Micellization and Phase Separation of Diblock and Triblock Model

Model for reversible nanoparticle assembly in a polymer matrix. Andrew J. Rahedi , Jack F. Douglas , Francis W. Starr. The Journal of Chemical Physics...
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Langmuir 2002, 18, 2940-2948

Micellization and Phase Separation of Diblock and Triblock Model Surfactants Athanassios Z. Panagiotopoulos* Department of Chemical Engineering, Princeton University, Princeton, New Jersey 08544-5263

M. Antonio Floriano† Department of Chemistry, Universita` della Calabria, 87036 Arcavacata di Rende (CS), Italy

Sanat K. Kumar Department of Materials Science and Engineering, Pennsylvania State University, University Park, Pennsylvania 16802 Received October 26, 2001. In Final Form: December 20, 2001 The phase behavior and micellization of several model lattice diblock and triblock surfactants have been investigated by histogram-reweighting grand canonical Monte Carlo simulations. By studying the systemsize dependence of the calculated phase diagrams, it has been found that for the cases studied (for which interactions are short ranged and temperature independent) each surfactant system either micellizes or phase separates, but never both. These results suggest that the experimentally observed behavior, where the same aqueous surfactant solution shows both phase separation and micellization under different conditions, is a consequence of the unusual solvation properties of water. The tendency to self-assemble is responsible for appreciable deviations from quasichemical theory even in systems that do not form micellar aggregates but are close to the boundary of macroscopic phase separation. For the micelle-forming systems, the surfactant volume fraction at the critical micellar concentration, φcmc, has been calculated from the point where a change of slope in the osmotic pressure versus surfactant volume fraction plots is observed. In all cases investigated, φcmc was found to increase with increasing temperature. As a consequence, positive values of the heat of micellization were obtained. For surfactant architectures close to macroscopic phase separation, the cluster size distributions are broad and extend to very large aggregation numbers indicating the presence of elongated micellar aggregates. This was also confirmed by an examination of typical configurations. Triblock systems, with symmetric architecture, behave in a similar manner, and architectures where the solvent-insoluble block is on the outside tend to phase separate over a broader range of parameter space than the triblock where the middle block is solvophobic. These results provide a growing understanding of the role of interactions and chain architecture on the self-assembly of surfactant systems and can be employed to benchmark existing theories in this area.

1. Introduction It is currently well appreciated that molecules with the right balance of solvophilic and solvophobic moieties can spontaneously self-assemble into a variety of nanostructures. Examples include (a) the assembly of surfactants into micelles, cylinders, and so forth,1,2 (b) multiblock copolymers assembling into spatially periodic structures such as spheres, cylinders, and lamellae,3 and (c) the assembly of associating polymers into amorphous gel phases, which possess solidlike properties over macroscopic time scales.4 In contrast, macroscopic phase separation will occur if one increases the solvophobic content of the molecule, while the heterochain molecule freely dissolves in the solvent when the solvophilic content of the molecule increases. While the behavior in these * To whom correspondence should be addressed. E-mail: azp@ princeton.edu. † Current address: Department of Physical Chemistry, University of Palermo, Parco d’Orleans II, Viale delle Scienze, 90128 Palermo, Italy. (1) Israelachvili, J. Intermolecular and Surface Forces, 2nd ed.; Wiley: New York, 1989. (2) Larson, R. G. The Structure and Rheology of Complex Fluids; Oxford University Press: New York, 1999. (3) Bates, F. S.; Fredrickson, G. H. Annu. Rev. Phys. Chem. 1990, 41, 525. (4) Kumar, S. K.; Douglas, J. F. Phys. Rev. Lett. 2001, 87, 188301.

extreme cases is apparent, it is unclear if a system can simultaneously show spontaneous self-assembly and simple phase separation in different temperature and concentration regions. For example, it is well-known in the surfactant literature that aqueous solutions of nonionic surfactants, denoted as CxEy, where Cx is a saturated hydrocarbon tail composed of x carbons and Ey corresponds to an ethylene oxide fragment of length y, can simultaneously show micellization and lower critical phase behavior. In some cases, the phase separation occurs between a micelle-enriched phase and a unimer phase. While the phase separation phenomenon in these cases has been generally accepted as being driven by the unique solvent properties of water, a theoretical argument which suggests that this is indeed possible for any solvent was presented by Semenov et al.5 These researchers considered the case of a telechelic copolymer, where the end blocks were solvent-phobic. Using a simple mean-field theory, they suggest that at low enough polymer concentration the chains form flowerlike micelles, with the chains forming looplike structures. However, with increasing concentration the chains begin to bridge across different micelles, which then leads to attractive interactions between the micelles. In specific cases, these authors (5) Semenov, A. N.; Nyrkova, I. A.; Cates, M. E. Macromolecules 1995, 28, 7879.

10.1021/la0156513 CCC: $22.00 © 2002 American Chemical Society Published on Web 02/12/2002

Micellization and Phase Separation of Surfactants

predict that the attractive interactions are strong enough to cause the system to “phase separate(s) into two macrophases one of them being a closed packed phase of micelles.” “The second phase is a very (exponentially) dilute solution of micelles and of free telechelic chains.” Several other theoretical attempts have been made in the past in this direction,6,7 and in particular we point to ref 7 for a discussion of the different terms in the free energy which can result in either phase separation or micellization. This paper focuses on this important topic and probes whether phase separation and micellization can occur under different conditions for the same surfactant architecture. In this context, it is important to highlight the fact that it is difficult to distinguish between phase separation and micellization by simulations, which are necessarily conducted on a finite-size system. Thus, while the formation of nanostructures in a simulation is a necessary condition for micellization, it is hard to simulate systems large enough to obtain multiple micelles, which is necessary to determine if these structures represent true self-assembly or simply the finite-size analogue for phase separation. Aggregates are often seen in simulation studies of triblock sequences.8,9 As a critical step to resolve this important issue, here we show that true micellization can be readily distinguished from phase separation in a simulation by observing (a) the strong system-size dependence of the apparent phase diagram that is obtained in the former case and (b) the lack of strong finite-size effects in the behavior of the osmotic pressure versus concentration curves in the case of micellizing systems. Once methods for making this distinction are available, it is reasonable to investigate possible connections between surfactant architecture and phase behavior, that is, the possible relationship between phase structure and molecular structure. This constitutes the second focus of our research, and in this paper we consider both diblock and triblock architectures so as to assess the role of chain architecture and interactions on these competing processes. There have been numerous simulation studies of model surfactant systems in recent years. Most researchers adopt highly simplified models for the surfactants and perform Monte Carlo simulations on lattice (e.g., refs 1018) and off-lattice systems (e.g., refs 19-23). Molecular (6) Israelachvili, J. Chem. Soc., Faraday Trans. 2 1976, 72, 1525. (7) Laughlin, R. G. The Aqueous Phase Behavior of Surfactants; Academic Press: London, 1994. (8) Timoshenko, E. G.; Basovsky, R.; Kuznetsov, Y. A. Colloids Surf., A 2001, 190, 129. (9) Timoshenko, E. G.; Kuznetsov, Yu. A. J. Chem. Phys. 2001, 112, 8163. (10) Xing, L.; Mattice, W. L. Macromolecules 1997, 30, 1711. (11) Talsania, S. K.; Rodrı´guez-Guadarrama, L. A.; Mohanty, K. K.; Rajagopalan, R. Langmuir 1998, 14, 2684. (12) Rodrı´guez-Guadarrama, L. A.; Talsania, S. K.; Mohanty, K. K.; Rajagopalan, R. Langmuir 1999, 15, 437. (13) Girardi, M.; Figueiredo, W. J. Chem. Phys. 2000, 112, 4833. (14) Bhattacharya, A.; Mahanti, S. D. J. Phys.: Condens. Matter 2001, 13, L861; 2000, 12, 6141. (15) Kim, S. H.; Jo, W. H. Macromolecules 2001, 34, 7210. (16) Pe´pin, M. P.; Whitmore, M. D. Macromolecules 2000, 33, 8644. (17) Maiti, P. K.; Kremer, K.; Flimm, O.; Chowdhury, D.; Stauffer, D. Langmuir 2000, 16, 3784. (18) Girardi, M.; Figueiredo, W. J. Chem. Phys. 2000, 112, 4833. (19) Von Gottberg, F. K.; Smith, K. A.; Hatton, T. A. J. Chem. Phys. 1997, 106, 9850; 1998, 108, 2223. Nelson, P. H.; Hatton, T. A.; Rutledge, G. C. J. Chem. Phys. 1999, 110, 9673. (20) Smit, B.; Hilbers, P. A. J.; Esselink, K.; Rupert, L. A. M.; van Os, N. M.; Schlijper, A. G. Nature 1990, 348, 624. Karaborni, S.; Esselink, K.; Hilbers, P. A. J.; Smit, B. Science 1994, 266, 254. (21) Wijmans, C.; Linse, P. Langmuir 1995, 11, 3748; J. Chem. Phys. 1997, 106, 328. (22) Rector, D. R.; Van Swol, F.; Henderson, J. R. Mol. Phys. 1994, 82, 1009.

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dynamics studies of realistic models for surfactants and solvent are starting to approach time scales relevant for micelle formation,24,25 but it is still too hard to sample equilibrium size distributions of micelles in relatively large systems. In earlier work by some of us,26 the grand canonical Monte Carlo [GCMC] methodology employed in the present paper was developed and the behavior of H3T3 and H4T4 surfactants was studied in detail. A preliminary report27 covering a few of the systems studied here has also been published. The main objective of the present study is to investigate the boundary between macroscopic phase separation and micellization as a function of surfactant architecture for relatively short diblock and triblock surfactants. In the notation we adopt, H represents the “solvent-soluble” head moieties, while T represents the tail groups. The solvent employed does not incorporate any of the special features that characterize water, that is, the potential to hydrogen bond, or the wellknown fact that its solvation power decreases with increasing temperature. Rather, we shall focus on a solvent that interacts with both the H and T moieties through short-ranged, temperature-independent potentials. This choice of solvent allows us to isolate the role played by the unique nature of water in determining the phase separation and nanostructure formation by these surfactant molecules. We have studied diblock surfactants of structure HxTy and triblock surfactants of structure TyHxTy and HxTyHx, where x is 1, 2, 4, or 8 and y is 1, 2, 4, 8, or 16. We find that each of these chain architectures shows either micellization or phase separation, but never both, regardless of the range of temperatures and concentrations that is surveyed. Thus, we conclude that the experimentally observed behavior for the CxEy surfactants,1 where phase separation and micellization can be observed in the same system, is a manifestation of the unusual temperature dependence of the solvation power of water. Thus, we also do not find any transition from flowerlike micelles to phase separation, as predicted by Semenov,5 in the case of the end-functionalized triblock copolymers. We have considered the sizes of the nanostructures as the amount of tail is increased and find that the micelle sizes increase as one proceeds toward chain compositions where phase separation is seen. These results provide strong benchmarks from which our understanding of the behavior of amphiphilic molecules can be developed and data against which theories can be checked. 2. Models and Methods The model on which this work is based consists of linear chains of beads on a simple cubic lattice of spacing l, originally proposed by Larson.28-30 Nearest-neighbor as well as next-nearest-neighbor interactions are present along the lattice vectors (0,0,1), (0,1,1), and (1,1,1) and their reflections along the principal axes, resulting in a (23) Milchev, A.; Bhattacharya, A.; Binder, K. Macromolecules 2000, 34, 1881. Viduna, D.; Milchev, A.; Binder, K. Macromol. Theory Simul. 1998, 7, 649. (24) Salaniwal, S.; Cui, S. T.; Cochran, H. D.; Cummings, P. T. Langmuir 2001, 17, 1773, 1784. (25) Bogusz, S.; Venable, R. M.; Pastor, R. W. J. Phys. Chem. 2001, 105, 8312. (26) Floriano, M. A.; Caponetti, E.; Panagiotopoulos, A. Z. Langmuir 1999, 15, 3143. (27) Kumar, S. K.; Floriano, M. A.; Panagiotopoulos, A. Z. Adv. Chem. Eng. 2001, 28, 260. (28) Larson, R. G.; Scriven, L. E.; Davis, H. T. J. Chem. Phys. 1985, 83, 2411. (29) Larson, R. G. J. Chem. Phys. 1989, 89, 1642. (30) Larson R. G. J. Phys. II 1996, 6, 1441.

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coordination number z ) 26. Allowable bonds between neighboring beads are the same set as the interaction vectors. Thus, the surfactant chains have higher conformational flexibility than would be the case for a simple cubic lattice with coordination number z ) 6 and correspond to effectively longer real surfactants. Interactions of strength HH occur between two head groups, a head group and solvent monomer, and two solvent monomers. Interactions of strength HT occur between head or solvent groups and tail groups, and interactions of strength TT occur between tail groups. The TT interaction was set to -2 (resulting in attractive interactions for nearestneighbor tail-tail contacts), and the HH and HT interactions were set to zero. While another interaction set proposed by us31 gives close agreement with the critical micelle concentration [cmc] of specific nonionic poly(ethylene oxide) alcohol surfactants, here we used the original Larson interaction set in order to allow for comparisons with earlier work on the micellization behavior26 and phase diagrams30 of diblock surfactants. All systems studied were binary mixtures of amphiphile and solvent. The lattice was fully occupied. The resulting two-component, fixed-density system is equivalent to a one-component, variable-density system of amphiphile. The pressure calculated from the simulations is equivalent to the osmotic pressure of the amphiphile in the twocomponent system. Reduced quantities (denoted by *) are defined in the standard fashion,26 using  ) HT - (1/2)HH - (1/2)TT and l as the reducing factors for energy and length, respectively. In particular, we have T* ) kBT/, where kB is Boltzmann’s constant and P* ) Pl 3/. We used the GCMC method with multihistogram reweighting as in previous work.26 This approach is based on performing a number of simulations for a given surfactant at selected temperatures and imposed chemical potentials in order to collect histograms of the probability of observation of a certain energy and number of particles. In brief, if the inverse temperature of a run is β ) 1/kBT and the chemical potential is µ, the entropy function is obtained from the observed frequencies f(N,U) of observation of N particles with total energy U:

S(N,V,U)/kB ) ln[f(N,U)] - βµN + βU + C

(1)

where C is an run-specific unknown constant. Histograms from multiple runs are combined using the technique of Ferrenberg and Swendsen32,33 to obtain the entropy function of the system over the range of temperatures and chemical potentials covered in the simulations. This is done essentially by determining the constants C in an optimal way from the regions of overlap of the histograms. From the global entropy function, properties can be calculated at any chemical potential and temperature, provided that the original simulation data covered energies and densities relevant for the new conditions. An estimate for the grand partition function, Ξ(µ,V,β), within an unknown multiplicative constant, C′, can be obtained from

Ξ(µ,V,β) ) C′

∑ exp[S(N,V,U) - βU + βµN]

(2)

U,N

Pressure can be obtained from

βPV ) ln Ξ(µ,V,β) ) ln C′ + exp(S(N,V,U) - βU + βµN)] (3) ln[

∑

U,N

The unknown constant in eq 3 can be obtained from

matching low-density results to the ideal gas equation of state, βPV ) N. GCMC simulations afford numerous advantages over the more commonly used constant volume (NVT) simulations. In particular, the GCMC method gives access to the chemical potential and free energy of the system and allows for unambiguous distinction between macroscopic phase separation boundaries and micellization into finite-size aggregates. It also allows determinations of cmc’s in systems that are much too small to contain even a single micelle. Since the cmc’s are less than 1% in volume fraction for many of the surfactants studied here, use of NVT simulations would have required very large system sizes that are hard to equilibrate reliably. Simulations were performed in cubic boxes under periodic boundary conditions. Determination of cluster distributions was performed following the convention used in our previous work.26,34 Two amphiphile molecules are considered to be in the same cluster if any tail segments of the first molecule are within the interaction range (26 nearest neighbors) of a tail segment of the second molecule. A typical mix of attempted moves was 70% particle creation/annihilation, 0.5% cluster moves, and the balance reptations. Cluster moves consist of displacing an aggregate as a whole in a random direction. To satisfy detailed balances, attempted cluster moves fail if the move has resulted in creation of new clusters. The cluster moves are necessary to achieve significant displacements of micellar aggregates but have a relatively high computational cost. Configurational-bias sampling methods35,36 were used to facilitate insertions and removals of surfactant molecules. Typical CPU time requirements on a 1 GHz Pentium III processor were 1 CPU hr/108 attempted Monte Carlo moves. Run lengths ranged from 107 to 109 attempted Monte Carlo moves, depending on density and system size. A long-period (>2 × 1018) random number generator37 was used. Statistical uncertainties were obtained by performing independent sets of runs (usually four) at identical thermodynamic conditions but using different random number sequences. Statistical uncertainties were obtained as the standard deviation of results from the independent simulations and are reported at the 2σ level in Tables 2-4. This procedure yields more conservative statistical uncertainty estimates than the common practice of dividing the production part of a simulation into supposedly independent subblocks. To provide a basis for comparisons with the simulations, Guggenheim’s quasichemical theory38 was used as outlined in earlier work.39 In Guggenheim’s mean-field approach, the chain segment connectivity is taken into account in the calculation of the system entropy and the energy, and the Bethe approximation is used for describing the local (31) Chatterjee, A. P.; Panagiotopoulos, A. Z. In Computer Simulation Studies in Condensed Matter Physics; Landau, D. P., Ed.; Springer Proceedings in Physics, Vol. 85; Springer, New York, 1999; p 211. (32) Ferrenberg, A. M.; Swendsen, R. H. Phys. Rev. Lett. 1988, 61, 2635. (33) Ferrenberg, A. M.; Swendsen, R. H. Phys. Rev. Lett. 1989, 63, 1195. (34) Mackie, A. D.; Panagiotopoulos, A. Z.; Szleifer, I. Langmuir 1997, 13, 5022. (35) Frenkel, D.; Mooij, G. C. A. M.; Smit, B. J. Phys.: Condens. Matter 1992, 4, 3053. (36) De Pablo, J. J.; Laso, M.; Siepmann, J. I.; Suter, U. W. Mol. Phys. 1993, 80, 55. (37) Press, W. H.; Teukolsky, S. A.; Vetterling, W. T.; Flannery, B. P. Numerical Recipes, 2nd ed.; Cambridge University Press: Cambridge, 1992. (38) Guggenheim, E. A. Mixtures; Clarendon Press: Oxford, 1952. (39) Mackie, A. D.; Panagiotopoulos, A. Z.; Kumar, S. K. J. Chem. Phys. 1995, 102, 1014.

Micellization and Phase Separation of Surfactants

Figure 1. Apparent phase envelope for HT4. Symbols are from simulations for L* ) 10 (2) and L* ) 15 (b). The line is from quasichemical theory.

Figure 2. Osmotic pressure, P*, versus volume fraction, φ, for HT4 at T* ) 11. Symbols are as for Figure 1, with dashed lines joining the points for visual clarity. The continuous line corresponds to ideal solutions.

mole fractions around a given segment. The theory assumes only homogeneous (not microstructured) phases and thus cannot be expected to give good results for micelleforming systems. More sophisticated mean-field theories have been developed40 that can be used to predict cmc’s, but their inclusion is beyond the scope of the present work. To illustrate the simulation approach used, in Figure 1 the calculated phase behavior is shown for HT4 for two system sizes, L* ) 10 and 15. The corresponding curves for the osmotic pressure, P*, as a function of volume fraction of amphiphile at a temperature below the critical point are shown in Figure 2. In a finite-size system, firstorder phase transitions are rounded, as can be seen from the finite slope of the osmotic pressure within the twophase region [Figure 2]. However, as the system size increases, the osmotic pressure curve approaches a horizontal line indicating a first-order transition between a dilute and a dense phase. Despite the strong finite-size effect on the osmotic pressure in Figure 1, the apparent phase envelope is essentially independent of simulation (40) Szleifer, I.; Carignano, M. A. Adv. Chem. Phys. 1996, 94, 165.

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Figure 3. Apparent phase transitions for H2T4. Symbols are from simulations for L* ) 10 (2), L* ) 15 (b), and L* ) 20 (+). The line is from quasichemical theory.

Figure 4. Osmotic pressure versus volume fraction for H2T4 at T* ) 8.5.

system size away from the critical point. The quasichemical theory describes the coexistence data very well, except near the critical point. For H2T4, in contrast, Figure 3 illustrates that there is a very strong finite-size effect on the phase envelope. As the system size is increased from L* ) 10 to 15 to 20, the apparent liquid density decreases quite sharply. We argue that this occurs since the system undergoes micellization and not macroscopic phase separation. To see this, consider the case where a hypothetical system forms a micelle consisting of N ) 100 monomers. As the chemical potential is increased, the system spontaneously forms a single micelle. If the system volume is V ) 1000, then the apparent volume fraction of the “liquid” phase would be φ ) N/V ) 0.1. On doubling the system volume, since only a single micelle will initially form, the apparent coexisting volume fraction would drop by a factor of 2, qualitatively consistent with the results presented in Figure 3. Another indicator that micellization has occurred is the behavior of osmotic pressure [Figure 4], which indicates that all systems have similar behavior, with the two larger system sizes being essentially identical. The slope of the osmotic pressure curve is related to the number of independent kinetic entities (aggregates) and is expected to be finite

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Panagiotopoulos et al. Table 1. Micellization versus Phase Separation for the Systems Studieda

Figure 5. Temperature, T*, versus critical micelle concentration, φcmc, for H8T. Filled circles are obtained from the osmotic pressure curves, and open triangles are from the aggregate size distributions.

and independent of system size for systems that form micelles. This lack of size dependence, which should be contrasted to the behavior of phase-separating systems, is a strong indication that there is no first-order transition in the system. In our work, we therefore employ the size dependence of the phase diagrams and the osmotic pressures to sensitively determine if a system undergoes micellization, phase separation, or both. As first suggested in our previous work,26 the location of the “break” in the osmotic pressure curves can be usefully defined as the cmc. There are other possible definitions of the cmc, for example, as the concentration at which the fractions of surfactant that exists as free monomer and as part of larger clusters are equal. As discussed in ref 26, the volume fraction at which the break in osmotic pressure occurs is the point at which the first micelle appears in the thermodynamic limit, corresponding to half the concentration obtained from the condition of equal concentration of free monomers and micelles. In practice, we obtained the cmc as the volume fraction at which the second derivative of the osmotic pressure curve is a maximum. In contrast to our earlier work,26 we do not place a restriction on the slope of the osmotic pressure line for identifying the cmc. We have found that for systems such as H8T that have long hydrophilic sections, the earlier criterion on slope < 10% is overly restrictive. An alternative approach to calculation of the cmc is to obtain histograms for the aggregate size distribution as a function of volume fraction of surfactant. The cmc would then be the concentration at which a micellar peak first appears. This approach has been frequently used in simulation studies of aggregation but suffers from the need to study very large systems when the cmc’s are low. In Figure 5, the critical micelle concentration is plotted for H8T. The two alternative methods for calculating the cmc agree very well at intermediate chemical potentials, giving us confidence that either method is appropriate to locate the cmc. At volume fractions greater than φcmc ≈ 0.2, the osmotic pressure approach starts breaking down, because the osmotic pressure is an increasing function of φ in the semidilute regime. Conversely, at volume fractions less than φcmc ≈ 0.1, the method based on aggregate size histograms requires increasingly large systems and long runs. Since we are primarily interested in micellization behavior at low volume fractions of surfactant, we have

a Shaded cells correspond to systems that form micelles, and clear cells correspond to phase-separating systems.

used the osmotic pressure approach for all the remaining calculations reported. At lower temperatures, equilibration becomes a lot harder, especially for larger systems. The free energy barrier for micellar aggregate breakup becomes impossible to overcome in a simulation of reasonable length, as evidenced by the lack of any aggregates of size intermediate between full micelles and oligomers in the aggregate size distributions. However, results for the critical micelle concentration as a function of temperature are still reliable, as they have been obtained by linking to states of higher temperatures for which equilibration is readily achieved. The choice of temperatures in Table 2 is dictated by two competing requirements, namely, the need to operate at sufficiently high temperatures so that equilibration is not impossible to achieve but also at sufficiently low temperatures so that micelles are well formed and intermediate states between full micelles and oligomers are not significantly populated. 3. Results and Discussion Table 1 lists all surfactants examined in this work. Micelle-forming systems are indicated by shaded cells, whereas systems undergoing macroscopic phase separation are indicated by the clear cells. The methodology employed to distinguish between micellization and phase separation was explained in detail in the previous section and is based primarily on the finite-size dependence of the calculated phase diagrams and also on the systemsize dependence of the osmotic pressure versus composition plots. There are roughly diagonal demarcation lines between phase-separating and micelle-forming systems. An important point we stress in our work is that we never see the same system show phase separation and micelle formation in different regions of parameter space. In general, decreasing the number of T segments or increasing the number of H segments in a molecule decreases the tendency to phase separate and leads to

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Table 2. Aggregation Properties for Diblock Systemsa system

T*

φcmc

d(ln φcmc)/d(1/T*)

φr

M(T*,φr)

HT H2T H4T H8T H2T2 H4T2 H8T2 H2T4 H4T4 H8T4 H4T8 H8T8 H4T16 H8T16

3.1 2.7 2.4 2.1 4.8 4.5 4.2 8.0 7.5 7.2 12 11.5 16 15

0.0312 0.0471 0.0702 0.0933 0.0221 0.0391 0.0672 0.0121 0.0171 0.0321 0.0141 0.0211 0.00491 0.00301

-181 -161 -151 -131 -351 -333 -291 -671 -662 -582 -1182 -1143 -2233 -2083

0.061 0.0745 0.101 0.131 0.0262 0.0493 0.0944 0.0172 0.0191 0.0625 0.0263 0.0353 0.0113 0.0175

988 602 414 414 662 463 363 12210 523 412 691 423 1108 915

a T* is the reference temperature for the data reported. The critical micelle concentration (volume fraction of surfactant) at T* is φcmc. Micellar aggregate size distributions were measured at T* and volume fraction of surfactant φr. M is the aggregation number corresponding to the maximum of the size distribution. Estimated 2σ statistical uncertainties in units of the last digit reported are given as subscripts: 0.0312 means 0.031 ( 0.002.

micellization. For example, when moving across the second row of Table 1, systems of architecture H2Ty form aggregates when y e 4 but phase separate when the number of tail segments is y g 8. When the number of T segments is kept at y ) 8 (fourth column), phase separation occurs for x e 2 and micellization for x g 4. The minimum ratio of hydrophilic to hydrophobic groups, H/T, required for formation of micelles decreases rapidly with increasing chain length. For HxTy diblock surfactants, this ratio ranges from 1 (for the shortest surfactants) to 1/4 (for the longest). Micelle sizes are functions of temperature and overall concentration of surfactant for a given system, making comparisons across different systems somewhat difficult. The maximum of the measured aggregate size distribution, M(T*,φr), at the reported temperature, T*, and overall volume fraction of surfactant, φr, is reported in Table 2. These conditions were selected in an attempt to enable comparisons across systems: the temperature is sufficiently low so that there is clear separation of micelles and oligomers in the aggregate size distributions, and the concentration is as near the cmc as is practical for the system sizes studied. It is clear from Table 2 that aggregation numbers increase when moving away from the boundary between phase separation and micellization, for example, by increasing the number of H groups with a fixed number of T groups. Micelle size also increases as the number of T groups is increased at a fixed number of H groups, but the effect is weaker. HTH triblock surfactants are slightly more prone to phase separation than diblocks with the same number of H and T segments. For example, H2T4 forms micelles, while HT4H phase separates; the same is true for H4T16 and H2T16H2. In other words, grouping together H segments of a molecule at one end protects against phase separation and favors formation of micellar aggregates. The minimum H/T ratio required for formation of micelles decreases from 1 to 1/2 as the chain length is increased. Interestingly, the behavior of HTH triblock surfactants can be mapped exactly to that of the diblocks that result by splitting them in the middle. For example, H2T16H2 has the same behavior as H2T8 (phase separation) while H4T16H4 and H4T8 both form micelles. Therefore, diblock surfactant dimerization at the T end does not change the basic micellization/phase separation behavior. The behavior of THT triblock surfactants is summarized in the bottom third of Table 1. The ordering of the columns

Figure 6. Phase behavior for systems with four T segments. Symbols are from simulations: (2) is for L* ) 10 (HT4, T4, T2H2T2) or L* ) 12 (HT4H, T2H4T2); (b) is for L* ) 15 (HT4), L* ) 18 (HT4H), or L* ) 20 (T2H4T2). Lines are from quasichemical theory.

in the table is by the total number of T groups; there are no entries in the first column for the THT architecture, as the minimum number of T groups is y ) 2. THT triblocks are significantly more prone to phase separation than either diblocks or HTH triblocks. In contrast to the other two architectures studied, THT surfactants can form “bridges” between aggregates with their hydrophobic end groups. A high ratio of H/T ) 2 is required to form micelles over the limited range of chain lengths studied. Systems located near the demarcation lines drawn in Table 1 will display evidence of impending crossover to the opposite behavior. In Figure 6 are reported phase diagrams for chains listed in Table 1 as phase-separating systems, all containing four T beads. T4, a previously reported27 homopolymer that cannot form aggregates, is also included. The progressive deviation from the quasichemical line on increasing the number of H beads indicates that the systems have some aggregation tendency though not sufficient to form well-defined micelles. In particular, a comparison between T2H2T2 and HT4H indicates that the latter has a greater tendency to aggregate and it is closer to the boundary line of Table 1 than the former. Figure 7 plots the volume fractions in the dilute phase against the reciprocal temperature for the same T4 systems as in Figure 6. Quasichemical theory predicts a linear trend with a decreasing slope on increasing the number of H beads, and the simulation data are in qualitative agreement with this prediction except for T2H4T2 and HT4H, the latter being, as mentioned, the closest system to cross over into the micellization region. In Tables 2-4, we report the aggregation properties of all micelle-forming systems. The reference temperature used was chosen as explained in the previous section. The volume fraction of surfactant at the cmc, φcmc, was calculated from the breakpoints of osmotic pressure versus surfactant volume fraction similar to those reported in Figure 2 and Figure 4. For all surfactants considered, φcmc increases with increasing temperature, as also observed in previous simulation studies of model systems.13,15,26 The heat of micellization is proportional to -d(ln φcmc)/d(1/T*). The slopes from the data in Tables 2-4 are negative, so that heats of micellization are positive. This is contrary to the observed behavior of real

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Figure 7. Volume fraction in the dilute phase for systems with four T segments. Symbols and lines are as in Figure 6. Table 3. Aggregation Properties for HTH Triblock Systemsa system

T*

φcmc

d(ln φcmc)/d(1/T*)

φr

HTH H2TH2 H4TH4 HT2H H2T2H2 H4T2H4 H2T4H2 H4T4H4 H2T8H2 H4T8H4 H4T16H4

2.1 1.7 1.4 4.0 3.2 2.8 5.8 5.2 9.5 8 13

0.0312 0.0493 0.0683 0.0211 0.0182 0.0322 0.00912 0.01334 0.00511 0.00221 0.00321

-151 -141 -111 -271 -271 -251 -552 -532 -1052 -1044 -1922

0.0453 0.0753 0.0801 0.0331 0.0281 0.0542 0.0141 0.0221 0.0101 0.0083 0.0081

a

Figure 8. Critical micelle concentration, φcmc, as a function of 1/T* for a series of surfactants with a single T group. Points are from our simulations, with dotted lines joining them for visual clarity. Solid lines are from quasichemical theory.

M(T*,φr) 332 241 201 342 231 191 271 191 382 311 332

Columns are as for Table 2. Table 4. Aggregation Properties for THT Triblock Systemsa

system T*

φcmc

TH4T 3.1 0.0381 TH8T 2.2 0.0121 T2H8T2 4.9 0.0262

d(ln φcmc)/d(1/T*) -291 -271 -562

φr

M(T,φr) f(T*,φr)

0.0501 372 0.0162 332 0.0301 272

0.732 0.812 0.693

a Columns are as for Table 2, except for the last one, f, which is the fraction of chains that form loops.

nonionic surfactants in water but in agreement with experimental measurements for nonionic surfactants in nonpolar solvents.41 In Figure 8 and Figure 9 are shown the temperature dependences of φcmc for a series of diblock surfactants containing a single T group and for a series of diblock surfactants with a H2 group. In both cases, lines obtained from quasichemical theory are also reported. The closest agreement between simulation and theory occurs for systems close to the demarcation line of Table 1, HT and H2T4. On the contrary, systems well into the micellization region, as in the case of H4T and H2T, show the greatest deviations. Considering that quasichemical theory does not take into account the formation of stable aggregates, the above observation is consistent with a progressive aggregation capability of the surfactants as the boundary between macroscopic phase separation and micellization is crossed. Systems close to this boundary, on both sides, will tend to display mixed characters (see also Figure 6). (41) Couper, A.; Gladden, O. P.; Ingram, B. T. Faraday Discuss. 1975, 59, 63.

Figure 9. Critical micelle concentration, φcmc, as a function of 1/T* for a series of surfactants with two H groups. Symbols and lines are as in Figure 8.

This is further confirmed by Figure 10 reporting cluster size distributions for H2T, H2T2, and H2T4 at the conditions listed in Table 2. On increasing the number of T beads, the micelle-forming systems are increasing their tendency to phase separate and, as a consequence, the average aggregate size increases and the size distribution broadens even covering very large M values. Typical configurations obtained under the conditions of Figure 10 are reported in Figure 11 showing the difference in the aggregate shape and size as the systems increase their phase separation tendency. While the volume fractions for the three systems are not the same, the systems are all at approximately 50% higher volume fraction than their respective cmc’s. As seen in Figure 11, the large aggregates that form in systems near the phase separation boundary tend to be elongated, “wormlike” micelles. The general trend in Table 2 is that for a given length of the tail section, the cmc values increase on increasing the number of H beads in the surfactant chain. The

Micellization and Phase Separation of Surfactants

Figure 10. Aggregate size distributions at the conditions of Table 2 for (0) H2T, (O) H2T2, and (×) H2T4. For the last two systems, the values have been multiplied by a factor of 5 for visual clarity.

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Figure 12. Critical micelle concentration, φcmc, as a function of 1/T* for a series of surfactants with four H groups and two or four T groups. Symbols and lines are as in Figure 8.

suggests that an increase of the length of the tail section or a decrease of the length of the head section produces surfactant architectures that are closer and closer to macroscopic phase separation. Similar conclusions can be derived for the triblock surfactants listed in Tables 3 and 4, although, as mentioned, in this case also geometry considerations become important. From Figure 12, it can be seen that by use of the above criteria based on comparisons with quasichemical theory, the surfactant H4T4 has a greater tendency to form micelles than H2T4H2 since the former shows the largest deviation from the theory and, indeed, the latter surfactant is closer to the boundary set in Table 1. In Table 4, the fraction of surfactant molecules, f, that form loops within micellar aggregates is listed for the THT systems. A loop is considered to form if any of the T segments of one hydrophobic end of the molecule is within the interaction range of a T segment of the other hydrophobic end. The fraction f is quite high at the conditions studied, indicating that most molecules participate in aggregates with both ends. However, a significant fraction, 1 - f, of the molecules have “free ends” which can eventually connect to other aggregates. This observation is consistent with the observed tendency of the THT systems in Table 1 to phase separate rather than aggregate and is consistent with findings from other simulation studies of THT surfactants.15 4. Conclusions

Figure 11. Typical configurations for the systems of Figure 10. Box sizes are L* ) 30, 40, and 60 for H2T, H2T2, and H2T4, respectively.

opposite trend is observed on increasing the length of the tail section for a fixed number of H beads. Since a lower cmc is usually interpreted as an indication of the formation of more stable aggregates, it can be inferred, at least for the systems here considered, that repulsions among groups of different natures are mainly responsible for the stabilization of the aggregate. At the same time, however, the classification of surfactants reported in Table 1

Our work, which has employed a simple surfactant model and also a solvent which does not display any unusual solvation effects, unequivocally suggests that a single surfactant-solvent mixture can display either macroscopic phase separation or micellization, but not both. These results appear to argue that the experimentally observed behavior for the CxEy surfactant, where both phase separation and micellization were observed, is a manifestation of the unusual solvation properties of water. Similarly, our results argue against a triblock copolymer system displaying a coexistence between a micelle-rich and a micelle-lean phase, as suggested recently by Semenov et al.5 and Bhatia and Russel.42 In (42) Bhatia, S. R.; Russel, W. B. Macromolecules 2000, 33, 5713.

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this context, it must be argued that the triblocks experimentally have been observed to aggregation numbers larger than M ) 50, that is, more than 50 chains can spontaneously assemble into a single micelle.43 Our results, shown in Table 4, for the THT surfactants which are most relevant to the experiments, only have M < 30. Thus, it is possible that our conclusions could be modified if one considers telechelic chains where larger aggregation numbers are found in the simulations. We have found that high values of the hydrophilic/ hydrophobic segment ratio favor phase separation and low values favor micellization. The minimum ratio required for micellization was found to be 1/4 for the longest diblock surfactant studied, H4T16. Triblock surfactants with the hydrophilic groups on the outside (HTH) have phase separation or micellization behavior identical to that of the constituent diblocks. Triblock surfactants with the hydrophobic groups on the outside (THT) are significantly more prone to phase separation than the other two architectures. The two definitions of micellization, that is, using a break in the osmotic pressure versus concentration plots and using the cluster size distributions, yield the same estimate for the cmc, at least over the concentration regimes where the two techniques overlap. The histogram method, which only utilizes snapshots of the system, cannot provide direct information on the lifetimes of these clusters; however, a sharp break in the osmotic pressure curve implies that intermediate states between full micelles and oligomers are not populated, thus ensuring (43) Serero, Y.; Aznar, R.; Porte, G.; Berret, J.-F.; Calvet, D.; Collet, A.; Viguier, M. Phys. Rev. Lett. 1998, 81, 5584.

Panagiotopoulos et al.

long-term stability of the micelles. Mere formation of clusters in a static snapshot of the system is generally insufficient to conclude that these nanostructures will be persistent enough to affect properties and to be observed experimentally. A final point relates to the criteria that determine the sizes and shapes of micelles which are formed in these solvents. In general, for nearly symmetric surfactant structures [i.e., HxTx] the surfactants will form nearspherical micelles. However, as one proceeds to reduce the size of the head, the surfactant will eventually phase separate. Very close to this boundary, the surfactant will tend to form elongated structures, for example, cylinders and worms, structures which are of great interest to modification of system viscosity as thickeners. There has been considerable interest in these issues recently, and we therefore suggest that the strategy of hugging the boundary between phase separation and micellization might yield interesting results in this context. Acknowledgment. Primary financial support for this work has been provided by the Office of Basic Energy Sciences, Department of Energy (DE-FG02-01ER15121). Additional support has been provided by the Petroleum Research Fund administered by the American Chemical Society and by the National Science Foundation (Grant CTS-9975625 to S.K.). We thank Dr. George Stan for many helpful discussions and for suggesting the study of “THT” architectures and Professor R. Nagarajan for many helpful discussions of this work and for placing it in its proper perspective. LA0156513