Research Article  Open Access
Daming Gao, Hui Zhang, Peter Lücking, Hong Sun, Jingyu Si, Dechun Zhu, Hong Chen, Jianjun Shi, "Computation of Isobaric VaporLiquid Equilibrium Data for Binary and Ternary Mixtures of Methanol, Water, and Ethanoic Acid from , , , and Measurements", Journal of Thermodynamics, vol. 2012, Article ID 641251, 13 pages, 2012. https://doi.org/10.1155/2012/641251
Computation of Isobaric VaporLiquid Equilibrium Data for Binary and Ternary Mixtures of Methanol, Water, and Ethanoic Acid from , , , and Measurements
Abstract
Vaporliquid equilibrium (VLE) data for the strongly associated ternary system methanol + water + ethanoic acid and the three constituent binary systems have been determined by the total pressuretemperatureliquidphase compositionmolar excess enthalpy of mixing of the liquid phase (, , , ) for the binary systems using a novel pump ebulliometer at 101.325 kPa. The vaporphase compositions of these binary systems had been calculated from and based on the function of molar excess Gibbs energy through an indirect method. Moreover, the experimental , data are used to estimate nonrandom twoliquid (NRTL), Wilson, Margules, and van Laar model parameters, and these parameters in turn are used to calculate vaporphase compositions. The activity coefficients of the solution were correlated with NRTL, Wilson, Margules, and van Laar models through fitting by leastsquares method. The VLE data of the ternary system were well predicted from these binary interaction parameters of NRTL, Wilson, Margules, and van Laar model parameters without any additional adjustment to build the thermodynamic model of VLE for the ternary system and obtain the vaporphase compositions and the calculated bubble points.
1. Introduction
New strategies for the correlation and accurate prediction of the vaporliquid equilibrium (VLE) data play a vital role in the distillation and separation process in chemical industry. The common technique for obtaining VLE data is by direct measurement on the system. That is to say, when the VLE is established, and phases are sampled and analyzed. Consequently, the experimental technique must be rather highly accurate to ensure meaningful results in the operation of equilibrium stills. Actually, when the vaporphase components are sampled and analyzed, the whole compositions of components in solution and vapor have been changed. Accordingly, the behavior of the systems has been changed with the amount of compositions. Moreover, it has been long realized that the analysis of vaporliquid composition for the infinite dilute solution is very difficult. In addition, for VLE measurements of mixtures containing a highly volatile compound, the accurate measurement of the vaporphase composition can be difficult. This fact, coupled with the necessity for much analytical work, tends to enhance interest in exploring new methods for the determination of equilibrium data that do not involve sampling and analysis of the vaporphase components.
Several methods have been explored for the calculation of component behavior from gross solution behavior. Van Ness and coworkers have suggested the classification of these methods into two categories, direct and indirect methods [1]. The direct methods involve calculation of vapor compositions by integration of the coexistence equation, a firstorder differential equation derived from the GibbsDuhem equation relating phase compositions at equilibrium. Hala and coworkers have given a detailed discussion of the basic direct method [2], and Van Ness and coworkers have discussed techniques for handling nonconstant temperature or pressure conditions as well as nonideal vapor phase behavior [1, 3]. Moreover, in the total pressure method one can calculate from , , measurements using an indirect method discussed by Mixon et al. [4].
Of all the methods, the indirect methods involve first the measurement, by some appropriate means, the temperature and total pressure of the system, the liquidphase compositions, and subsequent calculation of vaporphase compositions there from. These methods usually involve ascertaining which of selected solution equations to the GibbsDuhem equation lead to the best fit to the experimental data, and of the determination of the parametric values producing the best fit. For example, Barker has developed a procedure based on the assumption that the excess free energy can be represented as a polynomial function of composition [5].
There is a basic difference in the degree of rigor associated with the direct method and the indirect method of Barker. In the former, one makes no assumptions about the solution behavior for the nature of molecular interactions. Solution behavior is determined directly from the experimental data. The Barker method necessitates the assumption of a particular model and the estimation of its parameters, this deficiency in the method of Barker has been recognized by Tao, who has presented another indirect method in which the necessity for the a priori assumption of a particular functional form for the excess free energy has been removed. Tao’s procedure involves calculation of the activity coefficient essentially by integration of an equation resembling the coexistence equation. His procedure, though indirect, retains the rigor usually associated with the direct method [6]. The method of Tao appears specific to binary systems and does not seem to be easily generalized. However, an indirect method such as that of Barker is readily and easily generalized to ternary and higherordered systems, but this method retains the disadvantage of lacking rigor as compared with the direct method.
This paper presents the vaporliquid compositions calculated based on the measurements of VLE data for temperaturetotal pressureliquidphase compositionmolar excess enthalpy energy of mixing of the liquid phase (, , , ) at 101.325 kPa according to the function of molar excess Gibbs energy through an indirect method. We know that the reaction of methanol carboxylated with carbon monoxide is the most common and important technology for synthesis of ethanoic acid in the chemical industrial process. Modeling the thermodynamic properties and correlating and predicting the phase equilibria of a mixture involving associating components forming hydrogen bonding such as carboxylic acid remain a challenging problem, since such systems show extremely nonideal behavior and the formation of monomer, dimer, and even trimer in vapor and liquid phase. In addition, for the VLE measurements of vaporphase components containing a highly volatile compound such as carboxylic acid, the accurate measurement of the vaporphase composition can be difficult. Many attempts have been made to describe the vaporliquid equilibria of carboxylic acid containing mixtures using the concept of multiscale association [7]. Arlt reported the isothermal VLE data of a new apparatus for phase equilibria in reaction mixtures containing water with ethanoic acid and propanoic acid at (333 to 363) K [8]. Xu and Chuang have developed a new correlation for the prediction of the vaporliquid equilibrium of methyl acetatemethanolwaterethanoic acid mixtures [9]. Sawistowski and Pilavakis explored the vaporliquid equilibrium behavior of the quaternary system methyl acetatemethanolwaterethanoic acid modeled by using the Margules equation in combination with Marek’s method for the association of ethanoic acid [10]. Moreover, Guan et al. investigated that the isobaric vaporliquid equilibria for water + ethanoic acid + npentyl acetate, isopropyl acetate, Nmethyl pyrrolidone, or Nmethyl acetamide were correlated and predicted by both the nonrandom twoliquid (NRTL) and universal quasichemical activity coefficient (UNIQUAC) models used in combination with the HaydenO’Connell (HOC) method [11–14]. In our recent work, we have concluded that the VLE data for the associating systems containing the carboxylic acid system can be correlated and predicted [15, 16]. Although the VLE data of the mixture containing the associating systems were previously reported by the different research groups [17–21], respectively, the VLE data for the associating binary system containing the carboxylic acid have been still extensively studied because of the extensive association effects occurring in them and the difficulty of properly calculating the activity coefficients [22–25]. Nominally, the system is binary but in practice, it is multicomponent. Because the carboxylic acid monomer undergoes partial dimerization and even higher polymerization. This association is attributed to the formation of hydrogen bonds and occurs in both the vapor and liquid phase. Therefore, the challenge for the VLE data of the associating systems has evoked more and more researchers to focus on new strategies for exploring them. The VLE data for methanol + water + ethanoic acid ternary system and the constituent binary systems are indispensable in distillation separation process to the product of methanol carboxylation through the correlation and prediction by the new method, while some of the isobaric VLE data for these binary and ternary systems are correlated and predicted earlier [9–25]. To provide the new correlation and prediction for some necessary basic thermodynamic data on the separation process of methanol carboxylation, therefore, it is very indispensable for these systems studied on the VLE data of the constituent binary and ternary systems using the new method. This paper reports a novel correlation and prediction for the VLE data for these systems containing the associating component that has been developed. The VLE data for methanol + water + ethanoic acid system and constituent binary systems were measured by the total pressuretemperatureliquidphase compositionmolar excess enthalpy of mixing of the liquid phase (, , , ) for the binary systems using the novel pumpebulliometer at 101.325 kPa. Owing to the association of ethanoic acid molecules, Marek’s method in combination with HaydenO’Connell (HOC) model was used to deal with the associating properties of the liquid and vapor phase and the nonideality of vapor phase, respectively [26–28]. However, the nonideality of liquid phase was corrected by the calculation of its activity coefficient, which was obtained based on NRTL, Wilson, Margules, and van Laar models as the function of , through the nonlinear fit of the leastsquares method. NRTL, Wilson, Margules, and van Laar models were applied to correlate the VLE data for the three constituent binary systems, and the model parameters together with the deviations of temperature and vaporphase molar fractions calculated from , , , according to the function of molar excess Gibbs energy by the indirect method were obtained by the leastsquares method. The VLE data of the ternary system were well predicted from these binary interaction parameters of NRTL, Wilson, Margules, and van Laar models without any additional adjustment to build the thermodynamic model of VLE for the ternary system and obtain the vaporphase compositions and the calculated bubble points. The excess Gibbs free energies for these binary systems as the function of liquidphase composition and activity coefficient were calculated through the activity coefficient correlation to NRTL model parameters with the experimental data.
2. Modeling Section
There is an added complexity when working with carboxylic acids because they associate in the vapor and liquid phases. This association can be represented by assuming that the organic acid exists as monomer and dimer according to the Marek’s method [26, 27]. This fact, coupled with the necessity for much analytical work, tends to enhance interest in exploring the new strategies for correlation and prediction of the VLE data for the systems containing the associating carboxyl acid.
According to the Marek’s chemical theory, there are monomer and dimer carboxylic acid molecules in both liquid and vapor phases. The equilibrium constant of vaporphase dimerization, , is calculated by the expression and the equilibrium constant of liquidphase dimerization, , is calculated by the expression
In equations mentioned above, and are the mole fractions of dimers of ethanoic acid molecules in both vapor and liquid phases, respectively, and can be defined as the function of temperature by the expression obtained from the literature [27] where was presented in kPa^{−1} and in K.
When the dimers of ethanoic acid molecules mainly exist, the binary system for the methanol or water + ethanoic acid is nominally binary; however, it is actually ternary for the methanol or water + monomer ethanoic acid + dimer ethanoic acid system. In this system, the mole fractions of vaporliquid equilibrium phases are , , , , , and , respectively. Therefore, the VLE relations of the nonassociating and associating components are individually calculated by the expressions
In (4), the actual mole fractions can be denoted by the apparent mole fractions of easily determined components (, , , and multiplied by a modified coefficient. Likewise, the measured apparent vapor pressure saturated of ethanoic acid () multiplied by a modified coefficient can also denote the actual vapor pressures saturated of monomer and dimer ethanoic acid (, ). Consequently, the apparent compositions and vapor pressures saturated substitute for the actual ones, and the VLE relation of the nonassociating component, methanol or water , is expressed by where and for ethanoic acid , which is the associating component, its relation is expressed by where
In (5) to (11), and can be regarded as modified coefficients for the deviation from ideality in vapor phase on account of association. The fugacity of coefficients and is not negligible, and their values were obtained through the HOC model [28]. From another point of view, can be viewed as a modified coefficient for the vapor pressure of the associating component. Moreover, and denote modified coefficients of the deviation from ideality in the liquid phase by reason of the existence of association, and and express the deviation from ideality in liquid phase because of other factors. Herein, in liquid phase, the modified coefficients of deviation from ideality, and , can be incorporated to the activity coefficients, and , respectively. So (5) and (8) are obtained by the expressions
3. Experimental Section
3.1. Materials
Methanol (99.8 mass %) and ethanoic acid (99.8 mass %) were purchased from Sigma. The purities of the chemicals are provided by the manufacture’s specifications. Ultra sound was used to dispel the solvent air in the materials, which were dried on molecular sieve (pore diameter 30 nm from Fluka) to remove all possible traces of moisture before use, but no other treatments were applied. The densities and refractive indices at 298.15 K and normal boiling points at 101.325 kPa of the pure component were compared with the literature values of Riddick et al. [29]. The results show that the measured values are approximately in agreement with those of the literature, as presented in Table 1. The measurement method of the composition dependence of densities and refractive indices has been previously reported [30].
 
^{
a}Riddick et al. [29]. 
3.2. Apparatus and Procedure
A new type of magnetic pump ebulliometer described in detail by Qiu et al. [31] was used to measure the boiling points with different liquid phase compositions at the 101.325 kPa. The experimental main apparatus for pumpebulliometer is schematically shown in Figure 1. The recirculation still is entirely constructed from borosilicate glass. The main parts are a magnetic recirculating bump (F), a magnetic stirrer (G), a thermowell (filled with a conducting oil) provided to enable good measurement of equilibrium temperature (H), a partheating tube (resistance heating wire inserted) (K), and a tube window to observe in VLE status (I). The apparatus is an allglass dynamic recirculation still with total volume of about 1.00 × 10^{−4} m^{3}. During the run, to avoid the over heating, the still was submerged in a constant temperature bath at about 3°C below the equilibrium boiling point, which was obtained by the Nichrome wire in the partheating tube (K) to partially heat the known mass of the material. The equilibrium pressure was measured using a Fischer digital manometer with a precision of ±0.01 kPa. The pressure measurement for the manometer had an uncertainty of ±0.07 kPa, as provided by the manufacturer’s specifications. The total uncertainty of calibration and pressure measurement is estimated to be ±0.15 kPa because of the uncertainty of the calibration and measurement errors. Since the barometric pressure changed slightly, the experimental temperatures of the systems were automatically calibrated to that at 101.325 kPa with selfadjusted pressure system. The temperature was measured using a Heraeus QuaT100 quartz thermometer with a thermosensor, with an accuracy of ±0.01 K. The calibration of the thermometers was carried out at the accredited calibration laboratory (Quality and Technique Bureau, Anhui). The total uncertainty of calibration and temperature measurement is evaluated to be ±0.085 K because of the uncertainty of the calibration, the probe’s position, and the pressure fluctuations. The equilibrium temperature, , was measured by means of the thermosensor inserted into the thermowell (filled with conducting oil) (H). In each experiment, a known mass of the material was introduced from the injector into the still from feed inlet (B) and heated to equilibrium boiling point of the system at a fixed pressure of 101.325 kPa by an automatic pressure regulation system. The liquid mixtures of required composition were prepared gravimetrically, with the use of a Sartorius electronic analytic balance (model ER182A) with an accuracy of ±0.0001 g. The values of mole fraction were reproducible to ±0.0001 and have uncertainty of 0.01%. The ebulliometer was charged with the mixture of desired composition, and the boiler was then heated by nichrome wire wound around the boiler. After the liquid mixture started boiling, the bubbles along with the drops of liquid slowly spurted on the thermowell one by one through the tube window observed in VLE status (I). When the VLE state was attained by adjusting the pressure to 101.325 kPa, it remained constant for 20 min to ensure the stationary state, and then the boiling temperature was measured.
A flow microcalorimeter (model 2107, LKB produkter, Bromma, Sweden) was applied to measure the molar excess enthalpies, , of the mixtures. The electricalcalibration apparatus and its operating procedure have been described elsewhere by Francesconi and Comelli [34]. Two automatic burets (ABU, Radiometer, Copenhagen, Denmark) were used to pump liquids through the mixing cell of the calorimeter. The performance of the calorimeter was checked by measuring the of the test mixture hexane + cyclohexane system reported by Gmehling [35]. Agreement with literature is better than 0.5% over the central range of mole fraction of hexane.
The uncertainties in mole fraction and are estimated to be 0.0005 and 0.5%, respectively. The liquidphase mole fraction of component , , could be calculated from the known mass of the material added to the still. The vaporphase mole fraction of component , , was calculated from the experimental , , , and data based on the function (the function of molar excess Gibbs energy) by the indirect method.
4. Results and Discussion
4.1. Calculation of VaporPhase Mole Fraction for the Binary Systems from and
The vaporphase mole fraction of the components was calculated from and the fugacity of coefficients of the components was obtained by the expression where is the fugacity coefficient of component in the vapor mixture, is the fugacity coefficient of component at saturation, is the molar volume of component in the liquid phase, is the universal gas constant, is the experimental temperature, is the total pressure (101.325 kPa), and is the vapor pressure of pure component . The Antoine equations were applied to calculate the values of these vapor pressures. The Antoine constants , , , and the values of , , , and were obtained from Shi et al. [36], as shown in Table 2. The Poynting correction factor was also included in the calculation of fugacity at the reference state. The liquid molar volumes were evaluated from the modified Rackett equation [37].

According to the GibbsDuhem equation, any extensive molar thermodynamic property of a given phase, such as the Gibbs and Helmholtz energies, the enthalpy, and molar volume, must satisfy the following differential relation:
In (16) is a general molar property, the molar fraction of component in the phase under consideration, the pertinent set of compositions, the number of components, and the partial contribution of component to .
When considering VLE, the molar excess Gibbs energy, , can be evaluated from measurable (, , ) data using activity coefficient relations. Replacing by in the following equation yields the wellknown relation where and are the molar excess enthalpy and volume of mixing of the liquid phase. According to the thermodynamic principles, the activity coefficients of the components were calculated from the expression as follows: where is the function of molar excess Gibbs energy and combining (17) and (18) yields
Application to a binary system gives
Simultaneous solution of (18) and (20) yields
Equation (18) substituted into (21) reduced it to
For a binary system comprised of species 1 and 2 at VLE state, at constant pressure, , substitution into (22) reduced it to
The righthand side, , of (23) cannot be neglected. Proper use of (23) requires the availability of heats of mixing as a function of composition and temperature. The activity coefficients of the components as functions of the excess Gibbs energy are as follows:
From (14), this equation is rearranged to obtain
Because , for the binary system, the equation may be summed to give
In (26), solved for by difference method. Suppose that is subdivided into subintervals of equal step size by using for . In difference point, we obtain
Meanwhile, (23) may be shown as follows:
And (27) is linearized to obtain:
The number of linear equation from (28) is solved for by chasing method where is relaxation factor, finally, () is obtained by difference method. The block diagram for calculation procedure was detailedly shown in Figure 2.
4.2. Calculation of VaporPhase Mole Fraction for the Binary Systems from Model
There are many methods concerning the correlation and prediction of VLE data. The modelfree approach data treatment of vaporliquid equilibrium is also one of the best strategies for the correlation and prediction of VLE data. Wisniak’s group has developed that the novel modelfree computation techniques and limiting conditions have been applied to VLE data for azeotropic systems [42]. Moreover, Segura and coworkers reported that a modelfree approach dealt with VLE data in application of ternary systems [38, 43]. For the three binary systems of this study, the activity coefficients were correlated with the NRTL [39], Wilson [40], Margules [41], and van Laar [44] equations, respectively. The interaction parameters optimized were achieved by the objective function (OF) minimized using the leastsquares fitting as follows: where is the number of experimental points and are the liquidphase mole fraction of component calculated and experimental values from the (12) or (13) and from measured data, respectively.
Because carboxylic acids are always present in an associated form, like a dimer or trimer, in both the vapor and liquid phases even at low pressures, a significant deviation in fugacity coefficient may exist using the ideal gas assumption. To illustrate the deviation from ideal behavior, Marek’s chemical theory and HOC model were applied to deal with the associating component and modify the deviation from idealities of vapor phase [26–28], respectively. The Poynting correction factor was also included in the calculation of fugacity at the reference state. The liquid molar volumes were evaluated from the modified Rackett equation [37]. However, under isobaric conditions, the most volatile component cannot exist in the liquid state, only as superheated vapor. Hence, there is no way to calculate or measure this property for the molar volumes of the pure liquids. Therefore, the correct procedure for isobaric measurements is to calculate the overall values of and , as adjustable parameters, and not the values of the interaction excess energy. Hence, for Wilson model, and were reported as adjustable parameters.
The activity coefficients computed on the basis of NRTL model were used to evaluated dimensionless excess Gibbs function at 101.325 kPa for three binary systems over the overall range of composition. Liquidphase mole fraction , experimental boiling point temperature , calculated bubble point temperature , vaporphase mole fraction from the model, activity coefficients and using NRTL equation correlation, fugacity coefficients and , and the molar excess enthalpies of mixing of the liquid phase are included in Table 3. The  diagrams for the methanol + water , methanol + ethanoic acid , and water + ethanoic acid three binary systems at 101.325 kPa are shown in Figures 3, 4, and 5, respectively. The plot of excess Gibbs energy function versus liquidphase mole fraction for the three binary systems is shown in Figure 6. All the mixtures exhibit deviations from ideality with a range that may be attributed to interactions leading to the formation of various associated aggregates. Observed nonideal behavior is indicative of the magnitude of experimental activity coefficients , as well as of the variation of excess Gibbs energy function, , with composition, as depicted in Figure 6. The obtained absolute maximum values of for the methanol + water , methanol + ethanoic acid , and water + ethanoic acid three binary systems are 0.1703, 0.0180, and 0.0892, respectively. The values of excess Gibbs energy function are positive for methanol + water and water + ethanoic acid binary systems. However, for methanol + ethanoic acid system, the values of those are negative in the overall range of mole fraction. values follow the order methanol + water > water + ethanoic acid > methanol + ethanoic acid . The absolute maximum value of is approximate at an equimolar fraction in three binary systems. Figures 3–5 show that the comparison of the predicted vaporphase and experimental liquidphase compositions with those of the literature [10, 11, 32, 33]. Comparing with the values of vaporphase and liquidphase component from the literatures, the values of those from the paper are in good agreement with the literature, as shown in Figures 3–5. The results have demonstrated that the methods for and are appropriate for representing the experimental data of the three binary systems. The optimum model interaction parameter of liquid activity coefficient and the absolute average deviations for the different models, and from and are listed in Table 4. Herein, we obtained the results by the four different types of correlations for the prediction of activity coefficients in these systems, which reveal that the deviations of NRTL, Wilson, Margules, and van Laar equations are reasonably small in Table 4. For comparison, the mean deviations obtained by Gmehling and Onken [32, 33, 45, 46] are also shown in the Table 4. It can seem that two sets of deviation values are comparable. Since the superiority of one method over the others is not always obvious, practice must rely on experience and analogy. The comprehensive comparisons of four of the methods (NRTL, Wilson, van Laar, and Margules) were made in Table 4. From the data analysis, the temperature deviations between the experimental and calculated values of four different types of model are very similar in the three binary systems, and the vaporphase mole fraction deviations between calculated values from and , and from the model are very similar. Therefore, the activity coefficient models are appropriate for representing the experimental data of the three binary systems. In Table 4, the absolute average deviations of the difference between boiling point temperature from experiment and bubbling point temperature from calculation by NRLT model parameters for the three binary systems are 0.28°C, 0.51°C, and 0.33°C, respectively. And the absolute average deviations of the difference between vaporphase mole fraction from and calculation and from NRTL model calculation are 0.0072, 0.0054, and 0.0096, respectively.

 
^{
a}Wilson’s interaction parameters (J·mol^{−1}), NRTL’s interaction parameters (J·mol^{−1}). ^{b}Margules and van Laar interaction parameters (dimensionless). ; : number of data points; : calculated bubble point from model, K; : experimental boiling point temperature, K. ; : number of data points; : calculated vaporphase mole fraction from Tpx; : calculated vaporphase mole fraction from model. ^{c}The values in parentheses from the literatures [38–41]. 
4.3. Correlation and Prediction of VLE of Ternary System
The binary interaction parameters of the NRTL, Wilson, Margules, and van Laar model given in Table 4 were used to correlate and predict the VLE data of the ternary system. VLE data for methanol + water + ethanoic acid at 101.325 kPa included liquidphase mole fraction , , and , experimental boiling point temperature , calculated bubble point temperature , calculated vaporphase mole fraction , , and , activity coefficients , , and , the average deviation in the bubble temperatures of the ternary system using NRTL equation correlation listed in Table 5. The absolute average and maximum deviation between the boiling point from experimental data and the bubble point from NRTL model calculation are 0.77°C and 1.94°C, respectively. The average and maximum deviations using Wilson, Margules, and van Laar equation individually are 0.79°C, 1.90°C, 0.85°C, 1.89°C, and 0.96°C, 1.91°C. Diagram of VLE for the ternary system methanol + water + ethanoic acid at 101.325 kPa is shown in Figure 7.

5. Conclusions
VLE data for the ternary system methanol + water + ethanoic acid and three constituent binary systems at 101.325 kPa: methanol + water, methanol + ethanoic acid, and water + ethanoic acid were determined at different liquidphase compositions using a novel pump ebulliometer. The equilibrium composition of the vapor phase was calculated from , , , and based on the function of excess Gibbs free energy by the indirect method. The experimental data were correlated using the NRTL, Wilson, Margules, and van Laar equations. It was shown that the deviations of NRTL, Wilson, Margules, and van Laar equations are reasonably small. The VLE data of ternary system were predicted by NRTL, Wilson, Margules, and van Laar equation. The calculated bubble points accorded well with the experimental data. The results show that the calculated bubble point is fitted by the models, which satisfy the need for the design and operation of separation process in chemical industry. Moreover, the method will provide theoretical guidance for the research of VLE data of strongly associating system of vapor and liquid phase in nonideal behavior.
Acknowledgments
This work was supported by the Deutscher Akademischer Austausch Dienst (DAAD) (Ref. Code: A/11/06441), the National Natural Science Foundation of China (Grant no. 21075026), and the Natural Science Foundation of the Higher Education Institutions of Anhui Province (Grant no. ZD200902).
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