Apparent charge density

For given to value the apparent charge density cr(r, to) is available in terms of the extended PCM procedure with a complex-valued dielectric function s, namely, e(w) = s1((o) + is2((o) where e w) = 1 +4ttXi(oj) and s2(oj) = 4ttx2(m) with complex-valued susceptibilities defined in Equation (1.127). The complication that both a(r, to) and 0(r, to) become complex is inevitable. However, after applying the inverse Fourier transform, they become real in the time domain. This is warranted by the symmetry properties,... 

This approach, based on a complex-valued realization of the PCM algorithm, reduces to a pair of coupled integral equations for real and imaginary parts of apparent charge density for tr(f,to) [13]. An alternative technique avoiding explicit treatment of the complex permittivity has been also derived [14,15]. The kernel K(f,f, t) of operator K does not appear explicitly. However, its matrix elements can be computed for any pair of basis charge densities p1(r) and p2(r) px k p = Jp2(j) (r, f)d3r, where tp(r, t), given by Equation (1.137), corresponds to p(r) = p2(r). 

The apparent charge density, which is ultimately responsible for the double layer interaction, depends strongly on the hydration potential of ions. Although the double layer interaction is always repulsive, for some electrolytes o can become sufficiently large to prevent coagulation, while for other electrolytes sufficiently low to allow coagulation. Therefore, the ion-hydration interaction can provide a suitable explanation for the specific ion effects observed in double layer interactions. 

Manning assumes that condensation of counterions occurs to prevent the divergence of this partition function. The net result will be to reduce the apparent charge density on the polyion, until the value of after condensation will be equal to the critical value l/Z. In the case of DNA, two consecutive pairs of phosphates are separated by = 17Qpm, then = 4.2, so that any ion, regardless to its charge will condense on DNA, until the effective value of will be reduced to l/Z. 

Sample Shape Diameter (nm) Zeta Potential/ mV Apparent charge density/ mV/nm... 

We can use Eq. [94] to obtain a condition under which the Debye-Hiickel equation may be reliably applied to a planar system. As the ratio of the apparent charge density to the actual charge density approaches unity, the ADH surface potential approaches the PB (and DH) value. If we solve Eq. [94] for the ratio cja/cjo and, using Eqs. [27] and [11], insert this into the value of the surface potential given by Eq. [28], we obtain the expressions... 

Figure 10 displays both functions in Eq. [95] - the ratio a /ao (solid line, top frame) and the sruface potential (solid line, bottom frame) as a function of the ratio CTADH/f fl- (We assume that the svuface potential is positive, although this does not affect the analysis.) As aAOH approaches csg, the ADH potential approaches the PB potential, implying that the DH equation gives an increasingly more accruate description of the system. Let us assiune that we are satisfied with the DH representation if we are within a tolerance limit of > 0.95, that is, the apparent charge density is less than 5% smaller... 

By comparing the actual charge density to the apparent charge density in Fq. [94], we can define the fraction of surface charge neutralized by counterions, /iieutj according to the apparent Debye-Hiickel solution, as... 

In the low-charge-density Debye-Hiickel limit, the scaled apparent charge density si (and S2) reduces to the scaled charge density di of Eq. [201]. 

Figure 35 The apparent charge density as a function of actual linear charge density for a sphere of radius 20 A in a mixed 1 1-2 1 electrolyte with concentration 0.1-0.02 M within the NLDH (dashed lines Eq. [312] with the exact C or approximate C values) and PGC (solid lines Eq. [365] with exact and approximate 5dgc values) approximations. The PGC low and high charge density approximations (obtained from Eqs. [324]) are shown by dotted lines the infinite-charge density limit is shown by the horizontal dotted-dashed line.

Figure 40 The interaction potential for two spheres with radii 20 and 10 A and charge densities of 0.01 and 0.001 cqIA, respectively, in a 0.1 M 1 1 electrolyte as a function of surface separation according to the HHF (solid line Eqs. [347] and [345] without the Overbeek correction), mHHF (long dashed line Eqs. [347] and [349]), mHHF with apparent charge densities (short dashed line Eq. [350]), lowest-order asymptotic PB (dotted-dashed line first term in Eq. [332]), and extended PB (dotted line Eq. [352]) expressions.

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