C. A. Barlow

Discreteness‐of‐Charge Adsorption Micropotentials. I. Infinite Imaging

The Ershler infinite imaging model for adsorption of ions from electrolytes is discussed in detail. An incorrect derivation of the basic equations for the micropotential is critically examined, and a correct derivation provided. The potential arising from discrete adions and their images is determined exactly based upon a method which heretofore has only been applied to crystal field problems. Finally, the accurate results obtained using a digital computer are compared with the approximate results of other treatments.

Theory of Work‐Function Change on Adsorption of Polarizable Ions

We first point out and correct an error which occurred in an earlier, more approximate theory of the work‐function change accompanying ion adsorption. Next, certain of the approximations of this earlier work are lifted to produce a more accurate theory. In particular, the adsorbed ions and their images in the adsorbent are no longer approximated as ideal dipoles. We employ results of recent exact calculations to obtain quite accurate expressions for the electric fields due to these nonideal dipoles and thus for the induced polarization associated with these fields. Furthermore, the images of induced dipoles are explicitly taken into account in their effect upon the self‐consistent depolarizing field. This approach leads to a new expression for the effective dielectric constant of the adsorbed phase which takes cognizance of discreteness effects. Theoretical predictions are found to be in agreement with data for Cs+ and K+ on tungsten for small θ and may apply well up to θ≲0.6. Finally we discuss possible ...

A simple method for ionic and molecular adsorption electrical calculations

Abstract The usual methods for calculating potentials and fields arising from a regular, adsorbed plane array of ions or dipoles are complex and time consuming to apply. We present here a method based on a modification of Grahames cut-off model which allows several such quantities to be calculated accurately but rapidly from simple closed formulas. The method is applied to hexagonal arrays of ideal and non-ideal dipoles. Non-ideal dipoles are assumed to arise from imaging of an array of adions in a conducting adsorbent. Results of the simple, approximate formulas are compared in detail with very accurate results obtained from lengthy computer calculations. We believe the latter will be unnecessary hereafter, whenever the field or potential is desired on a line perpendicular to the plane adsorbent through a removed dipole or ion. With proper normalization, potential-distance curves for ideal and non-ideal (finite-length) dipoles are found to be nearly the same. Finally, the present results are employed to yield an improved formula for the change of work function of a conducting surface, when a hexagonal array of polarizable molecules or atoms is adsorbed on it. The formula is illustrated and shows that adsorbed uncharged elements, with sufficiently high but still physical polarizability, must either ionize or their polarizability decrease upon adsorption.

Note Concerning the Velocity of Gunn Domains

Using computer solutions of the relevant differential equations, many authors have determined the velocity for which traveling domains exist in a semiconductor with a region of negative differential mobility. Recently, Gunn presented a topological argument leading to a closed form equation for the velocity of a finite amplitude domain. For the case of a domain shape controlled by diffusion we show that Gunns argument is based on a necessary, but not sufficient, condition that a certain singular point be a center. This leads to an incorrect expression for the domain velocity for all but infinitesimal amplitude domains. The topological character of each solution as the velocity parameter is varied is also described.

Penetration Parameter for an Adsorbed Layer of Polarizable Ions

We present several improvements in the calculation of the penetration parameter f, which is related to ionic and electronic work functions. A regular, hexagonal array of adions which are imaged in a conducting adsorbent plane is considered. The first improvement is the treatment of discrete adion charges and their images as non‐ideal, finite‐length dipoles instead of ideal dipoles as in almost all earlier treatments. Next, nonzero polarizability of adions is considered, discrete induced dipole moments are calculated in a self‐consistent manner, and the effect of induced‐dipole images is included. When the polarizability is nonzero, previous expressions for f are inadequate even when generalized in an obvious way, and an improved method of calculating f is presented. The inclusion of polarization effects leads to significant changes in the dependence of f on surface coverage of adions. Finally, f is calculated with and without polarization contributions for the situation where redistribution of the adsorbed array upon removal of an adion affects the desorption energy. Inclusion of redistribution also leads to appreciable changes in f. Under some circumstances, either with or without redistribution contributions, it is found to be easier to desorb an adion from an array than to desorb an isolated adion. In this case, the potential at the site of a removed adion may exceed that far away from the surface. To illustrate this phenomenon, which can lead to the establishment of a potential barrier against electron emission, or to a potential well enhancing emission, we give some curves of potential vs distance from the electrode along a line perpendicular to the electrode and through a removed adion site.

EQUILIBRIUM DOUBLE-LAYER THEORY

Abstract Those electrolyte double-layer theories are reviewed and compared which involve detailed consideration of charge and potential conditions in the double-layer region. Separate consideration is given to those calculations where it may be a good approximation to treat all charges as continuously distributed and those where discreteness-of-charge effects must be invoked. Three conditions are analyzed: (1) that where there is no specific adsorption and the (solvent) molecules in the mono-molecular inner region of the double layer are all of the same type and close-packed; (2) specific ionic adsorption; and (3) specific adsorption of neutral substances. It is shown how previous treatments of (1) may be improved by including induced molecular polarization and average planar dipolar interaction effects. Careful attention is given to the calculation of the field which orients dipoles in the inner layer, and it is pointed out how the inclusion of planar depolarization alters the approach to dielectric saturation. Errors, defects, and simplifications in earlier continuous-charge treatments of specific ionic adsorption are discussed. Some difficulties arose from failure to distinguish properly mean fields, based on the continuous-charge approximation, and fields derived from the micropotential, a quantity derived from discreteness-of-charge considerations. It is concluded that presently available phenomenological treatments of specific ionic adsorption and the micropotential are inadequate since they generally involve improper calculations of the micropotential, ignore various charge discreteness effects, and employ oversimplified statistical mechanics. Comparison of previous treatments of specific adsorption of neutral substances again shows unwarranted simplification. In particular, it does not appear that the dipole moment and/or dielectric constant of adsorbed molecules can be extracted with any confidence by applying current theory to experimental data. An improved treatment of the problem is outlined which makes use of an approximate formula of Frumkin that treats unadsorbed and adsorbed regions in the double-layer separately but which is here generalized by coupling such regions together by planar interaction. Finally, a statistical treatment of the discrete adsorption problem is briefly described which is more nearly correct than the usual Boltzmann expressions in those situations where competition and saturation effects are important.