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Depletion effects in smectic phases of hard rod--hard sphere mixtures

It is known that when hard spheres are added to a pure system of hard rods the stability of the smectic phase may be greatly enhanced, and that this effect can be rationalised in terms of depletion forces. In the present paper we first study the effect of orientational order on depletion forces in this particular binary system, comparing our results with those obtained adopting the usual approximation of considering the rods parallel and their orientations frozen. We consider mixtures with rods of different aspect ratios and spheres of different diameters, and we treat them within Onsager theory. Our results indicate that depletion effects, and consequently smectic stability, decrease significantly as a result of orientational disorder in the smectic phase when compared with corresponding data based on the frozen--orientation approximation. These results are discussed in terms of the $τ$ parameter, which has been proposed as a convenient measure of depletion strength. We present closed expressions for $τ$, and show that it is intimately connected with the depletion potential. We then analyse the effect of particle geometry by comparing results pertaining to systems of parallel rods of different shapes (spherocylinders, cylinders and parallelepipeds). We finally provide results based on the Zwanzig approximation of a Fundamental--Measure density--functional theory applied to mixtures of parallelepipeds and cubes of different sizes. In this case, we show that the $τ$ parameter exhibits a linear asymptotic behaviour in the limit of large values of the hard--rod aspect ratio, in conformity with Onsager theory, as well as in the limit of large values of the ratio of rod breadth to cube side length, $d$, in contrast to Onsager approximation, which predicts $τ\sim d^3$.