Thermodynamics of mixing

For example, the solubility parameter of polystyrene may be estimated from Table 3.3. The structure is 

In an ideal solution, the circumstances are reversed, and the heat of mixing is zero, by definition. Raoult s law is obeyed. 

The free energy of mixing is given as the sum of the free energies of dilution per molecule, incompressibility of the mixture implicitly assumed  

When the polymer has x chain segments (mers), the entropy of mixing is given by 

In Section 6.5 polymer-polymer miscibility was considered to be an important parameter in the dispersion of the two polymer phases. In this respect a polymer-polymer system can be considered as miscible, immiscible, or partially miscible depending on the relative solubility of the two polymers. Total solubility characterizes a miscible system, insolubility an immiscible system, and partial solubility a partially miscible system. The degree of miscibility has important effects on the mechanical, physical, rheological, and optical properties of the resulting blend. 

Thermodynamically, mixing will take place when it is favored energetically, that is, when 

FIGURE 6.25 Loss angle, 5, as a function of temperature for various types of polymer blends (a) miscible (dashed line), (b) immiscible (solid line), and (c) partially miscible (dotted line). The glass transition temperature, Tg, is the temperature at the peak of the loss angle. 

In a similar way we can consider an integral mixing process for the formation of an ideal solution from the components, as illustrated in Eigure 14.2. The mixing process can be represented by the equation 

5 EQUILIBRIUM BETWEEN A PURE SOLID AND AN IDEAL LIQUID SOLUTION 

For some ideal solutions, the range of composition that can be attained is limited because of the limited solubility of one or both components. As an example, let us consider the solution of naphthalene in benzene. 

If the solution is ideal, the chemical potential p.2 of the dissolved solute at a fixed temperature and pressure is given by the expression [from Equations (14.6) and (14.7)] 

Above a specified concentration (at a given temperature and pressure) no more naphthalene will dissolve at equilibrium that is, the solution becomes saturated. When solid naphthalene is in equilibrium with the solution, as in Equation (14.41), 

Shortly thereafter, Scott (1949) offered a theoretical explanation of the Dobry and Boyer-Kawenoki results specifically, he investigated the partial molal free energy of mixing AF of polymer-polymer binary systems, and polymer (l)-polymer (2)-solvent (3) ternary systems. For binary polymer-polymer systems, AF is given by 

Scott set the first and second derivatives with respect to volume of AF or AF2 to zero, and found that when the two polymers are at their critical solution temperatures 

Since the heat of mixing usually is positive unless some specific interaction occurs, such as hydrogen bond formation, the above theory predicts two phases for most mixtures of interest.t Experiments by Slonimskii (1958) and by Tompa (1956) tended to confirm the above results. 

These early works must be viewed in the context of the time during which they were performed. The results obtained are undoubtedly correct, yet their combined conclusion of rather total incompatibility of polymer mixtures tended to discourage efforts to find and study compatible or semicompatible polymer pairs. As we shall see below, the fact that many polymers could be usefully mixed and partial compatibility achieved was stumbled upon partly as a result of other lines of investigation. (See also Section 13.4.) 

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We shall devote a considerable portion of this chapter to discussing the thermodynamics of mixing according to the Flory-Huggins theory. Other important concepts we discuss in less detail include the cohesive energy density, the Flory-Krigbaum theory, and a brief look at charged polymers. 

Clough, P.N. et al., 1987, Thermodynamics of Mixing and Final State of a Mixture formed by the Dilution of Anhydrous Hydrogen Fluoride with Most Air, Safety and Reliability Directorate, UKAEA, Wigshaw Lane, Culcheth, Warrington, Cheshire, England, WA3 4NE, SRD R 396. 

The finding that the assumptions of the regular solution approximation do not hold for the mixed micellar systems investigated here suggests a re-examination of how the thermodynamics of mixing enter the nonideal mixed micelle model. 

Calorimetric measurements can be used to obtain heats of mixing between different surfactant components in nonideal mixed micelles and assess the effects of surfactant structure on the thermodynamics of mixed micellization. Calorimetry can also be successfully applied in measuring the erne s of nonideal mixed surfactant systems. The results of such measurements show that alkyl ethoxylate sulfate surfactants exhibit smaller deviations from ideality and interact significantly less strongly with alkyl ethoxylate nonionics than alkyl sulfates. 

HOLLAND Thermodynamics of Mixed Micellization Legend of Symbols... 

Except for some anionic/cationic surfactant mixtures which form ion pairs, in a typical surfactant solution, the concentration of the surfactant components as monomeric species is so dilute that no significant interactions between surfactant monomers occur. Therefore, the monomer—mi celle equilibria is dictated by the tendency of the surfactant components to form micelles and the interaction between surfactants in the micelle. Prediction of monomer—micelle equilibria reduces to modeling of the thermodynamics of mixed micelle formation. 

For a binary system of surfactants A and B, the mixed micelle formation can be modeled by assuming that the thermodynamics of mixing in the micelle obeys ideal solution theory. When monomer and micelles are in equilibrium in the system, this results in ... 

A brief accounting of the thermodynamics of mixed micelle formation is given here primarily to clarify certain important issues which appear to have been previously overlooked. The necessity for measuring the monomer and micellar composition will be demonstrated. 

Most of the studies on thermodynamics of mixed micellar systems are based on the variation of the critical micellar concentration (CMC) with the relative concentration of both components of the mixed micelles (1-4). Through this approach It Is possible to obtain the free energies of formation of mixed micelles. However, at best, the sign and magnitude of the enthalpies and entropies can be obtained from the temperature dependences of the CMC. An Investigation of the thermodynamic properties of transfer of one surfactant from water to a solution of another surfactant offers a promising alternative approach ( ), and, recently, mathematical models have been developed to Interpret such properties (6-9). 

The chemical equilibrium model of Roux et al (6) is a powerful tool for the study of the thermodynamics of mixed micellar solutions. It can estimate the distribution constant of the surfactant 3 between water and micelles of the surfactant 2 and the thermodynamic properties of the surfactant 3 in the mixed micelles. For this it is necessary to obtain reliable data over a large concentration range of solute 2. 

The thermodynamics of mixing upon formation of the bilayered surface aggregates (admicelles) was studied as well as that associated with mixed micelle formation for the system. Ideal solution theory was obeyed upon formation of mixed micelles, but positive deviation from ideal solution theory was found at all mixture... 

Critical Micelle Concentration. In order to demonstrate the analogy between our treatment of mixed adsorption and earlier treatments of mixed micellization, we will briefly review the thermodynamics of mixed micelles. The thermodynamics of formation of ideal mixed micelles by two surfactants has been treated by Lange and Beck (9 ) and Cling (10). Rubingh ( ) extended the treatment to account for interactions between the surfactants, essentially by writing the cmc in the mixed surfactant solution as. 

Model Development. There is vast opportunity for development of fundamentally based models to describe the thermodynamics of mixed micelle formation. As discussed in Chapter 1, regular solution theory has yielded useful relations to describe monomer—mi cel 1e equilibrium. 

Thermodynamics of Mixed Monolayers. Goodrich (22) considered mixed monolayers as two-dimensional solutions and derived an expression for excess free energy of mixing, Gxs, which is given as... 

Enthalpies of Mixing of (Metal Oxide + Silica) Systems The thermodynamics of mixing of molten metal oxides (MO ) with silica (Si02) can be... 

Typical thermodynamic properties of the polysloxanes, such as heat capacities, solubility parameters, thermodynamic interaction parameters, and so on, have been extensively tabulated.46,182,212 Of particular interest in this area are the thermodynamics of mixing,... 

Before we begin to study reactions in detail we discuss the concept of mixing without a reaction taking place. We now give an example of the use of chemical potentials to study the thermodynamics of mixing for two ideal solutions designated A and B which can be either solid, liquid or gaseous ... 

Frame 37 considered the thermodynamics of mixing. The total molar Gibbs energy, Gm of a binary liquid mixture, is given (in terms of the amounts, a and nB, of the two components) by equation (37.9), Frame 37 ... 

The fundamentals of plasticiser selection have been listed (92). The thermodynamics of mixing PVC with phthalate esters have been studied (63). PVC-plasticiser interaction has been described using light transmission (186). 

Miscible blends of poly(vinyl methyl ether) and polystyrene exhibit phase separation at temperatures above 100 C as a result of a lower critical solution temperature and have a well defined phase diagram ( ). This system has become a model blend for studying thermodynamics of mixing, and phase separation kinetics and resultant morphologies obtained by nucleation and growth and spinodal decomposition mechanisms. As a result of its accessible lower critical solution temperature, the PVME/PS system was selected to examine the effects of phase separation and morphology on the damping behavior of the blends and IPNs. 

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