Onsager excluded-volume mechanism ($v_{\mathrm{excl}}\simeq 2L^{2}D|\sin \gamma |$)

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 $c{L}^{2}D$ of order one. Finite rod diameter, flexibility and interactions beyond the hard-core second-virial approximation change quantitative predictions.