Phase Behaviour of Colloidal Binary Hard Sphere Mixtures

Abstract

In the previous chapters we considered the effect of added nonadsorbing polymers on the phase behaviour (Chap. 4) and interface (Chap. 5) appearing in suspensions of spherical colloids. The depletion effect is also operational in other types of mixtures, such as binary mixtures composed of large and small (hard) spheres where two big spheres in a sea of small spheres are brought together (Fig. 6.1). As the big spheres get close, the smaller spheres can no longer enter the gap between the big ones. The small particles then push the big spheres together.

6.1 Introduction to Binary Mixtures of Hard Spheres

In the previous chapters we considered the effect of added nonadsorbing polymers on the phase behaviour (Chap. 4) and interface (Chap. 5) appearing in suspensions of spherical colloids. The depletion effect is also operational in other types of mixtures, such as binary mixtures composed of large and small (hard) spheres where two big spheres in a sea of small spheres are brought together (Fig. 6.1). As the big spheres get close, the smaller spheres can no longer enter the gap between the big ones. The small particles then push the big spheres together.

The addition of nonadsorbing small hard spheres to a dispersion of big hard spheres can be treated within free volume theory (FVT) [1,2,3]. Original FVT treatments [1, 2] were limited to a specific range of sufficiently asymmetric hard sphere mixtures, say, $0.05 \lesssim q \lesssim 0.2$, with $q= d_2 / d_1$. For larger q values, binary colloidal crystals AB$_2$ and AB$_{13}$ (consisting of large colloids A with diameter $d_{1}$ and small colloids B with diameter $d_{2}$) have been observed in the size range $0.425 \le d_{2} / d_{1} \le 0.60$ (AB$_2$) and $0.485 \le d_{2} / d_{1} \le 0.62$ (AB$_{13}$) [4,5,6], and larger size ratios [7,8,9]. The situation gets even more complex in the case of binary mixtures of charged spheres [10, 11]. Such structures cannot easily be treated within FVT.

The added small colloids may be of a similar colloid shape (i.e. spheres) or a different shape such as rod-like colloids, as will be discussed in Chap. 7. The focus in this chapter is on rather asymmetric binary hard sphere mixtures, i.e. $q \lesssim 0.2$, although extensions towards larger q values are possible [3]. In Sect. 6.2 two free volume theory approaches are outlined and compared to computer simulation results, followed by comparisons with experiments in Sect. 6.3.

Fig. 6.1

Illustration of the depletion effect in a mixture of two big hard spheres and small hard spheres in 2D. As the big spheres approach each other the small spheres are no longer able to enter the gap between them. As a consequence, the small spheres impose an effective attractive force between the big spheres

6.2 Free Volume Theory for Binary Hard Sphere Mixtures

In 1964 Lebowitz and Rowlinson [12] showed that within the Percus–Yevick treatment of hard sphere fluids [13], binary hard sphere mixtures are completely miscible for all concentrations and size ratios. This proof was later extended by Vrij [14] to hard sphere mixtures with an arbitrary number of components. Until 1990, it was generally accepted that hard sphere mixtures do not phase separate into two fluid phases. In 1991 Biben and Hansen [15] showed, on the basis of a thermodynamically self-consistent theory, that dense binary mixtures of hard spheres with diameters $d_1$ and $d_2$ with a size ratio $q \lesssim 0.2$ show a spinodal instability, while phase separation into two fluid phases was not predicted.

Two years later Lekkerkerker and Stroobants [1], guided by their work on colloid–polymer mixtures [16], conjectured that the addition of small hard colloidal spheres will lead to a fluid–solid phase separation, preempting a metastable gas–liquid phase separation that includes a spinodal instability. Such metastable gas–liquid phase transitions for polymer–colloid mixtures have already been discussed in Chap. 4. In 1999 this conjecture of a metastable gas–liquid phase coexistence was confirmed by computer simulations of Dijkstra, van Roij and Evans [17].

Exercise 6.1. The effective depletion interaction mediated by hard spheres compared to penetrable hard spheres (PHSs) was discussed in Sects. 2.1 and 2.3. Based upon the difference between these interactions, why can one expect that the physical properties of a binary hard sphere mixture are more complex than those of hard sphere–PHS mixtures?

The physical origin of phase separation in highly asymmetric hard sphere mixtures is the depletion interaction, similar to what we encountered in Chaps. 3 and 4. Throughout this chapter we refer to small hard spheres in the reservoir (R) as the depletants. The free volume treatment given in Chap. 3 for mixtures of hard spheres and PHSs can be extended to the case of (highly) asymmetric hard sphere mixtures [1, 2]. Here, we first present the most straightforward extension possible [1], followed by the more rigorous approach [18] of Opdam et al. [3].

6.2.1 Simple FVT Extension for a Binary Hard Sphere Mixture

The osmotic equilibrium system considered is depicted in Fig. 6.2. We assume the depletion layers are equal to the radii of the small hard spheres. As discussed in Chap. 3, the semi-grand potential of a system with volume V and temperature T of a mixture of $N_1$ colloidal particles and $N_\textrm{2}$ depletants with chemical potential $\mu _2$ can be obtained by applying the exact expression Eq. (3.20) to this case:

$$\begin{aligned} {} \varOmega (N_\textrm{1}, V, T, \mu _\textrm{2}) = F_0 (N_\textrm{1}, V, T) - \int _{-\infty }^{\mu _\textrm{2}} N_\textrm{2} (\mu _\textrm{2}') \textrm{d} \mu _\textrm{2}'. {}\end{aligned}$$

(6.1)

Here, $F_0(N_1, V, T)$ is the Helmholtz energy of the pure system of hard colloidal particles 1, while $\varOmega (N_\textrm{1}, V, T, \mu _\textrm{2})$ is the grand potential of a mixture of $N_\textrm{1}$ hard spheres 1 and $N_\textrm{2}$ hard spheres 2 in a volume V at given chemical potential of the depletant hard spheres 2.

Fig. 6.2

Reprinted with permission from Ref. [3]. Copyright AIP Publishing 2021

Osmotic equilibrium system for a dispersion of big and small hard spheres in the system in equilibrium with a reservoir that consists of a small hard sphere dispersion. The semi-permeable membrane (dashes) allows permeation of small hard spheres but is impermeable to the big hard spheres. The shells indicate the excluded volume surrounding the particles for the centres of the small hard spheres.

Using the Widom insertion theorem [19] (see Sect. 3.3.2), the chemical potential of small hard spheres 2 in the system can be written as

$$\begin{aligned} {} \mu _\textrm{2} = \mu _\textrm{2}^0 + kT \ln \frac{N_\textrm{2}}{\langle V_{\text {free}}(N_1, N_2)\rangle } , {}\end{aligned}$$

(6.2)

where $\langle V_{\text {free}}(N_1, N_2)\rangle$ is the ensemble-averaged free volume for the small hard spheres 2 in the system, containing hard spheres 1 and hard spheres 2.

For the reservoir,

$$\begin{aligned} {} \mu _\textrm{2} = \mu _\textrm{2}^0 + kT \ln \frac{N_\textrm{2}^\textrm{R}}{\langle V_{\text {free}}(N_{2}^\textrm{R})\rangle } , {}\end{aligned}$$

(6.3)

with $\langle V_{\text {free}}(N_{2}^\textrm{R})\rangle$ the ensemble-averaged free volume for the small hard spheres 2 in the reservoir of hard spheres 2.

By equating the chemical potentials of component 2 in the system (6.2) and in the reservoir (6.3), we obtain

$$\begin{aligned} {} N_{2} = N_{2}^\textrm{R} \frac{\langle V_{\text {free}}(N_1, N_2)\rangle }{\langle V_{\text {free}}(N_2^\textrm{R})\rangle } . {}\end{aligned}$$

(6.4)

We now make the following approximations:

$$\begin{aligned} {} \frac{\langle V_{\text {free}}(N_1, N_2)\rangle }{\langle V_{\text {free}}(N_2^\textrm{R})\rangle } \approx \frac{\langle V_{\text {free}}(N_1)\rangle }{V^\textrm{R}} \approx \frac{\langle V_{\text {free}}\rangle _0}{V^\textrm{R}} , {}\end{aligned}$$

(6.5)

where $\langle V_{\text {free}}\rangle _0$ is the undistorted free volume of an added small hard sphere in the system of $N_1$ large hard spheres in a volume V. The first approximation implies that the free volume available for hard spheres 2 in the reservoir, $\langle V_{\text {free}}(N_2^\textrm{R})\rangle$, equals the total reservoir volume $V^\textrm{R}$. Secondly, the free volume available for hard spheres 2 in the system, $\langle V_{\text {free}}(N_1, N_2)\rangle$, is assumed to only depend on the number of hard spheres 1, so it equals $\langle V_{\text {free}}(N_1)\rangle$. The third approximation says that the configurations of the hard spheres 1 are not affected by the hard spheres 2.

Combination of Eqs. (6.4) and (6.5) results in

$$\begin{aligned} {} N_{2} = n_{2}^\textrm{R} \langle V_{\text {free}} \rangle _0, {}\end{aligned}$$

(6.6)

with $n_{2}^\textrm{R}=N_{2}^\textrm{R} /V^\textrm{R}$. Insertion of the Gibbs–Duhem equation (see Eq. (A.12)),

$$\begin{aligned} {} n_\textrm{2}^\textrm{R} \textrm{d} \mu _\textrm{2} = \textrm{d}P^\textrm{R}, {}\end{aligned}$$

(6.7)

into Eq. (6.1) now leads to the following simple expression for the semi-grand potential of the asymmetric hard sphere mixture:

$$\begin{aligned} {} \varOmega (N_1, V, T, \mu _2) = F_0 (N_1, V, T)- P^\textrm{R} \langle V_{\text {free}}\rangle _0, {}\end{aligned}$$

(6.8)

with $P^\textrm{R}$ as the pressure of the small hard spheres in the reservoir. The quantity $\langle V_{\text {free}}\rangle _0$ can now be approximated by the same expression as the free volume of an added PHS (Eq. (3.38)),

$$\begin{aligned} {} \langle V_{\text {free}}\rangle _0 = \alpha V. {}\end{aligned}$$

(6.9)

For the free volume fraction $\alpha$, one can use expression Eq. (3.38):

$$\begin{aligned} {} \alpha = (1-\phi ) \exp \left[ ( - a y - b y^2 - c y^3) \right] , {}\end{aligned}$$

(6.10)

with

$$\begin{aligned} {} \begin{array}{lll} a &{}=&{} 3q + 3q^2 + q ^3 \\ b &{}=&{} \frac{9}{2} q^2 + 3q^3\\ c &{}=&{} 3q^3, \end{array} {}\end{aligned}$$

(6.11)

and

$$\begin{aligned} {} y = \frac{\phi _1}{1 - \phi _1} , {}\end{aligned}$$

(6.12)

with $\phi _1 = n_1 v_1 = n_1 \pi d_1^3 /6$ denoting the volume fraction of the large spheres. The volume of a hard sphere 1 is defined as $v_1$.

In dimensionless form, Eq. (6.8) can be written as

$$\begin{aligned} {} \widetilde{\varOmega } = \widetilde{F}_0 -\frac{\alpha }{q^{3}} \widetilde{P}^\textrm{R}, {}\end{aligned}$$

(6.13)

with $\widetilde{\varOmega }=\varOmega v_1 / (kTV)$, $\widetilde{F}_0 = {F}_0 v_1 / (kTV)$ and $kT \widetilde{P}^\textrm{R}={P}^\textrm{R} v_2$, with $v_2$ the volume of hard sphere 2. Basically, we account for the hard interactions between the small spheres via $P^\textrm{R}$. For the pressure $P^\textrm{R}$ in the reservoir (which for the case of PHSs is given by the ideal gas law) we now use the SPT expression Eq. (3.37),

$$\begin{aligned} {} \frac{P^\textrm{R}}{n_2^\textrm{R} kT} = \frac{1 + \phi _2^\textrm{R} + (\phi _2^\textrm{R})^2}{(1- \phi _2^\textrm{R})^3}. {}\end{aligned}$$

(6.14)

Here, $n^\textrm{R}$ is the number density of small hard spheres in the reservoir and $\phi ^\textrm{R} = n^\textrm{R} v_2 = n^\textrm{R} \pi d_2^3 / 6$ the volume fraction of the small spheres in the reservoir. Hence we can rewrite Eq. (6.14) as

$$\begin{aligned} {} {\widetilde{P}^\textrm{R}} = \frac{\phi _2^\textrm{R} + (\phi _2^\textrm{R})^2 + (\phi _2^\textrm{R})^3}{(1- \phi _2^\textrm{R})^3}. {}\end{aligned}$$

(6.15)

We now have all the ingredients that make up the semi-grand potential (Eq. (6.8)) of the asymmetric hard sphere mixture. From it we obtain the pressure of the system P and the chemical potential $\mu _1$ of the large hard spheres using standard thermodynamic relations:

$$\begin{aligned} {} P = - \left( \frac{\partial \varOmega }{\partial V} \right) _{N_1, T, \mu _2} = P^0 + P^\textrm{R} \left( \alpha - n_1 \frac{\partial \alpha }{\partial n_1} \right) , {}\end{aligned}$$

(6.16)

and

$$\begin{aligned} {} \mu _1 = \left( \frac{\partial \varOmega }{\partial N_1} \right) _{V,T,\mu _2} = \mu _1^0 - P^\textrm{R} \frac{\partial \alpha }{\partial n_1}, {}\end{aligned}$$

(6.17)

where $P^0$ and $\mu _1^0$ are the pressure and chemical potential of the pure (big) hard sphere system (for which we use the expressions derived in Chap. 3). The dimensionless forms of Eqs. (6.16) and (6.17) are given in Eqs. (3.45) and (3.46). We can now calculate the phase behaviour of the asymmetric hard sphere mixture from the coexistence equations

$$\begin{aligned} {} \mu _1^\textrm{I} (n_1^\textrm{I}, \mu _2) = \mu _1^{\textrm{II}} (n_1^{\textrm{II}},\mu _2) {}\end{aligned}$$

(6.18)

and

$$\begin{aligned} {} P^\textrm{I} (n_1^\textrm{I}, \mu _2) = P^{\textrm{II}} (n_1^{\textrm{II}}, \mu _2). {}\end{aligned}$$

(6.19)

Analogously to Eqs. (3.47) and (3.48), the expressions for $\mu$ and P can be simplified to

$$\begin{aligned} {} \widetilde{\mu } = \widetilde{\mu }^0 + \widetilde{P}^\textrm{R} \; g(\phi _1) {}\end{aligned}$$

(6.20)

and

$$\begin{aligned} {} \widetilde{P} = \widetilde{P}^0 + \widetilde{P}^\textrm{R} \; h(\phi _1). {}\end{aligned}$$

(6.21)

The fluid–solid binodal can be obtained from

$$\begin{aligned} {} \widetilde{P}^\textrm{R} = \frac{\widetilde{\mu }_s (\phi _{1,s}) - \widetilde{\mu }_f (\phi _{1,f})}{g (\phi _{1,f}) - g (\phi _{1,s})} = \frac{\widetilde{P}_s (\phi _{1,s}) - \widetilde{P}_{f} (\phi _{1,f})}{h(\phi _{1,f}) - h(\phi _{1,s})}. {}\end{aligned}$$

(6.22)

Fig. 6.3

Phase diagrams of big hard sphere–small hard sphere mixtures. Data points are redrawn Monte Carlo simulation results [17] guided by grey curves. Phase diagrams are given for $q = 0.05$ (a, d), $q = 0.1$ (b, e) and $q = 0.2$ (c, f) in the reservoir representation (a–c) and the system representation (d–f). The black curves show the phase coexistence concentrations predicted by FVT using the approach of [1]. Phase regions are indicated in (a–c): the fluid phase (F), the solid phase (S), the fluid–solid coexistence region (F+S) and the isostructural solid–solid coexistence  region (S+S)

From an experimental point of view, we are interested in phase diagrams in the ($\phi _1$, $\phi _2$) representation. By using the relation

$$\begin{aligned}{} n_2 = - \frac{1}{V} \left( \frac{\partial \varOmega }{\partial \mu _2} \right) _{N_1,V,T} = \alpha n_2^\textrm{R}, {}\end{aligned}$$

or

$$\begin{aligned}{} \phi _2 = \alpha \phi _2^\textrm{R}, {}\end{aligned}$$

we can directly convert the ($\phi _1$, $\phi _2^\textrm{R}$) phase diagram to the ($\phi _1$, $\phi _2$) representation. In Fig. 6.3 we give the results for q = 0.05, 0.1 and 0.2. The Monte Carlo computer simulation results of Dijkstra et al. [17] have been added to the figure for comparison. The agreement is reasonable, although not as good as the agreement between FVT and computer simulations for the hard sphere–PHS system. For low q, the FVT predictions actually start to deviate quite significantly, as we shall see in Fig. 6.8. A qualitative difference also is that computer simulations reveal solid–solid equilibria at high $\phi _1$ for small q, which are not predicted by FVT.

Note that for the small size ratios, for which the FVT of asymmetric hard sphere mixtures is applicable, gas–liquid demixing (also predicted by FVT, not shown) is metastable with respect to the fluid–solid transition. The presence of this metastable phase does, however, affect the physical properties of the mixtures. Similar to mixtures of colloidal hard spheres and nonadsorbing polymers [20,21,22], asymmetric hard sphere mixtures display interesting gel and glass states that are supposed to be connected with the metastable gas–liquid phase transition [23, 24].

It follows that the simple predictions of this free volume theory approach are reasonable, but become especially inaccurate at small q. This is mainly due to the fact that the expression for the free volume fraction (Eq. (6.10)) already deviates somewhat from computer simulations. This is shown in Fig. 6.4a, in which the predictions of Eq. (6.10) (dashed) are compared to computer simulation results (data points) by Dijkstra et al. [17].

The description of hard spheres as depletants has been accounted for in a limited manner for a number of aspects. Ideally, one would like to:

• Account for the fact that the configurations of the big hard spheres are distorted by the small hard spheres,

• Describe the free volume fraction in the solid since it fundamentally differs from that in a fluid phase, and

• Incorporate excluded volume interactions between the hard depletants in both reservoir and system.

The last aspect also accounts for accumulation effects, leading to repulsive contributions to the depletion interaction as we saw in Sect. 2.3, which are not incorporated in the theory described above. Parts of these improvements were incorporated by Opdam et al. [3] and are discussed next.

6.2.2 Rigorous FVT Approach for a Binary Hard Sphere Mixture

Consider again the osmotic equilibrium system depicted in Fig. 6.2. Careful inspection of this sketch and comparison with Fig. 3.5 shows that the small hard sphere depletants in Fig. 6.2 now also have a hard-core excluded volume interaction with each other. This implies that the FVT treatment should be adapted not only in the sense that the depletant osmotic pressure is larger than that given by Van‘t Hoff’s law, but also one would like to account for the excluded volume interactions between the hard sphere depletants. These hard-core interactions influence the thermodynamic properties already in the reservoir. This affects the free volume fraction of small hard spheres in both reservoir and system.

We therefore return to the general expression for the semi-grand potential (Eq. (6.1)) and focus on Eq. (6.4). Using the definitions $\phi _{2}=N_{2}v_{2}/V$, $\phi _{2}^\textrm{R}=N_{2}^\textrm{R}v_2/V^\textrm{R}$, $\alpha (\phi _1, \phi _2)= \langle V_{\text {free}}(N_1, N_2)\rangle /V$ and $\alpha ^{\textrm{R}}(\phi _\textrm{2}^{\textrm{R}})= \langle V_{\text {free}}(N_2^\textrm{R})\rangle /V^\textrm{R}$, the following implicit expressions [3] are obtained for the fluid phase:

$$\begin{aligned} {} \phi _\textrm{2} = \phi _\textrm{2}^{\textrm{R}} \, \frac{\alpha _{\textrm{f}}(\phi _\textrm{1}, \phi _\textrm{2})}{\alpha ^{\textrm{R}}(\phi _\textrm{2}^{\textrm{R}})}, {}\end{aligned}$$

(6.23)

and for the solid phase:

$$\begin{aligned} {} \phi _\textrm{2} = \phi _\textrm{2}^{\textrm{R}} \, \frac{\alpha _{\textrm{s}}(\phi _\textrm{1}, \phi _\textrm{2})}{\alpha ^{\textrm{R}}(\phi _\textrm{2}^{\textrm{R}})}. {}\end{aligned}$$

(6.24)

The free volume fractions $\alpha ^{\textrm{R}}$ in Eqs. (6.23) and (6.24) are no longer unity and $\alpha _{\textrm{f}}$ and $\alpha _{\textrm{s}}$ depend on the volume fractions of both the small hard sphere depletants $\phi _\textrm{2}$ and of large hard sphere $\phi _\textrm{1}$. The volume fraction of depletants in the system $\phi _\textrm{2}$, in coexistence with the reservoir with a certain depletant volume fraction $\phi _\textrm{2}^\text {R}$, can be found by solving Eq. (6.23) and/or (depending on the phase states involved) Eq. (6.24). Substituting Eqs. (6.23) and (6.24) into the definition of the semi-grand potential given by Eq. (6.1) and applying the Gibbs–Duhem relation (see Appendix A.2) finally yields expressions for the semi-grand potential of a binary hard sphere mixture. For the fluid phase it yields

$$\begin{aligned} {} \widetilde{\varOmega }_{\textrm{f}} = \widetilde{F}_{0 \textrm{,f}} - \int \limits _{0}^{\phi _\textrm{2}^{\textrm{R}}} \frac{\alpha _{\textrm{f}} }{\alpha ^{\textrm{R}} } \, \left( \frac{\partial \widetilde{P}^{\textrm{R}}}{\partial \phi _\textrm{2}^{\textrm{R}'}} \right) \, \textrm{d} \phi _\textrm{2}^{\textrm{R}'}, {}\end{aligned}$$

(6.25)

and for the solid phase it gives

$$\begin{aligned} {} \widetilde{\varOmega }_{\textrm{s}} = \widetilde{F}_{0 \textrm{,s}} - \int \limits _{0}^{\phi _\textrm{2}^{\textrm{R}}} \frac{\alpha _{\textrm{s}} }{\alpha ^{\textrm{R}} } \, \left( \frac{\partial \widetilde{P}^{\textrm{R}}}{\partial \phi _\textrm{2}^{\textrm{R}'}} \right) \, \textrm{d} \phi _\textrm{2}^{\textrm{R}'}, {}\end{aligned}$$

(6.26)

where the dimensionless quantities from Appendix A are applied and the integration variable d$\mu '_{\textrm{d}}$ in Eq. (6.1) is changed to the volume fraction of depletants in the reservoir d$\phi _\textrm{2}^{\textrm{R}'}$ using the Gibbs–Duhem relation.

The free volume fraction for depletants in the reservoir $\alpha ^\text {R}$ can be evaluated using the steps taken in Sect. 3.3.3. First, we apply Eq. (3.29) to relate $\alpha ^{\textrm{R}}$ to the work of inserting a hard sphere into a hard sphere dispersion in the reservoir $W^{\textrm{R}}$:

$$\begin{aligned} {} \alpha ^{\textrm{R}} = \textrm{e}^{-W^{\textrm{R}}/kT}. {}\end{aligned}$$

(6.27)

For $W^{\textrm{R}}$ Eq. (3.35) is used:

$$\begin{aligned} {} \frac{W^{\textrm{R}}}{kT} = - \ln [1-\phi _2] + \frac{6 \phi _2}{1- \phi _2} + \frac{9 \phi _{2}^2}{2(1 - \phi _{2})^2} + \frac{\pi d_{2}^3 P}{6kT}. {}\end{aligned}$$

(6.28)

One could take Eq. (3.37) for P, but we follow Opdam et al. [3] and use the more accurate Carnahan–Starling equation (Eq. (3.1)).

Next, expressions for the free volume fraction of hard sphere depletants in the binary system for both the fluid and solid phases are presented. The free volume fraction in the fluid phase of a binary hard sphere mixture can be determined using the work for depletant insertion in a binary mixture given by SPT [25] (see also Chaps. 3 and 4), resulting in

$$\begin{aligned} {} \alpha _{\textrm{f}} = \exp -\bigg [\textrm{ln}&\left( \frac{1}{1 - \phi _\textrm{1} - \phi _\textrm{2}}\right) + \frac{3 q \phi _\textrm{1} + 3 \phi _\textrm{2}}{1 - \phi _\textrm{1} - \phi _\textrm{2}}\nonumber \\ &\quad \; + \frac{1}{2} \left\{ \frac{6 q^{2} \phi _\textrm{1} + 6 \phi _\textrm{2}}{1 - \phi _\textrm{1} - \phi _\textrm{2}} + \left( \frac{3 q \phi _\textrm{1} + 3 \phi _\textrm{2}}{1 - \phi _\textrm{1} - \phi _\textrm{2}} \right) ^{2} \right\} +q^{3}\widetilde{P}\bigg ], {}\end{aligned}$$

(6.29)

where $\widetilde{P}$ is the osmotic pressure of the binary mixture of hard spheres. It is possible to use an SPT result for $\widetilde{P}$ [25]; however, we again follow [3] and use an expression for $\widetilde{P}$ given by the Boublík–Mansoori–Carnahan–Starling–Leland equation of state for binary hard sphere mixtures [26, 27]:

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

(6.30)

Exercise 6.2. Think of arguments to explain why inclusion of excluded volume interaction in the reservoir increases $\phi _\textrm{2}$ in the system at fixed $\phi _\textrm{2}^{\textrm{R}}$.

The above approach cannot be followed for the solid phase since the osmotic pressure of a hard sphere solid containing smaller hard spheres is not known. The free volume fraction in the solid phase is approximated here by considering an FCC crystal of the larger spheres and assuming that the small spheres behave as a fluid in the free space left by the large spheres, which is valid for highly asymmetric binary sphere mixtures [17, 28] with $q \lesssim 0.2$. With this assumption, the free volume fraction $\alpha _\text {s}$ can be approximated by a product of the free volume fraction of the hard sphere solid and the free volume fraction in the small sphere fluid that surrounds the larger spheres. This yields

$$\begin{aligned} {} \alpha _{\textrm{s}}(q,\phi _\textrm{1},\phi _\textrm{2}) = \alpha _{\textrm{s}}(q, \phi _\textrm{1}) \, \alpha _{\textrm{f}}(q=1, \phi _\textrm{2}^{\dagger }), {}\end{aligned}$$

(6.31)

where $\phi _\textrm{2}^{\dagger } = \phi _\textrm{2} / (1-\phi _\textrm{1})$ is the effective volume fraction of the small spheres in the space that is not occupied by large spheres, and $\alpha _{\textrm{s}}(q, \phi _\textrm{1})$ is given by the free volume fraction in the solid phase, which can be determined using geometrical arguments [29]. It reads

$$\begin{aligned} {} \alpha _{\textrm{s}} = {\left\{ \begin{array}{ll} 1-\phi _\textrm{1} \, \widetilde{v}_{\textrm{exc}}^{0} &{} \text {for }\phi _\textrm{1} < \phi _\textrm{1}^{*}\\ 1-\phi _\textrm{1} \, \widetilde{v}_{\textrm{exc}}^{*} &{} \text {for }\phi _\textrm{1}^{*} \le \phi _\textrm{1} < 2^{3/2} \, \phi _\textrm{1}^{*}\\ 0 &{} \text {otherwise}\text {.} \end{array}\right. } {}\end{aligned}$$

(6.32)

It is assumed that the centres of the spherical colloids are perfectly located on the FCC lattice points, where $\phi _\textrm{1}^{*} = \phi _\textrm{1}^{\textrm{cp}} \, / \,\widetilde{v}_{\textrm{exc}}^{0}$ denotes the volume fraction of large spheres above which the depletion zones overlap, with $\widetilde{v}_{\textrm{exc}}^{0} = (1 + q)^{3}$. Furthermore, the normalised excluded volume $\widetilde{v}_{\textrm{exc}}^{*}$ in Eq. (6.32) is given by

$$\begin{aligned} {} \widetilde{v}_{\textrm{exc}}^{*} = \widetilde{v}_{\textrm{exc}}^{0} - 6 \left[ 1 + q - \left( \frac{\phi _\textrm{1}^{\textrm{cp}}}{\phi _\textrm{1}} \right) ^{\frac{1}{3}} \right] ^{2} \left[ 1 + q + \frac{1}{2} \left( \frac{\phi _\textrm{1}^{\textrm{cp}}}{\phi _\textrm{1}} \right) ^{\frac{1}{3}} \right] . {}\end{aligned}$$

(6.33)

Equation (6.32) formally only holds for small q, since it assumes there is no multiple overlap of depletion zones.

Furthermore, this expression for the free volume fraction in the solid does not accurately account for the overlap between the depletion zones of large and small spheres. To take this overlap into account one can assume that

$$\begin{aligned} {} \alpha _{\textrm{f}}(q,\phi _\textrm{1},\phi _\textrm{2}) = \alpha _{\textrm{f}}(q, \phi _\textrm{1}) \, \alpha _{\textrm{f}}(q=1, \phi _\textrm{2}^{\dagger }). {}\end{aligned}$$

(6.34)

Combining Eqs. (6.34) and (6.31) gives

$$\begin{aligned} {} \alpha _{\textrm{s}}(q,\phi _\textrm{1},\phi _\textrm{2}) = \alpha _{\textrm{s}}(q, \phi _\textrm{1})\frac{\alpha _{\textrm{f}}(q,\phi _\textrm{1},\phi _\textrm{2})}{\alpha _{\textrm{f}}(q, \phi _\textrm{1})}. {}\end{aligned}$$

(6.35)

In Fig. 6.4a the predicted volume fraction of hard sphere depletants in the system $\phi _\textrm{2}$ is plotted as a function of the reservoir volume fraction $\phi _\textrm{2}^\text {R}$ for the fluid phase, given by $\alpha /\alpha ^\text {R}$ (solid curves). The dashed curves are predictions using Eq. (6.10), derived for hard spheres mixed with PHS depletants (Eq. (3.38)). Also shown in Fig. 6.4 are computer simulation data (symbols) by Dijkstra et al. [17]. It is clear that inclusion of excluded volume interactions between the hard sphere depletants in the reservoir gives a more accurate description of the computer simulation data.

In Fig. 6.4b the hard sphere depletant volume fraction in the system $\phi _\textrm{2}$ is plotted as a function of the reservoir volume fraction $\phi _\textrm{2}^\text {R}$ for the solid phase. As mentioned above, this relation is given by the ratio $\alpha /\alpha ^\text {R}$. Also shown in Fig. 6.4b is the prediction from the original simple FVT extension (dashed) and the computer simulation data (symbols) by Dijkstra et al. [17]. The results from the rigorous FVT approach follow the simulation data remarkably well, which confirms the validity of the equations obtained for the free volume fractions in the fluid phase and the solid phase given by Eqs. (6.29) and (6.35).

Fig. 6.4

Volume fraction of small spheres in the system $\phi _\textrm{2}$ versus their reservoir volume fraction $\phi _\textrm{2}^\text {R}$. a Fluid phase for q = 0.1. Solid curves are the result of Eqs. (6.23) and (6.29); symbols denote data from Monte Carlo simulations by Dijkstra et al. [17] for $\phi _\textrm{1}$ = 0.1 ( , 0.2 ( ), 0.3 ( ) and 0.5 ( ). b Solid phase for $\phi _\textrm{1}$ = 0.74. Solid curves are from Eqs. (6.24) and (6.35); symbols are from Monte Carlo simulations by Dijkstra et al. [17] for q = 0.05 ($\square$) and 0.1 ($\circ$). In both panels, the dashed curves are for penetrable hard spheres (Eq. (6.10) for fluid and Eq. (6.32) for solid)

All elements required to calculate the semi-grand potentials given in Eqs. (6.25 and (6.26) are now available. This enables one to compute the phase behaviour for binary mixtures of hard spheres by using the standard thermodynamics relations given in Appendix A and by solving Eqs. (6.18) and (6.19).

Fig. 6.5

Phase diagrams of binary hard sphere mixtures as in Fig. 6.3 for q’s as indicated in the reservoir representation (a–c) and the system representation (d–f). The black curves show the binodals determined with rigorous FVT, dashed curves indicate metastable phase coexistence, and the grey data points are the results of direct coexistence simulations from Dijkstra et al. [17]. The grey lines guide the eye. For $q = 0.05$ and $q = 0.1$, Eq. (6.35) was used for $\alpha _\text {s}$, and a numerical procedure was used for $q = 0.2$ [3]. Phase regions are indicated in (a–c): the fluid phase (F), the solid phase (S), the fluid–solid coexistence region (F+S) and the isostructural solid–solid coexistence region (S+S)

Predicted phase diagrams are shown in Fig. 6.5 for binary hard sphere mixtures with size ratios $q = 0.05, 0.1$ and 0.2 using the theory outlined in this subsection. The free volume fraction in the solid phase of the binary mixture is described using Eq. (6.35). For $q = 0.05$ and $q = 0.1$, the free volume fraction of the one-component solid $\alpha _\text {s}$ given by Eq. (6.32) is used. Multiple overlap of depletion zones is possible for $q = 0.2$ at high densities (for details see Ref. [3]). It is noted that the phase diagram for $q = 0.2$ determined with Eq. (6.32) showed no significant difference from the phase diagram calculated with the numerically computed $\alpha$, which is most likely due to the fact that the deviations between both methods are quite small for this size ratio (see Ref. [3]).

A comparison of the theoretical phase diagrams and phase coexistence data obtained from direct coexistence simulations from Dijkstra, Van Roij and Evans [17] for both the reservoir and the system depletant representation is shown in Fig. 6.5.

Exercise 6.3. Compared to using PHSs as depletants one could expect the phase-transition concentrations could shift to either higher or to smaller depletant concentrations. Give an argument for both based upon the depletion potentials provided in Sects. 2.1 and 2.3.

The theoretical binodals calculated using rigorous FVT are in semi-quantitative agreement with the computer simulation results. The binodals shift to lower depletant concentrations when the size ratio q becomes smaller and an isostructural solid–solid coexistence  region appears at high hard sphere densities and at low values of q. The solid–solid coexistence region in the theoretical phase diagram is metastable for $q = 0.1$, whereas a small stable isostructural solid–solid coexistence region was found in simulations. For $q = 0.2$ there is a slight underestimation of the fluid branch of the binodal compared to the simulation data. The agreement of the phase diagrams obtained with the FVT presented here with simulations [17] and previous perturbation and DFT studies [30, 31] indicates that the excluded volume of the depletants is now more accurately taken into account and FVT can be accurately applied to hard depletants.

6.3 Phase Behaviour of Mixed Suspensions of Large and Small Spherical Colloids

6.3.1 Phase Separation in Binary Mixtures Differing Only in Diameter

Sanyal et al. [32] and Van Duijneveldt [33] were the first to present experimental evidence for phase separation in binodal suspensions of colloidal spheres with a large size difference. Since then, several studies [11, 34,35,36,37,38] have appeared that present experimental phase diagrams for mixed suspensions of large and small colloids. It should be noted, however, that experimental model systems of mixtures in which both types of spherical colloidal particles are hard sphere-like do not (yet) exist, as far as we are aware.

In Fig. 6.6 we give the experimental phase diagram for $q \simeq 0.1$ by Imhof and Dhont [36], which is compared to free volume theory and Monte Carlo computer simulations. Rigorous FVT predictions (curves) and computer simulations (open symbols) overestimate the depletion activity of the small spheres at the binodal as compared to the experiments (closed symbols). The difference might be caused by charges on the colloidal particles in the experimental system not accounted for theoretically. Additional double layer repulsion does shift theoretical FVT binodals for fluid–solid coexistence at small q upwards [39].

Fig. 6.6

Fluid–solid coexistence curves established from experiments ($\blacksquare$, guided by dotted lines) with sterically stabilised silica spheres of $q \simeq 0.1$ dispersed in DMF with $10^{-2}$ M LiCl [36]. Monte Carlo simulations [17] ($\square$) and rigorous FVT [3] (solid curves) for $q=0.1$ predict phase transitions at lower $\phi _2$

Kaplan et al. [34] and Dinsmore et al. [35] observed crystallites at the sample walls at volume fractions of the small spheres significantly below the value required for the fluid–solid transition in the bulk (Fig. 6.7). This is a manifestation of the stronger depletion interaction between a colloidal sphere and a wall than the depletion interaction between two spheres (as was discussed in Chap. 2). This effect was also demonstrated using micrometre-sized silica spheres dispersed in cyclohexane in contact with hydrophobised silica substrates under the influence of nonadsorbing polymers by Ouhajji et al. [40] using confocal microscopy. A theoretical treatment for the wall phase behaviour based on the semi-grand potential of an adsorbed layer of colloids has been given by Poon and Warren [2]. Comparison with experiment [41] shows that this treatment also overestimates the depletion effect of the small spheres.

Fig. 6.7

Reprinted with permission from Ref. [41]. Copyright 1997 IOP Publishing, Ltd

Optical micrographs of polystyrene spheres ($d_1 = 0.8$ $\upmu$m) at a glass wall a without small spheres; b and c have small spheres of $d_2 = 70$ nm added of $\phi _2 = 0.08$ and $\phi _2 = 0.16$, respectively. The volume fraction of big spheres $\phi _1 =0.02$.

6.3.2 Mixtures of Latex Particles and Micelles

In 1980, Yoshimura, Takano and Hachisu [42] reported a fluid–solid phase separation in a dispersion of polystyrene latex ($d_1 = 510$ nm) spheres mixed non-ionic surfactant polyoxyethylene alkyl phenylether at KCl concentrations above 0.05 $\text {mol/l}$. Under these conditions the surfactants form spherical micelles. At a surfactant concentration of 2 wt$\%$ an iridescent bottom phase appeared, which increased in amount upon further increase of the surfactant concentration. At the same time, the latex concentration in the top phase decreased. The formation of colloidal crystals in the bottom phase, which causes the iridescence, could be confirmed by direct visual observation in the microscope.

A few years later, Ma [43] recognised that the origin of the phase separation is the depletion interaction between the latex particles caused by the micelles. Piazza and Di Pietro [44] have done quantitative measurements on the depletion-induced phase separation in mixtures of latex particles and micelles. In Fig. 6.8 we give their results for a mixture of colloidal polytetrafluoro-ethylene spheres with diameter $d_1 = 220$ nm and the non-ionic surfactant Triton X100 which forms globular micelles with diameters $d_2 =$ 6–8 nm. In Fig. 6.8 we compare these experimental results (closed symbols) with the fluid binodal branch predicted from FVT. The simple FVT extension described in Sect. 6.2.1 is given as the dotted curve, while the rigorous FVT approach that explicitly includes excluded volume interactions in the reservoir presented in Sect. 6.2.2 is given as the solid curve. Computer simulations [17] are presented as the dashed curve. Clearly, rigorous FVT is close to both computer simulations and experiments for $q=0.033$. Piazza et al. [45] have also shown that the fluid–solid phase transition induced by micellar depletants can be exploited to perform an efficient size fractionation of latex particles.

Fig. 6.8

Phase diagram for asymmetric colloidal sphere mixtures for $q=0.033$. Data points are experimental results from Piazza and Di Pietro [44], dotted curve is the simple FVT approach [1], solid curve is the rigorous FVT prediction [3] and the dashed curve represents Monte Carlo computer simulation results by Dijkstra et al. [17]

As early as 1952, Cockbain [46] observed the reversible aggregation and creaming of soap stabilised oil-in-water emulsion droplets at soap concentrations greater than the critical micelle concentration. Fairhurst et al. [47] suggested that this reversible aggregation and creaming arises from the depletion interaction between the oil droplets caused by the soap micelles. Quantitative measurements on depletion-induced phase separation by micelles were performed by Bibette and co-workers on silicone oil-in-water emulsions stabilised by sodium dodecylsulfate (SDS) [48]. Since the depletants now are charged it is more complicated to formulate simple models to quantify the effects, but similar phase diagrams to that in Fig. 6.8 have been measured. For small size ratios $q<0.03$ it is clear from experiments that the phase-transition points shift to lower depletant volume fractions. This is also predicted by rigorous FVT but not by classical FVT. In conclusion, rigorous FVT [3] is in good agreement with computer simulations and is more accurate than simple FVT.