# Fast low-temperature irradiation creep driven by athermal defect dynamics

Alexander Feichtmayer  Max Planck Institute for Plasma Physics, Boltzmannstr.
2, 85748 Garching, Germany  Technical University Munich, Boltzmannstr. 15,
85748 Garching, Germany  Max Boleininger<EMAIL_ADDRESS>UK Atomic
Energy Authority, Culham Centre for Fusion Energy, Oxfordshire OX14 3DB,
United Kingdom  Johann Riesch  Max Planck Institute for Plasma Physics,
Boltzmannstr. 2, 85748 Garching, Germany  Daniel R. Mason  UK Atomic Energy
Authority, Culham Centre for Fusion Energy, Oxfordshire OX14 3DB, United
Kingdom  Luca Reali  UK Atomic Energy Authority, Culham Centre for Fusion
Energy, Oxfordshire OX14 3DB, United Kingdom  Till Höschen  Max Planck
Institute for Plasma Physics, Boltzmannstr. 2, 85748 Garching, Germany
Maximilian Fuhr  Max Planck Institute for Plasma Physics, Boltzmannstr. 2,
85748 Garching, Germany  Technical University Munich, Boltzmannstr. 15, 85748
Garching, Germany  Thomas Schwarz-Selinger  Max Planck Institute for Plasma
Physics, Boltzmannstr. 2, 85748 Garching, Germany  Rudolf Neu  Max Planck
Institute for Plasma Physics, Boltzmannstr. 2, 85748 Garching, Germany
Technical University Munich, Boltzmannstr. 15, 85748 Garching, Germany  Sergei
L. Dudarev  UK Atomic Energy Authority, Culham Centre for Fusion Energy,
Oxfordshire OX14 3DB, United Kingdom

###### Abstract

The occurrence of high stress concentrations in reactor components is a still
intractable phenomenon encountered in fusion reactor design. We observe and
quantitatively model a non-linear high-dose radiation mediated microstructure
evolution effect that facilitates fast stress relaxation in the most
challenging low-temperature limit. In situ observations of a tensioned
tungsten wire exposed to a high-energy ion beam show that internal stress of
up to 2 GPa relaxes within minutes, with the extent and time-scale of
relaxation accurately predicted by a parameter-free multiscale model informed
by atomistic simulations. As opposed to conventional notions of radiation
creep, the effect arises from the self-organisation of nanoscale crystal
defects, athermally coalescing into extended polarized dislocation networks
that compensate and alleviate the external stress.

The central engineering challenge in realising commercially viable fusion
power-generating reactors lies in the degradation of materials in components
surrounding the fusion plasma1. Structural components inside the vacuum vessel
are exposed to extreme conditions2, involving high magnetic fields,
gravitational loads, plasma disruptions, and unprecedented levels of neutron
and gamma radiation3 that continuously generate microscopic defects, leading
to severe degradation of properties of materials critical to their intended
function. Furthermore, spatio-temporal variations in irradiation exposure and
temperature induce stress concentrations of sufficient magnitude to threaten
the integrity of load-bearing or physical barrier4, 5, 6 components.

The tendency of metals to undergo steady viscoplastic deformation when
subjected to mechanical stress at elevated temperatures, known as creep, is of
pivotal significance to the structural integrity of a reactor. Irradiation by
high-energy particles significantly accelerates creep, causing deformation
rates orders of magnitude higher than thermal creep under the otherwise
equivalent stress7, 8. However, mechanistic understanding of the phenomenon
proves elusive, partly because it is challenging to design experiments
enabling accurate measurements under simultaneous high irradiation fluence,
mechanical, and thermal loads.

In particular, the role of temperature remains undetermined. Conventional
argument suggests that creep rates increase with temperature due to the
enhanced thermal diffusion of microstructural defects, mediating the
viscoplastic flow. In the apparent contradiction with this argument,
measurements of irradiation creep in steels9 show that at low temperatures the
irradiation creep occurs at an anomalously high rate, decreasing as a function
of temperature, only to increase again at higher temperatures. Low
temperatures, and large temperature and radiation exposure gradients are
expected in the actively cooled reactor components, making a rational model
for the phenomenon a requirement for an expert fusion reactor design.

Figure 1: Time-resolved measurement of stress relaxation under irradiation. a,
A thin tungsten wire of length 15 mm and diameter 16 µm is loaded with tensile
stress, after which a central wire section of 4 mm length is exposed to a beam
of 20.3 MeV W6+ ions whilst monitoring the force required to maintain the
initial elongation. b, The ion beam damages a thin surface layer of about 2 µm
thickness, as seen in the simulated exposure profile in the cross-section. c,
Upon exposure of the wire to the beam, a rapid relaxation of load is measured.
d-e, While the wire is initially under uniform tensile stress, a multi-scale
simulation of the developing internal stress reveals that in the irradiated
surface layer (A), the tensile stress relaxes completely and even becomes
compressive, while in the unexposed region deeper in the wire (B), the tensile
stress increases to balance the expansion of the irradiated layer. f, The
final magnitude of induced stress relaxation is proportional to the magnitude
of the initial tensile stress, suggesting that the relaxation is strongly
biased by the external stress. Error bars indicate the standard error over
multiple experiments. The shaded region indicates the 3$\sigma$-confidence
interval of the simulation prediction, with uncertainty originating from the
stochasticity of the atomistic data underlying the surrogate model.

In this study, we develop a comprehensive quantitative treatise of the
anomalous low-temperature irradiation creep phenomenon by combining custom-
designed experiments with in situ measurement capabilities and a parameter-
free multiscale model. Fig. 1 illustrates a time-resolved observation of
stress relaxation in a 16 µm thin tungsten wire – thinner than a human hair –
under exposure to highly energetic ions. Initially, the wire is stretched
until a specified tensile load is reached. Subsequently, the wire is exposed
to the ion beam while the force required to maintain its length is monitored.
Despite the wire remaining at a temperature where no thermally-driven creep
occurs, our observations reveal an initial rapid relaxation of the force
occurring within minutes, followed by slower relaxation plateauing over a few
hours. Significantly, in Fig. 1f we observe the extent of relaxation to be
linearly proportional to the external load, which is in remarkable qualitative
and quantitative agreement with predictions derived from an in silico
replication of the experiment through a parameter-free model. Our analysis
shows that the relaxation is non-linear and involves the coarsening of
microscopic defects under external stress. Since only 4 % of the wire
materials is exposed to high-energy ions, the extent of the measured
relaxation suggests that locally in the irradiated surface layer, the applied
tensile loads of up to 2 GPa relax entirely and solely through the anomalous
fast low-temperature irradiation creep.

Time-resolved stress relaxation measurement
To measure the time-resolved irradiation-induced stress relaxation, it is
necessary to select an appropriate high-energy particle source. Neutrons
penetrate deep into materials, allowing for testing of bulk samples, but the
conditions in a typical materials test reactor rarely permit in situ
mechanical testing. Neutron activation of materials also complicates handling,
with low dose rate implying long exposure times. For these reasons, we chose
irradiation by high-energy heavy ions. These are readily available, can be
produced in the form of intense beams, and cause no activation. It is also
possible to incorporate a mechanical testing setup into an ion accelerator
beam line.

As high-energy heavy ions have a penetration depth of only a few
$\text{\,}\mathrm{\SIUnitSymbolMicro m}$, the exposed zone should span many
grains in order to replicate bulk-like irradiation conditions. We selected
potassium-doped (60–75 ppm) cold-drawn tungsten wires, as tungsten is the
prime candidate material for plasma-facing components due to its high melting
point and high thermal conductivity. The wires are industrially produced with
diameters as small as $16\text{\,}\mathrm{\SIUnitSymbolMicro m}$ and feature
nanoscale grains. No thermal creep is expected, as only low homologous
temperatures are reached ($T<0.1\,T_{\mathrm{melt}}$). Extensive studies of
mechanical properties10, 11, deuterium retention12, and post-irradiation
properties13 of this material are available. To prevent the implantation of
impurities, we employ a beam of $20.3\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$
W6+ ions. These ions penetrate about $2\text{\,}\mathrm{\SIUnitSymbolMicro m}$
into tungsten, exposing about $16\text{\,}\mathrm{\char 37\relax}$ of the wire
cross-section.

Figure 2: a, Microstructure of the 16 µm tungsten wire. Left: cross-section
images taken using a scanning electron microscope (SEM) in backscattered
electron contrast mode. Right: longitudinal section using secondary electron
contrast. b, Schematic overview of the ion beam path during the experiment.
From left to right: beam limiting aperture, first suppression electrode, 16 µm
wire sample, 102 µm measuring wire, second suppression electrode. c, Tensile
stress (blue) and ion current (orange) during the experiment. Black circles
indicate the points where experimental measurements were performed between the
irradiation intervals.

To enable a time-resolved measurement of the stress relaxation phenomenon, a
dedicated experiment comprising a tensile testing machine mounted in a vacuum
vessel was set-up at a beamline of the $3\text{\,}\mathrm{M}\mathrm{V}$ tandem
accelerator at the Max Planck Institute for Plasma Physics. The
$15\text{\,}\mathrm{m}\mathrm{m}$ long wire samples are mounted between two
crossheads of the machine and mechanically tensioned to a uniaxial stress
ranging from $0.5\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$ to
$2.0\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$, which is still in the elastic
region. While the strain is kept constant, a central wire section of
$4\text{\,}\mathrm{m}\mathrm{m}$ length is exposed to the ion beam. Since
almost all of the impacting ions’ kinetic energy is absorbed in the wire, the
sample’s temperature rises to at most
$100\text{\,}\mathrm{\SIUnitSymbolCelsius}$. To correct for the influence of
the resulting thermal expansion on the force measurement, the beam is
periodically turned off at intervals of
$5\text{\,}\mathrm{m}\mathrm{i}\mathrm{n}30\text{\,}\mathrm{m}\mathrm{i}\mathrm{n}$.
The sample cools down over these intervals, allowing for the measurement of
the actual force drop. The ion fluence is determined by measuring the electric
current flowing through the wire, from which the dose in “displacements per
atom” (dpa)14 is computed using the number of displacements per colliding ion
from the SRIM software15. The samples are exposed to a peak dose of up to
$6\text{\,}\mathrm{d}\mathrm{p}\mathrm{a}$ over a typical
$6\text{\,}\mathrm{h}$ duration experiment. The measured forces are adjusted
for shifting and relaxation of the load cell and measurement setup, and
further steps are taken to improve the signal-to-noise ratio of the measured
ion current, see the Methods sections. Figure 2b schematically illustrates the
irradiation part of the experiment.

Insights from in silico testing

While the extent of relaxation in the irradiated volume can be numerically
estimated, such estimates do not account for the transient and heterogeneous
nature of the stress field developing over the course of irradiation.
Similarly, even if microstructural evolution were precisely characterised
using imaging techniques, without understanding the physical principles
governing it, this insight would not be transferable to complex geometries
involving spatially or temporally varying stress fields.

Figure 3: Data-driven surrogate model. a, Atomistic simulations generate data
on deformation eigenstrains for a representative volume element under
irradiation and external uniaxial stress. Radiation damage is simulated by
assigning randomly chosen atoms in a tungsten crystal (white cutaway) recoil
energies, drawn from scattering events with 20.3 MeV W6+ ions, leading to
initiation of collision cascades that result in the formation of nanoscale
interstitial (red) and vacancy (blue) defects, shown here for 1 GPa.
Eigenstrains are extracted in directions parallel ($\parallel$) and
perpendicular ($\perp$) to the uniaxial stress direction. b, A surrogate model
is matched to the data using maximum likelihood estimation. Solid lines and
shaded areas indicate the mean value and standard deviation, respectively, of
eigenstrains from five repeated simulations per stress value. c, The surrogate
model, when adjusted to isotropic expansion at zero stress (see text), is in
agreement with in situ measurements of elongation strain of tungsten
irradiated by fission fragments at 20 K under tension.

We have opted to develop a virtual representation of the wire experiment based
on a small number of precisely defined and tested theoretical principles. The
stress relaxation mechanism is described by a surrogate model, derived from
the data obtained directly from atomistic simulations, with no additional
parameters adjusted to fit the experimental result. In this way, the model has
predictive capabilities over a well-defined range of experimental conditions.
By embedding the surrogate model into a finite element representation of the
wire, we bridge the gap between atomic and micro-mechanical scales, arriving
at a “digital shadow” of the experiment. Atomistic simulations also provide
full and precise information about the microstructure, enabling the in silico
identification of the mechanisms responsible for the observed relaxation.

Molecular dynamics (MD) simulations have been used to successfully predict
properties of materials exposed to irradiation without the need for adjustable
parameters16, 17, 18. While MD simulations enable direct and parameter-free
analysis of response of radiation effects to thermal and mechanical loads, it
would require over
${10}^{12}\text{\,}\mathrm{a}\mathrm{t}\mathrm{o}\mathrm{m}\mathrm{s}$ just to
represent a $1\text{\,}\mathrm{\SIUnitSymbolMicro m}$ long wire section. The
largest simulation reported in literature involved
$2\text{\times}{10}^{9}\text{\,}\mathrm{a}\mathrm{t}\mathrm{o}\mathrm{m}\mathrm{s}$19.
Consequently, with MD we cannot explicitly simulate the heterogeneous dose
profile attenuating on the micrometer scale. In what follows, we employ MD to
describe the behaviour of a representative volume element (RVE) under
irradiation. The simulation operates as a black box; given an external
homogeneous stress, we obtain data on dimensional changes of an RVE during
irradiation.

In our atomistic simulations, see Fig. 3, the initially pristine tungsten
single crystals were exposed to atomic recoils representative of those
initiated by $20.3\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$ W6+ ions, while
maintaining a specified external uniaxial load
$\sigma_{\mathrm{zz}}^{\mathrm{ext}}$. These recoils lead to the generation of
self-similar cascades of atomic displacements, ultimately resulting in the
formation of crystal defects that cause dimensional changes of the RVE, here
described as eigenstrains parallel and perpendicular to the uniaxial stress
direction, $\varepsilon^{*}_{\parallel}$ and $\varepsilon^{*}_{\perp}$,
respectively. While the simulated dose-rate of 20 dpa/µs is nine orders of
magnitude higher than in experiment, at a temperature of at most
$100^{\circ}$C, vacancies remain immobile with a mean migration time of 1
year. Interstitials, on the other hand, are mobile, driven by thermal and
stress fluctuations. As a result, microstructural evolution follows an
effectively athermal pattern, both in experiment and simulations18.

Because of the discrete nature of atomic recoils, MD simulations inherently
generate discrete datasets of eigenstrain
$\\{(\phi_{0},\bm{\mathrm{\varepsilon}}^{*}_{0}),(\phi_{1},\bm{\mathrm{\varepsilon}}^{*}_{1}),\,\cdots,(\phi_{N},\bm{\mathrm{\varepsilon}}^{*}_{N})\\}$
over dose $\phi_{i}$ in dpa. Moreover, independent MD simulations for the same
uniaxial stress yield different eigenstrain datasets due to the stochasticity
inherent to collision cascades20 and dislocation motion21, 22. Since these
eigenstrains are intended for the use in a homogenised finite element model,
we require a surrogate model representing the expectation value $\mu$ and
variance $s^{2}$ of eigenstrains as continuous functions of dose and stress.
We express the probability to draw a parallel or perpendicular eigenstrain
component $(\phi_{i}$, $\varepsilon^{*}_{i})$ from a normal distribution
$\mathcal{N}(\mu,s^{2})$ by

$p(\bm{\theta}|\varepsilon^{*}_{i})=\frac{1}{\sqrt{2\pi}s(\phi_{i};\bm{\theta})}\exp\left(-\frac{\left(\varepsilon^{*}_{i}-\mu(\phi_{i};\bm{\theta})\right)^{2}}{2s^{2}(\phi_{i};\bm{\theta})}\right),$
(1)

where $\mu(\phi;\bm{\theta})$ and $s(\phi;\bm{\theta})$ are cubic spline
functions with unknown knot heights $\bm{\theta}$. The likelihood to draw the
entire eigenstrain data for a common external stress
$\sigma_{\mathrm{zz}}^{\mathrm{ext}}$ from the model is

$\mathcal{L}(\bm{\theta}|\bm{\varepsilon}^{*})=\prod_{i=1}^{N}p(\bm{\theta}|\varepsilon^{*}_{i}).$
(2)

By maximising the log-likelihood over $\bm{\theta}$,

$\hat{\bm{\theta}}=\underset{\bm{\theta}\in\Theta}{\arg\max}\log\mathcal{L}(\bm{\theta}|\bm{\varepsilon}^{*}),$
(3)

we obtain a maximum likelihood estimate for $\hat{\bm{\theta}}$ best matching
the data. This process is applied to $\varepsilon_{\parallel}$ and
$\varepsilon_{\perp}$ components, independently for each simulated stress.
Finally, $\mu(\phi;\bm{\theta})$ is interpolated in stress-space using cubic
splines, resulting in continuous representations of the eigenstrains
$\varepsilon^{*}_{\mathrm{MD},\parallel}(\phi,\sigma_{\mathrm{zz}}^{\mathrm{ext}})$
and
$\varepsilon^{*}_{\mathrm{MD},\perp}(\phi,\sigma_{\mathrm{zz}}^{\mathrm{ext}})$.
From these, the diagonal eigenstrain tensor is formed with
$\bm{\mathrm{\varepsilon}}^{*}_{\mathrm{MD}}(\phi,\sigma_{\mathrm{zz}}^{\mathrm{ext}})=\mathrm{diag}(\varepsilon^{*}_{\mathrm{MD},\perp},\varepsilon^{*}_{\mathrm{MD},\perp},\varepsilon^{*}_{\mathrm{MD},\parallel})$.
The eigenstrain input data are obtained from five repeated MD simulations for
each stress $\sigma_{\mathrm{zz}}^{\mathrm{ext}}$ between
$-1.0\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$ and
$2.0\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$ in increments of
$0.5\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$. Additional details are given in
the Methods section.

It remains to formulate a model for the incremental accumulation of
eigenstrain $\bm{\mathrm{\varepsilon}}^{*}$ during irradiation. In experiment,
the wire begins in a uniform stress state that becomes non-uniform upon
exposure to the spatially attenuating ion beam. The surrogate model, however,
describes dimensional changes of a RVE irradiated under a homogeneous external
stress. The eigenstrain induced by a complex microstructure generally
comprises contributions with varying degrees of reversibility in response to a
changing stress, and the same is expected to hold true for microstructures
formed by irradiation. Atomistic simulations show that eigenstrain is largely
irreversible; a momentary change in external stress has little effect on the
eigenstrain. A similar observation was recently reported for aluminium
irradiated under stress23. Fig. 3a suggests that at high dose, interstitial
defects preferentially arrange themselves into extended structures, such as
dislocation networks or new crystal planes24, which cause dimensional change
that is no longer affected by external stress. Motivated by this observation,
we assume that eigenstrains accumulate irreversibly over the course of
irradiation, with the accumulation driven by the surrogate model evaluated for
the instantaneous and local elastic stress
$\bm{\mathrm{\sigma}}^{\mathrm{el}}(\bm{r},t)$ and dose $\phi(\bm{r},t)$
fields:

$\frac{\mathrm{d}}{\mathrm{d}t}\bm{\mathrm{\varepsilon}}^{*}(\bm{r},t)=\frac{\mathrm{d}}{\mathrm{d}t}\bm{\mathrm{\varepsilon}}_{\mathrm{MD}}^{*}[\phi,\bm{\mathrm{\sigma}}^{\mathrm{el}}](\bm{r},t),$
(4)

where
$\bm{\mathrm{\sigma}}^{\mathrm{el}}=\bm{\mathrm{\sigma}}-\bm{\mathrm{\sigma}}^{*}$
is the elastic stress, with
$\bm{\mathrm{\sigma}}^{*}\equiv\bm{\mathrm{C}}:\bm{\mathrm{\varepsilon}}^{*}$
using the elastically isotropic stiffness tensor $\bm{\mathrm{C}}$ of
tungsten25.

Lastly, because elastic fields have long range, variations in the local stress
are implicitly dependent on the spatially heterogeneous stress field involving
the entire wire. The gap in length scales between the nanoscale RVE and the
microscale wire is bridged by representing the eigenstrain as a body-force
acting in a linear elasticity framework5, expressed as
$\bm{f}^{*}=-\bm{\nabla}\cdot\bm{\mathrm{\sigma}}^{*}$:

$\bm{\nabla}\cdot\bm{\mathrm{\sigma}}(\bm{r},t)=\bm{\nabla}\cdot\bm{\mathrm{\sigma}}^{*}(\bm{r},t).$
(5)

We solve for the stress tensor field $\bm{\mathrm{\sigma}}$ using the finite
element method, with the time propagation scheme described in the Methods
section.

Figure 4: Stress relaxation under heterogeneous and homogeneous irradiation.
a, Comparison of four irradiation-induced stress relaxation experiments to the
parameter-free simulation using the virtual wire model under an initial
uniaxial tensile stress of 1 GPa. The model predicts a small degree of
relaxation, in agreement with experiment, due to the small volume fraction of
the irradiated volume. b, Predicted stress relaxation curves for a
hypothetical homogeneous irradiation scenario under various initial uniaxial
tensile and compressive stresses. Most of the relaxation is predicted to occur
within the first 0.3 dpa.

In Fig. 4a, we compare in situ measurements and in silico predictions of
stress relaxations for the initial applied stress of 1 GPa. Over time, we
observe in the stress profile shown in Fig. 1d, the emergence of mutually
counterbalancing compressive and tensile elastic stresses in the irradiated
and unirradiated regions, respectively. The material exposed to irradiation
deforms plastically to accommodate the external tensile stress, resulting in
local stress relaxation. This deformation likewise gives rise to a long-ranged
elastic tensile stress field extending well into the unexposed region,
increasing the tensile stress beyond its initial value, as depicted in Fig.
1e.

The measured and predicted final relaxations for the various initial stresses
are shown in Fig. 1f. The predictions are in good qualitative agreement with
experimental results, with an overestimation of approximately 20 %. The amount
of relaxation appears to obey a linear relationship with the initial external
stress $\sigma_{\mathrm{zz}}^{\mathrm{ext}}$. Specifically, the simulation
data follows the curve given by
$\Delta\sigma^{\mathrm{sim}}=14.9\,\mathrm{MPa}+0.050\,\sigma_{\mathrm{zz}}^{\mathrm{ext}}$,
whereas the experimental data follows
$\Delta\sigma^{\mathrm{exp}}=18.1\,\mathrm{MPa}+0.036\,\sigma_{\mathrm{zz}}^{\mathrm{ext}}$.
Assuming that at high dose, the elastic stress relaxes completely within the
irradiated region, the fraction of relaxed stress should be close to the
irradiated volume fraction $V_{\mathrm{irr}}/V=0.04$; this is in agreement to
the proportionality constants of the linear relations. Bearing this in mind,
the major source of uncertainty in our virtual wire model originates not from
the stochastic uncertainty of atomistic data, but rather from the dose profile
determining the irradiated volume fraction.

We observe that the irradiated region relaxes to a lightly compressive state
of $-200$ MPa instead of an entirely stress-free state. This is a finite-size
effect originating from atomistic simulations, where we used a simulation cell
with a shape elongated in the $z$-direction (32$\times$32$\times$80 nm3). In a
stress-free simulation, at low dose, the interstitial loops are in prismatic
orientation, with their habit planes oriented perpendicular to the Burgers
vectors, thus giving rise to isotropic volumetric swelling. As the loops grow
under continued irradiation, they interact with their own periodic image
across the shorter simulation cell sides, eventually forming extended
dislocation networks or new complete crystal planes; as a result the initially
isotropic expansion becomes preferentially polarised in $z$-direction. For the
same reason, the final induced relaxation, shown in Fig. 1f, does not vanish
at zero initial stress, as tensile eigenstrain develops under irradiation even
in the absence of external stress. The same effect is also observed in
experiment, where the grains are similarly elongated along $z$
($\sim$50$\times$50$\times$1000 nm3), suggesting a genuine irradiation growth
phenomenon caused by the anisotropic grain microstructure.

Finally, to assess the potential beneficial effect of fast low-temperature
stress relaxation under a realistic reactor operating scenario, we applied the
model to a hypothetical case involving a homogeneously irradiated wire,
illustrated in Fig. 4b. This scenario is experimentally inaccessible with
heavy ion irradiation due to their limited implantation range and is, instead,
more representative of neutron irradiation. Using the virtual wire, we can
also predict relaxation under a compressive stress, a condition that cannot be
tested in our experiments where the wire would buckle. Our predictions exhibit
the rapid complete relaxation of the initial stress within 0.3 dpa of
exposure. In the same figure, we also show relaxation as predicted by a
surrogate model constrained to expand isotropically under stress-free
conditions (dotted lines), which is more representative of tungsten with
isotropic grain microstructure.

Figure 5: Anomalous low-temperature irradiation creep. a, Irradiation creep is
driven by the coarsening and coalescence of highly mobile interstitial
clusters. At low dose, irradiated microstructure contains mostly interstitial-
type (red) and rarely vacancy-type (blue) dislocation loops, as well as
dispersed vacancies. With further exposure, the interstitial-type loops grow
and coalesce, eventually forming a complex, system-spanning dislocation
network of mixed character which enables plastic deformation of the crystal.
Shown are defect clusters (N$>$2) and dislocations (1/2$\left\langle
111\right\rangle$: green, $\left\langle 100\right\rangle$: red) as identified
with the Wigner-Seitz and DXA26 methods, respectively. The reference crystal
for Wigner-Seitz analysis is chosen to minimise the number of point defects,
allowing for direct visualisation of plastic deformation. b, Interstitial and
vacancy-type loops orient themselves into opposing directions relative to the
external uniaxial stress direction, which leads to a stress-bias in the
eventual formation of the dislocation network. Shown are mean alignments for a
given dose (dots) and spline-interpolated values (line) with standard error
(shaded). The dashed line indicates prismatic alignment. c, The extent of
volume swelling is independent of external stress and can be well estimated by
0.64 times the vacancy concentration, with the numerical factor arising from
comparison of interstitial and vacancy defect relaxation volumes, see text.
Shaded areas indicate the standard error over five repeated simulations.

Discussion

Inspecting the atomistic simulations underpinning the surrogate model, we have
identified three distinct mechanisms contributing to stress relaxation
occurring over some distinct dose and stress intervals. Up to the dose of
about 0.05 dpa, dislocations form a dispersed microstructure, primarily
consisting of $1/2\left\langle 111\right\rangle$-type interstitial loops shown
in Fig. 5a. In the absence of external stress, the prismatic loops generate
isotropic volumetric swelling. When external stress is applied, the loop habit
planes tilt to lower the elastic interaction energy27, resulting in a
polarised eigenstrain28, 24. This tilting is balanced by an increase in the
dislocation line tension energy29. Consequently, the degree of anisotropy
increases with increasing external stress.

At higher doses, in the transient phase occurring between approximately 0.05
and 0.10 dpa, the loops grow and coalesce, eventually forming extended
networks or even complete crystal planes28, 24. The planes form preferentially
normal to, or parallel to, the uniaxial tensile and compressive stress
directions, respectively, as illustrated in Fig. 5a (plastic). Plane formation
contributes significantly to the overall dimensional change. As dislocations
structures transform into crystal planes, the counterbalancing effect of line
tension vanishes, and their partially polarised relaxation volumes become
fully polarised.

If the external stress is large,
$\lvert\sigma_{\mathrm{zz}}^{\mathrm{ext}}\rvert$ $\gtrsim$ 1 GPa, the
dislocation network can additionally undergo slip; in an individual atomistic
simulation, this is manifested though a substantial change30 in eigenstrain
over a few millidpa. Among the three mechanisms, only slip can keep deforming
the crystal indefinitely under continued irradiation, while loop tilting and
plane formation eventually cease as the microstructure saturates with
vacancies, and no further interstitials are created18.

The above relaxation mechanisms conserve volume, keeping the amount of
swelling unchanged under application of external stress, see Fig. 5c.
Examining the atomistic simulations, we find that mono-vacancies represent
over 90 % of the vacancy content, while interstitials coalesce into very large
clusters. Therefore, the amount of swelling can be estimated using
$\varepsilon^{*}_{\mathrm{vol}}=\mathrm{tr}(\bm{\mathrm{\varepsilon}}^{*})\approx
c_{\mathrm{vac}}\left(1+\Omega_{\mathrm{vac}}\right)$, where
$c_{\mathrm{vac}}$ represents the mono-vacancy number concentration, including
those constituting vacancy clusters, and $\Omega_{\mathrm{vac}}$ $=$ $-0.36$
is the relaxation volume of a tungsten mono-vacancy31 in atomic volumes, close
to the ab initio value32 of $\Omega_{\mathrm{vac}}$ $=$ $-0.345$. The
simulated microstructures saturate to a vacancy concentration of
$c_{\mathrm{vac}}$ = (0.30 $\pm$ 0.01) % with a volume swelling of
$\varepsilon^{*}_{\mathrm{vol}}$ = (0.20$\,\pm$ 0.01) % (with uncertainty
indicating standard deviation), which is in close agreement with the swelling
estimate of 0.19 % based on the vacancy concentration alone.

As was already hypothesised by Grossbeck and Mansur9, the mechanisms
responsible for the anomalous, low-temperature irradiation creep are
distinctly different from the conventional thermal irradiation creep
mechanisms. Here, stress relaxation occurs entirely in the absence of thermal
diffusion of defects. We observe that dislocation loops preferentially align
themselves with respect to the external stress direction at low dose, see Fig.
5a at 0.04 dpa. This observation agrees with a recent study reaching a similar
dose23, supporting the stress-induced preferential nucleation hypothesis.
However, at higher doses, the dislocation microstructure evolves into a
system-spanning network33, enabling the plastic deformation of the entire
crystal24. This is correlated with the sudden increase in eigenstrain over
this dose range, as seen in Fig. 3b, and a simultaneous decrease in the
dislocation line density. This process is driven by the elastic interaction
between dislocation loops, ultimately leading to their coalescence into an
interconnected network. A microstructure with
$c_{\mathrm{vac}}=$0.3\text{\,}\mathrm{\char 37\relax}$$ and preferentially
tilted loops can at most attain
$\varepsilon_{\parallel}^{*}=$2.6\text{\times}{10}^{-3}\text{\,}$$, which is
already exceeded at
$\sigma_{\mathrm{zz}}^{\mathrm{ext}}=$0.5\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$$,
see Fig. 3b, illustrating that the crystal has undergone plastic deformation.

The peak eigenstrain value of 1 % is quantitatively comparable to low-
temperature irradiation creep measurements in steels9, where higher
temperatures were found to yield lower creep rates. Notably, we observed a
significant amount of relaxation to occur within 0.3 dpa of exposure, which
matches low-temperature in situ irradiation creep experiments conducted in
tungsten34. We conclude that in a high-fluence reactor environment, stresses
are expected to relax extremely quickly. Considering the operating conditions
in a fusion power plant 1, it takes about a week of full-time operation to
reach this dose in critically important components such as the first wall,
divertor, or cooling pipes. While structural materials close to the plasma may
be kept sufficiently hot to stimulate recombination of radiation defects, the
actively cooled materials, in which stress concentrations critical to
structural integrity are expected to develop, would relax through the
favourable low-temperature creep mechanism explored in this study.

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  * 38 www.csd3.cam.ac.uk.

Methods
Stimulated stress relaxation experiment. The stimulated stress relaxation
experiments were carried out using the General-Purpose Irradiated Fiber and
Foil Experiment (GIRAFFE) at the Max Planck Institute for Plasma Physics (IPP)
in Garching. GIRAFFE is a specialised device designed and built for the study
of synergistic effects in materials science, consisting of a tensile testing
machine that can be integrated into one of the beamlines of a
$3\text{\,}\mathrm{M}\mathrm{V}$ tandem accelerator.

The samples were fabricated out of potassium-doped, cold drawn tungsten wire
with a diameter of $16\text{\,}\mathrm{\SIUnitSymbolMicro m}$, which was
manufactured by _OSRAM GmbH_ in Germany. To prepare the samples, an
approximately $8\text{\,}\mathrm{c}\mathrm{m}$ long piece of wire was affixed
to a polypropylene (PP) frame using _UHU PLUS ENDFEST 300_ epoxy resin, which
significantly improved the handling and mounting of the sample. Defined
electrical contacts were established at both ends using adhesive copper tape,
leaving a central usable length of the sample measuring
$15\text{\,}\mathrm{m}\mathrm{m}$.

After the sample was mounted in the fixture of the tensile testing machine, it
was loaded with about $50\text{\,}\mathrm{m}\mathrm{N}$, and an x-y stage was
translated between the sample fixture and the load cell in both axes until a
minimum force was reached. This procedure ensured a uniaxial tensile load on
the wire during the test. To measure the ion current, the lower sample holder
was connected to a _Keithley 6487_ picoamperemeter. In addition, a tungsten
beam-limiting aperture with a vertical opening of
$4\text{\,}\mathrm{m}\mathrm{m}$ was installed, along with suppression
electrodes maintained at a voltage of $-500\text{\,}\mathrm{V}$ to minimise
the effect of secondary electron emission from the sample and aperture. An
additional $102\text{\,}\mathrm{\SIUnitSymbolMicro m}$ thin tungsten wire was
attached to the lower sample holder, which has no contact with the upper
sample holder to avoid influencing the mechanical measurement. This wire
performs two functions. During the experiment, it was positioned in the beam
path behind the sample, thus increasing the measured ion current by a factor
of $\frac{102}{16}$ due to its larger diameter, resulting in a significantly
improved signal-to-noise ratio. The ion fluence ($\Phi$) was then calculated
from the measured ion current ($I$), the ion charge state ($z$ = 6), the
diameter of the thick wire ($d$ = 16 µm), the aperture height
($l_{\mathrm{irr}}$ = 4 mm), and the elementary charge using equation

$\Phi(t)=\frac{1}{eZdl_{\mathrm{irr}}}\int_{t_{0}}^{t}I(t^{\prime})\,\textup{d}t^{\prime}.$
(6)

The second function of the $102\text{\,}\mathrm{\SIUnitSymbolMicro m}$ wire
was to protect the sample during accelerator start-up and adjustment. For this
purpose, the sample holders were rotated by
$180\text{\,}\mathrm{\SIUnitSymbolDegree}$ so that the sample lied in the beam
shadow of the thick wire, preventing the sample from being irradiated before
the start of the experiment.

Previous to the actual main experiment, a preliminary experiment was
performed. In this test, the sample is mechanically tensioned to
$110\text{\,}\mathrm{\char 37\relax}$ of the initial value of the main
experiment, and the force drop measured over a duration of about
$2\text{\,}\mathrm{h}$. This was done to quantify the relaxation of the entire
test setup. During the first $30\text{\,}\mathrm{m}\mathrm{i}\mathrm{n}$, an
exponential drop of the force was observed, followed by a linear decay. This
was likely due to settling effects in the adhesive bonding of the sample. For
this reason, we delayed the start of the main experiment by
$30\text{\,}\mathrm{m}\mathrm{i}\mathrm{n}$ after the application of the
mechanical load.

For the main experiment, the sample was again mechanically tensioned to
initial values of either $0.5\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$,
$1.0\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$,
$1.5\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$, or
$2.0\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$. After the aforementioned
$30\text{\,}\mathrm{m}\mathrm{i}\mathrm{n}$ delay, the irradiation with the
high-energy heavy ion beam began, while the position of the sample ends is
kept constant. Since almost all of the energy of the impacting ions was
deposited in the sample, it heats up by less than $100\text{\,}\mathrm{K}$,
resulting in thermal expansion and an immediate drop in the tensile force. To
compensate for this, the irradiation was carried out in intervals, and the
force drop was only measured during brief irradiation pauses after a short
cooling phase. At the start of the experiment, the interval duration was
$5\text{\,}\mathrm{m}\mathrm{i}\mathrm{n}$ to provide higher time resolution,
with the interval length gradually increased up to
$30\text{\,}\mathrm{m}\mathrm{i}\mathrm{n}$ during the experiment.

The $20.3\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$ W6+ ion beam used to
generate the damage in the sample had a beam spot with a diameter of about
$1.5\text{\,}\mathrm{m}\mathrm{m}$ and a Gaussian intensity distribution. To
ensure uniform irradiation along the sample axis, the ion beam was scanned
vertically at a frequency of $1\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$ and
limited to a height of $4\text{\,}\mathrm{m}\mathrm{m}$ using a tungsten
aperture. Due to the significant diameter ratio between the ion beam and the
sample wire, the intensity was assumed to be constant across the sample
diameter. The net ion current passing through both wires was initially
approximately $0.15\text{\,}\mathrm{n}\mathrm{A}$ to achieve high time
resolution and was gradually increased up to
$0.9\text{\,}\mathrm{n}\mathrm{A}$ during the course of the experiment to
reach the desired final dose. This resulted in an ion flux in the range of
$2.5\text{\times}{10}^{11}\text{\,}\mathrm{c}\mathrm{m}^{-2}\mathrm{s}^{-1}1.5\text{\times}{10}^{12}\text{\,}\mathrm{c}\mathrm{m}^{-2}\mathrm{s}^{-1}$.

The measured force values needed to be corrected for two sources of error. The
first source was a long-term drift of the zero point in the load cell. A
preparatory experiment showed that the zero point of the load cell shifted by
less than $1\text{\,}\mathrm{m}\mathrm{N}$ during the typical 6 hour duration
of the experiment. To quantify this effect, the sample was completely unloaded
at the end of the main experiment at time $t_{\mathrm{E}}$, allowing the
determination of the zero drift $F_{\mathrm{E}}$. Each measured value $F_{i}$
at time $t_{i}$ was then corrected using equation

$F_{i}^{\mathrm{corr,1}}=F_{i}-F_{\mathrm{E}}\frac{t_{i}}{t_{\mathrm{E}}}.$
(7)

The second correction pertained to the relaxation of the entire measurement
setup, particularly the epoxy resin embedding of the sample ends. To address
this, the stress-strain curve obtained from the preliminary experiment was
divided into intervals of 1000 data points, and the slope of each interval was
determined. The individual slopes were then divided by the average force in
the interval, and the overall slope average ($\alpha$) was calculated. The
measured values were then corrected using equation

$F_{i}^{\mathrm{corr,2}}=\sum_{j=1}^{i}(t_{j}-t_{j-1})\cdot\alpha\cdot\frac{F_{j}+F_{j-1}}{2}.$
(8)

Molecular dynamics. Simulations were run using the LAMMPS35 software,
employing an embedded atom model potential for tungsten31. This potential was
chosen because it accurately predicts the defect content of highly irradiated
material, in agreement with experiments36, 18. All simulations were
initialised as pristine single-crystal tungsten consisting of
100$\times$100$\times$250 body-centered crystal unit cells
(32$\times$32$\times$80 nm3), with periodic boundary conditions applied to all
three directions in order to describe a materials volume element
representative of a grain in a wire. Uniaxial stress was applied along the
$z$-direction by minimising the potential energy of the system with respect to
atom positions and box dimensions using the method of conjugate gradients,
under the constraint that all but the $\sigma_{\mathrm{zz}}$ component of the
internal stress tensor reach near zero. To introduce irradiation damage,
rather than simulating the entire track of incident high-energy tungsten ions
that extend several microns into the material, we leveraged the known property
of collision sequences in which damage-generating scattering events are
relatively rare and can be considered as independent and uncorrelated. We used
the srim software15, which is based on the binary collision approximation
method, to generate a set of primary recoil energies representative of those
produced by 20.3 MeV W6+ ions impacting solid bulk tungsten. The effect of
irradiation was then introduced by the following iterative procedure: First,
recoil energies are randomly drawn from the representative set of recoil
energies until a target dose increment of
$\sum_{i}N_{\mathrm{d}}(T_{\mathrm{d}}^{(i)})/N$ $>$ 0.0002 dpa is reached,
where $N_{\mathrm{d}}(T_{\mathrm{d}})$ is an estimate for the number of atoms
displaced by a recoil with damage energy $T_{\mathrm{d}}$ in the NRT model14;
a convention commonly used in the nuclear materials community. The damage
energy $T_{\mathrm{d}}^{(i)}$ is computed from the recoil energy $E_{R}^{(i)}$
using Lindhard’s formula14 to account for losses due to electronic stopping.
For each drawn recoil energy, a random atom in the system is selected and
assigned the corresponding kinetic energy with randomly-oriented velocity. We
avoided selecting atoms too close to one another to prevent any spurious
effects caused by overlapping regions where damage production occurs. Next,
the simulation is propagated in the NVE ensemble for a duration of 10 ps,
which was sufficient for the cascade dynamics to conclude. We including
damping terms describing energy loss due to electronic excitations; for more
details on this, we refer to Ref.18. After the propagation, the atom
velocities are set to zero, and the system is relaxed to a local energy
minimum while maintaining the uniaxial stress constraint. This procedure is
repeated, with every iteration incrementing the dose by approximately 0.0002
dpa, until a target dose of 0.5 dpa is reached. After each relaxation step,
the simulation cell dimensions were recorded in a file, from which we computed
the eigenstrain as a function of dose for the specified target stress using
the formula
$\bm{\mathrm{\varepsilon}}^{*}(\phi;\sigma_{\mathrm{zz}})=\bm{\mathrm{A}}_{i}\cdot\bm{\mathrm{A}}_{0}^{-1}-\mathds{I}$,
where $\bm{\mathrm{A}}_{i}$ is a matrix with columns represented by the
simulation cell vectors at dose $\phi_{i}$.

Virtual wire model. The full boundary value problem is given by

$\displaystyle\bm{\nabla}\cdot\bm{\mathrm{\sigma}}(\bm{r},t)$
$\displaystyle=\bm{\nabla}\cdot\bm{\mathrm{\sigma}}^{*}(\bm{r},t)$
$\displaystyle\bm{r}\in\Omega$ (9)
$\displaystyle\bm{\mathrm{\sigma}}(\bm{r},t)\cdot\bm{n}(\bm{r})$
$\displaystyle=\bm{\mathrm{\sigma}}^{*}(\bm{r},t)\cdot\bm{n}(\bm{r})$
$\displaystyle\bm{r}\in\partial\Omega_{\mathrm{t}}$
$\displaystyle\bm{u}(\bm{r},t)$ $\displaystyle=\bm{u}^{\mathrm{C}}(\bm{r})$
$\displaystyle\bm{r}\in\partial\Omega_{\mathrm{u}}.$

We solve for the displacement field $\bm{u}(\bm{r})$ within the region
$\Omega\subset\mathbb{R}^{3}$ representing the elastic body, subject to
traction and displacement conditions applied to the surfaces
$\partial\Omega_{t}$ and $\partial\Omega_{u}$, respectively, with
$\partial\Omega_{t}\cup\partial\Omega_{u}=\partial\Omega$. Vector field
$\bm{n}(\bm{r})$ represents the outward-facing normal vector of surface
$\partial\Omega$.

The boundary value problem was solved using the finite element method (FEM)
with the FEniCS37 solver. In practice, we exploit that the wire diameter ($d$
= 16 µm) is significantly smaller than both the initial wire length ($l$ = 15
mm) and the length of the irradiated wire section ($l_{\mathrm{irr}}$ = 4 mm),
and therefore only solve for the fields within a two-dimensional x-y slice of
the irradiated wire section, considering it as representative of the entire
irradiated region. In the simulations, we described the uniaxial tensile loads
applied in experiment as an external homogeneous stress
$\bm{\mathrm{\sigma}}^{\mathrm{ext}}$, in which only the
$\sigma^{\mathrm{ext}}_{\mathrm{zz}}$ component was non-zero. The wire
boundaries were subject to traction free conditions. Before simulating the
stimulated stress relaxation, we required a spatial map of the dose-rate
$\dot{\phi}(x,y)$ within the wire slice, originating by the ion beam
irradiation, as the wire is only irradiated from one side and the beam does
not penetrate beyond a surface layer of about 2 µm thickness. We used srim15
to generate the dose-rate profile $\dot{\phi}(x,y)$ shown in Fig. 1, with more
comprehensive details given in the supplemental material. Beginning with step
$n=0$, the initial dose and eigenstrain were set to zero: $\phi_{n}(x,y)=0$
and $\bm{\mathrm{\varepsilon}}^{*}_{n}(x,y)=0$. Additionally, the initial
external stress was set at
$\bm{\mathrm{\sigma}}^{\mathrm{ext}}_{0}(x,y)=\bm{\mathrm{\sigma}}^{\mathrm{ext}}$.
The stress relaxation was propagated using the following explicit method:
First, the FEM problem is solved for the total stress
$\bm{\mathrm{\sigma}}_{n}(x,y)$, from which we obtain the elastic stress in
the irradiated region by subtracting the eigenstrain contribution5,
$\bm{\mathrm{\sigma}}^{\mathrm{el}}_{n}(x,y)=\bm{\mathrm{\sigma}}_{n}(x,y)-\bm{\mathrm{C}}:\bm{\mathrm{\varepsilon}}^{*}_{n}(x,y)$.
Next, the dose profile is updated to the next time step,
$\phi_{n+1}(x,y)=\phi_{n}(x,y)+\dot{\phi}(x,y)\Delta t$, and the eigenstrain
is incremented using the previous elastic stress state, under assumption that
dimensional changes are accrued irreversibly:

$\bm{\mathrm{\varepsilon}}^{*}_{n+1}(x,y)=\bm{\mathrm{\varepsilon}}^{*}_{n}(x,y)+\bm{\mathrm{\varepsilon}}_{\mathrm{MD}}^{*}[\phi_{n+1},\bm{\mathrm{\sigma}}_{n}^{\mathrm{el}}](x,y)-\bm{\mathrm{\varepsilon}}_{\mathrm{MD}}^{*}[\phi_{n},\bm{\mathrm{\sigma}}_{n}^{\mathrm{el}}](x,y).$

The new eigenstrain is used to update body forces and surface tractions for
the next iteration of the FEM solver. Relaxation of the uniaxial stress arises
from the length change of the irradiated section, which follows entirely from
the eigenstrain5 through
$\langle\sigma^{\mathrm{el}}_{\mathrm{zz}}\rangle=\sigma^{\mathrm{ext}}_{\mathrm{zz}}-E\langle\varepsilon_{\mathrm{zz},n}^{*}\rangle
l_{\mathrm{irr}}/l$, where $E$ is the Young’s modulus of tungsten and
$\langle\varepsilon^{*}_{\mathrm{zz},n}\rangle$ is the component of the
eigenstrain along the wire length at step $n$, averaged over the wire cross-
section. This procedure is repeated until the simulation advances to the
desired exposure time. After every iteration, the profiles of dose,
eigenstrain, and elastic stress are exported.

Uncertainty quantification. The stochastic variation of the atomistic
eigenstrain data-sets gives rise to uncertainty on the surrogate model
parameters $\hat{\bm{\theta}}$, which propagates up the scale as uncertainties
on the predictions from the virtual wire model. Using the Metropolis-Hastings
(MH) algorithm, we obtain $10^{5}$ random samples of the parameter vector
$\bm{\theta}$. The Markov chain is started with an initial value of
$\bm{\theta}_{0}=\hat{\bm{\theta}}$, and moves are proposed according to the
protocol $\bm{\theta}_{n+1}=\bm{\theta}_{n}+x\cdot\bm{\mathcal{N}}(0,\bm{v})$,
where $\bm{v}$ is a vector with components $v_{i}=\hat{s}_{i}^{2}$ and $x$ is
a step-size between 0.02 and 0.03, chosen to maintain an acceptance ratio
between $10\text{\,}\mathrm{\char 37\relax}$ and $20\text{\,}\mathrm{\char
37\relax}$. A move is accepted if
$r\leq\mathcal{L}(\bm{\theta}_{n+1}|\bm{\varepsilon}^{*})/\mathcal{L}(\bm{\theta}_{n}|\bm{\varepsilon}^{*})$,
where $r\in[0,1]$ is a uniform random number. The initial $10^{4}$ samples are
discarded to allow for burn-in. This process is repeated for every eigenstrain
data-set for each common external stress, resulting in $10^{5}$ realisations
of surrogate model parameter vectors $\bm{\theta}$ per stress for both
parallel and perpendicular eigenstrains. Finally, every virtual wire
simulation is independently executed $5\text{\times}{10}^{3}\text{\,}$ times,
each time using a surrogate model
$\bm{\mathrm{\varepsilon}}^{*}_{\mathrm{MD}}(\phi,\sigma_{\mathrm{zz}}^{\mathrm{ext}})$
with parameter vectors randomly drawn from their respective Markov chains.
From this, a distribution of stress relaxation curves is obtained, allowing
construction of the confidence intervals shown in Figs. 1 and 4.

Data availability
The datasets generated during the current study will be made public on
acceptance of this paper.

Code availability
The open-source computer code LAMMPS used for MD simulations is available at
https://lammps.sandia.gov. The open-source Python package FEniCS used for
multiscale simulations is available at https://fenicsproject.org/.

Acknowledgements
We are grateful to Robert Lürbke for taking the photograph shown in Fig. 1a.
This work has been carried out within the framework of the EUROfusion
Consortium, funded by the European Union via the Euratom Research and Training
Programme (Grant Agreement No. 101052200 - EUROfusion), and by the RCUK Energy
Programme, Grant No. EP/W006839/1. To obtain further information on the data
and models underlying the paper please contact<EMAIL_ADDRESS>The views and opinions expressed herein do not necessarily reflect those of
the European Commission. The authors acknowledge the use of the Cambridge
Service for Data Driven Discovery (CSD3) and associated support services
provided by the University of Cambridge Research Computing Services 38 that
assisted the completion of this study.

Author contributions
A.F. and M.B. contributed equally to the formulation of the scientific problem
and conceptualisation of experimental and computational studies. A.F. designed
and conducted the experiments and analysed the stress relaxation data. M.B.
developed the multiscale simulation framework and conducted the simulations.
S.L.D. and J.R. contributed jointly to design of the research project. All
authors contributed to the data discussion and manuscript preparation.

Competing interests
The authors declare no competing interests.


# Deep Image Retrieval is not Robust to Label Noise

Stanislav Dereka
Tinkoff
<EMAIL_ADDRESS>Ivan Karpukhin
Tinkoff
<EMAIL_ADDRESS>Sergey Kolesnikov
Tinkoff
<EMAIL_ADDRESS>
###### Abstract

Large-scale datasets are essential for the success of deep learning in image
retrieval. However, manual assessment errors and semi-supervised annotation
techniques can lead to label noise even in popular datasets. As previous works
primarily studied annotation quality in image classification tasks, it is
still unclear how label noise affects deep learning approaches to image
retrieval. In this work, we show that image retrieval methods are less robust
to label noise than image classification ones. Furthermore, we, for the first
time, investigate different types of label noise specific to image retrieval
tasks and study their effect on model performance.

##  1 Introduction

Over the last decade, deep learning achieved impressive results in multiple
computer vision domains, including image classification [16], image retrieval
[2], and face verification [6]. One reason behind the success of deep learning
is the ability of deep neural networks to extract useful information from
large amounts of annotated data. It was shown that increasing the amount of
data along with model size leads to an improvement in prediction quality [13,
31]. To do so, datasets with tens of thousands and even millions of items were
collected for deep learning [8, 18]. While the majority of data can usually be
obtained from the Internet, annotating this data is the most laborious part of
the data preparation process [28]. When provided with millions of items, it is
almost impossible to manually annotate all of them. Multiple semi-supervised
approaches were created to reduce the amount of manual work required in data
annotation [24]. However, both manual annotation and semi-supervised
algorithms can introduce errors to final results. It was shown recently that
the annotation error rate in large-scale datasets can exceed 40% [33].

Previous research on label noise in computer vision was primarily focused on
image classification [26, 1]. In this paper, we study the performance of deep
convolutional neural networks in image classification and retrieval tasks when
working with label noise. The core contributions of the paper can be
summarized as follows:

  1. 1.

We study the effects of label noise in image retrieval on Stanford Online
Products [25], In-shop [21] and face recognition datasets [8, 11] along with
image classification on ImageNet [28]. Our results show that image retrieval
models are less robust to label noise than image classification models under
similar training conditions.

  2. 2.

We study different retrieval-specific noise types with respect to the number
of mislabeled samples and corrupted classes. Our findings show that the amount
of corrupted classes affects performance more than the proportion of corrupted
samples in each class.

##  2 Related Work

In deep computer vision, label noise was primarily studied in the context of
image classification [26, 1]. Many techniques were proposed to reduce the
effects of label noise on the final models’ performance. For example, one of
these methods involves automatically detecting mislabeled items and excluding
them from training [27, 29, 17]. Another way to deal with label noise is to
design specialized training algorithms [3, 20, 5]. While these methods can, to
some degree, reduce the destructive impact of annotation errors, they are
still affected by the labels’ quality. While we find these approaches valuable
for the community, we argue for deeper model robustness analysis in image
retrieval.

Several previous works addressed the general question of deep image
classification robustness to label noise [13, 15, 31, 37]. It was shown that
modern convolutional neural network (CNN) architectures, such as ResNet [10],
are robust to label noise in image classification. While the models’ final
accuracy reduces with the growing noise level, the model still trains even
with five times more noisy samples than clean ones. Similar studies were
recently performed for image retrieval and face recognition [33, 19, 12, 4],
showing that metric learning algorithms are more sensitive to label noise than
classification methods. While there is a fundamental difference between
classification and metric learning approaches when it comes to the effects of
label noise, the reason for this difference is still unclear. To address the
issue of earlier works using different network architectures, training
strategies, and noise types, we compared retrieval, classification, and face
recognition methods under similar training conditions. Also, we study novel
types of noising schemes designed for image retrieval.

##  3 Label Noise in Image Retrieval

Many image retrieval datasets are gathered using unreliable methods such as
search engine crawling [4, 33, 9], which leads to label noise and degrades
learning [33]. Manual cleaning of large-scale datasets with millions of
samples can be expensive and resource demanding [33]. According to previous
research, data annotation techniques used in image retrieval can produce
specific types of label noise [33, 19]. In particular, noise can be
concentrated in clusters of similar images leading to non-uniform distribution
of noisy labels among classes. Moreover, well-studied datasets, e.g. Labeled
Faces in the Wild (LFW) 111See known LFW errors on the official website:
http://vis-www.cs.umass.edu/lfw/ [11], include ”trash” classes which
completely consist of mislabeled samples. We hereby raise a question of how
the number of corrupted (trash) classes affects final model performance? We
call class corrupted if almost all items in this class have the wrong label.
This question differs from previous research in image classification, where
some amount of clean data needs to be provided for each class.

Figure 1: Class size distributions in popular image retrieval datasets.
Examples of classes affected by small and large class label noise are
highlighted in red and green respectively.

When modeling label noise patterns, there are multiple ways to choose classes
for corruption. As shown in Figure 1, image retrieval datasets are highly
class-imbalanced. Large classes can contain thousands of times more samples
than small ones. By selecting a fixed amount of noisy items, we can choose
either many small classes or a small number of large classes. Based on this
observation, we propose two new label noising schemes for image retrieval.

In the proposed large class label noise we select the most frequent classes
for a given proportion of samples in a dataset and shuffle the labels among
the samples of the selected classes. The number of classes to corrupt is
selected to match the expected number of corrupted samples. Small class label
noise is similar to large class label noise, but the rarest classes are
corrupted instead of large ones. The dataset parts which were corrupted by
each noise pattern are illustrated on histograms in Figure 1. The proposed
patterns are opposites regarding the number of totally corrupted classes.
Large class label noise minimizes the number of corrupted classes while the
number of corrupted samples is fixed. In contrast, small class label noise
maximizes the number of corrupted classes.

As a baseline, we study traditional uniform label noise [1], also known as
label flipping [33]. In this noise pattern, a given proportion of samples in a
dataset is randomly selected, and a random label from all the available labels
is assigned to the selected samples. This noise pattern slightly corrupts each
class in the dataset with a probability proportional to the number of samples
in a class.

##  4 Experiment Setup

###  4.1 Datasets and Evaluation Metrics

The scope of this paper covers both image retrieval and image classification
datasets. We include the image classification task in our work to compare its
robustness to label noise with image retrieval. Closed-set classification
datasets share the same set of labels between training and testing. The goal
of an algorithm in this case is to predict label for each input image.
Accuracy is usually used to evaluate classification performance. For image
classification we use the ImageNet [28].

The goal of image retrieval (IR) is to find images from the gallery collection
which are similar to the query image [25, 23]. Let’s call retrieval successful
for some query if the most similar image in the gallery has the same label. We
measure Recall@1 (R@1) during evaluation, which is equal to the fraction of
successful retrievals.

Datasets for image retrieval usually have class-disjoint training and testing
parts. This setup is also called open-set problem. In our experiments we use
Stanford online products (SOP) [30] and In-shop Clothes (InShop) [21]
datasets. Both of these datasets have severe class imbalance as shown in
Figure 1.

In addition to the above, we extend the scope of our work by adding face
recognition (FR) datasets. The majority of face recognition datasets has class
imbalance similar to other retrieval datasets. Furthermore, face recognition
tasks are open-set and can be subject to common image retrieval problems [35].
Finally, public FR datasets are large-scale and make it possible for the
experiment setup to be close to real-world practical applications. For
training, we use the MS-Celeb-1M [8] dataset, which contains over 10M images
of around 100K individuals. For face recognition evaluation, we chose the LFW
[11] dataset. In addition to computing R@1, we evaluate face verification
performance. LFW verification test set includes 6K image pairs labeled with 0
(same person) or 1 (different people). During testing, a similarity in model
embedding space is computed for each pair. For face verification, we then
compute TPR@FPR metric for 10-3 FPR by using a common FR approach [35, 14].

###  4.2 Training Details

In image retrieval and classification datasets (SOP, In-shop, ImageNet) we
preprocess images using approaches from previous papers [10, 23]. Images in
ImageNet, SOP, and InShop datasets are resized to 256 pixels by the shortest
side. During training, each image is randomly cropped to 224x224 pixels.
During testing, we use central crop. In MS-Celeb-1M v2 dataset [6], the size
of aligned images is 112x112 pixels in both training and testing. We use
random horizontal flip augmentation for all datasets.

In all experiments, we use ResNet50 [10] CNN architecture as a backbone
network. ResNet50 is initialized with ImageNet pretrain weights for all the
tasks, except for ImageNet and MS-Celeb-1M, where random initialization is
used. The embedding size is set to 512 in all experiments.

Our models are trained with the ArcFace [6] loss function. According to recent
works [23, 6], this is one of the best-performing loss functions in considered
benchmarks. We use the default ArcFace scale and margin hyperparameters from
the original paper for training on SOP, InShop, and MS-Celeb-1M. For ImageNet,
we use Normalized Softmax loss [34].

We use the stochastic gradient descent (SGD) optimizer with momentum 0.9 and
weight decay $0.0001$. The initial learning rate is set to 0.1. We follow the
pipeline from [6] and perform 16 epochs of training, decreasing the learning
rate by a factor of 0.1 at the 9th and 14th epochs. In all experiments we set
batch size to 256.

##  5 Experiment Results

###  5.1 Robustness Comparison of Classification and Retrieval Tasks

Dataset | Metric | Label noise level
---|---|---
|  | clean | 0.01 | 0.05 | 0.1
ImageNet | Accuracy | 67.32 | 67.09 | 66.42 | 65.80
InShop | R@1 | 84.61 | 82.09 | 74.70 | 72.70
SOP | R@1 | 63.49 | 63.68 | 60.72 | 58.78
LFW | R@1 | 67.22 | 67.36 |  $\approx$0.0 |  $\approx$0.0
LFW | TPR@10-3 | 99.27 | 99.23 |  $\approx$0.0 |  $\approx$0.0
Table 1: Uniform label noise effects on model performance for image
classification (ImageNet) and verification tasks compared to image retrieval
tasks. All metric values are shown in %. We label cases where the model was
unable to generalize on test set with $\approx$.

The goal of our experiments is to measure the robustness of image retrieval
tasks for comparison with image classification. We quantify robustness as a
drop in performance when a model is trained on a noisy dataset compared to
training on a clean one. We compare models trained with multiple noise levels
ranging from 0 (clean dataset) to 10%. The results are presented in Table 1.

Figure 2: Training curves for MS-celeb-1M dataset with different label uniform
noise levels.

On all retrieval tasks, we observe a rapid decrease of quality with the
increase of uniform label noise level. Even 5% of label noise causes a
relative performance drop of up to 10%. In contrast, classification accuracy
under equal conditions shows a much smaller drop. These results show that
image classification is more robust to uniform label noise than image
retrieval.

One of the most interesting findings from our experiments is that, when a
model is trained on the MS-Celeb-1M dataset with at least 5% label noise, its
score is close to that of an untrained model. In order to clarify this
phenomena, we show training curves for several noise levels in Figure 2. It
can be seen that at 5% label noise, loss function optimization still occurs,
which means that the model training still converges. These findings allow us
to conclude that image retrieval setups are extremely sensitive to label noise
and their performance can be affected drastically by only a small fraction of
label noise.

###  5.2 Effects of Noise Patterns

Noise type | Dataset | Noise level
---|---|---
|  | 0.01 | 0.05 | 0.1
|  | Corrupted classes, %
Uniform | InShop | 6.0 | 26.0 | 44.3
Large Classes | InShop | 0.05 | 0.5 | 1.6
Small Classes | InShop | 2.5 | 11.1 | 19.2
Uniform | SOP | 5.0 | 22.6 | 40.0
Large Classes | SOP | 0.4 | 2.2 | 4.4
Small Classes | SOP | 2.6 | 13.2 | 23.8
Uniform | MS-Celeb-1M | 45.5 | 88.0 | 95.5
Large Classes | MS-Celeb-1M | 0.2 | 1.4 | 4.2
Small Classes | MS-Celeb-1M | 7.3 | 19.5 | 29.1
Table 2: Percentage of corrupted classes for each dataset and noise pattern
for three noise levels. A class is considered corrupted if at least one label
in this class has been flipped during the noising procedure.

According to the previous experiment, image retrieval is not robust to label
noise. Therefore, we can raise a question of how the number of corrupted
classes affects retrieval performance. To answer this question, we corrupted a
training set using new types of label noise for image retrieval, previously
described in section 3. Given a fixed set of noisy items, we can either
corrupt a small number of large classes or many small classes. These noising
approaches are called large-class noise and small-class noise respectively.
The total amount of corrupted items and corrupted classes in our experiments
is given in Table 2.

It can be seen from Figure 3 that the more training set classes are affected
by label noise, the higher is the drop in performance, even if the total
amount of corrupted elements doesn’t change. For example, the noise level 5%
on the InShop dataset leads to a 10% quality drop for uniform noise with 26%
corrupted classes. On the other hand, large and small class noises with the
same 5% noise level produce only 2% and 4.5% quality drop, while the numbers
of corrupted classes for these noises are 0.05% and 2.5% respectively (Table
2).

As a conclusion of these experiments, small class label noise leads to more
significant degradation than large class label noise. The effect of uniform
label noise is the strongest, as it affects more classes than the other noise
types. It’s worth noting that thoroughly corrupted classes affect performance
less than a large amount of slightly damaged classes.

Figure 3: Label noise patterns’ effect on test performance.

##  6 Discussion and Future Work

According to our experiments, image retrieval is less robust to label noise
than image classification. We used similar models, training objectives, and
optimization methods for both tasks. The difference in robustness can lie in
contrast between closed-set and open-set tasks. Image retrieval models need to
generalize well to unseen classes, while classification tasks share the same
set of labels between training and testing. Our experiments show that while
models converge with all considered noise levels, the test set accuracy
completely drops even for 5% label noise. These results bring us to the
conclusion that image retrieval models fail to generalize to unseen classes in
this case, highlighting the need for further investigation on label noise
robustness in image retrieval and other open-set tasks.

The goal of this work is to highlight difference between classification and
retrieval robustness to label noise as well as to study retrieval-specific
types of annotation errors. We choose ArcFace as one of the best-performing
models for our experiments. Future work can extend presented results to a
variety of training objectives [36, 22] and model architectures [32, 7].

##  7 Conclusion

In this work, we studied image retrieval robustness to label noise. According
to our experiments, image retrieval is less robust to label noise than image
classification approaches. Therefore, more attention must be paid to training
data quality for image retrieval tasks. We further showed that the performance
is rather affected by the number of corrupted classes than by the proportion
of corrupted samples in each class.

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