Theory of Concentrated Solutions

Fuller and Newman [4] based their model of coupled proton and water transport in PEFC on the theory of concentrated solutions, wherein the effective diffusion constant was determined from the hydraulic permeability. Based on flux measurements of Fales and Vanderborgh [79], the model of Fuller and Newman used, practically, a -independent permeability. 

Complications of anothe kind will arise if there is an energetic interaction between neighbouring molecules of the sorbate, or if, in the case of 2-dimensionaI sorption, more than a monolayer of molecules is adsorbed on the surface. Several theories showing features similar to those introduced in the theories of concentrated solutions have been tried. However, no more than in the case of solutions, is it necessary to... 

Even though the Flory-Huggins theory predicts an upper critical solution temperature and allows a qualitatively correct phase diagram to be calculated, it cannot predict the experimentally observed lower critical solution temperature observed for virtually all polymer solutions. The fundamentally incorrect assumption in the theory is that the volume of mixing of the solution is assumed to be zero. To remedy this problem and improve the predictive power of the theory of concentrated solutions, a full fteory for... 

The thermodynamic properties of concentrated polymer solutions were studied by Floryi and independently by Huggins. The Flory-Huggins theory of polymer solutions still forms the basis for much discussion of these solutions in industry and even in academic research. Understanding this model is important for making coimections to much of the literature. Flory also substantially improved this model to include compressible fluids. The Flory-OrwoU theory of polymer solutions is still transparent and easily applicable, predicting both upper and lower critical solution temperatures. More-empirically adequate theories of concentrated solutions do not lend themselves to simple lecture presentation and often require detailed computer calculations to obtain any results. Concentrated solutions also introduce the phenomenon of viscoelasticity. An extensive treatment of the full distribution of relaxation times necessary to imderstand the dynamic properties of polymers in concentrated solution is presented. 

Because F(Y) is always less than unity, the value of the second virial coefficient predicted by the Flory-Krigbaum theory is necessarily lower than that predicted by the theory of concentrated solutions. 

The main progress in the theory of concentrated solutions came from two somewhat complementary directions of approach. A decisive step toward the understanding of the liquid state was made in 1937 by Lennard-Jones and Devonshire using a free volume theory (or cell model). Before Lennard-Jones and Devonshire, the cell model had been used by many authors (mainly by E3rring and his coworkers) to correlate the thermodynamic properties of liquids. However, Lennard-Jones and Devonshire were the first to use it to express the thermodynamic properties in terms of intermolecular forces (as deduced for example, from drial measurements). 

Edwards S F 1966 The theory of polymer solutions at intermediate concentration Proc. Phys. See. 88 265... 

In applying the Debye theory to concentrated solutions, we must extrapolate the results measured at different concentrations to C2 = 0 to eliminate the effects of solute-solute interactions. 

A finite time is required to reestabUsh the ion atmosphere at any new location. Thus the ion atmosphere produces a drag on the ions in motion and restricts their freedom of movement. This is termed a relaxation effect. When a negative ion moves under the influence of an electric field, it travels against the flow of positive ions and solvent moving in the opposite direction. This is termed an electrophoretic effect. The Debye-Huckel theory combines both effects to calculate the behavior of electrolytes. The theory predicts the behavior of dilute (<0.05 molal) solutions but does not portray accurately the behavior of concentrated solutions found in practical batteries. 

Battery electrolytes are concentrated solutions of strong electrolytes and the Debye-Huckel theory of dilute solutions is only an approximation. Typical values for the resistivity of battery electrolytes range from about 1 ohmcm for sulfuric acid [7664-93-9] H2SO4, in lead—acid batteries and for potassium hydroxide [1310-58-3] KOH, in alkaline cells to about 100 ohmcm for organic electrolytes in lithium [7439-93-2] Li, batteries. 

The physical picture in concentrated electrolytes is more apdy described by the theory of ionic association (18,19). It was pointed out that as the solutions become more concentrated, the opportunity to form ion pairs held by electrostatic attraction increases (18). This tendency increases for ions with smaller ionic radius and in the lower dielectric constant solvents used for lithium batteries. A significant amount of ion-pairing and triple-ion formation exists in the high concentration electrolytes used in batteries. The ions are solvated, causing solvent molecules to be highly oriented and polarized. In concentrated solutions the ions are close together and the attraction between them increases ion-pairing of the electrolyte. Solvation can tie up a considerable amount of solvent and increase the viscosity of concentrated solutions. 

P. G. De Gennes. Exponents for the excluded volume problem as derived by the Wilson method. Phys Lett 38A 339, 1972 J. des Cloiseaux. The Lagrangian theory of polymer solutions at intermediate concentrations. J Phys 26 281-291, 1975. 

The theory of concentration cells was first developed with great generality by Helmholtz (1878), who showed how the electromotive force could be calculated from the vapour pressures of the solutions, and his calculations were confirmed by the experiments of Moser (1878). 

Hence, the theory of electrolyte solutions subsequently developed in two directions (1) studies of weak electrolyte solutions in which a dissociation equilibrium exists and where because of the low degree of dissociation the concentration of ions and the electrostatic interaction between the ions are minor and (2) studies of strong electrolyte solutions, in which electrostatic interaction between the ions is observed. 

Van t Hoff introduced the correction factor i for electrolyte solutions the measured quantity (e.g. the osmotic pressure, Jt) must be divided by this factor to obtain agreement with the theory of dilute solutions of nonelectrolytes (jt/i = RTc). For the dilute solutions of some electrolytes (now called strong), this factor approaches small integers. Thus, for a dilute sodium chloride solution with concentration c, an osmotic pressure of 2RTc was always measured, which could readily be explained by the fact that the solution, in fact, actually contains twice the number of species corresponding to concentration c calculated in the usual manner from the weighed amount of substance dissolved in the solution. Small deviations from integral numbers were attributed to experimental errors (they are now attributed to the effect of the activity coefficient). 

Suspension Model of Interaction of Asphaltene and Oil This model is based upon the concept that asphaltenes exist as particles suspended in oil. Their suspension is assisted by resins (heavy and mostly aromatic molecules) adsorbed to the surface of asphaltenes and keeping them afloat because of the repulsive forces between resin molecules in the solution and the adsorbed resins on the asphaltene surface (see Figure 4). Stability of such a suspension is considered to be a function of the concentration of resins in solution, the fraction of asphaltene surface sites occupied by resin molecules, and the equilibrium conditions between the resins in solution and on the asphaltene surface. Utilization of this model requires the following (12) 1. Resin chemical potential calculation based on the statistical mechanical theory of polymer solutions. 2. Studies regarding resin adsorption on asphaltene particle surface and... 

Fig. 17 B/E-p dependence of the critical temperatures of liquid-liquid demixing (dashed line) and the equilibrium melting temperatures of polymer crystals (solid line) for 512-mers at the critical concentrations, predicted by the mean-field lattice theory of polymer solutions. The triangles denote Tcol and the circles denote T cry both are obtained from the onset of phase transitions in the simulations of the dynamic cooling processes of a single 512-mer. The segments are drawn as a guide for the eye (Hu and Frenkel, unpublished results)...

Does Surface Precipitation occur at Concentrations lower than those calculated from the Solubility Product As the theory of solid solutions (see Appendix 6.2) explains, the solubility of a constituent is greatly reduced when it becomes a minor constituent of a solid solution phase (curve b in Fig. 6.10).Thus, a solid species, e.g., M(OH)2 can precipitate at lower pH values in the presence of a hydrous oxide (as a solid solvent), than in its absence. 

For dilute solutions, Equations 4 and 5 reduce to the Bronsted-Guggenheim equations, and the parameters a23 and cu2 can be expressed in terms of the interaction parameters of tne Bronsted-Guggenheim theory. For concentrated solutions, Harned s rule is a simple empirical extension of the Brb nsted-Guggenheim theory. Thus, 1t 1s surprising how well the rule describes activity coefficients 1n highly concentrated solutions. 

Futerko and Hsing presented a thermodynamic model for water vapor uptake in perfluorosulfonic acid membranes.The following expression was used for the membrane—internal water activity, a, which was borrowed from the standard Flory—Huggins theory of concentrated polymer solutions ... 

Two different methods have been presented in this contribution for correlation and/or prediction of phase equilibria in ternary or mul> ticomponent systems. The first method, the clinogonial projection, has one disadvantage it is not based on concrete concepts of the system but assumes, to a certain extent, additivity of the properties of individiial components and attempts to express deviations from additivity of the properties of individual components and attempts to express deviations from additivity by using geometrical constructions. Hence this method, although simple and quick, needs not necessarily yield correct results in all the cases. For this reason, the other method based on the thermodynamic description of phase equilibria, reliably describes the behaviour of the system. Of cource, the theory of concentrated ionic solutions does not permit a priori calculation of the behaviour of the system from the thermodynamic properties of pure components however, if a satisfactory equation is obtained from the theory and is modified to express concrete systems by using few adjustable parameters, the results thus obtained are still substantially more reliable than results correlated merely on the basis of geometric similarity. Both of the methods shown here can be easily adapted for the description of multicomponent systems. 

In their investigation of polydimethylsiloxane and polyethylene oxide) in solution with various solvents, Tanner, Liu, and Anderson40 extrapolated the observed polymer diffusion coefficients to zero polymer concentration c. They applied Flory s theory of dilute solutions 45) to the case of diffusion ... 

A new theory of electrolyte solutions is described. This theory is based on a Debye-Hiickel model and modified to allow for the mutual polarization of ions. From a general solution of the linearized Poisson-Boltzmann equation, an expression is derived for the activity coefficient of a central polarized ion in an ionic atmosphere of non-spherical symmetry that reduces to the Debye-Hiickel limiting laws at infinite dilution. A method for the simultaneous charging of an ion and its ionic cloud is developed to allow for ionic polarization. Comparison of the calculated activity coefficients with experimental values shows that the characteristic shapes of the log y vs. concentration curves are well represented by the theory up to moderately high concentrations. Some consequences in relation to the structure of electrolyte solutions are discussed. 

In Fiery s theory of the excluded volume (27), the chains in undiluted polymer systems assume their unperturbed dimensions. The expansion factor in solutions is governed by the parameter (J — x)/v, v being the molar volume of solvent and x the segment-solvent interaction (regular solution) parameter. In undiluted polymers, the solvent for any molecule is simply other polymer molecules. If it is assumed that the excluded volume term in the thermodynamic theory of concentrated systems can be applied directly to the determination of coil dimensions, then x is automatically zero but v is very large, reducing the expansion to zero. 

The viewpoint parallels that of many other theories of condensed state behavior. The van der Waals theory develops an equation of state for dense gases from the assumption that each molecule moves in an average field provided by its neighbors and that the molecules contribute additively to the pressure. The Flory-Huggins thermodynamic theory of concentrated polymer solutions proceeds similarly. Chains select configurations on a lattice partially occupied by... 

Fundamental theories of transport properties for systems of finite concentration are still rather tentative (24). The difficulties are accentuated by the still uncertain effects of concentration on equilibrium properties such as coil dimensions and the distribution of molecular centers. Such problems are by no means limited to polymer solutions however. Even for the supposedly simpler case of hard sphere suspensions the theories of concentration dependence for the viscosity are far from settled (119,120). 

It has been remarked in the preceding sections that the equilibrium concentration of monomer in solution of its living polymer is affected by the nature of the solvent and by the polymer concentration, because these factors influence the activities of the components. A quantitative treatment of these effects, based on Scott s modification of the standard lattice theory of polymer solutions (33), has been outlined recently by Bywater (34). 

With increase in salt concentration the approximations involved in the Debye-Hiickel theory become less acceptable. Indeed it is noteworthy that before this theory was published a quasi-lattice theory of salt solutions had been proposed and rejected (Ghosh, 1918). However, as the concentration of salt increases so log7 ,7 being the mean ionic activity coefficient, appears as a linear function of c1/3 (the requirement of a quasi-lattice theory) rather than c1/2, the DHLL prediction (Robinson and Stokes, 1959). Consequently, a quasi-lattice theory of salt solutions has attracted continuing interest (Lietzke et al., 1968 Desnoyers and Conway, 1964 Frank and Thompson, 1959 Bahe, 1972 Bennetto, 1973) and has recently received some experimental support (Neilson et al., 1975). 

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