Migrating triplet precipitation bands of calcium phosphates in gelatinous matrix

Abstract

Outside the realm of biology and geology, one of the most beautiful and familiar examples of self-organized precipitate structure formation in nature can be found in reaction-diffusion systems. Though the patterned precipitations in reaction-diffusion systems was discovered by Raphael Eduard Liesegang more than a century ago [1], the mechanism of this nonlinear phenomenon is not yet fully understood and hence are still under discussion [2–5]. The formation of the banded precipitate patterns is due to the diffusion of an electrolyte A into a gelatinous matrix impregnated with another electrolyte B and the subsequent chemical reaction in a diffusion limited way [6]. Scientists speculated on the regular Liesegang banding framed three generic spatio temporal relations. The first one, called ‘time law’ connects the distance xn of the nth band measured from the junction point of the two electrolytes and the time tn of its formation through the relation x2 n = α tn . This result given by Morse and Pierce [7] reflects the diffusive behavior of the outer electrolyte into the gelatinous matrix. The second one, the ‘spacing law’ of Jablczynski [8] is a more intricate property of the bands, according to which the ratio between the positions of the adjacent bands follow a geometric series xn+1 = β xn, where β approaches a constant value when the order of the rings is sufficiently large. The spacing between the bands (xn+1 − xn) normally increases as n increases. Hence the constant β referred to as the spacing coefficient is usually greater than unity and is generally expressed as (1 + p) where 0.05 ≤ p ≤ 0.4 [9]. Finally the width wn of the nth band has observed to increase with n and obey the simple linear relation wn = γ xn where γ is another constant [10]. The authors have recently developed a new model for the band formation [11–16] based on the assumption that the boundary which separates the outer ions and inner electrolyte migrate virtually into the positive direction of the advancement of the A type ions. Once the moving boundary concept is introduced, we found that the theory straight away upholds the time law, spacing law [11] and width law [16] in a more efficacious manner. With the assumptions and approximations of the model, we have obtained modified versions of the conventional laws as