COMSOL Multiphysics Finite Element Software For Electrochemical Analysis
COMSOL Multiphysics Finite Element Software For Electrochemical Analysis
COMSOL Multiphysics Finite Element Software For Electrochemical Analysis
Electrochemistry Communications
journal homepage: www.elsevier.com/locate/elecom
Mini review
COMSOL Ltd, Broers Building, 21 JJ Thomson Avenue, Cambridge CB3 0FA, United Kingdom
COMSOL AB, Tegnrgatan 23, SE-111 40 Stockholm, Sweden
a r t i c l e
i n f o
Article history:
Received 13 December 2013
Received in revised form 16 December 2013
Accepted 16 December 2013
Available online 22 December 2013
Keywords:
Simulation
Finite element
COMSOL
Voltammetry
Electroanalysis
a b s t r a c t
We discuss the use of the commercial nite element software COMSOL Multiphysics for electrochemical analysis. Practical considerations relevant to nite element modelling are highlighted. A review of contemporary applications of this software is supplied; the subjects concerned reveal the particular suitability of general-purpose
nite element methods for non-standard geometries, complex reaction chemistry, hydrodynamic electrochemistry, and rapid verication of standard results.
2013 Elsevier B.V. All rights reserved.
Contents
1.
2.
3.
4.
5.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . .
Electrochemical theory . . . . . . . . . . . . . . . . . . . . . .
Modelling electrochemistry in COMSOL Multiphysics. . . . . . . . .
Using the nite element method: practical considerations . . . . . .
Historical review of nite element software for electrochemical analysis
5.1.
Complex geometry . . . . . . . . . . . . . . . . . . . . .
5.2.
Complex chemical mechanisms . . . . . . . . . . . . . . .
5.3.
Hydrodynamic electrochemistry . . . . . . . . . . . . . . .
5.4.
Miscellaneous . . . . . . . . . . . . . . . . . . . . . . .
6.
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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1. Introduction
2. Electrochemical theory
A short review does not allow a comprehensive exposition of electrochemical theory, but we shall discuss the most important standard
assumptions. Electrochemical phenomena are governed by the coupling
of the conservation of charge and current in the electrolyte and electrodes, together with the conservation of mass for each solute species
in an electrolyte. Prediction of a uid ow additionally involves the
conservation of momentum and total mass in a solution or mixture.
These phenomena are normally described mathematically using
partial differential equations (PDEs). Therefore, electrochemical theory
is developed by solving such PDEs on a suitable geometry and timescale.
Corresponding author.
1388-2481/$ see front matter 2013 Elsevier B.V. All rights reserved.
http://dx.doi.org/10.1016/j.elecom.2013.12.020
72
Except for simple cases, these equations lack exact mathematical solutions, and so computational solution is required.
In the most general case, charge conservation obeys Gauss's law
(Eq. (1)), and mass transport obeys the NernstPlanck equations
(Eq. (2), for an ideal solution) subject to mass continuity (Eq. (3)):
Ni Di ci zi ui ci ci u
ci
Ni Ri
t
zi ci 0
Assuming electroneutrality and small absolute concentration gradients of charge-carrying electrolyte species, the electrolyte current
obeys Ohm's law (Eq. (5)), subject to a (near-)constant conductivity
(Eq. (6)):
soln Q
soln F
zi ui ci
5
6
COMSOL Multiphysics is designed for multiphysics: the incorporation and coupling of diverse physical phenomena, expressed as PDEs,
within one model. The desired phenomena often originate from traditionally separate elds of applied physics and engineering. One electrochemical example of a multiphysics problem would be fuel cell
analysis, which combines uid dynamics, mass transport, heat transfer,
73
the tuning of solver tolerances to resolve voltammetric events, mitigating issues such as spiking of voltammetry due to fast follow-up
kinetics [4].
5. Historical review of nite element software for
electrochemical analysis
The literature from the last four years suggests that the majority of
recent applications of COMSOL Multiphysics in analytical electrochemistry involve one or more of the following: non-standard or complex
geometry; hydrodynamic electrochemistry; or, multiple or complex
reaction chemistry, whether hetero- or homogeneous.
All of these phenomena imply a theoretical description that cannot
necessarily be implemented straightforwardly in existing commercial
tools for electroanalysis, at standard geometries (planar electrode,
microdisc, etc.) or under standard conditions. Moreover, numerical concerns may make such theory challenging for the researcher to implement in code. In such cases, electrochemical research benets from
the exibility of general-purpose commercial software providing a
broad scope for numerical analysis across a range of physical conditions.
Where not stated otherwise, the theoretical work reported in this
section was achieved using COMSOL Multiphysics.
5.1. Complex geometry
The investigation of Godino et al. [8] into chronoamperometric transients at electrode arrays eschewed the usual diffusion domain approximation [9] (reducing a 3D problem to 2D axisymmetry; hereafter,
DDA) and treated the problem as three-dimensional. This is geometrically exact, but had traditionally been viewed as inaccessible for reasons
of computational problem size.
Lavacchi et al. [10] solved for inlaid and recessed microdisc arrays
with the DDA. They emphasised the need for care with meshing while
also highlighting the exibility of the approach. The recessed microdisc
and microwell problem was addressed more recently by Oleinick et al.
[11], who considered the action of a diffusional Faraday cage in a
generator-collector mode sensor design. Thin layer effects at gold
micropillars have also been studied [12] and compared to experiment
with good agreement. Odijk et al. [13] modelled differential cyclic voltammetry for redox cycling at interdigitated electrodes.
Other non-standard geometries in electroanalysis have included
crystals and electroactive nanoparticles. Claussen et al. [14] modelled
diffusionreaction systems (with DDA) to gauge the inuence of mass
transport effects on the efcacy of nanoparticle-catalysed biosensors.
Unwin and co-workers have considered dissolution kinetics of calcite
crystals in acidic solution [15,16] including the implications for local
surface measurement techniques such as scanning ion-conductance microscopy [17].
5.2. Complex chemical mechanisms
Sartin et al. [18] implemented a complex chemical kinetic scheme
in COMSOL Multiphysics to determine concentration proles in
electrochemiluminescence (ECL). The theory required the exibility to
introduce an arbitrary kinetic scheme and arbitrary transient voltage
loads. Klymenko et al. recently revisited ECL analysis [19], using
COMSOL Multiphysics to verify their KISSA software. The authors reported relatively poor runtime compared to their own, specialised
code, and re-iterated the need for care and electrochemical experience
in model setup.
Scanning electrochemical microscopy (SECM) combines a sometimes complex electrochemical system with a non-standard geometry.
Examples of theoretical work include the response to an EC mechanism
close to an SECM tip [20] and tip-position modulation with a moving
electron transfer site [21]. Zhou et al. simulated approach curves for
an electrochemically heterogeneous substrate to demonstrate the
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