A study of the structure and thermodynamics of non-ionic microemulsion droplets: integral equation methods (IEs) and molecular dynamics simulation (MD)

. This paper aims to explore the structural and thermodynamic properties of a bare neutral oil / water microemulsions (MEs), TX100 by using a combination of molecular dynamics simulations (MD) and Ornstein-Zernike integral equations (IEs) with the hypernetted chain closure relation (HNC), at di ff erent volume fractions ϕ (1 . 4%, 2 . 8%, 5%, 6 . 98%, 10%). The employed e ff ective pair potential is a combination of a hard sphere, the van der Waals and a Yukawa type potential. Structural properties were studied by examining the radial distribution function (RDF), g(r), as a function of ϕ ; increasing ϕ enhances order and correlation between droplets, manifesting as a narrow and pronounced correlation peak and a decrease in the average distance between MEs. Regarding the thermodynamic properties, as ϕ increase, the reduced pressure and internal energy increase exponentially.


Introduction
Microemulsions (MEs) are monophasic, transparent and thermodynamically stable mixtures [1], these systems are widely used in the pharmaceutical industry as drug delivery systems [2][3][4].Filali et al. combined Small-Angle Neutron Scattering (SANS) experiments with the IEMs to investigate the shape and interactions of neutral MEs droplets (TX100) with the addition of two types of telechelic polymers (PEO-m and PEO-2m), at various volume fractions (ϕ) and polymer quantities (Np).They demonstrated that the shape and size of the droplets remained unchanged regardless of ϕ and Np (R = 8.2 nm, polydispersity 20%).Concerning the impact of added polymers on MEs droplets interactions; in the dilute case (ϕ = 6.8%), the addition of PEO-m induced a steric repulsive interaction, while PEO-2m induced a bridging attractive interaction.Conversely, in the concentrated case (ϕ = 13.3%), both polymers amplified the repulsive interactions between microemulsion droplets [5][6][7][8][9].Similarly, Ahfir et al. examined the dynamics of the TX100 system, including relaxation modes and diffusion coefficients, as function of ϕ [10].Furthermore, the authors explored the structure and dynamics of mixed systems consisting of charged MEs covered with telechelic polymers (PEO-m, PEO-2m), using a combination of IEMs and MD simulations.It is important to note that the interaction between MEs droplets increases proportionally with ϕ.However, with an increase in ϕ, the average distance between MEs, isothermal compressibility, and diffusion coefficient decrease [11][12][13][14].
In this paper, we used MD simulations to examine the structural and thermodynamic properties of neutral oil/water MEs, TX100, for low concentrations ϕ (1.4%, 2.8%, 5%, 6.98%, 10%).The document is structured as follows: Sections 2 and 3 are reserved for droplets interaction potentials and MD simulations principles.Section 4 represents the results obtained.The last section concludes the paper.

Theoretical aspects 2.1 Pair-potential expression
For a system comprising a non-ionic microemulsion, the droplets are modelled as neutral hard spheres that interact through an effective potential U(r).This potential is the combination of a hard sphere potential, denoted as U HS , along with the van der Waals attractive contribution U V DW , and a Yukawa potential [7]:

With
• U HS is the hard sphere interaction potential: Here "r" represents the distance between centers, while "R" denotes the radius of the droplets.
• U V DW (r) is the Van der Waals attraction potential [15]: A H is the effective Hamaker constant, in this study, we consider A H = 1.1kB T [16].The strength of the van der waals attractions is determined by the difference in the refractive indices of the solvent and the colloids [17].
• U T X (r) is the Yukawa repulsion potential: where: V T X denotes the magnitude of the interaction strength and λ T X signifies its range.In this study, V T X = 6k B T , λ T X = 27Å [7].The Ornstein-Zernike integral equation is [18]: Where h(r) represents the total correlation function, and c(r) is the direct correlation function.The relationship between h(r) for two particles at a distance "r" is expressed through the pair distribution function g(r) as denoted by the relation: g(r) = h(r) + 1 [15] The Ornstein-Zernike equation involves two unknown functions, h(r) and c(r).To complete the system, an additional relation must be introduced.Various approximate relations exist in the literature, and for this study, we employed the hypernetted chain equation (HNC) [19].
where U(r) is the mean interaction potential.

Molecular Dynamics Simulation
In this study, molecular dynamics (MD) simulations were performed employing the LAMMPS simulation package [20].The simulations were carried out under the NVT thermodynamic ensemble, where N = 10 6 denotes the number of simulated microemulsions, V represents the volume of the simulation box, and T is the temperature.To mitigate edge effects and emulate an infinite system [12], three-dimensional periodic boundary conditions were implemented.The use of dimensionless units was adopted to enhance computational efficiency [21].Specifically, the distance scale was normalized by the microemulsion radius R, and the dimensionless temperature was set to T * = 1 .The equations of motion were numerically integrated using Verlet's algorithm [22], spanning one million steps with a time step of δt = 0.001.

Effect of volume fraction ϕ on the structural properties of microemulsions
Figure 1 represents the RDF, g(r), derived from a numerical study utilizing the HNC closure relation and MD simulation, as a function of dimensionless distance r σ , at various volume fractions ϕ (1.4%, 2.8%, 5%, 6.98%, 10%).Firstly, we observe that the first peaks of the correlation functions from the MD simulations are in good agreement with the corresponding values calculated from the integral equation.Furthermore, at very diluted cases (ϕ = 1.4%, ϕ = 2.8%, ϕ = 5%), a weak and broad correlation peak is observed, indicating that the microemulsion droplets are widely spaced and thus have very weak interactions.The position of the first peak indicates the average distance between the microemulsion droplets (ϕ = 1.4%: d ∼ 296Å, g max = 1.01, ϕ = 2.8%: d ∼ 288Å, g max = 1.04, ϕ = 5%: d ∼ 272Å, g max = 1.12).With the increase in ϕ, the correlation peak narrows and becomes more intense, indicating that the average distance between droplets decreases (ϕ = 6.98%: d ∼ 267Å, g max = 1.18, ϕ = 10%: d ∼ 259Å, g max = 1.40).At ϕ = 10%, a second, less intense, and relatively broad peak is observed, corresponding to interactions between second neighbors.All these observations indicate an increase in correlation between microemulsion droplets as the volume fraction ϕ increases.This behavior of the RDF has been observed in similar studies [12][13][14].

Effect of volume fraction ϕ on the thermodynamics properties of microemulsions
The objective of this section is to validate the findings from the preceding section through an examination of thermodynamic properties, specifically pressure and internal energy.Figure 2A shows the variation of the Virial pressure as a function of ϕ; we observe an exponential increase in pressure with ϕ.In fact, due to the increase in ϕ, repulsion between MEs droplets also increases.Consequently, the pressure increases progressively as ϕ increase.In Figure 2B, the variation of the internal energy, △E Nk B T with ϕ is examined.A discernible, monotonic increase in △E Nk B T is clearly visible as ϕ increases.This result is consistent with the expected behavior of the repulsive interaction between droplets.

Conclusion
In this study, we investigate the structural and thermodynamic properties of a bare neutral oil/water microemulsions (MEs), at different volume fractions ϕ (1%, 2.8%, 5%, 6.98%, 10%).Regarding structural properties, we compared the MD simulations curves of the RDF with the numerically solved integral equations (OZ) using the hypernetted chain (HNC) closure.We found that the structural properties obtained are in good agreement; increasing ϕ enhances correlations among MEs droplets (ϕ ↑↑, d ↓↓, and g max ↑↑).Regarding thermodynamic properties; the Virial pressure and the internal energy increases exponentially with ϕ.