Effects of type of adsorption isotherms on parallel diffusion of sulfonated dyes into porous cellulose membrane

Transport phenomenon of three sulfonated azo dyes, C.I. Acid Red 88, C.I. Direct Yellow 12, and C.I. Direct Blue 15 into water-swollen cellulose membranes has been analyzed on the basis of parallel transport theory by surface and pore diffusion. Langmuir equation was applied into the mass balance equation to estimate dye concentration in the pores. The results were compared with the results obtained by applying Freundlich equation in our previous papers. The surface diffusivity (D _s) and the pore diffusivity (D _p) for the parallel diffusion model obtained by applying Langmuir equation agreed with those obtained by applying Freudlich equation. The theoretical concentration profiles for parallel diffusion calculated usingD _s andD _p coincided accurately with the experimental data when we applied either Langmuir or Freundlich equations.     

One-dimensional models of polymer dynamics. I. Dashpot-spring models applied to a single rotor

The proposion of Clark and Zimm that a dashpot and a spring can be used in place of a set of rotational barriers in polymer dynamics is studied. The simplest possible case is examined here, that of a single rotor. Reasons for altering the Clark–Zimm diffusion equation are presented and an alternative diffusion equation is proposed. The results of both diffusion equations for the correlation function 〈exp[?iθ(0)] exp[iθ(t)]〉 (θ is the angular position of the rotor) are compared with the correlation function for a rotor in an n-fold cosine potential. Although the two diffusion equations differ, both agree well with the n-fold cosine model for barriers above a few k_BT. This agreement is obtained in the absence of adjustable parameters, and motivates the application of these two diffusion equations to polymeric systems.     

Diffusion of water in solutions of alkali metal bromides,tetraalkylammonium bromides and ammonium bromide

The diffusion coefficient of water D _w in aqueous solutions of the alkali metal bromides, tetraalkylammonium bromides (methyl, ethyl, n-propyl, and n-butyl) and ammonium bromide at 25°C is reported for concentrations up to 2 mol-dm^–3. In addition, values for D _w in 2 mol-dm^–3 solutions of CsBr, KBr, NaBr, LiBr, and fully deuterated methanol, acetonitrile, and acetone have been measured for temperatures in the range 5 to 50°C. The concentration dependence of the relative water diffusion coefficient D _w /D _o , where D _o is the self-diffusion coefficient of water, has been analyzed in terms of an equation analogous to the Jones-Dole equation for relative viscosity. The B-coefficient for diffusion is well correlated with the viscosity B-coefficient. For the structure-breaking electrolytes CsBr and KBr, D _w /D _o decreases rapidly with increasing temperature, whereas for the structure-makers NaBr and LiBr, the temperature dependence of D _w /D _o has the same sign but is much smaller in magnitude. For the nonelectrolyte solutions, the structure-making effect decreases with increasing temperature and the temperature coefficient of D _w /D _o is positive. It is apparent that, when diffusion of the solvent is being considered, the temperature must be taken into account in the classification of an electrolyte as a structure-breaker or structure-maker.     

Diffusion of protons in silica hydrogel

The effect of preparation pH of silica hydrogel on the effective diffusion coefficient of protons in silica hydrogel (D _e , m²/s), on surface area of silica gel (S, m²/s) and on particle size of silica gel (D _p , mm) was studied. Silica hydrosols were obtained by adding water glass to sulfuric acid. The effective diffusion coefficient of proton in silica hydrogel was determined by the method of diffusion from silica hydrogel plane sheet to a stirred solution of a limited volume. A numerical solution was obtained for the diffusion equation using the Regula Falsi method. Regression analyses of experimental data were conducted.Diffusion of protons in silica hydrogel is a complicated process due to a decelerating effect of the porous structure of silica hydrogel and to the accelerating effects of slow ions such as Na⁺ and surface diffusion. The effective diffusion coefficient increased with surface area of silica gel, indicating the diffusion of protons on the surface of the silica particles.     

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.     

Permeation,diffusion, and solution of cyclopropane in silicone rubber

Permeability, solubility, and diffusion coefficients have been determined for cyclopropane (c-C₃H₆) in silicone rubber at temperatures between ?8 and 70°C at relative pressures from 0.04 to 0.30. The permeability coefficients, , are of the order of 10^?6 cm³ (STP) · cm/(s · cm² · cmHg). increases slightly with increasing penetrant pressure and decreases with increasing temperature, the energy of activation for permeation being ?1.27 kcal/gmol at zero pressure. The solubility of cyclopropane in silicone rubber can be represented over the experimental concentration range by the Flory-Huggins equation. The solubility decreases with increasing temperature and the partial molar heat of solution is ?4.95 kcal/gmol. The solubility coefficient in the Henry's law limit, S(0), for cyclopropane and many other gases and vapors can be correlated with (T_c/T)², where T and T_c are the experimental and critical temperatures, respectively. The mutual diffusion coefficients, D, increase with increasing concentration and temperature, the energy of activation for diffusion being 3.68 kcal/gmol. The pressure dependence of &\[\bar P\] is described satisfactorily by a free-volume model proposed by Fujita and extended by Stern, Frisch, and coworkers. The permeability, diffusion, and solubility behavior of cyclopropane in silicone rubber is similar to that of propane (C₃H₈).     

Ionic contribution to the complex dielectric constant of a polymer under dc bias

A diffusion equation for the distribution of ionic impurities in a polymer under a dc field has been solved analytically for the conditions that neither injection nor absorption of impurity ions takes place at either electrode and that the ions do not interact. By using the solution of the diffusion equation, an ionic contribution to the complex dielectric constant is calculated. The most important results are as follows. (1) At high temperatures and low frequencies, the dielectric loss factor of the polymer decreases exponentially with the time elapsed after the application of the dc bias. (2) The dielectric constant is not greatly affected by the bias. (3) The ionic contribution to the loss factor in the absence of the dc bias is represented as ?_i^″ = 4πνDq²/ωkT, where ν is the density of mobile ions, D the diffusion coefficient, q the charge of an ion, ω the angular frequency, k the Boltzmann constant, and T the absolute temperature. (4) A method is presented for distinguishing between the apparent activation energy for diffusion and that for generation of mobile ions.     

Studies with model membranes. IX. Evaluation of thermodynamic parameters from the transition state theory of rate processes for electrolyte diffusion through silver chloride parchment-supported membranes

Diffusion rate of biologically important electrolytes through parchment-supported silver chloride membrane have been determined by the use of Nernst-Planck equation at various temperatures taking into account membrane resistance R_m, membrane potential E_m, etc. The diffusion rates were found to be primarily dependent upon the difference in the hydration energies of the counterion in the external solution. On the basis of Eisenman-Sherry theory the diffusion rate sequence of alkali metal cations point towards the weak field strengths of the fixed charge groups. The various thermodynamic parameters, namely, ΔH^?, ΔF^?, and ΔS^? were evaluated. The values of ΔS^? found to be negative, indicating that the diffusion is taking place with partial immobilization in the membrane phase. A formal relation between ΔH_hydration, ΔF_hydration, and ΔS_hydration of cations with the corresponding values of ΔH^?, ΔF^?, and ΔS^? for diffusion was also found to exist for the investigated system.     

A revised equation for estimating the vapour pressure of low-volatility substances from isothermal TG data

Previous investigators have used the Langmuir vaporisation relation to estimate the vapour pressures of low-volatility compounds from thermogravimetric data. However, this equation is strictly valid for evaporation into a vacuum only. For measurements conducted at finite pressures, molecular diffusion must be taken into account. A revised equation is proposed: dm _A/dt8M _A P _A D _A B/T. It is also shown that the proportionality between vapour pressure and vaporisation rate is very general. It arises from the assumptions of ideal gas behaviour, Raoult's law and a negligible concentration of the sample compound far from the sample surface. This revised version was published online in August 2006 with corrections to the Cover Date.     

Voltammetric Analysis of Iodide and Diffusion Coefficients Between 25 and 85°C

The oxidation wave of iodide in 0.075 mol-L^–1 H₂SO₄ was analyzed at 25, 40, 55, 70, and 85°C. The reversibility of the I₂/I^– system was checked using logarithmic transforms, half-wave potentials, and by studying I ^–1 = f(^–1/2). The limiting currents obtained enabled us to determine the diffusion coefficient of I^– using Newman's equation. These experimental results were compared with Nernst's limiting values. The Stokes–Einstein equation is not verified. Hydration numbers for I^– at different temperatures were established. An empirical equation is proposed to predict the evolution of diffusion coefficients in a sulfuric acid medium with temperature.     

The effect of pressure and temperature of self-diffusion coefficients in several concentrated deuterium oxide diamagnetic electrolyte solutions

The pressure dependences of the self-diffusion coefficients of deuterium oxide in 4.5m solutions of LiCl–D₂O and CsCl–D₂O (also 7m) and 3.06m CaCl₂–D₂O have been measured by the NMR spin-echo method at 30°C, 60°C, and 90°C. Shear viscosities and densities of these solutions have also been determined over the same range of experimental conditions. The experimental data show that the diffusion constantD decreases with the increasing structure-making ability of the electrolyte cation Ca⁺²>Li⁺. In contrast, the diffusion coefficient for D₂O in the 4.5 and 7m CsCl solutions is equal to that for pure D₂O at 30°C but lower at 60°C and 90°C. It has been found that the Stokes-Einstein equation relates well the diffusion coefficients to shear viscosity in these concentrated electrolyte solutions.     

Effect of solvent viscosity on the kinetics of reversible dimerization of phenoxy radicals

Flash photolysis technique has been used to obtain the rate and thermodynamic parameters of the reversible dimerization reactions of a range of ten phenoxy radicals (I–X) in a toluene–dibutylphthalate mixture (0.6 cP ≤η≤18.4 cP): The main reason for the difference in the k₁ values are the different steric hindrances in radicals. It has been found that the values of k₁ for 2,6-diphenyl-4-methoxy- (I), 2-phenyl-(III), and 2-methoxyphenoxy (IV) radicals are 3–5 times smaller than the respective diffusion constants calculated according to the Debye formula with regard to the spin-statistical factor: The resultant ΔH₁^≠values for these radicals in toluene and dibutylphthalate are close to the activation energies of the viscous flow of the solvents B. Linear relationships with a slope equal to unity are observed between log k₁ and log(T/η). The recombination of radicals I, III, and IV is limited by translational diffusion. The k₁ values for 2,6-diphenyl- (VII), 2,6-di-tert-butyl- (IX), and 2,6-di-tert-butyl-4-methylphenoxy (X) radicals are 10–60 times smaller than k_diff and Δ H^≠ B. In the case of radical X in toluene ΔH₁^≠ 0. The recombination of these three radicals includes an intermediate step of complex formation: For 4-phenyl- (II), 2,6- dimethoxy- (V), 2,4-diphenyl- (VI), and radicals VII, IX, and X the linear relationships between log k₁ and log (T/η) have a slope of from 0.5 ± 0.05 to 0.8 ± 0.05. The k₁^-1 versus η relationships for these radicals are not straight lines. The recombination of these six radicals is limited by translational and rotational diffusion. With the aid of theoretical models, the k₁ versus η relationships have been used to derive the steric factor f in radical recombination and the angle θ between the axis and the solid angle generatrix. The solid angle defines the reaction spot on the radical-sphere surface. The recombination of the 2,6-diphenyl-4-diphenylmethylphenoxy radical (VIII) takes place in the region intermediate between the diffusion and the kinetic ones, and the relationship between log k₁ and log (T/η) for this radical has a plateau portion. The log k_-1 versus log (T/η) relationships have precisely the same form as the corresponding k₁ relationships, which is quite in line with the theory of diffusion-controlled reversible recombination reactions.     

New approach to the effects of pressure dependence on sedimentation velocity experiments

A theory is developed for sedimentation velocity experiments when the sedimentation coefficient s_p depends on pressure P as s_p/s_o = (1 + γP)^?1, where γ is a constant. In contrast to the more usually analyzed from s_p/s_o = 1 ? γP, this model does not lead, in extreme cases, to a negative sedimentation rate. A theory is presented for homogeneous macromolecules sedimenting with no diffusion. It leads to estimations of s_o and γ from a knowledge of the point of maximum concentration gradient as a function of time. Results of these calculations are compared with accurate numerical solutions of the Lamm equation with diffusion included.     

Mass transport in gas channels of electrochemical cells with a solid-oxide electrolyte: Allowing for the electrode reaction rates in numerical calculations

The concentration fields, Nusselt number distributions along the electrode, and ratios of concentrations of electrochemically active components at the entrance into and exit out of an electrochemical cell are determined by numerical integration of the convective-diffusion equation for a broad range of rate constants of electrode reactions, Peclet numbers, and geometrical parameters of channels. Three regions of parameter values are revealed. At low reaction rates (K _l < 0.5), mass transfer is completely determined by electrode kinetics; at high rates (K _l > 20), it is completely determined by the diffusion in the gas phase; and in the intermediate region of values of K _l it is necessary to allow for both the diffusion processes and the electrode reactions.Translated from Elektrokhimiya, Vol. 41, No. 3, 2005, pp. 317–324.Original Russian Text Copyright © 2005 by Ezin, Somov.     

Low-frequency dielectric behavior of poly(vinylidene fluoride)

An explicit mechanism is described for the anomalous increase in dielectric constant and dielectric loss at low frequencies and high temperatures for poly(vinylidene fluoride) containing ionic impurities. Relations are proposed for the ionic contributions, ε_i″ and ε_i″, to the dielectric constant and dielectric loss: where v₀ and D₀ are the concentration and the diffusion coefficient of the mobile ions at infinite temperature, q is the charge of an ion (in cgs electrostatic units), l is the distance between electrodes, k is the Boltzmann constant, T is the absolute temperature, E_d is the apparent activation energy for diffusion of the ions, and W is the dissociation energy of the ionic impurities. From the slopes of curves of log εT′ versus 1/T and log ε″T versus 1/T for poly(vinylidene fluoride), energies E_d = 34 kcal/mole and W = 342 kcal/mole were obtained.     

Effect of radical scavenger on initiation efficiency

In order to analyze the effect of the scavenger on initiator efficiency f or ratio of combination between two radicals (1 ? f), an equation was derived by solving the modified Smoluchowski equation under the boundary condition based on the law of conservation of mass. At time t approaching infinity in a stationary state, this equation becomes: f = {D + a(Dk[S])^1/2}/{ak₀ + D + a(Dk[S])^1/2}, where D is the sum of the diffusion constants of two radicals, a is the diameter of the radical particle, k₀ is a specific rate of recombination in the cage between two radicals, k is the rate constant of reaction between the radical and the scavenger as a pair, and [S] is the scavenger concentration. When ak₀ ? D and k[S] is small, this equation approximates the previously derived Wijsman's equation. An equation of the same type as Noyes' equation is also derived from this equation, when [S] = 0. By using the equations derived in this paper, the effect of α,α′-diphenyl-β-picrylhydrazyl on the ratio of recombination of the (CH₃)-(CN)C^* radical and the effect of monomer on the initiation efficiency are satisfactorily explained.     

Nonlinear viscoelastic diffusion in concentrated polystyrene/ethylbenzene solutions

Nonlinear gradient-driven diffusion was studied in concentrated polystyrene (PS)/ethylbenzene (EB) solutions using vapor sorptions with a finite driving force. The nonlinear sorptions were carried out on thin films (≅2.05, 3.50 μ thick) at conditions where non-Fickian, “viscoelastic” effects appear. These data were modeled with the nonlinear diffusion equation studied by Tang. Four dimensionless material parameters in the model were determined from a limited amount of linear-response, differential sorption data on PS/EB mixtures measured in the same range of experimental conditions as for the nonlinear sorptions. The nonlinear model successfully predicts the observed nonlinear response either above or below the glass transition (T_g). In order to simultaneously capture the nonlinear response both above and below T_g, the abrupt change in the concentration dependence of physical properties at T_g must be accounted for. © 1997 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 35: 2103–2119, 1997     

Statics and dynamics of single chains in polymer blends and copolymer melts — results beyond the random phase approximation

We present new results on the behaviour of single chains in homo-and copolymer blends which are beyond the rendom phase approximation (RPA). The radius of gyration of a chain in the blend was found to depend strongly on the interactions as well as the effective diffusion constant of a single chain in the blend or copolymer melt. Note that these results cannot be obtained in the original formulation of the RPA. They are based on the effective Hamiltonian of the chain in the medium. In the equation found for the self diffusion constant of a single chain, D_s, the second term is directly the value of the effective potential at zero wave vector. Thus the measurement of D_s would detect the screened potential at q = 0. Experimental verification of these results is in progress.     

Sorption characteristics of an economical sorbent material used for removal radioisotopes of cesium and europium

Application study for the evaluation of sorption characteristics of sawdust as an economical sorbent material used for decontamination of radioisotopes cesium and europium from aqueous solution has been carried out in the present work. In this respect, sawdust (untreated and treated by HNO₃) has been prepared from the commercial processing of wood for furniture production. Pore properties of the activated carbon such as BET surface area, pore volume, pore size distribution, and pore diameter were characterized by N₂ adsorption and DFT software. Radiotracer method onto sawdust from aqueous solutions was studied in a batch technique with respect to pH, contact time, temperature. The kinetics of adsorption of Eu³⁺ and Cs⁺ have been discussed using five kinetic models namely, pseudo-first-order model, pseudo-second-order model, Elovich equation, intraparticle diffusion model, and modified Freundlich equation that have been tested in order to analysis the experimental data. Kinetic parameters and correlation coefficients were determined. It was shown that the second-order kinetic equation could describe the sorption kinetics for two metal ions. The metal uptake process was found to be controlled by intraparticle diffusion. Thermodynamic parameters, such as ΔH, ΔG and ΔS, have been calculated by using the thermodynamic equilibrium coefficient obtained at different temperatures. The obtained results indicated that endothermic nature of sorption process for both ^152+154Eu and ¹³⁴Cs onto sawdust.