Phase behavior of colloidal suspensions with critical solvents in terms of effective interactions T. F. Mohry,1,2, a) A. Macio lek,1,2,3, b) and S. Dietrich1,2, c) 1)Max-Planck-Institut f¨ur Intelligente Systeme, Heisenbergstraße 3, 70569 Stuttgart, Germany 2)Universit¨at Stuttgart, Institut f¨ur Theoretische und Angewandte Physik, Pfaffenwaldring 57, 70569 Stuttgart, Germany 3)Institute of Physical Chemistry, Polish Academy of Sciences, Kasprzaka 44/52, PL-01-224 Warsaw, Poland (Dated: 15 July 2018) We study the phase behavior of colloidal suspensions the solvents of which are considered to be binary liquid mixtures undergoing phase segregation. We focus on the thermodynamic region close to the critical point of the accompanying miscibility gap. There, due to the colloidal particles acting as cavities in the critical medium, the spatial confinements of the critical fluctuations of the corresponding order parameter result in the effective, so-called critical Casimir forces between the colloids. Employing an approach in terms of effective, one-component colloidal systems, we explore the possibility of phase coexistence between two phases of colloidal suspensions, one being rich and the other being poor in colloidal particles. The reliability of this effective approach is discussed. PACS numbers: 61.20.Gy, 64.60.fd, 64.70.pv,64.75.Xc, 82.70.Dd I. INTRODUCTION tional to the correlation length ξ which diverges upon approaching the critical point of the unconfined solvent (s) In colloidal suspensions the dissolved particles, typi- (without colloids) at the critical temperature Tc . Ex- cally micrometer-sized, interact via effective interactions posing the near-critical fluid to boundaries, e.g., by in- which are a combination of direct interactions, such as serting colloidal particles acting as cavities, perturbs the electrostatic or van der Waals interactions, and indirect, fluctuating order parameter near the surfaces of the col- effective ones due to the presence of other, smaller so- loids on the scale of ξ and restricts the spectrum of its lute particles or mediated by the solvent.1 For example, thermal fluctuations. Since these restrictions depend on by adding non-adsorbing polymers to the solution deple- the spatial configuration of the colloids, they result6 in tion interactions between colloidal particles can be in- an effective force between the particles which is called 2 duced which are predominantly attractive. The range the critical Casimir force (CCF) fC . Accordingly the and the strength of this entropy-driven attraction can range of the CCF is proportional to the bulk correla- be manipulated by varying the polymer-colloid size ratio tion length ξ. Therefore it can be tuned continuously by or the polymer concentration. The resulting phase dia- small changes of the temperature T . The range can also grams of the colloids are very sensitive to changes in the be controlled by varying the conjugate ordering field hb depletion-induced pair potential between colloids.1,3 It is of the order parameter of the solvent such as the chemi- also possible to induce such attractive depletion forces cal potential in the case of a simple fluid or the chemical by adding much smaller colloidal particles or by using potential difference of the two species forming a binary solvents with surfactants which form micelles acting as liquid mixture. The strength and the sign of fC can be depletion agents.4 manipulated as well. This can be achieved by varying the In contrast to depletion forces, solvent mediated in- temperature T or hb and by suitable surface treatments, teractions between colloidal particles can depend sensi- respectively.7–9 Compared with other effective forces be- arXiv:1201.5540v2 [cond-mat.soft] 27 Jan 2012 tively on the thermodynamic state of the solvent.5 This is tween colloid particles CCFs offer two distinct features. the case if the solvent exhibits fluctuations on large spa- First, due to the universality of critical phenomena, to tial scales such as the fluctuations near the surfaces of a large extent CCFs do not depend on the microscopic the colloidal particles associated with wetting phenom- details of the system. Second, whereas adding depletion ena near a first-order phase transition of the solvent or agents or ions changes effective forces irreversibly, the the thermal fluctuations of the solvent order parameter tuning of fC via T is fully and easily reversible. near a second-order phase transition. In the latter case, Although the CCFs between two colloidal particles de- critical fluctuations are correlated over distances propor- pend on the (instantaneous) spatial configuration of all colloids, one can consider dilute suspensions or tempera- (s) tures sufficiently far away from Tc , such that the range a)Electronic mail: [email protected] 2R+ξ of the critical Casimir interaction between the col- b)Electronic mail: [email protected] loids of radius R is much smaller than the mean distance c)Electronic mail: [email protected] between them. For these cases the assumption of pairwise 2 additive CCFs is expected to be valid. Using this pair- tal data7–9,28,29 are available. Based on the Derjaguin wise approximation we are able to map the actual system approximation30 the knowledge of the scaling function of of a mixture of (monodisperse) colloidal particles and sol- the CCF acting between two parallel surfaces can be used vent molecules to an effective, one-component system of to obtain the scaling function for fC between two spheres colloidal particles, in which the presence and influence and between a sphere and a planar wall.17,18 The Der- of the other components of the mixture enter through jaguin approximation assumes that the surface-to-surface the parameters of the effective pair potential. Adopt- distance D between two spheres is much smaller than ing this approach allows us to use standard liquid state their radius R. In many cases, however, this approxi- theory in order to determine the structure of an ensem- mation works surprisingly well8,31 even for D . R. We ble of colloidal particles immersed in a near-critical sol- shall use this approximation for temperatures which cor- vent and to study its sensitivity to changes of the critical respond to ξ . R, because under this condition the CCFs Casimir potential due to temperature variations. Since between the colloids act only at surface-to-surface dis- within this effective approach the feedback of the colloids tances D small compared with R. In order to handle on the solvent and its critical behavior is neglected, this the dependence of the CCF on hb within the Derjaguin way the phase behavior of the full many-component sys- approximation, we propose a suitable approximation for tem cannot be determined in all details. However, for the film scaling function of the CCF in d = 3. A neces- those parameters of the thermodynamic phase space for sary input for this latter approximation is the mean-field which this approach is applicable, we predict a colloidal scaling function for the CCF, which we have calculated “liquid”-“gas” phase coexistence, i.e., the coexistence of using the field-theoretical approach. two phases which differ with respect to their colloidal Experimental studies of the phase behavior of colloidal number densities. suspensions with phase separating solvents have encom- passed silica spheres immersed in water-2-butoxyethanol The necessary input for the approach employed in (C E ) mixtures and in water-lutidine mixtures.32,33 In the present study is the CCF f between two spheri- 4 1 C Ref. 34 both silica and polystyrene particles immersed in cal particles. It is known that at the bulk critical point these binary mixtures have been studied focusing on the f is long-ranged. In the so-called protein limit corre- C formation of colloidal crystals and its relation to aggrega- sponding to D/R, ξ/D ≫ 1, where R is the colloid ra- tion phenomena. A few theoretical35 and simulation36,37 dius and D the surface-to-surface distance between the attempts have been concerned with such kinds of col- colloids, the so-called small-sphere expansion10 renders loidal suspensions. (s) (s) d−2+η −(d−1+η) fC(D; Tc , R)/ kBTc ∼ R D , where d Our paper is organized such that in Sect. II we summa- is the spatial dimension and η is the standard bulk crit- rize the theoretical background of our analysis. In Sub- ical exponent for the two-point correlation function. In sect. II A we discuss colloidal suspensions and the effec- the opposite, so-called Derjaguin limit D ≪ R one has10 tive model we use, whereas Subsect. IIB provides infor- (s) (s) (d−1)/2 −(d+1)/2 mation concerning critical phenomena and the CCF. For fC(D; Tc , R)/ kBTc ∼ R D . Thus the parameters entering into the effective potential be- the CCF can indeed successfully compete with direct tween the colloids, in Subsect. IIIA we discuss the range dispersion11 or electrostatic forces in determining the sta- of their values corresponding to possible experimental re- bility and phase behavior of colloidal systems. Away alizations. In Subsect. IIIB the thermodynamics of the from the critical point, according to finite-size scaling considered colloidal suspensions is analyzed. A general theory (see, e.g., Ref. 12), the CCF exhibits scaling de- discussion of the phase diagram of the actual ternary scribed by a universal scaling function which is deter- mixtures consisting of the colloids and the binary solvent mined solely by the so-called universality class (UC) of is given in Subsects. IIIB1 and IIIB2. The reliability the phase transition occurring in the bulk, the geome- and the limitations of the effective, one-component ap- try, and the surface universality classes of the confin- proach are discussed in Subsect.