We have used coherent soft-x-ray dynamic light scattering at the Bragg peak to measure the thermally driven layer fluctuations in freely suspended films of five different smectic- $A$ liquid crystals: 4O.8, 7O.7, 8CB, 8OCB, and 10OCB. The measured relaxation times at a scattering wave vector corresponding to the interlayer spacing ( $2\pi q^{-1}\approx 30Å$) ranged from 2 to $60\mu s$ for films from 4 to $50\mu m$ thick. Thus, we have achieved the same time resolution as conventional laser dynamic light scattering, but with 100 times higher spatial resolution. The measured relaxation times at a scattering wave vector corresponding to the interlayer spacing ( $2\pi q^{-1}\approx 30Å$) ranged from 2 to $60\mu s$ for films from 4 to $50\mu m$ thick. Thus, we have achieved the same time resolution as conventional laser dynamic light scattering, but with 100 times higher spatial resolution.

In this paper, we study the dynamic density autocorrelation function $G(r,t)$ for smectic-$A$ films in the layer sliding geometry. We first postulate a scaling form for G, and then we show that our postulated scaling form holds by comparing the scaling predictions with detailed numerical calculations. We find some deviations from the scaling form only for very thin films. For thick films, we find a region of a bulklike behavior, where the dynamics is characterized by the same static critical exponent $\eta ,$ which was originally introduced by Caillé [C. R. Acad. Sci. Ser. B 274, 891 (1972)]. In the limit of very large distance perpendicular to the layer normal, or in the limit of very long time, we find that the decay of G is governed by the surface exponent $\chi {=k}_B{\mathrm{Tq}}_z^2/\left(4\pi \gamma \right),$ where $\gamma$ is the surface tension and the wave-vector component $q_z$ satisfies the Bragg condition. We also find an intermediate perpendicular distance regime in which the decay of G is governed by the time-dependent exponent $\chi \exp (-t/{\tau }_0),$ where the relaxation time is given by ${\tau }_0={\eta }_3\left(\mathrm{Ld}\right)/\left(2\gamma \right),$ where ${\eta }_3$ is the layer sliding viscosity, and $\mathrm{Ld}$ is the film thickness.

We present a simulation algorithm for a diffusion on a curved surface given by the equation $\phi \left(r\right)=0.$ The algorithm is tested against analytical results known for diffusion on a cylinder and a sphere, and applied to the diffusion on the P, D, and G periodic nodal surfaces. It should find application in an interpretation of two-dimensional exchange NMR spectroscopy data of diffusion on biological membranes.

We describe a model for the layer-thinning transition in free-standing liquid-crystal films based on the successive, spontaneous formation of dislocation loops. As the film temperature increases and the smectic order and layer compressional modulus decrease, the condition for creating a dislocation loop of critical radius is met and a thinning is nucleated. The resulting equation for N, the number of smectic layers, as a function of temperature yields good fitting results to the thinning transitions obtained from several fluorinated compounds.

The system of AB diblock copolymers forms many ordered structures, e.g., hexagonal, lamellar, cubic, and gyroid. These structures are formed owing to the incompatibility of the homopolymers A and B that constitute the copolymer. The A-B covalent bonds can prevent the incompatibility that results in macrophase separation. The system can separate locally (microphase separation) into ordered A-rich and B-rich spherical (cubic phase), layered (lamellar phase) or cylindrical (hexagonal phase) domains. The gyroid structure is special: it is bicontinuous and the A-rich and B-rich domains form channels of the Ia3d symmetry. Three methods used to study the systems are the mean-field model, self-consistent one-loop approximation, and the self-consistent field theory; each can be developed from the Edwards Hamiltonian. The single chain statistics in the disordered phase of a diblock copolymer is shown to deviate from the Gaussian statistics on account of fluctuations. In the one-loop approximations, the diblock copolymer chain is shown to stretch at the point where two incompatible blocks meet; each block shrinks close to the microphase separation transition. Stretching outweighs shrinking and the net result is the increase in the radius of gyration about the Gaussian value. Another example of the ordered structure is provided by liquid crystalline (LC) polymers. The LC polymer main chains (usually very stiff) form primarily nematic phases which are characterized by the orientational order. The long chains are ordered in one direction, breaking the rotational symmetry. Finally we show the general features of the Landau-Ginzburg model, which is the simplest model for the study of ordered polymer systems.

In ordinary liquids the size of a meniscus and its shape is set by a competition between surface tension and gravity. The thermodynamical process of its creation can be reversible. On the contrary, in smectic liquid crystals the formation of the meniscus is always an irreversible thermo-dynamic process since it involves the creation of dislocations (therefore it involves friction). Also the meniscus is usually small in experiments with smectics in comparison to the capillary length and, therefore, the gravity does not play any role in determining the meniscus shape. Here we discuss the relation between dislocations and meniscus in smectics. The theoretical predictions are supported by a recent experiment performed on freely suspended films of smectic liquid crystals. In this experiment the measurement of the meniscus radius of curvature gives the pressure difference , ∆p, according to the Laplace law. From the measurements of the growth dynamics of a dislocation loop (governed by ∆p) we find the line tension (∼ 8×10 −8 dyn) and the mobility of an elementary edge dislocation (∼ 4 × 10 −7 cm 2 s/g).

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)=(k_BTq²_z/4πγ)[1−exp(−t/τ⁽⁰⁾(0))]. Our results clearly show that the boundary conditions strongly affect the hydrodynamics of real smectics.

The radius of gyration, R, of polymer chains in homopolymer blends is studied in the framework of a self‐consistent one‐loop approximation. We show that the polymer chains can shrink or swell in comparison to the Gaussian chain. Swelling of the polymer chains in a region far away from the critical point is caused by the steric repulsive forces that were included in the model as the constraint of incompressibility. The chains shrink progressively, as we approach the critical region passing through the Gaussian limit, $ R_0 = \sqrt{\frac{N}{6}}l $, far away from the critical point (N − degree of polymerization, l − length of monomer). The correction responsible for the swelling and the shrinking is small when the concentrations of components ϕ are comparable ($ N = 1000,\bar\phi = 0.5,\frac{{R_0^2 – R^2}}{{R_0^2}} = \pm 0.02\%$). This effect, although small, leads to a local demixing, a sharp shrinking of chains in both components accompanied by a strong change of the global inter‐monomer distance, which should be observable experimentally. When the local demixing occurs there is a sudden increase in the scattering intensity (of the order of 30% for N = 1000, and ϕ_A = 0.2). The increase of the degree of polymerization of the same type of chains leads to an increase of the swelling‐shrinking effects. In addition, the critical concentration of the shorter chains component is smaller in comparison to the value obtained in the Flory‐Huggins theory. The self‐consistent determination of the radius of gyration and the upper wave‐vector cutoff make our model free from any divergences. In the limit of N → ∞ this theory reduces to the random phase approximation (RPA) of de Gennes.

The polarization vector in ferroelectric smectic- $C^{\ast }$ films preferably aligns along a dislocation line due to the Coulomb interaction between polarization charges. The electric field locally perpendicular to the dislocation line distorts it, and in the case of a dislocation loop an $n$-finger structure is formed, with $n$ depending on the applied voltage. This phenomenon has been observed in an experiment in which the screening effect of ion impurities has been partially lifted in a low-frequency (3 Hz) electric field. A characteristic length scale related to this phenomenon is of the order of ${10}^4Å$.

The polarization vector in ferroelectric smectic- $C^{\ast }$ films preferably aligns along a dislocation line due to the Coulomb interaction between polarization charges. The electric field locally perpendicular to the dislocation line distorts it, and in the case of a dislocation loop an $n$-finger structure is formed, with $n$ depending on the applied voltage. This phenomenon has been observed in an experiment in which the screening effect of ion impurities has been partially lifted in a low-frequency (3 Hz) electric field. A characteristic length scale related to this phenomenon is of the order of ${10}^4Å$.