The Scattering Function and Thermodynamics

In the preceding chapter we have, based on the Flory-Huggins theory, discussed the basis for the phase behavior of polymer blends. Miscible polymer blends and polymer solutions have, even in the mixed one-phase system, spatial variations in the polymer concentration. These concentration fluctuations reflect the thermodynamic parameters of the free energy, as described in the Flory-Huggins model. 

I 7 Characterization of Polymer Blends and Block Copolymers by Neutron Scattering rather than the concentrations themselves  

With the density constraint the fluctuation and 50g terms holds  

Usually, these correlations would be expect to have a relatively short range, since distant segments will be completely independent. The products in Eq. (7.45) will accordingly be nonzero only for relatively small values of r. 

The thermodynamics of the system is described in terms of the ensemble aver-age, yij(r), of the spatially correlated fluctuations  

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Note that in the hypothetical incompressible limit the form of both the scattering functions and spinodal condition are much simpler than the rigorous expressions of Eqs. (6.4)-(6.6). Although the IRPA can usually be fitted to low wavevector experimental scattering data, and an apparent chi-parameter thereby extracted, the literal use of the IRPA for the calculation of thermodynamic properties and phase stability is generally expected to represent a poor approximation due to the importance of density-fluctuation-induced compressibility or equation-of-state effects [2,65,66]. The latter are non-universal, and are expected to increase in importance as the structural and/or intermolecular potential asymmetries characteristic rrf the blend molecules increase. 

Note 1 An infinite number of molar-mass averages can in principle be defined, but only a few types of averages are directly accessible experimentally. The most important averages are defined by simple moments of the distribution functions and are obtained by methods applied to systems in thermodynamic equilibrium, such as osmometry, light scattering and sedimentation equilibrium. Hydrodynamic methods, as a rule, yield more complex molar-mass averages. 

In addition to the repulsive part of the potential given by Eq. (4), a short-range attraction between the macroions may also be present. This attraction is due to the van der Waals forces [17,18], and can be modelled in different ways. The OCF model can be solved for the macroion-macroion pair-distribution function and thermodynamic properties using various statistical-mechanical theories. One of the most popular is the mean spherical approximation (MSA) [40], The OCF model can be applied to the analysis of small-angle scattering data, where the results are obtained in terms of the macroion-macroion structure factor [35], The same approach can also be applied to thermodynamic properties Kalyuzhnyi and coworkers [41] analyzed Donnan pressure measurements for various globular proteins using a modification of this model which permits the protein molecules to form dimers (see Sec. 7). 

Here, (...) = J2pn ri. .. ri) is the appropriate combinedquantal and thermodynamic average (over the classical probabilities pn) related with the condensed matter system. M and n(p) are the mass and momentum distribution of the scattering nucleus, respectively, and ior = q2 /2 M is the recoil energy. For convenience, h = 1. Eq. (2) is of central importance in most NCS experiments, since it relates the SCS directly to the momentum distribution. Furthermore, n(p) is related to the nuclear wave function by Fourier transform and therefore, to the spatial localization of the nucleus. It takes into account the fact that, if the scattering nucleus has a momentum distribution in its ground state, the 5-function centered at uor will be Doppler broadened. 

The idealized symmetric blend model is not representative of the behavior of most polymer alloys due to the artificial symmetries invoked. Predictions of spinodal phase boundaries of binary blends of conformationally and interaction potential asymmetric Gaussian thread chains have been worked out by Schwelzer within the R-MMSA and R-MPY/HTA closures and the compressibility route to the thermodynamics. Explicit analytic results can be derived for the species-dependent direct correlation functions > effective chi parameter, small-angle partial collective scattering functions, and spinodal temperature for arbitrary choices of the Yukawa tail potentials. Here we discuss only the spinodal boundary for the simplest Berthelot model of the Umm W t il potentials discussed in Section V. For simplicity, the A and B polymers are taken to have the same degree of polymerization N. 

Further work with application to large molecules or particles in solution was done by Smoluchowski (J), Einstein (4), Debye (5, 6) and Zimm (7). For many years now the measurement of the intensity of the light scattered from dilute solutions of macromolecules as a function of concentration, and scattering angle has provided much important information on the size, shape, and thermodynamic properties of these... 

Statistical thermodynamics already provide an excellent framework to describe and model equilibrium properties of molecular systems. Molecular interactions, summarized for instance in terms of a potential of mean force, determine correlation functions and all thermodynamic properties. The (pair) correlation function represents the material structure which can be determined by scattering experiments via the scattering function. AU macroscopic properties of pure and mixed fluid systems can be derived by weU-estabhshed multiphase thermodynamics. In contrast, a similar framework for particulate building blocks only partly exists and needs to be developed much further. Besides equibbrium properties, nonequilibrium effects are particularly important in most particulate systems and need to be included in a comprehensive and complete picture. We will come back to these aspects in Section 4. 

All discussions of transport processes currently available in the literature are based on perturbation theory methods applied to kinetic pictures of micro-scattering processes within the macrosystem of interest. These methods do involve time-dependent hamiltonians in the sense that the interaction operates only during collisions, while the wave functions are known only before and after the collision. However these interactions are purely internal, and their time-dependence is essentially implicit the over-all hamiltonian of the entire system, such as the interaction term in Eq. (8-159) is not time-dependent, and such micro-scattering processes cannot lead to irreversible changes of thermodynamic (ensemble average) properties. 

By that procedure, an additional factor V l appears in the equation of motion of pep [Eq. (4.29)]. This factor leads to the fact that the four-particle processes accounted for in this manner are not real and may vanish in the thermodynamic limit. At least this is true for four-particle scattering states. However, in the limiting case that we have only two-particle bound states, that is, the neutral gas, we can obtain a kinetic equation for the atoms if we use the special definition of the distribution function of the atoms (4.17) and (4.24). Using the ideas just outlined, the kinetic equation (4.62) was obtained. 

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