Traveling wave solutions for case II diffusion in polymers

Case II diffusion of penetrant liquids in polymer films is characterized by constant-velocity propagation of a phase interface. We review the development of viscoelastic models describing case II diffusion and then present a phase plane analysis for traveling wave solutions. For simplified, piecewise-constant coefficient models we give closed-form analytic solutions showing the dependence on various physical parameters in both viscous and viscoelastic diffusive systems. We will also compare the results of our analysis with results from numerical simulations of more general models. © 1996 John Wiley & Sons, Inc.     

Effects of polymer molecular weight and temperature on case II transport

We investigated the effects of polymer molecular weight and temperature on Case II transport in the poly(methyl methacrylate)/methanol (PMMA/MeOH) system by a laser interferometric technique, using monodisperse polymer samples. Both the induction process and the steady-state front propagation were investigated. The data gave the volume fraction of MeOH in the swollen layer behind the moving front, ϕ, the steady state front speed, υ, and the characteristic induction time, t_ind. Values of ϕ separated into two groups, independent of molecular weight within each group. Significantly lower values of ϕ were found for polymers with molecular weight above the critical threshold for entanglement which can be explained by unrelaxed entanglements in the swollen layer. The Case II front velocity was independent of molecular weight for molecular weights at, or above, the critical weight for entanglement, suggesting that anelastic deformation processes other than simple viscous flow control the front propagation. Analysis of induction time data shows that the film surface properties differ from those of the bulk. © 1999 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 37: 3159–3171, 1999     

Numerical simulation of coupled‐stress case II diffusion in one dimension

This article is concerned with the robust and efficient numerical simulation of case II diffusion, which constitutes an important regime of solvent diffusion into glassy polymers. Even in the one‐dimensional case considered here, the numerical simulation of case II diffusion is made difficult by the extreme nonlinearities and coupling in the governing model equations due to a nonlinear flux law needed to produce sharp solvent fronts, a concentration‐dependent relaxation time of the polymer used to model the glass‐rubber transition, and coupling between the diffusion and deformation phenomena. Having an efficient and accurate solution to such equations is central to advancing a clear understanding of the meaning of such models. The difficulties due to coupling and nonlinearities are highlighted by the consideration of a specific, normalized, one‐dimensional case II diffusion model laid out in a general framework of balance laws. Issues such as the stiffness of the spatially discrete differential algebraic equations obtained from the finite element discretization of the governing equations and their bearing on the choice of time‐stepping schemes are discussed. The key requirements of numerical schemes, namely, robustness and efficiency, are addressed by the use of an implicit, adaptive, second‐order backward differentiation formula with error control for time discretization. Error control is used to maximize the step size to satisfy a target error and the radius of convergence requirements while nonlinear algebraic equations are solved at each time step. An example of an initial boundary value problem is solved numerically to show that the chosen model reproduces case II behavior and to validate that the stated objectives for the numerical simulation are met. Finally, the features and numerical implementation of this model are compared with those of a closely related case II diffusion model due to Wu and Peppas. © 2003 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 41: 2091–2108, 2003     

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.     

Non-fickian diffusion in thin polymer films

Diffusion of penetrants through polymers often does not follow the standard Fickian model. Such anomalous behavior can cause difficulty when designing polymer networks for specific uses. One type of non-Fickian behavior that results is so-called case II diffusion, where Fickian-like fronts initially move like √t with a transition to a non-Fickian concentration profile and front speed for moderate time. A mathematical model is presented that replicates this behavior in thin polymer films, and an analysis is performed that yields relevant dimensionless groups for study. An unusual result is derived: In certain parameter ranges, the concentration profile can change concavity, reflecting Fickian behavior for short times and non-Fickian behavior for moderate times. Asymptotic and numerical results are then obtained to characterize the dependence of such relevant quantities as failure time, front speed, and mass transport on these dimensionless groups. This information can aid in the design of effective polymer protectant films. © 1996 John Wiley & Sons, Inc.     

Stability of solutions to a reaction diffusion system based upon chemical reaction kinetics

In this paper, we deal with the stability problem to some mathematical models that describe chemical reaction kinetics. One is a set of ordinary differential equations induced by one reversible chemical reaction mechanism containing three chemical species. The other is a set of reaction diffusion equations based on the same chemical reaction. We show that all solutions of the model are asymptotically stable by applying the Liapunov method. We thus find that the concentration of each species has certain limits as time proceeds.      

Diffusion Kinetics at Liquid‐Glassy Polymer Interphases

Summary: We explored the diffusion mechanisms in a series of liquid/glassy polymer interphases. The diffusion experiments were performed in a unique way: the temperature range studied encompassed the glass transition temperature (T_g) of the glassy matrices. We observed that the diffusion behavior of the liquid polymer was remarkably continuous when passing through the matrix T_g, and that the diffusion modes at the liquid/glassy interphases were very similar to those observed in liquid/liquid polymer diffusion.

Diffusion profiles of liquid PS in glassy PPO obtained by confocal Raman spectroscopy. The sample was held at 160 °C for the times indicated in the plot.     

Acid Dissociation Behaviour and Complexation with Nickel(II), Copper(II) and Zinc(II) for Two Conformational Isomers of Dicyclohexylcyclam

Abstract

Acid dissociation constants for two conformational isomers of dicyclohexylcyclam, cis-anti-cis, (P) and cis-syn-cis, (N) have been determined at 25, 35 and 40°C, and thermodynamic data are estimated. It was found that (N) shows very different behaviour from (P). Stability constants of (P) and (N) toward Ni(II), Cu(II) and Zn(II) have been determined by pH-titration at 25°C by using a ligand exchange reaction. It is found that the (P) complex is more stable for Ni(II) and the (N) complex is more stable for Cu(II). Contributions of the cyclohexyl group to the macrocyclic effect (ME) have been also estimated by considering basicity corrections. It is found that substitution of the cyclohexyl group in cyclam increases ME only for the Ni(II) complex of (P).     

Copper(II) ion hydrolysis in aqueous solution

A copper(II) ion-selective-electrode potentiometric method was used to determine the first and second hydrolysis constants of Cu²⁺. Special techniques prevented copper(II) hydroxide precipitation, and copper(II) carbonate and cipper(II) organic complexation during the titration of the experimental solution over the pH range 6.8–8.4. The large change in the total copper concentration during the titration due to adsorption of copper onto the vessel walls was accounted for by measuring the total copper concentration at each pH by atomic absorption spectrophotometry. The two hydrolysis constants were determined at 25°C in 0.7 and 0.05m NaClO₄ media. The measured stability constants are independent of the copper concentration and yield similar zero ionic strength values. Also, the stepwise equilibrium constants decrease as the ligand number increases.     

Thermal, spectral and magnetic behaviour of 2,3,4-trimethoxybenzoates of Mn(II), Co(II), Ni(II) AND Cu(II)

Four new complexes of 2,3,4-trimethoxybenzoic acid anion with manganese(II), cobalt(II), nickel(II) and copper(II) cations were synthesized, analysed and characterized by standard chemical and physical methods. 2,3,4-Trimethoxybenzoates of Mn(II), Co(II), Ni(II) and Cu(II) are polycrystalline compounds with colours typical for M(II) ions. The carboxylate group in the anhydrous complexes of Mn(II), Co(II) and Ni(II) is monodentate and in that of Cu(II) monohydrate is bidentate bridging one. The anhydrous complexes of Mn(II), Co(II) and Ni(II) heated in air to 1273 K are stable up to 505–517 K. Next in the range of 505–1205 K they decompose to the following oxides: Mn₃O₄, CoO, NiO. The complex of Cu(II) is stable up to 390 K, and next in the range of 390–443 K it loses one molecule of water. The final product of its decomposition is CuO. The solubility in water at 293 K is of the order of 10^–3 mol dm^–3 for the Mn(II) complex and 10^–4 mol dm^–3 for Co(II), Ni(II) and Cu(II) complexes. The magnetic moment values of Mn²⁺, Co²⁺, Ni²⁺ and Cu²⁺ ions in 2,3,4-trimethoxybenzoates experimentally determined in the range of 77–300 K change from 5.64–6.57 μB (for Mn²⁺), 4.73–5.17 μB (for Co²⁺), 3.26–3.35 μB (for Ni²⁺) and 0.27–1.42 μB (for Cu²⁺). 2,3,4-Trimethoxybenzoates of Mn(II), Co(II) and Ni(II) follow the Curie–Weiss law, whereas that of Cu(II) forms a dimer.     

Change of the roginski “control band” in the case of surface diffusion of adsorbed particles

The effect of surface particle migration between different surface patches on the distribution of the catalytic activity of patches of a linearly inhomogenous surface has been studied. It has been shown that the diffusion can result in an equilibrium distribution of particles A_ads on the surface patches and, therefore, shift of the “control band” and significant change of the reaction rate.     

Lysozyme as diffusion tracer for measuring aqueous solution viscosity

Measuring tracer diffusion provides a convenient approach for monitoring local changes in solution viscosity or for determining viscosity changes in response to multiple solution parameters including pH, temperature, salt concentrations or salt types. One common limitation of tracer diffusion in biologically relevant saline solutions is the loss of colloidal stability and aggregation of the tracer particles with increasing ionic strength. Using dynamic light scattering to measure tracer diffusion, we compared the performance of two different types of tracer particles, polystyrene nanobeads vs. the small protein lysozyme, for viscosity measurements of saline solutions. Polystyrene beads provide reliable values for water viscosity, but begin flocculating at ionic strengths exceeding about 100 mM. Using lysozyme, in contrast, we could map out viscosity changes of saline solutions for a variety of different salts, for salt concentrations up to 1 M, over a wide range of pH values, and over the temperature range most relevant for biological systems (5–40 °C). Due to its inherently high structural and colloidal stability, lysozyme provides a convenient and reliable tracer particle for all these measurements, and its use can be readily extended to other optical approaches towards localized measurements of tracer diffusion such as fluorescence correlation spectroscopy.     

Complexes of Mn(II), Co(II), Ni(II), Cu(II) and Zn(II) with 4-chloro-2-methoxybenzoic acid anion

The complexes of 4-chloro-2-methoxybenzoic acid anion with Mn²⁺, Co²⁺, Ni²⁺, Cu²⁺ and Zn²⁺ were obtained as polycrystalline solids with general formula M(C₈H₆ClO₃)₂·nH₂O and colours typical for M(II) ions (Mn – slightly pink, Co – pink, Ni – slightly green, Cu – turquoise and Zn – white). The results of elemental, thermal and spectral analyses suggest that compounds of Mn(II), Cu(II) and Zn(II) are tetrahydrates whereas those of Co(II) and Ni(II) are pentahydrates. The carboxylate groups in these complexes are monodentate. The hydrates of 4-chloro-2-methoxybenzoates of Mn(II), Co(II), Ni(II), Cu(II) and Zn(II) heated in air to 1273 K are dehydrated in one step in the range of 323–411 K and form anhydrous salts which next in the range of 433–1212 K are decomposed to the following oxides: Mn₃O₄, CoO, NiO and ZnO. The final products of decomposition of Cu(II) complex are CuO and Cu. The solubility value in water at 293 K for all complexes is in the order of 10^–3 mol dm^–3. The plots of χ_M vs. temperature of 4-chloro-2-methoxybenzoates of Mn(II), Co(II), Ni(II) and Cu(II) follow the Curie–Weiss law. The magnetic moment values of Mn²⁺, Co²⁺, Ni²⁺ and Cu²⁺ ions in these complexes were determined in the range of 76−303 K and they change from: 5.88–6.04 μ_B for Mn(C₈H₆ClO₃)₂·4H₂O, 3.96–4.75 μ_B for Co(C₈H₆ClO₃)₂·5H₂O, 2.32–3.02 μ_B for Ni(C₈H₆ClO₃)₂·5H₂O and 1.77–1.94 μ_B for Cu(C₈H₆ClO₃)₂·4H₂O.     

Interface microstructure and diffusion kinetics in diffusion bonded Mg/Al joint

The interface microstructure, formation of diffusion bonded joint and regulation of atom diffusion were studied by means of scanning electron microscope (SEM), energy dispersion spectroscopy (EDS) and electron probe microanalyser (EPMA). The experimental results indicated that an obvious interfacial transition zone was formed between Mg and Al, and there are three intermetallic layers Mg₁₇Al₁₂, MgAl and Mg₂Al₃ in this zone. Diffusion activation energy of Mg and Al in the above layers was lower than that in the Mg and Al base metals. The thickness (x) of each layer can be expressed as x ² = 4.14exp(−28780/RT)(t−t ₀), x ² = 31.4exp(−25550/RT)(t−t ₀) and x ² = 0.6exp(−22600/RT)(t−t ₀) corresponding to Mg₁₇Al₁₂, MgAl and Mg₂Al₃ with heating temperature (T) and holding time (t).     

Tracer diffusion in fluctuating block copolymer melts

We present a theoretical investigation of the tracer diffusion of diblock copolymers and homopolymers in a thermally fluctuating block copolymer melt above the order-disorder transition (ODT) temperature. Entanglement effects and differences in monomeric friction coefficients are ignored; hence, the theory should be most applicable to short copolymers with rheologically similar blocks. Overall, we find that the diffusion rates of both tracer block copolymers and homopolymers in a block copolymer melt are suppressed when compared with diffusivities in a strictly homogeneous medium with the same average composition. This mobility suppression is due to thermally excited composition fluctuations in block copolymer melts near the ODT; the latter result in transient potential barriers to diffusion. We explore the dependence of the tracer diffusion coefficient on molecular weights and compositions of both matrix and tracer, as well as temperature. A comparison of our theoretical predictions to recent experiments by T. Lodge and coworkers shows qualitative agreement. © 1996 John Wiley & Sons, Inc.     

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     

氧化物体系中Eu2+的价态稳定和转换

采用光谱方法研究了氧化物体系中Eu2 的价态稳定和转换问题．研究表明体系中Eu2 的价态稳定与取代离子的半径、电荷及基质晶相等密切相关．运用晶格能理论和晶体结构变化对实验结果进行了计算和讨论．     

Application of molecular dynamics simulations in molecular property prediction II: diffusion coefficient

In this work, we have evaluated how well the general assisted model building with energy refinement (AMBER) force field performs in studying the dynamic properties of liquids. Diffusion coefficients (D) have been predicted for 17 solvents, five organic compounds in aqueous solutions, four proteins in aqueous solutions, and nine organic compounds in nonaqueous solutions. An efficient sampling strategy has been proposed and tested in the calculation of the diffusion coefficients of solutes in solutions. There are two major findings of this study. First of all, the diffusion coefficients of organic solutes in aqueous solution can be well predicted: the average unsigned errors and the root mean square errors are 0.137 and 0.171 × 10(-5) cm(-2) s(-1), respectively. Second, although the absolute values of D cannot be predicted, good correlations have been achieved for eight organic solvents with experimental data (R(2) = 0.784), four proteins in aqueous solutions (R(2) = 0.996), and nine organic compounds in nonaqueous solutions (R(2) = 0.834). The temperature dependent behaviors of three solvents, namely, TIP3P water, dimethyl sulfoxide, and cyclohexane have been studied. The major molecular dynamics (MD) settings, such as the sizes of simulation boxes and with/without wrapping the coordinates of MD snapshots into the primary simulation boxes have been explored. We have concluded that our sampling strategy that averaging the mean square displacement collected in multiple short-MD simulations is efficient in predicting diffusion coefficients of solutes at infinite dilution.     

The significance of the surface condition in solutions to the diffusion equation: explaining “anomalous” sigmoidal, Case II, and Super Case II absorption behavior

Absorption into polymers is frequently described by the terms Fickian, sigmoidal (S-shaped), Case II, or Super Case II. This terminology is used to describe absorption that is respectively, linear with the square root of time, has a slight delay or S-shape with the square root of time, is linear with linear time, or increases more rapidly than with linear time. Solutions to the diffusion equation, Fick’s second law, that include a potentially significant surface condition are shown to reproduce all of these. Sigmoidal absorption results when the surface condition is moderately significant for either a constant diffusion coefficient or exponential diffusion coefficients. Exponential diffusion coefficients and a lower surface mass transfer coefficient result in Case II type behavior, with Super Case II behavior resulting when the surface condition becomes still more significant. The results reported here are supported by extensive experimental data with reasonable and verifiable values for the diffusion coefficients and surface mass transfer coefficients.     

HPLC-UV analytical method for determination of pizotifen after in vitro transdermal diffusion studies

Pizotifen malate is an antihistamine and serotonin inhibitor used in the preventive treatment of migraine and eating disorders. A simple, rapid, accurate and precise high‐performance liquid chromatography (HPLC) method involving ultraviolet detection was validated for the quantitative analysis of pizotifen malate in samples from in vitro transdermal diffusion studies. The method was validated for specificity, linearity, accuracy, precision, limit of detection, limit of quantification and robustness. Drug stability in the solution was also determined under different conditions. Separation was carried out using a 250 × 4.0 mm Kromasil^® C₁₈ column at room temperature. The detector response, fitted at 254 nm, was found to be linear in a concentration range between 0.24 and 24.0 µg/mL. The limit of detection was 0.02 µg/mL and the limit of quantification was 0.07 µg/mL. Finally, in vitro transdermal diffusion of pizotifen malate was characterized using the validated HPLC method. Copyright © 2011 John Wiley & Sons, Ltd.