Prediction of liquid–liquid phase diagrams of aqueous salt+PEG systems using a thermodynamic model

Computer Coupling of Phase Diagrams and Thermochemistry 30 (2006) 296–300 www.elsevier.com/locate/calphad

Prediction of liquid–liquid phase diagrams of aqueous salt + PEG systems using a thermodynamic model Alireza Salabat ∗ Chemistry Department, Arak University, P.O. Box 38156-879, Arak, Iran Received 9 December 2005; received in revised form 1 March 2006; accepted 19 March 2006

Abstract In this study a thermodynamic model for the phase behavior of aqueous salt + polymer solutions is developed. The model is based on the solution theory of Hill, which included scaling laws for the polymer molecular mass dependence and Pitzer–Debye–H¨uckel theory. This model was tested for systems composed of two different molecular mass of polyethylene glycols (PEG) and five different inorganic salts. All the model parameters were determined from independent measurements. The agreement between the experimental and predicted phase diagrams by this model is good. c 2006 Elsevier Ltd. All rights reserved.

Keywords: Thermodynamic model; Aqueous two-phase system; Phase diagram; Polymer–salt system

1. Introduction Aqueous two-phase systems (ATPS) present a gentle, scalable and efficient procedure for the separation of various biological materials [1,2]. These systems are formed when two polymers or one polymer and one salt are dissolved in water above a threshold concentration. There are some reports on the thermodynamic properties and liquid–liquid equilibrium (LLE) of aqueous PEG/salt two-phase systems [3–7]. Some thermodynamic models exist for the correlation and prediction of phase diagrams of these systems. The model of Haynes et al. [8] is based on an additive combination of a hardsphere equation of state and mean spherical approximation-type theory. Edmond and Ogston [9] proposed a virial expansion to predict phase diagrams of aqueous mixtures of polyethylene glycol (PEG) and dextran. King et al. [10] have extended this model and used it to predict biomolecular partitioning. Another model based on the UNIQUAC equation had been used by Kang and Sandler [11,12] to describe the phase behavior of a PEG–dextran–water system. Haghtalab et al. proposed a UNIQUAC-NRF model [13] and a UNIFAC-NRF model [14] and applied them for aqueous two-phase systems. Baskir et al. ∗ Tel.: +98 861 3664691; fax: +98 861 2767306.

E-mail address: [email protected]. c 2006 Elsevier Ltd. All rights reserved. 0364-5916/$ - see front matter doi:10.1016/j.calphad.2006.03.002

[15,16] developed a lattice model for the partitioning of proteins in aqueous two-phase systems. Forciniti and Hall [17] proposed a model similar to that of Edmond and Ogston but based on the solution theory of Hill [18]. Cabezas et al. [19] also proposed a model based on the Hill formalism but included scaling laws for the polymer molecular mass dependence. Kabiri-Badr et al. [20] described a thermodynamic model for the phase behavior of aqueous salt–polymer solutions based on the solution theory of Hill [18] and fluctuation solution theory [21]. Recently, Wu et al. developed a modified NRTL equation to describe LLE of polymer–polymer and polymer–salt two-phase aqueous systems [22]. Jiang and Prausnitz [23] proposed a molecular thermodynamic model based on McMillan–Mayer solution theory [24] to calculate phase diagrams of these systems. This work is also intended to develop a thermodynamic model to predict the salt–polymer solution. In this model, the Kabiri-Badr model [20] is simplified by changing the Long Range part of the model by Pitzer–Debye–H¨uckel theory, with better or at least the same results. The basis of this model is the solution theory of Hill, that is extended to include the effect of electrostatic forces using Pitzer–Debye–H¨uckel (PDH) theory [25,26]. Scaling laws were used to develop this model for polymer molecular mass dependence of the polymer–polymer and the salt–polymer Hill osmotic virial coefficients. This model was tested for systems composed of

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two different molecular mass of polyethylene glycol (PEG 1000 and PEG 6000) and MgSO4 , Na2 SO4 , Na2 CO3 , (NH4 )2 SO4 , KH2 PO4 and K2 HPO4 at 298.15 K. The model parameters were evaluated using the water activity data. For these salts + PEG aqueous systems there are accurate water activity data in the literature [27–29]. 2. Thermodynamic framework Since the osmotic pressure is related to chemical potential, this model is partly based on the osmotic virial expansion of Hill [18]. The osmotic pressure, Π , is written as a sum of two terms. The first term is Π LR and comes predominantly from the long-range coulombic forces between the ions. The second term is Π SR and originates mainly from the shortrange repulsive and other non-electrostatic interactions between solute molecules and ions. The non-electrostatic interactions term, Π SR , is based on the solution theory of Hill that in terms of solute molality m 2 is Π SR = m 2 + C22 (T, P)m 22 + C222 (T, P)m 32 + · · · RTd1

(1)

Π SR = c2 + B22 (T, µ1 )c22 + B222 (T, µ1 )c23 + · · · (2) RT where c2 is the molar concentration of solute and B22 and B222 are the osmotic virial coefficients. The relationship between the two different types of osmotic virial coefficients for an incompressible solvent is given by Hill as 1 ∞ ∞ d1 (2B22 − V 2 − V 2 ) 2 ∞

(3) ∞

C222 = d12 (B222 − 2B22 V 2 + (V 2 )2 )

(4)

∞ where V 2

is the infinite dilution partial molar volume of solute. Hill’s solution theory can be extended for a mixture of salt (s), polymer ( p) and water (w) as Π SR = νs m s + m p + νs2 Css m 2s + Cpp m 2p + νs3 Csss m 3s RTdw + Cppp m 3p + 2νs Cps m p m s + 3νs Cpps m 2p m s + 3νs2 Cpss m p m 2s + · · ·

(5)

where m s is the molality of salt, m p is the molality of polymer, νs is the number of positive and negative ions in the salt, and dw is the density of pure water. The long-range interaction contribution term to the osmotic pressure is based on the Pitzer–Debye–H¨uckel equation [26]. The relation between salt activity coefficient γ± and osmotic pressure Π LR can be derived as ! Z ms ∂ ln γ±LR Π LR = υs ms dm s . (6) RTdw ∂m s 0 T,P

In this equation, Aφ is the Debye–H¨uckel parameter, which at 298.15 K is 0.391, I is ionic strength and b is a parameter with the value 1.2 kg1/2 mol−1 . Using the Gibbs–Duhem equation and the equality of the chemical potential cross derivatives the following expressions for the chemical potential of the salt (s), the polymer ( p), and the water (w) can be obtained: µs − µ◦s = νs ln m s + νs ln γ±LR + 2νs2 Css m s + 2νs Cps m p RT 3 3 + νs3 Csss m 2s + νs Cpps m 2p 2 2 + 3νs2 Cpss m p m s + · · · (8) µ P − µ◦p

where d1 is the mass density of pure solvent (kg L−1 ) and C22 and C222 are the Hill osmotic virial coefficients at constant temperature and pressure. On the other hand, the osmotic virial expansion from the McMillan–Mayer solution theory is as follows

C22 =

The Pitzer–Debye–H¨uckel equation for ln γ±LR has the following form # " √   √ I 2 ln γ±LR = − |Z + Z − | Aφ √ + ln 1 + b I . (7) b 1+b I

3 = ln m P + 2Cpp m p + 2νs Cps m s + Cppp m 2p 2 3 2 2 + 3νs Cpps m p m s + νs Cpss m s + · · · 2 ! Z ms ∂ ln γ±LR µw − µ◦w − = νs m s + νs ms dm s RTM w ∂m s 0 RT

(9)

T,P

+ m p + νs2 Css m 2s + Cpp m 2p + νs3 Csss m 3s + Cppp m 3p + 2νs Cps m p m s + 3νs Cpps m 2p m s + 3νs2 Cpss m p m 2s + · · · . (10) In these equations, Cpp and Cppp are the Hill osmotic virial coefficients for the interaction between polymer molecules. These coefficients can be related to McMillan–Mayer osmotic virial coefficients Bpp and Bppp using Eqs. (3) and (4). In this model the polymer chains can be assumed to be roughly spherical on average, and their interaction can be modeled as that between two hard spheres of effective diameter R, where R is the mean end-to-end distance of a chain. The virial coefficient for two hard spheres with equal effective diameter is [30]. 4π 3 R . (11) 3 Using polymer scaling laws [31,32], the mean end-to-end distance of a chain, R, can be related to the chain size under conditions of constant solvent chemical potential or McMillan–Mayer conditions. The relevant result is Bpp =

R ∝ Nν

(12)

where N is the degree of polymerization and ν is a universal scaling exponent valid for all linear polymers. Using Eqs. (11) and (12) the McMillan–Mayer second virial coefficient, Bpp becomes: Bpp = bpp N 3ν

(13)

where bpp is an effective monomer–monomer interaction coefficient. Substitution of Eq. (13) into Eq. (3) gives ∞

Cpp = dw (bpp N 3ν − V p )

(14)

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Table 1 Hill osmotic virial coefficient and standard deviations (σ ) for salts, infinite dilution of salts partial molar volumes and ion–polymer interaction coefficients at 298.15 K Css (kg mol−1 ) MgSO4 (NH4 )2 SO4 Na2 SO4 Na2 CO3 K2 HPO4

0.1728 −0.0516 −0.0292 0.0104 0.0037

∞

Csss (kg2 mol−2 )

σ

m Max. a

V s (L mol−1 )

bps (L mol−1 )

0.0174 0.0204 0.0150 −0.0013 0.0005

2.9 × 10−3

3.6 2.9 5.0 2.4 2.9

−0.0071 0.05066 0.01184 −0.0028 0.02470

0.087 0.063 0.069 0.072 0.075

1.9 × 10−3 3.4 × 10−3 3.9 × 10−3 7.1 × 10−3

a Maximum molality of salt solutions.

where dw = 0.99704 kg L−1 is the mass density of pure water ∞ at 298.15 K and V p is the partial molar volume of the polymer at infinite dilution in pure water, which can be approximated using group additive results of Zana [33]. ∞

∞

V p = N V MP

(15)

∞

where V MP is the infinite dilution partial molar volume of a monomer. The values of bpp and the exponent ν in Eq. (14) can be obtained from the experimental second virial coefficient data. These values for PEG have been found as 0.0208 L mol−1 and 0.593 respectively by Kabiri-Badr et al. [20]. For other long linear polymers the value of the scaling exponent, ν, is the same as for PEG but the value of the effective monomer–monomer interaction coefficient, bpp , will be different. The scaling law for Cppp can be obtained using the relation between the second (Bpp ) and third (Bppp ) McMillan–Mayer osmotic virial coefficients αp =

Bppp 2 Bpp

(16)

where α p = 0.075 is a universal constant for long linear polymers. Substitution of Bppp into Eq. (4) and using the Eqs. (16) and (13) gives 

     2 α 3ν 2 − 2 b N 3ν V ∞ N + V ∞ N Cppp = dw pp p bpp N MP MP

2 

.

(17) The values for the salt Hill osmotic virial coefficients (Css and Csss ), can be obtained from experimental data on salt–water mixtures. The expression for the fitting of activity coefficient data is exp

ln γ±

3 = ln γ±LR + 2νs2 Css m s + νs3 Csss m 2s 2

(18)

where the expression for ln γ±LR is given by Eq. (7). It was found that the best fitting to Eq. (18) can be obtained using b = 1.8 in the Pitzer–Debye–H¨uckel equation for MgSO4 , Na2 SO4 , Na2 CO3 , (NH4 )2 SO4 , and K2 HPO4 salts. The coefficients Css and Csss for these salts were obtained using the mean ionic activity coefficient data of Rard and Miller [34] and Robinson and Stokes [35]. These coefficients and corresponding standard deviations (σ ), which were obtained by the method of least squares are given in Table 1. The parameters Cps , Cpps and Cpss which are salt–polymer parameters, can be evaluated from experimental water activity

data for the polymer–salt–water systems by rearranging Eq. (10) into the following form: " 1 ln aw − + m p + νs m s m s m p Mw ! # Z ms ∂ ln γ±LR dm s + νs ms ∂m s 0 T,P i 1 h 2 − νs Css m 2s + Cpp m 2p + νs3 Csss m 3s + Cppp m 3p ms m p = 2νs Cps + 3νs Cpps m p + 3νs2 Cpss m s .

(19)

A plot of the left-hand side of Eq. (19) versus m p and m s will give 2νs Cps as the intercept and 3νs Cpps , and 3νs2 Cpss as the slopes. The uncertainty in the water activity did not allow us to calculate accurate values of Cpss and Cpps so these parameters were set to zero within the experimental uncertainty of ±0.02% in the water activity data. A scaling law for the dependence of the second McMillan–Mayer osmotic virial coefficient Bps on polymer molecular mass was developed by Kabiri-Badr et al. [20] as follows Bps = bps N τ

(20)

where τ is the scaling exponent and bps is the salt dependent interaction coefficient. The expression for the dependence of Cps on the polymer molecular mass using polymer-scaling laws is obtained by inserting Eq. (20) into Eq. (3) as shown below Cps =

1 ∞ ∞ dw (2bps N τ − V s − V MP N ) 2

(21)

∞

where V s is the partial molar volume of salts at infinite dilution which were reported by Zafarani et al. [36], and are given in Table 1. The values for bps and τ were obtained from the water activity data for salts + PEG systems [27–29]. The scaling exponent τ is 0.96 and the values for bps are listed in Table 1. 3. Phase diagram calculation The phase diagram of a salt(s)–polymer( p)–water(w) mixture at equilibrium is obtained from the following equation: β

µiα (T, P, m αp , m αs ) = µi (T, P, m βp , m βs ) i = p, s, w

(22)

where α and β indicate the two equilibrium phases. The standard state of the chemical potential is infinite dilution in the

A. Salabat / Computer Coupling of Phase Diagrams and Thermochemistry 30 (2006) 296–300

Fig. 1. Phase behavior of aqueous mixture of PEG1000 and Na2 SO4 at 298.15 K, (N Experiment, Model).

•

Fig. 2. Phase behavior of aqueous mixture of PEG6000 and Na2 SO4 at 298.15 K, (N Experiment, Model).

•

299

Fig. 4. Phase behavior of aqueous mixture of PEG6000 and K2 HPO4 at 298.15 K, (N Experiment, Model).

•

Fig. 5. Phase behavior of aqueous mixture of PEG1000 and (NH4 )2 SO4 at 298.15 K, (N Experiment, Model).

•

MathCAD software was used to solve these equations. We have calculated phase diagrams for ten different polymer–salt–water systems using the model. The calculated phase diagrams were compared with the experimental phase diagrams of Refs. [4– 7,20]. The comparisons of calculation to experiment for some systems are illustrated in Figs. 1–6. The average relative deviations of the calculated weight percent from the experimental values for the investigated systems have been obtained using the following equation: 1 X wi (ca) − wi (ex) δ% = × 100 N wi (ex)

Fig. 3. Phase behavior of aqueous mixture of PEG1000 and K2 HPO4 at 298.15 K, (N Experiment, Model).

•

pure solvent. The component chemical potentials are inserted on both sides of Eq. (22). The chemical potential model is given by Eqs. (8)–(10). The resulting three simultaneous equations β β contain four unknowns (m αp , m αs , m p , m s ). We set a value for one of the molalities and solve for the remaining three. The

where N is the number of experimental points, wi is the weight percent of the component i, and the ‘ex’ and ‘ca’ denote the experimental and calculated values respectively. The maximum average relative deviations (δ%) for the investigated systems have been obtained as δ1 = 1.2 δ2 = 1.0 and δ3 = 0.7 for polymer, water and salt, respectively. 4. Conclusion This work presents a new model for the phase behavior of aqueous mixtures of salt and polymer. In this model, the

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References

Fig. 6. Phase behavior of aqueous mixture of PEG6000 and (NH4 )2 SO4 at 298.15 K, (N Experiment, Model).

•

Kabiri-Badr model is simplified by changing the Long Range part of the model by Pitzer–Debye–H¨uckel theory, with better or at least the same results. The basis of this model is the solution theory of Hill, which is extended to include the effect of electrostatic forces using the Pitzer–Debye–H¨uckel equation. Scaling laws were used to develop the model for the polymer molecular mass dependence of the polymer–polymer and the salt–polymer Hill osmotic virial coefficients. The experimental values for the polymer–polymer and salt–salt Hill osmotic virial coefficients were obtained from binary polymer–water and salt–water data. The salt–polymer Hill osmotic virial coefficients were obtained from the water activities in aqueous salt–polymer mixtures. In this study it was found that when this model is used for binary aqueous salt systems, the best fitting to experimental activity coefficients of salt can be obtained by b = 1.8 in the Pitzer–Debye–H¨uckel equation. The model has been used to calculate phase diagrams for several aqueous salt–polymer systems. The phase diagram calculations are basically a priori since all the model parameters were determined from independent measurements not directly connected with the experimental phase compositions. The agreement between the predicted and the experimental phase diagrams is good.

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