Manning radius

Manning radius and Manning fraction can be found by plotting P as a function of ln(r) and localizing the point of inflection. 

In contrast to the clear tendency of the measured condensed fraction to decrease upon dilution, the behavior of the condensation radius Rm appears to be more complicated. There does not seem to exist a simple monotonic convergence of R towards Ru- Rather, for high densities the measured condensation distance is larger than the Manning radius, while for the investigated low densities it is smaller. Unfortunately, a clear-cut statement is difficult, since the localization of the point of inflection in P as a function of ln(r) is only possible with an error estimated to be of the order of 1%. 

If the measured Manning radius R can be smaller than the PB prediction, this has the following consequence. Within PB theory fe is monotonically increasing with Ru. Therefore / >/f, and R < RM is incompatible with PB theory. It is impossible to describe such a P(r) with a PB distribution... 

The resulting Manning radius is surprisingly close to the one determined by simulation, particularly for the system with = 3.32. The osmotic coefficients constructed from the PB formula are less accurate, 10-20% too large. It would be interesting to observe the effects of introducing more free parameters, similar to the procedure described above, in order to reduce the Poisson-Boltzmann bias. ... 

Before we obtain the potential, we notice that for systems with a linear charge density such that > 1 (DNA and most other biopolyelectrolytes fall into this category), there is a specific distance at which (r) = 1. This distance we denote by Rm (the Manning radius) and is found from Eq. [240] ... 

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.

Dem Mittelpunkte eines Atoms A (z. B. eines Cl-Atoms) umschreibt man eine Kugel vom Radius (zJa, [(Oci = f>55 A], und schneidet im Abstand, der dem mittleren Atomradius des kovalent gebundenen Atoms (nach Pauling) entspricht, eine Kugelhaube ab. (> Aa= 0,99 A). 

Tlhe Sedan thermonuclear detonation in July 1962 created the largest A man-made crater ever produced by a single explosive. Approximately 7.6 million tons of earth were moved by this detonation to produce a crater with a radius of 608 feet and a depth of 323 feet (9). Fifty-eight percent of the crater mass or 4.8 million tons of desert alluvium were ejected from the crater during the detonation and distributed somewhat symmetrically around the ground zero point to a distance of 6000-7000 feet (1). 

For hundreds of thousands of years man was limited in his activity to the immediate vicinity of available water sources, thus restricting his territorial distribution. The invention of water carrying vessels increased the radius of travel a little. With the passage to animal husbandry and farming in the Neolithic period, some 7000 years ago, the demand for water increased. This was the background for the development of the large river civilizations in... 

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