Nonlinear viscoelasticity of polystyrene solutions. II. Non-Newtonian viscosity

The steady shear viscosity η(k) and the stress decay function \documentclass{article}\pagestyle{empty}\begin{document}$ \tilde \eta \left({t,k} \right)$\end{document} (the shear stress divided by the rate of shear k after cessation of steady shear flow) were measured for concentrated solutions of polystyrene in diethyl phthalate. Ranges of molecular weight M and concentration c were 7.10 × 10⁵ to 7.62 × 10⁶ and 0.112–0.329 g/cm³, respectively. Measurements were performed with a rheometer of the cone-and-plate type in the range 10^?4 < k < 1 sec^?1. The Cox–Merz relation η(k) = |η^*(ω)|_ω=k was tested with the experimental result (|^*(ω)| is the magnitude of the complex viscosity). It was found to be applicable to solutions of relatively low M or c but not to those of high M and c. For the latter η(k) began to decrease at a lower rate of shear than |η^*(ω)|_ω=k did; the Cox–Merz law underestimated the effect of rate of shear. The stress decay function was assumed to have a functional form \documentclass{article}\pagestyle{empty}\begin{document}$\tilde \eta \left( {t,k} \right) = \sum {\eta _p \left( k \right)e^{ - t/\tau p\left( k \right)} } $\end{document} where τ₁ > τ₂ > …, and the values of τ₁, τ₂ η₁ and η₂ were determined for some solutions. The relaxation times τ₁ and τ₂ were found to be independent of k and equal to the relaxation times of linear viscoelasticity. At the limit of k → 0, η₁ and η₂ were approximately 60 and 20–30%, respectively, of η and the non-Newtonian behavior was due to large decreases of η₁ and η₂ with increasing k. It was shown that η₁(k) may be evaluated from the relaxation strength G₁(s) for the longest relaxation time of the strain-dependent relaxation modulus with a constitutive model for relatively high c–M systems as well as for low c–M systems.     

The chain length dependence of self-diffusion in melts of polyethylene and polystyrene

The self-diffusion coefficients in melts of polyethylene fractions and polystyrene standards were measured by the NMR pulsed field gradient technique and compared with those measured by other techniques. The data agree very well if one takes into account the molar mass distribution of the samples and the free volume of the matrix. For molar masses much higher than the critical molar massM _c, reptation is confirmed,D M ^–2 holds. BelowM _e=M_c/2 the self-diffusion coefficients corrected for constant free volume show approximately the dependenceD M ^–1 confirming Rouse-like diffusion. This result was also obtained by investigating the self-diffusion of the molecules with different molar masses of a polyethylene fraction with a rather broad molar mass distribution aroundM _e andM _c, i. e. diffusion in a constant matrix. In the molar mass region betweenM _c and about 3 ·M _c the observed molar mass dependence of self-diffusion can be explained by tube formation. The constraint release model of Graessley seems to slightly overestimate the self-diffusion coefficients.     

Rheological properties of poly(2,6-dimethylphenylene oxide)—polystyrene blends

The viscoelastic (VE) response of freeze-dried blends of polystyrene (PS) and poly-(2,6-dimethyl phenylene oxide) (PPO) has been studied as a function of composition, frequency, and temperature to examine the degree of rheological compatibility. When blended together, the relaxation processes of both molecular species exhibit the same temperature dependence. However, the temperature dependence of the VE response is a function of composition. It is shown that this behavior can be predicted from the measured glass transition temperatures by assuming the additivity of the free volumes of the components. The properties of the blends are compared at equal free volumes. The effective segmental friction factor is found to be independent of composition while the modulus of the rubbery plateau increases with PPO concentration. This result is interpreted as a change in the entanglement molecular weight M_e of the blends. When the changes in M_e are considered, the relationship between the zero-shear viscosity η0 and the 3.4 power of the weight-average molecular weight, commonly found for high molecular weight homopolymers, predicts the compositional dependence of η0 for the PPO–PS blends. It is concluded that the PPO–PS system forms a rheologically compatible blend.     

Generalized correlations for molecular weight and concentration dependence of zero-shear viscosity of high polymer solutions

Data are presented to show that two correlations of viscosity–concentration data are useful representations for data over wide ranges of molecular weight and up to at least moderately high concentrations for both good and fair solvents. Low molecular weight polymer solutions (below the critical entanglement molecular weight M_c) generally have higher viscosities than predicted by the correlations. One correlation is η_sp/c[η] versus k′[η], where η_sp is specific viscosity, c is polymer concentration, [η] is intrinsic viscosity, and k′ is the Huggins constant. A standard curve for good solvent systems has been defined up to k′[η]c ≈? 3. It can also be used for fair solvents up to k′[η]c ≈? 1.25· low estimates are obtained at higher values. A simpler and more useful correlation is η_R versus c[η], where η_R is relative viscosity. Fair solvent viscosities can be predicted from the good solvent curve up to c[η] ≈? 3, above which estimates are low. Poor solvent data can also be correlated as η_R versus c[η] for molecular weights below 1 to 2 × 10⁵.     

Molecular dimensions and melt rheology of an all‐aromatic liquid crystalline polymer

The molecular dimensions and melt rheology of a thermotropic all‐aromatic liquid crystalline polyester (TLCP) composed of p‐hydroxy benzoic acid, hydroquinone, terephthalic acid, and 2,4‐naphthalenedicarboxylic acid is examined. The Mark–Houwink exponent (α) of 0.95 is estimated for the TLCP. The persistence length estimated from molecular weight (M) and intrinsic viscosity ([η]) data using the Bohdanecky–Bushin equation is about 95 Å, whereas that estimated from light scattering data is 117 Å. These persistence lengths and the observed α value, both higher than those for flexible polymers, suggest that the present TLCP is a semirigid polymer. The zero shear melt viscosity (η₀) varies with approximately M⁶ for molecular weight M > 3 × 10⁴ g/mol; below this molecular weight, η₀ varies almost linearly with M. Widely different entanglement molecular weights (M_e) are predicted, depending on the method used; the plateau modulus estimates M_e of about 8 × 10⁵ g/mol, whereas the ratio of mean square end‐to‐end distance and molecular weight (〈R²〉₀/M) predicts M_e's either too small (0.33 g/mol) or too large (2.5 × 10⁶ g/mol), depending on the theory used. Although the change in the molecular weight dependency of melt viscosity appears to be associated with the onset of entanglement coupling of the semirigid molecules, its origin needs further investigation. © 2001 John Wiley & Sons, Inc. J Polym Sci Part B: Polym Phys 39: 2378–2389, 2001     

Synthesis, characterization, and ethylene polymerization behavior of [(RR′-admpzp)2Ti(OPr)2] complexes (RR′-admpzp = 1-dialkylamino-3-(3,5-dimethyl-pyrazol-1-yl)-propan-2-olate)

[(RR′-admpzp)₂Ti(OPrⁱ)₂] complexes (2a-c), synthesized from reaction of Ti(OPrⁱ)₃Cl (0.5 equiv) with 1-dialkylamino-3-(3,5-dimethyl-pyrazol-1-yl)-propan-2-ol compounds in the presence of triethylamine (0.5 equiv), are pseudo-octahedral with each RR′-admpzp ligand κ²-O,N(pyrazolyl) coordinated to the titanium center. In solution, 2a-c adopt isomeric structures that are in dynamic equilibrium. At 23 °C, 2a-c/1000 MAO catalyst systems furnished high molecular weight polymers with narrow molecular weight distributions (M_w/M_n = 2.7-2.8). At 100 °C, 2a-c/MAO catalyst systems exhibited increased polymerization activity and 2c/1000 MAO system furnished high molecular weight polyethylene with a molecular weight distribution (M_w/M_n = 2.1) that is close to that found for single-site catalysts.     

Viscosity of poly(vinyl acetate) and its concentrated solutions

Data on the viscosity of poly(vinyl acetate) (PVOAc) and its concentrated solutions in diethyl phthalate (DEP) and cetyl alcohol (CeOH) are examined over the molecular weight range 8 × 10³ < M < 1.3 × 10⁶, the range 0.15 < φ < 1.0 of the volume fraction φ of polymer, the temperature range 308 to 430°K for PVOAc and its solutions in DEP, and at 396°K in CeOH. The latter is the θ temperature for dilute solutions of PVOAc in CeOH. The data are analyzed with the relation η = KX_c(αφM/M_c)^a exp[1/β(T – T₀)], where a is 1 or 3.4 for αφM less than or greater than a constant M_c, respectively, and X_C, is a constant. The expansion factor α_φ of the chain dimension is found to be essentially unity for φ > ca. 0.25, increasing with decreasing φ for smaller φ. Both β and T₀ depend on φ, and T₀ also depends on M at low M.     

Chain dynamics in entangled polymers: Diffusion versus rheology and their comparison

This review article scrutinizes and reanalyzes the extensively available literature data on the tracer and self chain diffusion coefficients D_tr and D_s along with the corresponding zero‐shear viscosity η₀ to show that D_s ∝ M^?ν starts with ν > 2.0 and converges to the asymptotic scaling exhibited by D_tr ∝ M^?2.0 as the molecular weight M increases beyond M/M_e = 10–20, in contrast to the onset of the asymptotic scaling M³ for η₀ taking place typically for M/M_e ? 10–20. A coherent analysis of these observations leads to the suggestion that the observed crossover in D_s is due to the constraint release effect, which diminishes around M/M_e = 10–20 and is negligible in measurements of D_tr when the matrix molecular weight P is much greater than M. The contour length fluctuation (CLF) effect, which is believed to cause the molecular weight scaling of η₀ to deviate significantly from its limiting behavior of M³, has little direct influence on the chain diffusion. The absence of the CLF effect on D_s leads to a much stronger than linear dependence of the product η₀D_s on M, which has been observed previously. © 2003 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 41: 1589–1604, 2003     

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.     

Corresponding state relations for the Newtonian viscosity of polymer solutions. II. Further systems and concentrated solutions

In earlier work we have indicated a superposition principle for moderately concentrated mixtures (c ? 2/[η]) in good and poor solvents. By an examination of data on a number of vinyl polymers and cellulose derivatives in good as well as poor solvents, the validity of this principle is extended to concentrated solutions (c ? 50%). The characteristic concentration factor γ is proportional to M over the whole concentration range, with 0.47 ≤ a₁ ≤ 1.10 being larger for good than for poor solvents, the result obtained earlier. Significant deviations from this relationship are noted in good solvents for those low molecular weights at which deviations from the usual intrinsic viscosity relationship occur. This may be related to the expansion factor of the polymer coil. On the basis of these results, the concentration and molecular weight dependence of the viscosity in the concentrated solution can be related to each other in terms of the parameter a₁ and thus to thermodynamic characteristics. In this manner a bridge between the relatively dilute and concentrated regions is established. Currently used semiempirical expressions are analyzed in terms of these results. For the polystyrene–cyclohexane systems and θ ? 9 ≦ T ≦ θ + 3, γ can be identified with the critical concentration for phase separation. Provided an “entanglement” concentration c_e exists, in the neighbourhood of which the concentration dependence of the viscosity changes reapidly, γ can alternatively be shown to be proportional to c_e, or c_e ∝ M. The temperature reduction scheme suggested earlier remains to be investigated.     

Simple method for determining the critical molecular weight from the loss modulus

The normal concept is that the critical molecular weight (M_C) is about twice as large as the entanglement molecular weight (M_e). However, experimental data have shown considerable deviations from M_C ≈ 2M_e. Furthermore, a determination of M_C requires samples with a wide range of molecular weights, including weights lower than M_C and higher than M_C. In this article, we suggest a simple method for determining M_C from the loss moduli of nearly monodisperse linear polymers with M ? M_C. We consider two characteristic relaxation times, which correspond to the local maximum and minimum of the loss modulus. M_C is determined from the intersection of two phenomenological relaxation times as a function of the molecular weight. The method precisely agrees with M_C ≈ 2M_e, which is not shown by conventional methods. Moreover, our method provides a determination of relaxation time τ_e, at which chain segments first feel the constraints imposed by the conceptual tube, without the measurement of the tube diameter and the monomeric friction coefficient, which may be determined by complicated procedures with a lot of data. © 2004 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 42: 2724–2729, 2004     

Properties of amorphous and crystallizable hydrocarbon polymers. I. Melt rheology of fractions of linear polyethylene

The dynamic moduli G′(ω) and G″(ω) for two groups of linear polyethylene fractions (reported M _w/M _n < 1.2) were measured in the melt state using the eccentric rotating disk method. Values of zero shear viscosity η₀ were obtained and compared with published results on similar fractions. Molecular weight data were converted to a common basis through intrinsic viscosities in trichlorobenzene (TCB) at 135°C. With recent data on M _w (light scattering) vs. [η]_TCB, for linear polyethylene, the relationship at 190°C, η₀ = 3.40 × 10^?14(M _w)^3.60, was obtained. The flow activation energy E_a was 6.4 kcal (T = 140–195°C). The plateau modulus G at 190°C was determined from the area under the loss modulus peak in one high-molecular-weight sample. The value obtained, G = 1.5₈ × 10⁷ dyn/cm², corresponds to an apparent molecular weight between entanglements of 1850. The storage compliance J′(ω) becomes anomalously large at low frequencies. The recoverable compliance J could not be determined for any of the fractions.     

Steady-shear viscosity of polydimethylsiloxanes over the range from the lower to the upper Newtonian region

The mechanism of non-Newtonian behavior for flow from the lower to the upper Newtonian region is explained by a modification of Graessley's theory. In the theory proposed here, a viscosity η_fric, which is based on friction between polymer segments and is almost shear-independent, is introduced in addition to Graessley's entanglement viscosity η_ent, which decreases with increasing shear rate. The theory is applied to previously obtained data on steady flow of polydimethylsiloxanes of different molecular weights. The agreement between calculated and experimental results is good. In polymers with the molecular weight above the critical molecular weight for entanglement M_c, the major contribution to viscosity near zero shear rate is η_ent. As the shear rate increases, the flow curve has an inflection where η_fric cannot be disregarded in comparison with η_ent. In the upper Newtonian region, η_fric has more influence on the viscosity than η_ent. The theory can also explain the experimental results on flow of polymers with molecular weight below M_c, which were shown to be slightly non-Newtonian in the previous paper.     

Energy-adjustedab initio pseudopotentials for the second and third row transition elements: Molecular test for M2 (M=Ag,Au) and MH (M=Ru,Os)

Summary Recently published nonrelativistic and quasirelativistic energy-adjustedab initio pseudopotentials representing the M^(Z–28)+ cores of the second row transition metal atoms and the M^(Z–60)+ cores of the third row transition metal atoms have been tested in SCF, CI(SD) and CEPA1 calculations of the spectroscopic constants (R _e ,D _e , and _e ) of the ground states of the neutral and singly charged silver and gold dimers, and in state averaged CASSCF and multi-reference CI(SD) calculations of the spectroscopic constants (R _e ,D _e , _e , _e , /R). Comparison is made with experimental and reliable theoretical data where available; in the case of the hydrides, additional calculations with pseudopotentials published by other groups have been made for comparison.     

Synthesis and reactivity of μ-η2-carbamato complexes of triosmium decacarbonyl

The synthesis of (-H)(-²-RRNCO₂)Os₃(CO)₁₀ (R=R=CH₃ 2a; R=R=CH₂CH₃,2b; R=CH₃, R=CH₂CH₃,2c) and their cyclic analogs (-H)(-²--CO₂)Os₃(CO)₁₀(n=42d,n=5,2e) from carbon dioxide, secondary amine, and Os₃(CO)₁₀(CH₃CN)₂ (1) are reported. A solid-state structure of2c reveals a bonding mode for the carbamato ligand very similar to that observed for related carboxylato complexes. Compound2c crystallizes in the orthorombic space group Pbca witha=9.136 (3),b=15.310 (4) andc=30.361 (5) Å;V=4247 ų,Z=8. Least-squares refinement of 2405 observed reflections gave a final agreement factor ofR=0.043 (R _w =0.043). The reactivity of the complexes2a–2e was examined. Compound2c or2b give good yields of the cluster derivatives (-H)(-X)Os₃(CO)₁₀ (X=Cl,3; X=OCH₃,4; X=N(CH₃)₂,7) when reacted with HX. Reaction of2a with P(CH₃)₃ at 68°C gives good yields of the otherwise difficult to obtain 1,1,2-(P(CH₃)₃)₃Os₃(CO)₉ (5). Evidence is presented that suggests that2a–2e form by oxidative addition of preformed carbamic acids to1.     

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     

Structure evolution in the PbO-ZrO2-TiO2 sol-gel system: Part I — Characterization of prehydrolyzed precursors

In this investigation, several spectroscopic and analytical techniques were used to determine the chemical compositions and structures of the lead, zirconium, titanium, and Pb-(Zr, Ti) alkoxides involved in the sol-gel synthesis of PZT thin films. These techniques included ¹H, ¹³C, and ²⁰⁷Pb NMR; FT-IR; gas chromatography; Karl Fischer titration; and number-average molecular weights (M _n ) determined by cryoscopy. It was found that the titanium precursor had a M _n of 548 and a formula of [Ti(OCH₂CH₂OCH₃)₄]_1.6; the zirconium precursor had a M _n of 1015 and a formula of [Zr(OCH₂CH₂OCH₃)₄]_2.6; and the lead precursor had a formula Pb₆(OOCCH₃)₅(OCH₂CH₂OCH₃)₇. 4 H₂O and a molecular weight of 2131 (M _n =2113). It was observed that residual water from the incomplete dehydration of lead acetate trihydrate coupled with released water due to the esterification of acetic acid caused M-O-M (M=Pb, Zr, Ti) bonds in the Pb-(Zr, Ti) alkoxide. Two possible isomeric structures of the Pb-(Zr, Ti) alkoxide have been proposed. They are both cyclic and have a formula of Pb₂MMO₂(OR)₈(ROH)₂, (MM=Zr and/or Ti) and a molecular weight of 1336 (M _n =1386).     

Electron diffraction study of the molecular structure of gaseous 1,1-dibromo-3,3,5,5-tetramethyl-1-stanna-3,5-disila-4-oxacyclohexane,Br2Sn(CH2SiMe2)2O

The molecular structure of gaseous Br₂Sn(CH₂SiMe₂)₂O was studied by electron diffraction. The six-membered ring has a chair conformation whereas the entire molecule possessesC _s symmetry. The existence of a boat conformer cannot be completely excluded. The results of theoretical calculations for a twisted-boat conformation are at variance with the experimental data. Steric strain caused by mutual repulsion of the two axial methyl groups is reduced to the tilt of the Me₂Si fragments in opposite directions. This results in an increase (up to 26°C) in the angle formed by the bisector of the C_M-Si-C_M angle with the C_cSiO plane. The main geometrical parameters are as follows:r _g (Å): Si-O 1.708(20); Si-C_M 1.862(20); Si-C_c 1.882(9); Sn-C 2.108(26); Sn-Br 2.456(3); C-H 1.099(30); (degr.): C-Sn-C 105(2); Br-Sn-Br 107.9(1.2); Si-O-Si 129.6(3); C_M-Si-C_M 112; Si-C-H 113 (fixed value in accordance with experiment); C_c-Si-O 107(2); Sn-C-Si 109(2); torsion angles: (Si-C) 52(2); (Si-O) 62(1); (C_c-Sn) 54(1). The average amplitudes were fixed at the values calculated from the force field. Structural parameters of molecules with similar structures were analyzed and compared.Translated fromIzvestiya Akademii Nauk. Seriya Khimicheskaya, No. 2, pp. 384–387, February, 1993.