1 Introduction

A mixture of aqueous solutions of two incompatible polymers (polymer/polymer system) or a polymer and a salt (polymer/salt system) can form a stable two-phase liquid system called an aqueous two-phase system (ATPS) [1,2,3]. In aqueous two-phase systems, 70 to 90% w/w of each phase is water, in this regard, they are considered to be environmentally friendly systems [4, 5]. Moreover, ATPSs are biocompatible, nontoxic and nonflammable [6, 7]. Ease of scale-up, lower interfacial tension, viscosity and cost of materials are some advantages of polymer/salt systems over the polymer/polymer systems [8, 9]. ATPS can be used for separation and purification of biomolecules [10] and metal ions [11, 12]. Polymer/salt systems spontaneously separate into a polymer-rich top phase and salt-rich bottom phase. Among all the polymers, polyethylene glycol (PEG) [13, 14], polypropylene glycol (PPG) [15, 16] and polyvinylpyrrolidone (PVP) [17, 18] have been most used. There are considerable numbers of experimental studies on application of PEG and PPG for ATPS in the literature. As example, Voros et al. [19] studied the equilibrium of ATPSs containing PEG1000 or PEG2000 in the presence of Na2CO3 or (NH4)2SO4 at different temperatures and Mishima et al. [20] measured the LLE data of the PEG + K2HPO4 + H2O system at 288.15, 308.15 and 318.15 K.

In addition to experimental studies, it is important to have a good thermodynamic model to describe and predict liquid–liquid equilibrium conditions in engineering and design. To obtain global and reliable parameters for thermodynamic models usually phase equilibrium data is suitable. As there is polymer, electrolyte and water in polymer/salt systems, all different types of interactions should be taken into account. Up to now, several models have been used, including NRTL [21,22,23], Chen–NRTL [24,25,26], Wilson [27], UNIQUAC [28], NRTL–NRF [29] and UNIFAC–NRF [30]. It has been shown that, in all cases, the models were successful in reproducing tie-line data of polymer/salt aqueous two phase systems. In most of the previous works, the excess Gibbs functions have been used for modeling. In this way, Jimenez et al. [31] determined the phase diagram and LLE results for ATPS containing PEG4000 and NaClO4. They examined the Chen–NRTL, modified Wilson and UNIQUAC models to correlate the tie-line data in this system and reported that quality of fitting is better with the modified Wilson model. Recently, Valavi et al. [5] used the PHSC equation of state (PHSC EOS) for modeling of aqueous two phase systems. They found that PHSC EOS could correlate a considerable number of systems using just two adjustable parameters.

In this paper, in continuation of previous work [32,33,34,35,36], the experimental and thermodynamic behavior of system containing PPG, PEG, water and electrolyte is studied. The LLE data for the system containing PPG425 (polypropylene glycol with molecular weight of 425 kg·mol−1), NaClO4 and water at 288.15, 298.15 K is measured experimentally at atmospheric pressure. Considering industrial applications of ATPSs, the experiments were carried out at moderate temperatures. The data are then modeled using the E-UNIQUAC model considering two scenarios (A and B). In the Case A, water is solvent while in the Case B, water–polymer was treated as a pseudo-solvent. Finally, the performance of the model in representing of the experimental data of the ATPS is examined in each case.

2 Experimental

2.1 Chemicals Used

The polypropylene glycol (purity above 99.9%) and sodium perchlorate of analytical grade (purity greater than 98%) were obtained from Sigma–Aldrich. All chemicals were used as received without further purification. In this work doubly distilled deionized water was used in all experiments.

2.2 Apparatus and Procedure

Sodium perchlorate was dried before use in an oven at 378.15 K for 3 days. The experiments were carried out in 15 mL test tubes. Feed samples were prepared by mixing appropriate amounts of materials. The sample solutions were mixed for 1 h and then were placed in a water bath at constant temperatures for 36 h to reach equilibrium with clear phases. The temperature was controlled using a water bath with a precision of ± 0.1 K. Samples of both phases were taken with plastic syringes. In order to increase the accuracy of experiments, the samples of upper phase were taken 0.5 cm above the interface and the remaining of the top phase was discarded. Samples of both phases were taken for chemical analysis. Each experiment was carried out three time and average values of the results are reported. After separation of the two phases, the concentrations of sodium perchlorate in the top and bottom phases were determined by inductively coupled plasma atomic emission spectrophotometry (ICPS-7000, VER 2). The precision of the mass fraction of sodium perchlorate was better than ± 0.001 and this was checked by measuring standard salt solutions. The concentration of PPG in both phases was determined by refractive index measurements performed at 298.15 K using a refractometer (OPTECH) with a precision of ± 0.0001. To ensure the accuracy of the linear response, the concentration of sample solutions must be near the concentration of unidentified solutions. The relation between the mass fraction of polymer (w 1), salt (w 2) and the refractive index (n D) is given by:

$$ n_{\text{D}} = n_{\text{D}}^{\text{o}} + a_{1} w_{1} + a_{2} w_{2} $$
(1)

where a 1 and a 2 are adjustable parameters and \( n_{\text{D}}^{\text{o}} \) is refractive index of pure water. The values of the a 1 and a 2 can be obtained by measuring the refractive index of standard solutions. The \( n_{\text{D}}^{\text{o}} \) is 1.3325 [37] and the values of a 1 and a 2 for the system were obtained as 0.1385 and 0.105, respectively. It must be mentioned that the precision of the mass fraction of PPG achieved using Eq. 1 was better than 0.002.

3 Thermodynamic Modeling

Sander et al. [38] presented a model to be used for electrolyte solutions by adding an extended Debye–Hückel term to the UNIQUAC model in 1986. Using this model, the excess Gibbs energy, G ex, can be calculated as a sum of two contributions as follows [39]:

$$ G^{ex} = G_{\text{UQ}}^{\text{ex}} + {\text{G}}_{\text{DH}}^{\text{ex}} $$
(2)

In Eq. 2 \( {\text{G}}_{\text{UQ}}^{\text{ex}} \) is the original UNIQUAC equation and \( {\text{G}}_{\text{DH}}^{\text{ex}} \) is the extended Debye–Hückel term and accounts for the contribution of long-range interactions. Therefore, the activity coefficients of the solvents (\( \gamma_{i}^{{}} \), given in the symmetrical convention) and the activity coefficients of the ions (\( \gamma_{i}^{ *} = {{\gamma_{i} } \mathord{\left/ {\vphantom {{\gamma_{i} } {\gamma_{i}^{\infty } }}} \right. \kern-0pt} {\gamma_{i}^{\infty } }} \), given in the unsymmetrical convention) are also given as the sum of two terms as follows:

$$ { \ln }\gamma_{i} = { \ln }\gamma_{i}^{\text{UQ}} + { \ln }\gamma_{i}^{\text{DH}} $$
(3)
$$ { \ln }\gamma_{i}^{*} = { \ln }\gamma_{i}^{{ * {\text{UQ}}}} + { \ln }\gamma_{i}^{{ * {\text{DH}}}} $$
(4)

It must be mentioned that a value of unity is assigned to the activity coefficients at infinite dilution in the unsymmetrical convention and a value of unity is assigned to the activity coefficients in the pure state in the symmetrical convention. The UNIQUAC term for the activity coefficient can be calculated using Eq. 5:

$$ \ln \,\gamma_{i}^{\text{UQ}} = \ln \,\left( {\frac{{\phi_{i} }}{{x_{i} }}} \right) + 1 - \frac{{\phi_{i} }}{{x_{i} }} - \frac{z}{2}q_{i} \,\left[ {\ln \,\left( {\frac{{\phi_{i} }}{{\theta_{i} }}} \right) + 1 - \frac{{\phi_{i} }}{{\theta_{i} }}} \right] + q_{i} \left[ {1 - \ln \,\left( {\sum\limits_{k} {\theta_{k} \psi_{ki} } } \right) - \sum\limits_{k} {\frac{{\theta_{k} \psi_{ki} }}{{\sum {\theta_{k} \psi_{lk} } }}} } \right] $$
(5)

where the subscripts i and k are used to denote the components in the system, z is the coordination number and taken as z = 10. ϕ i and θ i are the volume fraction and the surface area fraction of component i, respectively, that can be calculated as Eqs. 6 and 7.

$$ \phi_{i} = \frac{{x_{i} r_{i} }}{{\sum {x_{i} r_{i} } }} $$
(6)
$$ \theta_{i} = \frac{{x_{i} q_{i} }}{{\sum {x_{i} q_{i} } }} $$
(7)

where q i and r i are the pure component area and volume parameters of the UNIQUAC model, respectively. Here, it is supposed that the sodium perchlorate in the aqueous phase is completely dissociated into ions; therefore, in Eq. 5, x i is the mole fraction of component i, which can be calculated as follows:

$$ x_{1} = \frac{{n_{1} }}{{n_{1} + 2n_{2} + n_{3} }} $$
(8)
$$ x_{2} = \frac{{2n_{2} }}{{n_{1} + 2n_{2} + n_{3} }} $$
(9)
$$ x_{3} = \frac{{n_{3} }}{{n_{1} + 2n_{2} + n_{3} }} $$
(10)

where n is mole number of species. In this work two scenarios (A and B) are considered in the modeling, in Case A, water (3) is solvent and the polymer (1) and ions (2) are solutes, while in Case B, water–polymer was treated as a pseudo-solvent, this case is known as the mixed solvent model.

Meanwhile, infinite dilution in pure water was taken as the reference state for the solute (sodium perchlorate and PPG), and the pure liquid state for water as described by Gao et al. [40]. The contribution of long–range interaction in the electrolyte solutions was modeled using the extended Debye–Hückel (DH) equation of Fowler–Guggenheim [41]. In Case B, the electrostatic term of the activity coefficient for neutral molecule i was calculated based on Eq. 11, while in the Case A, the electrostatic term is zero for the polymer species.

$$ \ln \,\gamma_{i}^{\text{DH}} = M_{i} \frac{2A}{{b^{3} }}\left[ {1 + bI^{{\frac{1}{2}}} - \frac{1}{{1 + bI^{{\frac{1}{2}}} }} - 2\,\ln \,\left( {1 + bI^{{\frac{1}{2}}} } \right)} \right] $$
(11)

where M i is molar mass of component i. The ionic activity coefficient, \( \gamma_{i}^{ *} \), of salt can be written as:

$$ \ln \,\gamma_{i}^{{ * {\text{DH}}}} = - z_{i}^{2} \frac{{AI^{{\frac{1}{2}}} }}{{1 + bI^{{\frac{1}{2}}} }} $$
(12)

The ionic strength (I) is calculated as:

$$ I = \frac{1}{2}\sum {Z_{i}^{2} m_{i} } $$
(13)

where Z i is the charge number of ion i, m i is the molal concentration and can be calculated using Eqs. 14 and 15 for Case A and Case B, respectively.

$$ m_{i} = \frac{{x_{i} }}{{M_{3} x_{3} }},\;{\text{Case}}\;{\text{A}} $$
(14)
$$ m_{i} = \frac{{x_{i} }}{{M_{1} x_{1} + M_{3} x_{3} }},\;{\text{Case}}\;{\text{B}} $$
(15)

where M i and x i represent the molecular weight and the mole fraction, respectively.

In the asymmetric convention, in both cases, the reference is taken as pure water.

The DH constants (A and b), can be calculated as Eqs. 16 and  17. A value of 4 Å is assumed for closest distance between ions [42].

$$ A = 1.327757 \times 10^{5} \frac{{d^{0.5} }}{{\left( {DT} \right)^{1.5} }} $$
(16)
$$ b = 6.359696\frac{{d^{0.5} }}{{\left( {DT} \right)^{1.5} }} $$
(17)

where T, D and d represent temperature, dielectric constant and density of the mixed solvent, respectively. The values for D and d can be calculated using the following relations:

$$ D = \sum {\varphi^{\prime}_{k} } D_{k} $$
(18)
$$ d = \sum {\varphi^{\prime}_{k} d_{k} } $$
(19)

where \( \varphi^{\prime}_{k} \) is the salt-free volume fraction of non-ionic species k in the liquid phase and is defined as:

$$ \varphi^{\prime}_{k} = \frac{{x_{k} V_{k} }}{{x_{1} V_{1} + x_{3} V_{3} }} $$
(20)

where V i is the molar volume of i. A group contribution method can be used to calculate the molar volume of polymer. In this way, the molar volume of PPG425 was obtained as V 1 = 425 × 10−6 (m3·mol−1) using the group contribution data reported by Zana [43]. The dielectric constant of PPG was calculated according to the method proposed by Van et al. [44]. For water, the value of D 3 = 82.22 at 288.15 K and D 3 = 78.34 at 298.15 K were used. The binary energy interaction parameters of the E-UNIQUAC model are defined as follows:

$$ \psi_{ij} = \exp \left( { - \frac{{\Delta U_{ij} }}{T}} \right) $$
(21)
$$ \Delta U_{ij} = U_{ij} - U_{jj} $$
(22)

where U ij is the interaction parameter between species i and j. These parameters are symmetrical and temperature dependent as follows:

$$ U_{ij} = U_{ij}^{\text{o}} + U_{ij}^{T} \left( {T - 298.15} \right) $$
(23)

It must be noted that in the recent equations U ij  = U ji . In this work, the interaction parameters of \( U_{kl}^{\text{o}} \) and \( U_{kl}^{T} \) were fitted to the experimental data. Therefore, the interaction parameters between polymer, salt and water were obtained using the E-UNIQUAC model.

4 Results and Discussion

The experimental LLE data of the ATPS containing PPG425 and NaClO4 was measured at different temperatures. The experimental data at 288.15 and 298.15 K are shown in Table 1.

Table 1 Experimental phase equilibrium compositions for the PPG425 (1) + NaClO4 (2) + H2O system at T = 288.15 and 298.15 K
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As can be seen, the bottom phase is rich in salt and the upper one is polymer rich. Also, in Fig. 1, binodal curves at different temperatures have been presented. It is obvious that, as the temperature increases, the two-phase region expands and tie-line slopes increase. This phenomenon usually is explained by the higher solubility of the phase forming components at higher temperatures. Recently, Sadeghi et al. [45] recognized that the hydrophobic nature of polymers can be increased by increasing temperature, which expand the two-phase region.

Fig. 1
figure 1

Binodal curves for ATPS containing PPG425 and NaClO4 at different temperatures

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To find the model parameters the ternary systems were modeled using E-UNIQUAC. In this way, the values of structural parameters (r and q) for water and polypropylene glycol have been taken from Larsen et al. [46], while the relevant values for the ions (Na+ and \( {\text{ClO}}_{4}^{ - } \)) have been extracted from Haghtalab et al. [47]. Considering the equality of component fugacity in two phases, the binary interaction parameters for PPG425 + NaClO4 + H2O system can be obtained using the binary LLE data (Table 1). To decrease the number of adjustable parameters, ion–water and ion–ion interaction parameters were obtained using experimental data from the literature, in this regard the ATPSs composed of sodium salts and PPG or PEG were used (Table 2). Meanwhile, it was assumed that a polymer with different molecular weight has the same interaction parameter with the ions; therefore, the effect of molecular weight of polymers can be considered in the UNIQUAC part because the structural parameters of UNIQUAC model are changed by changes in the molecular weights of polymers.

Table 2 Literature data for the PPG + salt + water and PEG + salt + water system
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The adjustable parameters were obtained by minimizing following objective function for all tie lines:

$$ {\text{OF}} = \mathop \sum \limits_{i = 1}^{M} \mathop \sum \limits_{j = 1}^{N} \left( {x_{ij}^{\text{I}} \gamma_{ij}^{\text{I}} - x_{ij}^{\text{II}} \gamma_{ij}^{\text{II}} } \right)^{2} $$
(24)

where the subscripts M and N represent the number of components and the number of tie lines, respectively, x i and γ i represent the experimental mole fraction and the activity coefficient of component i. The superscripts I and II represent the two liquid phases in equilibrium. The parameters obtained for the ATPS containing PEG are given in Tables 3 and 4 for Case A.

Table 3 The interaction parameters (\( U_{kl}^{0} = U_{lk}^{0} \)) of the UNIQUAC model for ATPS containing PEG (\( U_{kl} = U_{kl}^{o} + U_{kl}^{T} \left( {T - 298.15} \right) \))
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Table 4 The interaction parameters (\( U_{kl}^{T} = U_{lk}^{T} \)) of UNIQUAC model for ATPS containing PEG (\( U_{kl} = U_{kl}^{o} + U_{kl}^{T} \left( {T - 298.15} \right) \))
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The parameters for Case B are given in Tables 5 and 6.

Due to lack of temperature dependent data on the PEG + Na2SO4 + H2O system \( U_{{{\text{PEG}} - {\text{SO}}_{4}^{2 - } }}^{T} \) was set to zero in Tables 4 and 6. The interaction parameters of ATPS containing PPG and salt for Case A are reported in Tables 7 and 8; the same for Case B are given in Tables 9 and 10.

Table 5 The interaction parameters (\( U_{kl}^{0} = U_{lk}^{0} \)) of UNIQUAC model for ATPS containing PEG (\( U_{kl} = U_{kl}^{o} + U_{kl}^{T} \left( {T - 298.15} \right) \))
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Table 6 The interaction parameters (\( U_{kl}^{T} = U_{lk}^{T} \)) of UNIQUAC model for ATPS containing PEG (\( U_{kl} = U_{kl}^{o} + U_{kl}^{T} \left( {T - 298.15} \right) \))
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Table 7 The interaction parameters (\( U_{kl}^{0} = U_{lk}^{0} \)) of UNIQUAC model for ATPS containing PPG (\( U_{kl} = U_{kl}^{o} + U_{kl}^{T} \left( {T - 298.15} \right) \))
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Table 8 The interaction parameters (\( U_{kl}^{T} = U_{lk}^{T} \)) of UNIQUAC model for ATPS containing PPG (\( U_{kl} = U_{kl}^{o} + U_{kl}^{T} \left( {T - 298.15} \right) \))
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Table 9 The interaction parameters (\( U_{kl}^{0} = U_{lk}^{0} \)) of UNIQUAC model for ATPS containing PPG (\( U_{kl} = U_{kl}^{o} + U_{kl}^{T} \left( {T - 298.15} \right) \))
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Table 10 The interaction parameters (\( U_{kl}^{T} = U_{lk}^{T} \)) of UNIQUAC model for ATPS containing PPG (\( U_{kl} = U_{kl}^{o} + U_{kl}^{T} \left( {T - 298.15} \right) \))
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In these tables, the binary interaction parameters which previously were reported by Thomsen, are marked with an asterisk [39]. Due to lack of temperature dependent data on PPG + Na2CO3 + H2O and PPG + Na2SO4 + H2O systems, \( U_{{{\text{PPG}} - {\text{SO}}_{4}^{2 - } }}^{T} \) and \( U_{{{\text{PPG}} - {\text{CO}}_{3}^{2 - } }}^{T} \) were set as zero in Tables 8 and 10. In Figs. 2 and 3, the experimental and the calculated results (Case A), using reported binary interaction parameters, are compared at 288.15 and 298.15 K, respectively.

Fig. 2
figure 2

Experimental (dotted circle) and calculated liquid–liquid equilibrium tie-lines (solid line) for the PPG(1)–NaClO4 (2)–water(3) system at 288.15 K. Calculations have been performed using the extended UNIQUAC model (black circle) (Case A)

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Fig. 3
figure 3

Experimental (dotted circle) and calculated liquid–liquid equilibrium tie-lines (solid line) for the PPG (1)–NaClO4 (2)–water (3) system at 298.15 K. Calculations have been performed using the extended UNIQUAC model (black circle) (Case A)

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Furthermore, a comparison between the experimental and the calculated results (Case B), at 288.15 and 298.15 K are shown in Figs. 4 and 5.

Fig. 4
figure 4

Experimental (dotted circle) and calculated liquid–liquid equilibrium tie-lines (solid line) for the PPG (1)–NaClO4 (2)–water (3) system at 288.15 K. Calculations have been performed using the extended UNIQUAC model (black circle) (Case B)

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Fig. 5
figure 5

Experimental (dotted circle) and calculated liquid–liquid equilibrium tie-lines (solid line) for the PPG (1)–NaClO4 (2)–water (3) system at 298.15 K. Calculations have been performed using the extended UNIQUAC model (black circle) (Case B)

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As can be seen from these figures, there is good agreements between the calculated and experimental data at the studied temperatures.

The average absolute deviation (%ΔX) between calculated and experimental mole fractions is calculated as:

$$ \% \Delta X = 100 \times \,\sqrt {\frac{{\sum\limits_{i = 1}^{M} {\sum\limits_{j = 1}^{N} {\left\{ {\left( {x_{ij}^{ \exp } - x_{ij}^{\text{cal}} } \right)_{\text{I}}^{2} + \left( {x_{ij}^{ \exp } - x_{ij}^{\text{cal}} } \right)_{\text{II}}^{2} } \right\}} } }}{2MN}} $$
(25)

In Eq. 25, x exp and x cal represent the experimental and the calculated mole fractions, respectively. The %ΔX between calculated and experimental data using the E-UNIQUAC model in ATPS containing PEG for Case A are given in Table 11.

Table 11 Average absolute deviation (%ΔX) between calculated and experimental mass fractions for ATPS containing PEG in Case A
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The results (Table 11) show good agreement between calculated and experimental data. The %ΔX between calculated and experimental data in ATPS containing PEG for Case B are also reported in Table 12.

Table 12 Average absolute deviation (%ΔX) between calculated and experimental mass fractions for ATPS containing PEG in Case B
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A comparison between Tables 11 and 12 shows that using the pseudo-solvent approach (Case B) increases the accuracy of the model and the average error of 0.095% was obtained in this case.

In Table 13, the %ΔX in ATPS containing PPG for Case A are reported. The same for Case B are shown in Table 14.

Table 13 Average absolute deviation (%ΔX) between calculated and experimental mass fractions for ATPS containing PPG in Case A
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Table 14 Average absolute deviation (%ΔX) between calculated and experimental mass fractions for ATPS containing PPG in Case B
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As can be seen from Tables 13 and 14, the results are similar in both cases and the reported errors are almost equal. In Tables 11–14, %ΔX is the average absolute deviation between calculated and experimental data at a fixed temperature. It must be mentioned that the reported parameters were obtained using experimental data at all temperatures.

In this work the ability of single solvent (Case A) and pseudo-solvent (Case B) approaches in correlation of ternary liquid–liquid phase equilibrium data were studied and it was found that pseudo-solvent theory gives better results compared to single solvent in the systems containing PEG. Meanwhile, Case A and Case B showed similar results in the ATPS containing PPG.

5 Conclusions

In this work the liquid–liquid equilibrium of a ternary system composed of PPG425, NaClO4 and H2O were determined at 288.15 and 298.15 K. It was found that increasing temperature expands the two-phase region and tie-line slopes. This phenomenon can be explained through the solubility of the phase forming components at different temperatures. The experimental data were correlated using the E-UNIQUAC model. In this regard, two procedures of single solvent and pseudo-solvent were used and unknown binary interaction parameters were estimated for future applications. To present global parameters, the available liquid–liquid experimental data were collected from the literature and were modeled simultaneously. The results showed that the model can correlate the experimental data efficiently. It was found that both scenarios (Case A and Case B) are fairly equal and there is no significant difference between them in modeling of ATPSs containing PPG while in the case of PEG, the pseudo-solvent scenario showed better results. In overall it must be mentioned that good agreement with the experimental data was obtained in all cases, however the performance of Case B was slightly better than the Case A. Finally, it is worth mentioning that the results in this work can enhance the experimental data and thermodynamic modeling approach to polymer/salt aqueous two-phase systems.