Penetrable hard spheres

We note that in the original paper of Asakura and Oosawa [54], where expression (1.21) was first derived, the polymers were regarded as pure hard spheres. Vrij [40, 56] arrived at the same result by describing the polymer chains as penetrable hard spheres, see Sect. 2.1. Inspection of (1.21) and (1.22) reveals that the range of the depletion attraction is determined by the size 2S of the... 

Theoretical work on depletion interactions and their effects on macroscopic properties such as phase stability commenced along various routes. First, Vrij [40] considered the depletion interaction between hard spheres due to dilute non-ad-sorbing polymers such as penetrable hard spheres (see Sect. 1.2.5 and Sect. 2.1). Vrij [40] referred to the work of Vester [82], Li-In-On et al. [55] and preliminary experiments at the Van t Hoff Laboratory on micro-emulsion droplets mixed with free polymer [40] for experimental evidence of depletion effects. 

Until the end of the 1990s most theoretical approaches were based on describing polymer chains as ideal or as penetrable hard spheres. Especially at the turn of the last century a wealth of different approaches were proposed to describe colloid-polymer mixtures in which interactions between polymer segments were accounted for. Essential was the progress made in Monte Carlo computer simulation studies on depletion effects [172-179] to test such theories. 

As dimensionless concentration variable (f> is used throughout. In case of hard colloidal particles the quantity volume fraction. For polymers and penetrable hard spheres (j> refers to the relative concentration with respect to overlap (see (1.24))... 

In this chapter we consider the depletion interaction between two flat plates and between two spherical colloidal particles for different depletants (polymers, small colloidal spheres, rods and plates). First of all we focus on the depletion interaction due to a somewhat hypothetical model depletant, the penetrable hard sphere (phs), to mimic a (ideal) polymer molecule. This model, implicitly introduced by Asakura and Oosawa [1] and considered in detail by Vrij [2], is characterized by the fact that the spheres freely overlap each other but act as hard spheres with diameter a when interacting with a wall or a colloidal particle. The thermodynamic properties of a system of hard spheres plus added penetrable hard spheres have been considered by Widom and Rowlinson [3] and provided much of the inspiration for the theory of phase behavior developed in Chap. 3. 

Fig. 2.1 Schematic picture of two parallel flat plates in the presence of penetrable hard spheres dashed circles)...

Since the penetrable hard spheres behave thermodynamically ideally the osmotic pressure outside the plates is given by the Van t Hoff law... 

On the other hand, when the plate separation is less than the diameter of the penetrable hard spheres, no partieles can enter the gap and... 

Fig. 2.3 The overlap volume (hatched area) of depletion layers due to penetrable hard spheres between two parallel flat plates equals A a — h)...

Fig. 2.4 Two hard spheres in the presence of penetrable hard spheres as depletants. The PHS impose an unbalanced pressure P between the hard spheres resulting in an attractive force between them. The overlap volume of depletion layers between the hard spheres (hatched) has the shape of a lens with width a - h and height 2H = 2Rjtim 9o, where 6o is given by cos 6o = r/2Rj...

This (attractive) force originates from an uncompensated (osmotic) pressure due to the depletion of penetrable hard spheres from the gap between the colloidal particles. This is depicted in Fig. 2.4 from which we immediately deduce that the uncompensated pressure acts on the surface between 9 = 0 and Oq = arc cos r/2Rd). 

With respect to the depletion interaction the Derjaguin approximation becomes accurate when considering a depletion agent which is small compared to the radius of the colloidal spheres. For example, applying the Derjaguin approximation to (2.3), the case of penetrable hard spheres, using (2.28)... 

For the case of the penetrable hard sphere as depletion agent this leads to... 

Comparing the expression (2.21) for penetrable hard spheres and (2.58) for ideal chains reveals that we match the contact potentials for... 

The result a = 2.35Rg agrees closely (within 5%) with the value flat plates. Hence in the limit R Rg ideal polymers behave almost as penetrable hard spheres with a diameter c 2Rg. just as for ideal chains between flat plates. In the next section we will see that this picture changes when R Rg. 

This brings us to the conclusion that as far as the depletion interaction is concerned ideal polymer chains to a good approximation can be replaced by penetrable hard spheres with a diameter a = 2, where the depletion thickness bs now depends on the size ratio q = Rg/R. In dilute polymer solutions the ideal chain description suffices to describe depletion effects. In Chap. 4 we shall see that for polymers with excluded volume the depletion thickness not only depends on the size ration q but also on the polymer concentration, see also [36, 39-41]. Also the (osmotic) pressure is no longer given by the ideal (Van t Hoff) expression. Both features significantly affect depletion effects. 

We now consider the depletion interaction due to (small) colloidal hard spheres with diameter interaction between the spheres so the system can considered to be thermodynamically ideal, the results for the depletion interaction are identical to those for penetrable hard spheres. At higher concentrations, say at volmne fractions larger than a few percent, the interactions between the spheres cannot be neglected. This has two important consequences for the depletion interaction. First of all the pressure and chemical potential are no longer given by the ideal expressions. The corrections to ideal behaviour can be written in terms of the virial series (see textbooks on statistical thermodynamics, e.g.. Hill [42] or Widom [43]) ... 

Substituting (2.91) in (2.92) after some algebra yields (2.84). Note that in all cases considered so far (penetrable hard spheres, polymers) the quantity [r(h) — r(oo)] was always positive (or zero) for all values for h. Here we see that due to accumulation effects in the concentration profiles [r(/i) — r(oo)] is negative for a certain range of h values. This leads to a positive interaction energy as is clear from (2.92). 

Such a repulsive contribution to the depletion interaction originates from excluded volume interactions between the depletants in case of ideal polymers and penetrable hard spheres it is absent One might expect accumulation effects also in the case of interacting polymers. From Monte Carlo simulation studies [51] and numerical self-consistent field computations [52, 53] it follows that interacting polymers do contribute to repulsive depletion interactions but with a strength of the repulsion that is nearly imperceptible. 

The effective pair interactions measured with these techniques are the direct pair interactions between two colloidal particles plus the interactions mediated by the depletants. In practice depletants are poly disperse, for which there are sometimes theoretical results available. For the interaction potential between hard spheres we quote references for the depletion interaction in the presence of polydisperse penetrable hard spheres [74], poly disperse ideal chains [75], poly-disperse hard spheres [76] and polydisperse thin rods [77]. 

In this chapter we discuss the basics of the phase behaviour of hard spheres plus depletants. Phase transitions are the result of physical properties of a collection of particles depending on many-body interactions. In Chap. 2 we focused on two-body interactions. As we shall see, depletion elfects are commonly not pair-wise additive. Therefore, the prediction of phase transitions of particles with depletion interaction is not straightforward. As a starting point a description is required for the thermodynamic properties of the pure colloidal dispersion. Here the colloid-atom analogy, recognized by Einstein and exploited by Perrin in his classical experiments, is very useful. Subsequently, we explain the basics of the free volume theory for the phase behaviour of colloids -I- depletants. In this chapter we treat only simplest type of depletant, the penetrable hard sphere. 

The hard sphere fluid-crystal transition plays an important role as a reference point in the development of theories for the liquid and solid states and their phase behaviour [10]. We consider it in some detail in the next section here the phase behaviour is relatively simple as there is no gas-liquid (GL) coexistence. After that we discuss the phase behaviour under the influence of the attraction caused by the depletion interaction now there is such GL transition. We illustrate the enrichment of the phase behaviour in the somewhat hypothetical system consisting of an assembly of hard spheres and (non-adsorbing) penetrable hard spheres. 

In the early nineties of the last century a theory that accounts for depletant partitioning over the coexisting phases was developed [27], which nowadays is commonly referred to as free volume theory (FVT) [28]. This theory is based on the osmotic equilibrium between a (hypothetical) depletant and the colloid + depletant system. The depletants were simplified as penetrable hard spheres. A pictorial representation is given in Fig. 3.6. 

Fig. 3.6 A system (right) that contains colloids and penetrable hard spheres (phs) in osmotic equilibrium with a reservoir (left) only consisting of phs. A hypothetical membrane that allows permeation of solvent and phs but not of colloids is indicated by the dashed line. Solvent is considered as background .

The key step now is the calculation of the number of depletants in the colloid -I- depletant system as a function of the chemical potential imposed by the depletants in the reservoir. In the calculations presented below we model the colloidal particles as hard spheres with diameter 2R and the depletants by penetrable hard spheres with diameter a. 

In Fig. 3.9 we present a comparison of the free volume fraction predicted by SPT (3.36) and computer simulations [32] on hard spheres plus penetrable hard spheres for q = 0.5 as a function of . As can be seen the agreement is very good. In the limit of small depletants the 2 and ) terms of (3.30) can be omitted giving ... 

Fig. 3.9 Free volume fraction for penetrable hard spheres in a hard sphere dispersion for q = a/2R = 0.5 as function of the hard sphere concentration. Data points are redrawn from Meijer [32]. Curve is the SPT prediction (3.36)...

For non-interacting depletants such as penetrable hard spheres the f/s and P s in the phase coexistence equations (3.41) and (3.42) can be written such that binodal colloid concentrations follow from solving one equation in a single unknown [28]. We rewrite (3.38) and (3.39) as... 

In this chapter we have presented the free volume theory for hard spheres plus depletants and focused on the simplest possible case of hard spheres + penetrable hard spheres. In the next chapters we will extend the free volume theory to more realistic situations (Chap. 4 hard spheres + polymers. Chap. 5 hard spheres -I- small colloidal particles. Chap. 6 hard rods -I- polymers) and compare the results with experiments and simulations. 

In Chap. 3 we introduced the phase behaviour of hard spheres mixed with penetrable hard spheres (phs). This provides a starting point for describing the phase behaviour of colloid-polymer mixtures. In Sect. 4.1 we show that the phs-description using penetrable hard spheres is adequate for mixtures in the colloid-limit small q with polymer chains smaller than the particle radius. In Sect. 4.2 we treat the modifications for the case that the polymers are treated as ideal chains. More advanced treatments accounting for non-ideal behaviour of depletion thickness and osmotic pressure for interacting polymer chains enable to also describe intermediate and large q situations. This is the topic of Sect. 4.3. In Sect. 4.4 we qualitatively consider work available on the effects of polydispersity on... 

In Fig. 4.3 we also plot the (equilibrium) binodals using FVT outlined in Chap. 3 for hard spheres plus penetrable hard spheres with diameters of 2Rg. Qualitatively, the phase diagram topology is quite well predicted. For q = 0.08, only equilibrium fluid, crystal and fluid + crystal regions are found and predicted. Both for q = 0.57 and 1 the phase diagram contains fluid, gas, liquid and crystalline (equilibrium) phases. In the different unmixing regions one now finds gas-liquid coexistence with a critical point, three-phase gas-liquid-crystal and... 

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