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.     

Stress overshoot of polymer solutions at high rates of shear

Overshoot of shear stress, σ, and the first normal stress difference, N₁, in shear flow were investigated for polystyrene solutions. The magnitudes of shear corresponding to these stresses, γ_σm and γ_Nm, for entangled as well as nonentangled solutions were universal functions of γ˙τ_eq, respectively, and γ_Nm was approximately equal to 2γ_σm at any rate of shear, γ˙. Here τ_eq = τ_R for nonentangled systems and τ_eq = 2τ_R for entangled systems, where τ_R is the longest Rouse relaxation time evaluated from the dynamic viscoelasticity at high frequencies. Only concentrated solutions exhibited stress overshoot at low reduced rates of shear, γ˙τ_eq < 1. The behavior at very low rates, γ˙τ_eq < 0.2, was consistent with the Doi–Edwards tube model theory for entangled polymers. At high rates, γ˙τ_eq > 1, γ_σm and γ_Nm were approximately proportional to γ˙τ_eq. At very high rates of shear, the peak of σ is located at t = τ_R, possibly indicating that the polymer chain shrinks with a characteristic time τ_R in dilute solutions. © 2000 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 38: 1917–1925, 2000     

Flow properties of poly(methyl methacrylate) solutions

Viscosity and normal stress behavior were measured for poly(methyl methacrylate) samples of various average molecular weights in diethyl phthalate solution at 30 and 60°C. All samples conformed approximately to the most probable distriution (M?_w/M?_n = 2). Concentrations ranged from 0.113 to 0.38 g/ml, and M?_w from 53,800 to 1,620,000. Despite considerable evidence in the literature of unusual linear viscoelastic behavior for this polymer, its nonlinear properties appear to be rather conventional. The viscosity–shear rate master curve was similar to that found earlier for concentrated solutions of polystyrene and poly(vinyl acetate) of comparable molecular-weight distribution. The viscosity time constant τ_o parallels τ_R, the characteristic time of the Rouse model, although the residual dependence of τ_o/τ_R on concentration and molecular weight appears to be slightly different from that for polystyrene and poly(vinyl acetate). Similar conclusions apply to the recoverable compliance J_e,^o estimated from the normal stress behavior of each solution, and its relationship to the Rouse model compliance J_R.     

Dynamic viscosity of dilute aqueous solutions of poly(acrylic acid) at frequencies of 2–500 kHz

The dynamic viscosity of aqueous solutions of poly(acrylic acid) at a polymer concentration of ca. 0.15 g/100 ml has been measured at frequencies from 2 to 500 kHz as a function of degree of polymerization P, degree of neutralization α, and salt (NaCl) concentration C_s. Relaxation spectra have been obtained from the dynamic viscosity. The spectra in the short relaxation time region can be approximated by the Zimm theory for the conformational relaxation of nonionic polymers. The maximum relaxation time τ₁ of the Zimm spectra is proportional to P² and depends rather moderately on α and C_s. Increased deviation is found, however, in the long relaxation time region, in particular for high values of P and α and low values of C_s. The major part of the deviation is interpreted in terms of rotational relaxation of a molecule as a whole. The rotational relaxation time τ_R is proportional to P³ and increases with increasing α and decreasing C_s. The remaining part of the excess spectra located between τ₁ and τ_R is ascribed to the deviation of the conformational relaxation from the Zimm theory arising from ionization of the polymer.     

Effects of polymer chain disentanglement on NMR properties. II. 13C magnetic relaxation in polystyrene

The transverse magnetic relaxation of ¹³C_α nuclei has been studied in concentrated solutions of polystyrene. The magnetic relaxation rate was measured as a function of molecular weight at several temperatures (313,318, and 323 K) and at several concentrations (0.53, 0.43, and 0.34 g/cm³). The spin-system response of these nuclei in natural abundance exhibits a characteristic evolution from pseudosolid properties to liquidlike one, induced by decreasing the molecular weight of polymer molecules. This evolution is analogous to that already observed in protons attached to polyisobutylene or polydimethylsiloxane chains; it is assumed to be induced by an increase of the disentanglement rate of polymer chains. The spin-system response may be considered as reflecting single-chain magnetic properties, because of the low concentration of ¹³CC_α nuclei, although all chains are in dynamic interaction with one another. The NMR disentanglement transition is interpreted in terms of a two-step motional averaging effect involving submolecules. A numerical analysis of NMR properties is given using a model of polymer chain relaxation based on a multiple-mode relaxation process, characterized by (i)a terminal relaxation time τ^v₁ depending upon M³, the molecular weight, and approximately proportional to the polymer concentration C (like the reptation time); (ii)a relaxation-time spectrum analogous to a Rouse spectrum; (iii)a terminal relaxation time τ^v₁ = 2.5 × 10^?2s for M = 2.5 × 10⁵, C = 0.53 g/cm³ in carbon tetrachloride at 313 K.     

The dynamics of polymer melts as seen by neutron spin echo spectroscopy

Using the neutron spin echo spectroscopy, the internal segmental diffusion of chain molecules in polymer melts and concentrated solutions was studied. These investigations show that beyond a characteristic length d_t and after a cross over time τ_e(d_t) the segmental diffusion of the single chains is strongly impeded and deviates from the Rouse dynamics. d_t is polymer specific and depends on the temperature as well as on the polymer concentration. Within the framework of the reptation concept, where d_t is identified with the mean distance between intermolecular entanglements or with the tube diameter, the microscopically determined d_t-values agree quite well with those derived from related macroscopic measurements of the plateau modulus. A similar good agreement is also found with respect to the segmental friction coefficients obtained either from the Rouse regime of the NSE spectra or from Theological data of corresponding short chain systems, where entanglements are not yet effective.     

Molecular geometry and chain entanglement: parameters for the tube model

The tube diameter in the reptation model is the distance between a given chain segment and its nearest segment in adjacent chains. This dimention is thus related to the cross-sectional area of polymer chains and the nearest approach among chains, without effects of thermal fluctuation and steric repulsion. Prior calculated tube diameters are much larger, about 5 times, than the actual chain cross-sectional areas. This is ascribed to the local freedom required for mutual rearrangement among neighboring chain segments. This tube diameter concept seems to us to infer a relationship to the corresponding entanglement spacing. Indeed, we report here that the critical molecular weight, M_c, for the onset of entanglements is found to be M_c = 28 A/(〈R²〉₀/M), where A is the chain cross-sectional area and 〈R²〉₀ the mean-square end-to-end distance of a freely jointed chain of molecular weight M. The new, computed relationship between the critical number of backbone atoms for entanglement and the chain cross-sectional area of polymers, N_c = A^0,44, is concordant with the cross-sectional area of polymer chains being the parameter controlling the critical entanglement number of backbone atoms of flexible polymers.     

Mobility of Spherical Probe Objects in Polymer Liquids

We use scaling theory to derive the time dependence of the mean-square displacement ?Δr(2)? of a spherical probe particle of size d experiencing thermal motion in polymer solutions and melts. Particles with size smaller than solution correlation length ξ undergo ordinary diffusion (?Δr(2) (t)? ~ t) with diffusion coefficient similar to that in pure solvent. The motion of particles of intermediate size (ξ < d < a), where a is the tube diameter for entangled polymer liquids, is sub-diffusive (?Δr(2) (t)? ~ t(1/2)) at short time scales since their motion is affected by sub-sections of polymer chains. At long time scales the motion of these particles is diffusive and their diffusion coefficient is determined by the effective viscosity of a polymer liquid with chains of size comparable to the particle diameter d. The motion of particles larger than the tube diameter a at time scales shorter than the relaxation time τ(e) of an entanglement strand is similar to the motion of particles of intermediate size. At longer time scales (t > τ(e)) large particles (d > a) are trapped by entanglement mesh and to move further they have to wait for the surrounding polymer chains to relax at the reptation time scale τ(rep). At longer times t > τ(rep), the motion of such large particles (d > a) is diffusive with diffusion coefficient determined by the bulk viscosity of the entangled polymer liquids. Our predictions are in agreement with the results of experiments and computer simulations.     

Effects of chain length distribution on viscoelastic properties of entangled linear polymers: Blending laws for binary blends

The reptation idea of de Gennes and the tube model theory of Doi and Edwards are extended to explain the terminal viscoelastic properties of binary blends in the highly entangled state of two linear monodisperse polymers with different molecular weights M₁ and M₂. A modified tube model is proposed that considers the significance of the constraint release by local tube renewal in accounting for the relaxation process of the higher molecular weight chain. Its relaxation by both reptation and the constraint release is remodeled as the disengagement by pure reptation of an equivalent primitive chain. From knowledge of the longest relaxation times of the blend components, the stress equation is formulated from which blending laws of viscoelastic properties for the binary blends are derived. To force better agreement between theory and experiment at the pure monodisperse limits of the blends, a crude treatment to include the effect of contour-length fluctuation in the equivalent-chain model is also discussed. Theoretical predictions of the zero-shear viscosity and steady-state shear compliance are shown to be in good agreement with literature data on undiluted polystyrenes and polybutadienes over a wide range of the blend composition and M₂/M₁ ratio. The asymptotic of the laws for blends with M₂/M₁ → 1 and 0 are comparable to those from the relaxation spectrum proposed by others earlier on the basis of the tube model.     

Interdiffusion of polymers across interfaces

Neutron Reflection (NR) and Dynamic Secondary Ion Mass Spectroscopy (DSIMS) experiments were conducted on symmetrically deuterated polystyrene triblock bilayers (HDH/DHD) which directly probed the interdiffusion dynamics of the chains during welding. The HDH chains had their centers deuterated 50%, the DHD chains had their ends deuterated (25% at each end) such that each chain contained approximately 50% D. During welding, anisotropic motion of the chains produces a time-dependent oscillation (ripple) in the H and D concentration at the interface, which bears the characteristic signature of the polymer dynamics. These oscillations were compared with those predicted by Rouse, polymer mode coupling (PMC), and reptation dynamics. The following conclusions can be made from this study. (a) During the interdiffusion of high molecular weight HDH/DHD pairs, higher mobility of the chain ends caused a concentration oscillation which increased to a maximum amplitude, and eventually vanished at times, t > τ_D. The amplitude, or excess enrichment found, was appreciably more than that predicted by Rouse and PMC simulations, and was only slightly less than that predicted from reptation simulations. (b) The oscillations were completely missing in the 30 and 50K HDH/DHD polymers, which are only weakly entangled. The lack of oscillations for the 30 and 50K pairs may be due to a combination of surface roughness and fluctuations of order 30 Å. (c) It was found that the position of the maximum in this ripple stayed at the interface during its growth. This is also consistent with reptation and has not been explained by other theories. (d) All dynamics models for linear polymers produce ripples, many of which are qualitatively similar to that predicted for reptation. However, each ripple bears the fingerprint of the dynamics in terms of its time-dependent shape, position, and magnitude, and the models are clearly distinguishable. Our results, in summary, support reptation as a candidate mechanism of interdiffusion at polymer(SINGLEBOND) polymer interfaces and its uniqueness is being further pursued. © 1996 John Wiley & Sons, Inc.     

Reptation theory: geometrical and topological aspects

This paper discusses topological and geometrical aspects of reptation theory which are common to all versions of reptation theory. These are: the postulated existence of the tube, the functional relationship between the tube diameter a and the polymer/monomer density p, the crossover from the Rouse to reptation regime. Statistical mechanics of the geometrically confined polymer chain is reanalyzed by careful separation of the diffusive motion of the chain into the longitudinal and transversal parts. Connection between old results and the new formalism is established. It is shown that the longitudinal motion resembles that known for directed polymers. This provides a source of the effective rigidification of the reptating chain's backbone thus facilitating the viscosity exponent to be larger than 3. The transversal motion is also reanalyzed. It is shown that the diffusion on the Bethe lattice used before to describe the transversal (planar) motion (conformational statistics) of the trapped chain is actually the diffusion on the universal covering of the corresponding Riemannian surface. This fact allows to reanalyze the tube stability using topological arguments. Detailed numerical comparison of the obtained new theoretical results with available experimental and Monte Carlo data is provided. Very good agreement between theory and experiment is found. It is also shown that the emerging physical picture of the tube destruction is isomorphic to that which was developed earlier with the help of the quantum Hall effect analogy (J. Phys. I 4 , 843 (1994)). Remarkable connections between the reptation theory and the theory of quantum chaotic/mesoscopic systems are established thus making the reptation theory part of the more general theory of quantum chaotic systems.     

Molecular architecture and rheological characterization of novel intramolecularly crosslinked polystyrene nanoparticles

Novel polystyrene nanoparticles were synthesized by the controlled intramolecular crosslinking of linear polymer chains to produce well‐defined single‐molecule nanoparticles of varying molecular mass, corresponding directly to the original linear precursor chain. These nanoparticles are ideal to study the relaxation dynamics/processes of high molecular mass polymer melts, as the high degree of intramolecular crosslinking potentially inhibits entanglements. Both the nanoparticles and their linear analogs were characterized by measuring their intrinsic viscosity, hydrodynamic radius (R_h), and radius of gyration (R_g). The ratio R_g/R_h was computed to characterize the molecular architecture of the nanoparticles in solution, revealing a shift toward the constant density sphere limit with increasing crosslink density and molecular mass. Further, confirming particulate behavior, Kratky plots obtained from neutron scattering data show a shift toward particle‐like nature. The rheological behavior of the particles was found to be strongly dependent on both the extent of intramolecular crosslinking and molecular mass, with a minimal viscosity change at low crosslinking levels and a gel‐like behavior evident for a large degree of crosslinking. These and other results suggest the presence of a secondary mode of polymer relaxation/movement besides reptation, which in this case, is influenced by the total number of crosslinked loops present in the nanoparticle. © 2006 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 44: 1930–1947, 2006     

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.     

Investigation of Structure and Mobility of Silica Bonded Stationary Acridine Phases by Means of Fluorescence and NMR Spectroscopy

Ligand mobility of silica-based HPLC stationary phases modified by various surface coverages of acridine-9-carboxy(N-aminoethylaminopropyl)amide ligands was investigated by fluorescence spectroscopy, time-resolved fluorescence anisotropy measurements, as well as solid-state ¹³C-CP/MAS- and ¹H-MAS-NMR spectroscopy. Rotational correlation times, τ_R, of the bound acridine fluorophore obtained from fluorescence anisotropy measurements are significantly longer in the bound phase, than in solution. Also, in time-resolved experiments anisotropies do not decay to zero. These results are interpreted in terms of wobble-in-cone ligand motion. The mobility of the fluorophore in the presence of liquid phase correlates strongly with the solubility of the model compound acridine-9-carboxy-n-butylamide in the same solvent. In the good solvent acetonitrile τ_R = 3.2 ns is found, whereas in methanol, τ_R > 80 ns is obtained. NMR measurements of the dry phase yield large linewidths, cross polarization constants, T_CH, and spin-lattice relaxation times, T_1ρH, shifting around the minimum in the correlation time curve. Both fluorescence and NMR data indicate medium to low ligand mobility. No difference in the mobilities of alkyl spacer and aromatic group is observed, probably due to the rigidity of the amide group.     

Polydispersity effects on diffusion in polymers: Concentration profiles of d-polystyrene measured by forward recoil spectrometry

Forward recoil spectrometry is shown to be a useful technique for measuring diffusion of d-polymer chains in h-polymer melts. Concentration profiles of a deuterated diffusing species may be determined with a depth resolution of 80 nm and a sensitivity of 0.1 vol % d-polymer in h-polymer. Consequently diffusion coefficients as small as 10^?16 cm²/s can be readily measured. If polymer chains diffuse by a reptation mechanism, the concentration profile ø(x) of diffusing polydisperse polymer should be quite different from ø_m(x), the Fickian solution, which one obtains for monodisperse polymer. This idea was tested by measuring ø(x) of polydisperse d-polystyrene (d-PS) diffusing into h-PS. The results are in excellent agreement with the ø(x) predicted from the reptation model and the experimentally determined molecular weight distribution.     

Hole growth as a microrheological probe to measure the viscosity of polymers confined to thin films

We review recent hole growth measurements performed at elevated temperatures in freely-standing polystyrene (PS) films, using optical microscopy and a differential pressure experiment (DPE). In the hole growth experiments, which were performed at temperatures close to the bulk glass-transition temperature of PS, T = 97 °C, we find evidence for nonlinear viscoelastic effects, which markedly affect the growth of holes in freely-standing PS films. The hole radius R initially grew linearly with time t before undergoing a transition to exponential growth characterized by a growth time τ. The time scale τ₁ for the decay of the initial transient behavior prior to reaching steady state was consistent with the convective constraint release mechanism of the tube theory of entangled polymer dynamics, while the characteristic hole growth times τ of the holes were consistent with significant reductions in viscosity of over eight orders of magnitude with increasing shear strain rate due to shear thinning. DPE measurements of hole growth on very thin freely-standing films revealed that hole formation and growth occurs only at temperatures that are comparable to or greater than T, even for films for which the T_g value was reduced by many tens of degrees Celsius below the bulk value. © 2006 Wiley Periodicals, Inc. J Polym Sci Part B:Polym Phys 44: 3011–3021, 2006     

Weak collision effects: Non-steady-state thermal unimolecular reactions

Bench-mark calculations for non-steady-state thermal systems are presented. Although the equations and calculations were modeled for the isomerization of cyclopropane in the single-pulse shock tube, the results are qualitatively and semiquantitatively correct for any experimental technique in which a finite time is required to energize the reactant to the reaction temperature. A useful parameter in describing this nonequilibrium behavior is the induction time τ₉₅; the log of τ₉₅ is linearly related to the collision frequency ω. For the model cyclopropane system a steady state is reached in ～1 msec when ω ≈ 10⁸ sec^?1 and T ≈ 1500 K.     

Thermal relaxation of poled non-linear optical sidechain polymers: A new,semi-empirical model

Electro-Optic relaxation of a poled, Non-Linear Optical sidechain polymer with T_g 140°C, containing 4-dimethylamino-4′-nitrostilbene (“DANS”) in the sidechains, has been studied at 120°C with and without annealing at the same temperature. The time-dependence of the decaying EO coefficients r(t) shows a strong departure from the classical single-exponential Debye model, especially in the unannealed samples. This departure is attributed to physical ageing, slowing down the orientational relaxation of the sidechains. The Debye model with r(t)-r(0). exp -t/τ] is modified semi-empirically by introducing a time-dependent characteristic Debye relaxation time τ(t). Of several trial expressions, one is selected which fits the relaxation data. This is τ(τ)-τ_i+C.t^b