^{1,2}, Yan-Wei Li

^{1}, Zhao-Yan Sun

^{1,a)}and Li-Jia An

^{1,b)}

### Abstract

Despite their fundamental and practical interest, the physical properties of hard ellipses remain largely unknown. In this paper, we present an event-driven molecular dynamics study for hard ellipses and assess the effects of aspect ratio and area fraction on their physical properties. For state points in the plane of aspect ratio (1 ⩽ k ⩽ 9) and area fraction (0.01 ⩽ ϕ ⩽ 0.8), we identify three different phases, including isotropic, plastic, and nematic states. We analyze in detail the thermodynamic, structural, and self-diffusive properties in the formed various phases of hard ellipses. The equation of state (EOS) is shown for a wide range of aspect ratios and is compared with the scaled particle theory (SPT) for the isotropic states. We find that SPT provides a good description of the EOS for the isotropic phase of hard ellipses. At large fixed ϕ, the reduced pressure p increases with k in both the isotropic and the plastic phases and, interestingly, its dependence on k is rather weak in the nematic phase. We rationalize the thermodynamics of hard ellipses in terms of particle motions. The static structures of hard ellipses are then investigated both positionally and orientationally in the different phases. The plastic crystal is shown to form for aspect ratios up to k = 1.4, while appearance of the stable nematic phase starts approximately at k = 3. We quantitatively determine the locations of the isotropic-plastic (I-P) transition and the isotropic-nematic (I-N) transition by analyzing the bond-orientation correlations and the angular correlations, respectively. As expected, the I-P transition point is found to increase with k, while a larger k leads to a smaller area fraction where the I-N transition takes place. Moreover, our simulations strongly support that the two-dimensional nematic phase in hard ellipses has only quasi-long-range orientational order. The self-diffusion of hard ellipses is further explored and connections are revealed between the structure and the self-diffusion. We discuss the relevance of our results to the glass transition in hard ellipses. Finally, the results of the isodiffusivity lines are evaluated for hard ellipses and we discuss the effect of spatial dimension on the diffusive dynamics of hard ellipsoidal particles.

This work is subsidized by the National Basic Research Program of China (973 Program, 2012CB821500), and supported by the National Natural Science Foundation of China (NNSFC) (Grant Nos. 21222407, 21074137, and 50930001) programs.

I. INTRODUCTION

II. SIMULATION DETAILS

III. RESULTS AND DISCUSSION

A. Equation of state

B. Static structure

C. Self-diffusion

IV. CONCLUSIONS

### Key Topics

- Nematic liquid crystals
- 34.0
- Equations of state
- 31.0
- Self diffusion
- 24.0
- Glass transitions
- 18.0
- Diffusion
- 16.0

##### C09K19/00

## Figures

Reduced pressure p as a function of area fraction ϕ for (a) hard disks and hard ellipses with k = 1.1–2 and (b) hard ellipses with k = 3–9. The results are shifted up by 5 from the preceding small k for clarity. The dotted lines are simulation results and the solid lines are the results of the SPT [Eq. (5) ]. The green dashed line in (a) is the SPT prediction for the equation of state of hard disks in the isotropic state, i.e., Eq. (10) in Ref. 56 . The circles indicate the area fractions where the system starts to form stable plastic crystals in (a) and stable nematic crystals in (b), respectively (see the structural analysis in Subsection III B ). The semi-log plot in the inset of Fig. 1(b) highlights the change of dependence of p on ϕ with increasing k.

Reduced pressure p as a function of area fraction ϕ for (a) hard disks and hard ellipses with k = 1.1–2 and (b) hard ellipses with k = 3–9. The results are shifted up by 5 from the preceding small k for clarity. The dotted lines are simulation results and the solid lines are the results of the SPT [Eq. (5) ]. The green dashed line in (a) is the SPT prediction for the equation of state of hard disks in the isotropic state, i.e., Eq. (10) in Ref. 56 . The circles indicate the area fractions where the system starts to form stable plastic crystals in (a) and stable nematic crystals in (b), respectively (see the structural analysis in Subsection III B ). The semi-log plot in the inset of Fig. 1(b) highlights the change of dependence of p on ϕ with increasing k.

Reduced pressure p as a function of γ at fixed densities. The solid lines are the results of Eq. (5) .

Reduced pressure p as a function of γ at fixed densities. The solid lines are the results of Eq. (5) .

k-dependence of p at the highest densities studied. The dotted lines are a guide to the eye. For these densities, the system has a plastic phase for k ⩽ 1.4, a nematic phase for k ⩾ 3 and an isotropic phase in between.

k-dependence of p at the highest densities studied. The dotted lines are a guide to the eye. For these densities, the system has a plastic phase for k ⩽ 1.4, a nematic phase for k ⩾ 3 and an isotropic phase in between.

Pair correlation function g(r) of hard ellipses at ϕ = 0.8 for k ⩾ 1.5. The results are shifted up by 0.5 from the preceding small k for clarity. As there are two basic length scales in a system of hard ellipses, g(r) also peaks at r = 2a, as indicated by the arrows for k = 9.

Pair correlation function g(r) of hard ellipses at ϕ = 0.8 for k ⩾ 1.5. The results are shifted up by 0.5 from the preceding small k for clarity. As there are two basic length scales in a system of hard ellipses, g(r) also peaks at r = 2a, as indicated by the arrows for k = 9.

g 6(r)/g(r) of hard ellipses with k = 1.2 in the vicinity of the isotropic-plastic transition. The solid lines are the results of the OZ fittings and the dash dotted lines are the results of the power-law fittings. The green dashed line indicates g 6(r)/g(r) ∼ r −1/4. The results are similar for other aspect ratios with k ⩽ 1.4.

g 6(r)/g(r) of hard ellipses with k = 1.2 in the vicinity of the isotropic-plastic transition. The solid lines are the results of the OZ fittings and the dash dotted lines are the results of the power-law fittings. The green dashed line indicates g 6(r)/g(r) ∼ r −1/4. The results are similar for other aspect ratios with k ⩽ 1.4.

Isotropic-plastic transition point ϕ p as a function of k. The solid line is a guide to the eye.

Isotropic-plastic transition point ϕ p as a function of k. The solid line is a guide to the eye.

Nematic order parameter P 2 as a function of nematic director θ dir for hard ellipses with k = 6 for various ϕ. Note that the result is shown for a single sample because P 2(θ dir ) cannot be averaged among different samples since the nematic direction differs from one sample to another.

Nematic order parameter P 2 as a function of nematic director θ dir for hard ellipses with k = 6 for various ϕ. Note that the result is shown for a single sample because P 2(θ dir ) cannot be averaged among different samples since the nematic direction differs from one sample to another.

Upper: as a function of ϕ for various k. Lower: contour plot of in the plane of ϕ and k.

Upper: as a function of ϕ for various k. Lower: contour plot of in the plane of ϕ and k.

Angular correlation function g 2(r) for hard ellipses with k ⩽ 2 at ϕ = 0.8. The green dashed line indicates g 2(r) ∼ r −1/4.

Angular correlation function g 2(r) for hard ellipses with k ⩽ 2 at ϕ = 0.8. The green dashed line indicates g 2(r) ∼ r −1/4.

Angular correlation function g 2(r) for a system of hard ellipses with k = 6 in the vicinity of the isotropic-nematic transition. The solid lines are the results of the power-law fittings. The green dashed line indicates g 2(r) ∼ r −1/4. The results are similar for other aspect ratios with k ⩾ 3.

Angular correlation function g 2(r) for a system of hard ellipses with k = 6 in the vicinity of the isotropic-nematic transition. The solid lines are the results of the power-law fittings. The green dashed line indicates g 2(r) ∼ r −1/4. The results are similar for other aspect ratios with k ⩾ 3.

Isotropic-nematic transition point ϕ n as a function of k. The triangle and the squares are the MC results taken from Refs. 22 and 25 , respectively. The dashed and solid lines are the fits to the EDMD data by ϕ n = A/k with A = 2.90 and ϕ n = ϕ0/(k 0 + k) with ϕ0 = 6.37 and k 0 = 5.14, respectively.

Isotropic-nematic transition point ϕ n as a function of k. The triangle and the squares are the MC results taken from Refs. 22 and 25 , respectively. The dashed and solid lines are the fits to the EDMD data by ϕ n = A/k with A = 2.90 and ϕ n = ϕ0/(k 0 + k) with ϕ0 = 6.37 and k 0 = 5.14, respectively.

Angular correlation function g 2(r) for hard ellipses with k ⩾ 3 at ϕ = 0.8.

Angular correlation function g 2(r) for hard ellipses with k ⩾ 3 at ϕ = 0.8.

(a) Phase diagram of hard ellipses in the plane of aspect ratio k and area fraction ϕ. (b)–(e) Representative snapshots for a high-density isotropic phase with k = 1.5 and ϕ = 0.8, a low-density isotropic phase with k = 6 and ϕ = 0.5, a plastic phase with k = 1.2 and ϕ = 0.8, and a nematic phase with k = 9 and ϕ = 0.8. Note that there is visually no distinction for the orientation of an ellipse with θ and θ + π so that particles with θ are shown as the same color as those with θ + π in the snapshots.

(a) Phase diagram of hard ellipses in the plane of aspect ratio k and area fraction ϕ. (b)–(e) Representative snapshots for a high-density isotropic phase with k = 1.5 and ϕ = 0.8, a low-density isotropic phase with k = 6 and ϕ = 0.5, a plastic phase with k = 1.2 and ϕ = 0.8, and a nematic phase with k = 9 and ϕ = 0.8. Note that there is visually no distinction for the orientation of an ellipse with θ and θ + π so that particles with θ are shown as the same color as those with θ + π in the snapshots.

Time evolution of (a) translational and (b) rotational MSDs at ϕ = 0.8 for three aspect ratios. The system has a plastic phase for k = 1.1, an isotropic phase for k = 2, and a nematic phase for k = 6.

Time evolution of (a) translational and (b) rotational MSDs at ϕ = 0.8 for three aspect ratios. The system has a plastic phase for k = 1.1, an isotropic phase for k = 2, and a nematic phase for k = 6.

(a) Translational diffusion constant D T and (b) rotational diffusion constant D θ as a function of area fraction ϕ for various k.

(a) Translational diffusion constant D T and (b) rotational diffusion constant D θ as a function of area fraction ϕ for various k.

Rotational diffusion constant D θ as a function of translational diffusion constant D T for various k. The green dashed line indicates D θ = D T .

Rotational diffusion constant D θ as a function of translational diffusion constant D T for various k. The green dashed line indicates D θ = D T .

Isodiffusivity lines in the plane of aspect ratio k and area fraction ϕ. The solid lines are isodiffusivity lines from translational diffusion coefficients D T and the dashed lines are isodiffusivity lines from rotational diffusion coefficients D θ. The green squares and the red circles indicate the locations of the isotropic-plastic transition and the isotropic-nematic transition, respectively.

Isodiffusivity lines in the plane of aspect ratio k and area fraction ϕ. The solid lines are isodiffusivity lines from translational diffusion coefficients D T and the dashed lines are isodiffusivity lines from rotational diffusion coefficients D θ. The green squares and the red circles indicate the locations of the isotropic-plastic transition and the isotropic-nematic transition, respectively.

For hard ellipses with k = 2 and 4, effect of the system size N on (a) reduced pressure, (b) nematic order parameter, (c) translational diffusion constant, and (d) rotational diffusion constant. The results for k = 4 in (a) are shifted up by 5 for clarity, and the error bars in (b) correspond to the standard deviation over four independent samples. The system shows an isotropic phase in the whole density range in the case of k = 2, while the I-N transition occurs at ϕ ≈ 0.7 for k = 4.

For hard ellipses with k = 2 and 4, effect of the system size N on (a) reduced pressure, (b) nematic order parameter, (c) translational diffusion constant, and (d) rotational diffusion constant. The results for k = 4 in (a) are shifted up by 5 for clarity, and the error bars in (b) correspond to the standard deviation over four independent samples. The system shows an isotropic phase in the whole density range in the case of k = 2, while the I-N transition occurs at ϕ ≈ 0.7 for k = 4.

Effect of the system size N on the angular correlation function g 2(r) for k = 4 at ϕ = 0.7 and 0.8.

Effect of the system size N on the angular correlation function g 2(r) for k = 4 at ϕ = 0.7 and 0.8.

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