Description of Micellar Radii for Phase Behavior and Viscosity

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Description of Micellar Radii for Phase Behavior and Viscosity Modeling of Aqueous Surfactant Solutions and Microemulsions V. A. Torrealba,*,† H. Hoteit,† and R. T. Johns‡ †

King Abdullah University of Science and Technology, 3216 Building 5, Thuwal 23955, Saudi Arabia The Pennsylvania State University, 119 Hosler Building, University Park, Pennsylvania 16801, United States

‡

Langmuir Downloaded from pubs.acs.org by YORK UNIV on 12/12/18. For personal use only.

S Supporting Information *

ABSTRACT: The study of surfactant solutions and microemulsions is relevant for many industrial applications that include oil recovery, environmental remediation, and detergency. Understanding the coupled nature of microstructure, phase behavior, and rheology can help in the design of these applications. The microstructure of surfactant solutions and microemulsions determines their equilibrium properties, including phase composition and viscosity. Modeling microemulsion phase behavior by explicitly defining the shape and size of micelles allows for the consistent estimation of phase viscosity. We define a coupled microstructure−phase behavior−viscosity model by considering a twodimensional lattice, where micelles are assumed to be elliptical. We also define equivalent three-dimensional models, where micelles are assumed to be oblate and prolate spheroids. The compositional dependence of the micellar radii is defined such that the oil-internal−bicontinuous−water-internal structural transitions are consistent with experimental observations. The model offers physical insights by modeling the transition between oil-internal−bicontinuous−water-internal regions in ternary diagrams as critical events, which are then tied to the unusual behavior observed in viscosity and scattering.

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viscosity increases with increasing temperature because of structural transformations from globular micelles into rodlike micelles and then into networks;1 surfactant solutions also exhibit the viscosity double peak with increasing salinity.2,3

INTRODUCTION The study of surfactants in solution is of great interest for many industries for their role in detergents,1−3 well-stimulation and fracturing fluids,4−6 enhanced oil recovery,7,8 environmental remediation fluids,9 and drug delivery.10−12 For most of these applications, the ability to model and predict the solubilization capacity, rheological behavior, and micellar aggregation size and shape can be helpful in the design and optimization for the target process.

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MODELING APPROACHES Einstein29,30 proposed one of the earliest models for estimating the viscosity of emulsions in the limit of dilute hard spheres. Matsumoto and Sherman31 accounted for the increase in particle size distribution in microemulsions to describe the viscosity increase with an increase in dispersed-phase concentration. De Kruif et al.32 proposed a suspension viscosity model in the low-shear limit that consisted of a cubic polynomial relationship of the dispersed-phase volume fraction. This model was later improved by accounting for the temperature dependence in the polynomial coefficients.28 More recently, coupled phase behavior and viscosity modeling has been proposed using the net and average curvature framework33 with the hydrophilic−lipophilic deviation (H) concept34 under the constant geometry assumption of cylinders with hemispherical end caps35 and spheres.36 Until now, no attempt has been made to model the doublepeak viscosity behavior of surfactant solutions and microemulsions by explicitly capturing the change in micellar

EXPERIMENTAL OBSERVATIONS

The link among microstructure, microemulsion composition, and viscosity has been experimentally corroborated by various researchers. A change in the overall composition by decreasing the water−oil ratio for an amphiphilic oil−water system led to electrical, optical, NMR, and viscosity observations that were in agreement with the following structural transition mechanism: water-internal spheres, then waterinternal cylinders, then repeating lamellae of oil and water domains, then oil-internal cylinders, and oil-internal spheres.13,14 The size of the micellar domains can be determined using low-angle X-ray diffraction, light scattering, electron microscopy, and viscosity.15−21 Critical behavior, such as the divergence of correlation length, has been linked to sharp increases in viscosity for aqueous micellar solutions and microemulsion systems.22,23 An increase in salinity for a microemulsion system caused a double divergence of the correlation length,24 which was associated with consistent structural transitions as those for compositional changes13,14 and corresponded to the doublepeak behavior in viscosity measurements.25,26 The dependence of viscosity on correlation length near critical points has also been observed for changes in temperature.27,28 For surfactant solutions, © XXXX American Chemical Society

Received: August 21, 2018 Revised: November 14, 2018 Published: November 27, 2018 A

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thickness as L (see Figure 1), the volume of the interfacial layer (VL) is

geometry as a function of composition and surfactant affinity and giving importance to the divergence of correlation length corresponding to the viscosity peaks. In this paper, we present a lattice-based model for micelles. The model is used to define the correlation length, phase behavior, and viscosity in a coupled framework. Finally, the model is compared to scattering, phase behavior and viscosity data for both surfactant solutions and microemulsions.

VL = Niπ (raiL + rbiL + L2)d

where VL = Vsl + Vwl + Vol with Vil the volume of component i adsorbed on the interfacial layer. We assume that Vil = λiVsl for i = o, w with λo being the ratio of the lipophilic chain length to the surfactant length (L), and λo + λw = 1. The volume of the interfacial layer can be related to the surfactant volume as

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METHODOLOGY In this section, we develop the lattice elliptical model, which is used to capture the phase behavior, correlation length and viscosity experimental observations. Preliminaries. The solubilization ratio of component i is defined by σi ≡

Vim Vsm

VL = Vsl + Vwl + Vol = 2Vsl

σi =

(2)

where ν is the average micellar velocity and τ is the excess correlation time.37 Lattice Elliptical Model. Consider a two-dimensional lattice of thickness d. In this lattice, micelles are approximated using ellipses that can be defined by the radii rai and rbi, noting that rbi ≥ rai, as illustrated in Figure 1. For a component i-

(7)

where the superscript, bic, denotes bicontinuity. The corresponding equations for the three-dimensional (3D) micelles using prolate and oblate spheroid geometries are shown in the Supporting Information. For a ternary diagram, where only a two-phase region is present, any overall composition inside the two-phase region will separate into a microemulsion phase and a component j-rich excess phase, with j ≠ i and j = o, w. Here we assume that the excess phase is pure, so that Vj = Vjj, where Vj denotes the total volume of the phase. We consider that the two-phase region can be subdivided into three different structural domains: oil-internal, water-internal, and bicontinuous.13,14 The oil-internal regime occurs for a water-rich microemulsion, with σ°w ≥ σblw , where the ° superscript denotes that the water component is continuous in the microemulsion phase and the bl superscript denotes the limiting tie-line for the onset of bicontinuity. Similarly, the water-internal regime occurs for an oil-rich microemulsion with σo° ≥ σblo . Finally, the bicontinuous regime occurs for an intermediate composition microemulsion, where both σw° < σblw and σ°o < σblo hold. For emphasis, we set σ°i ≡ σbic i for i = o, w during bicontinuity. In the limit of σ°j → ∞ micelles are iinternal and we assume they approach a circular shape, with rbi → rai. As σj° → σblj we have rbi → ∞. On the basis of the assumption of pure excess phases, we consider both critical points to coincide with pure phase compositions. Therefore, as the overall composition approaches the limiting tie-line at the i,cp critical point (σi,cp = ∞. j ), we have rai → ∞, such that σi = σi By considering these limits, we found the following definitions for the radii to be appropriate

Figure 1. Oil-internal elliptical micelle.

internal microemulsion, the effective diameter of the micelle is given by (3)

The volume of component i in the microemulsion phase can be divided into the volume in the micellar core (Vic) and the volume in the interfacial layer (Vil). The volume in the micellar core is given by Vic = Niπrairbid

+ λi

We define a component to be continuous when rbi → ∞, and the microemulsion to be bicontinuous when rbi → ∞ holds for i = o, w simultaneously. Therefore, for a bicontinuous microemulsion, eq 7 simplifies to r σi bic = 2 ai + λi (8) L

E

ξi = 2 rairbi

ij L Vic + Vil L L2 yzz = 2jjj + + z j rbi Vsl rai rairbi zz{ k

−1

(1)

ξ + τE ν

(6)

where all surfactant in the microemulsion phase is assumed to be at the interfacial layer. Further, we assume λo = 1 for all cases considered. Then, the solubilization ratio of component i is expressed by

where Vim is the volume of component i in the microemulsion phase and s is the surfactant component. The correlation time from scattering (τ) can be expressed in terms of the correlation length or average diameter of a micelle (ξ) by τ=

(5)

rai =

(4)

rbi =

where Ni is the number of component i-internal micelles in the microemulsion phase. By considering the surfactant shell B

Ka L(σi j ,cp − λi) + 2 2(σj° − σji ,cp) Kb 2(σj° − σjbl)

+

L(σi j ,cp − λi) 2

(9) DOI: 10.1021/acs.langmuir.8b02828 Langmuir XXXX, XXX, XXX−XXX

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average diameter equation for i-internal micelles (eq 3) with eq 15 allows for an effective micellar diameter definition consistent with scattering data through eq 2 and viscosity data, as shown in what follows. Microemulsion Viscosity Model. The microemulsion viscosity must approach the excess phase viscosity when the microemulsion composition approaches the corresponding critical point (Cim → 1 with Cij ≡ ((Vij)/(Vj)) and i = o, w). A simple function that takes such limits into account is μm = Comμo + Cwmμw, with μj being the viscosity of phase j. Experimental data suggest that the viscosity of microemulsions and surfactant solutions reaches a local maximum at the onset of bicontinuity, which corresponds to ξ → ∞ in our lattice model. We then propose the following viscosity model

where Ka and Kb are adjustable parameters with dimensions of length to be determined from the phase behavior or scattering data. The solubilization of component i in a bicontinuous microemulsion is then obtained by substituting eqs 9 into 8 σi bic =

Ka L(σjbic

− σji ,cp)

+ σi j ,cp (10)

Equations 7−10 allow for modeling of two-phase regions (i.e., H ≤ HL for type I microemulsions and H ≥ HU for type II microemulsions). However, for HL < H < HU, we need to define the dependence of the invariant point (i.e., the threephase microemulsion composition) on H. At H = HL, the invariant point coincides with the pure-water critical point, so 3 w,cp that σ3w = σw,cp w = ∞ and σo = σo , where superscript 3 denotes the three-phase microemulsion. Analogously, at H = HU, the invariant point coincides with the pure-oil critical point, so that σ3o = σo,cp = ∞ and σ3w = σo,cp o w . Finally, when the three-phase microemulsion is in the bicontinuous regime, it must follow eq 10, which says (σo3 − σow,cp)(σw3 − σwo,cp) =

Ka L

2

μm = Comμo + Cwmμw + β(1 − e−γξ )

where β and γ are adjustable parameters. The model can be defined using a single β parameter, but we can define β as a function of microemulsion composition by considering the following conditions l 0, o o o o o o o βwbl , o o o o o o * β (X ) = m β , o o o o o o βobl , o o o o o o o 0, n

(11)

Considering the H limits and the bicontinuity condition (eq 11), we propose the following H dependence for the invariant point σo3 =

Ka L

σw3 =

Ka L

ij HU − H yz w,cp jj z jj H − H zzz + σo L{ k ij H − HL yz o,cp jj z jj H − H zzz + σw U k {

(12)

(13)

σobl = σobl,max (14)

where α is an adjustable parameter defined using phase behavior data such as salinity scans, and σbl,min ≡ σj,cp i i . Bicontinuous Diameter Model. In the bicontinuous regime, we define the effective micellar diameter as ξobic + ξwbic 2

(15)

where ξi bic = 2 rairbibic and rbibic =

Kb 2(σjbl − σjbic)

+

L(σi j ,cp − λi) 2

if X = 0 if X = Xobl if X ≥ X max

(18)

RESULTS AND DISCUSSION In this section, we demonstrate the ability of the proposed elliptical lattice model to capture the phase behavior, viscosity, and correlation time from dynamic light scattering. First, we consider a case of aqueous surfactant solutions at different salinities, where both correlation time and viscosity data are available. Then, we consider a case where the phase behavior and viscosity of microemulsions are available for various overall compositions at a fixed salinity. Finally, we consider two sets of salinity scans for a pure and a commercial surfactant formulation at a constant overall composition, with each set consisting of three different linear alkanes. Model parameters for all cases studied are in Tables 1−2. Modeling Correlation Time and Viscosity for Aqueous Surfactant Solutions as a Function of Salinity. Gaudino et al.38 studied the correlation time and viscosity of aqueous surfactant solutions as a function of salinity (see Figure 2). They considered the cetylpyridiniumchloride (CPyCl) surfactant and NaCl brine solutions. The figure shows both correlation time and viscosity correlated with their response to salinity. In the low-salinity regime (H < 0), the properties increased until they reached a local maximum value. After this, the properties decreased until they reached a local minimum close to H = 0. The behavior is qualitatively

whereas for H ≤ 0, we have

ξb =

if X = X wbl

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σobl = σobl,min + (σobl,max − σobl,min)e−αH

σwbl = σwbl,min + (σwbl,max − σwbl,min)e αH

if X ≤ −X max

where X ≡ Com − Cwm. This compositional definition for β allows for giving importance to the viscosity at both peaks and at optimum. For the composition range X ∈ [−Xmax, Xmax], we interpolate with a quadratic equation of the form β(X) = c1X2 + c2, where the constants, c1 and c2, are determined by the boundary values based on eq 18. Equation 17 can also be defined using a hypothetical phase behavior parameter to capture the viscosity of aqueous surfactant solutions.

We also consider the composition range of the bicontinuous region to be dependent on H. Near optimum (H = 0), the composition range for bicontinuity is maximum, whereas as | H|→ ∞, the composition range for bicontinuity is minimum. For H > 0, we have

σwbl = σwbl,max

(17)

(16)

Note that for the phase behavior model rbi = ∞; eq 16 defines a hypothetical radius so that we can define ξbic i to be finite in the bicontinuous region away from the σblj tie-lines. The C

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Hoffmann3 reviewed the origin of these viscosity peaks for viscoelastic aqueous surfactant solutions as a function of salt concentration. The complex behavior is a result of the various mechanisms for stress relaxation present in the networks of cylindrical micellar solutions. To the best of our knowledge, this is the first attempt to continuously model both viscosity and correlation time for such systems. Modeling Phase Behavior and Viscosity as a Function of the Overall Composition. Labrid39 studied the phase behavior and viscosity of microemulsions as a function of the overall composition (see Figure 3). The formulation consisted

Table 1. Phase Behavior Parameters for All Considered C ases Kb

case 1 2 3 4 5 6 7 8

HU 2.0 0.5 1 0.6 0.9 0.9 0.8 0.7

10Ka 5Ka Ka Ka Ka 0.5Ka Ka 0.5Ka

σj,cp i

σbl,max o

σbl,max w

0.1 0 1 1 1 1 1 1

2 × 10 4 24 10 7 41 32 30

3

α

1 × 10 13 9 11 7 14 31 31

3

5 1 5 5 1 5 5 5

Table 2. Viscosity Parameters for All Considered Cases γ

case 1 2 3 4 5 6 7 8

1 1 3 2 3 1 6 4

× × × × × × × ×

βblw −5

10 10−2 10−5 10−5 10−5 10−5 10−6 10−6

7 × 10 16 7 8 4 7 4 5

β* 4

βblo

3 × 10 5 5 2 2 10 6 4

3

4 × 10 5 4 3 3 9 3 2

Xmax 3

0.9 0.8 1 1 1 1 1 1

symmetric for the high-salinity regime (H > 0). Although our model relies on the definition of phase behavior for surfactant−oil−brine systems to define the correlation time and viscosity of microemulsion solutions, we define effective phase behavior parameters to estimate the corresponding properties for these aqueous surfactant solutions. Most notably, we want to show that the model’s definition of viscosity as a function of effective diameter is consistent with experimental observations, where both viscosity and scattering data are available. This system displays a very challenging behavior, where viscosity varies by about 5 orders of magnitude over the sampled salinity range; this behavior was adequately captured by the proposed model. In this case, we define effective phase behavior and viscosity for aqueous surfactant solutions by considering a high-salinity pseudocomponent to replace the oil component.

Figure 3. Case 2: microemulsion phase behavior and viscosity as a function of the overall composition. Experimental data from Labrid.39

of the commercial surfactant, TRS 10-80, from Witco Chemical Company, n-butanol, and nC10−nC14 paraffinic oil, and NaCl brine solutions. As the overall composition changes, the corresponding microemulsion composition shifts from water-rich (i.e., Xm < 0) to oil-rich (i.e., Xm > 0). The behavior shows a similar double-peak viscosity behavior in composition, which is modeled with each peak corresponding to transitions from oil-internal to bicontinuous micelles (in the case of Xm
0). For this case, the model is successful in

capturing the coupled behavior of both phase behavior and viscosity. E

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Figure 5. Phase behavior and viscosity data for the commercial surfactant system containing TRS 10-80/tAA and various linear alkanes as a function of brine salinity (rescaled through H), and the corresponding results using the new phase behavior−viscosity model. Experimental data from Bennett et al.40

Modeling Phase Behavior and Viscosity as a Function of Salinity. Bennett et al.40 studied the phase behavior and

viscosity of microemulsions as a function of salinity (see Figures 4 and 5). They considered both a pure (Figure 4) and F

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(3) Hoffmann, H. Structure and Flow in Surfactant Solutions; American Chemical Society, 1994; Chapter 1, pp 2−31. (4) Nasr-El-Din, H. A.; Samuel, M. M. Lessons Learned From Using Viscoelastic Surfactants in Well Stimulation. SPE Prod. Oper. 2007, 22, 112−120. (5) Huang, T.; Crews, J. Nanotechnology Applications in Viscoelastic Surfactant Stimulation Fluids. SPE Prod. Oper. 2008, 23, 512−517. (6) Li, L.; Nasr-El-Din, H. A.; Cawiezel, K. E. Rheological Properties of a New Class of Viscoelastic Surfactant. SPE Prod. Oper. 2010, 25, 355−366. (7) Huh, C. Interfacial tensions and solubilizing ability of a microemulsion phase that coexists with oil and brine. J. Colloid Interface Sci. 1979, 71, 408−426. (8) Miller, C. A.; Hwan, R. N.; Benton, W. J.; Fort, T. Ultralow interfacial tensions and their relation to phase separation in micellar solutions. J. Colloid Interface Sci. 1977, 61, 554−568. (9) Acosta, E. J.; Quraishi, S. Oil Spill Remediation; John Wiley & Sons, Inc: Hoboken, 2014; pp 317−358. (10) Sarciaux, J. M.; Acar, L.; Sado, P. A. Using microemulsion formulations for oral drug delivery of therapeutic peptides. Int. J. Pharm. 1995, 120, 127−136. (11) Malcolmson, C.; Lawrence, M. J. Three-component non-ionic oil-in-water microemulsions using polyoxyethylene ether surfactants. Colloids Surf., B 1995, 4, 97−109. (12) Lawrence, M. J. Microemulsions as drug delivery vehicles. Curr. Opin. Colloid Interface Sci. 1996, 1, 826−832. (13) Shah, D. O.; Tamjeedi, A.; Falco, J. W.; Walker, R. D. Interfacial instability and spontaneous formation of microemulsions. AIChE J. 1972, 18, 1116−1120. (14) Falco, J. W.; Walker, R. D.; Shah, D. O. Effect of phase-volume ratio and phase-inversion on viscosity of microemulsions and liquid crystals. AIChE J. 1974, 20, 510−514. (15) Schulman, J. H.; Matalon, R.; Cohen, M. X-ray and optical properties of spherical and cylindrical aggregates in long chain hydrocarbon polyethylene oxide systems. Discuss. Faraday Soc. 1951, 11, 117. (16) Schulman, J. H.; Riley, D. P. X-ray investigation of the structure of transparent oil-water disperse systems. I. J. Colloid Sci. 1948, 3, 383−405. (17) Schulman, J.; Friend, J. Light scattering investigation of the structure of transparent oil-water disperse systems. II. J. Colloid Sci. 1949, 4, 497−509. (18) Schulman, J. H.; Stoeckenius, W.; Prince, L. M. Mechanism of Formation and Structure of Micro Emulsions by Electron Microscopy. J. Phys. Chem. 1959, 63, 1677−1680. (19) Cooke, C. E., Schulman, J. H. Surface Chemistry; Academic Press; 1965, pp 231−251. (20) Shinoda, K.; Friberg, S. Microemulsions: Colloidal Aspects. Adv. Colloid Interface Sci. 1975, 4, 281−300. (21) Shah, D.; Walker, R.; Hsieh, W.; Shah, N.; Dwivedi, S.; Nelander, J.; Pepinsky, R.; Deamer, D. In Some Structural Aspects of Microemulsions and Co-Solubilized Systems, SPE Improved Oil Recovery Symposium, 1976. (22) Degiorgio, V.; Piazza, R.; Corti, M.; Minero, C. Critical properties of nonionic micellar solutions. J. Chem. Phys. 1985, 82, 1025. (23) Cazabat, A. M.; Langevin, D.; Sorba, O. Anomalous viscosity of microemulsions near a critical point. J. Phys., Lett. 1982, 43, 505−511. (24) Cazabat, A.; Langevin, D.; Meunier, J.; Pouchelon, A. Critical behaviour in microemulsions. J. Phys., Lett. 1982, 43, 89−95. (25) Kaler, E. W.; Davis, H. T.; Scriven, L. E. Toward understanding microemulsion microstructure II. J. Chem. Phys. 1983, 79, 5685− 5692. (26) Kaler, E. W.; Bennett, K. E.; Davis, H. T.; Scriven, L. E. Toward understanding microemulsion microstructure: A small-angle x-ray scattering study. J. Chem. Phys. 1983, 79, 5673−5684. (27) Sengers, J. Transport Properties of Fluids Near Critical Points. Int. J. Thermophys. 1985, 6, 203−232.

a commercial (Figure 5) surfactant formulation. The pure surfactant was sodium 4-(1-heptyl-nonyl)benzenesulfonate (SHBS), which is also known as SPHS or Texas No. 1. The commercial surfactant was TRS 10-80 from Witco Chemical Company. The cosolvents used were tertiary amyl alcohol (tAA) and isobutyl alcohol (iBA). For each formulation, the following linear n-alkane oils were used: nC8, nC10, and nC14. Brine solutions were made with NaCl. The viscosity behavior for both formulations as a function of salinity is consistent with those obtained in the case of viscoelastic surfactant solutions and for a composition scan. The mechanisms in this case are likely to be similar to those observed in the composition scan, where, as the salinity increases, the microemulsion transitions from a Winsor type I (i.e., water-rich microemulsion) to a Winsor type II (i.e., oil-rich microemulsion) passing through a Winsor type III. For all six cases, the model accurately captures both phase behavior and viscosity.

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SUMMARY AND CONCLUSIONS We presented a new model to simultaneously capture the phase behavior, viscosity, and scattering observations for surfactant solutions and microemulsions. The model defines micellar radii that determine whether the phase is waterinternal, bicontinuous, or oil-internal. These radii are assumed to be a function of both overall composition and surfactant affinity, and the resulting phase behavior and viscosity modeling results are consistent with experimental observations. Furthermore, the dependence of viscosity on the effective micellar diameter is corroborated with scattering observations for aqueous surfactant solutions. The impact of this approach is to allow for an integrated property description of microemulsion and aqueous surfactant solutions, which can be helpful in the design and optimization of industrial-scale processes.

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ASSOCIATED CONTENT

* Supporting Information S

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.langmuir.8b02828. Derivation of the 3D prolate and oblate spheroid models (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

V. A. Torrealba: 0000-0002-3361-4469 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors thank Xavier Pita, scientific illustrator at King Abdullah University of Science and Technology (KAUST), for producing the Table of Contents Graphic.

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REFERENCES

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H

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