Expanding the Solubility Parameter Method MOSCED to Pyridinium-, Quinolinium-, Pyrrolidinium-, Piperidinium-, Bicyclic-, Morpholinium-, Ammonium-, Phosphonium-, and Sulfonium-Based Ionic Liquids

Abstract

MOSCED (modified separation of cohesive energy density) is a solubility parameter method that offers an improved treatment of association interactions. Solubility parameter methods are well known for their ability to both make quantitative predictions and offer a qualitative description of the underlying molecular-level driving forces, lending themselves to intuitive solvent selection and design. Currently, MOSCED parameters are available for 130 organic solvents, water, and 33 imidazolium-based room temperature ionic liquids (ILs). In this work, we expand MOSCED to cover 66 additional ILs containing the pyridinium, quinolinium, pyrrolidinium, piperidinium, bicyclic, morpholinium, ammonium, phosphonium, and sulfonium cations using 10,052 experimental limiting activity coefficients. The resulting parameters may readily be used to predict the phase behavior in mixtures involving ILs.

Introduction

Separation processes are expensive. Separations are nonspontaneous processes and require an external “separating agent”, typically either energy or a solvent. It stands that even a minute improvement of existing separation process technologies could lead to significant cost savings. Central to the design and understanding of a separation process is the phase equilibrium thermodynamics of the system.¹ Conventional engineering design schemes to improve performance and reduce costs often neglect the molecular level details of the system of interest. However, it is these molecular level interactions upon which the entire process is built. Significantly greater improvements may be expected if molecular-level insight is incorporated in conventional design schemes.²

Room temperature ionic liquids (ILs) are a unique class of compounds that have drawn significant attention and research to improve existing separation processes. ILs are liquid at room temperature, consist of an organic cation paired with an organic or inorganic anion, and are stable over a wide range of temperatures.³⁻⁶ Due to the ability to alter the anion and cation virtually at will to tailor their physical and chemical properties, ILs may be “tuned” for a wide range of applications. For example, for physical solvent extraction, it has been demonstrated that ILs may be designed with a large affinity for the desired solute and with high selectivity.^4,7,8 As a consequence of their ionic nature, ILs have negligible vapor pressures. The combination of high tunability and negligible vapor pressure make ILs excellent candidates for entrainers in extractive distillation for the separation of azeotropic and close boiling compounds.^5,9,10

Nonetheless, the design of an optimal IL for a particular application is not without extreme challenge. Given the near infinite number of possible ILs, the chemical compound space that one must explore to find a (near) optimal IL is massive. For this reason, the development of predictive thermodynamic models is of utmost importance.¹¹

When modeling the phase behavior of mixtures with ILs, it is common to use an established local composition, binary interaction excess Gibbs free energy models such as Wilson’s equation, UNIQUAC (universal quasi-chemical theory), and NRTL (non-random, two-liquid).¹²⁻¹⁶ In this treatment, the neutral IL pair (cation plus anion) is modeled as a single component. UNIFAC (UNIQUAC functional-group activity coefficients),¹⁷ the group contribution analog of UNIQUAC, and modified-UNIFAC (Dortmund)¹⁸⁻²³ have long been the standard-bearer of predictive excess Gibbs free energy models. Many great efforts have been made, and successes were found in expanding UNIFAC to model mixtures with ILs.²⁴⁻³²

Nonetheless, while UNIFAC may be used to make accurate phase equilibria predictions, it is unable to elucidate the molecular-level details of the system. It is for this reason that, in this work, we continue to expand the MOSCED (modified separation of cohesive energy density) parameter database to model 66 additional ILs containing the pyridinium, quinolinium, pyrrolidinium, piperidinium, bicyclic, morpholinium, ammonium, phosphonium, and sulfonium cations.³³⁻⁴³ MOSCED is a solubility parameter-based method that can be used to both accurately predict limiting activity coefficients and help provide an explanation in terms of the responsible intermolecular interactions. The strength of MOSCED is in its treatment of association interactions.

To better understand MOSCED’s treatment of association, it is useful to briefly explore the evolution of MOSCED. Following the development of UNIQUAC, the limiting activity coefficient of component 2 in 1 (in a binary mixture), γ₂^∞, can be written as the sum of a combinatorial (COMB) and residual (RES) contribution:^12,13

1

The combinatorial term corresponds to the entropic contribution and is due to the size and shape dissimilarity of the components, and the residual term corresponds to the enthalpic contribution and results from intermolecular interactions. It is possible for a system to exhibit negative or positive deviations from Raoult’s law due to favorable or unfavorable intermolecular cross interactions, respectively. Therefore, to capture the correct physics of the system, it is important that our model be able to capture both negative and positive values of ln γ₂^∞, RES.

As in UNIQUAC, the combinatorial term can readily be modeled using the athermal Flory–Huggins equation (or one of its various extensions).^12,13,44 The residual term can be modeled using a solubility parameter-based method. The original Scatchard–Hildebrand regular solution theory (RST) takes the form^13,44

2

where R is the molar gas constant, T is the absolute temperature, v₂ is the molar volume of component 2, and δ_i is the solubility parameter of component i = {1,2}, defined as the square root of the component’s cohesive energy density. Unfortunately, for all cases, we find that ln γ₂^∞, RES is positive. This limitation stems from the fact that RST only accounts for dispersion interactions. Additional interactions may be taken into account by expanding the cohesive energy density as a sum of contributions (or partial solubility parameters).⁴⁴⁻⁴⁷ For example, the expansion used by the Hansen solubility parameter (HSP) method takes the form^46,48

3

where δ_i, D, δ_i, P, and δ_i, H correspond to the solubility parameter resulting from dispersion, polar, and association (or hydrogen bonding) interactions, respectively. We equivalently write our dispersion and polar solubility parameters as λ_i and τ_i, respectively, and write the association solubility parameter as the product of the contribution due to hydrogen bond acidity (α_i) and basicity (β_i). This results in

4

While HSP offers an improvement in accuracy, unfortunately we again find that for all cases ln γ₂^∞, RES is positive. Tijssen et al.⁴⁷ acknowledged the deficiency and suggested a splitting of the association term, leading to the following expression

5

With this update, the association term can be positive or negative, allowing ln γ₂^∞, RES to be positive or negative. We can rewrite the association term to better understand its physical meaning as³⁷

6

where we have defined _cross and _self to be the mean “cross” and “self” association term (or interaction energy), respectively:

7

If the system has favorable intermolecular cross-association interactions (relative to the self-association interactions), such that the two components would prefer to associate with each other, then the association term will be negative. We therefore see that the limitation of HSP is that it compares only the self-association interaction of the two components (see eq 4). The split association term given by eq 5 forms the basis of MOSCED, which has been shown to predict a range of phase equilibrium with a high level of accuracy.^{34−37,39,43,49}

To better understand the importance of splitting the association term and MOSCED’s improved treatment of association, let us consider, as an example, the vapor/liquid equilibrium in the binary system of acetone(1)/chloroform(2). This systems exhibits negative deviations from Raoult’s law (ln γ₁ and ln γ₂ < 0) and a maximum boiling azeotrope.^50,51 It has been confirmed using neutron diffraction⁵² and molecular simulation⁵³ that this results from association between acetone and chloroform. To be physically correct, it is important that our model be able to capture negative deviations from Raoult’s law due to the residual contribution (intermolecular interactions) and, more specifically, as a result of the association term. If we were to consider the use of RST or HSP, eq 2 or eq 4, the residual and association term would always be positive.

Next, we consider the MOSCED association parameters for acetone and chloroform and the MOSCED interpretation of this system. For acetone, we have α₁ = 0 MPa^1/2 and β₁ = 11.14 MPa^1/2, and for chloroform, we have α₂ = 5.80 MPa^1/2 and β₂ = 0.12 MPa^1/2. This leads to self-association interactions for acetone and chloroform, respectively, of α₁β₁ = 0 MPa and α₂β₂ = 0.7 MPa. As expected, we find that there is no self-association of acetone, and the propensity for chloroform to self-associate is extremely small. Considering next association between acetone and chloroform, we have α₁β₂ = 0 MPa and α₂β₁ = 64.61 MPa. MOSCED predicts association between acetone and chloroform, where chloroform is the proton donor and acetone is the proton acceptor, in agreement with the reference studies. Moreover, quantitatively, the association term in MOSCED would make a negative contribution to the residual term. We have shown in our recent work that for this system, the limiting activity coefficients predicted using MOSCED can be used to parameterize a binary interaction excess Gibbs free energy model to perform vapor/liquid equilibrium predictions in good agreement with experimental data.³⁷ The splitting of the association term as used by MOSCED is important. In fact, it has been suggested that HSP be improved by likewise splitting the association term.⁵⁴⁻⁵⁷

We expect that the separate treatment of hydrogen bond acidity and basicity is important for the modeling of ILs. In our recent work to regress MOSCED parameters for 33 imidazolium-based ILs, we found that in general the ILs had relatively large values of β. Of the 33 ILs, 17 had values of β greater than dimethyl sulfoxide (DMSO). This makes ILs promising candidates for entrainers in extractive distillation processes to separate azeotropic mixtures involving associating compounds.⁴³ This is agreement with our recent molecular simulation studies of the solubility of pharmaceuticals in imidazolium-based ILs. In that work, we observed hydrogen bonding wherein the solute was the hydrogen bond donor, and the IL anion was the hydrogen bond acceptor.⁵⁸⁻⁶⁰

However, a shortcoming of MOSCED has always been a limitation of available parameters. Development of MOSCED began with Eckert and co-workers in 1984.³³ There was a desire to relate the MOSCED parameters to molecular properties. It was found that λ could be related to the refractive index, but suitable relations were not found for τ, α, and β. These three parameters were therefore fit to reference limiting activity coefficients. Later in 1989, Eckert and co-workers presented an “improved” MOSCED equation.⁶¹ In that work, correlations were presented for the hydrogen bond acidity (α) and basicity (β) with solvachromatic hydrogen bond acidity and basicity scales. Later in 1993 Eckert and co-workers presented the related SPACE model, wherein τ was additionally related to solvochromatic polarity scales.⁶² However, these latter methods were limited to monofunctional compounds, and the parameters used in the correlations were specific to the chemical nature of the compound of interest. Most recently, MOSCED was subject to a “revision” in 2005.³⁴ This revised MOSCED model employed the same functional form as original presented in 1984, but λ, τ, α, and β were all taken to be adjustable and fit to reference limiting activity coefficients. This work builds upon the latest, revised MOSCED model.

Presently, MOSCED parameters are available for 130 organic solvents, water, and 33 imidazolium-based ILs.^34,39,43 Recently, many efforts have been focused on developing techniques to predict MOSCED parameters for organic compounds using molecular simulation,^41,63,64 electronic structure calculations,^{40,42,63−66} and group contribution methods.^38,40 Here, our efforts are focused on further expanding MOSCED to model ILs. Specifically, here we regress MOSCED parameters for 66 ILs containing the pyridinium, quinolinium, pyrrolidinium, piperidinium, bicyclic, morpholinium, ammonium, phosphonium, and sulfonium cations using 10,052 experimental limiting activity coefficients.

Computational Methods

Using MOSCED ln γ₂^∞, RES is calculated using the following system of equations^33,34

8

where v₂ is the (liquid) molar volume of the solute, λ_i, τ_i, α_i, and β_i are the solubility parameters due to dispersion, polarity, and hydrogen bond acidity and basicity, respectively, and the induction parameter, q_i, reflects the ability of the nonpolar part of a molecule to interact with a polar part, where i = {1,2}. The terms ψ₁ and ξ₁ are (solvent-dependent) asymmetry terms to modify the residual contribution for polar and hydrogen bonding interactions, whose functional form has been optimized for numerical predictions. While empirical in nature, the inclusion of ψ₁ and ξ₁ is physically based and very important. The solubility parameters are solute descriptors. The same parameters used to characterize a solute may not be appropriate to characterize a solvent.^67,68 This disparity is accounted for (or corrected) by including ψ₁ and ξ₁. These additional terms are not adjustable but are a function of the solvent solubility parameters. As emphasized by Park and Carr,⁶⁹ the introduction of the asymmetry terms was a major advancement in improving the predictive accuracy of MOSCED over other solubility parameter methods. R is the molar gas constant, and T is the absolute temperature. The superscript (T) is used to indicate temperature-dependent parameters, where the temperature dependence is computed using the empirical correlations provided in eq 8. As suggested by the equations, MOSCED adopts a reference temperature of 293 K (20 °C). The combinatorial contribution is calculated using a modified Flory–Huggins equation^33,34

9

where r is the size dissimilarity term, v₁ is the molar volume of the solvent, and aa₂ is an empirical (solute-dependent) term to modify the size dissimilarity for polar and hydrogen bonding interactions. The term aa₂ is not adjustable but is a function of the solubility parameters of the solute. For all cases aa₂ ≤ 0.953, effectively reducing the size dissimilarity and magnitude of the combinatorial contribution, with the value smaller for polar and associating compounds. An equivalent expression for the residual and combinatorial contribution to the limiting activity coefficient for component 1 in 2 (γ₁^∞, RES and γ₁) can be written by switching the subscript indices in eqs 8 and 9.

In the present study, MOSCED parameters were regressed for 66 ionic liquids (ILs) containing the pyridinium, quinolinium, pyrrolidinium, piperidinium, bicyclic, morpholinium, ammonium, phosphonium, and sulfonium cations using experimental limiting activity coefficients. Values were not available for the IL as the solute, so only data wherein the IL was a solvent was used. The IL molar volume was adopted from experimental reference data.⁷⁰⁻¹²³

Additionally, values for q were not regressed but were fixed at a value of 0.9. The parameter q corresponds to the induction parameter, and in the original MOSCED publications, for saturated molecules q = 1, and for aromatic compounds, q = 0.9. (This includes all compounds containing an aromatic group.) For alkenes, q, is computed as 1 minus the number of double bonds (or in general, the degrees of unsaturation) divided by twice the number of carbon atoms, and for halogenated molecules, q was varied for best fit and typically takes on values between 0.9 and 1.^33,34 Previous works on the calculation of thermophysical and transport properties of ILs using classical molecular simulation have demonstrated the importance of induction,^124,125 suggesting a value of q < 1. Here, we fix the value of 0.9 for simplicity.

The remaining IL MOSCED parameters (λ, τ, α, and β) were regressed by minimizing the objective function (OBJ)

10

the sum of the squared difference between the reference (ref) experimental value and that predicted using MOSCED, where the summation is over all N reference data points for each IL. This is the same objective function as used in the original MOSCED parameterization³⁴ and our recent IL work.⁴³ The objective function was minimized using the differential evolution global optimization method¹²⁶ as implemented in GNU Octave.¹²⁷

The experimentally measured value of the limiting activity coefficient is sensitive to the method with which it was measured.^34,128 Therefore, it is necessary to screen the reference data for suspect data points prior to the regression. This was accomplished here by using the compiled reference data set of Paduszyńsk¹²⁹ that was screened for suspect data when parameterizing his machine learning-based models for limiting activity coefficients of nonelectrolytes in ILs.^{30,70−73,75,80,95,96,100−106,113−116,119−123,130−161} The most accurate model parameterized by Paduszyński¹²⁹ was the least-squares support-vector machine learning (LSSVM) method, to which we will compare our MOSCED results. Note that MOSCED is restricted to solutes for which MOSCED parameters currently exist. For each IL, we regress four MOSCED parameters (λ, τ, α, and β). We did not regress parameters for ILs for which experimental data was not available for a chemically diverse set of solutes. Generally, this corresponds to ILs for which none of the solutes contained a nonzero value of α.

A summary of the 45 unique cations and 20 unique anions is provided in Tables 1 and 2, respectively. The resulting MOSCED parameters for the ILs containing the quinolinium, pyridinium, ammonium, and bicyclic cations are summarized in Table 3, and the parameters for the piperidinium, pyrrolidinium, sulfonium, morpholinium, and phosphonium cations are summarized in Table 4. In total, parameters are presented for 66 ILs. As a further reference, in Table S1 of the Supporting Information accompanying the electronic version of this manuscript, we tabulated the original set of MOSCED parameters for 130 organic solvents and water,^34,39 and in Table S2, we tabulated the MOSCED parameters for the 33 ILs with the imidazolium cation studied in our recent work.⁴³

Table 1. A Summary of the 45 Unique IL Cations with Their Preferred IUPAC Name and the Adopted Abbreviation^a.

cation IUPAC name	abbreviation
2-octylisoquinolin-2-ium	[C8iQuin]
2-hexylisoquinolin-2-ium	[C6iQuin]
1-(3-hydroxypropyl)pyridin-1-ium	[C3OHPy]
1-butyl-3-methylpyridin-1-ium	[C4C1(3)Py]
1-butyl-4-methylpyridin-1-ium	[C4C1(4)Py]
1-butylpyridin-1-ium	[C4Py]
1-ethyl-2-methylpyridin-1-ium	[C2C1(2)Py]
1-ethylpyridin-1-ium	[C2Py]
1-hexyl-3-methylpyridin-1-ium	[C6C1(3)Py]
1-hexylpyridin-1-ium	[C6Py]
1-pentylpyridin-1-ium	[C5Py]
4-cyano-1-butylpyridin-1-ium	[C4CCN(4)Py]
pyridinium	[Py]
(2-hydroxyethyl)trimethylazanium	[C2OH(C1)3N]
1,14-dihydroxy-9-methyl-9-tridecyl-3,6,12-trioxa-9-azatetradecan-9-ium	[C2O2OHC1O1O2OHC13C1N]
decyltrimethylazanium	[C10(C1)3N]
diethyl(2-methoxyethyl)methylazanium	[C2O1C2C2C1N]
methyltrioctylazanium	[(C8)3C1N]
tributyl(hexyl)azanium	[C6(C4)3N]
tributyl(methyl)azanium	[(C4)3C1N]
triethyl(octyl)azanium	[C8(C2)3N]
trimethyl(octyl)azanium	[C8(C1)3N]
1,3,4,6,7,8-hexahydro-1-methyl-2H-pyrimido[1,2-a]pyrimidine	[C1TBDH]
1-hexyl-1,4-diaza[2.2.2]bicyclooctanium	[DABCO6]
1-hexyl-1-azabicyclo[2.2.2]octan-1-ium	[C6Quinuc]
1-octyl-1-azabicyclo[2.2.2]octan-1-ium	[C8Quinuc]
1-butyl-1-methylpiperidin-1-ium	[C4C1Pip]
1-methyl-1-propylpiperidin-1-ium	[C3C1Pip]
1-(2-methoxyethyl)-1-methylpiperidin-1-ium	[C2O1C1Pip]
1-methyl-1-pentylpiperidin-1-ium	[C5C1Pip]
1-hexyl-1-methylpiperidin-1-ium	[C6C1Pip]
1-(2-methoxyethyl)-1-methylpyrrolidin-1-ium	[C2O1C1Pyr]
1-butyl-1-methylpyrrolidin-1-ium	[C4C1Pyr]
1-ethyl-1-methylpyrrolidin-1-ium	[C2C1Pyr]
1-hexyl-1-methylpyrrolidin-1-ium	[C6C1Pyr]
1-methyl-1-pentylpyrrolidin-1-ium	[C5C1Pyr]
1-methyl-1-propylpyrrolidin-1-ium	[C3C1Pyr]
1-octyl-1-methylpyrrolidin-1-ium	[C8C1Pyr]
triethylsulfanium	[(C2)3S]
4-(2-methoxyethyl)-4-methylmorpholin-4-ium	[C2O1C1Mor]
4-butyl-4-methylmorpholin-4-ium	[C4C1Mor]
4-(3-hydroxypropyl)-4-methylmorpholin-4-ium	[C3OHC1Mor]
tributyl(ethyl)phosphonium	[(C4)3C2P]
tributyl(methyl)phosphonium	[(C4)3C1P]
trihexyl(tetradecyl)phosphonium	[(C6)3C14P]

^aThe cations are organized based on their family.

Table 2. A Summary of the 20 Unique IL Anions with Their Preferred IUPAC Name and the Adopted Abbreviation.

anion IUPAC name	abbreviation
2-ethoxyethylsulfate	[C2O2SO4]
2-hydroxypropanoate	[LA]
4-methylbenzene-1-sulfonate	[TOS]
bis(2,4,4-trimethylpentyl)phosphinate	[C8(i)C8(i)PO2]
bis(pentafluroethylsulfonyl)azanide	[NPf2]
bis(trifluoromethylsulfonyl)azanide	[NTf2]
camphorsulfonate	[CS]
chloride	[Cl]
dicyanazanide	[DCA]
diethyl phosphate	[DEP]
ethyl sulfate	[C2SO4]
hexafluorophosphate	[PF6]
methyl sulfate	[C1SO4]
tetracyanoboranuide	[TCB]
tetrafluoroborate	[BF4]
tetraoxo-1,4,6,9-tetraoxa-5-boraspirono[4.4]nonan-5-uide	[BOB]
thiocyanate	[SCN]
tricyanomethide	[C(CN)3]
trifluoromethanesulfonate	[OTf]
trifluorotris(pentafluoroethyl)-λ5-phosphanuide	[FAP]

Table 3. MOSCED Parameters for the ILs Containing the Quinolinium, Pyridinium, Ammonium, and Bicyclic Cations Regressed Using Experimental Limiting Activity Coefficients^a.

ionic liquid	family	v	λ	τ	q	α	β
[C8iQuin][NTf2]	quinolinium	392.65	16.04	9.07	0.9	0.00	10.92
[C6iQuin][SCN]		251.95	18.03	10.37	0.9	2.15	28.83
[C4C1(4)Py][NTf2]	pyridinium	304.37	16.65	9.17	0.9	6.36	7.50
[C4C1(4)Py][SCN]		196.27	17.45	13.58	0.9	0.00	21.09
[C6C1(3)Py][TOS]		309.83	17.42	7.49	0.9	2.90	34.22
[Py][C2O2SO4]		194.52	12.15	10.16	0.9	1.12	82.51
[C4C1(3)Py][TOS]		288.80	16.94	6.38	0.9	8.96	22.19
[C4C1(3)Py][OTf]		255.82	17.32	12.81	0.9	0.15	18.45
[C3OHPy][NTf2]		270.50	16.03	11.37	0.9	14.51	7.16
[C2Py][NTf2]		252.67	13.03	9.33	0.9	8.60	2.83
[C4Py][NTf2]		286.73	13.19	9.23	0.9	8.84	2.43
[C5Py][NTf2]		302.79	13.38	8.97	0.9	10.51	1.77
[C4C1(4)Py][DCA]		207.92	16.80	13.17	0.9	0.12	19.42
[C4C1(4)Py][C(CN)3]		231.66	16.88	8.64	0.9	15.26	6.70
[C6Py][NTf2]		319.92	17.05	8.81	0.9	10.94	1.64
[C2C1(2)Py][C2SO4]		202.32	15.13	12.59	0.9	3.69	35.17
[C4CCN(4)Py][NTf2]		299.39	16.16	11.31	0.9	9.11	7.08
[C8(C2)3N][NTf2]	ammonium	399.36	15.73	9.66	0.9	0.26	10.44
[(C4)3C1N][NTf2]		381.07	14.49	8.40	0.9	4.58	7.00
[C8(C1)3N][NTf2]		355.91	14.89	8.02	0.9	4.44	6.95
[(C8)3C1N][NTf2]		586.05	15.37	6.05	0.9	0.64	8.97
[C10(C1)3N][NTf2]		391.73	14.81	6.83	0.9	4.39	6.54
[C6(C4)3N][NTf2]		462.02	15.31	7.70	0.9	3.47	4.25
[C2O1C2C2C1N][FAP]		355.24	14.79	11.33	0.9	6.70	2.57
[C2O2OHC1O1O2OHC13C1N][C1SO4]		504.47	16.59	4.29	0.9	3.64	19.43
[C2OH(C1)3N][NTf2]		253.79	14.88	11.99	0.9	17.13	7.46
[C2O1C2C2C1N][NTf2]		302.98	15.27	10.74	0.9	3.69	9.63
[C1TBDH][NPf2]	bicyclic	338.46	15.76	9.26	0.9	8.68	6.20
[DABCO6][NTf2]		349.81	16.01	9.91	0.9	4.48	7.67
[C6Quinuc][NTf2]		349.88	15.64	8.36	0.9	3.00	7.65
[C8Quinuc][NTf2]		388.45	15.37	7.69	0.9	1.02	9.59

^aThe molar volume (v) is in units of cm³/mol, q is dimensionless, and λ, τ, α, and β all have units of MPa^1/2 or (J/cm³)^1/2.

Table 4. MOSCED Parameters for the ILs Containing the Piperidinium, Pyrrolidinium, Sulfonium, Morpholinium, and Phosphonium Cations Regressed Using Experimental Limiting Activity Coefficients^a.

ionic Liquid	family	v	λ	τ	q	α	β
[C4C1Pip][SCN]	piperidinium	207.72	16.83	14.32	0.9	0.00	18.58
[C3C1Pip][NTf2]		298.23	16.55	10.31	0.9	6.98	6.25
[C4C1Pip][NTf2]		314.57	15.75	9.30	0.9	4.95	7.07
[C2O1C1Pip][NTf2]		305.93	15.48	10.35	0.9	6.65	7.57
[C5C1Pip][NTf2]		332.52	15.60	8.74	0.9	5.37	6.29
[C6C1Pip][NTf2]		349.49	15.58	8.25	0.9	4.44	6.53
[C4C1Pyr][BOB]	pyrrolidinium	266.48	12.50	12.76	0.9	0.00	13.75
[C4C1Pyr][TCB]		264.49	13.01	10.09	0.9	0.00	13.77
[C4C1Pyr][OTf]		231.95	17.04	9.10	0.9	13.15	10.08
[C4C1Pyr][SCN]		194.12	16.78	13.90	0.9	0.00	20.02
[C6C1Pyr][NTf2]		337.07	14.20	9.70	0.9	0.10	11.22
[C8C1Pyr][NTf2]		370.65	15.31	8.79	0.9	0.00	11.77
[C4C1Pyr][NTf2]		299.78	14.77	9.92	0.9	3.78	8.12
[C4C1Pyr][FAP]		366.13	15.13	10.76	0.9	5.55	0.62
[C2O1C1Pyr][NTf2]		291.06	15.53	10.63	0.9	6.25	4.39
[C2O1C1Pyr][FAP]		335.57	14.86	10.79	0.9	5.86	1.18
[C4C1Pyr][C(CN)3]		229.90	18.01	1.15	0.9	21.06	7.43
[C3C1Pyr][NTf2]		285.37	14.42	10.40	0.9	5.43	7.24
[C5C1Pyr][NTf2]		319.50	14.73	9.46	0.9	5.26	6.37
[C4C1Pyr][DCA]		205.51	15.85	12.35	0.9	1.23	26.84
[C2C1Pyr][LA]		186.29	16.38	10.72	0.9	4.40	43.54
[(C2)3S][NTf2]	sulfonium	272.24	15.98	10.54	0.9	7.46	6.80
[C2O1C1Mor][NTf2]	morpholinium	293.51	15.51	11.53	0.9	9.21	8.62
[C2O1C1Mor][FAP]		365.26	14.99	11.90	0.9	8.27	2.54
[C4C1Mor][C(CN)3]		231.19	16.18	13.52	0.9	0.25	18.78
[C3OHC1Mor][NTf2]		286.68	16.01	12.47	0.9	16.70	9.68
[(C6)3C14P][Cl]	phosphonium	579.23	14.93	2.53	0.9	0.00	12.62
[(C6)3C14P][BF4]		615.93	15.37	4.67	0.9	0.35	14.22
[(C6)3C14P][NTf2]		714.01	15.70	4.83	0.9	0.48	8.41
[(C4)3C1P][C1SO4]		319.81	16.51	8.66	0.9	1.07	27.85
[(C6)3C14P][PF6]		636.69	16.58	4.06	0.9	3.28	6.64
[(C6)3C14P][LA]		630.27	15.23	5.55	0.9	0.00	21.99
[(C6)3C14P][CS]		741.27	16.15	4.37	0.9	1.06	21.35
[(C4)3C2P][DEP]		386.11	15.35	5.03	0.9	0.99	47.98
[(C6)3C14P][C8(i)C8(i)PO2]		872.45	16.46	0.43	0.9	2.03	6.45

^aThe molar volume (v) is in units of cm³/mol, q is dimensionless, and λ, τ, α, and β all have units of MPa^1/2 or (J/cm³)^1/2.

Results and Discussion

The values of ln γ₂^∞ predicted using MOSCED are well correlated with the reference data as shown in the parity plot (Figure 1) with an R² value and slope close to 1. This is additionally confirmed by the random pattern in the residual plot (Figure 2). Note that the points that do appear correlated and are an artifact of the same solute/IL pair at different temperatures.

Figure 1.

Parity plot of ln γ₂^∞ predicted using MOSCED versus the reference data for all N = 10,052 systems used to regress MOSCED parameters. The dashed lines correspond to ±1 log unit and are drawn as a reference. The value of R² and slope correspond to the squared correlation coefficient and slope of the line of best fit. The root-mean-square error (RMSE) in ln γ₂ and the average absolute relative deviation (AARD) in γ₂^∞ is computed over all 10,052 points.

Figure 2.

Residual plot of the difference in ln γ₂^∞ predicted using MOSCED and the reference values versus the reference values for all N = 10,052 systems used to regress MOSCED parameters. The dashed lines correspond to ±1 log unit and are drawn as a reference.

The number of reference data points used to regress MOSCED parameters for each IL and the corresponding error are summarized in Tables 5 and 6. Statistics for the top performing LSSVM method of Paduszyński¹²⁹ are additionally provided for comparison. It is important to remember the context of the error. Both MOSCED and LSSVM were trained on the dataset for which comparison is made. The error for each IL is computed as the root-mean-square error (RMSE) in ln γ₂^∞

11

and the average absolute relative deviation in γ₂^∞ (AARD)

12

Table 5. A Summary of the Number of Data Points (N), Root-Mean-Square Error (RMSE) in the Log Limiting Activity Coefficient, and the Percent Average Absolute Relative Deviation (AARD) in the Limiting Activity Coefficient Predicted Using MOSCED and with LSSVM from the Work of Paduszyński¹²⁹ for the ILs Containing the Quinolinium, Pyridinium, Ammonium, and Bicyclic Cations^a.

		MOSCED			LSSVM
ionic liquid	family	N	RMSE	AARD	N	RMSE	AARD (%)
[C8iQuin][NTf2]	quinolinium	135	0.184	15.539	185	0.080	6.810
[C6iQuin][SCN]		219	0.246	20.245	320	0.121	9.200
average		177	0.215	17.892	253	0.101	8.005
[C4C1(4)Py][NTf2]	pyridinium	216	0.268	23.338	276	0.084	6.360
[C4C1(4)Py][SCN]		202	0.361	28.036	245	0.106	8.380
[C6C1(3)Py][TOS]		116	0.241	16.366	176	0.105	7.950
[Py][C2O2SO4]		48	0.293	24.421	56	0.298	24.500
[C4C1(3)Py][TOS]		56	0.196	16.546	76	0.105	7.950
[C4C1(3)Py][OTf]		167	0.288	21.828	206	0.091	6.250
[C3OHPy][NTf2]		272	0.434	31.001	314	0.129	8.600
[C2Py][NTf2]		96	0.332	27.863	138	0.180	13.800
[C4Py][NTf2]		96	0.230	18.640	138	0.097	7.590
[C5Py][NTf2]		96	0.214	17.375	138	0.089	6.590
[C4C1(4)Py][DCA]		212	0.339	28.959	317	0.118	9.360
[C4C1(4)Py][C(CN)3]		171	0.362	32.494	206	0.086	6.620
[C6Py][NTf2]		30	0.225	18.158	30	0.126	10.800
[C2C1(2)Py][C2SO4]		55	0.136	10.750	55	0.085	6.690
[C4CCN(4)Py][NTf2]		315	0.295	24.403	455	0.134	9.850
average		143	0.281	22.679	188	0.122	9.419
[C8(C2)3N][NTf2]	ammonium	270	0.194	15.446	378	0.100	7.780
[(C4)3C1N][NTf2]		93	0.303	21.776	135	0.190	12.300
[C8(C1)3N][NTf2]		93	0.296	21.568	135	0.171	11.500
[(C8)3C1N][NTf2]		114	0.292	20.967	182	0.275	19.800
[C10(C1)3N][NTf2]		93	0.322	24.688	141	0.166	11.700
[C6(C4)3N][NTf2]		90	0.539	43.841	130	0.309	22.800
[C2O1C2C2C1N][FAP]		240	0.279	23.507	390	0.145	10.800
[C2O2OHC1O1O2OHC13C1N][C1SO4]		64	0.183	13.437	100	0.048	3.590
[C2OH(C1)3N][NTf2]		226	0.292	20.737	376	0.157	11.000
[C2O1C2C2C1N][NTf2]		175	0.225	19.778	265	0.086	6.700
average		146	0.293	22.575	223	0.165	11.797
[C1TBDH][NPf2]	bicyclic	128	0.205	17.465	213	0.119	9.050
[DABCO6][NTf2]		148	0.192	14.661	224	0.349	20.500
[C6Quinuc][NTf2]		173	0.212	18.110	225	0.179	13.000
[C8Quinuc][NTf2]		171	0.268	22.559	215	0.125	8.910
average		155	0.219	18.199	219	0.193	12.865

^aIn the last row for each IL cation family, we compute the average of each column within that family.

Table 6. A Summary of the Number of Data Points (N), Root-Mean-Square Error (RMSE) in the Log Limiting Activity Coefficient, and the Percent Average Absolute Relative Deviation (AARD) in the Limiting Activity Coefficient Predicted Using MOSCED and with LSSVM from the Work of Paduszyński¹²⁹ for the ILs Containing the Piperidinium, Pyrrolidinium, Sulfonium, Morpholinium, and Phosphonium Cations^a.

		MOSCED			LSSVM
ionic liquid	family	N	RMSE	AARD	N	RMSE	AARD (%)
[C4C1Pip][SCN]	piperidinium	114	0.720	33.867	164	0.105	8.420
[C3C1Pip][NTf2]		149	0.251	21.950	218	0.083	5.920
[C4C1Pip][NTf2]		170	0.213	17.965	254	0.075	6.000
[C2O1C1Pip][NTf2]		234	0.225	18.503	354	0.095	7.320
[C5C1Pip][NTf2]		148	0.241	21.049	249	0.075	5.480
[C6C1Pip][NTf2]		168	0.213	18.278	250	0.070	5.170
average		164	0.310	21.935	248	0.084	6.385
[C4C1Pyr][BOB]	pyrrolidinium	115	0.267	22.681	150	0.176	13.200
[C4C1Pyr][TCB]		115	0.384	36.880	415	0.115	8.530
[C4C1Pyr][OTf]		187	0.429	38.759	249	0.106	8.250
[C4C1Pyr][SCN]		174	0.435	35.841	242	0.118	9.380
[C6C1Pyr][NTf2]		103	0.286	21.711	138	0.171	10.100
[C8C1Pyr][NTf2]		98	0.249	21.791	141	0.165	10.100
[C4C1Pyr][NTf2]		133	0.317	24.633	167	0.170	11.300
[C4C1Pyr][FAP]		227	0.352	25.656	358	0.145	11.000
[C2O1C1Pyr][NTf2]		234	0.345	25.302	366	0.102	7.840
[C2O1C1Pyr][FAP]		345	0.345	25.621	522	0.153	11.700
[C4C1Pyr][C(CN)3]		221	0.644	68.574	370	0.103	8.180
[C3C1Pyr][NTf2]		84	0.316	21.260	117	0.199	12.800
[C5C1Pyr][NTf2]		96	0.292	20.912	126	0.177	10.200
[C4C1Pyr][DCA]		110	0.251	21.697	356	0.134	10.300
[C2C1Pyr][LA]		216	0.503	36.729	356	0.134	10.300
average		164	0.361	29.870	272	0.145	10.212
[(C2)3S][NTf2]	sulfonium	181	0.319	28.026	250	0.072	5.430
[C2O1C1Mor][NTf2]	morpholinium	268	0.289	25.385	369	0.120	8.770
[C2O1C1Mor][FAP]		264	0.305	26.461	372	0.132	9.510
[C4C1Mor][C(CN)3]		252	0.296	24.923	366	0.140	9.530
[C3OHC1Mor][NTf2]		244	0.515	26.550	360	0.157	10.900
average		257	0.351	25.830	367	0.137	9.678
[(C6)3C14P][Cl]	phosphonium	36	0.131	10.468	48	0.092	7.250
[(C6)3C14P][BF4]		91	0.233	21.002	135	0.174	13.500
[(C6)3C14P][NTf2]		218	0.283	21.327	294	0.176	12.000
[(C4)3C1P][C1SO4]		30	0.088	6.859	38	0.057	4.450
[(C6)3C14P][PF6]		80	0.171	12.832	120	0.071	5.570
[(C6)3C14P][LA]		72	0.694	51.124	93	0.280	20.500
[(C6)3C14P][CS]		84	0.417	36.432	120	0.237	16.900
[(C4)3C2P][DEP]		156	0.265	19.783	516	0.167	12.500
[(C6)3C14P][C8(i)C8(i)PO2]		55	0.092	7.420	70	0.074	5.340
average		91	0.264	20.805	159	0.148	10.890

^aIn the last row for each IL cation family, we compute the average of each column within that family.

We additionally compute the average error for each cation family. The RMSE and AARD for each IL can be used as an estimate of expected error in MOSCED predictions involving each IL.

Comparing MOSCED to LSSVM, we notice two general trends: (i) LSSVM has a smaller error (RMSE and AARD), and (ii) it is able to model a larger number of systems. If your goal was to accurately model limiting activity coefficients for a wide range of organic solutes in ILs, then LSSVM is the clear winner. To model a solute, one needs only LSER (linear solvation energy relationship) parameters for the compound of interest. At present, LSER parameters are available for thousands of compounds. Moreover, LSSVM uses a group contribution method for the IL parameters, allowing a wider range of ILs to be modeled. On the other hand, MOSCED can only be used to model solutes for which MOSCED parameters are available. At present, this corresponds to 130 organic compounds and water. This is the reason why, in Tables 5 and 6, there is a discrepancy in the number of systems for which calculations were performed using LSSVM and MOSCED. We additionally point out that MOSCED parameters must be available for the IL of interest.

Nonetheless, there are a number of advantages of using MOSCED. First, while the IL MOSCED parameters were regressed using data wherein the IL was the solvent, MOSCED can additionally make predictions wherein the IL is the solute. When modeling the phase behavior of mixtures with ILs, it is common to use established local composition, binary interaction excess Gibbs free energy models such as Wilson’s equation, UNIQUAC (universal quasi-chemical theory), and NRTL (non-random, two-liquid).¹³⁻¹⁶ The values of γ₁^∞ and γ₂ can be used to calculate the binary interaction parameters (BIPs) for a binary system, which can in turn be used to calculate composition-dependent activity coefficients.^37,163,164 It is helpful to mention the similarity with the successful IL-UNIFAC (UNIQUAC functional-group activity coefficients) development efforts of Lei and co-workers.^24,25 In that work, only limiting activity coefficients for solutes in ILs were used in the regression of UNIFAC group interaction parameters for systems involving liquid solutes and ILs. The resulting UNIFAC model was demonstrated to model accurately both limiting activity coefficients and ternary vapor/liquid equilibrium. Similar performance is expected with MOSCED. As a similar example, in the parameterization of the group contribution equation of state, Breure et al.¹⁶⁵ used only limiting activity coefficients for alkanes in ILs to regress binary interaction parameters for IL–IL and IL–paraffin interactions.

Second, as a solubility parameter-based method, the parameters all have physical significance, which can be used to help explain underlying intermolecular interactions. In addition, solubility parameters readily lend themselves to intuitive solvent selection and formulation. Historically, solubility parameters were first introduced in the context of regular solutions 88 years ago.⁴⁴ As a testament of their importance and utility, the measurement of these same solubility parameters for ILs is an active area of research.^166,167 MOSCED further separates the intermolecular interactions leading to an improved description of the underlying intermolecular interactions and an advanced treatment of association.⁴³

While the error in the RMSE and AARD is in general greater than LSSVM, it useful to put this into perspective with other common methods. Recently, Brouwer and Schuur⁴⁹ performed an extensive evaluation of the ability of eight predictive methods to predict values of γ₂^∞ at 298.15 K for molecular solvents and ILs. This included the Hildebrand and Hansen solubility parameter methods,^44,46 MOSCED,³⁴ LSER,^67,162,168 UNIFAC and two of its modified versions,^{17−29,31,32,141} and the conductor-like screening model for realistic solvents (COSMO-RS).^169−171 For the case of molecular solvents, MOSCED was the top performer with an AARD of 16.2% followed by LSER, the UNIFAC-based methods, and COSMO-RS, which had comparable values of AARD over the range of 24.3–33.3%. For the case of ILs, the values of AARD were found to be much larger. LSER was the top performing model with an AARD of 65.1%. With the UNIFAC-based methods, the values of AARD were over the range of 86.2–122%. The reported errors with COSMO-RS and the Hildebrand and Hansen solubility parameter methods were even larger. It follows that, in the context of conventional methods, the observed performance of MOSCED is excellent. Computing the AARD over all 10,052 systems using eq 12, we get 25.62%.

Similar to LSSVM, as a group contribution method, UNIFAC is able to look at a wider range of solutes and ILs. A group contribution method to estimate MOSCED parameters is additionally possible. A group contribution method currently exists for organic compounds,³⁸ and a similar approach could be adopted for ILs. With the further expansion of the MOSCED parameter matrix for ILs, future work will be necessary to evaluate how best to divide the IL into groups. The three common strategies are described by Gani, Kontogeorgis and co-workers.^28,29 (i) The IL could be decomposed into an cation and anion group. (ii) The IL could be decomposed into several groups, which includes a single neutral group consisting of the cation skeleton (or base) and the anion. (iii) The IL could be decomposed into several groups including a separate group for the anion and the cation skeleton.

Last, it is useful to compare the IL MOSCED parameters to those of the 130 organic solvents for which parameters exist. Only 10 of the 130 organic solvents have values of β greater than 20 MPa^1/2.³⁴ 2-Pyrrolidone has the largest value with β = 27.59 MPa^1/2, and dimethyl sulfoxide (DMSO) is the second largest with β = 26.17 MPa^1/2. Of the 66 ILs parameterized here, 13 have values of β greater than 20 MPa^1/2. Of these systems, five have β values greater than 30 MPa^1/2, [Py][C_2O2SO₄] being the largest with a value of 82.51 MPa^1/2. Additionally, the ILs all have relatively large molar volumes (v), leading to a relatively large combinatorial (or entropic) contribution to γ₂^∞.

All of the reference data and MOSCED predictions along with the accompanying error and with comparison to the top performing LSSVM method of Paduszyński¹²⁹ are tabulated in the Supporting Information accompanying the electronic version of this manuscript. We additionally include a simple GNU Octave/MATLAB M-file that can be used to predict γ₁^∞ and γ₂ using MOSCED; a screen cast tutorial and additional references are available on the YouTube Channel of A. Paluch.¹⁷² P. Dhakal also maintains a website with freely available programs to make MOSCED predictions.¹⁷³ As already noted, in the Supporting Information accompanying the electronic version of this manuscript, we tabulate the MOSCED parameters for 130 organic solvents, water, and 33 imidazolium-based ILs in Tables S1 and S2.

Conclusions

MOSCED is a solubility parameter-based method that can be used to make accurate predictions of limiting activity coefficients. If composition-dependent activity coefficients are necessary, as in the modeling of phase equilibrium, the limiting activity coefficients may be used to parameterize a local composition, binary interaction excess Gibbs free energy model (i.e., Wilson’s equation, NRTL, or UNIQUAC). Additionally, the MOSCED parameters all have physical significance, which can be used to gain insight into the underlying intermolecular interactions, and additionally lend themselves to intuitive solvent selection and formulation. The advantage of MOSCED over the well-known Hildebrand and Hansen solubility parameter methods is MOSCED’s advanced treatment of association.

At present, MOSCED parameters are limited to 130 organic solvents, water, and 33 ILs containing the common imidazolium-based cation.^34,39,43 In the present study, we expand MOSCED to cover 66 additional ILs containing the pyridinium, quinolinium, pyrrolidinium, piperidinium, bicyclic, morpholinium, ammonium, phosphonium, and sulfonium cations using 10,052 experimental limiting activity coefficients. Over all 10,052 reference systems, the RMSE in ln γ₂^∞ and AARD in γ₂ with MOSCED is 0.343 and 25.62%, respectively. While the error is slightly larger than the machine learning-based method of Paduszyński,¹²⁹ following the recent work of Brouwer and Schuur,⁴⁹ it is superior to the error expected with conventional methods such as UNIFAC. With the continued expansion of the MOSCED parameter matrix and the development of methods to predict parameters, MOSCED stands as a promising model for process and product design applications.

Supporting Information Available

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acsomega.9b03087.

• Document containing MOSCED parameters for 130 organic solvents, water, and 33 imidazolium-based ILs (PDF)

• Reference limiting activity coefficients, MOSCED predictions, and error with MOSCED and LSSVM (XLSX)

• GNU Octave/MATLAB M-file to make MOSCED predictions (TXT)

The authors declare no competing financial interest.