Introduction

Abstract

According to IUPAC [1], the term colloidal refers to ‘a state of subdivision

1.1 Colloids

According to IUPAC [1], the term colloidal refers to ‘a state of subdivision, implying that the molecules or polymolecular particles dispersed in a medium have at least in one direction a dimension roughly between 1 nm and 1 $\upmu$m, or that in a system discontinuities are found at distances of that order’. This means that colloidal particles are submicrometre sized substances dispersed in a medium that can be a liquid or a gas [2,3,4,5,6,7,8,9,10]. This implies that colloids are much bigger than ‘normal’ molecules (though they may be comparable in size to macromolecules). The lower limit of the length scale for a colloidal particle is close to a nm. The medium of low molecular mass substances in a colloidal suspension can often be regarded as ‘background’ with respect to the colloidal size range and, in that case, this medium may be approximated as a continuum.

From a physics point of view, colloidal particles are characterised by observable Brownian motion, originating from a thermal energy of order of kT for each colloidal particle. Particles in a solvent are considered to be Brownian if sedimentation can be neglected with respect to thermal motion. This means that the sedimentation length, the ratio of thermal energy and gravity force should be larger than the colloid radius. The sedimentation length is defined as [11]:

$$\begin{aligned} {} \ell _\textrm{sed} = \frac{kT}{m^{*}g}. {}\end{aligned}$$

(1.1)

Here, the buoyant mass is given by $m^{*} = (4\pi /3) \Delta \rho R^{3}$ for a spherical colloid with radius R, where $\Delta \rho$ is the density difference between particle and solvent. Hence, the upper colloidal size corresponds to the condition where $\ell _\textrm{sed} \approx R$. For $\Delta \rho$ = 100 kg/m$^3$, this implies an upper limit of the radius of about 1 $\upmu$m at 300 K.

Perrin [12] studied dispersed resin colloids and detected Brownian motion as a visible manifestation of thermal fluctuations, verifying Einstein’s theoretical results [13]. The height distribution of the resin colloids in the field of gravity was shown to obey Boltzmann’s law for the sedimentation equilibrium. The picture emerged that colloids behave as big atoms in many respects. Later, Onsager [14, 15] and McMillan and Mayer [16] laid down a statistical mechanics  foundation for the colloid–atom analogy. They pointed out that the degrees of freedom of the solvent molecules in a colloidal dispersion can be integrated out, implying the solvent can be considered as ‘background’. The resulting description only involves colloidal particles interacting through an effective potential , the potential of mean force, that accounts for the presence of the solvent.

Often, the (rotationally averaged) interactions between small spherical atoms and some molecules can be reasonably described using the Lennard-Jones  interaction [17] (see Ref. [18] for an in-depth critical discussion). For many of these systems, the phase diagrams scaled by the critical values of temperature, pressure and molar volume appear similar as well. The fact that the thermodynamic properties of all simple gases exhibit basic similarities is expressed by the law of corresponding states of Van der Waals. A statistical mechanical derivation of this law was provided by Pitzer [19].

Just as the pressure of an atomic gas is affected by the interaction between the atoms, the physical properties of a colloidal dispersion depend on the potential of mean force between colloidal particles. An extended law of corresponding states has been conjectured [20], stating that knowledge of the potential of mean force between spherical colloidal particles enables prediction of the phase diagram (topology). Therefore, one may expect similarities between the phase diagrams of atomic and colloidal systems.

Apart from such similarities, there are also distinct differences between atoms and colloidal particles. In contrast to pair interactions between atoms, interactions between colloidal particles can be tuned by choosing particle type or solvent, by supplementing additives such as electrolytes, polymers or other colloidal particles, or by modifying the particle surface. Since the 1970s it has gradually become clear that adding small particles or polymers that do not adsorb onto the colloids opens up a wide variety of possibilities for tuning the phase behaviour of colloidal dispersions. The interactions mediated by such nonadsorbing species and the resulting phase behaviour are at the core of this book.

The science of colloids is important for applications ranging from drug delivery and dairying to coating technology and energy storage materials. Colloidal dispersions can be found in a wide range of environments and products. Industrial examples include emulsions (mayonnaise), foams (shaving cream), surfactant solutions (shampoo) or polymer latex dispersions (paint). Long-term stability of a colloidal dispersion is often desired, for example, in storage of paint [21] or food [22]; and this is regularly achieved by adjusting the particle surface chemically or via adsorption.

Unconsciously, humankind has long held an interest in colloidal stability. For example, carbon is the oldest ink material known, and its use for writing in Egypt can be dated back to 3400 BCE [23]. The carbon used for making ink was soot in most cases. By mixing it with gum arabic and water, soot was made into ink. Without understanding the underlying principles, the Egyptians effectively used the principle of stabilising dispersions by adsorbed macromolecules [3, 24]. This is nowadays recognised as an example of polymeric stabilisation (see Sect. 1.2.4). In this manner the Egyptians succeeded in engineering the soot particles, such that they can remain suspended for an indefinite period.

An example where the instability of colloids (clay) plays a role in nature is delta formation. Deltas [25] are formed due to precipitation of colloidal (clay) particles carried by the river as its flow meets the sea (or ocean), where the fresh river water mixes with salty sea water.  The delta formation process had already been described by Barton [26] in 1918 before a clear understanding of the role of salt on colloidal stability had been established.

Milk is a natural colloidal dispersion that contains casein micelles—self-assembled protein associates with a diameter of about 200 nm [27]. The casein micelles are protected against flocculation by an assembly of dense ‘hairs’ (often called a ‘brush’ ) at their surfaces. Polymer brushes can thus provide steric stabilisation  of colloids. For millennia man has used the fact that milk flocculates and gels when it is acidified, as in yogurt production. Below $\text {pH}$ = 5 macroscopic flocculation of the casein micelles in milk is observed [28]. This means that the interactions between casein micelles change from repulsive to attractive. The explanation is that acidification leads to collapse of the casein brushes [29]. In cheese-making the steric stabilisation is removed by enzymes that induce gelation into cheese curd.

Modest solvent composition changes can also affect the state of a colloidal dispersion. A charge-stabilised dispersion of polymer latex particles or gold colloids may flocculate irreversibly upon adding salt, while ion removal through dialysis may turn the dispersion into an ordered structure that exhibits Bragg reflection [30]. Obviously, the physical state of a colloidal dispersion is a function of the interactions between the colloidal particles.

In foods, paints and biological systems such as living cells, colloids and polymers are often present simultaneously. When the polymers do not adsorb onto the colloidal particles the result is a so-called depletion layer: a zone near the particle surface which contains a lower polymer concentration than the bulk of the solution. As we shall see, overlap of depletion layers leads to an attractive depletion interaction between the colloidal particles. The term depletion derives from Latin meaning ‘emptied out’. The verb ‘plere’ is ‘to fill’ [31]. Thus a ‘pletion’ force is due to accumulation of some substance between two colloids. The reversal, a ‘depletion’ force, is due to the expulsion of material. Feigin and Napper [32] were probably the first to introduce the term depletion.

Mixing colloids with polymers or other colloids can lead to phase transitions or aggregation resulting in, for instance, gelation, crystallisation, glass transition, flocculation, or fluid–fluid demixing of the dispersion. Figure 1.1 illustrates a colloid–polymer mixture and its tendency to phase separate into a phase enriched in colloids and a phase concentrated in polymers due to the attraction mediated by nonadsorbing polymer chains.

Fig. 1.1

Representation of a colloid–polymer mixture. Top: the system just after mixing. Bottom: the dispersion becoming inhomogeneous after mixing the colloidal spheres with a sufficiently high amount of nonadsorbing polymers

The type of instability depends on the range and strength of the particle interactions involved. The knowledge gained over the last decades on depletion effects in mixtures of colloidal particles and polymers is of great interest for designers of new products. Insight into the factors determining the stability of mixtures changes product development from trial-and-error towards knowledge-driven innovation. This book serves as a guide to help understand what happens when colloids are mixed with polymers or other colloids.

This chapter gives an introduction to colloidal interactions (including the depletion force in a historical context) and provides examples of the manifestations of depletion effects. First, we start with a brief overview of colloidal interactions in Sect. 1.2, including the basic concept of the depletion interaction. In Sect. 1.3, we outline the effects of unbalanced forces, addressing depletion forces in colloidal dispersions from a historical perspective, and including an overview of selected literature. Finally, a brief outline of the other chapters of this book are given in Sect. 1.4.

1.2 Colloidal Interactions

The basic understanding of colloidal interactions [10] commenced in the 1940s. Derjaguin and Landau [33] in the former USSR, and Verwey and Overbeek [34] in The Netherlands pointed out that, in a dispersion of charged colloids in an electrolyte solution, the Van der Waals attraction between two colloidal particles is opposed by a repulsion that originates from electrical double layers . This foundation for the stability of colloids is known as the DLVO theory and has been remarkably successful in explaining the results of a vast number and broad range of experiments, including direct force measurements [35]. Polymers, either depleted from or adsorbed or anchored to colloidal surfaces, also turned out to strongly influence colloidal interactions; these were not considered by DLVO.

In Sects. 1.2.1 and 1.2.2 we shall first consider Van der Waals and double layer interactions (the two contributions to the DLVO potential (Sect. 1.2.3), and then discuss (polymeric) steric stabilisation by end-attached polymers in Sect. 1.2.4. Finally, the depletion interaction will be addressed in Sect. 1.2.5.

1.2.1 Van der Waals Attraction

The attractive interaction between two colloidal particles is due to London–Van der Waals  attraction between their constituent atoms or molecules. For two atoms at a centre-to-centre distance r apart, the attraction has the form:

$$\begin{aligned} {} W (r) = - \frac{C}{r^6} {}\end{aligned}$$

(1.2)

(see Chap. 13 and Table 13.3 in Ref. [35]). According to London theory [36], C is given by the (approximate) expression

$$\begin{aligned} {} C = \frac{3}{4} \overline{E} \left( \frac{\alpha _\textrm{p}}{4\pi \varepsilon _0} \right) ^2. {}\end{aligned}$$

(1.3)

Here, $\overline{E}$ is a typical (average) electronic excitation energy, $\alpha _\textrm{p}$ is the static polarisability and $\varepsilon _0$ is the vacuum permittivity.

Hamaker [37] calculated the London–Van der Waals  attraction between two colloidal particles by summation of all atomic/molecular Van der Waals interactions between these two colloidal particles. For two colloidal spheres with radius R (Fig. 1.2) and closest distance between the spheres h, the resulting Van der Waals attraction without intervening medium reads (see for instance Ref. [38])

$$\begin{aligned} {} W_\textrm{VdW} (h) = - \frac{A}{6} f(h/R) , {}\end{aligned}$$

(1.4)

with

$$\begin{aligned}{} f(h/R) = \frac{2R^2}{h^2+4Rh}+\frac{2R^2}{h^2+4Rh+4R^2}+\ln \left( \frac{h^2+4Rh}{h^2+4Rh+4R^2}\right) , \nonumber {}\end{aligned}$$

Fig. 1.2

Two colloidal spheres with radii R at closest separation distance h

and

$$\begin{aligned} {} A = C \pi ^2 n^2 , {}\end{aligned}$$

(1.5)

where n is the number density of the atoms/molecules in the colloidal particles.

To make an estimate of the quantity A, which is known as the Hamaker constant we follow Israelachvilli [35] and use $C = 10^{-77}$ J$\cdot$m$^6$ and a number density n = 2$\cdot$10$^{28}$ m$^{-3}$ (corresponding to a diameter of 0.4 nm of the atoms/molecules in the colloidal particles). We then find

$$\begin{aligned} {} A = \pi ^2 \cdot 10^{-77} \cdot (3 \cdot 10^{28})^2 \simeq 10^{-19} \, \text {J}. {}\end{aligned}$$

(1.6)

Values for the Hamaker constants for different materials range between $(0.4-5)\cdot 10^{-19}$ J and can be found, for instance, in Table 13.2 in Ref. [35]. With an intervening medium between the colloidal particles the Hamaker constants are (significantly) lower than without intervening medium. From the values presented in Table 13.3 of [35] it turns out that the reduction may be as much as a factor of 2 or 3.

From Eq. (1.4) it follows that the Van der Waals attraction is very strong at short interparticle separations. For small h,

$$\begin{aligned} {} W_\textrm{VdW} (h) \simeq - \frac{AR}{12h}. {}\end{aligned}$$

(1.7)

To stabilise a colloidal dispersion, a significant repulsion that prevents the particles getting too close and aggregating irreversibly is needed. The Van der Waals interaction is shown schematically in Fig. 1.3 as the lower dashed curve.

Fig. 1.3

Common contributions to the interaction potential W(h) between colloidal particles as a function of separation distance h: typical double layer repulsion between charged colloidal  spheres (top), Van der Waals attraction (bottom) and their sum (solid curve), which is the DLVO interaction potential

1.2.2 Double Layer Interaction

A charged colloid is surrounded by a solution with an inhomogeneous distribution of ions. Co-ions (with the same charge as the colloids) are repelled from the colloid surface, whereas counterions (with opposite charge) accumulate at the surface. Far from the colloidal surface the concentrations of the two ion types attain a constant averaged value. The inhomogeneous layer is termed ‘double layer’, and its width depends on the ion concentration in the bulk solution: adding more ions screens the charges on the colloidal surfaces.

When two double layers overlap, a repulsive pair potential develops, which leads to a repulsive pressure. Dispersed like-charged colloids hence repel each other upon approach due to screened-Coulomb or double layer repulsion. The length scale over which this force is operational is set by the Debye screening length $\lambda _\textrm{D}$ which, for a simple 1–1 salt, reads

$$\begin{aligned} {} \lambda _\textrm{D} = \sqrt{\frac{1}{8 \pi \lambda _\textrm{B} n_\textrm{s}}}, {}\end{aligned}$$

(1.8)

where $n_\textrm{s}$ is the salt number density and $\lambda _\textrm{B}$ is the Bjerrum length,

$$\begin{aligned} {} \lambda _\textrm{B} = \frac{e^2}{4 \pi \varepsilon _0 \varepsilon _\textrm{r} kT}, {}\end{aligned}$$

(1.9)

with e the elementary charge and $\varepsilon _\textrm{r}$ the relative dielectric constant ($\approx 80$ in water). The Bjerrum length is the distance between two elementary charges at which their interaction equals kT. In water at room temperature its value is $\approx$ 0.7 nm. For the Debye length we can then use the expression $\lambda _\textrm{D} = 0.3 / \mathrm {\sqrt{c_\textrm{s}}}$ [35], with the salt concentration $c_\textrm{s}$ in mol/L and $\lambda _\textrm{D}$ in nm.

The interparticle separation dependence of double layer repulsion is approximately exponential for a thin double layer ($\lambda _\textrm{D} \ll R$) [34]

$$\begin{aligned} {} W_\textrm{DR} (h) = B \frac{R}{\lambda _\textrm{B}} \exp (- h / \lambda _\textrm{D}), {}\end{aligned}$$

(1.10)

which shows that the range of the screened double layer repulsion is $\lambda _\textrm{D}$, which depends on the salt concentration. The double layer interaction between two like-charged colloidal particles is represented in Fig. 1.3 (upper dashed curve).

The quantity B can be expressed in terms of the surface charge density $\sigma _\textrm{c}$ of the interacting colloids [34]

$$\begin{aligned} \frac{B}{kT} = \frac{8 p_\textrm{c}^2}{(1+q_\textrm{c})^2}, \end{aligned}$$

(1.11)

where $p_\textrm{c}=2\pi \lambda _\textrm{D} \lambda _\textrm{B} \left| \sigma _\textrm{c} / e \right|$ and $q_\textrm{c}=\sqrt{1+p_\textrm{c}^{2}}$, with $p_\textrm{c}$ the number of elementary charges e on a surface area $2\pi \lambda _\textrm{D} \lambda _\textrm{B}$. Given the fact that $\left| \sigma _\textrm{c} \right|$ varies roughly between 0.1 and 2 $e\mathrm {\cdot nm^{-2}}$, the value of $p_\textrm{c}$ ranges from 0.1 to 100 and thus B has a typical value of 0.1–8 kT. The quantity B can also be expressed as a function of the surface potential $\psi _0$ [34]:

$$\begin{aligned} \frac{B}{kT} = 8 \left[ \tanh \left( \frac{e \psi _0}{4 kT} \right) \right] ^2 . \end{aligned}$$

(1.12)

The surface potential of a charged colloidal particle typically varies from 10 to 100 mV, leading to B values in the same range as given above.

Fig. 1.4

Illustrative DLVO pair interactions (left) between two charged colloidal spheres (right) in an electrolyte solution as a function of increasing salt content from (i) $\rightarrow$ (iv)

1.2.3 DLVO Interaction

By assuming additivity of the interactions, the total DLVO potential is simply given by

$$\begin{aligned} {} W_\textrm{DLVO} = W_\textrm{VdW} + W_\textrm{DR}. {}\end{aligned}$$

(1.13)

In Fig. 1.3 the DLVO interaction potential $W_\textrm{DLVO}$ is represented alongside its two contributions. If the maximum of $W_\textrm{DLVO}$ is sufficiently high (larger than a few kT), flocculation is prevented. Flocculation does occur when the particles get very close together and reach the so-called primary minimum. This minimum is usually deep enough for irreversible flocculation.

For a given Van der Waals attraction and particle size the DLVO potential depends on the ionic strength. The DLVO potential is qualitatively represented in Fig. 1.4, from (i) towards (iv) by increasing the salt concentration. At low salt concentration (i) the double layer repulsion dominates, the maximum of $W_\textrm{DLVO}$ exceeds several kT, and a stable colloidal dispersion is expected. In situation (ii) the salt concentration is larger but there is still a local maximum that may be significant, preventing the particles from irreversibly sticking into the primary minimum. A shallow secondary minimum now manifests itself at large interparticle distances. If this local minimum is sufficiently deep (i.e. for large particles), weak flocculation can take place. Such weakly flocculated aggregates can be redispersed by shaking or by lowering the salt content. Adding still more salt (iii, iv) leads to irreversible aggregation: the Van der Waals attraction gets dominant and the colloidal dispersion will be unstable. DLVO theory is capable of accurately describing early stage aggregation of dilute charged colloidal spheres for $\lambda _\textrm{D} \gtrsim$ 3 nm [39].

Using the surface force apparatus, Israelachvili and Adams [40] measured a repulsive force between surfaces in aqueous solution at short separations that could not be interpreted in terms of DLVO theory. This interaction is due to hydration forces caused by the ordering of water molecules. Its range is very short, typically below 2 nm. For a discussion on the limitations of DLVO theory and possible improvements see Ref. [41].

In the above descriptions we concentrated on situations where a polar background solvent was implicitly assumed. In apolar solvents double layer repulsion is difficult to achieve because dissociation (which leads to charged surface groups) is less likely to occur, and it then becomes essential to stabilise colloids with polymers. In the first decades after the establishment of the DLVO theory, most papers on forces between colloidal particles focused on Van der Waals and double layer interactions. Forces of other origin, such as polymeric steric stabilisation [24], depletion [42], or effects of a critical solvent mixture [43], gained interest at a later stage.

Fig. 1.5

Polymers attached at a surface: a mushroom (left), a brush (middle) and a layer of adsorbed polymer (right)

1.2.4 Influence of Attached Polymers

Colloidal dispersions can be very well stabilised by attaching polymers to the particle surfaces [24]. Here, we consider polymer chains that are in a ‘good solvent’. This means that the chains are swollen and repel each other. As two colloidal particles protected with attached polymers approach each other, the local osmotic pressure increases dramatically due to mutual steric hindrance of the polymer chains on the particles. This competition between the chains for the same volume leads to a repulsive interaction, as was realised by Fischer [44].

Polymers can be attached to surfaces as, for instance, mushrooms, brushes or adsorbed chains (Fig. 1.5). In the case of mushrooms and brushes, the (nonadsorbing) chains are chemically bound to the surface by one chain end. When polymers adsorb at a surface many segments stick and densely pack at the interface. Attached polymers can contribute to a (significant) repulsive interaction between the particles. Upon overlap of the attached polymers the osmotic pressure between the surfaces strongly increases which leads to a repulsive interaction between the particles.

For polymer brushes—chains that are anchored to the surface by an end segment with a high anchor density—the chains are highly stretched. The Helmholtz (free) energy of interaction between brushes consists of two terms: an osmotic repulsion contribution and a stretching factor. The Alexander–De Gennes theory [45,46,47] considers the repulsive interaction of overlapping brushes of thickness $\delta _\textrm{b}$ (Fig. 1.5) in a good solvent. This thickness scales as $M \sigma _\textrm{b} ^{1/3}$, where M is the chain length and $\sigma _\textrm{b}$ the anchor density. For $h \le 2 \delta _\textrm{b}$ the pressure P between two parallel plates with anchored brushes at separation h reads:

$$\begin{aligned} {} \frac{P (h)}{kT \sigma _\textrm{b}^{3/2}} \approx \left( \frac{2 \delta _\textrm{b}}{h}\right) ^{9/4} - \left( \frac{h}{2\delta _\textrm{b}} \right) ^{3/4}. {}\end{aligned}$$

(1.14)

The first positive term on the right-hand side represents the osmotic repulsion between the brushes, and the second negative term originates from the elastic energy gain upon retraction of chains (less stretching). The repulsion dominates the interaction for $h<\delta _\textrm{b}$. As will become clear in Sect. 2.1, the pressure yields the interaction potential between two plates, from which the interaction between two spheres can also be derived.

Fig. 1.6

The influence of a brush repulsion on the interaction potential W(h) between two spheres with Van der Waals attraction

Figure 1.6 is a qualitative representation of the effect of adding a polymer brush to the interaction between two (uncharged) colloidal spheres subject to Van der Waals attraction. Commonly, one assumes the total interaction is the sum of all pair interactions:

$$\begin{aligned} {} W_\textrm{tot} = \sum _{i} W_{i}. {}\end{aligned}$$

(1.15)

So, in Fig. 1.6 the total interaction potential is $W_\textrm{tot} = W_\textrm{VdW} + W_\textrm{brush}$. Without the anchored polymer chains the particles would coagulate spontaneously since the Van der Waals attraction is very strong at small values of h. However, upon adding the polymer brush repulsion, the total interaction (solid curve) is repulsive for a wide h-range, with no significant attraction left.

The Van der Waals attraction can be reduced by choosing a solvent (mixture) that allows for refractive index matching of colloid and solvent. For model studies where one desires hard sphere-like particles, refractive index matching is combined with attaching short hairs (a thin brush; $\delta _\textrm{b} \ll R$) to the colloidal particles. This leads to absence of effective attractions and only a short-ranged repulsion between the particles, i.e. we now have a system of hard spheres (imagine submicrometer sized billiard balls). The pair interaction may then be approximated as

$$\begin{aligned} {}{} W(h) = {\left\{ \begin{array}{ll} \infty &{} h \le 0, \\ 0 &{} h > 0, \end{array}\right. } {}{} \end{aligned}$$

(1.16)

the hard sphere interaction plotted in Fig. 1.7 (left panel). In the next subsection we consider the effect of adding nonadsorbing polymers to such a hard sphere dispersion.

Fig. 1.7

Hard sphere (left) and square well or adhesive hard sphere (right) interaction

When the medium is a poor solvent for the attached polymers a rather different situation is encountered. The polymer chains then tend to assume collapsed configurations in order to minimise contact with solvent molecules and the polymer segments prefer to interact with each other. This results in (short ranged) attraction between colloidal particles covered with polymer chains in a poor solvent (see Sect. 5.5 in Ref. [48]). The interaction of such sticky spheres (billiard balls with a thin layer of honey [49]) is often described in a simple manner using the (square well or) adhesive hard sphere interaction  (see right panel in Fig. 1.7),

$$\begin{aligned} {}{} W(h) = {\left\{ \begin{array}{ll} \infty &{} h \le 0, \\ -\varepsilon &{} 0 < h \le \Delta , \\ 0 &{} h > \Delta , \end{array}\right. } {}{} \end{aligned}$$

(1.17)

where $\Delta$ is the range of the attraction set by the thickness of the polymer layers and $\varepsilon$ is the strength of attraction upon overlap of the polymer layers. For $\Delta \ll R$ the sticky sphere model of Baxter [50] can be employed providing simple expressions for the second osmotic virial coefficient and the equation of state. When the attractions are sufficiently strong, phase separation or aggregation occurs [51,52,53,54].

1.2.5 Depletion Interaction

Consider a room in a restaurant on two different occasions, as sketched in Fig. 1.8. On regular evenings the staff arranges the tables in a typical dinner set-up. Sometimes the room is booked for a cocktail party with many people present. In such a busy cocktail party the tables are laden with drinks and snacks and the configuration of the tables is rather different. Obviously, when the number of visitors exceeds a certain value, it is more efficient to push the tables close to each other and towards the wall in order to gain more translational freedom for the visitors.

Fig. 1.8

Left: dinner set-up in the restaurant on a quiet evening. Right: buffet set-up in the same restaurant after ‘phase separation’. Drawings by D. Frenkel (personal communication)

The ‘phase separation’ in Fig. 1.8 is driven by entropy only. The apparent attraction between the tables originates from purely repulsive people–people, people–table and table–table interactions: the visitors do not wish to be too close to each other (and can still fetch a drink from a table). It is, just like depletion, an example of what prof. Vrij [55, 56] referred to as ‘attraction through repulsion’. Below we explain the origin of the depletion effect, first by considering colloidal hard spheres in a solution of nonadsorbing polymer.

Suppose colloidal spheres are mixed with nonadsorbing polymers. The loss of configurational entropy of the polymer chains in the region near the surface results in negative adsorption. Hence the colloidal particles are surrounded by depletion layers: zones in which the polymer concentration is lower than in the bulk. The mechanism that is responsible for the depletion attraction originates from the presence of these depletion layers.

Fig. 1.9

Colloidal spheres in a polymer solution with nonadsorbing polymers. The depletion layers are indicated by the short dashes. When there is no overlap of depletion layers (upper two spheres) the osmotic pressure on the spheres due to the polymers is isotropic. For overlapping depletion layers (lower two spheres) the osmotic pressure on the spheres is unbalanced; the excess pressure is indicated by the arrows

Consider the depiction of a few colloidal spheres in a polymer solution shown in Fig. 1.9. Effective depletion layers are indicated by the (dashed) circles around the spheres. When the depletion layers overlap (lower two spheres) the volume available for the polymer chains increases. It follows that the free energy of the polymers is minimised by states in which the colloidal spheres are close together. The effect of this is just as if there were an attractive force between the spheres even though the direct colloid–colloid and colloid–polymer interactions are both repulsive [42]. For small depletant concentrations the attraction equals the product of the osmotic pressure and the overlap volume, indicated by the hatched region between the lower spheres in Fig. 1.9. The model illustrated above first became clear in the 1950s through the work of Asakura and Oosawa [57, 58] , and gained full attention only once Vincent et al. [59, 60] and Vrij [42] started systematic experimental and theoretical work on colloid–polymer mixtures.

Consider two colloidal spheres each with radius R, each surrounded by a depletion layer with thickness $\delta$. In that case the depletion potential can be calculated from the product of $P = n_\textrm{b} kT$, the (ideal) osmotic pressure of depletants with bulk number density $n_\textrm{b}$, multiplied by $V_\textrm{ov}$, the overlap volume of the depletion layers. Hence, the Asakura–Oosawa–Vrij (AOV) depletion potential equals [42, 57, 58]:

$$\begin{aligned} {}{} W_\textrm{dep}(h) = {\left\{ \begin{array}{ll} \infty &{} h < 0, \\ -P V_\textrm{ov} (h) &{} 0 \le h \le 2 \delta , \\ 0 &{} h \ge 2\delta , \end{array}\right. } {}{} \end{aligned}$$

(1.18)

with overlap volume $V_\textrm{ov} (h)$,

$$\begin{aligned} {} V_\textrm{ov} (h) = \frac{\pi }{6} (2\delta - h)^2 (3R+2\delta +h/2). {}\end{aligned}$$

(1.19)

This simple expression is often used for the depletion interaction and will be derived in more detail in Chap. 2.

The AOV interaction potential $W_\textrm{dep} (h)$ between two hard spheres in a solution containing free polymers is plotted in Fig. 1.10. The minimum value of the potential $W_\textrm{dep}$ is achieved when the particles touch ($h=0$).

Fig. 1.10

The depletion interaction W(h) between two hard spheres

We note that in the original paper of Asakura and Oosawa [57], where Eq. (1.18) was first derived, the polymers were regarded as pure (infinitely dilute) hard spheres. Vrij [42, 61] achieved the same result by describing the polymer chains as penetrable hard spheres (PHSs) (see Sect. 2.1). Inspection of Eqs. (1.18) and (1.19) reveals that the range of the depletion attraction is determined by the size $2 \delta$ of the depletant, whereas the strength of the attraction increases with the osmotic pressure and subsequently, with the depletant concentration. Depletion effects offer the possibility to independently modify the range and the strength of attraction between colloids. In dilute polymer solutions, the depletion thickness $\delta$ is close to the polymer’s radius of gyration $R_\textrm{g}$ (Sect. 2.2.1).

In a mixture of hard spheres and depletants, a phase transition occurs upon exceeding a certain concentration of colloidal spheres and/or depletants. This is the subject of Chaps. 3 and 4 and 6 and 7 in this book. This is extended to mixtures of anisotropic hard colloidal particles and depletants in Chaps. 8 to 10.

A key parameter in describing the phase stability of colloid–polymer mixtures is the size ratio q,

$$\begin{aligned} {} q = \frac{R_\textrm{g}}{R}. {}\end{aligned}$$

(1.20)

Throughout, colloid–polymer mixtures are described in terms of the volume fraction of colloids $\phi$ and the relative polymer concentration:

$$\begin{aligned} {} \phi _\textrm{p} = \frac{n_\textrm{b}}{n_\textrm{b}^*} = \frac{\varphi }{\varphi ^*}, {}\end{aligned}$$

(1.21)

which is unity at the (polymer coil) overlap concentration and can be regarded as the ‘volume fraction’ of polymer coils (and exceeds unity in the semidilute concentration regime). Here, $n_\textrm{b}$ is the bulk polymer number density and $n_\textrm{b}^*$ is its value at which the polymer coils overlap. In terms of the volume fraction of polymer segments $\varphi$ ($0 \le \varphi \le 1$), one then uses $\phi _\textrm{p} = \varphi / \varphi ^*$, with $\varphi ^*$ representing the segment volume fraction where the chains start to overlap:

$$\begin{aligned} {} \varphi ^* = \frac{M v_\textrm{s}}{v_\textrm{p}}, {}\end{aligned}$$

(1.22)

where M is the number of monomers per chain, $v_\textrm{s}$ is the monomer (segment) volume, and $v_\textrm{p} = (4\pi /3) R_\textrm{g}^3$ the coil volume. The overlap number density $n_\textrm{b}^*$ follows as $n_\textrm{b}^* = 3/ (4\pi R_\textrm{g}^{3})$.

It has actually become standard practice to normalise polymer concentrations in this way and use $\varphi / \varphi ^*$ (or $n_\textrm{b} / n_\textrm{b}^*$) as the parameter for ‘polymer concentration’. In terms of the more practically accessible concentration $c_\textrm{p}$ (in, e.g. kg/m$^3$ or g/L), the polymer coil overlap concentration is expressed as

$$\begin{aligned} {} c_\textrm{p}^{*}=\frac{3M_\textrm{p}}{4\pi R_\textrm{g}^{3} N_\textrm{Av}}, {}\end{aligned}$$

(1.23)

where $M_\textrm{p}$ is the polymer’s molar mass and $N_\textrm{Av}$ is Avogadro’s number. Note that $c_\textrm{p}^{*}=c_\textrm{p}/\phi _\textrm{p}$.

Exercise 1.1. Show that, when using the approximation $\delta = R_\textrm{g}$, the attractive part of Eq. (1.18) for ideal depletants can be written in normalised quantities as

$$\begin{aligned}{} \frac{W_\textrm{dep}(h)}{kT} = - \frac{\phi _\textrm{p}}{q^{3}} \left( q - \frac{h}{2R}\right) ^2 \left( \frac{3}{2} + q + \frac{h}{4R}\right) . \nonumber {}\end{aligned}$$

Hint: For ideal depletants the pressure P in Eq. (1.18) can be rewritten as $P v_\textrm{p} / kT= \phi _\textrm{p}$ by using $\phi _\textrm{p}= n_\textrm{b} v_\textrm{p}$.

Figure 1.11 shows the influence of a combined depletion attraction and a brush repulsion on the total interaction. The presence of brushes reduces the attraction and the minimum value of the attraction is found at $h>0$ [62].

Fig. 1.11

The total interaction potential between two spheres covered with polymer brushes in a good solvent. The solution contains nonadsorbing polymer chains

The fact that depletion forces enable the range and strength of attraction to be varied independently is helpful for studying fundamental properties of liquids, as well as crystallisation and gelation phenomena, using colloidal systems instead of low molar mass substances. Another advantage of colloid–polymer mixtures is that colloids can be investigated using microscopy. Aarts et al. [63] could even detect capillary waves at the colloidal gas–liquid interface. Observations of wetting phenomena can also be studied at the particle level [64, 65].

1.3 Historical Overview on Depletion

Depletion in colloidal dispersions is a central theme in this book. As we saw in Sect. 1.2.5 depletion effects in colloidal dispersions are caused by an unbalanced force. From a physics point of view, the depletion force between colloidal particles due to nonadsorbing polymer chains or small particles has common features with any other unbalanced force, whether of a colloidal nature or not. Before we focus on depletion effects on a mesoscopic level, we first give two classical examples of unbalanced forces.

1.3.1 Early Interest in Unbalanced Forces

1.3.1.1 The von Guericke Force

Halfway through the 17$\text {th}$ century a series of remarkable experiments were performed, initiated by Otto von Guericke. One took place in 1657 at the court of King Friedrich Wilhelm III of Brandenburg in Berlin, Germany. Two hollow copper hemispheres, each with a diameter of 51 cm, were joined together and air pumped out to create a partial vacuum. A team of horses was then harnessed to each hemisphere (Fig. 1.12). The teams, each pulling with a force of about 1500 N each, could not pull the two joined hemispheres apart, demonstrating the tremendous force of air pressure. This proved the existence of the nothing we now call a vacuum.

Exercise 1.2. Show that the force on the hemispheres due to air pressure is one order of magnitude larger than the force that can be produced by 24 horses.

Fig. 1.12

Caspar Schott’s (1602–1666) illustration of the vacuum force demonstration by two teams of horses attempting to separate the hemispheres

This experiment was the brainchild of scientist, inventor and politician Otto von Gericke (later spelled von Guericke), who lived between 1602 and 1686. The vacuum pump he used was invented by himself in 1650. His book on vacuum [66] reminds one of the difficulties in understanding vacuum at the time. Von Guericke abandoned established views and developed an independent vision on vacuum [67]. The result of this experiment showed that the surrounding air molecules push against the Magdeburg hemispheres (von Guericke was the mayor of Magdeburg). While it appears that there is an attractive force that pulls the hemispheres together, the vacuum in fact results from unbalanced repulsive forces.

Exercise 1.3. Explain why the osmotic pressure of the polymer solution in a colloid–polymer mixture plays a similar role to air pressure in von Guericke’s experiment.

1.3.1.2 Le Sage’s Gravitation Theory

In 1690, Nicolas de Fatio (and later in 1748 Georges-Luis Le Sage) proposed a mechanical theory for the explanation of both Newton’s gravitational force and cohesive forces in materials [68]. This theory assumes the existence of ‘ultramundane corpuscles’. Streams of such corpuscles are thought to impact on all materials from all directions. Now, if two bodies of materials are close to one another they can partially shield each other from the incoming ‘ultramundane corpuscles’; the bodies will be struck by fewer corpuscles from the side of the other body. This mutual shielding was then supposed to push the bodies together due to the unbalanced force of the colliding corpuscles. This line of reasoning was first formulated by de Fatio in a letter to Huygens [69], but Newton also had contact with de Fatio on this matter. While Huygens, Newton and Leibniz were interested, they never accepted de Fatio’s explanation as the driving force for gravity.

At a later stage Le Sage published a similar, more refined version of the theory [70]. He was in contact with some of the greatest physicists and mathematicians of his time, including Euler and Bernouilli, who found the theory rather speculative. Inconsistencies in the theory were later revealed by, for instance, Laplace, Lord Kelvin, Lorentz and, for didactic reasons, by Feynman [71] more recently. Occasionally, there is still interest in Le Sage’s theory [72]. In an interesting paper, Rowlinson [68] drew attention to the fact that the work of Le Sage has a remarkable similarity to the depletion force.

1.3.2 Experimental Observations on Depletion Before the 1950s

Long before Asakura and Oosawa rationalised the attractive interaction caused by depletants, the effects of depletion were already noted in various areas of specialisation. In this overview, we first give examples of such studies and try to interpret them with our current knowledge of depletion forces. Subsequently, we discuss several studies that were performed after the work of Asakura and Oosawa, often in light of the theoretical progress that is being made over the last decades especially. Although it is nearly impossible to cover all developments within the area of depletion phenomena in physics and chemistry, we aim to give the reader a broad overview here.

1.3.2.1 Clustering of Red Blood Cells

Red blood cells  (RBCs) are biconcave particles and their detailed shape and size depend on the RBC type. The human RBC may be considered a disc with a diameter $D=6.6$ $\upmu$m and a thickness of $L=2$ $\upmu$m, its volume thus being of the order of $10^2$ $\upmu \textrm{m}^3$. The RBCs occupy about 40–50 vol% of our blood.

Exercise 1.4. Demonstrate that stacking all red blood cells in a human being (having about 5 L of blood) in a single column provides an RBC cylinder with a height that is of the order of the earth’s circumference.

By the 18$\text {th}$ century it was already known that RBCs tend to cluster, preferably with their flat sides facing each other, like a stack of coins [73]. These structures are commonly denoted as ‘rouleaux’. In the blood of healthy human beings the tendency of RBCs to aggregate is low. Aggregation is found to be enhanced in pregnancy or a wide range of illnesses, giving rather pronounced rouleaux (Fig. 1.13). An impressive review on RBC clustering was written by Fåhraeus [73]. Thysegen [74] provides another historical review.

Fig. 1.13

Reprinted with permission from Ref. [73]. Copyright 1929 APS

Red blood cells in the blood of a pneumonia patient (left), in which rouleaux formation took place (strong aggregation), compared to weak aggregation in healthy blood (right) [73].

Enhanced RBC aggregation can be detected, for instance, by measuring the sedimentation rate. The sedimentation rate varies between 1–3 mm per hour for healthy blood and up to 100 mm per hour in the case of severe illnesses. The blood sedimentation test, based on monitoring the aggregation of red blood cells, became a standard method for detecting illnesses. The relationship between pathological condition, RBC aggregation and enhanced sedimentation rate has been known for at least two centuries, as described in Refs.  [73,74,75,76].

Fåhraeus [73, 75] related enhanced aggregation of RBCs longer and stronger rouleaux to the concentration of the blood serum proteins fibrinogen, globulin and albumin. The tendency to promote aggregation depends on the type of protein. Rouleaux formation is most sensitive to increased serum concentrations of fibrinogen (molar mass 340 kg/mol) compared to $\upbeta$- and $\upgamma$-globulins (90 and 156 kg/mol, respectively). The globulins in turn lead to RBC aggregation at lower protein concentrations than albumin proteins (69 kg/mol). Further, it has been shown that adding several types of macromolecules also promotes rouleaux formation [77]. Asakura and Oosawa [58] suggested that RBC aggregation might be caused by depletion forces between the RBCs induced by serum proteins. This is in line with the finding that the sedimentation rate is more sensitive to larger serum proteins.

Some authors interpret rouleaux formation as being caused by bridging of RBCs by serum proteins. There is, however, no evidence for protein adsorption onto RBCs. A study on rouleaux formation in mixtures of human RBCs (D = 6.6 $\upmu$m) and rabbit RBCs (D = 7.8 $\upmu$m) resulted in rouleaux structures that consisted (mainly) of only a single type of RBC [78]. This can be explained by a depletion effect (the overlap volume, hence entropy, is maximised if similar RBCs stack onto each other). However, the formation of mixed aggregates is expected if bridging were to occur; and since this is not observed, there is little support for the bridging hypothesis [79]. The general picture is that red blood cells tend to cluster at elevated concentrations of the blood serum proteins, which act as depletants [80, 81].

1.3.2.2 Demixing of Biopolymers in Solution

Another manifestation of segregative interactions leading to demixing was reported by the microbiologist Beijerinck [82] who tried to mix gelatin (denatured protein coil) with starch (polysaccharide) in aqueous solution in order to prepare new Petri dish growth media for bacteria. He reported that these biopolymers could not be mixed; emulsion droplets appeared instead. With current knowledge [83, 84] this can be regarded as an early detection of depletion-induced demixing. Tolstoguzov, Grinberg and co-workers extensively studied many mixtures of polysaccharides and proteins and concluded that such mixtures tend to segregate [85,86,87], unless there are specific interactions such as opposite charges. They further found that adding salt decreases the miscibility region in protein/polysaccharide mixtures [85]. It is obvious that, as well as pure depletion forces, double layer interactions play a role in such mixtures. The separate liquid phases in demixed protein–polysaccharide mixtures can sometimes be characterised by a sharp liquid–liquid interface. The interfacial tension between the coexisting phases in protein–polysaccharide mixtures has been determined and is of $\mathcal {O}(\upmu \mathrm {N/m})$ [88, 89], in agreement with Eq. 1.24 below.

1.3.2.3 Creaming of Particles in Latex and of Emulsion Droplets

In the beginning of the 20$\textrm{th}$ century, large scale production of latex for rubber and paint production commenced. The term ‘latex’ is nowadays identified with a stable dispersion of polymeric particles in an aqueous medium. In order to lower transport costs there was a significant interest in concentrating the polymeric latex. Centrifugation is highly energy consuming, and thus expensive.

Traube [90] showed that adding plant and seaweed polysaccharides led to a phase separation between an extremely dilute and a very concentrated phase. Since the particles are lighter than the solvent, the concentrated phase (with volume fraction $0.5 \le \phi \le 0.8$) floats on top. The lower phase is clear and hardly contains particles. Baker [91] and Vester [92] systematically investigated the mechanism that leads to what they called (enhanced) creaming.

Fig. 1.14

Reprinted from Ref. [91]

Optical micrographs of a rubber latex dispersion [91]. a 1% suspension without polymer. b 2 min after addition of 0.2% polysaccharide. c 10 min after addition of 0.2% polysaccharide. d Image of the creaming layer. Size of the images are about 130 by 100 $\upmu$m.

In Fig. 1.14 we show microscopy images of the latex dispersion investigated by Baker [91]. The images are for a 1% latex dispersion, first without added polymer (A). Images B and C were taken respectively 2 and 10 min after adding 0.3% of polymer (the polysaccharide tragon seed gum). After polymer addition, Baker reports an immediate deceleration of Brownian motion adjacent to particle aggregation. After about 10 min particle aggregation discontinues and the aggregates start creaming. The entire creaming process takes about 1 day. Image D was taken in the cream layer. Upon diluting the cream layer to 1% of latex particles Brownian motion restarts, suggesting that the flocs segregated into individual particles again. From the work of Baker [91] it can thus be concluded that the particles aggregate reversibly; upon dilution the latex particles can be resuspended. This suggests that bridging, which can also cause creaming [93], is not the driving force for enhanced creaming.

Fig. 1.15

Reprinted with permission from Ref. [92]. Copyright 1938 Springer

Optical micrographs of demixed dispersions of emulsion droplets (black) in a polymer solution before (left) and after (right) pressing the microscopy slide. Micrograph diameters correspond to 610 $\upmu$m.

Vester [92] reviewed ways to optimise the creaming speed of lattices by using nonadsorbing polymer chains as depletants. He found that polymer addition can also lead to formation of short-lived emulsion droplets with diameters of $\mathcal {O}(\text {10--100}\,\upmu \text {m})$ that are enriched in latex, while the continuous phase is dilute in latex particles. Nowadays, this is interpreted as a colloidal gas–liquid phase coexistence. A microscopy image of the resulting emulsion is given in Fig. 1.15 (left panel). The droplets deform very easily upon confining the emulsion (right panel Fig. 1.15). This must imply that the interfacial tension $\gamma$ is very low, say $\ll$ 1 mN/m. Indeed, we do expect a small interfacial tension. The order of magnitude of a surface tension can be estimated from

$$\begin{aligned} {} \gamma \approx \frac{kT}{d^2} , {}\end{aligned}$$

(1.24)

where d is the particle diameter [94]. For molecular systems this yields values for $\gamma$ of at least a few mN/m up to hundreds mN/m, which is indeed approximately the range of measured surface tensions. For particles in the colloidal size domain with, for instance, $d \approx 30$ nm, an interfacial tension of only 1 $\upmu$N/m is expected at room temperature. Indeed, this is the order of magnitude of the ultra-low interfacial tension measured in demixed colloid–polymer mixtures that are in colloidal gas–liquid equilibrium (see Chap. 5).

Cockbain [95] found that creaming of oil droplets in a surfactant-stabilised oil-in-water emulsion  is enhanced when the surfactant concentration exceeds the critical micelle concentration. This phenomenon was left unexplained at the time, but thirty years later Fairhurst et al. [96] made a connection with depletion interaction theories and suggested that the micelles play a similar role to nonadsorbing polymers or small colloidal particles in acting as depletants (see Chap. 6).

1.3.2.4 Precipitation and Isolation of Viruses

Cohen [97] demonstrated that adding less than a percent of heparin to solutions of rod-like viruses results in the precipitation of the virus particles. The isolated precipitate phase consists of ‘paracrystals’. The connection Cohen [97] makes with the work of Bernal and Farkuchen [98] suggests the phase appears liquid crystalline. This allows viruses to be isolated and concentrated [99, 100]. A microscopy image of clusters of tobacco mosaic virus (TMV) particles in a dispersion with 0.5 wt% of heparin [97] is shown in Fig. 1.16. In Chap. 8 we consider depletion effects in colloidal rod dispersions.

Fig. 1.16

Reprinted from Ref. [97] under the terms of CC-BY-4.0

Light microscopy image of TMV paracrystals upon adding heparin [97]. Size of image: 80 by 60 $\upmu$m.

1.3.3 1950–1969

The work of Fumio Oosawa (1922–2019) played a crucial role in the development of the insights into the depletion interaction. After finishing his education in physics in 1944 [102], Oosawa specialised in statistical mechanics because he wanted to do some ‘unorthodox physics’ [103]. Oosawa [101], who in the early 1950s was a young Associate Professor at Nagoya University in Japan, organised a winter symposium in Nagoya and invited a multidisciplinary group of Japanese scholars, mainly active in biology. He asked the participants to present work on phenomena in biological systems where statistical physics could be helpful to understand certain mechanisms. During the meeting the ‘aggregation’ of particles under the influence of macromolecules was a re-occurring theme [103]. It was observed, for instance, in suspensions of red blood cells [77], bacterial cells, soil powder and gum latex particles, as explained by Professor Oosawa during a symposium in 2014 [104]. During the winter symposium in 1953, professor Tachibana commented that these similar phenomena might originate from the same physical principle [103]. This inspired Oosawa to start work with Sho Asakura, then a graduate student, on the influence of polymers on the interaction between particles.

Fig. 1.17

Photograph courtesy of the family of Professor Fumio Oosawa

Oosawa (left) and Asakura (right) at Nagoya University in the 1960s [101].

Soon after, later Noble Prize (1974) winner P.J. Flory met Oosawa at a conference in Tokyo, organised by professor Yukawa. At this conference Oosawa invited professor Flory to come to Nagoya University [101]. During this visit Asakura and Oosawa (Fig. 1.17) reported unpublished theoretical results on two particles immersed in a solution containing nonadsorbing polymer chains, showing the chains impose an effective attractive interaction between the particles. The very positive response of Flory, at that time Associate Editor of J. Chem. Phys., resulted in submission of this work and lead to a seminal paper, in which Asakura and Oosawa [57] presented a derivation of the interaction between two particles immersed in a solution of other nonadsorbing species. The theory of Asakura and Oosawa is the first theoretical prediction of a depletion force.

The effective attraction is a purely entropic effect; the indirect attraction originates from purely repulsive interactions. In this short paper Askura and Oosawa [57] considered four cases. The first three concern the interaction between parallel flat hard plates mediated by dilute (i) hard spheres, (ii) thin rods  and (iii) ideal polymer chains. The segments of ideal polymer chains have no excluded volume; segments do not feel other segments. This will be explained in more detail in Sect. 2.2. Further, they also considered the interaction between hard spheres induced by small dilute hard spheres.

Not long after the publication of the work of Asakura and Oosawa, Sieglaff [105] demonstrated that a depletion-induced phase transition may occur upon adding polystyrene to a dispersion of microgel spheres in toluene. This demonstrated that the attractive depletion force is sufficiently strong to induce a phase separation. Sieglaff rationalised his findings in terms of the theory of Asakura and Oosawa. It took several years before subsequent work was done. This study of Sieglaff was later extended by Clarke and Vincent [106].

1.3.4 1970–1982

Early systematic experimental studies with respect to phase stability for colloid–polymer mixtures were performed by Vincent and co-workers [60, 107, 108]. At ICI and in Bristol, they concentrated on mixtures of colloidal spheres (latex particles) and nonadsorbing polymers such as polyethylene oxide (PEO).  The work started when Vincent worked at ICI in Slough (1970–1972), where he investigated the origin of the flocculation of pigment particles in paint dispersions with F. Waite. In the papers of Vincent et al. [107, 109,110,111] a lot of attention is given to properly qualifying the demixing phenomena in colloid–polymer mixtures. These experiments were ahead of a full theoretical understanding of the phase behaviour of colloid–polymer mixtures. One of the systems studied was polystyrene spheres with terminally attached PEO brushes dispersed in mixtures of free PEO and water for a wide range of concentrations. The spherical particles were stable in both pure water and pure PEO melts. However, in mixed solutions of PEO and water (for instance, 50% water and 50% of PEO) a ‘slow flocculation’ of the particles was observed. The maximum flocculation rate was measured and was found to shift to lower PEO concentrations upon increasing the molar mass. This restabilisation at very high polymer concentrations (reported in a series of papers [107, 109, 110, 112]) was also found in a dispersion of grafted  silica spheres mixed with polydimethyl siloxane (PDMS)  polymer chains. Only polymer melts that are sufficiently liquid-like allow systems at such high polymer concentrations to be studied because other polymers would be too viscous for a proper analysis of the phase behaviour.

Fig. 1.18

Reprinted with permission from Ref. [113]. Copyright 1976 Elsevier

Optical micrographs showing mixtures of monodisperse polystyrene latex particles with 185 mg/L sodium polyacrylate polymers after (top) 25 min and (middle) 55 min; and (bottom) with 370 mg/L sodium polyacrylate after 25 min.

In the same period, Hachisu et al. [114] investigated aqueous dispersions of negatively charged polystyrene latex particles that undergo a colloidal fluid-to-solid phase transition upon lowering the salt concentration using dialysis or increasing the particle concentration. Under conditions where the latex dispersion (particles with $R = 170$ nm) is not ordered (fluid-like), Kose and Hachisu [113] added sodium polyacrylate to polystyrene latex particles (both components are negatively charged) and observed crystallisation  of the colloidal spheres (Fig. 1.18). Since polymers and particles repel each other, the crystallisation process is probably induced by depletion interaction, although the authors themselves did not explicitly mention depletion. They do suggest that the ordering is due to ‘some attractive force’. When the polymer concentration is increased, crystallisation occurs faster (Fig. 1.18, bottom panel).

Theoretical work on depletion interactions and their effects on macroscopic properties such as phase stability commenced along various routes. First, Vrij [42] studied the polymer-mediated depletion interaction between hard spheres. He described the nonadsorbing polymers as PHSs (see Sects. 1.2.5 and 2.1). Vrij [42] referred to the work of Vester [92], Li-In-On et al. [60] and preliminary experiments at the Van ’t Hoff Laboratory on micro-emulsion droplets mixed with free polymer [42] for experimental evidence of depletion effects.

Progress on the quantification of the depletion layer thickness  was triggered by 1991 Noble Prize winner De Gennes. In his seminal book [115], De Gennes derived an expression for the density profile of a semidilute polymer solution near a hard wall and demonstrated that the depletion thickness equals the correlation length, the length scale over which the segments are correlated. In dilute polymer solutions (below coil overlap) it is close to the radius of gyration of the polymer chains. However, in semidilute solutions (above overlap) the correlation length becomes independent of the chain length and is a (decreasing) function of the polymer concentration. Hence, also the depletion thickness decreases with polymer concentration in the semidilute regime. De Gennes considered the depletion contact potential between two colloidal hard spheres in a semidilute polymer solution in a good solvent. For this case, where the only relevant length scales are the sphere radius R and the correlation length $\xi$, he derived the following scaling relation for the minimum of the interaction potential [116]:

$$\begin{aligned} {}{} \frac{W_\textrm{dep}(h=0)}{kT} \cong {\left\{ \begin{array}{ll} -\frac{R}{\xi } &{} R \gg \xi , \\ - \left( \frac{R}{\xi }\right) ^{4/3} &{} R \ll \xi , \end{array}\right. } {}{} \end{aligned}$$

(1.25)

with an unknown prefactor $\mathcal {O}(1)$.

Exercise 1.5. What is expected with respect to colloidal stability of large ($R \gg \xi$) and tiny ($R \ll \xi$) colloidal spheres in a semidilute polymer solution?

Depletion effects have been studied using mean-field methods since the end of the 1970s. Insights into polymer physics have increased tremendously through the development of mean-field theories. The advantage of these theories is that they simultaneously include excluded volume interactions and give insights into details of polymer configurations near interfaces. A detailed analytical mean-field treatment for depletion interaction was made by Joanny et al. [117] who calculated the polymer segment concentration profile between two plates in the semidilute regime, in agreement with De Gennes’ scaling prediction discussed above.

Using a Flory–Huggins-like mean-field model, Feigin and Napper [32] calculated the free energy of interaction between two flat plates mediated by nonadsorbing polymers and noted that a repulsive barrier is present for polymer concentrations in the concentrated regime. The potential at plate contact is, however, still attractive. The authors suggested that if the repulsive barrier is large enough this might lead to so-called depletion stabilisation; a colloidal dispersion is destabilised at low polymer concentrations but restabilised at high concentrations. A conceivable intuitive explanation is kinetic: at high polymer concentrations it is hard to push polymer chains out of the gap between two particles. The bulk osmotic pressure is very high in a concentrated polymer solution. The polymer chains between the particles thus need to be transported towards a very steep osmotic pressure gradient.

Scheutjens and Fleer  [118, 119] developed a numerical self-consistent field (SCF) method that enables the calculation of equilibrium concentration profiles near interfaces. This SCF method was applied to depletion effects in [120], showing that the depletion layer thickness is close to $R_\textrm{g}$ at low polymer concentrations but decreases with increasing polymer concentration in the semidilute regime. In the concentrated regime, very close to the melt concentration, the polymer concentration between two parallel plates oscillates around the bulk polymer concentration. This finding is supported by Monte Carlo computer simulations  of Broukhno et al. [121] The interaction potential between the plates was also calculated by Scheutjens and Fleer [120] using SCF. For dilute polymer solutions the range of the potential is close to 2$R_\textrm{g}$ and the depth of the potential increases with increasing solvent quality. When the volume fraction of polymer segments in the system is 0.1 (a very high polymer concentration, in practice) a weak repulsive part appears in the interaction potential, as was also found by Feigin and Napper [32]. This repulsion appears at lower concentrations for better solvent quality [24, 120].

A direct link between theoretical and experimental work on depletion-induced phase separation of a colloidal dispersion due to nonadsorbing polymers was made by De Hek and Vrij [61, 122]. They mixed sterically stabilised silica dispersions with polystyrene in cyclohexane and measured the limiting polymer concentration (phase separation threshold). Commonly, one uses the binodal or spinodal as the experimental phase boundary. A binodal  denotes the condition (compositions, temperature) in which two or more distinct phases coexist (see Chap. 3). A tie-line  connects two binodal points. A spinodal  corresponds to the boundary of absolute instability of a system to decomposition. At or beyond the spinodal boundary infinitesimally small fluctuations in composition will lead to phase separation.

De Hek and Vrij [61] used the pair potential of Eq. 1.18 to estimate the stability of colloidal spheres in a polymer solution by calculating the second osmotic virial coefficient  $B_{2}$:

$$\begin{aligned} {} B_2 = 2 \pi \int _{0}^{\infty } r^2 (1-\exp [-W(r)/kT]) \textrm{d}r, {}\end{aligned}$$

(1.26)

where we used the centre-to-centre distance r between the spheres, which equals $2R+h$. A simple argument was used to estimate the spinodal [61]. For colloid or polymer concentrations [123] exceeding the spinodal, phase separation occurs spontaneously. Therefore, at the spinodal

$$\begin{aligned} {} \frac{\textrm{d}P}{\textrm{d} \phi } = 0 . {}\end{aligned}$$

(1.27)

The virial expansion for the osmotic pressure P of a colloidal dispersion reads

$$\begin{aligned} {} \frac{Pv_0}{kT} = \phi + B_2^* \phi ^2 + \cdots . {}\end{aligned}$$

(1.28)

with $B_2^* = B_2 /v_0$. Here, $v_0$ is the volume of the colloidal sphere. In the limit of low $\phi$, Eqs. (1.27) and (1.28) provide

$$\begin{aligned} {} 1+ 2B_2^* \phi ^\textrm{sp} = 0. {}\end{aligned}$$

(1.29)

This relates the polymer activity (which determines $B_2$) to the colloid volume fraction $\phi ^\textrm{sp}$ at the spinodal. De Hek and Vrij [61] were able to give a good description of the phase line of mixtures of polystyrene chains and small volume fractions of (hard sphere like) octadecyl silica spheres dispersed in cyclohexane [122].

Fig. 1.19

Reprinted with permission from Ref. [61] Copyright 1981 Elsevier

a Photograph of a test tube containing a phase-separated mixture of polystyrene polymer chains (molar mass: 32.4 kg/mol) and sterically stabilised silica spheres ($R = 21$ nm) in cyclohexane. Initial concentrations: 1 wt % of silica spheres and 2.5 g/L polystyrene. The concentration below which no phase separation was found was 17 g/L. The two demixed phases are separated by a sharp interface. Reprinted with permission from Ref. [122]. Copyright 1979 Elsevier. b State diagram of 1 wt % silica spheres ($R = 46$ nm) in cyclohexane mixed with polystyrene polymer chains varying in molar mass $M_\textrm{p}$ [61]. The limiting polystyrene concentrations below which no phase separation occurred are indicated as the filled circles. Hatched region: theoretical limits between which the spinodal curve is situated.

Figure 1.19 depicts results obtained by de Hek and Vrij [61, 122] on a mixture of octadecyl silica spheres and polystyrene polymers in cyclohexane. Both separated phases are fluid. The limiting polymer concentration below which no phase separation occurs in a solution containing a given amount of silica is plotted versus the molar mass of the added polymer polystyrene. It was found that less polymer is required to induce a phase separation when the molar mass is larger. This experimental trend can be predicted by using the spinodal condition Eq. (1.29), but this is only a semi-quantitative test because, in fact, formally, one should compare the stability curve with the binodal, as this denotes the compositions of the coexisting phases in a demixed dispersion. The spinodal is, however, not too far off from the binodal and probably gives a good and simple estimate. For the smallest molar mass, the separated phase was gel-like instead of a fluid. A statistical mechanics calculation of ideal polymer chains between two walls by De Hek and Vrij [61] demonstrated that the range of attraction between two flat parallel plates due to ideal polymer chains is close to $2.25 R_\textrm{g}$, implying a depletion thickness at each plate of about $R_\textrm{g}$.

A static light scattering (SLS)  contrast variation study on elucidating the negative adsorption of polystyrene chains next to a silica sphere in cyclohexane solutions was described in another paper by de Hek and Vrij [124]. This negative adsorption can be converted to a depletion thickness, which is approximately the radius of gyration of polystyrene in cyclohexane. The second virial coefficient of the silica particles could be determined from the SLS experiments, and its value was shown to become negative when a sufficient amount of nonadsorbing polystyrene is added, which implies attraction between the spheres.

By mixing aqueous hydroxyethylcellulose (HEC)  with latex, Sperry and co-workers [125,126,127] observed phase separation and made a study on the effect of the structure of the colloid-rich phase as a function of the colloid–polymer size ratio $q = R_\textrm{g} / R$. The micrographs in Fig. 1.20 of phase separating mixtures demonstrate how the morphology of the segregating systems varies upon changing q and polymer concentration. Unstable systems at large q and not too high polymer concentrations are characterised by smooth interfaces, implying colloidal gas–liquid coexistence. For small q, demixed systems are characterised by irregular interfaces that indicate (colloidal) fluid–solid coexistence. This suggests that the width of the region where a colloidal liquid is found in colloid–polymer mixtures is limited. We return to this issue in Sect. 4.3. Irregular interfaces are also detected for $q > 1/3$ when the polymer concentration is substantially increased.

Fig. 1.20

Reprinted with permission from Ref. [127]. Copyright 1984 Elsevier

Optical micrographs of latex colloids mixed with HEC by Sperry [125,126,127].

1.3.5 1983–1999

The work of Sperry inspired Gast, Hall and Russel  to develop a theory which might explain the experimental phenomena. Gast et al. [128] used thermodynamic perturbation theory  (TPT) [129] to derive the free energy of a mixture of colloidal particles and polymers (described as penetrable hard spheres, PHSs), based on pair-wise additivity of the interactions between the colloids. This is an approach which is based upon a perturbation of the free energy of a pure colloidal dispersion due to depletion forces, with Eq. (1.18) as input. Using equations of state for the hard sphere fluid and the FCC crystal structure as references, they calculated the phase behaviour from the (perturbed) free energy. This made it possible to assign the nature (i.e. colloidal gas, liquid or solid) of the coexisting phases as a function of the size ratio q, the concentration of the polymers, and the volume fraction of colloids. For small values of q, say, $q = R_\textrm{g}/R < 0.3$, increasing the polymer concentration broadens the hard sphere fluid–solid coexistence region; a (stable) colloidal fluid–solid coexistence is expected if the polymer chains are significantly smaller than the colloidal spheres (low q). Inside the unstable regions a (metastable) colloidal gas–liquid branch is located. For intermediate values of q, the gas–liquid coexistence curve crosses the fluid–solid curve; and for large q-values, mainly gas–liquid coexistence is found for $\phi <0.49$, where $\phi$ is the volume fraction of colloids [123]. The results are in agreement with the findings of Sperry [125,126,127].

Exercise 1.6. Use the Gibbs phase rule and derive how many coexisting phases a system can assume when it consists of two components. For a discussion see Ref. [130].

Experimentally, Gast, Russel and Hall [131] later verified the predicted types of phase coexistence regions for a model colloid–polymer system. Colloid–polymer phase diagrams [123] are commonly plotted in terms of the volume fraction of colloids $\phi$ and the relative polymer concentration $\phi _\textrm{p}$, defined in Eq. 1.21. In both the descriptions by De Hek and Vrij and by Gast, Russel and Hall, the depletion thickness $\delta$ was assumed to be equal to the radius of gyration of the polymers. This assumption can be rationalised by calculating the density profile of polymer chains at a surface. This was done by Lépine and Caillé [132], who solved the Edwards equation for ideal polymer chains near a reflective, attractive and repulsive surface. Eisenriegler [133] also calculated the density profile of nonadsorbing ideal chains near a flat surface and from this density profile it follows that $\delta / R_\textrm{g} = 2 / \sqrt{\pi } \approx 1.13$ [134] (see Sect. 2.2). This agrees with an earlier result derived by Casassa and Tagami [135] using the end segment distribution at a nonadsorbing flat surface. Later, it was shown [136] that the depletion layer thickness is independent of the reference point (for instance, centre of mass, middle segment and end segments) used to describe the depletion density profile.

Experimental work on the determination of the depletion layer thickness commenced in this period, although these are indirect measurements. The depletion thickness $\delta$ of polystyrene in ethyl acetate at a nonadsorbing glass plate was measured using an evanescent wave technique by Allain et al. [137]. The value found for $\delta$ was indeed close to the radius of gyration of the polymer. Ausserré et al. [138] measured the depletion thickness of xanthan (a polysaccharide) in water at a quartz wall below and above the polymer overlap concentration.  In dilute solutions below overlap, $\delta$ was close to the radius of gyration of xanthan, whereas in the semidilute regime (i.e. above overlap ($\phi _\textrm{p} > 1$)) it decreases as $\delta \sim \phi _\textrm{p}^{-0.8}$. This is in accordance with what is expected theoretically (see Sects. 4.3.1 and 4.3.2). Pashley and Ninham [139] succeeded in measuring the depletion potential between mica plates (as induced by CTAB micelles) using the surface force apparatus.

The polymer density profile of nonadsorbing ideal chains next to a hard sphere for arbitrary size ratio q was first calculated by Taniguchi et al. [140] and later independently by Eisenriegler et al. [141] Eisenriegler also considered the pair interaction between two colloidal hard spheres for $R_\textrm{g} \ll R$ [142] and for $R_\textrm{g} \gg R$ [143], as well as the interaction between a sphere and a flat wall due to ideal chains [144]. Depletion of excluded volume polymer chains at a wall and near a sphere was considered by Hanke et al. [145] One of their results is that the ratio $\delta /R_\textrm{g}$ at a flat plate, which is 1.13 for ideal chains [133, 134], is slightly smaller for excluded volume chains (1.07).  The precise value for the depletion thickness is important. From Eq. (1.19) it follows that $V_\textrm{ov}$ scales with $\delta ^2$ for large colloidal spheres ($R \gg \delta$) and increases even more strongly for larger depletion zones.

Inspired by the work of De Gennes [115, 116], fundamental work commenced on colloid–polymer mixtures in which the polymers are relatively large compared to the colloids. This regime is relevant for mixtures of polymer or polysaccharides mixed with proteins and is often denoted as the protein limit. The opposite case (small q) is known as the colloid limit. We distinguish three regimes in colloid–polymer mixtures (Fig. 1.21): small q (also termed the ‘colloid limit’) of $q \lesssim 0.5$, ‘equal sized’ ($0.5 \lesssim q \lesssim 2$) and the large q regime (also termed the ‘protein limit’) of $q \gtrsim 2$.

Fig. 1.21

The size ratios in colloid–polymer mixtures in different regimes. Left: the ‘colloid limit’ of relatively small polymer chains. Middle: the equal size regime. Right: the ‘protein limit’ regime of relatively large polymer chains

Odijk [146,147,148,149] published a series of papers devoted to the protein limit $\xi \gg R$; he considered semidilute polymer solutions where the correlation length$\xi$ scales as $\phi _\textrm{p}^{-3/4}$. He first calculated the density profile of a small colloid in a semidilute polymer solution with $\xi \gg R$ and found a very simple shape of the density profile that is independent of $\xi$ and only depends on R [146]. By considering the second virial coefficient between a large polymer and a tiny colloid, he concluded that phase separation is not expected in this case. This was confirmed later by Eisenriegler [150], who from renormalisation group theory found that the second osmotic virial coefficient of small colloidal spheres, $B_2$, only marginally decreases with increasing polymer concentration up to the coil overlap concentration above which it increases. Odijk [147] also considered many-body effects by involving void–void correlations and statistical geometrical approaches [151]. He concluded that the depletion-induced interaction between small colloids due to large semidilute polymers levels off to a maximum attraction near a volume fraction $\phi \sim 0.3$. To mimic proteins, Odijk [149] and Eisenriegler [152,153,154,155] extended the approach of polymer depletion and small colloidal spheres to colloidal particles with ellipsoidal shape.

A semi-grand canonical treatment for the phase behaviour of colloidal spheres with nonadsorbing polymers was proposed by Lekkerkerker [156], who developed ‘free volume theory’ (also called ‘osmotic equilibrium theory’, see Chap. 3). The main difference with TPT [128] is that free volume theory (FVT) accounts for polymer partitioning between the phases and for multiple overlap of depletion layers, hence avoiding the assumption of pair-wise additivity, which becomes inaccurate for relatively thick depletion layers. These effects are incorporated through scaled particle theory  (see, for instance, [151] and references therein). The resulting free volume theory (FVT) phase diagrams calculated by Lekkerkerker et al. [157] revealed that for $q<0.3$ coexisting fluid–solid phases are predicted, whereas a gas–liquid coexistence is found for $q>0.3$ at low colloid volume fraction, as was predicted by TPT.

Fig. 1.22

Reprinted with permission from Ref. [160]. Copyright 1997 American Chemical Society

Photograph of a polystyrene latex dispersion (16 vol %) in 10 mM NaCl at pH 7 with (as indicated in wt %) added hydroxyethylcellulose (HEC) studied by Faers and Luckham [160]. In the lower photograph the tubes are tilted, demonstrating the difference between rigid colloidal solid–liquid and fluid colloidal gas–liquid interfaces for the three-phase coexistence at 0.3 wt % HEC.

A coexisting three-phase colloidal gas–liquid–solid region (not present in TPT phase diagrams) was predicted by FVT for $q>0.3$ and gained much attention. Experimental work [158, 159] demonstrated that this three-phase region indeed exists. Both Leal-Calderon [158] and Ilett et al. [159] measured phase diagrams of colloid–polymer mixtures as a function of the size ratio q. The topology of the phase diagrams corresponds well to FVT predictions, as long as q is below 0.6 (see Chap. 4).

As another example of a three-phase system, photographs of dispersions containing 16 vol$\%$ polystyrene latex spheres (with a diameter of 67 nm), published by Faers and Luckham [160], are reproduced in Fig. 1.22. The numbers shown represent the concentration (in wt$\%$) of the polysaccharide hydroxyethylcellulose (HEC). In the dispersion with 0.3 wt% of HEC three phases coexist. From top to bottom colloidal gas, liquid and solid phases can be recognised. The rigidity of the solid–liquid interface is demonstrated in the lower photographs where the tubes are tilted. The gas–liquid interface flows upon tilting the sample, for the gas–solid interface this is not the case. Using the theory of Lekkerkerker et al. [157] it is also possible to calculate the tie-lines along which the system demixes, enabling a comparison of the theory with experimental phase boundaries. The theory describes the experimental phase diagrams rather accurately [159] for small q. FVT for colloidal spheres mixed with PHSs was tested by Meijer and Frenkel [161]. Their Monte Carlo computer simulation results  on a dispersion of spheres immersed in a solution of ideal polymer chains showed that the agreement with the osmotic equilibrium theory of Lekkerkerker et al. [157] is very good for small values of q.

Faers and Luckham [160] also studied the effect of the amount of polymer grafted onto the colloid surfaces. Decreasing the amount of grafted polymer increased the phase separation concentration of polymers at fixed colloid concentration, demonstrating that it is worthwhile to investigate the effect of the presence of brushes in combination with nonadsorbing polymers.

Polymers are often added to oil-in-water emulsions in order to impose a certain emulsion viscosity. However, this may lead to instability problems, as is known in food emulsions [22, 162]. Bibette et al. [163,164,165] were the first to quantitatively relate phase transitions in emulsions due to nonadsorbing polymers to depletion-induced forces. They showed that it is possible to size fractionate an emulsion with a depletion-induced phase transition. An interesting aspect of (micro) emulsion droplets is that they are not hard spheres, as assumed in FVT [157]. Several groups [42, 166,167,168] studied the phase behaviour of droplets in a micro-emulsion mixed with nonadsorbing polymers. The phase behaviour could be explained by describing the micro-emulsion itself as a collection of sticky hard spheres rather than pure hard spheres. The colloid–polymer mixture is then described as a mixture of sticky spheres mixed with nonadsorbing polymers [166, 167]. The phase behaviour for the colloid limit has been studied extensively by, for instance, Meller and Stavans [169] for emulsions. The FVT of Lekkerkerker et al. [157] was found to agree well with these experimental studies. The $B_2$-approach of Vrij [42, 61] could explain the phase line measured for an aqueous mixture of casein micelles and nonadsorbing exocellular polysaccharides [170]. However, the polymer is often larger (protein limit) or has a similar size to the spherical droplets in polymer/micro-emulsion mixtures. Then phase transitions occur near or above the polymer overlap concentrations. Obviously, the assumption $\delta =R_\textrm{g}$ is then no longer correct. For a proper description of the phase behaviour in this case, the effect of interactions between the polymers must be taken into account: more accurate descriptions of the depletion thickness and osmotic pressure as a function of the polymer concentration are needed.

Free volume theory is an approximate approach; therefore, theoreticians worked on a more formal way of accounting for the influence of depletants on the properties of colloidal mixtures. The exact procedure of integrating out the degrees of freedom [171] was applied to a binary hard sphere mixture by Dijkstra, van Roij  and Evans [172]. They presented a method to derive an expression for the effective grand potential for the large hard spheres by formally integrating out the degrees of freedom of the small spheres.

For mixtures of hard spheres and PHSs as depletants, integrating out the depletant was laid out by Dijkstra, Brader and Evans [173], who formally derived the semi-grand canonical ensemble mixtures of hard spheres and PHSs and made a connection to FVT. They showed that for small $q<0.154$ only one and two body terms are needed. Although for larger q three and higher body terms are needed, integrating out is formally still possible and can be done numerically [174].

Depletion potentials were measured indirectly using scattering techniques [124, 175, 176] and methods such as atomic force microscopy [177,178,179] and total internal reflection microscopy [180,181,182] (Sect. 2.6). Work using the surface force apparatus was also extended (see, for instance, [183,184,185]). The structure factor of dispersed colloidal particles is sensitive to the details of the effective pair interactions. In colloidal dispersions the influence of added nonadsorbing polymers on the colloid structure factor was measured using neutron scattering by making the polymer chains invisible [175]. A characteristic feature of the structure factor is the upswing at small wave vectors (see Sect. 2.6.4). Depletion effects were also quantified by measuring the spin-spin nuclear resonance time. Cosgrove et al. [186] performed such a study using a dispersion of silica with added sodium polystyrene sulfonate  (NaPSS). The resonance time could be related to the depletion thickness, which decreased with increasing concentration of NaPSS.

When a colloid–polymer mixture phase separates into a colloid-rich and polymer-rich phase, an interface appears in between. For a colloidal gas–liquid interface it is possible to measure the interfacial tension using a number of techniques. The value of the interfacial tension [187] is interesting since it is related to phase separation kinetics (see Sect. 4.4). The spinning drop method was successfully used in the past to determine the interfacial tensions in demixed colloid–polymer mixtures [188, 189], yielding tensions with values of a few $\upmu$N/m, corroborating the relation between the interfacial tension expressed in Eq. (1.24). The order of magnitude of the data of De Hoog and Lekkerkerker [189] were comparable with the theoretical results of Vrij [187], Van der Schoot [190] and of Brader and Evans [191]. From the results of Chen et al. [192], it follows that the interfacial tension increases with the distance from the critical point, in agreement with scaling theory [94]. By analysis of the break-up of an elongated droplet in a centrifugal field De Hoog and Lekkerkerker [193] demonstrated that the value of the measured interfacial tension was independent of the method used. Overall, it can be concluded that the colloidal and the ‘molecular’ gas–liquid interface behave similarly. The difference is that the interfacial tension between a colloidal liquid and gas is ultra-low.

Systematic experimental studies were made on various aspects of colloid–polymer mixtures. The phase behaviour of hard sphere binary asymmetric mixtures gained attention from theoretical [194, 195] and experimental [196, 197] points of view (see Chap. 6). Detailed investigations were published on the phase behaviour [158, 159] of well-defined colloid–polymer mixtures using hard sphere-like colloids mixed with rather monodisperse flexible polymer chains. Studies also appeared on the role of depletion effects on the dynamics of colloid–polymer mixtures, such as the diffusion of colloids [198] or the rheology of colloidal dispersions [199, 200] in solutions containing nonadsorbing polymers.

Specific effects (such as the presence of polymer brushes [62, 160]) affect depletion phenomena, and studies on these themes were also initiated in the 1990s. The same holds for the influence of charges. Many theories and depletion studies with model systems are based on hard sphere like colloidal particles. In practice, many stable dispersions containing spherical colloids consist of particles that are not ‘pseudo-hard’, but can be characterised by a pair potential containing an additional soft repulsive tail. An example is a stable dispersion of charged colloids in a polar solvent [201]. Here, double layer interactions provide a soft repulsive interaction between the particles (Sect. 1.2.2).

When charged colloids are dispersed in an aqueous salt solution in the presence of neutral depletion agents, adjusting the salt concentration influences the stability of the dispersion [85, 202, 203]. Grinberg and Tolstoguzov [87] presented generalised phase diagrams of (globular) proteins mixed with neutral nonadsorbing polysaccharides in aqueous salt solutions. The miscibility or compatibility was shown to increase when the ionic strength of the solvent was lowered. Patel and Russel [204] studied the phase behaviour of mixtures of charged colloidal polystyrene latex spheres and dextran as (neutral) polymer chains, and reported a significant shift of the gas–liquid binodal curve towards higher polymer concentrations when compared to predictions for neutral polymer chains mixed with hard spheres.

An early theoretical study on polyelectrolytes as nonadsorbing polymers was made by Böhmer et al. [205], who used the self-consistent field method of Scheutjens and Fleer [24, 118,119,120]. For high salt concentrations, the polymer concentration dependence of the depletion layer thickness matches with that of an uncharged polymer in solution. Below a salt concentration of 1 mol/l, the depletion layer thickness  starts to decrease with increasing polyelectrolyte concentration at lower polymer concentration. At low salt concentrations a significant repulsive barrier in the potential between two uncharged parallel flat plates was found.

Walz and Sharma [206] proposed a force balance theory on the Derjaguin approximation level for the interaction between two spheres (regarded as hard spheres) dispersed in a solvent containing charged macromolecules. The magnitude of the interaction potential at contact increases as the Debye length increases or if the charge density on the large colloidal spheres (same sign as the ‘macromolecules’) increases. The range of the interaction potential also increases as the Debye length increases. At higher concentrations of the small particles a repulsive barrier in the interaction potential curve appears for sufficiently large size ratio of small and large colloid and sufficient Debye lengths. This might lead to the ‘depletion stabilisation’ that was also discussed for colloid–polymer mixtures by Napper [207]. In the model of Walz and Sharma [206], however, the polymers are modelled as charged hard spheres. It is therefore questionable as to whether this method applies to colloid–polymer mixtures, for which the polymer–colloid repulsion is soft.

Odijk [148] incorporated the effect of (like) charges on both polymer and colloid in theory [146] for two small colloidal spheres immersed in a polyelectrolyte solution. He related the effective depletion radius for small charged spheres, immersed in a solution with like-charged polyelectrolytes to the Debye length, the effective number of charges on the protein, the hard sphere radius and the Kuhn length [208]. When the effective depletion radius becomes larger than the correlation length of the polymer solution, phase separation due to depletion is expected.

Theoretical work was done on the influence of polydispersity on the depletion interaction and phase behaviour of colloid–polymer mixtures. Sear and Frenkel [209] investigated the phase behaviour of a colloid–polymer mixture by treating the polymers as PHSs using a distribution of polydisperse PHSs. Their calculations demonstrated that phase separation leads to size fractionation of the PHSs. FVT was extended to model polydispersity by replacing the monodisperse polymers with bidisperse polymers by Warren [210]. Warren found that polydispersity enhances the tendency to phase separate when a bidisperse polymer mixture is compared to a monodisperse mixture having identical number-averaged molar masses. It followed that the location of the binodals of the colloid–bidisperse polymer mixture is almost identical to that of a colloid–monodisperse polymer mixture when the weight-averaged molar mass of the bidisperse mixture is taken as the monodisperse molar mass.

1.3.6 2000–2022

Further progress was made on measuring depletion forces directly with high precision using a wide range of techniques [181, 182, 211, 212]. Using modern advances in microscopy techniques [213] or total internal reflection microscopy, it is possible to measure depletion forces [214] and analyse, for instance, the radial distribution function [215] (see Sect. 2.6). Confocal microscopy allows the potential of mean force between colloids in colloid–polymer mixtures to be measured via the radial distribution function, as explored by Royall et al. [213]. These techniques make it possible to directly test theoretical concepts at the level of the effective depletion-mediated pair interaction between colloidal particles.

Advances were also made using theoretical methods and computer simulations. Until the end of the 1990s most theoretical approaches were based on describing polymer chains as ideal or as PHSs. At the turn of the last century especially, a wealth of different approaches was proposed to describe colloid–polymer mixtures in which interactions between polymer segments were accounted for. Essential was the progress made in Monte Carlo computer simulation  studies on depletion effects [216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235] to test such theories. Below we first discuss some examples of theoretical developments.

1.3.6.1 Theoretical Equilibrium Approaches

(G)FVT

Despite the success of FVT in predicting the phase diagram of colloid–polymer mixtures for the colloid limit (small q) (semi-)quantitatively, in the protein limit (large q) the FVT predictions were far less convincing: it mainly provides useful qualitative information for large q. Quantitative deviations appear for $q>0.5$ when comparing FVT with Monte Carlo computer simulations  [220, 221], theory [236] and experiment [159, 168, 189, 198, 237,238,239,240,241] with realistic polymers. In short, classical FVT predicts binodal curves at too-small polymer concentrations for large q (see Sect. 4.1).

Under conditions where the polymer chains are much larger than the colloidal particles, such as in dispersions of proteins [242,243,244,245], tiny colloids [246,247,248] or micro-emulsion droplets [168, 240] mixed with large polymers (or polysaccharides), instability occurs at rather high polymer concentration. In such situations it does not suffice to stick to the classical Asakura–Oosawa–Vrij description. Van der Schoot [249] showed that polymer collapse can take place when adding small colloids to a polymer solution. He derived an expression for the free energy of a polymer solution in a good solvent in the presence of small colloidal spheres and showed that adding colloids decreases the conformational entropy of a polymer chain. Effectively, adding spheres thus turns the solvent quality from good to poor. As a consequence, a polymer chain is expected to collapse above a certain colloid concentration. This effect originates from the mutual exclusion of polymer segments and colloidal particles. Experimental work confirms shrinkage of a polymer chain caused by adding nanospheres [246,247,248]. Computer simulations of large polymer chains in a system with random small obstacles by Wu et al. [250] are in line with polymer shrinkage due to added nanospheres: the size of the polymers was found to decrease when small particles are added.

To better describe some of the phenomena mentioned above, FVT has also been extended to incorporate the effects of interactions between the polymer chains [251,252,253,254] (see Sect. 4.3). This generalised free volume theory (GFVT) includes the correct dependencies for the depletion thickness and osmotic pressure on the polymer concentration for interacting chains [255], and gives a good description of colloid–polymer phase diagrams of model systems up to large q [254]. GFVT is in (semi)quantitative agreement with experiments and computer simulations [253, 254] for a wide range of q values.

PRISM

Integral equation methods [17] are widely employed to understand structure and thermodynamics in atomic, colloidal and small molecule fluids, and have been generalised to treat macromolecular materials in the 1990s. Shaw and Thirumalai [256] applied the reference interaction site model (RISM) [257, 258] to the case of colloids and combined it with the Edwards model for polymers to explain depletion stabilisation effects [32, 259, 260]: at high polymer concentrations, repulsive contributions to the pair interactions appear. RISM was later extended to the polymer reference interaction site model (PRISM), a continuous space liquid state approach that allows computation of the equilibrium properties of polymeric systems [261].

The PRISM integral equation approach has been generalised to explicitly treat polymers and their conformational degrees of freedom in order to arrive at microscopic equilibrium theories of the thermodynamics of colloid–polymer dispersions [262, 263]. PRISM enables one to account for the role of particle size and polymer concentration to quantify the structure and phase stability of dilute and semidilute dispersions [264] and melts [265]. Although results on the structural and microscopic properties of colloid–polymer mixtures [262, 263, 266] heavily rely on the accuracy of approximate closure relations it can provide an accurate description of, for instance, the second osmotic virial coefficient of proteins with added nonadsorbing polymer chains [242].

Predicting complete phase diagrams (including binodals) using PRISM still requires extreme computational effort. PRISM can be used to predict fluid–fluid (colloidal gas–liquid) spinodal curves, which are close to binodals near the critical point. PRISM equilibrium predictions agree well with experiments [267]. The measured binodals of Ramakrishnan et al. [268] agree well with spinodals computed with PRISM.

GCM

Several liquid state theories have been developed that are based on effective potentials [269] from which the thermodynamic properties of many-body systems can be computed. Louis and Bolhuis et al. [216, 270,271,272,273,274] developed a Gaussian core model (GCM) to describe interacting polymer chains. In this model, the polymer chains are replaced with spherical particles with a soft repulsion between them. This model enables structure, depletion interactions and the full phase behaviour to be studied (in combination with Monte Carlo computer simulations). Basically, the theory is a liquid state approach. On the level of the depletion interaction mediated by interacting polymers, Louis and Bolhuis showed that their GCM agrees very well with Monte Carlo computer simulations [270, 274], except for a slight oscillation in the density profile in the case of Gaussian cores due to their ‘particle’ character. In the colloid limit GCM predictions for the phase diagram of colloid–polymer mixtures are similar to those for free volume theory [220]. For larger q-values FVT predicts phase separation at slightly smaller polymer concentrations compared to the GCM, although the trends are very similar.

DFT

Classical density functional theory (DFT)  [275] is a formal procedure that can be used to quantify thermodynamic properties of fluids. To apply DFT, approximations must be made. Fundamental measure theory [276, 277] is an accurate approximation for hard sphere mixtures and permits the study of the interactions in and structure and phase behaviour of, e.g. colloidal systems [278], colloid–polymer mixtures [191, 279, 280] and star polymer plus linear polymer mixtures [281].

Within DFT the polymers are commonly treated as PHSs. Oversteegen and Roth [282] discussed the close analogy and discrepancies between FVT and fundamental measure theory. For asymmetric additive hard sphere mixtures DFT can be exploited to study the influence of the degree of repulsive interaction (the ‘additivity’) between the small spheres and the interaction between the large spheres. From self-avoiding walk computer simulations, it follows that the degree of additivity of excluded volume polymers is very small [283]. DFT also allows the colloidal gas–liquid interface of a demixed dispersion to be studied [191, 284]. This made it possible to evaluate, for instance, the interfacial tension. For a review on the possibilities of DFT for studying colloid–colloid and colloid–polymer mixtures, we refer to Ref. [285]. These fundamental DFT studies [286] helped to quantify the effective interactions and microstructural effects [278] in hard particle mixtures.

Mortazavifar and Oettel [287] proposed a DFT for the Asakura–Oosawa model of colloid–polymer mixtures, describing both fluid and crystal phases. They find good agreement with available computer simulation studies. Also, they showed that the phase diagram is fairly sensitive to the specific (non-ideality) approximations within DFT.

1.3.6.2 Further Insights into the Large Polymer Chain Regime

The equilibrium phase diagram for $q \lesssim 0.3$ is much simpler than for larger q [158,159,160, 253, 288, 289]. For small q, the only effect of adding polymer chains to the pure hard sphere dispersion is the widening of the fluid–solid coexistence region. A gas–liquid phase transition occurs at larger polymer concentrations above the fluid–solid phase line and is therefore metastable (see Sect. 3.3.4). It is only above a certain range of attraction that the (colloidal) gas–liquid phase transition shifts to polymer concentrations below the fluid–solid coexistence curve. For a specific q value close to 1/3 the fluid–fluid critical point hits the fluid–solid coexistence curve. This critical point is known as the critical endpoint and denotes the boundary of stable gas–liquid phase coexistence. It is rather insensitive to the shape of the interaction potential used [290].

In the protein limit ($q \gtrsim 2$) the phase behaviour is dominated by the gas–liquid phase transition at low colloid volume fractions $\phi$. Colloidal gas–liquid coexistence concentrations have been determined using Monte Carlo simulations by Bolhuis, Meijer and Louis [221]. They studied mixtures of hard spheres and self-avoiding walk polymer chains consisting of segments with hard sphere interactions. For three q-values the phase coexistence data are shown in Fig. 1.23. Phase transitions then take place near and above the polymer overlap concentration ($\phi _\textrm{p} \ge 1$). In such cases, a more detailed description of the physics of polymer solutions is required in order to describe depletion forces and the resulting phase transitions. Rotenberg et al. [291] extended TPT to incorporate interactions between the polymer chains. This shifts the binodal curves for larger q to higher relative polymer concentrations, as was also predicted using GFVT [251, 254]. This is explained in some detail in Sect. 4.3. Mahynski et al. [292] performed more detailed simulations for large q. They found that the polymer osmotic pressures at the binodals collapse onto a single curve for various size ratios, even when far from the critical point. They also studied details of the structure of the mixture and of the potential of mean force between the small colloidal spheres in a solution of long polymer chains with excluded volume interaction [229].

Fig. 1.23

Monte Carlo computer simulation results for the gas–liquid coexistences of hard spheres mixed with excluded volume polymers for $q = 3.86$ ($\circ$), 5.58 ($\times$) and 7.78 ($\blacklozenge$), redrawn from Bolhuis et al. [221]. The binodal curves are drawn to guide the eye

Most of the models for depletion of polymer chains are based upon the assumption that the polymer segments are always much smaller than the colloidal spheres. Depletion of freely jointed chains near a spherical colloid can also be considered for arbitrary size of the segment, showing that depletion effects get weaker as the segments become longer (for a fixed value of the polymer’s radius of gyration) [293, 294]. This is in agreement with work of Paricaud, Varga and Jackson [295] who used Wertheim perturbation theory.

For large q many-body interactions become increasingly relevant. In the case of colloid–polymer mixtures, this means that multiple overlap of depletion zones play an important role in that high q regime. It is interesting to verify whether the effects of direct attractive forces can be compared to indirect depletion forces. For several types of direct attractions it has been shown that at the fluid–fluid (or gas–liquid) critical point, the $B_2^* \approx -6$ criterion [296] holds. However, in the case of a colloid–polymer mixture $B_2^*$ strongly depends on the sphere/depletant size ratio [297], highlighting the fundamental difference between the depletion-mediated interaction and direct attractions (see also Santos et al. [298]).

1.3.6.3 Structure of Colloid–polymer Mixtures

As mentioned, scattering techniques are very useful to indirectly measure the ensemble-averaged structure of colloidal and colloid–polymer mixtures, and here we mention a few further examples of the progress made. Mutch et al. [241] measured the structure factor of colloids in a polymer solution in the protein limit. Muratov et al. [299] presented a revised form of the Percus–Yevick approach to describe the scattering of colloid–polymer mixtures with short-range depletion attraction. An extensive small-angle neutron scattering (SANS)  and small-angle X-ray scattering (SAXS) study on the static and dynamic properties of silica spheres in semidilute solutions of high molar mass polystyrene in 2-butanone was performed by Poling-Skutvik et al. [300]. Their investigations revealed physical particle–polymer coupling on short length scales and long-ranged particle interactions, as well as sub-diffusive particle dynamics.

Kumar et al. [301] used SANS to study the influence that adding nonadsorbing polymers to charged silica spheres in an aqueous salt solution has on the structure factor. They found interesting re-entrant phase behaviour, which is reflected in the measured structure factors at small wave vectors. Peláez-Fernández et al. [302] used static light scattering (SLS) to study the effect of nonadsorbing polyelectrolytes on a dispersion of like-charged colloids. The experimental results for the colloid–colloid structure factor revealed that the main structure factor peak moves to higher wave vectors as the polyelectrolyte concentration increases. The authors interpret this in terms of an electrostatically enhanced depletion attraction. Later, Mehan et al. [303] showed that SANS enables measurement of the structure in multi-component (like-charged) systems.

The influence of the nonadsorbing polymers’ branching on the structure of mixtures of hard sphere-like colloids with star polymers (with varying number of arms but constant radius of gyration) was investigated by Stellbrink et al. [304]. They measured partial structure factors in mixtures of star polymers with colloids using SANS. The relative distance to the demixing transition was reflected in a change of the structure factor at small wave vectors.

Spin-echo SANS (SESANS)  is a SANS technique that was developed to probe length scales from 10 nm up to several tens of micrometres [305]. SESANS detects the polarisation of the neutron beam after scattering. Van Gruijthijsen et al. [306] and Washington et al. [307] used SESANS to quantify and describe polymer depletion-mediated structural effects in the dispersions, and showed that they can be interpreted using depletion forces. Van Gruijthijsen et al. [306] also compared SESANS to SAXS measurements conducted on the same experimental system, and found that similar structural information can be obtained. While SAXS has the advantage that it also provides form factors, SESANS can be applied more easily to study larger colloidal particles.

1.3.6.4 Polydispersity Effects

The particles and polymers in any real experiment have a finite polydispersity. The influence of polydispersity on depletion interaction and phase behaviour was investigated by extending existing approaches. Goulding and Hansen [308] computed the interaction potential between two spheres in a polydisperse bath of PHSs (polydisperse PHS model). When the polydispersity is characterised by a standard deviation of up to 30%, there is hardly an effect on the somewhat increased range and slightly deeper potential between the hard spheres. Above 30% polydispersity the effects become more significant. The original Asakura–Oosawa theory for two parallel plates immersed in a solution of nonadsorbing ideal polymer chains [57] could be extended to involve polymer size polydispersity [309], and would still provide analytical expressions for the interaction between the plates. This work was extended towards the interaction between two spheres in a solution of polydisperse ideal chains. It followed that the influence of polydispersity on the interaction is rather weak [309]. Even a polydispersity of 70% (standard deviation) only increases the attraction by less than 20%.

The phase behaviour of mixtures of monodisperse hard spheres and polydisperse ideal polymers has been investigated using original FVT [310]. At fixed mean polymer size, polydispersity favours gas–liquid coexistence and delays the onset of fluid–solid separation. On the other hand, systems with different size polydispersity but the same mass-averaged polymer chain length have nearly polydispersity-independent phase diagrams. The influence of polymer polydispersity on the colloidal gas–liquid phase coexistence of interacting polymers with spherical colloids is a complicated issue that has only been investigated using TPT by Paricaud et al. [311].

Nguyen et al. [312] derived an exact analytic expression for the many-body depletion interactions between the colloidal particles in the limit of long nonadsorbing polydisperse polymer chains. They also showed that depletion interactions in such systems can be described using mean-field theory.

The effect of particle polydispersity on the phase behaviour of mixtures of polydisperse hard spheres and ideal polymers has also been explored [313], also based on original FVT. Even modest polydispersities ($< 10\%$) can significantly change the phase diagram topology by introducing a host of new, multiphasic equilibria involving multiple solid phases. In practice, such multiphasic equilibria may show up as kinetic effects, preventing the system from reaching equilibrium. The nonequilibrium behaviour observed at higher polymer and particle concentrations may partly be due to this effect. Colloidal gas–liquid phase separation is, however, less sensitive to polydispersity [310].

1.3.6.5 Interfaces in Demixed Colloid–Polymer Mixtures

Since the 2000s the interface between coexisting phases has gained much more attention, as it was realised that it has special properties. For instance, more insight was gained on the ultra-low interfacial tension at the colloidal gas–liquid interface in demixed dispersions containing colloids and polymers. It became clear that this ultra-low interfacial tension affects the relevant characteristic length- and timescales [314]. The capillary length [315] decreases down to the order of micrometres, while the thermal length can become of the order of (sub)micrometres. This is special because other length scales (such as particle sizes) get bigger. The typical interface velocity in such systems is just a few micrometres per second. Inertial terms only become important at large length- and timescales. By means of confocal scanning laser microscopy, Aarts et al. [64] studied the influence of the ultra-low interfacial tension on wetting of colloid–polymer mixtures on a solid surface and on capillary waves [63] at the interface of a demixed colloid–polymer dispersion [316]. Studies on the bending rigidity of the colloidal gas–liquid interface in a demixed colloid–polymer dispersion have also been performed [317]. Interface physics in colloid–polymer mixtures has received ample attention. See, for instance, [213, 231, 318,319,320,321,322]. For more details, see Chap. 5.

1.3.6.6 Nonequilibrium Phenomena in Colloid–Polymer Mixtures

The influence of depletion effects on nonequilibrium phenomena in multi-component mixtures [323, 324] gained increasing interest from both theoreticians and experimentalists [325]. PRISM enabled calculations of the structural correlations, allowing the microscopic evaluation of slow colloid dynamics. Interest focused on arrested states of colloid–polymer mixtures: upon adding a significant amount of depletants the mixtures tend to assume space-spanning structures of aggregated colloidal particles; hence, gelation or glass formation occurs (see Sect. 4.4 or the reviews [325, 326]).

The structural relaxation time, formation of glasses and gels, nonlinear rheology and delayed gel collapse [327,328,329] were predicted quantitatively and compared to experimental results [330, 331]. Experimental studies on the rheology of (gel) networks made of dispersions [332, 333] and emulsions [334,335,336] at high concentrations of added nonadsorbing polymers also gained interest. Also the transition between gelation and glass formation was studied [337]. Wu et al. [338] experimentally studied the colloidal particle dynamics in colloid–polymer mixtures using polymers with different architectures: linear, subgranular cross-linked and branched microgels.

The use of relatively small polymeric depletants induced a short-range attractive interaction with a controlled strength. This enabled various research groups to study the glass transition and gelation of dense colloidal dispersions. In 2000, mode-coupling theory (MCT) was applied to predict the slow dynamics of the glass transition in colloid–polymer mixtures [339]. It revealed re-entrance of the repulsive-to-attractive glass transitions. The theoretical predictions were soon verified experimentally [323, 340].

Further progress was stimulated by computer simulation studies on the influence that depletion attraction has on the structural and dynamic behaviour of colloids. These include phenomena such as the onset of attractive and repulsive glasses and the occurrence of re-entrant melting when the range of the depletion attraction is very small [341, 342]. The idea to control the ‘stickiness’ of the interaction by changing the polymer concentration was also studied at lower colloid concentrations providing insights on colloidal gels [343, 344]. These studies gave an explanation of the route from the glass transition at high densities to gelation of colloidal particles by varying the concentration of nonadsorbing polymers. Percolation of rod-like colloids through nonadsorbing polymers also gained interest [226, 345].

The interplay of the attractive glass/gel line with phase separation [346] was also studied. A combination of computer simulations and confocal microscopy experiments showed that the gel line intersects the binodal at high densities, giving rise to a so-called arrested phase separation [344, 347] (see Sect. 4.4). Arrested states induced by nonadsorbing polymer chains have also been studied extensively in the context of hard sphere/star polymer [348] or star/star mixtures [349]. The introduction of soft interactions is found to enrich the phenomenology of glass transitions and the interplay between the two species [350] compared to binary mixtures of hard spheres [351].

1.3.6.7 Towards Complexity in Colloid–Depletant Mixtures

Interest in depletion phenomena began to broaden after the turn of the last century [352, 353]. A few selected items are briefly discussed below.

Influence of solvent quality on colloid–polymer mixtures

Although many initial approaches ignored details of the solvent, it became clearer that solvent quality [354,355,356,357,358] can play an important and sometimes complex [359] role. Hence, it can be important to properly treat the non-ideality of polymer chains in solution. As an example, we mention a study by Taylor, Evans and Royall [360] on the response to temperature of a well-known model colloid–polymer mixture. At room temperature they found that the critical value of the second virial coefficient for the colloidal gas–liquid phase transition for colloidal spheres can be described using the AOV concept. They could also accurately predict the onset of gelation observed experimentally. Upon cooling the system, the depletion attraction between colloids is reduced because the polymer radius of gyration decreases as the $\Theta$-temperature is approached. Paradoxically, this raises the ‘effective’ temperature and leads to ‘melting’ of the colloidal gels.

Depletion effects mediated by complex polymer mixtures

Work on less conventional depletants (other than, for instance, simple polymers or hard spheres) started to appear. Preisler et al. [361] performed SCF calculations and Monte Carlo computer simulations to analyse the depletion profiles of star-like and H-shaped polymers in a good solvent at a wall. The influence of polymer chain stiffness was also studied and it turned out that, at a fixed coil size, stiffer chains decrease the depletion thickness [293, 294, 362] for dilute polymer solutions. In the case of semidilute polymer solutions the depletion thickness goes through a maximum as a function of chain stiffness [294]. Lim and Denton [363] demonstrated that polymer shape distributions influence the resulting depletion-induced interaction potentials between colloidal particles. Depletion of ring polymers in solution next to hard nonadsorbing walls was also studied [364, 365]. The authors found more pronounced structuring of rings at a nonadsorbing hard surface as compared to linear chains. This structuring strongly affects the shape of the depletion potential between two hard walls.

Depletion forces between colloidal particles in a binary polymer blend were studied by Chervanyov [366], who showed that the relative contributions to the range and strength of the effective depletion attraction strongly depend on the mass fractions of the polymer species and their chain length ratio. Interactions between colloidal particles embedded in a polymer network were considered by Di Michele, Zaccone and Eiser [367]. They presented a theoretical framework to quantify the attractive interactions between the particles mediated by such a polymer network. These predictions agreed with Monte Carlo simulations performed by the authors.

Depletion effects mediated by complex colloids

The depletion forces between colloidal hard spheres mediated by self-assembling patchy particles were studied by García, Gnan and Zaccarelli [368]. They found that the depletion interaction is completely attractive and oscillatory. This may be relevant for understanding the behaviour of complex mixtures in crowded environments, or for targeted self-assembly aimed at building desired superstructures. For an overview of the use of complex depletants such as rods, platelets  and ellipsoids, see the review by Briscoe [369].

Surfactant micelles can also induce depletion attraction between colloidal particles [370]. Additionally, use can be made of the temperature dependence of the shape of self-assembled surfactants. Gratale et al. [371] showed that the strength and range of the depletion interaction can be tuned via shape anisotropy of the surfactant micelles. Depletion effects also can be encountered in mixtures of self-assembling block copolymers in a selective solvent under the influence of nonadsorbing polymers [372, 373]. This work showed that nonadsorbing polymers also induce attraction between block copolymer micelles.

A promising class of depletants involves a self-assembling medium forming either supramolecular chains [374,375,376] or clusters [377]. Also interesting are depletants in a solvent in the vicinity of a critical point [378, 379], which provide a connection between depletion interactions and so-called critical Casimir forces [43, 380] in colloidal dispersions. A solution of depletants near a gas–liquid critical point was also studied numerically, showing how critical solvent mixture forces can be used to effectively manipulate colloidal aggregation [381] and percolation [382]. For a surfactant solution near a fluid–fluid critical point, depletion forces can merge into critical Casimir forces [352].

Systems such as colloidal spheres with multi-component depletants [383,384,385,386,387] and dispersions of star polymers (soft colloids) and linear polymers [388, 389] received interest, as well as mixtures of different types of star polymers [390]. Also, microgel particles as depletants were studied [391]. A full understanding of such more complex mixtures is a topic of future research.

Non-hard colloids with depletants

Accounting for mixtures of colloidal spheres with a hard-core attractive or repulsive Yukawa interactions was studied within FVT [392]. It appeared that additional direct repulsive interactions between the colloids increased the single-phase stability region, whereas additional attractions reduce the stability regions. Rovigatti et al. [393] investigated the influence of brushes  anchored onto the colloidal particles in colloid–polymer mixtures.

Deposition of colloids via depletion forces

Linse and Wennerström [394] reported an interesting theoretical and simulation study on a mixture of particles interacting with each other and with a flat wall through a square well attractive potential. They found an interval of attraction strengths over which surface adsorption of the particles is significant, while bulk instability through nucleation remains negligible solely due to geometrical effects. In hindsight, this effect was already demonstrated experimentally by Dinsmore et al. [395] using mixtures of small and large colloidal spheres at a wall (see Sect. 6.3). Ouhajji et al. [396] realised the deposition of the silica spheres onto a glass plate mediated by adding PDMS. For mixtures of indented colloids, Ashton et al. [397] found conditions at which the particles crystallised at a wall by adding nonadsorbing polymers.

The depletion–adsorption transition

In 2015, Feng et al. [398] published intriguing results on the temperature-dependent phase stability of an aqueous mixture of relatively large silica spheres mixed with PEO polymer chains. They found that a reversible transition from depletion-induced crystallisation to adsorption-mediated bridging flocculation  occurs by changing the temperature. See also the study by Kwon et al. [399] on the temperature dependence in a similar system. It becomes clear that the classical depletion description in which it is assumed that the polymer concentration vanishes at the surface of a colloidal particle does not always suffice. Ouhajji et al. [396] studied dispersions of silica spheres in cyclohexane containing nonadsorbing PDMS polymers. They could only interpret the phase diagram by considering weak depletion effects [400], which is consistent with earlier force measurements on this system by Wijting, Besseling and Cohen Stuart [65]. The so-called depletion–adsorption transition (DAT) was theoretically studied in some detail on the level of pair interactions [401,402,403] and complete phase behaviour [402]. The predicted non-monotonic DAT temperature dependence [401] was confirmed experimentally [404]. Chen et al. [405] and Fantoni et al. [406] proposed models in which the DAT arises from the interactions with the solvent molecules.

Charged colloid–polymer mixtures

Insights into mixtures of charged colloids and nonadsorbing (charged) polymers also developed further. Studies of aqueous mixtures of proteins and nonadsorbing polymers such as polyethylene glycol (PEG) or (uncharged) polysaccharides yielded some interesting observations. Finet and Tardieu [202] studied the stability of solutions of the lens protein $\alpha$-crystallin. Adding an excess of salt to this system does not destabilise the protein dispersion. It follows that the effective attractions between the proteins are absent or are very weak in the case of screened charges. Adding PEG, however, induces significant attractions [202], and results in a shift of the fluid–fluid (in the protein field also termed liquid–liquid) phase transition to higher temperatures [407]. Adding excess salt and PEG induces instant phase separation [202]. A similar synergistic effect of salt and PEG was found in aqueous solutions of (spherical) brome mosaic virus particles [203] and lysozyme [408]. Adding PEG also influences protein crystallisation (see Sect. 11.2). Royall, Aarts and Tanaka [409] studied the influence of double layer repulsion on depletion forces using confocal microscopy. In conclusion, the trend found in experimental studies on mixtures of charged ‘colloids’ and neutral polymers is that the miscibility is, as expected, increased upon decreasing the salt concentration, i.e. increasing the range of the double layer repulsion.

There still are few theoretical studies on mixtures of colloids with a screened-Coulomb repulsion mixed with neutral or charged polymer chains. Ferreira et al. [410] made a PRISM analysis for mixtures of charged colloids and polyelectrolytes up to the level of the pair interaction and computed gas–liquid spinodal curves from the colloid–colloid structure factor. Denton and Schmidt [411] proposed a simple theory yielding the colloidal gas–liquid binodal curve for charged spheres mixed with free neutral polymer chains, described as PHSs. Fortini et al. [222] extended free volume theory to account for a short-ranged soft repulsion between the spherical colloids, allowing a description of the full phase diagrams. They also performed Monte Carlo simulations, and the theory was found to agree quite well with the extended free volume theory. It was found that the colloidal fluid–solid coexistence is especially sensitive to the screened-Coulomb repulsion.

Exercise 1.7. What happens to the miscibility region of a stable colloidal fluid with added nonadsorbing polymers upon adding a screened double layer repulsion between the spheres?

The work of Fortini et al. [222] was later extended towards highly screened, charged spheres mixed with interacting polymers [412]. Zhou, van Duijneveldt and Vincent [413] have shown that generalised free volume theory (GFVT), including short-ranged soft repulsion, is capable of quantitatively describing the depletion-induced phase separation in mixtures of charged silica particles and nonadsorbing polystyrene  polymer chains in dimethylformamide ($\Theta$-solvent conditions). They varied both the range of the double layer repulsion and the size ratio q.

Stradner et al. [414] and Sedgwick et al. [415] considered mixtures of charged spherical colloids with a long-ranged double layer repulsion mixed with very short polymer chains that induce a short-ranged depletion attraction. In such systems small equilibrium clusters are formed that can be described theoretically [416] or using Molecular Dynamics computer simulations [417]. The finite cluster size is a result of a competition between short-ranged depletion attraction and long-ranged repulsion.

Some aspects that could be relevant have not yet been incorporated into the theory for the phase behaviour. A first issue is the effect of gradients in permittivity. Croze and Cates [418] demonstrated that even the depletion zones caused by neutral polymers are affected by charged surfaces. The electrical field present between like-charged surfaces polarise the neutral polymer chains because of their (usually) low permittivity. Curtis and Lue [419] also showed dielectric discontinuities can be quite relevant for colloidal dispersions with added depletants and electrolytes in solution. These effects can enhance polymer depletion and increase the screening of double layer interactions.

The situation gets more complicated when the free polymers are (like-)charged as well [420]. Work of Israelachvili, Pincus and others [421] revealed that the addition of free polyelectrolyte mainly decreases the effective Debye length in aqueous salt solutions, leading to a decrease in the double layer repulsion. Grillo et al. [422] made an interesting study on aqueous mixtures of Pluronic F127 surfactants mixed with hyaluronic acid polyelectrolytes in the semidilute concentration regime. They found that the surfactant micelles and polyelectrolytes were homogeneously distributed in salt free solutions. By increasing the ionic strength the micelles start to cluster and the self-assembly is explained by depletion forces. The salt type is found to play an important role.

The interactions between charged particles mediated by like-charged polyelectrolytes were measured by Moazzami-Gudarzi et al. [423] (see also the review by Scarratt et al. [424]). The crystallisation of charged colloidal spheres mixed with like-charged polyelectrolytes was studied by Ioka et al. [425]. They found that the polyelectrolytes induced depletion forces but simultaneously cause screening of double layer forces. Experimental work was also done on binary asymmetric mixtures of charged colloids by Toyotama et al. [426], revealing a eutectic point. Colloidal probe atomic force microscopy measurements between large charged colloidal spheres mediated by small charged colloidal spheres were performed by Ludwig and von Klitzing et al. [427,428,429]. They observed an oscillatory force and analysed its wavelength in detail. The case of nonadsorbing polyelectrolytes near uncharged surfaces and the relation to the Donnan potential was explored theoretically [430].

In summary, it seems that at high salt concentrations like charges on polymers and colloids do not seem to strongly affect the depletion-induced attraction between colloids. At low ionic strength, however, the situation becomes quite complicated and detailed theories that enable a computation of the stability of such systems still have to be developed. For charged multi-component colloidal mixtures a rich phase behaviour is found. It is clear that there is much work left to be done on the role of (charged) depletants in (charged) systems before we obtain a complete picture.

1.3.6.8 Depletion Effects and Its Relevance for Biology

Nonadsorbing polymers or colloids can also be used to concentrate bacteria. This is very useful for water treatments, where one attempts to achieve the formation of bioflocs of bacteria. Schwarz-Linek et al. [431] studied the addition of nonadsorbing polymers on mixtures of Escherichia coli bacteria and found that concentration of these bacteria is possible in this way. Sun et al. [432] used rod-like nanoparticles to induce phase separation of suspensions containing Pseudomanas aeruginosa bacteria. They concluded that rod-shaped nanoparticles are very effective at inducing phase separation of suspensions containing bacteria. Phase transitions of dispersions containing rod-like viruses mediated by the addition of nonadsorbing polymers are discussed in Sect. 8.5.1 and other parts of Chap. 8.

Depletion effects play a role in protein dispersions [243, 244] similar to that in colloidal suspensions (as was made clear above) and can lead to protein aggregation and phase separation [433] in living matter. As summarised by Sapir and Harries [434, 435], excluded volume effects are thought to be of importance in explaining several intracellular processes [436, 437] and the appearance of membraneless organelles (MLOs). Hence, the resulting depletion effects that are operational are suggested to mediate several types of biological processes such as endocytose [438], microtubule bundling [439], protein dynamics [440,441,442], transcription and self-organisation of the molecules of life [443,444,445]. Crowding  of the subcellular environment by macromolecules is supposed to mediate conformational switches between active states of RNA [446], can influence the conformations of DNA [447,448,449], and may induce structural transitions in protein-like polymers [450, 451]. Quantifying the effects of entropic forces in biology remains a virgin area for additional research. See also the brief illustration on macromolecular crowding in Sect. 11.1.

1.3.6.9 Anisotropic Colloids and Depletion Effects

In this overview on the history of depletion in colloidal dispersions we have mostly focused on mixtures of colloidal spheres and nonadsorbing polymers, which also received much attention. At the beginning of the 21$\text {st}$ century, colloid synthesis evolved to such a degree [452,453,454,455,456] that it became possible to make colloidal particles of a wide range of shapes [457,458,459,460,461]. This, and the fact that anisotropic shapes occur in nature, has triggered experimental, theoretical and computer simulation studies on mixtures of non-spherical colloids in the presence of nonadsorbing polymers, and on binary colloidal mixtures containing anisotropic particles. Hence, insights have been obtained into the phase behaviour of mixtures containing, for instance, colloidal rods [130, 462,463,464,465,466,467,468,469], platelets [469,470,471,472,473,474,475,476,477], dendrimers [478], rocks (colloidal particles with an irregular surface) [479], boards [480], ‘golf-balls’ [481], ellipsoids [482] and cubes [483,484,485] with added polymers.

In Chaps. 8–10, the focus is on the phase behaviour of anisotropic colloidal particles and the influence of nonadsorbing polymers. An interesting feature of non-spherical hard colloidal particles [486] is that they can exhibit directionality purely based on (entropic) excluded volume interactions [452, 487], because flat faces tend to align. The interplay between orientational and excluded volume entropy enables (multiple) liquid-crystalline phases to occur. The addition of depletants can mediate entropic patchiness of anisotropic colloidal particles [488].

Fig. 1.24

Reprinted with permission from Ref. [488]. Copyright 2017 Elsevier

Illustration of the depletion zones around and overlap volumes between (i) spheres, (ii) and (iii) rods, and (iv) and (v) cubes with fixed particle volume, mediated by depletants that induce a fixed depletion thickness (indicated by the dashes).

In Fig. 1.24, overlap volumes are indicated for a few colloidal particles of different shapes. For isotropic spheres, the overlap volume reaches a maximum value as the spheres touch, see (i). For anisotropic particles, the overlap volume depends on their orientation. When comparing the overlap volumes in Fig. 1.24(ii) and (iii) it becomes clear that the overlap volume is maximised for rods as the particles align with their largest surface areas close to each other. Hence, aligned rod configurations are induced by the depletion effect. For cubes the overlap volume is maximised when two (flat) edges are aligned (see Fig. 1.24(iv) and (v)). This illustrates that particles tend to align due to the addition of depletants. As we shall see, the variation of particle shape, and the strength and range of depletion attraction yield a wide variety of self-assembled structures.

Besides studying the effects of nonadsorbing polymers as depletants, it is also of interest to treat colloids themselves as depletants when added to a dispersion of larger or different colloids (see Chaps. 6 and 7). Studies of depletion effects in colloidal dispersions reveal that mixing more complex particle shapes leads to increasingly exotic phase behaviour. Examples include mixtures of rods and platelets [489,490,491,492], rods and spheres [467, 492,493,494,495,496,497] (see Chap. 7), platelets and spheres [498,499,500,501,502,503,504,505,506,507,508], rods and cubes  [509] and bidisperse platelet mixtures [510,511,512,513]. Xie et al. [514] showed that mixing colloidal rods with surfactant micelles can lead to phase states in which the rods assume orientationally ordered nematic and smectic-like membrane superstructures.

Van der Schoot [515, 516] theoretically considered the interactions between hard spheres immersed in a dispersion of hard rods in the nematic phase state, and also studied their self-assembly [516]. Further, nonequilibrium phenomena are also quite relevant in such systems [226, 517]. Aggregation was found to occur in dispersions of spherical colloidal particles and worm-like micelles, leading to transient gels [518].

Depletion effects in dispersions of more complex shapes have also attracted attention. Krüger et al. [519] derived expressions for the depletion force between two arbitrarily shaped large convex colloidal particles immersed in a suspension of small spherical particles. Damasceno et al. [520] studied the thermodynamic self-assembly of a family of truncated tetrahedra, and reported several atomic crystal isostructures as the polyhedron shape varies from tetrahedral to octahedral. The self-assembled crystal structures can be understood as a tendency for polyhedra to maximise face-to-face alignment, which can be generalised as directional entropic forces. Interestingly, the self-assembled structures differ from the densest packing.

Although we cover several examples of mixtures of anisotropic particles mixed with (mainly polymeric) depletants in Chaps. 8 and 10, it is noted that binary mixtures of anisotropic colloids have also been investigated. Nakato et al. [521] studied pure titanate platelets (D = 7.1 $\upmu$m, L = 0.75 nm). They observed an isotropic–nematic phase transition at very small volume fractions $\phi _\textrm{I}=3.7 \cdot 10^{-5}$ and $\phi _\textrm{N}= 1.9 \cdot 10^{-3}$. The large width of the transition region indicates that the platelets are quite polydisperse. Upon adding small laponite platelets (D = 30 nm, L = 1 nm), Nakato et al. [512] observed both biphasic I–N and triphasic N$_1$–N$_2$–I equilibria. This triphasic region is observed for extremely small laponite volume fractions between $1.6 \cdot 10^{-7}$ and $3.6 \cdot 10^{-6}$. For higher laponite concentrations the depletion-induced attraction between the large platelets becomes so strong that the dispersion flocculates.

Free volume theory for binary platelets [513] confirms the observed equilibrium phase transitions. The calculations reveal that the biphasic regime (without added laponite) lies between titanate volume fractions $\phi _\textrm{I}=5.8 \cdot 10^{-4}$ and $\phi _\textrm{N}=8.4 \cdot 10^{-4}$, so in between the observed experimental values. The triphasic triangle region lies between laponite volume fractions of $4 \cdot 10^{-7}$ and $2 \cdot 10^{-6}$, close to the experimental volume fractions.

1.4 Outline

In this chapter we provided an introduction to colloidal interactions, a historical perspective on early observations, and a qualitative understanding of the basic depletion effects. Further, an overview was provided of the important developments in the field of the depletion interaction and the resulting phase behaviour of colloidal dispersions. In Chap. 2 we address the fundamentals of depletion interactions, including pair potentialsand the effects of anisotropic depletants. The focus will be on small depletant concentrations which allow simple treatments using both the force method and the adsorption method to arrive at depletion potentials. The basics of phase behaviour in colloidal dispersions with added depletants are set out in Chap. 3. This is followed in Chap. 4 by extending the model to also include a more detailed description of the polymer physics involved so that it can be applied to mixtures of spherical colloids and polymers. Experimental phase diagrams of well-defined colloid–polymer mixtures are discussed and compared to theories for colloid–polymer mixtures. Phase separation kinetics and nonequilibrium states in colloid–polymer mixtures are treated as well. Chapter 5 concerns the properties of the interface that appears between coexisting colloidal gas and colloidal liquid phases, induced by nonadsorbing polymers. Chapter 6 deals with the phase behaviour of binary colloidal sphere mixtures in the absence of nonadsorbing polymer; and we will discuss the effect of adding small rod-like colloids to a suspension with colloidal spheres in Chap. 7.

Rod-like colloids are considered in Chap. 8, first without polymer: the physics of the isotropic to nematic phase transition is discussed in some detail, followed by a treatment of charged rods. Next, polymer-induced depletion effects for rod-like colloids are discussed. At the end of the chapter it is shown how highly ordered phases (smectic and solid-like) can be treated, and the richness of the phase behaviour of rod–polymer mixtures is revealed. In Chap. 9 mixtures of platelets and depletants are discussed, and mixtures of cube-like colloids with added nonadsorbing polymers are discussed in Chap. 10. These anisotropic colloids have gained increasing attention in more recent years. Treatments of the equilibrium phase states of pure hard platelets involve the isotropic, nematic and columnar phase states, and lead to intriguing phase behaviour. Cube-like colloids are dispersions containing particles with superball-like shapes. Adding nonadsorbing polymers to such systems can promote the formation of ordered, simple, cubic crystalline structures. Throughout, the concepts will be illustrated by experimental and computer simulation results.

In Chap. 11 we highlight manifestations of depletion effects in more complex systems, in particular in biology and technology. This book ends with an Epilogue (Chap. 12), with reflections and an outlook on the possible areas where extensions of the current knowledge of depletion phenomena are needed.