Perspective: The Asakura Oosawa model: A colloid prototype for bulk and interfacial phase behavior

The physics of colloidal dispersions in the last decades has become a rapidly growing area of research in the science of soft matter. These dispersions are solutions of solid particles of mesoscopic size (linear dimensions typically ranging from 10 nm to 10 μm) dissolved in a suitable fluid. Such systems not only have ubiquitous applications in the chemical, pharmaceutical, and food industries (for the production of various pastes, paints, etc.) but also are very useful model systems for basic research, since one can tune their properties in a well-controlled way. For example, one can produce suspensions of monodisperse spherical silica particles or polymethylmetacrylate (PMMA) particles, where coagulation is prevented by sterical stabilization due to small surfactant molecules grafted at their surfaces, which basically realize an almost hard-sphere like interaction^1,2 (although one must not forget that this is an idealization^3,4). Recall that due to similar dielectric permittivities of the solvent and the particles the van der Waals attraction between the colloids is screened. In such systems, the occurrence of a fluid-solid two-phase coexistence region (predicted from computer simulations of hard-sphere fluids to occur^5–8 for colloid packing fractions η_c in the range 0.494 ⩽ η_c ⩽ 0.545) was observed,⁹ and it could also be shown that crystallization (into a face-centered cubic crystal structure) could be by-passed by formation of an amorphous glass-like structure.⁹ Analysis of the slow dynamics of the colloids near the glass transition¹⁰ provided one of the mostconvincing examples for a successful application of the mode coupling theory^11,12 of the glass transition, which still is one of the not yet fully understood grand challenge problems of condensed matter.^12,13 Experimental studies of either glassy dynamics or crystallization kinetics¹⁴ or melting¹⁵ are particularly illuminating, since due to the large size of colloidal particles one can study the structure and dynamics visualizing the arrangement and motions of the individual colloidal particles (e.g., Refs. 15 and 16). Thus, one can attempt to directly follow nucleation events,¹⁵ study the roughness of the crystal-liquid interface,¹⁶ etc. A key feature that makes such studies feasible is the fact that due to the large size of colloids (as compared to atoms or small molecules) the characteristic time of their motions is many orders of magnitude larger than the corresponding times for atoms or molecules in “ordinary” solids and liquids.

A particularly interesting phenomenon occurs when long, flexible, non-adsorbing polymer chains are added to the suspension; we assume here that the solvent is chosen such that no polymer-solvent phase separation^17,18 occurs. These macromolecules in solution then adapt a structure like a random walk (in a Theta solvent^17–19) or like a self-avoiding walk (in a good solvent).¹⁹ These polymers can overlap with each other with either negligible free energy cost (in the case of the Theta solvent, where the monomer-monomer repulsion is offset by a solvent-mediated attraction^17–19) or with a free energy cost of only of the order of the thermal energy k_BT.¹⁹ In the typical cases, which we wish to address here, the radius of these polymer coils will be in the range from 10 nm to 0.1 μm, i.e., comparable (but typically smaller) than the colloids. Being interested in the phenomena for which the colloidal diameter sets the scale, internal degrees of freedom of the polymers then need not be considered explicitly, we rather can deal with them by treating the polymers simply as soft spheres that may overlap with each other. Of course, there is also interest in the opposite limit when small nanoparticles occur in solutions of long polymers with linear dimensions much larger than the colloids, and then the configurational degrees of freedom of the polymer chains need to be explicitly considered.²⁰ But this case (which also often is called “the protein limit” of colloid-polymer mixtures^20,21) will not be further considered here.

The simplest model, originating from the seminal work of Asakura and Oosawa^22,23 (see also Vrij²⁴), describes the colloids as hard spheres of diameter σ_c and the polymers as ideal soft spheres of diameter, σ_p. The latter can overlap with zero energy, irrespective of their distance r,

while both the overlap between colloids and of colloids and polymers is forbidden,

and

Obviously, for the statistical mechanics of this Asakura-Oosawa (AO) model only packing constraints matter, there are no energy parameters whatsoever, and all the phase behavior that results is purely due to entropy. Of particular interest is the depletion effect. The polymers cause an effective attraction among the colloids, and for sufficiently large size ratio q = σ_p/σ_c a liquid-liquid phase separation in a polymer-rich phase and a colloid-rich phase occurs. Qualitatively, this is easily understood from the fact that according to Eq. (3) the center of a macromolecule can approach the surface of a colloidal particle only up to a distance σ_p/2, so around each colloidal particle there is a corresponding depletion zone. If the depletion zones of two colloids overlap, there is an increase in free volume for the polymers, and a corresponding gain in their translational entropy (Fig. 1).

Despite the extreme simplicity of this model, it does capture the main features of the phase behavior of experimental colloid-polymer mixtures (see, e.g., Refs. 25 and 26) surprisingly well: in the experimental phase diagrams, typically plotted in the plane of variables polymer packing fraction η_p (ordinate) versus colloid packing fraction η_c (abscissa), one finds for small enough q a single fluid-solid coexistence region, just as for a model of pure hard spheres, but which progressively widens when one increases η_p. For large enough q, however, there occur three two-phase coexistence regions (vapor-solid, vapor-liquid, liquid-solid) and a three-phase coexistence triangle. Note that η_p, η_c are defined in terms of the polymer and colloid particle numbers N_p, N_c (or respective densities ρ_p = N_p/V, ρ_c = N_c/V, V being the volume taken by the colloidal suspension) as

Clearly, it is a very attractive feature, from the experimental point of view, that with a parameter (q) that can be continuously varied one can tune the phase diagram. Such a possibility to vary an (effective) interparticle potential continuously normally is never found in small molecule systems. For the theorist, the model described by Eqs. (1)–(3) is particularly attractive as well due to its simplicity, which is an advantage both for analytical theory and simulation. Another simple model exhibiting vapor, liquid, and (crystalline) solid phases would be the Lennard-Jones system, of course.²⁷ But the long range of this interaction (which in simulations therefore often is cut off arbitrarily) causes some technical problems, particularly for the theoretical treatment of interfacial problems. Moreover, when one wishes to account for the properties of experimental systems to which this model approximately corresponds (noble gases such as Ne, Ar, etc.), quantum effects need to be considered in the crystal phase. For more complicated systems, e.g., molecules, such as CO₂, finding a good model potential is a debated problem.²⁸ For colloidal crystals quantum effects are completely negligible, of course.

However, one of the most interesting features of colloid-polymer mixtures, that has created a lot of attention due to pioneering experimental studies, is the possibility to investigate interfacial effects associated with the vapor-liquid type phase transition from macroscopic scales down to the single particle level, namely, capillary waves,^29,30 wetting layers,^31,32 early and late stages of phase separation kinetics,^32–34 etc. In parallel, there has been thorough theoretical work, starting out with initial mean-field type work on the bulk phase diagrams.^35,36 Both theory and simulations strongly profited from the idea to integrate out the polymer degrees of freedom. For small q, i.e., q ⩽ 0.1547,^35,37 one can map the AO model exactly to a one-component Hamiltonian, where colloids interact with the following pairwise Hamiltonian:

with z_p being the fugacity of the polymers, μ_p being their chemical potential

and Λ_p is the de Broglie wavelength of the polymers. For r > σ_c(1 + q) = σ_c + σ_p, we have U_eff(r) ≡ 0, and for r ⩽ σ_c , U_eff(r) = ∞, due to the underlying hard-core colloid-colloid interaction. Note that for theoretical calculations the use of a grand canonical ensemble for the polymers (where the chemical potential μ_p rather than their particle number N_p (or the packing fraction η_p, respectively) is used as a control variable, is more convenient even when the polymers are kept in the problem, rather than integrating them out: using Eq. (5) for

$q>q*=2/3−1≈0.1547$ would be a crude approximation to the original AO model: while one still can integrate out the polymer degrees of freedom (since they act like ideal gas particles and are lacking any direct polymer-polymer interaction) exactly, one would generate three-body, four-body interactions, etc. But Eq. (5) is particularly useful in the context of density functional theory (DFT), since this theory is much easier to work out for a one-component model system rather than for a two-component mixture. However, for small q the potential in Eq. (5) becomes “sticky,” and it is difficult to treat perturbatively. Therefore, we emphasize that DFT can also be worked out for the full (two-component) AO model. Brader et al.³⁸ have given a beautiful and thorough review of results obtained by this full DFT approach.

Of course, a caveat needs to be made when one wishes to address dynamical aspects of colloid-polymer mixtures, such as dynamics of interfacial fluctuations,²⁹ kinetics of phase separation,^32–34 etc. Then the effects of the solvent, not discussed in Eqs. (1)–(3), need to be included. In the simplest case, one still assumes that the solvent does not contribute anything to the static averages, but it affects the dynamics via friction forces on the particles, random forces causing their Brownian motion, and hydrodynamic interactions.³⁹ This problem is a challenge (particularly in the context of simulations⁴⁰) but also has great potential for the future. We also note that the structure and dynamics of colloidal suspensions can very strongly be influenced by electromagnetic fields, e.g., due to laser fields creating periodic potentials for the colloids. If the particles contain a magnetic core, also sensitivity to magnetic fields is created. Such fields can be used to move them far out of equilibrium.

The present article now attempts to illustrate, with a few examples, how the AO model can serve as a workhorse to explore bulk and interfacial phenomena in soft matter, but no comprehensive review is intended. As a first step, in Sec. II we comment on the phase behavior in the bulk (and the methodology to study it) in more detail. We shall also mention the extension to colloid-polymer mixtures where the colloids have rod-like^41–49 rather than sphere-like shape; in such systems phases with nematic order^50,51 also can occur.

A comprehensive review of phase behavior of colloid-polymer mixtures fills a whole book⁵² and hence clearly is not within the scope of this section, which rather only plans to give the reader the flavor of the basic methods and concepts, emphasizing the open problems, which may be the subject of future developments.

Dealing with phase transitions in condensed matter systems, the first step always is the formulation of a mean field theory. A simple approach of this kind is the free volume approach³⁶ where one treats the model in the grand canonical ensemble, assuming that the polymer translational degrees of freedom have been integrated out, such that the effective interaction between the N_c colloidal particles at positions

${r⃗i}$ is

where U_c is the bare colloid interaction potential (all the terms resulting from Eq. (2)), Π_p is the osmotic pressure of the pure polymer-solvent system (which depends on the polymer chemical potential μ_p), and V_free is free volume in which the polymer coils can exist. We emphasize that in general Eq. (7) is not just a sum of two-body terms (written in Eq. (5)), but contains many-body interactions. The key assumption then is to replace

$Vfree({r⃗i})$ by its average value in the corresponding unperturbed system of colloids, V_free = αV where V is the total volume, and the free volume fraction α(η_c) depends on the colloid packing fraction η_c {Eq. (4)}. The free energy then becomes a sum of a term corresponding to a pure colloidal suspension in the volume V, and a term corresponding to a pure polymer solution in the volume αV,

with F_p(N_p, αV) = k_BTN_pln (N_p/αV) + other terms, since the polymers are assumed to behave ideal gas-like (the terms not written do not contribute to the computation of the phase diagram). The colloid contribution F_c(N_c, V) is derived from the hard-sphere equation of state (for the fluid phases the Carnahan-Starling expression for the compressibility of a hard-sphere fluid⁵³ is used as an input; for the face-centered cubic (fcc) crystal expressions due to Hall⁵⁴ or from Frenkel and Ladd⁵⁵ were used^36,56). While these expressions are trivially compatible with the fcc close-packing limit (at

$ηc=π2/6$⁠), one can also choose the parameters such that the liquid-solid coexistence packing fractions (in the absence of polymers) are correctly reproduced.

The nontrivial feature now is to find an expression for the free volume fraction α(η_c). Motivated by scaled particle theory,⁵⁷ Lekkerkerker et al.³⁶ proposed the expression (γ ≡ η_c/(1 − η_c))

where the coefficients A, B, and C are polynomials in q.³⁶ From the condition that the colloid chemical potential μ_c as well as the total osmotic pressure

$Π(ηc,ηpr$⁠) must be equal in coexisting phases, one can construct the phase diagram in the plane of variables

$\eta _p^r$

$ηpr$ and η_c (Fig. 2). Here, the polymer reservoir packing fraction

$\eta _p^r$

$ηpr$ is basically the polymer fugacity z_p, Eq. (6), normalized such that it directly would give the polymer packing fraction η_p {Eq. (4)} in a reference system where in the same volume no colloids were present. Note that

$\eta _p^r$

$ηpr$ is an intensive thermodynamic variable, playing here a similar role as inverse temperature for a small molecule system (for which the phase diagram is normally plotted in the temperature-density plane: looking at Fig. 2 upside down the analogy is immediately present).

Fig. 2 intuitively makes it clear why the system is an attractive model system: changing the size ratio q of polymers to colloids (in principle, the size of both polymers and colloids can be varied over a wide range) one can vary the topology of the phase diagram. In real experiments, however,

$ηpr$ is not directly accessible, but rather η_p is controlled, and η_p is the density of an extensive thermodynamic variable. As a consequence, tie lines no longer are horizontal (Fig. 3), reflecting a nontrivial partitioning of the polymers between coexisting phases. The critical point is no longer at an extremal position of the vapor-liquid coexistence curve, and three-phase coexistence no longer corresponds to a triple line in the phase diagram, but rather to a triangle (Refs. 36 and 56, Fig. 3).

It is clear that for an experimental study of the phase diagram of an actual colloid-polymer mixture many different samples at varying state points (η_p, η_c) in this diagram need to be prepared; while qualitative evidence for both phase diagram topologies shown in Fig. 2 has been obtained,^25,26 one should not expect that quantitative comparisons between this simple theory and experiment yield agreement, for a variety of reasons:

• In the experiment, both polymers and colloids do not have strictly uniform size, but always are somewhat polydisperse.

• The colloidal particles (in the absence of polymers) do not strictly behave like hard spheres, and also asserting a very precise value of η_c in an experiment is a problem.^3,4

• It is not strictly true that the thickness Δ of the exclusion zone around a colloid precisely matches the polymer radius.⁵⁶ So the coefficients A, B, C in Eq. (8) should not be expressed in terms of the variable q = σ_p/σ_c, but rather in terms of a variable q′ = 2Δ/σ_c, where Δ actually depends on R_c, and whether one considers ideal polymers (i.e., for a Theta solvent) or polymers with excluded volume interaction (i.e., under good solvent conditions) in dilute solution.^58,59 Also F_p(N_p, αV) is no longer an ideal gas-like expression, when excluded volume matters, and more complicated results invoking renormalization group theory⁶⁰ then need to be used.^61,62 While Aarts et al.⁵⁶ show that taking such effects into account reduces somewhat the discrepancies between model calculations and experiment, it is clear that a perfect match cannot be expected. Thus, we feel that the claim made in Ref. 56 that the free volume approximation “qualitatively and for polymers much smaller than the colloids even quantitatively predicts the correct phase behavior as can be seen by comparison with experiments” is somewhat optimistic, and additional experiments which map out coexistence curves in great detail (as done in Ref. 63) clearly are needed.

Of course, a very clear-cut test of the theory is possible via computer simulation, since there precisely the same model as used in the theory {Eqs. (1)–(3)} is possible. Early simulation work^37,64,65 seemed to find almost perfect agreement with the phase diagrams of the AO model based on the free volume approximation,³⁶ and hence the early literature contains statements such as⁵⁶ “Extensive computer simulations validate the free volume approximation.” Of course, one must not forget that the free volume theory is a mean-field type approximation, as the step from Eqs. (7) to (8) clearly shows, and this is also true for DFT which contain free volume theory as a description of the bulk behavior.^38,66 Of course, DFT is much more powerful than free volume theory as it allows the calculation of structure factors S_cc(q), S_cp(q), and S_pp(q), which are the Fourier transform of correlation functions of concentration fluctuations of colloids (c) and polymers (p), and accessible to scattering experiments. Furthermore, DFT is a powerful approach to study concentration variations at walls, wetting phenomena, etc., as will be discussed in Sec. III. However, the phase separation of the AO model in gas-like

$(ηcg)$ and liquid-like

$(\eta _c^\ell )$

$(ηcℓ)$ phases (that occurs in equilibrium if q exceeds q_c ≈ 0.3³⁶) has a critical point that belongs to the Ising universality class,⁶⁷ and hence the coexistence curve does not have the simple parabolic shape near the critical point as shown in Fig. 2(b), but rather varies as^68,69

Also the extent of the region where phase separation occurs is overestimated as usually occurs in mean field-type theories, see Fig. 4 for an example.⁷⁰ The free volume theory underestimates the critical polymer fugacity by about 30% (recall that mean field theory overestimates the critical temperature of a simple cubic Ising model by about 25%⁶⁹). Only when either η_c → 0 or η_p → 0 the theory becomes quantitatively very accurate, as Fig. 4 shows.

This Ising-like behavior has also been verified for many other properties,⁷¹ including also the so-called “critical amplitude ratios,”⁶⁹ and actually the AO-model is one of the most carefully studied off-lattice system, as far as its critical behavior is concerned. While Fig. 4 refers to the full AO model, Eqs. (1)–(3), where polymers are explicitly present, interesting simulation work has also been done for the AO model using the effective pair potential, Eq. (5),⁷² where q = 0.4, 0.56, and 0.8 was studied (although Eq. (5) for such large values of q differs from the true multi-body potentials, as mentioned above), and compared to results for the full AO model. One finds that the model based on Eq. (5) phase separates at significantly smaller values of

$ηp,critr$⁠, and also the corresponding critical value of η_c is larger. However, in a reduced representation (versus variable t defined in Eq. (10)) there is not much difference. A particularly strong point of Ref. 72 is that for Eq. (5) also a refined analytical approach is presented, based on the so-called “hierarchical reference theory” (HRT)⁷³ of fluid criticality. This approach takes long-range fluctuations rather systematically into account and describes both the coexistence curve and response functions in very good agreement with the simulation results.

Of course, what is still lacking at this point is an extension of HRT to the full AO model and a combination with the successful theoretical descriptions of the liquid-solid transition, in the spirit of the free volume approach, to be able to develop a complete and coherent description of the phase diagram, containing all three phases (gas-like, liquid-like, and colloidal crystal phases). For these tasks, the AO model will be a very useful “testbed” in the future. We also mention that for the corresponding simulation work the AO model has been (and will continue to be) similarly fruitful to develop the “machinery” of computer simulation to its full power. Initial studies of the full phase diagram^65,74,75 are based on an exact mapping of the two-component system to the corresponding one-component system, where polymers are integrated out and define an effective multiparticle potential between the colloids, where more than two depletion layers around colloids may overlap. While this potential would be awkward to use in analytical work, it can be numerically implemented in the Monte Carlo simulation algorithm. This very beautiful idea is also useful to study the AO model in the presence of walls, focusing on possible wetting phenomena, as will be discussed in Sec. III. Dijkstra et al.^65,74,75 estimate the phase diagram by computing the free energy differences (relative to the reference hard sphere system) by standard thermodynamic integration methods^76,77 and the gas-liquid phase diagram is estimated via the “Gibbs ensemble”-technique.⁷⁶ In the latter, the equilibrium between two simulation boxes is simulated, one containing the gas, the other the liquid, exchanging both particles and volume between the boxes, to maintain the same chemical potential. However, it is known that this method becomes inaccurate near the critical point,⁷⁷ and the finite size scaling analysis^69,77 applied by Vink et al.^70–72 clearly is more reliable. While the pioneering work of Dijkstra et al.^65,74,75 estimated the full phase diagram for both q = 0.6 and q = 1.0, it suffers somewhat from inaccuracies due to finite size effects (only N_c = 108 colloidal particles were included throughout).

Vink et al.^70–72 carried out simulations of somewhat larger systems (e.g., cubic boxes with linear dimensions L up to 10.5 colloid diameters, and varying the linear dimensions systematically to exploit the size effects via a finite size scaling analysis) and included all the polymers. Due to the rather high polymer density (recall that the polymers can overlap with each other with no energy cost) a straightforward simulation of the system in the grand canonical ensemble of the colloid is inefficient. Therefore, Vink et al.^70–72 applied composite moves, where one attempts in a spherical region (with radius (σ_p + σ_c)/2) to take all polymers in that region out, and insert a colloid; or vice versa. Acceptance probabilities are constructed in such a way that they satisfy detailed balance. Using successive umbrella sampling,⁷⁸ the probability distribution P_L(η_c) in cubic boxes of linear dimension L at various values of

$ηpr$ is constructed, and phase coexistence is identified via the “equal weight rule”⁷⁹ of the two peaks representing the gas-like and liquid-like phases, respectively. Following then a path along the “rectilinear diameter”

$(\eta _c^d = (\eta _c^g+ \eta _c^\ell )/2)$

$(ηcd=(ηcg+ηcℓ)/2)$ one can apply the standard cumulant intersection method^77,80 to find reliable estimates for the location of the critical point.^70–72 From the minimum of P_L(η_c) at

$\eta _c^d$

$ηcd$ one can furthermore extract an estimate for the interfacial tension γ (a factor 1/k_BT is absorbed in the definition) between the coexisting phases^70,71 since⁸¹ 

$\ln [P_L(\eta _c^d)/P_L(\eta _c^v)]\break=\gamma L^{-2}$

$ln[PL(ηcd)/PL(ηcv)]=γL−2$⁠. Thus, the AO-model has served as a “laboratory” to develop further the tools to simulate strongly asymmetric fluid mixtures and analyze their phase diagrams and critical behavior.

In this spirit, also an extension to a version of the AO model with continuous rather than hard sphere potentials was proposed,⁸² using instead Weeks-Chandler-Anderson (WCA) potentials²⁷ (i.e., the Lennard-Jones potential cutoff in its minimum and shifted upward so that the minimum occurs at zero). Also an (arbitrarily chosen) smooth polymer-polymer potential was used (with U_pp(r = 0) = k_BT/2). This model (also applying a slight smoothing of the potentials at their cut off distances⁸²) is also suited for applying Molecular Dynamics simulation methods,⁷⁶ although it still is not a realistic model for the dynamics of the colloid-polymer mixture due to lack of explicit solvent (which, however, could be included via the multi-particle collision dynamics approach, see Ref. 40). To compare this model with the original AO model, one computes effective diameters for colloids and polymers with the Barker-Henderson^82,83 approach. One finds that the resulting phase diagram of this continuous Asakura-Oosawa model (CAO) in the (η_p, η_c) plane is almost indistinguishable from the original AO model, at least for the studied value of q = 0.8, apart from a slight shift of the critical colloid and polymer concentrations

$(ηccrit≈0.15$ rather than 0.13). This observation again shows that a successful “fitting” of a phase diagram to a model is not a unique procedure from which one can conclude that an accurate representation of the actual interactions among the particles has been found. The same conclusion has also emerged for simulation studies of the liquid-vapor transition of CO₂ and other small molecule fluids,^28,84 where highly simplified effective potentials did describe the phase boundaries accurately, although they clearly cannot describe the actual molecular interactions (and resulting pair correlations⁸⁵) on small scales. Therefore, the success of various generalized free volume theories, e.g., Ref. 86 to account (to a large extent) for experimental phase diagrams of colloid-polymer mixtures⁶² should not be taken as a validation of the underlying model for the effective colloid-colloid interactions. In the simulation work,⁸² the three types of structure factors S_cc(k), S_cp(k), and S_pp(k) were obtained over a wide range of wave numbers k and representative values of η_p, fixing η_c at its critical value. Systematic experimental studies of these structure factors for various experimental colloid-polymer mixtures would be highly desirable, and could yield considerable insight into the extent to which the AO model is an accurate “microscopic” model of the actual interactions in these systems. Such studies also could yield additional experimental confirmation of the Ising-like critical behavior.

So far, only the critical behavior of the order parameter and interfacial tension γ of one colloid-polymer system has been addressed,⁸⁷ studying grafted (i.e., sterically stabilized) silica particles (with a diameter of only 20 nm) in cyclohexane, with polydimethylsiloxane (of molecular weight 92 000) as depletant. It was found that

with exponents β^* = 0.371 ± 0.026, μ^* = 1.30 ± 0.08. These numbers should be compared to the theoretical values (taking into account “Fisher renormalization”⁸⁸) β^* = β/(1 − α) ≈ 0.365, μ^* = μ/(1 − α) ≈ 1.41, using the exponents^68,69 μ ≈ 1.26; α ≈ 0.11. We recall that Fisher renormalization (in strongly asymmetric systems like the present one) simply is expected due to the energy-like singularity of η_p, noting that η_p − η_{p, crit}∝(μ_p − μ_{p, crit})^1−α. Since it is expected⁸² that the renormalized exponents (such as β^* and μ^*) can only be seen in a very narrow environment of the critical point (note that the function η_p(μ_p) must be expected to also contain regular background terms, in addition to the singular relation quoted above) it is unclear whether this experiment⁸⁷ should be taken as evidence in favor of Fisher renormalization.⁸⁸ Also the results of large scale simulations⁸² (using at η_{c, crit} N_c = 5373 colloids and up to N_p = 22 734 polymers) were rather inconclusive in this respect. Finally, we mention that Royall et al.³⁰ attempt to estimate the “fractal dimension” of critical clusters of colloid particles from the direct optical observation of critical fluctuations in real space by confocal microscopy methods. Unfortunately, the accuracy of this study did not allow to extract a very precise number.

Bolhuis et al.⁸⁹ simulated another model where polymers were described as soft spheres constructed such that they yield the correct equation of state of the polymer solution under good solvent conditions.⁸⁹ Colloids were still taken as hard spheres, and the full phase diagrams were estimated for q = 0.34, 0.67, and 1.05, respectively. While the gas-liquid type phase separation was also studied using the Gibbs ensemble technique,⁷⁶ the fluid-solid coexistence was estimated from Kofke's⁹⁰ so-called “Gibbs-Duhem” thermodynamic integration technique, with the hard sphere fluid-solid coexistence as a starting point. The standard AO model was studied in the same way for comparison, and it was shown that for q = 0.34 both models give very similar results, while for q = 1.05 there are very significant differences. The case q = 0.34 also is of interest since it is close to the “critical endpoint” where the line η_{c, crit}(q) hits the fluid-solid first order transition (or two-phase coexistence, respectively) in the phase diagram.⁹¹ While the work of Bolhuis et al.⁸⁹ again is a nice example where the AO-model and its variants have served as a “laboratory” to test the tools of advanced simulation methodology, the quantitative accuracy of the results is somewhat limited, since it also uses no more than N_c = 108–200 colloidal particles in the simulation box. Thus, a detailed analysis of the interesting singularities expected (beyond mean field treatments) for the phase boundaries near a critical endpoint remains a future challenge.

Finally, we briefly address the very interesting extensions of the AO model to non-spherical shape of the colloidal particles, e.g., hard sphero-cylinders rather than hard spheres.^41–49 Note that polymer-induced depletion attractions between rod-like particles find interest in many different contexts, e.g., accounting for the experimentally observed phase separation in suspensions of bacteria,⁹² for understanding the electrical percolation conductivity of carbon nanotubes embedded in polymeric matrices,^93–95 etc. Both the free volume theory⁶³ and various of the simulation methods summarized above have been applied to this problem.^{42,46,48,49,96} Fig. 5 shows an example⁴⁸ for the polymer-induced liquid-liquid phase separation in the case of hard spherocylinders of aspect ratio L/D = 5 (D being the diameter and L the length of the spherocylinders, the polymer diameter was chosen to be D as well). In this system, there occurs a triple point between vapor-like and liquid-like fluid phases and a nematic liquid crystal phase (a solid crystalline phase, expected at still larger η_c, is beyond the range of the scale shown in Fig. 5). Also in this case, free volume theory overestimates the region of liquid-liquid phase separation significantly. Jungblut et al.⁴⁸ provided clear evidence for the Ising character of the critical behavior for the phase separation of these rod-coil suspensions, and showed that the depletion attraction also reduces the threshold concentration of rods above which percolation (defined in terms of a neighborhood criterion⁹³) occurs.

While for the colloid-polymer mixtures with spherical colloids extensive simulation studies of the full phase diagram exist and many model variants have been studied (including models that approximately account for the excluded volume interaction among the polymers in the solution), mixtures with rod-like colloids have found much less attention. In view of the numerous application that these systems find, it is likely that there will be much more work in such systems in the future.

At the end of this section, we emphasize that the whole treatment presented so far deliberately was restricted to the case of dilute concentrations of polymers in the suspension (i.e., their packing fraction η_p ⩽ 1). Then the polymer-colloid interaction involves a single length scale for the range of the depletion attraction among the colloids, namely, the polymer radius (Fig. 1). However, it is also possible to consider the problem of semi-dilute (or even concentrated) polymer solution, into which a small packing fraction of colloids (or nanoparticles) is added. In a semi-dilute solution under good solvent conditions, the correlation length ξ (which describes the length over which excluded volume interactions among the monomers of a coil are screened by monomers of other chains) arises as an important length scale, and one can show that the range of the depletion attraction no longer is given by σ_p (as written in Eq. (5)) but by the much smaller length ξ, and in the range ξ ≪ r < σ_p then the effective colloid-colloid interaction is repulsive!⁹⁷ This finding (sometimes termed “depletion repulsion” in the literature, see the recent seminal study by Shvets and Semenov⁹⁷ for a review) implies that also by free chains in the solution there can be a “stabilization” of colloidal suspensions (i.e., avoiding their aggregation by layers, which normally is done, of short end-grafted chains at the surface of the colloid particles^1,2). Of course, just as there is a breakdown of the AO model for q ≫ 1,²⁰ also these effective interactions in non-dilute polymer solutions are not at all captured by the AO model, and shall remain outside the focus of the present article.

As a rule of thumb, the scale for the interfacial tension γ in atomic and molecular systems is set by the ratio γ ≈ ε/σ², with ε being the depth of the minimum in the interaction potential between the particles and σ their diameter.⁹⁸ In a vapor-liquid or liquid-liquid phase separation, away from the critical point (where γ vanishes, cf. Eq. (11)) we expect ε to be of the order of the thermal energy k_BT. For molecular systems, σ is of the order of Å, and γ is of the order of 0.01 N/m. For colloidal systems, σ is of the order of nm, and γ is a factor of 10⁸ smaller, of the order of 0.1 nN/m. For rod-like colloids, we may have L = 200 nm, D = 10 nm and the estimate γ = k_BT/(LD) is then of the order of μN, in agreement with experiment.⁹⁹ Due to this ultra-low surface tension,³² fluctuations of interfaces such as capillary waves^98,100 have much larger amplitudes than for small molecule systems, and hence are directly observable with laser scanning confocal microscopy,²⁹ and studying the height-height correlation functions of the interface (taking gravity effects into account) good agreement with the predictions based on capillary wave theory was found. Note that γ has been measured independently by the spinning drop method.⁶³ This study then has motivated a simulation analysis of capillary waves in the AO model,¹⁰¹ which has the advantages that gravity effects are absent, and a very large range of wave numbers k in the spectrum of the Fourier components h_k of the interfacial fluctuations could be probed. The interfacial tension γ is found independently from the minimum in the probability distribution P_L(η_c), as mentioned in Sec. II, where L is the linear dimension of a finite simulation box with periodic boundary conditions. It is also interesting to note⁷⁰ that DFT overestimates γ not only near the critical point but makes an appreciable error along the whole vapor-like type phase boundary. Of course, one should recall that DFT calculations of interfacial profiles³⁸ neglect the capillary wave broadening of the profile completely, and rather obtain a profile that is interpreted as an “intrinsic” interfacial profile. The simulations, however, clearly show that identifying an intrinsic interfacial profile is a doubtful procedure:^101–103 When one performs a lateral coarse-graining along the interface, introducing cells of size B × B × L_z¹⁰⁴ the local height z = h(x, y) of the interface relative to a reference plane parallel to the average location of the planar interface can be found from the Gibbs equimolar dividing surface.^98,101–106 The mean-square width of the interface due to capillary waves then is

$wcw2=⟨h2⟩−⟨h⟩2$⁠, and the total mean square width

$w_L^2$

$wL2$ on the lateral scale L can be written as (absorbing a factor 1/k_BT in the definition of γ here)

where the “intrinsic width” w₀ clearly cannot be easily separated from the term containing the (non-unique) choice of B. This logarithmic variation of interfacial widths with lateral interfacial dimensions is readily seen for simulations of interfaces between coexisting vapor-like and liquid-like phases of colloid-polymer mixtures (Fig. 6(a)), but to a lesser extent also for solid-fluid interfaces (Fig. 6(b)).⁸ Since

$wL2$ depends on ln (L/B) = ln (n_B) only in Fig. 6(a), the (easier) test of a variation of B at fixed large L was only performed. It is clear that this variation of

$w_L^2$

$wL2$ with L (or B) can in principle be used to estimate γ from the slope of the straight line fit in Eq. (6), but the accuracy is not at all impressive. In particular, for the fluid-solid interface the statistical error of the final estimate

$(\tilde{\gamma } \approx 0.95 \pm 0.10k_BT/\sigma ^2$

$(γ̃≈0.95±0.10kBT/σ2$⁠) is rather unsatisfactory. Nevertheless, the result shown in Fig. 6(b) is of interest, because it clarifies that capillary wave-induced broadening is also present for (rough) fluid-solid interfaces, and hence the significance of an intrinsic interfacial width may be doubtful even for crystal-fluid interfaces. Already at this point we emphasize the fact that for fluid-crystal interfaces the interfacial tension is not isotropic, but depends on the orientation of the interface normal relative to the crystal axes. Then one must distinguish the case where the interface is not rough, and hence a size-independent interfacial profile exists and the intrinsic width w₀ hence is well-defined (this is the case for crystals forming facets) from the case with rough crystal surfaces. In the latter case, a formula such as Eq. (12) still holds, but with γ being replaced by the “interfacial stiffness”

$\tilde{\gamma }$

$γ̃$ .¹⁰⁷ In principle,

$\tilde{\gamma }$

$γ̃$ can be defined as the “coupling constant” describing the cost of long wavelength distortions of the interface in terms of the capillary wave Hamiltonian

$\mathcal {H}_{cw}$

$Hcw$⁠,^98,100–106

but for fluid-fluid interfaces one can show that

$γ̃=γ$⁠.¹⁰⁷ Of course, a crucial (and still controversial) aspect concerns corrections to Eq. (13).^{106,108–112} Note that Eq. (12) is derived by considering the Fourier components

$h(\vec{k})$

$h(k⃗)$ of h(x, y) to conclude that

$\langle |h(\vec{k})|^2 \rangle = (\tilde{\gamma }k^2)^{-1}$

$⟨|h(k⃗)|2⟩=(γ̃k2)−1$⁠, a factor 1/(k_BT) here has been absorbed in the normalization of

$\tilde{\gamma }$

$γ̃$⁠. Then Eq. (12) readily follows by integrating over all

$\vec{k}$

$k⃗$-values from 2π/L to 2π/B, i.e., the long wavelength description, Eq. (13), is used for all wavelengths, up to the shortest wavelength B which acts as a cutoff. However, one may argue that Eq. (13) no longer holds for small wavelength variations, e.g., one may postulate an additional term

$\frac{\kappa }{2} (\nabla ^2 h)^2$

$κ2(∇2h)2$ under the integral in Eq. (13). The parameter κ has the interpretation as a “bending stiffness” of the interface, such terms are familiar from the theory of membranes. However, simulations have shown^101,110 that κ is negative, if one relies on the concept of a local dividing surface (as conceived by Gibbs) to define h(x, y). This result has been confirmed in the seminal paper by Blokhuis¹⁰⁶ who derived an extended capillary wave model (including bending stiffness) from density fundamental theory, and used this approach for a detailed analysis of the simulation results of Vink et al.¹⁰¹ of the AO model with q = 0.8. Blokhuis¹⁰⁶ demonstrated that the structure factor describing the excess scattering due to the presence of an interface can be described excellently by the interpolation formula

where

$N$ is a normalization constant to ensure the correct limit S_cc(k → ∞) = 1. An analogous expression holds when the polymer density (rather than the colloid density) is used to define the dividing surface. Fig. 7 presents an example showing that the theory of Blokhuis¹⁰⁶ can describe the simulation results of Vink et al.¹⁰¹ almost perfectly. Of course, the caveat needs to be made that the normalization constant in Eq. (14) had to be chosen as an adjustable parameter to fit the simulation results. A consequence of this “capillary enhancement”¹⁰⁶ of short wavelength capillary wave excitations in comparison with the long wavelength extrapolation is also directly seen in Fig. 6(a), where the data deviate from the straight lines towards larger values when n_B gets very large (and hence wavelengths of a few σ_c are probed).

However, a very different view of interfacial fluctuations has been advocated by Tarazona et al.,^111,112 who define the instantaneous configuration of an interface avoiding any coarse-graining, but identify (in term of a suitable neighborhood criterion) a percolating network of particles that form the topmost layer of the liquid. It is argued that the intrinsic density profile ρ(z) is then defined at each site where such an interface particle occurs using the particle as the local origin of the z-axis perpendicular to the interface. However, an inevitable consequence then is that the intrinsic profile always shows pronounced layering oscillations. While DFT³⁸ predicts a weak layering on the liquid side of the interfacial profile for strongly segregated colloid-polymer mixtures as well, there is no δ-function singularity at z = 0, of course. For typical cases, DFT predicts smooth density profiles, almost having the tanh (z/w₀) form of Landau theory,^98,103 lacking any capillary wave broadening, of course. We note that Royall et al.³⁰ actually attempted to study both the critical fluctuations in real space and the local interfacial profile near criticality from confocal microscopy experiments. However, the large statistical fluctuations of the published data make quantitatively reliable estimations difficult. Near criticality, bulk fluctuations extend over the scale of the correlation length which then is much larger than the size of individual colloids, and hence identifying the fluctuating interface positions from the particle coordinates necessarily is rather ambiguous. For the lattice gas model, it is known that the lattice sites occupied by particles form (in d = 3 dimensions) a percolating cluster even at the vapor branch of the vapor-liquid coexistence curve, before the critical point is reached.¹¹³ Hence, using such a geometric neighborhood criterion to locate vapor-liquid interfaces has been shown to yield very misleading results near criticality.¹¹⁴ Thus, although the study of interfaces between gas-like and liquid-like phases of colloid-polymer mixtures by experiment, theory, and simulation has already given a lot of insight into the physics of interfacial fluctuations, there still is the need for more extensive and definitive studies.

We now consider the effect of (planar or curved) walls on colloid-polymer mixtures. While for small molecule systems the atomistic corrugation of solid surfaces is an important issue,¹¹⁵ on the size scale of a colloidal particle this corrugation clearly is negligible. Also for well-prepared surfaces (such as mica, silicon wafers, graphite, etc.) the roughness on medium length scales can also be neglected. Since colloidal particles as well as polymers cannot overlap a wall, the wall physically behaves like a very large colloidal particle, i.e., in the presence of polymers a depletion attraction is exerted on the colloids in the suspension. Thus, it is no surprise that the formation of wetting layers of the colloid-rich phase at walls indeed has been observed.^{31,32,116–118} Varying the distance of the considered state point

$(ηcv,ηpv)$ along the vapor branch from the critical point, it has become possible to see a transition from partial to complete wetting, from a study of the contact angle at glass walls.^117,118 However, a precise estimation of contact angles of interfaces of colloid-polymer mixtures at walls is very difficult (due to gravity effects the meniscus has a complicated shape¹¹⁶), and also the precise values of the packing fractions at the considered states are difficult to determine. Unfortunately, the intensive thermodynamic variable

$\eta _p^r$

$ηpr$ cannot be directly measured (or controlled) at all. Thus, Wijting et al.^117,118 could only conclude that in their system (silica particles with σ_c ≈ 28 nm, polydimethylsiloxane coils with σ_p ≈ 26 nm, dissolved in cyclohexane) the wetting transition occurs for

$0.35 < \eta _c ^\ell - \eta _c ^v < 0.38$

$0.35<ηcℓ−ηcv<0.38$⁠, and no statement on the order of the wetting transition was possible. Qualitatively, the behavior is in agreement with DFT calculations³⁸ and Monte Carlo simulations,⁶⁵ but one should note that the bulk phase diagram of this system deviates strongly from the AO model prediction.

While the colloid-colloid attraction can experimentally be tuned, if the size and concentration of polymers in the suspension is varied, it would also be interesting to find means to control the colloid-wall interaction. Theoretical studies,^119–122 suggested that coating the walls with a layer of long endgrafted flexible polymers (a “polymer brush”^123,124) one could selectively control the colloid-wall interaction (colloids are repelled by the brush, while polymers may enter the brush under good solvent conditions if it is not yet too strongly stretched). A notable attempt to experimentally modify the wall properties was made by Wijting et al.,¹¹⁸ who prepared a “soft” wall by also grafting a layer of polydimethylsiloxane on the glass surface. Indeed, they now found a drying rather than a wetting transition (polymers rather than colloids get enriched at the wall). However, this soft surface was not further characterized (grafting density and structure of this grafted polymeric layer were not known). Apart from this attempt, this suggestion has not been followed up in experimental work yet, and anyhow systematic experimental studies of surface and confinement effects on colloid-polymer mixtures to our knowledge are still lacking. An interesting alternative for confinement that acts on colloids only could be provided by an appropriate setup of laser tweezers to a thin sheet in space. Schmidt et al.¹²² suggested that in this way one can realize a selective confinement of colloids in a slab geometry using such laser fields, that would not constrain the polymers at all, which could still move in and out, passing through this confinement which would be fully permeable for them. This would provide a method to realize “capillary evaporation” (i.e., the vapor-like phase forms a layer adjacent to the confining boundary). Capillary evaporation is the counterpart to the more familiar “capillary condensation.”^98,125,126 Thus, the liquid-like phase is attracted to the wall, and this is expected if the confinement in a slab geometry is carried out by hard wall.¹²⁷ Both in the case of capillary condensation and in the case of capillary evaporation one expects a shift of the vapor-liquid transition relative to its bulk location. This shift is described by the Kelvin equation,

$μccoex(D)−μccoex(∞)∝1/D,D$ being the distance between the plates,

$\mu _c^{coex}(D)$

$μccoex(D)$ being the chemical potential at liquid-vapor coexistence in the thin film.^125,126 Early simulations^122,127 were roughly compatible with this relation, but due to the limited accuracy and due to the lack of knowledge on the required interfacial free energies a more quantitative assessment was not yet possible.

Vink et al.^120,128,129 obtained more precise data on the phase diagram of a confined polymer mixture and showed that in addition to the 1/D shift predicted by the Kelvin equation there also occurred a crossover in the critical behavior from the three-dimensional Ising behavior to the two-dimensional Ising behavior (i.e., critical exponents, β = 1/8, μ = 1 occur^67–69 but one needs to approach the critical point extremely closely even for film thicknesses D of only a few colloid diameters to observe this crossover). Thus, it is no surprise that experimental evidence for it is still lacking.

De Virgiliis et al.^119–121 also drew attention to the possibility to study the confinement between “competing walls,” i.e., one wall prefers the colloid-rich phase (e.g., an ordinary hard wall) and the other wall the polymer-rich phase (e.g., a wall coated with a polymer brush). At sufficiently large values of

$ηpr$ then always an interface, parallel to the confining walls and separating the two coexisting phases, is present. In this geometry, another novel transition is expected, the so-called interface localization-delocalization transition.^130–132 In the grand canonical ensemble of the colloids, a transition occurs at

$\mu _c^{coex}(\eta _p^r,D)$

$μccoex(ηpr,D)$⁠: there for

$\mu _c <\mu _c^{coex} (\eta _p^r,D)$

$μc<μccoex(ηpr,D)$ the planar slit is mostly filled by the polymers, apart from a precursor of a colloid-rich wetting layer that is attached to the wall that attracts the colloids. However, for

$\mu _c > \mu _c^{coex} (\eta _p^r,D)$

$μc>μccoex(ηpr,D)$ the slit is mostly filled by colloids, apart from a precursor of a polymer-rich wetting layer at the other wall. Practically, this transition would be observed if the two confining walls are immersed in a bulk colloid-polymer mixture, and adding polymers to it (so one enhances

$\eta _p^r$

$ηpr$⁠) one could reach the transition line

$\eta _p^r=\eta _{p,coex}(\mu _c,D)$

$ηpr=ηp,coex(μc,D)$ (which is the inverse function of

$\mu _c=\mu _c^{coex}(\eta _p^r, D)$

$μc=μccoex(ηpr,D)$⁠). This transition line in the (⁠

$\mu _c,\eta _p^r$

$μc,ηpr$⁠)-plane then again ends at a critical point that falls in the two-dimensional Ising class.¹²⁸ While simulations^119–121 have given evidence for this transition, corresponding experiments remain to be performed.

For a quantitative understanding of the wall effects on colloid-polymer mixtures, knowledge of the wall excess free energies

$γv(ηpr,ηc)$ and

$\gamma _\ell (\eta _p^r,\eta _c)$

$γℓ(ηpr,ηc)$ of the vapor-like and liquid-like phases is crucial. At phase coexistence, their difference enters Young's equation for the contact angle^98,132,133

When θ → 0 the wetting transition occurs, in a semi-infinite geometry: In the regime where a wall exhibits complete wetting by the colloid-rich phase, it is coated by an (ideally macroscopically thick) layer of the colloid-rich phase, and then we have

Likewise, when θ → π we encounter the “drying transition,” and in the regime where a wall exhibits complete drying, it is coated by an (ideally macroscopically thick) layer of the polymer-rich phase, and then we have

DFT is a powerful method suitable to compute both the packing fraction profiles near various walls and the associated wall tensions.^38,134–137 The predictions from DFT calculations have also been tested by Monte Carlo simulations on various cases (e.g., Ref. 138). Qualitative (and sometimes even quantitative) agreement has been found.^137,138 In cases where a precursor of a wetting layer at a wall is only very few colloid diameters thick, DFT captures the (oscillatory) colloid density profile very well, but the accuracy slightly but systematically deteriorates when the thickness of the layer increases. In fact, one must expect that the thicker the precursor of the wetting layer becomes, the more broadening of the profile due to capillary waves will occur. For Ising films (or films of binary polymer blends¹³⁹), it was first observed that the interface between two coexisting phases has a width w(D) that scales with the film thickness D as

$W(D)∝D$⁠. This anomalous broadening of wetting layers with film thickness is theoretically expected for wetting phenomena controlled by short range forces: the finite film thickness D cuts off capillary waves exceeding a correlation length ξ_|| that scales like ln (ξ_||) ∝ γ^−1/2D.¹³⁹ Hennequin et al.³¹ have observed this behavior experimentally for a colloid-polymer mixture. Of course, this capillary wave broadening is absent in DFT, and hence in many situations where rather thick layers occur, DFT predictions hence should be taken with caution. For example, Wessels et al.¹³⁶ observe that the growth of the colloid-rich layer occurs via a sequence of many layering transitions, but it is likely that higher order layering transitions are washed out by these capillary wave type fluctuations. In fact, a similar behavior is well-known for Ising (lattice gas) models, where two-dimensional layers are added step by step (“multilayer adsorption”) up to the bulk critical temperature, when one studies the problem by molecular field approximation.¹⁴⁰ However, including fluctuations one finds that the transitions of the individual layers (labelled by index n) end in layer critical points T_c(n) that converge to the roughening transition temperature T_R of the interface.¹⁴¹ For T > T_R, the lattice gas model despite its underlying lattice structure behaves like a continuum model, i.e., interfaces exhibit capillary wave excitations, and instead of multilayer adsorption one observes a continuous growth of a wetting layer when conditions of complete wetting are approached (usually preceded by a single pre-wetting transition, when the wetting transition at coexistence is of first order).^132,133 We expect that for the vapor-liquid type transition of colloid-polymer mixtures, where no underlying lattice is present, the multilayer adsorption transitions found in the DFT calculations by Wessels et al.¹³⁶ have to be viewed with great caution, since any height fluctuations of the local vapor-liquid interface in DFT are missing. A similar caveat was already made by Brader et al.³⁸ who suggested that low order layering transitions may persist even if high order layering transitions are washed out, and simulations⁶⁵ seem to confirm this expectation.

We have already seen that due to the neglect of fluctuations DFT calculations of γ are unreliable (and further systematic errors come in close to the bulk criticality either). Also the computation of the contact angle θ by DFT methods¹³⁶ will suffer from related errors. Unfortunately, high precision simulations studying precisely the same conditions as used by Wessels et al.¹³⁶ are still lacking.

We also note that while multilayer adsorption (via a sequence of first-order transitions) for the liquid-vapor transition is doubtful, it is more likely to occur for the vapor-solid transition (cf. Fig. 2). We speculate that for large enough

$ηpr$ the vapor-solid interface will be non-rough, and then crystals would grow from supersaturated vapor by a thermally activated layer-by-layer mechanism. If this is the case, one could search for the conditions of a roughening transition in colloid polymer mixtures, too. Studying the associated fluctuations by optical microscopy techniques on a single particle level clearly would be very illuminating.

An extensive simulation study of contact angles and wetting behavior has recently been carried out by Winkler et al.,^142,143 in the context of a study aiming to elucidate the consequences of spherical confinement on phase behavior. First, use is made of a developed method to calculate wall tensions, where a thermodynamic integration is performed along a path in phase-space connecting a system with two planar walls and an equivalent system but with periodic boundary conditions throughout (“ensemble switch method”^144,145). Varying then the range (σ_wc) of a wall potential (that was chosen to have hard-core form such as Eq. (2) as well), for the case q = 0.8 of the AO-model for which the bulk phase diagram is known very precisely,⁷⁰ as well as the interfacial tension,^70,71 both

$γv(ηpr,ηc)$ and

$\gamma _\ell (\eta _p^r,\eta _c)$

$γℓ(ηpr,ηc)$ were computed for

$\eta _p^r=0.94$

$ηpr=0.94$ as function of η_c, including coexistence conditions.^142,143 It was shown that varying σ_wc one could span the whole range of wetting behavior, from complete over partial wetting up to drying, using Young's equation {Eq. (15)} so that no direct “measurements” of droplets or inclined interfaces is needed. Then these data were used to analyze the behavior of the AO model under these conditions

$(q=0.8,\; \eta _p^r=0.94$

$(q=0.8,ηpr=0.94$⁠) under spherical confinement in spheres of various values of the radius R (Fig. 8). As expected, in such a confined geometry the transition from vapor to liquid is both rounded (i.e., no sharp singularities of any thermodynamic functions can occur) and shifted. In the grand canonical ensemble, where μ_c is used as an independent variable, the shift is dominated by surface effects (and scales like 1/R) while the rounding is a consequence of using a finite volume (and hence scales like 1/R³).¹⁴² In the canonical ensemble, one observes loops in the isotherms (which should not be confused with van der Waals loops, which are simply a mean field artefact!) but are due to surface/interfacial excess contributions to the thermodynamic potential, which are non-negligible in a finite system.

Since the shift of the transition is dominated by surface effects, one can actually use it to obtain information on wetting conditions as well: Using the Kelvin equation for spherical geometry implies for the shift

Note that the difference of wall tensions in the numerator of Eq. (18) just is the difference between the wall excess free energies of liquid-like and vapor-like bulk coexisting phases (which have packing fractions

$ηcℓ$ and

$\eta ^v_c$

$ηcv$⁠, respectively), and the factor (3/R) is just the surface to volume ratio for spherical geometry. Here, we have disregarded the fact that actually for complete wetting the radius R should be reduced by the thickness of the precursor of the wetting layer (that would form for Δμ_c → 0 in the limit of a planar wall, R → ∞.^199,200 However, the range of values for R available in the shown study (and the limited statistical accuracy) did not allow a clear identification of this correction term.

Fig. 9(a) tests the predicted 1/R behavior and Fig. 9(b) compares the obtained wall excess free energy difference with the independent prediction from the ensemble switch method. Since there is no adjustable parameter whatsoever, the agreement is significant, and it shows that sphere radii of the order of 5–10 colloid diameters suffice to reach the regime where the Kelvin equation, Eq. (18), holds. Note that in the regime of complete wetting the “measured” wall free energy difference is given by γ, Eq. (16), as expected.

Consideration of phase transitions of a fluid in a closed spherical container seems an extremely esoteric problem, but this is not so when one considers thermally driven systems (e.g., block copolymer melts or polymer blends confined in mini-emulsions¹⁴⁶) or chemically driven systems (e.g., phase separation in spaces confined by semipermeable cell membranes, etc.). For colloidal suspensions, one would need to create a hollow sphere with an opening such that colloidal particles could enter. Of course, for such geometries also the kinetics of filling the sphere would be an interesting problem to study; much interest has been paid to the related problem of colloid motion along one-dimensional channels, for instance.¹⁴⁷ However, we are not yet aware of any experiments on colloid-polymer mixtures in cylindrical geometry. Such studies would be interesting, since due to the quasi-one-dimensional character of the system interesting rounding phenomena of the vapor-liquid type phase transition are predicted.^148,149 While for small molecule systems the problem of condensation in cylindrical pores has been amply studied (e.g., Ref. 126), but is difficult to understand since the atomistic corrugation along the pore wall cannot be neglected, and chemically bound impurities, roughness, and other defects usually play an important role, such complications would be much less important on the scale of colloidal particles. Other geometries with interesting interfacial phenomena, studied so far for Ising (lattice gas) models, for instance,^150,151 are the “filling transitions” of cones and wedges. The experimental study of colloid-polymer mixtures in such confining geometries clearly would be illuminating.

While much effort has been devoted both by experiment, simulation, and analytical theory to clarify the bulk and interfacial behavior of colloid-polymer mixtures in the regime of moderate colloid packing fractions where the vapor-liquid type phase separation matters, much less work has been devoted to the regime where the colloidal suspension forms a colloidal crystal. One basic aspect on which experiments have focused is the problem of understanding the rate at which crystals form from supersaturated fluid states by homogeneous nucleation^152–157 for the simple hard sphere system. However, simulations (where the nucleation behavior is obtained from biased sampling¹⁵⁴) and experiments^152,153 are in rather strong disagreement. Recently, it has been suggested that hydrodynamic effects, that strongly depend on the packing fraction of the metastable fluid, significantly affect the kinetic prefactor of the nucleation rate,¹⁵⁷ which has not been regarded in the earlier simulations.

Another issue concerns the question whether the nucleation barrier can be understood in the simple manner suggested by nucleation theory,¹⁵⁸ namely, the formation free energy of the nucleus is described by the competition between the gain in bulk free energy (proportional to the volume) and the cost due to the surface free energy of the nucleus. While for nucleation of liquid droplets from a supersaturated vapor, the droplet is a sphere of radius R and hence the surface free energy simply is 4πR²γ, if a dependence of γ on R¹⁵⁹ is neglected, for crystal nucleation this approach in general is inadequate: the fluid-crystal interface free energy

$γ(n̂)$ depends on the orientation (⁠

$\hat{n}=$

$n̂=$ unit vector perpendicular to the interface) of the interface normal relative to the crystal axes. Then the shape of the nucleus is no longer spherical, but rather needs to be found from the Wulff construction.¹⁶⁰ Already for hard spheres there is evidence that this anisotropy of the interface free energy is non-negligible,^161–163 but estimates obtained from different groups (applying also different techniques) differ significantly from each other. Thus, an estimation of the total surface free energy of a (large) fcc crystallite coexisting with surrounding fluid still seems to be a difficult task, even for the simple hard sphere system!

In any case, this model has served as a “laboratory” to refine both simulation techniques^161–163 and the DFT approach¹⁶² for the study of crystal-fluid interfaces, and we expect that future applications of these techniques to the AO model will be very useful, since adding polymers one can systematically vary the interfacial free energies, as the study of the interfacial stiffness

$γ̃(100)$ of the 100 interface of the AO model⁸ already shows. We recall that the interfacial stiffness is defined as^107,164

for two directions

$n̂α$ and

$\hat{n}_\beta$

$n̂β$ that are orthogonal to

$\hat{n}$

$n̂$⁠. We emphasize that Eq. (19) only applies for rough interfaces, and the question whether for solid-liquid transitions in the AO model also non-rough interfaces can be observed is still unknown. If rough interfaces occur, it is

$\tilde{\gamma }_{\alpha \beta }(\hat{n})$

$γ̃αβ(n̂)$ rather than

$\gamma (\hat{n})$

$γ(n̂)$ that enters in the capillary wave description {Eq. (13)} as a prefactor.

Experimentally (at least in small molecule or atomic systems, e.g., metals¹⁶⁵), it is well known that in practice homogeneous nucleation is hard to observe, because heterogeneous nucleation (triggered by impurities, walls, etc.) involves greatly reduced nucleation barriers, and therefore occurs already for modest supersaturations of the fluid phase, where homogeneous nucleation is not yet observable. The conceptually simplest case is nucleation at a planar external wall, under conditions of incomplete wetting of the wall by the crystal (if complete wetting occurs upon approaching the fluid-crystal phase boundary from the fluid side, there would not be any barrier prohibiting the uniform growth of the crystalline film adjacent to the wall, when the fluid-crystal two-phase coexistence region is entered).

For atomic and small molecule systems, the corrugation of the (crystalline) wall acts like a periodic potential which in general has a misfit with the lattice structure of the crystal that forms, causing elastic distortions in the growing crystal, which complicate the understanding of the nucleation barrier enormously. In this respect, colloids would present a significant advantage; in fact, computer simulation studying heterogeneous nucleation of hard sphere crystals at walls have been performed,^166–168 but questions arise due to the suggestion¹⁶⁹ that complete wetting of the hard sphere fluid at hard walls occurs. This problem can be avoided when one studies wall-attached crystalline nuclei for the AO model (Fig. 10). Note that this simulation uses of the order of 60 000 colloidal particles, and the particle number in the crystalline nucleus is of the order of 3000; this means that also in this case the barrier that needs to be overcome in a heterogeneous nucleation event is so high that one has no chance to follow the kinetics of nucleation by straightforward simulation. But it is possible to stabilize a wall-attached nucleus by a suitable initial condition,¹⁴⁵ and equilibrate it and study its properties. Note that rather generally in finite systems one can study “critical nuclei” surrounded by supersaturated phase (which would be metastable in the thermodynamic limit, and the critical nucleus would represent a saddle point in the phase space of the system¹⁷⁰) in full thermal equilibrium; this type of finite size effect has been exploited (see, e.g., Ref. 159) to study nucleation barriers in supersaturated vapor described by the Lennard-Jones model, but (apart from Ref. 145) this method has not been applied to study crystal nucleation yet. From Fig. 10(b) one can see directly the “ordered” arrangement of the particles in the crystalline nucleus, and the layering of the fluid at the walls does not mean that there are crystalline precursor layers at the walls. Also the surface of the nucleus looks rough, its shape is still reminiscent of the simple sphere-cap shape that one expects for heterogeneous nucleation of fluid droplets at walls.¹⁷¹ If one tentatively adopts this shape for the snapshots such as shown in Fig. 10(b), one would extract a contact angle of about θ ≈ 70° ± 2°. Deb et al.¹⁴⁵ also succeeded to measure the pressure p^* of the fluid surrounding the nucleus. Ignoring the anisotropy effects of the crystal-fluid interface tension γ_fc completely, classical nucleation theory would predict

where the bulk coexisting volume fractions of the fluid

$(ηcf≈0.494)$ and crystal

$(\eta _c ^m \approx 0.64)$

$(ηcm≈0.64)$ and the estimate⁸ γ_vc ≈ 0.95 would yield p^* − p_coex ≈ 6.44/R^*, for R^* → ∞. Observations (such as R^* ≈ 14.4, p^* − p_coex ≈ 0.4¹⁴⁵) are roughly consistent with this equation. Here, we have again absorbed a factor of 1/k_BT in γ_fc and take the colloid diameter as the unit of length. It is also encouraging, that an estimate of θ based on Young's equation (γ_wf − γ_wc = γ_fccos θ), where the difference between the wall-fluid excess free energy (γ_wf) and the wall-crystal excess free energy (γ_fc), that also could be obtained by the ensemble switch method¹⁴⁵ agrees with the observed value, quoted above. Of course, this study can only be considered with caution, since it is well-known that in the case of an anisotropic interfacial tension Young's equation needs to be generalized.¹⁷² The accuracy of existing simulations for crystal-fluid interfaces in the AO model and homogeneous as well as heterogeneous nucleation probably is insufficient to study such anisotropy effects, and clearly more precise work in the future is desirable. The Ising model study¹¹⁴ indicates that for rough interfaces the effects of anisotropy on the nucleation barrier are of the order of 10% or less.

Finally, we draw attention to a study of very small crystalline clusters¹⁷³ of AO-models described by Eq. (5) for q = 0.05, 0.3, and 0.8, respectively (recall, however, that for the latter two of these choices multi-particle interactions beyond Eq. (5) would occur). The cluster size distribution has been studied, and the structure of the clusters has been analyzed in detail. Of course, also this work is a first explanatory step only, but this type of research also seems a promising future direction of research.

In this context, we also stress the need for more analytical work. DFT theories of heterogeneous crystallization mostly address hard spheres and hard disks,^174,175 but not yet the AO model. As emphasized above the AO model has the distinctive advantage that due to the presence of two control parameters

$(q,ηpr$⁠) a much more stringent test of theoretical concepts is possible.

Wall-attached nuclei (such as shown in Fig. 10) also clearly involve a line of three-phase contact (fluid-crystal-wall, in this case) and also the problem of understanding the associate excess free energy (the “line tension”⁹⁸) arises. This problem so far has found little attention for colloid-polymer mixtures, apart from a study in terms of a Landau-Ginzburg-type treatment¹⁷⁶ of the vapor-liquid-wall contact line. In this work, free volume theory is used to describe the bulk phases, and using a gradient square approximation an effective interface potential for the AO model is derived.

In this perspective article, we have given a brief introduction to the AO model for colloid-polymer mixtures, and have tried to give the reader a flavor of simulation work and analytical theories dealing with bulk equilibrium and interfacial properties for this model. We have sketched some of the key concepts and tried to make clear why this model is particularly valuable as a “laboratory” for theorists for testing and developing both analytical approaches and simulation methods. We have not covered experimental work on an equal footing, in view of the fact that a full book is available on the subject of colloid-polymer mixtures and the depletion interaction.⁵² 

Due to the short range of the attractive interactions in the model, and the flexibility it offers by suitable choices of its two control parameters

$(q,ηpr)$⁠, the AO model has become suitable to clarify some longstanding issues in the general theory of fluid systems. One example that we have not mentioned so far is the effect of quenched disorder on the vapor-liquid type critical behavior. The standard context for which this problem previously has been addressed is condensation of fluids (or phase separation of fluid binary mixtures) in random porous materials.¹²⁶ The basic idea that one wishes to test is the proposal due to de Gennes¹⁷⁷ that the problem near the critical point is equivalent to an Ising model in a random magnetic field.^178,179 Existing experiments on small molecule systems in silica aerogel¹⁸⁰ and corresponding simulations^181–183 did not succeed to provide evidence in favor (or against) of this issue. Vink et al.^184–186 clarified this problem by simulating an AO-model where a small fraction (of the order of 1%) of the colloids were held fixed at randomly chosen positions. Experimentally, fixing colloids in arbitrarily chosen positions could be done with laser tweezers, but only for a small number of particles, so in this respect simulation is ahead of experiment. Vink et al.^184–186 showed that very small (but nonzero) packing fractions of randomly fixed colloids significantly change the character of critical fluctuations as predicted.^177–179 In particular, the so-called “hyperscaling relation”^67–69 is violated, which relates standard critical exponents to the exponent ν describing the divergence of the order parameter correlation length and the dimensionality d(=3 here). For example, for the ordinary AO model we have the relation μ = (d − 1)ν = 2ν for the critical exponent of the interface tension {see Eq. (11) and subsequent discussion}. In the case of random-field type randomness, the exponent θ_R describing hyperscaling violation enters, μ_R = (d − 1 − θ_R)ν_R, with¹⁸⁶ θ_R ≈ 1.3, ν_R ≈ 2.1 implying μ_R ≈ 1.5 (which unfortunately is close to the mean field result). But another observable consequence would be that the order parameter distribution at criticality in a finite subsystem should exhibit very sharp peaks, unlike the smooth distribution observable³⁰ in standard colloid-polymer mixtures by confocal microscopy techniques.

Another field where quenched disorder matters are crystalline structures, of course, and colloid-polymer mixtures at high colloid packing fractions should provide there unique possibilities, because one can observe the local structure in crystals with defects on the single particle scale (e.g., Refs. 187 and 188). For example, defects in crystals are created due to the strain that builds up when one uses curved substrates as seeds.¹⁸⁸ Also the equilibrium concentration of vacancies and density distribution anisotropies in colloid crystals is of interest, and has been considered theoretically both by density functional methods and simulations.¹⁸⁹ So far, all such questions have been addressed for the limiting case of hard spheres only, but clearly the extension to the AO model should provide valuable additional insight.

Since colloidal crystals are very soft objects, the deformation of their structure by external fields (e.g., periodic laser fields, see, e.g., Ref. 190 and also Ref. 191 for a brief review) is an important direction of research. Much of this work, however, addresses the dynamical response of the colloidal particles, and their behavior far from equilibrium, and this subject is out of focus of the present article, which has focused on static equilibrium behavior. However, we do feel that far from equilibrium behavior of colloid-polymer mixtures has become an important field for experimental research: e.g., the suppression of thermally excited capillary waves by shear flow has been studied,¹⁹² as well as the response of the near-critical structure factor.¹⁹³ Very extensive studies have also addressed the kinetics of phase separation into gas-like and fluid-like structures via spinodal decomposition following quenching experiments (e.g., Refs. 32–34). Again the particular bonus of such studies is that one can follow the structural evolution from very early to very late stages, both by scattering experiments and by microscopic observations on the particle level.

Of course, an extension of theoretical studies, on the basis of the AO model, to the non-equilibrium behavior of colloid polymer mixtures is highly desirable, but then the effects of the solvent on the particles (namely, friction, random forces, and long-range hydrodynamic interactions) must be included. Zausch et al.⁸² included friction and random forces for a “soft” version of the AO model with continuous potentials, and Winkler et al.⁴⁰ building on this approach added hydrodynamic interaction effects via the multiparticle collision dynamics¹⁹⁴ method. Of course, the parameters had to be chosen such that the dynamics still is several orders of magnitude faster than in reality, to render such simulations feasible, but it is believed that there the behavior nevertheless is close to experiment. In this study, phase separation of a colloid-polymer mixtures confined by two planar parallel walls was studied, and it was shown⁴⁰ that the growth laws sensitively depend on the hydrodynamic boundary conditions on the confining walls: in case of stick boundary conditions hydrodynamic interactions are fully screened over large distances, and a purely diffusive growth law results. For slip boundary conditions, however, the theoretically predicted growth law for the domain size (ℓ(t) ∝ t^2/3 in d = 2, t being the time after the quench) was indeed observed.⁴⁰ 

Of course, this work⁴⁰ just is a first step, and more work along similar lines for related problems is likely to come. Also analytic methods for the dynamics of the AO model (such as dynamic density functional theories^195–198) would be valuable.

A very interesting issue concerns the interplay of metastable liquid-vapor type phase separation, crystallization, and glass formation (or irreversible gelation, respectively). Compare the phase diagrams of Figs. 2(a) and 2(b): for small q and large enough

$ηpr$ the volume fraction

$\eta ^f_c$

$ηcf$ where freezing sets in bends over from the region where hard spheres freeze (near

$\eta ^f_c \approx 0.49$

$ηcf≈0.49$⁠) to rather small values. The interpretation is that the vapor-liquid type phase separation (that exists for large enough q) now is “hidden” in the vapor-crystal two phase coexistence region, but may still be observed as a transient phenomena, by suitable preparation of initial states metastable liquid-liquid phase separation in a polymer-rich phase (at small η_c) and a colloid-rich phase (at a larger value of η_c, but still significantly smaller than η_m, where in equilibrium the crystal starts to melt) may occur.

This idea was explored by Fortini et al.,²⁰¹ for q = 0.15, where the vapor-liquid type phase separation occurs deeply in the metastable (or unstable) region, where stable equilibrium requires two-phase coexistence of vapor-like and crystal phases. Fortini et al.²⁰¹ found that in fact crystallization gets enhanced by liquid-liquid phase separation, because the formation of crystal nuclei via homogenous nucleation now takes place as a two-stage process: first, strong density fluctuations occur, and then crystal nucleation proceeds from high density regions. For certain volume fractions, also a gel-like irregular network of crystallites freezes into an irregular structure, a type of glass.²⁰¹ 

It might be hoped that metastable vapor-liquid type phase separation in the AO model might elucidate the suggestion that phase transitions between two liquids differing in density (and in local structure) might occur in molecular liquids like (supercooled) water, silica, triphenylphosphite, etc.²⁰² In fact, in all such cases a delicate interplay between phase separation, crystal nucleation, and glass formation may be expected on general grounds.²⁰³ However, unlike the AO model (where the colloid density in the vapor-like and liquid-like phases differ very much) the density difference between the two hypothetical phases in these molecular liquids are much smaller, and bond orientational order parameters may be needed to distinguish them.²⁰⁴ Nevertheless, it may be hoped that further studies of the AO model may elucidate this very controversial issue.

Finally, we mention that the AO model has seen many extensions in various other very interesting directions, that have not been mentioned so far: e.g., rather than mixtures of colloids with linear polymers one can consider polymers with other architecture, e.g., star polymers;²⁰⁵ also the exploration of the effects due to polydispersity (of colloids, of polymers, or both) is of interest (see, e.g., Ref. 206; and particularly fascinating is the consideration of colloids which exhibit intrinsic mobility (“active matter”), e.g., Ref. 207). The concept of depletion forces which was developed on the basis of the AO model, finds so widespread application in the context of biological soft matter, that we must refrain from trying to discuss this aspect here. In any case, we have tried to give the reader a flavor of the role of the AO model as a workhorse of research in a broad field of soft condensed matter, which will remain fruitful in the future.

One of us (K.B.) acknowledges fruitful collaborations on various aspects of colloid-polymer mixtures with D. Deb, G. Gompper, J. Horbach, S. Jungblut, H. Löwen, M. Oettel, T. Schilling, R. L. C. Vink, A. de Virgiliis, A. Winkler, R. G. Winkler, T. Zykova-Timan, and J. Zausch. He is grateful to numerous colleagues for illuminating discussions, in particular: A. van Blaaderen, C. Dellago, M. Dijkstra, S. U. Egelhaaf, R. J. Evans, D. Frenkel, H. N. W. Lekkerkerker, C. Likos, P. Nielaba, and T. Palberg. A.S. would like to acknowledge the MAINZ Graduate school of Excellence. All the authors would like to acknowledge funding from the DFG (VI237/4-3).