# Topological changes and deformation mechanisms of nanoporous Ta under
compression

N. Vazquez von Bibow  E.N. Millán  C.J. Ruestes  Facultad de Ciencias Exactas
y Naturales, Universidad Nacional de Cuyo, Mendoza 5500, Argentina  Instituto
de Física Enrique Gaviola, Universidad Nacional de Córdoba UNC-CONICET,
Facultad de Matemática, Astronomía, Física y Computación, Av. Medina Allende
s/n, Ciudad Universitaria, 5000, Córdoba, Argentina.  Facultad de Matemática,
Astronomía, Física y Computación, 5000, Córdoba, Argentina.  Instituto
Interdisciplinario de Ciencias Básicas (ICB), Universidad Nacional de Cuyo
UNCuyo-CONICET, Facultad de Ciencias Exactas y Naturales, Padre Contreras
1300, 5500, Mendoza, Argentina  IMDEA Materials Institute, C/Eric Kandel 2,
28906, Getafe, Madrid, Spain

(August 28, 2024)

###### Abstract

While the mechanical behavior of noble nanoporous metals has been the subject
of numerous studies, less is known about their recently developed refractory-
based counterparts. Here we report on the mechanical properties, deformation
mechanisms and topological changes of nanoporous tantalum, a prototypical
refractory metal, by means of atomistic simulations of compression tests. An
open-source multi-cpu and gpu-capable software is presented and used for the
generation of computational samples. The stress strain curves show a non-
linear elastic response, with early yielding. The plastic regime is first
characterized by a linear hardening followed by an exponential hardening at
large strains, associated with a high degree of densification. Plasticity is
dominated by dislocation activity, with twinning and vacancy formation
appearing as complementary deformation mechanisms. In order to study the
mechanical response from a topological perspective, we track the evolution of
the genus throughout the tests, finding direct correlations with each regime
of the stress strain curves. The results are in agreement with previous
studies of plasticity in nanoporous metals and highlight the importance of
using topological metrics, for gaining insights into complex aspects of the
deformation of nanoporous metals.

††journal: Arxiv

##  1 Introduction

Nanoporous (np) metals offer a vast range of applications. This includes
unique performance for catalytic and optical applications due to its high
surface-to-volume ratio [1, 2]. In addition, their development opened the door
for the fabrication of surface-chemistry-powered actuators and sensors, among
other cutting-edge technology [3, 4]. The range of potential applications
extends to materials for advanced fission and fusion reactors [5, 6, 7, 8].
Studies on the mechanical properties of nanoporous metals typically deal with
noble metals, most notably nanoporous gold (np-Au) [9, 10, 11, 12, 13, 14, 15,
16, 17, 18, 19, 20, 21, 22, 23, 24]. Despite the remarkable properties of np-
Au, its thermal instability [25] and propensity to coarsening [26] limits
practical applications.

The development of modern dealloying techniques opened the door for the
fabrication of new non-noble nanoporous metals, such those made of refractory
metals. Important examples from a technological application perspective
include np-W, np-Ta, np-Mo and np-Nb [27, 28, 29, 30, 31, 32, 33]. Despite
their potential, experiments on mechanical characterization of such materials
is generally lacking, with a notable exception after the work of Zhao et al.
[34, 35], who used nanoindentation testing to probe the mechanical properties
of np-W with relatively high solid fraction (above 0.484). Their studies show
that np-W is a high performance material, with a high hardness (several GPa),
highlighting its potential applications in harsh environments, such as nuclear
facilities.

The mechanical properties of macroporous foams are typically dictated by the
relative density and characteristics of its base material. In contrast, the
mechanical properties of nanoporous metals are not only dictated by the
former, but also by their ligament size [10], surface effects [36] and network
connectivity [37] among other topological aspects [38, 39].

Molecular dynamics (MD) simulations, despite timescale and lengthscale
limitations, have proven to be a valuable tool for studying the mechanical
response of nanoporous metals. Two notable contributions serve as examples.
The first one is that of Farkas et al. [13], who first reported on tension-
compression asymmetry for nanoporous gold, confirmed experimentally several
years later [19]. The second one is that of Bringa et al. [5], who showed that
nanoporous metals can display radiation tolerance depending on irradiation
conditions and ligament size, opening a new line of application for these
materials [6]. In addition, the MD technique has been used to provide
important insights into a variety of effects. A non-exhaustive list includes
ligament size effects [40, 41], strain-rate effects [42, 43], grain size
effects [40, 44], temperature effects [45], anomalous compliance and early
yielding [46], densification and hardening [16, 41] and load-bearing network
and topological effects [47, 48, 49]. The technique has even been used to
derive and inform scaling laws [12, 22, 50] and also to provide fundamental
explanations on np-Au fracture [51]. More recently, it has provided valuable
insights into the mechanical response of graded nanoporous structures [52] and
hierarchical nanoporous structures [53].

The knowledge on the mechanical properties of nanoporous refractory metals is
still in its infancy compared to that of noble metals, for which much has been
investigated using nanoporous gold [24]. Differences in its base material,
slip systems, dislocation interaction and storage are expected to play an
important role in the mechanical response. However, these aspects have not
been deeply investigated so far and are brought into focus in this study. In
fact, dislocation accumulation in nanoporous gold has been linked to its
hardening behavior under compression, together with uniform densification. In
can be expected that if densification takes place, the connectivity of the
foam would change and if so, the strain hardening behavior of nanoporous foams
could be correlated to changes in its topology. This leads to the conjecture
that alterations in the foam’s topology, stemming from the occurrence of
densification, may significantly influence the strain hardening behavior of
nanoporous foams. Here we test this hypothesis employing surface
reconstruction techniques and common topological metrics.

In this work, we explore the mechanical response, deformation mechanism and
topology evolution during compression of nanoporous tantalum, a prototypical
refractory metal. An open-source multi-cpu and gpu-capable software is
presented and used for the generation of computational samples. We perform MD-
simulated compression tests up to large strains, in excess of 60 percent,
fully exploring the elastic regime, linear hardening regime and exponential
densification regime. In addition to the inspection of the deformation
mechanisms, we present an analysis of the evolution of topology during
deformation. Finally, we correlate the different hardening stages with
notorious changes in the topology of the foams.

##  2 Computational modeling

###  2.1 Code for generation of nanoporous samples

Nanoporous tantalum produced by liquid-metal dealloying can be classified as
an open-cell foam with nanoscale features [29]. It shows striking similarities
with nanoporous gold produced by dealloying [1, 54, 6], resembling those
structures associated with spinodal decomposition [55]. Initial attempts to
study nanoporous gold with MD simulations relied on sample generation by means
of phase-field methods [56, 57]. More recently, the development of levelled-
wave methods based on Cahn’s equation [58, 59], and even fully molecular
dynamics methods based on Ag-Au demixing [60] resulted in a significant
increase of studies of np metals using MD simulations. In this work, we employ
an in-house implementation of the levelled-wave method presented by Soyarslan
et al. [58]. This method has been successfully applied in a number of studies
[49, 52, 48, 53, 26]. For completeness, the method is briefly described below.

In 1965, Cahn presented the theory of phase separation from a single phase
fluid by a spinodal mechanism. He showed that the predicted structure may be
described in terms of the superposition of an infinite number of sinusoidal
waves of a fixed wavelength, but random in amplitude, orientation, and phase
[61]. More recently, Soyarslan et al. [58] explore a modern implementation of
this random field $f$:

$f(\mathbf{x})=\sqrt{\frac{2}{N}}\sum_{i=1}^{N}\cos(\mathbf{q_{i}}\dotproduct\mathbf{x}+\varphi_{i}).$
(1)

where $\mathbf{x}$ is the position vector, $N$ is the number of waves
considered on the truncated series, while $\mathbf{q_{i}}$ and $\varphi_{i}$
denote the direction and phase of the wave $i$, respectively. The wave number
must be set to a constant value (i.e. $\absolutevalue{\mathbf{q_{i}}}=q_{0}$),
with directions uniformly distributed over a $4\pi$ solid angle, and phases
uniformly distributed between $0$ and $2\pi$. Under this constraints,
$f(\mathbf{x})$ is a random gaussian field with $\expectationvalue{f}=0$ and
$\expectationvalue{f^{2}}=1$. Provided N is sufficiently large, the value
$f(\mathbf{x)}$ for a given position $\mathbf{x}$ follows a Gaussian
distribution:

$P(f)=\frac{1}{\sqrt{2\pi}}e^{-f^{2}/2},$ (2)

and the mean wave direction

$\mathbf{\nu}=\frac{1}{N}\sum_{i=1}^{N}\mathbf{n_{i}}$ (3)

tends to zero as consequence of the central limit theorem. Therefore, $f$ is
an isotropic function.

For equation (1), the different phases of the system are obtained by means of
a cutoff value $\xi$:

$\begin{split}&\mathbf{x}\in\mathcal{B}\qquad\text{if}\,f(\mathbf{x})<\xi,\\\
&\mathbf{x}\in\partial\mathcal{B}\;\,\quad\text{if}\,f(\mathbf{x})=\xi,\\\
&\mathbf{x}\in\mathcal{P}\qquad\text{if}\,f(\mathbf{x})>\xi.\end{split}$ (4)

Here, $\mathcal{B}$ is the solid phase, $\mathcal{P}$ is the porous phase and
$\partial\mathcal{B}$ is the boundary between the two phases. $\xi$ can be
correlated to the solid fraction, defined as
$\phi_{\mathcal{B}}:=\absolutevalue{\mathcal{B}}/\absolutevalue{\mathcal{V}}$
(the ratio of the volume of the solid phase to the total volume of the
sample), thanks to the properties of the random Gaussian field:

$\xi(\phi_{\mathcal{B}})=\sqrt{2}\operatorname{erf}^{-1}(2\phi_{\mathcal{B}}-1),$
(5)

The structures generated using eqns. (1) and (4) are, in general, non
periodic. Periodicity can be obtained by choosing a finite number of waves
with an integer wave number in all directions, with constant modulus.

Starting from (1), with $\mathbf{e_{1},e_{2},e_{3}}$ unit vectors of an
orthonormal base in real space, $\mathbf{q_{i}}$ must be of the form

$\mathbf{q}=\frac{2\pi}{A}(h,k,l),$ (6)

where $h,k,l\in\mathbb{Z}$ correspond to Miller indexes and $A$ is a constant.
It can be verified that
$f(\mathbf{x})=f(\mathbf{x}+n_{1}A\mathbf{e_{1}}+n_{2}A\mathbf{e_{2}}+n_{3}A\mathbf{e_{3}}),$
with $n_{1},n_{2}$ y $n_{3}$ arbitrary integers. This implies that $f$ is
periodic with magnitude $A$ with respect to the base. Moreover, if (1) is
taken with identical $\phi_{i}$, then $f$ is also invariant with respect to an
axis interchange. So the resulting $f$ has cubic symmetry with grid parameter
$A$. Based on the aforementioned considerations, periodic stochastic
structures can be built by restricting the sum on eqn. (1) to a set of
$\mathbf{q_{i}}$ with a given value and constant $H$, so that

$H=\sqrt{h^{2}+k^{2}+l^{2}},$ (7)

with $\absolutevalue{\mathbf{q}}=q_{0}=2\pi H/A$.

For structures with equal fraction of the two phases, symmetry suggests that
the average ligament size is equal to the average pore size:

$L=1.23\frac{\pi}{q_{0}}$ (8)

For nanoporous structures with solid fraction different than 0.5, the average
ligament size of the solid phase is given by:

$L=\frac{A}{H}[0.53\phi_{B}+0.41].$ (9)

For more details on the method, the reader is referred to [61, 58].

While the implementation of the method is straightforward by means of nested-
loop programming, such approach is not practical given memory and wall-time
limitations. Therefore, we opted for a more elaborated approach. We define a
rank-3 tensor

$T_{ijk}:=(\mathbf{q\dotproduct x})_{ij}+\phi_{k},$ (10)

for all points $\mathbf{x}$ of the domain and for all $\mathbf{q}$ wave
vectors on the set of directions. The implementation makes use of NumPy
111https://numpy.org/ . In order to avoid memory limitations, the $T_{ijk}$
tensor was decomposed into smaller tensors. Such segmentation allows for
multi-core parallelization using joblib
222https://joblib.readthedocs.io/en/stable/ and
loky333https://github.com/joblib/loky libraries. Finally, the code was also
implemented on GPUs using Tensorflow 2 framework
444https://www.tensorflow.org/. Benchmarking of the code is presented in
section 8. The code is provided open-source
555https://github.com/nicolasvazquez95/Nanofoams_Creator.

###  2.2 Nanoporous samples and their relaxation

The method explained in the previous section was used to produce stochastic
bicontinuous structures of two phases, $\alpha$ and $\beta$. Equation 4 was
sampled at the atomic locations of a [100]-oriented Ta single crystal and all
atoms located in phase $\alpha$ – as determined by the threshold parameter
$\xi$ – were removed to create pores, with the nanoscale porous structure
resulting from the remaining phase $\beta$ atoms. The samples were generated
using periodic boundary conditions, with a lattice parameter
$a_{0}=0.3303\leavevmode\nobreak\ nm$, $N=20000$ waves, $H^{2}=161$ and a
solid fraction $\phi_{B}\approx 0.3$. We generated three virtual np
structures, each with a different ligament size, see Table 1.

| As-generated |  | After
---
relaxation
| Mechanical
---
properties
| Simulation
---
domain
Atoms |  | $L_{box}$
---
(nm)
$\phi_{B}$ |  | L
---
(nm)
$\phi_{B}$ |  | L
---
(nm)
| E
---
(GPa)
| $\sigma_{y}$
---
(MPa)
150 $a_{0}$ | 2.3 106 | 49.5 | 0.291 | 2.26 | 0.294 | 2.4 | 1.80 | 89.3
200 $a_{0}$ | 5.5 106 | 66 | 0.301 | 3.1 | 0.303 | 3.1 | 1.83 | 91.0
250 $a_{0}$ | 10.3 106 | 82.5 | 0.297 | 3.78 | 0.298 | 3.8 | 2.18 | 98.4
Table 1: Summary of sample characteristics and dimensions. These correspond to
the as-generated samples (i.e. before relaxation). Relative densities were
computed using the ConstructSurfaceMesh algorithm implemented in OVITO.
Average ligament sizes $L$ were computed using FoamExplorer [62].

Prior to the simulation of mechanical tests, each foam was first minimized by
the conjugate gradient method specifying a force tolerance of $10^{-8}$
eV/Angstrom and an energy tolerance of $10^{-6}$ eV. Then the structure was
thermally relaxed at 300 K for 500 ps and zero pressure using a NPT ensemble.
Periodic boundary conditions were imposed in all directions. The simulations
were performed using LAMMPS [63] and the Ta atomic interactions were modeled
by means of an extended Finnis-Sinclair potential (EFS) [64]. This potential
successfully reproduces lattice constants, cohesive energies, elastic
constants and vacancy formation energies of Ta [64]. The generalized stacking
faults energy curves and generalized twinning fault energy curves had been
thoroughly studied [65, 66, 67] and compared to other Ta potentials [67]. The
potential has proven useful in describing dislocation-mediated plasticity in
closed-cell nanoporous Ta under tension and compression [65, 66, 68, 69], as
well as bulk Ta under spherical nanoindentation [67].

As a result of the minimization and relaxation strategy, a small amount of
ligament coarsening and a reduction of the sample dimensions was verified, and
the updated dimensions are presented on Table 1. These correspond to converged
values, i.e. they remained unaltered during the last 200 ps of the relaxation
stage. Relaxation of the initially rigid and defect-free porous crystal in the
absence of mechanical loading led to samples with a number of dislocations
(Fig. 1). Similar defects had been reported after transmission electron
micrographs of nanoporous gold produced by dealloying [70].

As a result of the whole procedure, we were able to generate open-cell np-Ta
atomistic samples with pre-existing defects, that closely resembles np-Ta
foams produced by liquid-metal dealloying [29], combining nanoscale ligaments
and pores. According to [58], the generated microstructures can be considered
as representative volume elements (RVEs) given the parameters used on their
generation.

Figure 1: a) Plan view of the largest sample generated (Simulation domain 250
$a_{0}$, $L_{box}$ =82.5 nm, Scale bar 20 nm) compared to a nanoporous Ta
sample produced by liquid metal dealloying (Reprinted from [29], with
permission from Elsevier). b) Dislocation (green) nucleated during relaxation
on a ligament junction (circle). c) Another example of dislocation generated
during relaxation.

###  2.3 Simulation details

Uniaxial compression tests were simulated with the loading axis aligned to the
z-axis. A thermostat and barostat were used to maintain lateral pressure (x
and y directions) at approximately 0 Pa and to keep temperature constant at
300K. Mimicking strain-controlled compressive mechanical deformation tests,
each sample was uniaxially compressed along the [001]-direction by re-scaling
its z-dimension at a strain rate of 1 × 108 s-1 with a 2 fs timestep. Stresses
are presented in the form of von Mises stress. LAMMPS output of the stress
tensor of the simulation box is $\sigma_{ij}$ from which the scalar quantity
$\sigma_{VM}$ can be computed as:

$\sigma_{VM}=\sqrt{\sigma_{xx}^{2}+\sigma_{yy}^{2}+\sigma_{zz}^{2}-(\sigma_{xx}\sigma_{yy}+\sigma_{xx}\sigma_{zz}+\sigma_{yy}\sigma_{zz})+3(\sigma_{xy}+\sigma_{xz}+\sigma_{yz})}$
(11)

Post-processing was performed using OVITO [71] and the Crystal Analysis Tool
[72], thus allowing for detailed defect identification and tracking.

##  3 Results

###  3.1 Stress - strain curves

Figure 2 (top) presents the stress-strain curves for the three samples
studied. Figure 3 presents an extract of the stress- strain curve focusing on
the elastic regime and the early stages of plasticity. Noticeable features
included a non-linear elastic regime (3), as well as early-yielding. Both
behaviors had been previously reported for nanoporous gold [46, 24].

Table 1 presents the elastic modulus $E$ and yield stress $\sigma_{Y}$
obtained for our samples. The former was obtained from the stress strain
response during the first 0.01 strain. The latter was taken as the
intersection of the stress strain curve with a parallel line offset by 0.01
strain, its slope being $E$. Both elastic modulus and yield strength decrease
with ligament size, in a smaller is weaker fashion. This is attributed to the
ligament size range explored (below 5 nm), akin to an inverse Hall-Petch
effect.

Figure 2: Top von Mises stress - strain plot for the three samples studied,
according to their average ligament size. Bottom Dimensionless genus per
volume ratio ( $g_{V}/g_{V}^{0}$) for the three samples studied. Figure 3:
Early stage of the von Mises stress - strain plot for the three samples
studied, according to their average ligament size.

All simulations present a linear hardening regime for strains below 0.4. While
hardening is typically absent for macroporous foams [73], it is not new for
nanoporous metals. Linear hardening in nanoporous Au under compression is
known to be ligament size-dependent. This was first reported in experiments
[74], but can also be found in molecular dynamics simulation literature [41].
In our case, all three samples display a similar linear hardening stage,
almost overlapping, most likely due to ligament sizes within the same order of
magnitude, unlike in [74, 41]. Under compression, nanoporous metals typically
densify uniformly, leading to a continuous hardening response [74]. As a
consequence, the onset of densification regime is ill-defined for nanoporous
metals, and it is known to vary with ligament size. That is clearly seen in
our curves. Samples with smaller ligament size densify earlier than those with
larger ligament size, in agreement with experimental findings for np-Au [24].
Interestingly, the exponential hardening regime shows a noticeable ligament
size dependence, and will be further discussed in Sec. 3.3.

###  3.2 Deformation mechanisms

Initially and up to moderate strains, our simulations reveal that nanoporous
Ta deforms by ligament bending with plastic collapse at the ligament nodes or
close to them (Figure 4). Plasticity mainly proceeds by dislocation activity .
Nucleation takes place at free surfaces in the form of glissile dislocations
(Figure 4.a,b). They propagate across the ligaments and annihilate at nearby
surfaces. These corresponds to dislocations with Burgers vector
$\vec{b}=\frac{a_{0}}{2}\langle 111\rangle$. Despite the abundance of free
surfaces that act as dislocation sinks, deformation proceeds in a dislocation
accumulation scenario, in agreement with experimental and computational
studies on nanoporous gold [74, 16]. It must be noted that under creep
conditions, np-Au MD studies suggest that nanoporous metals can deform by
displacive-diffusion mechanisms [75]. We found no indications of such
mechanism, most probably due to the high strain-rate conditions and short
timescales explored.

Figure 4: Dislocation activity. (a,b) At a strain of 0.1, few dislocations are
noticeable on some ligaments. Most of the generated dislocations have reached
nearby surfaces, effectively annihilating. (c,d) At a strain of 0.4, a large
number of dislocations are seen, mainly on ligament junctions. These are
glissile dislocations (green - $\vec{b}=\frac{a_{0}}{2}\langle 111\rangle$)
and sessile dislocations (pink - $\vec{b}=a_{0}\langle 100\rangle$). Note that
the densification has started.

In addition to surface exhaustion mechanisms, a fraction of dislocations get
retained inside the ligaments. Such retention is favored at ligament
junctions, where dislocations have enough space for reaction with other
dislocations, promoting junction formation as well as the generation of
sessile dislocations ($\vec{b}=a_{0}\langle 100\rangle$) by the following
reaction,

$\frac{1}{2}\langle
1\bar{1}\bar{1}\rangle+\frac{1}{2}\langle\bar{1}\bar{1}1\rangle\to\langle
0\bar{1}0\rangle$ (12)

Prominent examples of the aforementioned mechanism can be seen on Figure 5.a.
It is also shown that vacancy formation also takes place due to the reaction
of dislocations (Figure 5.b). It is likely that self-interstitials get
generated too. However, we failed to identify them. In MD simulations, self-
interstitials are generally identified by means of a Wigner-Seitz analysis
[76] comparing a deformed configuration against an undeformed one. Such a
comparison was not possible here due to the significant amount of deformation.
It must also be noted that point defect generation might be influenced by
strain-rate effects [77].

Figure 5: Dislocation junctions in a region with high dislocation density.
Reaction of dislocations of glissile dislocations
($\vec{b}=\frac{a_{0}}{2}\langle 111\rangle$, green) lead to the generation of
sessile dislocations ($\vec{b}=a_{0}\langle 100\rangle$, pink). Dislocation
reactions also promote vacancy generation. Arrows indicate Burgers vector.
Reconstructed surfaces of the foam are not displayed to favor the
visualization of dislolcations. Red clusters correspond to vacancies.

Figure 6 presents an evolution of the total dislocation length for our largest
sample. In addition, the total length is discriminated considering the burgers
vector of the dislocations. It can be seen that $\langle 111\rangle$ glissile
dislocations dominate, followed by $\langle 100\rangle$ and $\langle
110\rangle$ sessile dislocations. Dislocations with different burgers vector
are also identified by the algorithm (type ’Other’). However, detailed
inspection of them reveals that the mostly have a burgers vector proportional
to one of the previous three categories, while others are assigned a burgers
vector not consistent with dislocations in bcc metals pointing to possible
artifacts in the identification of dislocations in heavily strained
configurations. Similar trends were observed for all the samples simulated
here.

Figure 6: Evolution of total dislocation length for our largest sample tested
(250 $a_{0}$, $L=3.8\leavevmode\nobreak\ nm$). Contributions to the total
length include $<111>-type$ glissile dislocations, $<110>-type$ and
$<100>-type$ sessile dislocations and ’Other’-type dislocations, as described
in Sec. 3.2.

A small fraction of twinning was also found, see Figure 7. Tantalum is known
to deform by twinning under high strain-rate compression [78]. High strain-
rate conditions are inherent to MD simulations when exploring uniaxial
compression of metals and therefore, this finding is not surprising.
Experimental studies of nanoporous Ta at low strain rates are necessary to
determine if twinning takes place in absence of high strain rates.

Figure 7: Example of a twin generated during deformation. (a) Atoms colored
according to local structural environment: blue (bcc), red (twin boundaries),
white (other). (b) Atoms colored according to degree of rotation. Continuous
lines indicate ligament surfaces. Dashed lines indicate twin boundaries. Note
the rotation of the atoms between them.

###  3.3 Topological evolution during deformation

A positive aspect of atomistic simulations is that they allow for detailed
tracking of atomic positions and microstructures. In this way, and making use
of surface reconstruction algorithms [79], it is posible to track the
evolution of the topology throughout the simulated compression test.

At all stages of deformation, the surface of the sample can be determined by
surface reconstruction techniques. Here we used the Gaussian density method,
as implemented in OVITO [80]. Detailed information on the reconstructed
surface mesh is readily available. This includes: number of vertices (V),
number of edges (E) and number of faces (F) forming the reconstructed closed
surface. This allows computing the Euler characteristic ($\chi$) as

$\chi=V-E+F$ (13)

For a closed surface, the Euler characteristic is equivalent to the genus
($g$) through:

$g=1-\frac{\chi}{2}$ (14)

This is a metric of the surface connectivity; it can be interpreted as the
number of continuous holes, or handles, on the resulting manifold made by the
closed surface.

The determination of $\chi$ and $g$ after the reconstructed surface of the
nanoporous sample is a valid procedure used both in experimental approaches,
through FIB-nanotomography methods [81], and in molecular dynamics studies
[60].

In order to take into account volume changes, the genus is often normalized by
the total volume of the foam and scaled using a characteristic length to
produce a non-dimensional quantity [58]. We normalized the genus by the sample
volume $g_{V}=g/V$. Then, by taking the ratio of genus $g_{V}/g_{V}^{0}$,
$g_{V}^{0}$ being the genus per volume in the absence of strain, we obtain the
results in a non-dimensional form.

Figure 2 presents a correlation of the stress strain curves (a), with the
$g_{V}/g_{V}^{0}$ ratio (b). Interestingly, the evolution of the genus ratio
for the three samples also overlaps for the same strain regime that the stress
strain curves do. Most importantly, the genus increases during the linear
hardening regime and that increase starts right after the onset of plasticity,
indicating that continuous densification is present through out this regime.

As the deformation of the foam enters the plastic regime, the foam experiences
a compaction where more and more ligaments become into contact. As more
ligaments become into contact, then one would expect that more cuttings would
be needed to produce a disconected manifold. This can be seen on Figure 4 by
comparing panels b) and d). Such behavior holds until strains of 0.4 (small
sample - $L=2.4\leavevmode\nobreak\ nm$) to 0.5 (large sample -
$L=3.8\leavevmode\nobreak\ nm$) are reached.

For larger strains, the evolution of the genus shifts and starts decreasing.
Notable differences in genus are also consistent with stress strain curves
differentiation in the exponential densification regime, indicating variations
in the densification rate. Note that the genus data is presented up to a
strain of the order of 0.55. After that strain, the reconstructed surface is
no longer closed, thus preventing the computation of genus over the whole
sample. Analysis of the reconstructed surfaces reveals that, at such large
strains, porosity is now distributed in the form of disconnected voids. The
reconstructed surfaces we obtain at very high strains are indeed disconnected,
as seen on Figure 8. This is correlated to a change of the foam from its
original open-cell nature, to a partially closed-cell structure.

Figure 8: Reconstructed surface for the largest sample tested (250 $a_{0}$,
$L=3.8\leavevmode\nobreak\ nm$). a) Undeformed state showing the free surfaces
of the ligaments. b) At a strain of 0.7, the porous phase is no longer
continuous. The sides of the box show a significant compaction of the foam,
yet not fully densified. In fact, the interior of sample presents isolated
voids in the material. The porous phase is now discontinuous and pores with
different volume can be appreciated. In order to enhance the visibility, a
number of interior surfaces were removed.

##  4 Discussion

The mechanical properties of nanoporous metals have often been studied using
nanoporous gold as a prototypical example. Our results also show significant
points of agreement with previous studies on that material. To begin with, the
response clearly depicts three regimes, namely an elastic regime, a linear
hardening regime, and an exponential regime, consistent with previous reports
on np-Au [41, 74, 46]. Deformation proceeds by ligament bending and plasticity
takes place by dislocation nucleation at the ligament surfaces, in a
dislocation accumulation scenario [74, 46, 16, 22, 47]. After yielding, some
np Au works display a stress plateau [13, 16] whereas others display linear
hardening [46, 41]. Our results show an agreement with the latter, probably
due to the ligament size and relative density explored.

Mathesan and Mordehai [48, 49] used MD simulations to explore the influence of
topology on the mechanical properties of np-Au nanopillars. In particular,
they focused on the load-bearing network. They show a strong influence on the
elastic properties [48]. Later on, by detailed analysis of the genus of the
yielded ligaments network, they conclude that the rate by which the strength
increases with strain decreases gradually, in correlation with a decrease in
the genus of a subnetwork of the unyielded ligaments [49]. Although our study
focuses on a different base material in the form of bulk np-W (in contrast to
np-Au nanopillars in [48, 49]), a general agreement can be found. That is, (i)
after the onset of the plastic regime the genus increases until significant
plastic strain has been reached; (ii) a correlation of the genus with the
stress strain response.

The linear hardening regime has been typically associated with dislocation
activity as well as densification, whereas the exponential hardening is
heavily influenced by latter. This has been shown in MD simulations of
nanoporous gold under compression. Ngo et al. [46] studies on np-Au provided
compelling evidence pointing to dislocation storage as responsible for
hardening behavior together with densification. Ruestes et al. [16] and
Saffarini et al. [41] later presented more evidence and provided dislocation
density-based equations relating dislocation density and solid fraction
evolution. Here we provide an alternative interpretation for the different
strain hardening regimes.

Figure 9 presents the evolution of the genus ration and the total dislocation
density for the largest sample tested (250 $a_{0}$, $L=3.8\leavevmode\nobreak\
nm$). For interpretation purposes, the linear hardening regime and exponential
hardening regime are indicated, as revealed by Fig. 2. During the linear
hardening regime, dislocation density rapidly increases, together with an
increase in the genus ratio. The former indicates a dislocation-accumulation
scenario, whereas the latter indicates a correlation of topology with linear
hardening. During the exponential hardening regime, dislocation density keeps
increasing in a less marked way due to dislocation densities approaching
saturation limits ($\rho_{D}\sim 10^{17}/\text{m}{{}^{-2})}$. Interestingly,
the genus ratio drops sharply during the exponential hardening regime. These
results only allow for qualitative conclusions. Specific studies are needed to
provide an adequate quantitative assessment of the respective contributions of
dislocations and topological changes for the two regimes.

With respect to refractory-based nanoporous metals, Xia and co-workers [82]
used MD simulations to study the mechanical properties of nanoporous tungsten,
another prototypical bcc metal with significant applications. By means of
simulated tensile tests, they explored the effects of relative density, grain
size , and temperature on the mechanical properties and deformation mechanisms
of np-W. They also report on dislocation mediated plasticity, as well as on a
contribution of twinning as a complementary deformation mechanism. Similar
deformation mechanisms were reported by Valencia and coworkers [83] in their
MD study of nanoporous W under nanoindentation. Our results are in agreement
with these contributions.

Figure 9: Comparative assessment of the evolution of the genus ratio and total
dislocation density for the largest sample tested (250 $a_{0}$,
$L=3.8\leavevmode\nobreak\ nm$) with respect to the different hardening
regimes.

##  5 Conclusions

In summary, we report on a computational study of the compression behavior of
nanoporous Ta using molecular dynamics simulations. Computational samples were
constructed using our own implementation of a levelled-wave method. This
software is released as an open-source tool, with multi-cpu and gpu
capabilities, allowing for fast and efficient generation of nanoporous
samples. Our main conclusions are:

  * 1.

Consistent with previous studies on nanoporous gold, we find that the stress
strain curves depict similar qualitative features. Namely, a non-linear
elastic regime, early yielding, and linear hardening followed by an
exponential hardening at large strains, associated with a high degree of
densification.

  * 2.

Plasticity takes place by dislocation activity in the expected bcc slip
systems. Even though dislocations nucleate at the surface and most of them
annihilate at a nearby surface after slip through the ligaments, deformation
proceeds in a dislocation accumulation scenario. Ligament junctions provide
enough volume for dislocation reaction, producing sessile dislocations, as
well as dislocation storage. Densification of the sample favors dislocation
retention. Twinning and vacancy formation was also found.

  * 3.

The evolution of the genus was succesfully correlated to the stress strain
behavior. Genus increases during the linear hardening regime, right after the
onset of plasticity, indicating continuous densification of the sample. In
contrast, the genus significantly drops at very large strains, that is during
the exponential hardening regime. This is due to significant densification of
the sample, shifting from an open cell foam to a closed cell foam.

  * 4.

Structural changes from open-cell foam to closed-cell foam are shown to be
ligament-size dependent. Our analysis indicates that, whenever possible,
tracking of topological invariants (here we use $g$) are effective measures to
characterize the evolution of the sample and could provide valuable insights
for the identification of deformation-induced changes.

As a closing remark, our work suggests that fundamental understanding gained
from np-Au studies is directly transferable to other nanoporous metals, albeit
with differences in their deformation mechanisms. Most importantly, our
analysis shows that topological metrics, genus being just one example, can
provide fundamental insights into complex aspects of the deformation of
nanoporous metals, such as the onset of densification and associated
exponential hardening regime.

##  6 Acknowledgements

The work by NVvB was supported by an EVC scholarship from Consejo
Interuniversitario Nacional - Argentina. NVvB is currently supported by
CONICET - Argentina. NVvB thanks F. Aquistapace for help on the use of FOAM
EXPLORER. This work used the Toko Cluster from FCEN, UNCuyo, which is part of
the SNCAD, MinCyT, Argentina. Support by ANPCyT PICT-2018-0773 is kindly
acknowledged. The authors acknowledge valuable discussions with M.G. Del
Pópolo, A. Lobos and E.M. Bringa during early stages of this manuscript. This
project has received funding from the European Union’s Horizon Europe research
and innovation programme under the Marie Sklodowska-Curie grant agreement no.
101062254. Funded by the European Union. Views and opinions expressed are
however those of the author(s) only and do not necessarily reflect those of
the European Union. Neither the European Union nor the granting authority can
be held responsible for them.

##  7 Data availability statement

The computer code required to reproduce the microstructures presented here are
available to download from
https://github.com/nicolasvazquez95/Nanofoams_Creator. All other information
is available from the authors upon a reasonable request.

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##  8 Appendix

###  8.1 Computational performance and microstructures

Figure 10 displays a limited set of microstructures generated by our
implementation of the levelled-wave method [58]. For a small solid fraction
$\phi_{B}$, or for ligament sizes within the order of magnitude of the lattice
parameter, it is possible to end up with a microstructure that it is below the
percolation limit. In other words, the solid phase is no longer continuous
[58]. For solid fractions above 0.2 and for $L/L_{\text{box}}$ above 0.04
($L_{\text{box}}$ being the simulation box size), our implementation produces
stochastic and fully bicontinuous microstructures.

Figure 10: Microstructures generated with our open-source implementation of
the levelled-wave method [58]. Note the percolation of the solid phase on the
left hand-side column.

To check the computational efficiency of our implementation, tests were
performed using 4 different implementations. Figure 11 presents the results of
the tests performed for the same foam input parameters ($100^{3}$ simple cubic
cells, 48 directions, 10000 waves) Blue bars indicate the execution “Walltime”
in seconds, while orange bars indicate the speedup. This is a metric used to
measure the scalability of a computational code, defined as
$\operatorname{speedup}=\frac{t_{1}}{t_{N}}$. $t_{1}$ corresponds to the time
required to run the code on a single processor, $t_{N}$ is the time required
on $N$ processors. The results show a clear acceleration after the multi-CPU
implementation. By recasting nested-loops into matrix operations, a GPU
implementation is straigthforward by means of TensorFlow packages. This last
implementation displays a notorious speedup.

Figure 11: Benchmarking results using four different code versions. Numpy/CPU
Tests were performed on an AMD Ryzen3 4000-series laptop. GPU tests were
performed on a NVidia K80 GPU (4.1 TFlops).

With the final CPU and GPU implementations, a last comparison of execution
times was performed. This was done on 1 core and 16 cores for the CPU
implementation. Tests were performed on TOKO FCEN UNCuyo computer cluster (1
node, AMD Ryzen 7, 64 GB Ram, 8 cores / 16 threads). The sample contained
$300^{3}$ unit cells and 20000 were used. The comparison includes walltime for
a test on a K80 NVidia GPU, used over GoogleColab. Excellent performance was
obtained (See Fig. 12).

Figure 12: (a) Comparison between three generation runs executed on a single
core (serial), parallely on 16 cores (multi-CPU) and on GPU. CPU
characteristics: 8 cores / 16 threads, AMD Ryzen 7 processor, 64 GB RAM). GPU:
NVidia K80 through GoogleColab.

Finally, our tests made use of 4GB of RAM (max) for our 16 threads CPU tests.
The GPU implementation made a peak memory usage of 16 GB on the GPU plus an
additional 3GB of RAM.