Figure 2.

The time dependence of excess bulk proton concentration. The concentration n(r,z,t) was calculated at three observation points r,z at equal distances ρ = 0.25L₀ from acid-injection point. The strong-coupling case with equal diffusion coefficients, D_s = D_b was assumed. Solid line 1, r = ρ, z = 0; dotted line 2 and dashed line 3, the 2D and 3D limiting forms of line 1, respectively; solid lines 4 and 5, r = z = ρ/√2 and r = 0, z = ρ, respectively. All curves are normalized to a given number of protons injected at t = 0 inside a hemisphere of radius r₀=1μm around the coordinate origin. The signals at three observation points (solid lines) all have the 2D shape (line 2) at t < t₀, and they all approach the 3D shape (line 3) at t > t₀. The surface proton concentration σ(r,t) is given by Eq. (8).