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.     

Form of multicomponent Fickian diffusion coefficients matrix

The form of multicomponent Fickian diffusion coefficients matrix in thermodynamically stable mixtures is established based on the form of phenomenological coefficients and thermodynamic factors. While phenomenological coefficients form a symmetric positive definite matrix, the determinant of thermodynamic factors matrix is positive. As a result, the Fickian diffusion coefficients matrix has a positive determinant, but its elements — including diagonal elements — can be negative. Comprehensive survey of reported diffusion coefficients data for ternary and quaternary mixtures, confirms that invariably the determinant of the Fickian diffusion coefficients matrix is positive.     

Mutual diffusion at polystyrene/poly(vinyl methyl ether) as measured by ATR‐FTIR and rheometry

Mutual diffusion at PS/PVME (polystyrene/polyvinyl methyl ether) interface both below and above the glass transition temperature, T_g, of polystyrene was probed by Attenuated total reflection (ATR) and rheometry techniques. The results were analyzed with a simple three layer model and a combination of Fickian and Case‐II diffusion models. The mutual diffusion coefficients obtained by the two techniques are compared and the importance of Case‐II diffusion is discussed.     

Plasticized microporous poly(vinylidene fluoride) separators for lithium‐ion batteries. II. Poly(vinylidene fluoride) dense membrane swelling behavior in a liquid electrolyte—characterization of the swelling kinetics

Some microporous poly(vinylidene fluoride) (PVdF) separators for lithium‐ion batteries, used in liquid organic electrolytes based on a mixture of carbonate solvents and lithium salt LiPF₆, were characterized by the study of the swelling phenomena on dense PVdF membranes. Various aspects of the kinetics of the carbonate solvents and the solvent mixture sorption in dense PVdF slabs were studied at different temperatures. Non‐Fickian behavior, characterized by S‐shaped sorption curves, was highlighted, and a salt effect, which resulted in two‐stage sorption, was studied. Diffusion coefficients and activation energies were calculated for the Fickian portions of the sorption curves, that is, at short times and low swelling ratios. A strong influence of the different interaction parameters was shown for the swelling kinetics. This study proved that the swelling of microporous PVdF membranes could be considered instantaneous. © 2003 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 42: 544–552, 2004     

Spinodal decomposition in asymmetric polymer mixtures

We present a preliminary numerical study of spinodal decomposition in an asymmetric polymer mixture, i.e., of polymer with different chain lengths, in three dimensions with full Flory-Huggins-de Gennes free energy, numerically integrating the time evolution equations for the conserved order parameter. For the sake of comparison, we also present a numerical study of the symmetric polymer mixture. The results indicate that the scaled structure factor for the asymmetric polymer mixture is much broader than that of a symmetric polymer mixture. It is interesting that the growth exponents are not symmetric around the critical quench, i.e., growth exponents on the two sides of the critical composition are different. In addition to that, the magnitudes of the pair correlation functions of asymmetric mixtures are very small for x larger than the characteristic domain size r_g and the oscillations seen in the symmetric mixture are almost absent. We have attributed this finding to the rough interfaces and broader domain size distribution in the phase separated asymmetric polymer mixtures. Therefore, the simulation reveals that the asymmetry plays an important role for the spinodal decomposition dynamics of polymer mixtures.     

Small-angle neutron scattering studies of dynamics and hierarchical pattern formation in binary mixtures of polymers and small molecules

We present the dynamics of the composition fluctuations and pattern formation of two-component systems in both single-phase and two-phase states as studied by time-resolved small-angle neutron scattering and light scattering. Two-component systems to be covered here include not only dynamically symmetric systems, in which each component has nearly identical self-diffusion coefficients, but also dynamically asymmetric systems, in which each component has different self-diffusion coefficients. We compare the dynamic behaviors of the two systems and illuminate their important differences. The scattering studies presented for dynamically asymmetric systems highlight that stress–diffusion coupling and viscoelastic effects strongly affect the dynamics and pattern formation. For dynamically symmetric systems, we examine the universality existing in both polymer systems and small-molecule systems as well as new features concerning the time evolution of hierarchical structures during phase separation via spinodal decomposition over a wide range of wave numbers (up to four orders of magnitude). For both systems, we emphasize that polymers provide good model systems for studying the dynamics and pattern formation. © 2004 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 42: 3027–3062, 2004     

Permeation of a hard sphere fluid into a quenched matrix

The permeation of a hard sphere fluid through a model membrane, composed of quenched (in space) hard spheres, is studied using molecular dynamics simulations. The fluid is initially placed outside the porous matrix and their initial intake is investigated and found to be non-Fickian. This non-Fickian behavior can be attributed to the high concentration difference between the fluid in the bulk and in the membrane. Once the system is equilibrated, the authors mark fluid particles that are outside the membrane and investigate their diffusion (color diffusion). Color diffusion is Fickian, and the mass intake and density profiles are well described by a continuum composite medium model with no adjustable parameters, i.e., with self-diffusion coefficients obtained from simulations. The matrix becomes impermeable when there are no percolating paths for the fluid.     

Diffusion in glassy polymers. I

Synopsis The diffusion of six solvents in three crosslinked, glassy epoxy polymers is studied. Case II swelling and Fickian sorption are observed as two simple limiting cases. The mechanism of diffusion changes from one limit to another as the nature of the solvent or the crosslink density of the polymer is altered. With mixed solvents, properly chosen, a superposition of Fickian diffusion and case II swelling is observed.     

Tracer diffusion properties of core–shell latex films studied by photoinduced grating relaxation

This article reports the application of the Photo‐Induced Grating Relaxation technique (also known as Forced Rayleigh Scattering) to investigate the dynamics of films prepared from structured core–shell latex particles via the transport property of the photochromic tracer molecule Aberchrome 540®. The core–shell particles were prepared with a fluoropolymer core (immiscible and impenetrable to the tracer) and a poly(butyl methacrylate) shell. The incompletely dried films (with residual water) manifest their spatial heterogeneity via non‐Fickian behavior (spatial scale‐ dependent apparent diffusion coefficient). The diffusion data was interpreted using the two‐state diffusion model, previously developed to describe the tracer diffusion in latex films without any core–shell structure. In contrast to dry latex films made from homogeneous particles, where one observes Fickian diffusion indicative of a homogeneous polymer film, we find that the lattice of fluoropolymer cores leads to a length scale dependent diffusion coefficient for the tracer. This effect can be interpreted as microscopic evidence for a strain hardening effect due to the presence of a hardened layer of matrix polymer (= shell) surrounding the core, which act as nanofillers. This strain hardening effect could be quantified within the two‐state diffusion model in terms of tracer diffusion coefficients and root mean squared displacements. © 2007 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 45: 2823–2834, 2007     

Guanidinium chloride molecular diffusion in aqueous and mixed water-ethanol solutions

Solutions containing guanidinium chloride (GdmCl), or equivalently guanidine hydrochloride (GdnHCl), are commonly used to denature macromolecules such as proteins and DNA in, for example, microfluidics studies of protein unfolding. To design and study such applications, it is necessary to know the diffusion coefficients for GdmCl in the solution. To this end, we use molecular dynamics simulations to calculate the diffusion coefficients of GdmCl in water and in water-ethanol solutions, for which no direct experimental measurements exist. The fully atomistic simulations show that the guandinium cation Gdm (+) diffusion decreases as the concentration of both Gdm (+) and ethanol in the solution increases. The simulations are validated against available literature data, both transformed measured viscosity values and computed diffusion coefficients, and we show that a prudent choice of water model, namely TIP4P-Ew, gives calculated diffusion coefficients in good agreement with the transformed measured viscosity values. The calculated Gdm (+) diffusion behavior is explained as a dynamic mixture of free cation, stacked cation, and ion-paired species in solution, with weighted contributions to Gdm (+) diffusion from the stacked and paired states helping explain measured viscosity data in terms of atom-scale dynamics.     

网络结构对多层聚乙烯共混物界面扩散行为的影响

通过溶液法制备超高分子量聚乙烯(UHMWPE)/乙烯-己烯共聚物(PEH)共混物,采用动态流变学方法研究了UHMWPE/PEH共混物多层膜在熔点以上的界面扩散行为,结合Double-Reptation理论计算得到了UHMWPE/PEH共混物多层膜的自扩散系数.研究表明UHMWPE在浓度c=1.0 wt%以上可以在PEH基体中形成网络结构.UHMWPE/PEH共混物多层膜界面扩散不符合严格的菲克扩散定律,扩散系数具有时间依赖性.扩散曲线显示扩散过程在到达平台值之前分为2个区域,区域I接近于菲克扩散;然而由于引入UHMWPE,区域II扩散显著偏离菲克扩散行为,特别是当UHMWPE在PEH基体中形成网络结构以后,偏离菲克扩散行为更加显著.     

含α-氨基酸功能基的聚苯乙烯-二乙烯苯不对称树脂的合成

为了应用配位体离子交换树脂拆分外消旋混合物,由氯甲基化苯乙烯-二乙烯苯共聚物经苯胺树脂中间体或相转移催化剂与各种光学活性的α-氨基酸进行胺缩合反应,可以得到含有α-氨基酸功能基的不对称离子交换树脂。红外光谱分析证实了这类树脂的结构。这类胺缩合反应受到各种条件和因素的影响,例如溶剂的性质,催化剂的用量,反应温度和反应时间等。采用上述两种方法合成的含L-脯氨酸功能基的不对称树脂的Cu(Ⅱ)络合物对DL-脯氨酸进行了拆分。     

甲醇-水混合溶剂后处理制备丝素蛋白膜及其药物释放研究

采用溶液浇注法制备丝素蛋白薄膜, 应用傅里叶红外光谱(FTIR)和X 射线衍射(XRD)研究了浓度不同的甲醇-水混合溶剂处理后丝素蛋白薄膜的结构变化, 并以罗丹明B 为模型药物与丝素蛋白构建药物缓释体系, 考察了丝素蛋白膜的结晶结构对药物释放动力学的影响. 结果显示, 在甲醇体积比浓度Φ_MeOH＝50%～90%的范围内, 丝素蛋白材料中以β-折叠为主的silk Ⅱ 结晶含量随着混合溶剂中甲醇浓度的增加而先增加后下降, 在Φ_MeOH＝80%附近出现最大值. 罗丹明B 从丝素蛋白膜的释放属于Fickian 扩散机理, 其扩散指数n 随着丝素蛋白膜中β-折叠含量的增加而增加, silk Ⅱ结晶是丝素蛋白材料药物释放的天然调节器.     

Differential permeation — Part I: A method for the study of solvent diffusion through membranes

We describe the differential permeation method for the study of the diffusion of solvents from a liquid (or liquid mixture) through flat or tubular membranes. This method consists of measuring the transient permeation rates through the membrane when one of its faces is suddenly put into contact with the liquid medium. The change in the transient rate with time is analyzed by numerical best fitting methods to determine the Fickian diffusion coefficient. A simplified equation is proposed for the fitting of the response of a tubular membrane. Deviations from the Fickian transport mechanism with concentration-independent diffusion coefficient can be evidenced and eventually analyzed by using other mechanistic models.     

Application of holographic interferometric microscopy to the study of diffusion in the polyvinylpyrrolidone–iodine system

Interferometric studies of the diffusion of molecular iodine into polyvinylpyrrolidone (PVP) are described. The diffusion process is observed by means of a holographic microscope which, with a camera, permits holograms to be recorded during the polymer–iodine interaction. The holograms are subsequently used to produce interferograms that monitor the diffusion process as a function of time. Measurements taken from the interferograms indicate that the diffusion mechanism is not purely Fickian and that the diffusing boundary of iodine is characteristic of case II diffusion. The interferograms yield values for average diffusion coefficients as a function of sample thicknesses that range from 2.32 to 9.53 × 10^?10 cm²/sec.     

Mixture diffusion in nanoporous adsorbents: equivalence of Fickian and Maxwell-Stefan approaches

Nanopore diffusion in multicomponent adsorption is described using different macroscopic theories: Onsager irreversible thermodynamics, Maxwell-Stefan, and Fickian approaches. A new equivalence between Fickian and Maxwell-Stefan formulations is described by [D]=[n(s)][B](-1)[Gamma][n(s)](-1). The elements of D and B are explicitly related to the Fickian and Maxwell-Stefan diffusivities, respectively. Only when the saturation loadings ni(s) for different components are the same can the matrix be reduced to the generally accepted equation [D]=[B](-1)[Gamma]. On the basis of the relationship between the irreversible thermodynamics and Maxwell-Stefan approaches, an equation is derived for a binary system with the symmetric form (1/Eth1 + theta2/Eth12)(1/Eth2 + theta1/ Eth21)=(L11L22)/(L12L21)(theta1theta2)/(Eth12Eth21) The Maxwell-Stefan binary exchange coefficients Ethij are shown to depend not only on the Maxwell-Stefan diffusivities, Ethi, but also on the Onsager coefficients. For a strong molecular interaction, that is, Ethi>Ethij , the ratio of Onsager coefficients will approach unity, giving the commonly used relation L12=square root L11L22 . In addition, the Maxwell-Stefan diffusivities, Ethi, are shown to depend on the interaction effects in mixtures, and Ethi in mixtures will not generally be equal to pure component values evaluated at the same total fractional loading.     

Diffusion of small molecules in a chitosan/water gel determined by proton localized NMR spectroscopy

Proton localized NMR spectroscopy (MRS) has been applied to study the diffusion of three small molecules, caffeine, theophylline and caprolactam, in chitosan gels with different concentration of water. This technique allows the non-destructive monitorization of diffusant concentration as a function of time and location. Concentration profiles were compared with theoretical curves based on solutions of Fick's diffusion equation for the best fitting, with the appropriate boundary conditions. The measured concentration profiles show a good agreement with the Fickian law. Values of the diffusion coefficients D ranging from 6.1×10(-6) to 3.4×10(-6)cm(2)s(-1) depending on chitosan concentration and type of diffusant molecule were determined. In addition, measurements of diffusion coefficients at equilibrium conditions with proton pulsed field gradient NMR methods supported the observed Fickian behavior and showed values of D in excellent agreement with those determined by proton MRS. All these facts demonstrate that proton MRS is an appropriate method for investigating diffusion process in complex systems, such as polymer gels.     

Development of a Unifying Framework for the Restrictions in Multi-component Diffusion Close to Equilibrium

In this work a unifying framework is developed for multi-component diffusion close to equilibrium by proposing additional restrictions for the Fickian diffusion coefficients such as the Onsager reciprocal relations. Moreover, it is shown that these additional restrictions can explain the discrepancy, reported in the literature, of calculating negative concentrations in Fickian ternary free diffusion observed even if the second thermodynamic law constraints and the phase stability criteria are satisfied. It is believed that this work could be used to further investigate multi-component diffusion.