Langmuir 2002, 18, 2063-2068
2063
Salt-Induced Ordering in Lamellar Colloids D. G. Rowan* and J.-P. Hansen Department of Chemistry, University of Cambridge, Lensfield Road, Cambridge CB2 1EW, United Kingdom Received September 11, 2001. In Final Form: December 7, 2001 Uncharged rods and disks have long been known to exhibit an isotropic to nematic phase transition on increasing density. In aqueous solution, however, such particles are generally highly charged, leading to repulsive electrostatic interactions. In this article the effect of adding salt to a monodisperse solution of thin charged colloidal disks is examined. The addition of microscopic salt ions more efficiently screens the largely quadrupolar electrostatic repulsion between the mesoscopic colloidal particles, while at the same time the electric double layer around each disk contracts. By treating each disk with its associated double layer as a new effective particle, the role of salt on the phase diagram of dilute suspensions is investigated.
Introduction The theory of the isotropic-nematic phase transition for hard rod-shaped particles has been well established since the pioneering work by Onsager.1 Here the interplay between excluded volume entropy and orientational entropy of the highly anisometric particles leads to a purely entropic first-order phase transition, which may be described quantitatively using only the second virial coefficient for large rod aspect ratios. At low particle concentration the solution is isotropic, but as the density increases, the rods start to align to maximize the free volume in which they may move, resulting in a nematic phase. Efforts to extend his theory to incorporate electrostatic effects2,3 have focused on the realization that, when immersed in a solution containing charged microions, the large charged rod acquires an electric double layer, the inclusion of which modifies significantly the aspect ratio of the rod. This leads to an effective particle diameter Deff which may be written as the sum of the bare diameter plus a contribution from the double layer, which is roughly proportional to the Debye screening length λD. At very low salt concentrations Deff can differ from the bare diameter by 2 orders of magnitude, and the second virial coefficient B2 ) πL2Deff/4 is thus greatly increased, leading to a much reduced critical concentration of rods at the isotropic-nematic transition. For disk-shaped particles the isotropic-nematic (I-N) transition has attracted much less attention, in particular because the particles have a tendency to form gels at low concentration. Some theoretical progress has been made by applying Onsager’s ideas to this problem,4 and via Monte Carlo simulation.5,6 A recent experimental study of the liquid crystal phases of hard sterically stabilized colloidal platelets has identified a liquid-phase isotropic to nematic phase transition7 for large aspect ratios (diameter/thickness =12). For charge-stabilized platelets the nature of the phase transition is more elusive. Studies (1) Onsager, L. Ann. N. Y. Acad. Sci. 1949, 51, 627. (2) Vroege, G. J.; Lekkerkerker, H. N. W. Rep. Prog. Phys. 1992, 55, 1241. (3) Lekkerkerker, H. N. W.; Vroege, G. J. Philos. Trans. R. Soc. London A 1993, 344, 419. (4) Forsyth, P. A.; Marcelja, J. S.; Mitchell, D. J.; Ninham, B. W. Adv. Colloid Interface Sci. 1978, 9, 37. (5) Eppenga, R.; Frenkel, D. Mol. Phys. 1984, 52, 1303. (6) Veerman, J. A. C.; Frenkel, D. Phys. Rev. A 1992, 45, 5632. (7) van der Kooij, F. M.; Lekkerkerker, H. N. W. Philos. Trans. R. Soc. London A 2001, 359, 985.
of Laponite (a small platelet for which D = 30 nm) show an I-N transition for which the nematic phase is progressively stabilized as the concentration of added salt is reduced,8 a trend also observed in the phase diagram of rodlike tobacco mosaic virus (TMV) particles,9 explained by the effective diameter theories of Vroege and Lekkerkerker.2 However, for bentonite, a significantly larger platelet (D = 300 nm), the opposite trend is observed,10 where the nematic phase is destabilized with respect to the isotropic phase in the gel. This trend may be explained solely on electrostatic grounds: at low salt concentrations the electrostatic interaction between disks is less efficiently screened and, as the electrostatic potential energy is a maximum for parallel orientations,11,12 the nematic phase is higher in energy. However, there are clearly two competing factors at work: (a) the swelling in the platelet’s dimensions due to the double layer, tending to push the I-N transition to lower disk concentrations at lower salt concentrations, and (b) the favored T-shaped alignment of platelets induced by the quadrupolar coupling, which tends to stabilize the isotropic phase. This second effect is envisaged to be dominant for larger platelets, for which the ratio λD/D is much smaller and the change in the second virial coefficient on changing salt concentration is much less pronounced. Reference System of Hard Particles The colloidal suspension under consideration is composed of Np infinitely thin platelets of diameter D which, when dispersed in water, each carry a total charge Zpe (Zp < 0) uniformly distributed on the surface of the disk. Associated with this ionization is the release of Nc monovalent counterions, where Nc ) ZpNp; charge neutrality is thus preserved. When monovalent salt is added, this condition may be simply expressed as ZpNp + N- ) N+, where N+(-) is the number of microscopic cations (anions), whether counterions or salt ions, present in the total volume V. (8) Mourchid, A.; Lecolier, E.; Van Damme, H.; Levitz, P. Langmuir 1998, 14, 4718. (9) Fraden, S.; Maret, G.; Caspar, D. L. D. Phys. Rev. E 1993, 48, 2816. (10) Gabriel, J. C.; Sanchez, C.; Davidson, P. J. Phys. Chem. 1996, 100, 11139. (11) Rowan, D. G.; Trizac, E.; Hansen, J. P. Mol. Phys. 2000, 98, 1369. (12) Meyer, S.; Levitz, P.; Delville, A. J. Phys. Chem. To be published.
10.1021/la011424p CCC: $22.00 © 2002 American Chemical Society Published on Web 02/23/2002
2064
Langmuir, Vol. 18, No. 6, 2002
Rowan and Hansen
The collection of platelets may, as a first approximation, be taken to comprise a system of large hard particles, in which the microions are assumed to occupy no volume and behave ideally. The platelets, which will in the most general case have a finite thickness L, have a second virial coefficient calculated by Onsager1 to be
1 1 1 B2 ) πD L2 + (π + 3)LD + πD2 4 2 4
[
]
(1) P2 )
This formula is general for all cylindrical objects, both rods (L > D) and disks (D > L). For the case of long thin rods (L/D f ∞), only the leading term in eq 1 remains, and all higher virial coefficients become vanishingly small, reflecting the infinitesimal probability that three rods will simultaneously intersect. For thin disks the chance that one disk has contact with two or more disks simultaneously is more significant, and the third and higher virial coefficients are nonnegligible (B3/B22 ) 0.445).5 Naturally, if the system is restricted to low disk densities, then it is to be expected that higher order interactions will not be significant. However, the Onsager second virial approximation (βP ) F + B2F2) yields pressures that are significantly lower than those obtained from Monte Carlo simulations5 at higher densities, and I-N transition densities4 that are large compared to those of experiments.13 While the machinery of the Onsager method of treating the isotropic-nematic transition is appealing, the inability of the second-order virial expansion to accurately describe the equation of state is problematic. This problem may be avoided though, by use of a recently published equation of state derived analytically14 from the PRISM15 integral equations, which gives excellent quantitative agreement with both experimental structure factors16 and numerical equation of state data.5 The equation of state, for infinitely thin disks (L ) 0) takes the simple form
βP ) F + BP2 F2 + BP3 F3
(2)
where BP2 should not be confused with a virial coefficient. Formally, it is the coefficient of the quadratic term in the PRISM equation of state,14 and although it is related to the exact second virial coefficient via BP2 /B2 ) 2x2/π, the PRISM theory result is not exact at low densities, although the agreement of eq 2 with simulation at higher densities is excellent.14 The coefficient of the cubic term is given exactly by BP3 ) 1/3(BP2 )2 within this theory. Following the method of Onsager,1 the free energy of the reference system of hard uncharged disks may be expressed as a functional of the orientational distribution function. To investigate the role of orientational order, the Onsager trial functional is adopted, where the distribution function for the platelet orientations is assumed to be of the functional form
R cosh(R cos θ) f(cos θ) ) 4π sinh R
the distribution peaks around θ ) 0 and π and the platelets are almost parallel (nematic phase), while, for low values, the angular distribution is spread out and the system may be said to be isotropic. The choice of this form of trial function is not important, and many different trial orientational functions give essentially the same result. The nematic order parameter is defined as
(3)
where θ is the angle the normal to a platelet makes with an arbitrary director axis, and R is a variational parameter used to minimize the free energy. For large values of R, (13) van der Kooij, F. M.; Lekkerkerker, H. N. W. J. Phys. Chem. B 1998, 102, 7829. (14) Harnau, L.; Costa, D.; Hansen, J. P. Europhys. Lett. 2001, 53, 729. (15) Schweizer, K.; Curro, J. Adv. Chem. Phys. 1996, 98, 1. (16) Kroon, M.; Vos, W. L.; Wegdam, G. H. Phys. Rev. E 1998, 57, 1962.
3〈cos2 θ〉 - 1 2
)1+
3 coth R 3 2 R R
(4)
in terms of the Onsager R parameter. Using this trial function as an input into the free energy functional, the total free energy per particle f ) F/Np may be written as the sum of an ideal gas term, an orientational entropy term, and a term arising from excluded volume interactions:
βf ) βfid + βforient + βfex
(5)
The ideal term is the familiar logarithm of the density, while the purely entropic orientational free energy is simply
βforient )
∫f(ω) log(4πf(ω)) dΩ
) log(R coth R) -1 +
arctan(sinh R) sinh R
(6)
the integration being performed over all possible orientations of the disk. This expression is identical to that found in the theory of the I-N transition for rods. For the excluded volume term we use the result derived from the PRISM equation of state (eq 2), which may be generalized to include the possibility of anisotropy in the solution according to
βfex ) BP2 F(1 - P2)x1 + 2P2 + 1 P 2 B F (1 - P2)2(1 + 2P2) (7) 2 3 The total free energy for the uncharged system of platelets is now given by
βf ) log(Λ3F) - 1 + log(R coth R) - 1 + arctan(sinh R) + BP2 F(1 - P2)x1 + 2P2 + sinh R 1 P 2 B F (1 - P2)2(1 + 2P2) (8) 2 3 This free energy depends only on the number density of disks and on the orientational distribution function, parametrized by R. It may now be minimized with respect to the orientational parameter R, the value of which may then be used to calculate an order parameter using eq 4. For this theory of uncharged disks, an isotropic-nematic phase transition is predicted to occur at the density FD3 = 3.2, in reasonable agreement with simulation.5 The purpose of this paper is, however, to investigate the role of electrostatics on the phase transition, and to do so, we use this free energy for uncharged disks (eq 8) as a reference free energy for the system, which shall be denoted f0. The highly angular-dependent electrostatic contribution to the free energy, f1, shall be treated as a
Salt-Induced Ordering in Lamellar Colloids
Langmuir, Vol. 18, No. 6, 2002 2065
first-order correction to the uncharged free energy f0 using thermodynamic perturbation theory f ) f0 + f1.17 This is addressed in the next section. Incorporation of Electrostatics The electrostatic interaction between two charged platelets at positions r1,2 and orientations ω1,2 immersed in a screening bath of microions is known at the level of linearized Poisson-Boltzmann theory11 to be a complex algebraic function of the particles’ separation r ) |r2 - r1|, their mutual angles, and the Debye screening length (1/ κ). This potential energy, Vel(r,ω1,ω2) is highly anisotropic and may be expressed as a series expansion in even multipoles of the charge distribution11 (the symmetry of the charge distribution forbidding odd multipoles such as the dipole moment). This is a particularly appealing method of calculating the electrostatic potential energy, for it highlights the large quadrupole-quadrupole interaction, which disfavors nematic ordering of platelets and rather prefers disks to orient themselves in the “T-shaped” house of cards structure.18,19 Unfortunately, no exact expression for the pair distribution function g(r,ω1,ω2) is available, because of the high degree of anisotropy in the interaction. To incorporate the electrostatic potential into the free energy calculation, a high-temperature approximation to the pair distribution function is used, whereby the distribution function of the disks is taken as that of the uncharged reference system of hard particles, g(r,ω1,ω2) ) g0(r,ω1,ω2). The electrostatic free energy is thus expressed as an integral of the electrostatic potential over all possible configurations of two platelets:
βf1(R,F) )
β 2
∫dr1 dr2 dω1 dω2 F(r1,ω1) g0(r,ω1,ω2) F(r2,ω2) Vel(r,ω1,ω2)
)
β 2
∫dr1 dr2 dω1 dω2 F(r1,ω1) F(r2,ω2) Vel(r,ω1,ω2)e-βU
hp
(9)
Here Uhp is the hard-particle interaction of the two platelets; Uhp ) ∞ if platelets intersect and zero otherwise, and thus the Boltzmann factor is simply one if configurations arising in the integral are allowed and zero if they are forbidden (corresponding to intersecting disks). The shorthand density F(r1,ω1) is simply the density of particles at position vector r1 with an orientation vector ω1 given by the Onsager R-distribution (eq 3). The calculation of this electrostatic contribution to the free energy necessitates numerical integration over all possible configurations of particles. However, the decay of the screened Coulomb potential is such that the integration grid need only be a modest number of particle diameters at the typical screening parameter κ. Equation 9 allows direct calculation of the potential energy for a dispersion of charged clay platelets at a given density and orientation. It is perhaps interesting to note that at any typical density 2.0 < FD3 < 6.0 the electrostatic potential energy of an isotropic arrangement of platelets is always lower than that of a nematically ordered structure, by around 10%. Procedure With the machinery to determine the phase behavior of the solution of charged disks in place, the total free (17) Hansen, J. P.; McDonald, I. R. Theory of Simple Liquids, 2nd ed.; Academic Press: London, 1986. (18) Dijkstra, M.; Hansen, J. P.; Madden, P. A. Phys. Rev. E 1997, 55, 3044. (19) Kutter, S.; Hansen, J. P.; Sprik, M.; Boek, E. J. Chem. Phys. 2000, 112, 311.
Figure 1. Schematic diagram of the new effective hard particles considered in this article, encompassing the bare colloid and its electric double layer, assumed to be of a height given by the Debye screening length.
energy of the system may be minimized with respect to the only unknown in our formalism, namely the Onsager R-parameter. However, before proceeding, we recall from the theory of and experiments on charged hard rods2,9 that the microions present in solution dress the bare colloid to such an extent that the physical dimensions of the hard particles are drastically altered. The central idea is that the electrostatic double layer is bound to the particle to such an extent that it defines a larger “hard” particle. No exact theory of this swelling exists for platelets, and while the arguments for rods are rather heuristic, they do explain the trends observed in the isotropic-nematic transition when the salt concentration is changed. If the double layer were to be ignored, both systems of charged rods and platelets would exhibit phase diagrams in which the nematic phase was stabilized as the salt concentration increased, in direct contradiction to experiment (except for very large platelets such as bentonite10). We therefore choose to examine swollen disk-shaped particles, where the particles retain their diameter, but the thickness of the disks is increased from zero to twice the Debye length, as shown in Figure 1. This may appear a rather arbitrary definition of the double layer, but it is in keeping with Onsager’s initial comment that the swelling should be a few “modest multiples of the Debye length”,1 and clearly of the correct order of magnitude. As the newly defined hard particle has greatly increased volume in comparison to the infinitely thin disk, the free energy of the reference system f0 will of course need to be adapted. The ideal gas and orientational entropy terms remain the same, being independent of the dimensions of the disk, but the excluded volume interaction fex will become much larger, because of the reduced free volume particles may move in. This manifests itself in a new value of the BP2 parameter which governs the equation of state obtained from PRISM theory (eq 2). This new value of BP2 has been modified to incorporate the finite thickness of the plate using the Onsager result for finite-thickness disks (eq 1) by replacing the thickness of the disk by twice the Debye length, and shall be referred to as the Onsager scaled BP2 parameter. This scaling is possible due to the algebraic relationship between BP2 and the exact second virial coefficient B2, and calculations of the equation of state using the new scaled BP2 parameter show good agreement with the equation of state obtained using PRISM for platelets of finite thickness.21 This agreement is illustrated in Figure 2, where the Onsager scaled PRISM result is compared with the PRISM result for finitethickness platelets, and the Onsager second virial result. Concomitant with the new definition of the hard particle, the total charge of the hard object must now be renormalized to include the contribution of the microions included within the volume Vd ) 2πR2λD. This is obtained (20) Trizac, E.; Hansen, J. P. Physica A 1997, 235, 257. (21) Rowan, D. G.; Hansen, J. P.; Harnau, L. To be published.
2066
Langmuir, Vol. 18, No. 6, 2002
Rowan and Hansen Table 1. Effective Charges for the Platelets with Their Double Layers, Calculated for Various Platelet Densities (Which Impose Counterion Densities) at Several Salt Concentrations disk density, F* ) FD3
salt conc (mol/L)
effective charge, Zeff
1.0 1.0 1.0 3.0 3.0 3.0 5.0 5.0 5.0
0.0 0.001 0.01 0.0 0.001 0.01 0.0 0.001 0.01
345.04 352.91 406.65 253.74 256.72 280.76 202.92 204.49 220.59
The calculations were performed within linearized Poisson Boltzmann theory, for platelets of diameter 30 nm and bare charge of Zp ) 1000, corresponding to Laponite.
Figure 2. Comparison of the equation of state for uncharged clay platelets at a finite thickness-to-diameter ratio L/D ) 1/20. The main figure shows the variation of the pressure with increasing density F/FM, where FM is the maximum density of disks, corresponding to close packing. The uppermost curve (solid squares) is the PRISM equation of state for finite thickness platelets,21 which diverges correctly as close packing is approached. Beneath it is the Onsager scaled BP2 -theory result discussed in the text (open circles), which agrees well over a large range of densities. For completeness, the PRISM equation of state for infinitely thin platelets is also presented (triangles) along with the Onsager second virial result, which predicts pressures that are markedly too low. In the inset the corresponding isothermal compressibility for each case is plotted against log(F/FM), this being the route by which the pressure is calculated in the integral equation theories.
by integration of the microion density, obtained from linearized Poisson-Boltzmann theory,20 over this volume Vd.
Zeff ) Zp +
∫V [F+(r) - F_(r)] d3r
(10)
d
where the microion densities are expressed in terms of their bulk values (F0() by F( ) F0((1 - βψ), where
ψ(r,z) )
2Ze R0
xk2 + κ2
-|z|
∫0R J1(kR) J0(kr)e
xk
2
+κ
dk
(11)
2
is the LPB potential (expressed in cylindrical coordinates) around one isolated platelet in an electrolyte providing the inverse screening length κ.20 The effective charge defined in this way is rather insensitive to the salt concentration for a given platelet density, as seen in Table 1. This is due to the large concentration of counterions, which at low salt concentrations dominate the microions present because of the addition of salt. The effective charge is thus observed to be much more sensitive to the platelet density, F* ) FD3, as this density fixes the concentration of counterions, which is typically much larger than the concentration of added salt. Results The total free energy for a system of charged disks of fixed diameter and surface charge density has been expressed in terms of a reference free energy for a system of hard disks plus a perturbation due to the microionscreened electrostatic repulsion between the like charges. This free energy, given by eqs 8 and 9, contains only one
Figure 3. Minimization of the free energy per particle by changing the orientational parameter R, for a suspension of Laponite (diameter D ) 30 nm) at a density given by FD3 ) 3.0. At low salt concentration (upper two lines) the solution is nematic, with a finite order parameter, while above 10-3 M salt the isotropic phase is stabilized.
unknown variable, the degree of order in the orientation of the disks, governed by the Onsager R parameter (eq 3). This parameter may be freely varied to minimize the free energy, a finite value of R at the minimum indicating that the system prefers nematic order, while the absence of a minimum in the free energy indicates that the system prefers isotropy, where both the orientational entropy is maximized and the electrostatic free energy is minimized simultaneously. In Figure 3, the free energy of a suspension of Laponite at a density of F* ) FD3 ) 3.0 is calculated for a number of salt concentrations as a function of orientational order. Here, and in what follows, the temperature is fixed at 300 K. As is evident at very dilute salt concentrations (where the majority of the screening is from counterions), the free energy is minimized by R ) 7.2, which indicates that the preferred phase is nematic. On increasing the salt concentration by a factor of 10 to 10-4 mol/L, the system remains nematic, but the minimizing value of R falls to 3.5, showing that the nematic phase is being destabilized. This is due to the contraction of the double layer (λD ∝ [salt]-1/2) rather than to electrostatic repulsion arguments, for the screening between the two disks is more efficient at higher salt concentration. When the salt concentration is increased further, the minimum completely disappears,
Salt-Induced Ordering in Lamellar Colloids
Figure 4. As for Figure 3 at the increased density of FD3 ) 5.0, a density where the solution is always nematic, independent of the salt concentration. On increasing the salt concentration (from top to bottom) the nematic order parameter is reduced (R decreases), indicating the destabilization of the nematic phase.
Figure 5. As for Figure 4 for a particle with twice the diameter of Laponite with the same charge density at FD3 ) 3.0. The trend is seen to be the same, though the isotropic phase is seen to be stabilized more than for Laponite.
showing that at high salt concentration the solution prefers to be isotropic, in good agreement with experiment.8,10 Increasing the disk density well into the nematic regime at F* ) 5.0 in Figure 4, we observe that the free energy is always minimized at large values of the R parameter, for any salt concentration. The same trend as before is still observed, however, with the nematic order parameter, given by eq 4, falling as the concentration of added monovalent salt is increased. This may again be interpreted in terms of the large change in aspect ratio of the disks, due to the contraction of the electric double layer. In Figures 5 and 6 the corresponding free energy profiles for clay platelets twice the diameter of Laponite (60 nm) are presented, where the surface charge density is the same as that of Laponite. Here the situation is qualitatively the same; at large salt concentrations the isotropic phase is stabilized, while when the concentration of added salt is decreased, the larger “swollen” particles tend to
Langmuir, Vol. 18, No. 6, 2002 2067
Figure 6. As for Figure 5 at the increased density of FD3 ) 5.0, above the I-N transition. The effect of increasing salt concentration is seen to hardly alter the order parameter at all.
favor nematic ordering, as seen in Figure 5. For the reduced density of F* ) 3.0 the nematic phase becomes stable at a much lower salt concentration than that for Laponite, reflecting the fact that while the microion concentration fixes the thickness of the new particle at 2λD, the larger diameter of this particle ensures that the aspect ratio L/D is less significantly changed than that for Laponite. This trend is again observed deep in the nematic phase at F* ) 5.0, where the value of the Onsager parameter which minimizes the free energy varies only very slightly over the range of salt concentration investigated, shown in Figure 6. On decreasing the salt concentration, the electrostatic repulsion disfavoring nematic ordering is enhanced, while the larger effective particles prefer to align to increase the system’s entropy. These competing factors thus almost balance in this case, and the effect of adding salt is minimal. In Figure 7 the proposed phase diagram for charged smectite clays of diameters 30 (Laponite) and 60 nm is presented. At high density the stable phase is well-known to be nematic. For any density in the range 2.5 < F* < 3.0, however, the concentration of added salt is observed to play a huge role in determining which phase is most stable. For Laponite the observed slope with increasing salt concentration is in semiquantitative agreement with that observed in experiments,8 and on increasing the size of the colloid, this slope is seen to decrease, becoming almost salt-independent for a particle twice the size of Laponite. This is interpreted as being due to the less dramatic change in the aspect ratio of the larger particle on changing the Debye screening length. Thus, for a given reduced density FD3, the larger particles are more free to rotate than the smaller particles, and the isotropic phase persists for higher densities. Extending this argument to the much larger bentonite system could explain why the observed slope in the I-N phase diagram is the opposite of that for Laponite, where the isotropic phase is stabilized at low salt concentrations. We believe in this case that the swelling effect is negligible and the phase diagram is dominated by long-range electrostatics, which disfavor nematic ordering. Conclusions Studies of the isotropic to nematic phase transition for charged clay platelets, interacting via screened Coulombic
2068
Langmuir, Vol. 18, No. 6, 2002
Figure 7. Proposed phase diagram for dilute dispersions of Laponite (solid line) and a particle of twice the diameter of Laponite with the same charge density. On reducing the salt concentration for both systems, the transition moves to lower density, reflecting the larger effective particles (and larger virial coefficients) present when fewer small ions are present. The slope of the line is reduced for the second particle, being almost vertical, and it is expected that if the particle size is further increased to that of bentonite, the electrostatic quadrupolar repulsion will dominate the phase diagram and the slope will reverse, as observed in experiments.10 The vertical arrow indicates the position of the phase transition for an uncharged system of disks within this theory.
repulsion and excluded volume constraints, have illuminated the role of added salt on the phase diagram. A subtle competition between the preference of diskshaped objects to align as density is increased to maximize system entropy, and the highly anisotropic electrostatic
Rowan and Hansen
interaction which favors isotropy over ordering, has been explained. By redefining the clay platelets as new effective hard particles incorporating both the bare disk and the associated electric double layer, which serves to define the thickness of the particle, the influence of salt on these competing interactions is readily interpreted. As the salt concentration is lowered, the extent of the double layer expands, and the effective particle size grows. This increased aspect ratio of the cylindrical object enhances the tendency to align. At the same time, however, the electrostatic repulsion between the particles increases, as the screening length drops, which acts to stabilize the isotropic phase. For small platelets, such as those considered here, the former of these two effects is always dominant, and the nematic phase is stabilized on decreasing the salt concentration. On increasing the platelet diameter, however, the stabilization is observed to be less pronounced. This is due to the fact that the relative change in the aspect ratio of the new particle (2λD/D) is less sensitive to salt concentration for large particle diameters, so the stabilization of the nematic phase is reduced while the electrostatic preference for isotropy remains. For larger platelets such as bentonite, which are beyond the scope of the numerical techniques used in this work, this may explain why the isotropic phase is stabilized at low salt concentrations. Work is currently in progress to treat these large platelets within a restricted orientation framework, similar to Zwanzig’s treatment of hard rods, which qualitatively reproduces the Onsager result.22 Acknowledgment. The authors would like to acknowledge useful discussions with L. Harnau, and thank the EPSRC for their support. LA011424P (22) Zwanzig, R. J. Chem. Phys. 1963, 39, 1714.
