# Dynamical correlations in simple disorder and complex disorder liquids

Bernarda Lovrinčević‡, Martina Požar${}^{\ddagger},$ Ivo
Jukić${}^{\dagger}{}^{,\ddagger}$
David Perera∤ and Aurélien Perera†<EMAIL_ADDRESS>School of
Biophysics, University of Split, Croatiacorresponding author
<EMAIL_ADDRESS>
###### Abstract

Liquids in equilibrium exhibit two types of disorder, simple and complex.
Typical simple disorder liquid are liquid nitrogen, or weakly polar liquids.
Complex liquids concern those who can form long lived local assemblies, and
cover a large range from water to soft matter and biological liquids. The
existence of such structures leaves characteric features upon the atom-atom
correlation functions, concerning both atoms which directly participate to
these structure and those who do not. The question we ask here is: does these
features have also characteristic dynamical aspects, which could be tracked
through dynamical correlation functions. Herein, we compare the van Hove
function, intermediate scattering function and the dynamical structure factor,
for both types of liquids, using force field models and computer simulations.
The calculations reveal the paradoxical fact that neighbouring atom
correlations for simple disorder liquids relax slower than that for complex
disorder liquids, while prepeak features typical of complex disorder liquids
relax even slower. This is an indication of the existence of fast kinetic
self-assembly processes in complex disorder liquids, while the lifetime of
such assemblies itself is quite slow. This is further confirmed by the
existence of a very low-$k$ dynamical pre-peak uncovered in the case of water
and ethanol.

†Laboratoire de Physique Théorique de la Matière Condensée (UMR CNRS 7600),
Sorbonne Université, 4 Place Jussieu, F75252, Paris cedex 05, France.

‡Department of Physics, Faculty of Science, University of Split, Ruđera
Boškovića 33, 21000, Split, Croatia.

∤ Laboratoire de Traitement et Communication de l’Information, Telecom Paris,
Institut Polytechnique de Paris, 19, place Marguerite Perey CS 20031 F-91123
Palaiseau, France.

##  1 Introduction

While liquids are known to be the archetype of disorder (excluding all liquid
crystalline materials), liquid nitrogen and water belong to two different
classes of liquids [1, 2]. The first one is characterised by simple disorder,
while the second is about complex disorder, due to the presence of hydrogen
connectivity [3, 4], which gives a local coherence that simple disorder
liquids do not have. Other complex disorder liquids, such as alcohols, show
chain like clustering of the hydroxyl head groups [5, 6, 7, 8, 9, 10]. These
are elementary forms of self-assembly, which is the major property of more
complex liquids, such as soft matter [11, 12, 13] or biological liquids [14].
Admittedly, disorder is not really a physical chemistry property, such as
energy or entropy. However, it is related to the structural properties, which
are a window into the microscopic arrangement of particles, from statistical
point of view. In that, it is much more specific of the nature of liquids than
properties like energy or entropy.

Perhaps the typical experimental evidence of such complex form of disorder is
the radiation scattering pre-peak [5, 6, 15, 16, 17, 18, 19, 8, 20] in the
scattering intensity $I(k)$, which witnesses the existence of local
aggregates. For typical liquids of small molecules, the relevant experiment is
wide angle x-ray scattering, which covers the range from $k=0$ until the atom-
atom peak which about $k\approx 1.5-2\mathring{A}^{-1}$. Interestingly, not
all complex disorder systems have a radiation scattering pre-peak [21, 22, 23,
24]. This is because the scattering pre-peak is the result of sum of pre-peaks
and anti-peaks in specific atom-atom structure factors [24, 25], which
sometimes tend to exactly cancel each other, thus leaving a net small-$k$
raise in $I(k)$, which is interpreted as the presence of large heterogeneity
[26].

$I(k)$ is however a static indicator, and it would be interesting to have
information about the kinetics and lifetime of the local heterogeneity and
labile structures, and also how they affect various dynamical quantities.
Unfortunately, there is no such thing as time dependent radiation scattering
intensity $I(k,t)$ for x-ray scattering. Spectroscopy methods provide
informations about the molecular relaxation, but these mostly focus on intra-
molecular relaxation, and the coupling with inter-molecular relaxation must be
obtained through interpretations.

Computer simulation techniques can provide a direct access to atom-atom
dynamical pair correlation functions and the corresponding structure factors.
However, this implies a necessary model dependence, which can be a drawback in
some cases, such as that of mixtures involving aggregate forming components,
as exemplified with the difficulties in computing the Kirkwood-Buff integrals
[27, 28, 29, 30, 31, 32, 33]. Despite these difficulties, it is interesting to
obtain dynamical correlation functions, since these could provide a good idea
of the molecular kinetics, as well an opportunity to check agreements with
dynamical quantities, such as transport coefficients for example.

In the present work, we compute atom-atom van Hove functions $G(r,t)$,
corresponding intermediate scattering functions $F(k,t)$ and dynamical
structure factors $S(k,\omega)$, hence covering the direct and reciprocal
spaces, as well time and frequency domains, and for various types of single
component liquids, both of simple and complex disorder types. The liquids
studied herein are carbon-tetrachloride, acetone, water and ethanol. In
addition to allowing us to find correspondances between typical structural
features associated with local clustering in various representation of
correlations, we expect to learn correspondence between the relaxation
mechanisms themselves, since these functions are microscopic probes of better
investigation power than $I(k,t)$ could access.

There has been previous report of dynamical correlations in the literature,
which we briefly review here. It is interesting to note that not, compared to
calculation from computer of static correlations covering many types of
liquids, there are remarkably much less coverage with dynamical ones. Early
calculations concern mostly simple liquids, and lately water, but in the
supercooled regime. Going into details, dynamical structure factors have been
calculated for liquid metals, such as rubidium [34, 35] and aluminum [36, 37],
and molten salts [38]. Similar analysis have been performed for polyisoprene
and their MD simulations gave a clearer interpretation of the experimentally
observed results [39]. Models for glass forming systems, like silica, were
studied via MD simulations by Sciortino _et al._ [40] and for liquid iron by
Wu _et al._ [41]. Dynamical correlations have lately been reported from
inelastic x-ray scattering for liquid water at ambient conditions and
inelastic neutron scattering at various temperatures [42]. MD results of the
coherent dynamic structure factor of liquid water at the mesoscale have been
presented by Alvarez _et al._ [43].

In view of this scarcity of coverage, the present study could, although
focused on the quality of disorder, could be equally seen as providing the
missing coverage other types of liquids than those studied so far.

The remainder of the manuscript is as follows. We first recall what dynamical
atom-atom correlations are, and how they are interrelated. We then explain how
they are calculated and the models and simulation details. Then, in the
results section, we study into some detail these correlation functions and
investigate what they can tell us about the molecular relaxations at different
levels. A final section gathers what perspectives this study inspires and
outlines our conclusions.

##  2 Atom-atom dynamical correlations

We make the choice of dealing with atom-atom correlation functions, even in
the case of molecular liquids, instead of orientational correlation functions.
It is generally believed that these latter functions are a better choice,
since they contain information irretrievably lost in the former [44]. In
addition, atom-atom correlation functions can be obtained from the
orientational ones, but not the opposite [45]. However, these orientation
correlations cannot be manipulated directly and necessitate and infinite
number of terms in the standard rotational invariant expansion [46]. For
instance, the atom-atom functions can be obtained only as a sum of such
infinite number of terms [47]. This raises practical issues about the
convergence, which can become crucial when specific orientations play an
important, such as the hydrogen bond, for instance. Since atom-atom
correlation functions contain both the intra-molecular and inter-molecular
correlations, we believe that these are much simpler functions to calculate,
and they always come in a finite number, even when large proteins are
concerned. There are additional two appealing reasons for using atom-atom
functions. Firstly, they implicitly assume that molecular liquids are like a
“soup of atoms”, some of which are grouped through intra-molecular
correlations, hence making a link between covalent binding and labile binding,
with open interesting possibilities to unify both representations into a
single one. Secondly, as shown below in this Section, the intra-molecular part
is in fact very naturally connected to the self part of the van Hove function,
thus becoming a necessary ingredient in a dynamical approach of molecular
liquids,of which the static representation is only the $t=0$ limit. We believe
that these 2 reasons enforce an overwhelming bias in choosing atom-atom
correlations over orientational ones.

For the above mentioned reasons, we dwell into details of these dynamical
functions from the atomic perspective, even though many of these details can
be found in most text books [47, 48].

###  2.1 The van Hove function

The time dependent microscopic density per particle of atomic species $a$ , at
position $\mathbf{r}$ and time $t$, is defined as:

$\rho_{i;a}(\mathbf{r},t)=\delta\left[\mathbf{r}-\mathbf{r}_{i;a}(t)\right]$
(1)

where $\mathbf{r}_{i;a}(t)$ is the time dependent position of particle $i$ or
atomic species $a$, in the lab fixed frame. This definition holds regardless
whether the atom belongs to a molecule or not, since we use the “molecular
liquid=soup of atoms” convention, where each atomic species is named
differently, even if it concerns the same atom type. In this convention, the
two hydrogen atoms of water are labeled differently (e.g $H_{1}$ and $H_{2}$,
for instance). From this per-particle microscopic density, one defines the
microscopic random variable which is the total density of atomic species $a$
by summing over all atom $i$ of species $a$:

$\rho_{a}(\mathbf{r},t)=\sum_{i}\rho_{i;a}(\mathbf{r},t)$ (2)

From these two random variables, one can compute various type of statistical
averages in selected statistical ensembles. For instance, in the constant
$NVT$ Canonical ensemble, the simple 1-body average give the trivial relation:

$<\rho_{a}(\mathbf{r},t)>=\frac{N_{a}}{V}=\rho_{a}$ (3)

where $N_{a}$ is the number of atoms of species a in the volume V,
$\rho_{a}$the number density of species a, and we have used the fact that we
consider liquids at equilibrium which are homogeneous and isotropic. In the
“molecular liquid=soup of atoms” convention, for given molecular species $m$ ,
since all atoms are uniquely represented, the number density $\rho_{a_{m}}$ of
any atomic species $a_{m}$ within that molecule is exactly that $\rho_{m}$ of
the molecular species itself $\rho_{a}=\rho_{m}$. For this reason, we will use
$\rho_{a}$without any ambiguity about its meaning.

Two body correlations can be defined in a similar way, such as
$\rho_{ab}^{(2)}(\mathbf{r}_{1},\mathbf{r}_{2},t)=<\rho_{a}(\mathbf{r}_{1},t)\rho_{b}(\mathbf{r}_{2},0)>$
which correlates two types of atoms $a$ and $b$ at respective positions
$\mathbf{r}_{1}$ and $\mathbf{r}_{2},$and taken at two different times, where
we set arbitrarily the origin of time the system of atomic species $b$. Using
the fact that one can use the variable change
$(\mathbf{r},\mathbf{r}^{\prime})\rightarrow(\mathbf{r},\mathbf{R})$, where
$\mathbf{r}=\mathbf{r}_{1}-\mathbf{r}_{2}$ and
$\mathbf{R}=(\mathbf{r}_{1}+\mathbf{r}_{2})/2,$whose Jacobian is $1$, using
the random variable in Eq.(1) we define the _dimensionless_ self van Hove
function for homogeneous and isotropic liquids as:

$G_{aa}^{(s)}(r,t)=\frac{1}{\rho_{a}^{2}V}\sum_{i}\int d\hat{\mathbf{r}}\int
d\mathbf{R}<\rho_{i;a}(\mathbf{r}_{1},t)\rho_{i;a}(\mathbf{r}_{2},0)>$ (4)

where we have integrated both on the irrelevant $\mathbf{R}$ variable, as well
as the unit vector $\hat{\mathbf{r}}$ representing the orientational part of
$\hat{\mathbf{r}}$ of $\mathbf{r}$, since the system is isotropic and the pair
correlation function depends only on
$r=|\mathbf{r}|=|\mathbf{r}_{1}-\mathbf{r}_{2}|$ .

Similarly, using the total microscopic density defined in Eq.(2), we define
the _dimensionless_ total van Hove function as:

$G_{ab}(r,t)=\frac{1}{\rho_{a}\rho_{b}V}\int d\hat{\mathbf{r}}\int
d\mathbf{R}<\rho_{a}(\mathbf{r}_{1},t)\rho_{b}(\mathbf{r}_{2},0)>$ (5)

Using Eq.(2), this latter equation can be split into two parts, a distinct van
Hove correlation function

$G_{ab}^{(d)}(r,t)=\frac{1}{\rho_{a}\rho_{b}V}\int
d\hat{\mathbf{r}}\sum_{i\neq j}\int
d\mathbf{R}<\rho_{i;a}(\mathbf{r}_{1},t)\rho_{j;b}(\mathbf{r}_{2},0)>$ (6)

and a _new_ self van Hove function:

$G_{ab}^{(s)}(r,t)=\frac{1}{\rho_{a}\rho_{b}V}\sum_{i}\int
d\hat{\mathbf{r}}\int
d\mathbf{R}<\rho_{i;a}(\mathbf{r}_{1},t)\rho_{i;b}(\mathbf{r}_{2},0)>$ (7)

which differs from that in Eq.(4) since it can now concern two atoms which do
not belong the same species. However, since, by definition, the self
correlation can only concern similar atoms, we see that one can now consider
as self part, the correlations between different atoms, but belonging to the
_same_ molecular species. In this case, the self van Hove function is nothing
else that the intra-molecular dynamical correlation function. Needless to say,
$G_{ab}^{(s)}(r,t)=0$ when atoms $a$ and $b$ belong to different molecules.
Eq.(4) is contained in Eq.(7), and we will from now on consider Eq.(7) as the
definition of the self van Hove function.

Several remarks can be made. First of all, these 3 functions hold for neat
liquids as well for mixtures, which is a convenient unified description of
both cases. Secondly, since we have defined dimensionless van Hove functions,
the two following static limits hold:

$G_{ab}^{(d)}(r,t=0)=g_{ab}(r)$ (8)

where $g_{ab}(r)$ is the usual static pair distribution function between atoms
$a$ and $b$, and

$G_{ab}^{(s)}(r,t=0)=w_{ab}(r)$ (9)

where $w_{ab}(r)$ is the intra-molecular correlation function which appears in
the RISM theory.

Thirdly, two types of dynamical Kirkwood-Buff integrals (dKBI) and running
dKBI (RdKBI) can be defined, one for the distinct function and one for the
self function. We first define the RdKBI as

$K_{ab}(r,t)=4\pi\int_{0}^{r}dss^{2}\left[G_{ab}^{(d)}(s,t)-1\right]$ (10)
$K_{ab}^{(s)}(r,t)=4\pi\int_{0}^{r}dss^{2}G_{ab}^{(s)}(s,t)$ (11)

The dynamical KBI are then defined as $dKBI(t)=K_{ab}(\infty,t)$ and the self
part as $dKBI^{(s)}(t)=K_{ab}(\infty,t)$. It turns out that both these
functions are time independent in ergodic equilibrium liquids. Since the self
van Hove function represent the correlation of one atom with itself across
time, in the infinite time limit, in an ergodic equilibrium liquid, the
trajectory of any atom should cover the entire system. Hence its integral is
just the volume occupied by the atomic species, which is the molar volume
$V_{a}$ scaled by the mole fraction $x_{a}$ for the molecular species
containing atom $a$:

$K_{ab}^{(s)}(\infty,t)=x_{a}V_{a}\delta_{ab}$ (12)

Similarly, for ergodicity reasons, the distinct dynamical correlation
function, $K_{ab}(\infty,t)$ is the same as the static KBI:

$K_{ab}(\infty,t)=K_{ab}$ (13)

It is important to underline that both dynamical KBI are the same for all
pairs of atoms belonging to same molecular species pairs.

###  2.2 The intermediate scattering function

The intermediate scattering functions are the spatial Fourier transforms of
the van Hove functions, and come in same 3 varieties, total, self and
distinct, and are dimensionless functions, generically defined as:

$F^{(\bullet)}(k,t)=\sqrt{\rho_{a}\rho_{b}}\int
d\mathbf{r}\exp(i\mathbf{\mathbf{k.r}})G^{(\bullet)}(k,t)$ (14)

where the superscript $(\bullet)$ could be either blank for the total
function, or $(d)$ for the distinct function and$(s)$ for the self function.

Interesting equalities occur for the static case. The self-part reduces to the
Fourier transform of the RISM theory w-matrix elements

$F_{ab}^{(s)}(k,t=0)=w_{ab}(k)=j_{0}(kd_{ab})$ (15)

where the second equality holds in case of rigid molecules, with
$d_{ab}=|\mathbf{r}_{a}-\mathbf{r}_{b}|$ is the distance between the two atoms
$a$ and $b$ inside the molecule, and with $j_{0}(x)=\sin(x)/x$ being the
zeroth-order spherical Bessel function.

The total part reduces to the total structure factor:

$F_{ab}(k,t=0)=w_{ab}(k)+\sqrt{\rho_{a}\rho_{b}}\int
d\mathbf{r}\exp(i\mathbf{k.r})\left[g_{ab}(r)-1\right]$ (16)

the latter which appears in the Debye expression for scattering intensity [49,
50]. We note that, for like atoms, this expression reduce to that of the usual
static structure factor

$S_{aa}(k)=1+\rho_{a}\int
d\mathbf{r}\exp(i\mathbf{k.r})\left[g_{aa}(r)-1\right]$ (17)

The relations to the dynamical KBI are directly derived from above relations
and Eqs.(12,13):

$F_{ab}^{(s)}(k=0,t)=V_{a}$ (18) $F_{ab}^{(d)}(k=0,t)=K_{ab}$ (19)

###  2.3 The dynamical structure factor

The dynamical structure factors are the time-Fourier transforms of the 3 types
of van Hove functions, generically defined as:

$S_{ab}^{(\bullet)}(k,\omega)=\int_{-\infty}^{+\infty}dt\exp(i\omega
t)F_{ab}^{(\bullet)}(k,t)$ (20)

where the ($\bullet$) stands for any of the (t,d,s) symbols.

The dynamical structure factors are interesting from several point of view.
Firstly and most importantly, they appear naturally in the theoretical
approaches, such as the Mori-Zwanzig formalism, since the operational form
involved both the $r$ Fourier transform and the time Laplace transform. The
resulting function $S(k,z)$ is related to the dynamical structure factor
$S(k,\omega)$ through the well known relation [47]:

$S^{(\bullet)}(k,\omega)=\lim_{\epsilon\rightarrow
0}\frac{1}{\pi}\mbox{Re}\left[S^{(\bullet)}(k,z=\omega+i\epsilon)\right]$ (21)

We will not expand further on these aspects in this work in the present
context.

Secondly, dynamical structure factors allow to make contact with molecular
hydrodynamics. This is perhaps the most intriguing aspect of liquids, that the
small $k$-range from k$=0$ to the main atom-atom peak around $k\approx
1.5-2\mathring{A}^{-1}$ covers in fact the entire spatial range from atom size
to macroscopic size. In computer simulation with system size $L$, the largest
k value corresponds to $k_{L}=2\pi/L$. For typical size $L\approx
40\mathring{A},$this gives $k_{L}\approx 0.12$, which is close enough to k=0.
In other words, it is not unexpected that hydrodynamic modes, such as the
Rayleigh and Brillouin sound modes could be covered within such tiny simulated
scales. With simulation times of $t=100$ps, the smallest frequency is
$\omega\approx 0.0$6Ghz , which is much smaller than $\omega\approx
0.8-1.2$Ghz where the Brillouin peak is found. In other words, one could
extract sound speed for model molecular liquids by finding if the
corresponding low $k$ low $\omega$ peak is obtained from calculated
$S(k,\omega)$.

Finally, it is $S(k,\omega)$ which is directly obtained from scattering
experiments, mostly through light scattering. Such experiments, however,
cannot detect small molecules such as water. They are more appropriate for
nano-sized molecules, hence present little interest in this work.

###  2.4 Dynamical scattering intensities

It is straightforward to extend the Debye formula for the static scattering
intensity $I(k)$ [49, 50] to dynamical equivalents $I(k,t)$ and $I(k,\omega)$.
This way, one can report calculated intensities for x-ray and neutron
experiments, using the static form atomic factors $f_{a}$, even if no real
experiment can obtain these intensities at present. For x-ray scattering in a
single component system, one has:

$I(k,t)=r_{0}^{2}\rho\sum_{a,b}f_{a}(k)f_{b}(k)F_{ab}(k,t)$ (22)
$I(k,\omega)=r_{0}^{2}\rho\sum_{a,b}f_{a}(k)f_{b}(k)S_{ab}(k,\omega)$ (23)

where $r_{0}=0.2818\mathring{A}$ is the electronic radius and $\rho$ the
molecular number density.

##  3 Molecular models, simulation details and methodology

The simulations were performed with the GROMACS program package [51]. For each
system we followed the same protocol. The initial random configurations of
2048 molecules were obtained by the Packmol program [52], which were
subsequently energy minimized and equilibrated in the NPT ensemble for 1 ns.
NPT production runs of nearly 1 ns were used to collect 10 000 independent
configurations for the calculation of all statistical properties. The liquids
were simulated in ambient conditions of T = 300 K and p = 1 bar, maintained by
the Nose-Hoover thermostat [53, 54] and Parinello-Rahman barostat [55, 56].
The former algorithm had the time constant of 0.1 ps, whereas the latter one
had the time constant of 1 ps. The integration algorithm of choice was leap-
frog REF, with the time step being 1 fs. The cut-off radius for short-range
interactions was 1.5 nm. For the long-range Coulomb interactions we employed
the particle mesh Ewald (PME) method [57], with FFT grid spacing of 0.12 nm
and interpolation order of 4. The constraints were handled with the LINCS
algorithm [58]. The forcefields used were: SPC/E for water [59], OPLS-UA for
ethanol [60], OPLS-AA for carbon tetrachloride [61] and a modified OPLS-UA for
acetone. The latter forcefield is based on the original OPLS-UA model by
Jorgensen and coworkers [62], which we modified to better reproduce some
thermodynamic and dynamic quantities. The details of the forcefield
modification are documented in detail in previous publications [63, 64].

###  3.1 Numerical evaluation of the dynamical functions

The LiquidLib package [65] was used to extract the total $G_{ab}(r,t)$ and
self $G_{ab}^{(s)}(r,t)$ atom-atom van Hove functions, directly from the
Gromacs trajectory file. We have modified the code in order to obtain
dimensionless functions, and to include the unlike atom self van Hove function
Eq.(7) among the usual like atom self van Hove function Eq.(4). This is an
important step, since it allows to separate the unlike atom distinct van Hove
function, which, in turn, allows to fully understand separately the intra-
molecular and inter-molecular dynamics of all atoms in molecular liquids.

The LiquidLib package allows to sample the van Hove functions in the same
$r$-grid as the static correlation function $g_{ab}(r)$, but with the user’s
choice for the time grid. In order to avoid data storage burden, we have
computed the time dependence in 3 different samplings. The first sampling
consists in 10 time-points from 0 to 1ps, spaced by 0.1ps. The second grid is
from 0 to 10ps, spaced by 1ps, and the final grid from 0 to 100ps, spaced by
10ps. This is motivated by the 3 facts. Firstly, the time decays are quite
fast and all functions are significantly decayed to their limits ($1$ for the
total function and $0$ for the self part) by 100ps. However, this is not true
for small r values, specially those within the core part. The reason for this
is detailed in the Results section below. Secondly, most interesting changes
occur within the first ps, and the second grid compensates for any slower
dynamics occurring beyond 1ps. Lastly, the sampling near $r=0$ are quite noisy
and demand excessively long runs. This is more serious at small times than
longer ones. In order to minimise computational times, we have fitted the
initial time decays to a Gaussian function.

While the $r$-Fourier transforms are made exactly as for the static case,
using fast Fourier techniques as in our previous work [2, 25], the time-
Fourier transforms require interpolating the time dependence from the
different 3 time grids discussed above. We have used a time grid of 0.1ps. In
addition, care must be taken for time functions for $r$-values inside the
core, since these are not decayed by t=100ps. In such case, we have used the
following trick. We use the observed empirical property that all time
correlations decay exponentially at large times, specifically those in the 3rd
window between 10ps and 100ps. The functions are fitted to an exponential
decay which covers mostly the large times. Then the difference between the
data and the exponential fit is a short ranged function which can be
numerically Fourier transformed. Then, the Lorentzian, corresponding to the
exact Fourier transform of the exponential, is added to the numerical
transform, in order to obtain the total time Fourier transform.

For large $r$ values, the van Hove data is generally noisy for all times.
Interestingly, in all such cases, the global $t$-decay is found to be
exponential. We therefore replaced these functions by their exponential fit.
When the fitting does not cover properly the small times, which happens in the
intermediate $r$-values, then only the tail region is replaced by an
exponential. This way the entire procedure can be automated.

##  4 Results

We are interested in reporting details of various dynamical correlation
functions, and find details which can relate to their local structural and
relaxation features.

###  4.1 Simple disorder liquids

In this section we present dynamical correlations for carbon tetrachloride and
acetone. Both are polar liquids, but CCl4 is considered less polar than
acetone. Both liquids have polar order, but they do not form labile clusters
such as that we consider for complex liquids. For each liquid we present the
self part and the distinct or total part of the dynamical correlation
functions, for typical 2 pairs of sites. For each function, we represent all
the 29 times, covering the range [0,1ps], [0,10ps] and [0,100ps], with 10
points in each interval. We observe that the times decays are always
monotonous, with no-crossover. Hence, the curves in the figures are not
labeled.

####  4.1.1 Apolar or weakly polar liquid: Carbon tetrachloride

CCl4 is a polar molecule, but the force field model uses small partial
charges, which mimic the fact that the molecule is polarisable. The central
carbon has a charge of +0.248, while the Cl sites share equally the opposite
charge. In terms of charge ordering, it become noticeable only when the charge
is above 0.6 [66]. Therefore CCl4 can be considered as simple disorder liquid.

Fig.1 shows the carbon-carbon self van Hove function $G_{CC}^{(s)}(r,t)$ in
the left panel and the corresponding self-intermediate scattering function
$F_{CC}^{(s)}(k,t)$ in the inset, as well as the corresponding functions for
the carbon tetrachloride dynamical correlations in the right panel for
$G_{CCl_{4}}^{(s)}(r,t)$ and its inset for $F_{CCl_{4}}^{(s)}(k,t)$. It can be
seen that the time decays are always monotonous in both representations,
representing the spreading of the curves as function of time.

Figure 1: CCl4: comparison of Self correlations $G^{(s)}(r,t)$ and
$F^{(s)}(k,t)$ (inset) for carbon-carbon (left panel) and carbon-chloride
(right panel) pairs of sites. The lines correspond to the 29 times selected
(see text for curve plotting conventions). Note the vertical log-scale for the
main panel.

The most notable features are the Dirac delta functions, one at $r=0$ for
$G_{CC}^{(s)}(r,t=0)$ (left panel), whose Fourier transform (FT) is the black
horizontal line in the inset, and the other at $r_{d}=2\mathring{A}$ for the
C-Cl intermolecular distance (right panel), whose FT is $j_{0}(kr_{d})$ leads
to the marked oscillations observed in the inset. Another interesting feature
is that, although the $G^{(s)}(r,t)$ look Gaussian-like, they cannot be well
fitted to Gaussians. This feature invalidates the usual Gaussian approximation
$G^{(s)}(r,t)=\exp(-r^{2}/(4Dt)/(4\pi Dt)^{3/2}$ [47, 67, 68]. This is equally
true of the small-$k$ Gaussian approximation of
$F^{(s)}(k,t)\approx\exp(-Dk^{2}t)$.

Fig.2 shows the distinct part of the van Hove function $G^{(d)}(r,t)$ between
the carbon atoms (left panel) and the carbon-tetrachloride in the right panel.
Basically, these curves look like usual $g(r)$, but damped in time, as all the
curves relax to 1 with time.

Figure 2: CCl4: comparison of distinct van Hove functions $G^{(d)}(r,t)$ for
carbon-carbon (left panel) and carbon-chloride (right panel) pairs of sites.

The most notable feature is the shift of the first peak of
$G_{CCl_{4}}^{(d)}(r,t)$ to larger $r$-values after $t=1$ps, while the second
peak is totally damped. It indicates a remarkable short time decorrelation of
the central carbon with respect to the 4 LJ sites within this time range of
$1$ps. It is equally worth noting that the replacement of the central particle
at the core is not fully relaxed even after 100ps. It shows a slow diffusion
of the molecule from its initial position.

Fig.3 shows the total intermediate scattering function $F^{(t)}(k,t)$ for the
same 2 previous cases. These curves show remarkable features. First of all, we
notice that at $t=0$, $F_{CC}^{(t)}(k,t=0)$ is exactly the static structure
factor $S_{CC}(k)$, with the standard definition
$S_{ab}(k)=\delta_{ab}+\rho\int
d\mathbf{r}\exp(i\mathbf{k.r)}\left[g_{ab}(r)-1\right]$. However, in the
present case, the asymptote term $\delta_{ab}$ is related to the intra-
molecular correlation, and since $G^{s}(r,t=0)=\delta(r)/\rho$, it is this
term which gives the $\delta_{ab}$ term, which is the asymptote 1 in case of
the CC correlations.

Figure 3: CCl4: comparison of total intermediate scattering functions
$F^{(s)}(k,t)$ for carbon-carbon (left panel) and carbon-chloride (right
panel) pairs of sites.

The most important feature is that the plot shows clearly the decay of the
asymptote with time. The large oscillations in the $F_{CCl_{4}}^{(t)}(k,t)$
are coming from the same self part $j_{0}(kr_{d})$ that is shown in the right
inset of Fig.1.

Fig.4 show the total dynamical structure factor $S^{(t)}(k,\omega)$ and the
self part in the inset, for the same atoms as above. The frequencies $\omega$
correspond to the times on the time grid described earlier, using the formula
$\omega=2\pi/t$, and they are in units of ps-1 which is also Ghz. The explicit
values are There are 2 remarkable features. The first is that the main peak of
$S(k,\omega)$ coincides with that of $F(k,t)$ in both cases. The second
feature is that the self part looks very much like the usual Lorentzian
approximation [47] $S^{(s)}(k,\omega)=Dk^{2}/\pi(\omega^{2}+(Dk^{2})^{2})$,
but cannot be fully fitted to this simple form.

Figure 4: CCl4: comparison of total dynamical structure factors
$S^{(s)}(k,\omega)$ and corresponding self parts $S^{(s)}(k,\omega)$ (inset)
for carbon-carbon (left panel) and carbon-chloride (right panel) pairs of
sites. The lines correspond to the different $\omega=2\pi/t$ frequencies
corresponding to the time grid in the $G(r,t)$ plots.

####  4.1.2 Polar liquid: acetone

Acetone is polar molecule, with dipole moment of $10\times
10^{20}$C$\mathring{A}$. Yet, there is no micro-structure formation in this
liquid, other than the usual fluctuations, some related in part to dipole
correlations. We can consider this liquid as a simple disorder liquid.

Fig.5 shows the oxygen-oxygen self van Hove function $G_{OO}^{(s)}(r,t)$ in
the left panel and the corresponding self-intermediate scattering function
$F_{OO}^{(s)}(k,t)$ in the inset, as well as the corresponding functions for
the oxygen-carbon dynamical correlations in the right panel for
$G_{OC}^{(s)}(r,t)$ and its inset for $F_{OC}^{(s)}(k,t)$. These curves have
strong similarities with those for CCL4 in Fig.1.

Figure 5: Acetone: comparison of Self correlations $G^{(s)}(r,t)$ and
$F^{(s)}(k,t)$ (inset) for oxygen-oxygen (left panel) and oxygen-carbon (right
panel) pairs of sites. The lines correspond to the 29 times selected (see text
for curve plotting conventions). Note the vertical log-scale for the main
panel.

They also show characteristic differences. The $r$-decay and the time decay in
$r$-space are faster for acetone than for CCl4. However, since all
$G^{(s)}(r,t)$ integrate to 1 (that is $F^{(s)}(k=0,t)=1),$it means that self
correlations are longer ranged for acetone than CCl4, and especially at larger
times. We can attribute this to dipole correlation with neighbours. However,
it can be seen from the inset that $F_{OO}^{(s)}(k,t)$ decay faster for
acetone than for CCl4. We speculatively attribute this to CCl4 being more
“spherical” than acetone, possibly because of the dipole moment of the latter.

Fig.6 shows the time decay of distinct van Hove correlations for OO and OC.
These are basically the equivalent of time dependent pair correlation
functions since $g(r)=G^{(d)}(r,t=0)$, which is why they are worth analyzing.
The comparison with CCl4 in Fig.2 shows remarkable differences. For acetone,
the main peaks are much smaller and the spatial oscillations much less
pronounced (faster decay of spatial correlations). This is surprising since
both liquids are dense. Again, this can be attributed to dipole correlations,
using the following argument. The maintenance of dipole alignments can be
achieved by several molecular positions, hence, for entropic reasons, there
would be a faster decorrelation between sites, while the dipole correlations
are maintained for longer times.

Figure 6: Acetone: comparison of distinct van Hove functions $G^{(d)}(r,t)$
for oxygen-oxygen (left panel) and oxygen-carbon (right panel) pairs of sites.

We note that the first peak spreading with time in the cross site correlation
is very similar to that of CCl4.

Fig.7 shows the total intermediate scattering function for the same two pairs
of sites. We observe the same faster temporal decay for acetone thane for
CCl4, as that observed for the self part in the insets of Fig.5.

Figure 7: Acetone: comparison of total intermediate scattering functions
$F^{(s)}(k,t)$ for oxygen-oxygen (left panel) and oxygen-carbon (right panel)
pairs of sites.

The oscillations of the OC cross-correlations are also larger for acetone than
for CCl4. But this can be interpreted as a Fourier transform mathematical
consequence of the core and first peak parts of the $G^{(d)}$function: the
higher the first peak, the tighter the oscillations in $k$-space.

Fig.8 shows the total and self(inset) dynamical structure factors. The most
prominent differences with CCl4 of Fig.4 are the absence of second peak around
$k\approx 2\mathring{A}$, and the small negative part near $k\approx 0$. The
negative part is in fact a numerical artifact coming from the small negative
part near $k=0$ of the $F(k,t)$ in Fig.7. This negative part is coming from a
known problem in g(r), hence G(r,t), asymptote in simulations: they do not
tend exactly to 1, but slightly below [69]. This problem is well known for the
Kirkwood-Buff calculation in computer simulations [27, 30, 31, 32, 33].

Figure 8: Acetone: comparison of the total dynamical structure factors
$S^{(s)}(k,\omega)$ and corresponding self parts $S^{(s)}(k,\omega)$ (inset)
for oxygen-oxygen (left panel) and oxygen-carbon (right panel) pairs of sites.

Aside from these, both sets of curves for CCl4 and acetone look quite similar.

###  4.2 Complex disorder liquids

Herein, we study water and ethanol, which are both archetypical examples of
hydrogen bonded liquids. We do not expect the self parts to show much
differences, unless indirectly, since most of differences are related to the
distinct part which contain the specific inter-molecular parts which make the
difference between strongly and weakly clustered liquids.

####  4.2.1 Water

Fig.9 shows the self van Hove correlations both in $r$ and $k$(inset) space.
The comparison with the simple disorder counter parts of CCl4 and acetone in
Figs[1,5], shows that there is quite a bit of resemblance with both of them.
Perhaps the most importance difference is the smaller height of the main
peaks, both at $r=0$ and $r=1\mathring{A}$ , and the faster decay in
$r$-space. This is an important feature, since, because $F^{(s)}(k=0,t)=1$, it
implies the existence of longer ranged $r$-correlations for water, which is an
indirect proof of the Hbond network persistence for this particular liquid,
over their absence for simple disorder liquids. Despite this difference, we
observe that the $F^{(s)}(r,t)$ are quite similar.

Figure 9: Water: comparison of Self correlations $G^{(s)}(r,t)$ and
$F^{(s)}(k,t)$ (inset) for oxygen-oxygen (left panel) and oxygen-hydrogen
(right panel) pairs of sites. The lines correspond to the 29 times selected
(see text for curve plotting conventions). Note the vertical log-scale for the
main panel.

This is, speculatively, an indication that water is a 2-state liquid: it
appears as a r-space long range correlated liquid, and at the same time a
disordered one in k-space. This is probably an typical difference, specially
when comparing with the $F^{(s)}$ for acetone in Fig.5, which also appear as
long ranged in r-space (as discussed previously sub-section 4.1.2), but
differs in k-space.

Fig.10 shows the distinct dynamical correlations. Since
$g_{OO}(r)=G_{OO}^{(d)}(r,t=0)$ we can observe the typical pair correlation of
water, which differs considerably from that of simple disorder liquids, such
as illustrated by CCl4 and acetone, and its decay with time. The most striking
difference is the rapid decay of the O-O contact peak at $r\approx
3\mathring{A}$ and the O-H Hbond peak at $r\approx 2\mathring{A}$ . This
appears at first quite contradictory, since on would think that the Hbond
related peaks should show more persistence than the contact peak of simple
disorder liquids. This in fact similar to that observed for the self-functions
above: what is important is not how rapidly the Hbond correlations are lost,
but how rapidly they are reestablished. The latter is better observed in
k-space (see below). Nevertheless, it appears that Hbonding has a very short
life time since it decays very fast, which we have already observed in our
previous work [70].

Figure 10: Comparison of distinct van Hove functions $G^{(d)}(r,t)$ for
oxygen-oxygen (left panel) and oxygen-hydrogen (right panel) pairs of sites.

The above analysis is further confirmed in Fig.11, which shows the total
intermediate scattering functions. We observe the typical split-peak feature
of the structure factor of water, since $S_{OO}(k)=F_{OO}^{(t)}(k,t=0)$. But
what seems interesting is that the shoulder peak at $k\approx
2\mathring{A}^{-1}$ ($r\approx 3\mathring{A}$) decays more slowly than the
Hbond peak at $k\approx 3\mathring{A}^{-1}$($r\approx 2\mathring{A}$). We have
recently proposed [71] that the shoulder peak is in fact the cluster peak for
water, which points to water as forming mostly a dimer, which is a direct
consequence of charge order. This is further confirmed by the fact that at
$r\approx 3\mathring{A}$we observe an anti-peak in the OH correlations.

Figure 11: Water: comparison of total intermediate scattering functions
$F^{(s)}(k,t)$ for oxygen-oxygen (left panel) and oxygen-hydrogen (right
panel) pairs of sites.

Interestingly, the asymmetry of the time decay of these 2 peaks is very
strongly reminiscent of their temperature dependence [72]: the Hbond peak is
lost more quickly by heating than the contact-cluster peak.

Fig.12 shows the dynamical structure factors for the OO and OH atom pairs. The
most striking feature is the small-k peak at $k\approx 0.3\mathring{A}^{-1}$,
which corresponds to $r\approx 20\mathring{A}$. This peak is even more
prominent for the OH correlations, indicating that its origin is mostly from
the Hbond correlations. This peak is more important than dual peak that we
observe at $k\approx 2\mathring{A}^{-1}$ and $k\approx 3\mathring{A}^{-1}$,
which are directly related to those in Fig.11. In fact, this new peak is the
dynamical equivalent of static pre-peak in the static structure factor.

Figure 12: Water: comparison of the total dynamical structure factors
$S^{(s)}(k,\omega)$ and corresponding self parts $S^{(s)}(k,\omega)$ (inset)
for oxygen-oxygen (left panel) and oxygen-hydrogen (right panel) pairs of
sites.

The distance size of $20\mathring{A}$ corresponds to a radius of
$10\mathring{A}$, which is about the range of the marked oscillations in the
$g_{OO}(r)$: the “3 peak structure” discussed in our previous work [73]. It
this represent a kinetic clustering, as opposed to the static clustering which
is observed through the pre-peak of the static structure factor. This analysis
is further confirmed in the the case of ethanol below.

####  4.2.2 Ethanol

Contrary to water, alcohols have a well defined scattering pre-peak [16, 18,
19, 17, 20, 8, 74, 75, 9]. Herein, we focus on the case of ethanol for
illustrative purposes.

Fig.13 shows the self parts of van Hove and intermediate scattering (insets)
functions for OO and OH correlations. The general features are not so much
different than that observed form the simple disorder liquids, which gives the
false impression that the decorrelation of a single ethanol molecule is
similar to that of CCl4 or acetone.

Figure 13: Ethanol: comparison of Self correlations $G^{(s)}(r,t)$ and
$F^{(s)}(k,t)$ (inset) for oxygen-oxygen (left panel) and oxygen-hydrogen
(right panel) pairs of sites. The lines correspond to the 29 times selected
(see text for curve plotting conventions). Note the vertical log-scale for the
main panel.

This is quite interesting, in the sense that water stands apart, even though
both water and alcohols are Hbonded liquids, since we already observed
distinct features at the level of the self dynamical correlations for this
particular liquid.

In order to find the specificity related to Hbonding, we turn towards the
distinct correlation functions in Fig.14. We observe several similarities with
water. The high first peak is a witness of Hbonding, just like water, and it
is also seen to decay very fast. A specificity of alcohols is the depletion
correlations after the main peak, which are at the origin of the alcohol pre-
peak [2, 25].

Figure 14: Ethanol: comparison of distinct van Hove functions $G^{(d)}(r,t)$
for oxygen-oxygen (left panel) and oxygen-hydrogen (right panel) pairs of
sites.

These depletion correlation of the second and higher neighbours are related to
the chain formation of the hydroxyl groups, and express the fact that the
corresponding neighbours are in reduced number with respect to full space
filling. It is seen that the depleted region re-emerge above 1 around
$r\approx 7-8\mathring{A}.$ These depletion correlations are seen to decay
more slowly, a feature more visible in the Fourier transform discussed below.

Fig.15 shows how the pre-peak at $k_{PP}\approx 0.75-0.8\mathring{A}^{-1}$
($r\approx 7-8\mathring{A}$) and Hbond peak at $k_{MP}\approx
3\mathring{A}^{-1}$ ($r\approx 2\mathring{A}$) of ethanol decay in time.
$k_{PP}$ corresponds to the depletion range observed in Fig.14, while the main
peak is exactly that of water and corresponds to the O-H Hbonding distance.
Just like water, the cluster pre-peak is seen to decay much more slowly than
the main peak. The rate of decay is seen to be even slower than the dimer
cluster peak of water.

Figure 15: Ethanol: comparison of total intermediate scattering functions
$F^{(s)}(k,t)$ for carbon-carbon (left panel) and carbon-chloride (right
panel) pairs of sites.

This is in line with the fact that alcohol have long lived chain clusters [76,
70].

Fig.16 shows the dynamical structure factors for the OO and OH atom pairs. We
observe a very high dynamical pre-peak at $k\approx 0.7-0.8\mathring{A}^{-1}$,
similar to water, but much higher in magnitude, which corresponds exactly to
the static pre-peak of ethanol observed above. In contrast to the case above,
we see that the peak at $k\approx 3\mathring{A}$ is much smaller. These
features highlight the dynamics of cluster formation in ethanol, and more
generally in alcohols.

Figure 16: Ethanol: comparison of the total dynamical structure factors
$S^{(s)}(k,\omega)$ and corresponding self parts $S^{(s)}(k,\omega)$ (inset)
for oxygen-oxygen (left panel) and oxygen-hydrogen (right panel) pairs of
sites.

This analysis reveals that the dynamical structure factors are better
indicators of the cluster dynamics than the static structure factors, and
highlight the underlying kinetics.

##  5 Conclusion

In this work, we have conducted an analysis of the dynamics of liquids are
revealed by the pair correlation functions, both in $r$ and $k$ space, as well
in time and frequency. To this end, the van Hove, intermediate scattering
functions and dynamical structure factors have been calculated in computer
simulations of model liquids. Having in mind the structural differences
between simple disorder liquids and clustering liquids, we have studied CCl4
and acetone for the first case, water and ethanol for the second case. Our
study reveals that many important structural differences between the two
categories of liquids exist, as exemplified by the analysis of the various
dynamical pair correlation functions. The most important difference is the
existence of kinetic processes due to clustering, which are prominently found
in the small $k$ part of atom-atom dynamical structure factors $S(k,\omega)$
of complex disorder liquids, and are totally absent from the simple disorder
liquids. This feature is not so much perceptible from the static structure
factor $S(k)$ and intermediate scattering functions $F(k,t)$, as exemplified
for the case of water. Since the x-ray radiation scattering experiments are
related to the static structure factors, it appear important to refer to
neutron scattering experiments in order to gain access to the dynamical
scattering intensity. We expect that this work will motivate research in this
direction.

## Acknowledgments

This work has been supported in part by the Croatian Science Foundation under
the project UIP-2017-05-1863 “Dynamics in micro-segregated systems”.

## References

  * [1] Perera, A.  From solutions to molecular emulsions.  _Pure and Applied Chemistry_ , 2016.  _88_ , 189.
  * [2] Požar, M.; Lovrinčević, B.; Zoranić, L.; Primorac, T.; Sokolić, F. and Perera, A.  Micro-heterogeneity versus clustering in binary mixtures of ethanol with water or alkanes.  _Physical Chemistry Chemical Physics_ , 2016.  _18_ , 23971.
  * [3] Dixit, S.; Crain, J.; Poon, W.; Finney, J. and Soper, A.  Molecular segregation observed in a concentrated alcohol-water solution.  _Nature_ , 2002.  _416_ , 829.
  * [4] Guo, J.H.; Luo, Y.; Augustsson, A.; Kashtanov, S.; Rubensson, J.E.; Shuh, D.K.; Ågren, H. and Nordgren, J.  Molecular structure of alcohol-water mixtures.  _Phys. Rev. Lett._ , 2003.  _91_ , 157401.
  * [5] Warren, B.  X-ray diffraction in long chain liquids.  _Phys. Rev._ , 1933.  _44_ , 969.
  * [6] Pierce, W. and MacMillan, D.  X-ray studies on liquids: the inner peak for alcohols and acids.  _Journal of the American Chemical Society_ , 1938.  _60_ , 779.
  * [7] Benmore, C. and Loh, Y.  The structure of liquid ethanol: A neutron diffraction and molecular dynamics study.  _The Journal of Chemical Physics_ , 2000.  _112_ , 5877.
  * [8] Tomšič, M.; Jamnik, A.; Fritz-Popovski, G.; Glatter, O. and Vlček, L.  Structural properties of pure simple alcohols from ethanol, propanol, butanol, pentanol, to hexanol: Comparing monte carlo simulations with experimental saxs data.  _The Journal of Physical Chemistry B_ , 2007.  _111_ , 1738.
  * [9] Cerar, J.; Lajovic, A.; Jamnik, A. and Tomšič, M.  Performance of various models in structural characterization of n-butanol: Molecular dynamics and x-ray scattering studies.  _Journal of Molecular Liquids_ , 2017.  _229_ , 346 .
  * [10] Perera, A.; Sokolić, F. and Zoranić, L.  Microstructure of neat alcohols.  _Physical Review E_ , 2007.  _75_ , 060502(R).
  * [11] Prévost, S.; Gradzielski, M. and Zemb, T.  Self-assembly, phase behaviour and structural behaviour as observed by scattering for classical and non-classical microemulsions.  _Advances in Colloid and Interface Science_ , 2017.  _247_ , 374 .
  * [12] Goddeeris, C.; Cuppo, F.; Reynaers, H.; Bouwman, W. and den Mooter, G.V.  Light scattering measurements on microemulsions: Estimation of droplet sizes.  _International Journal of Pharmaceutics_ , 2006.  _312_ , 187 .
  * [13] Teubner, M. and Strey, R.  Origin of the scattering peak in microemulsions.  _The Journal of Chemical Physics_ , 1987.  _87_(5), 3195.
  * [14] Kundu, N.; Banik, D. and Sarkar, N.  Self-assembly of amphiphiles into vesicles and fibrils: Investigation of structure and dynamics using spectroscopy and microscopy techniques.  _Langmuir_ , 2018.  _34_(39), 11637.  PMID: 29544249.
  * [15] Magini, M.; Paschina, G. and Piccaluga, G.  On the structure of methyl alcohol at room temperature.  _The Journal of Chemical Physics_ , 1982.  _77_ , 2051.
  * [16] Narten, A. and Habenschuss, A.  Hydrogen bonding in liquid methanol and ethanol determined by x-ray diffraction.  _The Journal of Chemical Physics_ , 1984.  _80_ , 3387.
  * [17] Vahvaselkä, K.S.; Serimaa, R. and Torkkeli, M.  Determination of liquid structures of the primary alcohols methanol, ethanol, 1-propanol, 1-butanol and 1-octanol by x-ray scattering.  _Journal of Applied Crystallography_ , 1995.  _28_ , 189.
  * [18] Sarkar, S. and Joarder, R.N.  Molecular clusters and correlations in liquid methanol at room temperature.  _The Journal of Chemical Physics_ , 1993.  _99_ , 2032.
  * [19] Sarkar, S. and Joarder, R.N.  Molecular clusters in liquid ethanol at room temperature.  _The Journal of Chemical Physics_ , 1994.  _100_ , 5118.
  * [20] Karmakar, A.; Krishna, P. and Joarder, R.  On the structure function of liquid alcohols at small wave numbers and signature of hydrogen-bonded clusters in the liquid state.  _Physics Letters A_ , 1999.  _253_ , 207.
  * [21] Nishikawa, K. and Iijima, T.  Small-angle x-ray scattering study of fluctuations in ethanol and water mixtures.  _The Journal of Physical Chemistry_ , 1993.  _97_ , 10824.
  * [22] Akiyama, I.; Ogawa, M.; Takase, K.; Takamuku, T.; Yamaguchi, T. and Ohtori, N.  Liquid structure of 1-propanol by molecular dynamics simulations and x-ray scattering.  _Journal of Solution Chemistry_ , 2004.  _33_ , 797.
  * [23] Takamuku, T.; Maruyama, H.; Watanabe, K. and Yamaguchi, T.  Structure of 1-propanol-water mixtures investigated by large-angle x-ray scattering technique.  _Journal of Solution Chemistry_ , 2004.  _33_ , 641.
  * [24] Almásy, L.; Kuklin, A.; Požar, M.; Baptista, A. and Perera, A.  Microscopic origin of the scattering pre-peak in aqueous propylamine mixtures: X-ray and neutron experiments versus simulations.  _Phys. Chem. Chem. Phys._ , 2019.  _21_ , 9317.
  * [25] Požar, M.; Bolle, J.; Sternemann, C. and Perera, A.  On the x-ray scattering pre-peak of linear mono-ols and the related microstructure from computer simulations.  _J. Phys. Chem. B_ , 2020.  _124_(38), 8358.
  * [26] Anisimov, D.S.D.A.I.I.K.Y.M.A. and Sengers, J.V.  Mesoscale inhomogeneities in aqueous solutions of 3-methylpyridine and tertiary butyl alcohol.  _Journal of Chemical and Engineering Data_ , 2011.  _56_(4), 1238.
  * [27] Perera, A. and Sokolić, F.  Modeling nonionic aqueous solutions: The acetone-water mixture.  _The Journal of Chemical Physics_ , 2004.  _121_ , 11272.
  * [28] Ploetz, E. and Smith, P.  A kirkwood-buff force field for the aromatic amino acids.  _Physical Chemistry Chemical Physics_ , 2011.  _13_ , 18154.
  * [29] Gupta, R. and Patey, G.N.  Aggregation in dilute aqueous tert-butyl alcohol solutions: Insights from large-scale simulations.  _The Journal of Chemical Physics_ , 2012.  _137_(3), 034509.
  * [30] Ganguly, P. and van der Vegt, N.  Convergence of sampling kirkwood-buff integrals of aqueous solutions with molecular dynamics simulations.  _Journal of Chemical Theory and Computation_ , 2013.  _9_(3), 1347.  PMID: 26587597.
  * [31] Krüger, P.; Schnell, S.; Bedeaux, D.; Kjelstrup, S.; Vlugt, T. and Simon, J.M.  Kirkwood-buff integrals for finite volumes.  _The Journal of Physical Chemistry Letters_ , 2013.  _4_(2), 235.  PMID: 26283427.
  * [32] Milzetti, J.; Nayar, D. and van der Vegt, N.  Convergence of kirkwood-buff integrals of ideal and nonideal aqueous solutions using molecular dynamics simulations.  _The Journal of Physical Chemistry B_ , 2018.  _122_(21), 5515.  PMID: 29342355.
  * [33] Dawass, N.; Krüger, P.; Schnell, S.; Bedeaux, D.; Kjelstrup, S.; Simon, J. and Vlugt, T.  Finite-size effects of kirkwood-buff integrals from molecular simulations.  _Molecular Simulation_ , 2018.  _44_ , 599.
  * [34] Rahman, A.  Density fluctuations in liquid rubidium. ii. molecular-dynamics calculations.  _Phys. Rev. A_ , 1974.  _9_ , 1667.
  * [35] Yoshida, F.  Dynamical structure factor in liquid rubidium.  _Journal of Physics F: Metal Physics_ , 1978.  _8_(3), 411.
  * [36] Sjogren, L.  Short-wavelength density fluctuations in liquid metals.  _Journal of Physics C: Solid State Physics_ , 1979.  _12_(3), 425.
  * [37] Ebbsio, I.; Kinell, T. and Waller, I.  The dynamical structure factor for liquid aluminium.  _Journal of Physics C: Solid State Physics_ , 1980.  _13_(10), 1865.
  * [38] Kahol, P.K.; Chaturvedi, D.K. and Pathak, K.N.  Dynamical structure factors in binary liquids. i. molten rbbr.  _Journal of Physics C: Solid State Physics_ , 1977.  _10_(21), 4181.
  * [39] Moe, N.E. and Ediger, M.D.  Calculation of the coherent dynamic structure factor of polyisoprene from molecular dynamics simulations.  _Phys. Rev. E_ , 1999.  _59_ , 623.
  * [40] Handle, P.H.; Rovigatti, L. and Sciortino, F.  $q$-independent slow dynamics in atomic and molecular systems.  _Phys. Rev. Lett._ , 2019.  _122_ , 175501.
  * [41] Wu, B.; Iwashita, T. and Egami, T.  Atomic dynamics in simple liquid: de gennes narrowing revisited.  _Phys. Rev. Lett._ , 2018.  _120_ , 135502.
  * [42] Noguere, G.; Scotta, J.P.; Xu, S.; Farhi, E.; Ollivier, J.; Calzavarra, Y.; Rols, S.; Koza, M. and Marquez Damian, J.I.  Temperature-dependent dynamic structure factors for liquid water inferred from inelastic neutron scattering measurements.  _The Journal of Chemical Physics_ , 2021.  _155_(2).  024502\.
  * [43] Alvarez, F.; Arbe, A. and Colmenero, J.  Unraveling the coherent dynamic structure factor of liquid water at the mesoscale by molecular dynamics simulations.  _The Journal of Chemical Physics_ , 2021.  _155_(24).  244509\.
  * [44] Kežić, B. and Perera, A.  Towards a more accurate reference interaction site model integral equation theory for molecular liquids.  _The Journal of Chemical Physics_ , 2011.  _135_(23).  234104\.
  * [45] Gray, C.G. and Gubbins, K.E.  _Theory of molecular fluids_.  Oxford University Press, New York, 1984.
  * [46] Blum, L. and Torruella, A., J.  Invariant expansion for two-body correlations: Thermodynamic functions, scattering, and the ornstein–zernike equation.  _J. Chem. Phys._ , 2003.  _56_(1), 303.
  * [47] Hansen, J.P. and McDonald, I.  _Theory of Simple Liquids_.  Academic Press, Elsevier, Amsterdam, 3rd edition, 2006.
  * [48] Hirata, F.  _Theory of Molecular Liquids_ , pages 1–60.  Springer Netherlands, Dordrecht, 2003.
  * [49] Debye, P.  Zerstreuung von rontgenstrahlen.  _Annalen der Physik_ , 1915.  _351_ , 809.
  * [50] Debye, P.  Scattering of x-rays.  In _The collected papers of Peter J.W. Debye_. Interscience Publishers, 1954.
  * [51] Pronk, S.; Páll, S.; Schulz, R.; Larsson, P.; Bjelkmar, P.; Apostolov, R.; Shirts, M.; Smith, J.; Kasson, P.; van der Spoel, D.; Hess, B. and Lindahl, E.  Gromacs 4.5: a high-throughput and highly parallel open source molecular simulation toolkit.  _Bioinformatics_ , 2013.  _29_ , 845.
  * [52] Martínez, J. and Martínez, L.  Packing optimization for automated generation of complex system’s initial configurations for molecular dynamics and docking.  _Journal of Computational Chemistry_ , 2003.  _24_ , 819.
  * [53] Nose, S.  A molecular dynamics method for simulations in the canonical ensemble.  _Molecular Physics_ , 1984.  _52_ , 255.
  * [54] Hoover, W.  Canonical dynamics: Equilibrium phase-space distributions.  _Physical Review A_ , 1985.  _31_ , 1695.
  * [55] Parrinello, M. and Rahman, A.  Crystal structure and pair potentials: A molecular-dynamics study.  _Physical Review Letters_ , 1980.  _45_ , 1196.
  * [56] Parrinello, M. and Rahman, A.  Polymorphic transitions in single crystals: A new molecular dynamics method.  _Journal of Applied Physics_ , 1981.  _52_ , 7182.
  * [57] Darden, T.; York, D. and Pedersen, L.  Particle mesh ewald: An n*log(n) method for ewald sums in large systems.  _The Journal of Chemical Physics_ , 1993.  _98_ , 10089.
  * [58] Hess, B.; Bekker, H.; Berendsen, H. and Fraaije, J.  Lincs: A linear constraint solver for molecular simulations.  _Journal of Computational Chemistry_ , 1997.  _18_ , 1463.
  * [59] Berendsen, H.J.C.; Grigera, J.R. and Straatsma, T.P.  The missing term in effective pair potentials.  _The Journal of Physical Chemistry_ , 1987.  _91_(24), 6269.
  * [60] Jorgensen, W.  Optimized intermolecular potential functions for liquid alcohols.  _The Journal of Physical Chemistry_ , 1986.  _90_(7), 1276.
  * [61] Duffy, E.; Severance, D. and Jorgensen, W.  Solvent effects on the barrier to isomerization for a tertiary amide from ab initio and monte carlo calculations.  _Journal of the American Chemical Society_ , 1992.  _114_(19), 7535.
  * [62] Jorgensen, W.; Briggs, J. and Contreras, M.  Relative partition coefficients for organic solutes from fluid simulations.  _The Journal of Physical Chemistry_ , 1990.  _94_(4), 1683.
  * [63] Požar, M. and Zoranić, L.  The structuring in mixtures with acetone as the common solvent.  _Physics and Chemistry of Liquids_ , 2020.  _58_(2), 184.
  * [64] Požar, M.; Jukić, I. and B., L.  Thermodynamic, structural and dynamic properties of selected non-associative neat liquids.  _Journal of Physics: Condensed Matter_ , 2020.  _32_(405101).
  * [65] Walter, N.; Jaiswal, A.; Cai, Z. and Zhang, Y.  Liquidlib: A comprehensive toolbox for analyzing classical and ab initio molecular dynamics simulations of liquids and liquid-like matter with applications to neutron scattering experiments.  _Computer Physics Communications_ , 2018.  _228_ , 209 .
  * [66] Perera, A. and Mazighi, R.  Simple and complex forms of disorder in ionic liquids.  _Journal of Molecular Liquids_ , 2015.  _210_ , 243.  Mesoscopic structure and dynamics in ionic liquids.
  * [67] Boon, J.P. and Yip, S.  _Molecular Hydrodynamics_.  Dover Publications, Inc., New York, 1980.
  * [68] Berne, B.J. and Pecora, R.  _Dynamic Light Scattering: With Applications to Chemistry, Biology, and Physics_.  John Wiley and Sons, Inc., 2000.
  * [69] Lebowitz, J. and Percus, J.  Long-range correlations in a closed system with applications to nonuniform fluids.  _Physical Review_ , 1961.  _122_(6), 1675.
  * [70] Jukić, I.; Požar, M.; Lovrinčević, B. and Perera, A.  Universal features in the lifetime distribution of clusters in hydrogen-bonding liquids.  _Phys. Chem. Chem. Phys._ , 2021.  _23_ , 19537.
  * [71] Lovrinčević, B.; Požar, M.; Jukić, I. and Perera, A.  On the role charge ordering in the dynamics of cluster formation in associated liquids.  _ChemRxiv. Cambridge: Cambridge Open Engage_ , 2022.  This content is a preprint and has not been peer-reviewed.
  * [72] Amann-Winkel, K.; Bellissent-Funel, M.C.; Bove, L.E.; Loerting, T.; Nilsson, A.; Paciaroni, A.; Schlesinger, D. and Skinner, L.  X-ray and neutron scattering of water.  _Chemical Reviews_ , 2016.  _116_(13), 7570.
  * [73] Perera, A.  On the microscopic structure of liquid water.  _Molecular Physics_ , 2011.  _109_(20), 2433.
  * [74] Vrhovšek, A.; Gereben, O.; Jamnik, A. and Pusztai, L.  Hydrogen bonding and molecular aggregates in liquid methanol, ethanol, and 1-propanol.  _The Journal of Physical Chemistry B_ , 2011.  _115_ , 13473.
  * [75] Sillrén, P.; Swenson, J.; Mattsson, J.; Bowron, D. and Matic, A.  The temperature dependent structure of liquid 1-propanol as studied by neutron diffraction and epsr simulations.  _The Journal of Chemical Physics_ , 2013.  _138_ , 214501.
  * [76] Fujii, A.; Sugawara, N.; Hsu, P.J.; Shimamori, T.; Li, Y.C.; Hamashima, T. and Kuo, J.L.  Hydrogen bond network structures of protonated short-chain alcohol clusters.  _Phys. Chem. Chem. Phys._ , 2018.  _20_ , 14971.