Front motion in an A+B→C type reaction-diffusion process: Effects of an electric field

Abstract

We study the effects of an external electric field on both the motion of the reaction zone and the spatial distribution of the reaction product, C, in an irreversible A- + B+ -->C reaction-diffusion process. The electrolytes A identical with (A+,A-) and B identical with (B+,B-) are initially separated in space and the ion-dynamics is described by reaction-diffusion equations obeying local electroneutrality. Without an electric field, the reaction zone moves diffusively leaving behind a constant concentration of C's. In the presence of an electric field which drives the reagents towards the reaction zone, we find that the reaction zone still moves diffusively but with a diffusion coefficient which slightly decreases with increasing field. The important electric field effect is that the concentration of C's is no longer constant but increases linearly in the direction of the motion of the front. The case of an electric field of reversed polarity is also discussed and it is found that the motion of the front has a diffusive as well as a drift component. The concentration of C's decreases in the direction of the motion of the front, up to the complete extinction of the reaction. Possible application of the above results to the understanding of the formation of Liesegang patterns in an electric field is briefly outlined.

Designed Patterns: Flexible Control of Precipitation through Electric Currents

“…C; therefore, the motion of the front is potentially affected by applying a constant external electric field [12][13][14][15][16][17]. These investigations indicate however that for a wide range of electric field intensities (for specifics see [16,17]) neither the locality (i), nor the diffusive nature (ii) of the front, which implies the time-law (1), are altered. Nevertheless, the electric field has an important effect on (iii) the production of C-s [16].…”

“…These investigations indicate however that for a wide range of electric field intensities (for specifics see [16,17]) neither the locality (i), nor the diffusive nature (ii) of the front, which implies the time-law (1), are altered. Nevertheless, the electric field has an important effect on (iii) the production of C-s [16]. Namely, for a forward field (i.e., a field that drives the ionic reagents towards each other), the concentration c increases in the direction of the front motion, while a backward field yields a decreasing concentration of C-s, eventually till the complete extinction of the reaction.…”

Formation of Liesegang patterns in the presence of an electric field

“…As already mentioned, this term models the production of C by the moving reaction front, and it is influenced by the presence of an external electric field. The effect of the electric field has been studied in detail in [32], and we summarize below the main results for the range of parameters that are relevant for typical experimental situations. One has to realize first that the reagents A and B are electrolytes which dissociate,…”

“…The modelling of the system in [32] was based on several simplifying assumptions: (i) the one-dimensional character of the system (i.e., all the relevant parameters are only x-dependent); (ii) the complete dissociation of the electrolytes A and B into their respective ions (that allows to eliminate the dynamics of the neutral A-s and B-s from our description); (iii) infinite reaction rate and irreversibility of the basic reaction A − +B + → C. This is justified by the fact that the characteristic reaction timescale is much smaller than any time-scale connected with diffusion and precipitation (pattern formation), and leads to a point-like reaction zone; (iv) the electroneutrality approximation (the local charge density is zero on space scales that are relevant to pattern formation), whose applicability for the systems under study was discussed in detail in [37]; (v) we considered monovalent ions; (vi) finally, we assumed equal diffusion coefficients D of the ions. As mentioned in [32], any of these assumptions may be relaxed without generating major changes of the conclusions of our study. The reaction-diffusion equations for the concentration profiles of the ions can be solved numerically (with boundary conditions that correspond to the presence of the two reservoirs of ions -of concentrations a 0 and b 0 , respectively -at the ends of the reaction cylinder).…”

“…Our aim with this paper is to improve the theoretical description of electric field effects and to bring it up to the level of quantitative predictions. We have shown recently [32] that there are problems with previous attempts [20,27,28,29,31] which incorporate the electric field by assuming that it results in the drift of the reacting ions as well as in the drift of the reaction zone. Treating the background ions properly by using the electroneutrality condition, we found that the main effect of the field is that the amount of reaction product left behind the reaction zone increases (decreases) linearly in space depending whether the field drives the reacting ions towards (away) from the reaction zone.…”