Highly conducting interfaces: equivalent conductivity vs. aspect ratio

The companion of 32_spheroid_effective_conductivity.jl, which treats the low-conducting (Kapitza) interface. Here the interface is highly conducting — a surface layer of zero thickness and finite surface conductance $\beta$:

\[\text{LC:}\quad [\![T]\!] = \rho\,q_n \qquad\text{vs.}\qquad \text{HC:}\quad [\![q_n]\!] = -\beta\,\mathrm{div}_S(\nabla_S T).\]

The two are duals: LC impedes the normal flux, HC adds a tangential short-circuit. Both are the imperfect-interface models of [54]; the confocal $N$-layer solution used here is [50], and the surface-conductive model goes back to [53].

What the script computes. For an insulating particle carrying an HC interface, the equivalent particle conductivity

\[\boldsymbol{k}^{eq} = \boldsymbol{B}_\Omega\cdot\boldsymbol{A}_\Omega^{-1},\]

where $\boldsymbol{A}_\Omega$ and $\boldsymbol{B}_\Omega$ are the volume-averaged concentration tensors of the particle, defined by $\langle\nabla T\rangle_\Omega = \boldsymbol{A}_\Omega\cdot\underline{H}$ and $\langle\boldsymbol{K}\cdot\nabla T\rangle_\Omega = \boldsymbol{B}_\Omega\cdot\underline{H}$ under a remote gradient $\underline{H}$. Its transverse and axial components are traced against the aspect ratio $\varpi = \rho_a/\rho_t$, over the oblate ($\varpi<1$) and prolate ($\varpi>1$) ranges — the HC counterpart of the four LC figures of the reference paper.

Because $\beta$ carries a length (it is a conductance per unit length, not a conductivity), $k^{eq}$ is size-dependent; the natural dimensionless group is $\beta/(k_m\,b)$ with $b$ a particle radius. The curves below are normalized so that this dependence is explicit.

Theory: Layered spheroid.

using MeanFieldHom
using TensND
using Printf
using Plots
gr()

Plots.GRBackend()

Setup

An insulating core (k₁ ≈ 0) isolates the interface contribution: whatever conductivity the particle ends up showing is carried entirely by its skin. The matrix is the unit-conductivity reference.

const KM = 1.0
const KCORE = 1.0e-8
const NSERIES = 6
const K0 = TensISO{3}(KM)
const KC = TensISO{3}(KCORE)

3×3 TensND.TensISO{2, 3, Float64, 1}:
 1.0e-8   ⋅       ⋅ 
  ⋅      1.0e-8   ⋅ 
  ⋅       ⋅      1.0e-8

Build a one-layer spheroid of aspect ratio ϖ at fixed transverse radius ρ_t = 1, carrying an HC interface of surface conductance β.

function _particle(ϖ, β; ρ_t = 1.0)
    ρ_a = ϖ * ρ_t
    return LayeredSpheroid(
        (ρ_a,), (ρ_t,), (KC,);
        interfaces = (MeanFieldHom.SurfaceConductiveInterface(β),),
        Nseries = NSERIES,
    )
end

_particle (generic function with 1 method)

k^eq from the volume-averaged concentration tensors. Both are transversely isotropic and diagonal in the spheroid frame, so the ratio is taken component-wise: index 1 is transverse, index 3 axial.

function _keq(ϖ, β)
    s = _particle(ϖ, β)
    A = gradient_gradient_loc(s, KC, K0)
    B = flux_gradient_loc(s, KC, K0)
    return real(B[1, 1] / A[1, 1]), real(B[3, 3] / A[3, 3])
end

_keq (generic function with 1 method)

Aspect-ratio sweep

Oblate side $\varpi \in [10^{-1}, 1]$ and prolate side $\varpi \in [1, 10]$, for a range of surface conductances.

# The sphere `ϖ = 1` is excluded from both ranges: it has zero focal distance,
# so it is not a confocal spheroid at all and `LayeredSpheroid` rejects it
# ("use LayeredSphere"). The sweeps stop just short of it on either side.
const BETAS = (0.25, 0.5, 1.0, 2.0)
const OBL = exp10.(range(-1, -0.02; length = 40))
const PRO = exp10.(range(0.02, 1; length = 40))

function _curves(ϖs)
    kt = Dict{Float64, Vector{Float64}}()
    ka = Dict{Float64, Vector{Float64}}()
    for β in BETAS
        vt = Float64[]; va = Float64[]
        for ϖ in ϖs
            t, a = _keq(ϖ, β)
            push!(vt, t); push!(va, a)
        end
        kt[β] = vt; ka[β] = va
    end
    return kt, ka
end

kt_obl, ka_obl = _curves(OBL)
kt_pro, ka_pro = _curves(PRO)

println("HC (surface-conductive) interface on an insulating spheroid")
println("─"^70)
@printf "%-8s %12s %12s %12s %12s\n" "ϖ" "k_t (β=0.25)" "k_a (β=0.25)" "k_t (β=2)" "k_a (β=2)"
for (i, ϖ) in enumerate(OBL)
    (i - 1) % 13 == 0 || continue
    @printf "%-8.3f %12.4f %12.4f %12.4f %12.4f\n" ϖ kt_obl[0.25][i] ka_obl[0.25][i] kt_obl[2.0][i] ka_obl[2.0][i]
end
println()
println("Sanity checks:")
@printf "  β → 0 recovers the insulating particle : k_t = %.2e (expect ≈ %.0e)\n" _keq(0.5, 1.0e-9)[1] KCORE
println("  k^eq grows monotonically with β at fixed shape :        ",
    all(kt_obl[0.25] .< kt_obl[2.0]) ? "OK" : "FAILED")
println()

HC (surface-conductive) interface on an insulating spheroid
──────────────────────────────────────────────────────────────────────
ϖ        k_t (β=0.25) k_a (β=0.25)    k_t (β=2)    k_a (β=2)
0.100          3.6376       0.1396      28.6958       1.0291
0.212          1.7613       0.2545      13.9257       1.9337
0.450          0.8967       0.3908       7.1326       3.0782
0.955          0.5142       0.4953       4.1136       3.9622

Sanity checks:
  β → 0 recovers the insulating particle : k_t = 1.33e-08 (expect ≈ 1e-08)
  k^eq grows monotonically with β at fixed shape :        OK

Graphical output

Four panels: transverse and axial equivalent conductivity, over the oblate and the prolate range. The flat-particle limit is where the effect is strongest — a thin oblate platelet has a large surface-to-volume ratio, so its skin dominates.

function _panel(ϖs, curves, ttl, ylab; xlog = true)
    plt = plot(;
        xscale = xlog ? :log10 : :identity, yscale = :log10,
        xlabel = "aspect ratio ϖ = ρ_a / ρ_t", ylabel = ylab, title = ttl,
        legend = :best, size = (620, 460),
        left_margin = 10Plots.mm, bottom_margin = 5Plots.mm,
    )
    for (i, β) in enumerate(BETAS)
        plot!(plt, ϖs, curves[β]; label = "β = $β", lw = 2, color = i)
    end
    hline!(plt, [KM]; color = :grey, ls = :dash, label = "matrix k_m")
    return plt
end

p1 = _panel(OBL, kt_obl, "Oblate — transverse", "k_t^eq / k_m")
p2 = _panel(PRO, kt_pro, "Prolate — transverse", "k_t^eq / k_m")
p3 = _panel(OBL, ka_obl, "Oblate — axial", "k_a^eq / k_m")
p4 = _panel(PRO, ka_pro, "Prolate — axial", "k_a^eq / k_m")

p_full = plot(p1, p2, p3, p4; layout = (2, 2), size = (1250, 900))
p_full

Effect on the effective conductivity of a composite

The equivalent particle is not just a diagnostic: it can be fed to any mean-field scheme as an ordinary homogeneous ellipsoid. Here a Mori–Tanaka estimate for aligned insulating platelets, with and without the HC skin.

const FRAC = 0.15
const OMEGA_PLATE = 0.1

function _khom(β)
    s = _particle(OMEGA_PLATE, β)
    rve = RVE(:M)
    add_matrix!(rve, Ellipsoid(1.0, 1.0, 1.0), Dict(:K => K0))
    add_phase!(rve, :I, s, Dict(:K => KC); fraction = FRAC)
    K = homogenize(rve, MoriTanaka(), :K)
    return real(K[1, 1]), real(K[3, 3])
end

βsweep = range(0.0, 3.0; length = 30)
kh_t = Float64[]; kh_a = Float64[]
for β in βsweep
    t, a = _khom(β)
    push!(kh_t, t); push!(kh_a, a)
end

@printf "Mori–Tanaka, aligned platelets ϖ=%.2f at f=%.2f, insulating core:\n" OMEGA_PLATE FRAC
@printf "  β = 0 :  k_11/k_m = %.4f   k_33/k_m = %.4f\n" kh_t[1]/KM kh_a[1]/KM
@printf "  β = 3 :  k_11/k_m = %.4f   k_33/k_m = %.4f\n" kh_t[end]/KM kh_a[end]/KM
println("  → the skin turns an obstacle into a conductor above β ≈ ",
    (idx = findfirst(kh_t .> KM); idx === nothing ? "never (in this range)" : @sprintf("%.2f", βsweep[idx])))

p_hom = plot(
    βsweep, kh_t ./ KM; label = "k₁₁ (in-plane)", lw = 2.5, color = :steelblue,
    xlabel = "surface conductance β", ylabel = "k^eff / k_m",
    title = "Mori–Tanaka: insulating platelets with an HC skin (f = $FRAC, ϖ = $OMEGA_PLATE)",
    legend = :topleft, size = (780, 480),
)
plot!(p_hom, βsweep, kh_a ./ KM; label = "k₃₃ (out-of-plane)", lw = 2.5, color = :darkorange)
hline!(p_hom, [1.0]; color = :grey, ls = :dash, label = "bare matrix")
p_hom

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