Scaling and structural properties of a polymer in a simple solvent: A study based on integral equations

We present a statistical mechanical theory for polymer–solvent systems based on integral equations derived from the polymer Kirkwood hierarchy. Integral equations for pair monomer–monomer, monomer–solvent, and solvent–solvent correlation functions yield polymer–solvent distribution, chain conformation in three dimensions, and scaling properties associated with polymer swell and collapse in athermal, good, and poor solvents. Variation of polymer properties with solvent density and solvent quality is evaluated for chains having up to 100 bonds. In good solvents, the scaling exponent v has a constant value of about 0.61 at different solvent densities computed. For the athermal solvent case, the gyration radius and scaling exponent decrease with solvent density. In a poor solvent, the chain size scales as N^v with the value of the exponent being about 0.3, compared with the mean field value of ⅓. © 1998 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 36: 3025–3033, 1998     

Static and dynamic scaling behavior of a polymer melt model with triple-well bending potential

We perform molecular-dynamics simulations for polymer melts of the coarse-grained poly(vinyl alcohol) model that crystallizes upon slow cooling. To establish the properties of its high temperature, liquid state as a reference point, we characterize in detail the structural features of equilibrated polymer melts with chain lengths 5 ≤ N ≤ 1000 at a temperature slightly above their crystallization temperature. We find that the conformations of sufficiently long polymers with N > 50 obey essentially the Flory's ideality hypothesis. The chain length dependence of the end-to-end distance and the gyration radius follow the scaling predictions of ideal chains and the probability distributions of the end-to-end distance, and form factors are in good agreement with those of ideal chains. The intrachain correlations reveal evidences for incomplete screening of self-interactions. However, the observed deviations are small. Our results rule out any preordering or mesophase structure formation that are proposed as precursors of polymer crystallization in the melt. Moreover, we characterize in detail primitive paths of long entangled polymer melts and we examine scaling predictions of Rouse and the reptation theory for the mean squared displacement of monomers and polymers center of mass. © 2018 Wiley Periodicals, Inc. J. Polym. Sci., Part B: Polym. Phys. 2018 , 56, 1376–1392     

Adsorption Transition of a Polymer Chain at a Weakly Attractive Surface: Monte Carlo Simulation of Off‐Lattice Models

A bead‐spring model of a polymer chain with one end attached to a wall is studied by Monte Carlo simulations for chain lengths 16 ≤ N ≤ 256. Two types of adsorption potentials, 9‐3 and 10‐4 Lennard‐Jones (LJ) potentials, between the effective monomers and the wall are assumed. For both cases the adsorption transition where the chain changes its asymptotic statistical properties from a three‐dimensional to a two‐dimensional configuration is located using a scaling analysis. It is shown that the crossover exponent φ = 0.50 ± 0.02 is the same for both LJ potentials. This value is compatible with recent theoretical predictions and simulation results for lattice models with short‐range wall potentials. The results of our study support the expectation that the exponents describing the adsorption transition are universal, i.e., they are not influenced by the precise form and the long‐range character of the adsorption potentials used. The technical aspects of the simulations (which use configurational bias methods as well as histogram re‐weighting) are also carefully discussed.

Snapshot pictures of a bead‐spring model of a polymer chain with N = 256 beads with one end anchored on the surface: (a)“mushroom configuration”, (b) ε_a ≈ ε_w at the adsorption transition, and (c)“pancake configuration” of a strongly adsorbed chain.     

Some remarks on the dynamics of polymer melts

Considering one long chain (N monomeric units) in a homodisperse melt of chemically identical, but shorter, “solvent” chains (P monomers per chain), we propose some tentative scaling laws for the self-diffusion constant D(N) and the relaxation time T(N) of the solute chain. We also discuss the viscosity increment δ_η due to a small volume fraction Φ of the long chains. We find three regimes of behavior, depending on N and P, and on the distance between entanglement points (assumed smaller than N and P): (A) reptation of the N chain; (B) Stokes–Einstein regime; the solute moves like a usual polymer coil in a viscous fluid of P chains; (C) mixed regime, where D(N) is controlled by reptation, while δ_η is of type B. Contrary to our earlier belief, we find no significant regime where the process of “tube renewal” could be dominant.     

Theory of relaxation properties of two‐dimensional polymer networks, 2. Local dynamic characteristics

Using normal mode transformation obtained in Part 1 of this series^[1], the exact analytical expressions for the mean‐square displacements of junctions and non‐junction beads, the autocorrelation functions of the end‐to‐end chain vectors between neighboring junctions, and those of subchain vectors of a two‐dimensional regular network consisting of "bead and spring" Rouse chains are obtained. Contributions of intra‐ and interchain relaxation processes to the local dynamic characteristics considered are compared. The time behavior of dynamic quantities obtained is estimated for different scales of motions. The possibility of describing long‐time relaxation of a two‐dimensional network by a simplified coarse‐grained network model is demonstrated. It is shown that the local relaxation properties of a two‐dimensional polymer network (as well as a three‐dimensional network) on scales smaller than the average distance between cross‐links are very close to those of a single Rouse chain. The large‐scale collective relaxation of the polymer networks having a two‐dimensional connectivity differs considerably from that of the three‐dimensional networks.     

Microstructure of Highly Syndiotactic “Living” Poly(propylene)s Produced from a Titanium Complex with Chelating Fluorine‐Containing Phenoxyimine Ligands (an FI Catalyst)

Highly syndiotactic “living” poly(propylene)s were synthesized at 25°C using a bis[N‐(3‐tert‐butylsalicylidene)‐2,3,4,5,6‐pentafluoroanilinato]titanium (IV) dichloride/MAO catalyst system, and microstructures of the polymer were analyzed by means of ¹³C NMR spectroscopy. The syndiotactic poly(propylene) contains isobutyl, isopentyl and propyl end groups, suggesting that the living polymerization of propylene was initiated via 1,2‐insertion, followed by 2,1‐insertion as the principal mode of polymerization. Pentad distribution analysis revealed that the syndiospecific polymerization proceeds under chain‐end control.     

Short‐time stretch relaxation of entangled polymer solutions investigated using full Rouse model predictions

We conduct a systematical investigation into the short‐time stretch relaxation behavior (i.e., shorter than the Rouse time but sufficiently longer than the glassy time) of entangled polymer liquid in single‐step strain flows, on the basis of theory/data comparisons for a broad series of type‐A entangled polymer solutions. First, within existing normal‐mode formulations, the Rouse model predictions on a full‐chain stretch relaxation in single‐step strain flows are derived for a popular 1‐D model proposed within the Doi–Edwards tube model, as well as for the original 3‐D model for nonentangled systems. In addition, an existing formula for the aforementioned 1‐D model that, however, rested upon a consistent‐averaging or the so‐called uniform‐chain‐stretch approximation is simultaneously examined. Subsequently, the previously derived formulas on chain stretch relaxation are directly incorporated into a reliable mean‐field tube model that utilizes the linear relaxation spectrum and the Rouse time constant consistently determined from linear viscoelastic data. It is found that the predictions of the 1‐D model differ substantially from that of the original 3‐D model at short times. Theory/data comparisons further indicate that the 1‐D model without approximations seems able to describe fairly well the nonlinear relaxation data under investigation. © 2006 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 44: 1199–1211, 2006     

Predictions of a recently proposed non-markovian model for polymer melts and concentrated solutions

Several predictions for a recently proposed mesoscopic model for polymer melts and concentrated solutions is presented. It is a single Kramers chain model in which elementary motions of the Orwoll-Stockmayer type are allowed. However, for this model, the bead jumps are no longer given by a Markovian probability, but rather are described by a fractal “waiting-time” distribution function, with a single adjustable parameter β, which describes the long-time behavior of the distribution: ∼ 1/t^1+β. We find that the model predicts D ∼ 1/N² and η₀ ∼ N^3.4 for β ≈︁ 1.4, where N is the degree of polymerization. The generalized model predicts that the relaxation spectrum has a plateau regime whose height is independent of N, but whose width is strongly N dependent, in agreement with experiment. The model also predicts that rings will diffuse somewhat more slowly than linear chains of the same molecular weight (about 80% as fast), with the same scaling dependence on N as linear chains, also in agreement with preliminary data.     

Facile one‐pot/one‐step technique for preparation of side‐chain functionalized polymers: Combination of SET‐RAFT polymerization of azide vinyl monomer and click chemistry

An azido‐containing functional monomer, 11‐azido‐undecanoyl methacrylate, was successfully polymerized via ambient temperature single electron transfer initiation and propagation through the reversible addition–fragmentation chain transfer (SET‐RAFT) method. The polymerization behavior possessed the characteristics of “living”/controlled radical polymerization. The kinetic plot was first order, and the molecular weight of the polymer increased linearly with the monomer conversion while keeping the relatively narrow molecular weight distribution (M_w/M_n ≤ 1.22). The complete retention of azido group of the resulting polymer was confirmed by ¹H NMR and FTIR analysis. Retention of chain functionality was confirmed by chain extension with methyl methacrylate to yield a diblock copolymer. Furthermore, the side‐chain functionalized polymer could be prepared by one‐pot/one‐step technique, which is combination of SET‐RAFT and “click chemistry” methods. © 2011 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem, 2012     

Intra- and inter-chain fluctuations in entangled polymer melts in bulk and confined to pore channels

It is known that topological restraints by “chain entanglements” severely affect chain dynamics in polymer melts. In this field-cycling NMR relaxometry and fringe-field NMR diffusometry study, melts of linear polymers in bulk and confined to pores in a solid matrix are compared. The diameter of the pore channels was 10 nm. It is shown that the dynamics of chains in bulk dramatically deviate from those observed under pore constraints. In the latter case, one of the most indicative signatures of the reptation model is verified 28 years after its prediction by de Gennes: The frequency and molecular mass dependencies of the spin-lattice relaxation time obey the power law T_! ∝ M⁰ v^3/4 on a time scale shorter than the longest Rouse relaxation time τ_R. The mean squared segment displacement in the pores was also found to be compatible to the reptation law < r²>∝ M^−1/2t^1/2 predicted for τ_R < t < τ_d, where τ_d is the so-called disengagement time. Contrary to these findings, bulk melts of entangled polymers show frequency and molecular mass dependencies significantly different from what one expects on the basis of the reptation model. The data can however be described with the aid of the renormalized Rouse theory.     

Equilibrium conformational dynamics of a polymer in a solvent

Molecular dynamics simulations were used to study the conformational dynamics of a bead-spring model polymer in an explicit solvent under good solvent conditions. The dynamics of the polymer chain were investigated using an analysis of the time autocorrelation functions of the Rouse coordinates of the polymer chain. We have investigated the variation of the correlation functions with polymer chain length N, solvent density rho, and system size. The measured initial decay rates gamma(p) of the correlation functions were compared with the predictions from a theory of polymer dynamics which uses the Oseen tensor to describe hydrodynamic interactions between monomers. Over the range of chain lengths considered (N = 30-60 monomers), the predicted scaling of gamma(p) proportional to N(-3nu) was observed at high rho, where nu is the polymer scaling exponent. The predicted gamma(p) are generally higher than the measured values. This discrepancy increases with decreasing rho, as a result in the breakdown in the conditions required for the Oseen approximation. The agreement between theory and simulation at high rho improves considerably if the theoretical expression for gamma(p) is modified to avoid sum-to-integral approximations, and if the values of (R(p)2), which are used in the theory, are taken directly from the simulation rather than being calculated using approximate scaling relations. The observed finite-size scaling of gamma(p) is not quantitatively consistent with the theoretical predictions.     

Small‐angle neutron scattering analysis of bottlebrush backbone and side chain flexibility

Bottlebrush polymers have densely tethered side chains grafted to a linear polymer backbone, resulting in stretching of both the side chains and backbone. Prior studies have reported that the side chains are only weakly stretched while the backbone is highly elongated. Here, scaling laws for the bottlebrush backbone and side chains are determined through small‐angle neutron scattering analysis of a systematic series of poly(lactic acid) bottlebrush polymers synthesized via a “grafting‐through” ring‐opening polymerization. Scattering profiles are modeled with the empirical Guinier–Porod, rigid cylinder, and flexible cylinder models. Side chains are found to be only weakly stretched, with an end‐to‐end distance proportional to N^0.55, while the overall bottlebrush increases in size proportional to N^0.77. These results demonstrate that the bottlebrush backbone is not fully extended and that both side chains and backbone have significant conformational flexibility in solution. © 2016 Wiley Periodicals, Inc. J. Polym. Sci., Part B: Polym. Phys. 2017 , 55, 104–111     

Synthesis and crosslinking behavior of a novel linear polymer bearing 1,2,3‐triazol and benzoxazine groups in the main chain by a step‐growth click‐coupling reaction

The click‐coupling reaction was applied to polycondensation, to synthesize a high‐molecular weight prepolymer having benzoxazine moieties in the main chain. For the polycondensation, a bifunctional N‐propargyl benzoxazine was synthesized from bisphenol A, propargylamine, and formaldehyde. The propargyl group was efficiently used for the copper(I)‐catalyzed alkyne‐azide “click” reaction with p‐xylene‐α,α′‐diazide, to give the corresponding linear polycondensate having 1,2,3‐triazole junctions. The polycondensation proceeded in N,N‐dimethylformamide (DMF) at room temperature. By this highly efficient “click‐” polycondensation reaction, the benzoxazine ring in the monomer was successfully introduced into the polymer main chain without any side reaction. The obtained polymer (=prepolymer) underwent thermal crosslinking to afford the corresponding product, which was insoluble in a wide range of organic solvents and exhibited higher thermal stability than the polymer before crosslinking. © 2008 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 46: 2316–2325, 2008     

Polymer chains confined into tubes with attractive walls: A Monte Carlo simulation

A bead-spring off-lattice model of a polymer chain with repulsive interactions among repeating units confined into straight tubes of various cross sections, D_T², is studied by Monte Carlo simulation. We are also varying the chain length from N = 16 to 128 and the strength of a short-range attractive interaction between the repeating units and the walls of the tube. Longitudinal and perpendicular static linear dimensions of the chains are analyzed, as well as the density profile of repeating units across the tube. These data are interpreted in terms of scaling concepts describing the crossover between three-dimensional and quasi-one-dimensional chain conformations and the adsorption transition of chains at flat infinite walls, respectively. We also study the time-dependent mean-square displacements of repeating units and obtain various relaxation times. It is shown that both relaxation times scaling proportional to N² and to N³ play a role in the reptative motion of the chain in the tubes.     

Structure factor of a self-avoiding polymer chain at high momentum transfer

The molecular weight of a Gaussian polymer chain can be obtained from the intercept of the asymptote in a plot of the inverse structure factor 1/S(q) as a function of the square of the wave vector q². Using an ϵ expansion and scaling arguments, we show that in a good solvent the situation is more complex and that the molecular weight determination is difficult to justify. The corrections to the asymptotic behavior of the structure factor at large wave vector involve several terms that are difficult to separate experimentally. This is qualitatively explained by the nonuniform swelling of a polymer chain in a good solvent due to the existence of the chain ends.     

Contribution of Polymer‐Solvent Interactions to Dynamics of Linear Polymers in Dilute Solutions

In this work a theoretical approach to dynamics of linear vinyl polymers in dilute solutions of high viscosity solvents is presented. The calculations for the relaxation time spectra, polymer intrinsic viscosity [η (ω)], complex elastic modulus G*(ω), total intrinsic viscosity [η^T (ω)] and specific heat capacity C (ω) were carried out in the non‐free‐draining limits. The relaxation time spectrum calculated for dynamics of low frequency modes exhibits a Rouse‐like character. Its position and shape corresponds to the ultrasonic relaxation time spectrum observed in the system at 10⁶ Hz. On the other hand, the relaxation time spectrum associated with moderate frequency mode dynamics is narrower and typical for ultrasonic relaxation observed at 10⁷ Hz. The polymer intrinsic viscosity [η (ω)] and elastic modulus G*(ω) are shown to be represented by the model within a low‐frequency range. In turn, the specific heat capacity C (ω) is displayed as a representation of the model in the acoustic region mentioned above. In the high‐frequency range the dynamics is described by the total intrinsic viscosity [η^T (ω)] tending to a plateau where the value is equal to the sum of the single‐bead intrinsic viscosity [η_N] and effective solvent viscosity [η_eff].     

Computer simulation of relaxation phenomena in star‐branched polymers. Temperature dependence

Dynamic Monte Carlo simulations of simple models of star‐branched polymers were conducted. A model star macromolecule consisted of f = 3 arms of equal length with a total number of polymer segments up to 800. The chain was confined to a simple cubic lattice with simple nearest neighbor attractive interactions. The relaxation phenomena were studied by means of autocorrelation functions in wide ranges of temperatures. Short‐time‐scale dynamic processes in the entire star‐branched chain were examined. It was found that under good solvent conditions the longest relaxation time of the end‐to‐center vector decreases with decreasing temperature. For low temperatures (below the Θ‐point) where the chain is collapsed, the dependence of the relaxation time on the temperature is opposite.     

Cellulose organic solutions: A nuclear magnetic resonance investigation

Four solvents of cellulose have been studied by using ¹³C-NMR spectroscopy. All these solvents, N-methyl morpholine-N-oxide, methylamine, hydrazine, and paraformaldehyde (PF), contained dimethyl sulfoxide (DMSO) as a cosolvent. Oligomers of cellulose of DP = 10 soluble in hot DMSO have been used as model compounds. ¹³C chemical shifts and line shapes show that three of the mentioned solvents are “true solvents” of cellulose. On the other hand, dissolution of cellulose in DMSO-PF system occurs by the formation of a statistical derivative of cellulose. Enriched ¹³C bacterial cellulose on C-1 and C-6 positions have been used to identify the ¹³C positions mainly in DMSO-N-methyl morpholine-N-oxide system. This solvent has been found to be degradative for the macromolecule when the solution is kept at 100°C over a long period. Viscosity measurements show a reduction of the molecular weight in these conditions. Polarimetry indicates that no glucose is present in solution and hence there is a statistical break of the chain. Enriched cellulose solution in DMSO–N-methyl morpholine-N-oxide has been also used for relaxation time (T₁) determination both of the solvent and of the enriched carbons of the polymer. Nuclear Overhauser enhancement (NOE) was found to be 1.8 for C-1 and 2.1 for C-6 showing that relaxation phenomenon is not purely dipolar. T₁ values of 97 and 65 msec are found for C-1 and C-6 of cellulose, in good agreement with the values known for polysaccharides. Determination of T₁ for the different carbon atoms of the solvent DMSO-N-methyl morpholine-N-oxide with and without cellulose shows a large reduction of T₁ for N-methyl morpholine-N-oxide molecule. This denotes a slower molecular motion of this molecule and a preferential interaction with the cellulose macromolecule.     

Surface diffusion and relaxation of partially adsorbed polymers

We studied by lattice simulation the surface diffusion and relaxation of isolated, self‐avoiding polymers partially adsorbed onto a flat surface. The key parameters describing the system are the number of segments in the chain, N, the adsorption energy of a segment, expressed as a dimensionless surface temperature T_s, and the segmental friction factor on the surface relative to that in the bulk, ζ_s/ζ_b. The simulation data indicate Rouse scaling of the surface diffusion coefficient, D_∥, and in‐plane relaxation time, τ, versus N for all values of T_s and ζ_s/ζ_b studied. A simple application of the Rouse model to a partially adsorbed chain, which ignores fluctuations in adsorbed trains, yields a formula for D_∥ with the correct N‐scaling. It can account for the effects of T_s when ζ_s/ζ_b is finite (≲10), but it fails when ζ_s/ζ_b diverges, predicting no surface diffusion at all, whereas simulations indicate finite surface mobilities facilitated by a caterpillar‐like motion. © 2000 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 38: 1146–1154, 2000