The Modified Poisson−Boltzmann Equation - American Chemical

Ind. Eng. Chem. Res. 1996, 35, 2823-2833

2823

Electrostatic Modeling of Surfactant Liquid-Crystalline Aggregates: The Modified Poisson-Boltzmann Equation John C. Blackburn and Peter K. Kilpatrick* Department of Chemical Engineering, North Carolina State University, Raleigh, North Carolina 27695-7905

A model of liquid-crystalline aggregates is developed to calculate areas per surfactant head group in high-concentration cylindrical and planar geometries. Two free energy contributions, electrostatic and interfacial, are summed to generate the total cell free energy for a range of head-group areas. The modified Poisson-Boltzmann equation (MPBE) including dielectric saturation and hydration energies is used to calculated the electrostatic free energy. A Stern layer is used to prevent counterions at the charged interface, the thickness of which is decreased for large head-group areas to simulate the intercalation of counterions between head groups. The hydration constant was decreased at high surfactant concentration to account for reduced ion hydration as water is depleted. The calculation is performed for a one-dimensional variation of physical properties. Solutions to the MPBE were achieved for higher surface charge densities than previously reported, using a finite difference to estimate the derivatives and solving the resulting nonlinear algebraic equations. The optimized head-group areas for lamellar phase samples are compared to published X-ray data to estimate the extent of ion binding. Introduction The ability to design or use a specific surfactant for a given application is diminished by a lack of knowledge concerning the relationship between molecular structure and physical properties. Studies performed in this laboratory have indicated a high sensitivity of the liquidcrystalline phases formed by surfactant-water mixtures to subtle changes in surfactant tail length, counterion type, electrolyte addition, and solubilization of different types of oil (Kilpatrick and Bogard, 1988; Blackburn and Kilpatrick, 1992, 1993; Kilpatrick et al., 1992). A key to the prediction of liquid-crystalline phase behavior is a theoretical model which takes into account molecular structural features and which can be used to predict phase diagrams. Thermodynamic modeling of surfactant aggregation has included the investigation of micelle formation (Israelachvili et al., 1976; Nagarajan and Ruckenstein, 1977, 1979; Tanford, 1980), the oil solubilization capability and the stability of microemulsions (Safran, 1983; Safran and Turkevich, 1983; Evans and Ninham, 1983; Leodidis and Hatton, 1989), and liquid-crystalline phase stability (Parsegian, 1966, 1973; Jonsson, 1981; Jonsson and Wennerstrom, 1981; Jonsson et al., 1980, 1984; Bogard, 1986). We have used the cell model developed by Jonsson et al. (1980, 1981, 1984) and adapted by Bogard (1986) as the basis for our modeling effort. The cell model is based on organizing the surfactant and solvent into a unit cell of a particular geometry, summing several individual contributions to the free energy, and then minimizing the total free energy to determine the equilibrium phase. Contributions to the free energy in the cell model typically include hydrophobic, entropic, electrostatic, and interfacial. The hydrophobic contribution is calculated from the energy of interaction between the hydrophobic tails and the aqueous surroundings (Tanford 1980; Jonsson and Wennerstrom, 1981; Bogard, 1986). We are investigating the effect of counterion variation on cell free energy in ionic surfac* Author to whom correspondence should be addressed. Telephone: (919) 515-7121. Fax: (919) 515-3465. E-mail: [email protected].

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tant-water mixtures and therefore expect the major contribution to the free energy to be electrostatic. We will also include the interfacial contribution, due to the interaction between hydrophobic tail groups and the aqueous surroundings at the aggregate surface. Because the calculations are for completely aggregated systems, the hydrophobic contribution is identical for all aggregate geometries and will be neglected. The entropic contribution, or chain-packing energy, while significant for the monomer to micelle aggregation, will be neglected for liquid crystal-liquid crystal transitions. Developing a theoretical basis to explain and predict phase behavior would be useful to reduce the amount of experimental work necessary to map the phase behavior of new surfactants. Modeling of the complete range of structures is beyond the scope of this study, but initial results can be used to identify important features of the model to consider. This study focuses on optimization of free energy with respect to aggregate surface charge density and aggregate size in order to investigate effects of counterion identity and aggregate geometry on (i) area per surfactant head group and (ii) counterion binding. The cell model (Jonsson, 1981; Jonsson and Wennerstrom, 1981; Jonsson et al., 1980, 1984; Bogard, 1986) is based on the assumption that the liquid-crystalline aggregates can be divided into two separate regions: the aggregate interior and the surrounding aqueous region. The aggregate interior consists of the surfactant tail groups with no counterions and no water molecules. The aqueous region between aggregates contains no surfactant molecules, and the relative volume of the aqueous region is calculated from the bulk density of water. The spatial dependence of electrostatic potential and ion concentration within the cell are assumed to be onedimensional for this work. The lamellar phase is modeled as parallel flat plates, and the hexagonal phase is modeled as a cylinder in an annular aqueous region. A schematic diagram of the two cell geometries used is shown in Figure 1. The aggregate dimension R1 is determined by the surfactant length, and the cell boundary R2 is determined by the surfactant concentration, that is the ratio of volume occupied by the water to that occupied by the surfactant. The surface charge © 1996 American Chemical Society

2824 Ind. Eng. Chem. Res., Vol. 35, No. 9, 1996 Table 1. Alkali-Metal Bare Ion Radii and Calculated Born Hydration Coefficients (Gur et al., 1978a,b; Booth, 1951) ion

bare radius (Å)

Ai (kcal/mol)

Li Na K Rb Cs

0.68 0.97 1.33 1.47 1.67

127.1 103.0 85.1 79.9 72.1

Cell Model Background Electrostatic Free Energy. The first contribution to the cell free energy, the electrostatic contribution, is calculated from a relationship developed for charged surfaces in contact with an electrolyte solution. The free energy associated with charging a surface is (Verwey and Overbeek, 1948; Mitchell and Ninham, 1983; Hunter, 1987):

∆Gelectrostatic )

∫φsurface dσ

(1)

In eq 1, φsurface is the electrostatic potential at the head group-water interface and σ is the surface charge density (C/cm2). With a surfactant aggregate, the charge density is dependent on the packing of surfactant head groups. The charges of the carboxylate head groups are assumed to be uniformly distributed over the head group-water interface. The calculation of φsurface is related to the ion distribution through Poisson’s equation, which describes the dependence of the electric potential on the charge density of a solution (Verwey and Overbeek, 1948).

∇‚0∇φ ) -F

Figure 1. Schematic representation of cell model geometries. (a) lamellar geometry; (b) cylindrical geometry. Circles with pluses signify counterions in the annular region, and circles with minuses signify the ionic head groups.

of the surfactant aggregates is assumed to be uniformly distributed across the aggregate surface, neglecting localized charge of the surfactant head group. To account for ion identity, we have used the modified Poisson-Boltzmann equation (MPBE) to calculate the electrostatic potenial. Results are presented in which the MPBE is solved for concentration and dielectric constant profiles as functions of surfactant concentration and surface charge density. The surface electrostatic potential and the area per surfactant head group were used to calculate cell free energies. Minimization of the free energy with respect to head-group area yields an optimized area per head group for a given surfactant concentration, counterion type, aggregate geometry, and surface charge. The parameters chosen were intended to represent experimental systems comprised of sodium or cesium carboxylates with water in cylindrical and planar geometries. The optimized areas per head group for different surface charges, corresponding to differing degrees of counterion binding, were compared to experimental values determined by X-ray diffraction (Boden et al., 1987; Gallot and Skoulios, 1966). By equating calculated areas per head group to experimental values for lamellar phase samples, counterion binding values were determined.

(2)

In eq 2, φ is the electric potential (V),  is the dielectric constant of the solution, 0 is the permittivity of a vacuum (8.854 × 10-14 C/V/cm), and F is the charge density (C/cm3) at a given location. A second relationship between the electric potential and the ion charge density can be derived from the electrochemical potential (µi) in an electrolyte solution (Hunter, 1987; Kim, 1988):

µi ) µi0 + RT ln ci + ziFφ +

Ai 

(3)

In eq 3, µi0 is the reference chemical potential (J/mol), R is the gas constant (J/mol/K), T is the absolute temperature (K), ci is the concentration of the ith component, zi is the valence of the ionic head group, F is Faraday’s constant (the charge on a mole of electrons, 96 485 C/mol), φ(r) is the electrostatic potential in the solution, and Ai is the hydration coefficient for ion i defined using the Born theory (Israelachvili, 1991). Gur et al. (1978a,b) performed one-dimensional calculations of ion and potential profiles in solutions near a charged surface. They used Booth’s equation (Booth, 1951) to evaluate the effect of electric field on solution permittivity. The free energy to move an ion from a vacuum ( ) 1) to a region with a dielectric permittivity  is given by (Israelachvili 1991; Gur et al., 1978a,b):

∆GBorn )

z2e2 1 1 - 1 ) Ai - 1 2r  

(

)

(

)

(4)

Given radius (r) and valence (z) data for an ion and the charge on an electron (e), the coefficient Ai can be

Ind. Eng. Chem. Res., Vol. 35, No. 9, 1996 2825

calculated from eq 4, and is collected in Table 1 for several alkali-metal ions. At thermodynamic equilibrium, the gradient of the electrochemical potential vanishes (Kim, 1988):

(1)

∇µ ) 0 ) RT∇(ln ci) + ziF∇φ + Ai∇

(5)

Solution of this differential equation (5), gives a second relationship between ci, the ion concentration, and the electric potential (φ).

( )]

[

-ziFφ Ai 1 1 ci ) ci ziF exp + RT RT  ∞ 0

|

)

(6)

aggregate surface

dφ dx

|

cell boundary

(7)

-σ 0

)0

(8)

Previous electrostatic calculations of liquid-crystalline phase behavior (Parsegian, 1966, 1973; Jonsson, 1981; Jonsson and Wennerstrom, 1981; Jonsson et al., 1980, 1984; Bogard, 1986) have treated the permittivity in solution as independent of electric field and equal to the zero-field value ( ) 78.3 for water at 25 °C) (Booth, 1951; Conway, 1981; Bogard, 1986). Under this assumption, eqs 2 and 6 can be combined to form the Poisson-Boltzmann equation (PBE) (Mitchell and Ninham, 1983; Bogard, 1986).

∇2φ )

F

-F )∞

{ }

zici0 exp ∑  i ∞

-ziFφ RT

∇2φ )

-F ) +E

d dE

∑i

-F

In eq 6, ci0 is the concentration of ion i at zero field and is set by the total solution ion concentration. The value of  at zero electric field, which is determined by the system temperature, is shown by ∞. Equations 2 and 6, along with the appropriate boundary conditions, can be solved simultaneously to calculate the ion and potential profiles within the cell. The boundary conditions for the cell determine the electric potential gradient at the two ends of the cell. The electric field at the aggregate surface is equal to the surface charge density (σ) divided by the local dielectric permittivity, which is the product of the dielectric constant and the dielectric permittivity of vacuum. The electric field vanishes due to symmetry at the cell boundary.

dφ dx

field dependence of the dielectric permittivity and for the associated hydration energy in eq 6, the modified Poisson-Boltzmann equation (MPBE) can be obtained as outlined in Appendix A.

(9)

Bogard (1986) solved the one-dimensional PBE to calculate the free energy of aggregation of surfactants into spherical, cylindrical, and lamellar aggregates. A finite difference scheme, with an adaptive mesh to concentrate the node points near the aggregate surface, was used to approximate the Laplacian. The exponential on the right-hand side of eq 9 was linearized around a trial solution and the computer subroutine BAND(J) (Atkinson, 1978; White, 1978; Bogard, 1986) was used to invert the resulting matrix. Successful prediction of micelle formation concentrations (cmc’s) was calculated using an optimization routine governing the aggregate size and fraction of surfactant molecules existing in aggregates (Bogard, 1986). Previous applications of the cell model (Jonsson, 1981; Jonsson and Wennerstrom, 1981; Jonsson et al., 1980, 1984; Bogard, 1986) did not account for ion identity in the electrostatic calculation. Accounting for the electric

zici0 exp

{

-ziFφ

( )}

Ai 1 -

RT

+E

1

-

RT  d

∞

(10)

dE

As in eq 9, φ is the electric potential, F is the charge density, F is Faraday’s constant, z is the valence of the ith component, and ci0 is the concentration of component i at zero field. The dependence of the dielectric constant on electric field and the derivative of the dielectric constant with respect to electric field were evaluated using Booth’s equation (Booth, 1951):

(βE3 )(tanh1 βE - βE1 ),

 ) n2 + (1 - n2)

with β )

5µd 2 (n + 2) (11) 2kT

In eq 11, n is the refractive index of the pure solvent, µd is the solvent dipole moment, k is Boltzmann’s constant, and E is the electric field. Booth’s equation is used here to describe the permittivity profile in a concentrated electrolyte solution, although it was derived for pure water. The dependence of  on salt concentration is weak when compared to its dependence on E (Gur et al., 1978a,b). The Booth equation has been used to model electrolyte solutions with reasonable success (Gur et al., 1978b; Cevc and Marsh, 1983; Kim, 1988; Leodidis and Hatton, 1989), as shown by comparing calculated and measured solution capacitance (Gur et al., 1978a,b). The development of the MPBE for onedimensional (r-direction) variation of the electric potential and ion concentration in cylindrical coordinates is shown in Appendix B. Interfacial Free Energy. The second contribution to the cell free energy is the interfacial term due to tail group contact with the aqueous surroundings. The driving force for aggregation is the large free energy of interaction between the hydrophobic tail groups and water in the surrounding aqueous region. When in an aggregated state, there is residual contact and a resulting interaction which still exists between the tails and the water. We have modeled this interaction free energy as the product of the area of interaction, the total area per head group less the actual physical area occupied by the head group. A literature value for the interfacial tension between hydrocarbon and water was used (γ ) 50 dyn/cm) (Israelachvili et al., 1976).

γ ) 50 dyn/cm ) 301 J/Å2/mol ∆Ginterfacial) (A - A0)γ ) (A - A0) × 301 J/Å2/mol (12) The area occupied by the ionic head group (A0, 21 Å2) (Tanford, 1980) is fixed by the surfactant and does not contribute to the interfacial energy. Given an electro-

2826 Ind. Eng. Chem. Res., Vol. 35, No. 9, 1996

static free energy contribution (eq 2) which favors large area per head group and an interfacial component (eq 12) which favors small area per head group, it is anticipated that there exists an optimal value of the head group area which minimizes the total free energy. Previous MPBE Solutions Gur et al. (1978a,b) were one of the first groups to report solutions of the MPBE. They calculated ion and dielectric constant profiles for a range of univalent salt concentrations (0.001-0.1 N NaF) and a range of surface potentials (100-200 mV). Their results were for conditions sufficient to reduce the dielectric constant from 78 for bulk water to 20-40 for the 200 mV surface potential. Frahm and Diekmann (1979) calculated solutions to the MPBE using a variation of Gur et al.’s backshooting method (Gur et al., 1978a). For multivalent ions they concluded that the deviations from the PBE case were significant for surface potentials greater than 100 mV. Leodidis and Hatton (1989) used the MPBE to calculate ion concentration in order to model the concentration of dissolved proteins in reversed micelles. They assumed that the surfactant head groups occupied a region around a mean location from the center of the micelle rather than a plane. By distributing the charges over a volume, they reduced the effective surface charge density. Dielectric constant values were reduced to ca. 40 for their calculations. Kim (1988) developed solutions to the MPBE, including dielectric constant profiles and ion hydration, for colloidal dispersions and mica sheets with surface charge densities of 1.0 × 10-6 C/cm2. The strategy for solving the set of equations was to use the BAND(J) approach of Newman (Atkinson, 1978; White, 1978; Bogard, 1986) to solve the linearized form of the equation. For more highly charged cases, this approach was very slow to converge and very sensitive to the initial guess, although solutions were achieved with dielectric constant values of 10 M) is the layer less than the radius of a bare cesium or sodium ion. It is therefore sensible to effectively reduce the surface concentration of counterions by imposing a Stern (or excluded-volume) layer. The thickness of the Stern layer is set equal to the bare ion radius of the counterion. The bare ion radius was chosen, rather than the hydrated ion radius, because it is reasonable to assume that the alkali-metal ions shed their waters of hydration in the high-field region near the surface. Within the Stern layer there are no counterions, so Poisson’s eq 2 reduces to Laplace’s equation, where the charge density of the solution is zero.

∇‚∇φ ) 0

(13)

The free energy of the Stern layer volume is essentially a capacitor energy obtained from the electric field (E ) -∇φ) and the dielectric permittivity within

Figure 2. Schematic diagram illustrating the decrease in the Stern layer (δ) as the area per surfactant head group (A) increases. When A increases, the counterions can fit between the head groups, decreasing the distance of exclusion of the center of the counterion from the charged head-group boundary.

the Stern layer. Stern ∆Gelectrostatic )

∫E‚E dV

(14)

As the area per surfactant head group was increased, due to a smaller R1, the cell boundary (R2) also decreased. When the available area per head group was larger than the actual size of the carboxylate head group, counterions can fit between the head groups. By approximating the head group by a circular region of influence at the aggregate surface and the counterion as a sphere, a geometric model was used to decrease the thickness of the Stern layer with increasing area per head group. For available head-group areas of 40 Å2 or greater, the Stern layer thickness was reduced to zero for the Cs counterion. A schematic representation of the decrease in the Stern layer with an increase in surfactant head-group area is illustrated in Figure 2. Hydration Constant. Studies of liquid-crystalline phase behavior have shown a decrease in counterion hydration at high surfactant concentration levels (Blackmore and Tiddy, 1990; Blackburn and Kilpatrick, 1992) (>0.1 mole fraction surfactant). At these concentrations there is no longer sufficient water in the system to fully hydrate head groups and counterions. To account for this, we have decreased the ion hydration coefficient at high surfactant concentration (Ai). The simplest functional form for Ai is a linear decrease from the bare ion value at xs ) 0.05 to zero at xs ) 1.0. The predominant influence of changing the hydration constant was for the higher concentration lamellar geometry calculations. Surface Charging. The surface charge for an alltrans n-dodecanoate tail in a planar (lamellar) geometry is σ ) 7.5 × 10-5 C/cm2. This is calculated by assuming a dodecanoate tail volume of 377 Å3 (Tanford, 1980), an extended all-trans tail length of 17.4 Å, and a totally ionized head group. When the BAND(J) algorithm was used to solve the MPBE for this surface charge, the system was unstable and therefore a different solution algorithm was attempted. Rather than linearizing the exponential portion of the equation, a nonlinear algebraic equation solver (DNEQNF) from the IMSL library was used to solve the set of nonlinear equations generated from the finite difference estimation of the derivatives. This approach was not sufficient to solve the problem, as the stability of the solution was very dependent upon the initial guess.

Ind. Eng. Chem. Res., Vol. 35, No. 9, 1996 2827 Table 2. Geometric Calculations of the Area/Head Group (A) and Cell Boundary (R2) lamellar case area per head group (cm2) R2 (cell boundary, cm)

cylindrical case

V/R1

2V/R1

(1 - xs)MWwR1 + R1 xsFwV

[

]

(1 - xs)MWwR1 + R12 2πxsFwV

1/2

To obtain an accurate initial guess, the MPBE was first solved for a more stable set of conditions. The surface charge was reduced to 1% of the total surface charge density and then increased using a pseudo-firstorder continuation algorithm. In this scheme, the solution at one step and the gradient in potential with increasing surface charge are used to generate the initial guess for the subsequently more charged case. The incremental charging of the surface was from 1 to 5% of the final charge density per step. The algorithm of stepping up the surface charge and using DNEQNF to solve the system of algebraic equations was more robust than linearizing the exponential portion of the MPBE and enabled solution for charge densities up to 7.5 × 10-5 C/cm2 for the lamellar case and 3.7 × 10-5 C/cm2 for the cylindrical case. This “charging” procedure facilitated the calculation of the electrostatic free energy which requires integrating the surface potential as a function of increasing surface charge. As the MPBE was solved for a range of surface charges, the free energy at intermediate surface charges was calculated. This is equivalent to calculating the free energy at differing degrees of “counterion binding”. By matching the calculated optimum area per head group to experimental data for a range of surface charges (40-100% of a fully charged aggregate, the completely charged aggregate case corresponding to a surface charge density of ca. 7 × 10-5 C/cm2), equivalent counterion binding values were obtained. Physical Parameters. The area per surfactant head group was calculated from the aggregate geometry, the volume of the tail group, and the radius of the aggregate (R1). The volume and the all-trans length of the surfactant tail group was based on the correlation of Tanford (1980). Given the area per surfactant head group and the surfactant composition, the value for R2, the edge of the cell, was calculated to match the ratio of surfactant and water volumes. The density of the aqueous solution was taken to be that of water. Forms for the area per head group and the cell boundary are shown in Table 2, for cylindrical and planar geometries. Other parameters used in the calculations, such as those used in the Booth equation, are shown in Table 3 (Tanford, 1980; Gur et al., 1978a; Kim, 1988). The electrostatic problem was posed in the form of the MPBE, and the electrostatic free energy was calculated for a range of R1 values. The area per head group for a given aggregate radius with the surface charge density is listed in Table 4. For the corresponding areas per surfactant head group, the interfacial energy was calculated to generate a curve of free energy vs head-group area. The minima in these curves determined the optimum area for a given surface charge, geometry, and counterion. Results and Discussion Electrostatic free energy calculations were performed for cesium and sodium n-dodecanoate. The algorithm

Figure 3. Counterion profiles calculated using MPBE for the lamellar geometry for varying surface charge. (a) Na profile, xs ) 0.1; (b) Cs profile, xs ) 0.1. Distance is measured from the aggregate interface. The Stern layer, where the counterion concentration is zero, is indicated at the aggregate surface.

followed in solving the problem is outlined in Appendix C. After fixing the geometry, counterion, temperature, and composition, the aggregate size R1 is fixed. The surface charge density (σ) and R2 are calculated from the above parameters. Using the surface charging approach along with the nonlinear equation routine (DNEQNF), solutions for both lamellar and cylindrical geometries were calculated. The total free energy of the cell was calculated from the surface potential, the Stern layer free energy, and the interfacial energy. Counterion and Dielectric Constant Profiles. Solutions to the MPBE for cesium and sodium in the lamellar geometry generated profiles of counterion concentration, dielectric constant, and electric field across the cell. Figure 3 shows a comparison of counterion profiles for sodium and cesium in the lamellar geometry and a range of surface charges. The 100% charged case (i.e., no ion binding) corresponds to σ ) 7.45 × 10-5 C/cm2 for the all-trans n-dodecanoate tail. The secondary maximum in ion concentration is a result of the ion hydration contribution to the electrochemical potential of the counterion. The mathematical solution became unstable when the secondary maximum reached the cell boundary. Figure 4 shows the dielectric constant profiles for sodium and cesium corresponding to the conditions in Figure 3 for the lamellar geometry. For the cases of 50% or greater surface charge, there is substantial reduction of the dielectric constant within the solution. Free-Energy Calculations. The cell free energy was calculated using eqs 1, 10, and 12. The surface potential calculated using the MPBE was integrated from zero charge to the desired charge level to calculate the electrostatic free energy. This integration was

2828 Ind. Eng. Chem. Res., Vol. 35, No. 9, 1996 Table 3. Surfactant (n-Dodecanoate) Ion Parameters Used in MPBE and the Booth Equation physical area of a carboxylate HG all-trans n-dodecanoate tail length Boltzmann’s constant tail volume charge on an electron refractive index of water dimensionless dipole moment of water interfacial tension (water/n-dodecane) permittivity of vacuum zero-field dielectric constant (25 °C) dipole moment of water surface charge density

A0 L k V e n β γ 0 ∞ µ σ

21 Å2 (Tanford, 1980) 17.4 Å (Tanford, 1980) 1.381 × 10-23 J/K 377 Å3 (Tanford, 1980) 1.6022 × 10-19 C 1.33 1.41 × 10-6 50 dyn/cm (Israelachvili et al., 1976) 8.542 × 10-14 V/C/cm 78.3 1.102 × 106 C‚cm e/A C/cm2

Figure 5. Electrostatic and interfacial contributions to the cell free energy as a function of area per surfactant head group and surface charging for the lamellar geometry. Cesium surfactant is at mole fraction 0.05.

Figure 4. Calculated dielectric constant profile () for the lamellar geometry for varying surface charge. (a) Na, xs ) 0.1; (b) Cs, xs ) 0.1. Distance is measured from the aggregate interface. The Stern layer, where  is a constant, is indicated at the aggregate surface. Table 4. Surface Charge Densities (σ) for Fully Charged Aggregates R1 (Å) 17 14 12 10 8 6

A (lamellar) σ (lamellar) A (cylinder) σ (cylinder) (Å2) (C/cm2 × 105) (Å2) (C/cm2 × 105) 22.2 26.9 31.4 37.7 47.1 62.8

7.22 5.95 5.10 4.25 3.40 2.55

44.4 53.9 62.8 75.4 94.3 125.7

3.61 2.97 2.55 2.12 1.70 1.28

performed using two IMSL routines, one (DCSAKM) to fit a cubic spline through the potential profile and DCSITG to evaluate the integral using a quadrature technique. The Stern layer free-energy calculation is an integral as well; however, for our case the integrand is a constant, reducing this to a simple multiplication. The interfacial free energy has a linear dependence on surfactant head-group area (eq 10). Figure 5 contains sample results of free-energy calculations for cesium n-dodecanoate in a lamellar geometry (xs ) 0.05). Curves corresponding to various surface charges are labeled as is the interfacial term. By summing the electrostatic and interfacial contributions, the total cell

Figure 6. Dependence of the total cell free energy on area per head group for cesium at xs ) 0.1. The interfacial and electrostatic contributions shown in Figure 5 are summed to these curves. The minimum in each curve indicates the optimum area per head group for a surface charge.

free energy can be determined (Figure 6). The minimum in the total free energy signifies the optimum area per head group for a given surfactant, geometry, and surface charge. The electrostatic free energy favors larger head-group areas, and with higher electrostatic free energy, the minimum in the sum of the two freeenergy contributions moves to higher head-group areas. Figure 7 contains calculated areas per head group as a function of surfactant mole fraction and surface charging for cesium and sodium in the lamellar geometry. The amount of surface charge is relative to the 100% charged case which is defined for a range of headgroup areas in Table 4. As expected, the area per head group increases with increased surface charge. With areas per head group larger than ca. 40 Å2, the Stern layer does not contribute to the free energy as the Stern thickness is reduced to zero. Areas per Head Group and Degree of Counterion Binding. Figure 8 is a plot of experimental area per head group data as a function of the surfactant to water mole ratio for several alkali-metal counterions.

Ind. Eng. Chem. Res., Vol. 35, No. 9, 1996 2829 Table 5. Calculated Areas per Surfactant Head Group and Corresponding Surface Chargesa mole fraction surfactant

Cs area (Å2)

% charge

Na area (Å2)

% charge

0.05 0.01 0.15

45.1 40.3 37.4

72 68 66

41.2 36.8

66 63

a The areas per surfactant head group were taken from Figure 7 [data reported by Boden et al. (1987) and Gallot and Skoulios (1966)], and the percent charged column was calculated so that the model optimum area per head group would match the experimental data.

Figure 7. Dependence of the optimized area per head group on surfactant concentration and surface charge for (a) Na and (b) Cs in the lamellar geometry. The 100% charge density case for different areas per head group is shown in Table 4.

Figure 8. Experimental correlation of area per head group as a function of water to surfactant ratio for alkali-metal counterions (from Boden et al. (1987)).

Boden et al. (1987) used previously published X-ray diffraction data (Gallot and Skoulios, 1966) of aggregate sizes in conjunction with a model for partial molar areas of the components to calculate these areas. As tail length effects were neglected, the curves in Figure 8 represent counterion effects only. We have compared calculated areas per head group from our model with the experimental data of Gallot and Skoulious (1966) to estimate the degree of counterion binding to the aggregates. We have assumed that the sole effect of counterion binding is to reduce the surface charge of the aggregate, i.e., as counterion binding increases from 0 to 100%, the % surface charge decreases from 100 to 0%. By fitting the counterion binding of our model calculations using the experimental area per head group data, we can present our data in two forms. By fitting a single concentration point, we can analyze the model

trends of head-group area with surfactant concentration for a constant surface charging and compare those to the experimentally measured trends. However, it is clear from Figure 7 that the MPBE yields an area per head group which is relatively insensitive to surfactant concentration at fixed percent surface charge at all values of surface charge except the totally charged case. For the fully charged case (100% dashed line in Figure 7), a decrease in the area per head group with increasing surfactant composition is observed. For cesium at 100% charge, the variation is from 76.3 (xs ) 0.05) to 70.0 Å2 (xs ) 0.15). This compares to 45-37.5 Å2 from Boden’s data. The change in area is ca. 7 Å2 in each case. Although the magnitudes of the areas are not equal, the change in each is quite similar. These similarities indicate that this model may be capturing the trend in area per head group with the appropriate physics but is missing the absolute magnitude. Possible reasons for this difference are the hydrophobic and interfacial free energy contributions chosen for the calculations and the molecular parameters used in the calculations. The trends in head-group areas with the existing model are in agreement with the experimental data and give us confidence that we are including sufficient free-energy contributions. The model results are most sensitive to changes in values used for the interfacial interaction (γ) and the ion radius. By increasing γ from 21 to 50 dyn/cm, the calculated area per head group decreased by ca. 10 Å2 for the 100% charged cesium case over the composition range xs ) 0.05 to xs ) 0.15. The choice for the ion radius and the corresponding Stern layer also has an influence on the calculated areas. However, the Stern layer free-energy contribution does not have an influence when the areas are greater than 40 Å2. When comparing sodium and cesium at the same charging, the model predicts a larger area per head group for sodium. This is due to the electrostatic freeenergy contribution. The sodium ion hydration constant (A+) is larger, producing larger electrostatic free energy which favors a larger area per head group. The second method of performing the match of experimental area per head group to our theoretical calculations is to match the area per head group for each composition, estimate the amount of counterion binding, and then observe the trend with composition. This comparison is shown in Table 5 for both cesium and sodium calculations. As the concentration of surfactant increases, the amount of surface charge of the aggregates seems to decrease for both Na and Cs, suggesting an increase in counterion binding. This is likely the result of dehydration of the counterions, which allows a closer approach of the counterions to the aggregate surface and subsequently a lower effective surface charge. A surface charge of 65-72% corresponds to an ion binding of 28-35% over the composi-

2830 Ind. Eng. Chem. Res., Vol. 35, No. 9, 1996

Figure 9. Optimum area per head group for the lamellar geometry using the PBE for electrostatic free-energy calculation. This plot illustrates the effect of no dielectric saturation or ion hydration on the electrostatic calculation. The areas per head group are significantly smaller than those for the same charge case with the MPBE.

tion range studied. This compares to binding values of 60% measured for sodium and cesium octanoate-water micelles (Rosenholm and Lindman, 1976; Gustavsson and Lindman, 1978). Our calculations are lower than the experimental values. One possible explanation for this is in differences in definition of “bound” ions between the two cases. In our model, ions are considered 'bound' if they are in sufficient proximity to the head groups to reduce the surface charge. It is likely that the bound ions from the micellar experiments include a larger number of ions. Areas per head group also were calculated for the PBE, with no dielectric constant saturation. The ion dependence is a result of the Stern layer thickness, which is different for sodium and cesium. Figure 9 is a plot of optimized area per head group as a function of composition, for three surface charges. The areas are significantly smaller than in the case of dielectric saturation, even for the 100% charged case. This is a result of considerably reduced electrostatic free energy when dielectric saturation is not included. The calculated electrostatic potentials were in the range of 400 mV for the PBE calculations, while for the MPBE the potentials were 1-2 V. The larger potential resulted in larger electrostatic free energy which favors larger areas per head group. Calculations using the PBE predict a small (1 Å2) decrease in area per head group over the composition range investigated. The trend in head-group area with counterion is opposite for the PBE calculations compared to the MPBE calculations. The areas per head group are markedly smaller for the sodium calculation compared to cesium. Both of these results are in contrast to the MPBE calculations and to the experi-

Figure 10. Area per head group calculations for the cylindrical geometry case: (a) Cs and (b) Na.

mental results presented in Figure 8, where there is a substantial decrease in area with an increase in composition and the cesium and sodium cases have areas in the range of 35-45 Å2. For the PBE, the dominating influence in the electrostatic free energy is the Stern layer, which is proportional to the counterion radius. As the cesium ion has a larger bare ion radius, the free energy is larger, and the area per head group is larger. Cylindrical Geometry Free-Energy Calculations. Calculations of counterion concentration, dielectric permittivity, and free energy were also performed for cylindrical aggregates of sodium and cesium surfactants over a range of concentrations using the MPBE. By minimizing the total cell free energy, optimized areas per surfactant head group were determined. The calculated areas per head group for several surface charges are shown in Figure 10. The areas per head group for the cylindrical case are substantially larger than those for the lamellar case. This is primarily due to the geometry of the aggregates shown in Table 2, where the area per surfactant head group is calculated. Conclusions Calculations were performed using the MPBE with a Stern layer for lamellar and cylindrical geometries of mixtures of water with cesium n-dodecanoate or sodium n-dodecanoate. The algorithm presented incorporates incremental charging of the aggregate surface with a nonlinear algebraic equation solver to solve the differential equation for the high surface charges encountered for the case of surfactant aggregates. Using the MPBE to determine ion concentration and cell free energy enabled the study of counterion influence on aggregate electrostatics. The calculated areas per head group were compared to experimental data to estimate counterion binding values, which were lower than previous values for

Ind. Eng. Chem. Res., Vol. 35, No. 9, 1996 2831

counterion binding to surfactant aggregates. For the fully charged aggregates, there was a reduction in headgroup area with increased surfactant concentration as observed experimentally. Calculations using the MPBE yield comparable head group areas for both cesium and sodium counterions. This is in contrast to the PBE calculation, in which there is a large difference between cesium and sodium due to the larger bare ion radius of cesium versus sodium and a consequent increase in Stern layer thickness.

∇‚∇φ )

( )

∂ ∂φ  +0+0 ∂x ∂z

Applying the product rule and the definition of the electric field (E) gives:

∇‚∇φ )

∂2φ ∂ ∂E ∂φ ∂2φ ∂ ∂φ + 2) + 2 ∂x ∂x ∂E ∂x ∂x ∂x ∂x

which can be rearranged to factor out the potential.

Acknowledgment

∇‚∇φ )

This work was supported in part by a grant from the National Science Foundation (CPE-8404599).

2

(∂E∂ E + ) ∂∂xφ 2

Using the form for F, as presented in eq 6 gives the MPBE

Nomenclature A ) area per surfactant head group Ai ) hydration coefficient of ion i A0 ) area occupied by surfactant head group ci0 ) bulk ion concentration (zero field) e ) electron charge E ) electric field F ) Faraday’s constant ∆G ) free energy k ) Boltzmann’s constant L ) surfactant tail length MWs ) surfactant molecular weight MWw ) water molecular weight n ) refractive index of water r ) bare ion radius R ) gas constant R1 ) aggregate surface distance R2 ) cell boundary T ) temperature V ) tail volume xs ) mole fraction surfactant zi ) ion valence

∇2φ )

) +E

d dE

∑i zici0 exp

-F

{

-ziFφ

( )}

Ai 1 -

RT

+E

1

-

RT 

∞

d dE

Appendix B This is the development of the MPBE in cylindrical coordinates, beginning with Poisson’s equation:

∇‚∇φ ) -F Using the definitions of the gradient and the divergence in cylindrical coordinates,

Greek Letters β ) dimensionless dipole moment o ) vacuum permittivity ∞ ) dielectric constant φ ) electrostatic potential φsurface ) surface electrostatic potential γ ) interfacial tension Fw ) density of water σ ) surface charge density µd ) dipole moment of water ∆µi ) chemical potential

-F

∇φ )

1 ∂φ ∂φ ∂φ e + e + e ∂r r r ∂θ θ ∂z z

∇‚v )

1 ∂rvr + r ∂r 1 ∂vθ ∂vz + r ∂θ ∂z

the left-hand side of Poisson’s equation in one dimension takes this form.

∇‚∇φ )

(

)

∂φ 1 ∂ r +0+0 r ∂r ∂r

Applying the product rule gives:

Appendix A This is the algebra to develop the MPBE in rectangular coordinates, beginning with Poisson’s equation (Bogard, 1986).

∇‚∇φ ) -F

∇‚∇φ )

(

)

∂ ∂φ ∂2φ 1 ∂φ +r + r 2  r ∂r ∂r ∂r ∂r

which can be expanded using the definition of the electric field.

Using the definition of the gradient of a scalar:

∂φ ∂φ ∂φ ∇φ ) e + e + e ∂x x ∂y y ∂z z and the definition of the divergence of a vector

∂vx ∂vy ∂vz + + ∇‚v ) ∂x ∂y ∂z the left-hand side of Poisson’s equation, in one dimension, becomes:

∇‚∇φ )

∂2φ  ∂φ ∂ ∂E ∂φ + + 2 r ∂r ∂E ∂r ∂r ∂r

and rearranged

∇‚∇φ )

∂2φ  ∂φ ∂ ∂2φ ∂φ + +  r ∂r ∂E ∂r2 ∂r ∂r2

The left-hand side of Poisson’s equation can be rearranged to several forms.

2832 Ind. Eng. Chem. Res., Vol. 35, No. 9, 1996

∇‚∇φ )

∂2φ ∂ ∂φ  ∂φ ∂2φ ∂ 2 ) ∇  + + φ + E ∂E ∂r r ∂r ∂r2 ∂r2 ∂E

(

)

Using the form for F, as presented in eq 6 gives the MPBE.

∇‚∇φ )

(

∂2φ ∂r2

+

)

∂ ∂φ ∂E ∂r

-F ) -F

 ∂φ )

+ r ∂r

∑i zici0 exp

[

-ziFφ RT

( )]

Ai 1 -

1

-

RT 

0

Appendix C The algorithm used to determine optimum area per head group is

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Received for review June 8, 1995 Revised manuscript received March 11, 1996 Accepted March 25, 1996X IE950343J

X Abstract published in Advance ACS Abstracts, August 15, 1996.