In 1954, Asakura and Oosawa1 explained that nonadsorbing macromolecules can induce attractive forces between colloidal particles in a single page paper in this journal. The effective pair interaction between the colloidal particles mediated by the nonadsorbing species is nowadays often termed the Asakura–Oosawa (AO) potential.2 Figure 1 illustrates such 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.
This special issue of The Journal of Chemical Physics contains a collection of papers related to the concept introduced by Asakura and Oosawa. We are very honored that the late professor Oosawa contributed to this issue with a historical overview3 on the origin of the Asakura–Oosawa theory. In his synopsis, he explains that he came to Nagoya to “do some unorthodox physics” and that experimental results have stimulated him to theoretically consider the effects of adding macromolecules to colloids.
In the same period, Vrij7 independently found that excluded volume interactions in a colloidal dispersion containing an additional component have important consequences for the effective interactions, structure,8 and phase stability.9,10 Vrij explicitly introduced the description of penetrable hard spheres (PHSs) to describe the nonadsorbing polymers, which was implicitly proposed by AO. In hindsight, the delayed recognition clarifies that Asakura and Oosawa were ahead of their time. Since the 1980s, especially in the 1990s, attention concerning the influence of nonadsorbing macromolecules on the interaction between colloidal particles and the resulting phase behavior of colloid–polymer and binary colloidal mixtures gained increasing interest (see Fig. 2 in Ref. 4).
In 1980, the term “depletion”11 was introduced to describe the effect of nonadsorbing species near a surface. In the field of colloid chemistry, the accumulation of species (ions, polymers, and proteins) at (colloidal) surfaces received quite some attention after the 1960s (see, for instance, Refs. 12 and 13). Terms such as positive and negative adsorption were common, and the latter was also termed depletion.
In the decades that followed the work of Vincent5,6,14,15 and Vrij7–10 and their co-workers, various theoretical tools were applied to better understand the microstructure, pair depletion interactions, and phase stability of colloid–polymer mixtures. We summarize these below with a focus on bulk properties. For some relevant references on the interface physics in colloid–polymer mixtures, see Refs. 16–22.
A key step was the application of thermodynamic perturbation theory by Gast, Hall, and Russel23 to predict the phase behavior of colloid–polymer mixtures to interpret systematic experiments24,25 that were performed in several laboratories. At the same time, field theoretical methods were employed to understand the detailed polymer physics related to depletion.26–28 Shaw and Thirumalai29 formulated a reference interaction site model for colloids and combined it with the Edwards model for polymers to explain depletion stabilization effects:11,30,31 at high polymer concentrations, repulsive contributions to the pair interactions appear.
Methods such as free volume theory (FVT),32 polymer reference interaction site model (PRISM) integral equation theory,33 integrating out the depletant using an effective one-component Hamiltonian,34 density functional theory (DFT),35–37 and a Gaussian core model38 were applied to gain insight into depletion effects. Many theories treat polymers as effective soft spheres, which can be useful when considering dilute solutions of polymers much smaller than the particles, the so-called “colloid limit.” New physics emerges in polymer semidilute solutions and melts, and it was found that the solvent quality also matters. Accounting for polymer physics is also relevant particularly in the “protein” limit,39–41 where the particles are small with respect to the polymer chains. The “protein” limit regime is of particular interest also for (cell) biology.
Free volume theory (FVT)32 (see also Ref. 42) is a simple, yet insightful and reasonably accurate, theory for the macroscopic phase behavior that also enables partitioning of colloids and depletants over the phases to be predicted. For colloid–polymer mixtures described as hard spheres plus PHS, the predicted phase diagrams correspond to computer simulation results.43,44 FVT also allows accounting for interactions between the depletants45,46 and to evaluate the rich phase behavior of anisotropic colloids mixed with nonadsorbing polymers.47,48
Microscopic equilibrium theories of thermodynamics, structure, and phase separation of polymer–particle suspensions that explicitly treat polymers and their conformational degrees of freedom were created by generalizing the PRISM integral equation approach.33,49 The role of particle size (from the protein to colloid limits), polymer concentration (dilute and semidilute good and theta solvent conditions50 and dense melts51), arbitrary particle volume fractions, and the full microstructural correlations were determined in a unified manner.
Experimental work focusing on measuring (i) the pair interaction (see, e.g., Refs. 52–55), (ii) the structure of the dispersion mediated by nonadsorbing polymers,56 and the phase behavior57,58 of well-defined colloid–polymer mixtures appeared. PRISM equilibrium predictions were successfully confronted against experiments.59,60
Soon, nonequilibrium phenomena in multi-component mixtures and the role of depletion effects61–64 gained interest from both theoreticians and experimentalists.65 Knowledge of the structural correlations computed using PRISM theory allowed microscopic dynamical theories of slow colloid dynamics to be constructed, at both the mode coupling and activated dynamics level. Quantitative predictions for the structural relaxation time, formation of glasses and gels, nonlinear rheology, and delayed gel collapse66–68 were made and compared with experiments.69,70
The short-range and controlled strength of attraction induced by the AO interactions also offers an ideal playground for studying the glass transition and gelation of dense colloidal suspensions. In 2000, the mode-coupling theory (MCT), which is one of the most successful first-principles theories to describe the slow dynamics of the glass transition, has been applied to the hard-sphere system with short-range attraction.71 It unveiled the existence of a series of singular dynamical ideal glass transitions as well as the re-entrance of the repulsive-to-attractive glass transitions. The theoretical prediction was soon verified by experiments for a mixture of colloidal particles interacting via the AO potential induced by depletant polymers.63,72
Progress on all the above elements was stimulated by computer simulation. Several works have focused on the influence of depletion-induced attraction on the structural and dynamical behavior of colloids, highlighting, for example, the onset of attractive and repulsive glasses and the occurrence of reentrant melting when the range of the depletion attraction is very small.73,74 The idea to tune the “sticky” interaction by changing the concentration of the depletant has been extended to colloidal suspensions with lower densities and sparked a series of experimental and numerical studies on colloidal gels (see Refs. 75 and 76 and references therein). These studies succeeded in explaining the route from the glass transition at high densities to gelation of colloidal particles by tuning the concentration of depletant.
Another important research topic concerns the fate of the attractive glass/gel line at lower densities and the interplay with phase separation.77 For the latter, an appropriate combination of simulations and confocal microscopy experiments was able to show that such a line intersects the binodal at high densities, giving rise to so-called arrested phase separation.76 Arrested states induced by depletion have also been studied extensively in the context of hard-sphere/star polymer78 or star/star mixtures.79 The introduction of soft interactions is found to enrich the phenomenology of glass transitions and the interplay between the two species80 compared to binary mixtures of hard spheres.81
As indicated above, the interest was broadening and systems such as colloidal spheres plus multi-component depletants82 and dispersions of star polymers (soft colloids) plus linear polymers dispersions83,84 received interest, as well as mixtures of different types of star polymers.85 Besides studying the effects of nonadsorbing polymers as depletants, it is also of interest to treat colloids themselves as depletants added to a dispersion of larger or different colloids. Depletion effects also can be encountered in mixtures of self-assembling block copolymers in a selective solvent under the influence of nonadsorbing polymers.86,87 The phase behavior of hard-sphere binary asymmetric mixtures gained attention in the 1990s.88–91 Fundamental studies using DFT37 helped to quantify the effective interactions and microstructural effects.35 Studies of depletion effects in mixtures of different particle shapes are also gaining interest.92–94 It also became clear that more specific effects, such as charges,95,96 the presence of polymer brushes,97,98 solvent quality,99,100 and polydispersity,101–104 are important.
A very interesting research direction involves systems where depletion attraction competes with a long-range repulsion, often of electrostatic nature, giving rise to the onset of equilibrium cluster phases and arrested states where clusters are dominant,105 with important implications to understand features of solutions of globular proteins. Recent reviews on this topic can be found in Refs. 106 and 107. In addition, simulations have been very useful to locate and characterize gas–liquid (colloid-rich/colloid poor) phase separation of the Asakura–Oosawa effective potential,108 clarifying that this belongs, as expected, to the Ising universality class. Investigations on binary mixtures of colloids and polymers109 where non-ideality effects of the polymer are taken into account have also been explored.
Another rich direction of work in simulations is to calculate effective interactions between colloids immersed in different kinds of solutes that are more complex than polymers. This can be achieved by umbrella sampling or by exploiting virtual moves in Monte Carlo simulations. With these methods, depletion induced by soft spheres or microgels has been investigated.110 The latter have also been recently used in experiments to modify the depletion interactions in situ, exploiting the thermoresponsive character of microgels.111,112 Furthermore, a promising class of depletants involves a self-assembling medium, such as a patchy co-solute, forming supramolecular chains113–115 or clusters116 or even in the vicinity of a critical point,117,118 thus providing a connection between depletion interactions and critical Casimir forces.
Colloid synthesis has evolved to such a degree119,120 that it is nowadays possible to make colloidal particles of a wide range of shapes.121–125 This, and the fact that anisotropic shapes occur in nature, has triggered studies on mixtures of non-spherical colloids plus added nonadsorbing polymers. Hence, insights have been obtained into the phase behavior in mixtures of rods,93,126–129 platelets,47,130–132 and cubes133–135 plus added polymers. In addition, nonequilibrium phenomena are quite relevant here.136,137
Insights into depletion effects inspired by the Asakura–Oosawa concepts have also gained attention in the life science field. Already at an early stage,138–140 it was appreciated that the large volume fraction occupied by the macromolecules in living cells has consequences. Walter and Brooks141 suggested that macromolecular crowding is the basis for microcompartmentation. As summarized a few years ago,142 excluded volume effects are thought to be of importance to explain several intracellular processes.143,144 Hence, depletion effects are suggested to mediate several types of biological processes, including dynamics.145,146
It is abundantly clear that macromolecular crowding affects all aspects of biological processes ranging from transcription to self-organization of the molecules of life. Nowhere is it more transparent than in crowding-driven structural transitions in protein-like polymers,147,148 conformational switches between active states of RNA,149 and depletion effects on the conformations of DNA.150–152 Depletion effects play a similar role in protein dispersions153,154 and dispersed bacteria155 as in colloidal suspensions. There are many more biological processes in which crowding effects, especially the consequences of polydispersity, have not been explored at all. Quantifying the effects of entropic forces in biology remains a virgin area for additional research.
In materials science, depletion effects were used in various ways to self-organize colloidal systems. An example is to select the strength of the attraction by introducing colloidal surface roughness,156–158 which allows the creation of colloidal micelles.159 The use of the different shapes can help to tune the strength of the depletion attraction. This can facilitate the use of depletion effects to make colloidal “key-lock” systems.160 An interesting element that is gaining interest in this field is the influence of colloidal shape on the self-assembly of colloidal particles119,161 and how shape can induce “entropic patchiness.”
It is clear that the depletion field has begun to develop in many different new directions, of which we mention a few that connect to the contributions in this special issue:
Macromolecular crowding. In this issue, some novel insights are presented (see Refs. 164–169).
Interesting findings on charged colloids, proteins, and bacteria upon addition of nonadsorbing polymers, while also specific effects of polyelectrolytes are considered (see Refs. 170–174).
Anisotropic colloids and depletion effects (see Refs. 182–185).
Glasses in colloid–polymer mixtures (see Refs. 186–188).
After almost 70 years, it turns out that the classical theory of Asakura and Oosawa is very much alive. This collection of papers highlights the relevance of the Asakura–Oosawa theory and shows its promise to further understand multi-component soft matter systems, with significant relevance for science, technology, and biology.
We thank associate editors John E. Straub, Francesco Sciortino, David R. Reichman, and Carlos Vega for enabling the assembly of this special edition. We also acknowledge the Editor-in-Chief Tim Lian and appreciate the help of Jenny Stein and Judith Thomas. Finally, we thank all authors who contributed.
The AO(V) potential is often specifically used for the effective depletion interaction between colloidal particles mediated by penetrable hard spheres (PHS) to describe the nonadsorbing species, which are also termed depletants. PHS are ghost spheres that do not interact with themselves but which are hard spheres for the colloidal particles. Here we use the AO potential in a general way to identify the effective potential mediated by the nonadsorbing species.