Local electric charge density potential

Here, pe is the local electric charge density in C/m3. With the Poisson equation, the potential distribution can be calculated once the exact charge distribution is known. The complication in our case is that the ions in solution are free to move. Before we can apply the Poisson equation we need to know more about their spacial distribution. This information is provided by Boltzmann5 statistics. According to the Boltzmann equation the local ion density is given by... 

Apart from the electroactive and electroinactive species in solution (A, B, M and X), we also need to include the description of the electric potential in the simulation. The Poisson equation relates the potential 4> with the local electric charge density, p (C cm ) ... 

The electrical state of the system is so sensitive to small changes in composition [9] that the gradient Vtp in Eq. (7) cannot be assumed, in general, to be a simple constant [12], and an additional equation is required to determine it. Given the slowness of particle motion in solution, it is justified to use the Poisson equation from electrostatics to relate the changes in electric potential to the local electric charge density pe... 

The ion and electrical potential distributions in the electrical double layer can be determined by solving the Poisson-Boltzmann equation [2,3]. According to the theory of electrostatics, the relationship between the eleetrieal potential ij/ and the local net charge density per unit volume at any point in the solution is deseribed by the Poisson equation ... 

Solving the Poisson-Boltzmann equation with proper boundary conditions will determine the local electrical double layer potential field y/ and hence, via Eq.(3), the local net charge density distribution. 

It is assumed that the electric charge density is not affected by the external electric fields due to the thin EDLs and small fluid velocity therefore, the charge convection can be ignored, and the electric field equation and the fluid flow equation are decoupled. Based on the assumption of local thermodynamic equihbrium, for small zeta potential, the electric potential due to the charged wall is described by the linear Poisson-Boltzmann equation which can be written in terms of dimensionless variables as... 

Below we present a well-known calculation of membrane potential based on the classical Teorell-Meyer-Sievers (TMS) membrane model [2], [3]. The essence of this model is in treating the ion-selective membrane as a homogeneous layer of electrolyte solution with constant fixed charge density and with local ionic equilibrium at the membrane/solution interfaces. In spite of the obvious idealization involved in the first assumption the TMS model often yields useful results and represents in fact the main tool for practical membrane calculations. We shall return to TMS once again in 4.4 when discussing the electric current effects upon membrane selectivity. In the case of our present interest, the simplest TMS model of membrane potential for a 1,2 valent electrolyte reads... 

Now we assume that only electric work has to be done. We neglect for instance that the ion must displace other molecules. In addition, we assume that only a 1 1 salt is dissolved in the liquid. The electric work required to bring a charged cation to a place with potential -ip is W+ = etf). For an anion it is W = -ertp. The local anion and cation concentrations c and c+ are related with the local potential ij) by the Boltzmann factor c = c0 ee /fesT and c+ = co e e /kBT. Here, cq is the bulk concentration of the salt. The local charge density is... 

Figure II 2 9a-s. The valence electron iso-density lines in the plane of B atoms (a-b plane) for equilibrium (a) and distorted structures (b-e). The electron density is localized at B atom positions for equilibrium structure (a). The B atoms displacements ( Af = 0.005) induce the alternating interatomic charge density delocalization, different for the particular types of the distortion (b-d). Nuclear microcirculation enables then effective charge transfer over the lattice in an external electric potential. The Fig (e) corresponds to the case of the distortion (d) over the larger lattice segment...

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