Unified Model To Predict Self-Assembly of Nonionic Surfactants in

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Unified Model To Predict Self-Assembly of Nonionic Surfactants in Solution and Adsorption on Solid or Fluid Hydrophobic Surfaces: Effect of Molecular Structure Nitin Kumar*,†,‡ and Robert D. Tilton‡ Departments of Physics and Chemical Engineering and Center for Complex Fluids Engineering, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213 Received July 18, 2003. In Final Form: February 12, 2004 We have developed a pseudo-phase model to predict the self-assembly of nonionic surfactants on hydrophobic solid or fluid interfaces and in bulk solution. The uniqueness of this model is that it provides the relationship between molecular structure and self-assembly in solution and on interfaces. This model requires the input of minimal new experimental data. The remaining model parameters may be calculated on the basis of the surfactant molecular structure. The validity of the model has been established by comparing predictions with a wide array of experimental data for nonionic surfactant adsorption at the hydrophobic solid-water interface and at the air-water interface. The same model is then used to predict the self-assembly in bulk solution. The model predictions for critical aggregation concentration, aggregate shapes, and adsorption isotherms of various surfactants are in good agreement with the experimental data available in the literature.

1. Introduction Surfactant amphiphilicity drives surfactants to selfassemble in bulk solution at surfactant concentrations exceeding the critical aggregation concentration (CAC). (The widely used nomenclature for this critical concentration is critical micelle concentration. In this paper we will use the term CAC because it envelops every kind of aggregate shape, including micelles.) Various aggregate configurations are possible in bulk solution (see Figure 1), and the particular form adopted by a given type of surfactant depends on the surfactant’s molecular structure as well as the bulk concentration. The amphiphilic nature of surfactant molecules is also responsible for their selfassembly on hydrophobic interfaces with its hydrophobic group oriented generally toward the surface and its hydrophilic group generally toward the aqueous solution. Over many decades, a great deal of experimental and theoretical research has been carried out to understand surfactant self-assembly both in bulk solution1-9 and on a wide variety of interfaces including solid hydrophobic10-15 * Address correspondence to Dr. Nitin Kumar, e-mail nitin@ mit.edu, or Prof. Robert D. Tilton, e-mail [email protected]. † Department of Physics, Carnegie Mellon University. ‡ Department of Chemical Engineering and Center for Complex Fluids Engineering, Carnegie Mellon University. (1) Israelachivili, J. N. Intermolecular and Surfaces Forces; Academic Press: New York, 1991. (2) Huibers, P. D. T.; Lobanov, V. S.; Katritzky, A. R.; Shah, D. O.; Karelson, M. Langmuir 1996, 12, 1462-1470. (3) Penfold, J.; Staples, E.; Tucker, I. J. Phys. Chem. B 2002, 106, 8891-8897. (4) Danino, D.; Talmon, Y.; Zana, R. J. Colloid Interface Sci. 1997, 186, 170. (5) Kim, J. H.; Domach, M. M.; Tilton, R. D. Colloids Surf., A 1999, 150, 55-68. (6) Tanford, C. The hydrophobic effect: formation of micelles and biological membranes; Wiley-Interscience: New York, 1980. (7) Nagarajan, R. Langmuir 2002, 18, 31-38. (8) Vasilescu, M.; Caragheorgheopol, A.; Caldararu, H. Adv. Colloid Interface Sci. 2001, 89-90, 169-194. (9) Puvvada, S.; Blankschtein, D. J. Phys. Chem. 1992, 96, 55675579. (10) Subramanyam, R.; Maldarelli, C. J. Colloid Interface Sci. 2002, 253, 377-392. (11) Velegol, S. B.; Tilton, R. D. Langmuir 2001, 17, 219-227.

Figure 1. (A) Structure of bulk and surface aggregates, showing spherical micelles and bilayer or cylindrical bulk aggregates. On hydrophobic surfaces, surfactants assemble as uniform layers. (B) Surface and bulk aggregates as different phases in contact with monomers in solution. Aggregates have hydrophobic (HC) and hydrophilic (PEO) regions. For clarity, the PEO chain of the surfactant is shown as a circle.

and fluid hydrophobic interfaces.10,16-20 Current models are mainly restricted to one of these self-assembly contexts. Furthermore, these models typically rely on a large (12) Thritle, P. N.; Li, Z. X.; Thomas, R. K.; Rennie, A. R.; Satija, S. K.; Sung, L. P. Langmuir 1997, 13, 5451-5458. (13) Gilchrist, V. A.; Lu, J. R.; Keddie, J. L.; Staples, E.; Garrett, P. Langmuir 2000, 16, 740-748. (14) Haidara, H.; Vonna, L.; Schultz, J. J. Adhes. Sci. Technol. 1999, 13, 1393-1403. (15) Kjellin, U. R. M.; Claesson, P. M.; Linse, P. Langmuir 2002, 18, 6745-6753. (16) Schick, M. J. Nonionic Surfactants; M. Dekker: New York, 1987. (17) Kumar, N.; Couzis, A.; Maldarelli, C. J. Colloid Interface Sci. 2003, 267, 272-285.

10.1021/la035311c CCC: $27.50 © 2004 American Chemical Society Published on Web 04/28/2004

Self-Assembly and Adsorption of Nonionic Surfaces

number of experimental data for parametrization.21 There is no single general and unified model to link self-assembly in bulk solution, on solid hydrophobic interfaces, and fluid hydrophobic interfaces. In some cases, progress has been made to relate the self-assembly in bulk with self-assembly at either the air-water (AW)22 or a solid-water (SW) interface.23,24 There is still no clear and predictive understanding of how the molecular structure of a surfactant molecule dictates its self-assembly behavior. Given the similarity of driving forces for the self-assembly processes in question, a unified model should be within reach for self-assembly in bulk or in contact with uncharged, hydrophobic surfaces. In this paper, we develop a unified model to simultaneously predict the self-assembling characteristics of nonionic surfactants on hydrophobic interfaces and in bulk solution. We compare the model predictions with available experimental data for nonionic polyethoxylate (PEO) surfactants (CIEJ) with varying molecular structures. The hydrophobic part of these surfactants consists of a linear hydrocarbon chain (HC) consisting of I carbons. The hydrophilic part is a PEO chain with J ethoxylate (EO, CH2CH2O) units. The PEO chain is usually terminated with a hydroxyl group. While there is no limitation on I and J, the common CIEJ surfactants have I ) 10-16 and J ) 3-12. In this paper, we limit our attention to surfactants with I and J in this range. Many models are available to predict the self-assembly of CIEJ surfactants in bulk.25-28 The adsorption of CIEJ surfactants on the AW interface has been extensively studied. Some models have also been developed to predict the adsorption of CIEJ surfactants on the AW interface 27,29,30 and on hydrophobic solid surfaces.24,31,32 For hydrophobic SW interfaces, the structure and chemical heterogeneity of the surface adds to the complexity. Experimental data for hydrophobic solid surfaces, especially for smooth and chemically homogeneous surfaces, have been rather limited compared to the AW interface. In the accompanying paper,33 we measured the selfassembling characteristics of CIEJ surfactants on model hydrophobic surfaces that are uniform and chemically homogeneous. In this paper, we compare the predictions of the model for self-assembly in bulk and at the AW and SW interfaces with the experimental data obtained from a number of papers in the literature. (18) Pan, R.; Green, J.; Maldarelli, C. J. Colloid Interface Sci. 1998, 205, 213-230. (19) Kumar, N.; Tilton, R.; Garoff, S. Langmuir 2003, 19, 53665373. (20) Mulqueen, M.; Stebe, K. J.; Blankschtein, D. Langmuir 2001, 17, 5196-5207. (21) Prosser, A. J.; Franses, E. I. Colloids Surf., A 2001, 178, 1-40. (22) Nikas, Y. J.; Puvvada, S.; Blankschtein, D. Langmuir 1992, 8, 2680-2689. (23) Johnson, R. A.; Nagarajan, R. Colloids Surf., A 2000, 167, 3746. (24) Johnson, R. A.; Nagarajan, R. Colloids Surf., A 2000, 167, 2136. (25) Nagarajan, R.; Ruckenstein, E. Langmuir 1991, 7, 2934-2969. (26) Puvvada, S.; Blankschtein, D. J. Chem. Phys. 1990, 92, 37103724. (27) Meguro, K.; Ueno, M.; Esumi, K. In Nonionic Surfactants; Schick, M. J., Ed.; Marcel Dekker: New York, 1987. (28) Jodar-Reyes, A. B.; Ortega-Vinuesa, J. L.; Martin-Rodriguez, A.; Leermakers, F. A. M. Langmuir 2002, 18, 8706-8713. (29) Hsu, C.-T.; Chang, C.-H.; Lin, S.-Y. Langmuir 1999, 15, 19521959. (30) Lin, S.-Y.; McKeigue, K.; Maldarelli, C. Langmuir 1991, 7, 1055. (31) Steinby, K.; Silveston, R.; Kronberg, B. J. Colloid Interface Sci. 1993, 155, 70-78. (32) Kronberg, B.; Costas, M.; Silveston, R. Pure Appl. Chem. 1995, 67, 897-902. (33) Kumar, N.; Garoff, S.; Tilton, R. D. Langmuir 2004, 20, 44464451.

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This paper is organized as follows. In section 2, we describe the conceptual development of the model that we mathematically describe in section 3. The comparison of the model predictions with the experimental data for SW and AW interfaces is presented in sections 4-6. In section 7, the predictions of self-assembly in the bulk solution are compared with the experimental results available in the literature. In section 8, we discuss the results and further explore the relationship between the surfactant molecular structure and self-assembly. 2. Development of the Concept The process of surfactant assembly in bulk solution and on interfaces is sketched schematically in Figure 1. In bulk solution, surfactants self-assemble into aggregates such that the hydrocarbon core region is nearly devoid of water molecules. The shape of the aggregates is such that the total free energy of the system is minimum.1,6 Aggregate shapes depend on surfactant molecular structure. We restrict this study to concentrations just above the CAC, where interaggregate interactions are negligible. On hydrophobic surfaces, the shape of the self-assembled structure is influenced by the shape of the interface. Surfactants tend to assemble into a uniform layer on hydrophobic surfaces, as shown in Figure 1. This has been confirmed experimentally by a number of researchers for SW interfaces.12,33-35 For the AW interface, CIEJ (J g 3) surfactants also adsorb as uniform monolayers.36 The planarity of the hydrophobic SW or AW interface confines surfactants to similar packing arrangements as would be assumed in lamellar bilayer aggregates in bulk solution. While all the hydrophobic interfaces have common important features, there are important differences between solid and fluid hydrophobic interfaces. On fluid surfaces, such as the AW interface, the hydrophobic chains are able to penetrate out of the aqueous environment. This cannot happen on a SW interface because hydrocarbon chains cannot escape the aqueous environment entirely. The hydrophobic region of the self-assembled structure on the SW interface is considered to be a mixture of water and surfactant hydrocarbon chains. The selfassembled hydrophobic region at a fluid interface is a mixture of hydrocarbon chain and molecules of the hydrophobic fluid (e.g., oil, air) and can be considered to be devoid of water molecules. We take the pseudo-phase model approach, where the surface aggregate or the bulk aggregate is considered as a phase distinct from the surrounding solution of monomeric surfactant molecules. We model the hydrophilic region as a mixture of PEO chains and water. The hydrophobic region of bulk aggregates is made of only hydrocarbon chains. The dilute solution of surfactant monomers can be considered as an ideal solution. With these considerations, a molecular thermodynamic approach has been developed to predict the self-assembly of CIEJ surfactants. The molecular thermodynamic approach to the self-assembly of surfactants in bulk solution was pioneered by Tanford6 and later developed and used by others.25,26 (34) Grant, L. M.; Tiberg, F.; Ducker, W. A. J. Phys. Chem. B 1998, 102, 4288-4294. (35) Fragneto, G.; Lu, J. R.; McDermott, D. C.; Thomas, R. K.; Rennie, A. R.; Gallagher, P. D.; Satija, S. K. Langmuir 1996, 12, 477-486. (36) Lu, J. R.; Thomas, R. K.; Penfold, J. Adv. Colloid Interface Sci. 2000, 84, 143.

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surfactant in the aggregate phase comes from the standard part µP0 in eq 3. Thus,

3. Molecular Thermodynamic Model for the Formation of Aggregates 3.1. Chemical Potential of the Surfactant. In the pseudo-phase model, the bulk or surface aggregates are in thermodynamic equilibrium with the dilute solution of surfactant monomers. The volume fraction of surfactant molecules in the aggregates is much higher than the volume fraction of surfactant monomers in solution. The hydrophilic and hydrophobic regions of bulk and surface aggregates are considered to be of uniform composition; that is, we do not consider gradients of composition within an aggregate. At equilibrium, the chemical potentials of the surfactant molecules in all aggregate phases and in the monomer solution are equal. The standard state of water is defined as pure liquid water. The standard state of surfactant is defined as an isolated molecule in infinitely dilute surfactant solution conditions. The intersurfactant interactions are assumed to be negligible for surfactant monomers in solution. The chemical potential of a surfactant molecule present as a monomer in solution is given by

µ1 ) µ01 + RT ln(x1)

(1)

where µ01 is the chemical potential of the surfactant at its standard state of infinite dilution and x1 is the surfactant monomer mole fraction in the solution. For CIEJ surfactants, CAC , 1. Thus, the monomer solution can be considered as ideal. The chemical potential of a bulk aggregate containing P surfactants can be written as25 agg,0 + RT ln(xagg µagg P ) µP P )

where µPagg,0 is the standard chemical potential for aggregate containing P surfactant molecules, xagg P is mole fraction of bulk aggregates in solution, and chemical potential of one surfactant molecule in aggregate is

µP ) µ0P +

RT ln(xagg P ) P

(2) the the the the

(3)

RT ln(xagg P ) P

(4)

or

µ01 - µ0P ln(xagg P ) ) - ln(x1) RT P

(5)

This is the pseudo-phase model relation that describes the population balance of the number of aggregates with different aggregation numbers for a given monomer concentration. It should be noted that the last term in eq 3 is the “inter-aggregate” entropic contribution to the chemical potential of the aggregate. This term is negligible because the aggregation number is typically large. Therefore, a significant part of the chemical potential of the

(6)

where ∆µ0P is the difference in the standard chemical potential between a surfactant molecule in a bulk aggregate (µP) and that as a monomer in solution. This is also called the standard free energy of transfer. In a similar fashion, we can also define the free energy of transfer from the bulk solution to a surface aggregate. For the surface aggregate, the aggregation number is limited only by the available interface area and, therefore, could be considered to be infinite. In a manner similar to eq 6, the expression for surfactant in a surface aggregate is 0 0 ) µsurface - µ01 ) µsurface - µ01 ∆µsurface

(7)

where µsurface is the average chemical potential of a surfactant molecule present in the surface aggregate. The same standard state of an isolated molecule in dilute solution (µ01) is used for surfactant in solution as monomers and surfactant in surface aggregates. In the next sections, we will calculate this difference in chemical potential as a function of the structure of the bulk and surface aggregates. 3.2. Modeling of Surfactant Molecules as Short Polymer Chains. The hydrocarbon chain of each surfactant molecule in an aggregate is taken to reside in the hydrophobic region of the aggregate. The average surface area of the hydrophobic region occupied per surfactant molecule is denoted as a. For the bulk aggregates, this is equal to the total surface area of the hydrophobic core divided by the aggregation number P. The surface aggregates are planar in shape. Thus, for surface aggregates, a is also denoted as the average area of the hydrophobic interface per molecule. This is related to the surface concentration Γ (mol per unit area) as

a ) 1/(ΓNAv)

µ0P ()µagg,0 /P) is the standard part of the chemical potenP tial of the surfactant molecule present in an aggregate having an aggregation number P. Equality of the chemical potential of surfactant molecules existing in aggregates and as monomers in solution dictates that

µ1 ) µ01 + RT ln(x1) ) µ0P +

∆µ0P ) µ0P - µ01 ) µP - µ01

(8)

where NAv is Avogadro’s number. The PEO and hydrocarbon regions of the surfactant are modeled using polymer solution theory.37-43 A lattice model representation is used to model the configuration of hydrocarbon chains and PEO chains in the hydrophilic and hydrophobic regions of the aggregate phases. The lattice model for polymers requires the polymer chains to be treated as chains composed of molecular segments. Each segment corresponds to the linear dimension of the lattice. For hydrocarbon chains, Dill et al.25,44-46 suggested a suitable length of the lattice dimension or segment length (37) Flory, P. J. Statistical mechanics of chain molecules; Oxford University Press: New York, 1989. (38) de Gennes, P. G. J. Phys. (Paris) 1976, 37, 1445-1452. (39) de Gennes, P. G. Macromolecules 1980, 13, 1069-1075. (40) de Gennes, P. G. J. Polym. Sci., Part B: Polym. Phys. 1978, 16, 1883-1885. (41) de Gennes, P. G. J. Phys., Lett. 1978, 39, L299-L301. (42) Brochard, F.; de Gennes, P. G. Macromolecules 1977, 10, 11571161. (43) Tirrell, M. In Interactions of surfactants with polymers and proteins; Goddard, E. D., Ananthapadmanabhan, K. P., Eds.; CRC Press: Boca Raton, 1993. (44) Flory, P. J.; Yoon, D. Y.; Dill, K. A. Macromolecules 1984, 17, 862-868. (45) Dill, K. A.; Flory, P. J. Proc. Natl. Acad. Sci. U.S.A. 1981, 78, 676-680. (46) Dill, K. A.; Flory, P. J. Proc. Natl. Acad. Sci. U.S.A. 1980, 77, 3115-3119.


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LHC to be 4.6 Å. The number of polymer segments in the hydrocarbon with I carbons can be calculated by

NHC ) vCH2I/LHC3

(9)

where vCH2 is the volume of one methylene group. The volume of the hydrocarbon chain can be calculated from the volumes of methyl and methylene groups using the following formulas:25 3

vCH2, Å ) 26.9 + 0.0146(T - 298) vCH3, Å3 ) 56.4 + 0.124(T - 298) vHC ) vCH3 + (I - 1)vCH2

(10)

(11)

where vEO is the volume of one EO unit. On the basis of density data,22,25,47 the value of vEO is 63 Å3. The length of one segment of a PEO chain, LEO, is same as that of the hydrocarbon chain, that is, 4.6 Å, to use the same lattice for both the hydrocarbon and the PEO headgroup. For an eight-EO-unit PEO chain, the value of NPEO is calculated to be 5.18. In these calculations, we have neglected the effect of the hydroxyl group present at the end of the PEO chain. The hydroxyl group volume is very small compared to that of the PEO chain. The hydrophilic region of the aggregates consists of PEO chains and water molecules, with the requirement that one end of each PEO chain is connected to the hydrocarbon chain. The thickness of the hydrophilic region, DPEO (Figure 1), can be taken to be the average length of PEO chains normal to the interface. Sarmoria and Blankschtein48 have shown that a PEO chain, whose one end is restricted to remain on an interface, behaves like a polymer chain in solution. The length of a polymer chain in a good solvent varies nearly as the square root of the number of segments.37 Water is a good solvent for PEO chains. Therefore, we assume that the PEO headgroups are not stretched as in a brush, and we calculate the average length of the PEO chains and the thickness of the hydrophilic region as

DPEO ) NPEO1/2LEO

DHC,surface ) NHC1/3LHC

(13)

The thickness of the hydrocarbon region can be compared with the length of a fully stretched hydrocarbon chain (Lmax). Lmax for a hydrocarbon chain with I carbons is calculated as1,6

Lmax, Å ) 1.5 + 1.265I

where T is absolute temperature and vHC is the total volume of the hydrocarbon chain. At 25 °C, vHC and NHC for a C12 chain are 350 Å3 and 3.31, respectively. In a similar fashion, the number of polymer segments (NPEO) for a PEO chain with J EO units is given by

NPEO ) vEOJ/LEO3

ness of the planar hydrophobic region of surface aggregates is equated to the average length of a hydrocarbon chain in the hydrophobic region. The average length of a polymer chain in a poor solvent scales as the cube root of the number of polymer segments.37,49 Thus, the thickness of the hydrophobic region for surface aggregates is given by

(14)

For a 12-carbon chain, the fully stretched length, the length in water (poor solvent), and the length in a good solvent are 16.68, 6.86, and 8.36 Å, respectively. Another characteristic length of a polymer chain could be defined as Lchar ) vHC1/3, which is calculated to be 7.04 Å. The realistic dimension of a hydrocarbon chain under any solvent condition would lie between the extreme cases for poor and good solvents, that is, between NHC1/3L and NHC1/2L. For the common CIEJ surfactants, the number of carbon atoms in the hydrocarbon chain varies from 10 to 16. For these short chains, the average lengths in poor and good solvents are very close to each other. Therefore, eq 13 is a good estimate for the thickness of the hydrophobic region of surface aggregates. 3.3. Volume Fraction of Surfactant in Aggregates. The volume fraction of the surfactant hydrocarbon chains and PEO chains in aggregates can be calculated from the thickness of the hydrophilic and hydrophobic regions and the area of the interface or the hydrocarbon core that is occupied by one surfactant molecule, a. The volume fraction of the hydrocarbon chain in the hydrophobic region of the surface aggregates is given by

φHC,surface )

vHC aDHC

(15)

In the case of the bulk aggregates, the density of the hydrocarbon core is close to that of liquid hydrocarbon. Thus,

φHC,bulk ) 1

(16)

Using the average thickness of the hydrophilic region, the volume fraction of the PEO chains in the hydrophilic region of bulk aggregates as well as surface aggregates is given as

(12)

The thickness or the size of the hydrophobic region can be estimated in a similar manner. For the bulk aggregates, the hydrophobic region consists entirely of the hydrocarbon chains. The size of the bulk aggregate core is related to the aggregation number (P) and the hydrocarbon chain volume as described later in this paper. The hydrophobic region for the surface aggregates is planar and is a mixture of water and hydrocarbon chains. Water is a poor solvent for the hydrocarbon chain. Similar to the PEO chain, one end of the hydrocarbon chain is restricted to remain in contact with the hydrophilic region. Therefore, the thick(47) Mulley, B. A. In Nonionic Surfactants; Schick, M. J., Ed.; Marcel Dekker: New York, 1967. (48) Sarmoria, C.; Blankschtein, D. J. Phys. Chem. 1992, 96, 19781983.

φPEO )

JvEO aDPEO

(17)

The overall volume fraction of surfactant molecules can be written as

φsurfactant )

JvEO + vHC JvEO/φEO + vHC/φHC

(18)

for either the bulk aggregate or the surface aggregate phases. 3.4. Free Energy of Transfer of the Surfactant. Various factors contribute to the change in chemical (49) de Gennes, P. G. Scaling concepts in polymer physics; Cornell University Press: Ithaca, NY, 1979.

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potential of the surfactant (eqs 6 and 7) as it is transferred from the dilute monomer solution to a bulk or surface aggregate phase. In this process, the hydrocarbon chain is transferred from solution, where it is exposed to water, to the hydrophobic region of the aggregates. The PEO chain is also transferred from dilute aqueous solution to the hydrophilic region with higher density of PEO chains. 0 can be estimated Various contributions to ∆µ0P or ∆µsurface using polymer solution theory and the surfactant molec0 ular structure. The main contributions to ∆µ0P or ∆µsurface are (i) ideal mixing of surfactants in the aggregate phase, (ii) transfer of hydrocarbon chains from the aqueous environment to the hydrophobic region, (iii) transfer of PEO chains from the dilute solution to the dense hydrophilic region, (iv) steric interactions among surfactant molecules, and (v) formation of a new hydrophobicaqueous interface. Each of these contributions is examined below and treated in a similar manner for both surface and bulk aggregates. 3.4.1. Ideal Mixing. The aggregate phase is considered as a phase that consists of surfactant and water. When a surfactant is transferred from its standard state as an isolated molecule in dilute solution to the surface or bulk aggregate, the free energy of mixing is 0 ∆µideal mixing/kT ) ln(φsurfactant)

U(R B )/kT ) φ(R)

(19)

The expression for φsurfactant for the surface and bulk aggregate phases is given in eq 18. In this equation, the same expression for φPEO is used for either bulk or surface aggregates, but different expressions are used for φHC in the case of bulk and surface aggregates, as shown in eqs 15 and 16. 3.4.2. Transfer of Hydrocarbon Chains from an Aqueous to a Hydrocarbon-Rich Environment. For bulk aggregates, the hydrophobic region is similar to liquid hydrocarbon. The free energy of transfer of a hydrocarbon chain with I carbons from the aqueous environment to a liquid hydrocarbon environment can be evaluated from the solubility and heat of vaporization data for hydrocarbons:25 0 /kT ) (5.85 ln T + 896/T - 36.25 ∆µHC,bulk 0.0056T)(I - 1) + 3.38 ln T + 4064/T 44.13 + 0.02595 T (20)

This contribution is negative and its absolute value increases with increasing length of the hydrocarbon chain, a result of the lower solubility of longer hydrocarbon chains in water. The situation for a surface aggregate is similar. 0 /kT is also expected to be negative, but its ∆µHC,surface value cannot be calculated from expressions similar to eq 20 because the hydrophobic region of the surface aggregates does not consist entirely of hydrocarbon chains. It can also contain other fluid molecules in the case of a fluid interface or water molecules in the case of solid 0 /kT remains unknown hydrophobic surfaces. ∆µHC,surface and is expected to be different for different types of hydrophobic surfaces. Thus, we write 0 /kT ) ln Ksurface ∆µHC,surface

a function of T for a given surfactant-surface system and independent of a. 3.4.3. Transfer of PEO Chains from the Aqueous Phase to the Aggregate Phase. The free energy change associated with the transfer of a polymer chain from its standard state in dilute solution to this hydrophilic region can be calculated using polymer solution theory.37,43 The contribution of the ideal mixing in the aggregate phase is already accounted for in eq 19. However, there are other contributions to the free energy of transfer that account for the excluded volume effects and polymer-solvent interactions. This contribution of the PEO chain to the total free energy of transfer of surfactant is estimated as follows. The volume fraction of PEO chains, φPEO (eq 17), is assumed to be uniform throughout the hydrophilic region. When a PEO chain is transferred from its standard state as an isolated molecule in dilute solution to the concentrated solution, it experiences a potential U(R B ) due to all other polymer segments and solvent molecules in the phase. R B is the position vector. In the mean field approach, the expression of U(R B ) is given by25,43,49

(21)

where Ksurface depends on the particular surfactantsurface system. ∆µHC,bulk0/kT depends only on the number of carbons in the hydrocarbon chain and T. It does not depend on a. In a similar way, ∆µHC,surface0/kT can be assumed to be only

(21 - χ ) + [φ(R6B)]

2

(22)

PEO

ℵPEO is the Flory-Huggins parameter that accounts for solvent-polymer interactions. The free energy of transfer of a PEO chain to the hydrophilic region is

∆µ0PEO/kT )

1 NLEO3

∫Vφ(RB ) U(RB )/kT dRB 3

(23)

where N is the total number of PEO chains in the hydrophilic region of the aggregate. The integral is taken over the entire volume of the hydrophilic region, and φPEO is assumed to be uniform over the entire hydrophilic region. U(R B ) is independent of the location; thus,

∆µ0PEO/kT )

∫Vφ(RB ) U(RB )/kT dRB 3 )

1 NLEO3

1 1 U(R B )VTPEO/kT ) U(R B )NJvEO/kT (24) 3 NLEO NLEO3 T is the total volume of all PEO chains in the where VPEO hydrophilic region, that is, the product of the PEO volume fraction and the total volume of the hydrophilic region. Using the definition of NPEO from eq 11,

{

)

}

2

φPEO 1 - ℵPEO + 2 6

(

∆µ0PEO/kT ) NPEO φPEO

(25)

The same expression for φPEO (eq 17) is used to calculate ∆µPEO for bulk and surface aggregates. Water is a good solvent for PEO at room temperature. For good solvent conditions, ℵ < 0.5, and the first term of eq 25 adds a positive contribution (i.e., unfavorable) to the free energy. The second term in eq 25 is always positive and, therefore, always adds a positive contribution to the free energy. In principle, the value of ℵ can be obtained from the thermodynamic properties of water-PEO solutions. The activity data of water-PEO solutions, represented in the framework of standard Flory-Huggins theory, indicates that the value of ℵPEO changes with the composition of the polymer solution. For very dilute solutions, a value of 0.1


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describes the activity data.50 However, for more concentrated polymer solutions, values in the range 0.1-0.4 have been reported.25,51,52 The explicit expression for the dependence of ℵPEO on the solution composition, temperature, and polymer chain length is still not available. In the absence of such information, we take the value of ℵPEO to be 0.3 at room temperature,25,31 independent of the concentration of the PEO chains in the hydrophilic region of the bulk and surface aggregates. This value of ℵPEO results in an overall positive contribution to ∆µPEO. This is the thermodynamic representation for the “repulsive” interaction among the PEO headgroups comprising the hydrophilic region of the aggregate. For the AW interface, the repulsive interactions among ethoxylated surfactant molecules have been observed experimentally and attributed to the interactions among the PEO chains.17,18 3.4.4. Steric Interactions. In the process of the formation of bulk or surface aggregates, the surfactant molecule is brought into the aggregate phase. This process results in crowding of the aggregate phase. If ah is the effective area of the surfactant headgroup, that is, the PEO chain, then as a surfactant molecule is transferred into the aggregate, the total area of the aggregate that is available for the translational motion of the surfactant molecule is reduced by an amount equal to ah.22,25,26 This generates a steric repulsion among the surfactant molecules. For uniform monolayers, the free energy associated with the steric repulsion is expressed as 0 ∆µsteric ) -ln(1 - ah/a)

(26)

The same expression is used for the bulk and surface aggregates. The effective area of the PEO chain in the plane of the interface can be calculated from the volume of the PEO chain and the average length of the PEO chain normal to the interfaces, which are proportional to J and J1/2, respectively. Thus, ah is proportional to J1/2.

ah ∝ J1/2

(27)

The experimental results suggest a value of 19 Å2 for the proportionality constant.33 Thus,

ah ) 19J1/2 Å2

(28)

The value of the prefactor, 19 Å2, is very close to the cross-sectional area of fully stretched PEO or hydrocarbon chains or the area of one EO group. The square-root scaling for the average length of the PEO chain is valid for J g 3; therefore, the above scaling for ah is valid for surfactants with three or more EO units.48 3.4.5. Aggregate Core-Water Interfacial Energy. As a surfactant molecule is transferred into a bulk aggregate, the aggregation number increases. Unlike the surface aggregates that occupy a finite area (the area of the SW interface), the volume and surface area of the hydrophobic region of bulk aggregates are not externally limited. To maintain φHC equal to unity in the hydrophobic region, the volume and area of the hydrophobic region increase by vHC and a, respectively, for each additional surfactant molecule. The generation of this new hydrocarbonaqueous interface between the hydrophobic region and the hydrophilic region also contributes to the free energy (50) Malcom, G. N.; Rawlinson, J. S. Trans. Faraday Soc. 1957, 53, 921. (51) Hurter, P. N.; Scheutjens, J.; Hatton, T. A. Macromolecules 1993, 26, 5592-5601. (52) de Bruijn, V. G.; van den Broeke, L. J. P.; Leermakers, F. A. M.; Keurentjes, J. T. F. Langmuir 2002, 18, 10467-10474.

of transfer of surfactant. For every surfactant molecule transferred from dilute solution to the bulk aggregate, some portion of the area increment a does not result in the formation of a new hydrocarbon-aqueous interface. This area, a0, is shielded from the aqueous environment because of the presence of the PEO chain attached to the hydrocarbon chain. We take this value to be 23 Å2, which is close to L2, which is the area spanned by one segment of a PEO chain.25,26 This value is also very close to the cross-sectional area of a fully stretched hydrocarbon or PEO chain. The change in free energy of a surfactant molecule associated with this increase in interfacial area of the hydrocarbon core in a bulk aggregate is bulk ∆µinterfacial /kT ) σcore-aqueous(a - a0)/kT

(29)

where σcore-aqueous is the interfacial energy of the hydrocarbon core-hydrophilic region interface. The hydrophilic region is a solution of water and PEO. We take a value of 45 mN/m for σcore-aqueous, estimated in a manner described in Nagarajan and Ruckenstein.25 We have assumed that σcore-aqueous does not change with the aggregate shape and size. Puvvada and Blankschtein have also considered the variation of the σcore-aqueous aggregate size and shape.26 Let us now consider the contribution of this interfacial free energy for surface aggregates. At any bulk concentration, the surfactant molecules occupy any available hydrophobic surface to form a planar and uniform layer. When a surfactant is transferred into this surface aggregate, there is no increase in the total area of the interface. The volume fraction of the hydrocarbon chain in the aggregate increases while maintaining the same total area of the interface. Thus, surface /kT ) 0 ∆µinterfacial

(30)

This is in contrast with the bulk aggregation, where the aggregate size has to increase to maintain φHC equal to unity. As discussed above, φHC for surface aggregates is not constrained to be unity. This is equivalent to assuming that adsorbed nonionic surfactant layers do not grow in a patchwise manner on uncharged hydrophobic surfaces. This is consistent with the adsorption isotherms presented in the accompanying paper.33 This is an important distinction between the selfassembly in the bulk and on interfaces. As we will show later in this paper, the contribution of this interfacial energy term plays a major role in the formation of bulk aggregates. The absence of this term for the surface aggregate explains why surfactants assemble on interfaces at concentrations below the CAC. 3.4.6. Net Free Energy of Surfactant Transfer. The total free energy change that occurs when a surfactant is transferred from its isolated state in dilute solution to the aggregate phase in bulk or on an interface is the sum of all the contributions described above. All of the individual contributions, except for the hydrocarbon transfer contribution, are functions of a. Thus, 0 0 ∆µideal ∆µ0HC ∆µ0PEO ∆µbulk or surface mix (a) ) (a) + + (a) + kT kT kT kT 0 0 ∆µinterfacial ∆µsteric (a) + (a) (31) kT kT

The same general expression is used for surface as well as bulk aggregates. The same standard state for a surfactant has been used for surface aggregates, bulk aggregates, and surfactants

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present as monomers in solution. By taking the chemical potential of this standard state (µ01) to be zero, the chemical potential of surfactant molecules in aggregates or as monomers in solution can be calculated (via eqs 6 and 7). For a monomer in solution,

µ1/kT ) ln(x1)

(32)

For a surfactant molecule in a surface or bulk aggregate, 0 µbulk or surface ∆µbulk or surface ) (a) kT kT

(33)

At thermodynamic equilibrium,

µbulk or surface (a) ) ln(x1) kT

(34)

As a decreases, φsurfactant and φPEO increase, while φHC for bulk aggregates remains equal to unity. The volume fractions are always less than 1. Therefore, 0 ∆µideal mix(a)/kT is always negative and increases as a decreases. Both the terms in the expression of 0 (a)/kT are positive and increase as a decreases. The ∆µPEO contribution to the chemical potential of the surfactant due to the transfer of the hydrocarbon chain from the aqueous environment to the aggregate core, ∆µ0HC/kT, is negative and is considered to be independent of a. The contribution of steric interaction among the PEO chains 0 (a)/kT is always positive and increases as a de∆µsteric creases. Using eqs 17, 27, and 28, it can be shown that at as a f ah, φPEO ≈ 1. Similarly, φsurfactant is also nearly 0 1 as a f ah. The maximum values of ∆µideal mix(a)/kT and 0 ∆µPEO(a)/kT are limited as a f ah. In comparison, 0 (a)/kT increases sharply and approaches infinity ∆µsteric as a f ah. 0 The common characteristic of ∆µideal mix(a)/kT, 0 0 ∆µPEO(a)/kT, and ∆µsteric(a)/kT is that all of them increase monotonically as a decreases. This is in contrast 0 (a)/kT, which for bulk aggregates dewith ∆µinterfacial creases monotonically as a decreases. This opposing 0 (a)/kT compared to other contrivariation of ∆µinterfacial butions to the chemical potential results in a minimum in ∆µbulk(a)/kT. Later in this paper, we will evaluate the CAC from this minimum. For surface aggre0 (a)/kT is 0; thus, ∆µsurface(a)/kT increases gates, ∆µinterfacial monotonically as a decreases. The effect of temperature could be included in this model, but currently, we use this model to predict self-assembly only at 25 °C. 4. Self-Assembly on Interfaces At equilibrium, surfactant molecules on the interface have the same chemical potential as those in solution. At concentrations below the CAC, all surfactant molecules in solution exist as monomers. ∆µsurface(a)/kT increases monotonically with increasing a. Thus, for any value of x1, there is a unique and real value of a that satisfies the thermodynamic equilibrium condition as given by eq 34. Consequently, the surface aggregate phase on a hydrophobic surface is formed at all concentrations below the CAC. Or in other words, surfactant adsorbs on hydrophobic surfaces at all concentrations. Thus, for surface aggregates and for x1 < xCAC, eq 34 can be written as

ln(x1) ) ln(φsurfactant) + ln K + NPEO{φPEO(1/2 ℵPEO) + φPEO2/6} + {-ln(1 - ah/a)} (35) Using eq 8 to relate the surface excess concentration of surfactant, Γ, to a, eq 35 is readily converted to an expression for the adsorption isotherm written in terms of Γ. This is a new adsorption isotherm that we propose for the assembly of CIEJ surfactants on hydrophobic interfaces. Because no distinction has been made for solid or fluid surfaces, this isotherm can be used for both types of interfaces. Various parameters implicit in eq 35 are obtained from the surfactant structure as described earlier. The only unknown is ∆µ0HC/kT or ln K. For the surface aggregates, ln K is expected to be different for different surfactantsurface systems. K is assumed to be independent of a and Γ. With all other parameters known for any surfactantsurface system, K can be obtained from the measurement of Γ at only one bulk surfactant concentration below the CAC. With this value of K, Γ at any other x1 can be obtained by solving eq 35. We will further show that for a particular hydrophobic interface the experimental measurement for one CIEJ surfactant and at only one bulk concentration below the CAC can be used to predict the adsorption isotherm of any CIEJ surfactant on that particular hydrophobic surface. The asymptotic behavior of eq 35 can be analyzed as follows. At very low surface concentrations, that is, Γ f 0 or a f ∞, all terms on the right-hand side of eq 36 become negligible except ln(φsurfactant) and ln K. Thus,

µsurafce/kT ) ln(x1) ) ln(φsurfactant) + ln K as Γ f 0 or a f ∞ (36A) Because φsurfactant ∝ Γ (eqs 8,15, 17,18), Γ ∝ x1 (36B) This proportional relation between Γ and x1 is the linear regime of the adsorption isotherm at low solution concentration. In this region, all the intermolecular interactions, including steric interactions, are negligible and the surface aggregate behaves like an ideal solution.53 At higher concentration, other terms in eq 35 also become significant. A particular value of Γ is obtained at a higher x1 compared to that expected from eq 36. As a result, the adsorption isotherm deviates from the linear behavior. Finally, as a f ah, the steric repulsions among the surfactant molecules increase sharply. In this region, a very large increase in x1 results in a very small change in Γ, and the Γ - x1 plot approaches a plateau. The increase in x1 cannot continue uninhibited. When x1 ) xCAC, bulk aggregates begin to form in the solution. The monomer concentration and chemical potential of the surfactant become almost constant. As a result, Γ does not increase beyond the CAC. The maximum surface concentration (Γmax) is obtained at the CAC. 5. Self-Assembly on the Hydrophobic Solid Surface In the accompanying paper, we presented the experimental adsorption isotherms for various CIEJ surfactants on hydrophobic silane monolayers. Thritle et al.12 and Fragneto et al.35 have also studied the self-assembly of CIEJ surfactants on similar silane monolayers. The hydrophobic surfaces prepared from silane monolayers (53) Adamson, A. W. Physical Chemistry of Surfaces; Wiley: New York, 1976; p 83.


Langmuir, Vol. 20, No. 11, 2004 4459

Figure 2. Adsorption isotherm of C12E8 on OTS monolayers; comparison of experiments (0) with model predictions.

are uniform and chemically homogeneous and, therefore, are often used as model hydrophobic surfaces.54,55 The self-assembly on a SW interface can be predicted using eq 35, provided the value of K is known. The value of K would be different for different hydrophobic surfaces. Figure 2 shows the adsorption isotherm of C12E8 on hydrophobic octadecyltrichlorosilane monolayers.33 The maximum Γ is obtained at the CAC. K is the only parameter that is not known in eq 35, which can be obtained from any one experimental data point. The value of ln(K) ) -15.2 is obtained from the data point just below the CAC. Using this value of K, Γ at any monomer concentration, x1, can be predicted. It might be preferable to obtain the value of K from the best fit to the model using all the data. But we demonstrate that only one experimental data point is sufficient to satisfactorily predict the adsorption isotherms of all surfactants. Note that we have changed the units of monomer concentration from mole fraction (x1) to the more familiar units of mol/ m3. The model predictions are in good agreement with the experimental data, except at low concentration, where the model underpredicts the surface concentration. Further ahead in this paper, we will discuss the possible reason for this deviation. The agreement of the model prediction with the experiments is quite remarkable considering that all parameters in the model are obtained from the molecular structure of the surfactant, with the exception of K. For any other CIEJ surfactant-silane monolayer system, the only unknown in eq 35 is again K. ln(K) is the change in free energy when a hydrocarbon chain is transferred from its standard state as an isolated molecule in solution to the hydrophobic region of the aggregate. Thus, ln(K) is independent of J and is assumed to vary linearly with I. Then, setting ln(K) ) -15.2 for C12E8, the following empirical expression can be used for any CIEJ surfactant on a hydrophobic silane monolayer:

ln(Ksilane monolayer) ) (-15.2/12)I ) -1.27I

(37)

Now, with all the terms in eq 35 determined, the selfassembly of any CIEJ surfactant on hydrophobic silane monolayers can be predicted. The maximum surface concentration, Γmax, and the corresponding minimum interface area per molecule (amin) are obtained at the CAC. At this stage, the CAC values are taken from the data compiled by Huibers et al.2 Figure 3 shows the comparison

Figure 3. amin for C12EJ surfactants on hydrophobic solid surfaces prepared using silane monolayers. Model predictions (solid line), experimental data12,33,35 (0).

Figure 4. amin for CIE6 surfactants on hydrophobic solid surfaces prepared using silane monolayers. Symbols represent experimental data.12,33,35 Dashed line is the model predictions with no constraint on φHC. Solid line is the model prediction with the maximum value of φHC limited to 1. Solid and dashed lines overlap between I from 10 to 14.

of the model prediction for amin for the C12EJ surfactant with the experimental values.12,33,35 For J from 4 to 12, the predicted values of amin are in good agreement with the experiments. The only exception is C12E3, for which the experimental value of amin is significantly less than the model prediction. The plot of the model prediction is not a smooth curve. This is because the experimental CAC values show a variation with J that is consistent yet not smooth.2 The experimental values of the CAC were obtained using different experimental techniques. The experimental values of amin for CIE6 surfactants on silane monolayers are also available.12,33 These values are shown in Figure 4. As I increases from 10 to 14, the experimental value of amin decreases. Further increasing I from 14 to 16 does not decrease amin. In fact, amin for C16E6 is slightly higher than that for C14E6. The values of amin for these surfactants can again be predicted in a manner similar to that of C12EJ. The model (54) Kumar, N.; Maldarelli, C.; Steiner, C.; Couzis, A. Langmuir 2001, 17, 7789-7797. (55) Ulman, A. Chem. Rev. 1996, 96, 1533-1554.

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water water Figure 5. nwater total (solid line), nHC (long dashed line), and nPEO (dotted line) for C12EJ surfactants. (b) Experimental data for total water.12

predictions are also shown in Figure 4. The dashed line in Figure 4 shows the predicted values of amin in which eq 15 is used for φHC. The value of φHC for C10E6, C14E6, and C16E6 at the respective CACs are 0.75, 0.99, and 1.1, respectively. Because a value of 1.1 for C16E6 is not possible, eq 15 should be constrained such that φHC e 1. The recalculated values of amin, including this constraint, are shown in Figure 4 as a solid line. Between C10E6 and C14E6, the solid and dashed lines coincide with each other because φHC,surface < 1 at amin. For C16E6, φHC,surface ) 1. The recalculated value of amin for C16E6 is now higher than that of C14E6. The model predictions after taking into account the constraint on φHC,surface are in very good agreement with the experimental results. In view of the necessity of the constraint on φHC, it is important to check φHC,surface for C12EJ surfactants as well, whose results are shown in Figure 3. For all these surfactants, φHC,surface does not exceed 1. For C12E3 and C12E4, it is nearly 1. When φHC,surface < 1, the hydrophobic region must contain some water molecules. The number of water molecules per surfactant in the hydrophobic (nwater HC ) and hydrophilic water ) can be calculated from the following regions (nPEO expressions:

nwater ) HC

aDHC - vHC aDPEO - JvEO , nwater , PEO ) vwater vwater water ) nwater + nwater ntotal HC PEO (38)

where vwater is the volume of one water molecule (30 Å3). water water , nwater Figure 5 shows the values of ntotal HC , and nPEO at the water 12 CAC. The experimental values of ntotal obtained by neutron reflectometry are in good agreement with the model predictions. This shows that even at Γmax, a significant number of water molecules are present in the hydrophobic and hydrophilic regions, and this number increases with increasing J. For C12E4, there are no water molecules present in the hydrocarbon region, because φHC,surface is approximately 1 at the CAC. 5.1. Common Adsorption Isotherm for Surfactants on Solid Surfaces. In the accompanying paper,33 we have measured the adsorption isotherms for a range of CIEJ surfactants on hydrophobic solid surfaces. It is observed that when Γ is scaled as Γ* ) Γ/Γmax and the surfactant

Kumar and Tilton

Figure 6. Rescaled isotherms on silane monolayers. Experimental results for various surfactants33 are shown by the same symbol ([). Solid lines with other symbols show rescaled predicted isotherms.

concentration in solution is scaled as C* ) C/CAC, then the adsorption isotherms for all CIEJ surfactants (I ) 1014 and J ) 3-8) collapse onto the same curve, even though the individual unscaled adsorption isotherms fell in widely different regions of C and Γ. The isotherms predicted from the model can also be rescaled in a similar manner. Figure 6 shows the experimental and predicted rescaled isotherms for CIEJ surfactants on hydrophobic silane monolayers. The predicted rescaled isotherms for all the surfactants also almost collapse onto each other in agreement with the experimental observations. It should be noted again that only one measurement of the surface concentration of C12E8 is used to predict the self-assembly of all the surfactants. 6. Self-Assembly on Air-Water Interfaces The AW interface is also a hydrophobic surface. Selfassembly of CIEJ surfactants on the AW interface can also be predicted in the same manner used as that for the solid surfaces. The value of ln(K) is again obtained from one single measurement of the surface concentration. Lu et al.56 have measured the surface concentration for various CIEJ surfactants using neutron reflectivity. Their results of C12E4, C12E8, and C12E12 are shown in Figure 7. The value of ln(K) is obtained from the maximum surface concentration of C12E8 at the CAC. This data point gives ln(K) ) -16.5. The adsorption isotherms of other C12EJ surfactants are predicted using the same value of K. The predicted isotherms are shown in Figure 7. The predicted isotherms for the three surfactants are in close agreement with the experimental data. Similar to the SW interface, the value of K is expected to depend only on the hydrocarbon chain length and not on the PEO chain. The value of K for any CIEJ surfactant on the AW interface can be written as

ln(Kair-water) ) (-16.5/12)I ) -1.375I

(39)

The predicted values of amin for various C12EJ surfactants are shown in Figure 8, which are in close agreement with the experimental results.36 The results for CIE6 are obtained after taking into account the constraint on φHC (56) Lu, J. R.; Su, T. J.; Li, Z. X.; Thomas, R. K.; Staples, E. J.; Tucker, I.; Penfold, J. J. Phys. Chem. B 1997, 101, 10332-10339.


Figure 7. Adsorption isotherms on the AW interface. Experimental data from Lu et al.:56 (b) C12E4 experimental; (dashed line) C12E4 model; (0) C12E8 experimental; (solid line) C12E4 model; (4) C12E12 experimental; (dashed-dotted line) C12E12 model.

Figure 8. amin for C12EJ surfactants on the AW interface. Model predictions (line), experimental data36 (symbols).

and are shown in Figure 9. For all surfactants, amin is lower for the AW interface than for the SW interface. The predicted and measured adsorption isotherms of various CIEJ surfactants at the AW interface can be rescaled as Γ* ) Γ/Γmax and C* ) C/CAC, as plotted in Figure 10. Similar to the case of the SW interface, these rescaled isotherms for all CIEJ surfactants also almost collapse onto each other. However, the model predictions are in better agreement with the experiments for the AW interface than for the SW interface. 7. Self-Assembly in Bulk Solution The model used for predicting self-assembly on interfaces can also be used for bulk aggregation. Eqs 32 and 35 can be used to calculate the chemical potential of a surfactant molecule in a bulk aggregate. There are, however, two main differences between the self-assembly in the bulk and that on interfaces. The hydrocarbon region of the bulk aggregate is similar to a liquid hydrocarbon. Therefore, φHC,bulk is always unity. For surface aggregates, eq 15 is used with the maximum value of φHC equal to unity. The second difference is related to the value of

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Figure 9. Predicted values of amin for CIE6 surfactants on the AW interface.

Figure 10. Rescaled isotherms on the AW interface. Solid lines with small symbols are model predictions. Large symbols are for experimental data.56 0 ∆µinterfacial (a)/kT, which is taken to be 0 for a surface 0 aggregate. For bulk aggregates, ∆µinterfacial (a)/kT is given 0 by eq 29. This variation of ∆µinterfacial(a)/kT results in a minimum in ∆µbulk(a)/kT at aopt, while ∆µsurface(a)/kT increases monotonically as a decreases. The values of aopt obtained by numerical solutions show that aopt mainly depends on J and varies little with I.

The equilibrium aggregation behavior of surfactants in bulk solution can be analyzed in a manner pioneered by Tanford6 and further developed by others.1,25,26 Starting from the free energy, the bulk aggregation behavior can be examined either by treating the surfactant solution as consisting of aggregates with a distribution of sizes or by treating the solution as consisting of a number of aggregates with the same size. The ensemble of bulk aggregates can be considered as a separate pseudo-phase if all the aggregates are of the same size. In this case, the minimum in free energy of the entire system corresponds to the minimum in the chemical potential of the surfactant in bulk aggregates, µopt. µopt is obtained by minimizing eq 31 with respect to a. The CAC is then given by

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ln(xCAC) )

µopt kT

Kumar and Tilton

(40)

For xtotal < xCAC, all surfactant exists as monomer. As the total surfactant concentration is increased above the CAC, the monomer concentration remains nearly fixed and the number of bulk aggregates increases. Below the CAC, the chemical potential of surfactants increases with concentration. At the CAC and above, it becomes constant. The CAC for a range of CIEJ surfactants predicted using eq 40 are shown in Figure 11. Note that we have converted the units of CAC from mole fractions to mol/m3. Figure 11 also shows experimental CAC values compiled from various sources (and various experimental techniques) by Huibers et al.2 There is a reasonable agreement with the experimental values. As expected, the CAC decreases as I increases because ∆µ0HC/kT is more negative for larger and more hydrophobic hydrocarbon chains. The CAC increases as J increases because of the larger 0 0 (a)/kT and ∆µPEO (a)/kT. The experimental values ∆µsteric of the CAC also show a similar variation with I and J. 27 While predicting the CAC, we do not consider any effect of the geometry of the aggregates. aopt and the CAC do not depend on the shape of the aggregates. The geometrical constraints associated with the structure of the surfactant molecule control the shape of aggregates.1,7 Many different aggregate shapes are possible, but only those aggregates that conform to these geometrical constraints will form. There are two constraints that along with the value of aopt determine the aggregate shape and size. The first constraint is φHC ) 1. The volume of the hydrophobic core is equal to the product of the aggregation number and the volume of the hydrocarbon chain. The second constraint is due to the length of the hydrocarbon chain. One end of the hydrocarbon chain is attached to the PEO chain; therefore, this end of the hydrocarbon chain is restrained to remain on the surface of the hydrophobic core. Therefore, the characteristic dimension of the hydrophobic core cannot exceed the length of the fully stretched hydrocarbon chain (eq 14). The shape factor (S) is defined as vHC/(aoptlmax). If 0 < S < 1/3, then spherical micelles are favored; for 1/3 < S < 1/2, cylindrical aggregates are favored; and for 1/2 < S < 1, disc-like bilayers are favored. This method for predicting aggregate shapes has been extensively utilized.1,7 The values of S as a function of J for C10EJ, C12EJ, and C14EJ are shown in Figure 12. The shape factor is almost independent of the hydrophobic group, and the three plots almost overlap each other. This is because vHC/lmax does not vary significantly with I and aopt is also almost independent of I. As the number of EO units in the surfactant decreases, S increases. On the basis of the condition described above, the surfactants with longer PEO chains (J > 6) would form spherical micelles. Surfactants with smaller J have a larger S. They would form cylindrical- or bilayer-type aggregates. The negligible effect of the hydrophobic group on S shows that CIEJ surfactants with the same J but different I would form bulk aggregates of similar shape and their phase diagrams would be very similar. These predictions regarding the aggregate shapes can be compared with experimental results. Sjo¨blom et al.57 have compiled the phase diagrams for various CIEJ surfactants. Near room temperature, C12E3 and C12E4 form lamellar phases consisting of cylindrical or bilayer lamellar (57) Sjo¨blom, J.; Stenius, P.; Denielsson, I. In Nonionic Surfactants; Schick, M. J., Ed.; Marcel Dekker: New York, 1987.

Figure 11. CAC (mol/m3) of different CIEJ surfactants (∼25 °C). Lines and symbols show the model predictions and experimental values,2 respectively. (dashed line) C14EJ; (solid line) C12EJ; (dashed line) C10EJ; (4) C14EJ; (0) C12EJ; (O) C10EJ.

Figure 12. Shape factor S as a function of J for C10EJ, C12EJ, and C14EJ surfactants. The three plots almost overlap with each other.

aggregates. This is in agreement with the model prediction which gives S > 0.4 for these surfactants. For C12E8, the predicted value of S is approximately 1/3, which predicts the formation of spherical micelles. The phase diagram of C12E8 shows that, at room temperature, spherical micelles are formed. For C12E5 and C12E6, the values of S are 0.4 and 0.36, respectively. This is very near 1/3, where the transition between spherical micelles and cylindrical aggregates is expected. Thus, for C12E5 and C12E6, the model does not unambiguously predict the shape of the aggregates. The experimental evidence shows that both C12E5 and C12E6 form spherical micelles at room temperature. As we have mentioned previously, our model predicts an insignificant effect of the hydrophobic group on the phase behavior and structure of the aggregates at low concentrations. This is in agreement with the phase diagrams of surfactants with different I but the same J. The phase diagrams of C12E3 and C10E3 are similar.57 At the onset of aggregation, both the surfactants form lamellar phases and bilayers. C12E4 and C16E4 also have very similar phase diagrams, and both surfactants form


Figure 13. Comparison of a at the CAC for bulk aggregates (3), on the AW interface (0), and for silane monolayers (O) and ah (9) for C12EJ surfactants. Dashed lines show fits with a ∼ J1/2.

lamellar phases with cylindrical or bilayer aggregates near room temperature. For C12E8 and C16E8, a similar observation is made. Both C16E8 and C12E8 form spherical micelles and have very similar phase diagrams. Similarly, C10E6 and C12E6 form spherical micelles at ∼25 °C and have similar, nearly identical phase diagrams. 8. Comparison of the Self-Assembly in the Bulk and on AW and SW Interfaces In this section, we will discuss the relationship between self-assembly, in the bulk or on interfaces, and the surfactant molecular structure. The predicted values of a for C12EJ surfactants at the CAC on the SW interface (amin,SW) and AW interface (amin,AW) and in bulk aggregates (aopt) are shown in Figure 13. This figure also shows the values of the PEO headgroup area, ah. All curves are almost parallel to each other, following the square-root relationship with J that is predicted for ah by eq 27. The proportionality constants (F) for amin,SW, amin,AW, and aopt are obtained by fitting the curves in Figure 13 with a ) FJ1/2. The fits are shown as dashed lines. F for amin,SW, amin,AW, aopt, and ah are 24.9, 20.9, 23, and 19 Å2. The squareroot scaling for amin has been experimentally observed for both SW33 and AW27,58 interfaces. No direct experimental data is available to verify the square-root scaling of aopt for bulk aggregates. However, if the aggregation number, P, is known, then aopt/J1/2 can be obtained for spherical aggregates.7 A large number of experimental data for P are available in the literature. Very different values of P have been measured by different researchers using different techniques (Herrington and Sahi).59 So for consistency, we compare data for different surfactants from the same research group. The values of aopt/J1/2 at 25 °C (Å2) for C12E8, C12E12, C12E12, and C12E23, measured by Bechar,60,61 are 17.1, 16.1, 15.3, and 14.7, respectively; for C12E4, C12E6, and C12E8 as measured by Herrington and Sahi59 are 24.9, 20.9, and 25.05, respectively; and for C16E9, C16E12, and C16E21 measured by Elworthy et al.61-63 are 15.9,15.6, and 15.2, respectively. The values aopt/J1/2 (58) Sedev, R. Langmuir 2001, 17, 562-564. (59) Herrington, T. M.; Sahi, S. S. J. Colloid Interface Sci. 1988, 121, 107. (60) Becher, P. J. Phys. Chem. 1960, 64, 1221. (61) Rosen, M. J. Surfactants and Interfacial Phenomena; John Wiley & Sons: New York, 1989.

Langmuir, Vol. 20, No. 11, 2004 4463

Figure 14. Surface concentration of C12E8 on the AW and SW interfaces. Solid and dashed lines are model predictions for AW and SW interfaces, respectively. Circles and squares are experimental data for the AW and SW interface, respectively.

from different research groups are different. Nevertheless, aopt/J1/2 for different surfactants from the same research group is nearly independent of J, suggesting that aopt is indeed proportional to J1/2. This is in agreement with the prediction of the model that aopt is proportional to J1/2. The value of the proportionality constant, F, is 23 Å2, which is in agreement with the experimental values. With the exception of ∆µHC0/kT or ln K, all model parameters are either obtained from the surfactant structure or the same for surface and bulk aggregation. For a C12 hydrocarbon chain, the values of ∆µ0HC/kT are -19.82, -16.5, and -15.2 for bulk aggregates, the AW interface, and the SW interface, respectively. A hydrocarbon chain is in an energetically more favorable condition in the bulk aggregate than on the AW interface. Furthermore, the AW interface is more favorable than the SW interface. The difference in ∆µ0HC/kT is the only distinction between the AW and SW interfaces as far as the model is concerned. As a consequence of the more negative ∆µ0HC/kT for the AW interface, the chemical potential of a surfactant molecule at a particular a or Γ is lower on the AW interface than on the SW interface. In other words, a particular Γ is achieved at a lower surfactant solution concentration on the AW interface than on the SW interface. Figures 3, 8, and 13 for C12EJ surfactants and Figures 4 and 9 for CIE6 surfactants confirm this inference at the CAC. The higher Γ on the AW interface is also expected at all concentrations. Figure 14 shows the predicted and experimental data for C12E8 on the AW and SW interfaces. Both the model and experiments show higher adsorption on the AW interface than for the SW interface over the entire concentration range. All other CIEJ surfactants considered, with the exception of C12E3, follow the similar behavior (compare Figures 3 and 8). The higher adsorption at the AW interface relative to the SW interface can be attributed to the ability of the hydrocarbon chain to penetrate the AW interface and escape more completely from the aqueous environment. In Figures 2, 6, 7, and 10, it can be observed that the predicted adsorption isotherms are in much better agree(62) Elworthy, P. H.; MacFarlane, C. B. J. Chem. Soc. 1963, 907. (63) Elworthy, P. H.; McDonald, C. Kolloid-Z. 1964, 195, 16.

4464


Figure 15. Experimental values of amin for C12EJ surfactants on AW (b) and SW (0).

ment with the experiments for the AW interface than for the SW interface. Predicted values of ΓSL are lower than the experimental values for the SW interface. Because the experimental data taken from different experimental techniques are in mutual agreement, this comparison is not likely due to experimental error. This deviation can be explained as follows. In the model, ∆µ0HC/kT is a significant contributor to the chemical potential of surfactant in a surface or bulk aggregate. While ∆µHC0/kT can be expected to vary with a, we have not considered this effect in the model. We use a constant value that is independent of Γ. For the bulk aggregates, a constant value is justified because the hydrophobic region of the bulk aggregate is always considered as liquid hydrocarbon with φHC equal to 1. On the AW interface, the hydrocarbon chains are able to come out of water. However, on SW interfaces hydrocarbon chains are not able to come entirely out of water, and the hydrocarbon region, therefore, also contains water molecules. The hydrocarbon chains are attracted toward each other in the presence of water molecules as a result of the hydrophobic effect. The increased attraction among the hydrocarbon chains would reduce the chemical potential of surfactant in the aggregate phase (in the same manner in which the chemical potential is increased due to repulsion among PEO chains). This interaction among the hydrocarbon chains can be expected to vary with Γ. As a consequence, ∆µ0HC/kT for the SW interface would also vary with Γ. We do not account for this hydrophobic interaction in the model. That could be the reason for the difference in the model prediction and experimental results on the SW interface. Strong hydrophobic attraction among hydrocarbon tails at the SW interface could also in principle lead to patchwise adsorption over some range of surfactant concentrations,10 an effect that is not allowed in the current model. Figure 15 shows the experimental values of amin for AW and SW interfaces for C12EJ surfactants. For all surfactants, amin on the AW interface is less than that on the SW interface, with the exception of C12E3. The difference between the amin on SW and AW interfaces decreases as J decreases. At the CAC, for all surfactants with J > 4, the model indicates that φHC < 1. For C12E3, φHC ) 1. The hydrophobic silane surface is a monolayer of dense hydrocarbon chains assembled in a plane. Therefore, the

Kumar and Tilton

hydrophobic region of C12E3 on the SW interface resembles the hydrophobic core of a bulk aggregate for C12E3. As a result, we should use the value of ∆µ0HC/kT as given by eq 20 rather than as given by eq 37 for C12E3 at the CAC. The value of amin calculated using eq 20 is 33.3 Å2, and it is in much better agreement with the experimental value (28.1 Å2) than the one calculated from eq 37 (47.3 Å2). This suggests that the hydrophobic region of surface aggregates of C12E3 on silane monolayers is similar to the hydrophobic region of the bulk aggregate and liquid hydrocarbon. A qualitative explanation is as follows. The shape factor of C12E3 is close to 1/2. The “natural tendency” of C12E3 is to form planar bilayerlike bulk aggregates. Because of this “natural tendency” to form bilayers, φHC for surface aggregates (which are constrained to be in a plane) is also equal to 1, and no water molecules are present 0 /kT for in the hydrophobic region. Consequently, ∆µHC,SW 0 C12E3 is nearly equal to ∆µHC,bulk/kT, and it adsorbs to a higher surface concentration on the SW interface. In comparison, the hydrophobic region of surfactants that have a natural tendency to form spherical micelles (e.g., C12E6, C12E8) would contain water molecules. Conse0 quently, their ∆µHC,SW /kT is less negative. This discussion can be summarized as follows: the surfactants that form planar aggregates in the bulk assemble to higher surface concentrations on hydrophobic solid-water interface than the surfactants that form spherical micelles. 9. Conclusion A molecular thermodynamic model based on the pseudophase approach has been developed to predict selfassembly of various CIEJ surfactants in solution, on hydrophobic SW, and on hydrophobic fluid-water interfaces. This model requires minimal input of experimental data. The model parameters are obtained from the surfactant molecular structure. The maximum surface concentration of surfactants on hydrophobic surfaces is controlled by the hydrophilic group and follows an inversesquare-root scaling with J. This scaling has been confirmed experimentally for hydrophobic silane monolayers and the AW interface. Similar square-root scaling is also predicted for the bulk aggregates, which is also confirmed by a number of experimental data available in the literature. The predicted CAC, bulk aggregate shapes, and adsorption isotherms are in reasonable qualitative and quantitative agreement with the experiments. Surfactants that form planar aggregates in bulk have a higher tendency to assemble on hydrophobic solid surfaces compared to the surfactants that form spherical aggregates in the bulk. The model also predicts the experimental observation that CIEJ adsorption isotherms on hydrophobic surfaces collapse onto a common isotherm when they are rescaled by the maximum surface concentration and CAC. This model successfully predicts the selfassembly of nonionic surfactants in solution and on hydrophobic solid or air interfaces with minimal experimental data, thereby providing the theoretical link between bulk aggregation and adsorption. Acknowledgment. The authors would like to thank Prof. Stephen Garoff for useful comments on this work and Air Products and Chemicals, Inc., for financial support. LA035311C

Unified Model To Predict Self-Assembly of Nonionic Surfactants in

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