PHYSICAL REVIEW E VOLUME 47, NUMBER 4 APRIL 1993 Power-law scattering in Auids with a nonscalar order parameter Apollo P. Y. Wong, Pierre Wiltzius, Ronald G. Larson, and Bernard Yurke A Tdk T Bell Laboratories, Murray Hill, Xew Jersey 07974 (Received 30 November 1992) We studied the coarsening behavior of two lyotropic liquid-crystal systems by static light scattering. The samples were quenched from the isotropic phase into either the nematic phase or a region of coex- istence between nematic and isotropic phases. In the coexistence region, we observed, in both two and three dimensions, Porod power-law tails of the scattering intensity. Such a behavior is described by S(q) -q '"+" in the limit of large wave vectors q, where S is the scattering intensity, q is the wave vec- tor, and d is the dimension of the system. In addition, the nernatic phases displayed novel power-law scaling behavior at large q, namely, S(q)-q ", where u=4 in two dimensions and u=6 in three dimen- sions. These results will be compared to recent theoretical predictions. PACS number(s): 05.70.Fh, 64.60. — i, 61.30. — v In x-ray scattering from porous materials with sharp interfaces, Debye, Anderson, and Brumberger concluded as long ago as 1957 that at large wave vector [q =(4~/A, )sin(9/2)], the structure factor should fall off as q [I]. This result was later generalized to include any binary system with sharp interfaces. In this case, the structure factor was expected to obey S(q)-q ' +", d being the dimension of the system. This is usually called Porod's law [2]; there are ample experimental evidences confirming this behavior for various fluid and magnet sys- tems [3,4]. Very recently, Porod's law was generalized still further to include systems with complicated order parameters [5,6]. It was predicted that, for a system with an n- component vector order parameter, the structure factor should obey S(q, t)-L(t) f(qL(t)) where L(t) is a time-dependent characteristic length and f is a scaling function that asymptotically approaches f (x)-x at large x. Therefore, for a system quenched into its or- dered phase, the large-q behavior of the structure factor should be S(q, t)-L (t)"(qL(t)) ' +"' and the usual Po- rod behavior for a phase-separated binary system with a scalar order-parameter is recovered as the special case at n =1. In the case of binary systems, the sharp interfaces be- tween the two components are the dominant scatterers and the characteristic length of the system can then natu- rally be taken as the characteristic domain size. Howev- er, for systems with complicated order parameters, the concepts of domains and domain walls are no longer applicable. The physical meaning of the characteristic length scale is not obvious, although one length scale in the system is related to the defect density. Nevertheless, the results below agree well with the generalized Porod form of scaling which indicates that there is such a characteristic length scale in the nematic regime. More- over, the sample thickness d* at which the cross over from two- to three-dimensional behavior occurs is surprisingly large (on the order of tens or hundreds of microns) compared to the "molecular" length scales, which are only on the order of 100 to 2000 A. Presum- ably d' is determined by the characteristic length L (t). We will argue that in the case of a nematic liquid crystal, the dominant source of scattering comes from the dis- clinations, and that the length scales L (t) and d* are indeed related to the distance between them. We report in this article static light-scattering results on two lyotropic liquid-crystalline systems, namely, poly y-benzyl-glutamate (PBG) in meta-cresol and cesium perAuoro-octanoate (CsPFO) in heavy water. These two lyotropes were chosen because of their slow ordering time scales and their small birefringence in the nernatic phase. The birefringence of a typical therrnotropic liquid crystal is one to three orders of magnitude higher than that of the above two systems, which makes it difticult to avoid multiple scattering. Moreover, both the PBG and CsPFO systems have been well studied, and their phase diagrams are readily available [7,8]. For the concentra- tions considered here, both systems have an isotropic phase at high temperatures, a nematic-isotropic coex- istence phase in an intermediate temperature range, and they are pure nematic liquid crystals at yet lower temper- atures. One difference between the two systems, howev- er, is that the coexistence region of the PBG solution spans approximately 40 C while that of the CsPFO sys- tem is only 0.5 C wide. The polymer chains of PBG form rods with lengths of approximately 2000 A and diameters of approximately 15 A [8]. On the other hand, the CsPFO molecules form disklike micelles that are about 40 A thick and 150 A in diameter [9]. Despite their differences, we show below that both systems display similar generalized Porod scal- ing behavior in the nernatic phase. This provides strong evidence that such a scaling behavior is independent of the molecular details and is a generic property of systems with complicated order parameters. We employ a new technique for gathering the static light-scattering data in which carefully positioned geometric optical elements project the light scattered be- tween 0 and 40' onto a charge-coupled-device (CCD) 47 2683