Devlog #69 – Wave 49: Membrane Transport, Van der Waals Gas & Grain Growth

Wave 49 at a Glance

Membrane Transport — The Goldman-Hodgkin-Katz Equation

Every resting neuron sits at roughly −70 mV because its membrane is selectively permeable: K⁺ leaks out faster than Na⁺ leaks in. Quantifying this requires the Goldman-Hodgkin-Katz (GHK) voltage equation, derived in 1943–1949 by independently solving the Nernst-Planck electrodiffusion equation for each permeant ion under the assumption of a constant electric field across the membrane:

V_m = (RT/F) · ln[(P_Na[Na]_out + P_K[K]_out + P_Cl[Cl]_in) / (P_Na[Na]_in + P_K[K]_in + P_Cl[Cl]_out)]

Note the asymmetric sign for Cl⁻: because it carries negative charge, inside and outside concentrations are swapped in the numerator/denominator relative to cations.

Ion Concentrations and Nernst Potentials

The resting-state physiological concentrations are canonical:

• Na⁺: 145 mM extracellular, 12 mM intracellular → E_Na ≈ +67 mV (inward driving force at rest)
• K⁺: 4 mM extracellular, 140 mM intracellular → E_K ≈ −94 mV (outward driving force at rest)
• Cl⁻: 120 mM extracellular, 4 mM intracellular → E_Cl ≈ −89 mV

The resting membrane sits at −70 mV because P_K ≫ P_Na at rest — the cell is almost a potassium electrode. During an action potential, voltage-gated Na⁺ channels open transiently and P_Na spikes, driving V_m toward +35 mV.

Animation and Particle Physics

The simulation renders a cross-section of the lipid bilayer with phospholipid heads (circle-line icons) forming two leaflets. Na⁺ particles (blue), K⁺ (gold), and Cl⁻ (green) bounce within extracellular and intracellular compartments. Each particle receives a small directional drift proportional to its electrochemical driving force (V_m − E_ion), creating net unidirectional flow through open channels whenever the membrane is displaced from the Nernst potential of that ion. The Na⁺/K⁺-ATPase pump is shown as a purple protein that exchanges 3 Na⁺ outward for 2 K⁺ inward per cycle, contributing a small electrogenic current.

Van der Waals Gas — Maxwell Construction and the Critical Point

The ideal-gas law PV = nRT ignores intermolecular attractions and finite molecular volume. Van der Waals (1873) corrected both:

(P + a/V_m²)(V_m − b) = RT

The term a/V_m² represents the reduction in pressure due to attractive forces; b is the excluded volume (four times the molecular volume). At high temperatures the isotherm is monotonically decreasing and gas-like. Below the critical temperature T_c, the isotherm develops an unphysical “van der Waals loop” — a region where ∂P/∂V > 0 — which must be replaced by a flat horizontal tie line (two-phase coexistence) found by the Maxwell equal-area rule.

Critical Point and Reduced Variables

Setting ∂P/∂V = 0 and ∂²P/∂V² = 0 simultaneously gives:

T_c = 8a/(27Rb),  P_c = a/(27b²),  V_c = 3b

In reduced variables (T_r = T/T_c, P_r = P/P_c, V_r = V/V_c) the VdW equation becomes universal: all substances satisfying VdW assumptions obey the same equation of state — the law of corresponding states. This is why CO₂ and H₂O at T/T_c = 0.9 look qualitatively similar on a P-V diagram.

Maxwell Construction Implementation

Finding the equal-area pressure P_eq for a given subcritical T requires solving a cubic equation. The VdW isotherm in V_m is a depressed cubic and below T_c has three real roots (V_liq, V_spinodal1, V_spinodal2, V_vap). The Maxwell pressure satisfies:

∫(V_liq to V_vap) P_VdW dV = P_eq · (V_vap − V_liq)

The simulator uses a binary search on P_eq (between P_min and P_max of the loop) and for each candidate P_eq solves the cubic (P + a/V²)(V − b) = RT via the trigonometric method of Cardano to find all three real roots, then evaluates the area balance. Convergence to 10⁻⁶ relative tolerance takes fewer than 60 iterations.

Grain Growth — Microstructure Kinetics and Hall-Petch

When a cold-worked metal is annealed, stored strain energy drives grain boundary migration. Boundaries move toward their centre of curvature, eliminating small grains in favour of large ones. The macroscopic law (Burke and Turnbull, 1952) is parabolic:

d̄² − d₀² = K(T) · t,  K(T) = K₀ exp(−Q/RT)

where Q is the activation energy for grain-boundary migration (depends on lattice diffusivity of the rate-limiting species), K₀ is a pre-exponential factor, and R = 8.314 J/mol·K. Steel has Q ≈ 280 kJ/mol; aluminium only 142 kJ/mol because Al has a higher stacking-fault energy and its boundaries migrate more easily.

Hall-Petch Strengthening

Grain boundaries block dislocation glide. In 1951, Hall and Petch independently showed that yield strength follows:

σ_y = σ₀ + k_HP / √d̄

where σ₀ is the bulk lattice friction stress (independent of grain size) and k_HP is a material constant that measures how difficult it is to transmit slip across a grain boundary. For steel k_HP ≈ 0.74 MPa·m^½; for aluminium only 0.07 because Al grain boundaries are weaker barriers.

This means every time you anneal steel at 1000 °C and double the mean grain diameter, you halve the Hall-Petch contribution to yield strength — a direct trade-off between processability (coarse grains are easier to form) and mechanical performance (finer grains are stronger).

Visual Microstructure

The left canvas uses a Voronoi tessellation seeded with N₀ random points to construct the initial grain structure. Each grain receives a distinct pastel colour; grain boundaries are rendered in near-black. During animation, the smallest grains are progressively merged into their nearest neighbours by reassigning their pixels, reproducing curvature-driven coarsening qualitatively. The right canvas shows the parabolic d̄(t) growth curve and the Hall-Petch σ_y(d) log-scale curve — an orange dot on each marks the current state of the simulation.

What’s Next

The four weakest categories — cell-biology, physical-chemistry, materials, and immunology — have each been strengthened in this wave. Future waves will continue addressing similar gaps, with planned simulations in: osmosis and turgor pressure (cell biology), reaction kinetics and Arrhenius plots (physical chemistry), fatigue crack propagation (materials), and complement cascade activation (immunology).