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Fluid Phase Equilibria 306 (2011) 67– 75
Contents lists available at ScienceDirect
Fluid Phase Equilibria
j our na l ho me page: www.elsev ier .com/ locate / f lu id
ePC-SAFT: Modeling of polyelectrolyte systems 2. Aqueous two-phase systems
hahbaz Naeem, Gabriele Sadowski ∗
aboratory of Thermodynamics, Technische Universität Dortmund, Emil-Figge-Str. 70, D-44227 Dortmund, Germany
r t i c l e i n f o
rticle history:eceived 27 October 2010eceived in revised form 24 February 2011ccepted 28 February 2011vailable online 8 March 2011
eywords:
a b s t r a c t
This work considers aqueous two-phase systems (ATPS) containing one polymer–polyelectrolyte as wellas one salt. To model the liquid–liquid equilibria (LLE) of these systems, the recently presented modelpePC-SAFT has been employed. ATPS containing poly(acrylic acid) of different degrees of neutralization orpoly(vinyl pyrrolidone), respectively, were considered. The binary interaction parameters used betweenwater–poly(acrylic acid) and water–poly(vinyl pyrrolidone) were adjusted to vapor–liquid equilibrium(VLE) data of these systems. ATPS consisting of poly(vinyl pyrrolidone)–water–sodium sulfate were pre-
olyelectrolytesegree of neutralizationounterion condensationqueous two-phase systemsiquid–liquid equilibriaodeling
dicted as function of temperature as well as of molar mass of the polymer. For poly(acrylic acid) systems,ATPS were predicted as function of charge density (degree of neutralization) for different types of salt. Forthese calculations, the polyelectrolyte model parameters were determined from the non-charged poly-mer whereas the effect of increasing charge density has been purely predicted by the model. Using thisapproach, it is possible to predict the shrinking of the liquid–liquid equilibrium region with increasingcharging of the polyelectrolyte.
C-SAFT
. Introduction
Aqueous two phase systems (ATPS) are used in biochemicalesearch for the separation and purification of macromolecules,ells and cell particles [1]. In recent years, ATPS were also applied forreparing polymeric micro particles without using organic solvents2]. ATPS are based on the phenomenon that systems consistingf water and either two hydrophilic (or even charged) polymersr one hydrophilic polymer and an inorganic salt separate intowo immiscible water-rich phases [3]. The high water content inqueous two-phase systems, typically greater than 80 wt.%, cou-led with a low interfacial tension provide a benign environmentor biomolecules not attained in solvent extraction. The ATPS offers
technically simple, energy efficient, easily scalable and mild sepa-ation technique for product recovery in biotechnology [4]. They areainly used for the concentration and purification of proteins and
n the extractive bioconversion of enzymes. The separation tech-ique is also becoming important in non-biotechnology areas suchs industrial waste remediation.
As a result of the development of aqueous two-phase systems,iquid–liquid extraction has gained an increasing importance iniochemical engineering for purification and isolation of macro-
olecules, such as proteins or antibiotics. This process has a lowerost compared with traditional biomolecules separations, due tose of traditional equipment of extraction and the small number
∗ Corresponding author. Tel.: +49 231 755 2635; fax: +49 231 755 2572.E-mail address: [email protected] (G. Sadowski).
378-3812/$ – see front matter © 2011 Elsevier B.V. All rights reserved.oi:10.1016/j.fluid.2011.02.024
© 2011 Elsevier B.V. All rights reserved.
of stages. It is possible to have an extremely selective separation ofsubstances using aqueous two-phase systems; they provide a gen-tle and protective environment for biological material, since bothphases are primarily composed of water.
ATPS of the type polymer–inorganic salt have several advan-tages over the polymer–polymer type: larger relative size of thedrops, larger density differences, larger selectivity, smaller viscos-ity and lower costs. Moreover, systems with two polymers are moredifficult to adapt to industrial scale due to high viscosity [5]. How-ever, the thermodynamic behavior of the polymer–salt systemsis more complicated and therefore needs to be investigated verycarefully by both, experiments as well as modeling.
Thermodynamic modeling of ATPS, especially of an aqueouspolymer–salt system, is challenging and complex. Compared toATPS composed of two polymers, additional electrostatic interac-tions as well as the electro neutrality of the coexisting phases hasto be considered. Several excess Gibbs energy models like NRTL(Renon and Prausnitz) [6], UNIQUAC (Abrams and Prausnitz) [7],ASOG (Derr and Deal) [8] and UNIFAC (Fredenslund et al.) [9] havebeen used in the past to describe polymer–polymer ATPS. But, tothe best of our knowledge, there exist only a few thermodynamicmodels (GE-models and equations-of-state; EOS) for polymer–saltATPS. One example is the Elbro-FV model of Sé and Aznar [10](based on the UNIFAC-Dortmund group contribution model) whichwas employed to calculate ATPS containing poly(ethylene glycol)
and sodium sulfate. Zafarani-Moattar et al. [11] calculated the LLEi.e., ATPS of aqueous poly(ethylene glycol)–sodium citrate systemusing the UNIQUAC model with an additional Debye–Hückel term.Recently, Grossmann et al. [12] presented a VERS Gibbs-energy
6 Phase Equilibria 306 (2011) 67– 75
maa
aptidicpotbNm(Tdseooa
tll
bm[ineF
tgmpaicta
tomikd
fltaa
av[pse
Sa lt
Pol yelectro lyte( > 0 )
S1L1
S1S2L2
S2L2
S1L1L2
L
L1L2
Wate r
Water Sal t
Polym era
b
( = 0)
S1L1
S1S2L2
S2L2
S1L1L2
S1L2
L
L1L2
Fig. 1. (a) General phase diagram of polyelectrolyte ( = 0)–salt ATPS system at con-stant temperature and pressure [15]. Shaded area represents the LLE region. S and
8 S. Naeem, G. Sadowski / Fluid
odel based on the Pitzer equation for small electrolytes andpplied the model to aqueous poly(ethylene glycol)–sodium sulfatend poly(vinyl pyrrolidone)–sodium sulfate systems.
Due to their better solubility, polyelectrolytes are a worthwhilelternative for polymers in ATPS. Although the charging of theolymer backbone increases the solubility of the respective sys-em components and shrinks (decreases) the region of liquid–liquidmmiscibility, enrichment of target concentrations can be obtaineduring extraction. Polyelectrolytes are polymers that dissociate
n aqueous solutions into a charged polymeric backbone and theorresponding number of mobile counterions. The properties ofolyelectrolyte solutions strongly depend upon (1) their degreef neutralization which is the ratio of charged monomer groupso the total number of monomers along the polyelectrolyte back-one, (2) the counterions used for neutralization, e.g. Na+, Li+, K+ orH4
+ etc., (3) the extent of counterion condensation which deter-ines the ultimate conformation of polyion chain in solution and
4) the molar mass and concentration of the polyelectrolyte [13].he phase behavior of polyelectrolyte–salt ATPS moreover stronglyepends on the choice of the salt. Due to the complex nature of theseystems, only very few theoretical studies exist to describe poly-lectrolyte ATPS. Lammertz et al. [14] extended the VERS modelf Grossmann et al. [12] to polyelectrolyte systems and calculatedsmotic coefficients of polyelectrolyte solutions with and withoutdditional sodium chloride but did not consider ATPS.
The general phase behavior of polyelectrolyte–salt ATPS as func-ion of degree of neutralization is shown in Fig. 1a–b where theiquid–liquid phase envelope is confined within solid–liquid equi-ibrium regions.
For polymer–polymer ATPS it is known that the shape androadness of the liquid–liquid immiscibility gap depends upon theolar masses of the respective polymers as well as on temperature
16]. However, in case of polyelectrolyte ATPS, the liquid–liquidmmiscibility envelope also strongly depends on the degree ofeutralization of the polyelectrolyte. Increasing of the poly-lectrolyte causes a general shift of the binodal curve as shown inig. 1b.
It is evident from Fig. 1b that by increasing ˛, the broadness ofhe binodal curve shrinks whereas at the same time the tie lineset longer which in turn means that the coexisting phases becomeore different and thus more valuable for extraction purposes. The
hysical explanation of this phenomenon is the presence of chargeslong the polyelectrolyte backbone which obviously leads to anncrease of the polyion solubility in water. Further increasing theharge-density of the polyelectrolyte causes a further shrinking ofhe miscibility gap and for a certain value, of ˛, it finally disappearsnd thus cannot be used for extraction purposes anymore.
The general phase behavior shown in Fig. 1a and b depends uponhe degree of neutralization, molar mass of polyelectrolyte, typef counterion, type of additional salt, and of temperature. Ther-odynamic modeling of polyelectrolyte systems is, therefore, very
mportant and at the same time very challenging. To the best of ournowledge, so far there exists no model which predicts or at leastescribes the LLE of polyelectrolyte–salt ATPS as function of ˛.
We recently presented a thermodynamic model pePC-SAFT forexible polyelectrolyte chains [13]. It was already used to predicthe vapor–liquid equilibria of aqueous solutions of polyelectrolytess function of molar mass, degree of neutralization ˛, counter ionnd temperature with and without additional salt.
pePC-SAFT uses a unique set of pure component parameters forll degrees of neutralization. It is obtained by fitting to density andapour–pressure data of solutions containing the neutral polymer
13]. The effect of polyelectrolyte’s charge density was explicitlyredicted by the model. The water activities in poly(acrylic acid)olutions of different degrees of neutralization were found to be inxcellent agreement with the experimental data.L stand for solid and liquid, respectively. (b) General phase diagram of aqueouspolyelectrolyte ( > 0)–salt ternary system at constant temperature. Shaded arearepresents LLE region. S and L stand for solid and liquid, respectively.
In this paper we apply the pePC-SAT to model ATPS systemscontaining [poly(vinyl pyrrolidone); PVP] and [poly (acrylic acid);PAA] of different degrees of neutralization at the one hand as wellas a salt on the other hand. We intend to predict the influence ofthe degree of neutralization as well as of different salts (NaCl andNa2SO4).
A brief introduction to pePC-SAFT is given in Section 2.
2. pePC-SAFT: the equation of state
As polymers in the PC-SAFT approach [17], polyelectrolytesare represented in pePC-SAFT by a linear chain of tangentiallyjointed hard spheres. However, in pePC-SAFT [13] the hard-chaincontribution of PC-SAFT was replaced by a charged-hard-spherecontribution based on the work of Jiang et al. [18]. Counterioncondensation [19] has been considered to determine the effectivepolyion charge as well as the number of free counterions in solution.Moreover, the corresponding counterions as well as the chargedpolyelectrolyte segments were considered as embedded in a dielec-tric continuum [13]. Compared to original PC-SAFT, pePC-SAFT doesnot introduce any new pure-component or binary parameters andreduces to original PC-SAFT for non-polyelectrolyte systems. Thus,all parameters determined earlier for original PC-SAFT or ePC-SAFT
can still be used within pePC-SAFT.In pePC-SAFT, the residual Helmholtz energy is given as a sumof a repulsive contribution (charged-hard-chain, chc), and attrac-tive contribution (disp) and association (assoc). The electrostatic
S. Naeem, G. Sadowski / Fluid Phase Equilibria 306 (2011) 67– 75 69
Table 1Pure-component parameters of the pePC-SAFT EOS.
Substance m/Mw [mol/g] � [Å] u/k [K] εassoc/k [K] � Reference
Water 0.0670 2.79a 353.94 2425.67 0.05 [21]PAA and PA− 0.0160 4.20 249.50 2035.00 0.34 [28]PVP 0.0217 2.63 244.49 – – This workNa+ 0.0435 1.70 250.38 – – [13]Cl− 0.0282 3.19 249.62 – – [13]
0.00 [29]
a
ctctH
a
t[fA
tatcene
x
wn
ttpeatm
as(Shdfwit
3
3
Ctrw
0.00
0.04
0.08
0.12
0.16
0.20
0 0.2 0.4 0.6 0.8 1
wwater [g/g]
p [bar]
Fig. 2. Vapor pressure of an aqueous PVP solution as function of water weightfraction at different temperatures. Solid lines are results for water vapor pressure
independent of molar mass of PVP but linearly depends upon thetemperature of the system (see Table 2). All pePC-SAFT calculationsin this work have been performed using the weight average molarmass Mw.
0.98
1.02
1.06
1.10
1.14
0.4 0.6 0.8 1wwater [g/g]
ρ [g cm-3]
SO42− 0.0104 2.45
�w = � + 10.11 · exp{
−0.01775 · T}
− 1.417 · exp{
−0.01146 · T}
.
ontribution (ion) describes the Coulomb interactions betweenhe effective counterions (which do not participate in counterionondensation), charged segments of the polyion as well as addi-ional inorganic salts as already applied in ePC-SAFT [20,21]. Theelmholtz energy for polyelectrolyte systems thus reads as:
res = achc + adisp + aassoc + aion (1)
The details of charged-hard-chain [13], dispersion and associa-ion contribution [22,23] as well as of the electrostatic contribution20] can be obtained elsewhere. The Helmholtz energy expressionsor all the contributions, described in Eq. (1), are summarized inppendix A.
Depending upon the degree of neutralization of a polyelec-rolyte, a certain number of polyelectrolyte segments is chargedccompanied by the same number of counterions in solution. Dueo high charge density of the polyion chains, a certain number ofounterions in solution is attracted to them to form ion pairs. Thextent of counterion condensation and the remaining (effective)umber of counterions as well as of charged polyion segments arestimated by Manning’s theory [24,25]:
effectivecounterion = (1 − ˇ)xtotal
counterion and zeffectivep = (1 − ˇ)ztotal
p (2)
here x is the mole fraction of counterions while zp represents theumber of charges on the polyion backbone.
It has been assumed throughout this work that only the coun-erions originally belonging to the polyelectrolyte condense backo the chain and that the cations of additional salts do not takeart in counterion condensation. As already shown for vapor–liquidquilibria of aqueous poly(sodium acrylate) + sodium chloride andqueous poly(ammonium acrylate) + sodium chloride systems [13],his assumption is reasonable and has no negative impact on the
odeling results.All together, five pure-component parameters are required for
ssociating components (solvents or polymers/polyelectrolytes):egment diameter (�), segment number (m), dispersion energyu/k), association energy (εassoc/k) and association volume (�).mall inorganic ions are considered as non-associating, singleard-spheres (m = 1) and thus require only segment diameter andispersion energy as pure ionic parameters. Simple combining rulesor cross association as suggested by Wolbach and Sandler [26]ere used within the scope of this work. Hence only one binary
nteraction parameter kij, for the correction of dispersive interac-ions, is permitted.
. Results and discussion
.1. pePC-SAFT parameters
Pure-component parameters for water were taken from
ameretti et al. [21] who fitted the model parameters for awo-association-site water model. Pure ion parameters, fitted toespective aqueous salt solutions within the scope of pePC-SAFT,ere taken from our previous work [13]. They were treated ascalculated by pePC-SAFT at respective temperatures using kij from Table 2. Symbols[298.15 K (�), 308.15 K (�), 318.15 K (�), 328.15 K (�)] represent experimental data[27]. Mw,PVP = 10,000 g mol−1.
charged-hard-spheres with hydrated diameter �j and dispersionenergy uj/k (when interacting with water).
The pure-component parameters of poly(vinyl pyrrolidone)(PVP) have been fitted within this work to aqueous binary VLE anddensity data of this system at constant temperature and molar massof PVP (see Table 1). Figs. 2 and 3 show the vapour pressures andsolution-density data, respectively, of a PVP (10,000 g mol−1) aque-ous solution. The kij between water and PVP has been found to be
Fig. 3. Density of aqueous PVP solutions as function of water concentration at differ-ent temperatures. Solid lines are results of solution density calculated by pePC-SAFTat respective temperatures using kij (water–PVP) = (0.00084·T) – 0.399446. Symbols[298.15 K (�), 308.15 K (�), 318.15 K (�), 328.15 K (�)] represent experimental data[27]. Mw,PVP = 10,000 g mol−1.
70 S. Naeem, G. Sadowski / Fluid Phase
Table 2Binary interaction parameters kij used within this work.
Binary system kij Reference
tu
fppp
aT
jungnau
bas
h
3s
3
(ckcpad
FtNs
Water–PAA −0.13 [13]Water–PA− −0.13 [13]Water–PVP −0.399446 + 0.00084 T This work
As it can be seen in Figs. 2 and 3, a very good agreement betweenhe experimental data and model calculations can be obtainedsing the parameters from Tables 1 and 2.
As already mentioned earlier, the pure-component parametersor polyelectrolytes have been taken from the respective neutralolymer. The pure-component parameters of poly(acrylate) i.e.,olyion [PA−], have been kept identical to poly(acrylic acid) PAAarameters taken from Kleiner et al. [28].
Detailed pure-component parameters and all binary inter-ction parameters, used in this work, are also summarized inables 1 and 2, respectively.
The dispersion energy between water i and an ion (or polyion) is calculated using the standard van der Waals mixing ruleij = (ui·uj)0.5(1 − kij). Dispersion between two inorganic ions iseglected in this work. Since Held et al. [29] fitted uij/k for inor-anic ions to salt–water systems, a kij between water and ion is noteeded. The only binary interaction parameters used in this work,s given in Table 2, are those between water and PAA (the same issed for water and PA−) as well as between water and PVP.
The kij between water and PVP, as given in Table 2, was found toe a linear function of temperature while for water and PAA as wells for water and PA−, kij remains constant for all the temperaturestudied in this as well as in our previous work [13].
The kij between all other binary systems, not given in Table 2,ave been kept equal to zero for all calculations.
.2. Aqueous two-phase systems containing polyelectrolyte andalt
.2.1. Effect of saltFig. 4 shows the liquid–liquid immiscibility region for PAA
= 0)–water–NaCl and PAA ( = 0)–water–Na2SO4 systems at aonstant temperature of 25 ◦C. The binary interaction parameterij, as estimated in our previous work [13] (Table 2), has been kept
onstant for these calculations. The molar mass of the PAA, the tem-erature of the system as well as the cation originating from thedditional salt, i.e., Na+ are the same. The only difference is theifferent anion of the salt, i.e., Cl− and SO42−. The results in Fig. 4
0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6 0.0
0.2
0.4
0.6
Salt
PAA
Water0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6 0.0
0.2
0.4
0.6
Salt
PAA
Water
ig. 4. Liquid–liquid equilibrium region of aqueous PAA–NaCl and PAA–Na2SO4
ernary systems at 25 ◦C. Solid symbols and lines represent results predicted foraCl while open symbols and dashed lines are pePC-SAFT predictions for the Na2SO4
ystem. Mw,PAA = 4300 g mol−1.
Equilibria 306 (2011) 67– 75
show that pePC-SAFT predicts a remarkable effect of the various saltanions on the ATPS. The different slope of the tie lines predicted bypePC-SAFT for different salt systems is evident. The polymer-richphase, i.e., top phase in Fig. 4, exhibits a minor difference in concen-trations for both salts while the salt-rich phase, i.e., bottom phasein Fig. 4, contains significantly different salt concentrations.
It can, thus, be interpreted from Fig. 4 that the type of the saltstrongly influences the concentration region where ATPS exists. Thepolymer-rich phase exhibits almost the same concentration of therespective components for either NaCl or Na2SO4 containing sys-tems. This could be attributed to the fact that the two salts containthe same cation (which is also the counterion) while the anions, i.e.,Cl− and SO4
2− carry different amounts of negative charges. There-fore, the concentrations in the first liquid phase, i.e., polymer-richphase is controlled by the polymer back bone (charged or non-charged) while the concentrations in the second liquid phase, i.e.,salt-rich phase can be attributed to the type and valence of theadditional salt anion. In case of the PAA–NaCl system, the salt-richphase contains equimolar concentrations of Na+ and Cl− while inPAA–Na2SO4 system the molar concentration of Na+ is twice theconcentration of SO4
2−. This difference in molar concentrations ofcation and anion of the additional salt, when translated into weight-based concentrations, results in a different overall concentration ofsalt in salt-rich phase. However, it is not only the stoichiometryof the ions which determines the distribution of salt between thecoexisting phases of an ATPS. Moreover, the type of the ions, themolar mass of the polyelectrolyte as well as the type of its coun-terions and temperature play significant roles in influencing theliquid–liquid partitioning in ATPS.
3.2.2. Effect of degree of neutralizationAs already established in the introduction, with increasing
charge density of the polyelectrolyte, the miscibility gap ofpolyelectrolyte–water–salt system shrinks and the length of tielines alters. The latter is favourable since a higher concentra-tion enrichment of the extract can be obtained compared toextraction with polymer–salt systems. This phenomenon of mis-cibility gap shrinking and tie-line elongation, from PAA( = 0.0) toPNaA( = 0.35), is well predicted by pePC-SAFT for Na2SO4 and NaClsystems as in shown in Fig. 5a and b, respectively. Temperatureand molar mass of the polyelectrolyte are the same as in Fig. 4. Theonly difference is an additional concentration of Na+ counterionsoriginating from the polyelectrolyte backbone for PNaA( = 0.35)in both Na2SO4 and NaCl systems. The concentrations of coexistingphases are in excellent agreement with the experimental data forPNaA ( = 0.35)–Na2SO4 system. It is worthwhile to mention thatthe results in Figs. 4 and 5 are pure predictions without fitting anyparameters and the polyion parameters [poly(acrylate)] were takenfrom pure non-charged PAA.
A comparison of Fig. 5a and b, further reveals that for all degreesof neutralization (e.g., = 0.0 and 0.35), pePC-SAFT significantlydifferentiates between different salts. The model again predicts adifferent result for Na2SO4 and NaCl systems where the only dif-ference is that of an anion, i.e., SO4
2− and Cl− coming from thesalt.
3.2.3. Estimation of limiting degree of neutralizationFurther calculations of the shrinking liquid–liquid immiscibil-
ity with increasing degree of neutralization of PNaA have beenperformed to determine the limiting degree of neutralization. Lim-iting degree of neutralization is defined here as the highest charge
density of the polyelectrolyte for which an ATPS exists. A limitingdegree of neutralization strongly depends upon the type and molarmass of the polyelectrolyte, type of additional salt and temperatureof the system.
S. Naeem, G. Sadowski / Fluid Phase Equilibria 306 (2011) 67– 75 71
0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6 0.0
0.2
0.4
0.6
Na2SO4 Water
PNaA
retaWlCaN
PNaA
0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6 0.0
0.2
0.4
0.6
0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6 0.0
0.2
0.4
0.6
Na2SO4 Water
a bPNaA
retaWlCaN
PNaA
0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6 0.0
0.2
0.4
0.6
Fig. 5. Liquid–liquid equilibrium region with increasing at 25 ◦C. Mw,PNaA = 4300 g.mol−1 (a) PAA( = 0.0)–Water–Na2SO4 and PNaA( = 0.35)–Water–Na2SO4, Open squaresj ons fof A( = 0d
agdior
Pa
0TltllptclNs
FPtoM
oined by solid lines and open triangles joined by dotted lines are pePC-SFT predictior aqueous PNaA( = 0.35)–Na2SO4 system. (b) PAA( = 0.0)–Water–NaCl and PNaashed lines are pePC-SFT predictions for = 0.0 and 0.35, respectively.
As already observed from Figs. 4 and 5, for constant temperaturend molar mass of the polyelectrolyte, the liquid–liquid miscibilityap starts to shrink towards right side of the triangular diagram, i.e.,epleting salt concentration and length of tie lines keeps increas-
ng. Following this trend, it is presumed that for limiting degreef neutralization the tie lines move towards the solubility of theespective salt in the system.
pePC-SAFT predictions for aqueous PNaA–NaCl andNaA–Na2SO4 systems at limiting degree of neutralizationre given in Fig. 6.
In Fig. 6, the liquid–liquid region disappears for a smaller of.39 for the Na2SO4 system compared to 0.41 for the NaCl system.his effect may be attributed to the fact that at 25 ◦C Na2SO4 has aower solubility in water (224 g l−1 of water, i.e., wwater = 0.817 g/g)han NaCl (359 g l−1 of water, i.e., wwater = 0.736 g/g). Since theimiting degree of neutralization marks the boundary betweeniquid–liquid and homogeneous liquid phase region, the salt-richhase should be very close to the solubility point of the respec-ive salt-system (see Figs. 1a and b). Upon closely examining the
oncentrations of salt-rich phases the trend of the decreasingiquid–liquid miscibility gap towards the solubilities of NaCl anda2SO4 in water can be observed. Therefore, the solubility of thealt in solvent (water) might be one of the contributing factors to
0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6 0.0
0.2
0.4
0.6
Salt Water
PNaA
0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6 0.0
0.2
0.4
0.6
Salt Water
PNaA
ig. 6. Liquid–liquid equilibrium region of aqueous PNaA( = 0.41) – NaCl andNaA(˛ = 0.39) – Na2SO4 ternary systems at 25 ◦C for limiting degree of neu-ralization. Solid symbols and lines represent predicted results for NaCl whilepen symbols and dashed lines are pePC-SAFT predictions for Na2SO4 system.w,PNaA = 4300 g mol−1.
r = 0.0 and 0.35, respectively. Solid circles are experimental cloud point data [30].35)–Water–NaCl Solid squares joined by solid lines and open triangles joined by
the limiting degree of neutralization of the polyelectrolyte. Owingto lower solubility of Na2SO4 in water, PNaA–Water–Na2SO4 sys-tem converges to homogeneous liquid phase at smaller comparedto NaCl system.
No liquid–liquid equilibria could be predicted for thePNaA–Water–NaCl and PNaA–Water–Na2SO4 systems for ˛-valuesgreater than 0.41 and 0.39, respectively.
3.3. Aqueous two-phase systems containing non-polyelectrolyteand salt
3.3.1. Effect of temperatureIn Fig. 7 experimental cloud-point data of the ATPS consisting
of PVP–Na2SO4 at 298.15 K and 338.15 K [15] is presented in com-parison with modeling results from pePC-SAFT. Cloud-point datarepresents the overall region where the ATPS exists but does notshow the tie lines. It is evident from the predicted results in Fig. 7that the model predicts the liquid–liquid immiscibility exactly inthe same region as found by the experimental data.
It can be seen from the experimental data that temperaturedoes not influence the liquid–liquid immiscibility of the aqueousPVP–Na2SO4 ATPS appreciably. Similar results were predicted bypePC-SAFT at 298.15 K as well as 338.15 K, as shown in Fig. 7. It isimportant to note that the results given in Fig. 7 have been obtainedwithout fitting adjustable parameters and thus are completely pre-dicted in excellent agreement with experimental data. Mw has beenused for all pePC-SAFT predictions.
3.3.2. Effect of molar massFigs. 8 and 9 compare the ATPS of water–Na2SO4–PVP for
two different molecular weights of PVP at 298.15 K and 338.15 K,respectively. It shows experimental cloud-point data as well asthe two-phase concentrations predicted with pePC-SAFT. It is evi-dent from the predicted results that model again predicts theliquid–liquid immiscibility in the same region as found by theexperimental data.
Experimental cloud-point data at 298.15 K and 338.15 K showthat increasing the molar mass of the polymer, i.e., PVP17 andPVP30 slightly increases the liquid–liquid immiscibility region.Although not very significant, there is an expansion of theliquid–liquid two-phase region (binodal curve) towards the water-
rich phase as seen by the experimental data of PVP17 and PVP30.A similar prediction is given by pePC-SAFT for both molecularweights of the polymer at both temperatures. A very small shiftof the tie lines towards the water-rich phase in Fig. 8 is observed
72 S. Naeem, G. Sadowski / Fluid Phase Equilibria 306 (2011) 67– 75
0.5 0.6 0.7 0.8 0.9 1.0
0.0
0.1
0.2
0.3
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0.5 0.0
0.1
0.2
0.3
0.4
0.5
0.5 0.6 0.7 0.8 0.9 1.0
0.0
0.1
0.2
0.3
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0.5 0.0
0.1
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Na2SO4
PVP17
Water Na2SO4
PVP1 7
Water
a) 298 .15K b) 338 .15K
Fig. 7. Liquid–liquid equilibrium of the aqueous PVP17–Na2SO4 ternary system at (a) 298.15 K and (b) 338.15 K. Solid symbols are experimental cloud point data [15] whileopen symbols joined by dashed lines represent pePC-SAFT predictions. PVP17; Mw = 9411 g mol−1 and Mn = 3882 g mol−1.
0.5 0.6 0.7 0.8 0.9 1.0
0.0
0.1
0.2
0.3
0.4
0.5 0.0
0.1
0.2
0.3
0.4
0.5
Na2SO4
PVP17
Water Na2SO4
PVP30
Water0.5 0.6 0.7 0.8 0.9 1.0
0.0
0.1
0.2
0.3
0.4
0.5 0.0
0.1
0.2
0.3
0.4
0.5T = 298.15K
0.5 0.6 0.7 0.8 0.9 1.0
0.0
0.1
0.2
0.3
0.4
0.5 0.0
0.1
0.2
0.3
0.4
0.5
Na2SO4
PVP17
Water Na2SO4
PVP30
Water0.5 0.6 0.7 0.8 0.9 1.0
0.0
0.1
0.2
0.3
0.4
0.5 0.0
0.1
0.2
0.3
0.4
0.5T = 298.15K
F Solid
b = 941
wSiippi(
Fb
ig. 8. Liquid–liquid equilibrium aqueous PVP–Na2SO4 ternary system at 298.15 K.y dashed lines represent pePC-SAFT predictions. PVP17; Mn = 3882 g mol−1 and Mw
hich is in qualitative agreement with the experimental data.imilarly, at 338.15 K an increase in concentrations of the coexist-ng phases especially in polymer-rich phase is observed as shownn Fig. 9. The water-rich phase in both cases contains almost noolymer which is quantitatively predicted without adjusting any
arameters for PVP17 as well as PVP30 systems. The only binarynteraction parameter used was the one between water and PVPTable 2).
0.5 0.6 0.7 0.8 0.9 1.0
0.0
0.1
0.2
0.3
0.4
0.5 0.0
0.1
0.2
0.3
0.4
0.5
Na2SO4
PVP17
Water
0.
T = 338.
0.5 0.6 0.7 0.8 0.9 1.0
0.0
0.1
0.2
0.3
0.4
0.5 0.0
0.1
0.2
0.3
0.4
0.5
Na2SO4
PVP17
Water
0.
T = 338.
ig. 9. Liquid–liquid equilibrium aqueous PVP–Na2SO4 ternary system at 338.15 K. Solid
y dashed lines represent pePC-SAFT predictions. PVP17; Mn = 3882 g mol−1 and Mw = 941
symbols represent experimental cloud points data [15] while open symbols joined1 g mol−1 PVP30; Mn = 17,750 g mol−1 and Mw = 57,980 g mol−1.
As evident from the experimental data, temperature does notinfluence the liquid–liquid immiscibility region appreciably asobserved from Fig. 7 as well as by comparing Figs. 8 and 9. Sim-ilar results were predicted by the pePC-SAFT at 298.15 K as well as338.15 K for both, PVP17 and PVP30.
Due to enhanced liquid–liquid immiscibility region, as evidentfrom the comparison of Figs. 7–9, the extraction of an inert com-ponent with the help of aqueous PVP–Na2SO4 ATPS would be
0.5 0.6 0.7 0.8 0.9 1.0
0.0
0.1
0.2
0.3
0.4
5 0.0
0.1
0.2
0.3
0.4
0.5PVP30
Na2SO4 Water
15K
0.5 0.6 0.7 0.8 0.9 1.0
0.0
0.1
0.2
0.3
0.4
5 0.0
0.1
0.2
0.3
0.4
0.5PVP30
Na2SO4 Water
15K
symbols represent experimental cloud points data [15] while open symbols joined1 g mol−1 PVP30; Mn = 17,750 g mol−1 and Mw = 57,980 g mol−1.
Phase
pacp
4
etsdPmaotnaahi
pwpfosip
LAaaaaBb
bbCcdegIKkkMMMMmmNNNnopTu
S. Naeem, G. Sadowski / Fluid
referred when a polymer of higher molar mass is used. However,t the same time the viscosity of polymer-rich phase has to be takenare of especially at high concentrations and high molar masses ofolymer.
. Conclusion
An extended PC-SAFT EOS, called pePC-SAFT, accounting for theffect of charge density of a polyelectrolyte in the reference con-ribution has been used as a predictive tool for ATPS containingalts as well as a polyelectrolyte or a non-charged polymer. Pre-ictions have been performed for polyelectrolyte (i.e., PAA andNaA) ATPS as a function of degree of neutralization of the poly-er and type of salt as well as for non-polyelectrolyte (PVP) ATPS
s function of temperature and molar mass of the polymer. Basedn the unique set of pure-component parameters for the neu-ral polymer, the liquid–liquid immiscibility in polyelectrolyte andon-polyelectrolyte ATPS could be predicted. The only binary inter-ction parameters used in this work are the ones between waternd the neutral polymers, i.e. water–PAA and water–PVP, whichave been kept constant throughout this work for all systems stud-
ed.The shift in the range of liquid–liquid immiscibility of
olyelectrolyte–salt ATPS for different degrees of neutralizationith different salts has been predicted almost quantitatively. Modelredictions reveal a slightly charged polyelectrolyte to be usefulor the extraction of inert biomolecules. In addition to aque-us polyelectrolyte–salt systems, ATPS of aqueous polymer–saltystems have also been predicted in good agreement with exper-mental data for different temperatures and molar masses usingePC-SAFT.
ist of symbols
i association site of type A on molecule i Helmholtz free energy per number of particles [J]i, aj Debye inverse screening length [Å−1]01, a02, a03 model constants defined in (A.20)i(m) function defined in (A.20)j association site of type B on molecule j
average distance between charged-segments of polyion[Å]
01, b02, b03 model constants defined in (A.21)i(m) function defined in (A.21)1 abbreviation for compressibility expressionc fraction of effectively charged-segments of polyion
temperature dependent segment diameter [Å] elementary charge ( = 1.60217646 × 10−19 [col])hs average radial distribution function of hard-sphere fluid
1, I2 abbreviations defined in (A.18 and A.19)a equilibrium constant of counterion condensation
Boltzmann constant, 1.38065 × 10−23 [J/K]ij binary interaction parameter between component i and jw mass average molecular weight [g/mol]n number average molecular weight [g/mol]neutralizedm molar mass of neutralized monomer [g/mol]non-neutralizedm molar mass of non-neutralized monomer [g/mol]
number of segments per chain¯ mean segment number in the system
Ai number of association site of A type on molecule iBj number of association site of B type on molecule j
AV Avogadro’s number, 6.023 × 10−23 mol−1
sites,i total number of association sites per molecule i
o fraction of non-charged segments of polyionpressure [bar]temperature [K]
dispersion-energy parameter [K]
Equilibria 306 (2011) 67– 75 73
w weight fraction [g/g]XAi fraction of association site of type A on molecule iXBj fraction of association site of type B on molecule jx mole fraction [mol/mol]ycc average cavity correlation function of charged hard-chain
fluidyoo average cavity correlation function of non-charged hard-
chain fluidZ compressibility factorz valence number of ions or polyions (number of unit
charges per ion or polyion) [Eq. (2)]
Greek letters˛ degree of neutralization of polyelectrolyte� temperature-independent segment diameter of molecule
[Å]�∗
w temperature-dependent segment diameter of water [Å] extent of counterion condensation
� abbreviation defined in (A.31)�AiBj association strength defined in (A.26)ε dielectric constant of medium, εrε0 [C/Vm]εassoc/k association-energy parameter [K]� packing fraction, � = �3� association-volume parameter�D Debye length [Å−1]� density [g cm−3]�N total number density of molecules [Å−3]˚c volume fraction of charged-segments of polyion�n abbreviation (n = 0,1,. . .,3) defined in (A.4 and A.5) scaling parameter
Superscriptsassoc associationchc charged-hard-chaindisp dispersioneffective effective (concentration or number)hs hard-sphereion ionicres residual [Eq. (1)]total total concentration of counterions or total number of
charges on polyion backbone [Eq. (2)]
Subscripts1,2 phase1, phase2i, j, k component indicescounterion cation belonging to polyion chaincp charged segments of polyionm monomerp polyelectrolyte
Acknowledgment
The authors thank BASF SE for financial support.
Appendix A.
This section provides a summary of equations for calculatingthe different Helmholtz-energy contributions of pePC-SAFT. Theresidual Helmholtz energy consists of the charged-hard-chain (chc)reference contribution and dispersion (disp), association (assoc) aswell as electrolyte (ion) contributions as different perturbations.
Thus, the Helmholtz energy for polyelectrolyte systems accordingto pePC-SAFT reads as:ares = achc + adisp + aassoc + aion (A.1)
7 Phase
A
a
wcpt
m
g
a
fl
y
ac
y
wdc
�t
�
an
d
c
wodsst
4 S. Naeem, G. Sadowski / Fluid
.1. Charged-hard-chain reference contribution
chc = mahs − kT∑
i
xi(mi − 1) · [(1 − cc)
· ln yoo(dii) + cc · ln ycc(dii)] (A.2)
here xi and mi are the mole fraction and the segment number ofomponent i, respectively. cc is the fraction of effectively chargedolyion segments given in (A.10). The mean segment number m ofhe mixture is defined as:
¯ =∑
i
ximi (A.3)
The hard-sphere contribution to the Helmholtz energy ahs isiven on a per-segment basis
hs = 6�N
kT
[�3
2 + 3�1�2� − 3�1�2�23
�3(1 − �3)2−(
�0 − �32
�23
)· ln(1 − �3)
](A.4)
The radial distribution function of the non-charged hard-sphereuid is:
00(dii) = ghsii (dii) = 1
(1 − �3)+(
didi
di + di
)3�2
(1 − �3)2
+(
didi
di + di
)2 2�22
(1 − �3)3(A.5)
nd cavity correlation function of two segments (either of themharged or both) of two different molecules of same type
cc(dii) = ghsii (dii) · exp
(− e2
4εkT�ii
zizi
(1 + �ii)2
)
·exp
(− zizie
2
4εkT�ii
)(A.6)
here zi is the valance of the charged-segment, �ii is the segmentiameter of one polyion segment and is scaling parameter cal-ulated using the analytical expression given as:
=√
1 + 2�ii�D − 1
2�ii(A.7)
D is inverse Debye screening length given in the electrolyte con-ribution (A.30).
�n is defined as:
n =
6NAV �
∑i
ximi · dni n = 0, 1, 2, 3 (A.8)
nd di is the temperature-dependent segment diameter of compo-ent i, used in (A.5),
i = �i
(1 − 0.12 · exp
(−3ui
kT
))(A.9)
The fraction cc of effectively charged polyion segments is:
c = (1 − ˇ) · Mp
mp· ˛
· Mnon-neutralizedm + (1 − ˛) · Mneutralized
m
(A.10)
here ˛, ˇ, Mp, mp, Mnon−neutralizedm and Mneutralized
m are degreef neutralization of polyelectrolyte, fraction of counterion con-
ensation, molar mass of polyelectrolyte, number of polyionegments, molar mass of non-neutralized monomer (with protontill attached) and molar mass of neutralized monomer (with pro-on replaced by an inorganic counterion), respectively.Equilibria 306 (2011) 67– 75
The fraction of counterion condensation is estimated by com-paring Manning’s dimensionless association constant Ka (fromcounterion condensation theory) with the law of mass action.
Ka = exp
(z2
pe2
4εkTb
)= 1
˚c
ˇ
(1 − ˇ)2(A.11)
where b and ˚c are average distance between charged-segmentsalong the polyion chain and volume fraction of the charged seg-ments, respectively.
b and ˚c are calculated as:
b = maximum length of polyion chainnumber of charged segments on polyion chain
= �pmp
mcp(A.12)
and
˚c = xpmcp�3p∑
ximi�3i
(A.13)
A.2. Dispersion contribution
The dispersion contribution to the Helmholtz energy is given by
adisp
kT= −2� · I1(�, m) · m2u�3 − � · m · C1 · I2(�, m) · m2u2�3
(A.14)
where C1 is the abbreviation for the compressibility expression,defined as:
C1 =(
1 + Zhc + �∂Zhc
∂�
)−1
=(
1 + m8� − 2�2
(1 − �)4+ (1 − m)
20� − 27�2 + 12�3 − 2�4
[(1 − �)(2 − �)]2
)(A.15)
and the abbreviation in (A.14) are as follows:
m2u�3 =∑
i
∑j
xixjmimj
(uij
kT
)�3
ij (A.16)
m2u2�3 =∑
i
∑j
xixjmimj
(uij
kT
)2�3
ij (A.17)
The integrals, I1(�, m) and I2(�, m), of the perturbation theoryhave been replaced by power series which depends only on densityand segment number
I1(�, m) =6∑
i=0
ai(m) · �i (A.18)
I2(�, m) =6∑
i=0
bi(m) · �i (A.19)
where the coefficients ai and bi are the functions of segment num-ber and given as:
ai(m) = a0i + m − 1m
a1i + m − 1m
m − 2m
a2i (A.20)
bi (m) = b0i + m − 1b1i + m − 1 m − 2
b2i (A.21)
m m mThe model constants a0i, b0i, a1i, etc., have been fitted to theexperimental data of alkane homologous series and are reported inthe original PC-SAFT publication of Gross and Sadowski [17].
Phase
ck
�
u
A
wtla
fo
X
hItbe
�
r
ε
�
A
i
[[
[
[[
[[[[[
[[
[[[[[
S. Naeem, G. Sadowski / Fluid
In order to describe mixtures, conventional Berthelot–Lorentzombining rules using one adjustable binary interaction parameterij, to correct the mixture dispersion energy if necessary, are used.
ij = 12
(�i + �j) (A.22)
ij =√
uiuj(1 − kij) (A.23)
.3. Association contribution
The associating mixtures are modeled by repulsive coresith a number of off-center, short-ranged attraction sites and
he Helmholtz-energy contribution, for molecules having aarge number of association sites, due to association is givens:
aassoc
kT=∑
i
xi
nsites,i∑Ai=1
NAi
(ln XAi − XAi
2+ 1
2
)(A.24)
The summation is over all association sites of the molecule i. Theraction of molecules not bonded at the association site A can bebtained by the mass balance and is given by the implicit equation
Ai =
⎛⎝1 + � ·
∑j
xj
∑Bj
NBj XBj · �AiBj
⎞⎠
−1
(A.25)
ere NAi denotes the number of association types on molecule i.t is important to note that the summation runs over the associa-ion types not over the association sites where �AiBj denotes theonding strength for square-well off-center bonding sites can bexpressed as:
AiBj = ghsij (dij) · �AiBj · �3
ij
[exp{
εAiBj
kT
}− 1]
(A.26)
The cross-association in the mixtures is calculated using mixingules:
AiBj = εAiBi + εAjBj
2for well depth (A.27)
AiBj = √�AiBi · �AjBj ·
(√�i�j
�ij
)3
for association volume (A.28)
.4. Electrolyte contribution
The Helmholtz energy contribution aion due to electrostaticnteraction between charged polyion segments and free counter
[[
[[
Equilibria 306 (2011) 67– 75 75
ions as well as between salt ions is calculated by
aion = − 14ε
∑i
ximi(1 − ˇ) · z2i
3· �D · �i (A.29)
becomes zero for non-polyelectrolyte systems reducing (A.29)to original Debye–Hückel expression.with
�2D = 1
4kTε
∑i
ximi(1 − ˇ) · z2i (A.30)
and
�i = 3
(�Dai)3
[32
+ ln(1 + �Dai) − 2(1 + �Dai) + 12
(1 + �Dai)2](A.31)
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