Monomeric Friction Factor Research Articles

Rouse–Bueche theory and the calculation of the monomeric friction coefficient in a filled system

ABSTRACTDirect experimental access to the monomeric friction coefficient (ζ0) relies on the availability of a suitable polymer dynamics model. Thus far, no method has been suggested that is applicable to filled systems, such as filled rubbers or microphase‐segregated A–B–A thermoplastic elastomers (TPEs) atTg,B < T < Tg,A. Building upon the procedure proposed by Ferry for entangled and unfilled polymer melts, the Rouse–Bueche theory is applied to an undiluted triblock copolymer to extractζ0from the linear behavior in the rubber‐glass transition region, and to estimate the size of Gaussian submolecules. When compared at constantT – Tg, the matrix monomeric friction factor is consistent with the corresponding value for the homopolymer melt. In addition, the characteristic Rouse dimensions are in good agreement with independent estimates based on the Kratky–Porod worm‐like chain model. These results seem to validate the proposed approach for estimatingζ0in filled systems. © 2016 Wiley Periodicals, Inc. J. Polym. Sci., Part B: Polym. Phys.2016,54, 1437–1442

Determination of molecular weight distribution and composition dependence of monomeric friction factors from the stress relaxation of ultrahigh molecular weight polyethylene gels

The stress relaxation of ultrahigh molecular weight polyethylene (UHMWPE) gels in a broad composition range ( 0.025≤ϕP≤1.0) was studied within the framework of the reptation model. The molecular weight distribution (MWD) was determined at an optimum gel concentration ( ϕP=0.365) by regularization analysis of the stress relaxation data. The calculated relaxation moduli [ G(t)] by using MWD evaluated at a certain composition (UHMWPE gels, ϕP=0.365) were consistent with the experimental G(t) of other compositions in a broad range ( 0.218≤ϕP≤1.0). That is, the MWD (regularization method) was self-consistent within the framework of the reptation model. The composition dependence of the monomeric friction factor ( ζeff) of the UHMWPE gels was examined by use of the double reptation (DR) model with the determined MWD. The Ferry's free volume model of the molecular weight dependence of ζeff can be extended to the composition dependence of ζeff. In the low-concentration gels ( 0.025≤ϕP≤0.17), the experimental G(t)...

A Combined Analysis of PVT and Rheological Measurements: Novel Results for Three Amorphous Polymers

AbstractThe fractional free volume f of polycarbonate (PC), polystyrene (PS) and poly(methyl methacrylate) (PMMA) is determined by pressure–volume–temperature (PVT) measurements using the Simha–Somcynsky equation of state. Results are compared with the free volume obtained by viscoelastic measurements. Temperature and pressure dependent viscosity and free volume results are correlated using the characteristic ratio C∞ and the parameter B of the monomeric friction factor ζ = ζexp B/f. This leads to obtain new results on the temperature and pressure dependence of C∞, as well as on the pressure dependence of B parameter.magnified image

Component Terminal Dynamics in Poly(ethylene oxide)/Poly(methyl methacrylate) Blends

The rheological response of high molecular weight tracer chains of poly(ethylene oxide) (PEO) and poly(methyl methacrylate) (PMMA), separately blended with low molecular weight PEO/PMMA matrices of varying composition, were obtained over a 120 deg temperature window. Monomer friction factors, ζ, for each component as a function of temperature and matrix composition were extracted by various means, all of which yielded consistent results. The tracer diffusivities of low molecular weight PEO chains in some of the same blends were obtained by forced Rayleigh scattering, and the resulting friction factors agree well with those obtained using rheology. The overall results show that the mobilities of both PMMA and PEO are strongly composition-dependent in PMMA-rich blends, their monomeric friction factors dropping precipitously upon addition of small amounts of PEO. The composition dependence of the PMMA ζ is strong over a wide composition range, whereas that of PEO is only significant at high PMMA content blends. Friction factors for the PEO component are significantly larger than those inferred from segmental dynamics measurements reported in the literature. The Lodge−McLeish self-concentration model, as usually applied, is unable to predict the observed behavior of either component. However, the data can be nearly quantitatively described using a simple but empirical mixing rule. Selected experiments were repeated using a PEO matrix with methoxyl rather than hydroxyl end groups; except for the change in glass transition temperature for the PEO, no significant effects were observed.

Effect of Short-Chain Branching on the Rheology of Polyolefins

A series of carefully synthesized ethylene/butene copolymers (0.2−0.85 mole fraction butene) were synthesized using metallocene catalysts to probe the relationship between the chemical architecture of polyolefins and their rheology. The dependence of , the plateau modulus, on mb, the molecular weight per backbone bond, which we have seen previously for polyolefins is obeyed by these copolymers. Moreover, the modulus at the frequency at which G‘ = G‘ ‘ also scales with copolymer composition. We also show that the van Gurp−Palmen plots (|G*| vs δ) for these copolymers show a universal behavior, which could allow for the characterization of copolymer composition by rheology. We demonstrate that combining small-amplitude oscillatory shear data with stress relaxation experiments the theory of linear viscoelasticity allows us to determine low-frequency behavior much more rapidly. The zero shear viscosity of these ethylene copolymers also shows a regular dependence on the copolymer composition, and we propose a new scaling relationship for zero shear viscosity in terms of mb. We show that this applies to a wide range of polyolefins and show similar relations for the equilibration time, τe, and monomeric friction factor, ζ.

Dynamics of a poly(ethylene oxide) tracer in a poly(methyl methacrylate) matrix: Remarkable decoupling of local and global motions

The tracer diffusion coefficient of unentangled poly(ethylene oxide) (PEO, M=1000 gmol) in a matrix of poly(methyl methacrylate) (PMMA, M=10 000 gmol) has been measured over a temperature range from 125 to 220 degrees C with forced Rayleigh scattering. The dynamic viscosities of blends of two different high molecular weight PEO tracers (M=440 000 and 900 000 gmol) in the same PMMA matrix were also measured at temperatures ranging from 160 to 220 degrees C; failure of time-temperature superposition was observed for these systems. The monomeric friction factors for the PEO tracers were extracted from the diffusion coefficients and the rheological relaxation times using the Rouse model. The friction factors determined by diffusion and rheology were in good agreement, even though the molecular weights of the tracers differed by about three orders of magnitude. The PEO monomeric friction factors were compared with literature data for PEO segmental relaxation times measured directly with NMR. The monomeric friction factors of the PEO tracer in the PMMA matrix were found to be from two to six orders of magnitude greater than anticipated based on direct measurements of segmental dynamics. Additionally, the PEO tracer terminal dynamics are a much stronger function of temperature than the corresponding PEO segmental dynamics. These results indicate that the fastest PEO Rouse mode, inferred from diffusion and rheology, is completely separated from the bond reorientation of PEO detected by NMR. This result is unlike other blend systems in which global and local motions have been compared.

A framework for predicting the viscosity of miscible polymer blends

A new model for the viscosity of miscible polymer blends is described. It combines a mixing rule for chain relaxation, based on the double reptation concept as adopted by Tsenoglou (1989), with a calculation of the concentration dependence of the monomeric friction factors, based on the self-concentration model of Lodge and McLeish (2000). The model is successful in anticipating the experimentally observed curvatures in plots of viscosity versus composition, and the predictions are in promising agreement with the data for several systems. The model introduces no freely adjustable parameters, and all the necessary parameters may be determined from measurements on pure components.

Composition and Temperature Dependence of Terminal and Segmental Dynamics in Polyisoprene/Poly(vinylethylene) Blends

The frequency-dependent shear moduli were measured over a range of temperature for polyisoprene (PI, M = 22 000 and 78 000), poly(vinylethylene) (PVE, M = 10 000 and 120 000), and their blends. The longest relaxation times of each component in the blends were extracted by a modified Tsenoglou mixing rule, and converted to monomeric friction factors via the reptation model. Dielectric relaxation and pulsed field gradient NMR were used to follow the terminal relaxation and diffusion, respectively, of a low molecular weight PI (M = 1300) in blends with perdeuterated PVE (M = 2300), for a range of temperature and composition. In this case monomeric friction factors were extracted via the Rouse model. The friction factors for PI via all three techniques agreed quantitatively, and the friction factors for both components were independent of molecular weight. Segmental dynamics of each component in PI/PVE blends by NMR methods, reported in the literature, could be compared directly with the terminal relaxation data. In all cases, the segmental and terminal dynamics exhibited equivalent dependences on temperature and composition. The predictions of the model of Lodge and McLeish were compared to the combined data, by calculating component self-concentrations and effective glass transition temperatures using a single temperature-independent length scale. As anticipated in the original model, this length scale is close to the Kuhn length of the particular component. In the case of PI, the model provides a quantitative description over the entire composition and temperature range studied. For PVE the agreement is not as quantitative, but it is still quite satisfactory.

Segmental dynamics of disordered styrene–isoprene tetrablock copolymers

The local segmental dynamics of four styrene-b-isoprene-b-styrene-b-isoprene (SISI) tetrablock copolymers with different styrene composition fs and constant total degree of polymerization N≈120 has been studied in the disordered state in the nano-picosecond time scale, by incoherent quasielastic neutron (QENS), and Brillouin (BS) and depolarized Rayleigh (DRS) light scattering. Far above the glass transition temperature, all three techniques demonstrate the presence of two distinct time scales from which the fast segmental relaxation was quantitatively resolved. This process is associated with the mobility of the polyisoprene (PI) component, and is moderately slower and possesses a broader distribution of relaxation times than in bulk PI. The comparison between the correlation times of DRS and the characteristic times of QENS suggest that segment (hydrogen nucleus) diffusion over a distance of ≈0.8 nm suffices for the loss of local orientation correlations. The faster times of the BS experiment correspond to shorter displacements, ≈0.3 nm. These results demonstrate that the segmental dynamics of the PI are much faster than would be inferred from the monomeric friction factor of PI previously extracted by diffusion and viscosity measurements in the same tetrablock matrices. This, in turn, indicates a substantial local spatial heterogeneity in the segmental dynamics. The slow process is due to the PS segments, which do not relax, appreciably on the time scales accessible here.

Composition and Temperature Dependence of Monomer Friction in Polystyrene/Poly(methyl methacrylate) Matrices

The composition (φ) and temperature dependence (T) of the monomeric friction factor (ζ) has been examined for styrene (S) and methyl methacrylate (MMA) in five SMMA diblock copolymers and the corresponding homopolymers. In all cases the degree of polymerization was approximately 150, sufficiently low to ensure that the chain dynamics follow the Rouse model and that the copolymers are far above the order−disorder transition. The five diblock copolymers had styrene compositions of 91, 70, 39, 19, and 9 wt %. Values of an effective friction factor, ζeff, were obtained from measurements of the steady flow viscosity via the Rouse model, whereby ζeff represents an average over the S and MMA contributions. Values of the component friction factors, ζPS and ζPMMA, were extracted from forced Rayleigh scattering tracer diffusion measurements of dye-labeled PS, PMMA, and SMMA chains. In accord with previous studies, ζPMMA in pure PMMA is significantly greater than ζPS in pure PS, whether compared at equal T or equal ...

A generalized method to calculate diffusion rates in polydisperse systems. Further results on Rouse dynamics in the concentrated regime

Experiments designed to thoroughly test a recently proposed generalized method to calculate diffusion rates in polydisperse systems have been carried out. Polydisperse polystyrene (PS) samples were allowed to diffuse in a poly(phenylene oxide) (PPO) matrix. Designed blends were made from anionically polymerized PS with molecular weights which cover most of the ranges where Rouse dynamics control the diffusion processes. The diffusion temperatures range from (T g - 1 K) to (T g + 105 K), causing the monomeric friction factor values for PS to change by up to seven orders of magnitude along the diffusion coordinate. Calculations performed with the above mentioned method agree with Raman and DMA experimental data.

Self-Concentrations and Effective Glass Transition Temperatures in Polymer Blends

In a miscible polymer blend the local environment of a monomer of type A will, on average, be rich in A compared to the bulk composition, φ, and similarly for B; this is a direct consequence of chain connectivity. As a result, the local dynamics of the two chains may exhibit different dependences on temperature and overall composition. By assigning a length scale (or volume) to particular dynamic mode, the relevant “self-concentration” φs can be estimated. For example, we associate the Kuhn length of the chain, lK, with the monomeric friction factor, ζ, and thus the composition and temperature dependences of ζ should be influenced by φs calculated for a volume V ∼ lK3. An effective local composition, φeff, can then be calculated from φs and φ. As lower Tg polymers are generally more flexible, the associated φ s is larger, and the local dynamics in the mixture may be quite similar to the pure material. The higher Tg component, on the other hand, may have a smaller φs, and thus its dynamics in the mixture w...

Diffusivity and Viscosity of Concentrated Hydrogenated Polybutadiene Solutions

We report measurements of diffusion (D) and viscosity (η) in hydrogenated polybutadiene (hPB)/alkane solutions. The volume fractions (φ) extend from 0.20 up to the melt and the molecular weights (M) from 4900 up to 440 000; the number of entanglements per chain, M/Me, ranges up to about 450. The temperature dependences of both D and η are similar, consistent with literature values, and independent of φ and M. We thus conclude that the monomeric friction factor is also independent of φ and M. For all samples with M/Me ≥ 3 the data are consistent with the scaling relations D ∼ M-2.4±0.1φ-1.8±0.2 and η ∼ M3.4±0.1φ3.8±0.2; within the precision of the data the M exponents do not depend on φ, and the φ exponents do not depend on M. The M exponent for D in solution is consistent with previously reported studies for φ ≤ 0.4 but apparently conflicts with the reptation value of −2.0 in the melt. However, by comparing with the literature melt self-diffusion data for hPB and six other polymers, it is clear that the exponent is, in fact, universally stronger than −2.0. Furthermore, if a single universal exponent is assumed, the best value based on all the data is −2.28 ± 0.05. Our hPB data resolve the longstanding anomaly that the longest relaxation times τ1 based on translation (∼Rg2/D) and orientational relaxation (∼η) appeared to have different M dependences. The data also indicate that for both solutions and melts the dynamics are determined primarily by the number of entanglements per chain. The data are compared with current models of polymer dynamics, both reptative and nonreptative. In the former case it is clear that a self-consistent theory incorporating both “constraint release” and “contour length fluctuations” is required.

A model for the zero shear viscosity

An empirical equation for the number of entanglements per molecule has been proposed, which applies over all the molecular weight range. On this ground a simple equation for the zero shear viscosity of monodisperse polymer melts, η 0 , has been worked out that appears able to properly take into account the sharp transition of viscosity between the monomeric and the entanglement regimes. The molecular parameters appearing in the new viscosity equation are: the monomeric molecular weight m 0 , the monomeric friction factor ζ 0 , the molecular weight M, the average molecular weight between entanglements M e , and the entanglement friction factor ζ e 3.4 , This last parameter was evaluated for a number of monodisperse polymers.

Styrene and isoprene friction factors in styrene-isoprene matrices

Monomeric friction factors, Ξ, for polystyrene (PS), polyisoprene (PI), and a polystyrene–polyisoprene (SI) diblock copolymer have been determined as a function of temperature in four poly(styrene-b-isoprene-b-styrene-b-isoprene) tetrablock copolymer matrices. The Rouse model has been used to calculate the friction factors from tracer diffusion coefficients measured by forced Rayleigh scattering. Within the experimental temperature range the tetrablock copolymers are disordered, allowing for measurement of the diffusion coefficient in matrices with average compositions determined by the tetrablock copolymers (23, 42, 60, and 80% styrene by volume). Remarkably, for a given matrix composition the styrene and isoprene friction factors are essentially equivalent. Furthermore, at a constant interval from the system glass transition temperature, Tg, all of the friction factors (obtained from homopolymer, diblock copolymer, and tetrablock copolymer dynamics) agree to within an order of magnitude. This is in marked contrast to results for miscible polymer blends, where the individual components generally have distinct composition dependences and magnitudes at constant T − Tg. The homopolymer friction factors in the tetrablock matrices were systematically slightly higher than those of the diblock, which in turn were slightly higher than those of the homopolymers in their respective melts, when all compared at constant T − Tg. This is attributed to the local spatial distribution of styrene and isoprene segments in the tetrablocks, which presents a nonuniform free energy surface to the tracer molecules. © 1998 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 36: 3079–3086, 1998

Self-diffusion of a polystyrene-polyisoprene block copolymer

Forced Rayleigh scattering (FRS) has been used to measure the self-diffusion coefficient, D, of a lamellar-forming polystyrene-polyisoprene diblock copolymer (M PS = 1.0 x 10 4 , M PI = 1.3 x 10 4 ) as a function of temperature. The measurements traverse the order-disorder transition (ODT), which occurs at 160°C. There is no obvious change in either D or the temperature dependence of D at the ODT, in agreement with measurements on several other systems. Electron microscopy confirms that the sample in the ordered state is quenched, with no long-range orientation of lamellae, and a typical grain size well below 1 μm. In contrast to previous measurements on a similar styrene-isoprene diblock, these FRS signals are well-described by single exponential decays ; this may be largely attributed to differences in average grain size. The temperature dependence of D is modeled with several empirical expressions, based on the known monomeric friction factors for pure polystyrene and pure polyisoprene, but without quantitative success. These results underscore the need for a greater understanding of the composition and temperature dependences of local friction in polymer mixtures.