= Onsager excluded-volume mechanism
{c}
{title2=$v_{\rm excl}\simeq2L^2D|\sin\gamma|$}
For slender hard rods of length $L$ and diameter $D\ll L$, the leading pair <excluded volume> is the displayed function of their mutual angle $\gamma$. Alignment lowers mutual obstruction and increases positional <entropy>, while narrowing the orientation distribution lowers orientational <entropy>. Their competition can drive an isotropic-to-nematic transition without attractive forces. The natural concentration scale is $cL^2D$ of order one. Finite rod diameter, flexibility and interactions beyond the hard-core second-virial approximation change quantitative predictions.
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