D. Simeone et al. Nucl. Instr. and Meth. in Phys. Res. B 250(2006)95-100 3. Links between the order parameters To study the sensitivity of this phase transition to the The estimated critical grain size c is 13.5 nm order parameters, very pure nanocrystals of tetragonal zir Moreover, the structural refinements allow one to study conia have been characterized and their evolution has been simultaneously the structure of the nanocrystalline tetrago- followed as a function of temperature with the neutron dif- nal phase. No significant change of the position zt( O)of the fraction technique. Moreover, many studies on nanocrys- 0 atom is observed in this phase at the different annealing tals have proved that the small grain size ensures that no temperatures. The tetragonal structures of micrometric and Schottky nor Frenkel defects can exist in these materials nanometric zrO2 are the same. Therefore the structural even at high temperature. The nanocrystals of zirconia per- instabilities in nanometric ZrO2 can be described using mit us to understand the impact of different order parame- the same Landau free energy expansion already used in micrometric Zro2 [5]. In nanometric zirconia, the existence The tetragonal phase can be observed at room tempera- of a large surface of free energy constrains the secondary ire in zirconia nanoparticles [15] of less than 30 nm diam- order parameters (strain field) to a fixed value which eter. Increasing the temperature, the sintering of tetragonal nanocrystals occurs, their grain size increases and the depends only on the observed grain size(Eq (1). The exis- tence of a coupling between secondary and primary order monoclinic phase appears at high temperature (above parameters leads to a pinning at a fixed value of the pri- 800 K). The accurate study of the transformation kinetics in zirconia nanocrystals from the tetragonal to the mond mary order parameters. This coupling is then responsible for the stability of the tetragonal phase in nanocrystal clinic phase as a function of the grain size(obtained apply- The Landau free energy, F, computed to describe all possi- ing the Hall Williamson methods to diffraction diagrams) ble couplings between order parameters [8]. associated with results and allows formulation of a model for the mecha- the tetragonal to monoclinic phase transition in micromet nism of the tetragonal to monoclinic phase transition mechanism of this phase transition in zirconia nanocrystals observed in pure samples. By this analysis, it is then possl- as a function of the nanoparticle's size. For this purpose, it ble to establish a detailed description of the evolution o is still necessary to use two invariants, I,=n+and the tetragonal phase versus the size of zirconia nanoparti- I2=n? 2, associated to Zr and O displacements, as the cles. These results are analysed with the Landau theory, basis for the Landau free energy expressed in the tetragonal and they can be understood by the mechanism of a size- ph induced phase transition where the phonon condensations The evolution of the unit cell parameters of the tetrago- F(Li,la)-(a+2/e,) are quenched by the particle size 21+(G-22)+5h2 nal phase versus the grain size allows one to compute e using Voigt notation), the only pertinent component of (F1-3/2l1)+-e3 the volume strain tensor within the grain, using a modified Laplace's law(Fig 3) where a, b, c, d, f are phenomenological coefficients and C33 is the isothermal elastic constant. The existence of two phases for different grain sizes dictates a positive value of the phenomenological coefficient f. 0.0030 This equation shows that the component of the strain tensor, es, is responsible for the decrease of the critical temperature associated with the phase transition [16]. This critical temperature, controlling the evolution of this lead ing term in F, exhibits a grain-size dependence. This critical temperature becomes inferior or of the same order of m nitude as room temperature for important values of the strain field, i.e. small grain sizes, and is responsible for 0.0005 the quenching of the tetragonal phase 4. Structural stability of monoclinic zirconia under irradiation -00010 To study in detail the mechanism associated with the structural evolution of pure monoclinic zirconia under irra Fig 3 of the strain field es(squares)as a function of the grain diation, ZrOz samples were irradiated at room temperature size. The e3 decreases with the grain size following a classical sing 400 keV Xe ions to maximize the creation of displace- Laplace line)( c=13.5 nm) ment cascades and then the defect concentrations. during3. Links between the order parameters To study the sensitivity of this phase transition to the order parameters, very pure nanocrystals of tetragonal zir￾conia have been characterized and their evolution has been followed as a function of temperature with the neutron dif￾fraction technique. Moreover, many studies on nanocrys￾tals have proved that the small grain size ensures that no Schottky nor Frenkel defects can exist in these materials even at high temperature. The nanocrystals of zirconia per￾mit us to understand the impact of different order parame￾ters on the tetragonal to monoclinic phase transition. The tetragonal phase can be observed at room tempera￾ture in zirconia nanoparticles [15] of less than 30 nm diam￾eter. Increasing the temperature, the sintering of tetragonal nanocrystals occurs, their grain size increases and the monoclinic phase appears at high temperature (above 800 K). The accurate study of the transformation kinetics in zirconia nanocrystals from the tetragonal to the mono￾clinic phase as a function of the grain size (obtained apply￾ing the Hall Williamson methods to diffraction diagrams) allows a straightforward interpretation of the experimental results and allows formulation of a model for the mecha￾nism of the tetragonal to monoclinic phase transition observed in pure samples. By this analysis, it is then possi￾ble to establish a detailed description of the evolution of the tetragonal phase versus the size of zirconia nanoparti￾cles. These results are analysed with the Landau theory, and they can be understood by the mechanism of a size￾induced phase transition where the phonon condensations are quenched by the particle size. The evolution of the unit cell parameters of the tetrago￾nal phase versus the grain size allows one to compute e3 (using Voigt notation), the only pertinent component of the volume strain tensor within the grain, using a modified Laplace’s law (Fig. 3): e3 / c ug uc : ð1Þ The estimated critical grain size uc is 13.5 nm. Moreover, the structural refinements allow one to study simultaneously the structure of the nanocrystalline tetrago￾nal phase. No significant change of the position zt(O) of the O atom is observed in this phase at the different annealing temperatures. The tetragonal structures of micrometric and nanometric ZrO2 are the same. Therefore, the structural instabilities in nanometric ZrO2 can be described using the same Landau free energy expansion already used in micrometric ZrO2 [5]. In nanometric zirconia, the existence of a large surface of free energy constrains the secondary order parameters (strain field) to a fixed value which depends only on the observed grain size (Eq. (1)). The exis￾tence of a coupling between secondary and primary order parameters leads to a pinning at a fixed value of the pri￾mary order parameters. This coupling is then responsible for the stability of the tetragonal phase in nanocrystals. The Landau free energy, F, computed to describe all possi￾ble couplings between order parameters [8], associated with the tetragonal to monoclinic phase transition in micromet￾ric zirconia [5], can be successfully used to formulate the mechanism of this phase transition in zirconia nanocrystals as a function of the nanoparticle’s size. For this purpose, it is still necessary to use two invariants, I1 = g2 + /2 and I2 = g2 /2 , associated to Zr and O displacements, as the basis for the Landau free energy expressed in the tetragonal phase: F ðI 1; I 2Þ ¼ ða þ 2fe3Þ 2 I 1 þ b 4 ðI 2 1 2I 2Þ þ c 2 I 2 þ d 6 ðI 3 1 3I 2I 1Þ þ C33 2 e2 3; ð2Þ where a, b, c, d, f are phenomenological coefficients and C33 is the isothermal elastic constant. The existence of two phases for different grain sizes dictates a positive value of the phenomenological coefficient f. This equation shows that the component of the strain tensor, e3, is responsible for the decrease of the critical temperature associated with the phase transition [16]. This critical temperature, controlling the evolution of this lead￾ing term in F, exhibits a grain-size dependence. This critical temperature becomes inferior or of the same order of mag￾nitude as room temperature for important values of the strain field, i.e. small grain sizes, and is responsible for the quenching of the tetragonal phase. 4. Structural stability of monoclinic zirconia under irradiation To study in detail the mechanism associated with the structural evolution of pure monoclinic zirconia under irra￾diation, ZrO2 samples were irradiated at room temperature using 400 keV Xe ions to maximize the creation of displace￾ment cascades and then the defect concentrations. During Fig. 3. Evolution of the strain field e3 (squares) as a function of the grain size. The parameter e3 decreases with the grain size following a classical Laplace law (solid line) (tc = 13.5 nm). D. Simeone et al. / Nucl. Instr. and Meth. in Phys. Res. B 250 (2006) 95–100 97