Gas diffusion in partially crystalline polymers part I: Concentration dependence

In this work, a phenomenological model for the gas diffusion in partially crystalline polymers using differential effective medium theory is presented. By making an analogy with the power law known as Archie's law which relates the d.c. conductivity of a brine saturated porous rock to its porosity; we show that gas diffusion through semicrystalline polymers can be described in a similar way. It is assumed that the diffusion coefficient in the crystalline region is zero, while in the amorphous region it is given by a free volume model, and an effective diffusion coefficient D^eff, is obtained using the mentioned analogy. The variation of D^eff upon concentration is analyzed through its free volume dependence. The crystallinity dependence is considered through an average chain immobilization factor 〈β〉 which is explicitely derived. Finally, the results of this model are compared with experimental data given by Kreituss and Frisch, obtaining a good agreement. © 1996 John Wiley & Sons, Inc.     

An application of an homogenization method to a model of diffusion in glassy polymers

The sorption of gases in polymers below their glass-transition temperature T_g is known in many cases to be described by the “dual sorption” theory, according to which the gas is held in accordance with both the Langmuir and Henry's laws. Based on this theory, expressions for the “effective diffusion coefficient” in the glassy polymers have been obtained by investigators in the past, notably by Paul and Koros.² The present analysis regards the glassy polymers as inhomogeneous with regions on which the gas sorption follows the Langmuir law. Assuming that the linear dimensions of these regions, which are often referred to as “microvoids” (although they are not space filled by vacuum), are small compared to the macroscopic length of interest but large compared to the mean free path of the penetrant gas molecules, we derive a rigorous relation between the average flux and the concentration gradient in the polymer and show that this relation can be expressed in terms of an “effective diffusion coefficient” D_eff which depends on the details of the microstructure, i.e., the size, shape and spatial distribution of the “microvoids.” This expression for D_eff is shown to reduce to that of Paul and Koros² in two situations: (1) when the “voids” consist of slabs running parallel to the concentration gradient, and (2) when the “voids” are spherical and the temperature of the polymer is not too different from T_g. The results of the present study lead to an alternative procedure for interpreting the experimental data on sorption and permeation which may have some advantages over the procedure currently employed. Finally, the analysis presented here is also applicable to polymers containing adsorptive fillers.     

The effect of plasticization on the transport of gases in and through glassy polymers

The effects of plasticization on the transport of gases and vapors in and through glassy polymers are examined from the viewpoint of the “dual-mode” sorption model with partial immobilization. The analysis assumes the existence of two penetrant populations with different mobilities in the Henry's law and Langmuir domains of the glassy polymers. These mobilities are characterized by their mutual diffusion coefficients D_D and D_H. The plasticization of the polymer by penetrant gases is reflected in the concentration dependence of D_D and D_H. Expressions for the effective (apparent) diffusion and permeability coefficients are derived assuming that D_D and D_H are exponential functions of the penetrant concentration in the polymers. The results of this study are compared with a similar analysis which assumed the existence of a single mobile penetrant population. The present analysis provides information on the effects of plasticization on the penetrant transport in the Henry's law and Langmuir domains separately. The effects of antiplasticization or clustering of penetrant molecules on the effective diffusion and permeability coefficients are also examined.     

Water sorption by the calcium chloride/silica gel composite: The accelerating effect of the salt solution present in the pores

The kinetics of isothermal water sorption by the CaCl₂/silica gel composite initiated by a small stepwise pressure rise over the sample has been investigated at a constant underlying plate temperature of 35°C. The initial portion of the kinetic curves is consistent with Fick’s diffusion model: the amount of sorbed water increases in proportion to the square root of the sorption time. This makes it possible to determine the effective diffusivity of water (D _eff). At small amounts of sorbed water (w < 0.19 g/g), D _eff changes slightly. The diffusivity of water in the composite pores (D) calculated for the same conditions is close to the Knudsen diffusivity of water vapor in mesopores. The D _eff value grows with an increasing water content of the composite; that is, sorbed water accelerates water transport in the pores. This is likely due to the appearance of an extra diffusion channel, namely, diffusion through the aqueous solution of the salt, whose formation begins on the silica gel surface at w > 0.1 g/g. The contribution from this channel increases markedly when the amount of adsorbed water is above 0.25 g/g. This can be explained by the formation of the “connected” phase of the solution in the pores.     

The transient response of electrodes coated with membrane-type polymer films under conditions of diffusion and migration of the redox ions

Membrane transport processes involving diffusion and migration of the electroactive species A and product B are analysed. The transport equations (Nernst-Planck equation) are solved by digital simulation techniques with boundary conditions appropriate for the system electrode/membrane-type polymer coating. A large potential step leads to a zero surface concentration of A at time t = 0. Time-dependent concentration gradients of A and B, and field strengths are derived. The current transients are shown to follow Cottrell's law; however, the observed “effective” diffusion coefficients, D_eff, differ from D_A.     

Effects of carbon filler on sorption and diffusion of gases through rubbery materials

Effects of carbon filler on the sorption and diffusion of carbon dioxide in natural rubber and in styrene-butadiene rubber have been studied. Sorption isotherms conform to Henry's law in unfilled rubber and to Langmuir's law in carbon black. The isotherms in filled rubber exhibit a combination of the two sorption modes. The Henry's law solubility parameter k_D increases with carbon filler content; the Langmuir saturation constant C′_A initially is constant with filler level, but then decreases abruptly when carbon particles begin to aggregate. The diffusion coefficient decreases with increasing filler content, presumably owing to geometric effects and to polymer chain immobilization in the interfacial regions.     

Effect of microheterogeneities on the diffusion of gases through glassy polymers

The diffusion of gases through glassy polymers is studied and the effective diffusion coefficient D^eff is represented as the result of the superposition of two fundamental mechanisms, namely slipping and hopping. D^eff is calculated by a two-point correlation method. Comparisons are made with experimental data of Meares for diffusion coefficients of Kr, O₂, He, and A in poly(vinyl acetate) in the glassy state. Good fits are obtained and yield significant parameters.     

On the diffusion of gases in partially crystalline polymers

The diffusion of gases through partially crystalline polymers is studied. The effective diffusion coefficient D^eff is obtained as the result of the averaged superposition of two fundamental mechanisms, namely, diffusion through the crystallites is considered to be zero, and diffusion through the rubbery fraction of the polymer obeys a Fujita-like free-volume theory. The predicted D^eff is compared with experimental data of Kreituss and Frisch. The behavior of the diffusion coefficient in terms of concentration and crystalline fraction is satisfactorily explained through the model.     

Influence of the structure and morphology of ultrathin poly(3-hydroxybutyrate) fibers on the diffusion kinetics and transport of drugs

Ultrathin fibers of a biodegradable polymer poly(3-hydroxybutyrate) with an encapsulated drug (dipyridamole, 0–5% of poly(3-hydroxybutyrate) mass) are obtained by electrospinning. Introduction of the drug substantially affects the geometric shape and crystallinity of individual filaments as well as the total porosity of the fibrillar film on their basis. As follows from the SEM data, in the absence of the drug or at its low concentration (<3%), poly(3-hydroxybutyrate) fibers appear as ellipse-like fragments alternating with cylindrical ones. At a higher content of the drug (3–5%), the abnormal ellipse-like structures are practically absent and the fiber acquires the cylindrical shape. A set of morphological and crystallinity characteristics of the fibers determines the absorption of water and the rate of the diffusion transport of the drug as well as the corresponding profiles of its controlled release. A simplified model of drug desorption from the fibrillar film is advanced which considers two sequential stages of the process: (i) diffusion of the drug in the polymer fiber with coefficient D _f ~ 10^–12 cm²/s and dimeter φ_f ~ 2–4 μm and (ii) transport of the drug in the interfibrillar porous space filled by solvent with diffusion coefficient D _w = 5.5 × 10^–6 cm²/s. Using the characteristics of porosity, crystallinity, and geometry of the fibers and diffusion effective coefficients D _eff calculated from the profile of drug release, it is shown that the limiting stage of the transport of the drug is its diffusion in the volume of the cylindrical fiber. The model makes it possible to turn from the experimental values of D _eff to partial diffusion coefficients D _f and to calculate the kinetic profile of drug release with allowance made for the above-listed factors.     

Kinetics of Antibiotic Sorption by Monocarboxyl Cellulose

Kinetics of the sorption of lincomycin and gentamicin from aqueous solutions of their salts and bases, as well as a cephalosporin group antibiotic cephalexin having the zwitterionic nature, by monocarboxyl cellulose is studied. It is disclosed that the sorption of the studied antibiotics is characterized by a combined diffusion type of kinetics. Effective diffusion coefficients (D _eff) are determined, and it is shown that they correspond to the times (t _1/2) of half-equilibrium establishment and increase with passing from lincomycin to gentamicin and, further, to cephalexin. The D _eff and t _1/2 values are found to depend on solution pH and the degree of cellulose sorbent swelling.     

Scattering properties of a single semiflexible polyelectrolyte

The structure factor of a single semiflexible polyelectrolyte has been calculated as a function of chain length, intrinsic backbone stiffness, and salt concentration. Because of the insignificant coupling of the intrinsic stiffness and electrostatic persistence length, we carry out our calculations in the flexible limit. Within the variational scheme adopted here, we obtain fractal dimensions consistent with our earlier calculations of the configurational properties. As the chain length is increased, the electrostatic interaction is progressively screened, leading to the crossover regions. In the first crossover, the effective fractal dimension, D_eff, is as low as 1, and in the second crossover D_eff is 5/3, although the radius of gyration exponent is 2/5. © 2001 John Wiley & Sons, Inc. J Polym Sci Part B: Polym Phys 39: 2644–2652, 2001     

Sorption and diffusion of gases in a polyimide

Sorption and diffusion of gases (CO₂, N₂, and He) in a polyimide (PI2080) film were measured by using an apparatus which gives the sorption rate curves from the initial state to the equilibrium state. Nonlinear isotherms observed for CO₂ sorption were interpreted successfully in terms of the dual-mode model for sorption in glassy polymers. Linear isotherms observed for N₂ and He seemed to obey Henry's law. Two diffusion coefficients (D_I and D_E) were obtained using the short-time method and the long-time method for a Fickian diffusion model, together with the equilibrium solubility (C_e) from each experiment. The initial sorption rate curves agreed with the calculated curves using D_I, however near sorption equilibrium the curves are in accord with the calculated curves using D_E. These observations suggest that some relaxation process is superimposed on the diffusion process. The non-Fickian transport data were correlated successfully with a model that combines time-dependent diffusion and the Fickian model.     

A Film-Pore-Surface Diffusion Model for the Adsorption of Acid Dyes on Activated Carbon

The sorption of acid dyes from aqueous effluents onto activated carbon has been studied. The effects of initial dye concentration and activated carbon mass on the rate of Acid Blue 80, Acid Red 114 and Acid Yellow 117 removal have been investigated. A three-resistance mass transport model based on film, pore and surface diffusion control has been applied to model the concentration decay curves. The model incorporates an effective diffusion coefficient D _eff, which is dependant on the equilibrium solid phase concentration or fractional surface coverage. The results of the film-pore-surface diffusion model are compared with the data obtained from the basic film-pore diffusion model. It has been found that the film-pore-surface diffusion model provides a major improvement over the data correlated by the film-pore diffusion model. Also, the relationship between surface diffusion and fractional surface coverage has been investigated for the adsorption of acid dyes on activated carbon.     

A Film-Pore-Surface Diffusion Model for the Adsorption of Acid Dyes on Activated Carbon

The sorption of acid dyes from aqueous effluents onto activated carbon has been studied. The effects of initial dye concentration and activated carbon mass on the rate of Acid Blue 80, Acid Red 114 and Acid Yellow 117 removal have been investigated. A three-resistance mass transport model based on film, pore and surface diffusion control has been applied to model the concentration decay curves. The model incorporates an effective diffusion coefficient D _eff, which is dependant on the equilibrium solid phase concentration or fractional surface coverage. The results of the film-pore-surface diffusion model are compared with the data obtained from the basic film-pore diffusion model. It has been found that the film-pore-surface diffusion model provides a major improvement over the data correlated by the film-pore diffusion model. Also, the relationship between surface diffusion and fractional surface coverage has been investigated for the adsorption of acid dyes on activated carbon.     

Cationic grafting of olefins from PVC: The effect of reaction conditions

The cationic grafting of isobutylene, styrene, α-methylstyrene, and β-pinene from a poly(vinyl chloride) (PVC) backbone was investigated. Grafting-from was induced by Et₂AlCl in 1,2-dichloroethane and methylene dichloride solutions from 20 to −70 °C. The effects of temperature and proton trap [2,6-di-tert-butylpyridine (DtBP)] on grafting-from efficiency (G_eff), extent of grafting, branch length (molecular weight), and number of branches per PVC molecule were determined. Reducing the temperature invariably increased the G_eff and the molecular weight of polyisobutylene, polystyrene, poly(α-methylstyrene), and poly(β-pinene) branches attached to PVC. The magnitude of the effects was different with the various olefins and depended on the reaction conditions. The effect of DtBP was examined in the 5 × 10⁻⁴–4 × 10⁻³ mol/L range. By increasing the DtBP concentration the G_eff increased; however, the number-average molecular weight of the grafted branches decreased. The lengths of the grafted branches can be controlled, and G_eff's close to 100% were obtained. The fact that the proton trap reduced the molecular weights of grafted branches suggests that besides proton scavenging, DtBP may also abstract protons from the growing carbenium ion site. © 2001 John Wiley & Sons, Inc. J Polym Sci A: Polym Chem 39: 1675–1680, 2001     

A study of solvent diffusion in thin polyimide films using laser interferometry

Using laser interferometry, we have determined in situ the thickness increase with time of thin supported polyimide films (4–8 μm in initial thickness) immersed in n-methyl-2-pyrrolidone (NMP) as a function of NMP temperature (22–120°C). Similar experiments were also performed in dimethyl sulfoxide (DMSO) at 22°C with polyimide films of 4.1 μm in thickness. For NMP, the equilibrium fractional thickness increase (about 20%) is independent of initial polyimide thickness and temperature. The time scale for reaching equilibrium sharply decreases with temperature from 2–3 days at 22°C to 30–60 min at 120°C. Compared with NMP, the rate of DMSO sorption is considerably faster, reaching equilibrium swelling of about 28% in about 5 h at 22°C. To describe the transport process, we applied a phenomenological model proposed by Astarita and Sarti¹ but reformulated in polymer fixed frame to enable straightforward comparison with the thickness data. Our analysis indicated that the transport of NMP is best described as anomalous, that is, intermediate between diffusion controlled and case II transport. The effective diffusion coefficient D_eff and the front velocity U₀ at 22°C were found to be 3–6 × 10^?12 cm²/s and 8 × 10^?9 cm/s, respectively. Our front velocity is in good accord with the value of 6 × 10^?9 cm/s obtained for a similar polyimide based on gravimetric measurements.² Both D_eff and U₀ show an activation energy of ～56 kJ/mol. For DMSO, however, the transport is clearly case II. The front velocity at 22°C was found to be about 6 × 10^?8 cm/s, which is about four times that obtained by Rutherford back-scattering spectrometry.³     

Sorption and transport of carbon dioxide in a polyimide from 3,3′4,4′-biphenyltetracarboxylic dianhydride and 4,4′-diaminodiphenyl sulfone

CO₂ sorption and transport were investigated for the polyimide prepared from 3,3′,4,4′-biphenyltetracarboxylic dianhydride (BPDA) and 4,4′-diaminodiphenyl sulfone (DDS). The morphology of films did not change on annealing above the glass transition temperature and remained amorphous unlike the polyimide prepared from BPDA and 4,4′-oxydianilline (ODA). This seems to be due to the strong hindrance to rotation of the sulfonyl linkage. Sorption and transport data were analyzed according to the dual-mode model. Solubility, diffusion, and permeability coefficients at 20 atm and 80°C for BPDA-DDS polyimide were substantially equal between as-cast and annealed films and were 1.7, 2.2, and 3.7 times greater, respectively, than for the as-cast films of the BPDA-ODA polyimide. The higher solubility was due to larger values of the Henry's law solubility constant k_D, Langmuir capacity constant C, and the Langmuir affinity constant b. The sorption and transport properties were compared with those for amorphous glassy aromatic polymers including other polyimides. The relation of k, C, b, and the diffusion coefficients in the Henry's law population and the Langmuir population (D_D and D_H) with other properties of the polymers were discussed. Values D_D and D_H for BPDA-DDS polyimide were much larger than expected from the estimated free-volume fraction.     

A Local D4h Symmetric Dysprosium(III) Single-Molecule Magnet with an Energy Barrier Exceeding 2000 K**

Three six-coordinate Dy^III single-molecule magnets (SMMs) [Dy(O^tBu)₂(L)₄]⁺ with local D_4h symmetry are obtained by optimizing the equatorial ligands. One of the compounds with L=4-phenylpyridine shows an energy barrier (U_eff) of 2075(11) K, which is the third largest U_eff, and the first U_eff>2000 K for SMMs with axial-type symmetry so far. Ab initio analysis indicates that the exceptional uniaxial magnetic anisotropy is deeply related to the axially compressed octahedral geometry. This work provides a new insight into the local D_4h symmetry for high-performance SMMs.     

A Branched Pore Model Analysis for the Adsorption of Acid Dyes on Activated Carbon

A new branched-pore adsorption model has been developed using an external mass transfer coefficient, K _f, an effective diffusivity, D _eff, a lumped micropore diffusion rate parameter, K _b, and the fraction of macropores, f, to describe sorption kinetic data from initial adsorbent-adsorbate contact to the long-term adsorption phase. This model has been applied to an environmental pollution problem—the removal of two dyes, Acid Blue 80 (AB80) and Acid Red 114 (AR114), by sorption on activated carbon. A computer program has been used to generate theoretical concentration-time curves and the four mass transfer kinetic parameters adjusted so that the model achieves a close fit to the experimental data. The best fit values of the parameters have been determined for different initial dye concentrations and carbon masses. Since the model is specifically applicable to fixed constant values of these four parameters, a further and key application of this project is to see if single constant values of these parameters can be used to describe all the experimental concentration-time decay curves for one dye-carbon system.The error analysis and best fit approach to modeling the decay curves for both dye systems show that the correlation between experimental and theoretical data is good for the fixed values of the four fitted parameters. A significantly better fit of the model predictions is obtained when K _f, K _b and f are maintained constant but D _eff is varied. This indicates that the surface diffusivity may vary as a function of surface coverage.     

十二烷基硫酸钠水溶液的动态表面吸附性质

利用MPTC型气泡压力张仪研究了十二烷基硫酸钠(SDS)溶液在不同NaCl 浓度下的动态表面吸附性质, 分析了离子型表面活性剂在表面吸附层和胶束中形成双电层结构产生表面电荷对动态表面扩散过程和胶束性质的影响. 结果表明, SDS在表面吸附过程中, 表面电荷的存在会产生5.5 kJ·mol^-1的吸附势垒(E_a), 显著降低十二烷基硫酸根离子(DS^-)的有效扩散系数(D_eff). 十二烷基硫酸根离子的有效扩散系数与自扩散系数(D)的比值(D_eff/D)仅为0.013, 这表明SDS与非离子型表面活性剂不同, 在吸附初期为混合动力控制吸附机制. 加入NaCl可以降低吸附势垒. 当加入不小于80 mmol·L^-1 NaCl后, E_a小于0.3 kJ·mol^-1, D_eff/D在0.8-1.2之间, 表现出与非离子型表面活性剂相同的扩散控制吸附机制. 同时, 通过分析SDS胶束溶液的动态表面张力获得了表征胶束解体速度的常数(k₂). 发现随着NaCl 浓度的增大, k₂减小, 表明SDS胶束表面电荷的存在会增加十二烷基硫酸根离子间的排斥力, 促进胶束解体.