Density Prediction of Ionic Liquids at Different Temperatures Using the Average Free Volume Model.

In this work, based on the average free volume model, the correlation between the molar volume and the van der Waals volume V w for the ionic liquids (ILs) was derived. With this model, the density of pure ILs and binary and ternary mixtures of ILs over a wide range of temperature (278-473.15 K) can be calculated with good accuracy only with the information of the chemical components. A total number of 1859 data points of 41 pure ILs and IL mixtures based on imidazolium, pyridinium, pyrrolidinium, phosphonium, and ammonium cations were used to verify the model. For pure ILs and IL mixtures, the average absolute relative deviations (ARDs) are 1.04 and 1.19%, respectively. The overall discrepancies are less than 4%.

Introduction

The ionic liquids[1] are organic salts with melting temperature lower than 100 °C. Due to the diversity of constituent ions, ILs may have a large number of different chemical structures (up to 1016 possible combinations[2]). A fast and accurate method to predict the density of ILs at different temperatures is crucial. Several methods have been developed to predict the liquid density of ILs, such as molecular dynamics simulation,[3,4] group contribution method (GCM),[5−8] and quantitative structure–property relationship (QSPR).[9−11] The molecular dynamics simulation method is time-consuming and difficult for practical application. For the GCM method, the contribution values of the groups are fitted parameters from experimental data, and the density of some ILs could not be predicted due to the lack of the parameters of new structures. The QSPR method can estimate the physical properties and activity of substances accurately through the information of molecular structure. However, because the characteristic descriptor is specific for a certain homologue, the QSPR method is difficult to apply as a universal method. This work supplies a model to fast predict the density of ILs at different temperatures with good accuracy and can be applied to new ILs only with the information of the chemical components.

The correlation between macroscopic physical properties and microscopic structural is a fundamental topic in material science. Free volume that exists due to static and dynamic disorder of the structure influences the substance properties such as mass density, viscosity, structural relaxation, physical aging, phase transition, conductivity, mobility, and permeability. For the polymers, there is a correlation between the crystalline volume Vc and the van der Waals volume Vw: Vc ∼ 1.435Vw.[12] For the ionic liquids, the correlation is Vc (= VM) ∼ 1.410 Vw (VM is the molecular volume).[13] The values 1.435 for the polymers and 1.410 for the ionic liquids are close to 21/2, which corresponds to the minimum energy in Lennard-Jones (or 6-12) potential (at the position r/σ = 21/6, then the volume ratio is (21/6)3 = 1.414).[13−15] In our previous work, the average free volume model—that is considering the atom or molecule as the hard core with attractive force, each particle is surrounded by the average free volume—was established.[16] According to this model, there is a correlation between the molar volume and the van der Waals volume of ILs in the liquid state. The specific volume and the density of the ILs over a wide range of temperature can be calculated with only the information of the components, which is the molecular weight and van der Waals volume.

Results and Discussion

From the experimental data,[17,18] under the atmospheric pressure, the molar volume Vmol displays a linear correlation with temperature T for all of the ionic liquids in the liquid state, which can be presented by the linear functionHere, VT0 is the volume when extrapolate the molar volume in the liquid state to absolute zero. The constant C1 corresponds to the temperature coefficient.

Based on 5 cations (imidazolium, pyridinium, pyrrolidinium, phosphonium, and ammonium), up to 14 ionic liquids were fitted to eq , and experimental data were taken from the references as listed in Table , which were collected from the NIST website.[17,19] The fitting results and the van der Waals volume for each sample are displayed in Table .

Table 1

Linear Fitting Results of eq for 14 Ionic Liquidsa

name		formula	Vw (cm3/mol)	C1 (cm3/mol/K)	VT0 (cm3/mol)	references
1	1-ethyl-3-methylimidazolium tetrafluoroborate	C6H11BF4N2	99.87	0.0894	127.65	(20)
2	1-butyl-3-methylimidazolium hexafluorophosphate	C8H15F6N2P	133.02	0.1304	168.89	(21)
3	1-hexyl-3-methylimidazolium hexafluorophosphate	C10H19F6N2P	153.82	0.1503	196.66	(22)
4	1-hexyl-3-methylimidazolium bis[(trifluoromethyl)sulfonyl]imide	C12H19F6N3O4S2	204.30	0.2175	261.47	(23)
5	1-heptyl-3-methylimidazolium bis(trifluoromethylsulfonyl)imide	C13H21F6N3O4S2	214.80	0.234	273.42	(24)
6	1-decyl-3-methylimidazolium bis(trifluoromethylsulfonyl)imide	C16H27F6N3O4S2	246.08	0.2721	314.57	(25)
7	1-butylpyridinium tetrafluoroborate	C9H14BF4N	119.09	0.1058	152.25	(26)
8	1-ethylpyridinium bis(trifluoromethylsulfonyl)imide	C9H10F6N2O4S2	161.14	0.1623	204.44	(27)
9	1-methyl-1-propylpyrrolidinium bis[(trifluoromethyl)sulfonyl]imide	C10H18F6N2O4S2	183.11	0.1837	231.17	(28)
10	trihexyl(tetradecyl)phosphonium acetate	C34H71O2P	390.26	0.4192	484.09	(29)
11	trihexyl(tetradecyl)phosphonium trifluoromethanesulfonate	C33H68F3O3PS	408.08	0.435	514.64	(29)
12	butyltrimethylammonium bis(trifluoromethylsulfonyl)imide	C9H18F6N2O4S2	179.47	0.1837	230.02	(30)
13	hexyltrimethylammonium bis[(trifluoromethyl)sulfonyl]imide	C11H22F6N2O4S2	200.34	0.1956	265.34	(31)
14	tributylmethylammonium bis(trifluoromethylsulfonyl)imide	C15H30F6N2O4S2	241.64	0.2582	303.52	(32)

Vw: van der Waals volume. C1: temperature coefficient. VT0: extrapolation of molar volume from the liquid state to absolute 0 K.

The idea of the average free volume model was explained in detail in our previous work; according to the model, both C1 and VT0 correlate with Vw as C1 ∝ Vw2/3 and VT0 ∝ Vw, respectively. The fitting results of VT0 and C1 from eq were compared with the van der Waals volume Vw of the ionic liquids. An obvious correlation between VT0, C1, and Vw can be discerned from Figure .

Figure 1

(a) Correlation between C1 and Vw; (b) correlation between VT0 and Vw. Data were fitted by linear function fixing intercept at 0. C1: temperature coefficient. VT0: extrapolation of molar volume from liquid state to absolute zero. Vw: van der Waals volume.

According to the fitting result, the molar volume and the density of pure ILs can be calculated by the following equationA total number of 27 ILs were used to verify the equation above. Detailed information of experimental data can be found in the Supporting Information. A comparison between the experimental and calculated density data is shown in Figure . Good agreement between calculated and experimental data can be discerned from the figure. The distribution of relative error is shown in Figure . For the pure ILs, the overall deviations are within ±3%, 53% data within ±1%, and 89% data within ±2%.

Figure 2

Comparison between experimental and calculated density data of pure ILs.

Figure 3

Histogram of relative error of the calculated density of pure ILs.

The density of binary ILs can be calculated from the law of mixture of liquidThe subscripts 1 and 2 represent different IL components. Mw (g/mol) is the molecular weight, R is the molar ratio, and Vmol (cm3/mol) is the molar volume that can be obtained from eq .

For the prediction of the density of the mixture composed of ILs and water, the water density under the desired temperature is required. Data of water density in this work were obtained from the NIST database website.[34,35]

For the mixture of one IL and water, the density isThis equation can expand theoretically to any number of components of a mixture, for example, the mixture of two ILs solved in water, the density isWhen different components are mixed, excess molar volume[36−38] exists due to various factors, such as shrinkage as a result of small molecules in voids between large molecules, effect of molecular structure, interaction between molecules, and polarity. Due to these factors, the estimated density will obviously deviates from the real density.

A total number of 14 groups of IL mixtures were used to verify the equations above. A comparison between the experimental and calculated density data is shown in Figure . The distribution of relative error is shown in Figure . For the IL mixtures in this work, the overall deviations are within ±4%, 61% data within ±1%, and 89% data within ±2%. Because of the excess molar volume, the relative error for the IL mixtures is larger than the pure ILs.

Figure 4

Comparison between experimental and calculated density data of IL mixtures.

Figure 5

Histogram of relative error of the calculated density of IL mixture.

Conclusions

The average free volume theory was used to deduce the relationship between the van der Waals volume Vw and the molar volume at different temperatures in the liquid state. The Vw used in this work was calculated from first principles theory. Density prediction data of ILs based on imidazolium, pyridinium, pyrrolidinium, phosphonium, and ammonium cations, including 27 pure ILs with 453 density data points, and 14 groups of binary and ternary mixtures with 1406 data points were compared with experimental density data. The AVDs of pure ILs and IL mixtures are 1.04 and 1.19%, respectively. According to this model, the density of ILs can be predicted using a simple equation with only the information of chemical components.

Computational Details of Vw

The van der Waals volume Vw is a well-defined physical quantity. The Vw is the space occupied by a molecule, which is impenetrable to other molecules with normal thermal energies.[15,39,40] It can be calculated by Bondi’s method,[40] first principles,[41] and fast estimation as a sum of atomic and bond contribution.[42] Many computational programs incorporate the function to calculate the Vw, such as HyperChem, Gaussian, and MaterialStudio. Normally, the absolute value of Vw from different methods or programs with different parameters would not be exactly the same. But the deviation should be systematically correlated. The van der Waals volume Vw is the key factor to the prediction of the mole volume. In this work, Vw of ionic liquids was calculated as the sum of the ion sizeTo calculate the ion volume, the density functional theory (DFT) calculation has been applied using the ORCA 4.2.0 package.[43,44] The geometry of the ions (except Cl– and Br–) was optimized with the B3LYP-D3/def2-TZVP method.[45−47] For the anions, the diffuse function was taken into account. The optimized structure was analyzed by the Multiwfn program[48−50] to get the ion volume. The ion volume was calculated as the volume enclosed by the isosurface with an electron density of 0.003 a.u. The resultant data of ion volume are listed in Table .

Table 2

Ion Volume Calculated in This Work

cation		Vw (cm3/mol)	Vw (Å3)
imidazolium	1,3-dimethylimidazolium (CMIM+)	57.78	95.97504
	1-ethyl-3-methylimidazolium (C2MIM+)	68.56	113.88178
	1-propyl-3-methylimidazolium (C3MIM+)	78.95	131.14084
	1-butyl-3-methylimidazolium (C4MIM+)	89.37	148.4581
	1-pentyl-3-methylimidazolium (C5MIM+)	99.82	165.82139
	1-hexyl-3-methylimidazolium (C6MIM+)	110.17	183.00708
	1-heptyl-3-methylimidazolium (C7MIM+)	120.67	200.4522
	1-octyl-3-methylimidazolium (C8MIM+)	131.12	217.80738
	1-decyl-3-methylimidazolium (C10MIM+)	151.95	252.4061
	1,3-diethylimidazolium (C2EIM+)	78.77	130.84546
	1-ethyl-2,3-dimethylimidazolium (C2MMIM+)	78.87	131.01158
	1-propyl-2,3-dimethylimidazolium (C3MMIM+)	89.35	148.42789
	1-butyl-2,3-dimethylimidazolium (C4MMIM+)	99.68	165.57754
pyridinium	pyridinium (Py+)	46.15	76.66169
	1-ethylpyridinium (C2Py+)	67.01	111.30742
	1-butylpyridinium (C4Py+)	87.78	145.81795
	2-methyl-1-propylpyridinium (C3CPy+)	87.61	145.5282
	1-butyl-4-methylpyridinium (C4CPy+)	98.12	162.99491
pyrrolidinium	1-methyl-1-propylpyrrolidinium (C3MPyr+)	88.98	147.80283
	1-butyl-1-methylpyrrolidinium (C4MPyr+)	99.55	165.3731
	1-hexyl-1-methylpyrrolidinium (C6MPyr+)	120.37	199.95269
phosphonium	trihexyl(tetradecyl)phosphonium (C6H13)3P(C14H29)+	355.06	589.79554
	triethyl(pentyl)phosphonium (C2H5)3P(C5H11)+	134.31	223.10099
ammonium	butyltrimethylammonium (Me3BuN+)	85.34	141.75326
	hexyltrimethylammonium (Me3HeN+)	106.21	176.43065
	N,N-dimethyl-N-propyl-1-butanaminium (Me2BuPrN+)	106.11	176.26241
	tributylmethylammonium (MeBu3N+)	147.51	245.03342
	triethylhexylammonium (Et3HeN+)	137.08	227.70087