Deconvolution

Background

The release rates of gases can be deconvoluted in order to estimate the contribution of different entities. For samples of activated carbons, e.g. this could be different oxygenated surface complexes, that decompose in overlapping temperature ranges, releasing the same gas(es).

The deconvolution is performed using the Scipy curve_fit function and the Trust Region Reflective algorithm. By default the sum of gaussian functions is used to represent the gas profiles. Each gaussian function can be characterized with three parameters:

• center (T_m) in °C, the mode of the function
• height in arbitrary units, depending on FTIR. The maximum height at peak center
• width in °C (commonly, the Half Width at Half Maximum, HWHM is used)

Initial values and bounds

Initial values and bounds for the optimization can be defined in the file Fitting_parameter.xlsx. For a proper functionality, solely the sheet with the name 'center_0' must be filled in. In the first column, the name of the parameter sets are listed that can be used as 'reference' for the fitting functions of the package.

The first two rows specify the group by name and the released gas of the different entities. More entities can be appended in new columns. In your 'reference' row, the initial position (Temperature) of the peaks can be defined. Only entities with supplied value are used for the deconvolution. Names of the groups may contain spaces. If an underscore '_' is used the end of the group's name is indicated, following numbers allow to use more than one peak to be assigned to the same group. Package functions support subsequent summarization (or condensation) of groups of the same name.

The other sheets of the file can be used to define individual initial values as well as upper and lower boundaries for center, HWHM and height for each entity. Blank cells are internally filled with the default values supplied in the settings.ini file.

• The initial value for 'height' is defined as a fraction of the maximum value of the relese rate.

• The Boundaries for 'center' are set as an symmetric interval with the width 2 * tol_center around the initial values.

  E.g. supplying center_0 = 300, tol_center: 30 results in the interval [center_0 - tol_center, center_0 + tol_center] = [270, 330]

• _0, _min and _max of 'hwhm' are given in absolute °C.

For all manually supplied parameters, _min < _0 < _max must hold!

Fits

To perform the TG_IR.fit() operation on several objects and calculate statistical values for samples of the same type, the tir.fits() fuction is used.

the objs-list may contain different samples, as they are sorted internally according to following mask applied to object's 'alias': sample_number where number has up to three digits.

E.g. calcium-oxalate-monohydrate_01 is recognized as calcium-oxalate-monohydrate with the number 01.

For all samples in the objs-list, the mean and standard deviation (stddev) are calculated for center, height, hwhm, area and mmol for each entity (gaussian function). For the mmol_per_mg, q, the deviation, Δq (dev), is calculated according to error propagation with Δn = LOD and Δm = mass resolution of thermo balance. For multiple measurements, the mean deviation is used.

Eq: deviation of molar amount

Eq: mean deviation

Robustness

To test the robustness of the fitting results by varying the fitting parameters center, center tolerance, initial height and HWHM as well as maximum HWHM, the tir.robustness() funtion is used.

For each parameter, initial center (center_0), center tolerance (tolerance_center), maximum HWHM (hwhm_max), initial height (height_0) and initial HWHM (hwhm_0) it was raised and lowered by the respective variance (var_T for center, tolerance center and maximum HWHM by default 10°C; var_rel for initial height and HWHM relative to maximum height and HWHM by default 30%). For center, the initial center is varied separately for each gaussian function.

For each obj in objs, this results in 1 + 2 * (4 + N(gaussian functions)) individual fits.

1 [initial fit] + 2 [due to raising and lowering] * (4 [tolerance_center, hwhm_max, height_0, hwhm_0] + N(gaussian functions))

For each of the five parameters, there are now two results for q in addition to the initial results, 11 in total. For a better overview these results are further summarized:

• mean: mean of all mean values for each sample
• meanstddev: standard deviation of all mean values for each sample
• stddev: standard deviation for all individual measurement of each sample
• min: minimum q from all 11 fitting results
• max: maximum q from all 11 fitting results

The results of the robustness testing are saved as figures and in an Excel file robustness_in_mmol_per_mg.xlsx. These results can be loaded, individually arranged and summarized by evaluation functions.