Chain conformation of rod-like polymers in the melt: Small-angle neutron scattering of poly(benzoyl paraphenylene)

The chain conformation of a rigid rod polymer, poly(benzoyl paraphenylene), is determined in the melt using small-angle neutron scattering. The coherent scattering cross-section from blends of partially deuterated and hydrogenous poly(benzoyl paraphenylene) agree well with ideal rod scattering for q > 0.02 Å⁻¹, indicating that the polymer chains are highly extended. Comparison of the results to a single chain-scattering function for chains of arbitrary stiffness yield a persistence length of ca. 130 Å. Pure component scattering at the lowest scattering lengths indicate that the melt is not molecularly homogenous, but is comprised of domains, potentially reflecting localized groupings of chains with similar backbone orientation. Furthermore, this mesoscopic structure depends on the processing history of the polymer in the melt state. © 1998 John Wiley & Sons, Inc. J. Polym. Sci. B Polym. Phys. 36: 2449–2459, 1998     

Determination of chain stiffness and polydispersity from static light-scattering

Chain stiffness is often difficult to distinguish from molecular polydisperity. Both effects cause a downturn of the angular dependence at large q² (q = (4π/λ)sin θ/2) in a Zimm plot. A quick estimation of polydisperity becomes possible from a bending rod (BR) plot in which lim (c → 0) qRθ/Kc is plotted against q(〈S²〉_z)^1/2 = u. Flexible and semiflexible chains show a maximum whose position is shifted from u_max = 1.41 for monodisperse chains towards larger values as polydispersity is increased, while simultaneously, the maximum height is lowered. Stiff chains display a constant plateau at large q, its value is πM_L where M_L is the linear mass density. Using Koyama's theory, the number of Kuhn segments can be determined from the ratio of the maximum height to the plateau height, if the polydispersity index z = (M_w/M_n ? 1)^?1 is known. Thus, if the weight-average molecular weight M_w, is known, the contour length L_w, the number of Kuhn segments (N_k)_w, the Kuhn segment length l_k and the polydispersity of the stiff chains can be determined. The influence of excluded volume is shown to have no effect on this set of data. The reliability of this set can be cross-checked with the mean-square radius of gyration 〈s²〉_z which can be calculated from the Benoit-Doty equation for polydisperse chains. Rigid and slightly bending rods exhibit no maximum in the BR plot, and the effect of polydispersity can no longer be distinguished from a slight flexibility if only static scattering techniques are applied.     

Simulations and scattering functions of polyelectrolyte–macroion complexes

Using Monte Carlo simulations of complex formation between a polyelectrolyte chain and an oppositely charged macroion, we calculated the scattering function of the polyelectrolyte chain. We investigated the case of the isolated polyelectrolyte chain and studied the effect and influence of key parameters such as the ionic concentration of the solution, polyelectrolyte length and intrinsic rigidity on the scattering function. Then, we focused on the polyelectrolyte–macroion complex by calculating the structure factor S(q) of the adsorbed polyelectrolyte chain. Typical conformations ranging from coils, extended chains to solenoids are revealed and the corresponding S(q) analysed. The effects of ionic concentration, chain length and intrinsic rigidity and relative size ratio between the polyelectrolyte and the macroion are investigated. Important effects on the structure factor of the adsorbed polyelectrolyte are observed when the macroion is partially or totally wrapped by the polyelectrolyte. Distance correlations between the polyelectrolyte monomer positions at the surface of the macroion induce the formation of peaks in the fractal regime of S(q). For semiflexible chains, when solenoid conformations are observed, the position of the peaks in the fractal regime corresponds directly to the separation distance between the turns. The formation of a protruding tail in solution is also observed through the formation in the fractal regime of a linear domain.This revised version was published online in November 2004 with corrections to the authors.     

Influence of molecular stiffness on the dynamic structure factor

Characteristic features of the influence of molecular stiffness on the dynamic structure factor of macromolecules are briefly outlined. The relaxation times characterizing the internal dynamics of the macro‐molecules exhibit a crossover from Rouse‐Zimm to bending modes with increasing mode number. As a consequence the dynamic structure factor is strongly influenced by the molecular stiffness. In particular, a stretched exponential relaxation of the dynamic structure factor at scattering vectors larger than the inverse persistence length is predicted and confirmed by a comparison with experimental data. Moreover, the influence of polydispersity is discussed.     

A monte carlo lattice study of living polymers in a confined geometry

We investigate the changes in the average chain length of a solution of semi-flexible living polymers between two hard repulsive walls as the width of the slit, D, is varied. Two different Monte Carlo models, that of the ‘slithering snake’ and of the ‘independent monomer states’ are employed in order to simulate a polydisperse system of chain molecules confined in a gap which is either closed (with fixed total density), or open and in contact with an external reservoir. It appears that the mean chain length L in a state of equilibrium polymerization depends essentially on the geometry constraints for sufficiently small D. We find that in the case of an open slit the mean length L(D) decreases with D → 0 for flexible chains whereas it grows if the chains are sufficiently stiff. As the width of a closed gap D is decreased, in a three-dimensional gap L(D) gradually decreases for absolutely flexible chains whereas for semi-rigid chains it goes through a minimum at D = 2 and then grows again for D = 1. In two dimensions, in a closed strip the average chain length L(D) for both flexible and rigid macromolecules goes through a sharp minimum and then grows steeply in compliance with a predicted divergence for semi-rigid polymers as D → 0. We attribute the observed discrepancies of our numeric experiments with some recent analytic predictions to the ordering effect of container walls on the polymer solution when chain stiffness and excluded volume interactions are taken into account.     

Conformational properties and dynamics of molecular bottle‐brushes: A cellular‐automaton‐based simulation

Extensive computer simulation was performed using the bond‐fluctuation model and cellular‐automaton (CA)‐based simulation technique to probe the equilibrium structure and dynamical behavior of comb‐branched polymers in which the flexible side chains of a given length are placed regularly along the backbone and the number of branches increases linearly with total molecular weight. By applying very efficient CA algorithm – the “lattice molecular dynamics” (LMD) method – we have been able to study the properties of sufficiently large structures (up to 5880 monomeric units). Depending on the length of main and side chains as well as on interbranch spacing, we have calculated mean chain dimensions, local fractal dimensionalities, particle scattering functions, time autocorrelation functions, etc. The following main conclusions may be drawn from the results presented in our study: (i) The critical exponent, governing the mean size of the main chain, remains unchanged from its value known for a 3d self‐avoiding walk (SAW). On the other hand, two‐dimensional branched macromolecules with one‐sided branches are effectively in a collapsed state even under conditions of a good solvent, forming specific helical superstructures. (ii) Comparison of the simulated data with the predictions of the scaling model indicates that the latter is valid in describing the mean dimensions of the backbone as a function of side‐chain length and interbranch spacing. (iii) The excluded volume interaction between side chains dramatically slows down the relaxation of the backbone chain.     

Solution properties of exocellular polysaccharides of rhizobium leguminosarum. Static and dynamic light-scattering: Characterization in dilute solution

Three polysaccharides, Rhizobium leguminosarum 8002 EPS(I), Rhizobium trifolii TA1-EPS (II), Rhizobium leguminosarum 127K87 EPS (III), produced by bacteria of Rhizobium genus have been investigated by static and dynamic light scattering combined with chirooptical measurements. All three polymers have the same backbone but differ in the length of the side chains and in the content of minor substituents. An isothermal conformational transition coil → helix was observed with I and II by adding salt (NaCl). The molecular parameters of the polysaccharides in the ordered state were determined by light-scattering data. Increasing the ionic strength a shrinking of the helix was observed accompanied by a corresponding decrease in the radius of gyration. An extraordinary chain stiffness in terms of Kuhn segment lengths was found in both cases, similar to that already observed for other microbial polysaccharides. In the case of III no disorder → order transition was induced by the salt, and the scattering behavior corresponds to that of a rather flexible polymer with a characteristic ratio C_∞ = 24. The incapability of III to form a helical structure is attributed to the effect of the very long side chain. The analysis of the time correlation functions revealed typical flexible chain behavior for all three polysaccharides. This behavior for the two ordered polymers is in agreement with a recent theory by Maggs and is due to bending modes of the rods. ©1995 John Wiley & Sons, Inc.     

Stiffness and excluded volume effects on conformation and dynamics of polymers: A simulation study

This work investigates the effects of the excluded volume and especially those of the chain stiffness on the structural and dynamical properties of a model polymer chain.The theoretical framework is the same as in the recent works by Steinhauser et al.,where a Rouse approach is adopted.Our model differs in that our chains have a finite average bending angle.As in the works by Steinhauser et al.,Langevin dynamic simulations were performed without hydrodynamic interactions.Whereas this doesn’t impact the static properties we obtain,it also allows us to compare our results on dynamic properties to those predicted by Rouse theory,where hydrodynamic interactions are also neglected.Our results show that the structural properties are very sensitive to the chain stiffness,whereas the dynamic scaling laws remain the same as those by Rouse theory,with the prefactor depending on the persistence length.     

Gamma radiation-initiated polymerization of styrene at high pressure. II. Chain termination

The pressure dependence of the termination rate constant k_t for the free radical polymerization of monomers such as styrene is a function of polymer chain length, chain stiffness, and monomer viscosity, all of which influence the rate of segmental diffusion of an active radical chain end out of the coiled polymer chain to a position in which it can react with a proximate radical. Although k_t is not sensitive to changes in chain length, the large increase in molecular weight is responsible for a significant reduction in k_t at high pressures. For most of the common vinyl polymers, which exhibit some degree of chain stiffness, k_t is inversely proportional to a fractional power of the monomer viscosity because it depends in part on the resistance of chain segments to movement and in part on the influence of viscosity in controlling diffusion of the chain ends. The fractional exponent appears to increase with pressure and this is interpreted as evidence that the polymer chains become more flexible in a more viscous solvent. Because the fractional exponent is higher for more flexible chains, the value of the activation volume for chain termination is an indication of the degree of flexibility of the polymer chains, provided that the monomer is a good solvent for the polymer and that chain transfer is negligible.     

Length scale hierarchy in sol,gel, and coacervate phases of gelatin

Length scale hierarchy in gelatin sol, gel, and coacervate (induced by ethanol) phases, having same concentration of gelatin in aqueous medium (13% w/v), has been investigated through small angle neutron scattering and rheology measurements. The static structure factor profile, I(q) versus wave vector q, was found to be remarkably similar for all these samples. This data could be split into three distinct q‐regimes: the low‐q regime, I_ex(q) = I_ex(0)/(1+q²ζ²)² valid for q < 3R_g^?1; the intermediate q‐regime, I(q) = I(0)/(1+q²ξ²) for 3R_g^?1 < q < ξ^?1; and the asymptotic regime, I(q) = (c/q) exp(?R_c²q²/2) for q > ξ^?1. Consequently, three distinct length scales could be deduced from structure factor data: (a) inhomogeneity of size, ζ = 20 ± 1 nm for all the three phases; (b) average mesh size, ξ₀ = 2.6 ± 0.2 nm for sol and gel, and smaller mesh size, ξ_os = 1.2 ± 0.2 nm for coacervate; and (c) cross section of gelatin chains, R_c = 0.35 ± 0.04 nm. In addition, the structure factor data obtained from coacervating solution analyzed in the Guinier region, I(q) = exp(?q²R_g²/3), yielded value of typical radius of gyration of clusters, R_g ≈ 69 nm that indicated existence of triple‐helices of length, L ≈ 239 nm; (d) Frequency and temperature sweep measurements conducted on coacervate samples revealed two other length scales: (e) viscoelastic length, ξ_ve = 14 ± 2 nm and (f) correlation length at melting, ξ_T = 500 ± 70 nm. Thus, existence of six distinct length scales, (a–f), ranging from 1.2 to 500 nm has been established in the coacervate phase of gelatin–ethanol–water system. Results are discussed within the framework of Landau‐Ginzburg treatment of dynamically asymmetric systems (Prog Theor Phys 1977, 57, 826; Phys Rev A 1991, 44, R817; J Phys II (France) 1992, 2, 1631). © 2006 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 44: 1653–1667, 2006     

Characteristic lengths and the structure of salt free polyelectrolyte solutions. A small angle neutron scattering study

We have studied salt free semi dilute polyelectrolyte solutions by small angle neutron scattering. Specific labelling associated with an extrapolation method has allowed the separation of the form factor of a single polyelectrolyte chainS ₁(q) and the structure factorS ₂(q). Two lengths are deduced from these two factors: the persistence lengthb _t which characterizes the electrostatic interactions along the chain by a fitting ofS ₁(q) with calculation of the scattering function for a wormlike chain, and fromS ₂(q),q _m ^–1 which characterizes the interactions between chains. These two lengths vary in the same way with the concentration of polyions (b _t C _p ^–1/2 ,q _m ^–1 C _p ^–1/2 ) and a constant relation exists between them: only one length is then necessary to describe the structure of polyelectrolyte soltuion on this semidilute concentration range.Laboratoire Commun CEA-CNRS.     

Simulation of Chain Organization in Encapsulated Polymers

Summary : The dimensional and structural properties of polymers confined into a cavity are computed by the Monte Carlo method as a function of the chain stiffness. The reduction of the size ratio <R² > / < R> close to 2, distinctive of compact spheres, is observed at squeezing of chains into a capsule. The plots of the static structure factor S(q) computed for stiff chains show characteristic humps attributed to the toroidal structure. The orientation correlation function is found to be a very sensitive indicator of the globule – toroid transition in encapsulated chains. Evidence is presented that the toroidal morphology is formed in stiff polymers when the capsule radius approaches the chain persistence length (D ∼ P).     

Polymerization processes at conditions of moderately short kinetic chains by calorimetry

A method has been developed for the determination of the heat of the acts of chain growth q_g and of the summary heats of the acts of chain initiation and termination q^** for the radical polymerization of vinyl monomers at the conditions of moderately short kinetic chains, when the length of the kinetic chain v < 100, but the degree of polymerization is ¯P > 20. The analysis of the experimental data, obtained by investigating the kinetics of polymerization of vinyl monomers in solution by calorimetry has led to the conclusion, that at v > 100 an approximation of the long material chains can be used (when the contribution of q^** is negligibly small) with a permissible experimental error; however, in the region of moderately short chains a correction for q^** must be introduced, in order to obtain the correct kinetics of the process.Translated from Teoreticheskaya i Éksperimental'naya Khimiya, Vol. 24, No. 6, pp. 695–701, November–December, 1988.     

Comb copolymer cylindrical brushes containing semiflexible side chains: a Monte Carlo study

The off‐lattice Monte Carlo method is applied to investigate the equilibrium conformations of isolated comb copolymer cylindrical brushes in an athermal solution. The molecules considered consist of a flexible backbone, which is densely grafted with semiflexible side chains. The study focuses on the influence of the degree of intrinsic stiffness, λ_side, of the side chains on the conformational behavior of the molecules. It is demonstrated that with a fixed side chain length, M, the local length scale conformational fluctuations of the backbone increase as a function of λ_side. However, the persistence length, λ, of the cylindrical brush increases considerably with the side chain stiffness, indicating that the backbone becomes more extended at the large length scale. Moreover, as a function of λ_side, there is an increase in the ratio λ/D of the persistence length and the diameter, D, of the brush. This behavior is in good agreement with recent theoretical predictions and provides important new insight in the latest experimental observations.     

Conformation of comb‐like liquid crystal polymers in isotropic solution probed by small‐angle neutron scattering

The conformational characteristics of a comb‐like side‐chain liquid crystal polysiloxane (SCLCP), dissolved in deuterated chloroform, were evaluated by small‐angle neutron scattering (SANS) measurements over a wide q range. SANS studies were carried out on specimens with constant backbone length (DP = 198) and variable spacer length (n = 3, 5, and 11), and with constant spacer length (n = 5) and variable DP (45, 72, 127, and 198). The form factor P(q) at high q was analyzed using the wormlike chain model with finite cross‐sectional thickness (R_c) and taking into account the molecular weight polydispersity. The analysis generated values of persistence length in the range l_p = 28–32 Å, considerably larger than that of the unsubstituted polysiloxane chain (l_p = 5.8 Å), with contour lengths per monomer comparable to the fully‐extended polysiloxane backbone (l_m = 2.9 Å). This indicates a relatively rigid SCLCP chain due to the influence of the densely attached mesogenic groups. The SCLCP with n = 11 is more flexible (l_p = 28 Å) than those with n = 3 and n = 5 (l_p = 32 Å). The cross‐sectional thickness increases with spacer length, R_c ～ n^0.21±0.02 (3 ≤ n ≤ 11), and the contour length per monomer decreases with increasing spacer length, l_m ～ n^?0.35±0.01. © 2006 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 44: 2412–2424, 2006     

Rheological and structural properties of aqueous solutions of a hydrophobically modified polyelectrolyte and its unmodified analogue

Rheological methods and small angle neutron scattering (SANS) were used in the characterization of dilute and semidilute aqueous solutions, both with and without added salt, of anionic poly(vinyl alcohol) (PVA⁻) and its hydrophobically modified analogue (HM-PVA⁻). The rheological measurements showed that the concentration induced viscosification effect and elastic responses are considerably stronger for solutions of HM-PVA⁻ than in solutions of the unmodified polyelectrolyte. Over the considered polymer concentration domain, the solutions of PVA⁻ exhibit virtually Newtonian behavior, whereas strong shear thinning effects are observed in the HM-PVA⁻ solutions. The SANS results for HM-PVA⁻ solutions reveal a pronounced peak in the plot of scattering intensity versus scattering wavevector q at intermediate q values and the position of the maximum scales with polymer concentration as q_max∼c^0.28±0.02. This peak is suppressed in solutions of the unmodified polyelectrolyte and merely a shoulder in the scattering curve appears. Additionally, an “upturn” in the scattered intensity is observed at small q values and the magnitude of this effect depends on polymer concentration, hydrophobicity and salt addition. At large q values, the SANS results from HM-PVA⁻ solutions suggest morphological changes, from rod-like chains to a network of semiflexible chains, as the polyelectrolyte concentration increases.     

The glass transition in polymer melts

This paper presents some results of a Monte Carlo simulation for the glass transition in two- and three-dimensional polymer melts. The melt was simulated by the bond-fluctuation model on a d-dimensional cubic lattice which was combined with a two-level hamiltonian favouring long bonds in order to generate a competition between the energetic and topological constraints in the system. This competition prevents crystallization and makes the melt freeze in an amorphous structure as soon as the internal relaxation times match the observation time of the simulation set by the cooling rate. The freezing point of the melt, i.e the glass transition temperature T_g, thus depends upon the cooling rate and additionally upon the chain length of the polymers. The dependence of the glass transition temperature on the cooling rate was closely analysed in three and that on the chain length in both two and three dimensions, resulting in a non-linear relationship between T_g and the logarithm of the cooling rate and a linear relationship between T_g and the inverse chain length, respectively. In addition to this behaviour of the melt during the cooling process an example for the relaxational properties of the three-dimensional model is provided by a quantitative analysis of the incoherent intermediate scattering function in the framework of the idealized mode coupling theory.     

On the dynamics of semiflexible fibrin gels

The internal dynamics of semiflexible fibrin gels has been investigated by means of dynamic light scattering (DLS). Fibrin gels, grown at room temperature from fibrinogen solutions at different NaCl concentrations, exhibit rather different structural features. High salt concentrations produce “fine” gels characterized by thin fibers and small mesh sizes, while low salt concentrations give rise to “coarse” gels with thick fibers and much larger mesh sizes. The observed dynamics of these two kinds of gels are quite different as well. “Fine” gels behave as typical semiflexible polymer gels, in which the dynamic structure factor f(q,t) shows an initial monoexponential decay followed by a stretched exponential decay, f(q,t) ∼ exp ‐(Γ^qt)^β. Conversely, “coarse” gels exhibit a highly arrested dynamics, in which the dynamic structure factor does not relax to zero, but decays to a plateau whose value depends on the scattering wavevector q. Moreover, only the stretched exponential decay is observed at the fastest decay times, with the exponent β = 0.63±0.07 being independently of q. This behaviour can be interpreted as given by the contributions of the internal elastic modes of many different length scales.     

Miscibility behavior and single chain properties in polymer blends: a bond fluctuation model study

Computer simulation studies on the miscibility behavior and single chain properties in binary polymer blends are reviewed. We consider blends of various architectures in order to identify important architectural parameters on a coarse grained level and study their qualitative consequences for the miscibility behavior. The phase diagram, the relation between the exchange chemical potential and the composition, and the intermolecular pair correlation functions for symmetric blends of linear chains, blends of cyclic polymers, blends with an asymmetry in cohesive energies, blends with different chain lengths, blends with distinct monomer shapes, and blends with a stiffness disparity between the components are discussed. For strictly symmetric blends the Flory‐Huggins theory becomes quantitatively correct in the long chain length limit, when the χ parameter is identified via the intermolecular pair correlation function. For small chain lengths composition fluctuations are important. They manifest themselves in 3D Ising behavior at the critical point and an upward parabolic curvature of the χ parameter from small‐angle neutron scattering close to the critical point. The ratio between the mean field estimate and the true critical temperature decreases like √χ/(ρb³) for long chain lengths. The chain conformations in the minority phase of a symmetric blend shrink as to reduce the number of energeticaly unfavorable interactions. Scaling arguments, detailed self‐consistent field calculations and Monte Carlo simulations of chains with up to 512 effective segments agree that the conformational changes decrease around the critical point like 1/√N. Other mechanisms for a composition dependence of the single chain conformations in asymmetric blends are discussed. If the constituents of the blends have non‐additive monomer shapes, one has a large positive chain‐length‐independent entropic contribution to the χ parameter. In this case the blend phase separates upon heating at a lower critical solution temperature. Upon increasing the chain length the critical temperature approaches a finite value from above. For blends with a stiffness disparity an entropic contribution of the χ parameter of the order 10^–3 is measured with high accuracy. Also the enthalpic contribution increases, because a back folding of the stiffer component is suppressed and the stiffer chains possess more intermolecular contacts. Two aspects of the single chain dynamics in blends are discussed: (a) The dynamics of short non‐entangled chains in a binary blend are studied via dynamic Monte Carlo simulations. There is hardly any coupling between the chain dynamics and the thermodynamic state of the mixture. Above the critical temperatures both the translational diffusion and the relaxation of the chain conformations are independent of the temperature. (b) Irreversible reactions of a small fraction of reactive polymers at a strongly segregated interface in a symmetric binary polymer blend are investigated. End‐functionalized homopolymers of different species react at the interface instantaneously and irreversibly to form diblock copolymers. The initial reaction rate for small reactant concentrations is time dependent and larger than expected from theory. At later times there is a depletion of the reactive chains at the interface and the reaction is determined by the flux of the chains to the interface. Pertinent off‐lattice simulations and analytical theories are briefly discussed.