Characterization of polymer molecular weight distribution by transient viscoelasticity: Polytetrafluoroethylenes

The relationship between the relaxation time spectrum H(τ) in the terminal zone and the volume-fraction differential molecular-weight-distribution function P(M) is derived by considering binary chain contacts for stress transmission, where β and λ are constants for a given chemical type. This is used to determine the molecular-weight-distribution curves from the stress relaxation modulus spectrum (above the crystal melting point) at 370°C for a number of commercial and experimental poly(tetrafluoroethylenes) (PTFEs). It is found that PTFEs typically have bimodal molecular-weight distributions. The lower-molecular-weight peak conforms essentially to the “most-probable” distribution, and the higher-molecular-weight peak to the binary coupling distribution. The entanglement molecular weight M_e is 5490, and the number of main-chain atoms between entanglement points is 110, consistent with a flexible chain. The zero-shear melt viscosity at 370°C is η₀ = 1.79 × 10^?13 M_w^2.94±0.13, where η₀ is in Pa.s and M_w/M_e = 2,000 to 12,000. The monomeric friction coefficient is also determined.     

Variation of periodic crazing based on polymer blends of an ultra‐high and a low molecular weight poly(methyl methacrylate)

Periodic crazes are caused in a polymer film by the unique mechanical method using bending. Generation of a craze depends on entanglements of the molecular chains of a polymer. Therefore, control of composite morphology of periodic crazes was attempted by varying the entanglements of molecular chains. An effective entanglement network became sparse by polymer blends of an ultra‐high molecular weight polymethylmethacrylate (PMMA) and a low molecular weight PMMA. Consequently, the composite morphology of periodic crazes caused in the blend film varied. In other words, the periodic craze can be used for the evaluation of the effective entanglements. In addition, it was figured out that PMMA of which the number‐average molecular weight (M_n) is less than twice of the effective entanglement molecular weight (M_e^*) works as a plasticizer in the blend film. And also, it was revealed that the mechanical properties of the blend film decreased dramatically at M_n ≒ 6M_e^*. © 2016 Wiley Periodicals, Inc. J. Appl. Polym. Sci. 2016 , 133, 44332.     

The dispersion of magnetic nanorods in poly(2‐vinylpyridine)

The dispersion of magnetic nanorods in poly(2‐vinylpyridine) (PVP) as a function of rod length, particle loading and molecular weight of PVP was investigated. The nanorods were organized into small spherical clusters at low particle loading. Further increasing the particle concentration caused an increase in the size of the aggregates. Additionally, the internal structure of the nanorods developed into a raft‐like structure, forming rectangular clusters. The incorporation of longer nanorods in the PVP amplified the magnetic interaction energy, which created conditions to induce extensive aggregation. The entanglement of the polymer also played an important role in the arrangement of the nanorods. This behavior could be categorized into two regimes, M_PVP > M_e and M_PVP < M_e, where M_PVP and M_e are the number‐average molecular weight and entanglement molecular weight of PVP, respectively: for M_PVP > M_e, PVP formed entanglements that prevented nanorods from extensive aggregation; for M_PVP < M_e, PVP could not form entanglements, and nanorods could move freely in the PVP, and thus significant rod aggregation occurred. Simple calculations to assess the contribution of the magnetic interaction, the van der Waals interaction and the free energy of mixing of the system to the arrangement of magnetic nanorods in the homopolymer are discussed. © 2013 Society of Chemical Industry     

Role of chain entanglements on fiber formation during electrospinning of polymer solutions: good solvent, non-specific polymer-polymer interaction limit

Chain entanglements are one of many parameters that can significantly influence fiber formation during polymer electrospinning. While the importance of chain entanglements has been acknowledged, there is no clear understanding of how many entanglements are required to affect/stabilize fiber formation. In this paper, polymer solution rheology arguments have been extrapolated to formulate a semi-empirical analysis to explain the transition from electrospraying to electrospinning in the good solvent, non-specific polymer-polymer interaction limit. Utilizing entanglement and weight average molecular weights (M_e, M_w), the requisite polymer concentration for fiber formation may be determined a priori, eliminating the laborious trial-and-error methodology typically employed to produce electrospun fibers. Incipient, incomplete fiber formation is correctly predicted for a variety of polymer/solvent systems at one entanglement per chain. Complete, stable fiber formation occurs at ≥2.5 entanglements per chain.     

Dependence of zero-shear viscosity and steady-state compliance on molecular weight between entanglements for ethylene-cycloolefin copolymers

Dynamic viscoelastic properties of a series of cyclic olefin copolymers have been investigated. The specimens differ in total molecular weight as well as molecular weight between entanglements. The angular frequency (ω) dependence curves of dynamic storage and loss moduli (G′ and G″, respectively) of the specimens have shown that G′ ∝ ω² and G″ ∝ ω in the terminal region, and a plateau region at high ω. On the basis of the experimental results, the dependence of total molecular weight as well as molecular weight between entanglements has been examined for zero-shear viscosity (η₀) and steady-state compliance (J_e). It is shown that for the melts of the copolymers in the entangled regime, M_w being the weight-average molecular weight and M_e the molecular weight between entanglements. The steady-state compliance J_e for the melts scales with M_e and M_w as .     

Entanglement network of chitin and chitosan in ionic liquid solutions

Concentrated solutions of a chitin from squid pens and of two commercial samples of chitosan were successfully prepared by using an ionic liquid 1‐butyl‐3‐methylimidazolium acetate as a solvent. The dynamic viscoelasticity data for the solutions exhibited rubbery plateaus, indicating the existence of entanglement network of chitin and chitosan in the solutions. To characterize the network, the values of the molecular weight between entanglements (M_e) for chitin and chitosan in the solutions were determined from the plateau moduli. Then the values of M_e in the molten state (M_e,melt), a material constant reflecting the inherent nature of polymer species, for chitin and chitosan were estimated to be 1.7 × 10³ and 3.0 × 10³, respectively. It was found that there was a significant difference in M_e,melt between chitin and chitosan. Compared with other polysaccharides such as cellulose and agarose in terms of the number of monosaccharide units between entanglements (N_unit), chitin had significantly smaller N_unit of 8, while chitosan had equivalent N_unit of 19. © 2013 Wiley Periodicals, Inc. J. Appl. Polym. Sci. 130: 2439–2443, 2013     

Polymer Network Mobility and Environmental Stress Cracking Resistance of High Density Polyethylene

Zero shear viscosity and molecular weight between entanglements (M _e ) are determined from dynamic oscillatory shear experiments. Lower M _e value means higher number of entanglements in the system and is associated with increasing strain hardening stiffness. With the understanding that strain hardening is related to environmental stress cracking resistance (ESCR) of high density polyethylene (HDPE), M _e is then related to the ESCR of several resins in this study. The inversely proportional relationship between M _e and ESCR indicates that low network mobility due to an increasing number of chain entanglements increases the ESCR of HDPE.     

An empirical model relating the molecular weight distribution of high-density polyethylene to the shear dependence of the steady shear melt viscosity

An empirical model has been developed to relate molecular weight distribution to the shear dependence of the steady shear viscosity in high-density polyethylene melts. It uses a molecular weight, M_c, which partitions molecular weights into two classes; those below M_c contribute to the viscosity as they do at zero shear, and those above M_c contribute to the viscosity as though they were of molecular weight M_c at zero shear. Each individual molecular weight species contributes on the basis of its weight fraction. M_c is proposed to be a unique function of the shear rate. Using this method of treating the molecular weight distribution, and the zero shear relation for relating η0 to molecular weight, the calculated steady shear viscosities at various shear rates for polyethylene samples of widely varying polydispersities agree well with experimental results. The model makes no judgment on the existence or importance of entanglements in non-Newtonian behavior since it has no specific parameters involving an entanglement concept. Use of the model suggests that for the samples studied, only the upper portion of the molecular weight distribution contributes toward the experimentally observed decrease of steady shear viscosity with shear rate for shear rates of up to 10,000 sec^?1. The lower molecular weight species are assumed to behave in a Newtonian manner.     

Craze microstructure and molecular entanglements in polystyrene-poly(phenylene oxide) blends

Polystyrene and poly(phenylene oxide) are miscible over the entire range of compositions. Thin films of five blends of high molecular weight polystyrene (PS) with high molecular weight poly(phenylene oxide) (PPO), and four blends of low molecular weight PS (whose molecular weight lies below its entanglement molecular weight M_e) with the same PPO have been prepared. Following bonding of these films to copper grids, crazes were grown by uniaxial straining in air. Suitable crazes were then observed by transmission electron microscopy. From microdensitometry of the image plates it is possible to measure the extension ratio λ_craze within crazes in the nine blends. These measured values are compared with predicted values of λ_max, computed from λ_max = I_ed, where I_e is the chain contour length between entanglements and d is the root mean square end-to-end distance for a chain of molecular weight M_e. For the high molecular weight PS blends λ_max depends on the entanglement properties of both PS and PPO chains. For the low molecular weight PS blends, the PS chains cannot form part of the entanglement network and the correct value of λ_max is obtained from appropriate scaling of the pure PPO value. Comparison of λ_craze and λ_max for both types of blends shows excellent agreement, demonstrating the importance of the entanglement network in determining craze parameters and hence the toughness of a given polymer.     

Characterization of cis-1,4-polyisoprene polymerized with lanthanide catalyst system (Ln-PIR)

The PIR raw rubber samples, Ln–PIR and Ti–PIR, were subjected to molecular characterization, which shows that Ln–PIR contains microgel particles, but the least branching in its macromolecular chains; its cis-1,4 content is about 96% and molecular weight distribution is rather broad. The value of α in the Mark-Houwink viscosity equation for molecular weight M_η is determined as 0.70. The abnormal stress–relaxation behavior can be normalized by the introduction of an entanglement reduction factor, e.g., M_η/M_c for maximum relaxation time. This reduction is subsequently verified by evaluating the molecular weight dependence on bulk viscosity with 3.45th power. The starting of a yield process is equivalent to that of a disentanglement process, since both processes have approximately the same activation energy, 8 kJ/mol. For the onset of yield or of entanglement, the critical molecular weight M_c as estimated independently by yield strength method or by relaxation spectrum is equal to (5.4 ± 0.2) × 10⁴. It is confirmed by the reduced yield strength method by calculating M_e from the equation M_e = 3g_NρRT/E_eN with g_N = 1.22 and then by extrapolation, where E_eN is the equilibrium modulus due to entanglement.     

Influence of molecular weight on the fracture of poly(methyl methacrylate) (PMMA)

A model is proposed to explain the dependence of fracture parameters on the molecular weight of glassy polymers. The model assumes that the fracture event occurs in two stages; the first involves the orientation of polymer chain segments between entanglement points and the second, the fracture itself. A value has been calculated, (～0.6J m^?2), for the fracture surface energy corresponding to the lower critical molecular weight between entanglements, M=M_e. Allowing for the simplifying assumptions made in its derivation, this value is in good agreement with that found experimentally. It is proposed that, after the chain segments between entanglements crossing a plane have been fully extended, two possible mechanisms are involved; chain ‘pull-out’ up to a maximum governed by the time scale of the local fracture event, or chain scission. Using the concept of a reptating chain it is proposed that above M ～2 M_e there is a relationship between the fracture energy (γ) and the molecular weight of the form γ∝ ∝M² up to a critical value of M, above which γ is constant. It has been shown that there is some agreement with experimental relationships determined independently.     

Solvent dependence of the viscous and viscoelastic properties of a high molecular weight polystyrene in moderately concentrated solution

The steady-flow viscosity and viscoelastic behaviour of two solutions of a sensibly monodisperse polystyrene of high molecular weight (M_w = 498 000) have been measured over a temperature range of 100°C for identical concentrations of 20.55 wt.%. Toluene and methyl ethyl ketone were chosen as the two low viscosity solvents having, respectively, good and marginal thermodynamic affinities. Dynamic viscoelastic measurements were made at a frequency of 41 kHz using travelling torsional waves. At this frequency, both solutions exhibit behaviour characteristic of the rubbery region, and the ratio of the dynamic viscosity normalised by dividing by the corresponding solvent viscosity is independent of the solvent until the onset of the glass transition region with decreasing temperature. The storage shear modulus of the toluene solution in the rubbery region is higher than for the MEK solution, indicating a higher entanglement density in the better solvent and a larger polymer radius. Some features of the results in the poor solvent (MEK) appear to indicate that, as the temperature decreases, partial exclusion of the solvent leads to the formation both of stronger entanglements and of macromolecular aggregates or bundles, as suggested by Dreval and others^6–8,11,22.     

Melt viscosity of oligomeric triblock copolymers (type PEP) of ethylene and propylene oxides

Melt viscosity η at 25°C of four oligomeric triblock copolymers consisting of a central block of ethoxamer units and two end blocks of propoxamer units (PEP) (M_n × 10^?3 = 1 and 2; mole fraction of ethoxamer units x_E = 0.41, 0.57, and 0.74) was analyzed in terms of the theory advanced by Berry and Fox. The structure-dependent factor F(X) was deduced from the intrinsic viscosity data, and the mean friction factor per friction center ζ was computed from η and F(X). At a fixed molecular weight, it increases with increasing x_E. The dependence of ζ versus x_E was compared with curves computed from the data for homopolymers. The best agreement was obtained if the following values of the inherent friction factor (log ζ₀) were used: ?10.25 for poly(ethylene oxide) PEO and ?10.65 for PPO. © 1994 John Wiley & Sons, Inc.     

Rheological quantification of molecular parameters: Application to a hydrogen bond forming blend

The component dynamics and molecular parameters were investigated for miscible poly(4‐vinyl phenol)/poly(ethylene oxide) (PVPh/PEO) blends. Global values of molecular weight between entanglements (M_e) were first estimated for the blends and were compared with existing athermal model predictions. Global interchain friction coefficients (ξ) of the blends were deduced from the zero‐shear viscosity. A maximum was observed at a composition of 20–30 wt % of PEO. Chain dimensions of this phase are estimated by using a relationship between the plateau modulus and a packing length (i.e., number of individual chains present in a given small volume of the melt). A slight increase in M_e is observed at low PEO weight fraction (before 0.20), followed by a sharp decrease in M_e values after this concentration. Values of ξ in PVPh/PEO blends show a maximum value at 20–30 wt % of PEO. © 2004 Wiley Periodicals, Inc. J Appl Polym Sci 93: 1623–1630, 2004     

Polymer rheology: Influence of molecular weight and polydispersity

Literature data on the non-Newtonian flow of bulk polymer and of polymer solutions are correlated on the basis of a four-parameter equation, η = η_∞ + (η₀ ? η_∞)/[1 + (τD)^m], η being the viscosity at shear rate D, and η₀ and η_∞ limiting values at D = 0 and D = ∞, respectively. The parameters η₀, η_∞, and τ all show dependence on molecular weight, and in general there is good correlation between τ and η₀. There is evidence that τ is related to a molecular weight higher than the weight-average. The exponent m shows dependence on molecular weight distribution and approaches an upper limit of unity for a monodisperse linear polymer. For linear unblended polymers it may be expressed empirically by m = (M?_n/M?_w)^1/5.     

Effect of soft segment molecular weight on tensile properties of poly(propylene oxide) based polyurethaneureas

Influence of soft segment molecular weight and hard segment content on the morphology, thermomechanical and tensile properties of homologous polyurethaneurea copolymers based on narrow molecular weight poly(propylene oxide)glycol (PPG) oligomers were investigated. A series of polyurethaneureas with hard segment contents of 12–45% by weight and PPG number average molecular weights <M_n> of 2000 to 11,800 g/mol were synthesized and characterized structurally by SAXS and mechanically by DMA and stress strain analysis. Bis(4-isocyanatocyclohexyl)methane and 2-methyl-1,5-diaminopentane were used as the diisocyanate and the chain extender respectively. All copolymers displayed microphase separation by SAXS and DMA. The critical entanglement molecular weight (M_e) of PPG is reported to be around 7700 g/mol. Our mechanical results suggest that when copolymers possess similar hard segment contents and are compared to those based on soft segments with number average molecular weights (M_n) greater than M_e, they generally displayed higher tensile strengths and particularly lower hysteresis and creep than those having soft segment molecular weights below M_e. These results imply that soft segment entanglements in thermoplastic polyurethaneureas may provide a critical contribution to the tensile properties of these copolymers – particularly in the range where the soft segment content is dominant.     

Influence of molecular weight on scaling exponents and critical concentrations of one soluble 6FDA-TFDB polyimide in solution

Polyimde (PI) samples with different molecular weights were synthesized. Based on SEC coupled with multidetectors measurement and Yamakawa-Fujii-Yoshizaki (YFY) model, eight soluble samples with absolute M_w from 40,600 g/mol to 197,000 g/mol are chosen and applied to investigate the influence of molecular weight on scaling exponents and critical concentrations at 20–45 °C in dilute, semidilute unentangled, and semidilute entangled solutions. Most of the scaling exponents are higher than the theoretical values in three concentration regions, and scaling exponent increases with molecular weight; overlap concentration (C*) increases and entanglement concentration (C_e) decreases with molecular weight. Considering bead-bead interaction, corrected bead-spring model can explain the related results. Finally, the relationship among C^*, C_e, and molecular weight is established at different temperatures (from 20 °C to 45 °C), and two linear equations are available at each temperature. Thus, both C^* and C_e are calculated at a fixed molecular weight. And from C/C^* and C/C_e ratios, the morphology of PI fiber during electrospinning can be controlled. These results are helpful to guide the preparation of polyimide solutions for different processing.

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Graphical abstract Different molecular weight soluble polyimide (PI) was synthesized and the relationship between scaling exponent (calculated from the relationship between specific viscosity and concentration) and molecular weight in different concentration ranges was established. It is found that most of the scaling exponents are higher than the theoretical values in three concentration regions, and scaling exponent increases with molecular weight. Moreover, overlap concentration (C^*) increases and entanglement concentration (C_e) decreases with molecular weight, and its reason is discussed. Finally, the relationship among C^*, C_e, and molecular weight is established, which is helpful in guiding the preparation of polyimide solution for different processing.

     

The viscosity of concentrated polymer solutions containing low molecular weight solvents

The zero shear rate viscosities of polystyrene/ethylbenzene solutions having polymer weight fractions ranging from 0.5 to 1.0 have been measured using a novel sealed rheometer cell over a temperature range of 50 to 200°C. The concentration and temperature dependence of the solution viscosity has been found to be well described by the relation η₀ = K c^aM_w ^3.4ζ(c, T) where the monomeric friction coefficient ζ is determined by the free volume of the solution. Following the procedure of Berry, the free volume parameters, α_f(c)/γ and T_∞ (c), and the fractional free volume, f(c,T)/γ, have been determined. After using these parameters to account for the concentration dependence of the friction coefficient, the concentration exponent a has been evaluated and found to be in reasonable agreement with the value of 3.4 obtained by Berry and Fox for other polymer/solvent systems. A comparison of the relative conributions made by the friction coefficient and the term c^3.4 to the overall concentration dependence of the viscosity of these highly concentrated solutions shows the friction coefficient to be the dominant factor     

Rheological properties of concentrated solutions of agarose in ionic liquid

The rheological properties of agarose solutions were examined under the effect of entanglement coupling between agarose chains. Agarose solutions were prepared by using an ionic liquid 1‐butyl‐3‐methylimidazolium chloride as a solvent. The concentration of agarose was varied from 1.1 × 10¹–2.1 × 10² kg m^?3. The master curves of the angular frequency (ω) dependence of the storage modulus (G′) and the loss modulus (G″) showed a rubbery region in the middle ω region and a flow region at low ω region, respectively. The molecular weight between entanglements (M_e) for agarose was calculated from the plateau modulus. Moreover, M_e for agarose melt was determined to be 2.3 × 10³ from the concentration dependence curve of M_e. By using well‐known empirical relations in polymer rheology, information on molecular characteristics of sample agarose was derived. © 2011 Wiley Periodicals, Inc. J Appl Polym Sci, 2012     

The determination of network chain density and the chemical stress relaxation of crosslinked polymers

Both initial network chain densities n_M(0) and n_s(0) of dicumyl peroxide-cured natural rubbers were determined from the tensile stress and swelling method, respectively. The difference between n_M(0) and n_S(0) was usually constant, independent of the magnitude of network chain density. That is, it was found that the number of entanglement network chains in the crosslinked natural rubber was usually constant, independent of network chain density. The entanglement network chain density n_II(0) was 0.7 × 10^?4 mole/cc. This led to the supposition that the molecular weight between entanglement points (M_e) would be about 9000. Although this value is far from exact, it does not differ too greatly from the value found for noncrosslinked natural rubber. Next, in order to calculate the number of main-chain scission of crosslinked polymers from their chemical stress relaxation, we proposed our modification of Tobolsky's equation. Using our equation, it was found that the scission of dicumyl peroxide-cured natural rubber occurred in the main chain only. Furthermore, this value agreed with the one obtained from the oxidation of toluene solution of noncrosslinked rubber under the same conditions.