^{1,a)}, Juho S. Lintuvuori

^{2,b)}, Mark R. Wilson

^{2}and Claudio Zannoni

^{1}

### Abstract

Classical molecular dynamics simulations have been used to explore the phase diagrams for a family of attractive–repulsive soft–core Gay–Berne models [R. Berardi, C. Zannoni, J. S. Lintuvuori, and M. R. Wilson, J. Chem. Phys.131, 174107 (2009)] and determine the effect of particle softness, i.e., of a moderately repulsive short–range interaction, on the order parameters and phase behaviour of model systems of uniaxial and biaxial ellipsoidal particles. We have found that isotropic, uniaxial, and biaxial nematic and smectic phases are obtained for the model. Extensive calculations of the nematic region of the phase diagram show that endowing mesogenic particles with such soft repulsive interactions affect the stability range of the nematic phases, and in the case of phase biaxiality it also shifts it to lower temperatures. For colloidal particles, stabilised by surface functionalisation, (e.g., with polymer chains), we suggest that it should be possible to tune liquid crystal behaviour to increase the range of stability of uniaxial and biaxial phases (by varying solvent quality). We calculate second virial coefficients and show that they are a useful means of characterising the change in effective softness for such systems. For thermotropic liquid crystals, the introduction of softness in the interactions between mesogens with overall biaxial shape (e.g., through appropriate conformational flexibility) could provide a pathway for the actual chemical synthesis of stable room–temperature biaxial nematics.

R.B. and C.Z. thank the EU–STREP project “Biaxial Nematic Devices” (BIND) FP7–216025 for financial support, and the CINECA computing centre for computer time. M.R.W. wishes to thank the CINECA computing centre and the HPC Europa2 initiative for financial support for a visit helping to make this work possible.

I. INTRODUCTION

II. SOFT–CORE MODEL AND COMPUTER SIMULATIONS

III. SIMULATION RESULTS

A. Uniaxial particles

B. Biaxial particles

IV. CONCLUSIONS

### Key Topics

- Liquid crystals
- 18.0
- Anisotropy
- 13.0
- Colloidal systems
- 13.0
- Molecular dynamics
- 9.0
- Nematic liquid crystals
- 9.0

## Figures

Representative profiles for the uniaxial GBSC potential with σ_{ x } = σ_{ y } = σ_{ c } = 1 σ_{0}, σ_{ z } = 3 σ_{0}, and ε_{ x } = ε_{ y } = 1 ε_{0}, ε_{ z } = 0.2 ε_{0}, μ = 1, ν = 3, and slopes , −80, −60, −40, −30, and −20. (See text for additional details.) The steepness for the logistic seaming function was , and a threshold cut–off value ‖1 − *f*‖ = 10^{−6} was used (see Ref. 29). The energy curves are relative to the *side–by–side* interaction of two parallel ellipsoids. A plot for the standard GB *side–by–side* interaction energy is also provided for comparison.

Representative profiles for the uniaxial GBSC potential with σ_{ x } = σ_{ y } = σ_{ c } = 1 σ_{0}, σ_{ z } = 3 σ_{0}, and ε_{ x } = ε_{ y } = 1 ε_{0}, ε_{ z } = 0.2 ε_{0}, μ = 1, ν = 3, and slopes , −80, −60, −40, −30, and −20. (See text for additional details.) The steepness for the logistic seaming function was , and a threshold cut–off value ‖1 − *f*‖ = 10^{−6} was used (see Ref. 29). The energy curves are relative to the *side–by–side* interaction of two parallel ellipsoids. A plot for the standard GB *side–by–side* interaction energy is also provided for comparison.

The average dimensionless potential energy ⟨*U*⟩/ε_{0} (plate A), pressure (plate B), compressibility factor ⟨*P*⟩/(ρ*k* _{B} *T*) (plate C), and the second virial coefficient *B* _{2} (plate D) plotted against the reduced temperature *T*/*T* _{IN} for uniaxial ellipsoids modelled either with the standard GB potential (see Ref. 34) or the GBSC parametrisation described in the text. The soft–core energy slopes were , −80, −60, −40, and −30, while the logistic function steepness was . The results from the *NVT* MD simulation of the standard uniaxial GB model (see Ref. 34) are also plotted. The rms errors computed, as described in the text, from a block average analysis of the simulation results (see supplementary material of Ref. 46) are also plotted as error bars, however, their size smaller than that of the symbols makes them hardly visible. For every SC parametrisation studied *T* _{IN} is the specific nematic–isotropic transition temperature, respectively, , 5.05, 4.55, 3.85, and 3.45. The thick grey lines join the I–N and N–S_{m} transition temperatures for the GBSC models. The standard GB model has instead . The state points have been computed from MD simulations in the *NVT* ensemble for *N* = 1024 particles samples at dimensionless density .

The average dimensionless potential energy ⟨*U*⟩/ε_{0} (plate A), pressure (plate B), compressibility factor ⟨*P*⟩/(ρ*k* _{B} *T*) (plate C), and the second virial coefficient *B* _{2} (plate D) plotted against the reduced temperature *T*/*T* _{IN} for uniaxial ellipsoids modelled either with the standard GB potential (see Ref. 34) or the GBSC parametrisation described in the text. The soft–core energy slopes were , −80, −60, −40, and −30, while the logistic function steepness was . The results from the *NVT* MD simulation of the standard uniaxial GB model (see Ref. 34) are also plotted. The rms errors computed, as described in the text, from a block average analysis of the simulation results (see supplementary material of Ref. 46) are also plotted as error bars, however, their size smaller than that of the symbols makes them hardly visible. For every SC parametrisation studied *T* _{IN} is the specific nematic–isotropic transition temperature, respectively, , 5.05, 4.55, 3.85, and 3.45. The thick grey lines join the I–N and N–S_{m} transition temperatures for the GBSC models. The standard GB model has instead . The state points have been computed from MD simulations in the *NVT* ensemble for *N* = 1024 particles samples at dimensionless density .

The orientational order parameter plotted against dimensionless temperature (plate A), and reduced temperature *T*/*T* _{IN} (plate B), for the GBSC uniaxial ellipsoids of various softness , −80, −60, −40, −30, and −20 as described in the text. (See the legend of Figure 2 for additional details.)

The orientational order parameter plotted against dimensionless temperature (plate A), and reduced temperature *T*/*T* _{IN} (plate B), for the GBSC uniaxial ellipsoids of various softness , −80, −60, −40, −30, and −20 as described in the text. (See the legend of Figure 2 for additional details.)

The radial correlation function *g*(*r*) for nematic phases with similar orientational order parameter formed by standard and GBSC uniaxial ellipsoids of various softness , −80, −60, −40, −30, and −20 at dimensionless temperatures , 3.5, 3.3, 3.1, 2.8, 2.6, and 2.3 as described in the text. The radial distribution functions parallel (*g*(*r* _{∥})), and perpendicular (*g*(*r* _{⊥})) to the director are shown in the insets. (See the legend of Figure 2 for additional details.)

The radial correlation function *g*(*r*) for nematic phases with similar orientational order parameter formed by standard and GBSC uniaxial ellipsoids of various softness , −80, −60, −40, −30, and −20 at dimensionless temperatures , 3.5, 3.3, 3.1, 2.8, 2.6, and 2.3 as described in the text. The radial distribution functions parallel (*g*(*r* _{∥})), and perpendicular (*g*(*r* _{⊥})) to the director are shown in the insets. (See the legend of Figure 2 for additional details.)

The phase diagrams plotted against dimensionless temperature for the uniaxial GBSC ellipsoids of various softness , −80, −60, −40, −30, and −20 as described in the text. Samples in the isotropic (I), nematic (N), and smectic (S_{m}) phases are represented as crosses, circles, and triangles. The I–N and N–S_{m} transition temperatures for the standard uniaxial GB model (see Ref. 34) are plotted as vertical grey bars. (See the legend of Figure 2 for additional details.)

The phase diagrams plotted against dimensionless temperature for the uniaxial GBSC ellipsoids of various softness , −80, −60, −40, −30, and −20 as described in the text. Samples in the isotropic (I), nematic (N), and smectic (S_{m}) phases are represented as crosses, circles, and triangles. The I–N and N–S_{m} transition temperatures for the standard uniaxial GB model (see Ref. 34) are plotted as vertical grey bars. (See the legend of Figure 2 for additional details.)

Snapshot of an N sample for the uniaxial GBSC ellipsoids of softness at , and .

Snapshot of an N sample for the uniaxial GBSC ellipsoids of softness at , and .

The average dimensionless potential energy ⟨*U*⟩/ε_{0} (plate A), pressure (plate B), compressibility factor ⟨*P*⟩/(ρ*k* _{B} *T*) (plate C), and second virial coefficient *B* _{2} (plate D) plotted as a functions of reduced temperature *T*/*T* _{IN} for biaxial ellipsoids modelled either with the standard GB potential (see Ref. 35) or the GBSC parametrisation described in the text. The soft–core energy slopes were , −80, −60, −40, and −30, while the logistic function steepness was . The biaxial system with does not form a nematic phase upon cooling. The points from the *NVT* MD simulation of the standard biaxial GB model (see Ref. 35) are also plotted. The rms errors computed, as described in the text, from a block average analysis of the simulation results (see supplementary material of Ref. 46) are also plotted as error bars, however their size smaller than that of the symbols makes them hardly visible. For every SC parametrisation studied *T* _{IN} is the specific nematic–isotropic transition temperature, respectively , 5.25, 4.45, 3.65, and 3.15. The thick grey lines join the I–N, N–N_{b}, and N_{b}–S_{b} transition temperatures for the GBSC models. The standard biaxial GB model has instead . The state points have been computed from MD simulations in the *NVT* ensemble for *N* = 1024 particles samples at dimensionless number density .

The average dimensionless potential energy ⟨*U*⟩/ε_{0} (plate A), pressure (plate B), compressibility factor ⟨*P*⟩/(ρ*k* _{B} *T*) (plate C), and second virial coefficient *B* _{2} (plate D) plotted as a functions of reduced temperature *T*/*T* _{IN} for biaxial ellipsoids modelled either with the standard GB potential (see Ref. 35) or the GBSC parametrisation described in the text. The soft–core energy slopes were , −80, −60, −40, and −30, while the logistic function steepness was . The biaxial system with does not form a nematic phase upon cooling. The points from the *NVT* MD simulation of the standard biaxial GB model (see Ref. 35) are also plotted. The rms errors computed, as described in the text, from a block average analysis of the simulation results (see supplementary material of Ref. 46) are also plotted as error bars, however their size smaller than that of the symbols makes them hardly visible. For every SC parametrisation studied *T* _{IN} is the specific nematic–isotropic transition temperature, respectively , 5.25, 4.45, 3.65, and 3.15. The thick grey lines join the I–N, N–N_{b}, and N_{b}–S_{b} transition temperatures for the GBSC models. The standard biaxial GB model has instead . The state points have been computed from MD simulations in the *NVT* ensemble for *N* = 1024 particles samples at dimensionless number density .

The orientational order parameters and plotted against dimensionless temperature (plate A, uniaxial; plate C, biaxial), and reduced temperature *T*/*T* _{IN} (plate B, uniaxial; plate D, biaxial), for the biaxial GBSC ellipsoids of various softness , −80, −60, −40, and −30 as described in the text. (See the legend of Figure 7 for additional details.)

The orientational order parameters and plotted against dimensionless temperature (plate A, uniaxial; plate C, biaxial), and reduced temperature *T*/*T* _{IN} (plate B, uniaxial; plate D, biaxial), for the biaxial GBSC ellipsoids of various softness , −80, −60, −40, and −30 as described in the text. (See the legend of Figure 7 for additional details.)

The phase diagrams plotted against dimensionless temperature for the biaxial GBSC ellipsoids of various softness , −80, −60, −40, and −30 as described in the text, and in the legend of Figure 7. Samples in the isotropic (I), nematic (N), biaxial nematic (N_{b}), and biaxial smectic (S_{b}) phases are represented as crosses, circles, squares, and triangles. The I–N, N–N_{b}, and N_{b}–S_{b} transition temperatures for the standard biaxial GB model (see Ref. 35) are plotted as vertical grey bars.

The phase diagrams plotted against dimensionless temperature for the biaxial GBSC ellipsoids of various softness , −80, −60, −40, and −30 as described in the text, and in the legend of Figure 7. Samples in the isotropic (I), nematic (N), biaxial nematic (N_{b}), and biaxial smectic (S_{b}) phases are represented as crosses, circles, squares, and triangles. The I–N, N–N_{b}, and N_{b}–S_{b} transition temperatures for the standard biaxial GB model (see Ref. 35) are plotted as vertical grey bars.

Snapshot of a N_{b} sample for the biaxial GBSC ellipsoids of softness at , , and .

Snapshot of a N_{b} sample for the biaxial GBSC ellipsoids of softness at , , and .

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