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Molecular dynamics simulation of the sodium octanoate micelle in aqueous solution
Chemical Physics Letters 411 (2005) 474–478 www.elsevier.com/locate/cplett
Molecular dynamics simulation of the sodium octanoate micelle in aqueous solution Andre´ Farias de Moura a, Luiz Carlos Gomide Freitas a
b,*
Instituto de Quı´mica de Sa˜o Carlos, Universidade de Sa˜o Paulo, CP 780, Sa˜o Carlos, SP, CEP 13560-970, Brazil b Departamento de Quı´mica, Centro de Cieˆncias Exatas e de Tecnologia, Universidade Federal de Sa˜o Carlos, Rodovia Washington Luiz, km 235, CP 676, Sa˜o Carlos, SP, CEP 13565-905, Brazil Received 15 March 2005; in final form 13 May 2005 Available online 11 July 2005
Abstract A 20 ns molecular dynamics simulation was performed with a realistic model system of sodium octanoate micelles in aqueous solution. The system comprised three micellar aggregates, each containing 15 monomers, and 15 free octanoate monomers. The initial configuration relaxed within 2 ns, mostly due to the fusion of aggregates and the exchange of monomers between the aggregates and the solution. The process led to a decrease in the total number of octanoate clusters and to an increase in the average aggregation number and micellar radius, observations in agreement with experimental results. 2005 Elsevier B.V. All rights reserved.
1. Introduction Self-assembling processes play a central role in research areas as diverse as nanotechnology and biophysics. However, the complex and extended nature of real systems makes it difficult to obtain a detailed picture of the aggregation phenomena involved. Instead, smaller systems have been used as models in both experimental and theoretical investigations of the origins of self-assembling. Aqueous solutions of amphiphilic molecules are among the most frequently used systems for this purpose, due to their ability to form several self-assembled structures, ranging from small spheroid aggregates (micelles) to large assemblies (e.g., monolayers, bilayers and vesicles) [1]. In the last two decades, molecular dynamics (MD) simulations have become a standard tool for the investigation of micellar systems [2–16]. MD simulations provide insight into structural and dynamical features of *
Corresponding author. Fax: +55 16 3351 8350. E-mail addresses: [email protected], [email protected] (A.F. de Moura), [email protected] (L.C.G. Freitas). 0009-2614/$ - see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2005.05.039
assembled molecules. The inherent complexity of the systems and the long relaxation times involved in selfassembling, impose some limitations on the detailed study of aggregation processes using MD simulations [2]. Consequently, nearly all micellar systems that have been studied to date have employed model systems containing a previously assembled aggregate in an aqueous solution [3–14]. Such studies can provide interesting insights into the structure of the aggregates, but an accurate description of the structural and dynamic features of self-assembling processes requires a more elaborate model. Recently, MD simulations of complex liquids have become feasible due to the increase in computational power and the development of more efficient algorithms. Two recent studies reported the self-assembling of cationic [15] and non-ionic [16] surfactants in aqueous solution, suggesting possible mechanisms for the formation of micelles. Both studies shed light on the subject, but present the same drawback – the amount of water and surfactant molecules had to be decreased in order to reduce the computational cost, resulting in small micelles within concentrated solutions. Experimental evidence indicates
A.F. de Moura, L.C.G. Freitas / Chemical Physics Letters 411 (2005) 474–478
that small aggregates should be found in dilute solutions, just above the critical micelle concentration (CMC), whereas concentrated solutions should have larger aggregates [1]. The sodium octanoate micellar system is appropriate for theoretical studies because it has a very high CMC value, making it possible to work with a representative (though relatively small) model system. The first MD simulations of this system employed only one small aggregate (N = 15) and its counter ions surrounded by 1094 water molecules [3], which is the micellar size equal to experimental results obtained by light scattering at this concentration [17]. Other subsequently performed MD studies have not addressed the role of micelle size or micelle–micelle and micelle–monomer interactions [4–9]. The aim of the present work is to fill this information gap, reporting on the results of a large-scale MD simulation of sodium octanoate micelles in aqueous solution. A system with three micelles and free octanoate anions and its structural evolution over a 20 ns simulation, shall be described.
2. Methodology The model system consisted of 60 octanoate anions and 60 sodium counter ions, surrounded by 3300 water molecules. Most anions were previously assembled into three replicas of a spheroid aggregate that comprised 15 octanoate molecules, taken from a previous simulation [9]. These aggregates were placed in the simulation box, followed by 15 free monomers, the water molecules and the counter ions. All molecules and aggregates were randomly rotated and randomly placed into the box, avoiding strong repulsive contacts. The OPLS-AA force field parameters were used for ˚ qvistÕs parameters for sodium the octanoate [18], the A [19] and the SPC model for water [20]. The efficiency of the combination of these force fields has been tested elsewhere [9]. The interactions were cut off at 1.5 nm, using the particle mesh Ewald method [21] to treat the long-range electrostatic potential beyond cutoff distance. The initial structure was optimized using both steepest descent and conjugated gradient algorithms to minimize the maximum force in the system below 50 kJ mol 1 nm 1. After energy minimization, six simulated annealing runs were performed between 0 and 50 K to further relax the structure. Simulations were performed using GROMACS software (version 3.1) [22,23]. The weak coupling scheme of Berendsen [24] was used for the temperature (300 K, sT = 0.1 ps) and the pressure (1 bar, sp = 1.0 ps), coupling octanoate, sodium and water molecules to three separate heat baths. The trajectory was integrated up
475
to 20 ns using a time step of 1.0 fs and the neighbor list was updated every tenth step.
3. Results and discussion As a general trend, the simulation box must be large enough to allow the development of all the structural patterns of the physical system under consideration. Concerning micelles in aqueous solution, there should be a collection of aggregates ranging from free monomers to large assemblies at the end of a simulation, resulting in a size distribution rather than a single micelle size. Besides the size of the system, simulations should be long enough to allow structural relaxation. A typical surfactant forms micelles in very dilute solutions but the medium-chain surfactant sodium octanoate forms small micelles at a rather high CMC value (ca. 0.4 mol L 1 [25]). Thus, sodium octanoate may be regarded as an adequate model system for theoretical investigations. Experimental evidence suggests that in 1.0 mol L 1 solutions of sodium octanoate there should be octanoate molecules in micelles and as free monomers in the ratio of 3:1 [26]. Bearing this in mind, a model system with three micelles and 15 free monomers (Fig. 1a) was assembled, thus explicitly taking into account the interactions between different micelles and also between micelles and free surfactant molecules. At the end of the 20 ns trajectory the system still contained three micellar aggregates, although the number of free monomers decreased (Fig. 1b). To obtain a quantitative description of the evolution of the number of aggregates in the system the reasoning of Marrink et al. [16] was used and a cutoff distance of 1.0 nm between the center of mass of two octanoate molecules was adopted as the criterion to decide to which cluster a given molecule should be assigned to. The total number of aggregates must lie between 1 (only one large micelle formed) and 60 (no association at all). It was found that the number of octanoate clusters decreased steadily during the first 2 ns and then reached an average value of ca. 7 (Fig. 2). During the evolution of the model system along its trajectory, a change in the micelle size distribution occurred. Two populations could be distinguished during the first 0.5 ns, one composed of monomers and small aggregates (N 6 5) and the other of micelles with an average aggregation number N = 15 (Fig. 3a). As the simulation proceeded, a completely different distribution arose. Free surfactant molecules and small aggregates are still present, but large clusters (N 30) along with smaller aggregates (N = 6–20) begin to prevail (Fig. 3b). The integration of the size distribution (Fig. 3b) gave the fraction of each range of micelle size – with 6% present as small clusters (N 6 5), 40% as
476
A.F. de Moura, L.C.G. Freitas / Chemical Physics Letters 411 (2005) 474–478
Fig. 1. (a) Initial structure (t = 0 ns); (b) final structure (t = 20 ns).
Fig. 2. Total number of octanoate aggregates during the simulation (including monomers and premicellar aggregates).
Fig. 3. Size distribution of octanoate aggregates: (a) during the first 0.5 ns; (b) from 2.0 to 20 ns.
medium clusters (N = 6–20) and 54% as larger micelles (N > 20). Graphical visualization demonstrated that a larger aggregate began to be formed at ca. 600 ps due to the fusion of two micelles. This aggregate remained stable during the simulation except for the exchange of a few monomers with the solution. On the other hand, the third micelle initially present did not keep its identity throughout the 20 ns trajectory, during which it broke up into smaller clusters. At the same time, the free monomers also began to associate into small aggregates and all these small clusters eventually formed new micelles, though the micelle breaking and forming continued. The size of the aggregates of sodium octanoate in aqueous solution was derived from diffraction experiments using small angle X-ray scattering (SAXS) [27], light scattering (LS) [17] and small angle neutron scattering (SANS) [28,29]. The average micelle size may be computed from the simulation, taking into account the aggregation number and the radius of gyration of the octanoate clusters at each step. Considering the three largest aggregates at each moment of the trajectory, a clear indication that a large structural rearrangement occurred in the first 2 ns of the simulation, characterized by an increase in the average values of the aggregation number (Fig. 4) and radius of gyration (Fig. 5), was again observed. The average radius of gyration (RG) presents a peak at around 1 ns because the fusion of two micelles led first to the formation of an elongated aggregate that relaxed to a spheroid micelle after a while. Assuming that the aggregates are nearly spherical, it follows that the micelle radius (R) is ÆRæ = (5/3)1/2ÆRGæ. With this assumption, the average micelle radius between 2 and 20 ns was 1.18 ± 0.06 nm while the average
A.F. de Moura, L.C.G. Freitas / Chemical Physics Letters 411 (2005) 474–478
Fig. 4. Average aggregation number of the micellar aggregates.
477
Fig. 5. Average radius of gyration of the micellar aggregates.
Table 1 Average aggregation number (N) and radius (R) of the sodium octanoate micelle Reference
Method
Octanoate mole fraction
ÆNæ
ÆRæ (nm)
Zemb et al. [17] Zemb et al. [17] Zemb et al. [17] Hayter and Zemb [28] Hayter and Zemb [28] Hayter and Zemb [28] Friman and Rosenholm [27] Hayter et al. [29] Jo¨nsson et al. [3] Watanabe et al. [4] Watanabe and Klein [5] Shelley et al. [6] Khun and Rehage [7] Laaksonen and Rosenholm [8] de Moura and Freitas [9] This work
LS LS LS SANS SANS SANS SAXS SANS MD MD MD MD MD MD MD MD
0.0095 0.0137 0.0205 0.0148 0.0182 0.0217 0.0205 0.0254 0.0135 0.0139 0.0205 0.0123 0.0169 0.0186 0.0068 0.0179
11 15 18 17 ± 1 19 ± 1 21 ± 1 – 24 15 15 15 15 15 15 15 17 ± 1
0.83 0.91 0.98 1.25 ± 0.03 1.29 ± 0.03 1.32 ± 0.03 2.43 1.37 1.19 ± 0.40 1.27 ± 0.20 1.05 ± 0.02 1.10 1.31 ± 0.24 1.39 ± 0.04 1.05 ± 0.04 1.18 ± 0.06
aggregation number was 17 ± 1. These values are in good agreement with the experimental data (Table 1), particularly those derived from LS and SANS experiments. The SAXS technique was also employed in experimental investigations, but it lacks contrast between the octanoate head groups and the surrounding water molecules, giving a biased structural picture (a very small hydrophobic core surrounded by a very large hydrophilic layer) [27]. Previous simulations employed only one aggregate and thus the micelle size should be considered as a constraint instead of a result. In these studies the estimation of micelle size was based on the average position of the head group atoms [2–8] or on the radius of gyration [9] and the results obtained also agree with experimental data (Table 1), particularly with SANS results. The fact that the large model system employed here yielded an average micelle size comparable to the results reported in previous studies shows that great care must be taken in interpreting the simulation results.
Regarding the micelle radius, sound conclusions may be drawn only if the size distribution is taken into account.
4. Conclusions Results from a large-scale molecular dynamics simulation of a sodium octanoate micellar system have been presented. Three micelles (containing 15 octanoate molecules each) and 15 free monomers formed the initial system. In this way, micelle–micelle and micelle– monomer interactions were taken explicitly into account. The micelle size distribution changed over the course of the simulation due to the exchange of monomers, which led lead to the presence of both small and large aggregates. The average micelle size obtained is in agreement with experimental results. The present results indicate that larger clusters are stable in the nanosecond regime but smaller aggregates are continually being
478
A.F. de Moura, L.C.G. Freitas / Chemical Physics Letters 411 (2005) 474–478
formed and broken in this time scale. Two conclusions seem to be well grounded: firstly, free monomers are not stable, they tend to associate into micelles or small clusters and second, small aggregates are short-lived when compared to larger clusters, they will either associate to form a small micelle or break up.
Acknowledgement The authors are indebted to FAPESP for the financial support.
References [1] J.N. Israelachvili, Intermolecular & Surface Forces, second edn., Academic Press, London, 1995. [2] J.C. Shelley, M.Y. Shelley, Curr. Opin. Colloid Interface Sci. 5 (2000) 101. [3] B. Jo¨nsson, O. Edholm, O. Teleman, J. Chem. Phys. 85 (1986) 2259. [4] K. Watanabe, M. Ferrario, M.L. Klein, J. Phys. Chem. 92 (1988) 819. [5] K. Watanabe, M.L. Klein, J. Phys. Chem. 93 (1989) 6897. [6] J. Shelley, M. Sprik, M.L. Klein, Langmuir 9 (1993) 916. [7] H. Kuhn, H. Rehage, Ber. Bunsen. Phys. Chem. 101 (1997) 1485. [8] L. Laaksonen, J.B. Rosenholm, Chem. Phys. Lett. 216 (1993) 429. [9] A.F. de Moura, L.C.G. Freitas, Braz. J. Phys. 34 (2004) 64. [10] T. Bast, R. Hentschke, J. Phys. Chem. 100 (1996) 12162. [11] T. Wymore, X.F. Gao, T.C. Wong, J. Mol. Struct. 485–486 (1999) 195.
[12] S. Bogusz, R.M. Venable, R.W. Pastor, J. Phys. Chem. B 104 (2000) 5462. [13] D.P. Tieleman, D. van der Spoel, H.J.C. Berendsen, J. Phys. Chem. B 104 (2000) 6380. [14] S. Balasubramanian, B. Bagchi, J. Phys. Chem. B 106 (2000) 3668. [15] J.B. Mailet, V. Lachet, P.V. Coveney, Phys. Chem. Chem. Phys. 1 (1999) 5277. [16] S.J. Marrink, D.P. Tieleman, A.E. Mark, J. Phys. Chem. B 104 (2000) 12165. [17] T. Zemb, M. Drifford, M. Hayoun, A. Jehanno, J. Phys. Chem. 87 (1983) 4524. [18] W.L. Jorgensen, D.S. Maxwell, J. Tirado-Rives, Am. Chem. Soc. 118 (1996) 11225. ˚ qvist, J . Phys. Chem. 94 (1990) 8021. [19] J. A [20] H.J.C. Berendsen, J.P.M. Postma, W.F. van Gusteren, J. Hermans, in: B. Pullman (Ed.), Intermolecular Forces, Reidel, Dordrecht, 1981. [21] U. Essman, L. Perela, M.L. Berkowitz, T. Darden, H. Lee, L.G. Pedersen, J. Chem. Phys. 103 (1995) 8577. [22] E. Lindahl, B. Hess, D. van der Spoel, J. Mol. Model. 7 (2001) 306. [23] H.J.C. Berendsen, D. van der Spoel, R. van Drunen, Comp. Phys. Commun. 91 (1995) 43. [24] H.J.C. Berendsen, J.P.M. Postma, A. DiNola, J.R. Haak, J. Chem. Phys. 81 (1984) 3684. [25] A. Gonza´lez-Pe´rez, J.M. Ruso, G. Prieto, F. Sarmiento, Langmuir 20 (2004) 2512. [26] B. Lindman, N. Kamenka, M.-C. Puyal, B. Brun, B. Jo¨nsson, J. Phys. Chem. 88 (1988) 53. [27] R. Friman, J.B. Rosenholm, Colloid Polym. Sci. 260 (1982) 545. [28] J.B. Hayter, T. Zemb, Chem. Phys. Lett. 93 (1982) 91. [29] B. Hayter, M. Hayoun, T. Zemb, Colloid Polym. Sci. 262 (1984) 798.
Molecular dynamics simulation of the sodium octanoate micelle in aqueous solution Andre´ Farias de Moura a, Luiz Carlos Gomide Freitas a
b,*
Instituto de Quı´mica de Sa˜o Carlos, Universidade de Sa˜o Paulo, CP 780, Sa˜o Carlos, SP, CEP 13560-970, Brazil b Departamento de Quı´mica, Centro de Cieˆncias Exatas e de Tecnologia, Universidade Federal de Sa˜o Carlos, Rodovia Washington Luiz, km 235, CP 676, Sa˜o Carlos, SP, CEP 13565-905, Brazil Received 15 March 2005; in final form 13 May 2005 Available online 11 July 2005
Abstract A 20 ns molecular dynamics simulation was performed with a realistic model system of sodium octanoate micelles in aqueous solution. The system comprised three micellar aggregates, each containing 15 monomers, and 15 free octanoate monomers. The initial configuration relaxed within 2 ns, mostly due to the fusion of aggregates and the exchange of monomers between the aggregates and the solution. The process led to a decrease in the total number of octanoate clusters and to an increase in the average aggregation number and micellar radius, observations in agreement with experimental results. 2005 Elsevier B.V. All rights reserved.
1. Introduction Self-assembling processes play a central role in research areas as diverse as nanotechnology and biophysics. However, the complex and extended nature of real systems makes it difficult to obtain a detailed picture of the aggregation phenomena involved. Instead, smaller systems have been used as models in both experimental and theoretical investigations of the origins of self-assembling. Aqueous solutions of amphiphilic molecules are among the most frequently used systems for this purpose, due to their ability to form several self-assembled structures, ranging from small spheroid aggregates (micelles) to large assemblies (e.g., monolayers, bilayers and vesicles) [1]. In the last two decades, molecular dynamics (MD) simulations have become a standard tool for the investigation of micellar systems [2–16]. MD simulations provide insight into structural and dynamical features of *
Corresponding author. Fax: +55 16 3351 8350. E-mail addresses: [email protected], [email protected] (A.F. de Moura), [email protected] (L.C.G. Freitas). 0009-2614/$ - see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2005.05.039
assembled molecules. The inherent complexity of the systems and the long relaxation times involved in selfassembling, impose some limitations on the detailed study of aggregation processes using MD simulations [2]. Consequently, nearly all micellar systems that have been studied to date have employed model systems containing a previously assembled aggregate in an aqueous solution [3–14]. Such studies can provide interesting insights into the structure of the aggregates, but an accurate description of the structural and dynamic features of self-assembling processes requires a more elaborate model. Recently, MD simulations of complex liquids have become feasible due to the increase in computational power and the development of more efficient algorithms. Two recent studies reported the self-assembling of cationic [15] and non-ionic [16] surfactants in aqueous solution, suggesting possible mechanisms for the formation of micelles. Both studies shed light on the subject, but present the same drawback – the amount of water and surfactant molecules had to be decreased in order to reduce the computational cost, resulting in small micelles within concentrated solutions. Experimental evidence indicates
A.F. de Moura, L.C.G. Freitas / Chemical Physics Letters 411 (2005) 474–478
that small aggregates should be found in dilute solutions, just above the critical micelle concentration (CMC), whereas concentrated solutions should have larger aggregates [1]. The sodium octanoate micellar system is appropriate for theoretical studies because it has a very high CMC value, making it possible to work with a representative (though relatively small) model system. The first MD simulations of this system employed only one small aggregate (N = 15) and its counter ions surrounded by 1094 water molecules [3], which is the micellar size equal to experimental results obtained by light scattering at this concentration [17]. Other subsequently performed MD studies have not addressed the role of micelle size or micelle–micelle and micelle–monomer interactions [4–9]. The aim of the present work is to fill this information gap, reporting on the results of a large-scale MD simulation of sodium octanoate micelles in aqueous solution. A system with three micelles and free octanoate anions and its structural evolution over a 20 ns simulation, shall be described.
2. Methodology The model system consisted of 60 octanoate anions and 60 sodium counter ions, surrounded by 3300 water molecules. Most anions were previously assembled into three replicas of a spheroid aggregate that comprised 15 octanoate molecules, taken from a previous simulation [9]. These aggregates were placed in the simulation box, followed by 15 free monomers, the water molecules and the counter ions. All molecules and aggregates were randomly rotated and randomly placed into the box, avoiding strong repulsive contacts. The OPLS-AA force field parameters were used for ˚ qvistÕs parameters for sodium the octanoate [18], the A [19] and the SPC model for water [20]. The efficiency of the combination of these force fields has been tested elsewhere [9]. The interactions were cut off at 1.5 nm, using the particle mesh Ewald method [21] to treat the long-range electrostatic potential beyond cutoff distance. The initial structure was optimized using both steepest descent and conjugated gradient algorithms to minimize the maximum force in the system below 50 kJ mol 1 nm 1. After energy minimization, six simulated annealing runs were performed between 0 and 50 K to further relax the structure. Simulations were performed using GROMACS software (version 3.1) [22,23]. The weak coupling scheme of Berendsen [24] was used for the temperature (300 K, sT = 0.1 ps) and the pressure (1 bar, sp = 1.0 ps), coupling octanoate, sodium and water molecules to three separate heat baths. The trajectory was integrated up
475
to 20 ns using a time step of 1.0 fs and the neighbor list was updated every tenth step.
3. Results and discussion As a general trend, the simulation box must be large enough to allow the development of all the structural patterns of the physical system under consideration. Concerning micelles in aqueous solution, there should be a collection of aggregates ranging from free monomers to large assemblies at the end of a simulation, resulting in a size distribution rather than a single micelle size. Besides the size of the system, simulations should be long enough to allow structural relaxation. A typical surfactant forms micelles in very dilute solutions but the medium-chain surfactant sodium octanoate forms small micelles at a rather high CMC value (ca. 0.4 mol L 1 [25]). Thus, sodium octanoate may be regarded as an adequate model system for theoretical investigations. Experimental evidence suggests that in 1.0 mol L 1 solutions of sodium octanoate there should be octanoate molecules in micelles and as free monomers in the ratio of 3:1 [26]. Bearing this in mind, a model system with three micelles and 15 free monomers (Fig. 1a) was assembled, thus explicitly taking into account the interactions between different micelles and also between micelles and free surfactant molecules. At the end of the 20 ns trajectory the system still contained three micellar aggregates, although the number of free monomers decreased (Fig. 1b). To obtain a quantitative description of the evolution of the number of aggregates in the system the reasoning of Marrink et al. [16] was used and a cutoff distance of 1.0 nm between the center of mass of two octanoate molecules was adopted as the criterion to decide to which cluster a given molecule should be assigned to. The total number of aggregates must lie between 1 (only one large micelle formed) and 60 (no association at all). It was found that the number of octanoate clusters decreased steadily during the first 2 ns and then reached an average value of ca. 7 (Fig. 2). During the evolution of the model system along its trajectory, a change in the micelle size distribution occurred. Two populations could be distinguished during the first 0.5 ns, one composed of monomers and small aggregates (N 6 5) and the other of micelles with an average aggregation number N = 15 (Fig. 3a). As the simulation proceeded, a completely different distribution arose. Free surfactant molecules and small aggregates are still present, but large clusters (N 30) along with smaller aggregates (N = 6–20) begin to prevail (Fig. 3b). The integration of the size distribution (Fig. 3b) gave the fraction of each range of micelle size – with 6% present as small clusters (N 6 5), 40% as
476
A.F. de Moura, L.C.G. Freitas / Chemical Physics Letters 411 (2005) 474–478
Fig. 1. (a) Initial structure (t = 0 ns); (b) final structure (t = 20 ns).
Fig. 2. Total number of octanoate aggregates during the simulation (including monomers and premicellar aggregates).
Fig. 3. Size distribution of octanoate aggregates: (a) during the first 0.5 ns; (b) from 2.0 to 20 ns.
medium clusters (N = 6–20) and 54% as larger micelles (N > 20). Graphical visualization demonstrated that a larger aggregate began to be formed at ca. 600 ps due to the fusion of two micelles. This aggregate remained stable during the simulation except for the exchange of a few monomers with the solution. On the other hand, the third micelle initially present did not keep its identity throughout the 20 ns trajectory, during which it broke up into smaller clusters. At the same time, the free monomers also began to associate into small aggregates and all these small clusters eventually formed new micelles, though the micelle breaking and forming continued. The size of the aggregates of sodium octanoate in aqueous solution was derived from diffraction experiments using small angle X-ray scattering (SAXS) [27], light scattering (LS) [17] and small angle neutron scattering (SANS) [28,29]. The average micelle size may be computed from the simulation, taking into account the aggregation number and the radius of gyration of the octanoate clusters at each step. Considering the three largest aggregates at each moment of the trajectory, a clear indication that a large structural rearrangement occurred in the first 2 ns of the simulation, characterized by an increase in the average values of the aggregation number (Fig. 4) and radius of gyration (Fig. 5), was again observed. The average radius of gyration (RG) presents a peak at around 1 ns because the fusion of two micelles led first to the formation of an elongated aggregate that relaxed to a spheroid micelle after a while. Assuming that the aggregates are nearly spherical, it follows that the micelle radius (R) is ÆRæ = (5/3)1/2ÆRGæ. With this assumption, the average micelle radius between 2 and 20 ns was 1.18 ± 0.06 nm while the average
A.F. de Moura, L.C.G. Freitas / Chemical Physics Letters 411 (2005) 474–478
Fig. 4. Average aggregation number of the micellar aggregates.
477
Fig. 5. Average radius of gyration of the micellar aggregates.
Table 1 Average aggregation number (N) and radius (R) of the sodium octanoate micelle Reference
Method
Octanoate mole fraction
ÆNæ
ÆRæ (nm)
Zemb et al. [17] Zemb et al. [17] Zemb et al. [17] Hayter and Zemb [28] Hayter and Zemb [28] Hayter and Zemb [28] Friman and Rosenholm [27] Hayter et al. [29] Jo¨nsson et al. [3] Watanabe et al. [4] Watanabe and Klein [5] Shelley et al. [6] Khun and Rehage [7] Laaksonen and Rosenholm [8] de Moura and Freitas [9] This work
LS LS LS SANS SANS SANS SAXS SANS MD MD MD MD MD MD MD MD
0.0095 0.0137 0.0205 0.0148 0.0182 0.0217 0.0205 0.0254 0.0135 0.0139 0.0205 0.0123 0.0169 0.0186 0.0068 0.0179
11 15 18 17 ± 1 19 ± 1 21 ± 1 – 24 15 15 15 15 15 15 15 17 ± 1
0.83 0.91 0.98 1.25 ± 0.03 1.29 ± 0.03 1.32 ± 0.03 2.43 1.37 1.19 ± 0.40 1.27 ± 0.20 1.05 ± 0.02 1.10 1.31 ± 0.24 1.39 ± 0.04 1.05 ± 0.04 1.18 ± 0.06
aggregation number was 17 ± 1. These values are in good agreement with the experimental data (Table 1), particularly those derived from LS and SANS experiments. The SAXS technique was also employed in experimental investigations, but it lacks contrast between the octanoate head groups and the surrounding water molecules, giving a biased structural picture (a very small hydrophobic core surrounded by a very large hydrophilic layer) [27]. Previous simulations employed only one aggregate and thus the micelle size should be considered as a constraint instead of a result. In these studies the estimation of micelle size was based on the average position of the head group atoms [2–8] or on the radius of gyration [9] and the results obtained also agree with experimental data (Table 1), particularly with SANS results. The fact that the large model system employed here yielded an average micelle size comparable to the results reported in previous studies shows that great care must be taken in interpreting the simulation results.
Regarding the micelle radius, sound conclusions may be drawn only if the size distribution is taken into account.
4. Conclusions Results from a large-scale molecular dynamics simulation of a sodium octanoate micellar system have been presented. Three micelles (containing 15 octanoate molecules each) and 15 free monomers formed the initial system. In this way, micelle–micelle and micelle– monomer interactions were taken explicitly into account. The micelle size distribution changed over the course of the simulation due to the exchange of monomers, which led lead to the presence of both small and large aggregates. The average micelle size obtained is in agreement with experimental results. The present results indicate that larger clusters are stable in the nanosecond regime but smaller aggregates are continually being
478
A.F. de Moura, L.C.G. Freitas / Chemical Physics Letters 411 (2005) 474–478
formed and broken in this time scale. Two conclusions seem to be well grounded: firstly, free monomers are not stable, they tend to associate into micelles or small clusters and second, small aggregates are short-lived when compared to larger clusters, they will either associate to form a small micelle or break up.
Acknowledgement The authors are indebted to FAPESP for the financial support.
References [1] J.N. Israelachvili, Intermolecular & Surface Forces, second edn., Academic Press, London, 1995. [2] J.C. Shelley, M.Y. Shelley, Curr. Opin. Colloid Interface Sci. 5 (2000) 101. [3] B. Jo¨nsson, O. Edholm, O. Teleman, J. Chem. Phys. 85 (1986) 2259. [4] K. Watanabe, M. Ferrario, M.L. Klein, J. Phys. Chem. 92 (1988) 819. [5] K. Watanabe, M.L. Klein, J. Phys. Chem. 93 (1989) 6897. [6] J. Shelley, M. Sprik, M.L. Klein, Langmuir 9 (1993) 916. [7] H. Kuhn, H. Rehage, Ber. Bunsen. Phys. Chem. 101 (1997) 1485. [8] L. Laaksonen, J.B. Rosenholm, Chem. Phys. Lett. 216 (1993) 429. [9] A.F. de Moura, L.C.G. Freitas, Braz. J. Phys. 34 (2004) 64. [10] T. Bast, R. Hentschke, J. Phys. Chem. 100 (1996) 12162. [11] T. Wymore, X.F. Gao, T.C. Wong, J. Mol. Struct. 485–486 (1999) 195.
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