169 research outputs found
Modeling of electrochemical double layers in thermodynamic non-equilibrium
We consider the contact between an electrolyte and a solid electrode. At first we formulate a thermodynamic consistent model that resolves boundary layers at interfaces. The model includes charge transport, diffusion, chemical reactions, viscosity, elasticity and polarization under isothermal conditions. There is a coupling between these phenomena that particularly involves the local pressure in the electrolyte. Therefore the momentum balance is of major importance for the correct description of the layers. The width of the boundary layers is typically very small compared to the macroscopic dimensions of the system. In a second step we thus apply the method of asymptotic analysis to derive a simpler reduced model that does not resolve the boundary layers but instead incorporates the electrochemical properties of the layers into a set of new boundary conditions. For a metal-electrolyte interface, we derive a qualitative description of the double layer capacitance without the need to resolve space charge layers
A new perspective on the electron transfer: Recovering the Butler--Volmer equation in non-equilibrium thermodynamics
Understanding and correct mathematical description of electron transfer reaction is a central question in electrochemistry. Typically the electron transfer reactions are described by the Butler-Volmer equation which has its origin in kinetic theories. The Butler-Volmer equation relates interfacial reaction rates to bulk quantities like the electrostatic potential and electrolyte concentrations. Since in the classical form, the validity of the Butler-Volmer equation is limited to some simple electrochemical systems, many attempts have been made to generalize the Butler-Volmer equation. Based on non-equilibrium thermodynamics we have recently derived a reduced model for the electrode-electrolyte interface. This reduced model includes surface reactions but does not resolve the charge layer at the interface. Instead it is locally electroneutral and consistently incorporates all features of the double layer into a set of interface conditions. In the context of this reduced model we are able to derive a general Butler-Volmer equation. We discuss the application of the new Butler-Volmer equations to different scenarios like electron transfer reactions at metal electrodes, the intercalation process in lithium-iron-phosphate electrodes and adsorption processes. We illustrate the theory by an example of electroplating
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A mixture theory of electrolytes containing solvation effects
In this work we present a new mixture theory of a liquid solvent
containing completely dissociated ions to study the space charge layer of
electrolytes in contact with some inert metal. We incorporate solvation shell
effects (i) in our derivation of the mixing entropy and (ii) in the pressure
model. Chemical potentials of ions and solvent molecules in the
incompressible limit are then derived from a free energy function. For the
thermodynamic equilibrium the coupled equation system of mass and momentum
balance, the incompressibility constraint and the Poisson equation are
summarized. With that we study the space charge layer of the electrolytic
solution for an applied half cell potential and compare our results to
historic and recent interpretations of the double layer in liquid
electrolytes. The novelties of the new model are: (i) coupling of momentum-
and mass-balance equations, (ii) calculation of entropic contributions due to
solvated ions and (iii) the potential and pressure dependence of the free
charge density in equilibrium
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Hysteresis and phase transition in many-particle storage systems
We study the behavior of systems consisting of ensembles of
interconnected storage particles. Our examples concern the storage of lithium
in many-particle electrodes of rechargeable lithium-ion batteries and the
storage of air in a system of interconnected rubber balloons. We are
particularly interested in those storage systems whose constituents exhibit
non-monotone material behavior leading to transitions between two coexisting
phases and to hysteresis. In the current study we consider the case that the
time to approach equilibrium of a single storage particle is much smaller
than the time for full charging of the ensemble. In this regime the evolution
of the probability to find a particle of the ensemble in a certain state, may
be described by a nonlocal conservation law of Fokker-Planck type. Two
constant parameter control whether the ensemble transits the 2-phase region
along a Maxwell line or along a hysteresis path or if the ensemble shows the
same non-monotone behavior as its constituents
Hysteresis in the context of hydrogen storage and lithium-ion batteries
The processes of reversible storage of hydrogen in a metal by
loading and unloading and of charging and discharging of
lithium-ion batteries have many things in common. The both
processes are accompanied by a phase transition and loading and
unloading run along different paths, so that hysteretic behavior
is observed.
For hydrogen storage we consider a fine powder of magnesium (Mg)
particles and lithium storage is studied for iron phosphate
(FePO) particles forming the cathode of a lithium-ion battery.
The mathematical models that are established in \cite{DGJ08} and
\cite{DGH09a}, describe phase transitions and hysteresis
exclusively in a single particle and on that basis they can
predict the observed hysteretic plots with almost horizontal
plateaus. Interestingly the models predict that the coexistence of
a 2-phase system in an individual particle disappears, if its size
is below a critical value. However, measurements reveal that this
is qualitatively not reflected by the mentioned hysteretic plots
of loading and unloading. In other words: The behavior of a
storage system consisting of many particles is qualitatively
independent of the fact whether the individual particles itself
develop a 2-phase system or if they remain in a single phase
state.
This apparent paradoxical observation will be resolved in this
article. It will be shown that if each of the individual particles
homogeneously distributes the supplied matter, nevertheless the
many particle ensemble exhibits phase transition and hysteresis,
because one of the two phases is realized in some part of the
particles while the remaining part is in the other phase
Theory and structure of the metal/electrolyte interface incorporating adsorption and solvation effects
In this work we present a continuum theory for the metal/electrolyte interface which explicitly takes into account adsorption and partial solvation on the metal surface. It is based on a general theory of coupled thermo-electrodynamics for volumes and surfaces, utilized here in equilibrium and a 1D approximation. We provide explicit free energy models for the volumetric metal and electrolyte phases and derive a surface free energy for the species present on the metal surface. This surface mixture theory explicitly takes into account the very different amount of sites an adsorbate requires, originating from solvation effects on the surface. Additionally we account for electron transfer reactions on the surface and the associated stripping of the solvation shell. Based on our overall surface free energy we thus provide explicit expressions of the surface chemical potentials of all constituents. The equilibrium representations of the coverages and the overall charge are briefly summarized. Our model is then used to describe two examples: (i) a silver single crystal electrode with (100) face in contact to a (0.01M NaF + 0.01M KPF6) aqueous solution, and (ii) a general metal surface in contact to some electrolytic solution AC for which an electron transfer reaction occurs in the potential range of interest. We reflect the actual modeling procedure for these examples and discuss the respective model parameters. Due to the representations of the coverages in terms of the applied potential we provide an adsorption map and introduce adsorption potentials. Finally we investigate the structure of the space charge layer at the metal/surface/electrolyte interface by means of numerical solutions of the coupled Poisson-momentum equation system for various applied potentials. It turns out that various layers self-consistently form within the overall space charge region, which are compared to historic and recent pictures of the double layer. Based on this we present new interpretations of what is known as inner and outer Helmholtz-planes and finally provide a thermodynamic consistent picture of the metal/electrolyte interface structure
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Bulk-Surface Electrothermodynamics and Applications to Electrochemistry
We propose a modeling framework for magnetizable, polarizable, elastic, viscous, heat conducting, reactive mixtures in contact with interfaces. To this end, we first introduce bulk and surface balance equations that contain several constitutive quantities. For further modeling of the constitutive quantities, we formulate constitutive principles. They are based on an axiomatic introduction of the entropy principle and the postulation of Galilean symmetry. We apply the proposed formalism to derive constitutive relations in a rather abstract setting. For illustration of the developed procedure, we state an explicit isothermal material model for liquid electrolyte|metal electrode interfaces in terms of free energy densities in the bulk and on the surface. Finally, we give a survey of recent advancements in the understanding of electrochemical interfaces that were based on this model
Bulk-surface electro-thermodynamics and applications to electrochemistry
We propose a modeling framework for magnetizable, polarizable, elastic, viscous, heat conducting, reactive mixtures in contact with interfaces. To this end we first introduce bulk and surface balance equations that contain several constitutive quantities. For further modeling the constitutive quantities, we formulate constitutive principles. They are based on an axiomatic introduction of the entropy principle and the postulation of Galilean symmetry. We apply the proposed formalism to derive constitutive relations in a rather abstract setting. For illustration of the developed procedure, we state an explicit isothermal material model for liquid electrolyte metal electrode interfaces in terms of free energy densities in the bulk and on the surface. Finally we give a survey of recent advancements in the understanding of electrochemical interfaces that were based on this model
Overcoming the shortcomings of the Nernst--Planck model
This is a study on electrolytes that takes a thermodynamically consistent coupling between mechanics and diffusion into account. It removes some inherent deficiencies of the popular Nernst-Planck model. A boundary problem for equilibrium processes is used to illustrate the new features of our model
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