## 1 Introduction

Water in oil micro-emulsions are thermodynamically stable systems which, under certain conditions, consist of discrete aqueous droplets dispersed in a continuous oil medium. One of the most well characterized droplet-type micro-emulsions is the one that formed in the ternary system sodium bis(2-ethylhexyl) sulfosuccinate AOT/water/oil. These systems have been studied by a great number of physicochemical methods [1,2]. It is now well established that AOT reverse micelle in isooctane consists of an approximately spherical water core surrounded by closed AOT palisade. An important and potential property of AOT reverse micelles is to confine various amounts of solubilized water. There is an apparent linear relationship between the molar ratio W_{0} = [H_{2}O]/[AOT] and the radius R_{m} of the water core. Reverse micelles and water in oil micro-emulsions can be used in order to solubilize biopolymers such as proteins, enzymes and genetic material in apolar solvents and to study their physicochemical properties in the presence of a limited amount of water [3]. The guest molecules are either dissolved in the water core or oriented to the AOT-water interface. They may acquire properties and reactivities that are different from those measured in the bulk aqueous phase.

It is known that by using a variety of physical techniques, the part of the water present in the water core of reverse micelles is bound to the AOT polar group as well as to the sodium counter-ions. Spectroscopic techniques, such as NMR, IR and Raman spectroscopies [4–8], have shown that the physicochemical properties of AOT reverse micelles change at low water content up to a molar ratio W_{0} = 10. In this regime, water is structured by its interaction with Na^{+} counter-ions and the strong dipole of the polar AOT. Above this threshold, the properties of the solubilized water approach progressively those of the bulk water. The water dynamics in reverse [9] or direct micelles [10] was investigated using molecular dynamics simulations. These works point out the existence of a slow orientational water dynamics near the surfactant interface. Similar trends, related to the water residence time, were found in various NMR studies of water molecules bonded to biological molecules such as DNA [11,12] or proteins [13,14].

Only few experimental tools are actually able to probe, on a very large extent, the liquid dynamics in confinement. A promising way is to measure the Nuclear Magnetic Resonance Dispersion (NMRD) of the proton or deuteron spin-lattice relaxation rate (R_{1}) on a large range of Larmor frequency [15]. The dynamical information is thus drawn out from these NMRD data according to a dynamical model. Several basic models and experiments have been proposed either for reorientational [16–18] and/or translational [19] molecular diffusion in interfacial confined media.

In an earlier work [20], we have studied the water dynamics in AOT reverse micelles for a single value of W_{0} (W_{0} = 50). An experimental NMRD study was conducted for the proton and/or the deuterium belonging either to the water, the surfactant or the isooctane molecule. A preliminary simulation of the NMRD dispersion curve was presented taking into account the intermittent Brownian dynamics of water inside reverse micelle [17,18]. At this stage, the Brownian rotational diffusion of the entire reverse micelle immersed in isooctane was not considered. This simulation was found to be qualitatively in agreement with the experiments but no serious attempt to fit the experimental data was conducted.

In the present work, ^{2}H NMRD measurements are obtained by varying W_{0} in the range 20 ≤ W_{0} ≤ 50. In parallel to the experimental investigation, we present a numerical simulation of the intermittent Brownian [16,17,19] dynamics of the water molecule inside a rotating reverse micelle, taking into account the internal intermittent Brownian motion of the water molecule and the overall Brownian motion of the reverse micelle. These simulations allow the computation of the NMRD dispersion curves, which are compared to the experimental data.

## 2 Experimental set-up

Sodium bis (2-ethylhexyl) sulfosuccinate, i.e. AOT of 99% purity and trimethyl-2,2,4 pentane (isooctane C_{8}H_{18}) have been purchased from Sigma Aldrich. Both compounds have been used without further purification. H_{2}O water has been double distilled and de-ionized (resistivity of 18 MΩ) and was used at pH = 6. D_{2}O water (99.9%) has been provided by CEA (Saclay). The appropriate amounts of D_{2}O have been injected into AOT solutions in isooctane by means of a Hamilton syringe. The amount of water incorporated into AOT reverse micelles corresponds to W_{0} = 20, 30, 40 and 50 with a micelle volume fraction ranging from 0.08 to 0.12. Table 1 gives the spherical reverse micelles radius R_{m} versus W_{0} [1,2]. The hydrodynamics radius R_{h} was measured by Dynamics Light Spectroscopy (DLS) using a nano sizer Malvern. R_{h} values are in good agreement with the data published by Zulauf and Eicke [1]. NMRD measurements have been performed, at a stabilized temperature of 298 K, on a fast-field-cycling spectrometer from Stelar Company.

**Table 1**

Variation of the spherical reverse micelles R_{m} versus the molar ratio W_{0} [1–2].

W_{0} | R_{m} (nm) |

20 | 3.5 |

30 | 5.1 |

40 | 6.5 |

50 | 7.9 |

## 3 Experimental results

Four deuteron NMRD profiles have been performed for (AOT/C_{8}H_{18}/D_{2}O) reverse micelles (W_{0} = 20, 30, 40 and 50). In these experiments, the longitudinal magnetizations exhibit a mono-exponential decay in all the Larmor frequency range studied (10 kHz–3 MHz). All profiles present a similar behavior for dispersion curves R_{1} (ω): a plateau up to a cut-off frequency followed the beginning of an algebraic decay (Fig. 1). The cut-off frequency and the spin-lattice relaxation rate of the plateau R_{1,plateau} decrease as W_{0} increases. As shown in Fig. 2, the spin-lattice relaxation rate of the plateau R_{1plateau} is inversely proportional to the micelle radius R_{m} (or to the molar ratio W_{0}). This result is due to the rapid exchange between bound water in interaction with the sulfonated polar surface and water in the internal micellar poll. Interesting enough, the interpolation to infinite radius R_{m} gives a relaxation rate very close to the spin-lattice relaxation rate of bulk water R_{1} (0.33 s^{−1}).

## 4 Discussion and numerical simulations

One of the main interests to look at deuteron NMRD is the knowledge of the magnetic interaction inducing the relaxation, mainly the quadrupolar interaction [21]. In the Larmor frequency range studied (10 kHz–3 MHz for deuteron), the spin-lattice relaxation rate is mainly due to two mechanisms: (i) the reorientation modulation coupled with translational diffusion of a quadrupolar interaction for ^{2}H [16]; and (ii) the Brownian rotational diffusion of the entire reverse micelle in isooctane [7]. Concerning the first mechanism, as shown elsewhere [17,18], a two-step model can be considered to describe the magnetic relaxation of the aqueous fluid nearby the interface. This model involves the superposition of fast motions (local rotations…) and slow dynamics depending on the time correlation of the spherical surface director probed by the molecule during its self-diffusion near the interface. We consider that fast and slow motions, in this two-step model, occur on different time-scales and are statistically independent [22]. At low frequencies, the spin lattice relaxation rate can be decomposed into a fast and a slow contribution such as ${R}_{1}(\omega )={R}_{1}^{slow}(\omega )+{R}_{1}^{fast}$. The fast contribution is related to local molecular dynamics and is almost constant at low frequency. For the slow contribution, the fluid trajectories are modelled as an alternate successions of adsorption steps (A) where the magnetic interaction I(t) is acting and Brownian bridge (B) in the confined bulk media where I(t) = 0. The slow motion contribution can be written as:

$${R}_{1}^{slow}(\omega )\propto J(\omega )+4J(2\omega )$$ | (1) |

Following [17,18], for a flat surface in the strong adsorption limit, J(ω) can be written as:

$$J(\omega )\propto 1/\left[{\left(\omega /{\omega}_{0}\right)}^{1/2}+\left(\omega /{\omega}_{0}\right)+1/2{\left(\omega /{\omega}_{0}\right)}^{3/2}\right]$$ | (2) |

_{0}is a characteristic frequency evolving as:

$${\omega}_{0}=\frac{{\delta}^{2}}{2D{{\tau}_{A}}^{2}}$$ | (3) |

δ is about the size of a water molecule, D is the self-diffusion of the water molecule nearby the interface (during the Brownian bridge) and τ_{A} is the average residence time on the interface. At low frequencies (ω << ω_{0}), J(ω) evolves as $1/\sqrt{\omega}$. Around ω_{0}, we have a (1/ω) regime. Finally, for (ω >> ω_{0}), J(ω) is dominated by the term ${(1/\omega )}^{3/2}$. Such a regime should be observed in reverse micelle at relatively high frequencies where, in the corresponding time scale, the water molecule probes an intermittent interaction with the sulfonate heads on a locally flat internal surface. At a longer time-scale, a cut-off should appear, linked to the finite size of the reverse micelle and the appearance of an upper terminal time for the bridge statistics. In parallel, the Brownian rotation of the entire reverse micelle in isooctane could impose its own orientation relaxation. A Lorentzian profile of J(ω) (or R_{1}(ω)) corresponds to this mechanism.

At this point, there is no trivial and obvious separation between these sources of orientation fluctuations.

In order to classify these relaxation processes mentioned, a Brownian dynamics simulation is performed by taking into account the two mechanisms: the Brownian intermittent dynamics of water inside the reverse micelle core and the Brownian rotational diffusion of the entire reverse micelle. Using the measured hydrodynamic radius R_{h} of the reverse micelle, a rotation diffusion coefficient is estimated using the Perrin's relationship, allowing one to simulate the rotational Brownian motion of the reverse micelle. The Y^{m}_{2}(Ω(t)) autocorrelation functions are computed in the laboratory framework. More numerical details can be found elsewhere [16].

In Fig. 3, we present our numerical results for the case W_{0} = 40. The self-diffusion of water is taken as D = 10^{−9} m^{2}/s. This value, of the same order that the bulk water diffusion, is in the same range that the molecular dynamics simulation proposed in [9]. An increase of D will shift the NMRD dispersion curve towards high frequencies. In a first computation, we have set the adsorption time of the water molecule to a very large value (1 ms). In such a situation, we suppress the intermittent Brownian motion. The only origin of the NMR dispersion is due to the rotation of the reverse micelle. As shown in Fig. 3, the dispersion curve exhibits a Lorentzian shape with a 1/ω^{2} evolution in the experimental frequency window (0.01–2 MHz). This evolution is far off the experimental trend. In a second computation, we take τ_{A} = 5 ns, an order of magnitude also found in some other hydrophilic surfaces [16,17,23]. An extended plateau is observed at low frequencies followed by the beginning of a decay (which finally evolves as 1/ω^{3/2} (in a range which was not experimentally probed in this present work)). This last part is related to the intermittent Brownian dynamics. Interesting enough, the renormalized experimental data exhibit a similar trend.

Our simulation shows that the rotation of the reverse micelle, almost for W_{0} = 40, is not the main origin of the deuteron spin-lattice relaxation. The high frequency part of the NMRD dispersion curves reveals the internal dynamics of the water molecule inside reverse micelle.

In a next future, we plan to analyse the NMRD in an extended frequency range from 5 MHz to 50 MHz.

## 5 Conclusion

In this work we have reported deuteron field-cycling relaxometry for the microemulsion phase in the (AOT/isooctane/D_{2}O) system. This study has allowed us to analyse the slow dynamics of water confined in reverse micelles (of nanometric sizes) and the Brownian rotational diffusion of the spherical micro-emulsion droplet (water and AOT) in isooctane. The typical relaxation features have been interpreted according to two processes: (i) the molecular reorientation of water molecules coupled with translation diffusion inside a spherical confinement; and (ii) the Brownian rotational diffusion of the entire reverse micelle in isooctane. A numerical simulation, for W_{0} = 40, is proposed to analyse the degree of coupling of these mechanisms. In a future work, we aim at extending our study to analyse the influence of the micellar size, the volume fraction and the temperature on the relaxation NMRD profiles in a more extended frequency range.

## Acknowledgements

This research was supported by the grant CNRS/DGRSRT N 18518. Grants from ANR project DYOPTRI is also acknowledged.