# Crystal Nucleation in Al-Ni Alloys:
an Unsupervised Chemical and Topological Learning Approach

Sébastien Becker  Univ. Grenoble Alpes, CNRS, Grenoble INP, SIMaP, F-38000
Grenoble, France  Univ. Grenoble Alpes, CNRS, Grenoble INP, LIG, F-38000
Grenoble, France  Emilie Devijver  CNRS, Univ. Grenoble Alpes, Grenoble INP,
LIG, F-38000 Grenoble, France  Rémi Molinier  Univ. Grenoble Alpes, CNRS, IF,
F-38000 Grenoble,France  Noël Jakse  Univ. Grenoble Alpes, CNRS, Grenoble INP,
SIMaP, F-38000 Grenoble, France

###### Abstract

Crystallization represents a fundamental process engendering solidification of
a material and determines its microstructure. Driven by complex phenomena at
the atomic scale, its understanding for alloys still remains elusive. The
present work proposes a large scale molecular dynamics simulation study of the
homogeneous crystal nucleation pathways of prototypical undercooled Al-Ni
binary alloys. An unsupervised topological learning analysis shows that the
nucleation sets in first from a chemical ordering, followed by a bond-
orientational ordering of the underlying crystal phase. Our results indicate
also a different polymorph selection that depends on composition. While the
nucleation pathway of Al50Ni50 displays a single step with the emergence of B2
short-range order, a step-wise nucleation toward the L12 phase is seen for
Al25Ni75. The influence of the nucleation of pure Al and Ni counterparts is
further discussed.

Homogeneous crystal nucleation, Unsupervised machine learning, Topological
data analysis, Persistence Homology, Molecular dynamics

##  I Introduction

Understanding homogeneous crystal nucleation during which an undercooled
liquid morphs into its underlying crystalline phase is of importance from a
fundamental point of view as well as on the application side for materials
manufacturing in industrial applications [1, 2]. Its intimate complex
mechanisms take their roots at the atomic level and involve local symmetry
breaking that can hardly be observed experimentally until very recently for
Fe-Pt binary metallic nanoparticles by atomic electron tomography [3].
However, especially for metallic materials, identifying the early stages of
nucleation from an experimental point of view is still merely out of reach,
and simulations at the atomic scale remain largely the dedicated tools,
applied to generic models [4, 5, 6, 7], pure metals [8, 9, 10, 11, 12, 5] and
alloys [14, 15, 16, 17, 18, 21, 19, 20] to name a few.

In classical nucleation theory, crystal nucleation and its rate can be
attributed essentially to two thermodynamic factors, namely, the enthalpy
difference between the crystal and liquid at the melting point, and effective
crystal-liquid interfacial free energy. They depend also on atomic transport
in the liquid state such as the diffusion [1, 2]. The seminal work of Turnbull
[22] on the ability to deeply undercool monatomic liquid metals led to
consider that the local atomic ordering of the undercooled melt might be
incompatible with the crystalline structure, thus impeding crystal nucleation.
It was later shown by Frank [23] that the icosahedral ordering is locally more
stable than the crystalline counterpart, which has triggered many theoretical
and experimental works [24, 25] in terms of the variety of polymorphs [26],
competing short-range orders [27, 12, 28], or an interplay between chemical
and five-fold symmetry orderings in the case of liquid alloys [29, 30, 21].
For the latter multi-component systems, this raises naturally the question of
the nature of the chemical and topological local orderings, and how they play
a role at the onset of nucleation, which still remains open.

In order to uncover the structural features prior and during homogeneous
crystal nucleation process in large scale molecular dynamics simulations
without a priori, an unsupervised learning approach [31] was proposed very
recently [5, 6]. The method is based on topological data analysis (TDA)
through persistent homology (PH) [33, 34], which emerged recently in the field
of materials science as a mean to identify local atomic structure as a post-
treatment in atomic scale simulations [35, 36, 37], or experimentally in
Scanning Electron Microscopy images of microstructure in aluminium alloys [38,
39]. The originality of our method was to use PH as a translational and
rotational invariant descriptor to encode the local structures. A Gaussian
Mixture Model (GMM) clustering method [40] is then applied, and estimated
through an Expectation Maximization (EM) algorithm [41]. This method called
hereafter TDA-GMM, was shown to be successful to identify and describe the
structural and morphological properties of the nuclei for various monatomic
metals, namely Al, Zr, Ta and Mg, revealing their specific nucleation pathways
[5, 6]. The application of the PH to multi-component alloys as a post-
treatment was the subject of very recent works on ice [36] imposing specific
constraints on O-H and O-O bonds, or on Pd-Si alloy [37] by removing the
central atom of a given type to discriminate local structures around Pd or Si.
However, in the latter case, while the topological ordering was successfully
captured, the chemical short-range order (CSRO) seems to be more difficult to
describe that way.

In the present work, large-scale molecular dynamics simulations (MD) are
carried out to investigate the homogeneous nucleation and nucleation pathways
of deep undercooling of Al-Ni alloys for two specific compositions, namely
Al50Ni50 and Al25Ni75, that possess different underlying stable crystalline
phase, namely B2 and L12, respectively. Interestingly, Al-Ni alloys are known
to be poor glass forming alloys [42, 21] with small crystallization times that
are reachable by brute force MD. From a methodological point of view, the PH
descriptor in our TDA-GMM scheme is extended here with the objective of
describing both the chemical ordering, focusing on the central atom and its
distance to the first neighborhood shell, and the topological local ordering
using edge-weighted persistent homology (EWPH), in a similar way as it was
used in biomolecular data analysis [7, 8, 9, 10] to capture information
between different elements and/or molecules. Then a GMM is built specifically
for each alloy by including in the training set samples of all possible
crystalline structures [1, 48], liquid configurations in the stable and
undercooled states, and out-of-equilibrium configurations at various stages of
the homogeneous nucleation process. Our results show that the nucleation is
initiated first from the chemical ordering, followed by progressive bond-
orientational ordering of the underlying crystal phase. Our findings further
indicate that the nucleation pathway depends on composition, with Al50Ni50
displaying a single step nucleation with the emergence of B2 short-range
order, and for Al25Ni75 a step-wise nucleation toward the L12 phase with the
emergence of fcc-, hcp-, and bcc-type polymorphs in a first stage.

##  II Unsupervised learning approach for alloys

###  II.1 Molecular dynamics simulations

In order to track homogeneous nucleation in Al-Ni alloys, large-scale MD
simulations were performed using the lammps code [49]. A number of atoms $N=$
1 024 000 atoms were considered, of which 512 000 of each specy for Al50Ni50,
and 256 000 Al and 768 000 Ni for Al25Ni75. They were placed randomly in a
cubic simulation box subject to the standard periodic boundary conditions
(PBC) in the three directions of space. Interactions were taken into account
through the semi-empirical potentials of Purja and Mishin [50] based on the
embedded atom model. Equations of motion are solved numerically with Verlet’s
algorithm in its velocity form with a time step of $1$ fs [51, 52]. Control of
the thermodynamic parameters was done with the Nosé-Hoover thermostat and
barostat [53, 54] and all the simulations were conduced at constant pressure.
Nucleation events were produced in undercooled states at constant temperature.
Properties of each alloy are summarized in Table 1, and Fig. 1 shows a
comparison between the classical MD simulation with the EAM potential [50] and
our previous ab initio MD simulations [55, 56] for the two compositions
considered here at $T=1795$ K taking the same densities. A reasonably good
agreement can be seen for both alloys, and more importantly, an excellent
match of the first peaks position for the three partials is found, indicating
that the Al-Al, Al-Ni and Ni-Ni bond lengths are well reproduced. Especially,
the first peak of Al-Ni partials are well reproduced indicating that the
strong chemical affinity between the two species is well predicted and that
the EAM potential is reliable.

Alloy |  $N$Al |  $N$Ni |  $T_{L}$ (K) |  $T_{g}$ (K) |  $T_{\textrm{iso}}$ (K) | $T_{rg}$ | $\Delta{T}$ |  $Q$ (K.s-1) |  $Q_{c}$ (K.s-1)
---|---|---|---|---|---|---|---|---|---
Al50Ni50 | $5.12\times 10^{5}$ | $5.12\times 10^{5}$ | 1780 | 915 | 1150 | 0.51 | 0.35 | $5.0\times 10^{12}$ | $1.4\times 10^{12}$
Al25Ni75 | $2.5\times 10^{5}$ | $7.5\times 10^{5}$ | 1678 | 815 | 1050 | 0.49 | 0.37 | $1.0\times 10^{11}$ | $9.0\times 10^{10}$
Table 1: Characteristic parameters of MD simulations. $N$Al and $N$Ni are the
number of Al and Ni in the simulation boxes. Liquidus temperatures $T_{L}$ are
taken from Ref. [50]. $T_{g}$ represents the glass transition temperature,
$T_{\textrm{iso}}$ the chosen isotherm for the analysis of the nucleation
events, $T_{rg}=T_{g}/T_{m}$, $\Delta T=(T_{m}-T_{iso})/T_{m}$, $Q$ the
cooling rate, and $Q_{c}$ the critical cooling rate, inferred from the nose of
the TTT curves. Figure 1: Partial pair-correlation functions for Al50Ni50 (a)
and Al25Ni75 (b) at $T=1750$ K. Classical MD simulation with the EAM potential
of Ref. [50] (solid lines) are compared with AIMD simulations of Ref. [55, 56]
(dashed lines).

For the study of the crystal nucleation, the alloys were equilibrated in the
liquid state, at $T=2500$ K and $T=2000$ K respectively for Al50Ni50 and
Al25Ni75, far above their liquidus temperatures $T_{L}$ (see Table 1). They
were subsequently brought in the undercooled states, using a ramp of
temperature with a cooling rate $Q=5.0\times 10^{12}$ K.s-1 and $Q=1.0\times
10^{11}$ K.s-1 respectively, to below the glass transition $T_{g}$ to avoid
crystallization during the quench. The resulting observed values of $T_{g}$
are given in Table 1. Configurations are recorded during the quench by
temperature steps of $50$ K. They were used to determine the time-temperature-
transformation (TTT) curve in the vicinity of the nose region, as shown in
Fig. 2 for both alloys. For each temperature, nucleation time was determined
from a sharp drop of the energy as was done in our preceding work [6], and is
averaged over five independent runs where initial velocity distribution were
randomized.

Figure 2: TTT curves in a temperature range near the nose for Al50Ni50 (a) and
Al25Ni75 (b). (c) Potential energy along $T_{\textrm{iso}}$ along a nucleation
process as a function of time for both alloys. The sharp drop of the potential
energy marks the nucleation events and the nucleation time is taken as the
time for which 30 % of the atoms are in a local crystalline order according to
the CNA analysis.

###  II.2 Data preparation for the training set

For the purpose of monitoring the homogeneous nucleation process with the TDA-
GMM method, a number of $4$ configurations of interest along $T_{iso}$ are
chosen for each alloy, spanning from the undercooled liquid state up to the
poly-crystalline state after the impingement of the nuclei. Each of these
configurations is first brought in its inherent structure (IS) by applying a
conjugate gradient minimization in the corresponding local minimum on the
potential energy surface [57].

The next step it to construct a training set from these IS configurations.
From the million atoms of each IS configuration, local atomic structures are
subsampled to construct a training set for the learning, using the procedure
developed in [6]. A local structure is composed of all atoms around a central
atom within a sphere of radius corresponding to the second minimum of
$g_{\alpha\beta}(r)$ with $\alpha,\beta\in\\{Al,Ni\\}$. It can be seen from
Fig. 1 that this radius, taken as $5.7$ Å and $5.5$ Å respectively for
Al50Ni50 and Al25Ni75, is similar for the three partials at the same
composition. This results in around $5600$ structures per IS configuration,
taking care that the proportion of the central atom species is in agreement
with the chemical composition of the corresponding alloy. Finally, to
guarantee that the TDA-GMM model is able to detect existing crystalline
polymorphs, local structures from all known Al-Ni crystal structures [1, 48]
(whose description are given in Table 2) were added through the perfect
crystal structures but also random distortions of them modeled by a Gaussian
noise (see Supplementary Information [58] for more details). Finally, the two
training sets for Al50Ni50 and Al25Ni75 contain respectively $61608$ and
$61775$ local structures.

Id | Composition | Number of sites | Sample size
---|---|---|---
mp-622209 | Al3NiS | 2Al \- 1Ni | 3000
mp-1057 | Al3Ni2S | 2Al \- 1Ni | 3000
mp-16514 | Al3Ni5S | 2Al \- 3Ni | 5000
mp-16515 | Al4Ni3S | 2Al \- 1Ni | 3000
mp-1487 | AlNiS | 1Al \- 1Ni | 2000
mp-2593 | AlNi3S | 1Al \- 1Ni | 2000
mp-1183232 | AlNi3M | 1Al \- 1Ni | 2000
mp-1025044 | AlNi2M | 1Al \- 2Ni | 3000
mp-1229048 | Al2Ni3M | 2Al \- 3Ni | 5000
mp-672232 | AlNi3M | 1Al \- 2Ni | 3000
mp-1228713 | Al4Ni15M | 2Al \- 4Ni | 6000
mp-1228854 | AlNiM | 1Al \- 1Ni | 2000
Table 2: Crystalline structure given by the Materials Project [1], composition
and number of distinct sites. Superscripts (S) and (M) correspond respectively
to a stable and metastable composition and superscripts (Al) and (Ni) to the
two types of central atom. For each distinct site, a sample of $1000$ randomly
distorted structures have been taken.

###  II.3 Topological and chemical descriptors

In our previous works for pure elements [5, 6], homogeneous crystal nucleation
pathways were successfully characterized using an unsupervised learning
approach (TDA-GMM), in which local structures were encoded by persitent
diagrams (PD) via persistence homology, a main tool in TDA. A topological
vector is built from a PD $D$: each coordinate is associated to a pair of
points $(x,y)$ for a fixed level of homology, except the infinite point, and
is calculated by

$m_{D}(x,y)=\min\\{\|x-y\|_{\infty},d_{\Delta}(x),d_{\Delta}(y)\\},$ (1)

where $d_{\Delta}(\cdot)$ denotes the $\ell^{\infty}$ distance to the
diagonal. The coordinates are sorted by decreasing order, with a vector length
equal to the number of points in the PD $D$ (for details we refer to [34, 6]).

For alloys, our descriptor is composed of two parts, one centered on the
chemical information and the other on the topological information.

The first part is a chemically oriented radial information encoding the
chemical ordering which is a very important aspect for the short-range order
of alloys [25, 29, 30]. More precisely, it is composed of two ingredients. The
first one is the vector of the distances from the central atom to the atoms of
the first neighbourhood shell and of the same type as the central one, sorted
in a decreasing order. The second one is the vector of the signed distances
from the central atom to the atoms again of the first neighbourhood shell but
of a different type from the central one, sorted again in a decreasing order.
The sign of the distance from the central particle and a given neighbouring
particle is chosen from a choice of a fixed total ordering on the type of
atoms: if the neighboring atom is of a higher type than the central atom, the
sign is positive and it is negative otherwise.

The second part of the descriptor is the concatenation of, for each type of
atom, one topological vector as in our previous work with only the
neighbouring atoms of the considered type. Only homology of dimension $1$ is
computed, being the only one discriminating. Those choices to reduce the
dimension are motivated by good performance on the reduced vector, as
illustrated in Supplementary Information [58].

Figure 3 gives a synthetic overview of the descriptor in the case of a binary
alloy. A more detailed description and motivation are given in Supplementary
Information [58].

Figure 3: Schematic view of the descriptor for a binary local atomic structure
with type A (cyan) and type B (orange) atoms. A slice view of the structure is
provided on the left, and decomposed onto two parts using transparency to
distinguish the atoms and the bounds that are in each part of the description.
For the chemically oriented vector (see text), as the type of the central
atoms is B, the two parts are $\mathbf{d}_{\text{BB}}$ and
$\mathbf{d}_{-\text{BA}}$ where $\mathbf{p}_{c}$ is the position of the
central atom and $\mathbf{p}_{i\in\mathcal{N}_{1,\text{A}}}$ and
$\mathbf{p}_{i\in\mathcal{N}_{1,\text{B}}}$ are, respectively, the position of
the atoms in the first neighbors of type A or B. The two components of the
topological part, for each persistence diagram, are computed using Eq. 1.

###  II.4 Gaussian mixture modeling

As in the TDA-GMM method introduced in our previous work [5], a GMM is used to
cluster the descriptors, with diagonal covariance matrices. An EM algorithm
[41] is used to estimate the parameters, through the scikit-learn package
[11]. The model is initialized with the best model among $3000$ $k$-means
runs. An elbow criterion on the likelihood is used to select the number of
clusters, which leads to exactly $50$ components for the two datasets.

As a posterior validation of the GMM model, more than $99.5$% of the
structures not used in the training set are assigned to a cluster with a
posterior probability larger than $0.999$. This gives a clear distinction
between clusters, and the remaining observations, coming from the simulated
configurations of nucleation, still have posterior probabilities higher than
$0.5$.

Figure 4: Log-likehood functions for the Al50Ni50 (a) and Al25Ni75 (b)
modeling, with the tangents to the curve in the vicinity of the inflection
point symbolized by the red dotted lines.

##  III Results and discussion

Figure 5: Snapshots of MD simulations at various stages of the homogeneous
nucleation for (a) Al50Ni50 and (b) Al25Ni75. Only Al (orange) and Ni (cyan)
atoms belonging to clusters with B2 and L12, respectively, are drawn (see
text).

The homogeneous nucleation pathways in Al50Ni50 and Al25Ni75 explored by MD
are illustrated in Fig. 5 by drawing simulation snapshots at specific times
during the process in deep undercooling conditions close to the nose of the
TTT at $T=1150$ K and $T=1050$ K, respectively. For each composition, all
atoms of the simulation box are first associated to one of the $50$ clusters
of the trained TDA-GMM model, as described in the previous Section II.4. The
evolution of the proportion of local structures in each cluster is monitored
during the whole process, and those for which a significant increase is
detected are deemed to participate to the nucleation, without prior knowledge
of their orientational and chemical orderings (see also Supplementary
Information [58]). For Al50Ni50, $16$ clusters with increasing proportions are
seen, and they all display a bcc ordering as analyzed by the CNA [2, 61] with
various chemical orderings in terms of the Cowley parameter [12]. For Al25Ni75
the situation appears to be more complex with $22$ clusters having an
increasing proportion of local structures with fcc, hcp, and bcc structures
with various chemical ordering. The relevant clusters proportions in both
cases are gathered in Table SIII and SIV in Supplementary Information [58]. In
Fig. 5, atoms are drawn only if they belong to clusters having a B2 structure
for Al50Ni50 with Cowley parameter $\alpha_{1}=-0.1429$, and L12 structure for
Al25Ni75 with $\alpha_{1}=-0.333$, which correspond to their respective
underlying crystalline phase. This highlights clearly for both alloys the
homogeneous nucleation stage up to the impingement followed by the grain
growth stage, also called the solidification. the latter is a much slower
process that can not be completely captured by brute force MD. For Al50Ni50
nucleation starts around $0.3$ ns with multiple nucleation events up to $1$ ns
where solidification starts. The same holds for Al25Ni75 $\alpha_{1}=-0.333$
with nucleation appearing at $10$ ns and a solidification stage starting
$1.75$ ns later.

Figure 6:  Evolution of the crystalline and five-fold symmetry of Al50Ni50
during nucleation at $T=1150$ K(a) for the bond-orientational ordering, and
(b) chemical short-range order with $\alpha_{1}=-0.1429$ corresponding to a B2
signature.

The bond-orientational and chemical orderings during nucleation is first
examined for Al50Ni50. In Fig. 6(a) the cumulative proportions of local
structures in clusters having a bcc structure are shown, as well as those
having a dominant five-fold symmetry (FFS) local ordering, also called
icosahedral ordering, distinguishing Al and Ni centered ones. It is worth
mentioning that local structures with fcc or hcp orderings are negligibly
small for this composition so they are not displayed in the plot. In the
undercooled liquid, the FFS ordering is slightly higher around Ni atoms than
Al ones, which is consistent with our former AIMD simulations [56]. The onset
of nucleation is characterized by a strong increase of bcc ordering, which
slows down in the grain growth stage, and a simultaneous strong decrease of
the FFS short-range order. While the bcc order is preponderant around Ni, Fig.
6(b) shows that both species are equally involved in the B2 ordering. More
interestingly, Fig. 6(b) indicates that in the liquid prior to nucleation, a
significant chemical ordering, corresponding to the B2 with
$\alpha_{1}=-0.1429$, exists in clusters with disordered local structures This
is in favor of a scenario in which the chemical ordering of the phase is
already present before the bond-orientational ordering sets in. As a matter of
fact, Table SIII in Supplementary Information [58] shows that clusters with B2
local structures at the early stages of nucleation have a higher proportion.

Figure 7:  Evolution of the crystalline and five-fold symmetry of Al25Ni75
during nucleation at $T=1050$ K (a) for the bond-orientational ordering, and
(b) chemical ordering with $\alpha_{1}=-0.333$ corresponding to a L12
signature.

For Al25Ni75, the bond-orientational and chemical orderings are displayed in
Fig. 7 as for the equiatomic composition above. Proportions of local
structures having a crystalline order are composed of various polymorphs,
namely bcc, hcp and fcc as can be seen in Fig. 7(a). It should be noticed that
the relative proportions of Al and Ni in all the bond-orientational orderings
are consistent with the alloy composition. The FFS is more pronounced than for
Al50Ni50, and dominant around Ni in the undercooled liquid, which is again
consistent with AIMD results [56], and essentially vanishes during the
nucleation stage. The onset of nucleation is characterized by a two-stage
process in which the bcc ordering is selected first. It transforms back to the
fcc and hcp orderings in a second stage when the fcc and hcp experience a
strong increase in equal proportions. They finally become more steady after
impingement when the grain growth stage sets in. A closer inspection shows
that the hcp local order participates in numerous stacking fault in the nuclei
as well as in grain boundaries as shown in Fig. S4 in Supplementary
Information [58]. A significant chemical ordering corresponding to the L12
with $\alpha_{1}=-0.333$ Fig. 7(b) preexists in the undercooled liquid, which
is again an indication for this composition, that the chemical ordering of the
underlying phase precedes the bond-orientational ordering at the onset of
nucleation (see also Table SIV in Supplementary Information [58]).

Figure 8:  Distribution of the chemical short-range order around all atoms for
the configurations of Fig. 5 during the nucleation phase for (a) Al50Ni50 and
(b) Al25Ni75. The values $\alpha_{1}=-0.333$ $\alpha_{1}=-0.1429$, and
$\alpha_{1}=0$ correspond to L12, B2, and disordered signatures, respectively.

Finally, our unsupervised learning approach enables us to show that the two
compositions studied possess two distinct nucleation pathways, namely a single
step nucleation for Al50Ni50, and a two-step nucleation for Al25Ni75 following
Ostwald’s rule [63]. In both cases, the preexisting chemical short-range order
of the underlying phase precedes the bond-orientational ordering. This can be
seen from the local chemical ordering distributions in Fig.8. For Al50Ni50,
beside the disordered peak, a preponderant peak is seen around the B2 value of
$\alpha_{1}$ before and during the whole nucleation stage. For Al25Ni75, the
distributions show a peak around the B2 at the early stage that subsequently
decrease at the expense of a peak at the L12 value of $\alpha_{1}$. For
Al50Ni50, our results are consistent with previous MD simulations with the
same EAM potential [20], in which a single step towards the B2 phase was
observed albeit with a lower bcc ordering. It was proposed [20] that the two-
step nucleation observed in pure Ni starting with the bcc local ordering could
be at the origin of the B2 polymorph selection in this alloy. Our results are
not in favor of this scenario as they rather indicate a symmetric role of Al
and Ni that can be explained by their strong affinity and similar diffusion
coefficients at that composition [56]. Moreover, it was shown in a preceding
work [5] that homogeneous nucleation in pure Al is characterized by a single
step with an initial fcc ordering. This is an additional counterargument. The
situation is different for Al25Ni75 for which the two-step nucleation can come
from that of pure Ni [20]. Then, our results might indicate for this
composition that the two-step pathway is a combined effect of that of pure Ni
and of the preexisting chemical ordering of the underlying phase at the onset
of nucleation.

##  IV Conclusion

In conclusion, this study was devoted to the investigation of homogeneous
nucleation and nucleation pathways of deep undercooled Al50Ni50 and Al25Ni75
alloys by means of large-scale MD simulations. For this purpose an
unsupervised topological and chemical learning approach was successfully
extended to alloys from our previous work based on the persistent homology
developed for pure elements [5, 6], by including the chemical ordering in the
PH descriptor for building a Gaussian mixture model. This allowed us to show
that the chemical short-range order of the underlying stable crystalline phase
of both alloys preexist in the undercooled liquid prior to the onset of
crystal nucleation, followed by an increase of the bond-orientational
ordering. Our findings further indicate that the nucleation pathway depends on
composition, with Al50Ni50 displaying a single step nucleation with the
emergence of B2 short-range order, and for Al25Ni75 a step-wise nucleation
toward the L12 phase with the emergence of bcc-type polymorph in the first
stage that might come from the combine effect of the chemical ordering and the
two-stage nucleation pathway of pure Ni.

## Acknowledgement

We acknowledge the CINES and IDRIS under Project No. INP2227/72914, as well as
CIMENT/GRICAD for computational resources. This work was performed within the
framework of the Centre of Excellence of Multifunctional Architectured
Materials CEMAM-ANR-10-LABX-44-01 funded by the "Investments for the Future"
Program. This work has been partially supported by MIAI@Grenoble Alpes
(ANR-19-P3IA-0003). Fruitful discussions within the French collaborative
network in high-temperature thermodynamics GDR CNRS3584 (TherMatHT) and in
artificial intelligence in materials science GDR CNRS 2123 (IAMAT) are also
acknowledged.

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Supplemental Information

## Data preparation: construction of a set of local crystalline structures

We built a set of local crystalline structures up to the second shell of
neighbors from the available known unit cells of AlNi crystal alloys [1]. Each
unit cell was reproduced in the three directions of space to obtain virtual
boxes of about 30 000 atoms. A Gaussian noise, centered and with a standard
deviation of $0.01$ Å was then applied to the atomic positions to perturb the
crystal lattices without losing the structural signature of the different
sites. We have checked that the partial radial distribution functions and the
common neighbor analysis (CNA) signatures [2] remained the same between the
noisy and initial crystalline configurations. The CNA was performed here with
a fixed cut off radius $r=3.5$ Å 111In the case of the CNA performed for the
structures in the MD configurations studied in the main text, we took three
cut-off radii, one different for each of the possible chemical bonds:
$r_{\text{Al}-\text{Al}}=3.57$ Å, $r_{\text{Al}-\text{Ni}}=3.49$ Å,
$r_{\text{Ni}-\text{Ni}}=3.33$ Å for Al50Ni50; and
$r_{\text{Al}-\text{Al}}=3.38$ Å, $r_{\text{Al}-\text{Ni}}=3.35$ Å,
$r_{\text{Ni}-\text{Ni}}=3.24$ Å for Al25Ni75., using the Python
implementation in the software ovito [3]. We identified for each composition
the distinct sites with two shells of neighbors by discriminating autonomously
the atomic structures with a CNA signature using the atomic bonds of the
central atom and the first neighbors of the same type as the latter,
normalized with the total number of bounds (central atom and first shell of
neighbors). We used the Python package pyscal [4] on each full configuration
to extract the coordinates of the atoms. Finally, a sample of 1000 randomly
distorted structured for each site is picked in each composition, which leads
to a set composed of 39 000 structures (21 000 with a central atom Ni and 18
000 with a central atom Al).

## Edge-weighted persistent homology, chemical and topological description

In our previous work [5, 6] the idea was to encode the topological information
using persistent homology. Since we are now working with an alloy, we have to
take into account the different types of atoms. Inspired by [7, 8, 9, 10], we
used Edge-Weighted Persistent Homology (EWPH). The idea is to adapt the
appearance of 1-simplices in the Vietoris-Rips complexes along the filtration
according to the types of the extremities. We computed several persistent
diagrams, one for each possible (non ordered) pair of types (in the case of an
alloy AB, we have three types A-A, A-B, B-B) letting only appearing
1-simplices corresponding to this pair. More precisely, as the Vietoris-Rips
complexes can be computed from the distance matrix
$M={\\{M_{ij}=d(x_{i},x_{j}),i,j\in\\{1,\dots,\mu\\}\\}}$ of the point cloud
$X=(x_{i})_{i=1,\dots,\mu}$, we define the components of a modified distance
matrix for each non ordered pair of type $(\alpha,\beta)$ as

$M^{\alpha,\beta}_{ij}=\begin{cases}d(x_{i},x_{j})&\text{if $(\text{type
}x_{i},\text{type }x_{j})=(\alpha,\beta)$ ;}\\\
\infty&\text{otherwhise.}\end{cases}$ (S1)

In practice, $\infty$ corresponds to a value larger than the maximum
filtration size. Thus, for a structure with two types of atoms $A$ and $B$,
three modified distance matrices are defined, which lead to three independent
filtration processes, each capturing the topological information given by one
of the possible chemical interactions. The generalization to higher order
alloys is straightforward and the case of a single-element atomic structure
corresponds to a single modified matrix, which strictly corresponds to the
distance matrix of the unweighted PH.

We can use this artefact to obtain only information from one or a set of
specific atoms in the structure. In other words, it is possible to divide a
structure into several substructures.

First, to focus on the chemical information, we consider a substructure with
the central atom and its neighbors in the first neighbors shell. We set
$M_{i,j}=\infty$ in S1 if $x_{i}$ or $x_{j}$ is not the central atom, and if
the other is not in the first neighbors shell. However, we only get
information in dimension 0, and a radial information contains much more
information with less parameters, which we encode in our chemically ordered
radial information vector. Its first components correspond to the distances,
arranged in increasing order, from the central atom to the neighboring atoms
of the same type as the central one. Remaining components are the signed
distances, still arranged in decreasing order, from the central atom to the
atoms with a different type from the central one. In these last components,
the sign is obtained by ordering the types of atoms: the distance is positive
if the type of the neighboring atom is higher than the one of the central one
and negative otherwise.

We can also obtain another substructure corresponding to the second shell of
neighbors by taking into account all the distances in (S1) except the ones
between the central atom and its first neighbors, setting $i,j\neq 1$. For
each pair $(\alpha,\beta)$ of possible types (so $(A,A)$, $(A,B)$ and $(B,B)$
for a binary alloy with atoms of type $A$ and $B$) we obtain a persistent
diagram. In the diagram associated to the pair $(\alpha,\alpha)$ we have
points in homological dimension 0, 1 and 2. In the diagram associated to the
pair $(\alpha,\beta)$ with $\alpha\neq\beta$ we have points in homological
dimension 0 and 1. Indeed, there is no point in homological dimension 2
because there are only 0-simplicies and 1-simplices arising in the filtration
process. Moreover, the points in homological dimension 1 are at infinity. In
the cases of NiAl alloy considered in this paper, they were several hundreds
of points at infinity for a given local structure which give rise in the end
to a topological vector with more than one thousand of components. This is too
much to deal with our procedure and we decided to forget this part of the
topological vector in our study.

The final descriptor constructed here has for first component the chemical
information, and the second component corresponds to 3 PDs with $H_{0}$, and
potentially $H_{1}$ and $H_{2}$, components.

## Supervised learning to select sufficient topological information

As described in the previous section, the full descriptor has a lot of
coefficients (about 300). To have a good clustering with a GMM, we would need
a huge number of local structures in the training set. As it was not the case
in our dataset, we needed to reduce the dimension of the descriptor. Also,
since we are doing several PDs on the same local structures, though with
different weights, it is reasonable to think that there is redundancy in the
topological information we obtain.

Using the set of local crystalline structures we constructed, we wanted to
select the necessary and sufficient components of the topological vector to
distinguish at least three kinds of information: the type of the central atom,
the chemical short-range ordering (CSRO, [12]), and the structural information
(e.g. B2 and L12). The CSRO parameter $\alpha_{1}$ is computed for each
structure on the first shell of neighbors by

$\alpha_{1}=1-\frac{\overline{\nu_{\beta}}}{\nu.\overline{c_{\beta}}}$ (S2)

where $\beta$ is the type of the central atom, $\overline{\nu_{\beta}}$ is the
number of neighbors of a type different from $\beta$, $\nu$ is the total
number of neighbors and $\overline{c_{\beta}}$ is the concentration of atoms
of a type different from $\beta$ in the original crystalline configuration
from which the structure comes from.

First, to check that the chemical oriented distance vector is able to retrieve
the chemical information, we use a classification Random Forest (implemented
in the Python package scikit-learn [11]) with 100 decision trees. We performed
two classification tasks: one for the type of the central atom, and one
related to the CSRO. As only 28 different values were obtained for the CSRO on
the local noisy crystalline structures (16 values for structures centered in
Ni and 12 values for structures centered in Al), with more than 1000
structures in each case, we consider those values as different classes. For
each classification task, we consider 100 stratified train test splits with
proportions 90%/10%. All the data are use for predicting the type of the
central atom, whereas the CSRO is differentiate by the central atom. Table SI
gives the accuracy, recall and precision of the predictions made on each test
set. The high values of these scores are a good indication that the chemical
oriented distances vector is a proper representative of the information in
terms of central atom and CSRO.

| Type of central atom | CSRO
---|---|---
|  | Central atom Ni | Central atom Al
Accuracy | $1.00\pm 0.00$ | $0.99\pm 0.01$ | $0.98\pm 0.01$
Recall | $1.00\pm 0.00$ | $0.99\pm 0.01$ | $0.99\pm 0.01$
Precision | $1.00\pm 0.00$ | $0.99\pm 0.01$ | $0.99\pm 0.01$
Table SI: Means and standard deviations of the accuracy, recall and precision metrics of the random forest over 100 train/test splits. | Structural information
---|---
| Central atom Ni | Central atom Al
| $\text{TV}_{0}$ | $\text{TV}_{1}$ | $\text{TV}_{2}$ | $\text{TV}_{0}$ | $\text{TV}_{1}$ | $\text{TV}_{2}$
Dimension | 156 | 86 | 19 | 156 | 86 | 19
Accuracy | $0.98\pm 0.01$ | $0.99\pm 0.01$ | $0.45\pm 0.01$ | $0.94\pm 0.01$ | $0.99\pm 0.01$ | $0.47\pm 0.01$
Recall | $0.99\pm 0.01$ | $0.99\pm 0.01$ | $0.43\pm 0.01$ | $0.94\pm 0.01$ | $0.99\pm 0.01$ | $0.41\pm 0.01$
Precision | $0.99\pm 0.01$ | $0.99\pm 0.01$ | $0.51\pm 0.02$ | $0.94\pm 0.01$ | $0.99\pm 0.01$ | $0.39\pm 0.01$
Table SII: Means and standard deviations of the accuracy, recall and precision
metrics computed from the predictions on test sets made by 100 different
random forest classifiers trained with the different constituent parts of the
topological vectors (TV) on labels representing structures with two shells of
neighbors for each subset of structures with a central Ni or Al atom.
$\text{TV}_{0}$ stands for the 0-dimensional part of the topological vectors:
$\text{TV}^{H^{\text{NiNi}}_{0}}$, $\text{TV}^{H^{\text{NiAl}}_{0}}$ and
$\text{TV}^{H^{\text{AlAl}}_{0}}$ ; $\text{TV}_{1}$ corresponds the
1-dimensional part of the topological vectors:
$\text{TV}^{H^{\text{NiNi}}_{1}}$ and $\text{TV}^{H^{\text{AlAl}}_{1}}$ 333As
mentioned above $H_{1}^{\text{NiAl}}$, with only points at infinity in the PD,
was not considered because it increases drastically the dimension of the
vector. ; and $\text{TV}_{2}$ gives the 2-dimensional part of the topological
vectors: $\text{TV}^{H^{\text{NiNi}}_{2}}$ and
$\text{TV}^{H^{\text{AlAl}}_{2}}$.

For the topological vectors, we applied the same supervised learning process.
We set a label for each distinct CNA structural signature. This leads to a
total number of 33 labels (17 labels for structures with a central atom Ni and
16 labels for structures with a central atom Al). Table 3 gives the means and
standard deviations of the accuracy, recall and precision metrics computed on
the predictions made on each test set, through the hundred random forest
models.

Based on these scores and to get a moderate dimension for the descriptor
(according to our training set), we choose to keep the topological vectors
constructed from $H_{1}^{AA}$ and $H_{1}^{BB}$ to encode the structural
information of our local structure.

## Additional tables and figures

We give here additional tables and figures that support the main text.

(%) | 0.3 ns | 0.5 ns | 0.7 ns | 0.9 ns | 2 ns | 5 ns
---|---|---|---|---|---|---
B2Al | $0.008^{1}$ | $0.236^{1}$ | $1.731^{1}$ | $6.018^{1}$ | $15.576^{2}$ | $18.721^{2}$
B2Ni | $0.016^{3}$ | $0.245^{3}$ | $1.758^{3}$ | $6.025^{3}$ | $15.626^{4}$ | $18.701^{4}$
BCCAl$\alpha_{1}\notin\\{-0.1429,0\\}$ | $0.003^{2}$ | $0.104^{2}$ | $1.082^{3}$ | $3.950^{3}$ | $7.875^{2}$ | $7.579^{2}$
BCCNi$\alpha_{1}\notin\\{-0.1429,0\\}$ | $0.001^{1}$ | $0.084^{5}$ | $0.843^{5}$ | $3.102^{5}$ | $5.817^{4}$ | $5.480^{4}$
BCCAl$\alpha_{1}=0$ | $0.004^{2}$ | $0.086^{2}$ | $0.851^{2}$ | $3.004^{3}$ | $5.602^{3}$ | $5.185^{3}$
BCCNi$\alpha_{1}=0$ | $0.007^{3}$ | $0.132^{3}$ | $1.197^{3}$ | $4.400^{3}$ | $9.158^{2}$ | $8.949^{2}$
Table SIII: Evolution of B2 ($\alpha_{1}=-0.1429$), BCC chemically ordered ($\alpha_{1}\notin\\{-0.1429,0\\}$) and BCC chemically disordered ($\alpha_{1}=0$) crystalline structures in Al50Ni50 during nucleation for central Al and Ni atoms. The superscripts correspond to the number of clusters in which the structures are distributed. Figure S1: Total evolution of B2 (green), BCC chemically disordered (blue) and BCC chemically ordered (red) in Al50Ni50 during nucleation for central (a) Al and (b) Ni atoms. We also display the evolution of the corresponding clusters. Figure S2: Total evolution of five-fold symmetry (icosahedral), with a B2 ordering (green), chemically disordered (blue) and chemically ordered (red) in Al50Ni50 during nucleation for central (a) Al and (b) Ni atoms. We also display the evolution of the corresponding clusters. (%) | 9.6 ns | 10 ns | 10.25 ns | 10.50 ns | 11 ns | 14 ns
---|---|---|---|---|---|---
BCCAl$\alpha_{1}=-0.333$ | $0.001^{2}$ | $0.006^{2}$ | $0.027^{2}$ | $0.029^{2}$ | $0.018^{2}$ | $0.019^{2}$
BCCNi$\alpha_{1}=-0.333$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
BCCAl$\alpha_{1}\notin\\{-0.333,0\\}$ | $0.005^{4}$ | $0.130^{5}$ | $0.460^{5}$ | $0.454^{4}$ | $0.301^{5}$ | $0.297^{5}$
BCCNi$\alpha_{1}\notin\\{-0.333,0\\}$ | $0.024^{4}$ | $0.620^{7}$ | $2.259^{8}$ | $2.734^{8}$ | $0.385^{5}$ | $0.414^{5}$
BCCAl$\alpha_{1}=0$ | $0.006^{1}$ | $0.096^{1}$ | $0.322^{1}$ | $0.422^{1}$ | $0.229^{1}$ | $0.230^{1}$
BCCNi$\alpha_{1}=0$ | $0.002^{2}$ | $0.002^{2}$ | $0.002^{2}$ | $0$ | $0$ | $0$
HCPAl$\alpha_{1}=-0.333$ | $0$ | $0$ | $0.005^{4}$ | $0.160^{4}$ | $0.557^{4}$ | $0.625^{4}$
HCPNi$\alpha_{1}=-0.333$ | $0.002^{2}$ | $0.004^{3}$ | $0.051^{5}$ | $1.349^{5}$ | $5.248^{5}$ | $5.793^{5}$
HCPAl$\alpha_{1}\notin\\{-0.333,0\\}$ | $0.002^{2}$ | $0.004^{3}$ | $0.086^{4}$ | $0.676^{4}$ | $2.427^{5}$ | $2.618^{5}$
HCPNi$\alpha_{1}\notin\\{-0.333,0\\}$ | $0.004^{4}$ | $0.027^{11}$ | $0.422^{12}$ | $2.263^{13}$ | $7.123^{11}$ | $7.699^{11}$
HCPAl$\alpha_{1}=0$ | $0$ | $0.002^{2}$ | $0.03^{3}$ | $0.235^{2}$ | $0.822^{2}$ | $0.860^{2}$
HCPNi$\alpha_{1}=0$ | $0.003^{3}$ | $0.008^{4}$ | $0.152^{5}$ | $1.739^{5}$ | $5.989^{5}$ | $6.489^{5}$
L12Al | $0$ | $0$ | $0.008^{4}$ | $0.122^{4}$ | $0.549^{4}$ | $0.653^{4}$
L12Ni | $0$ | $0.002^{2}$ | $0.033^{5}$ | $1.090^{5}$ | $5.243^{6}$ | $6.083^{6}$
FCCAl$\alpha_{1}\notin\\{-0.333,0\\}$ | $0$ | $0.004^{4}$ | $0.065^{4}$ | $0.564^{5}$ | $2.558^{6}$ | $2.858^{6}$
FCCNi$\alpha_{1}\notin\\{-0.333,0\\}$ | $0.002^{1}$ | $0.014^{8}$ | $0.320^{12}$ | $1.935^{11}$ | $7.346^{12}$ | $8.305^{12}$
FCCAl$\alpha_{1}=0$ | $0$ | $0$ | $0.0190^{2}$ | $0.190^{2}$ | $0.846^{2}$ | $0.931^{3}$
FCCNi$\alpha_{1}=0$ | $0$ | $0.005^{5}$ | $0.116^{5}$ | $1.441^{5}$ | $6.08^{5}$ | $6.948^{5}$
Table SIV: Evolution of the BCC, HCP, and FCC crystalline structures with a
L12 ordering $\alpha_{1}=-0.333$, chemically ordering
$\alpha_{1}\notin\\{-0.333,0\\}$, and chemically disordered ($\alpha_{1}=0$)
in Al25Ni75 during nucleation for central Al and Ni atoms. The superscripts
correspond to the number of clusters in which the structures are distributed.
Figure S3: Total evolution of BCC with a L12 ordering (green), BCC chemically
disordered (blue) and BCC chemically ordered (red) in Al25Ni75 during
nucleation for central (a) Al and (b) Ni atoms. We also display the evolution
of the corresponding clusters. Figure S4: Total evolution of HCP with a L12
ordering (green), HCP chemically disordered (blue) and HCP chemically ordered
(red) in Al25Ni75 during nucleation for central (a) Al and (b) Ni atoms. We
also display the evolution of the corresponding clusters. Figure S5: Total
evolution of L12 (green), FCC chemically disordered (blue) and FCC chemically
ordered (red) in Al25Ni75 during nucleation for central (a) Al and (b) Ni
atoms. We also display the evolution of the corresponding clusters. Figure S6:
Total evolution of five-fold symmetry (icosahedral), with a L12 ordering
(green), chemically disordered (blue) and chemically ordered (red) in Al25Ni75
during nucleation for central (a) Al and (b) Ni atoms. We also display the
evolution of the corresponding clusters.

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