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CU-Boulder PHYS 7450 - Depletion interactions in lyotropic nematics

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Depletion interactions in lyotropic nematicsPaul van der Schoota)Theoretical Polymer Physics Group, Department of Applied Physics, Eindhoven University of Technology,P.O. Box 513, 5600 MB Eindhoven, The Netherlands共Received 4 January 2000; accepted 1 March 2000兲A theoretical study of depletion interactions between pairs of small, globular colloids dispersed ina lyotropic nematic of hard, rodlike particles is presented. We find that both the strength and rangeof the interaction crucially depends on the configuration of the spheres relative to the nematicdirector, and on their size. The interaction is significantly stronger parallel than perpendicular to thedirector, explaining the emergence of stringlike aggregates observed in experiment. © 2000American Institute of Physics. 关S0021-9606共00兲51220-3兴I. INTRODUCTIONColloidal particles dispersed in uniaxial nematic mediaexhibit unusual interactions, brought about by the couplingto the surrounding anisotropic fluid. See, e.g., a recent re-view article by Poulin,1and work cited therein. At least threeimportant sources of effective interaction have been identi-fied between colloids in a nematic matrix, provided these aremacroscopically large on the scale of the nematogens.2Whenthe anchoring of the nematic to the surfaces of the particles isweak, an initially homogeneous director field will remainlargely unaffected by the presence of the foreign particles.What is likely to be perturbed, however, is the ordering ofthe nematogens in the vicinity of the colloids, as well as thespontaneous fluctuations of the director field between them.A freezing out of director fluctuation modes tends to giverise to a long-range, Casimir-type of attraction between thedispersed colloidal particles.3Inhomogeneities in the degreeof nematic order, including the possible induction of pre-smectic ordering,4also induce an effective interaction, actingwhen the zones of perturbed nematic order around two ap-proaching particles overlap.5,6A third type of medium-induced interaction arises when the anchoring conditions areso strong that they cause the director field to deform,7–9asituation which seems to predominate. The deformation ofthe director field allows the surface anchoring energy of thecolloids to be minimized, albeit at the cost of a concomitantgeneration of defects in the surrounding nematic fluid.10,11The stored elastic energy associated with the deformed di-rector field is thought to be the main driving force behind thehighly anisotropic effective colloid–colloid interactions ob-served in this kind of system.1It appears that the three medium-induced interactions de-scribed conspire to make it generally quite difficult to pro-duce thermodynamically stable colloidal dispersions in nem-atic liquids at weight fractions in excess of, say, a fewpercent.12It has to be borne in mind, however, that in lyo-tropic nematics one may encounter situations where the dis-persed colloidal particles are substantially smaller than theparticles that make up the nematic matrix. When this hap-pens to be the case, the foreign colloidal particles cannotdistort the director field, nor influence the director fluctua-tions, or the degree of nematic order. The reason is that thesequantities are only defined on a macroscopic scale, i.e., on alength scale large compared to the dimensions of thenematogens.13One would, therefore, perhaps naively expectthat adding small colloidal particles to a lyotropic nematicshould not be able to perturb this nematic in any significantway. Yet, recent experiments indicate that relatively smallamounts of submicron-sized colloidal particles do interferestrongly with a host nematic phase consisting of rodlike vi-ruses in water, leading to all kinds of novel, micro-phaseseparated phases.14,15Fraden et al. attributed the observed phase behavior inthis type of dispersion to so-called depletion interactions be-tween the spheres, which come into play when zones aroundthe spherical particles, depleted of rods due to a hard-coreexclusion interaction, interpenetrate.14The osmotic pressuredifference between the overlapping rod-free depletion vol-umes and the bulk nematic drive the particles together,thereby inducing an attractive interaction between them. Aspointed out by Fraden et al.14and by others,16since the over-lap volume of two approaching test spheres depends on theirrelative orientation to the nematic director, the depletion in-teraction arising in a lyotropic nematic must be highly aniso-tropic. Although available theoretical work on rod-spheremixtures rationalizes at least some the observed behavior,many aspects remain poorly understood.14,16–19In an effort to contribute to the understanding of aniso-tropic colloidal mixtures, we investigate theoretically thestrength, orientation dependence, and range of the depletioninteraction between colloidal spheres in a hard-rod nematic.It is shown that even when the spheres are small on the linearscale of rods, the depletion interaction can be very strongindeed. Depending on the size of the globules, the interactionstrength at contact can easily reach values of several tens ofkBT. This explains the sensitivity of the nematic to smallamounts of added colloidal particles, but also suggests thatsome of the observed behavior may well be kinetically de-termined. We find that the depletion attraction parallel to thenematic director becomes stronger with increasing concen-tration of rods, while that perpendicular to the director is nota兲Electronic mail: [email protected] OF CHEMICAL PHYSICS VOLUME 112, NUMBER 20 22 MAY 200091320021-9606/2000/112(20)/9132/7/$17.00 © 2000 American Institute of PhysicsDownloaded 22 Sep 2004 to 192.38.100.242. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsponly weaker at equal concentrations, but in addition reducesin strength the more concentrated the nematic. Perhaps themost important insight obtained is that both the range and thestrength of the depletion interaction depend crucially on thesize of the spheres relative to the lateral correlation length ofthe nematic medium.This paper is organized as follows. In Sec. II, we firstbriefly review the basics of the hard-rod nematic, guiding usto a simplified physical picture that emerges when the radiiof the colloids are large on the scale of the lateral correlationlength of the nematic, but small compared to the length ofthe rods. It allows us to calculate, in


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