Diffusion

Molecular diffusion is the transport phenomena caused by a concentration gradient [33]. It must not be mistaken for convection which is caused by the bulk motion.

The diffusive flux can be described by the Fick’s first law at steady state. It postulates that the flux (J in mol. m2.s-1) is proportional to the spatial concentration gradient (from large to low concentration areas):

dC

J = — D‘j dX (12.2)

Dj j is the diffusion coefficient (or diffusivity in m2.s-1) of the compound i in the solvent j.

In gas-liquid mass transfer operations, the removal efficiency depends directly on the mass transfer rate, the absorption rate (which accounts for the solubility) and on the contact time. According to various mass transfer theories (Higbie or Danckwerts theories for example), the mass transfer rate in both phases are pro­portional to the square root of the diffusion coefficient in each phase [34]. A low mass transfer rate in the liquid phase can significantly hamper the VOC removal and is therefore a key point to control for a liquid ionic selection.

With traditional solvents (water and organic solvents), the diffusion coefficient of non ionic solute increases with the temperature and decreases with the solvent dynamic viscosity (jxj) and the solute molar volume (Vm i):

Several theories are currently applied to calculate diffusion coefficients in traditional solvents (Stokes-Einstein, Wilke-Chang, Arnold, Hayduk-Laudie, Schreibel, etc.) [27, 33]. Except the Arnold theories, a = в = 1 and 1/3 < у < 0.6. The solvent molecule size is also sometime taken into account. Thereby, it is admitted that the diffusion coefficient increases with the loading of the solvent in solute. For mass transfer operations, the infinite dilution diffusion coefficient, with an order of magnitude of 10~9 m2.s-1, is generally used in the computation [35].

Several experimental techniques have been applied for the determination of the diffusion coefficients of various solutes (CO2, alkanes, alkenes and hydrofluor­ocarbons) in ionic liquids. All these techniques are based on two methods: the thermogravimetric method or the manometric method. These methods enable to investigate at the same time the solubility and the diffusion properties of a solute in a given solvent. They are usually conducted in static mode to avoid convection contributions. The manometric method is based on the measurement of the pressure decay in a thermo-controlled cell chamber which contains a layer of the investigated solvent. An accurate amount of gas, often provided by a pressur­ized feed chamber, is introduced rapidly at the beginning of the measurement. The diffusion coefficient is determined by fitting the pressure decay to a one-dimensional diffusion model for solute uptake into the liquid [28, 36, 37]. An alternative manometric method consists to immobilize the solvent in a membrane which separates the feed and the cell chambers [27, 29, 30]. This technique is often called the two-cells methods or the lag-time technique. The thermogravimetric method is a relatively recent method based on the measure­ment of the solute uptake in the investigated solvent using a microbalance [31, 3840]. Buoyancy corrections are necessary to take into account the expansion of the solvent due to the solute absorption during the experiment. The feed gas is usually pure.

For both techniques, vacuum is applied to the cell chamber before the gas sample introduction. Depending on the thickness of the solvent layer, several dimensional diffusion models have been used to determine by fitting the diffusion coefficient (thin-film model, semi-infinite model, etc.).

Diffusion coefficients of small solutes such as VOCs in ionic liquid are usually one or two order of magnitude lower than in traditional organic solvents (in the range 10~n-10~10 m2.s-1) (Table 12.1) mainly because of the high viscosity of the RTILs. Diffusivity drops more significantly in RTILs than in traditional solvents when the temperature decreases (lower molecular agitation and larger viscosity). Several studies demonstrate that this evolution follows the Arrhenius law, with activation energies typically larger than traditional solvents in the range 10-25 kJ. moP1 [27, 36, 38].

Table 12.1 Viscosity and molar volumes of the various solvent investigated by Scovazzo et al. and Camper et al. Values of the diffusivities at 303 K for several solutes [2730]

RTIL (solvent j)

Hj (cP)

V

Vmj

(cm3/

mol)

Оідтіь at 303 K (1010 m

2.s-1)

CO2

Ethene

Propene

1-Butene

Butadiene

Methane

Propane

[C2Mim][NTf2]

26

258

6.6

5.1

3.3

2.7

3.7

[C2Mim][NTf2]a

27.8

8.1

7.4

4.3

5.1

[C2Mim][TfO]

45

188

5.2

4.5

2.6

2.2

3.0

[C2Mim] [BETI]

77

294

4.5

2.5

1.7

1.2

1.7

[C4Mim][PF6]

176

209

2.7

2.0

1.1

0.8

1.2

[C4SO2Mim] [TfO]

554

210

1.1

0.7

0.5

0.3

0.4

[P2444][DEP]

207

323

3.5

2.1

1.2

0.78

1.46

[P(14)666][DCA]

213

578

3.0

2.2

1.36

1.08

1.67

[P(14)666][NTf2]

243

804

6.2

2.9

2.6

1.98

2.9

[P(14)666][Cl]

1,316

590

3.0

2.1

1.61

1.15

1.73

[P(14)444][DBS]

3,011

731

1.7

1.06

0.65

0.41

0.65

[N(4)111][NTf2]

71

289.6

4.87

2.29

1.73

1.37

1.95

NQb

1.16

[N(4)113][NTf2]

85

315.4

4.83

2.67

2.02

1.33

2.13

NQ

1.09

[N(6)111][NTf2]

100

324.5

4.38

3.19

1.39

1.39

2.14

NQ

1.02

[N(6)113][NTf2]

126

353.1

3.72

NQ

1.59

1.33

2.12

1.22

1.02

[N(6)222][NTf2]

167

365.8

4.68

1.90

1.56

1.02

1.46

NQ

0.88

[N(10)111][NTf2]

173

393.2

4.60

3.07

2.37

1.67

2.63

2.64

1.32

[N(10)113][NTf2]

183

426.4

3.78

3.44

1.73

1.59

2.06

[N(1)444][NTf2]

386

383.5

3.41

0.90

0.62

1.13

3.41

[N(1)888][NTf2]

532

600.6

3.43

2.70

1.58

1.37

2.33

NQ

1.01

aValues of Camper et al. [28] bNQ means Not Quantifiable

Scovazzo and coworkers investigated by the lag-time technique the solubility and diffusion of CO2 and several VOCs (ethylene, propylene, 1-butene, butadi­ene, methane, butane) at 303 K in five imidazolium, five phosphonium and nine ammonium based RTILs, covering a large range of viscosities (from 10 to 3,000 cP) [27, 29, 30]. They found that the value of a depends on the kind of cation (0.66 for the imidazolium, 0.35 for the phosphonium and 0.59 for the ammonium based RTILs) whereas the Stokes-Einstein and the Arnold theories predict respectively 1 and 0.5. у was equal to 1.04 for imidazolium, 1.26 for phosphonium and 1.27 for ammonium based RTILs. Therefore, diffusivity in RTILs is less dependent on viscosity and more dependent on solute size than predicted by the conventional Stokes-Einstein model. As mentioned by Morgan, the deviation between RTILs and traditional organic solvents may result from the physical situation of small solutes diffusing in an universe of large IL molecules solvents [27].

At identical viscosities, diffusivities are larger in phosphonium, then in ammo­nium, then in imidazolium based RTILs. This may be explained by the increased molar volume of the phosphonium based ILs («600 cm3.mol-1 compared to approximately 400 and 200 cm3.mol_1 for respectively ammonium and imidazolium based ILs) which can allow for faster diffusion rates. Skrzypczak and Neta suggest that this trend is due to the increased number and length of the aliphatic chains present on the phosphonium cation [41]. Because the chains are flexible and can move more rapidly than the whole cation, they enable a more rapid diffusion of solutes from one void to another in the phosphonium-based ILs [41]. Therefore, the amount of free volume in an anionic liquid could be a better indicator of diffusivity than viscosity.

1,3-butadiene diffuses faster for an identical molar volume than the other alkene due to a possible weak complexation of the conjugated double bonds with the positively charged cation which facilitates the transport [30]. All these conclusions suggest that finding a universal correlation to determine diffusivities for all classes of RTILs and solute is utopist, even if similar trends are observed for different kind of RTILs. Moreover, it is also important to consider the effect of other impurities and large solute concentrations on diffusion. It has been shown that water and other co-solvents increase very significantly the viscosity and consequently the diffusiv — ity. Therefore, the diffusivities should be measured for any couple RTIL-solute to assess the mass transfer rate.

Camper et al. measured the diffusivities of several VOCs (ethane, ethane, propane, propene) and CO2 on [C2Mim][NTf2] for various temperatures using the manometric method and a semi-infinite model [28]. They confirmed the same trends and the order of magnitude found by Morgan et al. with the same solute-RTIL couple, even if the values of Morgan were 20 % lower (Table 12.1).

Shiflett and Yokozeki investigated the solubility and the diffusion of hydrofluorocarbons in several RTILs (mainly imidazolium based RTILs) by the thermogravimetric method in isothermal mode for pressure up to 20 bar [31, 39, 40]. The effective measured diffusion coefficients increase with the pressure and the temperature applied in the chamber. Indeed, a larger pressure applied increases the solute uptake in the solution. Therefore, the determined diffusion coefficients are not determined at infinite dilution. The values found for the various diffusion coefficients are all included between 2 • 10~n and 8 • 10_11 m2.s-1 at 298.15 K.

Except for the few VOCs presented before, the most investigated solute remains CO2. Even if this compound is out of the scope of this review, this study is worthwhile since it enables a comparison between the different mea­surement techniques. Hou and Baltus investigated CO2 solubility and diffusivity in five imidazolium based RTILs by the manometric method and using a transient thin-liquid film model [36]. Their results are consistent with the results of Camper et al. and Morgan et al. deduced with manometric methods (Table 12.2) [27, 28]. However, the values found using the thermogravimetric method were considerably smaller (five times) [38]. Hou and Baltus suggest that the thermogravimetric method present a larger uncertainty due to several buoy­ancy corrections necessary and they questioned the fact that the measurements

Table 12.2 Values of CO2 diffusion coefficients in three RTILs reported in the literature

RTIL

Di, RTiL (1°10m2.s 1)

Refs.

Method

Water

22.5 (303 K)

[34]

[C2Mim][NTf2]

6.6 (303 K)

[27]

Lag-time technique (manometric method)

8.1 (303 K)

[28]

Pressure decay technique (semi-infinite model)

[C4Mim][BF4]

1.7 ± 0.6 (323 K, 1-10 bar)

[38]

Thermogravimetric method

4.8 (323 K)

[36]

Pressure decay technique (transient thin liquid film model)

1.8 (323 K, 20 bar)

[42]

Expansion measurement with a cathethometer

[C4Mim][PF6]

1.2 ± 0.3 (323 K, 1-10 bar)

[38]

Thermogravimetric method

0.8 (323 K, 20 bar)

[42]

Expansion measurement with a cathethometer

were performed at large pressure which might influence the density and the viscosity of the RTIL [36]. Whatever, the difference between the two methods remains controversial and poorly commented in the literature. Some systematic and/or random errors would be responsible for the discrepancy among research groups. Further investigations to try to understand the difference between both methods would be interesting.

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