A method of computation of the pressure effect on melt viscosity

Interrelationships of Pressure-Dependent Hole Fraction and Elongational Viscosity in Polymer Melts

The elongational flow behavior of polyethylene, polypropylene, polystyrene, poly(methyl methacrylate), and polycarbonate, temperatures from 70 to 290 °C and pressures up to 70 MPa, is examined with the Yahsi-Dinc-Tav (YDT) model and its particular case known as the Cross model. The viscosity data employed in the range of 3-405 s-1 elongational rates were acquired from the literature at ambient and elevated pressures. The predictions and the fitting results of the proposed YDT model with the same measurement data are compared with the Cross model. The average absolute deviations of the viscosities predicted by the YDT model range from 0.54% to 9.44% at ambient and 1.95% to 6.28% at high pressures. Additionally, the linear formulations derived from the YDT model are employed to relate the viscosity with temperature and hole fraction (“thermooccupancy” function) at zero level of elongational rate and constant elongational rate along with constant elongational stress. The effects of the four viscosity parameters (such as transmission and activation energy coefficients in these equations) on the elongational viscosity are analyzed in detail and some conclusions on the structural differences for the polymers are discussed.

Study of Shear and Extensional Viscosities of Biodegradable PBS/CO2 Solutions

The purpose of this research is to study the pressure drop profiles of biodegradable polybutylene succinate (PBS)/CO2 solutions in a slit die and to measure the rheological properties of the solutions as a function of the blowing agent concentration. A slit die with four pressure transducers has been designed to describe the effects of shear rate, temperature, pressure, and gas content on the shear viscosity and extensional viscosity of the flowing PBS/CO2 solutions. The low shear rate viscosity of the pure polymer was measured using a cone and plate rheometer. Extensive experiments were conducted to investigate the polymer/gas solution viscosities at five different shear rates, three temperatures and five gas contents. Cross-Carreau model and generalized Arrhenius equation were used to describe the shear-viscosity behaviors of PBS/CO2 solutions. The extensional viscosity of solution was modeled based on Cogswell’s equation.

Relationship of Fractional Free Volumes Derived from the Equations of State (EOS) and the Doolittle Equation

This research investigates the relationships of the pressure–volume–temperature (PVT) and the zero-shear viscosity of polymer melts through their correlations with the fractional free volume. Polystyrene (PS) has been used as a case example in this study. First, the fractional free volume is determined from the Simha–Somcynsky (SS) equation of state (EOS) or the Sanchez–Lacombe (SL) EOS using the PVT data of a polymer melt. Then the fractional free volume is also determined from the Doolittle equation (with respect to the occupied volume) using the zero-shear viscosity and PVT data. These two fractional free volumes are compared to check if the EOS and the Doolittle equation consistently describe the PVT behaviors and the zero-shear viscosity through the fractional free volume. Before comparison, the fractional free volume based on the Doolittle equation is recalculated with respect to the total volume to be consistent with other fractional free volumes defined with respect to the total volume.

Hole Fraction Dependence on Linear Viscosity of PS, PP and ABS

A linear dependency of zero shear, constant shear-rate and constant shear-stress viscosities with temperature and hole fraction (“thermo-occupancy” function) was derived for polyacrylonitrile-butadiene-styrene (ABS), polypropylene (PP) and polystyrene (PS). The relation of viscosity parameters, such as transmission coefficient and a measure of activation energy coefficient, with shear-rate and shear-stress was also investigated and some conclusions on the differences for the studied polymers were discussed. In particular, it was found that, for all materials, the derivative of logarithm of viscosities at zero shear, constant shear-rate and constant shear-stress decreases with decreasing rate with the hole fraction.

On the Pressure Dependency of the Bagley Correction

The effect of pressure on the viscosity of polymer melts is an often forgotten parameter due to the inherent difficulty to measure this quantity. Different experimental approaches have already been undertaken in literature in the past. A popular methodology to measure the pressure dependence of the viscosity is to use a capillary rheometer equipped with a counter pressure chamber in which the exit pressure can be controlled. In order to process the data, one of the key elements is the Bagley correction that is required to determine the correct entrance pressure at a specific shear rate. In all analysis approaches presented in literature on data at controlled exit pressure, the Bagley correction was always determined at atmospheric exit pressure, disgarding possible effects of an enhanced exit pressure. In this paper, a new analytical approach is presented that for the first time allows for a direct assessment of the entrance pressures obtained when capillary measurements are performed with controlled counter pressures. It is demonstrated, using polycarbonate, that the entrance pressure correction needed to obtain correct viscosity values under pressure is significantly different than the one needed to correct measurements performed at atmospheric exit pressure.

On PVT and Rheological Measurements of Polymer Melts

The relation between PVT and rheological measurements of several polymer melts including polyethylenes, polypropylene, polystyrene, poly(methyl methacrylate), and polycarbonate has been taken into investigation. Pressure-temperature dependent viscosities, determined on rotational and backpressure-modified capillary rheometers, were fitted through the Carreau-Yasuda model. PVT data was analyzed by the help of the Simha-Somcynsky equation of state (SS EOS). The thermodynamical parameters of the SS EOS were connected to constant-stress viscosity (experimental) and zero-shear viscosity (extrapolated). The Doolittle relationship was modified into the form of η   =   exp ( C 1   ln ( h ′ h ) ) . The relation was employed and tested for the data evaluation. It proved to be a good tool for linearization of PVT and rheological data.

Free Volume from Pressure and Temperature Dependent Viscosity and from PVT Measurements for Homo- and Copolymers

Pressure and temperature dependency of shear and elongation viscosity and the thermal expansivity and the compressibility determined from PVT data were investigated for propylene homo- and propylene-1-pentene, -1-hexene, 1-heptene and -nonene copolymers and ethylene homopolymer and ethylene-1-butene, 1-pentene and 1-hexene copolymers. The short branching degree dependence of thermal sensitivity and pressure coefficient and the thermal expansivity and the compressibility has been determined. The fractional free volumes were calculated from the viscosity and PVT curves and the thermal expansion coefficient and compressibility factor of fractional free volume were determined. The temperature, pressure and stress dependence of fractional free volume was investigated. The fractional free volume calculated from viscosity data were compared from values comes from PVT measurement. A conversion equation was suggested.

Free volume of an oligomeric epoxy resin and its relation to structural relaxation: evidence from positron lifetime and pressure-volume-temperature experiments

From positron annihilation lifetime spectroscopy analyzed with the new routine LT9.0 and pressure-volume-temperature experiments analyzed employing the equation of state (EOS) Simha-Somcynsky lattice-hole theory (SS EOS) the microstructure of the free volume and its temperature dependence of an oligomeric epoxy resin (ER6, M(n) approximately 1750 g/mol , T(g)=332 K ) of diglycidyl ether of bisphenol-A (DGEBA) have been examined and characterized by the hole free-volume fraction h, the specific free and occupied volumes V(f)=hV and V(occ)=(1-h)V, and the size distribution (mean, <nu(h)>, and mean dispersion, sigma(h)) and the mean density N(h)'=V(f)/<nu(h)>, of subnanometer-size holes. The results are compared with those from a previous work [G. Dlubek, Phys. Rev. E 73, 031803 (2006)] on a monomeric liquid of the same resin (ER1, M(n) approximately 380 g/mol, T(g)=255 K ). In the glassy state ER6 shows the same hole sizes as ER1 but a higher V(f) and N(h)'. In the liquid V(f), <nu(h)>, dV(f)/dT, and dV(f)/dP are smaller for ER6. The reported dielectric alpha relaxation time tau shows certain deviations from the free-volume model which are larger for ER6 than for ER1. This behavior correlates with the SS EOS, which shows that the unit of the SS lattice is more heavy and bulky and therefore the chain is less flexible for ER6 than for ER1. The free-volume fraction h in the liquid can be described by the Schottky equation h proportional to exp(-H(h)/k(B)T) , where H(h)=7.8 - 6.4 kJ/mol is the vacancy formation enthalpy, which opens a different way for the extrapolation of the equilibrium part of the free volume. The extrapolated h decreases gradually below T(g) and becomes zero only when 0 K is reached. This behavior means that no singularity would appear in the relaxation time at temperatures above 0 K. To quantify the degree to which volume and thermal energy govern the structural dynamics, the ratio of the activation enthalpies E(i)=R[(d ln tau/dT(-1))]i, at constant volume V and constant pressure P(E(V)/E(P)), is frequently determined. We present arguments for necessity to substitute E(V) by E(Vf), the activation enthalpy at constant (hole) free volume, and show that E(Vf)/E(P) changes as expected: it increases with increasing free volume, i.e., with increasing temperature, decreasing pressure, and decreasing molecular weight. E(Vf)/E(P) exhibits smaller values than E(V)/E(P), which leads to the general inference that the free volume plays a larger role in dynamics than concluded from E(V)/E(P). The same conclusion is obtained when scaling tau to T(-1)V(f)(-gamma) instead of to T(-1)V(-gamma), where both gamma's are material constants.

Free volume of an epoxy resin and its relation to structural relaxation: evidence from positron lifetime and pressure-volume-temperature experiments

The microstructure of the free volume and its temperature dependence in the epoxy resin diglycidyl ether of bisphenol-A (DGEBA) have been examined using positron annihilation lifetime spectroscopy (PALS, 80-350K, 10(-5) Pa) and pressure-volume-temperature (PVT, 293-470 K, 0.1-200MPa) experiments. Employing the Simha-Somcynsky lattice-hole theory (S-S eos), the excess (hole) free volume fraction h and the specific free and occupied volumes, Vf=hV and Vocc=(1-h)V, were estimated. From the PALS spectra analyzed with the new routine LT9.0 the hole size distribution, its mean, <Vh>, and mean dispersion, sigma h, were calculated. <Vh> varies from 35 130 A3. From a comparison of <Vh>with V and Vf, the specific hole number N'h was estimated to be independent of the temperature [Nh(300 K)=N'h/V=0.65 nm-3]. From comparison with reported dielectric and viscosity measurements, we found that the structural relaxation slows down faster than the shrinkage of the hole free volume Vf would predict on the basis of the free volume theory. Our results indicate that the structural relaxation in DGEBA operates via the free-volume mechanism only when liquidlike clusters of cells of the S-S lattice appear which contain a local free volume of approximately 1.5 or more empty S-S cells. The same conclusion follows from the pressure dependency of the structural relaxation and Vf. It is shown that PALS mirrors thermal volume fluctuations on a subnanometer scale via the dispersion in the ortho-positronium lifetimes. Using a fluctuation approach, the temperature dependency of the characteristic length of dynamic heterogeneity, xi, is estimated to vary from xi=1.9 nm at Tg to 1.0 nm at T/Tg>1.2. A model was proposed which relates the spatial structure of the free volume as concluded from PALS to the known mobility pattern of the dynamic glass transition at low (cooperative alpha-relaxation) and high (alpha-relaxation) temperatures. We discuss possible reasons for the differences between the results of our method and the conclusion from dynamic heat capacity.