Eficient computational methods for the characterization of the materials at the nanoscale

Investigators: Viktor Witkovský (ÚM SAV), Gejza Wimmer (MÚ SAV), Anna Charvátová Campbell (ČMI), Petr Klapetek (ČMI), Radek Šlesinger (ČMI) As part of the international bilateral SK-CZ project, we designed and implemented efficient computational methods for evaluating the mechanical properties of materials at the nanoscale using instrumented indentation (IIT) and atomic force microscopy (AFM). The OEFPIL algorithm (Optimal Estimation of Function Parameters by Iterative Linearization) was optimized and applied for the analysis of data with errors-in-variables and correlated measurements, enabling more accurate and robust estimates of parameters and their uncertainties. The algorithm was implemented as open-source libraries in C and MATLAB. Within the project, we also designed and demonstrated an automated calibration procedure for scanning thermal microscopy (SThM), which simplifies and accelerates experimental measurements while maintaining reliable results. The OEFPIL algorithm was further applied to the analysis of force–distance (FD) curves in AFM, calibration curves of nanoindenter tips in IIT, and to the evaluation of measurement uncertainties obtained using coordinate measuring machines (CMM). The results of the project were presented in several scientific articles published in peer-reviewed journals indexed in WoS and Scopus, as well as in refereed conference proceedings, documenting applications of the OEFPIL algorithm to FD curve analysis in AFM, nanoindenter tip calibration, SThM calibration, and CMM uncertainty evaluation. The project outcomes contributed to a better understanding and quantification of measurement uncertainties in nano- and micro-metrology, to the development of new algorithmic approaches suitable for implementation on field-programmable gate arrays (FPGAs), and to the simplification of experimental procedures for characterizing the mechanical and thermal properties of materials. The project thus fulfilled all its objectives and strengthened the link between fundamental research and metrological practice.

Fig.: An example of the resulting calibration curve of the scanning thermal microscope which relates the true values of the measurand Y (the difference between the voltage measured on the sample and the voltage measured on the reference sample SiO2) to the true value of the thermal conductivity. The relationship between the true values is expressed by equation Y = a·k / (b + k) + c, where a, b, and c are fitting parameters. The uncertainties shown in the image are expanded. We can clearly see that for high thermal conductivities the function changes only little, leading to high uncertainties in conductivity. The method will be the most useful in the range of the steep slope between approx. 0.1 and 10  W·m⁻¹·K⁻¹. The calibration sample set consisted of six reference samples: PMMA (polymethyl methacrylate), POM-C (polyoxymethylene copolymer), SiO₂ (silicon dioxide), glass,  Al₂O₃  (aluminum oxide), and p-doped silicon (silicon doped with positive charge carriers). The number of the calibration samples was chosen to keep the time requirements within one day of (automated) measurement.

International Partner: Český metrologický institut (ČMI), Okružní 772/31, 638 00 Brno, Česká republika, Related Projects: SK-CZ-RD-21-0109 / INTER-ACTION-LUASK22 “Efficient computation methods for nanoscale material characterization”. Publications
  • CHARVÁTOVÁ CAMPBELL, A. – KLAPETEK, P. – ŠLESINGER, R. – MARTINEK, J. – HORTVÍK, V. – WITKOVSKÝ, Viktor – WIMMER, G. Calibration of scanning thermal microscopes using optimal estimation of function parameters by iterated linearization. In International Journal of Thermal Sciences, 2025, vol. 218, art. no. 110080. ISSN 1290-0729. (5 – IF2024) Q1
  • CHARVÁTOVÁ CAMPBELL, A. – KLAPETEK, P. – ŠLESINGER, R. – WITKOVSKÝ, Viktor – WIMMER, G. Fitting the AFM force–distance curves the correct way. In Measurement Science and Technology, 2025, vol. 36, art. no. 015022. ISSN 0957-0233. (3.4 – IF2024) Q1
  • CHARVÁTOVÁ CAMPBELL, A. – ŠLESINGER, R. – WITKOVSKÝ, Viktor – WIMMER, G. – BURŠÍKOVÁ, V. Applications of iterated linearization for non-linear errors-in-variable regression to metrological data. In Measurement: Sensors, 2025, vol. 38, art. no. 101729. ISSN 2665-9174. Q2
  • WITKOVSKÝ, Viktor – WIMMER, G. – CHARVÁTOVÁ CAMPBELL, A. – KLAPETEK, P. – ŠLESINGER, R. Estimation of function parameters through iterated linearization for nonlinear errors-in-variable regression with correlated variables. In Measurement: Sensors, 2025, vol. 38, art. no. 101728. ISSN 2665-9174. Q2
  • WITKOVSKÝ, Viktor. Weighted estimation and uncertainty propagation in nonlinear errors-in-variables regression for metrological applications. In Proceedings of the 15th International Conference on Measurement. – Bratislava : Institute of Measurement Science, SAS, 2025, p. 69-72. ISBN 978-80-69159-01-3.
  • WITKOVSKÝ, Viktor. Metóda odhadu parametrov a ich neistôt v modeli merania s chybami v premenných (Method for estimating parameters and their uncertainties in a measurement model with errors in variables). In Metrológia, skúšobníctvo a technické normy, 2025, roč. 10, č. 1, s. 58-64. ISSN 2989-3178.