Allura Red Analysis — Stages 5–8

Dataset: 2026-08-31  |  Dye: Allura Red  |  Device: device_1  |  Channel: green

Empirical Beer–Lambert analysis of stepped-height microfluidic channel images acquired 2026-08-31 with 5 Allura Red concentrations (0.375, 1.0, 1.5, 2.5, 5.0 mM).

Transmittance:

Absorbance:

Forward model (empirical Beer–Lambert):

No Nanodrop spectrum or filter profile is used. The scale s absorbs all unknown proportionality factors (molar absorptivity, filter efficiency, camera sensitivity) and is determined purely from the capillary at known path length L = 80 µm.

Stage 5a — Green vs Blue Channel Comparison (Capillary)

The capillary (L = 80 µm, known path length) is used to compare measured absorbance for the green and blue camera channels against a purely linear Beer–Lambert model (no nonlinearity correction, no free parameters). Whichever channel shows smaller residuals is selected for all subsequent stages.

Left: measured vs linear-fit absorbance for both channels at the capillary. Right: residuals from the linear fit. Systematic curvature in both channels indicates mild nonlinearity that is accounted for in Stage 6.

Stage 6 — Capillary Calibration: Empirical (s, η) Fit

The effective absorptivity scale s [AU/(mM·µm)] and camera nonlinearity parameter η are fit jointly from the 5 capillary measurements at known L = 80 µm using Nelder–Mead minimisation.

s anchors the absolute path-length scale for all subsequent stages — knowing L = 80 µm exactly is what lets us convert from absorbance to height. η captures the systematic deviation from strict linearity; at the highest capillary condition (5 mM × 80 µm = 400 mM·µm) the correction is -8.44%.

Stage 6 fit results

Left: measured absorbance vs linear prediction for the green channel; the small positive deviation at high concentrations reflects η > 0. Right: residuals after fitting both s and η — the curve is removed and residuals are near zero.

Stage 7 — Joint Channel Height + Nonlinearity Fit

The 8 section heights h_i and the nonlinearity η are fit jointly from the full 8×5 channel absorbance matrix (8 sections × 5 concentrations = 40 measurements), using s from Stage 6 to anchor the absolute scale.

Degrees of freedom: 40 measurements − 9 unknowns (8 h_i + η) = 31 DoF. The 6.7× concentration span (0.375–5 mM) gives strong leverage on η because it produces a wide absorbance range; s from Stage 6 anchors the absolute height scale.

Stage 7 height fit results

s = 0.0008683 AU/(mM·µm) — Stage 6 capillary (fixed)
η_cap = -0.02111 — Stage 6, capillary-only fit
η_fit = -0.03570 — Stage 7, from 6.7× channel concentration span
RMS = 0.00685 AU over 40 channel measurements (31 DoF)

Left: residual heatmap (A_meas − A_pred). Blue = model overpredicts; red = model underpredicts. A systematic negative column at 5 mM is visible. Right: fitted vs nominal heights; all sections agree within a few µm except 200 µm, which lands at 212.7 µm.

Anomaly: the Stage 7 free fit places the 200 µm section at 212.7 µm, roughly 6% above the 200 ± 1 µm confirmed by direct measurement. All other sections (30–150 µm) fall within a few percent of their nominal depths. The residual heatmap also shows a systematic negative bias in the 5 mM column for all sections except 200 µm. Three sensitivity tests below investigate whether this reflects model error, a concentration outlier, or a true depth discrepancy.

Sensitivity Test 1 — 4 Concentrations (Excluding 5 mM)

The 5 mM column shows a clear systematic pattern in the residual heatmap: negative residuals for all sections except 200 µm (positive). One hypothesis is that the empirical quadratic correction extrapolates poorly to the highest Ch products (up to 1000 mM·µm at 5 mM × 200 µm, vs the capillary calibration pivot of 80 mM·µm). This test repeats the fit using only the four lower concentrations (0.375, 1.0, 1.5, 2.5 mM), reducing the maximum Ch to 500 mM·µm.

Degrees of freedom: 4 × 8 = 32 measurements − 9 unknowns = 23 DoF.

5-conc fit: η = -0.03570, RMS = 0.00685 AU (40 meas, 31 DoF)
4-conc fit: η = -0.04446, RMS = 0.00278 AU (32 meas, 23 DoF)

Left: 4-conc residuals. Centre: 5-conc residuals for comparison. Right: fitted heights for both fits — orange triangles (4-conc) sit systematically above blue circles (5-conc) for every section, including 200 µm.

Sensitivity Test 2 — Fix h_200 = 200 µm (Option 2)

A note on concentration errors and the dilution series. All five concentrations were prepared by diluting the same 5 mM stock solution. Any error in the stock propagates as the same relative factor to every concentration. A uniform scale factor on all concentrations aliases exactly through s and cannot produce a column-specific residual pattern. The 5 mM column's systematic negative residuals therefore cannot be explained by a stock error; they point to model inadequacy at large Ch products.

This test pins h_200 = 200 µm from direct measurement and re-fits the remaining 7 heights and η from all 40 channel measurements.

Degrees of freedom: 40 measurements − 8 unknowns (7 h_i + η) = 32 DoF.

Free fit: η = -0.03570, RMS = 0.00685 AU (40 meas, 31 DoF)
Fixed h_200: η = -0.03487, RMS = 0.00773 AU (40 meas, 32 DoF)
The 200 µm row residuals (blue annotations in centre heatmap) reveal whether the measured absorbances there are consistent with h = 200 µm under the fitted η.

Left: free-fit residuals. Centre: fixed-h_200 residuals (200 µm row annotated in blue). Right: height comparison — orange triangles (fixed) coincide with blue circles (free) for all sections except 200 µm.

Sensitivity Test 3 — Channel-Only Fit (No Capillary; h_200 = 200 µm Anchor)

What if we ignore the capillary calibration entirely and let s float as a free parameter, anchored instead by pinning h_200 = 200 µm from direct measurement?

Scale degeneracy: Beer–Lambert absorbance depends on the product s·C·h. Scaling s by k and all heights by 1/k leaves every prediction unchanged — the fit is under-determined without an anchor. Pinning h_200 = 200 µm breaks this degeneracy: the absolute height scale is now set by the direct measurement, not by the capillary.

Free parameters: s, η, and h₃₀–h₁₅₀ (7 heights) — 9 unknowns from 40 measurements → 31 DoF (same as Stage 7, but capillary data play no role).

Comparing the fitted s_nocap to s_emp from the capillary quantifies how much the two anchoring methods agree on the absolute absorptivity scale.

Free fit (Stage 7): s = 0.0008683 AU/(mM·µm), η = -0.03570, RMS = 0.00685 AU
Fix h_200 (Option 2): s = 0.0008683 AU/(mM·µm), η = -0.03487, RMS = 0.00773 AU
Channel-only: s = 0.0009215 AU/(mM·µm), η = -0.03806, RMS = 0.00685 AU

The ratio s_nocap / s_emp = 1.0613 — the two anchoring methods disagree on the absolute scale by 6.1%.

Left: channel-only residuals (200 µm row in blue). Centre: free-fit residuals for comparison. Right: all three fits on the same height scatter plot — green squares (channel-only) sit systematically below blue circles (free fit with capillary anchor).

Keyence 3-Point Anchor Fit

The Keyence confocal distance sensor directly measured three section heights. All three are above their nominal depths, consistent with the Stage 7 free-fit placing h₂₀₀ at 212.7 µm:

These three heights are pinned; s, η, and the remaining five heights (30, 40, 50, 70, 90 µm) are free — giving 40 − 7 = 33 DoF.

s_key3 = 0.0008561 AU/(mM·µm) (s_key3 / s_emp = 0.9859)
η_key3 = -0.03512
RMS = 0.00757 AU (33 DoF; cf. 0.00685 AU for Stage 7 free, 31 DoF)

Left: Keyence 3-pt residuals heatmap; blue annotations mark the three fixed rows, dashed lines separate them. Centre: Stage 7 free residuals for comparison. Right: height scatter with all fits overlaid — the three open-circle markers show Keyence-fixed points.

Keyence 4-Point Anchor Fit

A fourth Keyence measurement of the 70 µm section (67.7 ± 1.44 µm) is now available, providing a critical data point: unlike the 120–200 µm sections, the 70 µm section is below nominal (ratio 0.967).

These four heights are pinned; s, η, and h₃₀, h₄₀, h₅₀, h₉₀ are free — 40 − 6 = 34 DoF.

s_key4 = 0.0008569 AU/(mM·µm) (s_key4 / s_emp = 0.9868)
η_key4 = -0.03518
RMS = 0.00758 AU (34 DoF)

Left: Keyence 4-pt residuals; four fixed rows marked. Centre: Keyence 3-pt residuals for comparison. Right: height comparison across all four fits.

Fabrication depth pattern

Four independent Keyence measurements reveal a systematic fabrication crossover:

• ≤ 90 µm sections: fabricated depth is 2–4% below nominal (h/nom ≈ 0.962–0.978)
• ≥ 120 µm sections: fabricated depth is 2.5–3.5% above nominal (h/nom ≈ 1.025–1.036)

The crossover (where fabricated depth equals nominal) lies between 90 and 120 µm. This depth-dependent sign reversal is consistent with known behaviour in UV-photopolymer layer-by-layer fabrication, where thin features may be undercut and thick features overcut relative to nominal.

Recommended heights for future concentration calculations

Summary of Findings

Two self-consistent interpretations exist, differing by 6.1%:

1. Capillary anchor (Stage 7): s is set by the capillary at L = 80 µm; the fit then requires h_200 ≈ 213 µm. If the capillary path length and concentrations are accurate, the 200 µm section was fabricated ~6% deeper than its nominal height.
2. Direct-measurement anchor (channel-only): h_200 is fixed at 200 µm; s is then 6.1% higher than the capillary-derived value and all other heights are correspondingly lower. If the 200 µm measurement is accurate, the capillary L or the stock concentration carries a ~6% error.