Polychromatic Beer-Lambert Analysis
Allura Red × 510/10 nm Filter — Measured Nanodrop Spectra

Repeats the analysis from filter_update_analysis.py ("Allura Red × 510/10 Filter" section) using directly measured Nanodrop UV-Vis spectra recorded 2026-08-31, replacing the digitised + ε₅₀₈-scaled spectral shape.

2026-08-31 · Allura Red AC (FD&C Red 40) · 10 mM HEPES blank · Nanodrop L = 1.0 mm

1 · Measured Spectra Overview

Five Allura Red concentrations (0.375, 1.0, 1.5, 2.5, and 5 mM) were measured in the Nanodrop against a 10 mM HEPES blank at a 1.0 mm path length. The measured visible absorption peak is 493–496 nm, slightly blue-shifted from the ~506 nm value in the digitised spectrum used previously. The 510/10 filter passband (480.8–536.9 nm) overlaps this peak well.

All five measured Allura Red spectra overlaid with the 510/10 filter passband — **Figure 1.** All five measured Allura Red Nanodrop spectra (L = 1.0 mm, 10 mM HEPES blank) overlaid with the normalised 510/10 nm filter transmittance (right axis, purple). Concentrations above ~0.375 mM show saturation artefacts at the visible absorption peak.

2 · Reference Spectrum Selection

The polychromatic calculation requires A(λ) to be in the Beer-Lambert linear regime. At L = 1.0 mm the expected peak absorbance is approximately:

Concentration	Expected A₅₀₈ (L=1.0 mm)	Status
0.375 mM	~0.97	✓ linear
1.0 mM	~2.59	⚠ moderate saturation
1.5 mM	~3.88	✗ saturated
2.5 mM	~6.47	✗ severely saturated
5.0 mM	~12.9	✗ off-scale

The 0.375 mM spectrum is used as the reference. It has the lowest saturation risk in the 480–540 nm passband while still providing good signal. All higher concentrations are used only for the Beer-Lambert linearity check below.

Reference absorbance spectrum (0.375 mM) with 510/10 filter — **Figure 2.** Allura Red reference absorbance spectrum at 0.375 mM (L = 1.0 mm, left axis, green) overlaid with the 510/10 filter transmittance (right axis, purple). The filter passband (480–537 nm) sits across the visible absorption peak, maximising signal.

3 · Beer-Lambert Linearity Check

The filter-weighted polychromatic absorbance A_poly is computed for each measured concentration and compared against the ideal linear prediction anchored at 0.375 mM:

Beer-Lambert linearity check for Nanodrop spectra at L=1.0 mm — **Figure 3.** Filter-weighted absorbance `A_poly` for each measured concentration at L = 1.0 mm, versus the ideal Beer-Lambert line anchored to 0.375 mM (dashed). Strong negative deviations at 1.0 mM and above confirm Nanodrop saturation; only the 0.375 mM point is usable as a spectral reference.

Concentration (mM)	A_poly (measured)	Measured ratio / 0.375 mM	Expected ratio	Deviation (%)
0.375	0.6947	1.0000	1.0000	0.00 %
1.0	1.6614	2.3914	2.6667	−10.32 %
1.5	2.1795	3.1371	4.0000	−21.57 %
2.5	2.3458	3.3764	6.6667	−49.35 %
5.0	1.8024	2.5944	13.3333	−80.54 %

4 · Polychromatic Beer-Lambert Analysis

Using the 0.375 mM reference spectrum, the absorbance at any working condition (C, L) is scaled as:

A(λ; C, L) = A_ref(λ) × (C / 0.375 mM) × (L / 1000 µm)

The filter-weighted polychromatic transmittance and apparent absorbance are then:

T_poly = ∫ F(λ) · 10^−A(λ) dλ / ∫ F(λ) dλ → A_meas = −log₁₀(T_poly)

4.1 · Polychromatic transmittance at the reference condition

Reference condition: 0.375 mM · L = 1.0 mm
Filter-weighted mean A = 0.7022
T_poly (integrated) = 0.20196 → A_meas = 0.69475
T_naive (10^−Ā) = 0.19853 → A_mean = 0.70217
Polychromatic underestimate: 1.057 % — negligible for this dye–filter pair.

Polychromatic vs naive transmittance across the 510/10 passband — **Figure 4.** Monochromatic transmittance *T(λ)* across the 510/10 passband (blue fill) versus the polychromatic average T_poly (red dashed) and the naïve single-wavelength approximation T_naive (green dotted). The near-flat transmittance profile across the passband is why the polychromatic error is so small (<1.1 %).

4.2 · Absorbance vs. concentration fraction and deviation ratio

Scaling to a working concentration of 1.0 mM in the channels and a capillary calibration path length of 80 µm, the polychromatic absorbance is computed at four concentration fractions of the 1.0 mM stock:

A_poly vs A_naive and polychromatic deviation ratio — **Figure 5.** *Left:* Polychromatic A_meas (tomato) vs. naïve Beer-Lambert A_naive (green) at the 80 µm capillary for four concentration fractions of the 1.0 mM stock. *Right:* Deviation ratio A_poly / A_naive — values within 0.2 % of unity confirm that the polychromatic correction is negligible for Allura Red × 510/10.

4.3 · Effective molar absorptivity

Effective molar absorptivity vs concentration — **Figure 6.** Filter-weighted effective molar absorptivity ε_eff (tomato) versus true monochromatic ε (green dashed) for the four concentration fractions at L = 80 µm. The near-constant ε_eff = 1.871 L mol⁻¹ µm⁻¹ confirms Beer-Lambert linearity in the measurement range.

5 · Polychromatic A_meas Table

Computed A_meas for all eight channel heights (30–200 µm) at four concentration fractions of the 1.0 mM working stock. Reference calibration: 0.375 mM Nanodrop spectrum at L = 1.0 mm, scaled to (C, L) via Beer-Lambert.

Height (µm)	f = 1/8 (0.125 mM)	f = 1/4 (0.250 mM)	f = 1/2 (0.500 mM)	f = 1/1 (1.000 mM)
30	0.0070	0.0140	0.0281	0.0561
40	0.0094	0.0187	0.0374	0.0748
50	0.0117	0.0234	0.0468	0.0935
70	0.0164	0.0328	0.0655	0.1309
90	0.0211	0.0421	0.0842	0.1682
120	0.0281	0.0561	0.1122	0.2241
150	0.0351	0.0702	0.1402	0.2799
200	0.0468	0.0935	0.1868	0.3726

All values in absorbance units (AU). Working concentration in channels: 1.0 mM. Capillary calibration path length: 80 µm.

6 · Height Estimation Accuracy

Heights are estimated from measured A_meas using the capillary calibration at 80 µm: h_est = 80 µm × A_meas(h) / A_meas(80 µm). The error (h_est − h_true) / h_true quantifies the residual polychromatic bias.

Estimated vs true channel height and height estimation error — **Figure 7.** *Left:* Estimated height vs. true channel height for all four concentration fractions (plasma colormap). All curves lie on the ideal line (dashed), confirming negligible bias. *Right:* Height estimation error — all values are within ±0.2 % for all heights and concentrations, confirming that the polychromatic correction is not needed for this dye–filter combination.

Key conclusion: The polychromatic Beer-Lambert error for Allura Red × 510/10 is < 0.2 % across all channel heights (30–200 µm) and concentration fractions (1/8–1/1 of 1.0 mM). The height estimation is essentially linear, and no polychromatic correction is needed. This confirms the result from the digitised-spectrum analysis — the direct Nanodrop measurement produces consistent conclusions.

7 · Notes on Nanodrop Measurement Quality

Peak wavelength: The measured visible absorption peak is at 493–496 nm, slightly blue-shifted from the ~506 nm value in the digitised reference spectrum. This difference may reflect solvent composition, pH, or digitisation uncertainty in the literature source.
Saturation at L = 1.0 mm: Only the 0.375 mM spectrum is in the linear Beer-Lambert regime. For future Nanodrop measurements, use ≤ 0.4 mM or the shorter 0.1 mm Nanodrop path if available.
NaN values: The 2.5 mM and 5 mM spectra have NaN readings (5 % and 20 % respectively) in the 480–540 nm passband due to detector saturation. These spectra are not used for the polychromatic calculation.
Effective ε: ε_eff = 1.871 L mol⁻¹ µm⁻¹ from the measured spectrum versus 2.59 × 10⁴ L mol⁻¹ cm⁻¹ = 2.59 L mol⁻¹ µm⁻¹ from the literature value used previously. The 28 % difference likely reflects the digitisation error in the literature spectrum shape and/or concentration uncertainty in the 0.375 mM Nanodrop sample. The capillary calibration strategy eliminates this uncertainty from the height inference.

Generated 2026-08-31 from allura_red_nanodrop_polychromatic.py using measured Nanodrop data in 2026-08-31_allura_16bit/nanodrop/Allura_Red/. Figures produced with matplotlib; no JavaScript required.