Understanding Voltammetry Appendix
Appendix
Simulation of Electrode Processes
The purpose of this appendix is to provide an insight into how simple numerical simulations of one dimensional diffusion problems can be carried out.
A.1 Fick’s First and Second Laws
The voltammetry experiment conducted in quiescent solution considers mass transport of species by diffusion only. The flux of a species through a solution is mathematically described by Fick’s First Law [1] which is described in Chapter 3. This is as follows:
Fick’s First Law: (A1)
where j is the flux, D is the diffusion coefficient, c is the concentration of a species and x is the spatial coordinate. Equation (A1) allows us to calculate the passage of flux in a steady-stat system, where the concentration gradients are invariant in time. However the nature of the electrochemical systems is such that concentration gradients are usually constantly changing. Fick derived a second law (Chapter 3) to describe this change of concentration with time, t, in the form of a second order differential equation:
Fick’s Second Law:
(A2)
As stated, these laws only describe a one-dimensional system. Generalised to three Cartesian directions, the 2nd law becomes:
(A3)
A.2 Boundary Conditions
The solution to a second order differential equation may only be solved with the introduction of boundary conditions. In physical terms this represents the restrictions imposed on the electrolyte concentration (Dirichlet boundaries) or its derivative (Neumann boundaries- e.g. flux) in time and space by the experiment.
A.3 Finite Difference Equations
Mass transport equations are partial differential equations. The concentration is a function of both distance, x and time, t. Since our model of cyclic voltammetry (Chapter 4) demands only one dimension in space, we can approximate this function as discrete points in time and in space. This process is known as ‘discretisation’. Points in the x-direction are assigned values of j = 0, 1, 2, 3….NJ spaced Δx apart whilst those in the time are assigned values of l = 0, 1, 2, 3….Nl, spaced Δt apart. Hence any instantaneous point concentration is specified by values of l and j and we use the notation to emphasise this. Finite difference equations are approximations of partial derivatives in terms of discretised values. Those applicable to our model are:
(upwind differencing – concentration gradient at j + ½)
(downwind differencing – concentration gradient at j - ½)
and (central differencing – concentration gradient at j )
The upwind and downwind differencing equations may be combined to produce an approximation for the second derivative.
(A4)
Also for the derivative of concentration with time:
A.4 Backward Implicit Method
The Backward Implicit (BI) method [2,3] is a powerful system for calculating a set of concentration, in a one or two – dimensional spatial system. Because the concentration profile is solved vector-by-vector, only three nodes can be spanned within one spatial coordinate, so limiting general application. However, it is well suited to producing quick solution to 1D problems. The BI method involves the assimilation of a linear system of equation by discretisation of the mass transport equation, which can be re-arranged to form a matrix equation and subsequently solved.
To exemplify the use of the BI method, we will simulate the application of a potential step to a
(A5)
redox couple, such that at a time, t > 0, the A species at the electrode surface is fully oxidised to B. The A species occupies the x-coordinate between x = 0 and x = δ, where δ is a diffusion layer thickness large enough that semi-infinite diffusion operates, i.e.
This problem was considered in Chapter 3. The mass transport for the A species is:
The BI methods follows the following four steps:
Conversion of MT equations to finite difference form:
where and is assigned arbitrarily
2. Rearrangement to give a set of linear equations:
where
3. Application of boundary conditions:
The boundary conditions applicable to this simulation are
(a)
(b)
(c) (A species fully oxidised at surface)
The introduction of boundary conditions at the diffusion layer limit to the finite difference equations gives:
But since
At the electrode surface,
But since
4. Arrangement of linear equations into a matrix equation
With the initial values of [A] set to their initial bulk values, this matrix equation can be solved sequentially to give a concentration vector at every time node. After many iterations, the system will become steady state, i.e. the concentration profile will be invariant with each new iteration. Equations of the above form can be solved using The Thomas Algorithm [4,5] devised by Laasonen [6] before the invention of computer. This allows the solution of a tridiagonal matrix equation:
to be found implicitly, where is unknown, and being vectors of elements and being a tridiagonal matrix of the form:
In solution to the mass transport equations, represents the set of known concentrations , and , the unknown concentrations which are to be calculated in the next pass of the Thomas algorithm. Once calculated, the vector is set to , and the new matrix calculated. This process is repeated up to the end time of the simulation. In this way the concentration versus distance profile can be discussed as a function of time.
A6. Conclusion
This appendix has introduced some elementary theory behind the mathematical modelling of electro-active species in solution, as well as its implementation to produce computational simulations.
A7. References
[1] A. Fick, “Uber Diffusion”, Poggendorff's Annel. Physik. 94, (1855), 59
[2] J.L. Anderson, S. Moldoveanu, J. Electroanal. Chem., 179, (1984), 109
[3] R.G.Compton, M.B.G. Pilkington, G.M. Stearn, J. Chem. Soc., Faraday Trans 1, 84, (1988), 2155
[4] L.H. Thomas, Elliptical problems in linear difference equations over a network, Watson Sci. Comput. Lab. Rept. Columbia University, New York, 1949
[5] G.H.Bruce, D.W. Peaceman, H.H. Rachford, J.D. Rice. Trans. Am. Inst. Min. Engrs, 198, (1953), 79
[6] P. Laasonen, Acta Math., 81, (1949), 30917
11 GARCÍA LANDA – UNDERSTANDING MISREADING UNDERSTANDING MISREADING HERMENÉUTICA
13 UNDERSTANDING OF LIMITS AND DIFFERENTIATION AS THRESHOLD CONCEPTS
1595542239UNDERSTANDING_LEXICAL_STRUCTURE_(corrected)
Tags: appendix appendix, this appendix, appendix, simulation, understanding, electrode, voltammetry, processes