Counterion distribution functions

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...

If the counterion distribution function P is known, the condensed counterion fraction can be characterized in the following geometric way Eqs. 18 and 19 show that P viewed as a function of ln(r) is merely a shifted tangent function with its center of symmetry at (ln(RM) fe). Since, however, tan"(0)... 

FIG. 6 Counterion distribution functions P(r), as defined in Eq. 19, for various simulated densities. Note that an increasing cell radius corresponds to functions extending towards larger values of r. The heavy dots mark the points of inflection in P as a function of ln(r), while the crosses mark the positions at which those points would be located on the corresponding PB distribution functions. For the sake of clarity such a PB distribution is only plotted for the system with lowest density, i.e, R = 123.85cr (dotted line). 

FIG. 7 Counterion distribution functions P(r) (solid lines) for seven systems with the same dimensions as the ones in Figure 6, but with a Bjerrum length tB/Manning parameter = 0.959 < 1, counterion condensation is not expected to occur. This is borne out by the observation that the functions are convex up already at r = r0. In these weakly charged systems the predictions of PB theory (dotted lines) are excellent and can hardly be distinguished from the simulation results. 

FIG. 9 Counterion distribution functions P(r) versus r/r0. The high (low) density situation is shown in the left (right) frame. The lower three curves are for vCB/cr = 2, while the upper three correspond to xj b/ct = 3. The systems with multivalent counterions (solid lines) always show a stronger condensation than the complementary systems with monovalent ions (dashed lines), which themselves show a stronger condensation than PB theory (dotted lines). 

Thus, effects of the surfaces can be studied in detail, separately from effects of counterions or solutes. In addition, individual layers of interfacial water can be analyzed as a function of distance from the surface and directional anisotropy in various properties can be studied. Finally, one computer experiment can often yield information on several water properties, some of which would be time-consuming or even impossible to obtain by experimentation. Examples of interfacial water properties which can be computed via the MD simulations but not via experiment include the number of hydrogen bonds per molecule, velocity autocorrelation functions, and radial distribution functions. 

The use of capillary electrophoresis (CE) during the synthetic drug development is described from the preclinical development phase to the final marketed stage. The chapter comprises the determination of physicochemical properties, such as acid—base dissociation constants (pKJ, octanol—water distribution coefficients (logP), and analysis of pharmaceutical counterions and functional excipients. 

H. Gordon and S. Goldman,/. Pbys. Cbem., 96,1921 (1992). Simulations on the Counterion and Solvent Distribution Functions Around Two Simple Models of a Poly electrolyte. 

The most easily diagnosed finite size artifact is the presence of a concentration mismatch between the counter- and coion to water ratio (molality) and the actual electrolyte concentration at the edge of the simulation box. This is detected by calculating the macromolecule-counterion and macromolecule—coion radial distribution functions (RDFs) from an equilibrated trajectory. Examples of RDFs based on previous work on the... 

Figure 20.2 Illustration of how finite size artifacts lead to concentration mismatches in the bulk (A) RNA phosphate-counterion radial distribution functions for the Tar—Tar complex in an 80 A box of 800 mm NaCl. The anion concentration is 15% different than the cation concentration at large separations. (B) RNA phosphate-counterion radial distribution functions for the Tar—Tar complex in a 120-A box of 800 mm NaCl. The anion and cation concentrations agree with each other to within 1% at large radii. Taken from Chen et al. (2009).

Figure 20.6 Comparative cumulative distribution functions to assess the amount of counterion accumulation around the Tar—Tar complex, A-form RNA, and B-form DNA, respectively. Data were obtained from molecular dynamics simulations with explicit representations of water molecules and ions. The ordinate quantifies the number of counterions accumulated around the macroions within contours that have AG less than or equal to the value on the abscissa.

Fig. 3.16 Radial distribution function (RDF) for one monolayer in compression, using as reference the pyridine group in relation to the distance of the counterion Br (—) 4.7 A, (,..)3.8A. (From ref. [81])...

Within PB theory [2] and on the level of a cell model the cylindrical geometry can be treated exactly in the salt-free case [3, 4]. The Poisson-Boltzmann (PB) solution for the cell model is reviewed in the chapter in this volume on the osmotic coefficient. The PB approach can provide for instance new insights into the phenomenon of Manning condensation [5-7]. For example, the distance up to which counterions can be called condensed can be conveniently found via the inflection point in the log plot of the integrated radial distribution function P(r) of counterions [8, 9], defined as... 

The literature on high temperature fused salts is extensive [72-74], and we make only the most cursory review here. The first and most obvious statement about fused salts is that they are fundamentally different than molecular liquids, in that they retain a substantial degree of order on melting. The strength of interion Coulomb interactions mandates that ions be surrounded by counterions, and so maintain the most uniform possible charge distribution throughout the liquid. This expectation is bom out by X-ray [75, 76] and neutron diffraction [76, 77] experiments, which indicate that molten salts retain much of their solid-state structure in the liquid state a representative radial distribution function for a molten salt is given in Fig. 2. 

The radial distribution function of polymer-counterion as a function of distance from the polyion surface exhibits two peaks. These peaks are interpreted as reflecting two populations of counterions a diffuse Debye-Htickel cloud at a distance characteristic of the Debye screening length, and the condensed layer near the polymer surface where the condensed counterions reside [43]. 

It is interesting to know that the distribution of the ethylene oxide segments is rather independent of the salt concentration, but the point of the pistachio ripples was to investigate how the presence of the polymer affects the distribution of the counterions. To elucidate this in more detail, we employed the modeling technique used to fit the raspberry ripples [4], We recall that this essentially crystallographic technique is a method for producing a single-particle distribution function, p(z),... 

The use of difference methods offers a means whereby a detailed picture of ionic hydration can be obtained 22). For neutron diffraction, the first-order isotopic difference method (see Section III,A) provides information on ionic hydration in terms of a linear combination of weighted ion-water and ion-ion pair distribution functions. Since the ion-water terms dominate this combination, the first-order difference method offers a direct way of establishing the structure of the aquaion. In cases for which counterion effects are known to occur, as, for example, in aqueous solutions of Cu + or Zn +, it is necessary to proceed to a second difference to obtain, for example, gMX and thereby possess a detailed knowledge of both the aquaion-water and the aquaion-coun-terion structure. 

Figure 3. The counterion-counterion pair-distribution function as obtained by theory (lines) and MC simulations (symbols). The calculations apply to cp = 0.0001 and Ce = 0.005 mol dm 3. The results apply, from bottom to top, to zp zc ratios —20 +1, —30 +1,-40 +1 and-50 +1.

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