Screening by mobile charges

Next consider the implications of the existence of mobile charge carriers in the system. These can be ions in an electrolyte solution or in molten salts, electrons in metals and semiconductors, and electrons and ions in plasmas. For specificity we consider an ionic solution characterized by bulk densities n and of positive and negative ions. The ionic charges are 

Consider now such a semi-infinite system, confined on one side by an infinite planar surface, and assume that a given potential I .s is imposed on this surface. The interior bulk potential is denoted t g. Having h.s implies that the mobile 

In what follows we take that the direction perpendicular to the surface and pointing into the solution as the positive x direction. At any point the potential S (x) = (x) — may be found as the solution of the Poisson equation (1.226), written in the form 

In the second equality we have used Eq. (1.245). The densities rt+/ (x) are related to their bulk value by the Boltzmann equilibrium relations 

We continue with the assumption that the conditions for expanding the exponential Boltzmann factors to linear order as in (1.248) hold, and that the expansion to first order is valid. Using this together with (1.245) in (1.247) leads to 

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We observe curves of two different types, with and without a plateau region inside the polymer phase. Such a plateau must exist at all nonzero values of kx and kx or k x when the thickness d is large enough so that the electric field is screened by mobile charge carriers. At a finite thickness existence of the plateau depends on the ratio between d and the screening length of the polymer phase... 

J. N. Chazalviel, Coulomb Screening by Mobile Charges, Birkhauser, Boston, 1999. 

In accord with experiments on emulsions (Husband et al., 1997), the molecular configurations deduced from SCF calculations have demonstrated the crucial role of the cluster ( blob ) of 5 charged phosphoserine residues in p-casein in maintaining the steric stabilizing layer, whilst also preventing interfacial precipitation (multilayers). The mobility of this blob was demonstrated experimentally by P NMR measurements on P-casein-stabilized emulsions (ter Beek et al., 1996). It was inferred that, when the effective charge on the blob is reduced (by dephosphorylation) or screened (by salt addition), the macromolecular spring relaxes... 

Driven by a Boltzmann thermal-energy source kT and measured in kT units, coupled by a number of effective mobile ionic charges rs from Gibbs, the monopole-monopole correlation force is screened by Xnebye across the length 21, back and forth between point particles. At the same time its power-law dependence on length is measured in the natural thermal unit >.Bj. Boltzmann, Gibbs, Debye, Bjerrum—all at the same time. Can it get any prettier ... 

The seasoned Debye-Hiickel (D-H) theory, put forth in 1923 [33,34] takes into account the contribution of the ionic electrostatic interactions to the free energy of a solution and provides a quantitative expression for the activity coefficients. The basic concept of the D-H theory is that the long-range Coulomb interaction between two individual ions bathed in a salt solution is mediated by mobile ions from the solution. The effective charges of a certain ion are decreased as the result of charge screening by the mobile counterions it follows that, at sufficient distance, the interaction between two ions decays exponentially. We briefly outline the main considerations and assumptions of the D-H model ... 

The effective polarizability should still be given by Eq. (8-2) when an ac field is applied whose frequency co is fast compared to the inverse of the Maxwell-Wagner time fMW required for the mobile charges to screen the particle dipoles. When /mww 1> however, should be given instead by Eq. (8-2a). The Maxwell-Wagner time is given by (Parthasarathy and Klingenberg 1996) ... 

Erom the summary of the theoretical results presented above, it follows that the intramolecular volume of a star-branched PE, with a sufficiently large number of arms, is essentially electroneutral. That is, the bare charge of a star polymer is neutralized by mobile counterions. These counterions are predominantly retained inside the macroion volume, even if the star is immersed in a dilute salt-free solution. Moreover, if the intrinsic Debye length associated with the intramolecular concentration of entrapped counterions, = l paNis used as an upper estimate for the intramolecular electrostatic screening length, one finds that in the osmotic PE star, p a), the electrostatic interactions are screened at... 

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