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, 1977tetragonal volumic fraction is proportional to the oxygen vacancy concentration, n. To check this point, we have applied Eq. (4), which indeed predicts the correct depen￾dence 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 transi￾tion induced by irradiation is driven by the appearance of oxygen vacancies associated to Fa centers in pure mono￾clinic 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 mar￾tensitic 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 phe￾nomenological coefficient which embodies the correction to the free energy density associated with the oxygen vacan￾cies 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 tempera￾ture 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 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￾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 explains why the evolution of the tetragonal phase versus the fluence can be described by a simple rate equation tak￾ing 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 irra￾diation in pure zirconia. The phase transition induced by irradiation can be considered as a two step mechanism. The radiation damage creates a non-equilibrium concentra￾tion 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 impor￾tant 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 coal￾escence 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 devel￾oped 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 tran￾sitions 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