SANS Chi-parameter

The incompressible random phase approximation (IRPA) is routinely used by experimentalists to analyze small angle scattering data from polymer alloys. A common approach to empirically defining an apparent SANS chi-parameter, Xs. is based on the total scattering intensity extrapolated to k = 0 as [56,57] ... 

Substitution of Eq. (6.4) into Eq. (6.8) yields a neral expression for the SANS chi-parameter which can be emfdoyed in PRISM studies to make direct contact with experiments on spedfic blends. Sudi an approach to comparing theory and experiment has been emfdiasized by Dudowicz, Freed, and coworkers [2] as the most approfuiate procedure. The resultant empirical chi-parameter contains not only microscopic information concerning intermolecu-lar interactions in blends (i.e. some combination of the independent Cmm X also reflects all the errors made by the neglect of compressibility effects which depend on Nm, T, < > , etc. [2]. 

Note that the apparent SANS chi-parameter can acquire a k-dependence even in the small angle regime via a cross-term betw n the intramolecxilar and... 

The results of PRISM theory can be substituted in this relation to make contact with SANS experiments which extract an apparent chi-parameter via the k-dependent fitting procedure. Note, however, that in general the predictions of an apparent SANS chi-parameter using Eq. (6.15) will not be the same as the extrapolated zero angle intensity approach of Eq. (6.8). This point re-emphaazes the phenomenological nature of the single, effective chi-parameter approach. 

Most prior PRISM predictions [23, 59-63] for the effespecial cases and/or well-defined additional approximation for which Eqs. (6.6) and (6.12) reduce to the literal IRPA forms. As discussed in Sect. 8 and elsewhere [61,67,68], the one special case corresponds to the theoretically much studied, but experimentally unrealizable, symmetric polymer blen. More generally, the additional approximation required to recover the IRPA forms corresponds in integral equation language to the k = 0 statement [67] ... 

The rigorous spinodal boundary is given by Eq. (6.6) and generally strongly diflers from that predicted by the literal incompressiUe RPA approximation of Eqs. (6.10) or (6.18). Analytical rraults for the spinodal temperature can be derived bas on Eqs. (6.6), (7.1X (7.2X (8.1 IX and (8.14X Other properties such as the critical composition, SANS chi-parameter, firee energy of mixing, etc. can also be obtained as discussed in depth dsewhere [67]. 

The apparent SANS chi-parameter is also easily determined analytically for stiffness asymmetric Berthelot thread model with the R-MMSA or R-MPY/ HTA closure approximations. For algebraic simplicity we consider the neutron data analysis approach which leads to Eq. (6.IS). In the effectively incompressible regime, defined here as Cmm- > IPHmm- in Eq. (8.11), one easily obtains the result [67]... 

Experimental papers normally report a SANS chi parameter, defined by... 

The PRISM theory can be used to calculate the partial structure factors, which can be input into equation (31) to obtain the SANS chi parameters. In principle, if a molecular model for the polymers is available this comparison could be carried out with no adjustable parameters since the potential parameters can be obtained from fitting to thermodynamic data. Nearly symmetric isotopic blends, where the only difference between the two components is that some of the hydrogen atoms in one component have been substituted with deuterium, are a very good model system because they are fairly easy to model and show interesting behavior, including a strongly composition dependent In most cases... 

Here, ( ) = Pm Vm/tj is the volume fraction of sites of type M, Vm is the volume of a site of type M, Vq is a reference volume , and Sm is the total neutron scattering length of a site of species M. Note that this chi-parameter will generally diverge as < )m - 0 due to the unrealistic incompressibility assumption. The SANS chi is, by construction, equivalent to the incompressible Flory value at the spinodal... 

An alternative approach to extracting an apparent chi-parameter from SANS data is to fit the scattering curves to an IRPA form over the entire measured wavevector range. Using Eqs. (6.12) one obtains for the scattering profile in the effectively incompressible approximation ... 

Figure 16. Summary plot of activation parameters (enthalpy of activation, AH entropy of activation, AS ) for oxygen isotope exehange in siUeates reacted with either pure water or salt solntions. The Eyring-Polanyi relationship (Eqn. 102) was nsed with data described by Cole et al. (1983), Cole and Ohmoto (1986), Cole et al. (1987, 1992), and in Table 2 (see Appendix). Note that data falling on a hnear trend generally indieate that a conunon leaetion meehanism predominates, ab = albite, san = sanidine, eels = celsian, qtz = qnartz, mnsc = muscovite, bio = biotite, chi = chlorite, parag = paragonite, woll = wollastonite, diop = diopside, gran = granite, bas = basalt.

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