Surface potentials dimensionless

The dimensionless partition coefficient K is based on mole fractions. v, or number of moles In the literature, partition coefficients are more often defined as concentration ratios. At low solute concentration and when the adsorbed amounts become very small, the activity coefficients approach zero and the surface potential also becomes insignificant (ZiF fj —> 0) ... 

In Eq. 30, Uioo and Fi are the activity in solution and the surface excess of the zth component, respectively. The activity is related to the concentration in solution Cioo and the activity coefficient / by Uioo =fCioo. The activity coefficient is a function of the solution ionic strength I [39]. The surface excess Fi includes the adsorption Fi in the Stern layer and the contribution, f lCiix) - Cioo] dx, from the diffuse part of the electrical double layer. The Boltzmann distribution gives Ci(x) = Cioo exp - Zj0(x), where z, is the ion valence and 0(x) is the dimensionless potential (measured from the Stern layer) obtained by dividing the actual potential, fix), by the thermal potential, k Tje = 25.7 mV at 25 °C). Similarly, the ionic activity in solution and at the Stern layer is inter-related as Uioo = af exp(z0s)> where tps is the scaled surface potential. Given that the sum of /jz, is equal to zero due to the electrical... 

Using the dimensionless surface potential y0 = y(x = 0) integration constant... 

FIGURE 5.15 Dimensionless surface potential (y-axis, F< >S/RT) vs. pH U-axis) for three types of carbon electrode materials Cl, acidic carbon (pHIEP = pHPZC = 3.0) C2, typical as-received ( amphoteric ) carbon (pHIEP = pHPZC = 6.5) C3, basic carbon (pi In,. = pHPZC = 10.0). Based on Gouy-Chapman double-layer theory, for a maximum surface charge of 0.03 C/m2 and ionic strength of 10 3 M. 

FIGURE 4.2 Schematic illustration of the behavior of the electric potential function in the two models. The thin line is a sketch of < >(x) for the approximation thick line is a plot of an exponentially decaying function, for which d = 0.135s. The potential s and the Debye screening length 1/k, respectively. The dashed line at x = 2/k designates the midplane. 

FIGURE 6.2 Variation of the dimensionless surface potential <6(x) against the coordinate x whose origin is taken at the midpoint between plates with separation dt = 20 nm. The analysis is for the two-charged-plate system (N= 1). The concentration of the external soaking solution is c = 0.001 mol T1, and the surface potential is 4>s = -4.0 in the Dirichlet model, which corresponds to the surface charge density Z0 = -0.084 nmr2 in the Neumann model. The two plates are located at = -10 nm and x2 = 10 nm. 

Note, this equation is for a symmetric electrol5de. When the dimensionless surface potential, %(= ez o/ksT), is less than one (i.e., i]jq is less than 25 mV with z = 1) corresponding to the linearization approximation where sinh = T, we find the potential distribution... 

For electrostatically stabiUzed suspensions, this maximum volume fraction will depend primarily on the dimensionless double layer thickness, Ka, and the electrostatic interaction energy through the dimensionless surface potential, T g(= etpelksT). Because the dimensionless double layer thickness can be large (i.e., 10-100) when the salt concentration is low, the effective volume fraction at which the maximum volume fraction is reached can be very small. 

Figure 3.18. Surface charge as a function of surface potential according to some staUstlcal theories. GC = modified Gouy-Chapmein, S = HNC/MSA, P = modified "PB 5". = Monte Carlo simulations. (Redrawn from Carnie and Torrie, loc. cit. 196). The dimensionless surface charge a is scaled in such a way that for aqueous solutions 0.1 unit in (T corresponds to 8.8 iC cm". The concentration of the (1-1) electrolyte is indicated.

As an illustrative example taken from Russel et al. (1989), let us consider a 0.01 molar solution of sodium chloride in contact with a surface charged at a density of 5 x 10 negative charges per square meter at room temperature, 298°K. Equation (2-52) gives /c = 3 nm. The dimensionless surface potential exfJkrtT. obtained from Eq. (2-45), is —5.21, and Eqs. (2-46) and (2-49) give respectively the exact and the Debye-Huckel approximations for the potential as a function of distance from the surface. The results are plotted in Fig. 2-13. Note that since — s/ ks T > 1, the Debye-Huckel approximation is... 

Figure 2.13 Calculated concenfratinns of monovalent cations and aninns as a function nf distance from a surface with a charge of —8 /rC/cm (5 x lO charges/m ), yielding a dimensionless surface potential of —5.21 (—133.9 mV). The bulk electrolyte concentration is 0.01 M, T = 298 K, and the solvent dielectric constant is 80, that of water. Tlie Debye-Htickel theory [Eqs. (2-49) and (2-50)] fails close to the surface. The exact result is given by Eqs. (2-46) and (2-41). 4/ is defined as exfs/kBT. (From Russel et al. 1989, with permission of Cambridge University Press.) ...

Here, is the dimensionless surface potential and is the value of d>j for h o°. Equation 5.179 expresses the dependence riei(/ ) in a parametric form riei(0), hifd). Fixed surface potential or charge means that or s, does not depend on the film thickness h. The latter is important to be specified when integrating H(h) or f(h) (in accordance with Equations 5.162 to 5.165) to calculate the interaction energy. 

Figure 6.5.2 Dimensionless potential distribution across a cylindrical capillary for different values of the Debye length ratio A " and a constant surface potential = C = 2.79 (after Gross 8c Osterle 1968).

Equation (22) reveals the effect of salt otiEq when flie salt concentration increases, C2oo also increases, whereas the (dimensionless) surface potential 0 decreases (see Fig. 5,... 

Here r is the radius of the spheres and the dimensionless surface potential is... 

The dimensionless conductivity 0/0 and coupling coefficients depend linearly upon C The conductivity and electro-osmotic coefficients 6- = (cr/a - l)/c ) and are plotted in Figs 4a and 4b. respectively, versus the solid volume concentrations ( ) for the three reduced surface potentials = -1,0, +1. For = 0, one can see that 6- tends to -3/2 as (j) 0. Moreover, for uncharged particles, the... 

Fig. 10.5 Calculated order parameter at the surface So as a function of temperature. The numbers at the curves corresponds to different surface potential in dimensionless units W = 0 (1), 0.0056 (2), 0.008 (3), 0.01 (4), Wc = 0.01078 (5), 0.012 (6), 0.017 (7). Note that at Wc the discontinuity of the first order N-lso phase transition disappears (adapted from [7])...

FIGURE 6.12 The scaled interaction free energy between two equal-sized spheres, with Ka = 10 a being the sphere radius) for several values of the dimensionless surface potential and at various ratios of the surface potentials. Results are shown for (a) constant potential interaction with 2/ 1 = (b) constant charge interaction with = 3. 

From equation 2.55 we recover the Debye-Hiickel approximation for 4 j/4 << 1. The factor accounts for the numerical accuracy of the Debye-Huckel formula for dimensionless potentials somewhat larger than unity. For large positive surface potentials I j >> 1, and a > 0,... 

Thus, we need only modify the boundary condition at the surface to move from a specified surface potential value to a specified charge density. Modify your program from problem 6.B.3 to plot the dimensionless solution as a function of dimensionless charge density. 

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