* Corresponding author. Tel.: #1-212-650-6688; fax: #1-212-650- 6660. E-mail address: mauri@che-mail.engr.ccny.cuny.edu (R. Mauri). Chemical Engineering Science 55 (2000) 6109 } 6118 Two-dimensional model of phase segregation in liquid binary mixtures with an initial concentration gradient Natalia Vladimirova, Andrea Malagoli, Roberto Mauri* Department of Astronomy and Astrophysics, University of Chicago, Chicago, IL, USA Department of Chemical Engineering, The City College of CUNY, Convent Ave at 138th St., New York, NY 10031, USA Department of Chemical Engineering, Universita % di Pisa, Italy Received 29 November 1999; received in revised form 25 May 2000; accepted 26 May 2000 Abstract We simulate the phase segregation of a deeply quenched binary mixture with an initial concentration gradient. Our theoretical model follows the standard model H, where convection and di!usion are coupled via a body force, expressing the tendency of the demixing system to minimize its free energy. This driving force induces a material #ux much larger than that due to pure molecular di!usion, as in a typical case the Peclet number , expressing here the ratio of thermal to viscous forces, is of the order of 10. Integrating the equations of motion in 2D, we show that the behavior of the system depends on the values of the Peclet number  and the non-dimensional initial concentration gradient . In particular, the morphology of the system during the separation process re#ects the competition between the capillarity-induced drop migration along the concentration gradient and the random #uctuations generated by the interactions of the drops with the local environment. For large , the nucleating drops grow with time, until they reach a maximum size, whose value decreases as the Peclet number and the initial concentration gradient increase. This behavior is due to the fact that the nucleating drops do not have the chance to grow further, as they tend to move towards the homogeneous regions where they are assimilated.  2000 Published by Elsevier Science Ltd. All rights reserved. Keywords: Phase separation; convection-induced spinodal decomposition 1. Introduction In this work we simulate numerically and explain physically a phenomenon which was observed by Gupta, Mauri, and Shinnar (2000): when a low-viscosity liquid binary mixture with a strong initial concentration gradi- ent is quenched deeply into the unstable region of its phase diagram it phase separates without the appearance of large (i.e. larger than 10 m) drops. In general, when a binary mixture is quenched from its single-phase region to a temperature below the composi- tion-dependent spinodal curve, it phase separates through a process called spinodal decomposition (for a review on spinodal decomposition, see Gunton, San Miguel & Sahni, 1983), which is characterized by the spontaneous formation of single-phase domains which then proceed to grow and coalesce. Unlike nucleation, where an activation energy is required to initiate the separation, spinodal decomposition involves the growth of any #uctuations whose wavelength exceeds a critical value. Experimentally, the typical domain size R is de- scribed by a power-law time dependence, t, where n&1/3 when di!usion is the dominant mechanism of material transport, while n&1 when hydrodynamic, long-range interactions become important (Chou & Goldburg, 1979; Wong & Knobler, 1981; Guenoun, Gastaud, Perrot & Beysens, 1987; Gupta, Mauri & Shinnar, 1999). Theoretically, spinodal decomposition in #uids has been described within the framework of the Gin- zburg}Landau theory of phase transition (see Le Bellac, 1984, Chapter 2) by Cahn and Hilliard (1959), showing that during the early stages of the process, initial instabil- ities grow exponentially, forming, at the end, single-phase microdomains whose size corresponds to the fastest- growing mode   of the linear regime (Mauri, Shinnar & Triantafyllou, 1996). During the late stages of the 0009-2509/00/$ - see front matter  2000 Published by Elsevier Science Ltd. All rights reserved. PII: S 0 0 0 9 - 2 5 0 9 ( 0 0 ) 0 0 4 1 2 - 7