Uncertainty Evaluation of the Diffusive Gradients in Thin Films Technique

Although the analytical performance of the diffusive gradients in thin films (DGT) technique is well investigated, there is no systematic analysis of the DGT measurement uncertainty and its sources. In this study we determine the uncertainties of bulk DGT measurements (not considering labile complexes) and of DGT-based chemical imaging using laser ablation - inductively coupled plasma mass spectrometry. We show that under well-controlled experimental conditions the relative combined uncertainties of bulk DGT measurements are ∼10% at a confidence interval of 95%. While several factors considerably contribute to the uncertainty of bulk DGT, the uncertainty of DGT LA-ICP-MS mainly depends on the signal variability of the ablation analysis. The combined uncertainties determined in this study support the use of DGT as a monitoring instrument. It is expected that the analytical requirements of legal frameworks, for example, the EU Drinking Water Directive, are met by DGT sampling.


CALCULATION OF COMBINED UNCERTAINTY.
In a bottom-up uncertainty evaluation, all individual contributors to uncertainty are determined and a mathematical model is used for the propagation of all errors in the full analytical process. The measurement uncertainty can be determined rigorously using a function based on partial derivatives but for most applications this is impractical due to its complexity. Therefore a spreadsheet approach for the approximation of uncertainty can be used which also covers the full analytical process (Eq. S1) to calculate the combined standard uncertainty (u c ), characterizing the dispersion of the measurand during the experiment. In Eq. S1 u c is the combined uncertainty of the measurand (y) depending on its input quantities (x 1 , x 2 ,…, x n ) which is calculated by taking the square root of the sum of squares of the contributor uncertainties u(x 1 ), u(x 2 ), …, u(x n ). The Kragten spreadsheet approximation can handle simple, linear model equations, the parameters used have to be mutually independent and compensating errors have to be avoided. If dependent variables occur other equations have to be used for the estimation of uncertainty. 1 u c �y (x 1 , x 2 , … ) � = �u x 1 2 + u x 2 2 + … (Eq. S1) The variability of the observed values for the sources of uncertainty are transformed into normally distributed standard uncertainties (u x1 , u x2 , …) by assigning probability distributions. If the input quantity is estimated from independent repeated observations the arithmetic mean and standard deviation are used as the input estimate (Type A distribution). However, if an estimate of an input quantity has not been obtained from repeated observations, the associated estimated variance or the standard uncertainty is evaluated by scientific judgment based on all available information on the possible variability (Type B distribution). If a stated value and standard deviation are available from, e.g., a manufacturer's handbook, the quantity value and standard deviation are directly used. If a value is equally probable to lie anywhere within a range of ± a(x i ) (a uniform or rectangular distribution of possible values) an equivalent of the standard deviation is estimated by u 2 (x i ) = a(x i ) 2 /3. In a case where it is more realistic to expect that values near the bounds are less likely than those near the midpoint (a triangular distribution of possible values) an equivalent of the standard deviation is estimated by u 2 (x i ) = a(x i ) 2 /6. 2 The resulting combined uncertainty is normally distributed and multiplied with a coverage factor to give the expanded uncertainty (U) which is equivalent to a confidence interval of 68 % (coverage factor of k = 1), k = 2 is equivalent to a 95 % and k = 3 to a 99 % confidence interval.

CAUSE AND EFFECT DIAGRAM OF THE UNCERTAINTY SOURCES.
A Cause-and-effect diagram is used to identify all relevant contributors to uncertainty. This step aims at a comprehensive identification of uncertainty sources, but also to avoid including uncertainty contributors more than once. Figure SI 1

Eluate Dilution Factor and Eluate
Volume. An uncertainty of 1 % was used to account for the uncertainty of pipetting. This uncertainty value was assigned to the eluate volume (V s ) as well as of the eluate dilution factor (f dil ) and was considered to be triangularly distributed. Gel Swelling during Elution. The calculation of elution factors usually assumes that the resin gel disc volume (196 µL; 2.5 cm diameter, 0.4 mm thickness) consists largely of water and therefore dilutes the eluent volume in which it is placed. However, an increased resin gel weight of 332 ± 15 µg (n = 12) was observed after placing Chelex 100 gels in the eluent (1 mol L -1 HNO 3 , ρ = 1.027 g ml -1 ). Presumably, the known effect of gel-swelling in acid 9 leads to the removal of more volume from the solution causing a systematic overestimation of the eluate volume. If this is left unaccounted, e.g. by assuming a constant eluate volume which is directly used for the further sample preparation, this systematic error is introduced into the analysis. In case of large elution volumes, the bias is small (for 10 mL: ~1 %) but if the elution is performed in 1 mL of eluent and the loss of volume is not accounted for, this can cause a bias of ~ 10 %.

CRITICAL ASPECTS OF GEL ELUTION
As a systematic error, this can easily be corrected for by using defined eluate aliquots for sample preparation or by considering gel swelling during the elution process. Also, the use of larger eluate S-5 volumes (10 mL instead of 1 mL) minimizes systematic as well as random errors introduced by the elution procedure. Table S1.  Table S2. * Three values for Cd and Cu at ionic strengths from 0.1 -0.001 mol L -1 were reported. Here the average of these three values was used.

Measurement of Diffusion Coefficients and Calculation of Combined Uncertainty. A diffusion cell
comprises two compartments which have a circular, 1.5 cm diameter window. A 2.5 cm diameter gel disc is placed between these openings and the assembly is clamped together. One compartment contains a solution containing the target analyte (source solution) and the other compartment an analyte-free solution (receptor solution). 20 The solutions in both compartments (each compartment contained 100 mL solution) are well stirred with magnetic stirrers and solution temperature was monitored. Subsamples were taken every 20 minutes during at least 3 h from both compartments. The pH value was very similar in both compartments (pH ± 0.2) and ionic strength was controlled (I = 10 mmol L -1 ) by adding NaNO 3 (Sigma Aldrich, Reagent Plus).
The diffusion coefficient was calculated using Eq. S2 according to Scally et al. 20 Eq. S2 Where A (cm 2 ) is the exposure area, Δg is the diffusive layer thickness (cm) between the two compartments, C is the initial concentration of the analyte containing solution (ng cm -3 ), and a is the slope of the plotted mass flow versus timein ng min -1 . The factor '60' is correcting the value from minutes to seconds.

S-7
The uncertainty of the diffusion coefficients was calculated using the Kragten approach as described in the main paper based on Eq. S2. The parameters, their description, type, distribution and unit are shown in Table S2.  and LOQ (limit of quantification) of the instrumental performance as well as estimations of the MDL (method detection limit) and PQL (practical quantification limit) considering the DGT deployment were made. The PQL is the lowest concentration that can be reliably measured during routine laboratory operating conditions. This was done by the measurement of blank solutions (i.e. of DGT eluate samples of blank DGT gels) carried out on a ELEMENT XR (Thermo Fisher Scientific, Bremen, Germany) ICPMS instrument. LOD and LOQ were calculated as 3 times and 10 times the standard deviation of calibration blanks; MDL and PQL were calculated as 3 times and 10 times the standard deviation of DGT gel blank eluates.

EXAMPLE OF UNCERTAINTY CONTRIBUTORS IN DGT LA-ICP-MS.
The contributors to the uncertainty of a DGT based LA-ICP-MS measurement were investigated (see Table 2). The relative distributions of uncertainty contributors of a representative measurement are visualized in Fig. S2. S-9

FITS FOR DGT -LA-ICP-MS CALIBRATIONS.
Four calibrations for As, Cd, Cu and P were computed using the widely used ordinary least squares regression (OLS) and the bivariate weighted fitting (BWF). 22,23 The resulting data is shown in Table S5. Table S5. Calibration data for OLS and BWF fits for As, Cd, Cu and P in the form of y = ax + b with the corresponding uncertainties, for which the standard errors of the line parameter estimates (slope, intercept) were adopted.