Ionic concentration, distribution potential

Instead of an exact calculation, Gouy and Chapman have assumed that (4) can be approximated by combining the Poisson equation with a Boltzmann factor which contains the mean electrical potential existing in the interface. (This approximation will be rederived below). From this approach the distribution of the potential across the interface can be calculated as the function of a and from (2) we get a differential capacitance Cqc- It has been shown by Grahame that Cqc fits very well the measurements in the case of low ionic concentrations [11]. For higher concentrations another capacitance in series, Q, had to be introduced. It is called the inner layer capacitance and it was first considered by Stern [1,2]. Then the experimental capacitance Cexp is analyzed according to ... 

The effect of surface potential on interfacial ionic concentration is given by a Boltzmann distribution relating the total solution concentration to that at the interface. For charged, amphiphilic species, binding constants replace these ionic concentrations, and the expression... 

A related phenomenon occurs when the membrane in the above-mentioned experiment is permeable to the solvent and small ions but not to a macroion such as a polyelectrolyte or charged colloidal particles that may be present in a solution. The polyelectrolyte, prevented from moving to the other side, perturbs the concentration distributions of the small ions and gives rise to an ionic equilibrium (with attendant potential differences) that is different from what we would expect in the absence of the polyelectrolyte. The resulting equilibrium is known as the Donnan equilibrium (or, the Gibbs-Donnan equilibrium) and plays an important role in... 

Figure 13 shows the potential and concentration distributions for different values of dimensionless potential under conditions when internal pore diffusion (s = 0.1) and local mass transport (y = 10) are a factor. As expected the concentration and relative overpotential decrease further away from the free electrolyte (or membrane) due to the combined effect of diffusion mass transport and the poor penetration of current into the electrode due to ionic conductivity limitations. The major difference in the data is with respect to the variation in reactant concentrations. In the case when an internal mass transport resistance occurs (y = 10) the fall in concentration, at a fixed value of electrode overpotential, is not as great as the case when no internal mass transport resistance occurs. This is due to the resistance causing a reduction in the consumption of reactant locally, and thereby increasing available reactant concentration the effect of which is more significant at higher electrode overpotentials. 

Hung derived a general expression for calculating the distribution potential from the initial concentrations of ionic species, their standard ion transfer potentials, and the volumes of the two phases [19]. When all ionic species in W and O are completely dissociated, and the condition of electroneutrality holds in both phases, the combination of Nernst equations for all ionic species with the conservation of mass leads to... 

In Eq. (3), summations are taken for all ionic species. The only unknown in Eq. (3) is Aq 0- By knowing initial concentrations of ionic species and their standard ion transfer potentials, together with the volumes of W and O phases, the distribution potential can be calculated by solving Eq. (3). The concentration of each ionic component at a distribution equilibrium can then be obtained through Aq 0 using the relation... 

In this limiting case, the distribution potential, and hence the equilibrium concentrations of ionic species, do not depend on the volumes of W and O. Equation (7) is useful in calculating the distribution equilibria in systems involving small particles, e.g., emulsions and thin membranes. 

The derivation presented here gives a greatly oversimplified picture of the actual situation in a membrane with fixed ion exchange sites. Use of the Henderson equation implies that the diffusion process in the membrane has reached a steady state with a linear concentration distribution. This is certainly not the case for solid membranes such as glass which are thick with respect to the diffusion length of the ion. Moreover, it is probably not valid to assume that the ionic mobility is independent of position in the membrane. More complex models for the membrane potential have been developed but they lead to essentially the same result. More details can be found in monographs devoted to this subject [9]. 

The ion activities employed in Eq. (5) should be those established after equilibration of the two phases It is generally assumed that the equilibrium distribution values can be approximated by the initial ionic concentrations. The validity of this approximation is dependent on the standard transfer potentials of the counterions, and in practical cases it will only hold over a restricted range ( 0.1— 0.2 V [19]). The variance of such a system means that its Galvani potential difference is defined, over the above working range, through the ratio of activities of X. 

For many years, it has been known that many individual biological cells maintain different distributions in ionic concentrations and an electrical potential difference between their intracellular and extracellular phases at the resting state (Table 19). In some cells, upon application of an appropriate stimulus (electrical depolarization or chemical stimulus), the cells exhibit a time-dependent response via a potential difference across the cell membranes which does not necessarily follow Ohm s law. The former potential is called a resting membrane potential and the latter an excitation potential. We would like to review the origins of these electrical potential differences across the cell membrane. 

The functional depends on three types of fields, namely, surface fluctuations (R), electrostatic potential < (z, R), and the ionic concentrations rtf (z, R). Minimizing it with respect to < i and nf at a given f(R), one obtains Poisson-Boltzmann equations that describe the distribution of the electrostatic potential and ionic concentrations. Substituting the results into the density functional, one can minimize that with respect to f (R). This... 

Erom the analysis presented as above, it is apparent that the surface potential (or, equivalently, the C potential, in an approximate sense) plays a key role in determining the potential distribution within the EDL. The potential, in turn, depends on the bulk ionic concentration and the pH [1]. The influence of ionic concentration can be quantitatively assessed by rewriting the Boltzmann distribution of ionic charges as... 

To begin with the fundamental analysis of EDL interactions, let us first consider the distribution of potential between two charged infinite parallel plates at a distance of 2H apart and subjected to constant surface potentials (refer to Eig. 1). Assuming a Boltzmann distribution of the respective ionic concentrations, one can write, for a symmetric 1 1 electrolyte (see entry Electrical Double Layers ),... 

Fu et al. [4] presented a numerical model for electrokinetic dispensing in microfluidic chips where simple cross, double-T, and triple-T configurations were considered. In this model, the Nemst-Planck equation was employed to describe the ionic concentration instead of the Boltzmann distribution, which is a more general approach. The model was numerically solved using the finite difference method where the artificial compressibility method was employed to deal with the pressure term in the N-S equation. It is found that the applied potentials play an important role in crnitroUing the loaded and dispensed sample shape. The unique feature of this study is the concept of the multi-T injection system which can function as a simple cross, double-T, or triple-T injectimi unit. Their numerical results agreed well with their experimental results. More injection techniques were also developed by the same group later. 

The electrokinetic effects are completely different in the microchannels and the nanochannels. In the microchannels, the velocity field is independent of the size of the channels, the electric potential distributes linearly along the microchannel, and the bulk ionic concentrations of the co-ions and the counterions are the same. These conditions are not valid for the nanochannels. 

Equation 1 is the Poisson equation. This equation should be solved in order to obtain the electric potential distribution in the computational domain. On the right hand of this equation, the term F Z] iZ,c, shows the gradient influence of the co-ions and counterions on the electric potential inside the domain. The electric field is the gradient of the electric potential (Eq. 2). Equation 3 is the Nemst-Planck equation, where the definition of ionic flux is given by Eq. 4. On the right-hand side of this equation, (m c,), (D, Vc,), and (z,/t,c,V( ) represent flow field (the electroosmosis), diffusion, and electric field (the electrophoresis), respectively, which contribute to the ionic mass transfer. The ionic concentrations of each species can be found by solving these two equations. Equations 5 and 6 are the Navier-Stokes and the continuity equations, respectively, which describe the velocity field and the pressure gradient in the computational domain. 

Since the ionic concentration has a great effect on the electric potential distribution in microchannels [13], we can model the flow rate versus the ionic concentration in an electroosmotic micropump packed by nanoscale charged porous media when E = I kV/m, AP = 0 Pa/m, jjp = —50 mV, and dp = 108 nm. 

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