## Abstract

The Onsager theory for the liquid crystal phase transition of a gas of hard rods is known to be accurate if the rods are long enough. In order to quantify the necessary length, the first correction term (the third virial coefficient) to the Onsager theory is estimated numerically. On the basis of a study of the behavior of this function (for the cases L/D equals 10, 20, 40, and 100), a model function which approximates its angular dependence is proposed. This is used to estimate the corrections to the predictions of the Onsager theory arising from the finite length of the rods, in both isotropic and ordered phases. It is concluded that the Onsager approximation is not quantitatively accurate for L/D less than 100.

Original language | English |
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Pages (from-to) | 7-20 |

Number of pages | 14 |

Journal | Molecular Crystals and Liquid Crystals |

Volume | 24 |

Issue number | 1-2 |

DOIs | |

State | Published - 1973 |

### Bibliographical note

Funding Information:The gas of long hard rods exhibits a phase transition whereby at sufficiently large density a spontaneous alignment of the rods occurs.(') The phenomenon has been discussed as a model for liquid crystals,(2-6) of particular relevance to lyotropic systems such as Tobacco Mosaic Virus or poly ( y-benzyl-L-glutamate). Onsager(') has given a discussion of the theory of this system, wherein it was shown that in the limit that the ratio of length L to diameter D is large the ordering can be accurately discussed by a form of molecular field theory. Although several alternative treatments of the rod gas have been their predictions reduce to that of the Onsager theory in the large LID limit,@)a nd none has given a discussion of the equation, '' How long must the rods be for the theory to have quantitative validity? " The present work attempts to resolve this t Presented at the Fourth International Liquid Crystal Conference, Kent State University, August 21-25, 1972. 1 Supported by the National Science Foundation. 3 Current address: PhysicsDept., U.of Kentucky, Lexington,Ky. 40506USA 7

### Funding

The gas of long hard rods exhibits a phase transition whereby at sufficiently large density a spontaneous alignment of the rods occurs.(') The phenomenon has been discussed as a model for liquid crystals,(2-6) of particular relevance to lyotropic systems such as Tobacco Mosaic Virus or poly ( y-benzyl-L-glutamate). Onsager(') has given a discussion of the theory of this system, wherein it was shown that in the limit that the ratio of length L to diameter D is large the ordering can be accurately discussed by a form of molecular field theory. Although several alternative treatments of the rod gas have been their predictions reduce to that of the Onsager theory in the large LID limit,@)a nd none has given a discussion of the equation, '' How long must the rods be for the theory to have quantitative validity? " The present work attempts to resolve this t Presented at the Fourth International Liquid Crystal Conference, Kent State University, August 21-25, 1972. 1 Supported by the National Science Foundation. 3 Current address: PhysicsDept., U.of Kentucky, Lexington,Ky. 40506USA 7

Funders | Funder number |
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National Science Foundation (NSF) |

## ASJC Scopus subject areas

- General Engineering