =0.45$ (filled green pentagons), and octahedra at $\varphi =0.45$ (filled lavender triangles). The solid curves through the theoretical points are a guide to the eye.

Figure 8

(a) Dynamic free-energy barrier height as a function of the normalized coupling constant, $(\lambda -{\lambda }_c)/{\lambda }_c$, computed at fixed attraction strength and varying volume fraction (as in direction c of Fig. 1) for snub disphenoids at $\varepsilon =2.76$ (filled, dark blue triangles), hexagons at $\varepsilon =2.79$ (open, green squares), and eight-site disks at $\varepsilon =3.26$ (filled, red squares). Results are also presented at fixed volume fraction and varying attraction strength (as in direction d of Fig. 1) for six-site rods at $\varphi =0.20$ (open, pink circles), eight-site disks at $\varphi =0.25$ (filled, light blue circles), and snub disphenoids at $\varphi =0.45$ (open black triangles). (b) Same as (a) calculated by fixing volume fraction and varying attraction strength for cubes at $\varepsilon =1.58$ (open black circles), eight-site disks at $\varepsilon =1.62$ (filled blue triangles), octagons at $\varepsilon =1.65$ (filled lavender circles), eight-site rods at $\varepsilon =0.51$ (filled brown diamonds), eight-site disks at $\varepsilon =0.51$ (filled red squares), and octahedra at $\varepsilon =0.53$ (open green diamonds).

Figure 9

The natural logarithm of the normalized barrier hopping time as a function of the attraction strength at a fixed high volume fraction along the green trajectory that intersects the dotted soft transition line in Fig. 1 for spheres at $\varphi =0.49$ (filled black circles), eight-site disks at $\varphi =0.45$ (filled red squares), and snub disphenoids at $\varphi =0.53$ (filled green triangles). The open black circles illustrate the calculations for spheres at a volume fraction ($\varphi =0.59$) beyond the dotted line transition of Fig. 1. The inset presents the hopping diffusion constant normalized by its hard-core value for the same shapes and at the same conditions (assuming that the short time diffusion constant is independent of $\varepsilon$).