Local equilibrium self-consistent fields

In simple terms, eqn [52] are diffusion equations in the component densities, which take into account the noise in the system. Dynamics of the molecular ensemble is based on the assumption that for each type of site a the local flux is proportional to the local site concentration 4> . At equilibrium, = constant, which results in the familiar self-consistent field equations for an inhomogeneous polymer system. When the system is not in equilibrium, the negative gradient -V/i,j(r) represents an effective thermodynamic force that drives collective relaxation processes. The time integration of eqn [52] generates an ensemble of density fields with the Boltzmann distribution. 

As remarked in Section VI the treatment of local equilibrium must be modified for charged particles, since the long-range coulomb interaction induces long-range correlations and neighboring volume elements cannot be considered to be statistically independent. We will indicate how the coulomb interaction can be taken into account by means of a self-consistent field, as in the Debye-Huckel theory. The results are then valid under the same conditions as the Debye-Hiickel theory, namely when the Debye radius is large compared to the mean interparticle separation. 

The book by Hamley [65] is a good general resource for self-consistent mean field theory. This formalism is based on the assumptions that (a) every chain in the system obeys Gaussian statistics, (b) the fluid is incompressible, and (c) the interactions between different structural units are local so that they depend only on the chemical nature but not on the positions of the units along their respective chains. As a result, the equations describe an ensemble of ideal chains in an external field which, in turn, is determined self-consistently from the structural unit probability distributions. As illustrated by Matsen and Schick [66], solving the exact equations requires a significant amount of computational effort to determine the equilibrium... 

The equilibrium in this theory entails self-consistent solutions of the potentials. The potential fields are themselves functionals of local monomer densities, which, at the same time, are determined by the potential field. Hence, the corresponding equations are solved self-consistently, with the potentials being updated at every iteration step until convergence. This technique amounts to the assumption that a single-field configuration dominates, and consequently all other configurations (or fluctuations ) can be neglected. 

Within the mean-field approach to the many-molecule problem, the free energy is represented by considering an ensemble of independent molecules in an effeaive external field density profile as in the real system of interest. The problem is thus reduced to that of a single ideal (maao)molecule in a complex external mean field, which characterizes the many-body effects of the other (macro)molecules. Physically, this self-consistent calculation means that the molecular conformations are assumed to be in local equilibrium under the given density profiles (r). 

DDFT, which was developed by Fraaije et al. in 1997, is a field-based theoretical method for studying complex fluids, their kinetics and their equilibrium structures at micrometer length and microsecond time scales. DDFT has been applied to the study of the self-assembly of block copolymers in bulk, under shear and in confinement, " and to study polymer blend compatibility. Compared to the DPD method, DDFT is computationally extremely fast since larger elements can be modelled. Moreover, since the fiuid elements can freely penetrate, larger time steps can be used, and furthermore it is less likely to become trapped in a local minimum. Since DPD is a particle-based method, it can provide somewhat more detailed structural information. Nonetheless, they are both powerful tools in simulating phase separated phenomena that occurs at the mesoscale and the consistency of results from the two methods for the same coarse-grain model is evaluated in this work. 

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