The highest entropy production during diffusion

The highest entropy production during diffusion

Daria Serafin, Bartek Wierzba

Rzeszow University of Technology, Faculty of Mechanical Engineering and Aeronautics, al. Powstańców Warszawy 12, 35-959 Rzeszow, Poland.



In the present paper, the theory of the highest entropy production is discussed. It allows to predict, which phases will grow during the multiphase ternary interdiffusion process. Moreover, the phase with the highest entropy production value will nucleate as first in the reaction zone. To verify the theory the mathematical formula for calculating the entropy production was formulated. The formula bases on the generalized Darken method. Two diffusion couples: pure titanium with high purity copper-nickel alloys with different initial composition: Ni10Cu90-Ti and Ni90Cu10-Ti (at at. %) were annealed to obtain the thermodynamic and kinetic data. The integral diffusion coefficients for each component in each phase were determined using Wagner method. The reaction zones in ternary system have been analyzed: the microstructure was observed and chemical compositions were measured. Based on obtained results the entropy production was calculated. The results show that the theory of the highest entropy production can be applied in determining a proper diffusion path in multicomponent, multiphase system. 

Cite as:

Serafin, D., Wierzba, B. (2018). The highest entropy production during diffusion. Computer Methods in Materials Science, 18(1), 11 – 17.

Article (PDF):


Diffusion, Cu-Ni-Ti system, Entropy production, Diffusion path


Barin, I., 1995, Thermodynamical Data of Pure Substances,VCH Verlagsgesellschaft, Weinheim.Danielewski, M., Wierzba, B., 2010, Thermodynamicallyconsistent bi-velocity mass transport phenomenology,Acta Materialia, 58(20), 6717-6727.

Darken, L.S., 1948, Diffusion, mobility and their interrelationthrough free energy in binary metallic systems,Trans. AIME, 184.

Effenberg, G., Ilyenko S., 2006, Ternary Alloy Systems -Phase Diagrams, Crystallographic and ThermodynamicData: Light Metal Systems, Part 4: SelectedSystems from Al-Si-Ti to Ni-Si-Ti, Materials ScienceInternational Team, MSIT, IV/11A4, New Series,Landolt-Bornstein.

Gusak, A.M., 2010, Diffusion-controlled Solid State Reactionsin alloys, Thin Films and Nano Systems, Wiley-VCH Verlag GmbH & Co, WeinHeim.Ho, M.C., Lo, P.J., Liu, W.L., Hsieh, K.C., 2017,Relationship of Brazing Microstructure and Ti-Cu-NiPhase Diagram, Journal of Materials Science andEngineering, B 7, 7-8, 142-148.

Ishida, A., Sato, M., Gao, Z.Y., 2013, Properties and applicationsof Ti-Ni-Cu shape-memory-alloy thin films,Journal of Alloys and Compounds, 577, 184-189.

Kirkaldy, J.S., Brown, L.S., 1963, Diffusion Behaviour inTernary, Multiphase Systems, Canadian MetallurgicalQuarterly, 2, 89-111.

Kirkaldy, J.S., Young, D.J., 1987, Diffusion in the CondensedState, The Institute of Metals, London.Kizilyalli, M., Corish, J., Metselaar, R., 1999, Definitionsof Terms for Diffusion in the Solid State, Pure andApplied Chemistry, 71(7), 1307-1325.

Li, H. Qiu, K., Zhou, F.Y., Li, L., Zheng, Y., 2016, Designand development of novel antibacterial Ti-Ni-Cushape memory alloys for biomedical application, ScientificReports, 6, 37475.

Moberly, W.J., Proft, J.L., Duerig, T.W., Sinclair, R., 1990,Twinless Martensite in TiNiCu Shape Memory Alloys,Materials Science Forum, 56-58, 605-610.

Morral, J.E., 2012, Diffusion Path Theorems for TernaryDiffusion Couples, Metallurgical and MaterialsTransactions, A, 43A, 3462-3470.

Nam, TH., Saburi, T., Shimizu, KI., 1990, Cu-contentdependence of shape memory characteristics in Ti-Ni-Cu alloys, Materials Transactions, JIM, 31(11), 959-967.

Nernst, W., 1889, Die elektromotorische Wirkamkeit derIonen, Z. Phys. Chem., 4, 129.

Paul A., Ghosh, C., Boettinger, W.J., 2011, Diffusion Parametersand Growth Mechanism of Phases in the Cu-Sn system, Metallurgical and Materials Transactions,A, 42A, 952-963.

Plank, M., 1890, Ber die potentialdirenz zwischen zweivernnten Isungen binrer elektoryle, Ann. Phys. Chem.,40, 561.

Progogine, I., 1961, Thermodynamics of Irreversible processes,Interscience, New York.Wagner, C, 1969, The evaluation of data obtained withdiffusion couples of binary single-phase and multiphasesystems, Acta Metallurgica, 17(2), 99-107.

Wierzba, B., 2016, Phase competition in ternary Ti-Ni-Alsystem, Physica A: Statistical Mechanics and its Applications,454, 110-116.