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ID: past-exam-of-the-mphil-in-micro-and-nanotechnology-enterprise-course-of-the-university-of-cambridge/2023/paper-1/section-a-iii/7/i/solution

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Past exam of the MPhil in micro- and nanotechnology enterprise course of the University of Cambridge / 2023 / Paper 1 / Section A(iii) / 7 / i / Solution by

Codex  0 2026-10-20

At fixed discharge current, delivered voltage can be represented as

V = E_{eq} (Q) - I R - η_{act} - η_{conc} - η_{phase},

(1)

where

Q

is discharged charge and the terms denote distinct physical losses. A typical trace has an initial drop from open circuit, an approximately flat useful region and a final decline toward cutoff. Its exact shape depends on the cell chemistry and current; not every battery has an equally flat plateau.

Four distinguishable contributions are:

Ohmic polarization: electronic, ionic and contact resistance cause a rapid $I R$ drop on applying current. This need not wait for a concentration gradient to develop.
Activation overpotential: finite electrode-reaction kinetics require a departure from equilibrium to sustain charge transfer. It is important when the exchange current is small.
Concentration overpotential: electrolyte transport and diffusion within active particles can deplete reactants or accumulate products near reaction sites. These losses often grow near the end of discharge or at high current.
Phase-transformation polarization: nucleation and growth of a new phase can require extra driving force, produce an initial voltage dip or cause hysteresis. It is especially relevant to conversion and two-phase electrodes, rather than a universal independent loss in every chemistry.

Terminology sometimes groups solid-state transport or phase-change effects differently. The important distinction is the mechanism, not counting the same diffusion loss under two names. A falling equilibrium voltage as composition changes is also not, by itself, polarization.

Figure 1.
An illustrative discharge voltage combines equilibrium changes and distinct losses
. Original stipulated model, not battery measurements. The loss components show an instantaneous resistive contribution, activation loss, a transient nucleation-related contribution and increasing late-discharge transport loss. The equilibrium curve is plotted separately. **Details**.

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