Counterion condensation partition functions

Simple as it is, there is much to discuss about the condensed layer partition function Q. An approximation has been made in its derivation. We have assumed that Q does not depend on 9, the number of condensed counterions (in the derivative of the free energy with respect to 9, we have taken d In Q/39 as negligible). The physical meaning is that short-range correlations among the condensed counterions are not considered. Within this approximation, Eq. 4 indicates that the sole structural determinant of Q is the linear charge density of the polyion. Note that the result 9 = 1 - 1 for the equilibrium number of condensed counterions was derived more generally and is not affected by any such approximation. 

In this section, we discuss the interaction of a counterion and a rodlike segment of a polyion as a function of separation distance r between the two [59], We consider as well the coion-polyion [59] and polyion-polyion pair potentials [57,58]. In the latter case, the two polyions may be identically charged or oppositely charged. For each type of pair, there is a polyion selfenergy of the form of Eq. 1 (for the polyion-polyion pairs the factor P is replaced by 2P). The essential difference from Eq. 1 is that the number of condensed counterions 9 is now a function of the pair separation distance, 9 = 9(r). Similarly, in the transfer free energy Eq. 2, both 9 and the condensed layer partition function Q depend on r. In addition to the self-assem-... 

FIG. 3 The partition function of the condensed counterion layer as a function of the distance r from the polyion of another counterion that has been brought from infinity to r. Debye screening length 30 A (0.01 M NaCl) polyion charge spacing 1.7 A. 

The cause of the inverted attraction in the polyion-polyion case is the strong distance dependence of the condensed layer partition function Q(r) revealed in Figure 4. The entropic tendency toward attraction mentioned above in connection with Figure 4 turns out to dominate the balance of forces. Staying with a free volume interpretation of Q(r), we can observe that when two polyions approach, their electric fields merge, and the common field is relatively flat in the space between the polyions. The barrier to entropic expansion of the condensed layers having been removed, the condensed counterions flood into the large volume of space between the poly-... 

FIG. 8 A calculated cross-section of the cylindrically symmetric distribution of condensed counterions held in common by a pair of identical parallel rodlike polyions when the partition function for the condensed layer is interpreted as a free volume. The numerical scale is in A, and the polyions pierce the page at 15 A. The Debye length equals 30 A, so the theoretically calculated condensed layer lies inside the Debye atmosphere, as required on physical grounds. Polymer charge spacing 1.7 A. 

Manning assumes that condensation of counterions occurs to prevent the divergence of this partition function. The net result will be to reduce the apparent charge density on the polyion, until the value of after condensation will be equal to the critical value l/Z. In the case of DNA, two consecutive pairs of phosphates are separated by = 17Qpm, then = 4.2, so that any ion, regardless to its charge will condense on DNA, until the effective value of will be reduced to l/Z. 

The entropic contribution arising from the various distributions of the adsorbed counterions and coions is determined as follows. We note that for the general case of both mono- and divalent salts being present, there are N monomers. Mi adsorbed monovalent counterions (Na ), M2—M3 adsorbed divalent counterions (Ba ) with no coion (Cl ) condensation, and M3 ion-triplets ( monomer-Ba -Cl ) in the system. Therefore, N—Mi—M2 monomers remain with their bare charge uncompensated. Consequently, the partition function is... 

Monte Carlo Studies in Polyelectrolyte Solutions Structure and Thermodynamics on Monte Carlo studies in polyelectrolyte solutions structure and thermodynamics, this chapter discussing about, Monte Carlo studies of polyelectrolytes, theoretical approach of Monte Carlo studies, application level of Monte Carlo in polyelectrolyte, authors of this chapter are also trying to discuss more with many topics, such as coarse-grain model for poly electrolyte and small ions, ideal gas and excess contribution to the partition function of the system, metropolis Monte Carlo method, Monte Carlo trial moves, conformational and persistence length of a single polyelectrolyte chain, counterions condensation and end-chain effects and morphology of polyelectrolyte complex. 

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