Onsager Formalism Research Articles

Vacancy diffusion and its consequences for void growth at the interface of a stripping metal electrode and solid electrolyte

It is commonly observed that voids can nucleate and grow in the lithium anode of a solid state Li-ion battery at a location adjacent to the solid electrolyte during the stripping (discharge) phase of the battery; a similar phenomenon is observed in sodium-based batteries. It is hypothesised in the current literature that the formation of these voids is due to the coalescence of vacancies that have been generated at the electrode/electrolyte interface when metal atoms are oxidized and transported into the electrolyte: the slow diffusion of the vacancies away from the electrolyte interface into the adjacent electrode results in their coalescence and the consequent growth of voids. These hypotheses are challenged in the current study by using the Onsager formalism to generate a variational principle for vacancy diffusion. Our analysis reveals that no driving force exists for the diffusion of vacancies into a homogeneous metal electrode that thins by stripping. This finding is contrary to models in the literature which have mistakenly assumed that the vanishing flux at the current collector prevents rigid body motion (drift) of the electrode which in turn prevents thinning of the electrode during stripping. Based on our analysis, we conclude that vacancy diffusion within a homogeneous electrode is not responsible for the nucleation and growth of voids at the interface between a stripping metal electrode and a solid electrolyte.

Relating the artificial chemotaxis of catalysts to a gradient descent of the free energy.

Recent experiments suggest that mesoscale catalysts are active materials that power their motion with chemical free energy from their environment and also "chemotax" with respect to substrate gradients. In the present work, we explore a thermodynamic framework for relating this chemotaxis to the evolution of a system down the gradient of its free energy. This framework builds upon recent studies that have employed the Wasserstein metric to describe diffusive processes within the Onsager formalism for irreversible thermodynamics. In this work, we modify the Onsager dissipation potential to explicitly couple the reactive flux to the diffusive flux of catalysts. The corresponding gradient flow is a modified reaction-diffusion equation with an advective term that propels the chemotaxis of catalysts with the free energy released by chemical reactions. In order to gain first insights into this framework, we numerically simulate a simplified model for spherical catalysts undergoing artificial chemotaxis in one dimension. These simulations investigate the thermodynamic forces and fluxes that drive this chemotaxis, as well as the resulting dissipation of free energy. Additionally, they demonstrate that chemotaxis can delay the relaxation to equilibrium and, equivalently, prolong the duration of nonequilibrium conditions. Although future simulations should consider a more realistic coupling between reactive and diffusive fluxes, this work may provide insight into the thermodynamics of artificial chemotaxis. More generally, we hope that this work may bring attention to the importance of the Wasserstein metric for relating nonequilibrium relaxation to the thermodynamic free energy and to large deviation principles.

Ingress of Li into Solid Electrolytes: Cracking and Sparsely Filled Cracks

The growth of Li dendrites in a solid electrolyte is commonly idealized by a pressure‐filled crack. Recent observations in both garnet and sulfide electrolytes show that sparsely filled cracks exist prior to shorting of the cell, thereby invalidating this assumption. Herein, a variational principle that uses the Onsager formalism to couple Li deposition into the crack, elastic deformation of the electrolyte, and cracking of the electrolyte with the electrochemical driving forces and dissipation within the electrolyte and interfaces is developed. Consistent with observations, it is shown that Li ingress and cracking occur together for garnet electrolytes, but the cracks are sparsely filled. This sparse filling is a direct consequence of the mismatch between the elastic opening of the cracks and the deposition of Li into the cracks across the crack flanks. An increase in the resistance of Li ingress into the tips of Li filaments results in crack propagating ahead of the Li filaments, as observed for sulfide electrolytes. In such cases, the cracks are largely dry. The results provide a framework to model Li ingress into solid electrolytes and explain why the observations are qualitatively so different from dendrites in liquid electrolytes.

Void growth in metal anodes in solid-state batteries: Recent progress and gaps in understanding

Stripping of metal cations from the anode of a Li- or Na-ion cell into a ceramic electrolyte results in the formation of voids on the electrolyte/electrode interface. Such voids have been observed to grow to sizes in excess of 100 μm. Dendrites can nucleate and grow in the electrolyte from the vicinity of the voids during the plating phase of cycling of the cell, and lead to short-circuiting of the cell. Current theoretical understanding of the formation of these voids is in its infancy: the prevailing qualitative notion is that voids form within the metal anode when the stripping current density removes metal from the interface faster than it can be replenished. We review models that employ the Onsager formalism to develop a variational approach to model void growth by coupling power-law creep of the metal electrode and the flux of metal cations through a single-ion conductor solid electrolyte. These models, based on standard Butler-Volmer kinetics for the interfacial flux, predict that voids will shrink for realistic combinations of interfacial ionic resistance and electrolyte conductivity. Additional physics in the form of modified kinetics, such that the interfacial resistance is decreased by the presence of dislocations within the creeping metal electrode, are shown to give rise to initial growth of voids around impurity particles on the electrolyte/electrode interface. However, these voids ultimately collapse under the imposed stripping fluxes and no conditions have been identified for which isolated voids grow to more than 10μm in size. This is in contrast to the experimentally observed sizes of ∼ 100μm. The physical processes by which large voids form remain unclear but the current state-of-the-art understanding does provide clues of possible mechanisms that have not as yet been considered.

Void growth within Li electrodes in solid electrolyte cells

The growth of voids at the electrode/electrolyte interface of a solid state Li battery is analysed by establishing a framework that uses the Onsager formalism to couple the power-law creep deformation of the Li electrode and flux of Li+ through a single-ion conductor solid electrolyte. For realistic combinations of the interfacial resistance and electrolyte conductivity, standard Butler-Volmer kinetics for the interfacial flux does not provide sufficient flux focussing to initiate void growth and so a modified kinetics is adopted where the interfacial resistance is decreased by the presence of dislocations within the creeping Li electrode. Micron-sized pre-existing voids shrink under stripping conditions as flux focussing on the periphery of these voids is always low. However, spatially inhomogeneous creep in the electrode around a hemispherical impurity particle reduces the interfacial resistance with consequent significant flux focussing at the periphery of the impurity. This flux focussing results in void growth with two distinct regimes of behaviour: (i) at low currents stable but small voids form while (ii) at higher currents large voids form but these ultimately collapse. No conditions are identified for which isolated voids are predicted to grow larger than 10μm in size suggesting that cell failure does not occur by the growth of isolated voids. We therefore propose a hypothesis for the coalescence of voids that initiate around impurity particles being deposited on the interface during stripping of the electrode. The ensuing predictions are consistent with measurements of cell failure and provide clues of the failure mechanisms due to void growth.

Elastodiffusion calculation of asymptotic absorption efficiencies and bias factors of dislocations in Al, Ni and Fe

The elastic interactions between dislocations and lattice point defects (PDs) control the rate of PD elimination at dislocations, and thereby the PD spatial distribution within the microstructure. By relying on the Onsager formalism, we introduce a new partition of the PD diffusion driving force in a strain field. We present DFT-based calculations of straight and loop-form edge dislocation absorption efficiencies and the resulting biases of vacancy and self-interstitial elimination in pure Fe, Ni, and Al metals. We highlight the effects of the dislocation density, the temperature and the dislocation orientation on the elastodiffusion behavior and the resulting PD redistribution near dislocations. Based on a detailed study of the sensitivity of absorption efficiencies to the boundary conditions, i.e., the capture radius and the associated PD concentrations, we introduce the notion of asymptotic absorption efficiencies and bias.

Correlation between estimated diffusion coefficients from different types of diffusion couples in multicomponent system

The interdiffusion coefficients can be estimated now purely experimentally from multicomponent diffusion profiles utilizing the recently proposed pseudo-binary and pseudo-ternary diffusion couple methods. One can estimate the composition dependent main interdiffusion coefficients over the composition range and main intrinsic diffusion coefficients at the Kirkendall marker plane from pseudo-binary diffusion couple. On the other hand, main and cross interdiffusion coefficients can be calculated from the pseudo-ternary diffusion couples. In this article, we have first proposed the method of estimation of the tracer diffusion coefficients at the cross of diffusion paths in pseudo-ternary diffusion couples by extending the approach propose by Lane and Kirkaldy in a ternary system but without neglecting the cross phenomenological terms of Onsager formalism. Following, one can even estimate the main and cross intrinsic diffusion coefficients, which is otherwise not possible following the Kirkendall marker experiments in this type of couple. Following we have derived the relation for estimation of the tracer diffusion coefficients at the Kirkendall marker plane in a pseudo-binary diffusion couple. We have chosen NiCoFeCr system to demonstrate the utilization of these methods by comparing the tracer diffusion coefficients estimated directly following radiotracer method. The relations derived in this article help to understand the correlation between estimated data in different types of diffusion couples by neglecting and considering the vacancy wind effect. This method can be now extended to the Al, Ga, Si containing systems in which the estimation of tracer diffusion coefficients following the radiotracer method is difficult/impossible because of various reasons.

Exact phenomenological theory for thermotransport in a solid binary alloy

ABSTRACT In this paper, we study the theory of the diffusion behaviour of a binary alloy in a temperature gradient (thermotransport). Using the Onsager formalism of irreversible thermodynamics we develop exact relations for the heats of transport of each component in terms of various measurable diffusion quantities under steady state conditions and with the vacancies being at local equilibrium. Apparent inconsistencies with some earlier work are indicated. Further relations for heats of transport are then developed in terms of the random alloy model for the Darken and Manning approximations. Relations for the formal separation of the heats of transport are also developed.

Diffusion, defects and understanding the growth of a multicomponent interdiffusion zone between Pt-modified B2 NiAl bond coat and single crystal superalloy

Composition-dependent diffusion coefficients are determined in B2-Ni(CoPt)Al system following the pseudo-binary and pseudo-ternary diffusion couple methods, which would not be possible otherwise in a quaternary inhomogeneous material fulfilling the conditions to solve the equations developed based on the Onsager formalism. The end-member compositions to produce ideal/near-ideal diffusion profiles are chosen based on thermodynamic details. The pseudo-binary interdiffusion coefficients of Ni and Al decrease in the presence of Co but increase in the presence of Pt. The pseudo-ternary interdiffusion coefficients indicate that the main interdiffusion coefficients increase significantly in the presence of Pt. Marginal changes of the cross interdiffusion coefficients substantiate a minor change of the diffusional interactions between the components. The thermodynamic driving forces show opposite trends with respect to composition as compared to the changes of the interdiffusion coefficients advocating a dominating role of the Pt(Co)-induced modifications of point defect concentrations. DFT-based calculations revealed that Pt alloying increases the Ni vacancy concentration and decreases the activation energy for the triple defect diffusion mechanism. These findings explain the increase in the thickness of the interdiffusion zone between the B2-Ni(Pt)Al bond coat and the single crystal superalloy René N5 because of Pt addition. Furthermore, the EPMA and TEM analyses reveal the growth of refractory elements-enriched precipitates.

Solvate Cation Migration and Ion Correlations in Solvate Ionic Liquids.

Lithium salt-glyme mixtures are interesting candidates as electrolytes for battery applications. Depending on the type of glyme or anion and the salt concentration, they either show ionic liquid-like behavior with stable lithium-glyme complex cations or concentrated salt solution-like behavior. Here, we apply electrophoretic NMR (eNMR) to elucidate transport mechanisms by observing the migration of the molecular species in an electric field. We investigate two solvate ionic liquids, i.e., lithium bis(trifluoromethanesulfonyl)amide (LiTFSA) and lithium tetrafluoroborate (LiBF4), in tetraglyme (G4) at different glyme-salt molar ratios X. A field-induced migration of neutral glyme molecules is directly observed, which is due to stable solvate-Li complex formation. Transference numbers, effective charges, and ionicities are derived from electrophoretic mobilities and self-diffusion coefficients, respectively, for the nuclei 1H, 7Li, and 19F. The effective charges are the highest at the equimolar mixture, X = 1, they differ strongly for lithium and anion, and they show large differences between both systems. These findings are qualitatively interpreted in a speciation model, suggesting anionic clusters and solvate cations as the species dominating charge transport. The resulting effective charges can only be explained taking into account ion-ion anticorrelations in the framework of the Onsager formalism, where anticorrelations between the solvate cation and the anionic complexes arise due to momentum conservation. The contributions to the anticorrelation are most dominant at high salt concentrations and in the system with the LiBF4- anion due to its lower mass and ability to form larger asymmetric clusters with Li+. Thus, in either system, also the lithium transference number is influenced to a different extent by ion-ion anticorrelations.

A correlation between electron-hole pair radii and magnetomodulation of exciplex fluorescence in electron donor-electron acceptor organic systems

Electric field dependencies of electromodulated photoluminescence and magnitudes of the magnetic-field effect on photoluminescence have been measured in vacuum-evaporated films of m-MTDATA [4,4′,4″-tris(N-(3-methylphenyl)-N-phenylamino)triphenylamine]:bathophenanthroline, m-MTDATA:BCP (bathocuproine), as well as 4,4′,4″-tris[2-naphthyl(phenyl)amino]triphenylamine:BCP. The Sano-Tachiya-Noolandi-Hong extension of standard Onsager formalism was used to investigate the electron-hole pair dissociation process which allowed us to estimate the pair intercarrier distances and the final speed of carrier recombination. A distinct correlation between the pair radii and the magnitude of the magnetic-field effect has been found.

Modelling fluid deformable surfaces with an emphasis on biological interfaces

Fluid deformable surfaces are ubiquitous in cell and tissue biology, including lipid bilayers, the actomyosin cortex or epithelial cell sheets. These interfaces exhibit a complex interplay between elasticity, low Reynolds number interfacial hydrodynamics, chemistry and geometry, and govern important biological processes such as cellular traffic, division, migration or tissue morphogenesis. To address the modelling challenges posed by this class of problems, in which interfacial phenomena tightly interact with the shape and dynamics of the surface, we develop a general continuum mechanics and computational framework for fluid deformable surfaces. The dual solid–fluid nature of fluid deformable surfaces challenges classical Lagrangian or Eulerian descriptions of deforming bodies. Here, we extend the notion of arbitrarily Lagrangian–Eulerian (ALE) formulations, well-established for bulk media, to deforming surfaces. To systematically develop models for fluid deformable surfaces, which consistently treat all couplings between fields and geometry, we follow a nonlinear Onsager formalism according to which the dynamics minimizes a Rayleighian functional where dissipation, power input and energy release rate compete. Finally, we propose new computational methods, which build on Onsager’s formalism and our ALE formulation, to deal with the resulting stiff system of higher-order partial differential equations. We apply our theoretical and computational methodology to classical models for lipid bilayers and the cell cortex. The methods developed here allow us to formulate/simulate these models in their full three-dimensional generality, accounting for finite curvatures and finite shape changes.

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.

Thermoelectric efficiency in three-terminal graphene nano-junctions.

The thermoelectric efficiency of a thermal machine consisting of a triangular graphene nano-junction connected to three electrodes in the linear response regime is studied. Using the Onsager formalism and a combination of semi-empirical tight-binding calculations as well as Green's function theory, the efficiency at maximum output power which can be written in terms of generalized figures of merit is investigated. The results for a set temperature and chemical potential parameters have shown that adding a third terminal improves the efficiency at maximum output power compared to the two-terminal setup.

A novel concept of pseudo ternary diffusion couple for the estimation of diffusion coefficients in multicomponent systems

A pseudo ternary diffusion couple technique in a multicomponent system by simplifying the mathematical complications of Onsager formalism is proposed for the estimation of composition dependent values of the interdiffusion coefficients. This is otherwise impossible following the conventional method in a system with more than three components. Other alternative methods estimate the average diffusion coefficients over a composition range of random choice and lack physical significance. This method can be followed in a multicomponent system with any number of components on the condition that only three components develop diffusion profiles keeping others as constant.

Atomic-based phase-field method for the modeling of radiation induced segregation in Fe–Cr

We present a quantitative phase-field modeling of radiation-induced segregation in Fe–Cr alloys. The evolution of chemical and point defect concentration fields are described by an Onsager formalism combined to a Cahn–Hilliard like diffusion equation for the introduction of non-uniform driving forces. Both the Onsager transport coefficients and the driving force parameters are extracted from atomic Monte Carlo simulations with point defect diffusion models fitted on DFT calculations, in a composition range between 0 and 20at.% Cr and in a temperature range between 600 and 1000K. Phase-field simulations are able to quantitatively reproduce the evolution of segregation profiles obtained from direct atomistic kinetic Monte Carlo simulations, while being typically two orders of magnitude faster. It is shown that a precise parameterization of the concentration-dependent Onsager transport coefficients, thermodynamic factors, and equilibrium point defect concentrations is crucial for the phase-field method to be quantitative.

Nonlinear irreversible thermodynamics of single-molecule experiments.

Irreversible thermodynamics of single-molecule experiments subject to external constraining forces of a mechanical nature is presented. Extending Onsager's formalism to the nonlinear case of systems under nonequilibrium external constraints, we are able to calculate the entropy production and the general nonlinear kinetic equations for the variables involved. In particular, we analyze the case of RNA stretching protocols obtaining critical oscillations between different configurational states when forced by external means to remain in the unstable region of its free-energy landscape, as observed in experiments. We also calculate the entropy produced during these hopping events and show how resonant phenomena in stretching experiments of single RNA macromolecules may arise. We also calculate the hopping rates using Kramer's approach obtaining a good comparison with experiments.

Thermoelectric efficiency of three-terminal quantum thermal machines

The efficiency of a thermal engine working in the linear response regime in a multi-terminal configuration is discussed. For the generic three-terminal case, we provide a general definition of local and non-local transport coefficients: electrical and thermal conductances, and thermoelectric powers. Within the Onsager formalism, we derive analytical expressions for the efficiency at maximum power, which can be written in terms of generalized figures of merit. Furthermore, using two examples, we investigate numerically how a third terminal could improve the performance of a quantum system, and under which conditions non-local thermoelectric effects can be observed.

Physicochemical investigations on the aqueous solution of an ionic liquid, 1-butyl-3-methylimidazolium methanesulfonate, [bmim][MS], in a concentrated and dilute regime

The physicochemical properties of the ionic liquid (IL) 1-butyl-3-methylimidazolium methanesulfonate ([bmim][MS]) in combination with water were evaluated through density, viscosity, surface tension, conductance, cyclic voltammetry, absorption and emission spectroscopic measurements. Binary mixtures were studied both in the water-rich and [bmim][MS]-rich regions (0.01mM–4.96M). The static and dynamic properties of the binary combinations were evaluated through density and viscosity measurements. The [bmim][MS] decreased the surface tension of the water, from which the surface excess and area per molecule of the ionic liquid were determined at the air–liquid surface. The equivalent conductance of [bmim][MS] at an infinite dilution was determined from the conductance data because the system obeyed the Debye–Hückel–Onsager formalism. Cyclic voltammetry measurements revealed the formation of some metastable organized structures at specific compositions. The absorption and emission spectral behavior of the anionic dye eosin Y were found to be dependent on the concentration of [bmim][MS] in the water-rich region. A significant change in the steady state anisotropy and excited state lifetime of the fluorophore occurred above 1.0M [bmim][MS] in water, which correlated with the viscosity of the medium. It was concluded that the aggregation of [bmim][MS] into micelle-like aggregates occurred above the specific concentration, which significantly altered the different physiochemical parameters of the [bmim][MS] binary mixture.

Thermodynamic forces, flows, and Onsager coefficients in complex networks

In this paper the linear theory of nonequilibrium thermodynamics, developed by Onsager and others, is applied to random networks with arbitrary degree distribution. Using the well-known methods of nonequilibrium thermodynamics we identify thermodynamic forces and their conjugated flows induced in networks as a result of single node degree perturbation. The forces and the flows can be understood as a response of the system to events, such as random removal of nodes or intentional attacks on them. Finally, we show that cross effects (such as thermodiffusion, or thermoelectric phenomena), in which one force may not only give rise to its own conjugated flow, but to many other flows, can be observed also in complex networks.