Pond Street, Sheffield S1 1WB, United Kingdom
Abstract
A simple model is proposed for the liquid crystal matrix surrounding ‘soft’ colloidal particles whose separation is much smaller than their radii. We use our implementation of the Onsager approximation of density-functional theory [A. Chrzanowska, P. I. C. Teixeira, H. Ehrentraut and D. J. Cleaver, J. Phys.: Condens. Matter 13, 4715 (2001)] to calculate the structure of a nanometrically thin film of hard Gaussian overlap particles of elongations κ = 3 and κ = 5, confined between two solid walls. The penetrability of either substrate can be tuned independently to yield symmetric or hybrid alignment. Comparison with Monte Carlo simulations of the same system [D. J. Cleaver and P. I. C. Teixeira, Chem. Phys. Lett. 338, 1 (2001); F. Barmes and D. J. Cleaver (unpublished)] reveals good agreement in the symmetric case.
Symmetric alignment of the nematic matrix between close penetrable colloidal particles (232.3 KiB)
Introduction
A nematic colloid, sometimes called inverted nematic emulsion, is a dispersion of isotropic particles, either rigid (grains) or deformable (e.g., water droplets), in a nematic liquid crystal (LC) [1]. Because the nematic director is anchored on the surface of the (nearly always quasi-spherical) inclusions, distortions of the director field are induced which in turn give rise to effective interactions between the inclusions themselves. These can be manipulated by treating the walls of the container, applying fields, or coating the surfaces of the colloidal particles, in order to produce various states of aggregation, e.g., strings along the director lines [2]. Nematic colloids are therefore ideal model systems to study topological defects, and one important practical application appears to be the suspension of abrasive particles in lyotropic mesophases. In a newer variant, inclusions of size exceeding the cholesteric pitch (radius ~ 1 μm) are dispersed in a well-aligned cholesteric sample. Here the colloidal particles stabilise a network of defects by residing at its nodes, thereby transforming the cholesteric liquid into a new type of material exhibiting gel-like rheological properties [3]. This study was later extended to the simpler system consisting of smaller (radii 150 – 200 nm) particles in a nematic host: upon quenching the initial homogeneously-mixed colloid from the isotropic (I) to the nematic (N) state, the particles were seen to phase-separate and aggregate into thin walls, bounding domains of practically pure nematic LC [4]. The resulting metastable, but relatively long-lived, cellular structure exhibited dramatically enhanced mechanical strength, with an elastic modulus G′ ≥ 105 Pa and a well-defined yield stress, which are functions of particle concentration.
In all the above, the colloidal particles are much larger than the LC molecules, which can then be regarded as a continuum background. Effective interaction potentials between colloidal particles have been derived, analytically in some limits, more generally numerically, using either Frank theory (below the I–N transition) [2, 5] or Landau-de Gennes (LdG) theory (above the I–N transition) [6]; more recent work considered Van der Waals, steric, elctrostatic and LC-mediated contributions [7]. However, continuum-based approaches are expected to fail when the interparticle separation becomes of the order of a few LC molecular sizes, as in fully-formed aggregates. Furthermore, especially in the case of LdG theory, the solution method is heavy, and affords little insight into the physics of the problem. Finally, evaluation of the three- and more-particle contributions (these effective potentials being in general not pairwise additive) becomes prohibitively complicated [6]: one would like to be able to obtain the different terms as functions of the microscopic potential parameters, and in a systematic, well-controlled manner.
The complexity of the task at hand requires that we start our attack from the very beginning: take the simplest molecular model of a LC and squeeze it to nanometric thickness; then find the free energy dependence on particle separation, and hence the effective interaction. Here we carry out only the first part of this programme, leaving the second for future work. In a previous paper [8] we showed how the simple Onsager approximation of density-functional theory could provide a semi-quantitatively accurate description of the structure of a fluid of hard rods confined between two hard, impenetrable walls, provided allowance was made, in a phenomenological way, for the incorrect prediction of the location of the isotropic–nematic (I–N) transition. In the present paper we apply the same strategy to symmetric films confined between flat substrates of variable penetrability, in order to mimic different anchoring conditions. This is not unreasonable as a first approach, in view of the large disparity of the typical lengthscales of LC molecules – a few nm – and inclusions – hundereds of nm (but see [9] for an attempt to consider curved hard surfaces).
This paper is organised as follows: in section II we recapitulate the theory of [8] and extend it to the case of unequal anchorings at the confining walls. Then in section III we present results for the density and order parameter profiles of LC films subject to symmetrical anchoring conditions, and compare them with those obtained by Monte Carlo (MC) simulation [10, 11]. Finally in section IV we discuss the potential and limitations of our approach, and outline some directions for future research.
Conclusions
In this paper we have presented a density-functional treatment of a HGO fluid confined between parallel walls of tunable penetrability. Despite its simplicity, the Onsager approximation can in some cases yield semi-quantitative results for the density and orientational distribution of particles of elongation as small as κ = 5 (but not κ = 3). This simple model for the structure of the nematic matrix squeezed between tight-packed colloidal particles captures effects missed by the more current Frank and LdG theories, namely to do with layering. Moreover, the solution procedure also yields the free energy, thus making it possible to derive the effective interaction between walls/particles. This will be the subject of future work.
When comparing theory and simulation, account has to be taken of the fact that they yield rather different I–N transition densities and widths. This is due to our neglect of correlations of order higher than second-virial, which are relevant in the range of densities of interest. Greater predictive power would require a far more sophisticated approach, such as a weighted-density [25] or fundamental-measure [26] approximation. The development and implementation of such a scheme are, however, highly non-trivial. In keeping with our aim of assessing the validity and usefulness of the Onsager approach, we instead adapted a phenomenological scaling of the density first proposed by Allen and collaborators [24]. Agreement for symmetric films is fairly good, in spite of the smallness of κ, but its quality is strongly dependent on the accuracy of the isotropic and nematic coexistence densities as determined independently by either theory or simulation. We have not addressed the fact that these are (sometimes dramatically) shifted from their bulk values by both confinement and wall penetrability [11].
The theory can also be applied to hybrid films. It would be particularly interesting to see whether it is able to describe: (i) the more common uniform and bent-director structures already predicted [29] and observed [30]; (ii) the discontinuous transition between these two structures found by ourselves [10]; and (iii) the more exotic biaxial structure in which two strata of film, each with a uniform director orientation dictated by the nearest wall, are separated by a sharp interface [31]. Both (ii) and (iii) depend crucially on the anchoring strengths at the two substrates being large and dissimilar or large and similar, respectively. However, preliminary calculations suggest that our chosen mechanism of making the walls partially penetrable to particles produces much stronger homeotropic than parallel anchoring. This would need to be checked by a direct calculation of the anchoring energy, similar to that performed in [22]. A more ambitious aim would be to be able to shed some light on the observation by Vandenbrouck et al. [32] of spinodal dewetting of the nematogen 5CB spun-cast onto silicon wafers, where hybrid anchoring was enforced by conflicting boundary conditions: orthogonal at the free surface, and planar at the silicon substrate. Such behaviour was initially interpreted in terms of a competition between elasticity and van der Waals forces [32], but subsequent arguments have related it to the fluctuation-induced interactions that underlie the pseudo-Casimir effect [33].
The present theory can be straightforwardly generalised to more sophisticated surface interactions, and also to mixtures of two or more types of hard body. One can envisage a very rich behaviour of a confined binary mixture where the two components have different easy axes at either substrate.
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