Hard colloidal rods

So far we have considered the phase behaviour of colloidal spheres plus deple-tants. In Chap. 3 we considered the simplest type of depletant, the penetrable hard sphere. We then extended this treatment in Chap. 4 to ideal and excluded volume polymers and in Chap. 5 we considered small colloidal spheres (ineluding mieelles) and colloidal rods as depletants. In this chapter we consider the phase behaviour of mixtures of colloidal rods plus polymeric depletants. For an overview of several types of colloidal rods encountered in practice we refer to [1]. 

One model for rod-like colloids is tire tobacco mosaic vims (TM V), which consists of rods of diameter D about 18 nm and lengtli L of 300 nm [17,18]. These colloids have tire advantage of being quite monodisperse, but are hard to obtain in large amounts. The fd vims gives longer, semi-flexible rods (L = 880 nm, D = 9 nm) [18,19]. Inorganic boehmite rods have also been prepared successfully [20]. 

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. 

Thin colloidal disks provide another example of an anisometrie eoUoidal particle as an efficient depletion agent. This problem was first eonsidered by Pieeh and Walz [61]. At the end of this seetion, where we compare spheres, rods and disks as depletion agents, we will see that the disk is intermediate in effieieney to induce depletion attraction between spheres and rods. Here we consider disks of diameter D and thickness L, see Rg. 2.30. Notiee that for the simplest ease, i.e., infinitely thin hard disks, the exeluded volume of the disks with respect to each other is nonzero and only in limit of the concentration going to zero will the disks behave thermodynamically ideal. We restrict ourselves to this limiting case. 

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]. 

The concept potential of mean force was used by Onsager [3] in his theory for the isotropic-nematic phase transition in suspensions of rod-like particles. Since the 1980s the field of phase transitions in colloidal suspensions has shown a tremendous development. The fact that the potential of mean force can be varied both in range and depth has given rise to new and fascinating phase behaviour in colloidal suspensions [4]. In particular, stcricaUy stabilized colloidal spheres with interactions close to those between hard spheres [5] have received ample attention. 

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. 

Fig. 5.14 Colloidal gas-liquid coexistence of mixtures of rods plus hard spheres for L/

The N-N transition is predicted to occur at quite high volume fractions of rods. At these high volume fractions the N-N transition may be superseded by more highly ordered (liquid) crystal phases such as the colloidal smectic phase. Experimentally, this colloidal smectic phase has been observed [28, 44, 45] in suspensions of monodisperse rods. Simulations confirmed that hard rods can form a thermodynamically stable smectic phase [2-A]. 

Here we outline how these more highly ordered phases can be accounted for in the phase diagram of mixtures of rod-like colloids and flexible polymers using FVT and follow the work of Bolhuis et al. [46]. The FVT requires the pressure, the chemical potential of the hard spherocylinder (HSC) reference system, and the free volume fraction (cf. (6.40) and (6.41)) as input. The computer simulations presented in [2, 4] contain the necessary information on the pressure and chemical potential of the HSC reference system and in [46] the free volume fraction was obtained using the Widom insertion method [47]. In this method one attempts to insert the polymers (represented by phs with diameter cr) at random positions in the simulation box. The fraction of insertions that does not result in an overlap corresponds to the free volume fraction. The free volume fraction measured in this way at different volume fractions of the HSC was fitted to a functional form similar to the SPT expression for the free volume fraction and used in (6.40) and (6.41). In Fig. 6.21 we present the simulation results for L/D = 5 and q= 1.0, q = 0.65 and = 0.15 obtained in [46] using the method outline above. In the upper graph of Fig. 6.21 q = 1) we compare the results for the Ii—12 transition... 

The possibility of entropy-driven phase separation in purely hard-core fluids has been of considerable recent interest experimentally, theoretically, and via computer simulations. Systems studied include binary mixtures of spheres (or colloids) of different diameters, mixtures of large colloidal spheres and flexible polymers, mixtures of colloidal spheres and rods," and a polymer/small molecule solvent mixture under infinite dilution conditions (here an athermal conformational coil-to-globule transition can occur)." For the latter three problems, PRISM theory could be applied, but to the best of our knowledge has not. The first problem is an old one solved analytically using PY integral equation theory by Lebowitz and Rowlinson." No liquid-liquid phase separation... 

So far I have discussed the occurrence of nematic and smectic phases in dispersions of rod-like colloidal particles. Stroobants et dlM observed in their Monte Carlo simulations of systems of hard parallel spherocylinders for L/D > 3 a columnar phase intermediate between the smectic and crystalline phases. In their freeze-fracture electron microscopy study of dispersions of the rod like bacterial virus fd, Booy and Fowler may have observed the columnar phase. They report one texture (Figure 2b, in Ref. 69), where the nearly parallel fd particles are arranged randomly in the longitudinal direction but the lateral packing is regular. This is indeed characteristic of a columnar phase. 

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