Generic van der Waals equation of state, modified free volume theory of diffusion, and viscosity of simple liquids

The shear viscosity formula derived by the density fluctuation theory in previous papers is computed for argon, krypton, and methane by using the self-diffusion coefficients derived in the modified free volume theory with the help of the generic van der Waals equation of state. In the temperature regime near or above the critical temperature, the density dependence of the shear viscosity can be accounted for by ab initio calculations with the self-diffusion coefficients provided by the modified free volume theory if the minimum (critical) free volume is set equal to the molecular volume and the volume overlap parameter (alpha) is taken about unity in the expression for the self-diffusion coefficient. In the subcritical temperature regime, if the density fluctuation range parameter is chosen appropriately at a temperature, then the resulting expression for the shear viscosity can well account for its density and temperature dependence over the ranges of density and temperature experimentally studied. In the sense that once the density fluctuation range is fixed at a temperature, the theory can account for the experimental data at other subcritical temperatures on the basis of the intermolecular force only; the theory is predictive even in the subcritical regime of temperature. Theory is successfully tested in comparison with experimental data for self-diffusion coefficients and shear viscosity for argon, krypton, and methane.     

Modified free volume theory of self-diffusion and molecular theory of shear viscosity of liquid carbon dioxide

In previous work on the density fluctuation theory of transport coefficients of liquids, it was necessary to use empirical self-diffusion coefficients to calculate the transport coefficients (e.g., shear viscosity of carbon dioxide). In this work, the necessity of empirical input of the self-diffusion coefficients in the calculation of shear viscosity is removed, and the theory is thus made a self-contained molecular theory of transport coefficients of liquids, albeit it contains an empirical parameter in the subcritical regime. The required self-diffusion coefficients of liquid carbon dioxide are calculated by using the modified free volume theory for which the generic van der Waals equation of state and Monte Carlo simulations are combined to accurately compute the mean free volume by means of statistical mechanics. They have been computed as a function of density along four different isotherms and isobars. A Lennard-Jones site-site interaction potential was used to model the molecular carbon dioxide interaction. The density and temperature dependence of the theoretical self-diffusion coefficients are shown to be in excellent agreement with experimental data when the minimum critical free volume is identified with the molecular volume. The self-diffusion coefficients thus computed are then used to compute the density and temperature dependence of the shear viscosity of liquid carbon dioxide by employing the density fluctuation theory formula for shear viscosity as reported in an earlier paper (J. Chem. Phys. 2000, 112, 7118). The theoretical shear viscosity is shown to be robust and yields excellent density and temperature dependence for carbon dioxide. The pair correlation function appearing in the theory has been computed by Monte Carlo simulations.     

A nuclear magnetic resonance study of toluene-polyisobutylene solutions. 3. Spin-lattice relaxation: Rotational and segmental motion

The spin-lattice relaxation times are determined for the methylene carbon of polyisobutylene (PIB), as well as for the ortho carbon of toluene in toluene-polyisobutylene solutions. The Hall-Helfand correlation function combined with restricted anisotropic rotational diffusion was used to treat the T₁ data of the methylene carbon of PIB. A simple exponential correlation function was used to describe the local motion of toluene in the solutions which falls in the extreme narrowing limit for the solutions studied. Both models described satisfactorily the temperature and field dependence of the spin-lattice relation times. From the temperature dependence of the correlation times for the polymer segmental motion, the free volume of the solution at each concentration is extracted and compared with the values obtained from previous studies of the translational motion of the toluene penetrant. The free volume values extracted from the T₁ data for the methylene carbon of PIB and the self-diffusion data for the toluene were found to be in substantial agreement. The interrelationship of the timescale of segmental motion of the polymer and the translational diffusion of the toluene was also examined and it was found that the two types of motion seem to be correlated in high polymer concentrated solutions. The toluene reorientational motion was found to be much faster than both the polymer segmental motion and the toluene translational diffusion leading to the conclusion that the toluene reorientational motion is uncoupled from these two motions. ©1995 John Wiley & Sons, Inc.     

A nuclear magnetic resonance study of toluene-polyisobutylene solutions. 2. Vrentas-Duda theory

The self-diffusion coefficients of toluene in polyisobutylene have been analyzed using the Vrentas-Duda free volume diffusion model. The diffusion coefficients were determined at different temperatures and concentrations, using the pulsed field gradient nuclear magnetic resonance technique. The data were satisfactorily described by the model and the size of the polymer jumping unit was extracted. Comparisons were made with the Fujita free volume theory and the Fujita free volume parameters were extracted from the Vrentas-Duda free volume parameters. From the diffusion data that now available, it can be concluded that for most polymers the jumping unit is about 1.5 times the polymer monomer molecular weight. The activation energy of the toluene diffusion in polyisobutylene is compared with the activation energies of other penetrants in the same polymer. The diffusion data presented in this work show that the energy per mole required to overcome the attractive forces which constrain a diffusing species to its neighbors should be considered to be zero, in order to be able to extract the free volume parameters (from viscosity and diffusion data) with an acceptable uncertainty. ©1995 John Wiley & Sons, Inc.     

A model for predicting solvent self-diffusion coefficients in nonglassy polymer/solvent solutions

Cohen-Turnbull diffusion theory is used to develop a model for predicting solvent self-diffusion coefficients D₁ in nonglassy polymer/solvent solutions. Polymer molecules are envisioned as hindering solvent mobility by reducing the average free volume per unit mass in the system and through the lower mobility of polymer segments relative to solvent molecules. The concentration dependence of D₁ predicted by the model is in reasonable agreement with data for the solvents heptane, hexadecane, benzene, cyclohexane, and decalin in polyisobutylene (PIB), and for toluene in polystyrene, poly(methyl mothacrylate), and PIB. Although none of the data is for high concentrations of polymer (volume fractions ?≥0.9) it is anticipated the model will be less representative in this regime where the assumptions in its development are unsure. The model also demonstrates the correct temperature and concentration dependence of the apparent activation energy for diffusion. The only experimental data needed to use the model are the viscosity and critical volume of the pure solvent, and the specific volume of both the solvent and mixture. No binary transport data are required.     

Correlation between thermal diffusion and solvent self-diffusion in semidilute and concentrated polymer solutions

We have performed measurements of thermal diffusion coefficients DT and solvent self-diffusion coefficients Dss in semidilute to concentrated polymer solutions. Solutes of different glass transition temperatures and solvents of different solvent qualities have been used. The investigated systems are in detail: poly(dimethyl-siloxane) in toluene, tristyrene in toluene, polystyrene in toluene, polystyrene in tetrahydrofuran, polystyrene in benzene, and polystyrene in cyclohexane. The thermal diffusion data are compared to our data and literature data for solvent self-diffusion coefficients. In all systems the concentration dependence of DT closely parallels the one of Dss which may be viewed as a local probe for friction on a length scale of the size of one polymer segment. This identifies local friction as the dominating parameter determining the concentration dependence of DT. Solvent quality, in contrast, has no influence on DT.     

Toluene diffusion in butyl rubber

The sorption isotherm and the polymer mass-fixed diffusion coefficients, D, for toluene in butyl rubber have been measured by the incremental sorption method to concentrations of 130%, corresponding to a solvent volume fraction of 0.578. The increase in D with concentration is strongly exponential to a concentration of 30% and then begins to level out. Since the nature of the dimensional change occurring in vapor sorption was not known, the values of D were converted to solvent self-diffusion coefficients, D₁, assuming both swelling in the thickness direction (1D) and isotropically (3D). The free volume (FV) theory of Fujita was fitted to the resulting D₁ with the zero concentration diffusion coefficient and the self-diffusion coefficient of toluene as limiting values leaving only a single arbitrary parameter. In this form the FV theory was able to describe the trend of the experimental D₁ for the 1D and 3D cases equally well. Values of D were back-calculated from the FV relations for the 1D and 3D cases for comparison with the experimental results and with the diffusion coefficient determined by immersion in toluene. These comparisons favor the assumption that swelling is isotropic. It appears that the simple free volume relation is capable of providing a satisfactory representation of the experimental data with only a single fitting parameter, although there are moderate quantitative discrepancies. © 1994 John Wiley & Sons, Inc.     

Pair correlation functions and the self-diffusion coefficient of Lennard-Jones liquid in the modified free volume theory of diffusion

In this paper, we apply the Matteoli-Mansoori empirical formula for the pair correlation function of simple fluids obeying the Lennard-Jones potential to calculate reduced self-diffusion coefficients on the basis of the modified free volume theory. The self-diffusion coefficient thus computed as functions of temperature and density is compared with the molecular dynamics simulation data and the self-diffusion coefficient obtained by the modified free volume theory implemented with the Monte Carlo simulation method for the pair correlation function. We show that the Matteoli-Mansoori empirical formula yields sufficiently accurate self-diffusion coefficients in the supercritical regime, provided that the minimum free volume activating diffusion is estimated with the classical turning point of binary collision at the mean relative kinetic energy 3k(B)T/2, where k(B) is the Boltzmann constant and T is the temperature. In the subcritical regime, the empirical formula yields qualitatively correct, but lower values for the self-diffusion coefficients compared with computer simulation values and those from the modified free volume theory implemented with the Monte Carlo simulations for the pair correlation function. However, with a slightly modified critical free volume, the results can be made quite acceptable.     

Temperature-Dependent Diffusion Coefficients from 1D Raman Spectroscopy

A setup for the measurement of temperature-dependent diffusion coefficients has been developed based on 1D Raman spectroscopy. The presented setup was used to measure binary diffusion coefficients for mixtures of monoethanolamine + water, cyclohexane + toluene, and methanol + toluene for temperatures between T = 298.15 and 330.15 K. The experimental diffusion coefficients agree well with available literature data. For the mixture methanol + toluene, literature data was only found for T = 298.15 K. The novel setup was therefore used to determine temperature-dependent diffusion coefficients of methanol + toluene up to T = 328.15 K. The experimental data are also compared to a temperature correlation for diffusion coefficients in concentrated solutions. While the correlation describes the temperature dependence well for simple systems, measurement techniques remain indispensable for diffusion in complex mixtures.     

Concentration dependence of the diffusion coefficient of polydisperse polystyrene in dilute solution

The concentration dependence of the diffusion coefficients (D) of two different polystyrenes in toluene was measured. The concentration dependence of D of a standard monodisperse sample (M = 498,000) for concentrations up to 1 · 25 g/dl is not linear. The dependence is adequately described by the theory of dilute polymer solutions up to about 0·7 g/dl and the second virial coefficients of the osmotic pressure can be evaluated. For a polystyrene sample having a broad molecular weight distribution, the concentration dependence of four different average diffusion coefficients was determined so indirectly characterizing the molecular weight distribution. These dependences are not linear and differ from each other owing to the different sensitivity of the individual averages to high-molecular and low-molecular weight fractions. The apparent distribution of the diffusion coefficients becomes narrower with increasing concentration. When evaluating polydispersity from the free diffusion data obtained in good solvents, it is necessary to determine directly the differential diffusion coefficients; an extrapolation of the integral diffusion coefficients can be misleading.     

Transient entanglement behaviour in a poly(propylene glycole) melt: A field gradient NMR self-diffusion study

 We report measurements of the temperature dependence of the self-diffusion of a poly(propylene glycole) in the melt. Two kinds of magnetic field gradient NMR are used: pulsed field gradient NMR with large field gradients and static field gradient NMR in a specially designed cryomagnet. The emphasis is put on large field gradients. The (true) long-time self-diffusion coefficients are compared with those calculated from normal-mode relaxation times in dielectric spectroscopy using the Rouse model. They agree very well. For temperature below about 270 K, a time-dependent self-diffusion coefficient appears which is an indication of anomalous diffusion. At 253 K we observe complete transient entanglement behaviour of the PPG melt. These results are discussed in comparison with dielectric and other experimental data. Received: 6 September 1996 Accepted: 13 November 1996     

Temperature and concentration dependence of diffusion coefficients in ethylene–propylene-diene copolymers

Self-diffusion and partition coefficients were measured for two commercial ethylene–propylene-diene copolymers (EPDM) and five solvents at infinite dilution using inverse gas chromatography. Mutual diffusion coefficients for solvents in EPDM also were measured for finite concentration using gravimetric sorption for three of the solvents. From the inverse gas chromatography experimental values for self-diffusion coefficients were obtained. Free-volume parameters were obtained through regression of the self-diffusion coefficient as a function of temperature. Mutual diffusion coefficients as a function of concentration were predicted using free volume theory and compared with experimental data obtained using gravimetric sorption. © 1998 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 36: 1713–1719, 1998     

Self-Diffusion Coefficients for Pure and Mixed Adsorbate Fluids in Narrow Pores

The equilibrium distribution and the concentration dependence of the local and average self-diffusion coefficients for pure fluid and binary mixture components in narrow slitlike pores were analyzed. The coefficients were calculated using the lattice gas model in the quasi-chemical approximation on the assumption of a spherical shape and approximately equal sizes of the components. For the pure adsorbate, these calculations were compared with molecular dynamics simulations. Both methods gave similar concentration profile changes and dynamic characteristics of interlayer particle redistributions in strong nonuniform adsorption fields for dense fluids. A satisfactory agreement was obtained for the temperature dependences of the self-diffusion coefficients along the pore axis. The influence of the molecule–wall potential and of intermolecular interaction were considered. The self-diffusion coefficients of the adsorbate were shown to strongly depend on the density of the mixture and the distance from pore walls.     

Influence of the glass transition on solvent self-diffusion in amorphous polymers

The free-volume theory of diffusion is used to analyze the temperature dependence of solvent self-diffusion coefficients both above and below the glass transition temperature at concentrations removed from the pure polymer limit. The glass transition can have a pronounced effect on the temperature dependence of solvent self-diffusion coefficients at small solvent concentrations, but the theory predicts a decreased effect of the transition on the diffusion process with increasing solvent concentration.     

Influence of concentration on the activation energy for diffusion in polymer-solvent systems

The self-diffusion of benzene, toluene, and ethylbenzene in polystyrene have been analyzed using the Vrentas/Duda free-volume diffusion model. Diffusion coefficient predictions suggest an exponential concentration dependence of the activation energy required to overcome attractive forces, E. Without the use of any diffusion data approximating E as zero over the entire concentration range yields self-diffusion coefficient predictions which are in good agreement with experimental data. © 1992 John Wiley & Sons, Inc.     

Anomalous thermal expansion of thin cetane layer solidified at the inner surface of confining nanoporous silica gel

Silica gel with extremely small pores of 3 nm diameter was filled with liquid cetane. Samples with various coefficients of filling k of cetane were prepared. After the solidification of confined cetane, its free volume has been investigated. The positron annihilation lifetime spectroscopy at 24–300 K and differential scanning calorimetry (DSC) at 233–303 K measurements have been performed. Decreasing the temperature, only small changes in dimensions of silica gel matrix and pronounced contraction of confined cetane have been indicated. In the case of k < 0.2, however, via cooling below 180 K, the temperature dependence changes its sign and the free volume in confined cetane increases. Decreasing the cetane content, the negative apparent expansion coefficient dramatically grows. This anomalous temperature dependence is interpreted by the cracking of thin layer of solid cetane which is in contact with the walls of silica gel pores.     

Diffusion of radioactively tagged 1,1-diphenylethane and n-hexadecane through rubbery polymers: Dependence on temperature and dilution with solvent

The diffusion of 1,1-diphenylethane in trace amounts through eight rubbery polymers has been studied by radioactive tagging of this penetrant with ¹⁴C. For several polymers, the dependence on temperature and on dilution (swelling) by untagged diphenylethane was investigated. In the diluted systems, tagged n-hexadecane was also used as a trace penetrant. The temperature and concentration dependences were interpreted rather successfully in terms of the free volume. In comparing different polymers, with a 4000-fold range of diffusion coefficients, the translatory friction coefficient of 1,1-diphenylethane was found to be proportional to that of n-hexadecane to the power 1.06. This is interpreted qualitatively by the free volume concept to indicate a slightly less efficient mobility mechanism for the diphenylethane.     

Temperature and Concentration Dependence of the Electrical Conductance,Diffusion, and Kinetics Parameters of the Ions in Aqueous Solutions of Sulfuric Acid,Selenic Acid,and Potassium Tellurate

The temperature and concentration dependences of the electrical conductance of aqueous solutions of sulfuric acid, selenic acid, and potassium tellurate were studied. The coefficients of the corresponding empirical equations were determined, and the values of equivalent conductances of the anions were evaluated at infinite dilution at the experimental temperatures. The values of the coefficients in the Fuoss and Onsager equation were evaluated for the three electrolytes at 298 K. The values of the molecular and ionic coefficients of self-diffusion at infinite dilution were calculated in the temperature range 288–318 K. The change of the translational energy Δ E∘_tr. of water molecules in the ionic hydration sphere was determined. The number of water molecules participating in the ionic hydration sphere at 298 K and the changes of Gibbs free energy, enthalpy, and entropy of activation of ionic conductance were calculated. The results obtained were interpreted according to the Samoylov’s theory of positive and negative hydration of ions. The differences observed in the temperature dependences of the mentioned parameters were explained in terms of the different radii and hydration numbers of the ions.     

The use of the cohen–turnbull model for calculation of the self-diffusion coefficients of liquid dichloroalkanes

The paper presents the self-diffusion coefficients calculated for liquid dichloroalkanes C₆H₁₂Cl₂, C₈H₁₆Cl₂, C₁₀H₂₂Cl₂ and C₁₂H₂₄Cl₂, with the use of the Cohen and Turnbull model. Determination of self-diffusion coefficients permits a separate analysis of intra- and intermolecular motions and provides information on geometrical and dynamical properties of molecules. The self-diffusion coefficients of selected dichloroalkanes have been determined by X-ray diffraction and compared with the corresponding NMR results. The suitability of the Cohen–Turnbull model of the translating motion for prediction of self-diffusion coefficients for molecules whose shape significantly differs from the spherical symmetry is analysed. Angular distributions of X-ray scattered intensity were measured, and differential radial distribution functions of electron density (DRDFs) were calculated. The mean coordination numbers were obtained from the area delimited by the minima of the DRDFs, and their dependence on the length of the methylene chain is also presented subsequently. On the basis of the DRDFs the average free volume of the molecules and total free volume of the liquids were calculated. The activation volume of the diffusion was found to make about 0.6 of the van der Waals volume of the molecule. As expected the diffusion coefficients decrease with increasing molecular weight. The equation relating the self-diffusion coefficient with the volume of the coordination spheres in the liquid has been derived.     

Solvent self-diffusion,polymer NMR relaxation,and free volume in polymer solutions

The self-diffusion coefficient of chloroform in poly(isopropyl acrylate)—chloroform solutions has been studied as a function of concentration and temperature by using the pulsed-field-gradient spin-echo NMR method. It is found that the self-diffusion coefficient of the solvent can be adequately fitted by using a simple free-volume approach with either a concentration or temperature superposition. It was noted that the free-volume parameters derived from the self-diffusion data are the same as those derived from deuterium NMR transverse relaxation-time measurements of the polymer in the same system. The equality of these two sets of experiments suggests a fundamental relationship between the two different processes. The simplest explanation is that the free volume necessary for the local segmental motion of the polymer and the translation of the solvent are similar.