Small-angle X-ray scattering from salt-free solutions of star-branched polyelectrolytes

The dispersion state of sodium-sulphonated polystyrene ( NaPSS) star-branched polyelectrolytes was investigated in salt-free aqueous solutions, by use of the small-angle X-ray scattering technique. With respect to polystyrene (PS) star-branched polymers of identical functionality, the ordering phenomenon occurring in the neighborhood of the overlap concentration c^* is reinforced and observed in a larger range of concentrations. Moreover, the degree of order is no longer maximum at c^* and is improved as the concentration decreases. The dispersion state is then mainly controlled by the electrostatic interaction. A crystalline order should therefore be achieved with stars of lower functionality, provided the electrostatic interaction is added to the osmotic repulsion. On the other hand, unusual scattering patterns are measured for aqueous solutions of NaPSS star polyelectrolytes. Indeed, a diffuse scattering is revealed at high angles, in addition to the regular diffraction rings related to preferred interstar distances. It is similar to the broad scattering peak produced by semidilute solutions of NaPSS linear polyelectrolytes and associated to the electrostatic correlation hole within the isotropic model. In the dilute regime (c < c ^*), it is just an intramolecular characteristic and represents the electrostatic repulsion between arms belonging to the same star. In the semidilute regime (c > c ^*), it also reflects the electrostatic repulsion between arms of distinct stars. So, as the concentration increases, it is mainly caused by the interpenetration of NaPSS stars. Such an observation is in agreement with the composite structure earlier proposed by Daoud and Cotton for star semidilute solutions. For c > c ^*, NaPSS star aqueous solutions can therefore be pictured as effective stars immersed in a matrix formed by the overlap of the arm ends. With respect to the dilute regime, the effective stars are smaller; the higher the concentration the smaller the size. Received 14 May 1999 and Received in final form 15 March 2000     

Effects of ionic strength and charge annealing in star-branched polyelectrolytes

We have developed a scaling theory that describes the conformations of weak star-branched polyelectrolytes in dilute solutions. The dependences of the overall star size on the number of branches and on the ionic strength of the solution (tuned by the addition of low molecular weight salt) are analyzed. The intrinsic structure of the polyelectrolyte stars in salt-free and salt-added solutions is discussed in terms of concentration and elastic blobs. In contrast to neutral stars, the swollen corona of the polyelectrolyte star consists of blobs which are not closely packed. We have shown that the size of star polyelectrolytes is less sensitive to the variation in the ionic strength than the size of linear polyelectrolytes. The effects of the ionization-recombination balance in the star polyelectrolyte were taken into account. For polyelectrolytes with small ionization constant, the size of the star depends nonmonotonically on the number of branches and on the ionic strength of the solution due to recombination of counterions with charged monomers. Received: 10 November 1997 / Revised: 16 February 1998 / Accepted: 1st April 1998     

A neutron spin-echo study of the dynamics of star-shaped polymers (with polystyrene arms grafted to C60) in solutions and in the bulk

Methods of small-angle and inelastic neutron scattering were used to study star-shaped polymers obtained by grafting polystyrene to fullerene C₆₀ (six arms with an elementary mass M=5×10³). The behavior of stars in the solution and bulk was compared with the dynamics of free molecules of polystyrene with a mass equal to the mass of an arm. The slow dynamics of arms in the ranges of times t=0.01–20 ns, momentum transfers q=0.2–0.6 nm^?1, and temperatures T=20–60°C does not obey the simple models of Rouse and Zimm. The interaction of the rays with the C₆₀ fullerene under the conditions of the specific geometry of the macromolecule leads to unusual oscillations of time correlations in the arms of stars.     

Anomalous dynamical light scattering in soft glassy gels

We compute the dynamical structure factor S(q,τ) of an elastic medium where force dipoles appear at random in space and in time, due to “micro-collapses” of the structure. Various regimes are found, depending on the wave vector q and the collapse time θ. In an early time regime, the logarithm of the structure factor behaves as (qτ)^3/2, as predicted in (L. Cipelletti et al., Phys. Rev Lett. 84, 2275 (2000)) using heuristic arguments. However, in an intermediate-time regime we rather obtain a (qτ)^5/4 behaviour. Finally, the asymptotic long-time regime is found to behave as q ^3/2τ. We also give a plausible scenario for aging, in terms of a strain-dependent energy barrier for micro-collapses. The relaxation time is found to grow with the age t_w, quasi-exponentially at first, and then as t _w ^4/5 with logarithmic corrections. Received 15 April 2002     

Anomalous dynamical light scattering in soft glassy gels

We compute the dynamical structure factor S(q,τ) of an elastic medium where force dipoles appear at random in space and in time, due to “micro-collapses” of the structure. Various regimes are found, depending on the wave vector q and the collapse time θ. In an early-time regime, the logarithm of the structure factor behaves as (qτ)^3/2, as predicted in L. Cipelletti, S. Manley, R.C. Ball, D.A. Weitz, Phys. Rev. Lett. 84, 2275 (2000) using heuristic arguments. However, in an intermediate-time regime we rather obtain a (qτ)^5/4 behaviour. Finally, the asymptotic long-time regime is found to behave as q ^3/2τ. We also give a plausible scenario for aging, in terms of a strain-dependent energy barrier for micro-collapses. The relaxation time is found to grow with the age t _w, quasi-exponentially at first, and then as t _w ^4/5 with logarithmic corrections. Received 23 July 2001     

Ordering of urchin-like charged copolymer micelles: Electrostatic, packing and polyelectrolyte correlations

We report the results of small-angle neutron scattering (SANS) studies on aqueous solutions of spherical polyelectrolyte micelles formed by association of charged-neutral diblock copolymers. The neutral moieties are found to self-assemble into small dense spheres (cores of the micelles) whose sizes are independent of the polymer concentration c. In the dilute regime, c<c ^*, where c^* is the overlap concentration of the micelles, the conformation of the charged groups, which form the corona of the micelles, is found to be extended. A liquid-like order is observed over a wide concentration range spanning from the dilute regime to the concentrated regime. For c>c ^*, polyelectrolyte correlations appear at smaller spatial scales and coexist with the liquid-like order. These results suggest that for dense brushes, above c^*, the rod-like statistics of the charged chains begin to disappear due to contraction of corona arms or by interpenetration of coronae. For less dense brushes, the charged chains are found to be extended up to concentrations far above c^*, before the progressive development of polyelectrolyte correlations. Received 8 October 1999     

Effect of boundaries and impurities on electron–phonon dephasing

Electron scattering from boundaries and impurities destroys the single-particle picture of the electron–phonon interaction. We show that quantum interference between ‘pure‘ electron–phonon and electron–boundary/impurity scattering may result in the reduction as well as to the significant enlargement of the electron dephasing rate. This effect crucially depends on the extent, to which electron scatterers, such as boundaries and impurities, are dragged by phonons. Static and vibrating scatterers are described by two dimensionless parametersq_Tl and q_TL, where q is the wavevector of the thermal phonon, l is the total electron mean-free path, L is the mean-free path due to scattering from static scatterers. According to the Pippard ineffectiveness condition , without static scatterers the dephasing rate at low temperatures is slower by the factor 1 / ql than the rate in a pure bulk material. However, in the presence of static potential the dephasing rate turns out to be 1 / qL times faster. Thus, at low temperatures electron dephasing and energy relaxation may be controlled by electron boundary/impurity scattering in a wide range.     

Multiblock copolymers at interfaces

Summary We consider the configurations and the various regimes when an alternating, symmetric, multiblock copolymer made of successive sequences of A and B monomers with equal lengthZ is at the interface between two immiscible solvents that are respectively good for one of the sequences and poor for the other one. We find two semi-dilute regimes, when the copolymers overlap. For monomer concentrations belowГ ^**, we find a first range where the thickness of the copolymers is constant and equal to the radius of a sequence. For concentrations larger thanГ ^**, a second regime appears, where the seqeunces stretch our and adopt a conformation similar to what was reported for grafted polymers. Finally at a concentrationГ _sat, the interface is completely saturated, and the copolymers form micelles in the bulk: it becomes energetically more favourable to have copolymers in the bulk than at interface. Paper presented at the I International Conference on Scaling Concepts and Complex Fluids, Copanello, Italy, July 4‘8, 1994.     

Adsorption of a semiflexible wormlike chain

Adsorption of ideal polymers with stiff backbone onto a flat surface is considered theoretically. Both scaling approach and quantitative theory are developed. We predict a self-similar monomer concentration profile c(x) ∼ x ^-4/3 near the surface (when the distance to the surface x is much smaller than the chain persistence length l /2). The typical conformation of a weakly adsorbed chain can be viewed as a sequence of alternating flat (2-dimensional) trains of wormlike short loops (flat blobs) and coil-like (3-dimensional) loops forming a triple-layer structure: contact layer (x < Δ) of adsorbed fragments virtually laid on the surface, proximal layer (Δ < x < l) of flat blobs, and more dilute distal corona layer (x > l). Here Δ defines the range of monomer/surface attraction, Δ ≪ l. The adsorption transition is continuous. However, its relative width is small (T ^* is the adsorption temperature, ΔT is the relevant temperature interval): ∼ , i.e. a discontinuous transition in the limit Δ/l↦ 0. Received 10 October 2002 and Received in final form 22 November 2002 RID="a" ID="a"Permanent address: Physics Department, Moscow State University, Moscow 119992, Russia. e-mail: semenov@polly.phys.msu.ru     

Microphase separation in weakly charged hydrophobic polyelectrolytes

Aqueous solutions of a well-defined poly(N-isopropylacrylamide-co-sodium 2-acrylamido-methylpropanesulfonate) (NIPAM/NaAMPS in a 95/5 molar ratio) have been investigated by means of small-angle neutron scattering (SANS) and rheological experiments as a function of temperature ( 25^° C T 60^° C) and polymer concentration ( 0.5wt% C 12wt%). The solutions remain optically transparent and isotropic over the whole temperature range, in contrast with the homopolyNIPAM which precipitates above its lower critical solution temperature (LCST = 32^° C). Upon addition of salt, the systems undergo a micro-macrophase separation. At temperatures above 45^° C, the SANS spectra exhibit a sharp peak at a scattering wave vector, q _max, which increases slightly with temperature. At high temperature ( T∼ 60^° C), the scattered intensity follows a power law I(q) ∼q ^-4 in the asymptotic regime, characteristic of two-density media with sharp interfaces, and q _max is found to vary with polymer concentration as q _max∼C ^0.22. Estimates of the typical sizes give values between 40 ? and 200 ?. These results provide a strong evidence of a thermally induced microphase separation, which is corroborated by the very sharp increases of the viscosity (over 2 decades) and of the stress relaxation time of the solutions, occurring in the temperature range where the scattering peak is observed. The results are discussed and compared with the theoretical models proposed for weakly charged polyelectrolytes in a poor solvent. Received 1 October 2001     

The structure of the glass phase of Rb1−x(ND4)xD2PO4

X-ray scattering techniques have been used to study the diffuse scattering from a single crystal of Rb_1–x(ND₄)_xD₂PO₄ withx=0.65. This system has a structural glass phase at low temperatures resulting from the competing ferroelectric interactions of RbD₂PO₄ and antiferroelectric interactions of (ND₄)D₂PO₄. The diffuse scattering shows a broad peak with a maximum occurring at a wavevector of about 0.3a ^*, and the intensity of these peaks is surprisingly different for wavevectorsq on opposite sides of the Bragg reflections. A model of the D bonding is developed which suggests that the diffuse scattering arises from the interaction between ferroelectric displacements alongc, ferroelectric displacements alongb, and transverse acoustic modes polarized alongb andc. The model accounts for the incommensurate wavevector and, qualitatively, for the intensity of the diffuse scattering around different Bragg reflections. The temperature dependence of the scattering is also measured.     

Neutron scattering from equilibrium-swollen networks

Small-angle neutron scattering measurements were performed on end-linked poly (dimethylsiloxane) (PDMS) networks swollen to equilibrium with d-benzene. Comparison was made with equivalent concentration PDMS solutions. Equilibrium-swollen networks consistently displayed a linear scattering regime at low q followed by a good-solvent-like scaling regime at high q in agreement with the predictions of the Gel Tensile Blob (GTB) model. Data are fit using the unified function modified for the GTB model (3-parameter fit). Equilibrium-swollen networks display a base structural size, the gel tensile-blob size, ξ, that was found to be independent of the molecular weight between crosslinks for the series of molecular weights studied, consistent with the predictions of the model. The length of the extended tensile structure, L, can be larger than the length of the fully extended network strand. The predicted scaling relationship for L, L ∼ Q^1/2N_avg, where N_avg = (1/fN_c²+1/4N_e² , Q is the equilibrium swelling ratio, N_c is the molecular weight between crosslinks, N_e is the entanglement molecular weight and f is the crosslink functionality is in agreement with experimental results for the networks studied.     

Scattering properties of salt-free wormlike micellar solutions

Static and dynamic light scattering and conductivity experiments were carried out on salt-free aqueous micellar solutions of cetyltrimethylammonium n-hexane sulfonate (CTAC_6SO_3) and cetyltrimethylammonium n-heptane sulfonate (CTAC_7SO_3) as a function of surfactant concentration. This study confirms the analogy between the behavior in the semi-dilute regime of elongated micellar systems and “classical” polyelectrolyte solutions. Time-resolved scattering experiments performed after a variation of concentration from about twice the overlap volume fraction to less than half of it revealed the existence of a structural relaxation with a characteristic time of several hours. Received 21 December 1999     

Overlap properties and adsorption transition of two Hamiltonian paths

We consider a model of two (fully) compact polymer chains, coupled through an attractive interaction. These compact chains are represented by Hamiltonian paths (HP), and the coupling favors the existence of common bonds between the chains. We use a (n=0 component) spin representation for these paths, and we evaluate the resulting partition function within a homogeneous saddle point approximation. For strong coupling (i.e. at low temperature), one finds a phase transition towards a “frozen” phase where one chain is completely adsorbed onto the other. By performing a Legendre transform, we obtain the probability distribution of overlaps. The fraction of common bonds between two HP, i.e. their overlap q, has both lower () and upper () bounds. This means in particular that two HP with overlap greater than coincide. These results may be of interest in (bio)polymers and in optimization problems. Received 4 December 1998 and Received in final form 10 March 1999     

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     

Relating structure and flow of soft colloids

To relate the complex macroscopic flow of soft colloids to details of its microscopic equilibrium and non-equilibrium structure is still one big challenge in soft matter science. We investigated several well-defined colloidal model systems like star polymers or diblock copolymer micelles by linear/non-linear rheology, static/dynamic light scattering (SLS/DLS) and small angle neutron scattering (SANS). In addition, in-situ SANS experiments during shear (Rheo-SANS) revealed directly shear induced structural changes on a microscopic level. Varying the molecular architecture of the individual colloidal particle as well as particle-particle interactions and covering at the same time a broad concentration range from the very dilute to highly concentrated, glassy regime, we could separate contributions from intra- and inter-particle softness. Both can be precisely “tuned” by varying systematically the functionality, 6 ≤ f≤ 64, for star polymers or aggregation number, 30 ≤ N _agg ≤ 1000 for diblock copolymer micelles, as well as the degree of polymerization of the individual polymer arm 100 ≤ D _p ≤ 3000. In dilute solutions, the characteristic shear rate at which deformation of the soft colloid is observed can be related to the Zimm time of the polymeric corona. In concentrated solutions, we validated a generalized Stokes-Einstein approach to describe the increase in macroscopic viscosity and mesoscopic self diffusion coefficient on approaching the glassy regime. Both can be explained in terms of an ultra-soft interaction potential. Moreover, non-equilibrium structure factors are obtained by Rheo-SANS. All experimental results are in excellent quantitative agreement with recent theoretical predictions.     

Investigation of Ring and Star Polymers in Confined Geometries: Theory and Simulations

The calculations of the dimensionless layer monomer density profiles for a dilute solution of phantom ideal ring polymer chains and star polymers with f=4 arms in a Θ -solvent confined in a slit geometry of two parallel walls with repulsive surfaces and for the mixed case of one repulsive and the other inert surface were performed. Furthermore, taking into account the Derjaguin approximation, the dimensionless layer monomer density profiles for phantom ideal ring polymer chains and star polymers immersed in a solution of big colloidal particles with different adsorbing or repelling properties with respect to polymers were calculated. The density-force relation for the above-mentioned cases was analyzed, and the universal amplitude ratio B was obtained. Taking into account the small sphere expansion allowed obtaining the monomer density profiles for a dilute solution of phantom ideal ring polymers immersed in a solution of small spherical particles, or nano-particles of finite size, which are much smaller than the polymer size and the other characteristic mesoscopic length of the system. We performed molecular dynamics simulations of a dilute solution of linear, ring, and star-shaped polymers with N=300 , 300 (360), and 1201 (4 × 300 + 1-star polymer with four arms) beads accordingly. The obtained analytical and numerical results for phantom ring and star polymers are compared with the results for linear polymer chains in confined geometries.     

Amphiphilic telechelic poly(N-isopropylacrylamide) in water: From micelles to gels<Superscript>⋆</Superscript>

We report the first study of aqueous solutions (0.025 gL^-1 to 46 gL^-1) of a telechelic poly(N-isopropylacrylamide) with octadecyl termini (C₁₈-PNIPAM-C₁₈, M_w: 37000, 320 NIPAM units, M_w/ M_n = 1.07) obtained by reversible addition-fragmentation chain transfer (RAFT) free radical polymerization of N-isopropylacrylamide. Static and dynamic light scattering measurements and fluorescence spectroscopy, using 8-anilino-1-naphthalenesulfonic acid (ANS) as probe, yielded the concentration dependence of the size and aggregation number of C₁₈-PNIPAM-C₁₈ micelles in cold ( 20^°C) dilute aqueous solutions. Concentrated solutions ( c > 20gL^-1) form transient gels exhibiting an oscillatory shear behavior that can be approximated by a single-relaxation Maxwellian model. Aqueous solutions of C₁₈-PNIPAM-C₁₈ undergo a phase transition upon heating to 31.5^°C as determined by microcalorimetry. The heat-induced phase separation of dilute (0.025 gL^-1) C₁₈-PNIPAM-C₁₈ solutions yields a fluid that is colloidally stable at temperatures higher than 33^°C. The overall results are consistent with a model assuming the formation of flowerlike micelles in the dilute regime and a network of micelles connected by telechelic chains in the concentrated regime.     

Collective electronic excitations in a semiconductor superlattice in the Landau and Wannier-Stark ladder regime

Using a mean-field approximation, we have developed a systematic treatment of collective electronic modes in a semiconductor superlattice (SL) in the presence of strong electric and magnetic fields parallel to the SL axis. The spectrum of collective modes with zero wavevector along the SL axis is shown to consist of a principle magnetoplasmon mode and an infinite set of Bernstein-like modes. For non-zero wavevector along the SL axis, in addition to the cyclotron modes, extra collective modes are found at the frequencies |Nω _c±Mω _s|, which we call cyclotron-Stark modes (ω _c and ω _s are respectively the cyclotron and Stark frequencies, N and M are integer numbers). The frequencies of the modes propagating in “oblique” direction with respect to the SL axis show oscillatory behavior as a function of electric field strength. All the modes considered have very weak spatial dispersion and they are not Landau damped. The specific predictions made for the dispersion relations of the collective excitations should be observable in resonant Raman scattering experiments. Received 29 August 2002 / Received in final form 25 February 2003 Published online 4 June 2003 RID="a" ID="a"e-mail: 612033@inbox.ru     

Phase behaviour of a dispersion of charge-stabilised colloidal spheres with added non-adsorbing interacting polymer chains

We present a theory for the phase behaviour of mixtures of charge-stabilised colloidal spheres plus interacting polymer chains in good and θ -solvents within the framework of free-volume theory. We use simple but accurate combination rules for the depletion thickness around a colloidal particle and for the osmotic pressure up to the semi-dilute concentration regime. Hence, we obtain expressions for the free energy for mixtures of charged colloidal particles and non-adsorbing interacting polymers. From that, we calculate the phase behaviour, and discuss its topology in dependence on the competition between the charge-induced repulsion and the polymer-induced attraction. The homogeneous mixture of colloids and polymers becomes more stabilised against demixing when increasing the electrostatic repulsion. This charge-induced stabilisation is strongest for small polymer-to-colloid size ratios and is more pronounced for charged colloids mixed with polymers in a good solvent than for polymers in a θ -solvent. For the weakly charged regime we find that the phase diagram becomes salt-concentration-independent in the protein limit for charged colloids plus polymers in a θ -solvent. The liquid window, i.e., the concentration regimes where a colloidal liquid exists, is narrowed down upon increasing the charge-induced repulsion. Also this effect is more pronounced when charged colloids are mixed with polymer chains in a good solvent. In summary, we demonstrate that the solvent quality significantly influences the phase behaviour of mixtures of charged colloids plus non-adsorbing polymers if the range of the screened electrostatic repulsion becomes of the order of the range of the depletion-induced attraction.