Charge density linear

Manning s theory does not take the local effective dielectric constant into consideration, but simply uses the a value of bulk water for the calculation of E,. However, since counterion condensation is supposed to take place on the surface of polyions. Manning s 2, should be modified to E, by replacing a with aeff. The modified parameters E, is compared with E, in Table 1, which leads to the conclusion that the linear charge density parameter calculated with the bulk dielectric constant considerably underestimates the correct one corresponding to the interfacial dielectric constant. 

Linear charge density in units of (elementary charge)/nm... 

Another related phenomenon to be discussed in 2.3 is known in the polymer literature as counterion condensation. This term refers to a phase transition-like switch of the type of singularity, induced by a line charge to solutions of (2.1.2), occurring at some critical value of the linear charge density. Counterion condensation as a limiting property of the solutions of the Poisson-Boltzmann equation was studied in detail in [11]—[19]. Presentation of 2.3 follows that of [17]. 

Fig. 2.3.1. Scheme of the experimental dependence of the effective linear charge density on the structural linear charge density. 

Formula (2.3.41) implies that in a system with added electrolyte of a higher valency z, the magnitude of the effective linear charge density is determined by the latter. Upon reduction of the concentration of the added electrolyte (N —> 0) a transition occurs in the way prescribed by (2.3.41) to the effective linear charge as determined by the proper counterions of valency Q in a polyelectrolyte solution free from added low molecular electrolyte. 

Fig. 2.3.5a. Schematic dependence of the equilibrium order parameter rieq= (aj-a ")/o on the structural linear charge density a.

Fig. 2.3.5b. Schematic dependence of the system s free energy F on the effective linear charge density ae,=a-an-...

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

Fig. 10 Young s modulus for multilayer of PSS/(DADMAC/NMVA) random copolymers as a function of ambient humidity at room temperature. The DADMAC-content of the copolymer is varied between 75 and 100% (75% filled triangles, 89% filled circles, 100% filled squares and filled inverted triangles), indicating a clear dependency of the mechanical properties of the multilayer on the linear charge density. In all cases, a plasticizing effect of water is evident...

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]... 

Observe that the linear charge density of B-DNA is greater than one. If this constraint is satisfied, Manning and Onsager argued that the system is electrostatically unstable [17, 31]. To lower the total energy of the system, counterions from the bulk condense on the backbone of DNA to reduce the bare phosphate charges [17, 31]. 

We regroup those terms in Eq. (6) that are associated with lncs. This is so because In cs diverges as cs approaches zero. Hence, the coefficient of In cs must vanish. This leads to an explicit connection between 0 and the linear charge density [17, 31]... 

Despite the inherent weaknesses of the Poission-Boltzmann (PB) equation that includes neglect of polyion-polyion interactions and the size of mobile ions, it has nevertheless played an important role in elucidating the characteristics of electrostatic interactions and ion atmopshere in nucleic acids [12]. Two salient results have emerged from such an analysis. First, for cylindrical-shaped molecule of infinite length, there is a critical linear charge density beyond which counterion condensation takes place [33, 34]. Second, the phenomenon of counterion condensation takes place in one and two dimensions only [33, 35], An... 

A noteworthy conclusion is that for a polyion in aqueous solution, and in which kL is held fixed, the equilibrium state of the macromolecule is stable provided 0=1— c 1 [40]. Furthermore, the number of condensed counterions, as well as the threshold value for linear charge density, is unchanged for a polyion whose length is of the order of k 1 [40]. 

As before, the total free energy is minimized with respect to 0. Minimization leads to the following results [41]. (a) The minimum value of 0 is identical with Eq. (7) from the line model. This implies that the number of condensed ions is unaffected by the helix and is governed only by the linear charge-density parameter E,. (b) The partition function <2heiix is influenced by the helical structure of the phosphate charges... 

For fixed values of the linear charge densities, the counterion distribution can be studied by numerically solving Eqs. (31)—(33). Such an analysis reveals three distinct phases of counterion distributions labeled I, II, and III. 

Phase I corresponds to polyions that have small values of linear charge density y0 [47]. The electrostatic attractions in this region are unable to localize the counterions close to the polyion surface [47]. 

In Manning model, the slope off versus the Coulomb strength or the linear charge density parameter lB lB/l is discontinuous at 1 [31,48], This is a consequence of the fact, that for all less than or equal to unity, / is linear function of 1/s and saturates at unity [31, 35, 48]. The present model indicates a sigmoidal variation off with 1/e and saturates as /H > 0 [48],... 

The choice of a buffer system becomes the most significant variable in developing a separation of PCR products by CE. Smaller DNA components (such as dNTPs and primers) have been separated by means of a partitioning approach based on affinity for a micelle in solution (Cohen et al., 1987). In separations of larger dsDNA components by CE, a sieving mechanism is required in the capillary to separate species with the same linear charge density. In optimizing... 

Now, a linear charge density-potential relation is consistent with the law of superposition of potentials, which states that the electrostatic potential at a point due to an assembly of charges is the sum of the potentials due to the individual charges. Thus, when one uses an unlinearized P-B equation, one is assuming the validity of the law of superposition of potentials in the Poisson equation and its invalidity in the Boltzmann equation. This is a basic logical inconsistency which must reveal itself in the predictions that emage from the so-called rigorous solutions. This is indeed the case, as will be shown below. 

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