Transport through a slab membrane governed by a concentration-dependent diffusion coefficient: Part IV. D(C)/D0=1+(αC)−β(αC)

Using the specific functional form D(C)/D₀=1+(αC)−β(αC)² an investigation has been made of (isothermal) transport through a slab membrane under ‘simple’ boundary conditions and governed by a diffusion coefficient, D(C), which, with increasing concentration, at first increases, passes through a maximum value and finally decreases. The flux, integral diffusion coefficient and concentration profile characteristic of steady-state permeation have been evaluated; special attention has been paid to the positions of such profiles in relation to the corresponding linear distribution associated with a constant diffusion coefficient.The corresponding transient-state transport has been studied within a framework of the time-lag ‘early-time’ and ‘ ’ procedures. Expressions for the ‘adsorption’ and ‘desorption’ time-lags are given. The concentration-dependence of these time-lags, of the (four) integral diffusion coefficients derived from them and of the arithmetic-mean time-lag ratios have been considered in some detail. The ‘early-time’ and ‘ ’ finite-difference procedures have likewise been employed to derive four further integral diffusion coefficients, so enabling a comparison to be made of the nine integral coefficients pertaining to established experimental techniques.Particular interest attaches to the situation for which n≡β(αC₀)=1 (where C₀ is the ingoing or upstream concentration of diffusant) resulting in D(C₀) being symmetrical about C₀/2. Some consideration has been given, in general, to features of transient-state transport when governed by a symmetrical D(C).     

Transport properties of annealed,drawn low-density polyethylene (LDPE)

The specific concentration c_a of methylene chloride, the zero-concentration diffusion coefficient D₀, and the concentration coefficient γ_D of the diffusivity in drawn and annealed LDPE were measured. The influence of the drawing rate, of annealing with the ends of the sample free and fixed and the effects of time of standing at room temperature after annealing were investigated. The observed transport properties are in good agreement with the microfibrillar model of fibrous structure, its relaxation during annealing, and the slow crystallization of relaxed tie-molecules upon standing at room temperature.     

Investigation on extremely dilute solution of PEO by PFG-NMR

The technique of pulsed-field gradient nuclear magnetic resonance (PFG-NMR) was applied to study the solution properties of a series of low molecular weight poly(ethylene oxide). The self-diffusion coefficients of solutions from semi-concentrated to extremely diluted were measured, leading to a critical concentration. When the concentration of solution is higher than the value of critical concentration, the diffusion coefficient of the solute decreases as the concentration increases and remains the same when the concentration is lower than it. This critical concentration agrees well with the definition of dynamic contact concentration (C _s) and confirms indirectly the Flory's scaling law between the molecular weight and D ₀. In addition, the influences of molecular weight and terminal groups on C _s were discussed. All the diffusion coefficients determined at extremely dilute condition were equivalent to the diffusion coefficients at infinite concentration (D ₀), from which the polymer coil size was estimated.     

Voltammetry at microcylinder electrodes: Part III. Chronopotentiometry

Expressions for chronopotentiometry at stationary microcylinder electrodes are presented. The transition time, τ, is expressed as a function of a single parameter λ ( = nFc^*_RD / ia, where a is the radius of the electrode, D is the diffusion coefficient, i is the current density, n is the number of transferred electrons, F is the Faraday constant and c^*_R is the bulk concentration. When the values of λ are small, the transition-time constant, i√τ / c^*_R, depends linearly on λ, with the intercept predicted from the Sand equation. Conversely, when values of λ increase, the transition time also increases exponentially. Therefore transition necessarily occurs no matter how small a current flows. An approximate equation for the transition time is presented, from which one can evaluate the diffusion coefficient. Equations for the potential-time curve and the quarter-wave potential are also obtained. The equations were tested experimentally using carbon fiber electrodes (a = 4.1 μ m) and platinum wire electrodes (a = 10-100μ m). The transition times obtained experimentally were in good agreement with those predicted theoretically for various values of the applied current, for several different radii of the electrodes.     

Diffusion in polyacrylate aqueous systems at 25°C. Role of reference frame choice in interpreting diffusion coefficient data

Diffusion coefficients have been measured for the binary systems sodium polyacrylate-water and polyacrylic acid-water at 25°C as a function of concentration. Diffusion coefficients have been also measured for the ternary system sodium chloride-sodium polyacrylate-water at constant NaCl concentration and varying polyacrylate concentration. The experimental results have been compared with some limit expressions, available in literature, for the four D _ik diffusion coefficients of systems containing two electrolytes with a common ion. The ternary system shows strong interaction between flows: as the polyelectrolyte concentration, C₂, approaches zero, the main diffusion coefficient D₂₂ and the cross coefficient D₂₁ approach zero, while the cross coefficient D₁₂ reach quite high values. The water motion during the diffusion process is also discussed.     

Lipophilic counterion effect on aggregation and adsorption behavior of quaternary ammonium surfactants

The obvious different aggregation and adsorption behavior of six newly quaternary ammonium surfactants with different lipophilic counterions has been discoverd by measurements of equilibrium and dynamic surface tension, fluorescence and conductivity. Interestingly, the critical micelle concentration (CMC) and its surface tension γ_CMC decrease with the increasing counterion chain length. However, three methods have confirmed that an exception of CMC increases slightly from C₁₆NC₁ to C₁₆NC₂. According to experimental results, a balanced mechanism between hydrophobicity and electrostatic of counterion is proposed. Besides, the dynamic surface tension results show the diffusion coefficient increases with the increasing counterion length both at a short time (D_t?→?0) and long time (D_t?→?∞), which indicates a faster adsorption process. Unlike the inorganic counterion, the diffusion coefficient decreases with the increase of hydrophobic chain length. This is attributed to the strong electrostatic interactions between counterions and cationic headgroups.     

Transient permeation of organic vapors through elastomeric membranes

The permeation of benzene and acetone vapors through sulfur-cured natural rubber was studied by the time-lag method. The experimental results were analyzed by a method suggested by Meares. The zero concentration diffusion coefficient D₀ was obtained by the early-time method. The Frisch time-lag equation was utilized to estimate both the solubility coefficient s and the additional parameter b required to define the concentration dependence of the diffusion coefficient: D(c) = D₀ exp {bc}. This form of concentration dependence was manifested by the corresponding permeability coefficient values. At low entering penetrant pressure, where the transport coefficients are constant, indirect evidence was obtained that D₀ is the mechanistically correct diffusion coefficient. The solubility coefficient values calculated for benzene vapor in natural rubber are in reasonable agreement with published equilibrium sorption data for a similar rubber compound. At higher entering penetrant pressures, average diffusion coefficients obtained at steady state tended to be larger than the corresponding average diffusion coefficients derived from the time lags. This same effect has been detected by other experimental approaches. Permeation experiments designed for this rapid method of analysis appear capable of yielding information consistent with that obtained by more time-consuming traditional methods.     

Diffusion of additives and plasticizers in poly(vinyl chloride)—II: The self-diffusion of the stabilizers dibutyltin dilaurate and dibutyltin bismonobutylmaleate in plasticized poly(vinyl chloride)

The self-diffusion coefficient, D, of dibutyltin dilaurate and dibutyltin bismonobutylmaleate have been obtained at 35, 45 and 55° in samples of poly(vinyl chloride) plasticized with 34, 60 and 100 phr of di(2-ethylhexyl)phthalate. D at 2 phr of the laurate is 3–5 times larger than for the smaller maleate molecule. In all cases, D increases with increasing plasticizer concentration, an effect interpreted in terms of the free volume theory of diffusion. D for the laurate increases by a factor of about 2.7 when the laurate diffusant concentration is increased from 0 to 4 phr. The activation energies for diffusion, E_D, lie between 50 and 90 kJ mol^?1. They increase with increasing plasticizer concentration but become constant at higher plasticizer concentrations (60–100 phr). It is impossible to correlate all the known data on diffusion in plasticized PVC with an equation of the form log D₀ = C₁ + C₂ E_D/RT     

The effect of the electrolyte concentration in the solution on the voltammetric response of insertion electrodes

A solid-state redox reaction involving an insertion of ions is analyzed with respect to the influence of the concentration of inserting ions in the solution phase. The voltammetric response is independent of the mass transfer in the solution provided that z = (D _ss/D _aq)^1/2 ρ/[C⁺]* is smaller than 0.1 (D _ss: diffusion coefficient of the cation C⁺ in the crystal; D _aq: diffusion coefficient of the cation C⁺ in the solution; ρ: density of the solid compound; [C⁺]*: concentration of cations in the bulk of the solution). In real cases this condition will be satisfied at solution concentrations above 1 mol/l. Received: 15 December 1997 / Accepted: 5 March 1998     

Role of diffusion in gel permeation chromatography

A theory is developed that describes the diffusion of solute into the gel particles during a gel permeation chromatographic experiment. The particles are treated as homogeneous spheres of radius a, into which diffusion takes place with diffusion coefficient D_s. The concentration in the mobile phase at any level at any time is supposed to be uniform throughout the cross-section of the column. It is shown that in the usual columns the effect of diffusion in the mobile phase is unimportant. A determinative quantity in the process is the parameter a²/D_st, where t is the time. For large values of a²/D_st an explicit expression for concentration versus time in the mobile phase at the end of the column is derived [eq. (26) and Fig. 1]. It shows a relatively long tail at large efflux volumes V, where the concentration varies at V^?3/2. For arbitrary values of a²/D_st the first three moments of the concentration versus time curve are calculated [eqs. (33)–(37)]. Pronounced skewness of the curve is found unless a²/D_st is small.     

Diffusion of Methanol in Polydimethylsiloxane

The permeation and sorption of methanol in polydimethylsiloxane at 10 and 30°C. has been measured and the results analyzed to determine the concentration dependence of the steady-state diffusion coefficient D which is found to decrease as the total concentration C is increased. An analysis of the isotherms indicates that clustering of the methanol occurs in the polymer, becoming more predominant as the concentration is increased. A polymerization model used to describe the shape of the D versus C curve for water in polydimethylsiloxane has been modified and applied with some success to describe the shape of the isotherm and the D versus C curve for methanol. The linearity of the permeation rate with relative pressure in this and a number of water—polymer systems is briefly commented on.     

Small-Molecule Diffusion in Semicrystalline Polymers as Revealed by Experimental and Simulation Studies

Summary: Diffusion of n-hexane in poly(ethylene-co-1-hexene)s with 15–75 wt.% crystallinity was studied by desorption experiments analyzing data using the Fickian equations with a concentration dependent diffusivity. The effect of the impenetrable crystalline phase on the penetrant diffusivity (D) is described by D = D_a/(τβ), where D_a is the diffusivity of the amorphous polymer, τ is the geometrical impedance factor and β is a factor describing the constraining effect of the crystals on the non-crystalline phase. For a polymer with 75 wt.% crystallinity, τβ varied markedly with penetrant concentration (v_1a) in the penetrable phase: 1000 (v_1a = 0) and 10 (v_1a = 0.15). This penetrant-uptake had no effect on the gross crystal morphology, i.e. β must be strongly dependent on v_1a. Samples saturated in n-hexane exhibited a penetrant-induced loosening of the interfacial structure, as revealed by an increase in crystal density that require an increased mobility in the interfacial component and by a decrease in the intensity of the asymmetric X-ray scattering associated with the interfacial component. The geometrical impedance factor has been modelled by mimicking spherulite growth and τ was obtained as the ratio of the diffusivities of the fully amorphous and semicrystalline systems. The maximum τ obtained from these simulations is ca. ten, which suggests that β in the systems with v_1a = 0.15 takes values close to unity. The simulations showed that the geometrical impedance factor is insensitive to the ratio of the crystal width and the crystal thickness. A free path length scaling parameter characteristic of the amorphous phase correlated with τ.     

NMR study of diffusion of butane in linear polyethylene

The diffusion coefficient of butane in linear polyethylene at room temperature as a function of the vapor pressure of butane was measured by the spin-echo method with a pulsed magnetic field gradient. For the Special morphology of randomly oriented stacks of parallel lamellas the detour factor is 1/3. As long as the blocking factor and migration through the lamellas can be neglected, the local diffusion coefficient D_a of the small molecules through the amorphous layers in the direction parallel to the lamellas is three times the apparent diffusion coefficient D derived from the decay of the amplitude of the spin echo under the assumption of an infinitely extended homogeneous medium. The diffusion coefficient and the spin–spin relaxation time both increase exponentially with increasing pressure, i.e., butane concentration in the polymer, while the spin-lattice relaxation time is pressure independent and seems to be determined by interaction with the amorphous polyethylene matrix.     

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.     

Diffusion of99TcO4 - in compacted bentonite: Effect of pH, concentration, density and contact time

In order to assess radionuclide diffusion and transport properties in compacted bentonite, the “in-diffusion” method based on bentonite filled capillaries is used. The effect of ⁹⁹TcO₄ ^- concentration and pH value of the solution, the contact time and the dry density of compacted bentonite on the apparent diffusion coefficient (D _a) and on the distribution coefficient (K _d) values obtained from the capillary test was studied. The D _a and K _d values decrease with increasing of the bulk dry density of compacted bentonite. Ion exclusion influences the diffusion of ⁹⁹TcO₄ ^- in the same substance. As compared to literature data, the K _d values obtained from capillary tests are in most cases lower than those from batch tests, the difference between the two K _d values is a strong function of dry density of the compacted bentonite. This revised version was published online in July 2006 with corrections to the Cover Date.     

Comparison of the Diffusion of Aqueous Glycine Hydrochloride and Aqueous Glycine

A Taylor dispersion tube has been used to measure mutual diffusion in aqueous solutions of glycine hydrochloride at 25°C and concentrations from 0.0005 to 0.5 M. Analysis of the dispersion profiles shows that the diffusion of glycine hydrochloride (GlyHCl) produces a subtantial additional flow of hydrochloric acid that is liberated by the dissociation: GlyH⁺ + Cl^- Gly + H⁺ + Cl^-. Diffusion in this system is, therefore, a ternary process described by the equations J ₁(GlyHCl) = – D ₁₁C ₁ – D ₁₂C ₂ and J ₂(HCl) = –D ₂₁C ₁ – D ₂₂C ₂ for the coupled fluxes of total glycine hydrochloride (1) and hydrochloric acid (2) components. The ratio D ₂₁/D ₁₁ of measured diffusion coefficients indicates that up to two moles of HCl are cotransported per mole of GlyHCl. Although protonated glycine diffuses with relatively mobile Cl^– counterions, the main diffusion coefficient of glycine hydrochloride, D ₁₁, is lower than or nearly identical to the diffusion coefficient of aqueous glycine. A model for the diffusion of protonated solutes is developed to interpret this result and the large coupled flows of HCl. Diffusion coefficients are also reported for the aqueous hydrochlorides of 3- and 4-aminobenzoic acids.     

Effect of the solvent upon the diffusion coefficient of polydisperse polystyrene and styrene-acrylonitrile copolymer in dilute solution

A laser homodyne spectrometer was used to obtain translational diffusion coefficients for dilute polystyrene and styrene-acrylonitrile copolymer solutions at room temperature. Data were obtained in the concentration range from 0.01 to 2.0 g polymer per 100 cm³ solution for polystyrene in benzene and in decalin; and for copolymer in dimethyl formamide, in methyl ethyl ketone, and in benzene. The samples were polydisperse polystyrenes of weight average molecular weights between 80,000 and 350,000 and polydisperse copolymers of weight average molecular weights between 200,000 and 800,000. The SAN copolymers were random copolymer samples containing 24% by weight acrylonitrile. For each of the systems investigated the concentration dependence of the diffusion coefficient was linear over the concentration range studied, and was expressed as D(c) = D₀(1+k_Dc). Values of D₀ could be explained with a modified Kirkwood-Riseman expression. Values of the parameter k_D obtained from the slopes could be interpreted using the two-parameter theory approach as suggested by Vrentas and Duda. The value of k_D is positive for high-molecular-weight polymers and negative for low-molecular-weight polymers. For a particular polymer, the molecular weight at which k_D changes sign is greater for poor solvents than for good solvents. Observed values of D₀ were 1 × 10^?7 to 7 × 10^?7 cm²/sec.     

Radiation-induced grafting of styrene vapor to polyethylene

The radiation-induced grafting of styrene vapor to low-density polyethylene film of 0.063 mm thickness was studied at 23°C at a dose rate of 1.98 × 10⁴ rad/hr. The concentration C of monomer in the film was measured as a function of pre-irradiation exposure time to monomer vapor. The concentration-dependent diffusion coefficient of styrene in polyethylene was calculated to be 4.9 × 10^?9 exp {2.0C/C₀} cm²/sec, where C₀ is the saturation concentration of styrene in the film, and a linear boundary diffusion coefficient for styrene vapor into polyethylene film was found to be 2.0 × 10^?7 cm/sec. The rate of grafting was determined as a function of the concentration of styrene absorbed in the film. The maximum graft yield was obtained with an initial styrene concentration in the film of 4 wt-%. Under conditions of low initial monomer concentration, the grafting rate increases with irradiation time. The results are compared with previously published data on grafting of polyethylene from methanol–styrene solutions. They are explained in terms of the viscosity of the amorphous region as a function of styrene content and the resistance to the diffusion of monomer at the film–vapor interface.     

Diffusion mechanisms in solid and molten polyethylene

The diffusion coefficient D and solubility coefficient k of small molecules [C₃H₆, C₄H₁₀, (CH₃)₄C] are determined at very low solute concentrations in annealed linear polyethylene over a wide range of temperature above and below the melting point T_m. For measurements above T_m the specimen was lightly crosslinked by irradiation from a ⁶⁰Co source. The diffusion data fit equations of the form D = D₀ exp {–ΔH_D/RT}. An abrupt change in ΔH_D occurs at T_m: representative values (for C₄H₁₀) are 4.53 and 14.9 kcal/mole above and below T_m. At T_m, D₀ also changes abruptly: representative values (for C₄H₁₀) are log D₀ = ?2.65 above T_m and log D₀ = +2.70 below T_m. The mechanism of diffusion therefore changes at the melting point. The melt exhibits typical liquidlike characteristics (negative values of activation entropy ΔS_D). The ratio ΔS_D/ΔH_D = 4β (β denoting the isobaric coefficient of volume expansion) holds below but not above T_m. Equations of the form k = k₀ exp {–ΔH_k/RT} fit the solubility data. The log k versus T^?1 plots above and below T_m are parallel but separated by a step at T_m. If crystallization followed by annealing is assumed to leave a weight fraction of polymer α_k (the amorphous fraction) in which the solute can absorb and if the specific solubility coefficient of the amorphous fraction is identical to that of the melt, then log α_k Equals the magnitude of the step at T_m. Values of α_k determined from the observed step are very close to values of amorphous fraction determined by measurement of density. The solubility experiments support the concept of polyethylene as a two-phase solid with the amorphous fraction of specific volume equal to the extrapholated specific volume of the melt. The passage of a solute molecule from one potential well to another, however, occurs by processes in the melt and the amorphous fraction which are entirely different.     

Oxygen diffusion in poly(dimethyl siloxane) using fluorescence quenching. I. Measurement technique and analysis

The transport properties of oxygen in poly(dimethyl siloxane) have been measured using the method of quenching of fluorescence. This paper discusses the uniqueness of this method and its use in measuring the diffusion coefficient of oxygen in unfilled PDMS. The results show (1) a large value for the diffusion coefficient of oxygen in pure PDMS at 25°C, D = 3.55 × 10^?5 cm²/s, (2) a low value of the acitivation energy, E_D = 4.77 kcal/mole, which was not temperature dependent in the ranges evaluated, and (3) a large value of the preexponential term, D₀ = 0.115 cm²/s. The diffusion coefficient was found to be independent of both the oxygen concentration and fluorophor concentration in the pressure and temperature ranges used in these experiments. The import of these experiments lies in their application to a unique biomedical oxygen sensor which is fast, sensitive, and does not consume oxygen.