Hard-sphere theory modified

In Fig. 18 we also show the density dependence of the viscosity tj for argon as computed by this modified Enskog theory (MET) at 348 K, where x and b in Eq. (147a) are replaced by x and b as determined from Eqs. (151) and (154), respectively. We see that the MET agrees with experiment to within 10% over a much wider range of densities than does the simple Enskog theory discussed earlier. This improvement appears to be due to the fact that the MET takes into account the actual density dependence of the collision frequency, while the simple Enskog theory approximates the collision frequency by the hard-sphere theory, extrapolated from low densities. 

Having at our disposal accurate structural and thermodynamic quantities for HS fluid, the latter has been naturally considered as a RF. Although real molecules are not hard spheres, mapping their properties onto those of an equivalent HS fluid is a desirable goal and a standard procedure in the liquid-state theory, which is known as the modified hypemetted chain (MHNC) approximation. According to Rosenfeld and Ashcroft [27], it is possible to postulate that the bridge function of the actual system of density p reads... 

The flow properties of disordered micellar phases are now reasonably well understood. For spherical micelles the viscosity can be estimated from modified hard-sphere-suspension theories, while for disordered semidilute cylindrical micelles the Cates theory of entangled living polymers provides at least a good starting point, and in some cases nearly quantitative prediction of rheological properties. 

In the liquid phase also, the Enskog theory has also proved quite effective for the representation and prediction of the properties of pure and mixed substances. In this case, the theory has been modified using the results of computer simulations of hard spheres, which have indicated the limitations of the assumptions of entirely random motions at elevated densities. In combination, the Enskog theory corrected in this way provides a good description of the properties of some pure liquids. An even more general result of the Enskog theory is that the transport properties of a fluid or... 

In a recent contribution [87] we studied the catalytic effect in polyelectrolyte-electrolyte mixtures by various theoretical techniques. For an isotropic model where the macroions, co-ions and counterions are pictured as charged hard spheres, we employed the HNC approximation, the modified PB and symmetric PB theories. The results for k/k° were compared with the computer simulations for the same quantity. Note that this quantity is much more sensitive to the details of the model and theory than thermodynamic properties like osmotic pressure studied before. The conclusion was that these theories are not well-suited to treat the problem they were capable of reproducing MC values only qualitatively and even this merely for low-charged macroions. 

The first step in quantitative description of pure polyamorphic fluid is a selection of the model that can qualitatively describe a possible multiplicity of critical points in wide range of temperatures and pressures. A great many of explanations of multicriticality in monocomponent fluids (perturbation theory models semiempirical models lattice models, two-state models, field theoretic models, two-order-parameter models, and parametric crossover model has been disseminated after the pioneering work by Hemmer and Stell Here we test more extensively the modified van der Waals equation of state (MVDW) proposed in work and refine this model by introducing instead of the classical van der Waals repulsive term a very accurate hard sphere equation of state over the entire stable and metastable regions... 

Table I. Values of the Hard-Sphere g (r) Calculated from the Exact, the Modified Born-Green-Yvon (BGY2), the Unmodified Born-Green-Yvon (BGY), the Percus-Yevick (PY), and the Convolution-Hypernetted-Chain (CHNC) Theories ...

Helfand and Stillinger have recently employed this theory to obtain a suitable "local density correction of the superposition theory for the hard sphere system. While the numerical solution of the modified integral equation is not available at this time, the equation is unique among other approximate theories in )delding the exact value of the fourth virial coefficient. 

First we consider the virial expansion in density originally due to Boltzmann and derived from the kinetic theory of gases for hard-sphere molecules. This equation was modified by Hirschfelder and Roseveare, and covolume terms for product species were adjusted to high temperature by setting them equal to the high-temperature second virial coeflficientz. The equation is... 

The perturbed-hard-ehain (PHC) theory developed by Prausnitz and coworkers in the late 1970s was the first successful application of thermodynamic perturbation theory to polymer systems. Sinee Wertheim s perturbation theory of polymerization was formulated about 10 years later, PHC theory combines results fi om hard-sphere equations of simple liquids with the eoneept of density-dependent external degrees of fi eedom in the Prigogine-Flory-Patterson model for taking into account the chain character of real polymeric fluids. For the hard-sphere reference equation the result derived by Carnahan and Starling was applied, as this expression is a good approximation for low-molecular hard-sphere fluids. For the attractive perturbation term, a modified Alder s fourth-order perturbation result for square-well fluids was chosen. Its constants were refitted to the thermodynamic equilibrium data of pure methane. The final equation of state reads ... 

Computation of shear viscosity of hard spheres has been attempted using NEMD [11], Modified non-equilibrium molecular dynamics methods have also been developed for study of fluid flows with energy conservation [12], NEMD simulations have also been recently performed to compare and contrast the Poiseuille and Electro-osmotic flow situations. Viscosity profiles obtained from the two types of flows are found to be in good mutual agreement at all locations. The simulation results show that both type of flows conform to continuum transport theories except in the first monolayer of the fluid at the pore wall. The simulations further confirm the existence of enhanced transport rates in the first layer of the fluid in both the cases [13, 14]. 

Consider a dense gas of hard spheres, all with mass m and diameter a. Since the collisions of hard-sphere molecules are instantaneous, the probability is zero that any particle will collide with more than one particle at a time. Hence we still suppose that the dynamical events taking place in the gas are made up of binary collisions, and that to derive an equation for the single-particle distribution function /(r, v, /) we need only take binary collisions into account. However, the Stosszahlansatz used in deriving the Boltzmann equation for a dilute gas should be modified to take into account any spatial and velocity correlations that may exist between the colliding spheres. The Enskog theory continues to ignore the possibility of correlations in the velocities before collision, but attempts to take into account the spatial correlations. In addition, the Enskog theory takes into account the variation of the distribution function over distances of the order of the molecular diameter, which also leads to corrections to the Boltzmann equation. 

Fig. 18. A comparison of the experimental viscosity data for argon as a function of density with the predictions of the simple Enskog theory and of the modified Enskog theory. In the simple Enskog theory the effective hard-sphere diameter is determined by fitting the low-density limit of the viscosity to the Boltzmann equation result for hard spheres. Then x is calculated for all densities by using this diameter in Eq. (144). In the modified Enskog theory, both x nd the hard-sphere diameter are determined from the equation of state data. (Figure courtesy of J. V. Sengers.)...

The modified Enskog equation can easily be generalized to apply to dense mixtures of hard-sphere gases/ The transport coefficients that result satisfy the Onsager reciprocal relations. Thus the principal difficulty in generalizing the Enskog theory to mixtures has now been removed. 

Yu YX, WuJZ Structures of hard-sphere fluids from a modified fundamental-measure theory, J Chem Phys 117(22) 10156-10164, 2002. 

The decisive advantage of the original Elory-Huggins theory [1] lies in its simplicity and in its ability to reproduce some central features of polymer-containing mixtures qualitatively, in spite of several unrealistic assumptions. The main drawbacks are in the incapacity of this approach to model reality in a quantitative manner and in the lack of theoretical explanations for some well-established experimental observations. Numerous attempts have therefore been made to extend and to modify the Elory-Huggins theory. Some of the more widely used approaches are the different varieties of the lattice fluid and hole theories [2], the mean field lattice gas model [3], the Sanchez-Lacombe theory [4], the cell theory [5], different perturbation theories [6], the statistical-associating-fluid-theory [7] (SAET), the perturbed-hard-sphere chain theory [8], the UNIEAC model [9], and the UNIQUAC [10] model. More comprehensive reviews of the past achievements in this area and of the applicability of the different approaches are presented in the literature [11, 12]. 

This theory covers the complete density range from dilute gas to solidification. However, the hard sphere model is inappropriate for a real gas at low and intermediate densities where specific effects of intermolecular forces are significant. It is therefore necessary to modify the theory this has been done in different ways. In Section 5.2, a method is described for determination of the second viscosity virial coefficient and the translational part of the thermal conductivity coefficient. Although this approach is not rigorous, it does provide a useful estimate for these coefficients - especially for low reduced temperatures. 

For application to real gases, this theory has been modified (Enskog 1922 Hanley et al. 1972 Hanley Cohen 1976 Vogel etal. 1986 Ross etal. 1986). Although this has no rigorous theoretical basis, it does provide an alternative rqiresentation of the second viscosity virial coefficient and the translational part of the second thermal conductivity virial coefficient, which is particularly useful at reduced temperatures below T = 0.5, the lower limit of the coefficients in Table 5.1. On the basis that a real fluid differs from a hard sphere fluid mainly in the temperature dependence of the collision frequency, the pressure P of the hard-sphere fluid is replaced by the thermal pressure T(dP/dT)p of... 

Dense fluid transport property data are successfully correlated by a scheme which is based on a consideration of smooth hard-sphere transport theory. For monatomic fluids, only one adjustable parameter, the close-packed volume, is required for a simultaneous fit of isothermal self-diffusion, viscosity and thermal conductivity data. This parameter decreases in value smoothly as the temperature is raised, as expected for real fluids. Diffusion and viscosity data for methane, a typical pseudo-spherical molecular fluid, are satisfactorily reproduced with one additional temperamre-independent parameter, the translational-rotational coupling factor, for each property. On the assumption that transport properties for dense nonspherical molecular fluids are also directly proportional to smooth hard-sphere values, self-diffusion, viscosity and thermal conductivity data for unbranched alkanes, aromatic hydrocarbons, alkan-l-ols, certain refrigerants and other simple fluids are very satisfactorily fitted. From the temperature and carbon number dependency of the characteristic volume and the carbon number dependency of the proportionality (roughness) factors, transport properties can be accurately predicted for other members of these homologous series, and for other conditions of temperature and density. Furthermore, by incorporating the modified Tait equation for density into... 

Additional examples of equation of state models include the lattice gas model (Kleintjens et al, [33,34], Simha-Somcynsky hole theory [35], Patterson [36], the cell-hole theory (Jain and Simha [37-39], the perturbed hard-sphere-chain equation of state [40,41] and the modified cell model (Dee and Walsh) [42]. A comparison of various models showed similar predictions of the phase behavior of polymer blends for the Patterson equation of state, the Dee and Walsh modified cell model and the Sanchez-Lacombe equation of state, but differences with the Simha-Somcynsky theory [43]. The measurement and tabulation of PVT data for polymers can be found in [44]. 

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