Abstract
We shall briefly review some early nucleation models, and then examine some aspects of the subsequent evolution of their solutions. Such situation is characterised by the onset of comparatively large clusters that can diffuse into the medium and interact among themselves. We next discuss some situations where the aggregates being formed, whose actual shape is one of the major questions under consideration, do posses a filamentary nature, and can sometimes generate a percolating network. Finally, a particularly interesting case of such tree-like structures, that of vascular systems, will be addressed, and some facts (and open questions) concerning their simulation via reaction-diffusion equations will be discussed.
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References
Andreucci, D., Herrero, M.A. and Velazquez, J.J.L. (2001) The classical one-phase Stefan problem: a catalogue of interface behaviours. To appear in Surveys on Mathematics in Industry.
Andreucci, D., Herrero, M.A. and Velazquez, J.J.L. On the growth of filamentary planar structures. To appear.
Boal, A.K., Ilhan, F., De Rouchey, J., Albrecht, T.T., Russell, T.P. and Rotello, V.M. (2000) Self assembly of nanoparticles into structured spherical and network aggregates. Nature, 404, 746–748.
Banfield, J.F., Welch, S.A., Zhang, H., Ebert, T.T. and Penn, R.L. (2000) Aggregation-based crystal growth and microstructure development in natural iron oxyhydroxide biomineralization products. Science, 289, 751–754.
Chandrasekhar, S. (1943) Stochastic problems in physics and astronomy. Rev. Modern Physics, 15, 1, 1–91.
Escobedo, M., Herrero, M.A. and Velazquez, J.J.L. (2000) Radiation dynamics in homogeneous plasma. Physica D, 126, 236–260.
Flory, P. (1941) Molecular size distribution in three dimensional problems: I Gellation. J. Am. Chem. Soc., 63, 3083–3090.
Gierer, A. and Meinhardt, H. (1972) A theory of biological pattern formation. Kybernetik, 12, 30–39.
Herrero, M.A., Velazquez, J.J.L and Wrzosek, D. (2000) Sol-gel transition in a coagulation-difussion model. Physica D, 141, 221–247.
Luckhaus, S. (1991) The Stefan problem with the Gibbs-Thomson relation for the melting temperature. Europ. J. Appl. Math., 1, 101–111.
Lifshitz, I.M. and Slyozov, V.V. (1961) Kinetics of precipitation from supersaturated solid solutions. J. Phys. Chem. Sol., 19, 35–50.
Leyvraz, F. and Tschudi, H.R. (1981) Singularities in the kinetics of coagulation processes. J. Phys. A, 14, 3389–3405.
Meinhardt, H. (1982) Models of biological pattern formation. Academic Press.
Meinhardt, H. (1976) Morphogenesis of lines and nets. Differentiation, 6, 117–123.
Meinhardt, H. (1997) Biological pattern formation as a complex dynamic phenomenon. Int. J. Bifurcation and Chaos, 7, 1, 1–26.
McLeod, J.B. (1962) On an infinite set of nonlinear differential equations. Quart. J. Math. Oxford 2, 119–128.
Metzger, R.J. and Krasnow, M.A. (1999) Genetic control of branching morphogenesis. Science, 284, 1635–1639.
Niethammer, B. and Pego, R.L. (1999) Non self-similar behaviour in the LSW theory of Ostwald ripening. J. Stat. Phys., 95, 867–902.
Pelcé, P. (ed.) (1988) Dynamics of curved fronts. Perspectives in Physics, Academic Press.
Peng, G., Quiu, F., Guinzburg, V.V., Jasnow, D. and Balasz, A.C. (2000) Forming supramolecular networks from nanoscale rods in binary, phase-separating mixtures. Science, 288, 1802–1804.
Smoluchowski, M. (1916) Drei Vorträge über Diffusion, Brownische Bewegung und Koagulation von Kolloiden. Physik Z, 17, 557–585.
Turing, A.M. (1952) The chemical basis of morphogenesis. Phil. Trans. Royal Soc. London B, 237, 37–72.
Velâzquez, J.J.L. (1998) The Becker-Döring equations and the Lifshitz-Slyozov theory. J. Stat. Phys., 92, 195–236.
Velâzquez, J.J.L. (2000) On the effect of stochastic fluctuations in the dynamics of the Lifshitz-Slyozov-Wagner model. J. Stat. Phys., 99, 57–113.
Wales, D.J. and Scheraga, H.A. (1999) Global optimization of clusters, crystals and biomolecules. Science, 285, 1368–1372.
Yancopoulos, G.D., Davis, S., Gale, N.W., Rudge, J.S., Wiegand, S.J. and Holash, J. (2000) Vascular-specific growth factors and blood vessel formation. Nature, 407, 242–248.
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Herrero, M.A. (2002). From Nucleation to Large Aggregates: the Growth of Filamentary. In: Anile, A.M., Capasso, V., Greco, A. (eds) Progress in Industrial Mathematics at ECMI 2000. Mathematics in Industry, vol 1. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-04784-2_2
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DOI: https://doi.org/10.1007/978-3-662-04784-2_2
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