Counterions distribution

Surface forces measurement is a unique tool for surface characterization. It can directly monitor the distance (D) dependence of surface properties, which is difficult to obtain by other techniques. One of the simplest examples is the case of the electric double-layer force. The repulsion observed between charged surfaces describes the counterion distribution in the vicinity of surfaces and is known as the electric double-layer force (repulsion). In a similar manner, we should be able to study various, more complex surface phenomena and obtain new insight into them. Indeed, based on observation by surface forces measurement and Fourier transform infrared (FTIR) spectroscopy, we have found the formation of a novel molecular architecture, an alcohol macrocluster, at the solid-liquid interface. 

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. 

As expected, the D-H theory tells us that ions tend to cluster around the central ion. A fundamental property of the counterion distribution is the thickness of the ion atmosphere. This thickness is determined by the quantity Debye length or Debye radius (1/k). The magnitude of 1/k has dimension in centimeters, as follows ... 

Terao T. Counterion distribution and many-body interaction in charged dendrimer solutions. Mol Phys 2006 104 2507-2513. 

I. Rubinstein, Concentration polarization effects upon the counterion selectivity of an ion-exchange membrane with differing counterion distribution coefficients, J. Chem. Soc., Faraday Trans., 86 (1990), pp. 1857-1861. 

The ionic groups on the micellar surface and the counterions will give rise to a nonuniform electrostatic potential according to the Poisson equation. If furthermore the electrostatic effects dominate the counterion distribution the ion concentration is determined by following a Boltzmann distribution. These approximations lead to the Poisson-Boltzmann equation. 

Pabit, S. A., Qiu, X. Y., Lamb, J. S., Li, L., Meisburger, S. P., and Pollack, L. (2009). Both helix topology and counterion distribution contribute to the more effective charge screening in dsRNA compared with dsDNA. Nucleic Acids Res. 37, 3887—3896. 

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...

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. 

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. 

An experimental measurable quantity that is of fundamental importance in nucleic acid is the preferential interaction coefficient T. The parameter T is a statistical thermodynamic quantity that probes nonideality effects of nucleic acids in aqueous solution. This quantity also provides information on the characteristics of co-ions and counterions distributions in the vicinity of polyions [65-76]. 

As is well known, the counterions distributed close to the plates work to shield their surface charges. Figure 6.3 shows, however, that there exist counterions at the middle of the inner region that do not belong to either of the plates. It is those counterions that result in an attraction, a counterion-mediated attraction, between... 

Q The Counterion Distribution between Charged Plates in Solution... 

Theories of colloid stability based on electrostatics go way back beyond the DLVO theory, to the Gouy-Chapman theory of the electrical double layer proposed in the early 1910s and the Stem theory of counterion condensation proposed in 1924. There was much weighty speculation about the counterion distribution around colloidal particles throughout the 20th century, but nobody succeeded in measuring it until our work in 1997. This work is described in detail in Chapter 8. 

---