A comparison of equations for the effect of pressure on the viscosity of amorphous polymers

Several equations which are used to predict the pressure coefficient of viscosity for amorphous polymers have been examined on the basis of type of information required and equation reliability. These equations can be useful in accounting for pressure effects observed in tubular flow and in other shear geometries. The correlations of Penwell and Porter and of Miller are evaluated and in the perspective of expressions by Matheson and by Eyring. Data on linear amorphous polystyrene (PS), polyisobutylene, poly(vinyl acetate), poly(methyl methacrylate) (PMMA), natural rubber, and polycarbonate are examined and presented. Predictions from the Penwell-Porter and Miller equations are compared with experimental coefficients at one atmosphere for all data available. For PS and PMMA, it was found that Miller's equation tends to predict values somewhat higher than experiments but is closer to the data on PS and on a high molecular weight PMMA. The Penwell-Porter equation, on the other hand, tends to predict values somewhat lower than experiments and does a slightly better job for lower molecular weight PMMA. Both equations require WLF or Vogel coefficients and T_g-pressure-molecular weight data. Miller's equation also requires compressibilities at T_g and at the temperature of interest, although an alternate method can be used which only requires average “K” values without T_g-pressure or compressibilities at T_g.     

Estimation of the inherent friction factor per main-chain friction unit of liquid low-molecular-weight unsaturated polyesters

The temperature dependence (at 323–443 K) of the zero-shear viscosity η of about 30 nonfractionated samples of an unsaturated polyester (300 < M_n < 1500) was analyzed and the parameters of the Vogel equation describing the η(T) dependence were estimated. Their dependence on the molecular weight is discussed. The inherent friction factor per main-chain friction unit, ξ_o, has been evaluated for 15 < Z_n < 70 (where Z_n is the number-average number of main-chain friction units) and compared with the values available for other polymers.     

Rheological properties of phenophthalein poly(ether ether ketone)

The rheological properties of the novel engineering thermoplastic phenophthalein poly(ether ether ketone) (PEK-C) have been investigated using both a rotational and a capillary rheometer. The dependence of the viscosity on the shear rate and temperature was obtained. The activation energy was evaluated both from the Arrhenius and the Williams-Landel-Ferry (WLF) equation. An estimate for the proper E_η (dependent only on the chemical structure of the polymer) has been found from the WLF equation at temperatures about T_g + 200°C. Measurements of the die swell have been performed. The first normal stress differences were evaluated from the die swell results and compared with the values obtained from the rotational rheometer at low shear rates.     

Dynamics of polymeric fluids in transient-state theory

Transient-state theory recently proposed has enabled us to describe the chain length dependence of viscosity of polymeric melts from the Rouse to entangling regimes by the single equation which also takes the factor of temperature into account. On the basis of this theory, this contribution attempts to treat the effect of temperature on viscosity and provides a molecular explanation to the coefficients of M-dependence in the WLF equation, obtaining the activation energy ΔE₀ and elastic interaction parameter a for example selected. A reinterpretation from a molecular viewpoint directly leads to the common observation of the M-dependence of the glass transition temperature. The mathematical expressions are developed for diffusion coefficient D_s, showing the scaling behavior for special cases as M⁻¹ and M^−2.4 below and above the entanglement coupling mass M_e, respectively. Any deviation from the scaling can be accounted by the quantum confinement effect a. The terminal relaxation time τ_D behaves in the same way as η above the onset of entanglement. It is found that both D_s and τ_D scale on temperature in the way analogous to the WLF correlation. In addition, an expression for Young's modulus is presented by a molecular deduction. The predictions are in consistence with existing experimental data via the adjustment of a which can correlate more findings difficult to be accommodated into conventional theories. © 1996 John Wiley & Sons, Inc.     

Melt viscosity of poly(diethylene glycol-co-succinic acid): Molar mass dependence in the nonentangled regime

Experimental and theoretical results are reported on the dependence of melt viscosity on the molar mass of poly(diethylene glycol-co-succinic acid) (DEG-SA) in the nonentangled regime. A power law described the behavior with an exponent of 2.09 ± 0.10 at 30°C and of 2.59 ± 0.09 at 90°C. The modified Rouse theory was used to describe the relationship between melt viscosity and molar mass. The parameters related to the modified Rouse theory were determined by the WLF equation and the free-volume theory. The WLF parameters of various samples were checked at their T_g, which is the reference temperature. Results confirmed that the free volume and the monomeric friction coefficient (ξ₀) dominate the behavior of the melt. The agreement between the calculated and experimental melt viscosity is satisfactory. © 1994 John Wiley & Sons, Inc.     

Concentration dependence of zero-shear viscosity for polymer solutions: A test of the lyons-tobolsky equation

The empirical equation proposed by Lyons and Tobolsky (1) to describe the concentration dependence of zero-shear viscosity in polymer solutions was tested on a variety of data taken from the literature, covering different polymer-solvent systems, molecular weight, temperature, and the whole concentration range. This equation was fitted to each set of experimental viscosity-concentration data by a least squares method. The validity of this equation is verified. The dependence of the optimal values obtained for the three parameters intrinsic viscosity, [η], Huggins constant, k, and new empirical constant, b, on molecular weight, temperature, and solvent is also discussed.     

Viscosity Behaviour of Dilute and Moderately Concentrated Solutions. 1. Poly(vinylpyrrolidone) in 2-propanol

Abstract

The zero-shear viscosity of dilute to moderately concentrated poly(vinylpyrrolidone) solutions in 2-propanol were measured at different temperatures in the newtonian flow range. The viscosity data were analyzed in terms of Martin's and Fedor's equations. The viscosity-concentration data were generalized in terms of reduced variables. The relation between the K_M parameter for the rheological interaction and some characteristics of polymer solutions such as the flexibility of the PVP chain is discussed. The temperature dependence of viscosities was expressed by the Arrhenius-Frenkel-Eyring equation and the activation energy of viscous flow of polymer solutions (ΔGv) was calculated. It was shown that ΔGv increases with concentration and it is independent of the temperature.     

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.     

Relationship between melt viscosity and dielectric relaxation time for a series of epoxide oligomers

Melt viscosity has been investigated for a series of bisphenol-A type epoxide oligomers with different weight-average mol wts (M?_w), ranging from 388 to 2640. The temperature dependence of the melt viscosity is described by the Williams–Landel–Ferry (WLF) equation. The melt viscosity η is correlated with both the direct current (dc) conductivity σ and the dielectric relaxation time τ. The two relationships between these three properties, σ·η^κ = const (0.63 ≦ κ ≦ 1.12) and η/τ^? = const (0.73 ≦ ? ≦ 1.06), are experimentally derived. Both exponents, κ and ?, depend on the M?_w of the oligomer. The lower M?_w oligomer has the larger value of κ. The κ value is close to unity for the low M?_w oligomer, which agrees with Walden's rule, σ·.η = const, applicable to most low mol wt liquids. The ? value is near unity for the epoxide oligomer with higher M?_w than 2000, which means that the melt viscosity is proportional to the dielectric relaxation time. The low M?_w oligomer (M?_w < 2000), on the other hand, has a smaller value of ? below unity. The result indicates that the melt viscosity is not proportional to the dielectric relaxation time for the low M?_w epoxide oligomer, whose dielectric α-relaxation is not governed by the Debye equation. © 1993 John Wiley & Sons, Inc.     

A method of computation of the pressure effect on melt viscosity

Simha's equation of state provides the relation between reduced pressure, temperature, and volume (P?, T?, and ?, respectively) and the occupied site fraction, y = y (P?, T?). The latter theoretical parameter combines the P and T effects on the occupied and unoccupied (“free volume”) part of the model liquid. It can be computed for each liquid once the thermodynamic reducing parameters are known. Empirical correlation between published zero shear viscosity data, η = η (P, T), and y indicates that for n-paraffins and molten polymers η is a single parameter function: η = η (y). The mathematical form of this dependence was explicitly given for n-paraffins. However, for polymers the correlation depends on molecular weight, molecular weight distribution, branching, composition, etc. In Practical terms, η = η (y) should be determined for each polymer by measuring the temperature dependence of η in as wide a range of T as possible. Then pressure effect on η can be determined from η = η(y) plot, knowing the y = y(P?, T?) relation.     

Effect of temperature on creep fracture of polypropylene fibers

The tensile strength and ultimate strain of polypropylene fibers were measured by the creep fracture method at various temperatures. The tensile strength against time-to-break curves at various temperatures, which were plotted on log–log scales, were superposed by shifting the curves along the logarithmic time-to-break axis, and the composite curve of the tensile strength as a function of a reduced time to break was obtained. On the other hand, to construct the composite curve of ultimate strain from the ultimate strain against time-to-break curves at various temperatures, shifting the curves along the logarithmic ultimate strain axis was required in combination with shifting along the logarithmic time-to-break axis. The temperature dependence of the shift factor a_T followed an equation of the Williams–Landel–Ferry (WLF) form. The volume fraction of free volume at the glass transition temperature and the coefficient of thermal volume expansion, which were calculated from the WLF coefficients determined for the polypropylene fibers, are almost the same as those known as “universal values” for amorphous polymers.     

Prediction of solvent‐diffusion coefficient in polymer by a modified free‐volume theory

Several versions of free‐volume theory have been proposed to correlate or predict the solvent diffusion coefficient of a polymer/solvent system. The quantity of free volume is usually determined by the Williams–Landel–Ferry (WLF) equation from viscosity data of the pure component in these theories. Free volume has been extensively discussed in different equation‐of‐state models for a polymer. Among these models, the Simha–Somcynsky (SS) hole model is the best one to describe the crystalline polymer, because it describes it very approximately close to the real structure of a crystalline polymer. In this article, we calculated the fractions of the hole free volume for several different polymers at the glass transition temperature and found that they are very close to a constant 0.025 by the SS equation of state. It is quite consistent with the value that is determined from the WLF equation. Therefore, the free volume of a crystalline polymer below the glass transition temperature (T_g) is available from the SS equation. When above the T_g, it is assumed that the volume added in thermal expansion is the only contribution of the hole free volume. Thus, a new predictive free‐volume theory was proposed. The free volume of a polymer in the new predictive equation can be estimated by the SS equation of state and the thermal expansion coefficient of a polymer instead of by the viscosity of a polymer. The new predictive theory is applied to calculate the solvent self‐diffusion coefficient and the solvent mutual‐diffusion coefficient at different temperatures and over most of the concentration range. The results show that the predicted values are in good agreement with the experimental data in most cases. © 2000 John Wiley & Sons, Inc. J Appl Polym Sci 77: 428–436, 2000     

Rheological properties of liquid crystalline solutions of ethyl celluloses with different molecular weight in m-cresol

Steady-state shear rheological properties of liquid crystalline solutions of four ethyl celluloses (ECs) were determined at a low shear rate (1 s^?1) and at relatively high shear rates by using two rheometers (cone-plate and capillary types), and were compared with those of liquid crystalline hydroxypropyl cellulose (HPC). The effect of molecular weight (MW) on the viscoelastic behavior was also determined. The viscoelastic behavior was also determined. The viscometric behavior of EC solutions was similar to that of HPC solutions: (1) with respect to temperature, the shear viscosity (η) at shear rate of 1 s^?1 exhibited a minimum (η_min) and a maximum (η_max), and the concentration–temperature superposition for η could be applied; (2) the behavior of η at relatively high shear rates as a function of shear rate or polymer concentration was typical of lyotropic liquid crystals. The MW dependence of η_min was greater than that of η_max for EC solutions. The behavior of the elastic parameters such as Bagley correction factor (v), entrance pressure drop (ΔP_ent), and die swell (B) at relatively high shear rates for EC solutions was essentially similar to that for HPC solutions: (1) the shear rate or stress dependence of the elastic parameters was greatly dependent on whether the polymer solution was in a single phase or biphase; (2) with respect to concentration the elastic parameters showed a maximum and a minimum and the maximum or minimum point for each parameter was not always identical to each other. η for the isotropic or fully anisotropic solutions at a given concentration (C) increased, whereas η for the solutions in the vicinity of the biphasic region showed a minimum, with respect to MW. The slope of η at a given shear rate vs. CM _w depended on shear rate, and this slope for the isotropic solutions appeared to be greater than that for fully anisotropic solutions. ΔP_ent and v at a given concentration showed either a monotonical increase or a maximum or minimum with MW, and this behavior was not fully consistent with that of η. B for the isotropic solutions increased and B's for both biphasic and fully anisotropic solutions were almost constant, with MW.     

Viscosity–temperature relationships for cellulose acetate–acetone solutions

Viscosity measurements were made for dilute solutions of three grades of cellulose acetate (acetyl content 39.8%, molecular weight M?_v 30270 to 46250) in acetone in the temperature t range of 10° to 35°C. The data satisfied the Mark-Houwink equation, [η] = KM?, where [η] = limiting viscosity number and K and α are Mark-Houwink constants. The values of [η] and α decreased with increase in temperature, and straight-line correlations were obtained for ?d[η]/dt versus M?_v and log η versus 1/T (absolute temperature). The results are discussed in terms of solution properties of cellulose acetate in acetone and their possible relevance to reverse osmosis membrane science.     

Physical properties of perfluoropolyethers: Dependence on composition and molecular weight

Three copolymeric perfluoroethers with the structure CF₃[(OCF₂CF₂)_p(OCF₂)_q] OCF₃, having different p/q ratios, have been fractionated. The fractions obtained have been characterized by Gel Permeation Chromatography and ¹⁹F-NMR. The viscosity η the specific volume v and the glass transition temperature, T_g have been measured by standard techniques for all the above samples as well as for some other perfluorinated polyethers. The temperature dependence of viscosity of the unfractionated samples is described by the W.L.F. equation. The values of f_g (fractional free-volume at T_g) and of a_f (free-volume expansion coefficient) are independent of composition, for p/q ratios from 0.53 to 1.15. The critical molecular weight, M_c, is of the order of 8–9,000. From the molecular weight dependence of specific volume, the contribution to the molar volume of the in-chain CF₂ group and the excess molar free volume of the chain ends have been determined. The limiting value of T_g for an infinite molecular weight polymer was found to depend linearly on the compositional ratio O/C and the extrapolated values for polytetrafluoroethylene and for the homopolymer (CF₂O)_n were found to be respectively 200 K and 120 K.     

Plasticating single-screw extrusion of amorphous polymers: Development of a mathematical model and comparison with experiment

A mathematical model was developed for plasticating single-screw extrusion of amorphous polymers. We considered a standard metering screw design. By introducing a ‘critical flow temperature’ (T_cf), below which an amorphous polymer may be regarded as a ‘rubber-like’ solid, we modified the Lee-Han melting model, which had been developed earlier for the extrusion of crystalline polymers, to model the flow of an amorphous polymer in the screw channel. T_cf is de facto a temperature equivalent to the melting point of a crystalline polymer. The introduction of T_cf was necessary for defining the interface between the solid bed and the melt pool, and between the solid bed and thin melt films surrounding the solid bed. We found from numerical simulations that (1) when the T_cf was assumed to be close to its glass transition temperature (T_g), the viscosity of the polymer became so high that no numerical solutions of the system of equations could be obtained, and (2) when the value of T_cf was assumed to be much higher than T_g, the extrusion pressure did not develop inside the screw channel. Thus, an optimum modeling value of T_cf appears to exist, enabling us to predict pressure profiles along the extruder axis. We found that for both polystyrene and polycarbonate, T_cf lies about 55°C above their respective T_gs. In carrying out the numerical simulation we employed (1) the WLF equation to describe the temperature dependence of the shear modulus of the bulk solid bed at temperatures between T_g and T_cf, (2) the WLF equation to describe the temperature dependence of the viscosity of molten polymer at temperatures between T_cf and T_g + 100°C, (3) the Arrhenius relationship to describe the temperature dependence of the viscosity of molten polymer at temperatures above T_g + 100°C, and (4) the truncated power-law model to describe the shear-rate dependence of the viscosity of molten polymer. We have shown that the T_g of an amorphous polymer cannot be regarded as being equal to the T_m of a crystalline polymer, because the viscosities of an amorphous polymer at or near its T_g are too large to flow like a crystalline polymer above its T_m. Also conducted was an experimental study for polystyrene and polycarbonate, using both a standard metering screw and a barrier screw design having a length-to-diameter ratio of 24. For the study, nine pressure transducers were mounted on the barrel along the extruder axis, and the pressure signal patterns and axial pressure profiles were measured at various screw speeds, throughputs, and head pressures. In addition to significantly higher rates, we found that the barrier screw design gives rise to much more stable pressure signals, thus minimizing surging, than the metering screw design. The experimentally measured axial pressure profiles were compared with prediction.     

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.     

On the estimation of the Kauzmann temperature from relaxation data

The well known WLF equation describing the relaxation behaviour of glass-forming liquids near the glass transition temperature has been rederived on the basis of Adam and Gibbs' excess entropy model, making use of a novel expression for the entropy of undercooled liquids. It has been shown that C₂ in the WLF equation is a nearly constant fraction of (T_s - T₂), where T_s and T₂ are the reference and the Kauzmann temperatures, respectively. It is demonstrated that the values of T₂ obtained from the relaxation data agree well with those calculated from thermodynamic data. The arguments used provide an explanation for the universality of the WLF constant, C₂.     

Viscous flow in aqueous solution of quaternary ammonium salts: high concentration range

By introducing the condition that the viscosity of solutions η → η₀, the viscosity of the solvent as the concentration of the salt N → 0, we have recast Angell's equation in the following form, ln η_rel = kN/[N_o(N_o  N)]. This equation has been found to represent satisfactorily the viscosity behaviour of quaternary ammonium halides in aqueous solution up to high concentration and the values of N₀, the hypothetical concentration at which the solution becomes a glass, agree with those obtained from Angell's equation. The relationship with Vand's equation of fluidity is also discussed.     

Dielectric determination of cure state during non-isothermal cure

As a result of increased interest from industry in using dielectric cure monitoring, a need has arisen for simplifying frequency, cure, and temperature dependent data so that control decisions can be readily made. Techniques utilizing data covering several decades of frequency now exist for separating ionic conduction levels from dipole and electrode polarization responses. Ionic conduction levels are particularly useful since they can be correlated to both viscosity and extent of cure. In addition to being a function of extent of cure, dielectric properties are also influenced by temperature. This dependence often makes the dielectric response more difficult to interpret. This paper investigates two methods for overcoming the temperature dependence of the dielectric response during nonisothermal cure. The first method utilizes recent WLF modeling techniques and extends them with the end result of extracting T_g in real time during cure. The second technique involves measuring the temperature dependence of uncured and cured material. Utilizing the correlation between log ionic conductivity and extent of cure, which has been noted by previous researchers, the normalized conductivity can be converted to a cure index. Several examples including epoxy, polyurethane, and a UV cured photoresist are presented, showing data before normalization and after both T_g and cure index determination.