Dynamic Random Phase Approximation RPA

A multicomponent dynamic RPA for incompressible diblock copolymer mixtures was developed by Akcasu [270]. The details of the approach may be found in the above reference. Here we recall some basic ideas dealing with the calculation of the dynamic structure factor S Q,t) for Rouse chains. 

We consider an incompressible (m-l-1) multicomponent mixture of polymers consisting of m different types of polymer chains within a matrix referred to as 0 . Components may be either homopolymers of a given chemical species or, e.g. homopolymer sections in block copolymers. Hydrogenated and deuterated species of the same homopolymer are considered as different components. In this context a diblock copolymer is a two-component polymer system. A mixture of partially protonated diblock chains hA-dB with deuterated diblock chains is consequently regarded as a four-component system. 

We denote the fluctuations of the number density of the monomers of component j at a point r and at a time t as pj r,t). With this definition we have pj(r,t))=0. In linear response theory, the Fourier-Laplace transform of the time-dependent mean density response to an external time dependent potential U r,t) is expressed as  

In order to proceed further a reference system is introduced (bare system) where the interaction between the monomers is removed but the chain connectivity is preserved. The response matrix in the bare system is denoted as X Q,t). Then in the bare system the density response is written as  

The relation of with is analogous to Eq. 6.2. is related to x in introducing an interaction potential between monomers a and b . In the Laplace 

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