It is shown that linearized elasticity theory fails for nematic polymers in less than four dimensions. Instead, the polymer osmotic elastic modulus E and the elastic moduli ${\mathit{K}}_2$ and ${\mathit{K}}_3$ all become singular functions of wave vector q as ‖q‖→0, with E vanishing like ${\mathit{q}}^{\mathrm{\eta }\mathrm{\perp }}$ and ${\mathit{K}}_{2,3}$ diverging like ${\mathit{q}}_{2,3}^{\mathrm{-}\mathrm{\eta }}$. These exponents satisfy an exact scaling relation ${\mathrm{\eta }}_{\mathrm{\perp }}$+${\mathrm{\eta }}_2$+${\mathrm{\eta }}_3$=1 in three dimensions, and are calculated to second order in ɛ=4-d, yielding ${\mathrm{\eta }}_{\mathrm{\perp }}$=0.46±0.015, ${\mathrm{\eta }}_2$=0.28±0.015, and ${\mathrm{\eta }}_3$=0.21±0.015 in d=3.

• Received 9 October 1991

DOI:https://doi.org/10.1103/PhysRevLett.68.1331