ISSN 0869-6632 (Print)
ISSN 2542-1905 (Online)

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Belenkov R. N., Postnikov E. B. Approach to nonlinearity parameter in liquids calculation based on the scaling theory of thermodynamic fluctuations. Izvestiya VUZ. Applied Nonlinear Dynamics, 2023, vol. 31, iss. 1, pp. 45-62. DOI: 10.18500/0869-6632-003020, EDN: BMTNBQ

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Approach to nonlinearity parameter in liquids calculation based on the scaling theory of thermodynamic fluctuations

Belenkov Roman N, Kursk State University
Postnikov Eugene B, Kursk State University

The nonlinearity parameter B/A is a characteristic of liquids and soft matter, which gains growing attention due to its sensibility to the composition of materials. This makes it a prospective indicator for nondestructive testing applications based on the ultrasound sounding suitable for a variety of applications from physic chemistry to biomedical studies. At the same time, the thermodynamic definition of the nonlinearity parameter requires extensive measurements at elevated pressures that are not always available; in addition, there are known certain contradiction of such data with the data obtained by methods of nonlinear acoustics. Objective. In this work, we consider a recently proposed approach to the prediction of the speed of sound at high pressures, which uses the property of invariance of the reduced pressure fluctuations and the data obtained at normal ambient pressure only. The method generalises the classic Nomoto model, which however gives only a qualitative picture, and results in the quantitative correspondence to the experimental values within their range of uncertainty. Methods. Analytical methods of the theory of thermodynamic fluctuations applied to the parameters of equations of nonlinear acoustics as well as numerical simulation in the COMSOL Multiphysics® environment. Results. Expressions for calculating the nonlinearity parameter with acceptable accuracy were obtained using thermodynamic data obtained only at atmospheric pressure. Numerical calculations were performed for toluene. In addition, the discrepancy between values of the nonlinear parameter obtained via the thermodynamic and nonlinear acoustic routes is analysed based on the numerical solution of the Westervelt equation; it is revealed that this deviation emerges when the effects of absorption of finite-amplitude waves were not properly taken into account.

  1. Zarembo LK, Krasil’nikov VA. Some problems in the propagation of ultrasonic waves of finite amplitude in liquids. Sov. Phys. Usp. 1959;2(4):580–599. DOI: 10.1070/PU1959v002n04 ABEH003149.
  2. Fox FE, Wallace WA. Absorption of finite amplitude sound waves. Journal of the Acoustical Society of America. 1954;26(6):994–1006. DOI: 10.1121/1.1907468.
  3. Beyer RT. Lord Rayleigh and nonlinear acoustics. Journal of the Acoustical Society of America. 1995;98(6):3032–3034. DOI: 10.1121/1.414465.
  4. Lord Rayleigh OMFRS. XLII. On the momentum and pressure of gaseous vibrations, and on the connexion with the virial theorem. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science. 1905;10(57):364–374. DOI: 10.1080/14786440509463381.
  5. Beyer RT. Parameter of nonlinearity in fluids. Journal of the Acoustical Society of America. 1960;32(6):719–721. DOI: 10.1121/1.1908195.
  6. Shutilov VA. Fundamental Physics of Ultrasound. London: CRC Press; 1988. 394 p. DOI: 10.1201/ 9780429332227.
  7. Cobbold RSC. Foundations of Biomedical Ultrasound. Oxford: Oxford University Press; 2006. 832 p.
  8. Lauterborn W, Kurz T, Akhatov I. Nonlinear acoustics in fluids. In: Rossing T, editor. Springer Handbook of Acoustics. Springer Handbooks. New York: Springer; 2007. P. 257–297. DOI: 10.1007/978-0-387-30425-0_8.
  9. Panfilova A, van Sloun RJG, Wijkstra H, Sapozhnikov OA, Mischi M. A review on B/A measurement methods with a clinical perspective. Journal of the Acoustical Society of America. 2021;149(4):2200–2237. DOI: 10.1121/10.0003627.
  10. Duck FA. Nonlinear acoustics in diagnostic ultrasound. Ultrasound in Medicine & Biology. 2002;28(1):1–18. DOI: 10.1016/S0301-5629(01)00463-X.
  11. Gan WS. B/A nonlinear parameter acoustical imaging. In: Nonlinear Acoustical Imaging. Singapore: Springer; 2021. P. 37–48. DOI: 10.1007/978-981-16-7015-2_6.
  12. Dzida M, Zorebski E, Zoreebski M, Zarska M, Geppert-Rybcznska M, Chorazewski M, Jacquemin J, Cibulka I. Speed of sound and ultrasound absorption in ionic liquids. Chemical Reviews. 2017;117(5):3883–3929. DOI: 10.1021/acs.chemrev.5b00733.
  13. Tiwari RK, Verma V, Awasthi A, Trivedi SK, Pandey PK, Awasthi A. Comparative study of acoustic non-linearity parameter in binary mixtures of N,N-dimethylacetamide with Polyethylene Glycols at different temperatures. Journal of Molecular Liquids. 2021;343:117707. DOI: 10.1016/ j.molliq.2021.117707.
  14. Jordan PM. A survey of weakly-nonlinear acoustic models: 1910–2009. Mechanics Research Communications. 2016;73:127–139. DOI: 10.1016/j.mechrescom.2016.02.014.
  15. Kaltenbacher B, Rundell W. On the identification of the nonlinearity parameter in the Westervelt equation from boundary measurements. Inverse Problems & Imaging. 2021;15(5):865–891. DOI: 10.3934/ipi.2021020.
  16. Nomoto O. Nonlinearity parameter of the “Rao liquid”. Journal of the Physical Society of Japan. 1966;21(4):569–571. DOI: 10.1143/JPSJ.21.569.
  17. Sharma BK. Nonlinearity acoustical parameter and its relation with Rao’s acoustical parameter of liquid state. Journal of the Acoustical Society of America. 1983;73(1):106–109. DOI: 10.1121/ 1.388842.
  18. Rao MR. Velocity of sound in liquids and chemical constitution. Journal of Chemical Physics. 1941;9(9):682–685. DOI: 10.1063/1.1750976.
  19. Wada Y. On the relation between compressibility and molal volume of organic liquids. Journal of the Physical Society of Japan. 1949;4(4–6):280–283. DOI: 10.1143/JPSJ.4.280.
  20. Daridon JL, Coutinho JAP, Ndiaye EHI, Paredes MLL. Novel data and a group contribution method for the prediction of the speed of sound and isentropic compressibility of pure fatty acids methyl and ethyl esters. Fuel. 2013;105:466–470. DOI: 10.1016/j.fuel.2012.09.083.
  21. Gupta AK, Gardas RL. The constitutive behavior of ammonium ionic liquids: a physiochemical approach. RSC Advances. 2015;5(58):46881–46889. DOI: 10.1039/C5RA02391B.
  22. Zhang Y, Zheng X, He MG, Chen Y. Speed of sound in methyl caprate, methyl laurate, and methyl myristate: measurement by Brillouin light scattering and prediction by Wada’s group contribution method. Energy & Fuels. 2016;30(11):9502–9509. DOI: 10.1021/acs.energyfuels.6b01959.
  23. Praharaj MK, Misra S. Ultrasonic and conductometric studies of NaCl solutions and study of ionicity of the liquid solution through the Walden plot and various ultrasonic parameters. Journal of Thermal Analysis and Calorimetry. 2018;132(2):1089–1094. DOI: 10.1007/s10973-018-7038-9.
  24. Daridon JL. Predicting and correlating speed of sound in long-chain alkanes at high pressure. International Journal of Thermophysics. 2022;43(5):78. DOI: 10.1007/s10765-022-02999-x.
  25. Postnikov EB, Jasiok B, Melent’ev VV, Ryshkova OS, Korotkovskii VI, Radchenko AK, Lowe AR, Chorazewski M. Prediction of high pressure properties of complex mixtures without knowledge of their composition as a problem of thermodynamic linear analysis. Journal of Molecular Liquids. 2020;310:113016. DOI: 10.1016/j.molliq.2020.113016.
  26. Lu Z, Daridon JL, Lagourette B, Ye S. A phase-comparison method for measurement of the acoustic nonlinearity parameter B/A. Measurement Science and Technology. 1998;9(10): 1699–1705. DOI: 10.1088/0957-0233/9/10/009.
  27. Lagemann RT, Corry JE. Velocity of sound as a bond property. Journal of Chemical Physics. 1942;10(12):759. DOI: 10.1063/1.1723659.
  28. Schaaffs W. Molekularakustische Ableitung einer Zustandsgleichung fur Flussigkeiten bei hohen Drucken. Acustica. 1974;30:275–280 (in German).
  29. Kudryavtsev BB, Samgina GA. Use of ultrasonic measurements in the study of molecular interactions in liquids. Soviet Physics Journal. 1966;9(1):5–8. DOI: 10.1007/BF00818478.
  30. Aziz RA, Bowman DH, Lim CC. An examination of the relationship between sound velocity and density in liquids. Canadian Journal of Physics. 1972;50(7):646–654. DOI: 10.1139/p72-089.
  31. Lemmon EW, Span R. Short fundamental equations of state for 20 industrial fluids. Journal of Chemical & Engineering Data. 2006;51(3):785–850. DOI: 10.1021/je050186n.
  32. Diky V, Muzny CD, Lemmon EW, Chirico RD, Frenkel M. ThermoData Engine (TDE): Software implementation of the dynamic data evaluation concept. 2. Equations of state on demand and dynamic updates over the web. Journal of Chemical Information and Modeling. 2007;47(4): 1713–1725. DOI: 10.1021/ci700071t.
  33. Lafarge T, Possolo A. The NIST Uncertainty Machine. NCSLI Measure. 2015;10(3):20–27. DOI: 10.1080/19315775.2015.11721732.
  34. Shklovskaya-Kordi VV. An acoustic method of determining the internal pressure in a liquid. Acoustic Journal. 1963;9(1):107–111 (in Russian).
  35. Wu J. Handbook of Contemporary Acoustics and Its Applications. Singapore: World Scientific; 2016. 468 p. DOI: 10.1142/9470.
  36. Nonlinear Acoustics – Modeling of the 1D Westervelt Equation [Electronic resource]. Application ID: 12783. COMSOL Multiphysics®; 2022. Available from:
  37. Hamilton MF, Blackstock DT. Nonlinear Acoustics. San Diego: Academic Press; 1998. 455 p.
  38. Chien LD, Cormack JM, Everbach EC, Hamilton MF. Determination of nonlinearity parameter B/A of liquids by comparison with solutions of the three-dimensional Westervelt equation. Proceedings of Meetings on Acoustics. 2021;45(1):020003. DOI: 10.1121/2.0001563.
  39. Zarembo LK, Krasilnikov VA, Shklovskaya-Kordi VV. On the propagation of ultrasonic waves of finite amplitude in liquids. Acoustic Journal. 1957;3(1):29–36 (in Russian).
  40. Dukhin AS, Goetz PJ. Bulk viscosity and compressibility measurement using acoustic spectroscopy. Journal of Chemical Physics. 2009;130(12):124519. DOI: 10.1063/1.3095471.
  41. Ramires MLV, Nieto de Castro CA, Perkins RA, Nagasaka Y, Nagashima A, Assael MJ, Wakeham WA. Reference data for the thermal conductivity of saturated liquid toluene over a wide range of temperatures. Journal of Physical and Chemical Reference Data. 2000;29(2): 133–139. DOI: 10.1063/1.556057.
  42. Jasiok B, Postnikov EB, Pikalov IY, Chorazewski M. Prediction of the speed of sound in ionic liquids as a function of pressure. Journal of Molecular Liquids. 2022;363:119792. DOI: 10.1016/ j.molliq.2022.119792. 
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