Experimental Evidence of the Motion of a Single Out-of-Equilibrium Drop Dafne Molin, Roberto Mauri,* and Vincenzo Tricoli Department of Chemical Engineering, Industrial Chemistry and Material Science, UniVersita ` di Pisa, 56126 Pisa, Italy ReceiVed March 21, 2007. In Final Form: May 21, 2007 We show experimentally that when a single, neutrally buoyant drop is injected into a binary mixture either it remains quiescent or it moves, depending on whether the composition of the drop and that of the surrounding phase coincide with the equilibrium concentrations. In general, the movement of out-of-equilibrium drops, which is called diffusiophoresis, is induced by the Korteweg body force. This force is proportional to the chemical potential gradient and is therefore nonzero only when the system is in chemical nonequilibrium. In this letter, we show experimentally that this movement occurs for a single drop as well, even when the initial condition is (almost) isotropic. This instability, although it does not have a complete analytical explanation, has been predicted in the numerical simulations by Vladimirova et al. (Vladimirova, N.; Malagoli, A.; Mauri, R. Phys. ReV.E 1999, 60, 2037). Introduction In this work, we intend to document with a “clean” experiment that a single, neutrally buoyant drop under conditions of chemical nonequilibrium experiences a force that induces its movement, whereas at equilibrium such a drop remains still. The movement of out-of-equilibrium drops has been extensively observed in the study of the process of phase separation of liquid binary mixtures. In fact, it was observed that even in the viscous regime (i.e., in the absence of any buoyancy forces) the liquid-liquid phase transition is driven by convection. This conclusion was reached in early works by Chou and Goldburg 2 and Wong and Knobler 3 using light-scattering techniques and more recently by Guenoun et al., 4 Tanaka, 5 Tanaka and Araki, 6 and Poesio et al. 7 by direct observation. In particular, 10 μm droplets during phase separation are observed to move at a typical speed of 10/100 μm/s. 8 As a result, phase separation of partially miscible solvent mixtures occurs very rapidly, irrespective of the presence of emulsifying compounds within the solution. 9,10 However, no clean experiment has been conducted so far to document the movement of a single drop. Theoretically, phase transitions in fluid mixtures have been successfully described through the diffuse interface model, 11-13 also known as the H model under the taxonomy of Hohenberg and Halperin. 14 Here, mass and momentum transports are coupled via a Korteweg body force. This force, which arises from minimizing the free energy of the system, is proportional to chemical potential gradients; therefore, it is identically zero at thermodynamic equilibrium. As shown in many simulations, the Korteweg body force is responsible for diffusiophoresis, that is, the strong motion of the single-phase domains that is observed experimentally during the liquid-liquid phase transition. 15,16 As expected, when the system is composed of single-phase domains separated by sharp interfaces, the Korteweg force reduces to the more conventional Marangoni capillary force. 17,18 By imposing the condition that such a nonequilibrium capillary force balances viscous forces (assuming that inertial forces are negligible), Siggia 19 showed that the enhanced coalescence caused by such effects can explain the experimentally observed linear growth of the nucleating droplet size during phase separation. In related works, Karpov 20 and Karpov and Oxtoby 21 noticed that capillary forces drive the motion of nucleating droplets along a composition gradient, leading to particle clustering and direct coalescence. A similar phenomenon was also observed by Santonicola et al., 22 who noticed that convection starts to occur as soon as the temperature of the mixture reaches its critical value, well before the appearance of nucleating droplets. Here we are interested in the motion of a single drop under conditions of nonequilibrium. Such a phenomenon has been studied in other contexts as well. In particular, Kogi et al. 23 studied the behavior of a single micrometer-sized oil droplet that forms at the end of a capillary tube and is immersed in an aqueous surfactant solution. The drop appears to vibrate, with oscillation frequencies and amplitudes that depend on the droplet size and the surfactant concentration. Also, Magome and Yoshikawa 24 studied the self-movement in an oil/water system generated by chemically driven Marangoni instability. This effect was also studied theoretically by Tsemakh et al. 25 In these cases, though, (1) Vladimirova, N.; Malagoli, A.; Mauri, R. Phys. ReV.E 1999, 60, 2037- 2044. (2) Chou, Y. C.; Goldburg, W. I. Phys. ReV.A 1979, 20, 2105-2113 and references therein. (3) Wong, N. C.; Knobler, C. Phys. ReV.A 1981, 24, 3205-3211 and references therein. (4) Guenoun, P.; Gastaud, R.; Perrot, F.; Beysens, D. Phys. ReV.A 1987, 36, 4876-4890. (5) Tanaka, H.; Araki, T. Phys. ReV. Lett. 1998, 81, 389-392. (6) Tanaka, H.; Phys. ReV.E 1995, 51, 1313-1328. (7) Poesio, P.; Cominardi, G.; Lezzi, M.; Mauri, R.; Beretta, G. P. Phys. ReV. E 2006, 74, 011507/1-011507/13 and references therein. (8) Gupta, R.; Mauri, R.; Shinnar, R. Ind. Eng. Chem. Res. 1999, 38, 2418- 2424. (9) Ullmann, A.; Ludmer, Z.; Shinnar, R. AIChE J. 1995, 41, 488-500. (10) Gupta, R.; Mauri, R.; Shinnar, R. Ind. Eng. Chem. Res. 1996, 35, 2360- 2368. (11) Anderson, D. M.; McFadden, G. B.; Wheeler, A. A. Annu. ReV. Fluid Mech. 1998, 30, 139-165. (12) Lowengrub, J.; Truskinovsky, L. Proc. R. Soc. London, Ser. A 1998, 454, 2617-2654. (13) Vladimirova, N.; Malagoli, A.; Mauri, R. Phys. ReV.E 1999, 60, 6968- 6977. (14) Hohenberg, P. C.; Halperin, B. I. ReV. Mod. Phys. 1977, 49, 435-479. (15) Lamorgese, G.; Mauri, R. Phys. Fluids 2005, 17, 034107/1-034107/10. (16) ) Lamorgese, G.; Mauri, R. Phys. Fluids 2006, 18, 044107/1-044107/11. (17) Jasnow, D.; Vin ˜als, D. J. Phys. Fluids 1996, 8, 660-669. (18) Jacqmin, D. J. Fluid Mech. 2000, 402, 57-88. (19) Siggia, E. D. Phys. ReV.A 1979, 20, 595-605. (20) Karpov, V. G. Phys. ReV. Lett. 1995, 75, 2702-2705. (21) Karpov, V. G.; Oxtoby, D. W. Phys. ReV.E 1997, 55, 7253-7259. (22) Santonicola, G.; Mauri, R.; Shinnar, R. Ind. Eng. Chem. Res. 2001, 40, 2004-2010. (23) Kogi, O.; Yuya, K.; Kim, H.; Kitamura, N. Langmuir 2001, 17, 7456- 7458. (24) Magome, N.; Yoshikawa, K. J. Phys. Chem. 1996, 100, 19102-19105. BATCH: la7b42 USER: mcp29 DIV: @xyv04/data1/CLS_pj/GRP_la/JOB_i15/DIV_la700826z DATE: June 1, 2007 10.1021/la700826z CCC: $37.00 © xxxx American Chemical Society PAGE EST: 2.6 Published on Web 00/00/0000 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72