Theories of liquid mixtures

In Eq 2.39, ([)j is the volume fraction and Vj is the molar volume of the specimen i . The first two logarithmic terms give the combinatorial entropy of mixing, while the third term the enthalpy. For polymer blends Vj is large, thus the combinatorial entropy is vanishingly small — the miscibility or immiscibility of the system mainly depends on the value of the last term, 

For binary systems that contain an ingredient i = 1 or 2 (traditionally, for polymer solutions the subscript 1 indicates solvent, and 2 polymer) the Huggins-Flory, H-F, relation has been expressed in several equivalent forms  

it takes nine parameters to describe variation of concentration and temperature, at 

For less demanding thermodynamic calculations, Eq 2.41 can be simplified. Thus, to express conditions of miscibility in PS blends with poly(styrene-co-4-bromostyrene) the binary interaction parameter per one mer of styrene (within the T = 440-500 K region) was expressed as [Strobl et al, 1986]  

In this notation is expressed as composed of the enthalpic and entropic parts, Xh d 

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The UNIFAC (Unified quasi chemical theory of liquid mixtures Functional-group Activity Coefficients) group-contribution method for the prediction of activity coefficients in non-electrolyte liquid mixtures was first introduced by Fredenslund et al. (1975). It is based on the Unified Quasi Chemical theory of liquid mixtures (UNIQUAC) (Abrams and Prausnitz, 1975), which is a statistical mechanical treatment derived from the quasi chemical lattice model (Guggenheim, 1952). UNIFAC has been extended to polymer solutions by Oishi and Prausnitz (1978) who added a free volume contribution term (UNIFAC-FV) taken from the polymer equation-of-state of Flory (1970). 

UNIFAC UNIfied quasi chemical theory of liquid mixtures Functional-group Activity Coefficients 4.3.2... 

The realisation that lattice theories of liquids were getting nowhere came only slowly from about 1950 onwards. A key paper for chemists was that of Longuet-Higgins on what he called conformal solutions in 1951. In this he avoided the assumption that a liquid had a lattice (or any other particular) structure but treated the different strengths of the intermolecular potentials in a mixture as a first-order perturbation of the physical properties of one of the components. In practice, if not formally in principle, his treatment was restricted to molecules that could be assumed to be spherical, but it was so successful for many mixtures of non-polar liquids that this and later derivatives drove lattice theories of liquid mixtures from the field. 

Another difficulty in developing a molecular theory of liquid mixtures is the relatively poor knowledge of the intermolecular interactions between molecules of different species. While the intermolecular forces between simple spherical particles are well-understood, the intermolecular forces between molecules of different kinds are usually constructed by the so-called combination rules, the most well-known being the Lorentz and the Berthelot rules. 

The concept of an ideal mixture is central to the theory of liquid mixtures. This concept can be introduced by considering the partition function of a mixture of two ideal gases... 

Experimental excess functions of liquid mixtures are useful in that they provide data to test theories of liquid mixtures and provide a guide for the formulation of new theories. The data are also useful in the chemical and petroleum industries. This chapter does not contain a summary of experimental data since Chapter 9 of this volume consists of a bibliography of excess function and related measurements on binary mixtures of non-electrolytes. 

The activity coefficient is the most important and fundamental property in the thermodynamic study of liquid mixtures. It is a measure of the deviation of the behaviour of a component in a mixture from ideality and it has been interpreted by various theories of liquid mixtures. Gas-liquid elution chromatography offers a rapid method of determining this property at infinite dilution. Conder and Purnell have developed a method of determining activity coefiicients at finite concentrations and this has recently been used by other workers. " To do this, the elution technique must be supplemented by... 

The third phase of development corresponds to a renaissance in the theoretical understanding of the nature of critical phenomena. On the one hand, the realization that experimental critical indices are incompatible with those predicted by simple theories of the van der Waals type has led to a number of very careful studies of the behaviour of substances in the neighbourhood of a critical point. On the other hand, new theories of liquid mixtures such as the generalized van der Waals theories have given impetus to the study of systems suitable for examining the accuracy and usefulness of these theories. Also, recently, increased attention has been paid to phase equilibrium of mixtures for which the critical line between the two pure component critical points is not continuous. Such studies involve pressures considerably higher than the critical pressure of either of the pure components. 

Studies of the equations of state of pure substances have played a significant role in the development of our knowledge of intermolecular forces. Similar measurements on gas mixtures can provide important information concerning interactions between unlike molecules, the interactions that must be specified in all theories of liquid mixtures. 

The applicability of corresponding-states theories of liquid mixtures has been tested for all the dense gas systems for which the excess functions have been directly determined. 186-139 results are inconclusive. For many of the mixtures studied there is a large difference in the critical temperatures between the components. In such cases, as shown above, the excess functions at moderate pressures depend primarily on the properties of the component of higher critical temperature - they are insensitive to the details of the mixed interactions. At higher pressures, where the test of theory is more significant, the calculated properties depend markedly on the choice of reference fluid. Even simple... 

Theories of Liquid Mixtures 2.5.1 Lattice, Cell, and Hole Theories... 

Hermann(145) used a method based on the first order perturbation theory of liquid mixture in order to calculate the solubilities of hydrocarbons in water and the hydrophobic interaction. 

In order to construct a theory of liquid mixtures, it is necessary to know two types of information the structure of liquids (i.e. how the molecules in the liquid are disposed in space) and the intermolecular forces between similar or dissimilar molecules. However, information about the second type of information is insufficient and as a consequence all the theories must make simplifying h q)otheses to overcome this drawback [PRA 99]. Theoretical works have concerned liquid mixtures whose molecules are apolar and spherical in shape for example, the theory independently developed in 1933 by Scatchard and Hildebrand [PRA 99] frequently used... 

Several other empirical relations for diffusion coefficients have been suggested Olson and Walton (01) have devised a means for estimating diffusion coefficients of organic liquids in water solution from surface-tension measurements. Hill (H5) has proposed a method based on Andrade s theory of liquids which allows for the concentration dependence of the diffusion coefficient in a binary liquid mixture. The formula of Arnold (A2, T6, p. 102) does not seem generally useful inasmuch as it contains two constants ( abnormality factors ) characteristic of the solute and of the solvent. 

In these remarks, one can see a pioneering suggestion of a cluster mixture theory of liquids with short-range (exchange-like) forces, along the lines of Mayer cluster theory (Sidebar 13.5) or quantum cluster equilibrium theory (Section 13.3.4). 

Measurements of static light or neutron scattering and of the turbidity of liquid mixtures provide information on the osmotic compressibility x and the correlation length of the critical fluctuations and, thus, on the exponents y and v. Owing to the exponent equality y = v(2 — ti) a 2v, data about y and v are essentially equivalent. In the classical case, y = 2v holds exactly. Dynamic light scattering yields the time correlation function of the concentration fluctuations which decays as exp(—Dk t), where k is the wave vector and D is the diffusion coefficient. Kawasaki s theory [103] then allows us to extract the correlation length, and hence the exponent v. 

The development of a general theory of systems with non-central force fields can be divided into two parts. First the many types of directional interaction that may occur have to be classified within a general mathematical framework and then approximate methods of evaluating the partition function have to be devised. This paper summarizes some of the results of a method developed by the author 2 with particular reference to its application to the properties of liquid mixtures. 

We shall not attempt to review and compare critically various theories of liquid crystallinity in this chapter. Inasmuch as theory based on a lattice model has proved most successful in the treatment of liquid crystallinity in polymeric systems, we shall present an abbreviated account of that theory confined to its essential aspects. The versatility of this theory has permitted its extension to polydisperse systems, to mixtures of rodlike polymers with random coils and to some of the many kinds of semirigid chains. These ramifications of the theory will be discussed in this chapter... 

The Kirkwood-Buff theory of solutions and the local composition of liquid mixtures. 

The KB theory provides a unique opportunity to obtain information about the structure of liquid mixtures at a nanometer level from the excesses (zl y)- However, it took a long time to find the correct procedure to calculate the above excess (or deficit). For several decades the calculations of were based on the expression suggested by Ben-Naim [1] ... 

The Kirkwood—Buff Theory of Solutions and the Local Composition of Liquid Mixtures... 

The present paper is devoted to the local composition of liquid mixtures calculated in the framework of the Kirkwood—Buff theory of solutions. A new method is suggested to calculate the excess (or deficit) number of various molecules around a selected (central) molecule in binary and multicomponent liquid mixtures in terms of measurable macroscopic thermodynamic quantities, such as the derivatives of the chemical potentials with respect to concentrations, the isothermal compressibility, and the partial molar volumes. This method accounts for an inaccessible volume due to the presence of a central molecule and is applied to binary and ternary mixtures. For the ideal binary mixture it is shown that because of the difference in the volumes of the pure components there is an excess (or deficit) number of different molecules around a central molecule. The excess (or deficit) becomes zero when the components of the ideal binary mixture have the same volume. The new method is also applied to methanol + water and 2-propanol -I- water mixtures. In the case of the 2-propanol + water mixture, the new method, in contrast to the other ones, indicates that clusters dominated by 2-propanol disappear at high alcohol mole fractions, in agreement with experimental observations. Finally, it is shown that the application of the new procedure to the ternary mixture water/protein/cosolvent at infinite dilution of the protein led to almost the same results as the methods involving a reference state. 

One of the most important applications of the KB theory consists of its use to extract some microscopic characteristics of liquid mixtures from measurable macroscopic thermodynamic quantities. The excess (or deficit) number of molecules of... 

Shulgin, I. L. Ruckenstein, E. The Kirkwood-Buff theory of solutions and the local composition of liquid mixtures. J. Phys. Chem. B 2006, no, 12707-12713. 

In this chapter, we introduce the concepts of molecular distribution function (MDF), in one- and multicomponent systems. The MDFs are the fundamental ingredients in the modern molecular theories of liquids and liquid mixtures. As we shall see, these quantities convey local information on the densities, correlation between densities at two points (or more) in the system, etc. 

We start with detailed definitions of the singlet and the pair distribution functions. We then introduce the pair correlation function, a function which is the cornerstone in any molecular theory of liquids. Some of the salient features of these functions are illustrated both for one- and for multicomponent systems. Also, we introduce the concepts of the generalized molecular distribution functions. These were found useful in the application of the mixture model approach to liquid water and aqueous solutions. 

An indirect route has been developed mainly by Kirkwood, which involves molecular distribution functions (MDF) as an intermediate step. The molecular distribution function approach to liquids and liquid mixtures, founded in the early 1930s, gradually replaced the various lattice theories of liquids. Today, lattice theories have almost disappeared from the scene of the study of liquids and liquid mixtures. This new route can be symbolically written as... 

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