  The geometric properties of $\mathrm{Pb-MnO_2}$ composite electrodes are
studied, and a general formula is presented for the length of the triple phase
boundary (TPB) on two dimensional (2D) composite electrodes using sphere
packing and cutting simulations. The difference in the geometrical properties
of 2D (or compact) and 3D (or porous) electrodes is discussed. It is found that
the length of the TPB is the only reasonable property of a 2D electrode that
follows a 1/r particle radius relationship. Subsequently, sphere packing cuts
are used to derive a statistical electrode surface that is the basis for the
earlier proposed simulations of different electrochemical mechanisms. It is
shown that two of the proposed mechanisms (conductivity and a
two-step-two-material kinetic mechanism) can explain the current increase at
$\mathrm{Pb-MnO_2}$ anodes compared to standard lead anodes. The results show
that although $\mathrm{MnO_2}$ has low conductivity, when combined with Pb as
the metal matrix, the behaviour of the composite is not purely ohmic but is
also affected by activation overpotentials, increasing the current density
close to the TPB. Current density is inversely proportional to the radius of
the catalyst particles, matching with earlier experimental results. A
hypothetical two-step-two-material mechanism with intermediate
$\mathrm{H_2O_2}$ that reacts on both the Pb matrix and $\mathrm{MnO_2}$
catalyst is studied. It was found that assuming quasi-reversible generation of
$\mathrm{H_2O_2}$ followed by its chemical decomposition on $\mathrm{MnO_2}$,
results are obtained that agree with the experiments. It is further emphasised
that both the Pb matrix and $\mathrm{MnO_2}$ catalyst are necessary and their
optimum ratio depends on the used current density. Yet, additional experimental
evidence is needed to support the postulated mechanism.
