DYNAMIC CORRELATION-FUNCTIONS FOR FINITE AND INFINITE SMECTIC-A SYSTEMS - THEORY AND EXPERIMENT

In this paper, we present the dynamic layer displacement-layer displac ement and the dynamic density-density correlation functions-both for s mectic-A systems in the thermodynamic limit, and for real smectic-A fi lms that have finite size, nonzero surface tension acting al the two f ree surfaces, and nonzero layer sliding viscosity. We also present the results of our soft-x-ray photon correlation spectroscopy experiment, which we have used to directly measure the dynamic density-density co rrelation function for two different liquid crystals (40.8 and 70.7) i n the overdamped surface tension restoring force limit of our theory. We used linearized hydrodynamics to first calculate the behavior of sm ectic-A systems in the thermodynamic limit, and then to calculate the behavior for real, finite size, nonzero surface tension freely suspend ed liquid crystal films. For the real films, we used the linearized sm ectic-A hydrodynamic equations and the Gaussian model for the layer fl uctuations to compute the set of relaxation times for the displacement field in a finite smectic-il film bounded by two free surfaces. We fi nd that all of the relaxation times have maxima at nonzero values of t he transverse wave vector q(perpendicular to). For thicker films; the maxima shift towards q(perpendicular to)=0 and grow linearly with the number of smectic layers N+1. For finite N all of the relaxation times tend to zero as q(perpendicular to)-->0 except one that attains the f inite value tau((0))(0)=(N+1)eta(3)d/2 gamma, where eta(3) is the laye r sliding viscosity, d is the smectic period, and gamma is the surface tension. We find that the time-dependent scattering intensity integra ted over q(perpendicular to) has the simple scaling form S(q(z),t)simi lar to(a(0)/Lambda)(gamma(t)), where a(0) and Lambda are the molecular size cutoff and the instrument resolution cutoff, respectively, and t he time-dependent exponent y(t) = (k(B)Tq(z)(2)/4 pi gamma)[1 - exp(-t /tau((0))(0))]. Our results clearly show that the boundary conditions strongly affect the hydrodynamics Of real smectics.