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Polydispersity is believed to have important effects on the formation ofliquid crystal phases in suspensions of rod-like particles. To understand sucheffects, we analyse the phase behaviour of thin hard rods with lengthpolydispersity. Our treatment is based on a simplified Onsager theory, obtainedby truncating the series expansion of the angular dependence of the excludedvolume. We describe the model and give the full phase equilibrium equations;these are then solved numerically using the moment free energy method whichreduces the problem from one with an infinite number of conserved densities toone with a finite number of effective densities that are moments of the fulldensity distribution. The method yields exactly the onset of nematic ordering.Beyond this, results are approximate but we show that they can be madeessentially arbitrarily precise by adding adaptively chosen extra moments,while still avoiding the numerical complications of a direct solution of thefull phase equilibrium conditions.  We investigate in detail the phase behaviour of systems with three differentlength distributions: a (unimodal) Schulz distribution, a bidispersedistribution and a bimodal mixture of two Schulz distributions whichinterpolates between these two cases. A three-phase isotropic-nematic-nematiccoexistence region is shown to exist for the bimodal and bidisperse lengthdistributions if the ratio of long and short rod lengths is sufficiently large,but not for the unimodal one. We systematically explore the topology of thephase diagram as a function of the width of the length distribution and of therod length ratio in the bidisperse and bimodal cases.Comment: 18 pages, 16 figure

Simplified Onsager theory for isotropic–nematic phase equilibria of length polydisperse hard rods