^{1}, Kenneth R. Cox

^{1}and Walter G. Chapman

^{1,a)}

### Abstract

The ubiquitous use of surfactants in commercial and industrial applications has led to many experimental, theoretical, and simulation based studies. These efforts seek to provide a molecular level understanding of the effects on structuring behavior and the corresponding impacts on observable properties (e.g., interfacial tension). With such physical detail, targeted system design can be improved over typical techniques of observational trends and phenomenological correlations by taking advantage of predictive system response. This research provides a systematic study of part of the broad parameter space effects on equilibrium microstructure and interfacial properties of amphiphiles at a liquid-liquid interface using the interfacial statistical associating fluid theorydensity functional theory as a molecular model for the system from the bulk to the interface. Insights into the molecular level physics and thermodynamics governing the system behavior are discussed as they relate to both predictions qualitatively consistent with experimental observations and extensions beyond currently available studies.

The authors thank the following for their financial support: Robert A. Welch Foundation (Grant No. C-1241), the National Science Foundation (CBET-0756166), and the Petroleum Research Fund (Grant No. 47236-AC6).

I. INTRODUCTION

II. THEORY AND MODEL

A. Model system

B. Classical density functional theory formalism

C. Free energy functionals

D. Functional derivatives

E. Equilibrium fluid structure and interfacial tension

III. RESULTS AND DISCUSSION

A. Effect of chain length

B. Oil structure

C. Single vs. double tail surfactantstructure

D. Mixed surfactant synergism

E. Temperature effects

IV. CONCLUSIONS

V. FUTURE WORK

### Key Topics

- Surfactants
- 158.0
- Interface structure
- 41.0
- Density functional theory
- 37.0
- Solvents
- 25.0
- Free energy
- 22.0

## Figures

Schematic of spherical solvents and amphiphilic chain construction within the Telo da Gama and Gubbins (Ref. 18) oil-water-surfactant model.

Schematic of spherical solvents and amphiphilic chain construction within the Telo da Gama and Gubbins (Ref. 18) oil-water-surfactant model.

Schematic of chain formation from a mixture of associating spheres using Wertheim's TPT1 (Refs. 57–60) in the complete bonding limit (Refs. 50 and 53).

Schematic of chain formation from a mixture of associating spheres using Wertheim's TPT1 (Refs. 57–60) in the complete bonding limit (Refs. 50 and 53).

Effect of surfactant structure and bulk aqueous surfactant concentration on the scaled interfacial tension at *P** = β*P*/ρ = 2.66 for a series of linear single tail surfactants. Predictions made using *i*SAFT for an oil molecule with *m* = 1 (solid curves) and *m* = 8 (dashed curves). Scaling factor (γ_{0}) is the interfacial tension of the bare oil-water interface.

Effect of surfactant structure and bulk aqueous surfactant concentration on the scaled interfacial tension at *P** = β*P*/ρ = 2.66 for a series of linear single tail surfactants. Predictions made using *i*SAFT for an oil molecule with *m* = 1 (solid curves) and *m* = 8 (dashed curves). Scaling factor (γ_{0}) is the interfacial tension of the bare oil-water interface.

Effect of surfactant structure and bulk aqueous surfactant concentration on the scaled interfacial tension at *P** = β*P*/ρ = 2.66 for a series of diblock symmetric surfactants. Predictions made using *i*SAFT for an oil molecule with *m* = 1 (solid curves) and *m* = 8 (dashed curves). Scaling factor (γ_{0}) is the interfacial tension of the bare oil-water interface.

Effect of surfactant structure and bulk aqueous surfactant concentration on the scaled interfacial tension at *P** = β*P*/ρ = 2.66 for a series of diblock symmetric surfactants. Predictions made using *i*SAFT for an oil molecule with *m* = 1 (solid curves) and *m* = 8 (dashed curves). Scaling factor (γ_{0}) is the interfacial tension of the bare oil-water interface.

Equilibrium fluid microstructure from *i*SAFT for an *H* _{7} *T* _{7} surfactant using an *m* = 1 oil molecule with bulk mole fraction: (a) , (b) 10^{−18}, (c) 10^{−15}, and (d) 10^{−12}. Curves are for the oil (black), water (blue), tail (green), head (red), and total (purple) segment density profiles.

Equilibrium fluid microstructure from *i*SAFT for an *H* _{7} *T* _{7} surfactant using an *m* = 1 oil molecule with bulk mole fraction: (a) , (b) 10^{−18}, (c) 10^{−15}, and (d) 10^{−12}. Curves are for the oil (black), water (blue), tail (green), head (red), and total (purple) segment density profiles.

Detailed equilibrium segment density profiles for the symmetric diblock *H* _{7} *T* _{7} surfactant at bulk mole fraction (Figure 5(c)). Curves are the head group (red) and tail group (green) terminal (solid), central (dashed), and 5 internal (dotted) segment density profiles.

Detailed equilibrium segment density profiles for the symmetric diblock *H* _{7} *T* _{7} surfactant at bulk mole fraction (Figure 5(c)). Curves are the head group (red) and tail group (green) terminal (solid), central (dashed), and 5 internal (dotted) segment density profiles.

Collapsed interfacial packing of a chain-like diblock symmetric surfactant approximated by a diatomic molecule with equivalent volume and the resulting “trapped” solvent molecules.

Collapsed interfacial packing of a chain-like diblock symmetric surfactant approximated by a diatomic molecule with equivalent volume and the resulting “trapped” solvent molecules.

Equilibrium fluid microstructure from *i*SAFT for an *H* _{7} *T* _{7} surfactant using an *m* = 8 oil molecule with bulk mole fraction: (a) , (b) 10^{−36}, (c) 10^{−34}, and (d) 10^{−32}. Curves are for the oil (black), water (blue), tail (green), head (red), and total (purple) segment density profiles.

Equilibrium fluid microstructure from *i*SAFT for an *H* _{7} *T* _{7} surfactant using an *m* = 8 oil molecule with bulk mole fraction: (a) , (b) 10^{−36}, (c) 10^{−34}, and (d) 10^{−32}. Curves are for the oil (black), water (blue), tail (green), head (red), and total (purple) segment density profiles.

Structure permutations containing a single head segment and six tail segments for assessing the shift in head group location.

Structure permutations containing a single head segment and six tail segments for assessing the shift in head group location.

Performance effects on the scaled interfacial tension of (a) the bulk surfactant concentration in the aqueous phase (ρ* = ρ_{surfactant}σ^{3}) and (b) the number of surfactant molecules per unit area in the interface (*N** = *N* _{surfactant}σ^{2}/*A*) for the *H* _{1} *T* _{6} (blur), *T* _{1} *H* _{1} *T* _{5} (green), *T* _{2} *H* _{1} *T* _{4} (red), and *T* _{3} *H* _{1} *T* _{3} (black) structures depicted in Figure 9.

Performance effects on the scaled interfacial tension of (a) the bulk surfactant concentration in the aqueous phase (ρ* = ρ_{surfactant}σ^{3}) and (b) the number of surfactant molecules per unit area in the interface (*N** = *N* _{surfactant}σ^{2}/*A*) for the *H* _{1} *T* _{6} (blur), *T* _{1} *H* _{1} *T* _{5} (green), *T* _{2} *H* _{1} *T* _{4} (red), and *T* _{3} *H* _{1} *T* _{3} (black) structures depicted in Figure 9.

Effect of (a) the total bulk aqueous phase surfactant concentration and (b) the total number of surfactant molecules in the interface on the scaled interfacial tension. Curves are *i*SAFT predictions for mixtures of *H* _{1} *T* _{3} and *H* _{1} *T* _{7} (blue) with the labeled bulk mole fractions and the reference curves for the *H* _{1} *T* _{3} (red), *H* _{1} *T* _{5} (green), and *H* _{1} *T* _{7} (black). The scaling factor is the interfacial tension of the bare oil-water interface.

Effect of (a) the total bulk aqueous phase surfactant concentration and (b) the total number of surfactant molecules in the interface on the scaled interfacial tension. Curves are *i*SAFT predictions for mixtures of *H* _{1} *T* _{3} and *H* _{1} *T* _{7} (blue) with the labeled bulk mole fractions and the reference curves for the *H* _{1} *T* _{3} (red), *H* _{1} *T* _{5} (green), and *H* _{1} *T* _{7} (black). The scaling factor is the interfacial tension of the bare oil-water interface.

Equilibrium density profiles for an *H* _{1} *T* _{3} + *H* _{1} *T* _{7} mixed surfactant system from *i*SAFT for oil (black), water (blue), tail (green), head (red), and total (purple) segment densities. Dotted lines are *H* _{1} *T* _{3}, and dashed lines are *H* _{1} *T* _{7}. Fixed bulk mole fractions are ((a) and (b)), ((c) and (d)), ((a) and (c)), and ((b) and (d)).

Equilibrium density profiles for an *H* _{1} *T* _{3} + *H* _{1} *T* _{7} mixed surfactant system from *i*SAFT for oil (black), water (blue), tail (green), head (red), and total (purple) segment densities. Dotted lines are *H* _{1} *T* _{3}, and dashed lines are *H* _{1} *T* _{7}. Fixed bulk mole fractions are ((a) and (b)), ((c) and (d)), ((a) and (c)), and ((b) and (d)).

Magnified equilibrium density profiles from *i*SAFT for mixed surfactant system focusing on the head and tail segment profiles. See Figure 12 caption for description.

Magnified equilibrium density profiles from *i*SAFT for mixed surfactant system focusing on the head and tail segment profiles. See Figure 12 caption for description.

Effects on the *i*SAFT predicted scaled interfacial tension for the dimensionless temperature (*T** = *k* _{ b } *T*/ε) and surfactant concentration in the bulk aqueous phase for an *H* _{1} *T* _{5} surfactant. Plots show curves for (a) isotherms and (b) constant bulk compositions.

Effects on the *i*SAFT predicted scaled interfacial tension for the dimensionless temperature (*T** = *k* _{ b } *T*/ε) and surfactant concentration in the bulk aqueous phase for an *H* _{1} *T* _{5} surfactant. Plots show curves for (a) isotherms and (b) constant bulk compositions.

Effect of dimensionless temperature (*T**) and bulk oil phase mole fraction of surfactant on the *i*SAFT equilibrium density profiles for an *H* _{1} *T* _{5} surfactant. Curves are oil (black), water (blue), tail (green), head (red), and total (purple) segment density profiles at 1/*T** = 0.75 ((a), (b), and (c)), 1.00 ((d), (e), and (f)), and 1.25 ((g), (h), and (i), with ((a), (d), and (g)), 0.10 ((b), (e), and (h)), and 0.20 ((c), (f), and (i)).

Effect of dimensionless temperature (*T**) and bulk oil phase mole fraction of surfactant on the *i*SAFT equilibrium density profiles for an *H* _{1} *T* _{5} surfactant. Curves are oil (black), water (blue), tail (green), head (red), and total (purple) segment density profiles at 1/*T** = 0.75 ((a), (b), and (c)), 1.00 ((d), (e), and (f)), and 1.25 ((g), (h), and (i), with ((a), (d), and (g)), 0.10 ((b), (e), and (h)), and 0.20 ((c), (f), and (i)).

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