Dimensionless Equation of State to Predict Microemulsion Phase

Aug 9, 2016 - The invariant point marks the beginning of the “fish-tail” and the type IV region. .... The type III region can exist only when χ >...
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Dimensionless Equation of State to Predict Microemulsion Phase Behavior Soumyadeep Ghosh and Russell T. Johns* Department of Energy and Mineral Engineering, College of Earth and Mineral Sciences, The Pennsylvania State University, University Park, Pennsylvania 16802, United States ABSTRACT: Prediction of microemulsion phase behavior for changing state variables is critical to formulation design of surfactant−oil−brine (SOB) systems. SOB systems find applications in various chemical and petroleum processes, including enhanced oil recovery. A dimensional equation-of-state (EoS) was recently presented by Ghosh and Johns1 that relied on estimation of the surfactant tail length and surface area. We give an algorithm for flash calculations for estimation of three-phase Winsor regions that is more robust, simpler, and noniterative by making the equations dimensionless so that estimates of tail length and surface area are no longer needed. We predict phase behavior as a function temperature, pressure, volume, salinity, oil type, oil−water ratio, and surfactant/alcohol concentration. The dimensionless EoS is based on coupling the HLD-NAC (Hydrophilic Lipophilic Difference−Net Average Curvature) equations with new relationships between optimum salinity and solubility. An updated HLD expression that includes pressure is also used to complete the state description. A significant advantage of the dimensionless form of the EoS over the dimensional version is that salinity scans are tuned based only on one parameter, the interfacial volume ratio. Further, stability conditions are developed in a simplified way to predict whether an overall compositions lies within the single, two-, or threephase regions. Important new microemulsion relationships are also found, the most important of which is that optimum solubilization ratio is equal to the harmonic mean of the oil and water solubilization ratios in the type III region. Thus, only one experimental measurement is needed in the three-phase zone to estimate the optimum solubilization ratio, a result which can aid experimental design and improve estimates of optimum from noisy data. Predictions with changing state variables are illustrated by comparison to experimental data using standard diagrams including a new type of dimensionless fish plot. The results show that the optimum solubilization ratio and salinity using the tuned dimensionless EoS are within average errors of 2.44% and 1.17% of experimental values for the fluids examined. We then use the dimensionless equations and thermodynamic firstprinciples to derive the constant in Huh’s equation for interfacial tension prediction.

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INTRODUCTION AND BACKGROUND Microemulsions are homogeneous solutions consisting of surfactant, oil, and brine (SOB). The term was first introduced by Hoar and Schulman who found that, by titration of a milky emulsion containing potassium oleate (a soap) with a mediumchain alcohol (pentanol or hexanol), a stable oil-in-water emulsion was produced.2 Winsor later described microemulsions as swollen micellar solutions.3 Broadly, micelles (surfactant aggregates in aqueous solution), macroemulsions (or simple emulsions), and microemulsions (also known as micellar solutions) can be categorized by the dimensions of the dispersed phase. Microemulsions can have low interfacial tensions that result in negative Gibbs free energy of emulsion formation, which implies microemulsions are formed spontaneously and are thermodynamically stable.4 The affinity of a surfactant toward the oil or water phase is a function of the surfactant type, composition, and state of the system. Winsor introduced the concept of the R-ratio to explain the balance between the surfactant’s oil and water affinity.5 Three distinct regions near the interface are defined: an aqueous region (W), a nonpolar oleic region (O), and an © 2016 American Chemical Society

amphiphilic or bridging region (C). The interfacial zone (C) is of a fixed thickness separating the oil and water bulk phases.6 The R-ratio is the ratio of molecular interaction energies on the oil to the water side of the C-layer. For an R-ratio equal to unity, the surfactant has equal affinity toward the oil and water regions. This occurs at optimum conditions, where the surfactant solubilizes oil and water equally. However, the use of R-ratio is not practical, as interaction energies at the molecular level are not known. Several attempts have been made to predict optimum conditions, which are of special interest to chemical enhanced oil recovery applications. Salager and coauthors presented an empirical equation valid at optimum conditions for oil−waterbrine systems in the presence of an anionic surfactant.7 That relationship was extended to include the effect of pressure, Received: July 18, 2016 Published: August 9, 2016 8969

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The HLD-NAC model has been extended to allow for multiple surfactants, and also for the interaction of alkali with acidic crudes to form soaps.14 The average curvature (Ha) can be expressed by the average radii of oil (Ro) and water (Rw) micelles in the microemulsion phase and the characteristic or correlation length (ξ) as defined by DeGennes and Taupin.15 That is,

ln S* − K (EACN) − f (A) − α(T − Tref ) − β(P − Prref ) (1)

+ Cc = 0

where S* is the optimum salinity expressed as grams per 100 mL, EACN is the equivalent alkane carbon number of the oil, f(A) is a function of alcohol type and amount, and Cc is a surfactant-dependent parameter called the characteristic curvature.8 The parameter K is the slope obtained from the observed linear relationship between the logarithm of optimum salinity and the EACN. Similarly, constant α (or β) is derived from the observed linear dependence of 1n S* with temperature (or pressure) when all other formulation variables are fixed.1,8 Equation 1, which is valid for microemulsion systems at optimum conditions, is more useful when related to surfactant affinity difference (SAD), where SAD is the difference between the standard chemical potential of the surfactant in the water and oil phases. The value of SAD relates directly to the affinity of the surfactant between the water and oil phases.9,10 The dimensionless SAD (made so by dividing by RT) is defined as the hydrophilic−lipophilic difference or HLD. HLD is therefore a dimensionless state function that can be used in place of the R-ratio to quantify phase behavior changes as formulation variables change.11,12 The expression for HLD becomes

Ha =

Hn =

(2)

where eq 1 is obtained when HLD is zero at optimum. The values for the parameters in eq 2 can be experimentally determined and used to predict optimum salinity, but not optimum solubilization ratio. Thus, it is not a complete EoS. Acosta and coauthors used the net and average curvature (NAC) with the concept of HLD to fit microemulsion phase behavior from salinity scans.13 Their model uses three pseudocomponents oil (o), water (w), and surfactant (s) and assumes no change in total volume upon mixing. The model also assumes that solubilized micelles are spherical, which is likely true only away from the boundary between type III and type II−/II+ regions. However, micelles are considered to be spherical in all Winsor type regions as a first approximation. HLD-NAC has been shown to fit salinity scans well, but by itself is not an EoS because it cannot predict changes in phase behavior outside of the fitted data. As discussed by Acosta, a separate relationship is also needed for the correlation length parameter, as we discuss below. Still, the physically based HLDNAC model is a significant improvement over the widely used Hand’s model. The micelle radii containing component i (oil or water) in the HLD-NAC model can be obtained from the volume of component i solubilized (Vi,m) and the interfacial area occupied by the surfactant molecules at the interface (As). The parameter As is dependent on the moles of surfactant in the system (ns) and surface area per molecule of surfactant (as). For a mixture of surfactants, the term As is a summation of the area contributed by every surfactant molecule (Avogadro’s number Na times nsas) so that Ri =

3Vi ,m As

=

1 1 HLD − =− Ro Rw L

(5)

At optimum, the oil and water curvatures (hence radii) are equal to each other. This is consistent with the fact that oil and water solubilization ratios are equal to each other at optimum. Equation 5 further satisfies the condition that HLD is zero at optimum (see eq 2). In type III microemulsions, the correlation length is assumed to always be equal to a critical value ξ*. The critical correlation length is a function of the optimum solubilization ratio σ*. Ghosh and Johns derived a new empirical relationship between optimum salinity and solubilization ratio for salinity scans where other parameters such as pressure and temperature are held constant.1 1 = B1 ln S* + B2 σ*

(6)

The dimensionless constants B1 and B2 are determined by tuning to available experimental data based on standard salinity scans in glass pipets. Equation 6 is a key equation needed to complete the phase behavior description, as without its use the solubility at optimum would be constant as optimum salinity changes, which is inconsistent with experimental data. The relationships in eqs 2 and 6 are coupled equations that allow prediction of microemulsion phase behavior as a function of changing state variables. Equation 6 allows for estimation of solubilization ratios for water and oil, and hence the invariant point composition of the microemulsion phase, once the optimum salinity is known from eq 2. The Huh18 correlation is then used to predict interfacial tensions between the excess phase(s) and the microemulsion phase. This paper first develops a dimensionless equation-of-state from the dimensional EoS presented in by correctly accounting for the surfactant component volume in the EoS of Ghosh and Johns.1 New important relations for identifying how many

3Vi ,m Nansas

(4)

where the correlation length ξ is equivalent to the average micelle radius in simple microemulsions with spherical micelles.16 As a consequence, the average curvature describes the amount of oil and water solubilized in micelles in the microemulsion phase. The curvature of the surfactant interface is described by the net curvature (Hn). The net curvature is related to the radii of oil (Ro) and water (Rw) micelles, the surfactant length parameter (L), and to HLD.13,17 The surfactant length parameter resembles the thickness of the interface and is generally 1.2 times lc, the surfactant hydrophobe tail length. The net curvature was found to be proportional to the HLD of the system with 1/L as the proportionality constant. Therefore, the parameter HLD translates the changes in formulation variables into the NAC model as,

HLD = ln S − K (EACN) − f (A) − α(T − Tref ) − β(P − Pref ) + Cc

1⎛ 1 1 ⎞ 1 ⎜⎜ ⎟⎟ = + 2 ⎝ Ro Rw ⎠ ξ

(3) 8970

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Figure 1. Linear relationships of inverse of solubilization ratios as a function of HLD. Data from a single salinity scan experiment reported by Sheng.22 eqs 3, and using the definitions for solubilization ratios. After rearrangement, eqs 4 and 5 become

phases form at equilibrium are given next, and the tuning process is described for the new EoS, based on numerous experimental data. We demonstrate the predictive capability of the dimensionless EoS with experimental data and also show the importance of a new parameter called the I-ratio.

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6Vs 1 1 1 + = = σo σw A sξ ξD

(8)

1 1 − = − 3I HLD σo σw

(9)

and

EQUATIONS AND METHODOLOGY

The EoS assumes there are three pseudocomponents, oil, water, and an anionic surfactant. The surfactant pseudocomponent must be fixed, but can contain multiple surfactants including alcohol. The water component contains the dissolved salts. Ideal mixing is assumed and the surfactant is assumed present only in the microemulsion phase. Thus, the excess phases contain pure oil or water component. Equations 1−6 describe the complete set of dimensional equations used in this paper. The correlation length is updated from the EoS described by Ghosh and Johns1 to include the surfactant volume in the microemulsion phase by substitution of eq 3 into eq 4, and by using ΣCim = 1. That is,

ξ=

6VmeCwmCom 6Vsσwσo = A s(1 − Csm) A s(σw + σo)

where a new dimensionless parameter is defined as the interfacial volume ratio I=

Vs LA s

(10)

and

ξD =

σwσo ξ = 6IL σw + σo

(11)

is the dimensionless correlation length. The parameter I is a dimensionless group that is the ratio of the volume of the surfactant component without micelles to the volume of the interface (the Clayer volume occupied by surfactant) within a micelle. The I-ratio is determined by tuning to experimental data, but once known it is constant in the EoS. Hence, there is a significant advantage of making the equations dimensionless as only the group of parameters in eq 10 are tuned, not individual values. That is, the product LAs is estimated in the tuning process since Vs is known. Although eq 10 would appear to depend significantly on the surfactant volume, the I-ratio is not a strong function of the overall composition or the surfactant volume. The ratio can be rewritten in terms of surfactant density (ρ), Avogadro’s number (Na), area per surfactant molecule (as), and surfactant molecular weight (MW) as I = MW/ρLNaas. Unless these parameters change significantly with overall

(7)

The component volume fractions Cim in eq 7 are the ratio of the volume of a component (Vi) solubilized in the microemulsion to the total volume of the microemulsion phase (Vme). The correlation length in prior research,13,16,19 was not a function of the surfactant content in the microemulsion likely because small overall surfactant concentrations were used in typical formulations. The use of a more accurate expression for the correlation length (eq 7) allows for larger surfactant concentrations and more accurate modeling of the type IV region (single-phase microemulsion). Dimensionless Equation-of-State (EoS). Equations 1, 2, and 6 are already dimensionless, but eqs 3−5 and 7 are not. Equations 4 and 5 are made dimensionless by elimination of the micelle radii through 8971

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Type II+ Microemulsions. For type II+ microemulsions, Vo,m is always equal to the total Vo in the system. Analogous to type II−, we have

composition, the I-ratio will be constant and tuning of data measured at one overall composition is likely sufficient. Equations 1, 2, 6, and 8−11 form the basis of the dimensionless EoS. These equations could be solved simultaneously for solubilizations in type III systems if the correlation length is known. We show an elegant way to eliminate the correlation length to solve for the microemulsion composition in type III. Equation 11 is a definition and is used only in that regard. I-Ratio and the Aggregation Number/Packing Factor. The functional form of the dimensionless group I appears similar to that of the packing factor (PF) or aggregation number defined by Israelachvili, Mitchell, and Ninham.20 The packing factor is the ratio of the component volume solubilized within the micellar core (VH) to the product aslc, where lc is the surfactant-tail length. Contrarily, the I-ratio as described in this paper is the ratio of the surfactant component volume (different from the volume in the micellar core) to the total interfacial volume AsL. Hence, the I-ratio as we describe in our paper, is a macroscopic parameter significantly different from the packing factor (or aggregate number). Further, the I-ratio is estimated from experimental data, not theoretically. The packing factor changes as a function of HLD parameters (like salinity), as shown by Israelachvili and coresearchers. This is primarily due to the change in the volume solubilized within the micellar core due to a shift in HLD, which is explained by the dimensionless equation of state presented in this paper. However, the surfactant component volume remains constant irrespective of the aqueous phase salinity. Additionally, the compressibility of the surfactant component (with respect to pressure and temperature) is negligible. Hence, the change of surfactant molar volume over the range of temperatures and pressures considered in this paper were largely ignored. Such an assertion is reflected by the results in this paper, where we report mean I-ratios with small standard deviations. Furthermore, the total volume of surfactants at the interface is LAs. A large I indicates the surfactants are well packed at the interface, where complete packing gives I = 1. At complete packing, the total volume of the surfactant is equal to its minimum value (the total volume of surfactant component Vs). Thus, 0 ≤ I ≤ 1 with typical values at atmospheric pressure in the range of 0.15−0.35 indicating less than complete packing. However, the volume occupied by the hydrophobe (VH) is dependent on the shape of the micelle (and not the surfactant component volume). Hence, when VH is small compared to aslc, (PF < 1) the system is likely to form spherical or cylindrical micelles. Similarly, lamellar structures form when PF ∼ 1 and inverted micelles form when PF > 1.21 Hence, unlike the I-ratio, the packing factor is dependent on micellar shape and can hold values greater than 1. Type II− Microemulsions. The EoS for two-phase regions is a subset of the type III equations. Type II− consists of an excess oil phase and a microemulsion phase where water is the continuous phase. This implies Vw,m in a type II− microemulsion is constant and equal to Vw of the system, since all of the water component must be in the microemulsion phase. Therefore, in type II− microemulsions,

1 1 = − 3I HLD + 0 σo σw

1 1 = 3I HLD + 0 σw σo

where is based on the overall oil composition. For a salinity scan, eq 14 reduces to ⎛S ⎞ 1 1 = 3I ln⎜ ⎟ + ⎝ S* ⎠ σw σ*

(15)

These results are similar to eqs 12 and 13, but now the inverse of water solubilization ratio is linearly related to the HLD of the system in the type II+ region. Figure 1b presents experimental data reported by Sheng22 showing the linear relationship. This data comes from the same overall composition as used in Figure 1, but the slope of the linear relationship is +3I (as opposed to −3I for type II−). Solubilization Ratio Relationships in Type III Microemulsions. Type III phase behavior consists of three phases, an excess oil phase, an excess brine phase, and the middle phase microemulsion. At optimum, σw = σo = σ* and HLD = 0. In type III microemulsions, the correlation length is assumed relatively constant at its optimum value ξ*, which is reasonable when oil and water form continuous planar micelles (spheres with large radii of curvatures). Experimental validation of this assumption using small angle neutron scattering (SANS) can be found in Strey.23 Consequently, from eq 11, ξD* = σ*/ 2 and eq 8 is simplified to

1 1 2 + = σo σw σ*

(16)

Equation 16 gives an important new relationship. The optimum solubilization ratio is equal to the harmonic average of the three-phase solubility ratios even when not at an optimum. Experimental data in the results section demonstrates the high degree of accuracy of eq 16, and its importance to experimental measurement and design. Using eq 16, the optimum solubility can be estimated from a single pipet anywhere in the three-phase region, aiding in further experimental design. Equation 16 also constrains the solubilization ratios in the type III microemulsion region and that relationship is used to eliminate the correlation length completely from the equations. Equation 16 can be substituted into eq 9 to eliminate one of the solubility ratios. Thus, when the water solubility ratio is eliminated we have in the type III region:

1 3I HLD 1 =− + σo 2 σ*

(17)

and when oil solubility ratio is eliminated,

1 3I HLD 1 = + σw 2 σ*

(18)

HLD varies depending on the parameters and state variables in eqs 17 and 18. For example, a salinity scan with all other formulation variables constant gives

(12)

where σ0w is constant and is based on a fixed overall composition (overall water volume). HLD includes the change of all formulation variables, but for a salinity scan, eq 12 reduces to

1 S 1 = − 3I ln + 0 σo S* σw

(14)

σ0o

1 3I ⎛ S ⎞ 1 = − ln⎜ ⎟ + 2 ⎝ S* ⎠ σo σ*

(19)

1 3I ⎛ S ⎞ 1 = ln⎜ ⎟ + σw σ* 2 ⎝ S* ⎠

(20)

and

(13)

Hence, an important new result of this paper is that the inverse of oil solubilization ratio in type II− microemulsions for a fixed overall composition varies linearly with HLD. Further, the slope of this line is always equal to −3I. Numerous experimental data confirm the relationship of eq 12. Figure 1a, for example, shows a nearly linear trend for experimental data reported by Sheng.22

Equations 19 and 20 present important new relationships not published previously. That is, there is a linear correlation between the inverse of solubilization ratios and logarithm of salinity in a typical salinity scan within the three-phase region. The inverse of the solubilization ratios will also vary linearly as a function of other HLD inputs such as EACN, T, and P. 8972

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Langmuir Further, the slope of each linear relationship in eq 19 and 20 for type III behavior is either 1.5I or −1.5I. This slope is half of that in the two-phase regions (3I), giving a discontinuity in the derivative of the solubilities at the transition between two and three phases. This discontinuity arises owing to the assumption of spherical micelles in all Winsor regions, which is only an approximation of the shape of the micelles especially near these transitions. Experimental data confirm these trends of optimum solubilization ratios in the two- and three-phase regions. For example, Figure 1c shows a linear relationship between solubilization ratios and HLD in the type III region using the same data reported by Sheng.22 Figure 2

type IV microemulsions, where in type IV microemulsions only a single phase exists (all oil and water components are solubilized in the microemulsion phase leaving no excess phases). Hence, from eq 23 at the invariant point (ΔHLD = 0),

2σo0σw0 − σ *σw0 − σ *σo0 = 0

(24)

Equation 24 expressed in terms of component volume fractions in the microemulsion becomes,

σ* = 2

CwmCom Csm(Cwm + Com)

(25)

Equation 25 represents the locus of the invariant point of the tie triangle in the compositional space (Com, Cwm, Csm). This is the first time a mathematical equation for the invariant point composition has been presented. We use the equation for the invariant point to define a new parameter χ, which determines the presence of a three-phase region using the overall compositions as, χ=2

combines the regions shown in Figure 1 in one plot for an example using WOR = 1, σ* = 13.5 cc/cc, and I = 0.129. The intersection of the linear trends from eqs 19 and 20 gives the optimum salinity, which is a more robust approach compared to the traditionally used Hand’s model fits. Prediction of Phase Number and Winsor Type. The dimensionless EoS can be used to predict if an overall composition is within the type II−, II+, III, or IV region prior to solving for solubility ratios. The HLD limits at phase transitions are used to define the Winsor regions. Both the two- and three-phase equations are valid at these transitions at the same HLD. At the lower HLD limit (HLDL) where type II− and III merge, eq 12 substituted into eq 17 gives 2⎛1 1 ⎞ ⎜ 0 − ⎟ σ* ⎠ 3I ⎝ σw

HLDL * = −

HLDU * = (21)

2⎛ 1 1⎞ ⎜ − 0⎟ 3I ⎝ σ * σo ⎠

(22)

2⎛ 2 1 1⎞ ⎜ − 0 − 0⎟ 3I ⎝ σ * σo σw ⎠

(27)

2 ⎛⎜ 1 ⎞⎟ 3I ⎝ σ * ⎠

(28)

Unlike HLDL and HLDU, HLDL* and HLDU* are independent of the overall composition. Consequently, HLDL* and HLDU* represent the true HLD limits within which a three-phase system may exist. Furthermore, the maximum possible width of the three-phase region (present for any overall compositions within the ternary diagram) obtained by subtracting eq 27 from eq 28 is

Thus, the width of the three-phase region in terms of HLD for a fixed overall composition given by σ0w and σ0o becomes, ΔHLD = HLDU − HLDL =

2 ⎛⎜ 1 ⎞⎟ 3I ⎝ σ * ⎠

Similarly, a critical upper HLD limit (HLDU*) exists above which a type III microemulsion cannot exist. HLDU* is obtained by setting the inverse of σ0w to zero. Hence. From eq 22,

Similarly, for the upper HLD limit (HLDU) at the type III and II+ transition,

HLDU =

(26)

where the overall composition Ci is equal to the component volume fractions in the microemulsion phase at the invariant point (the invariant point is a single-phase mixture). When χ > σ*, the overall composition must split into two or three-phases (type II−, type II+, or type III). The existence of type III is determined from the overall composition and the invariant point. When χ ≤ σ*, type IV microemulsions exist (single-phase only). HLDL, as defined in eq 21 marks the transition from type II− to type III microemulsion for a particular overall composition with σ0w. Thus, HLDL gives the point of transition between the two-phase lobe (type II−) and the tie triangle (type III region). However, a critical lower HLD limit (HLDL*) exists below which a type III microemulsion cannot exist. Because the excess phases are assumed pure, the critical lower HLD limit (HLDL*) is obtained by finding HLDL as the overall surfactant concentration approaches zero. Therefore, HLDL* is obtained when the inverse of σ0w is set to zero. Hence, from eq 21,

Figure 2. Illustration of the linear relationships between the inverse of solubilization ratio and HLD. Red represents inverse of the oil solubilization ratio. Blue is the inverse of the water solubilization ratio. (WOR = 1, σ* = 13.5 cc/cc, and I-ratio = 0.129).

HLDL =

CwCo Cs(Cw + Co)

ΔHLD* = HLDU * − HLDL * =

4 ⎛⎜ 1 ⎞⎟ 3I ⎝ σ * ⎠

(29)

Equation 29 demonstrates the empirical observation by Bourrel and Schechter6 that the width of the three-phase region is inversely proportional to the optimum solubilization ratio. Equation 29 is also a special case of eq 23 when water and oil solubilities become very large, which occurs at small surfactant concentration. A flash calculation in the three-phase region can be expressed functionally by

(23)

Equation 23 shows that a decrease in the interfacial volume ratio (I) increases the width of the three-phase region (ΔHLD). For ΔHLD > 0, type II− exists when HLD < HLDL, while type II+ is present when HLD > HLDU. Between these limits three-phases exist (type III). For ΔHLD < 0, type III cannot exist and the phase behavior is type II−, type II+, or type IV (single-phase microemulsion). The invariant point is the composition of the type III microemulsion phase, and is also a point of transition between type III and

[σo , σw ] = f (T , P , EACN, f (A), Cc, I )

(30)

Once the solubility ratios are known, the saturation can be determined based on a simple volume balance, 8973

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Table 1. Summary of Optima for Experiments Using SDS (Sodium Dodecyl Sulfate) and AAS (Mixture of an Alkyl-Aryl Sulfonate and Dodecyl Ethoxy Sulfate) Surfactant Reported by Aarra et al.24 surfactant

salt

T, °C

S* in meq/mL reported by Aarra et al.24

SDS

NaCl

20.00 35.00 50.00 20.00 35.00 50.00 20.00 35.00 50.00 20.00 35.00 50.00 20.00 50.00 90.00 20.00 50.00 90.00 20.00 50.00 90.00 20.00 50.00 90.00

1.55 1.57 1.69 1.09 1.17 1.33 0.97 0.95 0.97 1.29 1.29 1.39 0.358 0.444 0.564 0.240 0.327 0.491 0.070 0.077 0.086 0.092 0.102 0.117

KCl

CaCl2

MgCl2

AAS

NaCl

KCl

CaCl2

MgCl2

S* in meq/mL predicted from eqs 19 and 20

σ* in cc/cc reported by Aarra et al.24

1.53 1.56 1.68 1.09 1.16 1.32 0.93 0.90 0.97 1.26 1.27 1.38 0.355 0.428 0.560 0.242 0.323 0.481 0.070 0.077 0.086 0.092 0.102 0.117

6.20 6.00 5.80 6.00 5.80 5.70 4.70 4.80 4.80 5.00 5.00 5.00 8.10 7.00 5.00 8.30 7.30 5.30 12.10 9.30 7.10 9.60 8.60 5.70

NP

Ci =

% relative error in S*

% relative error in σ*

6.18 5.91 5.73 6.23 5.94 5.85 4.88 4.93 4.82 5.19 5.14 5.03 7.90 7.13 5.03 7.93 7.43 4.93 11.89 9.16 6.65 9.64 8.39 5.79 avg

1.07 0.77 0.56 0.15 0.76 0.76 3.78 4.71 0.15 2.26 2.03 0.82 0.79 3.64 0.77 0.54 1.17 2.06 0.53 0.13 0.11 0.43 0.03 0.03 1.17 ± 0.26

0.28 1.52 1.23 3.87 2.46 2.68 3.78 2.68 0.45 3.82 2.79 0.62 2.49 1.82 0.66 4.51 1.75 7.02 1.77 1.46 6.36 0.47 2.48 1.62 2.44 ± 0.36

Figure 3 uses eq 37 to demonstrate that smaller interfacial volume ratios (I) gives a wider three-phase region (expressed as

∑ CijSj j=1

σ* in cc/cc predicted from eq 16

(31)

where Ci is the overall component concentration (volume fraction), Cij is the phase component concentration (volume fraction), and Sj is the phase saturation. For example, for type II+ where the brine phase is pure water component, we have Co = CoMSM and Cw = 1 − SM + CwMSM, and Sw = 1 − SM. Type II− is similar. In the three-phase region, we have Co = CoMSM + 1 − Sw − Sm and Cw = Sw + CwMSM. These equations are trivial to solve for the saturations when the phase compositions and overall compositions are specified.

■

EXAMPLE RESULTS

Example data are given in this section to illustrate the tuning process with the new dimensionless EoS, and its predictive capability. The dimensionless EoS eliminates the need to tune or estimate as and L separately. Instead, the interfacial volume ratio (I) can now be used as the single tuning variable. I-Ratios can be determined from linear correlations as shown in Figure 1. In this section, we present additional results using the dimensionless EoS. Calculating Optimum Salinities and Optimum Solubilization Ratios. Salinity scans reported by Aarra et al.24 used NaCl, KCl, CaCl2 and MgCl2 salts. Two different surfactant systems were considered, one with sodium dodecyl sulfate (SDS) and the other with a blend of an alkyl aryl sulfonate and dodecyl ethoxy sulfonate (AAS). Table 1 shows a comparison of measured data to calculated optimum values using the dimensionless EoS. The relative error in the predicted values is excellent, with average relative errors of 1.17 ± 0.26% for S* and 2.44 ± 0.36% for σ*.

Figure 3. Width of the three-phase region as a function of the optimum solubilization ratio and the interfacial volume ratio (I).

ΔHLD*). The width of the three-phase zone also decreases with increasing σ*. For good oil recovery, high solubilizations and a wide three-phase region are desirable. Hence, EOR formulations should be designed to have low interfacial volume ratios (I) with high σ*. The EoS developed here can be used for that design. Dimensionless Fish Diagrams. Fish diagrams are traditionally represented at a water−oil ratio of one (fixed overall composition). The three-phase limits are then expressed as a function of the total surfactant content in the system. The dimensionless parameter χ in eq 28 considers compositional effects of both WOR and surfactant volume. Hence, the new parameter χ is a more useful variable to generate fish plots. Figure 4 shows an example of a fish plot with an interfacial volume ratio (I) of 0.2 and σ* equal to 10 cc/cc. The 8974

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Langmuir

figure shows that a wider three-phase region exists when the optimum solubilization ratio is low. Invariant Point Locus. Equation 28 gives the locus of compositions of the invariant point for changing HLD. Figure 6

Figure 4. Example of a modified fish diagram (interfacial volume ratio I = 0.2). Red shows the upper salinity limit and blue the lower salinity limit. Type III microemulsions exist when χ is larger than σ* and HLD is within the upper and lower critical limits HLDU* and HLDL*.

composition at which χ becomes equal to σ* is the invariant point of the three-phase region. The invariant point marks the beginning of the “fish-tail” and the type IV region. Therefore, three-phase systems cannot exist when χ is less than σ*, as explained earlier. The figure also shows the critical upper (HLDU*) and lower HLD limits (HLDL*). A type III microemulsion can only exist when conditions for both χ and HLD are satisfied. HLDU and HLDL curves intersect and change their relative position (HLDU becomes less than HLDL) when χ is less than σ*. Figure 5 shows modified fish plots for different water−oil ratios and optimum solubilization ratios. At each WOR, χ was varied by changing the surfactant content in the system. The

Figure 6. Locus of invariant type III microemulsion composition. Black line σ* = 1 cc/cc. Green line σ* = 3 cc/cc. Red line σ* = 10 cc/ cc. Blue line σ* = 30 cc/cc.

shows the locus for σ* equal to 3 cc/cc, 10 cc/cc, and 30 cc/cc. The height of the locus decreases with increasing σ*. This is consistent with the observed physics that solubilization ratio is inversely proportional to the volume of surfactant in the microemulsion. Impact of Temperature and Pressure on Phase Behavior. Pressure and temperature scans from Austad and Strand25 using the surfactant dodecyl-orthoxylene sulfonate are examined next. They used dead (n-heptane) and live synthetic oils (n-heptane saturated with methane) in a PVT cell, where salinity and overall composition of the system were kept constant. Pressure (50−200 bar) and temperature (55−120 °C) were varied to quantify their effect on solubilization and optimum conditions. As described previously, the inverse of solubilization ratios should vary linearly with HLD, and HLD varies linearly with temperature and pressure depending on the value of α and β, respectively, in eq 3. Therefore, for a pressure scan of a type III microemulsion, 3Iβ(P − Pref ) 1 1 =− + σo 2 σ*

(32)

and 3Iβ(P − Pref ) 1 1 = + σw 2 σ*

(33)

Hence, the slope of 1/σ vs pressure is given by the product Iβ. Similarly, for a temperature scan (type III), 3Iα(T − Tref ) 1 1 =− + σo 2 σ*

(34)

and,

Figure 5. Dimensionless fish diagrams at different values of optimum solubilization ratio and WOR. I was fixed at 0.2. The width of the three-phase region is constrained by σ* only. The width decreases as σ* increases.

3Iα(T − Tref ) 1 1 = + σw 2 σ* 8975

(35) DOI: 10.1021/acs.langmuir.6b02666 Langmuir 2016, 32, 8969−8979

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Langmuir

Figure 7. Linear relationships of inverse of solubilization ratios with pressure at different temperatures using dead and live oil. Data from Austad and Strand.25 Results summarized in Table 2.

Figure 8. Linear relationships of inverse of solubilization ratios with temperature at different pressures using dead and live oil. Data from Austad and Strand.25 Results summarized in Table 2.

The slope of 1/ σ with temperature is dependent on Iα. Thus, the slopes are scaled by either α or β.

Figure 7 show the near perfect linear dependence of 1/ σ on pressure as predicted in this paper for both dead and live oil. 8976

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Langmuir Figure 8 also shows the linear dependence of 1/σ with temperature. A summary of the slopes obtained and the interfacial volume ratios for each case are presented in Table 2.

γ=

γ=

pressure scans T, °C 55 60 65 70 75 80 85 70 75 80 85 90

dead dead dead dead dead dead dead live live live live live

β from Ghosh and Johns8 7.70 7.60 7.30 7.10 8.30 9.80 1.00 2.50 3.80 3.00 3.70 2.90

pressure in bars

oil type

50 100 150 200 250 300 600 500 450 400 300 250 200 100

dead dead dead dead dead dead live live live live live live live live

× × × × × × × × × × × ×

tuned βI

10−04 2.48 × 10−04 2.60 × 10−04 2.65 × 10−04 2.90 × 10−04 2.82 × 10−04 2.89 × 10−03 2.96 × 10−04 1.18 × 10−04 1.26 × 10−04 1.23 × 10−04 1.29 × 10−04 1.15 × temperature scans

α from Ghosh and Johns8

10−04 10−04 10−04 10−04 10−04 10−04 10−04 10−04 10−04 10−04 10−04 10−04

tuned αI

3.55 8.90 × 10−03 8.30 × 10−03 3.19 7.70 × 10−03 2.97 6.40 × 10−03 2.60 5.90 × 10−03 2.39 5.00 × 10−03 2.33 3.20 × 10−03 1.62 5.10 × 10−03 1.49 4.20 × 10−03 1.66 4.40 × 10−03 1.72 6.40 × 10−03 2.00 5.40 × 10−03 2.00 7.30 × 10−03 2.75 8.40 × 10−03 2.69 avg interfacial volume

× 10−03 × 10−03 × 10−03 × 10−03 × 10−03 × 10−03 × 10−03 × 10−03 × 10−03 × 10−03 × 10−03 × 10−03 × 10−03 × 10−03 ratio

C=

0.21 0.23 0.24 0.27 0.23 0.20 0.20 0.31 0.22 0.27 0.23 0.26

0.266 0.256 0.257 0.271 0.270 0.311 0.338 0.195 0.264 0.260 0.208 0.247 0.251 0.214 0.250 (±0.007)

■

3Vi = 3ILσi As

(39)

Er 36πI 2L2

(40)

CONCLUSIONS

A new dimensionless EoS was developed for the first time to model microemulsion phase behavior to fit and predict phase amounts and compositions of type III regions. The procedure for flash calculation of an overall composition was outlined. The following conclusions are made: (1) A new ratio, the interfacial volume ratio or I-ratio, is presented to tune experimental data. Once tuned, the I-ratio remains constant, and is independent of temperature and pressure based on the data examined to date. The new tuning procedure is robust and more accurate than the current approach of fitting with Hand’s model, and does not require individual knowledge of surfactant tail-length and area parameters. These parameters are determined by tuning Iratio directly. (2) The inverse of oil and water solubilization ratios varies linearly with HLD and hence, ln S. The slopes of the linear relationships are determined by the interfacial volume ratio (I). The slope in the two-phase regions are twice that of the slopes in the three-phase region. (3) The optimum solubilization ratio σ* is the harmonic mean of σo and σw in the type III microemulsion. This new finding can be used to estimate optimum solubilization ratio based solely on one solubility measurement in the three-phase region. (4) A new dimensionless parameter, the χ factor, is derived based on the overall composition. The type III region can exist only when χ > σ*, while the single-phase region (type IV) is present otherwise. The type III microemulsion window is dependent on the upper and lower HLD limits as derived in this paper.

(36)

where Er is interfacial rigidity (same as ΔG). From eq 3, Ri =

C σi 2

Equation 39 has the same functional form of the Huh equation.18 This is a key result of this paper. We show that the constant C is dependent on the I-ratio and L, which also governs microemulsion phase behavior. The C constant in eq 40 is proportional to the inverse of the I-ratio squared, and can be easily estimated. Values of Er ranging from 0.33 to 1kBT (Botlzmann energy unit) have been reported by Acosta et al.13 For example, at a temperature of 400 K, with interfacial rigidity Er equal to one Boltzmann energy unit, and an I-ratio of 0.2 and L of 30 Å, the constant C is 0.31 mN/m, which is very close to the traditionally accepted value of 0.3 mN/m. Hence, the dimensionless EoS proposed here is consistent with Huh’s correlation and shows when it is valid (spherical micelles). Further, eqs 39 and 40 can be used to quantify the effect of surfactant properties and solubilization on interfacial tension for other variables.

interfacial volume ratio I

Er ΔG = A 4πR2

(38)

where

Interfacial volume ratio I

The mean interfacial volume ratio obtained is approximately 0.25 (±0.007). Table 2 shows that the tuned interfacial volume ratio is relatively constant with temperature and pressure. Theoretical Proof of Chun Huh’s Equation: Role of the I-Ratio. Gibbs was the first to define a planar interfacial region between two distinct immiscible bulk fluids.26 The Gibbs− Duhem equation applied to the Gibbs’ interfacial region model defines interfacial tension as free energy change per unit area.4 Acosta assumed spherical micelles in his model and hence defined interfacial tension as γ=

36πI 2L2σi 2

The interfacial tension can therefore be defined for spherical micelles as

Table 2. Summary of Interfacial Volume Ratios from Temperature and Pressure Scan Experiments Using Dodecyl-Orthoxylene Sulfonate As a Surfactant Reported by Austad and Strand.25 oil type

Er

(37)

From eq 36 and 37, we get 8977

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Langmuir (5) The χ factor can be used to generate dimensionless fish diagrams that incorporate the combined impact of WOR and Vs on phase behavior. (6) The locus of the type III microemulsion composition (invariant point) occurs when χ = σ*. An equation for the invariant point composition as HLD varies was given. (7) Huh’s constant is dependent on the I-ratio, interfacial rigidity and the length parameter L. A derivation of Huh’s equation was obtained from the definition of interfacial tension by assuming the geometry of micelles to be spherical.

■

Subscripts

U = Upper limit corresponding to a phase transition from type III to type II+ or vice versa L = Lower limit corresponding to a phase transition from type II− to type III or vice versa o = Oil component w = Water component s = Surfactant component m = Microemulsion phase Superscripts

AUTHOR INFORMATION

■

Notes

The authors declare no competing financial interest.

■

* = Optimum state unless mentioned otherwise 0 = Initial condition

REFERENCES

(1) Ghosh, S.; Johns, R. T. An Equation-of-State Model To Predict Surfactant/Oil/Brine-Phase Behavior. Soc. Pet. Eng. J. 2016, 21, 1106. (2) Hoar, T.; Schulman, J. Transparent Water-in-Oil Dispersions: The Oleopathic Hydro-Micelle. Nature 1943, 152, 102−103. (3) Winsor, P. A. Hydrotropy, Solubilisation and Related Emulsification Processes. Trans. Faraday Soc. 1948, 44, 376−398. (4) Tadros, T. F. Applied Surfactants: Principles and Applications; John Wiley & Sons: New York, 2006. (5) Winsor, P. A. Solvent Properties of Amphiphilic Compounds; Butterworths Scientific Publications: London, 1954. (6) Bourrel, M.; Schechter, R. S. Microemulsions and Related Systems: Formulation, Solvency, and Physical Properties; Editions OPHRYS: Paris, 2010; 140−142. (7) Salager, J. L.; Morgan, J. C.; Schechter, R. S.; Wade, W. H.; Vasquez, E. Optimum Formulation of Surfactant/Water/Oil Systems for Minimum Interfacial Tension or Phase Behavior. SPEJ, Soc. Pet. Eng. J. 1979, 19, 107−115. (8) Ghosh, S.; Johns, R. T. In A New HLD-NAC Based EoS Approach to Predict Surfactant-Oil-Brine Phase Behavior for Live Oil at Reservoir Pressure and Temperature; SPE Annual Technical Conference and Exhibition; Society of Petroleum Engineers: Richardson, TX, 2014. (9) Marquez, N.; Anton, R.; Graciaa, A.; Lachaise, J.; Salager, J. L. Partitioning of Ethoxylated Alkylphenol Surfactants in MicroemulsionOil-Water Systems. Colloids Surf., A 1995, 100, 225−231. (10) Marquez, N.; Graciaa, A.; Lachaise, J.; Salager, J. L. Partitioning of Ethoxylated Alkylphenol Surfactants in Microemulsion-Oil-Water Systems: Influence of Physicochemical Formulation Variables. Langmuir 2002, 18, 6021−6024. (11) Salager, J. L.; Marquez, N.; Graciaa, A.; Lachaise, J. Partitioning of Ethoxylated Octylphenol Surfactants in Microemulsion-Oil-Water Systems: Influence of Temperature and Relation Between Partitioning Coefficient and Physicochemical Formulation. Langmuir 2000, 16, 5534−5539. (12) Salager, J. L.; Anton, R. E.; Sabatini, D. A.; Harwell, J. H.; Acosta, E. J.; Tolosa, L. I. Enhancing Solubilization in Microemulsions - State of the Art and Current Trends. J. Surfactants Deterg. 2005, 8, 3− 21. (13) Acosta, E.; Szekeres, E.; Sabatini, D. A.; Harwell, J. H. NetAverage Curvature Model for Solubilization and Supersolubilization in Surfactant Microemulsions. Langmuir 2003, 19, 186−195. (14) Ghosh, S.; Johns, R. T. In A Modified HLD-NAC Equation of State to Predict Alkali-Surfactant-Oil-Brine Phase Behavior; SPE Annual Technical Conference and Exhibition; Society of Petroleum Engineers: Richardson, TX, 2015. (15) DeGennes, P. G.; Taupin, C. Microemulsions and The Flexibility of Oil-Water Interfaces. J. Phys. Chem. 1982, 86, 2294− 2304. (16) Buijse, M. A.; Tandon, K.; Jain, S.; Handgraaf, J.-W.; Fraaije, J. In Surfactant Optimization for EOR Using Advanced Chemical Computational Methods; SPE Improved oil recovery symposium; Society of Petroleum Engineers: Richardson, TX, 2012.

NOMENCLATURE

Roman

As = Total interfacial area occupied by surfactant (Å2) A1 = Constant slope for σ* vs 1/ΔHLD linear trend (dimensionless) A2 = Constant intercept for σ* vs 1/ΔHLD linear trend (dimensionless) B1 = Constant slope for ln S* vs 1/σ* linear trend (dimensionless) B2 = Constant intercept for ln S* vs 1/σ* linear trend (dimensionless) as = Area per surfactant molecule (Å2) Cij = Volume fraction of component i in the phase j (dimensionless) Cc = Characteristic curvature of surfactant (dimensionless) EACN = Equivalent alkane carbon number (EACN unit) f(A) = Function of alcohol type and concentration (dimensionless) Ha = Average curvature (Å−1) Hn = Net curvature (Å−1) HLD = Hydrophilic lipophilic difference (dimensionless) K = Slope of logarithm of optimum salinity as a function of EACN (per EACN unit) I = Interfacial volume ratio (dimensionless) L = Surfactant length parameter (Å) MW = Molecular weight (g mol−1) Na = Avogadro’s number (mol−1) ns = Moles of surfactant component (mol) P = Pressure (bars) Ri = Radius of component i in the microemulsion (Å) R = Universal gas constant (J mol−1 K−1) S = Salinity (g/100 mL) Sj = Saturation of phase j SAD = Surfactant affinity difference (J mol−1) T = Temperature (°C or K) Vi = Volume of component i (cm3) Greek

α = Temperature coefficient (°C −1 or K−1) β = Pressure coefficient (bar−1) σ = Solubilization ratio (dimensionless) ξ = Correlation length (Å) ξ* = Critical correlation length (Å) ξD = Dimensionless correlation length (dimensionless) χ = Overall composition parameter (dimensionless) ρ = Density (g/cm3) 8978

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Langmuir (17) Acosta, E. J.; Yuan, J. S.; Bhakta, A. S. The Characteristic Curvature of Ionic Surfactants. J. Surfactants Deterg. 2008, 11, 145− 158. (18) 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. (19) Acosta, E. J.; Bhakta, A. S. The HLD-NAC model for mixtures of ionic and nonionic surfactants. J. Surfactants Deterg. 2009, 12, 7−19. (20) Israelachvili, J. N.; Mitchell, D. J.; Ninham, B. W. Theory of SelfAssembly of Hydrocarbon Amphiphiles Into Micelles and Bilayers. J. Chem. Soc., Faraday Trans. 2 1976, 72, 1525−1568. (21) Mitchell, D. J.; Ninham, B. W. Micelles, vesicles and microemulsions. J. Chem. Soc., Faraday Trans. 2 1981, 77, 601−629. (22) Sheng, J. Modern Chemical Enhanced Oil Recovery: Theory and Practice; Gulf Professional Publishing: Houston, TX, 2010; 249−252. (23) Strey, R. Microemulsion Microstructure and Interfacial Curvature. Colloid Polym. Sci. 1994, 272, 1005−1019. (24) Aarra, M. G.; Hoiland, H.; Skauge, A. Phase Behavior and Salt Partitioning in Two- and Three-Phase Anionic Surfactant Microemulsion Systems: Part I, Phase Behavior as a Function of Temperature. J. Colloid Interface Sci. 1999, 215, 201−215. (25) Austad, T.; Strand, S. Chemical Flooding of Oil Reservoirs: 4. Effects of Temperature and Pressure on the Middle Phase Solubilization Parameters Close to Optimum Flood Conditions. Colloids Surf., A 1996, 108, 243−252. (26) Gibbs, J. W.; Bumstead, H. A.; Longley, W. R. The Collected Works of J. Willard Gibbs; Longmans, Green and Company: London, 1928; Vol. 1.

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