Sharp Diffusion Front Research Articles

On modeling the infiltration of water in a PEG-DA hydrogel and the resulting swelling under unconstrained and mechanically-constrained conditions

Polyethylene glycol-diacrylate (PEG-DA) hydrogels are the most extensively studied synthetic gels in biomaterial science, and mathematical modeling of the diffusion of water into this polymer and the resultant swelling of these hydrogels is of central importance for quantitatively predicting the coupled diffusion–deformation response of these materials. Lee (2011) in his Ph.D. thesis reported on some novel experiments on diffusion and swelling of initially dry rods of PEG-DA when put in contact with water at one end — under two different mechanical boundary conditions: (a) when the rods were free to swell, with no mechanical constraints, and (b) when the rods were mechanically-constrained in a glass tube. In the unconstrained case he observed a sharp diffusion “front” that separated the swollen rubbery part and the dry glassy part of the rod, while in the constrained case no sharp diffusion front was observed. His experimental results have not been mathematically modeled in the literature. Here, we build on an existing theory for gels (Chester and Anand, 2010; 2011; Chester et al., 2015) and modify it to account for a dependence of the diffusivity of the solvent on the polymer volume fraction, and a dependence of the Flory–Huggins interaction parameter on the pressure. We show that results from our numerical simulations based on the modified theory can satisfactorily reproduce the experimental observations of Lee (2011).

Tracking Water Sorption in Glassy Aerosol Particles using Morphology-Dependent Resonances.

Morphology-dependent resonances (MDRs) can serve as a sensitive probe of the size and composition of microspheres. While the utilization of MDRs to characterize homogeneous spheres is now routine, analysis of spherical particles with more complicated refractive index profiles can be extremely difficult and time consuming. In ultraviscous and glassy aerosol particles, the concentration profile of water during sorption often contains a sharp front that propagates from the particle surface to the particle center over time. Here we show that the MDR positions associated with this type of concentration profile closely match those of a spherical core-shell profile. Due to the similarities, a core-shell model can be used to simplify the analysis of MDR positions that are observed during water uptake by high-viscosity aerosol particles. We examined the applicability and limitations of this core-shell model in the tracking of water sorption by single particles. Overall, the core-shell model allows for the radial position of a sharp diffusion front to be readily found using MDR positions observed during water sorption, making the analysis of light-scattering measurements much faster and less error prone than previously used fitting schemes. Additionally, methods for calculating MDRs in spherical core-shell particles are also discussed.

Kinks in experimental diffusion profiles of a dissolving semi-crystalline polymer explained by a concentration-dependent diffusion coefficient.

The dissolution of polyethylene oxide (PEO) tablets in water has been followed in situ by neutron radiography. When in contact with water, the crystalline phase of semi-crystalline PEO melts once a certain water content is attained. Polymer concentration profiles obtained from the neutron transmission images exhibited a pronounced kink which corresponds to a sharp front in the images and which is related to the melting transition. Sharp diffusion fronts and phase transitions are often linked to non-Fickian behaviour. However, by considering the time evolution of the complete concentration profiles in detail it is shown that the dissolution process can be explained using Fickian diffusion equations with a concentration-dependent diffusion coefficient.

Lattice strain across Na-K interdiffusion fronts in alkali feldspar: an electron back-scatter diffraction study.

Cation exchange experiments between gem quality sanidine (X_mathrm{Or} = 0.85) and KCl melt produced chemical alteration of alkali feldspar starting at the grain surface and propagating inwards by highly anisotropic Na–K interdiffusion on the alkali sublattice. Diffusion fronts developing in b-direction are very sharp, while diffusion fronts within the a–c-plane are comparatively broad. Due to the composition dependence of the lattice parameters of alkali feldspar, the diffusion induced compositional heterogeneity induces coherency stress and elastic strain. Electron back-scatter diffraction combined with the cross-correlation technique was employed to determine the lattice strain distribution across the Na–K interdiffusion fronts in partially exchanged single crystals of alkali feldspar. The strain changes gradually across the broad fronts within the a–c-plane, with a successive extension primarily in a-direction conferring to the composition strain in unstressed alkali feldspar. In contrast, lattice strain characterised by pronounced extension in b-direction is localised at the sharp diffusion fronts parallel to b, followed by a slight expansion in a-direction in the orthoclase-rich rim. This strain pattern does not confer with the composition induced lattice strain in a stress-free alkali feldspar. It may rather be explained by the mechanical coupling of the exchanged surface layer and the mechanically strong substratum. The lattice distortion localised at the sharp diffusion front may have an influence on the diffusion process and appears to produce a self-sharpening feedback, leading to a local reduction of component mobilities.

Simulation of non-classical diffusion in polymers

The present contribution is concerned with the computational modelling of non-classical diffusion in amorphous polymers. Special attention is paid to the limiting case of Case II diffusion. Application of the dual-phase-lag concept to Fick’s first law leads to a description of Case II behaviour. The change in material properties during the glass transition is explicitly accounted for by a concentration dependent formulation of the material parameters. The proposed model is well suited for modelling the sharp diffusion front and linear uptake kinetics associated with Case II diffusion. Application of a concentration dependent diffusion coefficient reduces the concentration gradient behind the front to a minimum. For the solution procedure, a finite element scheme in space and a finite difference method in time are applied. Three-dimensional numerical results are presented for classical Fickian and non-classical Case II diffusion. This paper adds to the basic understanding of the computational modelling of the Case II diffusion phenomenon.

Sharp diffusion front in diffusion problem with change of state

We propose an alternative model of diffusion in polymers, which may explain formation of propagating sharp diffusion fronts to be observed in some experiments. Differently from typical Case II or Stefan problem models, plasticisation of the polymer matrix is not necessarily required. Instead, diffusing small molecules are assumed to be captured by some specific sites of the polymer matrix for certain retention times. For example, they may become adsorbed at the surface of the holes and microvoids that are considered to be present in glassy polymer regions according to the well-known dual-sorption model and are attributed to excess free volume frozen into the glassy matrix. On the other hand, the small molecules may become bound to polymer chains. In this adsorbed/bound state they have much smaller movability and thus are effectively excluded from the diffusion. This adsorption/binding is considered to be a reversible process of state changing, but with asymmetric probabilities for direct and reverse events. With this assumption, taking into account that the amount of space available for such immobilised molecules is limited (limited free volume, limited number of anchoring sites) the model naturally results in the formation of sharp diffusion fronts or strong concentration gradients. Such a diffusion front is a region of effective immobilisation (adsorption/binding, though reversible) of the free diffusing molecules. For the simplest prescribed boundary conditions the diffusion front propagates linearly with the square root of time. By means of imposing generalised boundary conditions, it is possible to model other regimes of propagation of the diffusion front, including constant velocity.

Temperature-induced transition of the diffusion mechanism of n-hexane in ultra-thin polystyrene films, resolved by in-situ Spectroscopic Ellipsometry

In-situ Spectroscopic Ellipsometry is used to study diffusion of liquid n-hexane in silicon wafer supported 150 nm thick polystyrene films, in the temperature range 16–28 °C. In the higher part of this temperature range Case II diffusion is shown to be dominant. In this case the temporal evolution of the ellipsometric spectra is adequately described by an optical model that divides the polystyrene into a swollen layer and a non-swollen layer, separated by a sharp diffusion front, i.e. mimicking the essential characteristics of the Case II diffusion process. This description is found to be applicable for tracking of the penetrant wave progressing through films with a thickness between 100 nm–1000 nm, covering much of the technologically important thin polymer film range. For temperatures below 20 °C transport is shown to occur by a combination of Fickian diffusion and polymer relaxation (Fickian Relaxation). The temperature-induced transition from Case II to Fickian Relaxation finds validation in the Spectroscopic Ellipsometry modeling and is related to the occurrence of penetrant induced glass transition of the system at around 20 °C. For the kinetics of swelling below the glass transition of the n-hexane – polystyrene mixture Berens – Hopfenberg formalism is applied to extract the penetrant diffusion coefficient and characteristic chain relaxation time.

Molecular Mechanisms of Interphase Evolution in the Liquid Polystyrene−Glassy Poly(phenylene oxide) System

We report a diffusion study on a series of interphases formed between a polystyrene-rich (PS) liquid layer and a poly(phenylene oxide) (PPO) glassy matrix. Diffusion was promoted by annealing the polymer pair at several temperatures below the glass transition temperature of the PPO matrix, in experiments where the liquid component was supplied from an unlimited source. Depth PS concentration profiles were obtained via optical sectioning through the glassy layer with confocal Raman microspectroscopy. The PS profiles measured in all the samples had sharp diffusion fronts followed by a region with relatively uniform PS concentration. From the time evolution of the PS front advance, we directly obtained the time-scaling laws for interphase kinetics which are interpreted in the context of the Fickean and case II diffusion theories. Despite some recent studies reporting the occurrence of case II in this particular system, our results and analysis show conclusively that interphase kinetics are, on the contrary, ...

Diffusion of water-acetic acid mixtures in epoxy

The solubility and diffusion of water–acetic acid solutions in epoxy resins has been studied by nuclear magnetic resonance imaging (MRI) and spectroscopy (NMR), Fourier transform infrared (FTIR) spectroscopy, and light microscopy (LM). These techniques revealed the progression of a sharp diffusion front at a rate proportional to the square root of time. Both the swelling and rate of diffusion are dependent on acid dilution. At higher acid concentrations, hydrogen bonds were present, which has been interpreted as formation of acid dimers. The results indicate that molecular interactions play a major role in controlling the solubility and diffusion processes. The observation that voids can be filled up with penetrant strongly support the interpretation that molecular interactions rather than concentration gradient are responsible for these effects. © 1999 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 37: 3328–3336, 1999

Limited-supply non-Fickian diffusion in glassy polymers

Non-Fickian diffusion of a flame-retardant plasticizer, resorcinol bis(diphenyl phosphate) (RDP), in a glassy poly(ether imide), Ultem TM, was measured with Rutherford backscattering spectroscopy (RBS). Volume fraction versus depth profiles were obtained as a function of time, temperature and externally applied stress in experiments where a limited supply of RDP was initially present on the surface of the Ultem. The profiles of the plasticizer in the glassy polymer in all samples had sharp diffusion fronts with constant volume fraction behind the front. The limited-supply boundary condition requires that the volume fraction, φ, of RDP in the plastized zone decreases as penetration depth increases. Isochronal values of φ decrease with increasing temperature. At long times, φ approaches a value such that the material in the plasticized zone has a glass transition temperature equal to the temperature of the experiment. At 140, 160 and 180°C, φ decreased in direct proportion to the logarithm of time. At 120°C, two regimes of diffusion behaviour were observed in a plot of φ approaches a value such that the material in the diffusion, and long-time low-φ behaviour corresponds to anomalous diffusion. Only the regime of anomalous diffusion was observed at the higher temperatures. Externally applied biaxial tensile or compressive stresses of order 10 to 40 MPa in the plane of the sample had no effect on diffusion normal to the plane in experiments at 120°C lasting 1 or 72h.

In situ Evaluation of Diffusion Mechanisms in Thin Polystyrene Films Used as Integrated Optical Structures

The use of optical waveguide techniques allows the in situ, nondestructive examination of diffusion in thin films. Through the eigenvalue equation, the changes in thickness and refractive index, corresponding to changes in the composition of the film, can be monitored. By assuming a Case-II-like sharp diffusion front, one can model the system as a two-layer film, whose individual layer thicknesses can be determined by the analysis of the observed eigenmodes. This model is used to judge the appropriateness of the Case II description for a given set of conditions. Thin films (on the order of 1–7 μm in thickness) of polystyrene were exposed to n-hexane, methylcyclopentane, and a mixture of the two. The analysis of the observed eigenmodes indicates that n-hexane diffuses into polystyrene via a Case II mechanism, but a Fickian leading edge of the diffusion front can also be observed. In general, this approach provides a simple technique to distinguish between Fickian and Case II diffusion in planar polymer films in the micron thickness regime.

Growth and diffusion of abrupt zinc profiles in gallium arsenide and heterojunction bipolar transistor structures grown by organometallic vapor phase epitaxy

The growth and diffusion of abrupt Zn profiles in undoped gallium arsenide (GaAs), silicon-doped GaAs, and heterojunction bipolar transistor structures grown by organometallic vapor phase epitaxy have been studied using secondary ion mass spectrometry depth profiling. The depth profiles indicate that abrupt (within 100 Å) turn-on of Zn doping to levels approaching 1020 cm−3 are obtainable, while abrupt turn-off is limited to about two orders of magnitude due to dopant tailing toward the surface resulting from residual Zn in the reactor. The sharp diffusion fronts resulting from post-growth anneals indicate that the Zn diffusion coefficient has a concentration dependence. However, the diffusion of Zn at high concentrations appears to be inhibited by crystal defect kinetics resulting in a relatively concentration-independent Zn diffusion coefficient. The V/III growth ratio did not have an effect on Zn diffusion in undoped or silicon-doped GaAs. The diffusion of Zn in heterojunction bipolar transistor structures differs in that the diffusion of Zn into the GaAs collector is larger by an order of magnitude and decreases with an increase in V/III growth ratio. In addition, the diffusion of Zn into the aluminum gallium arsenide (AlGaAs) emitter is significantly lower and more effectively inhibited by an increase in V/III ratio than the diffusion of Zn into the GaAs collector.

Pollution of groundwater through nonlinear diffusion

An analytical solution for the prediction of the spread of contaminant caused by nonlinear diffusion in an aquifer adjoining a polluted source is presented. The solution is based on the assumption of the existence of a sharp diffusion front and is similar to the solution of a two-dimensional unsaturated flow problem. The effect of concentration dependency of the diffusion coefficient on the contamination spread is discussed.

Cathodoluminescence on the diffusion front of CdS doped with trivalent metals

When CdS crystals are doped with In, Ga or Al by heating in the metal vapour, the trivalent metal diffusion is concentration dependent and a sharp diffusion front is observed. A cross-section of the diffusion specimen under electron irradiation in a scanning electron microscope shows very strong cathodoluminescence, highly localized at and near the diffusion front. The intensity profile across the diffusion front varies with concentration gradient and crystal orientation. The emission spectrum also varies with orientation and distance from the diffusion front and with dopant element but is close to the band-gap energy and is basically exciton and donor-acceptor pair recombination radiation.