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Surface properties, thermal expansion, and segregation in the U–Zr solid solution
Computational Materials Science 50 (2010) 447–453
Contents lists available at ScienceDirect
Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci
Surface properties, thermal expansion, and segregation in the U–Zr solid solution G. Bozzolo a,⇑, H.O. Mosca b, A.M. Yacout a, G.L. Hofman a, Y.S. Kim a a b
Argonne National Laboratory, 9700 S. Cass Ave., Argonne, IL 60439, USA Gerencia de Investigaciones y Aplicaciones CNEA, Av. Gral. Paz 1499, B165KNA, San Martín, Buenos Aires, Argentina
a r t i c l e
i n f o
Article history: Received 14 June 2010 Received in revised form 30 August 2010 Accepted 2 September 2010
Keywords: Segregation Atomistic modeling Uranium Zirconium Thermal expansion
a b s t r a c t Atomistic simulation results of the (cU, bZr) solid solution behavior are discussed, including the behavior of the lattice parameter and coefficient of thermal expansion as a function of concentration and temperature. Output from these calculations is used to study the surface structure of U–Zr for different crystallographic orientations, determining the respective concentration profiles, surface energy, and segregation behavior. The segregation of Zr to the surface and overall composition of the near-surface region is studied as a function of concentration and temperature. Ó 2010 Elsevier B.V. All rights reserved.
1. Introduction The main metallic fuels proposed for advanced nuclear energy systems, such as fast reactors, consist of metallic U–Zr based alloys, with additions of Pu and/or minor actinides (Am, Np, Cm) [1]. A starting point for understanding and predicting the performance of those advanced fuels during reactor operations is to understand the properties and behavior of the base U–Zr alloy. The U–Zr fuels have been used in the past in fast reactors, and one of the key issues that was found to have an impact on the performance of this base fuel is the way it interacts with the surrounding barrier (cladding). The cladding material prevents the release of fission products generated within the fuel during irradiation to the reactor coolant. Attempts to model fuel/cladding interaction would benefit from a better understanding of the fuel surface and migration of fuel elements to the surface during irradiation, as they can have significant effect on the diffusion of cladding elements into the fuel or the formation of an interaction layer with detrimental effects to the cladding and fuel. U–Zr fuels surfaces have not been characterized, although some evidence exists on the enhancement of Zr concentration in the surface region, mostly seen in experiments dealing with the interaction of cladding elements with the metallic fuel or in studies of general properties of U–Zr [1–6]. Partly, the formation of U- or Zr-rich phases may depend on the temperature gradient constituent redistribution, but in addition there could be
⇑ Corresponding author. Tel.: +1 410 995 4063. E-mail address: [email protected] (G. Bozzolo). 0927-0256/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.commatsci.2010.09.002
surface effects that have an additional impact on the resulting phase structure of the interaction layer. For example, the presence of impurities, such as N, where a Zr-rich Zr–N layer was observed in the inner surface of the cladding effectively reducing molten phase formation at the fuel/cladding interface [1,7]. As a result, Zr barriers have been proposed to reduce interdiffusion of fuel and cladding components. In addition to the numerous features that characterize the interaction between metallic fuels and cladding, the segregation behavior can have an important effect on the actual surface structure and composition, as it has the potential of drastically changing the concentration of the fuel elements in the surface region. Therefore, in this work, we present modeling results of the structure and composition of the cU–bZr solid solution and its bulk properties including the dependence of the lattice parameter and coefficient of thermal expansion on composition and temperature. Also included is a detailed discussion of the complex Zr segregation behavior observed by means of atomistic simulations using the Bozzolo– Ferrante–Smith (BFS) method for alloys [8]. Very few experimental results are readily available for any of these properties. Some values of the lattice parameter vs concentration were measured by Huber and Ansari [9] in their study of superconductivity of bcc U–Zr alloys, where a contraction of the lattice parameter for U-rich alloys was found, and dilatometric results for as cast U-rich U–Zr in the work of Basak et al. [10]. From a theoretical standpoint, recent work by Landa et al. [11] provides a thorough investigation of phase equilibria in the U–Zr system based on ab initio DFT calculations of several ground state properties and decomposition curve of the solid solution, with extensive analysis of the concentration
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dependence of the bulk modulus, heat of formation, Debye temperature, Grünesein constant, and volume. Therefore, this work is intended to fill the gap in the current knowledge of the surface characteristics and behavior of the U–Zr system, and set the stage for a better modeling approach of the complex interactions between fuels and cladding.
(a)
2. The BFS method for alloys The BFS method [8] is based on the concept that the energy of formation of a given atomic configuration is the sum of the individP ei . Each contribution by atom i, ual atomic contributions, DH ¼ ei, can be calculated as the sum of two terms: a strain energy, eSi , computed in the actual lattice as if every neighbor of the atom i were of the same atomic species i, and a chemical energy, eCi , computed as if every neighbor of the atom i were in an equilibrium lattice site of a crystal of species i, but retaining its actual chemical identity. The computation of eSi , using Equivalent Crystal Theory (ECT) [12], involves three pure element properties for atoms of species i: cohesive energy (Ec), equilibrium lattice parameter (ae) and bulk modulus (B0). These three parameters for each of the constituent elements are needed in the general derivative structure of the final alloy (bcc in the case of U–Zr alloy of interest). Additional parameters, a (average electron density) and k (screening length), can be easily derived from Ec, ae and B0 [12]. The chemical energy, eCi , accounts for the corresponding change in composition, considered as a defect in an otherwise pure crystal. The chemical ‘defect’ deals with pure and mixed bonds, therefore, additional perturbative parameters, DUZr and DZrU are needed to describe these interactions. A reference chemical energy, eCo i , is also included to insure a complete decoupling of structural and chemical features. Finally, the strain and chemical energies are linked with a coupling function gi, which ensures the correct volume dependence of the BFS chemical energy contribution. Therefore, the contribution of atom i to the energy of formation of the system is given by ei ¼ eSi þ gðeCi eCi 0 Þ. All the necessary parameters were computed using the Linearized Augmented Plane Wave method (LAPW) [13]. For the single elements, the total energy of the pure solids were computed in the bcc symmetry of the alloy, while the parameters D were computed from LAPW calculations of the equilibrium properties (lattice parameter and energy of formation) of the B2 UZr metastable structure. These parameters are universal and transferable to any situation dealing with U and Zr in the bcc symmetry, as is the case in the cU–bZr solid solution [14].
(b)
(c)
3. Results and discussion 3.1. Lattice parameter and coefficient of thermal expansion Simulations were performed on a bcc computational cell with uniform lattice parameter at each stage of the calculation. With this limitation, a solid solution for the whole range of concentration was found to be the equilibrium state above T = 1000 K and up to the melting point (as estimated from the approximate expression for the melting temperature by Guinea et al. [15]), thus ensuring that at least for the range of temperatures considered, the simulations agree with the phase diagram. The lattice parameter of the cU–bZr solid solution was obtained from a variant of Monte Carlo–Metropolis simulations [16] and was found to exhibit a non-uniform behavior, as can be seen in Fig. 1. For low Zr concentration (<20 at.%), the lattice parameter shows a minor contraction with respect to average values accompanied by a dip in the values for the energy of formation, as shown in Fig. 1c. For Zr concentrations above 20 at.%, an increasing expansion that peaks at approximately 70 at.% Zr is observed. These results are somewhat in
Fig. 1. (a) Lattice parameter (in Å) of the U–Zr solid solution for the temperature range in which it exists. (b) Deviation of the computed lattice parameter aBFS (in Å) with respect to Vegard’s law, aVeg, for different temperatures. (c) Energy of formation (in eV/atom) as a function of Zr concentration for T = 1000 and 1200 K.
agreement with experimental measurements of the lattice parameter [9], in that those also reflect a noticeable contraction for con-
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centrations below 40 at.% Zr. It should be noted, however, that no measurements were quoted for alloys with less than 25 at.% Zr. Theoretical results [11], validated by comparison with CALPHAD values for the heat of formation, do not predict such contraction. No such disagreement between experimental, ab initio, and BFS results, is seen for Zr-rich alloys. The apparent difference between ab initio results and those presented here might have its source on the fact that the BFS parameters, based in turn on ab initio calculations, might be sensitive to minor changes in the predicted properties (i.e., cohesive energy, bulk modulus, and lattice parameter of the single species). While this deserves further investigation, the fundamental nature of these properties might be an interesting feature to be thoroughly examined experimentally in the future. Based on these results, the evolution of the lattice parameter with temperature was computed using the finite temperature extension of the BFS method [17], and the same feature was found at every temperature for which the solid solution exists. The coefficient of thermal expansion (CTE) was also computed, and found to have a uniform behavior for the whole range of concentrations, slightly below the average values, as shown in Fig. 2. A simple expression describing a(xZr, T) for the range 1000 K < T < 1400 K is given by
aðxZr ; TÞ ¼ a1000 ðxZr Þð1 þ aðxZr ÞðT 1000ÞÞ
ð1Þ
where
a1000 ðxZr Þ ¼ 3:4728 1 þ 3:542 104 xZr þ 4:607 106 x2Zr 3:743 108 x3Zr
ð2Þ
is the lattice parameter (in Å) at T = 1000 K as a function of Zr concentration (in at.%) and where
aðxZr Þ ¼ 12:35 þ 1:045 102 xZr 4:924 105 x2Zr þ 4:614 107 x3Zr
ð3Þ 6
is the average coefficient of thermal expansion (in 10 aforementioned temperature range.
K
1
) in the
3.2. Surface energy It is known that the surface energy, r, for low Miller indices in bcc crystals, follows the order r110 < r111 < r100 (with other stepped surfaces in between). This was found to be true both for bcc U and Zr and the cU–bZr solid solution. For the pure elements, there is an interesting behavior in the surface energy as a function of temperature and crystallographic orientation, rarely found on other elements. Table 1 shows the values of the surface energy for pure U and Zr bcc surfaces for T = 1000 K and T = 1400 K. In addition, the values at T = 0 K are shown (although neither U nor Zr is present as a bcc phase at that temperature). In both unrelaxed and relaxed cases, some features are apparent: U has a lower surface energy than Zr for any crystallographic orientation, but this trend is reversed in the temperature range for which the solid solution exists, with Zr having a lower surface energy than U at 1000 K (in all three orientations), and even lower at T = 1400 K. In spite of the reversal in the gap between them at any temperature, and whether they favor U or Zr, the surface energies are always very similar. The small differences in surface energy as well as the peculiar variation of the lattice parameter of the cU–bZr solid solution (Fig. 1), together with the behavior of the coefficient of thermal expansion, suggest that the expected Zr segregation behavior to the surface will be highly dependent on crystallographic orientation, temperature, and concentration. Five specific alloys were modeled: U–12Zr (in at.%), U–22.45Zr, U–39.45Zr, U–72.27Zr, and U–85.94Zr (corresponding to 5, 10, 20, 50 and 70 wt.% Zr, respectively). These compositions were chosen because they correspond to different behaviors of the lattice parameter with respect to the average values, as depicted in Fig. 1b: U–12Zr corresponds to the case with maximum contraction relative to average values, U– 22.45Zr corresponds to the crossing point where the lattice parameter achieves exactly the average value, and the other three cases correspond to the increasing, maximum, and decreasing expansion relative to average values. With the surface energies being so close to each other, it is possible that the behavior of a (x, T) could be an important factor in deciding the segregation behavior.
Table 1 Surface energy (in ergs/cm2) for different U and Zr unrelaxed surfaces at T = 0, 1000, and 1400 K. Values for relaxed surfaces are shown in parenthesis. The lattice parameters for U at the different temperatures are a = 3.4350 Å, 3.4736 Å, and 3.4908 Å, respectively, while those for Zr are a = 3.5765 Å, 3.6189 Å, and 3.6383 Å.
Fig. 2. (a) Coefficient of thermal expansion (CTE) for the U–Zr solid solution for different temperatures. (b) Deviation of the CTE with respect to average values.
Face
T=0K
T = 1000 K
T = 1400 K
U(1 0 0) Zr(1 0 0) U(1 1 0) Zr(1 1 0) U(1 1 1) Zr(1 1 1)
1906.30 (466.34) 1924.00 (522.41) 859.70 (821.24) 870.42 (834.65) 1320.50 (1220.60) 1350.40 (1254.40)
2121.40 2094.00 1201.30 1143.90 1502.00 1498.30
2310.00 2245.50 1486.30 1374.60 1627.20 1601.20
(616.22) (560.70) (1129.30) (1081.90) (1388.60) (1393.10)
(757.56) (757.35) (1396.10) (1298.60) (1507.10) (1491.20)
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3.3. Segregation behavior The simulations were limited to the range of temperatures for which the solid solution exists, and the lattice parameter and plane spacing were kept unchanged during the simulation. In all cases, the competition between U and Zr segregation was more noticeable in the lower temperature range (1000–1200 K). At higher temperatures, while the segregation behavior remained unchanged, there was a slow return towards bulk concentrations, as thermal effects washed out any minor difference between U or Zr tendency to segregate. In what follows, Zr(nb) or U(nb) denote the concentration (in at.%) in the n-th layer below the surface (S). Only results for unrelaxed planes are shown. Calculations including relaxation of the near-surface planes were made and no qualitative change was made. The following comments therefore concentrate on unrelaxed surfaces, in order to avoid introducing another degree of freedom in the discussion. (1 0 0) face: For this face, Zr has lower surface energy than U for 1000 < T < 1400 K. However, the difference is very small and tends to disappear as the temperature increases. In U–Zr, as the concentration of Zr increases the lattice parameter expansion (faster than average) tends to favor Zr in the bulk reducing its strain energy (as the lattice parameter approaches the pure bcc Zr value). U, finding itself in an environment of high strain as the Zr concentration increases (as the lattice parameter of cU is further away from the equilibrium lattice value) is then driven to the surface layer. The segregation profiles for increasing Zr concentration are shown in Fig. 3. 12 at.% Zr (1 0 0): Zr segregates to the surface (S) depleting the first layer below (1b), nearly doubling its reference concentration. The pattern remains as the temperature increases, with no repopulation of 1b and rather steady surface Zr concentration (Zr(S)). 22.45 at.% Zr (1 0 0): The temperature-dependent concentration profiles show nearly no difference with the previous case, other than the fact that Zr(S) more than doubles the reference concentration. This increase in Zr segregation is also apparent in the 1b layer, as opposed to the previous case in which all surface Zr came from the layer below. This occurs at the expense of Zr(2b). Segregation of Zr peaks at 1200 K, when some reduction of Zr(S) starts. 39.45 at.% Zr (1 0 0): Zr segregation is dominant, now populating the S and 1b layers. Zr(1b) is more significant than in the lower Zr concentration cases, exceeding the nominal (bulk) value, while Zr(S) still remains at roughly twice the nominal value. Zr depletion now spreads to lower layers, as opposed to just the layer 1b (in the 12 at.% Zr case) or layer 2b (in the 22.45 at.% Zr case). 72.27 at.% Zr (1 0 0): It is at this high Zr bulk concentration that a change in segregation patterns occurs (corresponding to the peak in Fig. 1b). As mentioned above, the large Zr bulk concentration creates a high strain environment for U, favoring its segregation to the surface in spite of the lower surface energy for Zr in this surface orientation. As a result, there is Zr depletion in S for all temperatures and any Zr enrichment is seen only in subsurface layers. 85.94 at.% Zr (1 0 0): Segregation reversal is now complete, and there is strong Zr depletion in the top layer and some in the 1b layer, slightly and evenly enriching subsurface layers (2b and layers below). (110) face: This is the lowest surface energy face for either element, where once again Zr has lower surface energy than U. The gap, however, is wider than that seen in the (1 0 0) case, thus leading to a dominance of Zr segregation over U even at high Zr concentrations where, as noted above in the discussion of the (1 0 0) case, U segregation would be favored. In this case, this enhanced U segregation (driven by strain relief as the lattice parameter of the alloy increases relative to the equilibrium cU value) barely matches Zr segregation, leading to nearly bulk-like profiles at high Zr
Fig. 3. Concentration profiles (in at.%) for (from top to bottom) U100xZrx (x = 12, 22.45, 39.45, 72.27, and 85.94 at.% Zr) alloys in the (1 0 0) orientation. The horizontal line in each panel indicates the reference Zr bulk concentration.
concentration. The segregation profiles for increasing Zr concentration are shown in Fig. 4.
G. Bozzolo et al. / Computational Materials Science 50 (2010) 447–453
Fig. 4. Concentration profiles (in at.%) for (from top to bottom) U100xZrx (x = 12, 22.45, 39.45, 72.27, and 85.94 at.% Zr) alloys in the (1 1 0) orientation. The horizontal line in each panel indicates the reference Zr bulk concentration.
12 at.% Zr (1 1 0): Very similar to the (1 0 0) case, Zr enriches the S plane at the expense of Zr(1b). For this face, however, the profiles are more sensitive to increasing temperature.
451
22.45 at.% Zr (1 1 0): Zr(S) enrichment at the expense of Zr(1b) is dramatically increased and remains high even after the reduction brought upon by increasing temperature. 39.45 at.% Zr (1 1 0): As in the (1 0 0) case, Zr segregation is markedly increased, leading to enhanced Zr(1b). Also as in the (1 0 0) case, this is due to an overall depletion of Zr in subsurface layers deep into the bulk. 72.27 at.% Zr (1 1 0): While the (1 0 0) case showed a trend towards segregation reversal at this concentration, this is not the case for the (1 1 0) face, where Zr(S) is still much higher than its nominal value. The profiles, however, show substantial changes as the temperature increases. The subsurface depletion of Zr at T = 1000 K is rapidly replaced by a thick Zr-rich surface region extending to the first few layers at T = 1200 K. This effect is not sustained at higher temperatures, returning to nearly bulk values throughout the cell, except for a still Zr-rich surface layer. 85.94 at.% Zr (1 1 0): Once again, the segregation reversal seen at similar concentrations in the (1 0 0) face is not seen here. There is, however, a slight manifestation of this reversal as the competition between Zr and U is now even enough, leading to nearly bulklike profiles for all temperatures with a slight Zr enrichment of the surface planes. (1 1 1) face: This surface orientation, in terms of the magnitude of the surface energy, r, is an intermediate case between the (1 1 0) (lowest) and (1 0 0) (highest) faces. As shown in Table 1, the surface energy of Zr is slightly lower than that of U in the solid solution temperature range for unrelaxed surfaces, and switching from rZr > rU at T = 1000 K to rU > rZr at T = 1400 K in the relaxed case. However, the differences are truly negligible, so that segregation of either species will be driven by strain effects due to lattice size. The segregation results for alloys with increasing Zr concentration are shown in Fig. 5. 12 at.% Zr (1 1 1): Like in all other faces, there is Zr(S) enhancement at the expense of Zr(1b). The difference from nominal values, however, is very small as any additional Zr segregation would be driven by lattice expansion and not by the contraction in the lattice parameter values that characterize this range of Zr bulk concentration. 22.45 at.% Zr (1 1 1): Following the pattern seen in (1 0 0) and (1 1 0) for this Zr concentration, there is noticeable Zr segregation to the surface (not just S, but also the few planes below the surface), thus defining a rather thick Zr-rich surface region for all temperatures. 39.45 at.% Zr (1 1 1): The thickness of the Zr-rich surface is substantially enhanced with respect to alloys with lower Zr concentration. General Zr depletion in the bulk leads to enhanced Zr(S), Zr(1b), and Zr(2b). Unlike other cases, Zr(S) is smaller than Zr(1b), but still above the nominal level, this being the first indication of the segregation reversal that seems to characterize Zr-rich alloys, favoring U surface segregation. 72.27 at.% Zr: The pattern seen in the 39.45 at.% Zr case is greatly enhanced. However, as was the case in the other faces, U segregation becomes more noticeable leading Zr(S) below nominal values, but the Zr-enriched subsurface is now extensive, encompassing several layers below the surface at the expense of an extensive dip in Zr concentration in layers deeper in the bulk. 85.94 at.% Zr (1 1 1): At this high Zr concentration, the lattice parameter is still well above the average value (see Fig. 1b), so the strain-driven segregation of U continues to dominate, leading to strong Zr(S) depletion and the almost complete disappearance of the subsurface Zr-enrichment. Summarizing, in spite of some quantitative changes for different crystallographic orientations, there is a general trend favoring Zr segregation and the creation of a generally thick Zr-rich surface region for alloys with low Zr concentration, followed by a strain-
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ing temperature, neither thermal effects or surface energies seem to be the dominant factors in determining such behavior. 4. Conclusions In conclusion, the composition of the U–Zr solid solution surface is highly dependent on the particular behavior of the lattice parameter as a function of concentration. The computed values of the surface energy of each element are nearly identical, making finer details of the bulk to be the driving force for the resulting segregation behavior. It was found that at low Zr concentration, a slightly favorable Zr surface energy and lattice contraction lead to enhanced Zr segregation, as the low value of the bulk lattice parameter increases the BFS strain of Zr atoms. However, as the concentration of Zr increases and the lattice parameter expands above average values, where strain relief favors U segregation, leading to a final pattern that shows significant differences for different surface orientations. In addition, changes in temperature can also have an effect due to the changes in the coefficient of thermal expansion for different Zr concentrations as a function of temperature. Polycrystalline surfaces, as they are likely to be found in the actual system, would therefore display a combination of the features described above, with dominant (1 1 0) characteristics. In any case, the peculiar behavior of the bulk lattice parameter as a function of Zr concentration leads to the complex superposition of several effects, driven by the small difference in individual surface energies, change in segregation behavior (from Zr to U) with increasing Zr concentration, and interesting changes in the surface region depending on crystallographic orientation. Finally, this analysis suggests that the Zr enrichment of the surface, a desired and beneficial behavior in nuclear fuels (as it reduces interdiffusion between the Fe-based cladding and fuel), is an effect that is limited to alloys with rather low Zr concentration. Should that not be the case, high temperature U repopulation of the surface could have a deleterious impact on the Zr protective barrier and further interaction of U with cladding. This behavior is then important to be taken into account when considering the inclusion of other alloying elements in the fuel (such as Pu) as they could alter the fine balance that leads to the observed behavior. Acknowledgments Fruitful discussions with N. Bozzolo are gratefully acknowledged. Argonne National Laboratory’s work was supported under US Department of Energy contract DE-AC02-06CH11357. References
Fig. 5. Concentration profiles (in at.%) for (from top to bottom) U100xZrx (x = 12, 22.45, 39.45, 72.27, and 85.94 at.% Zr) alloys in the (1 1 1) orientation. The horizontal line in each panel indicates the reference Zr bulk concentration.
driven segregation reversal where U repopulates the surface layers for Zr-rich alloys. While some differences are observed with chang-
[1] G.L. Hofman, L.C. Walters, in: B.R.T. Frost (Ed.), Materials Science and Technology: A Comprehensive Treatment, vol. 10A, Nuclear Materials, Part 1, Wiley-VCH, Germany, 1994. [2] T. Ogata, M. Kurata, K. Nakamura, A. Itoh, M. Akabori, J. Nucl. Mater. 250 (1997) 171. [3] K. Nakamura, T. Ogata, M. Kurata, A. Itoh, M. Akabori, J. Nucl. Mater. 275 (1999) 246. [4] D.D. Keiser, M.A. Dayananda, J. Nucl. Mater. 200 (1993) 229. [5] C.T. Lee, H. Kim, T.K. Kim, C.B. Lee, J. Nucl. Mater. 395 (2009) 140. [6] T. Ogawa, T. Ogata, A. Itoh, M. Akabori, H. Miyamishi, H. Sekino, M. Nishi, A. Ishikawa, J. Alloys Compd. 271–273 (1998) 670. [7] M. Akabori, A. Itoh, T. Ogawa, J. Nucl. Mater. 289 (2001) 342. [8] G. Bozzolo, J. Ferrante, J.R. Smith, Phys. Rev. B 45 (1992) 493. [9] J.G. Huber, P.H. Ansari, Physica 135B (1985) 441. [10] C. Basak, G.J. Prasad, H.S. Kamath, N. Prabhu, J. Alloys Compd. 480 (2009) 857. [11] A. Landa, P. Söderlind, P.E.A. Turchi, J. Alloys Compd. 478 (2009) 103. [12] J.R. Smith, A. Banerjea, T. Perry, J. Ferrante, G. Bozzolo, Phys. Rev. B44 (1991) 6444. [13] P. Blaha, K. Schwarz, G.K.H. Madsen, D. Kvasnicka, J. Luitz, WIEN2K, An Augmented Plane Wave + Local Orbitals Program for Calculating Crystal Properties, Karlheinz Schwarz, Techn. Universität Wien, Austria, 2001 ISBN 3-9501031-1-2.
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Contents lists available at ScienceDirect
Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci
Surface properties, thermal expansion, and segregation in the U–Zr solid solution G. Bozzolo a,⇑, H.O. Mosca b, A.M. Yacout a, G.L. Hofman a, Y.S. Kim a a b
Argonne National Laboratory, 9700 S. Cass Ave., Argonne, IL 60439, USA Gerencia de Investigaciones y Aplicaciones CNEA, Av. Gral. Paz 1499, B165KNA, San Martín, Buenos Aires, Argentina
a r t i c l e
i n f o
Article history: Received 14 June 2010 Received in revised form 30 August 2010 Accepted 2 September 2010
Keywords: Segregation Atomistic modeling Uranium Zirconium Thermal expansion
a b s t r a c t Atomistic simulation results of the (cU, bZr) solid solution behavior are discussed, including the behavior of the lattice parameter and coefficient of thermal expansion as a function of concentration and temperature. Output from these calculations is used to study the surface structure of U–Zr for different crystallographic orientations, determining the respective concentration profiles, surface energy, and segregation behavior. The segregation of Zr to the surface and overall composition of the near-surface region is studied as a function of concentration and temperature. Ó 2010 Elsevier B.V. All rights reserved.
1. Introduction The main metallic fuels proposed for advanced nuclear energy systems, such as fast reactors, consist of metallic U–Zr based alloys, with additions of Pu and/or minor actinides (Am, Np, Cm) [1]. A starting point for understanding and predicting the performance of those advanced fuels during reactor operations is to understand the properties and behavior of the base U–Zr alloy. The U–Zr fuels have been used in the past in fast reactors, and one of the key issues that was found to have an impact on the performance of this base fuel is the way it interacts with the surrounding barrier (cladding). The cladding material prevents the release of fission products generated within the fuel during irradiation to the reactor coolant. Attempts to model fuel/cladding interaction would benefit from a better understanding of the fuel surface and migration of fuel elements to the surface during irradiation, as they can have significant effect on the diffusion of cladding elements into the fuel or the formation of an interaction layer with detrimental effects to the cladding and fuel. U–Zr fuels surfaces have not been characterized, although some evidence exists on the enhancement of Zr concentration in the surface region, mostly seen in experiments dealing with the interaction of cladding elements with the metallic fuel or in studies of general properties of U–Zr [1–6]. Partly, the formation of U- or Zr-rich phases may depend on the temperature gradient constituent redistribution, but in addition there could be
⇑ Corresponding author. Tel.: +1 410 995 4063. E-mail address: [email protected] (G. Bozzolo). 0927-0256/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.commatsci.2010.09.002
surface effects that have an additional impact on the resulting phase structure of the interaction layer. For example, the presence of impurities, such as N, where a Zr-rich Zr–N layer was observed in the inner surface of the cladding effectively reducing molten phase formation at the fuel/cladding interface [1,7]. As a result, Zr barriers have been proposed to reduce interdiffusion of fuel and cladding components. In addition to the numerous features that characterize the interaction between metallic fuels and cladding, the segregation behavior can have an important effect on the actual surface structure and composition, as it has the potential of drastically changing the concentration of the fuel elements in the surface region. Therefore, in this work, we present modeling results of the structure and composition of the cU–bZr solid solution and its bulk properties including the dependence of the lattice parameter and coefficient of thermal expansion on composition and temperature. Also included is a detailed discussion of the complex Zr segregation behavior observed by means of atomistic simulations using the Bozzolo– Ferrante–Smith (BFS) method for alloys [8]. Very few experimental results are readily available for any of these properties. Some values of the lattice parameter vs concentration were measured by Huber and Ansari [9] in their study of superconductivity of bcc U–Zr alloys, where a contraction of the lattice parameter for U-rich alloys was found, and dilatometric results for as cast U-rich U–Zr in the work of Basak et al. [10]. From a theoretical standpoint, recent work by Landa et al. [11] provides a thorough investigation of phase equilibria in the U–Zr system based on ab initio DFT calculations of several ground state properties and decomposition curve of the solid solution, with extensive analysis of the concentration
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dependence of the bulk modulus, heat of formation, Debye temperature, Grünesein constant, and volume. Therefore, this work is intended to fill the gap in the current knowledge of the surface characteristics and behavior of the U–Zr system, and set the stage for a better modeling approach of the complex interactions between fuels and cladding.
(a)
2. The BFS method for alloys The BFS method [8] is based on the concept that the energy of formation of a given atomic configuration is the sum of the individP ei . Each contribution by atom i, ual atomic contributions, DH ¼ ei, can be calculated as the sum of two terms: a strain energy, eSi , computed in the actual lattice as if every neighbor of the atom i were of the same atomic species i, and a chemical energy, eCi , computed as if every neighbor of the atom i were in an equilibrium lattice site of a crystal of species i, but retaining its actual chemical identity. The computation of eSi , using Equivalent Crystal Theory (ECT) [12], involves three pure element properties for atoms of species i: cohesive energy (Ec), equilibrium lattice parameter (ae) and bulk modulus (B0). These three parameters for each of the constituent elements are needed in the general derivative structure of the final alloy (bcc in the case of U–Zr alloy of interest). Additional parameters, a (average electron density) and k (screening length), can be easily derived from Ec, ae and B0 [12]. The chemical energy, eCi , accounts for the corresponding change in composition, considered as a defect in an otherwise pure crystal. The chemical ‘defect’ deals with pure and mixed bonds, therefore, additional perturbative parameters, DUZr and DZrU are needed to describe these interactions. A reference chemical energy, eCo i , is also included to insure a complete decoupling of structural and chemical features. Finally, the strain and chemical energies are linked with a coupling function gi, which ensures the correct volume dependence of the BFS chemical energy contribution. Therefore, the contribution of atom i to the energy of formation of the system is given by ei ¼ eSi þ gðeCi eCi 0 Þ. All the necessary parameters were computed using the Linearized Augmented Plane Wave method (LAPW) [13]. For the single elements, the total energy of the pure solids were computed in the bcc symmetry of the alloy, while the parameters D were computed from LAPW calculations of the equilibrium properties (lattice parameter and energy of formation) of the B2 UZr metastable structure. These parameters are universal and transferable to any situation dealing with U and Zr in the bcc symmetry, as is the case in the cU–bZr solid solution [14].
(b)
(c)
3. Results and discussion 3.1. Lattice parameter and coefficient of thermal expansion Simulations were performed on a bcc computational cell with uniform lattice parameter at each stage of the calculation. With this limitation, a solid solution for the whole range of concentration was found to be the equilibrium state above T = 1000 K and up to the melting point (as estimated from the approximate expression for the melting temperature by Guinea et al. [15]), thus ensuring that at least for the range of temperatures considered, the simulations agree with the phase diagram. The lattice parameter of the cU–bZr solid solution was obtained from a variant of Monte Carlo–Metropolis simulations [16] and was found to exhibit a non-uniform behavior, as can be seen in Fig. 1. For low Zr concentration (<20 at.%), the lattice parameter shows a minor contraction with respect to average values accompanied by a dip in the values for the energy of formation, as shown in Fig. 1c. For Zr concentrations above 20 at.%, an increasing expansion that peaks at approximately 70 at.% Zr is observed. These results are somewhat in
Fig. 1. (a) Lattice parameter (in Å) of the U–Zr solid solution for the temperature range in which it exists. (b) Deviation of the computed lattice parameter aBFS (in Å) with respect to Vegard’s law, aVeg, for different temperatures. (c) Energy of formation (in eV/atom) as a function of Zr concentration for T = 1000 and 1200 K.
agreement with experimental measurements of the lattice parameter [9], in that those also reflect a noticeable contraction for con-
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centrations below 40 at.% Zr. It should be noted, however, that no measurements were quoted for alloys with less than 25 at.% Zr. Theoretical results [11], validated by comparison with CALPHAD values for the heat of formation, do not predict such contraction. No such disagreement between experimental, ab initio, and BFS results, is seen for Zr-rich alloys. The apparent difference between ab initio results and those presented here might have its source on the fact that the BFS parameters, based in turn on ab initio calculations, might be sensitive to minor changes in the predicted properties (i.e., cohesive energy, bulk modulus, and lattice parameter of the single species). While this deserves further investigation, the fundamental nature of these properties might be an interesting feature to be thoroughly examined experimentally in the future. Based on these results, the evolution of the lattice parameter with temperature was computed using the finite temperature extension of the BFS method [17], and the same feature was found at every temperature for which the solid solution exists. The coefficient of thermal expansion (CTE) was also computed, and found to have a uniform behavior for the whole range of concentrations, slightly below the average values, as shown in Fig. 2. A simple expression describing a(xZr, T) for the range 1000 K < T < 1400 K is given by
aðxZr ; TÞ ¼ a1000 ðxZr Þð1 þ aðxZr ÞðT 1000ÞÞ
ð1Þ
where
a1000 ðxZr Þ ¼ 3:4728 1 þ 3:542 104 xZr þ 4:607 106 x2Zr 3:743 108 x3Zr
ð2Þ
is the lattice parameter (in Å) at T = 1000 K as a function of Zr concentration (in at.%) and where
aðxZr Þ ¼ 12:35 þ 1:045 102 xZr 4:924 105 x2Zr þ 4:614 107 x3Zr
ð3Þ 6
is the average coefficient of thermal expansion (in 10 aforementioned temperature range.
K
1
) in the
3.2. Surface energy It is known that the surface energy, r, for low Miller indices in bcc crystals, follows the order r110 < r111 < r100 (with other stepped surfaces in between). This was found to be true both for bcc U and Zr and the cU–bZr solid solution. For the pure elements, there is an interesting behavior in the surface energy as a function of temperature and crystallographic orientation, rarely found on other elements. Table 1 shows the values of the surface energy for pure U and Zr bcc surfaces for T = 1000 K and T = 1400 K. In addition, the values at T = 0 K are shown (although neither U nor Zr is present as a bcc phase at that temperature). In both unrelaxed and relaxed cases, some features are apparent: U has a lower surface energy than Zr for any crystallographic orientation, but this trend is reversed in the temperature range for which the solid solution exists, with Zr having a lower surface energy than U at 1000 K (in all three orientations), and even lower at T = 1400 K. In spite of the reversal in the gap between them at any temperature, and whether they favor U or Zr, the surface energies are always very similar. The small differences in surface energy as well as the peculiar variation of the lattice parameter of the cU–bZr solid solution (Fig. 1), together with the behavior of the coefficient of thermal expansion, suggest that the expected Zr segregation behavior to the surface will be highly dependent on crystallographic orientation, temperature, and concentration. Five specific alloys were modeled: U–12Zr (in at.%), U–22.45Zr, U–39.45Zr, U–72.27Zr, and U–85.94Zr (corresponding to 5, 10, 20, 50 and 70 wt.% Zr, respectively). These compositions were chosen because they correspond to different behaviors of the lattice parameter with respect to the average values, as depicted in Fig. 1b: U–12Zr corresponds to the case with maximum contraction relative to average values, U– 22.45Zr corresponds to the crossing point where the lattice parameter achieves exactly the average value, and the other three cases correspond to the increasing, maximum, and decreasing expansion relative to average values. With the surface energies being so close to each other, it is possible that the behavior of a (x, T) could be an important factor in deciding the segregation behavior.
Table 1 Surface energy (in ergs/cm2) for different U and Zr unrelaxed surfaces at T = 0, 1000, and 1400 K. Values for relaxed surfaces are shown in parenthesis. The lattice parameters for U at the different temperatures are a = 3.4350 Å, 3.4736 Å, and 3.4908 Å, respectively, while those for Zr are a = 3.5765 Å, 3.6189 Å, and 3.6383 Å.
Fig. 2. (a) Coefficient of thermal expansion (CTE) for the U–Zr solid solution for different temperatures. (b) Deviation of the CTE with respect to average values.
Face
T=0K
T = 1000 K
T = 1400 K
U(1 0 0) Zr(1 0 0) U(1 1 0) Zr(1 1 0) U(1 1 1) Zr(1 1 1)
1906.30 (466.34) 1924.00 (522.41) 859.70 (821.24) 870.42 (834.65) 1320.50 (1220.60) 1350.40 (1254.40)
2121.40 2094.00 1201.30 1143.90 1502.00 1498.30
2310.00 2245.50 1486.30 1374.60 1627.20 1601.20
(616.22) (560.70) (1129.30) (1081.90) (1388.60) (1393.10)
(757.56) (757.35) (1396.10) (1298.60) (1507.10) (1491.20)
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3.3. Segregation behavior The simulations were limited to the range of temperatures for which the solid solution exists, and the lattice parameter and plane spacing were kept unchanged during the simulation. In all cases, the competition between U and Zr segregation was more noticeable in the lower temperature range (1000–1200 K). At higher temperatures, while the segregation behavior remained unchanged, there was a slow return towards bulk concentrations, as thermal effects washed out any minor difference between U or Zr tendency to segregate. In what follows, Zr(nb) or U(nb) denote the concentration (in at.%) in the n-th layer below the surface (S). Only results for unrelaxed planes are shown. Calculations including relaxation of the near-surface planes were made and no qualitative change was made. The following comments therefore concentrate on unrelaxed surfaces, in order to avoid introducing another degree of freedom in the discussion. (1 0 0) face: For this face, Zr has lower surface energy than U for 1000 < T < 1400 K. However, the difference is very small and tends to disappear as the temperature increases. In U–Zr, as the concentration of Zr increases the lattice parameter expansion (faster than average) tends to favor Zr in the bulk reducing its strain energy (as the lattice parameter approaches the pure bcc Zr value). U, finding itself in an environment of high strain as the Zr concentration increases (as the lattice parameter of cU is further away from the equilibrium lattice value) is then driven to the surface layer. The segregation profiles for increasing Zr concentration are shown in Fig. 3. 12 at.% Zr (1 0 0): Zr segregates to the surface (S) depleting the first layer below (1b), nearly doubling its reference concentration. The pattern remains as the temperature increases, with no repopulation of 1b and rather steady surface Zr concentration (Zr(S)). 22.45 at.% Zr (1 0 0): The temperature-dependent concentration profiles show nearly no difference with the previous case, other than the fact that Zr(S) more than doubles the reference concentration. This increase in Zr segregation is also apparent in the 1b layer, as opposed to the previous case in which all surface Zr came from the layer below. This occurs at the expense of Zr(2b). Segregation of Zr peaks at 1200 K, when some reduction of Zr(S) starts. 39.45 at.% Zr (1 0 0): Zr segregation is dominant, now populating the S and 1b layers. Zr(1b) is more significant than in the lower Zr concentration cases, exceeding the nominal (bulk) value, while Zr(S) still remains at roughly twice the nominal value. Zr depletion now spreads to lower layers, as opposed to just the layer 1b (in the 12 at.% Zr case) or layer 2b (in the 22.45 at.% Zr case). 72.27 at.% Zr (1 0 0): It is at this high Zr bulk concentration that a change in segregation patterns occurs (corresponding to the peak in Fig. 1b). As mentioned above, the large Zr bulk concentration creates a high strain environment for U, favoring its segregation to the surface in spite of the lower surface energy for Zr in this surface orientation. As a result, there is Zr depletion in S for all temperatures and any Zr enrichment is seen only in subsurface layers. 85.94 at.% Zr (1 0 0): Segregation reversal is now complete, and there is strong Zr depletion in the top layer and some in the 1b layer, slightly and evenly enriching subsurface layers (2b and layers below). (110) face: This is the lowest surface energy face for either element, where once again Zr has lower surface energy than U. The gap, however, is wider than that seen in the (1 0 0) case, thus leading to a dominance of Zr segregation over U even at high Zr concentrations where, as noted above in the discussion of the (1 0 0) case, U segregation would be favored. In this case, this enhanced U segregation (driven by strain relief as the lattice parameter of the alloy increases relative to the equilibrium cU value) barely matches Zr segregation, leading to nearly bulk-like profiles at high Zr
Fig. 3. Concentration profiles (in at.%) for (from top to bottom) U100xZrx (x = 12, 22.45, 39.45, 72.27, and 85.94 at.% Zr) alloys in the (1 0 0) orientation. The horizontal line in each panel indicates the reference Zr bulk concentration.
concentration. The segregation profiles for increasing Zr concentration are shown in Fig. 4.
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Fig. 4. Concentration profiles (in at.%) for (from top to bottom) U100xZrx (x = 12, 22.45, 39.45, 72.27, and 85.94 at.% Zr) alloys in the (1 1 0) orientation. The horizontal line in each panel indicates the reference Zr bulk concentration.
12 at.% Zr (1 1 0): Very similar to the (1 0 0) case, Zr enriches the S plane at the expense of Zr(1b). For this face, however, the profiles are more sensitive to increasing temperature.
451
22.45 at.% Zr (1 1 0): Zr(S) enrichment at the expense of Zr(1b) is dramatically increased and remains high even after the reduction brought upon by increasing temperature. 39.45 at.% Zr (1 1 0): As in the (1 0 0) case, Zr segregation is markedly increased, leading to enhanced Zr(1b). Also as in the (1 0 0) case, this is due to an overall depletion of Zr in subsurface layers deep into the bulk. 72.27 at.% Zr (1 1 0): While the (1 0 0) case showed a trend towards segregation reversal at this concentration, this is not the case for the (1 1 0) face, where Zr(S) is still much higher than its nominal value. The profiles, however, show substantial changes as the temperature increases. The subsurface depletion of Zr at T = 1000 K is rapidly replaced by a thick Zr-rich surface region extending to the first few layers at T = 1200 K. This effect is not sustained at higher temperatures, returning to nearly bulk values throughout the cell, except for a still Zr-rich surface layer. 85.94 at.% Zr (1 1 0): Once again, the segregation reversal seen at similar concentrations in the (1 0 0) face is not seen here. There is, however, a slight manifestation of this reversal as the competition between Zr and U is now even enough, leading to nearly bulklike profiles for all temperatures with a slight Zr enrichment of the surface planes. (1 1 1) face: This surface orientation, in terms of the magnitude of the surface energy, r, is an intermediate case between the (1 1 0) (lowest) and (1 0 0) (highest) faces. As shown in Table 1, the surface energy of Zr is slightly lower than that of U in the solid solution temperature range for unrelaxed surfaces, and switching from rZr > rU at T = 1000 K to rU > rZr at T = 1400 K in the relaxed case. However, the differences are truly negligible, so that segregation of either species will be driven by strain effects due to lattice size. The segregation results for alloys with increasing Zr concentration are shown in Fig. 5. 12 at.% Zr (1 1 1): Like in all other faces, there is Zr(S) enhancement at the expense of Zr(1b). The difference from nominal values, however, is very small as any additional Zr segregation would be driven by lattice expansion and not by the contraction in the lattice parameter values that characterize this range of Zr bulk concentration. 22.45 at.% Zr (1 1 1): Following the pattern seen in (1 0 0) and (1 1 0) for this Zr concentration, there is noticeable Zr segregation to the surface (not just S, but also the few planes below the surface), thus defining a rather thick Zr-rich surface region for all temperatures. 39.45 at.% Zr (1 1 1): The thickness of the Zr-rich surface is substantially enhanced with respect to alloys with lower Zr concentration. General Zr depletion in the bulk leads to enhanced Zr(S), Zr(1b), and Zr(2b). Unlike other cases, Zr(S) is smaller than Zr(1b), but still above the nominal level, this being the first indication of the segregation reversal that seems to characterize Zr-rich alloys, favoring U surface segregation. 72.27 at.% Zr: The pattern seen in the 39.45 at.% Zr case is greatly enhanced. However, as was the case in the other faces, U segregation becomes more noticeable leading Zr(S) below nominal values, but the Zr-enriched subsurface is now extensive, encompassing several layers below the surface at the expense of an extensive dip in Zr concentration in layers deeper in the bulk. 85.94 at.% Zr (1 1 1): At this high Zr concentration, the lattice parameter is still well above the average value (see Fig. 1b), so the strain-driven segregation of U continues to dominate, leading to strong Zr(S) depletion and the almost complete disappearance of the subsurface Zr-enrichment. Summarizing, in spite of some quantitative changes for different crystallographic orientations, there is a general trend favoring Zr segregation and the creation of a generally thick Zr-rich surface region for alloys with low Zr concentration, followed by a strain-
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ing temperature, neither thermal effects or surface energies seem to be the dominant factors in determining such behavior. 4. Conclusions In conclusion, the composition of the U–Zr solid solution surface is highly dependent on the particular behavior of the lattice parameter as a function of concentration. The computed values of the surface energy of each element are nearly identical, making finer details of the bulk to be the driving force for the resulting segregation behavior. It was found that at low Zr concentration, a slightly favorable Zr surface energy and lattice contraction lead to enhanced Zr segregation, as the low value of the bulk lattice parameter increases the BFS strain of Zr atoms. However, as the concentration of Zr increases and the lattice parameter expands above average values, where strain relief favors U segregation, leading to a final pattern that shows significant differences for different surface orientations. In addition, changes in temperature can also have an effect due to the changes in the coefficient of thermal expansion for different Zr concentrations as a function of temperature. Polycrystalline surfaces, as they are likely to be found in the actual system, would therefore display a combination of the features described above, with dominant (1 1 0) characteristics. In any case, the peculiar behavior of the bulk lattice parameter as a function of Zr concentration leads to the complex superposition of several effects, driven by the small difference in individual surface energies, change in segregation behavior (from Zr to U) with increasing Zr concentration, and interesting changes in the surface region depending on crystallographic orientation. Finally, this analysis suggests that the Zr enrichment of the surface, a desired and beneficial behavior in nuclear fuels (as it reduces interdiffusion between the Fe-based cladding and fuel), is an effect that is limited to alloys with rather low Zr concentration. Should that not be the case, high temperature U repopulation of the surface could have a deleterious impact on the Zr protective barrier and further interaction of U with cladding. This behavior is then important to be taken into account when considering the inclusion of other alloying elements in the fuel (such as Pu) as they could alter the fine balance that leads to the observed behavior. Acknowledgments Fruitful discussions with N. Bozzolo are gratefully acknowledged. Argonne National Laboratory’s work was supported under US Department of Energy contract DE-AC02-06CH11357. References
Fig. 5. Concentration profiles (in at.%) for (from top to bottom) U100xZrx (x = 12, 22.45, 39.45, 72.27, and 85.94 at.% Zr) alloys in the (1 1 1) orientation. The horizontal line in each panel indicates the reference Zr bulk concentration.
driven segregation reversal where U repopulates the surface layers for Zr-rich alloys. While some differences are observed with chang-
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