Debye length defined

It is the overlap of the profiles that gives rise to a repulsive "osmotic pressure" which gives a form Force f(a) e, where f(a) is a fimction of surface charge (or potential) and k is the inverse Debye length, defined above. 

Just to complicate matters further, we remark on the range of double-layer forces. All theories take it as axiomatic that the force between surfaces or between charged molecules should decay as exp(-jd) where is the Debye length defined by... 

The barrier thickness yQ evidently has the character of a screening length. How, then, does it differ from the conventional Debye lei th The two calculations are based on somewhat different models. That for the Debye ler th sets no limit to the local carrier concentration the barrier calculation which leads to eqn. (1) does set such a limit, namely the charge density accommodated in donors, and this assumed to be constant throughout the barrier. Accordingly, the Debye length defines an exponentially decaying potential contour, whereas the present barrier system yields a parabolic system. 

The same effect happens inside a random flight chain where the close proximity of the polymer segments offers mutual screening from the bulk flow field. The idea of a chain being non-drained was first considered by Debye Bueche who introduced the concept of a shielding length defined as [46] ... 

With electrochemically studied semiconductor samples, the evaluation of t [relation (39)] would be more straightforward. AU could be increased in a well-defined way, so that the suppression of surface recombination could be expected. Provided the Debye length of the material is known, the interfacial charge-transfer rate and the surface recombination... 

One of the main assumptions of the Donnan partition model is that two well-defined phases (polymer and solution) exist and the electrostatic potential presents a sharp transition between them. This approximation is fulfilled when the typical decay length of the electrostatic potential (Debye length) is much shorter than the film thickness. The other limiting situation is that where all the redox sites are located in a plane and thus the Debye length is larger than the film thickness. This situation can be described by the surface potential model ... 

We shall now consider what happens when the film thickness is of the order of the Debye length. In such a situation, no analytical expressions can be derived and numerical calculations should be used [125]. The real situation could be even more complicated, since an ill-defined film thickness can exist, like the example in Figure 2.6. We can use the molecular theory to obtain a self-consistently determined electrostatic potential profile across the interface as was shown in Figure 2.7 (see... 

On the basis of this description, a relationship between the two lengths 8 and K can be established. Different 5 values are obtained by gradually increasing the amount of micelles and fitting the force profiles. The evolution of 5 as a function of the calculated Debye length is plotted in Fig. 2.8. The thickness 5 increases linearly with The inherent coupling between depletion and doublelayer forces is reflected by this empirical linear relationship which is a consequence of the electrostatic repulsion between droplets and micelles. The thickness 5 may be conceptually defined as a distance of closer approach between droplets and micelles and thus may be empirically obtained by writing ... 

Now, for convenience (and not arbitrarily, as will be seen later), let us replace the real distance x with a scaled distance X, such that X = Kx and K", which is called the Debye length, is defined as ... 

Of course, the borderline between the two pictures (the one-phase and the two-phase ones) becomes diffused when the second transforms into the first with the decrease of the typical pore radius. In fact, as will become clear in Chapter 6, 6.4, distinction between the phases becomes meaningless as soon as the typical pore radius becomes shorter than some typical electro-diffusional length scale—the Debye length—to be defined below. 

What are the correct values of the potentials In the metal the potential is the same everywhere and therefore 99 has one clearly defined value. In the electrolyte, the potential close to the surface depends on the distance. Directly at the surface it is different from the potential one Debye length away from it. Only at a large distance away from the surface is the potential constant. In contrast to the electric potential, the electroc/zmz caZpotential is the same everywhere in the liquid phase assuming that the system is in equilibrium. For this reason we use the potential and chemical potential far away from the interface. 

Rh is the hydrodynamic radius of the analyte, k is the inverse of the Debye length, r is the viscosity of the separation buffer, e is the fundamental unit of charge, and ft is a function that describes the effect of the molecule (or particle) on the electric field and is defined between two limits (i) the Htickel limit,/ = 1 when k,Rh < 1 (when the hydrodynamic radius is lower than the Debye length) and (ii) the Helmholtz-Smoluchovski limit, fi= /2 when k,Rh > 10 (when the hydrodynamic radius is higher than the Debye length). Between the limits / is calculated from the following equation ... 

Ramanathan-Woodbury carried out such an analysis and showed that if d, but in which kL = 0(1) [33], then there is critical value of a dimensionless charge density, defined as %0 = log(Ku) 1/log(L/d), beyond which counterions will condense. Furthermore, if the length of the polyion is of the order of the Debye length, then the charge density is independent of the Debye screening parameter [33]. These predictions are in accord with experimental results [37-39]. 

In Eq. (11) we have introduced the Debye screening length defined as... 

The quantity I /at has units of length and is called the Debye length it defines the extent of the double layer, i.e., the distance in which the potential decays to I je of its initial value k is called the Debye-Huckel parameter. Hence within validity of this approximation (low surface potentials < 25 mV) the potential decreases exponentially away from the surface. 

Here and t] are, respectively, the relative permittivity and the viscosity of the electrolyte solution. This formula, however, is the correct limiting mobility equation for very large particles and is valid irrespective of the shape of the particle provided that the dimension of the particle is much larger than the Debye length 1/k (where k is the Debye-Htickel parameter, defined by Eq. (1.8)) and thus the particle surface can be considered to be locally planar. For a sphere with radius a, this condition is expressed by Ka l. In the opposite limiting case of very small spheres (Ka 3> 1), the mobility-zeta potential relationship is given by Hiickel s equation [2],... 

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