Availableonlineatwww.sciencedirect.com DIRECT. NIM B Beam Interactions with Materials atoms ELSEVIER Nuclear Instruments and Methods in Physics Research B 250(2006)95-100 www.elsevier.com/locate/nimb Phase transition of pure zirconia under irradiation: a textbook example D. Sin G. Baldinozzi.d. gosset a.s. le caer CEAISaclay, DENDMNSRMAILA2M, F-91191 Gif sur Yvette, France b Laboratoire de Structures, Proprietes et Modelisation des Solides, UMR CNRS 8580 Ecole Centrale Paris, F-92295 Chatenay Malabry,france CEA/Saclay, DSMIDRECAMISCMIURA 331 CNRS, F./ Gif sur Yvette, Fran Available online 13 June 2006 Abstract One of the most important goals in ceramic and materials science is to be able to design materials with specific properties. Irradiation seems to be a powerful tool for the design of advanced ceramics because of its ability to modify over different scales the microstructure of solids. Nowadays, it is clearly proved that irradiation induces order-disorder phase transitions in metallic alloys and in some ceramics. In this paper, we show that a displacive phase transition can also be induced by irradiation. Based on many experimental facts, a micro- scopic model is proposed to explain the displacive phase transition observed in this material after irradiation. Defects, produced in the oxygen sublattice, induce important strain fields on a nanometric scale. This strain field can be handled as a secondary order parameter within the Landau theory approach, leading to a decrease of the phase transition temperature and thus quenching the high temperature tetragonal phase c 2006 Elsevier B.V. all rights reserved PACS:6180.Jh;64.70.Kb Keywords: Irradiation effect; Phase transition; Point defects 1. Introduction has then been the object of extensive investigations, and therefore it is a textbook example for describing the displa At high temperature, most solids are in a thermody- cive phase transition out of irradiation within the Landau namic equilibrium state. At low temperature this is not theory approach [8]. Under irradiation, it is quite surpris- always true since their relaxation timescales can be very ing that displacive transitions associated to correlated long. Therefore, a perturbation of the system will not movements of atoms can occur. In fact, displacement cas- always allow the system to reach the equilibrium ground cades and amorphous tracks, breaking the spatial coher state [1]. Sometimes, the system perturbation is so impor- ence of the crystal, should forbid such a mechanism tant that the phase transition sequence can be modified: Nevertheless, some authors [9] attemped to explain amor several examples, like the occurrence of order-disorder phisation of some oxides within the Landau theory frame- phase transitions driven by irradiation in metallic alloys, work. In this paper, the analysis of different experiments are discussed by many authors [2, 3]. A phase transition done on pure monoclinic zirconia out of and under irradi induced by irradiation has also been observed in zirconia ation allows us to build a simple microscopic model 4-6 where the monoclinic to tetragonal phase occurs. explaining the monoclinic to tetragonal phase transition Some authors [7] have also observed the effect of grain size observed in pure zirconia samples. To reach such a goal on radiation induced transformations in zirconia. Zirconia we have studied the phase transition triggered by the temperature in this solid The key parameters associated with this transition were described within the landau Corresponding author. Tel: +331 69 08 29 20 fax: +33 1 69 08 90 82. framework of phase transitions. On the other hand, to E-mail address: david simeone(@cea. fr(D. Simeone) understand the impact of different order parameters in this 0168-583XS. see front matter c 2006 Elsevier B v. All rights doi:l0.l016 i nimb200604.092
Phase transition of pure zirconia under irradiation: A textbook example D. Simeone a,*, G. Baldinozzi b , D. Gosset a , S. Le Cae¨r c a CEA/Saclay, DEN/DMN/SRMA/LA2M, F-91191 Gif sur Yvette, France b Laboratoire de Structures, Proprie´te´s et Mode´lisation des Solides, UMR CNRS 8580 Ecole Centrale Paris, F-92295 Chaˆtenay Malabry, France c CEA/Saclay, DSM/DRECAM/SCM/URA 331 CNRS, F-91191 Gif sur Yvette, France Available online 13 June 2006 Abstract One of the most important goals in ceramic and materials science is to be able to design materials with specific properties. Irradiation seems to be a powerful tool for the design of advanced ceramics because of its ability to modify over different scales the microstructure of solids. Nowadays, it is clearly proved that irradiation induces order–disorder phase transitions in metallic alloys and in some ceramics. In this paper, we show that a displacive phase transition can also be induced by irradiation. Based on many experimental facts, a microscopic model is proposed to explain the displacive phase transition observed in this material after irradiation. Defects, produced in the oxygen sublattice, induce important strain fields on a nanometric scale. This strain field can be handled as a secondary order parameter within the Landau theory approach, leading to a decrease of the phase transition temperature and thus quenching the high temperature tetragonal phase. 2006 Elsevier B.V. All rights reserved. PACS: 61.80.Jh; 64.70.Kb Keywords: Irradiation effect; Phase transition; Point defects 1. Introduction At high temperature, most solids are in a thermodynamic equilibrium state. At low temperature this is not always true since their relaxation timescales can be very long. Therefore, a perturbation of the system will not always allow the system to reach the equilibrium ground state [1]. Sometimes, the system perturbation is so important that the phase transition sequence can be modified: several examples, like the occurrence of order–disorder phase transitions driven by irradiation in metallic alloys, are discussed by many authors [2,3]. A phase transition induced by irradiation has also been observed in zirconia [4–6] where the monoclinic to tetragonal phase occurs. Some authors [7] have also observed the effect of grain size on radiation induced transformations in zirconia. Zirconia has then been the object of extensive investigations, and therefore it is a textbook example for describing the displacive phase transition out of irradiation within the Landau theory approach [8]. Under irradiation, it is quite surprising that displacive transitions associated to correlated movements of atoms can occur. In fact, displacement cascades and amorphous tracks, breaking the spatial coherence of the crystal, should forbid such a mechanism. Nevertheless, some authors [9] attemped to explain amorphisation of some oxides within the Landau theory framework. In this paper, the analysis of different experiments done on pure monoclinic zirconia out of and under irradiation allows us to build a simple microscopic model explaining the monoclinic to tetragonal phase transition observed in pure zirconia samples. To reach such a goal, we have studied the phase transition triggered by the temperature in this solid. The key parameters associated with this transition were described within the Landau framework of phase transitions. On the other hand, to understand the impact of different order parameters in this 0168-583X/$ - see front matter 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.nimb.2006.04.092 * Corresponding author. Tel.: +33 1 69 08 29 20; fax: +33 1 69 08 90 82. E-mail address: david.simeone@cea.fr (D. Simeone). www.elsevier.com/locate/nimb Nuclear Instruments and Methods in Physics Research B 250 (2006) 95–100 NIMB Beam Interactions with Materials & Atoms
D Simeone et al. /Nucl. Instr. and Meth. in Phys. Res. B 250(2006 )95-100 transition, the structural behavior of different nanocrystals of zirconia were studied. From this analysis, it seems clear that only a single order parameter controls the phase tran sition. All these facts explain the monoclinic to tetragonal phase transition observed in pure zirconia samples exposed to radiation damage, pointing out the key role of defect created by the radiation exposure and the way they couple to order parameters to induce the monocline to tetragonal phase transition 50472 2. The monoclinic to tetragonal phase transition of zirconia Zirconia undergoes two successive phase transitions on cooling. The former transition occurs at about 2573 K and it is second order or weakly first order [10, 11. The ubic structure becomes tetragonal(P42/nmc)and it is characterized in particular by the onset of a shear strain along the fourfold axis; the coordination polyhedron for Zr in the ideal fluorite structure(Zros unit) is only slightly modified On the other hand, the second phase transition, occurring at about 1330 K on cooling and at about 0.348 500 K on heating, involves important variations of unit 200400 00800100012001400 cell parameters. This transition, associated with an impor- tant volume change, is strongly first order. To understand Fig. 1. Variation of the coordinate of oxygen atoms along the the monoclinic to tetragonal phase transition versus tem- direction (squares)in the m These atomic displacements perature, we have collected several high resolution neutron the Landau framework of phase mperature as expected within difiraction patterns at different temperatures above and (Tc=1500K) below the tetragonal to monoclinic transition. These dia grams were analysed with the rietveld method to extract the behaviour of the structural parameters and to describe their evolution in the framework of the Landau theory of phase transition. Since neutron scattering lengths for O (bo= 5.80 fm)and Zr atoms (bzr=7.16 fm)are of the same order, the accuracies on the structural parameters for all atoms are better than those obtained from X-ray dif- fraction [12, 13] Only the tetragonal and the monoclinic phases were used to index all peaks of the diffraction patterns in the investigated temperature range. The anisotropic mean square displacements of Zr and O atoms associated to the tetragonal phase as well as atomic displacements of Zr and O atoms in the monoclinic phase(Fig. 1)display a square root evolution versus temperature. Moreover,an mportant strain field associated to the evolution of the monoclinic angle B versus the temperature(Fig. 2)appears during the phase transition. From these experimental facts, is possible to discribe this displacive phase transition ithin the Landau theory framework. The mechanism associated to the tetragonal to monoclinic phase transition is the softening of two phonon modes coupled with a strain 006008001000120 field, as expected for displacive phase transitions. The use T。T)(K of the group theory permits us to build a Landau free energy to describe all the possible couplings between these Fig. 2. Evolution of the monoclinic angle, i.e. the strain field, as a function phonon modes-the primary order parameters -and the of temperature(square points and open dots are collected during heating strain field - the secondary order parameter during th as expected within the Landau framework of phase transitions hase transition [14] (Te=1500K)
transition, the structural behavior of different nanocrystals of zirconia were studied. From this analysis, it seems clear that only a single order parameter controls the phase transition. All these facts explain the monoclinic to tetragonal phase transition observed in pure zirconia samples exposed to radiation damage, pointing out the key role of defects created by the radiation exposure and the way they couple to order parameters to induce the monoclinc to tetragonal phase transition. 2. The monoclinic to tetragonal phase transition of zirconia versus temperature Zirconia undergoes two successive phase transitions on cooling. The former transition occurs at about 2573 K and it is second order or weakly first order [10,11]. The cubic structure becomes tetragonal (P42/nmc) and it is characterized in particular by the onset of a shear strain along the fourfold axis; the coordination polyhedron for Zr in the ideal fluorite structure (ZrO8 unit) is only slightly modified. On the other hand, the second phase transition, occurring at about 1330 K on cooling and at about 1500 K on heating, involves important variations of unit cell parameters. This transition, associated with an important volume change, is strongly first order. To understand the monoclinic to tetragonal phase transition versus temperature, we have collected several high resolution neutron diffraction patterns at different temperatures above and below the tetragonal to monoclinic transition. These diagrams were analysed with the Rietveld method to extract the behaviour of the structural parameters and to describe their evolution in the framework of the Landau theory of phase transition. Since neutron scattering lengths for O (bO = 5.80 fm) and Zr atoms (bZr = 7.16 fm) are of the same order, the accuracies on the structural parameters for all atoms are better than those obtained from X-ray diffraction [12,13]. Only the tetragonal and the monoclinic phases were used to index all peaks of the diffraction patterns in the investigated temperature range. The anisotropic mean square displacements of Zr and O atoms associated to the tetragonal phase as well as atomic displacements of Zr and O atoms in the monoclinic phase (Fig. 1) display a square root evolution versus temperature. Moreover, an important strain field associated to the evolution of the monoclinic angle b versus the temperature (Fig. 2) appears during the phase transition. From these experimental facts, it is possible to discribe this displacive phase transition within the Landau theory framework. The mechanism associated to the tetragonal to monoclinic phase transition is the softening of two phonon modes coupled with a strain field, as expected for displacive phase transitions. The use of the group theory permits us to build a Landau free energy to describe all the possible couplings between these phonon modes – the primary order parameters – and the strain field – the secondary order parameter – during the phase transition [14]. Fig. 1. Variation of the fractional coordinate of oxygen atoms along the z direction (squares) in the monoclinic phase. These atomic displacements follow a square root law (solid line) versus temperature as expected within the Landau framework of phase transitions (Tc = 1500 K). Fig. 2. Evolution of the monoclinic angle, i.e. the strain field, as a function of temperature (square points and open dots are collected during heating and cooling respectively). This angle follows a square root law (solid line) as expected within the Landau framework of phase transitions (Tc = 1500 K). 96 D. Simeone et al. / Nucl. Instr. and Meth. in Phys. Res. B 250 (2006) 95–100
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. during
3. Links between the order parameters To study the sensitivity of this phase transition to the order parameters, very pure nanocrystals of tetragonal zirconia have been characterized and their evolution has been followed as a function of temperature with the neutron diffraction technique. Moreover, many studies on nanocrystals 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 permit us to understand the impact of different order parameters on the tetragonal to monoclinic phase transition. The tetragonal phase can be observed at room temperature in zirconia nanoparticles [15] of less than 30 nm diameter. 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 monoclinic phase as a function of the grain size (obtained applying the Hall Williamson methods to diffraction diagrams) allows a straightforward interpretation of the experimental results and allows formulation of a model for the mechanism of the tetragonal to monoclinic phase transition observed in pure samples. By this analysis, it is then possible to establish a detailed description of the evolution of the tetragonal phase versus the size of zirconia nanoparticles. These results are analysed with the Landau theory, and they can be understood by the mechanism of a sizeinduced phase transition where the phonon condensations are quenched by the particle size. The evolution of the unit cell parameters of the tetragonal 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 tetragonal 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 existence of a coupling between secondary and primary order parameters leads to a pinning at a fixed value of the primary 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 possible couplings between order parameters [8], associated with the tetragonal to monoclinic phase transition in micrometric 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 leading term in F, exhibits a grain-size dependence. This critical temperature becomes inferior or of the same order of magnitude 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 irradiation, ZrO2 samples were irradiated at room temperature using 400 keV Xe ions to maximize the creation of displacement 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
D Simeone et al. / Nucl. Instr. and Meth. in Phys. Res. B 250(2006 )95-100 the irradiation by low energy ions, displacement cascades occur. They produce point defects in the target. A balance between production and recombination of point defects occurs leading to a non-equilibrium defect population. At lution of the population of each defect $[17] d d=(vnoa-4Tr Nb a,5)-4 NDoe g722 where is the ions flux, ad the displacement cross section, n he collision efficiency, re the capture radius, N the number of atoms per unit volume in the samples, b the mean free length associated with recombination collision sequencies 101.52.02.530354.0455.05.560 CS), o, is the RCS cross section [2] and Do the diffusion coefficient of these defects in the target The concentration of defects after irradiation at low Fig 4. Evolution of the Kubelka-Munk function AF(R)as a function of temperature, i.e. where the atomic diffusion can be the photon energy for difierent samples of pure zirconia irradiated by 400 Xe ions at different fluences at room temperature (5x 10cm squares, eglected, is controlled by the first term. The second term 10 triangle). The peak near 2.55ev is identified with Fa color drives the thermal migration of defects. centers [20] Since the experiments were performed at room tempera ture, atomic diffusion can be neglected. Only the first term of Eq(3)has to be taken into account to model the behav s (p)=s tanh(4ItreNb-5) lour of the irradiated samples. In the most general case, an equation of this type must be solved for each kind of s defect. and it is then crucial to determine which kind of 4πrNbσ defect occurs in the material where t is the fluence, t the irradiation time, the As the thickness of damaged areas is relatively small flux. The asymptotic non-equilibrium concentration of de (around 0.5 um for low energy Xe), usual volume tech- fects s does not depend on niques, like NMR, EPR or optical absorption, are not While the diffuse reflexion experiments are sensitive to $ efficient. Nevertheless, optical spectroscopy in diffuse diffraction is sensitive to the symmetry change. Since the reflection mode [18] is very sensitive to the irradiated penetration of low energy ions is very small, we have to volume near the sample surface, and it is an efficient way enhance the response coming from the volume where this to probe the existence of these defects transformation occurs. Therefore, a grazing X-ray diffrac- Different diffuse reflectance spectra were thus collected tion setup [5] was used to probe specifically the irradiated ent fluences. From these experimental results, it is possible zirconia, irradiated at various fluences, were studied to sug to obtain the Kubelka-Munk function, F(R), which is pro- gest a direct relation between the defect concentration and portional to the optical absorption coefficient [18]. In all the observed volumic fraction, Vn, of the tetragonal phase cases,we have recorded the spectra, the non-irradiated Only two phases, the monoclinic and the tetragonal mple being the reference. This allows us to identify pre- ones, were observed in the irradiated samples. The lattice cisely the nature of point defects induced by irradiation. parameters of the tetragonal phase at all fluences for Fig 4 displays AF(R)versus the photon energy for different 400 keV Xe ions were fluence independent. These values irradiated samples. All samples present the same shape for (a,=5.13 A and c,=5.165 A) can be conveniently com- the AF(R) function: a single maximum is observed at about pared to those already measured in zirconia nanocrystals 2.55 eV which is the characteristic signature of Fa colour [16]. From these results, the estimated size of the tetragonal centers [19] already observed in yttria stabilized zirconia domains is about 7 nm. Rietveld refinements of these pat (YSZ) single crystals. These Fa centers, consisting of simple terns do not show any significant change of the position anionic vacancies, are electrically charged and responsible of the refined coordinates zr(O)of the O atom for different for the local strain field as observed in YSZ [20] irradiation fluences. The tetragonal structures of micromet- This analysis suggests that only one kind of defect is ric, nanometric and irradiated ZrO2 are then isostructural produced during the irradiation of zirconia and, conse- Fig. 5 gives the evolution of the tetragonal phase as a quently, only a single(Eq.(3))is necessary to quantify function of the fluence for Xe(400 keV) ions. At high flu- the evolution of defect concentration as a function of the ence. a saturation effect is also observed. because the sizes ion fluence. The analytical solution of Eq (3)at low tem- of the domains associated with the tetragonal phase do not perature IS. evolve versus the fluence. it is reasonable to assume that the
the irradiation by low energy ions, displacement cascades occur. They produce point defects in the target. A balance between production and recombination of point defects occurs leading to a non-equilibrium defect population. At low flux, the following rate equation, taking into account well known recombination processes [2], describes the evolution of the population of each defect n [17]: dn dt ¼ ðgrd 4prcNb2 rrn2 Þ/ 4prcND0e E kBT n2 ; ð3Þ where / is the ions flux, rd the displacement cross section, g the collision efficiency, rc the capture radius, N the number of atoms per unit volume in the samples, b the mean free length associated with recombination collision sequencies (RCS), rr is the RCS cross section [2] and D0 the diffusion coefficient of these defects in the target. The concentration of defects after irradiation at low temperature, i.e. where the atomic diffusion can be neglected, is controlled by the first term. The second term drives the thermal migration of defects. Since the experiments were performed at room temperature, atomic diffusion can be neglected. Only the first term of Eq. (3) has to be taken into account to model the behaviour of the irradiated samples. In the most general case, an equation of this type must be solved for each kind of defect, and it is then crucial to determine which kind of defect occurs in the material. As the thickness of damaged areas is relatively small (around 0.5 lm for low energy Xe), usual volume techniques, like NMR, EPR or optical absorption, are not efficient. Nevertheless, optical spectroscopy in diffuse reflection mode [18] is very sensitive to the irradiated volume near the sample surface, and it is an efficient way to probe the existence of these defects. Different diffuse reflectance spectra were thus collected on various zirconia samples irradiated by Xe ions at different fluences. From these experimental results, it is possible to obtain the Kubelka–Munk function, F(R), which is proportional to the optical absorption coefficient [18]. In all cases, we have recorded the spectra, the non-irradiated sample being the reference. This allows us to identify precisely the nature of point defects induced by irradiation. Fig. 4 displays DF(R) versus the photon energy for different irradiated samples. All samples present the same shape for the DF(R) function: a single maximum is observed at about 2.55 eV which is the characteristic signature of Fa colour centers [19] already observed in yttria stabilized zirconia (YSZ) single crystals. These Fa centers, consisting of simple anionic vacancies, are electrically charged and responsible for the local strain field as observed in YSZ [20]. This analysis suggests that only one kind of defect is produced during the irradiation of zirconia and, consequently, only a single (Eq. (3)) is necessary to quantify the evolution of defect concentration as a function of the ion fluence. The analytical solution of Eq. (3) at low temperature is: nðUÞ ¼ n1 tanhð4prcNb2 rr rd n1UÞ; n1 ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi grd 4prcNb2 rr r ; ð4Þ where U = /s is the fluence, s the irradiation time, / the flux. The asymptotic non-equilibrium concentration of defects n1 does not depend on /. While the diffuse reflexion experiments are sensitive to n, diffraction is sensitive to the symmetry change. Since the penetration of low energy ions is very small, we have to enhance the response coming from the volume where this transformation occurs. Therefore, a grazing X-ray diffraction setup [5] was used to probe specifically the irradiated volume near the sample surface. Different samples of pure zirconia, irradiated at various fluences, were studied to suggest a direct relation between the defect concentration and the observed volumic fraction, Vt, of the tetragonal phase. Only two phases, the monoclinic and the tetragonal ones, were observed in the irradiated samples. The lattice parameters of the tetragonal phase at all fluences for 400 keV Xe ions were fluence independent. These values (at = 5.13 A˚ and ct = 5.165 A˚ ) can be conveniently compared to those already measured in zirconia nanocrystals [16]. From these results, the estimated size of the tetragonal domains is about 7 nm. Rietveld refinements of these patterns do not show any significant change of the position of the refined coordinates zt(O) of the O atom for different irradiation fluences. The tetragonal structures of micrometric, nanometric and irradiated ZrO2 are then isostructural. Fig. 5 gives the evolution of the tetragonal phase as a function of the fluence for Xe (400 keV) ions. At high fluence, a saturation effect is also observed. Because the sizes of the domains associated with the tetragonal phase do not evolve versus the fluence, it is reasonable to assume that the Fig. 4. Evolution of the Kubelka–Munk function DF(R) as a function of the photon energy for different samples of pure zirconia irradiated by 400 Xe ions at different fluences at room temperature (5 · 1015 cm2 squares, 1016 cm2 triangle). The peak near 2.55 eV is identified with Fa color centers [20]. 98 D. Simeone et al. / Nucl. Instr. and Meth. in Phys. Res. B 250 (2006) 95–100
D. Simeone et al. Nucl. Instr. and Meth. in Phys. Res. B 250(2006)95-100 reaches the threshold value $0, the strain field associated to these defects is enough to lower the transition tempera ture Tefr(F)=T-2 below the laboratory temperature quenching the tetragonal phase. The local strain associated toO vacancies produced by irradiation acts locally as the strain induced by the surface free energy in zirconia nano- crystals [16]. Because the tetragonal to cubic phase transi- tion is not very sensitive to strain [24]. it is unlikely that the stabilization of a cubic phase via this mechanism is pos- 02 sible. Since O vacancies are associated with randomly ori- ented electric dipoles, their electric interactions prevent the coalescence and growth of the tetragonal domains These tetragonal regions are almost independent, and this oo explains why the evolution of the tetragonal phase versus FluenceΦ(x10°cm the fluence can be described by a simple rate equation tak ing into account only the concentration of the point defects Fig. 5. Evolution of the tetragonal volumic fraction in pure zirconia as a function of the fluence, for Xe ions(open circles) irradiated at room 5. Conclusion temperature at a given flux(100cm-2s-). Eq. (4)was used to predict these experimental results(solid lin This work suggests a microscopic mechanism to explain the appearance of a displacive phase transition under irra tetragonal volumic fraction is proportional to the oxygen diation in pure zirconia. The phase transition induced by vacancy concentration, $. To check this point, we have irradiation can be considered as a two step mechanism applied Eq.(4), which indeed predicts the correct depen- The radiation damage creates a non-equilibrium concentra dence of the tetragonal volume fraction versus the fluence tion of defects in this solid. The spectroscopic signature of (solid lines in Fig. 5) these defects clearly shows that these defects are mainly O All these facts lead us to introduce the central physical vacancies forming Fa color centers, as already observed assumption that the monoclinic to tetragonal phase transi- several other ionic compounds under irradiation. These tion induced by irradiation is driven by the appearance of particular defects generate a local strain field. This impor- oxygen vacancies associated to Fa centers in pure mono- tant local strain lowers the critical temperature associated clinic zirconia. Previous ab initio calculations have shown to the phase transition, and it quenches the tetragonal thatO vacancies generate an important strain field in their phase at room temperature in the irradiated samples. neighbourhood in the monoclinic phase of zirconia [21]. Because these defects also carry randomly oriented electric From this analysis, and previous investigations on mar- dipoles, a glass-like state is expected, preventing the coal tensitic phase transition [22], it is clear that O vacancies escence of these transformed domains. Therefore, these lead to the appearance of local elastic dipoles. Since the regions act almost independently, and a simple kinetic standard and irradiated tetragonal phases are isostructural, equation for the defect concentration successfully describes the Landau theory formalism, already used to describe the the structural stability under irradiation. The model devel- tetragonal to monoclinic phase transition in non-irradiated oped in this paper to explain the behaviour of pure zirconia zirconia [14], can be applied. Under irradiation, elastic under irradiation can describe the structural evolution of dipoles, proportional to $, must be included in the Landau solids, alloys or ceramics, presenting displacive phase tran- free energy density, f, which describes the phase transition sitions versus pressure or temperature out of irradiation in a non-homogeneous medium. Neglecting Ginzburg Moreover. in our model two characteristic timescales terms in this expression [23] and focussing our attention appear in this phase transformation induced by irradiation only on the leading terms, it is possible to write the free the first is associated to the production of stable defects in energy density as: the material, and the second is linked to the propagation of the displacive phase transition in well defined domains. 5UF2-7-H(5( 6)-50)41-1(2), (5) Although many authors have defined the observed states in metallic alloys as dynamical steady states resulting from where ar is a positive constant [8] and ag is a positive phe- a competition between thermal and ballistic diffusion,a nomenological coefficient which embodies the correction two step process also occurs under irradiation in pure to the free energy density associated with the oxygen vacan- monoclinic zirconia cies concentration <().s is a threshold concentration above which the phase is tetragonal, H is the step function References and I1 is the Landau invariant responsible for the displacive phase transition in pure zirconia [14]. Once the oxygen [1G. Nicolis, I. Prigogine, Self Organisation in Non-equilibrium vacancy concentration in some regions of the samples Systems, wiley, 1977
tetragonal volumic fraction is proportional to the oxygen vacancy concentration, n. To check this point, we have applied Eq. (4), which indeed predicts the correct dependence of the tetragonal volume fraction versus the fluence (solid lines in Fig. 5). All these facts lead us to introduce the central physical assumption that the monoclinic to tetragonal phase transition induced by irradiation is driven by the appearance of oxygen vacancies associated to Fa centers in pure monoclinic zirconia. Previous ab initio calculations have shown that O vacancies generate an important strain field in their neighbourhood in the monoclinic phase of zirconia [21]. From this analysis, and previous investigations on martensitic phase transition [22], it is clear that O vacancies lead to the appearance of local elastic dipoles. Since the standard and irradiated tetragonal phases are isostructural, the Landau theory formalism, already used to describe the tetragonal to monoclinic phase transition in non-irradiated zirconia [14], can be applied. Under irradiation, elastic dipoles, proportional to n, must be included in the Landau free energy density, f, which describes the phase transition in a non-homogeneous medium. Neglecting Ginzburg terms in this expression [23] and focussing our attention only on the leading terms, it is possible to write the free energy density as: f ðI 1Þ ¼ aT 2 T T c an aT HðnðUÞ n0 Þ I 1 1 4 ðI 2 1Þ; ð5Þ where aT is a positive constant [8] and an is a positive phenomenological coefficient which embodies the correction to the free energy density associated with the oxygen vacancies concentration n(U). n0 is a threshold concentration above which the phase is tetragonal, H is the step function and I1 is the Landau invariant responsible for the displacive phase transition in pure zirconia [14]. Once the oxygen vacancy concentration in some regions of the samples reaches the threshold value n0 , the strain field associated to these defects is enough to lower the transition temperature T effðF Þ ¼ T c an aT below the laboratory temperature quenching the tetragonal phase. The local strain associated to O vacancies produced by irradiation acts locally as the strain induced by the surface free energy in zirconia nanocrystals [16]. Because the tetragonal to cubic phase transition is not very sensitive to strain [24], it is unlikely that the stabilization of a cubic phase via this mechanism is possible. Since O vacancies are associated with randomly oriented electric dipoles, their electric interactions prevent the coalescence and growth of the tetragonal domains. These tetragonal regions are almost independent, and this explains why the evolution of the tetragonal phase versus the fluence can be described by a simple rate equation taking into account only the concentration of the point defects. 5. Conclusion This work suggests a microscopic mechanism to explain the appearance of a displacive phase transition under irradiation in pure zirconia. The phase transition induced by irradiation can be considered as a two step mechanism. The radiation damage creates a non-equilibrium concentration of defects in this solid. The spectroscopic signature of these defects clearly shows that these defects are mainly O vacancies forming Fa color centers, as already observed in several other ionic compounds under irradiation. These particular defects generate a local strain field. This important local strain lowers the critical temperature associated to the phase transition, and it quenches the tetragonal phase at room temperature in the irradiated samples. Because these defects also carry randomly oriented electric dipoles, a glass-like state is expected, preventing the coalescence of these transformed domains. Therefore, these regions act almost independently, and a simple kinetic equation for the defect concentration successfully describes the structural stability under irradiation. The model developed in this paper to explain the behaviour of pure zirconia under irradiation can describe the structural evolution of solids, alloys or ceramics, presenting displacive phase transitions versus pressure or temperature out of irradiation. Moreover, in our model two characteristic timescales appear in this phase transformation induced by irradiation: the first is associated to the production of stable defects in the material, and the second is linked to the propagation of the displacive phase transition in well defined domains. Although many authors have defined the observed states in metallic alloys as dynamical steady states resulting from a competition between thermal and ballistic diffusion, a two step process also occurs under irradiation in pure monoclinic zirconia. References [1] G. Nicolis, I. Prigogine, Self Organisation in Non-Equilibrium Systems, Wiley, 1977. Fig. 5. Evolution of the tetragonal volumic fraction in pure zirconia as a function of the fluence, for Xe ions (open circles) irradiated at room temperature at a given flux (1010 cm2 s 1 ). Eq. (4) was used to predict these experimental results (solid line). D. Simeone et al. / Nucl. Instr. and Meth. in Phys. Res. B 250 (2006) 95–100 99
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