Journal of Nuclear Materials 458 (2015) 56–63
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Journal of Nuclear Materials
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Density functional theory metadynamics of silver, caesium and palladium diffusion at b-SiC grain boundaries ⇑ Jeremy Rabone a, , Eddie López-Honorato b
a European Commission, Joint Research Centre, Institute for Transuranium Elements, D-76125 Karlsruhe, Germany b Centro de Investigación y de Estudios Avanzados del IPN (CINVESTAV), Unidad Saltillo, Industria Metalúrgica 1062, Parque Industrial, Ramos Arizpe 25900, Coahuila, Mexico
highlights
DFT metadynamics of diffusion of Pd, Ag and Cs on grain boundaries in b-SiC. The calculated diffusion rates for Pd and Ag tally with experimental release rates. A mechanism of release other than grain boundary diffusion seems likely for Cs.
article info abstract
Article history: The use of silicon carbide in coated nuclear fuel particles relies on this materials impermeability towards Received 22 May 2014 fission products under normal operating conditions. Determining the underlying factors that control the Accepted 9 November 2014 rate at which radionuclides such as Silver-110m and Caesium-137 can cross the silicon carbide barrier Available online 4 December 2014 layers, and at which fission products such as palladium could compromise or otherwise alter the nature of this layer, are of paramount importance for the safety of this fuel. To this end, DFT-based metadynam- ics simulations are applied to the atomic diffusion of silver, caesium and palladium along a R5 grain boundary and to palladium along a carbon-rich R3 grain boundary in cubic silicon carbide at 1500 K. For silver, the calculated diffusion coefficients lie in a similar range (7.04 10 19–3.69 10 17 m2 s 1) as determined experimentally. For caesium, the calculated diffusion rates are very much slower (3.91 10 23–2.15 10 21 m2 s 1) than found experimentally, suggesting a different mechanism to the simulation. Conversely, the calculated atomic diffusion of palladium is very much faster (7.96 10 11–7.26 10 9 m2 s 1) than the observed penetration rate of palladium nodules. This points to the slow dissolution and rapid regrowth of palladium nodules as a possible ingress mechanism in addi- tion to the previously suggested migration of entire nodules along grain boundaries. The diffusion rate of palladium along the R3 grain boundary was calculated to be slightly slower (2.38 10 11–8.24 10 10 - m2 s 1) than along the R5 grain boundary. Rather than diffusing along the precise plane of the boundary, the palladium atom moves through the bulk layer immediately adjacent to the boundary as there is greater freedom to move. Ó 2014 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/3.0/).
1. Introduction carbide as the barrier layers in TRISO (Tristructural Isotropic) fuel particles have been conducted for over forty years. There Understanding the atomic bases for processes such as diffu- are numerous complexities associated with such investigations, sion provides useful input for the interpretation of experimental implanting guest atoms without significantly altering the nature measurements. Even if a process cannot be adequately modelled of the silicon carbide prior to the experiment and controlling the in its entirety, the most common reason being the sheer size and microstructure of the silicon carbide layers are two difficulties. complexity of model that would be required at the atomic scale, Even when such factors are dealt with, there is the remaining atomistic models can highlight the underlying interactions and problem that diffusion measurements on such samples poten- help to narrow the number of possible interpretations of exper- tially include contributions from many competing mechanisms. imental observations. Experimental investigations of silicon Hence to identify the mechanisms that make the greatest contri- butions and so focus on improving these material properties requires information that can only be obtained through atomistic ⇑ Corresponding author. Tel.: +49 7247 951297. simulations. E-mail address: [email protected] (J. Rabone).
http://dx.doi.org/10.1016/j.jnucmat.2014.11.032 0022-3115/Ó 2014 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/3.0/). J. Rabone, E. López-Honorato / Journal of Nuclear Materials 458 (2015) 56–63 57
It seems clear from both experiments [1–4] and simulations [5– are computationally intractable using DFT. The changes in cell vol- 8] that grain boundary diffusion is an important mechanism for ume and especially cell shape that were observed during the sim- diffusion in silicon carbide and that the microstructure [9] of the ulations indicated the extreme stresses that would otherwise arise silicon carbide layer is critical for the design and operation of TRISO during the simulations. The Canonical Sampling through Velocity particle fuel. The potential for the escape of radionuclides such as Rescaling (CSVR) thermostat-barostat [24] was used with time 110mAg (a strong c emitter) and 137Cs (has high environmental constant of 10 fs for equilibration and 1000 fs for simulations. mobility) through the silicon carbide layer of TRISO is a factor in The metadynamics [25,26] methods available in CP2 K were their use in high-temperature reactors. In particular, Ag diffusion applied in exactly the same manner as used previously [6] in order has received special attention since is can apparently escape far to induce more rapid diffusion of silver, caesium and palladium more readily than any other element, even in intact fuel, thus atoms along the grain boundaries. The collective variables of the making it a radiological hazard. Additionally, the corrosion of the metadynamics were the y- and z-coordinates of the palladium, sil- silicon carbide layers of TRISO particles by radiogenic palladium ver or caesium atom in relation to the average y- and z-coordinates [10] is also an important consideration in the use of such fuels. of the silicon and carbon atoms more than 1.9 Å from the plane of Density function theory (DFT) molecular dynamics and metady- the grain boundary. The yz plane is parallel to the plane of the grain namics simulations were previously applied to the high-tempera- boundary and defining the coordinate origin in terms of the ture diffusion of Ag and Cs along the symmetric tilt (120) R5 coordinates of non-boundary atoms prevents the metadynamics grain boundary [6]. In the initial stages of these simulations, the translating the entire simulation cell in space while avoiding noise two atoms relaxed to two different sites and the dynamic simula- from the motion of the boundary atoms. Every 5 steps (femtosec- tions were started from these two configurations. Static calcula- onds) during the metadynamics a new Gaussian was added to tions confirmed that the configurations in which the positions of ! !! XNðtÞ 2 2 the atoms were exchanged were also feasible. Therefore it is desir- 1 ðyðiÞ yð0ÞÞ 1 ðzðiÞ zð0ÞÞ VðtÞ¼0:01 exp exp able to investigate the dynamics starting from this second pair of 2 0:01 2 0:01 i¼1 configurations to obtain further sampling of the diffusion energy barriers and to see whether the diffusion trajectories depend where V(t) is the history-dependent potential, in hartrees, at time strongly on the starting configuration. The diffusion of palladium step t and N(t) is the number of accumulated distance variables, y along the R5 boundary was also investigated for comparison with and z, in angstrom. As the metadynamics simulation progressed, silver diffusion and because on the basis of static DFT calculations this potential gradually forced the diffusing atom over the lower of palladium, silver, tin and caesium with silicon carbide [11], pal- energy barriers into previously unexplored regions of phase space. ladium is the more easily accommodated in the silicon carbide The sampled energy barriers were then used to calculate diffusion structure. The diffusion of palladium along an anti-phase R3 coefficients for the diffusion. Because the instantaneous potential (111) tilt grain boundary (the carbon rich boundary) was also energy of such a small system fluctuates significantly during investigated, to asses the importance of the grain boundary struc- dynamics and metadynamics, it was necessary to average the ture and free space on the diffusion rate of palladium. potential energy at any given instant in time to obtain meaningful energy barriers. Averaging over 200 fs windows is sufficient to 2. Methodology almost remove all of the energy fluctuations from 1500 K simula- tions of the grain boundaries alone and the same window size The computational methods used for these calculations were was used for the metadynamics simulations. The maxima of the identical to those used before [6]. In summary, the calculations standard deviation of the mean at each point in time was used to were carried out using the QUICKSTEP [12,13] module of CP2K give an indication of the uncertainty in the energies and hence (development version 2.2.214) [14] with a mixed Gaussian and the uncertainty in the energy barriers. Since this uncertainty plane waves basis [15]. Periodic boundary conditions were applied dominates the uncertainty in the calculated values for the diffusion in all three dimensions. The spin-polarised PBE functional was coefficients, it is the only source of uncertainty that has been used [16] with Goedecker–Teter–Hutter (GTH) pseudopotentials included. In particular the uncertainties from the diffusion step [17–19] incorporating scalar-relativistic core corrections. The orbi- lengths and atomic collision rates, which would be of the order of tal transformation method [20] was employed for an efficient ±10% of the final diffusion coefficients, are difficult to quantify well wavefunction optimization. In both carbon and silicon the outer from the timescales accessible using these simulations. 4 electrons (2s2 2p2/3s2 3p2) were treated as the valence shell Views of the simulation cells for the R5 and R3 grain boundaries while for palladium the outer 18 electrons (4s2 4p6 4d10), for silver are shown in Fig. 1, showing the two very different boundary struc- the outer 11 electrons (4d10 5s1) and for caesium the outer 9 elec- tures. The Cartesian directions shown in Fig. 1 is the same in all of trons (5s2 5p6 6s1) were treated as valence. Contracted Gaussian the following figures. The calculated surface energies of the R5 basis sets of DZVP quality were used with a grid cut off of 200 har- and R3 grain boundaries were 3.35 and 2.16 J m 2 (as there are trees [21]. Where required, basis set superposition contributions two different boundaries in R3, this energy is the average of the (BSSE) to the system energy were taken into account using the two boundaries) respectively and is reflected in the relative abun- counterpoise correction method of Boys and Bernardi [22]. The dances of CSL grain boundaries in polycrystalline silicon carbide magnitude of the BSSE on the interactions between the Pd, Ag [27]. An indication of the expected mobility along the boundaries and Cs and the silicon carbide was found to be very small, a few is given by the ‘‘free volume’’, which is the difference between the hundredths of an electron volt. volume of some sections of the grain boundary cell and the volume The molecular dynamics simulations were carried out in the that the equivalent number of atoms would occupy as bulk b-SiC. Born–Oppenheimer approximation using a timestep of 1 fs with The carbon rich grain R3 boundary occupies less volume than the the Always Stable Predictor–Corrector (ASPC) method [23] in a equivalent amount of bulk silicon carbide with a free volume of constant pressure and temperature ensemble, thus allowing full 2.0 10 2 mm3 m 2, while the silicon rich R3 boundary has a flexibility of the simulation cells. Although the use of a constant free volume of 5.1 10 2 mm3 m 2. The free volume of the pressure ensemble increases fluctuations in the instantaneous R5 boundary (both boundaries in the cell are identical) is potential energy of the system compared to a constant volume 3.1 10 2 mm3 m 2. ensemble, the greater flexibility of the simulation cell better paral- Note that these volumes are for static calculations at 0 K, and lels the damping of stresses over the extended length scales which therefore only give a rough guide to relative mobilities since the 58 J. Rabone, E. López-Honorato / Journal of Nuclear Materials 458 (2015) 56–63
(a) (b)
X=20.43, Y=10.09, Z=8.54, α=β=γ=90.0° X=19.95, Y=9.82, Z=8.81, α=β=90.0° γ=92.7° (c) y
z x
X=20.18, Y=10.54, Z=9.56, α=β=γ=90.0°
Fig. 1. DFT optimised grain boundary structures (without guest atoms) and dimensions (in Å) for (a) the initial and (b) post MD R5 boundary, compared with (c) the R3 boundary. freedom of movement of the boundary atoms can also influence silicon atom ahead of it for a distance of about 4.5 Å, in the process diffusion of atoms along the boundary. altering the structure of the grain boundary. This would suggest that as Pd diffuses it can modify the characteristics of the grain boundary that in turn could affect, possible increase, the diffusion 3. Results rate of other elements such as Ag. On the R3 grain boundary the Pd atom leaves the plane of the Figs. 2, 3, 5 and 6 show the metadynamics trajectories and boundary in the first 500 fs, crossing the first energy barrier as it potential energy profiles for the four sets of simulations. It is does so, and thereafter diffuses parallel to the plane of the bound- immediately apparent that the silver atom diffuses relatively ary. The reasons for this behaviour are that the relatively short car- slowly, making two transitions in the 3 ps of simulation time, bon–carbon bonds of the grain boundary discourage diffusion in and the palladium diffuses relatively quickly, making 6 transitions the plane of the boundary while less restricted diffusion pathways in 2 ps in both the R5 and the R3 grain boundaries, while the are present in the adjacent bulk. Diffusion along the pathways caesium atom remains in the initial site over the 6 ps simulation. immediately adjacent to the grain boundary is apparently easier The diffusion of the silver from site 2 occurs in a similar direc- than through the bulk and so the palladium atom remains near tion to the diffusion event starting from site 1 [6], the motion being the grain boundary. The metadynamics trajectory reveals why this primarily along the y-axis. After the initial, rate limiting diffusion is the case – the atoms at the grain boundary are slightly more step in the y-direction the silver atom continues along the z-axis flexible than those in the bulk and can move further aside as the to rest briefly in the next site. The diffusion in the z-direction palladium atom passes (Fig. 7). occurred unhindered because there was a relatively large void in The energy barriers jump lengths and approximate collision fre- the structure at that point in the simulation. Several grain bound- quencies that were obtained from the simulations are summarised ary atoms were therefore able to move around the silver atom as it in Table 1. made this transition. The uncertainties in the energy barriers in Table 1 are the worst- The caesium atom collides with several of the grain boundary case estimates taken as the maximum standard deviation of the atoms during the metadynamics simulation, knocking them into mean for the rolling 200 fs samples over the course of each simula- new positions. This strengthens a cage of atoms surrounding the tion (hence there is a single value for each of the metadynamics sim- caesium atom, thereby making diffusion along the grain boundary ulations conducted). This uncertainty arises from the intrinsic more difficult. DFT optimised structures from the start and after thermal oscillations of the system potential energy and would only 6 ps metadynamics are compared in Fig. 4. The radial distribution be reduced by increasing the cell size. The calculated energy barriers functions centred on the caesium atom are also shown and indicate compare well with other DFT calculations of silver [5,7], palladium that the silicon atoms were pushed outwards while the carbon [28] and caesium [8] in cubic silicon carbide. The jump distances are atoms moved inwards. The three closest carbon atoms form weak estimated from the distances between the start and end positions of bonds with the caesium atom that are evidenced by an increase in the diffusing atoms and are in general slightly longer than the dis- the Mulliken charges. The cell expanded slightly overall, the tances between the precise starts and ends of transitions shown in volume of the statically optimised cell was 1737 Å3 before the Figs. 2, 3, 5 and 6 since there is a certain degree of unhindered metadynamics simulation and increased to 1758 Å3 afterwards. motion of the atoms at these sites. The values for the collision rate The metadynamics of the palladium atom on the R5 grain are rather approximate because of the few collision events that boundary shows almost continual diffusion of this atom along are sampled and because they are influenced by the metadynamics. the boundary. After 1 ps the palladium atom rebounds off a silicon This is particularly true of palladium where the collision rate atom and diffuses back in roughly the reverse direction to near observed in a molecular dynamics simulation ( 1ps 1) is rather where it started. Towards the end of the trajectory and correspond- less than observed in the metadynamics ( 3ps 1). The former value ing with the final energy barrier at 3730 fs, the Pd atom pushes a gives a better indication of the actual collision rate and has been J. Rabone, E. López-Honorato / Journal of Nuclear Materials 458 (2015) 56–63 59
(a) (b) 14 12 12 10 11 8 10 6 z-axis x-axis 4 9 2 0 8 0 5 10 15 20 0 5 10 15 20 y-axis y-axis
Metadynamics potential energy (c) -21835.0 Ag atom 200 fs average -21841.3
-21847.5
-21853.8 Potential energy / eV
-21860.0 4000 4500 5000 5500 6000 6500 7000 Time / fs
Fig. 2. The trajectory and potential energy profile of Ag on the R5 grain boundary during metadynamics. (a) and (b) Show the y–z and x–y coordinates of the Ag atom; the blue dot indicates the start location of the atom, and the red dots indicate the start, midpoints, and end of barrier crossings. These points in time are indicated with vertical, black lines on the potential energy curve (c). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
(a) (b) 5 12
4 11 3 10 2 z-axis x-axis
1 9
0 8 0246810 0246810 y-axis y-axis Metadynamics potential energy (c) -21370.0 Cs atom 200 fs average -21377.5
-21385.0
-21392.5 Potential energy eV / Potential energy
-21400.0 4000 5000 6000 7000 8000 9000 10000 Time / fs
Fig. 3. The trajectory and potential energy profile of Cs on the R5 grain boundary during metadynamics. (a) and (b) Show the y–z and x–y coordinates of the Cs atom; the blue dot indicates the start location of the atom. The potential energy curve is shown in (c). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) 60 J. Rabone, E. López-Honorato / Journal of Nuclear Materials 458 (2015) 56–63
(a) 4 all C 3 Si
2
1
0 2 2.5 3 3.5 4 Radius / Å (b) 4 all 3 C Si
2
1
0 2 2.5 3 3.5 4 Radius / Å
Fig. 4. DFT optimised structures and associated radial distribution functions around the caesium atom (shown in blue in the left hand diagrams) for the structure before (a) and after (b) the metadynamics simulation. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
(a)10 (b) 12
8 11 6 10 4 z-axis x-axis 9 2
0 8 -4-20246810 -4-20246810 y-axis y-axis Metadynamics potential energy (c) -24280.0 Pd atom 200 fs average -24287.5
-24295.0
-24302.5 Potential energy eV / Potential energy
-24310.0 2000 2500 3000 3500 4000 Time / fs
Fig. 5. The trajectory and potential energy profile of Pd on the R5 grain boundary during metadynamics. (a) and (b) Show the y–z and x–y coordinates of the Pd atom; the blue dot indicates the start location of the atom, and the red dots indicate the start, midpoints, and end of barrier crossings. These points in time are indicated with vertical, black lines on the potential energy curve (c) with the taller lines representing the midpoints. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
used in place of the value from the metadynamics simulations, 1 D ¼ r2m e DE=kT where the reduction in rebounds from the energy barriers has arti- 2 ficially reduced the time between collisions. Diffusion coefficients were calculated using the sampled energy where r is the distance the atom travelled between sites when an barriers and treating the diffusion as a two-dimensional random energy barrier is crossed, DE is the height of the energy barrier, k walk [6]. The diffusion coefficients using this model are given by: is Boltzmann’s constant, T the temperature and m is the atom J. Rabone, E. López-Honorato / Journal of Nuclear Materials 458 (2015) 56–63 61
(a) (b) 8 11 6 10 4
2 9
z-axis 0 x-axis 8 -2
-4 7 -10 -5 0 5 10 -10 -5 0 5 10 y-axis y-axis
Metadynamics potential energy (c) -28468.0 Pd atom 200 fs average -28475.1
-28482.2
-28489.3 Potential energy eV / Potential energy
-28496.5 2000 2500 3000 3500 4000 Time / fs
Fig. 6. The trajectory and potential energy profile of Pd on the R3 grain boundary during metadynamics. (a) and (b) Show the y–z and x–y coordinates of the Pd atom; the blue dot indicates the start location of the atom, and the red dots indicate the start, midpoints, and end of barrier crossings. These points in time are indicated with vertical, black lines on the potential energy curve (c) with the taller lines representing the midpoints. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
order as calculated for bulk diffusion [8] and does not explain the measured rates (2.1 10 17–2.4 10 18 m2 s 1 near 1500 K) [1,2,29]. The further diffusion events sampled for silver increase slightly the uncertainty but shift the median to a slightly higher value (rate calculated from first diffusion event 1.5 10 18±0.84), bringing it closer to the experimentally measured rate (1.6– 7.0 10 17 m2 s 1 near 1500 K) [4,30]. Experiments of the thinning of silicon carbide by palladium led to a linear relationship between the penetration depth and time [31], rather than proportionality to the square root of time expected of a simple diffusion process. The observed trend of penetration depth against time gives a palladium reaction rate of 1.11 10 12 ms 1 at 1500 K. Comparing this rate with the calcu- lated atomic diffusion coefficient is somewhat difficult, however Fig. 7. Plot of the 2 ps metadynamics trajectory showing the diffusion of the the faster atomic diffusion rate of 1.4 10 10±0.77 m2 s 1 is sup- palladium atom parallel to the R3 grain boundary (the boundary plane is shown with a dashed line). The palladium atom is shown in blue, silicon atoms in yellow ported by experimental observations of palladium travelling well and carbon atoms in grey. (For interpretation of the references to colour in this beyond the reaction front and the presence of palladium within figure legend, the reader is referred to the web version of this article.) silicon carbide without the formation of nodules at the PyC/SiC interface [32,33]. Our work and other reported results [4–6] sup- collision frequency. As already noted, the uncertainty in the energy port the experimental evidence [4] that silver can diffuse by its barriers dominates the uncertainty in the calculated diffusion coef- own depending of the microstructure and without the aid of other ficients and uncertainties from the distances travelled by the atoms elements. However, our results also suggest that as palladium dif- and the atom collision frequency, are not included owing the lim- fuses it can modify the characteristics of the grain boundary. This ited number of diffusion events sampled. Fig. 8 gives a plot of the could mean that the rapid diffusion of this element could alter diffusion coefficients for each of the sampled energy barriers in each grain boundary structures and lead to higher diffusivities for the of the metadynamics simulations (the first sampled barriers for Ag otherwise less mobile atoms such as caesium or silver. Further- and Cs are from the previous calculations [6]). Averaging the med- more, our results show that even for elements as mobile as palla- ian values and the extrema give the following effective diffusion dium, the characteristics of the grain boundary, or in other words coefficients at 1500 K in m2 s 1:PdR3 = 1.4 10 10±0.77,Pd the microstructure, can still influence the diffusion behaviour R5 = 7.6 10 10±0.98,AgR5 = 5.1 10 18±0.86,CsR5 = 2.9 (palladium diffuses 5 times faster along the R5 grain boundary 10 22±0.87. As reported previously [6], the calculated diffusion coef- compared to the R3 boundary). Although recent work in actual ficient of Cs for diffusion along the R5 grain boundary is of the same irradiated fuel did not show clear evidence of Ag diffusing by the 62 J. Rabone, E. López-Honorato / Journal of Nuclear Materials 458 (2015) 56–63
Table 1 Energy barriers (DE), jump lengths (r), and approximate collision frequencies (m) for all of the sampled diffusion events.
Pd R3 DE (eV) 1.90 ± 0.23 1.54 ± 0.23 1.22 ± 0.23 0.72 ± 0.23 0.52 ± 0.23 1.88 ± 0.23 r (Å) 1.7 4.4 0.8 0.9 2.9 3.0 m (ps 1)111111 Pd R5 DE (eV) 0.63 ± 0.29 0.47 ± 0.29 0.17 ± 0.29 0.48 ± 0.29 0.87 ± 0.29 1.18 ± 0.29 r (Å) 1.2 2.5 1.5 1.5 2.5 2.2 m (ps 1)111111 Ag R5 DE (eV) 3.35 ± 0.25 3.18 ± 0.26 3.14 ± 0.26 r (Å) 5.2 6.3 5.8 m (ps 1)212 Cs R5 DE (eV) 4.65 ± 0.26 r (Å) 11.0 m (ps 1)2
25 Calculated diffusion coefficients at 1500K
20 ) -1
s 15 2 (D / m
10 10 -log
5
0 Ag (Σ5) 7ps Cs (Σ5) 11ps Pd (Σ5) 2ps Pd (Σ3) 2ps
Fig. 8. Minus the logarithm of the calculated diffusion coefficients for the diffusion events sampled in these and previous metadynamics simulations. The error bars represent the uncertainties based on the maxima of the observed oscillations (which include contributions from the thermal motion and perturbations of the accelerating potentials) in the instantaneous potential energies. formation of a Pd–Ag compound [33] the effect of elements such as slower than the release rates measured in the laboratory. The dif- palladium or strontium cannot be totally disregarded considering fusion mechanism of caesium remains to be identified, since the the complexity of the system (temperature, irradiation, presence simulated grain boundary diffusion is as energetically unfavour- of other elements, variations in microstructure) [34]. Further work able as bulk diffusion and neither mechanism explains the experi- is needed in order to understand how these variables affect the dif- mental release of caesium from silicon carbide. What is clear from fusion of fission products. the simulations is that caesium has an extremely low solubility in bulk silicon carbide as well as a strongly repulsive interaction with 4. Conclusions the grain boundary. For palladium, our results suggest that not only it can diffuse readily but also that it can modify the character- At the atomic scale and lengths of time accessible using DFT, istics of the grain boundary as it progresses, possibly affecting the diffusion is complicated by the potential for the boundary struc- diffusion rates of other elements such as Cs and Ag. ture to change during, sometimes with changes induced by the dif- To fully model the diffusion processes occurring in the experi- fusing atom itself. Variation of the ground boundary structure and ments would require more sophisticated models that can combine thermal oscillations also influence the nature of the energy barriers atomic diffusion with segregation rates into different phases as that a diffusing atom must pass in order to move, leading to a range well as modelling the evolution of grain boundary structure over of energy barriers and therefore a range of diffusion coefficients. time. The scale of these models precludes the direct use of DFT Since the diffusion simulations presented in this manuscript spe- calculations but the relevant parameters for such models could cifically target diffusion of a single atom at a grain boundary, they be obtained from DFT and applied to simulations that would allow provide insights into the diffusion mechanisms responsible for dif- the various contributing processes to be identified. fusion rates observed experimentally. Where other mechanisms dominate, for example involving different microstructures or References phases, discrepancies between the simulated and experimental diffusion rates are to be expected. [1] H.-J. Allelein, Spaltproduktverhalten – Speziell Cs-137 – in Htr Triso The simulated rate of silver diffusion along the R5 grain bound- Brennstoffteilchen. Berichte der Kernforschungsanlage Jülich, 1980, p. 148. [2] W. Amian, D. Stover, Nucl. Technol. 61 (1983) 475–486. ary is in close agreement with the experimentally measured diffu- [3] E. Friedland, J.B. Malherbe, N.G. Van der Berg, T. Hlatshwayo, A.J. Botha, E. sion rates. It is therefore highly likely that atomic grain boundary Wendler, W. Wesch, J. Nucl. Mater. 389 (2009) 326–331. diffusion is the responsible mechanism, since bulk diffusion is cal- [4] E. López-Honorato, H. Zhang, Y. DaXiang, P. Xiao, J. Am. Ceram. Soc. 94 (9) (2011) 3064–3071. culated to be much slower and escape through cracks would be a [5] S. Khalil, N. Swaminathan, D. Shrader, A.J. Heim, D.D. Morgan, I. Szlufarska, lot faster. For caesium, the calculated atom diffusion rate is much Phys. Rev. B 84 (2011) 214104. J. Rabone, E. López-Honorato / Journal of Nuclear Materials 458 (2015) 56–63 63
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