Bjerrum, length

In addition, it is convenient to define the Bjerrum length parameter Lb = e z ) / f.eikTi (where is the universal dielectic constant) and the so-called quenching parameter, defined as g = EqTq/eiTi. 

Thus, counterion condensation occurs if b is smaller than the Bjerrum length. 

The fraction of condensed counterions is equal to 1-Hence, the Bjerrum length appears to be the smallest distance between elementary charges. The fraction of uncondensed ions is then —. 

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] ... 

Figure 6 gives the comparison of the osmotic coefficient predicted by the PB-theory to simulated data [26, 60]. The simulation system is not strictly a cell system, rather we considered an infinite array of parallel aligned rods which sit on a hexagonal lattice. The rod diameter a was of the same size as the counterions a, the line charge density X had the value 1=0.9593 e0 o, and the density and the Bjerrum length was varied. For details of the simulations we refer to Ref. [26, 60]. 

The first set of simulation has been done for monovalent counterions of size a and three values of the reduced Bjerrum length XBlo= 1, 2, 3. Several findings may be noted The osmotic coefficient from the simulations is always smaller than the PB prediction but for low density both values converge. This also illustrates that the Manning limiting law becomes asymptot-... 

Fig. 6 Osmotic coefficient 0 versus reduced density n/a3 for monovalent counterions. Heavy dots mark the measurements, while the solid lines are fits which merely serve to guide the eye. The dotted lines are the prediction of PB theory. From top to bottom the Bjerrum length lB/o varies as 1,2,3. The errors in the measurement are roughly as big as the dot size [29]...

Fig. 1 Counterion distribution function P(r) from Eq. (1) for two cylindrical cell models with R/b= 123.8,1=0.959 e0/b and the values for Bjerrum length and valence as indicated in the plots. The solid line is the result of a molecular dynamics simulation [9] while the dotted line is the prediction from Poisson-Boltzmann theory. The increased counterion condensation visible in the simulation is accurately captured by the extended Poisson-Boltzmann theory (dashed line) using the DHHC correction from Ref. [18]. An approach using the DHH correction from Ref. [16] (dash-dotted line) evidently fails to correctly describe the ion distribution...

Fig. 3 Effective potential between two counterions (left) and two monomers (right), respectively for various Bjerrum lengths /B. The chain length is N= 80 and the density 77= 1(T2...

Fig. 5 Electrostatic persistence length of a flexible polyelectrolyte chain as function of k/v b for the Bjerrum lengths lB/b=0.1, 0.5, 1.0 (bottom to top). The chain length is N=1000. The slopes of the straight lines are -1 and -2, respectively...

Fig. 8 Monomer-monomer (gmmir)) pair correlation function for the packing fractions 77=1(T3, 10-4, and 10-5 (left to right) of rod-like (...) and flexible chains (—). The chain length is N=63 and the Bjerrum length ZB=0.5 b...

Now, consider B-DNA molecules in an aqueous solution. Manning showed that the electrostatic stability of the DNA is controlled by the dimensionless linear charge density parameter , [17, 31]. The ratio of the Bjerrum length ZB to the charge spacing b of the phosphates defines the linear charge density [17, 31]... 

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