David A. Edwards

Non-fickian diffusion in thin polymer films

SYNOPSIS 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 I1 diffusion, where Fickian-like fronts initially move like fi 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. 0 1996 John Wiley & Sons, Inc.

Characterizing Molecular Diffusion in the Lens Capsule

The lens capsule compartmentalizes the cells of the avascular lens from other ocular tissues. Small molecules required for lens cell metabolism, such as glucose, salts, and waste products, freely pass through the capsule. However, the lens capsule is selectively permeable to proteins such as growth hormones and substrate carriers which are required for proper lens growth and development. We used fluorescence recovery after photobleaching (FRAP) to characterize the diffusional behavior of various sized dextrans (3, 10, 40, 150, and 250 kDa) and proteins endogenous to the lens environment (EGF, gammaD-crystallin, BSA, transferrin, ceruloplasmin, and IgG) within the capsules of whole living lenses. We found that proteins had dramatically different diffusion and partition coefficients as well as capsule matrix binding affinities than similar sized dextrans, but they had comparable permeabilities. We also found ionic interactions between proteins and the capsule matrix significantly influence permeability and binding affinity, while hydrophobic interactions had less of an effect. The removal of a single anionic residue from the surface of a protein, gammaD-crystallin [E107A], significantly altered its permeability and matrix binding affinity in the capsule. Our data indicated that permeabilities and binding affinities in the lens capsule varied between individual proteins and cannot be predicted by isoelectric points or molecular size alone.

Trace element disequilibria and magnesium isotope heterogeneity in 3655A: Evidence for a complex multi-stage evolution of a typical Allende Type B1 CAI

We used the Panurge ion microprobe to measure concentrations of the rare earth elements (REEs), Ba, Hf, and Sr in melilite, clinopyroxene, plagioclase, and perovskite and Mg isotopes in plagioclase, spinel, melilite, fassaite, hibonite, grossular, and monticellite from the Allende Type B1 calcium-, aluminum-rich inclusion (CAI), USNM 3655A. The distribution and concentration of Ba and the REE in melilite from the melilite-rich mantle of 3655A are unlike those predicted from melilite-melt REE partitioning experiments for closed system crystal fractionation. REE concentrations are lower than expected in the first crystallized gehlenitic melilite, increase rapidly to higher than expected concentrations in melilite with intermediate akermanite contents (Ak30–Ak40), and decrease as expected only during the late stage of mantle crystallization. Barium concentrations in melilite are 10–50 times those expected, and the LREE/HREE ratio increases continuously rather than remaining constant. The unexpected distribution of trace elements in melilite reflects a progressive enrichment of trace elements in the melt during the early stages of crystallization. A partial explanation for this observation is the dissolution of precursor perovskite that contained half or more of the total REE budget of the inclusion. In addition, there are large trace element enrichments adjacent to included spinel in melilite and similar but smaller enrichments adjacent to spinet in clinopyroxene. These enrichments are consistent with the existence of trace element enriched boundary layers at the mineral/melt interfaces. The fact that kinetic processes partially control trace element abundances and distributions suggests rapid cooling during crystallization of the melilite-rich mantle. Similar trace element signatures are ubiquitous in Type B1 CAI, suggesting that each experienced a similar thermal history. The Mg isotope record of 3655A is distinguished by four salient features: (1) large ^(26)Mg excesses correlated with the respective AI/Mg ratios in plagioclase, melilite, and hibonite, (2) F_(Mg), the mass-dependent fractionation of Mg, is positive, with enrichment of the heavier Mg isotopes in all primary phases, (3) a heterogeneous distribution of F_(Mg) values, with F_(Mg) in melilite systematically greater than in either spinel or fassaite, and (4) isotopically normal Mg in the secondary alteration phases, grossular and monticellite. The occurrence of ^(26)Mg^∗, the decay product of ^(26)Al, in anorthite implies early formation of 3655A, while ^(26)Al was extant at nearly the canonical solar system value of ∼5 × 10^(−5). The trace element and Mg isotope heterogeneities suggest a formation scenario for 3655A which includes (1) flash heating to partially melt a solid precursor, (2) rapid cooling to allow survival of relict phases, (3) diffusive exchange of Mg between melilite and a nebular reservoir, and (4) alteration at low temperature. Although this model explains most of the trace element and isotopic characteristics of 3655A and other Type BI CAIs, it does not provide an explanation for the rapid change of LREE/HREE ratios of melilite during crystallization.

Constant front speed in weakly diffusive non-Fickian systems

In certain polymer-penetrant systems, the effects of Fickian diffusion are dominated by nonlinear viscoelastic behavior. Consequently, such systems often exhibit concentration fronts unlike those seen in classical Fickian systems. These fronts not only are sharper than in standard systems but also propagate at constant speed. The mathematical model presented is a moving boundary-value problem, where the boundary separates the polymer into two distinct states, glassy and rubbery, where different physical processes dominate. The moving boundary condition that results is not solvable by similarity solutions but can be solved by integral equation techniques. In the case under consideration, namely, one where the standard Fickian diffusion coefficient is small, asymptotic solutions where a comparatively sharp front moves with constant speed are obtained.

An unusual moving boundary condition arising in anomalous diffusion problems

In the context of analyzing a new model for nonlinear diffusion in polymers, an unusual condition appears at the moving interface between the glassy and rubbery phases of the polymer. This condition, which arises from the inclusion of a viscoelastic memory term in our equations, has received very little attention in the mathematical literature. Due to the unusual form of the moving-boundary condition, further study is needed as to the existence and uniqueness of solutions satisfying such a condition. The moving boundary condition which results is not solvable by similarity solutions, but can be solved by integral equation techniques. A solution process is outlined to illustrate the unusual nature of the condition; the profiles which result are characteristic of a dissolving polymer.

A Mathematical Model for Trapping Skinning in Polymers

When saturated polymer films are desorbed, a thin skin of glassy polymer can form at the exposed surface, inhibiting desorption. In addition, trapping skinning, in which an increase in the force driving the desorption decreases the accumulated flux, can also occur. These behaviors cannot be described by the simple Fickian diffusion equation. The mathematical model presented for the system is a moving boundary-value problem with a set of coupled partial differential equations that cannot be solved by similarity variables. Therefore, integral equation techniques are used to obtain asymptotic estimates for the solution. It is shown that although increasing the driving force will increase the instantaneous flux, the time of accumulation will decrease, thus reducing the overall flux. In addition, the model is shown to exhibit sharp fronts moving with constant speed, another distinctive feature of non-Fickian polymer-penetrant systems.

Desorption Overshoot in Polymer-Penetrant Systems: Asymptotic and Computational Results

Many practically relevant polymers undergoing desorption change from the rubbery (saturated) to the glassy (nearly dry) state. The dynamics of such systems cannot be described by the simple Fickian diffusion equation due to viscoelastic effects. The mathematical model solved numerically is a set of two coupled PDEs for concentration and stress. Asymptotic solutions are presented for a moving boundary-value problem for the two states in the short-time limit. The solutions exhibit desorption overshoot, where the penetrant concentration in the interior is less than that on the surface. In addition, it is shown that if the underlying time scale of the equations is ignored when postulating boundary conditions, nonphysical solutions can result.

Testing the validity of the effective rate constant approximation for surface reaction with transport

Abstract When one incorporates transport effects into a surface-volume reaction, an integrodifferential equation for the bound state concentration occurs. Such a form is inconvenient for data analysis. An effective rate constant approximation for the solution is correct to O(Da2) as the Damkohler number Da → 0. A numerical simulation of the integrodifferential equation is performed which shows that the effective rate constant approximation is useful even outside this regime.

Biochemical reactions on helical structures

Analyses of biochemical surface-volume reactions often focus on the reaction-domi- nated case, where the free-floating analyte is well-mixed and the binding kinetics are unaffected by transport. In actuality, transport effects often play a role. A mathematical model is formulated for a cylinder with a helical reacting strip on its surface, which is a good model for a helical biopolymer such as DNA that interacts with molecules that bind to periodic structures along its backbone. Per- turbation techniques are used to analyze the concentration of the reacting species for association and dissociation kinetics when the cylinder is immersed in a quiescent medium. In the case of an insulat- ing boundary for the analyte region, a multiple-scale expansion must be used. The secularity which necessitates such an expansion is significantly different from canonical examples. The expressions for the bound state provide a direct way to estimate the rate constants from raw data. Remarks on the calculation of effective rate constants for general transport-affected systems are presented.

A spatially nonlocal model for polymer-penetrant diffusion

Abstract. Diffusion of a penetrant in a polymer entanglement network cannot be described by Ficks Law alone; rather, one must incorporate other nonlocal effects. In contrast to previous viscoelastic models which have modeled these effects through hereditary integrals in time, a new model is presented exploiting the disparate lengths of the polymer in the glassy (dry) and rubbery (saturated) states. This model leads to a partial integrodifferential equation which is nonlocal in space. The system is recast as a moving boundary-value problem between sets of coupled partial differential equations. Using singular perturbation techniques, sorption in a semi-infinite polymer is studied on several time scales with varying exposed interface conditions. Though some of the results match with those from viscoelastic models, new physically relevant behaviors also appear. These include the formation of stopping fronts and overshoot in the pseudostress.