Hamiltonian Edwards

Continuum models go one step frirtlier and drop the notion of particles altogether. Two classes of models shall be discussed field theoretical models that describe the equilibrium properties in temis of spatially varying fields of mesoscopic quantities (e.g., density or composition of a mixture) and effective interface models that describe the state of the system only in temis of the position of mterfaces. Sometimes these models can be derived from a mesoscopic model (e.g., the Edwards Hamiltonian for polymeric systems) but often the Hamiltonians are based on general symmetry considerations (e.g., Landau-Ginzburg models). These models are well suited to examine the generic universal features of mesoscopic behaviour. 

The Edwards Hamiltonian is an appealing but most formal object. To mention a simple fact, shrinking to zero the segment size of the discrete chain model as done in the continuous chain limit, we in general get a continuous but not differentiable space curve. Strictly speaking the first part, of Vj, does not exist. Further serious mathematical problems are connected to the (5-function interaction. Hie question in which sense Ve is a mathematically well defined object beyond its formal perturbation expansion is ari interesting problem of mathematical physics. 

The fifth term describes the free energy of a polyelectrolyte chain associated with M dipoles and N-M charges. The chain is in aqueous solution of N-M counterions and salt ions [48]. This free energy is obtained from the Edwards Hamiltonian [49]... 

In what follows, H defined by either eq 3.23 or 3.26 is called the Edwards Hamiltonian. 

It can be shown that the Edwards Hamiltonian extended to include the ternary... 

It is now possible to account qualitatively for the dumbbell formation seen in Fig. 7.38, by a theory that computes effective renormalized interactions from the Edwards Hamiltonian. However, a quantitative description of these data presented in Figs 7.17(a) and 7.38 is still lacking. While experimental results have also been interpreted in terms of chain stretching, it is important to recall that experiments up to now could measure S coii( ) only an independent measurement of both q and Rg for the same material has thus far not been feasible. This fact again illustrates one advantage of simulation, namely that it makes a much more detailed insight possible in that all quantities of interest can be measured simultaneously from one model system. 

The free energy of the isolated and flexible polyelectrolyte chain is obtained by the variational method [1, 21] starting from the Edwards Hamiltonian,... 

There is a variety of computer simulation methodologies that have been applied to study the properties of grafted polymer layers. They include Monte-Carlo (MC) simulations, Molecular Dynamics (MD) simulations, and Brownian Dynamics (BD) simulations. These methodologies have been applied on a variety of model systems, including lattice chains, off-lattice chains, and Edwards Hamiltonians. A recent excellent review of the application of simulation methods has been written by Grest and Murat. Here we just discuss the applicability of the simulations methods in general, their advantages and limitations. 

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