Small surface charge densities

Figure 5. Ratio between the double layer force with site exclusion (modified Poisson-Boltzmann) and the double layer force provided by the Poisson-Boltzmann equation (a)constant (small) surface potential (Vs = 0.02 V) (b) constant (small) surface charge density (<7S = 0.032 C/m2), n/V = 1.0 M and T = 300 K.

Fundamental studies on such systems ate scarce, but some attention has been paid to dispersion in hydrocarbon solutions of surface-active agents for which the ionic concentrations are extremely small, e.g. 10 ° mol dm corresponding to I/k 10 psn. Since 1/k is large the capacity of the double layer is small and only a small surface charge density is necessary to obtain an appreciable surface potential Furthermore, the slow decay in potential from the surface means that the zeta potential, readily obtained from electrophoresis experiments, may be equated with considerable accuracy to the surface potential. 

Interestingly, the form of these expressions is similar to that of the expressions for the bent brush, that is, /0/15 for K and K and /o/io for Kcq. Indeed, the free energy per unit area of a flat double layer is, in the limit of small surface charge density, given by... 

For small surface charge densities the free energy of interaction per unit area, between flat surfaces is given by the following (107) ... 

Sufficient charge must be available to supply the required critical surface charge density. Calculations show that 50-pound plastic bags and plastic lined 55-gallon drums are too small to represent a credible risk. 

The electrosorption valence can be related to the dipole moment of an adsorbed species introduced in Chapter 4. For this purpose consider an electrode surface that is initially at the pzc and free of adsorbate. When a small excess charge density o is placed on the metal, its potential changes by an amount A given by ... 

Fig. 5.5. Geometrical structure of a close-packed metal surface. Left, the second-layer atoms (circles) and third-layer atoms (small dots) have little influence on the surface charge density, which is dominated by the top-layer atoms (large dots). The top layer exhibits sixfold symmetry, which is invariant with respect to the plane group p6mm (that is, point group Q, together with the translational symmetry.). Right, the corresponding surface Brillouin zone. The lowest nontrivial Fourier components of the LDOS arise from Bloch functions near the T and K points. (The symbols for plane groups are explained in Appendix E.)...

Figure 4.4 shows the calculated relationship between the surface potential and surface charge for different concentrations of a monovalent salt. We see that for small potentials the surface charge density is proportional to the surface potential. Depending on the salt concentration, the linear approximation (dashed) is valid till ipo 40...80 mV. At high salt concentration more surface charge is required to reach the same surface potential than for a low salt concentration. 

With alkali metal cryptates, cations and anions are well separated (except for the KNCS complex of [2.2.1] in which the ligand is too small to effectively shield the cation). Indeed the cryptated cation may be regarded as a very large, spheroidal cation (—10 A in diameter) of low surface charge density. 

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