Journal of Thermal Analysis and Calorimetry

, Volume 116, Issue 3, pp 1141–1151 | Cite as

Optimizing fitting parameters in thermogravimetry

  • Matilde Ríos-Fachal
  • Javier Tarrío-Saavedra
  • Jorge López-BeceiroEmail author
  • Salvador Naya
  • Ramón Artiaga


This study presents an alternative to simple estimation of parametric fitting models used in thermal analysis. The addressed problem consists in performing an alternative optimization method to fit thermal analysis curves, specifically TG curves and their first derivatives. This proposal consists in estimating the optimal parameters corresponding to fitting kinetic models applied to thermogravimetric (TG) curves, using evolutionary algorithms: differential evolution (DE), simulated annealing and covariance matrix adapting evolutionary strategy. This procedure does not need to include a vector with the initial values of the parameters, as is currently required. Despite their potential benefits, the application of these methods is by no means usual in the context of thermal analysis curve’s estimation. Simulated TG curves are obtained and fitted using a generalized logistic mixture model, where each logistic component represents a thermal degradation process. The simulation of TG curves in four different scenarios taking into account the extent of processes overlapping allows us to evaluate the final results and thus to validate the proposed procedure: two degradation processes non-overlapped simulated using two generalized logistics, two processes overlapped, four processes non-overlapped and four processes overlapped two by two. The mean square error function is chosen as objective function and the above algorithms have been applied separately and together, i.e., taking the final solution of the DE algorithm is the initial solution of the remaining. The results show that the evolutionary algorithms provide a good solution for adjusting simulated TG curves, better than that provided by traditional methods.


TG DTG Model-fitting Overlapping Global optimization Evolutionary algorithms 



This research has been partially supported by the Spanish Ministry of Science and Innovation, Grant MTM2011-22392 (ERDF included).


  1. 1.
    Bicerano J. Prediction of polymer properties, 3rd ed., rev. and expanded. New York: Marcel Dekker; 2002.Google Scholar
  2. 2.
    Barbadillo F, Fuentes A, Naya S, Cao R, Mier JL, Artiaga R. Evaluating the logistic mixture model on real and simulated TG curves. J Therm Anal Calorim. 2007;87:223–7.CrossRefGoogle Scholar
  3. 3.
    Naya S, Cao R, de Ullibarri IL, Artiaga R, Barbadillo F, García A. Logistic mixture model versus Arrhenius for kinetic study of material degradation by dynamic thermogravimetric analysis. J Chemom. 2006;20:158–63.CrossRefGoogle Scholar
  4. 4.
    Rios-Fachal M, Gracia-Fernández C, López-Beceiro J, Gómez-Barreiro S, Tarrío-Saavedra J, Ponton A, et al. Effect of nanotubes on the thermal stability of polystyrene. J Therm Anal Calorim. 2013;113:481–7.CrossRefGoogle Scholar
  5. 5.
    Cao R, Naya S, Artiaga R, García A, Varela A. Logistic approach to polymer degradation in dynamic TGA. Polym Degrad Stab. 2004;85:667–74.CrossRefGoogle Scholar
  6. 6.
    R Development Core Team. R: A language and environment for statistical computing. Viena: R Foundation for statistical computing; 2009. Accessed 05 Oct 2013.
  7. 7.
    Nelder JA, Mead R. A simplex method for function minimization. Comput J. 1965;7:308–13.CrossRefGoogle Scholar
  8. 8.
    Deuflhard P. Newton methods for nonlinear problems: affine invariance and adaptive algorithms. New York: Springer; 2011.CrossRefGoogle Scholar
  9. 9.
    Price KV. Differential evolution: a practical approach to global optimization. Berlin: Springer; 2005.Google Scholar
  10. 10.
    Storn R, Price K. Differential evolution—a simple and efficient heuristic for global optimization over continuous spaces. J Glob Optim. 1997;11:341–59.CrossRefGoogle Scholar
  11. 11.
    Storn R. Differential Evolution Homepage. International Computer Science Institute, University of California, Berkeley. 2013. Accessed 05 Oct 2013.
  12. 12.
    Kirkpatrick S, Gelatt CD, Vecchi MP. Optimization by simulated annealing. Science. 1983;220:671–80.CrossRefGoogle Scholar
  13. 13.
    Černý V. Thermodynamical approach to the traveling salesman problem: an efficient simulation algorithm. J Optim Theory Appl. 1985;45:41–51.CrossRefGoogle Scholar
  14. 14.
    Gubian S, Xiang Y, Suomela B, Hoeng J. R functions for generalized simulated annealing. R Package ‘GenSA’; 2013. Accessed 05 Oct 2013.
  15. 15.
    Hansen N, Ostermeier A. Completely derandomized self-adaptation in evolution strategies. Evol Comput. 2001;9:159–95.CrossRefGoogle Scholar
  16. 16.
    Hansen N. The CMA Evolution strategy: a comparing review. In: Lozano JA, Larrañaga P, Inza I, Bengoetxea E, editors. New evolutionary computation. Berlin: Springer; 2006. p. 75–102.CrossRefGoogle Scholar
  17. 17.
    Ghalanos A. Parma: portfolio allocation and risk management. R package version 1.03. 2013. Accessed 05 Oct 2013.
  18. 18.
    Dennis JE Jr, Moré JJ. Quasi-Newton methods, motivation and theory. SIAM Rev. 1977;19:46–89.CrossRefGoogle Scholar
  19. 19.
    Gay DM. A trust-region approach to linearly constrained optimization. In: Griffiths DF, editor. Numerical analysis. Springer: Heidelberg; 1984. p. 72–105.CrossRefGoogle Scholar
  20. 20.
    Levenberg K. A method for the solution of certain problems in least squares. Quart Appl Math. 1944;2:164–8.Google Scholar
  21. 21.
    Marquardt D. An algorithm for least-squares estimation of nonlinear parameters. SIAM J Appl Math. 1963;11:431–41.CrossRefGoogle Scholar
  22. 22.
    Christopoulos A, Lew MJ. Beyond eyeballing: fitting models to experimental data. Crit Rev Biochem Mol Biol. 2000;35:359–91.CrossRefGoogle Scholar
  23. 23.
    Chambers JM, Hastie. Statistical models in S. Boca Raton, Fla.: Chapman & Hall/CRC; 1992.Google Scholar
  24. 24.
    Dennis JE, Gay DM, Walsh RE. An adaptive nonlinear least-squares algorithm. ACM Trans Math Softw. 1981;7:348–68.CrossRefGoogle Scholar
  25. 25.
    Kingsland SE. Modeling nature: episodes in the history of population ecology. 2nd ed. Chicago: University of Chicago Press; 1995.Google Scholar
  26. 26.
    Román-Román P, Torres-Ruiz F. Modelling logistic growth by a new diffusion process: application to biological systems. Biosystems. 2012;110:9–21.CrossRefGoogle Scholar
  27. 27.
    López-Beceiro J, Gracia-Fernández C, Artiaga R. A kinetic model that fits nicely isothermal and non-isothermal bulk crystallizations of polymers from the melt. Eur Polym J. 2013;49:2233–46.CrossRefGoogle Scholar
  28. 28.
    Francisco-Fernández M, Tarrío-Saavedra J, Mallik A, Naya S. A comprehensive classification of wood from thermogravimetric curves. Chemom Intell Lab Syst. 2012;118:159–72.CrossRefGoogle Scholar
  29. 29.
    López-Beceiro J, Pascual-Cosp J, Artiaga R, Tarrío-Saavedra J, Naya S. Thermal characterization of ammonium alum. J Therm Anal Calorim. 2011;104:127–30.CrossRefGoogle Scholar
  30. 30.
    Pato-Doldán B, Sánchez-Andújar M, Gómez-Aguirre LC, Yáñez-Vilar S, Lopez-Beceiro J, Gracía-Fernandez C, et al. Near room temperature dielectric transition in the perovskite formate framework [(CH3) 2NH2][Mg (HCOO) 3]. Phys Chem Chem Phys. 2012;14:8498–501.CrossRefGoogle Scholar
  31. 31.
    Lopez-Beceiro J, Gracia-Fernández C, Gomez-Barreiro S, Castro-García S, Sánchez-Andújar M, Artiaga R. Kinetic study of the low temperature transformation of Co (HCOO) 3 [(CH3) 2NH2]. J Phys Chem C. 2012;116:1219–24.CrossRefGoogle Scholar
  32. 32.
    Artiaga R, Cao R, Naya S, González-Martín B, Mier JL, García A. Separation of overlapping processes from TGA data and verification by EGA. In: Pan WP, Judovits L, editors. Techniques in thermal analysis: hyphenated techniques, thermal analysis of the surface, and fast rate analysis. West Conshohocken, PA: ASTM Special Technical Publications; 2007. p. 60–71.Google Scholar
  33. 33.
    Tarrío-Saavedra J, Francisco-Fernández M, Naya S, López-Beceiro J, Gracia-Fernández C, Artiaga R. Wood identification using pressure DSC data: wood identification from PDSC. J Chemom. 2013;. doi: 10.1002/cem.2561.Google Scholar
  34. 34.
    Cai J, Liu R. Application of Weibull 2-mixture model to describe biomass pyrolysis kinetics. Energy Fuels. 2008;22:675–8.CrossRefGoogle Scholar
  35. 35.
    Adnađević B, Janković B, Kolar-Anić L, Minić D. Normalized Weibull distribution function for modelling the kinetics of non-isothermal dehydration of equilibrium swollen poly(acrylic acid) hydrogel. Chem Eng J. 2007;130:11–7.CrossRefGoogle Scholar
  36. 36.
    Tarrio-Saavedra J, López-Beceiro J, Naya S, Artiaga R. Effect of silica content on thermal stability of fumed silica/epoxy composites. Polym Degrad Stab. 2008;93:2133–7.CrossRefGoogle Scholar

Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2014

Authors and Affiliations

  • Matilde Ríos-Fachal
    • 1
  • Javier Tarrío-Saavedra
    • 2
  • Jorge López-Beceiro
    • 2
    Email author
  • Salvador Naya
    • 2
  • Ramón Artiaga
    • 2
  1. 1.CPI Cruz do Sar. Consellería de Cultura, Educación e Ordenación UniversitariaBergondoSpain
  2. 2.Escola Politécnica SuperiorUniversidade da CoruñaFerrolSpain

Personalised recommendations