Simulación atomística de la producción y evolución de ...
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Simulación atomística de la producción y evolución de defectos en aleaciones basadas en Fe debido a la irradiación María José Aliaga Gosálvez
Transcript of Simulación atomística de la producción y evolución de ...
Simulación atomística de la producción y evolución de defectos en aleaciones basadas en Fe debido a la irradiación
María José Aliaga Gosálvez
Departamento de Física Aplicada – Instituto Universitario de Materiales Facultad de Ciencias
Simulación atomística de la producción y evolución de defectos en aleaciones basadas en Fe debido a
la irradiación
María José Aliaga Gosálvez
Tesis presentada para aspirar al grado de
DOCTORA POR LA UNIVERSIDAD DE ALICANTE
MENCIÓN DE DOCTORA INTERNACIONAL
DOCTORADO EN CIENCIA DE MATERIALES
Dirigida por la doctora: María José Caturla Terol
Financiada por el programa de Ayudas para becas y contratos destinados a la formación
de doctores del Vicerrectorado de Investigación, Desarrollo e Innovación (Resolución 22
de diciembre de 2011)
A toda mi familia
A todos mis amigos/as
A mi directora de tesis María José
A Ben
y a Max
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Resumen
Esta tesis doctoral se centra en el estudio a nivel atómico de la producción de
defectos en aleaciones basadas en Fe, particularmente aleaciones de Fe-Cr,
debido a la interacción con partículas energéticas. Uno de los grandes retos a
los que se enfrenta la energía de fusión es el desarrollo de materiales
resistentes a los altos niveles de radiación a los que se verán sometidos [1]. La
falta de fuentes de radiación que reproduzcan las condiciones que existirán en
los reactores de fusión hace necesario el desarrollo de modelos que permitan
extrapolar los resultados experimentales obtenidos con fuentes de radiación
como iones o neutrones de fisión, a las condiciones de fusión. Tales modelos
sólo son posibles si se conoce el comportamiento del material bajo irradiación a
nivel fundamental, desde la formación de defectos a nivel atómico y en tiempos
de picosegundos, hasta la evolución de estos defectos a tiempos de horas o
incluso años. Este tipo de simulaciones requiere de la unión de varios métodos
de cálculo distintos en lo que se denomina modelización multiescala [2].
Además, el desarrollo de estos modelos requiere de una validación de los
resultados obtenidos a través de comparaciones con experimentos. La
radiación de los materiales con iones y su caracterización mediante
microscopía electrónica de transmisión (TEM por sus siglas en inglés,
Transmission Electron Microscopy), son las técnicas de referencia para validar
los modelos. Es importante tener en cuenta las diferencias entre irradiación con
iones e irradiación con neutrones, especialmente cuando la irradiación con
iones se lleva a cabo mientras el material está siendo observado con TEM
(TEM in-situ), ya que requiere que la muestra sea una lámina muy fina, en
comparación con la irradiación con neutrones que sucede en el interior del
material (bulk).
Uno de los materiales de mayor interés para la fusión son las aleaciones de
Fe, consideradas como el principal candidato para los materiales estructurales
de estos reactores [3]. Debemos así pues comprender los cambios en las
propiedades mecánicas de estos materiales debido a la producción de
defectos.
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En esta tesis doctoral se han utilizado tres técnicas de simulación para estudiar
los procesos de formación y de evolución de defectos en hierro y en Fe-Cr,
centrándonos en estudiar las diferencias entre irradiación en capas finas y en
muestras en volumen . En primer lugar se ha utilizado la técnica de Dinámica
Molecular con potenciales empíricos [4] para estudiar cómo se produce el daño
en los primeros picosegundos después de la irradiación. Con esta técnica
hemos calculado el número y tipo de defectos que se producen al irradiar con
iones de Fe láminas finas de este mismo material para TEM in-situ, así como
muestras de bulk. La base de datos obtenida de cascadas de desplazamiento
se ha utilizado a continuación como entrada al método de simulación de Monte
Carlo Cinético para analizar la evolución del daño primario por la interacción y
difusión de los defectos. Finalmente hemos utilizado algunas de las cascadas
para simular imágenes de microscopía electrónica de transmisión (TEM). Los
resultados obtenidos con cada técnica se han comparado con resultados
experimentales de otros grupos de investigación del mismo proyecto y con
experimentos diseñados y realizados por este mismo grupo.
El informe que sigue a continuación consiste en un resumen global del trabajo
realizado, los artículos publicados y los que se encuentran en vía de
publicación. Su estructura es la siguiente. El capítulo 1 es una introducción al
daño por radiación en materiales para fusión y su modelización. El capítulo 2
resume la metodología de la modelización multiescala utilizada en esta tesis.
En el capítulo 3 se discuten los resultados más relevantes de los trabajos
publicados y no publicados que se encuentran en los capítulos 4 y 5. El
capítulo 6 concluye el informe con las conclusiones globales de la tesis.
Los trabajos presentados en los capítulos 4 y 5 son:
M.J. Aliaga, A. Prokhodtseva, R. Schaeublin, M.J. Caturla (2014),
Molecular dynamics simulations of irradiation of α-Fe thin films with
energetic Fe ions under channeling conditions, Journal of Nuclear
Materials, 452, 453.
M.J. Aliaga, M.J. Caturla, R. Schaeublin (2015), Surface damage in TEM
thick α-Fe samples by implantation with 150 keV Fe ions, Nuclear
Instruments and Methods in Physics Research B, 352, 217.
iii
M.J. Aliaga, R. Schaeublin, J.F. Löffler, M.J. Caturla (2015), Surface-
induced vacancy loops and damage dispersión in irradiated Fe thin
films, Acta Materialia, 101, 22.
M.J. Aliaga, I. Dopico, I. Martin-Bragado, M.J. Caturla (2016), Influence
of free surfaces on microstructure evolution of radiation damage in Fe
from molecular dynamics and object kinetic Monte Carlo calculations,
Physica Status Solidi A, enviado.
M.J. Aliaga, I. Dopico, I. Martin-Bragado, M. Hernández-Mayoral, L.
Malerba, M.J. Caturla, Insights on loop nucleation and growth in α-Fe
thin films under ion implantation fron atomistic models.
A.E. Sand, M.J. Aliaga, M.J. Caturla, K. Nordlund, Surface effects and
statistical laws of defects in primary radiation damage: tungsten vs. Iron.
S. García-González, A. Rivera, M.J. Aliaga, M.J. Caturla, I. Martín-
Bragado, OKMC study of differences between MD and BCA cascades in
neutron irradiated Fe simulations.
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v
Abstract
This thesis is focused on the study, at the atomic level, of the production of
defects in Fe based alloys, particularly FeCr alloys, due to the interaction with
energetic particles. One of the biggest challenges of fusion energy is the
development of resistant materials to the high level of radiation that they will
have to withstand in the nuclear reactor [1]. The lack of radiation sources able
to reproduce the exact conditions in a future fusion plant makes it necessary the
development of models that will permit the extrapolation of experimental results
of radiation with ions or fission neutrons, to fusion conditions. These models are
only possible if we know the behaviour of the material under irradiation at a
fundamental level, from the formation of defects at the atomic level and
picoseconds time scale, to the evolution of these defects at a level of hours or
even years. These kinds of simulations require the union of several methods in
what is known as multiscale modelling [2]. Importantly, these models need to be
validated with experiments. The reference technique to validate the models and
characterize the damage produced by irradiation is the Transmission Electron
Microscopy (TEM). It is important to have into account the differences between
ion and neutron irradiation, particularly when the ion irradiation takes place
while the material is being observed under the TEM, what is called in-situ TEM,
because the sample needs to be a thin film, in comparison with neutron
irradiation, that takes place in the bulk.
Some of the materials of major interest to fusion are Fe base alloys,
considered as the main candidates for structural materials of the reactor [3]. In
this way, it is essencial to know the changes in the mechanical properties of
these materials due to production of defects.
In this thesis three simulation techniques have been used to study the formation
and evolution of defects in Fe and FeCr. The focus has been in studying the
differences between irradiation in thin films and irradiation in bulk. First, we
have used molecular dynamics with empirical potentials [4] to study how
damage is produced during the first picoseconds after irradiation. With this
technique we have calculated the number and type of defects produced while
irradiating Fe thin films with ions for in-situ TEM, as well as bulk samples. The
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obtained database has then been used as input for the Monte Carlo technique
to analyse the evolution of the primary damage by interaction and diffusion of
defects. Finally, we have used some of the cascades to simulate TEM images.
Results obtained with this technique have been compared to experimental
results of other groups in the same project and with experiments designed and
performed by this group.
This report consists of a global summary of the objectives, methods and main
results of the thesis. It is structured as follows. Chapter 1 is an introduction to
radiation damage in materials for fusion and its modelling. Chapter 2 gives a
description of the multiscale methodology used in this thesis. Chapter 3
discusses the most relevant results of the papers, which are in chapters 4 and
5. Finally, chapter 6 closes the report with the conclusions.
The papers in chapters 4 and 5 are:
M.J. Aliaga, A. Prokhodtseva, R. Schaeublin, M.J. Caturla (2014),
Molecular dynamics simulations of irradiation of α-Fe thin films with
energetic Fe ions under channeling conditions, Journal of Nuclear
Materials, 452, 453.
M.J. Aliaga, M.J. Caturla, R. Schaeublin (2015), Surface damage in TEM
thick α-Fe samples by implantation with 150 keV Fe ions, Nuclear
Instruments and Methods in Physics Research B, 352, 217.
M.J. Aliaga, R. Schaeublin, J.F. Löffler, M.J. Caturla (2015), Surface-
induced vacancy loops and damage dispersión in irradiated Fe thin
films, Acta Materialia, 101, 22.
M.J. Aliaga, I. Dopico, I. Martin-Bragado, M.J. Caturla (2016), Influence
of free surfaces on microstructure evolution of radiation damage in Fe
from molecular dynamics and object kinetic Monte Carlo calculations,
Physica Status Solidi A, enviado.
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M.J. Aliaga, I. Dopico, I. Martin-Bragado, M. Hernández-Mayoral, L.
Malerba, M.J. Caturla, Insights on loop nucleation and growth in α-Fe
thin films under ion implantation fron atomistic models.
A.E. Sand, M.J. Aliaga, M.J. Caturla, K. Nordlund, Surface effects and
statistical laws of defects in primary radiation damage: tungsten vs. Iron.
S. García-González, A. Rivera, M.J. Aliaga, M.J. Caturla, I. Martín-
Bragado, OKMC study of differences between MD and BCA cascades in
neutron irradiated Fe simulations.
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ÍNDICE
RESUMEN i
ABSTRACT v
1 INTRODUCCIÓN 1
1.1. Daño por radiación en materiales para fusión 4
1.2. Modelización del daño por radiación: modelos multiescala 6
1.3. Propósito de este trabajo 9
2 METODOLOGÍA 11
2.1 Dinámica Molecular 11
2.1.1. Potenciales de interacción 12
2.1.2. El modelo del átomo embebido 13
2.1.3. Métodos de integración 15
2.1.4. Condiciones de contorno 17
2.1.5. El código MDCASK 20
2.1.6. Simulación de irradiación de Fe en Fe: antecedentes 21
2.2 Monte Carlo Cinético 22
2.2.1. La teoría del estado de transición 25
2.2.2. El código MMonCa 26
2.3. Simulación de imágenes TEM 28
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2.3.1. El método multicapa 28
2.3.2. El código EMS 30
3 RESULTADOS Y DISCUSIÓN 32
3.1. Daño en escala de picosegundos: iones frente a neutrones 32
3.2. Evolución del daño primario en Fe y en FeCr 35
3.3. Comparación con imágenes TEM 36
4 TRABAJOS PUBLICADOS O ACEPTADOS 41
I. Molecular dynamics simulations of irradiation of α-Fe thin films with
energetic Fe ions under channeling conditions
43
II. Surface damage in TEM thick α-Fe samples by implantation with
150 keV Fe ions
49
III. Surface-induced vacancy loops and damage dispersión in
irradiated Fe thin films
55
IV. Influence of free surfaces on microstructure evolution of radiation
damage in Fe from molecular dynamics and object kinetic Monte
Carlo calculations
67
5 TRABAJOS NO PUBLICADOS 75
V. Surface effects and statistical laws of defects in primary radiation
damage: tungsten vs. iron
77
VI. OKMC study of differences between MD and BCA cascades in
neutron irradiated Fe simulations
87
VII. Insights on loop nucleation and growth in α-Fe thin films under ion 95
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implantation fron atomistic models
CONCLUSIONES 107
CONCLUSIONS 109
BIBLIOGRAFÍA 111
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1
1. INTRODUCCIÓN
El cambio climático es un tema que preocupa a la mayoría de la población y a
sus gobiernos, y se están tomando muchas medidas desde hace años para
intentar frenarlo o atenuar al menos sus efectos. El principal objetivo de estas
medidas es la sustitución del petróleo, principal causante del gas de efecto
invernadero CO2, por energías alternativas que no contribuyan al cambio
climático. Una de estas energías es la energía de fusión.
Llegar a utilizar comercialmente la energía de fusión para producir electricidad
es un gran reto, ya que los requerimientos para que se produzcan reacciones
de fusión son muy exigentes, y a diferencia de las reacciones de fisión, si estas
condiciones no se cumplen el proceso simplemente se detiene, haciendo
inexistente el riesgo de las reacciones en cadena. Estos requerimientos
específicos son parte de la razón por la cual no existe todavía a día de hoy un
reactor de fusión que produzca más energía de la que consume.
Para conseguir la fusión se debe forzar a los núcleos de los átomos de deuterio
(D) y tritio (T) a unirse formando helio (He) y neutrones con una energía de 14
MeV. Esto se consigue calentando el combustible a unos 100 millones de
grados para que alcance el estado de plasma. El confinamiento de este plasma
es el parámetro más crítico y el más desafiante. Hoy día existen dos métodos
para conseguir el confinamiento, uno de ellos es mediante la utilización de
láseres de alta potencia, conocido como fusión por confinamiento inercial,
como NIF en los EEUU [25] o el LaserMegaJoule en Francia [26]. Un segundo
método consiste en la utilización de campos magnéticos, conocido como fusión
por confinamiento magnético. Este es el sistema en el que se enmarca el
proyecto de investigación de esta tesis.
El dispositivo utilizado en el confinamiento magnético es el llamado tokamak
(que en ruso significa “cámara toroidal con espirales magnéticas”). En un
reactor tokamak, el plasma se mantiene confinado mediante líneas de campo
magnético en una cámara de vacío con forma de toroide [6].
2
Los neutrones altamente energéticos que se forman en la reacción de fusión
del D+T serán absorbidos en agua y el vapor de agua generado moverá una
turbina como sucede en los reactores de fisión convencionales. Pero partículas
del plasma escaparán inevitablemente al confinamiento y golpearán las
paredes del reactor. Los materiales que conforman la primera pared del reactor
en contacto directo con el plasma sufrirán erosión, pero además, los altamente
energéticos neutrones pueden penetrar en los materiales estructurales del
reactor.
Las propiedades mecánicas de estos materiales se alteran debido a los
defectos inducidos por la radiación, siendo la fragilización y el aumento de
volumen las principales consecuencias [9]. Las condiciones de fusión son
mucho más agresivas para los materiales del reactor que las condiciones de
fisión, por lo que se hace necesaria la utilización de nuevos materiales que
alargue el tiempo de vida del reactor. Con esta finalidad, diferentes escenarios
y materiales están siendo testeados tanto en reactores existentes como en
instalaciones experimentales. Por ejemplo, diferentes tipos de aceros
ferríticos/martensíticos especiales de baja activación se están desarrollando
por considerarse los mejores candidatos para ser utilizados como materiales
estructurales. Sin embargo, una evaluación experimental completa es difícil de
conseguir, debido por un lado a la gran variedad de materiales, y por otro a la
imposibilidad de reproducir las condiciones exactas del plasma en el reactor de
la próxima generación ITER [6]. A día de hoy no existe ningún dispositivo
experimental capaz de producir neutrones de 14 MeV como en un reactor de
fusión. Por esta razón, las técnicas de simulación por ordenador, tales como las
empleadas en este trabajo, son una herramienta vital para proporcionar
información y comprensión fundamentales del comportamiento de un material
específico para un futuro reactor de fusión.
En este sentido comenzó en el año 2002, dentro del programa europeo de
fusión conocido como EFDA (European Fusion Development Agreement),
transformado hoy en EUROfusion, una tarea de modelización de materiales
para fusión. Esta tarea, dentro de la cual se enmarca esta tesis, incluye la
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validación de los modelos desarrollados a través de distintos métodos
experimentales, en particular, la irradiación con iones.
Figura 1: Una ilustración del reactor de fusión de próxima generación ITER [6]
resaltando sus principales componentes. El volumen del plasma es de 850 m3 y
el radio principal del toroide es de 6,2 m. El objetivo de ITER es conseguir un
factor de ganancia energética de Q = 10 y producir 500 MW para un pulso de
unos 100 s. Está siendo construido en Cadarache, Francia. Cortesía de ITER.
1.1. Daño por radiación en materiales para fusión
En los reactores de fusión tanto inercial como magnética, la radiación procede
de las partículas emergentes de la reacción de fusión: neutrones, iones y rayos
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X, siendo la contribución neutrónica la más importante. De los neutrones
emitidos, sólo una pequeña fracción será absorbida por la primera pared, lo
que significa que la mayor parte de los 14MeV de energía serán transferidos a
los materiales estructurales. Aquí producirán defectos debidos a la radiación
que pueden alterar profundamente las propiedades de los materiales, tanto por
procesos inelásticos como por colisiones elásticas [7]. Los procesos inelásticos
incluyen interacciones con los electrones o transmutaciones de (n,α) o (n,p), en
los cuales se forman He o H (o renio en W). La cantidad de He e H que se
produce de este modo en un reactor de fusión se espera que sea alrededor de
un orden de magnitud mayor que en los reactores de fisión [8], provocando
problemas de hinchamiento debido a la formación de burbujas, así como
fragilización por la segregación de He en los bordes de grano. Por ejemplo, la
producción de He en Be cuando ITER sea sometido a un flujo de neutrones de
1 MWa/m2 es de 3500 partes atómicas por millón (appm) y en los aceros de
150 appm [9].
Otro gran problema son las colisiones elásticas entre los neutrones y los
núcleos atómicos, que resultan en cascadas de desplazamiento. El daño
resultante en la red cristalina consiste en defectos puntuales, clusters de
defectos, loops de dislocaciones, precipitados y/o agujeros (agregados de
vacantes). El daño se expresa a menudo en desplazamientos por átomo (dpa),
que describe con qué frecuencia cada átomo en el material es en promedio
desplazado de su posición en la red. Debido a la recombinación de defectos, el
número final de desplazamientos es a menudo sólo una pequeña fracción del
valor inicial de dpa.
Como ejemplo de defectos inducidos por la radiación, se muestran en la figura
2 las imágenes de microscopía electrónica de transmisión (TEM) del daño
producido en aleaciones de Fe con Cr por bombardeo de iones de Fe3+.
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Antes de irradiar
Después de irradiar
Figura 2: Imágenes de TEM de una aleación de Fe-5Cr antes de ser irradiada y
después de la irradiación con iones de Fe3+ con una dosis de 0,45 dpa.
Cortesía de Anya Prokhodtseva, École Polytechnique Fédérale de Lausanne,
Centre de Reserches en Physique des Plasmas, 2011.
El aumento de defectos debido a la radiación muestra claramente cómo la
irradiación con partículas tiene un gran impacto en la microestructura del
material.
Puesto que las propiedades físicas y mecánicas de un material están
gobernadas por su microestructura, éstas pueden sufrir grandes cambios
debido al daño en la red. Por ejemplo, la agrupación de defectos formando
nanoestructuras de mayor tamaño, así como los precipitados, dificultan el
movimiento de las dislocaciones, provocando fragilización inducida por la
radiación [10]. Esto incluye endurecimiento, pérdida de ductilidad, resistencia a
Fe-5Cr 0.45 dpa 790 appm He Fe-5Cr 0.45 dpa
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la fractura y creep por irradiación. Un cambio en la temperatura de transición
dúctil-frágil hacia temperaturas más altas es también común.
El conocimiento a nivel atómico de los defectos de la red cristalina es así pues
de vital importancia para poder predecir la respuesta a la irradiación de los
materiales estructurales.
1.2. Modelización del daño por radiación: modelos multiescala
Los fenómenos que ocurren en los materiales debido a la radiación tienen lugar
en diferentes escalas de tiempo y de espacio, son fenómenos jerárquicos y
típicos multiescala, desde la producción de defectos a escala atómica (unos 10-
15 s, 10 -10 m) hasta la evolución de la microestructura durante los años de
operación (unos 109 s, 10-3 m). Esto hace que la predicción a largo término de
los cambios inducidos por la radiación en las propiedades mecánicas de los
materiales se convierta en un desafío importante. Una aproximación consiste
en encadenar modelos existentes en las diferentes escalas de manera que los
resultados de un modelo sirven como datos de entrada para el siguiente
modelo de orden superior.
Seguidamente se analizan brevemente los procesos que tienen lugar a las
diferentes escalas arriba mencionadas y el modelo computacional utilizado
para su estudio, mostrados también en la figura 3. Para una información más
detallada de los procesos de daño por irradiación se pueden consultar las
siguientes referencias [11-16].
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Figura 3: Diagrama de los fenómenos relevantes en daño por irradiación y los
métodos computacionales típicamente empleados [17]
1. Átomo primario de retroceso o PKA (Primary knock-on atom): El proceso
de daño comienza con la creación de un átomo de alta energía desplazado de
su posición original por la partícula incidente. Las secciones eficaces y los
modelos cinemáticos necesarios para el cómputo del espectro de PKAs están
implementados en códigos como SPECTER. SPECTER permite calcular el
espectro energético de PKAs producido por un neutrón de una determinada
energía.
2. Estructura de defectos: La estructura electrónica de defectos puntuales
se estudia con modelos de mecánica cuántica, denotados como primeros
principios, los cuales resuelven por aproximaciones la ecuación de Schrödinger
para un conjunto de átomos. Este método es computacionalmente muy costoso
por lo que el número de átomos que es posible estudiar es reducido. Códigos
como VASP [18] o SIESTA [19] son los más utilizados en el campo del daño
por radiación. Estos métodos de simulación se utilizan fundamentalmente para
obtener información sobre las energías de formación y de migración de
defectos puntuales y pequeñas agrupaciones de defectos.
3. Producción de defectos primarios en cascadas de desplazamiento: La
alta energía del PKA es rápidamente transferida a otros átomos mediante una
cadena de colisiones atómicas generando una cascada de vacantes y átomos
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intersticiales. Este proceso se simula mediante códigos de dinámica molecular
(MD) que integran las ecuaciones del movimiento de millones de átomos. El
tiempo máximo de simulación que puede alcanzarse, sin embargo, es sólo de
nanosegundos.
4. Evolución de cascadas y migración de defectos de largo rango: Estudia
como los defectos migran e interactúan entre ellos modificando así la
microestructura. En general los átomos intersticiales son muy móviles migrando
a gran velocidad desde la zona de daño, mientras que las vacantes son más
lentas, agrupándose en la mayoría de los casos y permaneciendo cerca del
lugar en el que la cascada fue creada. Esta difusión se expande a lo largo de la
vida del material irradiado. En general, los coeficientes de difusión de las
especies que difunden son hallados mediante técnicas ab initio o con MD, y
utilizados por modelos de cinética de Monte Carlo (KMC) como el object kinetic
Monte Carlo (OKMC) para estudiar la evolución y acumulación de dichos
defectos. Del mismo modo es posible utilizar modelos cinéticos de ecuaciones
de tasas para estudiar la evolución de la microestructura y la difusión de
defectos. Grandes conjuntos de ecuaciones diferenciales se integran para
predecir la evolución de los defectos. Este método es mucho más eficiente que
los métodos Monte Carlo pero pierden la resolución atómica de estos.
5. Incremento en el stress de fluencia inducido por irradiación: las
heterogeneidades de la microestructura provocan un aumento substancial en el
estrés de fluencia (Δσy ) actuando como obstáculos dispersos para el
movimiento de las dislocaciones. Dicho estudio de la interacción entre una
dislocación y un obstáculo se han realizado tradicionalmente con modelos de
dinámica molecular [20].
6. Dinámica de Dislocaciones (DD): Estudia el movimiento de las
dislocaciones y su interacción entre ellas y con la posible introducción de
obstáculos como precipitados que modifican el mapa de estreses e impiden el
movimiento de dichas dislocaciones libremente. En este modelo las
dislocaciones son discretizadas resolviendo la formulación elástica y
avanzando la estructura a través de una función de movilidad que relaciona la
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fuerza en un segmento de dislocación con su velocidad. Normalmente, esta
función de movilidad se obtiene mediante simulaciones con MD.
7. Curvas de cambio de la resistencia a la fractura con la temperatura:
modificaciones en las propiedades constitutivas resultan en una degradación
de la resistencia a la fractura del material que se manifiesta como un cambio
(ΔT) en la temperatura de transición dúctil-frágil.
Los modelos teóricos utilizados necesitan de la validación experimental de
cada uno de ellos, en las diferentes fases del proceso de daño que se
describen, para garantizar que ofrecen resultados fiables. Los datos
experimentales para validar los procesos a nivel microscópico proceden en
gran medida de la irradiación de los materiales con iones, ya que es mucho
menos costosa y más versátil que la irradiación con neutrones, y la posterior
caracterización microestructural del daño con diferentes técnicas, siendo la
Microscopía Electrónica de Transmisión (TEM) una de las más utilizadas.
1.3. Propósito de este trabajo
En el marco de la modelización multiescala del daño por irradiación descrita en
el apartado anterior un punto importante que ha sido resaltado por varios
autores [21] es que la evolución del daño a largo plazo es especialmente
sensible a la distribución inicial de defectos obtenida en la cascada de
desplazamiento. Esta importancia de la distribución del daño primario ya fue
resaltada en los años 80 en los primeros cálculos de dinámica molecular de
cascadas de desplazamiento [22, 23].
El objetivo principal de este estudio ha sido obtener, utilizando dinámica
molecular, Monte Carlo cinético y simulación de imágenes TEM, una
descripción detallada del daño primario producido al irradiar películas finas de
hierro puro con iones de hierro de entre 50 y 500 kev de energía. Las películas
finas tienen un espesor de entre 40 y 80 nm para reproducir el espesor de las
muestras utilizadas en las medidas de TEM durante los experimentos de
irradiación que se están llevando a cabo en Jannus (Francia) y los ya
10
publicados de Mercedes Hernández Mayoral y colaboradores [24]. En las
simulaciones se ha estudiado principalmente el efecto de las superficies en la
producción y evolución de daño y su comparación con la radiación en volumen
producida por neutrones. El estudio de las diferencias entre el daño que
produce un neutrón y el daño que produce un ión a nivel fundamental es clave
para la construcción de modelos predictivos de más alto nivel en el marco de la
simulación multiescala.
11
2. METODOLOGÍA
Para la realización de este trabajo se han utilizado tres métodos de simulación:
dinámica molecular con potenciales empíricos, Monte Carlo cinético y cálculo
de imágenes de TEM. En los siguientes apartados se describen brevemente
cada una de estas técnicas.
2.1. DINÁMICA MOLECULAR
La técnica de dinámica molecular consiste en el estudio de la evolución en el
tiempo de un sistema de N-cuerpos. Está basado en una interpretación
determinista de la naturaleza donde el comportamiento de un sistema se puede
realizar de forma computacional si conoces las condiciones iniciales y las
fuerzas de interacción.
La metodología de una simulación de dinámica molecular consiste en la
integración de las ecuaciones de movimiento para todos los átomos en una
celda computacional. Las trayectorias de las partículas se obtienen de la
integración de la ecuación de Newton:
(1)
Para un potencial conservativo la fuerza es una función de las coordenadas y
se puede obtener a partir del gradiente del potencial:
(2)
La integración numérica de estas ecuaciones nos proporciona la trayectoria de
las partículas, siendo el tiempo el paso de integración. El paso de tiempo se
elige para minimizar los errores y maximizar la eficiencia computacional.
12
Normalmente es del orden de 0.5-1.0 femtosegundo (fs), ya que debe ser
menor a la frecuencia de vibración de los átomos del material.
Aunque en principio es un método muy simple de implementar, sus
fundamentos residen en:
Mecánica clásica,
Dinámica clásica no lineal,
Teoría cinética,
Mecánica estadística,
Principios de conservación y
Física del estado sólido.
Las partes fundamentales de un código de dinámica molecular son el potencial
interatómico, el algoritmo de integración y las condiciones de contorno.
2.1.1. Potenciales de interacción
En la simulación de un material real por dinámica molecular el potencial
interatómico es la parte fundamental. Es en él donde está incluida toda la
información sobre la interacción entre las partículas y la precisión con la que
reproduzca las propiedades del material estudiado determina en su mayor
parte la fiabilidad de la simulación. El principal reto en la construcción de
potenciales es hacerlos suficientemente simples para permitir una velocidad de
cálculo aceptable, y a su vez capaces de reproducir fielmente las propiedades
del material. Los potenciales más extendidos en la actualidad son de carácter
semiempírico.
El desarrollo empieza buscando una forma del potencial adecuada que
depende de una cierta cantidad de parámetros. A continuación estos
parámetros se ajustan a una base de datos relativamente extensa. La base de
datos puede contener propiedades calculadas experimental o teóricamente. El
cálculo teórico de propiedades se lleva a cabo por primeros principios.
Dependiendo del fenómeno que se quiera simular, se valora en mayor o menor
13
medida la calidad del ajuste a cada propiedad. Es importante destacar que, el
hecho de reproducir fielmente una gran cantidad de propiedades no garantiza
que el valor calculado de una propiedad nueva sea el correcto.
2.1.2. El modelo del átomo embebido
En el caso de los metales, el modelo del átomo embebido (EAM de embedded
atom method) es el potencial de uso más extendido [27]. Se basa en la idea de
que la energía de un átomo en un sólido cristalino es igual a la energía
necesaria para embeberlo en la nube electrónica de la red. En otras palabras,
la energía de cohesión de un átomo en la red es un funcional de la densidad
electrónica en ese punto de la red. Si ρi es la densidad electrónica en torno al
átomo i, su valor se obtiene por la superposición de las densidades electrónicas
de los N átomos del sistema en la posición del átomo i.
(3)
Existen varias parametrizaciones del EAM. Originalmente, la forma funcional de
las densidades electrónicas ρa se tomaba de cálculos previos de las funciones
de onda de Hartree-Fock:
(4)
Donde ns y nd son el número de electrones s y d externos, y ρs y ρd son las
densidades asociadas con las funciones de onda de los orbitales s y d.
Conocida la densidad electrónica en la posición de cada átomo, la expresión de
la energía total de un sistema de N átomos es la suma de las energías de
embebimiento de todos los átomos, más la suma de la suma de las repulsiones
nucleares (que se representa mediante un potencial de pares)
(5)
14
En su forma más general, suponiendo la repulsión electrostática entre dos
núcleos, uno de un átomo tipo A y otro de un átomo tipo B, φ toma la forma
(6)
Donde Z es la carga efectiva, una función de la distancia R. La forma funcional
de la energía de embebimiento F no se puede expresar de forma analítica.
Para determinarla, se usa la ecuación de estado universal, descrita por Rose,
en la que la energía de cohesión de un átomo puede expresarse en función de
la energía de sublimación Esub y la constante de red a
(7)
Donde
(8)
Siendo B el bulk modulus, Ω el volumen atómico y Esub la energía de
sublimación, todos ellos valores de equilibrio. Entonces, basándose en la
fórmula (7), la función de embebimiento se puede tabular variando el parámetro
de red.
Otra variante de los potenciales EAM es el potencial interatómico de Finnis-
Sinclair [28]. Esta familia de potenciales parte del modelo EAM y lo deriva
utilizando una aproximación de segundo orden de la teoría de tight-binding de
sólidos [31]. Desde el punto de vista de cálculo, es más simple todavía, puesto
que la función F es una raíz cuadrada.
15
Durante los últimos años, el EAM ha demostrado ser un potencial con
excelentes prestaciones en el caso de los metales fcc [29].
Desafortunadamente, la forma funcional del potencial predice
empaquetamientos compactos, por lo que no es un buen candidato para las
simulaciones de metales bcc. EAM es incapaz de describir la fuerte
direccionalidad de los enlaces en los metales bcc. En cualquier caso, a falta de
mejores alternativas, sigue siendo enormemente utilizado en metales bcc
también.
Los potenciales utilizados en este trabajo, en el que el metal simulado (α-Fe)
tiene un empaquetamiento bcc, han sido el potencial de Dudarev y Derlet (DD)
[30] y el potencial de Ackland, Mendelev y Srolovitz et al. (AM) [33]. Ambos
potenciales fueron modificados en el rango de distancias cortas, esto es, en la
parte repulsiva del potencial, de acuerdo con el procedimiento descrito en [32].
Fueron elegidos porque reproducen con bastante exactitud la estabilidad de
diferentes defectos puntuales en comparación con cálculos obtenidos con la
teoría del funcional de la densidad (DFT). El potencial AM es un potencial multi-
cuerpo de tipo Finnis-Sinclair y está ajustado a valores de ab initio de energías
de defectos puntuales y de propiedades de bulk [33]. El potencial DD es
también un potencial multi-cuerpo de tipo Finnis-Sinclair, pero incluye los
efectos del magnetismo en la energía de interacción de los átomos de Fe
utilizando los modelos de Stoner y de Ginzburg-Landau [30].
2.1.3. Métodos de integración
La estabilidad del sistema, exactitud del cálculo y eficacia del código
determinan la elección de uno u otro algoritmo. En general, estos algoritmos
resuelven las ecuaciones de movimiento de un sistema calculando las
posiciones de las partículas como una función del tiempo. Suelen asumir que la
fuerza aplicada sobre cada partícula se mantiene constante durante el
incremento temporal Δt. Las condiciones ideales de un buen algoritmo son:
Rápido en el cálculo,
16
Estable con incrementos temporales relativamente grandes,
Buena conservación de energía en el conjunto microcanónico.
El algoritmo utilizado en este trabajo es el predictor-corrector de cuarto grado.
Dadas la posición, velocidades, y demás información en el instante t, este
algoritmo calcula las nuevas posiciones, velocidades y demás en el instante t +
δt. Como la trayectoria es continua, estas cantidades se pueden expresar en
forma de serie de Taylor.
(9)
(10)
(11)
(12)
Donde el superíndice p significa predicho. El siguiente paso es calcular las
nuevas fuerzas para los valores predichos de las posiciones, con lo que se
obtienen las aceleraciones corregidas . Por lo tanto, el error en la
predicción es
(13)
Entonces, este error se incluye en los valores predichos de las posiciones,
velocidades y demás, para calcular los valores corregidos
(14)
(15)
(16)
17
(17)
Los valores óptimos de los coeficientes c0, c1, c2 y c3 para conseguir máximas
estabilidad y precisión en las trayectorias fueron obtenidos por Gear. En la
tabla 1 se muestran los valores para diferentes órdenes de truncamiento de la
serie de Taylor.
Orden C0 C1 C2 C3 C4 C5
3 0 1 1
4 1/6 5/6 1 1/3
5 19/20 ¾ 1 ½ 1/12
6 3/20 251/360 1 11/18 1/6 1/60
Tabla 1: Coeficientes para el algoritmo predictor-corrector
El número de pasos correctores puede ser el que se desee aunque, por
cuestiones de eficiencia computacional, sólo se suele usar un paso corrector.
Este es el algoritmo que suele ofrecer la mejor conservación de energía en el
conjunto microcanónico. Por lo tanto, es muy utilizado siempre que el número
de pasos de integración sea alto. Un criterio para la selección del paso
temporal es considerar que Δt << 1/ donde es la frecuencia de Debye (1013
s-1). El valor elegido, exceptuando aplicaciones de alta energía, suele ser del
orden del femtosegundo. De esta forma, el error en la conservación de energía
se mantiene inferior a 0.1 eV/ns.
2.1.4. Condiciones de contorno
Las condiciones de contorno más simples son considerar superficies libres.
Pero por lo general, interesa simular sólidos macroscópicos, por lo que se
recurre a las condiciones periódicas de contorno. Estas consisten en la
repetición de la celda MD básica infinitas veces en las tres direcciones del
espacio. Cada una de las réplicas de la celda básica recibe el nombre de celda
imagen. Las celdas imagen son del mismo tamaño y contienen las mismas
18
partículas que la celda original. Considerando una celda MD de lado L con N
partículas, las ecuaciones básicas que definen las condiciones periódicas de
contorno son
(18)
Donde i´ es la partícula imagen de i en la celda de índices (n1,n2,n3).
Centrándose ahora en cómo afectan las condiciones periódicas a la celda
original, suponiendo una celda de simulación cúbica, toda partícula que
abandone la celda reentra por la cara opuesta con la misma velocidad (figura
4).
Por lo tanto, las condiciones periódicas de contorno conservan el número de
partículas y la cantidad de movimiento, pero no el momento cinético (aunque sí
su valor medio).
Figura 4: esquema del funcionamiento de las condiciones periódicas [34].
19
Además, las condiciones de contorno periódicas suponen que para calcular las
fuerzas sobre un átomo en el borde de la celda principal, se consideran las
celdas imágenes, de forma que no hay átomos de superficie, todos los átomos
está rodeados de otros átomos en las tres dimensiones. De esta forma se
simula un sólido infinito.
En este trabajo, para simular la irradiación con iones de las películas finas
analizadas con TEM, se han utilizado condiciones periódicas únicamente en las
direcciones del espacio x e y, dejando en el eje z las superficies libres.
2.1.5. El código MDCASK
El código MDCASK, dedicado específicamente a simular cascadas de
desplazamiento, ha sido el empleado en este estudio. Está basado en el código
MOLDY desarrollado por Finnis y colaboradores en el laboratorio Harwell del
Reino Unido [35]. Este programa ha sido implementado con características
especiales para estudios de implantación de iones de energías entre eV y keV
en metales y blancos de silicio. Además ha sido modificado para funcionar en
todo tipo de plataformas en paralelo.
Su implementación en paralelo fue realizada en el laboratorio Lawrence
Livermore National Laboratory de California en los EEUU. Para el cálculo de
vecinos de cada uno de los átomos utiliza el algoritmo de las listas enlazadas
[36] que se explica a continuación.
En cualquier código de dinámica molecular, la mayor parte del tiempo se
consume en calcular las distancias de cada átomo a sus vecinos. Suponiendo
potenciales de alcance infinito, el problema sería de orden N2.
Afortunadamente, al ser los potenciales de alcance limitado, no es necesario el
cálculo de todas las distancias. Por potencial de alcance limitado se entiende
aquel en el que la posición de un átomo i no influye en la fuerza sobre el átomo
j si la distancia que los separa (rij) es mayor que un cierto radio de corte rc.
20
Entonces, para calcular la fuerza en el átomo j sólo se necesita calcular su
distancia con todos los átomos dentro de una esfera de radio rc. En la figura 4,
la energía del átomo 1 depende sólo de la posición de los átomos en el círculo
oscuro. El algoritmo de las listas enlazadas es el algoritmo que menos
distancias innecesarias calcula y organiza las listas de vecinos de forma
eficiente. Este método divide la celda de simulación en pequeñas celdas de
arista ligeramente superior a rc. De esta forma, se garantiza que dos átomos
que no estén en celdas adyacentes no interaccionan entre sí. Entonces,
cuando se quiera calcular los vecinos de un átomo, basta con calcular las
distancias con los átomos de su propia celda y las adyacentes.
2.1.6. Simulación de irradiación de Fe con iones de Fe: antecedentes
El concepto de cascada de desplazamiento como proceso fundamental en la
producción de daño por irradiación data de los años 50, cuando se introdujeron
por primera vez los términos displacement spike [37] y thermal spike [38]. En
esos años ya se conocían bastante bien los aspectos básicos de este
fenómeno, aunque de forma cualitativa [39]. Pronto se reconoció que para
obtener más información sobre este proceso – que experimentalmente es
imposible de observar – el uso de métodos numéricos era especialmente
adecuado. Las primeras simulaciones de colisiones en cristales mediante
técnicas computacionales se hicieron en cobre con estructura fcc [40] y poco
después, debido a su importancia en aplicaciones prácticas, en hierro bcc [41].
La aproximación de colisiones binarias (BCA), desarrollada en paralelo a lo que
ahora conocemos como dinámica molecular (MD), era en esos tiempos la única
posibilidad para explorar eventos de energías relativamente altas [42], por lo
que permaneció como técnica de referencia hasta que los avances en la
ciencia computacional de los años 80 permitió el uso generalizado de la MD,
demostrado el hecho de que BCA predecía una cantidad de defectos
demasiado alta comparada con los experimentos [43].
Desde entonces se han realizado una gran cantidad de simulaciones con MD
de irradiación de iones en α-Fe, pero debido a su intención de simular el daño
21
por irradiación de neutrones en los materiales del reactor, prácticamente todos
los cálculos se han realizado en el interior del material aplicando condiciones
periódicas en las tres direcciones del espacio. Los efectos de la irradiación en
el interior de un material son sin embargo muy difíciles de observar
experimentalmente, y de hecho, la mayoría de técnicas como la microscopía
electrónica de transmisión (TEM), examinan defectos cercanos a las
superficies. Trabajos recientes [44-50] indican que la presencia de superficies
cercanas puede influir en la formación del daño primario, por lo que las
simulaciones con sólidos infinitos no se corresponden con los experimentos de
irradiación con iones de las películas finas observadas por TEM. De estos
trabajos, los únicos realizados en Fe al comienzo de la tesis [48, 49] habían
utilizado potenciales interatómicos que han demostrado tener una configuración
errónea para los átomos intersticiales, además de emplear energías muy bajas
(10kev). La motivación de este trabajo de tesis ha sido por tanto estudiar
mediante simulación multiescala las diferencias entre la irradiación en láminas
finas de Fe y en el interior del material de manera detallada y cuantitativa,
utilizando potenciales interatómicos recientes y condiciones homólogas a las
experimentales.
2.2. MONTE CARLO CINÉTICO
Las simulaciones de Monte Carlo se caracterizan por utilizar números
aleatorios en sus algoritmos. Su nombre se debe precisamente al famoso
casino de la ciudad de Mónaco. El desarrollo sistemático del método comenzó
en el Laboratorio Nacional de Los Alamos en los años 40, con los trabajos de
Ulam y Metropolis [51], desarrollándose ampliamente desde entonces. Hoy en
día los métodos de Monte Carlo más usados son el Metrópolis Monte Carlo [52]
y el Monte Carlo Cinético [53]. El método de Metrópolis Monte Carlo se
desarrolló en los cincuenta para estudiar propiedades en equilibrio de un
sistema. Diez años más tarde, en los sesenta, se introdujo el método de Monte
Carlo Cinético que, a diferencia del Metrópolis Monte Carlo, permite estudiar
propiedades dinámicas de un sistema.
22
En nuestro problema de simular una cascada de desplazamiento hemos
utilizado dinámica molecular clásica. La dinámica molecular sigue de manera
natural la evolución de un sistema de átomos a través de la propagación en el
tiempo de las ecuaciones clásicas del movimiento. Sin embargo, como se ha
comentado anteriormente, la integración exacta de las ecuaciones de Newton
requiere pasos de tiempo lo suficientemente cortos (~10-15 s) como para
resolver las vibraciones atómicas. Esto hace que el tiempo total de simulación
que se puede alcanzar con Dinámica Molecular Clásica sea menor de un
microsegundo. El método de Monte Carlo Cinético salva esta limitación
utilizando el hecho de que la dinámica de este tipo de sistemas consiste
típicamente en procesos de difusión (saltos atómicos) que ocurren en escalas
de tiempo mucho mayores. De este modo, se puede asumir que la evolución de
un sistema se caracteriza por transiciones ocasionales de un estado a otro,
donde cada estado corresponde a un pozo de potencial o mínimo local y dos
estados adyacentes están separados por una barrera de energía, tal como
indica la figura 5.
Figura 5: Contorno esquemático de una superficie de energía potencial para la
transición de un sistema entre dos estados. Después de muchos períodos de
vibración, el sistema pasa de un estado a otro. Los puntos indican las barreras
de energía [54].
Este tipo de sistemas se catalogan dentro de los llamados “infrequent-event
systems”. Para que un suceso sea considerado como “rare event” la barrera de
23
energía potencial tiene que ser mucho mayor que KBT. La propiedad clave de
este tipo de sistemas es que, después de una transición, y debido al largo
tiempo de permanencia en cada estado, la partícula pierde la memoria de cómo
llegó hasta allí. Esto hace que la probabilidad de escape hacia un nuevo estado
sea independiente de los anteriores eventos y sólo dependa del estado actual y
del posible nuevo estado adyacente. Esta característica es lo que define a los
procesos Markovianos [55] y permite obtener a priori todas las probabilidades
de transición. En los procesos estocásticos las transiciones se describen a
menudo con procesos de Poisson [55, 56]. Si consideramos un suceso con una
frecuencia r, la densidad de probabilidad de la transición f(t) que nos dará la
probabilidad de que la transición ocurra a un tiempo t será:
(19)
Generalizando a N procesos independientes de Poisson, con frecuencia ri,
obtenemos un proceso de Poisson con una frecuencia acumulada
(20)
y una densidad de probabilidad de la transición igual a:
(21)
Podemos determinar el tiempo medio entre eventos sucesivos:
(22)
La evolución de un sistema se puede describir eligiendo al azar eventos con
una probabilidad proporcional a su frecuencia tal como muestra la figura 6.
24
Figura 6. Esquema del procedimiento para elegir el camino de reacción.
Los eventos se colocan uno al lado del otro con un tamaño proporcional a su
frecuencia. A continuación se genera un número aleatorio s en el intervalo (0,1)
y se elige el evento cuya frecuencia cumple:
(23)
Para actualizar el reloj del sistema, el tiempo se debe incrementar en (el
tiempo medio entre eventos sucesivos). De manera general, el reloj del sistema
se puede incrementar en un tiempo aleatorio , de acuerdo a la ecuación:
(24)
Donde s’ es un número aleatorio en el intervalo (0,1). El uso de números
aleatorios para evolucionar el tiempo de un sistema da una mejor descripción
de la naturaleza estocástica de los procesos que intervienen y se justifica
porque . Este procedimiento es el más usado en Monte Carlo
Cinético y se conoce como el algoritmo del tiempo de residencia [57].
r1
r2
r3
rn
...
sR
25
2.2.1. La teoría del estado de transición
Para calcular las frecuencias de transición entre un estado i y un estado j, el
método de Monte Carlo Cinético se basa en la aproximación armónica del
formalismo de la teoría del estado de transición [58, 59]:
(25)
Donde Pij es el llamado prefactor, y Ea es la energía de activación del proceso
tal como muestra la figura 7. Los prefactores normalmente se encuentran en el
rango de 1012 s-1 – 1013 s-1, por lo que una práctica común es utilizar un valor
fijo en este rango para ahorrar tiempo de computación. Las energías de
activación se obtienen de cálculos de DFT o en su defecto de dinámica
molecular.
Figura 7. Esquema de la cinética de transición de un sistema entre dos
estados. En la teoría del estado de transición la reacción se produce siempre
del estado A al estado B.
26
2.2.1. El código MMonCa
En el presente trabajo se ha utilizado el código de Monte Carlo Cinético
MMonCa [60]. MMonCa es un simulador multi-material creado por el Dr.
Ignacio Martín-Bragado en el Instituto IMDEA Materiales de Madrid. Está
escrito en C++ y ha sido integrado en la librería TCL para la interfaz de usuario.
Contiene dos módulos independientes para conseguir diferentes niveles de
simulación:
Lattice kinetic Monte Carlo (LKMC). Este módulo utiliza la red del
material y se usa por ejemplo para simular cambios de fase tales como la
recristalización epitaxial en estado sólido, donde la orientación cristalina es
fundamental.
Object kinetic Monte Carlo (OKMC). Este módulo no utiliza la red del
material (off-lattice en inglés) y se usa para estudiar la evolución de defectos en
un sólido.
La estructura simplificada de MMonCa se muestra en la figura 8. El usuario
lanza una simulación a través de un script escrito con comandos TCL y otros
comandos especiales para inicializar una simulación, simular un recocido, etc.
27
Figura 8. Estructura por bloques del simulador MMonCa.
En esta tesis hemos utilizado el módulo OKMC para estudiar la evolución del
daño implantado en las simulaciones de dinámica molecular en láminas finas y
en bulk de Fe irradiando con iones de Fe de 100 keV. En OKMC los defectos
se definen como objetos. En particular, en MMonCa se definen como
interfases, partículas libres, clusters, defectos extendidos y multiclusters. Para
cada uno de estos objetos es necesario definir el número de eventos asociados
con el objeto, la frecuencia asociada a cada evento, y las funciones para llevar
a cabo el evento una vez es elegido por el algoritmo de MMonCa.
User input script
TCL
library
KMC kernel:Space managerTime manager
LKMC OKMC
Operating System
User
interface C+
+
28
2.3. SIMULACIÓN DE IMÁGENES TEM
La simulación de imágenes de microscopía electrónica de transmisión (TEM)
se utiliza, entre otros motivos, para relacionar las imágenes de dinámica
molecular con las imágenes reales de experimentos de irradiación en
materiales observados por TEM. Como se ha comentado anteriormente, la
mayor limitación de la dinámica molecular es la escala de tiempo en la que
opera, la cual alcanza como máximo los nanosegundos, mientras que la escala
real de laboratorio comienza en los segundos. Esto hace imposible una
correlación directa entre la dinámica molecular y los experimentos. La
simulación de imágenes TEM de las configuraciones obtenidas con dinámica
molecular permite relacionar ambas técnicas. Así, es posible, por ejemplo,
averiguar si los clusters de defectos obtenidos en dinámica molecular serán
estables y visibles en la escala real de los experimentos, identificar la
naturaleza (intersticial o vacante) de defectos observados en el microscopio, o
saber si los clusters observados se han formado en la cascada de
desplazamiento o mediante migración y recombinación de defectos.
2.3.1. El método multicapa
El método multicapa [61] es el más usado en los programas de simulación de
imágenes TEM y consiste básicamente en seccionar la muestra en múltiples
capas perpendiculares al haz incidente. Para ello existen varias
aproximaciones entre las que destacan:
El formalismo del espacio recíproco.
El formalismo FFT.
La aproximación del espacio real.
La aproximación de las funciones de Bloch.
El fundamento de todos ellos es el mismo. En primer lugar, el potencial del
cristal se divide en capas y se proyecta en un plano perpendicular a la dirección
29
de observación. Al potencial proyectado en cada capa se le denomina “phase
grating” o “rejilla de fase”. A continuación se calculan las amplitudes y fases de
todos los haces generados por la interacción de la función de onda del haz de
electrones incidente con el primer plano proyectado. Estos haces se propagan
por el vacío (el microscopio virtual) hasta la segunda capa y se repite el
proceso. Así hasta la última capa de la muestra (figura 9).
Figura 9 Representación gráfica de la metodología multicapa para obtener
imágenes TEM simuladas [62].
Principalmente el método multicapa considera tres componentes:
Ψ describe la función de onda del electrón.
P es la propagación de la función de onda electrónica en el vacío: el
microscopio.
Q es el “fase grating” o “rejilla de fase”: la muestra.
En el formalismo del espacio recíproco este proceso se puede describir por la
ecuación:
(26)
30
Donde es la función de onda en el espacio recíproco a la salida de la
capa n+1, es la función de propagación, es la función de
transmisión, y (x) es la convolución. Las tres funciones descritas son funciones
del espacio recíproco, de ahí el nombre del formalismo. En el formalismo FFT
la ecuación (26) se transforma en:
(27)
Donde es la transformada de Fourier de la función en el
espacio real.
2.3.2. El código EMS
El programa que hemos utilizado en esta tesis, el código EMS [63] utiliza el
formalismo FFT para maximizar la eficiencia computacional del método
multicapa. La figura 10 representa esquemáticamente la implementación del
método en el programa EMS.
31
Figura 10. Esquema de la implementación del método multicapa en el software
EMS [64].
Con este software hemos simulado imágenes TEM de campo claro de clusters
de defectos a 200 kV. La muestra de dinámica molecular se divide
perpendicularmente al haz de electrones en láminas de 0.2 nm de espesor. Las
láminas han de ser periódicas para realizar la transformada de Fourier. Para
cumplir con este requisito se elige una dirección cristalográfica para realizar el
corte, la cual determina la condición de difracción. A continuación se obtiene la
función de onda de cada una de las láminas, y finalmente, se obtiene la imagen
TEM simulada haciendo pasar cada función de onda por el “microscopio
virtual”, un subprograma del código EMS.
32
3. RESULTADOS Y DISCUSIÓN
En este apartado se resumen brevemente los principales resultados del trabajo
realizado durante la tesis. Estos resultados se discuten en mayor profundidad
en los trabajos presentados en los apartados 4 y 5.
3.1. Daño en escala de picosengundos: iones frente a neutrones
En los trabajos I y II se muestra que el daño por irradiación en thin films es
totalmente diferente al daño en bulk. En el trabajo III se cuantifican las
diferencias y se muestra la formación de loops visibles de vacantes <100>
directamente en la cascada de desplazamiento. Estos resultados explican los
experimentos de irradiación in-situ de láminas finas de Fe y FeCr a bajas dosis
donde se observa la formación de este tipo de loops.
En el trabajo I estudiamos con dinámica molecular el daño producido al irradiar
láminas finas de Fe puro con iones de Fe de 50, 100 y 150 keV bajo
condiciones de channeling y con el potencial de DD. Los resultados se
comparan con los resultados de Björkas [65] en cascadas de bulk. En este
trabajo obtenemos dos conclusiones importantes. En primer lugar mostramos
que, a diferencia de las cascadas en bulk que producen el mismo número de
vacantes e intersticiales, en las cascadas en láminas finas, aún en condiciones
de channeling, el número de vacantes es siempre mayor al número de
intersticiales. Esto es debido a la morfología del daño producido en las
cascadas en láminas finas. La segunda conclusión importante es que, debido a
esta morfología que produce diferentes tipos de eventos según dónde colisione
el PKA con mayor energía, la dispersión en el número de defectos en láminas
finas es mucho mayor que en las simulaciones en el bulk debido a la influencia
de las superficies. Además esta dispersión aumenta con la energía del PKA.
En el trabajo II nos centramos en las cascadas de mayor energía, 150 keV, y
estudiamos el efecto del ángulo de incidencia del ión y la morfología de la
cascada, también con el potencial de DD. El resultado más relevante es que,
por el contrario de lo ocurre a menores energías donde sólo se daña la lámina
33
superior, a esta energía y por extrapolación a energías superiores, tanto la
lámina superior como la inferior sufren un daño considerable (Figura 11).
Figura 11. Las figuras (a-c) muestran tres cascadas representativas de
irradiación de Fe por un ión de Fe de 150 keV al cabo de 25 ps. Las esferas
verdes son vacantes, las esferas rojas son intersticiales y las amarillas átomos
de superficie. Se representan ambas superficies. La flecha indica la posición de
implantación del átomo de Fe. En (a) y (c) el átomo de Fe impacta con un
ángulo de 10º y en (b) el ángulo de implantación es de 22º respecto a la
normal. La figura (d) es una imagen aumentada del daño producido en la
superficie inferior del caso (c). Del trabajo II.
En el trabajo III hacemos un estudio estadístico detallado comparando en
paralelo cascadas en láminas finas y cascadas de bulk con dos potenciales
distintos, el de DD y el AM, dos ángulos distintos, 10º y 22º, y dos energías
distintas, 50 keV y 100 keV. En este trabajo cuantificamos las diferencias entre
las dos geometrías y estudiamos en detalle la morfología de las cascadas. La
diferencia más remarcable es la creación de loops de vacantes <100>. Tanto
su tamaño como su frecuencia son mucho mayores en las cascadas en
láminas finas, con loops que alcanzan los 4 nm de tamaño. Además, los
34
clusters de intersticiales producidos en la cascada son en promedio más
pequeños en las láminas finas, especialmente en los casos en los que se
produce un cluster de vacantes de gran tamaño (Figura 12).
Figura 12. Microestructura de dos cascadas de 50 keV de Fe en Fe a los 25 ps.
En ambos casos el ángulo de impacto es de 10º pero en (a) el ángulo acimutal
es de 10º y en (b) de 80º. La distribución inicial de velocidades es también
distinta. Las esferas rojas son intersticiales, las verdes vacantes y las amarillas
átomos de superficie. Del trabajo III.
En el trabajo V comparamos los resultados de dinámica molecular de daño en
Fe con el daño que se produce en el otro material clave en fusión, el W.
Encontramos que en ambos materiales el tamaño de los defectos sigue una ley
de escala tanto en bulk como en láminas finas. Sin embargo, la pendiente de la
ley de escala en Fe bulk es marcadamente diferente, y se ve afectada por la
presencia de superficies en las irradiaciones en láminas finas. Esto explica las
diferencias en los experimentos de TEM en Fe y W. En W bulk las cascadas
son más compactas y con mayor densidad de energía que las cascadas en Fe
bulk, por lo que en W bulk se producen clusters grandes tanto de vacantes
como de intersciales, que no se producen en Fe bulk. El efecto de las
superficies en ambos materiales es el mismo, pero debido al diferente
comportamiento en bulk, este efecto es mucho más evidente en Fe.
En el trabajo VI estudiamos las diferencias entre cascadas de dinámica
molecular y cascadas de colisiones binarias (BCA) en simulaciones de
35
irradiación de Fe con neutrones, y la evolución de la microestructura con
OKMC. En cuanto a la morfología de las cascadas, en ambos casos el número
de pares de Frenkel generado aumenta con la energía del PKA de manera
proporcional, y las cascadas tienen el mismo rango de longitud. Sin embargo,
las cascadas de BCA, debido a que no sufren recombinación, producen un
mayor número de defectos que las cascadas de dinámica molecular, y
muestran una distribución espacial diferente. Las consecuencias de estas
diferencias se discuten en el siguiente apartado.
3.2. Evolución del daño primario en Fe y en FeCr
La influencia de las superficies en la evolución del daño en Fe irradiado se
muestra en el trabajo IV. Utilizamos como datos de entrada las cascadas de
dinámica molecular de 100 keV en láminas finas y en bulk irradiadas con iones
de Fe con un ángulo de 22º. Para las energías de enlace y de migración de
defectos utilizamos datos de cálculos de DFT y asumimos que los loops de
intersticiales <100> se forman por reacción de dos loops de intersticiales <111>
de tamaño similar. En este trabajo nos centramos principalmente en los efectos
de los obstáculos y de la distribución inicial del daño en la cascada. Los
principales resultados son, por un lado, que el efecto de la superficie depende
en gran medida de la concentración de obstáculos, y por otro, y aún más
interesante, que la distribución del daño inicial es clave tanto para la evolución
de la concentración de defectos como para su naturaleza. En este modelo la
irradiación cercana a la superficie favorece la formación de loops <100>.
En el trabajo VII abordamos el problema de la nucleación y crecimiento de
loops en Fe y en FeCr bajo irradiación mediante OKMC. Utilizamos como en el
trabajo IV datos de DFT para la energética de los defectos y las cascadas de
100 keV en láminas finas con el potencial de AM. Para la formación de los
loops de intersticiales <100> cosideramos dos posibles modelos. Uno de ellos
es el utilizado en el trabajo IV, según el cual los loops de intersticiales <100> se
producen por reacción de dos loops <111> se tamaño similar (modelo A), y un
segundo modelo, según el cual los clusters pequeños de intersticiales pueden
transformarse en loops <111> o <100> al crecer (modelo B). La principal
36
conclusión es que el modelo B reproduce más fielmente los resultados
experimentales, ya que reproduce la tendencia de crecimiento de los loops con
la dosis y la proporción de loops <100> y <111>.
3.3. Comparación con imágenes TEM
Finalmente la comparación con imágenes TEM no ha dado lugar a un trabajo
preparado para publicar en la tesis, de manera que se discutirán en este
apartado los principales resultados obtenidos hasta el momento.
Hemos realizado imágenes TEM simuladas de las cascadas producidas por la
irradiación de Fe en Fe en láminas finas con energías de 30 keV, 50 keV y 100
keV. Como se ha explicado en la metodología el software empleado ha sido el
código EMS desarrollado por Robin Schaublin y las simulaciones se han
llevado a cabo en el cluster Atsimat del departamento de Física Aplicada de la
Universidad de Alicante. Cada imagen simulada de toda la muestra, unos 50
nm, necesita hasta 1500 horas de CPU.
La figura 13 muestra la imagen TEM simulada de una cascada de Fe en Fe de
30 keV. También se muestra la posición de todos los átomos en la superficie,
así como la localización de los defectos. Los cálculos revelan que incluso para
esta energía tan baja, el daño producido puede observarse experimentalmente
ya que sus dimensiones alcanzan unos pocos nanómetros.
37
Figura 13. Imagen TEM simulada (izquierda) de una cascada de dinámica
molecular de 30 keV de Fe en Fe (derecha). La figura muestra la posición de
todos los defectos y de ambas superficies a los 25 ps del inicio de la cascada.
Las esferas verdes son vacantes, las rojas intersticiales y las amarillas átomos
de la superficie.
Hemos realizado el mismo tipo de cálculo para energías más altas, 50 keV y
100 keV. En la figura 14 se muestra el caso de una cascada de 50 keV. En este
caso se forman dos loops de vacantes bajo la superficie, uno mucho mayor que
el otro. En la imagen TEM simulada a los 7 nm de espesor se observan dos
puntos brillantes que muestran la localización de estos loops de vacantes, con
dimensiones de unos pocos nanómetros. Sin embargo, en la imagen TEM
simulada de toda la muestra sólo se observa un único punto, lo cual significa
que el contraste del loop de mayor tamaño eclipsa el contraste del de menor
tamaño, hecho corroborado experimentalmente.
38
Figura 14. Imágenes TEM simuladas (superior) de una cascada de dinámica
molecular de 50 keV de Fe en Fe (inferior). La figura muestra la posición de
todos los defectos y de la superficie superior a los 35 ps del inicio de la
cascada. Las esferas verdes son vacantes, las rojas intersticiales y las
amarillas átomos de la superficie. El loop de vacantes tiene unos 5 nm de
diámetro y el daño en la celda se extiende hasta los 11 nm de espesor.
Además de las imágenes TEM simuladas, realizamos experimentos de
irradiación de Fe ultrapuro con átomos de Ga de 30 keV utilizando un sistema
FIB (Focused Ion Beam) y una dosis entre 0.02 iones/nm y 0.32 iones/nm. El
análisis de TEM de las muestras muestran la formación de defectos de tamaño
considerable cerca de la superficie, como los mostrados en la Figura 15.
39
Figura 15. Imagen de microscopía electrónica de transmisión de una lámina de
Fe ultrapuro irradiado con iones de Ga+ de 30 keV mediante un sistema FIB.
La forma de estos defectos, con tres lóbulos oscuros alrededor de un punto
brillante, es semejante a la forma que observamos en las imágenes TEM
simuladas de nuestras cascadas, por lo que podrían ser loops de vacantes. Sin
embargo, estos resultados son preliminares y deberán ser estudiados en mayor
profundidad en el futuro.
20 nm
40
4. TRABAJOS PUBLICADOS O ACEPTADOS
I
Journal of Nuclear Materials 452 (2014) 453–456
Contents lists available at ScienceDirect
Journal of Nuclear Materials
journal homepage: www.elsevier .com/ locate / jnucmat
Molecular dynamics simulations of irradiation of a-Fe thin films withenergetic Fe ions under channeling conditions
http://dx.doi.org/10.1016/j.jnucmat.2014.05.0770022-3115/ 2014 Published by Elsevier B.V.
⇑ Corresponding author. Tel.: +34 965903400x2056; fax: +34 965909726.E-mail address: [email protected] (M.J. Caturla).
M.J. Aliaga a, A. Prokhodtseva b, R. Schaeublin b, M.J. Caturla a,⇑a Departamento de Física Aplicada, Facultad de Ciencias, Fase II, Universidad de Alicante, Alicante E-03690, Spainb Ecole Polytechnique Féderale de Lausanne (EPFL), Centre de Recherches en Physique des Plasmas, Association Euratom-Confédération Suisse, 5232 Villigen PSI, Switzerland
a r t i c l e i n f o
Article history:Received 18 November 2013Accepted 31 May 2014Available online 10 June 2014
a b s t r a c t
Using molecular dynamics simulations with recent interatomic potentials developed for Fe, we havestudied the defects in thin films of pure bcc Fe induced by the displacement cascade produced by Featoms of 50, 100, and 150 keV impinging under a channeling incident angle of 6 to a [001] direction.
The thin films have a thickness between 40 and 100 nm, to reproduce the thickness of the samples usedin transmission electron microscope in situ measurements during irradiation. In the simulations we focusmostly on the effect of channeling and free surfaces on damage production. The results are compared tobulk cascades. The comparison shows that the primary damage in thin films of pure Fe is quite differentfrom that originated in the volume of the material. The presence of near surfaces can lead to a variety ofevents that do not occur in bulk collisional cascades, such as the production of craters and the glide ofself-interstitial defects to the surface. Additionally, in the range of energies and the incident angle used,channeling is a predominant effect that significantly reduces damage compared to bulk cascades.
2014 Published by Elsevier B.V.
1. Introduction
One of the main challenges facing the use of nuclear fusion as afuture energy source is related to the reactor materials. In thefuture fusion reactors, a large amount of 14 MeV neutrons will beproduced, which will deteriorate the reactor vessel as theycontinuously collide with the first wall or plasma facing compo-nents. In order to have long-lived nuclear reactors that permitfusion to be an economically competitive energy source, it isnecessary to understand radiation damage in the harshest condi-tions expected for these materials. However, experiments withhigh energy neutrons are very expensive and there are only afew facilities around the world where they can be performed. Inaddition, neutrons produce radioactive isotopes in the irradiatedmaterial, which complicates its analysis. For these reasons, ionirradiation appears to be an excellent tool for understanding defectproduction in materials. However, the correlation to neutron irra-diation induced damage is not trivial. Therefore, substantial effortis devoted to modeling these effects, mainly using moleculardynamics, and to model-oriented experiments using ion implanta-tion. This would increase our knowledge of the correlation toneutron irradiation and, eventually, it will expand our capacity topredict these effects.
Experiments are currently being performed in JANNuS, France,where while a target is being irradiated with Fe ions it is observedin situ in a transmission electron microscope (TEM). For theseexperiments to be performed the samples can only be a few tensof nanometers thick to be transparent to electrons. There are manystudies of ion irradiation in a-Fe using molecular dynamics simu-lations (see Ref. [1] for a review), but the majority of these calcula-tions are performed in bulk specimens in view of reproducingneutron damage in reactor materials. The conditions in an ion irra-diation experiment are significantly different. On the one hand, forlow ion irradiation energies, the damage will occur near thesurface. On the other hand, in TEM in situ irradiation, the effectof surfaces cannot be neglected because the specimen thicknessis between 40 and 200 nm. The influence of surfaces on theprimary damage was studied in detail during the 1990s usingmolecular dynamics simulations for low energies in f.c.c. materials[2–4]. However, the only studies in Fe including surfaces are thosefrom Refs. [5,6]. Those calculations were performed for very lowion energies (10–20 keV) and using interatomic potentials thatare now known to provide the wrong self-interstitial configuration.Nonetheless, the influence of the surface in terms of damageproduction and defect distribution was already shown.
Our objective is to further study how damage is produced inthese thin films and how it relates to bulk specimens. In this articlewe study, using molecular dynamics simulations with recent inter-atomic potentials, the very first stages of the damage produced by
Fig. 1. Defect distribution for a series of 20 simulations of cascades in thin filmsproduced by irradiation with a 100 keV Fe atom.
454 M.J. Aliaga et al. / Journal of Nuclear Materials 452 (2014) 453–456
Fe ions with energies between 50 and 150 keV in thin films of purea-Fe under channeling conditions.
2. Method
Calculations were performed using the molecular dynamicscode MDCASK, developed at Lawrence Livermore National Labora-tory, with the interatomic potential of Dudarev and Derlet [7] fora-Fe. This interatomic potential was developed to include themagnetic character of Fe by adding a new term in the embeddingfunction of the potential [7]. For short range interactions thepotential is connected to the Universal potential as described inRef. [8]. Displacement cascades were simulated injecting an exter-nal Fe atom, or primary knock-on atom (pka), in the top freesurface of a thin film with a [001] normal. The energy of theincident Fe atom is 50, 100, and 150 keV. The Fe atom incident
Fig. 2. Three snapshots of the time evolution of the damage produced by a 150 keV Finterstitials, (a) 0.06 ps after the initiation of the recoil, (b) 0.5 ps and (c) 10 ps. (For interweb version of this article.)
angle was tilted 6 from normal incidence, lower than the criticalangle for channeling which, according to the Lindhard expression[9] is 21, 15, and 12 for the 50, 100, and 150 keV respectively.Cell dimensions are 40 nm 40 nm 40 nm (140 140 140lattice parameters) with a total of 5,154,801 atoms. We used theLindhard model to include a friction force proportional to thevelocity for all atoms with a kinetic energy greater than 5 eV tomimic the inelastic energy losses produced by collisions with theelectrons. Periodic boundary conditions were applied to directionsperpendicular to that of the incident ion to emulate an infinite sys-tem, whereas in the direction of the incident ion free surfaces areconsidered. The conditions used are intended to reproduce theexperiments already published by Yao et al. [10].
The simulations were run for the duration needed for the eventto be completed and the number of defects stabilized, which occursin less than 15 ps in most cases. The temperature was kept near 0 Kto minimize thermal atomic vibrations and thus facilitate the iden-tification of defects. Before the recoil atom was started, the cell wasequilibrated for 1 ps to the desired temperature of the simulation.A thermal bath was imposed to dissipate the energy deposited bythe ion in the solid by rescaling the velocity of two atom layerssituated at the end of the [100] and [010] directions.
In order to identify the defects, vacancies and interstitials, weused Wigner–Seitz cells centered in each (perfect) lattice positionso that an empty cell corresponds to a vacant and a doubleoccupied cell corresponds to an interstitial defect. Variability wasintroduced to obtain statistical results by changing the value forthe impact angle orientation from 0 to 180 for up to a total of20 cases for the 50 and 100 keV pka, and 30 cases for the150 keV pka.
3. Results and discussion
Results are presented here and compared to the results byBjörkas for bulk cascades with energies of 50 keV and 100 keV [11].
Fig. 1 represents the final number of defects (vacancies andinterstitials) that results for the series of 20 simulations ofirradiation of thin films with a pka of 100 keV. It appears that the
e ion in a 40 nm thick Fe sample. Light dots are vacancies and dark dots are self-pretation of the references to color in this figure legend, the reader is referred to the
Fig. 3. Average number of defects produced in thin films (vacancies and intersti-tials) compared to bulk cascades (Frenkel pairs) [11]. (For interpretation of thereferences to color in this figure legend, the reader is referred to the web version ofthis article.)
M.J. Aliaga et al. / Journal of Nuclear Materials 452 (2014) 453–456 455
dispersion in the number of defects is much wider in cascadesaffected by free surfaces than in cascades of the same energy pro-duced in the bulk of the material [11]. In the latter case, the meanvalue of Frenkel pairs produced is 159 with a standard deviation of7 [11]. In our case, we observe, on the one hand, that the averagenumber of vacancies does not match that of interstitials, with 52vacancies and 34 self-interstitials. On the other hand, the standarddeviation for vacancies is 17 and for interstitials is 11. This is inagreement with the results of Stoller et al. [5,6] that show both ahigher production of vacancies than self-interstitials as well as lar-ger standard deviations in their results for surface damage. It isinteresting to note that recent simulations of cascades close tograin boundaries by Bai et al. also show a larger production ofvacancies than interstitials close to the grain boundary [12]. In thatcase, with the use of temperature accelerated dynamics (TAD), theauthors show how at later times self-interstitials trapped at grainboundaries can annihilate those remaining vacancies in the bulk.
The higher number of vacancies is mainly the result of the migra-tion of interstitials to the surface. Differences in the energy spatialdeposition account for the dispersion in the results, with some casesin which the energy is deposited very close to the surface, creatingsurface damage [2], and other cases in which damage is deeper,being more similar to the results obtained in bulk cascades.
Fig. 2 shows three snapshots of the damage produced by a150 keV Fe ion in a 40 nm thick Fe sample. Dark dots show the loca-tions of self-interstitials ions while light dots are the locations ofvacancies. Fig. 2(a) shows the distribution of defects at 0.06 ps,before the ion reached the back surface. Fig. 2(b) shows thedamage at 0.5 ps which corresponds to the time of maximum num-ber of displacements in the lattice, that is the peak of the collisioncascade. Much of the damage is recovered during the next stage,reaching an almost constant number of defects. The final distribu-tion at 10 ps is shown in Fig. 2(c). Note that the damage producedin the top and bottom surfaces is mostly vacancies since the intersti-tials are ejected to the surface and stay as adatoms (not shown in thefigure), again because of the creation of surface damage. It should benoted that in this case the pka traveled through the sample andescaped through the bottom surface, producing little damage, whichshows that the impact angle used (6) is within channeling condi-tions. Therefore the damage produced consists of Frenkel pairs andsmall vacancy and self-interstitial clusters. In all cases studied inthese conditions there was transmission of the pka. We have ana-lyzed the damage produced as a function of the distance to the frontand back surfaces. For the case of Fig. 2(c), the total number of vacan-cies produced in this cascade is 137 while the total number ofself-interstitials produced is 95. The total number of vacancies at adistance of 3 nm from either the back or the front surface is 56, whileonly 6 interstitials are found at this distance. Below 3 nm we thenfind 89 self-interstitials and 81 vacancies, that is, similar values forthe two types of defects, getting closer to bulk calculations.
Comparing the different energies studied, we observe sometrends. The maximum production of defects occurs when eventsvery near the surface take place, leading to the formation of surfacedamage either at the top or the bottom surface. On the other hand,in the cases with the minimum number of defects, events very nearthe surface are not important and the damage created is moresimilar to that produced in bulk material.
Table 1Average number of stable defects and their standard deviation produced in thin films and
Energy (keV) Thin films vacancies Standard deviation Thin films inter
50 47 8 32100 52 17 34150 73 28 46
a Ref. [11].
The results for our simulations in thin films and in bulk cas-cades [11] are summarized in Table 1. Fig. 3 graphically representsthese values. Unlike the results of Stoller et al. [5,6], the averagenumber of defects is significantly lower in our simulations thanin the case of bulk cascades. This is due to the channeling condi-tions used in the calculations, which allow the ion to travelthrough the whole sample thickness without depositing all itsenergy in the target. The total energy deposited by the ion can becalculated from the difference between the initial energy and theenergy of the ion after it crosses the film. On average, the percent-age of the energy deposited for the 50 keV, 100 keV, and 150 keVcascades is 44%, 32%, and 29% respectively. However, even underthese conditions, where little damage should be produced, thereare cases where a high number of vacancies is created, as can beseen in Fig. 1. These are related to cases where surface damage isformed in the front or back surfaces, or in both.
4. Conclusions
In this paper we have studied damage produced by Fe ions withenergies between 50 and 150 keV in thin films of pure bcc Fe usingmolecular dynamics simulations under channeling conditions. Theresults are also compared with those obtained in bulk cascades.Conclusions can be summarized as follows:
1. Unlike cascades simulated in the bulk that produce the samenumber of vacancies and interstitials, cascades in thin filmssimulated by external ion irradiation produce more vacanciesthan interstitials, even under channeling conditions such asthose studied here.
2. Dispersion in the number of defects in ion irradiated thin filmsis greater than in simulations in bulk materials due to the vari-ety of events that can occur because of the surface influence,and this dispersion increases with the pka energy.
bulk cascades.
stitials Standard deviation Bulka Frenkel pairs Standard deviation
7 91 511 159 714 – –
456 M.J. Aliaga et al. / Journal of Nuclear Materials 452 (2014) 453–456
These conclusions show that to be able to reproduce andunderstand ion irradiation experiments in thin films analyzed byTEM, a detailed description of the primary damage is needed. Thisimplies the inclusion of surfaces in the simulations because thedamage produced is completely different from that originated inbulk materials. The time scale in MD simulations is too short tobe able to make a direct comparison with TEM experimentalmeasurements. Kinetic Monte Carlo (kMC) simulations using theinformation obtained from these MD calculations could be usedto make such comparison. If the initial damage distribution hasan impact on microstructure evolution under irradiation, asshown in previous works, the concentration of defects with doseas well as the cluster size distribution obtained by kMC shouldbe different when using bulk cascades or surface cascades.These results could then be contrasted with the experimentallymeasured values.
Acknowledgements
We thank Cristian Denton from the UA and Carolina Björkasfrom the University of Helsinki for helpful discussions. This workwas supported by the FPVII projects FEMaS, GETMAT and PERFECT
and by the MAT-IREMEV program of EFDA. We acknowledge thesupport of the European Commission, the European Atomic EnergyCommunity (Euratom), the European Fusion Development Agree-ment (EFDA) and the Forschungszentrum Jülich GmbH, jointlyfunding the Project HPC for Fusion (HPC-FF), Contract NumberFU07-CT-2007-00055. The views and opinions expressed hereindo not necessarily reflect those of the European Commission.
References
[1] L. Malerba, J. Nucl. Mater. 351 (2006) 28–38.[2] M. Ghaly, R.S. Averback, Phys. Rev. Lett. 72 (1994) 364.[3] K. Nordlund, J. Keinonen, M. Ghaly, R.S. Averback, Nature 398 (1999) 48.[4] K. Nordlund et al., Nucl. Instrum. Methods Phys. Res. B 148 (1999) 74–82.[5] R.E. Stoller, S.G. Guiriec, J. Nucl. Mater. 329 (2004) 1238.[6] R.E. Stoller, J. Nucl. Mater. 307–311 (2002) 935.[7] S. Dudarev, P. Derlet, J. Phys. Condens. Matter. 17 (2005) 1–22.[8] C. Björkas, K. Nordlund, Nucl. Instrum. Methods Phys. Res. B 259 (2007) 853–
860.[9] L.-P. Zheng et al., Nucl. Instrum. Methods Phys. Res. B 268 (2010) 120–122.
[10] Z. Yao, M. Hernández Mayoral, M.L. Jenkins, M.A. Kirk, Philos. Mag. 88 (2008)2851.
[11] C. Björkas, (Private communication).[12] X.-M. Bai, A.F. Voter, R.G. Hoagland, M. Nastasi, B.P. Uberuaga, Science 327
(2010) 1631–1634.
II
Nuclear Instruments and Methods in Physics Research B 352 (2015) 217–220
Contents lists available at ScienceDirect
Nuclear Instruments and Methods in Physics Research B
journal homepage: www.elsevier .com/locate /n imb
Surface damage in TEM thick a-Fe samples by implantationwith 150 keV Fe ions
http://dx.doi.org/10.1016/j.nimb.2014.11.1110168-583X/ 2014 Elsevier B.V. All rights reserved.
⇑ Corresponding author.
M.J. Aliaga a,⇑, M.J. Caturla a, R. Schäublin b
a Dept. Física Aplicada, Facultad de Ciencias, Fase II, Universidad de Alicante, Alicante E-03690, Spainb Metal Physics and Technology, Department of Materials, ETH Zürich, HCI G515, Vladimir-Prelog-Weg 5, 8093 Zürich, Switzerland
a r t i c l e i n f o
Article history:Received 11 July 2014Received in revised form 21 October 2014Accepted 29 November 2014Available online 17 December 2014
Keywords:Molecular dynamicsDefectsIon irradiationSurface damageTransmission electron microscopy
a b s t r a c t
We have performed molecular dynamics simulations of implantation of 150 keV Fe ions in pure bcc Fe.The thickness of the simulation box is of the same order of those used in in situ TEM analysis of irradiatedmaterials. We assess the effect of the implantation angle and the presence of front and back surfaces. Thenumber and type of defects, ion range, cluster distribution and primary damage morphology are studied.Results indicate that, for the very thin samples used in in situ TEM irradiation experiments the presenceof surfaces affect dramatically the damage produced. At this particular energy, the ion has sufficientenergy to damage both the top and the back surfaces and still leave the sample through the bottom. Thisprovides new insights on the study of radiation damage using TEM in situ.
2014 Elsevier B.V. All rights reserved.
1. Introduction
Ion irradiation experiments are being used extensively tounderstand the fundamental aspects of the damage produced inmetals and alloys by irradiation [1]. In the current nuclear powerplants and experimental fusion reactors the damage is producedby neutrons. However, the study of neutron irradiation is difficultsince conditions cannot be easily controlled, samples are activatedand experiments are very costly. For this reason, ion irradiation isnowadays being used to gain basic understanding of the effects ofradiation in the structural materials of the reactors.
Iron is the main element of the reactor vessel, and although ithas been studied for many years, there are still many issues underdebate considering its radiation damage. Both neutrons and Fe ionsproduce damage in cascades, but their damage profile is very dif-ferent. Neutrons have a long range of penetration and producedamage quite homogeneously, whereas ion damage is more super-ficial. However, 100 keV Fe ions have been used in the last fewyears to approach indirectly the study of neutron irradiation inFe, since the first Fe atom that is hit by a neutron in a lattice (pri-mary knocked-on atom o PKA) has around 100 keV energy [2]. Theirradiated samples can then be examined by in situ TEM [3] infacilities such as Jannus at CEA in France [4] or the IVEM-TandemFacility at Argonne National Laboratory [5]. Using this character-ization technique the sample can be observed while it is being
irradiated. The requirement is that it has to be between 40 and100 nm thick to be transparent to electrons.
As it has been demonstrated in previous works [6–13] the pres-ence of surfaces in metals affect the damage produced in the mate-rial, being quite different from the damage in bulk. Earlier, it hasbeen observed that irradiation at room temperature of pure Fe inthe form of a transmission electron microscopy (TEM) thin filmleads to a0 h100i dislocation loops while in the bulk form irradi-ated Fe exhibits mainly ½a0 h111i [14]. This has been attributedto so-called elastic ‘image forces’ due to the free surfaces but neverquantified. This effect is confirmed in recent works [15–18]. In thispaper we continue our work from an atomistic point of view withthe study of the primary damage produced by Fe ions of 150 keV inthin films of pure Fe using molecular dynamics.
2. Methodology
Calculations were performed using the molecular dynamicscode MDCASK with the interatomic potential of Dudarev and Der-let [19]. This potential was modified for short range interactionsfollowing the procedure described in [20]. This potential wasselected since it reproduces fairly well the energetics of pointdefects compared to density functional theory calculations [20].It is also the only interatomic potential for Fe that includes anexplicit term for the magnetic contribution to the interatomicpotential energy. Displacement cascades were simulated sendingan Fe atom with an energy of 150 keV towards the top free surface
218 M.J. Aliaga et al. / Nuclear Instruments and Methods in Physics Research B 352 (2015) 217–220
of a [001] thin film of a-Fe. Two impinging angles have been used,10 and 22, being this second angle the one used in the TEM in situanalysis of ion irradiation experiments at Orsay JANNUS facility [4].20 cases were run for each angle. Variability was introducedchanging the azimuthal angle from 0 to 200. Simulation cellscontained 10,076,401 atoms with a size of 180a0 180a0 180a0,where a0 is the lattice parameter for Fe (a0 = 2.8665 Å). This sizecorresponds to thin films of about 50 nm. The setup described cor-responds to the energies used in the experiments for low doses ofYao et al. [15]. Temperature was kept close to 0 K in order to avoidthermal vibrations and thus facilitate the identification of defects.The excess energy deposited by the injected atom is dissipated byadding a thermal bath that scales the velocity of two atom layers atthe border of the simulation cell. Inelastic energy losses wereincluded by the Lindhard model [21], which introduces a frictionforce proportional to the velocity. This force was introduced onlyfor those atoms with a kinetic energy larger than 5 eV. Periodicboundary conditions are imposed in two axes, while free surfacesare considered in the third axis. Simulations were run until thenumber of defects reached a stable population (25 ps in mostcases). Wigner–Seitz cells were used to identify the defects. Thenthe defects are grouped in clusters considering that two defectsbelong to the same cluster when the distance between them isbetween first and second nearest neighbours.
3. Results
Fig. 1 shows the primary damage of three representative casesof the events found in the simulations after the irradiation with a150 keV Fe atom. The location of vacancies (light circles) andself-interstitials (dark circles) are shown for the three cases afterthe simulation had run for 25 ps. The arrow indicates approxi-mately the initial position of the energetic atom. Both surfacesand adatoms are also represented in the figure. In Fig. 1(a) the Fe
Fig. 1. (a–c) show three representative 150 keV cascades of Fe implantation in Fe after 2also represented. The arrow indicates the implanted Fe atom. In (a) and (c) the Fe atom imbottom surface damage produced in (c) is shown.
atom is launched with an angle of 10 impacting heavily anddepositing most of its energy near the top surface. The damage isdivided into 3 displacement subcascades. The first and most ener-getic one occurs at the surface, creating a large vacancy cluster of1070 vacancies, another vacancy cluster of 115 vacancies and493 adatoms and 764 sputtered atoms (not shown in the figure)above the surface. The other two subscascades are around the cen-tre of the simulation cell where the ion stops. The total number ofvacancies is 1557 and the total number of interstitials is 155. InFig. 1(b) the Fe atom is sent with an angle of 22 and travelsthrough the whole sample leaving the film at the bottom surfacewith 20% of its initial energy, so depositing 80% of the total150 keV energy. In this case both top and bottom surfaces are dam-aged, but the back surface more strongly, with the creation of a 196vacancy cluster. Finally Fig. 1(c) shows an event in which the atomimpinges with a 10 angle and again goes through the entire sam-ple, but in this case it stops just before it escapes the film due to astrong collision with the back surface. In this case the top surfacebarely suffers any damage, but in the bottom surface a huge craterof 3441 vacancies is created, as well as large islands of adatoms.Fig. 1(d) shows a close-up of the crater created at the bottom sur-face of the case represented in Fig. 1(c).
Fig. 2 represents the histograms of the total number of vacan-cies and interstitials that results from the 20 cases simulated foran impact angle of 10 (two of these cases shown in Fig. 2(a) and(c)) and the other 20 cases for the impact angle of 22 (one exam-ple in Fig. 2(b)). As already shown in previous works [6,12] thenumber of vacancies for cascades with near surfaces is greater thanthe number of self-interstitials. This is due to the attraction theself-interstitials suffer by the surface. Also, the dispersion of resultsis larger than in bulk cascades. The main difference with [6] for thesame energy is that the increase in the angle results in some events(3 for 10 and 2 for 22) with a huge number of vacancies. Two ofthese events for the impact angle of 10 are the ones represented inFig. 1(a) and (c). These type of events correspond to cases where
5 ps. Light circles are vacancies and darker circles are interstitials. Both surfaces arepact angle is 10 and in (b) the impact angle is 22 off normal. In (d) a close-up of the
Fig. 2. Defect distribution for a series of 20 simulations of cascades in thin films produced by irradiation with a 150 keV Fe atom impacting with an angle of (a) 10 and (b) 22of the normal.
Fig. 3. Statistical analysis of ion ranges for the 150 keV Fe atom impacting with an angle of (a) 10 and (b) 22 of the normal.
Table 1Average number of vacancies and interstitials and their cluster fractions after the 150 keV cascades. Large clusters contain more than 55 defects. The ion range is also shown.
Angle () Number ofvacancies
Number ofinterstitials
% V in clusters % V in largeclusters
% I in clusters % I in largeclusters
Ion range (nm)
10 132 73 43 12 24 0 4222 225 166 47 16 27 0 30
M.J. Aliaga et al. / Nuclear Instruments and Methods in Physics Research B 352 (2015) 217–220 219
the energetic Fe atom injected has a strong collision near the top orthe back surface.
The histograms of the ion ranges for both angles are presentedin Fig. 3. Comparing both angles it is clear, on one hand, that in themajority of cases the PKA has enough energy to escape the samplethrough the bottom surface, and, on the other hand, that anincrease in the impact angle from 10 to 22 leads to a reductionin the ion range, as expected. As with the number of defects, thereis a wide spread in the results for the different cases. The caseswhere the ion range is very short coincide usually with the caseswhere the Fe atom has collided very strongly with or near thetop surface. Indeed, in one of the two events for 22 with manyvacancies, 1441 in this case, the PKA is backscattered after collidingwith an atom near the top surface, creating a large h100i loop with
521 vacancies. After this, the bottom surface is also badly damagedby secondary subcascades. On average, the energy deposited for10 impacts is 67%, and 86% for 22.
Table 1 summarizes the mean values obtained from fitting theabove histograms for the number of vacancies, self-interstitialsand the ion range to either lognormal or Gaussian distributions.The percentage of vacancies and self-interstitials in clusters andin large clusters (more than 55 defects) are also shown. It can beseen on the table, as mentioned above, that the ion range for 22with a value of 30 nm is shorter than the value for 10 which is42 nm. The SRIM values [22] are 50 nm for the ion range at animpact angle of 10 and 47 nm for the ion range at an impact angleof 22. This is reasonable because SRIM assumes on one hand a ran-dom target, and on the other hand these events where the ion
Fig. 4. Histograms of number of vacancy (a) and interstitial (b) clusters of different sizes normalized to the number of cascades.
220 M.J. Aliaga et al. / Nuclear Instruments and Methods in Physics Research B 352 (2015) 217–220
interacts very energetically with the surfaces shortening its rangedo not happen.
The increase in the angle also has an effect, as expected, on themean value of vacancies and interstitials. The angle of 10 is stillslightly below the Lindhard critical angle for channeling [23] which,for 150 keV is 12 and this results in a lower number of defects as amean value. Fig. 4 represents the distribution of clusters normal-ized by the number of cascades for both angles. It can be seen a ten-dency to larger clusters of vacancies and interstitials from 10 to22, but this increase is not remarkable because at 10 there areevents where the back surface is profoundly damaged.
4. Conclusions
The primary damage produced in thin films of pure bcc Fe byirradiation with150 keV Fe ions using two different implantationangles has been studied. Results show that, differently from lowerenergies where only the top surface is quite affected, at this energyregime front and back surfaces can be damaged. When the ionimpacts strongly with or near one of the surfaces the creation oflarge vacancy clusters is observed. This is produced by the attrac-tion the self-interstitials suffers when they are close enough tothe surface. These results have important implications for higherenergies, because they indicate that at high irradiation energies,like the range of energies used mainly in the in situ TEM irradiationexperiments, the damage will be produced mostly at the back sur-face of the film. Moreover, these effects should be taken intoaccount in models that predict the latter evolution of damageand damage accumulation, such as kinetic Monte Carlo or rate the-ory calculations.
These calculations provide new insights on the study of radia-tion damage using TEM in situ irradiation experiments, providingthe fundamental background needed to use the data from TEM insitu experiments to understand damage in bulk specimens.
Acknowledgements
We would like to thank Drs. A. Prokodtseva, M. Hernández-Mayoral, Z. Yao and S. Dudarev for fruitful discussions. Simulations
were carried out in the computer cluster of the Dept. of AppliedPhysics at the UA, the HPC-FF supercomputer of the Jülich Super-computer Center, Germany, and the HELIOS supercomputer inJapan. MJA thanks the UA for support through an institutionalfellowship. This work was supported by the European FusionDevelopment Agreement (EFDA), the VII EC framework throughthe GETMAT and MATISSE projects, and the Generalitat ValencianaPROMETEO2012/011.
References
[1] J.L. Boutard, A. Alamo, R. Lindau, M. Rieth, C.R. Phys. 9 (2008) 287.[2] Hernández-Mayoral Mercedes, Estudio por Microscopía Electrónica de
Transmisión del efecto de la irradiación iónica y neutrónica en hierro puro yaleaciones modelo de los aceros de vasija de los reactors nucleares. Madrid,Centro de Investigaciones energéticas, Medioambientales y Tecnologicas(CIEMAT). Departamento de Tecnología, División de Materiales, 2007.
[3] Y. Matsukawa, S.J. Zinkle, Science 318 (2007) 959.[4] <http://jannus.in2p3.fr/spip.php?rubrique15>.[5] C.W. Allen, L.L. Funk, E.A. Ryan, Mater. Res. Soc. Symp. Proc. 396 (1996) 641.[6] M.J. Aliaga, A. Prokhodtseva, R. Schaeublin, M.J. Caturla, J. Nucl. Mater. 452
(2014) 453–456.[7] M. Ghaly, R.S. Averback, Phys. Rev. Lett. 72 (1994) 364.[8] K. Nordlund, J. Keinomen, M. Ghaly, R.S. Averback, Nature 398 (1999) 48.[9] K. Nordlund, J. Keinonen, M. Ghaly, R.S. Averback, Nucl. Instr. Meth. Phys. Res.
B 148 (1999) 74–82.[10] S.V. Starikov, Z. Insepov, J. Rest, Phys. Rev. B 84 (2011) 104109.[11] P.D. Lane, G.J. Galloway, R.J. Cole, M. Caffio, R. Schaub, G, J. Ackland Phys. Rev. B
85 (2012) 094111.[12] R.E. Stoller, J. Nucl. Mater. 307–311 (2002) 935.[13] R.E. Stoller, S.G. Guiriec, J. Nucl. Mater. 329 (2004) 1238.[14] B. Masters, Nature 200 (1963) 254.[15] Z. Yao, M. Hernández Mayoral, M.L. Jenkins, M.A. Kirk, Phil. Mag. 88 (2008)
2851.[16] M.H. Mayoral, Z. Yao, M.L. Jenkins, M.A. Kirk, Phil. Mag. 88 (2008) 2881.[17] A. Prokhodtseva, B. Décamps, R. Schäublin, J. Nucl. Methods 442 (2013) S786–
S789.[18] A. Prokhodtseva, B. Décamps, A. Ramar, R. Schäublin, Acta Mater. 61 (2013)
(2013) 6958–6971.[19] S. Dudarev, P. Derlet, J. Phys.: Condens. Matter 17 (2005) 1–22.[20] C. Björkas, K. Nordlund, Nucl. Instr. Meth. Phys. Res. B 259 (2007) 853–860.[21] J. Lindhard, M. Sharff, Phys. Rev. 124 (1961) 128.[22] J.F. Ziegler, J.P. Biersack, The Stopping and Range of Ions in Matter, SRIM-2003,
Ó1998, 1999 by IB77M co.[23] L.P. Zheng, Z.-Y. Zhu, Y. Li, F.O. Goodman, Nucl. Instr. Meth. Phys. Res. B 268
(2010) 120–122.
III
Acta Materialia 101 (2015) 22–30
Contents lists available at ScienceDirect
Acta Materialia
journal homepage: www.elsevier .com/locate /actamat
Surface-induced vacancy loops and damage dispersionin irradiated Fe thin films
http://dx.doi.org/10.1016/j.actamat.2015.08.0631359-6454/ 2015 Published by Elsevier Ltd. on behalf of Acta Materialia Inc.
⇑ Corresponding author.E-mail address: [email protected] (M.J. Caturla).
M.J. Aliaga a, R. Schäublin b, J.F. Löffler b, M.J. Caturla a,⇑a Dept. Física Aplicada, Facultad de Ciencias, Fase II, Universidad de Alicante, Alicante E-03690, Spainb Laboratory of Metal Physics and Technology, Department of Materials, ETH Zurich, 8093 Zurich, Switzerland
a r t i c l e i n f o a b s t r a c t
Article history:Received 20 May 2015Revised 21 July 2015Accepted 28 August 2015
Keywords:Molecular dynamics simulationsIon irradiationIn situ transmission electron microscopyDefectsMicrostructure
Transmission electron microscopy (TEM) in situ ion implantation is a convenient way to study radiationdamage, but it is biased by the proximity of the free surfaces of the electron transparent thin sample. Inthis work this bias was investigated by performing irradiation of Fe in thin foil and bulk form with ions ofenergies between 50 keV and 100 keV using molecular dynamics simulations. The damage resulting fromthe subsequent displacement cascades differs significantly between the two sample geometries. Themost remarkable difference is in the resulting h100i vacancy loops. Both their size and frequency aremuch greater in thin films, with loops reaching 4 nm in size. This is due to an imbalance between thenumber of vacancies and self-interstitials produced, since the faster self-interstitials can escape to thesurfaces and remain there as ad-atoms. In addition, the self-interstitial clusters are smaller for thin foilsand there is a larger dispersion of the induced damage in terms of defect number, defect clustering anddefect morphology. The study discusses the impact of these results on the study of radiation effectsduring in situ experiments.
2015 Published by Elsevier Ltd. on behalf of Acta Materialia Inc.
1. Introduction
Ion implantation is a common way to produce defects in mate-rials in a controlled manner, in terms of particle type and energy,irradiation dose, dose rate, and temperature. It is therefore a veryvaluable tool for investigating radiation effects in materials witha high degree of control of all involved variables [1]. The damageproduced by implantation can be characterized using transmissionelectron microscopy (TEM), which provides information about theirradiation-induced defects, their number density, size, and possi-bly their type for those defects that are larger than about 1 nm[2]. In some facilities, such as JANNuS at CEA in France [3] or theIVEM-Tandem Facility at Argonne National Laboratory [4], it is pos-sible to perform ion irradiation experiments in situ in a TEM, thusallowing observation while irradiating [5]. This technique is beingused to validate simulation models of radiation effects to predictthe damage produced by the 14 MeV neutrons expected in futurefusion reactors [1], with a particular focus on ferritic materialsbecause of their good resistance to irradiation relative, for instance,to austenitic stainless steels.
The difficulty of extrapolating the results of such in situ exper-iments to the effects of radiation in bulk materials was recognizedlong ago [6]. This difficulty arises because, to be able to use con-ventional TEM (CTEM) for the characterization of defects viadiffraction contrast, the sample must be thin enough to minimizeelectron absorption and inelastic scattering that blur the image,but thick enough to reduce the effect of the free surfaces, implyinga thickness of at least a few tens of nanometers but less than about80 nm [7]. Free surfaces have indeed an impact on the elastic fieldsof defects because of the so-called image forces, which alter theconfiguration of the defects, and their interaction and migration.Surfaces affect the irradiation-induced microstructure also simplybecause defects produced in their vicinity can annihilate there ifthe defects migrate to the free surfaces. It is known experimentallythat in the case of ferritic steels irradiation generates 1/2 h111i andh100i dislocation loops [8]. Because the activation energy formigration of 1/2 h111i loops along h111i directions is low, asshown both experimentally [9] and in computer simulations[10,11], they migrate rapidly and can thus easily meet a surfacewhere they will disappear. Note that if the foil normal is orientedclose to a h111i or h100i direction all 1/2 h111i loops can disap-pear at surfaces, while only half of them will escape if the orienta-tion of the foil is h110i [12]. The nature, vacancy or self-interstitial,of the loops observed after irradiation at low doses is controversialdue to the difficulty of analyzing them when they are small. For
M.J. Aliaga et al. / Acta Materialia 101 (2015) 22–30 23
larger sizes (beyond a few nanometers) obtained at higher doses,they have been identified as interstitial loops [13–16]. At low dosesbut with heavy ions of low energy, large h100i vacancy loopslocated close to the surface have also been identified [13,16].
The damage produced by energetic recoils in bulk bcc Fe hasbeen extensively studied by different research groups [17–20]using molecular dynamics (MD) simulations; for a review, seeRef. [21]. There is a good apparent understanding of the formationof interstitial clusters. For instance, recent simulations by Calderet al. [22] have shown how large interstitial clusters are producedin the early stages of the collision cascade in bulk Fe due to theinteraction between shock waves. Sand et al. [23] have also shownthat a scaling law can be obtained for the size of self-interstitialclusters as a function of recoil energy in bulk tungsten. However,much less is known regarding vacancy clustering and the effectof surfaces in bcc metals, which is relevant to thin foil irradiation.
The effect of free surfaces in fcc metals has been studied by sev-eral groups. The work performed in the 1990s by Ghaly and Aver-back [24–26] revealed how the presence of the surface can changethe morphology of the damage produced by a collision cascadewith respect to the bulk. Later, Nordlund et al. [25] showed thatin Cu and Ni new mechanisms of defect production occur whendamage is close to a surface. In bcc materials, recent MD simula-tions in bcc Mo revealed the formation of large vacancy loops justbelow the surface [27] and Osetsky et al. have described the evolu-tion of cascades close to the surface in bcc Fe [28].
In this work we focus on the differences between irradiation inFe thin film and bulk material which has not been studied in detailup to now. In particular we focus on the formation of vacancy clus-ters, and their orientation, size and frequency. For this comparisonwe study the early stage of the damage produced by recoils ofenergies between 50 keV and 100 keV in thin films and bulk sam-ples of bcc Fe at 0 K using MD simulations with 2 different empir-ical interatomic potentials. We assess and compare the number ofvacancies and interstitials, and the size and morphology of theirclusters immediately after the cool-down of the displacement cas-cade in the thin foil and bulk irradiation conditions, in order toidentify the impact of the free surfaces. We intend to understand,in particular, whether in Fe thin foils vacancy loops of sizes visiblein TEM could be formed directly in the cascade. We also discuss thevarious implications of extrapolating TEM in situ irradiation analy-sis to model radiation effects in bulk materials.
Table 1Irradiation conditions. The table shows the number of runs for each energy of the Feion in dependence of incidence angle, DD or AM interatomic potential, and thin filmor bulk sample geometry used. The sample thickness is 51.4 nm except for ⁄ and ⁄⁄,where it is 45.7 nm and 70 nm respectively.
Energy (keV) Incidence angle () Thin foilNumber ofcascadesperformed
BulkNumber ofcascadesperformed
DD AM DD AM
50 10 14⁄ 14 14 1422 17 30 14 14
100 10 20 20 – –22 20 30 – 14⁄⁄
2. Methodology
The parallel MD code MDCASK developed at LLNL [29] was usedfor the calculations. Two different types of interatomic potentialswere considered for comparison: one developed by Dudarev andDerlet [30] and one by Ackland et al. [31], cited respectively asDD and AM. Both potentials were modified for the high energy-short range interactions according to the procedure used in [32].These potentials were also used previously to study cascade dam-age in bulk samples [33]. They were selected because they repro-duce fairly well the stability of different point defects as obtainedby density functional theory (DFT) calculations. The cluster sizedistribution obtained with these potentials also seems to repro-duce experimental observations better than other potentials [33].
Simulations were performed in bcc Fe at constant volume. Twotypes of calculations were performed: one in a crystalline thin filmoriented along a h001i direction, and the other inside a crystal,quoted as ‘bulk’, with h001i axes. The crystal thin foil is con-structed with periodic boundary conditions along x and y axes,while both surfaces along the z direction were free. An energeticFe atom was launched from the outside of the crystal towardsthe surface with the selected implantation angle and energy. For
the bulk crystal periodic boundary conditions were applied in alldirections and one atom near the top of the simulation box wasselected as the high-energy atom, or primary knock-on atom(PKA), and was given the selected angle and energy.
In order to minimize defect migration and to focus on the effectof surfaces on defect production, the temperature in all simulationswas kept low, close to 0K. This was achieved by means of a thermalbath located at the border of the simulation box. The thermal bathconsisted of two atomic layers where the velocity of the atoms wasscaled every time step to correspond to the desired temperature of0 K. Calculations were followed for a period sufficient to reach aconstant value for the number of defects produced. In this waysimulated time amounts to about 20–40 ps depending on the case.Between 14 and 30 simulations were performed for each conditionstudied in order to obtain statistically significant results. The inci-dent angle, i.e., the angle of the incoming energetic ion withrespect to a line perpendicular to the surface (which would corre-spond to the polar angle in a spherical coordinate system) is keptconstant, while the azimuthal angle in that same coordinate sys-tem is varied via increasing it by 10 for each different case. Theinitial random distribution of velocities of the atoms in the simula-tion box is also different for each case. Two energies, 50 keV and100 keV, were considered, and two incidence angles, 10 and 22,for each energy. We should mention that the critical angle forchanneling according to the Lindhard expression [34] is 21 for50 keV and 15 for 100 keV ions. The simulation conditions aresummarized in Table 1. The simulation volume for most conditionswas 180 a0 180 a0 180 a0, where a0 = 2.8665 Å is the latticeparameter for the Fe potentials considered here, which corre-sponds to a cube of about 50 nm a side. This size is comparableto the thickness of the thin films that are used experimentally inTEM in situ irradiations [12,13].
The analysis of the resulting damage was first conducted usingthe Wigner–Seitz cell method, which gives the number of vacan-cies and self-interstitials in the crystal. Secondly, once pointdefects were identified, their clusters were established usinganother method, which considers that two defects belong to thesame cluster when the distance between them is between the firstand second nearest neighbor of the bcc Fe lattice. Finally, the radiusR of a loop with N defects was calculated according to the relation-
ship [35] R ¼ a0
ffiffiffiffiffiffiffiffiffiN
21=2p
q.
3. Cascade damage: defect type and size
We first present results for the thin foil geometry. Fig. 1 showsthe damage microstructure of the Fe thin film 25 ps after launchingthe energetic ion for two different runs (with different azimuthalangles) under the same condition, i.e. 0 K with a 50 keV Fe atomat an incidence angle of 10 and with the AM potential. Green
Fig. 1. Snapshot of the Fe thin film microstructure 25 ps after launching a 50 keV Fe ion at 10 incidence angle and at 0 K for two different azimuthal angles, 10 in (a) and 80in (b), and initial velocity distributions (a and b). Green/light spheres: vacancies, red/dark spheres: interstitials, yellow spheres: surface atoms. (For interpretation of thereferences to color in this figure legend, the reader is referred to the web version of this article.)
24 M.J. Aliaga et al. / Acta Materialia 101 (2015) 22–30
spheres mark the location of vacancies, while red spheres repre-sent self-interstitials. The arrow indicates the approximate initiallocation of the energetic atom. Surface atoms are also representedin the figure with yellow spheres. In Fig. 1(a) the damage consistsof isolated interstitials and vacancies, two neighboring relativelylarge clusters of 23 and 27 interstitials and two significantly largerclusters of 108 and 148 vacancies. The total number of vacancies inthis case is 448, while the total number of self-interstitials is 140.79% of the vacancies are in clusters and 57% of them are in clusterswith more than 55 defects, which corresponds to a loop of about1 nm in radius. 58% of the self-interstitials are in clusters but nocluster with more than 55 interstitials was found. As can beobserved in the figure, most of the damage is located within12 nm of the surface, while the displacement cascade reached amaximum depth of 20 nm. Note that there are 308 ad-atoms atthe free surface, as seen in Fig. 1(a). The number of missing self-interstitial atoms corresponds to these ad-atoms.
Fig. 1(b) exhibits a clear difference to Fig. 1(a) even though theinitial condition is the same, except for the azimuthal angle and theinitial random distribution of velocities of the atoms in the simula-tion box. The difference in Fig. 1(b) consists of the absence of largeinterstitial clusters and the larger size of the vacancy loops; there isindeed one loop with 317 vacancies, much larger than any of thosein Fig. 1(a). This is a trend we observed in the rest of the condi-tions: when a large vacancy loop is produced then only small inter-stitial clusters are obtained.
Fig. 2(a) shows the damage in an Fe thin film 20 ps after a dis-placement cascade induced by a 100 keV Fe ion with a 10 incidenceangle, with the AM potential. The total number of resulting vacan-cies is 662, while the total number of self-interstitials is 187. As inFig. 1, a significant number of atoms end up at the surface as ad-atoms (469), while 6 atoms are sputtered away. A large vacancy loopof 480 defects can be observed right below the surface. Fig. 2(b)shows a closer view of this large cluster. Its dimensions are approx-imately 9 a0 5 a0 14 a0, or 3 nm 1 nm 4 nm3. It takes theform of a rectangular parallelepiped with a (010) habit plane. Thisrectangular shape correlates well with the calculations by Gilbertet al. [36,37], showing that rectangular h100i vacancy loops areenergetically more stable than e.g. circular ones. Fig. 2(c) shows across section of the specimen through the vacancy cluster, confirm-ing the presence of a vacancy loop. The use of the RHFS rule gives a
Burgers vector b = a0[0–10] (Fig. 2(c)). Note that the presence ofh100i vacancy loops following displacement cascades in Fe wasalready shown by Soneda et al. [19] and Kapinos [38], but in bulkspecimens. They also indicated that their formation was rare, consti-tuting only 1% of all clusters.
We now present results for the bulk geometry. Fig. 3 shows thedamage in bulk Fe 30 ps after a 50 keV recoil with a 22 angle of inci-dence, with the AM potential. The differences in the damage distri-bution and configuration between the bulk and thin foil conditionsare clearly seen when compared with Fig. 1. Note that as expectedfor the bulk the number of vacancies is the same as the number ofself-interstitials, and equals 151. A cluster with 38 vacancies and arelatively large cluster above it with 37 interstitials are observed.
Fig. 4 shows the damage in bulk Fe 22 ps after a 100 keV recoilwith an angle of 22 and the AM potential for two different runs.There are differences: in the first case (Fig. 4(a)) small vacancyand self-interstitial clusters are observed; in the second case(Fig. 4(b)) slightly larger vacancy clusters are found, together witha few small self-interstitial clusters. We should note that, accord-ing to previous studies by Stoller and others [39], the breakdowninto sub-cascades for Fe occurs at around 20 keV. Both cases hereexhibit sub-cascade formation, visible with the different branchesin the cascade.
3.1. Statistical analysis
As mentioned above, several authors have already reportedsome large vacancy loops following displacement cascades in thebulk [19,33,38]. Our simulations indicate that the frequency of for-mation of these large vacancy loops is greater when damage is pro-duced in thin films, as seen in Figs. 1 and 2. In order to quantify thiseffect, we performed a statistical analysis of the data, focusing onthe impact of the two different interatomic potentials. The numberof vacancies and self-interstitials as well as their percentage inclusters were calculated for all conditions. Table 2 provides themean values of the number of point defects, Table 3 lists their clus-tering fraction, and Table 4 provides the ion range projected alongthe [001] direction (i.e. the one perpendicular to the surface) foreach condition. These values were obtained by fitting the differenthistograms to either a lognormal or a Gaussian distribution.
Fig. 2. (a) Snapshot of the Fe thin film microstructure 20 ps after a 100 keV cascade at 0 K and at 10 incidence angle after 20 ps. (b) Closeup of the h100i large vacancycluster. Green/light spheres: vacancies, red/dark spheres: interstitials, yellow spheres: surface atoms. (c) Cross section of the specimen through the h100i cluster showing allatoms with the RHFS circuit (in blue) used to identify its Burgers vector. (For interpretation of the references to color in this figure legend, the reader is referred to the webversion of this article.)
Fig. 3. Snapshot of the Fe bulk microstructure 30 ps after a 50 keV cascade at 0 Kand at 22 incidence angle. Green/light spheres: vacancies, red/dark spheres:interstitials. (For interpretation of the references to color in this figure legend, thereader is referred to the web version of this article.)
M.J. Aliaga et al. / Acta Materialia 101 (2015) 22–30 25
We first present the results for the thin film geometry. It can beseen in Table 2 that the number of self-interstitials is always lowerthan the number of vacancies. As already noted above, the missingself-interstitials correspond to atoms that have moved onto thesurface, remaining as ad-atoms, as seen in Figs. 1 and 2. The num-ber of atoms that are sputtered away is small. The maximumoccurs for the 50 keV Fe ion and 22 incidence with the AM poten-tial and constitutes 6% of the total number of interstitials. Table 2shows that when the energy increases from 50 keV to 100 keV thedamage exhibits an increase of between 50% and 200% for thenumber of vacancies and of between 10% and 110% for interstitials.
Fig. 5 presents the scatter in the data. It shows the histograms ofthe number of vacancies and self-interstitials obtained for the thin
foil. Results for the 50 keV Fe ion with an incidence angle of 10 aregiven in Fig. 5(a), and results for 100 keV and 22 incidence angle inFig. 5(c). These were obtained with the AM potential. Fig. 5(b) and (d) present the results for the DD potential for the 50 keVFe ion and 10 incidence and the 100 keV Fe ion and 22 incidence,respectively. Several common features emerge from all of thesecases.
It appears that the scatter in the number of defects from one runto another under the same irradiation condition is extensive. Someof the cases exhibit a significantly larger number of vacancies rel-ative to the average of 300–400 vacancies. For example, for thecondition of the 50 keV Fe ion, there is one case with close to3000 vacancies (Fig. 5(a)), while for 100 keV and the same poten-tial (Fig. 5(c)) one case has more than 1000 vacancies. Fig. 6 showsthe resulting damage configuration for one of these cases exhibit-ing a large number of vacancies: that with the AM potential, 50 keVcascade and 10 incidence. This damage is significantly differentfrom that shown in Figs. 1–3. Even though the damage appearsas a large vacancy cluster, it actually consists of a dislocated vol-ume of the crystal with a crystalline structure in the center, whichcorresponds to a short dislocation array similar to that obtained byGhaly and Averback [23]. We should point out that these cases arerare when using the AM potential but more frequent with the DDpotential. For 50 keV and 22 incidence, 60% of the vacancy clusterswith more than 20 vacancies correspond to surface damage in thecase of the DD potential, whereas this is only 7% for the AM poten-tial. As seen in Fig. 5(a)–(d), even if the extreme cases with a verylarge number of vacancies are not considered, the spread in thenumber of vacancies is still large, varying from 100 up to 600, i.e.there is still a variation of between 50% and 100%.
We now present the results for the bulk geometry. Note that theslight differences between the mean value of vacancies and self-interstitials (see Table 2) result from an inaccuracy in the methodof calculating defects when large clusters are formed. In bulk cas-cades the number of defects produced does not seem to depend onthe recoil direction. This applies to both interatomic potentials. Inthe case of the DD potential, the numbers of vacancies are 135 and129 for recoil angles of 10 and 22, respectively. The valuesobtained for the AM potential are slightly higher: 159 and 164
Fig. 4. Snapshot of the Fe bulk microstructure following 100 keV cascades at 0 K and at an incidence angle of 22 for two different azimuthal angles and initial velocitydistributions (a and b). Case (a) shows the formation of small clusters only, while larger vacancy type clusters appear in case (b). Green/light spheres: vacancies, red/darkspheres: interstitials. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Table 2Average number of vacancies/interstitials following the displacement cascade in bccFe at 0 K, depending on the irradiation and simulation condition.
Energy(keV)
Incidenceangle ()
Thin foil(number of vacancies/interstitials)
Bulk(Number of vacancies/interstitials)
DDpotential
AMpotential
DDpotential
AMpotential
50 10 173/135 190/172 135/125 159/16422 130/125 237/160 129/122 164/168
100 10 257/148 450/350 – –22 380/267 364/285 – 333/334
26 M.J. Aliaga et al. / Acta Materialia 101 (2015) 22–30
respectively for the same angles. These values are consistent withthose obtained by Björkas [33], with 130 Frenkel pairs with the AMpotential and 131 with the DD potential, with calculations per-formed for random recoil directions and 300 K. The larger numberof defects obtained in our work compared to the values of Björkaset al. could be due to the higher temperature used by the latter,because in this case more recombinations would be expected. For100 keV recoils the number of vacancies that we obtain is 333.Stoller [38] reported 330 Frenkel pairs for the same recoil energybut calculated using a different interatomic potential. It shouldbe noted that the scatter in the data, or the difference in the totalnumber of point defects between runs for the same condition, is atmost 5%, which is less than that observed in the thin film (Fig. 5).
As seen in Table 3, for the bulk material the percentage ofvacancies and interstitials in clusters of any size is very similardespite the different angles of incidence, and this is also true for
Table 3Percentage of vacancies/interstitials in clusters following the displacement cascade inbcc Fe at 0 K, depending on the irradiation and simulation condition.
Energy(keV)
Incidenceangle ()
Thin foil(% in clustersvacancies/interstitials)
Bulk(% in clustersvacancies/interstitials)
DDpotential
AMpotential
DDpotential
AMpotential
50 10 52/34 55/38 40/53 39/3222 68/42 62/38 38/52 45/43
100 10 67/44 42/37 – –22 53/35 54/38 – 46/99
the two potentials studied, with values between 38% and 45% ofvacancies in clusters and between 32% and 53% for self-interstitials in clusters. Note, however, that the fraction of intersti-tials in clusters for the DD potential is slightly larger than the frac-tion of vacancies in clusters. In the case of thin films, the fraction ofvacancies in clusters is much larger than the fraction of self-interstitials in clusters in all cases. In addition, the increase in inci-dent angle from 10 to 22 increases the clustering of vacancies by13–30%.
The examples of the damage produced in thin foils (Figs. 1 and2) and in the bulk (Figs. 3 and 4) visually indicate that in thin films,compared to the bulk, larger vacancy clusters and smaller self-interstitial clusters are produced. Fig. 7 shows the cluster size dis-tribution obtained for the 50 keV cascades and 22 incidence anglewith the AM potential for thin films and bulk samples. Fig. 7(a) and (b) give the size distributions of clusters of vacancies andinterstitials, respectively. They show clearly that thin film irradia-tion produces much larger vacancy clusters and slightly smallerinterstitial ones than bulk material irradiation, with an averagesize of 140 vacancies and 30 interstitials for thin foil and 44 vacan-cies and 40 interstitials for the bulk. The clusters in the thin foilreach sizes that are visible in TEM.
Table 4 gives the mean value of the range of the initial energeticatom for the different conditions, excluding channeling cases. Forthe 50 keV ion in the Fe thin film with an incidence angle of 10the mean value is 20 nm for the AM potential and 21 nm for theDD potential. Both have a standard deviation of 12 nm. The valueof the range obtained from the usual SRIM code [40] for this energyand angle is 18 nm. Note that SRIM does not account for the crys-talline structure of the target. For the 22 impact angle and the DDpotential the ion range increases by as much as 200% when their
Table 4Ion range, in nm, depending on the irradiation and simulation condition. Highchanneling cases are not included in the average.
Energy(keV)
Incidenceangle ()
Thin foilIon range in nm
BulkIon range in nm
DDpotential
AMpotential
DDpotential
AMpotential
50 10 21 20 10 1122 7 11 12 8
100 10 22 26 – –22 21 27 – 32
Fig. 5. Distribution of the number of point defects (vacancies and self-interstitials) in Fe thin film following (a) a 50 keV Fe ion, 10 incidence angle and the AM potential; (b)the same condition as in (a) but with the DD potential; (c) a 100 keV Fe ion in thin films, 22 incidence angle and the AM potential; and (d) the same condition as in (c) butwith the DD potential.
Fig. 6. (a) Microstructure resulting from a 50 keV cascade in Fe thin film at 0 K and at a 10 incidence angle, with the AM potential illustrating the extension of the surfacedamage into the thin film; (b) image of the same cascade, slightly tilted to show the damage directly at the surface. Green/light spheres: vacancies, red/dark spheres:interstitials, yellow spheres: surface atoms. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
M.J. Aliaga et al. / Acta Materialia 101 (2015) 22–30 27
energy is increased from 50 keV to 100 keV. As expected, the meanvalue of the ion range decreases when the incidence angleincreases. Fig. 8 shows the distribution of the ion range obtainedfor the 50 keV ion in the Fe thin film (Fig. 8(a) and (b)) and in
the bulk (Fig. 8(c) and (d)), for all cases. Fig. 8(a) corresponds toan incidence angle of 10 with both the AM potential and DDpotential. As for the number of point defects, there is a wide spreadin the values obtained, with some ions reaching only 5 nm while
Fig. 7. Cluster size distribution of (a) vacancies and (b) self-interstitials comparing bulk cascades with those of thin films for 50 keV and 22 incidence angle with the AMpotential.
Fig. 8. Distribution of ion ranges for the two interatomic potentials studied (AM and DD) and for (a) 50 keV cascades in thin films and 10 incidence angle, (b) 50 keV cascadesin thin films and 22 incidence angle, (c) 50 keV cascades in bulk and 10 incidence angle, and (d) 50 keV cascades in bulk and 22 incidence angle.
28 M.J. Aliaga et al. / Acta Materialia 101 (2015) 22–30
others cross the whole sample thickness. In some cases theimplanted ion is backscattered after colliding with the surface.Fig. 8(b), presenting the results for 50 keV ions at 22 incidenceangle, again shows extensive scatter in the ion range, in particularfor the AM potential. The ion range is on average 11 nm for the AMpotential, with a standard deviation of 8 nm, and 7 nm for the DDpotential, with a standard deviation of 2 nm.
Fig. 8(c) and (d) show the range distribution of the 50 keV Feion in bulk Fe for incidence angles of 10 and 22, respectively.The ion range distribution is narrower than for the thin film.
However, some channeling can be observed (particularly in thecase of 10 implantation), as expected from the Lindhard relation-ship [34], which leads to a greater ion range. As seen in Table 4,the mean values for the range obtained with the DD potentialare similar for the two incidences studied: 10 nm with astandard deviation of 5 nm at 10 incidence and 12 nm with astandard deviation of 8 nm at 22 incidence. Similar values areobtained with the AM potential: 11 nm with a standard deviationof 9 nm at 10 incidence and 8 nm with a standard deviation of3 nm at 22 incidence.
M.J. Aliaga et al. / Acta Materialia 101 (2015) 22–30 29
4. Comparison to experimental observations
The first observations of dislocation loops formed in iron byself-irradiation were performed by Masters [6] and further workwas done in the 1980s by Robertson et al. [41,42]. In ultrahigh-purity (UHP) Fe these loops are mostly of h100i type [12–16].For high doses (>2 1018 ions m2) they are considered to be ofinterstitial type [14,15]. However, the nature of the loops is diffi-cult to assess when they are smaller than 5 nm. Jenkins et al.[14] showed the formation of a0h100i vacancy loops in Fe closeto the surface after irradiation performed with Ni+, Ge+, Kr+, Xe+
and W+ and energies between 40 keV and 240 keV. More recently,Yao et al. [13] studied dislocation loops induced by 30 keV Ga+ ionsin Fe–11%Cr, which produced damage within 10 nm from the sur-face. They were able to determine that the loops (at least thoseclosest to the surface) are of vacancy type.
For comparison with this experimental evidence we simulated a30 keV Ga+ ion implanted in an a-Fe matrix. The AM interatomicpotential was used for Fe–Fe interactions, while the interactionbetween the Ga+ ion and the Fe atoms was calculated using a purerepulsive potential, the so-called Universal potential described in[43]. In this way the damage produced by this ion as it travelsthrough the lattice is described well. Fig. 9 shows the defects pro-duced by the 30 keV Ga ion after 17 ps. The total number of vacan-cies in this case is 799 and the total number of interstitials is 107. Alarge ad-atom island at the surface with 627 ad-atoms is seen. Theformation of a large h100i vacancy loop close to the surface is alsoclearly observed. It has 692 vacancies and is approximately 15a0 9 a0 21 a0, or 4 nm 3 nm 6 nm. This cluster is compara-ble to those observed experimentally by Yao et al. [13], who per-formed these experiments with Fe–11%Cr. As shown, however, inMD calculations by Malerba et al. [44] the damage produced inthe cascade in a-Fe and FeCr alloys is not significantly different.The only difference is in the self-interstitial loops, which in FeCralloys can be a mixture of Fe and Cr atoms.
We should point out that experiments by Robertson et al.[41,42] using low-energy Fe ions (50 keV and 100 keV) in Fe showyields for the formation of loops much lower than those found inthe simulations presented here. A probable cause for this discrep-ancy is the difference in time scales between simulations andexperiments. The simulations have been performed for tens ofpicoseconds. For longer time scales, it is, however, conceivable thatvacancy loops close to the surface are able to climb and disappearby recombination, while self-interstitial clusters may coalesce andform larger loops, resulting in a yield lower than the one obtained
Fig. 9. Microstructure resulting from a cascade in an Fe thin film
in simulations. On the other hand, one should also keep in mindthat for loops smaller than 1 nm the contrast in CTEM is reducedand the image size saturates (because of the diffraction-limitedresolution), making the observation of loops more difficult, orimpossible [45], which may also explain the lower yields observedin the experiment.
5. Conclusions
The calculations presented here show that the damage pro-duced by ion implantation in Fe thin film using 50 keV and100 keV ions is significantly different from that produced in bulkFe by recoils of the same energy. In thin films, results show the for-mation of h100i vacancy loops with sizes visible in the TEM. Thisresults from the imbalance in the number of vacancies with respectto self-interstitials, due to the trapping of the latter at the surfacewhere they remain as ad-atoms.
Statistical analysis reveals a large dispersion in the defects pro-duced: while bulk results present a narrow dispersion in terms ofthe total number of defects or the percentage of defects in clusters,in thin foil the total number of defects varies significantly from onecascade to another, as does the morphology of the damage pro-duced. Two types of structures were identified in the thin foil. Onthe one hand, there are those exhibiting small self-interstitial clus-ters and large vacancy clusters right below the surface in the formof large h100i. On the other hand, there are structures presenting anarray of dislocations and ad-atoms. The latter are, however, rare.Both the AM and DD potentials induce equivalent results in termsof damage, although morphologically the DD potential producesmore frequent surface damage of the type shown in Fig. 6.
A larger fraction of vacancies in clusters are found in thin filmsthan in the bulk, independently of the energy, angle of incidence orinteratomic potential used. These vacancy clusters are also larger.The inverse behavior is observed for self-interstitials: in mostcases, the fraction of self-interstitials in clusters is higher in thebulk and their sizes are larger than in the thin film. Increasingthe energy from 50 keV to 100 keV shows the formation of sub-cascades in the case of bulk irradiation, resulting in smaller self-interstitial clusters for the higher energy. Sub-cascade formationat 100 keV in the thin film can be seen only in some cases.
The formation of large h100i vacancy loops directly in the cas-cade revealed here for Fe thin film agrees well with experiments.However, the nature of these loops has been experimentally iden-tified only in the case of irradiation of Fe–11%Cr with Ga+ ions oflower energy (30 keV) [13], or irradiation of Fe with heavy ions
at 0 K induced by a 30 keV Ga ion at 22 incidence angle.
30 M.J. Aliaga et al. / Acta Materialia 101 (2015) 22–30
(40–240 keV) [16]. Our simulations show that these loops alsoform when irradiating with Fe ions of 50 keV and 100 keV. Theseloops are, however, smaller than those produced in the Ga irradia-tion experiment at lower energy, and are therefore presumablymore difficult to observe and analyze experimentally.
In previous works [33,35] it is shown that the initial damage inthe cascade together with defect mobilities define how damagewill grow with dose. This has consequences for the modeling ofirradiation effects. In the quest to develop models that are ableto describe neutron damage in the bulk, ion implantation experi-ments using thin films are often used for validation. Our resultsshow that one should carefully account for the effect of free sur-faces in these models.
Acknowledgments
We would like to thank Drs. A. Prokhodtseva, M. Hernández-Mayoral, Z. Yao and S. Dudarev for fruitful discussions. Simulationswere carried out using the computer cluster of the Dept. of AppliedPhysics at the UA, the HPC-FF supercomputer of the Jülich Super-computer Center (Germany) and the Helios supercomputer at Rok-kasho (Japan). MJA thanks the UA for support through aninstitutional fellowship. The research leading to these results ispartly funded by the European Atomic Energy Community’s (Eura-tom) Seventh Framework Programme FP7/2007–2013 under Grantagreement No. 604862 (MatISSE project) and in the framework ofthe EERA (European Energy Research Alliance) Joint Programmeon Nuclear Materials and the Generalitat Valenciana PROME-TEO2012/011. This work has been carried out within the frame-work of the EUROfusion Consortium and has received fundingfrom the Euratom research and training programme 2014–2018under Grant agreement No. 633053. The views and opinionsexpressed herein do not necessarily reflect those of the EuropeanCommission.
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IV
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Influence of free surfaces on
microstructure evolution of radiation
damage in Fe from molecular dynamics and object kinetic
Monte Carlo calculations
Maria J. Aliaga *,1
I. Dopico2, I. Martin-Bragado
2, Maria J. Caturla
,1
1 Dept. Física Aplicada, Facultad de Ciencias, Fase II, Universidad de Alicante, Alicante E-03690
2 IMDEA Materials Institute, C/Eric Kandel 2, 28906 Getafe, Madrid, Spain
Received ZZZ, revised ZZZ, accepted ZZZ
Published online ZZZ (Dates will be provided by the publisher.)
Keywords Monte Carlo, molecular dynamics, fusion materials, radiation damage
* Corresponding author: e-mail [email protected], Phone: +34 96590 3400, Fax: +34 965909726
The influence of surfaces on the evolution of damage of
irradiated Fe is studied using object kinetic Monte Carlo
with input from molecular dynamics simulations and ab
initio calculations. Two effects are analysed: the influ-
ence of traps and the initial distribution of damage in the
cascade. These simulations show that for a trap concen-
tration of around 100appm, there are no significant dif-
ferences between defect concentrations in bulk and thin
films. However, the initial distribution of defects plays an
important role not only on total defect concentration but
also on defect type, for the model used in this study.
20 n
m
34 nm
100 keVFe ion
22o
Free surface
Interstitials
Vacancies
Damage produced by a 100keV Fe ion impinging a Fe
thin film. Blue (dark) spheres are self-interstitials, red
(light) spheres are vacancies.
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R2
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1 Introduction
Ion implantation is used to understand the effect of
damage production in materials for applications in fusion
and fission energy production [1]. Unlike neutron irradia-
tion, ion implantation allows for better control of variables
such as irradiation temperature, dose, energy and, to some
extent, dose rate. However, the extrapolation of results
from ion implantation to neutron irradiation regarding de-
fect production and microstructure evolution is not
straightforward. Differences in dose rate or sample thick-
ness could affect significantly the evolution of the damage.
The influence of the surface on defect production, defect
distribution and damage evolution needs to be understood
in order to develop reliable models that can extrapolate the
results from ion implantation to neutron irradiation condi-
tions.
Atomistic models have proven to be an appropriate
technique to predict the evolution of the irradiation damage
in materials. In this context, we present here a study, using
a recently developed object kinetic Monte Carlo (OKMC)
code, of the effect of surfaces on defect evolution in irradi-
ated Fe. We consider, on one hand, the interplay between
trapping sites for defects and surfaces,. and on the other
hand the influence of the initial defect distribution of de-
fects. It is well known from the early 1990s, that damage
produced close to the surface by an energetic ion in f.c.c.
metals gives rise to defect structures that are significantly
different from those produced in the bulk [2]. Recent simu-
lations in Fe [3] and in Mo [4] have shown that this is also
the case in b.c.c. metals, where large vacancy loops close
to the surface have been identified. In order to study how
this initial damage distribution (during the first few picose-
conds) affects the long term defect evolution, we have used
two databases of cascades, one obtained in bulk Fe and an-
other one obtained in thin films with free surfaces [3].
OKMC simulations have been performed to calculate the
defect concentration, type and size as a function of dose for
different conditions: i) bulk cascades with periodic bound-
ary conditions, to consider the case of recoils within a bulk
sample, ii) bulk cascades in the presence of free surfaces,
which would correspond to implantation at high energies
such that the initial damage is not affected by the surfaces,
and iii) surface damage in a thin film, that takes inot ac-
count the low enery irradiation close to the surface. In sec-
tion 2 we describe the model for damage accumulation and
growth used in this work. Section 3 describes the results
for the different conditions studied which are finally dis-
cussed in section 4.
2 Model for damage accumulation
Values obtained from ab initio calculations [5] and molec-
ular dynamics simulations [6] are used for migration and
binding energies of vacancies and self-interstitials, as
shown in table 1. All mobile defects in table 1 are consid-
ered to move in three dimensions except for ½<111> loops
which move in one-dimension.
All clusters can grow by addition of other defects of the
same type. Recombination occurs between vacancy and in-
terstitial type of defects, whether isolated or in clusters. In
this particular model, the formation of <100> loops occurs
through the interaction between ½<111> loops, following
the atomistic simulations of Marian et. al [7] and, more re-
cently, Terentyev et al [8]. This assumption is not the only
one possible for considering the formation of <100> loops.
Recently it was proposed that <100> loops can be formed
from the nucleation of C15 clusters formed in the collision
cascade [9, 10]. However, the aim of this work is not to
discuss about the model for loop formation but about the
effect of surfaces on a particular microstructure evolution
model. For such a study we have selected the first model of
loop formation; that of coalescence of ½<111> loops. This
model considers that a <100> loop can be formed as long
as the two ½<111> loops interacting have similar sizes,
with a maximum difference of 5%. Once the <100> loop is
formed it can grow by adding new ½<111> loops, <100>
loops or small self-interstitial clusters.
Table 1 Migration energies (Em) and binding energies (Eb) con-
sidered for different defects in the OKMC model (n is the number
of defects in a cluster).
Defect type Em (eV) Eb (eV)
V 0.67*
V2 0.62* 0.3*
V3 0.35* 0.37*
V4 0.48* 0.62*
Vn>4 Immobile **
I 0.34*
I2 0.42* 0.8*
I3 0.43* 0.92*
I4 0.3* 1.64*
In>4 ½<111> 0.06+0.11/n1.6 **
In>4 <100> Immobile **
* [5] ** [6]
Self-interstitial clusters larger than 4 defects are considered
to be ½<111> loops and therefore move with a very low
migration energy barrier and in one-dimension. As a result,
unless some traps are considered in the matrix, these loops
quickly migrate to the surfaces and recombine leaving no
residual damage. This, although a desirable situation from
a radiation resistance point of view, it is not a realistic sce-
nario. In a real sample, there is always some amount of
impurities such as carbon. Therefore, it is necessary to in-
clude traps in the simulation, to consider the effect of dis-
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persed carbon. In this model ½<111> loops can be stopped
when they find a trap, distributed randomly in the sample,
with a binding energy of 1 eV. This trapped ½<111> loop
is then immobile and can grow by addition of other loops
or small self-interstitial clusters or monointerstitials.
The initial damage distribution produced by the energetic
particle is obtained from MD simulations [3]. We use two
databases of cascade damage: one for damage produced by
a recoil in the bulk and a second set of cascades produced
by launching an energetic Fe ion on a Fe thin film (of
about 50 nm thickness). In the first type of simulations pe-
riodic boundary conditions (PBC) were applied in all three
directions while in the second type PBC were applied only
in two directions and free surfaces are considered in the
third one. All simulations were performed with the intera-
tomic potential of Ackland and Mendelev [11].
The object kinetic Monte Carlo simulator MMonCa [13]
was used in this study. Unlike other OKMC codes, where
clusters are defined as an entity with a particular capture
radius and number of defects, where the individual defect
position is not considered.MMonCa tracks the position of
each individual defect within the cluster. The shape of the
cluster is defined as a property and determines how defects
are distributed within the cluster. Shapes can then be
spherical, two dimensional disks or amorphous. These
shapes can be selected according to experiments, when
known, or other simulations. As a consequence, there is
not a single capture radius associated to a cluster, but each
defect has its own capture radius. In order to select the
most appropriate value for this capture radius we have
compared the defect morphology as obtained from the MD
simulations with the one in the OKMC calculations just be-
fore any diffusion event occurs Figure 1(a) shows the dis-
tribution of vacancies and self-interstitials obtained from a
MD simulation of a 50 keV Fe recoil in Fe. Red (light)
dots correspond to the location of vacancies while blue
(dark) dots correspond to self-interstitials. The position of
these defects is calculated using a Wigner-Seitz cell algo-
rithm. Figure 1(b) shows the location of those vacancies
and self-interstitials as given by the OKMC code when a
capture radius of 0.4 nm is used. This capture radius has
been calibrated to agree with the number of monovacan-
cies and monointerstitials from the MD simulation (con-
sidering that two defects that are at first-nearest neighbours
distance belong to the same cluster). If a larger capture ra-
dius is used in the OKMC, immediate recombination be-
tween vacancies and self-interstitials occurs and the total
number of mono-defects is lower than in the MD results. If
a shorter capture radius is used, the number of defects
identified as monovacancies or monointerstitials is larger
than those used to identify clusters from the MD simula-
tions. Nevertheless, one must keep in mind that this criteri-
on for clustering of defects is somehow arbitrary, since
other cut-off distances could be used. An evaluation of the
damage distribution of defects from MD and OKMC, such
as those shown in figure 1, reveals a good agreement for
the 0.4 nm cut-offradius .
Figure 1 Defect distribution as obtained from MD simulations of
a 50keV Fe ion in Fe (a) and as initial conditions for the OKMC
calculations (b). Red (light) spheres represent the location of va-
cancies, blue (dark) spheres are self-interstitials.
3 Results
Using the model described above, we have studied the
evolution of the damage produced by energetic recoils in
Fe. We have analysed both the influence of surfaces on de-
fect concentration and defect size and the influence of the
initial defect distribution. As mentioned above, recent sim-
ulations of cascades in Fe have shown that damage pro-
duced by ion implantation with low energies (~100keV)
results in defect structures significantly different from
those produced by the same energy recoils but in the bulk
of the material [3]. One important difference is the for-
mation of large (> 1nm) vacancy clusters of <100> type
when the damage is very close to the surface, together with
smaller self-interstitial clusters as compared to bulk dam-
age. Those results, however, only consider the first few pi-
coseconds after the energy is transferred from the recoil to
the lattice. Here, we follow the evolution of those defects
produced in the cascade over longer times and under con-
tinuous irradiation with the use of the OKMC model.
Two data bases for cascade damage were used for the-
se calculations, both obtained with the same interactomic
potential [11]: one of 100keV recoils in bulk Fe and anoth-
er one of 100keV Fe ion implantation in Fe thin films [3].
In order to decouple the effect of surfaces from the effect
of the initial damage distribution we have performed three
types of calculations: (1) bulk cascades with periodic
boundary conditions, (2) bulk cascades in a thin-, and (3)
surface cascades in a thin-film. Thin films have a thickness
of 50 nm, similar to those used in in situ transmission elec-
tron microscopy (TEM) studies. The simulation box was
200 nm x 200 nm x 50 nm and cascades were located ran-
domly within this box with a dose rate of 8 x 1014
ions m-2
s-1
. The same calculation is performed introducing two
concentrations of traps in the lattice: 1 appm and 118 appm,
corresponding to a highly pure sample and a high concen-
tration of carbon sample. Self-interstitial defects bind to
4 Author, Author, and Author: Short title
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these traps with a binding energy of 1eV, which should
mimic the effect of having carbon in the sample [13].
Figure 2 shows the total concentration of visible clus-
ters as a function of dose for the case of 1 appm traps and
the three different simulations: bulk cascades with PBC,
bulk cascades in thin films and surface cascades in thin
films. Figure 1(a) represents the concentration of ½ <111>
loops while figure 1(b) is the total number of <100> loops.
Clusters are considered visible if they have more than 100
defects (~1nm radius). As expected, the total number of
clusters when PBC are imposed is higher than when we
consider a thin film. Also, visible clusters appear at lower
doses when PBC are used. This is due to the fast migration
of ½<111> loops to surfaces in the thin film, that lower the
total defect concentration. Notice that no sinks were in-
cluded in the simulations with PBC. Comparing the con-
centration of ½<111> loops for the two cases with free sur-
faces, bulk cascades and surface cascades, no significant
differences are observed, the total concentration is quite
similar in these two examples. However, there is an im-
portant difference regarding the formation of <100> loops.
When bulk cascades are considered almost no <100> loops
are formed but, when surface cascades are used, the con-
centration of <100> loops is comparable to that of ½<111>
loops. This is surprising at first since the size of the self-
interstitial clusters in surface cascades is smaller than in
the case of bulk cascades. Considering that the model that
we are using here for <100> loop formation is the recom-
bination of two ½ <111> loops, the reason for this differ-
ence has to be the higher probability of two small self-
interstitial loops finding each other before reaching the sur-
face or getting trapped by a carbon in the case of surface
cascades.
Figure 2 Concentration of visible (> 100 defects) self-interstitial
clusters as a function of irradiation dose for 1 appm concentration
of traps (a) ½<111> loops and (b) <100> loops. Three different
cases are considered: irradiation in bulk (squares), irradiation in a
thin film with bulk cascades (circles) and irradiation in a thin film
with surface cascades (triangles).
This, in fact, can be rationalized in terms of the distri-
bution of defects within the cascade in the case of bulk
damage or surface defects. Figure 3 shows one example of
a cascade in the bulk (figure 3(a)) and a surface cascade
(figure 3(b)). As it can be seen, bulk cascades are spread
over a longer range while surface cascades are more local-
ized, in this particular case confined to a region of only 34
x 20 x 26 nm. Therefore, there is a much higher probability
for two self-interstitial clusters to interact and form a
<100> loop before reaching the surface or a trap in the case
of surface cascades than in the case of bulk damage.
(a)
(b)70
nm
30 nm
20
nm
34 nm
Figure 3 Defect distribution as obtained from MD simulations
for 100 keV and 22o angle (a) Fe recoil in bulk Fe and (b) Fe ion
implanted in an Fe matrix. Red (light) spheres represent the loca-
tion of vacancies, blue (dark) spheres are self-interstitials.
Calculations were also performed for higher concentration
of trapping sites, 118 appm. Figure 4 shows the concentra-
tion of ½<111> loops (figure 4(a)) and <100> loops (fig-
ure 4(b)) for the three cases considered in this study. Now,
the concentration of ½ <111> loops is almost the same for
the three cases. That is, the effect of the surface is negligi-
ble since all loops were trapped before they can reach the
surface. However, the difference in the concentration of
<100> loops is still clear between bulk and surface cas-
cades, since this is an effect of cascade damage distribution
and therefore almost independent of trapping concentration.
Figure 4 Concentration of visible (> 100 defects) self-interstitial
clusters as a function of irradiation dose for 118 appm concentra-
tion of traps (a) ½<111> loops and (b) <100> loops. Three dif-
ferent cases are considered: irradiation in bulk (squares), irradia-
tion in a thin film with bulk cascades (circles) and irradiation in a
thin film with surface cascades (triangles).
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The different behaviour of <100> loops with respect to ½
<111> loops in this particular model are also observed in
the average size of these loops as a function of dose, pre-
sented in figure 5. Results are shown for the cases of thin
films with bulk cascades and with surface cascades. In the
case of surface cascades, for any given dose, the average
size of the clusters is larger than for the case of bulk cas-
cades. Loops of the ½ <111> type have a constant average
size for low doses but after a certain dose, which is lower
for surface cascades, the average size of these loops in-
creases rapidly with dose. For the case of <100> loops, the
average size remains constant for all doses studied in this
work. The reason for these differences is the nucleation
mechanism considered by the model. <100> loops grow
through the reaction between ½ <111> loops of similar
size, therefore, nucleation sites for these loops are con-
stantly forming as long as the material is being irradiated.
Since ½<111> loops are highly mobile, they can either mi-
grate to the surface, interact to form <100> loops or be-
come trapped. When all trapping sites are saturated, no
more nucleation sites for ½ <111> loops can be created
and the ones that are already trapped can rapidly grow by
the addition of new ½<111> loops. That gives rise to the
rapid increase of the average size of these loops at high
doses.
Figure 5 Average cluster size as a function of dose for the cases
of bulk cascade in thin films and surface cascades in thin films.
Values for ½<111> and <100> loops.
4 Conclusions This work shows the importance of
surfaces on the microstructure evolution of damage pro-
duced by irradiation. Surfaces act as sinks for defects, and,
as expected, lower the total concentration of defects com-
pared to bulk irradiation. This effect, however, will depend
strongly on the purity of the sample and the presence of
traps. More interestingly, if damage is produced very close
to the surface, the distribution of this damage differs from
that of bulk irradiation, resulting in, not only a different
concentration of defects, but also differences in the type of
damage that can be observed. In this particular model the
differences are mostly related to the ratio of ½<111> to
<100> loops. Irradiation close to the surface favours the
formation of <100> loops due to the localization of the
damage within the cascade.
Acknowledgements Simulations were carried out using the
computer cluster of the Dept. of Applied Physics at the UA, the
HPC-FF supercomputer of the Jülich Supercomputer Center
(Germany) and the Helios supercomputer at Rokkasho (Japan).
MJA thanks the UA for support through an institutional
fellowship. The research leading to these results is partly funded
by the European Atomic Energy Community’s (Euratom)
Seventh Framework Programme FP7/2007-2013 under grant
agreement No. 604862 (MatISSE project) ) and in the framework
of the EERA (European Energy Research Alliance) Joint
Programme on Nuclear Materials. This work has been carried out
within the framework of the EUROfusion Consortium and has
received funding from the Euratom research and training
programme 2014-2018 under grant agreement No 633053. The
views and opinions expressed herein do not necessarily reflect
those of the European Commission.
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Norman, V. V. Stegailov, A. V. Yanikin, Radiation-induced
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104109 (2011). [5]C. C. Fu, J. Dalla Torre, F. Willaime, J.-L.
Bouquet, A. Barbu, Nature Materials (2004)
[6] Soneda, N. & Díaz de la Rubia, T. Phil. Mag. A 78, 995-1019
(1998)
[7] J. Marian and Brian D. Wirth PRL 88, 25 (2002).
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Terentyev, PRL 110, 265503 (2013)..
[9] M.-C. Marinica, F. Williaime, J.-P. Crocombette, PRL 108,
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Bulent Biner, Scripta Materialia 98 (2015) 5-8.
[11] G.J. Ackland, M.I. Mendelev, D.J. Srolovitz, S. Han, A.V.
Barashev, Development of an interatomic potential for
phosphorus impurities in alpha-iron, J. Phys.: Condens. Matter 16
(2004) S2629.
[12] I. Martin-Bragado, Antonio Rivera, Gonzalo Valles, Jose
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5. TRABAJOS NO PUBLICADOS
V
epl draft
Surface effects and statistical laws of defects in primary radiationdamage: tungsten vs. iron
A. E. Sand1, M. J. Aliaga2, M. J. Caturla2 and K. Nordlund1
1 Department of Physics - P.O. Box 43, FI-00014 University of Helsinki, Finland2 Dept Fisica Aplicada - Facultad de Ciencias, Fase II, Universidad de Alicante, Alicante E-03690, Spain
PACS nn.mm.xx – First pacs descriptionPACS nn.mm.xx – Second pacs descriptionPACS nn.mm.xx – Third pacs description
Abstract – We have investigated the effect of surfaces on the statistics of the primary radiationdamage formed in the bcc metals iron (Fe) and tungsten (W). Through molecular dynamicssimulations of collision cascades we show that both interstitial and vacancy cluster sizes followscaling laws in these materials, in bulk as well as in thin foils. However, the slopes of the power lawdistributions in bulk Fe are markedly different from those in W, and furthermore the slope of thevacancy cluster size distribution in Fe is clearly affected by the surface in thin foil irradiation, whilein W mainly the overall frequency is affected. The distinct behaviour of the statistical distributionsuncovers different defect production mechanisms effective in the two materials, and provides insightinto the underlying reasons for the differing behaviour observed in TEM experiments of low-doseion irradiation in these metals.
Introduction. – One of the main challenges on theroad to commercial fusion power is presented by the needfor materials that can withstand the harsh conditions ina fusion reactor. Energetic fusion neutrons will cause sig-nificant damage to the wall materials of future reactors,leading to swelling, hardening and embrittlement. The de-velopment of materials that can withstand this irradiationand retain the structural integrity of the reactor requiresa thorough understanding of the radiation damage pro-cesses.
Two materials of prime interest in current reactor de-signs are iron (Fe), in steels for structural components, andtungsten (W) for plasma-facing components. These twometals, though both have bcc structure, exhibit markeddifferences in their response to radiation. While self-ionirradiation produces primary defects in W which are im-mediately visible in TEM experiments [1], in Fe nothingvisible is produced in either neutron or ion irradiation ex-periments until significant dose levels are reached [2–4].Nevertheless, indirect observations of low-dose radiationdamage using a combination of electron irradiation andneutron irradiation indicate that sub-microscopic defectclusters are initially formed also in Fe [4].
Molecular dynamics (MD) simulations confirm the for-mation of clusters directly from collision cascades in Fe
(see, e.g., [5–7]). In W, MD simulations have furthershown that the size-frequency distribution of interstitialclusters in bulk material follows a power law [8], a resultsupported by experiments [9]. The formation of clustersdirectly in cascades has a significant impact on the furtherevolution of the damage, and is therefore an importantfactor in microstructural evolution models.
While ion irradiation experiments serve as a usefulproxy for neutron irradiation, the close proximity of ma-terial surfaces in the former must be taken into account.The surface affects the evolution of the damage via imageforces, and by acting as a sink for defects, but also theinitial formation of defects is known to be affected by anearby surface [10]. As a result the accumulated damagein thin foils and bulk samples shows significant differences[11]. Surface effects are particularly important in the caseof in-situ TEM ion implantation experiments, since theirradiated sample must be less than 100 nm thick [12] tobe transparent to the electrons. They also play a majorrole in low energy (a few tens of keV) irradiation experi-ments, due to the shallow penetration depth of the inci-dent ions. In Fe, for example, vacancy loops have beenidentified close to the surface when irradiating with heavyions of low energy and at low doses [2, 13].
In this work, we investigate the effect of surfaces on the
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A. E. Sand et al.
statistics of the defects constituting the primary radiationdamage in Fe and W. We also consider the differences inthe formation of the damage underlying the dissimilarityin observations of defects in the two materials. We use MDsimulations to study the experimentally invisible defectsize range. With a statistical analysis of results, we areable to shed light also on events which may occur toorarely to be directly captured by MD, due to the limitednumber of simulations that can be performed, in contrastto the thousands of impacts which are recorded in typicalTEM experiments.
Simulation methodology. – We have simulated fullcollision cascades in W and α-Fe using molecular dynamicsmethods. Simulations of bulk cascades were performed us-ing periodic boundary conditions in all directions, and bychoosing the primary knock-on atom (PKA) from amongthe lattice atoms. Thin foil irradiation was simulated withperiodic boundaries in two directions, and free surfaces inthe z-direction. An incident ion was placed above the sur-face, and given the desired kinetic energy in a chosen angletowards the surface.
Cascades in W were simulated with the MD code PAR-CAS [14], using the interatomic potential by Derlet et al.[15], with the short range part fitted by Bjorkas et al. [16].Bulk simulations were performed in a cubic cell of 48 nmto a side. The direction of the PKA in the bulk was var-ied randomly, with a uniform distribution over the unitsphere. Foil simulations were performed in a cell with di-mensions 48×48×65 A, where the lattice was oriented togive a (014)-surface. The incident ion trajectory formeda 15 degree angle with the surface normal. This geom-etry corresponds to that used in recent in-situ TEM ex-periments [9]. Simulations in Fe were performed usingthe MD code MDCASK, with the interatomic potentialof Dudarev and Derlet [17], modified for short range in-teractions following the procedure described in [18]. Bulksimulations were performed in a cubic cell of 34 nm oneach side, where the polar and the azimuthal angles of thePKA were varied for the different cases. Thin film simu-lations were performed in a cubic cell of 40 nm to a side,oriented along a 〈001〉 direction. The incident angle in thiscase was 22 degrees, which corresponds to the geometryused in [11].
Electronic stopping Se in the form of a friction termwas included in the simulations in W, since recent resultsindicate an effect of the dynamic treatment of electronicenergy losses on the residual damage [19]. The frictionterm follows the Lindhard model [20], and is independentof position, with the magnitude determined by SRIM cal-culations [21]. In W simulations it was applied to all atomswith a kinetic energy larger than 10 eV [19]. In Fe, tradi-tionally no electronic energy losses have been included incollision cascade simulations [5–7,18], and here we presentresults using that same convention. The effect on the de-fect statistics of including or excluding Se was neverthelessinvestigated for chosen conditions in each material, and is
presented in the last part of the results section. The wayof introducing Se in the Fe simulations is similar to that inW, with the friction term applied to atoms with a kineticenergy larger than 5 eV.
In the ballistic scenario of the binary collision approxi-mation, the number of defects NNRT produced from cas-cades depends on the initial PKA energy EPKA, the elec-tronic energy losses Eel, and the threshold displacementenergy (TDE) Ed, according to the Norgett-Robinson-Torrens (NRT) formula [22]
NNRT =0.8(EPKA − Eel)
2Ed, (1)
where the term in parentheses equals the damage energyEdam, i.e. the energy available to the ionic system. Inorder to compare cascade simulations in different materi-als, and those performed with and without electronic en-ergy losses, it is therefore reasonable to consider them interms of the reduced damage energy Er = Edam/Ed [23].We calculate the reduced damage energy from the TDEpredicted by the interatomic potential, determined as theminimum energy needed to displace an atom in a givendirection Emind (θ, φ), averaged over all directions [24]
Eavd =
∫ 2π
0
∫ π0Emind (θ, φ) sin θ dθ dφ∫ 2π
0
∫ π0
sin θ dθ dφ. (2)
For the potentials used here, Eavd is 84.5 eV for W [16]and 35 eV for Fe [18].
Residual defects were identified using a Wigner-Seitzcell method which determines the location of vacanciesand self-interstitials in a crystal lattice. Defects werethen grouped into clusters: two vacancies were consid-ered to be in the same cluster if the distance betweenthem was within the 2nd nearest-neighbor distance, whilethe 3rd nearest-neighbor distance was assumed for self-interstitials. Size-frequency distributions of defect clus-ters were determined by binning the data on the numberof occurrences of each cluster size into roughly logarith-mic bins, taking care that the bin width was sufficient toinclude at least a few data points in each bin.
Results. –
Scaling laws in Fe and W. Our results show that inboth Fe and W, the frequency f(N) per ion of the occur-rence of defect clusters of size N closely follows power lawsof the form
f(N) =A
NS, (3)
where N is the size of the defect in terms of the number ofpoint defects included in the cluster, and A is a frequencyscaling factor, in agreement with earlier work in W [8].However, when considering defects down to the smallestsizes, including single point defects, we find that two powerlaws emerge in several cases.
The size-frequency distribution of single point defectsand smaller clusters of size N . 10 follow scaling laws with
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Surface effects and statistical laws of primary radiation damage
Fig. 1: (Color online) Comparison of scaling laws for SIA defectsizes in bulk and foil, W and Fe.
the same slope in Fe as in W. In addition, the frequencyof the defects is approximately the same in the two mate-rials, when comparing defects from cascades with similarreduced damage energy Er (see Figs 1 and 2).
The difference between Fe and W becomes apparent inthe distribution of the larger clusters. For bulk cascadesin Fe, larger clusters of both vacancy and interstitial typefollow the same scaling law as small clusters, while in bulkW, both vacancy and interstitial type clusters of size N &10 follow a scaling law with a lower value of S ≈ 1.6.Parameters for the best fit of the power laws are given inTable 1.
Surface effects. The effect of the surface on the dis-tribution of interstitial-type defects in both materials isminimal, but discernible as a slight preference for the for-mation of smaller defects, leading to a steeper slope inthe distributions. This preference arises from the portionof cascades which occur very close to the surface. Whenthe liquid core of the heat spike extends to the surface, itcauses the cascade to erupt, ejecting large amounts of ma-terial in the form of sputtered atoms and atom clusters.Such cascades form only very few and small interstitial-type clusters.
In the case of vacancy-type defects, the difference be-tween the bulk and foil cascade damage is clear, and es-pecially pronounced in Fe. Near-surface cascades readilyform large vacancy clusters, due to the ejection of material,and material flow to the surface causing an underdense re-gion to form in the core of the cascade. The size-frequecydistribution of these surface-induced vacancy clusters alsofollows a power law, with a slope that is roughly the sameas that for vacancy clusters in bulk W. Thus, in W, thesurface has the effect of simply increasing the overall fre-quency of vacancy-type defect clusters, maintaining thesame slope for the power law. In Fe, however, the surface
Fig. 2: (Color online) Vacancy cluster size distributions, fittedto power laws, for Fe and W in bulk and foil.
mechanism gives rise to a new scaling law for the largervacancy clusters, with S ≈ 1.6. In both materials, thesmallest vacancy-type defects still follow the same powerlaw as in bulk cascades, with S ≈ 3.0. Parameters forthe best fit of the power law to the distributions of defectclusters in thin foils are given in Table 2.
Electronic energy losses. We find that simulationswith and without electronic stopping in Fe result in thesame distributions for both vacancy- and interstitial-typedefects, as shown in Fig. 3 and Table 2. Only an over-all scaling of the frequency occurs due to the difference indamage energy with the two methods, from PKAs withthe same initial energy.
In W, however, the treatment of electronic energy lossesaffects the slope of the frequency-size distribution for bothinterstitial and vacancy clusters. Fig. 4 shows the distri-butions from 200 keV bulk cascades including Se in thesimulations, and for bulk cascades without Se with thesame total damage energy, Edam = 140 keV. The effect ofthe dynamic energy losses can be seen in the distributionof the larger clusters, which shows a decrease in the slopewhen Se is excluded (see Table 1). This effect is espe-cially apparent in the vacancy cluster distribution. Thedistributions of small clusters remains roughly the same.
Discussion. – The different scaling laws appearingin the defect distributions in W and Fe, and in bulk andthin foils, indicate the presence of different defect forma-tion mechanisms. The size-frequency distributions of thesmallest defects, in both bulk and foil cascades, follow thesame power laws in Fe and W. Furthermore, the frequencyof occurrence of these defects is similar for both materialsin simulations with similar reduced cascade energy, indi-cating a connection to the ballistic phase of the cascade.
In bulk W, a separate mechanism for the formation of
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A. E. Sand et al.
Material EPKA (keV) Ered (keV) type A S No. cascades noteW bulk 150 1.3 SIA (small) 24.1 ± 1.0 2.3 ± 0.1 38 Se ≈ 43 keVW bulk 150 1.3 SIA (large) 5.7 ± 1.6 1.6 ± 0.1 38 Se ≈ 43 keVW bulk 150 1.3 vac (small) 59.4 ± 5.2 2.9 ± 0.1 38 Se ≈ 43 keVW bulk 150 1.3 vac (large) 1.0 ± 0.2 1.5 ± 0.1 38 Se ≈ 43 keVW bulk 200 1.7 SIA (small) 26.9 ± 1.6 2.2 ± 0.1 10 Se ≈ 60 keVW bulk 200 1.7 SIA (large) 9.3 ± 2.4 1.5 ± 0.1 10 Se ≈ 60 keVW bulk 200 1.7 vac (small) 149.3 ± 3.8 3.0 ± 0.05 10 Se ≈ 60 keVW bulk 200 1.7 vac (large) 23.4 ± 8.3 2.0 ± 0.2 10 Se ≈ 60 keVW bulk 140 1.7 SIA (small) 27.3 ± 2.3 2.5 ± 0.2 5 no SeW bulk 140 1.7 SIA (large) 2.0 ± 0.2 1.1 ± 0.03 5 no SeW bulk 140 1.7 vac (small) 152.0 ± 6.0 3.0 ± 0.1 5 no SeW bulk 140 1.7 vac (large) 0.6 ± 0.1 1.1 ± 0.1 5 no SeFe bulk 50 1.4 SIA (all) 53.1 ± 1.3 2.6 ± 0.2 18 no SeFe bulk 50 1.4 vac (all) 58.2 ± 1.2 2.7 ± 0.1 18 no Se
Table 1: Power law parameters for defects in bulk cascades.
Material EPKA (keV) type A S No. cascades noteW foil 150 SIA (small) 25.7 ± 2.0 2.4 ± 0.1 49W foil 150 SIA (large) 9.9 ± 1.5 1.8 ± 0.1 49W foil 150 vac (small) 135.1 ± 11.4 3.0 ± 0.1 49W foil 150 vac (large) 6.7 ± 1.0 1.7 ± 0.1 49Fe foil 50 SIA (all) 58.1 ± 1.2 2.8 ± 0.1 20Fe foil 50 vac (small) 104.7 ± 1.1 3.1 ± 0.1 20Fe foil 50 vac (large) 8.7 ± 1.3 1.7 ± 0.1 20Fe foil 100 SIA (all) 56.0 ± 1.4 2.3 ± 0.1 20Fe foil 100 vac (small) 153.8 ± 1.3 3.0 ± 0.2 20Fe foil 100 vac (large) 8.3 ± 2.4 1.7 ± 0.2 20Fe foil 100 SIA (all) 85.5 ± 1.2 2.9 ± 0.1 20 Se ≈ 33 keVFe foil 100 vac (small) 122.4 ± 1.3 3.0 ± 0.2 20 Se ≈ 33 keVFe foil 100 vac (large) 4.1 ± 2.8 1.6 ± 0.2 20 Se ≈ 33 keV
Table 2: Power law parameters for defects in foil cascades.
large defect clusters is apparent, which is absent in Fe.This formation mechanism is likely related to the energydensity of cascades, which is higher in W than in Fe, dueto the lower mass and lower subcascade splitting thresholdof the latter. A dependence on energy density is furtherdemonstrated by the sensitivity of the scaling law to themethod of treating electronic energy losses in W. The dif-ference is especially apparent in the vacancy cluster distri-bution, and cannot be ascribed to different cooling rates,since the Se energy losses take place exclusively during theinitial ballistic phase of the cascade [19], and thus do notaffect the rate of cooling of the heat spike. In fact the sizeof the liquid, in terms of the number of atoms with ener-gies exceeding the melting point, evolves similarly in bothcases. Rather, it is likely that the initially higher energy ofthe PKA and subsequent recoils in simulations where Se isincluded results in an increased probability for the energyto be deposited in a more wide spread region, as com-pared to simulations which do not include Se but instead
initiate the PKA with a kinetic energy corresponding toonly the damage energy. In total the energy deposited inthe ionic system is the same with the two methods, butthe higher likelyhood for compact energy deposition whenSe is excluded translates into an increase in large defects,and thus a decrease in the slope of the scaling law. Amechanism of defect formation depending on the cascadeenergy density is in agreement with experimental obser-vations [25] as well as MD simulations [6], showing thatlarger defects are formed from heavier projectiles, whichdeposit their kinetic energy in a more compact region.
A third mechanism for vacancy defect production occursin near-surface cascades, and involves flow of material tothe surface, leaving large underdense regions in the cas-cade core. This mechanism has been reported in previousstudies [10], and is present in both W and Fe. The effect ofthis surface-induced mechanism is especially pronouncedin Fe, since it introduces a different distribution, as com-pared to bulk, for large vacancy-type defects.
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Surface effects and statistical laws of primary radiation damage
Fig. 3: (Color online) Scaling laws for vacancy (left) and SIA (right) defect sizes in Fe with and without electronic stopping.
The slope of the scaling law for large vacancy-type de-fects formed by the latter two mechanisms is the same.The size-frequency distribution is thus likely a result ofthe recrystallization processes taking place in the core ofthe heat spike, once the conditions for a depleted zonehave been met by the removal of material.
The different defect formation mechanisms present in Wand Fe means that cascade simulations respond differentlyto electronic energy losses. On the one hand, the effect ofelectronic energy losses that we observed in W indicatesthe importance of including electronic stopping in thesesimulations. In Fe, on the other hand, the standard prac-tice of excluding electronic energy losses is supported byour results, which indicate that the main factor affectingdamage production in Fe is the total damage energy, withlittle effect of the dynamics of energy removal.
The scaling laws found in this work show that no defectclusters large enough to be seen in TEM are likely to formin Fe directly from collision cascades in bulk. Thus visi-ble defects in bulk samples have likely formed as a resultof the thermal evolution of the invisible primary damage.In thin foil irradiation, however, the flow of material tothe surface in a heat spike causes the in-cascade forma-tion of large vacancy clusters. In MD simulations of cas-cades in bcc metals, such as α-Fe and W, SIA-type defectsgenerally cluster in 2-dimensional configurations, in otherwords as dislocation loops, while vacancies mainly form3-dimensional clusters. Such vacancy clusters are oftennot perfect voids, but rather form depleted zones, whichhave been directly observed in W [26,27] as a result of ionirradiation. The large vacancy clusters in Fe observed inour simulations may nevertheless become visible in TEMmicrographs after collapse due to cascade overlap, as hasbeen speculated in the literature (see, e.g., [25]). Corre-sponding large SIA defects do not form from this process,and thus SIA defects in Fe large enough to be visible musthave formed from coalescense and aggregation of smallerdefects.
On the other hand, in W, the slope of the scaling law in
the limit of large defect sizes gives a fairly large probabilityof in-cascade formation of visible defects also in the bulk,of both SIA and vacancy type. Since the proximity of thesurface in W foil irradiation also gives rise to the sameprocess as that present in Fe, which is reponsible for thecreation of additional large vacancy-type defects, the fre-quency of vacancy-type defects in foil irradiation is higherthan that in bulk, while SIA defects are formed with sim-ilar frequency as that in bulk. However, the formation oflarge SIA defects happens only in cascades which do noterupt through the surface, and thus only from ions thathave penetrated deeply into the sample. Thus small SIAdefect clusters are favoured in foil irradiation, since onlya percentage of ions penetrate past the surface to producelarge clusters, while all cascades produce small defects.
Conclusions. – We have shown that the size-frequency distribution of defects in the primary damageof both Fe and W follows power laws. A mechanism forthe production of large SIA and vacancy clusters depend-ing on the energy density of cascades is effective in self-ionor neutron irradiation in bulk W, but not in Fe. The ef-fect of nearby surfaces is the same in the two materials,but due to the different bulk behaviour, the impact of thesurface on statistics in Fe is more evident. The surfaceaffects the formation of large vacancy clusters, while thedistribution of single vacancies and small clusters remainslargely unaffected. The formation of SIA clusters is onlyslightly affected by the surface, with a preference for smallclusters in foil irradiation of both W and Fe.
∗ ∗ ∗
The authors thank Sergei Dudarev for valuable dis-cussions. This work has been carried out within theframework of the EUROfusion Consortium and has re-ceived funding from the Euratom research and trainingprogramme 2014-2018 under grant agreement No 633053.The views and opinions expressed herein do not necessarilyreflect those of the European Commission. MJA thanks
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A. E. Sand et al.
Fig. 4: (Color online) Scaling laws for SIA (top) and vacancy(bottom) defect sizes in W with different treatment of elec-tronic stopping, from cascades with damage energy Edam =140 keV.
the UA for support through an institutional fellowship.Simulations were carried out using the computer clusterof the Dept. of Applied Physics at the UA, the HPC-FFsupercomputer of the Jlich Supercomputer Center (Ger-many), the Helios supercomputer at Rokkasho (Japan)and the supercomputers at CSC - IT Center for Science(Finland).
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[2] English C. A., Eyre B. L. and Jenkins M. L., Nature,263 (1976) 400401 10.1038/263400a0.
[3] Robertson I. M., King W. E. and Kirk M. A., ScriptaMetall., 18 (1984) 317.
[4] Yoshiie T., Satoh Y., Taoka H., Kojima S. and Kir-itani M., Journal of Nuclear Materials, 155157, Part 2(1988) 1098 .
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[6] Calder A., Bacon D., Barashev A. and Osetsky Y.,Phil. Mag., 90 (2010) 863.
[7] Terentyev D., Lagerstedt C., Olsson P., Nord-lund K., Wallenius J., Becquart C. and MalerbaL., J. Nucl. Mater., 351 (2006) 65 .
[8] Sand A. E., Dudarev S. L. and Nordlund K., EPL,103 (2013) 46003.
[9] Yi X., Sand A. E., Mason D. R., Kirk M. A.,Roberts S. G., Nordlund K. and Dudarev S. L.,EPL, 110 (2015) 36001.
[10] Ghaly M., Nordlund K. and Averback R. S., Phil.Mag. A, 79 (1999) 795.
[11] Prokhodtseva A., Dcamps B. and SchublinR., Journal of Nuclear Materials, 442 (2013)S786 FIFTEENTH INTERNATIONALCONFERENCE ON FUSION REACTORMATERIALS.
[12] Stobbs W. and Sworn C., Philos. Mag., 24 (1971) 1365.[13] Yao Z., Mayoral M. H., Jenkins M. and Kirk M.,
Philos. Mag., 88 (2008) 2851.[14] Nordlund K., parcas computer code. The main princi-
ples of the molecular dynamics algorithms are presentedin [10, 28]. The adaptive time step is the same as in [29](2006).
[15] Derlet P. M., Nguyen-Manh D. and Dudarev S. L.,Phys. Rev. B, 76 (2007) 054107.
[16] Bjorkas C., Nordlund K. and Dudarev S. L., Nucl.Instr. Meth. B, 267 (2009) 3204.
[17] L. D. S. and P. D., J. Phys. Condens. Matter., 17 (2005)1.
[18] Bjorkas C. and Nordlund K., Nucl. Instr. and Meth.B, 259 (2007) 853.
[19] Sand A., Nordlund K. and Dudarev S., J. Nucl.Mater., 455 (2014) 207 proceedings of the 16th Interna-tional Conference on Fusion Reactor Materials (ICFRM-16)Proceedings of the 16th International Conference onFusion Reactor Materials (ICFRM-16), Beijing, China,20th - 26th October, 2013.
[20] Lindhard J. and Sharff M., Phys. Rev., 124 (1961)124.
[21] Ziegler J. F., SRIM-2008.04 software package, availableonline at http://www.srim.org. (2008).
[22] Norgett M. J., Robinson M. T. and Torrens I. M.,Nucl. Eng. Des., 33 (1975) 50.
[23] Setyawan W., Selby A. P., Juslin N., Stoller R. E.,Wirth B. D. and Kurtz R. J., Journal of Physics: Con-densed Matter, 27 (2015) 225402.
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Surface effects and statistical laws of primary radiation damage
[28] Nordlund K., Ghaly M., Averback R. S., CaturlaM., Diaz de la Rubia T. and Tarus J., Phys. Rev. B,57 (1998) 7556.
[29] Nordlund K., Computational Materials Science, 3(1995) 448 .
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VI
OKMC study of differences between MD and BCA cascades in neutronirradiated Fe simulations
S. Garcıa-Gonzaleza, A. Riverab, M.J. Aliagac, M.J. Caturlac, I. Martin-Bragadoa
aIMDEA Materials Institute, c/ Eric Kandel 2, 28906 Getafe, Madrid, SpainbInstituto de Fusion Nuclear, Universidad Politecnica de Madrid, Spain
cDept. Fısica Aplicada, Universidad de Alicante, Spain
Abstract
In this work we compare the evolution of neutron damage cascades generated by Molecular Dynamics (MD) andBinary Collision Approximation (BCA) techniques. Differences and similarities are discussed.
Keywords: OKMC, damage generation, BCA, MD
1. Introduction
One of the purposes of this work is to compare theevolution of neutron damage cascades generated byMolecular Dynamics (MD) and Binary Collision Ap-proximation (BCA) techniques.
2. Simulation methods
2.1. OKMCIn this study, radiation damage evolution has been
simulated using MMonCa [1, 2]– an Object KineticMonte Carlo code capable of modeling the atomistic-scale evolution of a system by assuming, essentially,that the transition rate between two different states (ri j)is independent of time and determined as following:
ri j = Pi j · e−Ei j/kBT , (1)
where Pi j and Ei j represent, according to the HarmonicTransition State Theory [3], the prefactor and the activa-tion barrier energy of the transition, that is computed asthe difference in formation energies between final andinitial states plus a barrier energy (typically a migrationenergy):
Ei j = E fj − E f
i + Ebi j. (2)
The OKMC algorithm is based on calculate the cu-mulative functions given by
Ri =
i∑j=1
ri j i = 1, ...,N, (3)
Email address: [email protected] (I.Martin-Bragado)
being N the total number of transitions in the system.After that, two random numbers, r and s, are computedin the interval (0,1] and the event that complies thatRi−1 < rRN ≤ Ri is performed. Lastly, the total sim-ulated time is increased by
∆t =ln(1/s)
RN(4)
and the affected transitions rates are recalculated tocompute the new cumulative functions and repeat thatprocess until the total simulated time has been reached.
2.2. Damage cascades: MD and BCA50 keV and 100 keV PKA (primary knock-on atoms)
damage cascades have been obtained as described be-low.
2.2.1. MDMD cascades have been provided by M. J. Caturla
and M. J. Aliaga and were performed using the molec-ular dynamics code MDCASK, developed at LawrenceLivermore National Laboratory [4], with the interatomicpotential developed by Ackland et al. [5] for α-Fe.This potential was modified for short range interactionsto connect to the Universal potential as described inRef. [6]. Cell dimensions for the 50 keV cascades are180 a0×180 a0×180 a0 and 250 a0×250 a0×250 a0 forthe 100 keV cascades, where a0 is the lattice parameterfor Fe (a0 = 2.8665 Å). Periodic boundary conditionswere used in all directions.
For the identification of defects, vacancies and in-terstitials are quantified and located using Wigner-Seitz
Preprint submitted to Journal of Nuclear Materials May 1, 2016
cells centered in each (perfect) lattice position so that anempty cell corresponds to a vacant and a double occu-pied cell corresponds to an interstitial defect.
2.2.2. BCABCA cascades have been simulated with
SRIM/TRIM [7, 8] considering a displacementenergy, different from the default value (20 eV), of 40eV [9]. This code only computes vacancy coordinates ingenerated cascades, thus corresponding self-interstitialdefects must be added considering that the probabilityof its final position is determined by a Gaussian distri-bution whose range and struggling depend on the recoilenergy. Hence other TRIM simulations are needed todetermine the values that characterize this distribution.Finally, each self-interstitial position is obtained con-sidering the aforesaid distance and a direction randomlygenerated from the vacancy position.
3. Model used in Iron
Depending on the material, different types of defectsare defined in MMonCa. Moreover, allowed migration,annihilation, clustering and emission of each type of de-fect and how they occur are included in the code.
In the particular case of Fe, the current model consid-ers the following defects /table 1): vacancies (V) andself-interstitial atoms (I), irregular interstitial vacancyand atom clusters and <100> and <111> interstitialloop clusters. Because neither ions from radiation norimpurities are dealt with, no other kind of defects is in-cluded. The energy values considered in the model arereferenced in table 2.
Defect Max. size Migration Transform toI & V – Yes –Iclust 9 Yes <111>clust (size>5)Vclust 500 size<5 –
<111>clust 500 Yes Iclust (size<5)<100>clust 500 No –
Table 1: Considered defects and its allowed maximum sizes, migra-tions and transformations.
The reactions between defects are defined as follows:if two same-type clusters (or point defects) react, the re-sult is a same-type cluster except when two <111> clus-ters with similar size interact, those produce a <100>cluster. If two different-type clusters react, generally,the cluster final type is the same as the largest one, butthere are some exceptions: if a vacancy cluster and anatom cluster with the same size react, they annihilate
independent of the interstitial cluster type; if an irreg-ular interstitial cluster reacts with a <111> cluster ora <100> cluster, the final cluster type is each one ofthese respectively independent of the cluster sizes; ifa <100> cluster interacts with a <111> cluster with asimilar size, a <100> cluster is produced.
Concerning interfaces, they are considered as idealsinks where every mobile defect will disappear if anyreach it.
I defects V defectsEI
m = 0.34 eV EVm = 0.67 eV
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .EI2
b = 0.80 eV EV2b = 0.30 eV
EI3b = 0.92 eV EV3
b = 0.37 eV
EI4b = 1.64 eV EV4
b = 0.62 eV. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
n > 4 : E(I|V)nb = EI|V
f +n
23 − (n − 1)
23
223 − 1
(E(I|V)2
b − EI|Vf
)Table 2: Iron model parametrization: migration and binding energyvalues (Em and Eb) and formula utilized to generate binding energiesfor clusters formed by more than four I or V.
4. Simulations set-up
In order to compare MD and and BCA cascades,OKMC simulations have been carried out introducingneutron cascades into a 300 × 300 × 300 nm3 box. Thevalue of the neutron main free path in Fe is ∼12 cm[REFERENCE], in consequence, neutron-caused dam-age takes place in the bulk. To simulate that effect, pe-riodic boundary conditions have been imposed to everyface of the simulation box and the cascades have beenrandomly introduced into this volume. Due to that mainfree path value, it is necessary to overestimate the neu-tron flux coming from a hypothetical fusion reactor. Ifnot, damage cascades would be introduced into the sim-ulation volume every too long time so nothing wouldoccurs during the main part of the simulation and itscomputational time would increase excessively. Thesesimulations have been done considering an average flu-ence rate of 2.5 neutron-cascades/(300 × 300 nm2· s).Depending on the used method to generate the damagecascades and its energies, different amount of defects isintroduced (Table 3). Distinct type of damage simula-tions have been done at temperatures in the range of 0 Kto 800 K considering a dose of 50 cascades introducedduring an average time of 20 s i. e. a dose of 5.556·1010
cm−2.The simulation characteristics of the second part of
this work partially differ from the previous ones. To
2
MD BCA50 keV 100 keV 50 keV 100 keV∼170 ∼340 ∼660 ∼1260
Table 3: Aproximated number of Frenkel pairs in damage cascadesdepending on method and PKA energy.
compare neutron with emulated proton damage, inter-faces have been added to all boundaries so defects canbe annihilated there modeling an ideal material grain. Itmakes possible to reach higher simulated volumes sim-ilar to iron grain sizes. In particular, a 500-nm-side cu-bic simulation box is considered. The fluence rate is thesame as the aforesaid but, in this case, the physical sim-ulated time is 50 s. Temperatures in the range of 0 K to800 K are also considered.
5. Results and discussion
5.1. BCA versus MD cascadesThe distribution of the defects depends on the method
used to calculated cascades. As expected, the numberof Frenkel pairs generated increases with PKA energy –besides, for each method, the damage seems to be pro-portional with energy (Table 3 ) – but, more important,the BCA method generates cascades in the same rangeof length as the MD method but with a higher number ofdefects and a different spatial distribution. In addition,self-interstitial defects in MD cascades are further apartfrom vacancies than in BCA cascades. In consequence,there is not a good agreement between interstitial distri-bution in both methods (Figure 1). The impact of thesefacts is explained as follows.
Some of the main consequences generated by thedifferences between both generating techniques are in-ferred observing the V and V clusters evolution whentemperature increases. In the case of vacancy point de-fects (figure 2 a)), despite the fact that the introduceddamage is ∼4 times higher in BCA than in MD cas-cades (table 3), a good agreement in the number of gen-erated vacancy defects exists between both methods foreach energy value. This point might be interpreted asthat vacancies and self-interstitial defects further anni-hilate in BCA cascades due to its spatial distribution.In connection with vacancy clusters (figure 2 b)), thereis a wide variation in the number of vacancies in clus-ters depending on the method but not in its average size.Furthermore, it is worth mentioning that the fraction ofthe total number of particles corresponding to vacanciesand clusters does not depend on the cascade energy, butit strongly does on the method used to compute them(figure 2 c)).
MD BCA
Figure 1: Example of 100 keV MD (left) and BCA (right) damagecascades appearance. (Colour code: purple – self-interstitial defects,grey – vacancy.)
Concerning self-interstitial point defects and clusters,the two methods reproduce the same evolution withtemperature of the first ones independent of energy al-though it is not very relevant because they vanish dueto recombination at ∼200 K and, this value is out of thetemperature range of a fusion-reactor structural mate-rial. The two methods generate a distinct evolution ofthe I clusters proportion with temperature until ∼400 Kwhen irregular interstitial clusters disappear. Further-more, this evolution also depends on the energy in caseof MD cascades but not in case of BCA ones. Despiteof that, the two methods reproduce in good agreementthe cluster average size growing with temperature, thatis independent with cascade energy (Figure 3).
Figure 3: I particles in spherical clusters proportion (left axis) andcluster average size (right axis) evolution vs. temperature.
Similar to the vacancy clusters, the fact that the BCA
3
Figure 2: V defects and clusters evolution vs. temperature for the different damage cascades : a) Number of V defects, b) Number of V clustereddefects (right axis) and V clust. average size (left axis), c) V defects and defects in V clusters proportions compared to total particles.
method generates a higher number of Frenkel pairs hasas a consequence that <111> clusters grow bigger whenthe temperature increases. In addition, as it is logical,the higher the energy of the cascade, the higher the num-ber of particles (figure 4 a)). The growing of the clustersize is described similarly by the two methods (figure 4b)). Despite there are differences in this plot, they aresmaller than the standard deviation values respect to theaverage, being open for interpretation that each methodreproduces a distinct evolution. A completely differentbehaviour is observed concerning the fraction of parti-cles in <111> clusters respect to the total number: thetwo generation techniques reproduce the same results inall temperature range except at 200 K, when this typeof clusters starts forming (figure 4 c)). The existenceof <100> clusters is also reflected in this graph as thedecrease in the total particles fraction at high tempera-tures but it is not very significant due to the low statisticassociated with this type of defects.
References
[1] I. Martin-Bragado, A. Rivera, G. Valles, J. L. Gomez-Selles, M. J.Caturla, Computer Physics Communications 184 (2013) 2703 –2710.
[2] MMonCA webpage, www.materiales.imdea.org/MMonCa, 2014.[3] G. H. Vineyard, Journal of Physics and Chemistry of Solids 3
(1957) 121 – 127.[4] T. Diaz de la Rubia, M. W. Guinan, Phys. Rev. Lett. 66 (1991)
2766–2769.[5] G. J. Ackland, M. I. Mendelev, D. J. Srolovitz, S. Han, A. V. Bara-
shev, Journal of Physics: Condensed Matter 16 (2004) S2629.[6] C. Bjorkas, K. Nordlund, Nuclear Instruments and Methods in
Physics Research Section B: Beam Interactions with Materialsand Atoms 259 (2007) 853 – 860.
[7] J. F. Ziegler, M. Ziegler, J. Biersack, Nuclear Instruments andMethods in Physics Research Section B: Beam Interactions withMaterials and Atoms 268 (2010) 1818 – 1823. 19th InternationalConference on Ion Beam Analysis.
[8] SRIM/TRIM webpage, www.srim.org/, 2014.[9] G. S. Was, Fundamentals of Radiation Materials Science: Metals
and Allowys, Springer, 2007.
4
Figure 4: <111> clusters evolution vs. temperature for the different damage cascades : a) Number of particles in <111> clusters, b) <111> clustersaverage size, c) Clusters proportion compared to total particles.
5
VII
Insights on loop nucleation and growth in α-Fe thin films under ion implantation from atomistic models
M. J. Aliaga1, I. Martin-Bragado2, I. Dopico2, M. Hernández-Mayoral3,
L. Malerba4, M. J. Caturla1* 1Dep. Física Aplicada, Universidad de Alicante, Spain 2IMDEA Materials Institute, Getafe, Madrid, Spain 3CIEMAT, Madrid, Spain 4SCK-CEN, Belgium
Abstract
The outstanding question of loop growth in a-Fe under irradiation is
addressed using object kinetic Monte Carlo with parameters from
molecular dynamics and density functional theory calculations. Two models are considered for the formation of <100> loops, both based
on recent atomistic simulations. In one model <100> loops are formed by the interaction between ½ <111> loops. In a second
model small interstitial clusters can grow as <100> or ½ <111> loops. Comparing results from the two models to experimental
measurement of loop densities, ratios and sizes produced by Fe
irradiation of Fe thin films, the validity of the models is addressed.
Introduction
An outstanding question in the field of radiation damage effects of
materials is the nucleation and growth of loops in a-Fe under irradiation. Experimentally it is well known since the 1950s that two
types of loops are formed: <100> and 1/2<111> loops [Jenkins, papers from CompMatSci]. However, the type, concentration and
ratio of one loop type vs the other type differs considerably
depending on the experimental conditions. Zhang et al. [Zhang2015] have reported the transformation of C15 clusters to both <100> and
<111>/2 loops in bcc iron, being the <100> loops more probable
(70%).
Model parametrization
We have used our database of 100 keV cascades of Fe irradiation of
Fe thin films as input for the Object Kinetic Monte Carlo code
MMonCa, developed by I. Martin-Bragado [Bragado2013]. This code
is open-source and available from [Bragado]. The simulation box we
have used is also a thin film of 50nm, reproducing a typical TEM
sample. In our code, small self-interstitial atom (SIA) clusters up to
size 4 have irregular shape and are considered mobile, with the
migration energies given in table I obtained from density functional theory calculations [Fu2004]. These self-interstitial clusters are
considered to move in three dimensions. From size 5, they can grow
according to one of this two models:
Model A: In this model all interstitial clusters above size 4 transform
in <111> loops with mobilities given also in table I and obtained from classical molecular dynamics simulations [Soneda01]. These loops
move one-dimensionally, unlike vacancies or smaller SIA clusters.
The interaction between <111> loops results in the formation of
<100> loops, <111> + <111> = <100> when the size of the two <111> loops is similar, with a maximum difference of 5%. Once the
<100> loops are formed, they can grow by addition of other <100> or <111> loops, and small interstitial clusters ( 4). The same occurs
for <111> loops. In these interactions between <100>, <111> loops and small interstitial clusters, the larger species absorbs the
smaller one. And when a <100> loop reacts with a <111> loop of
approximately the same size (5% maximum difference) the resulting
loop is also considered to have a <100> Burgers vector.
Model B: In this model <111> and <100> loops form and grow independently. SIA clusters from size 5 can either transform into
<100> loops with an initial ratio of 5%, or into <111> loops with a
ratio of 95%. These ratios are based on the work by Marinica et al [Marinica2012] according to the proportion of C15 clusters found in
cascades. Once formed, <111> loops grow by addition of other <111> loops or SIA clusters < 5, and <100> loops grow by addition of other <100> loops or SIA clusters < 5 but do not grow by addition
of ½<111> loops of any size .
In both models <111> loops can be stopped by interaction with carbon atoms, with a binding energy of 1.3 eV. These immobile C-
<111> loops can then grow by addition of <111> loops or SIA
clusters < 5. Also, <100> vacancy loops have been included in the
models. The Gilbert equation in [Gilbert2008] has been used for the binding energy of the vacancies in the loop. In this equation the
radius of the loop is calculated using the size and the density of the
loop. The density of the loop has been calculated fitting the equation to figure 4 in [Gilbert2008]. For the binding energies of Vn > 4 and
In>4 clusters, we have used the usual extrapolation law [Soneda98]:
Eb(n)=Ef+[Eb(2)-Ef][n2/3-(n-1)2/3]/(22/3-1). For the smaller species
up to 4 DFT values have been used [Fu2004]. These small vacancy
clusters are considered mobile, with a 3D mobility, while larger
vacancy clusters are immobile. Table 1 summarizes the most
important parameters of the species involved.
Table 1
Type of defect, migration and binding energies of the objects defined
in our OKMC model. Last column corresponds to the dimensionality of
migration. For the mono-defects, V and I, the formation energy is
taken from ab initio calculations [Fu204], Ef(V)=2.07 eV and Ef(I)=3.77 eV.
Defect Migration
Barrier (eV) Binding energies (eV) Migration
type
V 0,67 3D
V2 0,62 0,3 3D
V3 0,35 0,37 3D
V4 0,48 0,62 3D
Vn > 4 immobile As in ref. [Soneda98]
V 100 loops Immobile As in ref. [Gilbert08]
I 0,34 3D
I2 0,42 0,8 3D
I3 0,43 0,92 3D
I4 0,3 1,64 3D
In>4, I111 loops 0.06+0.11/n1.6 As in ref. [Soneda98] 1D
In > 4, I100 loops Immobile As in ref. [Soneda98]
C-I111 loops Immobile 1.3
In MMonCa, it is possible to specify, optionally, the capture radius for
a particular interaction. The interaction will happen when the distance between two particles belonging to each defect is smaller or equal
than the specified capture radius. If not specified, the default value of
lambda is used. In our work, and after a detailed study, we have used the capture radius for individual defects of 0.4 nm, selected to
reproduce the isolated number of defects obtained in MD.
Results
Main results:
- If we consider that all <100> vacancy loops formed in the MD simulations do not migrate or recombine with the surface, the
concentration of <100> loops is extremely high and in complete
disagreement with the experimental observations (Figure 1). Therefore, <100> vacancy loops formed close to the surface must
migrate or recombine with the surface. We have used in the OKMC
simulations a migration energy of 0.5 eV in both models.
Figure 1: Total concentration of defects as a function of dose (a)
<100> vacancy loops are immobile, compared to experiments and
(b) concentrations for different migration energy values for <100> vacancy loops.
- MD simulations of vacancy loops close to the surface are being performed to study this issue.
- Without carbon: all loops are <100> type in both models since <111> loops escape to the surface or recombine to form <100>
(Model A). In both cases, concentrations at any given dose are much higher than in the experiments (see Figure 2). For the case of model
A, the dose dependence follows the same trend as the experiment,
however, for model B the dose dependence does not look like the experiments. The concentration is very high from very low doses and
it remains almost constant.
Figure 2: Models A and B compared to experimental data.
Figure 3: Model B compared to experiments for different distribution
of <100> to <111> loops.
- Model B: changing the distribution of <100> and <111> can reduce
the total concentration of defects. If we consider that only 0.1% of all
self-interstitial clusters are of type <100> and 99.9% are of type <111> the concentrations are close to the only value that we have
experimentally for (001) orientation. Still much higher than for the
(111) orientation (see Figure 3). Things to check: orientation of the
sample. I have already tried this but I do not see any difference in
the total concentration of defects. Calculate the slope of the curves to
see if it fits the experimental power law.
- Model B: including carbon, that is, including traps for <111> self-
interstitial clusters with a trapping energy of 1eV (CHECK) results in both <100> and <111> clusters, with <100> being predominant at a
dose of ~ 4e17 ions/cm2 (Figure 4). Similar experimental trend in
terms of dose. High concentrations. Carbon concentration is 0.1%. Check other concentrations.
Figure 4: Model B including trapping sites for <111> loops.
- Model B: effect of <111> mobility. If the mobility of <111> loops is
reduced with a migration energy of 1 eV again we have two populations of loops, with more <111> loops than <100> loops at
any given dose, and higher total concentrations that for a faster
mobility of loops and similar to those concentrations obtained
experimentally for FeCr alloys (Figure 5). One problem with this
model is that we do not see saturation at high doses. Loop-loop interactions or long range interactions must be reviewed.
Figure 5: Model B including a low migration for <111> loops.
- Model A: effect of the minimum size to produce a <100> loops. Checked cases where loops are formed for any size of <111>
clusters, for clusters with size > 15 and > 30. The total concentration
is reduced slightly as the cut-off is increased. However, the trend does not change: the dose dependence is wrong, with large clusters from very low concentrations and almost no dose dependence (Figure
6).
Figure 6: Model A, effect of minimum size to produce a <100> loop.
- Model A: including Carbon as a trapping site with a binding energy
of 1 eV. Like in Model B, now we have two populations of loops,
<100> and <111> with a higher concentration of <100> than <111> at any given dose. Still wrong dose dependence (Figure 7).
Carbon concentration 0.1% like before.
Figure 7: Model A with Carbon.
- Model A: effect of <111> mobility. Now we also have two population of loops <100> and <111> with <111> at higher
concentration. And the dose dependence follows the experimental
observations (Figure 8). So, it seems the problem with Model A in terms of the dose dependence is that <111> loops migrate so fast (if
we consider the MD migration energies) that the immediately form
large <100> loops, even at very low doses, which is not observed experimentally. Can we modify this model to fit the experiments with
some reasonable assumptions?. Same problem as with Model B with
the saturation of loops.
Figure 8: Model A, effect of <111> mobility.
Conclusions
Both models reproduce the right trend for the effect of carbon (<100> and <111> loops with higher concentration of <100> loops)
and the effect of Cr (<111> and <100> loops with higher concentration of <111> loops, and higher total concentration of
defects). That is, trapping defects or reducing their mobility do not give rise to the same results, at least for a particular concentration of
trapping sites. We should check also as a function of trapping concentration (at higher concentrations there will be also more
<111> than <100> loops). These results are not new, in terms of total concentrations, but have never been shown, to the best of my
knowledge, with the distribution of <100> and <111> loops.
The main conclusion, in view of these simulations, is that Model B
(two independent population of loops) follows closer the experimental
results than Model A (<100> loop formation from <111> loop recombination), since it reproduces the right trend of loops growth
with dose. However, we should check more thoroughly all possible
parameters before making such a conclusion. Also, a model that combines both Model A and Model B should be considered for
completeness.
Bibliography
[Jenkins, papers from CompMatSci] buscar ref
[Zhang2015] Y. Zhang et al. Scripta Materialia 98 (2015) 5-8.
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Conclusiones
En esta tesis se han desarrollado modelos de simulación multiescala para
explicar los efectos de la radiación en materiales para fusión, en particular Fe y
en FeCr, con especial atención al estudio de la utilización de iones como
sustitutos del daño por radiación con neutrones. Se describen a continuación
las conclusiones más significativas de este trabajo.
El estudio estadístico detallado de las cascadas de dinámica molecular muestra
que el daño primario en láminas finas de Fe irradiado con iones de Fe es
totalmente distinto de la irradiación en el interior del material. Esto tiene
importantes implicaciones a la hora de comparar y utilizar la irradiación con
iones para “simular” la irradiación con neutrones en un reactor nuclear de
fusión, así como en el uso de otras técnicas de simulación que utilizan
cascadas de dinámica molecular como datos de entrada. El análisis del daño
primario también ha permitido mostrar que la ley de escala para la producción
de defectos bajo irradiación obtenida para W por Sand y colaboradores [5],
también ocurre en Fe. Este resultado es significativo ya que la ley de escala
obtenida podría emplearse con el fin de generar distribuciones de daño
primario más detalladas para modelos de evolución de la microestructura como
los códigos de Monte Carlo Cinético.
Además, mediante Object Kinetic Monte Carlo hemos simulado el crecimiento
de loops en Fe y en FeCr asumiendo dos modelos diferentes y hemos
mostrado que uno de ellos reproduce la dependencia con la dosis observada
experimentalmente para irradiación de láminas finas.
Por último, las simulaciones de imágenes TEM muestran, por un lado, que los
loops de vacantes obtenidos llegan a tamaños visibles al microscopio
electrónico de transmisión y, por otro lado, que la proximidad de dos loops
cercanos produce un solapamiento en la imagen que hace que se vean como
un único cluster de vacantes.
La conclusión global y más importante de esta tesis es que, para reproducir y
comprender los experimentos de irradiación con iones que se llevan a cabo en
láminas finas de material para ser analizadas por TEM, es necesaria una
descripción detallada del daño inicial producido. Esto implica la inclusión de las
superficies en las simulaciones, ya que, como se ha puesto de manifiesto en
este trabajo, el daño producido es completamente diferente al que se produce
en el interior del material, y los resultados no pueden extrapolarse de un caso
al otro.
Los resultados y conclusiones aquí obtenidos sirven de base para el desarrollo
y mejora de modelos en el programa de fusión, además de para la selección de
experimentos a realizar, con el fin último de hacer viable la fusión como energía
alternativa.
Conclusions
In this thesis we have developed multiscale simulation methods to explain the
effects of irradiation in fusion materials, in particular Fe and FeCr, focusing in
the use of ions as substitutes of neutron damage.
Detailed statistical study of the cascades produced by molecular dynamics
shows that the primary damage in thin films of Fe irradiated with Fe ions is
completely different to irradiation in bulk. This has important consequences
when comparing and using irradiation with ions to ‘simulate’ neutron irradiation
in a fusion reactor, and also in the use of other simulation techniques that use
molecular dynamics cascades as input. By analysis of the primary damage we
have as well demonstrated that the scaling law obtained for W by Sand and
coworkers [5], also occurs in Fe. This result is significant because the scaling
law obtained can be used to generate detailed primary damage distributions as
input for microstructure evolution models as Kinetic Monte Carlo.
In addition, using Object Kinetic Monte Carlo we have simulated the growth of
loops in Fe and FeCr assuming two different models, and we have
demonstrated that one of them reproduces the dose dependence observed
experimentally in thin film irradiations.
Finally, TEM image simulations show, on one hand, that vacancy loops can
grow to visible clusters under the transmission electron microscope and, on the
other hand, that the proximity of two close vacancy loops produces an overlap
in the image that causes the two clusters to be seen as only one.
The global and most important conclusion of this thesis is that, to be able to
reproduce and understand ion irradiation experiments in thin films for in-situ
TEM, it is necessary a detailed description of the primary damage produced.
This implies the inclusion of surfaces in the simulations, because, as it has been
shown in this work, the damage produced is totally different than the damage
produced in the bulk of the material, therefore, results cannot be extrapolated
from one case to the other.
The results and conclusions obtained here can be used as basis for the
development and improvement of the models in the fusion project, as well as for
the selection of new experiments, with the ultimate goal of making fusion a real
alternative energy.
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