11institutetext: H.H. Wills Physics Laboratory, University of Bristol,
Bristol, BS8 1TL, United Kingdom 22institutetext: Centre for Nanoscience and
Quantum Information, Bristol BS8 1FD, United Kingdom 33institutetext: Institut
für Physik, Johannes Gutenberg-Universität Mainz, Staudingerweg 7-9, 55128
Mainz, Germany 44institutetext: School of Chemistry, University of Bristol,
Bristol BS8 1TS, United Kingdom

# Structural-dynamical transition in the Wahnström mixture

Francesco Turci 1122 Thomas Speck 33 C. Patrick Royall 112244

(Received: date / Revised version: date)

###### Abstract

In trajectory space, dynamical heterogeneities in glass-forming liquids
correspond to the emergence of a dynamical phase transition between an  active
phase poor in local structure and an  inactive phase which is rich in local
structure. We support this scenario with the study of a model additive mixture
of Lennard-Jones particles, quantifying how the choice of the relevant
structural and dynamical observable affects the transition in trajectory
space. We find that the low mobility, structure-rich phase is dominated by
icosahedral order. Applying a nonequilibrium rheological protocol, we connect
local order to the emergence of mechanical rigidity.

###### pacs:

PACS-64.70.Q-theory and modelling of glasses and PACS-83.10.Rscomputer
simulation of molecular dynamics in rheology

##  1 Introduction

Supercooled liquids show emergent dynamical and structural heterogeneities
when cooled towards the glass transition cavagna2009 ; berthier2011 ;
royall2015physrep ; barrat2011 . The relation between slow dynamics and some
form of short-range (local) order, however, is still poorly understood. On the
one hand, the efficient filling of space with atoms of different sizes
requires a certain degree of topological order torquato2000  and the dynamic
slowdown can rigorously be linked to emerging static lengthscales
montanari2006 ; on the other hand, computer simulations have shown that the
correlation between local structural features and slow dynamics is strongly
model dependent hocky2014pre ; royall2015 . In experiments, colloidal
royall2008 ; wochner2009 ; leocmach2012 ; royall2018jcp  and metallic glasses
sheng2006 ; chen2011 ; hirata2013  provide evidence for emerging local order
as well as, on the contrary, support for purely dynamical scenarios where
local structure has limited influence on the dynamics gokhale2014 ;
chikkadi2014 .  Historically, the study of local structure with complex higher
order metrics has played a decisive role in understanding amorphous systems
and packings since the times of Bernal and Finney bernal1959 ; bernal1960 ;
finney1970 ; finney1970mc  and has contributed to a geometric and
thermodynamic interpretation of the emerging frustration in glasses nelson1983
; tarjus2005 . However, alternative approaches which disregard structural
features and focus on dynamical chandler2010  or vibrational/elastic aspects
dyre2006  of the problem or relaxation have been proposed, in striking
contrast with established thermodynamic theories of the glass transition
lubchenko2007 ; lubchenko2015theory . It is therefore important to understand
what drives strong or weak coupling between structure and dynamics in
different supercooled liquids.

A major difficulty encountered in the investigation of the role of structural
changes in dynamic arrest is the fact that particle-resolved studies (and in
particular conventional computer simulations) can only access a limited
dynamic range of slow relaxation. Typically, this encompasses 4 to 5 orders of
magnitude in time, meaning that such studies mainly capture the onset of the
mechanisms that characterise the deeply supercooled and glassy regimes (when
the relaxation times are 10 to 20 orders of magnitude larger with respect to
the liquid regime) berthier2016 . Therefore, alternative sampling routes to
explore the deeply supercooled regime from a structural and/or dynamical point
of view have been developed in recent years, including pinning fields
cammarota2012 ; kob2013 ; kob2014nonlinear , particle-swap Monte-Carlo on
particular models berthier2016prl ; ninarello2017  or biased dynamical
ensembles chandler2010 ; hedges2009 ; speck2012 ; turci2017prx .

A potential route to study dynamical and structural heterogeneities in
glassformers is provided by efficient sampling methods in trajectory space,
where novel dynamical phase transitions have been uncovered and connected to
the dynamical slowdown observed in supercooled liquids chandler2010 . The
study of trajectory space in glassy systems has been originally promoted in
the context of the dynamical facilitation theory of slow dynamics garrahan2007
; keys2011 ; hedges2009 . Within this framework, on-lattice idealised models
garrahan2007 ; elmatad2009 ; turci2011 ; nemoto2017  as well as more realistic
models of structural glasses hedges2009 ; speck2012 ; jack2011 ;
pitard2011dynamic ; isobe2016  have been shown to undergo a first-order
dynamical phase transition in trajectory space between an active phase with
high mobility (fast relaxation) and an inactive phase with low mobility (slow
dynamics).

However, this purely dynamical picture has been more recently complemented by
a structural aspect: active/inactive phases correspond to trajectories
particularly poor/rich in local structure speck2012jcp ; pinchaipat2017  and
can be seen as representative of the low temperature state of the supercooled
liquid turci2017prx . Dynamical transitions are therefore understood to
correspond to a structural-dynamical transitions, where the slowdown of the
dynamics becomes intimately related to the growth of short-range-order
domains.

Still, much of the evidence for structural-dynamical phase transitions in
atomistic models of glassformers up to now is restricted to only two model
systems (the Kob-Andersen mixture hedges2009 ; jack2011 ; speck2012jcp ;
speck2012 ; turci2017prx , a popular Lennard-Jones mixture with weak
structural dynamical correlations hocky2014 , and the moderately polydisperse
hard spheres pinchaipat2017 ). In order to understand how system-dependent
this picture is, it is important to extend the scope of these studies to other
model systems.

In the present numerical work, we consider the case of a popular atomistic
glassformer originally introduced by Göran Wahnström as a simple model for
supercooled liquids wahnstrom1991 . It consists in a binary mixture of
Lennard-Jones particles whose parametrization has been found to provide a good
model of fragile glasses, with a particularly strong coupling between its slow
dynamics and the emergence of local geometrical motifs hocky2014 ;
malins2013jcp ; pinney2015 . These are typically icosahedra, a very common
arrangement in simple models of glass-forming liquids composed of spherically
symmetric particles.

The article is structured as follows: in Section 2 we present the model
studied and the importance sampling technique employed for trajectory
sampling; in Section 3 we introduce the relevant observables and the phase
transitions in trajectory space that can be probed through the dynamical
$s$-ensemble and the structural-dynamical $\mu$-ensemble; in Section 4 we show
that it is possible to connect the structural-dynamical transition to the
emergence of rigidity in the glass, as the icosahedra-rich phase presents
distinctive rheological properties; finally, we conclude the article with an
overview of the results and their implications.

##  2 Model and sampling technique

###  2.1 The Wahnström binary mixture

We study the Wahnström binary mixture of Lennard-Jones particles. The model is
a 50:50 mixture of large (A) and small (B) particles with parameters
$\sigma_{A}/\sigma_{B}=1.2$, $m_{A}/m_{B}=2$, $\epsilon_{A}/\epsilon_{B}=1$
and cutoff $r_{\rm cut}=2.5\sigma$ at number density $\rho=1.296$. Lengths,
temperature and times are reported in units of $\sigma_{A}$,
$\epsilon_{A}/k_{B}$ , and $(m_{A}\sigma_{A}^{2}/\epsilon_{A})^{1/2}$,
respectively. The mixing rule for the interaction is additive, i.e. it follows
the Lorentz-Berthelot rules

$\sigma_{AB}=\frac{\sigma_{A}+\sigma_{B}}{2},\quad\epsilon_{AB}=\sqrt{{\epsilon_{A}\epsilon_{B}}}.$
(1)

This atomistic supercooled liquid has been extensively studied since its
original design wahnstrom1991 . The model reproduces to a good degree the
relaxation behaviour of so-called fragile glasses, as its structural
relaxation time $\tau_{\alpha}$ (as measured from the decay of the
intermediate scattering function malins2013jcp ) undergoes a non-Arrhenius
(super-exponential) increase when the system is cooled below the crossover or
onset temperature $T_{\rm onset}=1.0$ coslovich2011 ; malins2013jcp .
Furthermore, as the temperature is decreased, the disordered structure of the
liquid changes with the formation of five-fold symmetric domains and in
particular of local particle motifs with icosahedral coordination
malins2013jcp ; coslovich2011 ; coslovich2007  which contribute to the
emergence of strong frustration turci2017prl . Equilibration of the liquid in
conventional simulations around and below the so-called mode-coupling
temperature $T_{MC}=0.56$ is computationally expensive, making the low
temperature, activated regime (crucial for testing theoretical
predictionsberthier2011 ) unreachable. Divergence of the relaxation times, if
modelled by the super-Arrhenius Vogel-Fulcher-Tamman law
$\ln\tau_{\alpha}/\tau_{\infty}=DT_{0}/(T-T_{0})$, is predicted at temperature
$T_{0}\approx 0.46$. For reference, we report in Fig. 1(a) the temperature
dependence of both the structural and dynamical properties of the model.

Beyond local structural order, the model has been shown to crystallise, under
suitable conditions, into a $\rm MgZn_{2}$ Laves phase formed by icosahedral
motifs and so-called Frank-Kasper bonds pedersen2010  but in the supercooled
liquid regime the contribution of such a large unit cell to the increased
degree of local order has been shown to be limited malins2013jcp .

###  2.2 Replica exchange in trajectory space

As in previous work speck2012 ; turci2017prx ; speck2012jcp ; pinchaipat2017 ,
in order to sample large fluctuations of the time-integrated observables we
employ an importance sampling technique that extends equilibrium replica
exchange methods to ensembles of trajectories.

We sample space and time extensive observables $O_{x}$ on systems of $N=512$
evolving for a finite observation time $t_{\rm obs}$. A generic time-
integrated observable $O_{x}$ is defined as a double sum over the number of
particles and a discretization of time into $L$ intervals for a total of
$K=L+1$ points:

$O_{x}=\sum_{t=-L/2}^{+L/2}\sum_{i=1}^{N}f_{t,i}$ (2)

where $f_{t,i}$ is a specific microscopic observable (e.g. a single particle
indicator function).

The goal of the importance sampling technique is to efficiently measure the
probability distribution $P(O_{x};T_{\rm s})$ for a given value of the
thermostat temperature $T_{\rm s}$. In particular, we are interested in the
large deviations from the typical value of the probability distribution. In
order to calculate such rare fluctuations in trajectory space, new
trajectories are generated through shifting and shooting moves (inspired by
Transition Path Sampling bolhuis2002 ). Hence, the algorithm performs a random
walk in trajectory space with acceptance probability determined by a
Metropolis rule

$p_{\rm acceptance}=\min\left\\{1,\frac{e^{-\Psi(O_{x}^{\rm
new}})}{e^{-\Psi(O_{x}^{\rm old})}}\right\\},$ (3)

which ensures detailed balance and where $\Psi(O_{x}^{\rm
new}),\Psi(O_{x}^{\rm old})$ are the values of a biasing pseudo-potential
which is a function of the extensive observable $O_{x}$ computed over old and
new trajectories. We choose $\Psi$ to have a parabolic form

$\Psi_{j}(O_{x})=\frac{1}{2}\omega(O_{x}-O_{0}^{j})^{2},$ (4)

where $O_{0}^{j}$ is the reference (typical) value associated to replica $j$.
Depending on the observable, we take a number of distinct replicas varying
from 8 to 16, with equally spaced values of $O_{j}$ and values for the
harmonic constant $\omega$ that ensure good mixing of neighboring replicas.
Mixing is also enhanced by 2500 swap attempts among all (not-necessarily
neighboring) replicas.

The Monte-Carlo algorithm in trajectory space simulation starts with an
equilibrated trajectory assigned to all replicas at temperature $T_{\rm s}$. A
new trajectory is then generated via Transition Path Sampling moves (1/4
shifting, 3/4 shooting bolhuis2002 ) independently for every replica, accepted
or rejected according to Eq. 3. Swap attempts between different replicas are
then performed, completing the cycle. During the sampling, we employ a
velocity-Verlet integrator with timestep $dt=0.005$ to resolve the equation of
motion and the Andersen thermostat to keep the temperature constant.

Figure 1:  (a) Structural relaxation time $\tau_{\alpha}$(circles, logarithmic
scale) and population of particles in icosahedral motifs $n$ (pentagons,
linear scale) as a function of the inverse temperature. Continuous lines are
the Vogel-Fulcher-Tammann (VFT) fit
$\ln\tau_{\alpha}/\tau_{\infty}=DT_{0}/(T-T_{0})$ and an empirical
$n=[1+(T/T_{1/2})^{\gamma}]^{-1}$ fit for $\tau_{\alpha}$ and $n$
respectively, with $\gamma=6.6,T_{1/2}=0.47$ from Ref. jenkinson2017 .
Vertical lines also indicate the location of relevant temperatures in the
Wahnström model: the onset of slow dynamics $T_{\rm onset}$, the mode-coupling
transition temperature $T_{\rm MC}$ and the VFT temperature $T_{0}$. (b)
Three-dimensional rendering of the four local motifs considered in this work:
the icosahedron, the defective icosahedron (10B) and two nine-particle motifs
unrelated to five-fold symmetry. For the icosahedron and 10B, a pentagonal
ring of particles is highlighted in gold.

We perform several tens of thousands of cycles and collect statistics and
block averages from three to eight non-overlapping blocks of data whose size
ranges between $1.2\cdot 10^{4}$ and $3\cdot 10^{4}$ trajectories, which
deliver estimates for the averages and standard errors. Crucially, depending
on the sampling temperature $T_{\rm s}$, correlations may be very long-lived
and the number of Monte-Carlo cycles spent during the equilibration in
trajectory space can be very large ($\sim 6\cdot 10^{4}$) Monte-Carlo sweeps),
as shown in Fig. 2. We then discard trajectories produced during equilibration
and collect data from the converged, late steps of the Monte-Carlo.

Figure 2: Example of a Monte-Carlo run in trajectory space. At temperature
$T_{\rm s}=0.71$ we collect trajectories for the time-integrated number of
icosahedra from 12 distinct replicas corresponding to the different colours. A
very long transient of $6\cdot 10^{4}$ Monte-Carlo cycles (vertical dashed
line) is observed and statistically relevant data are only collected from the
remaining $4\cdot 10^{4}$ trajectories.

From the collected ensemble of trajectories, we calculate distributions and
expectation values using the Multistate Bennett Acceptance ratio (MBAR) method
extended to ensembles of trajectories minh2009 . This technique allows us to
obtain the unbiased probability distribution $P(O_{x};T_{\rm s})$ and
expectation values for any quantity $A$ as

$\langle A\rangle_{y}=\frac{\langle Ae^{yO_{x}}\rangle}{\langle
e^{yO_{x}}\rangle}$ (5)

where $y$ is the conjugated field to the observable $O_{x}$ and
$\langle\rangle$ indicate averages according to the unbiased distribution
$P(O_{x};T_{\rm s})$. Notice that the denominator in Eq. 5 corresponds to the
moment generating function of the probability distribution and is a
generalization of the partition sum to trajectory spaces. In this work, we
focus on two particular ensembles: the $s$-ensemble, where $y=-s$ and the
observable $O_{x}$ is the time-integrated mobility of the particles; and the
$\mu$-ensemble, where $y=\mu$ and the relevant observable is a time-integral
over the number of particles in a particular local motif (here the
icosahedron).

In the presence of transitions in trajectory space, we expect to measure
probability distributions for the time-integrated observables that are not
Gaussian and display long, eventually exponential tails. For variables that
follow a Gaussian probability distribution, the kurtosis
$\kappa_{4}=\langle(O_{x}-\langle
O_{x}\rangle)^{4}\rangle/\langle(O_{x}-\langle O_{x}\rangle)^{2}\rangle^{2}$
(i.e. the ratio between the fourth central moment and the squared second
moment) has value $\kappa_{4}=3$. Therefore, the excess kurtosis $\kappa_{\rm
exc}=\kappa_{4}-3$ is often employed as a benchmark for the deviations from a
Gaussian distribution. So-called leptokurtic (fat-tailed) distributions
correspond to positive $\kappa_{\rm exc}$ while platykurtic (thin-tailed)
distributions correspond to negative $\kappa_{\rm exc}$.

##  3 Dynamical and structural phase transitions

Figure 3: Probability distributions as a function of the mobility-conjugated
field $s$ for dynamical trajectories of the Wahnström model of duration $K=20$
at temperature $T=0.67$. The several observables $x$ are: the time-integrated
population of mobility excitations, the inherent state energy (ISE) of the
central configuration of every trajectory, and the time-integrated population
of particles in icosahedral motifs, 9A motifs and 10B motifs. The blue, green,
yellow and red datasets refer to trajectories of length $K=20,40,60,100$
respectively. The dashed black lines are fits to a Gaussian probability
distribution centred at the peak value.

K |  Excitations (Mean, Variance, $\kappa_{\rm exc}$) |  13A (Mean, Variance, $\kappa_{\rm exc}$) |  10B (Mean, Variance, $\kappa_{\rm exc}$) |  9A (Mean, Variance, $\kappa_{\rm exc}$) |  BCC9 (Mean, Variance, $\kappa_{\rm exc}$) |  ISE (Mean, Variance, $\kappa_{\rm exc}$)
---|---|---|---|---|---|---
20 | 0.09 0.0005 -0.04 | 0.09 0.0007 0.5 | 0.6 0.002 -0.1 | 0.07 7e-05 -0.06 | 0.7 0.0003 -0.5 | -7 0.0008 0.02
40 | 0.1 0.0003 -0.08 | 0.09 0.0006 0.8 | 0.6 0.001 0.1 | 0.07 4e-05 0.08 | 0.7 0.0002 -0.5 | -7 0.0008 0.003
60 | 0.1 0.0003 -0.03 | 0.09 0.0005 0.9 | 0.6 0.001 0.2 | 0.07 3e-05 0.06 | 0.7 0.0002 -0.4 | -7 0.0007 -0.03
100 | 0.1 0.0002 0.1 | 0.09 0.0004 2.0 | 0.6 0.0008 0.6 | 0.07 2e-05 0.07 | 0.7 0.0002 -0.3 | -7 0.0007 -0.005

Table 1: Expectation values for the mean, the variance and the excess kurtosis
$\kappa_{\rm exc}$ for several observables as measured via trajectory sampling
for different values of the trajectory length at temperature $T_{\rm s}=0.67$.

###  3.1 Observables

We analyse the emergence of phase transitions in trajectory space by
monitoring a variety of observables. We perform importance sampling in
trajectory space according to time-integrated observables that are either
dynamical (such as the mobility excitations) or structural (a selection of
geometrically different structural motifs, see Fig. 1(b)). Furthermore, in
order to relate the trajectory-space picture back to the thermodynamic
picture, we also monitor the inherent state energy of the selected
configurations, whose statistics in the trajectory ensemble has been proven to
closely reproduce the equilibrium properties. Structures are detected
employing the Topological Cluster Classification algorithm and we refer to
Reference malins2013tcc  for a more detailed discussion of the geometries
considered here.

In particular, for the time-integrated quantities we have:

  * •

number of excitations: To quantify the number of mobile particles, we compute
the observable

$O_{\rm
excitation}=\sum_{t=-L/2}^{L/2}\sum_{i=1}^{N}|\Theta(\Delta\bm{r}_{i}(t)-a)|,$
(6)

where $\Theta(\Delta\bm{r}_{i}(t)=\bm{r}_{i}(t)-\bm{r}_{i}(t-\Delta t)$ is the
single particle displacement, $\Theta$ is the Heaviside function and $a$ is a
scale for cage motion, here set to $a=0.3\sigma_{\rm large}$.

  * •

number of particles in icosahedral motifs: Given the important role of
icosahedral order in the Wahnström mixture, we track this specific local motif
along the trajectories. Additionally, we perform importance sampling according
to the number of icosahedra. The corresponding time-integrated extensive
structural-dynamical observable is then

$O_{\rm ico}=\sum_{t=-L/2}^{L/2}\sum_{i=1}^{N}h_{i}^{\rm ico}(t),$ (7)

where $h_{i}^{\rm ico}$ is an indicator function, which takes value 1 if a
particle is found in an icosahedral environment or 0 if it is not.  With a
certain abuse of language, we will interchangeably refer to the population of
icosahedra or the population of particles in icosahedral motifs when
considering the intensive quantity $O_{\rm ico}/NK$.

  * •

number of particles in 9A motifs: We compute $O_{9A}$ performing the summation
as in Eq.7, but with a different indicator function $h_{i}^{\rm 9A}$. In this
case, we consider the 9A structure of the Topological Cluster Classification,
which is composed of six particles combined to form three four-folded rings,
surrounded by three further spindle particles on each quadrangular facet
(forming a tricapped trigonal prism). According to previous studies
malins2013jcp , we do not expect this motif to be a good predictor of
structural-dynamical heterogeneity for the Wahnström mixture. However, in the
case of other simple liquids dominated by five-fold symmetric local order,
such as moderately polydisperse hard-sphere, 9A motifs have been shown to be
complementary to local icosahedral order, becoming less frequent when the
packing fraction (and the population of icosahedra) increase royall2017 .

  * •

number of particles in BCC motifs: As a further test, we compute the time-
integrated observable $O_{BCC}$ considering a nine particle structure that
(weakly) correlates with body centered cubic local order and anti-correlates
strongly with icosahedral and five-fold symmetric order.

  * •

number of particles in five-fold symmetric motifs: Finally, to track five-fold
symmetric local order that is not fully icosahedral, we consider the defective
icosahedron structure 10B, composed of three interlaced pentagonal rings. This
structure is characteristic of hard-sphere mixtures, and has been shown both
in simulations and experiments to drive a clear structural-dynamical phase
transition pinchaipat2017 .

We also measure a static observable, i.e. not time-integrated. This is the
inherent state energy (ISE) of configurations located at the centre of each
trajectory, chosen in order to avoid finite-time effects on the statistics
coslovich2016 . Inherent state energies are obtained minimising the potential
energy of the system for a maximum of 1000 iterations of the FIRE algorithm
bitzek2006 .

###  3.2 $s$-ensemble

First, we consider the response of the system to a dynamical bias. This means
that we collect trajectories according the observable $O_{\rm exc}$, i.e. the
time-integrated number of mobility excitations. We employ the large deviation
formalism and notation, and we define $s$ as the dynamical conjugate field
related to the excitations, so that positive/negative values of $s$ correspond
to atypically small/large densities of mobility excitations, hence the name of
$s$-ensemble garrahan2007 . As we sample the mobility large deviations, we
track all the other dynamical and static order parameters.

In Fig. 3 and in Table 1 we summarise our findings for a particular thermostat
temperature $T_{\rm s}=0.67T_{\rm onset}\approx 1.2T_{\rm MC}$, where $T_{\rm
MC}$ and $T_{\rm onset}$ indicate the transition temperature predicted by the
power-law fit to the relaxation of mode-coupling theory and the onset of the
two-step relaxation dynamics respectively. In Fig. 3 we compare the (scaled)
logarithm of the probability distributions (i.e. the rate function) of the
considered observables $O_{x}$ for increasing values of the trajectory length
$t_{\rm obs}=1.9,3.8,5.7,9.5\tau_{\alpha}$ ($K=20,40,60,100$). At the
considered temperature, we expect to observe deviations from Gaussian
fluctuations in the tails (i.e. large deviations) of the probability
distributions. With this comparative analysis, we want to stress that the
choice of the observable is non-trivial and different observables present
characteristic features.

First we notice that the population of excitations (which is the reaction
coordinate along which we perform importance sampling) shows mostly Gaussian
fluctuations around the mean value $\langle O_{x}\rangle/KN=0.096(2)$ for all
the sampled trajectory lengths. However, the variance computed at different
trajectory lengths appears to slowly converge to smaller values, with the
tails of the probability distributions gradually narrowing. This indicates
that very short trajectories of length $t_{\rm obs}=1.9,K=20$ are affected by
finite size effects that enhance the observation of large fluctuations.

Higher order moments converge even more slowly but point to the emergence of
non-Gaussian features. For example, the excess kurtosis is negative for short
trajectories and becomes mildly positive for the longest trajectories $K=100$.
This underlines that even longer trajectories are needed to obtain more marked
signatures of a dynamical phase transition in terms of population of
excitations at the relatively high temperature considered here, with an non-
negligible increase of the computational cost. Notice that it is only in the
long time limit that a large deviation principle holds and rate functions
converge touchette2017 , and therefore it is only in this limit that a formal
phase transition in trajectory space is expected.

Given the weak response in the mobility excitations, what signatures do we
observe in the other observables measured on the same trajectories produced in
the $s$-ensemble? In the following, we analyse them one by one.

For the time-integrated population of particles in icosahedral motifs, we
observe that average values do not depend on $K$; however, higher order
moments show a dependence on the trajectory length. The values of the excess
kurtosis $\kappa_{\rm exc}$ show a marked increase in non-Gaussian features of
the trajectory probability distribution, as confirmed by direct inspection of
the probability distribution. The excess kurtosis is positive (i.e. fat-tails)
and goes approximately from 0.48 to 2.0 when the trajectory length increases
from K=20 to K=100. For a comparison, notice that for a common leptokurtic
distribution of positive random variables such as the Rayleigh distribution,
the excess kurtosis is $\kappa_{\rm exc}=-(6\pi^{2}-24\pi+16)/(4-\pi)^{2}$
hence $\kappa_{\rm exc}\approx 0.24$ showing that the distribution for the
icosahedra is even more leptokurtic. Compared to the response of the mobility
excitations, the time-integrated population of icosahedra provide a much
stronger signature for a dynamical phase transition. In particular we observe
that populations of icosahedra of order $0.2$ are only two orders of magnitude
less likely than the converged typical value $\langle O_{\rm
ico}\rangle/NK=0.09$, with a strong exponential tail in the probability
distribution. Non-Gaussian fluctuations are therefore stronger when tracking
the time-integrated population of icosahedra than in the case of excitations.

These results are consistent with previous literature malins2013jcp ;
turci2017prl ; hocky2014  where the role of icosahedral motifs as locally
favoured structures (LFS) of the Wahnström mixture as been discussed and their
strong correlation with dynamical heterogeneities measured. They also confirm
the scenario originally suggested for another popular glass-former (the Kob-
Andersen mixture), whereby trajectories sampled according to time-integrals of
the LFS delivered stronger signatures for a dynamical transition that mobility
excitations speck2012 ; turci2017prx .

Figure 4: Dynamical transition at $T_{\rm s}=0.67$ for trajectories of length
$K=60$. Excitations and icosahedra clearly anticorrelate while mobility and
inherent state energies correlate. BCC9 and 9A structures are negatively
correlated with the icosahedra while the five-fold symmetric 10B are
positively correlated.

An icosahedral motif is detected in the TCC via the combination of seven five-
fold symmetric rings malins2013tcc , and the statistics of the number of
icosahedra appears to strongly indicate the presence of non-Gaussian
fluctuations related to a structural-dynamical phase transition in the system.
How does such transition change if we take into account a less restrictive
observable that still identifies five-fold symmetry? To answer this question,
we consider the so-called defective icosahedron structure 10B (see Sec.3.1
above). We first notice that the average population of particles in 10B per
trajectory is much larger than the population of icosahedra (0.59 vs 0.089)
and the variance again slowly converges with increasing $t_{\rm obs}$.
However, The excess kurtosis is much smaller in absolute values, changing sign
from negative towards positive values (leptokurtic distributions) as the
trajectory length is increased. This matches the dynamical notion of locally
favoured structures: icosahedra are not only the minimum energy structure for
the Wahnström interaction, they also are the individual motif (among the
several options of the Topological Cluster Classification) that displays the
longest persistence time malins2013jcp . The indicators for a structural-
dynamical transition in terms of 10B motifs are much weaker than in the case
of icosahedral order. Yet, they confirm that the inactive (low population of
excitations) regime is dominated by long-lived five-fold symmetric motifs.

Is it possible to detect signs of the transition in other structural
observables? We consider the two exemplary cases of the 9A and BCC9
structures. These motifs both correspond to arrangements of 9 particles with
different symmetries which are not minimum energy clusters of the potential.
The average populations of the two motifs are very different ($\sim 0.07$ for
9A and $0.74$ for BCC9). The 9A probability distribution is well approximated
by a Gaussian for all the trajectory lengths considered here, and the
corresponding excess kurtosis are (in absolute value) the smallest among all
the considered structures. The BCC9 motif, conversely, presents relatively
large but negative excess kurtosis, indicating that the tails of the
distributions decay more rapidly than in the case of a Gaussian distribution.

For a given trajectory length, we consider the $s$-ensemble averages as a
function of the field $s$ to highlight correlations and anticorrelations
between the observables. With trajectories of length $K=60$, we show in Fig. 4
that for negative $s\ll 0$ we sample trajectories characterised by large
densities of excitations (active phase) while for $s\gg 0$ we have
trajectories with low densities of excitations (inactive phase). These
correspond respectively to trajectories that are poor and rich in icosahedra.
The anticorrelation between mobility and five-fold symmetry is reflected also
in the negative correlation between mobility and 10B structures. On the other
hand, mobility positively correlates with the remaining motifs (9A and BCC9).

Finally, we consider how the active/inactive transition is translated in terms
of the energy landscape of the system. To do so we also track the inherent
state energy (ISE) of the central configuration of every single trajectory and
plot the corresponding probability distribution. This (as expected) does not
show dependence in the trajectory length and it is well reproduced by a
Gaussian fit, see Fig. 3. Normal fluctuations are confirmed by the analysis of
the respective excess kurtosis, which are by far the smallest measured
throughout our analysis (as small as $\kappa_{\rm exc}=-0.003$). In Fig. 4, we
do observe a transition to trajectories whose central configurations display
typically much more negative energies with respect to the equilibrium typical
value at $s=0$. This is consistent with the finding that in a different binary
mixture (Kob-Andersen) low mobility is a good predictor of low inherent state
energies coslovich2016 .

###  3.3 $\mu$-ensemble

The direct route to access structural-dynamical phase transitions is to sample
trajectories according to a relevant time-integrated structural observable.
From the previous discussion, and in particular from the magnitude of the non-
Gaussian fluctuations as measured by the excess kurtosis, it is evident that
icosahedral motifs are well suited to this purpose.

Therefore we perform additional trajectory sampling according to the time-
integrated number of icosahedral motifs. As  in the case of the $s$-ensemble,
we sample trajectories following the replica exchange scheme, with quadratic
pseudo-potentials for the replicas with suitable spring constant $\omega$.

In the new ensemble of trajectories, the conjugate field related to the number
of particles in icosahedral motifs is termed $\mu$. Consistently with previous
works in the literature speck2012 ; turci2017prx , averages of any arbitrary
quantity in the $\mu$-ensemble are defined as

$\langle A\rangle_{\mu}=\frac{\langle Ae^{\mu O_{ico}}\rangle}{\langle e^{\mu
O_{ico}}\rangle}.$ (8)

In the previous section, we have shown that in the $s$-ensemble an emergent
active/inactive transition is mirrored by a rapid increase of the population
of particles in icosahedral motifs. In the $\mu$-ensemble we sample such
structural transition directly. In Fig. 5(a,b) we plot the $\mu$-dependence of
the average mobility $\langle O_{\rm exc}\rangle_{\mu}/NK$ and the average
population of icosahedra $\langle O_{\rm ico}\rangle_{\mu}/NK$ for several
thermostat temperatures $T_{\rm s}$, from $T_{\rm s}=0.72$ to $T_{\rm
s}=0.65$. At different temperatures, we perform simulations of different
trajectory lengths $t_{\rm obs}$. Since the relevant time-scale for the
dynamics is the structural relaxation time $\tau_{\alpha}$, we plot the first
moments as a function of the non-dimensional scaled conjugate field
$\mu\tau_{\rm obs}:=\mu t_{\rm obs}/\tau_{\alpha}$. Just below the onset
temperature we observe signs of a phase transition at large $\mu\tau_{\rm
obs}$ between  trajectories poor in icosahedra with high mobility and
trajectories rich in icosahedra with low mobility. As we reduce the thermostat
temperature, the transition moves to values closer to $\mu=0$. Through a
spline fit and the estimate of the maximum in the derivative, we obtain the
value $\mu^{\ast}\tau_{\rm obs}$ at which the transition takes place.

Figure 5: Structural-dynamical phase transition. (a-b) Time-integrated density
of excitations (a) and of icosahedra (b) vs the rescaled conjugate field
$\mu\tau_{\rm obs}$ for different sampling temperatures $T_{\rm s}$. The
observation time expressed in units of the structural relaxation time
$\mu\tau_{\rm obs}=\mu t_{\rm obs}/\tau_{\alpha}$ is also reported. (c-d)
Probability distributions of the population of icosahedra per trajectory: (c)
at $\mu=0$ and (d) at $\mu=\mu^{\ast}(T)$, shifted by the mean value and the
standard deviation $\sigma_{ico}$ in order to highlight the tail behaviour.
Figure 6: Structural-dynamical phase transition in the $T-\mu$ plane. The dots
correspond to the couples $(T_{\rm s},\mu^{\ast})$ determined from trajectory
sampling. The blue and red dashed lines are a linear and power law fit
respectively.

The very small values of $\mu^{\ast}\tau_{\rm obs}$ at relatively high
temperatures compared to $T_{0}$ obtained from the Vogel-Fulcher-Tammann fit
or the mode-coupling $T_{MC}$ temperatures suggest that trajectories with an
exceptionally high population of icosahedra should be highly likely, and
signatures of bi-modality in the probability distribution of the time-
integrated observables should become accessible even to conventional
simulations as the temperature is reduced.

In Fig. 5 (c,d) we plot such probability distributions both for $\mu=0$ and
the critical value $\mu=\mu\tau_{\rm obs}^{\ast}$, shifting and rescaling the
abscissa axis by the mean and the standard deviation. We observe that, as
temperature decreases, the structure-rich tail of the probability
distributions raises of several orders of magnitude. Signs of bimodality are
weak at low temperatures, due to the relatively short observation time $t_{\rm
obs}$, but clearer at higher temperatures. Moreover, if we evaluate the
probability distributions at coexistence $\mu=\mu^{\ast}$, Fig.5(d), a peak at
high population of structures emerges more clearly.

The knowledge of $\mu^{\ast}\tau_{\rm obs}(T_{\rm s})$ allows us to draw an
approximate structural-dynamical phase diagram, Fig. 6, identifying the locus
of points where the transition from icosahedra-poor to icosahedra-rich
trajectories occurs. For the considered temperatures, we observe that most of
the data points lie on a straight line. An extrapolation of the line to
$\mu\tau_{\rm obs}=0$ would imply that at temperature $T=0.64$ coexistence
between the two structural-dynamical phases would be observable at $\mu=0$,
i.e. in conventional simulations with no need for importance sampling.
Previous numerical studies of the model kim2013 ; malins2013jcp  managed to
equilibrate the supercooled liquid down to temperature $T=0.58$, with no
signature of a transition while decreasing the temperature, but with a rapid
increase of the population of icosahedra. This excludes the possibility of a
transition at $\mu=0$ for at least $T>0.58$. As discussed in turci2017prx ,
several alternative scenarios can be obtained with different extrapolations at
low temperatures, including ones where the transition asymptotically reaches
$\mu=0$ only in the $T\rightarrow 0$ limit jack2010 . Here we notice that as
we reduce the temperature, the critical field $\mu^{\ast}\tau_{\rm obs}$ is
reduced by progressively smaller amounts for the successive temperatures.
Lower temperature sampling is partly hindered by the long convergence times of
the Monte-Carlo in trajectory space, see Fig. 2.

In the icosahedra-rich regime, approximately 50% of the particles can be found
in a local icosahedral environment. However, a complex unit cell formed by
several icosahedra and Frank-Kasper bonds has been shown to drive the system
towards crystallisation pedersen2010 . We check this possibility by monitoring
the concentration of Frank-Kasper bonds, here defined as pairs of large A
particles surrounded by six common B particles. In Fig. 7 we plot the average
fraction of particles involved in Frank-Kasper bonds for the increasing
reference concentration of icosahedra $O_{\rm ico}^{j}$ in the replica-
exchange scheme at an exemplary temperature $T_{\rm s}=0.67$. We observe a
rapid increase in the number of Frank-Kasper bonds as we consider replicas
with very high concentrations of icosahedra. This is consistent with the
overall behaviour of the Wahnström supercooled liquid at low temperatures,
where Frank-Kasper bonds are very common pedersen2010 . However, in order to
form a crystalline phase, four-fold Frank-Kasper bonds between the large
particle species are necessary. If we focus on the fraction of A particles in
four-fold bonds, this increases very mildly across all of the replicas, and
stays below 5% in the highest bias replica, excluding crystal formation in the
icosahedra-rich phase.

In conclusion, both the $s$ and the $\mu$ ensemble calculations provide
evidence for an inactive and icosahedra-rich dynamical phase that becomes
progressively more likely to be observed for $T<T_{\rm onset}$. We now study
the icosahedral phase more in detail to understand its relation with the
emergence of rigidity in the glass.

Figure 7: Analysis in terms of Frank-Kasper (FK) bonds for a representative
temperature $T_{\rm s}=0.67$. The fraction of particles involved in FK bonds
increases as we consider $\mu$-ensemble replicas with larger typical values of
the concentration of icosahedra $O_{\rm ico}^{j}/NK$ (orange diamonds).
However, the fraction of large particles involved in four-fold FK bonds
(necessary for crystallisation) is below 5% even for the replicas with the
largest $O_{\rm ico}^{j}$ (blue circles). Errorbars indicate standard
deviations and lines are guides to the eye.

##  4 Rheological response of the inactive/icosahedra-rich phase

As a supercooled liquid is cooled down, it eventually undergoes an
experimental glass transition where the relaxation time $\tau_{\alpha}$
exceeds the available observation time by many orders of magnitude. Such a
phenomenological transition is accompanied by the emergence of solid
behaviour: the glass behaves like a solid, in the sense that it can be probed
through rheological measurements, proving a finite elastic response and shear
modulus.

We have shown that as the temperature is decreased, the Wahnström mixture
explores more and more frequently trajectories that are exceptionally rich in
structure. Moreover, the icosahedra-rich trajectories not only are
characterised by low mobility (inactive trajectories) but they also tend to
have configurations with low inherent state energies. Is it possible to
connect these structural and dynamical changes to the emergence of solidity,
i.e. to the rheological response of the system?

We test this idea realising an ensemble of configurations extracted from the
trajectories produced in the $\mu$ ensemble at the thermostat temperature
$T_{\rm s}=0.65$. From every umbrella $i$ of the replica-exchange algorithm we
extract a population of configurations that are representative of the
fluctuations, in trajectory space, around a specific value of the population
of icosahedra

$n_{i}^{0}=O_{\rm ico}/NK.$ (9)

In particular, we produce a discrete group of 8 sets with 75 initial
configurations each at the following typical population of icosahedra
$n_{i}^{0}=[0.09,0.17,0.29,0.33,0.38,0.43,0.49,0.55]$. According to the
available data and the extrapolation of the fit shown in Fig. 1, these
populations would be typical in the equilibrium supercooled at temperatures
[0.67, 0.60, 0.54, 0.52, 0.50, 0.49, 0.46]. In our simulations, we take
averages for every set of initial conditions extracted from distinct replicas.

To understand the purely mechanical response of the different sets of
configurations we study the linear shearing of the system in the Athermal
Quasi-static limit (AQS) maloney2004 ; maloney2006 . Under this protocol, the
system is slowly deformed in a chosen direction at a fixed shear rate
$\dot{\gamma}=0.005$ for a small time interval $\Delta t=0.005$; subsequently,
the FIRE energy minimisation algorithm bitzek2006  is employed to lead the
particles to the closest inherent state. The two steps are repeated until the
system reaches a maximum total strain of $\gamma=0.5$

Figure 8: Average shear-stress as a function of shear strain for different
values of the typical initial population of icosahedra in $\mu$-ensemble
configurations $n_{i}^{0}$. The vertical dashed line corresponds to the yield
strain.  Figure 9: Overall average population of icosahedra as a function of
shear strain for different values of the typical initial population of
icosahedra in $\mu$-ensemble configurations $n_{i}^{0}$. The vertical dashed
line corresponds to the yield strain.

In Fig. 8 and Fig. 9 we plot the response of the system in terms of shear
stress $\sigma^{xy}$ and fraction of particles in icosahedral domains for
different typical values of the initial population of icosahedra $n_{i}^{0}$.
The first striking result is that the yield stress $\sigma_{\rm
yield}=\max_{\gamma}\sigma^{xy}(\gamma)$ strongly depends on $n_{i}^{0}$, and
it approximately doubles as the typical population of icosahedra quadruples.
The yield strain $\gamma_{\rm yield}$ (the value of strain at which the
maximum stress is reached) is not sensitive to the different starting
conditions and is located at approximately $\gamma_{\rm yield}\approx 0.12$
for the chosen strain rate. At the same time, we notice that the shear
protocol induces a sudden increase of the population of icosahedra at very
early times (very small strains) and a progressive decay of the population
which accelerates as the yield strain is reached. The overall, instantaneous
increase of the population of icosahedra can be understood as a consequence of
the minimisation procedure, which destroys thermal fluctuations present in the
initial configurations and promotes the formation of local minimum energy
motifs, such as the icosahedron. This implies that the overall population of
icosahedra $n_{\rm ico}(\gamma)$ can be distinguished into two families: the
first refers to the subset of particles that are located in icosahedral
domains in the original starting configurations produced in the $\mu$
ensemble, and it is identified by the boolean vector $\bm{\eta}_{\mu}$ of
length $N$; the second refers to all the remaining particles in icosahedral
domains, resulting from the AQS protocol, identified by the vector
$\bm{\eta}_{\bar{\mu}}$.

As the system is sheared, the number of icosahedra changes very mildly for
strains below the yield strain, and only later declines, supporting the idea
that the population of icosahedra is related to the rigid, elastic response of
the system. Having defined two subpopulations of icosahedra, we now quantify
their respective differences in the mechanical response.

Figure 10: Auto-correlation of the probability for a particle to be in
icosahedral domain for the $\eta_{\mu}$ and the $\eta_{\bar{\mu}}$ populations
(see main text for definition) for different values of the typical initial
population of icosahedra.The $\bm{\eta}_{\bar{\mu}}$ family is not defined at
$\gamma=0$ so, we take the smallest $\gamma$ as the reference state. The
vertical dashed line corresponds to the yield strain.

To do so, we compute separate auto-correlation functions
$\langle\bm{\eta}_{x}(\gamma)\bm{\eta}_{x}(0)\rangle$ for the
$\bm{\eta}_{\mu}$ and the $\bm{\eta}_{\bar{\mu}}$ populations, Fig. 10(a,b).
We notice that only for large initial populations of icosahedra the
autocorrelation functions start close to unity. This shows that the
reorganisation induced by the AQS protocol not only forms new icosahedral
motifs, but it also initially destroys a fraction of them. The two families of
autocorrelation functions show distinctively different behaviours: the
icosahedra present in the initial $\mu$-ensemble, Fig. 10(a), show a long
plateau that terminates only when the yield strain is attained; the icosahedra
generated via AQS, Fig. 10(b), continuously decorrelate at earlier times
(smaller strains).

A further confirmation of the different responses between the
$\bm{\eta}_{\mu}$ and the $\bm{\eta}_{\bar{\mu}}$ icosahedra is provided by
the distribution of the potential energy of the individual particles
constituting the two families. In Fig. 11 we plot the overall energy
distributions, for the population, collected all along the shearing protocol.
We clearly observe that not all icosahedral motifs are energetically the same:
particles located in icosahedral motifs purely emerging from energy
minimisation have energies that are typically higher than particles identified
in icosahedral motifs in the original $\mu$ ensemble configurations. The
energy gap between the two families widens as we consider initial
configurations with larger concentrations of icosahedra: at very high initial
concentrations (55%), the $\bm{\eta}_{\mu}$ subpopulation has energies that
are 8% lower than the $\bm{\eta}_{\bar{\mu}}$ subpopulation.

Figure 11: Distribution of potential energies for particles belonging to the
$\eta_{\mu}$ (filled distributions) and the $\eta_{\bar{\mu}}$ (dashed curves)
families (see main text for definition). The distributions are computed over
the whole duration of the shearing protocol and averaged over 75 initial
configurations.

##  5 Conclusions

Through numerical simulations, we have discussed a third example of
structural-dynamical phase transition in a model of atomistic glassformer,
after the previously considered cases of the Kob-Andersen mixture hedges2009 ;
jack2011 ; speck2012jcp ; speck2012 ; turci2017prx  and the moderately
polydisperse hard-spheres pinchaipat2017 .

A quantitative analysis of the probability distributions of time-integrated
observables demonstrates that well-chosen time-integrated structural motifs
can be used to perform efficient importance sampling. In particular, it makes
possible to explore structure-rich trajectories (representative of colder
temperature states turci2017prx ) that are otherwise hard to reach. At the
same time, we find confirmation for a sharp (first-order) transition in
trajectory space, that becomes measurable below the onset temperature, between
a structure-rich and a structure poor dynamical phase, the former becoming
more and more likely as the temperature is reduced, similarly to what has been
previously observed in the Kob-Andersen mixture turci2017prx . Within the
range of the explored temperatures, it is impossible to assess what the low
temperature fate of the transition may be: the reduction of the critical
conjugate field value $\mu^{\ast}$ suggests that, as the temperature
decreases, the structure-rich phase would prevail. However, it is unclear
whether the previously reported crystallisation into complex Laves phases
pedersen2010  would interfere with the emergence of the icosahedra-rich phase.
In our simulations, we monitored the evolution Frank-Kasper bonds (an
essential element of the complex crystalline phase) and do not find a
significative increase in the icosahedra-rich phase compared to the
icosahedra-poor phase.

In Reference turci2017prx , the study of the alternative Kob-Andersen mixture
at low temperature indicated possible scenarios for the temperature dependence
of the structural-dynamical transition. In the present case of the Wahnström
mixture, we observe that at $0.72T_{\rm onset}$ the structure-rich phase is
highly metastable, while a relatively modest decrease of the temperature to
$0.67T_{\rm onset}$ makes the exploration of the structure-rich basin 5 to 7
times more likely, see Fig. 5 (d). The temperature dependence of the critical
value $\mu^{\ast}\tau_{\rm obs}$ shows a decrease towards $\mu=0$ which
becomes less pronounced as the temperature is decreased. This is accompanied
by strong correlations between successive steps in the trajectory-space Monte-
Carlo that slow down equilibration and make lower temperature sampling
particularly challenging. The present data support the narrowing of the free-
energy gap (in trajectory space) between the structure-rich and structure-poor
states when the temperature is decreased, and do not exclude the possibility
that the transition terminates at a lower critical point at finite
temperature, as in kinetically constrained models with additional softness
elmatad2010 .

In order to better understand the importance of the structure-rich phase,
configurations obtained through trajectory sampling have been probed through
an out-of-equilibrium rheological protocol, effectively regarding these
configurations as samples of an amorphous material at $T=0$. Consistently with
previous studies of the Wahnström mixture based on conventional simulations
pinney2016 , we find that icosahedra play a major role in the emergence of
rigidity: icosahedra-rich configurations display much larger yield stresses
than icosahedra-poor ones. However, we nuance this statement, as we are able
to split the overall family of icosahedral motifs according to the preparation
protocol: well-thermalised configurations from trajectory sampling have
icosahedral regions that are more robust to shear and with lower energies than
the icosahedral domains obtained via energy minimisation. This highlights that
the requirement of sampling long-lived structural motifs (implicit in
trajectory sampling) allows us to explore metabasins that are not just richer
in structure, but more stable as well.

###### Acknowledgements.

CPR acknowledges the Royal Society for funding. FT and CPR acknowledge the
European Research Council (ERC consolidator grant NANOPRS, project number
617266). This work was carried out using the computational facilities of the
Advanced Computing Research Centre, University of Bristol. FT contributed to
the generation and analysis of the data and the writing of the manuscript. FT,
TS and CPR contributed to the editing of the manuscript.

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