Free-volume Expression Research Articles

Extrapolation/interpolation of infinite dilution, activity coefficient as well as liquid and solid solubility between solvents: Part 1. Alkane solvents

Infinite dilution activity coefficients play a vital role in modeling of phase and chemical equilibrium behavior. From the dependence of the limiting activity coefficient of a solute on the solvent, important properties like liquid and solid solubility in these solvents can be deduced. Although a large amount of experimental information is available in the open literature as well as from thermophysical data banks such as the Dortmund Data Bank (DDB), the behavior of high boiling solutes in low boiling solvents is often not known, partly due to difficulties in experimental determination.A method for the extrapolation of infinite dilution activity coefficients and thus the liquid and solid solubility of any solute in different solvents within one solvent family is under development. Results of a method applicable to any solute in a large variety of different saturated hydrocarbon compounds are presented in this paper.In this method, the activity coefficient is assumed to consist of a combinatorial and residual contribution. In the case of alkane solvents, only this combinatorial contribution changes when going from one solvent to another. Several well known combinatorial expressions were evaluated and were found to produce poor extrapolations in many instances. Quite surprisingly, free-volume combinatorial expressions performed best even for some of the rather low molecular weight compounds used in this test.A new empirically modified free-volume expression is proposed which allows for accurate extrapolation. Additional activity coefficient data have been determined via GLC (gas–liquid chromatography) to validate the method. In most cases extrapolations were within 10% of the experimental findings and safely within or close to the scatter of literature data.The method allows for example the safe estimation of the infinite dilution behavior as well as liquid and solid solubility of high boiling components in hydrocarbons from data for high boiling solvents usually determined easily by gas–liquid chromatography. Additionally, the newly developed combinatorial expression for infinite dilution data should be able to greatly improve other predictive mixture models, e.g. UNIFAC, mod. UNIFAC and COSMO-RS type models in the cases when large molecules in low molecular weight solvents are considered. This was verified for a larger number of experimental limiting activity coefficient data from literature.

First principles investigation of temperature and pressure dependent elastic properties of ZrC and ZrN using Debye–Gruneisen theory

We investigated the effects of temperature and pressure on the elastic properties of ZrC and ZrN using first-principles calculations and quasi-harmonic Debye–Gruneisen theory. Three expressions, the Slater, Dugdale–MacDonald (DM), and free volume expressions, were employed in the Debye–Gruneisen theory to obtain the Gruneisen parameter, γ. The DM expression agreed well with the experimental measurements of the ZrN volume expansion. The volume expansion of ZrC, on the other hand, did not agree well with any expression investigated here. ZrC calculations, therefore, require a modified expression. Variations in the Debye temperature, the Gruneisen parameter, the bulk (B), shear (G), and Young’s (E) moduli were calculated as a function of temperature and pressure. The temperature-dependent properties at room pressure agreed well with previous experimental results and other calculated results; however, the pressure effects at the various temperatures were not easily compared with the results from other experimental and computational results because their effects on the elastic properties of ZrC and ZrN remain largely unknown.

Viscosity of argon to 5 GPa and 673 K

A rolling-sphere technique has been used to measure shear viscosities of (supercritical) fluid argon in the diamond-anvil cell between the temperatures of 294 and 673 K, up to a maximum pressure of 5 GPa. At these pressures, the viscosities can be fit to a modified free-volume expression. A single correlation between reduced viscosity and reduced residual entropy is shown to give a good account of the current high pressure data, data at lower pressures and those for the sub-critical liquid.

Dielectric analysis of chlorinated polyvinyl chloride stabilized with di‐n‐octyltin maleate

AbstractDielectric relaxation of chlorinated polyvinyl chloride stabilized with di‐n‐octyltin maleate has been studied in the frequency range 1 kHz–1 MHz at various temperatures between 300 and 450 K. A clear dielectric relaxation band (α‐type) associated with dipolar polarization has been observed in the studied range. The stabilizer molecules increase the segmental mobility of the polymer, and specific interaction occurs between both the CCl and CO polar groups. The temperature dependence of the relaxation process was analyzed by combining the critical free volume expression by Cohen and Turnbull with the Williams‐Landel‐Ferry (WLF) model of the thermal expansion of free and occupied volumes. © 2009 Wiley Periodicals, Inc. J Appl Polym Sci, 2009

Ultraviolet light absorber mobility in crosslinked coatings: Experiments and modeling

The diffusion of an ultraviolet absorber (UVA) across a concentration step change in an acrylic melamine clearcoat was measured via micro-ultraviolet spectroscopy. Concentration profiles were obtained after diffusion times of 4–200 h at steady temperatures of 70, 80, and 90 °C. The glass transition temperature of the coating was measured via dynamic mechanical analysis for each time–temperature pair, and T g was observed to increase with thermal aging, with a faster rate of increase at higher aging temperatures. The change in T g was modeled using a first-order dependence on the deviation from a long-time T g asymptote. The diffusion coefficient was then represented by a free volume expression, and the combination provided an accurate model of the measured concentration profiles. The time-dependent increases in glass transition temperature helped to explain why UVA diffusion does not alleviate concentration gradients created by photooxidation under natural weathering conditions.

The shear viscosity of supercritical oxygen at high pressure

Shear viscosities of supercritical oxygen have been measured up to a pressure of 5.7 GPa at 294 K. A modified free-volume expression fits the data within 6% between the limits of the tenuous gas and 4.8 times the critical density. Nitrogen viscosities were found to correspond to those of oxygen through a simple scaling by critical constants. Viscosities were measured in the high-pressure diamond-anvil cell with a rolling-ball technique. The dynamics of a sphere rolling on an inclined plane were investigated in the context of these experiments. The effect of a second surface, situated above the sphere, was experimentally determined.

Mutual Diffusion Coefficients of Some Aqueous Alkanolamines Solutions

The diffusion coefficients of alkanolamines in water at infinite dilution and the mutual diffusion coefficients of aqueous diglycolamine, triethanolamine, 2-amino-2-methyl-l-propanol, and 2-piperidineethanol solutions have been studied by using the Taylor dispersion method. Experimental results for temperatures from (30 to 70) °C and at atmospheric pressure as functions of the concentration of alkanolamines from {0.5 to 3 (or 4)} kmol·m-3 are reported. The uncertainty of the measurement is estimated to be ±2.0%. The diffusion coefficients of alkanolamines in water at infinite dilution were found to vary systematically with the molar mass of the alkanolamines. The obtained mutual diffusion coefficients were correlated by using both a simple relation and the UNIDIF equation. The measured mutual diffusivity data are also well represented by a free-volume expression. The results of this study can be used to develop the kinetic model for gas absorption using aqueous alkanolamine solutions as absorbents.

Generalised free-volume theory for transport properties and new trends about the relationship between free volume and equations of state

In this work, a new generalised model for transport properties is given, the theoretical background of some free-volume equations is revealed and the analysis of various free-volume expressions in the literature is presented. Firstly, we develop the free-volume equation for the self-diffusion coefficient, which embodies the hybrid principles of the real fluids, since it considers both the concepts of molecular free volume and activation energy. Secondly, it is shown that the relevant expressions presented in the literature are particular cases of our generalised equation. With this it is possible to clarify their validity and range of applicability, since the expressions arise by introducing approximations duly established in the fundamental model. Moreover, the relationship between free-volume expressions and the equations of state (EoS) of van der Waals (vdW) type is discussed. Some representative free-volume expressions are given and compared according to the effects of density, temperature and molecular shape. Following this comparison, some expressions are recommended.

A modified free-volume model: correlation of ion-conduction in strongly associating polymeric materials

Ions tend to aggregate in media of low dielectric constant, and this tendency to form strong associations causes small amounts of ionic functionality to have a significant influence on the properties of ionic polymers. As a consequence of ionic functionality and strong interactions ionomers exhibit unique transport characteristics compared to conventional polymers. While it has been observed that ion mobility is related to viscosity, a comprehensive theory like the existing theories for non-associating systems has not been developed for associating systems. In this work, free-volume expressions are developed for self-diffusion of ions in ionomers by treating the polymer chain as a two-component system. The ion–counterion pair and the non-ionic part that is involved in ion transport are treated as species 1 while the remaining non-ionic part of the polymer is treated as species 2. The free-volume of the system is obtained by fitting the viscosity–temperature relationship using the Williams–Landel–Ferry (WLF) equation [Viscoelastic Properties of Polymers, Wiley, New York, 1970]. However, since the flow behavior of the ionomer consists of contributions from both the ionic and non-ionic parts, the definitions of the free-volume parameters have been modified. The expressions developed are then used to evaluate data for polymer systems with different molecular weights, and type and size of cation in the ionic group. The results of the correlation are generally consistent with the free-volume concepts and assumptions employed to describe ion-conduction. Using the approach developed, an estimate of the relative size of the jumping units responsible for conduction and diffusion in polymeric materials can be obtained. The free-volume expressions demonstrate that the size of the jumping unit is a complex function of the ionomer composition and the size of the ion–counter ion pair; a finding consistent with the values obtained from correlation of experimental data. The relatively large size of the ionic jumping units compared to conventional polymeric materials agrees with the ion-hopping mechanism used to describe mobility in ionic polymers.

Modified combinatorial factor in the hole theory

To describe the behavior of liquid state by hole theory, Flory's combinatorial factor used in the partition function of the hole theory is modified in a general form by assuming that unit segment of chain molecule can occupy more than one unit cell or lattice. From this combinatorial factor and free volume expression, a new equation of state is derived, and is used to calculate the PVT behaviors of 10 pure low molecular weight liquid systems. In addition, the PVT behavior of carbontetrachloride/cyclohexane binary system is calculated. Good agreements between the theory and experimental data are obtained.

A new equation of state in hole theory based on a modified Flory's combinatorial factor

To describe the behavior of liquid state by hole theory, Flory's combinatorial factor used in the partition function of the hole theory is modified in a general form by assuming that unit segment of chain molecule can occupy more than one unit cell or lattice. From this combinatorial factor and two kinds of free volume expressions, two equations of state are derived, and are used to calculate the PVT behaviors of 10 pure polymer melts to test these EOS's. Good agreements between the theory and experimental data are obtained.

Concentration-dependent viscoelastic scaling models for polydimethysiloxane melts with dissolved carbon dioxide

Recent measurements of the viscosity of polydimethylsiloxane (PDMS) melts swollen with dissolved carbon dioxide at 50 and 80°C by Gerhardt et al. have shown that classical viscoelastic scaling factors can be employed to superpose the viscosity curves for CO2-swollen melts onto the viscosity curve for pure PDMS at the same temperature and pressure. Here a free volume expression for the viscosity of a diluted polymer melt is combined with equation-of-state theories for the volumetric properties of PDMS-CO2 mixtures to develop models for the CO2 concentration-dependent viscoelastic scaling factors. Both the Sanchez-Lacombe and Panayiotou-Vera equations-of-state are investigated for this purpose. The predicted viscoelastic scaling factors are found to be in very good agreement with the data of Gerhardt et al. The model predictions show that mixing of gaseous carbon dioxide into the swollen PDMS phase is highly non-ideal, and that significant free volume is added to the melt upon carbon dioxide dissolution. The free volume added to the melt by dissolved carbon dioxide is the principal mechanism for viscosity reduction, relative to pure PDMS, in this system. © 1998 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 36: 1911–1918, 1998

A new equation of state based on hole theory

A new equation of state (EOS) based on hole theory was developed. The free volume expression used in this equation of state did not use the notion of conventional linear superposition assumption. Free volume developed in this paper is composed of two parts: free volume by cell and free volume by hole. The intrinsic cell free volume is derived from square well potential and the free volume by hole was derived by equally partitioning hole volume into each segment. When applied to PVT data of polymeric melts, this new equation of state shows better agreement over the SS model, especially at low pressure.

Accurate correlations for the self-diffusion coefficients of CO 2, CH 4, C 2H 4, H 2O, and D 2O over wide ranges of temperature and pressure

In this work, two models are proposed for the correlation of self-diffusivities over the entire ranges of gas and liquid phases. The first model is an empirical equation containing 12 adjustable parameters and is based on the density expansion for self-diffusivities presented by Kawasaki and Oppenheim. The second model is a modification of the free-volume expression by Macedo and Litovitz, with seven adjustable parameters. Self-diffusion coefficient data for CO 2, CH 4, C 2H 4, H 2O, and D 2O, which are known to be very important supercritical fluids employed in extraction processes, were correlated with the proposed equations. The average absolute deviations obtained are smaller than 5% for wide ranges of both temperature and pressure. Therefore, these equations can be recommended for accurate predictions of self-diffusivities for these substances in the covered density ranges.

Prediction of flux and selectivity in pervaporation through a membrane

Flux and selectivity in pervaporation through a membrane can be predicted via the solution-diffusion mechanism by the use of solubility and diffusivity models. The solubility model developed takes into account the free energy contributions in the mixture system of multicomponent penetrants and a polymer from combinatorial-entropy, free-volume, interactional-enthalpy, and elastic factors. The diffusivity model developed is the hybrid model that combines the key features of the free-volume and molecular models. In the hybrid model, penetrant molecular thickness and polymer molecular parameters are used to determine the term equivalent to the preexponential factor of the free-volume model, and the free-volume expression is used to relate penetrant diffusivity to penetrant size and the free volume of the mixture system including concentration contributions from the multicomponent penetrants. Both the solubility and diffusivity models have been verified with the sorption of single-component aromatic vapors in a semicrystalline polyethylene film. Based on the single-component solubility and diffusivity model parameters determined from the sorption, the prediction of flux and selectivity is in line with experimental data obtained from the perva-poration of a mixture of toluene, p-xylene, and mesitylene through the polyethylene film at various temperatures and different permeate pressures. The effects of temperature on flux and selectivity are investigated via both the model prediction and experiments. Also predicted are the effects of permeate vapor pressure on flux and selectivity, and the concentration and diffusivity profiles across the membrane.

The effect of stoichiometry on chain segment and ion mobility in partially polymerized epoxy systems

AbstractThe temperature dependence of steady‐shear viscosity and ionic conductivity were measured for a series of unreacted mixtures and partially cured, ungelled samples of diglycidyl ether of bisphenol‐A (DGEBA) and an amine cross‐linking agent, diamino diphenyl sulfone (DDS). Six stoichiometric ratios of epoxide groups to amine hydrogens were examined. Free volume expressions were used to model the temperature dependence of the conductivity and viscosity for the unreacted DGEBA‐DDS mixtures. In addition, these expressions were combined to successfully correlate changes in viscosity and conductivity during the DGEBA‐DDS polymerization prior to gelation. It also was demonstrated that the change in weight average molecular weight during polymerization could be interpreted from the dielectric data. Through studying variations in the stoichiometry, it was possible to examine the effects of changes in chemical structure and ion concentration on the fitted parameters in the free volume models. The inherent ion transport factor (ζ0) was found to be inversely proportional to the concentration of ions in the test samples. The fractional free volume for segmental motion (B) was found to increase with an increase in the glass transition temperature and to be a function of the rigidity of the polymer. ©1995 John Wiley & Sons, Inc.

Prediction of solvent activities in polymer solutions

The Holten-Andersen equation of state and the UNIFAC-FV (Free Volume) activity coefficient models are compared using a polymer solution data base. Both the Holten-Andersen equation of state and the UNIFAC-FV models have a foundation in the equation of state developed by Flory for polymer solutions. The Holten-Andersen equation of state is presented here with some corrections from the original publication. The UNIFAC-FV model is an extension of the UNFAC model for polymer solutions and includes the free volume expression from the Flory equation of state. The Holten-Andersen group contribution equation of state and the UNIFAC-FV activity coefficient model are the only predictive models available that were developed specifically for polymer solutions. Unlike the older correlative methods, these two models have not been previously tested and compared using an experimental data base.

Prediction of solvent activities using the UNIFAC-FV model

A UNIFAC group contribution method with an explicit temperature dependent free volume term has been developed. The systems used in the development were benzene-polyethylene oxide from 319 to 423 K and n-hexane—n-hexadecane from 303 to 333K. The addition of temperature dependence to the proportionality factor in the free volume expression decreased the average error in prediction of the solvent activities from 15% to 6%.

GAS CHROMATOGRAPHIC DETERMINATION AND CORRELATION OF WEIGHT-FRACTION HENRY'S CONSTANTS FOR HYDROCARBON GASES AND VAPORS IN MOLTEN POLYMERS

In the polymer industry, separation equipment is required to separate unreacted monomers and solvents from polymers. In order to design such equipments, solubilities of gases and vapors in molten polymers are needed as fundamental data. It is very helpful if the weight-fraction Henry's constant, which gives solubilities of a solute at concentrated regions of a polymer, can be estimated by a predictive method. In the present study, a new expression for the free volume term has been derived and is coupled with the UNIFAC model. The present UNIFAC-FV model was applied to common polymer systems. The weight-fraction Henry's constants were calculated for several volatile hydrocarbons and the calculated results were compared with the experimental data determined by a gas chromatographic technique. The present UNIFAC-FV model with a new free volume expression was found to be helpful in predicting the weight-fraction Henry's constants of hydrocarbon solutes in molten polystyrene, in polypropylene, and in low-density polyethylene. It is advantageous that our model is applicable both supercritical gases and subcritical vapors with no adjustable parameters.

A pulsed field gradient NMR study of diffusion in semicrystalline polymers: self-diffusion of alkanes in polyethylenes

By means of the pulsed field gradient NMR technique the self-diffusion of six alkanes (from n-butane to n-pentadecane) in three low density polyethylenes and one high density polyethylene differently thermally treated was examined. The concentration dependence could be described very satisfactorily with the free volume theory in the form of Fujita (Adv. Polymer Sci. 3(1961) 1). The parameter B of the diffusants and the fractional free volumef 2 of the polyethylenes were determined from the experimental data. The fractional free volumesf 2 show a strong dependence on the type of polyethylene, the main influence results from the different content of CH3 groups or short chain branches. The diffusion coefficient extrapolated to zero diffusant concentration is proportional to the eighth power of the amorphous content. This strong dependence shows that the free volumes of the amorphous parts of the polyethylenes are intimately connected with crystallinity, both determined by the different degrees of short chain branching. The pre-exponential factor in the free volume expression decreases with increasing amorphous content of the polyethylenes and increases with increasing length of the diffusants. It was found that the spherulite boundaries in the polyethylenes do not act as diffusion barriers.