VOLUME 60, NUMBER 4 PHYSICAL REVIEW LETTERS 25 JANUARY 1988 Nonequilibrium Structure during Phase Separation Jess P. Wilcoxon and Dale W. Schaefer Sandia National Laboratories, Albuquerque, New Mexico 87185 and Eric W. Kaler Department of Chemical Engineering, University of Washington, Seattle, Washington 98l85 (Received 2 November 1987) To establish the nature of the early-stage structures that occur during phase separation, we studied solutions of a nonionic surfactant micellar solution (C~2E6 in D20). We report the existence of long- lived nonequilibrium structures during shallow quenches along the critical isochore. The power-law scattering profile of these structures is consistent with ramified clusters with a fractal dimension D - l.6+0.l. PACS numbers: 64.60.Ht, 61.12.Ex, 64.80.6d Recent simulations of cluster formation by both nu- cleation and growth and spinodal decotnposition' sug- gest that ramified fractal structures may exist in the ear- liest stages of phase separation. Heermann and Klein, for example, found that droplets for deep quenches in the nucleation and growth regime are not, in general, initial- ly compact (i.e. , they do not grow by uniform addition of "molecules" at the cluster surface). Desai and Denton, ' on the other hand, studied the spinodal decomposition characterized by critical quenches and also found that individual clusters were fractals in the early stages. Some hints of ramified growth can be found in existing scattering curves, but to our knowledge the existence of ramified clusters during phase separation has never been demonstrated experimentally. Although we did find nonequilibrium fractal structures on the critical isochore, we believe that these structures are remnants of critical fluctuations, not the result of phase separation itself. In the nucleation regime, on the other hand, we observe only normal compact droplets. Experimental observation of nonequilibrium structures requires a sufficiently slow approach to thermodynamic equilibrium to allow time for measurement of the angu- lar distribution of scattered radiation (light and/or neu- trons). For simple fluids, surface tension drives nonclas- sical structures very rapidly to smooth, compact droplets, precluding the observation of the early-stage structure by most techniques. To obviate this problem we choose a system which has both low surface tension and long time scales associated with structural rearrangement by dif- fusion. We choose a solution of a nonionic surfactant, n- dodecylhexaoxyethylene glycol monether (C&2E6), in D20. Because C~2E6 is a surfactant, the surface tension between the phases is low. Also, in a micellar system the elementary units are aggregate of ca. 100 molecules and so the rate of decay of the critical concentration fluctua- tions is typically 10-100 times slower than in simple fluids. Thus the ability to observe nonequilibrium struc- tures in such a system is substantially enhanced. In the C~2E6 micellar system, Wilcoxon and Kaler previously demonstrated that mean micelle size and po- lydispersity are invariant when one approaches the criti- cal point along the critical isochore. This observation in- sures that the amphiphile concentration is proportional to the number density of micelle aggregates and both are conserved order parameters along this path. Combined with the observations of universal critical exponents which fall into the same universality class as binary fluids, this makes this micellar system indistinguish- able from a binary mixture of small molecules from the standpoint of critical behavior. In contrast to binary fluids, tnicelle systems possess asymmetric coexistence curves due to the large size difference between aggregate (micelle) and solvent (D20 or H20). In the present system the critical point is a lower critical consolute point (i. e. , as the temperature is raised the homogeneous single-phase system separates into two homogeneous phases of different densities). Al- though the system shows substantial diff'erences in the critical temperature T, when D20 is substituted for H20 (see inset in Fig. 1), no difference in the static or dynam- ic scaling behavior is found when quantities are ex- pressed in terms of the reduced temperature e = (T, — T)/T, . Thus the features reported are likely to be universal (although not necessarily easily observable) as- pects of nonequilibrium structures characteristic of criti- cal quenches. The experiment consists of measurements of the static and dynamic structure factor on two isochores: the criti- cal one, where phase separation proceeds by spinodal decomposition, and an off-critical isochore, where, for a shallow quench, a nucleation and growth mechanism is expected. We demonstrate the existence of ramified 1988 The American Physical Society 333