Thermodynamic aspects of

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1 Thermodynamic aspects of nanomaterials Advanced nanomaterials H.HofmannHofmann EPFL-LTP 2011/2012

2 ÉCOLE POLYTECHNIQUE FÉDÉRALE DE LAUSANNE Thermodynamic properties p of nanosized materials Total no. of atom ms Surfa ace atoms (% %) Cluster size (nm) Size of cluster (nm)

3 Density is an intensive property p of a substance because it does not depend on the amount of that substance; mass and volume, which are measures of the amount of the substance, are extensive properties. This is not the case in nanoscaled system! V is a non-extensive variable, Density is a non-intensive variable

4 ÉCOLE POLYTECHNIQUE FÉDÉRALE DE LAUSANNE Buffat et al.

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6 Model (1) n = total number of atoms N = number of atoms at the surface E 0 cohesive energy per atom (2) With E b = AE 0

7 It is known that both the cohesive energy and the melting temperature are parameters to describe the bond strength of materials, and it is reported that the cohesive energy is linear relation to the melting temperature for a material [1,2]. Since the cohesive energy of a nanosolidis the function of N/n, its melting temperature t should follow a relation similar il to Eq. (2), T mp = melting temperature of the nanosized crystal T mb melting temperature t of the bulk material [1] J.H. Rose, J. Ferrante, J.R. Smith, Phys. Rev. Lett. 47 (1981) 675. [2] J. Ferrante, J.H. Rose, J.R. Smith, Appl. Phys. Lett. 44 (1984) 53.

8 Mli Melting temperature of fsn nanoparticles as a function of particle size. The solid line is the calculated results by Eq. (3), where the melting temperature of bulk Sn is 505 K

9 Melting temperature of In nanowires as a function of wire diameter. The solid line is the calculated results by Eq. (3), where the melting temperature of bulk In is 429.8K Melting temperature of In nanofilms as a function of film height.

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11 Thermodynamics in Small Particle Systems. According to Pawlow[P. Pawlow : Z. Phys. Chem. Vol.65 (1909), p 1.] the chemical potentials of pure element X in liquid and solid small particles with radius r, Tanaka

12 Equilibrium: Approximation: Δ S = m, X ΔH T mx, mx, Tanaka

13 Experimental Value: Tanaka

14 Binary Alloy Phase Diagrams of Small Particle Systems consisting of Pure Solid Phases and Liquid Phase. The phase diagrams can be evaluated by only the information on the Gibbs energy in the bulk and the surface tension of liquid phase, which can be obtained as functions of temperature. Example : Cu-Bi Au-Si L = interaction ti parameter Tanaka

15 Example Tanaka

16 Correction factor because surface tension of solid at T m is 25 larger than surface tension of liquid Molar volume of the liquid alloy: Molar volume of the solid : Tanaka

17 Tanaka

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26 Ge in SiO 2 Large Melting-Point Hysteresis of Ge Nanocrystals Embedded in SiO2 QX Q. Xu, I. IDSh D. Sharp, CWY C.W. Yuan, DOYiCYLi D. O. Yi, C.Y. Liao, AMGl A. M. Glaeser, AMMi A. M. Minor, JW J.W. Beeman, M. C. Ridgway, P. Kluth, J.W. Ager III, D. C. Chrzan, and E. E. Haller PHYSICAL REVIEW LETTERS, 97, , Oct. 2006

27 ÉCOLE POLYTECHNIQUE FÉDÉRALE DE LAUSANNE Transient liquid phase sintering of CeO 2 with Co Gauckler et al. Adv. Mat. 2001,

28 ÉCOLE POLYTECHNIQUE FÉDÉRALE DE LAUSANNE Gauckler et al. Adv. Mat. 2001,

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30 ÉCOLE POLYTECHNIQUE FÉDÉRALE DE LAUSANNE Inhomogeneous State of Nanoparticles Crystal lattice γ θ κ Segregation * s Δ a/ a = r 2 l ' = θ Δ θ = 5; θ 1; θ 3.5; Δs 3nm s κ volume compressibility coefficient l radiusof the particle γ surface energy

31 Inhomogeneous State of Nanoparticles 2 3 r θ a δ s a G G rs ( ) * θ a Ι = 6. kt 1/3 κ volume compressibility coefficient l radius of the particle γ surface energy δ s = (a f / a 1) a f is the effective radius of a foreign atom. G shear modulus

32 ÉCOLE POLYTECHNIQUE FÉDÉRALE DE LAUSANNE Free energy of two-component nanoparticles γ S γ S γ SΙ F F F i v 1. f 1 + v2. f < v. f +...(15) 0 ' V1 V V S1 l r f 2,V 2 S 2 f 0 V = V + V f S1 Fig. 2. Schematic transformation of two-element nanoparticle with initially hamogeneous structure into the heterogeneous phase. f 1 and f 2 =1-f 1 are the concentrations of component A in nanoparticles

33 ÉCOLE POLYTECHNIQUE FÉDÉRALE DE LAUSANNE Example: Alumina b) 35 nm 40 kev 90 nm 60 kev

34 ÉCOLE POLYTECHNIQUE FÉDÉRALE DE LAUSANNE Phase stability F γ S + < F + V γ S v2, f2 v1, f1 V 2 1 Example: m t Zirconia; TiO2: Anatase - Rutil T γ S γ S t Δ T t =. λ V2 V1 λ is the heat of phase transition per unit volume T t phase transition temperature With decreasing nanoparticle size the phase with lower surface energy (packed more tightly) become energetically favoured. For example, in the case of fthe common body-centered d cubic (BCC) and face-centered cubic (FCC) crystal lattice the latter may become energetically more favourable

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36 ÉCOLE POLYTECHNIQUE FÉDÉRALE DE LAUSANNE Phase diagramme I F F l F L F F l 1 F L l 1 > l 2 F F a) b) T + ( T m ) Tm T T + ( Tm ) T T T F F L Tm l l l 1 2 F 2 ( Tm ) L F c) ( T m ) Tm (T + ) T + d) 1 / l

37 ÉCOLE POLYTECHNIQUE FÉDÉRALE DE LAUSANNE Phase diagramme II W.A.Jesser, Mat Res Innovat (1999) 2;

38 Materials properties are not constant Calorimetric measurements show that the energy dependence d of supported dpb particles vary much more quickly than predicted by the Gibbs-Thompson relationship. This shows that the surface energy increases substantially as the radius decreases below 3 nm. C.T Campbell et. al. Science 298 (2002)

39 Equilibrium Thermodynamics A Different Approach to Nanothermodynamics Terrell L. Hill, Nano Letters Vol 1 (2001) In contrast to macrothermodynamics, the thermodynamics of a small system will usually be different in different environments.

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43 NANOLETTERS 2001 Vol. 1, No Thermodynamics for macroscopic systems: du = TdS pdv du = TdS pdv + μi dni Gibbs For example, if the system is a small one-component spherical aggregate that has a nonnegligible surface free energy term proportional to N 2/3, above equation is no longer applicable.

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47 E is a kind of system (rather than molecule) chemical potential, called the sub-division potential. If the systems of the ensemble are macroscopic and are subdivided in order to increase N in eq 4 (e.g., in an extreme case, each system is cut in half to double N ), this will not have a noticeable effect on E t if S t, V t, and Ni t are all held constant. That is, surface effects, edge effects, system rotation and translation, etc., are all negligible for macroscopic systems, and hence E is essentially zero: the term E d N does not contribute t appreciably to the equation. But the effects just mentioned are not negligible if an ensemble of small systems is subdivided to increase N in eq 4. This is because such an increase in N implies a decrease in V and N i (V t, and N it are held constant): unlike macroscopic systems, size effects are significant in small systems.

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49 S= f(translation, rotation, vibration) inert if μ so that N = N in the T,N case, S in the μ,t case is larger (additional fluctuation) not inert T, μ T N N Variables:N,P,T N: P can be ignored (incompressibility. Variables:μ,P,T

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