Structural properties of charged diblock copolymer solutions

Aqueous micellar solutions of ionic/neutral block copolymers have been studied by light scattering, small angle neutron scattering and small angle X-ray scattering. We made use of a polymer comprised of a short hydrophobic block (polyethylene-propylene) PEP and of a long polyelectrolytic block (polystyrene-sulfonate) PSSNa which has been shown previously to micellize in water. The apparent polydispersity of these micelles is studied in detail, showing the existence of a few large aggregates coexisting with the population of micelles. Solutions of micelles are found to order above some threshold in polymer concentration. The order is liquid-like, as demonstrated by the evolution with concentration of the peak observed in the structure factor (), and the degree of order is found to be identical over a large range of concentrations (up to 20 wt%). Consistent values of the aggregation number of the micelles are found by independent methods. The effect of salt addition on the order is found to be weak. Received: 19 June 1997 / Received in final form: 4 September 1997 / Accepted: 9 October 1997     

Kinetics of microphase segregation in one-dimensional symmetric diblock copolymer systems

An analytical one-dimensional model of the microphase separation in symmetric diblock copolymers is developed. Three stages of the process of the microphase segregation of a quenched diblock copolymer system into a lamella structure are predicted. The first stage involves a fast increase of the amplitude of the quasi-periodical lamella structure (with the average wave vector q₀) up to a nearly equilibrium value; the second stage is a slow phase diffusion process which is characterized by increasing coherency of the lamella structure; the third stage is the slow process of the lamella swelling, which is driven by the thermally activated process of the spontaneous deletion of excessive lamellas, the lamella size increasing from the initial value 2π/q ₀ to the equilibrium 2π/q _eq > 2π/q ₀ during the process. The last two stages are described with the specially introduced coarse-grained “amplitude-phase” approximation. It is shown that the relaxation of the gradient of the phase of the lamellar structure is the slowest relaxation process and, thus, can be used as an effective order parameter of the lamellar structure at the later stages of the microphase segregation. Received 10 March 2000 and Received in final form 5 June 2000     

Low temperature properties of the random field Potts chain

The random field q-states Potts model is investigated using exact groundstates and finite-temperature transfer matrix calculations. It is found that the domain structure and the Zeeman energy of the domains resembles for general q the random field Ising case (q = 2). This is also the expected outcome based on a random-walk picture of the groundstate. The domain size distribution is exponential, and the scaling of the average domain size with the disorder strength is similar for q arbitrary. The zero-temperature properties are compared to the equilibrium spin states at small temperatures, to investigate the effect of local random field fluctuations that imply locally degenerate regions. The response to field perturbations (`chaos') and the susceptibility are investigated. In particular for the chaos exponent it is found to be 1 for q = 2,..., 5. Finally for q = 2 (Ising case) the domain length distribution is studied for correlated random fields. Received 27 August 2002 Published online 19 December 2002 RID="a" ID="a"e-mail: rieger@lusi-sb.de     

Some exact results on the ultrametric overlap distribution in mean field spin glass models (I)

The mean field spin glass model is analyzed by a combination of exact methods and a simple Ansatz. The method exploited is general, and can be applied to others disordered mean field models such as, e.g., neural networks. It is well known that the probability measure of overlaps among replicas carries the whole physical content of these models. A functional order parameter of Parisi type is introduced by rigorous methods, according to previous works by F. Guerra. By the Ansatz that the functional order parameter is the correct order parameter of the model, we explicitly find the full overlap distribution. The physical interpretation of the functional order parameter is obtained, and ultrametricity of overlaps is derived as a natural consequence of a branching diffusion process. It is shown by explicit construction that ultrametricity of the 3-replicas overlap distribution together with the Ghirlanda-Guerra relations determines the distribution of overlaps among s replicas, for any s, in terms of the one-overlap distribution. Received 14 February 2000     

Block persistence

We define a block persistence probability p _l (t) as the probability that the order parameter integrated on a block of linear size l has never changed sign since the initial time in a phase-ordering process at finite temperature T<T _c . We argue that in the scaling limit of large blocks, where z is the growth exponent (), is the global (magnetization) persistence exponent and f(x) decays with the local (single spin) exponent for large x. This scaling is demonstrated at zero temperature for the diffusion equation and the large-n model, and generically it can be used to determine easily from simulations of coarsening models. We also argue that and the scaling function do not depend on temperature, leading to a definition of at finite temperature, whereas the local persistence probability decays exponentially due to thermal fluctuations. These ideas are applied to the study of persistence for conserved models. We illustrate our discussions by extensive numerical results. We also comment on the relation between this method and an alternative definition of at finite temperature recently introduced by Derrida [Phys. Rev. E 55, 3705 (1997)]. Received: 25 February 1998 / Revised: 24 July 1998 / Accepted: 27 July 1998     

Phase equilibria in random multiblock copolymers

A mean-field theory for domain structures in random multiblock copolymer melts is developed. We focus on the finite molecular weight effects resulting in a competition between macroscopic phase separation and microdomain formation in the system. We identify an essential parameter N ε controlling the phase behavior of the system, where N is the number of blocks per chain and ε is the composition asymmetry parameter (= the difference between the mean copolymer composition and its critical value). The phase diagram involving N ε and the reduced temperature as variables is obtained. The regions of coexistence of two or more phases are identified. We show that a superstructure formation on cooling is always pre-empted by a macroscopic phase separation of the macroscopically homogeneous (disordered) system yielding two homogeneous phases: H ₀↦H ₁ + H ₂. The third (lamellar) phase separates on further cooling. Then hexagonal and body-centred-cubic phases take over if N ε 1. As the Flory interaction parameter χ increases further, the standard transitions BCC↦HEX↦LAM take place. Received 13 July 2001     

Can one hear the shape of an electrode? II. Theoretical study of the Laplacian transfer

The flux across resistive irregular interfaces driven by a force deriving from a Laplacian potential is computed on a rigorous basis. The theory permits one to relate the size of the active zone to the derivative of the spectroscopic impedance with respect to the surface resistivity r through: . It is shown that the macroscopic transfer properties through a system of arbitrary shape are determined by the characteristics of a first-passage interface-interface random walk operator . More precisely, it is the distribution of the harmonic measure (or normalized primary current) on the eigenmodes of this linear operator that controls the transfer. In addition, it is also shown that, whatever the dimension, the impedance of a weakly polarizable electrode for any irregular geometry scales under a homothety transformation as L^d-1, L being the size of the system and d its topological dimension. In this new formalism, the question addressed in the title is transformed in a open mathematical question: “Knowing the distribution of the harmonic measure on the eigenmodes of the self-transport operator, can one retrieve the shape of the interface?” Received 3 November 1998     

Ionic distribution and polymer conformation, near phase separation, in sodium polyacrylate/divalent cations mixtures: small angle X-ray and neutron scattering

Anomalous small angle X-ray scattering experiments show that before demixion in sodium polyacrylate/cobalt and sodium polyacrylate/calcium mixtures all the divalent counterions are in the close vicinity of the polyacrylate chain. The present results are consistent with previous UV/VIS spectroscopy, which have shown that all cobalt ions are chemically associated with acrylate groups. The chemical association dehydrates the acrylate monomers. However, the hydrophobicity of the complexed monomers is not strong enough to induce a collapse of the polymer chain at small spatial scale before the demixion. Indeed, the scattered intensity (X-ray and neutron scattering) decreases with the scattering vector q as q^-x with for q > 0.1 nm ^-1 which indicates that the local conformation of the chain is Gaussian. Received 21 January 1999     

Free energy of semiflexible polymers and structure of interfaces

The free energy of semiflexible polymers is calculated as a functional of the compositional scalar order parameter and the orientational order parameter of second-rank tensor S_ij on the basis of a microscopic model of wormlike chains with variable segment lengths. We use a density functional theory and a gradient expansion to evaluate the entropic part of the free energy, which is given in a power series of .The interaction term of the free energy is derived with a random phase approximation. For the rigid rod limit, the nematic-isotropic transition point is given by , N and w being the degree of polymerization and the anisotropic interaction parameter, respectively, and the degree of ordering at the transition point is 0.33448. We also find that the contour length of polymer chains becomes larger in a nematic phase than in an isotropic phase. Interface profiles are obtained numerically for some typical cases. In the neighborhood of isotropic-isotropic interfaces, polymer chains tend to align parallel to the interface on the polymer-rich side and perpendicular on the poor side. When an isotropic region and a nematic region coexist, orientational order parallel to the interface is preferred in the nematic region. Received: 28 May 1998 / Revised: 12 August 1998 / Accepted: 8 September 1998     

Inclusions induced phase separation in mixed lipid film

The effect of rigid inclusions on the phase behavior of a film containing a mixture of lipid molecules is investigated. In the proposed model, the inclusion-induced deformation of the film, and the resulting energy cost are strongly dependent upon the spontaneous curvature of the mixed film. The spontaneous curvature is in turn strongly influenced by the composition of film. This coupling between the film composition and the energy per inclusion leads to a lateral modulation of the composition, which follows the local curvature of the membrane. In particular, it is shown that inclusions may induce a global phase separation in a film which would otherwise be homogeneously mixed. The mixed film is then composed of patches of different average composition, separated by the inclusions. This process may be of relevance to explain some aspects of lipid-protein association in biological membranes. Received 8 April 1999 and Received in final form 4 October 1999     

Reordering kinetics of ion-disordered Ni Al

A study of the reordering kinetics of ion-irradiated Ni₃Al is presented. The development of long-range order during annealing treatments is measured by quantitative electron diffractometry. According to the temperature dependence of the observed kinetics, the reaction is dominated by non-equilibrium vacancies. As a consequence, the kinetics can be calibrated versus the absolute number of atomic jumps necessary to establish the observed degree of order. In order to analyse the experimental data, Monte Carlo simulations are performed. It is shown that, beside the temperature dependence of the driving force, the ordering efficiency of the vacancy jumps itself is temperature dependent due to different mobilities of the atomic species. Received 6 December 1999 and Received in final form 23 June 2000     

Aging kinetics of porous media due to freezing-thawing cycles

The two-dimensional Ausloos et al. model of fluid invasion, freezing and thawing in a porous medium is elaborated upon and investigated in order to take into account the pore volume redistribution and conservation during freezing. The results are qualitatively different from previous work, since the damaged pore sizes are found to be much less than the possible maximum value and is reached after a large number of invasion-freezing-thawing cycles, e.g. the material is “slowly damaged”. The pore size distribution is thus found in better agreement with expected practical findings. The successive invasion percolation clusters are still found to be self-avoiding with aging. The cluster size decreases with a power law as a function of invasion-frost-thaw iterations. The aging kinetics is also discussed through the normalized totally invaded pore volume. Received 24 September 1999 and Received in final form 5 January 2000     

Phases of wave functions and level repulsion

Avoided level crossings are associated with exceptional points which are the singularities of the spectrum and eigenfunctions, when considered as functions of a complex coupling parameter. It is shown that the wave function of one state changes sign but not the other, if the exceptional point is encircled in the complex plane. An experimental setup is suggested where this peculiar phase change could be observed. Received: 7 January 1999 / Received in final form: 15 March 1999     

Local environment and electron correlation effects on the magnetic properties of clusters

The electronic and magnetic properties of clusters are investigated in the framework of the Hubbard model by treating electron correlations effects in a saddle-point slave-boson approximation. The size dependent single-particle spectrum is calculated using a third moment real-space expansion of the local density of states. Results for the magnetic moments, magnetic order, average number of double occupations and hopping renormalizations are given as a function of the local coordination number z, for different representative values of the Coulomb interaction strength U/t and band filling n. Several transitions between paramagnetic, ferromagnetic and antiferromagnetic behaviors are obtained as a function of z. The environment dependence of the magnetic behavior and of the degree of electron delocalization is analyzed. Advantages and limitations of the present approach are discussed. Received: 8 January 1998 / Revised: 22 June 1998 / Accepted: 6 August 1998     

Analysis of long crack lines in paper webs

A differential cluster variation method (DCVM) is proposed for analysis of spinoidal decomposition in alloys. In this method, lattice symmetry operations in the presence of an infinitesimal composition gradient are utilized to deduce the connection equations for the correlation functions and to reduce the number of independent variables in the cluster variation analysis. Application of the method is made to calculate the gradient energy coefficient in the Cahn-Hilliard free energy function and the fastest growing wavelength for spinodal decomposition in Al-Li alloys. It is shown that the gradient coefficient of congruently ordered Al-Li alloys is much larger than that of the disordered system. In such an alloy system, the calculated fastest growing wavelength is approximately 10 nm, which is an order of magnitude larger than the experimentally observed domain size. This may provide a theoretical explanation why spinodal decomposition after a congruent ordering is dominated by the antiphase boundaries.Received: 17 November 2003, Published online: 2 April 2004PACS: 64.75. + g Solubility, segregation, and mixing; phase separation - 81.30.-t Phase diagrams and microstructures developed by solidification and solid-solid phase transformations - 05.70.Ln Nonequilibrium and irreversible thermodynamics     

Scaling concepts for polymer brushes and their test with computer simulation

After a brief review of the scaling concepts for static and dynamic properties of polymer brushes in good solvents and Theta solvents, the Monte Carlo evidence is discussed. It is shown that under typical conditions the diameter of the last blob is of the order of 10-20% of the brush height, and therefore pronounced deviations from the self-consistent field predictions occur. In bad solvents, lateral microphase separation occurs leading to an irregular pattern of “dimples”. Particularly interesting is the response of brushes to shear deformation, and the interaction between two interpenetrating brushes. Recent attempts to understand the resulting shear forces via molecular-dynamics simulations are briefly described, and an outlook on related experiments is given. Dedicated to Prof. H.E. Stanley on the occasion of his 60th birthday Received 11 March 2002 and Received in final form 3 June 2002     

The origin of the redshift in Brillouin spectra of silica films containing tin nanoparticles

The abrupt change of velocity in surface acoustic waves in thin films of amorphous SiO_x containing nanometre scale -Sn crystals is shown to be directly associated with the size-dependent melting of the nanoparticles, confirming preliminary experiments. High resolution thin film powder diffraction using synchrotron radiation shows that the abrupt redshift in the Brillouin spectra satellites occurs at the same temperature as the melting of the nanoparticles, evident for the loss of the Bragg peaks. Effective medium theory is used to explain the origin of the anomaly. A central peak in the Brillouin spectrum, the intensity of which shows a maximum at the melting temperature, can be interpreted in terms of overdamped fluctuations in the dielectric function. The melting temperature as a function of particle size is in agreement with theoretical predictions. No evidence for strain could be found on the X-ray diffraction profiles; the a- and c-axis thermal expansion coefficients are the same as those in bulk tin. Received 30 March 2000 and Received in final form 24 July 2000     

Local impurity phase pinning and pinning force in charge density waves

Starting from the static Fukuyama-Lee-Rice equation for a three-dimensional incommensurate charge density wave (CDW) in quasi one-dimensional conductors a solvable model for local phase pinning by impurities is defined and studied. We find that average CDW energy and average pinning force show critical behaviour with respect to the pinning parameter h. Specifically the pinning force exhibits a threshold at h=1 with exponent . Our model exemplifies a general concept of local impurity pinning in which the force exerted by the impurity on the periodic CDW structure becomes multivalued and metastable states appear beyond a threshold. It is found that local impurity pinning becomes less effective at low temperatures and may eventually cease completely. These results are independent of spatial dimensionality as expected for local impurity pinning. Comparison with Larkin's model is also made. Received 8 July 1998     

Elastic property of bilayer membrane in copolymer-homopolymer mixtures

We study the domain morphology in a phase separated state of diblock copolymer-homopolymer mixtures. In the situation that one of the blocks of copolymers is incompatible with homopolymers, formation of a bilayer membrane of block copolymers is shown to be possible in the matrix of homopolymers. Starting with the free energy functional in terms of the local volume fractions of each monomer, we derive the bending and curvature rigidities in the Helfrich free energy for a membrane. It is found that the curvature modulus is negative only in a limited region of the parameters, where a closed shape of a membrane like a vesicle is possibly formed. We establish a method to calculate the rigidities without a molecular picture in a consistent way with the field theoretic model free energy. Stability of a bilayer membrane compared with micelles is also investigated. Received: 22 July 1997 / Received in final form: 1 December 1997 / Accepted: 22 January 1998