Hard particle theories

Reiss H 1977 Scaled particle theory of hard sphere fluids Statistical Mechanics and Statistical Methods in Theory and Application ed U Landman (New York Plenum) pp 99-140... 

Reiss H and Hammerich ADS 1986 Hard spheres scaled particle theory and exact relations on the existence and structure of the fluid/solid phase transition J. Phys. Chem. 90 6252... 

Alternative integral equations for the cavity functions of hard spheres can be derived [61,62] using geometrical and physical arguments. Theories and results for hard sphere systems based on geometric approaches include the scaled particle theory [63,64], and related theories [65,66], and approaches based on zero-separation theorems [67,68]. These geometric theories have been reviewed by Stell [69]. 

Scaled Particle Theory for Wormlike Hard Spherocylinders.93... 

In the present article, we focus on the scaled particle theory as the theoretical basis for interpreting the static solution properties of liquid-crystalline polymers. It is a statistical mechanical theory originally proposed to formulate the equation of state of hard sphere fluids [11], and has been applied to obtain approximate analytical expressions for the thermodynamic quantities of solutions of hard (sphero)cylinders [12-16] or wormlike hard spherocylinders [17, 18]. Its superiority to the Onsager theory lies in that it takes higher virial terms into account, and it is distinctive from the Flory theory in that it uses no artificial lattice model. We survey this theory for wormlike hard spherocylinders in Sect. 2, and compare its predictions with typical data of various static solution properties of liquid-crystalline polymers in Sects. 3-5. As is well known, the wormlike chain (or wormlike cylinder) is a simple yet adequate model for describing dilute solution properties of stiff or semiflexible polymers. 

Fig. 7. Comparison of experimental phase boundary concentrations between the isotropic and biphasic regions for various liquid-crystalline polymer solutions with the scaled particle theory for wormlike hard spherocylinders. ( ) schizophyllan water [65] (A) poly y-benzyl L-glutamate) (PBLG)-dimethylformamide (DMF) [66-69] (A) PBLG-m-cresoI [70] ( ) PBLG-dioxane [71] (O) PBLG-methylene chloride [71] (o) po y(n-hexyl isocyanate) (PHICH°Iuene at 10,25,30,40 °C [64] (O) PHIC-dichloromethane (DCM) at 20 °C [64] (5) a po y(yne)-platinum polymer (PYPt)-tuchIoroethane (TCE) [33] ( ) (hydroxypropyl)-cellulose (HPC)-water [34] ( ) HPC-dimethylacetamide (DMAc) [34] (N) (acetoxypropyl) cellulose (APC)-dibutylphthalate (DBP) [35] ( ) cellulose triacetate (CTA)-trifluoroacetic acid [72]...

Fig. 8. Comparison of experimental phase boundary concentrations between the biphasic and liquid crystal regions for various liquid crystalline polymer solutions with the scaled particle theory for hard wormlike spherocylinders. The symbols are the same as those in Fig. 7...

Fig. 10. Theoretical ternary phase diagram calculated from the scaled particle theory for worm-like hard spherocylinders with (Ni, N2) = (0.930,0.070), d = 1.52nm, q = 200nm, and ML = 2150 nm-1 [17]...

In concluding this section, we should touch upon phase boundary concentration data for poly(p-benzamide) dimethylacetamide + 4% LiCl [89], poly(p-phenylene terephthalamide) (PPTA Kevlar)-sulfuric acid [90], and (hydroxy-propyl)cellulose-dichloroacetic acid solutions [91]. Although not included in Figs. 7 and 8, they show appreciable downward deviations from the prediction by the scaled particle theory for the wormlike hard spherocylinder. Arpin and Strazielle [30] found a negative concentration dependence of the reduced viscosity for PPTA in dilute Solution of sulfuric acid, as often reported on polyelectrolyte systems. Therefore, the deviation of the Ci data for PPTA in sulfuric acid from the scaled particle theory may be attributed to the electrostatic interaction. For the other two systems too, the low C] values may be due to the protonation of the polymer, because the solvents of these systems are very polar. 

Recognizing that the extensive art in rubber toughening teaches that only adherent rubbers toughen (nonadherent rubber particles act like voids), we nevertheless tested the above conclusions of the dilatation theory by attempting to toughen styrene acrylonitrile copolymer (S/AN) with dispersions of voids and with dispersions of hard particles. 

Further, the Goodier equations predict that hard particles and voids produce higher stress concentrations (i.e., stronger craze nucleation) than rubbers, and thus hard particles and voids should toughen even better than rubbers if nucleation were the operative mechanism. This is not observed experimentally. The nucleation theory is thus seen to have substantial drawbacks. 

The molecular size of a solvent can be characterized in several ways. One of them is to assign the solvent a molecular diameter, as if its molecules were spherical. From a different aspect, this diameter characterizes the cavity occupied by a solvent molecule in the liquid solvent. From a still further aspect, this is the mean distance between the centers of mass of two adjacent molecules in the liquid. The diameter plays a role in many theories pertaining to the liquid state, not least to those treating solvent molecules as hard spheres, such as the scaled particle theory (SPT, see below). Similar quantities are the collision diameters a of gaseous molecules of the solvent, or the distance characterizing the minimum in the potential energy curve for the interaction of two solvent molecules. The latter quantity may be described, e.g., according to the Lennard-Jones potential (Marcus 1977)... 

B. A. Cosgrove and J. Walkley, Can.]. Chem., 60,1896 (1982). Scaled Particle Theory of Gas Solubility and Inclusion of the Temperature Dependent Hard Sphere Term. 

The case of finite magnetic rigidity of the particles is addressed in Section V the theory becomes much more complicated and, hence, cumbersome in form. Meanwhile, the case of magnetically hard particles suspended in a fluid is formally very close to the case of isotropic magnetic particles in a solid matrix. That is why we present it here. 

The only approximate analytical solution for the RSA of a binary mixture of hard disks was proposed by Talbot and Schaaf [27], Their theory is exact in the limit of vanishing small disks radius rs — 0, but fails when the ratio y = r Jrs of the two kinds of disk radii is less than 3.3 its accuracy for intermediate values is not known. Later, Talbot et al. [28] observed that an approximate expression for the available area derived from the equilibrium Scaled Panicle Theory (SPT) [19] provided a reasonable approximation for the available area for a non-equilibrium RSA model, up to the vicinity of the jamming coverage. While this expression can be used to calculate accurately the initial kinetics of adsorption, it invariably predicts that the abundant particles will be adsorbed on the surface until 6=1, because the Scaled Particle Theory cannot predict jamming. 

The binary diffusion coefficient of liquid extract in supercritical C02 is calculated with correlations based on the rough-hard-sphere-theory [7], Within the particle structure diffusion is determined by various effects. First, the diffusion can occur only in the void fraction of the particle. Secondly, the diffusion path is given by the contorsion of the pores. 

This section deals with a general theory of electrophoresis of soft particles and approximate analytic expressions for the mobility of soft particles [30-51]. This theory unites the electrophoresis theories of hard particles [1-29] and of polyelec-trolytes [52], since a soft particle tends to a hard particle in the absence of the polymer layer and to a polyelectrolyte in the absence of the particle. 

Electrokinetic equations describing the electrical conductivity of a suspension of colloidal particles are the same as those for the electrophoretic mobility of colloidal particles and thus conductivity measurements can provide us with essentially the same information as that from electrophoretic mobihty measurements. Several theoretical studies have been made on dilute suspensions of hard particles [1-3], mercury drops [4], and spherical polyelectrolytes (charged porous spheres) [5], and on concentrated suspensions of hard spherical particles [6] and mercury drops [7] on the basis of Kuwabara s cell model [8], which was originally applied to electrophoresis problem [9,10]. In this chapter, we develop a theory of conductivity of a concentrated suspension of soft particles [11]. The results cover those for the dilute case in the limit of very low particle volume fractions. We confine ourselves to the case where the overlapping of the electrical double layers of adjacent particles is negligible. 

The essence of the scaled particle theory is that formation of a cavity in a fluid requires work. The theory for hard spheres has been well developed from statistical mechanics, and the work, W(R, p), can be calculated as follows ... 

Scaled particle theory has not yet been discussed. Equation 13.4.39] is taken from Baumer and Findenegg but originally dates back to Helfand et al. The equation is rigorous for hard disk-like molecules it is combined with a mean field lateral Lennard-Jones pair interaction. In this equation their = na lA if a is the diameter of the disk, is the depth of the Lennard-Jones pair interactions (i.e. the minimum in fig. 1.4.1.a). In this case 6 = a F its maximum in a close-packed monolayer corresponds to (max) = 0.906. The accent emphasizes this different scaling. Baumer and Findenegg applied [3.4.39] to dilute monolayers of 1-chlorobutane, perfluorohexane and fluorobenzene, adsorbed on water from the gas phase. 

The hard part. The first term in (7.249), AG H, corresponds to turning on the hard part of the protein-solvent interaction potential. This is the same as the free energy of creating a cavity of suitable size at some fixed position in the solvent. We assume that the globular protein is spherical with an effective diameter dp, so we can calculate AG H using the scaled particle theory (see section and Appendix N). If we choose dw = 2.8 A as the diameter of a water molecule, then the cavity suitable to accommodate the protein has a radius of... 

The scaled particle theory (SPT) was developed in the late 1950s and the early 1960s. It started with the quest for the probability of creating a cavity, or a hole, in the liquid (Hill 1958). It was developed as a theory that was initially designed for hard spheres, and then applied for more realistic fluids and mixtures (Reiss et al. 1959, 1960, 1966 Helfand et al. 1960). 

---