Effects of fluctuations on correlation functions in inhomogeneous mixtures

Approximate expressions for correlation functions in binary inhomogeneous mixtures are derived in a framework of the mesoscopic theory [A. Ciach, Mol. Phys. {\bf 109}, 1101 (2011)]. 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 periodic arrangement of alternating species A and B. However, the optimal distance between the species A and B is significantly different in the two models. Theoretical results for different temperature and volume fractions of the two components are compared with 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.


I. 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 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 means in fact that the distribution of the components in majority of the 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, leading 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 of the macroscopic volume, the density is either significantly larger or significantly smaller than the average density, when an aggregate either enters or leaves the window. This means very large fluctuations of the local density. 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 the repulsion, there is 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). Because of that, the energy of the majority of states in the inhomogeneous disordered phase is significantly 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 different density inside the clusters and between them, i.e. the variance of the local density or concentration, have to be taken into account to restore the stability of the disordered inhomogeneous phase and to lead to 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. Similar approach was used for studies of ionic systems in [5,6,26].
On 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 [16,24,27]. The contribution to the grand potential associated with the presence of inhomogeneities is calculated on 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 then leads to a negative contribution to the internal energy, and to stabilization of the disordered phase for 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 on 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 but the dependence of the parameters on thermodynamic state is significantly different: there is no spurious divergence of A or 1/α 0 . Eq.(1) should describe the asymptotic decay of correlations when the correlation function in Fourier representation,G(k), takes a maximum for k = k 0 > 0 even in the exact theory. It can be obtained from the pole analysis ofG(k), by taking into account the pair of poles iα 0 ± α 1 with the smallest imaginary part. Simulations show, however that equation (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 Sec. 2, we briefly summarize the Brazovskii-type formalism generalized to mixtures in [25]. Next, we limit ourselves to binary mixtures and derive approximate equations for the correlation functions in Fourier representation,G αβ (k).
We limit ourselves to equal sizes of the hard cores of the particles, and consider two particular examples of a binary mixture. In the first example, Sec. 3, we assume interactions leading to inhomogeneities on a length scale of ∼ 10σ, where σ is a molecular size. We choose the 'mermaid' potential between like particles, but a pair of different particles interacts with a 'peacoc' potential with attractive tail and repulsive head. The tails of the interactions correspond to screened electrostatic potentials (repulsion between like-and attraction between opposite charges) and the short-range part of the interactions favours phase separation of the two species. 'Two mermaids and a peacoc' effective interactions can occur between oppositely charged hydrophilic and hydrophobic colloid particles suspended in a near-critical mixture with ions [29].
In Sec. 4, we assume short-range interactions of the square-well form only between differ-ent species. This potential can lead to gas-liquid phase transition, as well as to the ordered phase resembling an ionic crystal, with the structure on the length scale of σ [25]. We calculate the three correlation functions in the disordered phase for several state points for the 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 considered models. We conclude in Sec. 5.

II. FORMALISM
The theory is based on the mesoscopic formalism developed for inhomogeneous mixtures with n components in [25]. Instead of the number density ρ α of the α-component, we consider the volume fraction in the mesoscopic region around r, ζ α (r), and deviations ∆ζ α (r) from the average value,ζ α (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 significantly larger than the size of the particles, then the volume fraction and the number density of the particles with the volume v α are related by ζ α ≈ ρ α v α .
where the elementV αβ (r) of the matrixV(r) is the interaction potential between the species α, β, 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 = |r 1 − 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 −T where F h is the free-energy of the hard-core reference system. We assume here local-density approximation.
In this work we study effects of fluctuations on the correlation functions G, with the matrix elements G αβ (r α , r β ) = ∆ζ α (r α )∆ζ β (r β ) representing the correlations between the volume fractions of the species α and β at the positions r α and r β respectively. In order to include fluctuations, we introduce the functional βF [{ζ}] of the form where φ α (r) is the local fluctuation of the volume fraction of the component α, and The functional (2) becomes equal to the grand potential, when {ζ} = {ζ}, with {ζ} = (ζ α 1 (r), ...,ζ αn (r)) that satisfy the minimum condition for (2). By definition, φ α = 0 when Note that from (2) it follows that the vertex functions (related to the direct correlation functions) defined by 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 .
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, In the lowest-order nontrivial approximation the matrix elements of the inverse correlation function in Fourier representation are given by [25], where the first term is the function defined in (3) with j = 2 in Fourier representation, summation convention is used in the whole article, and In the case of the disordered phase,C co αβ (k) = βṼ αβ (k) + A αβ . The matrixṼ(k) with the ele-mentsṼ αβ (k) is the interaction potential between the species α, β in Fourier representation, and A α 1 ....α j is given by Equations (4)-(6) have to 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 α, β = 1, 2. Inhomogeneities at the mesoscopic length scale indicate that the correlation functions in Fourier representation take a maximum for 0 < k < 2π. We assume that the inhomogeneities occur on a well-defined length scale, and the peak ofG γδ (k) is high and narrow.
Let us consider where For the considered functions with a high, narrow peak, the main contribution to the integral comes from the vicinity of the maximum. In general,C αβ (k)/ detC(k) can take the maximum for different values of k for different pairs of α, β. Here we focus on the case where the maximum of all the integrands in (7) is very close to the minimum at k = k 0 of detC(k), and we can make the approximation In the Brazovskii-type theory considered here, the k dependence ofC αβ comes only from V αβ (k), and we can write (5) in the form where the elements c αβ of c are We introduce the notation with Close to a deep minimum, we can make the expansion where k 0 is determined by the equation and From the approximation (12) and (8), we obtain [25,27] In order to obtain an explicit equation for k 0 , we take into account that from (9) it follows that c =C(k 0 ) − βṼ(k 0 ). From the above and (13), (11) we obtain the equatioñ that contains the 3 unknownsC αβ (k 0 ). D 0 is expressed in terms of the above unknowns (see (14)), and finallỹ In order to determine C in this approximation from (9), we need 3 equations in addition to equation (16), because we have 4 unknowns, k 0 andC αβ (k 0 ). Using (5), (15), (14) and (17) we obtain 3 equations with the unknowns k 0 andC αβ (k 0 ), 8 Equations (16) and (18) form a closed set of 4 equations for 4 unknowns, when the expressions (14) and (17) for D 0 andW are used, and [C γδ (k 0 )] are defined below (7).
OnceC αβ (k 0 ) are determined, the correlation functions can be obtained from the equatioñ The approximate expression (15) is valid only when the correlation functions in Fourier representation have a pronounced maximum at k = k 0 . In such a case, in (19) 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- For equal sizes, we have 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.

III. THE CASE OF INTERACTION POTENTIALS
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'. Like particles attract each other at short distances due to a similar chemistry, but at large distances 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 like particles is known as a 'mermaid' or SALR potential. Note that V 12 is repulsive at short-and attractive at large distances, i.e. it has a repulsive head and 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 like surfaces, and repulsive between the hydrophilic and the hydrophobic surface. The range of the Casimir potential, equal to the bulk correlation length, ξ b , can be tuned by temperature. When ξ 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 peacoc ' potential can be created.

A. Theory
To simplify the calculations, we assume that the interactions between like particles are the same in this model, V 11 = V 22 = V , and for different kind of particles we assume 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, Here we do not try to model any particular system. Our aim is to verify the predictions of the mesoscopic theory in 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, κ 1 = 1, κ 2 = 0.5 that leads to relatively large clusters, with diameter ∼ 5σ. The amplitude of the attractive part of V sets the energy unit, and we use the dimensionless temperature T * = k B T /K 1 . The high symmetry of the interaction potentials significantly simplifies the calculations.
In MF, we obtain after some algebra the following expressions for the correlation functions in Fourier representation (proportional to structure factors) where the constant term leading to the Dirac delta function in real space is is the interaction potential (20) in Fourier representation with the explicit expression given for example in [30], and we have introduced c = ζ 1 − ζ 2 . In MF, the disordered phase becomes unstable with respect to oscillatory modulations of the volume fractions at the λ-surface given by therefore we limit ourselves to T * above the λ-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,G co αβ (k), are shown in figure 1 for T * = 0.28, total volume fraction of the particles ζ = ζ 1 + ζ 2 = 0.1, and the difference in the volume fractions of the two species c = ζ 1 − ζ 2 = 0.02. The maximum ofG αα (k) or a minimum ofG 12 (k) is assumed for k = k 0 ≈ 0.609 that corresponds to the minimum ofṼ (k). The period of damped oscillations in real space is ∼ 2π/k 0 ≈ 10.3.
In figure 2 we showG co αβ (k 0 ). Due to the symmetry,G co 11 =G co 22 for c = 0. When c = 0, the structure factor of the majority component is larger than the structure factor of the minority component, and the difference increases with increasing asymmetry (see figure 2).
Somewhat surprisingly, the maximum ofG co 11 (k 0 ) as a function of c is assumed for slightly different volume fractions of the two components, i.e. for c = 0.01 that corresponds to The correlation functions in the presence of fluctuations in the Brazovskii-type approximation after some algebra take the forms where D(k) = P [P (βṼ (k) − βṼ (k 0 )) + D 0 ] with D 0 defined in (14). In order to calculate the correlation functions, we solved numerically the set of equations (18)  It is also interesting to consider the correlation function for the total volume fraction, G(r 1 − r 2 ) = ∆ζ(r 1 )∆ζ(r 2 ) . Equations (22)-(23) givẽ This equation and figure 3 show that the correlations for the total volume fraction increase with increasing asymmetry that in this case is induced by increasing c. In the fully symmetrical case,G(k) vanishes in this theory. If the assumptions leading to the fluctuation contribution obtained in the Brazovskii-type approximation are valid, we can make the approximatioñ where we took into account thatG αβ (k) must be even functions of k. In this approximation, the correlation functions in real-space representation can be calculated analytically by inverse where P and D 0 are defined in (21) and (14)   We have computedG αβ for the thermodynamic states corresponding to figures 5 and 6.
In the first case, we obtained quite flat maximum ofG αα (k). 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 ofG αα (k) is better developed, but the peak is not as high and narrow as for ζ = 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. g αβ = G αβ +1, where G αβ is given in (1) with α 1 determined in our theory, but with α 0 = 0.1 about 5 times smaller than predicted by the theory. The amplitude A = 8.5 in equation (1) for g 11 (r) and A = −8.5 for g 12 (r) is also significantly 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) is very good. The theory developed in this work is based on the Brazovskii approximation. In [31] additional correction to the MF inverse correlation function has been taken into account. This correction term in one-component systems is proportional to −A 3 (ζ) 2 , and leads to a smaller value of the inverse correlation function than in the Brazovskii approximation. Hence, larger correlations can be expected.
Satisfactory agreement with exact results has been obtained in [31] for an equation of state in a one-dimensional model when the correction proportional to −A 3 (ζ) 2 has been taken into account. Since A 3 (ζ) = 0 for the critical volume fraction ζ ≈ 0.129, the Brazovskii approximation works well for ζ ≈ 0.1. A 3 (ζ) increases to large values for ζ decreasing from ζ ≈ 0.129, and the accuracy of the Brazovskii approximation should decrease for decreasing ζ. Our results show that for ζ ≈ 0.05, the Brazovskii-type approximation is not sufficiently accurate. The period of damped oscillations, however, remains close to 2π/k 0 , and the formula (1) is still valid. can cancel each other approximately, and the attraction between the particles of different types can be doubled.

A. Theory
As a 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 where a > 1 is the range of the potential, ε is the interaction strength at contact of the two unlike particles and r is in σ units. We are interested in periodic structure on the length scale of σ, 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 (26) for a = 2 is shown in figure 9. The MF analysis showed [25] that the model undergoes two types of instability: one connected with k = 0 and another one with k 0 ≈ 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 2π/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.
First, we focus on the MF approximation. In this case, the correlation functions in Fourier representation obtained for the model are of the form: where D co (k) = A 11 A 22 − (A 12 + βṼ 12 (k)) 2 andṼ 12 (k) is the Fourier transform of the interaction potential (26). From the equation D co (k) = 0 one can get the expressions for the gas-liquid spinodal and for the λ-surface, respectively where the dimensionless temperature is defined as T * = k B T /ε. In figure 10, we present the T * -ζ-plots of the MF boundaries of stability determined by equations (28) for different values of c = ζ 1 − ζ 2 .  Using equations (27), we calculate the MF correlation functionsG co αβ (k) above the λsurface. 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 ≥ 0. The correlation functionsG co αβ (k) are shown in figure 11 (left panel) for T * = 2.5 and c = 0.06. The main maximum (minimum) ofG co αβ (k) corresponds to the maximum of the interaction potentialṼ 12 (see figure 9). We observe a very small difference in the peak heights ofG co 11 (k) andG co 22 (k) in this case. In figure 12 (left panel), we show the dependence ofG co α,β (k 0 ) on c. As a result, equations (19) reduce to the form: where ∆Ṽ 12 (k) =Ṽ 12 (k)−Ṽ 12 (k 0 ). The correlation functionsC αβ (k 0 ) are obtained by solving equations (18). Finally, correlation functionsG αβ (k) can be presented as follows: where D(k) =C 11 (k 0 )C 22 (k 0 ) − (C 12 (k 0 ) + β∆Ṽ 12 (k)) 2 . In the above equations, for ∆Ṽ 12 (k) we use the approximation (24). expected.
In figure 14, the correlation functions in real space are presented. As is seen, G αβ (r) has a similar qualitative behaviour 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 ≈ 2σ.

B. Simulations
A binary mixture of particles interacting with the potentials (25)  show oscillatory decay with the period of damped oscillations λ ≈ 2 and the maxima and minima occur for very similar r in theory and simulations.

V. 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 periodic structure on the length scale of the size of the particles, and on the lenght 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 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 'peacoc' 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 periodic arrangement of alternating species A and Our results can be valid only in 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 remains stable. Our theory agrees with simulations on 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, the periodic ordering be enhanced when the size asymmetry increases.
Increasing tendency for clustering with an increase of size asymmetry was observed in ionic systems in mesoscopic theory [25,33] and in simulations [34,35]. 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.