ElectrochemicalKinetics.jl

This package implements a variety of models for rates of electrochemical reactions. The goal is to have a generic interface where different model types can be easily swapped in.

Usage/Examples

Building and Evaluating Models

We can construct a simple Butler-Volmer model with symmetric electron transfer as follows:

julia> bv = ButlerVolmer()
ButlerVolmer(A=1.0, α=0.5)

To evaluate the model and find the reaction rate at a given voltage is as simple as:

julia> bv(0.1)
6.865409234059785

(note that this is equivalent to rate_constant(0.1, bv), but this "callable" syntax is often convenient)

By default, this computes the net reaction rate. We can also get the rate only in the oxidative direction by passing the ox flag:

julia> bv(0.1, true)
7.0081012366762145

Let's try a fancier model now. The MarcusHushChidseyDOS model requires information about the density of states of the electrode, which we can read from a file:

julia> mhcd = MarcusHushChidseyDOS(20, 0.2, "data/DOSes/Cu_111_dos.txt")
MarcusHushChidseyDOS(A=20.0, λ=0.2)

In this case, since MarcusHushChidseyDOS<:IntegralModel, we can't directly evaluate the object as a callable because it's ambiguous if we want the integrand or the full integrated rate constant, and we must specify:

julia> rate_constant(0.1, mhcd)
3.426690873746331

By default, the integration bounds are chosen to be the energy bounds over which the DOS is defined, which we can easily check:

julia> mhcd.dos
DOSData: avg value 6.7 from energy -8.6 to 18.0

The DOSData struct stores information about the DOS constructed from file. In particular, it houses an interpolation function that is used in the quadrature when evaluating the integral to compute the rate constant.

Another feature we support for MarcusHushChidseyDOS models is incorporating the effects of quantum capacitance. In this case, we simply need to pass a few more arguments...

julia> mhcd = MarcusHushChidseyDOS(0.82, "data/DOSes/bernal_graphene.txt")
MarcusHushChidseyDOS(A=1.0, λ=0.82)

julia> rate_constant(0.15, mhcd, true; calc_cq=true, Vq_min=-0.3, Vq_max=0.3, E_min = -0.6, E_max=0.6)
2.8017566534572208e-5

Fitting and Comparing Models

The package also supports fitting model parameters to experimental data. This is where the generic syntax and the magic of Julia multiple dispatch really shine! Let's say we want to try fitting a few different types of models to some experimental data...the following script is enough to do it:

# load in some dependency packages
using DelimitedFiles
using Statistics
using ElectrochemicalKinetics

# Source the data...feel free to try other options included in the `data/` folder of the repo!
dosfile = joinpath("data", "DOSes", "Cu_111_dos.txt")
expfile = joinpath("data", "exp_data", "ecdec.txt")
exp_data = readdlm(expfile, ',')

# set the types of models to fit
model_types = [ButlerVolmer, Marcus, MarcusHushChidseyDOS]

# this is all it takes to fit the models!
models = [fit_model(exp_data, model_type; dos_file=dosfile) for model_type in model_types]

# There's a convenience function to compare results
plot_exp_and_models(exp_data, models)

The resulting plot looks like:

But we might also want to see the higher-voltage behavior and compare the models absent the experimental data. There's another function for that (by default it plots to +/- 1V)...

julia> plot_models(models)

Electrochemical Phase Diagrams

For more on this, see upcoming paper: [arxiv link placeholder]

Supported Models

Butler-Volmer

Probably the most basic (and largely empirical) kinetic model. The rate constants are given by:

$$k_{\text{ox}}^{\text{BV}}(\eta)=A\exp\left(\frac{\alpha\eta}{k_{\text B}T}\right)\ k_{\text{red}}^{\text{BV}}(\eta)=A\exp\left(\frac{(1-\alpha)\eta}{k_{\text B}T}\right)$$

It is implemented in the package as ButlerVolmer, possessing two fields: the prefactor A and the transfer coefficient α.

Marcus

One step up with a bit of mechanism...

$$k_{\text{red/ox}}^{\text{Marcus}}(\eta) = A\exp\left(-\frac{(\lambda\pm\eta)^2}{4k_{\text B}T}\right)$$

Implemented in the package as Marcus, with parameters for prefactor A and reorganization energy λ.

Marcus-Hush-Chidsey

Originated from Chidsey's 1991 paper.

$$k_{\text{red/ox}}^{\text{MHC}}(\eta) = A\int_{-\infty}^\infty \exp\left(-\frac{(\varepsilon-\lambda\mp\eta)^2}{4k_{\text B}T}\right)f_{\text{FD}}(\varepsilon, T)\mathrm d\varepsilon$$

Implemented as MarcusHushChidsey, with prefactor A, reorganization energy λ, and also an average_dos parameter (which defaults to 1.0), which is useful for making direct comparisons against MarcusHushChidseyDOS (see below).

Asymptotic Marcus-Hush-Chidsey

Originated from Zeng et al. 2014 (equation 17). We have corrected it to recover the correct temperature scaling, [paper link placeholder]

Implemented as AsymptoticMarcusHushChidsey with parameters A and λ.

Marcus-Hush-Chidsey + DOS

Originated from Kurchin and Viswanathan 2020.

$$k_{\text{ox}}^{\text{MHCKV}}(\eta)=A\int_{-\infty}^\infty\mathcal{D}(\varepsilon)\exp\left(-\frac{(\lambda-\eta+\varepsilon)^2}{4\lambda k_{\text{B}}T}\right)(1-f_{\text{FD}}(\varepsilon,T))\mathrm{d}\varepsilon\\ k_{\text{red}}^{\text{MHCKV}}(\eta)=A\int_{-\infty}^\infty\mathcal{D}(\varepsilon)\exp\left(-\frac{(\lambda+\eta-\varepsilon)^2}{4\lambda k_{\text{B}}T}\right)f_{\text{FD}}(\varepsilon,T)\mathrm{d}\varepsilon$$

Implemented as MarcusHushChidseyDOS, with parameters A, λ, and dos, which must be a DOSData object (can be automatically constructed from a two-column text file of energies and DOS values).

Contributing

We are happy to take pull requests for new features, new models, etc. by pull request! It is suggested (though not required) that you first open an issue to discuss.