Statistical mechanical model for diffusion of simple penetrants in polymers. II. Applications—nonvinyl polymers

The theory developed in Part I of this series is applied to a number of nonvinyl “smooth” chained homopolymers. The agreement between predicted and observed activation energies of diffusion for simple penetrants is generally good, particularly for polyethylene. Discrepancies observed for the smallest penetrants, He and H₂, in some polymers may be rationalized in terms of atomic scale irregularities on the polymer chain surface. It is shown that in favorable cases the theory may permit diffusion data to be used as an additional check on the accuracy of conformational energy maps for polymers.     

Statistical mechanical model for diffusion of simple penetrants in polymers. I. Theory

Following Di Benedetto it is proposed that noncrystalline polymer regions possess an approximate semicrystalline order with chain bundles that are locally parallel along distances of several nanometers. Packing with on-average four nearest neighbors is assumed. A spherical molecule may move through such a substrate in two distinct ways: (a) along the axis of a “tube” formed by locally parallel chains or (b) perpendicular to this axis by two polymer chains separating sufficiently to permit passage of the molecule. The first process is relatively fast, generally requires little activation energy, and determines the effective jump length in diffusion. The second is responsible for the activation energy of diffusion, which is taken as the minimum energy necessary to produce a symmetrical chain separation which allows transfer of a molecule. This is calculated as a function of the penetrant diameter d and parameters Γ and β which characterize the interchain cohesion and chain stiffness, respectively. Γ is estimated from the polymer density and cohesive energy density by suitably linearizing a relation given by Di Benedetto for the potential between two polymer chains approximated as infinite strings of Lennard-Jones force centers. β is shown to be approximately obtainable from the polymer chain backbone geometry and bond rotation potentials. An expression for the diffusion coefficient D is developed which contains only one disposable parameter, the effective jump length.     

Statistical mechanical model of diffusion of complex penetrants in polymers. II. Applications

The theory of Part I is applied to the diffusion of several aromatic diffusants in two “smooth chained” polymers: poly(ethylene terephthalate) (PET) and natural rubber. Modifications of the theory necessary to accommodate vinyl polymers are discussed and applied to benzene in PMA and PEA. In all cases the theory agrees satisfactorily with the experimental D(0,T) and D(c,T)/D(0,T) data, and the values of the disposable r_g and Δ parameters are of the expected order. The limiting Arrhenius behavior of benzene in natural rubber appears to be correctly predicted. The cell model is definitely more appropriate than the free volume model for the calculation of enfolding chain effects in highly crystalline PET. For the three amorphous polymers the two models give comparable results, the cell model being somewhat superior for natural rubber and PMA.     

Statistical mechanical model of diffusion of complex penetrants in polymers. I. Theory

A previously developed model of simple penetrant diffusion is extended to encompass complex penetrants of idealized molecular shape, characterized by dimensions of length, width, and thickness. Expressions are obtained for D(0,T), the diffusion coefficient at zero penetrant concentration (c), and the fractional increase in D(0,T) as a function of c and temperature (T). The model predicts that D(0,T) will exhibit Arrhenius behavior at temperatures well above T_g and gives the limiting activation energy as a function of penetrant thickness and the polymer energy/distance constants used previously. For T_g < T ? T_g + 150 K the model requires two new disposable parameters, in addition to the jump-length parameter of the simple penetrant theory. These parameters, however, have precise physical meanings (all are lengths) and together with the penetrant dimensions and polymer constants determine the absolute magnitude of the diffusion coefficient as well as its relative dependence on c and T. For T ? T_g + 40 the relative concentration dependence may be calculated in terms of the penetrant dimensions and polymer constants only.     

Mass and momentum transfer in polymers. III. Effects of liquid penetrants on tensile stress relaxation of amorphous polymers

Sorption, diffusion, swelling, and tensile stress relaxation measurements were made at room temperature (23°C) for the systems poly(n-butyl methacrylate) (PBMA) with liquid methanol and ethanol, and poly(methyl acrylate) (PMA) with liquid water. Stress relaxation curves for the fully swollen polymers could be superimposed approximately with those for the dry polymers by appropriate shifting along the long axes. For PMA–water the measured curve for stress relaxation with concurrent sorption could be predicted accurately by using a moving boundary theory with data measurements of stress relaxation of the unswollen and swollen polymer combined with sorption data. The modified moving boundary theory is generalized to include the effects of dimension changes through swelling and the larger effects of plasticization associated with sorption of liquids. This improved theory accurately predicts measured curves of stress relaxation with concurrent sorption for the PBMA–alcohol systems from individual stress relaxation, sorption, diffusion and swelling data. The general approach should be applicable to other amorphous polymer–liquid swelling agent systems. The anisotropic nature of swelling of polymer films and its effect on calculated diffusion coefficients are discussed briefly.     

Statistical mechanical estimation of the entropy of fusion of linear vinyl polymers

The entropy of fusion of linear vinyl polymers [polypropylene, polystyrene, and poly(vinyl chloride)] having asymmetric carbon atoms was calculated on the basis of the rotational isomeric state model. The results are found to be in good agreement with the experimental entropies of fusion per bond for these polymers.     

Short branching in poly(vinyl alcohol). III. Some properties of model polymers of short-branched poly(vinyl alcohol)

Melting point, the iodine color reaction, and foam fractionation were studied on model poly(vinyl alcohol) (PVA) having short branches of one or two monomer units in length. An increase in the amount of short branching units caused a marked decrease in color intensity of the PVA–iodine reaction and in the melting point. These tendencies were more remarkable when the short branching was two monomer units in length than when it was one monomer unit. It was also found that foam fractionation of an aqueous PVA solution produced PVA fractions with different degree of short branching, the degree increasing with increase in the fraction number. The color intensity of the PVA–iodine reaction has been confirmed to decrease with increase in the fraction number, but this result cannot be explained solely in terms of the short branching. It is concluded that the phenomenon of foam fractionation of PVA and the iodine color reaction of the fraction appear to be governed by many factors such as molecular weight, stereoregularity, and short branching.     

A mathematical model for stress-driven diffusion in polymers

A model for case II diffusion into polymers is presented. The addition of stress terms to the Fickian flux is used to produce the characteristics progressive front. The stress in turn obeys a concentration-dependent evolution equation. The model equations are analyzed in the limit of small diffusivity for the problem of penetration into a semiinfinite medium. Provided that the coefficient functions obey two monotonicity conditions, the solvent concentration profile is shown to have a steep front that progresses into the medium. The formulas governing the progression of the front are developed. After the front decays away, the long time behavior of the solution is shown to be a similarity solution as in Fickian diffusion. Two techniques for approximating the solvent concentration and the front position are presented. The first approximation method is a series expansion; formulas are given for the initial speed and deceleration of the front. The second approximation method uses a portion of the long time similarity solution to represent the short time solution behind the front.     

Polymer fracture—A simple model for chain scission

A simple model for calculating the fracture process for a single extended-chain molecule such as polyethylene is considered. The model consists of a chain of N coupled Morse oscillators. There exists a critical overall extension ΔL_c below which the fracture is energetically unfavorable but above which fracture is favored both energetically and kinetically. This elongation ΔL_c scales as N^1/2. For the critically stretched chain, the activation energy for rupture increases with N. Long chains must be stretched beyond this critical value to fail within experimentally meaningful times. Chains of all lengths subjected to the same force will fail with the same activation energy, provided this force is large enough to stretch each chain to ΔL > ΔL_c. Observed activation energies are less than 1/3D_e, where D_e is the bond energy.     

Segmental mobility model for diffusion of gases in polymers

Diffusion of small molecules in polymers is described quantitatively in terms of segmental mobility processes. The diffusion coefficient depends on a diffusive jump length, which is characteristic of the polymer, and a jump frequency, which is equated to the segmental mobility rate. The presence of a particular solute increases mobility of the surrounding polymer segments by a predictable amount, which is related to the partial molar volume of the solute. The theory is fit to experimental diffusion data, and partial molar volumes are calculated from the fitting parameters. Good agreement with experimental partial molar volumes is obtained.     

Gas sorption and diffusion in hydrogen-bonded polymers. I. Vinyl alcohol/vinyl butyral copolymers

A series of vinyl alcohol/vinyl butyral copolymers was examined to assess the effect of internal hydrogen bonding on gas sorption and diffusion. Sorption and permeation measurements for carbon dioxide and methane were performed on four vinyl alcohol/vinyl butyral copolymers. Upon comparing the various data, it was found that hydrogen-bonded copolymers exhibit a much wider variation in diffusion coefficient than non-hydrogen-bonded copolymers. The fractional free volumes of the studied copolymers were considerably lower than expected based on values of the diffusion coefficient. This may be due to the fact that predicted occupied volumes are too large and that the effect of internal hydrogen bonding is not accounted for properly. Using a relationship between infrared spectral shift and hydrogen-bond length, fractional free volumes in the hydrogen bonded copolymers were correlated with the interatomic spacing associated with the hydrogen bond. This implies that the average length of a hydrogen bond can be used as a measure of chain packing in hydrogen-bonded polymers. © 1993 John Wiley & Sons, Inc.     

Multifractal model of gas diffusion in polymers

A probabilistic (multifractal) model of gas diffusion in polymer membranes is proposed. This model has a simple and clear physical meaning and provides fairly accurate predictions of the diffusivities of various gases. For the latter purpose, a minimum set of structural characteristics is required.     

Statistical mechanical theory for steady-state systems. III. Heat flow in a Lennard-Jones fluid

A statistical mechanical theory for heat flow is developed based upon the second entropy for dynamical transitions between energy moment macrostates. The thermal conductivity, as obtained from a Green-Kubo integral of a time correlation function, is derived as an approximation from these more fundamental theories, and its short-time dependence is explored. A new expression for the thermal conductivity is derived and shown to converge to its asymptotic value faster than the traditional Green-Kubo expression. An ansatz for the steady-state probability distribution for heat flow down an imposed thermal gradient is tested with simulations of a Lennard-Jones fluid. It is found to be accurate in the high-density regime at not too short times, but not more generally. The probability distribution is implemented in Monte Carlo simulations, and a method for extracting the thermal conductivity is given.     

Morphology and mechanical properties of crystalline polymers. III. Deformation of spherulites in polyethylene

The behavior of individual spherulites on uniform deformation of bulk polyethylene is studied with a model consisting of a spherical spherulite and a homogeneous matrix having the properties of the polymer. On the basis of this model, the nonaffine or inhomogeneous deformation in the spherulite is found to be primarily due to the curvilinearly anisotropic property or, more specifically, the spherically isotropic property of the spherulite. In the case of uniaxial stretching, mechanical interactions between spherulites are also to some extent responsible for the microinhomogeneous deformation. Details of the predicted deformation mode are compared with experimental observations reported in the literature and reasonable agreement is attained between theory and experiment for lightly drawn samples.     

Statistical mechanical treatment of protein conformation. III. Prediction of protein conformation based on a three-state model.

The method proposed for the evaluation of statistical weights in paper I, and the three-state model [alpha-helical (alpha), extended (epsilon), and other (c) states] formulated in paper II, have been used to develop a procedure to predict the backbone conformations of proteins, based on the concept of the predominant role played by shortrange interactions in determining protein conformation. Conformational probability profiles, in which the probabilities of formation of three consecutive alpha-helical conformations (triad) and of four consecutive extended conformations (tetrad) have been defined relative to their average values over the whole molecule, are calculated for 19 proteins, of which 16 had been used in paper I to evaluate the set of statistical weights of the 20 naturally occurring amino acids. By comparing these conformational probability profiles to experimental x-ray observations, the following results have been obtained: 80% of the alpha-helical regions and 72% of the extended conformational regions have been predicted correctly for the 19 proteins. The percentage of residues predicted correctly is in the range of 53 to 90% for the alpha-helical conformation and in the range of 63 to 88% for the extended conformation for the 19 proteins in the two-state models [alpha-helical (alpha) and other (c) states, and extended (epsilon) and other (c) states]. In the three-state model, the percentage of residues predicted correctly is in the range of 47% to 77 for 19 proteins. These results suggest that the assumption of the dominance of short-range interactions, on which the predictive scheme is based, is a reasonable one. The present predictive method is compared with that of other authors.     

The alpha–beta transformation in polypivalolactone. II. A simple mechanical model

A simple mechanical model based on the equilibrium rate theory of Burte and Halsey is presented in order to explain the stress–strain curve of polypivalolactone (PPL) fibers. A semiquantitative application of this model indicates that the PPL alpha–beta transformation implies the cooperative deformation of at least thrity chain units and that the free energy difference between the alpha and beta states is of the order of 2 kcal/mole of monomer.     

Pyrolysis of polyacrylonitrile and related polymers—VIII. Copolymers of acrylonitrile with vinyl acetate,vinyl formate,acrolein and methyl vinyl ketone

Vinyl acetate exerts some blocking effect on the nitrile group oligomerization which dominates the initial pyrolytic processes in polyacrylonitrile, but the other comonommes studied allow more extensive nitrile reaction and do not effectively block the reaction. Decomposition of the vinyl esters, taking place concurrently with the nitrile reaction, tends to obscure any interaction with the nitrile reaction. Acrolein and methyl vinyl ketone do participate in the cyclization reactions and the former exhibits a small accelerating influence. The bulkiness of the comonomer appears to be an important factor in determining the thermogravimetric behaviour of the polymers. Thus vinyl acetate and methyl vinyl ketone copolymers are more liable to chain scission and fragmentation during the initial stages of pyrolysis than the vinyl formate and acrolein copolymers. These different behaviours are discussed in terms of the effect of the stereochemistry of the comonomer unit on the chemistry of the interaction between the comonomer structure and the propagating nitrile reaction.