Spinodal equations

The spinodal Equation can be obtained by applying the spinodal condition (Eq. 9) to the chemical potential of a polymer in a mixture, (obtained from Eq. (27) by including... 

The critical point Equation is derived from the spinodal Equation using the critical condition given earlier. The differential of the spinodal Equation with respect to is zero at the critical point. This calculation is perfectly feasible but so far no one has made any attempt to use these Equations to predict the critical point. In the past it has usually been approximated using the simple Flory-Hi jns expression for the critical point. 

I do not believe this difference to have general meaning, so have used the consistent spinodal equations obtained from FEOS (1) ... 

To describe the solubility of sodium poly(styrene sulfonate) in the presence of z-valent counterions, the electrostatic model [18,19] was introduced. This model takes into account a short-range electrostatic attraction between negatively charged monomers (1) and those carrying condensed multivalent counterions that are positively charged (z 1). Phase diagrams have been calculated using two techniques the polymer reference interaction site model (PRISM) [18] and the random phase approximation (RPA) [19]. Both techniques lead to the spinodal equation... 

For a given monomer concentration, the spinodal equation gives two solutions corresponding to the precipitation and the resolubilization limits as the salt concentration increases. The salt concentration at the resolubilization limit increases with the valency z of counterion [18,19], As we can see in Figures 4-6, the electrostatic model allows the experimental phase diagrams to be explained. 

As spinodal equations in the context of RPA can be obtained using a Flory mean field approach [24], this approach was used to describe the stability of the NaPA/CaCl2 system when chemical associations between charged monomers and Ca2+ are taken into account. The spinodal line was found to be [22,30]... 

FIG. 12 Variation of interaction terms of the spinodal equation as a function of the added calcium (the monomer concentration is fixed to 0.01 M) (thin full line) repulsive electrostatic term, (thick full line) attractive term due to the dicomplexation in absolute value, (dashed line) attractive electrostatic term in absolute value. 

A more general approach will give the following form of the spinodal equation for a one-component system (see Equation 1.1.2-12)... 

Near the spinodal, the second term in the right-hand side of Equation 53 predominates, since on the spinodal (Equations 1.2 52, 1.3-19) dAfii/dx2 = 0, therefore, dApi/dc = 0, and, according to Equation 1.2-27,... 

The spinodal equation in the P-f-LMWL system results from state equation 50 and dAp /dni = 0 (Figure 3.9)... 

So, wc are able to convert the x, axis (sec Figure 3.9) into a temperature one which is necessary to a comparison between the theoretical spinodals (Equation 239) and binodals (further) and the experimental ones. 

Substitution of Equations 259 and 261 to spinodal equation 239 leads to expressions for the left-hand Vjjp/ and right-hand Vijp// branches of the spinodal... 

For a polynary u -f l)-component system, the spinodal equation is expressed (Stock-mayer, 1949 Flory, 1953 Prigogine and Dcfay, 1954 Tompa, 1.956 Koningsveld, 1968, 1970b Koningsveld and Staverman, 1968b Kennedy ct al., 1975 Vink, 1976) as (see Equations 1.2. 58.. .61)... 

The condition /d j, o = 0 means the spinodal (Equation 3.3 10), hence, according to Equation 47, / = 0 defines the spinodal as well. Then, on the basis of Equation 48, the spinodal condition is expressed by... 

I he spinodal (Equation 3.2 -2) and critical point (Equations 3.2-2,-3) conditions lead, in this case, to... 

Naturally, a relationship between the parameters of the two-body and thrcc-body interactions, other than Equation 53, will lead to a different spinodal equation. 

For a binary polymer-solvent mixture, the spinodal equation reduces to... 

For ternary polymer (3)/solvent (2)/non-solvent (1) systems which apply to the FRRPP fluid systems, the spinodal equation becomes (Altena and Smolders, 1982)... 

In this first approximation we have introduced the factors p. which deviate from unity because the bromination changes the numbers of occupied sites. Consequently, the 50/50 (v/v) composition is no longer a critical concentration, and the measured A value does not refer to a spinodal situation. Ignoring this source of error we may use the spinodal equation to derive values for g. In the present approximation it reads... 

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