# The influence of charge ordering in the microscopic structure of monohydroxy
alcohols

Martina Požar‡, Bernarda Lovrinčević‡, and Aurélien Perera† Corresponding
author<EMAIL_ADDRESS>
###### Abstract

While radiation scattering data provides insight inside the microstructure of
liquids, the Debye relation relating the scattering intensity $I(k)$ to the
atom-atom structure factors $S_{ab}(k)$ shows that, ultimately, it is these
individual structure correlation functions which contain the relevant
information about the micro-structure. However, these quantities are not
observables, except in few cases where one can invert the Debye relation in
order to obtain the structure functions. In the majority of other cases, the
need for model dependent computer simulations is unavoidable. The resulting
calculations reveal that the scattering pre-peak is the result of
cancellations between positive pre-peaks and negative anti-peaks contributions
from the atom-atom structure factors. What of systems where this cancellation
is such that it entirely suppresses the scattering pre-peak? One would be
tempted to falsely conclude that there is no uderlying micro-heterogeneity.
Hence, the structure functions appear as hidden variables, and it is important
to understand the relation between their features and the micro-structure of
the system. Through the computer simulation study of various mono-ols, ranging
from methanol to 1-nonanol, as well as the branched octanols, we show how the
features of the atom-atom pair correlation function $g_{ab}(r)$ affect that of
the structure factors $S_{ab}(k)$, and reveal that the micro-structure is
ultimately the result of the charge ordering between different atoms in the
system.

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

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

##  1 Introduction

Radiation scattering provides an experimental way of probing the micro-
structure of liquids and mixtures [1, 2, 3], but with disputable insights
about the exact nature of this microstructure. For instance, the intensity
$I(k)$ of scattering experiments on colloids are usually interpreted in terms
of the sole colloid structure factor $S(k)$ using the expression [4]:

$I(k)=\left[f_{c}(k)\right]^{2}S(k)$ (1)

where $f_{c}(k)$ is the form factor of the colloid, reflecting its shape, and
neglecting all other other molecules present in the solution, such as the
solvent, for example. In the case when the colloid is a micelle, the proper
expression would be the Debye formula [5, 6]:

$I(k)=\sum_{i,j}f_{i}(k)f_{j}(k)S_{ij}(k)$ (2)

where the sum would run over _all pairs of atoms_ $i$ and $j$ present in the
molecules which make the colloidal solution, and where the $f_{i}(k)$ are the
atoms form factors and $S_{ij}(k)$ the atom-atom structure factor, including
the intra-molecular contributions from the various molecules in the solution.
As to how Eq.(1) emerges from Eq.(2) is an open problem. For instance, one
could separate in Eq.(2) the sum over atoms which making up the molecules
found in the micellar colloids, and would require to neglect the other atom
contributions, namely that of the solvent, ions and co-ions, as well as the
cross-terms between these molecules and those of the colloidal micelles. It is
difficult to imagine how this could justified, and moreover how the “shape” of
the micelle would emerge from those terms retained in Eq.(2). Yet, it is an
experimental fact that both expressions are found equivalent in many cases.
Clearly, in view of the massive amount of data confirming Eq.(1), it would be
important to show the equivalence of these 2 equations.

Another similar problem concerns the radiation scattering data from aqueous
mixtures of _small_ surfactant like molecules, such as nonionic polyoxy-
ethylene monoalkyl ether surfactants. The experiments [7, 8] show a variety of
behaviour for the $I(k)$ at small $k$, namely Ornstein-Zernike (OZ) type
behaviour and Teubner-Strey (TS) type behaviour. The former has a large peak
at $k=0$, while the latter has a well defined pre-peak. One might conclude
that the corresponding latter mixtures may have aggregated structures which
are responsible for the pre-peak, while the former may not have such
structures. Yet, computer simulations of aqueous mixtures with simpler
alcohols, such as tertbutyl alcohol, for example, shows micro-domain
separation [9, 10, 11, 12, 13]. In this context, it is quite clear that, while
various types of aggregative structure may exist, these may not necessarily
lead to the pre-peak feature, hence posing the question of the exact origin of
the presence or absence of the pre-peak in Eq.(2).

In the present paper, we study the more modest problem of how the various
atom-atom correlations in neat monools combine to produce the experimentally
observed x-ray scattering intensity, which are well known to be characterised
by the pre-peak feature [14, 15, 16, 17, 18, 19, 20, 21, 22, 23].

##  2 Theoretical and simulation details

###  2.1 Charge order and pre-peak

Charge order in liquids have been documented since the first studied of ionic
melts [24, 25], and more recently in room temperature ionic liquids (RTILs)
[26, 27, 28, 29, 30], particularly in relation to the radiation scattering
pre-peak. Model calculations [31] show that the pre-peak, which does not exist
in simple ionic melts, comes from the charge ordering in presence of uncharged
groups attached to them. Alcohols also have such a scattering pre-peak [18,
32, 33, 34, 16, 19, 17, 35, 36, 37, 38]. Since charges in alcohol are tied
into the hydroxyl head group, while these are detached in RTILs into anion and
cation groups, it is clear that charge ordering has nothing to do with
polar/apolar order. It is a very simply the basic Coulomb fact that unlike
charges attract each other while like charges repell. With complex molecules
bearing partial charges, this leads to several near neigbour positioning
possibilities, thus sampling a variety of micro-states. The presence of
uncharged groups introduces a supplementary constraint, since both types of
charged groups try to avoid them. In absence of bonds tying the uncharged
groups to the charged ones, one would quite simply observe a phase separation,
with pre-transitional scattering raise at $k=0$. The scattering pre-peak can
be interpreted as a consequence of the avoided phase separation, induced by
the covalent bonding of the various groups. In fact, we have demonstrated [39]
that charge order with bond constraints leads to a non-uniform near neighbour
positioning of the charged group, often in form of a chain of oppositely
charges sitting in alternance. This local molecular conformation leads to
typical features in the pair correlation functions $g_{ab}(r)$, between 2
charged atomic sites of opposite valences, which we have described elsewhere:
the strong pair Coulomb interaction raises the first contact peak in the
$g_{ab}(r)$, while the lack of uniform spherical distribution of next and
higher order neighbours leads to a depletion correlation feature below one for
medium distance in $g_{ab}(r)$. The first feature gives, by Fourier transform,
a wide and high $k=0$ peak in the corresponding structure factor $S_{ab}(k)$
while the second feature leads to a narrow and small negative anti-peak at
$k=0$. The total contribution to $S_{ab}(k)$ is then a positive pre-peak.
Anti-peaks in $S_{ab}(k)$ happen in a similar way, but for pair of sites that
are weakly correlated or fully uncorrelated, such as a charged and an
uncharged site. For instance, the case of alcohols, the hydroxyl head group
atoms tend to be uncorrelated with the hydrophobic tail atoms. In this case
the first neighbour correlations are depleted, and mostly below $1$, and raise
again above one at medium distances. This combination leads to a total
negative anti-peak, as we will discuss in the Results section below.

The total radiation scattered intensity $I(k)$ by molecular liquids is given
by the Debye expression similar to Eq.(2), in terms of all the site-site
structure factors:

$I(k)=\alpha\sum_{i,j}\sqrt{x_{i}x_{j}}\sum_{a_{i},b_{j}}f_{a_{i}}(k)f_{b_{j}}(k)F_{a_{i}b_{j}}^{(t)}(k)$
(3)

where the first sum runs over all the molecular species $i$ and $j$, and the
second sum over all the atoms $a_{i}$ of molecular species $i$ and $b_{j}$ of
molecular species $j$, the $f_{c_{n}}(k)$ are the form factor for atom type
$c_{n}$ of molecular species $n$, and are radiation dependent, $\alpha$ is a
radiation dependent coefficient, $x_{i}$ is the mole fraction of species $i$,
and the total _static_ intermediate scattering function
$F_{a_{i}b_{j}}^{(t)}(k)$ (also called total structure factor
$S_{ab}^{(T)}(k)$ in our previous papers [22, 40]) is defined as:

$F_{a_{i}b_{j}}^{(t)}(k)=F_{a_{i}b_{j}}^{(s)}(k)+F_{a_{i}b_{j}}^{(d)}(k)$ (4)

where the self-part $F_{a_{i}b_{j}}^{(s)}(k)$ is in fact the intra-molecular
structure factor, and the distinct part $F_{a_{i}b_{j}}^{(d)}(k)$ is defined
in terms of the atom-atom structure factors $S_{a_{i}b_{j}}(k)$ as

$F_{ab}^{(d)}(k)=\rho\sqrt{x_{i}x_{j}}S_{a_{i}b_{j}}(k)$ (5)

with

$S_{a_{i}b_{j}}(k)=\int
d\mathbf{r}\exp(i\mathbf{k.r})\left[g_{a_{i}b_{j}}(r)-1\right]$ (6)

The self intermediate scattering function, defined only for atoms of the same
species $i$, has a simple expression for rigid molecules, in terms of the
atom-atom intra-molecular distances $d_{a_{i}b_{i}}$, as:

$F_{a_{i}b_{j}}^{(s)}(k)=\delta_{ij}j_{0}(kd_{a_{i}b_{i}})$ (7)

where $\delta_{ij}$ is a Kronecker symbol and $j_{0}(x)=\sin(x)/x$ is the
zeroth-order spherical Bessel function. For non-rigid-molecules, one must
compute the self intra-molecular function $g_{a_{i}b_{i}}^{(s)}(r)$ by
computer simulations, much the same way one computes the inter molecular pair
distribution function $g_{a_{i}b_{j}}(r),$ by building the histogram of
neighbouring atoms:

$g_{ab}(r)=\frac{H_{ab}(r,\delta r)}{N_{i}^{2}\delta V(r,\delta r)}$ (8)

where $H_{ab}(r,\delta r$) is the number of atoms $b$ in a spherical shell of
radii $r$ and thickness $\delta r$, centered on atom $a$, $N_{i}$ is the
number of molecules of species $i$ in a volume V, and $\delta V(r,\delta
r)=(4\pi r^{2}\delta r)/V$ is the normalised volume of the shell.

Eq.3 shows that positive pre-peaks and negative anti-peaks can cancel each
other, to leave either no peak, or a residual pre-peak in $I(k)$. It is this
residual pre-peak which is interpreted in terms of the existence of clusters
in scattering experiments. However, it is clear that it is the $g_{ab}(r)$ and
the $S_{ab}(k)$ which are the true indicators of the preferred molecular
dispositions inside the liquid. Interestingly, it is the $g_{ab}(r)$ which is
the ultimate structure indicator, since, as discussed above, the pre-peak of
the structure factor $S_{ab}(k)$ is itself the result of a cancellation of 2
features of the $g_{ab}(r).$ Therefore, it is perhaps more important to
analyse these functions, rather than the scattering intensity $I(k)$.
Molecular simulation allows to do that, albeit with the price of representing
the true liquids by molecular models, with all the bias that it introduces.

###  2.2 Molecular models and simulation details

The simulations were performed with the program package Gromacs [41, 42],
using the following protocol for all alcohols. The Packmol program [43] was
used to generate random configurations of 2048 molecules for all systems. This
system size is sufficient for studying the structure of monools with long
alkyl tails, as shown in our previous work [22]. After minimizing the energy,
we equilibrated the systems by doing two consecutive runs of 1 ns, first in
the canonical (NVT) and then in the isobaric-isothermal (NpT) ensemble. The
latter ensemble was used in the following production runs, which lasted 5 ns
and sampled at least 2000 congurations for the subsequent calculations of the
site-site correlation functions $g_{ab}(r).$

All alcohols were simulated in ambient conditions, at the temperature of T =
300 K and pressure p = 1 bar. The temperature was maintained with the Nose-
Hoover thermostat (time constant of 0.1 ps) [44, 45], whereas the pressure was
achieved with the Parrinello- Rahman barostat (time constant of 1 ps) [46,
47]. The time step of 2 fs was used for all systems and the integration
algorithm was leap-frog [48]. For short-range interactions the cut-off radius
was 1.5 nm. For the long-range Coulomb interactions we used the particle mesh
Ewald (PME) method [49], with FFT grid spacing of 0.12 nm and interpolation
order of 4. The LINCS algorithm [50] was utilized for the constraints.

All of the alcohols were modeled as united-atom, using the classic OPLS
forcefield for alcohols [51].

##  3 Results

###  3.1 Non-branched mono-ols

Fig.1 shows the oxygen-oxygen pair correlation functions $g_{OO}(r)$ in the
left panel and the corresponding structure factors $S_{OO}(k)$ in the right
panel, for simulated monohydroxy alcohol ranging from methanol to 1-nonanol.
The left panel shows very cleary how the O-O correlations increase
dramatically from methanol to 1-nonanol, even though the partial charges on
the oxygen and hydrogen sites remain the same across models, and the contact
position remains the same. The reason for this increase is that, for the same
packing fraction of atoms, there are much less hydroxyl groups in 1-nonanol
than in methanol. Hence, their contact probability increases in inverse
proportion to their rarity, which is just a statistical effect. In contrast,
the depletion part, shown in the inset, increases both in width an strength
when going from methanol to 1-nonanol, suggesting longer hydroxyl chains in
1-nonanol than in methanol.

Figure 1: Left panel: oxygen-oxygen pair correlation functions $g_{OO}(r)$ for
methanol up to 1-nonanol. The inset is a zoom on the depletion correlation
part (see text). Right panel: corresponding structure factors $S_{OO}(k)$ (the
color codes are shown on the side).

Both observations are corroborated in the analysis of the structire factors.
The Hbond peak at $k_{HB}\approx 3\mathring{A}^{-1}$ is related to the
distance hydrogen bonding distance between oxygen and hydrogen
$r_{HB}=2\pi/k_{HB}\approx 2\mathring{A}$) (see Fig.2 for an explanation). The
peak at $k\approx 1.5\mathring{A}^{-1}$ , which is quite visible for methanol,
and barely marked for the other alcohols, corresponds to the distance
$r\approx 3.4\mathring{A}$ which is the diameter of the oxygen atom in the
OPLS model [51]. The weakness of this peak indicates that the direct, non-
bonding, contact between the two oxygen is not very likely, since it is the
Hbond mediated contact which is preferred. The prominent feature is the pre-
peak, which is at $k_{PP}\approx 1.25\mathring{A}$ up to $k_{PP}\approx
0.5\mathring{A}^{-1}$ for 1-nonanol, shows the increase of the hydroxyl chains
with alcohol size. This is confirmed by the snapshots from the simulations,
which show clear long chains in the case of longer alcohols. However, the
direct cluster analysis [22] shows that the _average_ cluster size remain more
or less constant across all monohydroxy alcohols, with about 5 hydroxyl groups
per chain. We have no explanation for this puzzling fact.

Fig.2 shows the oxygen-hydrogen correlations for the same cases as in previous
figure 1. Although the first peak of the $g_{OH}(r)$ and the pre-peak of the
$S_{OH}(k)$ offer strong similarities with the previous case, we note that the
$g_{OH}(r)$ are positioned at the O-H bonding distance, as expected. But the
most important feature is the charge order between like and unlike charges.
This is seen from the peak/anti-peak features which characterise charge order
correlations, as demonstrated in our previous work [31, 52].

Figure 2: Left panel: oxygen-hydrogen pair correlation functions $g_{OH}(r)$
for methanol up to 1-nonanol. The inset is a zoom on the depletion correlation
part (see text). Right panel: corresponding structure factors $S_{OH}(k)$ (the
color codes are shown on the side).

Accordingly, we note that the second neighbour OH peak at $r\approx
3.5\mathring{A}$ in Fig.2 coincides with the first minimum of the OO in Fig.1.
This is even more striking in k-space, when the peak at $k\approx
3\mathring{A}^{-1}$ coincides with the minimum at the same k-value in Fig.2
right panel. This would be more apparent if both plots would be superimposed,
but aside from this, it would be also unreadable. These features are simple
translation of the alternation of the O and H atoms along the hydroxyl group
chain aggregates.

It is instructive to look at the oxygen atom correlation with the first carbon
of the alkyl chain. In all force field model, this atom is weakly charged,
essentially to neutralize the total charge on the molecule. The valence is too
weak to speak of charge ordering. Nevertheless, it is important to confirm if
this is true. Fig.3 shows the $g_{OC_{1}}(r)$ and $S_{OC_{1}}(k)$. We note
that the generic patterns of the correlations are very similar to that of OO
correlations in Fig.1, albeit with reduced magnitudes. This is compatible with
the weak charge of the $C_{1}$ carbon. In addition, since this carbon is the
closest to the hydroxyl head group, it seem natural that the correlations are
slaved to that between the oxygens. This would happen even of the carbon was
not charged.

Figure 3: Left panel: Pair correlation functions $g_{OC_{1}}(r)$ between the
oxygen and the carbon atom in the first methylene group for methanol up to
1-nonanol. Right panel: corresponding structure factors $S_{OC_{1}}(k)$ (the
color codes are shown on the side).

The left panel shows the corresponding structure factors $S_{OC_{1}}(k)$. We
observe again features very similar to that in the left panel of Fig.1, albeit
with lesser magnitude.

Now, we look at the inter-molecular correlations between the head oxygen atom
$O$ and the last (uncharged) carbon atom $C_{n}$ of the methyl group of the
alkyl tail. Fig.4 shows the $g_{OC_{n}}(r)$ and $S_{OC_{n}}(k)$. What we
observe in the left panel is that, when the alcohol is short, from methanol
until 1-butanol, there is a dual first neighbour/second neighbour contact high
probability, and the first contact decays rapidly, while the second survives
until 1-butanol. Ethanol is interesting since it shows equal probablity for
both contacts. For longer alcohols, the correlations are essentially depleted.
We note that the 3rd correlation peak is in the range $7-9\mathring{A}$, and
is more pronounced for longer alkyl chains than for the shorter ones. This is
somewhat surprizing, since one would imagine exactly the opposite behaviour.
We interpret this as an indication that the self-assembled objects have better
defined shapes and in particular well defined boundaries for longer alkyl
chains. This would be consistent with the “inverse line micelle” picture, with
the center of the cylinder occupied by the hydroxyl chain, and the alkyl tails
protruding all around. The resulting alcohol would then appear as a melt of
such self-assembled line-micelles.

Figure 4: Left panel: Pair correlation functions $g_{OC_{n}}(r)$ between the
oxygen and the last methyl group for methanol up to 1-nonanol. Right panel:
corresponding structure factors $S_{OC_{n}}(k)$ (the color codes are shown on
the side).

The structure factors in the right panel are a good illustration of the
generic anti-peak description provided in Section 2.1. We observe the negative
anti-peaks, which are the consequence of the depletion correlations described
in the left panel. These are deeper and narrower for longer alcohols, in
proportion of the depth and width of the depletion correlations.

We turn finally towards the carbon-carbon correlations, and we have picked
those between the last and one before the last carbons of the alkyl tails.
Fig.5 shows the corresponding $g_{C_{n-1}C_{n}}(r)$ and $S_{C_{n-1}C_{n}}(k)$
functions. We observe that these correlations are strikingly reminiscent of
those in a Lennard-Jones liquid for ethanol and for dimer Lennard-Jones for
all others. Since these sites do not bear any charges, these results appear
consistent with the charge order hypothesis that the main features of the
correlations are slaved to the presence or absence of charge order. However,
for longer alcohols we observe a small depression of high order neighbours.

Figure 5: Left panel: Pair correlation functions $g_{C_{n-1}C_{n}}(r)$ between
the last methyl group and the one before, for ethanol up to 1-nonanol. Right
panel: corresponding structure factors $S_{C_{n-1}C_{n}}(k)$ (the color codes
are shown on the side).

This latter trend is further confirmed by the pre-peak in the corresponding
structure factors in the right panel. In fact, the structure factors bears
quite some resemblance with the $S_{OO}(k)$ in Fig.1 , at the exception that
now both the pre-peak and main peak are about the same height, at least for
the longer alcohols above ethanol. This is very interesting because it
witnesses the fact that the tail atoms share both the simple disorder
distribution of Lennard-Jones liquids, while keeping a memory that they are
slaved to the self-assembly of the head groups. Hence, the role of the tails
appears as important in some aspect. This is further confirmed when looking at
branched alkyl tails, where entropic effects are expected to play an important
role on the self-assembly of the charged head groups.

###  3.2 Branched octanolsoctanol

The n-octanol molecules are a good example to study alkyl tail branching. The
chains are sufficiently long to expect to observe chain packing entropic
effects. The computer simulations reveal that most of the hydroxyl groups are
clustered into chain-like aggregates. Hence, a good picture of this liquid is
that of either a melt of line-like micelles, or chains “floating” in a
“solvent” of methyl/methylene super atoms. We expect two types of effects: the
influence of the branching of the tails on the hydroxyl chains formation, and
the way the irrepressible chain formation of the hydroxyl groups affect the
packing of the alkyl tails. Correlation functions are an excellent observable
to answer these questions.

Fig.6 shows the O-O correlations (left panel) and the corresponding structure
factors (right panel). Perhaps the most intringuing feature is the non-linear
behaviour of these curves as a function of branching. 2-octanol, which is the
“less branched” of these octanols, stands out as very different from the
others. This is particularly striking in the first peak of the $g_{OO}(r)$,
which is smaller for 2-octanol, while all others coincide almost exactly. In
contrast, both 3-octanol and 4-octanol offer strong similarities (similar
depletion correlations and pre-peak heights). The depletion correlation for
the branched octanols are more marked: the more the branching, the wider and
deeper the depletion. This is a perfect illustration of the packing entropic
problems posed by branching.

Figure 6: Left panel: oxygen-oxygen pair correlation functions $g_{OO}(r)$ for
1-octanol up to 4-octanol. The inset is a zoom on the depletion correlation
part (see text). Right panel: corresponding structure factors $S_{OO}(k)$ (the
color codes are shown on the side).

We tentatively explain the case of 2-octanol as the difficulty of aggregating
the hydroxyl groups versus the rigidity/flexibility of the alkyl tail. Indeed,
if we consider the idea of a “sea” of methyl/methylene “solvent”, this poses
the problem of packing these super atoms, while keeping the hydroxyl groups
attached together. The correlation seems to indicate that these problems are
maximised for the case of 2octanol.

Fig.7 shows the O-H correlations for the same cases as in Fig.7. The charge
order based alternation that we observed in Figs.1 and 2 is more pronounced
with increasing branching. This is directly reflected in the structure factor
pre-peak for 4-octanol being higher than the others. However, the positions of
the pre-peaks change in a non-linear manner: the k-value decreases from
1-octanol to 3-octanol, and then seems to saturate since it is nealy the same
between 3-octanol and 4-octanol.

Figure 7: Left panel: Pair correlation functions $g_{OH}(r)$ between the
oxygen and the hydrogen for the n-octanol. Right panel: corresponding
structure factors $S_{OH}(k)$ (the color codes are shown on the side).

The influence of the tail length and branching can be appreciated as before by
comparing the correlations from the head oxygen to the last carbon. This is
shown in Fig.8. We observe that there are no net pre-peaks. Instead, the
shoulder structure at small $r$ value $r\approx 5\mathring{A}$ for 1-octanol,
tends to disappear, while the correlation heights increase from 1-octanol to
4-octanol, suggesting that these correlations are quite Lennard-Jones like. We
tentatively interpret this trend as as randomisation of correlations between
the oxygen atom and the tail atoms because of increased branching order. We
observe again this saturation of curves at 3-octanol and 4-octanol, since they
appear quite similar.

.

Figure 8: Left panel: Pair correlation functions $g_{OC_{n}}(r)$ between the
oxygen and the last methyl group for the n-octanol. Right panel: corresponding
structure factors $S_{OC_{n}}(k)$ (the color codes are shown on the side).

The structure factors show well marked anti-peaks, but of smaller magnitude
than those observed for the shorter and medium lenght non-branched monools in
Fig.2. Basically, these results reflect the decorrelation between head groups
and tail atoms.

Fig.9 shows the $g_{C_{n-1}C_{n}}(r)$ and $S_{C_{n-1}C_{n}}(k)$ in order to
appreciate the correlations between the tail atoms that the farthest from the
charge order constraints. The figures show quite clearly that the correlations
are both Lennard-Jones like and depleted in the $r$-range $8-14\mathring{A}$.
This is confirmed by a net pre-peak in the corresponding structure factors.

Figure 9: Left panel: Pair correlation functions $g_{C_{n-1}C_{n}}(r)$ between
the last methyl group and the one before, for the n-octanol. Right panel:
corresponding structure factors $S_{C_{n-1}C_{n}}(k)$ (the color codes are
shown on the side).

If we recoup these observations with that of the previous figure, we are
forced to conclude that, while head and tail atoms appear randomized, the tail
atoms themselves are still constrained by the charge order imposed by the head
atoms. This is not very surprizing, but it shows that “internal” correlations
between aggregated atoms are different from those outside. This enforces the
picture of a coherent self-assembled object. We can conjecture that self-
assembled objects with better shape visibility, such as micelles will show
similar change of correlations between atoms belonging to the “inside” of the
object and those outside.

##  4 Calculated scattering intensities

The X-ray scattering experiments provide a biased information on the
microscopic structure of the complex disorder liquids, such as those with
aggregates. This is because of the cancelling contributions from the various
structure factors in the k-range $0<k<k_{max}$, where $k_{max}\approx
2\pi/<\sigma>$ corresponds to the mean atomic diameter size $<\sigma>$ of all
the atoms in the molecular system, and which corresponds also to the main peak
in the wide angle x-ray scattering experiments.

X-ray spectra calculated from simulations using Eq.(3) are also biased for
several reasons. First of all, the model dependence is strong, as illustrated
in our previous studies [22, 23]. Secondly, The intra-molecular correlations
$W_{ab}(k)$are an important ingredient since they affect the small-$k$ region.
Fig.10 illustrates this. The left panel shows a comparison the $W_{OC_{8}}(k)$
between the oxygen $O$ and last carbon atom$C_{8}$ in the case of 1-octanol,
between the rigid model (in red) and that (in black) obtained by explicit
calculation in the course of the simulation using the flexible OPLS model. We
see that there is quite a strong dephasing between the 2 calculations.

Figure 10: Intra-molecular correlations $W_{OC_{8}}(k)$ between the head group
oxygen atom $O$ and last carbon atom $C_{8}$ of the tail. Left panel:
$W_{OC_{8}}(k)$ for 1-octanol; black line for flexible model and red line for
rigid model (see text). Right panel: influence of chain branching on
$W_{OC_{8}}(k)$ for the branched octanol.

In case of long molecules, the rigid model approximation becomes a problem. To
avoid that, we sample the mean atom-atom distance $d_{ab}$, as obtained from
the simulations, and use the rigid atom approximation with
$W_{ab}(k)=j_{0}(kd_{ab}).$ The right panel shows the intra-molecular
correlations $W_{OC_{8}}(k)$ of the flexible models of various branched
octanols. A larger dephasing corresponds to a shorted distance $d_{OC_{8}}$,
which is consistent in what concerns 1-octanol, but is less trivial for the
others.

Fig.11 illustrates the charge order induced pre-peak anti-peak compensations
for 1 and 2-octanol. It shows that the anti-peak does not quite compensate the
pre-peak, in both cases, which is the reason why a net pre-peak would appear
in the scattering intensity $I(k)$. But, there are cases where these peaks
exactly compensate, and this happen for more spherical aggregative shapes,
where the depletion gains more depth.

Figure 11: Charge order signature on partial structure factors for like-atoms
($OO$) and cross-atom ($OC_{8}$) correlations. Left panel: 1octanol; right
panel: 2octanol.

Fig.12 shows the x-ray scattering intensities, from experiments (red lines)
and calculated (black) for 1 and 2-octanol. The net like-atom (green) and
cross-atom (blue) contributions are equally shown. We see that both
contributions are positive for the main peaks ($k_{max}\approx
1.5\mathring{A}$), while they have opposite signs for the pre-peaks
($k_{PP}\approx 0.5\mathring{A}$). These opposting contributions are directly
related to those illustrated in Fig.11 in the particular case of $O$ and
$C_{8}$ atoms.

.

Figure 12: x-ray scattering intensities for 1octanol (left panel) and 2octanol
(right panel). Red lines: experimental data from [22, 23], black lines:
calculated spectra; green lines: total like-atoms contributions to $I(k)$;
blue lines: total corss-atoms contributions to $I(k)$.

The agreement between the calculated and experimental spectra have been
discussed in our previous work [22, 23]. It is mostly qualitative,
particularly for the pre-peak part which is very sensitive to local atomic
labile positionings and force field details. This is less sensitive for the
main peak, which is mostly controled by the Lennard-Jones interactions between
sites, hence the the part of the interaction which is not responsible for
labile self-assembly.

##  5 Conclusion

In this study, we have analyzed the local structure of hydrogen bonded
monohydroxy alcohols, both with non-branched and branched alkyl chains,
through the analysis of the pair correlation functions and associated
structure factors, in order to apprehend the interplay between self-assembly
of the hydroxyl groups in the midst of the alkyl chains. We have shown that
the atom-atom pair correlation functions are the ideal observables of the
molecular micro-structure inside liquids, and that this micro-structure is
chiefly ruled by charge order, including non-charged sites. We have shown that
the pre-peaks and anti-peaks in the structure factors are a consequence of
special highly inhomogeneous local molecular dispositions induced by charge
ordering between the different charged or uncharged atomic groups. What
remains unclear at present is relationship between the peak/antipeak position,
heights and width, in relation to the supra structure of the self-assembly.
Perhaps this is not so well defined for the small self-assembly such as in
these mono-ols, and it becomes more relevant to discuss these issues with
higher forms of self-assembly, such as with spherical and cylindrical
micelles, vesicles, lamella and others. This poses the problem of the interest
of studying in detail precursor self-assembling systems. Our answer is that
such systems are similar to pre-transitional fluctuations in spinodal and near
critical points of lines. While one can focus solely on well defined self-
assembled shapes, such micelles, and underestimate the influence of atom-level
arrangements and correlations, systems such as those we have studied herein
allow to explore precursor self-assembling systems, and pay attention to
molecular details down to atom level.

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