Dynamic correlation functions for finite and infinite smectic-Asystems: Theory and experiment

In this paper, we present the dynamic layer displacement--layer displacement and the dynamic density-density correlation functions---both for smectic-A systems in the thermodynamic limit, and for real smectic-A films that have finite size, nonzero surface tension acting at the two free 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 correlation function for two different liquid crystals (4O.8 and 7O.7) in the overdamped surface tension restoring force limit of our theory. We used linearized hydrodynamics to first calculate the behavior of smectic-A systems in the thermodynamic limit, and then to calculate the behavior for real, finite size, nonzero surface tension freely suspended liquid crystal films. For the real films, we used the linearized smectic-A hydrodynamic equations and the Gaussian model for the layer fluctuations to compute the set of relaxation times for the displacement field in a finite smectic-A film bounded by two free surfaces. We find that all of the relaxation times have maxima at nonzero values of the transverse wave vector q_⊥. For thicker films the maxima shift towards q_⊥=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_⊥→0, except one that attains the finite value τ⁽⁰⁾(0)=(N+1)η₃d/2γ, where η₃ is the layer sliding viscosity, d is the smectic period, and {γ} is the surface tension. We find that the time-dependent scattering intensity integrated over q_⊥ has the simple scaling form S(qz,t)~(a₀/Λ)ʸ⁽ᵗ⁾, where a₀ and {Λ} are the molecular size cutoff and the instrument resolution cutoff, respectively, and the time-dependent exponent y(t)=(kBTqz²/4πγ)[1-exp(-t/τ⁽⁰⁾(0))]. Our results clearly show that the boundary conditions strongly affect the hydrodynamics of real smectics.