Several notes on the lattice thermal conductivity of fractal-shaped nanoparticles
DOI:
https://doi.org/10.31489/2022No3/10-17Keywords:
thermoelectric materials, thermal conductivity, nanoparticles, phonons, fractal dimensionAbstract
Using the additive technologies in the production of nanoparticle-fabricated three-dimensional materials has become one of the most promising ways of obtaining effective and low-cost thermoelectric energy converters. Nanostructuring provides a route to modifying selectively the transport properties which determine the materials thermoelectric efficiency. In this paper, we have shown one more effect which consists in a significant dependence of the contribution of lattice vibrations to the thermal conductivity coefficient of a nanoparticle (its reducing is required in practice) on its morphology for nanoparticles of a pure substance. The particle morphology has been specified by the values of its effective diameter, fractal dimension and surface roughness.Using nanoparticles of pure bismuth at low temperatures as an example, we have demonstrated a notable decrease in the lattice thermal conductivity in “complicating” the particle morphology. In the final section, we have presented methods of calculating characteristics of nanoparticle ensembles, the methodology of measuring the fractal dimension experimentally also being discussed.
References
Rowe D.M. (ed.) Thermoelectric handbook macro to nano. Boca Raton, CRC Press. 2006. 1008 p.
Tambasov I.A., Voronin A.S., Evsevskaya N.P., et al. Thermoelectric properties of low-cost cost
transparent single wall carbon nanotube thin films obtained by vacuum filtration. Phys E. 2019. Vol. 114. 113619.
https://doi.org/10.1016/j.physe.2019.113619.
Hu J.-Z., Liu B., Zhou J., et al. Enhanced thermoelectric cooling performance with
graded thermoelectric materials. Jpn. j. appl. phys. 2018. Vol. 57, pp. 71801 – 71806.
www.doi.org/10.7567/JJAP.57.071801.
Li Z., Miao N., Zhou J., et al. High thermoelectric performance of few-quintuple Sb2Te3 nanofilms. Nano
energy. 2018. Vol. 43, pp. 285 – 290. www.doi.org/10.1016/j.nanoen.2017.11.043.
Erofeeva I.V., Dorokhin M.V., Lesnikov V.P., et al. Thermoelectric effects in nanoscale layers of manganese
silicide. Semiconductors. 2017. Vol. 51, No. 11, pp. 1403 – 1408. www.doi.org/10.1134/S1063782617110112.
Caballero-Calero O., Martín-González M. Thermoelectric nanowires: A brief prospective.Scripta mater. 2016.
Vol. 111, pp. 54 – 57. www.doi.org/10.1016/j.scriptamat.2015.04.020.
Dorokhin M.V., Erofeeva I.V., Kuznetsov Yu.M., et al. Investigation of the initial stages of spark-plasma
siltering of Si-Ge-based thermoelectric materials. Nanosyst.: phys., chem., math. 2018. Vol. 9, No. 5, pp. 622 – 630.
www.doi.org/10.17586/2220-8054-2018-9-5-622-630.
Kuznetsov Yu.M., Bastrakova M., Dorokhin M.V., et al. Molecular dynamics studies on spark-plasma
sintering of Si-Ge-based thermoelectric materials. AIP adv. 2020. Vol.10, No.6, 065219.
https://doi.org/10.1063/5.0011740.
Bulat L.P., Drabkin I.A., Karatayev V.V., et al. Effect of boundary scattering on the thermal conductivity of a
nanosctructured semiconductor material based on the BixSb2-xTe3 solid solution. Phys. solid state. 2010. Vol. 52, No. 9,
pp. 1836 – 1841. https://doi.org/10.1134/S1063783410090088.
Shishulin A.V., Fedoseev V.B., Shishulina A.V. Phonon thermal conductivity and phase equilibria of fractal
Bi-Sb nanoparticles. Tech. phys. 2019. Vol. 64, No. 4, pp. 512 – 517. www.doi.org/10.1134/S1063784219040200.
Shishulin A.V., Potapov A.A., Shishulina A.V. Fractal nanoparticles of phase-separating solid solutions:
nanoscale effects on phase equilibria, thermal conductivity, thermoelectric performance. 14th Chaotic modeling and
simulation, Springer proceedings in complexity. www.doi.org/10.1007/978-3-030-96964-6_30.
Fedoseev V.B., Shishulin A.V. Shape effect in layering of solid solutions in small volume: bismuth-antimony
alloy. Phys. solid state. 2018. Vol. 60, No. 7, pp. 1398 – 1404. www.doi.org/10.1134/S1063783418070120.
Lee S., Esfarjani K., Mendoza J., et al. Lattice thermal conductivity of Bi, Sb, and Bi-Sb alloy from first
principles. Phys. rev. B. 2014. Vol. 89, pp. 85206 – 85215. www.doi.org/10.1103/PhysRevB.89.085206.
Shishulin A.V., Fedoseev V.B.Stratifying polymer solutions in microsized pores: phase transitions induced by
deformation of a porous material. Tech. phys. 2020. Vol.65, No.3, pp. 340 – 346.
www.doi.org/10.1134/S1063784220030238.
Shishulin A.V., Fedoseev V.B. Thermal stability and phase composition of stratifying polymer solutions in
small-volume droplets. J. eng. phys. thermophys. 2020. Vol. 93, No. 4, pp. 802 – 809. www.doi.org/10.1007/s10891-
-02182-9.
Shishulin A.V., Fedoseev V.B.Features of the influence of the initial composition of organic stratifying
mixtures in microsized pores on the mutual solubility of components. Tech. phys. lett. 2020.
Vol. 46, No. 9, pp. 938 – 941. www.doi.org/10.1134/S1063785020090291.
Geoffrion L.-D., Guisbiers G. Chemical ordering in Bi1-xSbx nanostructures: alloy, janus or core-shell. J. phys.
chem. C. 2020. Vol. 124, No.25, pp. 14061 – 14068. www.doi.org/10.1021/acs.jpcc.0c04356.
Guisbiers G. Advances in thermodynamic modeling of nanoparticles. Adv. phys. X. 2019. Vol. 4, No. 1,
www.doi.org/10.1080/23746149.2019.1668299
Hill T.L. Thermodynamics of small systems. Parts 1 & 2. Mineola, New York, Dover Publications.
416 p.
Qi W.H., Wang M.P. Size and shape-dependent melting temperature of metallic nanoparticles. Mater. chem.
phys. 2004. Vol. 88, pp. 280 – 284. www.doi.org/10.1016/j.matchemphys.2004.04.026.
Rose J.H., Ferrante J., Smith J.R. Universal features of bonding in metals. Phys. rev. B. 1983.
Vol. 28, No. 4, pp. 1835 – 1845. https://doi.org/10.1103/PhysRevB.28.1835.
Guinea F., Rose J.H., Smith J.R. et al. Scaling relations in the equation of state, thermal expansion and
melting of metals. Appl. phys. lett. 1984. Vol. 44, pp. 53 – 55. https://doi.org/10.1063/1.94549.
Fedoseev V.B., Potapov A.A., Shishulin A.V., Fedoseeva E.N. Size and shape effect on the phase transitions in
a small system with fractal interphase boundaries. Eurasian phys. tech. j. 2017. Vol. 14, No.1, pp. 18 – 24.
Shishulin A.V., Fedoseev V.B. Peculiarities of phase transformations of polymer solutions in deformable
porous matrices. Tech. phys. lett. 2019. Vol. 45, No. 7, pp. 697 – 699. https://doi.org/10.1134/S1063785019070289.
Potapov A.A. On fractal dimension spectrum of new lightning discharge types in ionosphere: elves, jets and
sprites. Eurasian phys. tech. j. 2016. Vol. 13, No. 2, pp. 5 – 11.
Shishulin A.V., Fedoseev V.B. On some peculiarities of stratification of liquid solutions within pores of fractal
shape. J. mol. liq. 2019. Vol. 278, pp. 363 – 367. www.doi.org/10.1016/j.molliq.2019.01.050.
Shishulin A.V., Fedoseev V.B., Shishulina A.V. Melting behavior of fractal-shaped nanoparticles (the example
of Si-Ge system). Tech. phys. 2019. Vol. 64, No. 9, pp. 1343 – 1349. https://doi.org/10.1134/S1063784219090172.
Potapov A.A. On the issues of fractal radio electronics: Processing of multidimensional signals, radiolocation,
nanotechnology, radio engineering elements and sensors. Eurasian phys. tech. j. 2018. Vol. 15, No. 2, pp. 5 – 15.
Vopson M.M., Rogers N., Hepburn I. The generalized Lindemann melting coefficient. Solid state commun.
Vol. 218. 113977. www.doi.org/10.1016/j.ssc.2020.113977.
Post E.J. On the characteristic temperatures of single crystals and the dispersion of “Debye heat waves”. Can.
j. phys. 1953. Vol. 31, No. 1, pp. 112 – 119. https://doi.org/10.1139/p53-010.
Regal A.R., Glazov V.M. Entropy of melting of semiconductors. Semiconductors. 1995. Vol. 29, No. 5,
pp. 405 – 417.
Magomedov M.N. On the statistical thermodynamics of a free-standing nanocrystal: silicon. Cryst. rep. 2017.
Vol. 63, No. 3, 480 – 496. https://doi.org/10.1134/S1063774517030142.
Zimann J.M. Electrons and phonons. Oxford, Clarendon Press. 1960. pp. 58, 288, 396, 456.
Soyez G., Eastman J.A., Thompson L.J. et al. Grain-size-dependent thermal conductivity of nanocrystalline
yttria-stabilized zirconia films grown by metal-organic chemical vapor deposition. Appl. phys. lett. 2000. Vol. 77, No.
, pp. 1155 – 1157. https://doi.org/10.1063/1.1289803.
Khvesyuk V.I. Skryabin A.S. Heat conduction in nanostructures. High temp. 2017. Vol. 55, No. 3, pp. 434 –
https://doi.org/10.1134/S0018151X17030129.
Goyal M. Shape, size and phonon scattering effect on the thermal conductivity of nanostructures. Pramana: j.
phys. 2018. Vol. 91, pp.87. www.doi.org/10.1007/s12043-018-1660-8.
Fedoseev V.B. The use of fractal geometry for the thermodynamic description of the free-dimensional crystal
structure elements. Lett. mater. 2012. Vol. 2, pp. 78 – 83.
Shishulin A.V., Fedoseev V.B., Shishulina A.V. Variation of the Curie temperature in porous materials. Tech.
phys. lett. 2020. Vol. 46, No. 7, pp. 680 – 682. https://doi.org/10.1134/S106378502007024X.
Shishulin A.V., Potapov A.A., Shishulina A.V. On the transition between ferromagnetic and paramagnetic
states in mesoporous materials with fractal morphology. Eurasian phys. tech. j. 2021. Vol. 18, No.2, pp. 6 – 11.
https://doi.org/10.31489/2021NO2/6-11.
Fedoseev V.B., Shishulin A.V. On the size distribution of dispersed fractal particles. Tech. phys. 2021. Vol. 66,
No. 1, pp. 34 – 40. www.doi.org/10.1134/S1063784221010072.
Shishulin A.V., Potapov A.A., Shishulina A.V. The initial composition as an additional parameter determining
the melting behavior of nanoparticles (a case study on Six-Ge1-x alloys). Eurasian phys. tech. j. 2021. Vol. 18, No.4,
pp. 5 – 13. https://doi.org/10.31489/2021NO4/5-14.
Potapov A.A., Pakhomov A.A., Potapov A.A. (Jr.), Potapov V.A. Processing by robust fractal-topological
methods of flows of optical texture images on the Martian surface. Eurasian phys. tech. j. 2019. Vol. 16, No.2, pp. 5 – 10. https://doi.org/10.31489/2019NO2/5-10.
Li J., Du Q., Sun C. An improved box-counting method for image fractal dimension estimation Pattern
recognit. 2009. Vol. 42, pp. 2460 – 2469. www.doi.org/10.1016/j.patcog.2009.03.001.
Fedoseeva E.N., Fedoseev V.B. Interaction of chitosan with benzoic acid in solution and films. Polymer sci.
Ser. A. 2011. Vol. 53, No. 11, pp. 1040 – 1046. https://doi.org/10.1134/S0965545X1110004X.
