diff --git a/doc/sphinx/reference/kinetics/reaction-rates.md b/doc/sphinx/reference/kinetics/reaction-rates.md index becff62574..273c63f536 100644 --- a/doc/sphinx/reference/kinetics/reaction-rates.md +++ b/doc/sphinx/reference/kinetics/reaction-rates.md @@ -106,6 +106,113 @@ reactions: - [](sec-plog-rate) - [](sec-chebyshev-rate) + +(sec-electrochemical-reactions)= +## Electrochemical Reactions + +In an electrochemical reaction (one that moves electrical charge from one phase of +matter to another), the electric potential difference $\Delta\phi$ at the phase boundary +exerts an additional "force" on the reaction that must be accounted for in the rate +expression. + +The free energy of the reaction equals the electrochemical potential change: + +$$ \Delta\tilde{\mu}_{\rm rxn} = \Delta\mu_{\rm rxn} + n_{\rm elec}F\Delta\phi $$ + +where $\mu_{\rm rxn}$ is the chemical potential, $n_{\rm elec}$ is the total electrical +charge moved across the phase boundary (in other words, the charge moved from "phase 1" +to "phase 2", where $\Delta\phi = \phi_2 - \phi_1$), and $F$ is Faraday's constant +(96,485 Coulombs per mole of charge). + +Cantera's charge transfer treatment assumes a reversible reaction with a linear energy +profile in the region of the transition state. From above, for any $\Delta\phi$ the +free energy of the reaction changes by $n_{\rm elec}F\Delta\phi$. The transition state +energy, meanwhile, changes by a fraction of this, $\beta n_{\rm elec}F\Delta\phi$, where +the "symmetry parameter" $\beta$ is a number between 0 and 1. + +This means that the activation energy for the reaction changes: +- The barrier height for the forward reaction increases by + $\beta n_{\rm elec}F\Delta\phi$. +- The reverse reaction barrier height decreases by + $\left(1-\beta\right) n_{\rm elec}F\Delta\phi$. + +Note that $n_{\rm elec}$ and $\Delta \phi$ both have a sign, so the terms "increase" and +"decrease" are relative; the forward barrier height might increase by a negative amount +(that is, decrease), for instance. + +From transition state theory, the forward and reverse reaction rates are therefore +calculated as: + +$$ R_f = k_f\exp\left(-\frac{\beta n_{\rm elec}F\Delta\phi}{RT} \right)\prod_k +C_{{\rm ac},\,k}^{\nu_k^\prime} $$ + +and + +$$ R_r = k_r\exp\left(\frac{\left(1-\beta\right)n_{\rm elec}F\Delta\phi}{RT} \right) +\prod_k C_{{\rm ac},\,k}^{\nu_k^{\prime\prime}}, $$ + +respectively, where $k_f$ and $k_r$ are the normal chemical rate coefficients in the +absence of any electric potential difference (such as those calculated using Arrhenius +coefficients), $C_{{\rm ac},\,k}$ is the +{ct}`activity concentration ` of species $k$, +$\nu_k^\prime$ and $\nu_k^{\prime\prime}$ are the reactant and product stoichiometric +coefficients, respectively, for species $k$ for this reaction, and $R$ and $T$ are the +universal gas constant and temperature, respectively. + +Note that Cantera's actual software implementation looks quite different from the +description above, which is meant solely to give a clearer understanding of the science +behind Cantera's calculations. + +```{admonition} YAML Usage +:class: tip +- Electrochemical reactions only occur at phase boundaries and therefore use the + standard [``interface``](sec-yaml-interface-Arrhenius) reaction rate implementation. +- Charge transfer is automatically detected, and $n_{\rm elec}$ automatically + calculated. If no value for `beta` is provided, an + [``electrochemical``](sec-yaml-electrochemical-reaction) reaction assumes a default of + ``beta = 0.5``. +``` + +(sec-butler-volmer)= +### The Butler-Volmer Form + +Cantera's electrochemical reaction rate calculation is equivalent to the commonly-used +Butler-Volmer rate form. In Butler-Volmer, the net rate of progress, +$R_{\rm net} = R_f - R_r$, can be written as: + +$$ R_{\rm net} = \frac{i_\circ}{n_{\rm elec}F}\left[\exp\left(-\frac{\beta +n_{\rm elec}F\eta}{RT} \right) - \exp\left( +\frac{\left(1-\beta\right)n_{\rm elec}F\eta}{RT}\right) \right]$$ + +where the kinetic rate constant $i_\circ$ is known as the "exchange current density" and +$\eta$ the "overpotential" -- the difference between the actual electric potential +difference and that which would set the reaction to equilibrium: + +$$ \eta = \Delta\phi - \Delta\phi_{\rm equil} $$ + +To convert between the two forms, the exchange current density varies with the chemical +state and can be calculated as: + +$$ i_\circ = n_{\rm elec}Fk_f^{\left(1-\beta\right)}k_r^\beta\prod_k +C_{{\rm ac},\,k}^{\left(1-\beta\right)\nu_k^\prime} +\prod_k C_{{\rm ac},\,k}^{\beta\nu_k^{\prime\prime}}. $$ + + +````{admonition} YAML Usage +:class: tip +One can explicitly provide an exchange current density, rather than the $k_f$ value, +by setting the optional ``exchange-current-density-formulation`` field to ``true``. + +```yaml +- equation: LiC6 <=> Li+(e) + C6 + rate-constant: [5.74, 0.0, 0.0] + beta: 0.4 + exchange-current-density-formulation: true +``` +Here, the rate constant Arrhenius parameters will be used to calculate the exchange +current density. +```` + (sec-reaction-orders)= ## Reaction Orders diff --git a/doc/sphinx/yaml/reactions.rst b/doc/sphinx/yaml/reactions.rst index db9c114410..73da0bef86 100644 --- a/doc/sphinx/yaml/reactions.rst +++ b/doc/sphinx/yaml/reactions.rst @@ -518,7 +518,8 @@ Example:: ``electrochemical`` ------------------- -Interface reactions involving charge transfer between phases. +Interface reactions involving :ref:`charge transfer ` +between phases. Includes the fields of an :ref:`sec-yaml-interface-Arrhenius` reaction, plus: