Reciprocal Debye radius

Dimensional Analysis The constant /I in Equation 1.31 is the reciprocal Debye radius. 1116 Dimensional analysis of this term is instructive because it involves a number of often needed constants and occurs frequently in a variety of contexts. 

When charged colloidal particles in a dispersion approach each other such that the double layers begin to overlap (when particle separation becomes less than twice the double layer extension), then repulsion will occur. The individual double layers can no longer develop unrestrictedly, as the limited space does not allow complete potential decay [10, 11]. The potential v j2 half-way between the plates is no longer zero (as would be the case for isolated particles at 00). For two spherical particles of radius R and surface potential and condition x i <3 (where k is the reciprocal Debye length), the expression for the electrical double layer repulsive interaction is given by Deryaguin and Landau [10] and Verwey and Overbeek [11],... 

K has the dimension of reciprocal length. This is the reason for referring to k as the Debye radius. As we shall soon see, this may be interpreted as the effective radius of the ionic atmosphere. 

Figure 2.31. The origin of the powder diffraction eone as the result of the infinite number of the completely randomly oriented identical reciprocal lattice vectors, d hki, forming a circle with their ends placed on the surface of the Ewald s sphere, thus producing the powder diffraction cone and the corresponding Debye ring on the flat screen (film or area detector). The detector is perpendicular to both the direction of the incident beam and cone axis, and the radius of the Debye ring in this geometry is proportional to tan20.

A Debye-Scherrer camera consists of a metal cylinder provided with a photographic film. The primary beam is perpendicular to its axis. The distance between two symmetrical lines, produced by the intersection of a cone with the cylinder, is 46R, 6 being the Bragg angle (in radians) and R the radius of the camera. The interval is derived from Bragg s law. The powder method gives us only the norms of the reciprocal vectors. The set of norms corresponds to the projection of the reciprocal lattice onto a straight line. 

A key quantity in the Debye-Huckel theory, leading to the values of the constants A and B, is the screening length, k, the average reciprocal of the radius of the ionic atmosphere surrounding an ion in the solution, made up essentially by ions of the opposite charge. The square of this quantity is proportional to the ionic strength of the solution and also to the reciprocal of the product w T ... 

Quantity k is the reciprocal radius of the unperturbed Debye ion cloud it is given by the relationships (22). 

In equation (2), is the relative dielectric constant of the continuous phase, is the absolute permittivity of free space, a is the particle radius, tjf is the surface potential, and d is the closest surface-to-surface separation. The quantity k, the reciprocal of the Debye length, is defined by... 

N is the number of charged groups per particle of hydrodynamic radius R e is the electron charge K is the reciprocal of Debye length thickness n is the viscosity of the medium. 

The quantity p (r), the fraction of charge between the spherical shells of radius r and r + dr, is plotted against r in Fig. 7. We can see from this figure that a central ion is surrounded by an atmosphere of ions of opposite sign, and that the thickness of this ionic atmosphere is 1/K, the reciprocal of the Debye-Huckel parameter. 

In addition to neglecting ion correlation, using the mean electrostatic potential has the undesirable consequence that the (nonlinear) PB equation no longer satisfies a reciprocity condition that use of the potential of mean force would obey. Linearization of the equation by Debye and Hiickel regained this condition. These considerations led Outhwaite and others to propose modifications of the PB equation to treat these problems. Within this modified Poisson-Boltzmaim (MPB) theory, the effect of ion correlation is expressed in terms of a fluctuation potential for which a first-order (local) expression, written as an activity coefficient, can be derived. Their result for bulk hard-sphere electrolyte ions of valence z, and common radius a gives the formula ... 

As is well known, this is the Debye reciprocal distance k which describes the radius within which the counter-ion can be expected to be located and the ionic strength of the system which determines the value of k. 

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