Generalized depletion potentials

We propose that the behaviour of asymmetric binary fluid mixtures with a large class of attractive or repulsive interparticle interactions can be understood by mapping onto effective non-additive hard-sphere models. The latter are best analysed in terms of their underlying depletion potentials which can be exactly scaled onto known additive ones. By tuning the non-additivity, a wide variety of attractive or repulsive generalized depletion potential shapes and associated phase behaviour can be ‘engineered’, leading, for example, to two ways to stabilize colloidal suspensions by adding smaller particles. Varying the interactions between the mesoscopic constituent particles in colloidal dispersions—examples include proteins, micelles, polymeric composites, ceramic materials etc in polar or non-polar solvents—results in a broad range of equilibrium and non-equilibrium fluid behaviour. It is this tunability which has led to the widespread industrial and biological applications of colloidal suspensions [1]. Some very promising recent experimental advances allow for an exquisite control over the colloidal interactions, leading, for example, to the design of complex self-assembled materials such as photonic band-gap crystals by use of templates [2]. Concurrently, new measurement techniques are being developed that directly determine these interactions with a greatly increased accuracy [3]. Designing colloidal fluids with certain desired properties requires direct control over the interparticle interactions. These interactions are typically effective, i.e. they are a combination of direct interactions (such as Coulomb forces) with indirect interactions mediated through the solvent and the other solute particles [4–6]. One of the best known is the indirect depletion interaction, where one set of (typically smaller solute or solvent) particles induces an effective interaction between another set of particles. Depletion potentials were first calculated for mixtures of polymers and colloids [7] and, with the advent of new experimental and theoretical techniques, they have been the subject of intensive recent interest [8–12, 14]. In this letter we show how a generalization of the depletion potential concept leads to new ways to tune and understand the properties of asymmetric binary colloid mixtures. 3 Present addresses: Max-Planck Institut für Metallforschung, Heisenbergstrasse 1, D-70569 Stuttgart, Germany, and ITAP, University of Stuttgart, Pfaffenwaldring 57, D-70569 Stuttgart, Germany. 0953-8984/01/330777+08$30.00 © 2001 IOP Publishing Ltd Printed in the UK L777 L778 Letter to the Editor Theoretical work has often focused on the binary hard-sphere (HS) model, for which a depletion-induced phase separation for size ratios q = σss/σbb < 0.2 was suggested [8] (here σαα is the diameter of the big (subscript b) or small (subscript s) particles). An important advance was made by Dijkstra et al [10], who used an effective one-component depletion potential picture to show that the fluid–fluid phase separation found with a two-component integral equation technique by Biben and Hansen [8] was metastable w.r.t. a fluid–solid phase transition. More generally, their approach added to the growing consensus that a carefully derived effective pair potential is a powerful tool for analysing the behaviour of an asymmetric binary mixture, at least for size ratios q 0.3 where many-body interactions are not thought to be important (see e.g. [4–6] for some recent reviews). The key step in all these approaches is integrating out the smaller component of a binary mixture to leave a new one-component system with an effective interaction between the big particles. Most theories of depletion have considered only hard-core interactions leading to purely entropic depletion potentials. Their range varies with σss, while increasing the small-particle density ρs or packing fraction ηs = πρsσ 3 ss/6 increases the depth of the (always) attractive well at contact, and possibly adds enhanced oscillations at larger separations r [7, 9, 14]. There have been a number of recent attempts to go beyond purely entropic depletion by including extra interactions between the particles of a binary HS mixture [15–19]. Of course many different kinds of extra interaction can be added, leading to a seemingly enormous increase in complexity. However, in this letter we propose that the effect of a wide variety of these extra interactions on depletion potentials can be understood by a simple mapping onto a non-additive HS mixture model, for which the depletion potentials can be calculated by a second exact mapping or scaling onto those of an additive system, which are well understood [14]. The phase behaviour of many asymmetric binary fluids (with independent components) can be understood on the basis of these depletion potentials [4–6, 10, 11], which then implies that our (double) mapping can be used to analyse a wide variety of interacting asymmetric binary mixtures4. These ideas can also be turned around, leading to the possibility of explicitly engineering a wide variety of generalized depletion potential shapes, including potentials that are repulsive at contact by tuning the interparticle interactions to vary the non-additivity. Non-additive binary HS models are defined by specifying the cross-diameter [21] σbs = 1 2 (σss + σbb)(1 + ). (1) When = 0, the model follows the Lorentz mixing rule, and is traditionally called additive (not to be confused with pairwise additivity of potentials); that is, the cross-diameter is simply the sum of the two radii, exactly what one would expect on purely geometric grounds. If > 0 or < 0 the system shows positive or negative non-additivity respectively. As shown in figure 1, each big particle excludes a volume vb = πσ 3 bs/6 from the centres of the smaller particles. When the depletion layers of the two big particles (width defined as l = σbs − σbb/2 = 1 2 (σss + (σss + σbb))) begin to overlap, then the small particles can gain free volume v , leading to a depletion interaction. To calculate these potentials we first note that the depletion potential βVeff (r) depends only on the big–small and small–small interactions (βVbs(r) and βVss(r) respectively), but not on any direct big–big interaction βVbb(r), which can simply be added to the depletion potential [6, 14]. For non-additive systems at fixed ρs this means that the depletion potential is determined by σbs and σss, and is equivalent to an additive one with the same parameters! 4 We restrict ourselves here to mesoscopic binary colloidal mixtures where the so-called volume terms do not contribute to the phase behaviour [10]. If one were instead to integrate out the microscopic coand counter-ions, this might lead to volume terms which affect phase behaviour. Letter to the Editor L779