Phase Behaviour of Colloidal Platelet–Depletant Mixtures

Abstract

Colloidal platelets  are encountered in a wide range of systems in nature and technology. Examples are hydroxides , smectite clays and exfoliated inorganic nanosheets. Suspensions of these platelets have been found to exhibit liquid crystal ordering, including gibbsite  [1,2,3,4], nickel hydroxide [5], layered double hydroxides [6, 7], nontronite [8,9,10], beidellite [11, 12], fluorohectorite [13, 14], solid phosphatoantimonate acid [15, 16], zirconium phosphate [17,18,19], niobate [20, 21] and titanate [22].

9.1 Introduction to Colloidal Platelets

Colloidal platelets  are encountered in a wide range of systems in nature and technology. Examples are hydroxides and layered double hydroxides, smectite clays and exfoliated inorganic nanosheets. Suspensions of these platelets have been found to exhibit liquid crystal ordering, including gibbsite  [1,2,3,4], nickel hydroxide [5], layered double hydroxides [6, 7], nontronite [8,9,10], beidellite [11, 12], fluorohectorite [13, 14], solid phosphatoantimonate acid [15, 16], zirconium phosphate [17,18,19], niobate [20, 21] and titanate [22]. TEM micrographs of dispersions made from some of these particles are displayed in Fig. 9.1, giving some insight into the morphology. Table 9.1 provides the chemical composition of the platelets.

Fig. 9.1

Examples of plate-like colloids: a–c synthetic platelets, d–f smectite clay particles, g–i exfoliated inorganic nanosheets. Scale bars are as indicated. Reprinted with permission from a Ref. [2], copyright 2000 Nature; b Ref. [5], copyright 1999 Springer; c Ref. [6], copyright 2003 the American Chemical Society (ACS); d Ref. [9], copyright 2008 ACS; e Ref. [12], copyright 2011 ACS; f Ref. [14], copyright 2010 the Royal Society of Chemistry (RSC); g P. Davidson; h Ref. [17], copyright 2012 the American Physical Society (APS); i Ref. [22], copyright 2014 RSC

Upon increasing the concentration of platelets a dispersion of initially isotropic platelets can become liquid crystalline, just as for rods, see Chap. 8. Concentrated dispersions of platelets may display nematic [1, 6, 8, 14, 16, 18, 20, 22], columnar [5, 24] and smectic [4, 7, 15, 16, 19] phase states, illustrated in Fig. 9.2.

Table 9.1 Chemical composition of colloidal plate-like particles

While each of these nematic (N), columnar (C) and smectic (Sm) phases exhibits long-range orientational order, they differ by the positional correlations between the particles. Long-range positional order is absent in the N phase. The C phase has a two-dimensional lattice of columns, which consist of liquid-like stacks of particles. The Sm phase is characterised by a one-dimensional periodic array of layers of particles.

Fig. 9.2

Structure of the three main classes of liquid crystals made from disc-like particles: the nematic phase (N), the columnar phase (C) and the smectic phase (Sm). Reprinted with permission from Ref. [2]. Copyright 2000 Nature

Fig. 9.3

Tubes containing gibbsite suspensions [2] in toluene at varying platelet concentrations photographed between crossed polarisers. Volume fractions from left to right: 0.19 (I–N), 0.28 (N), 0.28 (N–C) and 0.47 (C). The tube to the right depicts a monophasic columnar sample illuminated by white light. The colours of its Bragg reflections (visible as small bright spots) vary from yellow to green as the angle between the incident light and viewing direction is in the range of 50°–70°. Reprinted with permission from Ref. [2]. Copyright 2000 Nature

Figure 9.3 depicts gibbsite dispersions which exhibit these liquid-crystalline phase states [2]. These images illustrate the phases and phase transitions that can be detected when using crossed polarisers for samples varying in platelet concentration. The Sm phase is only rarely observed [4, 5, 7, 15, 16, 19, 25]. The smectic phases observed in Refs. [7, 25] and [4], probably originated from the high surface charge of the platelets. The large polydispersity [19, 26] of the diameter of the platelets probably explains the Sm phase observed in Ref. [19] . These smectic phases can show bright iridescence, see Fig. 9.4, depending on the spacing between the smectic layers [4].

Fig. 9.4

Silica-coated gibbsite platelets suspended in DMF without added salt show iridescence upon illumination with white light in a 10 mm wide capillary, indicating a lamellar phase

Disc-like colloidal particles are also found in biological systems such as the red blood cells in blood (already discussed in Sect. 1.3.2) and plate-like proteins such as kinetochore [27]. Smectite clays, which are inorganic plate-like nanoparticle mixed suspensions, are ubiquitous on Earth. Mixed suspensions of colloidal platelets and inorganic nanoparticles display interesting rheological [28] and electronic properties [29].

In this chapter we consider the phase behaviour of mixtures of colloidal plates and depletants. The focus is on nonadsorbing polymers as depletants, although experimental examples of added small colloidal particles are also considered. First, a treatment of the phase behaviour of pure platelets is given.

9.2 Phase Diagram of Hard Colloidal Platelets

Hard colloidal platelets are theoretically described here in a simplified way as monodisperse discs with diameter D and thickness L, with a focus on $L \ll D$. The volume of a colloidal platelet is given by $v_\textrm{0}={\pi }D^2 L/{4}$ and the volume fraction of colloidal platelets  $\phi = n v_\textrm{0}$, with n the number density N/V. It is noted that in experimental dispersions of these platelets there is often size dispersity in D and/or L. In this section theoretical and computer simulation results on the phase diagram of pure monodisperse hard platelets are reviewed.

9.2.1 Computer Simulations

In previous chapters it was identified that colloidal phase transitions of hard spheres and hard rods are governed by entropy. That is also the case for hard platelets . In Sect. 8.2 it was explained that, as was realised by Onsager, a theory based on the second virial coefficient suffices to accurately predict the thermodynamic properties of dilute long and thin rods. This includes the rod concentrations at the isotropic–nematic (I–N) phase transition. The reason for that is that the higher virial coefficients for long thin rods are very small compared to the second virial coefficient, as was first rationalised by Onsager [30] on the basis of geometric arguments. Using Monte Carlo simulations, Frenkel [31, 32] showed that the longer and thinner the rods the smaller the higher virial coefficients become.

As was pointed out by Onsager [30], there are no geometrical arguments that the higher virial coefficients become small even in the limit of infinitely thin hard platelets. This can be illustrated by comparing the results of the second virial approach for the I–N transition of infinitely thin hard platelets with simulations. Forsyth et al. [33] used the second virial approach for the I–N transition and found $n_\textrm{iso}D^3=5.3$ and $n_\textrm{nem}D^3=6.8$ as coexisting platelet concentrations. Using Monte Carlo simulations on infinitely thin hard platelets , Eppenga and Frenkel [34] found $n_\textrm{iso}D^3=4.04$ and $n_\textrm{nem}D^3=4.12$. Clearly the second virial coefficient theory does not accurately predict the I–N phase transition of hard platelets.

Veerman and Frenkel [35] extended the simulations of Eppenga and Frenkel to hard platelets of finite thickness. The phase diagram that results from their computer simulations is reproduced in Fig. 9.5a. Note that the platelets were simulated as cut spheres (see the inset of Fig. 9.5a).

Fig. 9.5

a Phase diagram from computer simulations of dispersions of hard cut spheres to mimic platelets. Inset: side view of a cut sphere with diameter D and thickness L [35]. b Experimentally measured relative volume of the nematic and columnar phases as a function of the volume fraction of gibbsite platelets [2]. The platelets have $D/L = 13$ and a polydispersity in D of 17% ($\vartriangle$) and 25% ($\circ$). The predicted phase states from panel a are indicated below the abscissa of (b). Reprinted with permission from a Ref. [35], copyright 1992 APS and b Ref. [2], copyright 2000 Nature

In Fig. 9.5a $\rho ^*$ is the density relative to the close packed density. For $L/D < 0.15$ (i.e. $D/L > 6.7$) the simulations of Veerman and Frenkel reveal an isotopic phase, an I–N phase transition, an N phase, a subsequent nematic–columnar (N–C) transition and finally a pure C phase upon increasing the platelet concentration. This simulation result was confirmed experimentally by using model systems of plate-like gibbsite particles with $D/L=13$ with different polydispersities by Van der Kooij, Kassapidou and Lekkerkerker [24] (see the observed phases in Fig. 9.5b).

The computer simulations of Veerman and Frenkel also predicted that coexistence between isotropic and columnar phases (i.e. without forming a nematic phase) is possible for thicker platelets with $L/D > 0.15$ (Fig. 9.5a). Brown et al. [5] studied nickel hydroxide platelets with $D/L =3.5$ and indeed found this direct I–C transition.

Fig. 9.6

a Snapshot of the configuration of a system containing 1728 cut hard spheres with $D/L=5$ at a reduced density of $\rho ^*=0.575$. The platelets spontaneously assemble in short stacks containing four or five particles. Neighbouring stacks tend to be approximately perpendicular [35]. Reprinted with permission from Ref. [35]. Copyright 1992 APS. b Cryo TEM image of a suspension of nickel hydroxide platelets at 20 wt$\%$. Reprinted with permission from Ref. [36]. Copyright 1984 Elsevier

Veerman and Frenkel [35] also observed an unexpected region in the phase diagram, which they refer to as the cubatic phase (CUB). In this CUB phase the cut spheres are assembled in short columns. The columns themselves have a random orientation, and hence there is appreciable interaction between different columns. The interaction between the columns eventually becomes so severe that the column segments try to order in a manner that minimises the packing problems. This is illustrated in the computer simulation snapshot in Fig. 9.6a. Qazi, Karlsson and Rennie [36] have presented experimental evidence for cubatic order in a dispersion of plate-like colloids (see Fig. 9.6b).

In the next subsection, an approximation is outlined that aims to develop a theoretical prediction for the phase diagram of hard platelets .

9.2.2 Theoretical Account

9.2.2.1 Onsager–Parsons–Lee Theory for the Isotropic and Nematic Phase States

In Sect. 8.3 we used scaled particle theory (SPT) to incorporate higher virial coefficients in the treatment of the isotropic–nematic phase transition of hard rods. An approach which is similar to SPT (in the fact that it may be considered a renormalised two-particle theory ) has been given by Parsons [37] and was used by Lee [38] to calculate the isotropic–nematic transition in solutions of hard spherocylinders. This approximate theory [39, 40] may be considered an extension of the Carnahan–Starling equation [41] for hard spheres (see Sect. 3.2.1).

The Helmholtz free energy F within the Onsager–Parsons–Lee approach [37, 38] can be expressed as

$$\begin{aligned} {} \frac{F}{NkT} = \frac{\widetilde{F}}{\phi } = \ln \left( \frac{\Lambda ^3}{v_0}\right) + \ln \phi -1 + \mathfrak {s} [f] + \frac{2}{\pi }\frac{D}{L}\phi G_\textrm{P}(\phi )\langle \langle \tilde{v}_\textrm{excl}(\gamma ) \rangle \rangle ,{} \end{aligned}$$

(9.1)

We further focus on the excess free energy $\widetilde{F}^{\textrm{ex}}$, defined through $\widetilde{F} = \widetilde{F}_{\textrm{id}} + \widetilde{F}_{\textrm{exc}}$, with $\widetilde{F}_{\textrm{id}} = \phi \ln (\phi {\Lambda ^3}/{v_0}) - \phi$. As before, $\widetilde{F}=Fv_{0}/kT V$. The orientational entropy of the platelets  is (see Section 8.2.1) related to $\mathfrak {s} [f]$ and can be calculated using

$$\begin{aligned} {} \mathfrak {s} [f] = \int f (\boldsymbol{\Omega }) \ln [4 \pi f (\boldsymbol{\Omega })] \textrm{d} \boldsymbol{\Omega },{} \end{aligned}$$

(9.2)

which includes the orientational distribution function $f(\boldsymbol{\Omega })$, which is normalised according to

$$\begin{aligned} {} \int f(\boldsymbol{\Omega }) \textrm{d} \boldsymbol{\Omega } = 1,{} \end{aligned}$$

(9.3)

where $\boldsymbol{\Omega }$ is the solid angle (see Sect. 8.2.1). In Eq. 9.1, $\tilde{v}_\textrm{excl}(\gamma )$ is the excluded volume ${v}_\textrm{excl}(\gamma )$ between two hard platelets divided by $D^3$ at fixed interparticle angle $\gamma$:

$$\begin{aligned} {} \begin{aligned} \tilde{v}_\textrm{excl}(\gamma ) &= \frac{\pi }{2}|\sin (\gamma ) |+\frac{L}{D} \left\{ \frac{\pi }{2}+2E[\sin (\gamma )]+\frac{\pi }{2}\cos (\gamma )\right\} \\ &\qquad + 2\left( \frac{L}{D}\right) ^2 |\sin (\gamma ) |, \end{aligned}{} \end{aligned}$$

(9.4)

including the complete elliptic integral of the second kind E[x]. The average $\langle \langle \tilde{v}_\textrm{excl}(\gamma ) \rangle \rangle$ is defined as

$$\begin{aligned} {} \langle \langle \tilde{v}_\textrm{excl}(\gamma ) \rangle \rangle =\int \int \textrm{d}\boldsymbol{\Omega } \textrm{d}\boldsymbol{\Omega }' f(\boldsymbol{\Omega }) f(\boldsymbol{\Omega }') \tilde{v}_\textrm{excl}(\gamma ).{} \end{aligned}$$

(9.5)

The effects of higher order virial terms are incorporated via a Parsons–Lee  scaling factor $G_\textrm{P}$:

$$\begin{aligned} {} G_\textrm{P}(\phi ) = \frac{4-3\phi }{4(1-\phi )^2},{} \end{aligned}$$

(9.6)

The factor $G_\textrm{P}$ ensures that the ratios of the third and higher virial coefficients to the second virial coefficient are the same as for hard spheres.

At low concentrations the system is isotropic (I). In this isotropic phase, all platelets are oriented randomly and $f^\textrm{I}=(4\pi )^{-1}$ so that $\mathfrak {s}^\textrm{I}=0$. Within the Parsons–Lee  approximation, the isotropic excess free energy (Eq. (9.1)) becomes

$$\begin{aligned} {} \frac{{F}^{\textrm{exc}}_{\textrm{I}}}{NkT} = \frac{\widetilde{F}^{\textrm{exc}}_{\textrm{I}}}{\phi } = \frac{2}{\pi }\frac{D}{L}\phi G_\textrm{P}(\phi ) \tilde{v}_\textrm{excl}^\textrm{I},{} \end{aligned}$$

(9.7)

with $\tilde{v}_\textrm{excl}^\textrm{I}=\langle \langle \tilde{v}_\textrm{excl}(\gamma ) \rangle \rangle$, which becomes

$$\begin{aligned} {} \tilde{v}_\textrm{excl}^\textrm{I} \approx \frac{\pi ^2}{8}+\left( \frac{3\pi }{4}+\frac{\pi ^2}{4} \right) \frac{L}{D}+\frac{\pi }{2}\left( \frac{L}{D}\right) ^2,{} \end{aligned}$$

(9.8)

where the last term (of order $(L/D)^2$) is usually omitted because the focus is often on thin platelets  ($L/D \lesssim 0.1$), for which its magnitude is negligible.

The Helmholtz energy (Eq. (9.7)) directly provides the (dimensionless) osmotic pressure ($\widetilde{P}=Pv_{0}/k T$) and chemical potential ($\widetilde{\mu }=\mu /k T$) of the platelets in suspension (see Appendix A):

$$\begin{aligned} {} \widetilde{P}_{\textrm{I}} = \phi + \frac{2}{\pi }\phi ^2 \frac{D}{L} \frac{1-\phi /2}{(1-\phi )^3} \tilde{v}_\textrm{excl}^{\textrm{I}},{} \end{aligned}$$

(9.9)

$$\begin{aligned} {} \widetilde{\mu }_\textrm{I} =\ln \left( \frac{\Lambda ^3}{v_0}\right) + \ln \phi + \frac{2}{\pi } \frac{D}{L} \frac{8\phi -9\phi ^2+3\phi ^3}{4(1-\phi )^3} \tilde{v}_\textrm{excl}^\textrm{I},{} \end{aligned}$$

(9.10)

with the reference chemical potential $\ln (\Lambda ^3/v_\text {c})$.

Above a certain concentration, the platelets  spontaneously assume a preferred orientation, the nematic state. One may then compute the orientational distribution function (ODF) at each concentration numerically by minimising the Helmholtz free energy expression (Eq. (9.1)), while using the condition of Eq. 8.6. Since the nematic phases we consider are uniaxial in symmetry the solid angle $\boldsymbol{\Omega }$ only depends on the polar angle $\theta$ between a nematic director and the orientation of the platelet.

As in Chap. 8, Odijk’s Gaussian approximation  $f_\textrm{G}$ for the ODF $f(\theta )$ [42] is used:

$$\begin{aligned} {} {f_\textrm{G}(\theta )} = \frac{\kappa }{4 \pi } \,\exp \left[ -\frac{1}{2} \kappa \theta ^2 \right] ,{} \end{aligned}$$

(9.11)

which applies to angles $-\pi /2 \le \theta \le \pi /2$. The prefactor of the Gaussian ODF follows from Eq. (8.6). Insertion of Eq. (9.11) into Eq. (9.2) gives

$$\begin{aligned} {} \mathfrak {s}^\textrm{N} \approx \ln \kappa - 1.{} \end{aligned}$$

(9.12)

The normalised excluded volume in the nematic phase follows as [39]

$$\begin{aligned} {} \tilde{v}_\textrm{excl}^\textrm{N} = \langle \langle \tilde{v}_\textrm{excl}(\gamma ) \rangle \rangle ^\textrm{N} = 2\pi \frac{L}{D}+\frac{\pi }{2}\sqrt{\frac{\pi }{\kappa }} .{} \end{aligned}$$

(9.13)

Using the Gaussian ODF in the free energy (Eq. (9.1)), the excess nematic state free energy can now be written as

$$\begin{aligned} {} \frac{{F}^{\textrm{exc}}_\textrm{N}}{NkT} = \frac{\widetilde{F}^{\textrm{exc}}_\textrm{N}}{{\phi }} = \ln \kappa -1 + \phi G_\textrm{P}(\phi ) \left[ \frac{D}{L} \sqrt{\frac{\pi }{\kappa }}+4 \right] .{} \end{aligned}$$

(9.14)

The chemical potential and osmotic pressure in the nematic state can be easily obtained:

$$\begin{aligned} {} \widetilde{\mu }_\textrm{N} = \ln \left( \frac{\Lambda ^3}{v_0}\right) + \ln \phi + \ln \kappa -1 + \frac{8\phi -9\phi ^2+3\phi ^3}{4(1-\phi )^3} \left[ \frac{D}{L} \sqrt{\frac{\pi }{\kappa }} +4 \right] ,{} \end{aligned}$$

(9.15)

and

$$\begin{aligned} {} \widetilde{P}_\textrm{N} = \phi + \phi ^2 \frac{1-\phi /2}{(1-\phi )^3} \left[ \frac{D}{L} \sqrt{\frac{\pi }{\kappa }} +4 \right] .{} \end{aligned}$$

(9.16)

Once the variational parameter $\kappa$ is known, the free energy and various thermodynamic properties can be calculated explicitly. The $\kappa$ parameter follows from the minimisation of the free energy w.r.t. $\kappa$ (Eq. (8.24)). Applying this to Eq. (9.14) yields

$$\begin{aligned} {} \kappa = \frac{\pi }{4}\left( \frac{D}{L}\right) ^{2} \phi ^2 G^{2}_\textrm{P}(\phi ) .{} \end{aligned}$$

(9.17)

Insertion of this result into Eq. (9.14) gives

$$\begin{aligned} {} \frac{\widetilde{F}^\textrm{exc}_\textrm{N}}{\phi } = 2\ln \left( \frac{D}{L}\frac{\sqrt{\pi }}{2}\phi G_\textrm{P}(\phi )\right) + 4\phi G_\textrm{P}(\phi ) + 1,{} \end{aligned}$$

(9.18)

for the excess Helmholtz energy. The chemical potential of the hard platelets in the nematic phase follows as

$$\begin{aligned} {} \begin{aligned} \widetilde{\mu }_\textrm{N} = &\ln \left( \frac{\Lambda ^3}{v_0}\right) + 1+ 2\ln \left( \frac{D}{L}\frac{\sqrt{\pi }}{2} \right) + \ln \phi + 2\ln [\phi G_\textrm{P}(\phi )] \\ & + 4\phi G_\textrm{P}(\phi ) +\frac{2-\phi -\phi ^2+\phi ^3 /2}{(1-3\phi /4)(1-\phi )^3} , \end{aligned}{} \end{aligned}$$

(9.19)

and their osmotic pressure becomes

$$\begin{aligned} {} \widetilde{P}_\textrm{N} = \phi + \frac{2\phi -\phi ^2 - \phi ^3 + \phi ^4 /2}{(1-3\phi /4)(1-\phi )^3}.{} \end{aligned}$$

(9.20)

9.2.2.2 Isotropic–Nematic Phase Transition of Hard Platelets

It is now possible to compute the coexisting isotropic and nematic concentrations of hard platelets within the Parsons–Lee  approximation using the Gaussian form for the ODF. In general, coexisting concentrations (the binodals) follow from solving the concentrations for which the chemical potentials $\mu$ and osmotic pressures P are equal (see also Appendix A).

Theoretical Parsons–Lee predictions (curves) for the I–N phase coexistence concentrations are plotted in Fig. 9.7 as a function of L/D and are compared to Monte Carlo computer simulation results (data points). Two ‘flavours’ of the Parsons–Lee predictions are shown, using the Gaussian form of the ODF (dotted curves) and a numerical optimisation of the ODF (solid curves).

Fig. 9.7

I–N phase coexistence for pure hard platelets . Platelet concentration is given in terms of the quantity $c =N D^3/V = (4/\pi ) \phi D/L$. Curves are theoretical predictions using the Parsons–Lee (P–L) approximation  combined with a numerical minimisation (solid curves) and minimisation using the Gaussian approximation to the ODF (dotted curves). Data points are computer simulation results by Eppenga and Frenkel [34] for $L/D \rightarrow 0$ (Onsager limit) and Veerman and Fenkel [35] for other L/D values. Cut spheres were used to simulate the platelets , which slightly differs from the theoretical description of cylindrical platelets. Reprinted with permission from Ref. [43]. Copyright 2015 Taylor & Francis

Compared to the numerical approach, the Gaussian approximation predicts a wider coexistence region and slightly higher coexisting platelet concentrations, especially on the nematic side. This is similar to the situation for the I–N phase transition of hard spherocylinders as was discussed in Chap. 8 (see Fig. 8.4). Still, the relatively simple Gaussian ODF approach—combined with the Parsons–Lee —provides a reasonable description for the I–N phase transition of hard platelets. The Gaussian ODF approach deviates most near the limit $L/D \rightarrow 0$.

9.2.2.3 Lennard-Jones–Devonshire Cell Theory for the Columnar Phase

To predict the thermodynamic properties of a columnar phase, an extended cell theory by Lennard-Jones and Devonshire (LJD) can be used (see Wensink [44]). Within this model the columnar phase is described as a superposition of a 1D liquid and a 2D solid. The configurational (excess) free energy associated with the LJD cell theory is given by

$$\begin{aligned} {} \frac{F_{\textrm{LJD}}^\textrm{exc} }{NkT} = \frac{\widetilde{F} _{\textrm{LJD}}^\textrm{exc} }{\phi } = 2 \ln \left( \frac{ \bar{\Delta }_\textrm{C}^{-1} }{1- \bar{\Delta }_\textrm{C}^{-1}} \right) , {} \end{aligned}$$

(9.21)

where $\bar{\Delta }_\textrm{C}=\Delta _\textrm{C}/D$ is the (lateral) spacing, with $\Delta _\textrm{C}$ the nearest-neighbour distance (see Appendix C for details). Near close packing densities, Eq. (9.21) is expected to provide an accurate description of a 2D solid. Per column, it is assumed that the particles assume liquid-like configurations in one direction only. By applying the condition of single-occupancy, $\bar{\Delta }_\textrm{C}$ provides

$$\begin{aligned} {} \phi ^{*} \bar{\Delta }_\textrm{C}^{2} = \widetilde{\rho } , {} \end{aligned}$$

(9.22)

which relates the plate volume fraction $\phi = Nv_{0} / V$ (with $v_{0} = (\pi /4) L D^2$ as the particle volume) to the reduced linear density $\widetilde{\rho }$, where the reduced packing fraction $\phi ^{*} =\phi /\phi _{\textrm{cp}}$ with $\phi _{\textrm{cp}}=\pi /2\sqrt{3}\approx 0.907$ for the area fraction of discs at close packing.

The excess free energy of the columnar state now follows from adding the fluid and LJD contributions:

$$\begin{aligned} {} \begin{aligned} \frac{F^{\textrm{exc}}_\textrm{col}}{NkT} = \frac{\widetilde{F}_{\textrm{exc}}^\textrm{col}}{\phi } & = 2 \ln \left\{ \frac{3}{2} \frac{D}{L} \left( \frac{\phi ^{*} \bar{\Delta }_\textrm{C}^{2} }{1 - \phi ^{*} \bar{\Delta }_\textrm{C}^{2} } \right) \right\} \\ &\qquad - \ln (1 - \phi ^{*} \bar{\Delta }_\textrm{C}^{2})( 1 - \bar{\Delta }_\textrm{C}^{-1} )^2 . \end{aligned} {} \end{aligned}$$

(9.23)

The final step is to minimise the total free energy with respect to $\bar{\Delta }_\textrm{C}$ using

$$\begin{aligned} {} \frac{\partial F}{\partial \bar{\Delta }_\textrm{C}}= 0 ,{} \end{aligned}$$

(9.24)

which leads to

$$\begin{aligned} {} \bar{\Delta }_\textrm{C} =\frac{{{2^{1/3}}{\mathcal {K}^{2/3}} - {3^{1/3}}4\,{\phi ^{*}}}}{{{6^{2/3}}\,{\phi ^{*}}\,{\mathcal {K}^{1/3}}}},{} \end{aligned}$$

(9.25)

in which $\mathcal {K}$ is defined by

$$\begin{aligned} {} \mathcal {K} = 27 (\phi ^{*}) ^2 + [3 (\phi ^{*})^3 (32 + 243 \phi ^{*})]^{1/2} .{} \end{aligned}$$

(9.26)

With this, the free energy for the columnar state is fully specified. Unlike the nematic free energy, the columnar free energy is entirely algebraic and does not involve any implicit minimisation condition to be solved (see Eq. (C.6)). The pressure and chemical potential can be found in the usual way (see Appendix A). The nematic free energy can also be recast in closed algebraic form using a simple variational form for the ODF, similar to Eq. (C.7) (see Ref. [39]).

9.2.2.4 Theoretical Prediction of the Phase Diagram of Hard Platelets

Using the free energies for the different phase states for the isotropic, nematic and columnar phase states discussed above, standard thermodynamic relations can be applied to calculate the osmotic pressure and chemical potential of the pure platelet suspension for every phase state.

This enables the phase diagram for a system of hard platelets to be resolved, as is presented in Fig. 9.8. The relatively high excluded volume between thin platelets explains the I–N phase transition occurring at very low packing fractions for very small values of the aspect ratio ($L/D \rightarrow 0$). See also Fig. 9.7, which illustrates that $c \sim D/L$ at the I–N coexistence hardly varies. With increasing L/D, the I–N phase coexistence widens and its boundaries shift towards higher packing fractions. From Fig. 9.8 it also follows that the N–C phase coexistence concentrations barely depend on L/D. For sufficiently thick discs ($L/D \gtrsim 0.16$), transitions from an isotropic to a columnar phase occur without an intermediate nematic phase: thick discs are not sufficiently anisotropic to stabilise the occurrence of a nematic phase [39]. The grey vertical line in Fig. 9.8 at $L/D \approx 0.16$ indicates an I–N–C triple coexistence for hard colloidal platelets. Computer simulation data (symbols) from Refs. [35, 45,46,47] have been added to have an idea of the accuracy of the equations of states used. Qualitative agreement is found but quantitatively the theoretical phase transitions occur at somewhat higher disc concentrations as predicted theoretically. The phase diagram presented in Fig. 9.8 constitutes the pure platelet reference point for calculating the thermodynamics of platelet–depletant mixtures.

Fig. 9.8

Phase diagram of a monodisperse hard disc suspension. The grey triple line indicates the platelet aspect ratio of the I–N–C triple phase coexistence. For $L/D > 0.16$ there is only isotropic–columnar (I–C) phase coexistence. For $L/D < 0.16$ there are I–N and nematic–columnar (N–C) phase coexistences. Curves are computed using the Gaussian approximation for the nematic phase state; data points are Monte Carlo computer simulation results ($\circ$, [35, 45]), ($\bullet$, [46]), ($\blacksquare$, [47]). Figure is based upon Refs. [39, 48]

Exercise 9.1. What are the fundamental differences between the theoretical description of hard plates in this chapter and hard rod-like particles in Chapter 8?

9.3 Phase Behaviour of Hard Platelet–Penetrable Hard Sphere Mixtures

To predict the phase behaviour of hard platelets mixed with penetrable hard spheres (PHSs), the same steps are followed as outlined in Sect. 3.3 for hard spheres mixed with PHSs and in Sect. 8.4 for hard rods mixed with PHSs (see also Appendix A). The system of interest contains hard platelets (modelled as cylinders with diameter D and thickness L) in osmotic equilibrium with a reservoir that only contains PHSs (with diameter $\sigma$). The size ratio $q = \sigma / D$ and the depletion thickness of the PHS is constant ($\delta = \sigma /2$). The FVT expression for the semi-grand potential in case of ideal depletants can be written as [49]

$$\begin{aligned} {} \widetilde{\Omega } = \widetilde{F}_{0}- \frac{v_\textrm{0}}{v_\textrm{d}}\alpha \widetilde{P}^\textrm{R}. {} \end{aligned}$$

(9.27)

Here, $F_{0}$ is the free energy of the pure hard platelet dispersion, $v_0$ is the volume of the platelets, $v_\textrm{d}$ is the volume of the depletants, $\alpha$ is the free volume fraction for the depletants, and

$$\begin{aligned}{} \widetilde{P}^\textrm{R} = \frac{P^\textrm{R}v_\textrm{d}}{kT} =\phi _\textrm{d}^\textrm{R},{} \end{aligned}$$

is the dimensionless Van ’t Hoff osmotic pressure of an ideal solution of depletants in reservoir R having a volume fraction $\phi _\textrm{d}^\textrm{R}$. The depletant volume fraction in the system follows from

$$\begin{aligned}{} \phi _\textrm{d}&=\alpha \phi _\textrm{d}^\textrm{R}.{} \end{aligned}$$

The free volume fraction available for depletants in the system $\alpha = \langle V_\text {free}\rangle _0 / V$ and can be calculated using the relation $\alpha = e^{-W/(kT)}$, where W is the reversible work for inserting the PHSs in the hard platelet suspension [50]. Following scaled particle theory (SPT) [51, 52], W is calculated by scaling the size of the PHSs as $\lambda \sigma$. In the case $\lambda \ll 1$, it is unlikely that the platelets and PHSs overlap. Hence,

$$\begin{aligned} {} W(\lambda ) = - kT \ln [1- n v_\text {excl}(\lambda )]\qquad \text {for } \lambda \ll 1,{} \end{aligned}$$

(9.28)

where $v_\text {excl}$ is the excluded volume between a PHS and a hard platelet, given by

$$\begin{aligned} {} v_\text {excl}(\lambda ) = \frac{\pi }{4} D^2 \lambda \sigma + \frac{\pi }{4} L(D + \lambda \sigma )^2 + \frac{\pi ^2}{8} D (\lambda \sigma )^2 + \frac{\pi }{6} (\lambda \sigma )^3.{} \end{aligned}$$

(9.29)

In the opposite limit $\lambda \gg 1$, the inserted PHS is very large; in good approximation, W is then equal to the work required to create a cavity with volume $\frac{\pi }{6}(\lambda \sigma )^3$ against the pressure P of the hard platelets:

$$\begin{aligned} {} W = \frac{\pi }{6}(\lambda \sigma )^3 P \qquad \text {for } \lambda \gg 1.{} \end{aligned}$$

(9.30)

In SPT, these two limiting cases are connected by expanding W as a series in $\lambda$ (Eqs. (3.32) and (8.50)), which yields an expression for $W(\lambda = 1)$ [48, 53, 54]:

$$\begin{aligned} {} \alpha = (1-\phi )\exp [-Q] , {} \end{aligned}$$

(9.31)

with

$$\begin{aligned} {} \begin{aligned} Q &= q \left( \frac{D}{L}+\frac{\pi q D}{2\,L}+q+2\right) y \\ & \qquad + 2 q^2 \left[ \frac{1}{4}\left( \frac{D}{L}\right) ^2 +\frac{D}{L}+1\right] y^2 + \frac{2}{3}\frac{D}{L} q^3 \widetilde{P}, \end{aligned}{} \end{aligned}$$

(9.32)

where y is defined by Eq. (3.39e). It should be noted that Zhang, Reynolds and Van Duijneveldt [55] obtained $\alpha$ from SPT for a mixture of cut spheres and PHSs. For infinitely thin discs, the expressions for $\alpha$ obtained by Zhang et al. [55] and obtained here are identical. González García et al. [48] compared Eq. (9.31) (with Q given by Eq. (9.32)) with computer simulation results of Refs. [46, 56] and found good agreement between theory and simulations.

With all the components required to calculate the grand potential at hand, determination of phase coexistence is straightforward in principle (see Appendix A). This theoretical approach is now compared to phase diagrams computed by Zhang et al. [46] for $L/D=0.1$ and $q=0.2$. They calculated the phase diagram also using FVT, but employed Monte Carlo simulations to obtain the thermodynamic properties of the pure platelet system and measured the free volume fraction in such simulations by a trial insertion method. In Fig. 9.9 the phase diagrams are plotted in terms of a dimensionless fugacity $z = n_\textrm{d}^\textrm{R} D^3$ of PHSs versus platelet concentration $nD^3$. The overall topology of the phase diagram from theory agrees with the one obtained from the hybrid simulation method.

Fig. 9.9

Comparison of phase diagrams for mixtures of plate-like particles and PHSs for $L/D=0.1$ and $q=0.2$. Left: phase diagram for hard discs mixed with PHSs obtained from FVT with theoretical expressions for the thermodynamic properties of the pure plate system and SPT results for the free volume fractions [48]. Right: phase diagram for cut sphere–PHS mixtures by Zhang et al. [46] obtained from Monte Carlo simulations for the thermodynamic properties of pure plate systems combined with FVT with free volume fractions measured by a trial insertion method. Reprinted with permission from Ref. [48]. Copyright 2018 Taylor & Francis

The details of the phase diagrams depend on L/D and q. A few typical representative phase diagrams, calculated using the theory outlined above, are presented in Fig. 9.10a (in depletant reservoir concentrations along the ordinate) and Fig. 9.10b (ordinate plotted as system depletant concentrations). In Fig. 9.10a, the phase diagram for $L/D=0.15$ and $q=0.158$ shows that the I–N and N–C biphasic regions in the phase diagrams join in an I–N–C three-phase coexistence upon increasing the depletant concentration due to the widening of both I–N and N–C phase coexistences. By increasing q we encounter an I$_1$–I$_2$–N three-phase coexistence, in addition to a I$_1$–N–C three-phase coexistence for $L/D=0.15$ and $q=0.25$. Note that these triple lines lead to triple coexistences (see Fig. 9.10b).

For smaller L/D and q a nematic–nematic phase coexistence also becomes possible, for instance for $L/D=0.02$ and $q=0.04$. Now there are I–N$_1$–N$_2$ and I–N$_2$–C triple lines. As can be seen in the lower panels (b) the triple regions may be accessible experimentally because the concentrations are realistic and the regions are not too narrow. For a full overview of the possible phase diagrams, including a four-phase coexistence , see Refs. [48, 53].

Fig. 9.10

Phase diagrams for platelet–polymer mixtures for various L/D and various q values as indicated [48]. a Diagrams in the $\{\phi ,\phi _\textrm{d}^\textrm{R}\}$ polymer reservoir phase space. Horizontal lines mark multiple-phase coexistence. b As in a, but in the $\{\phi ,\phi _\textrm{d}^\textrm{S}\}$ phase space. The coloured triangles indicate the system representation of the triple point lines on the top panels. The inset plot zooms into the low depletant concentration regime. Reprinted with permission from Ref. [48]. Copyright 2018 Taylor & Francis

9.4 Experimentally Observed Phase Behaviour of Mixtures Containing Colloidal Platelets

In this section selected experimental examples of phase behaviour that includes colloidal platelets and depletants are discussed. The focus is on nonadsorbing polymers as depletants in Sects. 9.4.1 and 9.4.2, although we also illustrate some studies on colloidal mixtures of platelets and added colloidal spheres in Sects. 9.4.3–9.4.5, 9.4.4.

9.4.1 Sterically Stabilised Gibbsite Platelets Mixed with Polymers

As we have seen in Sects. 9.1 and 9.2 sterically-stabilised gibbsite platelets dispersed in toluene display a rich liquid crystal phase behaviour. With increasing concentration the isotropic phase, isotropic–nematic phase coexistence, the nematic phase, nematic–columnar phase coexistence and the columnar phase are observed [2]. Van der Kooij et al. [24] found that the depletion attraction, brought about by the addition of nonadsorbing polymer, enriches the phase behaviour of these platelet suspensions even further. They used sterically stabilised gibbsite platelets with an average diameter of 208 nm and thickness (including the thickness of the stabilising grafted polymer layer) of 14 nm, leading to an aspect ratio L/D of $1/15 \approx 0.067$.

The nonadsorbing polymer used by Van der Kooij et al. [24] is a trimethylsiloxy terminated polydimethylsiloxane (PDMS) with a weight-averaged molar mass $M_\textrm{w}=4.2 \cdot 10^5$ g/mol. The radius of gyration$R_\textrm{g}$ of this polymer is estimated as 33 nm. Hence, the ratio of the polymer coil diameter over the plate diameter is about 0.3.

Fig. 9.11

Experimental phase diagram of gibbsite platelet–PDMS polymer mixtures [24] in toluene. Phase boundaries are indicated by solid curves, their shape and position being based on the data points they enclose, and on the consistency with surrounding phase regions. Curves are dashed in cases where the location of the phase boundary is not known precisely due to local scarcity of data points. Reprinted with permission from Ref. [24]. Copyright 2000 APS

The observed phase behaviour is presented in Fig. 9.11. The overall topology of the plate–polymer phase diagram is characterised by a wealth of one-, two- and three-phase equilibria and even a four-phase equilibrium. Each of these phase regions can be rationalised, based on possible combinations of the I$_1$, I$_2$, N, and C phases.

In Fig. 9.12 examples of phase-separated plate–polymer mixtures are shown as observed between crossed polarisers. A calculated phase diagram for L/D $=$ 0.0673 and $2\delta /D$ $=$ 0.317 using the theoretical approach outlined in Sect. 9.3 is shown in Fig. 9.13.

Fig. 9.12

Phase separated gibbsite platelet–PDMS polymer mixtures in toluene as observed between crossed polarisers [24]. Depicted are a triple phase coexistence I$_1$–I$_2$–N (top to bottom: dilute isotropic, concentrated isotropic I$_2$ and nematic N at the composition $\phi _\textrm{plate} = 0.31$ and $c_\textrm{pol}= 2.0$ g/L); b triple phase coexistence I$_1$-N-C (dilute isotropic, nematic, columnar for $\phi _\textrm{plate} = 0.44$ and $c_\textrm{pol} = 2.2$); and c four-phase coexistence I$_1$–I$_2$–N–C (dilute isotropic, concentrated isotropic, nematic and columnar phase for $\phi _\textrm{plate} = 0.31$ and $c_\textrm{pol} = 5.3$). Reprinted with permission from Ref. [24]. Copyright 2000 APS

Fig. 9.13

Predicted phase diagram using free volume theory for L/D = 0.0673 and $2\delta /D$ $=$ 0.317, calculated using the approach outlined in Sect. 9.3, following Ref. [48]. The I$_2$–N and single-phase N regions are magnified in the upper inset, while the lower inset shows the N–C region

The richness of the observed phase diagram of colloidal plate dispersions with added nonadsorbing polymer chains (see Fig. 9.11) raises the question of how to explain the observed topology, including the four-phase region and the three-, two- and single-phase regions surrounding it.  It contrasts to some degree with the theoretical prediction of Fig. 9.13. At first sight, such four-phase coexistence seems to conflict with the phase rule of Gibbs, which (at a given temperature) limits the maximum number of coexisting phases to three for a system of effectively two components (platelet–polymer). See Refs.  [57, 58] for further discussion.

An explanation for the observed phase diagram can be found by considering the effect of gravity [59]. The height distribution of colloidal particles in a dispersion is influenced by gravity, particularly when the sedimentation length $l_\text {sed}$ (Eq. (1.1)) is much smaller than the sample height. At each height the system is thermodynamically different because locally, at each position, the external gravity field provides a different potential energy to the system. This implies that gravity has an influence on the system over the length scale of the sample, and therefore mediates the number of coexisting phases present, as well as their stacking. One can account for this using a so-called local density approximation (LDA), which assumes that at any height there is a local equilibrium.

Exercise 9.2. Argue why very rich apparent multi-phase coexistence is expected for a polydisperse colloidal dispersion in the field of gravity in case of a significant solvent-particle density difference.

Fig. 9.14

Three-phase sedimentation equilibrium in a system of sterically stabilised gibbsite platelets [47]. a The complete sample between crossed polarisers, where the upper right part is digitally enhanced to visualise the I–N interface. The columnar phase contains a dark region at the upper right of the phase, probably due to the orientation of the platelets along the sample walls. Although not clearly visible, the N–C interface is horizontal and sharp. b Columnar phase illuminated with white light to capture the red Bragg reflections. Reprinted with permission from Ref. [47]. Copyright 2004, with permission from AIP Publishing

We first consider the effect of gravity in a system of platelets without added polymer. Van der Beek et al. [47] observed that a suspension of sterically stabilised gibbsite platelets, which is initially an isotropic–nematic biphasic sample, develops a columnar phase on the bottom after prolonged standing (Fig. 9.14). By employing the theoretical approach of Wensink and Lekkerkerker [59], Van der Beek et al. [47] presented a simple calculation of the heights of the phases based on the sedimentation–diffusion equilibrium using the LDA. Consider a suspension of monodisperse, hard discs with number density n(z) at position z and buoyant mass $m^*=v_0 \Delta \rho$, where $v_0$ is the colloid particle volume and $\Delta \rho$ is the difference in the mass densities of the colloidal particles and solvent. The condition for sedimentation–diffusion equilibrium reads as

$$\begin{aligned} {} -\left( \frac{\partial \Pi }{\partial n}\right) _{T,\mu _\text {solvent}}\frac{\partial n}{\partial z}= m^* g n .{} \end{aligned}$$

(9.33)

It is convenient to use reduced quantities. The osmotic pressure of platelets is given by $\widetilde{\Pi }=\Pi D^3/k T$, the reduced concentration by $\tilde{n} = nD^3$, and positions can be scaled with the sedimentation length $\ell _\textrm{sed}$. Substituting these expressions in Eq. (9.33) yields

$$\begin{aligned} {} -\frac{1}{\tilde{n}}\left( \frac{\partial \widetilde{\Pi }}{\partial \tilde{n}}\right) _{T,\mu _\text {solvent}}\textrm{d}\tilde{n} = \frac{\textrm{d}z}{\ell _\textrm{sed}} .{} \end{aligned}$$

(9.34)

The height $H=z_\textrm{top}-z_\textrm{bottom}$ can be found for a single-phase state by integrating (Eq. (9.34)) from the bottom to the top of that phase:

$$\begin{aligned} {} H= \int _{z_\textrm{bottom}}^{z_\textrm{top}} \textrm{d}z = - \ell _\textrm{sed} \int _{\tilde{n}_\textrm{bottom}}^{\tilde{n}_\textrm{top}} \frac{1}{\tilde{n}}\left( \frac{\partial \widetilde{\Pi }}{\partial \tilde{n}}\right) _{T,\mu _\text {solvent}} \textrm{d}\tilde{n}.{} \end{aligned}$$

(9.35)

The average concentration $\overline{n}$ of this phase now follows as

$$\begin{aligned} {} \overline{n}=\frac{\int _{{z}_\textrm{bottom}}^{{z}_\textrm{top}} \tilde{n}(z) \textrm{d}z}{\int _{z_\textrm{bottom}}^{z_\textrm{top}} \textrm{d}z} =\frac{1}{H} \int _{\tilde{n}_\textrm{bottom}}^{\tilde{n}_\textrm{top}} \tilde{n}(z)\left( \frac{\partial z}{\partial \tilde{n}}\right) \textrm{d}\tilde{n}.{} \end{aligned}$$

(9.36)

Using (Eq. (9.34)), this yields

$$\begin{aligned} {} \overline{n} =\frac{\ell _\textrm{sed}}{H} \left[ \widetilde{\Pi }(\tilde{n}_\textrm{bottom}) - \widetilde{\Pi }(\tilde{n}_\textrm{top}) \right] .{} \end{aligned}$$

(9.37)

For a sedimentation equilibrium that includes multi-phase coexistences, Eqs. (9.35) and (9.37) apply to every phase. The total sample height $H_\textrm{sample}$ can be written as the sum of all individual phase heights $H^{i}$:

$$\begin{aligned} {} H_\textrm{sample} = \sum _{i} H^{\,i\,}.{} \end{aligned}$$

(9.38)

The average overall sample concentration $\overline{n}_\textrm{sample}$ can now be written as

$$\begin{aligned} {} \overline{n}_\textrm{sample} =&\frac{1}{H_\textrm{sample}} \sum _{i} H^{\,i\,}\, \overline{n}^{\,i} , \end{aligned}$$

(9.39a)

$$\begin{aligned} =& \frac{\ell _\textrm{sed}}{H_\textrm{sample}} \sum _{i} \left[ \widetilde{\Pi }(\tilde{n}_\textrm{bottom}^{\,i}) - \widetilde{\Pi }(\tilde{n}_\textrm{top}^{\,i}) \right] , {} \end{aligned}$$

(9.39b)

where the average phase concentration of phase i is given by $\overline{n}^{\,i}$. For two coexisting phases A and B (where A is on top of B) the osmotic pressures are equal:

$$\begin{aligned} {} \widetilde{\Pi } (\tilde{n}_\textrm{bottom}^\textrm{A}) = \widetilde{\Pi } (\tilde{n}_\textrm{top}^\textrm{B});{} \end{aligned}$$

(9.40)

so, Eqs. (9.39b) and (9.37) lead to

$$\begin{aligned} {} \overline{n}_\textrm{sample}= \frac{l_\textrm{sed}}{H_\textrm{sample}} \left[ \widetilde{\Pi }(\tilde{n}_\textrm{bottom}^{\textrm{sample}}) - \widetilde{\Pi }(\tilde{n}_\textrm{top}^{\textrm{sample}}) \right] .{} \end{aligned}$$

(9.41)

Using the above equations and computer simulation data for the equation of state  for cut spheres (L/D = 1/20) from Zhang, Reynolds and Van Duijneveldt [46] the phase diagram for colloidal platelets with [59] can be calculated. Results are plotted in Fig. 9.15. The calculated heights of the I, N and C phases are in reasonable agreement with the experimental data for the sample shown in Fig. 9.14.

Fig. 9.15

a Phase diagram for colloidal platelets with L/D = 0.05 in a gravitational field. Plotted is the reduced sample height $H/\ell _\textrm{sed}$ versus the overall plate volume fraction $\phi _0$. The three-phase region opens up at $H=11.15 \ell _\textrm{sed}$. b Concentration profile of a sample with overall volume fraction $\phi _0$= 0.157 and vessel height $H=30 \ell _\textrm{sed}$ (corresponding to the open dot in a). Plotted is the relative height z/H as a function of $\phi$. The I–N and N–C phase boundaries are indicated by the horizontal dotted lines. Reprinted with permission Ref. [59]. Copyright 2004 Institute of Physics (IOP)

The role of gravity on the phase behaviour of mixtures of colloidal plates with nonadsorbing polymer is more complicated but follows the same lines. Wensink and Lekkerkerker [59] performed calculations for L/D = 1/20 and a ratio of the polymer coil diameter over the plate diameter of 0.355 to mimic the experimental system of Ref. [24]. They obtained the phase diagram shown in Fig. 9.16 for a sample with a height of 15 mm, a gravitational length of $\ell _\textrm{sed}$ = 0.9 mm for the platelets, and $\ell _\textrm{sed} \rightarrow \infty$ for the polymers. All the experimentally observed multi-phase equilibria shown in Fig. 9.11 appear—even the four-phase equilibrium.

Fig. 9.16

Phase diagram under gravity of a plate–polymer mixture with $L/D=1/20$ and $q=0.355$ in the representation of dimensionless polymer fugacity $z_\textrm{p} D^3 = {n_\textrm{d}^\textrm{R}}^3 D^3$ versus volume fraction representation for a vessel height of 15 mm ($H= 16.67 \ell _\textrm{sed}$). Reprinted with permission from Ref. [59]. Copyright 2004 IOP

Fig. 9.17

Hypothetical bulk phase diagram of a binary mixture of two components 1 and 2 in terms of the chemical potentials of these components $\mu _1$ and $\mu _2$. Two phases are possible in this diagram: A (left region) and B (right region). The solid curve is a binodal at which two phases A and B coexist. The dashed line is the sedimentation path along the sample with height H. Reprinted with permission from Ref. [61]. Copyright 2015 IOP

De las Heras and Schmidt [60] also used the local density approximation (LDA) but applied it to account for multiple sedimenting components. The LDA implies that at any z there is a chemical potential $\mu _i (z)$ for each component that can be expressed as [61, 62]

$$\begin{aligned} {} \mu _i (z) =\mu ^{\textrm{bulk}}_i - z g m_{i,\textrm{eff}},{} \end{aligned}$$

(9.42)

where $\widetilde{\mu }^{\textrm{bulk}}_i$ is the bulk chemical potential of each component i. In Eq. (9.42) an external potential due to gravity is given by $zgm_{i,\textrm{eff}}$, with $m_{i,\textrm{eff}}$ denoting the buoyant mass of component i. This means that along the sample height there is a spectrum of chemical potentials for every component, termed the sedimentation path. A possible scenario for a binary mixture composed of components 1 and 2 is sketched in Fig. 9.17. Using an appropriate model for the thermodynamics of the bulk (e.g. DFT, FVT, TPT), De las Heras and Schmidt related the bulk phase diagram to its phase stacking in the field of gravity [60, 61, 63]. This enables one to predict the phase states in the field of gravity.

Along the dashed line of Fig. 9.17 the chemical potentials of both components from top to bottom are now position-dependent due to the external field of gravity. In this hypothetical example, two-phase transitions occur along the sedimentation path. At the bottom there is a certain phase A of mixed components 1 and 2. However, for a certain range of chemical potentials, phase B is the preferred phase state. Close to the top, phase A is again the preferred phase state. This explains the possibility of a floating phase (B in this case) between two A phases [63, 64]. The chemical potential differences $\Delta \mu _i$ are directly related to the height of phase B:

$$\begin{aligned} {} \Delta \mu _i = -h_\textrm{B} g m_{i,\textrm{eff}}.{} \end{aligned}$$

(9.43)

In this case the stacking sequence is ABA.

De las Heras and Schmidt [61] applied the LDA approach to various colloidal mixtures, including mixtures of sterically stabilised gibbsite platelets (D $=$ 208 nm, L $=$ 14 nm; $L/D \approx 0.067$) with PDMS polymers ($R_\textrm{g}$ $=$ 33 nm; $M_\textrm{w}=4.2 \cdot 10^5$ g/mol, so $q \approx 0.32$) [24] as were discussed earlier (see the experimental phase diagram in Fig. 9.11). The difference with the approach presented above is that De las Heras and Schmidt [61] took into account both sedimenting components explicitly, which is especially essential for the description of multi-component colloidal mixtures [65, 66].

The bulk diagram for this mixture was computed by De las Heras and Schmidt [61] using the perturbation approach of Zhang, Reynolds and van Duijneveldt [46] for $L/D = 0.05$ and $q=0.35$. It is presented in Fig. 9.18i in terms of a chemical potential plot ($\widetilde{\mu }_\textrm{p}$, $\widetilde{\mu }_\textrm{c}$), where $\widetilde{\mu }_\textrm{p}$ is the (normalised) chemical potential of the nonadsorbing polymers and $\widetilde{\mu }_\textrm{c}$ is the chemical potential of the colloidal platelets. This phase diagram is relatively simple. There are two isotropic phases, a nematic phase and a columnar phase. Additionally, there are two triple points (I$_1$–I$_2$–N and I$_1$–N–C; $\blacktriangle$) and an isostructural isotropic critical point ($\circ$).

Fig. 9.18

Bulk phase diagrams for mixtures of colloidal platelets and nonadsorbing polymers in terms of (i) ($\widetilde{\mu }_\textrm{p}$, $\widetilde{\mu }_\textrm{c}$) plots and (ii) concentrations {$c_\textrm{p}$, $\phi$}. Reprinted with permission from Ref. [61]. Copyright 2015 IOP

Each sedimentation path has an associated stacking sequence, for instance, CNI$_1$. The complete set of paths can be represented in a stacking diagram, e.g. in the plane of average chemical potential along the path. Each point in the stacking diagram corresponds to a sedimentation path in bulk. Boundaries in the stacking diagram between different stacking sequences are connected to paths that cross a binodal. A tiny change of such a path can alter the stacking sequence.

In Fig. 9.18(ii) the bulk phase diagram is presented, which follows from (i) but is now given in terms of the polymer concentration ($c_\textrm{p}$ (g/l)) and volume fraction of platelets $\phi$. In terms of concentrations the triple points now become (small) regions and the coexistence lines are now wide regions.

In Fig. 9.19 the stacking diagrams are plotted at heights of $H =$ 1, 2 and 4 cm. These phase diagrams of finite heights are much richer and also include a quadruple region (I$_1$–I$_2$–N–C). The phase diagrams for various heights are shown in Fig. 9.19(iv) and reveal a quadruple region (I$_1$–I$_2$–N–C) that is absent in (iii). Comparison with Fig. 9.11 shows that the phase diagrams in Fig. 9.19, and especially those for H = 2 and 4 cm, are much closer to what is observed experimentally. These phase diagrams of finite heights exhibit a quadruple region (I$_1$–I$_2$–N–C) that was observed in Ref. [24].

A full quantitative comparison requires exact knowledge of the height distributions of both particles in the field of gravity. This is a challenge since there are differences in sample preparation methods and solvent evaporation is possible. Further, polydispersity also affects the details of the experimental phase diagram.

Fig. 9.19

Sample height-dependent phase diagrams for mixtures of (both) sedimenting colloidal platelets and nonadsorbing polymers in terms of (iii) ($\widetilde{\mu }_\textrm{p}$, $\widetilde{\mu }_\textrm{c}$) plots and (iv) concentrations {$c_\textrm{p}$, $\phi$}. Reprinted with permission from Ref. [61]. Copyright 2015 IOP

9.4.2 Mixtures of Magnesium Aluminide Layered Double Hydroxide Platelets and Polymers

Liu et al. [6] observed isotropic–nematic phase coexistence in aqueous suspensions of $\text {Mg}_{2}\text {Al}$-layered double hydroxide platelets (D = 120 nm, L = 3.2 nm). The same research group [67] also studied the phase behaviour of mixtures of $\text {Mg}_{2}\text {Al}$-layered double hydroxide platelets and nonadsorbing polyvinylpyrrolidone (PVP, $M_\textrm{p}=630$ kg/mol). The radius of gyration of the PVP used in water is 42 nm, hence, in the dilute polymer concentration regime, one estimates a depletion thickness (see Chaps. 2 and 4) $\delta \approx 1.13R_\textrm{g}=47$ nm. The observations of the phase behaviour are shown in Fig. 9.20.

Fig. 9.20

Phase states and coexistence region observed in mixtures of $\text {Mg}_{2}\text {Al}$ layered double hydroxide platelets and PVP. The solid curves represent estimated phase boundaries between different regions: liquid phase (L), dilute isotropic region (I$_1$), concentrated isotropic region (I$_2$), faint birefringent (dilute nematic) phase (N$_1$), concentrated nematic phase (N$_2$) and sediment phase (S). Data points are experimental observations. Reprinted with permission from Ref. [67]. Copyright 2009 ACS

In Fig. 9.21 the transient phase transition for a sample is shown as a function of time.

Fig. 9.21

Illustration of the evolution of the phase states of a mixture of $\text {Mg}_{2}\text {Al}$ layered double hydroxide platelets (20 wt %) and PVP (0.3 wt %), as observed using crossed polarisers [67]. a Just after preparation, b after 2 days, c after 30 days and d after 55 days. e Schematic representation of the multi-phase coexistence regions in the sample. Reprinted with permission from Ref. [67]. Copyright 2009 ACS

The predicted theoretical phase diagram for $L/D=3.2/120$ and $q=94/120=0.78$ is plotted in Fig. 9.22. Calculations were done using FVT as outlined in this chapter.

Fig. 9.22

Phase diagram of hard platelets ($L/D=3.2/120=0.0267$) mixed with nonadsorbing polymer chains modelled as PHSs; $q=94/120=0.78$, calculated using the approach outlined in Sect. 9.3, following Refs. [48, 54]. Inset: magnified I$_2$–N region

This predicted theoretical phase diagram reveals an expected I$_1$–I$_2$–N three-phase coexistence region, but not a four-phase I$_1$–I$_2$–N$_1$–N$_2$ equilibrium, as is observed (see Fig. 9.20). The difference can probably be explained by effects of gravity and polydispersity.

9.4.3 Gibbsite Plate—Silica Sphere Mixtures

Doshi et al. [63, 68, 69] studied the phase behaviour of aqueous suspensions containing gibbsite platelets mixed with alumina-coated silica spheres with diameter $\sigma$. See Fig. 9.23 for an illustration of the alumina coating, chemically bound to a silica sphere.

Fig. 9.23

Sketch of the alumina coating at the silica surface. Reprinted from Ref. [69] with permission of the authors, copyright 2011

To inhibit double layer repulsions between the particles 5 mM NaCl was added, and commercially available stabilisers (Solplus D450 and Solsperse 41,000) were adsorbed on the particle surface to create near-hard particle interactions. The dimensions of the particles studied are given in Table 9.2; bare and effective dimensions are quoted. The effective dimensions of the spheres and plates take into account the additional layer that is due to steric stabilisation or the Debye length of 4 nm at 5 mM of NaCl.

Table 9.2 Mean dimensions of the plates and spheres discussed in this section. Dimensions include size dispersity from TEM/AFM and average bare and effective dimensions of plates and spheres from scattering data

Figure 9.24a presents a TEM micrograph of the gibbsite plates and Klebosol 30CAL25 alumina-coated silica spheres, while panels (b) and (c) show phase diagrams of these gibbsite plate–silica sphere mixtures. The effective size ratios for a mixture of gibbsite platelets with Klebosol 30CAL25 silica spheres are $L_\textrm{eff}/D_\textrm{eff}$ = 0.08 and $\sigma _\textrm{eff}/D_\textrm{eff}$ = 0.21, so on theoretical grounds we only expect an I–N transition. This is confirmed experimentally, as can be seen in Fig. 9.24d, e.

For a similar mixture of gibbsite  platelets and silica particles but with the larger Klebosol 30CAL50 silica spheres the phase behaviour is considerably richer. Now the ratio of $d_\textrm{eff}$/$D_\textrm{eff}$ = 0.47 and, in addition to an isotropic–nematic phase, both an isotropic–isotropic and an isotropic–isotropic–nematic phase transitions are expected. The latter is indeed observed, as is shown in Fig. 9.24e.

De las Heras et al. [61, 63] have shown that sedimentation–diffusion equilibria of binary colloidal mixtures can involve phase transitions, which can lead to complex phase stacks, such as the sandwich of a floating nematic layer between top and bottom isotropic phases. This may explain what is observed in mixtures of silica spheres and gibbsite platelets.

Fig. 9.24

Plate–sphere mixtures composed of gibbsite platelets and Klebosol silica spheres. Particle dimensions: a, b, d silica $\sigma$ $=$ 30 nm, c, e silica $\sigma$ = 74 nm. Platelet dimensions $\langle D \rangle$ = 203 nm ± 20 %, $\langle L \rangle$ = 5 nm ± 20 %. d, e $\phi _\text {silica} = 0.05$, gibbsite platelet volume fractions as indicated. Reprinted with permission from a, b Ref. [68], copyright 2012 IOP; c Ref. [69], copyright 2011; d, e Ref. [63], copyright 2012 Springer Nature

9.4.4 Mixtures of Zirconium Phosphate Platelets and Silica Spheres

Chen et al. [70, 71] studied the phase behaviour of aqueous mixtures of zirconium phosphate (ZrP) platelets with silica spheres. In Ref. [71] they used ZrP (D $=$ 704.3 nm, L $=$ 2.68 nm) and added silica spheres ($\sigma$ $=$ 162.7 nm).

Exercise 9.3. In aqueous dispersion platelets are typically charged. How would the I–N phase transition be affected upon adding salt?

The size ratio $q=\sigma /D$ of the diameters of the silica spheres and ZrP plates is 0.23; so we expect—in addition to the biphasic equilibria I$_1$–I$_2$, I$_2$–N and I$_1$–N—a triphasic phase triangle region I$_1$–I$_2$–N. The latter is indeed observed (see the left panel of Fig. 9.25).

Fig. 9.25

Left: experimental phase diagram of aqueous mixtures of ZrP platelets and silica spheres [71]. Right: photographs of the (red) state points A and C in the phase diagram. The numbers on the tubes indicate the number of hours lapsed after preparing the samples. Reprinted with permission from Ref. [71]. Copyright 2017 RSC

Again, a nematic phase is observed floating between two isotropic phases. Note that the amounts of the phases I$_1$, I$_2$, and N reflect the positions of the samples A, B and C denoted in the three-phase triangle denoted by points O (I$_1$), P (I$_2$) and Q (N). From applying the lever rule [72] to the triangle region of the phase diagram (see left panel of Fig. 9.25) it follows that state point A, which is close to the I$_1$ vertex, will have a relatively large amount of phase I$_1$. Similarly, state point C, which is close to the I$_2$ vertex, has a relatively large amount of phase I$_2$. This is indeed seen experimentally (Fig. 9.25, right panel). Chen et al. [70] also studied the phase behaviour of a mixture of ZrP platelets and silica spheres with a size ratio $q=\sigma /D = 0.013$. As expected, an I–N$_1$–N$_2$ phase equilibrium is now observed (Fig. 9.26).

Fig. 9.26

Triphasic I–N$_1$–N$_2$ equilibrium observed for an aqueous mixture of ZrP platelets and silica spheres with a sphere–platelet size ratio $q=\sigma /D$ of 0.013. The ZrP volume fraction is 0.0063 and the silica concentration is 2 wt % (corresponding to a volume fraction of 0.015) Reprinted with permission from Ref. [70]. Copyright 2015 RSC

9.4.5 Effect of Added Silica Nanoparticles on the Nematic Liquid Crystal Phase Formation in Beidellite Suspensions

While virtually all smectite clays dispersed in water form gels at very low concentrations, aqueous suspensions of beidellite [11] as well as nontronite [8] exhibit a unique behaviour with a first order isotropic–nematic phase transition before gel formation. Landman et al. [73] studied the modification of the phase behaviour of beidellite suspensions upon addition of colloidal silica spheres. TEM images of their beidellite platelets are reproduced in Fig. 9.27a. Figure 9.27b shows a TEM micrograph of the silica spheres. Images giving some indications of the phase behaviour of the pure beidellite suspensions are presented in Fig. 9.28.

Fig. 9.27

TEM micrographs of a beidellite platelets and b silica Ludox AS-40 spheres used. Reprinted with permission from Ref. [73]. Copyright 2014 ACS

Fig. 9.28

Aqueous beidellite suspensions observed between crossed polarisers one month after preparation. Volume fractions of clay ($\phi _\textrm{clay}$) are as follows: a 0.27%; b 0.32%; c 0.35%; d 0.37%; e 0.40%; f 0.41%. Note that the tiny bright layer just at the glass bottom of the vial in a is not due to a nematic phase but is a light reflection artefact. Reprinted with permission from Ref. [73]. Copyright 2014 ACS

Fig. 9.29

Mixed beidellite/silica suspensions observed between crossed polarisers one month after preparation: $\phi _\textrm{clay}$ = 0.41% and (from left to right) $\phi _\textrm{silica}$ = 0, 0.034 and 0.138%. Reprinted with permission from Ref. [73]. Copyright 2014 ACS

Note that sample (a) ($\phi = 0.27\%$) is still in the isotropic phase sample, (e) ($\phi = 0.40\%$) is completely nematic, and sample (f) ($\phi = 0.41\%$) is a nematic gel. Adding silica nanoparticles to the nematic gel sample (f) leads to an isotropic–nematic phase equilibrium as is displayed in Fig. 9.29.

Fig. 9.30

Experimental phase diagram of aqueous beidellite/silica suspensions. (${\circ }$) Isotropic, ( ) biphasic and ($\star$) gelled states are indicated. The boundary between the isotropic and the biphasic samples was obtained from naked-eye observations of the test-tubes, while the sol–gel transition line was determined by rheological measurements. Reprinted with permission from Ref. [73]. Copyright 2014 ACS

The modification of the beidellite phase diagram due to the addition of silica nanoparticles is indicated in the images of Fig. 9.30. Note that the addition of a tiny amount of silica nanoparticles (volume fraction of $10^{-3}$) has a significant effect on the phase diagram.