Demixing/mixing of polystyrene, with poly(methylphenylsiloxane) in a two-step cooling/heating process: Jump spinodal specification method

M. Graca, S. A. Wieczorek, M. Fiałkowski, and R. Hołyst

Macromolecules 2002, 35, 24, 9117–9129

We present experimental studies of the mixing process of a homopolymer blend of poly(methylphenylsiloxane) (PMPS) with polystyrene (PS). The system is first allowed to decompose spinodally at low temperature for few minutes and next is heated to a higher temperature to the one-phase or two-phase (metastable or unstable) region. In all cases the intensity drops initially after the temperature jump. In the one-phase region the intensity drops to the base scattering. In the metastable region it drops initially, and later on it starts to grow. In this region the peak position shifts strongly toward small wavevectors, and the intensity drops considerably. Finally, in the spinodal (unstable) region the peak position shifts toward smaller wavevectors, but the intensity of the peak hardly changes before increasing again. The decrease of the peak intensity is exponential with the characteristic decay time which approaches infinity when the temperature of the jump approaches that of the quench. The mixing process mainly involves the interdiffusion of polymers, without global movement of the interface. Small domains disappear faster than larger domains, and therefore the peak position (indicating an average size of the domains) shifts toward smaller wavevectors. In the metastable region the average wavevector as a function of time has a characteristic minimum, which shifts toward zero as we approach the spinodal. If the jump is made to the unstable region, the average wavevector monotonically decreases with time. The average wavevector, after temperature jump, allows to predict the location of the binodal and spinodal. We call this new method of spinodal location a jump spinodal specification method (JSS method). The generic features of this method have been confirmed in the computer simulations of the Flory−Huggins−de Gennes model with the Langevin dynamics for the mixture of polybutadiene and deuterated polybutadiene.

Photonic properties of multicontinuous cubic phases

V. Babin, P. Garstecki and R. Hołyst

Phys. Rev. B 2002, 66, 235120

We present a systematic study of the photonic properties (band structures) of periodic multicontinuous cubic phases based on the PDGIWP, FRD, and C(P) triply periodic minimal surfaces. We investigate the structures with up to five separate interwoven subvolumes. The influence of the dielectric constant modulation at different spatial scales is discussed. The lowest dielectric constant contrasts required to observe the full three-dimensional photonic band gaps are stated.

Application of the Euler characteristic to the study of homopolymer blends and copolymer melts

A. Aksimentiev and R. Hołyst

POLIMERY 2001, 6, 5, 307-323

Scaling of the Euler Characteristic, Surface Area, and Curvatures in the Phase Separating or Ordering Systems

M. Fiałkowski, A. Aksimentiev and R. Hołyst

Phys. Rev. Lett. 2001, 86, 240

We present robust scaling laws for the Euler characteristic and curvatures applicable to any symmetric system undergoing phase separating or ordering kinetics. We apply it to the phase ordering in a system of the nonconserved scalar order parameter and find three scaling regimes. The appearance of the preferred nonzero curvature of an interface separating ± domains marks the crossover to the late stage regime characterized by the Lifshitz-Cahn-Allen scaling.

Liquid‐Crystalline Order in Polymer Systems: Basic Models

R. Hołyst and P. Oswald

Macromolecular Theory and Simulations, 2001, 10, 1, 1-16

The liquid crystalline (LC) order appears in a variety of polymer systems, such as solutions of rod‐like molecules (DNA, TMV), solutions of semiflexible molecules (long fragments of DNA), block‐copolymer melts, main‐chain and side‐chain liquid crystalline polymer melts etc. Many LC phases have been observed in these systems; the most common being: the nematic, cholesteric, smectic or lamellar, hexagonal, and double gyroid (in block copolymers) phases. We will discuss in detail some of them and give their quantitative description in terms of order parameters. We will also present various theoretical models used to study LC ordering in the systems. The models discussed in this paper are as follows: Onsager model and its extension within the Density Functional Theories (DFT), Khohlov‐Semenov model for semiflexible polymers, Kratky‐Porod model, combination of the Kratky‐Porod model and Maier‐Saupe model, self‐consistent field theoretical model and finally the Landau‐Ginzburg models and their connection with the Edwards model for polymer systems. We will also briefly discuss the elasticity of polymer systems in the case of nematic ordering. We present in a pedagogical manner the general ideas which are behind various models and give references to the papers which contain the technical details.

Entropy-driven phase transitions and the Kauzmann entropy crisis

R. Hołyst

Physica A: Statistical Mechanics and its Applications 2001, 292, 1–4, 255-258

We briefly present the Kauzmann entropy crisis and show that it may have a simple explanation in terms of the entropy-driven phase transitions. We discuss in a pedagogical manner why the enthalpy change at the freezing transition is positive even for a hard sphere fluid, despite the fact that the transition is driven by the increase of the entropy. We use this observation to explain qualitatively the Kauzmann entropy crisis.

Intermediate Scaling Regime in the Phase Ordering Kinetics

M. Fialkowski and R. Holyst

ACTA PHYSICA POLONICA B. 2001, 32, 1579-1588

We have investigated the intermediate scaling regime in the phase ordering/separating kinetics of the three-dimensional system of the non-conserved scalar order parameter. It is demonstrated that the observed scaling behavior can be described in terms of two length scales LH(t) ∼ t2/5 and LK(t) ∼ t3/10. The quantity LH(t) is related to the geometrical properties of the phase interface and describes time evolution of the characteristic domain size, surface area, and the mean curvature. The second length scale, L)K(t), determining the Gaussian curvature and the Euler characteristic, can be regarded as the topological measure of the phase interface. Also, we have shown that the existence of the two length scales has a simple physical interpretation and is related to the domains-necks decoupling process observed in the intermediate regime.

Approach to equilibrium of particles diffusing on curved surfaces

D. Plewczyński and R. Hołyst

Physica A: Statistical Mechanics and its Applications 2001, 295, 3–4, 371-378

We present a simple numerical analysis of the diffusion on a curved surface given by the equation φ(r)=0 in a finite domain D⊂R3. The first non-vanishing eigenvalue of the Beltrami–Laplace operator with the reflecting boundary conditions is determined in our simulations for the P, D, G, S, S1 and I-WP, nodal periodic surfaces, where D is their respective cubic unit cell. We observe that the first eigenvalue for the surfaces of simple topology (P,D,G,I-WP) is smaller than for the surfaces of complex topology (S,S1).

Mechanisms for facilitated target location and the optimal number of molecules in the diffusion search process

K. Burdzy and R. Hołyst

Phys. Rev. E 2001, 64, 011914

We investigate the number N of molecules needed to perform independent diffusion in order to achieve bonding of a single molecule to a specific site in time t0. For a certain range of values of t0, an increase from N to kN molecules (k>1) results in the decrease of search time from t0 to t0/k. In this regime, increasing the number of molecules is an effective way of speeding up the search process. However when N>~N0 (optimal number of N) the reduction of time from t0 to t0/k can be achieved only by an exponentially large increase in the number of molecules [from N to Nexp(ck) for some c>0].

Scattering patterns of self-assembled gyroid cubic phases in amphiphilic systems

P. Garstecki and R. Hołyst

J. Chem. Phys. 2001, 115, 1095-1099

We present scattering patterns (with surface contrast) for five triply periodic minimal surfaces of the Ia3̄dIa3̄d cubic symmetry. We obtain a very good agreement between the numerically obtained spectrum and experimental patterns for the simple gyroid G structure. We show the scattering patterns for four gyroid GX1, GX2, GX3, and GX5 structures of a complex topology. We show how the scattering patterns change with increasing complexity of the unit cell of the structure. The spectra of the complex structures can give wrong estimates about the cubic cell parameter and even wrong establishment of the space symmetry group. Thus the correct recognition of the structure present in the system requires the analysis of the intensities of the peaks and comparison with numerically obtained spectra.

Periodic surfaces of simple and complex topology: Comparison of scattering patterns

P. Garstecki and R. Hołyst

Phys. Rev. E 2001, 64, 021501

We compute scattering patterns for six triply periodic minimal surfaces formed in oil/surfactant/water solutions: Three surfaces of a simple topology, Schwarz P (Im3¯m), Schwarz D–diamond (Pn3¯m), and Schoen G–gyroid (Ia3¯d), and three surfaces of a complex topology, SCN1 (Im3¯m), CD (Pn3¯m), and GX6 (Ia3¯d). We show that in the case of the complex structures, scattering intensity is shifted towards the higher hkl peaks. This might cause their misidentification and wrong estimates about the cell size of the structure.

Spinodal Decomposition of Homopolymer Blends: Geometrical Properties of the Interface

A. Aksimentiev, K. Moorthi and R. Hołyst

Progress of Theoretical Physics Supplement, 2000, 138, 398–399

The spinodal decomposition (SD) of the homopolymer blends has been studied by the numerical integration of the Cahn-Hilliard-Cook equation with the Flory-Huggins-de Gennes free energy functional. The scaling dependences for the surface area density, Euler characteristics density, average Gaussian and mean curvatures of the blend interface have been found. In the shallow and asymmetric quenches the topological transformation from the bicontinuous to droplet morphology has been observed. The influence of the thermal fluctuations on the curvature distribution has been investigated.

Reduction of dimensionality in a diffusion search process and kinetics of gene expression

R. Hołyst, M. Błażejczyk, K. Burdzy, G. Góralski and L. Bocquet

Physica A: Statistical Mechanics and its Applications 2000, 277, 1–2, 71-82

In order to activate a gene in a DNA molecule a specific protein (transcription factor) has to bind to the promoter of the gene. We formulate and partially answer the following question: how much time does a transcription factor, which activates a given gene, need in order to find this gene inside the nucleus of a cell? The estimate based on the simplest model of diffusion gives a very long time of days. We discuss various mechanisms by which the time can be reduced to seconds, in particular, the reduction of dimensionality, in which diffusion takes place, from three-dimensional space to two-dimensional space. The potential needed to keep the diffusing particle in 2D (i.e, at the surface of size L2 in a volume of size L3) should scale as U∼kBTlnL. For aL=1μm and a target size a=10 Å we find U=8kBT, i.e., it is a potential strength of the order of the strength of ionic interactions in water.

Scaling properties of the morphological measures at the early and intermediate stages of the spinodal decomposition in homopolymer blends

A. Aksimentiev and K. Moorthi

J. Chem. Phys. 2000, 112, 6049

The spinodal decomposition of the homopolymer blends has been studied by the numerical integration of the Cahn–Hilliard–Cook equation. We have investigated the time evolution of the morphological measures that characterize quantitatively the interface in the system. For symmetric blends we have found that the Euler characteristic of the interface is negative and increases with time as τ0.75 (connectivity of the domains decreases) regardless of the final quench temperature. The homogeneity index of the interface is constant in this case. This suggests that at the level of the integral geometry quantities (Minkowski functionals), the dynamic scaling hypothesis holds for the evolution of the interface morphology in quenched critical systems. The nonuniversal morphological evolution of the asymmetric blends have been studied. Also, we have shown that the thermal fluctuations can modify significantly the curvature distribution.

Topological Lifshitz Line, Off-Specular Scattering, and Mesoporous Materials

R. Hołyst and B. Przybylski

Phys. Rev. Lett. 2000, 85, 130

Ordered phases formed by surfactants in water solutions, and used in technological processes as templates for the synthesis of mesoporous materials, exhibit topological fluctuations. From the results of the Monte Carlo simulations of the lamellar phase we have established a relation between topological fluctuations and the behavior of the off-specular scattering intensity. We have defined the topological Lifshitz line. At this line the peak position in the off-specular scattering intensity moves from the zero (lamellar phase with fixed topology) to the nonzero value of the scattering wave vector (lamellar phase with fluctuating topology).

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