Add electrochemical reactions to Science section

Resolves Cantera/cantera-website#172

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 <ThermoPhase::getActivityConcentrations>` 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
 

doc/sphinx/yaml/reactions.rst

@@ -518,7 +518,8 @@ Example::
 ``electrochemical``
 -------------------
 
-Interface reactions involving charge transfer between phases.
+Interface reactions involving :ref:`charge transfer <sec-electrochemical-reactions>`
+between phases.
 
 Includes the fields of an :ref:`sec-yaml-interface-Arrhenius` reaction, plus: