Analysis of transport behavior in polymer powders

Vapor sorption studies on powder samples of glassy polymers have provided data which supplement results obtained on conventional film specimens and aid in the elucidation of glassy-state transport mechanisms. For uniform spherical particles of sub-micron size, sorption kinetics at very low activities of organic vapors follow a simple Fickian diffusion model. The short diffusion path in such samples allows determination of the very low diffusivities characteristic of the glassy state in experiments of conveniently short duration. Deviations from the Fickian, uniform-sphere model are observed in several circumstances: Particle size non-uniformity retards the approach to diffusion equilibrium. Sorption data at substantial vapor activities show an apparently similar slow approach to equilibrium which can be related to the contribution of a relaxation-controlled mode of sorption. The effects of particle non-uniformity and of relaxation processes can be distinguished by appropriate experimental design, and models for both have been developed. Sorption rate data obtained under Fickian diffusion conditions can be used to characterize particle size distribution. Sorption kinetics on uniform-sphere powders, conversely, can be analyzed through a diffusion-plus-relaxation model to distinguish and quantify the roles of the two transport modes more clearly than is possible with polymer film specimens. Polymer powder vapor solubility isotherms show significant variations with sample history which can be interpreted in terms of free volume changes and glassy state relaxations. This discussion, based on a study of vapor sorption by poly(vinyl chloride) samples, indicates that powder sorption measurements are also likely to be of general value in the study of other glassy polymers.     

A detailed study of the viscoelastic nature of vapor sorption and transport in a cellulosic polymer. I. Origin and physical implications of deviations from Fickian sorption kinetics

Various aspects of the kinetics of sorption of acetone vapor by cellulose acetate films at 30°C have been studied in detail, the principal aim being to understand more thoroughly the physical nature and causes of non-Fickian behavior in this and other similar polymer-micromolecular penetrant systems. Particular attention was given to the changes in sorption (including absorption, desorption, and resorption) kinetics caused by (a) systematic variation of the vapor pressure of acetone in different ways and (b) changes in membrane thickness. It has been shown that both viscous volume swelling relaxation and longitudinal differential swelling stress effects must be invoked, in order to explain fully the observed behavior. Detailed analysis of two-stage sorption kinetics indicated (1) reasonable agreement between estimates of the diffusion coefficient reported by different authors, as long as a consistent analysis of the first stage is used, although the significance of the values given is open to some doubt, because the said first stage is found not to be free of non-Fickian features; and (2) reasonable conformity of the second stage to a first-order volume relaxation process (except a long times), with a relaxation frequency strongly dependent on the width of the concentration interval covered by the sorption experiment (and hence on the applied “osmotic stress”). The close similarity of second-stage sorption to nonlinear viscoelastic creep behavior, previously found in the cellulose-water system was confirmed and is taken further here, by demonstrating semiquantitative agreement between the corresponding “elastic swelling” and mechanical bulk moduli. ©1995 John Wiley & Sons, Inc.     

Analysis of the kinetics of vapor absorption/desorption in/from silicone rubber and cellulose acetate membranes in the presence of stagnant boundary layers

The application of an interferometric technique (optical thickness meter, OTM) to the measurement of vapor sorption kinetics in both rubbery and glassy polymers is presented. In this technique, the membrane is formed by casting on a suitable glass surface and interferometry is applied in situ. The use of a carrier gas loaded with penetrant vapor introduced a stagnant boundary layer (SBL) effect which had to be corrected, in order to determine true sorption kinetics. The said SBL effect was estimated, on the basis of existing theory for the silicone rubber–methylene chloride (SR/MC) system and found to be more pronounced in the case of desorption. Upon correction for this effect, Fickian sorption curves were obtained; which yielded nearly constant values of the diffusion coefficient, not materially different for absorption and desorption, in line with theoretical expectation.Cellulose acetate–methylene chloride (CA/MC) was then studied as an example of a glassy polymer–vapor system, where the SBL effect distorts the absorption kinetic curve in the same way as the non-Fickian mechanism of sorption inherent in this kind of polymer–penetrant system. Here, the vapor sorption data were corrected using the results obtained from the Fickian SR/MC system. The corrected results were checked by comparison with independent data reflecting the true kinetic behavior of CA/MC, obtained with a vacuum balance apparatus (VBA), which is free of SBL effects. It is shown that this novel method of applying the SBL correction was reasonably successful in favorable circumstances, while a criterion is provided to identify cases where reasonably reliable correction is not possible.     

Use of a magnetic suspension microbalance to measure organic vapor sorption for evaluating the impact of polymer converting process

A gravimetric analyzer consisting of a magnetic suspension balance was set up to study the sorption kinetics of benzaldehyde in various forms of polypropylene (PP), including resin, sheet and thermoformed container. Instrument noise was determined throughout the experiments over a maximum of 4000 min or until equilibrium was reached. The sorption of benzaldehyde by PP resin exhibited Fickian sorption kinetics at low vapor activities (0.1–0.5). However, at high vapor activities (0.7 and 0.9), a two-stage sorption involving Fickian diffusion during the initial sorption, followed by a protracted polymer relaxation, which enhanced the sorption capacity, was observed. The converting process of PP resin into sheets and thermoformed containers had a significant impact on the benzaldehyde sorption profile. Benzaldehyde exhibited a much higher solubility in extruded sheet and thermoformed container than in the resin.     

Experimental and theoretical investigation of anomalous sorption kinetics in the methanol vapor–poly(methyl methacrylate) system at 35 °C

Non‐Fickian sorption kinetics of methanol vapor in a poly(methyl methacrylate) film of 8 μm, at 35 °C, are presented. The behavior of the system was studied in series of interval absorption runs. The relevant diffusion and viscous relaxation processes were studied by kinetic analysis of the sorption kinetic curves, using the relaxation‐dependent solubility model. The sorption isotherm concaves upward at high activities, typical to Florry–Huggins behavior, while it exhibits a convex‐upward curvature at low methanol vapor activities, indicating sorption in the excess free volume of the polymer matrix. Thermodynamic diffusivity presents a complex functional dependence on the concentration, while relaxation rate is found to be a function of concentration as well as of concentration interval. Relaxation rate becomes increasingly concentration‐dependent as the effective glass transition of the system is approached. © 2006 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 44: 3173–3184, 2006     

Interval kinetic gravimetric sorption of acetone in random copolymers of poly(ethylene terephthalate) and poly(ethylene 2,6-naphthalate)

Interval sorption kinetics of acetone in solvent cast films of random poly(ethylene terephthalate)-co-(ethylene 2,6-naphthalate) (PET-co-PEN) are reported at 35°C and at acetone pressures ranging from 0 to 7.3 cm Hg. Polymer composition is varied systematically from 0% to 50% poly(ethylene 2,6-naphthalate). Equilibrium sorption is well described by the dual-mode sorption model. Interval sorption kinetics are described using a two-stage model that incorporates both Fickian diffusion and protracted polymer structural relaxation. The incorporation of low levels of PEN into PET significantly reduces the excess free volume associated with the glassy state and, for these interval acetone sorption experiments in ∼ 5 μm-thick films, decreases the fraction of acetone uptake controlled by penetrant-induced polymer structural relaxation. © 1999 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 37: 2973–2984, 1999     

Water sorption and transport in blends of polyethyloxazoline and polyethersulfone

Water sorption and transport properties for a series of homogeneous blends of hydrophobic polyethersulfone and hydrophilic polyethyloxazoline are reported. Only blends that remained homogeneous after exposure to liquid water were studied in detail. Equilibrium solubility of water in the blend films increases with increasing hydrophilic polymer content. For all materials, equilibrium sorption isotherms show dual-mode behavior at low water vapor activities and swelling behavior at high activities. The sorption/desorption kinetics for PES are generally Fickian, but two-stage behavior is evident in blends containing 10 and 20% polyethyloxazoline. Diffusion coefficients decrease with increasing polyethyloxazoline content, owing to a decrease in the fractional free volume. For all materials, the diffusion coefficient shows a positive dependence on water vapor activity or concentration due to plasticization of the material by high levels of sorbed water, but it becomes a greater function of activity as the composition of hydrophilic polymer in the blend is increased. Since the decrease in the diffusion coefficient is greater than the increase in the solubility coefficient, the permeability coefficient decreases with increasing hydrophilic polymer content. © 1997 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 35 : 993–1007, 1997     

Incremental vapor sorption in a phase-segregated polyurethane elastomer

Incremental vapor sorption and desorption runs have been carried out with o-dichlorobenzene (ODCB) a strongly swelling solvent, in 2, 6, and 10 mil polyether polyurethane films. Two-stage sorption behavior occurs at intermediate and higher concentrations but is generally absent in the desorption runs. Analysis of the two-stage curves, using the Berens-Hopfenberg model of independent Fickian diffusion and first-order relaxation processes, leads to apparent diffusion coefficients which increase with thickness and show a pronounced maximum with concentration, whereas the relaxation rate constant decreases with concentration. Correction for the pressure drift during the runs, due to the low vapor pressure of ODCB, reduces the thickness dependence. The negative concentration dependence of the relaxation rate constant is related to the distribution of microdomain stabilities. Calculated values of the self-diffusion coefficient show that the maximum in the apparent diffusion constant with concentration can be accounted for largely, but not entirely, by the thermodynamic contributions. It is proposed that the additional factor is relaxation-controlled swelling which arises from the strong coupling between the matrix and hard-segment responses.     

Sorption of a finite amount of swelling solvent in a glassy polymer

We study the time history of a diffusing front when a polymer is exposed to a finite amount of penetrant which becomes used up. A class of polymers is considered for which slow molecular relaxation occurs only at or near the glass-gel interface with instantaneous relaxation both ahead of and behind the progressing front. We show that the position of the penetrant front versus time undergoes a long smooth transition from standard Fickian t^1/2 behavior to exponential time decay onto a final equilibrium position attained when all the penetrant is used up.     

The influence of transverse differential swelling stresses on the kinetics of sorption of penetrants by polymer membranes

The influence of transverse differential swelling stresses on the kinetics of sorption of a penetrant in a polymer membrane exhibiting linear viscoelasticity is described by a model developed from the much simpler one of Crank. Sorption and transverse swelling kinetic curves are computed numerically. The character of absorption and desorption curves is examined systematically mainly as a function of (i) the magnitude of the stresses set up and of the stress-dependence of the diffusion coefficient, (ii) the relative rates of stress relaxation and of diffusion, and (iii) the degree of plasticization or “softening” of the polymer by the penetrant. It is shown that important general features of experimental sorption kinetic curves can be reproduced satisfactorily under well defined conditions. Attention is also given to transverse swelling kinetic curves. Their correlation with the corresponding sorption curves is examined briefly but systematically and discussed with reference to experimental data.     

Modeling of penetrant diffusion in glassy polymers with an integral sorption deborah number

A mathematical model was developed to explain the anomalous penetrant diffusion behavior in glassy polymers. The model equations were derived by using the linear irreversible thermodynamics theory and the kinematic relations in continuum mechanics, showing the coupling between the polymer mechanical behavior and penetrant transport. The Maxwell model was used as the stress–strain constitutive equation, from which the polymer relaxation time was defined. An integral sorption Deborah number was proposed as the ratio of the characteristic relaxation time in the glassy region to the characteristic diffusion time in the swollen region. With this definition, an integral sorption process was characterized by a single Deborah number and the controlling mechanism was identified in terms of the value of the Deborah number. The model equations were two coupled nonlinear differential equations. A finite difference method was developed for solving the model equations. Numerical simulation of integral sorption of penetrants in glassy polymers was performed. The simulation results show that (1) the present model can predict Case II transport behavior as well as the transition from Case II to Fickian diffusion and (2) the integral sorption Deborah number is a major parameter affecting the transition. © 1993 John Wiley & Sons, Inc.     

Characterization of water vapor transport in glassy polyacrylonitrile by combined permeation and sorption techniques

Diffusion time lags, steady state permeabilities and sorption/desorption kinetics are reported for water vapor in biaxially oriented, solvent cast polyacrylonitrile (PAN) films. A wide range of vapor activities was studied at 15°C, 30°C, and 45°C. The transient and steady state permeation behavior at low and intermediate upstream vapor activities suggests that Fickian transport occurs under most of the conditions studied. Specifically, time lags predicted by Fick's law using the concentration-dependent diffusion coefficient derived from steady state permeation measurements agree reasonably well with experimentally measured values in most cases. p]Integral sorption/desorption kinetics at low and intermediate vapor activities also appear to be Fickian with a concentration-dependent diffusion coefficient. The form of the concentration dependency, evaluated from the “long time” solution of the diffusion equation for sorption experiments, is consistent with the form established for the diffusion coefficient from the steady-state permeation data. The diffusion coefficient exhibits a maximum near the concentration at which clustering is initiated. Presumably, the effective diffusion coefficient of water increases initially due to plasticizing or dual mode sorption effects associated with gap filling in the glassy matrix. As clustering becomes significant, the effective mobility of water is substantially reduced; therefore, the diffusion coefficient decreases at higher activities as clustering becomes the dominant mode of sorption. p]A tendency of the “early time” sorption/desorption kinetic data to exhibit concavity to the square root time axis at high activities suggests that time-dependent reductions in the diffusion coefficient may be occurring. Such reductions could be related to the kinetics of cluster formation at the higher vapor activities during sorption and to slow polymer consolidation during desorption. Any such non-Fickian effects, related to chain segment relaxations occurring over time scales similar to those of a diffusional jump, appear to be of importance only at short times. The short time nature of any such processes is suggested by the fact that diffusion coefficients evaluated from the “long time” solution to the diffusion equation for sorption are consistent with coefficients evaluated from steady state permeation data, in which case all time-dependent relaxation phenomena should be absent.     

NMR spectroscopy and free volume analysis of the effects of copolymer composition on the swelling kinetics and chain dynamics of highly crosslinked copolymers of acrylic acid with PEG‐containing multiacrylates

Novel ionizable polymer networks were prepared from oligo(ethylene glycol) (OEG) multiacrylates and acrylic acid (AA), employing bulk radical photopolymerization techniques. The properties of these materials exhibited a complex dependence on the network structure and composition. Penetrant sorption experiments demonstrated that the crosslinked structure of the copolymers depended very strongly on the AA content as well as the number of ethylene glycol groups. The impact of varying the AA content and the oligo(ethylene glycol) chain length on the polymer chain dynamics was examined using diffusion and ¹³C NMR relaxation studies. The penetrant uptake studies indicated a coupling of Fickian and relaxation‐driven contributions to the swelling behavior. The effect of increasing the AA content on the characteristic chain relaxation time was reversed as the oligo(ethylene glycol) chain length was varied, indicating that chain relaxation is controlled by structural considerations, for shorter oligo(ethylene glycol) chains, and by compositional considerations, for longer oligo(ethylene glycol) chains. Measured compositional effects on solid state ¹³C NMR relaxation times supported these conclusions. © 1999 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 37: 1953–1968, 1999     

Sorption and transport of linear alkane hydrocarbons in biaxially oriented polyethylene terephthalate

Equilibrium sorption and uptake kinetics of n‐butane and n‐pentane in uniform, biaxially oriented, semicrystalline polyethylene terephthalate films were examined at 35 °C and for pressures ranging from 0 to approximately 76 cmHg. Sorption isotherms were well described by the dual‐mode sorption model. Sorption kinetics were described either by Fickian diffusion or a two‐stage model incorporating Fickian diffusion at short times and protracted polymer structural relaxation at long times. Diffusion coefficients increased with increasing penetrant concentration. n‐Butane solubility was lower than that of n‐pentane, consistent with the more condensable nature of n‐pentane. However, n‐butane diffusion coefficients were higher than those of n‐pentane. Infinite‐dilution, estimated amorphous phase diffusion and solubility coefficients were well correlated with penetrant critical volume and critical temperature, respectively. © 2001 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 39: 1160–1172, 2001     

Paradoxes of thermodynamics of aqua-vapor-polymer interface equilibrium

Schroeder’s paradox is the different equilibrium degree of swelling for a hydrophilic polymer in saturated aqueous vapor and in liquid water; it is experimentally verified by the registration of increasing volume of spherical polymer samples upon the processes of vapor and liquid water sorption. Specimens of three main classes of hydrophilic cross-linked polymers were tested. These had gel, hypercrosslinked, and macroporous structures that differed by the scale of the Schroeder effect. A similar effect is typical also for hydrophobic polymers upon swelling in liquid organic solvents and their saturated vapor. The paradox is explained by the lower activity of a sorbate in a saturated vapor compared to its activity in a liquid phase. The ability of many samples of cross-linked polymers with different degrees of saturation with a sorbate to be at equilibrium with the saturated vapor of a sorbate is explained by the differences in the inner structure of these samples, i.e., by differing in the swelling combinations and the intensity of the interchain interactions in a polymer network.     

Dilation kinetics of glassy,aromatic polyimides induced by carbon dioxide sorption

Over the past years, the equilibrium sorption of gases in polymers has been intensively studied. Mostly, glassy polymers were investigated because of their excellent selective mass transport properties. This work does not focus on the equilibrium sorption but on the kinetics to reach the equilibrium. We developed a new experimental method measuring the sorption-induced dilation kinetics of a polymer film. Carbon dioxide and glassy, aromatic polyimides were chosen as model systems. Low-pressure experiments demonstrate that the measured dilation kinetics represent the sorption kinetics. A significant delay between the sorption and dilation kinetics is based on the fact that dilation kinetics occurs simultaneously with the concentration increase in the center of the polymer film. High-pressure experiments reveal significant differences in dilation kinetics compared to low-pressure experiments. Generally, three regimes can be distinguished in the dilation kinetics: a first, fast volume increase followed by two much slower regimes of volume increase. The magnitude of fast and slow dilation kinetics strongly depends on the swelling history of the polymer sample. The results of the experiments are analyzed in the light of a model relating the fast dilation kinetics to a reversible “Fickian” dilation and the slower dilation kinetics to an irreversible, relaxational dilation. © 1995 John Wiley & Sons, Inc.     

Anomalous transport of penetrants in glassy polymers

Solutions are presented for the Fickian and non-Fickian equations describing the case of penetrant transport in a glassy polymer. Due to associated macromolecular relaxation, a sharp penetrant front is observed which separates the glassy core from the rubbery (gel-like) layer at the surface. Concentration profiles are compared and general comments about Fickian versus non-Fickian transport in polymers are made.     

Influence of the organic compounds addition in the polymer free volume,gas sorption and diffusion

Three low molecular compounds were added to a glassy polymer, poly(amino-ether) resin, in order to change its free volume and to study the subsequent effect in transport properties. Free volume characterization was carried out using the positron annihilation lifetime spectroscopy technique. Two electrobalances were used to obtain carbon dioxide sorption kinetic. Diffusion coefficient variation is in good agreement with the additive antiplasticizer character, i.e. the reduction of the free volume fraction. However, solubility is affected by other variables too, as the balance of the interactions between penetrant, additive and polymer as well as the effect of coming closer the sorption temperature to the glass transition temperature in the sorption mode.     

Optical vs. direct sorption and swelling measurements for the study of stiff-chain polymer-penetrant interactions

This work involves interferometric ‘optical thickness’ and refractive index measurements performed in an optical thickness meter (OTM), on supported cellulose acetate (CA) films equilibrated with various activities of methylene chloride (MC) vapor. The relevant equilibrium sorption and volume swelling isotherms were determined by application of the Claussius-Mossotti equation on the assumption that these films swell unidimensionally along the thickness direction, and were compared with corresponding direct equilibrium sorption (weight gain), elongation and thickness dilation measurements on similar free films performed in a vacuum sorption/swelling apparatus (VSA) and complemented with refractive index data. Combined elongation and thickness dilation data from the VSA showed that free glass-cast CA films exhibit pronounced swelling anisotropy. The said anisotropy, although it cannot be completely eliminated, by conditioning at high degrees of swelling, does not appear to affect the extent of volume swelling significantly, thus permitting quantitative comparison of sorption and swelling isotherms determined by the VSA and the OTM. Such comparison showed satisfactory agreement between these two sets of results up to an MC uptake of ca. 0.4 gMC/cm³ of dry CA corresponding to a degree of swelling of ca. 0.2. Increasing discrepancies are observed at higher MC concentrations, which are attributable to breakdown of the assumption used that the supported films swell unidimensionally along the thickness direction. The present CA-MC volume swelling data exhibit the negative deviation from volume additivity on mixing typical glassy polymers.     

Sharp fronts due to diffusion and stress at the glass transition in polymers

We derive and analyze a model for sharp fronts in glassy polymers. We take the major effect of a diffusing penetrant on the polymer entanglement network to be the inducement of a differential viscoelastic stress. This couples diffusive and mechanical processes through a viscoelastic response where the strain depends upon the amount of penetrant present. Analytically, the major effect is to produce explicit delay terms, via a relaxation parameter, to account for the fundamental difference between a polymer in its rubbery state and the polymer in its glassy state, namely the finite relaxation time in the glassy state owing to slow response to changing conditions. We produce concentration profiles in good agreement with observations on sharp front formation. In addition the model can account for the phenomenon of sorption overshoot.