week ending VOLUME 90.NUMBER 5 PHYSICAL REVIEW LETTERS 7 FEBRUARY 2003 Mechanisms of Radiation-Induced Viscous Flow:Role of Point Defects S.G.Mayr,*Y.Ashkenazy,"K.Albe,and R.S.Averbacks Department of Materials Science and Engineering,University of Illinois at Urbana-Champaign. 405 West Green Street,Urbana,Illinois 61801 (Received 10 July 2002:published 6 February 2003) Mechanisms of radiation-induced flow in amorphous solids have been investigated using molecular dynamics computer simulations.It is shown for a model glass system,CuTi,that the radiation-induced flow is independent of recoil energy between 100 ev and 10 keV when compared on the basis of defect production and that there is a threshold energy for flow of 10 eV.Injection of interstitial-and vacancylike defects induces the same amount of flow as the recoil events,indicating that point-defect- like entities mediate the flow process,even at 10 K.Comparisons of these results with experiments and thermal spike models are made. DOI:10.1103/PhysRevLett90.055505 PACS numbers:61.80.Az,61.80.Jh,61.82.Bg Experimental investigations of stress relaxation and interatomic potentials for Cu-Ti developed by Sabochick surface smoothing have illustrated that many amorphous and Lam [12].Two types of boundary conditions were solids undergo Newtonian flow during ion irradiations at used.One employed periodic boundary conditions in the temperatures far below their respective glass tempera- x-y direction and open surfaces in the z direction,while tures.This phenomenon,which occurs in materials with the other was periodic in three dimensions to avoid covalent [1],ionic [2-5],and metallic [6-8]bonding, surfaces.The first type had fixed boundaries in the x-y provides new opportunities for processing materials at direction and monitored the relaxation of the initial state the nanometer length scale [9]as well as having signifi- of stress,oo,as a function of the number of recoil events. cance for radiation waste storage through glass encapsu- The completely periodic cell used the same applied lation.While radiation-induced viscous flow has been stress,o,in the x-y direction and zero applied stress in widely explored experimentally,a comprehensive theo- the z direction.After each recoil event,performed under retical understanding of this behavior has been more constant volume conditions,the initial state of stress was difficult to achieve.For heavy ions in the regime when restored and the strain in the z direction obtained.Many electronic excitation dominates the stopping,thermal such events were then averaged.The fully periodic cell spike models provide a satisfactory description of the contained 2.56 X 105 atoms for all events,while the flow process since the energy dissipation is uniform and number of atoms in the cell with free surfaces increased typically greater than =1 kev/nm.For lower energy approximately linearly with recoil energy,using a ratio of ions,thermal spike models have also been invoked more than 25 atoms/eV.For both cells the atoms in the [5,10],but in this energy regime,the energy loss is much periodic boundaries were damped to simulate the loss of smaller and the energy distribution tends to be rather energy in infinite solids.The irradiations proceeded by inhomogeneous along the path of the ion.Thermal spike alternating between Cu and Ti recoils. models are therefore less attractive in this situation,since The viscosity during irradiation was determined in the the flow occurs within a few picoseconds or less of the case of periodic boundary conditions using the expression recoil event and large gradients in energies,densities,and stresses are present.Over the past decade,it has been E- -Edef, (1) recognized that molecular dynamics (MD)computer 67 simulations provide a realistic alternative approach to treat such many-body problems,and we apply this where e is the total biaxial strain,o is the applied biaxial method here to explore the mechanisms of radiation- stress,and n is the viscosity.The second term in Eq.(1) induced viscous fow.Our results show that radiation- represents possible changes in strain due to the introduc- induced flow does not,in fact,require thermal spikes tion of "defects"in the amorphous structure during an and that the creation of point defects is an equally,or, irradiation with flux,.φ.Dividing by中yields in many cases,more efficient mechanism.Within this framework,a number of experimental results from differ- deoH dede! (2) ent systems are quantitatively explained. do 6 do Irradiation-induced flow was obtained by simulating the response of an amorphous (a-)CuTi alloy to an ap- whereH=1/(no)is the radiation-induced fluidity plied stress during a series of monoenergetic events at (RIF),and the fundamental quantity in this study.For 10 K.The MD code PARCAS [11]was employed with the calculations of stress relaxation,we use 055505-1 0031-9007/03/90(5)/055505(4)$20.00 2003 The American Physical Society 055505-1
Mechanisms of Radiation-Induced Viscous Flow: Role of Point Defects S. G. Mayr,* Y. Ashkenazy,† K. Albe,‡ and R. S. Averbackx Department of Materials Science and Engineering, University of Illinois at Urbana-Champaign, 405 West Green Street, Urbana, Illinois 61801 (Received 10 July 2002; published 6 February 2003) Mechanisms of radiation-induced flow in amorphous solids have been investigated using molecular dynamics computer simulations. It is shown for a model glass system, CuTi, that the radiation-induced flow is independent of recoil energy between 100 eV and 10 keV when compared on the basis of defect production and that there is a threshold energy for flow of 10 eV. Injection of interstitial- and vacancylike defects induces the same amount of flow as the recoil events, indicating that point-defectlike entities mediate the flow process, even at 10 K. Comparisons of these results with experiments and thermal spike models are made. DOI: 10.1103/PhysRevLett.90.055505 PACS numbers: 61.80.Az, 61.80.Jh, 61.82.Bg Experimental investigations of stress relaxation and surface smoothing have illustrated that many amorphous solids undergo Newtonian flow during ion irradiations at temperatures far below their respective glass temperatures. This phenomenon, which occurs in materials with covalent [1], ionic [2–5], and metallic [6–8] bonding, provides new opportunities for processing materials at the nanometer length scale [9] as well as having signifi- cance for radiation waste storage through glass encapsulation. While radiation-induced viscous flow has been widely explored experimentally, a comprehensive theoretical understanding of this behavior has been more difficult to achieve. For heavy ions in the regime when electronic excitation dominates the stopping, thermal spike models provide a satisfactory description of the flow process since the energy dissipation is uniform and typically greater than 1 keV=nm. For lower energy ions, thermal spike models have also been invoked [5,10], but in this energy regime, the energy loss is much smaller and the energy distribution tends to be rather inhomogeneous along the path of the ion. Thermal spike models are therefore less attractive in this situation, since the flow occurs within a few picoseconds or less of the recoil event and large gradients in energies, densities, and stresses are present. Over the past decade, it has been recognized that molecular dynamics (MD) computer simulations provide a realistic alternative approach to treat such many-body problems, and we apply this method here to explore the mechanisms of radiationinduced viscous flow. Our results show that radiationinduced flow does not, in fact, require thermal spikes and that the creation of point defects is an equally, or, in many cases, more efficient mechanism. Within this framework, a number of experimental results from different systems are quantitatively explained. Irradiation-induced flow was obtained by simulating the response of an amorphous (a-)CuTi alloy to an applied stress during a series of monoenergetic events at 10 K. The MD code PARCAS [11] was employed with the interatomic potentials for Cu-Ti developed by Sabochick and Lam [12]. Two types of boundary conditions were used. One employed periodic boundary conditions in the x-y direction and open surfaces in the z direction, while the other was periodic in three dimensions to avoid surfaces. The first type had fixed boundaries in the x-y direction and monitored the relaxation of the initial state of stress, 0, as a function of the number of recoil events. The completely periodic cell used the same applied stress, 0, in the x-y direction and zero applied stress in the z direction. After each recoil event, performed under constant volume conditions, the initial state of stress was restored and the strain in the z direction obtained. Many such events were then averaged. The fully periodic cell contained 2:56 105 atoms for all events, while the number of atoms in the cell with free surfaces increased approximately linearly with recoil energy, using a ratio of more than 25 atoms=eV. For both cells the atoms in the periodic boundaries were damped to simulate the loss of energy in infinite solids. The irradiations proceeded by alternating between Cu and Ti recoils. The viscosity during irradiation was determined in the case of periodic boundary conditions using the expression _ 6 _ def; (1) where is the total biaxial strain, is the applied biaxial stress, and is the viscosity. The second term in Eq. (1) represents possible changes in strain due to the introduction of ‘‘defects’’ in the amorphous structure during an irradiation with flux, _ . Dividing by _ yields d d H 6 ddef d ; (2) where H 1= _ is the radiation-induced fluidity (RIF), and the fundamental quantity in this study. For calculations of stress relaxation, we use PHYSICAL REVIEW LETTERS week ending VOLUME 90, NUMBER 5 7 FEBRUARY 2003 055505-1 0031-9007=03=90(5)=055505(4)$20.00 2003 The American Physical Society 055505-1
week ending VOLUME 90.NUMBER 5 PHYSICAL REVIEW LETTERS 7 FEBRUARY 2003 1.2 500eV recoils o with surface 108 x without surface 0.8 1.0 (edp 0.6 0.5 工 0.0 109 0.4 6 -0.5 0.2 -1.0 Number of recoils 0 204060801001 10~10 0.0 0.001 0.0100.1001.000 10.000 -1.0 -0.5 0.0 0.5 1.0 Damage energy [keV] [GPa] FIG.2.The dependence of the RIF,H,on recoil energy for FIG.1. The RIF H is calculated from the relaxation curves simulation cells with open surface or periodic boundaries. for different stresses and signs of the stresses,as show in the inset.The independence of the initial stress oo indicates pairlike defect.This process,therefore,very much re- Newtonian flow. sembles the concept of "flow defects,"which was devel- oped to explain thermally induced flow in amorphous 0=- +(+- (3) alloys [13].Within the picture that point-defect-like en- tities mediate the radiation-induced flow,the dependence of flow on energy follows directly from the Kinchin- where a deder/do const.and Y is the biaxial Pease model of defect production,viz.,no defect produc- modulus.Here H is obtained by fitting the simulation tion below Er,followed by a rapid increase in the number data to Eq.(3).Typical results from the stress relaxation of defects just above Er,and finally a linear dependence simulations are shown in Fig.I for compressive and of defects on energy above =2.5-10ET [14,15]. tensile stresses.They illustrate that the stress decays We have further examined this defect model of exponentially with dose,that the radiation-induced vis- radiation-induced flow by simply injecting or removing cosity is independent of stress,and from the symmetry of atoms in the stressed a-CuTi matrix at randomly chosen the response to tensile and compressive stresses,that a is sites and following the stress relaxation.The results of small.Similar behavior was observed for the other recoil these simulations for defect creation at temperatures of 10 energies. and 400 K are illustrated in Fig.3,where the residual The principal results of the simulations are shown in stress is plotted versus the number of added or removed Fig.2 where the dependence of RIF is plotted as a func- atoms.The radiation-induced flow is a factor of 3 larger at tion of recoil energy.At the lowest energies,a well- the higher temperature,which is 0.7 times the glass defined threshold is observed at Er10 ev,below temperature of our alloy.At 100 K(not shown)the flow which H is extremely small 1 X 10-11(Padpa)-1 is twice as large as at 10 K.Similarly,small changes in the dpa:displacements per atom].Just above this threshold, RIF with temperature have been observed experimentally between 10 and 100 ev,the flow increases rapidly with [5,8].A surprising result observed in Fig.3 is that the energy.Ignoring for the moment the data above I keV for induced flow per defect is the same for injecting or the simulation cells with free surfaces (open symbols),we removing atoms,ie.,each operates as a flow defect and see that above the transition regime,the flow is indepen- dent of energy up to the highest energy examined,10 keV. 1.2「 When free surfaces are available,the pressure produced 米10K within cascades induces flow of mass onto the surface, 1.0 口400K thereby creating tensile stress.For films initially under 0.8 compressive stress,this effect enhances the reduction in stress in the film,as seen in Fig.2.The influence of the 0.6 surface increases as the size and energy density in cas- cades increase.An interesting point here is that such 0.4 "surface"effects can strongly influence the state of stress 0.2 in thin films,and they act on crystalline as well as amorphous films [8. 0.0t The behavior at the lowest energies clearly indicates -400 -200 0 200 400 Number of defects that radiation-induced flow commences only when the recoil energy is sufficiently high to displace an atom FIG.3.The biaxial residual stress o versus the number of from its local equilibrium site and thus create a Frenkel injected or removed atoms. 055505-2 055505-2
6 H 0 6 H exp YbH 6 ; (3) where ddef=d const. and Yb is the biaxial modulus. Here H is obtained by fitting the simulation data to Eq. (3). Typical results from the stress relaxation simulations are shown in Fig. 1 for compressive and tensile stresses. They illustrate that the stress decays exponentially with dose, that the radiation-induced viscosity is independent of stress, and from the symmetry of the response to tensile and compressive stresses, that is small. Similar behavior was observed for the other recoil energies. The principal results of the simulations are shown in Fig. 2 where the dependence of RIF is plotted as a function of recoil energy. At the lowest energies, a welldefined threshold is observed at ET 10 eV, below which H is extremely small [ 1 10 11 Pa dpa 1, dpa: displacements per atom]. Just above this threshold, between 10 and 100 eV, the flow increases rapidly with energy. Ignoring for the moment the data above 1 keV for the simulation cells with free surfaces (open symbols), we see that above the transition regime, the flow is independent of energy up to the highest energy examined, 10 keV. When free surfaces are available, the pressure produced within cascades induces flow of mass onto the surface, thereby creating tensile stress. For films initially under compressive stress, this effect enhances the reduction in stress in the film, as seen in Fig. 2. The influence of the surface increases as the size and energy density in cascades increase. An interesting point here is that such ‘‘surface’’ effects can strongly influence the state of stress in thin films, and they act on crystalline as well as amorphous films [8]. The behavior at the lowest energies clearly indicates that radiation-induced flow commences only when the recoil energy is sufficiently high to displace an atom from its local equilibrium site and thus create a Frenkel pairlike defect. This process, therefore, very much resembles the concept of ‘‘flow defects,’’ which was developed to explain thermally induced flow in amorphous alloys [13]. Within the picture that point-defect-like entities mediate the radiation-induced flow, the dependence of flow on energy follows directly from the KinchinPease model of defect production, viz., no defect production below ET, followed by a rapid increase in the number of defects just above ET, and finally a linear dependence of defects on energy above 2:5–10ET [14,15]. We have further examined this defect model of radiation-induced flow by simply injecting or removing atoms in the stressed a-CuTi matrix at randomly chosen sites and following the stress relaxation. The results of these simulations for defect creation at temperatures of 10 and 400 K are illustrated in Fig. 3, where the residual stress is plotted versus the number of added or removed atoms. The radiation-induced flow is a factor of 3 larger at the higher temperature, which is 0:7 times the glass temperature of our alloy. At 100 K (not shown) the flow is twice as large as at 10 K. Similarly, small changes in the RIF with temperature have been observed experimentally [5,8]. A surprising result observed in Fig. 3 is that the induced flow per defect is the same for injecting or removing atoms, i.e., each operates as a flow defect and -1.0 -0.5 0.0 0.5 1.0 σ0 [GPa] 0.0 0.2 0.4 0.6 0.8 1.0 1.2 H [a.u.] 0 20 40 60 80 100 -1.0 -0.5 0.0 0.5 1.0 σ/|σ0 | 500eV recoils Number of recoils FIG. 1. The RIF H is calculated from the relaxation curves for different stresses and signs of the stresses, as show in the inset. The independence of the initial stress 0 indicates Newtonian flow. 0.001 0.010 0.100 1.000 10.000 Damage energy [keV] 10-10 10-9 10-8 H [(Pa dpa)-1] with surface without surface FIG. 2. The dependence of the RIF, H, on recoil energy for simulation cells with open surface or periodic boundaries. -400 -200 0 200 400 Number of defects 0.0 0.2 0.4 0.6 0.8 1.0 1.2 σ [GPa] 10 K 400 K FIG. 3. The biaxial residual stress versus the number of injected or removed atoms. PHYSICAL REVIEW LETTERS week ending VOLUME 90, NUMBER 5 7 FEBRUARY 2003 055505-2 055505-2
VOLUME 90.NUMBER 5 PHYSICAL REVIEW LETTERS week ending 7 FEBRUARY 2003 with equal strength.It is also noteworthy that the stress mal spikes play any significant role in the RIF process. relaxation for creation of either defect at 10 K occurs. Previous theories of RIF have assumed that thermal more or less,in steps.After an initially slow response to spikes are the primary mechanism of stress relaxation, adding defects,a rapid relaxation occurs,followed by and indeed,ion beam mixing in metals is usually domi- another slow response.This avalanchelike behavior was nated by thermal spike diffusion.We address this question also observed for the low-energy recoil events,but not for by first noting that the time (i.e.,dose)constant for the high-energy events.Presumably,the higher density of stress relaxation in the defect injection simulations is defects produced in each of the higher-energy events,or 0.03 dpa,which corresponds in our defect production local heating,suppresses this behavior.As seen in Fig.3 at calculations to 0.75 eV/atom.We consider,therefore,the 400 K this behavior is also suppressed.While we have not average energy required to raise an atom above a charac- yet examined the cause of this behavior,the fact that the teristic temperature,T,at which the viscosity is suffi- width of the step is 30 injected defects in a cell con- ciently low for relaxation to occur during a thermal spike. taining 16000 atoms suggests that a concentration of For simulated CuTi,T=800 K.Thus,for a 500 eV recoil 0.2%defects leads to an instability and subsequent event,for example,there need be over 700 such atoms to relaxation. compete with the defect mechanism.Results for the A quantitative comparison was made between the flow number of liquid atoms produced in recoil events are caused by defect creation at 10 K with that by recoil events illustrated in Table I An atom is considered liquid if it, by calculating the flow per Frenkel pair.For the injection and its neighbors,each has a kinetic energy equal to simulations this is straightforward since the number of 3/2kgT.A peak in this number is observed at short defects is known.For the recoil events we assume that the time 0.1-0.2 ps,followed by a long tail.The peak is threshold energy for defect production is 10 ev,as indi- due to a rapid outward expansion,or microexplosion, cated in Fig.2,and employ the Kinchin-Pease expression surrounding the recoil site,while the tail is due to local to calculate the number of Frenkel pairs per recoil event heating.Both numbers scale with energy above 500 ev [16].We find the rather remarkable result that,to within with the ratio of liquid atoms to energy being 1.2 eV per the uncertainties,simulations of both injection of defects atom and the ratio of atoms in the microexplosion to and the recoil events yield H=3x 10-9(Padpa)-1 recoil energy =0.8 ev per atom.Both ratios are close where dose is measured in normalized units of displace- to that deduced for the inserted defects. ments per atom,i.e.,the atomic fraction of Frenkel pairs. These several results indicate that three separate While this extremely close agreement must be somewhat mechanisms may contribute to stress relaxation,creation fortuitous,owing to the uncertainties in the threshold of point defects,atomic reorganization around a micro- energy and the Kinchin-Pease model,it illustrates that explosion,and local melting.In metals they appear the creation of point defects in recoil events is sufficient to operate with nearly the same efficiency and above in itself to induce the observed stress relaxation. 500 eV;they all scale linearly with recoil energy. We next compare our value of H=3x Below 500 eV,the thermal spike mechanism does not 10-9(Padpa)-obtained by MD simulation,with pub- appear viable since the cooling rate is too fast,less than lished experimental measurements of RIF on the metallic 0.3 ps;consequently,either defect production or the mi- glasses,Ni7sB14Sis and Zr65Cu27.5Al75,and glassy SiO2: croexplosion mechanisms must dominate.Since the mi- for 6.3 MeV proton irradiation of the Ni-based metallic croexplosion,ie.,a local excitation at the site of the glass H=4X 10-9(Padpa)[6];for 700 keV Kr irra- newly created Frenkel pair,always accompanies the ad- diation of the Zr-based glass,H2.6 X 10-(Padpa)-1 dition of a Frenkel pair in a recoil event,we refer to the [7,8];and for a series of MeV ion irradiations of different two simply as defect production.We find that,despite the mass projectiles on SiO2,H=2.8 x 10-9(Padpa)-1 [5]. different mechanisms of RIF its value remains constant For the normalization to dpa in these irradiations,we over the entire energy range of our simulations,100 eV to have used the displacement energy of 10 ev in the 10 keV.This further illustrates that the local melting Kinchin-Pease model.Remarkably,H has nearly the mechanism can be no more efficient than defect produc- same value for these very disparate irradiations as for tion.For this reason,RIF in the 6.4 Mev p-irradiated the simulations.Notice,for example,that the primary recoil spectrum for 6.3 MeV proton irradiation of the Ni- based glass is strongly weighted toward low energies;half TABLE I.Number of atoms during the microexplosion NM. the Frenkel pairs are produced in recoils below a charac- number of liquid atoms N and lifetime r as a function of teristic energy,T1/2,of 400 eV.For the 700 keV bombard- damage energy Ep. ment of Zr6sCu27.5Al75,T1/2 exceeds 20 keV.Moreover, Ep [keV] 0.1 0.5 1.0 3.0 10 the glass temperature for these metallic glasses is NM 80 763 1709 4549 13865 ≈400°C,whereas it is over1l00°C in SiO2 N 0 320 840 2450 8050 While a defect model appears to explain a variety of experimental results for RIF we consider whether ther- [fs] 180 1100 2100 3000 6000 055505-3 055505-3
with equal strength. It is also noteworthy that the stress relaxation for creation of either defect at 10 K occurs, more or less, in steps. After an initially slow response to adding defects, a rapid relaxation occurs, followed by another slow response. This avalanchelike behavior was also observed for the low-energy recoil events, but not for the high-energy events. Presumably, the higher density of defects produced in each of the higher-energy events, or local heating, suppresses this behavior. As seen in Fig. 3 at 400 K this behavior is also suppressed. While we have not yet examined the cause of this behavior, the fact that the width of the step is 30 injected defects in a cell containing 16 000 atoms suggests that a concentration of 0:2% defects leads to an instability and subsequent relaxation. A quantitative comparison was made between the flow caused by defect creation at 10 K with that by recoil events by calculating the flow per Frenkel pair. For the injection simulations this is straightforward since the number of defects is known. For the recoil events we assume that the threshold energy for defect production is 10 eV, as indicated in Fig. 2, and employ the Kinchin-Pease expression to calculate the number of Frenkel pairs per recoil event [16]. We find the rather remarkable result that, to within the uncertainties, simulations of both injection of defects and the recoil events yield H 3 10 9 Pa dpa 1, where dose is measured in normalized units of displacements per atom, i.e., the atomic fraction of Frenkel pairs. While this extremely close agreement must be somewhat fortuitous, owing to the uncertainties in the threshold energy and the Kinchin-Pease model, it illustrates that the creation of point defects in recoil events is sufficient in itself to induce the observed stress relaxation. We next compare our value of H 3 10 9 Pa dpa 1 obtained by MD simulation, with published experimental measurements of RIF on the metallic glasses, Ni78B14Si8 and Zr65Cu27:5Al7:5, and glassy SiO2: for 6.3 MeV proton irradiation of the Ni-based metallic glass H 4 10 9 Pa dpa 1 [6]; for 700 keV Kr irradiation of the Zr-based glass, H 2:6 10 9 Pa dpa 1 [7,8]; and for a series of MeV ion irradiations of different mass projectiles on SiO2, H 2:8 10 9 Pa dpa 1 [5]. For the normalization to dpa in these irradiations, we have used the displacement energy of 10 eV in the Kinchin-Pease model. Remarkably, H has nearly the same value for these very disparate irradiations as for the simulations. Notice, for example, that the primary recoil spectrum for 6.3 MeV proton irradiation of the Nibased glass is strongly weighted toward low energies; half the Frenkel pairs are produced in recoils below a characteristic energy, T1=2, of 400 eV. For the 700 keV bombardment of Zr65Cu27:5Al7:5, T1=2 exceeds 20 keV. Moreover, the glass temperature for these metallic glasses is 400 C, whereas it is over 1100 C in SiO2. While a defect model appears to explain a variety of experimental results for RIF, we consider whether thermal spikes play any significant role in the RIF process. Previous theories of RIF have assumed that thermal spikes are the primary mechanism of stress relaxation, and indeed, ion beam mixing in metals is usually dominated by thermal spike diffusion.We address this question by first noting that the time (i.e., dose) constant for stress relaxation in the defect injection simulations is 0:03 dpa, which corresponds in our defect production calculations to 0:75 eV=atom. We consider, therefore, the average energy required to raise an atom above a characteristic temperature, Tc, at which the viscosity is suffi- ciently low for relaxation to occur during a thermal spike. For simulated CuTi, Tc 800 K. Thus, for a 500 eV recoil event, for example, there need be over 700 such atoms to compete with the defect mechanism. Results for the number of liquid atoms produced in recoil events are illustrated in Table I. An atom is considered liquid if it, and its neighbors, each has a kinetic energy equal to 3=2kBTc. A peak in this number is observed at short time 0:1–0:2 ps, followed by a long tail. The peak is due to a rapid outward expansion, or microexplosion, surrounding the recoil site, while the tail is due to local heating. Both numbers scale with energy above 500 eV, with the ratio of liquid atoms to energy being 1.2 eV per atom and the ratio of atoms in the microexplosion to recoil energy 0:8 eV per atom. Both ratios are close to that deduced for the inserted defects. These several results indicate that three separate mechanisms may contribute to stress relaxation, creation of point defects, atomic reorganization around a microexplosion, and local melting. In metals they appear to operate with nearly the same efficiency and above 500 eV; they all scale linearly with recoil energy. Below 500 eV, the thermal spike mechanism does not appear viable since the cooling rate is too fast, less than 0.3 ps; consequently, either defect production or the microexplosion mechanisms must dominate. Since the microexplosion, i.e., a local excitation at the site of the newly created Frenkel pair, always accompanies the addition of a Frenkel pair in a recoil event, we refer to the two simply as defect production. We find that, despite the different mechanisms of RIF, its value remains constant over the entire energy range of our simulations, 100 eV to 10 keV. This further illustrates that the local melting mechanism can be no more efficient than defect production. For this reason, RIF in the 6.4 MeV p-irradiated TABLE I. Number of atoms during the microexplosion NM, number of liquid atoms NL and lifetime as a function of damage energy ED. ED [keV] 0.1 0.5 1.0 3.0 10 NM 80 763 1709 4549 13 865 NL 0 320 840 2450 8050 [fs] 180 1100 2100 3000 6000 PHYSICAL REVIEW LETTERS week ending VOLUME 90, NUMBER 5 7 FEBRUARY 2003 055505-3 055505-3
week ending VOLUME 90.NUMBER 5 PHYSICAL REVIEW LETTERS 7 FEBRUARY 2003 calculations are in excellent agreement with experimental o with surface 1015 -x without surface results.Thermal spike models of RIF were carefully analyzed by determining the number of atoms in the liquid and were shown to be at most as efficient as point defects,and then in only some cases. 0.9 10~16 3 0.8 We are grateful for grants of computer time from 0. NCSA and NERSC.The research was supported by the 0.6 0.5 U.S.Department of Energy,Division of Materials a Science,under Grant No.DEFG02-91-ER45439,and 1017 0.001 100 through the University of California under subcontract 0.001 0.0100.1001.00010.0 100.0 B341494. Damage energy [keV] FIG.4.The dependence of the average radiation-induced diffusivity g on the damage energy.The inset shows how the ratio of the diffusivities of Ti and Cu change with damage *Electronic address:smayr@uiuc.edu energy. Electronic address:yinona@uiuc.edu Permanent address:Institut fur Materialwissenschaft, TU Darmstadt,64287 Darmstadt,Germany. Ni7sB4Sis glass,despite its low recoil energy spectrum, Electronic address:averback@uiuc.edu is equally efficient as 700 kev Kr irradiation of the [1]C.Volkert,J.Appl.Phys.70,3521 (1991). Zr6sCu27.sAl7.5 glass.Additional experiments on other [2]E.Snoeks,T.Weber,A.Cacciato,and A.Polman,J.AppL metallic glasses using different ions to determine the Phys.78.4723(1995) universality of the value of H=3 x 10-9(Padpa) [3]E.Snoeks,A.Polman,and C.A.Volkert,Appl.Phys. would be valuable.While the simulations have been per- Lett65,2487(1994). formed only on metallic glasses,the results appear to [4]M.L.Brongersma,E.Snoeks,T.van Dillen,and explain why the value for RIF is the same in SiO2,whose A.Polman,J.Appl.Phys.88,59(2000). value of Te greatly exceeds that for the metallic glasses, [5]M.L.Brongersma,E.Snoeks,and A.Polman,Appl. and for which thermal spikes are not expected.It is Phys.Lett71,1628(1997). [6]P.Jumg,J.Appl.Phys.86,4876(1999). simply because the energy to create a Frenkel pairs [7]S.G.Mayr and R.S.Averback,Phys.Rev.Lett.87, does not vary by more than a factor of 2 in most materials 196106(2001). and defect production is at least as efficient a flow mecha- [8]S.G.Mayr and R.S.Averback,unpublished results nism as the thermal spike.Perhaps our result that defect (2002). production can account for RIF should not be considered [9]E Snoeks,A.van Blaaderen,T.van Dillen,C.M. surprising since a relaxation dose of 0.03 dpa (see Fig.3) van Kats,and M.L.B.A.Polman,Adv.Mater.12,1511 implies that a volume equal to 33 atomic volumes,or a (2000). sphere of radius 0.33 nm surrounding the Frenkel pair, [10]H.Trinkaus,J.Nucl.Mater.223,196 (1995). undergoes relaxation,as the Frenkel pair relaxes.In [11]PARCAS was written by K.Nordlund;see,e.g., crystalline materials the spontaneous relaxation volume K.Nordlund and R.S.Averback,Phys.Rev.B 56,2421 is≈100 atomic volumes or r≈0.5nm[171. (1997). As a final consideration of thermal spikes,we plot the [12]M.J.Sabochick and N.Q.Lam,Scr.Metall.Mater.24, 565(1990). mean square displacement per atom,per dpa,as a [13]F Spaepen,in Physics of Defects,edited by R.Balian, function of recoil energy;see Fig.4.The value is seen M.Kleman,and J.P.Poirier (Elsevier,Amsterdam, to increase with energy up to I kev and then it be- 1981). comes constant.Similar behavior is observed for the ratio [14]M.J.Norgett,M.T.Robinson,and IM.Torrens,Nucl. of the Cu and Ti diffusion coefficients,Dri/Dcu(Fig.4). Eng.Des.33,50(1975). We know from our simulations,moreover,that this ratio is [15]A.J.E.Foreman,C.A.English,and W.J.Phythian, near unity in liquid CuTi.We conclude,therefore,that the Philos.Mag.A 66,671(1992). thermal spike becomes increasingly important with en- [16]While the Kinchin-Pease expression is known to over- ergy up to 1 keV,when local melting becomes significant. estimate the number of stable Frenkel pairs by a factor of The behavior for RIF,in contrast,becomes independent of 3 owing to stimulated recombination in the thermal energy by 100 eV suggesting that the increasing impor- spike,these unstable defects,as well as the stable ones, contribute to the viscous flow. tance of the thermal spike has little influence on the RIF [17]H.J.Wollenberger,in Vacancies and Interstitials in In conclusion,the MD simulations illustrate that Metals,edited by A.Seeger,D.Schumacher, radiation-induced viscous flow results primarily by the W.Schilling,and J.Diehl (North-Holland, creation of point defects,which act as flow defects.The Amsterdam,1970). 055505-4 055505-4
Ni78B14Si8 glass, despite its low recoil energy spectrum, is equally efficient as 700 keV Kr irradiation of the Zr65Cu27:5Al7:5 glass. Additional experiments on other metallic glasses using different ions to determine the universality of the value of H 3 10 9 Pa dpa 1 would be valuable. While the simulations have been performed only on metallic glasses, the results appear to explain why the value for RIF is the same in SiO2, whose value of Tg greatly exceeds that for the metallic glasses, and for which thermal spikes are not expected. It is simply because the energy to create a Frenkel pairs does not vary by more than a factor of 2 in most materials and defect production is at least as efficient a flow mechanism as the thermal spike. Perhaps our result that defect production can account for RIF should not be considered surprising since a relaxation dose of 0.03 dpa (see Fig. 3) implies that a volume equal to 33 atomic volumes, or a sphere of radius 0.33 nm surrounding the Frenkel pair, undergoes relaxation, as the Frenkel pair relaxes. In crystalline materials the spontaneous relaxation volume is 100 atomic volumes or r 0:5 nm [17]. As a final consideration of thermal spikes, we plot the mean square displacement per atom, per dpa, , as a function of recoil energy; see Fig. 4. The value is seen to increase with energy up to 1 keV and then it becomes constant. Similar behavior is observed for the ratio of the Cu and Ti diffusion coefficients, DTi=DCu (Fig. 4). We know from our simulations, moreover, that this ratio is near unity in liquid CuTi. We conclude, therefore, that the thermal spike becomes increasingly important with energy up to 1 keV, when local melting becomes significant. The behavior for RIF, in contrast, becomes independent of energy by 100 eV, suggesting that the increasing importance of the thermal spike has little influence on the RIF. In conclusion, the MD simulations illustrate that radiation-induced viscous flow results primarily by the creation of point defects, which act as flow defects. The calculations are in excellent agreement with experimental results. Thermal spike models of RIF were carefully analyzed by determining the number of atoms in the liquid and were shown to be at most as efficient as point defects, and then in only some cases. We are grateful for grants of computer time from NCSA and NERSC. The research was supported by the U.S. Department of Energy, Division of Materials Science, under Grant No. DEFG02-91-ER45439, and through the University of California under subcontract B341494. *Electronic address: smayr@uiuc.edu † Electronic address: yinona@uiuc.edu ‡ Permanent address: Institut fu(r Materialwissenschaft, TU Darmstadt, 64287 Darmstadt, Germany. x Electronic address: averback@uiuc.edu [1] C. Volkert, J. Appl. Phys. 70, 3521 (1991). [2] E. Snoeks, T. Weber, A. Cacciato, and A. Polman, J. Appl. Phys. 78, 4723 (1995). [3] E. Snoeks, A. Polman, and C. A. Volkert, Appl. Phys. Lett. 65, 2487 (1994). [4] M. L. Brongersma, E. Snoeks, T. van Dillen, and A. Polman, J. Appl. Phys. 88, 59 (2000). [5] M. L. Brongersma, E. Snoeks, and A. Polman, Appl. Phys. Lett. 71, 1628 (1997). [6] P. Jung, J. Appl. Phys. 86, 4876 (1999). [7] S. G. Mayr and R. S. Averback, Phys. Rev. Lett. 87, 196106 (2001). [8] S. G. Mayr and R. S. Averback, unpublished results (2002). [9] E. Snoeks, A. van Blaaderen, T. van Dillen, C. M. van Kats, and M. L. B. A. Polman, Adv. Mater. 12, 1511 (2000). [10] H. Trinkaus, J. Nucl. Mater. 223, 196 (1995). [11] PARCAS was written by K. Nordlund; see, e.g., K. Nordlund and R. S. Averback, Phys. Rev. B 56, 2421 (1997). [12] M. J. Sabochick and N. Q. Lam, Scr. Metall. Mater. 24, 565 (1990). [13] F. Spaepen, in Physics of Defects, edited by R. Balian, M. Kleman, and J. P. Poirier (Elsevier, Amsterdam, 1981). [14] M. J. Norgett, M.T. Robinson, and I. M. Torrens, Nucl. Eng. Des. 33, 50 (1975). [15] A. J. E. Foreman, C. A. English, and W. J. Phythian, Philos. Mag. A 66, 671 (1992). [16] While the Kinchin-Pease expression is known to overestimate the number of stable Frenkel pairs by a factor of 3 owing to stimulated recombination in the thermal spike, these unstable defects, as well as the stable ones, contribute to the viscous flow. [17] H. J. Wollenberger, in Vacancies and Interstitials in Metals, edited by A. Seeger, D. Schumacher, W. Schilling, and J. Diehl (North-Holland, Amsterdam, 1970). 0.001 0.010 0.100 1.000 10.0 100.0 Damage energy [keV] 10-17 10-16 10-15 ξ [cm2/dpa] with surface without surface 0.001 100 0.5 0.6 0.7 0.8 0.9 D(Ti)/D(Cu) FIG. 4. The dependence of the average radiation-induced diffusivity on the damage energy. The inset shows how the ratio of the diffusivities of Ti and Cu change with damage energy. PHYSICAL REVIEW LETTERS week ending VOLUME 90, NUMBER 5 7 FEBRUARY 2003 055505-4 055505-4
