Incompressible random phase

To analyze the stability of the ordered microphases, the simplest incompressible random-phase approximation [132] can be employed. Using this approach, the critical value of the Flory-Huggins parameter, x > and the corresponding spinodal temperature, T = l/x > can be determined by the condition that the scattering intensity S(q) reaches its maximum value at a nonzero wave vector q. Within the RPA the scattering intensity is given by [132,142]... 

Using the simplest incompressible random phase approximation, the critical value of the Flory-Huggins parameter, / , and the corresponding transition temperature, T are... 

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] ... 

The model most often used for interpreting SANS data is based on the incompressible random phase approximation (RPA), because it allows to obtain the thermodynamic interaction parameter %. Once x is obtained, other features of the blend can be obtained, such as for example the concentration fluctuation of the single-phase state," and the thermal correlation length." ... 

In order to obtain the specific form of T(q), we now apply the random phase approximation (RPA) [22-26] to our system. The RPA provides a classical treatment of concentration fluctuations for incompressible mixtures of very large molecular weight molecules. It assumes a self-consistent potential uniformly acting on all species of monomers to ensure the incompressibility condition. The details of the RPA method, as applied to our polydisperse block copolymer blend, are given in Appendix 5.B. The result leads to... 

The WSL approach for the description of the order-disorder transition, ie, the transition between the microphase-separated block copolymer and the disordered melt, where the two blocks mix with each other, has been developed (74,88,89) using the random phase approximation. This transition is ofl en called the microphase separation transition (MST), and Toot is the temperature at which the order-disorder transition occurs. In this picture the system is described by a so-called order parameter, which is related to the space-dependent volume fraction or segment density of one of the components, say, component A. Again, the system is considered to be incompressible. The order parameter is then given by the deviation of the local segment density from the mean composition value. 

The structure factor S(Q) of an incompressible non-ideal mixture of two polymers with an enthalpy of mixing x has been derived by deGennes [3] within the random phase approximation (RPA) according to... 

---