Computer Simulations of Solute Exchange Using Micelles by a

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Computer Simulations of Solute Exchange Using Micelles by a Collision-Driven Fusion Process Shuangyang Li,† Xianren Zhang,*,† Wei Dong,‡ and Wenchuan Wang† DiVision of Molecular and Materials Simulation, Key Laboratory for Nanomaterials, Ministry of Education, Beijing UniVersity of Chemical Technology, Beijing 100029, China, and Laboratoire de Chimie, UMR 5182, CNRS, Ecole Normale Supe´rieure de Lyon, 46, Alleé d’Italie, 69364 Lyon Cedex 07, France ReceiVed May 17, 2008. ReVised Manuscript ReceiVed June 13, 2008 In this work, the kinetic process of collision-driven solute exchange in an aqueous phase in which micelles are used as solute carriers is investigated by dissipative particle dynamics simulations. Here, we try to answer two questions about the exchange process of hydrophobic solute molecules: How the solute molecules are exchanged and what factors affect the process. For the first question, the simulation results indicate that, after a stage of intermittent collision between two neighboring aggregates, there are roughly three sequential events in a coalescence stage: (1) molecular contact, (2) neck formation, and (3) neck growth. The coalescence stage is followed by a stage of solute transfer and diffusion. It is found that there are two rate-limiting steps in the whole process of solute exchange, i.e., the break of the water film between two neighboring aggregates and the nucleation of a pore between two surfactant films. For the second question, the effects of the collision velocity, the surface tension, the repulsive interaction between the surfactant films of the colliding aggregates, as well as the steric repulsion are examined. For example, the simulation results show that the depletion force plays an important role during the coalescence stage, while the initial collision velocity basically does not change the fusion ratio. The results also demonstrate that the surface tension and interaction show different effects on the different stages of a solute exchange process.

1. Introduction A surfactant contains two parts: a hydrophobic tail and a hydrophilic head. The tail is composed of one or more hydrocarbon chains, while the head is composed of chemical groups with high affinity to water. Such a composition of amphiphilic molecules results in many interesting properties and extensive applications. For example, micelles self-assembled by surfactants are extensively used as mobile carriers for the transport of different solutes with poor water solubility across an aqueous phase.1–10 In the so-called emulsion and microemulsion systems, water-insoluble solute molecules are solved inside the core of micelles, and the involving micelles play a role as mobile carriers of the solute molecules. In the applications, collision-driven solute exchange processes give rise to the migration of insoluble molecules between different aggregates. Apart from the conventional applications in chemical engineering, emulsion and microemulsion systems with solute dispersion and exchange are involved in a number of newly developed techniques. For example, for a process of oil extraction, an important fraction of crude oil (residual one) is trapped in the small porous spaces of the rock reservoir and can not be extracted with conventional techniques because of high capillary force. Injection of surfactant-like solutions can reduce this interfacial tension and hence facilitate residual oil recovery. The use of micelles for the transport of different solutes has also been used * Corresponding author. E-mail: [email protected]. † Beijing University of Chemical Technology. ‡ Ecole Normale Supe´rieure de Lyon.

(1) Xenakis, A.; Tondre, C. J. Phys. Chem. 1983, 87, 4737. (2) Tondre, C.; Xenakis, A. Faraday Discuss. Chem. Soc. 1984, 77, 115. (3) Derouiche, A.; Tondre, C. J. Chem. Soc., Faraday Trans. I 1989, 85, 3301. (4) Tondre, C.; Derouiche, A. J. Phys. Chem. 1990, 94, 1624. (5) Plucinski, P.; Nitsch, W. J. Colloid Interface Sci. 1992, 154, 104. (6) Plucinski, P.; Nitsch, W. J. Phys. Chem. 1993, 97, 8983. (7) Plucinski, P.; Nitsch, W. Langmuir 1994, 10, 371. (8) Plucinski, P.; Nitsch, W. Langmuir 1995, 11, 4691. (9) Nitsch, W.; Plucinski, P.; Ehrlenspiel, J. J. Phys. Chem. B 1997, 101, 4024. (10) Steyler, D. C.; Towey, T. F.; Robinson, B. H. Langmuir 2001, 17, 417.

for the preparation of self-assembled monolayers (SAMs).11,12 In a recently developed method to prepare SAMs,13–16 waterinsoluble solute molecules are transferred to a physisorbed surface through the aqueous phase by using micellar particles as mobile carriers. Such solute exchange processes also play important roles in encapsulated drug delivery (for example, see refs 17–19). Micelles are particularly attractive for drug delivery because they do not necessarily change the chemical identity of the drug. In the application, the drug is loaded into the core of micelles formed by surfactants or copolymers with surfactant nature, and the corona of the micelles can be adapted to enhance the solubility and stability of the drug and its permeability to biological membrane. As mentioned above, understanding the mechanism of the insoluble solute exchange under the help of micelles is of both fundamental and practical importance. Although extensive experimental and theoretical studies have been carried out in literature in order to understand the phase behaviors of microemulsion systems and their corresponding static structures, much less is known about the kinetic evolution of these structures. 20 Recently, Rharbi et al.21,22 measured the solute exchange by employing pyrene probes. Three mechanisms of solute exchange between micelles are usually considered: (i) through bulk (aqueous) phase, (ii) during collisions between micelles, namely the collision-exchange-separation mechanism, and (iii) via fragmentation-growth processes. The first mechanism involves (11) Ulman, A. Chem. ReV. 1996, 96, 1533. (12) Poirier, G. E. Chem. ReV. 1997, 97, 1117. (13) Liu, J.; Kaifer, A. E. Isr. J. Chem. 1997, 37, 235. (14) Yan, D.; Saunders, J. A.; Jennings, G. K. Langmuir 2000, 16, 7562. (15) Yan, D.; Saunders, J. A.; Jennings, G. K. Langmuir 2002, 18, 10202. (16) Yan, D.; Jordan, J. L.; Burapatana, V.; Jennings, G. K. Langmuir 2003, 19, 3357. (17) Bader, H.; Ringsdorf, H.; Schmidt, B. Angew. Makromol. Chem. 1984, 123-124, 457. (18) Savic´, R.; Luo, L.; Eisenberg, A.; Maysinger, D. Science 2003, 300, 615. (19) Otsuka, H.; Nagasaki, Y.; Kataoka, K. AdV. Drug. DeliV. ReV. 2003, 55, 403. (20) Gradzielski, M. Curr. Opin. Colloid Interface Sci. 2003, 8, 337.

10.1021/la801521b CCC: $40.75  2008 American Chemical Society Published on Web 08/01/2008

Collision-DriVen Solute Exchange

the exit of solute from a micelle into aqueous phase and its re-entry into another micelle. This mechanism is important when the solubility of solute in water is sufficiently high. In the second mechanism, a solute exchange process involves the coalescence of two neighboring micelles and exchange of solutes, followed by their separation. The third mechanism involves fragmentation of a micelle into two smaller entities followed by their growth into micelles with normal sizes. However, until now, it has not been possible to observe a solute exchange process directly. Experiments that measure relaxation kinetics provide evidence for a change of mechanism, but must rely on the proposed models for different mechanisms to try to distinguish them. Thus there is no direct evidence to confirm that only the particular mechanism is taken for the solute exchange in a given system. More importantly, two fundamental questions, namely, how solute molecules are transferred in molecular details and how molecular structures of surfactants and other microscopic properties would affect the solute exchange, are little known and difficult to probe experimentally. In the pioneering works of Karaborni et al.,23 the solubilization of nonpolar molecules into aqueous surfactant solutions has been studied by computer simulations rather than experiments. They identified three mechanisms through which oil molecules are transferred from the oil phase to the micelles. In light of the above, it would certainly be desirable to examine the exchange mechanism of the insoluble solute in a system whose components are described by a microscopic model. Such examination is readily possible now. For example, molecular simulations have been performed on the formation of vesicles24,25 and the fusion mechanism of membranes.26–28 A coarse-grained simulation of fusion between two contacted vesicles has been studied.29 In their simulations, the authors found that fusion is initiated when some of these tilted lipids splay their aliphatic tails. Shillcock and Lipowsky30 and Andrea et al.31 recently studied the fusion between vesicle and bilayer membranes. They showed that the fusion process depends on the membrane tension,30 which can be directly controlled in computer simulations with explicit water. Furthermore, several time scales of the fusion process were found to depend exponentially on the membrane tension.31 Recently, Rekvig and Frenkel32 studied droplet coalescence in oil/water/surfactant systems using dissipative particle dynamics (DPD) simulation along with forward flux sampling (FFS). They showed that the rate-limiting step in coalescence is the rupture of the surfactant film, and the rupture rate showed a nonmonotonic variation with natural curvature of surfactants. The above-mentioned works provide important insights to assess the mechanism of solute exchange. However, in all above literature, two aggregates were presumably placed close together. Thus, the fusion process in their studies is only controlled by surface tension29–32 and bending rigidity,32 and no dynamic aspect of collision-driven fusion was considered. For an individual fusion event, the fusion ratio may depend only on the interface properties. However, in a collision-driven solute (21) Rharbi, Y.; Winnik, M. A.; Hahn, K. G. Langmuir 1999, 15, 4697. (22) Rharbi, Y.; Li, M.; Winnik, M. A.; Hahn, K. G. J. Am. Chem. Soc. 2000, 122, 6242. (23) Karaborni, S.; van Os, N. M.; Esselink, K.; Hilbers, P. A. J. Langmuir 1993, 9, 1175. (24) Yamamoto, S.; Maruyama, Y.; Hyodo, S. J. Chem. Phys. 2002, 116, 4842. (25) Marrink, S. J.; Mark, A. E. J. Am. Chem. Soc. 2003, 125, 15233. (26) Noguchi, H.; Takasu, M. Biophys. J. 2002, 83, 299. (27) Mu¨ller, M.; Gompper, G. Phys. ReV. E. 2002, 66, 041805. (28) Mu¨ller, M.; Katsov, K.; Schick, M. Biophys. J. 2003, 85, 1611. (29) Stevens, M. J.; Hoh, J. H.; Woolf, T. B. Phys. ReV. Lett. 2003, 91, 18. (30) Shillcock, J. C.; Lipowsky, R. Nat. Mater. 2005, 4, 225. (31) Grafmu¨ller, A.; Shillcock, J. C.; Lipowsky, R. Phys. ReV. Lett. 2007, 98, 21810. (32) Rekvig, L.; Frenkel, D. J. Chem. Phys. 2007, 127, 134701.

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exchange process as is showed below, the efficiency of solute exchange depends on not only the surface tension, but also the dynamic aspects of collision process. The remainder of the paper is organized as follows. In Section 2, the simulation method and model are presented. In Section 3, we describe the preparations of initial configurations in detail. Then, the results and discussion are presented in Section 4. In this section, we first describe the mechanism of solute exchange in molecular details, and then discuss the effects of the initial collision velocity, the surface tension, the head-head repulsion, and the steric repulsive interaction between the colliding aggregates on the kinetics of the solute exchange, respectively. This is followed by a brief summary of the main conclusions.

2. Model and Simulation Method The lattice Monte Carlo (LMC) method developed by Larson33,34 was extensively used to simulate the behaviors of surfactants35–51 and very efficient at the self-assemblies of surfactants and equilibrium structures.35–48 However, the LMC is mainly constrained to investigate equilibrium properties of surfactant systems, although some efforts have been applied to simulate dynamic behaviors.49–51 In contrast to the Monte Carlo method, molecular dynamics (MD) is more suitable to study the dynamic properties of surfactant systems. Nevertheless, for the complex system we studied in this work, the relevant length and time scales almost exceed the capabilities of fully atomistic MD. During the past few years, a kind of “coarse-grained” simulation technique, DPD, has been proposed to overcome this problem. The DPD method was invented initially to study the hydrodynamic behaviors of complex fluids.52–54 Recently, it has been applied to study a variety of amphiphilic systems.55–61 In the present work, the model system contains solvent particles (denoted by W), surfactant molecules (head bead is denoted by (33) Larson, R. G.; Scriven, L. E.; Davis, H. T. J. Chem. Phys. 1985, 83, 2411. (34) Larson, R. G. J. Chem. Phys. 1989, 91, 2479. (35) Mackie, A. D.; Panagiotopoulos, A. Z.; Szleifer, I. Langmuir 1997, 13, 5022. (36) Floriano, M. A.; Caponetti, E.; Panagiotopoulos, A. Z. Langmuir 1999, 15, 3143. (37) Lisal, M.; Hall, C. K.; Gubbins, K. E.; Panagiotopoulos, A. Z. J. Chem. Phys. 2002, 116, 1171. (38) Panagiotopoulos, A. Z.; Floriano, M. A.; Kumar, S. K. Langmuir 2002, 18, 2940. (39) Kim, S.-Y.; Panagiotopoulos, A. Z.; Floriano, M. A. Mol. Phys. 2002, 100, 2213. (40) Lisal, M.; Hall, C. K.; Gubbins, K. E.; Panagiotopoulos, A. Z. J. Chem. Phys. 2002, 116, 1171. (41) Scanu, L. F.; Hall, C. K.; Gubbins, K. E. Langmuir 2004, 20, 514. (42) Siperstein, F. R.; Gubbins, K. E. Langmuir 2003, 19, 2049. (43) Bhattacharya, B.; Gubbins, K. E. J. Chem. Phys. 2005, 123, 134907. (44) Zhang, X. R.; Chen, B. H.; Dong, W.; Wang, W. C. Langmuir 2007, 23, 7433. (45) Zheng, F. X.; Zhang, X. R.; Wang, W. C.; Dong, W. Langmuir 2006, 22, 11214. (46) Zhang, X. R.; Chen, G. J.; Wang, W. C. J. Chem. Phys. 2007, 127, 034506. (47) Zheng, F. X.; Zhang, X. R.; Wang, W. C. J. Phys. Chem. C 2007, 111, 7144. (48) Zhang, X. R.; Cao, D. P.; Wang, W. C. J. Phys. Chem. C 2008, 112, 2943. (49) Arya, G.; Panagiotopoulos, A. Z. Phys. ReV. E 2004, 70, 031501. (50) Arya, G.; Panagiotopoulos, A. Z. Comput. Phys. Commun. 2005, 169, 262. (51) Zhang, X. R.; Chen, B. H.; Wang, Z. H. J. Colloid Interface Sci. 2007, 313, 414. (52) Hoogerbrugge, P. J.; Koelman, J. M. V. A. Europhys. Lett. 1992, 19, 155. (53) Groot, R. D.; Warren, P. B. J. Chem. Phys. 1997, 107, 4423. (54) Espanõl, P.; Warren, P. B. Europhys. Lett. 1995, 30, 191. (55) Shillcock, J. C.; Lipowsky, R. J. Chem. Phys. 2002, 117, 5048. (56) Malfreyt, P.; Tildesley, D. J. Langmuir 2000, 16, 4732. (57) Kranenburg, M.; Venturoli, M.; Smit, B. Phys. ReV. E 2003, 67, 060901. (58) Kranenburg, M.; Venturoli, M.; Smit, B. J. Phys. Chem. B 2003, 107, 11491. (59) Kranenburg, M.; Laforge, C.; Smit, B. Phys. Chem. Chem. Phys. 2004, 6, 4531. (60) Kranenburg, M.; Vlaar, M.; Smit, B. Biophys. J. 2004, 87, 1596. (61) Kranenburg, M.; Smit, B. J. Phys. Chem. B 2005, 109, 6553.

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Figure 1. An example demonstrating how to get an initial configuration in three steps (a-c). See text for details. The red beads represent head groups, gray ones represents tail groups, and the yellow particles in (b) and (c) are oil molecules. Water particles are not displayed.

H and tail bead denoted by T), oil molecules (oil bead is denoted by O), and two solid surfaces (solid bead is denoted by S). For simplicity, we consider the case that the tail bead is identical to the oil bead. A oil molecule is composed of five oil beads, i.e., T5, and two kinds of surfactant are studied, i.e., HT5 and H2T5. Like in a MD simulation, the time evolution of a DPD system is obtained by solving the equation of motion. The interparticle force exerted on a particle is composed of conservative, dissipative, and random force. The conservative force is a soft repulsion acting along the line of the particle centers and is given byFijC ) aijrîj max{1 - rij/rc,0}, where aij is the maximum repulsive force between particles i and j, rij ) rj- ri (ri and rj are the positions of particle i and particle j), rij ) |rij|, rîj ) rij/|rij|, and rc is the interaction range. Besides this main component of the conservative force, in this work there are also some other conservative forces exerted between the neighbor beads of a chain molecule (surfactant and oil molecule). They are the forces constraining the bond length and the bond angle, respectively. The bond length can be constrained by a spring force, FS ) KS(rij - req)rîj, where KS ) 128 and req ) 0.7rc.57 The force constraining the variation of the bond angle is given by Uφ ) Kφ(1 - cos (φ - φ0)), where Kφ ) 10 and φ0 is set to be π.55 In order to increase the number of collisions and hence the number of fusions between two aggregates, we chose that two full bilayers are supported by two solid surfaces, which provide the largest collision area, and confine the oil-bearing micelle. The solid surfaces are modeled by the arrangement of 6084 solid sites on a square lattice. The wall sites are attached to their equilibrium positions by a spring force, FW ˆ ij, where the spring constant KW is i ) KW(|ri - req,i|)r set to be equal to -50.56 The time evolutions of systems were obtained from a modified version of the velocity-Verlet algorithm53 with a time step of ∆t ) 0.025. Further details of the simulation technique are given in the Supporting Information.

3. Initial Configurations Since our main purpose here is to study the solute transfer from an oil-bearing micelle to a bilayer, we first prepared full bilayers covering solid surfaces. For this purpose, we used a system confined by two solid slabs of the size 24 × 24 and the box length in the direction perpendicular to the surfaces was set to 31. The simulation box contains 28340 solvent beads, 6084 solid particles forming the slabs, and 4500 H1T4 molecules, which are in larger excess so that bilayers can be formed on the solid surfaces. From a random initial configuration, a long DPD simulation of more than 300 000 time steps was performed. The simulation results indicate that the aggregation of surfactants takes place first on the surface. As the number of the adsorbed surfactants increases, the size of the aggregate increases, and the

morphology changes from adsorbed spherical micelles to cylinders and then to bilayers covering the surfaces totally. At the end of the simulation run, the surfactants self-assemble into bilayers on both solid surfaces. Since the surfactants are in large excess, a sheet of bilayer is also formed in the bulk. Figure 1a shows the final configuration of this simulation in which the solvent beads are not shown in order to get a clear presentation. Second, we prepared a solute-bearing micelle. A simulation box of the size 24 × 24 × 24 was used in which 450 H1T4 surfactants, 550 oil molecules (T5), and 36432 solvent molecules were randomly arranged. Then a DPD simulation was performed until the surfactants self-assembled into a single micelle wrapping all the oil molecules inside. Figure 1b shows the final configuration of this simulation. For studying the effects of surface tension and the surfactant architecture, several similar simulations were carried out for different numbers of surfactants and surfactant architectures, i.e., H1T4 and H2T4. In order to study the solute exchange, we need to combine the oil-free bilayers adsorbed on the solid surfaces with the oilbearing micelle together. First, we removed a portion containing the bulk bilayer from the system shown in Figure 1a. Then, we cut out a suitable portion of the system shown in Figure 1b, which contains the micelle, and then inserted this portion into the system in which the bilayer in the bulk has been removed. All these operations were carried out in such a way that the structures of the micelle and bilayer covering the solid surfaces were not affected, and the overall density of the system was not changed. The obtained configuration, which contains both oilfree bilayers and an oil-bearing micelle (see Figure 1c), was used as the initial configuration for the simulation of the solute exchange from the micelle to the bilayer.

4. Results and Discussion 4.1. Solute Exchange Mechanism. To study the solute exchange mechanism, we performed several simulation runs from the obtained initial configurations. At the beginning of the simulations, the velocity of each bead was randomly added, and then all the velocities were rescaled to fix the system and to maintain the given temperature (the kinetic energy of the system). The simulation runs are long enough for the solute exchange to occur. Simulation results show that solute-bearing micelles are continuously kicked by the fast-moving water beads, and diffuse to collide with precovered solid surfaces. During the intermittent collision stage, it is found that, after collision has taken place, the micelle is adsorbed on the bilayer for a long time until a desorption event occurs, and disappears into the solvent if no fusion occurs. In order to describe the collisions between the micelle and the bilayer before fusion


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Figure 3. Schematic drawings of (a) the two separate surfactant films, (b) the overlapped head film, and (c) the formation of a neck. Figure 2. Distributions of different contacts between the micelle and the bilayer during the intermittent collision stage before fusion.

occurs, Figure 2 records the number of contacts for beads belonging to different aggregates. First, we need to pay attention to Figure 2c, which shows the number of contacts between the tail beads belonging to the bilayer and the tail(oil) beads inside the micelle. It is found that during the whole collision stage, the number of the tail-tail contacts is always less than two, which means that both the surfactant film of the micelle and that of the bilayer keep their structures and do not rupture. Second, the head-head and head-tail(oil) contacts (see Figure 2a,b) show the behavior of intermittent collision. The depletion force between the two objects is believed to be responsible for the intermittent adsorption behavior. However, under the appropriate conditions, a coalescence stage takes place after both the surfactant film of the micelle and that of the bilayer break. And this is followed by the stage of solute transfer and exchange. In the coalescence stage, our results of computer simulations show that there are, in general, three sequential events. (i) Molecular contact: For a coalescence to be at all possible, the two surfactant films must come into molecular contact. If the micelle, which is in the close vicinity of the bilayer, has relative high surface tension and sufficiently long contact time (see below), collision-driven fusion of the two objects may occur. (ii) Neck formation: A characteristic of the event is the formation of a neck connecting the two objects. This event contains two steps: the formation of a single overlapped head film and its rupture, i.e., the formation of a pore. When the two aggregates get close together, the head groups of the two films may insert into each other, and a single film of head groups forms (see Figure 3b), which is called the overlapped head film. It is easy to understand that the overlap of two approaching surfactant films generates a repulsive barrier in this step. After the formation of the film, the two hydrophobic parts belonging to different aggregates approach each other and are separated only by a single film of headgroup. Then, the following step involves the formation of a pore in the film due to disturbance or instability, which makes it possible for hydrophobic beads (tail or oil) belonging to different aggregates to contact directly. The formation of such a pore (neck) is affected by the number density of surfactant on the separating film. Note that the neck at the stage is unstable and may disappear as an unstable nucleus. (iii) Neck growth: the size of the pore will grow continuously if it is larger than its critical value and makes the solute exchange possible. In contrast to the previous event, the growth is spontaneous, and there appears to be no resistance for the neck to grow irreversibly.

The coalescence stage is followed by the stage of solute transfer and diffusion. In this stage, oil (solute) molecules are transferred from the micelle to the bilayer and then diffuse inside the bilayer. In general, our simulation results show that the whole process can be roughly divided into three different stages. They are the intermittent collision stage, the coalescence stage, and the solute transfer and exchange stage. As an example, several snapshots for coalescence and transfer stages are shown in Figure 4. Correspondingly, the numbers of different contacts between different objects during the two stages are shown in Figure 5. According to our above discussion, Figures 4 and 5 show that there are three sequential events during the coalescence stage, i.e., molecular contact (a-b), neck formation (b-c) and neck growth (c-d), which is followed by the transfer and diffusion stage (d-g). During the waiting time for coalescence to occur, although the two objects keep contact for a long time, the number of tail-tail contacts is less than two, and both of them keep their structure and shape (see Figure 4a and Figure 5), just like that in the intermittent collision stage. And during the contact time, the two films of head groups belonging to different objects overlap and form a single layer (the overlapped head film), as is shown in Figure 3b. The existence of the film can also be recognized from the time evolution of molecular contacts, in which it is found that the number of head-tail(oil) contacts between different objects can be as large as 40 before the formation of a pore. The following rupture of the film (formation of a pore) is conceived as an activated process due to thermal fluctuation or a certain kind of collective behavior of surfactants inside the film. The formed pore is not stable and may be disappear due to fluctuation until it reaches a critical size. Figure 4b shows the formed neck. Once the pore reaches the certain critical size, i.e., its critical nucleus, a stable neck connecting the two aggregates is formed. To approximately recognize the energy barriers for the formation of the overlap film and its rupture, we define the variable called excess interaction energy. Imagine that the two aggregates in our system are composed solely of the solvent rather than surfactant and solute. As a result, after their first molecular contact, they can be mixed in the absence of surfactant without any significant energy barrier. Therefore, it seems reasonable to consider that the difference of the interaction between the two aggregates and that if the contacting groups are replaced by solvent can be used to approximate the barrier of coalescence.

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Figure 4. Typical snapshots at coalescence stage (a-d) and the stage of solute transfer and exchange (d-f). (a) t ) 244 000, molecular contact between the two aggregates. (b) t ) 245 000, a neck forms. (c) t ) 246 000, the neck grows. (d) t ) 248 000, the neck reaches its largest size. (e) t ) 250 000, oil molecules are transferring. (f) t ) 250 000, only oil molecules are shown. (g) t ) 390 000, oil molecules are diffusing. (h) t ) 390 000, only oil molecules are shown.

Figure 5. Numbers of tail-tail(oil) contacts between the micelle and the bilayer after the intermittent collision stage.

The difference, which is called here excess interaction energy, is written as

∆U ) UHH ×

aHH - aWW aHT(O) - aWW + UHT(O) × + aWW aWW aTT(O) - aWW UTT(O) × (10) aWW

where UHH, UHT(O), UTT(O) are total interaction energies between the two aggregates due to head-head, head-tail(oil), and tail-tail(oil) contacts, respectively. The evolution of excess interaction energy for a typical coalescence stage is shown in Figure 6, from which one can clearly find that the formation of the overlapped head films generates a repulsive barrier. More interestingly, the rupture of the film does not result in higher excess interaction energy. This indicates that the rupture is in fact a collective process. After reaching a critical value the neck would grow (see Figure 4c) in a spontaneous way since the coalescence reduces the surface tension of the micelle. From the snapshot at t ) 248 000 (see

Figure 6. Time evolution of the different molecular contacts and the corresponding excess interaction energy during the formation of an overlapped head film and its rupture.

Figure 4d), corresponding to point d in Figure 5, one can find that the neck reached its largest size, and oil molecules begin to transfer into the bilayer. During the range of time steps from d to e in Figure 5, the number of tail-tail contacts increases sharply with time. This stage is identified as the solute transfer stage since, in this stage, most oil molecules cross the neck and transfer from the micelle to the bilayer. At the state e, the transfer stage finishes. But due to the slow diffusion of oil inside the bilayer, all oil molecules still assemble as a cluster, as is shown in Figure 4f). Since then, the increase of number of tail-tail(oil) contacts slows down, and the rather slow diffusion of oil molecules inside the bilayer dominates the stage. Note that the different stages we defined above in some cases can not be recognized clearly. For example, the growth of a neck and the transfer of oil molecules are sometimes in progress almost simultaneously rather than in sequence. And their duration times may vary at different runs. In this work, since the solute molecules show a negligible solubility in water, only the collision-exchange-separation and fragmentation-growth mechanisms are possible to lead to the


exchange of solutes between micelles.21,22 Moreover, due to the very small headgroup size and the weak head-head repulsion, the rupture of a micelle is suppressed. These are the reasons why the fragmentation-growth process was not observed in this work, and also why after the fusion of two neighboring micelles, the intermediate does not break into two solute-bearing micelles, but remains as a bigger solute-bearing micelle. However, the increase of headgroup size or head-head repulsion achieved by changing the temperature or addition of slat to the ionic surfactant solution, would increase the possibility of the rupture of the micelle. On the other hand, the fusion of neighboring micelles becomes more difficult when the head size or head-head repulsion increases, as will be shown below. Thus, for larger head size or stronger head-head repulsion, the fusion of micelles is suppressed, and the fragmentation-growth mechanism would dominate. Below, the effects of different factors on the efficiency of solute exchange, including the collision velocity, the surface tension, the head-head repulsive interaction, and the molecular architecture of surfactants, are discussed. 4.2. Effects of the Initial Collision Velocity. Now, we need to define the efficiency of collision-driven solute exchange. Note that the time needed for fusion is not a good measure of the efficiency. First, a micelle in solvent moves rather randomly, and thus the time for the molecular contact between the two objects is strongly related to the initial condition (for example, the velocity and position distributions). Second, the inherently stochastic nature of fusion determines the stochastic variation of waiting time before fusion. Therefore, the time for fusion obtained from a single simulation run is not a suitable variable to represent the efficiency of collision-driven solute exchange. In this work, we introduced another variable, i.e., fusion ratio, to characterize the efficiency of the solution exchange. The fusion ratio is defined here as the ratio of the number of successful fusions to the total number of collisions, in which a collision is defined as when a micelle in solvent moves to a molecular contact with a bilayer. In principle, the fusion ratio is a statistical average value related to the height of the energy barrier preventing the two objects from fusion. Because of the stochastic nature of fusion, a large number of fusion events are required to get statistically meaningful results. Considering the fact that the most of computer time in the whole simulated process is spent on the Brownian motion of the micelle in the solvent, here we design a method to directly simulate the micelle-bilayer collisions rather than to consider the whole process including Brownian motion in the solvent. For this purpose, we first need to prepare initial configurations in which the micelles are located nearest the bilayer without molecular contact. To obtain the exact relation between the number of molecular contacts and the locations of the center of mass (COM) of the micelles, constrained DPD simulations were performed. During a constrained DPD simulation, the COM of a micelle is fixed by adding a virtual force on each bead i of the micelle. The virtual force is given by Fvirtual ) -∑i∑j*ifij/Nmicelle, where fij is the interaction force between the beads i belonging to the micelle and its neighbor j, and Nmicelle is the total number of beads for the micelle. Other than the fix of the COM, the evolution of the system is the same as the normal DPD simulation. For the micelle with 250 surfactants, we performed a number of constrained DPD simulation runs corresponding to different locations of its COM. The obtained molecular contact between the two objects is shown in Figure 7a as a function of the COM distance from the solid surface. The results show that the average number of head-head contacts increases as the distance between

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Figure 7. (a) Average values of head-head contacts for micelles at different locations in the z axis. (b) Average number of water beads in the unit cell of the water film between two colliding aggregates. The unit cell is of a cuboid shape with unit area of a square cross section centered at the bead of the micelle nearest to the bilayer.

the micelle and bilayer deceases. The locations where the micelles are nearest the bilayer is determined to be 13.4, at which the number of head-head contacts is first close to zero, and there are approximately three water beads in the unit cell of the water film between two colliding aggregates (see Figure 7b). The unit cell is of a cuboid shape with unit area of a square cross section centered at the bead of the micelle nearest to the bilayer. Because of Brownian motion of a micelle in water, the micelle and the bilayer collide with a random relative velocity. One may expect therefore that the initial collision velocity would play an important role for the fusion of two aggregates. For example, the high initial collision velocity may favor the fusion between the micelle and the bilayer. To investigate its effects, different simulation runs at different initial collision velocities were performed. Here the initial collision velocities, which are perpendicular to the bilayer, were set to 0.01, 0.05, 0.1, 0.2, 0.3, 0.4, and 0.5, respectively. To save computer time, we directly simulated the micelle-bilayer collisions. In such a simulation run there are two repeated steps. In the first step, the COM of the micelle with 250 surfactants was fixed at z ) 13.4, and 1500 time steps in the constrained DPD method were performed to generate a new starting configuration for the following collision. The obtained configuration is also saved as the returned point after the collision step finishes. In the collision step, the micelle moves to and then collides with the bilayer at the given initial collision velocity. In this step, if the numbers of tail-tail(oil) contacts are more than 20, it is considered that a fusion event occurs. However, if the evolution lasts 2 × 104 time steps and no fusion occurs, it is considered that the collision does not lead to fusion. Regardless of the occurrence of the fusion, the collision

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Figure 9. Distribution of collision duration time before fusion occurs for the micelle formed by 250 H1T4 surfactants. Three different curves correspond to three different initial collision velocities.

Figure 8. (a) Fusion ratio for the micelles formed by different numbers of surfactants and at different initial collision velocities. (b) Results from the five individual simulations to give average and statistical uncertainty for the micelles with 750 surfactants.

step completes, and the system returns to the saved configuration for next repeated steps. Again 1500 time steps of constrained DPD were performed to generate a new initial configuration, and then the collision step was repeated. In this way, this simulation method not only greatly enhances the sampling of collision stage, but produces a large number of initial configurations. A single simulation run in this method lasts, in general, 1.5 × 106 time steps. Furthermore, in this work, several individual simulations lasting 1.5 × 106 time steps were performed to given reasonable average and statistical uncertainties of the fusion ratio. Note that, during the process to generate the given collision velocity, several iterative steps may be required to ensure the system at the given temperature and the given relative collision velocity. Note that Rekvig et al. have developed a computer technique combining DPD with grand canonical ensemble Monte Carlo (GCMC) method, and they used the approach to study the force between surfactant-covered drops and the rupture of the oil-water-surfactant film.62 As to the rupture of the water film between the two drops, this approach is more effective than our method, which directly simulates the micelle-bilayer collisions. However, the insertion and deletion of water molecules at a fixed chemical potential in the frame of the GCMC method may spoil the dynamic behavior. Thus, in this work we directly simulated the micelle-bilayer collisions. Figure 8 shows the fusion ratio for the micelles of 250 surfactants with different initial collision velocities. Unexpectedly, the obtained fusion ratio does not change basically as the initial collision velocity increases. The independence of fusion ratio from the initial collision velocity is ascribed to the fact that the (62) Rekvig, L.; Hafskjold, B.; Smit, B. Langmuir 2004, 20, 11583.

initial collision velocity is quickly dissipated during the process of the intermittent collision. This again indicates that the fusion is in fact a collective process. In addition, the distribution of the duration time for collisions obtained from the simulations is shown in Figure 9. Agreeing with the fusion ratio, the distribution of collision time does not depend on the initial collision velocity. In the above simulations, the intermittent collision is observed. As is well-known, it is the depletion force that results in the intermittent collision. Hence, to find out the effects of depletion force, we also performed a series of simulation runs in which no intermitted collision is allowed. In these simulations we also first equilibrate the initial configuration in which the COM of a micelle is fixed by using constrained DPD. Then we set the micelles with the fixed collision velocities to collide with the bilayer, and directly reset it to the initial configuration after they finish a collision or fusion. Repeatedly, the initial configuration is relaxed by 500 time steps at the fixed COM to generate a new initial configuration for next collision process. In this way, the simulation is repeated until it reaches 1.5 × 106 time steps. The simulation results show that there is no fusion at all, even after more than 2000 times of collisions at the low velocities such as 0.01 0.05, 0.1, 0.2, and 0.3. Even at the higher velocities, only a small number of fusions occur, and the fusion ratios are much less than those in the case where the intermittent collisions are allowed. Because it is the depletion force that results in the intermittent collision, we conclude that the depletion force plays an important role during the coalescence stage. It not only results in the intermittent collision, but also lengthens the contact time, which therefore promotes the fusion ratio. In contrast, the initial collision velocity shows negligible effects on the fusion ratio. In the above studies, the initial location of the micelle is set to be nearest the bilayer but without direct molecular contact. In that case, the water film between the two aggregates is considered broken or nearly broken. Here we also investigated the effects of the thickness of water films by comparison of above results and that with a thicker water film. Another location of the COM of the micelle is set to be 13.9, which is of a thicker water film than the case of 13.4. The obtained fusion ratios for the two different water films are compared in Figure 10. Considering the scattered results of the fusion ratio, the fusion ratio with the thicker water film leads to nearly the same fusion ratio. The inset of Figure 10 shows the fraction of collision trials leading to molecular contact. The much lower fraction for the case of 13.9 indicate that the thicker water film significantly increase the difficulty of the first collision. In general, the thickness of the water film between the two objects shows a significant effect on the fraction of collision trial leading to molecular contact,


Figure 10. Fusion ratio for the micelle at different distances from the solid surface. The inset shows the fraction of collision trials leading to molecular contact.

Figure 11. Numbers of different contacts between the two aggregates after intermittent collision stages for three micelles with different numbers of surfactants on their surfaces.

and hence the rupture of water film is a rate-limiting step. However, because of the similar dynamic behavior (see Figure 9), the fusion ratio shows a similar value for the two cases. Note that the obtained fraction of collision trials leading to molecular contact, and the fusion ratio for the case of 13.9 suffers from the stochastic nature of coalescence. Several additional simulation runs are required to get more meaningful results with statistical average. 4.3. Effects of Surface Tension. In this section, we investigate the effects of the surface tension on the solute exchange process. Shillcock and Lipowsky30 used the area per surfactant molecules to control the surface tension. In a similar manner, we use the number of surfactants at the oil-water interface to control the surface tension. In this work, the numbers of H1T4 surfactants forming the micelle were set to 250, 450, 650, 750, respectively, to control the surface tension of the micelles bearing 550 oil molecules in their core. Keeping the other factors the same for these four cases, we carefully tuned the number of solvent beads to generate initial configurations with the same overall density. Several simulations including the whole solute exchange process were performed, and the corresponding molecular contacts after the collision stage for the different surfactant tensions are shown in Figure 11. We found that the number of the tail-tail(oil) contacts for the micelle with fewer surfactants (higher surface tension) grows faster than that with more surfactants, especially in the solute transfer and exchange stage. However, in the coalescence stage, the increase of tail-tail(oil) contacts for the micelle with 450 surfactants is seemed to be faster than that for 250 surfactants. This observation seems to be at odds with our

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expectation that more surfactants on the surface of a micelle would reduce surface tension and hence stabilize the micelle. As is shown below, it is the stochastic nature of the coalescence stage that biases the results. In other words, a large number of individual fusion events are required to obtain more meaningful results. To study the effects of surface tension on the fusion ratio, we directly simulated the micelle-bilayer collisions, as is described in the above section. First we need to obtain the exact locations of the micelles nearest to the bilayer. The micelles with 450, 650, and 750 surfactants were put at different positions in the z direction, and then constrained DPD simulations were performed, as in the case of the micelle with 250 surfactants. The simulation results are shown in Figure 7. The locations where the micelles are nearest the bilayer before their molecular contacts are determined to be 14.0 for the micelle with 450 surfactants, 14.6 for 650 surfactants, and 14.8 for 750 surfactants. Similar to the case of 250 surfactants, we adopted the simulation method including two repeated steps, namely, the equilibration and collision steps. The collisions and thus the fusion events between the bilayer and the micelle with 450, 650, and 750 surfactants, respectively, were efficiently sampled in this method. For each micelle, we performed several simulations to give the average value and statistical uncertainty of its fusion ratio. The corresponding fusion ratios are shown in Figure 8. In general, adding more surfactants on the surface of solute substantially reduces the fusion ratio, especially for the case with higher surface tension. This is because more surfactants on the surface of a micelle reduce surface tension and hence stabilize the micelle. Additionally, for the different micelles, the distribution of collision duration time before coalescence are shown in Figure 12a. Interestingly, the distributions of collision duration times are nearly the same for the different micelles and even at different initial collision velocities. Figure 12b shows the distribution of the interval time between collisions. Again, different micelles result in almost the same distribution of the interval time, although larger micelles show somewhat higher probability for shorter interval time because of the depletion force. The fact indicates that different micelles show almost the same dynamic behavior at the intermittent collision stage. However, decreasing the surfactant density in films of micelles substantially reduces the energy barrier of fusion, and, correspondingly, the probability of fusion ratio increases, as is shown in Figure 8. In general, we can draw a conclusion that the increase of the number of surfactants at an oil-water interface does not affect the dynamic behavior at the intermittent collision stage, but reduces the surface tension, which decreases the fusion ratio substantially. At the same time, it also results in the slower stage of the solute transfer and exchange. 4.4. Effect of Head(Micelle)-Head(Bilayer) Interaction. In the previous section, the higher energy barrier for coalescence stage origins from the increase of the surfactant number on the solute surface. A better wrapping of the solute-bearing micelle slows down both the coalescence and the solute exchange stage. In this section, the effect of head(micelle)-head(bilayer) repulsion is investigated. To find out how it affects solute exchange process, a series of simulation runs using the same parameters as in the above case with 250 H1T4 surfactants in the micelle were performed, except that the repulsive head-head interaction is increased from 25 to 35. First, several simulations including the whole solute exchange process is performed. In the case with stronger head-head repulsion, the time evolution of tail-tail contacts during a typical solute exchange process is shown in Figure 13, in which the data

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Figure 14. Comparison of the fusion ratios for the micelles with different head-head interactions. Both micelles are of 250 surfactants but having different head(micelle)-head(bilayer) interactions.

Figure 12. (a) Distribution of collision duration time before fusion occurs for different micelles. (b) Distribution of interval time between collisions.

Figure 13. Time evolutions of tail-tail(oil) contacts for two different micelles after the intermittent collision stage. The inset shows the corresponding time evolution of head-head contacts. Both micelles are of 250 surfactants but have different head (micelle)-head (bilayer) interactions.

for aHH ) 25 are also given for comparison. Because of the stochastic nature of the coalescence stage, we concentrate on the transfer and diffusion stage. It is found that at this stage, the increase of tail-tail contact, namely, the rate of the solute transfer and diffusion, follows almost the same trend for the two cases. However, the stronger head-head repulsive interaction between the head groups results in a much slower mixing of different surfactant molecules (see the inset of Figure 13). To consider the effects of head-head interaction on the fusion ratio during the coalescence stage, a series of DPD runs with

direct simulation of the micelle-bilayer collisions were performed. The fusion ratio for the system with aHH ) 35 is shown in Figure 14, in which the data for aHH ) 25 are also given for comparison. Again, initial collision velocity shows negligible effects on the fusion ratio. The comparison indicates that the head-head interaction between two aggregates results in different fusion ratios. The fusion ratio for the case of the head-head interaction being 35 is significantly lower than that of its counterpart. It means that it becomes more difficult for aggregates to fuse when the interaction between them is more repulsive. 4.5. Effects of Steric Repulsion between the Two Films. In all the results shown above, the surfactant is always modeled as a chain with five beads, i.e., H1T4, which is used to form the bilayer on the solid surface as well as the micelle containing oil molecules. To consider the effects of surfactant architecture, we also studied the case in which the bilayer is again formed with H1T4, while the solute-bearing micelle is formed with H2T4. The interaction between different head beads is set to 25, and the micelle contains 250 surfactants. Several simulation runs including the whole process of solute exchange were performed. Our simulation results show that the steric repulsion plays an important role in solute exchange process. For several trial simulation runs of 2 × 106 time steps, the coalescence of the micelle with the bilayer did not take place at all, and thus no solute exchange was observed. Obviously, the micelle formed with H2T4 induces a stronger steric repulsive interaction with the bilayer during their contact since it contains more head beads. Hence we can anticipate that the solute exchange process is much more difficult to happen than the case of H1T4, even if it can take place. This is because the surfactants with a larger number of head groups on the outside of micelles would generate the stronger overlap repulsion and prevent fusions from occurrence. In general, the increase of the number of head groups leads to an increase in the steric repulsion between the two films and thus makes the solute exchange process considerably slow down. The larger head size can be equivalently described by the increase of the number of the head groups in our study. Unfortunately, our simulation method fails in simulating the micelle-bilayer coalescence for large head size. This is because successful fusion sharply declines with the increase of head size. Therefore, no meaningful statistical results can be obtained because of the rare fusion events. In this situation, more sophisticated computer simulation techniques, such as the FFS32 method, are required to sample the rare events. FFS is a effective method that enhances the sampling of rare events, such as the


crossing of a high free-energy barrier that separates two (meta)stable states.

5. Conclusions In this work, we show that the kinetic process of solute exchange in an aqueous phase can be studied by DPD simulations, and we try to answer two questions about the solute exchange with the appearance of surfactants, i.e., how the solute is exchanged and what factors affect the process. In general, our results of computer simulations show that there are three sequential stages in the solute exchange process. After a intermittent collision stage, a coalescence stage may take place after both the surfactant film of the micelle and that of the bilayer break, and this is followed by a stage of solute transfer and diffusion. The simulation results indicate that there are roughly three sequential events during the coalescence stage: (i) Molecular contact: For a coalescence to be at all possible, the two surfactant films must come into molecular contact. In this stage, the depletion force plays an important role. (ii) Neck formation: A characteristic of the stage is the formation of a neck connecting the two objects. This stage contains two steps, the formation of a single overlapped head film and its rupture, i.e., the formation of a pore. The simulation results show that the rupture is in fact a collective process. (iii) Neck growth: The size of the pore will grow continuously if it is larger than its critical value. In contrast to the previous event, the growth is spontaneous and irreversible. For the second question, we study the effects of collision velocity, surface tension, the interaction between the two films, and the steric repulsion. It is found that there are two rate-limiting steps in the process of solute exchange, i.e., the breakup of the water film between two neighboring aggregates and the nucleation of a pore between the two surfactant films. Because of the stochastic nature of fusion, a large number of fusion events are required to get statistically meaningful results. Because most of the computer time in the whole simulated process is spent on the Brownian motion of the micelle in the solvent, in this work we designed a method to directly simulate the micelle-bilayer collisions rather than to consider the whole Brownian motion. For this purpose, we first need to prepare initial configurations in which the micelles are located nearest the bilayer without molecular contact. Then we directly simulated the micelle-bilayer collisions. In such a simulation run there are two repeated steps: equilibration step and collision step. The

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simulation method not only enhances the sampling of fusion events substantially, but also produces a large number of initial configurations. For the effects of the collision velocity on the efficiency of solute exchange, the simulation results show that the initial collision velocity basically does not change the fusion ratio in the coalescence stage. Instead, the depletion force plays an important role during the coalescence stage. It causes the two objects in contact for a long time and thus promotes the probability of fusion. The simulation results show that the increase of the number of surfactants at the oil-water interface does not significantly affect the dynamic behavior at the intermittent collision stage. However, adding more surfactants on films of micelles substantially reduces the surface tension and hence increases the activation energy of fusion. As a result, fusion ratio decreases, and at the same time, the stage of solute transfer and diffusion also slows down. As to the head-head interaction between two colliding aggregates, the simulations indicate that it affects not only the coalescence stage, but also the stage of solute transfer and diffusion. The fusion ratio decreases when the head-head interaction becomes more repulsive, which means that it becomes more difficult for two aggregates to fuse. For the stage of solute transfer and diffusion, the more repulsive head-head interaction seems to slightly affect the rate of solute transfer, which results in a much slower mixing of different surfactant molecules. We also investigated the effects of the different surfactant architectures. The results indicate that the increase of the number of head groups leads to an increase of the steric repulsion between the two colliding surfactant films, and thus makes the solute exchange process considerably slow down. Acknowledgment. The work is supported by the National Natural Science Foundation of China (Grant No. 20736005). X.Z. acknowledges the support of the Research Foundation for Young Researchers of BUCT. Generous allocations of computer time by the “Chemical Grid Project” of BUCT, and the Supercomputing Center, CNIC, CAS, are acknowledged. Supporting Information Available: The details of our DPD simulations. This material is available free of charge via the Internet at http://pubs.acs.org. LA801521B

Computer Simulations of Solute Exchange Using Micelles by a

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