^{1}, W. K. den Otter

^{1,a)}, J. K. G. Dhont

^{2}and W. J. Briels

^{3}

### Abstract

The isotropic-nematic spinodals of solutions of rigid spherocylindrical colloids with various shape anisotropies in a wide range from 10 to 60 are investigated by means of Brownian dynamics simulations. To make these simulations feasible, we developed a new event-driven algorithm that takes the excluded volume interactions between particles into account as instantaneous collisions, but neglects the hydrodynamic interactions. This algorithm is applied to dense systems of highly elongated rods and proves to be efficient. The calculated isotropic-nematic spinodals lie between the previously established binodals in the phase diagram and extrapolate for infinitely long rods to Onsager’s [Ann. N. Y. Acad. Sci.51, 627 (1949)] theoretical predictions. Moreover, we investigate the shear induced shifts of the spinodals, qualitatively confirming the theoretical prediction of the critical shear rate at which the two spinodals merge and the isotropic-nematic phase transition ceases to exist.

This work is part of the SoftLink research program of the Stichting voor Fundamenteel Onderzoek der Materie (FOM), which is financially supported by the Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO).

I. INTRODUCTION

II. SIMULATION METHOD

A. Event-driven algorithm

B. Validation of the algorithm

C. Start configurations

III. RESULTS AND DISCUSSIONS

A. Rotational diffusion coefficients

B. Spinodals

C. Spinodal shift due to shear

IV. CONCLUSION

### Key Topics

- Brownian dynamics
- 20.0
- Shear flows
- 15.0
- Diffusion
- 13.0
- Self diffusion
- 11.0
- Colloidal systems
- 9.0

## Figures

A cartoon of the trajectories of three interacting rods during one time step . On the left-hand side, , , and mark the start positions of the rods. On the right-hand side, , , and denote the prospective positions of the rods at the end of the step for the given initial set of random forces and torques. At the collision moments and new random forces and torques are drawn for the two colliding particles. The resulting modified final positions are indicated by (double) primed indices. The solid lines denote the traversed trajectories; dotted lines are extrapolations.

A cartoon of the trajectories of three interacting rods during one time step . On the left-hand side, , , and mark the start positions of the rods. On the right-hand side, , , and denote the prospective positions of the rods at the end of the step for the given initial set of random forces and torques. At the collision moments and new random forces and torques are drawn for the two colliding particles. The resulting modified final positions are indicated by (double) primed indices. The solid lines denote the traversed trajectories; dotted lines are extrapolations.

The rotational self-diffusion coefficients of rigid rods with aspect ratios of and 68 as functions of the scaled volume fraction . The open symbols denote results calculated with the new event-driven algorithm for hard rods, while the solid symbols are obtained by our previous algorithm in which the interaction potential between rods was proportional to their overlap volume.

The rotational self-diffusion coefficients of rigid rods with aspect ratios of and 68 as functions of the scaled volume fraction . The open symbols denote results calculated with the new event-driven algorithm for hard rods, while the solid symbols are obtained by our previous algorithm in which the interaction potential between rods was proportional to their overlap volume.

The rotational self-diffusion coefficients of rigid rods with various aspect ratios as functions of the scaled volume fraction.

The rotational self-diffusion coefficients of rigid rods with various aspect ratios as functions of the scaled volume fraction.

The rotational self-diffusion coefficients of rigid rods with various aspect ratios as functions of the scaled volume fraction on log-log scale. Here is the number density. The solid line is a linear fit based on the results for .

The rotational self-diffusion coefficients of rigid rods with various aspect ratios as functions of the scaled volume fraction on log-log scale. Here is the number density. The solid line is a linear fit based on the results for .

The simulated self (circles) and collective (squares) rotational diffusion coefficients as functions of the scaled volume fraction for solutions of rods with .

The simulated self (circles) and collective (squares) rotational diffusion coefficients as functions of the scaled volume fraction for solutions of rods with .

The scalar order parameter vs the scaled volume fraction for solutions of rods with an aspect ratio of . The closed circles and open squares denote stationary order parameters obtained when starting the simulations with perfectly aligned and isotropic boxes, respectively. The collective rotational diffusion coefficients are plotted as triangles. The closed triangles are calculated from the decay of an initially aligned state; the open triangles are obtained by autocorrelating thermal fluctuations, see Eq. (28). The dashed line is a fit with Eq. (29), reaching zero at the isotropic-nematic spinodal indicated by an arrow.

The scalar order parameter vs the scaled volume fraction for solutions of rods with an aspect ratio of . The closed circles and open squares denote stationary order parameters obtained when starting the simulations with perfectly aligned and isotropic boxes, respectively. The collective rotational diffusion coefficients are plotted as triangles. The closed triangles are calculated from the decay of an initially aligned state; the open triangles are obtained by autocorrelating thermal fluctuations, see Eq. (28). The dashed line is a fit with Eq. (29), reaching zero at the isotropic-nematic spinodal indicated by an arrow.

The scalar order parameters and collective rotational diffusion coefficients as functions of the scaled volume fraction for aspect ratios of 10 (a), 15 (b), 30 (c), and 60 (d). The arrows and numbers refer to the INS spinodals obtained by using Eq. (29). The values of in (a)–(d) are multiplied by 5, 8, 10, and 20, respectively.

The scalar order parameters and collective rotational diffusion coefficients as functions of the scaled volume fraction for aspect ratios of 10 (a), 15 (b), 30 (c), and 60 (d). The arrows and numbers refer to the INS spinodals obtained by using Eq. (29). The values of in (a)–(d) are multiplied by 5, 8, 10, and 20, respectively.

The INS (solid squares) and NIS (solid circles) spinodals as functions of . The open squares and circles are the binodals calculated in Ref. 19. The use of , rather than the shape anisotropy , facilitates the comparison with theoretical predictions. The theoretical binodals (open triangles) and spinodals (solid triangles) at infinite aspect ratio are plotted on the axis for ; these points were not included when fitting the lines.

The INS (solid squares) and NIS (solid circles) spinodals as functions of . The open squares and circles are the binodals calculated in Ref. 19. The use of , rather than the shape anisotropy , facilitates the comparison with theoretical predictions. The theoretical binodals (open triangles) and spinodals (solid triangles) at infinite aspect ratio are plotted on the axis for ; these points were not included when fitting the lines.

(a) The theoretical scalar order parameter as a function of the scaled volume fraction at various shear rates for . At low shear rates, the flow induces a small paranematic alignment in the isotropic phase and increases the alignment of the nematic phase. The end points of these lines are the INS and NIS spinodals, which are plotted in (b) as a function of the Peclet number . At the critical shear rate, corresponding to , the spinodals coalesce and end; hence the two phases merge into a single phase.

(a) The theoretical scalar order parameter as a function of the scaled volume fraction at various shear rates for . At low shear rates, the flow induces a small paranematic alignment in the isotropic phase and increases the alignment of the nematic phase. The end points of these lines are the INS and NIS spinodals, which are plotted in (b) as a function of the Peclet number . At the critical shear rate, corresponding to , the spinodals coalesce and end; hence the two phases merge into a single phase.

The simulated scalar order parameters as functions of the scaled volume fraction at various shear rates for (a) and (b) .

The simulated scalar order parameters as functions of the scaled volume fraction at various shear rates for (a) and (b) .

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