Segment Density Distribution Research Articles

A Gram-Charlier Analysis of Scattering to Describe Nonideal Polymer Conformations.

Theories of interpreting polymer physics and rheology at the molecular level from experiments, including small-angle scattering, typically rely on the assumption that polymer chains possess a Gaussian configuration distribution. This assumption frequently fails to describe features of real polymer molecules both at equilibrium (when polymers have nonlinear topology or heterogeneous chemistry) and out of equilibrium (when they are subjected to nonlinear deformations). To better describe non-Gaussian polymer conformation distributions, we propose a moments analysis based on the Gram-Charlier expansion as a natural framework for describing structure and scattering from non-Gaussian polymers. The expansion describes the conformation distribution in terms of cumulants (equivalent to moments of the distribution) of the underlying segment density distribution function, providing low-dimensional descriptors that can be inferred directly from measured scattering in a way that is agnostic to a polymer's topology, chemistry, or state of deformation. We use this framework to show that cumulants can be used to "fingerprint" non-Gaussian conformation distributions of polymers either at equilibrium (applied to sequence-defined heteropolymers) or out of equilibrium (applied to polymers experiencing nonlinear deformation due to flow). We anticipate that this new analysis method will provide a general framework for examining nonideal polymer configurations and the properties that arise from them.

Liquid-Crystal Ordering and Microphase Separation in the Lamellar Phase of Rod-Coil-Rod Triblock Copolymers. Molecular Theory and Computer Simulations.

A molecular model of the orientationally ordered lamellar phase exhibited by asymmetric rod-coil-rod triblock copolymers has been developed using the density-functional approach and generalizing the molecular-statistical theory of rod-coil diblock copolymers. An approximate expression for the free energy of the lamellar phase has been obtained in terms of the direct correlation functions of the system, the Flory-Huggins parameter and the Maier-Saupe orientational interaction potential between rods. A detailed derivation of several rod-rod and rod-coil density-density correlation functions required to evaluate the free energy is presented. The orientational and translational order parameters of rod and coil segments depending on the temperature and triblock asymmetry have been calculated numerically by direct minimization of the free energy. Different structure and ordering of the lamellar phase at high and low values of the triblock asymmetry is revealed and analyzed in detail. Asymmetric rod-coil-rod triblock copolymers have been simulated using the method of dissipative particle dynamics in the broad range of the Flory-Huggins parameter and for several values of the triblock asymmetry. It has been found that the lamellar phase appears to be the most stable one at strong segregation. The density distribution of the coil segments and the segments of the two different rods have been determined for different values of the segregation strength. The simulations confirm the existence of a weakly ordered lamellar phase predicted by the density-functional theory, in which the short rods separate from the long ones and are characterized by weak positional ordering.

Validation and Optimisation of Common Clear Sky Models with a use case for North-East Bulgaria

One of the most significant error contributors to preliminary design tools for Photovoltaic power systems is related to the simple parametric Clear Sky models. Therefore, this paper focuses on providing a methodology and a more sophisticated open-source tool for 3 commonly used Clear Sky models. This includes all relevant steps involved in the process - from filtering the raw meteorological data, identification of Clear Sky regions, data redistribution to genetic optimization of selected model parameter, etc.use case is built upon a multiyear dataset obtained from TU Varna meteorological station between 2012-2016. A significantly higher density distribution of Clear sky segments was identified during the summer through the Clear Sky Identification algorithm. To avoid the risk of overfitting the models to purely summer months and poor model fits in winter months, which was found to be the case with the legacy model, the underrepresented clear sky regions (based on θ) were replicated until uniform distribution is attained. Subsequently, a genetic optimization was applied to selected parameters in the Clear Sky algorithms and the updated models showed a significant improvement in low winter months (θ) and even overall performance boost RMSE / MAE /R2. Furthermore, such validations and optimizations are recommended prior to any design or real-time PV-system analysis for the specific location.

Tethered poly(2-isopropyl-2-oxazoline) chains: temperature effects on layer structure and interactions probed by AFM experiments and modeling.

Thermoresponsive polymer layers on silica surfaces have been obtained by utilizing electrostatically driven adsorption of a cationic-nonionic diblock copolymer. The cationic block provides strong anchoring to the surface for the nonionic block of poly(2-isopropyl-2-oxazoline), referred to as PIPOZ. The PIPOZ chain interacts favorably with water at low temperatures, but above 46 °C aqueous solutions of PIPOZ phase separate as water becomes a poor solvent for the polymer. We explore how a change in solvent condition affects interactions between such adsorbed layers and report temperature effects on both normal forces and friction forces. To gain further insight, we utilize self-consistent lattice mean-field theory to follow how changes in temperature affect the polymer segment density distributions and to calculate surface force curves. We find that with worsening of the solvent condition an attraction develops between the adsorbed PIPOZ layers, and this observation is in good agreement with predictions of the mean-field theory. The modeling also demonstrates that the segment density profile and the degree of chain interpenetration under a given load between two PIPOZ-coated surfaces rise significantly with increasing temperature.

Electrokinetics as an alternative to neutron reflectivity for evaluation of segment density distribution in PEO brushes.

Unravelling details of charge, structure and molecular interactions of functional polymer coatings defines an important analytical challenge that requires the extension of current methodologies. In this article we demonstrate how streaming current measurements interpreted with combined self consistent field (SCF) and soft surface electrokinetic theories allow the evaluation of the segment distribution within poly(ethylene oxide) (PEO) brushes beyond the resolution limits of neutron reflectivity technique.

Collapse from the top: brushes of poly(N-isopropylacrylamide) in co-nonsolvent mixtures

Using a combination of ellipsometry and friction force microscopy, we study the reversible swelling, collapse and variation in friction properties of covalently bound poly(N-isopropylacrylamide) (PNIPAM) layers on silicon with different grafting densities in response to exposure to good solvents and co-nonsolvent mixtures. Changes in the thickness and segment density distribution of grafted films are investigated by in situ ellipsometry. Based on quantitative modelling of the ellipsometry spectra, we postulate a structural model, which assumes that collapse takes place in the contacting layer between the brush and the co-nonsolvent and the top-collapsed brushes remain hydrated in the film interior. Using the structural model derived from ellipsometry spectra, we analyse the AFM based friction force microscopy data, which were obtained by silica colloidal probes. Results show a large increase of the friction coefficient of PNIPAM grafts when the grafts swollen by water are brought in contact with co-nonsolvents. For instance, the value of the friction coefficient for a medium density brush in water is four times lower than the value observed in a water-methanol (50% v/v) mixture. This increase of friction is accompanied by an increase in adherence between the PNIPAM chains and the silica colloidal probes, and is a result of chain collapse in the graft when contacted by a co-nonsolvent mixture in agreement with the model postulated on the basis of ellipsometric characterisation. The kinetic behaviour of the collapse is assessed by measuring the temporal variation of friction in situ as a function of elapsed time following contact with the co-nonsolvent as a function of graft density. In conclusion, the effect of co-nonsolvency influenced both the thickness of the PNIPAM brushes and the tribological behavior of the brush surfaces.

Theoretical Investigation on the Steric Interaction in Colloidal Deposition

A theoretical investigation was conducted upon the steric interaction between a spherical particle and a flat plate. The effects of curvature on both the segment density distribution of polymer and on the steric interaction energy between a particle and a plate were examined. It was found that the conventional approach using Derjaguin approximation may overestimate the interaction energy, especially for very small particles. Based on the results obtained from applying the Flory-Krigbaum theory with the accurate geometry change during the interaction, simple approximate expressions of exponential form were proposed for both the osmotic (mixing) and elastic contributions to the steric interaction energy for a sphere-plate interaction. The proposed model was validated against a set of experimental results from a reported study employing atomic force microscopy. Also investigated was the steric interaction between a spherical particle coated with polymer and an uncoated flat surface. It was found that the steric interaction energy from the osmotic contribution is significantly higher than that from the elastic contribution and that steric interaction is considerably weaker when the polymer exists on only one of the interacting surfaces.

Microphase Separation and Phase Diagram of Concentrated Diblock Copolyelectrolyte Solutions Studied by Self-Consistent Field Theory Calculations in Two-Dimensional Space

The self-consistent field theory (SCFT) is applied to the microphase separation of concentrated solutions of weakly charged polyelectrolytes. The generalized Poisson–Boltzmann equation describing the electrostatic interactions at the mean-field level is numerically solved by a full multigrid algorithm, which enables one to solve the SCFT equations of polyelectrolyte systems in real space as efficient as neutral polymer systems. To demonstrate the power of the real-space numerical scheme, we consider a diblock copolyelectrolyte consisting of a charged block and a neutral block in two-dimensional space. The phase diagram in the Flory–Huggins interaction parameter—the composition space is constructed by numerical calculations. The density distribution of polymer segments, the counterions, and the net charge of the ordered structures, namely the lamellar phase and the hexagonal phase, are intensively examined. The effects of the interaction parameter and the degree of ionization are examined carefully. The numerical scheme can be easily extended to 3D calculations, various chain architectures, various charge distribution models, and other statistics chain models such as the worm-like chain model without losing any computational efficiency.

Interrelations between charging, structure and electrokinetics of nanometric polyelectrolyte films

Streaming current, surface conductivity and swelling data of poly(acrylic acid) (PAA) and poly(ethylene imine) (PEI) thin films are analyzed on the basis of the theory for diffuse soft interfaces (J.F.L. Duval, R. Zimmermann, A.L. Cordeiro, N. Rein, C. Werner, Langmuir 25 (2009) 10691). Focus is put on ways to unravel the electroosmotic and migration contributions of the measured surface conductivity, which is crucial for appropriate electrokinetic analysis of films carrying high densities of dissociable groups. Results demonstrate that the osmotically-driven swelling of the PAA films with increasing pH is accompanied by an increase in diffuseness for the interphasial polymer segment density distribution. This heterogeneity is particularly marked at low ionic strength with a non-monotonous dependence of the streaming current on pH and the presence of a maximum at pH∼6.5. The analysis of the PEI films evidences heterogeneous swelling with lowering pH, i.e. upon protonation of the amine groups. The characteristic decay length in the interphasial PEI segment density distribution is found to be nearly independent of the pH, which is in line with the moderate swelling determined by ellipsometry. A critical discussion is given on the strengths and limitations of electrokinetics/surface conductivity for quantifying the coupled electrohydrodynamic and structural properties of moderately to highly swollen polyelectrolyte thin films.

Penetration of Polymer Brushes by Chemical Nonidentical Free Polymers

We report on the reaction of surface-attached polymer brushes with polymer molecules, which are present in a contacting solution. More specifically, we synthesize copolymer brushes containing N-hydroxysuccinimide active ester groups and study their reaction with poly(ethylene glycol) molecules, which carry an amino group at one end. Qualitatively, highly “pegylated” surfaces are obtained with an equivalent thickness of the PEG layers of up to 200 nm. Yield and kinetics of the grafting reaction are studied as a function of size and concentration of the incoming molecules, brush graft density, and brush molecular weight. It is observed that the completion of the grafting reaction is surprisingly fast, the molecular weight of the incoming molecules has a strong, while graft density and molecular weight of the brush have only a weak influence on the extent of functionalization. We develop a model for the grafting reaction, where due to limited penetration of the brush by the incoming chains the reaction occurs only at the outer periphery of the surface-attached layer. Reactive groups present in the inner parts of the brush do not take part in the attachment reaction. In the outer parts of the brush, however, partial penetration is possible so that the attachment reaction is fast and essentially only governed by the details of the shape of the segment density distribution and the size of the incoming chains.

A free-space density functional theory for polymer adsorption: Influence of packing effect on conformations of polymer

We propose a free-space density functional theory for polymer adsorption. The derivation within the framework of density functional theory leads to the splitting of the intrinsic free energy into an ideal-gas term and a residual term responsible for the intrinsic energy and the nonbonded interactions between monomers, respectively. A more reasonable treatment is adopted for the residual free energy to count for the monomer-monomer correlation underestimated by the local density approximation. An approach using propagators is proposed to calculate the single-chain partition function and the segment-density distributions, the three adsorption conformations as trains, loops, and tails are further described by propagators. Dirac's bra-ket notation used makes the derivation simpler and provides clearer physical meanings. The theoretical calculations for the adsorption of hard-sphere chains onto a nonadsorbing and an adsorbing hard wall show that the structure of the adsorption layer is strongly affected by the packing effect which has been underestimated by the previous lattice adsorption theory.

Steered molecular dynamics simulation of the detaching process of two parallel surfaces glued together by a single polyethylene chain

The detaching process of two parallel surfaces glued together by a single polyethylene chain in vacuo was investigated with a steered molecular dynamics method. Various statistical properties were analyzed in detail, including the mean-square end-to-end distance; parallel and perpendicular mean-square radii of gyration; shape factor; segment density distribution; average percentages of the microstructure of the chain of the tail, train, bridge, and loop; average surface adsorption energy; average total energy; and average pulling force (〈f〉). All these properties depended strongly on the pulling velocity (v). There existed a peak in the curve of 〈f〉 as a function of the detaching distance. Further, the relation between the maximum value of 〈f〉 and v showed three distinctive regions: a region of weak dependence at v < 10−2 Å/ps, a region of strong dependence at 10−2 Å/ps < v < 6.50 Å/ps, and a region of weak dependence at v > 6.50 Å/ps. These investigations may provide some insight into the microcosmic principle of the failure process of polymeric adhesives. © 2009 Wiley Periodicals, Inc. J Appl Polym Sci, 2010

Thickness dependence of elastic modulus of polycarbonate interphase

The oligomer of bis-phenol A (oligo-PC) with M w = 1300 and bis-phenol A polycar-bonate (PC) with M w = 20 000 were deposited onto E-glass surface using SiCl4 as the grafting and cross-linking agent. Thickness of the deposited layers was varied from 30 to 106 nm and the layers were investigated as prepared and after thermal annealing at 245°C for 10 min in the air. Vibrational piezoelectric resonator technique and the speed of Rayleigh wave measurement were used to determine elastic moduli of the ultra thin layers deposited on flat E-glass substrate as a function of their thickness. In all cases, increase of the Young modulus of the interphase, E i, with decreasing layer thickness, t i, was observed. At a given thickness, the E i of PC layer was significantly lower than that for the oligo-PC layer. Thermal annealing of the deposited PC layer resulted in a significant increase of its E i compared to the as received layer. No significant change was observed for oligo-PC interphases. Increase of the shear strength of the interface, τ a, with reducing interphase thickness, t i, was observed. The observed increase of E i with the decreasing t i was ascribed to the reduction of the molecular mobility of chains near solid surface compared to their mobility in the bulk. Most probably, the observed increase of E i after thermal annealing of PC was caused by rearrangement of both segment density distribution in individual PC coils near the solid surface and cooperative rearrangements of multiple PC chains. Since the oligomers attached to the surface attained presumably more regular extended conformations with lower conformation entropy compared to the PC random coils, the effect of thermal annealing was negligible. In agreement with theoretical predictions, increase of E i at the same extent of interfacial interactions resulted in the observed increase of the τ a measured using the single embedded fiber test.

Conformational structures and thermodynamic properties of compact polymer chains confined between two parallel plane boundaries

AbstractIn this paper, the authors investigated the adsorption phenomenon of compact chains confined between two parallel plane boundaries using a pruned‐enriched Rosenbluth method. The authors considered three cases with different adsorption energies of ε = 0, −1, and −3 (in units of kBT) for the confined compact chains of different chain lengths N, respectively. Several parameters were employed to describe the size and shape of compact chain, and some special behaviors in the conformational structures were investigated for the first time. For example, the size and shape of confined compact chains undergo distinct changes in the adsorption cases of ε = −1 and −3, and pass through the maximum values at the characteristic distances Dc. The authors found that this characteristic distance Dc could be scaled as Dc∼ (N + 1)ν (ν = 0.56 ± 0.01) in the case of ε = −3. In addition, the microstructures of chains were investigated, and several significant results were obtained by analyzing the segment density distribution and the mean fractions of segment in tails, trains, bridges, and loops structures. On the other hand, the thermodynamic properties were also investigated for the confined compact chains, such as average energy per bond, Helmholtz free energy per bond, and elastic force per bond. Results show that elastic forces f have different behaviors in three cases, indicating that it is not necessary to exert an external force on the boundaries in the nonadsorption case. At the same time, the average contact energy of compact chain obviously changes when the distance between the two parallel boundaries D increases, which is similar to those of the size and shape parameters. The authors also conclude that these thermodynamic properties of compact chains depend strongly on not only the adsorption energies but also the chain lengths and the confined condition. In addition, several results of the conformational and thermodynamic parameters, such as the segment density distribution and free energy, were compared with the results from the self‐consistent field theory. These investigations may help us to deepen the knowledge about the adsorption phenomenon of confined compact chains. © 2006 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 44: 2888–2901, 2006

Shape of Fluid Vesicles Anchored by Rigid Rod

A method combined the self-consistent field theory (SCFT) for the rigid rod with the Helfrich curvature elasticity theory for the vesicle has been developed for studying the shape of vesicles anchored by rigid rod. Both the deformation of the vesicle and the density distribution of rod segments can be obtained. Because of the vesicle's impenetrability for the rod segments and the decrease of the available space for the rod orientational configurations, the anchored rod segments exert the inhomogeneous entropic pressure on the vesicle and induce the change of vesicle shape. The interaction between the rod segments and the vesicle membrane exerts an extra tension to the membrane. Thus the interaction between the vesicle membrane and the rod segments, the rod length, and the bending rigidity of vesicle are investigated as the important factors to the shape transformation of the vesicle and the density distribution of rod segments. This method can be extended to more complicated and real biological systems, such as polymers with different topological architectures/vesicle, multiple chains/vesicle, protein inclusions, etc.

Hydrodynamics and electrokinetics of spherical liposomes with coatings of terminally anchored poly(ethylene glycol): Numerically exact electrokinetics with self-consistent mean-field polymer

A detailed theoretical model is presented to interpret electrokinetic experiments performed on colloids with uncharged polymer layers. The methodology removes many of the degrees of freedom that otherwise have to be accounted for by adopting multiple empirical fitting parameters. Furthermore, the level of detail provides a firm basis for future studies examining liposome surface chemistry and charge, surface-charge mobility, and the dynamics of adsorbed polymer on fluidlike membranes. The model predictions are compared with experimental measurements of the electrophoretic mobility of stealth liposomes with molecular weights of terminally anchored poly(ethylene glycol) (PEG) in the range 0.35-10 kg mol(-1) [J.A. Cohen and V.A. Khorosheva, Colloids Surf. A 195, 113 (2001)]. The experimental data are interpreted by drawing upon self-consistent mean-field calculations of the polymer segment density distributions and numerically exact solutions of the governing transport equations [R.J. Hill, D.A. Saville, and W.B. Russel, J. Colloid Interface Sci. 258, 56 (2003)]. The approach leads to excellent agreement between theory and experiment with one adjustable parameter--the hydrodynamic size (Stokes radius) a(s) approximately equal to 0.175 A of the statistical PEG segments with (Kuhn) length l=7.1 A . The remarkably small Stokes radius is demonstrated to be consistent with other applications of the well-known Debye-Brinkman model and, consequently, this work reveals important limitations of the mean-field hydrodynamic model. Despite such limitations, the "full" electrokinetic model is robust in its predictive capacity. The molecular weights of the terminally anchored PEG span the range where the coatings undergo a transition from mushroomlike to brushlike conformations, and the hydrodynamic size and electrophoretic mobility of the liposomes are demonstrated to be sensitive to the PEG chain length and the effects of double-layer polarization.

Bottle-brush polymers as an intermediate between star and cylindrical polymers.

We present a theoretical study of a single bottle-brush molecule, which consists of multiarmed polymer stars grafted densely onto a stiff backbone. Mean-field approximation and a variational approach are used to calculate the dominant trajectories of the grafted chains, the shape of the molecule, and the segment density distribution around the backbone. All these properties are calculated for an arbitrary relationship between the size of the backbone and that of a grafted star. Hence cylindrical comb copolymer brushes and spherically symmetric polymer stars are considered as the limiting cases of the present problem.

Molecular characterization of a hyperbranched polyester. I. Dilute solution properties

AbstractA hyperbranched polyester was fractionated by precipitation to produce 10 fractions with molecular weights between 20 × 103 and 520 × 103 g mol−1. Each of these fractions was examined by size exclusion chromatography, dilute‐solution viscometry, intensity, and quasi‐elastic light scattering in chloroform solution at 298 K. High‐resolution solution‐state 13C NMR was used to determine the degree of branching; for all fractions this factor was 0.5 ± 0.1. Viscometric contraction factors, g′, decreased with increasing molecular weight, and the relation of this parameter to the configurational contraction factor, g, calculated from a theoretical relation suggested a very strong dependence on the universal viscosity constant, Φ, on the contraction factor. A modified Stockmayer–Fixman plot was used to determine the value of (〈r2〉o/Mw)1/2, which was much larger than the value for the analogous linear polymer. The scaling relations of the various characteristic radii (Rg, Rh, RT, and Rη) with molecular weight all had exponents less than 0.5 that agreed with the theoretical predictions for hyperbranched polymers. The exponent for Rg was interpreted as fractal dimension and had a value of 2.38 ± 0.25, a value that is of the same order as that anticipated by theory for branched polymers in theta conditions and certainly not approaching the value of 3 that would be associated with the spherical morphology and uniform segment density distribution of dendrimers. Second virial coefficients from light scattering are positive, but the variation of the interpenetration function, ψ, with molecular weight and the friction coefficient, ko, obtained from the concentration dependence of the diffusion coefficient suggests that chloroform is not a particularly good solvent for the hyperbranched polyester and that the molecules are soft and penetrable with little spherical nature. © 2003 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 41: 1339–1351, 2003

Small-Angle Neutron Scattering Investigations of Layer−Block Dendrimers in Aqueous Solutions

Structures of poly(trimethyleneimine)/mono(amido amine) dendrimer with hydroxyl end groups and poly(amido amine) dendrimer with galactose having glycopeptide end groups were characterized in aqueous solutions and compared with that of poly(amido amine) dendrimer with hydroxyl end groups reported before (Langmuir 1999, 15, 4076). External contrast variation results of small-angle neutron scattering suggested the segment density distribution in dendrimer. The segment density distribution and solvent penetration amount in the fifth generation dendrimers were evaluated on the basis of the five concentric layers model. Those differed among three dendrimers, depending on the hydrophilic or hydrophobic interior segment and the small or bulky terminal group. Moreover, the profile of segment density distribution is not necessarily consistent with that of solvent penetration amount.

Probability density distribution of random line segments inside a convex body: Application to random media

We study the probability density distribution of random line segments entirely contained inside a convex body of Rn. Relations between this distribution function and other distribution functions of importance in geometric probability are shown. The expression for the probability density distribution of random line segments is given for several simple geometric bodies in R2, R3, and Rn, application to random media is then presented.