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Sedimentation equilibria of colloidal hard rod dispersions Rosalind Allen, David Goulding and Jean-Pierre Hansen Department of Chemistry, University of Cambridge, Lensfield Road, Cambridge, UK CB2 1EW Received 22nd June 1999, Accepted 14th July 1999, Published 19th July 1999 The concentration profiles of suspensions of cylindrical rods in a gravitational field are calculated within Onsager theory and its Parsons-Lee extension. The calculated profiles are very sensitive to the aspect ratio, for a given volume of the rods. In dispersions of mesoscopic colloidal particles, the gravitational length, defined in terms of the thermal energy kBT, the buoyant mass m and the acceleration of gravity g, by l = kBT/mg is comparable to the characteristic size of the particles. Once sedimentation equilibrium has been reached, the inhomogeneous dispersion is characterised by concentration profiles that vary significantly with altitude.For dispersions of interacting spherical particles, such concentration profiles have been calculated from density functional theory (DFT),1 and measured by light scattering techniques.2 Except at high altitudes, where the dispersion becomes sufficiently dilute for particles to behave like an ideal solution, the calculated and measured concentration profiles are highly non-exponentional, and may be "inverted’’ to yield the osmotic equation of state.3 Interesting sedimentation profiles have been predicted for more complex colloidal systems, like bidisperse suspensions,4 or dispersions of charge-stabilised colloids.5 In this letter we consider the case of dispersions of elongated, rod-shaped particles, like the tobacco mosaic virus (TMV), or boehmite crystals, coated with polyisobutene.Such mesoscopic rods have typical lengthto-diameter (aspect) ratios of the order of 10-20, and are observed to form lyotropic liquid crystal phases. In particular it is known, since Onsager’s classic work,6 that long cylindrical rods will undergo a first order phase transition from an isotropic to a nematic phase, where rods are aligned, on average, along a director, at sufficiently high concentration. The rods considered here are cylinders of length L, diameter D and buoyant mass m; the volume of a rod will be denoted by u0 = pD2L/4. In the gravitational field, the rods are subjected to the external potential: (1) where z is the vertical coordinate.This field, and the anisotropic nature of the particle shape, induce a translational and orientational inhomogeneity characterised by a local density r(z, W), where W = ( q, f) determines the orientation of a rod with respect to a laboratory-fixed frame. The local number concentration of rods is: (2) and is normalized such that: where H is the height of the vessel containing the suspension, and N is the number of rods per unit area of the base. Any orientational order of the rods may be characterised by the local order parameter: where P2 denotes the second order Legendre polynomial. Our objective is to calculate r(z, W), and more specifically its moments n(z) and S(z).This is achieved within DFT, by considering the grand potential functional per unit area:7 where m is the fixed chemical potential of the rods, y(z) = m - f(z), and F is the intrinsic free energy functional, which is traditionally split into an ideal part, Fid, corresponding to non-interacting rods, and an excess part, Fex, which accounts for the excluded volume correlations. The excess part is conveniently cast in the form: where h is an unknown functional. The local density is the solution of the variational problem: It is convenient to rewrite r(z, W) as: In practice two approximations were made in order to arrive at a tractable form of Fex: PhysChemComm, 1999, 7 (3) (4) (5) (6) (7) (8) (9)(a) Assuming that the concentration profile n(z) varies slowly on the scale of the rod length L, a local density approximation (LDA) was adopted which has two aspects: the orientational distribution function f in eqn.(9) is assumed to depend on z only through the local concentration, i.e.: (10) Secondly, the unknown functional h in eqn.(7) is assumed to be local in z, so that: (11) (b) For this local h, Onsager’s second virial coefficient approximation is adopted neglecting end effects, which is justified for the aspect ratios considered in this paper; h then reduces to the function: (12) where g = g ( W, W) is the angle between the axes of two rods.Onsager’s approximation is known to be exact in the limit where the aspect ratio x = L/D ® ¥.Improvements over the local approximation (a) and the second virial approximation (b) will be considered later in the paper. Meanwhile the intrinsic free energy functional has been cast in the local Onsager form: (13) where the dependence of the orientational distribution function on the local concentration is implicitly understood. Due to the factorization [eqn. (9)] and subsequent LDA assumption, the variational principle [eqn. (8)] leads to the following two extremum conditions: (14a) (14b) where l is a z-dependent Lagrange multiplier ensuring the proper normalisation of f( W) at any altitude. The variational problem may be further simplified by adopting Onsager’s normalized trial distribution function: (15) which reduces eqn.(14b) to the simple condition: (16) The choice [eqn. (15)] allows the angular integrals in eqn.(13) to be carried out analytically. For any value of n, the variational problem [eqn. (16)] is now identical to that for a homogeneous system of rods, which yields an isotropic solution ( a = 0), as well as a nematic branch ( a � 0) beyond a threshold concentration. The root a is determined numerically by a Newton-Raphson method, and the resulting functions a (n) and d a/dn are used as input in the extremum condition [eqn. (14a)], which reduces to a complicated algebraic expression for n, at any given altitude z, and for a fixed chemical potential m. This is solved once more by a Newton-Raphson method, starting from the top (z = H).By varying z at fixed m, two concentration profiles are mapped out corresponding to the isotropic (i) and nematic (n) branches. At each altitude, the stable phase is that of the lowest local grand potential W(z) = f(z) - n(z) y(z), where f(z) denotes the integrand in eqn.(13) (i.e. the intrinsic free energy per unit volume). Coexistence is achieved at an altitude zc such that (17) c the lower density isotropic phase is stable, while c, the nematic phase prevails. The equilibrium For z < z below z concentration profile exhibits a discontinuity at z = zc; this discontinuous jump in concentration is a consequence of the LDA, and would turn into a rapid but continuous variation in concentration if non-local corrections to the free energy functional were added.For any given m, the number N of rods per unit area is finally determined a posteriori from condition (3). Onsager’s second virial theory is quantitatively accurate only for aspect ratios x > 100. For physically more realistic values of x, higher order terms in the virial expansion become significant.8 A semi-empirical way of accounting for higher order terms, put forward by Parsons9 and Lee,10 has proved surprisingly successful in describing the isotropic-nematic transition in a homogeneous system of rods.11 The Parsons-Lee rescaling procedure amounts to replacing the function h in eqn.(12) by: (18) where h(z) = n(z) u0 is the local packing fraction.The correction factor is the Carnahan-Starling12 expression for the excess part of the free energy for a hard sphere fluid of packing fraction identical to that of the rods, divided by the same hard sphere free energy in the second virial limit. The free energy minimisation procedure then carries through along the same lines as for the Onsager functional. Packing fraction profiles obtained with the Onsager and Parsons-Lee functionals are compared in Fig. 1 for several aspect ratios, keeping the volume u0 of the rods constant. The results differ considerably, except at the largest aspect ratio, as expected. The second virial approximation underestimates the excluded volume, so that the Onsager profiles rise tosharply at low altitudes. Note the great sensitivity of the profiles, and in particular of the location of the interface, to the aspect ratio.Fig. 2 shows the Onsager and Parsons-Lee concentration and order parameter profiles [eqn. (4)], for three different overall concentrations N, under conditions appropriate for a suspension of TMV virus in water. Note the almost linear variation of the Parsons-Lee concentration profiles in the nematic phase, while the Onsager profiles, which rise toFig. 2 Packing fraction and order parameter profiles, h(z) and S(z), versus z/H, for N = 1016, 2 x 1016 and 2.6 x 1016 cm-2, for rods with aspect ratio x = 16.7, n0= 116 640 nm3 and m = 6.64 x 10-17 g, appropriate for TMV particles. H = 10 cm. unphysically high concentrations at z = 0, appear to decay exponentially in that phase.In fact a simple analysis using Onsager’s analytic high density expression for a(n) (ref. 6), leads to the surprisingly simple profile: (19) The slope of log n(z) differs by a factor 1/3 from that expected for a suspension of non-interacting particles. The first correction to the LDA used in this work is of the square-gradient form, which leads to a coupling between the concentration gradient and the local orientation in the nematic phase. The square-gradient contribution to the free energy functional will have two important consequences. First, it will lead to a continuous isotropic-nematic interface of finite thickness. Secondly, while the orientation of the director is indeterminate in the nematic phase within the LDA, the non-local correction is expected to lead to a preferred orientation in the vertical gravitational field.These issues will be addressed in detail elsewhere. Acknowledgements DG acknowledges the financial support of EPSRC. References 1 T. Biben, J. P. Hansen and J. L. Barrat, J. Chem. Phys.,1993, 98, 7330. Fig. 1 Packing fraction profiles h(z) versus z/H, for aspect ratios x = 10, 20, 40 and 90, keeping the volume n0 constant. N = 1016 cm-2 for x = 10, 20, 40, while N = 5 x 1015 cm-2 for x = 90. m = 6.64 x 10-17 g; u0 = 116 640 nm3; H = 10 cm. The dotted and full curves are for the Onsager and Parsons-Lee functionals, respectively.2 R. Piazza, T. Bellini and V. Degiorgio, Phys. Rev. Lett., 1993, 71, 4267. 3 S. Hachisu and K. Tokano, Adv. Colloid. Interface Sci., 1982, 16, 233. 4 T. Biben and J. P. Hansen, Molec. Phys., 1993, 80, 853. 5 T. Biben and J. P. Hansen, J. Phys. Cond. Matter, 1994, 6, A345; H. Löwen, J. Phys. Cond. Matter, 1998, 10, L479. 6 L. Onsager, Ann. NY Acad. Sci., 1949, 51, 627. 7 See e.g. R. Evans, in Fundamentals of Inhomogeneous Fluids, ed. D. Henderson, Marcel Dekker, New York, 1992. 8 For a review of improvements to Onsager's theory, see G. J. Vroege and H. N. W. Lekkerkerker, Rep. Prog. Phys., 1992, 55, 1241. 9 J. D. Parsons, Phys. Rev. A, 1979, 19, 1225. 10 S. D. Lee, J. Chem. Phys., 1987, 87, 4972. 11 See e.g. P. J. Camp, C. P. Mason, M. P. Allen, A. A. Khare and D. A. Kofke, J. Chem. Phys., 1996, 105, 2837. 12 N. F. Carnahan and K. E. Starling, J. Chem. Phys., 1969, 51, 635. Paper 9/05089B PhysChemComm © The Royal Society of Chemistry 1999

ISSN:1460-2733    

DOI:10.1039/a905089b

出版商:RSC    

年代:1999

数据来源: RSC