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Monte Carlo Simulation of Mixed Lennard-Jones Nonionic Surfactant Adsorption at the Liquid/Vapor Interface A. J. Howes† and C. J. Radke* Department of Chemical Engineering, UniVersity of California, Berkeley, California 94720-1462 ReceiVed May 17, 2007. In Final Form: July 30, 2007 New Monte Carlo simulations are presented for nonionic surfactant adsorption at the liquid/vapor interface of a monatomic solvent specifically investigating the roles of tail attraction and binary mixtures of different tail lengths. Surfactant molecules consist of an amphiphilic chain with a solvophilic head and a solvophobic tail. All molecules in the system, solvent and surfactant, are characterized by the Lennard-Jones (LJ) potential. Adjacent atoms along the surfactant chain are connected by finitely extensible harmonic springs. Solvent molecules move via the Metropolis random-walk algorithm, whereas surfactant molecules move according to the continuum configurational bias Monte Carlo (CBMC) method. We generate thermodynamic adsorption and surface-tension isotherms and compare results quantitatively to single-surfactant adsorption (Langmuir, 2007, 23, 1835). Surfactant tail groups with attractive interaction lead to cooperative adsorption at high surface coverage and higher maximum adsorption at the interface than those without. Moreover, adsorption and surface-tension isotherms with and without tail attraction are identical at low concentrations, deviating only near maximum coverage. Simulated binary mixtures of surfactants with differing lengths give intermediate behavior between that of the corresponding single-surfactant adsorption and surface-tension isotherms both with and without tail attraction. We successfully predict simulated mixture results with the thermodynamically consistent ideal adsorbed solution (IAS) theory for binary mixtures of unequal-sized surfactants using only the simulations from the single surfactants. Ultimately, we establish that a coarse-grained LJ surfactant system is useful for understanding actual surfactant systems when tail attraction is important and for unequal-sized mixtures of amphiphiles.
1. Introduction Surfactants strongly assemble at fluid/fluid interfaces, even in dilute concentration, due to their amphipathic moieties, one moiety preferring water and the other preferring oil (or gas). Surfaceactive agents are essential in industrial applications. Almost universally, mixtures of surfactants are employed to optimize performance of commercial products. Because of the importance of surfactant mixtures, treatises1-3 and numerous papers4-37 are * To whom correspondence should be addressed. Phone: 510-642-5204. Fax: 510-642-4778. E-mail:
[email protected]. † Present address: Chevron Oronite, Richmond, CA 94802. (1) Abe, M.; Scamehorn, J. F. Mixed Surfactant Systems; Marcel Dekker: New York, 2005. (2) Scamehorn, J. F. Phenomena in Mixed Surfactant Systems; American Chemical Society: Washington, D.C., 1986. (3) Rubingh, D. N.; Holland, P. M. Mixed Surfactant Systems; American Chemical Society: Washington, D.C., 1992. (4) Ariel, G.; Diamant, H.; Andelman, D. Langmuir 1999, 15, 3574-3581. (5) Fainerman, V. B.; Miller, R. Colloids Surf., A 1995, 97, 65-82. (6) Fainerman, V. B.; Miller, R.; Wustneck, R.; Makievski, A. V. J. Phys. Chem. 1996, 100, 7669-7675. (7) Fainerman, V. B.; Miller, R.; Wustneck, R. J. Phys. Chem. B 1997, 101, 6479-6483. (8) Fainerman, V. B.; Miller, R. Langmuir 1997, 13, 409-413. (9) Fainerman, V. B.; Lucassen-Reynders, E. H. AdV. Colloid Interface Sci. 2002, 96, 295-323. (10) Franses, E. I.; Siddiqui, F. A.; Ahn, D. J.; Chang, C. H.; Wang, N. H. L. Langmuir 1995, 11, 3177-3183. (11) Hines, J. D.; Thomas, R. K.; Garrett, P. R.; Rennie, G. K.; Penfold, J. J. Phys. Chem. B 1997, 101, 9215-9223. (12) Hua, X. Y.; Rosen, M. J. J. Colloid Interface Sci. 1982, 90, 212-219. (13) Hua, X. Y.; Rosen, M. J. J. Colloid Interface Sci. 1988, 125, 730-732. (14) Huber, K. J. Colloid Interface Sci. 1991, 147, 321-332. (15) Janczuk, B.; Bruque, J. M.; Gonzalezmartin, M. L.; Doradocalasanz, C. Colloids Surf., A 1995, 104, 157-163. (16) Janczuk, B.; Zdziennicka, A.; Wojcik, W. Colloids Surf., A 2003, 220, 61-68. (17) Miller, R.; Fainerman, V. B.; Aksenenko, E. Colloids Surf., APhysicochemical and Engineering Aspects 2004, 242, 123-128. (18) Miller, R.; Fainerman, V. B.; Leser, M. E.; Michel, A. Colloids Surf., A 2004, 233, 39-42. (19) Mulqueen, M.; Blankschtein, D. Langmuir 1999, 15, 8832-8848. (20) Mulqueen, M.; Blankschtein, D. Langmuir 2000, 16, 7640-7654.
available, mainly compiling salient properties, such as equilibrium and dynamic tensions, interfacial elasticities, emulsion and foaming strengths, critical micelle concentrations, and phase behavior. Application of mixtures of surfactants relies upon the successful prediction of fluid/fluid interfacial properties, including surfacetension and adsorption isotherms. A common approach is to extend the single-solute Langmuir adsorption isotherm toward mixtures.38 However, for muticomponent mixtures, the Langmuir model is thermodynamically inconsistent, except when the maximum adsorption density of each single surfactant is equal.10 The Langmuir mixture model is not applicable here, since our single surfactants exhibit different maximum adsorption densities. Similarly, the interfacial layer model39,40 does not appear to be thermodynamically consistent for mixtures of surfactants, (21) Mulqueen, M.; Datwani, S. S.; Stebe, K. J.; Blankschtein, D. Langmuir 2001, 17, 7494-7500. (22) Mulqueen, M.; Stebe, K. J.; Blankschtein, D. Langmuir 2001, 17, 51965207. (23) Mulqueen, M.; Blankschtein, D. Langmuir 2002, 18, 365-376. (24) Nikas, Y. J.; Puvvada, S.; Blankschtein, D. Langmuir 1992, 8, 26802689. (25) Puvvada, S.; Blankschtein, D. J. Phys. Chem. 1992, 96, 5567-5579. (26) Puvvada, S.; Blankschtein, D. J. Phys. Chem. 1992, 96, 5579-5592. (27) Rosen, M. J.; Hua, X. Y. J. Colloid Interface Sci. 1982, 86, 164-172. (28) Rosen, M. J.; Zhou, Q. Langmuir 2001, 17, 3532-3537. (29) Shiloach, A.; Blankschtein, D. Langmuir 1997, 13, 3968-3981. (30) Shiloach, A.; Blankschtein, D. Langmuir 1998, 14, 4105-4114. (31) Shiloach, A.; Blankschtein, D. Langmuir 1998, 14, 1618-1636. (32) Siddiqui, F. A.; Franses, E. I. Langmuir 1996, 12, 354-362. (33) Siddiqui, F. A.; Franses, E. I. AIChE J. 1997, 43, 1569-1578. (34) Talbot, J.; Jin, X.; Wang, N. H. L. Langmuir 1994, 10, 1663-1666. (35) Talbot, J. J. Chem. Phys. 1997, 106, 4696-4706. (36) Wustneck, R.; Miller, R.; Kriwanek, J. Colloids Surf., A 1993, 81, 1-12. (37) Wustneck, R.; Miller, R.; Kriwanek, J.; Holzbauer, H. R. Langmuir 1994, 10, 3738-3742. (38) Adamson, A. W.; Gast, A. P. Physical Chemistry of Surfaces; John Wiley & Sons: New York, 1997; Chapters 13 and 17. (39) Fainerman, V. B.; Miller, R.; Aksenenko, E. V. AdV. Colloid Interface Sci. 2002, 96, 339-359. (40) Fainerman, V. B.; Lucassen-Reynders, E.; Miller, R. Colloids Surf., A 1998, 143, 141-165.
10.1021/la701452g CCC: $37.00 © 2007 American Chemical Society Published on Web 10/05/2007
Surfactant Adsorption at Liquid/Vapor Interface
again unless the maximum single-surfactant adsorption densities are equal.41 Further, when the surfactant adsorption sizes are equal, the interfacial layer model collapses to the multicomponent Langmuir adsorption isotherm.41 Two currently available models for binary mixtures of unequalsized surfactants are thermodynamically consistent:10 ideal adsorbed solution (IAS) theory42 and scaled particle theory (SPT) of hard disks.24,34 SPT uses model parameter values from the single-surfactant results, whereas IAS requires only experimental single-surfactant adsorption isotherms and is model-free. Unfortunately, SPT also does not describe our simulated single surfactant results.43 Apparently, hard disks do not adequately represent our simulated chain surfactants at a gas/liquid interface. This work investigates tail interactions and binary mixtures of Lennard-Jones (LJ) surfactants using molecular simulation. We desire not only a description of the molecular architecture at the interface but also a quantitative assessment of the adsorption and surface-tension isotherms. This study builds upon our previous work43 and develops adsorption and surface-tension isotherms for single-surfactant systems with tail attraction and mixtures of surfactants with and without tail attraction. We also model quantitatively the behavior of surfactants with tail attraction and mixtures of those surfactants using IAS theory.
2. Monte Carlo Simulations The same simulation molecules and techniques are utilized as in our previous work,43 changing only one parameter in the case of tail attraction. We use an in-house Fortran code built on those in the texts of Allen and Tildesley44 and Frenkel and Smit45 with incorporation of configurational bias Monte Carlo (CBMC).46-49 The LJ surfactants consist of eight to ten molecular segments, each connected by a harmonic spring. To make the chain molecule amphiphilic, the first block of beads on the chain is solvophilic (H or head groups), while the remaining block consists of solvophobic beads (T or tail groups). A schematic of a four-head group, four-tail group (H4T4) chain molecule appears in Figure 1. Surfactant is dissolved in a monatomic LJ solvent designated by the symbol S. Each molecular bead in the system has equal size, meaning that all collision diameters, σij, in the LJ potential are equal. As previously discussed,43 nonvolatile but surfactant-like behavior is attained by judicious trial-and-error adjustment of the LJ interaction parameters ij/k, where k is the Boltzmann constant. ij/k parameter values, in units of kelvin (K), are given in Table 1 for mixtures without tail attraction and in Table 2 for mixtures and single-surfactant systems with tail attraction, as well as listing the cutoff radii in units of σ. Parameters for mixtures without tail attraction do not change from our previous work on singlecomponent surfactants. Note that for tail-tail interactions in Table 1, the cutoff radius (rcut.;TT) is 21/6σ, so that only the repulsive part of the LJ potential is utilized. Initially, we implemented this cutoff length to prevent excessive aggregation of surfactant tail (41) Howes, A. J. Towards a Molecular Understanding of Surfactant Adsorption at Fluid/Fluid Interfaces Using Monte Carlo Simulation of Lennard-Jones Amphiphiles. Ph.D. Thesis, University of California, Berkeley, 2007; Chapter 5. (42) Myers, A. L.; Prausnitz, J. M. AIChE J. 1965, 11, 121-127. (43) Howes, A. J.; Radke, C. J. Langmuir 2007, 23, 1835-1844. (44) Allen, M. P.; Tildesley, D. J. Computer Simulation of Liquids; Clarendon Press: Oxford, U.K., 1987. (45) Frenkel, D.; Smit, B. Understanding Molecular Simulation: From Algorithms to Applications; Academic Press: New York, 2002. (46) Harris, J.; Rice, S. A. J. Chem. Phys. 1988, 88, 1298-1306. (47) Frenkel, D.; Mooij, G.; Smit, B. J. Phys.: Condens. Matter 1992, 4, 3053-3076. (48) Siepmann, J. I.; Frenkel, D. Mol. Phys. 1992, 75, 59-70. (49) de Pablo, J. J.; Laso, M.; Suter, U. W. J. Chem. Phys. 1992, 96, 61576162.
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Figure 1. Thick-film simulation box. Table 1. Lennard-Jones Parameters for Mixtures without Tail Attraction i
j
ij/k (K)
rcut; ij(σ)
S S S H H T
S H T H T T
300 400 50 300 300 300
2.5 2.5 2.5 2.5 2.5 21/6
Table 2. Lennard-Jones Parameters for Mixtures and Single-Surfactant Systems with Tail Attraction i
j
ij/k (K)
rcut; ij(σ)
S S S H H T
S H T H T T
300 400 50 300 300 3
2.5 2.5 2.5 2.5 2.5 2.5
groups in the liquid. In this work, however, we evaluate the role of a small attraction among tail groups by using the parameters in Table 2, where the cutoff radius is 2.5σ to include the attractive portion of the LJ potential. The energy of tail-tail interaction, set at 3 K, is much lower than the other two-body interactions in the system. Any appreciable increase in the energy of interaction between tails from this value results in catastrophic surfactant aggregation in the liquid phase and almost no surface activity. The simulation cell, illustrated in Figure 1, consists of a rectangular film of monatomic solvent molecules at liquid density surrounded on both sides by vapor phase. Dimensions of the simulation box are Lx ) 20σ by Ly ) 20σ by Lz ) 60σ, where the z direction is perpendicular to the liquid/vapor interface. We find that films greater than or equal to 18σ are thick enough so that the liquid at the center has identical properties to that of a bulk liquid without an interface. For all simulations, the LJ dimensionless temperature is 0.9 (all dimensionless units are based upon the monatomic-solvent LJ parameters). The liquid volume fraction in the simulation box is approximately 0.3; 5039 monatomic solvent molecules and from 10 up to 200 surfactant molecules are employed. Adsorption amount and surface tension are calculated as in our previous work.43
3. Results and Discussion We investigate three different surfactant systems. For single surfactant with tail attraction, we simulated H4T4, H4T5, and H4T6 amphiphiles. We present here only results on H4T5 surfactants with tail attraction in comparison to H4T5 surfactants without tail attraction,43 as the remaining systems display similar
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Figure 2. Gibbs excess adsorption isotherms for all H4T5 surfactants with and without tail attraction (where, for example, the notation 2.E-03 represents 2.0 × 10-3). Points represent simulation data: closed squares for the simulation without tail attraction43 and closed diamonds for simulations with tail attraction. An adjacent line represents the Langmuir isotherm fit: a dashed line for simulations without tail attraction43 and a solid line for simulations with tail attraction. The critical aggregation concentration (CAC) is shown by vertical arrows for each surfactant. Table 3. Parameters for Langmuir Isotherm and the CAC for H4T5 Surfactant with and without Tail Attraction attraction
Γm (molecules/σ2)
area (σ2/molecule)
KL
CAC (molecules/σ3)
With Without
0.15 0.12
6.9 8.5
2080 2080
4.6 × 10-4 2.0 × 10-3
behavior. For binary mixtures of surfactants with and without tail attraction, we simulated a 50-50 mixture (by surfactant number) of H4T4-H4T5, H4T4-H4T6, and H4T5-H4T6. Again, we present here only the H4T5-H4T6 mixture results. 3.1. Role of Tail Attraction in Single-Surfactant Adsorption. Net attraction among surfactant tails leads to significant aggregation in the liquid phase at very low surfactant concentration. Such aggregation obviates simulation of surfactant adsorption and surface tension.43 Accordingly, we adopt the small tail-tail attraction listed in row 7 of Table 2. Density profiles for systems with this tail attraction are similar to those without tail attraction.43 Simulated adsorption isotherms for H4T5 surfactant molecules with (closed diamonds) and without (closed squares) tail attraction appear in Figure 2, where Γ is the Gibbs surface excess and F∞ is the liquid-phase surfactant concentration. Surprisingly, they have the general shape of a simple Langmuir isotherm
KLF∞/Fs Γ ) Γm 1 + KLF∞/Fs
(1)
where Γm is the maximum adsorption at the interface in units of molecules/σ,2 KL is the Langmuir equilibrium constant, and Fs is the solvent density at the center of a bulk film (0.7 molecules/ σ3). Traditionally, surfactant adsorption with tail attraction is fit with a Frumkin-type isotherm which includes a lateral-interaction correction term in the Langmuir framework.38 However, the Frumkin isotherm does not describe our results more accurately than does the Langmuir adsorption isotherm. We are unable to fit the single-surfactant-adsorption data with scaled particle theory (SPT)10,24,34,35,50,51 because SPT predicts continuing adsorption near high values of interfacial coverage, not seen in our results. (50) Talbot, J.; Tarjus, G.; Van, Tassel, P. R.; Viot, P. Colloids Surf., A 2000, 165, 287-324. (51) Jin, X. Z.; Ma, Z. D.; Talbot, J.; Wang, N. H. L. Langmuir 1999, 15, 3321-3333.
Figure 3. Schematic depiction of the surfactant architecture near the Gibbs dividing surface (GDS). The uppermost surfactant, which has no tail attraction, adsorbs relatively closer to the liquid phase as compared to the lowermost surfactant, which has tail attraction and is at high values of coverage.
In Figure 2, we label with arrows the first appearance of surfactant aggregates in the bulk portion of our liquid film as the critical aggregation concentration (CAC), as discussed below. As before,43 we establish Γm and KL from two single simulations: one in the linear region of the isotherm and one at the maximum surfactant interfacial packing density. The effect of tail attraction is negligible in the Henry region (i.e., low concentration) and, therefore, we fit both data sets (with and without tail attraction) with the same value of KL. Results of the calculations for Γm and KL for H4T5 surfactants with and without tail attraction appear in Table 3. We also include the minimum area per molecule, or the inverse of Γm, which can be thought of as the footprint of the molecule at the interface. Table 3 reveals several important points. First, the minimum area per adsorbed molecule is approximately 20% smaller for H4T5 surfactant with tail attraction than for H4T5 without tail attraction. In the Frumkin framework, lateral attraction among adsorbed molecules increases the surface coverage compared to no lateral interactions. However, the maximum packing is unchanged. Here, the opposite occurs. We understand this behavior as follows.43 The solvophilic head moiety adsorbs in a semicoiled configuration, whereas the solvophobic tail groups adsorb aligned perpendicular to the interface and almost fully extended. Both tail alignment and extension are independent of coverage. We attributed these behaviors to the “solvophobic” effect,43 namely, the adsorbed-surfactant configuration arises from the extreme dislike of the tails for the solvent phase. To understand the role of attractive-tail interaction, we calculate the architecture of the surfactant near the Gibbs dividing surface (GDS) set by the monatomic solvent, as illustrated in Figure 3. Adsorbed surfactant molecules with tail attraction and at higher coverage penetrate farther into the vapor phase than the corresponding surfactant with no tail attraction. Quantitatively, the junction head group (i.e., the head bead that is connected to a tail bead) on the surfactant without tail attraction lies on the average 1.8σ to the left of the GDS, whereas the junction head group on the surfactant with tail attraction lies 0.3σ to the right of the GDS. Therefore, the surfactant with tail attraction resides 2.1σ units closer to the vapor phase. Cooperative tail attraction collectively draws the surfactant closer toward the vapor phase, effectively enhancing the solvophobic effect. Penetration of the surfactant farther into the vapor phase permits them to pack closer together at the interface.43 Thus, the decrease in minimum area per molecule with tail attraction comes from less solvent
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γ ) γo - kTΓm ln(1 + KLF∞/Fs)
Figure 4. Surface-tension isotherms for the H4T5 surfactant with and without tail attraction (where, for example, the notation 1.E-05 represents 1.0 × 10-5). Closed symbols represent simulation data: squares for the simulation without tail attraction and diamonds for simulations with tail attraction. Adjacent lines represent the Langmuir isotherm fit: a dashed line for simulations without tail attraction and a solid line for simulations with tail attraction. Commencement of horizontal lines locates the critical aggregation concentration (CAC).
existing between the adsorbed surfactant molecules. We quantify this effect by calculating the local solvent density around the adsorbed surfactant molecules with and without tail attraction. The average local solvent density at the tail bead connected directly to the junction head bead is 0.35 molecules/σ3 for surfactants without tail attraction and 0.30 molecules/σ3 for surfactants with tail attraction. For reference, the average bulk liquid solvent density is 0.70 molecules/σ.3 Thus, there are on average fewer monatomic solvent molecules surrounding a surfactant with tail attraction as compared to the same surfactant without tail attraction. Local lower solvent density permits closer packing at the interface for surfactants with tail attraction. Apparently, the tail-attraction energy adopted in Table 2 is not strong enough to require a Frumkin-like correction to the adsorption isotherm. Only the maximum packing density is impacted. Second, Figure 2 and Table 3 report the CAC of H4T5 with and without tail attraction. There is almost an order of magnitude lowering for H4T5 with tail attraction compared to no tail attraction. This strong an effect is perhaps surprising, in view of the small tail-tail attraction energy. In addition, the interfacial adsorption continues to increase above the CAC. Initially, small aggregates form in the liquid film; it is not until higher values of bulk concentration, and thus interfacial adsorption, that we find large aggregates occupying the entire bulk liquid film.43 We attribute the appearance of smaller aggregates at lower concentration to the attraction among tail groups. Specifically, tail attraction allows surfactant in the bulk liquid to complex before the whole bulk fluid film is overwhelmed with surfactant and transforms to a single large aggregate. Finally, note that adsorption data for the surfactant with tail attraction (closed diamonds) are not simulated for very high coverages. This is due to excessive aggregation in the liquid phase.43 Essentially, the central portion of the liquid film becomes a bilayer; bulk fluid properties no longer exist in the central portion of the film. Figure 4 shows simulated surface tension as a function of bulk concentration for H4T5 surfactant with (closed diamonds) and without (closed squares) tail attraction. The Langmuir-Szyszkowski equation52 well fits both data sets (52) Defay, R.; Prigogine, I.; Bellemans, A.; Everett, D. H. Surface Tension and Adsorption; Longmans, Green & Co. Ltd.: London, 1966, Chapter 1.
(2)
where γo is the surface tension of the monatomic solvent determined by a separate simulation (0.27 /σ2).53 Γm and KL are known from the fitting procedure above and appear in Table 3. Hence, the Langmuir-Szyszkowski surface-tension isotherms, shown as lines in Figure 4, are predicted without additional fitting. Figure 4 reveals that the surface tension is nominally constant at higher values of surfactant concentration as expected for aggregate-forming solutes. Following common experimental practice,38 we determine the CAC from the intersection of the constant-tension line and the Langmuir-Szyszkowski isotherm (values are listed in Table 3). More specifically, when a plateau appears in the simulated surface-tension isotherm, we construct a best horizontal line and extrapolate that line to intersect the Langmuir-Szyszkowski curve. In addition, we generate a snapshot of the final surfactant configuration in the film to confirm aggregation (e.g., see Figure 6 of ref 43). Thus, the CAC is determined both from tension simulation and from visual inspection of aggregation in the film. All aggregates at high coverage in our simulations are bilayers that take up the entire breadth of the film. Because our films do not represent a bulk phase above the CAC, values in Table 3 may not represent either bulk CAC values or bulk aggregate structures.43 We also note in Figure 2 that adsorption continues to increase after the CAC is reached for surfactants with tail attraction. This suggests that the chemical potential of the surfactant in the bulk liquid is not constant above the CAC. However, the surface tension remains relatively constant. To explain this apparent inconsistency, we speculate that above the CAC there is a small decline in tension within the statistical uncertainty of our simulations. However, we can only justify drawing horizontal tension lines for surfactant concentrations above the CAC. Surface tensions for surfactants with and without tail attraction are identical at low concentrations and only deviate at higher concentration, beyond the critical aggregation concentration for surfactants with tail attraction. We would expect the surfacetension isotherms for surfactants with and without tail attraction to deviate more due to their differing values of Γm. But, as discussed above for the adsorption isotherm, the CAC is much lower for the surfactants with tail attraction than for those without. Therefore, the surface tension reaches a constant value before large deviations in the surface-tension isotherm behavior are evidenced. Ultimately, tail attraction acts as a tuning parameter. Small tail-tail attraction energy increases the adsorption and decreases the CAC as compared to the same surfactant with no tail-tail attraction. 3.2. Mixtures. 3.2.1. No Tail Attraction. Figure 5 presents a density profile for a mixture of 50 H4T5 surfactants (open squares) and 50 H4T6 surfactants (open triangles) each with no tail attraction. All other mixtures studied have qualitatively similar behavior. We omit the solvent density profile and focus on the surfactant center-of-mass density profile for both H4T5 (species 1) and H4T6 (species 2). Bulk liquid concentrations are as follows: H4T5, F∞1 ) (1.1 ( 0.2) × 10-3 molecules/σ3; H4T6, F∞2 ) (4.0 ( 1.0) × 10-4 molecules/σ3; total concentration, F∞ ) F∞1 + F∞2 ) (1.5 ( 0.2) × 10-3 molecules/σ3. The corresponding solvent-free mole fraction of H4T6 is 0.27. Gibbs excess adsorption values are Γ1 ) 0.053 molecules/σ2 for H4T5, Γ2 ) 0.059 molecules/σ2 for H4T6, and Γ ) Γ1 + Γ2 ) 0.11 molecules/σ2 for total adsorption. As expected, the surfactant with the additional solvophobic tail group, H4T6, exhibits larger (53) Holcomb, C. D.; Clancy, P.; Zollweg, J. A. Mol. Phys. 1993, 78, 437459.
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Figure 5. Center-of-mass density profiles for 50 H4T5 and 50 H4T6 molecules, at a total bulk concentration of (1.5 ( 0.2) × 10-3 molecules/σ3 or, equivalently, a H4T6 solvent-free mole fraction of 0.27. Open squares outline the H4T5 center-of-mass density profile, and the open triangles outline the H4T6 center-of-mass density profile. The vertical dashed line represents the Gibbs dividing surface.
Figure 6. Total adsorption versus total bulk concentration for 5050 H4T5-H4T6 mixtures of surfactants without tail attraction (where, for example, the notation 1.E-03 represents 1.0 × 10-3). Closed diamonds represent simulation-determined points for the mixture. The dotted line represents a model of the mixture results with ideal adsorbed solution (IAS) theory using an average value of the H4T6 solvent-free mole fraction of 0.17.42,54 Boxes adjacent to simulation points detail the actual H4T6 solvent-free mole fraction for the mixtures. For reference, Langmuir adsorption isotherms for singlesurfactant simulation without tail attraction are added for H4T5 (solid line) and H4T6 (dashed line). Table 4. Parameters for IAS Theory for a Mixture of H4T5 and H4T6 Surfactant without Tail Attraction surfactant
surfactant no.
Γm (molecules/σ2)
KL
H4T5 H4T6
1 2
0.12 0.15
2080 5380
adsorption at the interface and, therefore, the smaller bulk concentration. Clearly, the surfactant with more tail groups is the more surface active, all other factors being equal. Also, H4T6 surfactant resides slightly closer to the vapor phase. These two phenomena lead to the larger adsorption of H4T6 relative to H4T5, as shown by the single-surfactant isotherms in Figure 6. Because of the strong “solvophobic bond”, H4T6 penetrates the vapor-phase relatively more, as discussed earlier. Consequently, the local solvent density around H4T6 is less than that around H4T5 (which is less than that of H4T4). Basically, the H4T6 molecules compact closer together due to the absence of solvent. Simulated total adsorption amounts for the H4T5-H4T6 mixture (50% H4T5 surfactant and 50% H4T6 surfactant by
Figure 7. Species adsorption versus species concentration for 5050 H4T5-H4T6 mixtures of surfactants without tail attraction (where, for example, the notation 5.E-04 represents 5.0 × 10-4). Closed squares represent simulation-determined points for H4T5 (1), and closed circles are for H4T6 (2). The dotted line represents a model of the results with ideal adsorbed solution (IAS) theory for H4T5, while the solid line is for H4T6, using a value of the H4T6 solventfree mole fraction set to 0.17.42,54
number) appear in Figure 6 as closed diamonds. Both adsorption and concentration in Figure 6 refer to the total amounts: Γ ) Γ1 + Γ2 and F∞ ) F∞1 + F∞2. For reference, the single-surfactant isotherms are included: H4T5 (solid line) and H4T6 (dashed line). Boxes values next to each of the mixture points designate the H4T6 solvent-free mole fraction. As expected, mixture data fall between those of the single-surfactant isotherms (see Franses et al.,10 references within, and Blankschtein and co-workers21,23,24 for mixture data of nonionic surfactants at the air/water interface). Figure 6 also reports the IAS-mixture-theory predictions as a dotted line.42,54 IAS uses only the single-surfactant Langmuir parameters Γm and KL listed in Table 4 to quantify the mixture. The specific set of IAS equations used in this work can be found in the work by Franses and co-workers (eqs 31 and 32 in ref 10).10 IAS theory predicts the simulation results extremely well for an H4T6 average solvent-free mole fraction of 0.17. Although this H4T6 solvent-free mole fraction does not exactly match the actual H4T6 solvent-free mole fraction for all of the data points, we find the IAS line produced fits the results exactly if we use the specific values of H4T6 solvent-free mole fraction. In addition, plots of Γ1 versus F∞1 (closed squares) and Γ2 versus F∞2 (closed circles) are fit well with IAS (dotted line for H4T5, species 1, and solid line for H4T6, species 2), as seen in Figure 7. Here too, using the exact solvent-free mole fractions in Figure 7 shows excellent agreement with IAS theory. Figure 8 reports the surface tension for a binary mixture of 50% H4T5 surfactant and 50% H4T6 surfactant again by number. Closed diamonds are the simulation-determined values of the surface tension for the mixture. Boxed values near simulation data represent the H4T6 solvent-free mole fraction. Again, we report concentration values as totals: F∞ ) F∞1 + F∞2. Also shown in Figure 8 are the surface-tension results predicted by IAS theory (dotted line) for an average value of the H4T6 solvent-free mole fraction of 0.17.42,54 Again, the specific equations used for surface tension may be found in the works of Franses et al.10 and LeVan and Vermeulen.55 IAS theory calculates the mixture results extremely well. 3.2.2. Tail Attraction. For surfactants with tail attraction, we again present only mixture simulations of 50% H4T5 and 50% (54) Radke, C. J.; Prausnitz, J. M. AIChE J. 1972, 18, 761-768. (55) LeVan, M. D.; Vermeulen, T. J. Phys. Chem. 1981, 85, 3247-3250.
Surfactant Adsorption at Liquid/Vapor Interface
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Figure 8. Surface tension versus total bulk concentration for 5050 H4T5-H4T6 mixtures of surfactants without tail attraction (where, for example, the notation 1.E-05 represents 1.0 × 10-5). Closed diamonds represent simulation-determined points for the mixture. The dotted line represents a model of the results with the IAS theory for an H4T6 solvent-free mole fraction of 0.17.42,54 Boxes below simulation points detail the actual H4T6 solvent-free mole fraction.
Figure 10. Species adsorption versus species concentration for 5050 H4T5-H4T6 mixtures of surfactants with tail attraction (where, for example, the notation 2.E-04 represents 2.0 × 10-4). Closed diamonds represent simulation-determined points for H4T5 (1), and closed triangles are for H4T6 (2). The dotted line represents IAS theory for H4T5, and the solid line represents H4T6, using an H4T6 solvent-free mole fraction of 0.26.42,54
Figure 9. Total adsorption versus total bulk concentration for 5050 H4T5-H4T6 mixtures of surfactants with tail attraction (where, for example, the notation 1.E-03 represents 1.0 × 10-3). Closed circles represent simulation-determined points for the mixture. The dotted line represents the IAS theory using an H4T6 solvent-free mole fraction of 0.26.42,54 Boxes adjacent to simulation points detail the actual H4T6 solvent-free mole fraction. For reference, Langmuir adsorption isotherms for single-surfactant simulation with tail attraction are added for H4T5 (solid line) and H4T6 (dashed line).
Figure 11. Surface tension versus total bulk concentration for 5050 H4T5-H4T6 mixtures of surfactants with tail attraction (where, for example, the notation 1.E-05 represents 1.0 × 10-5). Closed circles represent simulation-determined points for the mixture. The dotted line represents the IAS theory at an H4T6 solvent-free mole fraction of 0.26.42,54 Boxes above simulation points detail the H4T6 solvent-free mole fraction.
Table 5. Parameters for IAS Theory for a Mixture of H4T5 and H4T6 Surfactant with Tail Attraction surfactant
surfactant no.
Γm (molecules/σ2)
KL
H4T5 H4T6
1 2
0.15 0.19
2080 5380
H4T6 (by number), as other surfactant combinations (i.e., H4T4H4T5 and H4T4-H4T6) have qualitatively similar behavior. Figure 9 graphs the total adsorption amounts for a binary mixture of 50% H4T5 and 50% H4T6 (by number) as closed circles. As before, adsorption and concentration values are total amounts. Adjacent boxes show the H4T6 solvent-free mole fractions for each simulation point. For reference, the single-surfactant isotherms are included: H4T5 (solid line) and H4T6 (dashed line). As before, mixture data fall between the single-surfactant isotherms. The dotted line in Figure 9 highlights IAS theory for an average H4T6 solvent-free mole fraction of 0.2642,54 based on the singlesurfactant adsorption isotherms with tail attraction in Figure 2 (Γm and KL are listed in Table 5). IAS theory predicts the simulation results extremely well for an H4T6 solvent-free mole
fraction of 0.26. In addition, plots of Γ1 versus F∞1 (closed diamonds) and Γ2 versus F∞2 (closed triangles) are fit well with IAS theory (dotted line for H4T5, species 1, and solid line for H4T6, species 2), as seen in Figure 10. One can calculate an enrichment or separation factor of H4T6 relative to H4T5, R ) (Γ2/F∞2)/(Γ1/F∞1).56 At high coverage R ) 3.2 for surfactants without tail attraction and 3.6 for surfactants with tail attraction. For both surfactants with and without tail attraction, R is much larger than unity, meaning H4T6 is highly preferred at the interface over H4T5. Further, R is larger for mixtures of surfactant with tail attraction than mixtures without tail attraction. This is no surprise, as H4T6 has an additional tail group that can cooperate as compared to H4T5. In addition, at high coverages, both H4T5 and H4T6 surfactant with tail attraction reside relatively closer to the vapor phase than the same mixture of surfactants without tail attraction. Nevertheless, we find it unnecessary to employ mixture models that explicitly account for tail-tail interactions, such as those of Frumkin57 and Wustneck and co-workers.7,36,37 Only the minimum area occupied is affected (56) Young, D. M.; Crowell, A. D. Physical Adsorption of Gases; Butterworths: Washington, D.C., 1962; Chapter 11. (57) Frumkin, A. Z. Phys. Chem. 1925, 116, 466-484.
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by the tail attraction. Again, we believe this result is related to the small tail-tail interaction energy chosen in Table 2. Figure 11 graphs the surface tension for a binary mixture with tail attraction of 50% H4T5 and 50% H4T6 (by number). Closed circles are the simulation-determined values of the surface tension for the mixture. Boxed values near the simulation data represent the H4T6 solvent-free mole fraction. Concentrations values are totals. IAS theory is shown in Figure 11 as a dotted line for an average H4T6 solvent-free mole fraction of 0.26.42,54 IAS theory predicts the mixture results extremely well.
Howes and Radke
Surfactants with tail attraction shed solvent and adsorb relatively closer to the vapor phase. Although both surface-tension and adsorption behavior are identical at low concentrations, surfactants with tail attraction exhibit lower CAC’s, not allowing simulation of high concentration. Binary mixtures of surfactants exhibit behavior that is intermediate to their counterpart single surfactants. We successfully predict simulated mixture results with IAS theory. Ultimately, a LJ coarse-grained surfactant model is successful at mimicking nonionic-surfactant adsorption behavior at the liquid/vapor interface in mixtures and for surfactants with and without tail attraction.
4. Conclusions In this work, we study model LJ surfactant molecules in singlesurfactant simulations with tail attraction and in binary-mixture simulations with and without tail attraction. Even though the classic premises of the Langmuir adsorption model are not strictly met, the Langmuir isotherm well fits the simulated adsorption and surface-tension isotherms for single surfactants with and without tail attraction. A Frumkin-like isotherm with an additional term explicitly accounting for tail attraction proves unnecessary. Our simulation results for mixtures are well fit with IAS theory. Surfactants with tail attraction attain higher coverages at the interface due to cooperation among attractive tail groups.
Acknowledgment. This work was funded by a National Science Foundation Graduate Research Fellowship, an Arkema Chemical Co. Fellowship, and the Sam Ruben and Irv Fatt Memorial Fund for Graduate Education in the College of Chemistry at UC Berkeley. In addition, all of the simulation work was conducted on the cluster of the Molecular Graphics and Computation Facility in the College of Chemistry at UC Berkeley. We are grateful to Divesh Bhatt for helpful comments and for some of the original computer code. LA701452G
