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Langmuir 1996, 12, 1098-1100
Entropy-Driven Vertical Phase Separation in a Binary Mixture of Amphiphiles at the Air-Water Interface Debashish Chowdhury* Institut fu¨ r Theoretische Physik, Universita¨ t zu Ko¨ ln, D-50923 Ko¨ ln, Germany, and Department of Physics, Indian Institute of Technology, Kanpur 208016, India Received July 3, 1995. In Final Form: October 16, 1995
1. Introduction Amphiphilic molecules, e.g., soap and phospholipid molecules, are known to form various types of selfassemblies, e.g., micelles, vesicles, etc.1-4 Self-assemblies of binary mixtures of amphiphiles have received attention in recent years.5 We are interested in binary mixtures of amphiphiles of two different sizes at the air-water interface.6-10 The aim of this paper is to introduce a microscopic lattice model of a binary mixture of long-chain and short-chain amphiphilic molecules in a system where water is separated from the air above it by a sharp welldefined interface and to report an interesting entropydriven phenomenon of phase separation observed in a set of computer experiments on this model. More specifically, so far as the interaction energies are concerned, the two types of molecules do not differ in any respect; therefore, they may be regarded as chemically identical. The hydrophilic heads of the two types of amphiphiles have identical size but the lengths of their hydrophobic tails are different. In the computer experiments the heads of all the amphiphiles are initially put randomly on the sites of the same lattice plane immediately below the air-water interface and the tails extend fully into the air. If the initial surface density of the amphiphiles is sufficiently high, most of the short amphiphiles are found to have moved out of the monolayer by a few molecular layers into water, when the system attains equilibrium, thereby exhibiting a “phase separation” in a direction perpendicular to the interface. We interpret this “out-of-plane” phase separation of the chemically identical amphiphiles of different lengths at the air-water interface as a phenomenon of steric exclusion of short amphiphiles from the monolayer of long amphiphiles. 2. The Model Our model is based on the Larson model of microemulsions.11,12 Some modifications and generalizations of the * Address correspondence to Indian Institute of Technology, e-mail address
[email protected]. (1) Tanford, C. The Hydrophobic Effect, 2nd ed.; John Wiley: New York, 1980. (2) Israelchvili, J. N. Intermolecular and Surface Forces; Academic Press: New York, 1985. (3) Gelbert, W., Ben Shaul, A., Roux, D., Eds. Micelles, Membranes, Microemulsions and Monolayers; Springer: Berlin, 1994. (4) Gompper, G.; Schick, M. Self-assembling amphiphilic systems. Phase Transitions and Critical Phenomena; Domb, C., Lebowitz, J. L., Eds.; Academic Press: New York, 1994; Vol. 16. (5) Seifert, U. Phys. Rev. Lett. 1993, 70, 1335 and references therein. (6) Moehwald, H. Annu. Rev. Phys. Chem. 1990, 41, 441; Rep. Prog. Phys. 1993, 56, 653. (7) Knobler, C. M. Adv. Chem. Phys. 1990, 77, 397. (8) McConnell, H. M. Annu. Rev. Phys. Chem. 1991, 42, 171. (9) Knobler, C. M.; Desai, R. C. Annu. Rev. Phys. Chem. 1992, 43, 207. (10) Lu, J. R.; Simister, E. A.; Thomas, R. K.; Penfold, J. J. Phys. Condens. Matter 1994, 6, A403. (11) Larson, R. G.; Scriven, L. E.; Davis, H. T. J. Chem. Phys. 1985, 83, 2411.
0743-7463/96/2412-1098$12.00/0
Larson model have been made in recent years.13 The system is modeled as a simple cubic lattice of size Lx × Ly × Lz. Each lattice site is occupied by a classical Ising spin; each Ising spin has only two possible allowed values, namely, +1 and -1. If the ith lattice site is occupied by water, it is represented by Si ) 1, whereas if it is occupied by air it is represented by Si ) -1; Si being the classical Ising spin at the ith site. Each amphiphile, denoted by the symbol HmTn, consists of a string of m head units and n tail units; each of these units occupies a lattice site and the distance between any two successive units is precisely one lattice spacing. If a lattice site, say the jth, is occupied by a head unit, the corresponding value of the jth Ising spin (Sj) is assumed to be +1 whereas Sj ) -1 if the jth site is occupied by a tail unit. The total length of each of the amphiphiles is l ) m + n. We shall refer to each unit of an amphiphile, regardless of whether it is a part of the head or tail, as a monomer. The intermolecular interactions are taken into account through the interaction between the corresponding pair of Ising spins. Two spins interact provided the spins are located on the nearest-neighbor sites on the lattice; the interaction is attractive if both the spins are in identical states and repulsive otherwise. The initial states are always constructed in such a way that air occupies the upper part while water occupies the lower part of the lattice. During the updating of the states of the system in our MC simulation, air and water molecules were not allowed to exchange position as dispersions of air and water inside each other are not possible in our model. The moves allowed for the amphiphiles in our model are as follows: (i) reptation, this move effectively mimics the reptile-like slithering of the amphiphile along its own contour by one lattice spacing, and hence the name; (ii) spontaneous chain-buckling, a portion in the middle of the chain is allowed to buckle spontaneously;14 (iii) kink movement, a kink formed by buckling or reptation can move to a new position;14 (iv) lateral diffusion at the interface, those amphiphiles whose heads are located no deeper than the molecular layer at the interface are allowed to move laterally where one of the four possible directions is chosen randomly and each of the units of the amphiphile is moved in that direction by one lattice constant.15 Each of these moves is possible only if the new positions are not occupied by any other amphiphile. We measure the temperature of the system in the unit in which the monomer-monomer interaction J ) 1.0 and the Boltzmann constant kB ) 1.0. We follow the standard “Metropolis algorithm”: each of the above mentioned moves takes place certainly if ∆E < 0 and with a probability proportional to exp(-∆E/T) if ∆E g 0, where ∆E is the change in energy that would be caused by the proposed move of the amphiphile under consideration. Each amphiphile is allowed to try each of the above mentioned moves once during each MC step. Time is measured in the units of the MC steps per molecule. Notice that because the monomers of the same chain as well as of different chains are not allowed to occupy the same lattice site, there is a hard-core intrachain as well as interchain repulsion whenever the chain separations are smaller than one lattice spacing. Moreover, at any nonvanishing temperature, during the out-of-line thermal (12) Larson, R. G. J. Chem. Phys. 1988, 89, 1642; J. Chem. Phys. 1989, 91, 2479. (13) Jan, N.; Stauffer, D. J. Phys. I 1994, 4, 345 and references therein. (14) Bernardes, A. T.; Liverpool, T. B. J. Phys. II 1995, 5, 1457. (15) Chowdhury, D. J. Phys. II 1995, 5, 1469.
© 1996 American Chemical Society
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fluctuations of the chains, the hard-core repulsion leads to steric repulsion between the chains. Furthermore, in the present formulation, there are no potential energies associated with the bending and torsion of isolated amphiphiles in water. But, we plan to extend our model to include these extra energies in a future publication. Since both types of amphiphiles are represented by the Ising spin variables and since the strength of the interaction is assumed to be identical for both, the only difference between them lies in the two different lengths l. In the HmTn representation of the amphiphiles, the size of the heads of both types of amphiphiles is m ) 2, but the lengths of the tails of the long and the short amphiphiles are, respectively, n ) 15 and n ) 5. Thus, the lengths of the long and short amphiphiles in our computer experiments are chosen to be 17 and 7, respectively. The gross features of the spatiotemporal organization of the constituent molecules can be expressed through the time-dependent profiles of concentrations of air, water, and amphiphiles in the Z-direction. Thus, N(z,t) is the number of amphiphilic molecules at a depth z, measured from the top, at a time t. So far as the concentration profiles for the amphiphiles are concerned, one can calculate two different quantities: at each layer one can count the number of monomers constituting the amphiphiles or one can count just the number of amphiphilic heads in that layer. 3. Results and Discussion We begin with a mixture of 405 long amphiphiles and 405 short amphiphiles; the heads of all the amphiphiles were initially put randomly on the sites in a horizontal lattice plane immediately below the air-water interface in a lattice of size 30 × 30 × 100 in such a way that each of the molecules was initially fully extended in the vertical direction. The system was then equilibrated at a temperature T ) 2.5 using the dynamics described above. The concentration profiles of the monomers of the long as well as the short amphiphiles are shown in Figure 1a. The two profiles are shifted with respect to each other in a direction perpendicular to the air-water interface thereby indicating a vertical phase separation of the two types of the amphiphiles. This is, perhaps more clearly, also visible in the corresponding profile of only the heads shown in Figure 1b. The majority of the short amphiphiles have been forced out of the monolayer, which is now dominantly populated by the long amphiphiles. We shall refer to this experiment as the first experiment. The effect disappears at higher temperatures, e.g., at T ) 4.0, as a fraction of the amphiphiles then moves into water spontaneously. Besides, no “in-plane” phase separation of the two components of the mixture of the amphiphiles was observed at any temperature. Next we make further experiments to establish the physical mechanism of this phase separation. We reduce the concentrations of the amphiphiles of both types by a factor of 9, but keeping the relative concentration ratio of the two types 1:1, as before, without changing the size of the system and the temperature. In other words, we repeat the first experiment now with 45 long amphiphiles and 45 short amphiphiles in a system of size 30 × 30 × 100 at T ) 2.5. Unlike Figure 1b, no vertical phase separation is now observed in the equilibrium profiles plotted in Figure 2. Notice that the total number of amphiphiles (90) is one-tenth of the total number of lattice sites available in the plane immediately below the airwater interface. We refer to this experiment as the second experiment. Do Figures 1 and 2 imply that phase separation seen in the first experiment occurs only if the surface density
Figure 1. (a) The equilibrium concentration profile of the monomers of the amphiphiles along the veritical direction (i.e., perpendicular to the air-water interface). The system consists of 30 × 30 × 100 latice sites and contains a mixture of 405 long amphiphiles (l ) 17) and 405 short amphiphiles (l ) 7) equilibrated at a temperature T ) 2.5 The solid lines correspond to the long amphiphiles whereas the dashed lines correspond to the short amphiphiles. (b) Same as in part a, except that the number of only the hydrophilic heads of the amphiphiles in each layer is shown.
Figure 2. Equilibrium concentration profile of the hydrophilic heads of the amphiphiles along the vertical direction (i.e., perpendicular to the air-water interface). The system consists of 30 × 30 × 100 lattice sites and contains a mixture of 45 long amphiphiles (l ) 17) and 45 short amphiphiles (l ) 7) equilibrated at a temperature T ) 2.5. The solid lines correspond to the long amphiphiles whereas the dashed lines corresponds to the short amphiphiles.
of the amphiphiles is sufficiently high (i.e., the interfacial area per amphiphile is suffficiently small)? In order to
1100 Langmuir, Vol. 12, No. 4, 1996
Figure 3. Same as in Figure 2, except that the system now consists of 10 × 10 × 100 lattice sites.
Notes
If the surface density is high, the tails of the amphiphiles encounter stronger steric repulsion. If the short chains move down by one or two molecular layers, that leaves more empty space for the unconstrained movement of the tails of the neighboring long amphiphiles. The same would not be possible if the longer amphiphiles, instead of the shorter ones, were selectively pushed into water by the same amount. It has been argued also earlier in a different context16 that the short amphiphiles, by placing themselves in between long amphiphiles, create more space for the movement of the long amphiphiles. How far are the small amphiphiles pushed out into water? The long amphiphiles tend to push out the short amphiphiles so as to gain entropy. But, as the small amphiphiles keep moving deeper into water, the system keeps losing more and more energy because of the increased contact between the tails of the short amphiphiles and water although it keeps gaining entropy. The equilibrium configuration is attained when the loss of the energy is compensated by the gain of entropy. 4. Summary and Conclusion
Figure 4. Equilibrium concentration profile of the hydrophilic heads of the amphiphiles along the vertical direction. The system consists of 30 × 30 × 100 lattice sites and contains 810 long amphiphiles (l ) 17), but no short amphiphiles, equilibrated at a temperature T ) 2.5.
check this possibility, we again repeat the experiment keeping now the numbers of amphiphiles of both varieties fixed at 45 but reducing the air-water interfacial area by a factor of 9, i.e., we repeat the experiment on a lattice of size 10 × 10 × 100. Notice that now the actual numbers of amphiphiles are identical to those in the second experiment but the surface density of the amphiphiles is 9 times larger; in fact, intentionally we have now chosen the surface density of the amphiphiles to be identical to that in the first experiment. In this third experiment we again observe a phase separation (Figure 3) which is very similar to that observed in the first experiment. This establishes that the vertical phase separation takes place only if the surface density of the amphiphiles is sufficiently high. If high surface density leads to the forced exclusion of some amphiphiles from the monolayer constructed initially then, in principle, the same phenomenon should be observed also if in the first experiment all the 810 amphiphiles had the same size. That, indeed, is true as shown in Figure 4. This observation, naturally, leads to the next question: Why is the majority of the short amphiphiles in the binary mixture so selectively pushed out of the original monolayer? The answer to this question is very closely related to the physical mechanism that leads to the phenomenon of vertical phase separation.
We have developed a novel microscopic model of binary mixtures of chemically identical amphiphilic molecules, of different sizes, in water and demonstrated an “out-ofplane” phase segregation at the air-water interface at sufficiently low temperatures provided the surface density of the amphiphiles is sufficiently high. Since the two components of the mixture of the amphiphiles differ only in their sizes but not energetically, the phase separation seen in our computer experiments should be regarded as entropy-driven, as explained above. Somewhat similar effects of entropy on the structure of bidisperse polymer brushes have also been observed earlier.17 The phenomenon of entropy-driven phase separation has become an active area of research in recent years.18,19 Detailed investigation of the phenomenon reported above would in future involve addressing the following questions: what are effects of varying (a) the ratio of the surface densities of the two components in the mixture, (b) the ratio of the lengths of the two types of amphiphiles, and (c) the strength of the intermonomer interaction. Moreover, the competition between “out-of-plane” segregation and “in-plane” segregation of two chemically nonidentical species of amphiphiles at the air-water interface should also be investigated. The phenomenon observed in our computer experiments is similar to those observed in recent experiments on similar systems.20,21 For the best demonstration of this “out-of-plane” phase separation, the monomer-monomer interaction energies of the two types of amphiphiles should be as close to each other as possible; this would minimize the tendency towards “in-plane” phase separation of the two components. Acknowledgment. I thank D. Stauffer and J. Zittartz for hospitality at the University of Koln. I also thank A. T. Bernardes, P. Y. Lai, T. B. Liverpool, D. Stauffer, and D. Woermann for useful discussions and D. Stauffer for a critical reading of the manuscript. This work was supported by SFB341 Aachen-Ju¨lich-Ko¨ln. LA950536B (16) Szleifer, I.; Kramer, D.; Ben-Shaul, A.; Gelbert, W. M.; Safran, S. A. J. Chem. Phys. 1990, 92, 6800. (17) Lai, P. Y.; Zhulina, E. B. Macromolecules 1992, 25, 5201. (18) Biben, T.; Hansen, J. P. Phys. Rev. Lett. 1991, 66, 2215. (19) Frenkel, D. J. Phys. Condens. Matter 1994, 6, A71 and references therein. (20) Runge, F. E.; Yu, H. Langmuir 1993, 9, 3191. (21) Fang, J. Y.; Uphaus, R. A. Langmuir 1994, 10, 1005.
