Resumen

Durante la irradiación neutrónica de aleaciones de U-Mo se produce un fenómeno de ordenamiento de cavidades cúbico

de caras centradas (fcc), coherente con la estructura cúbica de cuerpo centrado (bcc) de su matriz. En la aleación U-

10wt%Mo, las cavidades tienen un diámetro de unos 30 Å y un parámetro de superred de aproximadamente 120 Å.

Muchos trabajos de la literatura, implícitamente vinculan la sobrepresurización de las cavidades con gases de fisión

(Xe,Kr) como la responsable de la interacción que conduce a este ordenamiento. Sin embargo, observaciones recientes

indican que, en las primeras etapas del quemado del combustible, las cavidades que componen dicha superred se

encuentran prácticamente vacías. En este trabajo, realizamos simulaciones de dinámica molecular tendientes a estudiar la

morfología de una cavidad vacía y caracterizar su interacción mutua con la distancia y el ordenamiento. Se encuentra que

las cavidades son facetadas y sólo pueden interactuar a distancias muy cercanas, de unos pocos planos atómicos.

that, in the early burnup stages of the fuel, the cavities that make up this superlattice are practically empty. In this work,

we perform molecular dynamics simulations aimed at studying the morphology of an empty cavity (void) and

characterizing its mutual interaction with distance and ordering. The cavities are found to be faceted and can only interact

at very close distances, of a few atomic planes. Keywords: molecular dynamics, U-Mo, void superlattices.

1. INTRODUCTION

and Kr at the rate of one gas atom for every four fission events [2]. Inert gas atoms have a strong tendency to precipitate

in small bubbles due to their low solubility.

TABLA 1: Lattice parameter a, energy per atom Ea and elastic constants Cij of the bcc U-10wt%Mo alloy at T = 500 K

given by the homogeneous (AA) and the heterogeneous material (hetero).

elastic interaction, one-dimensional migration of self-interstitial atoms, and dislocation punch out. Several theoretical

works can be found in the literature [2, 5-8] exploring such mechanisms mostly under the assumption that they may be

due to overpressurized bubbles. Nevertheless, Salvato et al. [9] recently found that, at low burnup, i) nanobubble lattice

forms preferentially at grain boundaries and extends to the grain interior with increasing irradiation dose, and ii) bubbles

are extremely underpressurized, suggesting its void-like nature.

studied for the U-10wt%Mo alloy. The molecular dynamics technique, at T = 500 K (estimated working temperature in a

nuclear reactor) and periodic boundary conditions in all directions, is chosen. Embedded atom method (EAM) interatomic

potentials derived from the work of Smirnova et al. [10] for U-Mo system are employed. All simulations are performed

system, e.g. stabilizing the γ phase (bcc) only at relatively high temperatures (T > 400 K). Table 1 reports some of the

structural and elastic properties presently calculated for the studied alloy using each potential. In what follows simulations

performed with the AA potential are referred to as homogeneous (material), while those using the original one on purely

random mixtures, as heterogeneous.

2.2. Void faceting

Å (1984 vacancies) is carved in a cubic heterogeneous cell of around 144 Å edge 22at% Mo (149216 atoms, equivalent

to 10wt% Mo). The cell is run for 3000 ps at constant NPT at a high temperature (T = 800 K) in order to accelerate atomic

movements, in particular, at the surface of the void. The resultant surface diffusion is not homogeneous and depends on

position on the surface.

characterize the defect. This time, simulations are performed at constant NVT with T = 500 K and lattice parameters taken

from Table 1. In the case of the heterogeneous system, 100 short runs of 25 ps with different random configurations are

averaged out to obtain a representative energy. The first 10 ps in each run are discarded to avoid any contribution from

transient states. The formation energy E f of a vacancy cluster is calculated as:

FIG. 1: Void morphology in U-10%wtMo. a) (100) projection of the bcc lattice with a void of r = 20 Å showing atomic

displacements (black arrows) after 3000 ps at T = 800 K. Colored circles are U/Mo atoms. Traces of two {110} planes

are also shown. Ideal Wulff shapes of the void: b) {100} truncated and c) regular rhombic dodecahedrons. Miller indices

of the planes are indicated.

E f =EN+def −NEa (1)

where EN+def is the energy of the cell with N atoms and the defect, and Ea the corresponding energy per atom appearing in

species differ by more than 12% and the enthalpy of mixing is about 4 kJ/mol at 22at%Mo [17]. As a result, the U-Mo

mixture has important structural distortions, which are absent in a homogeneous medium such as the AA. It is also worth

noting that the AA is strictly a method for calculating the average energy at 0 K. At finite temperatures the (average)

thermodynamic energy of the heterogeneous medium will not necessarily match the AA result [14]. However, we believe

that presently the temperature effect is less important than that of size. We must keep in mind these limitations when using

the AA method for the UMo alloy.

energies calculated for the most compact planes for the homogeneous and the heterogeneous system are reported in Table

for the {110} compact bcc planes. On the other hand, differences between surface energies for homogeneous and

smallvacancy clusters remaining from a collision cascade adopt polyhedral shapes with {110} facets. Other authors

indirectly suggest that faceting is related to the gas pressure exerted on the bubble surface [2, 8, 20, 21]. Simulations

performed by Shenyang Hu et al [2] in U-10wt%Mo with the same interatomic potential show that bubbles become

faceted, but in {111} planes, when they emit dislocation loops by Xe overpressurization. On the other hand, Beeler et al.

made on facet orientation. In transmission electron microscopy, faceting may be difficult to observe because out-of-focus

conditions are required to measure bubble diameters and distances. On He+ ion implanted Fe-Cr alloys, Snoeck et al. [20]

at distance d between their centroids in a large cell of 240 Å × 125 Å × 120 Å (185858 atoms) of homogeneous material.

The bcc lattice was oriented such that one of the {110} facets of each void are directly facing each other along a <110>

crystallographic direction. Distances were varied from almost contact up to 120 Å. As it is assumed that the bcc lattice do

not change its bulk equilibrium volume by addition of these defects, simulations were performed at constant NVT. Fig. 2

shows the E f dependence on the distance d between voids, calculated by means of eq. 1. It can be seen a rather strong

repulsion at very short distances and a rapid decay to a constant value corresponding to twice the formation energy for

single void. In particular, the curve hardly shows any minimum that might set the superlattice parameter of experimental

findings.

2.4. Void superlattices

FIG. 4: Formation energies E f of voids (r = 15 Å) as a function of the first neighbor distance d1nn arranged in fcc, bcc