Insight into the Viscoelasticity of Self-Assembling Smectic Liquid Crystals of Colloidal Rods from Active Microrheology Simulations

The rheology of colloidal suspensions is of utmost importance in a wide variety of interdisciplinary applications in formulation technology, determining equally interesting questions in fundamental science. This is especially intriguing when colloids exhibit a degree of long-range positional or orientational ordering, as in liquid crystals (LCs) of elongated particles. Along with standard methods, microrheology (MR) has emerged in recent years as a tool to assess the mechanical properties of materials at the microscopic level. In particular, by active MR one can infer the viscoelastic response of a soft material from the dynamics of a tracer particle being dragged through it by external forces. Although considerable efforts have been made to study the diffusion of guest particles in LCs, little is known about the combined effect of tracer size and directionality of the dragging force on the system’s viscoelastic response. By dynamic Monte Carlo simulations, we apply active MR to investigate the viscoelasticity of self-assembling smectic (Sm) LCs consisting of rodlike particles. In particular, we track the motion of a spherical tracer whose size is varied within a range of values matching the system’s characteristic length scales and being dragged by constant forces that are parallel, perpendicular, or at 45° to the nematic director. Our results reveal a uniform value of the effective friction coefficient as probed by the tracer at small and large forces, whereas a nonlinear, force-thinning regime is observed at intermediate forces. However, at relatively weak forces the effective friction is strongly determined by correlations between the tracer size and the structure of the host fluid. Moreover, we also show that external forces forming an angle with the nematic director provide additional details that cannot be simply inferred from the mere analysis of parallel and perpendicular forces. Our results highlight the fundamental interplay between tracer size and force direction in assessing the MR of Sm LC fluids.


S1. Details of the Spherical Tracers and Hard Rods Systems
In this work, the systems studied consisted of one spherical particle with diameter d t embedded in a bath of N r = 1400 rod-like particles in smectic (Sm) phase with length-to-diameter ratio L * ≡ L/σ = 5.In Table S1, we report the external force intensity and direction with respect to the nematic director n, the tracer size d t , elementary time steps δt MC,r and δt MC,t in units of τ , and acceptance rates A r and A t of rods and tracer particles, respectively.
Table S1: Details of the systems studied in this work.The external force intensity, diameter d t , MC time steps δt MC,r and δt MC,t of the rods and tracer particles are presented with their acceptance rates A r and A t , respectively.
where ûi (v, t) refers to the unit orientation vectors of N r (v, t) rods in volume v at time t.If the rods are parallel to n, then E 2 (v) ≈ 1.On the other hand, if the rods orientations are perpendicular to the nematic director then E 2 (v) ≈ −1/2.

S3. Trajectories of a Spherical Tracer in a Bath of Rods in Smectic Phase
Bath density distribution for a system of hard rods and a tracer particle are shown in Figures S2 and S3 for external forces oriented 45 • and 90 • with respect to the nematic director n, respectively.While rods are monodisperse and have length-to-diameter ratio L * = 5, the tracers are modeled as hard spheres with diameters between 1σ and 3σ.

S4. Friction Tensor in Smectic Phases of Rod-Shaped Particles Probed by a Spherical Tracer
The effective friction tensor of a bath of rods probed by a spherical tracer reads: where the elements γ eff, and γ eff,⊥ represent the frictions experienced by the tracer along and transverse to the nematic director, repectively.The non-diagonal elements, γ eff, ⊥ and γ eff,⊥ , refer to the correlations between the effective frictions in the principal directions of motion.While γ eff, and γ eff,⊥ are calculated from the analysis of the effect of F and F ⊥ at low intensities on the tracer particle, the non-diagonal elements of γ eff are derived by decomposing the parallel and perpendicular contributions of F 45 • on the tracer.Table S2 shows the diagonal and non-diagonal effective frictions of a bath of hard rods in the Sm phase as probed by tracer particles of different sizes at low forces.

F
Figure S1: (colour on-line) Orientational correlation functions of a system of hard rods in Sm phase at φ = 0.51 around a tracer with diameter 1σ (left column), 2σ (center column), and 3σ (right column).The tracer particle is pulled by an external force F parallel to the nematic director n.The color palette is shown at the bottom of the figure.
Figure S2: (colour on-line) Density maps of a system of hard rods in Sm phase at φ = 0.51 around a tracer with diameter 1σ (left column), 2σ (center column), and 3σ (right column).The tracer particle is pulled by an external force F 45 • oriented 45 • with respect to the nematic director n. color palette is shown at the bottom of the figure.

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Figure S3: (colour on-line) Density maps of a system of hard rods in Sm phase at φ = 0.51 around a tracer with diameter 1σ (left column), 2σ (center column), and 3σ (right column).The tracer particle is pulled by an external force F ⊥ perpendicular to the nematic director n.The color palette is shown at the bottom of the figure.