Diffuse double layer properties

J. Frahm and S. Diekmann,/. Colloid Interface Set., 70,440 (1979). Numerical Calculation of Diffuse Double Layer Properties for Spherical Colloidal Particles by Means of a Modified Nonlinearized Poisson-Boltzmann Equation. 

Rheology. Flow properties of latices are important during processing and in many latex appHcations such as dipped goods, paint, inks (qv), and fabric coatings. For dilute, nonionic latices, the relative latex viscosity is a power—law expansion of the particle volume fraction. The terms in the expansion account for flow around the particles and particle—particle interactions. For ionic latices, electrostatic contributions to the flow around the diffuse double layer and enhanced particle—particle interactions must be considered (92). A relative viscosity relationship for concentrated latices was first presented in 1972 (93). A review of empirical relative viscosity models is available (92). In practice, latex viscosity measurements are carried out with rotational viscometers (see Rpleologicalmeasurement). 

According to the Gouy-Chapman model, the thickness of the diffuse countercharge atmosphere in the medium (diffuse double layer) is characterised by the Debye length k 1, which depends on the electrostatic properties of the... 

Although a family of OgS - Jig8 values are allowed under Equation 7 the actual equilibrium state of the oxide/solution interface will be determined by the dissociation of the surface groups and the properties of the electrolyte or the diffuse double layer near the surface. For surfaces that develop surface charges by different mechanisms such as for semiconductor, there will be an equation of state or charge-potential relationship that is analogous to Equation 7 which characterizes the electrical response of the surface. 

However, it is interesting to note that the theory of the diffuse double layer was presented independently by Gouy and Chapman (1910) 13 years before the Debye-Hiickel theory of ion ion interactions (1923). The Debye-Hiickel theory was immediately discussed and applied to the diffuse charge around an ion, doubtless owing to the preoccupation of the majority of scientists in the 1920s with bulk properties rather than those at surfaces. 

The charge density, Volta potential, etc., are calculated for the diffuse double layer formed by adsorption of a strong 1 1 electrolyte from aqueous solution onto solid particles. The experimental isotherm can be resolved into individual isotherms without the common monolayer assumption. That for the electrolyte permits relating Guggenheim-Adam surface excess, double layer properties, and equilibrium concentrations. The ratio u0/T2N declines from two at zero potential toward unity with rising potential. Unity is closely reached near kT/e = 10 for spheres of 1000 A. radius but is still about 1.3 for plates. In dispersions of Sterling FTG in aqueous sodium ff-naphthalene sulfonate a maximum potential of kT/e = 7 (170 mv.) is reached at 4 X 10 3M electrolyte. The results are useful in interpretation of the stability of the dispersions. 

Hyperbolic functions often occur in diffuse double layer theory. Some of their properties are summarized in appendix 2. 

As a consequence of the selective adsorption of ions with a higher affinity for the stationary phase than their counterions electrostatic theories assume the formation of a surface potential between the bulk mobile phase and stationary phase. The adsorbed ions constitute a charged surface, to which is attracted a diffuse double layer of strongly and weakly bound oppositely charged ions equivalent in number to the adsorbed surface charges to maintain electrical neutrality. Because of repulsion effects the adsorbed ions are expected to be spaced evenly over the stationary phase surface and at a concentration that leaves the properties of the stationary phase largely unaltered except for its electrostatic potential. The transfer of solutes from the bulk mobile phase to the... 

The method developed here for the description of chemical equilibria including adsorption on charged surfaces was applied to interpret phosphate adsorption on iron oxide (9), and to study electrical double-layer properties in simple electrolytes (6), and adsorption of metal ions on iron oxide (10). The mathematical formulation was combined with a procedure for determining constants from experimental data in a comparison of four different models for the surface/solution interface a constant-capacitance double-layer model, a diffuse double-layer model, the triplelayer model described here, and the Stem model (11). The reader is referred to the Literature Cited for an elaboration on the applications. 

The properties (the effective viscosity and density) of the liquid layer in close vicinity to the interface can differ from their bulk values. There are various reasons for these phenomena. For example, the properties of a thin liquid layer confined between solid walls are determined by interactions with the solid walls [58,59]. In electrochemical system the structuring of a solvent induced by the substrate and a nonuniform ion distribution in the diffuse double layer can significantly influence the properties of the solution at the interface. The nonuniform distribution of species, which influences the properties of the liquid near the electrode, also occurs in the case of diffusion kinetics. The latter was considered in Ref. 60, where the ferro/ferri redox system was studied by the EQCM. This was the case where the velocity decay length ( >25 pm) was much less than the thickness of the diffusion layer ( >100 pm), in which the composition of the solution is different from the bulk composition. 

An attempt to describe the response of the EQCM in the double-layer region only on the basis of the properties of the diffuse double layer was undertaken in Ref. 61. In doing so, the specific adsorption of ions and the potential-dependent specific interactions of the solvent with the metal were entirely ignored. Under these assumptions one could think of three reasons leading to the observed dependence of frequency on potential (1) dependence of the surface tension on potential, (2) electrostatic adsorption of charged species, and (3) a local change in viscosity in the diffuse double layer. 

The properties of the diffuse double layer depend directly on the surface charge density and not on the potential. In order to correct for this effect quantitatively, one needs to convert the dependence of frequency on potential A/( ), observed experimentally, to its dependence on charge density, A/( q). Having the analogous dependence A/o( q) for the supporting electrolyte, it is possible to evaluate the real response of the EQCM to specific adsorption, 5f q) = A/( ) — A ( ), and use this response for interpretation of the data obtained. This approach was taken in [74, 108,111] for several systems as seen in Figs. 9 and 10. For all cases studied, the surface excess was known from independent electrochemical experiments. 

The final charge on the surface is balanced by counterions from the solution, establishing the so-called diffuse double layer. The solution of the non-linear Poisson-Boltzmann (PB) equation yields the density of counterions, the potential and the electric field at any point [19]. The characteristic length or thickness of the diffuse double layer is the so-called Debye length, which determines the exponential decay of the counterion density away from the surface. It can also be viewed as a screening length and depends solely on the properties of the liquid such as the concentration of the electrolyte, and not on any property of the surface. For a 1 mM solution of a 1 1 electrolyte it is approximately 10 nm [19]. 

Note that in practice, experimental spectra — especially at low frequencies — may also contain contributions originating from the electrode/electrolyte interface. A typical example is electrode polarizaticHi arising from the formation of a diffuse double layer of ions close to a charged surface. Partly, such features depend on the dielectric properties of the electrolyte solution (our focus), but in essence, they are specific to the interface and thus are not topic of this contribution. However, such electrode processes are intensively studied in electrochemistry using, e.g., impedance spectroscopy. 

The description of the double layer properties by the Stem-Gouy model is a very crude one. A veiy weak point is the assumption that the dielectric contact suddenly changes from that of the solution to that of the Helmholtz double layer. The main information comes, therefore, from the minimum which indicates the potential of zero excess charge on the metal. This is, however, only correct in the absence of specific adsorption of ions. If ions are adsorbed, the counter charge for the diffuse double layer is the sum of the surface charge in the metal and of the adsorbed ions. Since the concentration of adsorbed ions also varies with the applied potential, this effect increases the apparent capacity of the Helmholtz double layer. 

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