- Email: [email protected]
Water activities in binary and ternary aqueous systems of poly(ethylene glycol), poly(propylene glycol) and dextran
European Polymer Journal 37 (2001) 1487±1492
www.elsevier.nl/locate/europolj
Water activities in binary and ternary aqueous systems of poly(ethylene glycol), poly(propylene glycol) and dextran A. Eliassi a, H. Modarress b,* b
a Iranian Research Organization for Science and Technology (IROST), Tehran, Iran Department of Applied Chemistry, Faculty of Chemical Engineering, Amir-Kabir University of Technology, 424 Hafez Ave Tehran, Iran
Received 1 February 2000; received in revised form 19 September 2000; accepted 17 November 2000
Abstract The vapor pressure osmometry (VPO) measurements were used to obtain activities of water for aqueous solutions of poly(ethylene glycol)/dextran and poly(ethylene glycol)/poly(propylene glycol) at temperatures 35°C, 45°C, 55°C and 65°C. The weight fractions of water were varied from 70% to 98%. To increase the accuracy, the calibration constants of the VPO apparatus for each measurement were also determined. The experimental results were used in Flory±Huggins theory to calculate the interaction parameters for the mixtures and the results were discussed in terms of Flory±Huggins theory for evaluating other thermodynamic properties. Ó 2001 Elsevier Science Ltd. All rights reserved. Keywords: Activity of water; Vapour pressure osometry; Polyethylene glycol; Dextran; Flory±Huggins theory
1. Introduction Aqueous polymeric two-phase systems are increasingly used in biochemistry and cell biology for separation of macromolecules from cell organells [1±4]. The properties of polymeric two-phase systems also make them very interesting for large scale industrial applications [5]. Poly(ethylene glycol) (PEG) is used in almost all applications of aqueous phase partitioning. For biochemical separations on the laboratory scale the most commonly used aqueous phase system is composed of PEG and dextran (DX) [1]. For large scale separations in biotechnical industry PEG/salt systems have been the most widely used, primarily for large scale enzyme extractions. Some binary mixtures of other polymers with PEG were also studied. These polymers were starch, poly(vinyl alcohol) (PVA) and poly(propylene glycol) (PPG). Modeling of the thermodynamic properties of polymer aqueous systems have been the interest of several
*
Corresponding author. Tel.: +98-911-222-0454; fax: +98021-640-5847. E-mail address: [email protected] (H. Modarress).
workers from dierent aspects. For example Edmond and Ogston [6,7] have used osmotic virial expansion model for studying the phase behavior of these systems. Connemann et al. [8] extended the osmotic virial expansion model to the polydisperse polymers. A combination of this model with HillÕs theory [9] was also used by Cabezas et al. [10] for predicting polymer/polymer/ water phase diagrams. Diamond et al. [11,12] applied the modi®ed Flory±Huggins theory to model the partitioning of proteins in PEG/DX/water two-phase systems. Hartounian and Sandler [13] utilized UNIQUAC, Debye±Huckel and Bronsted±Guggenheim equations for studying thermodynamic behavior of polymer/polymer/electrolyte/water systems. Also, Haynes et al. [14] have used integral equation theory to model aqueous polymeric systems. By experimental measurements various thermodynamic properties of aqueous PEG systems and with other polymers such as PPG and DX have been determined. For example Albertsson [1] and Zaslavsky et al. [15] have determined the phase diagrams of PEG/water systems. Hasse et al. [16] have obtained the second and third osmotic virial coecients and Florin et al. [17] have studied the salt eects on the cloud point of PEG/ water systems. Densities of PEG/water systems have
0014-3057/01/$ - see front matter Ó 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 0 1 4 - 3 0 5 7 ( 0 0 ) 0 0 2 6 2 - 7
1488
A. Eliassi, H. Modarress / European Polymer Journal 37 (2001) 1487±1492
been measured by Taw®k et al. [18] and Eliassi et al. [19]. Viscosities of PEG/salt/water systems have been determined by Mei et al. [20]. Schild and Tirrell [21] obtained the lower critical solution temperatures of PPG/water system. Phase behavior of PEG/PPG/water and PEG/ DX/water systems have been investigated by Malmsten et al. [22] and Sjoberg and Karlstrom [23], respectively. The vapor pressure data for binary PEG or DX aqueous systems have been obtained by Hatnes et al. [24]. Mixing enthalpy of PPG/water system have been measured by Carisson et al. [25]. Grossman et al. [26] and Eliassi et al. [27] have obtained the activity for PEG and DX aqueous solutions. In this work the activity of water measured by vapor pressure osmometry in PEG/PPG and PEG/DX aqueous solutions are reported. Based on the measured activities, the Flory±Huggins interaction parameters for binary and ternary aqueous polymer solutions are calculated.
gibly small, dT =dt 0 and a steady state is achieved. According to Kamide [32] the temperature dierence in steady state, T T0 s can be expressed as a power series of the concentration: DT T
T 0 s
2 VPO 3 Ks c=M AVPO 2;v c A3;v c
1
VPO where AVPO 2;v and A3;v are second and third virial coecients and Ks is a calibration parameter (cm3 K mol 1 ). Also, in Eq. (1) c (g cm 3 ), M (g mol 1 ) are the concentration, molecular weight of solute respectively. It is usual to express the activity of solvent in the form of virial expansion in terms of power series of concentration: 2 VPO 3 ln a1 V 0 c=M AVPO 2;v c A3;v c
2
where V 0 (cm3 mol 1 ) is the molar volume of solvent and the other symbols have been de®ned in Eq. (1). Therefore Eq. (1) can be written as: ln a1 V0
2. Experimental
DT Ks
There are some experimental methods for solvent activity measurements for polymer solution [28] such as dierential vapor pressure measurements [29], isopiestic methods [30] and membrane osmometry [31]. In this work, a VPO manufactured by Knauer (Germany) was used for measuring the activities of water in aqueous polymer systems. VPO method is based on the principle that at a given temperature the vapor pressure of a solution is less than that of the pure solvent. In a VPO, the solution and the solvent droplet are placed on two thermistors arranged in a wheatstone bridge circuit in such a way that the temperature rise can be measured very accurately as a function of the bridge imbalance output voltage, DV . Thermistors are placed in a chamber ®lled with a saturated vapor of solvent. Since the vapor pressures of solvent and polymer solution dier from each other, the solvent vapor condenses on the polymer solution droplet. Therefore, solution droplet starts getting diluted as well as heated up by the latent heat of condensation due to the solvent condensing on it. The instrument indicates the temperature dierence between the two thermistors (DT ) as the voltage dierence (DV / DT ) which is measured by a digital voltmeter. Kamide [32] has pointed out that actually there is a steady state and not an equilibrium state in VPO measurement. He has derived an equation for the rate of temperature change dT =dt of the liquid droplet on the thermistor in the VPO measurements. With the assumption that the temperature T of the solution droplet is very close to that of the solvent T0 , and changes in concentration, volume and density of the solution drop due to the condensation of solvent vapor are all negli-
On the other hand according to Kamide Ks is related to an equilibrium calibration constant Ke in the following form: Ks Ke g
3
4
where for simpli®cation g is used here to represent all the parameters introduced in the original Kamide equation and Ke has been given as [32]: Ke RT02 V 0 =DH
5
where DH is the molar heat of condensation which is DH vap and T0 is the temperature setting of apparatus in the VPO measurements. From Eqs. (3)±(5) the following equation can be obtained: ln a1
DH vap =RT02 gDT
6
Similar equation has been derived by Gaube et al. [33] and it is claimed that the eciency coecient g can correct for the in¯uences of heat losses in the apparatus. In the VPO apparatus the temperature dierence DT is related to the measured voltage DV . Brown [34], on the basis of equilibrium considerations, assumed that the measured voltage is proportional to the chemical potential dierence, that is; DV / Dl1 . The chemical potential dierence Dl1 for the solvent is expressed in terms of activity of the solvent as; Dl1 RT0 ln a1 . Therefore by introducing the relativity constant j the following equation can be obtained: DV jRT0 lna1
7
Brown [34] has introduced a molal calibration constant K as:
A. Eliassi, H. Modarress / European Polymer Journal 37 (2001) 1487±1492
K
jRT0 M1 1000
8
where M1 is the molecular weight of solvent. Thus the ®nal equation for activity of solvent is: ln a1
DV M1 1000K
9
As in VPO measurement DT / DV and also by considering the Eqs. (1), (6) and (9), it is obvious that these equations can be presented in the following general form: ln a1
b DV
10
where, b is a calibration parameter used to represent the coecients of DV in Eqs. (1), (6) and (9). On substituting for ln a1 from Eq. (2) in Eq. (10) and ignoring the higher order terms and then rearranging the ®nal equation, we have: DV K 0 Ac c
11
0 where K 0 V 0 =bM and A AVPO 2;v V =b. It is worth noting that in BrownÕs equation V 0 1000 cm3 . In this work from VPO measurements for a set of calibrating solutions, K 0 has been evaluated from Eq. (11) by plotting variations of DV =c versus c and extrapolating to in®nite dilution (c ! 0). In this way, K 0 is evaluated at each measurement and therefore, it is reasonable to believe that it includes any heat loss as well as any other de®ciencies mentioned by other workers in VPO measurements [32,33]. The calibrating compound used in our VPO measurements was aqueous solution of urea M 60. It is usual to de®ne a calibration constant for the VPO instrument as the product of K 0 by molecular weight of the calibrating sample Mcalib [35]:
K K 0 Mcalib
12
Now, by using the value of K in Eq. (9) the activity of the solvent will be determined. In a similar way, Fernandez et al. [36] have used VPO measurements to determine the activity of benzene in polyisobutylene/ benzene mixtures. They have concluded that VPO can be a suitable method for solvent activity measurements of solutions with concentrations of up to 40% in polymer weight. In this work, using the above described method, the activity of water in dierent solutions of PEG400/ DX9300, PPG425/PEG400 and PPG425/PEG4000 have been measured. PEG400 and PEG4000 were obtained from Merck (Germany), PPG425 were obtained from Fluka (Swiss) and DX9300 from Sigma (UK). Urea is the calibration sample recommended by the VPO manufacturer (Knauer, Germany) to be used for
1489
aqueous solution measurements. The urea used in this measurement was obtained from Merck (Germany). All materials were used without further puri®cation. The aqueous polymer solutions were made by mass using double-distilled water. The weighing was done by an analytical balance with 0.1 mg accuracy (Shimadzu, Japan, model AEU 210). In VPO technique, as mentioned above, when a droplet of solvent and a droplet of solution are placed on each thermistors, DV increases and reaches a maximum and remains constant. The required time (3±5 mins is adequate) for reaching this condition, however, depends on the type of solution and its temperature. To examine the accuracy and reproducibility of the measurements, each measurement was repeated ®ve to eight times, and then the average voltage dierence was used in the calculations by Eq. (9). The accuracy of the DV measurements is estimated to be better than 1%.
3. Results The values of calibration constant of the instrument K for three dierent set of experiments are reported in Table 1. As it is seen from Table 1 K is an increasing function of temperature. The values of water activity in PEG400/DX9300/ water, PEG400/PPG425/water and PEG4000/PPG425/ water systems are respectively reported in Tables 2±4. It is worth noting that the activities have been evaluated from Eq. (9). Since, polymers are considered as nonvolatile species and DV in Eq. (9) represents the eect of nonvolatile species on the volatility of the solvent in the mixture which is measured by DV in the VPO method. These results indicate that with increasing temperature, Table 1 Calibration constants (K in Eq. (9)) of VPO determined for a typical urea aqueous solutions Temperature (°C)
35
45
55
65
Urea (0.12 wt.%)
105.66
121.24
129.59
131.70
Table 2 Experimental results of activity of water for weight fractions of water (w1 ) and PEG (w2 ) in solutions of PEG400/DX9300/H2 O at various temperatures w1
w2
Temperature (°C) 35
45
55
65
0.9592 0.8886 0.8287 0.7488 0.7021
0.0215 0.0928 0.1542 0.2262 0.2719
0.9989 0.9945 0.9895 0.9815 0.9745
0.9989 0.9947 0.9900 0.9824 0.9763
0.9989 0.9948 0.9899 0.9831 0.9772
0.9989 0.9949 0.9901 0.9837 0.9786
1490
A. Eliassi, H. Modarress / European Polymer Journal 37 (2001) 1487±1492
Table 3 Experimental results of activity of water for weight fractions of water (w1 ) and PEG (w2 ) in solutions of PEG400/PPG425/H2 O at various temperatures w1
w2
Temperature (°C) 35
45
55
0.9715 0.9317 0.8875 0.8548 0.8239 0.7594
0.0151 0.0327 0.0580 0.0724 0.0869 0.1281
0.9982 0.9961 0.9935 0.9911 0.9887 0.9834
0.9865 0.9910 0.9928 0.9947 0.9968 0.9987
0.9885 0.9916 0.9932 0.9948 0.9968 0.9987
Table 5 Flory±Huggins interaction parameter (v12 ) for aqueous solutions of PEG400, PEG4000 and PPG425 at 35°C and 45°Ca Temperature (°C)
Concentration range of polymer
v12
Eb
n
35
PEG400 (0.0120±0.3150) PEG400 (0.0120±0.3150) PEG4000 (0.049±0.3949) PEG4000 (0.049±0.3949) PPG425 (0.0103±0.0951) PPG425 (0.0103±0.0951)
0.4148
0.0242
8
0.4207
0.0701
8
0.4857
0.0862
4
0.5147
0.0859
4
0.5766
0.0160
5
0.5768
0.0120
5
45 35 45 35
Table 4 Experimental results of activity of water for weight fractions of water (w1 ) and PEG (w2 ) in solutions of PEG4000/PPG425/ H2 O at various temperatures w1
w2
0.9788 0.9428 0.9109 0.8775
0.0114 0.0286 0.0446 0.0612
Temperature (°C) 35
45
55
0.9995 0.9986 0.9977 0.9965
0.9996 0.9988 0.9979 0.9969
0.9995 0.9987 0.9980 0.9970
the activity of water increases, but with increasing molecular weight of the polymers the activity of water decreases. 4. Modeling
where, a1 is the activity of solvent, u2 is the volume fraction of polymer, v12 is the interaction parameter and r2 is number of polymer segments. In general, the number of segments ri is de®ned as follows: molar volume of polymer i molar volume of solvent
On rearranging Eq. (13) we have: 1 1 u2 u2 v12 u2 ln a1 ln u1 r2
a
E is the relative error and n is the number of data points. P EXP E 1n i aEXP acal 100, aEXP is experimental i i =ai i activity of solvent, acal is calculated activity using reported v12 . i b
culations (E) are reported. The reported average errors between calculated and experimental activity data (in Table 5) indicate that the concentration dependency of v12 for the studied systems is negligible. Also, the results in Table 5 show that v12 for PEG/ water system is positive which means a positive value for enthalpy change upon mixing DHmix . This is consistent with the de®nition of DHmix in the original Flory±Huggins theory as represented by the following expression: DHmix RT v12 u1 u2
One of the well known models for activity of solvent in binary systems of solvent(1)/polymer(2) is the Flory± Huggins model which can be represented by the following equation [37]: 1 ln a1 ln 1 u2 1 u2 v12 u22 13 r2
ri
45
14
15
Therefore, the variation of the left-hand side of Eq. (15) versus u2 is linear and the slope is equal to v12 . The measured activity data in Tables 2±4, were treated by Eq. (9), and the values of v12 are reported in Table 5. Also, in the same table, the average errors in the cal-
16
where symbols have been de®ned before. For PEG8000/water system Bae et al. [38] have reported positive value for v12 by VLE measurements which means DHmix > 0. Also for PEG6000/water and PEG35000/water systems Grossmann et al. [26] have observed endothermic behavior DHmix > 0 by calorimetric measurements. These results are consistent with our results for PEG4000/water system by VPO measurements, whereas Table 5 shows, the treatment of measured activities by Flory±Huggins theory, Eq. (16), gives positive values for v12 , which means DHmix > 0 for this system. For PEG400/water system the values of v12 as reported in Table 5 are again positive while, for the same system Carisson et al. [39] have observed exothermic behavior. The similar behavior is seen for PPG400/water system, where the results in Table 5 shows v12 > 0, while for the same system Carisson et al. [39] have reported DHmix < 0. The discrepancies in the values and sign of v12 have been noticed by several workers [40] and have been attributed to the shortcomings of the Flory±Huggins theory. To improve the theory for calculating DHmix various temperature and concentration functionality for
A. Eliassi, H. Modarress / European Polymer Journal 37 (2001) 1487±1492
v12 has been proposed [40,41]. Both PEG/water and PPG/water systems are very sensitive to temperature variations and usually above a certain temperature, for each system, depending on the molecular weight of polymer they separate into two phases. Therefore a careful and extensive measurements on a wide temperature-range variation should be done to obtain the necessary data for proposing an exact temperature and concentration functionality for v12 and then improving the Flory±Huggins model for activity. For a three component mixture, where, component 1 is the solvent and components 2 and 3 are polymers, Eq. (13) can be presented as [37]: 1 1 ln a1 ln u1 1 u1 u2 u3 r2 r3 v12 u2 v13 u3 u2 u3 1 v23 u2 u3 r2
17
Determining the Flory±Hugins interaction parameters for a ternary system is not simple. Therefore, it is usual to use a mixed interaction parameter v1;23 for the ternary system of solvent(1)/polymer(2)/polymer(3). With this assumption, based on Flory±Huggins theory the activity of solvent can be expressed as: r1 ln a1 ln u1 1 u23 v1;23 u223 18 r23 Panayiotou and Vera [42] obtained the following expression for v1;23 : v1;23 v12 u2 v13 u3 1
u1
v023 u2 u3 =u223
19
Table 6 Flory±Huggins interaction parameter (v1;23 ) for aqueous solutions of PPG425/PEG400 and PPG425/PEG4000 at 35°C and 45°Ca Temperature (°C)
Concentration range of polymer
v1;23
E
n
35
PPG425 (0.0151± 0.1281)/PEG400 (0.0134±0.1125) PPG425 (0.0151± 0.1281)/PEG400 (0.0134±0.1125) PPG425 (0.0114± 0.0612)/PEG4000 (0.0098±0.0613) PPG425 (0.0114± 0.0874)/PEG4000 (0.0098±0.0857)
0.5616
0.2376
6
0.5515
0.0555
6
0.4821
0.0100
4
45 35 45
a
0.5093
0.0368
6
E is the relative error and n is the number of data points.
1491
where u23
r23
n X ui 1 i2
n X x i ri
u1
20
21
i2
and v023
r1 v r2 23
22
The measured activities are used in Eq. (18) and the calculated v1;23 are reported in Table 6. As it is seen from Table 6 the average error in calculating the solvent activity is less than 0.3%. 5. Conclusion By considering the VPO technique and the speci®cations of the apparatus used in the activity measurements a calibration constant can be introduced to minimize any possible error in the measurements. The measured activities for water by VPO method for the binary aqueous systems of PEG and PPG and ternary aqueous systems of PEG/PPG and PEG/DX can be used to calculate the interaction parameter as introduced in the original Flory±Huggins theory. However, on the basis of sign and value of the calculated interaction parameters, certain ambiguities may arise in evaluating and interpreting the thermodynamic properties such as DHmix , due to the shortcomings of the theory, Considering the small error in the calculated interaction parameters (see Tables 5 and 6) they can be used with reliability, at least as an adjustable parameter, for calculating activities of water for the system studied in the concentration range indicated in our measurements.
Acknowledgements The authors thank Dr. M. Nekoomanesh and Ms. Roghieh Jamjah of the Polymer Research Center of Iran for their valuable assistance, and Polymer Research Center of Iran for providing access to research facilities. References [1] Albertsson PA. Partition of cell particles and macromolecules. 3rd ed. New York: Wiley; 1986. [2] Soane DS. Polymer applications for biotechnology. Englewood Clis (NJ): Prentice Hall; 1992. [3] Kula MR, Kroner KH, Hustedt H. Advances in biochemical engineering, vol. 24. Berlin; 1982.
1492
A. Eliassi, H. Modarress / European Polymer Journal 37 (2001) 1487±1492
[4] Zaslavsky BY. Aqueous two-phase partitioning. Physical chemistry and bioanalytical applications. New York: Marcel Dekker Inc.; 1995. [5] Hustedt H, Kroner KH, Kula MR. Partition in aqueous two-phase systems: theory, methods, uses and applications to biotechnology. New York: Academic Press; 1985. [6] Edmond E, Ogston AG. Biochem J 1968;109:569. [7] Edmond E, Ogston AG. Biochem J 1970;117:85. [8] Connemann M, Gaub J, Lerang U, Muller S, Fenngs AP. J Chem Eng Data 1991;36:446. [9] Hill TL. J Am Chem Soc 1957;79:4885. [10] Cabezas H, Evans J, Szlag DC. Fluid Phase Equilibria 1989;53:453. [11] Diamond AD, Hsu JT. Bio Technol Bio Eng 1989;34:1000. [12] Diamond AD, Hsu JT. AIChE J 1990;36:7. [13] Hartounian H, Sandler SI, Kaler EW. Ind Eng Chem Res 1994;33:2288. [14] Haynes CA, Benitz FJ, Blanch HW, Prauznitz JM. AIChE J 1985;39:1539. [15] Zaslavsky BY, Gulaeva ND, Djafarov S, Masimov EA, Miheeva LM. J Coll Interf Sci 1990;137(1):147. [16] Hasse H, Kany HP, Tintinger R, Maurer G. Macromolecules 1995;28:3540. [17] Florin E, Kjellander R, Eriksson JC. J Chem Soc Faraday Trans 1984;80:2889. [18] Taw®k WY, Teja AS. Chem Eng Sci 1989;44:921. [19] Eliassi A, Modarress H, Mansoori GA. J Chem Eng Data 1998;43:719. [20] Mei LH, Lin DQ, Zhu ZQ, Han ZX. J Chem Eng Data 1995;40:1168. [21] Schild HG, Tirrell DA. J Phys Chem 1990;94:4352. [22] Malmsten M, Linse P, Zhang KW. Macromolecules 1993;26:2905. [23] Sjoberg A, Karlstrom G. Macromolecules 1989;22:1325.
[24] Haynes CA, Beynon RA, King RS, Blanch HW, Prausnitz JM. J Phys Chem 1989;93:5612. [25] Carisson M, Hallen D, Linse P. J Chem Soc Faraday Trans 1995;91:2081. [26] Grossman C, Tintinger R, Zhu J, Maurer G. Fluid Phase Equilibria 1995;106:111. [27] Eliassi A, Modarress H, Mansoori GA. J Chem Eng Data 1999;44:52. [28] Modarress H, Eliassi A. Iranian Polym J Sci Tech 1994:234 [Persian Edition]. [29] Krigbaum WR, Gymer DO. J Am Chem Soc 1959;81:1859. [30] Lin DQ, Mei LH, Zhu ZQ, Han ZX. Fluid Phase Equilibria 1996;118:241. [31] Morawetz H. Macromolecules in solutions. 2nd ed. Interscience Publishers Inc; 1975. [32] Kamide K. Comprehensive polymer science. vol 1. Oxford; 1989 [Chapter 4]. [33] Gaube J, Pfennig A, Stumpf M. J Chem Engng Data 1993;38:163. [34] Brown W. J Appl Polym Sci 1967;11:2381. [35] Gowariker VR, Viswanathan NV, Sreedhar J. Polymer science. New Age International Limited Publisher; 1996. [36] Fernandez MJ, Eguiazabal JI, Elorza JM. J Polym Sci 1983;21:859. [37] Flory PJ. Principles of polymer chemistry. Ithaca, New York: Cornell University Press; 1953. [38] Bae YC, Shim JJ, Soane DS, Prausnitz JM. J Appl Polym Sci 1993;47:1193. [39] Carisson M, Hallen D, Linse P. J Chem Soc Faraday Trans 1995;91:2081. [40] Taimoori M, Modarress H, Mansoori GA. J Appl Polym Sci 2000, in press. [41] Qian C, Mumby SJ, Eichinger BE. J Polym Sci Part B 1991;29:635. [42] Panayiotou C, Vera JH. Polym J 1984;16:89.
www.elsevier.nl/locate/europolj
Water activities in binary and ternary aqueous systems of poly(ethylene glycol), poly(propylene glycol) and dextran A. Eliassi a, H. Modarress b,* b
a Iranian Research Organization for Science and Technology (IROST), Tehran, Iran Department of Applied Chemistry, Faculty of Chemical Engineering, Amir-Kabir University of Technology, 424 Hafez Ave Tehran, Iran
Received 1 February 2000; received in revised form 19 September 2000; accepted 17 November 2000
Abstract The vapor pressure osmometry (VPO) measurements were used to obtain activities of water for aqueous solutions of poly(ethylene glycol)/dextran and poly(ethylene glycol)/poly(propylene glycol) at temperatures 35°C, 45°C, 55°C and 65°C. The weight fractions of water were varied from 70% to 98%. To increase the accuracy, the calibration constants of the VPO apparatus for each measurement were also determined. The experimental results were used in Flory±Huggins theory to calculate the interaction parameters for the mixtures and the results were discussed in terms of Flory±Huggins theory for evaluating other thermodynamic properties. Ó 2001 Elsevier Science Ltd. All rights reserved. Keywords: Activity of water; Vapour pressure osometry; Polyethylene glycol; Dextran; Flory±Huggins theory
1. Introduction Aqueous polymeric two-phase systems are increasingly used in biochemistry and cell biology for separation of macromolecules from cell organells [1±4]. The properties of polymeric two-phase systems also make them very interesting for large scale industrial applications [5]. Poly(ethylene glycol) (PEG) is used in almost all applications of aqueous phase partitioning. For biochemical separations on the laboratory scale the most commonly used aqueous phase system is composed of PEG and dextran (DX) [1]. For large scale separations in biotechnical industry PEG/salt systems have been the most widely used, primarily for large scale enzyme extractions. Some binary mixtures of other polymers with PEG were also studied. These polymers were starch, poly(vinyl alcohol) (PVA) and poly(propylene glycol) (PPG). Modeling of the thermodynamic properties of polymer aqueous systems have been the interest of several
*
Corresponding author. Tel.: +98-911-222-0454; fax: +98021-640-5847. E-mail address: [email protected] (H. Modarress).
workers from dierent aspects. For example Edmond and Ogston [6,7] have used osmotic virial expansion model for studying the phase behavior of these systems. Connemann et al. [8] extended the osmotic virial expansion model to the polydisperse polymers. A combination of this model with HillÕs theory [9] was also used by Cabezas et al. [10] for predicting polymer/polymer/ water phase diagrams. Diamond et al. [11,12] applied the modi®ed Flory±Huggins theory to model the partitioning of proteins in PEG/DX/water two-phase systems. Hartounian and Sandler [13] utilized UNIQUAC, Debye±Huckel and Bronsted±Guggenheim equations for studying thermodynamic behavior of polymer/polymer/electrolyte/water systems. Also, Haynes et al. [14] have used integral equation theory to model aqueous polymeric systems. By experimental measurements various thermodynamic properties of aqueous PEG systems and with other polymers such as PPG and DX have been determined. For example Albertsson [1] and Zaslavsky et al. [15] have determined the phase diagrams of PEG/water systems. Hasse et al. [16] have obtained the second and third osmotic virial coecients and Florin et al. [17] have studied the salt eects on the cloud point of PEG/ water systems. Densities of PEG/water systems have
0014-3057/01/$ - see front matter Ó 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 0 1 4 - 3 0 5 7 ( 0 0 ) 0 0 2 6 2 - 7
1488
A. Eliassi, H. Modarress / European Polymer Journal 37 (2001) 1487±1492
been measured by Taw®k et al. [18] and Eliassi et al. [19]. Viscosities of PEG/salt/water systems have been determined by Mei et al. [20]. Schild and Tirrell [21] obtained the lower critical solution temperatures of PPG/water system. Phase behavior of PEG/PPG/water and PEG/ DX/water systems have been investigated by Malmsten et al. [22] and Sjoberg and Karlstrom [23], respectively. The vapor pressure data for binary PEG or DX aqueous systems have been obtained by Hatnes et al. [24]. Mixing enthalpy of PPG/water system have been measured by Carisson et al. [25]. Grossman et al. [26] and Eliassi et al. [27] have obtained the activity for PEG and DX aqueous solutions. In this work the activity of water measured by vapor pressure osmometry in PEG/PPG and PEG/DX aqueous solutions are reported. Based on the measured activities, the Flory±Huggins interaction parameters for binary and ternary aqueous polymer solutions are calculated.
gibly small, dT =dt 0 and a steady state is achieved. According to Kamide [32] the temperature dierence in steady state, T T0 s can be expressed as a power series of the concentration: DT T
T 0 s
2 VPO 3 Ks c=M AVPO 2;v c A3;v c
1
VPO where AVPO 2;v and A3;v are second and third virial coecients and Ks is a calibration parameter (cm3 K mol 1 ). Also, in Eq. (1) c (g cm 3 ), M (g mol 1 ) are the concentration, molecular weight of solute respectively. It is usual to express the activity of solvent in the form of virial expansion in terms of power series of concentration: 2 VPO 3 ln a1 V 0 c=M AVPO 2;v c A3;v c
2
where V 0 (cm3 mol 1 ) is the molar volume of solvent and the other symbols have been de®ned in Eq. (1). Therefore Eq. (1) can be written as: ln a1 V0
2. Experimental
DT Ks
There are some experimental methods for solvent activity measurements for polymer solution [28] such as dierential vapor pressure measurements [29], isopiestic methods [30] and membrane osmometry [31]. In this work, a VPO manufactured by Knauer (Germany) was used for measuring the activities of water in aqueous polymer systems. VPO method is based on the principle that at a given temperature the vapor pressure of a solution is less than that of the pure solvent. In a VPO, the solution and the solvent droplet are placed on two thermistors arranged in a wheatstone bridge circuit in such a way that the temperature rise can be measured very accurately as a function of the bridge imbalance output voltage, DV . Thermistors are placed in a chamber ®lled with a saturated vapor of solvent. Since the vapor pressures of solvent and polymer solution dier from each other, the solvent vapor condenses on the polymer solution droplet. Therefore, solution droplet starts getting diluted as well as heated up by the latent heat of condensation due to the solvent condensing on it. The instrument indicates the temperature dierence between the two thermistors (DT ) as the voltage dierence (DV / DT ) which is measured by a digital voltmeter. Kamide [32] has pointed out that actually there is a steady state and not an equilibrium state in VPO measurement. He has derived an equation for the rate of temperature change dT =dt of the liquid droplet on the thermistor in the VPO measurements. With the assumption that the temperature T of the solution droplet is very close to that of the solvent T0 , and changes in concentration, volume and density of the solution drop due to the condensation of solvent vapor are all negli-
On the other hand according to Kamide Ks is related to an equilibrium calibration constant Ke in the following form: Ks Ke g
3
4
where for simpli®cation g is used here to represent all the parameters introduced in the original Kamide equation and Ke has been given as [32]: Ke RT02 V 0 =DH
5
where DH is the molar heat of condensation which is DH vap and T0 is the temperature setting of apparatus in the VPO measurements. From Eqs. (3)±(5) the following equation can be obtained: ln a1
DH vap =RT02 gDT
6
Similar equation has been derived by Gaube et al. [33] and it is claimed that the eciency coecient g can correct for the in¯uences of heat losses in the apparatus. In the VPO apparatus the temperature dierence DT is related to the measured voltage DV . Brown [34], on the basis of equilibrium considerations, assumed that the measured voltage is proportional to the chemical potential dierence, that is; DV / Dl1 . The chemical potential dierence Dl1 for the solvent is expressed in terms of activity of the solvent as; Dl1 RT0 ln a1 . Therefore by introducing the relativity constant j the following equation can be obtained: DV jRT0 lna1
7
Brown [34] has introduced a molal calibration constant K as:
A. Eliassi, H. Modarress / European Polymer Journal 37 (2001) 1487±1492
K
jRT0 M1 1000
8
where M1 is the molecular weight of solvent. Thus the ®nal equation for activity of solvent is: ln a1
DV M1 1000K
9
As in VPO measurement DT / DV and also by considering the Eqs. (1), (6) and (9), it is obvious that these equations can be presented in the following general form: ln a1
b DV
10
where, b is a calibration parameter used to represent the coecients of DV in Eqs. (1), (6) and (9). On substituting for ln a1 from Eq. (2) in Eq. (10) and ignoring the higher order terms and then rearranging the ®nal equation, we have: DV K 0 Ac c
11
0 where K 0 V 0 =bM and A AVPO 2;v V =b. It is worth noting that in BrownÕs equation V 0 1000 cm3 . In this work from VPO measurements for a set of calibrating solutions, K 0 has been evaluated from Eq. (11) by plotting variations of DV =c versus c and extrapolating to in®nite dilution (c ! 0). In this way, K 0 is evaluated at each measurement and therefore, it is reasonable to believe that it includes any heat loss as well as any other de®ciencies mentioned by other workers in VPO measurements [32,33]. The calibrating compound used in our VPO measurements was aqueous solution of urea M 60. It is usual to de®ne a calibration constant for the VPO instrument as the product of K 0 by molecular weight of the calibrating sample Mcalib [35]:
K K 0 Mcalib
12
Now, by using the value of K in Eq. (9) the activity of the solvent will be determined. In a similar way, Fernandez et al. [36] have used VPO measurements to determine the activity of benzene in polyisobutylene/ benzene mixtures. They have concluded that VPO can be a suitable method for solvent activity measurements of solutions with concentrations of up to 40% in polymer weight. In this work, using the above described method, the activity of water in dierent solutions of PEG400/ DX9300, PPG425/PEG400 and PPG425/PEG4000 have been measured. PEG400 and PEG4000 were obtained from Merck (Germany), PPG425 were obtained from Fluka (Swiss) and DX9300 from Sigma (UK). Urea is the calibration sample recommended by the VPO manufacturer (Knauer, Germany) to be used for
1489
aqueous solution measurements. The urea used in this measurement was obtained from Merck (Germany). All materials were used without further puri®cation. The aqueous polymer solutions were made by mass using double-distilled water. The weighing was done by an analytical balance with 0.1 mg accuracy (Shimadzu, Japan, model AEU 210). In VPO technique, as mentioned above, when a droplet of solvent and a droplet of solution are placed on each thermistors, DV increases and reaches a maximum and remains constant. The required time (3±5 mins is adequate) for reaching this condition, however, depends on the type of solution and its temperature. To examine the accuracy and reproducibility of the measurements, each measurement was repeated ®ve to eight times, and then the average voltage dierence was used in the calculations by Eq. (9). The accuracy of the DV measurements is estimated to be better than 1%.
3. Results The values of calibration constant of the instrument K for three dierent set of experiments are reported in Table 1. As it is seen from Table 1 K is an increasing function of temperature. The values of water activity in PEG400/DX9300/ water, PEG400/PPG425/water and PEG4000/PPG425/ water systems are respectively reported in Tables 2±4. It is worth noting that the activities have been evaluated from Eq. (9). Since, polymers are considered as nonvolatile species and DV in Eq. (9) represents the eect of nonvolatile species on the volatility of the solvent in the mixture which is measured by DV in the VPO method. These results indicate that with increasing temperature, Table 1 Calibration constants (K in Eq. (9)) of VPO determined for a typical urea aqueous solutions Temperature (°C)
35
45
55
65
Urea (0.12 wt.%)
105.66
121.24
129.59
131.70
Table 2 Experimental results of activity of water for weight fractions of water (w1 ) and PEG (w2 ) in solutions of PEG400/DX9300/H2 O at various temperatures w1
w2
Temperature (°C) 35
45
55
65
0.9592 0.8886 0.8287 0.7488 0.7021
0.0215 0.0928 0.1542 0.2262 0.2719
0.9989 0.9945 0.9895 0.9815 0.9745
0.9989 0.9947 0.9900 0.9824 0.9763
0.9989 0.9948 0.9899 0.9831 0.9772
0.9989 0.9949 0.9901 0.9837 0.9786
1490
A. Eliassi, H. Modarress / European Polymer Journal 37 (2001) 1487±1492
Table 3 Experimental results of activity of water for weight fractions of water (w1 ) and PEG (w2 ) in solutions of PEG400/PPG425/H2 O at various temperatures w1
w2
Temperature (°C) 35
45
55
0.9715 0.9317 0.8875 0.8548 0.8239 0.7594
0.0151 0.0327 0.0580 0.0724 0.0869 0.1281
0.9982 0.9961 0.9935 0.9911 0.9887 0.9834
0.9865 0.9910 0.9928 0.9947 0.9968 0.9987
0.9885 0.9916 0.9932 0.9948 0.9968 0.9987
Table 5 Flory±Huggins interaction parameter (v12 ) for aqueous solutions of PEG400, PEG4000 and PPG425 at 35°C and 45°Ca Temperature (°C)
Concentration range of polymer
v12
Eb
n
35
PEG400 (0.0120±0.3150) PEG400 (0.0120±0.3150) PEG4000 (0.049±0.3949) PEG4000 (0.049±0.3949) PPG425 (0.0103±0.0951) PPG425 (0.0103±0.0951)
0.4148
0.0242
8
0.4207
0.0701
8
0.4857
0.0862
4
0.5147
0.0859
4
0.5766
0.0160
5
0.5768
0.0120
5
45 35 45 35
Table 4 Experimental results of activity of water for weight fractions of water (w1 ) and PEG (w2 ) in solutions of PEG4000/PPG425/ H2 O at various temperatures w1
w2
0.9788 0.9428 0.9109 0.8775
0.0114 0.0286 0.0446 0.0612
Temperature (°C) 35
45
55
0.9995 0.9986 0.9977 0.9965
0.9996 0.9988 0.9979 0.9969
0.9995 0.9987 0.9980 0.9970
the activity of water increases, but with increasing molecular weight of the polymers the activity of water decreases. 4. Modeling
where, a1 is the activity of solvent, u2 is the volume fraction of polymer, v12 is the interaction parameter and r2 is number of polymer segments. In general, the number of segments ri is de®ned as follows: molar volume of polymer i molar volume of solvent
On rearranging Eq. (13) we have: 1 1 u2 u2 v12 u2 ln a1 ln u1 r2
a
E is the relative error and n is the number of data points. P EXP E 1n i aEXP acal 100, aEXP is experimental i i =ai i activity of solvent, acal is calculated activity using reported v12 . i b
culations (E) are reported. The reported average errors between calculated and experimental activity data (in Table 5) indicate that the concentration dependency of v12 for the studied systems is negligible. Also, the results in Table 5 show that v12 for PEG/ water system is positive which means a positive value for enthalpy change upon mixing DHmix . This is consistent with the de®nition of DHmix in the original Flory±Huggins theory as represented by the following expression: DHmix RT v12 u1 u2
One of the well known models for activity of solvent in binary systems of solvent(1)/polymer(2) is the Flory± Huggins model which can be represented by the following equation [37]: 1 ln a1 ln 1 u2 1 u2 v12 u22 13 r2
ri
45
14
15
Therefore, the variation of the left-hand side of Eq. (15) versus u2 is linear and the slope is equal to v12 . The measured activity data in Tables 2±4, were treated by Eq. (9), and the values of v12 are reported in Table 5. Also, in the same table, the average errors in the cal-
16
where symbols have been de®ned before. For PEG8000/water system Bae et al. [38] have reported positive value for v12 by VLE measurements which means DHmix > 0. Also for PEG6000/water and PEG35000/water systems Grossmann et al. [26] have observed endothermic behavior DHmix > 0 by calorimetric measurements. These results are consistent with our results for PEG4000/water system by VPO measurements, whereas Table 5 shows, the treatment of measured activities by Flory±Huggins theory, Eq. (16), gives positive values for v12 , which means DHmix > 0 for this system. For PEG400/water system the values of v12 as reported in Table 5 are again positive while, for the same system Carisson et al. [39] have observed exothermic behavior. The similar behavior is seen for PPG400/water system, where the results in Table 5 shows v12 > 0, while for the same system Carisson et al. [39] have reported DHmix < 0. The discrepancies in the values and sign of v12 have been noticed by several workers [40] and have been attributed to the shortcomings of the Flory±Huggins theory. To improve the theory for calculating DHmix various temperature and concentration functionality for
A. Eliassi, H. Modarress / European Polymer Journal 37 (2001) 1487±1492
v12 has been proposed [40,41]. Both PEG/water and PPG/water systems are very sensitive to temperature variations and usually above a certain temperature, for each system, depending on the molecular weight of polymer they separate into two phases. Therefore a careful and extensive measurements on a wide temperature-range variation should be done to obtain the necessary data for proposing an exact temperature and concentration functionality for v12 and then improving the Flory±Huggins model for activity. For a three component mixture, where, component 1 is the solvent and components 2 and 3 are polymers, Eq. (13) can be presented as [37]: 1 1 ln a1 ln u1 1 u1 u2 u3 r2 r3 v12 u2 v13 u3 u2 u3 1 v23 u2 u3 r2
17
Determining the Flory±Hugins interaction parameters for a ternary system is not simple. Therefore, it is usual to use a mixed interaction parameter v1;23 for the ternary system of solvent(1)/polymer(2)/polymer(3). With this assumption, based on Flory±Huggins theory the activity of solvent can be expressed as: r1 ln a1 ln u1 1 u23 v1;23 u223 18 r23 Panayiotou and Vera [42] obtained the following expression for v1;23 : v1;23 v12 u2 v13 u3 1
u1
v023 u2 u3 =u223
19
Table 6 Flory±Huggins interaction parameter (v1;23 ) for aqueous solutions of PPG425/PEG400 and PPG425/PEG4000 at 35°C and 45°Ca Temperature (°C)
Concentration range of polymer
v1;23
E
n
35
PPG425 (0.0151± 0.1281)/PEG400 (0.0134±0.1125) PPG425 (0.0151± 0.1281)/PEG400 (0.0134±0.1125) PPG425 (0.0114± 0.0612)/PEG4000 (0.0098±0.0613) PPG425 (0.0114± 0.0874)/PEG4000 (0.0098±0.0857)
0.5616
0.2376
6
0.5515
0.0555
6
0.4821
0.0100
4
45 35 45
a
0.5093
0.0368
6
E is the relative error and n is the number of data points.
1491
where u23
r23
n X ui 1 i2
n X x i ri
u1
20
21
i2
and v023
r1 v r2 23
22
The measured activities are used in Eq. (18) and the calculated v1;23 are reported in Table 6. As it is seen from Table 6 the average error in calculating the solvent activity is less than 0.3%. 5. Conclusion By considering the VPO technique and the speci®cations of the apparatus used in the activity measurements a calibration constant can be introduced to minimize any possible error in the measurements. The measured activities for water by VPO method for the binary aqueous systems of PEG and PPG and ternary aqueous systems of PEG/PPG and PEG/DX can be used to calculate the interaction parameter as introduced in the original Flory±Huggins theory. However, on the basis of sign and value of the calculated interaction parameters, certain ambiguities may arise in evaluating and interpreting the thermodynamic properties such as DHmix , due to the shortcomings of the theory, Considering the small error in the calculated interaction parameters (see Tables 5 and 6) they can be used with reliability, at least as an adjustable parameter, for calculating activities of water for the system studied in the concentration range indicated in our measurements.
Acknowledgements The authors thank Dr. M. Nekoomanesh and Ms. Roghieh Jamjah of the Polymer Research Center of Iran for their valuable assistance, and Polymer Research Center of Iran for providing access to research facilities. References [1] Albertsson PA. Partition of cell particles and macromolecules. 3rd ed. New York: Wiley; 1986. [2] Soane DS. Polymer applications for biotechnology. Englewood Clis (NJ): Prentice Hall; 1992. [3] Kula MR, Kroner KH, Hustedt H. Advances in biochemical engineering, vol. 24. Berlin; 1982.
1492
A. Eliassi, H. Modarress / European Polymer Journal 37 (2001) 1487±1492
[4] Zaslavsky BY. Aqueous two-phase partitioning. Physical chemistry and bioanalytical applications. New York: Marcel Dekker Inc.; 1995. [5] Hustedt H, Kroner KH, Kula MR. Partition in aqueous two-phase systems: theory, methods, uses and applications to biotechnology. New York: Academic Press; 1985. [6] Edmond E, Ogston AG. Biochem J 1968;109:569. [7] Edmond E, Ogston AG. Biochem J 1970;117:85. [8] Connemann M, Gaub J, Lerang U, Muller S, Fenngs AP. J Chem Eng Data 1991;36:446. [9] Hill TL. J Am Chem Soc 1957;79:4885. [10] Cabezas H, Evans J, Szlag DC. Fluid Phase Equilibria 1989;53:453. [11] Diamond AD, Hsu JT. Bio Technol Bio Eng 1989;34:1000. [12] Diamond AD, Hsu JT. AIChE J 1990;36:7. [13] Hartounian H, Sandler SI, Kaler EW. Ind Eng Chem Res 1994;33:2288. [14] Haynes CA, Benitz FJ, Blanch HW, Prauznitz JM. AIChE J 1985;39:1539. [15] Zaslavsky BY, Gulaeva ND, Djafarov S, Masimov EA, Miheeva LM. J Coll Interf Sci 1990;137(1):147. [16] Hasse H, Kany HP, Tintinger R, Maurer G. Macromolecules 1995;28:3540. [17] Florin E, Kjellander R, Eriksson JC. J Chem Soc Faraday Trans 1984;80:2889. [18] Taw®k WY, Teja AS. Chem Eng Sci 1989;44:921. [19] Eliassi A, Modarress H, Mansoori GA. J Chem Eng Data 1998;43:719. [20] Mei LH, Lin DQ, Zhu ZQ, Han ZX. J Chem Eng Data 1995;40:1168. [21] Schild HG, Tirrell DA. J Phys Chem 1990;94:4352. [22] Malmsten M, Linse P, Zhang KW. Macromolecules 1993;26:2905. [23] Sjoberg A, Karlstrom G. Macromolecules 1989;22:1325.
[24] Haynes CA, Beynon RA, King RS, Blanch HW, Prausnitz JM. J Phys Chem 1989;93:5612. [25] Carisson M, Hallen D, Linse P. J Chem Soc Faraday Trans 1995;91:2081. [26] Grossman C, Tintinger R, Zhu J, Maurer G. Fluid Phase Equilibria 1995;106:111. [27] Eliassi A, Modarress H, Mansoori GA. J Chem Eng Data 1999;44:52. [28] Modarress H, Eliassi A. Iranian Polym J Sci Tech 1994:234 [Persian Edition]. [29] Krigbaum WR, Gymer DO. J Am Chem Soc 1959;81:1859. [30] Lin DQ, Mei LH, Zhu ZQ, Han ZX. Fluid Phase Equilibria 1996;118:241. [31] Morawetz H. Macromolecules in solutions. 2nd ed. Interscience Publishers Inc; 1975. [32] Kamide K. Comprehensive polymer science. vol 1. Oxford; 1989 [Chapter 4]. [33] Gaube J, Pfennig A, Stumpf M. J Chem Engng Data 1993;38:163. [34] Brown W. J Appl Polym Sci 1967;11:2381. [35] Gowariker VR, Viswanathan NV, Sreedhar J. Polymer science. New Age International Limited Publisher; 1996. [36] Fernandez MJ, Eguiazabal JI, Elorza JM. J Polym Sci 1983;21:859. [37] Flory PJ. Principles of polymer chemistry. Ithaca, New York: Cornell University Press; 1953. [38] Bae YC, Shim JJ, Soane DS, Prausnitz JM. J Appl Polym Sci 1993;47:1193. [39] Carisson M, Hallen D, Linse P. J Chem Soc Faraday Trans 1995;91:2081. [40] Taimoori M, Modarress H, Mansoori GA. J Appl Polym Sci 2000, in press. [41] Qian C, Mumby SJ, Eichinger BE. J Polym Sci Part B 1991;29:635. [42] Panayiotou C, Vera JH. Polym J 1984;16:89.