We consider unsupervised learning methods for characterizing the disordered microscopic structure of supercooled liquids and glasses. Specifically, we perform dimensionality reduction of smooth structural descriptors that describe radial and bond-orientational correlations and assess the ability of the method to grasp the essential structural features of glassy binary mixtures. In several cases, a few collective variables account for the bulk of the structural fluctuations within the first coordination shell and also display a clear connection with the fluctuations of particle mobility. Fine-grained descriptors that characterize the radial dependence of bond-orientational order better capture the structural fluctuations relevant for particle mobility but are also more difficult to parameterize and to interpret. We also find that principal component analysis of bond-orientational order parameters provides identical results to neural network autoencoders while having the advantage of being easily interpretable. Overall, our results indicate that glassy binary mixtures have a broad spectrum of structural features. In the temperature range we investigate, some mixtures display well-defined locally favored structures, which are reflected in bimodal distributions of the structural variables identified by dimensionality reduction.

1.
C. P.
Royall
and
S. R.
Williams
,
Phys. Rep.
560
,
1
(
2015
).
2.
H.
Tanaka
,
H.
Tong
,
R.
Shi
, and
J.
Russo
,
Nat. Rev. Phys.
1
,
333
(
2019
).
3.
D.
Coslovich
,
J. Chem. Phys.
138
,
12A539
(
2013
).
4.
P. J.
Steinhardt
,
D. R.
Nelson
, and
M.
Ronchetti
,
Phys. Rev. B
28
,
784
(
1983
).
5.
M.
Tanemura
,
Y.
Hiwatari
,
H.
Matsuda
,
T.
Ogawa
,
N.
Ogita
, and
A.
Ueda
,
Prog. Theor. Phys.
58
,
1079
(
1977
).
6.
B. J.
Gellatly
and
J. L.
Finney
,
J. Non-Cryst. Solids
50
,
313
(
1982
).
7.
J. D.
Honeycutt
and
H. C.
Andersen
,
J. Phys. Chem.
91
,
4950
(
1987
).
8.
A.
Malins
,
S. R.
Williams
,
J.
Eggers
, and
C. P.
Royall
,
J. Chem. Phys.
139
,
234506
(
2013
).
9.
E. A.
Lazar
,
J.
Han
, and
D. J.
Srolovitz
,
Proc. Natl. Acad. Sci. U. S. A.
112
,
E5769
(
2015
).
10.
D.
Coslovich
and
G.
Pastore
,
J. Chem. Phys.
127
,
124504
(
2007
).
11.
A.
Malins
,
J.
Eggers
,
C. P.
Royall
,
S. R.
Williams
, and
H.
Tanaka
,
J. Chem. Phys.
138
,
12A535
(
2013
).
12.
A.
Malins
,
J.
Eggers
,
H.
Tanaka
, and
C. P.
Royall
,
Faraday Discuss.
167
,
405
(
2014
).
13.
G. M.
Hocky
,
D.
Coslovich
,
A.
Ikeda
, and
D. R.
Reichman
,
Phys. Rev. Lett.
113
,
157801
(
2014
).
14.
F.
Arceri
,
F. P.
Landes
,
L.
Berthier
, and
G.
Biroli
, “
A statistical mechanics perspective on glasses and aging
,” in
Encyclopedia of Complexity and Systems Science
, edited by
R. A.
Meyers
(
Springer Berlin Heidelberg
,
2020
), pp.
1
68
.
15.
P.
Mehta
,
M.
Bukov
,
C.-H.
Wang
,
A. G. R.
Day
,
C.
Richardson
,
C. K.
Fisher
, and
D. J.
Schwab
,
Phys. Rep.
810
,
1
(
2019
).
16.
B.
Cheng
,
R.-R.
Griffiths
,
S.
Wengert
,
C.
Kunkel
,
T.
Stenczel
,
B.
Zhu
,
V. L.
Deringer
,
N.
Bernstein
,
J. T.
Margraf
,
K.
Reuter
, and
G.
Csanyi
,
Acc. Chem. Res.
53
,
1981
(
2020
).
17.
E.
Boattini
,
M.
Dijkstra
, and
L.
Filion
,
J. Chem. Phys.
151
,
154901
(
2019
).
18.
R.
van Damme
,
G. M.
Coli
,
R.
van Roij
, and
M.
Dijkstra
,
ACS Nano
14
,
15144
(
2020
).
19.
S.
Becker
,
E.
Devijver
,
R.
Molinier
, and
N.
Jakse
,
Phys. Rev. E
105
,
045304
(
2022
).
20.
E.
Boattini
,
S.
Marín-Aguilar
,
S.
Mitra
,
G.
Foffi
,
F.
Smallenburg
, and
L.
Filion
,
Nat. Commun.
11
,
5479
(
2020
).
21.
J.
Paret
,
R. L.
Jack
, and
D.
Coslovich
,
J. Chem. Phys.
152
,
144502
(
2020
).
22.
V. L.
Deringer
,
M. A.
Caro
,
R.
Jana
,
A.
Aarva
,
S. R.
Elliott
,
T.
Laurila
,
G.
Csányi
, and
L.
Pastewka
,
Chem. Mater.
30
,
7438
(
2018
).
23.
B.
Monserrat
,
J. G.
Brandenburg
,
E. A.
Engel
, and
B.
Cheng
,
Nat. Commun.
11
,
5757
(
2020
).
24.
A.
Offei-Danso
,
A.
Hassanali
, and
A.
Rodriguez
,
J. Chem. Theory Comput.
18
,
3136
(
2022
).
25.
W.
Kob
and
H. C.
Andersen
,
Phys. Rev. Lett.
73
,
1376
(
1994
).
26.
27.
R. G.
Della Valle
,
D.
Gazzillo
,
R.
Frattini
, and
G.
Pastore
,
Phys. Rev. B
49
,
12625
(
1994
).
28.
D.
Coslovich
and
G.
Pastore
,
J. Phys.: Condens. Matter
21
,
285107
(
2009
).
29.
Y. Q.
Cheng
,
H. W.
Sheng
, and
E.
Ma
,
Phys. Rev. B
78
,
014207
(
2008
).
30.
L.
Berthier
,
G.
Biroli
,
D.
Coslovich
,
W.
Kob
, and
C.
Toninelli
,
Phys. Rev. E
86
,
031502
(
2012
).
31.
W.
Lechner
and
C.
Dellago
,
J. Chem. Phys.
129
,
114707
(
2008
); arXiv:0806.3345.
32.
J. A.
van Meel
,
L.
Filion
,
C.
Valeriani
, and
D.
Frenkel
,
J. Chem. Phys.
136
,
234107
(
2012
); arXiv:1202.5281.
33.
W.
Mickel
,
S. C.
Kapfer
,
G. E.
Schröder-Turk
, and
K.
Mecke
,
J. Chem. Phys.
138
,
044501
(
2013
).
34.
A. P.
Bartók
,
R.
Kondor
, and
G.
Csányi
,
Phys. Rev. B
87
,
184115
(
2013
).
35.
J.
Behler
and
M.
Parrinello
,
Phys. Rev. Lett.
98
,
146401
(
2007
).
36.
B.
Parsaeifard
,
D. S.
De
,
A. S.
Christensen
,
F. A.
Faber
,
E.
Kocer
,
S.
De
,
J.
Behler
,
O. A.
von Lilienfeld
, and
S.
Goedecker
,
Mach. Learn.: Sci. Technol.
2
,
015018
(
2021
).
37.
L.
Himanen
,
M. O. J.
Jäger
,
E. V.
Morooka
,
F.
Federici Canova
,
Y. S.
Ranawat
,
D. Z.
Gao
,
P.
Rinke
, and
A. S.
Foster
,
Comput. Phys. Commun.
247
,
106949
(
2020
).
38.
S.
De
,
A. P.
Bartók
,
G.
Csányi
, and
M.
Ceriotti
,
Phys. Chem. Chem. Phys.
18
,
13754
(
2016
).
39.
E.
Boattini
,
F.
Smallenburg
, and
L.
Filion
,
Phys. Rev. Lett.
127
,
088007
(
2021
).
40.
C. C.
Aggarwal
,
A.
Hinneburg
, and
D. A.
Keim
, in
Database Theory—ICDT 2001
, edited by
J.
Van den Bussche
and
V.
Vianu
(
Springer Berlin Heidelberg
,
2001
), pp.
420
434
.
41.
I. T.
Jolliffe
and
J.
Cadima
,
Philos. Trans. R. Soc., A
374
,
20150202
(
2016
).
42.
T.
Hastie
,
R.
Tibshirani
, and
J.
Friedman
,
The Elements of Statistical Learning: Data Mining, Inference, and Prediction
, 2nd ed. (
Springer
,
New York
,
2016
).
43.
S.
Goldt
,
M.
Mézard
,
F.
Krzakala
, and
L.
Zdeborová
,
Phys. Rev. X
10
,
041044
(
2020
).
44.
T.
Mendes-Santos
,
X.
Turkeshi
,
M.
Dalmonte
, and
A.
Rodriguez
,
Phys. Rev. X
11
,
011040
(
2021
).
45.
J.
Paret
and
D.
Coslovich
,
J. Open Source Software
6
,
3723
(
2021
).
46.
D.
Coslovich
,
R. L.
Jack
, and
J.
Paret
(
2022
). “
Dimensionality reduction of local structure in glassy binary mixtures
,” Zenodo.
47.
48.
A.
Engel
and
C.
Van den Broeck
,
Statistical Mechanics of Learning
(
Cambridge University Press
,
Cambridge
,
2001
).
49.
M.
Leocmach
and
H.
Tanaka
,
Nat. Commun.
3
,
974
(
2012
).
50.
J.
Paret
, “
Hidden order in amorphous materials
,” Ph.D. thesis,
Université de Montpellier
,
2021
.
51.
J. A.
Hartigan
and
P. M.
Hartigan
,
Ann. Stat.
13
,
70
(
1985
).
52.
H.
Tong
and
H.
Tanaka
,
Phys. Rev. X
8
,
011041
(
2018
).
53.
54.

The range of the components in the reduced SOAP descriptor scales with n [see Eq. (14)], which creates a trivial increase of the variance across features. The features with the largest variance would thus dominate the PCA, hence the need for feature scaling to bring all the features to the same range.

55.

Note that without feature scaling, the PC1 direction gathers instead a large variance, which further increases with increasing σ. Very likely, this reflects the lack of normalization of the descriptor, which gives a strong weight to Q0, i.e., the local coordination number.

56.

The closest approach distance in our mixtures is typically half of the cutoff distance rcut.

57.
M. O. J.
Jäger
,
E. V.
Morooka
,
F.
Federici Canova
,
L.
Himanen
, and
A. S.
Foster
,
npj Comput. Mater.
4
,
37
(
2018
).
58.
M.
Ceriotti
,
M. J.
Willatt
, and
G.
Csányi
, “
Machine learning of atomic-scale properties based on physical principles
,” in
Handbook of Materials Modeling: Methods: Theory and Modeling
, edited by
W.
Andreoni
and
S.
Yip
(
Springer International Publishing
,
Cham
,
2018
), pp.
1
27
.
59.

We found that energy-minimized configurations provide somewhat more robust results upon changes of hyper-parameters.

60.
A.
Widmer-Cooper
,
P.
Harrowell
, and
H.
Fynewever
,
Phys. Rev. Lett.
93
,
135701
(
2004
).
61.
L.
Berthier
and
R. L.
Jack
,
Phys. Rev. E
76
,
041509
(
2007
).
62.
D.
Richard
,
M.
Ozawa
,
S.
Patinet
,
E.
Stanifer
,
B.
Shang
,
S. A.
Ridout
,
B.
Xu
,
G.
Zhang
,
P. K.
Morse
,
J.-L.
Barrat
,
L.
Berthier
,
M. L.
Falk
,
P.
Guan
,
A. J.
Liu
,
K.
Martens
,
S.
Sastry
,
D.
Vandembroucq
,
E.
Lerner
, and
M. L.
Manning
,
Phys. Rev. Mater.
4
,
113609
(
2020
).
63.
H.
Tong
and
H.
Tanaka
,
Nat. Commun.
10
,
5596
(
2019
).
64.
R. L.
Jack
,
A. J.
Dunleavy
, and
C. P.
Royall
,
Phys. Rev. Lett.
113
,
095703
(
2014
).
65.
E. D.
Cubuk
,
S. S.
Schoenholz
,
J. M.
Rieser
,
B. D.
Malone
,
J.
Rottler
,
D. J.
Durian
,
E.
Kaxiras
, and
A. J.
Liu
,
Phys. Rev. Lett.
114
,
108001
(
2015
).
66.
S. S.
Schoenholz
,
E. D.
Cubuk
,
D. M.
Sussman
,
E.
Kaxiras
, and
A. J.
Liu
,
Nat. Phys.
12
,
469
(
2016
).
67.
V.
Bapst
,
T.
Keck
,
A.
Grabska-Barwińska
,
C.
Donner
,
E. D.
Cubuk
,
S. S.
Schoenholz
,
A.
Obika
,
A. W. R.
Nelson
,
T.
Back
,
D.
Hassabis
, and
P.
Kohli
,
Nat. Phys.
16
,
448
(
2020
).
68.
R. M.
Alkemade
,
E.
Boattini
,
L.
Filion
, and
F.
Smallenburg
,
J. Chem. Phys.
156
,
204503
(
2022
).
69.
F.
Pedregosa
,
G.
Varoquaux
,
A.
Gramfort
,
V.
Michel
,
B.
Thirion
,
O.
Grisel
,
M.
Blondel
,
P.
Prettenhofer
,
R.
Weiss
,
V.
Dubourg
,
J.
Vanderplas
,
A.
Passos
,
D.
Cournapeau
,
M.
Brucher
,
M.
Perrot
, and
E.
Duchesnay
,
J. Mach. Learn. Res.
12
,
2825
(
2011
).
70.
71.
P.
Linardatos
,
V.
Papastefanopoulos
, and
S.
Kotsiantis
,
Entropy
23
,
18
(
2021
).
72.
M.
Abadi
,
A.
Agarwal
,
P.
Barham
,
E.
Brevdo
,
Z.
Chen
,
C.
Citro
,
G. S.
Corrado
,
A.
Davis
,
J.
Dean
,
M.
Devin
,
S.
Ghemawat
,
I.
Goodfellow
,
A.
Harp
,
G.
Irving
,
M.
Isard
,
Y.
Jia
,
R.
Jozefowicz
,
L.
Kaiser
,
M.
Kudlur
,
J.
Levenberg
,
D.
Mané
,
R.
Monga
,
S.
Moore
,
D.
Murray
,
C.
Olah
,
M.
Schuster
,
J.
Shlens
,
B.
Steiner
,
I.
Sutskever
,
K.
Talwar
,
P.
Tucker
,
V.
Vanhoucke
,
V.
Vasudevan
,
F.
Viégas
,
O.
Vinyals
,
P.
Warden
,
M.
Wattenberg
,
M.
Wicke
,
Y.
Yu
, and
X.
Zheng
, “
TensorFlow: Large-scale machine learning on heterogeneous systems
,” software available from tensorflow.org,
2015
.
73.
X.
Glorot
and
Y.
Bengio
, in
Volume 9: Proceedings of the Thirteenth International Conference on Artificial Intelligence and Statistics
, edited by
Y. W.
Teh
and
M.
Titterington
(
Proceedings of Machine Learning Research, PMLR
,
Chia Laguna Resort, Sardinia, Italy
,
2010
), pp.
249
256
.
74.
T. R.
Kirkpatrick
and
D.
Thirumalai
, “
Random first-order phase transition theory of the structural glass transition
,” in
Structural Glasses and Supercooled Liquids
(
John Wiley and Sons
,
2012
), pp.
223
236
.
75.
G.
Biroli
and
J. P.
Bouchaud
, “
The random first-order transition theory of glasses: A critical assessment
,” in
Structural Glasses and Supercooled Liquids
(
John Wiley and Sons
,
2012
), pp.
31
113
.
76.
G.
Tarjus
,
S. A.
Kivelson
,
Z.
Nussinov
, and
P.
Viot
,
J. Phys.: Condens. Matter
17
,
R1143
(
2005
).
77.
M.
Ozawa
and
L.
Berthier
,
J. Chem. Phys.
146
,
014502
(
2017
).
78.
P.
Patel
,
M. K.
Nandi
,
U. K.
Nandi
, and
S.
Maitra Bhattacharyya
,
J. Chem. Phys.
154
,
034503
(
2021
).
79.
A.
Siffer
,
P.-A.
Fouque
,
A.
Termier
, and
C.
Largouët
, in
Proceedings of the 24th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, KDD ’18
(
Association for Computing Machinery
,
2018
), pp.
2210
2218
.
80.
Z.
Zhang
and
W.
Kob
,
Proc. Natl. Acad. Sci. U. S. A.
117
,
14032
(
2020
).
81.
S.
Plimpton
,
J. Comput. Phys.
117
,
1
(
1995
).
82.
R.
Soklaski
,
Z.
Nussinov
,
Z.
Markow
,
K. F.
Kelton
, and
L.
Yang
,
Phys. Rev. B
87
,
184203
(
2013
).
83.
R.
Soklaski
,
V.
Tran
,
Z.
Nussinov
,
K. F.
Kelton
, and
L.
Yang
,
Philos. Mag.
96
,
1212
(
2016
).
84.
R. E.
Ryltsev
,
B. A.
Klumov
,
N. M.
Chtchelkatchev
, and
K. Y.
Shunyaev
,
J. Chem. Phys.
145
,
034506
(
2016
).
You do not currently have access to this content.