Devlog #67 – Wave 47: Electrode Kinetics, Polymer Chain & Magnetic Domains

Wave 47 at a Glance

Electrode Kinetics — Butler-Volmer in Action

The rate at which an electrochemical reaction proceeds at an electrode surface is governed by the Butler-Volmer equation, which encodes two competing exponential processes — oxidation and reduction — each accelerated by the applied overpotential η:

j = j₀[exp(αFη/RT) − exp(−(1−α)Fη/RT)]

where j₀ is the exchange current density (the baseline current flowing in both directions at equilibrium), α is the transfer coefficient (asymmetry between forward and reverse activation energies), and η = E − E_eq is the overpotential.

Three Visualisation Modes

The right-hand canvas switches between three diagnostic plots that electrochemists routinely use:

• Polarization curve (j vs η): the familiar S-shaped current-voltage response, saturating toward the limiting current at large η.
• Tafel plot (log|j| vs η): linearises the Butler-Volmer equation in the high-η regime. The Tafel slope b = RT/((αF) for the anodic branch; kinetic parameters can be extracted graphically.
• Cyclic Voltammetry: a linear potential sweep is applied, sweeping forward then back, tracing out the characteristic duck-bill CV curve. Anodic (forward) scans are drawn in red, cathodic (reverse) in blue.

The left-hand electrode animation shows ion particles drifting toward and away from the electrode surface, with electron-transfer events spawning at a rate proportional to the instantaneous |j|. At high anodic overpotential the animation swarms with outbound electron transfers; at cathodic potentials the direction reverses.

Polymer Chain — From Random Walks to Flory Scaling

The simplest polymer model is the freely-jointed chain (FJC): N segments of equal length b, each oriented isotropically. In the absence of excluded-volume interactions (the “theta solvent” limit) the chain statistics are identical to a random walk, giving root-mean-square end-to-end distance R_ee = b√N and radius of gyration R_g = b√(N/6).

Once excluded-volume interactions are switched on (good solvent), the chain swells. Flory theory predicts the power-law scaling R_g ∝ N^ν with ν ≈ 0.588 in three dimensions. In a poor solvent the chain collapses and ν = 1/3.

Pivot Monte Carlo

Pure thermal diffusion is extremely slow at sampling polymer configuration space. The simulator uses pivot moves: a random bond along the chain is chosen, and the entire sub-chain on one side is rotated by a random angle about that pivot. The Metropolis criterion accepts the new configuration based on the Boltzmann weight of the energy change. For solvent quality, the energy model biases toward Rg changes — penalising expansion in poor solvent and compression in good solvent.

The right-hand histogram accumulates R_g samples over thousands of accepted moves, overlaying a Gaussian fit in yellow. The Gaussian approximation is exact at theta conditions; it deviates measurably in good and poor solvent regimes, as the distribution acquires a non-Gaussian tail.

Magnetic Domains — Ising Model & Hysteresis

Ferromagnetic materials below the Curie temperature T_c spontaneously magnetise, but the bulk sample may have zero net magnetisation because it is divided into Weiss domains — regions of aligned spins separated by domain walls. The 2D Ising model, exactly solved by Onsager in 1944, captures the essential physics:

H = −J ∑<ij> sisj − μH ∑i si

Each lattice site carries a spin s = ±1. Nearest neighbours interact with exchange coupling J > 0 (ferromagnetic). The Curie temperature is k_BT_c = 2J/ln(1+√2) ≈ 2.269J.

Real-Time Spin Lattice Rendering

The 80×60 spin lattice is rendered every frame using ImageData on a small off-screen canvas (80×60 pixels), then scaled to the full 480×300 display canvas via drawImage with imageSmoothingEnabled = false. This gives crisp pixel-art domain boundaries at negligible GPU cost. Up spins are indigo (#3f51b5); down spins are red (#e53935). Domain walls appear as sharp colour boundaries when T is well below T_c; they blur and fluctuate as T approaches T_c, and the entire lattice becomes a disordered salt-and-pepper pattern above T_c.

Animated B-H Loop

In Auto sweep mode the applied field H ramps linearly from −1 to +1 and back, recording (H, M) pairs at each step. The accumulated trace is drawn on the right-hand canvas: forward sweep in red, reverse in blue. The characteristic hysteresis loop emerges — its width reflects the coercive field H_c, its height the saturation magnetisation M_s, and its area the energy dissipated per cycle (Steinmetz’s law ∝ f B_max^1.6).

Five material presets capture the range from hard magnets (Cobalt, J=2.0, wide loop) to soft magnets (Permalloy, J=0.8, and soft ferrite, J=0.6, with narrow, nearly linear loops optimal for power transformers).

Companion Content

Wave 47 is accompanied by two blog posts on adjacent topics:

• Spotlight #48 — Materials Science & Engineering: a deep dive into crystal structures, alloy phase diagrams, stress-strain curves, shape memory alloys, band structure, superconductivity, and surface adsorption — all with links to on-platform simulators.
• Learning #36 — Fluid Mechanics & Hydrogeology: a nine-step guided exposition covering the Navier-Stokes equations, Reynolds number, Darcy’s law, the groundwater flow equation, and advection-diffusion transport.

What’s Next

The platform continues to fill gaps in coverage. High-priority additions under consideration include tribology (friction, wear and lubrication physics), seismic wave propagation (P and S waves in layered media), and quantum dot spectroscopy (quantum confinement and size-dependent fluorescence). On the blog side, a Spotlight covering earth-science simulations and a Learning entry on statistical mechanics and phase transitions are both planned.