The liquid–liquid coexistence of binary mixtures of the room temperature ionic liquid 1-methyl-3-hexylimidazolium tetrafluoroborate with alcohols

Precise coexistence curves are reported for the liquid–liquid phase transition of binary solutions of the room temperature ionic liquid (RTIL) 1-methyl-3-hexylimidazolium tetrafluoroborate (C6mim+BF4) in a series of alcohols (1-butanol, 1-pentanol, 2-butanol, and 2-pentanol).

The phase diagrams are determined by measuring the temperature dependence of the refractive index in the two phases of samples of critical composition.

The critical data of the systems are in the region predicted for the model fluid of equal-sized, charged, hard spheres in a dielectric continuum, the so-called restricted primitive model (RPM).

Therefore, the phase transition can be classified as essentially driven by Coulomb interactions.

The effective exponents βeff determined are close to the universal Ising value, where the deviations are found to be negative, when the volume fraction or the mass fraction are chosen as concentration variable.

The negative values of the first Wegner correction indicate non-uniform crossover from Ising to mean-field criticality.

The diameter of the coexistence curves shows the non-analytic temperature dependence typical for Ising systems.


Recently we have reported a survey on the location of the liquid–liquid phase transition1 in binary solutions of room temperature ionic liquids (RTIL) in water and in a series of alcohols.

The RTILs considered contain a 1-methyl-3-alkylimidazolium cation (Cnmim+, n = 4, 6, 8) and the PF6 or BF4 anion.

A first detailed investigation of critical properties of such systems concerned the viscosity in binary mixtures of the RTIL C6mim+BF4 with 1-pentanol.2

Ising criticality with crossover to regular behaviour was observed in accordance with viscosity measurements on other solutions of low melting salts.3–5

In this work, we continue investigating critical properties of binary mixtures of RTILs with non-ionic fluids analysing coexistence curves of solutions in alcohols.

While in ref. 1, separation temperatures were determined by visual inspection when cooling down homogenous mixtures, in this work, we investigate flame sealed samples of critical composition by a laser technique.

We determine the refractive index of the coexisting phases and of the one-phase region as function of the temperature applying the minimum beam deflection method.

This highly accurate method6–10 allows determining critical exponents, which cannot be achieved by the visual method applied in ref. 1 and in the work of other authors, who reported phase diagrams of other RTILs in alcohol solutions.11–13

Coexistence curves of solutions of RTILs are of technical interest in view of applications in chemical engineering as reaction media and in separation processes.14,15

Reactions have been proposed that, taking advantage of phase transitions, enable elegant separation of products, catalyst and solvent by small changes of temperature or composition.16

Clearly, the knowledge of coexistence curves is essential for designing such processes.

From a scientific point of view, RTILs are of interest for studies of the nature of the critical point in liquid–liquid phase transitions driven by Coulomb interactions.

We recall that Ising criticality, generally observed in fluid phase transitions,30 requires short-range rn interactions with n > 4.97 (ref. 17,18) as driving potential.

Different critical behaviour is expected when long-range interactions17–20 drive the phase transition.

Mean field criticality of the van der Waals (vdW) type was conjectured in the case of Coulomb forces.21

First measurements of the coexistence curve and of the turbidity of the solution of triethylhexyl ammonium triethylhexyl ammonium borate (N2226+B2226) in biphenyl ether yielded indeed vdW mean field criticality,22,23 which stimulated theoretical24,25 and experimental work.

However, later experiments on this system, using samples that were tempered for some days before the measurement, did not confirm the observations of mean field criticality4,7 but reported Ising behaviour.

Measurements on other Coulomb systems, e.g. solutions of tetrabutyl ammonium picrate8,26 (N4444Pic) or ethyl ammonium nitrate27–29 (EAN) in higher alcohols also yielded Ising exponents for all properties.8,10,26

With the exception of the work refs. 22 and 23 all experiments3–5,7,8 and most simulations30–33 indicate that phase transitions driven by Coulomb interactions also belong to the Ising universality class.

Nevertheless, this matter is still under discussion.34,35

The salts investigated have a rather low melting point if compared to typical inorganic salts.

The critical temperatures Tc of the solutions are near room temperature.

Thus, mK-accuracy can be achieved easily.

For a review, see .refs. 36–38

However, all experiments remain suspect, because of the limited chemical stability of the organic salts.

The so-called Pitzer salt N2226+B2226 is a notoriously unstable compound.4,7

Consequently; rather different figures for Tc have been reported.

Instability of Tc during the measurements may also cause erroneous conclusions.7

The solutions of the picrates (NR4+Pic) also cannot expected to be perfectly stable, because, after all, the picrates are explosives.

The EAN–octanol solution decomposes already 20 K above the consolute temperature.27

Therefore, it is worthwhile to investigate the critical properties of solutions of RTILs, which now are commercially available in good quality, chemically stable, and therefore well suited for accurate measurements.

Corresponding states analysis of the location of the consolute point enables to distinguish phase transition that are driven by Coulomb interactions from those determined by solvophobic interactions.39

We have carried out such an analysis of the location of the consolute point based on a survey of more than 200 mixtures of RTILs.1

The reduced variables are defined by the restricted primitive model (RPM), a model fluid of equal sized, charged, hard spheres in a dielectric continuum.

Phase transitions with reduced critical data near the RPM figures are termed Coulombic, because the phase transitions are expected to be driven by Coulomb forces.36,39

The energy scale defining the reduced temperature T* is set by the Coulomb energy of the charges q± at the contact separation σ in a continuum with the dielectric constant ε.

The reduced density is defined by the total number density of the ions ρ = (N+ + N)/V and the volume σ3.

Monte Carlo simulations, which are accompanied by finite-size scaling techniques, yield the critical point of the RPM at Tc* = 0.049, ρc* = 0..0830–33

Binary solutions of organic salts in solvents of small ε, e.g. in higher alcohols have their consolute point in that region.

In water, where the phase transition is driven by hydrophobic interactions, which are short-range, the reduced critical data (Tc* = 0.6 and ρc* =0.1) become much larger, which is in the region typical for phase transitions driven by solvophobic interactions.1,24,36,39

Similar values apply for phase transitions of non-ionic systems,24 where vdW forces set the energy scale.

The remarkable result in ref. 1 was a nearly linear relation between Tc* and ε of the solvents.

Including water and higher alcohols, this observation suggests a continuous change from Coulomb phase transitions to such driven by solvophobic interactions with ε as the determining parameter.

The nature of the critical point in ionic systems is a puzzling problem.

To start with, the thermodynamic limit does not exist in systems with particles interacting by r−1 Coulomb forces.

The thermodynamic limit exists only for rn potentials with n > .317

However, in ionic systems, long-range Coulomb interactions become effectively short-range40,41 because of shielding due to Debye–Hückel charge ordering, so that the thermodynamic limit exists.42

Monte Carlo simulations of the RPM yield Ising critical exponents30,32,33 or, at least, are consistent with Ising values.31

Ising critical behaviour is also obtained for the general primitive model, where both, size and charges of the ions may be different.43,44

However, in ionic solutions other long-range interactions may influence the criticality.

Charge-induced dipole interactions and the so-called charge cavity interactions that vary as r−4 are present.45

In 2.1 we will outline that in cases when long-range rn interactions with 3 < n < 5 determine the phase transition the critical exponents will deviate from the Ising values.

However, Debye–Hückel charge ordering can be expected to shield all electrostatic interactions, so that all long-range interactions, e.g. the mentioned charge-induced dipole interactions may become effectively short range.40,41

Therefore, the conservative expectation is: Ising criticality with crossover to vdW mean field behaviour at larger separation from the critical point.26,46,47

The crossover is determined by the Ginzburg temperature.48

Theory predicts a Ginzburg temperature for Coulomb phase transitions, which is large if compared to non-ionic systems and therefore implies a non-classical Ising region that is even larger than in non-ionic systems.49,50

In variance to this prediction, experiments indicate a crossover to vdW mean field criticality in a smaller temperature region above the critical temperature than in normal non-ionic systems.26

Semi-empirical crossover theory allows describing the experiments,46–49 but a physical explanation is not available.

A hypothesis, which could explain the reported crossover at rather small separation from the critical point, is a scenario involving a tricritical point.

A tricritical point arises, when a line of second order transitions cuts the coexistence curve at the critical consolute point.17,51

Even if this condition is not exactly met, the coupling of the two fluctuations is expected to change critical properties, e.g. the shape of the coexistence curve.52

In ionic fluids, order transitions between an insulating and a conducting state or between a uniform fluid and a charge ordered state53 might be thought of.

Therefore, precise measurements of coexistence curves are required to judge the validity of those theoretical reasoning.

In this work, we present measurements of the coexistence curves of solutions of C6mim+BF4 in the alcohols 1-butanol, 1-pentanol, 2-butanol and 2-pentanol.

In samples of critical composition we determine the refractive index in the homogeneous phase above Tc and in the two phases below Tc using the minimum beam deflection method.6

The coexistence curves are calculated from the refractive index data and compared with the results obtained by determining the separation temperatures in a set of mixtures of given concentration1.

Theoretical background

Critical exponents for long-range potentials

The power n of rn potentials may be written n = d + s, where d is the dimension of the system.

For s ≥ 2 − ηsr the potential is termed short range and phase transitions determined by such potential belong to the Ising universality class17,51.η is the so-called Fisher exponent, which corrects the classical Ornstein–Zernicke correlation function and assumes the value ηsr = 0.03 in (d = 3)-Ising systems.

The common scenario for fluid phase transitions driven by short-range interactions is Ising criticality in the asymptotic region with crossover to vdW mean field behaviour at large separations from the critical point.48

For s < 0 the thermodynamic limit does not exist.17

Potentials with 0 < s < 2 − ηsr are termed long-range potentials.

Theory predicts for potentials with 0 < s < d/2 the following set of critical exponents19,20ν = 1/s, η = 2 − s, γ = 1,which are termed mean field exponents.

The exponents ν and γ determine the temperature dependence of correlation length ξ and susceptibility χ, respectively.

The vdW mean field exponents, conventionally called mean field exponents, result from a mean field theory for fluids with particles interacting by a short-range potential.

The exponents given in eqn. (2) become identical with the vdW mean field exponents for s = 2.

Based on the hypothesis that thermodynamic functions are homogeneous functions, relations between the various critical exponents have been derived,17,51 which served as a guide in the development of renormalisation group theory of critical phenomena.

For convenience, we give the four relations and apply them to calculate the other mean field exponents:γ = (2 − η)ν, γ = β(1 − δ), γ = 2 − α − 2β, = 2 − α.

The last relation of eqn. (3), termed the hyperscaling relation, applies only for d ≤ 4.

Therefore, d = 4 is termed critical dimension dc.

The critical exponent of the specific heat is denoted by α, the exponent δ relates the field to the order parameter at critical temperature.

In the context of the phase transitions of fluids, the exponent δ determines the divergence of the osmotic susceptibility χ, when, at critical temperature, the variable X of the composition approaches the critical value:χ ∼ |XXc|1−δ.

The osmotic susceptibility can be determined by measuring the scattering intensity.

The mean field exponents (eqns. (2)) satisfy the relation γ = (2 − η)ν.

The other two relations determining γ and the hyperscaling relation can be used to calculate the exponents α, β and δ yieldingα = 2 − d/s, β = (ds)/2s, δ = (d + s)/(ds).For s = d/2 the exponents α, β, γ, and δ agree with the vdW mean field coefficients 0, 1/2, 1, 3, respectively.

Only if d = 4 the exponents ν and η also agree with the vdW mean field values.

However, theoretical analysis requires that for long-range interactions the critical dimension dc depends also on the power of the potential according dc = 2s.17

Furthermore, the dimension d in the hyperscaling relation has to be replaced by the critical dimension.

With this setting, the coefficients α, β, γ and δ agree with the vdW mean field values whenever long-range interactions drive a phase transition, while ν, η assume the vdW mean field values only for s = 2.

First simulations54,55 of fluids with long range potentials 0 < s < 2 yield β/ν = 0.8 for s = 1 and d = 3.

This result is between the estimates β/ν = 1 resulting from eqns. (5) and β/ν = 1/2 obtained with dc = 2, while the Ising value is β/ν = 0.515.

The power laws and corrections to scaling

Phase transitions in fluids that are driven by short-range interactions belong to the Ising universality class.

However, the simple power laws involving the universal critical Ising exponents are valid only in the asymptotic region near the critical point.

In general a crossover theory48 should be applied to analyse the date in a wide temperature region.

In fluid mixtures the asymptotic power laws commonly hold in the region τ = |TTc|/Tc < 10−3.

In the region 10−3 < τ < 10−2 corrections to scaling56,57 by power series in τ may suffice.

Considering the coexistence curve, the difference ΔX of the composition in the two coexisting phases vanishes as the critical temperature Tc is approached according to the following scaling law termed Wegner expansion:|XuXl| = β (1 + B1τΔ + B2τ2Δ + ⋯).

Xu and Xl represent the compositions in the upper and in the lower phase, respectively.

While the exponents β = 0.325 and Δ = 0.51 are universal for an Ising critical point, the amplitudes B of the coexistence curve and the amplitudes of the correction terms are specific to the system, but not independent.

There are rather strict conditions on size and sign of the terms in the Wegner expansion.

However, regular terms, which are not part of the Wegner expansion, may also contribute to the fits of the experimental data.47,54,55

Therefore, we apply the expansion (6) just as a tool to fit the data and do not claim to get Wegner coefficients in its strict sense from the data analysis.

The diameter Xm = (Xl + Xu)/2 is also a non-linear function of τ and may be represented56,57 by the seriesXmXc = + 2β + 1−α (1 + D1τΔ + …)involving the critical exponent α of the specific heat with α = 0.11.

Note, that in a vdW mean field system with α = 0 and β = 1/2 the diameter becomes a linear function of τ, thus satisfying the rule of the rectilinear diameter.

The temperature dependence of the diameter has been a matter of controversy for a long time.58

The 2β term is commonly regarded as a spurious contribution, which occurs when a “wrong” concentration variable56,57 is chosen for the data analysis.

This qualification of the 2β term, however, has been questioned recently.59

Furthermore; the deviation from rectilinear diameter is often small and not observable in many cases.

Therefore, it is difficult to determine uniquely the various coefficients in eqn. (7) in a numerical analysis of experimental data.

The composition variable

On experimental grounds, many choices for the composition variable X may be used with equal validity e.g. the mole fraction x, the mass fraction w or the volume fraction φ.

There is no a priory reason to choose one variable over the other.

Japas and Levelt Sengers named some criteria for selecting the best variable:60 simple scaling laws should hold over the largest range, where β assumes the Ising value; the critical composition should be near 0.5 and the coexistence curve should be almost symmetrical.

The asymmetry should be determined by the (1 − α) anomaly only.

Therefore, in the data analysis various choices of the concentration variables may be considered.

If, e.g. the coexistence curve is determined by measuring the transition temperatures in a series of samples of given composition, the mole fraction x or the mass fraction w are known and appear to be the natural choices to represent the composition.

In general, any concentration variable X can be transformed into a certain desired new variable Y by a transformation61 of the formIf the mole fraction x is transformed into the mass fraction w, the parameter p becomes p = M2/M1, where M1 is the molar weight of the compound with mole fraction x.

In general the parameter p may depend on temperature and composition as, e.g., in the transformation of the mole fraction x into the volume fraction φ, where p = V2/V1 is the ratio of the partial molar volumes.

As an approximation to the volume fraction an ideal volume fraction φ0 may be defined, in which the excess volume is neglected and p = V20/V10 is given by the molar volumes of the pure components.

The transformation eqn. (8) may also be applied to construct a symmetrical coexistence curve represented in terms of a new variable.

The transformation of the mole fraction x into a variable, which fixes the critical composition to Y = 0.5 requires p = xc/(1 − xc).

The thermodynamic analysis of Anisimov et al62. appears to remove the arbitrariness of the composition variable.

In the Landau theory the free energy density is expanded.

In this thermodynamic potential, the variable is the density and the corresponding field is the chemical potential.

Therefore, in mixtures the number density (concentration) of one of the components should be chosen as variable, which is ρ1 = φ1/V10.

Because in the investigated temperature range the molar volume V10 is changing very little, the volume fraction is also an appropriate concentration variable.

The density is not identical with the order parameter M.

The order parameter M is a linear combination of density and entropy density, where the (1 − α) term in eqn. (7) represents the entropy density2Mu,l = ±β (1 + B1τΔ + …) − 1−α.Eqns.

(6) and (7) result as difference or sum of the two branches given by eqn. (9).

The order parameter M is the variable in the crossover theory.47,48

The application of this rather involved approach is outside the scope of this work.

Determination of the composition from refractive index data

If the phase diagrams are determined by measurements of the refractive index of the phases in the sample, the refractive index n or the Lorenz–Lorentz function (n2 − 1)/(n2 + 2) may directly be taken as measure for the composition.

Otherwise, the Lorenz–Lorentz relation can be used to determine the concentration.

The Lorenz–Lorentz relation connects the averaged polarizability 〈αi〉 and the number densities ρi = Ni/V of the components of a mixture to the refractive index, where Ni is the number of particles of the component labelled i.

For a binary mixture we have

The averaged polarizabilities 〈α1〉 and 〈α2〉 are nearly independent of composition and temperature.

With the total number density ρ = ∑iNi/V and ρi = xiρ the Lorenz–Lorentz relation may be reformulated in terms of the mole fractions xi or in terms of the volume fractions φi = ρiVi.

For a binary mixture the Lorenz–Lorentz relation reads in terms of the mole fraction x of the component labelled 1The density may be written in terms of the partial molecular volumes vi = Vi/Ni asFor pure compounds eqn. (10) becomeswhere 〈α10 and vi0 denote the average molecular polarizability and the volume per molecule in the pure components, respectively.

In many applications, the Lorenz–Lorentz function eqn. (10) is identified with the ideal expression based on the parameters of the pure compounds:

In our analysis, it turns out to be necessary to supplement the ideal expression by an excess term, which takes care of excess volume and excess contributions to the polarizabilties.

With the Porter-Ansatz for the excess function and the assumption that the excess function assumes their maximum value at critical composition we getK = Kid[1 + 4Y (1 − Y)(KcKcid)/Kcid],where Kc is the value of the Lorenz–Lorentz function at critical composition.

The variable Y was defined in eqn. (8), where p = xc/(1 − xc).

Eqn. (15) is used in the final data analysis.


Sample preparation

Samples of critical composition were prepared for solutions of the RTIL C6mim+BF4 in alcohols (1-butanol, 2-butanol, 1-pentanol and 2-pentanol).

The alcohols with certified purity (1-pentanol (Fluka) > 99%, 1-butanol (Fluka) (HPLC), 2-butanol (Fluka) > 99.5%, and 2-pentanol (Sigma-Aldrich) > 99%) were used without further purification.

The ionic liquid C6mim+BF4 was purchased from Solvent Innovation.

Standard NMR-, MS- and chromatographic analysis did not show impurities.

Traces of water were removed from the salt by keeping it for three days at 60 °C under oil-pump vacuum and storing it in a desiccator.

Solutions were made up by weight with a precision of ±0.1 mg.

The critical compositions for the systems have been determined before and are given in .ref. 1

The samples were prepared in standard square 10 mm cells (Hellma, PY 221).

The critical solutions were filled into the cells using a syringe and a septum in order to prevent condensation of moisture in the sample.

The samples were flame sealed under vacuum after a pump and freeze procedure.

The criticality of the samples was checked employing the equal-volume criterion.

The cloud points were determined visually by repeated cooling the homogeneous solution in a thermostat (Schott) with glass windows filled with water.

The temperature was determined with an accuracy of 0.01 °C using a Quartz thermometer (Hereus QUAT200).

Refractive index measurements

The refractive index was measured in the uniform phase above the critical temperature and in the two-phase region by means of the minimum beam deflection method.6

The coexistence curves of the system were determined from the refractive indices of the upper and of the lower phase.

The optical arrangement and further details are described in .refs. 7–9

In order to prevent the formation of meta-stable states the temperature steps were increased when lowering the temperature in the two-phase region.

A waiting time of about 8 h was necessary to achieve a complete phase separation.

Equilibrium was assumed when the two phases were no longer opalescent, and the position of the laser beam on the screen did not change any more.

By checking the critical temperatures before and after the refractive index measurements, we tested the stability of the critical temperatures.

A shift of the critical temperatures of −8 × 10−8 K s−1 was observed during the measuring time and taken into account in the data evaluation.

As the critical temperature, we choose the average between the last measurement in the homogeneous phase and the first point in the two-phase region, which limits the accuracy of the critical temperature to 0.005 K. The data analysis was carried out using the Origin 6 and Mathematica 4 program packages.

Experimental results and data evaluation

Phase diagrams with the Lorenz–Lorentz function as variable

The coexistence curves of the binary solutions of C6mim+BF4 in 1-butanol, 2-butanol, 1-pentanol, and 2-pentanol were obtained by refractive index measurements.

In addition, the refractive index of the pure compounds was determined by the same method.

The temperatures and the corresponding refractive index data in the one phase region and in the coexisting phases below Tc are given in the electronic supplement. The upper phase is the alcohol-rich phase and has the lower refractive index.

The small differences of the refractive indices between salt and solvents limit the relative accuracy of the measurements.

In fact, the refractive indices of the alcohols and of the salts are so similar that it is difficult to see the meniscus.

The coexistence curves obtained from the refractive index measurements are shown in Fig. 1.

The reduced temperature τ is plotted as function of KKc.

We employ the Lorenz–Lorentz function K as variable instead of the often-used refractive index8,22 because it is more directly related to thermodynamic quantities than the refractive index.

At first, we check the accuracy of the critical composition deduced from the equal volume criterion.

It can be seen that the Lorenz–Lorentz function K in the homogeneous phase and the diameter Km, which is the mean of the K values in the coexisting phases, meet at the critical temperature.

No offset is noticeable, which proves that our samples have the critical composition.

The coexistence curves with K as composition variable are strongly skewed like the top of a banana.

Due to the banana shape, two temperatures correspond to the same K value.

In the upper phase (alcohol-rich phase with the lower refractive index), the variation of K with the temperature is very small.

The variation of the refractive index in the salt-rich, lower phase is much larger.

The coexistence curves of the solutions in 1- and 2-butanol are wider than those in 1- and 2-pentanol.

Obviously, the more polar alcohol can mix with the salt better.

There is little difference between the isomers; the width of the coexistence curves of the secondary alcohols is only slightly larger than that of the primary alcohols.

In the one-phase region, the Lorenz–Lorentz function is reduced linearly with increasing temperature.

This can be expected from the linear decrease of the density with temperature raise.

No indication of a non-analytic critical contribution63 can be seen.

In contrast, the diameter shows a marked nonlinear temperature dependence: the rectilinear diameter rule clearly does not apply.

Estimates of the critical composition based on the rectilinear diameter rule are necessarily far off.

The slope of the diameter has the same sign as the slope in the one-phase region.

The temperature dependence of the diameter is slightly stronger than the temperature dependence of K in the homogeneous phase.

Both, the temperature variation of K in the one phase region and of Km are almost the same for different mixtures.

The curves, which concern the mixture with 1-butanol, are the best fits using eqns. (6) and (7), which will be dicussed in what follows.

In Table 1, we give parameters obtained from fitting different functions to the experimental data of the coexistence curve.

As a routine we apply first a simple power law, where the exponent βeff is a free parameter, and then use the Wegner-type expansions for the fit with up to two coefficients.

In the expansion the exponents β and Δ are fixed, while the amplitudes B, B1, and B2 are the fitting parameters.

The effective exponents βeff are found to be smaller than the Ising value, which may be taken as indication of non-monotonous crossover to mean field criticality.

Accordingly, the first correction term in the Wegner-type expansion comes out negative.

However, conclusions based on the size of the coefficients are difficult, because their values change, when the second correction is included into the fit.

The statistical error estimated for the parameters becomes then unduly large, which shows that the parameters are not independent.

Hence, the expansion with two correction terms is not appropriate, therefore not included in Table 1.

Comparing the deviations from the asymptotic Ising behaviour for the different alcohols no obvious systematic can be seen.

The diameters Km of the coexistence curves are analysed by fitting different approximants of eqn. (7) to the data, where the amplitudes A, C, D and Kc are the fitting parameters, while the exponents are fixed to their universal values.

At first, we employ one-term expansions with Kc and one of the amplitudes A, C, or D as fit-parameters.

In a next step, a two-term expansion is used containing the linear term A and either the 1 − α term D or the 2β terms C.

The relevant results of the fits are given in Table 2.

The fits, which consider only the 1 − α term, are already rather good.

Including the linear term improves the fit although the statistical uncertainty becomes too large.

The value of the corresponding parameter D changes only little when the linear term is included in the fit.

The fits which include the linear and the 2β term are not as good as those, which include the linear and the 1 − α term and in most cases even worse than those that involve the 1 − α term only.

In Fig. 2a we show the log–log plots of ΔK = KlKuvs. the reduced temperature τ for the four systems.

The sets of data points are shifted by an offset for visual clarity.

The points appear to follow straight lines.

Deviations for τ < 10−4 are due to the rather small difference of the refractive index between the phases and due to the limited accuracy of the critical temperature, which was only ±0.005 K. Above τ = 10−4 no change of the direction can be seen although the measurements cover a temperature range up to 10 K from the critical point.

The lines in Figs. 2a and 1 represent the best fits obtained with eqn. (6) with one Wegner correction.

Fig. 2b shows the diameters of all investigated samples as function of the reduced temperature τ in a logarithmic scale.

Again, the curves are shifted by arbitrary factors for a better view.

The curves in Figs. 1 and 2b are the fits with Kc, A and D as fitting parameter.

Phase diagrams with thermodynamic concentration variables

The banana shape of the coexistence curves obtained shown in Fig. 1 indicates that K is not a good choice of a variable.

Furthermore, in the homogeneous phase the refractive index varies with temperature, although the relative composition of the components is unchanged.

Therefore, composition variables, which are constant in the homogeneous phase, like the mole fraction x or the mass fraction w appear more appropriate.

From the theoretical point of view, the volume fraction φ is the best choice of a concentration variable.

Dividing by the molar volume, which is almost independent of temperature in the critical region, it can easily be transformed into the density required in the advanced theoretical analysis.47,48

In order to investigate the influence of the choice of the variables on the parameters in eqns. (6) and (7), we transform the refractive index data into the concentration variables x, w, and φ.

The data are reanalysed in terms of those variables.

For estimating the mole fraction of a mixture from refractive index data it is necessary to know the refractive indices and the densities of the pure components.

The refractive indices of the pure components were also determined by the minimum deflection method.

Density data of the alcohols are collected from standard sources.64

The densities of the salt have been measured using a pycnometer.2

The resulting molar volume of the salt is Vs = (224.76 + 0.13324 ΔT) cm3 mol−1 agrees rather well with Vs = (227.116 + 0.1575ΔT) cm3 mol−1T = T − 318.15 K) obtained by linear interpolation of the volumes of the salts C4mim+PF6, C8mim+PF6 and C8mim+BF4,65 which was assumed in the analysis in .ref. 1

The data of the pure compounds required for the transformations are summarized in Table 3.

In the table, we give the refractive index n298, the polarizability α298, the mass densities ρm298, and the linear temperature coefficients n1, α1 and ρ1 of the pure compounds.

In order to check the accuracy of the transformation, we calculate the value of K at the critical point assuming ideal mixing properties and vice versa recalculate the mole fraction of the critical sample from the refractive index data.

As can be seen in Table 4, the relative accuracy of the estimate of K is 0.001, while relative accuracy of the mole fraction is only 0.1.

Both figures are not sufficient for our purpose.

In order to achieve the required accuracy it is necessary to take into account the excess of the mole refraction.

Therefore we apply the correction for non-ideal contributions as given in eqn. (15), which ensures consistent figures for Kc and xc within the accuracy of the measurements.

The excess of the Lorenz–Lorentz function is positive, while pycnometric measurements2 point towards a positive excess volume, which would account for a negative excess of the Lorenz–Lorentz function.

This observation indicates a positive excess of the averaged polarizabilities that overcompensates the effect of the small but noticeable positive excess volume.

Finally we compare the coexistence curves, which are based on the estimates for the composition obtained from the refractive index measurements with the phase diagrams, which are obtained by direct observation of the appearance of the meniscus in samples of given concentration.1

Fig. 3 shows the phase diagrams of the investigated systems with the weight fraction as concentration variable.

As can be seen, there is no substantial difference between the separation curves obtained by the two methods.1

This proves not only the reliability of our analysis, but it shows also that the compounds are sufficiently pure: The maximum of the phase diagram, determined by visual observation of the phase separation in a set of samples of different composition, is identical with the critical concentration.

In three-component systems, this is usually not the case.9

We mention that preliminary measurements with salts, which were not dried, yielded a critical point different from the maximum of the phase diagram.

In Fig. 3, we have included data of other solutions (water, 1-propanol and 1-hexanol) that were obtained by the visual method1 to emphasize the corresponding-state similarity of the phase diagrams.

The coexistence curves in terms of the new variables are shown in Figs. 4.

Representing the coexistence curves in terms of the variables x, w, and φ the banana shape that was found in the representation in terms of the Lorenz–Lorentz relation, see Fig. 1, disappears.

The representation in terms of the mole fraction is still rather skewed, while the mass fraction gives the most symmetrical shape.

In all figures, the coexistence curves of all four systems are very similar.

Only a minor difference between the solutions in the 1-alcohols and the 2-alcohols is noticeable.

The curves of the mixtures with the 1-alcohols appear slightly more narrow and symmetrical than those of the mixtures with the secondary alcohols.

The coexistence curves represented by the concentration variables mole fraction x, weight fraction w and the volume fraction φ are analysed in the same way as done in Section 4.1, where the Lorenz–Lorentz function was the concentration variable.

The results of the fits are given in Table 5.

In the fits, the critical temperatures Tc were fixed to the experimental values.

We start with a fit to a simple exponential with the exponent βeff as a fitting parameter.

With the exception of the temperature dependence of Δx, the analysis of the coexistence curves in terms of the other variables (Δw or Δφ) yield values for the exponent β that are smaller than the Ising value.

When the coexistence curves are represented by the mole fraction, we get figures slightly above or below the Ising value.

The results of the fits by a Wegner-type expansion with one correction term correspond to this analysis: small positive or negative values of B1, when x is the variable, or larger negative figures of B1 when w or φ is chosen.

As can be seen in Table 5, the fits to a single exponential, where the exponent β is a free parameter and to a one-term Wegner expansion with fixed exponents are equally good.

No systematic is noticeable, when the deviations from the Ising value are compared for the different solvents.

The representation of the coexistence curve by the mass fraction requires larger corrections than the volume fraction.

Fits involving two Wegner corrections are not included in the table because of the large uncertainty of the parameters obtained.

In Fig. 5a, we show the log–log plots of Δφ drawn as function of τ for the investigated systems.

The data follow straight lines that are almost parallel.

This indicates a very good representation of the data by an effective exponent, which is used to draw the lines.

It remains to analyse, how the choice of the variables influences the diameter of the coexistence curves.

As can be seen in Figs. 3 and 4 the diameters (Xu + Xl)/2 of the coexistence curves clearly show a nonlinear temperature dependence.

Again, the curves are not detailed enough to allow free fits of all the coefficients in eqn. (7).

Therefore, we start with one correction assuming either an exponent 2β or 1 − α.

As in the analysis of Km, the critical compositions Xc are treated as parameter in the fits.

In Table 6, we display the results of the fits with D as parameters and the fits with the two parameters A and D.

The results of the other fits are given in the electronic supplement. The agreement among the resulting values for Xc between the various fits and the estimates based on the known composition is of the order O(0.01).

Two terms were required to yield a reasonable fit.

In almost all cases, the combination of the linear term with the 1 − α term is superior to that with the 2β term, although the difference is not impressive.

The linear term is always negative.

The linear and the 1 − α term are always of the same order of magnitude but differ in the sign, which indicates that in the fit the parameters are coupled.

In fact, in the two-parameter fit the amplitude D is always about four times of that obtained in the one-parameter fit.

The Fig. 5b shows the diameter with the fit function when the volume fraction is chosen as concentration variable.

Concluding, we state that, independent of the concentration variable; the temperature dependence of the diameter is consistent with the assumption of a linear term and a non-analytic contribution with an exponent 1 − α.

The additional presence of a non-analytic 2β term cannot be excluded.


In this work we have investigated the coexistence curves of solutions of the RTIL C6mim+BF4 in 2-pentanol, 1-pentanol, 2-butanol, and 1-butanol.

For this purpose, we have measured the refractive index in critical samples of critical composition by applying the minimum beam deflection method.

Measurements on mixtures with higher alcohols turned out to be unfeasible because the difference between the refractive indices of the RTIL C6mim+BF4 and the alcohols became too small to allow for reliable measurements.

We first discuss the location of the critical points of the investigated solutions in the RPM phase diagram.

Assuming that the anion is located above the centre of the imidazolium ring and using vdW radii, we estimate for the distance of the ions in an ion pair 4.6 Å, which is in agreement with simulation66 and scattering67 results.

Using the reduced variables of the RPM, the values for the reduced critical temperature Tc* vary from 0.09 to 0.14.

We find a monotonous increase with the dielectric permittivity ε of the solvents, which is 10.2, 12.6, 15.2, and 16.6 for the alcohols 2-pentanol, 1-pentanol, 2-butanol, and 1-butanol, respectively.

The figures estimated for Tc* are above the value of 0.049, predicted by simulations of the RPM.

However, they are much below 0.65, found for this salt in water, where the phase transition is caused by hydrophobic interactions.1

Therefore, we expect that the phase transition is mainly driven by Coulomb interactions.

We note, that the variation of Tc* is in agreement with the first simulation results on ionic solutions.

The reduced critical temperature in mixtures of charged hard spheres and hard spheres68,69 comes out higher than in the RPM and increases further, when the hard spheres are replaced by dipolar hard spheres.68

Considering the reduced density, we find figures below the RPM value of ρc* = 0.08.

No simple correlation exists between the critical density and the dielectric permittivity of the solvents.

The primary and secondary alcohols form a group, where the critical reduced density is higher for the n-alcohols (1-pentanol: 0.063, 1-butanol: 0.065); the corresponding figures for the secondary alcohols are 0.056, and 0.060, respectively.

This indicates that packing effects modify the interactions of the ions with the solvent and influence the location of the critical point.

Such dependence on the structure of the solvent was already noted in the investigation of solutions of N4444+Pic in alcohols.8

We now turn to a discussion of the nature of the critical point.

The investigation of the coexistence curve is particularly appropriate for this purpose as, the relative change of the exponent β is larger than for the exponents ν or γ, when going from mean field to Ising behaviour.

Therefore, it appears promising to investigate the coexistence curve in order to trace eventual differences in the critical properties of ionic and non-ionic systems.

For our systems, we find effective exponents βeff for the coexistence curve that are near to the Ising value.

Within the accuracy of our measurements, we see no change of βeff with temperature.

The log-log plots Figs. 2a and 5a appear perfectly linear.

No curvature or change of the slope can be seen indicating a sharp crossover within a small temperature range.

A quantitative analysis of the corrections to scaling depends on the choice of the variable chosen to represent the coexistence curve.

Therefore, we will discuss this matter using the criteria given by Japas and Levelt Sengers.60

The mass faction is the variable, which yields critical points near w = 0.3, which is the value nearest 0.5 obtained for the set of variables considered.

The mole fraction gives the most asymmetric location of the critical point, which is xc = 0.11 for 2-butanol and 0.14 for 1-pentanol.

With K as variable, the critical compositions are Kc = 0.24 for all alcohols.

The volume fraction yields similar figures.

Clearly, the symmetry criterion supports the mass fraction, where wc = 0.3.

Judging, however, by the value found for the effective exponent βeff the mole fraction appears to be the best choice followed by the mass fraction.

The quantitative analysis of the diameter anomaly also rests on the chosen variable.

Here, the best choice appears to be the Lorenz–Lorentz function, because in the fit with one variable, the 1 − α term gives by far the best representation of the data and, furthermore, the amplitude D is almost unchanged when the linear temperature dependence of the diameter is taken into account.

From the phenomenological point of view, it can be said, there is no clear evidence why one of the variables should be preferred.

However, according to the theoretical analysis of Anisimov et al62. there is no ambiguity about the choice of the variable: The density of one component is the appropriate variable, which is the volume fraction divided by the molar volume.

Because the molar volume is changing very little in the investigated temperature range, it is trivial matter to transform the fit results for the volume fraction into such of the concentration.

Unfortunately, the analysis in terms of the crossover theory47,48 is not trivial and merits a separate paper.

Accepting the volume fraction as the best variable βeff is found to be significantly smaller than the Ising value.

The figures are in general agreement with the behaviour observed on the alcohol solutions of N4444+Pic.8 However, for solutions of N4444+Pic in long-chain alcohols like tetradecanol positive deviations from the Ising value were found, which became smaller with increasing value of ε and negative for 2-propanol.

2-propanol was the only alcohol with chain length <10 considered in this investigation.

In the work reported here, the polarities of the alcohols are between that considered in ref. 8 but near to that of 2-propanol.

Accordingly, we find negative deviations when the variables K, w, or φ are used.

Negative deviations indicate non-uniform crossover to mean field behaviour at higher temperatures.

No regular variation of βeff with the dielectric permittivity ε of the solvent can be observed.

Quite likely, the ε-range of the solvents used in this work is too small to establish such dependence.

The non-analytic temperature dependence of the diameter of the coexistence curves is a general property of phase transitions in fluids belonging to the Ising universality class.

The amplitude, however, depends on specific properties of the system.

In many systems, the amplitude is too small to be seen.

Substantial deviations from the rectilinear diameter rule are common in systems where the intermolecular interactions themselves depend on the density.

This is the case, e.g. in liquid-gas phase transition of metals, where one phase is an insulator and the other a molten metal.70

In a fluid of charged hard spheres the low-density phase can be pictured as a gas of ion pairs while the high-density phase appears as an expanded melt containing essentially free ions.53

In this picture the effective interactions are indeed density dependent, which may explain the observed irregularity.

With the exception of Pitzer′s system (N2226+B2226 in biphenylether)7 the diameter anomaly observed in the C6mim+BF4 solutions was observed in all other ionic systems investigated.8–10

A substantial diameter anomaly appears to be a signature of ionic systems.

Concluding, we state that the coexistence curves of the investigated systems agree remarkably well in the corresponding state representation.

We state Ising critical behaviour for a set of new nearly Coulomb systems and confirm the earlier results obtained on systems with rather limited chemical stability.

The coexistence curves show Ising critical behaviour with deviations.

No indication of a second maximum in the coexistence curves is noticeable, which is predicted for phase transition with strong coupling to a tricritical point.52

The deviations of the exponent βeff from the Ising value depend on the choice of the order parameter chosen for the analysis.

With the exception of the mole fraction, the analysis in terms of all other possible variables yields a negative deviation from Ising criticality, which may be taken as an indication of non-monotonous crossover.

Deviations from the asymptotic Ising exponent are noticeable in the complete temperature range investigated and not only in the region with τ > 10−2, where deviations from Ising critical behaviour are common.

A sharp crossover, reported from the turbidity measurements58 of solutions of picrates in alcohols, is not observed in the phase diagrams.

Measurements of higher accuracy in the mK-region are desirable, which are expected to show the change of the effective critical exponent from the asymptotic Ising value to the region with negative deviations, while investigations in a wider temperature range should show the increase of βeff towards the vdW mean field value.

Finally, we mention a result, which may be important in respect to applications of the RTILs in separation processes.

Already 10 K below the critical points the refractive index of the alcohol-rich phase is almost identical with that of the pure solvent.

Using the coefficients obtained for describing the coexistence curve, we estimate that the salt content is below 1%, while the equilibrium concentration of the alcohol in the salt-rich phase is substantial.