Single-chain Structure Factor Research Articles

Quantifying molecular deformation in polymer melts by a generalized Zimm plot approach

The Zimm plot has been widely used to characterize the molecular dimensions of polymers from small-angle scattering experiments, where the reciprocal intensity is analyzed as a function of the square of the magnitude of the scattering wavevector Q. This work explores the benefits of analyzing the reciprocal scattering intensity from deformed polymers, extending the original Zimm plot to anisotropic materials. In the small-angle limit, a tensorial extension of the Guinier law is found for the gyration tensor and the reciprocal single-chain structure factor. In the high-Q limit, application of the spherical harmonic expansion technique to the reciprocal structure factor permits direct model-independent analysis of spatially dependent molecular deformation of polymers. Additionally, the contributions from high-order spherical harmonics become insignificant in the reciprocal-intensity representation. The proposed generalized Zimm plot approach is demonstrated computationally with the affine deformation model and the Rouse model, and experimentally with small-angle neutron scattering measurements of deformed polystyrene melts.

Polymers: When S(Q,t) is not what you think! The role of dynamic RPA

Investigating a (polymeric) soft-matter system with several components often is aimed at the observation of the motions/mobilities of one of the components. The normal approach to this request in the context of quasielastic neutron scattering (e.g. NSE) is to put contrast on the targeted component, typically by leaving it hydrogenated and have the rest (surrounding, matrix) deuterated. In simple systems with only one molecular species as a homo-polymer melt with a few of the molecules contrasted (e.g. 10% h in 90% d) -provided the h-and d-molecular varieties behave sufficiently equalthis strategy yields a valid single chain (molecule) structure factor S(Q,t)/S(Q), even for a 50/50% mixture. However, if the component of interest and the matrix are of different kind, unexpected distortions of the observed S(Q,t) may occur. Interpretation of such results in terms of the single chain structure factor would then lead to erroneous conclusions. In this contribution the conditions, under which these distortions will occur, are discussed and how dynamic RPA may help to cope with them is explained. A practical method to apply this correction to polymer and similar problems is presented and an experimental verification is discussed [1, 12].

Quantitative examination of a fundamental assumption in small-angle neutron scattering studies of deformed polymer melts

It has long been established that the coherent scattering intensity of neutrons by isotropic mixtures of hydrogenous and deuterated polymers of matching molecular weights is, to the first approximation, proportional to the single-chain structure factor of the polymer chain. The validity of this fundamental relation for equilibrium, undeformed polymer melts is well supported by the extensive experimental and theoretical investigations over the past several decades. The generalization of this relation to the case of nonequilibrium, deformed polymer melts, however, is not a trivial one. Despite its widespread usage in small-angle neutron scattering (SANS) studies of deformed polymer melts, the assumed proportionality between coherent scattering intensity and single-chain structure factor has received very little experimental scrutiny. This work quantitatively examines this issue through spherical harmonic expansion analysis of the anisotropic SANS spectra of deformed polystyrene melts of different levels of deuterium labeling. It is shown that the classical assumption works extremely well over a wide range of scattering wavevectors, where the isotropic component of the SANS spectrum and the leading term of structural anisotropy vary by more than two orders of magnitude.

Conformation of Flexible and Semiflexible Chains Confined in Nanoposts Array of Various Geometries.

The conformation and distribution of a flexible and semiflexible chain confined in an array of nanoposts arranged in parallel way in a square-lattice projection of their cross-section was investigated using coarse-grained molecular dynamics simulations. The geometry of the nanopost array was varied at the constant post diameter dp and the ensuing modifications of the chain conformation were compared with the structural behavior of the chain in the series of nanopost arrays with the constant post separation Sp as well as with the constant distance between two adjacent post walls (passage width) wp. The free energy arguments based on an approximation of the array of nanopost to a composite of quasi-channels of diameter dc and quasi-slits of height wp provide semiqualitative explanations for the observed structural behavior of both chains. At constant post separation and passage width, the occupation number displays a monotonic decrease with the increasing geometry ratio dc/wp or volume fraction of posts, while a maximum is observed at constant post diameter. The latter finding is attributed to a relaxed conformation of the chains at small dc/wp ratio, which results from a combination of wide interstitial volumes and wide passage apertures. This maximum is approximately positioned at the same dc/wp value for both flexible and semiflexible chains. The chain expansion from a single interstitial volume into more interstitial volumes also starts at the same value of dc/wp ratio for both chains. The dependence of the axial chain extension on the dc/wp ratio turns out to be controlled by the diameter of the interstitial space and by the number of monomers in the individual interstitial volumes. If these two factors act in the same way on the axial extension of chain fragments in interstitial volumes the monotonic increase of the axial chain extension with the dc/wp in the nanopost arrays is observed. At constant wp, however, these two factors act in opposite way and the axial chain extension plotted against the dc/wp ratio exhibits a maximum. In the case of constant post diameter, the characteristic hump in the single chain structure factor whose position correlates with the post separation is found only in the structure factor of the flexible chain confined in the nanopost array of certain value of Sp. The structure factor of the flexible chain contains more information on the monomer organization and mutual correlations than the structure factor of the semiflexible chain. The stiffer chain confined in the nanopost array is composed of low number of statistical segments important for the presence of respective hierarchical regimes in the structure factor.

Structural Behavior of a Semiflexible Polymer Chain in an Array of Nanoposts.

The structural properties of a flexible and semiflexible circular chain confined in an array of parallel nanoposts with a square lattice cross-sectional projection were studied using coarse-grained molecular dynamics simulations. To address the effect of the circular topology, a comparison with linear analogs was also carried out. In the interpretation of the chain structural properties, the geometry of the post array is considered as a combination of a channel approximating the interstitial volume with the diameter dc and a slit approximating the passage aperture with the width wp. The number of interstitial volumes occupied by a chain monotonically increases with the decreasing ratio dc/wp regardless of the way the geometry of the post array is varied. However, depending on how the array geometry is modified, the chain span along the posts displays a monotonic (constant post separation) or a non-monotonic behavior (constant passage width) when plotted as a function of the post diameter. In the case of monotonic trend, the width of interstitial spaces increases with the increasing chain occupation number, while, in the case of non-monotonic trend, the width of interstitial spaces decreases with the increasing chain occupation number. In comparison with linear topology, for circular topology, the stiffness affects more significantly the relative chain extension along the posts and less significantly the occupation number. The geometrical parameters of the post arrays are stored in the single-chain structure factors. The characteristic humps are recognized in the structure factor which ensue from the local increase in the density of segments in the circular chains presented in an interstitial volume or from the correlation of parallel chain fragments separated by a row of posts. Although the orientation correlations provide qualitative information about the chain topology and the character of confinement within a single interstitial volume, information about the array periodicity is missing.

Fingerprinting Molecular Relaxation in Deformed Polymers

The flow and deformation of macromolecules is ubiquitous in nature and industry, and an understanding of this phenomenon at both macroscopic and microscopic length scales is of fundamental and practical importance. Here, we present the formulation of a general mathematical framework, which could be used to extract, from scattering experiments, the molecular relaxation of deformed polymers. By combining and modestly extending several key conceptual ingredients in the literature, we show how the anisotropic single-chain structure factor can be decomposed by spherical harmonics and experimentally reconstructed from its cross sections on the scattering planes. The resulting wave-number-dependent expansion coefficients constitute a characteristic fingerprint of the macromolecular deformation, permitting detailed examinations of polymer dynamics at the microscopic level. We apply this approach to survey a long-standing problem in polymer physics regarding the molecular relaxation in entangled polymers after a large step deformation. The classical tube theory of Doi and Edwards predicts a fast chain retraction process immediately after the deformation, followed by a slow orientation relaxation through the reptation mechanism. This chain retraction hypothesis, which is the keystone of the tube theory for macromolecular flow and deformation, is critically examined by analyzing the fine features of the two-dimensional anisotropic spectra from small-angle neutron scattering by entangled polystyrenes. We show that the unique scattering patterns associated with the chain retraction mechanism are not experimentally observed. This result calls for a fundamental revision of the current theoretical picture for nonlinear rheological behavior of entangled polymeric liquids.Received 23 March 2017DOI:https://doi.org/10.1103/PhysRevX.7.031003Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasPolymer conformation changesPolymer entanglementRheologyPhysical SystemsPolymer meltsTechniquesRheology techniquesSmall angle neutron scatteringTube theoriesPolymers & Soft MatterNonlinear DynamicsFluid Dynamics

Description of poly(ethylenepropylene) confined in nanopores by a modified Rouse model.

A recent model for unentangled polymer chains in confinement [M. Dolgushev and M. Krutyeva, Macromol. Theory Simul. 21, 565 (2012)] is scrutinized by small-angle neutron scattering (SANS) with respect to its static prediction, the single-chain structure factor. We find a remarkable agreement although the model simplifies the effect of the confinement to a harmonic potential. The effective confinement size from fits of SANS data with the model agrees well with the actual pore size. Starting from this result we discuss the possibility of an experiment on the dynamic structure factor predicted by the model. It turns out that such an experiment would need a large ratio polymer dimension/pore size which is difficult but not impossible to achieve.

Monte Carlo studies of two-dimensional polymer\u2013solvent systems

The static properties of two-dimensional athermal polymer solutions were studied by performing Monte Carlo lattice simulations using the cooperative motion algorithm (CMA) and taking into account the presence of explicit solvent molecules. The simulations were performed for a wide range of polymer chain lengths N (16–1024) and concentrations φ (0.0156–1). The results obtained for short chains (N < 256) were in good agreement with those given by previous simulations. For the longest chains (512 or 1024 beads), some unexpected behavior was observed in the dilute and semidilute regimes. A pronounced change in the concentration dependence of chain size and shape was observed below a certain critical concentration (0.6 for the longest chains under consideration, consisting of 1024 beads). Longer chains became more extended below this concentration. The behavior of the single-chain structure factor confirmed these changes in the fractal dimension of the chain as a function of the concentration. The observed phenomena are related to the excluded volume of solvent molecules, which causes the chain statistics to be modified in the vicinity of other chains; this effect is important in strictly 2D systems.Graphical abstractExtended long chains at moderate density with solvent molecules inside coils.

The long-chain dynamics in a model homopolymer blend under strong flow: small-angle neutron scattering and theory

We use small-angle neutron scattering (SANS) measurements to provide a detailed picture of the non-linear dynamics of the long chains in a model polystyrene blend. By a weighted subtraction of SANS measurements from two otherwise identical blends with different deuteration fractions, we isolate the single-chain form factor of the long-chain component of a model blend flowing through a 4 : 1 contraction–expansion flow. Complementary flow-birefringence also provides a measure of chain deformation on finer length-scales. In addition, higher flow Weissenberg numbers than in previous studies on monodisperse melts were achieved, leading to greater anisotropy in the measured single-chain structure factor. The short residence time inside the slit means that the chains are still oriented in the flow direction as they enter the contraction exit, leading to a rapid reversing flow. We compare these data to a simple generalisation of a non-linear tube model. Our model predictions are entirely ab initio, with all model parameters being determined from independent equilibrium measurements. The model shows very good agreement with the experimental data across the full range of length-scales for the contraction entrance and subsequent relaxation within the slit. However, there is conspicuous disagreement between theory and experiments at the contraction exit, in both the SANS and birefringence predictions, which we attribute to the reversing flow that occurs in this region.

End-to-end distribution for a wormlike chain in arbitrary dimensions

We construct an efficient methodology for calculating wormlike chain statistics in arbitrary D dimensions over all chain rigidities, from fully rigid to completely flexible. The structure of our exact analytical solution for the end-to-end distribution function for a wormlike chain in arbitrary D dimensions in Fourier-Laplace space (i.e., Fourier-transformed end position and Laplace-transformed chain length) adopts the form of an infinite continued fraction, which is advantageous for its compact structure and stability for numerical implementation. We then proceed to present a step-by-step methodology for performing the Fourier-Laplace inversion in order to make full use of our results in general applications. Asymptotic methods for evaluating the Laplace inversion (power-law expansion and Rayleigh-Schrödinger perturbation theory) are employed in order to improve the accuracy of the numerical inversions of the end-to-end distribution function in real space. We adapt our results to the evaluation of the single-chain structure factor, rendering simple, closed-form expressions that facilitate comparison with scattering experiments. Using our techniques, the accuracy of the end-to-end distribution function is enhanced up to the limit of the machine precision. We demonstrate the utility of our methodology with realizations of the chain statistics, giving a general methodology that can be applied to a wide range of biophysical problems.

Statistical mechanics of worm-like polymers from a new generating function.

We present a mathematical approach to the worm-like chain model of semiflexible polymers. Our method is built on a novel generating function from which all the properties of the model can be derived. Moreover, this approach satisfies the local inextensibility constraint exactly. In this paper, we focus on the lowest order contribution to the generating function and derive explicit analytical expressions for the characteristic function, polymer propagator, single chain structure factor, and mean square end-to-end distance. These analytical expressions are valid for polymers with any degree of stiffness and contour length. We find that our calculations are able to capture the fully flexible and infinitely stiff limits of the aforementioned quantities exactly while providing a smooth and approximate crossover behavior for intermediate values of the stiffness of the polymer backbone. In addition, our results are in very good quantitative agreement with the exact and approximate results of five other treatments of semiflexible polymers.

Diffusion of hard sphere fluids in disordered media: a molecular dynamics simulation study.

Molecular dynamic simulations are reported for the static and dynamic properties of hard sphere fluids in matrices (or media) composed of quenched hard spheres. The effect of fluid and matrix density, matrix structure, and fluid to matrix sphere size ratio on the static and dynamic properties is studied using discontinuous molecular dynamics. The matrix density has a stronger effect on the self-diffusion coefficient than the fluid density, especially at high matrix densities where the geometric constraints due to the quenched spheres are significant. When the ratio of the size of the fluid spheres to that of the matrix spheres is equal to or greater than one, the diffusion increases as the fluid density is increased, at constant total volume fraction. This trend is however reversed if the ratio is smaller than one. Different methods of generating the matrix have a very strong effect on the dynamic properties even though the static correlations are similar. An analysis of the single-chain structure factor of the particle trajectories shows a change in the particle diffusive behavior at different time scales, suggestive of a hopping mechanism, although normal diffusion is recovered at long times. At high matrix densities, there is considerable heterogeneity in the diffusion of the fluid particles. The simulations demonstrate that the correlations in the matrix play a significant role on the diffusion of fluid spheres. For example, the diffusion constant in matrices constructed by different methods can be an order of magnitude different even though the pair correlation functions are almost identical.

Neutron-Mapping Polymer Flow: Scattering, Flow Visualization, and Molecular Theory

Flows of complex fluids need to be understood at both macroscopic and molecular scales, because it is the macroscopic response that controls the fluid behavior, but the molecular scale that ultimately gives rise to rheological and solid-state properties. Here the flow field of an entangled polymer melt through an extended contraction, typical of many polymer processes, is imaged optically and by small-angle neutron scattering. The dual-probe technique samples both the macroscopic stress field in the flow and the microscopic configuration of the polymer molecules at selected points. The results are compared with a recent "tube model" molecular theory of entangled melt flow that is able to calculate both the stress and the single-chain structure factor from first principles. The combined action of the three fundamental entangled processes of reptation, contour length fluctuation, and convective constraint release is essential to account quantitatively for the rich rheological behavior. The multiscale approach unearths a new feature: Orientation at the length scale of the entire chain decays considerably more slowly than at the smaller entanglement length.

Microscopic theory of linear, entangled polymer chains under rapid deformation including chain stretch and convective constraint release

A refined version of the Doi and Edwards tube model for entangled polymer liquids is presented. The model is intended to cover linear chains in the full range of deformation rates from linear to strongly nonlinear flows. The effects of reptation, chain stretch, and convective constraint release are derived from a microscopic stochastic partial differential equation that describes the dynamics of the chain contour down to the length scale of the tube diameter. Contour length fluctuations are also included in an approximate manner. Predictions of mechanical stresses as well as the single chain structure factor under flow are shown. A comparison with experimental data is made in which all model parameters are fixed at universal values or are obtained from linear oscillatory shear measurements. With no parameter modification the model produces good agreement over a wide range of rheological data for entangled polymer solutions, including both nonlinear shear and extension.

Silica-filled elastomers: polymer chain and filler characterization by a SANS-SAXS approach

A study of composites based upon commercially available silica fillers and networks of blends of protonated and deuterated anionically prepared polyisoprene is presented. The extraction of the single chain structure factor for SANS in the polymeric soft phase in isotropic and deformed state has been performed for the first time. The quasi three-component system could not be compositionally matched due to the internal structures of the activated fillers. For this, a parallel SAXS investigation provided the neccesary information on the filler structure which was lacking in the SANS analysis. Whereas mechanically clear reinforcement at low strains and filler-networking can be observed, the microscopic characterization of the chain deformation in the framework of the network tube model agrees with the estimates for hydrodynamic reinforcement of fractal fillers.

A Monte Carlo Study of Mixtures of Charged and Neutral Polymers in Solution

Mixtures of neutral and charged polymers are studied by Monte Carlo simulation techniques. The charges on the chains are assumed to interact with a Debye–Hückel potential. The monomer density and the screening are kept constant, while the proportion between charged and neutral polymer is changed. The single-chain structure factor of each species is analyzed, and drastic changes are observed with the variation of the fraction of charged chains. Changes entail modifications of the radius of gyration, which suggest the existence of attractive interactions. Results agree with experimental evidence of incompatibility in mixtures at low polyelectrolyte concentrations.

A Monte Carlo Study of Mixtures of Charged and Neutral Polymers in Solution

Mixtures of neutral and charged polymers are studied by Monte Carlo simulation techniques. The charges on the chains are assumed to interact with a Debye–Hückel potential. The monomer density and the screening are kept constant, while the proportion between charged and neutral polymer is changed. The single-chain structure factor of each species is analyzed, and drastic changes are observed with the variation of the fraction of charged chains. Changes entail modifications of the radius of gyration, which suggest the existence of attractive interactions. Results agree with experimental evidence of incompatibility in mixtures at low polyelectrolyte concentrations.

Self-consistent integral equation theory for semiflexible chain polyelectrolyte solutions

A self-consistent integral equation theory is presented for the conformational behavior and static structure of polyelectrolyte solutions composed of semiflexible chain molecules. The free energy of the solution is expressed as the sum of an elastic, a Coulombic, and a medium-induced contribution, and is minimized with respect to the single chain structure factor. In the flexible limit, the theory is accurate for the chain conformations and static structure factor when compared to Monte Carlo simulations of flexible chains. For semiflexible chains near the flexible limit, the chain conformations are sensitive to the bare persistence length in dilute solutions but not in the semidilute regime. In contrast, the effect of the bare persistence length on the static structure factor is not as significant. The theory predicts that charged flexible chains exhibit more pronounced liquidlike order in solution than their rigid-rod counterparts, and the length scale of these liquidlike correlations is also shorter in flexible chains.

Monte Carlo simulations of hydrophobic polyelectrolytes: Evidence of complex configurational transitions

Off-lattice Monte Carlo simulations, including electrostatic and Lennard-Jones potentials, are performed to investigate at the limit of counterion condensation the dilute conformations of quenched polymer chains in poor solvent conditions. The Monte Carlo (MC) search procedure is improved to achieve dense conformations in the limit of both strong attractive and repulsive interaction potentials. Configurational properties such as the radius of gyration, and single chain structure factors are calculated as a function of attractive monomer–monomer interactions, ionic concentration, and monomer number. It is observed that hydrophobic polyelectrolytes exhibit a large variety of conformations compared to flexible or semiflexible polyelectrolytes without hydrophobic groups. MC results demonstrate that there is a range of electrolyte concentration and hydrophobicity for which polymers exhibit exotic but stable conformations, namely the pearl necklace and the cigar-shape conformation. It is shown that by gradually increasing the monomer hydrophobicity of a strong polyelectrolyte, it undergoes a cascade of transitions from an extended structure to a pearl necklace, a pearl necklace to a cylinder, and a cylinder to a coil, successively. Smooth transitions are observed when competition between hydrophillic and hydrophobic interactions is important, whereas sharp transitions are obtained in the other cases. A good agreement is observed with the theoretical description of weakly charged polyelectrolytes in poor solvent conditions. To have an insight of the spatial organization of the chain monomers, scattering functions are calculated for each characteristic conformation. It is shown that large changes in the scattering curve profiles can be expected when the hydrophobicity of the backbone has a definite influence on the chain dimensions.

Inter-Chain Structure Factors of Flexible Polymers in Solutions: A Monte Carlo Investigation

Off-lattice Monte Carlo simulations of both the single chain structure factor h(q) and the inter-chain structure factor HD(q) of flexible polymers in solutions are presented over a wide range of both wavenumber q and concentration c from the dilute to the concentrated regime, for chain lengths up to N = 256. The single chain properties $\{$gyration radius 〈Rg2〉, $h(q)\}$ are in reasonable agreement with the expected theoretical behavior, showing a crossover from swollen chains $\{\langle R_{\rm g}^2\rangle \propto N^{2\nu} ,~ h(q) \propto q^{-1/\nu}\}$ to Gaussian chains, and the data comply with a scaling description, with a correlation length ξ∝c-ν/(3ν-1). However, the inter-chain structure factor HD(q) disagrees with the corresponding predictions, we find a behaviour HD(q)∝q-3 only in an intermediate range but this is accidental: rather it is found that HD(q) smoothly bends over from its saturation value at small q to a behavior close to q-4 at q≈1/l, l being the length of effective bonds. This failure is traced back to the condition that the law HD(q)∝q-3 should only be observed for ξ-1≪q ≪l-1, a condition reached neither in the simulation nor in experiments. We also compare our results for HD(q) with the random phase approximation and find strong deviations.