Znd. Eng. Chem. lies. 1995,34, 4150-4160
4150
Development of User-Friendly Computer Programs To Predict Solution Properties of Single and Mixed Surfactant Systems Nancy J. Zoeller and Daniel Blankschtein* Department o f Chemical Engineering and Center for Materials Science and Engineering, Massachusetts Znstitute of Technology, Cambridge, Massachusetts 02139
We present a description of two “user-friendly” computer programs designed for the prediction of surfactant solution properties: (1)program PREDICT for single surfactants, and (2) program M E for binary surfactant mixtures. We also present a n overview of the molecularthermodynamic theories on which these programs are based. Program PREDICT can be utilized to predict a broad spectrum of properties, including (i) the critical micellar concentration, (ii) the optimal micellar shape, size, and size distribution, (iii)phase behavior characteristics, and (iv) surface tensions, Program MIX can be utilized to predict the critical micellar concentration of a binary surfactant mixture, and the BABinteraction parameter characterizing the nonidealities of mixing at the micellar level. Examples of the predictive capabilities of programs PREDICT and MIX are presented and compared with available experimental data. We believe that programs PREDICT and M E can facilitate the design of new surfactants and surfactant mixtures by alleviating the need for a priori synthesis and characterization of the new chemicals, as well as by reducing the level of experimentation required to evaluate their performance.
I. Introduction Surfactants are molecules composed of a polar hydrophilic group, the “head”, attached to a nonpolar hydrophobic group, the “tail”. Typically, the head can be anionic (negatively charged),cationic (positively charged), zwitterionic (dipolar), or nonionic (uncharged). Often, the tail consists of a linear hydrocarbon chain. This unique molecular structure leads to a rich spectrum of complex self-assembling phenomena when surfactants are dissolved in polar or nonpolar solvents [Mittal (19771, Tanford (1980)l. For example, when dissolved in water, surfactants can form a monolayer at the water-air interface, with the polar heads oriented toward the water and the nonpolar tails oriented toward the air. In addition, surfactants can self-assemble into aggregate microstructures, k n o w n as micelles, in which the polar heads remain exposed to water while the nonpolar tails are shielded inside the micellar core. Micellization occurs beyond a threshold surfactant concentration, k n o w n as the critical micellar concentration (cmc),below which the surfactant molecules are predominantly dispersed as monomers and above which they predominantly form micelles. Micelles can appear in sizes ranging from tens to thousands of monomers. Typically, the smaller micelles are spheroidal in shape, while the larger ones can form cylindrical or discoidal aggregates or infinite bilayers [Mittal (19771, Tanford (1980)l. A salient feature of micelles is that their shape and size are not necessarily fixed, and dramatic morphological changes can be induced by varying solution conditions such as overall surfactant concentration, surfactant composition, temperature, pressure, ionic strength, and pH [Mittal (19771,Tanford (19801,Blankschtein et al. (1986a)l. For example, when cylindrical micelles exhibit significant one-dimensional growth, they can overlap beyond a threshold surfactant concentration, k n o w n as the crossover surfactant concentration, t o form an entangled micellar solution phase [Carale and Blankschtein (1992)l. Moreover, micelles are dynamic entities which are
* To whom correspondence should be addressed.
continually and reversibly exchanging monomers with each other, a process which can generate an entire distribution of micellar sizes [Blankschtein et al. (1986a,b)l. At low surfactant concentrations above the cmc, typically below 20 wt %, micellar solutions often exist as homogeneous isotropic liquid phases. For many surfactants, phase separation can be induced in this concentration range by varying solution conditions such as temperature and ionic strength [Blankschtein et al. (1986a,b)l. The correspondingcoexistence or cloud-point curve, delineating the boundary between the one-phase and two-phase regions of the temperature-surfactant concentration phase diagram, usually exhibits a pronounced asymmetry between the dilute and concentrated branches and can display lower andlor upper critical (consolute)points. Typically, the observed critical points occur at very dilute surfactant concentrations, for example, in aqueous solutions of nonionic Surfactants at concentrations below 5 wt % [Blankschtein et al. (1986a,b)l. When a binary surfactant mixture (for example, anionic-nonionic or cationic-anionic) is dissolved in water, the resulting mixed micellar solution can exhibit new desirable properties which are not attainable in the correspondingsingle surfactant cases [Scamehorn (19861, Rosen (1989)l. For example, in aqueous solutions of anionic-nonionic surfactants, synergistic (attractive) interactions between the two surfactant species can result in mixture cmc’s which are substantially lower than those corresponding to the single anionic o r single nonionic surfactant cases. This and other synergisms can be, and have been, exploited [Scamehorn (19861, Rosen (198911 by the surfactant technologist to design solutions of surfactant mixtures which display unique desirable properties. The wide range of complex surfactant self-assembling behavior described above determines the actual performance of these systems in the various practical applications in which they are utilized [Karsa (1987)l. For example, the practical importance of surfactant monolayer formation at the water-air interface is related to the reduction of surface tension, which may cause
0888-5885/95/2634-4150$09.00/00 1995 American Chemical Society
Ind. Eng. Chem. Res., Vol. 34, No. 12,1995 4151 liquids to spread on a solid surface instead of beading on it, a feature which is of great importance in applications involving film coating, adhesives, detergency, drycleaning, and water-proof materials. Furthermore, the practical importance of micelle formation is related primarily t o the solubilization and emulsification capabilities of the micelles. For example, in the pharmaceutical industry, micelles serve to encapsulate drugs for more effective delivery in the body. In food, cosmetic, and photographic industries, micelles are utilized to solubilize and emulsify polar and nonpolar materials. Micelles also play a central role in several biomedical processes, including the solubilization and transport of cholesterol in the gastrointestinal tract during digestion mediated by bile salt-lecithin mixed micelles [Staggers et al. (1990)l. The fundamental challenges associated with developing a molecular-level understanding of the behavior of surfactant solutions, coupled with the tremendous practical importance of these complex fluids, undoubtedly indicate the need for developing a quantitative theoretical description of these systems capable of predicting their unusually rich behavior. Preferably, the desired theory should incorporate explicitly the detailed molecular structures of the surfactants involved, their composition, and the desired solution conditions. With this in mind, we have recently developed molecular-thermodynamic theories t o describe and predict micellization and micellar solution phase behavior of aqueous solutions of single surfactants [Puwada and Blankschtein (1990a,b, 19911, Briganti et al. (1991)l as well as of binary surfactant mixtures [Puwada and Blankschtein (1992a-c), Sarmoria et al. (1992)l. It is noteworthy that the predictions of these theories were found to be in good agreement with available experimental data. [Carale and Blankschtein (19921, Puwada and Blankschtein (1990a),Sarmoria et al. (199211 These molecular-thermodynamic theories combine molecular models of micelle formation with thermodynamic freeenergy descriptions of phase behavior and phase separation of micellar solutions. The molecular models of micellization account [Puwada and Blankschtein (1990a, 1992~1,Briganti et al. (19911, Sarmoria et al. (1992)l explicitly for the effects of surfactant molecular structure, composition, and solution conditions on the physical driving forces which control micelle formation and growth. The free-energy descriptions account [Blankschtein et al. (1986b),Puwada and Blankschtein (1992a)l explicitly for the effects of intermicellar interactions (described a t a mean-field level) and multiple chemical equilibrium on the micellar size (and composition) distribution as well as on the equilibrium bulk thermodynamic properties of the solution. In addition to predicting bulk surfactant solution properties, we have also developed [Nikas et al. (1992)l a molecularthermodynamic theory t o predict surface tensions of aqueous solutions containing nonionic surfactants. The predictions of this theory for single as well as for binary nonionic surfactant systems were found to be in good agreement with the experimentally measured surface tensions [Nikas et al. (199211. In order t o make our recent theoretical advances accessible to all those interested in surfactant design, manufacturing, and formulation, we have recently begun to incorporate some of the theoretical predictive capabilities into two “user-friendly” computer programs: (1)program PREDICT (for single surfactants)
and (2) program M E (for binary surfactant mixtures). These programs can be utilized with relative ease to predict a broad spectrum of surfactant solution properties for a variety of surfactant types and solution conditions (see sections I1 and VI). The remainder of the paper is organized as follows. In section 11, we discuss the predictive capabilities of programs PREDICT and MIX. In section 111,we review the central elements of the molecular model of micellization for single surfactants. In section lV,we briefly review the thermodynamic free-energy description, including predictive capabilities, for single surfactant systems. In section V, we briefly outline the main components of the molecular-thermodynamic theory for binary surfactant mixtures aimed at predicting the mixture cmc as well as the AB interaction parameter. In section VI, we present several examples of the predictive capabilities of programs PREDICT and MIX, along with a comparison with available experimental data. Finally, conclusions are presented in section VII.
11. Predictive Capabilities of Programs Predict and Mix Program PREDICT can be utilized to predict micellar solution properties of nonionic, ionic, and zwitterionic hydrocarbon-based surfactants under a variety of solution conditions. Note that program PREDICT can also be utilized in the case of fluorocarbon-based surfactants, but this case will not be addressed in this paper. Given the surfactant molecular structure and solution conditions, the following properties can be predicted using program PREDICT: 1. Bulk solution properties such as the critical micellar concentration (cmc). 2. Equilibrium micellar characteristics such as the optimal micellar shape, size, and size distribution. 3. Phase behavior characteristics such as the critical surfactant concentration signaling the onset of phase separation, and the crossover surfactant concentration marking the transition from the dilute (nonentangled) to the semidilute (entangled) micellar solution regimes. 4. Surface tensions of aqueous solutions containing hydrocarbon-based nonionic surfactants. Program MIX can be utilized to predict certain solution properties of aqueous solutions containing binary mixtures of ionic-nonionic, ionic-ionic, and nonionic-nonionic hydrocarbon-based surfactants. Predicted properties include the mixture critical micellar concentration,cmcmh,as a function of mixture composition, as well as the BABinteraction parameter characterizing the nonidealities of the A-B surfactant mixture at the micellar level, for various temperatures and salt concentrations. In addition t o the solution conditions, the required inputs are the surfactant molecular structures, as required for program PREDICT, and the critical micellar concentrations of the pure surfactants, which may be obtained either from program PREDICT or from available experimental values. Therefore, in principle, the variation of the mixture cmc as a function of mixture composition for various temperatures and salt concentrations can be predicted without performing a single experiment. Programs PREDICT and MIX are designed to be userfriendly both to those interested solely in predicting solution properties of surfactant types already incorporated into the programs and to those who are interested in incorporating new surfactant structures which are relevant to their specific needs. For both types of users,
4162 Ind. Eng. Chem. Res., Vol. 34, No. 12, 1995 Surfactant Molecular Architecture
-
-
Solution Conditions
Hydrophilic Moiety: Nonionic Zwitterionic *Ionic
Temperature Gd.CIM1
Cddltirr
Hydrophobic Moiety: Hydrocarbon
i
I J
I
I
1 geiec
0 Minimizes Solution Free Energy
Calculates Micellar Solution Properties
Micellar Characteristics
Properties I
Figure 1. Flow diagram of program PREDICT. Surfactant Molecular Architecture Surfactants Hydrophilic Moiety: Nonionic Ionic
L Solution Conditions Temperature
Hydrophobic Moiety: Hydrocarbon
GddiUve
I
Ii
II
g tr + g int +
pack'
g st+
elec
Figure 3. Schematic representation of the thought process to visualize the formation of a micelle from free monomers.
molecular parameters or solution conditions. For later use, the outputs are saved in a data file.
Program MIX 0
Minimizes Solution Free Energy Calculates Micellar Solution Properties
1I
I
Mixture CMC as a Function of Mixture Composition 1
gmic=
I
Interaction Parameter, PAB
Figure 2. Flow diagram of program MIX.
minimal knowledge of the underlying theoretical details is required. Instead, what is required is knowledge of the surfactant molecular structure and the solution conditions, which serve as inputs to the programs (see section VI.A). This greatly reduces the level of expertise and amount of computational effort required to make predictions of surfactant solution properties. For flow diagrams of programs PREDICT and MIX, see Figures 1 and 2. In order to facilitate the use of programs PREDICT and MIX, the programs were written in FORTRAN for use on a typical personal computer. The necessary calculations are performed in a matter of seconds. The operation of programs PREDICT and MIX is interactive; that is, the programs lead the user through a series of questions in order to gather the relevant data and determine which properties are to be predicted. The output is in a tabular format. First, the inputs are listed, then the free-energy calculations are presented, and finally the desired predicted properties are displayed. After the output is displayed, the user has the option of repeating the calculations using new adjusted
111. Molecular Model of Micellization for Single Surfactants The molecular model of micellization for single surfactants enables the calculation of the free energy of micellization, g m i c , given the molecular structure of the surfactant and the solution conditions. Specifically, gmic(n,Z,,sh) represents the free-energy change when a surfactant molecule is transferred from the aqueous solvent to a micelle (characterized by aggregation number, n, core-minor radius, Z, and shape, sh) present in the aqueous solvent. The magnitude of g m i c can be evaluated using a thought process which describes the formation of a micelle from individual surfactant monomers as a series of reversible steps, each associated with a physicochemical contribution to the micellization process [Puwada and Blankschtein (199111. For a schematic representation of the thought process, see Figure 3. The following six steps are involved [Puwada and Blankschtein (1990a)l: 1. Breaking the bonds between the heads and the tails. The free-energy change associated with this step is considered to be equal in magnitude and opposite in sign to the free-energychange associated with reforming the bonds between the heads and the tails, once the surfactant monomers have been incorporated into the micelle. Consequently, these two contributions cancel each other out and do not need to be quantified. 2. Transferring the hydrophobic tails, now separated and independent of the heads, from bulk aqueous solvent to bulk hydrocarbon, representative of the micellar core. Associated with this step, there is an
Ind. Eng. Chem. Res., Vol. 34, No. 12,1995 4153 attractive free-energy contribution that can be evaluated using experimental data on the solubility of hydrocarbons in an aqueous solvent. The resulting temperature-dependent free-energy contribution, gt,, is given by
tions have been extended to nc = 4-18 in order to further expand the predictive capabilities of program PREDICT.] This yields
gtJkT = [3.04 - 1.05(nc- 1)](298/T) [5.06 0.44(nc - 113 (1)
where A o , AI, and A2 are polynomial coefficients which depend on micellar shape as well as on the total number of carbon atoms in the hydrocarbon tail, nc. Detailed tables of Ai (i = 0, 1 , 2 ) values can be found in Naor et al. (1992). 5. Moving the uncharged heads to the micellar coreaqueous solvent interface and reforming the bonds between the heads and the tails. Note that the freeenergy contribution associated with moving the charges onto the micellar surface will be accounted for in the next step. As mentioned earlier, the free-energy change associated with reforming the headhail bonds is canceled out by the free-energy change associated with breaking these bonds. However, the presence of the uncharged heads a t the micellar interface results in repulsive steric interactions. The free-energy contribution associated with these steric repulsions is estimated by treating the heads present at the micellar interface as an ideal localized monolayer. The resulting freeenergy contribution, g,t, is given by
+
where k is the Boltzmann constant, Tis the temperature in degrees Kelvin, and n, is the total number of carbon atoms in the tail. 3. Creating an interface between the micellar core and the surrounding aqueous solvent. The free-energy contribution associated with this step is evaluated using the concept of a macroscopic interfacial free energy of a hydrocarbon-aqueous solvent interface, including its dependence on interfacial curvature. The resulting interfacial free-energy contribution, gint, is given by
where 00 is the interfacial tension between bulk hydrocarbon and the aqueous solvent, S is a shape factor (3 for spheres, 2 for infinite-sized cylinders, and 1 for infinite-sized disks or bilayers), a is the total interfacial area available per monomer a t the micellar coreaqueous solvent interface, and ao is the interfacial area screened from contact with the solvent by the chemical bond between the head and the tail (typically, 21 A2) when the heads are "reattached". [Note that eq 2 for gint represents a series expansion in powers of 6/2, (assumed to be 4)of a more general expression for the surface tension of a curved interface derived by: Tolman, R. C. The Effect of Droplet Size on Surface Tension. J . Chem. Phys. 1949,17,333-337. We have recently observed that eq 2 is not valid for certain values of I, and are currently investigating the use of alternative expressions for gint. However, this inherent limitation of eq 2 does not affect any of the predictions made with the version of program PREDICT presented in this paper.] In eq 2 , d is the Tolman distance, a measure of the interfacial thickness, and can be estimated [Puwada and Blankschtein (1990a)l as a function of n, using the following simple scaling relation based on nc = 12, d(nC)= 2.25Zma.y(nc)/Zm=(nc = 12) (in A), where Zma.y(nc) = 1.54 1.265(n, - 1) (in A) is the fully-extended (alltrans) tail length. Note that use of this scaling relation, although very convenient from a computational viewpoint, can lead to an overestimation of d(nc),especially for n, =. 12. 4. Anchoring the tails a t one end to the micellar core-aqueous solvent interface. Due to this positional constraint, the tails lose some of their conformational degrees of freedom within the micellar core. The resulting free-energy contribution associated with the packing of the tails inside the micellar core, g p a & is estimated utilizing a single-chain mean-field model in the context of the rotational isomeric state approximation [Puwada and Blankschtein (1990a)l. This packing contribution was originally calculated numerically as a function of I,. However, for computational efficiency, the numerically-generated functions of gpack(2,) were fitted to a second-order polynomial in Z.Jlma.y for a range of ncvalues between 4 and 18 [Naor et al. (199213. [Note that Naor et al. (1992) presents chain-packing calculations only for nc = 6-16. Recently, the packing calcula-
+
gJkT = -ln(l
- uh/u)
(4)
where ah is the effective cross-sectional area of the surfactant head. 6. Recharging the heads by bringing the charges to the micellar surface. Note that, in the case of nonionic surfactants, this is, of course, unnecessary, resultihg in an electrostatic free-energy contribution, gelec, which is equal to zero. On the other hand, in the case of ionic surfactants, where the polar heads carry a charge, we estimate gelec utilizing an analytical expression derived [Mitchell and Ninham (1983,1989), Evans and Ninham (198311 from a second-order approximate solution of the nonlinear Poisson-Boltzmann equation, which is very close to the exact solution. For details on the calculations of gelec in the case of ionic micelles, see Mitchell and Ninham (1983, 1989) and Evans and Ninham (1983). In the case of zwitterionic surfactants, the electrostatic free-energy contribution, gelec, associated with charging the dipolar heads at the spherical, cylindrical, or planar micellar surfaces can be estimated using electrostatic descriptions of increasing complexity. A particularly useful description involves the use of a capacitor model [Nagarajan and Ruckenstein (1991)l resulting in simple analytical expressions for gelec which are well-suited for the computations performed by program PREDICT. For details on the capacitor model and the calculation of gelec in the case of zwitterionic micelles, see Nagarajan and Ruckenstein (1991). The sum of the five free-energy contributions described above (see also Figure 3) yields an expression for the free energy of micellization,gmic(n,Zc,sh). Specifically, gmic
= g t r + gpack + g i n t + g s t + gelec
(5)
For each of the three regular micellar shapes (sh = sphere, infinite-sized cylinder, or infinite-sized disk or bilayer), gmic is then minimized [Puwada and Blankschtein (1990a)l with respect to I, to determine the optimal micellar-core minor radius,,:Z and the corresponding minimum value, gZ,(Z,*,sh), for that particu-
4164 Ind. Eng. Chem. Res., Vol. 34, No. 12, 1995
lar shape, sh. The optimal micellar shape, sh*, corresponds to that associated with the lowest gzi,(Z,*,sh*) value. Once the optimal free energy of micellization, gz,(Z,*,sh*), has been determined, one can predict a broad spectrum of micellar solution properties as discussed in Briganti et al. (1991),Carale and Blankschtein (19921, Puwada and Blankschtein (1990a, 1992b1, Nikas et al. (19921, and Sarmoria et al. (1992). The calculation of these micellar solution properties is presented in the next section, along with an overview of the thermodynamic free-energy description for single surfactants.
IV. Thermodynamic Free-Energy Description for Single Surfactant Systems As emphasized in section I, a complementary thermodynamic free-energy description has been developed [Blankschtein et al. (1986a,b)l to describe the phase behavior and phase separation of single surfactants in solution. This description incorporates the salient features of the micellar solution, including (i)a distribution of micellar aggregates in chemical equilibrium with each other and with the monomers, (ii) the entropy of mixing micelles, surfactant monomers, and solvent molecules, and (iii) intermicellar interactions modeled at a mean-field level. The resulting Gibbs free energy of the micellar solution, G, is given by [Blankschtein et al. (1986b), Puwada and Blanschtein (1990b)I
where N, and Nn are the total number of solvent molecules and n-mers, respectively,N, = &=lnNnis the total number of surfactant molecules, X, and Xn are the mole fractions of solvent and n-mers, respectively, p; and py are the standard-state chemical potentials of a solvent molecule and a surfactant monomer, respectively, C is a mean-field interaction parameter reflecting the magnitude of the effective intermicellar attraction, and r$ is the total surfactant volume fraction. The expression for G in eq 6 can be utilized t o determine the chemical potentials of the n-mers, pn = (3G/3yn)T,yNwq; (for n L 1). To determine the micellar size distribution, {X,} , the requirement of multiple chemical equilibrium, pn = np1, is utilized in conjunction with the mass balance relation, X = &lnX, (where X is the total surfactant mole fraction). The resulting micellar size distribution is given by [Briganti et al. (199111
(7) where XI is the monomer mole fraction, and p = 11kT. Note that eq 7 does not depend explicitly on the value of the interaction parameter, C . Using the distribution given in eq 7, all the equilibrium micellar solution properties associated with it can be computed. In particular, the cmc is given by [Puwada and Blankschtein (1990a)l cmc x expvg$, - 11
(8)
where ggic(Z,*,sh*)is the optimum value of g m i c obtained as explained in section 111. Several characteristics of the micellar size distribution can be computed [Puwada and Blankschtein (1990a), Blankschtein et al. (1986b)l, using eq 7. These include (i)the number-average aggregation number, (n)n, (ii)the weight-average aggregation number, (n),, and (iii) the relative variance, Var. Specifically,
Note that the relative variance, Var, constitutes a measure of micellar-size polydispersity (Var = 0 for spherical micelles, and Var = 0.5 for highly elongated cylindrical micelles). The crossover surfactant concentration, P, signaling the transition from the dilute (nonentangled) to the semidilute (entangled) micellar solution regimes can be computed using eq 7 in the context of the theory developed in Carale and Blankschtein (1992) to which the reader is referred for details of the calculations. The critical point, which signals the onset of phase separation, is characterized by the critical surfactant concentration, X,,and the critical temperature, T,. At the critical point, thermodynamic stability requires that the two conditions, (#g/rn>T,P= 0 and (@g/m)T,P= 0, should be satisfied, where g = G/(N, NA. By simultaneously solving these two equations, it is possible to deduce the values of X, and the critical interaction parameter, C,, for a given value of T,. Examples of the predictive capabilities of program PREDICT, utilizing the theoretical results described in this section, along with a comparison with available experimental data, are presented in section VI.
+
V. Molecular-ThermodynamicTheory for Binary Surfactant Mixtures We have recently developed [Puwada and Blankschtein (1992a-c)] a general molecular-thermodynamic theory to predict micellization, phase behavior, and phase separation of aqueous solutions containing binary surfactant mixtures. In principle, this theory [Puwada and Blankschtein (1992a)l is capable of quantitatively describing the behavior of binary mixtures containing ionic, zwitterionic, and nonionic surfactants. In practice, this theory has been successfully utilized [Puwada and Blankschtein (1992b,c)l to predict micellization, phase behavior, and phase separation of aqueous solutions containing binary mixtures of alkyl poly(ethy1ene oxide), CiEj, nonionic surfactants. More recently, we have developed [Sarmoria et al. (1992)l a simplified version of the original theory aimed at predicting the mixture cmc, cmcmk, as well as the AB interaction parameter of nonideal binary surfactant mixtures. This simplified version requires three inputs: (i) the molecular structures of both surfactant species, (ii) the cmc's of the pure surfactants, which can be either estimated from eq 8 or measured experimentally, and (iii) the overall surfactant composition and other solution conditions such as temperature and salt concentration. We have found [Sarmoria et al. (199211 that the simplified version of the theory yields reasonable quantitative predictions for aqueous solutions containing anionic-nonionic and cationic-nonionic sur-
Ind. Eng. Chem. Res., Vol. 34,No. 12, 1995 4155 Table 1. Required Surfactant Molecular Inputs for Program PREDICTQ nonionic Head Tal
ionic
zwitterionic
nc
nc
nc
ah
ah 2
ah dsep
dcharge
4
i Nonionic
Ionic
1.
n, is the total number of carbon atoms in the tail, a h is the effective cross-sectional area of the head, z is the valence of the head, dcharge is the distance between the position of the charge in the head and the beginning of the tail (including the length of the CH2 group adjacent to the head), and dsepis the distance between the two charges comprising the dipole in the zwitterionic head. a
Zwitterionic
Figure 4. Illustration of surfactant molecular parameters.
factant mixtures of hydrocarbon-based surfactants. In the case of anionic-cationic surfactant mixtures, the theory could not be tested to the same extent due to the scarcity of reliable experimental mixture cmc data. In view of these recent successful developments, we have implemented the simplified version of the theory into a user-friendly computer program, program MIX, the essence of which was described in section I1 (see also Figure 2). In the next section, we present examples of the predictive capabilities of programs PREDICT and MIX, which are based on the theoretical developments presented in sections 111-V, along with a comparison with available experimental data.
VI. Examples of Predictive Capabilities of Programs PREDICT and MIX and Comparison with Experimental Data
A. Inputs to Programs PREDICT and MIX. In order to operate program PREDICT, the user needs to input information about the surfactant molecular structure and the desired solution conditions. Regarding the surfactant molecular structure, descriptions of the head and the tail are needed. To describe the tail of a hydrocarbon-based surfactant, the user needs to simply input the total number of carbon atoms, nc, comprising the tail. To describe the head, the user needs to input the effective cross-sectional area of the head, a h , for all classes of surfactants. If the head is nonionic, no additional inputs are required. However, if the head is ionic, the distance between the position of the charge in the head and the beginning of the tail (including the length of the CHz group adjacent to the head), dcharge, and the valence, z, constitute additional required inputs. In addition, if the head is zwitterionic, the distance between the two charges comprising the dipole in the head, dsep, is a required input. The surfactant head molecular parameters, ah, dcharge, and dsep, can be calculated from the known bond lengths and bond angles of the various chemical groups comprising the head. If the head is particularly complex, it is often convenient to make use of commercially available molecular-simulation software to carry out these calculations. For a summary of the molecular parameters which need to be input in the case of nonionic, ionic, and zwitterionic surfactants, see Figure 4 and Table 1. Regarding solution conditions, the user needs to input the temperature, the total concentration of surfactant, and the type and concentration of any additive. Additional inputs may be needed to predict certain properties, including the critical surfactant concentration for phase separation, X,,the crossover surfactant
Table 2. Additional Required Inputs To Predict the Critical Surfactant Concentration for Phase Separation, X,,the Crossover Surfactant Concentration,P, and the Surface Tension, u(aa property required input
xc
Tc
X* surface tension
lhg,
6
ao,X
a Tcis the critical temperature for phase separation, lhg is the length of the head including the first CH2 group adjacent to the head, 6 is the micellar “chain” persistence length, and dXl is a surface tension value a t a total surfactant concentration, X.
Table 3. Required Surfactant Molecular Inputs for Program MIXa nonionic ionic cmc cmc nc nc Z
dcharge
a
The notation is the same as that in Table 1.
concentration for entanglements, P,and the surface tension, u. For a summary of the additional required inputs, see Table 2. The information needed to operate program MIX is very similar to that required for the operation of program PREDICT. To describe the molecular structures of both surfactants, the required inputs include (i)nc, (ii)a h , and (iii)z and dcharge if the surfactants are ionic. In addition, the cmc’s of the pure surfactants, which may be determined either by utilizing program PREDICT or experimentally, are required inputs. Finally, temperature and salt concentration need t o be input to describe solution conditions. The required inputs to program MIX are summarized in Table 3. Using the inputs described above, programs PREDICT and MIX can be utilized to predict a wide range of surfactant solution properties. Several examples of these predictive capabilities, including a comparison with available experimental data, are presented next. B. Examples of Predictive Capabilities of Pro. grams PREDICT and MIX. 1. Critical Micellar Concentration. Program PREDICT was utilized t o predict cmc’s of some widely-used, representative nonionic, ionic, and zwitterionic surfactants. Below, we present some examples. The nonionic surfactants examined belong to the alkyl poly(ethy1ene oxide), CiEj, family. These surfactants possess a polar head consisting ofj ethylene oxide (CHzCHzO = EO)groups, and a hydrocarbon tail consisting of i carbon atoms. The effective cross-sectional areas of the Ej heads, UhG), were estimated as a function o f j (=3-8) in the following manner. For the relatively short E3 head, Uh was estimated by assuming a fullyextended (all-trans) conformation of the PEO head and
4156 Ind. Eng. Chem. Res., Vol. 34, No. 12, 1995 100
I
0.Wl
Number of EO Groups
Figure 5. Predicted cmc as a function of the number of ethylene oxide (EO) groups, j , in the head, for aqueous solutions of CeE, (-1, CloE, (- - -1, and ClzEj at 25 "C. Experimental values are denoted by circles for CeE,, triangles for CloE,, and squares for C12E, [Mukherjee and Mysels (19711, Becher (1967), Meguro et al. (198111. (.a*)
100
30
0
10
I
1
I
11
12
13
14
Number of Carbon Atoms in the Tail Figure 7. Predicted cmc as a function of the number of carbon atoms in the hydrophobic tail, i , for aqueous solutions of alkyl betaines (C~-N+(CH~~ZCHZCOO-) at 25 "C. The line represents the theoretical predictions, and the circles denote experimental values [Nagarajan and Ruckenstein (1991)l.
Table 4. Examples of cmc Predictions at 25 "C, for Aqueous Solutions of Four Representative Commercial Surfactantsa predicted experimental commercial surfactant type crnc (mM) crnc (mM) Zwittergent-8 284 330 Zwittergent- 14
lo
0.1
0.1-0.4
C~~HZ~-N+(CH~)Z(CHZ)~SO~-
0 I
9
CeHi7-Nf(CH3)z(CHz)3S03-
5-
-E
1
a
MEGA-8
3
23
19-25
C~H~~-CON(CH~)CHZ(CH(OH))~OH MEGA-9
1
6
6-7
C~H~~-CON(CH~)CHZ(CH(OH))~OH 0.3 0.1 8
io
12
14
16
Number of Carbon Atoms in the Tail Figure 6. Predicted cmc as a function of the number of carbon atoms in the hydrophobic tail, i , for aqueous solutions of sodium alkyl sulfates (CiSOdNa) at 25 "C. The line represents the theoretical predictions, and the circles denote experimental values [Nagarajan and Ruckenstein (1991)l.
then utilizing the known bond lengths and bond angles of an EO group, resulting in a value of ah(j=3) = 26.9 A2. For the longer and more flexible "polymer-like" Ej heads (j 2 41, ah was estimated as a function of j by utilizing a recently developed [Puwada and Blankschtein (1990a)l scaling law based on an E6 head. Specifically,ah(j14) = ah(j=6)(j/6)'.*, with ah(j=6) = 38.1 A2. Figure 5 shows predicted cmc's a t 25 "C of aqueous solutions of CiEj surfactants as a function of the number of EO groups, j , for i = 8 (solid line), i = 10 (dashed line), and i = 12 (dotted line). The circles, triangles, and squares denote experimental values for CsEj, CloEj, and ClzEj, respectively [Mukherjee and Mysels (19711, Becher (1967), Meguro et al. (198l)l. The ionic surfactants examined belong to the sodium alkyl sulfate, CiS04Na, family. Figure 6 shows predicted cmc's (line) at 25 "C of aqueous solutions of CiS04Na as a function of the number of carbon atoms, i, in the tail. The circles denote experimental values [Nagarajan and Ruckenstein (199113. To make these predictions, values of z = 1,dchwge = 3.7 A, and ah = 25 A2 were input into program PREDICT. The zwitterionic surfactants examined belong to the alkyl betaine, Ci-N+(CH&CH&OO-, family. Figure 7 shows predicted cmc's (line) at 25 "C of aqueous solutions of Ci-N+(CH&CH2COO- as a function of the
a For details on the values of the molecular parameters input into program PREDICT in order to make these predictions, see section VI.B. The experimental crnc values were taken from: Neugebauer, J. A Guide to the Properties and Uses of Detergents in Biology and Biochemistry; CALBIOCHEM Corp.: San Diego, 1990.
number of carbon atoms, i, in the tail. The circles denote experimental values [Nagarajan and Ruckenstein (1991)l. To make these predictions, values of dsep = 2.5 A and Uh = 32 A2 were input into program PREDICT. In Figures 6 and 7, the observed systematic deviation of the predicted cmc's from the experimental ones as n, increases can be attributed to an overestimation of the Tolman distance, 6(n,). This overestimation, which becomes more pronounced as ncincreases, results from the use of the simple scaling relation to estimate d(n,) presented in section 111. In addition t o analyzing reagent-grade surfactants, program PREDICT can be utilized to estimate cmc values of commercial surfactants, where impurities (chemical heterogeneity) are typically present. Table 4 shows examples of predicted cmc's at 25 "C of aqueous solutions of four representative commercial surfactants, including zwitterionic surfactants from the Zwittergent family and nonionic surfactants from the MEGA family. In view of the chemical heterogeneity of these surfactants, their average molecular structure was utilized as an input to program PREDICT. For Zwittergent, the molecular parameters input into program PREDICT include dsep= 5.03 A and a h = 32.2 A2. For the MEGA surfactants, a value of Uh = 40 A 2 was input into program PREDICT. As can be seen from Table 4, program PREDICT provides reasonable estimations of the cmc's of these chemically heterogeneous commercial surfactants.
Ind. Eng. Chem. Res., Vol. 34,No. 12, 1995 4157 8
0.5
t
I
C,,E,
8 0.4 -
.-2
' >
.
0.3
.-
i i
I)
-
i .
2 0.2 0.1
-
n
0.2
0.4
0.6
08
Salt Concentration (M) Figure 8. Predicted crnc as a function of salt (NaC1)concentration for aqueous solutions of sodium dodecyl sulfate (C12S04Na) at 25 "C. The line represents the theoretical predictions, and the circles denote experimental values [Sarmoria et al. (19921, Stellner and Scamehorn (1986)l.
Changes in solution conditions can have marked effects on the cmc of the surfactant solution. With the aid of program PREDICT, the user can manipulate solution conditions by varying, for example, the temperature or additive type and concentration. As an illustration of this capability, Figure 8 shows the effect of adding sodium chloride, NaC1, on the cmc of an aqueous solution of sodium dodecyl sulfate, C12S04Na. The line corresponds to the predicted cmc at 25 "C as a function of NaCl concentration over the range 0-1 M. The circles denote experimental values [Sarmoria et al. (19921, Stellner and Scamehorn (1986)l. As can be seen, the cmc predictions presented in Figures 5-8 and Table 4 constitute a reasonably good representation of the experimental cmc data. 2. Characteristics of the Micellar Size Distribution. A very challenging and still controversial aspect of micellar solution phase behavior involves the extent of micellar growth and associated degree of polydispersity of C,E, nonionic micelles in aqueous solutions. As stressed in section IV, the relative variance of the micellar size distribution, Var, constitutes a quantitative measure of polydispersity. In particular, elongated, polydisperse cylindrical micelles are characterized by Var = 0.5, whereas small, monodisperse spherical micelles are characterized by Var = 0. As illustrated in Figure 9, program PREDICT can be utilized t o predict the temperature variation of the relative variance of the micellar size distribution for &E, surfactants in aqueous solutions, wherej = 5, 6, 7, and 8. In particular, f o r j = 6, 7, and 8,the narrow temperature range over which the relative variance changes rapidly from 0 to 0.5 corresponds to a sphereto-cylinder micellar shape transition. The experhentally determined shape transition temperatures (see the dashed arrows in Figure 9) are 18 "C [Brown and Rymden (198711, 34 "C [Fujimatsu et al. (198811, and 50 "C [Zana and Weill (1985)l for C12E6, C12E7, and C12E8, respectively. As can be seen, program PREDICT is capable of predicting the micellar shape transition behavior quite accurately. 3. Crossover Surfactant Concentration. At certain temperatures, increasing surfactant concentration may cause the micelles present in aqueous solutions of C,E, surfactants to grow into elongated cylindrical microstructures. As described in section N,these cylindrical micelles may elongate sufficiently t o overlap and form an entangled mesh. This, in turn, can
0
10
20
30
40
i I
I
50
60
70
Temperature ('C) Figure 9. Predicted relative variance of the micellar size distribution of C12E, 0' = 5,6, 7, and 8)micelles in aqueous solution as a function of temperature (solid lines). The dashed arrows denote the experimentally determined shape transition temperatures for j = 6, 7, and 8 [Brown and Rymden (1987), Fujimatsu et al. (1988). Zana and Weill (198511.
15
20
25
30
3s0
40
45
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Temperature ( C) Figure 10. Predicted crossover surfactant concentration as a function of temperature for aqueous solutions of C12E6. The line represents the theoretical predictions for a micellar persistence length of 200 A. The circles (squares) denote experimental values deduced from light scattering (viscosity) measurements [Carale and Blankschtein (199211.
dramatically alter the rheological behavior of the micellar solution. In order to quantitatively characterize the relatively broad surfactant concentration region separating the nonentangled and entangled micellar solution regimes, it is customary to single out a crossover surfactant concentration, X*,based on excludedvolume considerations, associated with the initial contact of micellar volumes [Carale and Blankschtein (1992)l. Program PREDICT is capable of predicting the crossover surfactant concentration, as illustrated in Figure 10. In addition to the inputs regarding surfactant molecular structure and solution conditions described in section VI.A, the prediction ofX* requires the user to input the micellar persistence length, 6, which constitutes a measure of micellar flexibility. Figure 10 illustrates the prediction o f F as a function of temperature for C12E6, where a typical value of 6 = 200 A was input into program PREDICT. Experimentally, X* can only be deduced indirectly from observed changes in certain solution properties with surfactant concentration as the solution traverses the nonentangled to entangled transition region. Since this transition region is relatively broad, the experimentally deduced x* values are meaningful only to within about 20%. With this in mind, Figure 10 shows experimental deductions of P as a function of temperature for Cl2E6
4158 Ind. Eng. Chem. Res., Vol. 34, No. 12, 1995
I
0
Figure 11. Predicted critical surfactant concentration for aqueous solutions of various CiE, surfactants. The empty bars denote theoretical predictions, and the cross-hatched bars denote experimental values [Fujimatsu et al. (1988), Mulley and Metcalf (1962), Lang and Morgan (1980), Corti et al. (19841, Strey and Pakusch (1986)l.
based on determinations of micellar diffusion coefficients using quasi-elastic light scattering (circles) and viscosity measurements (squares) [Carale and Blankschtein (1992)l. Considering the theoretical and experimental limitations discussed above, the F versus T predictions presented in Figure 10 provide a fair representation of the experimental data and, as such, provide a useful practical guideline. 4. Critical Surfactant Concentration. Program PREDICT can also predict characteristics of the critical point, signaling the macroscopic separation of the micellar solution into two coexisting micellar solution phases. Specifically, if the user inputs the critical temperature, T, (in addition to the required inputs regarding surfactant molecular structure and solution conditions), program PREDICT can be utilized to calculate the critical surfactant concentration, Figure 11illustrates the prediction of X,in the case of several CiEj nonionic surfactants. In this example, the critical temperature values input into program PREDICT are 44 and 58 "C for Q o E ~and C10E6, respectively, and 23, 50, 67, and 77 "C for C12E5, C12E6, C12E7, and C12E8, respectively [Puwada and Blankschtein (1990a)l. As can be seen, the X, predictions (empty bars) compare favorably with the experimental data (cross-hatched bars) [Fujimatsu et al. (1988), Mulley and Metcalf (1962), Lang and Morgan (1980), Corti et al. (19841, Strey and Pakusch (1986)l. 5. Surface Tension. For nonionic hydrocarbonbased surfactants in aqueous solutions, program PREDICT can be utilized to predict surface tensions as a function of surfactant concentration. The user needs to input a single surface tension value and its corresponding surfactant concentration, and then program PREDICT can predict the surface tension a t any surfactant concentration requested by the user. Figure 12 illustrates the predicted surface tension (line) at 25 "C as a function of surfactant concentration for aqueous solutions of C12E6. In order to make these predictions, a surface tension value of 35 dyn/cm a t a surfactant concentration ofX = 0.0556 mM was input into program PREDICT. As can be seen, the predictions compare favorably with the experimentally measured [Nikas et al. (199211 surface tension values (see circles in Figure 12). 6. Predictions for Binary Surfactant Mixtures. As stressed in section 11, program MIX can be utilized
x,.
0.05
0.15
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0.2
Surfactant Concentration (mM) Figure 12. Predicted surface tension as a function of surfactant concentration at 25 "C for aqueous solutions of C12E6. The line represents the theoretical predictions, and the circles denote experimental values [Nikas et al. (1992)l.
0
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Mixture Composition Figure 13. Predicted cmcmh as a function of overall surfactant composition, a, for a mixture of SDS (a= 1) and C8Elz (a = 0) in aqueous solution at 25 "C. The line represents the theoretical predictions for which .B E d = -4.07kT, and the circles denote the experimental cmcmuvalues for which = -4.36kT [Lange and Beck (1973)l.
~-
r&*
to predict two important properties of aqueous solutions of binary surfactant mixtures: (1)the mixture cmc, cmcmh,as a function of mixture composition and (2) the @ABinteraction parameter. Examples of CmCmix predictions are shown in Figure 13 for an anionic-nonionic surfactant mixture and in Figure 14 for a cationicnonionic surfactant mixture. Figure 13 shows the predicted CmCmix as a function of mixture composition for the anionic-nonionic mixture of SDS (C12S04Na)and C8El2 a t 25 "C, where the line represents the theoretical predictions and the circles denote experimental values [Lange and Beck (1973)l. The predicted and experimentally-deduced interaction parameters are Pzd= -4.36kT and = -4.07kT [Lange and Beck (1973)1,respectively. To make these predictions, the following parameters were input into SDS program MM: cmcSDS= 8.00 mM, ncSDs = 12, dcharge 3.7 A,zSDS= -1, cmcCBEIZ= 9.00 mM, ncCaE1z= 8, and dF{fiie= zCBEl2 = 0. Figure 14 shows the predicted CmCmix as a function of mixture composition for the cationic-nonionic mixture of decyl trimethylammonium bromide, CloTAB (GON(CH&Br), and CsE4 in 0.05 M NaBr at 23 "C, where the line represents the theoretical predictions and the circles denote experimental values [Holland and Rubingh (198311. The predicted and experimentally-deduced interaction parameters are @Ed = -2.24kT and
,8z
Ind. Eng. Chem. Res., Vol. 34, No. 12, 1995 4169
Acknowledgment We are grateful to Ayal Naor and Dr. Yvonne Nikas for their assistance with various aspects of incorporating the molecular-thermodynamic theories of micellization into programs PREDICT and MIX. We are also gratell to Pak Yuet for his assistance with various aspects of the chain packing calculations. This research was supported in part by the National Science Foundation (NSF) Presidential Young Investigator (PYI)Award to D.B. D.B. is also grateful t o Kodak and Unilever for providing PYI matching funds and to Exxon and Witco for providing partial support of this work. 0
0.2
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Mixture Composition
Figure 14. Predicted cmc- as a function of the overall surfactant composition, a, for a mixture of CloTAB (a= 1)and C8E4 (a = 0) in aqueous solution containing 0.05 M NaBr a t 23 "C. The line represents the theoretical predictions for which /?Ed = -2.24kT, and the circles denote the experimental cmcmixvalues for which = -1.8kT [Holland and Rubingh (1983)l.
/3zp
BEp= -1.8KT
[Holland and Rubingh (1983)1, respectively. To make these predictions, the following param= 48.51 eters were input into program MIX: cmcCloTAB mM, nCeloTAB = 10, = 3.2781, zC10Tm = 1,cmcCsE4= = zCaE4= 0. 6.2 mM, n","E,= 8, and As can be seen from Figures 13 and 14, program MIX provides accurate predictions of the mGtur; cmc for both anionic-nonionic and cationic-nonionic surfactant mixtures, as well as reasonable predictions of synergistic interactions through the interaction parameter.
e:'rye
VII. Conclusions As the need for a detailed understanding of surfactant solution behavior increases, the surfactant technologist is faced with the challenge of modeling the complex behavior of these systems. With this need in mind, we have recently developed comprehensive molecularthermodynamic theories of micellar solution behavior, for both pure surfactants and binary surfactant mixtures. These theories combine molecular models of micellization with thermodynamic descriptions of micellar solution phase behavior. To further enhance the practical utility of these theories, we have incorporated them into two user-friendly computer programs, PREDICT and MIX. As shown in section VI, program PREDICT is fairly accurate in predicting a wide range of surfactant solution properties, including the cmc, the micellar shape, size, and size distribution, the crossover surfactant concentration, and phase separation characteristics. In addition, program PREDICT can also quantify quite accurately the surface behavior of aqueous solutions containing nonionic surfactants. Program MIX is reasonably accurate in predicting the mixture cmc as a function of mixture composition, as well as in quantifying synergistic interactions between the surfactant species in terms of the interaction parameter. We hope that the availability of programs PREDICT and MIX will facilitate the design of new surfactants and surfactant mixtures possessing desirable properties by alleviating the need for a priori synthesis and characterization of the new chemicals, as well as by reducing the level of experimentation required to evaluate the performance of the new surfactants and surfactant mixtures.
Literature Cited Becher, P. Micelle Formation in Aqueous and Nonaqueous Solutions. In Nonionic Surfactants; Shick, M. J., Ed.; Arnold: London, 1967; pp 478-515. Blankschtein, D.; Thurston, G. M.; Benedek, G. B. Theory of Phase Separation in Micellar Solutions. Phys. Rev. Lett. 1986a, 54, 955-958. Blankschtein, D.; Thurston, G . M.; Benedek, G. B. Phenomenological Theory of Equilibrium Thermodynamic Properties and Phase Separation of Micellar Solutions. J . Chem. Phys. 1986b, 85, 7268-7288 and references cited therein. Briganti, G.; Puwada, S.; Blankschtein, D. Effect of Urea on Micellar Properties of Aqueous Solutions of Nonionic Surfactants. J. Phys. Chem. 1991,95,8989-8995 and references cited therein. Brown, W.; Rymden, R. Static and Dynamical Properties of a Nonionic Surfactant (C12E6) in Aqueous Solution. J . Phys. Chem. 1987,91, 3565-3571. Carale, T. R.; Blankschtein, D. Theoretical and Experimental Determinations of the Crossover from Dilute to Semidilute Regimes of Micellar Solutions. J . Phys. Chem. 1992,96,459467 and references cited therein. Corti, M.; Minero, C.; Degiorgio, V. Cloud Point Transition in Nonionic Micellar Solutions. J . Phys. Chem. 1984, 88, 309317. Evans, D. F.; Ninham, B. W. Ion Binding and the Hydrophobic Effect. J . Phys. Chem. 1983,87, 5025-5032. Fujimatsu, H.; Ogasawara, S.; Kuroiwa, S. Lower Critical Solution Temperature (LCST) and Theta Temperature of Aqueous Solutions of Nonionic Surface Active Agents of Various Polyoxyethylene Chain Lengths. Colloid Polym. Sci. 1988,266, 594-600. Holland, P. M.; Rubingh, D. M. Nonideal Multicomponent Mixed Micelle Model. J . Phys. Chem. 1983, 87, 1984-1990. Karsa, D. R. Industrial Applications of Surfactants; The Royal Society of Chemistry: London, 1987. Lang, J. C.; Morgan, R. D. Nonionic Surfactant Mixtures. I. Phase Equilibria in CloE4-H20 and Closed-Loop Coexistence. J . Chem. Phys. 1980, 73, 5849-5861. Lange, V. H.; Beck, K. H. Zur Mizellbildung in Mischlosungen homologer und nichthomologer Tenside. Kolloid 2.2.Polym. 1973,251,424-431. Meguro, K.; Takasawa, Y.; Kawahashi, N.; Tabata, Y.; Ueno, M. Micellar Properties of a Series of Octaethylene glycol-n-alkyl Ether with Homogeneous Ethylene Oxide Chain and Their Temperature Dependence. J . Colloid Interface Sci. 1981, 83, 50-56. Mitchell, D. J.; Ninham, B. W. Electrostatic Curvature Contributions to Interfacial Tension of Micellar and Microemulsion Phases. J . Phys. Chem. 1983, 87, 2996-2998. Mitchell, D. J.; Ninham, B. W. Curvature Elasticity of Charged Membranes. Langmuir 1989,5, 1121-1123. For an introduction to the field of micellar solutions, see: Micellization, Solubilization, and Microemulsions; Mittal, K. L., Ed.; Plenum: New York, 1977; Vols. 1 and 2 and references cited therein. Mukherjee, P.; Mysels, K. J. Critical Micelle Concentrations of Aqueous Surfactant Systems;National Standard Reference Data Series (National Bureau of Standards) No. 36; U.S. Department of Commerce: Washington, DC, 1971. Mulley, B. A.; Metcalf, A. D. Nonionic Surface-Active Agents Part IV.The Critical Micelle Concentration of Some Polyoxyethylene glycol Monohexyl Ethers in Binary and Ternary Systems. J . Colloid Sci. 1962, 17, 523-530.
4160 Ind. Eng. Chem.Res., Vol. 34, No. 12, 1995 Nagarajan, R.; Ruckenstein, E. Theory of Surfactant Self-Assembly: A Predictive Molecular Thermodynamic Approach. Langmuir 1991, 7, 2934-2969 and references cited therein. Naor, A.; Puwada, S.; Blankschtein, D. An Analytical Expression for the Free Energy of Micellization. J. Phys. Chem. 1992,96, 7830-7832. Nikas, Y. J.; Puwada, S.; Blankschtein, D. Surface Tensions of Aqueous Nonionic Surfactant Mixtures. Langmuir 1992, 8, 2680-2689. Puwada, S.; Blankschtein, D. Molecular-Thermodynamic Approach to Predict Micellization, Phase Behavior, and Phase Separation of Micellar Solutions. I. Application to Nonionic Surfactants. J. Chem. Phys. 1990a, 92,3710-3724 and references cited therein. Puwada, S.; Blankschtein, D. Molecular-Thermodynamic Approach to Predict Micellar Solution Properties. In MRS Symposium Proceedings; Materials Research Society: Pittsburgh, PA, 1990b; Vol. 177, pp 129-134. Puwada, S.; Blankschtein, D. Molecular Modelling of Micellar Solutions. In Proceedings of the 8th International Symposium on Surfactants in Solution; Mittal, K. L., Shah, D. O., Eds.; Plenum: New York, 1991; Vol. 11, pp 95-111 and references cited therein. Puwada, S.; Blankschtein, D. Thermodynamic Description of Micellization, Phase Behavior, and Phase Separation of Aqueous Solutions of Surfactant Mixtures. J. Phys. Chem. 1992a, 96, 5567-5579. Puwada, S.; Blankschtein, D. Theoretical and Experimental Investigations of Micellar Properties of Aqueous Solutions Containing Binary Mixtures of Nonionic Surfactants. J . Phys. Chem. 1992b, 96, 5579-5592. Puwada, S.; Blankschtein, D. Molecular-Thermodynamic Theory of Mixed Micellar Solutions. In Mixed Surfactant Systems; Holland, P. M., Rubingh, D. N., Eds.; ACS Symposium Series 501; American Chemical Society: Washington, DC, 1992~;pp 96-113.
Rosen, M. J. Surfactants and Interfacial Phenomena, 2nd ed.; Wiley: New York, 1989; and references cited therein. Sarmoria, C.; Puwada, S.; Blankschtein, D. Prediction of Critical Micelle Concentrations of Nonideal Binary Surfactant Mixtures. Langmuir 1992,8,2690-2697. For a comprehensive experimental and theoretical survey of mixed micellar solutions, see: Phenomena in Mixed Surfactant Systems; Scamehorn, J. F., Ed.; ACS Symposium Series 311; American Chemical Society: Washington, DC, 1986. Staggers, J. E.; Hernell, 0.;Stafford, R. J.; Carey, M. C. Physical Chemical Behavior of Dietary and Biliary Lipids during Intestinal Digestion and Absorption. 1. Phase Behavior and Aggregation States of Model Lipid Systems Patterned aRer Aqueous Duodenal Contents of Healthy Adult Human Beings. Biochemistry 1990,29,2028-2040. Stellner, K. L.; Scamehorn, J. F. Surfactant Precipitation in Aqueous Solutions Containing Mixtures of Anionic and Nonionic Surfactants. J . Am. Oil Chem. SOC.1986, 63, 566-574. Strey, R.; Pakusch, A. Critical Fluctuations, Micelle Kinetics and Phase Diagram of Water-Nonionic Surfactant, H20-C&6. In Surfactants in Solution; Mittal, K. L., Bothorel, P., Eds.; Plenum: New York, 1986; pp 465-472. Tanford, C. The Hydrophobic Effect;Wiley: New York, 1980. Zana, R.; Weill, C. Effect of Temperature on the Aggregation Behavior of Nonionic Surfactants in Aqueous Solutions. J . Phys. Lett. (Paris) 1985, 46, L953-L960.
Received for review April 3, 1995 Revised manuscript received J u n e 28, 1995 Accepted July 7, 1995@ IE950227G
Abstract published in Advance A C S Abstracts, November 1, 1995. @
