Thermodynamic Equilibria in Systems with Nanoparticles

  • Jindřich LeitnerEmail author
  • David Sedmidubský
Part of the Hot Topics in Thermal Analysis and Calorimetry book series (HTTC, volume 11)


Thermodynamic description of systems with nanoparticles in the frame of the Gibbs theory of interfaces is presented. Although much attention has been paid to thermodynamic modelling of nanosystems, the calculation of phase diagrams of nanoalloys as well as the assessment of effects of surface-related phenomena on the solubility of nanoparticles and gas–solid reactions, some discrepancy still remains dealing with the expression of the surface contribution to molar Gibbs energy and chemical potential of components. It is shown that due to the non-extensive nature of the surface area, these contributions are different for molar and partial molar quantities. The consistent expressions for molar Gibbs energy and chemical potentials of components of spherical nanoparticles are put forward along with the correct forms of equilibrium conditions. Moreover, the applicability of the shape factor α = A non-spherical/A spherical (V non-spherical = V spherical) which is used in the expressions involving surface-to-volume ratio of non-spherical particles is addressed. A new parameter, the differential shape factor α′ = dA non-spherical/dA spherical (V non-spherical = V spherical, dV non-spherical = dV spherical), is proposed which should be used in equilibrium conditions based on the equality of chemical potentials. The enhanced solubility of paracetamol nanoparticles in water and thermal decomposition of GaN nanowires are demonstrated as examples of size effect in nanosystems.


Gibbs Energy Homogeneous Function Surface Term Molar Gibbs Energy Regular Polyhedron 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



This work was supported by Czech Science Foundation, grant number No. 13-20507S.


  1. 1.
    Goswami GK, Nanda KK (2012) Thermodynamic models for the size-dependent melting of nanoparticles: different hypotheses. Current Nanosci 8:305–311CrossRefGoogle Scholar
  2. 2.
    Xue YQ, Zhao MZ, Lai WP (2013) Size-dependent phase transition temperatures of dispersed systems. Phys B 408:134–139CrossRefGoogle Scholar
  3. 3.
    Li ZH, Truhlar DG (2014) Nanothermodynamics of metal nanoparticles. Chem Sci 5:2605–2624CrossRefGoogle Scholar
  4. 4.
    Yang CC, Mai YW (2014) Thermodynamics at the nanoscale: a new approach to the investigation of unique physicochemical properties of nanomaterials. Mater Sci Eng R 79:1–40CrossRefGoogle Scholar
  5. 5.
    Šesták J (2015) Kinetic phase diagrams as a consequence of sudden changing temperature or particle size. J Ther Anal Calorim 120:129–137CrossRefGoogle Scholar
  6. 6.
    Eichhammer Y, Heyns M, Moelans N (2011) Calculation of phase equilibria for an alloy nanoparticle in contact with a solid nanowire. CALPHAD 35:173–182CrossRefGoogle Scholar
  7. 7.
    Garzel G, Janczak-Rusch J, Zadbyr L (2012) Reassessment of the Ag-Cu phase diagram for nanosystems including particle size and shape effect. CALPHAD 36:52–56CrossRefGoogle Scholar
  8. 8.
    Guisbiers G, Mejia-Rosales S, Khanal S, Ruiz-Zepeda F, Whetten RL, José-Yacaman M (2014) Gold-copper nano-alloy, “Tumbaga”, in the era of nano: phase diagram and segregation. Nano Lett 14:6718–6726Google Scholar
  9. 9.
    Sim K, Lee J (2014) Phase stability of Ag–Sn alloy nanoparticles. J Alloys Compd 590:140–146CrossRefGoogle Scholar
  10. 10.
    Sopoušek J, Vřešťál J, Pinkas J, Brož P, Buršík J, Stýskalík A, Škoda D, Zobač O, Lee J (2014) Cu–Ni nanoalloy phase diagram—prediction and experiment. CALPHAD 45:33–39CrossRefGoogle Scholar
  11. 11.
    Kroupa A, Káňa T, Buršík J, Zemanová A, Šob M (2015) Modelling of phase diagrams of nanoalloys with complex metallic phases: application to Ni-Sn. Phys Chem Chem Phys 17:28200–28210CrossRefGoogle Scholar
  12. 12.
    Ghasemi M, Zanolli Z, Stankovski M, Johansson J (2015) Size- and shape-dependent phase diagram of In–Sb nano-alloys. Nanoscale 7:17369–17387CrossRefGoogle Scholar
  13. 13.
    Bajaj S, Haverty MG, Arróyave R, Goddard FRSCWA III, Shankare S (2015) Phase stability in nanoscale material systems: extension from bulk phase diagrams. Nanoscale 7:9868–9877CrossRefGoogle Scholar
  14. 14.
    Kaptay G (2012) On the size and shape dependence of the solubility of nano-particles in solutions. Int J Pharm 430:253–257CrossRefGoogle Scholar
  15. 15.
    Du J, Zhao R, Xue Y (2012) Thermodynamic properties and equilibrium constant of chemical reaction in nanosystem: an theoretical and experimental study. J Chem Thermodyn 55:218–224CrossRefGoogle Scholar
  16. 16.
    Murdande SB, Shah DA, Dave RH (2015) Impact of nanosizing on solubility and dissolution rate of poorly soluble pharmaceuticals. J Pharm Sci 104:2094–2102CrossRefGoogle Scholar
  17. 17.
    Navrotsky A (2011) Nanoscale effects on thermodynamics and phase equilibria in oxide systems. ChemPhysChem 2011:2207–2215CrossRefGoogle Scholar
  18. 18.
    Chung SW, Guliants EA, Bunker CE, Jelliss PA, Buckner SW (2011) Size-dependent nanoparticle reaction enthalpy: oxidation of aluminum nanoparticles. J Phys Chem Solids 72:719–724CrossRefGoogle Scholar
  19. 19.
    Kang SY, Mo Y, Ong SP, Ceder G (2014) Nanoscale stabilization of sodium oxides: implication for Na-O2 batteries. Nano Lett 14:1016–1020CrossRefGoogle Scholar
  20. 20.
    Li M, Altman EI (2014) Cluster-size dependent phase transition of Co oxides on Au(111). Surf Sci 619:L6–L10CrossRefGoogle Scholar
  21. 21.
    Cui Z, Duan H, Li W, Xue Y (2015) Theoretical and experimental study: the size dependence of decomposition thermodynamics of nanomaterials. J Nanopart Res 17:321 (11 pp)Google Scholar
  22. 22.
    Hill TL (2001) A different approach to nanothermodynamics. Nano Lett 1:273–275CrossRefGoogle Scholar
  23. 23.
    García-Morales V, Cervera J, Pellicer J (2005) Correct thermodynamic forces in Tsallis thermodynamics: connection with Hill nanothermodynamics. Phys Lett A 336:82–88CrossRefGoogle Scholar
  24. 24.
    Turmine M, Mayaffre A, Letellier P (2004) Nonextensive approach to thermodynamics: analysis and suggestion, and application to chemical reactivity. J Phys Chem B 108:18980–18987CrossRefGoogle Scholar
  25. 25.
    Letellier P, Mayaffre A, Turmine M (2007) Solubility of nanoparticles: nonextensive thermodynamics approach. J Phys: Condens Matter 19:436229 (9 pp)Google Scholar
  26. 26.
    Letellier P, Mayaffre A, Turmine M (2007) Melting point depression of nanosolids: nonextensive thermodynamics approach. Phys Rev B 76:045428 (8 pp)Google Scholar
  27. 27.
    Qi WH, Wang MP (2004) Size and shape dependent melting temperature of metallic nanoparticles. Mater Chem Phys 88:280–284CrossRefGoogle Scholar
  28. 28.
    Qi WH, Wang MP, Liu QH (2005) Shape factor of nonspherical nanoparticles. J Mater Sci 40:2737–2739CrossRefGoogle Scholar
  29. 29.
    Tanaka T, Iida T (1994) Application of a thermodynamic database to the calculation of surface tension for iron-base liquid alloys. Steel Res 65:21–28CrossRefGoogle Scholar
  30. 30.
    Tanaka T, Hack K, Iida T, Hara S (1996) Application of thermodynamic databases to the evaluation of surface tensions of molten alloys, salt mixtures and oxide mixtures. Z Metallknd 87:380–389Google Scholar
  31. 31.
    Picha R, Vřešťál J, Kroupa A (2004) Prediction of alloy surface tension using a thermodynamic database. CALPHAD 28:141–146CrossRefGoogle Scholar
  32. 32.
    Tanaka T, Kitamura T, Back IA (2006) Evaluation of surface tension of molten ionic mixtures. ISIJ Int 46:400–406CrossRefGoogle Scholar
  33. 33.
    Nakamoto M, Kiyose A, Tanaka T, Holappa L, Hämäläinen M (2007) Evaluation of the surface tension of ternary silicate melts containing Al2O3, CaO, FeO, MgO or MnO. ISIJ Int 47:38–43CrossRefGoogle Scholar
  34. 34.
    Hanao M, Tanaka T, Kawamoto M, Takatani K (2007) Evaluation of surface tension of molten slag in multi-component systems. ISIJ Int 47:935–939CrossRefGoogle Scholar
  35. 35.
    Egry I, Ricci E, Novakovic R, Ozawa S (2010) Surface tension of liquid metals and alloys—recent developments. Adv Colloid Interface Sci 159:198–212CrossRefGoogle Scholar
  36. 36.
    Cahn JW (1980) Surface stress and the chemical equilibrium of small crystals—I. The case of the isotropic surface. Acta Metall 28:1333–1338CrossRefGoogle Scholar
  37. 37.
    Jesser WA, Shneck RZ, Gile WW (2004) Solid-liquid equilibria in nanoparticles of Pb-Bi alloys. Phys Rev B 69:144121 (13 pp)Google Scholar
  38. 38.
    Cammarata RC (1997) Surface and interface stress effects on interfacial and nanostructured materials. Mater Sci Eng A 237:180–184CrossRefGoogle Scholar
  39. 39.
    Cammarata RC (2008) Generalized surface thermodynamics with application to nucleation. Phil Mag 88:927–948CrossRefGoogle Scholar
  40. 40.
    Cammarata RC (2009) Generalized thermodynamics of surfaces with applications to small solid systems. In: Egrenreich H, Spaepen F (eds) Solid state physics, vol 61. Elsevier, Amsterdam, p 1Google Scholar
  41. 41.
    Espeau P, Céolin R, Tamarit JL, Perrin MA, Gauchi JP, Leveiller F (2005) Polymorphism of paracetamol: relative stabilities of the monoclinic and orthorhombic phases inferred from topological pressure-temperature and temperature-volume phase diagrams. J Pharm Sci 94:524–539CrossRefGoogle Scholar
  42. 42.
    Hendriksen BA, Grant DJW (1995) The effect of structurally related substances on the nucleation kinetics of paracetamol (acetaminophen). J Cryst Growth 156:252–260CrossRefGoogle Scholar
  43. 43.
    Prasad KVR, Ristic RI, Sheen DB, Sherwood JN (2001) Crystallization of paracetamol from solution in the presence and absence of impurity. Int J Pharm 215:29–44CrossRefGoogle Scholar
  44. 44.
    Omar W, Mohnicke M, Ulrich J (2006) Determination of the solid liquid interfacial energy and thereby the critical nucleus size of paracetamol in different solvents. Cryst Res Technol 41:337–343CrossRefGoogle Scholar
  45. 45.
    Lerk CF, Schoonen AJM, Fell JT (1976) Contact angles and wetting of pharmaceutical powders. J Pharm Sci 65:843–847CrossRefGoogle Scholar
  46. 46.
    Duncan-Hewitt W, Nisman R (1993) Investigation of the surface free energy of pharmaceutical materials from contact angle, sedimentation, and adhesion measurements. J Adhesion Sci Technol 7:263–283CrossRefGoogle Scholar
  47. 47.
    Heng JYY, Bismarck A, Lee AF, Wilson K, Williams DR (2006) Anisotropic surface energetics and wettability of macroscopic form I paracetamol crystals. Langmuir 22:2760–2769CrossRefGoogle Scholar
  48. 48.
    Alander EM, Rasmusson AC (2007) Agglomeration and adhesion free energy of paracetamol crystals in organic solvents. AIChE J 53:2590–2605CrossRefGoogle Scholar
  49. 49.
    Sedmidubský D, Leitner J (2006) Calculation of the thermodynamic properties of AIII nitrides. J Cryst Growth 286:66–70CrossRefGoogle Scholar
  50. 50.
    Sedmidubský D, Leitner J, Svoboda P, Sofer Z, Macháček J (2009) Heat capacity and phonon spectra of AIIIN. Experimental and calculation. J Therm Anal Calorim 95:403–407CrossRefGoogle Scholar
  51. 51.
    Moon WH, Kim HJ, Choi CH (2007) Molecular dynamics simulation of melting behaviour of GaN nanowires. Scripta Mater 56:345–348CrossRefGoogle Scholar
  52. 52.
    Wang Z, Zu X, Gao F, Weber WJ (2007) Size dependence of melting of GaN nanowires with triangular cross sections. J Appl Phys 101:043511 (4 pp)Google Scholar
  53. 53.
    Guisbiers G, Liu D, Jiang Q, Buchaillot L (2010) Theoretical predictions of wurtzite III-nitride nano-materials properties. Phys Chem Chem Phys 12:7203–7210CrossRefGoogle Scholar
  54. 54.
    Antoniammal P, Arivuoli D (2012) Size and shape dependence of melting temperature of gallium nitride nanoparticles. J Nanomater 2012:415797 (11 pp)Google Scholar
  55. 55.
    Reeber RR, Wang K (2000) Lattice parameters and thermal expansion of GaN. J Mater Res 15:40–44CrossRefGoogle Scholar
  56. 56.
    Assael MJ, Armyra IJ, Brillo J, Stankus SV, Wu J, Wakeham WA (2012) Reference data for the density and viscosity of liquid cadmium, cobalt, gallium, indium, mercury, silicon, thallium, and zinc. J Phys Chem Ref Data 41:033101 (16 pp)Google Scholar
  57. 57.
    Gomes MC, Leite DMG, Sambrano JR, Dias da Silva JH, de Souza AR, Beltrán A (2011) Thermodynamic and electronic study of Ga1–xMnxN films. A theoretical study. Surf Sci 605:1431–1437CrossRefGoogle Scholar
  58. 58.
    Mills KC, Su YC (2006) Review of surface tension data for metallic elements and alloys: part 1—pure metals. Int Mater Rev 51:329–351CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2017

Authors and Affiliations

  1. 1.Department of Solid State EngineeringUniversity of Chemistry and Technology PraguePrague 6Czech Republic
  2. 2.Department of Inorganic ChemistryUniversity of Chemistry and Technology PraguePrague 6Czech Republic

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