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Langmuir 2006, 22, 6514-6522
Effect of Stiffness on the Micellization Behavior of Model H4T4 Surfactant Chains Vanessa Firetto and Michele A. Floriano* Dipartimento di Chimica Fisica “F. Accascina”, UniVersita` di Palermo, Viale delle Scienze, Parco d’Orleans, Ed. 17, 90128 Palermo, Italy
Athanassios Z. Panagiotopoulos Department of Chemical Engineering, Princeton UniVersity, Princeton, New Jersey 08544-5263 ReceiVed February 9, 2006. In Final Form: April 28, 2006 The micellization behavior of a series of model surfactants, all with four head and tail groups (H4T4) but with different degrees of chain stiffness, was studied using grand canonical Monte Carlo simulations on a cubic lattice. The critical micelle concentration, micellar size, and thermodynamics of micellization were examined. In all cases investigated, the critical micelle concentration was found to increase with increasing temperature as observed for nonionic surfactants in apolar or slightly polar solvents. At a fixed reduced temperature and increasing chain stiffness, in agreement with previous observations, it was found that the critical micelle concentration decreased and the average micelle size increased. This behavior is qualitatively consistent with that experimentally observed when comparing hydrogenated and homologous fluorinated surfactants. Thermodynamic considerations based on the analysis of the temperature dependence of the critical micelle concentration indicated that both effects could be interpreted as arising from an increased number of heterocontacts between hydrophobic portions of stiff surfactants and a lower entropic cost on packing rigid chains. Structural analysis that was also based on considering the inner micellar radial dependence of the surfactant head and tail site fraction distributions suggested that, on stiffening the molecular backbone, the resulting micellar aggregates grew, without appreciable deviations from spherical symmetry. Stiffer surfactants led to a slightly denser micellar core because of better packing.
Introduction Interest in the self-assembly properties of amphiphilic molecules is motivated by numerous applications in industry and science. A variety of products such as detergents, emulsifiers, catalysts, and vesicles have a range of useful properties because of the ability of amphiphilic molecules to self-assemble in solution into organized structures. The self-assembly process is driven by the amphiphilic character of the molecular building blocks (i.e., different chemical groups in the molecules exhibit different solvent affinities). The simplest organized structure is a micelle, which is formed in order to minimize unfavorable interactions between the solvent medium and the poorly solvated moieties of the amphiphile. A considerable number of studies have concentrated on the micellization behavior of surfactant solutions either by experiments,1-10 theories,11-16 or computer simulations17-42 with the aim of elucidating the connection between surfactant molecular architecture and the structure of the resulting aggregates. The vast majority of studies have been focused on the effects of changing the surfactant chain length. For example, Maibaum et al.16 formulated a theory for micelle assembly. These theoretical * Corresponding author. E-mail:
[email protected]. (1) Corkill, J. M.; Goodman, J. F.; Harrold, S. P. Trans. Faraday Soc. 1964, 60, 202. (2) Shinoda, K.; Yamanaka, T.; Kinoshita, K. J. Phys. Chem. 1959, 63, 648. (3) Kon-no, K.; Jin-no, T.; Kitahara, A. J. Colloid Interface Sci. 1974, 49, 383. (4) Rosen, M. J.; Cohen, A. W.; Dahanayake, M.; Hua, X. Y. J. Phys. Chem. 1982, 86, 541. (5) Kwan, C.-C.; Rosen, M. J. J. Phys. Chem. 1980, 84, 547. (6) Herrington, T. M.; Sahi, S. S. Colloids Surf. 1986, 17, 103. (7) Ravey, J. C.; Gherby, A.; Ste´be´, M. J. Prog. Colloid Polym. Sci. 1988, 76, 234. (8) Matos, L.; Ravey, J. C.; Serratrie, G. J. Colloid Interface Sci.1989, 128, 341. (9) Ravey, J. C.; Ste´be´, M. Colloids Surf., A 1994, 84, 11.
predictions for the temperature and surfactant chain length dependence of the critical micelle concentrations (cmc’s) for nonionic surfactants were in good agreement with experimental observations. Rodriguez et al.32 have obtained the cmc’s of different amphiphiles by Monte Carlo simulations. These were compared with the experimental results reported in the literature for alkyl ethoxylate surfactants, and it was seen that the free (10) Ste´be´, M. J.; Istranov, V.; Langenfeld, A.; Vasnev, V. A.; Babak, V. G. J. Fluorine Chem. 2003, 119, 191. (11) Stillinger, F. H. J. Chem. Phys. 1983, 78, 4654. (12) Puvvada, S.; Blankschtein, D. J. Chem. Phys. 1990, 92, 3710. (13) Nagarajan, R.; Ruckenstein, E. Langmuir 1991, 7, 2934. (14) Israelachvili, J. N. Intermolecular and Surface Forces, 2nd ed.; Academic Press: London, 1992. (15) Mackie, A. D.; Panagiotopoulos, A. Z.; Szleifer, I. Langmuir 1997, 13, 5022. (16) Maibaum, L.; Dinner, A. R.; Chandler, D. J. Phys. Chem. B 2004, 108, 6778. (17) Larson, R. G. J. Chem. Phys. 1988, 89, 1642. (18) Larson, R. G. J. Chem. Phys. 1989, 91, 2479. (19) Larson, R. G. J. Chem. Phys. 1992, 96, 7904. (20) 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. (21) Smit, B.; Esselink, K.; Hilbers, P. A. J.; Van Os, N. M.; Rupert, L. A. M.; Szleifer, I. Langmuir 1993, 9, 9. (22) Smit, B.; Hilbers, P. A. J.; Esselink, K.; Rupert, L. A. M.; Van Os, N. M.; Schlijper, A. G. Nature 1990, 348, 624. (23) Wu, D.; Chandler, D.; Smit, B. J. Phys. Chem. 1992, 96, 4077. (24) Wang, Y.; Mattice, W. L.; Napper, D. H. Macromolecules 1992, 25, 4073. (25) Brindle, D.; Care, C. M. J. Chem. Soc., Faraday Trans. 1992, 88(15), 2163. (26) Adriani, P.; Wang, Y.; Mattice, W. L. J. Chem. Phys. 1994, 100, 7718. (27) Nguyen-Misra, M.; Mattice, W. L. Macromolecules 1995, 28, 1444. (28) Linse, P.; Wijmans, C. M. Langmuir 1995, 11, 3748. (29) Desplat, J.-C.; Care, C. M. Mol Phys. 1996, 87, 441. (30) Mackie, A. D.; Onur, K.; Panagiotopoulos, A. Z. J. Chem. Phys. 1996, 104, 3718. (31) Huibers, P. D. T.; Lobanov, V. S.; Katritzky, A. R.; Shah, D. O.; Karelson, M. Langmuir 1996, 12, 1462. (32) Rodrı´guez-Guadarrama, L. A.; Talsania, S. K.; Mohanty, K. K.; Rajagopalan, R. Langmuir 1999, 15, 437.
10.1021/la060386c CCC: $33.50 © 2006 American Chemical Society Published on Web 06/21/2006
Micellization BehaVior of H4T4 Surfactant Chains
energies were similar to those obtained from the simulations. Bhattacharya et al.42 reported the results of a detailed investigation of the dependence of the cmc on temperature for different lattice models of amphiphile self-assembly. They showed that for amphiphiles of different chain lengths and different head-to-tail length ratios the cmc decreased rapidly on increasing the chain length, in agreement with the experimental results. In our previous studies,35,39 a methodology was developed that allows a clear distinction between micellization and macroscopic phase separation. The dependence of cmc on chain length was obtained for fully flexible model surfactants. In the present study, we consider another parameter that controls the architecture of the amphiphile chain, namely, the degree of molecular flexibility. In fact, changes in the chemical composition of the surfactant molecules, because of internal geometric constraints and different intramolecular interactions, might possibly vary the bending ability of the surfactant backbone, and this in turn might influence its aggregation behavior and the structure of the resulting micellar aggregates. In fact, different solution behavior between hydrogenated and perfluorinated surfactants has been observed.7-10 This has been attributed to the differences in the nature of the hydrophobic moiety. Perfluorinated molecules show weaker hydrophilic character and higher thermal and chemical stability than the hydrogenated molecules, and these properties facilitate the formation of micellar aggregates and microemulsions. Moreover, at a given temperature, the critical micelle concentrations of fluorinated nonionic surfactants are smaller than those of their hydrogenated counterparts with the same number of carbon atoms in the hydrophobic tail. There are only a few prior studies on the effects of chain stiffness on the aggregation properties. Molecular simulations have been useful in understanding the influence of changes in molecular architecture on micellization. For example, Adriani et al.26 studied the influence of chain stiffness on the micellization of block copolymers in a selective solvent by Monte Carlo simulations. It was found that, when the stiffness of the insoluble block increases, the critical micelle concentration decreases and the average micelle size increases. However, an increase in the stiffness of the soluble block does not affect the micellization process to any appreciable extent. The decrease in the cmc when the stiffness of the insoluble block increases is attributable to an increased number of heterocontacts between the insoluble block and the solvent molecules. Liu et al.41 studied the effects of surfactant stiffness on the formation of aggregates. It was found that enhancing the stiffness of the surfactant chain leads to an increase in the mean end-to-end length. As a result, the aggregation degree increases. However, the correlation between molecular architecture and the details of the aggregation phenomenon has not yet been completely elucidated. In this work, we present a detailed Monte Carlo simulation study on a model surfactant in which systematic changes in the degree of stiffness were introduced both in the (33) Girardi, M.; Figueiredo, W. J. Chem. Phys. 1999, 112, 4833. (34) Care, C. M.; Dalby, T. Europhys. Lett. 1999, 45, 38. (35) Floriano, M. A.; Caponetti, E.; Panagiotopoulos, A. Z. Langmuir 1999, 15, 3143. (36) Nagarajan, R.; Wang, C.-C. Langmuir 2000, 16, 5242. (37) Marrink, S. J.; Tieleman, D. P.; Mark, A. E. J. Phys. Chem. B 2000, 104, 12165. (38) Kusaka, I.; Oxtoby, D. W. J. Chem. Phys. 2001, 115, 4883. (39) Panagiotopoulos, A. Z.; Floriano, M. A.; Kumar, S. K. Langmuir 2002, 18, 2940. (40) Pool, R.; Bolhuis, P. G. J. Phys. Chem. B 2005, 109, 6650. (41) Liu, H.-Y.; Zou, X.-W.; Yuan, Y.-Q.; Jin, Z.-Z. Eur. Phys. J. E 2002, 8, 373. (42) Bhattacharya, A.; Mahanti, S. D. J. Phys.: Condens. Matter 2001, 13, L861.
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whole chain and, independently, either in the solvophobic or in the solvophilic portion of the chain. The simulations were performed in the grand canonical ensemble and were combined with histogram reweighting techniques.43 The above methodology has proven itself capable of providing, in an efficient way, critical micelle concentrations and micellar thermodynamic and structural properties on varying the chain stiffness.
Model and Methods The simulations were performed on a cubic lattice of dimension L3 with periodic boundary conditions in all three directions. Solvent molecules, S, occupy single sites, and surfactant molecules, denoted by H4T4 and consisting of 4 solvophilic head units, H, and 4 solvophobic tail units, T, occupy 8 connected sites. Each of these molecules interacts with nearest and nextnearest neighbors at (x, y, z) ) (0, 0, 1), (0, 1, 1), and (1, 1, 1) plus all vectors resulting from symmetry operations along the x, y, and z axes, resulting in a coordination number, z, of 26. This surfactant model was originally proposed by Larson.17 Each bead pair is assigned an interaction energy, ij, where i, j ) H, T, S. The relevant energy scale, , is
1 ) ij - (ii + jj) 2
(1)
HH, HT, HS, SS, and TS were assumed to be zero, and TT was set equal to -2. We use the dimensionless reduced temperature, T*, defined with respect to the unit of interaction energies as T* ) -2kBT/TT, where kB is Boltzmann’s constant. These reduced units are the same as those used in previous studies of surfactant self-assembly in our group.35 The possibility of adjusting the chain stiffness was implemented, as previously done for homopolymer chains,44 by adding to the configurational energy a term, Ubend, defined by the following equation
Ubend ) R(1 + cos θ)
(2)
where θ is the angle between three consecutive beads of the molecular backbone and R is a parameter controlling the stiffness of the chain; R ) 0 corresponds to a fully flexible chain whereas progressively larger values make the chains stiffer, all the way to rodlike molecules. All simulations were started from a random initial configuration containing two chains, corresponding to a very low surfactant volume fraction. The moves employed in our simulations to sample phase space efficiently were transfers and reptations. We also implemented cluster moves to relocate micellar aggregates efficiently. In these kinds of moves, whole clusters are shifted by one site in a random direction in a manner that maintains detailed balance. An amphiphile is considered to be part of a cluster if any of its tail segments is in a neighboring site of a tail segment belonging to another amphiphile. Configurational-bias sampling methods45 were used to facilitate insertions and removals of surfactant molecules. Run lengths ranged from 106 to 109 Monte Carlo steps, depending on density, system size, and chain stiffness. CPU time requirements on 1.8 GHz Pentium 4 processors were 1CPU h/107 attempted Monte Carlo moves. (43) Ferrenberg, A. M.; Swendsen, R. H. Phys. ReV. Lett. 1988, 61, 2635. (44) Firetto, V; Floriano, M. A.; Panagiotopoulos, A. Z. Macromolecules 2005, 38, 2475. (45) Frenkel, D.; Mooij, G. C. A. M.; Smit, B. J. Phys.: Condens. Matter 1992, 4, 3053.
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Firetto et al.
The statistical uncertainties were calculated by averaging results from four independent simulations performed under the same conditions for different initial seeds of the random sequence. Simulation runs were performed at a series of different temperatures, T*, and chemical potentials, µ, for each R value. Histograms of the frequencies of occurrence of total amphiphile numbers and total system energy were combined and reweighted to obtain estimates of the grand partition function and hence the compressibility factor, as a function of the temperature and imposed chemical potential
PV kBT
) ln Ξ ) ln
(∑
)
e(Nµ - E)/kBT
N,E
(3)
where Ξ is the grand canonical partition function. More details on this methodology are given in refs 35 and 39. As previously suggested,35 critical micelle concentrations (φcmc) were identified in osmotic pressure versus overall surfactant volume fraction plots as the point at which the ideal line of unit slope and the second straight segment intersect. Furthermore, as previously suggested,35 with increasing temperature, aggregates become quite small and unstable. As a consequence, in this work a critical micelle temperature has been arbitrarily introduced and defined, as previously done,35 as the temperature where the slope of the second straight segment in the regions where P* versus φ is equal to 0.1.35
Results and Discussion In this work, the micellization behavior of the surfactant H4T4 was studied at R ) 0, 10, 20, 50, 100, 300, and 500. The R values were chosen in such a way as to increase the chain stiffness gradually and thereby change the average chain conformations from fully flexible chains to semiflexible ones all the way to stiff rods. Moreover, to further elucidate the effect of stiffness on the aggregation behavior and to make the model surfactant more realistic, the possibility of changing the degree of rigidity of the H and T portions of the chain separately has been introduced. To achieve this objective, two stiffness parameters, RH and RT, were used for the head and tail portions, respectively. We will indicate with RH-T the two different RH and RT values. For example, R0-20 indicates RH ) 0 and RT ) 20. For these systems, the following stiffness parameters were used: RH-T ) 0-10, 0-20, 0-50, 0-500, 20-0, 50-0, and 500-0. An estimate of the chain stiffness can be obtained from the mean end-to-end distance, 〈l〉, calculated for representative dilute configurations as
〈l〉 )
1 n
n
∑x(x8 - x1)2 + (y8 - y1)2 + (z8 - z1)2
(4)
where n is the number of different chains in the considered configuration and x, y, and z are the lattice coordinates of the last (8) and first (1) beads, respectively, in each chain. The values of 〈l〉, calculated at the different R values at temperatures T* ) 0.95T/cm, are reported in Figure 1. It can be observed that, on increasing the stiffness parameter, the chains become elongated and their length approaches a constant value. The lengths of completely extended chains consisting of eight beads and oriented along the (0, 0, 1), (0, 1, 1), and (1, 1, 1) directions are (8-1), (8-1)x2, and (8-1)x3 lattice units, respectively. Assuming that all 26 possible orientations are equally likely, the resulting effective length, lfe, of a fully extended chain can be considered to be the average of the
Figure 1. Mean end-to-end distance, 〈l〉, at different R (9) and R0-T (O) values. The lines are simply guides for the eye.
three possible lengths, that is, lfe ) 9.9. As shown in Figure 1, for R > 100 the mean end-to-end distance is equal to the predicted fully extended length; therefore, the surfactant chains can be considered to be stiff rods. A constant value is also obtained for R0-T > 100, but this value is smaller than lfe ) 9.9 because only one portion of the chain has been stiffened. To demonstrate the induced changes in the conformations, in Figure 2 are shown typical configurations obtained at the extreme R and RH-T values. The T/cm and φcmc values, obtained as described in the previous section at different reduced temperatures for all considered R and RH-T values, are reported in Tables 1 and 2. It has been demonstrated in previous work39 that the box size (L ) 15) used in the present study is sufficient to minimize finite size effects. For large R values and, consequently, longer chains, proportionally larger simulations cells (L ) 20 and 25) were also used. Statistical uncertainties were estimated by averaging results from four independent simulations performed under the same conditions and different initial seeds of the random sequence. The four φcmc values were averaged, and the reported uncertainties represent deviations from the mean value. Uncertainties arising from the estimate of the φcmc values as intersections of the two linear portions of P* versus φ plots as reported in the previous section were about 1 order of magnitude smaller than statistical uncertainties in all cases. Statistical uncertainties in the critical micelle temperatures were obtained as described in the previous section by considering regions of P* versus φ with a slope equal to 0.100 ( 0.004. These were calculated by averaging results from four independent sets of simulations obtained using different initial seeds of the random sequence; the uncertainties in Tables 1 and 2 represent deviations from the mean value. Statistical uncertainties are reported in the Tables in parentheses in units of the last decimal figure shown. Increasing surfactant stiffness shifts up the temperature range in which micellization occurs. It is well known46 that below a certain temperature a surfactant solution will tend to phase separate; therefore, if just the micellization process is of interest, the temperature cannot be lowered too much. However, on increasing the temperature, micelles become progressively less thermodynamically stable, and above T/cm, it is no longer possible to detect stable aggregates. As a consequence, micelles can be observed and studied only over a rather limited temperature range. From Figure 3, it can be seen that the critical micelle temperature increases with R and RH-T and reaches a constant (46) Hiemenz, P. C.; Rajagolpalan, R. Principles of Colloid and Surface Chemistry, 3rd ed.; Marcel Dekker: New York, 1997.
Micellization BehaVior of H4T4 Surfactant Chains
Langmuir, Vol. 22, No. 15, 2006 6517
Figure 2. Sample snapshot configurations for H4T4 obtained under the following conditions: R ) 500 (left) and RH-T ) 0-500 (right). In both cases, the simulation cell contained about 100 chains at T* ) 0.95T/cm. Table 1. Critical Micelle Concentration, φcmc, at Different Reduced Normalized Temperatures T/r ) T*/T/cm Where T/cm Is the Critical Micelle Temperaturea for H4T4 at Different Values of the Stiffness Parameter r φcmc R
T/cm
T/r ) 0.76
T/r ) 0.78
T/r ) 0.80
T/r ) 0.82
T/r ) 0.84
T/r ) 0.86
T/r ) 0.88
T/r ) 0.90
0 10 20 50 100 300 500
8.25(6) 9.10(1) 9.63(5) 10.65(6) 11.16(6) 11.42(3) 11.36(5)
0.0032(1) 0.0039(4) 0.0051(4) 0.0069(5) 0.0085(2) 0.0089(2) 0.0091(3)
0.0042(2) 0.0051(7) 0.0064(4) 0.0086(1) 0.0104(3) 0.0107(1) 0.0111(1)
0.0053(6) 0.0064(7) 0.0079(5) 0.0106(2) 0.0128(4) 0.0134(2) 0.0139(1)
0.0069(3) 0.0080(1) 0.0099(6) 0.0128(4) 0.0155(6) 0.0159(2) 0.0170(2)
0.0090(3) 0.0099(2) 0.0123(8) 0.0156(6) 0.0187(6) 0.0192(3) 0.0200(2)
0.0112(7) 0.0124(1) 0.0149(8) 0.0186(7) 0.0220(6) 0.0228(3) 0.0236(5)
0.0135(8) 0.0149(1) 0.0181(8) 0.0222(6) 0.0259(8) 0.0265(2) 0.0275(3)
0.0163(4) 0.0184(5) 0.0215(7) 0.0266(2) 0.0306(8) 0.0328(1) 0.0329(6)
a
See the text. Table 2. Critical Micelle Concentration, φcmc, at Different Reduced Normalized Temperatures T/r ) T*/T/cm Where T/cm Is the Critical Micelle Temperaturea for H4T4 at Different Values of the Stiffness Parameter rH-T φcmc
RH-T
T/cm
0-10 0-20 0-50 0-500 20-0 50-0 500-0
8.76(4) 9.11(3) 9.69(1) 10.06(4) 8.77(5) 9.11(3) 9.34(1)
a
T/r
) 0.76
0.0034(1) 0.0038(1) 0.0056(1) 0.0070(1) 0.0033(1) 0.0040(1) 0.0046(1)
T/r
) 0.78
0.0043(1) 0.0049(1) 0.0071(1) 0.0088(1) 0.0043(1) 0.0051(1) 0.0059(1)
T/r
) 0.80
0.0057(1) 0.0063(2) 0.0090(1) 0.0110(1) 0.0055(1) 0.0065(1) 0.0074(1)
T/r
) 0.82
T/r ) 0.84
T/r ) 0.86
T/r ) 0.88
T/r ) 0.90
0.0071(1) 0.0079(2) 0.0112(2) 0.0134(1) 0.0070(1) 0.0080(1) 0.0092(1)
0.0090(1) 0.0099(2) 0.0137(1) 0.0164(1) 0.0088(1) 0.0100(1) 0.0114(1)
0.0111(1) 0.0123(2) 0.0165(3) 0.0197(2) 0.0109(1) 0.0122(1) 0.0140(1)
0.0136(2) 0.0153(1) 0.0198(7) 0.0236(2) 0.0134(1) 0.0151(1) 0.0170(1)
0.0165(2) 0.0186(3) 0.0240(1) 0.0280(3) 0.0164(2) 0.0187(1) 0.0205(3)
See the text.
value at R and RH-T > 100. This trend is similar to that shown by 〈l〉 versus R (Figure 1). The observed increase in critical micelle temperature, then, is directly connected to the progressive stiffening of the surfactant chains. The same upward trend is also displayed by the lower temperature limit for micellization. For all R and RH-T values, φcmc increases on increasing the temperature. The effect of increasing the stiffness of the surfactant chains on φcmc can be viewed from two different perspectives. At a given temperature, in agreement with experimental observations on hydrogenated and fluorinated nonionic surfactants with the same number of carbon atoms in the hydrophobic tail,7-10 the φcmc values are smaller for stiffer chains. For example, this is evident from Figure 4 where the φcmc values as a function of R at T* ) 8.60 are reported. For the reasons discussed above,
it was not possible to obtain completely overlapping temperature ranges for all systems, and as a consequence, R ) 0 and 10 are not included in this plot. The same decreasing φcmc trend is displayed by the mixedflexibility systems reported in Figure 5. Furthermore, it can be noticed that stiffening the tail segment leads to a larger effect on φcmc than stiffening the heads. In both cases, consistent with the results reported in Figure 1, a constant value is obtained when either the tail section or the heads are completely stiff. Because of the different temperature ranges, in an attempt to compare different systems that are the same “distance” from some reference state, it might be instructive to compare the micellization behavior as a function of chain stiffness at the same normalized reduced temperature, T*/T/cm. In so doing, the
6518 Langmuir, Vol. 22, No. 15, 2006
Figure 3. Critical micelle reduced temperature, T/cm, at the different R values considered in the simulations for H4T4 (O). The line is simply a guide for the eye.
Firetto et al.
Figure 6. Linear fits of ln φcmc vs 1/T* for H4T4 at R ) 0 (O), 10 (b), 20 (4), 50 (2), 100 (0), 300 (9), and 500 (3).
at T*/T/cm ) 0.76, φcmc at R ) 20 is 0.0051 and at R ) 500 is 0.0091. In the former case, T* ) 7.32 whereas T* ) 8.63 for the latter. Increasing the temperature by the same amount for the R ) 20 system would raise φcmc to 0.0215. Therefore, raising T* produces a larger increase in φcmc than reaching the same temperature by stiffening the chain. In conclusion, despite the different operational temperature ranges, it is probably more correct to discuss the effect of stiffness on the aggregation properties at a given reduced temperature T*. Thermodynamic considerations37 suggest that the critical micelle concentration depends on temperature according to
ln φcmc )
Figure 4. Critical micelle concentration, φcmc, at T* ) 8.6 and at different R values for H4T4 (O). The line is simply a guide for the eye.
Figure 5. Critical micelle concentrations, φcmc, at T* ) 7.65 for H4T4 at different values of the tail stiffness parameter R0-T (O) and different values of the head stiffness parameter RH-0 (b). The lines are simply guides for the eye.
opposite trend is observed, and φcmc is larger for stiffer chains. At the same T*/T/cm, because T/cm increases with the rigidity parameter, φcmc of stiffer surfactants will correspond to a higher T* than that of flexible ones. Considering that φcmc increases on increasing temperature, the variation of φcmc with R at the same T*/T/cm also includes a temperature effect. To elucidate the individual influence of the two parameters, the data reported in Table 1 can be used to discriminate their effects. For example,
∆G°mic ∆H°mic ∆S°mic ) kBT kBT kB
(5)
where ∆G°mic, ∆H°mic, and ∆S°mic are, respectively, the standard Gibbs free energy, enthalpy, and entropy variations upon micellization per amphiphile molecule. This equation implies that plots of ln φcmc versus 1/T are linear. In Figures 6 and 7, it is shown that the above relation holds for all R and RH-T values. The slopes of linear fits through the results correspond, according to eq 5, to the micellization enthalpies, ∆H°mic/kB, whereas the intercepts are equal to ∆S°mic/kB. The resulting thermodynamic micellization parameters are reported in Tables 3 and 4. For completely flexible chains (R ) 0), the calculated standard enthalpy of micellization, ∆H°mic/kB, is in excellent agreement with previous studies.35 The negative sign of this parameter indicates that the transfer of unimers from solution to micelles is an exothermic process. At the same time, a negative entropy change is observed. Therefore, micellization is enthalpy-driven for our systems. In aqueous solutions of nonpolar surfactants, micellization is an endothermic process; therefore, the spontaneous formation of thermodynamically stable micelles is entropydriven.47 On the contrary, micellization in nonpolar solvents is dominated by favorable interactions between the surfactant hydrocarbon segments and the solvent.48 As previously suggested,35,39 our model is more suitable for solutions of nonionic surfactants in nonpolar solvents rather than in aqueous systems. The effect of making the surfactant chains progressively stiffer by increasing the R parameter is to make the enthalpic contribution more negative whereas the entropy change is progressively less negative. As a consequence, both the larger number of favorable (47) Alexandridis, P.; Holzwarth, J. F.; Hatton, T. A. Macromolecules 1994, 27, 2414. (48) Price, C. Pure Appl. Chem. 1983, 55, 1563.
Micellization BehaVior of H4T4 Surfactant Chains
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Figure 7. Linear fits of ln φcmc vs 1/T* for H4T4. (A) RH-T ) 0-0 (O), 0-10 (4), 0-20 (0), 0-50 (3), and 0-500 (]). (B) RH-T ) 0-0 (b), 20-0 (9), 50-0 (1), and 500-0 ([). Table 3. Micellization Enthalpies, ∆H°mic/kB, Micellization Entropies, ∆S°mic/kB, and Micellization Free Energies, ∆G°mic/kB at T* ) 8.6 and for H4T4 at Different Values of the Stiffness Parameter r R
-∆H°mic/kB
-∆S°mic/kB
-∆G°mic/kB T* ) 8.60
0 10 20 50 100 300 500
65.9(3) 68.5(4) 68.6(5) 69.6(3) 69.7(5) 69.9(7) 70.2(8)
4.8(1) 4.4(1) 4.1(1) 3.6(1) 3.5(1) 3.3(1) 3.4(1)
29.2 34.8 37.2 42.1 42.9 44.7 44.2
Table 4. Micellization Enthalpies, ∆H°mic/kB, Micellization Entropies, ∆S°mic/kB, and Micellization Free Energies, ∆G°mic/kB at T* ) 7.65 and for H4T4 at Different Values of the Stiffness Parameter rH-T RH-T
-∆H°mic/kB
-∆S°mic/kB
-∆G°mic/kB T*)7.65
0-0 0-10 0-20 0-50 0-500 20-0 50-0 500-0
65.4(3) 68.2(4) 70.7(4) 68.4(4) 68.5(1) 68.9(2) 68.0(7) 68.1(2)
4.7(1) 4.6(1) 4.6(1) 4.1(1) 4.0(1) 4.6(1) 4.3(1) 4.2(1)
29.4 33.0 35.5 37.0 37.9 33.7 35.1 36.0
intermolecular interactions between stiffer surfactant chains and the lower entropic cost lead to a more negative free-energy change (Table 3 and Figure 8). By analogy to the φcmc versus R trends at a fixed temperature (Figure 4), changing the stiffness parameter has an appreciable effect on the micellization thermodynamics only when significant changes in surfactant molecular architectures are induced, that is, for R < 100 (Figure 1). The same behavior is observed on varying the stiffness parameter RH-T. In these systems, stiffening only the tails leads to a more negative free-energy micellization than stiffening the heads (Table 4). However, ∆G°mic values as small as those observed when both the head and tail segments are stiff were never reached. Because progressively stiffer surfactant chains micellize in different temperature ranges (Tables 1 and 2), if the micellization enthalpy were temperature-dependent, then the above conclusions would be somewhat uncertain. In fact, it has been found that for aqueous solutions of nonionic surfactants3 the heat of micellization slightly decreases with temperature according to
Figure 8. Micellization enthalpies, ∆H°mic/kB (O), and micellization entropies ∆S°mic/kB (b), at different R values for H4T4. The lines are simply guides for the eye.
∆H°1 ) ∆H°0 + ∆Cp∆T
(6)
where ∆H°1 and ∆H°0 are the standard micellization heats at T1 and T0, respectively, ∆Cp is the difference between the heat capacity at constant pressure of a surfactant molecule in the micelle and that of the unimer, and ∆T ) T1 - T0. For some nonionic surfactants in water,3 ∆H° is on the order of 10-20 kJ mol-1, and ∆Cp is ∼ -0.5 kJ K-1mol-1. As a consequence, the temperature dependence of ∆H°mic/kB can be totally wiped out by experimental uncertainties in a ln φcmc versus 1/T plot.49 Therefore, we can safely assume that the above microscopic considerations regarding the effect of stiffness on micellization behavior should not be affected by neglecting the enthalpy temperature dependence.50-53 In addition, as immediately evident from Figure 8, the most dramatic effect is observed for relatively small values of the stiffness parameters where the micellization temperature ranges are almost the same. Of course, it is well known that entropy has an even weaker temperature dependence compared to that shown by enthalpy. Structural information on the effect of stiffening the backbone chain can be derived by examining the cluster distribution (49) Molina- Bolı´var, J. A.; Aguiar, J.; Peula-Garcia, J. M.; Ruı´z Carnero, C. J. Phys. Chem B 2004, 108, 12813. (50) Kim, H.-U.; Lim, K.-H. Colloids Surf., A 2004, 235, 121. (51) Kresheck, G. C.; Hargraves, W. A. J. Colloid Interface Sci. 1974, 48, 481. (52) Kresheck, G. C. In Water: A ComprehensiVe Treatise; Franks, F., Ed.; Plenum Press: New York, 1975; Chapter 2, pp 95-167. (53) Kira´ly, A.; Deka´ny, I. J. Colloid Interface Sci. 2002, 242, 214.
6520 Langmuir, Vol. 22, No. 15, 2006
Firetto et al.
Table 5. Mean Micelle Aggregation Numbers, M h , at Different Reduced Temperatures, T*, and Volume Fractions, φr ) φ/φcmc, for H4T4 at Different Values of the Stiffness Parameter ra T*/T/cm
T*
M h
W
R)0 -47.50 0.91(1) -47.40 2.40(2) -47.30 4.18(7)
70(1) 70(1) 72(1)
µ
φr
T*
µ
φr
M h
W
12(2) 12(1) 12(1)
7.70
-4.60 -4.50 -4.40 -4.10
R ) 10 1.4(2) 2.3(1) 3.3(2) 6.1(5)
89(1) 93(2) 95(1) 103(1)
14(1) 14(1) 14(1) 14(1)
-2.80 -2.70 -2.50 -2.40 -1.20 -1.10 -1.00 -0.90
1.9(5) 2.5(6) 4.1(4) 4.4(2) 2.2(2) 2.7(1) 3.2(4) 4.13(7)
76(2) 79(1) 83(1) 83(1) 71(1) 73(1) 73(1) 75(1)
15(1) 14(1) 15(1) 14(1) 14(1) 14(1) 14(1) 14(1)
73.80 74.00 74.20 74.40
R ) 50 2.0(4) 3.1(8) 4.0(1) 4.9(3)
127(1) 134(3) 141(2) 146(1)
19(1) 18(1) 18(1) 19(1)
78.30 78.65 78.70 79.10
1.33(8) 3.1(8) 4.0(1) 4.9(3)
92(1) 104(1) 108(2) 116(1)
18(1) 19(1) 19(1) 19(1)
84.30 84.60 84.80 85.00
1.9(2) 2.80(8) 3.1(2) 3.3(2)
89(1) 94(1) 97(2) 101(1)
19(1) 19(1) 19(1) 19(1)
130.00
R ) 300 1.9
196
30
144.20 144.60 145.20
1.81(5) 2.7(1) 3.08(1)
162(2) 165(1) 168(1)
31(3) 32(3) 32(1)
155.30 155.80
2.94(7) 3.63(2)
149(1) 162(1)
30(1) 30(1)
0.85
7.00
0.90
7.50
-46.35 -46.00 -45.95
1.59(8) 4.1(4) 5.0(8)
60(1) 65(1) 66(1)
13(1) 13(1) 13(1)
8.20
0.95
7.90
-45.30 -45.10 -44.90
2.2(4) 3.5(2) 4.1(3)
57(1) 58(1) 60(0)
13(1) 12(1) 12(1)
8.60
0.85
8.20
24.40 24.50 24.80
2.6(7) 3.7(8) 5.5(7)
124(1) 124(1) 133(2)
17(1) 16(1) 16(1)
0.90
8.70
26.85 26.95 27.00 27.20
1.9(2) 2.8(5) 4.6(4) 5.1(5)
89(1) 94(2) 97(1) 101(1)
17(1) 16(1) 16(1) 16(1)
0.95
9.10
28.80 29.00 29.30
2.1(4) 2.7(5) 4.4(4)
80(1) 84(1) 88(1)
17(1) 16(1) 16(1)
R ) 100 110.40 1.79(9) 110.50 3.03(9) 110.70 3.77(7)
205(4) 205(4) 220(1)
22(1) 22(2) 22(2)
117.80 118.00 118.30 119.20
2.2(2) 2.41(5) 2.7(1) 3.90(4)
125(2) 129(1) 140(3) 168(1)
22(1) 22(1) 23(1) 23(1)
111(1) 117(1) 138(1)
21(1) 21(1) 21(1)
9.60
125.00 2.37(3) 125.20 2.59(6) 126.20 3.75(2) R ) 500 130.00 2.1
201
30
10.30
144.20 144.60
1.97(5) 2.78(2)
165(3) 166(1)
30(1) 30(1)
10.80
155.30 155.80
2.64(5) 3.3(2)
147(2) 160(2)
27(1) 27(1)
R ) 20
0.85
9.50
0.90 10.10
0.95
a
10.60
9.10
9.60
10.20
9.6
10.30
10.80
µ represents the chemical potential, and W is the width of the Gaussian fits. (See the text.) Box sizes range from L ) 40 to 80.
functions obtained, as previously done,35 by considering two amphiphile molecules to be in the same cluster if any tail segments of the first molecule are within the interaction range of a tail segment of the second molecule. From previous studies,35,39 it is known that the aggregation numbers depend on temperature and, slightly, on the surfactant volume fraction. Therefore, when comparing surfactants of different stiffnesses, the above effects must be kept in mind. Aggregation results, summarized in Table 5, were obtained, in simulation cells of sizes ranging from L ) 40 to 80 depending on the R value at three different normalized reduced temperatures T*/T/cm (0.85, 0.9, and 0.95) and over a range of surfactant volume fractions, φ. In all cases, the cluster size distribution (frequency of occurrence of clusters of a given size) was obtained from a GCMC run equilibrated at the relevant values of T* and µ, and the peak was fitted to a Gaussian function. Statistical uncertainties were estimated by averaging results from four independent simulations performed under the same conditions for different initial seeds of the random sequence. The four M
values were averaged, and the uncertainties represent standard deviations from the mean. In Figure 9 are reported volume fractions of aggregates of a given size as functions of aggregation number at different values of the stiffness parameter at the same normalized reduced temperature and at roughly the same distance from the corresponding φcmc. For comparison purposes, for each system the volume fractions were normalized by dividing them by the maximum value. Furthermore, representative micelle configurations are also shown for selected conditions. As a general rule, stiffening the chain leads to larger aggregates although, as verified by monitoring the micellar radii of gyration,15 no deviations from spherical symmetry were observed. Even more pronounced micellar growth on stiffening the surfactant chain can be observed if the comparison is made at the same T*. This is easily understood if one considers that the aggregation number decreases on increasing the temperature for surfactants with the same flexibility. The same behavior is observed for H4T4 at different RH-T values. In Table 6, mean micelle aggregation numbers, M h , are
Micellization BehaVior of H4T4 Surfactant Chains
Langmuir, Vol. 22, No. 15, 2006 6521
Figure 10. Mean micelle aggregation numbers, M h , for H4T4 at different values of the R (9), R0-T (O), and RH-0 (b) parameters. The lines are simply guides for the eye.
Figure 9. Normalized (see text) volume fraction of clusters, φ/φmax vs aggregation number, M, for H4T4 obtained at T*/T/cm ) 0.90, φ/φcmc ) 2.0 ( 0.5, and different chain flexibility parameters R ) 0 (O), 10, 20, 50, 100, and 300 (3). Continuous lines represent Gaussian fits of the simulated points. For systems corresponding to intermediate R values, symbols are not reported for clarity reasons, and increasing R values correspond to lines from left to right. Results for RH-T ) 0-500 (0) are also reported. In addition, sample configurations under the conditions indicated by the arrows are also shown. Table 6. Mean Micelle Aggregation Numbers, M h , at Different Reduced Temperatures, T*, for H4T4 at Different Values of the Stiffness Parameter rH-Ta RH-T
T*
µ
M
W
0 0-10 0-20 0-50 0-500 20-0 50-0 500-0
7.50 7.86 8.20 8.72 9.05 7.89 8.20 8.41
-46.35 -24.50 -10.20 -13.70 38.40 -12.05 9.25 30.00
60(1) 74(1) 82(1) 82(1) 92(1) 73(1) 82(1) 84(1)
13(1) 13(1) 13(1) 15(1) 18(1) 14(1) 17(1) 16(1)
a
µ represents the chemical potential, and W is the width of the Gaussian fits. (See the text.) In all cases, T*/T/cm ) 0.90. Box sizes range from L ) 40 to 65.
reported. The effects on micellar size of stiffening different sections of the surfactant chain are compared in Figure 10. Overall, the largest aggregation numbers are obtained when both the head and tail sections are rigid. Stiffening only the head or the tail section produces micelles of roughly the same dimensions, although slightly larger aggregates are obtained for chains with stiff tails. This is particularly evident at small stiffness parameter values, where the trend in M h versus R0-T is similar to that obtained when uniform R values are used, in agreement with the conclusions derived above from thermodynamic considerations. To clarify the effects leading to larger aggregates when stiffer chains are packed, in Figure 11 site fraction distributions are reported at different selected values of the stiffness parameter. This function is defined as
Figure 11. Site fraction distributions, 〈s(r)〉 of the surfactant head, surfactant tail, and solvent vs (A) distance r and (B) normalized (see text) distance r/rmax from the micelle center of mass for H4T4 at R ) 0 (-) and 300 (-‚-) and RH-T ) 0-500 (- - -) and 500-0 (‚‚‚) within micelles of size M h ) 60 ( 1, 165 ( 2, 92 ( 1, and 84 ( 1, respectively.
〈si(r)〉 )
〈Ni(r)〉 NT(r)
(7)
where si is the site fraction distribution of each species i (surfactant head, surfactant tail, and solvent) at a given distance r from the center of mass of the micelle, Ni(r) is the number of sites of species i, and NT is the total number of sites at a distance r from the micelle center of mass. In all cases, each function displays three different regimes. At small distances, within the micellar core, the tail distribution function is near its maximum value of 1, and the head and solvent site concentrations are close to zero. At intermediate distances, while the solvent and head site distributions increase, the tail content decreases. Finally, at larger distances, on crossing the
6522 Langmuir, Vol. 22, No. 15, 2006
micellar shell and reaching the outer environment, the head as well as the tail distribution functions drop to 0, and as a consequence, only solvent sites are present. It has been demonstrated above that the mean end-to-end distances of stiff chains are greater than those of flexible ones and thus the radius of their micellar aggregates will be greater. The resulting larger inner micellar volume can be filled with additional chains; this is also facilitated by new favorable tailtail contacts. As a consequence, micelles made of stiff chains are larger and contain a larger number of monomers (i.e., they have a larger aggregation number). This is clearly demonstrated by the surfactant tail site fraction distributions of Figure 11A showing that, within the core and at the same (not too small) distance from the micellar center of mass, the fraction of tails is appreciably larger for stiff chains. Recall that the micellization enthalpy was more negative for large R values, and this has been attributed to an increased number of T-T interactions. For surfactants with mixed-flexibility parameters, the same ideas should hold. However, although rigid tails facilitate the packing of a large number of chains, the presence of flexible and folded head portions hinders the process. The result is that micelles can grow only slightly with respect to those constituted of completely flexible surfactants. The solvent distribution functions can be interpreted following the same reasoning because solvent is present in all sites not occupied by tails or heads. The maximum present in the head site distribution arises from the initial increase due to chain packing in the micellar core followed by a decrease associated with the rarefied shell. The peak is located at larger distances for aggregates constituted of stiff chains, and as a consequence, the position of the maximum is related to micellar size. The shape of this peak is somewhat dependent on surfactant chain stiffness, being lower and broader for stiffer chains. To compare differences in micellar structure due to changes in chain flexibility, independently of the micellar radius, in Figure 11B the same density profiles are reported as functions of the normalized dimension, r/rmax, with rmax being the distance at which the maximum of each head distribution is located. It is evident that, on this normalized scale, the only remaining difference among all curves is a lower and slightly broader peak in the head site distribution for rigid chains, presumably arising from the inability of stiff rods to bend on themselves and thereby originating a more rarefied shell.
Conclusions In the present study, the effect of gradual changes in chain flexibility of the model surfactant H4T4 on its self-aggregation behavior has been investigated by cubic lattice Monte Carlo simulations assisted by histogram reweighting. By properly tuning the stiffness parameter R, it was possible to change the flexibility
Firetto et al.
either of the entire chain or, selectively, of the solvophilic (H) or solvophobic (T) portions. On increasing R, the average conformations ranged from completely flexible chains all the way to stiff rods as shown by the trends in the mean end-to-end distances. It was found that the most immediate consequence of changing the surfactant ability to bend was shifting up the temperature range in which the micellization process occurs, as demonstrated by the increase in T/cm with R. This made it difficult to compare the aggregation behavior at the same temperature. When this could be done, the present results were in qualitative agreement with the available experimental observations and some previous simulation results. For example, the observed lowering of the φcmc on stiffening the H4T4 chain is analogous to the lower critical micelle concentration of fluorinated surfactants with respect to that of the hydrogenated counterparts. Thermodynamic considerations suggest that micellization is an exothermic process that is favored by a larger number of heterocontacts available when surfactant chains are stiffer. Presumably, the same effect also plays a role in facilitating the phase-separation process that takes place at low temperatures. The overall result is that the temperature interval that is useful for observing micellar aggregation depends on chain flexibility. In agreement with experimental observations, it was found that the aggregation number was greater for stiffer surfactants, regardless of whether the comparison was made either at the same T* or at the same T*/T/cm. Once again, this can be attributed to the easier and energetically more favorable packing of stiff rods. Further support for the above conclusion has been derived by analyses of the surfactant head, surfactant tail, and solvent site fraction distributions within the micelle. It was possible to conclude that the larger inner micellar volume made available by the increased micellar radius deriving from packing effectively longer chains was filled by additional surfactants, as facilitated by favorable T-T interactions and thereby leading to an increase in the micellar aggregation number. In the case of mixed-flexibility surfactants, growth was limited either by folded tail portions or, for surfactants with stiff tails, by disordered head portions. Acknowledgment. Financial support for this work has been provided by Universita` di Palermo (grants ex 60% to M.A.F.), the Department of Energy, Office of Basic Energy Sciences (grant no. DE-FG02-01ER15121 to A.Z.P.), and ACS-PRF (grant 38165-AC9 to A.Z.P.). This work was planned and partially performed during visits by M.A.F. and V.F. to Princeton University. We are grateful to A.Z.P. for making these visits possible and for making computational facilities available. LA060386C
