Evaluation of the free‐volume theory of diffusion

General characteristics of the free‐volume theory of diffusion are discussed, and a recent data‐theory comparison involving free‐volume theory is critically evaluated. © 2003 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 41: 501–507, 2003     

Prediction and estimation of solvent diffusivities in polyacrylate and polymethacrylates

The relationship between polymer side‐chain length and the hole free volume that is effective for solvent diffusion was investigated for polyacrylates and polymethacrylates on the basis of free‐volume theory. Measurements of a polymer's viscoelasticity and solvent diffusivity provided experimental evidence for polymer segment mobility, and the results indicated that hole free volume in a linear polymer increases with hydrocarbon side‐chain length. Because the molecular mechanisms of polymer viscoelasticity and diffusivity are identical, the free‐volume parameters obtained for polyacrylates and polymethacrylates by measuring the polymer viscoelastic‐temperature dependence can reliably be used in predicting the solvent diffusion coefficient. © 2003 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 41: 1393–1400, 2003     

Application of free‐volume theory to self diffusion of solvents in polymers below the glass transition temperature: A review

Theories based on free‐volume concepts have been developed to characterize the self and mutual‐diffusion coefficients of low molecular weight penetrants in rubbery and glassy polymer‐solvent systems. These theories are applicable over wide ranges of temperature and concentration. The capability of free‐volume theory to describe solvent diffusion in glassy polymers is reviewed in this article. Two alternative free‐volume based approaches used to evaluate solvent self‐diffusion coefficients in glassy polymer‐solvent systems are compared in terms of their differences and applicability. The models can correlate/predict temperature and concentration dependencies of the solvent diffusion coefficient. With the appropriate accompanying thermodynamic factors they can be used to model concentration profiles in mutual diffusion processes that are Fickian such as drying of coatings. The free‐volume methodology has been found to be consistent with molecular dynamics simulations. © 2011 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys, 2011     

Impact of average free‐volume element size on transport in stereoisomers of polynorbornene. I. Properties at 35 °C

This article demonstrates that transport of gases through glassy polymers is significantly influenced not only by the absolute amount but also by the distribution of free volume. Two stereoisomers of polynorbornene with nearly equivalent total free volume, but markedly different average free‐volume sizes, were evaluated. The free‐volume element size was probed with positron annihilation lifetime spectroscopy, wide‐angle X‐ray scattering, gas sorption, and molecular modeling. The permeation, sorption, and diffusion of light gases were measured in each stereoisomer at 35 °C. All analytical techniques indicated that one isomer (labeled as Architecture II) had a larger average free‐volume element size but fewer elements. This isomer also had a very slightly higher bulk density (1.000 vs 0.992 g/cm³ for the other stereoisomer). Architecture II also had gas sorption and diffusion coefficients that were two to three times those of the less dense counterpart. These differences have been attributed to differences in the free‐volume element size available within the polymer matrix. © 2003 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 41: 2185–2199, 2003     

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     

Solubility and diffusivity of methylmethacrylate and butylacrylate monomers in a MMA–BA copolymer

Mutual diffusion coefficients and sorption isotherms of methyl methacrylate (MMA) and butyl acrylate (BA) monomers in methyl methacrylate‐butyl acrylate copolymer (MMA‐BA) have been measured by gravimetric sorption. MMA is found to have higher solubility and diffusion rates in the copolymer than BA. Sorption data for MMA were interpreted using classical Flory‐Huggins thermodynamic theory with a constant interaction parameter (χ). A modified version of this theory has been applied to correlate the sorption data of BA, which exhibit a temperature and concentration‐dependent χ parameter. For MMA, the isotherm data reveal enhanced polymer‐solvent interactions with increasing temperature, while for BA the data indicate a drive toward phase separation with increasing temperature. Despite the difference in thermodynamic behavior, both monomers are found to exhibit Fickian diffusion and the diffusivity data are correlated reasonably well with the Vrentas‐Duda free volume theory. Some deviation between the free‐volume correlation and the experimental data is observed at the lowest temperature and BA concentration examined. © 2007 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 45: 1996–2006, 2007     

The size and number of local free volumes in polypropylenes: A positron lifetime and PVT study

The temperature dependence of the mean size of local free volumes in an amorphous atactic (aPP) and a semicrystalline syndiotactic polypropylene (sPP), and an amorphous ethylene‐propylene copolymer (E‐co‐P48) has been studied. Pressure‐volume‐temperature (PVT) experiments were performed for aPP, from which the hole fraction h of the Simha‐Somcynsky theory and the number density of holes were estimated. © 2003 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 41: 3089–3093, 2003     

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     

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     

Effect of blend composition on diffusivity and solubility of small molecules in polystyrene/poly(vinyl methyl ether) blends

The capillary column inverse gas chromatography technique was used to determine diffusivity and solubility data for several solvents in polymer blends composed of polystyrene and poly(vinyl methyl ether) (PVME). Diffusivity behaved as expected, increasing as the concentration of PVME increased in the blend. Knowing only the free‐volume parameters for the pure polymers, the free‐volume theory was successfully applied to predict the dependence of the diffusion coefficients on the blend composition. Transport in blends above the glass transition temperature is controlled by free volume, and the effect of concentration fluctuations is minimal at the temperatures studied. Experimental data show an increase in the partition coefficient of some solvents in the blends with respect to the pure polymers. © 2007 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 45: 2071–2082, 2007     

Intramolecular screening effects in nondilute polymer solutions: An equation‐of‐state approach

In a previous article, we presented a simple modification of the traditional Flory–Huggins theory that took intramolecular screening effects (or same chain contacts) into account. In this article, we present a natural extension of that work, in which free‐volume effects are also explained with an equation‐of‐state model. The predictions of the interaction parameter, χ, for several polymer–solvent systems are presented, over the entire concentration range, in θ solvents and good solvents. A geometric mean assumption is applied to the calculation of an exchange energy interaction term. The predictions of χ are successful to various degrees when internal pressures are used, whereas the use of solubility parameters in most cases produces fairly good agreement with experimental results. © 2003 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 41: 2911–2922, 2003     

Impact of average free‐volume element size on transport in stereoisomers of polynorbornene. II. Impact of temperature on solubility

This article evaluates the influence of temperature on the sorption of gases in two isomers of polynorbornene. The subject polymers were stereoisomers with nearly identical bulk density and total free volume. Because of differences in the mobility of the polymer backbone, the isomers packed differently resulting in differences in the average free‐volume element size within the matrix. The influence of these differences on free‐volume element size was characterized by the heat of sorption of gases in the matrix. The most pronounced differences were observed in the isosteric heats of sorption of condensable carbon dioxide and methane in the polymer isomers. This analysis suggests that the relative space available for sorption into free‐volume elements is higher in the methyl II isomer relative to methyl III. These conclusions support the physical characterizations reported in Part I of this series suggesting that the methyl II isomer has larger average free‐volume elements but fewer of them than the methyl III isomer. © 2003 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 41: 1939–1946, 2003     

Gas transport in vulcanized natural rubber‐cellulose. II. Composites

This work reports the transport of carbon dioxide, oxygen, and nitrogen in amorphous membranes of vulcanized natural rubber reinforced with regenerated cellulose. The values of the permeability coefficient of carbon dioxide, oxygen, and nitrogen in the composites with 25% of cellulose, measured at 25 °C and 15 cmHg of pressure, are roughly one‐third of those measured in the same conditions for these gases in natural rubber. The isotherms representing the variation of both the permeability and diffusion coefficients of the gases with pressure present a relatively sharp increase in the region of low pressures, attributed to changes in the free volume. The analysis of the permeability characteristics of the membranes in terms of the free‐volume theory suggests that gas transport is severely hindered in both the cellulose phase and the cellulose–rubber interphase of the composites. © 2000 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 38: 393–402, 2000     

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     

Contributions of side groups to the water diffusion in cured epoxy resins (2) study on physical aging

Novolac epoxy resins cured with novolac resin, novolac acetate resin, novolac butyrate resin, and novolac phenylacetate resin named as EP, EPA, EPB, and EPP, respectively, were prepared. Their physical aging behavior at a T_g‐30 °C (30 °C below glass‐transition temperature) was examined by positron annihilation lifetime spectroscopy and differential scanning calorimetry. The ortho‐positronium annihilation lifetime τ₃ variation extent of EP is less apparent than that of the other three esterified samples during physical aging. The time dependence of ops intensity I₃ agreed with the Kohlrausch‐Williams‐Watts (KWW) equation. The relaxation time (τ₀) and nonexponential parameter were calculated. The free volume and enthalpy relaxation rate characterized by the reciprocal of τ₀ and ?ΔH/?logt, respectively, exhibit the same order—EPP > EPB > EPA > EP. These results suggest that the extend and rate of relaxation are not only related to the frozen free volume produced by quenching but also significantly influenced by segmental mobility of the network that attributed to the side‐group flexibility and their interaction with networks. This work also supports the fact that side‐group flexibility and the free‐volume fraction and distribution act in concert to control the water‐diffusion behavior in epoxy networks. © 2003 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 41: 1135–1142, 2003     

The influence of the polymer chain stiffness on tracer diffusion in polymeric matrices

The diffusion of penetrants in polymers is of technological importance in many areas including chromatography and fuel cell membranes. In this work, the effect of chain conformations on tracer diffusion is studied using molecular simulations and a percolation theory. The polymeric matrix is composed of tangent hard sphere chains that are fixed in space; conformations are changed by tuning the stiffness of the chains. The tracer diffusion coefficient is relatively insensitive to the chain stiffness when polymer chains are frozen as in polymer glasses with the local chain dynamics switched off. An analysis of the matrix using percolation theory shows that the polymer volume fraction at the free volume percolation threshold is also relatively insensitive to the chain stiffness, consistent with the diffusion results. This is surprising because the site‐site intermolecular pair correlation functions in the matrix are quite sensitive to the chain stiffness. In contrast, the tracer diffusion coefficient in a melt of mobile chains decreases significantly as the chain stiffness is increased. We conclude that tracer diffusion is only weakly correlated with the chain conformations and local chain dynamics plays an important role. © 2011 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys, 2011     

Effect of orientation on the free volume and oxygen transport of a polypropylene copolymer

The effect of uniaxial orientation on the free‐volume and oxygen‐transport properties of a propylene copolymer with 4.5 wt % ethylene was examined. The free‐volume hole size and hole density were measured with positron annihilation lifetime spectroscopy. Subsequently, the free‐volume characteristics were correlated with the oxygen‐transport properties. Orientation had only a small effect on the total amount of free volume: a small increase in the hole density was offset by a small decrease in the hole size. As a result, the oxygen solubility and amorphous‐phase density were unchanged by orientation. However, a pronounced decrease in the oxygen diffusivity when the draw ratio exceeded 6 indicated a change in the dynamic free volume. This was attributed to an increasing number of taut tie chains, which retarded oxygen diffusion. The reduced amorphous chain mobility was also manifest in the increased glass‐transition temperature, decreased bulk thermal expansivity, and decreased expansivity of free‐volume holes. © 2005 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 43: 1230–1243, 2005     

Permeation measurements in ethylene‐1‐hexene,ethylene‐1‐octene,and ethylene‐1‐dodecene copolymers synthesized with metallocene catalysts

The permeation behavior of different ethylene‐1‐hexene, ethylene‐1‐octene, and ethylene‐1‐dodecene copolymers synthesized with metallocene catalysts has been analyzed. These copolymers cover a wide range of comonomer contents, so their crystallinities display rather considerable variations. The results for the permeability to oxygen of the different ethylene copolymers show that the main factor influencing the permeability is the noncrystalline fraction, although some influence of the kind of comonomer may also be present, which may be explained by the fact that when the alkyl branch of the α‐olefin is longer, there is an increase in the free volume in the amorphous and interfacial regions, causing slightly higher values of the permeability coefficient. From the results with different gases, it follows that, in general, an increase in the size of the penetrant (as expressed by its kinetic diameter or critical molar volume) leads to an increase in the solubility and a decrease in the diffusion coefficient. A wide range of permeability values is covered by these ethylene copolymers, depending basically on the crystallinity of the sample, but the permselectivity of CO₂ with respect to oxygen (and probably between other pairs of gases) does not differ very much among the different copolymers. © 2003 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 41: 2174–2184, 2003     

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     

Tensile creep of thermoplastics: Time‐strain superposition of non‐iso free‐volume data

The dimensional stability of thermoplastics is characterized by their tensile compliance D(t,σ,T) as a function of time t, stress σ, and temperature T. Creep retardation times are controlled by the free volume available for underlying molecular (segmental) motions. Tensile deformation of polymeric materials, whose Poisson ratio is smaller than 0.5, is accompanied by volume dilatation that can be identified with an increase in available free volume. Consequently, a steady increase in strain with time during tensile creep experiments accounts for shortening of the retardation times. The superposition of as‐received tensile compliance curves is difficult because any point of a curve requires a shift factor along the time axis that differs from those of other points. In this article, tensile creep at a constant stress and temperature is viewed as a non‐iso free‐volume process. A procedure is proposed to transform as‐received data to a pseudo‐iso free‐volume state that eliminates this deficiency and permits construction of a generalized compliance curve for the pseudo‐iso free‐volume state. This curve can be used for calculation of real‐time‐dependent compliance for any selected stress in the range of reversible deformations. As the superposed curve can be generated with several short‐term creep tests (e.g., 100 min) for a series of stresses, the proposed procedure saves experimental time. The effects of physical aging on tensile compliance (observed previously by other researchers) are interpreted in terms of the proposed approach in appendix A . © 2003 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 41: 736–748, 2003