Simha-Somcynsky Equation Of State Research Articles

Determination of Solubilities of CO2 in Linear and Branched Polypropylene Using a Magnetic Suspension Balance and a PVT Apparatus

Using a PVT apparatus for high pressure and temperature combined with a magnetic suspension balance, the solubility of carbon dioxide in linear and branched polypropylene (PP) was measured at temperatures from (453 to 493) K and at pressures of up to 31 MPa. The solubility of CO2 in both molten polymers increased linearly with pressure and decreased with temperature. However, above 20 MPa, the solubility−pressure relationship was no longer linear. This might be due to a significant hydrostatic effect on the swelling of the polymer that results from gas absorption above 20 MPa, so that swelling is no longer linearly related to pressure. At a high pressure, swelling significantly affects solubility, which is then no longer linearly related to pressure. It was noted that linear PP absorbs more gas than branched PP, due to the branched PP’s chain entanglement. The solubility of CO2 in the PP melts was compared with semiempirical data (determined by empirically measuring gas uptake and theoretically predicting swelling) and theoretical values calculated from the Simha−Somcynsky (SS) and Sanchez−Lacombe (SL) equations of state (EOSs). The Simha−Somcynsky equation of state (SS-EOS) was observed to have a better prediction capacity of the swelling effect and to thus provide better solubility predictions for both semiempirical and theoretical cases than the Sanchez−Lacombe equation of state (SL-EOS).

Pressure-dependent viscosity and free volume of atactic and syndiotactic polystyrene

The effect of pressure on viscosity is an important but often overlooked aspect of the flow properties of polymeric materials. In this work, two polymers (an atactic and a syndiotactic Polystyrene) were characterized to determine the effect of pressure on viscosity. In particular, a device was adopted to increase the exit pressure of a standard capillary rheometer, thus obtaining data of viscosity under high pressure and high shear rates. The Simha-Somcynsky equation of state was applied to the pressure–volume–temperature experimental data of both materials to obtain the dependence of free volume on temperature and pressure. The Doolittle equation was eventually employed to verify the dependence of viscosity on free volume. It was found that, for both materials, a linear relationship holds between the logarithm of zero-shear-rate viscosity (at several temperatures and pressures) and the inverse of free volume.

The free volume in two untreated, pressure-densified, and CO2 gas exposed polymers from positron lifetime and pressure-volume-temperature experiments

AbstractChanges in the microstructure of the free volume and its temperature dependence in ethylene-norbornene copolymer and bisphenol-A polycarbonate due to densification under pressure and swelling with CO2 gas have been examined using positron annihilation lifetime spectroscopy (PALS) and pressurevolume- temperature (PVT) experiments. Employing the Simha-Somcynsky equation of state the specific hole free and occupied volumes were estimated. From the PALS spectra analyzed with the new routine LT9.0 the size distribution of subnanometre holes and its mean and mean dispersion were calculated. Above Tg, the dispersion mirrors the thermal fluctuations in the free volume. From comparison of PALS and PVT data the specific number of holes was estimated. It was found that the occupied volume has a constant and identical compressibility in the glassy and rubbery state. It shows no memory for the history of the glass and mirrors only the pressure and temperature during the measurement. The change in the total volume due to the pre-treatments of the polymers occurs exclusively in the hole free volume Vf and the relative change in Vf is one order of magnitude larger than in the total volume. PALS data show that the mean hole size and its dispersion in the glassy state is decreased due to densification and increased due to swelling. PVT data show that the volume changes are frozen in the polymer glasses and that, when heating the samples, the volume begins to recover at temperatures ca. 50 K in gas swollen and 20 K in densified polymers below Tg. The PALS data show a corresponding behaviour.

Measurement of gas solubility in linear/branched PP melts

AbstractA magnetic suspension balance was employed in experiments to measure gas solubility in the polymer melts. The gas solubilities of CO2 and N2 in both linear and branched Polypropylene (PP) were investigated. The swollen volumes predicted by the Sanchez–Lacombe equation of state (EOS) and Simha–Somcynsky EOS were applied to incorporate the buoyancy effect, which is essential for the accurate measurement of solubility data. The effects of the branched structure on the swollen volume and gas solubility were discussed. It was observed that the long chain branched PP exhibited less expandability than the linear PP, due to the entangled molecular chain structure. Therefore, the total amount of gas that was able to dissolve into the long chain branched PP turned out to be less. © 2007 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 45: 2497–2508, 2007

Linking the Viscous and Vacancy Behavior of Mixtures of High-Molecular-Weight Hydrocarbons

A zero-shear viscosity model has now been extended to mixtures of three-branch alkanes with and without cyclopentyl or cyclohexyl attachments. The model links the equilibrium theory of Simha–Somcynsky (SS) statistical thermodynamics of liquids and the Eyring significant structure (ESS) transport theory. The predicted quantity of the hole fraction as a measure of the free volume, computed from the SS equation of state, plays a primary role in the approach to combine both theories. Experimental PVTη data from 310 to 408 K and 0.1 to 344.5 MPa have been used. The proportionality constant of the activation energy and the transmission coefficient have been calculated for each species separately, and they are in the expected trend with the fraction of mixtures. Satisfactory agreement in the prediction of viscosity from the hole fraction has been obtained. The volumetric and viscous data of physical mixtures are predicted from and compared with their chemical correspondence.

Investigating the Solubility of CO2 in Polypropylene Using Various EOS Models

Generally, in order to obtain reliable gas solubility data in a polymer melt, one must determine the swollen volume of the polymer with dissolved gas. The swollen volume is typically estimated by an equation of state (EOS). This paper reports on the application of three different equations of state (the Sanchez-Lacombe (SL) EOS, the Simha-Somcynsky (SS) EOS, and the Perturbed Chain Statistical Associating Fluid Theory (PC-SAFT) EOS) to a polymer/gas mixture system for the study of phase equilibria and gas solubility determination. The apparent solubility was obtained using a magnetic suspension balance (MSB). Approaches based on the SL, SS, and PC-SAFT EOS's are herein proposed for predicting the swelling behavior of polymer/gas mixtures and correlating the swollen volume to the apparent solubility.

Pressure and Temperature Dependence of LDPE Viscosity and Free Volume: The Effect of Molecular Structure

Abstract Temperature and pressure dependencies of shear and elongational viscosities were examined using rotational and capillary rheometers. Two different batches of the same polymer grade have shown that the molecular structure diversion significantly influences the magnitudes of pressure coefficients, which vary more than the temperature coefficients. The pressure effect on viscosity notably depends on the amount of long-chain branching in polymer. Further, the paper shows that pVT data analysis via the Simha-Somcynsky equation of state can be employed to reveal differences in temperature and pressure viscosity dependencies through the free volume fraction.

Temperature and Pressure Dependence of?-Relaxation and Free Volume in Poly(vinyl acetate)

Summary: We analysed reported specific volume V from pressure-volume-temperature (PVT) experiments for poly(vinyl acetate) (PVAc) using the Simha-Somcynsky equation of state (S-S eos) and calculated the specific occupied volume, Vocc, and hole free volume, Vfh We found that Vocc shows almost no thermal expansion but is distinctly compressible (T > Tg). Due to this, Vfh = V − Vocc is not a unique function of V but can depend on the specific values of P and T giving rise to the particular value for V. We observed that with decreasing temperature the α-relaxation detected via dielectric measurements slows down faster than the shrinkage of hole free volume Vfh would predict on the basis of the free volume theory of Cohen and Turnbull. Vfh becomes zero at approximately 65 K below the Vogel temperature T0, . Plots of the α-relaxation frequency log ωα versus 1/Vfh can be linearised when taking into account this discrepancy by substituting 1/Vfh by 1/(Vfh − ΔV) with , or by including an energy term of the type exp[−E′/R(T − T0)] in the pre-exponential factor of the frequency-volume equation. Our results indicate that the α-relaxation in PVAc operates via the Cohen-Turnbull mechanism only when liquid-like clusters of cells of the S-S lattice appear which contain a hole free volume of three or more empty S-S cells. For regions with a smaller number of empty cells an activation energy is required for allowing segmental motion via the cooperative rearranging of cells. The β-(Johari-Goldstein) relaxation which is a non-cooperative, more local, process follows completely the free volume theory. The same seems to be true for the primitive (uncoupled) relaxation of the coupling model.

Linking the viscous and vacancy behavior of high molecular weight hydrocarbons

A new theoretical model that was recently purposed for zero-shear viscosity has now been extended to the three-branch alkanes with pentyl, hexyl, or phenyl attachments. The model links the equilibrium theory of Simha-Somcynsky (SS) statistical thermodynamics of liquids and the transport theory of Eyring Significant Structure (ESS). The predicted quantity of hole fraction as a measure of the free volume, computed from the SS equation of state, plays a primary role in the theory to combine the both theories. The experimental PVTη data from 310 to 408 K temperature and 0.1 to 344.5 MPa pressure range have been used. The proportionality constants of the activation energy and the transmission coefficient have been calculated for each species separately and they are in expected trend with the substituted ring structure. Satisfactory agreement in prediction of viscosity from the hole fraction has been obtained.

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 (Tg) is available from the SS equation. When above the Tg, 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

Composition-Dependent Properties of Polyethylene/Kaolin Composites: VI. Thermoelastic behavior in the melt state

The equilibrium pressure-volume-temperature properties and thermoelastic behavior of kaolin-filled composites of the injection molding grade of high density polyethylene (HDPE) were studied in the temperature interval 423–473 K and in thepressure range 30–100 MPa. It was established that the HDPE melts in filled composites existed in a somewhat expanded, more compressible state. This effect was quantitatively accounted for by the increased number of external degrees of freedom derived from the Simha-Somcynsky equation of state. The effect of the coupling agent on thermal diffusivities and specific heat capacities of filled samples turned out negligibly small.

Precise measurement of the PVT of polypropylene and polycarbonate up to 330�C and 200 MPa

An apparatus for measuring the pressure—volume—temperature (PVT) properties of polymers using a metal bellows has been developed at temperatures from 313 to 623 K and pressures up to 200 MPa. A calibration of the device was performed by measuring PVT of mercury and water. The experimental uncertainty of specific volumes was estimated to be within ±0.2%, while that of temperatures was within ±0.1 K below 300°C and ±0.3 K above 300°C. The estimated uncertainty of pressures was ±0.1 MPa below 100 MPa and ±0.25 MPa above 100 MPa. The PVT properties of polypropylene and polycarbonate were measured by the apparatus in the temperature ranges from 40 to 300°C and from 40 to 330°C, respectively, and pressures from 10 to 200 MPa. The effects of a sample cup and sample forms were investigated. The use of the sample cup showed a little effect on the measurments of PVT properties for both samples. The shape (pellet and pillar) of the samples caused a small difference in the specific volumes only under high temperatures and low pressures. The PVT properties in a melt state were correlated by the Simha-Somcynsky equation of state, showing a good agreement with measurements. © 1997 John Wiley & Sons, Inc. J Appl Polym Sci 66: 141–150, 1997

Simplified hole theory equation of state for polymer liquids

Abstract Zhong, C., Wang, W. and Lu, H., 1993. Simplified hole theory equation of state for polymer liquids. Fluid Phase Equilibria, 86: 137-146 A simple exponential relation between the occupied-site fraction in the lattice and the reduced temperature is proposed, based on the statistical theory for the thermal defects of an imperfect crystal. Following the approach of Simha and Somcynsky (1969), [Simha, R. and Somcynsky, T., 1969. On the statistical thermodynamics of spherical and chain molecule fluids. Macromolecules, 2: 342-351], a simplified hole theory equation of state (EOS) for polymer liquids has been derived. For 11 polymer liquids, the calculated P-V-T results show that the equation proposed is of the same quality as the Simha-Somcynsky equation of state. Moreover, with much less computing effort, this equation can be easily extended to polymer solutions.

Effect of pressure on the thermal conductivity of polymers

The thermal conductivity of polyolefins and halogen-substituted polymers was studied in a broad temperature interval spanning both solid and melt states, in the range of pressures from 0.1 up to 100 MPa with the aid of a high-pressureλ-calorimeter in the continuous heating regime. Treatment of data on the pressure dependence of the thermal conductivity of melts in terms of Barker's equation yielded the values of ‘quasilattice’ Grueneisen parameter γB which exhibited the same dependence on molecular structure of a polymer as the parameter 3C/p from the Simha-Somcynsky equation of state (number of external degress of freedom per chain repeat unit). Analysis of the dependence of the thermal conductivity of polyethylene on the degree of crystallinity revealed the inadequacy of the current two-phase model which does not account for the microheterogeneity of the ‘amorphous phase’. It was concluded that interchain heat transfer makes the dominant contribution to the thermal conductivity of polymers both in amorphous and in crystalline states.

Thermodynamics of statistical copolymers of N-vinylcarbazole and alkyl methacrylates

Heat capacity at atmospheric pressure (between 163 and 493 K) and specific volume (between 325 and 455 K and whithin the pressure interval of 15 to 210 MPa) have been measured for equimolar statistical copolymers of N-vinylcarbazole and a series of alkyl methacrylates (from methyl to lauryl). Parameters in the Simha-Somcynsky equation of state were determined for the copolymers in the liquid state, and the individual contributions to the discontinuity in heat capacity at the glass-transition temperature were calculated according to Di-Marzio. It has been demonstrated for the first time that the number of degrees of freedom of the repeat unit in the liquid state coincides with the number of “flexible” bonds contributing to internal rotation.