Debye-Hiickel potential

Meeron60 62 first pointed out how the terms in S(Jt> in the solution theory can be arranged in a form much more compact than that above, which is of the form of a virial expansion in which the coefficients involve the Debye Hiickel potential of average force rather than the unscreened potential. Similar manipulations can be made in the present case, but we shall omit the details, which are very simple, and quote only the final result. It is found using Meeron s form of S that the activity coefficient of defect number s can be written... 

Here R is actually R, — Ry. We take the potential /y as the Debye-Hiickel potential. 

The predicted necklace conformations have been confirmed by simulations using only the Coulomb repulsion of the backbone charges [75], a Debye-Hiickel potential [75] and by simulations using the full Coulomb potential and explicit counterions [76]. In the following we will only review the simulations using explicit counterions. 

Since neutral block copolymer micelles are a convenient model system for the investigation of the steric stabilization potential, polyelectrolyte block copolymer micelles may serve to study electro-steric interaction. Similar to the case of neutral block copolymer micelles, the interaction potential u(r) can be probed by measuring the shear modulus of micellar gels as a function of distance. Assuming a simple Debye-Hiickel potential yields for the shear modulus by taking the second derivative... 

Here, T is the absolute temperature, e is the bulk dielectric constant of the solvent, P is the number of phosphate charges, k is the inverse of the Debye screening length, kB is the Boltzmann constant, qna is the renormalized charge, the interaction between the charges is screened Debye-Hiickel potential, and — j b is the distance between a pair of charges labeled i and /... 

At a large distance R from the micelle where (x0) << 1, the Debye-Hiickel potential (R) is adopted in order to start the calculation... 

The Debye-Hiickel potential needs modification for high values of ze. The second virial coefficient suitable for this case has been evaluated... 

Figure 13. The screened Coulomb potentials [11c] The Naive Approximation (full curve) is compared with the Debye-Hiickel potential (dashed curve), both at c=50mM. The unscreened potential (wide full curve) is also shown for comparison.

The critical charge density (13) is only half of the value provided in publications by Wiegel and Muthukumar [40,41]. This is related to the calculation of the surface potential, which is quite complicated in general [76, 154], We use the solution of the Poisson-Boltzmann equation in the limit of a strong electrolyte [154] instead of the large separation Debye-Hiickel potential [40, 41], The latter underestimates the potential at contact by a factor of two. 

The critical p presented in Fig. 4 is determined using the Hulthen potential rather than the Debye-Htickel potential. The above discussion demonstrates that, in the small curvature limit, the difference between the Debye-Hiickel potential and the Hulthen potential vanishes. From the following considerations, it will... 

Again, the first two terms are satisfied by the function (23) and flie remaining part disappears at r = a for = 2. Hence, the critical value p = 2is identical for the Hulthen and the Debye-Hiickel potential in the limit ka I and r — a)/a 1. 

The exact solution for the Debye-Hiickel potential predicts a linear dependence of the critical colloid charge density on the inverse Debye screening length for Kfl 1 (Fig. 5). This is different from the predicted dependence IcTcI based on... 

The electrostatic potential energy between two ions given by Equations 3.19 and 3.20 is called the Debye-Hiickel potential energy. The collective effect of the ions in the solution is to screen the Coulomb interaction between a pair of ions given by Equation 3.5 resulting in the screened electrostatic interaction given by Equation 3.19. 

The electric potential given by Equation 3.33 is called the extended Debye-Htickel potential and is plotted in Figure 3.3b. For comparison, the result of Equation 3.18 is included in the figure. The finite size of the ion modifies the prefactor by a factor of exp(iefl)/(1+ku), and the beginning of the decay of the potential is shifted to the ion diameter. The distance dependence is the same as the screened Coulomb potential for r > a. For a 0, Equation 3.33 reduces to the Debye-Hiickel potential. 

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