Poisson equation equilibrium

It will be assumed that, precisely as in the equilibrium situation, these potentials may be calculated with help of the Poisson equation ... 

The Poisson-Boltzmann equation. Equilibrium is a steady state without macroscopic fluxes. As we pointed out in the Introduction, under these conditions the equations of electro-diffusion reduce to... 

Let us show that no fluid equilibrium is compatible with the electrolyte concentration distribution given by (4.4.55d). Indeed, associated with this concentration distribution, there is an electric potential distribution, prescribed by (4.4.53). This latter is in turn inseparable from an electric space charge, whose density may be found from the Poisson equations as... 

Starting from the Poisson equation and assuming Gaussian spatial distributions for ions and intrablob electrons with the dispersions and ae, prove that ae — cn <3C a and estimate A a = ae —on numerically. Assume the initial number of ion-electron pairs in the blob are no 30 and a, 40 A and e - is the dielectric permitivity of the medium. Note that at r a, out-diffusion flux of electrons is compensated for by their drift in the electric field of the ions (quasi-equilibrium condition). 

By combining the Poisson equation (6.5) and Eq. (6.87), we obtain the following Poisson-Boltzmann equation for the equilibrium electric potential i//(r) at position r, r being the distance from the particle center (r>a) ... 

Under a depletion condition, an inversion layer is formed when near the surface the minority carriers accumulate and are in equilibrium with those in the bulk (that is, the consumption rate of the carriers in the electrochemical reactions is low). When an inversion layer occurs, the density of minority carriers near the surface may exceed that of the bulk majority carriers. Under such a condition, when n x) < p x) within the space charge layer equation, the Poisson equation [Eq. (1.16)] becomes... 

Here the index I is used to denote a particular charge distribution (i.e. a particular electronic state of the system). The displacement field P/(r) represents a charge distribution p/(r) according to the Poisson equation V T>i = npi. In (16.79) D(r ) and the associated p(r) represent a fluctuation in the nuclear polarization, defined by the equilibrium relationship between the nuclear polarization and the displacement vector (cf. Eqs (16.14) and (16.15))... 

The elliptic equation describes a steady-state or an equilibrium process within the region. Physically, steady state is not attained unless the net rate of generation is balanced with the net flux into the region. The physics is manifested in the existence of the solution. Thus, the elliptic equation does not admit a solution unless the condition for the existence of a steady state is satisfied. For p = 0, the condition for the existence of a solution of the Poisson equation is... 

The result for u is then obtained simply by integrating the momentum equation (Eq. 6.5.7) subject to the no-slip boundary condition u=0 at r=a and the symmetry condition duldr = 0 at r = 0. Note that the Poisson equation is not used here it is replaced by the condition of equilibrium, that is, overall charge neutrality. Carrying out the integration, we get... 

If we substitute the latter expression into Poisson equation (2.112), we will obtain equality of the electrostatic equilibrium between concentration of ions and internal charge of the interface at the distance x... 

The solvation dynamics experiment of interest here is different Here at time t = 0 the charge distribution p(r) is switched on and is kept constant as the solvent relaxes. In other words, the dielectric displacement D, the solution of the Poisson equation VD = 4jip that corresponds to the given p is kept constant while the solvent polarization and the electrostatic field relax to equilibrium. To see how the relaxation proceeds in this case we start again firom... 

Taking into account Poisson equation relating the volume charge density p(r) to the equilibrium potential distribution, r) (recall that the geometry of the double layer is assumed to be the same as in the absence of applied field hence, P only depends on the radial coordinate r). Equation... 

Instead of the Nemst-Plank equation for the flux of ionic transportation, the equilibrium Poisson equation can be selected as the starting point for describing the electrical potential because the time required for cellular motion, electrophoresis, coagulation, and deposition is much longer than the diffusion equihbrium of ionic profile in interacting double layers [70-72]. The spatial variation in the electrostatic potential can be described by the Poisson equation... 

The basic assumption is that the excess non-equilibrium potential also satisfies the Poisson equation even if the equilibrium state is not established. It means that ... 

A particle charge of 1 has been assumed here. Three regimes for this equation are possible. First, if there is no current [j(r, ) = 0], Eq. [52] along with the Poisson equation [which determines the potential < )(r)] lead to the PB equation this is the equilibrium situation. Second, if the particle densities are not changing [p(r, t) = constant], the steady-state (Nernst-Planck) case is obtained ... 

The solution of the Poisson equation dramatically increases the complexity and duration of the numerical simulations and requires numerical shortcuts. In addition, Watson and Postlethwaite [2] mentioned that the time required for electrical neutrality to be reached appears to be less than 10 s. This is very short compared with the time required for chemical equilibrium, and it is not clear whether the simple assumption of electrical neutrality could significantly change the results of such models. 

The above discussion is based on the thermodynamics of equilibrium associations and on the reasonable assumptions. However, the Poisson equation can also be derived strictly mathematically. Consider a random distribution of r balls in q cells, where both of them are independent and indistinguishable. The probability P(i) that a specified cell contains exactly i balls is given in the form ... 

V its seen to relax from its initial value Sq to its final equilibrium value e with the characteristic relaxation time to. The experimental realization of this situation is, for example, a capacitor in which a dielectric solvent fills the space between two planar electrodes and a potential difference between the electrodes is suddenly switched on, then held constant while the solvent polarization relaxes. This relaxation proceeds at constant electric field (determined by the given potential difference divided by the distance between the electrodes). To keep the field constant as the solvent polarization changes the surface charge density on the electrodes must change—the needed charge is supplied by the voltage source. The Poisson equation,... 

These do not contain the variable t (time) exphcitly accordingly, their solutions represent equihbrium configurations. Laplace s equation corresponds to a natural equilibrium, while Poisson s equation corresponds to an equilibrium under the influence of an external force of density proportional to g(x, y). 

---