A nuclear magnetic resonance study of dynamics in toluene-polyisobutylene solutions: 1. Penetrant diffusion and Fujita theory

The self-diffusion coefficients of toluene in polyisobutylene (PIB) solutions were determined using the pulsed field gradient nuclear magnetic resonance technique. The volume fraction of toluene in the polymer was varied from 0.045 up to 0.712 and the temperature was varied from 225 K up to 368 K. The concentration dependence of the data was interpreted using the Fujita free volume theory and the temperature dependence was interpreted with the WLF equation. These models describe separately the concentration and temperature dependencies of the toluene self-diffusion coefficients very well and the resulting free volume parameters are in good agreement with the ones extracted from the analysis of viscosity data on the same system. ©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.     

环己烷在聚乙烯和聚异丁烯中无限稀释扩散系数 的测定

采用填充柱的反相气相色谱方法,基于Hadj-Romdhane-Danner色谱过程的数学模型,在323.2～363.2K温度范围内测定了环己烷在聚异丁烯和聚乙烯膜中的无限稀释扩散系数。从实验数据可以看出,环己烷更容易在PIB中发生扩散。对实测扩散数据进行了自由体积理论关联,结果表明,采用空穴自由体积替代Vrentas-Duda自由体积理论方程中自由体积项的修正方程能很好地描述溶剂分子扩散系数随温度的变化关系。     

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.     

Diffusion in polymer—solvent systems. I. Reexamination of the free-volume theory

The free-volume theory describing diffusion in polymer–solvent systems is reexamined. Calculation of the specific free volume for such systems is discussed, and equations are presented for the determination of the self-diffusion coefficients of the polymer and the solvent. Conditions under which the mutual diffusion coefficient can be deduced solely from free-volume considerations are clarified, and a more general version of the free-volume diffusion theory proposed by Fujita is presented. The further restrictions needed for the theory of Fujita are discussed, and it is concluded that these additional restrictions are responsible for failures of the Fujita theory.     

1,3,5-triisopropylbenzene diffusion in polystyrene solutions

Self-diffusion coefficients of 1,3,5-triisopropylbenzene (TIB) in binary solutions with polystyrene and ternary mixtures comprised of TIB, toluene, and polystyrene have been measured by static gradient nuclear magnetic resonance (SG-NMR) techniques. These data, as well as mutual-diffusion coefficient data measured by capillary column inverse gas chromatography (CCIGC), have been analyzed here with the Vrentas-Duda free-volume diffusion model. Although both binary and ternary diffusion coefficient data can be accurately correlated with the model, the results contradict those of an earlier investigation, which suggested that TIB diffuses as a single unit in polystyrene solutions. The new data suggest that either: 1) TIB diffuses by piece-wise displacements in polystyrene solutions, or 2) a diffusive activation energy ceiling value has been reached. The implications of both possibilities are discussed in the context of established free-volume principles. © 1996 John Wiley & Sons, Inc.     

苯与DOP在LDPE与PEMA基体中无限稀释扩散系数的测定及研究

用逆流气相色谱法测定了苯、邻苯二甲酸二辛酯(DOP)在293～373K间在低密度聚乙烯(LDPE)、聚甲基丙烯酸乙酯(PEMA)基体中的无限稀释扩散系数,拟合了扩散系数与温度倒数的对数线性关系,并求得扩散活化能,发现它在玻璃化温度附近出现转折,且两者之间存在0．6倍关系。通过检验Vrentas-Duda自由体积扩散模型,发现在小分子无相变情况下该模型比较准确。     

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.     

Phase equilibrium and diffusion of solvents in polybutadiene: A capillary‐column inverse gas chromatography study

The capillary‐column inverse gas chromatography method was used to measure the diffusion and partition coefficients of ethylbenzene, styrene, and acrylonitrile in polybutadiene (PBD) at infinite dilution of the solvents. Experiments were performed over a temperature range of 50–125 °C. At temperatures well above the glass‐transition temperature of PBD, the diffusivities were correlated using an Arrhenius expression. The Arrhenius parameters in turn were intercorrelated and shown to be a function of the occupied volume, thus providing a method for predicting the diffusion of other solvents in the same polymer. Further, the activation energy was predicted using the Duda‐Vrentas free‐volume approach. The activation energy thus obtained was compared with the activation energy of the Arrhenius approach. The weight‐fraction activity coefficient data were compared to the predictions of the group contribution, lattice‐fluid equation‐of‐state, and the UNIquac Functional‐group Activity Coefficient (UNIFAC) free‐volume models. © 2002 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 40: 1046–1055, 2002     

Diffusion and solubility of methane in polyisobutylene

Diffusion coefficients and solubilities of methane in polyisobutylene have been measured at four temperatures between 102 and 188°C. in the pressure range 23–341 atm. Diffusion coefficients extrapolated to atmospheric pressure range from 1.72 × 10^?6 cm.²/sec. at 102°C. to 1.5 × 10^?5 cm.²/sec. at 188°C. corresponding to an activation energy for diffusion of 8.7 ± 0.4 kcal./mole. Solubilities are small, about one molecule of methane for every forty carbon atoms in the polyisobutylene at 300 atm. partial pressure of methane. Solubilities vary little with temperature, but show an apparent minimum between 127 and 188°C. With improved methods of data analysis, diffusion coefficients and solubilities have been recalculated from previously reported studies on nitrogen in branched polyethylene and methane in branched polyethylene, linear polyethylene, and polystyrene. Recalculated diffusion coefficients are essentially the same as those reported previously, but the recalculated solubilities are decreased from 2 to 30%. The solubilities of all five systems show strong deviations from Henry's law, i.e., increases in partial pressure of methane and nitrogen with respect to solubility exceed linearity. The partial pressure (or fugacity)—solubility data may be interpreted in terms of a sorption model in which sorbed molecules are accommodated in widely dispersed, unoccupied volumes or sites in the polymer. An almost equivalent, solution model in which the first sorbed molecules to enter the polymer are accommodated to a large extent in existing volumes in the polymer, with successively sorbed molecules swelling the polymer to a greater extent (i.e., partial molal volume of sorbed molecules, V ₁, increasing with concentration) can also account for these data.     

Liquid-liquid phase separation by nucleation and growth in solutions of poly(2,6 dimethyl-1,4 phenylene oxide) in toluene

In solutions of poly(2,6 dimethyl-1,4 phenylene oxide) in toluene, the nucleation of the newly formed phase during liquid-liquid phase separation takes place after induction periods which vary between several minutes (at temperatures close to the spinodal) and several hours (at temperatures close to the cloudpoint). The growth of the nuclei in the initial stages is diffusion controlled. The diffusion coefficients and the activation energy of diffusion were calculated. From these values, together with the calculated volume free energies and the experimental induction times, an estimate could be made of the size of a critical nucleus and of the surface free energy of the nuclei.     

Solvent diffusion in amorphous glassy polymers

In this article, a mathematical model is proposed for predicting solvent self‐diffusion coefficients in amorphous glassy polymers based on free volume theory. The basis of this new model involves consideration of the plasticization effects induced by small molecular solvents to correctly estimate the hole‐free volume variation above and below the glass‐transition temperature. Solvent mutual‐diffusion coefficients are calculated using free volume parameters determined as in the original theory. Only one parameter, which can be predicted by thermodynamic theory, is introduced to express the plasticization effect. Thus, this model permits the prediction of diffusion coefficients without adjustable parameters. Comparison of the values calculated by this new model with the present experimental data, including benzene, toluene, ethyl benzene, methyl acetate, and methyl ethyl ketone (MEK) in polystyrene (PS) and poly(methyl methacrylate) (PMMA), has been performed, and the results show good agreement between the predicted and measured values. © 2000 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 38: 846–856, 2000     

Finite concentration inverse gas chromatography: Diffusion and partition measurements

Inverse gas chromatography (IGC) is a very fast, accurate, and reliable technique to measure diffusion coefficients. This technique however, has been limited to measurements in the infinite dilution region, i.e., in the region of negligible amount of solvent in the polymer. We have extended the scope of inverse gas chromatography to measure diffusion coefficients at finite concentrations of the solvent. This involves doping the carrier gas with a solvent of interest to achieve finite concentrations of solvent in the carrier gas and hence in the polymer. The carrier gas is passed through a saturator maintained at constant temperature to achieve this purpose. Diffusion coefficients for polyvinyl acetate–toluene, and polystyrene–toluene systems were determined at finite concentrations. The results were compared with the traditional gravimetric sorption and piezoelectric sorption measurements reported in the literature. The data are in excellent agreement with the values reported, correlate well with the Vrentas–Duda free volume theory, and can also be predicted from infinitely dilute data using the free volume theory. © 1997 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 35 : 1279–1290, 1997     

Multicomponent diffusion of methyl ethyl ketone and toluene in polyisobutylene from vapor sorption FTIR-ATR spectroscopy

In this study, diffusion coefficients of toluene/methyl ethyl ketone (MEK) mixtures in polyisobutylene were measured at 50°C using vapor sorption FTIR-ATR (Fourier Transform Infrared Attenuated Total Reflectance) spectroscopy. For three mixture compositions, the diffusion coefficients were determined using a diffusion framework for ternary systems. The “crossterm” diffusion coefficient for MEK was found to be very small under the experimental conditions studied here, while that for toluene was found to increase with increasing MEK concentration. On the basis of this finding, a binary diffusion model was used to determine diffusion coefficients for MEK over a wide range of mixture compositions and the results compared well with those determined from pure MEK transport data. Relative transport rates during integral sorption experiments with mixtures were used to explain the results. © 1998 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 36 : 337–344, 1998     

Effects of substituent groups of polyimides on sorption and diffusion of gases

Sorption rate curves of CO₂, N₂, and He gases below 1 atm were measured for polyimide films prepared from benzophenone tetracarboxylic dianhydride (BTDA) with 3,5-diaminotoluene trifluoride (DATF), 2,4-diaminotoluene (DAT), m-phenylenediamine (MPD), and diaminobenzoic acid (DABA). The molecular structures of these four polyimides differ only in the substituent groups of the diamine structure. These polyimides exhibit dualmode type sorption isotherms for carbon dioxide that are concave to the pressure axis, typical of glassy polymer/gas system. The apparent diffusion coefficients below 1 atm pressure of carbon dioxide for this series of compounds decrease in the order: BTDA-DATF > BTDA-DAT > BTDA-MPD > BTDA-DABA. A linear relation between the logarithm of the apparent diffusion coefficient and the reciprocal of free volume, calculated by the method of Bondi using density data, is found for these polyimides. However, this tendency is not observed for the other two gases. The activation energies of the apparent diffusion coefficients at 20 cmHg pressure of carbon dioxide increase with increasing cohesive energy density of the polyimides. The energy per mole of free volume elements in a liquidlike structure in each cohesive energy density may be equated to the activation energy and used to calculate the free volume. The values from the activation energy are almost the same as those from Bondi's method.     

Diffusion coefficients and activation energies of diffusion of low molecular weight migrants in Poly(ethylene terephthalate) bottles

Poly(ethylene terephthalate) (PET) is nowadays the packaging material of choice for beverages. Therefore knowledge about the diffusion coefficients at a certain temperature or activation energies of diffusion of potential migrants in the polymer is of interest, especially for the definition of the basic parameter set for migration modelling of PET. In this study, the diffusion coefficients of acetaldehyde, benzene and tetrahydrofuran in PET bottle materials were determined from kinetic migration experiments at four different temperatures. The activation energies for tetrahydrofuran and benzene were determined to be 106.8 kJ mol⁻¹ and 101.4 kJ mol⁻¹, respectively. The activation energy for acetaldehyde is significantly lower (75.7 kJ mol⁻¹) which is due to the lower molecular weight of this molecule compared to benzene and tetrahydrofuran. The results were compared with literature data of diffusion coefficients of other low molecular weight molecules in PET. From the results it is evident that the current migration model with the default modelling parameters for PET does not describe realistically the diffusion coefficients in PET. The migration of small molecules like acetaldehyde will be underestimated whereas higher molecular weight compounds will be overestimated by the current migration model. Whereas the overestimation is useful for compliance evaluation of PET bottles, for more realistic migration calculations, e.g. for exposure estimations, such overestimation is not desirable. Therefore, more accurate modelling parameters should be established. The key parameters for more realistic migration modelling are the activation energies of diffusion in the polymer.     

Free volume and oxygen transport in cold‐drawn polyesters

Poly(ethylene terephthalate) (PET), poly(ethylene terephthalate‐co‐4,4′‐bibenzoate) (PETBB55), and poly(ethylene 2,6‐naphthalate) (PEN) were cold‐drawn to achieve uniform extension without crystallization or stress whitening, and oxygen transport properties were studied at temperatures from 10 to 40 °C. Correlation of oxygen solubility and polymer specific volume made it possible to consider the oriented polyester as a one‐phase densified glass. Orientation was viewed as decreasing the amount of excess‐hole free volume and bringing the nonequilibrium polymer glass closer to the equilibrium condition. Between 10 and 40 °C, the amount of excess‐hole free volume in PET decreased as the polymer approached the glass transition temperature. In contrast, temperature changes in this range had little effect on the excess‐hole free volume in PETBB55 and PEN, which were well below their glass transition temperature. Gas diffusion was viewed as discrete jumps of the oxygen molecule between holes of excess‐free volume. The jump length was extracted from the activation energy for diffusion according to a channel‐formation model. The result agreed well with the hole spacing estimated from a simple lattice model using the hole density reported in the literature. Extending the lattice model to estimate the mean excess‐free volume hole radius from the fractional free volume resulted in good correlation with the hole radius obtained from positron annihilation lifetime spectroscopy. © 2003 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 42: 493–504, 2004     

Prediction of solvent diffusion coefficient in amorphous polymers based on an equation‐of‐state

This study develops a modified free‐volume model to predict solvent diffusion coefficients in amorphous polymers by combining the Vrentas–Duda model with the Simha–Somcynsky (S‐S) equation‐of‐state (EOS), and all the original parameters can be used in the modified model. The free volume of the polymer is estimated from the S‐S EOS together with the Williams‐Landel‐Ferry fractional free volume, and the complex process of determining polymer free‐volume parameters in the Vrentas–Duda model and measuring polymer viscoelasticity can be avoided. Moreover, the modified model includes the influence of not only temperature but also pressure on solvent diffusivity. Three common polymers and four solvents are employed to demonstrate the predictions of the modified model. The calculation results are generally consistent with the experimental values. It is reasonable to expect that the modified free‐volume model will become a useful tool in polymer process development. © 2006 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 44: 1000–1009, 2006     

Transport of toluene/methyl ethyl ketone mixtures in polyisobutylene

In this study, the diffusion behavior of methyl ethyl ketone (MEK)/toluene mixtures, with MEK molar fractions (x₁) in the liquid source lower than 0.2, in polyisobutylene (PIB) was investigated with vapor‐sorption Fourier transform infrared attenuated total reflectance (FTIR‐ATR) spectroscopy. FTIR‐ATR Fickian diffusion models for both binary and ternary systems were used to determine diffusion coefficients. Obtained diffusion coefficients for MEK from the binary diffusion model did not agree with those determined previously for the diffusion of MEK/toluene mixtures, with x₁ > 0.2, in PIB. When the ternary diffusion model was used, the main‐term and cross‐term diffusion coefficients of MEK were comparable. © 2001 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 39: 908–914, 2001     

Self‐diffusion of small molecules into rubbery polymers: A lattice free‐volume theory

In the framework of the free‐volume (FV) theory, a new equation was derived for the evaluation of self‐diffusion coefficients of small molecules in polymers above the mixture glass transition temperature. The derivation of the equation turned out to be straightforward once the equivalence between the free volume and the unoccupied volume given by thermodynamic lattice theories is assumed. A parameter evaluation scheme is proposed, which is substantially simpler compared with the conventional Vrentas–Duda approach, even without losing generality. The key assumption is discussed, and its consistency is verified from a numerical viewpoint. A comparison with experimental solvent self‐diffusion coefficients for several solvent/polymer binary systems confirmed that the proposed theory presents good correlative ability over wide temperature and composition ranges. Moreover, the introduced thermodynamic foundation allows one to easily include the pressure effect too. In the frame of the proposed lattice free volume theory, the sizes of the polymer jumping units decrease with temperature and increase with pressure. Such behavior converges with theoretical expectations and opens the way for a predictive FV theory. © 2010 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 48: 529–540, 2010