Concentration Dependence Of Diffusion Coefficient Research Articles

Physics-based modeling of sodium-ion batteries part II. Model and validation

Sodium-ion batteries (SIBs) have recently been proclaimed as the frontrunner 'post lithium' energy storage technology. This is because SIBs share similar performance metrics with lithium-ion batteries, and sodium is 1000 times more abundant than lithium. In order to understand the electrochemical characteristics of SIBs and improve present-day designs, physics-based models are necessary. Herein, a physics-based, pseudo-two-dimensional (P2D) model is introduced for SIBs for the first time. The P2D SIB model is based on Na3V2(PO4)2F3 (NVPF) and hard carbon (HC) as positive and negative electrodes, respectively. Charge transfer in the NVPF and HC electrodes is described by concentration-dependent diffusion coefficients and kinetic rate constants. Parametrization of the model is based on experimental data and genetic algorithm optimization. It is shown that the model is highly accurate in predicting the discharge profiles of full cell HC//NVPF SIBs. In addition, internal battery states, such as the individual electrode potentials and concentrations, can be obtained from the model at applied currents. Several key challenges in both electrodes and the electrolyte are herein unraveled, and useful design considerations to improve the performance of SIBs are highlighted.

Physical Origin of the Differential Voltage Minimum Associated With Lithium Plating in Li-Ion Batteries

The main barrier to fast charging of Li-ion batteries at low temperatures is the risk of short-circuiting due to lithium plating. In-situ detection of Li plating is highly sought after in order to develop fast charging strategies that avoid plating. It is widely believed [1,2,3] that Li plating after a single fast charge can be detected and quantified by using a minimum in the differential voltage (DV) signal during the subsequent discharge, which indicates how much lithium has been stripped. In this work, [4] a pseudo-2D physics-based model is used to investigate the effect on Li plating and stripping of concentration-dependent diffusion coefficients [5] in the active electrode materials. A new modelling protocol is also proposed, in order to distinguish the effects of fast charging, slow charging and Li plating/stripping. The model predicts that the DV minimum associated with Li stripping is in fact a shifted and more abrupt version of a minimum caused by the stage II-stage III transition in the graphite negative electrode. Therefore, the minimum cannot be used to quantify stripping. Using concentration-dependent diffusion coefficients yields qualitatively different results to previous work [2,3]. This knowledge casts doubt on the utility of DV analysis for detecting Li plating.[1] Mathias Petzl and Michael A. Danzer, J.Power Sources vol. 254, pp. 80-87, 2014.[2] Xiao-Guang Yang et al., J. Power Sources vol. 395, pp. 251-261, 2018.[3] Dongsheng Ren et al., J. Electrochem. Soc. vol. 165 (10), pp. A2167-A2178, 2018.[4] Simon E. J. O’Kane et al., J. Electrochem. Soc. vol. 167, pp. 090540, 2020.[5] Madeleine Ecker et al., J. Electrochem. Soc. vol. 162 (9), pp. A1836-A1848, 2015.[6] Ian D. Campbell et al., J. Electrochem. Soc. vol. 166 (4), pp. A725-A739, 2019. Figure 1

Protein Transport upon Advection at the Air/Water Interface: When Charge Matters.

The formation of dense protein interfacial layers at a free air-water interface is known to result from both diffusion and advection. Furthermore, protein interactions in concentrated phases are strongly dependent on their overall positive or negative net charge, which is controlled by the solution pH. As a consequence, an interesting question is whether the presence of an advection flow of water toward the interface during protein adsorption produces different kinetics and interfacial structure of the adsorbed layer, depending on the net charge of the involved proteins and, possibly, on the sign of this charge. Here we test a combination of the following parameters using ovalbumin and lysozyme as model proteins: positive or negative net charge and the presence or absence of advection flow. The formation and the organization of the interfacial layers are studied by neutron reflectivity and null-ellipsometry measurements. We show that the combined effect of a positive charge of lysozyme and ovalbumin and the presence of advection flow does induce the formation of interfacial multilayers. Conversely, negatively charged ovalbumin forms monolayers, whether advection flow is present or not. We show that an advection/diffusion model cannot correctly describe the adsorption kinetics of multilayers, even in the hypothesis of a concentration-dependent diffusion coefficient as in colloidal filtration, for instance. Still, it is clear that advection is a necessary condition for making multilayers through a mechanism that remains to be determined, which paves the way for future research.

Characterisation and Modelling of Moisture Gradients in Polyamide 6.

Polyamide 6 (PA6) is able to absorb water from the surrounding air and bond to it by forming hydrogen bonds between the carbonamide groups of its molecular chains. Diffusion processes cause locally different water concentrations in the (component) cross-section during the sorption process, resulting in locally different mechanical properties due to the water-induced plasticisation effect. However, the water content of PA6 is usually specified as an integral value, so no information about a local water distribution within a component is provided. This paper shows a method to characterise moisture distributions within PA6 samples using low-energy computer tomography (CT) techniques and comparing the reconstructed results with a developed finite elements (FE) modelling method based on Fick’s diffusion laws with concentration-dependent diffusion coefficients. For this purpose, the ageing of the samples at two different water bath temperatures as well as at different integral water contents are considered. The results obtained by CT reconstruction and FE modelling are in very good agreement, so that the concentration distributions by water sorption of PA6 calculated by FEM can be regarded as validated.

Techniques of Tracer Diffusion Measurements in Metals, Alloys and Compounds

Tracer diffusion is one of most reliable techniques for providing basic kinetic data in solids. In the present review, selected direct methods, in particular the radiotracer measurements as a superior technique due to its high sensitivity, Secondary-Ion-Mass-Spectroscopy (SIMS) profiling, X-Ray Diffraction measurements and Rutherford Backscattering Spectrometry are presented and discussed. Special attention is put on the radiotracer technique describing the currently used sectioning techniques in detail with a focus on the experimental applications and complications. The relevant experimental results are exemplary shown. Furthermore, the most recent developments and advances related to the combined tracer/inter-diffusion measurements are highlighted. It is shown that this approach offers possibilities to provide the concentration-dependent tracer diffusion coefficients of the constituting elements in multi-component alloys in high-throughput experiments. Possibilities of estimating the tracer diffusion coefficients following different types of diffusion couple methods in binary and multicomponent systems are briefly introduced. Finally, specificity of SIMS analysis of diffusion in fine-grained materials are carefully analyzed. If applicable, a direct comparison of the results obtained by different techniques is given.

Influence of Li Concentration-Dependent Diffusion Coefficient and Modulus Hardening on Diffusion-Induced Stresses in Anisotropic Anode Particles

Particle cracking caused by diffusion-induced stresses (DISs) is an important reason for lithium-ion battery (LIB) capacity fading. In this study, concentration-dependent material properties are introduced to model the distribution of the concentrations and evolution of DISs in anisotropic active particles. The concentration-dependent diffusion coefficient increases the concentration gradient and thus the DISs, and the concentration-dependent elastic modulus hardening increases the internal DISs and thus the stress-enhanced diffusion of Li ions. Diffusion in the direction of a large diffusion coefficient enhances the diffusion in the direction of a small diffusion coefficient, which leads to an anisotropic concentration, concentration gradient and DISs. The greater the anisotropic difference within the particles is, the more obvious the decrease in the radial stress and hoop stress. The results can be comparable with many published experimental results of graphite and indicate that the role of concentration-dependent material properties and anisotropy in the particles cannot be ignored.

Modeling cross-flow ultrafiltration of permeable particle dispersions.

Cross-flow ultrafiltration is a pressure-driven separation and enrichment process of small colloidal particles where a colloidal feed dispersion is continuously pumped through a membrane pipe permeable to the solvent only. We present a semi-analytic modified boundary layer approximation (mBLA) method for calculating the inhomogeneous concentration-polarization (CP) layer of particles near the membrane and the dispersion flow in a cross-flow filtration setup with a hollow fiber membrane. Conditions are established for which unwarranted axial flow and permeate flow reversal are excluded, and non-monotonic CP profiles are observed. The permeate flux is linked to the particle concentration on the membrane wall using the Darcy-Starling expression invoking axially varying osmotic and trans-membrane pressures. Results are discussed for dispersions of hard spheres serving as a reference system and for solvent-permeable particles mimicking non-ionic microgels. Accurate analytic expressions are employed for the concentration and solvent permeability dependent dispersion viscosity and gradient diffusion coefficient entering into the effective Stokes flow and advection-diffusion equations. We show that the mBLA concentration and flow profiles are in quantitative agreement with results by a finite element method. The mBLA results are compared with predictions by an earlier CP layer similarity solution, showing the higher precision of the former method.

Determination of carbonization depths in Ti substrates in diamond growth atmospheres and modified diffusion models

Behaviors of diamond coated Ti-related parts are significantly affected by carbonized interlayers, so it is essential to determine or predict the carbonization depth during diamond growth. Following the Fick’s law of diffusion with carbon concentration-dependent diffusion coefficients, it is roughly predicted that carbonization depths in Ti and Ti–6Al–4V substrates hardly reach 30 μm within a long duration of 16 h. After trying some characterization methods (XPS, EDS, metallographic analysis and ToF-SIMS), the EDS line scanning is selected as the practical one with sufficient accuracy and convenience. Influences of some critical factors, including growth duration, substrate temperature, reactant pressure and acetone (carbon source) concentration, are studied and discussed. Considering all of the factors, especially those related to the diamond coverage that can suppress carbon diffusion, the modified diffusion models for describing the carbon diffusion, are constructed, defining carbon concentration and diamond coverage (duration, pressure, carbon concentration)-dependent diffusion coefficients. Such the models can be used for well predicting the carbonization depths under most conditions, which are valuable for controlling the carbonized layers during fabrications of diamond coated Ti-related components facing to various applications.

Measurement and verification of concentration-dependent diffusion coefficient: Ray tracing imagery of diffusion process**Project supported by the National Natural Science Foundation of China (Grant No. 11804296), the Joint Key Project of Yunnan Province, China (Grant Nos. 2018FY001-020 and 2018ZI002), and the Fund from the Educational Department of Yunnan Province, China (Grant No. 2016CYH05).

Ray tracing method is used to study the propagation of collimated beams in a liquid–core cylindrical lens (LCL), which has dual functions of diffusion cell and image formation. The diffusion images on the focal plane of the used LCL are simulated by establishing and solving both linear and nonlinear ray equations, the calculated results indicate that the complex imaging results of LCL in inhomogeneous media can be treated by the law of ray propagation in homogeneous media under the condition of small refractive index gradient of diffusion solution. Guided by the calculation conditions, the diffusion process of triethylene glycol aqueous solution is experimentally studied at room temperature by using the LCL in this paper. The spatial and temporal concentration profile Ce(z, t) of diffusion solution is obtained by analyzing diffusion image appearing on the focal plane of the LCL; Then, the concentration-dependent diffusion coefficient is assumed to be a polynomial D(C) = D0 × (1 + α1C + α2C2 + α3C3 + ⋅). The finite difference method is used to solve the Fick diffusion equation for calculating numerically the concentration profiles Cn(z, t). The D(C)of triethylene glycol aqueous solution is obtained by comparing the Cn(z, t) with Ce(z,t). Finally, the obtained polynomial D(C) is used to calculate the refractive index profiles nn(z,t)s of diffusion solution in the used LCL. Based on the ray propagation law in inhomogeneous media and the calculated n(z,t), the ray tracing method is used again to simulate the dynamic images of the whole experimental diffusion process to varify the correctness of the calculated D(C). The method presented in this work opens up a new way for both measuring and verifying the concentration-dependent liquid diffusion coefficients.

Drying kinetic measurements of polymer nanolayers – Experimental results with a model-based validation and interpretation of solvent diffusion

This contribution presents a method to examine the drying process of thin films of polymer gravimetrically, ranging from a few micrometers to below 20nm using a quartz crystal microbalance (QCM). Polyvinyl acetate (PVAc) and toluene (tol) serve as the model system to validate the method by drying experiments, simulations and the determination of the concentration dependent solvent diffusion coefficient. The results show that even on the nanometer scale, the diffusion coefficient is not a function of film thickness, which is contrary to the observations that can be found in literature. A thin polymer film, still loaded with residual solvent, is placed into a flow channel with controlled drying conditions. The QCM gravimetrically measures the drying curve and precisely detects changes of mass fractions on the scale of a hundredth of a percent over the course of hours, which translates to changes in layer thickness as small as to 0.1nm per hour. The presented method is a general-purpose tool for measuring concentration-dependent diffusion coefficients, which is relevant to all sorts of thicknesses and coatings. Since nanolayers are employed, in a reasonable amount of time diffusion coefficients in the order of 10−18m2s−1 are accessible.

Simulation of chloride diffusion in fly ash and limestone-calcined clay cement (LC3) concretes and the influence of damage on service-life

Numerical simulation of chloride diffusion through high-performance binding systems such as limestone-calcined clay cement (LC3) and its comparison to control and fly ash modified concretes is discussed. The simulation framework considers the pore structure of concrete, the concentration-dependence of diffusion coefficient, and Freundlich binding. The LC3 concrete and the companion fly ash concrete exhibit similar service lives (~8× control mixture), despite the LC3 system having a reduced clinker factor than the fly ash concrete (~0.5, as opposed to 0.7). The diffusion model is augmented with a scalar isotropic damage variable that accounts for random distribution of microcracks under fatigue loading (e.g., in a bridge deck). The impact of damage on service life at different stress levels, for the different concretes is elucidated. The modeling approach can be used to evaluate the influence of binder composition and damage on effective service life of chloride-exposed concrete structures, thereby aiding in binder selection.

Application of concentration-dependent HSDM to the lithium adsorption from brine in fixed bed columns

Adsorption of lithium ions from simulated brine using layered lithium-aluminum hydroxides was investigated at 303 K in fixed-bed columns. The lithium adsorption isotherms in simulated brine were conducted using batch system, and the Langmuir isotherm model was found to fit the equilibrium data well. The effects of initial lithium concentration (C0, 200–400 mg/L), bed height (H, 30–200 cm), and feed flow rate (Q, 6–18 mL/min) on lithium breakthrough curves in packed bed columns were investigated systematically. In order to simulate the practical operation process, the breakthrough time was defined when the lithium recovery rate dropped down to 0.6. The column adsorption performance was improved with the increasing of bed height and the decreasing of the initial lithium concentration, feed flow rate. Furthermore, the Homogeneous Surface Diffusion Model (HSDM) with concentration-dependent surface diffusion coefficient was developed to simulate and predict the lithium adsorption curves. The numerical model results showed good agreement with the experimental data in simulated brine and factual brine circumstance, which proved that the model could successfully simulate the column adsorption process of lithium recovery. These results provided important information to design, optimize and scale-up the column adsorption process for lithium recovery from brine.

Gradient diffusion in dilute suspensions of hard spheroidal particles

The renormalization method proposed by Batchelor is used to derive gradient diffusion coefficients in Brownian suspensions of hard spheroidal particles with aspect ratio in the range . The theory is based on pairwise steric and hydrodynamic interactions, and the results are therefore valid for dilute suspensions such that , where is the particle volume fraction. The driving force for gradient diffusion, i.e. the gradient in chemical potential, is larger for suspensions of spheroidal particles than for spheres at the same volume fraction. The hydrodynamic resistance also increases with aspect ratio, but the increase is weaker than that in the driving force. Consequently, at the same particle volume fraction, the increases in rates of gradient diffusion are greater for spheroidal particles than for spheres. The concentration-dependent gradient diffusion coefficient is shown to be closely approximated by , which reduces to the result for spheres when . Here, is the Stokes–Einstein diffusivity of a spherical particle with its radius equal to the longer dimension of the spheroidal particle, and is the orientation-averaged diffusivity of an isolated spheroidal particle.

Physical Origin of the Differential Voltage Minimum Associated with Lithium Plating in Li-Ion Batteries

The main barrier to fast charging of Li-ion batteries at low temperatures is the risk of short-circuiting due to lithium plating. In-situ detection of Li plating is highly sought after in order to develop fast charging strategies that avoid plating. It is widely believed that Li plating after a single fast charge can be detected and quantified by using a minimum in the differential voltage (DV) signal during the subsequent discharge, which indicates how much lithium has been stripped. In this work, a pseudo-2D physics-based model is used to investigate the effect on Li plating and stripping of concentration-dependent diffusion coefficients in the active electrode materials. A new modelling protocol is also proposed, in order to distinguish the effects of fast charging, slow charging and Li plating/stripping. The model predicts that the DV minimum associated with Li stripping is in fact a shifted and more abrupt version of a minimum caused by the stage II-stage III transition in the graphite negative electrode. Therefore, the minimum cannot be used to quantify stripping. Using concentration-dependent diffusion coefficients yields qualitatively different results to previous work. This knowledge casts doubt on the utility of DV analysis for detecting Li plating.

A Hybrid Backward Euler Control Volume Method to Solve the Concentration-Dependent Solid-State Diffusion Problem in Battery Modeling

Several efficient analytical methods have been developed to solve the solid-state diffusion problem, for constant diffusion coefficient problems. However, these methods cannot be applied for concentration-dependent diffusion coefficient problems and numerical methods are used instead. Herein, grid-based numerical methods derived from the control volume discretization are presented to resolve the characteristic nonlinear system of partial differential equations. A novel hybrid backward Euler control volume (HBECV) method is presented which requires only one iteration to reach an implicit solution. The HBECV results are shown to be stable and accurate for a moderate number of grid points. The computational speed and accuracy of the HBECV, justify its use in battery simulations, in which the solid-state diffusion coefficient is a strong function of the concentration.

Dissolution of surfactant mixtures investigated through hyperspectral imaging and multivariate curve resolution

Linear solvent penetration experiments are used to study the dissolution of mixed surfactant pastes consisting of two surfactants and water by means of confocal Raman microscopy. During the experiment, Raman spectra were collected along with space and over time. The resulting hyperspectral data were then processed by means of Multivariate Curve Resolution – Alternating Least Square (MCR-ALS) technique to estimate the evolution of the spatial profile of the surfactant concentrations over time. We show that hyperspectral imaging coupled with MCR-ALS can provide a detailed description of the dissolution process. The experimental concentration profiles are analyzed using a diffusive model with concentration-dependent diffusion coefficient, showing a quantitative agreement. We show that the additional information extracted by confocal Raman microscopy enables a more sophisticated analysis of linear penetration experiments when compared to the traditional approach of surfactant mesophases front propagation tracking even in absence of a detailed phase diagram.

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.

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.

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.