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Macrophase and Microphase Separations for Surfactants Adsorbed on Solid Surfaces: A Gauge Cell Monte Carlo Study in the Lattice Model Fengxian Zheng, Xianren Zhang,* and Wenchuan Wang DiVision of Molecular and Materials Simulation, Key Lab for Nanomaterials, Ministry of Education, Beijing UniVersity of Chemical Technology, Beijing 100029, China ReceiVed January 7, 2008. In Final Form: February 5, 2008 By combining the gauge cell method and lattice model, we study the surface phase transition and adsorption behaviors of surfactants on a solid surface. Two different cases are considered in this work: macrophase transition and adsorption in a single-phase region. For the case of macrophase transition, where two phases coexist, we investigate the shape and size of the critical nuclei and determine the height of the nucleation barrier. It is found that the nucleation depends on the bulk surfactant concentration. Our simulations show that there exist a critical temperature and critical adsorption energy, below which the transition from low-affinity adsorption to the bilayer structure shows the characteristic of a typical first-order phase transition. Such a surface phase transition in the adsorption isotherm is featured by a hysteresis loop. The hysteresis loop becomes narrower at higher temperature and weaker adsorption energy and finally disappears at the critical value. For the case where no macrophase transition occurs, we study the adsorption isotherm and microphase separation in a single-phase region. The simulation results indicate that the adsorption isotherm in adsorption processes is divided into four regions in a log-log plot, being in agreement with experimental observations. In this work, the four regions are called the low-affinity adsorption region, the hemimicelle region, the morphological transition region, and the plateau region. Simulation results reveal that in the second region the adsorbed monomers aggregate and nucleate hemimicelles, while adsorption in the third region is accompanied by morphological transitions.
1. Introduction Fundamental and practical aspects of surfactant adsorption at the solid/aqueous interface have received considerable attention due to their importance in many processes, such as detergency, mineral flotation, and oil recovery. To characterize the adsorption behavior, the surface excess, the rate of adsorption, and the structure of the adsorbed surfactant layers are mainly investigated. The surface excess and the rate of adsorption can be gained by adsorption isotherms. Therefore, the adsorption isotherms were determined by many researchers1-20 for the study of equilibrium surfactant adsorption. In general, a typical adsorption isotherm of surfactants observed by experimental methods can be divided into four regions, when it is plotted on a log-log scale.1-7 In * To whom correspondence should be addressed. E-mail: zhangxr@ mail.buct.edu.cn. (1) Somasundaran, P.; Fuerstenau, D. W. J. Phys. Chem. 1966, 70, 90. (2) Koopal, L. K.; Lee, E. M.; Bo¨hmer, M. R. J. Colloid Interface Sci. 1995, 170, 85. (3) Scamehorn, J. F.; Schechter, R. S.; Wade, W. H. J. Colloid Interface Sci. 1982, 85, 463. (4) Fuerstenau, D. W. J. Colloid Interface Sci. 2002, 256, 79. (5) Fan, A.; Somasundaran, P.; Turro, N. J. Langmuir 1997, 13, 506. (6) Goloub, T. P.; Koopal, L. K. Langmuir 1997, 13, 673. (7) Harwell, J. H.; Rorberts, B. L.; Scamehorn, J. F. Colloids Surf. 1998, 32, 1. (8) Chandar, P.; Somasundaran, P.; Turro, N. J. J. Colloid Interface Sci. 1987, 117, 148. (9) Chandar, P.; Somasundaran, P.; Waterman, K. C.; Turro, N. J. J. Phys. Chem. 1987, 91, 148. (10) Atkin, R.; Craig, V. S. J.; Wanless, E. J.; Biggs, S. AdV. Colloid Interface Sci. 2003, 103, 219. (11) Paria, S.; Khilar, K. C. AdV. Colloid Interface Sci. 2004, 110, 75. (12) Tiberg, F.; Jo¨nsson, B.; Tang, J.; Lindman, B. Langmuir 1994, 10, 2294. (13) Tiberg, F.; Jo¨nsson, B.; Tang, J.; Lindman, B. Langmuir 1994, 10, 3714. (14) Brinck, J.; Tiberg, F. Langmuir 1996, 12, 5042. (15) Kiraly, Z.; Bo¨rner, R. H. K.; Findenegg, G. H. Langmuir 1997, 13, 3308. (16) Kiraly, Z.; Findenegg, G. H. Langmuir 2000, 16, 8842. (17) Atkin, A.; Craig, V. S. J.; Biggs, S. Langmuir 2000, 16, 9374. (18) Atkin, A.; Craig, V. S. J.; Biggs, S. Langmuir 2001, 17, 6155. (19) Fleming, B. D.; Biggs, S.; Wanless, E. J. J. Phys. Chem. B 2001, 105, 9537. (20) Brinck, J.; Jo¨nsson, B. Langmuir 1998, 14, 1058.
the first region, the amount of adsorbed surfactants is very low and the interaction between adsorbed surfactants is negligible. The second region shows a sudden increase of adsorption due to the formation of primary aggregates, known as hemimicelles, when the critical surface aggregation concentration (CSAC) is reached. In the third region, the rate of adsorption is slow, compared to that in the second region. The fourth region shows a plateau near or above the critical micelle concentration (CMC). In addition to the experimental research, the adsorption behaviors were also studied by computer simulation. For example, Wijmans and Linse21 studied the self-assembly of surfactants on hydrophilic surfaces by using the lattice Monte Carlo method. They found that the surfactants formed almost spherical micellar aggregates or a prolatelike structure on hydrophilic surfaces at different adsorption energies or head/tail ratios. Our recent computer simulations22-24 also show that the adsorption behaviors of surfactants are strongly affected by the nature of solid surfaces. Besides the adsorption mechanism, the surface phase transition also has fundamental importance for aqueous surfactant systems. The surface phases coexisting at the interface are in thermodynamic equilibrium with both the bulk and each other. Recently, there has been growing interest in the phase behavior of the adsorbed monolayers of surfactants at the air-water interface. It was proved by experiments that the first-order phase transition can occur for amphiphilic molecules in adsorbed monolayers.25,26 This kind of phase transition is characterized by the surface pressure-time isotherm, which shows a conspicuous cusp point when the pressure reaches a certain value.27-33 It was also reported (21) Wijmans, C. M.; Linse, P. J. Phys. Chem. 1996, 100, 12583. (22) Zheng, F. X.; Zhang, X. R.; Wang, W. C.; Dong, W. Langmuir 2006, 22, 11214. (23) Zheng, F. X.; Zhang, X. R.; Wang, W. C. J. Phys. Chem. C. 2007, 111, 7144. (24) Zhang, X. R.; Chen, G. J.; Wang, W. C. J. Chem. Phys. 2007, 127, 034506. (25) Vollhardt, D.; Melzer, V. J. Phys. Chem. B 1997, 101, 3370. (26) Melzer, V.; Vollhardt, D.; Weidemann, G.; Brezesinski, G.; Wagner, R.; Mo¨hwald, H. Phys. ReV. E 2000, 57, 901.
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that no indicative feature of phase transition in the adsorbed monolayer is observed above a certain temperature and below a corresponding bulk concentration.27,33,34 By analogy with the first-order phase transitions at the air/ water surface, we assume that first-order surface phase transitions may also take place at solid surfaces for aqueous surfactant systems. Different from the case for the air/water interface, the presence of a solid surface provides not only geometric constraints on the surfactants, but also energetic interactions on the molecules adsorbed, which makes the question even more complicated. If the first-order surface phase transition does occur near the solid surface, the transition must start with heterogeneous nucleation. Nucleation is an activated process, and the barrier for the formation of a nucleus must depend on the properties of the solid surface. In the present work, we attempt to understand nucleation of the surface phase transition (if it exists) and microphase transition in a single-phase region (if no macrophase transition exists) by combining the lattice Monte Carlo (LMC) and gauge cell methods at the molecular level. The remainder of the paper is organized as follows. In section 2, the models and the simulation methods are described. Then, the simulated results for the adsorption mechanism and phase behaviors of surfactants on a hydrophilic surface are discussed in detail in section 3. This is followed by a brief summary of the main conclusions. 2. Model and Methods In this work, the LMC method developed by Larson35-37 was combined with the gauge cell method to simulate the phase behaviors of surfactants on hydrophilic surfaces. Larson’s lattice model uses a fully occupied simple cubic lattice. In this model, solvent molecules (water), represented by W, occupy single sites, while a surfactant molecule, HxTy, consists of x hydrophilic head units, H, and y hydrophobic tail units, T, and occupies x + y connected sites. Each of these sites can only interact with its 26 neighbors that lie within one lattice spacing in each of the three directions. Monte Carlo simulations performed with the model are very efficient at simulating the formation of surfactant aggregate structure. By combining the model with other computer simulation techniques,38-45 the Larson model has been widely used to study the micellization and phase equilibria of surfactant systems,38-41 micelle formation dynamics,42,43 and micelle behaviors in shear flow.44,45 The models were also used to study phase separation and self-assembly in supercritical solventsurfactant46-48 and surfactant-inorganic solvent systems.49,50 Using this method, we recently studied the equilibrium22 and kinetic51 (27) Vollhardt, D.; Fainerman, V. B.; Emrich, G. J. Phys. Chem. B 2000, 104, 8536. (28) Melzer, V.; Vollhardt, D. Phys. ReV. Lett. 1996, 76, 3770. (29) Melzer, V.; Vollhardt, D.; Brezesinski, G.; Mo¨hwald, H. J. Phys. Chem. B 1998, 102, 591. (30) Hossain, M. M.; Yoshida, M.; Iimura, K.; Suzuki, N.; Kato, T. Colloids Surf., A 2000, 171, 105. (31) Hossain, M. M.; Suzuki, T.; Kato, T. Langmuir 2000, 16, 9109. (32) Hossain, M. M.; Yoshida, M.; Kato, T. Langmuir 2000, 16, 3345. (33) Hossain, M. M.; Kato, T. Langmuir 2000, 16, 10175. (34) Islam, M. N.; Kato, T. J. Colloid Interface Sci. 2002, 252, 365. (35) Larson, R. G.; Scriven, L. E.; Davis, H. T. J. Chem. Phys. 1985, 83, 2411. (36) Larson, R. G. J. Chem. Phys. 1989, 91, 2479. (37) Larson, R. G. J. Chem. Phys. 1992, 96, 7904. (38) Floriano, M. A.; Caponetti, E.; Panagiotopoulos, A. Z. Langmuir 1999, 15, 3143. (39) Lisal, M.; Hall, C. K.; Gubbins, K. E.; Panagiotopoulos, A. Z. J. Chem. Phys. 2002, 116, 1171. (40) Panagiotopoulos, A. Z.; Floriano, M. A.; Kumar, S. K. Langmuir 2002, 18, 2940. (41) Kim, S.-Y.; Panagiotopoulos, A. Z.; Floriano, M. A. Mol. Phys. 2002, 100, 2213. (42) Kopelevich, D. I.; Panagiotopoulos, A. Z.; Kevrekidis, I. G. J. Chem. Phys. 2005, 122, 044907. (43) Kopelevich, D. I.; Panagiotopoulos, A. Z.; Kevrekidis, I. G. J. Chem. Phys. 2005, 122, 044908. (44) Arya, G.; Panagiotopoulos, A. Z. Phys. ReV. E 2004, 70, 031501. (45) Arya, G.; Panagiotopoulos, A. Z. Comput. Phys. Commun. 2005, 169, 262.
Zheng et al. behaviors of surfactant adsorption on solid surfaces, morphological transition23 and the CMC shift24 in a confined space, and the recognition of surface heterogeneities with surfactant adsorption.52 The gauge cell method was introduced by Neimark and Vishnyakov53-55 to study the confined phase transition for simple fluids. In this method, two simulation cells are adopted. One of the cells represents the pore, and the other is the gauge cell of a limited capacity. Mass exchange can be allowed between the two cells to maintain the chemical equilibrium, but the cell volumes are kept unchanged. The limited capacity of the gauge cell suppresses the density fluctuations in the pore and allows the fluid in the pore to be in a thermodynamically unstable state.53-55 For example, the gauge cell is capable of preventing thermodynamically unstable nucleation from decay or undesired growth. The gauge cell MC method has widely been used to study the phase transitions in confined systems.53-61 Neimark and his co-workers used the method to study the behavior of Lennard-Jones (LJ) fluids, e.g., vapor-liquid transition in confined systems,53-56 nucleation of condensation and evaporation in cylindrical pores,57 bubble formation confined in a spherical pore,58 and droplet nucleation of the fluids.59 The gauge cell MC method was also used to study the capillary phase transitions of short linear alkanes (from C1 to C4)60 and long linear and branched alkanes61 confined in a carbon nanotube. In the present paper, the gauge cell was a cubic box with triply periodic boundary conditions. The size of the gauge cell was adjusted to prevent the micellization of surfactants in the cell, at the same time maintaining a sufficient number of surfactant molecules for good statistics. The number of surfactant molecules in the gauge cell varied from 20 to 100. The pore cell, which is called the interface cell in this work, unless pointed out, was set to the size of 30 × 30 × 20. For the interface cell the periodic boundary conditions were implemented in x and y dimensions. The bottom of the interface cell at z ) 0 represents a smooth hydrophilic solid surface, S, while the upper surface of the cell at z ) 21 is an impenetrable wall made of water sites.62 Two surfactants, H2T4 and H3T4, were studied in this work at the reduced temperatures of 5.0, 6.0, and 7.0. The interaction energies between two neighboring sites (head, tail, and solvent) are given as follows: HH ) 0, TT ) 0, HT ) 1, HW ) 0, TW ) 1, WW ) 0. The interactions of surfactant chain sites and water with the solid surface are represented by SH, ST, and SW, respectively. These parameters were set to SH ) -9, ST ) 0, and SW ) 0. The initial configurations were generated by putting amphiphilic chains in place of water sites at random on a lattice, in which the initial chain conformations were obtained by using the Rosenbluth and Rosenbluth chain growth algorithm.63 An amphiphile was considered to be part of a cluster if any of its tail segments were in contact with any tail of another surfactant in the cluster. Normally, a total of 1 × 106 MC cycles were used in the simulation, with the first 5 × 105 MC cycles for equilibration and the second 5 × 105 cycles for the ensemble (46) Lisal, M.; Hall, C. K.; Gubbins, K. E.; Panagiotopoulos, A. Z. J. Chem. Phys. 2002, 116, 1171. (47) Scanu, L. F.; Hall, C. K.; Gubbins, K. E. Langmuir 2004, 20, 514. (48) Chennamsetty, N.; Bock, H.; Scanu, L. F.; Siperstein, F. R.; Gubbins, K. E. J. Chem. Phys. 2005, 122, 094710. (49) Siperstein, F. R.; Gubbins, K. E. Langmuir 2003, 19, 2049. (50) Bhattacharya, B.; Gubbins, K. E. J. Chem. Phys. 2005, 123, 134907. (51) Zhang, X. R.; Chen, B. H.; Wang, Z. H. J. Colloid Interface Sci. 2007, 313, 414. (52) Zhang, X. R.; Chen, B. H.; Dong, W.; Wang, W. C. Langmuir 2007, 23, 7433. (53) Neimark, A. V.; Vishnyakov, A. Phys. ReV. E 2000, 62, 4611. (54) Neimark, A. V.; Ravikovitch, P. I.; Vishnyakov, A. Phys. ReV. E 2002, 65, 031505. (55) Vishnyakov, A.; Neimark, A. V. J. Phys. Chem. B 2001, 105, 7009. (56) Vishnyakov, A.; Neimark, A. V. J. Phys. Chem. B 2006, 110, 9403. (57) Vishnyakov, A.; Neimark, A. V. J. Chem. Phys. 2003, 119, 9755. (58) Neimark, A. V.; Vishnyakov, A. J. Chem. Phys. 2005, 122, 054707. (59) Neimark, A. V.; Vishnyakov, A. J. Chem. Phys. 2005, 122, 174508. (60) Jiang, J. W.; Sandler, S. I.; Smit, B. Nano Lett. 2004, 4, 241. (61) Jiang, J. W.; Sandler, S. I. Langmuir 2006, 22, 7391. (62) Reimer, U.; Wahab, M.; Schiller, P.; Mo¨gel, H.-J. Langmuir 2001, 17, 8444. (63) Rosenbluth, M. N.; Rosenbluth, A. W. J. Chem. Phys. 1955, 23, 356.
Macrophase and Microphase Separations
Figure 1. Adsorption isotherm of H2T4 on a 30 × 30 hydrophilic surface at a reduced temperature of 6.0. The open circles correspond to the states with cylindrical structures. The spinodals SV and SL represent the limit of the metastability of the low-affinity adsorption structure and that of the bilayer structure, respectively. Statistical uncertainties are smaller than or comparable to the symbol size. average. One MC cycle was defined as each surfactant moving once on average. Four types of trail moves for surfactant molecules, including reptation, regrowth, interbox molecular transfer, and cluster moves, were adopted in this work. A typical mix of the Monte Carlo moves used was 50% reptation, 29.9% regrowth, 20% interbox molecular transfer, and 0.1% cluster moves. To obtain statistical uncertainties, most data in this work are averaged over at least four independent runs in the same thermodynamic conditions but with different seeds of random number. The statistical uncertainties were obtained as the standard deviations of the results from those independent runs.
3. Results and Discussion Our simulations show that surfactant adsorption on a solid surface yields macrophase transition or microphase transition in a single-phase region, depending on the temperature, adsorption energy HS, and surfactant architecture. Below, we use H2T4 and H3T4, respectively, to investigate the surface macrophase separation and microphase separation in a single-phase region. 3.1. Surface Phase Transition. The simulated adsorption isotherm for surfactant H2T4 on a 30 × 30 hydrophilic surface at a reduced temperature of 6.0 is shown in Figure 1. At relatively low bulk surfactant concentrations, the adsorbed surfactant molecules mainly exist in the form of monomers on the hydrophilic surface, and the average adsorption amount of the surfactants (the surface coverage) is very low. This structure is called the low-affinity adsorption structure. While at the high bulk surfactant concentration, the surface coverage reaches saturation and the fully developed bilayer structure is formed. The whole adsorption isotherm shows a sigmoid, van der Waalstype shape, which is related to a phase transition from the lowaffinity adsorption structure to the bilayer structure. The adsorption branch is the ascending region of the isotherm limited by the spinodal SV, while the desorption branch is the descending region of the isotherm limited by the spinodal SL. The spinodals SV and SL (see Figure 1) represent the limit of the metastability of the low-affinity adsorption structure and that of the bilayer structure, respectively. Due to the different pathways for the adsorption and desorption process, there exists a hysteresis loop inside the adsorption isotherm. The existence of the hysteresis loop indicates that the phase transition is first order. The two spinodals are connected by the backward trajectory of unstable states of negative compressibility, which would be subject to collapse, if the constraints on density fluctuations were removed.56 At the spinodal SV, the structure of the surfactants on a hydrophilic surface still corresponds to the low-affinity adsorption.
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The increase of the total surfactant concentration above the spinodal SV leads to a sharp increase of the surface coverage. Snapshots in Figure 2 show that the corresponding morphologies formed by the adsorbed surfactants change from low-affinity adsorption to spherical admicelles before and after SV. Although the total surfactant concentration increases, the strong tendency of self-assembly on the surface results in a decrease of the bulk surfactant concentration. Thus, a backward region composed of unstable states is formed with a negative slope between SV and SL. As the bulk surfactant concentration decreases along the backward trajectory, the morphologies of the adsorbed aggregates in the region vary from spherical to cylindrical structures, as seen in Figure 2. A typical snapshot for the cylindrical morphology, which corresponds to state E in Figure 1, is shown in Figure 2. Cylindrical structures of the adsorbed surfactants are followed by the perforated bilayers as the bulk surfactant concentration further decreases. Finally, at the liquidlike spinodal, SL, the surface coverage reaches saturation and the fully developed bilayer structure is formed. According to the arguments presented by Neimark and Vishnyakov,59 the aggregates stabilized in a closed system by the gauge cell method correspond to the critical nucleus in an open system with the same bulk fluid concentration. Thus, the backward region of the adsorption isotherm in Figure 1 corresponds to a collection of the states of critical nuclei. As discussed above, the nucleation is bulk surfactant concentration dependent. For the adsorption branch, the low-affinity phase is found to nucleate with the critical nucleus of a spherical admicelle, while for the desorption branch, the critical nucleus from a mestastable bilayer usually has the shape of a perforated bilayer. These morphologies of the critical nuclei can be not only recognized from the typical snapshots in Figure 2, but also proved by their principal moments of inertia. Figure 3 shows the three principal moments of inertia, I1, I2, and I3 (I1 > I2 > I3), for the most frequently observed structure with respect to its center of mass. Similar to our previous work,22 it is noted that I1/I2 ≈ 1 and I2/I3 ≈ 1 for spherical micelles, I1/I2 ≈ 1 and I2/I3 . 1 for cylindrical micelles, and I1/I2 ≈ 2 and I2/I3 ≈ 1 for a bilayer. When the surface coverage is lower than 1 (see Figure 3), the average values of I1/I2 and I2/I3 are both close to 1. This confirms that the adsorbed surfactants form a spherical or spheroidic structure. With an increase of the surface coverage, the average value of I1/I2 remains nearly constant, while the average value of I2/I3 increases very rapidly, up to 3.5 at a surface coverage of 1.5. According to the criterion mentioned above, the corresponding adsorbed structures change to cylindrical morphologies. As the surface coverage increases further, the average value of I1/I2 increases slowly, while the average value of I2/I3 descends gradually. For a surface coverage above 5.5, the average values of I1/I2 and I2/I3 are almost 2 and 1, respectively. Combined with the corresponding snapshots, the adsorbed aggregates are determined as perforated bilayers for surface coverage from 3 to 5.5 and bilayer structures when the surface coverage is greater than 5.5. In summary, the analysis of the principal moments of inertia confirms that the critical nuclei along SV-E-SL change from spherical to cylindrical admicelles and then to a perforated bilayer, depending on the bulk surfactant concentration. The continuity of the constructed isotherm makes the calculation of the grand thermodynamic potential, Ω(F,T), possible by thermodynamic integration along the isotherm starting from a reference ideal gas state,56 Ω(F,T) - Ω(Fr,T) ) -∫uµrN(F,T) dF, in which N is the surface coverage and F is the bulk surfactant concentration. Since no significant micellization occurs in the gauge cell, the chemical potential, µ, can be given as that of an
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Figure 2. Typical snapshots of morphologies on the 30 × 30 hydrophilic surface at the corresponding points (SV, A, B, E, C, D, SL) along the isotherm in Figure 1. Except for SL-H, in all snapshots only tail groups are shown for clarity. Both SL and SL-H snapshots correspond to the state of SL in Figure 1, but the head groups of the surfactants are also shown in the SL-H snapshot. The snapshot of BT corresponds to an adsorption state on the large size surface of 40 × 40 (see Figure 6).
Figure 3. Principal moments of inertia of the most frequently observed structures as a function of the average surface coverage along the isotherm in Figure 1.
Figure 4. Grand thermodynamic potentials obtained by using the thermodynamic integration along the adsorption isotherm. For denotations (points SV, SL, and E) see Figure 1.
ideal gas. Ω(Fr,T), the grand potential for the reference ideal gas at a sufficiently low bulk surfactant concentration, is obtained according to Ω(Fr,T) ) -kNrT. The bulk surfactant concentration, Fe, which corresponds to the two-phase coexistence point, can be evaluated by using the Maxwell rule of equal areas, IµN dF
) 0. Equivalently, the location of equilibrium phase transition corresponds to the intersection of the grand thermodynamic potentials between the low-affinity and the bilayer branches. Figure 4 gives the corresponding grand thermodynamic potentials for the states shown in Figure 1. The construction of the grand
Macrophase and Microphase Separations
Figure 5. Nucleation barriers for adsorption (right branch) and desorption (left branch) calculated using the thermodynamic integration along the interpolated isotherm of the unstable states by the gauge cell method.
potential allows one to predict the free energy barriers of nucleation as the difference in the grand thermodynamic potentials of the different states at the same bulk concentration and temperature. The calculated nucleation barriers are given in Figure 5. The analysis of the nucleation barriers gives an illuminative insight into the hysteresis on the isotherm in Figure 1. The right branch corresponds to the adsorption process, while the left branch corresponds to the desorption process. The nucleation barrier vanishes at the spinodals and achieves a maximum at the bulk concentration, where the equilibrium phase transition occurs. The maximum nucleation barrier in Figure 5 is approximately 8.1 kT. Such a high nucleation barrier is impossible to cross within the length of our simulation; thus, the hysteresis loop is prominent. In general, the backward trajectory connects a number of critical nuclei at different bulk concentrations, taking the shape of the spherical structure, cylinder structure, and perforated bilayer, respectively. As the bulk concentration deviates from that for the equilibrium phase transition, the height of the nucleation barrier separating the metastable and critical nuclei decreases, as shown in Figure 5. In the vicinity of SV and SL, the energy barrier between them is small enough to be crossed during a simulation run. Thus, there exist two growth mechanisms which can be understood as being separated by the surface spinodals. From SV to SL (the phase transition region), the layer growth proceeds through the nucleation mechanism, whereas above the limits this mechanism is not available. To investigate systematically the phase behaviors of surfactants on solid surfaces, the effects of the size of the hydrophilic surface, temperature, and surfactant structure were studied, and the results are given below. 3.1.1. Effects of the Surface Size. To study the finite size effects, simulations were also performed for adsorption of surfactant H2T4 on a large surface size of 40 × 40 at a reduced temperature of 6.0. The adsorption isotherm is shown in Figure 6. It is found that, similar to that for the smaller surface size of 30 × 30, the isotherm has a sigmoid shape, which is characteristic of the first-order phase transition. Moreover, the morphological evolution of nuclei along the backward trajectory follows the same trend. Namely, in the branch of the adsorption process, the critical nuclei have morphologies of spherical and cylindrical structures, and the desorption process initiates from the formation of the holes (the perforated bilayer structure). We also compare the nucleation barriers for the 30 × 30 and 40 × 40 cells in Figure 5. In general, the results for the two cell sizes agree well, although for the 40 × 40 cell the nucleation barrier at the concentration
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Figure 6. Adsorption isotherm of H2T4 on a 40 × 40 hydrophilic surface at a reduced temperature of 6.0.
of equilibrium phase transition is somewhat higher than that for the 30 × 30 cell. These similarities between the case with a small surface and that with a large surface are not sufficient to indicate that the surface size of 30 × 30 is large enough to exclude the finite size effects. On both the 30 × 30 and 40 × 40 hydrophilic surfaces, the critical nuclei near the location of equilibrium phase transition possess a cylindrical shape, which may be caused by the boundary conditions we adopted. For the substrate of 30 × 30 size, two spherical nuclei (one is the spherical nucleus in our simulation box, and the other is its image due to the periodical boundary conditions used) coalesce into one cylindrical nucleus (see the snapshots in Figure 2, points B and E). However, on the large 40 × 40 surface, the spherical nuclei prefer to grow into a large disk-shaped micelle before the formation of the cylindrical structure (see the snapshot in Figure 2, point BT). Thus, one may question whether the cylindrical structure can still be formed by the coalescence of two spherical nuclei if the distance from one spherical nucleus to another spherical nucleus is longer than 40. It is believed that if the studied substrate is large enough, the nuclei are more likely to grow isotropically into the large disk-shaped structure, until they approach each other and coalesce into a patchy bilayer. Although, due to the restriction of the computer condition, existence of the disk-shaped structures in the box larger than 40 has not yet been demonstrated. Due to the hydrophobic effect, transfer of a surfactant molecule from the aqueous phase to a disk-shaped aggregate reduces the free energy of the system. The free energy gain is proportional to the number of surfactants in the disk. At the same time, the creation of the edge of the disk-shaped structure requires an input of energy that is proportional to the length of the edge boundary. In the framework of classical nucleation theory, the nucleation barrier, which depends only on the size of the diskshaped nucleus, is determined by a balance between the free energy gain due to surfactant aggregation and the energy penalty due to the edge boundary. On the basis of the above consideration, the critical nucleus and nucleation barrier seem to be independent of the size of the interface cell as long as the size is large enough. As is shown in Figure 5, the corresponding nuclear barrier at the equilibrium phase transition becomes so high that it is impossible to cross. Near the equilibrium phase transition, the nucleation mechanism may depend on the nature of the substrate. If the substrate is free from surface heterogeneities, the system would remain in the metastable state until the phase transition takes place at deep supersaturation. If the substrate exhibits either physical or chemical heterogeneities, however, the heterogeneities of the solid surface would induce a number of small nuclei on
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Figure 7. Adsorption isotherms of H2T4 surfactants on a 30 × 30 hydrophilic surface at different temperatures: squares, 5.0; circles, 6.0; triangles, 7.0. The upper inset shows the phase diagram in the temperature-bulk concentration panel.
Figure 8. Nucleation barriers at different temperatures: squares, 5.0; circles, 6.0. See Figure 7 for details.
the substrate cooperatively. If the small nuclei are close to each other, they can coalesce into a critical nucleus which can grow into a bilayer. 3.1.2. Effects of Temperature. In this section, we explore the effects of temperature on the phase behaviors of H2T4 surfactants on a hydrophilic surface of 30 × 30. Adsorption isotherms at three different temperatures, namely, 5.0, 6.0, and 7.0, are shown in Figure 7. At lower temperatures of 5.0 and 6.0, it is found that the adsorption isotherms show hysteresis loops, which indicates the occurrence of first-order phase transitions from the lowaffinity adsorption to the bilayer structure, as is discussed in the previous section. Moreover, the obtained hysteresis loop shrinks in size with an increase of temperature. At T ) 7.0, the isotherm is almost vertical, which indicates that the hysteresis loop vanishes basically. The corresponding nucleation barriers for different temperatures are given in Figure 8. It can be seen from Figure 8 that the peak height of the barrier for the equilibrium phase transition decreases greatly with an increase of temperature from 5.0 to 6.0 and to 7.0. Note that the energy barriers at 7.0 are too low to be exactly determined. The characteristics of the hysteresis loop and nucleation barrier indicate that there exists a critical temperature for the surface phase transition, and only below the critical temperature, the characteristics of the first-order phase transition appear. The critical temperature is defined as the temperature that separates the region of first-order phase transition and a single-phase region. An analysis of the stability condition, ∂N/∂F g 0, in which N is the surface coverage and F is the bulk surfactant concentration, shows that, at lower temperature T ) 5.0 and 6.0, ∂N/∂F < 0 inside the loop of hysteresis. This means that those states are unstable, and thus, phase separation occurs and two phases coexist. However, for the system at T > 7.0, ∂N/∂F > 0, which means
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Figure 9. Adsorption isotherms of H2T4 surfactants at T ) 6.0 on a 40 × 40 hydrophilic surface with different adsorption energies, HS: squares, -4.5; circles, -6; upright triangles, -9.0; inverted triangles, -12. The upper inset shows the phase diagram in the adsorption energy-bulk concentration panel.
Figure 10. Adsorption isotherms of H3T4 surfactants on a 30 × 30 hydrophilic surface with adsorption energy HS ) -9 at different temperatures: squares, 5.0; circles, 6.0; triangles, 7.0.
that those states are stable and only a single phase exists. Hence, we estimate approximately that the critical temperature for H2T4 on the surface is 7.0. Above the critical temperature, the isotherm does not have a hysteresis loop and becomes reversible; thus, there is no macrophase transition. 3.1.3. Effects of the Adsorption Energy. We also investigated the effects of the adsorption energy (i.e., head-surface interaction) on the phase behaviors of H2T4 surfactants on a hydrophilic 40 × 40 surface. The calculated adsorption isotherms at T ) 6.0 with different adsorption energies from -4.5 to -12 are shown in Figure 9. As seen in Figure 9, the adsorption isotherms for the stronger adsorption energies (HS ) -12 and -9) possess hysteresis loops and the size of the hysteresis loop becomes narrow with the increase of the adsorption energy. When the adsorption energy increases to -6.0, the hysteresis loop vanishes basically. It is noticed that there are small backward regions in the adsorption isotherms for HS ) -6 and -4.5. This phenomenon is induced by the micellization process of surfactants rather than a signal of phase transition. In these systems, some small clusters are formed on the bare surfaces (micellization) with an increase of the surface coverage. As a result, the decrease of the bulk surfactant concentration spontaneously causes the small backward region due to the limited capacity of the gauge cell. Figure 9 indicates that the first-order phase transition disappears for the system in the limit of a weak hydrophilic surface. In general, the adsorption energy strongly affects the phase behaviors of surfactants on a hydrophilic surface. As the adsorption energy increases, the location of the equilibrium phase transition shifts
Macrophase and Microphase Separations
Figure 11. Adsorption isotherm of H3T4 at a temperature of 7.0 in a log-log plot. Region I, region II, region III, and region IV represent the region of low-affinity adsorption, the hemimicelle region, the morphological transition region, and a plateau region, respectively.
to a higher value of the bulk surfactant concentration (see the inset of Figure 9). Above a certain critical value of HS, the transition is no longer first order but continuous. From Figure 9, it is found that the critical adsorption energy is approximately -4.5 for the H2T4 surfactants at T ) 6.0. 3.2. Microphase Separation in a Single Phase. As shown above, at lower temperatures (below the critical temperature) and with strong head-surface attractive interactions (stronger than the critical value for HS), the macrophase separates for the H2T4 system and a low-affinity phase coexists with the bilayer phase. However, in most cases, adsorption of surfactants on surfaces occurs in a single phase with microphase separation, rather than macrophase separation. Below, we use surfactant
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H3T4 as an example to investigate the morphological evolution in a single-phase region during the adsorption process. Adsorption isotherms for the H3T4 surfactants on a 30 × 30 hydrophilic surface at different temperatures are presented in Figure 10. As is seen in Figure 10, the adsorption isotherms do not possess the hysteresis loop even at a low temperature of 5.0. Thus, for these systems the microphase transition rather than the macrophase transition occurs in the adsorption processes. For these systems we are concerned about the morphological evolutions in the adsorption process. To compare qualitatively the simulated data with the experimental observations, the simulated adsorption isotherm at a temperature of 7.0 in Figure 10 is shown in Figure 11 again on a log-log scale. The simulated adsorption isotherm in Figure 11 can be divided into four regions, being in good agreement with the experimental observations.1-7 We call the four regions the low-affinity adsorption region, the hemimicelle region, the morphological transition region, and the plateau region. The typical snapshots for the four regions are shown in Figure 12. In the first region, isolated surfactant molecules are adsorbed on the surface, as shown by the typical snapshot (see Figure 12a). The amount of the surfactants adsorbed in the region is very low because the adsorption is solely dominated by the attraction between the heads and the surface, and no significant self-assembly occurs. The snapshot for the second region (see Figure 12b) shows that there exist some surface aggregates of H3T4 molecules on the solid surface. In this region the cooperative adsorption mechanism dominates since adsorption is promoted by not only the attraction between the surfactants and the solid surface but also the self-assembly of the adsorbed surfactants. This is the reason why the rate of adsorption in this region rises suddenly. Because of strong attraction between the surface and
Figure 12. Typical snapshots of adsorbed surfactants on the hydrophilic surface at different bulk surfactant concentrations: (a) C ) 0.00208; (b) C ) 0.0034; (c) C ) 0.00467; (d) C ) 0.00489; (e) C ) 0.00726. The concentrations correspond to the points A, B, C, D, and E along the isotherm in Figure 11, respectively.
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Figure 14. Distributions of the heads of H3T4 surfactants in the interface cell for states B and C along the adsorption isotherm in Figure 11.
Figure 13. (a) Principle moments of inertia of the most frequently observed structure as a function of the bulk surfactant concentration. (b) Surfactant aggregation numbers as a function of the bulk surfactant concentration.
heads, the head groups of the adsorbed surfactants face toward the solid surface, while the tail groups protrude into the solution at relatively low bulk surfactant concentrations, which creates hydrophobic patches on the surface. This type of aggregate is in agreement with the hemimicelle structure observed experimentally by Mane et al.64 Morphological transition occurs in the third region. In this region, due to the increase of the bulk surfactant concentration, adsorption continues with an increase of the aggregation number. When two aggregates locate near each other, they coalesce, which leads to a morphological transition. From the snapshots in Figure 12 one can see that the surface morphologies in the third region are different from those in the second region. The above mechanism is supported by the aggregation number and density profile. Figure 13 shows the aggregation numbers and the principal moments of inertia for the most frequently observed structures. It is found from Figure 13 that the aggregation number is in the range of 16-35 for region II, 37-53 at the lower concentration of region III, and 357-624 at the higher concentrations of region III. Obviously, a morphological transition occurs in the third region. As shown in Figure 13b, it is found that I1/I2 and I2/I3 are roughly equal to each other over the range of the cluster size (16-53), which indicates that the surface morphologies are approximately spherical or hemispherical for region II and the low concentration part of region III. To further distinguish the surface morphologies for region II and those for the low concentration part of region III, the density profiles of the heads of states B (C ) 0.0034) and C (C ) 0.00467) in Figure 11 are shown in Figure 14. Obviously, the density profile of the heads for C ) 0.0034 in region II shows only one peak value on the surface. However, besides the peak near the surface, the density profile of the heads for C ) 0.00467 in region III shows another peak value (local maximum) at the layer LZ ) 5. This indicates that the aggregates of the surfactants on the surface are of hemimicelle morphology (hemisphere) in region II while full micelles (sphere) in region III. (64) Manne, S.; Cleveland, J. P.; Gaub, H. E.; Stucky, G. D.; Hasma, P. K. Langmuir 1994, 10, 4409.
It is observed from Figure 13 that in region III the aggregation number increases suddenly from 53 to 357 near the bulk concentration of 0.00489 (C point in Figure 11), and correspondingly, I2/I3 increases from 1.256 to 17.26. This observation indicates that the surface morphologies change from spheres to cylinders in this region. Due to a slowing of the new surface morphology formed, there is a reduction in the slope of the isotherm for this region. In region IV, saturation of adsorption is reached and the adsorption amount of the surfactants remains unchanged with a further increase of the bulk concentration. The equilibrium surface morphologies are in the shape of regular cylinders (see Figure 12). To summarize, our simulation results show that, for the adsorption process of surfactant H3T4 on a hydrophilic surface, there exist four regions for morphology evolution, which are in good agreement with the experimental observation and our previous simulation results.51 Our simulations reveal that adsorption in the second region corresponds to the nucleation of hemimicelles, while adsorption in the third region features morphological transitions, which are accompanied by an increase of the aggregation number and a reduction of the number of aggregates. Undoubtedly, the range and the rate of the adsorption isotherm in region III also give insight into the mechanism of morphological transition.
4. Conclusions In this work, we combined the gauge cell MC method with the lattice model to study the phase behaviors of adsorbed surfactants on a hydrophilic surface. The simulation results show that there exist two types of surface phase behaviors: macrophase transition and microphase transition, i.e., adsorption in a singlephase region. The two surface behaviors were studied by using surfactant H2T4 and H3T4 as examples, respectively. For surfactant H2T4, the full isotherm at low temperature shows obviously a van der Waals-type shape and has a hysteresis loop, which is characteristic of first-order phase transitions. We determined the nucleation barriers and studied the shape and the structure of the critical nuclei. It is found that the nucleus morphology and nucleation barrier depend on the bulk surfactant concentration. The temperature and adsorption energy have strong effects on the phase behaviors of surfactants adsorbed on solid surfaces. It is found that the hysteresis loop becomes narrower at higher temperature and weaker adsorption energy and finally disappears at certain values. Therefore, there exist a critical temperature and a critical adsorption energy, above which no macrophase transition occurs.
Macrophase and Microphase Separations
For surfactant H3T4, no macrophase transition is found; in other words, adsorption takes place in a single-phase region in the temperature range we studied. The morphological evolution with an increase of the bulk surfactant concentration exhibits four regions, which is in agreement with the experimental observations. They are the low-affinity adsorption region, the hemimicelle region, the morphological transition region, and the plateau region. Simulation results reveal that in the second region the adsorbed monomers aggregate and nucleate hemimicelles, while adsorption in the third region features morphological transitions. The morphological transition is accompanied by an
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increase of the aggregation number and a reduction of the number of aggregates. Acknowledgment. This work is supported by the National Natural Science Foundation of China (Grant No. 20736005). X.Z. acknowledges the support of the Research Foundation for Young Researchers of the Beijing University of Chemical Technology (BUCT). Generous allocations of computer time by the “Chemical Grid Project” of BUCT and the Supercomputing Center, CNIC, CAS, are acknowledged. LA800046S
