The spinodal decomposition of a binary mixture has been studied within several mesoscopic models. It has been found that the form of the equilibrium free energy has a crucial effect on the morphological development in asymmetric blends. We have shown that the principal quantity that determines the topology of the interface (and type of morphology) is the equilibrium minority phase volume fraction, while the transition from bicontinuous to droplet morphology can be treated as a percolation. The concentration dependence of the square gradient coefficient attributed for the Flory-Huggins–de Gennes free energy has no significant influences on the average domain growth, but can be distinguished experimentally from its constant-coefficient alternative by measuring the maximum wave vector of the scattering intensity as a function of the minority phase volume fraction for spinodally decomposing asymmetric blends. The concentration dependence of the Onsager coefficient has the weak, systematic effect of slowing down the morphological development. The local shape of the interface is not affected considerably by the concentration dependence of the square gradient and Onsager coefficient.
We consider a model with a large number N of particles which move according to independent Brownian motions. A particle which leaves a domain D is killed; at the same time, a different particle splits into two particles. For large N, the particle distribution density converges to the normalized heat equation solution in D with Dirichlet boundary conditions. The stationary distributions converge as N→∞ to the first eigenfunction of the Laplacian in D with the same boundary conditions.
The kinetics of the droplet formation during the spinodal decomposition (SD) of the homopolymer blends has been studied by numerical integration of the Cahn‐Hilliard‐Cook equation. We have found that the droplet formation and growth occurs when the minority phase volume fraction, fm , approaches the percolation threshold value, fthr = 0.3 ± 0.01. The time for the formation of the disperse droplet morphology (coarsening time) depends only on the equilibrium minority phase volume fraction, fm . fm approaches its equilibrium value logarithmically at the late SD stages, and, therefore, the coarsening time decreases exponentially as the average volume fraction or the quench depth decrease. Since the temporal evolution of the total interfacial area does not depend on the quench conditions and blend morphology, the average droplet size and the droplet number density is determined by the coarsening time. Within the time scale studied, the droplet number density decreases with time as t –0.63±0.03; the average mean curvature decreases as t –0.35±0.05; the average Gaussian curvature decreases as t –0.42±0.03, and the average droplet compactness ˜V/S3/2 where S is the surface area and V is the volume) approaches a spherical limit logarithmically with time. The droplets with larger area have lower compactness and in the low compactness limit their area is a parabolic function of compactness. The size and shape distribution functions have been also investigated.
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 ( 2πq−1≈30Å) ranged from 2 to 60μs for films from 4 to 50μm 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 ( 2πq−1≈30Å) ranged from 2 to 60μs for films from 4 to 50μm 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 η, 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 χ=kBTq2z/(4πγ), where γ is the surface tension and the wave-vector component qz 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 χexp(−t/τ0), where the relaxation time is given by τ0=η3(Ld)/(2γ), where η3 is the layer sliding viscosity, and Ld is the film thickness.
We present a simulation algorithm for a diffusion on a curved surface given by the equation φ(r)=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.
A theory which describes a local structure and global properties of a diblock copolymer melt has been developed in the framework of the one‐loop self‐consistent approximation. We have derived expressions for the sizes of a single diblock macromolecule and its parts. Two different behaviors of single macromolecule conformations in the disordered melt have been obtained depending on the asymmetry of chains and morphologies occurring in ordered states after the order‐disorder transition (ODT). In the nearly symmetric melt, 0.35 < f ⪇ 0.5 (f is a composition), the blocks of both types shrink a little initially as the temperature decreases and then, at some temperature, they begin to swell. In strongly asymmetric melts, f < 0.35, the block of a macromolecule which consists of the monomers of minority type shrinks monotonically, while the other block monotonically swells. We have found nearly Gaussian behavior of the individual blocks and stretching near the chemical bond joining the blocks. Near the ODT the chains are stretched with a magnitude which is of the order of a few percent of their Gaussian sizes. We have calculated the peak position in the scattering curve as a function of the Flory‐Huggins interaction parameter, composition and degree of polymerization. Less then 5% change in the size of copolymer molecules lead to a 25% shift of the scattering peak in comparison to the Gaussian limit. We have found a good quantitative agreement of our theoretical results with the experimental neutron scattering data.
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).