Nucleation and Growth on Curved Space

L. R. GÓMEZ; N. A. GARCÍA; D. A. VEGA

Carlos Paz

Congreso; LAWNP2013; 2013

Nucleation and growth (NG) is by far the most common mechanism leading the dynamics of first order phase transitions. Between other examples, this process directs the formation of a crystal phase from an initial liquid (crystallization), and the dynamics of order-order phase transitions in a huge variety of hard and soft condensed matter systems [P. G. Debenedetti, Metastable Liquids: Concepts and Principles (Princeton University Press, Princeton, NJ, 1998).].In its classical picture, NG starts with local structural uctuations of the initial phase whichlead to the formation of small nuclei of the new phase. In general, the formation of a nucleiinvolve changes in dierent free energy contributions, which are competing in nature, and control the dynamical evolution of the phase transition. On one hand, the birth of a nucleus of the new phase produces an interphase, leading to an increase of free energy due to surface tension. On the other hand, there is a decrease in the total free energy due to the local formation of the less energetic phase. Here the competition of these surface and volume terms produce an activated dynamics, such that the only nuclei which can propagate are the ones whose size overpass a critical value.The aim of the present study is to provide a general physical description of NG in twodimensional crystalline systems lying on curved surfaces. Such curved crystals are not only commonly found in nature in systems like viral capsids, insect eyes, pollen grains, and radiolaria, but also can be grown in the laboratory by using colloidal particles, liquid crystals, block copolymers, and possibly other self-assembled condensed matter systems [V. Vitelli et. al, Proc. Natl Acad. Sci. USA 103, 12323 (2006); W. T. M. Irvine et. al, Nature (London) 468, 947 (2010)]; [García et. al, Phys. Rev. E 88, 012306 (2013)].Here, by using theoretical calculations and simulations, we show how the curvature of thesubstrate aects the critical size of propagating nuclei and the minimum energy path towardsthe equilibrium crystal phase. The model is based on a coarse-grained Guinzburg-Landau free energy functional and a phase-eld evolution equation. This approach can be combined with polar geodesic coordinates and conformal mapping to obtain analytical expressions for the size of the critical nuclei as a function of degree of undercooling and local substrate´s curvature.