# Non-equilibrium view of the amorphous solidification
of liquids with competing interactions

Ana Gabriela Carretas-Talamante1, Jesús Benigno Zepeda-López1, Edilio Lázaro-
Lázaro1,
Luis Fernando Elizondo-Aguilera2, and Magdaleno Medina-Noyola1,(a) 1 Instituto
de Física “Manuel Sandoval Vallarta”, Universidad Autónoma de San Luis Potosí,
Álvaro Obregón 64, 78000 San Luis Potosí, SLP, México  2 Instituto de Física,
Benemérita Universidad Autónoma de Puebla, Apartado Postal J-48 72570, Puebla,
México  (a<EMAIL_ADDRESS>
###### Abstract

The interplay between short-range attractions and long-range repulsions (SALR)
characterizes the so-called liquids with competing interactions, which are
known to exhibit a variety of equilibrium and non-equilibrium phases. The
theoretical description of the phenomenology associated with glassy or gel
states in these systems has to take into account both the presence of
thermodynamic instabilities (such as those defining the spinodal line and the
so called $\lambda$ line) and the limited capability to describe genuine non-
equilibrium processes from first principles. Here, we report the first
application of the non-equilibrium self-consistent generalized Langevin
equation theory to the description of the dynamical arrest processes that
occur in SALR systems after being instantaneously quenched into a state point
in the regions of thermodynamic instability. The physical scenario predicted
by this theory reveals an amazing interplay between the thermodynamically
driven instabilities, favoring equilibrium macro- and micro-phase separation,
and the kinetic arrest mechanisms, favoring non-equilibrium amorphous
solidification of the liquid into an unexpected variety of glass and gel
states.

###### pacs:

05.40.-a, 05.70.Ln, 64.70.P-.

##  I Introduction.

This work reports the first systematic application of the non-equilibrium
self-consistent generalized Langevin equation theory nescgle1 ; nescgle2 ;
nescgle3 , to the description of non-equilibrium arrested states in fluids
with competing short-range attractions and long-range repulsions (SALR)
LiuXi2019 ; liuJPCB . Such arrested states may prevent the fluid from reaching
the experimentally elusive zhuangcharbonneauJCP16  ordered phases expected at
thermodynamic equilibrium conditions in the low-density low-temperature regime
of these systems ruizzaccarelli .

As a context, let us first recall that van der Waals (vdW) molecular
explanation of the gas-liquid coexistence had in mind a model fluid of
spherical particles interacting by an excluded-volume repulsion plus a weaker
short-ranged attraction vdw ; widom . In choosing this simple model of a
fluid, van der Waals was fortunate enough since, for example, innocently
adding a soft longer-ranged repulsive interaction immediately leads to a
different class of systems, whose equilibrium phase behavior turns out to be
far more complex. This more general class, characterized by the excluded-
volume potential plus the competing SALR interactions, bears an enormous
relevance in colloidal and soft materials. Familiar physical realizations of
SALR conditions may be represented, for example, by the effective interaction
between two charged particles (colloids, proteins, or macroions) in aqueous
solution verweyoverbeek ; derjaguinlandau  or between two colloids in colloid-
polymer mixtures asakura .

Different approaches have been applied to determine the equilibrium phase
diagram of SALR systems, including theoretical methods (integral equations
seargelbart ; pinijealinparolareatto ; imperioreatto04 ; LiuChenChenJCP05 ,
density functional theory archerpinievansreatto07 ; archerionescu3Djpcm08 ;
archer2Dpre08 ; chackochalmersarcher2Djcp15 , and field-theoretical models
described by Ginzburg-Landau Hamiltonians SchmalianWolynesPRL00 ;
TarziaConiglioPRL06 ; TarziaConiglioPRE07 ) and Monte Carlo computer
simulations candiaconiglio ; imperioreato ; archerwilding ;
zhuangzhangcharbonneau ; godfrin . As recently and thoroughly reviewed
LiuXi2019 ; ruizzaccarelli , the picture that emerges predicts a rich and
diverse phenomenology, which includes the appearance at low densities of an
equilibrium fluid phase of finite-sized clusters, the coexistence between
disordered equilibrium (gas and liquid) phases and ordered (“modulated”)
inhomogeneous phases, and the possibility that equilibrium microphase
separation preempts the usual vdW gas-liquid coexistence.

These different phenomenological features correspond in general to different
combinations of the short-range attraction and the long-range repulsion. A
major challenge is, of course, to establish which salient features correspond
to which regime in the four-dimensional parameter space spanned by the
intensity and range of the two competing interactions. After careful analysis
of experimental, simulation and theoretical results, Liu and Xi LiuXi2019
suggested that most of SALR systems can be grouped in three different regimes,
identified by the ratio between ranges of the attractive and repulsive
interactions. They noticed that most model experimental SALR systems studied,
belong to a similar type, referred to as type I SALR systems, whereas many
simulation works are type II SALR systems. As discussed in detail in LiuXi2019
, the difference between different types of SALR systems has strong impact on
the final equilibrium phase diagram. As specified below, however, the present
work will focus on still a third SALR regime, referred to in LiuXi2019  as
type III SALR systems.

Given this equilibrium scenario, the next most important issue is its actual
experimental observability. This concern arises from the possible emergence of
kinetic barriers to equilibration, which may lead to conditions of dynamic
arrest, as suggested by early theoretical considerations groenewoldkegel2001 ;
groenewoldkegel2004 ; wuliuchencao ; LiuChenChenJCP05  and by molecular
dynamics (MD) studies conigliodearcangelis ; sciortinomossa ; mossasciortino ;
sciortinotartaglia ; charbonneaureichmanPRE07 ;
BollingerTruskettSoftMatter2017 . The latter have confirmed, for example, the
presence of a phase of stable, freely diffusing clusters of particles and of
non-equilibrium phases of disordered arrested states. Beyond its fundamental
relevance, this knowledge is important for many practical purposes, such as
designing rules for the assembly of porous mesophases
LindquistTruskettSoftMatter2017 . In contrast with Montecarlo simulations,
molecular dynamics (and Brownian dynamics (BD)) simulations mimic realistic
trajectories in phase space, whose ensemble averages directly relate with the
real dynamical phenomenology, actually observed since early and careful
experimental studies sedgwick ; stradner ; bordi ; campbell ;
DibbleKoganSolomonPRE06 ; KlixRoyalTanakaPRL2010 .

Regarding the theoretical efforts to understand the formation of amorphous
arrested states in SALR systems, let us mention the analysis aimed at
determining if a glass transition exists in field-theoretical models. These
efforts include the use of the replica approach mezardparisiPRL99 ;
SchmalianWolynesPRL00 ; TarziaConiglioPRL06 ; TarziaConiglioPRE07  and of the
Langevin-equation version of these models GroussonKrakoviackPRE02 ;
GeisslerReichmanPRE04 . Early applications of mode coupling theory (MCT)
goetze1 ; goetze2  to SALR systems were also reported by Grousson et al.
GroussonKrakoviackPRE02  and by Geissler and Reichman GeisslerReichmanPRE04 ,
although only for field-theoretical models.

However, developing a general fundamental microscopic theory of dynamical
arrest in structural glasses, which explains the glass transition and extends
the van der Waals picture to non-equilibrium conditions and to more complex
interactions (such as SALR systems), has been the purpose of relevant work
over the last half a century berthierbiroli . In this context, one should
highlight the first-principles description of the dynamic properties of fluids
near conditions of dynamic arrest provided by MCT goetze1 ; goetze2 , which
predicts the location of the transition from equilibrium-fluid to dynamically-
arrested states. As early as in 2004, Wu et al. wuliuchencao  discussed the
application of MCT to the hard-sphere plus double Yukawa SALR type II
interaction, predicting its fluid-to-glass transition diagram in the high-
density high-temperature regime. Unfortunately, this work did not explore in
detail the opposite (low-density low-temperature) regime.

Starting this exploration is precisely the purpose of the present work,
motivated by the need to understand if the emergence of kinetic arrest and
non-ergodicity might be the source of the experimental elusiveness of the
predicted ordered (or “modulated”) equilibrium phases, which characterizes one
specific SALR regime. In contrast with the work of Wu et al., however, our
work will not be based on MCT, but on the more recent theory referred to as
the self-consistent generalized Langevin equation (SCGLE) theory of colloid
dynamics scgle1 ; scgle2 ; scgle3 ; scgle4 ; scgle5  and dynamical arrest
arrest1 ; arrest2 ; arrest3 , which in most aspects is analogous to MCT
thvg_elizondo .

In reality, however, the SCGLE theory, just like MCT, bears a fundamental
constraint to equilibrium conditions, thus impeding the description of
essential fingerprints of dynamic arrest, such as the aging of glass- and gel-
forming liquids. The route of escape from this limitation, however, was
provided in Ref. nescgle0 , which proposed a far-from-equilibrium extension of
the Onsager theory of irreversible processes onsager1 ; onsager2  and the
Onsager-Machlup theory of thermal fluctuations onsagermachlup1 ;
onsagermachlup2 , leading to the general theory of irreversible processes in
liquids, referred to as the non-equilibrium self-consistent generalized
Langevin equation (NE-SCGLE) theory nescgle1 . This more general approach
contains as a particular case the original equilibrium SCGLE theory, which is
thus enriched by the non-equilibrium kinetic perspective required to describe
non-stationary processes peredo1 .

Therefore, the present study will be based on the NE-SCGLE, a theory that has
solidly demonstrated its ability to predict some of the most relevant
universal signatures of both, the glass and gel transitions, including aging
effects, as well as very specific features reflecting the particular role of
the molecular interactions involved in the explicit systems considered so far.
For instance, for systems involving only excluded volume interactions, this
theory accurately describes the process of formation of high-density hard-
sphere-_like_ glasses nescgle2 ; nescgle3 ; gabriel ; nescgle5 , whereas for
liquids with excluded volume plus attractive interactions (i.e., those in
vdW’s mind) it predicts the formation of sponge-like gels and porous glasses
by arrested spinodal decomposition olais1 ; olais2 ; nescgle8 ; zepeda .
Extended to multi-component systems nescgle4 ; nescgle5  the NE-SCGLE theory
opens the possibility of describing the aging of “double” and “single” glasses
in mixtures of neutral voigtmanndoubleglasses ; arrest2 ; lazaro  and charged
arrest3 ; portadajcppedro  particles; the initial steps in this direction are
highly encouraging lazaro . Similarly, its extension to liquids formed by
particles interacting by non-radially symmetric forces gory1 ; gory2 ; peredo2
, accurately predicts the non-equilibrium coupled translational and rotational
dynamic arrest observed in simulations gory3 .

In this work we start a systematic application of the theoretical
infrastructure of the NE-SCGLE theory to the same model of SALR colloidal
fluid studied by Wu et al. wuliuchencao , namely, a three-dimensional fluid of
hard spheres (HS) of diameter $\sigma$, interacting through a total pair
potential $u(r)=u^{HS}(r)+u^{DY}(r)$, where $u^{HS}(r)$ is the hard-sphere
potential and $u^{DY}(r)$ is the sum of two competing Yukawa interactions,

$u^{DY}(r)=-\epsilon_{1}\frac{\exp[-z_{1}(r/\sigma-1)]}{r/\sigma}+\epsilon_{2}\frac{\exp[-z_{2}(r/\sigma-1)]}{r/\sigma}.$
(1)

This model system will be referred to as the “hard-sphere plus double Yukawa”
(HSDY) fluid, whose equilibrium phase diagram was outlined by Archer and
collaborators archerpinievansreatto07 ; archerionescu3Djpcm08  using density
functional theory within a random phase approximation for the free energy, an
approximation also employed in this work.

In its simplest version, the NE-SCGLE theory is summarized by a set of
equations describing the irreversible evolution of the non-equilibrium
structural and dynamical properties of an instantaneously quenched glass-
forming liquid. These properties include the non-equilibrium structure factor
(NESF) $S(k;t)$, and the _collective_ and _self_ intermediate scattering
functions (NEISF) $F(k,\tau;t)$ and (self-NEISF) $F_{S}(k,\tau;t)$, from which
the diffusion coefficient, relaxation times, and rheological properties can be
derived. Each of these properties depend on the final density and temperature,
and on the waiting time $t$, after the quench, as well as on the various
parameters characterizing the system (such as the interaction parameters
$\epsilon_{1},\ \epsilon_{2},\ z_{1}$, and $z_{2}$). Thus, the analysis of
this multidimensional dependence, in its various regimes, will require more
than one detailed report, and is completely out of the scope of the present
work. Hence, here we shall only summarize the conceptual and practical
infrastructure needed for such analysis and will illustrate its use focusing
mostly on the structural properties, represented by the NESF $S(k;t)$ and by
the corresponding non-equilibrium radial distribution function (NERDF)
$g(r;t)$.

For clarity, in Section II we summarize the relevant aspects of the NE-SCGLE
theory and briefly explain the procedure for its application to this SALR
model fluid. This includes introducing the mean field approximation of the
free energy density functional, which determines the main features of the
_equilibrium_ structure and phase behavior of the system, and is also the
fundamental thermodynamic input of the NE-SCGLE equations. Starting with
Section III, we restrict ourselves to the particular regime of long-ranged
repulsions ($z_{1}\geq 1>z_{2}$). This election of parameters will limit this
work to exploring only type III SALR system LiuXi2019 , for which we will
discuss what we refer to as the “glass transition diagram”; this is the NE-
SCGLE complement of the concept of equilibrium phase diagram. In Section IV we
discuss the various density and temperature regimes of the behavior of the
long-$t$ asymptotic limit $S^{a}(k)$ of the NESF $S(k;t)$, whereas in Section
V this analysis is extended to the finite waiting-time regime, to illustrate
the predicted scenario of the aging of the structural properties. In Section
VI we summarize the main results of the present paper.

##  II The NE-SCGLE theory.

The fundamental origin of the NE-SCGLE was laid down in detail in Refs.
nescgle1 ; nescgle2 ; nescgle3 . However, a practical summary can be found in
the supplementary material of Ref. nescgle8 , which highlights the main
simplifying approximations leading to the version of this theory employed in
this and in all its previous concrete applications. The present study will be
a straightforward extension of the work reported in Refs. olais1 ; olais2 ;
nescgle8 ; zepeda , which describes the predicted scenario of arrested
spinodal decomposition in liquids with excluded-volume plus attractive
interactions. The reader is invited to visit these references, which describe
in all detail the conceptual and practical challenges found in applying the
theory to quenches inside the spinodal region or inside other
thermodynamically unstable regions of the state space. These references also
illustrate the wealth of information provided by the NE-SCGLE on the time-
dependent physical properties of an attractive system under these non-
equilibrium conditions. In particular, the novel concept of time-dependent
non-equilibrium phase diagram that emerges from these applications is
carefully explained in Ref. zepeda .

###  II.1  The NE-SCGLE equations.

In its simplest version, the NE-SCGLE theory is summarized by a set of
equations that describe the irreversible evolution of the non-equilibrium
structural and dynamical properties of a model glass-forming liquid, formed by
$N$ identical spherical particles in a volume $V$ that interact through a
radially-symmetric pair potential $u(r)$. It starts with the time evolution
equation for the NESF $S(k;t)\equiv\overline{\delta n({\bf k};t)\,\delta
n(-{\bf k};t)}$, where the over-line indicates the average over a (non-
equilibrium) statistical ensemble, and where $\delta n({\bf k};t)$ is the
Fourier transform of the fluctuations in the local particle number density
$n({\bf r};t)$. For a system that is instantaneously quenched at time $t=0$
from initial bulk density and temperature $(n_{i},T_{i})$ to new final values
$(n,T)$, constrained to remain spatially uniform ($\overline{n}({\bf
r};t)=n\equiv N/V$), such an equation reads, for $t>0$,

$\frac{\partial S(k;t)}{\partial
t}=-2k^{2}D^{0}b(t)n\mathcal{E}(k;n,T)\left[S(k;t)-1/n\mathcal{E}(k;n,T)\right],$
(2)

with $D^{0}$ being the short-time self-diffusion coefficient shorttimed0 .

In this equation the time-dependent mobility function $b(t)$ is defined as
$b(t)\equiv D_{L}(t)/D^{0}$, with $D_{L}(t)$ being the long-time self-
diffusion coefficient at evolution time $t$. This function couples the
structural relaxation described by Eq. (2) with the non-equilibrium relaxation
of the dynamic properties of the fluid. Such coupling is established by the
following exact expression for $b(t)$,

$b(t)=[1+\int_{0}^{\infty}d\tau\Delta{\zeta}^{*}(\tau;t)]^{-1},$ (3)

in terms of the $t$-evolving, $\tau$-dependent friction function
$\Delta{\zeta}^{*}(\tau;t)$, for which the NE-SCGLE theory derives the
following approximate expression nescgle1 ,

$\begin{split}\Delta\zeta^{*}(\tau;t)=\frac{D_{0}}{24\pi^{3}n}\int d{\bf k}\
k^{2}\left[\frac{S(k;t)-1}{S(k;t)}\right]^{2}\\\ \times
F(k,\tau;t)F_{S}(k,\tau;t),\end{split}$ (4)

in terms of the NESF $S(k;t)$ and of the NEISF $F(k,\tau;t)\equiv
N^{-1}\overline{\delta n(\mathbf{k},t+\tau)\delta n(-\mathbf{k},t)}$, where
$\delta n(\mathbf{k},t)$ is the FT of the thermal fluctuations $\delta
n(\mathbf{r},t)\equiv n(\mathbf{r},t)-n$ of the local number density
$n(\mathbf{r},t)$ at time $t$. The self-NEISF $F_{S}(k,\tau;t)$ is defined as
$F_{S}(k,\tau;t)\equiv\overline{\exp\left[i\mathbf{k}\cdot\Delta\mathbf{r}_{T}(t,\tau)\right]}$,
with
$\Delta\mathbf{r}_{T}(t,\tau)\equiv\left[\mathbf{r}_{T}(t+\tau)-\mathbf{r}_{T}(t)\right]$
being the displacement of one particle considered as a tracer. As before, the
over-line indicates an average over a corresponding non-equilibrium
statistical ensemble.

The previous equations are complemented by the memory-function equations for
$F(k,\tau;t)$ and $F_{S}(k,\tau;t)$, written approximately, in terms of their
Laplace transforms (LT) $F(k,z;t)$ and $F_{S}(k,z;t)$, as

$\displaystyle
F(k,z;t)=\frac{S(k;t)}{z+\frac{k^{2}D^{0}S^{-1}(k;t)}{1+\lambda(k)\
\Delta\zeta^{*}(z;t)}},$ (5)

and

$\displaystyle F_{S}(k,z;t)=\frac{1}{z+\frac{k^{2}D^{0}}{1+\lambda(k)\
\Delta\zeta^{*}(z;t)}},$ (6)

where $\Delta\zeta^{*}(z;t)$ is the LT of $\Delta\zeta^{*}(\tau;t)$. In these
equations $\lambda(k)\equiv 1/[1+(k/k_{c})^{2}]$ is an “interpolating
function” arrest1 , with $k_{c}$ being an empirically determined cutoff wave-
vector. In the present work we use $k_{c}=1.305(2\pi)/\sigma$, with $\sigma$
being the hard-core particle diameter of our HSDY model fluid, which
guarantees that the hard-sphere liquid will have its dynamic arrest transition
at a volume fraction $\phi_{c}=0.582$, in agreement with simulations gabriel .

Eqs. (2)-(6) constitute the mathematical summary of the NE-SCGLE theory. Since
the function $\mathcal{E}(k;n,T)$ is considered known, they constitute a
closed system of equations for the time-dependent SF $S(k;t)$ and for the non-
equilibrium dynamic properties $b(t)$, $\Delta{\zeta}^{*}(\tau;t)$,
$F(k,\tau;t)$, and $F_{S}(k,\tau;t)$. Thus, Subsection II.2 defines the
thermodynamic function $\mathcal{E}(k;n,T)$, explains its role in determining
stability and structural properties.

###  II.2 Thermodynamic stability function $\mathcal{E}(k;n,T)$.

The function $\mathcal{E}(k;n,T)$ is defined as the FT of $\mathcal{E}[r;n,T]$
which, in turn, is the second functional derivative of the Helmholtz free
energy density-functional $\mathcal{F}[n,T]$,

$\mathcal{E}[\mid\mathbf{r}-\mathbf{r}^{\prime}\mid;n,T]\equiv\left(\frac{\delta^{2}\mathcal{F}[n,T]/k_{B}T}{\delta
n(\mathbf{\mathbf{r}})\delta n(\mathbf{\mathbf{r}^{\prime}})}\right),$ (7)

evaluated at the uniform (bulk) density and temperature fields
$n(\textbf{r},t)=n\equiv N/V$ and $T(\textbf{r},t)=T$. We refer to
$\mathcal{E}(k;n,T)$ as the thermodynamic stability function, since it
provides a criterion for the thermodynamic stability of the system. For
example, the state function $\mathcal{E}(k;n,T)$, evaluated at $k=0$, is
related with the thermodynamic derivative $(\partial p/\partial n)_{T}$, where
$p$ is the pressure, as

$\beta(\partial p/\partial n)_{T}=n\mathcal{E}(k=0;n,T),$ (8)

which is the so-called compressibility equation mcquarrie  ($\beta^{-1}\equiv
k_{B}T$). As we see in Subsection II.3, this equation of state allows us to
determine the main features of the gas-liquid coexistence region, including
the spinodal line, obtained from the condition $\mathcal{E}(k=0;n,T)=0$, which
separates the state space $(n,T)$ in the stable ($\mathcal{E}(k=0;n,T)>0$) and
the unstable ($\mathcal{E}(k=0;n,T)<0$) equilibrium domains.

More generally, the function $\mathcal{E}(k;n,T)$ must be positive for all
wave-vectors for the system to be stable, since if $\mathcal{E}(k;n,T)<0$ for
a finite $k$, the system would be unstable to density fluctuations of
wavelength $2\pi/k$. Under some conditions, a domain $k^{-}\leq k\leq k^{+}$
may exist in which $\mathcal{E}(k;n,T)<0$. Then, in analogy with the spinodal
line, we define a $\lambda$ line in the state space $(n,T)$ by the threshold
condition $k^{-}=k^{+}=k_{\lambda}$, at which
$\mathcal{E}(k_{\lambda};n,T)=0$.

The stability function $\mathcal{E}(k;n,T)$ also plays a relevant role in
determining the structural properties of the system. For example, the
_equilibrium_ static structure factor (SF) $S^{eq}(k)$ is given by

$S^{eq}(k)=1/n\mathcal{E}(k;n,T)$ (9)

which is just the well-known Ornstein-Zernike (OZ) equation mcquarrie , as can
be seen by writing $S^{eq}(k)=1+nh^{eq}(k)$ and $n\mathcal{E}(k;n,T)=1-nc(k)$,
with $h^{eq}(k)$ being the FT of the equilibrium _total_ correlation function
(TCF) $h^{eq}(r)$ and $c(k)$ being the FT of the _direct_ correlation function
(DCF) $c(r)$. Clearly, the SF $S^{eq}(k)$ of a uniform system does not exist
when $\mathcal{E}(k;n,T)<0$.

From the kinetic perspective of Eq. (2), the OZ equation in Eq. (9) is just
the equilibrium condition for the NESF $S(k;t)$. This is, in fact, the obvious
_stationary_ solution of the kinetic equation for $S(k;t)$ in Eq. (2),
asymptotically attained at long times in the kinetic process of equilibration,
i.e., $S(k;t\to\infty)=S^{eq}(k)=1/n\mathcal{E}(k;n,T)$. However, under
general non-equilibrium conditions, such as during the transient that follows
an instantaneous quench, the NESF $S(k;t)$ is certainly NOT given by the OZ
equation $S(k;t)=1/n\mathcal{E}(k;n,T)$, but by the solution of Eq. (2), which
provides a general manner to process the information contained in
$\mathcal{E}(k;n,T)$ to determine $S(k;t)$. Thus, one can say that Eq. (2) is
the non-equilibrium extension of the OZ equation. Notice that, as a
thermodynamic input of this kinetic equation, negative values of
$\mathcal{E}(k;n,T)$ are perfectly physical, leading in fact to some of the
most remarkable predictions of the NE-SCGLE theory.

###  II.3 Mean field free energy functional $\mathcal{F}[n,T]$.

In practice, we must identify a pertinent strategy to approximate
$\mathcal{F}[n,T]$ for our three-dimensional HSDY model fluid. With the aim of
adopting the same level of approximation as in the early theoretical
discussions of the equilibrium phase diagram of SALR systems, we follow Refs.
seargelbart ; archerpinievansreatto07 ; archerionescu3Djpcm08  in writing
$\mathcal{F}[n,T]$ as the sum
$\mathcal{F}[n,T]=\mathcal{F}^{HS}[n,T]+\mathcal{F}^{DY}[n,T]$ of the exact
hard-sphere free energy $\mathcal{F}^{HS}[n,T]$ plus the contribution
$\mathcal{F}^{DY}[n,T]$ of the double Yukawas. A simple explicit expression
for the latter is provided by its _random phase approximation_ (RPA), within
which $\mathcal{F}[n,T]$ is given by

$\mathcal{F}[n,T]=\mathcal{F}^{HS}[n,T]+\frac{1}{2}\int d\mathbf{r}\int
d\mathbf{r}^{\prime}n(\textbf{r})u^{DY}(\mid\mathbf{r}-\mathbf{r}^{\prime}\mid)n(\textbf{r}^{\prime}),$
(10)

and the function $\mathcal{E}[\mid\mathbf{r}-\mathbf{r}^{\prime}\mid;n,T]$ of
Eq. (7) is given by

$\mathcal{E}[\mid\mathbf{r}-\mathbf{r}^{\prime}\mid;n,T]=\mathcal{E}^{HS}[\mid\mathbf{r}-\mathbf{r}^{\prime}\mid;\phi]+\beta
u^{DY}(\mid\mathbf{r}-\mathbf{r}^{\prime}\mid),$ (11)

whose FT reads

$\mathcal{E}(k;n,T)=\mathcal{E}^{HS}(k;\phi)+\beta u^{DY}(k).$ (12)

Writing $\mathcal{F}^{HS}[n,T]$ as
$\mathcal{F}^{HS}[n,T]=\mathcal{F}_{id}^{HS}[n,T]+\mathcal{F}^{HS}_{ex}[n,T]$,
where $\mathcal{F}_{id}^{HS}[n,T]$ is the ideal-gas value and
$\mathcal{F}_{ex}^{HS}[n,T]$ is the “excess” HS contribution evans , allows us
to write Eq. (7) for the pure HS system as

$\mathcal{E}^{HS}[\mid\mathbf{r}-\mathbf{r}^{\prime}\mid;n,T]=\delta(\mathbf{r}-\mathbf{r}^{\prime})/n-c^{HS}(\mid\mathbf{r}-\mathbf{r}^{\prime}\mid;n,T)$
(13)

with $c^{HS}(\mid\mathbf{r}-\mathbf{r}^{\prime}\mid;n,T)$, defined by
$c^{HS}(\mid\mathbf{r}-\mathbf{r}^{\prime}\mid;n,T)\equiv-\beta\left(\delta^{2}\mathcal{F}_{ex}^{HS}[n,T]/\delta
n(\mathbf{\mathbf{r}})\delta n(\mathbf{\mathbf{r}^{\prime}})\right)$, being
the exact HS DCF. In Fourier space this equation reads
$\mathcal{E}^{HS}(k;\phi)=1/n-c^{HS}(k;\phi)$. As in Refs.
archerpinievansreatto07 ; archerionescu3Djpcm08 , here we will also
approximate $c^{HS}(k;\phi)$ by its Percus-Yevick (PY) approximation
percusyevick , but complemented with its Verlet-Weis (VW) correction
verletweis , $c^{HS}(k;\phi)\approx c^{HS}_{PYVW}(k;\phi)$. Thus, the
approximate thermodynamic input $\mathcal{E}(k;n,T)$ that we shall employ in
this work is finally written as

$\mathcal{E}(k;n,T)\approx 1/n-c^{HS}_{PYVW}(k;\phi)+\beta u^{DY}(k).$ (14)

Figure 1: Behavior of (a) the SF $S(k;n,T)=1/n\mathcal{E}(k;n,T)$ and (b) the
corresponding RDF $g(r;\phi,T)$, as provided by the RPA (Eq. (14)) for the
HSDY fluid, with parameters $z_{1}=1$, $z_{2}=0.5$ and $A=0.5$, for fixed
volume fraction $\phi=0.15$ and final temperatures above the $\lambda$ line,
$T>T_{\lambda}(\phi)=0.6733$, as indicated. (c) Equilibrium gas-liquid phase
diagram of the same HSDY fluid, exhibiting the spinodal (dashed line) and the
$\lambda$ (solid line) curves.

###  II.4 Equilibrium properties of the HSDY model.

We may now use this approximate expression for the thermodynamic input
$\mathcal{E}(k;n,T)$ of the NE-SCGLE equations, Eqs. (2)-(6), whose solution
will describe the non-equilibrium response of the liquid after the
instantaneous quench. For reference, however, let us first summarize some
purely equilibrium properties that directly derive from the RPA for the
thermodynamic input $\mathcal{E}(k;n,T)$ in Eq. (14), applied to the HSDY
potential. For this, let us rewrite the DY term in Eq. (1) in dimensionless
form as

$\beta
u^{DY}(r)=-\epsilon\frac{\exp[-z_{1}(r/\sigma-1)]}{r/\sigma}+A\frac{\exp[-z_{2}(r/\sigma-1)]}{r/\sigma}.$
(15)

with $\epsilon\equiv\epsilon_{1}/k_{B}T$ and $A\equiv\epsilon_{2}/k_{B}T$.

Following Refs. archerpinievansreatto07 ; archerionescu3Djpcm08 , we have in
mind a colloid-polymer mixture, and consider the (“athermal”) conditions in
which the physical temperature $T$ is kept fixed and the strength of the
attraction is controlled by varying the polymer concentration, i.e.,
$\epsilon^{-1}$ plays the role of an effective temperature
$T_{eff}=\epsilon^{-1}$, which for simplicity in notation will be denoted as
$T$. Thus, if the particle charge and ionic strength can be assumed fixed, we
can treat the parameters $A$ and $z_{2}$ (the intensity and range of the
electrostatic repulsion) as fixed parameters. As a result, the function
$\mathcal{E}(k;n,T)$ will actually depend on the dimensionless concentration
$n^{*}\equiv n\sigma^{3}$, which we shall continue denoting by $n$ (or in
terms of the packing fraction $\phi=\pi n/6$), and on the effective
temperature T. This dependence is additional to the dependence of
$\mathcal{E}(k;\phi,T)$ on the potential parameters $A$, $z_{1}$ and $z_{2}$.

###  II.5 Equilibrium structure of the HSDY model.

The first relevant property to discuss is, of course, the SF $S^{eq}(k;n,T)$
given by Eq. (9), which for notational convenience we will denote as
$S(k;n,T)$. From now on, we will consider a HSDY fluid with fixed parameters
$z_{1}=1$, $z_{2}=0.5$ and $A=0.5$, which according to Ref. LiuXi2019 ,
corresponds to a type III SALR system. Fig. 1 exhibits the main distinctive
feature of the SF, previously discussed by Sear and Gelbart seargelbart  and
by Archer et al. archerpinievansreatto07  for the same HSDY model within the
RPA. We refer to the development of a large peak centered at a small but
finite wave vector $k=k_{\lambda}$ (referred to as the “cluster” peak
archerpinievansreatto07 ). This peak is associated with the propensity to the
spontaneous formation of clusters in the equilibrium fluid, with its height
$S(k_{\lambda};\phi,T)$ increasing, and its position $k_{\lambda}$ decreasing,
as the temperature $T$ is reduced along a given isochore ($\phi=0.15$, in the
present case).

According to the RPA results in Fig. 1, $S(k_{\lambda};\phi,T)$ actually
diverges as the temperature $T$ reaches a singular value $T_{\lambda}(\phi)$
from above. The curve $T=T_{\lambda}(\phi)$, defined by the condition
$\mathcal{E}(k_{\lambda};\phi,T)=1/S(k_{\lambda};\phi,T)=0$, is referred to as
the _$\lambda$ line_ (for the isochore of the figure,
$T_{\lambda}(\phi=0.15)=0.6733$)). Taking the inverse FT of $[S(k)-1]/n$
yields the total correlation function $h(r)$, and in Fig. 1 we plot the same
information of Fig. 1, but in terms of the radial distribution function
$g(r)=1+h(r)$. The results in this figure exhibit a notorious structural
feature as $T$ approaches $T_{\lambda}(\phi)$ from above, namely the emergence
of a long-range minimum in $g(r;\phi,T)$ at $r=r_{min}$ ($\approx 6\sigma$ in
the RDF corresponding to the deepest quench, $T=0.68$, in the figure). As we
shall see later, this structural feature is an essential fingerprint of the
process of cluster formation and eventual dynamic arrest due to cluster-
cluster repulsions. Finally, notice that Figs. 1 and 1 report the same
information as Figs. 1 and 2 of Archer et al. archerpinievansreatto07 ,
although at a different value of the parameter $A$ (= 0.5 here, and = 0.082 in
Ref. archerpinievansreatto07 ).

At this point it is pertinent to clarify that the appearance of a low-$k$ peak
in the SF is a characteristic hallmark of many SALR systems (whether type I,
II and III) and it is generally associated with the emergence of additional
ordering at a length scale much larger than the particle diameter $\sigma$. As
discussed in Ref. liuJPCB , particularly referring to lysozyme protein
solutions, this low-$k$ peak is not necessarily related to the formation of
clusters with optimal sizes and hence, should more appropriately be referred
to as an _intermediate range order_ (IRO) peak liuJPCB . Notwithstanding this
clarification, we can say, following the discussion of Refs.
archerpinievansreatto07 ; archerwilding , that in our present type III SALR
model system, it is safe to refer to $S(k_{\lambda};\phi,T)$ as a “cluster”
peak.

###  II.6 Equilibrium phase diagram of the HSDY model.

The $\lambda$ line $T_{\lambda}(\phi)$ of the system described in Figs. 1 and
1 is represented in Fig. 1. For reference, the spinodal line is also plotted
and it was calculated from the approximate function $\mathcal{E}(k;n,T)$ in
Eq. (14). This figure, which is our version of Fig. 6 of Ref.
archerpinievansreatto07 , summarizes the equilibrium phase diagram of the HSDY
model predicted by the RPA, which indicates that the gas-liquid transition is
preempted by the occurrence of the singular $\lambda$ line, below which,
uniform disordered equilibrium states do not exist, since
$S(k;\phi,T)=1/n\mathcal{E}(k;\phi,T)$ is negative (and hence, non-physical)
for wave-vectors in an interval around $k_{\lambda}$.

With the aim of understanding this puzzling feature of the equilibrium phase
behavior of the type III HSDY model, Archer and Wilding archerwilding  carried
out Monte Carlo simulations, showing that, under some circumstances, the
repulsive part of the pair potential could lead to the replacement of the
liquid-vapor coexistence by two first-order phase transitions. The first
involves the coexistence of the vapor with a fluid of spherical liquid-like
clusters, and the second involves the coexistence of a phase of spherical
voids with the homogeneous liquid. These two transition lines meet with the
vapor-liquid transition at a triple point. Later on, using density-functional
theory, Archer et al. archerionescu3Djpcm08  concluded that below the
$\lambda$ line, the equilibrium states consisted of non-uniform spatially-
ordered (or “modulated”) phases (see Fig. 5 of Ref. archerionescu3Djpcm08 ),
thus theoretically supporting in part the equilibrium scenario advanced by
Monte Carlo calculations.

##  III Glass transition diagrams.

Unfortunately, careful experimental work in colloidal systems with competing
interactions does not report the observation of these equilibrium ordered
structures sedgwick ; stradner ; bordi ; campbell ; DibbleKoganSolomonPRE06 ;
KlixRoyalTanakaPRL2010 . Instead, as also suggested by extensive molecular
dynamics simulations sciortinomossa ; mossasciortino ; sciortinotartaglia ,
what seems to surround or substitute the predicted spatially-ordered phases
are equilibrium fluids of finite-sized clusters and percolating gel-like
states. So far, the experimental elusiveness in observing the aforementioned
modulated phases in SALR systems remains unresolved. One possible reason, of
course, is the fact such phases are theoretically predicted for type III SALR
liquids while – according to Ref. LiuXi2019  – most model experimental SALR
systems studied are of type I. A second possibility, however, is related to
the fact that the cluster fluid and percolated gel-like states are able to
evolve into genuinely arrested non-equilibrium states, whose nature is
controlled by the interplay between repulsions and attractions, and whose
fundamental understanding poses even more complex technical and fundamental
challenges.

The theoretical description of such dynamically-arrested states, in fact, is
typically out of the scope of conventional equilibrium theories and
simulations, and hence, must be described from a genuinely non-equilibrium
perspective, such as that provided by the solution of the NE-SCGLE equations
(2)-(6). In analyzing the solution of this set of equations, it is instructive
to start by discussing their asymptotic long-time stationary limits. For this,
we can consider two strategies: either we first take the stationary limit of
Eqs. (2)-(6) and then analyze its solutions, or else, we first formally
determine the full time-dependent solution of Eqs. (2)-(6) and then take the
long-time stationary limit. Let us describe the results of these two
strategies.

Figure 2: (a) _Glass transition_ diagram of the HSDY fluid with $z_{1}=1$,
$z_{2}=0.5$ and $A=0.5$, provided by the equilibrium SCGLE theory using Eq.
(16). The blue solid line is the $\lambda$ line, and the red solid one is the
fluid-glass transition line, with both meeting at the (“bifurcation”) point
$(\phi_{b},T_{b})=(0.365,0.452)$ (open circle). Region I identifies the
ergodic region and region II identifies the glass region. The predicted
equilibrium mobility at each state points $(\phi,T)$ is represented by a color
code (colored region) above the $\lambda$ line, with the black color
representing the limiting condition $b^{eq}(\phi,T)=0$. In the light empty
region below the $\lambda$ line, Eq. (16) cannot be used since $S(k;\phi,T)$
is non-physical. (b) Non-equilibrium glass transition diagram provided by the
NE-SCGLE theory using Eq. (17). Above the $\lambda$ line, it coincides with
the glass transition diagram in (a), but adds the information that the
$\lambda$ line itself is a “type B” ergodic-to-non-ergodic transition line,
and that the fluid-glass transition line continues from above to below the
$\lambda$ line as a “type B” non-ergodic to non-ergodic transition line.

###  III.1 MCT-like “glass transition” diagram.

The first strategy only considers stationary solutions that correspond to
thermodynamic equilibrium states, and only requires the knowledge of the
thermodynamic function $\mathcal{E}(k;n,T)$. Its main result is the MCT-like
“glass transition” diagram of the system zepeda ; sperl1 , which determines
the borderline of the region of the state space where no kinetic barriers will
impede the system from reaching thermodynamic equilibrium.

To explain the concept of _glass transition diagram_ (GTD), let us notice that
Eqs. (2)-(6) contain the description of the dynamic properties of _equilibrium
fluids_ as the particular limit in which the solution $S(k;t)$ of Eq. (2) has
reached its equilibrium limit
$S(k;t\to\infty)=S(k;n,T)=1/n\mathcal{E}(k;n,T)$. In this limit one recovers
the original _equilibrium_ SCGLE theory of dynamical arrest arrest1 ; arrest2
; arrest3 , that describes the dynamics of _equilibrium_ fluids. The
corresponding SCGLE equations follow from replacing $S(k;t)$,
$\Delta{\zeta}^{*}(\tau;t)$, $F(k,\tau;t)$, and $F_{S}(k,\tau;t)$ in Eqs.
(4)-(6) by, respectively, $S(k)$,
$\Delta{\zeta}^{*eq}(\tau)\equiv\Delta{\zeta}^{*}(\tau;t\to\infty)$,
$F^{eq}(k,\tau)\equiv F(k;\tau;t\to\infty)$ and $F_{S}^{eq}(k,\tau)\equiv
F_{S}(k;\tau;t\to\infty)$. As in MCT, the SCGLE theory provides equations for
the so-called non-ergodicity parameters, which are the long-$\tau$ asymptotic
limits $f^{eq}(k)\equiv F^{eq}(k,\tau\to\infty)$, and $f^{eq}_{S}(k)\equiv
F_{S}^{eq}(k,\tau\to\infty)$.

In the case of the SCGLE theory, $f^{eq}(k)$ and $f_{S}^{eq}(k)$ can be
written in terms of the SF, $S(k;n,T)$, and the squared localization length
$\gamma^{eq}(\phi,T)$ (Eqs. (9) and (10) of Ref. scgle3 ), the latter being
solution of the equation

$\begin{split}\frac{1}{\gamma^{eq}(\phi,T)}=\frac{1}{6\pi^{2}n}\int_{0}^{\infty}dkk^{4}\left[S(k;n,T)-1\right]^{2}\lambda^{2}(k)\times\\\
\frac{1}{\left[\lambda(k)S(k;n,T)+k^{2}\gamma^{eq}(\phi,T)\right]\left[\lambda(k)+k^{2}\gamma^{eq}(\phi,T)\right]}.\end{split}$
(16)

The dynamic order parameter $\gamma^{eq}(\phi,T)$ diverges at equilibrium, and
has a finite value at non-ergodic states. Thus, it partitions the state space
$(\phi,T)$ into its ergodic and non-ergodic regions, with the ideal glass
transition line being the borderline between them. The monotonically
increasing (red) solid line of Fig. 2 is such fluid-glass transition line
$T=T_{c}(\phi)$ for our HSDY model, obtained within the RPA for $S(k;n,T)$ in
Eqs. (14) and (9). This liquid-glass transition line originates at high-
temperatures and high-densities at the well-known hard-sphere glass
transition, occurring at $\phi_{c}\approx 0.58$. At lower $T$ and $\phi$, it
meets the $\lambda$ line at the state point $(\phi_{b},T_{b})=(0.365,0.452)$.

Upon crossing this transition line from the ergodic domain (region I of Fig.
2) to the non-ergodic domain (region II), the order parameter
$\gamma^{eq}(\phi,T)$ changes discontinuously from its equilibrium infinite
value to a finite non-equilibrium value, and this discontinuous behavior
characterizes what is referred to (borrowing MCT language) as a “type B” glass
transition. An alternative dynamic order parameter is the normalized
equilibrium long-time self-diffusion coefficient
$D_{L}(\phi,T)/D_{0}=b^{eq}(\phi,T)=b(t\to\infty;\phi,T)$, also provided by
the solution of the SCGLE equations. This dynamic state function is finite and
positive in region I and vanishes in region II and along the ideal fluid-glass
transition line $T=T_{c}(\phi)$. Its value is represented in Fig. 2 by a color
code, with the darkest color corresponding to the arrested region II.

Clearly, since $S(k;n,T)$ does not exist below the $\lambda$ line (non-
monotonic (blue) solid line in Fig. 2), the SCGLE theory cannot provide any
predictions there. Thus, this method can identify the ergodic and non-ergodic
regions only above the $\lambda$ line, leaving the (light) region below, empty
of information. This limitation is, of course, shared by MCT. In fact, our
glass transition diagram in Fig. 2 is the analog of the MCT “kinetic phase
diagrams” reported in Fig. 2 of Ref. wuliuchencao  for different values of
$A$, $z_{1}$ and $z_{2}$.

###  III.2 Non-equilibrium “glass transition” diagram (NEGTD).

The second strategy provides a route of escape from the limitation of the
SCGLE theory to equilibrium conditions. For this we must return to the full
set of equations (2)-(6) that summarize the more general NE-SCGLE formalism.
Following the approach explained in detail in Ref. olais1 , one can extend the
method based on the solution of Eq. (16) for $\gamma^{eq}(\phi,T)$, by
introducing the more general parameter $\gamma^{*}(u)$, defined in Ref. olais1
as the solution of its Eq. (3.6), namely,

$\begin{split}\frac{1}{\gamma^{*}(u)}=\frac{1}{6\pi^{2}n}\int_{0}^{\infty}dkk^{4}\left[S^{*}(k;u)-1\right]^{2}\lambda^{2}(k;u)\times\\\
\frac{1}{\left[\lambda(k;u)S^{*}(k;u)+k^{2}\gamma^{*}(u)\right]\left[\lambda(k;u)+k^{2}\gamma^{*}(u)\right]},\end{split}$
(17)

where $S^{*}(k;u)$ is given by

$\begin{split}S^{*}(k;u)\equiv[n\mathcal{E}(k;\phi,T)]^{-1}+\left\\{S_{i}(k)-[n\mathcal{E}(k;\phi,T)]^{-1}\right\\}\times\\\
e^{-2k^{2}D_{0}n\mathcal{E}(k;\phi,T)u},\end{split}$ (18)

with $S_{i}(k)=S^{*}(k;u=0)$ being an (arbitrary) initial condition.

Figure 3: Dependence of the non-equilibrium dynamic order parameter
$\gamma_{a}(\phi,T)$ on the final temperature $T$, for a sequence of quenches
of the HSDY fluid ($z_{1}=1$, $z_{2}=0.5$ and $A=0.5$) at fixed volume
fraction $\phi$ and with initial condition $S_{i}(k)=S^{eq}_{HS}(k;\phi)$ for,
(a) $\phi=0.15$, to the left of the bifurcation point
$(\phi_{b},T_{b})=(0.365,0.452)$ (see the vertical dashed line in the inset),
and (b) $\phi=0.45$, to the right of the bifurcation point. In (a), the
vertical dotted lines indicate the three temperatures corresponding to the
$\lambda$ and spinodal lines, and the glass-to-glass transition, thus
highlighting the discontinuous jumps of $\gamma_{a}$ at the two temperatures
$T_{\lambda}$ and $T_{c}$, and also the fact that $T_{s}$ passes completely
unnoticed for the non-ergodicity parameter. In (b), the vertical lines
emphasize that $\gamma_{a}$ only jumps discontinuously at $T_{c}$, and crosses
the two lines of thermodynamic instabilities without noticing them.

For a given state point $(\phi,T)$ we use $\mathcal{E}(k;\phi,T)$ to solve Eq.
(17), which determines $\gamma^{*}(u)$ as a function of $u$. If we find that a
finite value $u_{a}$ of the parameter $u$ exists, such that $\gamma^{*}(u)$ is
infinite in the finite interval $0\leq u<u_{a}$ and finite for $u_{a}\leq
u\leq\infty$, then we conclude that the system will become kinetically
arrested. If, instead, $\gamma^{*}(u)=\infty$ for $0\leq u\leq\infty$, we
conclude that the state point $(\phi,T)$ lies in the ergodic region of state
space. Thus, the functions $u_{a}(\phi,T)$ and
$\gamma_{a}(\phi,T)\equiv\gamma^{*}(u_{a}(\phi,T))$, are dynamic order
parameters, which allows us to draw what we refer to as the _non-equilibrium
glass transition diagram_ (NEGTD). In Fig. 2 we present the results
corresponding to our specific HSDY model. Note that, contrary to the algorithm
based on Eq. (16), whose input is the SF $S(k;n,T)$ (non-physical below the
$\lambda$ line), the input of this non-equilibrium algorithm is the
thermodynamic stability function $\mathcal{E}(k;\phi,T)$, whose negative
values simply indicate conditions of thermodynamic instability. As a result,
this more general criterion to determine dynamic arrest is far more powerful
than that based on Eq. (16), since it is also applicable to quenches to
thermodynamically unstable regions of state space, as illustrated by the
results in Fig. 2.

The NEGTD of Fig. 2 completely agrees with the GTD of Fig. 2, in that both
algorithms partition the portion of the plane $(\phi,T)$ _above_ the $\lambda$
line, into the two well-defined regions: region I of ergodic fluid states and
region II of non-ergodic glass states. However, in the complementary portion
of state space, i.e., at and below the $\lambda$ line, the NE-SCGLE theory
reveals a remarkable non-equilibrium scenario, whose first unexpected feature
is the existence of two new dynamic arrest transitions for volume fractions
below that of the meeting point $(\phi_{b},T_{b})$. The first of them occurs
at a dynamic arrest temperature that virtually coincides with the $\lambda$
temperature $T_{\lambda}(\phi)$, and constitutes the boundary between the
(ergodic) region I and a new region (region III) of dynamically arrested
states, whose nature is expected to be revealed by their non-equilibrium
structural and dynamical properties, provided by the full solution of the NE-
SCGLE equations.

The second is a dynamic arrest transition occurring along the temperature
$T=T_{c}(\phi)\ (<T_{\lambda}(\phi))$, which defines the boundary between the
arrested states in region III and the glass states in the low-temperature–low-
density extension of region II. These two arrest lines merge at the meeting
point $(\phi_{b},T_{b})$, which is now revealed to actually be a point of
bifurcation of the fluid-to-glass transition line of Fig. 2. The latter
clearly appears to continue below the $\lambda$ line (i.e., for
$\phi\leq\phi_{b}$) as the arrested-glass transition line $T=T_{c}(\phi)$.

###  III.3 NE-SCGLE dynamic order parameter $\gamma_{a}^{*}(\phi,T)$.

Most of the previous conclusions directly derive from the density and
temperature dependence of the dynamic order parameters $u_{a}(\phi,T)$ and
$\gamma_{a}(\phi,T)$ which, in turn, depend on $(\phi,T)$ through the function
$\mathcal{E}(k;\phi,T)$ in Eq. (18). The parameter $\gamma_{a}(\phi,T)$ bears
a more transparent physical significance, better explained by its original
_equilibrium_ definition: it is the long-time asymptotic value
$\gamma^{eq}\equiv\lim_{\tau\to\infty}W^{eq}(\tau)$ of the mean squared
displacement (MSD)
$W^{eq}(\tau)\equiv\langle(\Delta\mathbf{r}(\tau))^{2}/3\rangle^{eq}$, of
individual particles. This parameter is infinite if the particles diffuse, and
is finite if they are immobilized, in which case it defines the square
localization length. This definition extends to non-equilibrium conditions by
explicitly including the waiting time $t$ in the definition of the MSD, which
is now defined as
$W(\tau;t)\equiv\langle(\mathbf{r}(t+\tau)-\mathbf{r}(t))^{2}/3\rangle$, thus
allowing us to define the non-equilibrium square localization length
$\gamma_{a}$ as
$\gamma_{a}\equiv\lim_{\tau\to\infty}\lim_{t\to\infty}W(\tau;t)$.

The use of the function $\gamma_{a}(\phi,T)$ is better illustrated in Fig. 3,
which plots the inverse of $\gamma_{a}(\phi,T)$ as a function of the final
temperature $T$ along two isochores, $\phi=0.15$ (Fig. 3) and $\phi=0.45$
(Fig. 3). In Fig. 3, the (red) dashed horizontal line on the right
($\gamma_{a}^{-1}(\phi,T)=10^{-2}$) indicates, in reality, that the function
$\gamma_{a}(\phi=0.15,T)$ actually remains infinite for all final temperatures
above a critical temperature of about $0.673$ (which, within the resolution of
the figure, coincides with the temperature of the $\lambda$ line for that
isochore, $T_{\lambda}(\phi=0.15)=0.6733$). At this singular temperature,
$\gamma_{a}(\phi,T)$ jumps _discontinuously_ to a finite value
$\gamma_{a}(T_{\lambda}^{-})\approx 40$ (localization length
$\sqrt{\gamma_{a}}\approx 6.3$), remaining finite for $T<T_{\lambda}(\phi)$.
This implies that the $\lambda$ line, besides being the threshold of the
thermodynamic stability of uniform equilibrium states (leading to the
conjectured equilibrium modulated phases below the $\lambda$ line), also seems
to be an ergodic to non-ergodic transition line. In fact, since
$\gamma_{a}(\phi,T)$ jumps discontinuously at $T=T_{\lambda}(\phi)$, it is a
“type B” singularity (in MCT terminology).

Examining now temperatures below the $\lambda$ line, we see that the parameter
$\gamma_{a}(T)$ decreases continuously, and exhibits a second discontinuity at
a lower temperature $T_{c}(\phi=0.15)=0.165$. This discontinuity implies the
existence of still a second dynamic arrest transition, now corresponding to a
non-ergodic–to–non-ergodic (or “glass-glass”) “type B” transition, in which
the dynamic order parameter $\gamma_{a}(T)$ changes discontinuously by about
one order of magnitude, from a value $\gamma_{a}(T_{c}^{+})\approx 4$ to
another finite value $\gamma_{a}(T_{c}^{-})\approx 0.025$ (localization length
$\sqrt{\gamma_{a}}\approx 0.15$, typical of hard-sphere glasses). Besides
these two discontinuities, we could not identify any other dynamically
singular temperature. In fact, we found that the function $\gamma_{a}(\phi,T)$
is perfectly continuous through the spinodal curve $T=T_{s}(\phi)$, implying
that this line of thermodynamic instability would be dynamically irrelevant if
dynamical arrest were the only kinetic pathway available to the system.

Performing these calculations at other isochores allows us to determine the
two glass transition lines ($T=T_{\lambda}(\phi)$ and $T=T_{c}(\phi)$) shown
in Fig. 2, which merge at the bifurcation point $(\phi_{b},T_{b})$ into a
single fluid-glass transition line, and which coincides with that determined
by the method of subsection III.1 (see Fig. 2). For completeness, Fig. 3
illustrates the behavior of the function $\gamma_{a}(\phi=0.45,T)$,
corresponding to an isochore at the right of the bifurcation
$(\phi_{b},T_{b})$. In this case, one finds that $\gamma_{a}(\phi=0.45,T)$ is
infinite for temperatures above the fluid-glass transition line
$T=T_{c}(\phi=0.45)=0.656$, and jumps discontinuously to a finite value
$\gamma_{a}(\phi=0.45,T_{c}^{-})\approx 0.025$. To the right of the
bifurcation ($\phi>\phi_{b}$), the function $\gamma_{a}(\phi,T)$ is continuous
through both, the $\lambda$ line and the spinodal curve, implying that neither
of these lines have any dynamical relevance in this regime $\phi>\phi_{b}$.

Figure 4: Behavior of the stationary (a) NESF $S^{a}(k;\phi,T)$ and (b) NERDF
$g^{a}(r;\phi,T)$, of the HSDY fluid ($z_{1}=1$, $z_{2}=0.5$ and $A=0.5$), at
fixed volume fraction $\phi=0.45$ and for different (final) temperatures $T$
above and below the liquid-glass transition $T_{c}(\phi=0.45)=0.656$, as
indicated. The inset of (a) exhibits the $T$ dependence of the height
$S^{a}_{main}(\phi,T)\equiv S^{a}(k_{main};\phi,T)$ of the main peak of
$S^{a}(k;\phi,T)$ (darkest solid line) and of the equilibrium structure factor
$S(k;\phi,T)$ (dashed line). The solid line in the inset of (b) represents the
effective hard-sphere volume fractions $\phi_{HS}(\phi=0.45,T)$, determined as
described in the text.

Let us emphasize that the dynamic arrest scenario summarized by Fig. 2 appears
at first sight to be essentially identical to that of the same model system in
the absence of the repulsive Yukawa interaction (i.e., with $A=0$, see Refs.
olais1 ; olais2 ; nescgle8 ; zepeda ). This is illustrated by comparing Fig. 2
above, with Fig. 4 of Ref. olais1 . There are, however, remarkable fundamental
differences. The most important of them is the nature of the dynamic arrest
transition line $T=T_{\lambda}(\phi)$, along with the functions
$u_{a}(\phi,T)$ and $\gamma_{a}(\phi,T)$, which jump, from infinite values
above the $\lambda$ line, to finite values below (type B transition). In the
absence of the repulsive term, $A=0$, there is no $\lambda$ line, and the type
B arrest line $T=T_{\lambda}(\phi)$ is replaced by an arrest line
$T=T_{s}(\phi)$ lying along the spinodal curve. This transition happens to be
of type A, i.e., the function $\gamma_{a}(\phi,T)$ passes continuously from
infinite values above the line $T=T_{s}(\phi)$, to finite values below.

This qualitative difference between these two cases is, of course, associated
with a remarkable _structural_ difference between the arrested phases
predicted immediately below each transition. Thus, in the absence of
repulsions ($A=0$), the NE-SCGLE theory predicts the formation, below
$T=T_{s}(\phi)$, of spinodal-decomposition heterogeneities, whose growth is
halted only by dynamic arrest mechanisms, as established and discussed in
detail in Refs. olais1 ; olais2 . In contrast, in the presence of repulsions
($A>0$), below $T=T_{\lambda}(\phi)$ the NE-SCGLE theory predicts the dynamic
arrest due to the repulsion between finite-size clusters, as we now establish
by analyzing the structural properties predicted by the NE-SCGLE theory.

Figure 5: Behavior of the stationary (a) NESF $S^{a}(k;\phi,T)$ and (b) NERDF
$g^{a}(r;\phi,T)$, of the HSDY fluid ($z_{1}=1$, $z_{2}=0.5$ and $A=0.5$), at
fixed volume fraction $\phi=0.15$ and for different temperature quenches below
$T_{\lambda}(\phi=0.15)=0.6733$. The insets are auxiliary zooms to emphasize
both the amplification of structural correlations around $k_{\lambda}$ (or
$r_{\lambda}$, according to the case) , and the shift of $k_{\lambda}$
($r_{\lambda}$) towards smaller (larger) values with decreasing $T$. In both
figures, one notices two qualitatively different behaviors: for shallow
quenches, strong amplifications at low wave vector $k_{\lambda}$ (or
$r_{\lambda}$), and for deeper quenches, only a moderate amplification at
$k\approx 2\pi/\sigma$.

##  IV Stationary structure factor $S^{a}(k)$.

Let us now discuss the structural properties of the HSDY model predicted by
the NE-SCGLE theory, as described by the solution of Eq. (2) for $S(k;t)$. As
carefully discussed in Ref. olais1 , the time-dependent NESF $S(k;t)$ is given
by $S(k;t)=S^{*}(u(t))$, where the function $S^{*}(u)$ is defined in Eq. (18),
and with the variable $u(t)$ being the “material” time
$u(t)\equiv\int_{0}^{t}b(t^{\prime})dt^{\prime}$ peredo1 . Before discussing
the structural aging described by the full $t$-dependence of $S(k;t)$,
however, it is useful to consider its stationary, long-time asymptotic limit
$S^{a}(k)\equiv\displaystyle{\lim_{t\to\infty}}S(k;t)$, given by olais1

$\begin{split}S^{a}(k;\phi,T)\equiv[n\mathcal{E}(k;\phi,T)]^{-1}+\left\\{S_{i}(k)-[n\mathcal{E}(k;\phi,T)]^{-1}\right\\}\times\\\
e^{-2k^{2}D_{0}n\mathcal{E}(k;\phi,T)u_{a}(\phi,T)},\end{split}$ (19)

with the initial condition $S_{i}(k)$ chosen here as the SF at infinite $T$,
i.e, $S_{i}(k)=S(k;\phi,T=\infty)=S^{HS}(\phi)$.

Clearly, when the system is quenched to a state point $(\phi,T)$ in the
ergodic region (infinite $u_{a}(\phi,T)$ and $\gamma_{a}(\phi,T)$), the system
will equilibrate, and $S(k;t)$ will asymptotically attain its equilibrium
value $S^{a}(k;\phi,T)=S(k;\phi,T)$. The salient features of $S(k;\phi,T)$
have been discussed thoroughly in the literature seargelbart ;
LiuChenChenJCP05 ; archerpinievansreatto07 , and were briefly illustrated in
Fig. 1. The case of interest that the NE-SCGLE theory now allows us to discuss
is, of course, when the state point $(\phi,T)$ lies in one of the non-ergodic
regions (finite $u_{a}(\phi,T)$ and $\gamma_{a}(\phi,T)$), in which the system
will no longer be able to equilibrate. Instead, $S(k;t)$ is expected to attain
the non-equilibrium value $S^{a}(k;\phi,T)$ given by Eq. (19).

Figs. 4 and 5 illustrate the dependence on the depth of the quench of both,
the NESF $S^{a}(k;\phi,T)$ and the corresponding NERDF $g^{a}(r;\phi,T)\equiv
1+h^{a}(r;\phi,T)$, with the NETCF $h^{a}(r;\phi,T)$ being the inverse FT of
$[S^{a}(k;\phi,T)-1]/n$. To discuss the high-density–high-temperature
scenario, Figs. 4(a) and 4(b) exhibit the behavior predicted for these
structural properties along the isochore $\phi=0.45$, which lies well to the
right of the bifurcation point $(\phi_{b},T_{b})$, as a function of $T$. In
Fig. 4(a) we notice first the non-monotonic $T$-dependence of the hard-
sphere–like correlations, represented by the height
$S^{a}_{main}(\phi,T)\equiv S^{a}(k_{main};\phi,T)$ of the main peak of
$S^{a}(k;\phi,T)$, located at $k=k_{main}\approx 2\pi/\sigma$. As summarized
in the inset of this figure, this property (described there by the solid line)
first increases when $T$ is decreased still in the ergodic regime,
$T>T_{c}(\phi)$, but as $T$ crosses the liquid-glass line, it decreases with
smaller $T$. Note that the SF $S(k;\phi,T)$ still exists for $T\leq
T_{c}(\phi)$ and the height $S(k_{main};\phi,T)$ of its main peak continues to
increase as $T$ decreases below $T_{c}(\phi)$ (dashed line). The NE-SCGLE
theory, however, predicts that below $T_{c}(\phi)$, this equilibrium
$S(k;\phi,T)$ will be unreachable in practice due to kinetic considerations
nescgle3 .

Let us notice also the qualitative resemblance of the non-equilibrium
structure represented by $S^{a}(k;\phi,T)$ with the equilibrium structure of a
fluid of particles with only excluded volume interactions. This is also
observed in the dependence of the NERDF $g^{a}(r;\phi,T)$ illustrated in Fig.
4(b). To quantify this resemblance, let us borrow the notion of structural
equivalence between equilibrium soft and hard-sphere fluids proposed in Ref.
dynamicequivalence ; prl ; pre , and let us apply it to establish the
structural equivalence between our non-equilibrium HSDY system and the
equilibrium HS liquid. For this, we define an effective HS fluid whose RDF
$g_{HS}(r;\phi_{HS})$ matches $g^{a}(r;\phi,T)$ at least in the height
$g^{a}_{2}(\phi,T)\equiv g^{a}(r_{2};\phi,T)$ of its second maximum, located
at $r_{2}\approx 2\sigma$.

This procedure assigns an effective HS volume fraction $\phi_{HS}$ to each
state point $(\phi,T)$. In the inset of Fig. 4(b) we plot the function
$\phi_{HS}=\phi_{HS}(\phi,T)$ for $\phi=0.45$, which reveals that in the
ergodic region $\phi_{HS}(\phi,T)$ increases with the depth of the quench
until reaching a maximum similar in magnitude to the well-known hard-sphere
glass transition volume fraction $\phi^{HS}_{c}=0.58$ at $T\approx T_{c}$.
Below the liquid-glass transition line $T=T_{c}(\phi)$, however, we observe
that $\phi_{HS}(\phi,T)$ remains relatively constant, reminding the
$T$-dependence of the inverse square localization length
$\gamma_{a}^{*-1}(\phi,T)$ below $T_{c}$ (see Fig. 3(b)). These results
provide non-equilibrium evidences to conclude that, when the fluid-glass
transition is crossed along isochores in the high-density–high-temperature
regime, to the right of the bifurcation point $(\phi_{b},T_{b})$, and far from
the $\lambda$ line, dynamical arrest is mainly driven by the excluded volume
interactions and caging mechanisms. As a side observation, let us recall that
the definition of the function $\phi_{HS}=\phi_{HS}(\phi,T)$ was based on the
mapping of the non-equilibrium structure of the HSDY system onto the purely HS
equilibrium structure. Such mapping is characteristic of the so-called “HS
dynamic universality class” prl ; pre . This leaves open the provocative
possibility that, within adequate scalings, the asymptotic dynamics in this
regime could also be mapped onto the equilibrium dynamics of the HS system.

We now move to the complementary (low-density–low-temperature) regime, and
consider the same isochore as in Fig. 1, $\phi=0.15$, which lies well to the
left of the bifurcation point. Going back to that figure, we recall that,
along this low-density isochore, the salient feature, amply discussed in the
literature seargelbart ; LiuChenChenJCP05 ; archerpinievansreatto07 ;
ruizzaccarelli , was the emergence of the low-$k$ (“cluster”) peak of the SF
$S(k;\phi,T)$, centered at the wave vector denoted as $k_{\lambda}$. There, we
learned that the height $S(k_{\lambda};\phi,T)$ of this low-$k$ peak increases
monotonically as the temperature is decreased until reaching the $\lambda$
line, where it diverges. Fig. 5(a) now complements this equilibrium story with
its non-equilibrium continuation below the $\lambda$ line, where the SF
$S(k;\phi,T)$ ceases to exist, but the second, NESF $S^{a}(k;\phi,T)$ given by
Eq. (19), emerges. This new non-equilibrium stationary solution represents, of
course, the formation of amorphous dynamically arrested materials.

Fig. 5(a) illustrates how $S^{a}(k;\phi,T)$ changes as the final temperature
is lowered, now below $T_{\lambda}(\phi)$. The main trend observed is
highlighted in the inset, and refers to the notorious decrease with the quench
depth, of the height $S^{a}_{\lambda}(\phi,T)\equiv S^{a}(k_{\lambda};\phi,T)$
of the low-$k$ peak of $S^{a}(k;\phi,T)$, starting from its maximum value
attained at $T\approx T_{\lambda}$ (e.g.,
$S^{a}_{\lambda}(\phi=0.15,T=0.67)\approx 500$), down to a value of $\approx
45$ for $T$ approaching the glass-glass transition $T_{c}=0.165$ from above.
Decreasing $T$ below $T_{c}$, we observe that the height
$S^{a}_{\lambda}(\phi=0.15,T)$ of the small-$k$ peak drops precipitously, down
to values smaller than the height of the main (“particle-particle”) peak at
$k\approx 2\pi/\sigma$. While this happens to the height
$S^{a}_{\lambda}(\phi,T)$, we also observe a shift of the position
$k_{\lambda}$ of this peak, to larger wave-vectors, as $T$ decreases below
$T_{\lambda}$. This indicates that the size $\xi=2\pi/k_{\lambda}$ of the
corresponding spatial heterogeneities are largest at $T\approx T_{\lambda}$
(e.g., $\xi\approx 11\sigma$ at $T=0.67$, slightly below $T_{\lambda}$), and
decreases with the depth of the quench, down to approximately $6\sigma$ at
$T=0.17$ (slightly above $T_{c}$).

Figure 6: (a) $T$-dependence of the height $S^{a}_{\lambda}(\phi,T)\equiv
S^{a}(k=k_{\lambda};\phi,T)$ of the NESF $S^{a}(k:\phi,T)$ along the isochore
$\phi=0.15$, covering temperatures from below the glass transition point
$T_{c}$ up to temperatures above the fluid-glass-transition temperature
$T_{a}=0.67336>T_{\lambda}=0.67331$ (see the text). The inset considers an
amplification around $T_{a}$, used to emphasize the deviations of
$S^{a}_{\lambda}(\phi,T)$ (solid line) from the equilibrium value
$S(k_{\lambda};\phi;T)$ (dashed line) which diverges at $T_{\lambda}$. (b)
Comparison of the behavior of the asymptotic value of the NERDF
$g^{a}(r;\phi=0.15,T=0.67)$ (red solid line) and the effective hard-sphere RDF
$g(r;\phi_{HS})$ (black solid line) used to match the height of the second
maximum (see the text). (c) T- dependence of the volume fraction $\phi_{eff}$
of the effective hard-sphere fluid used to match the RDF for temperatures
below the $T_{a}$ transition line and slightly above the $T_{c}$ transition
(see the text).

The same information is presented in Fig. 5(b), but in terms of the stationary
NERDF $g^{a}(r;\phi,T)$. There we see that the long-range minimum in the RDF
$g(r;\phi,T)$ located at a distance $r_{min}\approx 7\sigma$, recalling the
lower temperature in Fig. 1(b) ($T=0.68>T_{\lambda}$), now continues to appear
in the NERDF $g^{a}(r;\phi,T)$ for quenches below $T_{\lambda}$. This is
illustrated by the shallowest quench of Fig. 5(b), with final temperature
$T=0.67$, whose NERDF $g^{a}(r;\phi,T)$ shows a much more pronounced long-
range minimum at $r_{min}\approx 7\sigma$, and a high long-range second
maximum $g_{2}^{a}(\phi,T)\equiv g^{a}(r_{\lambda};\phi,T)$ at
$r_{\lambda}\approx 12\sigma$. These long-ranged minimum and maximum describe
the cluster layering around a central particle (itself in a cluster). One
notices, in addition, that the relevance of these features, measured by the
heights $S^{a}(k_{\lambda};\phi,T)$ or $g_{2}^{a}(r_{\lambda};\phi,T)$, is
largest in the neighborhood of the $\lambda$ line and decreases with the
quench depth.

To have a more comprehensive summary of these trends in the behavior of
$S^{a}(k;\phi,T)$, in Fig. 6(a) we plot $S^{a}_{\lambda}(\phi=0.15,T)$ as a
function of $T$, covering temperatures from below the glass-glass transition
temperature $T_{c}$ up to temperatures above $T_{\lambda}$. There we see that
it is mostly in the temperature interval $T_{c}<T<T_{\lambda}$ where
$S^{a}_{\lambda}(\phi,T)$ attains very large values, decreasing
catastrophically outside this interval. Regarding the decay of
$S^{a}_{\lambda}(\phi,T)$ with $T$ in the close neighborhood of $T_{\lambda}$,
if the dynamic arrest transition occurred exactly at $T_{\lambda}$, we should
have a mathematical discontinuity right at $T_{\lambda}$, since
$S^{a}_{\lambda}(\phi,T)$ remains finite as $T$ approaches $T_{\lambda}$ from
below, but diverges when $T$ approaches $T_{\lambda}$ from above.

A close inspection of such a singular prediction, however, revealed a rather
unexpected scenario, illustrated in the inset of Fig. 6(a): as it turns out,
the dynamic arrest transition does not occur at $T=T_{\lambda}$, but at a
different, slightly larger temperature, which we denote $T_{a}$. More
precisely, $T_{\lambda}=0.67331$, whereas $T_{a}=0.67336$ (clearly, to detect
this difference we needed a very high numerical resolution). However, the
consequence is that the dynamic arrest transition preempts the occurrence of
the divergence of $S^{a}_{\lambda}(\phi,T)$, thus explaining the practical
impossibility of its observation. Hence, although quantitatively modest, this
observation may have far-reaching fundamental implications regarding the
theoretical understanding of the accurate descriptions of cluster formation
and dynamic arrest in SALR systems. In fact, although scanning the state space
spanned by the potential parameters ($z_{1}$, $z_{2}$, and $A$) is out of the
scope of the present work, let us mention that preliminary extensions of the
present NE-SCGLE calculations reveal that the minute difference between
$T_{\lambda}$ and $T_{a}$, becomes more significant when the range of the
attraction decreases. We thus expect that a more detailed study in this
direction will allow a more direct contact with the phenomenology of cluster
formation revealed by simulations sciortinomossa ; charbonneaureichmanPRE07 ,
performed for much narrower attractive wells than that discussed here. In this
paper, however, we shall continue neglecting this quantitative difference
between $T_{a}$ and $T_{\lambda}$, and will continue describing the dynamic
arrest transition as occurring at the temperature $T_{\lambda}$.

On a different subject, let us now refer to the early and sharp analysis by
Sciortino et al. sciortinomossa , who modeled the effective long-range
cluster-cluster interaction as the superposition of the particle-particle
Yukawa repulsion between the particles that form two interacting clusters.
This led to a renormalized cluster-cluster Yukawa repulsion, capable of
inducing dynamic arrest into a Wigner glass of clusters, very much as the
experimentally observed 82LindsayChaikinJCP ; 89SirotaPRL  Wigner glasses
formed by individual charged colloidal particles. Along this line of thought,
the long-range minimum and maximum of $g^{a}(r;\phi,T)$, in Fig. 5(b) should
correspond to the first minimum and maximum of the RDF of a repulsive Yukawa
fluid. However, since the repulsive Yukawa fluid is structurally equivalent to
the hard-sphere fluid prl ; pre , we can also pursue this idea using the
structural equivalence of the fluid of clusters with the hard sphere liquid.

This is best explained in Fig. 6(b), which compares the NERDF
$g^{a}(r;\phi,T)$ (red solid line) corresponding to the shallowest quench in
Fig. 5(b) (final temperature $T=0.67$, slightly below $T_{\lambda}=0.6733$),
with the RDF $g(r;\phi_{HS})$ (black solid line) of an effective hard-sphere
system that matches the height $g^{a}_{2}(\phi,T)\equiv g^{a}(r_{2};\phi,T)$
of the long-range second maximum at $r_{2}$ of $g^{a}(r;\phi,T)$. This
comparison determines the HS volume fraction $\phi_{HS}(\phi,T)$ of an
effective HS system structurally equivalent to the cluster-cluster
correlations in the RDF $g^{a}(r;\phi,T)$ of the SALR system at the final
state point $(\phi,T)$. Fig. 6(c) exhibits the value
$\phi_{eff}\equiv\phi_{HS}(\phi=0.15,T)$ thus computed, as a function of the
final temperature $T$.

As observed in Fig. 6(c), $\phi_{eff}$ can be determined even in the
equilibrium regime ($T>T_{\lambda}$), but only in the immediate neighborhood
of $T_{\lambda}$, where it has a sharp increase as $T$ approaches
$T_{\lambda}$ from above, and where it reaches its maximum value (slightly
larger than 0.5). For $T<T_{\lambda}$, we have that $\phi_{eff}$ decreases
with decreasing $T$, down to a value $\phi_{eff}\lesssim 0.45$ in the
neighborhood of the glass-glass transition temperature $T_{c}$. Thus, in the
interval $T_{c}<T<T_{\lambda}$, the quantitative value of $\phi_{eff}$ in Fig.
6(c) is clearly smaller than the HS glass transition volume fraction
$\phi_{c}\approx 0.58$, and hence, cannot account for the dynamic arrest
condition. This means that the cluster-cluster caging mechanism contributes to
the arrest, but it is not sufficient. Hence, other mechanisms must come into
play in this regime.

Those additional mechanisms are also suggested by the results for
$g^{a}(r;\phi,T)$ in Fig. 5(b). There we see that, as $T$ decreases
sufficiently, the true hard-sphere interactions manifest themselves in the
emergence of a local maximum in $g^{a}(r;\phi,T)$ at $r\approx 2$, indicating
growing particle-particle (not cluster-cluster) hard-sphere correlations. As
the results in Fig. 5(b) indicate, these become the dominant correlations for
$T$ below the glass-glass transition temperature $T_{c}$.

In summary, we can say that, in contrast with the high-density regime
illustrated in Fig. 4, whose dynamic arrest was strongly dominated by the
excluded volume forces, in the low-density–low-temperature regime, illustrated
by the isochore $\phi=0.15$, the dynamic arrest is the result of a much
subtler and complex interplay between the three components of the interaction:
the excluded volume, the long-range repulsion and the shorter-ranged
attraction.

##  V Structural aging: $t$-dependence of $S(k;t)$ and $g(r;t)$.

At this point, it is important to emphasize that the equilibrium structure
factor and phase diagram of Fig. 1, as well as the glass transition diagrams
of Fig. 2 and the structural properties $S^{a}(k;\phi,T)$ and
$g^{a}(r;\phi,T)$ just discussed, only refer to the infinitely long time
asymptotic limit, whose experimental observability is only possible, in
practice, if we wait longer than the longest relaxation times of the system.
This may be almost trivial when the system is able to relax to equilibrium,
which obviously makes the asymptotic equilibrium scenario verifiable within
practical waiting times (and hence, much more familiar and representative of
ordinary experience). Under conditions of dynamical arrest, however, the
longest relaxation times may actually be too long (ideally, infinite), and the
system might then remain in non-equilibrium conditions within any practical
observation time. This, in turn, may render the asymptotic glass transition
scenario impossible to verify in practice.

This practical impossibility could impede us to appreciate the value of these
asymptotic predictions. Such impediment, however, can be removed by
considering the full $t$-dependent solutions of the NE-SCGLE equations
(2)-(6), which predict what one would measure at the finite and practical
waiting times $t$ involved in any realistic experiment or simulation. From
this kinetic perspective, the fundamental role of the ideal asymptotic
scenario may be best appreciated, as we now illustrate with the aging of the
structural properties. Of course, the full $t$-dependent solution also yields
a wealth of information regarding, for example, the dynamic and rheological
properties, thus providing a more powerful resource to understand the nature
of the glassy phases described by the asymptotic glass transition scenario.

Figure 7: (a) Sequence of snapshots describing the $t$-evolution of the NESF
$S(k;t)$ for a quench of the HSDY system, at $\phi=0.15$, from the same
initial value as before, and towards the final temperature $T=0.67$ (right
below $T_{\lambda}(\phi=0.15)$). The inset compares the time evolution of the
two peaks $S_{main}(t)$ (solid squares) and $S_{\lambda}(t)$ (solid circles).
(b) T-dependence of $S_{\lambda}$ for a sequence of values of waiting times
$t$, as indicated, along with the asymptotic long time value
$S_{\lambda}(t\to\infty)$, also shown in Fig. 6(a). (c) Corresponding sequence
of snapshots for the NERDF $g(r;t)(\phi,T)$, along with the asymptotic value
$g(r)^{a}(\phi,T)$.

To illustrate the structural aging predicted for the HSDY system, Fig. 7(a)
plots the snapshots of the NESF $S(k;t)$ as a function of $k$, for a sequence
of representative values of the waiting time $t$, after its instantaneous
quench to the final temperature $T=0.67$ along the isochore $\phi=0.15$. These
results correspond to the shallowest quench illustrated in Fig. 5, and exhibit
the gradual enhancement of the correlations, starting from the chosen initial
state $S(k;t=0;\phi,T)=S^{eq}(k;\phi,T=\infty)=S^{HS}(k;\phi)$ (black solid
line), and ending at the asymptotic solution
$S^{a}(k;\phi,T)=S(k;t=\infty;\phi,T)$ (red dashed line). They also illustrate
the kinetic build-up of a rather fast, but very modest, increase around the
main peak at $k=k_{main}\approx 2\pi/\sigma$ of $S(k;t)$, along with the
appearance of the small-$k$ peak at $k=k_{\lambda}\approx 1/\sigma$ which
exhibits, in contrast, a slower but far more spectacular enhancement.

The main peak around $k_{main}$ describes short-ranged correlations, of the
order of one HS diameter, and implies rather fast processes of nearest
neighbors moving close to each other in the initial stage of cluster
formation. The small-$k$ peak at $k_{\lambda}$ describes, instead, longer-
ranged correlations, of the order of six HS diameters, and its slower kinetics
is associated with the build-up of cluster-cluster correlations, ultimately
leading to dynamic arrest. This striking kinetic difference is visualized more
precisely in the inset of Fig. 7, which compares $S_{main}(t;\phi,T)\equiv
S(k_{main},t;\phi,T)$ with $S_{\lambda}(t;\phi,T)\equiv
S(k_{\lambda},t;\phi,T)$ for the same $t$ sequence.

Let us stress again that the information provided by the full $t$-dependent
solutions of the NE-SCGLE equations (2)-(6) is too abundant to be reviewed in
one individual report, even if we restricted ourselves only to one specific
property, such as to $S(k,t;\phi,T)$. The main reason is that this function
depends on its four arguments, $(k,t;\phi,T)$, and it is not a simple task to
scan this four-dimensional parameter space. Still, partial views of this
dependence are particularly instructive. For example, Fig. 6(a) plotted
$S(k,t;\phi,T)$ as a function of $T$, keeping the other three arguments fixed,
namely, $t=\infty$, $k=k_{\lambda}$, and $\phi=0.15$. Since we now are
analyzing the dependence of $S(k,t;\phi,T)$ on the waiting time $t$, we may
now extend such analysis to finite $t$. This is done in Fig. 7(b), which plots
$S_{\lambda}(t;\phi,T)$ for fixed $\phi=0.15$, as a function of $T$ for a
sequence of values of the waiting time $t$.

The main message of Fig. 7(b) is that the NE-SCGLE theory not only predicts
the results for $S_{\lambda}(t=\infty;\phi,T)=S^{a}_{\lambda}(\phi,T)$, which
one would measure in an idealized and impossible experiment. A far more
relevant practical prediction refers to what one should measure at the
_finite_ waiting times involved in real specific experiments or simulations.
For instance, Fig. 7(b) illustrates the transient buildup of the structural
correlations at $k_{\lambda}$, from which we learn that, although this is a
slow process compared with the buildup of the correlations at $k_{main}$, it
only takes a finite time, $t\approx 10^{3}$, for $S(k_{\lambda},t;\phi,T)$ to
saturate to its asymptotic value $S^{a}_{\lambda}(\phi,T)$ within the
resolution of the figure. Hence, the temptation to identify this process with
one of equilibration is enormous. Fortunately, the corresponding analysis of
dynamic properties, such as the $\alpha$-relaxation time (not discussed in
this paper) allows us to clearly discriminate between equilibration and arrest
processes. Let us also mention a similarly interesting transient process,
whose description can be drawn from Fig. 7(a). We refer to the fact that
$k_{\lambda}$ becomes increasingly smaller with $t$, thus describing the
(time-dependent) cluster growth process.

The same structural information that we have just discussed can also be cast
in terms of the NERDF $g(r,t;\phi,T)$. Here we shall not make any effort to
review the resulting scenario, but only discuss Fig. 7(c), which is the real
space counterpart of Fig. 7(a). This figure describes the aging of
$g(r,t;\phi,T)$ (along $\phi=0.15$), whose most relevant feature is the slow
emergence of the long-ranged correlations associated with cluster-cluster
correlations, the main mechanism for dynamic arrest in the shallow quench
considered. From the aging of $g(r,t;\phi,T)$, the non-equilibrium evolution
of other relevant structural quantities (such as coordination numbers,
effective cluster size, and volume fraction) could be derived, each of which
adds a distinct perspective to the general scenario of cluster formation and
dynamic arrest. As said above, however, in our present contribution we only
aimed to pave the way to a more systematic discussion of both, structural and
dynamical properties. Such discussion is left for future work.

##  VI Discussion and conclusions.

In summary, in this paper, we have carried out the first systematic
application of the non-equilibrium self-consistent generalized Langevin
equation (NE-SCGLE) theory to the description of dynamic arrest in liquids
with competing short-ranged attractions and long-ranged repulsions (SALR). For
this, we have analyzed the non-equilibrium structural behavior in a model
hard-sphere plus double Yukawa (HSDY) fluid, quenched into its region of
thermodynamical instability.

For clarity, we have restricted ourselves to the study of the non-equilibrium
structural evolution of the model type III SALR after a quench into the
regions of thermodynamical instability. The most fundamental quantity to
understand such evolution is the thermodynamic stability function
$\mathcal{E}(k;n,T)$. When interpreted as a familiar thermodynamic state
function, it allows us to determine stability conditions, unstable domains and
equilibrium structural properties. Its appearance in the kinetic equation (2),
however, implies that $\mathcal{E}(k;n,T)$ bears a much deeper significance in
determining more general non-equilibrium properties, such as the time-evolving
structure factor $S(k;t)$. In this context, $\mathcal{E}(k;n,T)$ no longer
represents only an equilibrium property. Instead, it becomes a fundamental
input that determines the kinetics of the time-dependent structural
correlations $S(k;t)$ and $g(r;t)$ (and of any other related property, such as
those describing the non-equilibrium dynamics). In this sense, Eq. (2) can be
viewed as an innovative proposal to extend the Ornstein Zernike equation to
non-equilibrium conditions.

Restricted to the very specific set of parameters chosen in this study to
represent the type III SALR interactions ($z_{1}=1$, $z_{2}=0.5$ and $A=0.5$),
the physical scenario predicted by the NE-SCGLE equations reveals a stunning
and complex interplay between the thermodynamic instability represented by the
so called $\lambda$ line $T=T_{\lambda}(\phi)$, and different underlying
mechanisms for dynamical arrest. This interplay materializes in three distinct
types of non-ergodic discontinuous (“type B”) transitions: _(i)_ a “ _fluid to
arrested-cluster_ ” (F-AC) transition, occurring for concentrations below a
threshold value $\phi_{b}=0.365$ and moderately low temperatures, which in
practice coincides with the $\lambda$ line $T=T_{\lambda}(\phi)$; _(ii)_ an “
_arrested-clusters to glass_ ” (AC-G) transition $T=T_{c}(\phi)$, occurring in
the same $\phi$-regime but at lower temperatures, and; _(iii)_ a _fluid to
glass_ (F-G) hard-sphere–like transition, observed for $\phi>\phi_{b}$, also
denoted as $T=T_{c}(\phi)$, to emphasize its smooth continuation inside the
region of thermodynamic instability, as the previous AC-G transition (see Fig.
2(b)).

This asymptotic dynamic arrest transition scenario was greatly enriched by the
discussion of the non-equilibrium structural properties represented by
$S(k,t;\phi,T)$ and $g(r,t;\phi,T)$. We first discussed the asymptotic
stationary limits $S^{a}(k;\phi,T)$ and $g^{a}(r;\phi,T)$, and then considered
the non-equilibrium transient, provided by the full $t$-dependent solutions of
the NE-SCGLE equations, Eqs. (2)-(6). Besides describing in some detail the
most salient features, we also highlighted some seemingly innocuous
observations. The most relevant of them refers to the fact that at first
glance, and for the specific HSDY model liquid considered in this work, the
locus of the F-AC transition coincides _“in practice”_ with the $\lambda$ line
$T=T_{\lambda}(\phi)$. These quotation marks needed a clarification, which was
provided by the zoom in the inset of Fig. 6(a): a closer inspection with the
lens of the NE-SCGLE theory revealed that, in fact, the F-AC transition occurs
not exactly at the $\lambda$ temperature, but also at a dynamic arrest
temperature $T_{(F-AC)}$ slightly higher,
$T_{(F-AC)}(\phi)>T_{\lambda}(\phi)$. This immediately implied that the F-AC
transition preempts the $\lambda$-instability, thus avoiding the divergence of
the equilibrium structure factor expected to occur at the $\lambda$ line. As a
consequence, when the model SALR system is quenched right below $T_{\lambda}$,
the dynamical arrest conditions may prevent it from reaching full equilibrium
conditions.

This observation suggests the impossibility to experimentally observe the
ordered inhomogeneous (“modulated”) equilibrium phases predicted for the type
III SALR systems in the region of thermodynamic instability. As just
explained, the development of these phases is expected to become interrupted
by non-equilibrium dynamical arrest barriers. Unfortunately, to the best of
our knowledge no experimental results exist that can confirm (or disregard)
these important predictions. Thus, our work could be a guide for future
experiments on type III SALR systems, carried out to test the predicted non-
equilibrium scenario just outlined. Similarly, the existing experimental data
for type I and II SALR liquids surely will serve as a motivation for further
characterizations within the theoretical framework provided by the NE-SCGLE.
We leave such tasks for further work.

The present work was not aimed at scanning the parameter space ($z_{1}$,
$z_{2}$, $A$) of the HSDY potential, that is, to explore the physical scenario
for type I and type II SALR systems. Instead, it was meant to pave the way to
develop further systematic investigations with that and other purposes. An
important pending issue refers to the fact that, within the NE-SCGLE
description of arrested states, the specific evolution of the structural
correlations is intimately related, via Eq.(2), to the aging of the dynamics,
with the former being represented in (2) by the mobility function $b(t)$.
Since the specific kinetics of $S(k;t)$ determines the non-equilibrium
relaxation of the dynamics, it is a pending task to characterize the aging of
the dynamics along each of the transitions outlined in this work, and to
establish its connection to the non-equilibrium structural behavior discussed
here. From previous work on attractive systems (i.e., in the absence of the
long-ranged Yukawa repulsion) we anticipate an intricate dynamic landscape,
originated in the complex interplay between thermodynamic instabilities and
dynamical arrest, leading to distinct relaxation laws for the dynamics
approaching each transition. All these aspects, however, shall be addressed in
future work.

ACKNOWLEDGMENTS: This work was supported by the Consejo Nacional de Ciencia y
Tecnología (CONACYT, México) through Postdoctoral Fellowship Grant No.
I1200/224/2021 (L.F.E.-A.) and through Grants Nos. 320983, CB A1-S-22362, and
LANIMFE 314881. The authors acknowledge the anonymous referees for their
encouraging and constructive criticisms.

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