Theory for the Battery with Binary Electrolyte Interface
The Battery with Binary Electrolyte Interface defines the current balance in the electrolyte, the current balances in the electrodes, the mass balance for a salt, and the mass balance of an intercalating species such as hydrogen in a nickel-metal hydride battery.
The electrolyte in the modeled batteries has to be a quiescent alkaline binary 1:1 electrolyte, containing a cation (Cat+) and an anion (An-).
The physics interface solves for five dependent variables:
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s, the electric potential,
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l, the electrolyte potential
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Δs, film, the potential drop due to a resistive film on the electrode particles in the porous electrodes
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cs, the concentration of an intercalating species in the electrode particles
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cl, the electrolyte salt concentration
In the electrolyte and pore electrolyte, two variables are defined: l and cl. Assuming electroneutrality, cl denotes both the Cat+ concentration and the An- concentration.
The domain equations in the electrolyte are the conservation of current and the mass balance for the salt according to the following:
where il denotes the electrolyte current density, σl the electrolyte conductivity, f the activity coefficient for the salt, t+ the transport number for Cat+ (also called transference number), itot the sum of all electrochemical current sources, c0 the solvent concentration, and Ql denotes an arbitrary electrolyte current source. In the mass balance for the salt, Nl denotes the flux of the cation, εl the electrolyte volume fraction, Dl the electrolyte salt diffusivity, and Rl the total Cat+ source term in the electrolyte.
In the electrode, the current density, is, is defined as
where σs is the electrical conductivity.
The domain equation for the electrode is the conservation of current expressed as
where Qs is an arbitrary current source term.
Reactions occur on the surface of small solid spherical host particles of radius rp. The reactions can either be electrochemical or chemical adsorption/desorption reactions not involving electrons.
The electrochemical reactions involve cations or anions and are written generally as
where Θs is a free reaction site and SΘs is an occupied reaction site at the solid particle surface. Additional product species (X, …) are not handled by this physics interface.
The absorption/desorption chemical reactions that do not involve charged species and are written generally as:
with a reaction rate k (SI unit: mol/(s·m2)). The signs νs is here positive, and the reaction rate is defined as positive for reactions going from left to right.
The concentration of Θs does not have to be solved for because the total concentration of reaction sites, cs, max, is assumed to be constant, implying that
An important parameter for intercalation electrodes is the state-of-charge variable soc for the solid particles, defined as
The equilibrium potentials Eeq of intercalation electrodes reactions are typically functions of the soc.
The reactions occur on the particle surface only, but the intercalant species can be transported within the particles by diffusion. Within the particles the mass balance can be written as
where cs is the concentration of the intercalating species. This equation is solved locally by this physics interface in a 1D extra (pseudo) dimension, using a finite element discretization with the solid phase concentration as dependent variable. The divergence and gradient operator in the above equation are be applied using either spherical, cylindrical or Cartesian coordinates, depending on the particle type (spheres, cylinders, or flakes).
The boundary conditions are as follows:
where Rs, tot is the total surface molar flux of the intercalating species due to the electrochemical and chemical reactions.
The stoichiometric notations used in the physics interface are according to the general electrochemical reaction as expressed below:
where the stoichiometric coefficients, νi, are positive (νox) for products and negative (νred) for reactants in a reduction reaction. From this definition, the number of electrons, n, in the electrode reaction can be calculated according to
where zi denotes the charge of species i.
In the porous electrodes, itot denotes the sum of all charge transfer current density contributions according to:
where Av denotes the specific surface. The source term in the mass balance is calculated from:
It is also possible to specify additional reaction sources, Rl, src, that contribute to the total species source according to:
At the surface of the solid particles you have that
where the last factor (normally equal to 1) is a scaling factor accounting for differences between the surface area (Av,m) used to calculate the volumetric current density, and the surface area of the particles in the solid lithium diffusion model. Nshape is 1 for Cartesian, 2 for cylindrical, and 3 for spherical coordinates.
The surface area is commonly derived from the electrode volume fraction, particle size and shape according to
If the solid phase diffusion coefficient is very large and/or if the spatial concentration gradients in the particle can be neglected, the solid phase concentration evolution in time can be calculated from
The molar source, Rv, tot, due to the electrochemical and chemical reactions at the positive and negative electrodes is given as follows:
A resistive film (also called solid-electrolyte interface, SEI) might form on the solid particles resulting in additional potential losses in the electrodes. To model a film resistance, an extra solution variable for the potential variation over the film, Δs,film, is introduced in the physics interface. The governing equation is then according to
where Rfilm (SI unit: Ω·m2) denotes a generalized film resistance. The activation overpotentials, ηm, for all electrode reactions in the electrode then receives an extra potential contribution, which yields
It is also possible to model an electrode reaction at the interface between an electrolyte and a solid conductor. Typically a reaction of interest could be
where Y could be some metal deposited on the electrode surface. Because this is not an insertion reaction, cs is of no relevance at this boundary. The stoichiometric coefficients for the above reaction are:
This results in the following boundary condition for the species flux at the electrode - electrolyte interface
and the following condition for the currents:
where the normal vector n points into the electrolyte domain.
Initial Charge Distribution in the Battery Cell
The number of parameters in battery models are many, but especially setting the charge distribution in the cell (that is, the intercalating species concentration in each electrode material) is not always straightforward because it often requires more detailed information than just cell voltage and capacity.
It is, however, possible to compute the initial charge distribution taking into account that initially, when no current is applied on a battery cell and no sources of polarization apply, it is only the difference between the positive and negative electrode material equilibrium potentials that dictates the cell voltage. Two constraints can be set up with the battery cell capacity and voltage as inputs for this computation:
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The battery cell capacity, Qcell,0 (SI unit: C), is equal to the sum of the charge of cyclable species, Qcycl, in the positive and negative electrodes:
The cyclable species charge in an electrode is defined as:
where εs denotes the electrode volume fraction and cs,avg,cycl,electrode is the local average cyclable species concentration defined as:
cs,avg is the average species concentration, which initially, when no concentration gradients are present within the electrode particles, is equal to the concentration at the surface of the electrode particles, cs,surf. socmin is the minimum allowed state-of-charge in the electrode material.
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The electrode potential is constant in each electrode and the difference between the positive and negative electrode potentials is equal to cell voltage.
Initially, when no polarization is present in the cell, the expression is equal to difference in the open-circuit potential of the electrode materials, Eeq:
The cell voltage is restricted to the open-circuit potential of the electrode materials and the cell voltage should be set within the following range:
where the subscripts max and min of the electrode state-of-charge indicate the maximum and minimum allowed amount of intercalated species in terms of state-of-charge in the electrode materials.
Alternatively, the second constraint can be replaced with another to allow the initial cell voltage input to be replaced with initial cell state-of-charge:
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The cell state-of-charge, soccell,0 (dimensionless), relates the battery cell capacity to the charge of cyclable species in each electrode.
The cell state-of-charge ranges between 0 and 1. A state-of-charge of 0 indicates that the cell is completely discharged and if it is 1 the cell is fully charged.
Balancing the Electrodes
The balancing of the electrodes in the cell means that the amount of electrode active material in each electrode is designed after the battery cell capacity. In other words, the cyclable species capacity can be fully hosted either in the positive or negative electrode without having too much unused excess material and to keep the concentration of intercalating species within the specified state-of-charge window. For batteries this is of paramount importance to maximize energy density and life-time, and sometimes also for safety reasons.
The battery interface can supply electrode volume fractions that balance the electrodes. These are calculated by connecting the amount of active host material — that is, the maximum amount of cyclable species in the electrode — to the cell capacity initial. Here, the active host material in the positive electrode is set equal to the cell capacity. In some battery chemistries, for instance lithium-ion batteries, the host material amount in both electrodes deviate. Especially, negative carbon-based electrodes are often set in excess compared to the positive electrode to account for irreversible losses in the cell during operation. Cyclable species can in some cases be lost directly after cell assembly. The following relations therefore apply:
where Qhost (SI unit: C) is the amount of active host material, fcycl,loss the fractional loss of cyclable species, and fhost,neg,ex the fractional excess of negative active host material.
To calculate the electrode volume fraction, the fact that the amount of active host material can be computed from the following equation needs to be considered:
where Δsoc is the allowed state-of-charge window of the electrode material.
The expression for the electrode volume fraction in each electrode is therefore:
From the electrode volume fraction it is shown that the battery cell capacity should be selected carefully, because the capacity is limited by the electrode material and size. The capacity should never be set so that the electrode volume fraction is larger than 1.