Counterion fluctuation

Attractive forces arise from dipole interaction, a result of the fluctuations in the cloud of counterions. Although the mean distribution of counterions is uniform along the length of the polyion, there are fluctuations in the cloud of counterions which induce transient dipoles. When two polyions approach each other counterion fluctuations become coupled and enhance the attractive force. Since polyions have a high polarizability these attractive forces can be considerable. 

High concentration, however, is not the only means to obtain LC phases of DNA in aqueous solutions. DNA has been found to collapse upon adding in the solutions various condensing agents, which introduce effective attractive interhelical interactions. This is what happens with alcohols and other solvents, which reduce DNA solubility [40], or with multivalent cations like spermidine, spermine, and cobalt hexamine, which are thought to establish correlated counterion fluctuations with... 

Solutions of many proteins, synthetic polypeptides, and nucleic acids show large increases in permittivity c (u>) over that of solvent, normally aqueous, at sufficiently low frequencies f = w/2ir of steady state AC measurements, but with dispersion and absorption processes which may lie anywhere from subaudio to megahertz frequencies. Although our interest here is primarily in counterion fluctuation effects as the origin of polarization of aqueous DNA solutions, we first summarize some relevant results of other models for biopolymers. 

A field theoretical description of counterion fluctuations has also been formulated. In this approach, fluctuating electrostatic potential is introduced in which the surface charge density is treated as a variational parameter [54]. The methodology captures the nonlinearity of the counterion distributions of highly charged systems. 

Oosawa F. Counterion fluctuation and dielectric dispersion in linear polyelectrolytes. Biopolymers 1970 9 677-688. 

Another effect was introduced by Oosawa concerning the role of the mean square dipole

due to the counterion fluctuation along the DNA polyion structure. The polarizability of the DNA along the double helix axis is given by... 

F. Oosawa, Counterion Fluctuation and Dielectric Dispersion in Linear Polyelectrolytes, Biopolymers 9, 677-689 (1970). 

Flory PJ (1953) Principles of polymer chemistry. Cornell University Press, Ithaca Freed KF (1987) Renormalization group theory of macromolecules. Wiley, New York Freed KF, Dudowicz J, Stukalin EB, Douglas JF (2010) General approach to polymer chains confined by interacting boundaries. J Chem Phys 133 094901 Fuoss RM, Katchalsky A, Lifson S (1951) The potential of an infinite rod-like molecule and the distribution of the counter ions. Proc Natl Acad Sci USA 37 579-589 Golestanian R, Kardar M, Liverpool TB (1999) Collapse of stiff polyelectrolytes due to counterion fluctuations. Phys Rev Lett 82 4456-4459 Guggenheim FA (1952) Mixtures. The Clarendon Press, Oxford... 

The bundle formation in systems of stiff polyelectrolytes in the presence of divalent counterions was simulated by Stevens [175,176]. Figure 20 shows the ordering of stiff polyelectrolytes into a network of bundles. The number of divalent counterions condensed on the chain is extremely large and clearly drives the aggregation. The bundling is due to attraction between two chains caused by correlated counterion fluctuations [171,172]. The precise... 

The binding of multivalent counterions decreases the repulsion and causes attraction between polyions. This attraction is the result of the fluctuation of the counterion distribution and is equivalent to a multivalent counterion bridge between polyions. 

Cations can be seen as acting as ionic crosslinks between polyanion chains. Although this may appear a naive concept, crosslinking can be seen as equivalent to attractions between polyions resulting from the fluctuation of the counterion distribution (Section 4.2.13). Moreover, it relates to the classical theory of gelation associated with Flory (1953). Divalent cations (Zn and Ca +) have the potential to link two polyanion chains. Of course, unlike covalent crosslinks, ionic links are easily broken and re-formed under stress there could therefore be chain slipping and this may explain the plastic nature of zinc polycarboxylate cement. 

In the years 1910-1917 Gouy2 and Chapman3 went a step further. They took into account a thermal motion of the ions. Thermal fluctuations tend to drive the counterions away form the surface. They lead to the formation of a diffuse layer, which is more extended than a molecular layer. For the simple case of a planar, negatively charged plane this is illustrated in Fig. 4.1. Gouy and Chapman applied their theory on the electric double layer to planar surfaces [54-56], Later, Debye and Hiickel calculated the potential and ion distribution around spherical surfaces [57],... 

In a third step the counterions are released from the surface. Stimulated by thermal fluctuations, they partially diffuse away from the surface and form the diffuse double layer. The entropy and, at the same time, the energy increases. One can show that both terms compensate, so that in the third step no contribution to the Gibbs free energy results. 

This process is a relatively fast one and the diffusion coefficient determined from the correlation time of the fluctuations in the scattering can be too large202) giving too small radii, tm- To avoid this effect, Mazer et al.33) made measurements on sodium dodecyl sulphate in high salt concentration to contract the counterion charge cloud. 

The counterions are pinned in a spatial location in a time-averaged sense (there are thermally induced fluctuations about the mean position). 

Finally, in this section, the possible role of counterion displacements in relaxation of globular proteins should be mentioned. These can result in fluctuating dipole moments in addition to permanent moments p. along principal molecular axes. If their relaxations are independent and separately exponential with rate constants k and k, response theory formulation gives the complex permittivity in the form... 

The Manning counterion condensation line model provides insights into the mean-square number fluctuations of the condensed counterions, ((A0)2), where 0 is the number of condensed counterions per polyion charge [50], Denote by 0O the value of 0 for which the total polyelectrolyte free energy G per charge is a minimum, that is, (0G/00)e=e vanishes. Expanding G in powers of 0 0O to quadratic order leads to [50]... 

Now, let us consider the fluctuations of a segment of M charges. The segment is located at least one Debye length away from the ends of the polyion. If Ag denotes the deviation of the free energy of the segment from its minimum value, then Ag = MAG. But, the thermal average of AG is, by definition, (AG) = kBT/2. Consequently, the mean-square number fluctuations of condensed counterions, ((A0)2), is [50]... 

The contribution of the term in the square brackets is greater than unity and is due to electrostatic interactions between the polyion and the condensed counterions, as well as between the condensed counterions. Consequently, the mean-square number fluctuations are smaller in the presence of interactions [50]. 

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