%0 Book Section %1 statphys23_0165 %A Antal, T. %A Bena, I. %A Droz, M. %A Martens, K. %A Rácz, Z. %B Abstract Book of the XXIII IUPAP International Conference on Statistical Physics %C Genova, Italy %D 2007 %E Pietronero, Luciano %E Loreto, Vittorio %E Zapperi, Stefano %K bands cahn-hilliard equation formation fronts liesegang non-autonomous pattern reaction statphys23 topic-4 %T Precipitation patterns generated by phase separation in a guiding field %U http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=165 %X It has been recently recognized that reaction-diffusion processes may be used for engineering bulk patterns on mesoscopic and microscopic scales. The related problem we are concerned with is the control of Liesegang structures~1, where the precipitation patterns emerge in the wake of a moving reaction-diffusion front. The patterns consist of bands (or rings, shells, depending on the geometry), clearly separated in the direction of the motion of the front. The positions of the bands usually obey simple laws, e.g. they form a geometric series with increasing distance between consecutive bands. This is the so-called regular banding, which has been explained~2 using phase separation in the presence of a moving front as the underlying mechanism. There are, however, experimental observations of patterns with decreasing distance between successive bands. This is the so-called inverse banding~3 whose understanding would clearly help in gaining control on pattern formation. Here we present a possible scenario for the formation of inverse banding. The proposal is still based on the phase separation mechanism, to which we add a guiding field such as for example a temperature or a pH field. The phase separation is modeled through a non-autonomous Cahn-Hilliard equation whose spinodal line is controlled by the guiding field. As a consequence, the dynamics of the guiding field controls the velocity of the instability front and, since the wavelength of the emerging pattern is largely determined by this velocity, it also controls the distance between the successive bands. The design of a guiding field appropriate for generating a given pattern turns out to be a nontrivial problem which we studied both analytically and numerically. We found that a simple physically realizable guiding field, such as the one arising from the dynamics of a temperature field due to a temperature jump at the boundary, is sufficient to generate inverse patterns. The spacing characteristics of the patterns are determined from simulations, and we show that the result can be explained by relating the velocity of the front in the temperature field to the velocity resulting from a linear stability analysis of the phase separation process.\\ 1) H. K. Henisch, Crystals in Gels and Liesegang Rings, Cambridge University Press, Cambridge (1988).\\ 2) T. Antal, M. Droz, J. Magnin, and Z. Rácz, Formation of Liesegang patterns: A spinodal decomposition scenario, Phys. Rev. Lett. 83, 2880 (1999).\\ 3) N. Kanniah, F. D. Gnanam, P. Ramasamy, and G. S. Laddha, Revert and direct type Liesegang phenomenon of Silver Iodid, J. Coll. and Int. Sci. 80, 369 (1980).

@incollection{statphys23_0165, abstract = {It has been recently recognized that reaction-diffusion processes may be used for engineering bulk patterns on mesoscopic and microscopic scales. The related problem we are concerned with is the control of Liesegang structures~[1], where the precipitation patterns emerge in the wake of a moving reaction-diffusion front. The patterns consist of bands (or rings, shells, depending on the geometry), clearly separated in the direction of the motion of the front. The positions of the bands usually obey simple laws, e.g. they form a geometric series with increasing distance between consecutive bands. This is the so-called {\it regular banding}, which has been explained~[2] using {\it phase separation in the presence of a moving front} as the underlying mechanism. There are, however, experimental observations of patterns with decreasing distance between successive bands. This is the so-called {\it inverse banding}~[3] whose understanding would clearly help in gaining control on pattern formation. Here we present a possible scenario for the formation of inverse banding. The proposal is still based on the {\it phase separation} mechanism, to which we add a {\it guiding field} such as for example a temperature or a {\it pH} field. The phase separation is modeled through a non-autonomous Cahn-Hilliard equation whose spinodal line is controlled by the guiding field. As a consequence, the dynamics of the guiding field controls the velocity of the instability front and, since the wavelength of the emerging pattern is largely determined by this velocity, it also controls the distance between the successive bands. The design of a guiding field appropriate for generating a given pattern turns out to be a nontrivial problem which we studied both analytically and numerically. We found that a simple physically realizable guiding field, such as the one arising from the dynamics of a temperature field due to a temperature jump at the boundary, is sufficient to generate inverse patterns. The spacing characteristics of the patterns are determined from simulations, and we show that the result can be explained by relating the velocity of the front in the temperature field to the velocity resulting from a linear stability analysis of the phase separation process.\\ 1) H. K. Henisch, {\em Crystals in Gels and Liesegang Rings}, Cambridge University Press, Cambridge (1988).\\ 2) T. Antal, M. Droz, J. Magnin, and Z. R\'acz, {\it Formation of Liesegang patterns: A spinodal decomposition scenario}, Phys. Rev. Lett. {\bf 83}, 2880 (1999).\\ 3) N. Kanniah, F. D. Gnanam, P. Ramasamy, and G. S. Laddha, {\it Revert and direct type Liesegang phenomenon of Silver Iodid}, J. Coll. and Int. Sci. {\bf 80}, 369 (1980).}, added-at = {2007-06-20T10:16:09.000+0200}, address = {Genova, Italy}, author = {Antal, T. and Bena, I. and Droz, M. and Martens, K. and R\'acz, Z.}, biburl = {https://www.bibsonomy.org/bibtex/22621dcb7fdcf1a3d1b1a21eaf28e3fe9/statphys23}, booktitle = {Abstract Book of the XXIII IUPAP International Conference on Statistical Physics}, editor = {Pietronero, Luciano and Loreto, Vittorio and Zapperi, Stefano}, interhash = {fad6abedfeb30578312c49ebfb060ddb}, intrahash = {2621dcb7fdcf1a3d1b1a21eaf28e3fe9}, keywords = {bands cahn-hilliard equation formation fronts liesegang non-autonomous pattern reaction statphys23 topic-4}, month = {9-13 July}, timestamp = {2007-06-20T10:16:13.000+0200}, title = {Precipitation patterns generated by phase separation in a guiding field}, url = {http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=165}, year = 2007 }