The formation of stationary striations in an argon plasma due to the development of an instability is studied using the equations for the competing plasma population species. The nonlinear dynamic stability of the system is analyzed using a two-time perturbation procedure. The unstable uniform steady state bifurcates to a new state with a sinusoidal density variation in space leading to the formation of balls of glowing plasma. This is analogous to phenomena of structure formation that appear in chemical reactions and biological processes.

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In this paper, we continue our re−examination of the role of the continuum approximation in the theory of electrolytes and explore further aspects of the possible generalization of this approximation, introduced in the previous paper. In particular, we present a comparative study of the results obtained on calculating ion−distribution profiles, both for symmetrical (1−1,2−2) and asymmetrical (1−2,2−1) electrolytes. In that ion−distribution curves are extremely sensitive to small changes in the potential (computed via a generalized version of the nonlinear differential equation derived in the previous paper), more detailed information can be obtained regarding the consequences of relaxing the continuum approximation. Specifically, our studies enable us to assess the importance of changing the concentration or varying the charge on the central ion in influencing the distribution of ions about a given central ion. The overall conclusion which can be drawn from our investigation is that, relative to curves generated using the continuum approximation, ion−distribution curves generated when this approximation is relaxed tend to be more localized about the central ion and tend to exhibit an accelerated damping to zero (random distribution) as one moves away from the immediate neighborhood of the central ion. Following Bjerrum, we interpret this behavior as evidence for ’’association,’’ and from our studies we conclude that this effect is important for all electrolyte systems, and becomes more pronounced with increased concentration, or with an increase in the charge number of ionic species present in the system. Finally, we report the results of calculations performed to assess the internal consistency, with respect to charge number, of the nonlinear Poisson−Boltzmann equation, and we find that the internal consistency of this equation (and hence the associated model) is improved upon relaxation of the continuum approximation.

In this paper, the role of the continuum approximation in the theory of electrolytes is re-examined, and a possible generalization of this approximation is considered. For several reasons (cited in the manuscript), the model of Debye and Hiickel is used as the starting point in our analysis. We proceed by postulating a functional form for the dependence of the permittivity e(r) on the distance from the central ion. When this expression for the permittivity is used within the context of Poisson-Boltzmann theory, there results a nonlinear differential equation whose analytic properties are investigated in detail; in particular, it is proved that solutions of this equation exist and are unique. By construction of the Green's function, we obtain the associated nonlinear integral equation and solutions to this equation for a choice of parameters corresponding to an electrolyte system considered previously by Guggenheim. Our main conclusions follow from a comparison of our results with those obtained previously using the continuum approximation. We find that the relaxation of this approximation leads to a significant enhancement of the potential felt by a counterion in the immediate neighborhood of the central ion, with an attendent accelerated damping of the potential as one moves away from the central ion. A second, rather unexpected result which emerges from our study is that the computed potential is surprisingly insensitive to the explicit down-range behavior of the function postulated to describe the change in permittivity as a function of distance. This paper concludes with some remarks on future problems to be studied. In the following paper detailed ion-distribution profiles are reported, and the question of the internal consistency of the augmented Poisson-Boltzmann equation, introduced in this paper, is examined.

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