Hostname: page-component-77c89778f8-cnmwb Total loading time: 0 Render date: 2024-07-21T15:29:56.089Z Has data issue: false hasContentIssue false
  • English
  • Français

Transition of Liesegang Precipitation Systems: Simulations with an Adaptive Grid PDE Method

Published online by Cambridge University Press:  20 August 2015

Paul A. Zegeling ,
István Lagzi  and
Ferenc Izsák
Paul A. Zegeling*
Affiliation:
Department of Mathematics, Utrecht University, P.O. Box 80010, 3508 TA Utrecht, The Netherlands
István Lagzi*
Affiliation:
Department of Meteorology, Eötvös University, H-1117 Budapest, Pázmány sétány 1/A, Hungary
Ferenc Izsák*
Affiliation:
Department of Applied Analysis and Computational Mathematics, Eötvös University, H-1117 Budapest, Pázmány sétány 1/C, Hungary Department of Applied Mathematics, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands
*
Email:p.a.zegeling@uu.nl
Email:lagzi@nimbus.elte.hu
Corresponding author.Email:izsakf@cs.elte.hu
Get access

Abstract

The dynamics of the Liesegang type pattern formation is investigated in a centrally symmetric two-dimensional setup. According to the observations in real experiments, the qualitative change of the dynamics is exhibited for slightly different initial conditions. Two kinds of chemical mechanisms are studied; in both cases the pattern formation is described using a phase separation model including the Cahn-Hilliard equations. For the numerical simulations we make use of an adaptive grid PDE method, which successfully deals with the computationally critical cases such as steep gradients in the concentration distribution and investigation of long time behavior. The numerical simulations show a good agreement with the real experiments.

Keywords

65N5065M5035B36Liesegang phenomenonnumerical simulationsadaptive grid technique
Type
Research Article
Information
Communications in Computational Physics , Volume 10 , Issue 4 , October 2011 , pp. 867 - 881
Copyright
Copyright © Global Science Press Limited 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Alikakos, N. D., Bates, P. W. and Fusco, G. K., Slow motion for the Cahn-Hilliard equation in one space dimension, J. Differ. Equations, 90 (1991), 81–135.Google Scholar
[2]Antal, T., Droz, M., Magnin, J. and Rácz, Z., Formation of Liesegang patterns: a spinodal decomposition scenario, Phys. Rev. Lett., 83 (1999), 2880–2883.Google Scholar
[3]Antal, T., Droz, M., Magnin, J., Pekalski, A. and Rácz, Z., Formation of Liesegang patterns: simulations using a kinetic Ising model, J. Chem. Phys., 114 (2001), 3770–3775.Google Scholar
[4]Bates, P. W. and Fife, P. C., Spectral comparison principles for the Cahn-Hilliard and phase-field equations, and time scales for coarsening, Physica D, 43 (1990), 335–348.Google Scholar
[5]Bates, P. W. and Fife, P. C., The dynamics of nucleation for the Cahn-Hilliard equation, SIAM J. Appl. Math., 53 (1993), 990–1008.Google Scholar
[6]Budd, C. J., Huang, W. and Russell, R. D., Adaptivity with moving grids, Acta. Numer., 18 (2009), 111–241.Google Scholar
[7]Cahn, J. W. and Hilliard, J. E., Free energy of a nonuniform system I: interfacial free energy, J. Chem. Phys., 28 (1958), 258–267.Google Scholar
[8]Chacron, M. and L’Heureux, I., A new model or periodic precipitation incorporating nucle-ation, growth and ripening, Phys. Lett. A, 263 (1999), 70–77.CrossRefGoogle Scholar
[9]Dam, A. van and Zegeling, P. A., A robust moving mesh finite volume method applied to 1D hyperbolic conservation laws from magnetohydrodynamics, J. Comput. Phys., 216 (2006), 526–546.Google Scholar
[10]van Dam, A. and Zegeling, P. A., Balanced monitoring of flow phenomena in moving mesh methods, Commun. Comput. Phys., 7 (2010), 138–170.Google Scholar
[11]Doster, F., Zegeling, P. A. and Hilfer, R., Numerical solutions of ageneralized theory for macroscopic capillarity, Phys. Rev. E, 81 (2010), 036307.Google Scholar
[12]Droz, M., Magnin, J. and Zrinyi, M., Liesegang patterns: studies on the width law, J. Chem. Phys., 110 (1999), 9618–9622.Google Scholar
[13]Elliott, C. M. and French, D. A., A nonconforming finite-element method for the two-dimensional Cahn-Hilliard equation, SIAM J. Numer. Anal., 26 (1989), 884–903.Google Scholar
[14]Elliott, C. M., French, D. A. and Milner, F. A., A 2nd-order splitting method for the Cahn-Hilliard equation, Numer. Math., 54 (1989), 575–590.Google Scholar
[15]Feeney, R., Schmidt, S. L., Strickholm, P., Chandam, J. and Ortoleva, P., Periodic precipitation and coarsening waves-application of the competitive particle growth model, J. Chem. Phys., 78 (1983), 1293–1311.Google Scholar
[16]Feng, W. M., Yu, P., Hu, S. Y., Liu, Z. K., Du, Q. and Chen, L., Spectral implementation of an adaptive moving mesh method for phase-field equations, J. Comput. Phys., 220 (2006), 498–510.CrossRefGoogle Scholar
[17]Grzybowski, B. A., Bishop, K. J. M., Campbell, C. J., Fialkowski, M. and Smoukov, S. K., Microand nanotechnology via reaction-diffusion, Soft Matter, 1 (2005), 114–128.Google Scholar
[18]Izsák, F. and Lagzi, I., A new universal law for the Liesegang pattern formation, J. Chem. Phys., 122 (2005), 184707.CrossRefGoogle ScholarPubMed
[19]Jablczynski, K., La formation rythmique des pecipites: Les anneaux de Liesegang, Bull. Soc. Chim. Fr., 33 (1923), 1592–1597.Google Scholar
[20]Keller, J. B. and Rubinow, S. I., Recurrent precipitation and Liesegang rings, J. Chem. Phys., 74 (1981), 5000–5007.Google Scholar
[21]Krug, H-J. and Brandtstädter, H., Morphological characteristics of Liesegang rings and their simulations, J. Phys. Chem. A, 103(39) (1999), 7811–7820.Google Scholar
[22]Lagzi, I., Volford, A. and Büki, A., Effect of geometry on the time law of Liesegang patterning, Chem. Phys. Lett., 396 (2004), 97–101.Google Scholar
[23]Liesegang, R. E., Uber einige Eigenschaften von Gallerten Naturwiss, Wochenschr., 11 (1896), 353–362.Google Scholar
[24]Lovas, R., Kacsuk, P., Lagzi, I. and Turänyi, T., Unified development solution for cluster and Grid computing and its application in chemistry, Lect. Notes. Comput. Sci., 3044 (2004), 226–235.Google Scholar
[25]Petzold, L. R., A Description of DASSL: A Differential/Algebraic System Solver, in: IMACS Transactions on Scientific Computation, eds.: Stepleman, R. S.et al. (North-Holland, Amsterdam, 1993), 65–75.Google Scholar
[26]Qiao, Z., Numerical investigations of the dynamical behaviors and instabilities for the Gierer-Meinhardt system, Commun. Comput. Phys., 3 (2008), 406–426.Google Scholar
[27]Rácz, Z., Formation of Liesegang patterns, Physica A, 274 (1999), 50–59.Google Scholar
[28]Ripszám, M., Nagy, A., Volford, A., Izsäk, F. and Lagzi, I., The Liesegang eyes phenomenon, Chem. Phys. Lett., 414 (2005), 384–388.CrossRefGoogle Scholar
[29]Sanderson, A. R., Meyer, M. D., Kirby, R. M. and Johnson, C. R., A framework for exploring numerical solutions of advection-reaction-diffusion equations using a GPU-based approach, Comput. Visual. Sci., (2008), DOI 10.1007/s00791-008-0086-0.Google Scholar
[30]Scheel, A., Robustness of Liesegang patterns, Nonlinearity, 22 (2009), 457–483.Google Scholar
[31]Sultan, R. and Panjarian, S., Propagating fronts in 2D Cr(OH)3 precipitate systems in gelled media, Physica D, 157 (2001), 241–250.Google Scholar
[32]Sultan, R., Propagating fronts in periodic precipitation systems with redissolution, Phys. Chem. Chem. Phys., 4 (2002), 1253–1261.CrossRefGoogle Scholar
[33]Tan, Z., Tang, T. and Zhang, Z., A simple moving mesh method for one- and two-dimensional phase-field equations, J. Comput. Appl. Math., 190 (2006), 252–269.CrossRefGoogle Scholar
[34]Tang, H.-Z. and Tang, T., Adaptive mesh methods for one- and two-dimensional hyperbolic conservation laws, SIAM J. Numer. Anal., 41 (2003), 487–515.Google Scholar
[35]Wells, G. N., Kuhl, E. and Garikipati, K., A discontinuous Galerkin method for the Cahn-Hilliard equation, J. Comput. Phys., 218 (2006), 860–877.Google Scholar
[36]Zegeling, P. A., Tensor-product adaptive grids based on coordinate transformations, J. Comput. Appl. Math., 166 (2004), 343–360.CrossRefGoogle Scholar
[37]Zegeling, P. A., Theory and Application of Adaptive Moving Grid Methods, Chapter 7 in Adaptive Computations: Theory and Algorithms, ed. Tang, T.et al., Science Press, Beijing, 2007.Google Scholar
[38]Zhang, Z. R. and Tang, T., Resolving small-scale structures in Boussinesq convection by adaptive grid methods, J. Comput. Appl. Math., 195 (2006), 274–291.CrossRefGoogle Scholar
[39]Zhang, Z. R. and Tang, H. Z., An adaptive phase field method for the mixture of two incompressible fluids, Comput. Fluids, 83 (2007), 1307–1318.Google Scholar