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Modeling Electrochemical Electron Transfer Rates
 uprous-cupric
electron transfer is believed to play a role in limiting rates of stress corrosion
cracking in some engineering environments-particularly water cooled nuclear reactors,
which have incorporated brass components that provide a source of copper. In these
environments, cracks in the steel of the reactor are believed to grow by anodic dissolution
of metal from the crack interior, which leaves electrons behind. The dissolution
process will stop unless there is electron transfer from the region of the crack
to ions in the surrounding water. Indirect evidence suggests that this electron transfer
step is rate limiting for crack growth.
A program to model and better understand the electron transfer process was undertaken
by quantum chemist Larry Curtiss and electrochemist Zoltan Nagy, at Argonne National
Laboratory, and J.W. Halley of the Supercomputing Institute and the physics department
of at the University of Minnesota. The group is working to elucidate the mechanisms
of electron transfer as part of an effort to control the cracking problem.
When the project began, the goal was to model an electron transfer process between
solution and an electrode surface at the microscopic, chemically specific level.
The ferrous-ferric electron transfer reaction, not specifically implicated in crack
growth, was selected for the first analysis because of its relative simplicity. The
group constructed models using molecular dynamics techniques coupled to quantum chemical
calculations of potentials for the forces on the reactant and product ions. These
permitted calculation of barrier heights, transfer coefficients, and equilibrium
potentials. These compared well with the results of experiments that were carried
out for the first time at up to 300º C, the temperature at which reactors operate.
Though work continues on the ferrous-ferric reaction rate, it is clear that the barrier
to this reaction arises from the fact that the six neighboring water molecules around
the ferric ion are closer to the ion than they are when the ion has accepted an additional
electron and becomes a ferrous ion. The energy cost of moving the water away from
the ferric ion until the ion can accept an electron is the main origin of the barrier
to electron transfer from the electrode. This is called solvent rearrangement. Another
important conclusion of the ferrous-ferric study was that the reaction is nonadiabatic
(i.e., diabatic), meaning that the two ionic states experience only weak quantum
mechanical mixing near the electrode, so that the problem can be treated in perturbation
theory.
Most recently, the group has undertaken study of the more corrosion-relevant cupric-cuprous
electron transfer. Measurements indicated a barrier height of 32 ± 5 kJ/mol
for the reaction. Calculations were made assuming that the reaction is nonadiabatic,
like the ferrous-ferric one, and yielded a predicted barrier of 100 ± 20 kJ/mol.
Because the molecular dynamics model predicted solvation energies within a few 10s
of kJ/mol of the experimental ones, the discrepancy with the barrier heights was
not thought to be attributable to calculational error. In seeking a cause for the
discrepancy, Sean Walbran of the Halley group suggested that the reaction might be
adiabatic.
The accompanying figure contrasts calculated cuprous-cupric free energy surfaces
in the adiabatic and nonadiabatic cases. The plunge near the electrode surface is
a manifestation of the plating out of copper on the surface which does, in fact,
occur. The predicted barrier height, assuming adiabaticity, is 50 ± 40 kJ/mol,
which is in better agreement with experiment. In this adiabatic scenario, the cuprous-cupric
transfer mechanism is also different than in the ferrous-ferric case. In the adiabatic
case, the barrier is due to the need for the copper ion to approach the electrode,
whereas in the ferric-ferrous case it was due to the energy required for solvent
rearrangement. These two reaction path directions correspond to the z and DE directions
in the figure, respectively. Equilibrium occurs for the ferric-ferrous case when
there is a large charge on the electrode. In this case, the easiest way for the transfer
to occur is solvent rearrangement with the ion far from the electrode where mixing
is weak. In the copper case, a small charge on the electrode allows closer approach
and strong coupling. In the future the large uncertainty in the calculated barrier
height, which arises from uncertainties in the matrix element coupling reactant and
product, will be reduced by using direct dynamics techniques that recalculate electronic
structure at each stage of a molecular dynamics calculation. The enhancement of the
electron transfer rate by chloride adsorbed on the surface is also being studied
by the same methods. |
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Figure 1: Free energy surfaces
for the cuprous-cupric electron transfer reaction calculated from molecular dynamics
simulation. The surface at the left is calculated assuming that the reaction is nonadiabatic,
while at the right it is assumed to be adiabatic. The z axis is the distance of the
ion from the electrode surface, and DE is a measure of the amount of solvent rearrangement.
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