Fickian Transport Research Articles

Enhanced Supersaturation via Fusion-Assisted Amorphization during FDM 3D Printing of Crystalline Poorly Soluble Drug Loaded Filaments.

Filaments loaded with griseofulvin (GF), a model poorly water-soluble drug, were prepared and used for 3D printing via fused deposition modeling (FDM). GF was selected due to its high melting temperature, enabling lower temperature hot-melt extrusion (HME) keeping GF largely crystalline in the filaments, which could help mitigate the disadvantages of high HME processing temperatures such as filament quality, important for printability and the adverse effects of GF recrystallization on tablet properties. Novel aspects include single-step fusion-assisted ASDs generation during FDM 3D printing and examining the impact of tablet surface areas (SA) through printing multi-mini and square-pattern perforated tablets to further enhance drug supersaturation during dissolution. Kollicoat protect and hydroxypropyl cellulose were selected due to their low miscibility with GF, necessary to produce crystalline filaments. The drug solid-state was assessed via XRPD, DSC and FT-IR. At 165 °C HME processing temperature, the filaments containing ~80% crystalline GF were printable. Fusion-assisted 3D printing led to GF supersaturation of ~153% for cylindrical tablets and ~293% with the square-pattern perforated tablets, indicating strong monotonous impact of tablet SA. Dissolution kinetics of drug release profiles indicated Fickian transport for tablets with higher SA, demonstrating greater SA-induced drug supersaturation for well-designed 3D printed tablets.

Enhanced Floating and Intragastric Release of 5-Flourouracil through Sesame Oil Entrapped Gellan Composite Hydrogel Beads

Background: 5-Fluorouracil is an anti-metabolite compound used for several years as an anti-tumor drug. The development of a gastroretentive drug delivery system of 5-Fluorouracil may have advantages since they retain in the stomach for an extended time and release the drug in a sustained manner, which ultimately enhances the absorption of the drug and consequently the bioavailability. Objective: The objective of the present work was to prepare a sesame oil-entrapped gellan gum hydrogel bead for controlled stomach specific delivery of 5-Fluorouracil. Methods: Sesame oil-entrapped gellan gum hydrogel bead was prepared by ionotropic gelation method. The developed hydrogels were characterized by SEM, FTIR, DSC, and XRD. The entrapment efficiency, floating ability, swelling and drug release in vitro were also determined. Results: Electrostatic interaction between the carboxylic group of polymers and Ca++ was confirmed by FTIR analysis. The SEM photograph of the hydrogel beads portrayed an approximately spherical shaped structure. DSC thermogram and XRD spectra exhibited the molecular dispersion of the drug inside the hydrogel beads. The developed beads of 5-Fluorouracil floated in pH 1.2 buffer solutions for a prolonged time period and the duration of floating was improved significantly with increasing the concentration of oil. The developed formulations showed controlled release of drug and incorporation of oil retarded the release of the drug. Fickian and non-Fickian mechanism of drug transport was observed from the prepared hydrogel beads. Conclusion: Overall, the oil-entrapped gellan matrices could be used for the intragastric delivery of 5-Fluorouracil to treat stomach cancer.

Macroscopic Model for Passive Mass Dispersion in Porous Media Including Knudsen and Diffusive Slip Effects

In this work, a macroscopic model for incompressible and Newtonian gas flow coupled to Fickian and advective transport of a passive solute in rigid and homogeneous porous media is derived. At the pore-scale, both momentum and mass transport phenomena are coupled, not only by the convective mechanism in the mass transport equation, but also in the solid-fluid interfacial boundary condition. This boundary condition is a generalization of the Kramers-Kistemaker slip condition that includes the Knudsen effects. The resulting upscaled model, applicable in the bulk of the porous medium, corresponds to: 1) A Darcy-type model that involves an apparent permeability tensor, complemented by a dispersive term and 2) A macroscopic convection-dispersion equation for the solute, in which both the macroscopic velocity and the total dispersion tensor are influenced by the slip effects taking place at the pore-scale. The use of the model is restricted by the starting assumptions imposed in the governing equations at the pore scale and by the (spatial and temporal) constraints involved in the upscaling process. The different regimes of application of the model, in terms of the Péclet number values, are discussed as well as its extents and limitations. This new model generalizes previous attempts that only include either Knudsen or diffusive slip effects in porous media.

Microsponges Loaded Topical Drug Delivery System for the Effective Management of Rheumatoid Arthritis

Rheumatoid arthritis (RA) is an autoimmune disease associated with severe joint pain. Herein, we report aceclofenac (ACF)-loaded cellulosic microsponge gel formulation with sustained anti-inflammatory effects that are required to manage arthritic pain. The microsponges were formulated using quasi-emulsion solvent diffusion method. The optimized formulation was dispersed into Carbopol gel. Further, the optimized formulation was evaluated for pH, viscosity and spreadability, drug content, and kinetic drug release profile. The optimized microsponges loaded topical gel showed R2 value of 0.9930 which was nearest to one which revealed that formulation obeyed Higuchi kinetic model and n value was found to be 0.48 which depicts Fickian transport of drug diffusion. The prepared optimized formulation showed no signs of irritation when applied on hairless skin of rats along with inhibitory effect on inflammation was observed in in vivo studies in animals, hence, ACF-loaded topical drug delivery is considered as a potential platform for the management of RA.

Preparation and Evaluation of Gliptin Liposomes for Targetting Ocular Region in Neurodegeneration

Background: Neurodegeneration shows the distressing effects of miscommunications between brain cells. Insulin signaling dysregulation and small vessel disease in the base of diabetes may be important contributing factors in Alzheimer’s disease and vascular dementia. Gliptins play a role in neurodegeneration due to its neuroprotective effects. Eye is an extension of the brain. Ocular route reduces drug’s adverse effects offering an advantage in minimizing risk by targeted delivery to the brain. Conventional ophthalmic formulations exhibit poor bioavailability. Liposomes serve as promising active carriers of drugs to posterior segment eye disorders due to improvement in intravitreal half-life and targeted sustained drug delivery to the retina. Liposomes act as drug carriers for entrapment of hydrophilic and hydrophobic drugs. Objective: The study aimed to formulate and evaluate sitagliptin liposomal formulation for sustained effect in individuals suffering from neurodegeneration owing to high patient compliance, especially in geriatric patients. Methods: Sitagliptin liposomes were prepared by the ethanol injection method and were evaluated for various physicochemical properties such as visual appearance, particle size distribution, zeta potential, % drug entrapment efficiency, % drug loading capacity and in vitro drug release studies. Results: The optimized formulation (L-3) showed round-shaped distinct particles with good stability. The L-3 shows average diameter (281.9 nm); zeta potential (-11.9 mV); % entrapment efficiency (82.7 ± 0.89%); % drug loading (33.11 ± 0.67%). L-3 followed Korsmeyer- Peppas model with fickian diffusion transport of drug release giving n’ (0.3094), r2 (0.9753), with 83.78 ± 0.97% of sustained drug release. The L-3 passes the sterility test indicating its safe use in ophthalmic purposes. Conclusion: Thus, liposomal drug delivery is a highly effective approach for sustained drug delivery and minimizing the side effects of gliptins for effective therapy in neurodegeneration.

Curcumin Niosomes Prepared from Proniosomal Gels: In Vitro Skin Permeability, Kinetic and In Vivo Studies.

Curcumin is a poorly water-soluble drug that is used for the treatment of inflammations, tumors, wound healing antioxidant and other diseases. In the current manuscript, it is successfully formulated into proniosome gels. The proniosomes are readily hydrated into niosomal formulations using warm water. Proniosomes were prepared using nonionic surfactants (tween 80, span 60) either solely or in combinations with cholesterol. The produced niosomal formulations were homogenous in size with vesicular sizes >343 and <1800 nm. The encapsulation efficiency percentage “EE%” of curcumin in niosomal formulations was different according to niosomal composition. It increased up to 99.74% in formulations of tween 80/Chol of 200 μmole/mL lipid concentration. Span 60/chol niosomes showed decreased curcumin EE%. Niosomal formulations showed increased SSTF and PC with enhancement ratios of more than 20-fold compared with curcumin suspension form. Kinetically, niosomes fitted to the Korsemeyer-Peppas model with non-Fickian transport according to their calculated n-values where curcumin suspension form showed Korsemeyer-Peppas kinetics with Fickian transport. Niosomal formulations deposited higher curcumin amounts in the skin compared with the suspension form. The best niosomal formulation (F9) was used for niosomal gel and emulgel fabrication. Finally, the anti-inflammatory activity of curcumin in various formulations was evaluated using a rat hind paw edema method and the % of swelling was 17.5% following 24 h in group treated with curcumin niosomal emulgel. In conclusion, this study suggests that the developed niosomal emulgel could significantly enhance the anti-inflammatory effect of curcumin and be an efficient carrier for the transdermal delivery of the drug.

Mean Flow Direction Modulates Non‐Fickian Transport in a Heterogeneous Alluvial Aquifer‐Aquitard System

AbstractRegional‐scale groundwater quality degradation from nonpoint source pollution threatens the long‐term sustainability of major alluvial aquifer‐aquitard systems worldwide. Models are needed to simulate regionally upscaled transport processes in heterogeneous geology that produces non‐Fickian (anomalous) transport. In this investigation, we show the impact of nonhorizontal and nonvertical mean flow direction on non‐Fickian transport within a 3D, kilometer‐scale, typical alluvial aquifer‐aquitard system. Notably, vertical anisotropy in K and seasonal pumping and recharge can fundamentally change hydraulic gradients and shift the mean flow direction between mostly horizontal and mostly vertical flow. Detailed 3D flow and transport simulations under varying mean flow directions indicate that alterations to hydraulic gradients which control the mean flow direction can lead to increasingly non‐Fickian transport. Under mostly horizontal flow, diffusion and slow advection dominant low‐K facies slow mass transfer rates from low‐K material, and preferential flow along connected high‐K networks causes increased spatial spreading along the mean flow direction. Conversely, predominantly vertical flow caused by spatially distributed pumping and recharge shifts mass transfer processes in low‐K material from diffusion and slow advection dominant to advection dominant, resulting in vertically oriented particle trajectories that compactly migrate through high‐K and low‐K facies alike, and increasingly Fickian transport. Thus, mean flow direction transience driven by vertical anisotropy in K and seasonal pumping and recharge can create oscillating transport patterns, that range from persistently non‐Fickian to more Fickian. Results illustrate the regional‐scale hydrogeologic factors that explain why upscaled transport models fail to capture non‐Fickian effects resulting from mean flow direction transience

SOLUTE TRANSPORT MODELLING IN LOW-PERMEABILITY HOMOGENEOUS AND SATURATED SOIL MEDIA

Fickian and non-Fickian behaviors were often detected for contaminant transport activity owed to the preferential flow and heterogeneity of soil media. Therefore, using diverse methods to measure such composite solute transport in soil media has become an important research topic for solute transport modeling in soil media. In this article, the continuous-time random walk (CTRW) model was applied to illustrate the relative concentration of transport in low-permeability homogeneous and saturated soil media. The solute transport development was also demonstrated with the convection-dispersion equation (CDE) and Two Region Model (TRM) for comparison. CXTFIT 2.1 software was used for CDE and TRM, and CTRW Matlab Toolbox v.3.1 for the CTRW simulation of the breakthrough curve. It was found that higher values of determination coefficient (R2) and lower values of root mean square error (RMSE) concerning the best fits of CDE, TRM, and CTRW. It was found that in the comparison of CDE, TRM, and CTRW, we tend to use CTRW to describe the transport behavior well because there are prevailing Fickian and non-Fickian transport. The CTRW gives better fitting results to the breakthrough curves (BTCs) when β has an increasing pattern towards 2.00. In this study, the variation of parameters in three methods was investigated and results showed that the CTRW modeling approach is more effective to determine non-reactive contaminants concentration in low-permeability soil media at small depths.

PH responsive chitosan-coated microemulsions as drug delivery systems

Oil-in-water microemulsions (o/w MEs) were coated with chitosan to obtain new curcumin delivery systems. First, ternary phase diagrams were constructed using mixtures of Tween 20, Tween 80, ethanol, and monopropyleneglycol (MPG) to determine the macroscopic phase region of MEs and their capacity to incorporate clove oil. Particularly, MEs with size less than 20 nm could be formed using low amount of the surfactant Tween 20. The prepared MEs were subjected to diverse stability tests and characterized by dynamic light scattering (DLS), zeta potential (Z potential), conductivity, pH, and transmission electron microscopy (TEM). Stable, spherical uncoated and polymer-coated MEs were obtained with low droplet size (Dp), <50 nm, for the water/Tween 20/clove oil/ and water/Tween80/MPG/clove oil systems using a chitosan solution of 1 mg/mL. For all polymer-coated MEs, an increase of Z potential (17–54 mV) was observed due to the presence of chitosan. The loading with curcumin did not affect the Dp of MEs (e.g., for ME with Tween 20, Dp unloaded 12.35 ± 1.86 nm and Dp loaded 11.13 ± 0.32 nm) but Z potential and conductivity values were increased (typically for coated ME prepared with Tween 20 17.00 ± 0.36 mV ≤Z potential ≤41.5 mV, 162.20 ± 8.95 μS/cm ≤ conductivity ≤183.8 ± 0.56 μS/cm). Drug delivery studies for the chitosan-coated MEs were carried out at different pH (7.4, 6.5, and 5.8). The release of curcumin from the chitosan-coated MEs, formed with Tween 20, showed a dependence of the pH with a maximum release at the acidic pH of 5.8 (58%). The release of curcumin from all MEs was best described by the Korsmeyer- Peppas model with a Fickian transport mechanism.

Preparation and Evaluation of Extended Release Trilayered Matrix Tablets of Ramipril Using Design of Experiment

The current work is aimed to design, prepare, and evaluate the trilayer matrix tablets incorporated with ramipril for extended drug release. Twenty-seven formulations (RF1-RF27) for active layer (middle layer) were prepared by direct compression method using 33 Response surface method of polymers of different HPMC K4M, HPMC K15M and xanthan gum (low, middle, and high concentrations) by using Design of experiment software. Based on physico-chemical properties and drug release one formulation was chosen and formulated into extended release trilayered matrix tablets were by varying proportions of polymers by direct compression method and they were evaluated. The best optimized formulation was characterized for FTIR studies. Out of 27 active layer formulations, RF23 was chosen as best based with maximum drug release of 98.11% and was formulated into extended release trilayered matrix tablets (ARF23-HRF23) by varying proportions of polymers. The range of swelling index for all batches (ARF23-HRF23) was 82.34 to 97.46%, similarly the range of drug content was 95.49 to 99.11% and drug released in 24 hours sustainably over an extended period was 84.98 to 98.26% with all highest values exhibited by GRF23. The release order kinetics data indicate the zero-order release with highest (R2) = 0.983 for GRF23 better when compared to marketed product which followed first order showed R2 value 0.962. Further the formulation (GRF23) showed best fit to Korsmeyer-Peppas plots i.e., 0.957 indicating non Fickian (anomalous) transport coupled diffusion and erosion. The FT-IR data assure the compatibilityof drug and excipients. GRF23 was found to be stable for 180 days at accelerated conditions.

Hydrogel-based matrices for controlled drug delivery of etamsylate: Prediction of in-vivo plasma profiles

ObjectivesTo design oral controlled release (CR) hydrogel matrix tablets of etamsylate using various hydrophilic polymers. Additionally, to predict plasma concentration-time profiles of etamsylate released from different CR matrices. MethodsCharacterization of the in-vitro release rate was performed by various model dependent and model independent approaches. A simple numerical convolution strategy was adopted to predict the in-vivo performance of all matrices from their in-vitro percent released data. The statistical analysis was conducted utilizing a student t-test and ANOVA. ResultsThe release of etamsylate from all matrices showed a deviation from Fickian transport mechanism except; F2 followed Case II release whereas, F9 and F11 obeyed Fickian diffusion. CR hydrogel based-matrices (F4 and F11) demonstrated the maximum drug retardation and satisfied the USP release limits. Concentration–time profiles of etamsylate were predicted successfully from the in-vitro release data of all prepared matrices. Pharmacokinetic parameters of etamsylate CR hydrogel matrices were significantly changed with comparison to reference product except F1. ConclusionThe designed (F2-F11) matrices had the capability to extend the plasma level of etamsylate for an adequate time. However, F4 and F11 were considered the most ideal formulations for once daily application of etamsylate. The prediction of in-vivo pharmacokinetics of etamsylate was very useful to assess the rationality of the designed matrices for the practical application in humans.

Nonlocal transport models for capturing solute transport in one‐dimensional sand columns: Model review, applicability, limitations and improvement

AbstractModelling pollutant transport in water is one of the core tasks of computational hydrology, and various physical models including especially the widely used nonlocal transport models have been developed and applied in the last three decades. No studies, however, have been conducted to systematically assess the applicability, limitations and improvement of these nonlocal transport models. To fill this knowledge gap, this study reviewed, tested and improved the state‐of‐the‐art nonlocal transport models, including their physical background, mathematical formula and especially the capability to quantify conservative tracers moving in one‐dimensional sand columns, which represents perhaps the simplest real‐world application. Applications showed that, surprisingly, neither the popular time‐nonlocal transport models (including the multi‐rate mass transfer model, the continuous time random walk framework and the time fractional advection‐dispersion equation), nor the spatiotemporally nonlocal transport model (ST‐fADE) can accurately fit passive tracers moving through a 15‐m‐long heterogeneous sand column documented in literature, if a constant dispersion coefficient or dispersivity is used. This is because pollutant transport in heterogeneous media can be scale‐dependent (represented by a dispersion coefficient or dispersivity increasing with spatiotemporal scales), non‐Fickian (where plume variance increases nonlinearly in time) and/or pre‐asymptotic (with transition between non‐Fickian and Fickian transport). These different properties cannot be simultaneously and accurately modelled by any of the transport models reviewed by this study. To bypass this limitation, five possible corrections were proposed, and two of them were tested successfully, including a time fractional and space Hausdorff fractal model which minimizes the scale‐dependency of the dispersion coefficient in the non‐Euclidean space, and a two‐region time fractional advection‐dispersion equation which accounts for the spatial mixing of solute particles from different mobile domains. Therefore, more efforts are still needed to accurately model transport in non‐ideal porous media, and the five model corrections proposed by this study may shed light on these indispensable modelling efforts.

Aurora: A non-Fickian (and Fickian) particle tracking package for modeling groundwater contaminant transport with MODFLOW

We introduce the Aurora software, which performs continuous time random walk (CTRW) particle tracking (PT) of dissolved contaminants on macroscopic groundwater flow fields computed using MODFLOW. This is a natural and practical approach to modeling transport at field scale: deterministic, explicitly resolved flow information is used directly while unresolved small-scale physical and chemical heterogeneity that may cause non-Fickian transport is captured by the CTRW. Existing particle trackers are limited to streamline tracing or to homogeneous velocity fields, and no existing software handles subsurface non-Fickian transport, so this represents a substantial technical advance. Fickian transport simulation can also benefit from Aurora's PT approach, which lacks numerical dispersion and other artifacts. We outline the equations Aurora implements and explain the technical and practical details of MODFLOW interoperation. Two demonstrations are presented of PT simulations using Aurora to model physics that would otherwise be difficult to treat analytically or numerically.

Detemplated and Pillared 2-Dimensional Zeolite ZSM-55 with Ferrierite Layer Topology as a Carrier for Drugs.

The present studies were conducted to show the potential of 2D zeolites as effective and non-toxic carriers of drugs. Layered zeolites exhibit adjustable interlayer porosity which can be exploited for controlled drug delivery allowing detailed investigation of the drug release because the structure of the carrier is known exactly. This study was conducted with model drugs ciprofloxacin and piracetam, and ZSM-55 with ca 1 nm thick layers, in detemplated and pillared forms. The release profiles differed from the commercial, crystalline forms of drugs—the release rate increased for ciprofloxacin and decreased for piracetam. To understand the dissolution mechanisms the release data were fitted to Korsmeyer-Peppas equation, showing Fickian (for pillared) and anomalous (for detemplated sample) transport. FT-IR studies showed that strong interaction carrier-drug may be responsible for the modified, slowed down release of piracetam while better solubility and faster release of ciprofloxacin was attributed to formation of the protonated form resulting in weaker interaction with the zeolite than in the pure crystalline form. Two independent tests on L929 mice fibroblasts (ToxiLight and PrestoBlue) showed that ZSM-55, in moderate concentrations may be safely used as a carrier of drug molecules, not having negative effect on the cells viability or proliferation rate.

Formulate and evaluate once daily sustained release tablet of highly soluble drug of metformin HCL

The aim of the present study was to design an oral sustained release matrix tablet of highly water soluble biguanide anti diabetic drug. The matrix tablets are prepared by melt granulation method using HPMC K 200M as hydrophilic drug release retarding polymer, and stearic acid as melt able binder as well as hydrophobic carrier. The drug and excipients compatibility was studied by FT – IR. The formulated matrix tablets were characterized for physical parameters and in vitro dissolution profile. FT – IR spectra revealed the absence of drug excipients interaction. The physical parameters of the tablets were found within the limits. The drug release kinetics demonstrated that by increasing the concentration of hydrophilic polymer and hydrophobic carrier the drug release rate was retarded proportionally. Kinetic modelling of in vitro release profile revealing that the drug release from the matrix tablets following first order kinetics, and the drug release mechanism of optimized (F7) formula following non fickian transport mechanism. Accelerated stability studies were performed according to ICH guide lines. Temperature 40±20 c and relative humidity 75±5% RH to study physical and chemical changes of formulation. No physical or chemical changes were observed after t accelerated stability studies.

Kinetic analysis of pH influence on the fluorescein release from chitosan-coated mesoporous silica nanoparticles

The development of drug delivery systems that are able to respond to the different pH gradients found in the human body is a research field of vast scientific and technological interests. With such a responsive system, a specific pH condition can be used as a stimulus to trigger the local drug-releasing, increasing the efficiency of treatments in comparison to the traditional ones of systemic drug administration. In this study, a responsive drug nanocarrier to acid pH was produced by coating mesoporous silica nanoparticles with an outer chitosan layer that was covalently bonded to the nanoparticle. Fluorescein dye (FL) was loaded into the nanocarrier and used as a spectroscopic probe to evaluate the kinetics of release in buffered solutions with pH 6.0, 7.2 and 8.0. The kinetics analysis using mathematical models demonstrated higher rates of release in pH 6.0 than in pH 7.2 or 8.0. However, the experimental procedure used to load the dye into the nanocarrier had a huge influence on the kinetic mechanisms of delivering. When the dye was loaded into the nanocarrier before the chitosan coating had been made, the FL release was sustained and controlled through a Fickian's transport mechanism. When the dye was loaded after the nanocarrier coating a burst release effect was observed in every pH evaluated, which indicated that the dye was loaded only in the external coating of CS and did not inside of the silica mesopores.

TransPoly: A theoretical model to quantify the dynamics of water transfer through nanostructured polymer films

Mass transfer of penetrants is among the most important physical properties of polymer films to properly ensure function during service. In the case of interaction between penetrants and polymer, non-Fickian diffusion can be observed. This phenomenon is described by numerous reports and a few models have been proposed. Nevertheless, few of them quantify Fickian and non-Fickian behaviour using parameters with clear physical meaning. The present work develops a new non-Fickian mass transport model: TransPoly (Transfer through Polymers). Based on coupled heat and mass transfer, it accounts for simultaneous Fickian transport and molecular relaxation. This model is applied to data of water vapour mass transfer in nanocomposites of polylactide and cellulose nanocrystals. The contribution of the non-Fickian mass transfer on water vapour sorption is highlighted. TransPoly can be applied to several kinds of polymers and used for prediction purposes once the parameters are determined on a learning database.

Is There a Representative Elementary Volume for Anomalous Dispersion?

The concept of the representative elementary volume (REV) is often associated with the notion of hydrodynamic dispersion and Fickian transport. However, it has been frequently observed experimentally and in numerical pore-scale simulations that transport is non-Fickian and cannot be characterized by hydrodynamic dispersion. Does this mean that the concept of the REV is invalid? We investigate this question by a comparative analysis of the advective mechanisms of Fickian and non-Fickian dispersions and their representation in large-scale transport models. Specifically, we focus on the microscopic foundations for the modeling of pore-scale fluctuations of Lagrangian velocity in terms of Brownian dynamics (hydrodynamic dispersion) and in terms of continuous-time random walks, which account for non-Fickian transport through broad distributions of advection times. We find that both approaches require the existence of an REV that, however, is defined in terms of the representativeness of Eulerian flow properties. This is in contrast to classical definitions in terms of medium properties such as porosity, for example.

Ensemble and effective dispersion in three-dimensional isotropic fractal media

We determine the time-dependent behavior of the dispersion coefficient for transport in formations with isotropic log-conductivity fields showing fractal behavior.We consider two different dispersion coefficients for point-like injection: (1) the ensemble dispersion coefficients, defined as half the rate of change of the second central moments of the ensemble-averaged concentration distribution and (2) the effective dispersion, which is half the rate of change of the expected second central moments. Our results show, that the two longitudinal macrodispersion coefficients steadily grow with time and remain different at all times in a fully fractal regime, indicating that no Fickian transport regime is ever reached. The resulting effective longitudinal transport model is consequently a fractional advection–dispersion equation. In the semi-fractal regime, a Gaussian transport regime is reached eventually. However, compared to the case of a classic non-fractal regime, the transient non-Gaussian regime lasts much longer. In the transverse direction, the two dispersion coefficients approach the same large-time limit also in fractal media highlighting the fundamental difference between longitudinal and transverse dispersion.

Effect of Stress on Li Transport in High Energy Density Electrode Materials

Solid-state diffusion of lithium through the active material is a crucial aspect of lithium-ion battery operation. Quantitative analysis of transport properties is important to understand diffusion mechanisms and designing effective electrode architectures for Li-ion battery. There is a large body of literature exists on Li diffusion coefficient measurement techniques in various electrodes, among them, galvanostatic intermittent titration technique (GITT) and the potentiostatic intermittent titration technique (PITT) are the most widely used methods. However, none of the existent studies considered electrode stresses and the effect of stresses on the measured chemical diffusion coefficient. In this study, sputter deposited Ge films (as a model high energy electrode material) were subjected to potentiostatic intermittent titration technique (PITT) and galvanostatic intermittent technique (GITT) conditions while simultaneously measuring the stress evolution in the electrodes. It was observed that the stresses varied significantly in a single titration step during GITT experiment, which violates the assumptions of Fickian transport model where the electrode stresses are usually neglected. Therefore, only PITT data was analyzed to obtain the chemical diffusion coefficient of Li in Ge. As expected, the diffusion coefficient value increased considerably with Li concentration; however, the values obtained during delithiation are at least two times higher than those obtained during lithiation at any given concentration, with the difference becoming significantly higher with Li concentration. This difference is attributed to the state of stress, i.e., tensile state of stress enhances diffusion and compressive state of stress impedes the diffusion process. The data and observations presented here will be helpful in developing and using electro-chemo-mechanical models in producing optimized electrode microstructures.