Mean field electrostatics approximation

The concentration of salt in physiological systems is on the order of 150 mM, which corresponds to approximately 350 water molecules for each cation-anion pair. Eor this reason, investigations of salt effects in biological systems using detailed atomic models and molecular dynamic simulations become rapidly prohibitive, and mean-field treatments based on continuum electrostatics are advantageous. Such approximations, which were pioneered by Debye and Huckel [11], are valid at moderately low ionic concentration when core-core interactions between the mobile ions can be neglected. Briefly, the spatial density throughout the solvent is assumed to depend only on the local electrostatic poten-... 

In addition to the nearest-neighbor interaction, each ion experiences the electrostatic potential generated by the other ions. In the literature this has generally been equated with the macroscopic potential 0 calculated from the Poisson-Boltzmann equation. This corresponds to a mean-field approximation (vide infra), in which correlations between the ions are neglected. This approximation should be the better the low the concentrations of the ions. 

The ASEP/MD method, acronym for Averaged Solvent Electrostatic Potential from Molecular Dynamics, is a theoretical method addressed at the study of solvent effects that is half-way between continuum and quantum mechanics/molecular mechanics (QM/MM) methods. As in continuum or Langevin dipole methods, the solvent perturbation is introduced into the molecular Hamiltonian through a continuous distribution function, i.e. the method uses the mean field approximation (MFA). However, this distribution function is obtained from simulations, i.e., as in QM/MM methods, ASEP/MD combines quantum mechanics (QM) in the description of the solute with molecular dynamics (MD) calculations in the description of the solvent. 

Two infinite-size plates are immersed in a semidilute solution of polyelectrolyte in a good solvent which also contains the small ions of a salt. One of the plates is located at x=0 and the other one at x=D. The system is considered to be in contact with a large reservoir, which contains a polyelectrolyte/salt solution. In addition to the electrostatic interactions, the segments of the polymer have a van der Waals interaction —UkT with the plates. In the mean field approach, the intra- and interchain interactions together with the electric field induced by the surface charges of the plates and polyelectrolyte molecules are expressed as an external potential. Within the mean-field approximation, the free energy of the system with respect to that in the reservoir can be expressed as the sum of three contributions, the polymer contribution Fpol, the salt ions contribution Fion and the electrostatic field contribution Fels,... 

ASEP/MD, acronym for average solvent electrostatic potential obtained from molecular dynamics data, is a sequential QM/MM method that makes extensive use of the mean field approximation (MFA) [24], In solution, any static property A of the system must be calculated by averaging over the configurational space A defined by all the configurations thermally accessible to the system ... 

In ideally one-dimensional systems, only intrachain electron-phonon and spin-phonon couplings are, within mean-field approximation, at the origin of electronic-Peierls and/or spin-Peierls transitions, respectively. In real systems, such as the TCNQ salts under concern here, it is clear, however, that one should take properly into account the coupling of the electrons to external potentials also and, in the first case, to the periodic electrostatic cation potential. 

It should be appreciated that in contrast to the simple free electron models used in much of our discussion of metals and semiconductors, a treatment of screening necessarily involves taking into account, on some level, the interaction between charge carriers. In the Thomas-Fermi theory this is done by combining a semiclassical approximation for the response of the electron density to an external potential with a mean field approximation on the Hartree level—assuming that each electron is moving in the mean electrostatic potential of the other electrons. 

If one accepts the continuum, linear response dielectric approximation for the solvent, then the Poisson equation of classical electrostatics provides an exact formalism for computing the electrostatic potential (r) produced by a molecular charge distribution p(r). The screening effects of salt can be added at this level via an approximate mean-field treatment, resulting in the so-called Poisson-Boltzmann (PB) equation [13]. In general, this is a second order non-linear partial differential equation, but its simpler linearized form is often used in biomolecular applications ... 

There have been many studies of the effect of added electrolytes on ET rates [171, 234, 235], The main effect of ionic atmosphere is electrostatic screening, which is usually accounted for in terms of Debye-Huckel theory (mean-field, low-concentration approximation). At sufficiently low ionic strength, the corresponding component of the activation energy is simply proportional to the ratio of the Onsager radius (also referred to as the... 

One approach is a discrete multi-Stem layer model [117-121], where the system is placed on a lattice whose sites can be occupied by a monomer, a solvent molecule, or a small ion. The electrostatic potential is determined self-consistently (mean-field approximation) together with concentration profiles of the polymer and small ions. 

In summary, the computer simulation of micelles with shells formed by an annealed PE is a combination of MC with the self-consistent field treatment of electrostatic forces. However, it goes beyond the mean-field approximation. It is also evident from simulation snapshots (see below) that the a priori assumption of the spherical symmetry of the electrostatic field (which is an inherent feature of studied micelles) does not impose a strong constraint on instantaneous chain conformations. 

The BBB model is a means of macroscopic approximation to the system on an exclusively electrostatic basis it describes the solvation effects with the aid of classical field theories, primarily electrostatics and hydrodynamics. The nomenclature BBB, an abbreviation of the expression Brass Balls in a Bathtub , originates from Frank [Fr 65]. The more important classical theories included in this group have led to a result only for dilute solutions nevertheless, their refinement has continued up to the present [Ab 79, Be 78, Kr 79, Li 79]. 

The redox species can move about their equilibrium position of the irreversible attachment with the polymer (in the three-dimensional network the redox species are either covalently or electrostatically bound), which is referred to as bounded diffusion. In the opposite extreme (free diffusion), rapid molecular motion thoroughly rearranges the molecular distribution between successive electron hops, thus leading a mean-field behavior. The mean-field approximation presupposes that k > k, and leads to Dahms-RufF-type behavior for freely diffusing redox centers, but the following corrected equation should be applied [28] ... 

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