External Potential Dynamics EPD

Instead of propagating the composition in time, we can study the time evolution of the exchange potential W. In equilibrium the density variable (pA - Pb and the field variable W = i A - i B are related to each other via Pa- Pb = - /x y see Eq. 25. We also use this identification to relate the time evolution of the field W to the time dependence of the composition. Since the composition is a conserved quantity and it is linearly related to the field variable, we also expect the field W with which we are now describing our system to be conserved. Therefore, we can use the free energy functional M[W] from Eq. 42 and describe the dynamics of the field W through the relaxational dynamics of a model B system, referring to the classification introduced by Hohenberg and Halperin [94]. 

Aepd is a kinetic Onsager coefficient, and rj denotes noise that satisfies the fluctuation-dissipation theorem. The Fourier transform of this new diffusion equation is simply  

Comparing the diffusion equations of the dynamic SCF theory and the EP Dynamics, Eqs. 120 and 124, and using the relation W q, t) = - 2xN(f A(q, t), we obtain a relation between the Onsager coefficients within RPA  

In particular, the non-local Onsager coefficient Arousc that mimics Rouse-like dynamics (cf. Eq. 114) corresponds to a local Onsager coefficient in the EP Dynamics 

Generally, one can approximately relate the time evolution of the field W to the dynamic SCF theory [31 ]. The saddle point approximation in the external 

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The approach is commonly referred to as external potential dynamics (EPD). A related approach was originally introduced by Maurits and Fraaije [31]. However, these authors do not determine U W) exactly, but only approximately by solving separate Langevin equations for real fields Wa and Wb. This amounts to introducing a separate Langevin equation for a real field iU (i.e., an imaginary U in our notation) in addition to Eq. 97. 

Here, is the volume fraction of A block in diblock copolymer. To study the dynamics of phase separation, the polymeric external potential dynamics (EPD) method can be employed, which was proposed by Maurits and Fraaije [23] in dynamic density functional theory (DDFT) method (bead-string model). In EPD, the monomer concentration is a conserved quantity, and the polymer dynamics is inherently of Rouse type. The external dynamical equation in terms of the potential field m,- is expressed as... 

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