Unperturbed surface potential

The phase-11 islands have several interesting properties. First, they have a positive surface potential relative to the surrounding unperturbed water film. The potential is highest immediately after formation and decays with time to zero. Another interesting property is the shape of the islands. Their boundaries are often polygonal, bending in angles of 120°,... 

For two identical spheres 1 and 2 of radius a carrying unperturbed surface potential i/ o separated by R, one may choose the intermediate plane at z = 0 between the spheres as an arbitrary plane enclosing sphere 1 (Fig. 8.8). Here we use the cylindrical coordinate system (r, z) and take the z-axis to be the axis connecting the centers of the spheres and r to be the radial distance from the z-axis. Equation (8.34) can be rewritten by using the cylindrical coordinate (r, z) as... 

FIGURE 9.3 Potential distribution i/f(x) across two interacting plates 1 and 2 at constant surface potential ipo calculated with Eqs. (9.54) and (9.56) for Kh— (upper) and 2 (lower). The dotted lines, which stand for unperturbed potentials, correspond to the case of infinite... 

The calculation of the potential energies V" h) and h) of the double-layer interaction between two parallel plates requires numerical solutions to transcendental equations (9.105) and (9.137), respectively. In the following we give approximate analytic expressions for V (h), which does not require numerical calculation. The obtained results, which are correct to the order of the sixth power of the unperturbed surface potential i//q, are applicable for low and moderate potentials. 

FIGURE 9.7 Reduced potential energy V = K/64nkT)V of the double-layer interaction per unit area between two parallel similar plates with constant surface potential as a function of the reduced distance Kh between the plates for several values of the scaled unperturbed surface potential ya = ze j/olkT. Solid lines are exact values and dotted lines represent approximate results calculated by Eq. (9.177). The exact and approximate results for To = 1 agree with each other within the linewidth. (From Ref. 13.)... 

We introduce the unperturbed surface potential j/o, that is, the surface potential j/(0) in the absence of interaction ath oo. The surface charge density cr is related... 

FIGURE 10.1 Schematic representation of the potential distribution (soUd line) across two interacting parallel dissimilar plates 1 and 2 at separation h. Dotted line is the unperturbed potential distribution ath = oo. [Pg.242]

FIGURE 11.2 Scaled potential distribution y x) across two likely charged plates 1 and 2 at scaled separation Kh = 2 with unperturbed surface potentials yoi = 2 and >>o2 = 1-5. yi(x) and y2ix) are, respectively, scaled unperturbed potential distributions around plates 1 and 2 in the absence of interaction at Kh = oo. 

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]. 

We apply the Derjaguin s approximation (Eq. (12.3)) to the low-potential approximate expression for the plate-plate interaction energy, that is, Eqs. (9.53) and (9.65), obtaining the following two formulas for the interaction between two similar spheres 1 and 2 of radius a carrying unperturbed surface potential ij/f, at separation H at constant surface potential, V (H), and that for the constants surface charged density case, V (//) ... 

In Eq. (12.9) the unperturbed surface potential is related to the surface charge density a hy j/o = gIs oK. Note that Eq. (12.9) ignores the influence of the internal electric fields induced within the interacting particles. 

For the mixed case where sphere 1 carries a constant surface potential i/ oi and sphere 2 carries a constant surface charge density cr (or the corresponding unperturbed surface potential ij/oi), Kar et al. [12] derived the following expression... 

For the case where two similar spheres carrying unperturbed surface potential at separation H are immersed in a symmetrical electrolyte of valence z and bulk concentration n, we obtain from Eq. (11.14)... 

For two dissimilar spheres of radii a and 02 carrying scaled unperturbed surface potentials yoi and yo2 in a 2-1 electrolyte solution of concentration n, we obtain fromEq. (11.43)... 

If cylinder 1 has a constant surface potential i/ oi and cyhnder 2 has a constant surface charge density interaction energy per unit length between two parallel cyhnders 1 and 2 at separation H is given by... 

In Fig. 13.3, we plot the potential distribution i/ (x) between two parallel similar ion-penetrable membranes with i/tdoni = iAdon2 = don (or >Aoi = o2 = >Ao) for Kh = 0, 1,2, and oo. In Fig. 13.3, we have introduced the following scaled potential y, scaled unperturbed surface potential y, and scaled Donnan potential yooN-... 

Figures 13.4 and 13.5 give the results for the interaction between two dissimilar membranes, showing changes in the potential distribution y x) = e J/(x)/kT due to the approach of two membranes for the cases when the two membranes are likely charged, that is, Zj > 0 and Z2 > 0 (Fig. 13.4) and when they are oppositely charged, that is, Zi > 0 and Z2 < 0 (Fig. 13.5). Here we have introduced the following scaled unperturbed surface potentials yo, and scaled Donnan potentials yDONi and yooN2 ...

The interaction V"f /f) depends only on the unperturbed surface potential ipoi of sphere i (/ = 1, 2) and can be interpreted as the interaction between sphere i and its image with respect to sphere j ( j= 1, 2 j i). When both spheres are hard, the interaction energy is given by... 

In case (c), the image interaction energy carries both characters of cases (a) and (b). When the unperturbed surface potentials and the radii of the two spheres become similar, these two contributions from cases (a) and (b) tend to cancel each other so that the total image interaction for case (c) becomes small, as shown in Fig. 14.6, in which the interacting spheres are identical Ka = ku2 = 5 and i/ oi = 4 02)- In the opposite case where the difference in the two unperturbed potentials is large, the image interaction for case (c) is determined almost only by the larger unperturbed surface potential. 

It is interesting to note that i/ oi and 1//02, are, respectively, the unperturbed surface potentials of membranes 1 and 2 (i.e., the surface potentials at /i = 00) and that Eq. (16.9) states that the interaction force is proportional to the product of the unperturbed surface potentials of the interacting membranes. This is generally true for the Donnan potential-regulated interaction between two ion-penetrable membranes in which the distribution of the membrane-fixed charges far inside the membranes is uniform but may be arbitrary in the region near the membrane surfaces (see Eq. (8.28)). 

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