Concentration-dependent Diffusion Research Articles

Thermo-chemical-mechanical simulation of low temperature nitriding of austenitic stainless steel; inverse modelling of surface reaction rates

A multi-physics thermo-chemical-mechanical 1-dimensional implicit finite difference model is presented to simulate the evolution of composition and residual stress profiles over the expanded austenite case developing during low temperature nitriding of austenitic stainless steels. The model combines 1-dimensional diffusion of nitrogen in the depth direction with a concentration-dependent diffusivity, elasto-plastic accommodation of the lattice expansion, stress gradient-induced diffusion of nitrogen, solid solution-strengthening by nitrogen and trapping of nitrogen by chromium atoms. The rate of the surface reaction governing the transfer of nitrogen from the gas to the solid is unknown and was evaluated by inverse modelling. The modelling was applied adopting the surface reaction rate as the only fitting parameter and taking mass-uptake curves (thermogravimetry) as the constraint, while all other data were taken from established literature values. Very good agreement is achieved between the predicted and experimental composition-depth profiles. Further, the applicability of the present model to plasma nitriding was verified by simulating (not fitting) the evolution of composition-depth profiles obtained after plasma nitriding of stainless steel. The good to very good agreement of the present model's predictions with experimental data for gaseous and plasma nitriding, indicates that the essential multi-physics influences and parameters are taken into account, with a minimum of adjustable parameters.

Batch and fixed-bed mode mercury uptake by a modified adsorbent

The current study deals with the removal of mercury ions from aqueous solutions by sulfurized and original activated carbons (ACs) in the batch and fixed bed modes. The modified sample containing sulfur groups showed higher uptake capacities and larger breakpoints. The XPS and EDS analyses confirmed the introduction of carbon–sulfur bonds onto the AC’s surface after impregnation with dimethyl-disulfide. In the work, the diffusional models named, “Film Pore Concentration-Dependent Surface Diffusion (FPCDSD)” and “Axial dispersion Film Pore Concentration-Dependent Surface Diffusion (AFPCDSD)” were respectively applied to explain the batch and bed-mode kinetic data to clarify the rate-controlling mechanisms and estimate the diffusional/mass transfer properties of adsorbate–adsorbent systems. In all cases, the sulfurized AC showed greater Ds (surface diffusivity) values attributed to the higher mercury loading on this sorbent which facilitates ion hopping on the surface sites. The internal and film diffusions both played notable roles in mercury adsorption mechanism.

Optical measurement of concentration-dependent diffusion coefficients of binary solution using an asymmetric liquid-core cylindrical lens

An optical method for measuring concentration-dependent diffusion coefficients of binary solution is introduced in this paper. The concentration spatial and temporal profiles (Ce(z, t)s) are obtained from a single experiment using an asymmetric liquid-core cylindrical lens. The concentration-dependent diffusion coefficient is assumed to be a polynomial D(C) = D0 × (1 + α1C + α2C2 + α3C3 + …), where D0 is diffusion coefficient in infinite dilute solution, α1, α2 and α3 are under-determined parameters. The finite difference method (FDM) is applied to solve numerically Fick diffusion equation, the calculated concentration profiles (Cn(z, t)s) by varying the under-determined parameters are compared with the profile Ce(z, t), the parameters(α1, α2, α3, …) corresponding the minimum concentration standard deviation are selected to determine D(C). To verify the proposed method, we have carried out the diffusion experiment of ethylene glycol aqueous solution, the obtained concentration-dependent diffusion coefficients are in good agreement with literature values. On the other hand, the obtained polynomial D(C) has been used to calculate refractive index profiles (nn(z, t)s). Based on the nn(z, t)s, ray tracing method has been applied to simulate the diffusion images, which fit the experimental images very well over a wider range of diffusion time, demonstrating that the proposed method is reliable in rapidly measuring concentration-dependent liquid diffusion coefficients with high accuracy and stability.

Diffusion equation under swelling stresses

Water diffusing into silica glass surfaces reacts with the SiO2 structure under hydroxyl generation. This reaction must cause a volume expansion that is already reported in literature. As a consequence of volume swelling and restriction of free expansion by the bulk material, swelling stresses are caused. Since the diffusivity is a function of stress, the consequence is a diffusivity that depends on the local water concentration. Then the solution of the diffusion equation is complicated and makes numerical computations necessary. Disadvantage of numerical computations is the fact that the used extend of the depth range must be finite and, consequently, the semi-infinite body can only be approximated. In the following considerations we will discuss analytical diffusion solutions for constant diffusivity as well as under swelling conditions. Based on this, exact and semi-analytical solutions for diffusion problems in the half-space are given. Further, the effect of swelling affected diffusivity on resulting concentration profiles is shown. We used an analytical solution of the diffusion problem that is known from literature by Gardner and Singh, where the concentration-dependent diffusivity had to be approximated by an exponential function. In case of swelling stresses in silica the diffusivity is given by an exponential function, so that the obtained solution is exact. Due to the compressive swelling stresses, steeper diffusion profiles appear compared to the stress-free state. In the discussion section it will be shown that the water-affected diffusivity correctly predicts the astonishing time-dependency of effective diffusivities observed in literature on silica.

Spatial analog of the two-frequency torus breakup in a nonlinear system of reactive miscible fluids.

We present a theoretical study on pattern formation occurring in miscible fluids reacting by a second-order reaction A+B→C in a vertical Hele-Shaw cell under constant gravity. We have recently reported that the concentration-dependent diffusion of species coupled with a frontal neutralization reaction can produce a multilayer system where low-density depleted zones could be embedded between the denser layers. This leads to the excitation of chemoconvective modes spatially separated from each other by a motionless fluid. In this Rapid Communication, we show that the layers can interact via a diffusion mechanism. Since diffusively coupled instabilities initially have different wavelengths, this causes a long-wave modulation of one pattern by another. We have developed a mathematical model which includes a system of reaction-diffusion-convection equations. The linear stability of a transient base state is studied by calculating the growth rate of the Lyapunov exponent for each unstable layer. Numerical simulations supported by phase portrait reconstruction and Fourier spectra calculation have revealed that nonlinear dynamics consistently passes through (i) a perfect spatially periodic system of chemoconvective cells, (ii) a quasiperiodic system of the same cells, and (iii) a disordered fingering structure. We show that in this system, the coordinate codirected to the reaction front paradoxically plays the role of time, time itself acts as a bifurcation parameter, and a complete spatial analog of the two-frequency torus breakup is observed.

A Comprehensive Brownian Dynamics Approach for the Determination of Non-ideality Parameters from Analytical Ultracentrifugation.

Brownian dynamics (BD) has been applied as a comprehensive tool to model sedimentation and diffusion of nanoparticles in analytical ultracentrifugation (AUC) experiments. In this article, we extend the BD algorithm by considering space-dependent diffusion and solvent compressibility. With this, the changes in the sedimentation and diffusion coefficient from altered solvent properties at increased pressures are accurately taken into account. Moreover, it is demonstrated how the concept of space-dependent diffusion is employed to describe concentration-dependent sedimentation and diffusion coefficients, in particular, through the Gralen coefficient and the second virial coefficient. The influence of thermodynamic nonideality on diffusional properties can be accurately simulated and agree with well-known evaluation tools. BD simulations for sedimentation equilibrium and sedimentation velocity (SV) AUC experiments including effects of hydrodynamic and thermodynamic nonideality are validated by global evaluation in SEDANAL. The interplay of solvent compressibility and retrieved nonideality parameters can be studied utilizing BD. Finally, the second virial coefficient is determined for lysozyme from SV AUC experiments and BD simulations and compared to membrane osmometry. These results are in line with DLVO theory. In summary, BD simulations are established for the validation of nonideal sedimentation in AUC providing a sound basis for the evaluation of complex interactions even in polydisperse systems.

Sequential Slope and Intercept Method for Estimation of Gas Absorption and Diffusion Coefficients in Binary Gas–Liquid Systems

A novel geometric method based on a sequential slope–intercept approach is derived for estimation of concentration-dependent diffusion, gas absorption, and gas dissolution concentration in gas–liquid binary systems. The gas absorption and diffusion are modeled using an inverse free boundary problem governed by Fick’s second law of diffusion and Henry’s absorption law. An unknown gas–liquid interface is governed by the Stefan’s type boundary condition. Implementation of the estimation method involves piecewise linear approximation of the transformed concentration data and sequential application of the estimated slopes and intercepts from this approximation. Application and validation of the developed estimation method is given for synthetic concentration profiles and for real concentration measurements for a dimethyl ether–bitumen binary system obtained using computer tomography. The developed estimation method overcomes the existing estimation methods derived from polynomial approximations that significantly underestimate or overestimate the diffusion coefficients at very low dissolved gas concentrations.

Experimental study of zeolitic diffusion by use of a concentration-dependent surface diffusion model

Surface diffusivity in adsorption and ion exchange processes is probably the most important property studied expensively in the literature but some aspects, especially its dependence on solid phase concentration, is still an open subject to discussion. In this study a new concentration-dependent surface diffusion model, equipped with a flexible double selectivity equilibrium relationship is applied on the removal of Pb2+, Cr3+, Fe3+ and Cu2+ from aqueous solutions using a natural zeolite. The model incorporates the Chen-Yang surface diffusivity correlation able to deal with positive and negative dependence with surface coverage. The double selectivity equilibrium relationship successfully represents the experimental equilibrium data, which follow Langmurian isotherm type for Pb2+, sigmoidal for Cr3+ and Fe3+ and linear for Cu2+. The concentration-dependent surface diffusion model was compared with the constant diffusivity surface diffusion model and found to be moderately more accurate but considerably more useful as it provides more insights into the diffusion mechanism. The application of the model resulted in an average deviation of 8.56 ± 6.74% from the experimental data and an average solid phase diffusion coefficients between 10−9 and 10−10 cm2/s. The results showed that the diffusion of metal ions in the zeolite structure is unhindered following the surface diffusion mass transfer mechanism.

On the Relation between Ionic Transport Properties and Viscoelastic Response of Solid Polymer Electrolytes

The need for high energy density and micro-scaled power sources is pushing researchers and industries towards the development of three-dimensional architecture micro-batteries and structural batteries. In this perspective, solid polymer electrolytes (SPEs) represents an appealing alternative to conventional (liquid) electrolyte. Due to their intrinsic multifunctionality, SPEs can serve as the structural matrix of fiber-reinforced composites and as the electrolyte of a battery. In addition, the improved safety features, excellent flexibility and processability, allow to cast SPEs into complex three-dimensional geometries without leakage. Experimental studies indicate that mechanical and electrochemical responses of SPEs are related. In particular, the segmental motion of the polymer molecules is crucial for both the viscoelastic properties and the ionic conductivity of the material. A limited polymer segmental motion generally results in a stiffer mechanical response and a reduced ionic conductivity, while the opposite applies when the polymer segmental motion is enhanced. This results in a trade-off between mechanical and electrical performance from a material design point of view [1] and a deformation-dependent ionic transport mechanism in the polymer during service [2]. In this work we present a coupled electrochemo-mechanical model for SPEs. The polymer is modeled as a solid through which the transport of multiple charged species takes place and is described in terms of molar concentration of ions, electric potential and displacement field. Since the molar concentration of ions in the SPE is notably non-uniform during battery service [3,4], the model explicitly accounts for concentration-dependent diffusion coefficients and viscoelastic properties. Results of a finite element method formulation are presented, and the impact of the viscoelastic mechanical behavior on the overall battery response is discussed.

Modelling phase separation in amorphous solid dispersions

Much work has been devoted to analysing thermodynamic models for solid dispersions with a view to identifying regions in the phase diagram where amorphous phase separation or drug recrystallization can occur. However, detailed partial differential equation non-equilibrium models that track the evolution of solid dispersions in time and space are lacking. Hence theoretical predictions for the timescale over which phase separation occurs in a solid dispersion are not available. In this paper, we address some of these deficiencies by (i) constructing a general multicomponent diffusion model for a dissolving solid dispersion; (ii) specializing the model to a binary drug/polymer system in storage; (iii) deriving an effective concentration dependent drug diffusion coefficient for the binary system, thereby obtaining a theoretical prediction for the timescale over which phase separation occurs; (iv) calculating the phase diagram for the Felodipine/HPMCAS system; and (iv) presenting a detailed numerical investigation of the Felodipine/HPMCAS system assuming a Flory-Huggins activity coefficient. The numerical simulations exhibit numerous interesting phenomena, such as the formation of polymer droplets and strings, Ostwald ripening/coarsening, phase inversion, and droplet-to-string transitions. A numerical simulation of the fabrication process for a solid dispersion in a hot melt extruder is also presented. Statement of SignificanceSolid dispersions are products that contain mixtures of drug and other materials e.g. polymer. These are liable to separate-out over time – a phenomenon known as phase separation. This means that it is possible the product differs both compositionally and structurally between the time of manufacture and the time it is taken by the patient, leading to poor bioavailability and so ultimately the shelf-life of the product has to be reduced.Theoretical predictions for the timescale over which phase separation occurs are not currently available. Also lacking are detailed partial differential equation non-equilibrium models that track the evolution of solid dispersions in time and space. This study addresses these issues, before presenting a detailed investigation of a particular drug-polymer system.

Cochlear pharmacokinetics - Micro-computed tomography and learning-prediction modeling for transport parameter determination

Inner ear disorders such as sensorineural deafness and genetic diseases may one day be treated with local drug delivery to the inner ear. Current pharmacokinetic models have been based on invasive methods to measure drug concentrations, limiting them in spatial resolution, and restricting the research to larger rodents. We developed an intracochlear pharmacokinetic model based on an imaging, learning-prediction (LP) paradigm for learning transport parameters in the murine cochlea. This was achieved using noninvasive micro-computed tomography imaging of the cochlea during in vivo infusion of a contrast agent at the basal end of scala tympani through a cochleostomy. Each scan was registered in 3-D to a cochlear atlas to segment the cochlear regions with high accuracy, enabling concentrations to be extracted along the length of each scala. These spatio-temporal concentration profiles were used to learn a concentration dependent diffusion coefficient, and transport parameters between the major scalae and to clearance. The LP model results are comparable to the current state of the art model, and can simulate concentrations for cases involving different infusion molecules and different drug delivery protocols. Forward simulation results with pulsatile delivery suggest the pharmacokinetic model can be used to optimize drug delivery protocols to reduce total drug delivered and the potential for toxic side effects. While developed in the challenging murine cochlea, the processes are scalable to larger animals and different drug infusion paradigms.

Hydrogen diffusivity of Nb56-xWxTi23Co21 alloys under non-dilute conditions and its analysis based on hydrogen chemical potential

Microstructure and hydrogen permeation of Nb56-xWxTi23Co21 (x = 0, 5, 10) alloys have been investigated. The alloying effect of W on the hydrogen diffusivity of Nb56-xWxTi23Co21 alloys was discussed from a perspective of the chemical potential of hydrogen under non-dilute condition. All as-cast ingots were consisted of primary bcc-(Nb) and eutectic (bcc-(Nb)+B2-TiCo) structure. The relation between the hydrogen permeation flux J and the PCT factor fPCT is linear, from which the hydrogen diffusion coefficients were evaluated. The substituting Nb by W leads to improved intrinsic hydrogen diffusion coefficient D∗. With increasing W content, the potential barrier for hydrogen atoms hopping between stable interstitial sites gradually decreases. The concentration-dependent diffusion coefficient D increases with increasing hydrogen concentration.

Interface sharpening in miscible and isotopic multilayers: Role of short-circuit diffusion

Atomic diffusion at nanometer length scale may differ significantly from bulk diffusion, and may sometimes even exhibit counterintuitive behavior. In the present work, taking Cu/Ni as a model system, a general phenomenon is reported which results in sharpening of interfaces upon thermal annealing, even in miscible systems. Anomalous x-ray reflectivity from a Cu/Ni multilayer has been used to study the evolution of interfaces with thermal annealing. Annealing at 423 K results in sharpening of interfaces by about 38%. This is the temperature at which no asymmetry exists in the inter-diffusivities of Ni and Cu. Thus, the effect is very general in nature, and is different from the one reported in the literature, which requires a large asymmetry in the diffusivities of the two constituents [Z. Erd\'elyi et al., Science 306, 1913 (2004).]. General nature of the effect is conclusively demonstrated using isotopic multilayers of 57Fe/naturalFe, in which evolution of isotopic interfaces has been observed using nuclear resonance reflectivity. It is found that annealing at suitably low temperature (e.g. 523 K) results in sharpening of the isotopic interfaces. Since chemically it is a single Fe layer, any effect associated with concentration dependent diffusivity can be ruled out. The results can be understood in terms of fast diffusion along short-circuit paths like triple junctions, which results in an effective sharpening of the interfaces at relatively low temperatures.

Concentration-dependent diffusion kinetics of Ti-Al binary system at elevated temperatures: Experiments and modeling

The aim of the present work is to investigate the dependence of concentration gradient on diffusion kinetics of Ti-Al binary system at high temperatures, which is a long-standing physical problem that cannot easily addressed by tradition diffusion couple experiments due to the ultralow melting point of Al and complicated compositions of various titanium aluminides. We conceived and established a simplified Fick-based diffusion model, making use of successive concentration gradient, in combination with experiments, to reveal the concentration-governed diffusion behavior. Our results showed an excellent agreement with currently available diffusion models. Therefore this approach was expected to be extensively applied to other binary system with large deviations in the melting points.

Water Uptake and Loss in Viscous Aerosol Particles with Concentration-Dependent Diffusivities.

An accurate understanding of the equilibration timescale of organic aerosol particles with surrounding water vapor is difficult because of the strong concentration-dependent diffusivities that are present in these systems. We examine this problem along with the closely related problem of the time-dependent radius of a binary aerosol particle during the uptake or loss of water. The governing equations and boundary conditions are discussed and a boundary value problem is formulated and solved. The resulting expressions are applied to water uptake and loss in two systems of atmospheric importance: aqueous-inorganic particles and high-viscosity organic particles. Accuracy is evaluated through a comparison with numerical solutions. For particles whose diffusivity has a strong dependence on water concentration and whose viscosity remains above 1 Pa·s during water uptake or loss, the expression for the characteristic equilibration time is found to be in excellent agreement with numerical results. Moreover, it provides physical insights into mass transport processes.

Drug Delivery From Polymer-Based Nanopharmaceuticals-An Experimental Study Complemented by Simulations of Selected Diffusion Processes.

The success of medical therapy depends on the correct amount and the appropriate delivery of the required drugs for treatment. By using biodegradable polymers a drug delivery over a time span of weeks or even months is made possible. This opens up a variety of strategies for better medication. The drug is embedded in a biodegradable polymer (the “carrier”) and injected in a particular position of the human body. As a consequence of the interplay between the diffusion process and the degrading polymer the drug is released in a controlled manner. In this work we study the controlled release of medication experimentally by measuring the delivered amount of drug within a cylindrical shell over a long time interval into the body fluid. Moreover, a simple continuum model of the Fickean type is initially proposed and solved in closed-form. It is used for simulating some of the observed release processes for this type of carrier and takes the geometry of the drug container explicitly into account. By comparing the measurement data and the model predictions diffusion coefficients are obtained. It turns out that within this simple model the coefficients change over time. This contradicts the idea that diffusion coefficients are constants independent of the considered geometry. The model is therefore extended by taking an additional absorption term into account leading to a concentration dependent diffusion coefficient. This could now be used for further predictions of drug release in carriers of different shape. For a better understanding of the complex diffusion and degradation phenomena the underlying physics is discussed in detail and even more sophisticated models involving different degradation and mass transport phenomena are proposed for future work and study.

TiO2 Facets Shaped by Concentration-Dependent Surface Diffusion of Dopamine.

Facet engineering highlights the fundamental understanding of kinetic growth of facets with capping agents. Here, we provide a roadmap for modulating TiO2 facets using dopamine as a capping agent inspired by density functional theory calculations and molecular dynamics simulations. Our calculations revealed that the surface diffusion of dopamine and their facet-specific affinity direct the kinetic growth of TiO2{100} and {101} facets into a nonequilibrium crystal shape. Our TiO2 synthesis agreed well with the theoretical predictions, suggesting that the concentration-dependent diffusion is central in accurately tuning a desirable ratio of mixed facets. Our findings shed new light on the diffusion-limited kinetically controlled facet growth mechanism, and this fine-tuning of mixed facets on a single crystal provides a general approach to design and fabricate facets on metal oxides.

Simulation of multicomponent gas transport through mixed-matrix membranes

We extend the Maxwell-Stefan (M-S) formulation of irreversible thermodynamics to multicomponent transport in mixed-matrix membranes (MMMs), using a simulation-based rigorous modeling approach (SMA) through finite-element method (FEM) solution of the three-dimensional (3-d) transport problem in full-scale MMMs. In the new approach, we generalize the dual-mode/partial immobilization (DM/PI) theory for the local permeability in glassy polymers to describe multicomponent permeation in pure glassy polymer membranes and MMMs, by reformulating the M-S constitutive equations in the Onsager formalism considering concentration-dependent transport diffusivities and non-uniform concentration gradients across the MMM. In this way, the new M-S formulation explicitly considers effects of intrinsic MMM features such as finite filler particle size and isotherm nonlinearity in the MMM constituent phases, as well as mixture-related effects, such as competitive adsorption and friction amongst permeants, on the calculation of the mixture fluxes (permeabilities). This is achieved without introduction of empirical fitting parameters in the MMM permeability calculation and only requiring single-gas experimental or simulation-based adsorption and permeation data on the individual MMM materials to predict the mixture perm-selectivity in the MMM as a whole. Further, we validate the new approach by using available experimental permeation data for the separation of an equimolar binary mixture of propylene (C3H6) and propane (C3H8) in ZIF‑8/PIM-6FDA-OH MMMs, with the rigorous simulation results showing very good agreement with both experimental single and mixed-gas permeabilities and perm-selectivities.

Ab initio molecular dynamics simulation of supercritical aqueous ionic solutions: Spectral diffusion of water in the vicinity of Br− and I− ions

Supercritical aqueous ionic solutions containing bromide (Br−) and iodide (I−) ions have been studied using Car-Parrinello molecular dynamics simulation at two different concentrations and at two different densities. The time dependent trajectory has been used for calculating vibrational spectral diffusion, structural properties and, for other dynamical properties such as hydrogen bond dynamics, orientational relaxation and residence dynamics. The vibrational spectral diffusion of fluctuating OD stretch mode inside the solvation shell of the ions and in the bulk has been studied using frequency-frequency time correlation function. Hydrogen bond dynamics, orientational relaxation and residence dynamics of Sc-water have been investigated using population correlation function approach. The water molecules inside the solvation shell of Br− and I− ions are found to show complex molecular dynamics in comparison to the water-water dynamics. The concentration dependent spectral diffusion has been observed and it occurs in multiple time scales in the solvation shell of Br− and I− ions. At supercritical condition, the ion-water hydrogen bond shows faster dynamics with respect to the same at ambient condition and slower dynamics with respect to the water-water hydrogen bond dynamics at supercritical condition.

Effect of Velocity and Diffusion Functionality on Nonlinear Mass Transfer Mechanisms in Solvent Oil Recovery

Literature has indicated that, experimentally, solvent fronts in hybrid solvent recovery processes progress more rapidly than what can be predicted using current approximations and more rapidly than thermal processes alone. Research using finite differences to model the nonlinear advection, diffusion and dispersion (ADD) equation suggests that nonlinear mass transfer effects are important in predicting the rate of solvent advance. Nonlinearities can be ascribed to both diffusion and flow velocity functionality. Earlier work using linear concentration dependent diffusion and log-linear velocity behaviour confirmed the importance of nonlinear effects when compared to linear theory that uses constant diffusion, dispersion and velocity coefficients. The mathematical nature of the nonlinear ADD equation further suggests that the shape of concentration dependent diffusion and flow velocity will affect the shape of the solvent concentration profiles, and influence the rate of propagation of the solvent front. This research focuses on results obtained using finite differences to explore the effects of various diffusion and velocity functionalities that affect the solvent rate propagation using a nonlinear ADD equation. The results obtained from this analysis indicate that these functionalities determine the shape of the solvent concentration profile. The concentration dependent diffusion and velocity functions were chosen according to recent literature which proposes experimentally obtained functions to more accurately model solvent penetration in the media. Preliminary results from this study suggest that the velocity functionality has more influence on the process at both the lab and field scales for the parameters considered in this study. The shapes of the concentration profiles are affected by both diffusion functionality and velocity functionality.