Thermodynamic Limitations of the Reptation Model

Summary: The paper deals with the question whether the tube/reptation model of polymer chain dynamics is compatible with general laws of statistical physics. Based on a relation between the mean squared fluctuation of the number of segments in a given volume element and the isothermal compressibility of the polymer system, it follows straightforwardly that the tube/reptation model predicts fluctuations larger than permitted by thermodynamics on the time scale t ≳ τ_R, where τ_R is the Rouse relaxation time.     

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.     

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     

Polymer dynamics in solution from rayleigh line profile spectroscopy

The relaxation time, τ₁, of the first mode in the Rouse—Zimm analysis of intramolecular motion in polymers has been evaluated from an analysis of Rayleigh scattered light using intensity fluctuation spectroscopy. Linear polystyrene in the molecular weight range of 5 × 10⁶ to 20 × 10⁶ has been investigated in theta and non-theta dilute solution conditions. Values compatible with the free-draining Rouse model are obtained and a molecular weight dependence approximately as τ₁ α M^1.5 is deduced.     

Monte Carlo Simulation of Thin Film Polymer Melts

We present Monte Carlo simulation data on conformations and dynamics of polymer melts confined in narrow slits of different widths and compare with data of bulk systems. We find that in confined geometries the chains swell laterally; they retain and even expand their spatially long-range correlations compared to bulk polymers and in contrast to the assumption of a complete screening of excluded volume. Long chains in bulk melts show entangled dynamics with a clear signature of a t^1/4-power law for the mean square displacements of innermost monomers at intermediate time scales. This behavior is gradually lost by confining the melts in slits with decreasing width. For ultra-thin films, the dynamics appears to follow a Rouse-like behavior over the entire subdiffusive regime. However, the terminal relaxation time is significantly increased compared to Rouse relaxation. This interesting observation was not reported previously and is the focus of our ongoing research.     

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.     

Dynamic mechanical properties of moderately concentrated polystyrene solutions

The storage (G′) and loss (G″) shear moduli have been measured in the frequency range from 0.04 to 630 Hz for solutions of narrow distribution polystyrenes with molecular weights (M) 19,800 to 860,000, and a few of poly(vinyl acetate), M = 240,000. The concentration (c) range was 0.014–0.40 g/ml and the viscosities of the solvents (diethyl phthalate and chlorinated diphenyls) ranged from 0.12 to 70 poise. Data at different temperatures (0–40°C) were combined by the method of reduced variables. Two types of behavior departing from the usual frequency dependence describable by the Rouse-Zimm-Tschoegl theories were observed. First, for M ? 20,000, the ratio (G″ ? ωη_s)/G′ in the neighborhood of ωτ₁ = 1 was abnormally large and the steady-state compliance J was abnormally small, especially at the lowest concentrations studied. Here ω is circular frequency, η_s solvent viscosity, and τ₁ terminal relaxation time. Related anomalies have been observed by others in undiluted polymers at still lower molecular weights. Second, at the highest concentrations and molecular weights, a “crossover” region of the logarithmic frequency scale appeared in which G″ ? ωη_s < G′. The width of this region is a linear function of log c; the frequency dependence under these conditions can be represented by a sequence of Rouse relaxation times grafted on to a sequence of Zimm relaxation times. For each molecular weight, the terminal relaxation time τ₁ was approximately a single function of c for different solvents of widely different η_s. At lower concentrations, τ₁ was close to the Rouse prediction of 6ηM/π²cRT, where η is the steady-flow viscosity; but at higher concentrations, τ₁ was proportional to η/c² and corresponded, according to a recent theory of Graessley, to an average molecular weight of 20,000 between entanglement coupling points in the undiluted polymer.     

Cooperative chain relaxation in miscible polymer blends

Orientation relaxation of dissimilar chains in the molten miscible blends, poly(methyl methacrylate)/poly(vinylidene fluoride) and poly(methyl methacrylate)/poly(vinylidene fluoride-co-trifluoroethylene), were investigated by measuring (1) the change of infrared dichroic ratio with time after the uniaxial stretching of film specimens, (2) the shear stress relaxation spectrum, and (3) birefringence relaxation in shear. The dissimilar polymers showed an identical time variation of the normalized Hermans orientation function. The blend showed a relaxation spectrum with a single characteristic relaxation time τ_c, depending on the blend composition. The birefringence relaxed monotonically, remaining positive. These results suggest that the dissimilar polymers do not relax independently but cooperatively. This behavior may be induced by a constraint due to the specific interactions between the dissimilar polymers, e.g., weak hydrogen bonding. For the cooperative chain relaxation, a third power relationship was found; τ_c/τ_e vprop; (M/M_e),³ where τ_e and M_e are the relaxation time and molecular weight of entanglement strand, respectively, and M is the number average molecular weight in the blend.     

Diffuse interface between polymers: Structure and kinetics

The interfacial structure and diffusion kinetics of two compatible polymers, poly(methyl methacrylate) and poly(vinylidene fluoride) are studied in the melt. The interdiffusion rates of the two components are found to be unequal, giving unequal diffusion coefficients, a net mass flow across the interface, and an asymmetric interfacial composition profile. The structure and kinetics confirm the predictions of the reptation theory. The interfacial thickness d grows with t^1/2, and the interdiffusion coefficient is proportional to M^?2, where t is the time and M is the molecular weight. The scaling law for the interfacial thickness is therefore d ∝ M^?1t^1/2. The number of chains per unit area crossing the original interface reaches a constant value independent of diffusion time after a short induction time on the order of the tube disengagement time (about 0.1–10 s in the present cases depending on the molecular weights). The adhesive bond strength σ is scaled by σ ∝ t^1/4M^?1/2 and σ/σ∞ ∝ t^1/4M^?1/2 [1- (M_c/M)]^?1, where σ _∞is the σ at infinite molecular weight and M_c is the entanglement molecular weight.     

Stress overshoot of polymer solutions at high rates of shear; Polystyrene with bimodal molecular weight distribution

Overshoot of shear stress, σ, and the first normal stress difference, N₁, in shear flow was investigated for dilute solutions of polystyrene with very high molecular weight in concentrated solution of low M PS. In the case that the matrix was a nonentangled system, behavior of overshoot was similar to that of dilute solution of high M PS in pure solvent. The magnitudes of shear, γ_σm and γ_Nm, corresponding to the peaks of σ and N₁ lay on the universal functions of γ˙τ_R, respectively, proposed for dilute solutions in pure solvent. Here τ_R is the Rouse relaxation time for high M PS in the blend evaluated from dynamic modulus at high frequencies. In the case that the matrix was an entangled system, an additional σ peak was observed at high rates of shear at times corresponding to γ_σm = 2–3. This peak can be assigned to the motion of low M chains in entanglement network. When the matrix was entangled, stress overshoot was observed even at relatively low rates of shear, say γ˙τ_R < 10⁻². This is probably due to the motion of high M chains in entanglement of all the chains. In this case the γ_σm and γ_Nm values were higher than those expected for entangled chains of monodisperse polymer in pure solvent. © 2000 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 38: 2043–2050, 2000     

Relaxation dynamics of salt‐free polyelectrolyte solutions using flow birefringence and rheometry

Relaxation dynamics of salt‐free, aqueous solutions of sodium poly(styrene sulfonate) (NaPSS) were investigated by mechanical rheometry and flow birefringence measurements. Two semidilute concentration regimes were studied in detail for a range of polymer molecular weights. At solution concentrations c < 10 mg mL, limiting shear viscosity η₀ was found to scale with molecular weight and concentration as η₀ ∼ c^0.5M_w over nearly two decades in concentration. At higher solution concentrations, c > 10 mg mL, a change in viscosity scaling was observed η₀ ∼ c^1.5M, consistent with a change from simple Rouse dynamics for unentangled polyions to near‐perfect reptation dynamics for entangled chains. Characteristic relaxation times τ deduced from shear stress and birefringence relaxation measurements following start‐up of steady shearing at high rates reveal very different physics. For c < 10 mg mL, both methods yield τ ∼ c^−0.42M and τ ∼ c⁰M for c > 10 mg mL. Curiously, the concentration scalings seen in both regimes are consistent with theoretical expectations for salt‐free polyelectrolyte solutions undergoing Rouse and reptation dynamics, respectively, but the molecular weight scalings are not. Based on earlier light scattering studies using salt‐free NaPSS solutions, we contend that the unusual relaxation behavior is likely due to aggregation and/or coupled polyion diffusion. Simultaneous stress and birefringence measurements suggest that in concentrated solution, NaPSS aggregates are likely well permeated by solvent, supporting a loose collective of aggregated chains rather than the dense polymer aggregates previously supposed. Nonetheless, polyion aggregates of either variety cannot account for the inverse dependence of relaxation time on polymer molecular weight for c < 10 mg mL. © 1999 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 37: 825–835, 1999     

Oscillatory shear measurements on polystyrene melts in the terminal region

Oscillatory shear measurements have been made on a range of anionic polystyrene melts of molecular weights 1000--500,000. For M < 5000 the polymer chain is too short to act as a Gaussian coil and hence the compliance of the melt is very low. For 10,000 < M < 100,000 the compliance of the melt follows the Rouse model of the elasticity of isolated polymer molecules. It is necessary to use the Ferry, Landel and Williams extension of the Rouse theory for M > 40,000 to allow for the effect of entanglements on the complex modulus. For M > 200,000 the entangelment network dominates the compliance and the Rouse theory is no longer applicable.     

Nonlinear viscoelasticity of polystyrene solutions. I. Strain-dependent relaxation modulus

Strain-dependent relaxation moduli G(t,s) were measured for polystyrene solutions in diethyl phthalate with a relaxometer of the cone-and-plate type. Ranges of molecular weight M and concentration c were from 1.23 × 10⁶ to 7.62 × 10⁶ and 0.112 to 0.329 g/cm³. Measurements were performed at various magnitudes of shear s ranging from 0.055 to 27.2. The relaxation modulus G(t,s) always decreased with increasing s and the relative amount of decrease (i.e.,–log[G(t,s)/G(t,0)]) increased as t increased. However, the detailed strain dependences of G(t,s) could be classified into two types according to the M and c of the solution. When cM < 10⁶, the plot of log G(t,s) versus log t varied from a convex curve to an S-shaped curve with increasing s. For solutions of cM > 10⁶, the curves were still convex and S-shaped at very small and large s, respectively, but in a certain range of s (approximately 3 < s < 7) log G(t,s) decreased rapidly at short times and then very slowly; a peculiar inflection and a plateau appeared on the plot of log G(t,s) versus log t. The strain-dependent relaxation spectrum exhibited a trough at times corresponding to the plateau of log G(t,s). The longest relaxation time τ₁(s) and the corresponding relaxation strength G₁(s) were evaluated through the “Procedure X” of Tobolsky and Murakami. The relaxation time τ₁(s) was independent of s for all the solutions studied while G₁(s) decreased with s. The reduced relaxation strength G₁(s)/G₁(0) was a simple function of s (The plot of log G₁(s)/G₁(0) against log s was a convex curve) and was approximately independent of M and c in the range of cM <10⁶. This behavior of G₁(s)/G₁(0) was in agreement with that observed for a polyisobutylene solution and seems to have wide applicability to many polymeric systems. On the other hand, log G₁(s)/G₁(0) as a function of log s decreased in two steps and decreased more rapidly when M or c was higher. It was suggested that in the range of cM < 10⁶, a kind of geometrical factor might be responsible for a large part of the nonlinear behavior, while in the range of cM > 10⁶, some “intrinsic” nonlinearity of the entanglement network system might be important.     

Molecular bases of stress relaxation and viscosity of polymers, 1a) New empirical relations

Experiments with several polymers led to new relations which describe stress relaxation at small shears and show the influence of the molecular structure on relaxation. Bubble-free melts of anionically polymerized polystyrenes with the molecular masses (relative molecular mass is referred to as molecular mass throughout this paper)M=6 10² to 1.8 10⁶ g mol^–1 and their blends are studied at several temperatures and at timest>3 ms. As the stress (10² to 10⁶ Pa) deforms the rheometer, a new mathematical method was developed to correct the relaxations. The energy-elastic stress is separated from the entropy-elastic stress. The relaxations of the latter do not confirm the Rouse theory as is proportional to exp (–¦t ^1/2¦) forM<4000. For M>4000 the relaxations deviate from the theoretical functions as well. The initial modulus G(0) then depends onM, and is not proportional to ¦t ^–1/2¦. A new and simple function describes the relaxation of melts with entangled molecules at long times. The influences of concentration and chemical structure on relaxation are formulated for weak intermolecular forces. Published data for the constantD of self-diffusion are used in calculating values ofD for a constant free volume making use of the initial velocity of relaxation and its dependence on temperature. ThenDM ^–1 holds forM<1.3 10⁴ andDM ^–2 for largerM. Correct values of viscosity are calculated from the empirical functions of stress relaxation.Part 2 cf.In memoriam Professor Dr. Drs. h.c. Otto Bayer     

Polymer liquid dynamics studied via dielectric normal mode spectroscopy

Dielectric spectroscopy was applied to study dynamics of cis-polyisoprenes (PI), used as type-A probe, in blends with polybutadiene (PB) and in block copolymers with polystyrene (PS) of SI- and SIS-type. For dilute high-molecular weight (M) PI/low- M PB blends we identified Rouse mode with M²-dependent relaxation time τ, while for low- M Pi/high- M PB blends, we identified pure reptation mode with M³ -dependent τ. In between τ ∞ M^α with the exponent α varying from 2 to 3 as M_B was increased, as suggested by Graessley with constraint release via tube renewal mechanism. For the blends with the MW ratio M_I/M_B = 2.5, we found bulk polymer behaviour with τ ∞ M^3.5, in which competition between pure reptation and tube renewal appear to be essential and the contribution of contour length fluctuation may be ruled out. For SI-diblock copolymers between Tg(I) < T < Tg(S) we observed normal modes of I-block chains tethered on rigid S-domains. The mode distribution as judged from the dielectric loss ε” curves was dependent on the domain morphology, reflecting restricted motions of crowded I-tethered chains. For SIS-triblock copolymers normal modes became appreciable, even below their critical solution temperature, in the range of T > Tg(S), exhibiting broadening due presumably to their micro-phase-separated structure. The relaxation mechanisms for such end-capped I-chains in SIS-triblock copolymers could be junction hopping in those with isolated S-domains but chain rotating in those with S continuous morphologies.     

Cooperative polymer dynamics under nanoscopic pore confinements probed by field-cycling NMR relaxometry

Reptational dynamics of bulk polymer chains on a time scale between the Rouse mode relaxation time and the so-called disengagement time is not compatible with the basic thermodynamic law of fluctuations of the number of segments in a given volume. On the other hand, experimental field-cycling NMR relaxometry data of perfluoropolyether melts confined in Vycor, a porous silica glass of nominal pore dimension of 4 nm, closely display the predicted signatures for the molecular weight and frequency dependences of the spin-lattice relaxation time in this particular limit, namely T1 proportional M-1/2nu1/2. It is shown that this contradiction is an apparent one. In this paper a formalism is developed suggesting cooperative chain dynamics under nanoscopic pore confinements. The result is a cooperative reptational displacement phenomenon reducing the root-mean-squared displacement rate correspondingly but showing the same characteristic dependences as the ordinary reptation model. The tube diameter effective for cooperative reptation is estimated on this basis for the sample system under consideration and is found to be of the same order of magnitude as the nominal pore diameter of Vycor.     

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.     

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.