Intrinsic Diffusion Coefficients Research Articles

An experimental estimation method of diffusion coefficients in ternary and multicomponent systems from a single diffusion profile

Until recently, it was textbook knowledge that the diffusion coefficients could not be estimated in a multi-component system following the widely practised diffusion couple method. The recently proposed constrained diffusion couple method largely solved the problems. The need to produce two intersecting diffusion paths in a multi-component space with very well-controlled compositions of the diffusion couple end members may be tricky in certain situations, depending on the complications of diffusion paths. In this study, we have shown the estimation of all types of diffusion coefficients from a single diffusion profile without neglecting Onsager's cross terms in concentrated ternary Ni-Co-Fe, Fe-rich quaternary Fe-Ni-Co-Cr and concentrated Ni-Co-Fe-Cr-Al quinary alloys. This is applicable in ternary and multicomponent component systems (n≥3) at a stable Kirkendall marker plane position. One can even estimate the impurity diffusion coefficients utilizing the composition profiles at the ends of the diffusion couples following the Hall method, which has been rarely practised in multicomponent systems until now. We have further demonstrated the importance of estimating the tracer and intrinsic diffusion coefficients in a concentrated or multi-principal element alloy in which interdiffusion coefficients can be vague and misleading for understanding the elements' diffusional interactions and relative mobilities. We have also shown the importance of considering the vacancy wind effect in concentrated alloys. The method proposed in this study will help to generate a mobility database with relative ease in various multicomponent systems important for various applications, which was considered impossible until recently.

A scaling rule for power output of salt hydrate tablets for thermochemical energy storage

Salt hydrates are thermochemical materials capable of storing and releasing heat through reversible reaction with water vapor. In a heat battery, salt hydrate tablets of millimeter size are necessary to ensure a sufficient permeability of the packed bed. A profound understanding of the hydration process of these tablets is required to improve their kinetic performance. In this study we show that the hydration timescale of salt tablets is transport limited and that it depends primarily on the porosity and on the driving force (Δp). From gravimetric measurements done on SrBr2·6H2O and CaC2O4 we derived the intrinsic reaction and effective diffusion coefficients (k and Deff) and found that they validate a front-diffusion limited hydration hypothesis. In particular, the obtained Deff values (0.8–4.5 mm2 s−1) only depend on the tablets' porosities. Based on these parameters, we calculated the second Damköhler number (DaII) and proved that many other hydration reactions are diffusion limited. In the case of identical structures, the power output is therefore controlled only by the driving force. Its variation could be predicted by calculation of a so-called power scaling factor (Λ) for a selection of salts. This power scaling factor depends on the enthalpy (ΔH) and entropy (ΔS) of the reaction. For a temperature output of 40 °C and at 12 mbar most hydration reactions fall in the interval 0<Λ<30 and Λ exceeds 30 only in very few cases. This parameter establishes therefore another important constraint to the selection of the most ideal salt. Suitable strategies to circumvent the diffusion limitation will lead to the development of next generation salt hydrate tablets for thermochemical energy storage.

Intrinsic Water Transport in Moisture-Capturing Hydrogels.

Moisture-capturing hydrogels have emerged as attractive sorbent materials capable of converting ambient humidity into liquid water. Recent works have demonstrated exceptional water capture capabilities of hydrogels while simultaneously exploring different strategies to accelerate water capture and release. However, on the material level, an understanding of the intrinsic transport properties of moisture-capturing hydrogels is currently missing, which hinders their rational design. In this work, we combine absorption and desorption experiments of macroscopic hydrogel samples in pure vapor with models of water diffusion in the hydrogels to demonstrate the first measurements of the intrinsic water diffusion coefficient in hydrogel-salt composites. Based on these insights, we pattern hydrogels with micropores to significantly decrease the required absorption and desorption times by 19% and 72%, respectively, while reducing the total water capacity of the hydrogel by only 4%. Thereby, we provide an effective strategy toward hydrogel material optimization, with a particular significance in pure-vapor environments.

Quantifying the energy landscape in weakly and strongly disordered frictional media.

We investigate the "roughness" of the energy landscape of a system that diffuses in a heterogeneous medium with a random position-dependent friction coefficient α(x). This random friction acting on the system stems from spatial inhomogeneity in the surrounding medium and is modeled using the generalized Caldira-Leggett model. For a weakly disordered medium exhibiting a Gaussian random diffusivity D(x) = kBT/α(x) characterized by its average value ⟨D(x)⟩ and a pair-correlation function ⟨D(x1)D(x2)⟩, we find that the renormalized intrinsic diffusion coefficient is lower than the average one due to the fluctuations in diffusivity. The induced weak internal friction leads to increased roughness in the energy landscape. When applying this idea to diffusive motion in liquid water, the dissociation energy for a hydrogen bond gradually approaches experimental findings as fluctuation parameters increase. Conversely, for a strongly disordered medium (i.e., ultrafast-folding proteins), the energy landscape ranges from a few to a few kcal/mol, depending on the strength of the disorder. By fitting protein folding dynamics to the escape process from a metastable potential, the decreased escape rate conceptualizes the role of strong internal friction. Studying the energy landscape in complex systems is helpful because it has implications for the dynamics of biological, soft, and active matter systems.

Diffusion simulation based on atomistic model and mobility determination via Kirkendall experiment for multi-component systems

Currently the multi-component diffusion kinetic databases, one of the key inputs for successful predictions of microstructural evolution, are by no means reliable since they are developed mostly by only seeking interdiffusion properties via diffusion-couple experiments but without considering the indispensable tracer or intrinsic diffusion coefficients. Estimating interdiffusion coefficients becomes challenging with the increasing number of components. In the present work, a mobility-extraction method is presented to directly determine atomic mobilities, which is in turn used to develop kinetic databases and then calculate all sorts of diffusion coefficients. It requires only one single-phase diffusion couple with Kirkendall markers to obtain the atomic mobilities in the marker position, with no limit on the number of components. This novel idea is realized on the basis of a new simulation scheme to solve diffusion equations in line with the atomistic model, which takes the Kirkendall effect into account intrinsically and efficiently.

Estimation of diffusion coefficients in NiCoFeCrAl multi-principal element alloy following an inventive design strategy of diffusion couples

An inventive design strategy of producing the diffusion couples for intersecting the diffusion paths in multicomponent space is described in Ni-Co-Fe-Cr-Al multi-principal element alloy. This, for the first time, facilitates the estimation of the tracer, intrinsic, and interdiffusion coefficients of all the elements purely experimentally in this quinary system. This was not possible until now following any other experimental methods. The tracer diffusion coefficients estimated in this quinary system are compared with the impurity and tracer diffusion coefficients in pure Ni, and at (or near) equiatomic compositions of binary Ni-Co, ternary Ni-Co-Cr, and quaternary Ni-Co-Fe-Cr systems. The relative mobilities of the elements are found to follow the trend DNi*≤DCo*<DFe*≤DCr*<DAl* in all the systems. However, the tracer diffusion coefficients of the elements first decrease and then increase with the increasing order of the systems. The contribution of the vacancy wind effect on certain cross-intrinsic diffusion coefficients is found to be very significant and cannot be ignored. The influence of the vacancy wind effect on interdiffusion coefficients is less significant than the intrinsic diffusion coefficients since interdiffusion coefficients are the average of multiple intrinsic diffusion coefficients. This may lead to the wrong statement that the vacancy wind effects are negligible. We have further shown that describing the relative diffusion rates of the elements with main interdiffusion coefficients instead of the tracer diffusion coefficients can be misleading or confusing. Different elements as the dependent variables of interdiffusion coefficients indicate a very different or even opposite trend of relative diffusivities in this system.

A novel method to design Nb(W)–Ti–Co(Fe) alloy membrane with high hydrogen permeability and strong resistance to hydrogen embrittlement

A hydrogen permeable Nb43W5Ti27Co20Fe5 alloy membrane was designed and its microstructure and hydrogen transportation performances were systematically studied. W mainly dissolves into primary body-centered cubic (bcc)-(Nb) phases, while Fe is captured in eutectic structures. Substituting Nb with W leads to a substantial reduction in hydrogen solubility and substituting Co with Fe negligibly affects hydrogen solubility. The W and Fe substitutions contribute to an enhanced intrinsic hydrogen diffusion coefficient D∗. The designed Nb43W5Ti27Co20Fe5 alloy membrane exhibits lower hydrogen solubility and higher hydrogen permeability at temperatures below 623 K than that of the original Nb48Ti27Co25 alloy membrane. No hydrogen-induced failures were found for the designed Nb43W5Ti27Co20Fe5 alloy membrane during cooling to room temperature and under long-term hydrogen permeation at 673 K for 72 h, whereas the original alloy membrane failed at 444 K during cooling to room temperature and after 19 h during hydrogen permeation at 673 K. The designed Nb43W5Ti27Co20Fe5 alloy membrane exhibits high hydrogen permeability and strong resistance to hydrogen embrittlement.

Utilising physics-informed neural networks for optimisation of diffusion coefficients in pseudo-binary diffusion couples

ABSTRACT We propose a numerical inverse method based on physics-informed neural networks (PINN) for calculating composition-dependent diffusion coefficients in pseudo-binary (PB) diffusion couples in multicomponent alloys. Traditional methods rely solely on experimental diffusion profiles as design targets, which can lead to unreliable estimates. In contrast, PINN uses a combination of available data and physics-based constraints to obtain optimised design parameters and exact solutions for constrained governing differential equations. The constraints in PINN include governing partial differential equations, initial and boundary conditions, and any other equality/inequality relations obeyed by physical parameters. Our study shows the necessity of experimentally estimated intrinsic diffusion coefficients for the prediction of reliable composition-dependent mobility data. In the absence of such data, different combinations of unknown intrinsic diffusion coefficients can also produce reasonable approximations of diffusion profiles and interdiffusion coefficients while wrongly predicting more fundamental quantities (i.e. intrinsic diffusivities). Our method utilises PINN to simultaneously obtain optimised design parameters (diffusion coefficients) and exact solutions for governing diffusion equations. The implementation of PINN uses experimentally estimated diffusion coefficients obtained by the PB diffusion couple method, in addition to diffusion profiles, as design targets. The method is extended to non-ideal PB diffusion profiles, such as conventional diffusion couples in which all the components develop diffusion profiles, by incorporating additional constraints such as zero composition gradient of certain component(s) and experimentally estimated interdiffusion coefficients at extrema in the composition profiles. PINN is found to be a promising approach for obtaining reliable estimates of diffusion coefficients in multicomponent alloys.

Quantitative estimation of coarsening behavior of L12 domains in Ni–Al–V alloys by a first-principles calculation

Ni–Al–V based alloys possess superior creep resistance at higher temperatures because of the sluggish interdiffusion of constituent elements within the L12-Ni3(Al, V) matrix. In this study, intrinsic diffusion coefficients of Ni, Al and V in quasi-binary Ni3Al1-xVx (x = 0, 0.125, 0.25, 0.5) were quantified based on a first-principles calculation for elucidating the effect of V concentration on the coarsening behavior of L12 domains. From the results of the estimated formation energies, migration energies and activation energies, it was found that intrinsic diffusion coefficients of Ni and Al are decreased by the addition of V, while intrinsic diffusion coefficient of V remains two or three orders of magnitude smaller than them. Such a suppressed diffusion of Ni and Al by V addition well explains the experimentally obtained lower coarsening rates of L12 domains in Ni3Al0.58V0.42 and Ni3Al0.48V0.52 alloys.

Assessment of Thermodynamic and Structural Properties of Al–Er liquid Alloy at Different Temperatures

Theoretical expressions of quasi–lattice model were used to assess the thermodynamic and structural properties of Al–Er liquid alloy at different temperatures. The model fit parameters were optimised using the available literature data of the thermodynamic properties for the system at 1873 K. The computed values of thermodynamic properties such as excess Gibbs free energy of mixing, enthalpy of mixing, entropy of mixing and activities of Er and Al using the model fit parameters were found to be consistent with the reference data–set. Therefore, the same parameters were used to compute the structural properties such as concentration fluctuation in long wavelength limit, Warren–Cowley short range–order parameter and ratio of mutual to intrinsic diffusion coefficients. Assuming the model fit parameters to be linear temperature–dependent, the above mentioned properties were also assessed in the temperature range 1873–2173 K. The compound forming tendency of the system was found to decrease gradually with increase in temperature.

Space–time upscaling of reactive transport in porous media

Reactive transport in saturated/unsaturated porous media is numerically upscaled to the space–time scale of a hypothetical measurement through coarse-grained space–time (CGST) averages. The reactive transport is modeled at the fine-grained Darcy scale by the actual number of molecules involved in reactions which undergo advective and diffusive movements described by global random walk (GRW) simulations. The CGST averages verify identities similar to a local balance equation which allow us to derive expressions for the flow velocity and the intrinsic diffusion coefficient in terms of averaged microscopic quantities. The latter are further used to verify the CGST-GRW numerical approach. The upscaling approach is applied to biodegradation processes in saturated aquifers and variably saturated soils and the CGST averages are compared to classical volume averages. One finds that if the process is characterized by slow variations in time, as in homogeneous reaction systems, the differences between the two averages are negligible. Instead, the differences are significant and can be extremely large in simulations of time-dependent biodegradation processes in both variably saturated soils and saturated aquifers.

Method for extracting the intrinsic diffusion coefficient from grain boundary diffusion depth profile

Numerical research is done on grain boundary diffusion of Cu-Ni bilayer thin films. Together, the composition-dependent interdiffusion coefficient and the position of the grain boundary are considered. Fick’s second law provides a quantitative assessment of the effects of grain size, volume, grain boundary diffusion coefficient, diffusion temperature, and time on the depth profile. Regarding the situation where the plateau values of the Ni concentration in the Cu layer and the Cu concentration in the Ni layer are almost the same for the case where the Ni intrinsic diffusion coefficient is significantly bigger than that for Cu, a detailed explanation is provided. We may produce a profile that is like the direct volume diffusion profile with a relative inaccuracy of 7.53% by using a method that is provided to eliminate the impacts of grain boundaries and grain size on the grain boundary diffusion profile. For most of the circumstances covered in this study, the technique is effective. The transformation of the grain boundary diffusion profile yields a composition-dependent interdiffusion coefficient, and the calculation’s outcome is consistent with the theoretical conclusion drawn from the Darken equation. Thin films of Cu (150 nm)/Ni (150 nm)/SiOx/Si (100 nm) were produced as examples, and Auger electron spectroscopy was used to determine the respective depth profiles. The calculated diffusion parameters, namely, the pre-exponential factor and the activation energy, are well within the bounds of the available data.

Mechanism of tungsten strengthening hydrogen transportation in Nb48Ti27Co25 hydrogen permeable alloy membrane

Substitution of Nb by W in Nb48Ti27Co25 dual phase hydrogen permeable alloy was carried out. The elemental W replacing Nb mainly dissolved into the primary body-centered cubic (bcc)-(Nb) phase to modify its composition and crystal lattice. The W alloying effect on hydrogen transportation was systematically discussed based on hydrogen chemical potentials and first principle calculations. The substitution of Nb by W reduced the hydrogen solubility, and led to an improved intrinsic hydrogen diffusion coefficient, especially at temperature below 573 K, which promoted the hydrogen permeation flux of Nb48Ti27Co25 alloy membrane at 523 K. According to the theoretical study based on density function theory of the Nb–W system, the reduced hydrogen solubility can be attributed to the weakened binding strength between hydrogen atoms and the bcc-(Nb) lattice and the reduced charge gained by hydrogen atoms after W substitution for Nb. In addition, the W occupying the Nb sublattice can change the diffusion path of hydrogen atoms in the bcc-(Nb) lattice, thereby reducing the diffusion energy barrier for high hydrogen diffusivity.

Equation governing the probability density evolution of multi-dimensional linear fractional differential systems subject to Gaussian white noise

Stochastic fractional differential systems are important and useful in the mathematics, physics, and engineering fields. However, the determination of their probabilistic responses is difficult due to their non-Markovian property. The recently developed globally-evolving-based generalized density evolution equation (GE-GDEE), which is a unified partial differential equation (PDE) governing the transient probability density function (PDF) of a generic path-continuous process, including non-Markovian ones, provides a feasible tool to solve this problem. In the paper, the GE-GDEE for multi-dimensional linear fractional differential systems subject to Gaussian white noise is established. In particular, it is proved that in the GE-GDEE corresponding to the state-quantities of interest, the intrinsic drift coefficient is a time-varying linear function, and can be analytically determined. In this sense, an alternative low-dimensional equivalent linear integer-order differential system with exact closed-form coefficients for the original high-dimensional linear fractional differential system can be constructed such that their transient PDFs are identical. Specifically, for a multi-dimensional linear fractional differential system, if only one or two quantities are of interest, GE-GDEE is only in one or two dimensions, and the surrogate system would be a one- or two-dimensional linear integer-order system. Several examples are studied to assess the merit of the proposed method. Though presently the closed-form intrinsic drift coefficient is only available for linear stochastic fractional differential systems, the findings in the present paper provide a remarkable demonstration on the existence and eligibility of GE-GDEE for the case that the original high-dimensional system itself is non-Markovian, and provide insights for the physical-mechanism-informed determination of intrinsic drift and diffusion coefficients of GE-GDEE of more generic complex nonlinear systems.

Conformational Parameters and Hydrodynamic Behavior of Poly(2-Methyl-2-Oxazoline) in a Broad Molar Mass Range.

In this work, we report our results on the hydrodynamic behavior of poly(2-methyl-2-oxazoline) (PMeOx). PMeOx is gaining significant attention for use as hydrophilic polymer in pharmaceutical carriers as an alternative for the commonly used poly(ethylene glycol) (PEG), for which antibodies are found in a significant fraction of the human population. The main focus of the current study is to determine the hydrodynamic characteristics of PMeOx under physiological conditions, which serves as basis for better understanding of the use of PMeOx in pharmaceutical applications. This goal was achieved by studying PMeOx solutions in phosphate-buffered saline (PBS) as a solvent at 37 °C. This study was performed based on two series of PMeOx samples; one series is synthesized by conventional living cationic ring-opening polymerization, which is limited by the maximum chain length that can be achieved, and a second series is obtained by an alternative synthesis strategy based on acetylation of well-defined linear poly(ethylene imine) (PEI) prepared by controlled side-chain hydrolysis of a defined high molar mass of poly(2-ethyl-2-oxazoline). The combination of these two series of PMeOx allowed the determination of the Kuhn-Mark-Houwink-Sakurada equations in a broad molar mass range. For intrinsic viscosity, sedimentation and diffusion coefficients, the following expressions were obtained: η=0.015M0.77, s0=0.019M0.42 and D0=2600M-0.58, respectively. As a result, it can be concluded that the phosphate-buffered saline buffer at 37 °C represents a thermodynamically good solvent for PMeOx, based on the scaling indices of the equations. The conformational parameters for PMeOx chains were also determined, revealing an equilibrium rigidity or Kuhn segment length, (A) of 1.7 nm and a polymer chain diameter (d) of 0.4 nm. The obtained value for the equilibrium rigidity is very similar to the reported values for other hydrophilic polymers, such as PEG, poly(vinylpyrrolidone) and poly(2-ethyl-2-oxazoline), making PMeOx a relevant alternative to PEG.

Assessing the effect of moderately increasing medium viscosity on the intrinsic dissolution rate and diffusion coefficient of ibuprofen particles

This study investigated the effects of moderately increasing medium viscosity through two viscosity enhancing agents (VEA) (HPMC and sucrose) on the intrinsic dissolution of a BCS class II drug (ibuprofen). The differences in intrinsic dissolution were characterized through the estimation of a diffusion coefficient (D) of ibuprofen in different media with the Levich equation. Increasing viscosity decreased the intrinsic dissolution rate of ibuprofen with both VEAs. The calculated D from intrinsic dissolution data overestimated the experimental value, but the results suggest that intrinsic dissolution could be used as a tool to estimate a relative change in D in different viscosities.&amp;nbsp;

Diffusion Coefficient of Intracrystalline Water in Intrinsic Hen Egg-White Lysozyme Crystals Determined by Confocal Raman Spectroscopy.

Protein crystals composed of protein molecules are expected as a novel porous material. They have high porosity, and the knowledge of the diffusion of intracrystalline water is important. In this study, the diffusion coefficient of intracrystalline water in intrinsic hen egg-white lysozyme (HEWL) crystals was determined by a method that combines confocal Raman spectroscopy and air convection with controlled relative humidity. Similar to common porous materials, the drying process of the protein crystals includes three periods: constant-rate drying, falling-rate drying, and equilibrium state. During the falling-rate drying period, the drying rate depends on the diffusion of intracrystalline water in the protein crystal. The gradient of the water content was measured using confocal Raman spectroscopy. The diffusion coefficient of the intrinsic HEWL crystals was determined as 3.1 × 10-7 cm2/s with a water content of 36.3 vol %. The estimated diffusion coefficients of the intrinsic HEWL crystals without cross-linking were in close agreement with those of the cross-linked protein crystals. This study is timely as the knowledge of the intrinsic diffusion coefficient is useful not only for understanding the mechanism of hydration of proteins but also in practical applications such as porous materials, drug binding, and cryoprotectant soaks.

Tracer diffusion under a concentration gradient: A pathway for a consistent development of mobility databases in multicomponent alloys

Diffusion in the CoCrFeMnNi high-entropy alloy is investigated using an augmented tracer / interdiffusion couple approach. The tracer diffusion coefficients are measured along the diffusion path developed between the Co17Cr20Fe20Mn20Ni23 and Co23Cr20Fe20Mn20Ni17 end-members allowing an elaboration of a consistent composition-dependent diffusion database. Both tracer and chemical concentration profiles are successfully reproduced to a tolerable level using a pair-diffusion model avoiding the reference element concept. Up-hill diffusion of Co in the experimental profile is not reproduced by the existing kinetic and thermodynamic databases. The reasons for this mismatch are discussed. The calculation of intrinsic and interdiffusion coefficients from the estimated tracer diffusion coefficients utilizing the thermodynamic parameters is demonstrated. The calculated values of the interdiffusion coefficients produced a reasonable match with the interdiffusion flux profiles developed in the multicomponent diffusion couple. The calculation of the vacancy wind effect indicates its strong influence on several intrinsic cross diffusion coefficients.

Comparative Analysis of LiMPO4 (M = Fe, Co, Cr, Mn, V) as Cathode Materials for Lithium-Ion Battery Applications-A First-Principle-Based Theoretical Approach.

The rapidly increasing demand for energy storage has been consistently driving the exploration of different materials for Li-ion batteries, where the olivine lithium-metal phosphates (LiMPO4) are considered one of the most potential candidates for cathode-electrode design. In this context, the work presents an extensive comparative theoretical study of the electrochemical and electrical properties of iron (Fe)-, cobalt (Co)-, manganese (Mn)-, chromium (Cr)-, and vanadium (V)-based LiMPO4 materials for cathode design in lithium (Li)-ion battery applications, using the density-functional-theory (DFT)-based first-principle-calculation approach. The work emphasized different material and performance aspects of the cathode design, including the cohesive energy of the material, Li-intercalation energy in olivine structure, and intrinsic diffusion coefficient across the Li channel, as well as equilibrium potential and open-circuit potential at different charge-states of Li-ion batteries. The results indicate the specification of the metal atom significantly influences the Li diffusion across the olivine structure and the overall energetics of different LiMPO4. In this context, a clear correlation between the structural and electrochemical properties has been demonstrated in different LiMPO4. The key findings offer significant theoretical and design-level insight for estimating the performance of studied LiMPO4-based Li-ion batteries while interfacing with different application areas.

Intrinsic Carrier Diffusion in Perovskite Thin Films Uncovered by Transient Reflectance Spectroscopy.

Carrier diffusion and surface recombination are key processes influencing the performance of conventional semiconductor devices. However, the interplay of photon recycling together with these processes in halide perovskites obfuscates our understanding. Herein, we discern these inherent processes in a thin FAPbBr3 perovskite single crystal (PSC) utilizing a unique transient reflectance technique that allows accurate diffusion modeling with clear boundary conditions. Temperature-dependent measurements reveal the coexistence of shallow and deep traps at the surface. The inverse quadratic dependence of temperature on carrier mobility μ suggests an underlying scattering mechanism arising from the anharmonic deformation of the PbBr6 cage. Our findings ascertain the fundamental limits of the intrinsic surface recombination velocity (S) and carrier diffusion coefficient (D) in PSC samples. Importantly, these insights will help resolve the ongoing debate and clarify the ambiguity surrounding the contributions of photon recycling and carrier diffusion in perovskite optoelectronics.