Scaled particle models

Figure 8.13 The function G (A) for water at 300 K at the liquid saturation conditions (Ashbaugh and Pratt, 2004). The points are obtained by direct Monte Carlo calculation, and the solid line by matching an empirical thermodynamic model for the large solute case. The dashed lines are the classic scaled-particle model (Pierotti, 1976) predictions for several solvent hard-sphere diameter parameters between cr = 2.6 A and 3.0 A in 0.1 A increments. Notice that the parameter value that provides the best fit of the classic scaled-particle model for small radii is not the same as that for the large radii results.

Pt = —(l/V )(9V /9p)r the isothermal compressibility ITM = information theory model PC = Pratt-Chandler theory RSPM = revised scaled particle model SPM = scaled particle model. 

The scaled particle model (SPM) was the first essentially molecular theory of hydrophobicities (see Scaled Particle Theory), It derived from an earlier scaled particle theory, a successful theoretical calculation of the thermodynamic properties of the hard sphere liquid. Pierotti then adapted the scaled particle theory to produce a solubility model for realistic liquids by a natural replacement of the hard sphere pressure with the measured pressure of the real solvent of interest. With attractive solute-solvent interactions treated perturba-tively this scaled particle model was remarkably successful. The SPM is the molecular theory of hydrophobicities most widely considered among biomolecular modelers. However, its success is somewhat fortuitous.For example, though the SPM predicts a reasonable value for the surface tension for the water liquid-vapor interface at room temperature, the predicted temperature dependence is wrong. Since entropies and temperature dependencies are special goals of theories of hydrophobic effects, this incorrect temperature dependence is important. 

Stillinger analyzed the SPM and proposed a revised scale particle model (RSPM) that incorporates the measured surface tension in addition to the measured pressure, in this way correcting the most direct error of the SPM. The RSPM deserves note because it is more accurate for primitive hydrophobic effects than the SPM but not more difficult to use. 

Equilibrium theories of classical fluids are often formulated as integral equations for the distribution functions, e.g., AA(r) giving the probability distribution of AA pairs at a specified separation in water. The Pratt-Chandler (PC) theory was the first theory of hydrophobic effects largely built on that conventional basis. A less conventional feature was that the PC theory avoided calculating separately the structure of water in the absence of hydrophobic solutes, much in the spirit of the scaled particle models. Instead, the measured goo(r) was used as input to the theoretical calculation of gAA(r). The idea was to use what is known about pure water to make predictions about hydrophobic effects. The ITM makes this heuristic point of view explicit. Moreover, the success of the simplest two-moment ITM provides important support for... 

Stillinger F 1973 Structure in aqueous solutions from the standpoint of scaled particle theory J. Solution Chem. 2 141 Widom B 1967 Intermolecular forces and the nature of the liquid state Sc/e/ ce 375 157 Longuet-Higgins H C and Widom B 1964 A rigid sphere model for the melting of argon Mol. Phys. 8 549... 

Tlic cavity and van der Waals contributions may also be modelled as separate terms. In som implementations an estimate of the cavity term may be obtained using scaled particle theor [Eierotti 1965 Claverie et al. 1978], which uses an equation of the form ... 

Fitzgerald et al. (1984) measured pressure fluctuations in an atmospheric fluidized bed combustor and a quarter-scale cold model. The full set of scaling parameters was matched between the beds. The autocorrelation function of the pressure fluctuations was similar for the two beds but not within the 95% confidence levels they had anticipated. The amplitude of the autocorrelation function for the hot combustor was significantly lower than that for the cold model. Also, the experimentally determined time-scaling factor differed from the theoretical value by 24%. They suggested that the differences could be due to electrostatic effects. Particle sphericity and size distribution were not discussed failure to match these could also have influenced the hydrodynamic similarity of the two beds. Bed pressure fluctuations were measured using a single pressure point which, as discussed previously, may not accurately represent the local hydrodynamics within the bed. Similar results were... 

The methods used for modeling pure granular flow are essentially borrowed from that of a molecular gas. Similarly, there are two main types of models the continuous (Eulerian) models (Dufty, 2000) and discrete particle (Lagrangian) models (Herrmann and Luding, 1998 Luding, 1998 Walton, 2004). The continuum models are developed for large-scale simulations, where the controlling equations resemble the Navier-Stokes equations for an ordinary gas flow. The discrete particle models (DPMs) are typically used in small-scale simulations or... 

Finally we address the issue of contributions. In our view it is unbalanced to concentrate on a converged treatment of electrostatics but to ignore other effects. As discussed in section 2.2, first-solvation-shell effects may be included in continuum models in terms of surface tensions. An alternative way to try to include some of them is by scaled particle theory and/or by some ab initio theory... 

This article reviews the following solution properties of liquid-crystalline stiff-chain polymers (1) osmotic pressure and osmotic compressibility, (2) phase behavior involving liquid crystal phasefs), (3) orientational order parameter, (4) translational and rotational diffusion coefficients, (5) zero-shear viscosity, and (6) rheological behavior in the liquid crystal state. Among the related theories, the scaled particle theory is chosen to compare with experimental results for properties (1H3), the fuzzy cylinder model theory for properties (4) and (5), and Doi s theory for property (6). In most cases the agreement between experiment and theory is satisfactory, enabling one to predict solution properties from basic molecular parameters. Procedures for data analysis are described in detail. 

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. 

Before proceeding to a review of both scaled particle theory and fuzzy cylinder model theory, it would be useful to mention briefly the unperturbed wormlike (sphero)cylinder model which is the basis of these theories. Usually the intramolecular excluded volume effect can be ignored in stiff-chain polymers even in good solvents, because the distant segments of such polymers have little chance of collision. Therefore, in the subsequent reference to wormlike chains, we always mean that they are unperturbed . 

In this article, we have surveyed typical properties of isotropic and liquid crystal solutions of liquid-crystalline stiff-chain polymers. It had already been shown that dilute solution properties of these polymers can be successfully described by the wormlike chain (or wormlike cylinder) model. We have here concerned ourselves with the properties of their concentrated solutions, with the main interest in the applicability of two molecular theories to them. They are the scaled particle theory for static properties and the fuzzy cylinder model theory for dynamical properties, both formulated on the wormlike cylinder model. In most cases, the calculated results were shown to describe representative experimental data successfully in terms of the parameters equal or close to those derived from dilute solution data. 

Certainly a number of aspects are not covered by this overview, such as ultrafine particle or secondary organic aerosol formation processes and their roles on air quality degradation, urban-scale dispersion models for air quality modelling or the... 

Bridging the gap between micro- and macro-scale is the central theme of the first contribution. The authors show how a so-called Energy-Minimization Multi-Scale (EMMS) model allows to do this for circulating fluid beds. This variational type of Computational Fluid Dynamics (CFD) modeling allows for the resolution of meso-scale structures, that is, those accounting for the particle interactions, and enables almost grid-independent solution of the gas-solids two-phase flow. 

Theoretical models based on first principles, such as Langmuir s adsorption model, help us understand what is happening at the catalyst surface. However, there is (still) no substitute for empirical evidence, and most of the papers published on heterogeneous catalysis include a characterization of surfaces and surface-bound species. Chemists are faced with a plethora of characterization methods, from micrometer-scale particle size measurement, all the way to angstrom-scale atomic force microscopy [77]. Some methods require UHV conditions and room temperature, while others work at 200 bar and 750 °C. Some methods use real industrial catalysts, while others require very clean single-crystal model catalysts. In this book, I will focus on four main areas classic surface characterization methods, temperature-programmed techniques, spectroscopy and microscopy, and analysis of macroscopic properties. For more details on the specific methods see the references in each section, as well as the books by Niemantsverdriet [78] and Thomas [79]. 

Example 5.4 Design a geometrically similar laboratory-scale cold model fluidized bed to simulate the hydrodynamics of a large-scale fluidized bed combustor. Also specify the operating conditions for the cold model. The combustor is a square cross section column with a width of 1.0 m and a height of 6 m. The fluidized bed combustor is operated at a temperature of 1,150 K, a superficial gas velocity of 1.01 m/s, and a bed height of 1.06 m. Particles with a density of2,630 kg/m3 and a diameter of677ptm are used for the combustor. The cold model is operated at a temperature of 300 K. Air is used for both the cold model and hot model fluidized beds. The physical properties of air are... 

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