PHYSICAL REVIEW X 7, 031019 (2017) 1 1,2 1 3 2,4 1,* N. M. Anoop Krishnan, Bu Wang, Yingtian Yu, Yann Le Pape, Gaurav Sant, and Mathieu Bauchy Laboratory for the Physics of Amorphous and Inorganic Solids (PARISlab), Department of Civil and Environmental Engineering, University of California, Los Angeles, California 90095, USA 2 2 Laboratory for the Chemistry of Construction Materials (LC ), Department of Civil and Environmental Engineering, University of California, Los Angeles, California 90095, USA Oak Ridge National Laboratory, P.O. Box 2008, Oak Ridge, Tennessee 37831-6148, USA California Nanosystems Institute (CNSI), University of California, Los Angeles, California 90095, USA (Received 9 January 2017; revised manuscript received 14 April 2017; published 28 July 2017) Under irradiation, minerals tend to experience an accumulation of structural defects, ultimately leading to a disordered atomic network. Despite the critical importance of understanding and predicting irradiation- induced damage, the physical origin of the initiation and saturation of defects remains poorly understood. Here, based on molecular dynamics simulations of α-quartz, we show that the topography of the enthalpy landscape governs irradiation-induced disordering. Specifically, we show that such disordering differs from that observed upon vitrification in that, prior to saturation, irradiated quartz accesses forbidden regions of the enthalpy landscape, i.e., those that are inaccessible by simply heating and cooling. Furthermore, we demonstrate that damage saturates when the system accesses a local region of the enthalpy landscape corresponding to the configuration of an allowable liquid. At this stage, a sudden decrease in the heights of the energy barriers enhances relaxation, thereby preventing any further accumulation of defects and resulting in a defect-saturated disordered state. DOI: 10.1103/PhysRevX.7.031019 Subject Areas: Chemical Physics, Computational Physics, Materials Science I. INTRODUCTION observed in conventional glasses, i.e., when materials are quenched from the liquid state fast enough to avoid Understanding irradiation-induced damage is of great crystallization . After exposure to a critical radiation importance for applications including nuclear waste immo- energy, the amount of defects saturates, and no further bilization, nuclear plant safety, and nuclear fuel form changes in the structure are observed . Although a glass integrity, as well as for the controlled doping of semi- could be assumed to represent the upper (structural) limit conductors and in fusion reactors [1–5]. Irradiation induces of irradiation-induced disordering, glasses themselves can the formation and accumulation of defects in crystals’ also show damage upon irradiation [9,11]. In addition, such atomic networks [1,6]. Eventually, this can result in a loss comparisons are not obvious as glassy materials are out of of long-range order in the network, although some degree equilibrium; as such, their properties depend on their thermal of short-range order is preserved, e.g., interatomic distances history . As a result, several questions remain unclear or coordination numbers [6–9]. This is similar to what is [1,2], including the following: (1) Does irradiation-induced disordering fundamentally differ from the amorphization that The United States Government retains and the publisher, is induced via heating and fast quenching? (2) Does defect by accepting the article for publication, acknowledges that the saturation occur when the system reaches a glassy state?. United States Government retains a nonexclusive, paid-up, Numerous empirical and phenomenological models have irrevocable, worldwide license to publish or reproduce the been proposed to answer these questions . For example, published form of this manuscript, or allow others to do so, the resistance to irradiation of materials has been shown to for United States Government purposes. The Department of be related, amongst others, to the atomic structure [3,14] and Energy will provide public access to these results of federally topology [15–17], the ability of the network to accommodate sponsored research in accordance with the DOE Public Access lattice disorder [2,18], bond energies , glass-forming Plan (http://energy.gov/downloads/doe‑public‑access‑plan). ability , melting point , or other physical properties Corresponding author. . In turn, the saturation of damage (defects) observed at email@example.com high dosage has been suggested to arise when the network Published by the American Physical Society under the terms of loses its crystalline order [6,13] or becomes unable to the Creative Commons Attribution 4.0 International license. accommodate defect-induced swelling . Nevertheless, Further distribution of this work must maintain attribution to it is widely acknowledged that none of these models can the author(s) and the published article’s title, journal citation, and DOI. simultaneously explain all observations [1,5]. 2160-3308=17=7(3)=031019(10) 031019-1 Published by the American Physical Society N. M. ANOOP KRISHNAN et al. PHYS. REV. X 7, 031019 (2017) defects , interface-controlled amorphization , Taking an alternative viewpoint, all of these properties multiple cascade overlap , in-cascade amorphization ultimately depend on the enthalpy landscape of the material. , or direct impact . In most cases, amorphization is Hence, such a framework can provide a unifying and likely to occur through a combination of these processes. physically sound basis to describe irradiation effects . However, the final irradiated state does not significantly Here, via reactive molecular dynamics (RMD) simulations depend on the specific mechanism of amorphization of α-quartz as an example mineral, we demonstrate the [28,29,34,35]. However, it should be noted that there are crucial role of the topography of the enthalpy landscape in certain caveats associated with MD simulations of radiation controlling irradiation-induced amorphization. By character- damage. On the one hand, because of the limited time scale izing the different configurations of the enthalpy landscape accessible to MD simulations, the damage flux imposed that are accessed upon irradiation, we show that, prior to here is higher than those typically observed in radiation defect saturation, irradiated quartz explores “forbidden experiments. On the other hand, the incident energies used states” of the enthalpy landscape, i.e., those that cannot in our radiation simulations—300 eV, 600 eV, and 1000 eV be achieved by simply cooling a silica melt. Importantly, we (see Ref. )—are lower than experimental values, which establish that irradiation-induced disordering saturates when can be of the order of 10–100 keV. However, although the the system achieves the atomic configuration of an “allow- simulations represent time scales lower than experiments, able liquid,” whose enthalpy landscape features low-energy because of the infinitesimally low diffusion coefficient of barriers, thereby facilitating the relaxation of defects and oxygen atoms in quartz and silica at low temperatures , preventing any further damage. structural defects are unlikely to migrate through the system, even over extended periods of time. II. METHODS Because of the large region that is affected during each A. Irradiation simulations ballistic cascade, large system sizes are required to avoid potential spurious self-interactions arising from the periodic Realistic RMD simulations of irradiation-induced boundary conditions. Herein, the system size is determined by damage in α-quartz are carried out using LAMMPS . repeatedly knocking each atom of the quartz primitive cell, To this end, we follow a well-established methodology with the target radiation energy and in random directions. The [6,16,25,26]. First, a randomly chosen atom is accelerated maximum distances traveled by each of the impacted atoms with a kinetic energy equivalent to that of the targeted are then recorded. We eventually choose the size of the system incident neutron. Here, an incident energy of 600 eV is to be at least twice as large as the maximum distance among used. Note that weighted probabilities based on the neutron all the recorded ones. In the present case, the initial system cross sections of silicon and oxygen atoms are considered consists of a 10 × 10 × 9 α-quartz supercell comprising 8100 when choosing the incident atom. Once the atom is atoms. It is worth noting that, to offer realistic results, RMD accelerated with the desired incident energy, it collides simulations require the use of accurate interatomic potentials. with other atoms, thereby resulting in a ballistic cascade. A In particular, in the case of irradiation simulations, the spherical region is then created around the impacted zone, interatomic potential must (1) correctly describe both the outside which atoms are kept at a constant temperature pristine and disordered structures of the relevant system with a of 300 K by a Nosé-Hoover thermostat . In contrast, fixed set of parameters, (2) provide a realistic description of to avoid any spurious effects of the thermostat on the high-energy collisions, wherein pairs of atoms potentially dynamics of the cascade, the dynamics of the atoms inside explore the short-distance part of the potential, and (3) be able the sphere is treated in the NVE ensemble. The radius of to handle the formation of atomic species with defective the NVE sphere is fixed as 10 Å. Note that a variable time local environments—e.g., overcoordinated or undercoordi- step is used during the ballistic cascade to avoid potential nated atoms—whichare likelytoform uponirradiation. To numerical errors associated with excessive collisions and this end, we use the ReaxFF potential , with parameters overlapping of atoms due to the high velocities of the taken from Manzano et al. , as it can correctly describe primary and secondary knock-on atoms; a time step of the structure of both pristine α-quartz and glassy silica and 0.5 fs is used otherwise. The dynamics of the cascade is features robust potential forms that can dynamically adjust simulated for 20 ps, which was found to be long enough to the potential energy based on the local atomic environment ensure the convergence of both temperature and energy. of atoms . Furthermore, thanks to the shielding of short- Finally, after each collision, the system is further relaxed in range interactions [32,33], ReaxFF can be used without any the NPT ensemble at 300 K and zero pressure for another modifications to simulate high-energy collisions and ion 5 ps. This enables the system to adjust its density upon bombardment. irradiation. Such an iterative process is then repeated, with different knock-on atoms, until the system exhibits B. Glass preparation a saturation in terms of both enthalpy and density. The irradiation-induced amorphization can occur via In order to compare the structure of irradiated quartz to different mechanisms [28,29], e.g., accumulation of point that of its glassy counterpart, a silica glass of similar size 031019-2 ENTHALPY LANDSCAPE DICTATES THE IRRADIATION- … PHYS. REV. X 7, 031019 (2017) (8100 atoms) is prepared using RMD simulations following ensemble. The ground-state enthalpy is eventually com- the melting-quenching method . Note that, to ensure a puted for each configuration. meaningful comparison with irradiated quartz, we rely on the same potential and timestep. First, an initial system is D. Ring-size distribution generated by randomly placing atoms in a cubic box, while The distribution of Si–O ring sizes in irradiated quartz, ensuring the absence of any unrealistic overlap. The system liquid silica, and glassy silica is computed using the RINGS is then melted at 4500 K at zero pressure for 1 ns in the NPT package , wherein, starting from a given Si atom, a ring ensemble, which ensures that the memory of the initial is defined as the shortest closed path that comes back to this configuration is lost. The system is subsequently gradually atom. The size of the ring is then defined by the number of cooled from 4500 K to 300 K at zero pressure with a connected Si atoms. Therefore, a ring size of n contains a cooling rate of 1 K=ps in the NPT ensemble. The formed total of 2n atoms, where n corresponds to the number of glass is finally equilibrated at 300 K and zero pressure for Si (or O) atoms. The cutoff used to define an active Si–O 1 ns in the NPT ensemble to ensure complete relaxation bond (2.1 Å) is obtained from the partial PDF. A maximum of the structure. Note that, although the cooling rate can ring search size of 20 is used (the convergence of the ring- significantly affect the thermodynamic conditions of glassy size distribution was ensured by considering higher maxi- silica such as its density, it weakly affects its short- and mum ring sizes). medium-range order structure [37,38]. In particular, the density of glassy silica can vary from 2.3 g=cm to E. Roughness of the local enthalpy landscape 2.2 g=cm depending on the cooling rate [37,38]—lower For a given configuration, the height of the energy cooling rates yielding lower densities. However, the com- barriers at the vicinity of the local equilibrium position putational cost of reactive potentials like ReaxFF prevents of the system is estimated using the following methodol- the usage of cooling rates that are significantly lower than ogy. First, we perform multiple energy minimizations on the one used herein. The resulting density of silica glass is the pristine, partially irradiated, and fully irradiated quartz found to be slightly overestimated, around 2.3 g=cm . samples. This is to ensure that the samples are at their However, the final structure of the simulated glass shows ground state, corresponding to a temperature of 0 K. Then, a good agreement with experimental data . we provide a sudden energy bump to the system corre- sponding to a temperature of 1500 K. The temperature C. Ground-state enthalpy computation is chosen to be high enough to allow potential motion In this study, we aim to meaningfully compare the between low-energy barriers but low enough to avoid enthalpy of irradiated quartz to that of glassy silica. glass transition or melting of the system. Finally, we However, if calculated at finite temperature using RMD, allow the system to evolve in the microcanonical ensemble the energy of the system comprises the contributions of (NVE) for 100 ps. The MSD (r ) is obtained from ! ! the random vibrations of the atoms. As such, the random 2 N 2 r ¼ð1=NÞ ðr − r Þ , where N is the number of i¼1 i i;0 sampling of the configurational space will result in some atoms, r are the positions of each atom after 100 ps of uncertainty in the instantaneous potential energy of the dynamics following the energy bump, and r are the i;0 system. To overcome this issue, starting from the configu- positions in the inherent structure (0 K). rations obtained at a given temperature, we compute the ground-state enthalpy, that is, the enthalpy of the inherent III. RESULTS structure. This is achieved by performing an energy To establish our conclusions, we rely on RMD simu- minimization while enforcing a zero pressure, following lations, which allow us to assess the effect of irradiation the method presented elsewhere . This ensures that all on the structure and enthalpy landscape of α-quartz. The atoms reach a local minimum of potential energy, thereby removing any thermal contribution from the computed specifics of the simulations are provided in Sec. II. enthalpy. Note that this method provides the local—i.e., not Figure 1(a) shows the evolution of the enthalpy and density absolute—minimum of enthalpy of the system and, as such, of α-quartz upon irradiation. First, we observe that the can be used to obtain the value of the ground-state enthalpy density of quartz decreases monotonically upon irradiation at a given temperature. Such evolutions are investigated, and eventually converges to around 2.2 g=cm , in agree- both for α-quartz and glassy silica, by the following ment with experimental data [7,8,39,40]. Second, we note procedure. Starting from structures equilibrated at 300 K that the enthalpy increases upon irradiation, before plateau- and zero pressure, the system is gradually heated at a rate of ing. This arises from the formation and accumulation of 1 K=ps under zero pressure in the NPT ensemble up to a energetically unfavorable structural defects in the atomic temperature of 4500 K, that is, when both the crystal and network, as presented elsewhere . Note that, here, any glass melt. A posteriori, independent atomic configurations deviation from the initial pristine structure is referred to as are selected every 100 K, instantaneously cooled to 1 K, a defect, which somewhat differs from the typical crystallo- and further relaxed for 50 ps at this temperature in the NPT graphic viewpoint of “defects,” e.g., point, interstitial, 031019-3 N. M. ANOOP KRISHNAN et al. PHYS. REV. X 7, 031019 (2017) (a) (b) FIG. 1. Irradiation-induced disordering of quartz. (a) Enthalpy (left axis) and density (right axis) of irradiated α-quartz as a function of the deposited energy (DE). (b) Evolution of the structure of α-quartz following different amounts of (radiation) energy deposition, namely, 0 eV=atom, 1.8 eV=atom, and 18 eV=atom. vacancy defects, etc. The defects formed herein include, loss of periodicity, in the limit, is in agreement with electron diffraction analysis of irradiated quartz [8,16]. amongst others, overcoordinated and undercoordinated Si Furthermore, as shown in Fig. 2(b), irradiation also results and O species, stretched bonds or angles, edge-sharing Si tetrahedra, mis-sized silicate rings, etc. . As shown in in a decrease of the Si–O–Si bond angle, in agreement with Fig. 1(b), the percolation of such defects results in the experiments . Altogether, the methodology and the complete disordering of the network—note that, in such reactive interatomic potential used herein (see Sec. II) offer amorphous networks, the very notion of a traditional a realistic description of the influences of irradiation on crystallographic defect eventually becomes ill-defined. α-quartz and, as such, provide a reliable basis to assess the To ensure that the properties of the irradiated quartz topography of the enthalpy landscape and compare irradi- samples do not significantly depend on the incident energy ation-induced disordering to that observed upon thermal per neutron, but rather on the total deposited energy, the vitrification. We now compare the structure of irradiated quartz, after present methodology was repeated for varying system sizes the saturation of its density and enthalpy, to that of glassy and neutron energies. We observe that the obtained results silica. First, we note that, as expected, the density of fully do not exhibit any significant dependencies on either the irradiated quartz is fairly similar to that of glassy silica , system size or the incident energy (see Ref. ), which i.e., around 2.2 g=cm suggests a self-similar nature of the damage cascade. It is . Second, we observe that the PDF worth noting that irradiated quartz samples are out of of irradiated quartz shows the typical features of a glassy equilibrium and can therefore relax toward more stable structure, that is, well-defined short-range order and poorly configurations upon annealing . However, because of defined long-range order. Nevertheless, the PDF of irradi- the high viscosity of the system at room temperature, ated quartz differs from that of glassy silica: (1) At short relaxation is kinetically frozen at low temperatures—as distances, the peaks appear broader than in glassy silica, (2) a peak around 2 Å that is absent in glassy silica is observed in the case of silicate glasses . The relaxation observed (N.B.: the existence of such a peak has been of irradiated quartz at different annealing temperatures is suggested experimentally ), and (3) at larger distances presented elsewhere . A visual inspection of the obtained atomic structures (>3 Å), the peaks are not as well defined as in glassy silica. reveals that any structural periodicity reminiscent of that of Points (1) and (2) arise from the existence in the atomic quartz is lost after a deposited energy of around 3 eV=atom. network of undercoordinated or overcoordinated Si and O A detailed analysis of such disordering in terms of the species, which are virtually absent in glassy silica [25,31], short-range and medium-range defects has been presented while point (3) suggests that the medium-range order—i.e., elsewhere . The disordering is apparent from the structural correlations at distances larger than the first examination of the pair distribution functions (PDFs) of coordination shells—of irradiated quartz is less pronounced quartz, which, with increasing dosage of deposited energy, than that of glassy silica. Similarly, as shown in Fig. 2(b), denote a gradual loss of long-range order [Fig. 2(a)]. This both the O–Si–O and Si–O–Si bond angle distributions are 031019-4 ENTHALPY LANDSCAPE DICTATES THE IRRADIATION- … PHYS. REV. X 7, 031019 (2017) (a) (b) FIG. 2. Comparison between irradiated quartz and glassy silica. (a) Pair distribution functions of pristine α-quartz, irradiated quartz with increasing dosages of DE, and glassy silica. (b) Intratetrahedral (O–Si–O) and intertetrahedral (Si–O–Si) bond angle distributions in pristine α-quartz, glassy silica, and fully irradiated quartz. broader in irradiated quartz. Note that a detailed discussion showed that only a small fraction of the enthalpy landscape on the origin of these defects and how they differ from the basins are accessible to glasses, regardless of their thermal typical structure of glassy silica has been presented else- history—e.g., their cooling rate or annealing duration . where . Overall, although irradiated quartz features a The configurations associated with these accessible basins disordered structure, the short- and medium-range order are called “allowable glasses.” In contrast, all the other disordered configurations, which are not accessible via of irradiated quartz appear to be more disordered than in glassy silica. traditional thermal pathways, are referred to as “forbidden Having established that irradiated quartz differs from glasses” . Accessing these forbidden configurations glassy silica, as obtained by quenching a silica melt with a requires the application of an external stimulus, such as finite cooling rate, we now investigate to what extent this high pressure  or ion-exchange treatment [42,50]. From result depends on the thermal history of the glass. Indeed, a practical standpoint, these systems exhibit properties as nonequilibrium materials, the properties of glasses at that are notably different from those of their “allowable” fixed thermodynamic conditions depend on the thermody- counterparts (e.g., density, hardness, and heat capacity ). In the following, we investigate if irradiated quartz, namic path followed since their last equilibrium configu- ration, e.g., their thermal and pressure histories [10,46]. before or after defect saturation, lies in forbidden or Note that, although the short- and medium-range order allowable regions of the enthalpy landscape. structure (bond lengths, coordination numbers, PDF, struc- Figure 3(a) shows the ground-state enthalpy H (see ture factor, etc.) only weakly depends on the thermal Sec. II)of α-quartz and glassy silica as a function of history, the thermodynamic properties, such as the density temperature. As expected, we observe that, in the case of the crystal, H remains fairly constant at low temperatures, or thermal expansion, are found to be highly sensitive to the until the melting temperature T is reached. At T , a first- thermal path followed upon cooling [37,38]. The proba- m m order transition is observed, which manifests as a disconti- bility of each final configuration can be assessed from the nuity of the enthalpy and density. Similarly, H also remains knowledge of the enthalpy landscape, that is, the hyper- fairly constant at low temperatures in glassy silica. In agree- surface of the enthalpy as a function of the volume and atomic positions [41,47]. The enthalpy landscape contains ment with the original postulate of Zachariasen ,the various energy basins, or local minima, that are connected enthalpy of glassy silica has the same order of magnitude as that of its crystalline counterpart, being only slightly higher via channels. At high temperatures, in the liquid state, the because of the absence of long-range order. However, as system can explore several basins and, as such, is ergodic expected, no clear phase transition is observed in glassy silica. and at equilibrium. However, as the temperature decreases, Rather, in the vicinity of the glass transition temperature T , the probability of transitioning between distinct basins decreases, which ultimately results in a loss of ergodicity. H continuously increases and gradually reaches the same Finally, once in the glassy state, the system is trapped in a slope as that observed for liquid silica, as melted from the local minimum of the enthalpy landscape. Mauro et al. crystal. In this regime (T ≈ 2500 K <T <T ≈ 4000 K), g m 031019-5 N. M. ANOOP KRISHNAN et al. PHYS. REV. X 7, 031019 (2017) (a) (b) -6.2 -6 .2 Supercooled silica Irradiat ed quart z Ent halpy of irradiat ed quart z Liquid silica -6.3 -6 .3 F orbidde n st at es Liqu id Silica glass Inc reasing Prist ine quart z -6.4 irrad iat ion -6 .4 Irradiat ed quart z dosage Supercoole d liquid -6.5 -6 .5 Glass A llow able glasse s Cryst al Cry stal -6.6 -6.6 2.1 2.2 2 .3 2.4 2 .5 2.6 2 .7 0 1000 2000 3000 4000 5000 Densit y ( g/ cm ) Tem perat ure ( K ) (c) (d) Liquid silica Irradiat ed quart z 1 23456 Dist ance ( Å ) FIG. 3. Comparison of the effects of irradiation and temperature. (a) Ground-state enthalpy of pristine α-quartz and glassy silica as a function of temperature, compared with that of fully irradiated quartz at 300 K. The colored areas indicate the extent of the crystalline, glassy, supercooled liquid, and liquid states. (b) Positions of partially irradiated quartz samples in the ground-state enthalpy-density space, for increasing deposited energies. These data are compared to the ground-state enthalpy or density of α-quartz and liquid or supercooled silica at different temperatures. The light-blue and red background colors indicate the extent of the regions of allowable glasses and forbidden states, respectively. (c) Pair distribution function, gðrÞ, of fully irradiated quartz, compared to that of a silica liquid hyperquenched from 4400 K. (d) Ring-size distribution of fully irradiated quartz, compared to that of a silica liquid hyperquenched from 4400 K and glassy silica. (For reference, the ground-state enthalpy of quartz with respect to the deposited energy is presented in Fig. S1 of Ref. .) silica reaches a supercooled liquid state, which is ergodic and describe the relaxation state of a glass; namely, a glass at metastable equilibrium. with a given T shows the same structure as an equilibrium As shown in Fig. 3(a), the H of fully irradiated quartz is supercooled liquid at this temperature . Namely, glass significantly higher than that of α-quartz and glassy silica formed with lower cooling rates can relax to lower energy at room temperature, as it is indeed more disordered than states and, as such, tend to feature lower values of T . In the glassy silica. We note that the H of fully irradiated quartz following, we investigate whether the irradiation-induced corresponds to that of a silica melt at around 4400 K. This disordering of quartz is equivalent to glasses featuring suggests that irradiated quartz is equivalent to a “frozen increasing T . liquid” with a “fictive temperature” T of 4400 K. Note that To answer this question, we track the states of irradiated and vitrified SiO in the enthalpy–density (H –ρ) space. such concepts of fictive temperature are typically used to 2 0 031019-6 G round-state ent halpy (eV / at om ) P air dist rib ut ion funct ion g( r) G round-state enthalpy (eV / at om ) ENTHALPY LANDSCAPE DICTATES THE IRRADIATION- … PHYS. REV. X 7, 031019 (2017) Although we note that mere knowledge of H and ρ is not irradiated quartz is compared to that of a silica liquid at sufficient to fully characterize out-of-equilibrium materials, 4400 K. To enable a meaningful comparison, the liquid is this choice is motivated by the fact that short-range instantaneously quenched to zero temperature (i.e., hyper- interactions constitute the main contribution to H ,soit quenched) before the PDF is calculated. As shown in offers a good metric to characterize the short-range order of Fig. 3(c), we observe a good agreement between the PDFs the system. Conversely, ρ is a more complex property that of fully irradiated quartz and liquid silica. In particular, we depends on fine structural arrangements, including longer- note that the first two peaks show a similar position and range features such as dihedral angle, ring-size distribution, shape, which shows that the defected short-range order of fully irradiated quartz is similar to that of the silica liquid. etc. As such, density can be used as a metric to discriminate We also observe that the peaks at larger distance show good systems featuring a similar short-range order but different agreement, which suggests that fully irradiated quartz and medium- and long-range orders . liquid silica share similar medium- and long-range order, Figure 3(b) shows the states of liquid and supercooled or absence thereof. The medium- and long-range order liquid silica with varying T in the enthalpy-density space. are more accurately captured by the ring-size distribution (For reference, the ground-state enthalpy of quartz with (RSD) within the silicate network (where rings are defined respect to the deposited energy is presented in Fig. S1 in as the shortest closed paths made of Si–O bonds within the Ref. ). We observe that these configurations form a line atomic network, see Sec. II), which is shown in Fig. 3(d) for in the H –ρ space; that is, H decreases linearly with ρ. 0 0 fully irradiated quartz, liquid silica, and glassy silica. We As such, this line—and its extrapolation to lower and observe that glassy silica exhibits a relatively sharp RSD higher temperatures—indicates the extent of the allowable centered around an average size of 6.5, in agreement with domain, that is, the set of the atomic configurations of experiment . In contrast, the RSDs of irradiated quartz allowable glasses with varying T , which can be formed and liquid silica significantly differ from that of glassy by simply cooling a melt. The two theoretical limits of silica as they show broader distributions, centered around this domain correspond to the following: (1) at high larger average rings (around 8). In turn, the RSDs of temperatures—a hyperquenched liquid, i.e., a SiO melt irradiated quartz and liquid silica share a large degree of that is instantly quenched to zero temperature, and (2) similarity, which further demonstrates the striking struc- at low temperatures—an “ideal glass” , i.e., a tural concordance between irradiated quartz and liquid supercooled liquid that reaches its theoretical Kauzmann silica. Altogether, this shows that fully irradiated quartz is temperature . effectively a frozen allowable liquid, that is, a disordered We now assess the configurational states explored by system that is obtained by instantaneously hyperquenching irradiated quartz upon different dosages of irradiation. As a melt. It is worth noting that, from the viewpoint of shown in Fig. 3(b), irradiation induces an increase of H topological constraint theory, irradiation results in the and a decrease of ρ. We observe that, as irradiation-induced breakage of the weak Si–O–Si bond-bending constraints, defects start to accumulate, the thermodynamic states which are also thermally broken in silicate liquids . achieved by irradiated quartz lie outside of the region of This further supports the equivalence between irradiated allowable glasses. This shows that, prior to defect satu- quartz and a hyperquenched liquid. ration, irradiated quartz explores forbidden states of the enthalpy landscape that cannot be accessed by cooling a IV. DISCUSSION melt. More specifically, for a given ρ, irradiated quartz shows a higher H than supercooled liquid silica, in We now discuss the origin of the saturation of damage agreement with the fact that irradiated quartz contains as irradiated quartz achieves the structure of an allowable more short-range order defects than glassy silica. It should frozen liquid. Such behavior can be understood by con- be noted that before the saturation of defects, the system sidering how irradiation affects the configurational position does not exhibit a two-phase behavior—that is, partly of the system in the enthalpy landscape of SiO , as depicted crystalline and partly disordered—and becomes fully dis- in Fig. 4(a). The energy basin corresponding to the ordered well before it reaches an allowable glass (see crystalline state is typically very narrow . The system Ref. ). Altogether, this demonstrates that irradiation- can exit this energy basin if the deposited radiation energy induced disordering fundamentally differs from that reaches a threshold value (28.9 eV for oxygen and 70.5 eV observed upon vitrification and cannot be understood as for silicon in α-quartz ) permitting one to overcome the simply an increase of fictive temperature. surrounding energy barriers, thereby allowing the system to Interestingly, we observe that, eventually, irradiation- reach another basin corresponding to a defective structure. induced damage saturates when the irradiated system At this stage, because of its vicinity to the initial crystalline reaches the state of an allowable silica liquid experiencing configuration, the local enthalpy landscape remains rough a temperature of 4400 K in the H –ρ space [see Fig. 3(b)]. and features high-energy barriers. Additional deposited To ascertain whether fully irradiated quartz indeed presents energy allows the system to reach higher energy basins a structure similar to that of a frozen liquid, the PDF of fully associated with other forbidden configurations. Finally, as 031019-7 N. M. ANOOP KRISHNAN et al. PHYS. REV. X 7, 031019 (2017) FIG. 4. Topography of the enthalpy landscape. (a) Schematic of the enthalpy landscape explored upon irradiation. The x axis represents all the configurational states. The red box indicates the extent of the forbidden region that is not accessible by conventional thermal pathways. (b) Atomic MSD explored by the atoms during 100 ps of dynamics following a similar energy bump in pristine, partially irradiated, and fully irradiated quartz samples with respect to the ground-state enthalpy. The inset shows the same values of MSD with respect to the deposited energy. The shaded region indicates the extent of the liquidlike region of the enthalpy landscape, for which irradiation-induced damage saturates. the system experiences enough deposited energy to reach constitutes the upper limit of irradiation-induced damage. an energy state comparable to that of a liquid, it eventually From a practical standpoint, the equivalence between encounters lower-energy barriers . irradiated materials and hyperquenched liquids could be To establish this mechanism, we estimate the height of used as a basis to predict the swelling of mineral when the energy barriers by computing the atomic mean-square subjected to irradiation, that is, by comparing the density of distance (MSD) between irradiated quartz configurations at the pristine crystal and of its isochemical liquid. Hence, this 1500 K and the corresponding inherent structures, i.e., after approach can provide a means to identify radiation-resistant enthalpy minimization (see Sec. II). By capturing the extent materials. of the thermally induced atomic motion, such a metric has been shown to accurately describe the height of the energy V. OUTLOOK barriers locally sampled in the enthalpy landscape . Overall, our analysis highlights the fundamental impor- Namely, lower-energy barriers result in higher atomic tance of the underlying enthalpy, which provides a con- motion and, as such, higher values of MSDs. As shown sistent explanation for the accumulation and saturation of in Fig. 4(b), we observe a sharp increase of the MSD as the structural defects in quartz. In a similar fashion as the system reaches a frozen-liquid configuration. This denotes roughness of the enthalpy landscape explored in the a sudden change of the topography of the enthalpy land- supercooled liquid state controls the fragility of glasses scape, that is, a decrease in the height of the energy barriers. (i.e., the extent of deviation from an ideal Arrhenius In turn, the roughness of the enthalpy landscape controls behavior for the viscosity at the vicinity of T ), it is the extent of possible reorganizations for the network. At expected that the roughness of the enthalpy landscape low deposited energy, high-energy barriers prevent any around the crystalline state will control the value of the transition to other energy basins and, thereby, significantly minimum threshold energy that can induce some perma- limit the extent of possible structural relaxation. As such, nent damage, as well as the ability for the system to irradiation-induced defects accumulate. In contrast, once recrystallize or not upon thermal annealing. As such, the system reaches a frozen-liquid state, lower-energy elucidating the topography of the enthalpy landscape, barriers facilitate transitions between basins so that any i.e., the positions of basin minima and the height of saddle further deposited energy will significantly move the system points, appears to be key to understanding and predicting a away from its original energy basin. This ability to freely mineral’s resistance to irradiation. sample many energy basins allows the system to relax, that is, to feature local structural reorganizations. Therefore, the ACKNOWLEDGMENTS system is able to relax any additional unfavorable structural defects that would otherwise form because of further This research was performed using funding received irradiation. Consequently, this liquidlike configuration from the Department of Energy Office of Nuclear Energy’s 031019-8 ENTHALPY LANDSCAPE DICTATES THE IRRADIATION- … PHYS. REV. 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