Many lyotropic liquid crystals are composed of mesogens that display aconsiderable spread in size or shape affecting their material properties andthermodynamics via various demixing and multi-phase coexistence scenarios.Starting from a generalized Onsager theory we formulate a generic frameworkthat enables locating spinodal polydispersities as well as identifying thenature of incipient size fractionation for arbitrary model potentials and sizedistributions. We apply our theory to nematic phases of both hard rods anddisks whose main particle dimension is described by a unimodal log-normaldistribution. We find that both rod-based and discotic nematics become unstableat a critical polydispersity of about 20 %. We also investigate the effect ofdoping nematic assemblies with a small fraction of large species and highlighttheir effect on the stability of the uniform nematic fluid. We find that whilerod-based are only weakly affected by the presence of large species, dopingdiscotic nematics with very large platelets leads to a remarkable suppressionof the spinodal instabilities. This could open up routes towards controllingthe mechanical properties of nematic materials by manipulating the localstability of nematic fluid and its tendency to undergo fractionation-drivenmicrophase separation.
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