The presented analysis of the temperature course of a phase transformation of nanoparticles offers a powerful tool to obtain data for the particle size distribution and the enthalpy of transformation. The analysis starts with the temperature distribution within an ensemble of nanoparticles and a functional relation between particle size and transformation temperature. In contrast to the Maxwell-Boltzmann distribution, in case of nanoparticles, the distribution of the temperature of the particles follows a normal distribution. Finally a reliable characterization of the specimen and the transformation in question are presented. These results, particularly the particle size distribution, represent – in many cases – a statistically more significant result as compared to the outcome, obtained by evaluation of electron micrographs. Looking at the problems of microcalorimetric determination of reaction enthalpies, one may realize comparable problems. By comparison of the experimental data with the calculated ones, one may obtain additional information about the interaction of the particles within the ensemble.
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Arblaster JW (2016) Thermodynamic properties of gold. JPEDAV 37:229–245
Bostanjoglo O, Röhkel K (1972) Superferromagnetism in gadolinium films. Phys Status Solidi A 11:161–166
Clusius K, Schachinger L (1952) Ergebnisse der Tieftemperaturforschung IX. Die Atomwärme des Kobalts zwischen 15° und 270° K., L, Z. Naturforsch A 7:185–191
Horowitz M, Silvidi AA, Malaker SF, Daunt JG (1952) The specific heat of lead in the temperature range 1 K to 75 K. Phys Rev 88:1182–1186
http://hyperphysics.phy-astr.gsu.edu/hbase/Solids/chrlen.html. Accessed September 23,2019.
Jaynes ET (1965) Gibbs vs Boltzmann entropies. Am J Phys 33:391–398
Li WH, Yang CC, Tsao FC, Lee KC (2003) Quantum size effects on the superconducting parameters of zero-dimensional Pb nanoparticles. Phys Rev B 68(184507):1–6
London H (1940) The high-frequency resistance of superconducting tin. Proc R Soc Lond A 176:522–533
Meschede D (2004) Gehrtsen Physik. Springer, Berlin Heidelberg pp 883–888
Mørup S, Madsen MB, Franck J, Villadsen J, Koch CJW (1983) A new interpretation of Mössbauer spectra of microcrystalline goethite: "super-ferromagnetism" or "super-spin-glass" behavior? J Magn Magn Mater 40:163–174
Neeleshwar S, Chen YY, Wang CR, Ou MN, Huang PH (2004) Superconductivity in aluminum nanoparticles. Physica C 408–410:209–210
Pawlow PN (1909) Über die Abhängigkeit des Schmelzpunktes von der Oberflächenenergie eines festen Körpers. Z Phys Chem 65:545–548
Rupp J, Birringer R (1987) Enhanced specific-heat-capacity (cp) measurements (150–300 K) of nanometer-sized crystalline materials. Phys Rev B 36:7888–7890
Shan J, Chen Nuopponen M, Tenhu H (2004) Two phase transitions of poly(N-isopropylacrylamide) brushes bound to gold nanoparticles. Langmuir 20:4671–4676
Tinkham M (2000) Limits on superconductivity in nanoparticles and nanowires. J Supercond Nov Magn 13:801–804
Vojta G, Vojta M (2000) Teubner Taschernbuch der statistischen Physik. B.G. Teubner, Stuttgart, pp 161–163
Vollath D (2011) Estimation of particle size distributions obtained by gas phase processes. J Nanopart Res 13:3899–3909
Vollath D (2019) Energy distribution in an ensemble of nanoparticles and its consequences. Beilstein J Nanotechnol 10:1452–1457
Vollath D, Fischer F-D, Holec D (2018) Surface energy of nanoparticles – influence of particle size and structure. Beilstein J Nanotechnol 9:2265–2276
von Delft J (2001) Superconductivity in ultrasmall metallic grains. Ann Phys (Leipzig) 10(3):219–276
Woods SI, Kirtley JR, Sun S, Koch RH (2001) Direct investigation of superparamagnetism in Co nanoparticle films. Phys Rev Lett 87(137205):1–4
Zhu M-Q, Wang L-Q, Exarhos GJ, Li ADQ (2004) Thermosensitive gold nanoparticles. J Am Chem Soc 126:2656–2657
The author thanks F. D. Fischer and D. Holec for important discussions.
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Vollath, D. Determination of particle size distributions and transformation enthalpies from the temperature course of a phase transformation. J Nanopart Res 22, 33 (2020) doi:10.1007/s11051-019-4732-x
- Energy distribution
- Phase transformation
- Particle size distribution
- Enthalpy of reaction