derivative coupling constant). On the other hand, if we assume the maximum
number of derivatives is $2(2s-1)$, as suggested by the cubic Lagrangian
(102), we observe $s(6s-1)$ off-shell operators of the type
$(D...D\Phi)^{\star}(D...D\Phi){F_{\;\\!\\!-}}$. This overcounting can be
explained by the possibility to generate trivial interactions via field
redefinitions $\Phi\to\Phi+(D...DF_{-})(D...D\Phi)$, which we have not tried
to rule out here. In other words, the present analysis gives us a
significantly _overcomplete_ set of off-shell operators. At each multiplicity
$n$, however, we may make a suitable choice, such as the three-point action
(102), which will automatically include certain higher-order contributions,
but the same analysis at the next order will once again give us access to the
entire space of $(n+1)$-point couplings.

At the quartic level, there are three types of contact terms that are given by
the singlets in the following tensor products:

$(2s-2k+n,n)\otimes(2s-2k^{\prime}+n^{\prime},n^{\prime})\otimes(m+2,m)\otimes(2,0)$
(247)

for terms of the type
$(D...D\Phi)^{\star}(D...D\Phi)(D...D{F_{\;\\!\\!-}}){F_{\;\\!\\!-}}$,

$(2s-2k+n,n)\otimes(2s-2k^{\prime}+n^{\prime},n^{\prime})\otimes(m+2,m)\otimes(0,2)$
(248)

for $(D...D\Phi)^{\star}(D...D\Phi)(D...D{F_{\;\\!\\!-}}){F_{\;\\!\\!+}}$, and
finally

$(2s-2k+n,n)\otimes(2s-2k^{\prime}+n^{\prime},n^{\prime})\otimes(m,m+2)\otimes(0,2)$
(249)

$(D...D\Phi)^{\star}(D...D\Phi)(D...D{F_{\;\\!\\!+}}){F_{\;\\!\\!+}}$.

The cubic Lagrangian (102) suggests that only a limited number of possible
contact terms should be relevant if we are interested in modeling black holes,
for which a certain minimality concept is believed to exist. In particular,
Ochirov:2022nqz  explains that no terms of the third type
$(D...D\Phi)^{\star}(D...D\Phi)(D...D{F_{\;\\!\\!+}}){F_{\;\\!\\!+}}$ are
needed at all. In fact, the spin-1 action (97a) contains only certain terms of
the first type
$(D...D\Phi)^{\star}(D...D\Phi)(D...D{F_{\;\\!\\!-}}){F_{\;\\!\\!-}}$ and none
of the second type
$(D...D\Phi)^{\star}(D...D\Phi)(D...D{F_{\;\\!\\!-}}){F_{\;\\!\\!+}}$. More
generally, we need terms of the first type to restore parity and cannot
entirely rule out the terms of the second type.

Let us illustrate this general method on a few lower-spin cases. To bound this
problem, we need to formulate a power-counting assumption applicable in the
chiral formalism. Note that we cannot use our intuition from the non-chiral
actions, as even the three-point assumption (PC) of having at most $2s{-}1$
derivatives is explicitly violated by the cubic chiral Lagrangian (102), which
contains $4s{-}1$ derivatives, including those in the field strength. Taking
this $4s{-}1$ counting as a given, we can minimally extend it to four points
by assuming that

(PC3)

the number of derivatives in higher-point vertices contributing to an
amplitude should not exceed the momentum counting of the cubic diagrams in
this amplitude.

This assumption is consistent with the all-plus amplitudes, such as (109a),
that follow strictly from the minimal-coupling part of the Lagrangian (102)
and thus require that all additional terms involve at least one $F^{-}$. For
instance, the cubic vertex for positive helicity has $1$ derivative, so the
cubic diagrams contributing to ${\cal
A}(\bm{1}^{s}\\!,\bar{\bm{2}}^{s}\\!,3^{+}\\!,4^{+})$ have $1+1-2=0$
derivatives, if we take into account $-2$ derivatives from the propagator.
Since we insist that all higher-point vertices depend on the four-potential
$A_{\mu}$ through the field-strength, such a counting forbids any vertices
that only involve $F^{+}$.

Applying this to the opposite-helicity case, we observe that ${\cal
A}(\bm{1}^{s}\\!,\bar{\bm{2}}^{s}\\!,3^{-}\\!,4^{+})$ contains only up to
$1+(4s-1)-2=4s-2$ derivatives, judging by the trivalent diagrams. Since two
derivatives are contained within the two corresponding field strengths $F^{-}$
and $F^{+}$, there are a maximum of $n+n^{\prime}+m=4s-4$ derivatives that
need to be addressed when classifying singlets in the tensor-product space
(248). The counts produced by our code are given in table 7. As explained
above, these counts only give an upper bound for the number of independent
operators under the described assumptions.

$s$ | 0 | $\,{}^{1}\\!/\\!_{2}$ | $1$ | $\,{}^{3}\\!/\\!_{2}$ | $2$ | $\,{}^{5}\\!/\\!_{2}$ | $3$
---|---|---|---|---|---|---|---
count | 0 | 0 | 0 | 15 | 84 | 279 | 714
Table 7: Number of singlets in the four-point chiral operator product space
(248)

##  Appendix C Details on factorization poles

In this appendix, we give some useful formulae for checking the factorization
channels of the $\sqrt{\text{Kerr}}$ Compton amplitudes, both in the quantum
and classical cases.

First we note that the residues of the physical poles in the abelian Compton
amplitude (5.1) can be shown to match those of the corresponding AHH amplitude
Arkani-Hamed:2017jhn ,

$\Big{(}\frac{\langle
3|1|4]^{2}(U+V)^{2s}}{m^{4s}t_{13}t_{14}}-\frac{\langle\bm{1}3\rangle\langle
3|1|4][4\bm{2}]}{m^{4s}t_{13}}P_{2}^{(2s)}\Big{)}\Big{|}_{\rm
poles}=\frac{\big{(}\langle\bm{1}3\rangle[4\bm{2}]+\langle\bm{2}3\rangle[4\bm{1}]\big{)}^{2s}}{t_{13}t_{14}\langle
3|1|4]^{2s-2}}\Big{|}_{\rm poles}\,,$ (250)

where the two massive poles are located at $t_{13}=(p_{1}+p_{3})^{2}-m^{2}=0$
and at $t_{14}=0$. While the AHH amplitude on the right-hand side also
contains so-called spurious poles, $\langle 3|1|4]=0$, the left-hand side is
manifestly free of them. The absence of spurious poles is, of course, a
consequence of working with amplitudes that originate from a local Lagrangian
(122).

Note that there is an obvious asymmetry between the appearance of the two
denominators, $t_{13}$ versus $t_{14}$, on on the left-hand-side of eq. (250),
but not on the right-hand side. However, this is an artifact of our chosen
compact presentation of the amplitude (5.1). The following non-trivial
identity holds for generic kinematics and spin:

$\\!\\!\frac{\langle
3|1|4]^{2}(U+V)^{2s}}{m^{4s}t_{13}t_{14}}-\frac{\langle\bm{1}3\rangle\langle
3|1|4][4\bm{2}]}{m^{4s}t_{13}}P_{2}^{(2s)}=\frac{\langle
3|1|4]^{2}(U-V)^{2s}}{m^{4s}t_{13}t_{14}}-\frac{\langle\bm{2}3\rangle\langle
3|1|4][4\bm{1}]}{m^{4s}t_{14}}P_{2}^{(2s)}\,.$ (251)

Thus, in principle, one could average over these two presentations of the pole
terms, and obtain a manifestly symmetric presentation, at the price of a
slightly longer formula.

For the non-abelian color-ordered amplitude (5.2), the physical pole structure
again match those of the corresponding AHH amplitude Arkani-Hamed:2017jhn ,

$\displaystyle\bigg{[}\frac{\langle
3|1|4]^{2}(U+V)^{2s}}{m^{4s}s_{12}t_{14}}-\frac{\langle\bm{1}3\rangle\langle
3|1|4][4\bm{2}]}{m^{4s}s_{12}}P_{2}^{(2s)}+\frac{\langle\bm{1}3\rangle\langle
3\bm{2}\rangle[\bm{1}4][4\bm{2}]}{m^{4s}s_{12}}t_{13}P_{2}^{(2s-1)}$ (252)
$\displaystyle\quad-\frac{\langle\bm{1}3\rangle\langle
3\bm{2}\rangle[\bm{1}4][4\bm{2}]}{m^{4s-2}s_{12}}\big{(}t_{13}\langle\bm{12}\rangle[\bm{12}]P_{4}^{(2s-1)}\\!+2VP_{4}^{(2s)}\big{)}\bigg{]}\Big{|}_{\rm
poles}=\frac{\big{(}\langle\bm{1}3\rangle[4\bm{2}]+\langle\bm{2}3\rangle[4\bm{1}]\big{)}^{2s}}{s_{12}t_{14}\langle
3|1|4]^{2s-2}}\Big{|}_{\rm poles},$

where the $t_{14}=0$ residue matching follows from the abelian case (250), but
the $s_{12}=0$ residue matching is more non-trivial as it requires several
many terms.

It is interesting to also exhibit the factorization properties of the
classical all-spin Compton amplitude (6.5). On the massless factorization pole
the classical Gram determinant (143) reduces to

$0=(w-x-y)(w-x+y)\,,$ (253)

where the two solutions correspond to the two choices of helicity for the
propagating gluon. On the massless pole, the only surviving term in the
classical amplitude (6.5) comes from the interplay between the functions ${E}$
and ${\tilde{E}}$,

$\displaystyle\frac{w^{2}{-}z^{2}}{2}\Big{(}$
$\displaystyle{E}(x,y,z)+\frac{i\epsilon(q_{\perp},p,q,a)}{p\cdot
q_{\perp}}{\tilde{E}}(x,y,z)\Big{)}\xrightarrow[q^{2}\to
0]{}\frac{w^{2}{-}z^{2}}{2}\Big{(}{E}(x,y,z)+(w{-}x){\tilde{E}}(x,y,z)\Big{)}$
$\displaystyle=\frac{x^{2}-y^{2}-z^{2}}{2}\Big{(}{E}(x,y,z)\pm
y{\tilde{E}}(x,y,z)\Big{)}=e^{\pm y}+{\cal O}(e^{x+z},e^{x-z})\,,$ (254)

where the terms proportional to ${\cal O}(e^{x+z},e^{x-z})$ cancel against the
first two terms in eq. (6.5). The remaining exponential factor corresponds to
the product of the classical three point amplitudes, $e^{\pm y}=e^{\pm
k_{3}\cdot a}\times e^{\pm k_{4}\cdot a}$.

On the massive pole $t_{14}$, the cut kinematics enforces all momentum dot
products to be at least of order ${\cal O}(\hbar^{2})$:

$p_{1}\cdot k_{3}+p_{1}\cdot k_{4}+k_{3}\cdot k_{4}=0\qquad\Rightarrow\qquad
p_{1}\cdot k_{3}=-\frac{1}{2}q^{2}={\cal O}(\hbar^{2})\,.$ (255)

Therefore, we should take $w,z\to 0$ and $p\cdot q_{\perp}=\frac{1}{2}q^{2}$,
such that the classical amplitude (6.5) on the cut simplifies to

$\displaystyle t_{14}{\cal A}(\bm{1},\bm{2},3^{-}\\!,4^{+})\big{|}_{t_{14}\to
0}$
$\displaystyle=-\frac{(p\cdot\chi)^{2}}{q^{2}}\left([T^{c_{3}},T^{c_{4}}]-\frac{1}{4}\\{T^{c_{3}},T^{c_{4}}\\}\right)e^{x}$
(256)
$\displaystyle=(p\cdot\varepsilon_{3}^{-})(p\cdot\varepsilon_{4}^{+})\left([T^{c_{3}},T^{c_{4}}]-\frac{1}{4}\\{T^{c_{3}},T^{c_{4}}\\}\right)e^{x}\,.$

This is once again the expected product of three-point amplitudes
$e^{x}=e^{k_{4}\cdot a}\times e^{-k_{3}\cdot a}$, where the mismatch in
reference vector for $a^{\mu}$ is irrelevant on the cut.

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# Camel back shaped Kirkwood-Buff Integrals

Aurélien Perera†, Martina Požar‡ and Bernarda Lovrinčević‡

###### Abstract

Some binary mixtures, such as specific alcohol-alkane mixtures, or even water-
tbutanol, exhibit two humps “camel back” shaped KBI. This is in sharp contrast
with usual KBI of binary mixtures having a single extremum. This extremum is
interpreted as the region of maximum concentration fluctuations, and usually
occurs in binary mixtures presenting appreciable micro-segregation, and
corresponds to where the mixture exhibit a percolation of the two species
domains. In this paper, it is shown that two extrema occur in binary mixtures
when one species forms "meta-particle" aggregates, the latter which act as a
meta-species, and have their own concentration fluctuations, hence their own
KBI extremum. This "meta-extremum" occurs at low concentration of the
aggregate-forming species (such as alcohol in alkane), and is independant of
the other usual extremum observed at mid volume fraction occupancy. These
systems are a good illustration of the concept of the duality between
concentration fluctuations and micro-segregation.

†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.

##  1 Introduction

The so-called Kirkwood-Buff integrals (KBI)[1, 2] are defined as the integrals
of the species-species pair correlation functions$G_{ab}=\frac{1}{\Omega}\int
d\mathbf{X}_{1}d\mathbf{X}_{2}\left[g_{ab}(\mathbf{X}_{1},\mathbf{X}_{2})-1\right]$
where $\mathbf{X}_{i}$ is the set of position, and if required, the
orientations, of molecule $i$, where $a$ and $b$ designates the species
indexes ($\Omega=V\omega^{2}$, where $\omega$ is the angular integral, equal
to $4\pi$ or $8\pi^{2}$, depending on the symmetry of the molecules). It can
be shown [3, 4] that this integral is identical to that involving the pair
correlation between any two atoms belonging to each molecules. Following this,
the KBI are more simply defined as

$G_{ab}=4\pi\int_{0}^{\infty}drr^{2}\left[g_{i_{a}j_{b}}(r)-1\right]$ (1)

where $i_{a}$and $j_{b}$ designate any two atoms $i$ and $j$ of ,
respectively, species $a$ and $b$, and $g_{i_{a}j_{b}}(r)$ the atom-atom pair
correlation function.The Kirkwood-Buff theory relates specific thermodynamic
properties, such as the compressibility, or the partial molar volumes, for
example, to be related to these integrals, thus providing an appealing link
between macroscopic measurable properties to the microscopic structure, albeit
in an integrated form, where all microscopic details are lost. Ben-Naim [5],
Matteoli and Lepori [6] and other authors [7, 8, 9, 10, 11] have managed to
invert these relations, providing a way to calculate these integrals from the
experimentally available data on compressibility, partial molar volume and
vapour pressure or Gibbs free-energy. The corresponding expressions for a
binary mixture ($a,b$) are well known [12]:

$G_{aa}=G_{ab}+\frac{1}{x_{b}}\left[\frac{\bar{V}_{b}}{D}-V\right]$ (2)
$G_{ab}=\frac{\chi_{T}}{k_{B}T}-\frac{\bar{V}_{a}\bar{V}_{b}}{VD}$ (3)

were $x_{b}$ is the mole fraction of component $b$, $\bar{V}_{c}$ is the
partial molar volume of component $c$ ($c=a$ or $b$), $V$ is the molar volume,
$\chi_{T}$is the isothermal compressibility (with $T$ the temperature and
$k_{B}$ the Boltzmann constant) and $D$ is related to the concentration
fluctuations and given by

$D=x_{i}\left(\frac{\partial\beta\mu_{i}}{\partial x_{i}}\right)_{TP}$ (4)

where $\mu_{i}$ is the chemical component of species $i$ and $\beta=1/k_{B}T$
is the Boltzmann factor.

Fig.1 of the seminal Matteoli and Lepori paper [6] displays KBI for various
types of aqueous mixtures, and it can be seen that most of them have a single
extremum, which is positive for like KBI $G_{aa}$ and negative for then unlike
$G_{ab}$ with $(a\neq b)$. This extremum has its origin in the shape of the
coefficient $D$ above, which is often a U-like shaped curve with a single
minimum. An example of such typical KBI and $D$ are given in Fig.1, and other
similar examples can be found in various place in the literature such as [13,
14, 15, 16, 12]. Note that, while for a given mixture, the $D$ coefficient has
a single extremum, the corresponding extrema in the KBI are not necessarily at
the same x-position, as illustrated in Fig.1, as Eqs.(2,3) tend to alter this
position from that of D.

Figure 1: Examples of single extremum KBI. Red curves experimental KBI curves
for aqueous-acetone from Ref.[17]; blue curves for aqueous-2propanol from
Ref.[12]. Circles for Water-water KBI, squares for cross water-solute KBI, and
triangles for solute-solute KBI. The inset shows the corresponding D (Eq.(4))
with same color codes, both showing a single minimum.

A single extremum for $D$ is expected for a curve which reaches the same value
at both end points. In fact, Eq.(4) can be shown (see Appendix) to be related
to the stability limit criteria for a binary mixture, that is precursor for
demixing, namely

$D\propto C_{11}C_{22}-C_{12}^{2}$ (5)

where $C_{ij}=\delta_{ij}-\sqrt{\rho_{i}\rho_{j}}\tilde{c}_{ij}(k=0)$ , with
the $\tilde{c}_{ij}(k=0)$ being the integrals of the direct correlation
functions and i,j refer to species index.Indeed, when $D\rightarrow 0$ the
mixture becomes mechanically unstable [3], such when approaching a spinodal or
a critical point. Therefore, a single extremum in $D$ can be physically
interpreted as the point where concentration fluctuations are maximal, and
with the eventuality that when these grow beyond the mechanical stability, the
mixture undergoes demixing. Subsequently, the extremum in the KBI is a
signature of maximal concentration fluctuations. Since many binary mixtures,
such as aqueous mixtures, for example, show stable local nano-segregation of
constituents [18, 19, 20, 21, 22], the extremum can be interpreted as maximal
amplitude in local segregation.

In this context, it came as a surprise that Jan Zielkiewicz published
experimental KBI for various alcohols-heptane mixtures [23], which clearly
showed the existence of two extremum. In Fig.2 of his paper, he shows KBI
written as $\rho_{a}G_{ab}$ where $\rho_{a}=N_{a}/V$ is the partial density of
species $a$ ($N_{a}$ is the number of particle in volume $V$), which amplifies
the second extremum, the resulting KBI looking like “camel back” shaped, as
opposed to the usual single extremum KBI which are “dromadary camel back”
shaped.

The purpose of this paper is to understand the origin of this dual extremum in
terms of fluctuations. The driving idea is the following. It is well known
that alcohols generally tend to cluster their hydroxyl groups into linear
patterns, forming chains, loops, etc… [24, 25, 26, 27, 28, 29] When put in
small concentration in an inert alkane solvent, such as heptane, for example,
small “chain-micelles” of alcohols form and act as independant meta-particles.
Consequently, they exhibit their own concentration fluctuations, which
corresponds to one of the observed extrema in the corresponding KBI. With
increasing concentration, the frontier between well separated micelles and
alcohol micro-segregated domains become loose, and the mixture becomes a
simple micro-segregated mixture, which has its own independent maximum in
concentration fluctuations. It is not obvious that such scenario should
happen, and the purpose of this paper is to use computer simulation to provide
evidence that this scenario is correct, principally through the analysis of
pair correlation functions and the KBI.

##  2 Theoretical and simulation details

The study of equilibrium locally micro-segregated mixtures has proven to be
very difficult in the past decades, particularly in the case of aqueous
mixtures. Perhaps a canonical example is that of aqueous _tert_ -butanol (TBA)
mixtures, for which the force field induced slow demixing at very small TBA
concentrations around $x_{\mbox{TBA}}\approx 0.01$ was shown [30, 31, 32, 33,
34] to rule out many classically robust force field models such as OPLS [35,
36] or TraPPE [37, 38]. One may even consider this problem to remain unsolved.
Similarly, the water-tetrahydrofuran mixture remains difficult to simulate, in
particular for putting into evidence the loop coexistence phase diagram [39,
40, 41]. One of the principal obstacle for successfully simulating such
mixtures is the fact that both water and the alcohol tend to form hydrogen
bonded clusters, and it would seem that it is precisely the competition
between these cluster formation which tend to drive the simulated mixture into
demixing, often after unusually very long times.

These problems might be avoided if water is replaced by a more inert solvent
such as an alkane. In a previous work [42], we have studied ethanol-benzene
mixtures and found that the description of the strong micro-heterogeneity (MH)
in the low ethanol content region posed statistical issues which necessitated
large size simulations. We expect to find similar issues in the present study.

Herein, we focus on the ethanol-heptane mixtures, simulated in the following
range of ethanol mole fractions: 0.02, 0.05, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6,
0.7, 0.8 and 0.9. All of the mixtures contain N=16000 particles, in order to
properly describe extented MH. The program package Gromacs was used to perform
molecular dynamics simulations [43, 44]. The simulation protocol has been the
same for all mole fractions of ethanol. The initial random configurations of
16000 molecules were created with the program Packmol [45]. These initial
configurations were first energy minimized and then equilibrated in the NpT
ensemble 5 ns. The length of the production runs varied for each mole fraction
of ethanol and can be found in Table 1. The shortest production runs of 10 ns
sampled on average 1000 configurations, whereas the longest runs of 35 ns
sampled more than 3500 configurations.

Mole fractions xETH | Production run times | Mole fractions xETH | Production run times
---|---|---|---
0.02 | 10 ns | 0.5 | 25 ns
0.05 | 35 ns | 0.6 | 20 ns
0.1 | 35 ns | 0.7 | 15 ns
0.2 | 30 ns | 0.8 | 10 ns
0.3 | 35 ns | 0.9 | 10 ns
0.4 | 20 ns |  |
Table 1: Production run times for the simulated ethanol-heptane mixtures.

The TraPPE forcefield for heptane [37] and ethanol [38] was used throughout
the range of ethanol mole fractions. The mixtures were simulated at T = 300 K
and p = 1 bar. Temperature was maintained constant using mostly the v-rescale
[46] or Nose–Hoover [47, 48] thermostat, while pressure was maintained with
the Parrinello–Rahman barostat [49, 50]. The temperature algorithms had a time
constant of 0.2 ps, while the pressure algorithm was set at 2 ps. The
integration algorithm leap-frog [51] was used at every time-step of 2 fs. The
short-range interactions were calculated within the 1.5 nm cut-off radius. The
electrostatics were handled with the PME method [52], and the constraints with
the LINCS algorithm [53].

The snapshots were made with VMD [54].

##  3 Results

###  3.1 Experiments

The experimental KBI data for the ethanol-heptane mixtures scanned from
Zielkiewicz paper are reported as lines in the main panel of Fig.2.

Figure 2: Experimental “camel back shaped” KBI from Ref.[23] for the ethanol-
heptane mixtures, reproduced as lines, with symbols for recalculated values
(see text) for consistency check.

In order to make sure that these results are fully consistent with one
another, we have inverted Eqs(2,3) in order to extract $D$, using the
following assumptions. The partial molar volumes have been replaced by the
molar volume of each species, thus making these quantities independent of the
concentrations. The isothermal compressibility term has been neglected
(assuming the incompressibility of the liquid mixtures). Finally, the excess
volume of the mixture has been set to be a linear function of the pure liquid
volumes, hence neglecting the excess volumes. These are usually found to be
less that 0.3cm^3/mol. All 3 assumptions can be justified only by the
resulting KBI (as was proven by us for aqueous alcohol mixtures in Ref. [12,
21]). From Eqs(2,3) we have extracted the following equivalent expressions for
$D$

$D_{ab}=-\frac{V_{a}B_{b}}{VG_{ab}}$ (6)
$D_{aa}=\frac{x_{b}V_{b}-x_{a}V_{a}}{f_{1}-f_{2}}$ (7)

with $f_{1}=x_{a}x_{b}\left(G_{aa}-G_{bb}\right)$ and $f_{2}=V(x_{a}-x_{b})$.
The resulting values are reported in Fig.3.

Figure 3: Dual minima D for the ethanol-heptane mixtures, as calculated from
the experimental results of Ref.[23] (see Eqs.(6,7) in the text).

It is seen that they are quite similar, with the denominator in Eq.(7)
producing a singularity at $x_{\mbox{Eth}}\approx 0.7$. The dots represent the
values which have been selected to best represent a good compromise between
the two sets. The most prominent feature is the very apparent existence of the
2 extrema, one at $x_{\mbox{Eth}}\approx 0.2$ and the other at
$x_{\mbox{Eth}}\approx 0.5$. Interestingly, both evaluations of $D$ coincide
almost perfectly in the entire range of the minimum part of $D$. Using these
values of the extracted D function, we return to evaluate back the 3 KBI
integrals, by still preserving the 3 approximations mentioned above. The
resulting values are reported in Fig.2 as dots, and are seen to nicely
superpose to the original data. This simple exercise proves that a single form
of $D$, which must be quite close to the one reported in the inset, must have
served in Ref.[23] to calculate the KBI, and moreover, that the use of true
partial molar volumes, compressibilities and excess volume did not affect much
the final shapes. We also prove that the 2 extrema of the KBI originate from
the 2 extrema observed in $D$.

###  3.2 Computer simulations

####  3.2.1 Snapshots

When mixtures present strong local heterogeneity, it is often instructive to
look at snapshots in order to visually appreciate the nature and the extent of
the concentration dependence of the spatial segregation. Fig.4 shows snapshots
of 4 typical concentrations of ethanol, with different styles highlighting the
morphological changes in the ethanol clustering in heptane solvent.

Figure 4: Snapshots of the ethanol-heptane mixtures for 4 typical ethanol
concentrations. Ethanol oxygen, hydrogen and carbon groups, in red, white and
cyan, respectively, and heptane carbon groups in blue. Heptane is shown as
ghost pale blue for $x=0.1$ and $0.2$. Ethanol carbon groups are also shown as
ghost pale blue for $x=0.$2\. (see text for details).

For ethanol concentration $x=0.1$, full ethanol molecules are shown (oxygen
atom in red, hydrogen in white and carbon groups in cyan) while heptane
molecules are shown in transparent mode. It can be seen that small droplets of
ethanol float in the midst of heptane solvent. For $x=0.$3, only the hydroxyl
groups are fully shown, with all carbon groups of both species shown in
transparent mode. This way, hydroxyl clusters and their distribution are
highlighted. For $x=0.5$ and $x=0.$8, ethanol molecules are fully shown like
for $x=0.1$, and heptane molecules are shown in blue. These last 2 snapshots
highlight the clear micro-segregation of both species.

Fig.5 shows a zoom on the hydroxyl group clustering for the same 4 ethanol
concentrations highligted in Fig.4. For $x=0.1$ one sees that the small
ethanol droplets of Fig.4 consist in fact in pentameric rings of hydroxyl
groups, forming ethanol pentameric “micelles”. This structure vanishes for the
other higher ethanol concentrations shown in Fig.5, replaced by more complex
cluster structures, such as globules and various type of chains
conformations..

Figure 5: Zoom over the ethanol hydroxyl group clusters for the cases shown in
Fig.4. (Box edges can be seen as blue lines).

Detailed studies of snapshots for the entire concentration range shows that
ethanol molecules start to spontaneously form these pentameric micelles at the
smallest ethanol concentrations studied herein, which is $x=0.02$. These
pentameric micelle structures persist until $x=0.3$, after which larger
aggregated structures take over. The visual inspection of formed structure
tends to confirm that there are 2 regime of clustering, pentameric ethanol
micelles for $x<0.3$ and larger clusters for higher concentrations. We now
confirm this through cluster and pair correlation function studies.

####  3.2.2 Cluster structure

The cluster size probability distributions were calculated as:

$P_{n}=\frac{\sum_{k}s\left(n,k\right)}{\sum_{k}\sum_{n}s\left(n,k\right)}$
(8)

where $P_{n}$ is the probability for the cluster formed of _n_ sites,
$s\left(n,k\right)$ represents the number of clusters of the size $n$ in the
configuration $k$. $P_{n}$ is obtained by averaging the number
$s\left(n,k\right)$ of clusters of size _n_ over several such configurations.
The cut-off distance ($r_{c}$) for a calculation depends on the first minimum
in the pair correlation function of a particular site. For the oxygens in
ethanol, the $r_{c}$ = 3.7 Å.

Fig.6 shows the size $s$ dependence of the cluster distributions $P(s)$ in 2
different ways. Left panel (a) shows the distribution as function of the
cluster size, and for different ethanol concentrations. What is apparent is
the pentamer peak, which exists for all concentrations. In addition, we
observe the appearance of a shoulder-like feature for large clusters, showing
that the probability of larger clusters is not negligible for ethanol
concentrations above $x>0.3$. Moreover, while the probability of monomers is
higher than that of pentamers for $x<0.3$, and becomes comparable or lower for
$x>0.3$.

Figure 6: Cluster distribution probability function $P(s)$ as function of (a)
cluster size $s$ and (b) ethanol concentration. (See text for details).

The right panel (b) shows $P(s)$ as function of ethanol concentration, and for
different types of $n$-mers. What is strikingly apparent here is that, for the
entire concentration range, there is a sharp fall of P(s) from monomer to
dimers and 3-mers, followed by a second dramatic increase for $4$-mers and
$5$-mers, followed by $6$-mers, while higher $n$-mers are the same level as
monomers and dimers. An important feature is that the maximum of the curves
seems to saturate around $x=0.3$, indicating a turnover of cluster structures
above this concentration.

The cluster study complements the visual inspection of Fig.5 by showing more
details, such as for example the existence of 4-mers at small concentrations,
and not only pentamers, as the visual inspection may suggest.

####  3.2.3 Atom-atom pair correlation functions

In order to further confirm the sharp separation of cluster structure
highlighted in the previous sections, we show in Fig.7 the evolution of 3
typical pair correlation functions as a function of ethanol concentration. The
right panel shows the ethanol oxygen-oxygen correlation function $g_{OO}(r)$
for all the ethanol concentrations calculated. Focusing on the second peak,
which represent second neighbour correlations, hence the influence of
clustering beyond the first neighbour peak, we clearly a difference between
two successive concentrations: the gap is wider for $x<0.3$. In order to
facilitate this observation, the curve for x=0.3 is marked in thicker orange
color.

Figure 7: Ethanol concentration dependance of selected atom-atom pair
correlation functions. (a) Ethanol oxygen-oxygen pair correlation functions
$g_{OO}(r)$; (b) ethanol oxygen heptane 1st carbon group cross correlations
$g_{OM_{1}}(r)$: (c) heptane 1st carbon group correlations
$g_{M_{1}M_{1}}(r)$. The ethanol concentration $x=0.3$ is marked at thicker
orange line.

The narrower gap for $x>0.3$ can be interpreted as lesser difference in the
second neighbour correlations as a mark of insensitivity for the concentration
dependence. This is consistent with hydroxyl clusters being of more varied
shapes. In contrast, the almost similar gap for x<0.3 shows a linear
dependence in concentration, which would be expected if clustering was the
same and would only depend on the concentration of the pentamers.

In order to further confirm this trend, we examine in the middle panel of
Fig.7 the cross species correlations $g_{OM_{1}}(r)$ between the ethanol
oxygen and the heptane first(last) carbon group termed M1. These correlations
are seen to be less concentration dependent than the previously examined
$g_{OO}(r)$ correlations. This is expected, since heptane site are not
charged, hence only Lennard-Jones like correlations exist, which are less
prominent then for those between charged groups [42]. Nevertheless, we observe
that for $x<0.3$ all correlation functions are nearly superposed (below the
thick orange curve), and start to show appreciable differences only when
$x>0.3$. Again, this is fully consistent with the existence of same type of
clusters for x<0.3. Indeed, is clustered objects are the same, the
correlations between the carbon sites of heptane and the meta objects would be
nearly similar, and very weakly dependent on x. In contrast, if the clustered
objects are very different in shape, and if this depends strongly on ethanol
concentration, we would indeed expect a larger concentration dependence of
$g_{OM_{1}}(r)$.

The last right panel Fig.7 shows a weak concentration correlations between the
heptane last/first carbon site $g_{M_{1}M_{1}}(r)$, very similar to that
observed for $g_{OM_{1}}(r)$. First of al these are Lennard-Jones like
correlations, and correlations below x<0.3 are also nearly superimposed,
witnessing the same insensitivity to concentration in presence of same type of
aggregates of the ethanol.

If we reconsider now the experimental KBI in the light of this sharp
separation of the behaviour of the correlations for $x<0.3$ and for $x>0.3$,
we obtain the result shown in Fig.8, where the thick orange vertical line at
x=0.3 perfectly separates the

Figure 8: Illustration of the two ethanol concentration regions for each
extremum of the KBI, separated by $x=0.3$ (vertical orange line). Left part
for $x<0.3$ concerns ethanol micellar entities, while right part $x>0.3$
concerns standard micro-segregated ethanol-heptane domains.

concentrations under the first KBI extremum for x$<0.3$, where mostly
pentameric ethanol micelles are observed, from the concentrations under the
second weaker peak at right for $x<0.3$, for which usual micro-segregation of
the 2 species is observed. In order to fully confirm this picture of sharp
separation between a meta-object mixture and a usual mixture, we need to
calculate the KBI from the simulations and reproduce the same KBI shapes as
observed in experiments.

###  3.3 Kirkwood-Buff integrals from simulations

The evaluation of the KBI through computer simulation requires that the
asymptotes of the various site-site correlations are well defined and
converging to 1 as expected. Recent investigations have shown that there are 2
major obstacles. The first obstacle is that computer simulations conducted in
the Canonical or Isobaric ensemble cannot lead to the proper asymptote 1, this
value being reached only in the Grand Canonical ensemble [55, 56, 57]. The
second obstacle concerns the existence of micro-segregation, whose spatial
extent and slow kinetics alters the statistics of the correlations at large
atom-atom separations. Both cases are illustrated below, where we examine both
the tail of the correlation functions $g_{ab}(r),$but also the so-called
running KBI defined as [1]:

$G_{ab}(r)=4\pi\int_{0}^{r}ds\,s^{2}\left[g_{ab}(s)-1\right]$ (9)

which, when $r$ is large enough, is expected to converge to the KBI defined in
Eq.(1).

In a first example, we examine the case of ethanol concentration $x=0.7$, for
which we have shown above that micro-segregation is dominant. The typical
order parameter for micro-segregation is $g_{OM_{1}}(r)$, the cross species
pair correlation between the ethanol oxygen atom and the first/last carbon
group atom of heptanol. Fig.9 shown this function in the main panel for 3
different runs of 5ns each. The strong depletion between adverse ethanol and
heptane nano-domains is clearly visible, as all 3 curves stay below 1 until
$20\mathring{A}$ or so.

Figure 9: Long range tail contribution for ethanol-heptane cross correlation
function $g_{OM_{1}}(r)$ for ethanol concentration $x=0.7$ shown in the main
panel for 3 different runs of 5ns each. The upper inset is a close-up of the
tail part. The lower inset shows the respective RKBI $G_{OM_{1}}(r)$
functions, along with the experimental KBI (in blue).

On this scale, all 3 curves seem nearly similar, which is what is expected
from statistics. However, the zoom on tail region in the upper inset shows
that there are visible differences between the 3 runs, as the segregation of
domain is not the same in each of the 3 runs. What is observed is the typical
domains oscillatory correlations, with half-period about $20\mathring{A}$,
which is the depletion extent. All 3 curves oscillated about the asymptote 1,
hence the first obstacle mentioned is not really apparent. The lower inset
shows the corresponding running KBI $G_{OM_{1}}(r)$, as well as the expected
experimental value. It is seen that the various runs have an oscillatory
asymptotic feature, witnessing the domains alternation, but oscillate around
the experimental value. This result shows the rather excellent agreement
between the experimental KBI and the calculated one.

In the second example, we examine the case of ethanol concentration $x=0.2$,
for which we expect an homogeneous distribution of ethanol micelles in the
midst of heptane solvent. In this case, micro-segregation is very different in
that it generates these micelles. Fig.10 shows the same order parameter
$g_{OM_{1}}(r)$ in the main panel, and for 5 different runs of 5ns each.

Figure 10: Long range tail contribution for ethanol-heptane cross correlation
function $g_{OM_{1}}(r)$ for ethanol concentration $x=0.2$ shown in the main
panel for 5 different runs of 5ns each. The upper inset is a close-up of the
tail part. The lower inset shows the respective RKBI $G_{OM_{1}}(r)$
functions, along with the experimental KBI (in purple).

This function again shows appreciable depletion, but of the size of the
ethanol micelles, which is more about $12\mathring{A}$. But now, the zoom of
the tail in the upper inset shows that there are no domain oscillations, which
is indeed expected if the ethanol droplets are homogeneous distributed.
However, we observe that all the asymptote are clearly shifted upwards from 1.
This is a direct manifestation of the first obstacle mentioned above, which
occurs here since our calculations are in the isobaric ensemble instead of the
Grand canonical ensemble. It can be shown that the asymptote of the
correlation functions has the following form

$\lim_{r\rightarrow\infty}g_{ab}(r)=1-\frac{\epsilon_{ab}}{N}$ (10)

where $N$ is the number of particles in the simulation box, and the value of
$\epsilon_{ab}$ depend both on species $a$,$b$ and the statistical ensemble
[55, 56, 57]. For like correlations the shift is downwards from 1, and for
unlike it is upwards [42]. The RKBI are shown in the lower inset, and are seen
to lie above the experimental value, and not to have the expected flat
asymptote for a proper defintition of the KBI value for $G_{OM_{1}}$. In order
to correct for this “spurious” behaviour, we simply multiply the pair
correlation function with the appropriate coefficient
$\gamma_{ab}=1/\left(1-\frac{\epsilon_{ab}}{N}\right)$ before applying Eq.(10)
to $\bar{g}_{OM_{1}}(r)=\gamma_{OM_{1}}g_{OM_{1}}(r)$, which now has the
correct asymptote 1. The corresponding results are shown in Fig.11.

Figure 11: Tail corrected version of pair correlation functions shown Fig.10
(see text)

Due to the extreme smallness of $\frac{\epsilon_{ab}}{N}$, $\gamma_{ab}$ is
nearly 1 and the main panel shows no difference with respect for Fig.10.
However, while the upper inset confirms that the correct asymptote 1 has been
set by the operation above, the lower inset shows that all the RKBI have now
flat asymptotes, even though it is not quite the value set by the experiments.
This discrepancy is expected, since the ethanol aggregation in the real
mixtures may not be well represented by the model simulations.

For the calculations of the KBI, from the cross KBI $G_{OM_{1}}$ we extract D
from Eq.(6) and calculate the like KBI $G_{OO}$ and $G_{M_{1}M_{1}}$ using
Eq.(7). Indeed, the simulations results for these KBI do not often converge to
the consistency for D in Eqs.(7,6)), and the above trick is one way to get
around this problem. The origin of the inconsistency is due to system size
problems, since the like domain statistics require larger simulations, with
N=32000 particles or more [32, 33, 34]. When these operations are conducted
for all ethanol concentrations, we obtain the result shown in Fig.12, which
shows a remarkable agreement with the experimental results.

Figure 12: Comparison between the calculated KBI from simulations versus the
experimental ones [23].

The very low ethanol concentrations for $x=0.02$ and $x=0.05$ pose particular
problems for the proper evaluation of the asymptotes. Indeed, the
corresponding RKBI have very distorted shapes, which cannot be arranged to
look flat by simple shifting of the asymptotes of the pair correlation
functions. These distortions, which vary considerably from one run the other,
witness the strong kinetics of the ethanol micelle formation. Since these
micelles are about $15\mathring{A}$ in diameter, while the half box length
extends to $70\mathring{A}$, if the micelles were replaced by soft spheres of
same diameter, the statistics on the tail of the soft sphere correlations
would be much better. Therefore, it is really the micelle-monomer exchange
which affect those statistics at very small ethanol concentrations.

##  4 Discussion and Conclusion

It is not a priori obvious that the segregation of ethanol in small
concentration in oil-like solvent should evolve in a discontinuous manner into
the micro-segregation as this concentration is increased, leading to the
separation illustrated in Fig.8. For instance, in a previous study of ethanol-
benzene mixtures [42], we have found a single extremum shaped KBIs. Yet, this
study has shown that small micelle-like aggregates at low ethanol mole
fraction mixtures. This difference suggests that closed carbon group
molecules, such as benzene, affect differently the cluster structural changes
from micelle to domain, than chain-like shaped alkanes. Indeed, these latter
molecules can more easily merge with the micellar alkane corona than the disc-
shaped benzene molecules. In order to explain this apparent discrepancy, we
hypothetize here that it is solvent shape induced depletion entropic effects
[58, 59, 60] which affect the structural transition between the ethanol
micelles regime and the micro-segregated domain regime. Following this
hypothesis, chain-like solvent would allow a smoother transition between
micelles and domains, hence clearly separating the two manifestations in terms
of fluctuations, thus leading to separate fluctuation regimes.

In the abstract, we mention that water tert-butanol mixtures equally show two
extrema in KBI. This was indeed reported our previous work[12] , where the KBI
calculated by us (see Fig.6b in Ref.[12]) were obtained by Y. Koga from the
vapour pressure measurements of the chemical potentials (see the plot of $D$
in Fig.6a of Ref.[12]) . The double extrema is absent from the KBI data from
other authors (see Fig.6b in Ref.[12]), but present in the SANS data obtained
in the same work. Interestingly, in boths cases, the double extrema occurs
clearly only for the water-water KBI. In addition, the equivalent of the
micelle extrema - in the present case it would be tbutanol micellar
aggregates, occurs for very small tbutanol concentrations in the interval
$0.1<x<0.2$, which is similar to that observed in the present work for an
entirely different system. Interestingly, the aggregation in aqueous tbutanol
is not clearly understood to date, despite several experimental [61, 62] and
simulation [31, 32, 34, 63] investigations.

From these two points, it appears that the topic of the existence of camel
back shaped KBI deserves further investigations as to the conditions where it
might occur.

To conclude, the present study illustrates the difference between
concentration fluctuations and micro-segregation both in the microscopic and
macroscopic thermodynamic level. It is generally believed that micro-
segregation would correspond to a more microscopic $k\neq 0$ part of the
concentration fluctuations, related to the finite extent of segregated
domains, while concentration fluctuations would be their thermodynamic limit,
which correspond to $k=0$. The present study shown that the KBI, which are
related to the macroscopic $k=0$ limit of the structure factors, themselves
contain the difference in both manifestations. This was illustrated in Fig.8,
through the sharp separation between concentration fluctuations of meta-
objects (the ethanol micelles) for $x<0.3$ and the micro-segregation regime
for $x>0.3$. To be more specific, while micro-segregation concerns the initial
microscopic objects, namely ethanol and heptane molecules, the concentration
fluctuations described here concern the meta-objects formed by the ethanol
micelles, themselves floating in molecular heptane solvent. This study has
been possible, precisely because heptane is an inert solvent, due to its
uncharged carbon group sites, which cannot form associated entities. In this
context, heptane concentration fluctuations are very small, and the neutrality
of this oil-solvent enhances the charge association of ethanol molecules. This
is not possible to observe with water-solvent and small alcohol molecules,
because both species tend to associate. In order to observe a similar
phenomena in water, one requires much larger solutes molecules, such as
surfactants, which can form micelles and other self-assembled objects.

The present study also illustrates the importance of the pair correlation
functions and their asymptotes, and through these quantities, the issues
related to domain segregation and concentration fluctuations in finite size
simulations, in particular through the Lebowitz-Percus shift of the
asymptotes, illustrate here for the fluctuations between meta-objects, as
opposed to the usual illustration for molecular objects.

## Appendix

Eq.(2) in the Kirkwood-Buff paper [1] expresses the derivative of the chemical
potential $\mu_{i}$of species i with respect to its mole fraction $x_{i}$ in
terms of the KBI, which we rewrite below using a trivial rearrangement of the
original equation as:

$\left(\frac{\partial\beta\mu_{i}}{\partial
x_{i}}\right)_{T,P}=\frac{1}{x_{i}\left[1+\rho_{j}x_{i}\left(G_{11}+G_{22}-2G_{12}\right)\right]}$

The Ornstein-Zernike equation for a binary mixture can be written in a matrix
equation as

$\left(\begin{array}[]{cc}S_{11}&S_{12}\\\
S_{12}&S_{22}\end{array}\right)\left(\begin{array}[]{cc}C_{11}&C_{12}\\\
C_{12}&C_{22}\end{array}\right)=I$

where $I$ is the identity matrix, and where the structure matrix elements are
$S_{ij}(k)=\delta_{ij}+\sqrt{\rho_{i}\rho_{j}}\tilde{h}_{ij}(k)$ and the
C-matrix elements are defined in terms of the direct correlation functions as
$C_{ij}(k)=\delta_{ij}-\sqrt{\rho_{i}\rho_{j}}\tilde{c}_{ij}(k)$, with the
Fourier transform notation and definition $\tilde{f}(k)=\int
d\vec{r}f(r)\exp(i\vec{r}.\vec{k})$. Since the KBI are defined as in Eq.(1),
one has $G_{ij}=\tilde{h}_{ij}(k=0)$, and
$S_{ij}=\delta_{ij}+\sqrt{\rho_{i}\rho_{j}}G_{ij}.$ Therefore, one can rewrite
the equation above as (using $\rho_{i}=x_{i}\rho)$

$\left(\frac{\partial\beta\mu_{i}}{\partial
x_{i}}\right)_{T,P}=\frac{1}{x_{i}\sqrt{x_{1}x_{2}}\left[S_{11}+S_{22}-2S_{12}\right]}$

which can be rewritten by using the OZ equation as

$\left(\frac{\partial\beta\mu_{i}}{\partial x_{i}}\right)_{T,P}=\frac{\det
C}{x_{i}\sqrt{x_{1}x_{2}}\left[C_{11}+C_{22}+2C_{12}\right]}$

Using Eq.(4) which defines the KBI coefficient $D$, one has the relation
relating D to the determinant of C, which controls the stability limit of a
mixture [3], as written in Eq.(5)

$D=\alpha\left[C_{11}C_{22}-C_{12}^{2}\right]$

with $\alpha=1/(\sqrt{x_{1}x_{2}}\left[C_{11}+C_{22}+2C_{12}\right]).$

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