Scattering on triply periodic minimal surfaces—the effect of the topology, Debye–Waller, and molecular form factors

P. Garstecki and R.Hołyst

J. Chem. Phys. 2000, 113, 3772

We compute scattering patterns for four triply periodic surfaces (TPS). Three minimal—Schwarz P (Im3̄m), Schwarz D—diamond (Pn3̄m), Schoen G—gyroid (Ia3̄d), and one nodal S1 (Ia3̄d). Simple approximations are adopted to examine the influence of the molecular form factor, and the Debye–Waller factor on the scattering pattern. We find that the Debye–Waller factor has a much smaller influence on the scattering intensities of TPS than on the intensities of the lamellar structure consisting of parallel surfaces. This is caused by an almost spherelike distribution of normal vectors for TPS. We give a simple formula that allows a comparison of the experimental scattering data with the data for the P, D, G mathematical surfaces. Finally, the spectra of the two surfaces G and S1 of the same space group symmetry and different topologies are compared. It is found that in the case of the more complex S1 structure the intensities of the first two peaks are very small.

Influence of the free-energy functional form on simulated morphology of spinodally decomposing blends

A. Aksimentiev and R. Hołyst

Phys. Rev. E 2000, 62, 6821

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.

A Fleming–Viot Particle Representation of the Dirichlet Laplacian

K. Burdzy, R. Hołyst and P. March

Communications in Mathematical Physics 2000, 214, 679–703

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.

Kinetics of the droplet formation at the early and intermediate stages of the spinodal decomposition in homopolymer blends

A. Aksimentiev, R. Hołyst and K. Moorthi

Macromolecular Theory and Simulations 2000, 9, 8, 661-674

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.

Reorientational angle distribution and diffusion coefficient for nodal and cylindrical surfaces

D. Plewczyński and R. Hołyst

J. Chem. Phys. 2000, 113, 9920

We present a catalogue of diffusion coefficients and reorientational angle distribution (RAD) for various periodic surfaces, such as I-WP, F-RD, S, and S1 nodal surfaces; cylindrical structures like simple, undulated, and spiral cylinders, and a three-dimensional interconnected-rod structures. The results are obtained on the basis of a simulation algorithm for a diffusion on a surface given by the general equation φ(r)=0φ(r)=0 [Hołyst et al., Phys Rev. E 60, 302 (1999)]. I-WP, S, and S1 surfaces have a spherelike RAD, while F-RD has a cubelike RAD. The average of the second Legendre polynomial with RAD function for all nodal surfaces, except the F-RD nodal surface, decays exponentially with time for short times. The decay time is related to the Euler characteristic and the area per unit cell of a surface. This analytical formula, first proposed by B. Halle, S. Ljunggren, and S. Lidin in J. Chem. Phys. 97, 1401 (1992), is checked here on nodal surfaces, and its range of validity is determined. RAD function approaches its stationary limit exponentially with time. We determine the time to reach stationary state for all surfaces. In the case of the value of the effective diffusion coefficient the mean curvature and a connectivity between parts of surfaces have the main influence on it. The surfaces with low mean curvature at every point of the surface are characterized by high-diffusion coefficient. However if a surface has globally low mean curvature with large regions of nonzero mean curvature (negative and positive) the effective diffusion coefficient is low, as for example, in the case of undulated cylinders. Increasing the connectivity, at fixed curvatures, increases the diffusion coefficient.

Coherent Soft-X-Ray Dynamic Light Scattering from Smectic- A Films

A. C. Price, L. B. Sorensen, S. D. Kevan, J. Toner, A. Poniewierski and R. Hołyst

Phys. Rev. Lett. 1999, 82, 755

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 ( 2πq130) 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 ( 2πq130) 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.

Periodic surfaces and cubic phases in mixtures of oil, water, and surfactant

A. Ciach and R. Hołyst

J. Chem. Phys. 1999, 110, 3207

We study a ternary mixture of oil, water, and surfactant in the case of equal volume fractions of oil and water using the Landau–Ginzburg model derived from a lattice model of this ternary mixture. We concentrate on a phase region close to a coexistence line between microemulsion and cubic phases. In our model the bicontinuous cubic phases exist in a narrow window of the volume fraction of surfactant ρs≈0.6.ρs≈0.6. The sequence of phase transitions is L→G→D→P→CL→G→D→P→C as we increase ρsρs along the cubic-microemulsion bifurcation line. Here L stands for the lamellar phase and C for the cubic micellar phase. The gyroid G, primitive P, and diamond D phases are all bicontinuous. The transitions weakly depend on the temperature. The increase of ρsρs is accompanied by the increase of the surface area per unit volume. In the case of fluctuating monolayers the interface is diffused and the average area of the monolayer per unit volume is larger than the “projected area,” i.e., the area of the surface describing the average position of the monolayer, per unit volume. The effect is the strongest in the L and the weakest in the C structure.

Dynamic critical behavior of the Landau-Peierls fluctuations: Scaling form of the dynamic density autocorrelation function for smectic- A films

A. Poniewierski, R. Hołyst, A. C. Price, and L. B. Sorensen

Phys. Rev. E 1999, 59, 3048

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.

Diffusion on curved, periodic surfaces

R. Hołyst, D. Plewczyński, A. Aksimentiev, and K. Burdzy

Phys. Rev. E 1999, 60, 302

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.

Single chain conformations in the system of symmetric and asymmetric diblock copolymers

A. Aksimentiev and R. Hołyst

Macromolecular Theory and Simulations, 1999, 8, 4, 328-342

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.

Single-chain statistics in polymer systems

A. Aksimentiev and R. Hołyst

Progress in Polymer Science 1999, 24, 7, 1045-1093

In this review we study the behavior of a single labelled polymer chain in various polymer systems: polymer blends, diblock copolymers, gradient copolymers, ring copolymers, polyelectrolytes, grafted homopolymers, rigid nematogenic polymers, polymers in bad and good solvents, fractal polymers and polymers in fractal environments. We discuss many phenomena related to the single chain behavior, such as: collapse of polymers in bad solvents, protein folding, stretching of polymer brushes, coil–rod transition in nematogenic main-chain polymers, knot formation in homopolymer melts, and shrinking and swelling of polymers at temperatures close to the bulk transition temperatures. Our description is mesoscopic, based on two models of polymer systems: the Edwards model with Fixman delta interactions, and the Landau–Ginzburg model of phase transitions applied to polymers. In particular, we show the derivation of the Landau–Ginzburg model from the Edwards model in the case of homopolymer blends and diblock copolymer melts. In both models, we calculate the radius of gyration and relate them to the correlation function for a single polymer chain. We discuss theoretical results as well as computer simulations and experiments.

Thinning transitions in free-standing liquid-crystal films as the successive formation of dislocation loops

S. Pankratz, P. M. Johnson, R. Hołyst, and C. C. Huang

Phys. Rev. E 1999, 60, R2456(R)

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.

Phase behavior of gradient copolymers

A. Aksimentiev and R. Hołyst

J. Chem. Phys. 1999 111, 2329

Phase properties of gradient AB copolymer melts which consist of chains with the specified chemical distribution of A and B monomers have been studied within the Landau–Ginzburg model. All the melts with the linear distribution of the monomers exhibit only a direct continuous phase transition from disordered to the lamellar phase. The hexagonal, body-centered-cubic, double-gyroid (G), and lamellar-ordered structures have been found in the melts with the monotonic but nonlinear distribution of the monomers. The G structure has been also found in the gradient copolymer melts with the distribution function of monomers similar to the A–B–AA–B–A triblock copolymers.

Mesophases in polymer system: structure and phase transitions

R. Hołyst

Polimery 1998, 43, 9, 523-530

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.

SMECTIC MENISCUS AND DISLOCATIONS

Jean-Christophe Géminard, R. Hołyst and P. Oswald

Acta Physica Polonica B 1998, 28, 1737-1747

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).

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