Phase Behaviour of Colloidal Cubes Mixed with Depletants

Abstract

In Chaps. 8 and 9 it was shown that the phase behaviour of anisotropic hard particles is considerably richer than that of hard spheres (see Sect. 3.2). Recent breakthroughs in colloidal synthesis allow the control of particle shapes and properties with high precision. This provides us with a constantly expanding library of new anisotropic building blocks, thus opening new avenues to explore colloidal self-assembly at a higher level of complexity [1, 2]. One of these intriguing novel systems is cube-like colloids. In this chapter, a selective overview is given on the current knowledge of the phase behaviour of cube-like colloids with and without added depletants.

In Chaps. 8 and 9 it was shown that the phase behaviour of anisotropic hard particles is considerably richer than that of hard spheres (see Sect. 3.2). Recent breakthroughs in colloidal synthesis allow the control of particle shapes and properties with high precision. This provides us with a constantly expanding library of new anisotropic building blocks, thus opening new avenues to explore colloidal self-assembly at a higher level of complexity [1, 2]. One of these intriguing novel systems are cube-like colloids. In this chapter, a selective overview is given on the current knowledge of the phase behaviour of cube-like colloids with and without added depletants.

This chapter commences with an introduction to some experimental cube-like systems that have been developed, followed by an outline of the basic thermodynamics of dispersions comprising hard superballs, which are often used to model cubic colloids. Subsequently, an explanation is provided about how free volume theory can be applied to predict the phase stability of cube–polymer mixtures. We conclude by discussing the experimental work available on the phase stability of dispersions containing colloidal cubes mixed with nonadsorbing polymers.

10.1 Introduction to Colloidal Cubes

Recent improvements in the synthesis of a wide range of different types of colloidal particles have led to the realisation of well-defined building blocks that enable the structuring of matter (see also Sects. 8.1 and 9.1). The anisotropic shape of inorganic particles can be explained by the variation in growth rates of different crystal facets that develop during colloid synthesis [3, 4]. The shape can be tuned by the adsorption of additives, which inhibit or accelerate the growth of certain crystal facets [5]. The inhibition or acceleration of crystal facet growth also evolved into the development of various types of cube-like colloidal particles. The ability to tune size, shape and chemistry of cube-like colloids [6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31] extends the range of possible applications. Some examples are shown in Fig. 10.1.

Fig. 10.1

Microscopy images of examples of cube-like colloidal particles. a Micrometre-sized hollow silica cubes prepared by growing a shell around iron oxide particles according to the method of Rossi et al. [20]; image kindly provided by L. Rossi. b Sharp Fe₃O₄ nanocubes from Ref. [32], synthesised according to the method by Park et al. [13]. c Pd nanocubes; inset: magnification [19]. d, e SiO₂-coated Cu₂O nanocubes [27, 33], images kindly supplied by F. Dekker. The CuO₂ core is removed in e. f Ag cubes [11]. Images b, c and f reprinted with permission from: b Ref. [32] (CC-BY); c Ref. [19], copyright 2011 the American Physical Society [19]; f Ref. [11], copyright 2005 Wiley

Many of the synthesised cube-like colloidal particles have a shape that lies in between that of a cube and a sphere. To a certain degree the shape is also tunable within that range [27, 34]. A common way to quantify the shape of a cube-like particle with rounded edges is by describing it as a superball. Formally, superballs are a subset of a family of geometric shapes called superellipsoids [35]. The following expression describes the shape of a superball in Cartesian coordinates $\{x,y,z\}$ [36]:

$$\begin{aligned} {} \mathfrak {f}(x,y,z) = \left|\frac{x}{R}\right|^m + \left|\frac{y}{R}\right|^m + \left|\frac{z}{R}\right|^m \le 1 \text {,} {}\end{aligned}$$

(10.1)

where R is the radius of the superball (the shortest distance from the centre of the superball to its surface), which is related to the edge length $R_\text {el} \equiv 2R$. The quantity m is the shape parameter. The surface of the superball is described for $\mathfrak {f}(x,y,z)=1$, whereas the location of the material inside the superball is given by $\mathfrak {f}(x,y,z)<1$. For $m=2$ a sphere is recovered and $m=\infty$ corresponds to a cube. To describe cube-like colloidal particles the focus here is on $m \ge 2$. In Fig. 10.2 a collection of superballs is depicted for several m-values.

Fig. 10.2

Three-dimensional (top panel) and two-dimensional (bottom panel) representations of a superball for various values of the shape parameter (left to right): $m=2,3,5, \text { and } 10$. The superball radius (R) and the maximum distance of the superball surface to the centre ($\mathfrak {r}_\text {max}$) are indicated. Adapted from Ref. [38] under the terms of CC-BY-4.0

Exercise 10.1. Compute the maximum distance $\mathfrak {r}_\text {max}$ between the particle centre and its surface in terms of R for the shapes in Fig. 10.2. Hint: see the Appendix of Ref. [37].

It is noted that experimentally prepared cubes may be even more accurately described as a sphube [39], or by using the Minkowski sum of a cube and a sphere [32]. Also the Minkowski particle and sphube shapes interpolate smoothly from perfect sharp cubes to perfect spheres. The advantage of the superball shape however is that it provides approximate analytic expressions for various thermodynamic properties of the particle dispersions, enabling the theoretical prediction of phase diagrams.

10.2 Equations of State of Hard Colloidal Superballs

In this section, equations of state are presented for a fluid and two solid phase states composed of hard colloidal superball dispersions. These approximate results will be used in subsequent sections to predict phase diagrams of hard superballs and superballs mixed with depletants.

10.2.1 Second (Osmotic) Virial Coefficient of Superballs

The first step is to quantify the second (osmotic) virial coefficient for superballs. To quantify the second (osmotic) virial coefficient  ($B_2$) of hard superballs, the orientationally averaged excluded volume between two particles [40] should be calculated because the particles are anisotropic for $m>2$. For convex particles (hence, also for superballs with $m\ge 2$), a general expression for $B_2$ reads [41, 42]

$$\begin{aligned} {} \frac{B_2}{v_\text {c}}\equiv B_2^* = 3\varsigma + 1 \text {,} {}\end{aligned}$$

(10.2)

with the so-called asphericity $\varsigma$, defined through

$$\begin{aligned}{} \varsigma = \frac{s_\text {c}c_\text {c}}{3v_\text {c}} \text {,} {}\end{aligned}$$

which contains the geometrical superball characteristics volume $v_\text {c}$, surface area $s_\text {c}$ and surface integrated mean curvature $c_\text {c}$. In [38, 43] it is explained in detail how $\varsigma$ can be calculated as a function of the shape parameter m from numerical computation of $v_\text {c}$, $s_\text {c}$ and $c_\text {c}$. This yields $B_2^{*}$ via Eq. (10.2). The numerically obtained $B_2^*$ values for hard superballs [38] are plotted in Fig. 10.3. It follows that $B_2^*$ smoothly increases with m from the hard sphere (hs) limit ($m=2$, $B_2^*=4$) to the cube limit ($m=\infty$, $B_2^*=5.5$). The numerical data can be described using the following closed expression for $B_2$ (solid curve in Fig. 10.3):

$$\begin{aligned} {} B_2^*\approx \frac{1}{0.42\sqrt{1-\left( \frac{1-2/m}{1.83}\right) ^2}-0.17}\text {.} {}\end{aligned}$$

(10.3)

Fig. 10.3

Normalised second virial coefficient $B_2^*$ as a function of the shape parameter m. Numerical solutions are given by the grey dots (see Ref. [38] for details). The black curve shows a fit through the data points following Eq. (10.3). Adapted from Ref. [38] under the terms of CC-BY-4.0

10.2.2 Fluid Phase State of Superballs

To find an expression for the fluid state of superballs, consider a collection of $N_\text {c}$ hard superballs in a volume V, so the volume fraction of superballs $\phi _\textrm{c}=N_\text {c} v_\text {c}/{V}$. An accurate equation of state (EOS) for a fluid of hard convex particles  was proposed by Boublík [44,45,46] in terms of the reduced osmotic pressure of the pure hard superball dispersion $\widetilde{P}_\textrm{f}$:

$$\begin{aligned} {} \widetilde{P}_\textrm{f}= \frac{P v_\text {c}}{kT} =\frac{\phi _\textrm{c}+\mathfrak {Q}\phi _\textrm{c}^2+\mathfrak {R}\phi _\textrm{c}^3-\mathfrak {T}\phi _\textrm{c}^4}{(1-\phi _\textrm{c})^3} \text {,} {}\end{aligned}$$

(10.4)

with

$$\begin{aligned} {} \begin{aligned} \mathfrak {Q} &= 3\varsigma -2, \\ \mathfrak {R} &= 1-3\varsigma + 3 \varsigma ^2, \\ \mathfrak {T} &= 6\varsigma ^2-5\varsigma \text {.} \end{aligned} {}\end{aligned}$$

(10.5)

It is noted that Gibbons [47] derived an earlier, less accurate, EOS using scaled particle theory.

Exercise 10.2. Show that Eq. (10.4) equals the Carnahan–Starling  prediction Eq. (3.1) in the limit of hard spheres [48]. Hint: use Eq. (10.2) to find $\varsigma$ for hard spheres.

Using the relation between $B_2$, $\varsigma$ and $\widetilde{P}_\textrm{f}$ of Eqs. (10.2)–(10.4), the EOS for a fluid of hard superballs is completely defined for a given value of m. Computer simulations have shown the accuracy of the Boublík EOS for a wide range of m-values [49, 50]. In Fig. 10.4 computer simulation results for the limits of hard cubes ($m=\infty$) and hard spheres ($m=2$) are compared to predictions using Eq. (10.4).

Fig. 10.4

Volume fraction dependence of the osmotic pressure of hard cubes [50] ($\blacksquare$) and of hard spheres [51] ( ). Curves: Eq. (10.4)

The chemical potential of the superballs is related to the osmotic pressure through the Gibbs–Duhem relation Eq. (A.12) for a single-component system at constant temperature, which can also be written as

$$\begin{aligned} {} \textrm{d}\widetilde{\mu }_\textrm{f} = \frac{1}{\phi _\textrm{c}}\frac{\textrm{d}\widetilde{P}_\textrm{f} }{\textrm{d}\phi _\textrm{c}}\textrm{d}\phi _\textrm{c}\text {;} {}\end{aligned}$$

(10.6)

so, the chemical potential follows from combining Eqs. (10.4) and (10.6) (see also Eq. (3.5)):

$$\begin{aligned} {} \begin{aligned} \widetilde{\mu }_\textrm{f} &= \ln \left( \frac{\varLambda ^3}{v_\textrm{c}}\right) +(\mathfrak {T}-1)\ln (1-\phi _\textrm{c})+\ln \phi _\textrm{c} \\ &\qquad + \frac{(10+4\mathfrak {Q}+2\mathfrak {T})\phi _\textrm{c}-(13+3\mathfrak {Q}-3\mathfrak {R}+5\mathfrak {T})\phi _\textrm{c}^2}{2(1-\phi _\textrm{c})^3} \\ & \qquad + \frac{(5+\mathfrak {Q}-\mathfrak {R}+\mathfrak {T})\phi _\textrm{c}^3}{2(1-\phi _\textrm{c})^3} \text {,} \end{aligned} {}\end{aligned}$$

(10.7)

with $\ln (\varLambda ^3/v_\text {c})$ the reference chemical potential of a superball fluid. The free energy follows from the chemical potential and the osmotic pressure through the thermodynamic relation $\widetilde{F}_\textrm{f} = \phi _\textrm{c}\widetilde{\mu }_\textrm{f}-\widetilde{P}_\textrm{f}$ (see Eq. (A.11)).

10.2.3 Solid Phase States of Superballs

To approximate the free energy of the solid phases of hard superballs, the cell theory (see also Chaps. 3 and 9) proposed by Lennard-Jones and Devonshire (LJD) for hard spheres [52] was modified. Each particle is considered to be contained in a closed region whose shape is determined by neighbouring particles fixed at their lattice positions [53], as illustrated in Fig. 10.5. The free energy of the solid is computed from the number of configurations determined by the volume $v^{*}$ that the centre of the particle explores, provided it does not overlap with its nearest neighbours. This leads to the following normalised free energy for a solid:

$$\begin{aligned} {} \widetilde{F}_\textrm{s} = \phi _\textrm{c}\ln \left( \frac{\varLambda ^3}{v^{*}}\right) \text {.} {}\end{aligned}$$

(10.8)

The free volume $v^{*}$ depends on the shape parameter m and the volume fraction $\phi _\textrm{c}$ of the superballs. It is, however, also dependent on the structure of the solid phase state, because this structure affects the relative position of the nearest neighbours. Solids that appear in dispersions of hard superballs are the face-centred cubic (FCC), the simple cubic (SC) structure and two families of other more complex lattice packings: the $\mathbb {C}_0$-lattice and $\mathbb {C}_1$-lattice [36, 50, 54].

These packings possess twofold ($\mathbb {C}_0$) and threefold ($\mathbb {C}_1$) rotational symmetries. Both packings can be considered as a continuous deformation of the FCC lattice for a sphere (m = 2) to a SC lattice for a perfect cube ($m \rightarrow \infty$). Since these solids have a distorted structure the corners of the superballs can be closer to one another. Hence, the voids between the particles are smaller, which increases the maximum packing density. The $\mathbb {C}_0$-lattice provides the densest packing for small m, while for $m \gtrsim 2.308$ the $\mathbb {C}_1$-lattice is the most efficient. The FCC and SC structures are the thermodynamically preferred structures in the limits of hard spheres and hard cubes, respectively [50]. Since there are no analytic expressions available for the $\mathbb {C}_0$- and $\mathbb {C}_1$-lattice structures we consider only the FCC and SC phase states. A schematic view of the FCC and SC structures of superballs for several m-values is shown in Fig. 10.5.

Fig. 10.5

Representations of the face-centred cubic (FCC) crystal lattice (top panel) and simple cubic (SC) lattice (bottom panel) at $\phi _\textrm{c} = 0.45$ for (left to right) $m=2,3,5$, and 10. The free volume is illustrated as the shaded regions. As an illustration, a particle that just touches a nearest neighbour is also depicted. The small arrows indicate the superball radius R and the large arrows specify the nearest-neighbour distance r. Adapted from Ref. [38] under the terms of CC-BY-4.0

For the FCC crystal the free volume depends on the shape of the Wigner–Seitz cell [53], which for an FCC crystal has a rather complicated geometry [55, 56]. Usually it is approximated as a sphere (see Chap. 3). If one considers that this is still reasonable for superballs for small values of $m-2$, the free volume $v^{*,\textrm{FCC}}$ is then given by

$$\begin{aligned} {} v^{*,\textrm{FCC}} = \frac{4\pi }{3}\left( r-r_\textrm{cp}\right) ^3\text {,} {}\end{aligned}$$

(10.9)

where r is the distance between the centres of a superball and its nearest neighbours, and $r_\textrm{cp}$ is r at close packing. For the FCC crystal, a ‘frozen’ crystal can be considered in which the particles are perfectly aligned. For the FCC lattice $r_\textrm{cp}$ then is two times the distance between the edges of the superballs, $2\mathfrak {r}_\text {max}^\text {2D}$, in the two-dimensional representation. With $\phi _\textrm{cp}$ as the close packing fraction, the distance r at a certain volume fraction can be determined from $r_\textrm{cp}$:

$$\begin{aligned} {} r = r_\textrm{cp}\left( \frac{\phi _\textrm{cp}}{\phi _\textrm{c}}\right) ^{1/3}\text {.} {}\end{aligned}$$

(10.10)

Combining $r_\textrm{cp}=2\mathfrak {r}_\text {max}^\text {2D} = 2 R \sqrt{2} \left( 1/2\right) ^{1/m}$ with Eqs. (10.8)–(10.10) provides the free energy for the FCC phase. Taylor expansion of $(\phi _\textrm{cp}^\text {FCC}/\phi _\textrm{c})^{1/3}-1$ was used in the original LJD approach for hard spheres [52]. The chemical potential and (osmotic) pressure are calculated from the free energy via the thermodynamic relations given in Appendix A, leading to the following closed, m-dependent, expressions for the FCC phase:

$$\begin{aligned} {} \widetilde{F}_\text {FCC} &= \phi _\textrm{c}\ln \left( \frac{\varLambda ^3}{v_\text {c}}\right) +\phi _\textrm{c}\ln \left[ \frac{3^42^{3/m}f(m)}{\pi 2^{7/2}}\right] -3\phi _\textrm{c}\ln \left( \frac{\phi _\textrm{cp}^\text {FCC}}{\phi _\textrm{c}}-1\right) \text {,} \end{aligned}$$

(10.11a)

$$\begin{aligned} \widetilde{\mu }_\text {FCC} &= \widetilde{\mu }_\text {0} +\ln \left[ \frac{3^42^{3/m}f(m)}{\pi 2^{7/2}}\right] -3\ln \left( \frac{\phi _\textrm{cp}^\text {FCC}}{\phi _\textrm{c}}-1\right) +\frac{3}{1-\phi _\textrm{c}/\phi _\textrm{cp}^\text {FCC}}\text {,} \end{aligned}$$

(10.11b)

$$\begin{aligned} \widetilde{P}_\text {FCC} &= \frac{3\phi _\textrm{c}}{1-\phi _\textrm{c}/\phi _\textrm{cp}^\text {FCC}}\text {,} {}\end{aligned}$$

(10.11c)

with $\widetilde{\mu }_\text {0}=\ln {\varLambda ^3}/{v_\text {c}}$. The m-dependency of the close packing volume fraction in an FCC crystal is given by [38, 43]

$$\begin{aligned} {} \phi _\textrm{cp}^\text {FCC} = \frac{1}{2}f(m)2^{3/m}\text {,} {}\end{aligned}$$

(10.12)

with

$$\begin{aligned} {} f(m) = \frac{\left[ \varGamma _\textrm{E}(1+1/m)\right] ^3}{\varGamma _\textrm{E}(1+3/m)}\text {.} {}\end{aligned}$$

(10.13)

Here, $\varGamma _\textrm{E}(x)$ is the Euler Gamma function of x. For $m=2$, Eqs. (10.11)–(10.13) recover the result for hard spheres in the FCC phase [52], given in Sect. 3.2.2.

Following a similar procedure as for the FCC phase state, the thermodynamic functions of the SC phase read

$$\begin{aligned} {} \widetilde{F}_\text {SC} &= \phi _\textrm{c}\ln \left( \frac{\varLambda ^3}{v_\text {c}}\right) +\phi _\textrm{c}\ln f(m) -3\phi _\textrm{c}\ln \left[ \left( \frac{\phi _\textrm{cp}^\text {SC}}{\phi _\textrm{c}}\right) ^{1/3}-1\right] \text {,} \end{aligned}$$

(10.14a)

$$\begin{aligned} \widetilde{\mu }_\text {SC} &= \widetilde{\mu }_0+\ln f(m)-3\ln \left[ \left( \frac{\phi _\textrm{cp}^\text {SC}}{\phi _\textrm{c}}\right) ^{1/3}-1\right] +\frac{1}{1-(\phi _\textrm{c}/\phi _\textrm{cp}^\text {SC})^{1/3}}\text {,} \end{aligned}$$

(10.14b)

$$\begin{aligned} \widetilde{P}_\text {SC} &= \frac{\phi _\textrm{c}(\phi _\textrm{cp}^\text {SC}/\phi _\textrm{c})^{1/3}}{(\phi _\textrm{cp}^\text {SC}/\phi _\textrm{c})^{1/3}-1} + \frac{\phi _\textrm{c}}{1-(\phi _\textrm{c}/\phi _\textrm{cp}^\text {SC})^{1/3}} \text {,} {}\end{aligned}$$

(10.14c)

with the close packing fraction in the SC phase given by

$$\begin{aligned} {} \phi _\textrm{cp}^\text {SC} = \frac{\left[ \varGamma _\textrm{E}(1+1/m)\right] ^3}{\varGamma _\textrm{E}(1+3/m)}\text {.} {}\end{aligned}$$

(10.15)

In this simple approach [38] of estimating $v^{*}$, effects of particle rotations are only accounted for approximately. See also [57] for a comparison of cell theory of cubes and other methods. Algebraic expressions for nonaxisymmetric hard particles are scarce. No analytic expressions could even be derived for biaxial hard particles, and one must rely on computational approaches [58]. Table 10.1 provides close packing volume fractions for perfect spheres ($m=2$) and perfect cubes ($m=\infty$) and for a limiting intermediate case. SC arrangements can pack closer for large m and FCC packings are more efficient for small m. It follows from Eqs. (10.12) and (10.15) that $\phi _\textrm{cp}$ attains the same value for both the FCC and the SC phase states at $m=3$ (see Table 10.1).

Table 10.1 Close packing volume fractions for superballs with $m=2$ (spheres), $m=3$ and $m=\infty$ (cubes). At $m=3$ both crystals have the same close packing fraction

Cell theory is known to give accurate results for FCC and SC crystals of hard spheres [55, 56], but extending cell theory to other crystal structures is not straightforward. For a body-centred-cubic crystal of hard spheres, cell theory deviates from computer simulation results [56]. Still, the approach outlined above gives some semi-quantitative insight into the effects of m on the free energy of FCC and SC solid phase states, of which the latter has been found for micrometre-sized superball-like colloids upon adding nonadsorbing polymer chains [20]. Interestingly, experiments reveal that superballs will form FCC (rotator phase) and dense $\mathbb {C}_1$ -lattices upon sedimentation for $m \lesssim 3$ [59].

10.3 Phase Behaviour of Hard Colloidal Superballs

The complete phase diagram of hard superballs can now be calculated using the expressions of the chemical potential and osmotic pressure of the fluid, FCC and SC phase states. The resulting theoretical phase diagram for a suspension of pure hard superballs is presented in Fig. 10.6 (left panel). Results obtained using Monte Carlo computer simulations are plotted in the right panel of Fig. 10.6. The fluid–FCC coexistence for hard spheres [60] (discussed in Chap. 3) is recovered for $m=2$. The theoretically predicted fluid–FCC equilibrium gradually shifts to a higher volume fraction upon increasing m. A similar shift is found using computer simulations [54]. The forbidden region (grey) identifies volume fractions beyond close packing. A discontinuity at $m=3$ along the border of the forbidden region (left panel) corresponds to the transition between FCC to SC phase states. The preferred solid phase is related to the largest close packing volume fraction.

A triple F–FCC–SC point is found at $m\approx 3.71$ (left panel). Simulations also indicate a triple point at a somewhat higher m value [54] (right panel of Fig. 10.6), to be discussed later. Between $m=3$ and $m\approx 3.71$, theory predicts SC–FCC coexistence (left panel). Above $m\approx 3.71$, only F–SC coexistence is found theoretically, which shifts towards lower packing fractions with increasing m, also in qualitative agreement with simulations [54]. In the cube limit ($m=\infty$), the simple theory presented here predicts $\phi _\textrm{c}^\text {F}\approx 0.36$ and $\phi _\textrm{c}^\text {SC}\approx 0.54$. Computer simulation studies (not shown here) by Agarwal and Escobedo [50] indicate that phase coexistence between a fluid and a cubatic liquid crystal takes place at $\phi _\textrm{c}^\text {F}\approx 0.47$ and $\phi _\textrm{c}^\text {SC}\approx 0.58$ for perfect cubes.

Exercise 10.3. What is the theoretical maximum number of coexisting phases in a dispersion of hard superballs?

Fig. 10.6

Left: phase diagram for a suspension of superballs presented in terms of volume fraction and shape parameter. Two-phase coexistences take place in the regions bounded by two single-phase regions as indicated. The vertical dashed grey lines hold for the F–FCC–SC coexistence. Right: theoretical predictions based upon Monte Carlo computer simulations by Ni et al. [54]. Adapted from Ref. [38] under the terms of CC-BY-4.0

The overall topology of the theoretical phase diagram (left panel) corresponds roughly to computer simulations results (see Refs. [49, 50, 54]; right panel of Fig. 10.6). Differences can be justified because theory does not account for the same solid phases for superballs as in simulations, as mentioned earlier. The solid $\mathbb {C}_0$ and $\mathbb {C}_1$ phases can be accounted for in computer simulation studies [49, 54]. Not surprisingly, the triple point from simulations is a fluid–plastic FCC–$\mathbb {C}_1$ [54] and lies at a larger m value than the theoretical triple point. Due to the limitations inherent to the simple theory used here, the $\mathbb {C}_0$ and $\mathbb {C}_1$ phases are not accounted for. The FCC phase features (and their coexistences) roughly match those of the plastic FCC. The role played in simulations by the $\mathbb {C}_1$ phase state is mimicked by the SC phase in the simpler model applied in the theoretical description.

10.4 Theory for the Phase Behaviour of Colloidal Superballs Mixed with Polymers

The presence of the nonadsorbing polymers leads to depletion zones around the cube-like colloidal particles. In Fig. 10.7, overlap volumes of depletion layers (indicated by the dashed zones) are illustrated for superballs with shape parameters $m = 2, 4$, and $\infty$. The volume of overlapping depletion zones $V_\textrm{ov}$ (given by Eq. 1.19 for $m=2$) increases with m when the superballs align their flat faces, under the condition that the superball radius R and polymer size (and thus, the depletion thickness) are fixed. At those configurations a maximum depletion attraction is achieved [61].

Upon adding nonadsorbing polymer (or colloidal particles) to a colloidal dispersion, one expects that the entropic patchiness effect, discussed in Sect. 1.3.6, leads to an enhanced depletion attraction for cube-like particles as compared to spheres.

Fig. 10.7

Sketches of the maximum overlap of depletion zones for three types of hard superballs in nonadsorbing polymer solutions for fixed polymer size and constant particle radius R. The hatched areas reflect overlap volumes of depletion zones. The examples are given for superballs with shape parameters $m = 2$, 4 and $\infty$. These overlap volumes are drawn next to each other for comparison in the lower right section of the figure. Reprinted from Ref. [62] under the terms of CC-BY-4.0

When considering colloidal superball–polymer mixtures, the phase diagrams would only enrich upon refinements of the method. The liquid-crystalline and crystalline coexistence regions are found in simulations in a broader range of m-values [54].

10.4.1 Free Volume Theory

Adding depletants to dispersions of hard superballs is accounted for here in a semi-grand canonical fashion via free volume theory (FVT), as in the previous chapters. Within FVT, the superball–polymer system ($\text {S}$) is considered to be in equilibrium with a reservoir ($\text {R}$) of polymers. In $\text {R}$ and $\text {S}$ the solvent is treated as background as before (see Sect. 3.3.4), and system and reservoir are connected through a membrane permeable for the polymers and the common solvent but impermeable for the superball or cube-like particles. The relative volume available for depletants in the system, the free volume fraction $\alpha$, relates the polymer concentrations in $\text {R}$ and $\text {S}$. Following original FVT, it is assumed that this key quantity $\alpha$ is independent of the chemical potential of the depletants in $\text {R}$.

The simplest description is used here for polymeric depletants, namely the penetrable hard sphere (PHS) model: depletants are treated as ghost-like spheres (with radius $\delta$) that can freely interpenetrate each other but do not overlap with the superballs. Further, the approximation is made again that the ensemble-averaged free volume $\langle V_\textrm{free}\rangle$ is independent of the concentration of depletants. This implies that, similar to what was discussed in the previous chapters, $\alpha =\langle V_\textrm{free}\rangle /V \approx \langle V_\textrm{free}\rangle _0 /V$, resulting in the following (normalised) expression for the grand potential of the system:

$$\begin{aligned} {} \widetilde{\varOmega } = \frac{\varOmega v_\textrm{c}}{kTV} = \widetilde{F}-\widetilde{P}_\text {d}^\text {R}\alpha \frac{v_\textrm{c}}{v_\textrm{d}} \text {,} {}\end{aligned}$$

(10.16)

with V as the volume of the system, and $v_\textrm{d}$ as the volume of the depletant ($v_\textrm{d}=4\pi \delta ^3/3$). Since the depletants are considered to behave ideally, the osmotic pressure in the reservoir $\text {R}$ is simply given by Van‘t Hoff’s law:

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

The depletant concentration in the system $\phi _\text {d}$ is again given by $\phi _\text {d}=\alpha \phi _\text {d}^\text {R}$.

From the grand potential, the chemical potential and osmotic pressure of superball–PHS mixtures are obtained through the standard thermodynamic relations given in Appendix A, where it is also indicated how to calculate binodals, critical points, spinodals and multi-phase coexistences. If coexistence between three phases takes place a triple point (TP) arises, and a four-phase coexistence is denoted as a quadruple point (QP) (see also the previous two chapters). Colloidal systems may exhibit isostructural phase coexistence (such as gas–liquid equilibrium) when attractive interactions between particles are present. In such a case, the low density phase will be entropically favourable and the high-density phase will be stabilised by attractive interactions between the particles. The limit of isostructural phase coexistence is defined via the critical point (CP).

The conditions of Eq. (A.20) enable one to determine the topology of the phase diagrams as a function of the system parameters, which are the colloidal shape (through m) and the relative depletant size trough:

$$\begin{aligned} {} q = \frac{\delta }{R}, {}\end{aligned}$$

(10.17)

where $\delta$ is the radius of the PHSs.

10.4.2 Free Volume Fraction

The only unknown parameter in Eq. (10.16) is the free volume fraction for depletants in the system, $\alpha$. Widom’s insertion theorem [63] relates the free volume fraction $\alpha$ to the work (W) required to bring a depletant from $\text {R}$ to $\text {S}$ via

$$\begin{aligned} {} \alpha = \frac{\langle V_\textrm{free} \rangle _{\text {o}}}{V} = e^{-W/kT} , {}\end{aligned}$$

(10.18)

where ${\langle V_\textrm{free} \rangle _{\text {o}}}$ is the free volume for depletants in the undistorted (depletant-free) system. This work (W) is obtained via Scaled Particle Theory (SPT) [64, 65], by connecting the limits of inserting a very small depletant (up to second order) and a very big depletant in the system of interest, followed by scaling back to the actual size of the depletant. As the depletants are considered to be spherical here, a single scaling factor ($\lambda$) enables this work of insertion to be expressed as

$$\begin{aligned} {} W &= \lim _{\lambda \rightarrow 1} W (\lambda ) \text {,} \nonumber \\ W (\lambda ) &= \underbrace{ \phantom { \left( \frac{a^{0.3}}{b}\right) } W(0) + \left. \frac{\partial W}{\partial \lambda }\right| _{\lambda = 0} \lambda + \frac{1}{2} \left. \frac{\partial ^2\,W}{\partial \lambda ^2}\right| _{\lambda = 0} \lambda ^2 }_{\lambda \ll 1} + \underbrace{ \phantom { \left( \frac{a^{0.3}}{b}\right) } v_\textrm{d}{P}}_{\lambda \gg 1} \text {,} {}\end{aligned}$$

(10.19)

where P is the osmotic pressure of the pure hard superball dispersion to which the depletants are added (see Sect. 10.2).

In the limit of depletants with a vanishing size ($\lambda \rightarrow 0$) there is no overlap of depletion zones. Hence, the free volume fraction can then be written as a function of the excluded volume between a superball and a depletant ($v_\text {exc}$):

$$\begin{aligned} {} \alpha (\lambda \rightarrow 0) &= 1 - \phi _\textrm{c}\left( \frac{v_\text {exc}(\lambda )}{v_\textrm{c}}\right) \text {;} {}\end{aligned}$$

(10.20)

so, W becomes

$$\begin{aligned} {} W (\lambda \rightarrow 0) &= - kT \ln {\left[ 1 - \phi _\textrm{c} \left( \frac{v_\text {exc}(\lambda )}{v_\textrm{c}}\right) \right] } \text {.} {}\end{aligned}$$

(10.21)

In the limit of big depletants, one finds

$$\begin{aligned} {} \frac{W(\lambda \gg 1)}{kT}=\frac{\pi }{6f(m)}(\lambda q)^3\widetilde{P} \text {.} {}\end{aligned}$$

(10.22)

By combining Eqs. (10.19) and (10.22), a general expression for W in terms of $\widetilde{v}_\text {exc}(\lambda )={v_\text {exc}(\lambda )}/{v_\textrm{c}}$ can be derived:

$$\begin{aligned} {} \begin{aligned} \frac{W}{kT} &= -\ln (1-\phi _\textrm{c}) + y\left. \frac{\partial \widetilde{v}_{\text {exc}}(\lambda )}{\partial \lambda }\right| _{\lambda =0} \\ &\qquad +\frac{1}{2}y^2 \left. \left[ \frac{\partial \widetilde{v}_{\text {exc}}(\lambda )}{\partial \lambda }\right| _{\lambda =0}\right] ^2 \\ &\qquad + \frac{1}{2}y \left. \frac{\partial ^2\widetilde{v}_{\text {exc}}(\lambda )}{\partial \lambda ^2}\right| _{\lambda =0} + \frac{\pi q^3}{6 f(m)}\widetilde{P} . \end{aligned} {}\end{aligned}$$

(10.23)

In Eq. (10.23), we have used

$$\begin{aligned}{} y = \frac{\phi _\textrm{c}}{1-\phi _\textrm{c}} \text {,} {}\end{aligned}$$

which is similar to Eq. (3.39e).

A simple expression is available for the (normalised) excluded volume between general, convex bodies and spheres [66]. For the excluded volume superball–sphere it reads:

$$\begin{aligned} {} \widetilde{v}_\text {exc}= 1+ \frac{3\widetilde{s}_{\textrm{c}}q+ 6 \pi \widetilde{c}_{\textrm{c}} q^2 +{\pi }q^3}{6f(m)}, {}\end{aligned}$$

(10.24)

where $\widetilde{s}_{\textrm{c}}=s_{\textrm{c}}/R_\text {el}^2$, $\widetilde{c}_{\textrm{c}}=c_{\textrm{c}}/R_\text {el}$ and f(m) is defined in Eq. 10.13. Due to the linear relationship between $\delta$ and q ($\delta = q R$), $\widetilde{v}_\text {exc}(\lambda )$ is simply obtained from Eq. (10.24) by making the substitution $q \rightarrow \lambda q$.

It is not possible to solve Eq. (10.24) analytically (see Ref. [38] for details on the calculation of $\widetilde{s}_{\textrm{c}}$ and $\widetilde{c}_{\textrm{c}}$). Via interpolation of $\widetilde{s}_{\textrm{c}}$ and $\widetilde{c}_{\textrm{c}}$ it is possible to obtain an expression for Eq. (10.23). González García et al. [38] found that the depletion zone is accurately described by the following approximate, but accurate, algebraic expression:

$$\begin{aligned} {} \begin{aligned} \widetilde{v}_\text {exc}f(m) &= 454.337 + 216.356 (1 - 2/m) + 308.593 q \\ & \qquad + \exp [-0.005 (1-2/m) + 8.178] \\ & \qquad \times \sin [0.0604 (1-2/m) + 0.087 q -3.014] \text {.} \end{aligned} {}\end{aligned}$$

(10.25)

This enables straightforward calculation of $\alpha$ by inserting Eq. (10.25) into Eq. (10.23). Now the grand potential $\widetilde{\varOmega }$ (Eq. (10.16)) is completely defined.

10.5 Phase Diagrams of Mixtures of Hard Superballs and Polymers: Theoretical Predictions

Firstly, superball–polymer mixtures with (hard) superballs are considered, whose shape is still close to a sphere. Phase diagrams of superballs with $m=2.5$ and added depletants are presented in Fig. 10.8 for three relative size ratios q. For pure superballs ($\phi _\text {d}^\text {R}=0$) the fluid–FCC coexistence corresponds to the densities shown in Fig. 10.6 (left panel) and hardly differs from the fluid–solid phase coexistence concentrations of hard spheres (see Sect. 3.2). Upon addition of depletants the FCC phase at coexistence gets denser, and the coexisting fluid phase becomes more dilute in order to maximise the total free volume available for the depletants in the system. For sufficiently large q-values ($q=0.4$ and $q=0.6$ in Fig. 10.8), an isostructural colloidal F$_1$–F$_2$ (also termed gas–liquid) coexistence appears (metastable for low q-values). For $q=0.4$ a triple line is found (upper panel), which becomes a region in the system representation (lower panels of Fig. 10.8). For larger q, the F–S coexistence narrows and the F$_1$–F$_2$ critical point shifts to higher depletant volume fractions.

Fig. 10.8

Phase diagrams for a mixture of colloidal superballs for $m=2.5$ and penetrable hard sphere depletants for several relative depletant sizes q. The top panels are in the reservoir representation and the bottom panels are in the system representation. The F$_1$–F$_2$ critical point is indicated by a black dot. Triple-phase coexistences are shown as a horizontal black line in the reservoir representation, and as a black area bounded by black lines in the system representation. All coexisting phases present are indicated in the reservoir representation. Insets in the system representation zoom in on the low depletant concentration region, and some of the coexistence regions are indicated. A few illustrative tie-lines are shown as dashed grey lines for $q=0.2$. Above each phase diagram, a 2D illustration of the superball (black) and its depletion zone (grey) are shown, and the m and q-values are indicated. Reprinted from Ref. [38] under the terms of CC-BY-4.0

Fig. 10.9

Phase diagrams of superball–polymer mixtures as in Fig. 10.8, but for $m=5$. Reprinted from Ref. [38] under the terms of CC-BY-4.0

The coexistence regions in the system representation (bottom panels in Fig. 10.8) show that the fluid phase with a low concentration of superballs has a high concentration of depletants, whereas the FCC phase has a high concentration of superballs but a low concentration of depletants. The system representation also shows that for a superball–depletant mixture a single solid phase (without a coexisting fluid phase) only occurs at quite small depletant concentrations. Figure 10.8 reveals no special features compared to FVT for hard spheres mixed with PHSs (see Sect. 3.3), even though the colloidal shape considered is not perfectly spherical.

Phase diagrams for more cubic particles that have $m=5$ (see Fig. 10.2) are presented in Fig. 10.9, for which only a SC state is present in the pure hard superballs system (see left panel of Fig. 10.6) at high particle concentrations. Similar qualitative trends as in Fig. 10.8 are observed, but with the F–SC coexistence instead of the F–FCC equilibrium. For small q-values the broadening of the coexistence lines occurs at lower depletant concentrations with respect to the superball–polymer mixture with $m=2.5$ in Fig. 10.8: the overlap of depletion zones is larger for particles with an increased cubicity (see Fig. 10.7), which results in a stronger depletion attraction. For $m=5$ and $q=0.4$, there is no F$_1$–F$_2$ equilibrium phase coexistence, whereas F$_1$–F$_2$ coexistence was found at this q-value for spheres (superballs with $m=2$, see Sect. 3.3.4 or [67]), and for $m=2.5$. Due to the tendency of flat faces to align upon addition of depletant into the system, stable F$_1$–F$_2$ coexistence shifts to higher q-values: for larger m-values, longer ranges of attraction are required to induce a (stable) F$_1$–F$_2$ coexistence. The trends in Fig. 10.9 hold for larger m and F$_1$–F$_2$ occurs at even larger q-values. Quantitatively, comparison of Figs. 10.8 and 10.9 reveals that the stable single-phase fluid region for colloid volume fractions of, say, smaller than 0.4, is larger for smaller m. This means that dispersions of particles which have a more cube-like shape, are expected to undergo phase transitions at smaller depletant concentrations. The increased overlap volume of the depletion zones with increasing m, as illustrated in Fig. 10.7, drives this earlier onset of the phase transitions.

Based on Fig. 10.6, a physically interesting region is expected near $m=3.65$, since here fluid, FCC and SC phase states are predicted upon increasing the colloid volume fraction. In particular, one may wonder what the effect of adding depletants is near the transitions between these phases. A few phase diagrams for $q=0.4$ and a few selected m-values in that interesting region are plotted in Fig. 10.10. For $m=3.4$, a F$_1$–FCC–SC triple coexistence occurs at a higher $\phi _\text {d}^\text {R}$ than the F$_1$–F$_2$–FCC triple line (F$_1$–F$_2$–SC coexistence is metastable). For $m= 3.65$, however, the F$_1$–F$_2$–FCC triple point becomes metastable and an F$_1$–FCC–SC triple point arises at lower $\phi _\text {d}^\text {R}$ than the F$_1$–F$_2$–SC isostructural coexistence. This is explained by the fact that the stability of the FCC decreases as m increases.

The condition at which the F$_1$–F$_2$–FCC and the F$_1$–F$_2$–SC coexistences merge results in a quadruple coexistence (F$_1$–F$_2$–FCC–SC). This four-phase coexistence is present for a range of m-values (at different q-values). As a consequence of the enhanced alignment of the flat faces upon the addition of depletants, F–SC coexistence takes place at m-values below those of the depletant-free system. Hence, depletion-mediated entropic patchiness promotes the appearance of the SC phase. As a F–FCC–$\mathbb {C}_1$ triple point has been detected experimentally [59] and in a Monte Carlo computer simulation study [54] for the depletant-free superball system, the corresponding quadruple phase coexistence may be found from simulations or experiments with the $\mathbb {C}_1$ phase instead of the SC phase used here.

Fig. 10.10

Phase diagrams of mixtures of superball–polymer mixtures in the reservoir depletant concentration representation for $q=0.4$ and various m-values as indicated. Reprinted from Ref. [38] under the terms of CC-BY-4.0

At low q and high m-values an isostructural SC$_1$–SC$_2$ coexistence appears. A few illustrative phase diagrams are depicted in Fig. 10.11, where small isostructural SC$_1$–SC$_2$ coexistence regions appear. The single fluid phase and simple cubic regions get smaller upon decreasing q. For $m=10$ the binodals shift towards lower $\phi _\text {d}^\text {R}$-values with decreasing q. As can be observed in the rightmost panel of Fig. 10.11, the m-value tunes the depletant concentration at which SC$_1$–SC$_2$ coexistence is found: SC$_1$–SC$_2$ equilibria are driven by the alignment of the flat faces, and thus for more curved particles (decreasing m) the SC$_1$–SC$_2$ coexistence requires a higher depletant concentration. This leads to demixing of the crystal state into two coexisting solids, as is expected for short-ranged attractions in colloidal systems [68, 69]. With more accurate models or in experimental systems, these SC$_1$–SC$_2$ coexistences may be replaced, for example, by a $\mathbb {C}_1$ coexisting with a SC phase. The absence of a stable FCC$_1$–FCC$_2$ coexistence can be rationalised by the non-optimal overlap of depletion zones between the flat faces of the superballs in an FCC state.

Fig. 10.11

Illustrative phase diagrams of superball–polymer mixtures, in which isostructural SC phase coexistences appear. Reprinted from Ref. [38] under the terms of CC-BY-4.0

10.6 Phase Stability of Cubes Mixed with Polymers: Experiments

In Sects. 10.4 and 10.5 theoretical predictions [38] were outlined for the rich phase behaviour of colloidal cubes mixed with nonadsorbing polymers. A thorough verification of this phase behaviour is still underway; experimental studies on the bulk phase behaviour of mixtures containing cubes and nonadsorbing polymers (or other nonadsorbing components) are scarce. Depletion effects in dispersions containing cubes were studied by Park et al. [70]. They mixed gold rods and cubes, and added nonadsorbing polymers to separate them. For more details, see Sect. 11.3.

Another early demonstration of the effects of adding nonadsorbing polymers to dispersions of cube-like particles was performed by Rossi et al. [20, 34], who studied aqueous mixtures of polymers and micrometre-sized hollow silica  superball-shaped particles (typically with m between 3 and 4). The experiments were performed in 10 mM NaCl, so the Debye length $\lambda _\textrm{D} \approx 3$ nm. Images of the cube-like particles were mapped onto the so-called superball shape to determine m. These particles are very suitable for experiments: their size enables them to study their shape and configurations using an optical microscope. However, the combination of their size and silica shells makes them susceptible to gravity, so that equilibrium studies of the bulk properties are challenging. Hence, it is noted that these experimental observations correspond to colloid–polymer mixtures confined at a surface, whereas equilibrium theoretical results presented earlier hold for bulk systems.

Still, by studying their properties at a glass plate after sedimentation, Rossi et al. [20, 21, 34] could investigate the influence of nonadsorbing polymers on the structures that appeared. Their experimental observations [20, 34] revealed that the depletion attraction mediated by the nonadsorbing polymers leads to a phase transition towards a preferred simple cubic phase state, see Fig. 10.12. The appearance of such a solid phase state can be understood by the fact that the volume of depletion zones that is overlapping is then maximised (leaving space for the depletants to fit in the voids of the respective lattices). The size of the nonadsorbing polymers was shown to play a crucial role [20]. The cubic structures appear both in plane as well as in 3D.

Fig. 10.12

a Hollow silica cubes with a total edge length $2R=1.3$ $\upmu$m, silica shell thickness of 100 nm and shape parameter $m = 3.5$. b Adding nonadsorbing PEO (molar mass 600 kDa, $R_\textrm{g} \approx 57$ nm) drives depletion-mediated self-assembly of the cubes to a simple cubic symmetry. Size ratio $q = R_\textrm{g}/R \approx 0.085$. Reprinted with permission from Ref. [20]. Copyright 2011 Royal Society of Chemistry

The authors also studied adding nonadsorbing poly(N-isopropylacrylamide) (pNIPAM)  particles with a radius of 65 nm. The size of these particles is highly temperature-responsive between 20 and 45 $^{\circ }$C. Upon increasing the temperature the pNIPAM particles shrink in water as the solvency changes from good to poor. This enabled Rossi et al. [20] to reversibly induce thermoresponsive cube crystallisation.

The experiments on dispersions of these hollow micrometre-sized silica particles with added nonadsorbing polymers revealed a rich phase behaviour. It was demonstrated that the obtained phase states of the sediment depend on the colloid–depletant size ratio and the details of the shape of the cube-like colloidal particles [34] (Fig. 10.13).

Fig. 10.13

Solid phases prepared from hollow silica cubes similar to those in Fig. 10.12 but with $m = 3.9$. a Solid structures obtained upon adding depletants larger than the pockets available in an SC arrangement. b SC arrangement that appears upon using smaller depletants than for. Reprinted with permission from Ref. [34]. Copyright 2015 the National Academy of Sciences

Recently, it was shown that hollow silica nanocubes display effective hard-core interactions [29], which makes them promising particles for studying the effect of nonadsorbing polymers on the bulk phase behaviour of model anisotropic particles. Based on the estimated phase-transition points from light scattering, Dekker et al. [62] constructed an experimental phase diagram for mixtures of hollow silica (nano)shells ($R_\text {el}$ = 129 nm and m = 4.1) and polystyrene polymers (PS) in DMF (40 mM LiCl). Three different regions can be discerned in the phase diagram depicted in Fig. 10.14: (1) a concentration range where the mixture is stable ( ), (2) a concentration range where the mixture clearly phase separates ($\blacksquare$) and (3) an intermediate transition region where no clear phase separation occurs, but where the scattering studies indicated that significant attraction is present ($\blacktriangle$).

Theoretical predictions for the phase stability threshold are also plotted in Fig. 10.14 to compare with the experimental data. Particle volume fractions were calculated using the specific volumes obtained earlier [29] and the overlap concentration of polystyrene ($M_\text {w} = 600$ $\text {kg/mol}$, $R_\text {g} = 21.5 \pm 1.1$ nm) was 24 g/L. The curves correspond to the fluid branch of the fluid–solid coexistence binodal for superballs with $m = 2$, $m=4.1$ and $m=10^4$ added nonadsorbing polymers with size ratio $q = 0.32$. The dashed curves are phase coexistence lines for superballs with $m = 4.1$ and size ratios $q = 0.29$ and 0.35, representing the lower and upper limit of the polymer polydispersity. The experimental data are in remarkable agreement with the theoretical predictions, indicating that the theory is able to predict the depletion effects in experimental model systems and that dispersions of hollow silica cubes in DMF with 40 mM LiCl and polystyrene is such a model system.

Fig. 10.14

Experimental phase diagram of hollow silica (nano)shells and polystyrene in DMF with 40 mM LiCl compared to theoretical predictions (curves) of hard superballs with nonadsorbing polymers. The curves are predicted fluid–solid coexistence binodal curves for superballs with m-values equal to 2, 4.1 and $10^4$ in solution containing nonadsorbing polymers with size ratio $q = 2R_\text {g} / R_\text {el} = 0.32$. The dashed curves are binodal curves for superballs with $m = 4.1$ and size ratios $q = 2R_\text {g}/R_\text {el} = 0.29$ (bottom) and $q = 2R_\text {g}/R_\text {el} =0.35$ (top). Reprinted from Ref. [62] under the terms of CC-BY-4.0

Fig. 10.15

Phase diagrams of a Sn:In₂O₃ spheres and b F,Sn:In₂O₃ cubes mixed with nonadsorbing polymers. Comparison of theoretical spinodal boundaries (curves) and experimental observations (data points). The polymer concentration is normalised by the overlap concentration. ($\circ$) Single phase mixtures and ($\bullet$) gelled samples are indicated. Reprinted with permission from Ref. [71]. Copyright 2020 American Chemical Society

Saez Cabezas et al. [71] compared the influence of nonadsorbing polymers on the phase stability of tiny spherical and cubic nanocrystals. The prepared spherical particles were composed of magnetite, iron oxide Fe₃O₄, following the method of Yu et al. [72]. The cubic (F,Sn:In₂O₃) cubes were made using the procedure of Cho et al. [30]. As followed from SAXS measurements, both particles were rather monodisperse in size and the diameters were similar: 8–9 nm. PEG nonadsorbing polymers with $M = 1$ kg/mol, $R_\textrm{g} =1.0$ nm were added to the aqueous dispersions containing the colloidal particles and the observations are indicated in Fig. 10.15. The open symbols refer to a single-phase mixture, whereas the closed symbols refer to instability (the authors observed gelation). The curves are calculated spinodal points, computed from Eq. (A.15). It is clear that adding nonadsorbing polymer to the cubic particles leads to instability at much lower polymer concentrations, as is the case for spherical particles. This is confirmed by the predicted spinodal curves, which were computed using the theory outlined in Sect. 10.4. It is noted that both particles have similar charge densities at the surface in aqueous solution, so double layer interactions also play a role here.