Mean-field random phase approximation

Polymer blends. Although it is well known that the mean-field Flory-Huggins theory of the thermodynamics of polymer systems is not a rigorously accurate description, especially for polymer blends, it is sufficiently valid that its use does not incur serious errors. Furthermore, de Gennes [16] used the mean-field random-phase approximation to obtain the scattering law for a binary polymer blend as ... 

There are various ways of extracting the polymer-polymer interaction parameters from SAN S data. One of these is based on the mean-field random phase approximation (RPA) derived by de Gennes [66, 67] and Binder [68], according to which the differential scattering cross section is given by ... 

From the theoretical viewpoint, much of the phase behaviour of blends containing block copolymers has been anticipated or accounted for. The primary approaches consist of theories based on polymer brushes (in this case block copolymer chains segregated to an interface), Flory-Huggins or random phase approximation mean field theories and the self-consistent mean field theory. The latter has an unsurpassed predictive capability but requires intensive numerical computations, and does not lead itself to intuitive relationships such as scaling laws. 

Note that on the mean field level at %c a second-order transition is predicted for f=H2, while taking fluctuations into account renders the transition first order [192,210,211], as also found experimentally [231]. Although the nature of surface effects on this transition is quite different for the first-order case [6] than for the second-order case [12], we discuss mostly the second-order case here. The constant pc in Eq. (43) is the density of the chains (pc=l/N if a Flory-Huggins lattice with lattice spacing unity is invoked), and the constants e0 and u0 can be derived [197] from the random phase approximation as... 

The shortcoming of the mean field method is that it admits no correlation between the motions of the individual particles. This correlation can be introduced by means of the random phase approximation (RPA) or time-dependent Hartree (TDH) method. In order to formulate this method, we introduce excitation operators (Ep) which replace f) p by when applied to the mean field ground state of the crystal when applied to any other state, they yield zero. Then, we write the Hamiltonian as a quadratic form in the excitation operators (Ep)+ and their Hermi-tean conjugates Ep... 

As in general all the y-coefficients do not vanish one has to assume a more general reference state than the single determinant SCF state. This is the rather well-known problem of finding the consistent reference state for the Random Phase Approximation (RPA). It also means that the field operator basis can be enlarged and can for instance include the iV-electron occupation number operators (in this discussion, electron field operators and their adjoints are used referring to a basis of spin orbitals that are the natural spin orbitals of the reference state, as will be discussed below, i.e., the spin orbitals that diagonalize the one-matrix)... 

To test this prediction of the crossover from mean-field to Ising-t5q)e behavior, very precise small-angle neutron scattering measurements were completed on blends of deuterated polystyrene (d-PS) and poly(vinyl methyl ether) (PVME) at the critical concentration for a series of temperatures as the one-phase mixture approaches the temperature of phase separation (ie, the critical point Tc in Fig. 7) (61). The data were analyzed by fitting the measured I (q) to the random phase approximation to estimate l(g=0) for each temperature, where the temperature is controlled to 0.01 K... 

It allows to calculate S(concentrated system based on the known stracture factor of the reference ideal system, Sid(random phase approximation which is equivalent to the mean-field approximation adopted in Sections 1.02.5.1 and 1.02.5.2 its region of validity is defined by the condition [98]. 

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