Molecular Dynamics Simulations of Surfactant Aggregation on

To begin with, we verify that the simple model can capture, at least qualitatively, the basic characteristics of water/solid interfaces and micellar s...
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Langmuir 1999, 15, 578-586

Molecular Dynamics Simulations of Surfactant Aggregation on Hydrophilic Walls in Micellar Solutions Hiroyuki Shinto, Satoshi Tsuji, Minoru Miyahara, and Ko Higashitani* Department of Chemical Engineering, Kyoto University, Yoshida, Sakyo-ku, Kyoto, 606-8501 Japan Received May 27, 1998. In Final Form: October 16, 1998 In this paper, we investigate surfactant aggregation at solid/liquid interfaces using a molecular dynamics (MD) simulation with a simple description of molecules and interfaces. To begin with, we verify that the simple model can capture, at least qualitatively, the basic characteristics of water/solid interfaces and micellar solutions. Thereafter, we simulate the surfactant aggregation on the smooth hydrophilic walls in micellar solutions. The mechanisms of the surfactant adsorption and aggregation are proposed on the molecular level, and how they are influenced by the concentration and structure of surfactant molecules is discussed in detail.

I. Introduction Surfactant molecules play an important role in the vast region of industrial processing: for example, (1) surfactants assemble spontaneously in solutions to form micelles with various morphology, bilayers, and vesicles; (2) they adsorb at oil/water interfaces to reduce the interfacial tension; and (3) they adsorb onto solid surfaces to change the nature of the surfaces. These characteristics of surfactants have been investigated mainly by experimental methods, but much of their underlying physics is still poorly understood on the molecular level. Recently as computational power has advanced, computer simulations, based on the molecular dynamics (MD) and the Monte Carlo (MC) methods, have been gradually applied to the amphiphilic behavior under various conditions and environments.1-6 In an atomistic description, molecules are modeled as real as possible via stretching, bending, and torsion potentials between the bonding atoms, and Lennard-Jones and Coulomb potentials between the nonbonding atoms. By use of this atomistic description for amphiphilic systems, a number of MD simulations have been implemented on systems such as micelles,7-14 monolayers at * To whom correspondence should be addressed: Phone +8175-753-5562; Fax +81-75-753-5913; E-mail [email protected]. (1) Klein, M. L. J. Chem. Soc., Faraday Trans. 1992, 88, 1701. (2) Pastor, R. W. Curr. Opin. Struct. Biol. 1994, 4, 486. (3) Bandyopadhyay, S.; Tarek, M.; Klein, M. L. Curr. Opin. Colloid Interface Sci. 1998, 3, 242. (4) Smit, B. In Computer Simulation in Chemical Physics; NATO ASI Series C, Vol. 397; Allen, M. P., Tildesley, D. J., Eds.; Kluwer: Dordrecht, The Netherlands, 1993; Chapter 12. (5) Karaborni, S.; Smit, B. Curr. Opin. Colloid Interface Sci. 1996, 1, 411. (6) Larson, R. G. Curr. Opin. Colloid Interface Sci. 1997, 2, 361. (7) Watanabe, K.; Ferrario, M.; Klein, M. L. J. Phys. Chem. 1988, 92, 819. (8) Watanabe, K.; Klein, M. L. J. Phys. Chem. 1991, 95, 4158. (9) Wendoloski, J. J.; Kimatian, S. J.; Schutt, C. E.; Salemme, F. R. Science 1989, 243, 636. (10) Laaksonen, L.; Rosenholm, J. B. Chem. Phys. Lett. 1993, 216, 429. (11) Shelley, J. C.; Sprik, M.; Klein, M. L. Langmuir 1993, 9, 916. (12) Bo¨cker, J.; Brickmann, J.; Bopp, P. J. Phys. Chem. 1994, 98, 712. (13) MacKerell, A. D. J. Phys. Chem. 1995, 99, 1846. (14) Griffiths, J. A.; Heyes, D. M. Langmuir 1996, 12, 2418.

air/water interfaces,15-19 monolayers at immiscible liquid/ liquid interfaces,19,20 self-assembled monolayers,21,22 and bilayers23-25 to obtain quantitative and detailed molecular information about the real systems. Unfortunately these atomistic model simulations are limited to a few nanoseconds and this time scale is too short to examine the important behavior of micellar solutions, such as the surfactant exchange or micellar formation and breakdown that may occur on the time scale of milliseconds. An alternative approach is to mimic an oil/water/ surfactant system by using “waterlike” particles, “oillike” particles, and “surfactantlike” molecules composed of these different particles.26-28 On the basis of this simple idea, Larson performed the lattice MC simulations of oil/water/ amphiphile systems and observed that the amphiphiles form spontaneously spheres, cylinders, lamellas, and more complex phases.29,30 Smit and co-workers performed a series of MD simulations and reported as follows: (1) the surfactant molecules adsorb at the oil/water interface and consequently reduce the interfacial tension, where this reduction becomes larger as the surfactant concentration (15) Bareman, J. P.; Klein, M. L. In Interface Dynamics and Growth; MRS Symposium Proceedings, Vol. 237; Liang, K. S., Anderson, M. P., Bruinsma, R. F., Scoles, G., Eds.; Materials Research Society: Pittsburgh, PA, 1992; p 271. (16) Bo¨cker, J.; Schlenkrich, M.; Bopp, P.; Brickmann, J. J. Phys. Chem. 1992, 96, 9915. (17) Siepmann, J. I.; Karaborni, S.; Klein, M. L. J. Phys. Chem. 1994, 98, 6675. (18) Tarek, M.; Tobias, D. J.; Klein, M. L. J. Phys. Chem. 1995, 99, 1393. (19) Schweighofer, K. J.; Essmann, U.; Berkowitz, M. J. Phys. Chem. B 1997, 101, 3793. (20) Urbina-Villalba, G.; Landrove, R. M.; Guaregua, J. A. Langmuir 1997, 13, 1644. (21) Mar, W.; Klein, M. L. Langmuir 1994, 10, 188. (22) Siepmann, J. I.; McDonald, I. R. Mol. Phys. 1993, 79, 457. (23) Marrink, S.-J.; Berendsen, H. J. C. J. Phys. Chem. 1994, 98, 4155. (24) Heller, H.; Schaefer, M.; Schulten, K. J. Phys. Chem. 1993, 97, 8343. (25) Shinoda, W.; Namiki, N.; Okazaki, S. J. Chem. Phys. 1997, 106, 5731. (26) Telo da Gama, M. M.; Gubbins, K. E. Mol. Phys. 1986, 59, 227. (27) Smit, B. Phys. Rev. A 1988, 37, 3431. (28) Smit, B.; Schlijper, A. G.; Rupert, L. A. M.; van Os, N. M. J. Phys. Chem. 1990, 94, 6933. (29) Larson, R. G. Chem. Eng. Sci. 1994, 49, 2833. (30) Larson, R. G. J. Phys. II France 1996, 6, 1441.

10.1021/la9806193 CCC: $18.00 © 1999 American Chemical Society Published on Web 12/31/1998

MD Simulations of Surfactant Aggregation

is higher or the surfactants have a longer tail chain;27,28,31 (2) the molecular structure of surfactants influences the shape of self-assembled micelles;32-34 and (3) an oil droplet is solubilized in a micellar solution.35 The point to note is that the size distribution of surfactant aggregates and the critical micelle concentration (cmc) have been also evaluated by these simulations,36-42 while the atomistic model simulations are not accessible to these properties because of the lack of the computational power. All the results support that this simple description of amphiphilic systems is able to capture, at least qualitatively, much of their underlying physics. On the other hand, recent in situ measurements by the atomic force microscope (AFM) have shown that ionic surfactants form aggregates at solid/liquid interfaces as well as in bulk solutions when the dosed surfactant concentration is near the cmc, and the interfacial aggregates exhibit various morphology of spheres, cylinders, half-cylinders, and bilayers, depending on the surface chemistry and the surfactant geometry.43 Even in sufficiently dilute solutions (≈cmc/105), where micelles are believed not to form stably in the bulk, islandlike aggregates of cationic surfactants were formed on the mica surface.44 Why and how the surfactant aggregation at solid/ liquid interfaces occurs, however, have not been settled as yet. In the present study, our attention is focused on the MD simulation of the surfactant aggregation at the solid/ liquid interfaces using the simple model. We test whether the simple description of molecules and interfaces can capture the basic characteristics of water/solid interfaces and micellar solutions. Thereafter, we simulate the surfactant aggregation on the smooth hydrophilic walls in micellar solutions. The mechanisms of the adsorption and aggregation of surfactants on the surface are proposed on the molecular level, and how the mechanisms are influenced by the concentration and structure of surfactant molecules is discussed in detail. II. Methods A. Model of Oil/Water/Surfactant Systems. The simple description of an oil/water/surfactant system has been developed by Telo da Gama and Gubbins26 and by Smit and co-workers.27,28 The starting point of this model is to consider four types of particles, that is, waterlike, oillike, headlike, and taillike particles, which are referred to as w, o, h, and t particles, respectively. The w and h particles are hydrophilic, while the o and t particles are hydrophobic. By use of these particles, water, oil, and surfactant molecules are modeled as illustrated in Figure 1: a water molecule consists of one w particle; an oil molecule consists of one o particle; a surfactant molecule is composed of a headgroup of h particles (31) Smit, B.; Hilbers, P. A. J.; Esselink, K.; Rupert, L. A. M.; van Os, N. M.; Schlijper, A. G. J. Phys. Chem. 1991, 95, 6361. (32) Smit, B.; Hilbers, P. A. J.; Esselink, K. Tenside Surf. Det. 1993, 4, 287. (33) Esselink, K.; Hilbers, P. A. J.; van Os, N. M.; Smit, B.; Karaborni, S. Colloids Surf. A 1994, 91, 155. (34) Karaborni, S.; Esselink, K.; Hilbers, P. A. J.; Smit, B.; Kartha¨user, J.; van Os, N. M.; Zana, R. Science 1994, 266, 254. (35) Karaborni, S.; van Os, N. M.; Esselink, K.; Hilbers, P. A. J. Langmuir 1993, 9, 1175. (36) Larson, R. G. J. Chem. Phys. 1992, 96, 7904. (37) Smit, B.; Esselink, K.; Hilbers, P. A. J.; van Os, N. M.; Rupert, L. A. M.; Szleifer, I. Langmuir 1993, 9, 9. (38) Rector, D. R.; van Swol, F.; Henderson, J. R. Mol. Phys. 1994, 82, 1009. (39) Brindle, D.; Care, C. M. J. Chem. Soc., Faraday Trans. 1992, 88, 2163. (40) Desplat, J.-C.; Care, C. M. Mol. Phys. 1996, 87, 441. (41) Wang, Y.; Mattice, W. L.; Napper, D. H. Langmuir 1993, 9, 66. (42) Halilogˇlu, T.; Mattice, W. L. Chem. Eng. Sci. 1994, 49, 2851. (43) Manne, S. Prog. Colloid Polym. Sci. 1997, 103, 2226, and references therein. (44) Fukuda, K. Masters’ Thesis, Kyoto University, Japan, 1996.

Langmuir, Vol. 15, No. 2, 1999 579

Figure 1. Illustration of the oil/water/surfactant model. Symbols, w, o, H, and t denote waterlike, oillike, headlike, and taillike particles, respectively. Two surfactant molecules used are displayed. and a tail chain of t particles, whose neighboring pairs are connected together by the harmonic potentials, Uij, with spring length dS and spring constant k:

Uij(r) ) 1/2k(r - dS)2

(1)

where r is the distance between particles i and j. The value of k used is 1.4 × 104 in unit S/dS2, where S is the energy parameter of the Lennard-Jones potential as described below. It is confirmed that this spring is stiff enough to constrain the separations of all the connected particles to be dS ( 0.02dS. In addition to the harmonic potentials, pairs i and j between four species of particles (w, o, h, and t) interact each other via the shifted Lennard-Jones (12-6) potentials with the energy parameter ij, core diameter dij, and cutoff radius Rijcut:

Φij(r) )

{

φij(r) - φij(Rijcut) r e Rijcut

(2a)

r > Rijcut

0

[( ) ( ) ]

φij(r) ) 4ij

dij r

12

-

dij r

6

(2b)

The values of parameters used are as follows: ij/kB ) S/kB ) 119.8 K (kB is Boltzmann’s constant) and dij ) dS ) 0.3405 nm for all interactions, and mS ) 39.948 g/mol for the mass of all particles, which correspond to the parameters for argon. To represent the hydrophilicity and hydrophobicity, different values of Rijcut, 2.5dS and 21/6dS, are chosen depending on the types of pairs. The intraspecies forces include the repulsion and attraction using Rijcut ) 2.5dS, while the interspecies forces are completely repulsive using Rijcut ) 21/6dS. Nonionic and ionic surfactants are modeled using Rijcut ) 2.5dS and 21/6dS for h-h interaction, respectively. In the latter case, only the repulsion exists between particles of the headgroups, which are designated by H instead of h. This is analogous to the electrostatic repulsion between charged headgroups of real ionic surfactants. By connecting different number of particles h (or H) and t in different arrangement, this simple model is able to create surfactants with various molecular architectures. In the present study, two types of the surfactant molecules, H3Ht3 and H3Ht5, are considered as illustrated in Figure 1, which are different in chain length but the same in repulsive headgroup. B. Model of Solid Walls. We introduce the hydrophilic and hydrophobic walls by truncating the structureless (10-4-3) potentials at the cutoff separation Zijcut:

Ψij(z) )

{

ψij(z) - ψij(Zijcut) z e Zijcut

ψij(z) ) 2πFWWSdWS2∆ ×

[(

(3a)

z > Zijcut

0

) ( )

2 dWS 5 z

10

-

dWS z

4

-

dWS4

]

3∆(z + 0.61∆)3

(3b)

where z is the perpendicular distance between the center of a Lennard-Jones particle and the outermost plane of the solid wall, FW is the number density of atoms in the solid wall, and ∆ is the

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Table 1. Systems of Molecular Dynamics Simulations

a

system

N

dimensionsa

wall

I II III IV V VI

4000 4000 19652 19652 19652 19652

17.9 × 17.9 × 19.9 17.9 × 17.9 × 19.9 30.4 × 30.4 × 30.4 30.4 × 30.4 × 30.4 30.4 × 30.4 × 32.1 30.4 × 30.4 × 32.1

hydrophilic hydrophobic hydrophilic hydrophilic

surfactant

Xsurfb

T*

H3Ht5 H3Ht3 H3H′t5 H3H′t3

0 0 0.92-15.2 0.71-12.5 2.66, 9.75 2.07, 7.73

1.00 1.00 2.40 1.35 2.40 1.35

In unit dS. b Volume fraction of dosed surfactants in unit volume %.

separation between lattice planes.45 WS and dWS are the crossparameters for wall-solution interaction, which are written by

WS ) (WS)1/2

dWS ) 1/2(dW + dS)

(4)

The parameters for the graphite surface are used following Steele:45,46

W/kB ) 28.0 K

dW ) 0.340 nm

FW ) 114 nm-3

∆ ) 0.335 nm

(5)

We apply the idea of the simple model also to the particle-wall interaction; that is, when a particle and a wall are of the same kind, their interaction includes repulsive and attractive forces using Zijcut ) ∞ (i.e., no truncation), but they interact only through the repulsion using Zijcut ) 0.994dS () Zmin) when they are not of the same kind. When a basic cell with the dimensions of Lx × Ly × Lz is given, a particle located at z experiences the external potential Ψijext(z), given by the superposition of Ψij from the two planar walls, whose outermost planes are positioned at z ) -Lz/2 and +Lz/2:

Ψijext(z) )Ψij(z + 1/2Lz) + Ψij(1/2Lz - z)

(6)

where the origin of coordinates is taken at the center of the cell. The effective width of this cell in the z direction is defined as the separation between the points at z ) -Lz/2 + Zmin and Lz/2 Zmin, where each Ψij on the right-hand side of eq 6 has a minimum. We find that as far as the potentials described above are used for surfactant-wall interactions, the surfactants of H3Ht3 and H3Ht5 do not adsorb onto the hydrophilic walls in a waterlike fluid. Therefore, to introduce the driving force of the surfactant adsorption, we assume that the H particle centered in the headgroup of surfactants, which is denoted by H′, interacts with the wall 10 times more strongly than the other H particles and w particles; that is, the energy parameter for H′-wall interaction is given by 10WS. Although our assumption is very rough, it mimics the real systems where a charged headgroup of surfactant molecules interacts with the hydrophilic surfaces more strongly than a water molecule. C. Definition of Micelles. When micellar solutions are treated, two surfactants are considered to belong to the same micelle if the minimum distance between their tail sites is less than 1.4dS. The different micelles are then identified by the standard clustering procedure.47 D. Simulation Details. Six systems are considered as summarized in Table 1, which are classified roughly into following three systems: (I, II) a waterlike fluid confined within two planar walls; (III, IV) surfactant/water mixtures, that is, bulk micellar solutions; and (V, VI) micellar solutions between hydrophilic walls. When the first and third systems are simulated, the periodic boundary conditions are applied in the two directions parallel to the walls. The second systems are simulated with the periodic boundary conditions imposed in all the directions. The dimensions of the cell with N ) 4000 or 19 652 particles are chosen such that the reduced density of particles F* () FdS3) is equal to 0.7. The equation of motion is integrated by the leapfrog method with a time step of ∆t ) 0.00464τ0, where τ0 ) dS(mS/S)1/2. To compute (45) Steele, W. A. Surf. Sci. 1973, 36, 317. (46) Steele, W. A. The interaction of gases with solid surfaces; Pergamon Press: Oxford, England, 1974; Chapter 2. (47) Stoddard, S. D. J. Comput. Phys. 1978, 27, 291.

interactions between a large number of particles efficiently, the layered link cell procedure connecting with the Verlet neighbor list is employed on the Cray T-94/4128 vector computer.48 The neighbor list is updated automatically following Chialvo and Debenedetti.49 The reduced temperature T* () kBT/S) is kept at the constant values listed in Table 1 by velocity scaling if necessary. The simulation procedures are provided as follows: (1) systems I and II; (2) systems III and IV; and (3) systems V and VI. (1) Four thousand waterlike particles are equilibrated at T* ) 1.00 in the cubic cell with a side length of 17.9dS, where the periodic boundary conditions are applied in all the directions. Then two planar walls are inserted at z ) (9.93dS and the periodic boundary conditions in the z direction are removed. The system is allowed to reequilibrate over 4 × 104 time steps by velocity scaling, and the simulation is performed for 1 × 103 time steps without scaling. (2) After 19 652 waterlike particles are equilibrated at the given temperature, some of the particles are connected at random by the springs of eq 1 such that the desired number of the surfactants, either H3Ht3 or H3Ht5, are constructed. But all the particles of surfactant molecules are still regarded as w particles to prevent the surfactants from self-assembling, which are referred to as pseudosurfactants. To implement the stable computation, the spring constant in unit S/dS2 starts from an initial value of 1.4 and gradually increases to its final value of 1.4 × 104 for 3 × 104 time steps, and then the system is equilibrated over 4 × 104 time steps, in which no self-assembly of the pseudosurfactants occurs. Thereafter, the head and tail particles are treated as H and t particles, respectively, and the surfactants are allowed to assemble for 1.5 × 105 time steps with temperature scaling at every 100th step. The equilibrated micellar solution is simulated by the successive calculation over 1.5 × 105 time steps with scaling at every 200th step. Through this procedure, which seems to be intricate, we can observe the surfactant self-assembly. (3) Since the initial configurations of surfactants may influence the surfactant aggregation under the limited numbers of molecules, two types of configurations are prepared: (I) the solvent and pseudosurfactant mixture confined within the walls is equilibrated over 7 × 104 time steps as mentioned above, and (II) the same procedure as type I but the surfactants with no driving force of adsorption (inert surfactants), that is, the H′ particles of their headgroups are treated as H particles. During these preparations, the pseudosurfactants neither form micelles nor adsorb on the surfaces in the type I configuration; on the other hand, the inert surfactants aggregate in the bulklike region but never adsorb on the surfaces in the type II configuration. Thereafter substituting the pseudosurfactants or inert surfactants by the true surfactants, they are allowed to both aggregate and adsorb for 3 × 105 time steps with scaling at every 200th step. The initial configurations of type I are favorable for simultaneous observation of both the self-assembly and the adsorption of surfactants, and by using those of type II, on the other hand, the surfactant adsorption from equilibrated micellar solutions becomes observable.

III. Results and Discussion First of all, we have to find whether the simple model used can represent the basic characteristics of water/solid (48) Grest, G. S.; Du¨nweg, B.; Kremer, K. Comput. Phys. Commun. 1989, 55, 269. (49) Chialvo, A. A.; Debenedetti, P. G. Comput. Phys. Commun. 1990, 60, 215.

MD Simulations of Surfactant Aggregation

Langmuir, Vol. 15, No. 2, 1999 581

RR(t) ) N(0)N(t)/N2(0)

(8)

where N(t) is a vector (n1, ‚‚‚, n4000) with ni ) 1 or 0 depending on whether particle i resides in layer R or not at time t. Assuming that the particles behave as Brownian particles and the particle distribution in a layer is homogeneous at t ) 0, RR(t) is given by the following analytical expression derived from the diffusion equation:

RR(t) ) erf(τ) + [exp(-τ2) - 1]/π1/2τ τ)

Figure 2. Density profiles of waterlike particles confined within two planar walls. Solid and dotted lines represent the profiles for systems I and II, that is, near the hydrophilic and the hydrophobic surfaces, respectively.

interfaces and micellar solutions. In the following subsections A and B, the former problem is examined by using systems I and II, and the latter is investigated by using systems III and IV. In subsection C, the surfactant aggregation on the smooth hydrophilic wall in micellar solutions is explored by using systems V and VI. A. Waterlike Fluid near Solid Walls. Figure 2 shows the density profiles of a waterlike fluid between two planar walls, which are hydrophilic and hydrophobic for systems I and II, respectively. About four layers of the particles form near the hydrophilic surface. On the other hand, the hydrophobic wall repels the particles toward the middle region of the cell such that no distinct layer forms anywhere. Similar features have been observed also in the atomistic model simulations.50-54 To probe the motion of particles near the surfaces, the fluid lamina is divided into three interfacial layers (L1, L2, and L3) and the bulklike region (LB), as defined in Figure 2 and Table 2. The self-diffusion coefficients of Dξ in the direction ξ () x, y, z) are evaluated within these layers in the following way. The lateral diffusion coefficients of Dx and Dy are calculated from the mean square displacement and are given in Table 2:

Dξ ) lim tf∞

〈∆ξ2〉 (ξ ) x, y) 2t

(7)

Using eq 7 we attempt to calculate the diffusion coefficients of Dz perpendicular to the walls and find that they are nearly zero in the thin layers of L1, L2, and L3, but they are almost equal to the values of Dx and Dy in LB with the thickness of 3.5dS. This implies that eq 7 is not available to evaluate the vertical diffusion coefficients of particles within the thin film. Alternatively, the residence autocorrelation function is introduced to quantify Dz appropriately following Sonnenschein and Heinzinger:55 (50) Lee, C. Y.; McCammon, J. A.; Rossky, P. J. J. Chem. Phys. 1984, 80, 4448. (51) Lee, S. H.; Rossky, P. J. J. Chem. Phys. 1994, 100, 3334. (52) Boek, E. S.; Briels, W. J.; van Eerden, J.; Feil, D. J. Chem. Phys. 1992, 96, 7010. (53) Shinto, H.; Sakakibara, T.; Higashitani, K. J. Phys. Chem. B 1998, 102, 1974. (54) Raghavan, K.; Foster, K.; Berkowitz, M. Chem. Phys. Lett. 1991, 177, 426. (55) Sonnenschein, R.; Heinzinger, K. Chem. Phys. Lett. 1983, 102, 550.

(9a)

{

2h/(4Dzt)1/2 for half side closed layer (9b) h/(4Dzt)1/2 for open layer

where h is the thickness of the layer, the half side closed layer corresponds to L1 and the open layer corresponds to L2, L3, and LB.55 We fit eq 9 to the simulated values of RR(t) using eq 8 as shown in Figure 3 and obtain Dz in each layer as summarized in Table 2. It is found that the diffusion coefficients of waterlike particles in L1, L2, and L3 decrease near the hydrophilic surface compared with those in LB, while they increase near the hydrophobic surface. The tendency becomes more significant as the particles are closer to the surfaces. This behavior is in qualitative agreement with the atomistic model simulations.51-54 Thus the results of the density profiles and the diffusion coefficients manifest that the simple model used can mimic, at least qualitatively, water/solid interfaces with respect to the static structure and the diffusional motion of layered particles. B. Self-Assembly of Surfactants. We investigate the size distribution of surfactant aggregates and the critical micelle concentration (cmc) in detail using system III. The volume fraction of surfactant monomers in aggregates of size n, X(n), is depicted in Figure 4 for various values of the dosed surfactant concentration, Xsurf. The point to observe is that the distribution has a maximum and a minimum when Xsurf becomes larger than the value of about 3 vol % and otherwise it decreases monotonically. This behavior coincides with the other simulation results,36,38,40,42 the predictions by the mass-action-models,56,57 and one of the basic assumptions in the theory of the micelle formation dynamics.58 Figure 5 shows the free monomer concentration, X(1), as a function of Xsurf with the dotted line showing the relation X(1) ) Xsurf. X(1) is directly proportional to Xsurf up to Xsurf ≈ 1 vol %, implying that the surfactants do not form aggregates. This is consistent with the monotonic decrease of X(n) in Figure 4. At Xsurf > 3 vol %, X(1) becomes nearly constant, from which the cmc is taken to be about 3 vol % for an H3Ht5 surfactant solution at T* ) 2.40. Thereafter X(1) starts to fall at the point of Xsurf ≈ 9 vol % and becomes nearly constant again. Similar behavior was reported in other studies and was explained in the way that the micelles formed interact with each other when Xsurf is above the cmc.39,40 This remains, however, to be examined more extensively. The same features as described above appear in system IV and the cmc of an H3Ht3 surfactant solution is estimated to be about 2 vol % at T* ) 1.35, as shown in Figure 5. In addition to these “ionic” surfactants of H3Ht5 and H3Ht3, micellar solutions of “nonionic” surfactants, h2t5, are (56) Hoeve, C. A. J.; Benson, G. C. J. Phys. Chem. 1957, 61, 1149. (57) Wennerstro¨m, H.; Lindman, B. Phys. Rep. 1979, 52, 1. (58) Lang, J.; Zana, R. In Surfactant Solutions: New Methods of Investigation; Surfactant Science Series, Vol. 22; Zana, R., Ed.; Marcel Dekker: New York, 1987; Chapter 8.

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Table 2. Diffusion Coefficients D* a of Waterlike Particles in the Different Layers for Systems I and II system I (hydrophilic wall)

system II (hydrophobic wall)

layerb

L1

L2

L3

LB

L1

L2

L3

LB

Dx* c Dy* c Dx* d (Dx*)c

0.09 0.08 0.01 (≈0.00)

0.09 0.09 0.03 (≈0.00)

0.11 0.11 0.07 (≈0.00)

0.12 0.13 0.11 (0.12)

0.24 0.21 0.26 (≈0.00)

0.14 0.13 0.12 (0.01)

0.11 0.11 0.10 (0.01)

0.10 0.10 0.10 (0.09)

a D* ) D/( d 2/m )1/2. b The range of |z| in unit d is 8.40∼ for L1, 7.42-8.40 for L2, 6.57-7.42 for L3, and 0.00-3.50 for LB (see Figure S S S S 2). c Values are calculated from the mean square displacement. d Values are calculated from the residence autocorrelation function.

Figure 3. Residence autocorrelation function, RR(t*), of waterlike particles near the hydrophobic wall (system II). Symbols, + and ×, represent the values for layer R ) L1 and LB, respectively. Solid lines show the fitting curves given by eq 9. Here t* ) t/(mSdS2/S)1/2.

Figure 4. Size distributions of surfactant aggregates for system III. X(n) denotes the concentration of surfactant monomers in aggregates of size n. Dotted, dashed, and solid lines represent the distributions at dosed surfactant concentrations of Xsurf ) 1.79, 3.48, and 6.73 vol %, respectively. Each distribution was obtained by averaging over 150 configurations taken every 1000th time step.

simulated separately, implying that the cmc of model nonionic surfactant solutions is much lower. This agrees qualitatively with the behavior of real surfactant systems. These results confirm that the simple model used can capture, at least qualitatively, the characteristics of micellar solutions. Note that the temperature is kept at relatively high values following Smit and co-workers,34,35,37 such that the surfactant solutions have the typical micellar size distribution and the cmc. C. Surfactant Aggregation on Hydrophilic Walls in Micellar Solutions. We simulate the surfactant aggregation on the smooth hydrophilic surface using systems V and VI listed in Table 1. The types of surfactants used are H3H′t5 and H3H′t3, which are different in chain length but the same in headgroup. Two values of the dosed

Figure 5. Free monomer concentration, X(1), as a function of the dosed surfactant concentration, Xsurf. Solid and open circles indicate the plots for systems III and IV, respectively. Dotted line represents the relation of X(1) ) Xsurf. Each plot was obtained in the same way as in Figure 6.

surfactant concentration Xsurf are chosen as given in Table 1. The results presented here are obtained by using the initial configurations of type I unless specified. At the end of this subsection, we propose the surfactant aggregation mechanisms on the molecular level and consider how the mechanisms are influenced by the concentration and structure of surfactant molecules. C.1. H3H′t5 at Low Concentration. Figure 6 displays the snapshots of the H3H′t5 surfactant solution between the hydrophilic walls at Xsurf ) 2.66 vol %. As expected, it is found that surfactant molecules in the initial configurations neither form micelles nor adsorb on the surfaces. Figure 6a shows that some surfactants are adsorbed on the walls with their headgroups and tail chains pointing toward and away from the surface, respectively, at 1 × 104 time step. This is because the hydrophilic surface favors hydrophilic particles more than hydrophobic particles. At 2.2 × 105 time step of Figure 6b, more than five surfactants form the small aggregates on the surface, which are referred to as the 2D aggregates. One may suppose that an aggregate formed in the bulklike region adsorbs directly on the bare surface to construct the 2D aggregate, which is called by direct aggregate adsorption. However, such 2D aggregate formation never occurs during the present simulation, which starts from the configurations of type I. To explore the surfactant aggregation on the surface extensively, we focus on the surfactants in the rectangular box drawn in Figure 6b, and their time evolutions are displayed in Figure 7. As shown in Figure 6a, free monomers in the bulklike region adsorb at the unoccupied points of the surface, which are named direct monomer adsorption. In addition to this behavior, the monomers frequently adsorb near the points occupied by the other surfactants because of the favorable tail-tail interactions, which are named tail-induced monomer adsorption. For example, as shown in Figure 7a,b, the free monomer

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Figure 6. Snapshots of system V at Xsurf ) 2.66 vol %. White and black spheres display the head and tail particles, respectively, and two grid planes represent the hydrophilic walls. All solvent particles are not shown for clarity. (a) 1 × 104 time step; (b) 2.2 × 105 time step; (c) 3 × 105 time step.

approaches the surfactants bound on the surface to make tail-to-tail contact with them. This surfactant moves around keeping tail-to-tail contact during about 5 × 103 time steps from part b to d of Figure 7. After that, the surfactant puts the headgroup closer to adsorb on the surface and consequently joins into the 2D aggregate, as displayed in Figure 7e,f. Another point to observe in Figure 7 is that the small 2D aggregates and the isolated surfactants fluctuate (or diffuse) transversely on the surface, assembling by the tail-tail attraction to grow into the larger 2D aggregate shown in Figure 7f, which is called by surface-diffusion growth. On the other hand, during the simulation using the initial configuration type II, the 2D aggregates form not only through these two mechanisms but also through the direct aggregate adsorption, where the former mechanisms are faster than the latter mechanism. The reason why the aggregate adsorption never occurs in the simulation of type I is

Figure 7. Time evolutions of the H3H′t5 surfactants in the rectangular box shown in Figure 6b. Notice the movement of the surfactant whose head and tail particles are displayed by the light and dark gray spheres, respectively. (a) 2.05 × 105 time step; (b) 2.08 × 105 time step; (c) 2.13 × 105 time step; (d) 2.14 × 105 time step; (e) 2.16 × 105 time step; (f) 2.2 × 105 time step.

probably because free monomers adsorb onto the surface rapidly before they aggregate in the bulklike region as expected in subsection II.D. Figure 6c shows that after 3 × 105 time steps most of the surfactants adsorbed on the surface are incorporated into the 2D aggregates; on the other hand the surfactants in the bulklike region exist as free monomers. The 2D aggregates diffuse laterally on the surface, assembling and splitting repeatedly as described above. This indicates that they are not absolutely stable. It is also found that

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Figure 8. Size distributions of 2D aggregates on the surface. N(n2D) denotes the number of surfactants in 2D aggregates of size n2D. Dotted and solid lines indicate the distributions at low and high concentrations, respectively. (a) System V; (b) system VI. Each distribution was obtained by averaging over 50 configurations taken every 1000th time step of the last 5 × 104 configurations.

surfactant monomers migrate from one 2D aggregate to another at the time scale of 1 × 104 steps. To quantify the equilibrated 2D aggregates on the surface, their size distribution is evaluated as shown in Figure 8. A surfactant molecule is regarded as adsorbed on the surface only when the distance between the H′ particle and the surface is less than Zmin + 0.5dS, and thereafter the 2D aggregates are identified in the same way as described in subsection II.C. The interesting point to note is that the distribution has a minimum and a maximum. This behavior is found also in the size distribution of aggregates in the bulk as shown in Figure 4. Similar results were reported by Wijmans and Linse, where the adsorption of h10t10 surfactants at hydrophobic interfaces was simulated by the lattice MC method.59 C.2. H3H′t5 at High Concentration. Figure 9 depicts the snapshots of the H3H′t5 surfactant solution at Xsurf ) 9.75 vol %, which is higher than the last case of Xsurf ) 2.66 vol %. Figure 9a displays the initial configurations of surfactant molecules. At 1 × 104 time step of Figure 9b, it shows that some surfactants are adsorbed on the walls, some form micelles in the bulklike region, and others are dissolved in the solvent as free monomers. After 3 × 105 time steps most surfactants adsorb on the walls to form 2D aggregates as shown in Figure 9c, which are larger in size and located closer each other, compared with those at Xsurf ) 2.66 vol % shown in Figure 6c. Figure 8a also shows the influence of Xsurf on the 2D aggregate size quantitatively. These explain the observation that the 2D aggregates at Xsurf ) 9.75 vol % fluctuate less on the surface, that is, are more stable compared with those at Xsurf ) 2.66 vol %. Our careful observation manifests that the 2D aggregates at Xsurf ) 9.75 vol % form through the tail-induced monomer adsorption, the surface-diffusion growth, and the indirect aggregate adsorption that differs from the (59) Wijmans, C. M.; Linse, P. J. Chem. Phys. 1997, 106, 328.

Figure 9. Snapshots of system V at Xsurf ) 9.75 vol %. Symbols are the same as in Figure 6. (a) 0 time step; (b) 1 × 104 time step; (c) 3 × 105 time step.

direct aggregate adsorption as follows. An aggregate, which forms in the bulklike region, remains fluctuating above the bound surfactants without adsorbing directly onto the bare area during 5 × 103 to 1 × 104 time steps, because the surface has been already occupied by surfactants in parts. Thereafter a few surfactants are pulled out of the aggregate because of the tail-tail attraction between themselves and surfactants bound on the surface, and after fluctuating for a while they adsorb completely on the surface. The rest return to the bulklike region because of steric repulsion. This mechanism is named tailinduced aggregate adsorption. This behavior of the aggregates is, however, hardly found in the simulations of the lower concentration of Xsurf ) 2.66 vol %. C.3. H3H′t3 at Low and High Concentrations. The aggregation of H3H′t3 surfactants with a shorter tail chain than H3H′t5 is examined at Xsurf ) 2.07 and 7.73 vol %. Figure 10 shows the snapshots of the surfactant solutions after 3 × 105 time steps. Comparison of Figures 8b and 10 suggests that the total number of surfactants adsorbed

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Figure 11. Mechanisms of the adsorption and aggregation of surfactants on the hydrophilic surface. (a) Direct monomer adsorption; (b) tail-induced monomer adsorption; (c) surfacediffusion growth; (d) tail-induced aggregate adsorption; (e) direct aggregate adsorption. The frequency of appearance of the five mechanisms decreases from panel a to e.

Figure 10. Final snapshots of system VI after 3 × 105 time steps. Symbols are the same as in Figure 6. (a) Xsurf ) 2.07 vol %; (b) Xsurf ) 7.73 vol %.

on the surface and the size of 2D aggregates formed increase as the dosed surfactant concentration is higher. On the other hand, comparison between parts a and b of Figure 8 indicates that the larger 2D aggregates form as the tail chain of surfactants is longer. This characteristic is also observable when one make a comparison between Figures 6c and 10a and between Figures 9c and 10b. It is found that the 2D aggregates of H3H′t3 surfactants fluctuate assembling and splitting much more actively than those of H3H′t5 surfactants, although the H3H′t3 solutions are lower in temperature than the H3H′t5 solutions. The decreases in size and stability of the H3H′t3 aggregates are explained by the weaker tail-tail attraction. At each concentration the adsorption and aggregation mechanisms are almost the same in both the cases of H3H′t3 and H3H′t5 surfactants. C.4. Mechanisms of Adsorption and Aggregation of Surfactants on the Surface. On the basis of the present simulations, we propose five mechanisms of surfactant adsorption and aggregation on the hydrophilic surface, as illustrated in Figure 11, where the frequency of appearance of the mechanisms decreases from parts a to e of Figure 11. The two mechanisms shown in Figure 11a,e are promoted by the strong head-wall interactions, while the rest depicted in Figure 11b-d are induced by the favorable tail-tail interactions. Hence, the latter three mechanisms are more frequent as the tail chain of surfactants is longer. We emphasize that the two mechanisms shown in Figure 11b,c are dominant in the surfactant aggregation on the surface, and those in Figure 11d,e are secondary. The latter two are still less frequent as the surfactant concentration is lower, because the surfactants hardly aggregate in the bulk. As the number of surfactants adsorbed on the surface increases, the direct

adsorption of monomers and aggregates shown in Figure 11a,e is less frequent. Finally we consider why the islandlike aggregates were observed in AFM measurements even though the concentration of dosed surfactants was much below the cmc.44 Figures 8a and 4 show that at the low concentration of about 3 vol % (≈cmc) the size distribution of H3H′t5 aggregates on the surface has a peak at n2D ) 4, while the corresponding peak is absent in the bulk. In the case of the H3H′t3 solutions near the cmc, on the other hand, neither on the surface nor in the bulk do the size distributions have similar peaks, as shown in Figure 8b. These indicate that the solid/liquid interface is as favorable as the bulk for the H3H′t3 aggregation, whereas the interface is more favorable for the H3H′t5 aggregation. Hence, this demonstrates that the interface becomes more favorable for the surfactant aggregation than the bulk as the surfactant tail chain is longer. Supposing that surfactants have longer tails than the surfactants used here, the aggregation may be able to occur at the interface even when the dosed surfactant concentration is well below the cmc. IV. Conclusion In the present study, we investigate whether the simple description of molecules and interfaces can capture the basic characteristics of (1) water/solid interfaces and (2) micellar solutions, using MD simulations. Thereafter, (3) the surfactant aggregation on the smooth hydrophilic surfaces in micellar solutions is simulated by the simple model. The following conclusions are drawn from these MD simulations. (1) As for water/solid interfaces, the simulations of waterlike particles confined within the two planar walls verify that the simple model can mimic, at least qualitatively, water molecules near the hydrophilic and hydrophobic surfaces with respect to the static structure and the diffusional motion. About four adsorbed layers of waterlike particles form near the hydrophilic surface, while no distinct layer forms near the hydrophobic surface. As the particles come closer to the surfaces, their diffusion coefficients decrease in the case of the hydrophilic surface, while they increase in the case of the hydrophobic surface. (2) Even in simple-modeled micellar solutions, the cmc can be determined and the micelles have the optimum size depending on the temperature, concentration, and molecular architecture of surfactants. These verify that

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the simple model simulations can reproduce, at least qualitatively, the characteristics of real micellar solutions. (3) We propose five mechanisms of the adsorption and aggregation of surfactants on the hydrophilic surface: (a) direct monomer adsorption, (b) tail-induced monomer adsorption, (c) surface-diffusion growth, (d) tail-induced aggregate adsorption, and (e) direct aggregate adsorption. The frequency of appearance of the five mechanisms decreases from mechanisms a to e. Mechanisms a and e are induced by strong head-wall interactions, while the rest are induced by favorable tail-tail interactions. Hence, the latter three mechanisms are more frequent as surfactants have a longer tail chain. Mechanisms b and c are dominant in the aggregate formation on the surface, and those of d and e are secondary. The latter two are still less frequent as the dosed surfactant concentration is lower, because the surfactants hardly aggregate in the bulk. As

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the number of surfactants adsorbed on the surface increases, the direct adsorption of monomers and aggregates, that is, mechanisms a and e, are less frequent. The size distributions of the 2D aggregates on the surface demonstrate that the interface is more favorable for the surfactant aggregation than the bulk as the surfactant tail chain is longer. This implies that even well below the cmc the aggregation is able to occur at the interface if the surfactants have a sufficiently long tail chain. Acknowledgment. Computation time of the Cray T-94/4128 was provided by the Supercomputer Laboratory, Institute for Chemical Research, Kyoto University. Graphic visualization of the MD simulation data was performed at the Data Processing Center, Kyoto University. LA9806193