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Competing Ranges of Attractive and Repulsive Interactions in the Micellization of Model Surfactants Sumeet Salaniwal,† Sanat K. Kumar,*,†,§ and A. Z. Panagiotopoulos‡ Department of Materials Science and Engineering, The Pennsylvania State University, University Park, Pennsylvania 16801, and Department of Chemical Engineering, Princeton University, Princeton, New Jersey Received June 14, 2002. In Final Form: November 19, 2002 Using Monte Carlo simulations, we show that the range of repulsive head-head interactions has to be greater than or equal to the range of attractive tail-tail interactions to facilitate micellization in model surfactants. These results represent the first quantitative verification of a well accepted conjecture that the range of repulsions has to be much larger than the range of attractions for micellization to occur. Our results verify this conjecture but add to it by suggesting that micellization can even occur when the ranges of the two interactions are comparable. Studies of micelle structure show that we proceed from macroscopic phase separation, when the attractions are longer range than the head-head repulsions, to extended, wormlike micelles, which form when the two interactions have comparable range, to compact spheres, when the attractions are short range. These results provide new insights into the factors affecting the sizes and shapes of micelles and suggest that tuning the relative ranges of the repulsive and attractive interactions [e.g., through the addition of salt to a solution of ionic surfactant] can provide a facile means of controlling these structures.
Introduction It is now well-known that amphiphilic molecules spontaneously self-assemble into myriad nanostructures when their solution concentration exceeds a critical value, the critical micelle concentration (φcmc).1 Structurally, the basic ingredients of any amphiphilic molecule include a solvophilic functionality (known as the head group, denoted by H in this work) that is covalently bonded to a solvophobic moiety (known as the tail group, denoted by T). In many amphiphiles the head group is an ionic or polar species, while the tail group is a hydrocarbon chain. When the aqueous solution concentration of amphiphiles exceeds φcmc, tail groups of several of these amphiphiles self-assemble into a hydrophobic micellar core to minimize contact with the surrounding solvent, while the hydrophilic head groups, due to their propensity for the solvent, form the corona. Such an arrangement of amphiphiles within a micelle facilitates their stable existence as soluble entities in the solvent medium. More interestingly, the micellar core provides a suitable microenvironment for solubilization of solutes that are otherwise insoluble in the surrounding solvent. This property of micelles offers several applications in areas of detergency, molecular transport, and enhanced oil recovery. On a molecular level, it is accepted that micellization in aqueous environments is driven by the delicate balance between hydrophobic and hydrophilic interactions. In pioneering work, Pratt and co-workers2,3 concluded that the range of the repulsive interactions between head groups must be longer than the range of attractive tailtail interactions for micellization to occur. If this condition is not met, the amphiphilic solution would undergo †
The Pennsylvania State University. Princeton University. § Current address: Department of Chemical Engineering, Rensselaer Polytechnic Institute, Troy, NY. ‡
(1) Myers, D. Surfactant Science and Technology; VCH Publishers: New York, 1992. (2) Owenson, B.; Pratt, L. R. J. Phys. Chem. 1984, 88, 2905-2915. (3) Pratt, L. R.; Owenson, B.; Sun, Z. Adv. Colloid Interface Sci. 1986, 26, 69-97.
macroscopic phase separation, such as that observed for hydrocarbons in water.4 To substantiate these claims, Owenson and Pratt2 simulated a H1Ty [y > 1] surfactant with a long ranged head-head repulsion and showed that these molecules spontaneously self-assembled into micelles. As a corollary, these workers also suggest that long ranged head-head repulsions are necessary for micellization. This last claim has been refuted by Smit et al.,5 who considered HxTy with x, y > 2, with only short ranged repulsion between head groups, and showed that micelles are spontaneously formed. Several other simulation studies6-15 have found results consistent with the Smit et al. findings. In a recent work Kapila et al.16 have resolved the apparent discrepancy between the Owen and Pratt picture and the results of Smit et al. by considering a H1T12 surfactant with long ranged head-head repulsions and a H7T12 surfactant with short ranged head-head repulsions. While both of these molecules self-assemble into micelles, in contrast, a H1T12 surfactant with short ranged head-head repulsions did not form micelles. These results argue that, even if the H-H interactions are short (4) Tanford, C. The Hydrophobic Effect: Formation of Micelles and Biological Membranes; Wiley-Interscience: New York, 1991. (5) Smit, B.; Esselink, K.; Hilbers, P. A.; van Os, N. M.; Rupert, L. A. M.; Szleifer, I. Langnuir 1993, 9, 9-11. (6) Panagiotopoulos, A. Z.; Wong, V.; Floriano, M. A. Macromolecules 1998, 31, 912. (7) Floriano, M. A.; Caponetti, E.; Panagiotopoulos, A. Z. Langmuir 1999, 15, 3143. (8) Panagiotopoulos, A. Z.; Floriano, M. A.; Kumar, S. K. Langmuir 2002, 18, 2940. (9) Rodriguez-Guadarrama, L. A.; Talsania, S. K.; Mohanty, K. K.; Rajgopalan, R. Langmuir 1999, 15, 437. (10) Talsania, S. K.; Wang, Y.; Rajgopalan, R.; Mohanty, K. K. J. Colloid Interface Sci. 1997, 190, 92. (11) Mackie, A. D.; Onur, K.; Panagiotopoulos, A. Z. J. Chem. Phys. 1996, 104, 3718. (12) Mackie, A. D.; Panagiotopoulos, A. Z.; Szleifer, I. Langmuir 1997, 13, 5022. (13) Goetz, R.; Lipowsky, R. J. Chem. Phys. 1998, 108, 7397. (14) Maiti, P. K.; Lansac, Y.; Glaser, M. A.; Clark, N. A.; Rouault, Y. Langmuir 2002, 18, 1908. (15) Bernardes, A. T.; Henriques, V. B.; Bisch, P. M. J. Chem. Phys. 1994, 101, 645. (16) Kapila, V.; Harris, J. M.; Deymier, P. A.; Raghavan, S. Langmuir 2002, 18, 3728.
10.1021/la026076l CCC: $25.00 © 2003 American Chemical Society Published on Web 05/16/2003
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ranged, having a long head group, that is, HxTy with x > 1, increases the range of repulsive interactions sufficiently to facilitate micellization. Our objective in this paper is to address the central prediction of the work of Owenson and Pratt,2,3 who had suggested that the range of the repulsive interactions must be larger than the range of attractive intertail interaction for micellization to occur. We have considered lattice chains of architecture HxT4, with short ranged repulsions between head groups. However, we have varied the length of the head groups in the range 1 e x e 4 in a series of simulations. Note that each system is comprised of surfactant with a single x value, and hence no polydispersity effects are studied.2,3,5,16 We have then varied the range of the attractive interaction between tail beads and found through the aid of the lattice-based Monte Carlo simulations that micellization only occurs when the length of the head group is greater than or equal to the effective length scale of tail-tail attraction. For all cases where the tail-tail attraction is longer range, we find that the surfactant phase separates from the solvent. These statements quantify the earlier conjectures of Pratt and co-workers but show that it is sufficient to match the range of repulsive and attractive interactions to achieve micellization. More precisely, we find that one goes progressively from phase separation to micellization into wormlike objects as the range of the attractive interaction is varied from being longer range than the repulsive interhead interaction to being equal to the repulsive length scale. Further decreases in the attractive length scale then yield progressively more compact structures, culminating with the formation of spherical micelles. These results allow us to begin to understand complicated effects in surfactant physical chemistry, such as the additon of salt to an ionic surfactant, which serves to decrease the range of repulsive head-head interaction, thus yielding very dramatic structural changes. Simulation Model and Methodology Following past work,6,7 we employed grand canonical ensemble Monte Carlo (GCMC) simulations on model lattice amphiphiles of structure denoted by HxT4. We simulated cubic lattices of size L3 with a coordination number z ) 26, and utilized periodic boundary conditions in all three directions. Each lattice site is either occupied by a surfactant moiety or is vacant. The vacant lattice sites are assumed to represent the structureless solvent (denoted by s). Each amphiphilic molecule is represented as a fully flexible string of beads connected by bonds. The allowable bond lengths between successive monomers are l ) 1, x2, and x3, allowing some degree of bond length flexibility. No bond length potential is utilized. No overlap of monomeric units is permitted to account for the excluded volume effects. The only nonzero interaction is between two tail beads; that is, HH ) HT ) sH ) sT ) ss ) 0. Thus, the head-head repulsion is only a short ranged excluded volume interaction. The chain molecules behave as surfactants, since the tail-tail interaction is attractive. Thus, micellization is caused by the preference for the tail moieties to aggregate with other tails rather than with solvent or with head groups. This preference is enumerated by calculating the interchange energy parameter,
χij )
2ij - ii - jj kBT
(1)
which describes the normalized energetic cost of taking a pure i pair and a pure j pair and forming two i-j pairs.
Figure 1. Effective length scale of attraction between the head groups as a function of λ, as computed following eq 3. The line is a linear fit to the data points.
Clearly, χHs is zero, while χHT ) χTs > 0, restressing the fact that, on average, a tail group would not like to aggregate near head groups or solvent molecules. In contrast, the head and solvent molecules are athermal relative to each other. The tail-tail attraction is long ranged,
U(r) )
TT r
3
( λr)
exp -
(2)
where TT ≡ -2 denotes the bare interaction energy between any two T monomers, and r is the distance (in lattice units) between them. The reduced temperature, T*, is defined as T* ) kBT/TT, where kB is Boltzmann’s constant. The parameter λ, which is a positive quantity, is called the screening length, and its magnitude determines the characteristic range of the attractive interaction between tail monomers. By increasing λ, the range of the tail-tail interaction can be increased. We estimate the effective length scale of the pairwise interaction potential (eq 2) for a given λ value, 〈r〉, as
∫1∞r3U(r) dr 〈r〉 ) ∞ ∫1 r2U(r) dr
(3)
Since the integrals cannot be evaluated analytically, we have solved them numerically. In particular, note that the form of the potential (eq 2) will have well defined, finite integrals, for all λ < ∞, a fact that is also useful in the evaluation of long range interactions (see below). The 〈r〉 values obtained by this procedure are shown in Figure 1. It is immediately clear that 〈r〉 tracks λ but is not equal to it. Below we shall describe all of the results using λ, but it must be realized that the relevant variable for understanding surfactant behavior is actually 〈r〉. The initial configuration of the chains was generated at random, ensuring that no bead was placed on a previously occupied site. Elementary moves involved chain growth or elimination. We thus simulated in the grand canonical ensemble and used the configurational bias (CB) method to enhance the probability of chain growth.17 In the CB method, we only use the excluded volume criterion in chain growth, and the attractive tail-tail energetic interactions were used to calculate the acceptance probability following the standard Metropolis Monte Carlo method. All interactions were truncated at half the box size, and long range contributions to the energy were (17) Frenkel, D.; Smit, B. Understanding Molecular Simulation: From Algorithms to Applications; Academic Press: New York, 1996.
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included using standard means.18 That is, we assumed g(r) ) 1 for all distances past the cutoff in the calculation of long range corrections. In this context, we note that the form of the potential in eq 2 is particularly useful, since all integrals are bounded, thus facilitating the inclusion of long range interations. Depending on the simulation parameters, a typical simulation run consisted of 50-70 million trial configurations. We applied the histogram reweighting technique19,20 to probability distributions p(N, E) of system states with energy E and number of particles N obtained by GCMC simulations. This information then yielded temperaturecomposition (T-φ) phase envelopes of model amphiphilic systems, as well as other thermodynamic properties such as the heat capacity and the osmotic pressure of the solutions. The interested reader is referred to refs 6 and 7 for details of the simulation methods and techniques used to distinguish micellization from phase separation. To summarize: the micellization of these surfactant solutions was determined by monitoring the dependence of the osmotic pressure on composition. A relatively sudden decrease in the slope of this plot indicates either the formation of aggregates or phase separation, as has been discussed previously.6,7 For macroscopic phase separation, the slope after the break is a function of system size, decreasing to zero in the thermodynamic limit. In contrast, for micellization, the slope is independent of overall system size. The true thermodynamic character of a transition is easily revealed by performing an analysis of the system size dependence of the transition location. In particular, except near critical points, the effect of system size on the location of a first-order transition is only minor. In contrast, for systems forming finite aggregates, there is a strong dependence of the apparent location of the transition. This can be understood as follows. Consider a surfactant system that preferentially forms aggregates of N ) 100 molecules at very low concentrations of free molecules. Simulation in a system of volume V ) 1000 will show a “phase” of mole fraction N/V ) 0.1 corresponding to a single aggregate forming in the simulation cell. Doubling the system volume would not change the preferred aggregation number, and thus the apparent transition will move to half the previous mole fraction. Thus, by examining the system size dependence of the location of (apparently) first-order phase transition boundaries, we can reliably distinguish between micellization and macroscopic phase separation. We shall therefore use these methods to distinguish between phase separation and micellization, and for micellization we examine cluster size distributions and micellar shapes to understand more about the resulting aggregates. Results Crossover from Phase Separation to Micellization. Figure 2 presents the T-φ phase envelopes for the H4T4 amphiphile obtained from simulation for λ values of 2, 5, 7, and 10, respectively. For each value of λ, simulations on cubic lattices of two different sizes (three system sizes for λ ) 2) were performed to investigate the system size dependence of the phase envelopes. For λ ) 10 we used L ) 12 and 14, for λ ) 5 and 7 we employed L ) 10 and 12, and we employed L ) 8, 10, and 12, respectively, for λ ) 2. These results indicate that, for λ ) 2, 5, and 7, the (18) Allan, M. P.; Tildesley, D. J. Computer Simulation of Liquids; Oxford University Press: 1987. (19) Ferrenberg, A. M.; Swendsen, R. H. Phys. Rev. Lett. 1988, 61, 2635. (20) Ferrenberg, A. M.; Swendsen, R. H. Phys. Rev. Lett. 1989, 63, 1195.
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Figure 2. Temperature-composition phase diagrams for the H4T4 amphiphile. For λ ) 2, 5, and 7, circles denote the phase envelopes for L ) 10, squares for L ) 12, and traingles for L ) 8 (only for λ ) 2). For λ ) 10, circles denote the phase envelope for L ) 12, while squares for L ) 14.
Figure 3. Temperature-composition phase diagrams for the H3T4 system. Circles denote the phase envelopes for L ) 10, while squares for L ) 12.
model amphiphilic solution exhibits a predicted “phase” behavior that strongly depends on the system size, that is, L. As discussed above, this is a direct signature of micellization. In contrast, for λ ) 10, the system size independence of the phase boundaries indicates a firstorder phase transition, that is, macroscopic phase separation. These observations clearly show that, for this model amphiphilic system, there is a characteristic λ value at which there is a crossover from micellization to phase separation. Using Figure 1, we recognize that λ ) 7, that is, 〈r〉 ) 4, at the onset of phase separation. Thus, we conjecture that micellization occurs when the range of the tail-tail attraction is comparable to the length of the head group in this surfactant (i.e., x ) 4). To test the general validity of this result, we have considered three other model amphiphiles, that is, H3T4, H2T4, and H1T4, respectively. Figure 3 presents the T-φ phase envelopes for the H3T4 amphiphile obtained from simulation for λ ) 2 and 5, respectively. As seen in the figure, for λ ) 2 the phase envelopes for the two different lattice sizes exhibit a strong system size dependence, indicating micellization. However, for λ ) 5 the phase envelopes for the different lattices exactly overlap, indicating phase separation. The crossover from phase separation to micellization thus occurs for λ ) 4, which corresponds to a value 〈r〉 slightly smaller than 3. Similarly, for the H2T4 surfactant we find a crossover from micellization to phase separation at 〈r〉 ) 2. For the H1T4 case we find phase separation for all λ values we have tried, and we find micellization only occurring for a tail-tail potential which is only nonzero at contact, that is, for a very short ranged potential. While this trend is consistent with the results obtained from the three longer surfactants, we do not include it in Figure 4, which plots the 〈r〉 at the crossover between micellization and phase separation as
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Figure 4. Plot of the critical value of 〈r〉 as a function of the length of the head group for the HxT4 with x > 1 surfactants. The line is a fit to the data.
Figure 5. Simulation estimates of ln[φcmc] versus 1/T* for the H4T4 systems which exhibit micellization. The circles represent λ ) 2, squares λ ) 5, and triangles λ ) 7. The lines are linear fits to the data
a function of the length of the head group of the surfactant. While we have only three points on our plot, they are sufficient to strongly suggest that the length scales of attractive interactions have to match the length of the head group with short range repulsions to induce micellization. In conjunction with past work,16 we argue that a similar effect may be achieved by considering a short head group with a long ranged head-head repulsion. While our results are qualitatively consistent with the conjectures of Pratt and co-workers,2,3 we assert that it is not necessary for the range of repulsion to be much larger than the range of attractions to facilitate micellization. Rather, a crossover between phase separation and micellization occurs when these two scales are roughly matched to each other. Structural Details of Micelles. We now consider the structural aspects of the micelles formed. We consider the H4T4 system for the three smaller λ values of 2, 5, and 7, respectively. In all these cases, we found “phase diagrams” which change with system size, a feature we have argued as being a signature of micellization. Following past work, we have examined the osmotic pressure as a function of volume fraction surfactant, to locate the critical micelle concentration, φcmc. We do not show any examples, since the procedure follows that reported in refs 6 and 7. Figure 5 shows a plot of ln[φcmc] versus 1/T* for H4T4 at the three different λ values where micelles form. First of all, note that, for a given φcmc value, the effect of increasing λ results in an increase in the micellization temperature. Put another way, at a given temperature, reducing the λ increases the cmc concentration, suggesting that the propensity for micellization decreases. Also note that the slope of each curve, which corresponds to the enthalpy change on mixing, increases with increasing λ value. These enthalpy values become effectively constant when normalized by 〈r〉, suggesting
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Figure 6. Volume fraction distribution of micellar aggregation number (Nagg) for H4T4 model amphiphiles. The circles correspond to the system with λ ) 2, T* ) 2.4, βµ ) -6.9 [β ) 1/kBT, is the thermodynamic temperature, and µ is the chemical potential]; squares for λ ) 5, T* ) 4.0, βµ ) -6.7; and triangles for λ ) 7, T* ) 4.6, βµ ) -6.6.
that the increase in enthalpy of micellization with increases in λ might merely reflect the increasing range of attractions. To emphasize the role of λ on the structure of the micelles which form, in Figure 6 we present the distributions of micellar size for the H4T4 amphiphile with λ ) 2, 5, and 7, respectively. All the curves shown correspond to temperatures at which φcmc ≈ 0.0135 in each case. As expected for an amphiphilic system exhibiting micellization, a bimodal aggregate size distribution is obtained for all three λ values considered. The primary peaks (Nagg ∼ 1), which simply represents unaggregated surfactant, coincide for all the three curves. However, the secondary peak, which indicates the preferred micellar aggregation number, shifts to higher values as λ increases. Also, the width of the secondary distribution increases as the value of λ increases. This implies that, as the attractive interaction between the tail groups becomes increasingly long ranged, amphiphiles self-assemble into larger-sized looser aggregates, which reflect the fact that these objects are looking more like truly macrophase separated objects. To investigate the effect of increases in micellar size on their shape, we have examined snapshots of typical micellar aggregates formed in the H4T4 model amphiphilic system with λ ) 2, 5, and 7 (see Figures 7-9). The figures show that, as the value of λ increases, the shape of micelles changes from compact spherical to large spherical to wormlike. For λ ) 10, macroscopic phase separation occurs. Thus, the transition from micellization to macroscopic phase separation is quite gradual, involving changes to the morphological properties of micelles. However, we also now understand that as we reduce the range of attractive interactions, we progress from macroscopic phase separation to spherical micelles, while traversing elongated wormlike objects which occur at the borderline of phase separation. Discussion The simulations presented here investigated the role of the characteristic range of tail-tail attraction on the solution behavior of model diblock amphiphiles. In previous work6,7 on amphiphilic molecules with short ranged tail-tail attraction and head-head repulsion (squarewell potential with width of x3), we investigated the effect of the relative lengths of the head and tail groups on micellization. The most important observation there was that, as the length of the tail groups increased, the minimum ratio of the length of the hydrophilic to hydrophobic groups (H/T) required for micelle formation
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Figure 7. Snapshot of a micellar aggregate of the H4T4 amphiphile with λ ) 2. The red denotes the hydrophilic moieties while the black denotes hydrophobic monomers. The figure corresponds to T* ) 2.4, βµ ) -6.9, φ ) 0.0268, and φcmc ) 0.0135.
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Figure 9. Snapshot of a micellar aggregate of the H4T4 amphiphile with λ ) 7. The figure corresponds to T* ) 4.6, βµ ) -6.6, φ ) 0.0526, and φcmc ) 0.0137.
of tail groups also plays an important role in determining whether an amphiphilic solution undergoes micellization or macroscopic phase separation. Kapila et al.16 also reached a qualitatively similar conclusion regarding the role of head-head repulsion and the length of the head groups. Our present simulations do not attempt to address the role of tail length on micellization versus phase separation. A quantitative description of these factors and their complex interplay requires further simulations on a series of HxTy model amphiphiles. Conclusions
Figure 8. Snapshot of a micellar aggregate of the H4T4 amphiphile with λ ) 5. The figure corresponds to T* ) 4.0, βµ ) -6.7, φ ) 0.0423, and φcmc ) 0.0134.
decreased quite rapidly. For example, H1T2 phase separated, while H2T2 exhibited micellization. On the other hand, both H1T8 and H2T8 amphiphiles underwent phase separation, while H4T8 formed stable micellar aggregates. Clearly, besides the range of tail-tail attraction, the length
Our primary conclusion is that the range of attractive interactions has to be comparable to or smaller than the range of repulsive interactions to facilitate micellization by these model surfactants. This conclusion substantiates the earlier conjectures of Pratt and Owensen and extends these ideas by suggesting that micellization can even occur when the repulsions and attractions are comparable in range. A simple corollary, which seems to be well appreciated now, is that the range of repulsions is defined by the length of the hydrophobic chain, and thus long range repulsions can occur for long head groups even though the bare interactions between two H moieties are short ranged. In agreement with intuition, we find that, as λ is reduced, one progresses from phase separation to elongated micelles, which then become more compact with smaller ranges of the attractive interactions. Acknowledgment. A.Z.P. thanks the National Science Foundation (Grant CTS-9975625) and ACS-PRF (Grant 38165-AC9) for financial support. LA026076L
