Scaled surface charge density

FIGURE 1.6 Scaled surface potential yo = ze JkT as a function of the scaled surface charge density tr = zeals oKkT for a positively charged planar plate in a S3fmmetrical elec-trol3de solution of valence z. Solid line, exact solution (Eq. (1.41)) dashed line, Debye-Htickel linearized solution (Eq. (1.26)). 

FIGURE 4.3 Scaled potential y x) as a function of the scaled distance kx across an ion-penetrable surface charge layer of the scaled thickness Kct for several values of Kcl kcI = 0.5, 1, and 2). The scaled charge amount contained in the surface layer is kept constant at (AVn) Kd — 5. The vertical dotted line stands for the position of the surface of the particle core for the respective cases. The curve with kcI Q corresponds to the limiting case for the charged rigid surface with the scaled surface charge density equal to Nlri)Kd = 5. From Ref. [4]. 

Figure 17 The error in the PGC solution of Eq. [152] according to Eq. [158] for a cylinder as a function of the scaled radius Ki,a for several values of the scaled surface charge density a /ao curves obtained using the exact (Eq. [154]) and approximate (Eq. [155]) Debye-Gouy-Chapman lengths are grouped by arrows.

Figure 29 The scaled condensation radius determined from Eq. [269] as a function of the scaled surface charge density cta/cto for a plane (solid line), cylinder (dashed Hnes Kd

We compare the exact numerical solution to the Poisson-Boltzmann equation (6.6) and the approximate results, Eq. (6.37) for case 1 (low surface charge density case) and Eq. (6.50) for case 2 (high surface charge density case) in Fig. 6.3, in which the scaled surface potential jo = zeij/JkT is plotted as a function of the scaled... 

FIGURE 9.4 Reduced potential energy V (h) — Kl64nkT)V h) as a function of scaled plate separation Kh for the constant surface charge density model calculated with Eq. (9.125) for a = 0, 0.1, and 1 in comparison with V h) = (K/64nkT)V h) for the constant surface potential model calculated with Eq. (9.141). 

FIGURE 9.8 Reduced potential energy V = Kl64nkT)V of the double-layer interaction per unit area between two parallel similar plates with constant surface charge density u as a function of the reduced distance Kh between the plates for several values of the scaled unperturbed surface potential yo = ze JkT. Solid lines are exact values and dotted lines represent approximate results calculated by Eq. (9.197) as combined with Eq. (9.201). The exact and approximate results for To = 1 agree with each other within the linewidth. (From Ref. 14.)... 

FIGURE 11.4 Scaled double-layer interaction energy = K/64nkT)V h) per unit area between two parallel similar plates as a function of scaled separation Kh at the scaled unperturbed surface potential >>o = 1. 2, and 5 calculated with Eq. (11.14) (dotted lines) in comparison with the exact results under constant surface potential (curves 1) and constant surface charge density (curves 2). From Ref. [5]. 

Figure 5.33. Adsorbed amount 0 of a strong negative polyelectrolytc as a function of the salt concentration on an uncharged surface = 0) and on a positively charged surface at two values of the surface charge density a° (indicated) U). Note that the abscissa scale is not linear in but in. c. Parameters N = 500, = 10, = L. IT =...

Figure 5 Diffusion of charged spherical macroion of radius 10 A and a uniform surface charge density of le per 93 A2 on mixed membranes. The panels show the local surface charge densities after 0.6 ps of simulations (shades) and the entire macroion trajectories in that time (connected lines) for binary (71 29 PC/PS) mixture, D =10 (a), for ternary (74 25 1 PC/PS/PIP2) mixture, D =10 (c), for binary (PC/PS) mixture, D =2 (b), and for ternary (PC/PS/PIPJ mixture, D = 2 (d). The dashed circles on each panel represent the projected size of the macroion with arrows indicating the starting position for the macroion center of mass. For clarity, the figures zoom on the relevant membrane surface region explored by the macroion, and a scale bar of 20 A is shown for reference.

Hartley, P.G. and Scales, P.J., Electrostatic properties of polyelectrolyte modified surfaces studied by direct force measurement, Langmuir, 14. 6948, 1998. Rodrigues, E.A., Monteiro, P.J.M., and Sposito, G., Surface charge density of silica suspended in water-acetone mixtures, J. Colloid Interf. Sci.. 211, 408, 1999. Abraham, T. et al.. Asphaltene-silica interactions in aqueous solutions Direct force measurements combined with electrokinetic studies, Ind. Eng. Chem. Res., 41, 2170, 2002. 

Such two-dimensional pair correlation functions for mobile ions adsorbed on charged planes have recently been investigated theoretically in Ref. 42. The mean separation d, = ( < 0/s )l/2 of w-valent ions on the completely neutralized plane of surface charge density s is identified as the important scaling length. The main predictions for the strongly coupled regime =... 

In practice, a set of curves developed by Kemper and Quirk (Fig. 8.10), yields approximate electric potentials as a function of distance from the colloid surface. Such potentials can then be substituted directly into the Boltzmann equation to infer cation and anion distributions. 7/, is the scaled electric potential (equal to —Zef/fkT of Eq. 8,15) at the midplane between interacting colloids, T is the surface charge density in coulombs m-2 (96.5 times the ratio of CEC, in mmoles charge kg-1, divided by the specific surface, in m2 kg-1), Z is the valence of the exchangeable cation, Co is the molar salt concentration in the bulk solution, and x is the distance (in nm) from the midplane between colloids to the plane at which the ion concentration is to be calculated. 

All alkali ions and Ca +, Sr + and Ba + are very labile as a consequence of their relatively low surface charge density. The only direct experimental data on water exchange in some of these ions comes from incoherent quasi-elastic neutron scattering (IQENS) [74-76]. IQENS has an observation time scale fobs 1 ns and allows for the calculation of limits for ion to water-proton binding times Tj (Table 4.4). Mean lifetimes of H2O in the first shell of Ca + and Sr " can be estimated to 0.2 ns from the chemically similar Eu " ion (see Sect. 4.3.2). 

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