# Highly efficient visible and near-IR photon pair generation with thin-film
lithium niobate

Nathan A. Harper  1,${\dagger}$ Emily Y. Hwang  2,${\dagger}$ Ryoto Sekine  3
Luis Ledezma  3 Christian Perez  4 Alireza Marandi  2,3 and Scott K. Cushing1
1Division of Chemistry and Chemical Engineering, California Institute of
Technology, Pasadena, California 91125, USA
2Department of Applied Physics and Materials Science, California Institute of
Technology, Pasadena, California, 91125, USA
3Department of Electrical Engineering, California Institute of Technology,
Pasadena, California 91125, USA
4California State University, Los Angeles, 90032, USA
${\dagger}$These authors contributed equally to this work.
<EMAIL_ADDRESS>
###### Abstract

Efficient on-chip entangled photon pair generation at telecom wavelengths is
an integral aspect of emerging quantum optical technologies, particularly for
quantum communication and computing. However, moving to shorter wavelengths
enables the use of more accessible silicon detector technology and opens up
applications in imaging and spectroscopy. Here, we present high brightness
($(1.6\pm 0.3)\times 10^{9}$ pairs/mW/nm) visible-near-IR photon pair
generation in a periodically poled lithium niobate nanophotonic waveguide. The
degenerate spectrum of the photon pairs is centered at 811 nm with a bandwidth
of 117 nm. The measured on-chip source efficiency of $(2.3\pm 0.5)\times
10^{11}$ pairs/mW is on par with source efficiencies at telecom wavelengths
and is also orders of magnitude higher than the efficiencies of other visible
sources implemented in bulk crystal or diffused waveguide-based technologies.
These results represent the shortest wavelength of photon pairs generated in a
nanophotonic waveguide reported to date by nearly an octave.

††journal: opticajournal††articletype: Research Article

##  1 Introduction

Spontaneous parametric downconversion (SPDC) has been used for decades to
produce quantum entanglement in various photonic degrees of freedom, serving
as a workhorse in emerging quantum optical technologies. Compared to most
nonlinear processes, SPDC is relatively inefficient, requiring over one
million pump photons to produce one pair of entangled photons in even the
highest performing crystals. However, recent advances in nanophotonics,
particularly in thin-film lithium niobate (TFLN), have enabled significantly
more efficient frequency conversion and quantum state generation[1, 2, 3]
through sub-$\mu$m interaction areas, high nonlinearities, and low material
losses[4, 5]. By exploiting this platform, many recent demonstrations of SPDC
in TFLN [6, 7, 8, 9] have achieved efficiencies three orders of magnitude
greater than that of bulk crystal-based sources[10] and one order of magnitude
greater than that of large diffused waveguide-based sources[11]. To date, most
TFLN-based photon pair sources are designed for SPDC at telecom wavelengths
because of the low losses in optical fibers at 1550 nm[12, 13] and back-
compatibility for applications such as quantum communication[14],
computing[15], and a globally connected quantum network[16].

Although telecom photons are preferred for quantum information applications,
visible and near-infrared photons are generally better suited for imaging and
spectroscopy. Experiments at these wavelengths can take advantage of multi-
pixel detectors such as electron-multiplying charge coupled devices (EM-CCD)
and single photon avalanche detector (SPAD) arrays, enabling the measurements
needed for imaging[17, 18, 19] and characterization of high-dimensional
entangled states[20, 21]. Furthermore, the electronic transitions of molecules
and atoms are accessible at near-IR wavelengths, allowing for fluorescence
lifetime measurements[22, 23, 24], compatibility with quantum memories[25],
and fundamental studies of few-photon nonlinearities[26, 10, 27]. More
generally, near-IR and visible photons can be detected with high quantum
efficiency and low dark noise using existing mature silicon technology at room
temperature, compared to near-IR-IR detectors which require cryogenic
cooling[28]. Despite these advantages, all demonstrations of nanophotonic pair
production have resided in the telecom region, and the best near-IR and
visible photon pair sources are still large-area waveguides[29, 30, 31, 32]
and bulk periodically poled crystals[29, 33, 34, 35, 10, 36, 26]. Potential
reasons for this discrepancy stem from the difficulty in fabricating visible
nonlinear circuits on thin-film lithium niobate due to factors such as the
ultra-short poling periods required for quasi-phase matching and losses from
material absorption[37, 38, 39] and scattering[40]. In spite of these
difficulties, high-performance visible devices in thin-film lithium niobate
are becoming increasingly common for classical applications such as electro-
optic modulation and second harmonic generation[41, 42, 43, 44, 45].

Figure 1:  (a) Schematic of the periodically poled lithium niobate
nanophotonic waveguide. (b) Second harmonic microscopy image of the periodic
poling. The poling electrode period in this image is 2 $\mu$m. (c) Mode
profiles and waveguide geometry of the fundamental quasi-TE modes at the
designed pump and SPDC center wavelengths. (d) Refractive indices and
corresponding poling periods for a range of SPDC wavelengths.

Here, we extend TFLN-based SPDC sources by nearly an octave in frequency to
produce high-brightness photon pairs in the visible and near-IR. The device
produces an on-chip efficiency of $(2.3\pm 0.5)\times 10^{11}$ pairs/mW, which
corresponds to a per-photon conversion efficiency of more than 1 photon pair
converted in every 10,000 pump photons ($(1.1\pm 0.2)\times 10^{-4}$
pairs/photon). This efficiency is nearly two orders of magnitude better than
visible-light diffused waveguide SPDC sources[31] and is on par with other
TFLN sources in the telecom regime[8]. The SPDC from this device exhibits a
broad spectrum centered at 811 nm with a degenerate FWHM bandwidth of 117 nm
and an average brightness of $(1.6\pm 0.3)\times 10^{9}$ pairs/mW/nm, a number
limited by at least an order of magnitude by the pump laser linewidth (0.8
nm). Consistent with this bandwidth, we measure an ultrashort coherence time
of $\sim$40 fs for the entangled photons with an indistinguishability of
$100.\pm 1\%$. Our results therefore show that, although pumped at wavelengths
near what would usually be considered its cutoff range, TFLN can equally be a
platform for visible-near-IR entangled photon applications as it is at telecom
wavelengths.

##  2 Device design and fabrication

The periodically poled lithium niobate waveguides (Figure 1a) were simulated
in Lumerical MODE to determine the quasi-phase matching poling period. The
guided modes at the design pump wavelength (406 nm) and SPDC center wavelength
(812 nm) were simulated using the bulk Sellmeier coefficients of lithium
niobate[46] and silicon dioxide[47] with the geometric parameters shown in
Figure 1c and a 60°sidewall, which is consistent with the fabrication process.
To take advantage of lithium niobate’s largest nonlinear tensor element
($d_{33}$ = 28 pm/V)[48], only the fundamental quasi-transverse electric (TE)
modes were considered. An etch depth of 420 nm and a top width of 1.5 $\mu$m
with a total LN thickness of 600 nm were targeted for ease of optical
coupling, fabrication, and $\geq 2~{}\mu$m poling period while providing high
performance. For these parameters, the effective refractive indices
($n_{\text{eff,pump}}$ = 2.29, $n_{\text{eff,SPDC}}$ = 2.09) result in a
quasi-phase matching poling period of $\Lambda=\lambda_{\text{pump}}/\Delta
n_{\text{eff}}=2.03~{}\mu\text{m}$ at the target pump wavelength of 406 nm
(Figure 1d).

Figure 2:  Schematic for coupling and detection of the photon pairs. I,
isolator; ND, variable neutral density filter; HWP, half-wave plate; L,
aspheric lens; LPF, long-pass filter; emICCD, electron-multiplying intensified
charge-coupled device; BS, beamsplitter; D, single-photon avalanche detector;
TT, time-tagger; M, mirror. (a) Optical setup to couple into the TFLN
waveguide. (b) Characterization scheme to measure the photon pair spectra. (c)
Characterization scheme for coincidence counting. (d) Optical setup for the
Michelson interferometer.

The devices were fabricated from a 5% MgO-doped X-cut thin-film lithium
niobate on insulator wafer (NanoLN), which consists of 600 nm of lithium
niobate bonded to 2 $\mu$m of silicon dioxide on a 0.4 mm silicon substrate.
To quasi-phase match the SPDC sources, poling electrodes (7 mm long) were
first fabricated by performing a metal lift-off through electron beam
lithography with bilayer polymethyl methacrylate (PMMA) resist followed by
electron beam evaporation of titanium (15 nm) and gold (55 nm). The electrodes
were poled with a 490 V and 70 $\mu$s square wave, and the poled domain
formation was monitored with second harmonic microscopy[49] (Figure 1b). After
poling, waveguides were defined through an aligned electron beam lithography
step with hydrogen silesoxquiane (HSQ) resist followed by argon inductive
coupled plasma reactive ion etching to achieve an etch depth of 420 nm,
verified through atomic force microscopy. The chip facets were manually
polished to increase coupling efficiency. A broadband oscillator was used to
verify the phase matching wavelength through second harmonic generation and
was found to match with the computationally predicted second harmonic
wavelength within $\pm$3 nm. This small discrepancy was likely due to
fabrication tolerances, particularly in the etch depth and film thickness.

##  3 Device characterization

The spectrum, generation rate, and coherence properties of the entangled
photon pairs produced from the fabricated device are characterized as shown in
Figure 2. In these experiments, the room-temperature periodically poled
waveguide is pumped with a free-running laser diode (Coherent OBIS LX 405 nm)
to produce entangled pairs (Figure 2a). An antireflection-coated aspheric lens
(NA = 0.58, Thorlabs C140TMD-A) couples the free-space pump beam to the
fundamental TE mode of the waveguide. The photon pairs produced in the
fundamental TE mode are collected off-chip and collimated using a similar
aspheric lens (NA = 0.58, Thorlabs C140TMD-B).

The spectra of the entangled photon pairs are measured to assess the phase-
matching properties and tunability of the device (Figure 3). Pairs collected
from the waveguide are transmitted to a grating spectrometer and measured
using an electron-multiplying intensified camera (Figure 2b). To tune the SPDC
emission, the center wavelength of the pump wavelength is varied from
405-406.4 nm by changing the drive current of the laser diode. Variable
neutral-density filters are used to keep the pump power for all the collected
spectra at a consistent 10 $\mu$W. In doing so, three distinct phase matching
regions are explored (Figure 3a): 1) at long pump wavelengths, the phase
matching condition is not satisfied and emission is not observed; 2) from
405.6-405.9 nm, the degenerate wavelengths are phasematched; and 3) at short
pump wavelengths, the spectrum splits and nondegenerate emission extending to
the cutoff wavelength of the filter is observed. The dip in intensity in
Figure 3a around 760 nm is likely due to impurities in the thin film,
contamination during the device fabrication process, or ambient absorption.
Due to the linewidth of the laser used in the experiment (0.8 nm FWHM), the
spectra are considerably broadened compared to the spectra expected from a
single-frequency pump laser (Supplemental Figure S3). Nevertheless, experiment
and theory reach qualitative agreement by factoring the pump linewidth into
the calculations (Figure 3b). For all subsequent experiments, a laser center
wavelength of 405.7 nm is used for degenerate phase matching. The resulting
spectrum (Figure 3c) is centered at 811.4 $\pm$ 0.7 nm with a FWHM bandwidth
of 117 nm (53 THz) that accounts for 85% of the overall flux.

Figure 3:  (a) Measured and (b) theoretical SPDC spectra as a function of pump
wavelength. (c) Lineouts of the measured and theoretical SPDC spectra at a
pump wavelength of 405.7 nm, corresponding to the dashed lines in (a) and (b).

The pair generation efficiency of the device is measured through coincidence
counting between two SPADs (Figure 2c). In this experiment, the pairs from the
chip were split at a 50:50 broadband beamsplitter (Thorlabs BS014) and coupled
to multimode optical fibers connected to the detectors. Coincidence detection
events between the SPADs were recorded with a time-tagger (Picoquant PicoHarp
300). Figure 4a shows a representative coincidence histogram recorded at 45 nW
of pump power. The temporal correlation in this graph (3.4 ns FWHM) is given
by the response time of the SPADs and not the entangled photon correlations
(see Figure 5 later in the text). The coincidence counts are corrected by
background subtraction of the number of counts in a 9.5 ns window at the
histogram peak from the number of counts in another 9.5 ns window in a
background region far from the peak. Sweeping the laser power with a neutral
density filter yields the curves in Figure 4b-c, which are linearly fit to
determine the pair generation efficiency. To account for the wavelength
dependence of the SPAD quantum efficiency (Supplemental Equation S11), all
wavelengths in the spectrum are integrated over to calculate the average
detection efficiency for single photons ($\eta_{1}=0.52$) as well as the
average joint pair detection probability ($\eta_{12}=0.27$). Including a
factor of 2 due to the probability of splitting pairs at the beamsplitter
yields Equation 1 for the measured efficiency of the source:

$E=\frac{m_{1}m_{2}}{m_{c}}\frac{\eta_{12}}{2\eta_{1}^{2}}$ (1)

Here $E$ is the pair generation efficiency, $m_{1}$ and $m_{2}$ are the
singles rates at the two detectors, and $m_{c}$ is the rate of coincidences,
all in units of counts/mW. Accounting for the 10.2 dB transmission loss of the
pump laser into the waveguide, a pair generation efficiency of $(2.3\pm
0.5)\times 10^{11}$ pairs/mW is measured, which is equivalent to a per-pump-
photon efficiency of $(1.1\pm 0.2)\times 10^{-4}$ pairs/photon. Over the 117
nm FWHM bandwidth of the spectrum, this efficiency translates to an average
brightness of $(1.6\pm 0.3)\times 10^{9}$ pairs/nm/mW. The uncertainties here
and throughout this work are reported as one standard deviation, derived from
the standard error in the fits of Figure 4b-c and the uncertainty in the
detector quantum efficiency. The ratio of singles counts to coincidence counts
suggests that the transmission of the SPDC from the waveguide to each of the
two detectors is 8.4 dB and 8.0 dB, respectively, which includes losses out of
the waveguide and of the free-space optics. Our theoretical efficiency
(Supplemental), including the FWHM of the pump laser, is $2.66\times 10^{11}$
pairs/mW, in close agreement with our experimental results.

Figure 4:  (a) Coincidence histogram at an input pump power of 45 nW including
accidentals. Measured (b) coincidence and (c) singles counts while sweeping
the input power. The fitted slopes (solid lines) produce the pair generation
efficiency of our device.

Finally, the two-photon interference is measured (Figure 2d) to demonstrate
the non-classical behaviour of the produced photon pairs. Figure 5 shows the
measured two-photon interferogram (Figure 5a) obtained from the device without
subtracting accidentals, as well as the one-photon interferogram (Figure 5c)
for comparison. Due to the aforementioned temporal resolution of the SPADs
used here, the four unique paths through the interferometer are
indistinguishable and combine to yield the interference pattern. The important
features of the interferogram are as follows: 1) Near the zero path length
difference, photons are delayed within the coherence length of the source and
exhibit both one- and two-photon interference. A visibility of 100.$\pm$1% is
measured within the coherence length, with an uncertainty derived assuming
Poissonian statistics. This near-perfect visibility indicates good mode
overlap in the interferometer and indistinguishability of the photons within
the pair. 2) Far from the coherence length of the source (delays greater than
20 fs), interference between two photons taking different paths disappears,
which explains why the single-photon interference disappears in Figure 5c and
5e. Notably, interference between pairs of photons that travel together
through the interferometer persists with a fringe period at half the pair
wavelength and is shown in more detail in Figure 5d. This feature suggests
quantum interference due to the energy-time entanglement of the pairs, and
would not be observed if the light was generated from a coherent or a thermal
source with a similar spectrum[50]. A fringe visibility of $43\pm 3\%$ is
observed in this region far from the coherence length, which is close to the
theoretical maximum of 50% for this experiment due to the temporal resolution
of the detectors. The uncertainty in visibility is given from the standard
error in the fit of Figure 5d. The qualitative agreement between the measured
and theoretical two-photon interferogram (Figure 5a-b) suggests that SPDC and
genuine energy-time entanglement are being produced.

Figure 5:  The (a) measured and (b) simulated (Supplemental) two photon
Michelson interferogram for the device. (c) Corresponding singles counts out
of the interferometer, demonstrating single photon interference within the
coherence length of the source. The (d) coincidence counts and (e) singles
counts far from the coherence length of the source. Note that all measured
coincidence counts ((a) and (d)) do not have accidentals subtracted.

##  4 Discussion

Compared to the state of the art for visible photon pair sources, the device
presented in this work exhibits substantially improved brightness and
efficiency due to the small effective area of the waveguide. The device’s
performance against reported literature values is plotted in Figure 6 using
efficiency and wavelength as the figures of merit. Telecom and MIR TFLN
devices are also included, demonstrating comparable or improved performance.
At short wavelengths, the phase-matching bandwidth decreases due to group
velocity dispersion in lithium niobate, but the efficiency remains high
because the downconversion spectral power density scales with the inverse
fifth power of the SPDC wavelength ($\lambda_{s}^{-5}$)[29]. In addition to
the benefits of higher-energy photon pairs, the strategy of decreasing the
SPDC wavelength to access higher efficiencies could enable single-photon
nonlinearities when integrated with a resonator[51]. The high efficiency of
this device, which is limited here by the linewidth of the pump laser, has
significant implications for practical uses of entangled photons, including
reducing integration times, allowing the use of low-power laser diodes for
pair generation, and reducing fluorescence and stray light. Even with the
losses present in this setup, high signal to noise coincidence measurements
are acquired with nanowatts of laser power, highlighting the potential of this
device.

The efficiency and brightness of the device can be further improved by
narrowing the bandwidth of the pump laser, as discussed further in the
Supplemental. A single-frequency pump is estimated to shrink the phase-
matching bandwidth from 117 nm to 35 nm, increasing the brightness by a
comparable factor. Furthermore, the SPDC process is most efficient near
degeneracy because the group velocity of the signal and idler are equal to
first order, so the efficiency could potentially increase by a factor of 5.
However, one benefit of using a multifrequency pump is that the device
sensitivity to the laser wavelength is reduced, yielding a higher stability
average response with greater bandwidth.

Figure 6:  Comparison of relevant literature SPDC sources to this work.
Horizontal error bars represent the reported bandwidth of the sources. Data
for the efficiency, center wavelength, and bandwidth are taken from Refs. [29,
33, 35, 52, 36, 53, 26, 10] for bulk crystal sources, Refs. [29, 30, 32, 31,
54, 55, 56, 57, 11, 58, 59] for large waveguide sources (including
micromachined and diffused waveguides), and Refs. [8, 7, 6, 9] for TFLN
sources. The yellow shaded region represents the typical telecommunication
wavelength window.

##  5 Conclusion

Efficient photon pair generation has been demonstrated with an integrated
thin-film lithium niobate waveguide at visible and NIR wavelengths (720-900
nm). An on-chip SPDC efficiency of $(2.3\pm 0.5)\times 10^{11}$ pairs/mW,
which is on-par with reported literature at telecom wavelengths, has been
produced near the usually associated cut-off wavelengths for the pump (406 nm)
in TFLN. The photon pair spectra has an average brightness of $(1.6\pm
0.3)\times 10^{9}$ pairs/mW/nm, centered at 811 nm with a 117 nm bandwidth. To
date, these results are the shortest wavelength photon pairs generated in a
thin film platform by nearly an octave. The work opens up opportunities to
exploit the quantum advantage of integrated entangled photon circuits beyond
telecom to imaging and spectroscopy applications.

Funding Air Force Office of Scientific Research (FA9550-20-1-0040); Army
Research Office (W911NF-23-1-0048); National Science Foundation (EECS 1846273,
DGE-1745301); U.S. Department of Energy (DE-SC0020151).

Acknowledgments The authors gratefully acknowledge the critical support and
infrastructure provided for this work by The Kavli Nanoscience Institute (KNI)
and the Beckman Biological Imaging Facility at Caltech. This work was
additionally supported by the KNI-Wheatley Scholar in Nanoscience and the
Rothenberg Innovation Initiative. N.H. was supported by the Department of
Defense (DoD) through the National Defense Science and Engineering Graduate
(NDSEG) Fellowship Program. E.H. was supported by the National Science
Foundation Graduate Research Fellowship Program under Grant no. DGE-1745301.
Any opinion, findings, and conclusions or recommendations expressed in this
material are those of the authors(s) and do not necessarily reflect the views
of the National Science Foundation.

Disclosures The authors declare no conflicts of interest.

Data Availability Statement Data underlying the results presented in this
paper are not publicly available at this time but may be obtained from the
authors upon reasonable request.

Supplemental document See Supplement 1 for supporting content.

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# Effects of fluctuations on correlation functions in inhomogeneous mixtures

A. Ciach${}^{\textsf{\footnotesize{\mbox{?}}}}$, O.
Patsahan${}^{\textsf{\footnotesize{\mbox{?}}}}$, A.
Meyra${}^{\textsf{\footnotesize{\mbox{?},\penalty 1000\mbox{?}}}}$

(Received January 14, 2020, in final form February 16, 2020)

###### Abstract

Approximate expressions for correlation functions in binary inhomogeneous
mixtures are derived in a framework of the mesoscopic theory [Ciach A., Mol.
Phys., 2011, 109, 1101]. Fluctuation contribution is taken into account in a
Brazovskii-type approximation. Explicit results are obtained for two model
systems. In the two models, the diameters of the hard cores of particles are
equal, and the interactions favour a periodic arrangement of alternating
species A and B. However, the optimal distance between the species A and B is
much different in the two models. Theoretical results for different
temperature and volume fractions of the two components are compared with the
results of Monte Carlo simulations, and the structure is illustrated by
simulation snapshots. Despite different interaction potentials and different
length scale of the local ordering, properties of the correlation functions in
the two models are very similar.

Key words: correlation functions, inhomogeneous mixtures, mesoscopic theory

###### Abstract

Â ðàìêàõ ìåçîñêîïíî¿ òåîð¿ [A. Ciach, Mol. Phys., 2011, 109, 1101] âèâåäåíî
íàáëèæåí âèðàçè äëÿ êîðåëÿöéíèõ ôóíêöé â áíàðíèõ íåîäíîðäíèõ ñóìøàõ.
Ôëóêòóàöéíèé âíåñîê âðàõîâóòüñÿ â íàáëèæåíí Áðàçîâñüêîãî. Îòðèìàíî ÿâí
ðåçóëüòàòè äëÿ äâîõ ìîäåëüíèõ ñèñòåì. Â îáîõ ìîäåëÿõ, äàìåòðè òâåðäîãî êîðó
àñòèíîê îäíàêîâèìè, à âçàìîä¿ ñïðèÿþòü ïåðîäèíîìó åðãóâàííþ â ðîçòàøóâàíí
ñîðòâ A B. Ïðîòå, îïòèìàëüíà âäñòàíü ìæ ñîðòàìè A B ñóòòâî ðçíèòüñÿ â äâîõ
ìîäåëÿõ. Òåîðåòèí ðåçóëüòàòè äëÿ ðçíèõ çíàåíü òåìïåðàòóðè îá’ìíèõ ôðàêöé äâîõ
êîìïîíåíò ïîðâíþþòüñÿ ç ðåçóëüòàòàìè ìîäåëþâàííÿ ìåòîäîì Ìîíòå Êàðëî, à
ñòðóêòóðè ïðîëþñòðîâàí ìèòòâèìè çíèìêàìè ñèñòåìè. Íåçâàæàþè íà òå, ùî
ïîòåíöàëè âçàìîä¿ ìàñøòàáè ëîêàëüíîãî âïîðÿäêóâàííÿ â äâîõ ìîäåëÿõ ðçíèìè,
âëàñòèâîñò êîðåëÿöéíèõ ôóíêöé öèõ ìîäåëåé äóæå ïîäáíèìè.

Ключов слова: êîðåëÿöéí ôóíêö¿, íåîäíîðäí ñóìø, ìåçîñêîïíà òåîðÿ

##  1 Introduction

Biological and soft-matter systems are typically multicomponent and
inhomogeneous. For different systems, the inhomogeneities in density or
concentration may appear on different length scales, ranging from the scale
set by the size of molecules through a few- to a few tens or even hundreds of
molecular diameters. For example, in ionic systems or in two-component
mixtures of highly charged colloid particles, the concentration difference
between the positively and negatively charged ions or particles oscillates in
space in the crystalline phases on the length scale set by the size of the
ions or particles [1, 2]. In ionic liquids (IL) or molten colloidal crystals,
the inhomogeneities remain present, and are reflected in the oscillatory decay
of the correlation function for the concentration difference (or charge
density) on the same length scale [3, 4, 5, 6]. In this case, the
inhomogeneity or local order actually means that the distribution of the
components in the majority of microscopic states in the disordered phase
differs significantly from the random distribution.

In the case of weakly charged colloid particles with solvent-induced effective
short-range attraction, clusters of particles of various sizes and shapes or
other assemblies are formed, which leads to inhomogeneities on a mesoscopic
length scale [7, 8, 9, 10, 11, 12]. Similar inhomogeneities or self-assembly
into different aggregates can occur in other systems with interactions between
the particles of the form of the so-called ‘mermaid’ or SALR potential [13],
for example in globular proteins in water [7, 14, 15]. In the SALR class of
potentials, the interactions consist of short range attraction (SA) and long-
range repulsion (LR). At low temperature, a universal sequence of the ordered
phases (lyotropic liquid crystals) was predicted [16, 17, 18, 19] and was
found in simulations [20] for increasing the volume fraction of the particles.
When the density increases, there is observed the formation of a cluster
crystal of cubic symmetry, hexagonal arrangement of cylindrical clusters,
gyroid network of particles, lamellas, gyroid network of voids, cylindrical
voids and spherical voids. When the ordered phases melt, the aggregates start
to move freely leading to position-independent density. The local order is
reflected in the oscillatory decay of correlations on the length scale set by
the size of the aggregates. Importantly, even though the average density is
position independent, almost all simulation snapshots show the formation of
aggregates [9, 12, 11]. The random distribution of particles is not observed
in practice. Only at high temperature, the aggregates become disintegrated,
and the random structure domintes in the microscopic states of the disordered
phase. In a fixed mesoscopic window of the macroscopic volume, the density is
either much larger or much smaller than the average density, when an aggregate
either enters or leaves the window. This means that very large fluctuations of
the local density take place. As a result, the internal energy obtained by a
proper averaging of the energy of the microscopic states differs significantly
from the energy calculated in the mean-field (MF) approximation, i.e., for the
average density. This is because when the aggregates of a size determined by
the range of attraction are separated by distances larger than the range of
repulsion, there are much more pairs of attracting particles and much less
pairs of particles that repel each other than in the state with a position-
independent density (homogeneously distributed particles). That is why the
energy of the majority of states in the inhomogeneous disordered phase is much
lower than the energy calculated for the disordered phase in MF. The latter is
the same for homogeneous and inhomogeneous systems. The much overestimated
internal energy leads to instability of the disordered phase with respect to
density modulations, and to a continuous phase transition in MF that in
reality is absent. The spurious instability of the disordered inhomogeneous
phase in MF leads to divergent correlation functions that in reality do not
diverge. Mesoscopic fluctuations of the local density, representing a
different density inside the clusters and between them, i.e., the variance of
the local density or concentration, should be taken into account to restore
the stability of the disordered inhomogeneous phase and to lead correct
correlation functions and the first-order phase transition to the ordered
phases at lower temperature.

The effects of mesoscopic fluctuations can be taken into account within the
field-theory developed by Brazovskii [21]. This theory was adapted to
amphiphilic systems in [22, 23]. A mesoscopic density-functional theory with
the variance of the local density taken into account in the approach based on
the Brazovskii theory was developed in [16, 17, 24] for one-component systems,
and in [25] for mixtures. A similar approach was used for the study of ionic
systems in [5, 6, 26].

At the formal level, the variance of the local density can be obtained by
solving the self-consistent equation for the inverse correlation function in
the one-loop Hartree approximation [16, 27, 24]. The contribution to the grand
potential associated with the presence of inhomogeneities is calculated at the
same level of approximation. From the physical point of view, the variance of
the local volume fraction describes the average deviation of the volume
fraction inside the clusters or between them from the volume fraction averaged
over the whole volume. This average inhomogeneity leads to a negative
contribution to the internal energy, and to stabilization of the disordered
phase for a much larger part of the phase diagram than in MF. In a one-
component system, this theory leads to a correct high-$T$ part of the phase
diagram at a semi-quantitative level [24]. The mathematical form of the
correlation function for the volume fraction in this theory is the same as in
MF, namely

$G(r)=A\mathrm{e}^{-\alpha_{0}r}\sin(\alpha_{1}r+\theta)/r,$ (1.1)

but the dependence of the parameters on thermodynamic state is much different:
there is no spurious divergence of $A$ or $1/\alpha_{0}$. Equation (1.1)
should describe the asymptotic decay of correlations when the correlation
function in Fourier representation, $\tilde{G}(k)$, takes a maximum for
$k=k_{0}>0$ even in the exact theory. It can be obtained from the pole
analysis of $\tilde{G}(k)$, by taking into account the pair of poles
$\mathrm{i}\alpha_{0}\pm\alpha_{1}$ with the smallest imaginary part. However,
simulations show that equation (1.1) very well describes the correlation
function for the volume fraction or for the charge density in the SALR or IL
systems already for $r$ larger than one period of the damped oscillations [12,
28].

Much less attention was paid to inhomogeneous mixtures. In [25], only the
general formalism has been developed, and a few examples were considered only
in MF. In this article we focus on the structure of the disordered
inhomogeneous phase described by correlation functions for different
components. In section 2, we briefly summarize the Brazovskii-type formalism
generalized to mixtures in [25]. Next, we restrict ourselves to binary
mixtures and derive approximate equations for the correlation functions in
Fourier representation, $\tilde{G}_{\alpha\beta}(k)$. We restrict ourselves to
equal sizes of the hard cores of the particles, and consider two particular
examples of a binary mixture. In the first example, section 3, we assume
interactions leading to inhomogeneities on a length scale of $\sim 10\sigma$,
where $\sigma$ is a molecular size. We choose the ‘mermaid’ potential between
the like particles, but a pair of different particles interact with a
‘peacock’ potential having an attractive tail and a repulsive head. The tails
of the interactions correspond to screened electrostatic potentials (repulsion
between the like-particles, and attraction between the opposite charges) and
the short-range part of the interactions favours a phase separation of the two
species. ‘Two mermaids and a peacock’ effective interactions can occur between
oppositely charged hydrophilic and hydrophobic colloid particles suspended in
a near-critical mixture with ions [29].

In section 4, we assume that short-range interactions of the square-well are
formed only between different species. This potential can lead to a gas-liquid
phase transition, as well as to the ordered phase resembling an ionic crystal,
with the structure on the length scale of $\sigma$ [25]. We calculate three
correlation functions in the disordered phase for several state points for two
models both above and below the MF instability. In order to verify the
theoretical predictions and to visualize the structure for different state
points, we have performed MC simulations for the models considered. We
conclude in section 5.

##  2 Formalism

The theory is based on the mesoscopic formalism developed for inhomogeneous
mixtures with $n$ components in [25]. Instead of the number density
$\rho_{\alpha}$ of the $\alpha$-component, we consider the volume fraction in
the mesoscopic region around ${\bf r}$, $\zeta_{\alpha}({\bf r})$, and
deviations $\Delta\zeta_{\alpha}({\bf r})$ from the average value,
$\bar{\zeta}_{\alpha}({\bf r})$. The volume fraction is more suitable in the
mesoscopic theory, particularly for unequal sizes of the particles. When the
ordering occurs on the length scale much larger than the size of the
particles, then the volume fraction and the number density of the particles
with the volume $v_{\alpha}$ are related by
$\zeta_{\alpha}\approx\rho_{\alpha}v_{\alpha}$.

Fixed forms of the local volume fractions of all the components,
$\\{\zeta\\}=(\zeta_{1},\ldots,\zeta_{n})$, can be considered as a constraint
imposed on the microscopic states. Grand potential in the presence of the
above mesoscopic constraint can be written in the form
$\Omega_{\text{co}}=U-TS-\mu_{\alpha}N_{\alpha}$, where $U,S,N_{\alpha}$ and
$\mu_{\alpha}$ are the internal energy, entropy, the number of molecules and
the chemical potential of the species $\alpha$ respectively in the system with
the constraint of fixed $\\{\zeta\\}$ imposed on the microscopic densities.
$U$ is given by the expression

$U[\\{\zeta\\}]=\frac{1}{2}\int_{\bf r_{1}}\int_{\bf
r_{2}}\bar{V}_{\alpha\beta}(|{\bf r}_{1}-{\bf r}_{2}|)\zeta_{\alpha}({\bf
r}_{1})\zeta_{\beta}({\bf r}_{2}),$

where the element $\bar{V}_{\alpha\beta}(r)$ of the matrix $\bar{\bf V}(r)$ is
the interaction potential between the species $\alpha,\beta$, properly
rescaled when the volume fraction instead of the number density is considered.
In addition, the interaction potentials are multiplied by the function that
vanishes for $r=|{\bf r}_{1}-{\bf r}_{2}|$ smaller than the sum of the hard-
cores radii, and is equal to 1 otherwise. This way we avoid contributions to
the internal energy from the states with overlapping cores of the particles.
We further assume that the entropy $S$ satisfies the relation
$-TS=F_{\text{h}}$, where $F_{\text{h}}$ is the free-energy of the hard-core
reference system. We assume here a local-density approximation.

In this work we study the effects of fluctuations on the correlation functions
${\bf G}$, with the matrix elements $G_{\alpha\beta}({\bf r}_{\alpha},{\bf
r}_{\beta})=\langle\Delta\zeta_{\alpha}({\bf
r}_{\alpha})\Delta\zeta_{\beta}({\bf r}_{\beta})\rangle$ representing the
correlations between the volume fractions of the species $\alpha$ and $\beta$
at the positions ${\bf r}_{\alpha}$ and ${\bf r}_{\beta}$, respectively. In
order to include fluctuations, we introduce the functional $\beta
F[\\{\zeta\\}]$ of the form

$\beta
F[\\{\zeta\\}]=\beta\Omega_{\text{co}}[\\{\zeta\\}]-\log\left[\int\text{D}\phi_{1}\ldots\int\text{D}\phi_{n}\mathrm{e}^{-\beta
H_{\text{fluc}}}\right],$ (2.1)

where $\phi_{\alpha}({\bf r})$ is the local fluctuation of the volume fraction
of the component $\alpha$, and

$H_{\text{fluc}}[\\{\zeta\\},\\{\phi\\}]=\Omega_{\text{co}}[\\{\zeta\\}+\\{\phi\\}]-\Omega_{\text{co}}[\\{\zeta\\}].$

The functional (2.1) becomes equal to the grand potential, when
$\\{\zeta\\}=\\{\bar{\zeta}\\}$, with
$\\{\bar{\zeta}\\}=[\bar{\zeta}_{\alpha_{1}}({\bf
r}),\ldots,\bar{\zeta}_{\alpha_{n}}({\bf r})]$ that satisfy the minimum
condition for (2.1). By definition, $\langle\phi_{\alpha}\rangle=0$ when
$\\{\zeta\\}=\\{\bar{\zeta}\\}$.

Note that from (2.1) it follows that the vertex functions (related to the
direct correlation functions) defined by

${\cal C}_{\alpha_{1},\ldots,\alpha_{j}}({\bf r}_{1},\ldots,{\bf
r}_{j})=\frac{\delta^{j}\beta
F[\\{\bar{\zeta}\\}]}{\delta\bar{\zeta}_{\alpha_{1}}({\bf
r}_{1})\ldots\delta\bar{\zeta}_{\alpha_{j}}({\bf r}_{j})}$

consist of two terms: the first one is the contribution from the fluctuations
on the microscopic length scale with frozen fluctuations on the mesoscopic
length scale. This term is

${\cal C}^{\text{co}}_{\alpha_{1},\ldots,\alpha_{j}}({\bf r}_{1},\ldots,{\bf
r}_{j})=\frac{\delta^{j}\beta\Omega_{\text{co}}[\\{\bar{\zeta}\\}]}{\delta\bar{\zeta}_{\alpha_{1}}({\bf
r}_{1})\ldots\delta\bar{\zeta}_{\alpha_{j}}({\bf r}_{j})}.$ (2.2)

The second term is the contribution from the fluctuations on the mesoscopic
length scale.

We focus on the two-point inverse correlation function that satisfies the
analog to the Ornstein-Zernicke equation,

${\bf C}={\bf G}^{-1}.$ (2.3)

In the lowest-order nontrivial approximation, the matrix elements of the
inverse correlation function in Fourier representation are given by [25],

$\displaystyle\tilde{C}_{\alpha\beta}(k)=\tilde{C}^{\text{co}}_{\alpha\beta}(k)+\frac{A_{\alpha\beta\gamma\delta}}{2}{\cal
G}_{\gamma\delta}\,,$ (2.4)

where the first term is the function defined in (2.2) with $j=2$ in Fourier
representation, summation convention is used in the whole article, and

$\displaystyle{\cal G}_{\gamma\delta}=\int\frac{\mathrm{d}{\bf
k}}{(2\piup)^{3}}\tilde{G}_{\gamma\delta}(k).$ (2.5)

In the case of the disordered phase,
$\tilde{C}^{\text{co}}_{\alpha\beta}(k)=\beta\tilde{V}_{\alpha\beta}(k)+A_{\alpha\beta}$.
The matrix $\tilde{\bf V}(k)$ with the elements $\tilde{V}_{\alpha\beta}(k)$
is the interaction potential between the species $\alpha,\beta$ in Fourier
representation, and $A_{\alpha_{1}\ldots\alpha_{j}}$ is given by

$A_{\alpha_{1}\ldots\alpha_{j}}=\frac{\partial^{j}\beta
f_{\text{h}}(\\{\zeta({\bf r})\\})}{\partial\zeta_{\alpha_{1}}({\bf
r})\ldots\partial\zeta_{\alpha_{j}}({\bf r})}.$

Equations (2.3)–(2.5) should be solved self-consistently, which is not an easy
task, especially for periodic structures in multicomponent mixtures.

In this work we focus on a disordered inhomogeneous phase in a binary mixture,
with $\alpha,\beta=1,2$. Inhomogeneities at the mesoscopic length scale
indicate that the correlation functions in Fourier representation take a
maximum for $0<k<2\piup$. We assume that the inhomogeneities occur on a well-
defined length scale, and the peak of $\tilde{G}_{\gamma\delta}(k)$ is high
and narrow.

Let us consider

$\displaystyle{\cal G}_{\alpha\beta}=\int\frac{\mathrm{d}{\bf
k}}{(2\piup)^{3}}\frac{[\tilde{C}_{\alpha\beta}(k)]}{\det\tilde{\bf C}(k)}\,,$
(2.6)

where $[\tilde{C}_{\alpha\alpha}(k)]=\tilde{C}_{\beta\beta}(k)$ and
$[\tilde{C}_{\alpha\beta}(k)]=-\tilde{C}_{\alpha\beta}(k)$ for
$\alpha\neq\beta$. By symmetry, $\tilde{C}_{12}=\tilde{C}_{21}$. For the
considered functions with a high, narrow peak, the main contribution to the
integral comes from the vicinity of the maximum. In general,
$\tilde{C}_{\alpha\beta}(k)/\det\tilde{\bf C}(k)$ can take the maximum for
different values of $k$ for different pairs of $\alpha,\beta$. Here, we focus
on the case where the maximum of all the integrands in (2.6) is very close to
the minimum at $k=k_{0}$ of $\det\tilde{\bf C}(k)$, and we can make the
approximation

${\cal G}_{\alpha\beta}=[\tilde{C}_{\alpha\beta}(k_{0})]{\cal G},$

where

$\displaystyle{\cal G}=\int\frac{\mathrm{d}{\bf
k}}{(2\piup)^{3}}\frac{1}{\det\tilde{\bf C}(k)}.$ (2.7)

In the Brazovskii-type theory considered here, the $k$ dependence of
$\tilde{C}_{\alpha\beta}$ comes only from $\tilde{V}_{\alpha\beta}(k)$, and we
can write (2.4) in the form

$\displaystyle\tilde{\bf C}(k)=\beta\tilde{\bf V}(k)+{\bf c}\,,$ (2.8)

where the elements $c_{\alpha\beta}$ of ${\bf c}$ are

$c_{\alpha\beta}=A_{\alpha\beta}+\frac{A_{\alpha\beta\gamma\delta}}{2}{\cal
G}_{\gamma\delta}.$

We introduce the notation

$\displaystyle\det\tilde{\bf C}(k)=\det{\bf c}+\beta\tilde{W}(k),$ (2.9)

with

$\displaystyle\beta\tilde{W}(k)=\beta\Big{[}\beta\det\tilde{\bf
V}(k)+\tilde{V}_{11}(k)c_{22}+\tilde{V}_{22}(k)c_{11}-2\tilde{V}_{12}(k)c_{12}\Big{]}.$
(2.10)

Close to a deep minimum, we can make the expansion

$\displaystyle\det\tilde{\bf
C}(k)=D_{0}+\frac{\beta\tilde{W}^{\prime\prime}(k_{0})}{2}(k-k_{0})^{2}+\ldots\,,$
(2.11)

where $k_{0}$ is determined by the equation

$\displaystyle\beta\tilde{W}^{\prime}(k_{0})=0$ (2.12)

and

$\displaystyle D_{0}=\det\tilde{\bf C}(k_{0}).$ (2.13)

From the approximation (2.11) and (2.7), we obtain [25, 27]

$\displaystyle{\cal
G}\approx\frac{k_{0}^{2}}{\piup\sqrt{2\beta\tilde{W}^{\prime\prime}(k_{0})D_{0}}}.$
(2.14)

In order to obtain an explicit equation for $k_{0}$, we take into account that
from (2.8) it follows that ${\bf c}=\tilde{\bf C}(k_{0})-\beta\tilde{\bf
V}(k_{0})$. From the above and (2.12), (2.10) we obtain the equation

$\displaystyle\tilde{V}_{11}^{\prime}(k_{0})\tilde{C}_{22}(k_{0})+\tilde{V}_{22}^{\prime}(k_{0})\tilde{C}_{11}(k_{0})-2\tilde{V}_{12}^{\prime}(k_{0})\tilde{C}_{12}(k_{0})=0$
(2.15)

that contains the 3 unknowns $\tilde{C}_{\alpha\beta}(k_{0})$. $D_{0}$ is
expressed in terms of the above unknowns [see (2.13)], and finally

$\displaystyle\tilde{W}^{\prime\prime}(k_{0})=2\beta\det\tilde{\bf
V}^{\prime}(k_{0})+\tilde{V}_{11}^{\prime\prime}(k_{0})\tilde{C}_{22}(k_{0})+\tilde{V}_{22}^{\prime\prime}(k_{0})\tilde{C}_{11}(k_{0})-2\tilde{V}_{12}^{\prime\prime}(k_{0})\tilde{C}_{12}(k_{0}).$
(2.16)

In order to determine ${\bf C}$ in this approximation from (2.8), we need 3
equations in addition to equation (2.15), because we have 4 unknowns, $k_{0}$
and $\tilde{C}_{\alpha\beta}(k_{0})$. Using (2.4), (2.14), (2.13) and (2.16)
we obtain 3 equations with the unknowns $k_{0}$ and
$\tilde{C}_{\alpha\beta}(k_{0})$,

$\displaystyle\tilde{C}_{\alpha\beta}(k_{0})=\beta\tilde{V}_{\alpha\beta}(k_{0})+A_{\alpha\beta}+\frac{k_{0}^{2}A_{\alpha\beta\gamma\delta}[\tilde{C}_{\gamma\delta}(k_{0})]}{2\piup\sqrt{2\beta\tilde{W}^{\prime\prime}(k_{0})D_{0}}}.$
(2.17)

Equations (2.15) and (2.17) form a closed set of 4 equations for 4 unknowns,
when the expressions (2.13) and (2.16) for $D_{0}$ and
$\tilde{W}^{\prime\prime}$ are used, and $[\tilde{C}_{\gamma\delta}(k_{0})]$
are defined below (2.6).

Once $\tilde{C}_{\alpha\beta}(k_{0})$ are determined, the correlation
functions can be obtained from the equation

$\displaystyle\tilde{C}_{\alpha\beta}(k)=\tilde{C}_{\alpha\beta}(k_{0})+\beta[\tilde{V}_{\alpha\beta}(k)-\tilde{V}_{\alpha\beta}(k_{0})].$
(2.18)

The approximate expression (2.14) is valid only when the correlation functions
in Fourier representation have a pronounced maximum at $k=k_{0}$. In such a
case, $\tilde{V}_{\alpha\beta}(k)-\tilde{V}_{\alpha\beta}(k_{0})$ in (2.18)
can be expanded about $k_{0}$, and the expansion can be truncated. We should
stress that the theory is not valid at high temperature, where no
inhomogeneities at a well defined length scale are present.

In the next two sections we consider a binary mixture and assume the same size
of the spherical hard cores of the particles of the two kinds. For the free
energy of the hard-sphere reference system we assume
$F_{\text{h}}=\int\mathrm{d}{\bf r}f_{\text{h}}[\zeta_{1}({\bf
r}),\zeta_{2}({\bf r})]$ with the Carnahan-Starling approximation for the free
energy density
$f_{\text{h}}(\zeta_{1},\zeta_{2})=\zeta_{1}\ln\zeta_{1}+\zeta_{2}\ln\zeta_{2}+f_{\text{ex}}(\zeta)$,
where $\zeta=\zeta_{1}+\zeta_{2}$ and

$f_{\text{ex}}(\zeta)=\frac{6\zeta}{\piup}\left[\frac{4\zeta-3\zeta^{2}}{(1-\zeta)^{2}}-1\right].$

For equal sizes, we have $A_{12}=A_{2}(\zeta)$,
$A_{1112}=A_{1122}=A_{1222}=A_{4}(\zeta)$, and
$A_{\alpha\alpha}=A_{2}(\zeta)+1/\zeta_{\alpha}$,
$A_{\alpha\alpha\alpha\alpha}=A_{4}(\zeta)+2/\zeta_{\alpha}^{3}$, where
$A_{n}(\zeta)=\mathrm{d}^{n}f_{\text{ex}}(\zeta)/\mathrm{d}^{n}\zeta$. We use
the diameter of the particles as the length unit.

We assume different interaction potentials that can lead to different
inhomogeneities on different length scales.

##  3 The case of interaction potentials $V_{11}=V_{22}=-V_{12}=V$

In this section we assume interaction potentials that are a simplified version
of the interactions between charged colloid particles of equal sizes but of
different sign of the charge, and different chemistry of the two species. For
example, the species 1 can be ‘hydrophilic’, while the species 2
‘hydrophobic’. The like-particles attract each other at short distances due to
a similar chemistry, but at large distances they repel each other because of
the screened electrostatic repulsion between like charges. Particles of a
different kind attract each other at large distances due to the screened
electrostatic potential between opposite charges. We assume the short-range
repulsion to enhance the tendency for demixing of uncharged particles. This
kind of interactions between the like-particles is known as a ‘mermaid’ or
SALR potential. Note that $V_{12}$ is repulsive at short distances and
attractive at large distances, i.e., it has a repulsive head and a attractive
tail (like a peacock). In experiment, the ‘two mermaids and a peacock’
effective interactions can be obtained when the oppositely charged hydrophilic
and hydrophobic colloid particles are immersed in a near-critical mixture of
water and oil, for example lutidine. Critical concentration fluctuations lead
to the Casimir potential between confining surfaces [29]. This potential is
attractive for the like-surfaces, and repulsive between the hydrophilic and
the hydrophobic surface. The range of the Casimir potential, equal to the bulk
correlation length, $\xi_{\text{b}}$, can be tuned by temperature. When
$\xi_{\text{b}}$ is smaller than the Debye length of the screened
electrostatic potential, and the charge of the particles is properly chosen,
the ‘two mermaid and a peacock’ potential can be created.

###  3.1 Theory

To simplify the calculations, we assume that the interactions between the
like-particles are the same in this model, $V_{11}=V_{22}=V$, and for a
different kind of particles we assume $V_{12}=V_{21}=-V$. In our mesoscopic
theory $V(r)=0$ for $r<1$, and for $r>1$ we assume the double-Yukava potential
with short-range attraction and long-range repulsion,

$V(r)=-\frac{K_{1}}{r}\mathrm{e}^{-\kappa_{1}r}+\frac{K_{2}}{r}\mathrm{e}^{-\kappa_{2}r}.$
(3.1)

Here, we do not try to model any particular system. Our aim is to verify the
predictions of the mesoscopic theory for the case of inhomogeneities present
at the length scale larger than the size of the particles. We choose
$K_{1}=1,K_{2}=0.2,\kappa_{1}=1,\kappa_{2}=0.5$ that leads to relatively large
clusters, with diameter $\sim 5\sigma$. The amplitude of the attractive part
of $V$ sets the energy unit, and we use the dimensionless temperature
$T^{*}=k_{\text{B}}T/K_{1}$. The high symmetry of the interaction potentials
greately simplifies the calculations.

In MF, after some algebra we obtain the following expressions for the
correlation functions in Fourier representation (proportional to structure
factors)

$\displaystyle\tilde{G}_{11}^{\text{co}}$
$\displaystyle=G_{0}+\frac{(\zeta+c)^{2}[1+2A_{2}(\zeta-c)]^{2}}{D^{\text{co}}(k)}\,,$
$\displaystyle\tilde{G}_{22}^{\text{co}}$
$\displaystyle=G_{0}+\frac{(\zeta-c)^{2}[1+2A_{2}(\zeta+c)]^{2}}{D^{\text{co}}(k)}\,,$
$\displaystyle\tilde{G}_{12}^{\text{co}}$
$\displaystyle=G_{0}-\frac{(\zeta^{2}-c^{2})\left[(1+A_{2}\zeta)^{2}-A_{2}^{2}c^{2}\right]}{D^{\text{co}}(k)}\,,$

where the constant term leading to the Dirac delta function in real space is

$G_{0}=\frac{\zeta(1-c^{2}/\zeta^{2})}{4[1+A_{2}\zeta(1-c^{2}/\zeta^{2})]}\,,$
$D^{\text{co}}(k)=4(1+A_{2}\zeta)^{2}\zeta\left[1-\frac{A_{2}c^{2}}{\zeta(1+A_{2}\zeta)}\right]\left\\{1+\zeta\left[1-\frac{A_{2}c^{2}}{\zeta(1+A_{2}\zeta)}\right]\right\\},$
(3.2)

$\tilde{V}(k)$ is the interaction potential (3.1) in Fourier representation
with the explicit expression given for example in [30], and we have introduced
$c=\zeta_{1}-\zeta_{2}$.

Figure 1: (Colour online) Correlation functions in Fourier representation for
$T^{*}=0.28$, $\zeta=0.1$ and $c=0.02$ in MF (left-hand panel) and with the
effect of fluctuations taken into account (right-hand panel). Upper solid
lines: $\tilde{G}_{11}(k)$, dashed lines: $\tilde{G}_{22}(k)$, and lower solid
lines: $\tilde{G}_{12}(k)$.

In MF, the disordered phase becomes unstable with respect to oscillatory
modulations of the volume fractions at the $\lambda$-surface given by

$T^{*}(\zeta,c)=-\tilde{V}(k_{0})\zeta\left\\{1-\frac{A_{2}(\zeta)c^{2}}{\zeta[1+A_{2}(\zeta)\zeta]}\right\\},$

therefore, we restrict ourselves to $T^{*}$ above the $\lambda$-surface, i.e.,
to the stability region of the disordered phase. Because of the symmetry of
interactions, we assume that the majority component is the species 1, and
consider only $c>0$. The correlation functions,
$\tilde{G}_{\alpha\beta}^{\text{co}}(k)$, are shown in figure 1 for
$T^{*}=0.28$, total volume fraction of the particles
$\zeta=\zeta_{1}+\zeta_{2}=0.1$, and the difference in the volume fractions of
the two species $c=\zeta_{1}-\zeta_{2}=0.02$. The maximum of
$\tilde{G}_{\alpha\alpha}(k)$ or a minimum of $\tilde{G}_{12}(k)$ is assumed
for $k=k_{0}\approx 0.609$ which corresponds to the minimum of $\tilde{V}(k)$.
The period of damped oscillations in real space is $\sim 2\piup/k_{0}\approx
10.3$. In figure 2 we show $\tilde{G}_{\alpha\beta}^{\text{co}}(k_{0})$. Due
to the symmetry, $\tilde{G}_{11}^{\text{co}}=\tilde{G}_{22}^{\text{co}}$ for
$c=0$. When $c\neq 0$, the structure factor of the majority component is
larger than the structure factor of the minority component, and the difference
increases with an increasing asymmetry (see figure 2). Somewhat surprisingly,
the maximum of $\tilde{G}_{11}^{\text{co}}(k_{0})$ as a function of $c$ is
assumed for slightly different volume fractions of the two components, i.e.,
for $c=0.01$ which corresponds to $\zeta_{1}=0.055$ and $\zeta_{2}=0.045$. For
larger asymmetries, all the correlations decrease with an increasing $c$.

Figure 2: (Colour online) Left-hand panel: $\tilde{G}_{11}(k_{0})$ (upper
solid line), $\tilde{G}_{22}(k_{0})$ (dashed line) and $\tilde{G}_{12}(k_{0})$
(lower solid line) in MF. Right-hand panel: $\tilde{G}_{11}(k_{0})$ with the
effect of fluctuations taken into account. $\tilde{G}_{22}(k_{0})$ and
$\tilde{G}_{12}(k_{0})$ are monotonous functions of $c$. $T^{*}=0.28$,
$\zeta=0.1$ and $c$ is the difference in the volume fraction of the species 1
and 2.

Let us focus on the effects of fluctuations. Because of the assumed symmetry
of interactions, we have $\det\tilde{\bf V}(k)=0$, and from equations
(2.9)–(2.12) we can see that the minimum of $\det\tilde{\bf C}(k)$ is taken
for the same value of $k$ as the minimum of $\tilde{V}(k)$. Thus, $k_{0}$ is
not shifted in this model in the presence of fluctuations. Moreover, equation
(2.16) simplifies to
$W^{\prime\prime}(k_{0})=\tilde{V}^{\prime\prime}(k_{0})P$, where

$P=\tilde{C}_{11}(k_{0})+\tilde{C}_{22}(k_{0})+2\tilde{C}_{12}(k_{0}).$ (3.3)

The correlation functions in the presence of fluctuations in the Brazovskii-
type approximation after some algebra take the forms

$\displaystyle\tilde{G}_{11}$
$\displaystyle=P^{-1}+\frac{[\tilde{C}_{22}(k_{0})+\tilde{C}_{12}(k_{0})]^{2}}{D(k)}\,,\quad\tilde{G}_{22}=P^{-1}+\frac{[\tilde{C}_{11}(k_{0})+\tilde{C}_{12}(k_{0})]^{2}}{D(k)}\,,$
(3.4) $\displaystyle\tilde{G}_{12}$
$\displaystyle=P^{-1}-\frac{\tilde{C}_{11}(k_{0})\tilde{C}_{22}(k_{0})+\tilde{C}_{12}(k_{0})[\tilde{C}_{11}(k_{0})+\tilde{C}_{22}(k_{0})+\tilde{C}_{12}(k_{0})]}{D(k)}\,,$
(3.5)

where $D(k)=P\\{P[\beta\tilde{V}(k)-\beta\tilde{V}(k_{0})]+D_{0}\\}$ with
$D_{0}$ defined in (2.13). In order to calculate the correlation functions, we
numerically solved the set of equations (2.17) for $\tilde{C}_{11}(k_{0})$,
$\tilde{C}_{22}(k_{0})$, and $\tilde{C}_{12}(k_{0})$.

The correlation functions in this approximation, equations (3.4)–(3.5), are
compared with the MF result in figure 1 for $T^{*}=0.28$, $\zeta=0.1$, and
$c=0.02$. As expected, fluctuations lead to a much smaller amplitude and range
of the correlation functions, especially near the $\lambda$-surface, where
$\tilde{C}_{\alpha\beta}^{\text{co}}(k_{0})$ diverge in MF. This is manifested
by a smaller value and a larger width of the peak of
$\tilde{C}_{\alpha\beta}(k)$ compared to the MF result. The property that the
correlations between particles of the majority component are stronger is still
present. In figure 2 we show $\tilde{G}_{11}(k_{0})$ for $T^{*}=0.28$,
$\zeta=0.1$ and for a range of $c$, with the fluctuation contribution
included. The nonmonotonous dependence on $c$ is enhanced in the presence of
fluctuations, and the maximum occurs for $c=0.02$. The magnitude of the other
correlation functions decreases monotonously with an increasing $c$.

The peak of the correlation functions becomes higher and narrower for an
increasing density and/or a decreasing temperature, signaling correlations
stronger and of larger range. The effect of temperature can be seen by
comparing figure 1 for $T^{*}=0.28$ with figure 3 for $T^{*}=0.12$, both for
$\zeta=0.1$. In figure 3, the correlation functions are shown for $c=0.02$ and
$c=0.06$. We can see a strong effect of the difference in concentrations of
the species $1$ and $2$ on the correlation functions. All correlations
decrease, but especially the minority component becomes much less correlated
when its volume fraction decreases.

It is also interesting to consider the correlation function for the total
volume fraction, $G({\bf r}_{1}-{\bf r}_{2})=\langle\Delta\zeta({\bf
r}_{1})\Delta\zeta({\bf r}_{2})\rangle$. Equations (3.4)–(3.5) give

$\tilde{G}(k)=4P^{-1}+\frac{\left[\tilde{C}_{22}(k_{0})-\tilde{C}_{11}(k_{0})\right]^{2}}{D(k)}.$

This equation and figure 3 show that the correlations for the total volume
fraction increase with an increasing asymmetry which in this case is induced
by increasing $c$. In the fully symmetrical case, $\tilde{G}(k)$ vanishes in
this theory.

Figure 3: (Colour online) Correlation functions in Fourier representation for
$T^{*}=0.12$ and $\zeta=0.1$ with the effect of fluctuations taken into
account. Upper solid lines: $\tilde{G}_{11}(k)$, dashed lines:
$\tilde{G}_{22}(k)$ and lower solid lines: $\tilde{G}_{12}(k)$. Left-hand
panel: $c=0.02$. Right-hand panel: $c=0.06$.

If the assumptions leading to the fluctuation contribution obtained in the
Brazovskii-type approximation are valid, we can make the approximation

$\tilde{V}(k)-\tilde{V}(k_{0})\approx\frac{V^{\prime\prime}(k_{0})}{8k_{0}^{2}}(k^{2}-k_{0}^{2})^{2}\,,$
(3.6)

where we took into account that $\tilde{G}_{\alpha\beta}(k)$ should be even
functions of $k$. In this approximation, the correlation functions in real-
space representation can be calculated analytically by inverse Fourier
transforming equations (3.4)–(3.5). The results have the form (1.1) with
$\theta=0$, and the inverse lengths $\alpha_{0},\alpha_{1}$ are given by

$2\alpha_{0}^{2}=\left\\{8k_{0}^{2}\left[\frac{k_{0}^{2}}{8}+\frac{T^{*}D_{0}}{P\tilde{V}^{\prime\prime}(k_{0})}\right]\right\\}^{1/2}-k_{0}^{2}\,,\qquad
2\alpha_{1}^{2}=\left\\{8k_{0}^{2}\left[\frac{k_{0}^{2}}{8}+\frac{T^{*}D_{0}}{P\tilde{V}^{\prime\prime}(k_{0})}\right]\right\\}^{1/2}+k_{0}^{2}\,,$

where $P$ and $D_{0}$ are defined in (3.3) and (2.13), respectively. In figure
4 we show the correlation functions in real space for $T^{*}=0.12$,
$\zeta=0.1$ and $c=0.02$. The negative extremum of $\tilde{C}_{12}(k)$ at
$k=k_{0}$ leads to the opposite signs of $C_{12}(r)$ and $C_{\alpha\alpha}(r)$
for the same value of $r$. This means that the regions enriched in one
component are depleted in the other component.

Figure 4: (Colour online) Correlation functions in real space for
$T^{*}=0.12$, $\zeta=0.1$ and $c=0.02$ with the effect of fluctuations taken
into account. $G_{11}(r)$ (green upper solid line on the left), $G_{22}(r)$
(dashed line) and $G_{12}(r)$ (red lower solid line on the left).

###  3.2 Simulations

A binary mixture of particles interacting with the SALR potential (3.1) with
$K_{1}=1,K_{2}=0.2,\kappa_{1}=~{}1,\newline \kappa_{2}=0.5$ was simulated by
applying a Monte Carlo technique in the $NVT$ ensemble. The particles had the
same diameter ($\sigma_{1}=\sigma_{2}=\sigma=1.0$), and were placed in a cubic
box of edge length $50\sigma$, with periodic boundary conditions applied to
the system in the three directions. Cut-off radius was $15\sigma$. Each system
had run $10^{6}$ Monte Carlo steps for equilibration and $10^{5}$ for
production.

Figure 5: (Colour online) Pair distribution functions and a representative
configuration for $T^{*}=0.28$, and volume fractions
$\zeta_{1}=\zeta_{2}=0.026$.

We present the pair distribution functions,
$g_{\alpha\beta}=\frac{G_{\alpha\beta}}{\zeta_{\alpha}\zeta_{\beta}}+1$, and
representative configurations for $\zeta=0.052$ and $c=0$ in figures 5 and 6
for $T^{*}=0.28$, and $T^{*}=0.12$, respectively. In both cases, an
oscillatory decay with the period of damped oscillations $\lambda\approx 10$
can be seen, but the amplitude is much larger for $T^{*}=0.12$. This larger
amplitude and range of correlations is reflected in much more ordered
configuration shown in figure 6, as compared with the configuration shown in
figure 5. At $T^{*}=0.12$, we can see the chains of well assembled clusters,
where clusters of different species are attached to each other. At
$T^{*}=0.28$, the inhomogeneities are much less visible.

Figure 6: (Colour online) Pair distribution functions and a representative
configuration for $T^{*}=0.12$, and volume fractions
$\zeta_{1}=\zeta_{2}=0.026$.

We have computed $\tilde{G}_{\alpha\beta}$ for the thermodynamic states
corresponding to figures 5 and 6. In the first case, we obtained a quite flat
maximum of $\tilde{G}_{\alpha\alpha}(k)$. As far as a pronounced maximum is
assumed in our theory and the theory should be self-consistent, for this
thermodynamic state our theory is not accurate enough. In the second case, the
maximum of $\tilde{G}_{\alpha\alpha}(k)$ is better developed, but the peak is
not as high and narrow as for $\zeta=0.1$ at the same $T^{*}$. The
corresponding pair distribution functions are shown in figure 7 for $r>8$,
because in the mesoscopic theory the results for the correlation function
cannot be valid for distances smaller than one period of the damped
oscillations.

Figure 7: (Colour online) Left-hand panel: pair distribution functions in real
space for $T^{*}=0.12$, $\zeta_{1}=\zeta_{2}=0.026$ with the effect of
fluctuations taken into account in the mesoscopic theory.
$g_{11}(r)=g_{22}(r)$ (solid line), and $g_{12}(r)$ (dashed line). Right-hand
panel: Functions $g_{\alpha\beta}=G_{\alpha\beta}+1$, where $G_{\alpha\beta}$
is given in equation (1.1) with $\alpha_{1}=0.714$, $\alpha_{0}=0.1$,
$\theta=0$ and $A=\pm 8.5$.

By comparing figure 7 with the simulation results (figure 6) we can see that
in the theory the amplitude is too large and the decay length is too small. In
figure 7 (right-hand panel) we plot $g_{\alpha\beta}=G_{\alpha\beta}+1$, where
$G_{\alpha\beta}$ is given in (1.1) with $\alpha_{1}$ determined in our
theory, but with $\alpha_{0}=0.1$ about 5 times smaller than predicted by the
theory. The amplitude $A=8.5$ in equation (1.1) for $g_{11}(r)$ and $A=-8.5$
for $g_{12}(r)$ is also much smaller. We did not try to find the best fit to
the simulation results, but it is clear that for the chosen parameters the
agreement between simulations and equation (1.1) is very good. From this
agreement, it immediately follows that in simulations the structure factor for
the like (different) particles should assume a maximum (minimum) for $k\approx
k_{0}$, and near the extremum it behaves as
$|S_{\alpha\beta}|\propto[\text{const}+(k^{2}-k_{0}^{2})^{2}]^{-1}$. The
theory developed in this work is based on the Brazovskii approximation. In
[31], additional correction to the MF inverse correlation function is taken
into account. This correction term in one-component systems is proportional to
$-A_{3}(\zeta)^{2}$, and leads to a smaller value of the inverse correlation
function than in the Brazovskii approximation. Hence, larger correlations can
be expected. A satisfactory agreement with the exact results was obtained in
[31] for an equation of state in a one-dimensional model when the correction
proportional to $-A_{3}(\zeta)^{2}$ was taken into account. Since
$A_{3}(\zeta)=0$ for the critical volume fraction $\zeta\approx 0.129$, the
Brazovskii approximation works well for $\zeta\approx 0.1$. $A_{3}(\zeta)$
increases to large values for $\zeta$ decreasing from $\zeta\approx 0.129$,
and the accuracy of the Brazovskii approximation should decrease for a
decreasing $\zeta$. Our results show that for $\zeta\approx 0.05$, the
Brazovskii-type approximation is not sufficiently accurate. The period of
damped oscillations, however, remains close to $2\piup/k_{0}$, and the formula
(1.1) is still valid.

Figure 8: (Colour online) Pair distribution functions and a representative
configuration for $T^{*}=0.12$, and volume fractions $\zeta_{1}=0.042$,
$\zeta_{2}=0.01$.

The effect of asymmetry of the volume fractions of the two species is shown in
figure 8 for $T^{*}=0.12$, $\zeta=0.052$ and $c=0.032$. Correlations between
the majority species are stronger, in agreement with theoretical predictions.
Clusters are of a similar size as in the symmetrical case. However, there are
not enough particles of the minority component to fill the space between two
clusters of the majority component, and this space remains empty. Recall that
due to the long-range repulsion between the like particles, clusters of the
same kind repel each other. For this reason, the correlation function for
total volume fraction oscillates, in contrast to the symmetrical case. This
result also agrees with theoretical predictions.

##  4 The case of interaction potentials $V_{11}=V_{22}=0$ and $V_{12}\neq 0$

Now, we consider a particular case of the model binary mixture in which there
are only hard-core repulsions between the like-particles while the particles
of different types interact through the attractive potential $V_{12}(r)$
beyond the hard core. The mixture can be perceived as a crude model of charged
colloid particles in the solvent containing depletion agents. When charges are
switched off, colloid particles attract each other due to the presence of
depletants. If the particles are charged, screened electrostatic attraction
between different particles appears, while the like-particles repel each
other. For a suitably chosen size of the depletant and the screening length
and charge, the attraction and repulsion between the like-particles can cancel
each other approximately, and the attraction between the particles of
different types can be doubled.

###  4.1 Theory

As the first step, we assume equal sizes of particles for all species. For
$V_{12}(r)$, we choose the square well potential. Thus, the model is
characterized by the following interaction potentials beyond the hard core

$\displaystyle V_{11}(r)$ $\displaystyle=V_{22}(r)=0,$ (4.1) $\displaystyle
V_{12}(r)$ $\displaystyle=-\varepsilon\theta(r-1)\theta(a-r),$ (4.2)

where $a>1$ is the range of the potential, $\varepsilon$ is the interaction
strength at contact of the two unlike particles and $r$ is in $\sigma$ units.
We are interested in periodic structure on the length scale of $\sigma$, not
in any particular system. Since in [25] the MF phase diagram was obtained for
$a=2$, we assume $a=2$ here too. The Fourier transform of the potential (4.2)
for $a=2$ is shown in figure 10. The MF analysis showed [25] that the model
undergoes two types of instability: one connected with $k=0$ and the other one
with $k_{0}\approx 2.78$. The former is related to the gas-liquid phase
separation which occurs at lower volume fractions while the latter is related
to the appearance of local inhomogeneity at the length scale $2\piup/k_{0}$.
In this model, inhomogeneous structures can occur when the Fourier transform
of the interaction potential between unlike particles has positive maximum for
$k>0$.

Figure 9: Interaction potential between particles of different kinds (equation
(4.2) for $a=2$) in Fourier representation. The wave-number $k$ is in
$\sigma^{-1}$ units.

Figure 10:  Gas-liquid spinodals and $\lambda$-lines for the model (4.1)–(4.2)
at $a=2$ for different $c^{*}=(\zeta_{1}-\zeta_{2})/\zeta$: $c^{*}=0$ (solid
lines), $0.2$ (dashed lines), $0.4$ (dotted lines), $0.6$ (dash-dotted lines),
and $0.8$ (short-dashed lines).

First, we focus on the MF approximation. In this case, the correlation
functions in Fourier representation obtained for the model are of the form:

$\displaystyle\tilde{G}_{11}^{\text{co}}(k)=\frac{A_{22}}{D^{\text{co}}(k)}\,,\quad\tilde{G}_{22}^{\text{co}}(k)=\frac{A_{11}}{D^{\text{co}}(k)}\,,\quad\tilde{G}_{12}^{\text{co}}(k)=-\frac{A_{12}+\beta\tilde{V}_{12}(k)}{D^{\text{co}}(k)}\,,$
(4.3)

where
$D^{\text{co}}(k)=A_{11}A_{22}-\left[A_{12}+\beta\tilde{V}_{12}(k)\right]^{2}$
and $\tilde{V}_{12}(k)$ is the Fourier transform of the interaction potential
(4.2). From the equation $D^{\text{co}}(k)=0$ one can get the expressions for
the gas-liquid spinodal and for the $\lambda$-surface, respectively

$\displaystyle
T_{\text{sp}}^{*}=\frac{\tilde{V}_{12}(0)}{\sqrt{A_{11}A_{22}}+A_{12}}\,,\qquad
T_{\lambda}^{*}=\frac{\tilde{V}_{12}(k_{0})}{\sqrt{A_{11}A_{22}}-A_{12}}\,,$
(4.4)

where the dimensionless temperature is defined as
$T^{*}=k_{\text{B}}T/\varepsilon$. In figure 10, we present the
$T^{*}$–$\zeta$-plots of the MF boundaries of stability determined by
equations (4.4) for different values of $c^{*}=(\zeta_{1}-\zeta_{2})/\zeta$.

Figure 11:  (Colour online) Correlation functions in Fourier representation
for $T^{*}=2.5$, $\zeta=0.3$ and $c=0.06$ in MF approximation. The inset
magnifies the region between $k=2.6$ and $k=3$ (left-hand panel) and with the
effect of fluctuations taken into account (right-hand panel). Upper solid
lines: $\tilde{G}_{11}(k)$, dashed lines: $\tilde{G}_{22}(k)$, and lower solid
lines: $\tilde{G}_{12}(k)$.

Using equations (4.3), we calculate the MF correlation functions
$\tilde{G}^{\text{co}}_{\alpha\beta}(k)$ above the $\lambda$-surface. Due to
the proximity of the gas-liquid phase separation we chose the volume fraction
equal to $0.3$ which is higher than in the case of the ‘two mermaids and a
peacock’ model. As before, we assume that the majority component is the
species 1, and consider only $c\geqslant 0$. The correlation functions
$\tilde{G}^{\text{co}}_{\alpha\beta}(k)$ are shown in figure 11 (left-hand
panel) for $T^{*}=2.5$ and $c=0.06$. The main maximum (minimum) of
$\tilde{G}^{\text{co}}_{\alpha\beta}(k)$ corresponds to the maximum of the
interaction potential $\tilde{V}_{12}$ (see figure 10). We observe a very
small difference in the peak heights of $\tilde{G}^{\text{co}}_{11}(k)$ and
$\tilde{G}^{\text{co}}_{22}(k)$ in this case.

Figure 12:  (Colour online) Left-hand panel: $\tilde{G}_{11}(k_{0})$ (upper
solid line), $\tilde{G}_{22}(k_{0})$ (dashed line) and $\tilde{G}_{12}(k_{0})$
(lower solid line) in the MF approximation. The inset magnifies the region
between $c=0.11$ and $c=0.15$. Right-hand panel: $\tilde{G}_{11}(k_{0})$ and
$\tilde{G}_{22}(k_{0})$ with the effect of fluctuations taken into account.
$T^{*}=2.5$, $\zeta=0.3$ and $c=\zeta_{1}-\zeta_{2}$.

In figure 12 (left-hand panel), we show the dependence of
$\tilde{G}^{\text{co}}_{\alpha,\beta}(k_{0})$ on $c$. We see a monotonous
dependence of the three correlation functions on $c$. At the same time, the
difference in the peak heights of $\tilde{G}^{\text{co}}_{11}(k_{0})$ and
$\tilde{G}^{\text{co}}_{22}(k_{0})$ increases a bit with an increase of $c$.

Now we consider the effect of fluctuations on the correlation functions. Like
the ‘two mermaids and a peacock’ model considered in the previous section,
$k_{0}$ in our model is not shifted when the fluctuations are included. For
this model,
$W^{\prime\prime}(k_{0})=-2\beta\tilde{V}_{12}^{\prime\prime}(k_{0})\tilde{C}_{12}(k_{0})$.
As a result, equations (2.18) reduce to the form:

$\displaystyle\tilde{C}_{\alpha\alpha}(k)$
$\displaystyle=\tilde{C}_{\alpha\alpha}(k_{0}),\quad\alpha=1,2,$
$\displaystyle\tilde{C}_{12}(k)$
$\displaystyle=\tilde{C}_{12}(k_{0})+\beta\Delta\tilde{V}_{12}(k),$ (4.5)

where $\Delta\tilde{V}_{12}(k)=\tilde{V}_{12}(k)-\tilde{V}_{12}(k_{0})$. The
correlation functions $\tilde{C}_{\alpha\beta}(k_{0})$ are obtained by solving
equations (2.17). Finally, correlation functions $\tilde{G}_{\alpha\beta}(k)$
can be presented as follows:

$\displaystyle\tilde{G}_{11}(k)=\frac{\tilde{C}_{22}(k_{0})}{D(k)}\,,\quad\tilde{G}_{22}(k)=\frac{\tilde{C}_{11}(k_{0})}{D(k)}\,,\quad\tilde{G}_{12}(k)=-\frac{\tilde{C}_{12}(k_{0})+\beta\Delta\tilde{V}_{12}(k)}{D(k)}\,,$
(4.6)

where
$D(k)=\tilde{C}_{11}(k_{0})\tilde{C}_{22}(k_{0})-[\tilde{C}_{12}(k_{0})+\beta\Delta\tilde{V}_{12}(k)]^{2}$.
In the above equations, for $\Delta\tilde{V}_{12}(k)$ we use the approximation
(3.6).

Figure 13:  (Colour online) Correlation functions in Fourier representation
for $T^{*}=0.8$ and $\zeta=0.3$ with the effect of fluctuations taken into
account. Upper solid lines: $\tilde{G}_{11}(k)$, dashed lines:
$\tilde{G}_{22}(k)$ and lower solid lines: $\tilde{G}_{12}(k)$. Left-hand
panel: $c=0.06$. Right-hand panel: $c=0.18$.

Using equations (4.5)–(4.6) and taking into account (2.17), we calculate the
correlation functions in Fourier representation for $\zeta=0.3$ and for the
temperature above and below the $\lambda$-surface. In figure 11 (right-hand
panel), $\tilde{G}_{\alpha\beta}(k)$ are compared with the MF result for the
same values of the temperature, total volume fraction, and $c$. It is seen
that the maxima (minimum) of the correlation functions for the temperature
above the $\lambda$-surface become flat when the fluctuations are taken into
account. We observe a nonmonotonous dependence of $\tilde{G}_{11}(k_{0})$ on
$c$ with a maximum at $c\approx 0.03$ (see figure 12, right-hand panel) and
this behaviour is kept above and below the $\lambda$-surface. The correlation
functions $\tilde{G}_{22}(k_{0})$ and $\tilde{G}_{12}(k_{0})$ show a
monotonous behaviour with an increasing $c$. In figure 13, we present
$\tilde{G}_{\alpha\beta}(k)$ for $T^{*}=0.8$ and $c=0.06$ and $c=0.18$
(thermodynamic states below both the $\lambda$-surface and the gas-liquid
spinodal). It is seen that the correlation functions have the same trend with
an increase of $c$ as in the case of the ‘two mermaids and a peacock’ model:
all correlations become weaker, especially the correlations between the
particles of a minority component.

We should mention that the obtained extrema of the correlation functions in
Fourier representation for $T^{*}=2.5$, $\zeta=0.3$ and $c=0.06$ as well as
for $T^{*}=0.8$, $\zeta=0.3$ and $c=0.18$, are flat. However, we assumed that
high, narrow peaks are present. Since the theory should be self-consistent,
our findings signal that for these thermodynamic states our results cannot be
accurate. For $T^{*}=0.8$, $\zeta=0.3$ and $c=0.06$, the extrema are more
pronounced, and for this thermodynamic state better agreement with simulations
can be expected.

In figure 14, the correlation functions in real space are presented. As is
seen, $G_{\alpha\beta}(r)$ shows the qualitative behaviour similar to the
behaviour of the corresponding correlation functions of the ‘two mermaids and
a peacock’ model. The main difference is a period of oscillations which for
this model is $\approx 2\sigma$.

Figure 14:  (Colour online) Correlation functions in real space for
$T^{*}=0.8$, $\zeta=0.3$ and $c=0.06$ with the effect of fluctuations taken
into account. $G_{11}(r)$ (green upper solid line on the left), $G_{22}(r)$
(dashed line) and $G_{12}(r)$ (red lower solid line on the left).

###  4.2 Simulations

A binary mixture of particles interacting with the potentials (4.1)–(4.2) was
simulated using Monte Carlo technique in the $NVT$ ensemble. Particles
($N_{1}+N_{2}=15470$) are placed in a cubic box of the edge length $30\sigma$
with periodical boundary conditions in three directions. The volume fraction
is $\zeta=0.3$. Particles of both species are of the same diameter
($\sigma_{1}=\sigma_{2}=\sigma=1.0$). A cut-off radius of $2\sigma$ is used,
which is the interaction range of the square well potential. Each system has
run $10^{6}$ Monte Carlo steps for equilibration and $10^{5}$ for production.

In figures 15–17, the pair distribution functions and representative
configurations for $T^{*}=0.8$, $\zeta=0.3$ and for three values of $c$ are
presented. The visualization suggests that for $c=0$ and $0.06$ the dilute gas
phase coexists with the dense inhomogeneous liquid (figures 15–16). For the
strongly asymmetric case ($c=0.18$), the dense inhomogeneous liquid with
approximately equal densities of the two components coexists with the one-
component, very dense gas (figure 17). In all cases, the inhomogeneous liquid
phase mainly consists of stripes where the neighboring stripes are formed by
the particles of different type. However, this picture is less pronounced for
$c=0.18$.

Figure 15: (Colour online) Pair distribution functions and a representative
configuration for $T^{*}=0.8$, and volume fractions $\zeta_{1}=\zeta_{2}=0.15$
($c=0$).

Figure 16: (Colour online) Pair distribution functions and a representative
configuration for $T^{*}=0.8$, and volume fractions $\zeta_{1}=0.18$ and
$\zeta_{2}=0.12$ ($c=0.06$).

Figure 17: (Colour online) Pair distribution functions and a representative
configuration for $T^{*}=0.8$, and volume fractions $\zeta_{1}=0.24$ and
$\zeta_{2}=0.06$ ($c=0.18$).

Figure 18:  (Colour online) Concentration-concentration distribution function
at $T^{*}=0.8$, $\zeta=0.3$ and $c=0.06$. Left-hand panel: theoretical
results, right-hand panel: results of simulations.

It should be noted that the structure of the disordered inhomogeneous phase at
$\zeta=0.3$ is affected by the packing of hard spheres. In order to separate
the effect of hard-sphere packing we calculate the concentration-concentration
distribution function
$g_{cc}(r)=c_{1}^{2}c_{2}^{2}(g_{11}(r)+g_{22}(r)-2g_{12}(r))$, where
$c_{i}=\zeta_{i}/\zeta$ [32]. The results are presented in figure 18. Here, we
compare $g_{cc}(r)$ obtained from the theory (left-hand panel) and from the
simulations (right-hand panel) for $T^{*}=0.8$, $\zeta=0.3$ and $c=0.06$. One
can see a rather good agreement between the theoretical and simulation
results: the distribution functions $g_{cc}(r)$ show an oscillatory decay
around zero with the period of damped oscillations $\lambda\approx 2$, and the
maxima and minima occur for very similar $r$ in theory and simulations. It is
worth noting that in molten alkali halides, $g_{cc}(r)$ shows oscillations
around zero, which extend over distances of at least 10 Å [33]. Since the
simulation results for $g_{cc}(r)$ are well reproduced by equation (1.1), the
corresponding structure factor takes a maximum for $k=k_{0}$, and the peak
shape agrees with our predictions.

##  5 Conclusions

We have studied the effect of fluctuations on the correlation functions of a
binary inhomogeneous mixture by using the mesoscopic density-functional
theory. The theory is based on the Brazovskii-type approximation and allows
one to take into account the fluctuation contribution. For a binary mixture in
the disordered inhomogeneous phase, we have derived approximate equations for
the partial correlation functions in Fourier representation. Using the above-
mentioned approximations, we have calculated the correlation functions for
several state points above and below the MF instability for two particular
models of a binary mixture of species A and B. We have chosen models leading
to a periodic structure on the length scale of the size of the particles, and
on the length scale 10 times larger. We did so in order to compare the
ordering and to verify the accuracy of the mesoscopic description for
different length scales.

We have considered the two models in the MF and when the fluctuations are
taken into account. In the two models, we have limited ourselves to an equal
size of the hard cores of the particles of both species. In the first mixture,
the particles of the same type interact with the ‘mermaid’ potential $V(r)$
(it has an attractive head and a repulsive ‘tail’) while A and B particles
interact with a ‘peacock’ potential $-V(r)$ which has an attractive ‘tail’ and
a repulsive ‘head’. In the second mixture, there are only short-range
attractive interactions between species A and B beyond the hard core which is
chosen in the square-well form. In both models, the attractions lead to a
periodic arrangement of alternating species A and B. However, the
characteristic size of the regions rich in the particles of species A and B
differ significantly in the two models. In addition, the gas-liquid phase
separation can occur in the second model.

Despite different interaction potentials and different length scale of the
local ordering, the correlation functions of the two models demonstrate
similar properties. In order to verify the theoretical results and to
visualize the structure for different state points, we have performed MC
simulations. It is shown that the theoretical predictions are consistent with
the simulation results.

Our results can be valid only for the case of well-defined inhomogeneities.
Well developed short-range order occurs when the correlation functions in
Fourier representation have high, narrow peaks at $k_{0}>0$. It is the case in
the part of the phase diagram where the disordered phase looses stability in
MF, but in reality it remains stable. Our theory agrees with simulations at a
semiquantitative level for such state-points.

In the present paper, we have assumed equal size of the particles of different
species. It is expected, however, that the periodic ordering should be
enhanced when the size asymmetry increases. An increasing tendency for
clustering with an increase of size asymmetry was observed in ionic systems in
mesoscopic theory [25, 34] and in simulations [35, 36]. Further work is
necessary for the study of the correlation functions in mixtures within the
framework of the theory when the size asymmetry is taken into account.

## Acknowledgements

We would like to thank Vyacheslav Vikhrenko, Ruslan Lasovsky and Yaroslav
Groda for discussions and hospitality at the Belarusian State Technological
University, where a part of this work was done. This project has received
funding from the European Union Horizon 2020 research and innovation programme
under the Marie Skłodowska-Curie grant agreement No 734276 (CONIN). An
additional support in the years 2017–2020 has been granted for the CONIN
project by the Polish Ministry of Science and Higher Education. Financial
support from the National Science Center under grant No. 2015/19/B/ST3/03122
is also acknowledged.

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