Conformal solution parameters

As the distance of separation of the terminal functional groups increases, the sensitivity of the conformation towards solution parameters decreases. Attractions between side-chains become more dominant in determining conformations of longer peptides these interacting side-chains do not necessarily have to be close together if they are polar, but may be bridged by water molecules. 

Scattering techniques for measuring various static and thermodynamic properties of polymers, such as molar mass, size, conformations, interaction parameters, etc. were described in experimental sections of Chapters 1-5. In addition to static properties, scattering can provide important information about dynamic properties of polymeric systems. This section focuses on dynamic scattering from dilute solutions, but similar methods are used in semidilute and concentrated solutions."... 

It is obvious from the foregoing discussion that the enthalpies of mixing for charge-unsymmetrical systems do not follow the simple conformal solution theory. When the anion in a strontium halide-alkali metal halide mixture from chloride to bromide and from bromide to iodide is changed, the enthalpy of mixing is decreasing. For all systems, the enthalpy interaction parameter, k, is a linear function of 512 with the usual exception for lithium-containing systems. Two important features of the k versus 512 plot should be emphasized ... 

The transport parameters for this model were evaluated using a corresponding state conformal solution procedure (Henderson and Leonard, 1971 Hanley, 1976) to predict the transport coefficient of fluids and mixtures. This is based upon the work of Ely and Hanley... 

A general method of predicting the effective molecular diameters and the thermodynamic properties for fluid mix-tures based on the hard-sphere expansion conformal solution theory is developed. The method of Verlet and Weis produces effective hard-sphere diameters for use with this method for those fluids whose intermolecular potentials are known. For fluids with unknown potentials, a new method has been developed for obtaining the effective diameters from isochoric behavior of pure fluids. These methods have been extended to polar fluids by adding a new polar excess function, to account for polar contributions in a mixture. A new set of pseudo parameters has been developed for this purpose. The calculation of thermodynamic properties for several fluid mixtures including CH —C02 has been carried out successfully. 

It is important to realize that the diameters needed for thermodynamic calculations do not necessarily represent a true minimum attainable separation distance between molecules. The objective is rather to determine optimal or effective diameters which give best results when used with a particular method of dealing with the contributions of molecular attraction. In this chapter the effective diameters sought are to be used specifically with the hard-sphere expansion (HSE) conformal solution theory of Mansoori and Leland (3). This theory generates the proper pseudo parameters for a pure reference fluid to be used in predicting the excess of any dimensionless property of a mixture over the calculated value of this property for a hard-sphere mixture. The value of this excess is obtained from a known value of this type of excess for a pure reference fluid evaluated at temperature and density conditions made dimensionless with the pseudo parameters. For example, if Xm represents any dimensionless property for a mixture of n nonpolar constituents at mole fractions xu x2,. . . x -i at temperature T and density p, then ... 

The GHBL perturbation procedure is remarkably accurate and the HSE-VW method is only slightly better in its overall agreement with the machine-calculated results. This comparison is not completely valid in that the conformal solution theory uses pure component data while in the perturbation theory each term is calculated from molecular parameters. 

Three-Parameter, Corresponding-States Conformal Solution Mixing Rules for Mixtures... 

The method used here for considering conformal solution models for fluids with molecular anisotropies is based on the method used by Smith (4) for treating isotropic one-fluid conformal solution methods as a class of perturbation methods. The objective of the method is to closely approximate the properties of a mixture by calculating the properties of a hypothetical pure reference fluid. The characterization parameters (in this case, intermolecular potential parameters) of the reference fluid are chosen to be functions of composition (i.e., mole fractions) and the characterization parameters for the various possible molecular pair interactions (like-like and unlike-unlike). In principle, all molecular anisotropies (dipole-dipole, quadrupole-quadrupole, dipole-quadrupole, and higher multipole interactions, as well as overlap and dispersion interactions ) can be included in the method. Here, the various molecular anisotropies are lumped into a single term, so that the intermolecular potential energy uy(ri2, on, a>2) between Molecules 1 and 2 of Species i and / can be written in the form... 

The extension of the isotropic mixture conformal solution method of Smith (4) to the case of anisotropic molecular systems can be made easily in the following manner. The quantities aijy biiy and Cy are defined by the relations an = 8ykeylcrym. fcy = 8y%Vyr, c j = 8y%vconfigurational Helmholtz free energy A for an anisotropic mixture then can be expanded about the configurational Helmholtz free energy of a hypothetical pure reference fluid, Ax, with characterization parameters BXy eXy and ax (or ax, bXy cx)y... 

The study presented herein has a number of implications. First, this study implies that it is possible to obtain accurate predictions of the thermodynamic behavior of mixtures within a multiparameter, corresponding-states framework using empirically determined exponents for the characterization parameters of the reference system in a first-order truncation of the conformal solution method expression for the Helmholtz free energy. This result is important to the continuing effort to develop a highly accurate multiparameter, corresponding-states framework for correlation of fluid properties, and to the industrial use of such a correlation. Second, this study demonstrates that there is a need to study separately rather than collectively (as herein) the errors introduced by the various major approximations introduced into the correlation meth-... 

Figure 1. Temperature-dependent macrostate dissection of a two-dimensional potential-energy landscape, (a) Potential V as a function of two coordinates, (b) Gibbs-Boltzmann distribution p at low (left), medium (middle), and high (right) temperatures, (c) Corresponding p at each temperature constructed from solutions to the characteristic packet equations, (d) Characteristic packet solution parameters R° and 0 for each macrostate (labeled with indices a, / , 6, y, and s). (e) Trajectory diagram of macrostate conformational free energies Fa as a function of temperature. (Reproduced from Church et al. [17] with permission obtained.)...

Helmholtz free energy (1.2.8) molar Helmholtz free energy parameter m the conformal solution theory (4.3.1)... 

Numerical simulations are designed to solve, for the material body in question, the system of equations expressing the fundamental laws of physics to which the dynamic response of the body must conform. The detail provided by such first-principles solutions can often be used to develop simplified methods for predicting the outcome of physical processes. These simplified analytic techniques have the virtue of calculational efficiency and are, therefore, preferable to numerical simulations for parameter sensitivity studies. Typically, rather restrictive assumptions are made on the bounds of material response in order to simplify the problem and make it tractable to analytic methods of solution. Thus, analytic methods lack the generality of numerical simulations and care must be taken to apply them only to problems where the assumptions on which they are based will be valid. 

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