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

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

Phys. Rev. E 1998, 58, 2027

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)=(kBTq2z/4πγ)[1−exp(−t/τ(0)(0))]. Our results clearly show that the boundary conditions strongly affect the hydrodynamics of real smectics.

Fluctuating Euler characteristics in lamellar and microemulsion phases

R. Hołyst

Current Opinion in Colloid & Interface Science, 1998, 3, 4, 422-427

Swelling and shrinking of polymer chains in homopolymer blends

A. Aksimentiev and R. Holyst

Macromolecular Theory and Simulations, 1998, 7, 5, 447-456

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 ϕA = 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.

Confined complex liquids: Passages, droplets, permanent deformations, and order–disorder transitions

R. Hołyst, A. Poniewierski, P. Fortmeier and H. Stegemeyer

Phys. Rev. Lett. 1998, 81, 5848

The polarization vector in ferroelectric smectic- C 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 104.

Coupling of Polarization and Dislocation in Ferroelectric Smectic Liquid-Crystal Films

R. Hołyst, A. Poniewierski, P. Fortmeier, and H. Stegemeyer

Phys. Rev. Lett. 1998, 81, 5848

The polarization vector in ferroelectric smectic- C 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 104.

Meniscus and Dislocations in Free-Standing Films of Smectic-A Liquid Crystals

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

Phys. Rev. Lett. 1997, 78, 1924

A flat, freely suspended film of smectic-A liquid crystal supports a pressure difference, Δp, across its two free surfaces. The size of its meniscus is about 10 μm, 2 orders of magnitude smaller than the capillary length, and its profile is predicted to be circular, in accordance with our measurement. The measurement of its radius of curvature gives Δp. We nucleate ex nihilo an elementary edge dislocation loop, and from its critical radius and growth dynamics (governed by Δp), we find the line tension (∼8×10−7dyn) and the mobility of an elementary edge dislocation (∼4×10−7cm2s/g).

Fluctuating Euler characteristics, topological disorder line, and passages in the lamellar phase

R. Hołyst and W. T. Góźdź

J. Chem. Phys. 1997, 106, 4773

We introduce a concept of topological disorder line for systems with ordered internal surfaces. At one side of the line the ordered structure exhibits strong topological fluctuations, accompanied by changes in the Euler characteristics. At the other side topological fluctuations are rare. In a system of oil-water-surfactant, in the lamellar phase, the crossover between two regimes is marked by the appearance of thin wormhole passages and their further proliferation. Close to the lamellar-microemulsion phase boundary thin wormhole passages merge leading to the formation of large channels between lamellas pierced with holes. The lamellar phase with many large “torus-like” passages strongly resembles the microemulsion phase. In order to illustrate these concepts we perform Monte Carlo simulations of the one scalar order parameter Landau–Ginzburg model of microemulsions. We show how the Euler characteristics can be effectively used in such simulations to identify different ordered phases and count the number of wormhole passages.

Distribution functions for H-2 nuclear magnetic resonance band shapes for polymerized surfactant molecules forming triply periodic surfaces

W. T. Góźdź and R. Hołyst

J. Chem. Phys. 1997, 106, 9305

We present theoretical predictions of the distribution functions for 2H NMR bandshape for polymerized surfactant monolayers in triply periodic surfaces formed in ternary mixtures. We have calculated the distribution function for many triply-periodic structures of different topology, geometry, and symmetry. We have investigated applicability and usefulness of this new experimental technique to study the microstructures formed by surfactant molecules. The results presented in this paper can help experimentalists in better interpretation and analysis of nuclear magnetic resonance (NMR) bandshape experiments.

Confinement Induced Topological Fluctuations in a System with Internal Surfaces

R. Hołyst and P. Oswald

Phys. Rev. Lett. 1997, 79, 1499

The lamellar phase between two parallel walls, in the water, oil, surfactant system, exhibits strong topological fluctuations. As we change the distance between the walls we observe the formation of two layers, then the microemulsion between two layers, and finally four layers. The transition is marked by the peaks in the average Euler characteristics and in its variance. The topological fluctuations may be responsible for attractive background force found in force apparatus measurements of the system.

Polydispersity and Ordered Phases in Solutions of Rodlike Macromolecules

A. M. Bohle, R. Hołyst, and T. Vilgis

Phys. Rev. Lett. 1996, 76, 1396

We apply density functional theory to study the influence of polydispersity on the stability of columnar, smectic, and solid ordering in the solutions of rodlike macromolecules. For sufficiently large length polydispersity (standard deviation σ>0.25) a direct first-order nematic-columnar transition is found, while for smaller σ there is a continuous nematic-smectic and first-order smectic-columnar transition. The length distribution of macromolecules changes neither at the nematic-smectic nor at the nematic-columnar transition. In the binary mixtures the nematic-smectic transition is also continuous. Demixing in the smectic phase is preempted by transitions to solid or columnar phases.

From the plateau problem to periodic minimal surfaces in lipids, surfactants and diblock copolymers

W. Gòz̀dz̀ and R. Holyst

Macromolecular Theory and Simulations 1996, 5, 2, 321-332

A novel method is presented for generating periodic surfaces. Such periodic surfaces appear in all systems which are characterized by internal interfaces and which additionally exhibit ordering. One example are systems of AB diblock copolymers, where the internal interfaces are formed by the chemical bonds between the A and B blocks. In these systems at least two bicontinuous phases are formed: the ordered bicontinuous double diamond phase and the gyroid phase. In these phases the ordered domains of A monomers and B monomers are separated by a periodic interface of the same symmetry as the phases themselves. Here we present a novel method for the generation of such periodic surfaces based on the simple Landau‐Ginzburg model of microemulsions. We test the method on four known minimal periodic surfaces, find two new surfaces of cubic symmetry, and show how to obtain periodic surfaces of high genus and n‐tuply continuous phases (n > 2). So far only bicontinuous (n = 2) phases have been known. We point out that the Landau model used here should be generic for all systems characterized by internal interfaces, including the diblock copolymer systems.

High Genus Periodic Gyroid Surfaces of Nonpositive Gaussian Curvature

W. Gòźdź and R. Hołyst

Phys. Rev. Lett. 1996, 76, 2726

In this paper we present a novel method for the generation of periodic embedded surfaces of nonpositive Gaussian curvature. The structures are related to the local minima of the scalar order parameter Landau-Ginzburg Hamiltonian for microemulsions. The method is used to generate six unknown surfaces of Ia¯3d symmetric (gyroid) of genus 21, 53, 69, 109, 141, and 157 per unit cell. All of them but that of genus 21 are most likely the minimal surfaces. The Schoen-Luzzati gyroid minimal surface of genus 5 (per unit cell) is also obtained.

Configurational transition in a Fleming – Viot-type model and probabilistic interpretation of Laplacian eigenfunctions

K. Burdzy, R. Holyst, D. Ingerman and P. March

Journal of Physics A: Mathematical and General, 1996, 29, 11, 2633-2642

We analyse and simulate a two-dimensional Brownian multi-type particle system with death and branching (birth) depending on the position of particles of different types. The system is confined in a two-dimensional box, whose boundaries act as the sink of Brownian particles. The branching rate matches the death rate so that the total number of particles is kept constant. In the case of m types of particle in a rectangular box of size  and elongated shape  we observe that the stationary distribution of particles corresponds to the mth Laplacian eigenfunction. For smaller elongations a > b we find a configurational transition to a new limiting distribution. The ratio a/b for which the transition occurs is related to the value of the mth eigenvalue of the Laplacian with rectangular boundaries.

The structure and phase transitions in polymer blends, diblock copolymers and liquid crystalline polymers: The Landau‐Ginzburg approach

R. Hołyst and T. A. Vilgis

Macromolecular Theory and Simulations 1996, 5, 4, 573-643

The polymer systems are discussed in the framework of the Landau‐Ginzburg model. The model is derived from the mesoscopic Edwards Hamiltonian via the conditional partition function. We discuss flexible, semiflexible and rigid polymers. The following systems are studied: polymer blends, flexible diblock and multi‐block copolymer melts, random copolymer melts, ring polymers, rigid‐flexible diblock copolymer melts, mixtures of copolymers and homopolymers and mixtures of liquid crystalline polymers. Three methods are used to study the systems: mean‐field model, self consistent one‐loop approximation and self consistent field theory. The following problems are studied and discussed: the phase diagrams, scattering intensities and correlation functions, single chain statistics and behavior of single chains close to critical points, fluctuations induced shift of phase boundaries. In particular we shall discuss shrinking of the polymer chains close to the critical point in polymer blends, size of the Ginzburg region in polymer blends and shift of the critical temperature. In the rigid‐flexible diblock copolymers we shall discuss the density nematic order parameter correlation function. The correlation functions in this system are found to oscillate with the characteristic period equal to the length of the rigid part of the diblock copolymer. The density and nematic order parameter measured along the given direction are anticorrelated. In the flexible diblock copolymer system we shall discuss various phases including the double diamond and gyroid structures. The single chain statistics in the disordered phase of a flexible diblock copolymer system is shown to deviate from the Gaussian statistics due to fluctuations. In the one loop approximation one shows that the diblock copolymer chain is stretched in the point where two incompatible blocks meet but also that each block shrinks close to the microphase separation transition. The stretching outweights shrinking and the net result is the increase of the radius of gyration above the Gaussian value. Certain properties of homopolymer/copolymer systems are discussed. Diblock copolymers solubilize two incompatible homopolymers by forming a monolayer interface between them. The interface has a positive saddle splay modulus which means that the interfaces in the disordered phase should be characterized by a negative Gaussian curvature. We also show that in such a mixture the Lifshitz tricritical point is encountered. The properties of this unusual point are presented. The Lifshitz, equimaxima and disorder lines are shown to provide a useful tool for studying local ordering in polymer mixtures. In the liquid crystalline mixtures the isotropic nematic phase transition is discussed. We concentrate on static, equilibrium properties of the polymer systems.

Triply periodic surfaces and multiply continuous structures from the Landau model of microemulsions

W. T. Góźdź and R. Hołyst

Phys. Rev. E 1996, 54, 5012

We present a method for the generation of periodic embedded surfaces of nonpositive Gaussian curvature and multiply continuous phases. The structures are related to the local minima of the scalar order parameter Landau-Ginzburg Hamiltonian for microemulsions. In the bicontinuous structure the single surface separates the volume into two disjoint subvolumes. In some of our phases (multiply continuous) there is more than one periodic surface that disconnects the volume into three or more disjoint subvolumes. We show that some of these surfaces are triply periodic minimal surfaces. We have generated known minimal surfaces (e.g., Schwarz primitive P, diamond D, and Schoen-Luzatti gyroid G and many surfaces of high genus. We speculate that the structure of microemulsion can be related to the high-genus gyroid structures, since the high-genus surfaces were most easily generated in the phase diagram close to the microemulsion stability region. We study in detail the geometrical characteristics of these phases, such as genus per unit cell, surface area per unit volume, and volume fraction occupied by oil or water in such a structure. Our discovery calls for new experimental techniques, which could be used to discern between bicontinuous and multiply continuous structures. We observe that multiply continuous structures are most easily generated close to the water-oil coexistence region.

(+48) 22 343-31-23

Kasprzaka 44/52, 01-224 Warsaw, Poland