Systems scaled particle theory

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]. 

The above formulation for a monodisperse polymer system by the scaled particle theory can be readily extended to a polydisperse polymer system, as described in [17]. The result is... 

Now we compare the above osmotic pressure data with the scaled particle theory. The relevant equation is Eq. (27) for polydisperse polymers. In the isotropic state, it can be shown that Eq. (27) takes the same form as Eq. (20) for the monodisperse system though the parameters (B, C, v, and c ) have to be calculated from the number-average molecular weight M and the total polymer mass concentration c of a polydisperse system pSI in the parameters B and C is unity in the isotropic state. No information is needed for the molecular weight distribution of the sample. On the other hand, in the liquid crystal state2, Eq. (27) does not necessarily take the same form as Eq. (20), because p5I depends on the molecular weight distribution. 

Now we compare the isotropic-liquid crystal phase boundary concentrations for various polymer solution systems with the scaled particle theory for the wormlike spherocylinder. If the equilibrium orientational distribution function f(a) in the coexisting liquid crystal phase is approximated by the Onsager trial... 

The theoretical IA binodals successfully reproduce the experimental bi-nodals for both systems. Furthermore, the theoretical tie lines correctly predict the fractionation effect found by experiment. Thus, the scaled particle theory predicts the IA binodals and tie lines more accurately than the Abe Flory theory. The success owes much to incorporating chain flexibility into the theory. 

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. 

Fig. 12a-c. Polymer concentration dependence of the orientational order parameters S for three liquid-crystalline polymer systems a PBLG-DMF [92,93] b PHIC-toluene [94] c PYPt-TCE [33], Marks experimental data solid curves, theoretical values calculated from the scaled particle theory. The left end of each curve gives the phase boundary concentration cA... 

Finally the minimum observed for the AG°t function of n-Bu4NBr in both chemical systems may be interpreted as follows the variation of AG°t(+) and AG°t(-) with solvent composition are generally not linear, so the minimum can be looked upon as the consequence of the opposite behavior of anion and cation towards the solvent molecules. However this interpretation does not take into account the fact that at low organic solvent mole fraction one has AH°t < T AS°t. This observation means that a structural (nonspecific) effect predominates in the water-rich region. This effect might be related to the cavity effect as can be evaluated from Pierrotti s scaled particle theory because the large size of the... 

The main method used in PCM for Gcav is based on Pierotti s elaboration (Pierotti 1963 1965 1976) of the Scaled Particle Theory which was introduced by Reiss et al. (1959, 1960). This model which belongs to the family of physical descriptions of liquid systems defines Gcav as an expansion in powers of Rms, i.e. the radius of a sphere excluding the centers of solvent molecules ... 

The scaled particle theory was extended to mixture of hard spheres by Lebowitz et al. (1965). In a one-component system of hard spheres of diameter a, placing a hard particle of radius RHs produces a cavity of radius r= RHs + a/2. When there is a mixture of hard spheres of diameters a the radius of a cavity produced by a hard sphere of radius RHs depends on the species i, i.e., Y = Rhs + ai/2. 

The behavior of one-dimensional mixtures of hard spheres of different diameters a- and follows directly from the exact validity of the quasi-chemical approximation for the one-dimensional combinatorial factor for particles interacting only with their nearest neighbois. Unfortunately, this result is valid only for one-dimensional systems. It would be of considerable interest to extend the scaled particle theory to deal with mixtures of hard spheres. 

Specializing to planar walls for the moment, one has the exact relation [45] that p/kT = where is the local density of the adsorbate in contact with the wall when the pressure of the hard sphere fluid is p. For hard sphere mixtures [46-49] n . is the sum of the individual densities for each of the components in the fluid. Thus, the pressure of the fluid can be obtained from estimates of the intercept of the curve of n z) versus z for example. Fig. 1 indicates that palg/kT is between 8 and 9 for this system. This result, taken together with the calculation of F at a given n from Eq. (10), allows one to construct the isotherm TO). Figure 2 shows the adsorption of a hard sphere fluid on a hard wall as a function of the bulk-phase density [44]. The simulation points compare well with results of two theoretical calculations based on the scaled particle theory. 

FIG. 2 Adsorption (molecules adsorbed per unit area) for the hard sphere-hard wall system plotted in reduced units as a function of the bulk density r) = (7r/6)na j,. The points show the simulations, and the lines show theoretical calculations based on the scaled particle theory. (From Ref. 44.)... 

FIG. 4 The gas-solid surface tension in reduced units ( 3 = /kT) for the hard sphere-hard wall system plotted as a function of the reduced density p = (rt/6) a g. The points are simulated, and the two curves are calculated from different scaled particle theory expressions, (From Ref 44.)... 

An expression for the work of insertion W can be obtained from scaled particle theory (SPT) [31]. SPT was developed to derive expressions for the chemical potential and pressure of hard sphere fluids by relating them to the reversible work needed to insert an additional particle in the system. This work W is calculated is by expanding (scaling) the size of the sphere to be inserted from zero to its final size the size of the scaled particle is Act, with X running from 0 to 1. In the limit 2 0, the inserted sphere approaches a point particle. In this limiting case it is very unlikely that the depletion layers overlap. The free volume fraction in this limit can therefore be written as... 

FIGURE 2.6 Comparison between the observed distribntion constants of halobenzenes and those predicted from the scaled particle theory (solid line) in 1-octanol/water system. 

FIGURE 2.8 Distribution eonstants of metal (11, 111, or fV)-acetylacetone eomplexes in a benzene/water system. The solid line denoted SPT is the predicted Kp from the scaled particle theory. 

Other potentially useful transfer functions are the transfer heat capacity, SCI, transfer volume, and the transfer internal energy, SUl (Abraham, 1974). At present, measurements of these transfer functions are available for only a few systems, but they can in certain instances be predicted using scaled-particle theory (Desrosiers and Desnoyers, 1976). 

This chapter will not deal with theories of liquids per se. Instead we shall present only general relations between thermodynamic quantities and molecular distribution functions. The latter are fundamental concepts which play a central role in the modern theoretical treatment of liquids and solutions. Acquiring familiarity with these concepts should be useful in the study of more complex systems such as aqueous solutions, treated in Chapters 7 and 8. As an exception, a brief outline of the scaled particle theory is presented in section 5.11. This theory, although originally aimed at studying hard-sphere systems, has been used in systems as complex as aqueous protein solutions. The main result that will concern us is the work required to create a cavity in a fluid. This quantity is fundamental in the study of solvation phenomena of simple solutes, as well as very complex ones such as proteins or nucleic acids. 

Brusatori and Van Tassel [20] presented a kinetic model of protein adsorption/surface-induced transition kinetics evaluated by the scale particle theory (SPT). Assuming that proteins (or, more generally, particles ) on the surface are at all times in an equilibrium distribution, they could express the probability functions that an incoming protein finds a space available for adsorption to the surface and an adsorbed protein has sufficient space to spread in terms of the reversible work required to create cavities in a binary system of reversibly and irreversibly adsorbed states. They foimd that the scale particle theory compared well with the computer simulation in the limit of a lower spreading rate (i.e., smaller surface-induced unfolding rate constant) and a relatively faster rate of surface filling. 

Cotter is based on the scaled particle theory. The alternative way is to use the so-called y-expansion of the hard-core free energy proposed by Gelbart and Barboy [44]. This is an expansion in powers of 7] /(1 - rj) which is much more reliable at high densities compared with the usual virial expansion in powers of rj. The y-expansion has also been used by Mulder and Frenkel [51 ] in the interpretation of the results of computer simulations for a system of hard ellipsoids. 

The molecular thermodynamic theory for micelle formation has heen worked out with increasing sophistication following the pioneering work of Israelachvili, Mitchell, and Ninham.26 The most comprehensive reports on micelle formation are those of Nagarajan and Ruckenstein and of Shiloah and Blankschtein. Many other theoretical approaches have been used in recent years to account for the formation of micelles and their properties thermod3mamics of small systems, the self-consistent field lattice model, the scaled particle theory, and Monte-Carlo and molecular dynamics MD simulations. MC and DC simulations are presently much in favor due to the increased availability of fast computers. A prediction common to all these theories is that micelles represent a thermodynamically stable state and that micellar solutions are single-phase systems. Several recent results of MD and (MC) simulations are in agreement with experimental results. ... 

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