Manning parameter

A linear charge density parameter (Manning parameter) has been defined as ... 

In general, one of the characteristics of rod-like polyelectrolytes is the charge (Manning) parameter c which for monovalent counterions is defined through the ratio of the Bjerrum length Xp to the contour distance per unit charge b [22-24] ... 

These results have been generalized to planar and spherical surfaces [54]. For the later case, at high salt concentration, the charge renormalization is also given by Eq. (49) [54]. For low salt, however, and for id [Pg.157]

The Hamiltonian is transformed exactly to one having planar symmetry, and defined on the (u, 4>) strip, where it = log(r/a) and varies between zero and L = log (R/a), and 4> is between zero and 2n [56]. The canonical partition function is explicitly evaluated in the strip. In the mean-field approximation, the threshold for countercondensation is the same as that predicted by Manning [56]. As the Manning parameter E, is increased, transitions that reflect condensation due to a single ion is observed [56]. The unique feature has been verified by Monte Carlo simulations [57]. 

For the important strongly charged case, in which the product of the Manning parameter and the valence v is larger than 1, the Debye length is... 

This implies the normalization of the potential to be [Pg.68]

The right hand side of Eq. 16 is monotonically increasing with Manning parameter Hence y increases with and decreases with cell radius R. Also, from Eq. 15 it is seen that RM increases with y. 

For this particular system the Manning parameter is = 2.88 > 1, and the implied condensation of counterions is clearly visible. Note that most of the systems studied here contain many more particles, but for the purpose of illustration a small system is more informative. 

FIG. 7 Counterion distribution functions P(r) (solid lines) for seven systems with the same dimensions as the ones in Figure 6, but with a Bjerrum length tB/counterion condensation is not expected to occur. This is borne out by the observation that the functions are convex up already at r = r0. In these weakly charged systems the predictions of PB theory (dotted lines) are excellent and can hardly be distinguished from the simulation results. 

Up to now only monovalent ions have been investigated. For multivalent ions the prediction of the PB theory is that for the distribution function P(r) only the product of the Manning parameter and the counterion valence v matters. Therefore a system of monovalent ions at ln = 3a is claimed to have the same distribution function as a system of trivalent ions at B = lo It will now be shown that this statement is an artifact of the PB approximation. Figure 9 shows examples of systems that are complementary in the described sense. Not only is the condensation enhanced as compared to PB theory, but the enhancement is stronger for the case involving multivalent ions. Two different reasons may be suggested to explain this effect ... 

Qualitative deviations from Poisson-Boltzmann theory are sometimes very important for real systems. For instance, the zeta potential turns out to be a non-monotonic function of the Manning parameter. This has pronounced influence on the interpretation of electrophoresis experiments. 

FIG. 16 Ion distribution function P(r) (left) and mean electrostatic potential if/j) (right) for DNA-like systems (see Table 1) with 0.5 mol/L added 2 2 salt. The six curves differ in the line charge density of the rod, producing Manning parameters between 1.05 and 10.5 as indicated in the key. The value 4.2 corresponds to DNA. Notice that the radial distance is only plotted up to one third of the cell radius. 

FIG. 17 Distribution functions P(r) for the systems with Manning parameter 4.2 and 10.5 from Figure 16. The solid lines are the results from the molecular dynamics simulations while the dashed lines are the predictions from hypemetted chain theory. (See Ref. 36.)... 

Indeed, f is found first to increase with t, but from a certain value on it decreases and would finally even become negative. Note that this would reverse the drift direction in electrophoresis measurements. While nonlinear and linearized PB theory coincide with the data and with each other for small Manning parameter, they completely fail to predict the back-bending, which already sets in at comparatively small values of t Hypemetted chain... 

FIG. 18 Zeta potential = //(/, + cr/2) as a function of Manning parameter for the DNA-like models with 0.5 mol/L added 2 2 salt. The dotted line is the prediction of PB theory, the dash-dotted line is from bulk Debye—Htickel theory, and the dashed line is the result from a hypernetted chain calculation [36]. The solid line is a fit that merely serves to guide the eye. 

FIG. 19 Ion distribution function P(r) (left) and mean electrostatic potential i/4r) (right) for systems with Manning parameter = 4.2. The nine curves correspond to different numbers Ns of 2 2 salt molecules added to the box Ns 8, 17, 34, 68, 135, 270, 380, 540, 760. In the distribution function the salt content increases from bottom to top in the mean electrostatic potential it increases from top to bottom. 

Tabulated are polyelectrolyte concentration c, temperature /. Manning parameter t. ion diameter cr, distance of closest approach dca between ions and the rod, and cell radius R corresponding to the given concentration. The experiments have been performed under salt-free aqueous conditions with monovalent counterions. Source Ref. 34. 

With the help of the condensation criterion from Sec. IV.B, the extent of the correlation-enhanced condensation can be further quantified. The Manning fraction from the molecular dynamics simulation increases only by a fairly small amount. It is at most 4% larger than the PB value. This translates to an effective Manning parameter eff = 1/(1 — fe) being 5-10% larger than the bare one. This increase is very accurately captured by the Debye-Hiickel hole-cavity theory. Its prediction for fe is at most 1% smaller than the value obtained in the simulation. Interestingly, it is even independent of density. This, however, is not a feature to be generally expected and should therefore not be overinterpreted. 

From a fit to the structure factor the radius r of the rod has been determined. The integration constant y and the Manning parameter R fa the follow from the PB theory. The obtained values are listed in Table 4. 

FIG. 24 Two-dimensional pair correlation function g(r) for the projection of counterions within a close condensed layer onto the cylinder of closest approach see text and Figure 23. The four functions belong to the most strongly charged systems in Figure 16. From solid to dotted the Manning parameter decreases as 10.5, 8.4, 6.3, and 4.2. The last value corresponds to DNA. 

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On a sufficiently short length scale, the polyion can thus be considered to be linear (Fig. 4). The radial distribution of the dissociated counterions around such a linear charged chain can be computed by a Poisson-Boltzmann approach." The dielectric permittivity of the solvent as well as temperature can be considered by an electrostatic screening length called the Bjerrum length Ib. The electrostatic potential of the polyion is then described by the Manning parameter where b is... 

The ratio between the distances Iq and Ig, called the Manning parameter = Igl Iq [35], plays a central role in the condensation of counterions and will be used in the present analysis. In the conditions of the present work the Bjerrum length has a value of Ig = 0.71 nm. 

Figure 23 The radial Manning parameter (r) (top frame, cell model only Eq. [244]) and potential c )(r) (bottom frame Eq. [245]) in the PB cell and bulk models for a charged cylinder of radius <3 = 10 A and surface charge density = —0.094 e(jk (corresponding to an average charge spacing of h = 1.69 A as in B-DNA). A site concentration (corresponding to a phosphate concentration in DNA) of 0.1 M has been chosen, giving a Manning radius of Rm = 29.6 A and a cell radius of R = 56 A. The PB cell model potential profile (solid lines) is compared to the bulk PB (dotted-dashed line Eq. [389]), DH cell model (dashed lines Eq. [256]), DH bulk model (dotted line Eq. [259]), and no-ion (circles Eq. [362]) values.

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