Dynamic properties of decane/water microemulsions decorated with hydrophobically modified PEO Polymer (PEO-C 12 ): A molecular dynamics simulations study

. This work aims to probe the dynamics of Microemulsions/PEO complexes by using Molecular dynamics simulations (MD). The studied system is a highly diluted O/W charged MEs (Φ = 2.8%) , covered by significant amounts of PEO polymer chains, with a short sticky block PEO-C 12 (Np = 0, 4, 8, 12, 16, 24 and 32). The employed effective pair potential is a combination of a hard sphere, the van der Waals and a Yukawa type potential. I order to shed light on the dynamic properties of this system, we analyzed the mean-squared displacement (MSD), the diffusion coefficients (D c ) and the velocity autocorrelation function (VACF) as a number N p of added PEO-C 12 ; Increasing the polymer concentration Np slows down the diffusion of the system.

(from 4 to 12) [14][15][16].P. Malo de Molina et al combined SANS, DLS, high frequency rheology measurements, and fluorescence correlation spectroscopy (FCS) to investigate dynamic and structural properties of (TDMAO/decane/water) MEs, with telechelic polymers, poly (N, N-dimethylacrylamide) [17], and (C18-PEO150-C18) [18].An important increase in viscosity was noted when the polymers are added to the MEs.For a high polymer concentration, the diffusion coefficient measured by DLS and FCS show a slow relaxation mode, also, for a mixture of nonionic oil-in-water MEs (TX100/Brij30/decane/D2O) and hydrophobically modified polymers with functionalities f (2,3,4); the attraction between droplets increases with the number of polymers .[19].In the present manuscript, we used Molecular Dynamics to probe the dynamics of highly diluted (Φ = 2.8%) charged oil-in-water (O/W) MEs, covered by significant amounts of PEO-C12 polymer.The mean-squared displacement (MSD), the diffusion coefficients (Dc) and the velocity autocorrelation function (VACF) are discussed in terms of number Np of added PEO-C12.It is organized as follows: Sections 2 and 3 are reserved for the appropriate interaction potentials between droplets in the presence of PEO-C12, and the principles of MD simulation.In section 4, the obtained results are exposed.The last section concludes the article.

Pair-potential expression
The total interaction potential between droplets with PEO-C12 is [12]: (1) ▪ The pair potential UHS(r) between hard spheres of diameter 2R is given by: Here, 'r' is the center-to-center distance and R is the radius of the nanodroplets.▪ In the case of two dispersed spheres, Uvdw(r) potential is approximately [20]: AH is the effective Hamaker constant, here, we consider HB A = 1.1 k T , this value is appropriate for decane nanodroplets interacting through water [21].▪ In the Derjaguin-Landau-Verwey-Overbeek (DLVO) theory [22], the screened electrostatic repulsion between the charged colloidal spheres is described by the effective pair potential: ( ) The dielectric constant of solvent is r ε = 80 and temperature T=298K , with a Bjerrum length of k .In this study, -1 -1 k =6.77Å , which corresponds to a concentration of small ions added to the solution, Zeff is the number of charge effective per nanodroplets, it determines the strength of the electrostatic repulsions [9].
▪ To the bare MEs, we add a telechelic polymer with one end (PEO-C12), which induces a steric repulsive potential V(r) between nanodroplets, defined in eq. ( 5),  These results are consistent with a similar study examining the effect of adding polymer with one hydrophobic end functions, PEO-m, on the interaction potential between charged MEs droplets [15].Molecular dynamics (MD) simulation is based on determining the trajectories and velocities of atoms through the numerical solution of the equations of motion.In this study, NVT MD simulations were performed, using the LAMMPS simulation package [23], in a cubic box of volume V, with N = 10 6 MEs in the simulation box, the equations of motion are integrated by using the Verlet algorithm [24].The quantities are calculated in reduced units.In order to eliminate bord effect and simulate an infinite system, three-dimensional periodic boundary conditions are applied [25].

Impact of Polymers on the Dynamic Properties of Microemulsions:
In this section, we will examine the effect of the addition of PEO-C12 polymers on the dynamic properties of the ME in the dilute case =2.8%  .Fig. 2 shows the reduced MSD, coefficient.This means that the random walker always undergoes at normal diffusion in the dilute case.We notice that when the number of added PEO-C12 increases from 0 to 32 the MSDs decrease, i.e., the diffusion coefficient decreases with increasing parameter Np, which means that the presence of PEO-C12 slightly decreases the dynamics of MEs.  1. Due to the presence of the steric repulsion, we observe that, as Np increases, Dc decreases linearly.
We have: 2906 .This behaviour of Dc indicates that the correlation between the MEs becomes important, as Np increases.This result has been observed in previous similar work [15,18].Fig. 4 shows the variation of reduced VACF as a function of dimensionless time.We notice that VACF is always positive, this explains that the movement of the microemulsions does not undergo displacement, because there is no interpenetration between the MEs droplets in the diluted case.It is observed that the VACF decreases rapidly with the addition of PEO-m polymers, which means that the addition of PEO-m to the MEs makes the diffusion a little more difficult than that of a bare MEs.In addition, the same plots indicate that the backs of the VACFs asymptotically approach zero.

Fig. 1
Fig. 1 illustrates the dimensionless pair potential U(r)/kBT, as a function of the dimensionless distance r/σ.We note that the presence of the electrostatic repulsion causes the presence of an energy barrier.For the bare ME, this barrier is very low indicating weak repulsive interactions between ME nanodroplets.When the PEO-C12 is added with different amounts (Np = 4, 8, 16, 32), the energy barrier also increases (Np from 4 to 32), this proves that the addition of PEO-C12 introduces a repulsive interaction, the potential barrier increases proportionally to the number of added POE-m per ME.The parameters of plot are: steric B p steric B p steric B p steric B p V =0.7k T (N =4), V =1.4k T (N =8), V =3.4k T(N =16), V =10.4k T (N =32)

Fig. 2 .Fig. 3 ,
Fig. 2. MSD versus time on a log-log scale, obtained from MD simulation, for MEs with different numbers Np of added PEO-C12 (from 0 to 32), at =2.8%  Fig. 3, presents the diffusion coefficient Dc as a function of the number of PEO-m added to the MEs Np, see Table1.Due to the presence of the steric repulsion, we observe that, as Np increases, Dc decreases linearly.

Fig. 3 .
Fig. 3. Diffusion coefficient Dc as a function of the polymer concentration Np, at Φ=2.8%

Fig. 4 .
Fig. 4. VACF versus reduced time obtained from MD simulation, for MEs with different polymer concentration Np (from 0 to 32), at Φ=2.8% We observe the presence of two time-dependent regimes, the first regime MSDs are represented by straight lines of the same slope and superimposable independently of the parameter Np, and the second one is for longer times 2 r (t), as a function of reduced time t * , on a log-log scale, in all studied cases (Np=0, 4, 8, 12, 16, 24, and 32). 1 t t ,  called the normal diffusion, in this regime, the MSDs are of linear form 2 c r = 6.D .t , with Dc the normal diffusion