- Email: [email protected]
Micellar dynamics and water–water hydrogen-bonding from temperature-jump Monte Carlo simulations
Chemical Physics Letters 550 (2012) 83–87
Contents lists available at SciVerse ScienceDirect
Chemical Physics Letters journal homepage: www.elsevier.com/locate/cplett
Micellar dynamics and water–water hydrogen-bonding from temperature-jump Monte Carlo simulations G. Heinzelmann a, W. Figueiredo b, M. Girardi c,⇑ a
School of Physics, University of Sydney, NSW 2006, Australia Departamento de Física, Universidade Federal de Santa Catarina, 88040-900 Florianópolis, Santa Catarina, Brazil c Universidade Federal de Santa Catarina, 88900-000 Araranguá, Santa Catarina, Brazil b
a r t i c l e
i n f o
Article history: Received 10 June 2012 In final form 6 September 2012 Available online 14 September 2012
a b s t r a c t We perform Monte Carlo simulations of a model for water–amphiphiles solution to observe the interplay between the formation of micelles and the hydrogen bond (HB) dynamics of the solvent. By performing temperature jumps from a non-micellized to a highly micellized state, we show that the dynamics can be described by three relaxation times. The analysis of HBs reveals an opposite behavior from the water molecules in the first hydration shell of the micelles and their total number indicating that the system is sacrificing HBs locally in order to increase them as a whole. Ó 2012 Elsevier B.V. All rights reserved.
Amphiphilic molecules, or surfactants, have the property of selfassembling into micelles in aqueous solutions mainly due to the hydrophobic effect [1–3]. They have a wide variety of applications, ranging from biological structures to domestic detergents and industrial processing [4]. Throughout the years, there has been a great number of experimental and theoretical studies on the equilibrium properties of micellar solutions, such as the size distribution curve, the shape of the micelles and the critical micelle concentration (CMC) [5–12]. Experiments, simulations and analytical studies have also been extensively used to probe the micellar dynamics [13–23]. Chemical relaxation experiments like ultrasonic relaxation, as well as concentration and temperature jump experiments, have been performed with different ionic and non-ionic surfactants. In such experiments, one applies a sudden change in the temperature or surfactant concentration and monitors the micelle behavior by measuring a property of the system, like conductivity, light scattering, fluorescence or surface tension. Most results show that, when a micellar system is perturbed, the path to equilibrium involves two distinct relaxation times: a fast one, in the order of microseconds, related to the entering/exiting of individual monomers from micelles, and a slow one, in the order of milliseconds, that involves the formation and breakdown of entire aggregates by cooperative condensation or dissolution of monomers. This result has also been corroborated by theoretical studies by Aniansson and Wall, and is known as the AW mechanism [24,25]. Nonetheless, other interpretations for the slow relaxation time have been suggested, like the formation and subsequent ⇑ Corresponding author. E-mail addresses: [email protected] (G. Heinzelmann), wagner@ fisica.ufsc.br (W. Figueiredo), [email protected] (M. Girardi). 0009-2614/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.cplett.2012.09.011
breakdown of a transient supermicelle, and the temporary fragmentation of a normal micelle into two submicelles, which grow back by fusing together or by incorporating free monomers in the solution [14,21]. Recent experimental and theoretical studies have also shown the possibility of a third relaxation time that, depending on the system, can be ultraslow, ultrafast or in between the fast and slow relaxation times [15–17]. Similarly, the hydrophobic effect and related phenomena have received much attention through the study of hydrophobic and amphiphilic molecules immersed in water [26–34]. These studies indicate that, for small hydrophobic molecules, water can reorganize around the solute without sacrificing hydrogen bonds, and the entropic cost of this process leads to their low solubility. For a small number of these molecules, water tends to segregate such species rather than drive them together, and the free energy of solvation grows linearly with the solvent volume. On the other hand, close to a larger hydrophobic object, the maintenance of a hydrogen-bond network is geometrically impossible. In this case the free energy of solvation grows with the solvent accessible surface area (SASA). In other words, the free energy of solvation of a large number of separated hydrophobic molecules in water exceeds that of a large cluster of the same species. This is the force behind the oil–water phase separation. The crossover between the two regimes happens when the SASA is larger than the nm2 scale. Maibaum et al. [34] have demonstrated that the same principle can be applied to the aggregation of surfactants in solution and the formation of micelles. Here, due to the restriction of having a polar group right next to a hydrophobic one, the growth of the amphiphilic assemblies is limited to mesoscopic domains, as opposed to the macroscopic oil–water separation. An expression for the value of the CMC as a function of the temperature and
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amphiphilic tail length is obtained, with good agreement with experimental results. Even though the dynamics of both solvent hydrogen bonding and amphiphilic aggregation are closely related, it is very hard to observe the interplay between them due to the contrasting time scales involved. While the time scales related to hydrogen bonding are in the order of picoseconds [35,36], the relaxation times of micelle formation/breakdown are in the microseconds and milliseconds scale, well above what can be obtained using all-atomic molecular dynamics. Thus, in the present Letter, we attempt to combine these two phenomena through temperature-jump simulations, observing how the solvent hydrogen-bonding is related to the micelle dynamics when we go from a non-micellized to a highly micellized state. We make use of a well established lattice model previously considered in Refs. [37–40], which allows for the orientational freedom of solvent molecules as well as the formation of hydrogen bonds between them. Our model system considers a body-centered cubic (BCC) lattice with periodic boundary conditions, filled by single-site bonding solvent molecules and by self-avoiding chains of connected sites representing the amphiphiles. We use a four-site flexible amphiphile chain with three sites representing the hydrophobic tail and one site representing the hydrophilic head (H1T3). Concerning the water molecules, each one has four bonding arms in a tetrahedral geometry, trying to mimic the actual water geometry. When two bonding arms from neighboring water molecules are pointing to each other we have a hydrogen bond, with a energy value of c ¼ 2. There is also a repulsive interaction of e ¼ 1 for all pairs of nearest-neighboring water molecules. In the case of amphiphiles, the interaction parameters apply to pairs of nearest-neighbors regardless of the orientation, and are set to obtain micellized and non-micellized systems in the range of temperatures considered. We have the following set of interaction energies: EHH ¼ 7e; EHT ¼ 3e, EHS ¼ 10e; ETS ¼ ETT ¼ e; subscripts are (S) for solvent molecules, (T) for a hydrophobic tail segment and (H) for a hydrophilic head. The system has a linear lattice size of L ¼ 30 with a volume fraction of amphiphiles at U ¼ 0:05 and the remaining sites filled with water. The simulations were carried out by using the Metropolis algorithm [41] where, at each Monte Carlo step (MCs), we randomly attempt to change the orientation of all the water molecules, as well as move each amphiphile through the reptation movement. The goal of the present Letter is to use temperature-jump simulations to observe the interplay between the micellar dynamics and the hydrogen-bond dynamics, in order to study the hydrophobic effect and its influence on amphiphilic aggregation. To do so, we equilibrated the system in a non-micellized state at temperature T ¼ 2:8 (in units of e=kB ) for 5 105 MCs, and then performed a quick cooling to T ¼ 2:3, in which we have a highly micellized state, measuring the properties of the solution as a function of time. Our results are obtained from the averages over eighty samples, and we also performed the heating process for comparison. All the dynamic system properties are well fitted to single or multiple exponentials, in which we use the equation:
AðtÞ ¼ A0 þ A1 expðt=s1 Þ þ A2 expðt=s2 Þ þ A3 expðt=s3 Þ;
Figure 1. Average number of amphiphiles per micelle (squares) and total solvent accessible surface area of the micelles (triangles) as a function of time after a jump from T = 2.8 to 2.3. The fitting lines (Eq. (1)) are drawn over the data points.
the aggregates Scool) as a function of time, after a temperature jump from T = 2.8 to 2.3. The total SASA is directly proportional to the percentage of water molecules in the first hydration shell of the solute, so we obtain the former by calculating this percentage and multiplying it by 20 for a convenient visualization. We see a close relation between the two properties, i.e., the larger the micelles the smaller the sum of their surface areas exposed to the solvent. We have fitted both curves to the third-order exponential expression of Eq. (1), with the fitting line drawn in Figure 1, and the relaxation times shown in Table 1. We always attempted to fit all the data with three exponentials. However, we have seen that in some cases, two or more relaxation times were so close together that the resultant fit is adjusted with a single or two exponentials. Indeed, a given fit is accepted only when the coefficient of correlation is larger than 0.99. The error bars were omitted in the plots for better visualization and are typically around 1%. The errors in the properties are reported in the values of the corresponding relaxation times in Table 1. The three time constants are comparable in the two cases, confirming the observation that these two properties share the same dependence over time. Table 1 Relaxation times ðsi Þ in MCs, obtained by the fit with Eq. (1) for the temperaturejump simulations. The average number of amphiphiles per micelle during the cooling (mcool ðtÞ) and the heating (mheat ðtÞ) processes, the SASA of the solute (Scool ðtÞ and Sheat ðtÞÞ and the fraction of isolated amphiphiles (Icool ðtÞ and Iheat ðtÞ) for a temperature jump between T ¼ 2:3 and T ¼ 2:8. The average number of hydrogen bonds per water molecule in the bulk (nblk ðtÞ), second hydration shell (nss ðtÞ), pure water (npw ðtÞ), first hydration shell (nfs ðtÞ), and for all the water molecules of our system (ntot ðtÞ) for a temperature jump between T = 2.3 and 2.8. (–) indicates that two or three relaxation times are the same within the statistical errors.
ð1Þ
where t is the time in MCs, AðtÞ is the property being measured, and si are the relaxation times. Here, A0 ¼ Að1Þ and A1 þ A2 þ A3 ¼ Að0Þ Að1Þ. We calculate the average number of amphiphiles per micelle, the total solvent accessible surface area of the solute, and the number of hydrogen bonds (HBs) per water molecule in different regions of the solution, like the hydration shell of the micelles and in the solvent bulk. In Figure 1 we exhibit the average number of amphiphiles per micelle (mcool), and the total solvent accessible surface area of
a
Property
s1
s2
s3
mcool ðtÞ Scool ðtÞ mheat ðtÞ Sheat ðtÞ Icool ðtÞ Iheat ðtÞ Icool ðtÞa Iheat ðtÞa Cm nblk ðtÞ nss ðtÞ npw ðtÞ nfsh ðtÞ ntot ðtÞ
277 ± 35 149 ± 15 247 ± 16 262 ± 31 77 ± 6 203 ± 19 107 ± 27 82 ± 21 – 3.60 ± 0.05 4.10 ± 0.07 3.60 ± 0.09 4.50 ± 0.08 4.00 ± 0.06
6331 ± 518 3233 ± 232 3260 ± 129 4924 ± 107 787 ± 98 3691 ± 135 3242 ± 530 3246 ± 490 2704 ± 140 – – – 1594 ± 175 1339 ± 193
56969 ± 6272 38500 ± 2595 – – 9549 ± 882 – – – 40724 ± 2076 – – – 18665 ± 1875 33572 ± 4577
Between T ¼ 2:5 and T ¼ 2:6
G. Heinzelmann et al. / Chemical Physics Letters 550 (2012) 83–87
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We have also calculated the average lifetime of an isolated monomer in solution for T ¼ 2:3, and obtained a value of 115 MCs, which is comparable to the relaxation time of the fastest component. The latter time is consistent with the exchange of single monomers between aggregates, which is in accordance with the AW formalism. The reason behind the appearance of two other slower components, instead of the one predicted by the AW theory, is the large perturbation that we apply to the system. De Maeyer et al. [16] have performed simulations of a strong cooling from T = 80 to 25 °C, also from a non-micellized to a highly micellized state, and observed the appearance of a third, ultraslow component, in addition to the two already present in the AW model. This component is believed to arise from a transient state of the system after a large reduction in temperature, in which the concentration of monomers in solution quickly drops by their incorporation into the already existing micelles. The reassembly of normal sized micelles from these monomers would then take much longer due to their low availability in solution. This is supported by Figure 2a, where we show the behavior of the free amphiphiles during the cooling and heating processes for the same two temperatures T ¼ 2:3 and T ¼ 2:8. For instance, looking at the time interval between t = 0 and 200 MCs, we note that the decrease in the number of the free amphiphiles during the cooling process is significantly larger than its increase in the heating process. We observe in Figure 2b the result of a small
perturbation in the number of free amphiphiles. Now, the temperature changes from T = 2.5 and 2.6, and the third component is very small during the cooling process. The amplitude A3 is less than 0:5% and can be disregarded. These results corroborate the idea that the ultraslow time scale is due to the large jump in temperature. A recent calculation based on a free energy formalism [42] describes the kinetic of the surfactant micellization with three time scales. They assume a very large jump from infinite temperature, where all amphiphiles are free, to a state where the system becomes micellized. For this cooling process three different stages are identified: formation of the critical nuclei, time scale (s1 ), fast growth stage with the formation of intermediate aggregates, the so called premicellar state, time scale (s2 ), and final relaxation to thermodynamic equilibrium, time scale (s3 ). Thus, we propose that the two fastest components from our exponential fit come from the well-known AW mechanism, and the slowest one as a consequence of a sudden cooling of our system as seen in Ref. [16]). To confirm this hypothesis, we heated our system from equilibrium at T = 2.3 back to T ¼ 2:8, and noticed that, in this case, we could adjust both curves using a second order exponential fit, with relaxation times in the same order of magnitude of the two AW times of the cooling process (Table 1). In order to elucidate the larger timescales of the micellization dynamics, we have also determined the time evolution of intermediate size aggregates in the micellar formation for a large temperature jump. The intermediate aggregates are found in the region between the minimum and the maximum in the aggregate-size distribution curve (not shown). For the temperature T ¼ 2:3, the minimum is located at the size m ¼ 6, while the maximum is at m ¼ 20. As can be seen in Figure 3, the summed concentration C m , which represents the contributions to the concentration from aggregates in the range m = 9–13, increases with a characteristic time s2 reaching a maximum value, and then decreases to its equilibrium value with a timescale s3 . These characteristic times for a cooling between T = 2.8 and 2.3 are reported in Table 1, and are in agreement with the mechanism proposed by Hadgiivanova and coworkers [42] for the presence of the intermediate aggregates. We now turn to the hydrogen bonding of the water molecules of our system, following the temperature jump from T = 2.8 to 2.3. In the plot of Figure 4, we sketch two properties as a function of time: the average number of hydrogen bonds per water
Figure 2. Average volume fraction of isolated amphiphiles as a function of time for heating (squares) and cooling (triangles) between (a) T = 2.3 and 2.8, and between (b) T = 2.5 and 2.6.
Figure 3. Summed concentration of intermediate size aggregates (9 6 m 6 13) as a function of time for a cooling from T = 2.8 to 2.3 (squares), and the fit with Eq. (1) (continuous line).
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Figure 4. Average number hydrogen bonds in the solvent bulk (squares) and in the second hydration shell (triangles) as a function of time after a jump from T = 2.8 to 2.3. The fitting lines (Eq. (1)) are drawn over the data points, and both curves can be fitted to a single exponential.
Figure 5. Average number of hydrogen bonds for the overall water molecules (squares), and in the first hydration shell (triangles) as a function of time after a jump from T = 2.8 to 2.3. The fitting line is drawn over the data points, and both curves were fitted to a third-order exponential.
molecule in the solvent bulk ðnblk Þ, and the same for the second hydration shell of the solute ðnss Þ. Both are very similar, and can be fitted to a single exponential, with the relaxation times displayed in Table 1. The neat water hydrogen-bonding dynamics has time scales that are much faster than the aggregation process, with both relaxation times shorter than 5 MCs. In these two regions, water does not seem to be affected by the solute molecules, except for the time constant of second shell water molecules, which is slightly longer. Doing the jump in pure water, we obtained a time constant that is closer to the one from bulk water (Table 1). The average number of hydrogen bonds per water molecule in the first hydration shell of the micelles ðnfs Þ, and the average for all the water molecules ðntot Þ, are shown in Figure 5 as a function of time. They exhibit a contrasting behavior once they reach the time scales dominated by the micellar dynamics (t > 100 MCs), with an increase in the total number of HBs and a decrease in the HBs for the first-shell water molecules. The origin of this discrepancy is rooted in the hydrophobic effect, which is the main
drive responsible for aggregation. When the system is cooled, there is a reduction in the entropic term of the free energy, which leads to a lower enthalpy through an increase in the total number of hydrogen bonds in the system. There is a further increment in ntot ðtÞ brought by reducing the SASA, which is seen as the micellization process. When the aggregates are still small, first shell water molecules reorganize themselves, making hydrogen bonds around these structures, which gives us the peak in nfs ðtÞ around 100 MCs. As time goes by, the system starts sacrificing these hydrogen bonds locally in order to maximize them as a whole by aggregating the amphiphiles, and that is the reason for the reduction in nfs ðtÞ after 100 MCs. The three relaxation times obtained from the fitting of the two curves are shown in Table 1. The fast ones are very similar to the single relaxation times obtained for the bulk and second shell water molecules, so they are not likely to be connected to the micelle dynamics. The two slow relaxation times of each fit are of the same order of magnitude as the two slow times of the micellization process, which confirms that they are directly related. The fastest relaxation time of micellization has a small relative amplitude compared to the other two, being around 15%, so this is the reason they cannot be clearly seen in the fitting of the two curves of Figure 5. In conclusion, we have performed Monte Carlo simulations of a solution of amphiphiles in water, and observed the water hydrogen bonding and micellar dynamics after a quick cooling from a non-micellized to a micellized state. The micellization process is fitted by a third-order exponential expression with three distinct relaxation times. The two fastest ones can be related to the two relaxation times from the AW theory, whereas the ultraslow component is shown to be a consequence of the large perturbation caused by the temperature-jump on the system. The average number of hydrogen bonds in the first hydration shell of the micelles decreases with the micellar volume, since there is a reduction in the HBs with the growth of the micelles. On the other hand, the total number of hydrogen bonds increases with the total solvent accessible surface area of the solute, and this is the main driving force behind the aggregation of the amphiphiles. These mechanisms are consistent with the available theories on the hydrophobic effect, which also apply to micellar solutions. Our results give new insights into the dynamics of micellar aggregation in water, as well as the role of the hydrophobic effect on the self-assembly of amphiphilic molecules. As far as we know this the first study showing that the dynamics of micellization is closely related with the dynamics of the hydrogen bonds for a nonequilibrium dilute solution. Despite the most studies regarding the hydrophobic effect refer to thermodynamic equilibrium, our results give new insights into the dynamics of micellar aggregation in water, clarifying the role played by the hydrogen bonds, near and far the micellar aggregates, as the system evolves towards the equilibrium state. Acknowledgments The authors would like to acknowledge the Brazilian agency CNPq, and the University of Sydney, for the financial support. This Letter is also partially supported by INCT-FCX (FAPESP-CNPq) and the Australian Research Council. References [1] T.F. Tadros, Surfactants, Academic Press, London, 1985. [2] C. Tanford, Hydrophobic Effect: Formation of Micelles and Biological Membranes, Wiley, New York, 1980. [3] R. Zana (Ed.), Dynamics of Surfactant Self-Assemblies: Micelles, Microemulsions, Vesicles and Lyotropic Phases, CRC Press, Boca Haton, 2005. [4] D.R. Karsa, Industrial Applications of Surfactants, Royal Society of Chemistry, London, 1988.
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Contents lists available at SciVerse ScienceDirect
Chemical Physics Letters journal homepage: www.elsevier.com/locate/cplett
Micellar dynamics and water–water hydrogen-bonding from temperature-jump Monte Carlo simulations G. Heinzelmann a, W. Figueiredo b, M. Girardi c,⇑ a
School of Physics, University of Sydney, NSW 2006, Australia Departamento de Física, Universidade Federal de Santa Catarina, 88040-900 Florianópolis, Santa Catarina, Brazil c Universidade Federal de Santa Catarina, 88900-000 Araranguá, Santa Catarina, Brazil b
a r t i c l e
i n f o
Article history: Received 10 June 2012 In final form 6 September 2012 Available online 14 September 2012
a b s t r a c t We perform Monte Carlo simulations of a model for water–amphiphiles solution to observe the interplay between the formation of micelles and the hydrogen bond (HB) dynamics of the solvent. By performing temperature jumps from a non-micellized to a highly micellized state, we show that the dynamics can be described by three relaxation times. The analysis of HBs reveals an opposite behavior from the water molecules in the first hydration shell of the micelles and their total number indicating that the system is sacrificing HBs locally in order to increase them as a whole. Ó 2012 Elsevier B.V. All rights reserved.
Amphiphilic molecules, or surfactants, have the property of selfassembling into micelles in aqueous solutions mainly due to the hydrophobic effect [1–3]. They have a wide variety of applications, ranging from biological structures to domestic detergents and industrial processing [4]. Throughout the years, there has been a great number of experimental and theoretical studies on the equilibrium properties of micellar solutions, such as the size distribution curve, the shape of the micelles and the critical micelle concentration (CMC) [5–12]. Experiments, simulations and analytical studies have also been extensively used to probe the micellar dynamics [13–23]. Chemical relaxation experiments like ultrasonic relaxation, as well as concentration and temperature jump experiments, have been performed with different ionic and non-ionic surfactants. In such experiments, one applies a sudden change in the temperature or surfactant concentration and monitors the micelle behavior by measuring a property of the system, like conductivity, light scattering, fluorescence or surface tension. Most results show that, when a micellar system is perturbed, the path to equilibrium involves two distinct relaxation times: a fast one, in the order of microseconds, related to the entering/exiting of individual monomers from micelles, and a slow one, in the order of milliseconds, that involves the formation and breakdown of entire aggregates by cooperative condensation or dissolution of monomers. This result has also been corroborated by theoretical studies by Aniansson and Wall, and is known as the AW mechanism [24,25]. Nonetheless, other interpretations for the slow relaxation time have been suggested, like the formation and subsequent ⇑ Corresponding author. E-mail addresses: [email protected] (G. Heinzelmann), wagner@ fisica.ufsc.br (W. Figueiredo), [email protected] (M. Girardi). 0009-2614/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.cplett.2012.09.011
breakdown of a transient supermicelle, and the temporary fragmentation of a normal micelle into two submicelles, which grow back by fusing together or by incorporating free monomers in the solution [14,21]. Recent experimental and theoretical studies have also shown the possibility of a third relaxation time that, depending on the system, can be ultraslow, ultrafast or in between the fast and slow relaxation times [15–17]. Similarly, the hydrophobic effect and related phenomena have received much attention through the study of hydrophobic and amphiphilic molecules immersed in water [26–34]. These studies indicate that, for small hydrophobic molecules, water can reorganize around the solute without sacrificing hydrogen bonds, and the entropic cost of this process leads to their low solubility. For a small number of these molecules, water tends to segregate such species rather than drive them together, and the free energy of solvation grows linearly with the solvent volume. On the other hand, close to a larger hydrophobic object, the maintenance of a hydrogen-bond network is geometrically impossible. In this case the free energy of solvation grows with the solvent accessible surface area (SASA). In other words, the free energy of solvation of a large number of separated hydrophobic molecules in water exceeds that of a large cluster of the same species. This is the force behind the oil–water phase separation. The crossover between the two regimes happens when the SASA is larger than the nm2 scale. Maibaum et al. [34] have demonstrated that the same principle can be applied to the aggregation of surfactants in solution and the formation of micelles. Here, due to the restriction of having a polar group right next to a hydrophobic one, the growth of the amphiphilic assemblies is limited to mesoscopic domains, as opposed to the macroscopic oil–water separation. An expression for the value of the CMC as a function of the temperature and
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amphiphilic tail length is obtained, with good agreement with experimental results. Even though the dynamics of both solvent hydrogen bonding and amphiphilic aggregation are closely related, it is very hard to observe the interplay between them due to the contrasting time scales involved. While the time scales related to hydrogen bonding are in the order of picoseconds [35,36], the relaxation times of micelle formation/breakdown are in the microseconds and milliseconds scale, well above what can be obtained using all-atomic molecular dynamics. Thus, in the present Letter, we attempt to combine these two phenomena through temperature-jump simulations, observing how the solvent hydrogen-bonding is related to the micelle dynamics when we go from a non-micellized to a highly micellized state. We make use of a well established lattice model previously considered in Refs. [37–40], which allows for the orientational freedom of solvent molecules as well as the formation of hydrogen bonds between them. Our model system considers a body-centered cubic (BCC) lattice with periodic boundary conditions, filled by single-site bonding solvent molecules and by self-avoiding chains of connected sites representing the amphiphiles. We use a four-site flexible amphiphile chain with three sites representing the hydrophobic tail and one site representing the hydrophilic head (H1T3). Concerning the water molecules, each one has four bonding arms in a tetrahedral geometry, trying to mimic the actual water geometry. When two bonding arms from neighboring water molecules are pointing to each other we have a hydrogen bond, with a energy value of c ¼ 2. There is also a repulsive interaction of e ¼ 1 for all pairs of nearest-neighboring water molecules. In the case of amphiphiles, the interaction parameters apply to pairs of nearest-neighbors regardless of the orientation, and are set to obtain micellized and non-micellized systems in the range of temperatures considered. We have the following set of interaction energies: EHH ¼ 7e; EHT ¼ 3e, EHS ¼ 10e; ETS ¼ ETT ¼ e; subscripts are (S) for solvent molecules, (T) for a hydrophobic tail segment and (H) for a hydrophilic head. The system has a linear lattice size of L ¼ 30 with a volume fraction of amphiphiles at U ¼ 0:05 and the remaining sites filled with water. The simulations were carried out by using the Metropolis algorithm [41] where, at each Monte Carlo step (MCs), we randomly attempt to change the orientation of all the water molecules, as well as move each amphiphile through the reptation movement. The goal of the present Letter is to use temperature-jump simulations to observe the interplay between the micellar dynamics and the hydrogen-bond dynamics, in order to study the hydrophobic effect and its influence on amphiphilic aggregation. To do so, we equilibrated the system in a non-micellized state at temperature T ¼ 2:8 (in units of e=kB ) for 5 105 MCs, and then performed a quick cooling to T ¼ 2:3, in which we have a highly micellized state, measuring the properties of the solution as a function of time. Our results are obtained from the averages over eighty samples, and we also performed the heating process for comparison. All the dynamic system properties are well fitted to single or multiple exponentials, in which we use the equation:
AðtÞ ¼ A0 þ A1 expðt=s1 Þ þ A2 expðt=s2 Þ þ A3 expðt=s3 Þ;
Figure 1. Average number of amphiphiles per micelle (squares) and total solvent accessible surface area of the micelles (triangles) as a function of time after a jump from T = 2.8 to 2.3. The fitting lines (Eq. (1)) are drawn over the data points.
the aggregates Scool) as a function of time, after a temperature jump from T = 2.8 to 2.3. The total SASA is directly proportional to the percentage of water molecules in the first hydration shell of the solute, so we obtain the former by calculating this percentage and multiplying it by 20 for a convenient visualization. We see a close relation between the two properties, i.e., the larger the micelles the smaller the sum of their surface areas exposed to the solvent. We have fitted both curves to the third-order exponential expression of Eq. (1), with the fitting line drawn in Figure 1, and the relaxation times shown in Table 1. We always attempted to fit all the data with three exponentials. However, we have seen that in some cases, two or more relaxation times were so close together that the resultant fit is adjusted with a single or two exponentials. Indeed, a given fit is accepted only when the coefficient of correlation is larger than 0.99. The error bars were omitted in the plots for better visualization and are typically around 1%. The errors in the properties are reported in the values of the corresponding relaxation times in Table 1. The three time constants are comparable in the two cases, confirming the observation that these two properties share the same dependence over time. Table 1 Relaxation times ðsi Þ in MCs, obtained by the fit with Eq. (1) for the temperaturejump simulations. The average number of amphiphiles per micelle during the cooling (mcool ðtÞ) and the heating (mheat ðtÞ) processes, the SASA of the solute (Scool ðtÞ and Sheat ðtÞÞ and the fraction of isolated amphiphiles (Icool ðtÞ and Iheat ðtÞ) for a temperature jump between T ¼ 2:3 and T ¼ 2:8. The average number of hydrogen bonds per water molecule in the bulk (nblk ðtÞ), second hydration shell (nss ðtÞ), pure water (npw ðtÞ), first hydration shell (nfs ðtÞ), and for all the water molecules of our system (ntot ðtÞ) for a temperature jump between T = 2.3 and 2.8. (–) indicates that two or three relaxation times are the same within the statistical errors.
ð1Þ
where t is the time in MCs, AðtÞ is the property being measured, and si are the relaxation times. Here, A0 ¼ Að1Þ and A1 þ A2 þ A3 ¼ Að0Þ Að1Þ. We calculate the average number of amphiphiles per micelle, the total solvent accessible surface area of the solute, and the number of hydrogen bonds (HBs) per water molecule in different regions of the solution, like the hydration shell of the micelles and in the solvent bulk. In Figure 1 we exhibit the average number of amphiphiles per micelle (mcool), and the total solvent accessible surface area of
a
Property
s1
s2
s3
mcool ðtÞ Scool ðtÞ mheat ðtÞ Sheat ðtÞ Icool ðtÞ Iheat ðtÞ Icool ðtÞa Iheat ðtÞa Cm nblk ðtÞ nss ðtÞ npw ðtÞ nfsh ðtÞ ntot ðtÞ
277 ± 35 149 ± 15 247 ± 16 262 ± 31 77 ± 6 203 ± 19 107 ± 27 82 ± 21 – 3.60 ± 0.05 4.10 ± 0.07 3.60 ± 0.09 4.50 ± 0.08 4.00 ± 0.06
6331 ± 518 3233 ± 232 3260 ± 129 4924 ± 107 787 ± 98 3691 ± 135 3242 ± 530 3246 ± 490 2704 ± 140 – – – 1594 ± 175 1339 ± 193
56969 ± 6272 38500 ± 2595 – – 9549 ± 882 – – – 40724 ± 2076 – – – 18665 ± 1875 33572 ± 4577
Between T ¼ 2:5 and T ¼ 2:6
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We have also calculated the average lifetime of an isolated monomer in solution for T ¼ 2:3, and obtained a value of 115 MCs, which is comparable to the relaxation time of the fastest component. The latter time is consistent with the exchange of single monomers between aggregates, which is in accordance with the AW formalism. The reason behind the appearance of two other slower components, instead of the one predicted by the AW theory, is the large perturbation that we apply to the system. De Maeyer et al. [16] have performed simulations of a strong cooling from T = 80 to 25 °C, also from a non-micellized to a highly micellized state, and observed the appearance of a third, ultraslow component, in addition to the two already present in the AW model. This component is believed to arise from a transient state of the system after a large reduction in temperature, in which the concentration of monomers in solution quickly drops by their incorporation into the already existing micelles. The reassembly of normal sized micelles from these monomers would then take much longer due to their low availability in solution. This is supported by Figure 2a, where we show the behavior of the free amphiphiles during the cooling and heating processes for the same two temperatures T ¼ 2:3 and T ¼ 2:8. For instance, looking at the time interval between t = 0 and 200 MCs, we note that the decrease in the number of the free amphiphiles during the cooling process is significantly larger than its increase in the heating process. We observe in Figure 2b the result of a small
perturbation in the number of free amphiphiles. Now, the temperature changes from T = 2.5 and 2.6, and the third component is very small during the cooling process. The amplitude A3 is less than 0:5% and can be disregarded. These results corroborate the idea that the ultraslow time scale is due to the large jump in temperature. A recent calculation based on a free energy formalism [42] describes the kinetic of the surfactant micellization with three time scales. They assume a very large jump from infinite temperature, where all amphiphiles are free, to a state where the system becomes micellized. For this cooling process three different stages are identified: formation of the critical nuclei, time scale (s1 ), fast growth stage with the formation of intermediate aggregates, the so called premicellar state, time scale (s2 ), and final relaxation to thermodynamic equilibrium, time scale (s3 ). Thus, we propose that the two fastest components from our exponential fit come from the well-known AW mechanism, and the slowest one as a consequence of a sudden cooling of our system as seen in Ref. [16]). To confirm this hypothesis, we heated our system from equilibrium at T = 2.3 back to T ¼ 2:8, and noticed that, in this case, we could adjust both curves using a second order exponential fit, with relaxation times in the same order of magnitude of the two AW times of the cooling process (Table 1). In order to elucidate the larger timescales of the micellization dynamics, we have also determined the time evolution of intermediate size aggregates in the micellar formation for a large temperature jump. The intermediate aggregates are found in the region between the minimum and the maximum in the aggregate-size distribution curve (not shown). For the temperature T ¼ 2:3, the minimum is located at the size m ¼ 6, while the maximum is at m ¼ 20. As can be seen in Figure 3, the summed concentration C m , which represents the contributions to the concentration from aggregates in the range m = 9–13, increases with a characteristic time s2 reaching a maximum value, and then decreases to its equilibrium value with a timescale s3 . These characteristic times for a cooling between T = 2.8 and 2.3 are reported in Table 1, and are in agreement with the mechanism proposed by Hadgiivanova and coworkers [42] for the presence of the intermediate aggregates. We now turn to the hydrogen bonding of the water molecules of our system, following the temperature jump from T = 2.8 to 2.3. In the plot of Figure 4, we sketch two properties as a function of time: the average number of hydrogen bonds per water
Figure 2. Average volume fraction of isolated amphiphiles as a function of time for heating (squares) and cooling (triangles) between (a) T = 2.3 and 2.8, and between (b) T = 2.5 and 2.6.
Figure 3. Summed concentration of intermediate size aggregates (9 6 m 6 13) as a function of time for a cooling from T = 2.8 to 2.3 (squares), and the fit with Eq. (1) (continuous line).
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Figure 4. Average number hydrogen bonds in the solvent bulk (squares) and in the second hydration shell (triangles) as a function of time after a jump from T = 2.8 to 2.3. The fitting lines (Eq. (1)) are drawn over the data points, and both curves can be fitted to a single exponential.
Figure 5. Average number of hydrogen bonds for the overall water molecules (squares), and in the first hydration shell (triangles) as a function of time after a jump from T = 2.8 to 2.3. The fitting line is drawn over the data points, and both curves were fitted to a third-order exponential.
molecule in the solvent bulk ðnblk Þ, and the same for the second hydration shell of the solute ðnss Þ. Both are very similar, and can be fitted to a single exponential, with the relaxation times displayed in Table 1. The neat water hydrogen-bonding dynamics has time scales that are much faster than the aggregation process, with both relaxation times shorter than 5 MCs. In these two regions, water does not seem to be affected by the solute molecules, except for the time constant of second shell water molecules, which is slightly longer. Doing the jump in pure water, we obtained a time constant that is closer to the one from bulk water (Table 1). The average number of hydrogen bonds per water molecule in the first hydration shell of the micelles ðnfs Þ, and the average for all the water molecules ðntot Þ, are shown in Figure 5 as a function of time. They exhibit a contrasting behavior once they reach the time scales dominated by the micellar dynamics (t > 100 MCs), with an increase in the total number of HBs and a decrease in the HBs for the first-shell water molecules. The origin of this discrepancy is rooted in the hydrophobic effect, which is the main
drive responsible for aggregation. When the system is cooled, there is a reduction in the entropic term of the free energy, which leads to a lower enthalpy through an increase in the total number of hydrogen bonds in the system. There is a further increment in ntot ðtÞ brought by reducing the SASA, which is seen as the micellization process. When the aggregates are still small, first shell water molecules reorganize themselves, making hydrogen bonds around these structures, which gives us the peak in nfs ðtÞ around 100 MCs. As time goes by, the system starts sacrificing these hydrogen bonds locally in order to maximize them as a whole by aggregating the amphiphiles, and that is the reason for the reduction in nfs ðtÞ after 100 MCs. The three relaxation times obtained from the fitting of the two curves are shown in Table 1. The fast ones are very similar to the single relaxation times obtained for the bulk and second shell water molecules, so they are not likely to be connected to the micelle dynamics. The two slow relaxation times of each fit are of the same order of magnitude as the two slow times of the micellization process, which confirms that they are directly related. The fastest relaxation time of micellization has a small relative amplitude compared to the other two, being around 15%, so this is the reason they cannot be clearly seen in the fitting of the two curves of Figure 5. In conclusion, we have performed Monte Carlo simulations of a solution of amphiphiles in water, and observed the water hydrogen bonding and micellar dynamics after a quick cooling from a non-micellized to a micellized state. The micellization process is fitted by a third-order exponential expression with three distinct relaxation times. The two fastest ones can be related to the two relaxation times from the AW theory, whereas the ultraslow component is shown to be a consequence of the large perturbation caused by the temperature-jump on the system. The average number of hydrogen bonds in the first hydration shell of the micelles decreases with the micellar volume, since there is a reduction in the HBs with the growth of the micelles. On the other hand, the total number of hydrogen bonds increases with the total solvent accessible surface area of the solute, and this is the main driving force behind the aggregation of the amphiphiles. These mechanisms are consistent with the available theories on the hydrophobic effect, which also apply to micellar solutions. Our results give new insights into the dynamics of micellar aggregation in water, as well as the role of the hydrophobic effect on the self-assembly of amphiphilic molecules. As far as we know this the first study showing that the dynamics of micellization is closely related with the dynamics of the hydrogen bonds for a nonequilibrium dilute solution. Despite the most studies regarding the hydrophobic effect refer to thermodynamic equilibrium, our results give new insights into the dynamics of micellar aggregation in water, clarifying the role played by the hydrogen bonds, near and far the micellar aggregates, as the system evolves towards the equilibrium state. Acknowledgments The authors would like to acknowledge the Brazilian agency CNPq, and the University of Sydney, for the financial support. This Letter is also partially supported by INCT-FCX (FAPESP-CNPq) and the Australian Research Council. References [1] T.F. Tadros, Surfactants, Academic Press, London, 1985. [2] C. Tanford, Hydrophobic Effect: Formation of Micelles and Biological Membranes, Wiley, New York, 1980. [3] R. Zana (Ed.), Dynamics of Surfactant Self-Assemblies: Micelles, Microemulsions, Vesicles and Lyotropic Phases, CRC Press, Boca Haton, 2005. [4] D.R. Karsa, Industrial Applications of Surfactants, Royal Society of Chemistry, London, 1988.
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