Classical and recent free-volume models for polymer solutions: A comparative evaluation

Fluid Phase Equilibria 257 (2007) 63–69

Classical and recent free-volume models for polymer solutions: A comparative evaluation Hamid Reza. Radfarnia a,b , Georgios M. Kontogeorgis c,∗ , Cyrus. Ghotbi a , Vahid. Taghikhani a a

b

Department of Chemical and Petroleum Engineering, Sharif University of Technology, Tehran, Iran Process Department, Petrofac International Ltd., Petrofac House, Room 104, Al-Soor Street, P.O. Box 23467, Sharjah, United Arab Emirates c Center for Phase Equilibria and Separation Processes (IVC-SEP), Department of Chemical Engineering, Technical University of Denmark, DK-2800 Lyngby, Denmark Received 30 November 2006; received in revised form 11 May 2007; accepted 14 May 2007 Available online 18 May 2007

Abstract In this work, two “classical” (UNIFAC-FV, Entropic-FV) and two “recent” free-volume (FV) models (Kannan-FV, Freed-FV) are comparatively evaluated for polymer–solvent vapor–liquid equilibria including both aqueous and non-aqueous solutions. Moreover, some further developments are presented here to improve the performance of a recent model, the so-called Freed-FV. First, we propose a modification of the Freed-FV model accounting for the anomalous free-volume behavior of aqueous systems (unlike the other solvents, water has a lower free-volume percentage than polymers). The results predicted by the modified Freed-FV model for athermal and non-athermal polymer systems are compared to other “recent” and “classical” FV models, indicating an improvement for the modified Freed-FV model for aqueous polymer solutions. Second, for the original Freed-FV model, new UNIFAC group energy parameters are regressed for aqueous and alcohol solutions, based on the physical values of the van der Waals volume and surface areas for both FV-combinatorial and residual contributions. The prediction results of both “recent” and “classical” FV models using the new regressed energy parameters are significantly better, compared to using the classical UNIFAC parameters, for VLE of aqueous and alcohol polymer systems. © 2007 Elsevier B.V. All rights reserved. Keywords: Free-volume; Polymer solutions; Aqueous solutions

1. Introduction Phase equilibria for polymer solutions are required for the efficient design of processes in the polymer-related industries, e.g. removal of solvents and additives after polymerization and formulation of paints and coatings including drying and control of emissions. Many successful models have been developed [1] but UNIFAC [2]-based activity coefficient models are still particularly suitable for low-pressure applications and in cases

Abbreviations: FV, free-volume; LLE, liquid–liquid equilibria; PEO, polyethylene oxide; PPO, propylene oxide; PIB, polyisobutylene; PBMA, poly(n-butyl methacrylate); PVAc, poly(vinyl acetate); PS, polystyrene; PEMA, poly(ethyl methacrylate); PMA, poly(methyl acrylate); PEA, poly(ethyl acrylate); PAA, poly(acrylic acid); PMAA, poly(methacrylic acid); VLE, vapor–liquid equilibria; UNIFAC, UNIQUAC Functional Activity Coefficients ∗ Corresponding author. E-mail address: [email protected] (G.M. Kontogeorgis). 0378-3812/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.fluid.2007.05.015

where many complex solvents and polymers are present. The case of the paints and coatings industry can be mentioned as a typical example. UNIFAC-based models which are successful for polymers account for the free-volume differences between polymers and solvents and numerous such models have been presented in the literature [3–14]. In these models, typically the UNIFAC energetic parameters of the residual term are not reestimated using the new combinatorial/FV terms; thus original UNIFAC parameter tables are used. The first model in the family of UNIFAC-based free-volume models is the UNIFAC-FV proposed by Oishi and Prausnitz [4], which successfully predicts solvent activities in many non-aqueous polymer solutions, but the results are less satisfactory for aqueous solutions, longchain solute activities and LLE. The shortcomings for aqueous solutions are due to the fact that free-volume expression in UNIFAC-FV is always an additive term with a positive effect, while water has a lower free-volume percentage than most polymers [9]. Kannan et al. [5] have recently developed, based on

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H.Reza. Radfarnia et al. / Fluid Phase Equilibria 257 (2007) 63–69

the generalized van der Waals partition function, a new FV term coupled to UNIFAC in an attempt to solve this problem. Much improved results are obtained compared to the original UNIFACFV. Another successful model presented in our previous work [8], the Freed-FV, was shown to be flexible and provide satisfactory results for a large range of athermal and non-athermal asymmetric systems. Less satisfactory results were obtained for aqueous solutions and thus a modification in the line of KannanFV is proposed here. An additional modification of FV models is carried out in this work, and though originally developed for the Freed-FV model, it is used here for several FV models. It has been indicated in the literature that the use of the regressed group parameters of the surface volume and areas (R and Q) for OH, COO and water in optimizing the group interaction parameters of UNIFAC table might be one of the reasons why UNIFAC prediction is often poor for polymeric systems containing alcohols and water. The fitted parameters are necessarily used in both classical and recent FV models, since all of them are based on existing UNIFAC tables. However, the FV expressions have a clear theoretical origin and it is tempting to re-estimate UNIFAC parameters, at least for systems with water and alcohols, using the “physical” R and Q values, as reported by Bondi [15]. This task is undertaken here for Freed-FV and applied to several other FV models as well.

where is φifv the free-volume fraction of component i: xi vfv φifv =  i fv j xj vj

(4)

xi is the mole fraction of component, i and vfv i is the free-volume of component, i is defined as the molar volume of component i, vi minus the the van der Waals volume of component i (hard-core . The van der Waals and molar volumes of comvolume), vvdW i ponent i can be found for solvents and polymers in the literature [16–18]. 2.2. Kannan-FV model The free-volume combinatorial term proposed by Kannan et al. [5] is: ln γicomb+fv = ln γicomb + ln γifv

(5)

where ln γicomb is the Guggenheim-Staverman combinatorial expression (used in original UNIFAC), while the free-volume term is given by:     φihc − φifv φifv fv ln γi = ln + (6) xi φihc

2. Literature free-volume models xv where φihc = i i

hc

The free-volume models have been extensively presented in the literature and thus will be only briefly discussed here. All of them are of the form: ln γi = ln γicomb−fv + ln γires

(1)

One or two terms are used for the combinatorial and freevolume contributions. The original form of the UNIFAC residual [2] term is used with all FV models using temperature independent interaction parameters, which between m–n groups are given as:  a  mn ψmn = exp − (2) T where amn is the interaction parameter between groups m and n. This function enters the residual term of the activity coefficient models and for more information the reader is referred to the original UNIFAC publications, e.g. [2]. The differences between the various models lie in their combinatorial/FV term.

xj vhc j

j

vhc i is the hard-core volume of component i taken equal to the van der Waals volumes. The free-volume fractions are defined as in Eq. (4). 2.3. Freed-FV model The combinatorial-FV term is given by the equations [19,20,8]:     φifv φifv comb−fv ln γi +1− + fifreed−fv = ln (7) xi xi where the Freed-FV correction term, fifreed−fv , is given as: fifreed−fv

⎡ ⎤   fv fv fv fv fv ⎦ = rifv ⎣ βji φj (1 − φjfv ) − 0.5 βjk φj φkj j

j=i k=i

(8)

2.1. Entropic-FV model For the Entropic-FV model, the expression proposed by Elbro et al. [14] is used:   φfv φifv comb−fv +1− i ln γi = ln (3) xi xi

.

fv = α with βji ji

1 rjfv

−

1 rifv

The parameter αji is set equal to 0.2 while rjfv is the nonrandomness parameter between i–j pair, equal to the ratio of vfv j fv refers to the free-volume of the smallest solvent to vfv where v i i in mixture.

H.Reza. Radfarnia et al. / Fluid Phase Equilibria 257 (2007) 63–69

Table 1 Physical and regressed Rk and Qk parameters for the OH, COO and water groups

3. Two developments for improving results for aqueous and alcohol solutions We propose in this work two different approaches for improving the performance of FV models (especially of the Freed-FV but also others) for aqueous and other strongly polar polymer solutions. The first, described in Section 3.1, is a modification of the Freed-FV expression, which we will hereafter call “modified Freed-FV”. In the modified Freed-FV, interaction parameters and R and Q values are the same as in the original model, but the combinatorial/free-volume term is changed. In the second approach, described in Section 3.2, we have re-estimated the interaction parameters involving water and hydroxyl groups, using the “physical” R and Q values. This parameter re-estimation has been done with the original FreedFV model described in Section 2 (Eqs. (1), (7) and (8)) but the new parameters have been subsequently used with the other FV models as well. The purpose of studying both approaches is to investigate whether the results for aqueous and alcohol polymer solutions can be best improved by changes in the combinatorial/FV term or re-estimation of the energetic interaction parameters using physical R and Q values.

OH Water COO

Following the Kannan et al. [5] approach, we present here a modified version of the Freed-FV expression [19,20,8] towards improving the results for aqueous polymer solutions. This is done upon subtracting from Freed-FV the original Freed FH combinatorial expression based on the hard-core volume [21]:

ln

= ln

φih xi



 +1−

φih xi

Rk

Qk

Rk

Qk

1.000 0.92 1.380

1.200 1.40 1.200

0.52999 0.8154 1.002

0.5840 0.9040 1.200

j

j=i k=i

(11) Finally, the new modified Freed-FV expression is, for binary systems, given as:  ln

γifv

= ln

φifv φih 



 +

φih − φifv xi





φfv + 0.2 1 − i xi

2

2 (12)

The new modified FV model is based on Eq. (12) while the combinatorial and residual terms are those of original UNIFAC [2]. 3.2. Estimation of new interaction parameters for FV activity coefficient models

(9)

In the Freed FH model, hard-core volume should be used instead of free-volumes: 

Physical parameters

and fifreed,FH is the Freed’s modified expression for the original Flory-Huggins model [21]: ⎡ ⎤    h⎦ fifreed,FH = rih ⎣ βji φjh (1 − φjh ) − 0.5 βjk φjh φkj

3.1. The modified Freed-FV model

γicomb

Regressed parameters

φh − 0.2 1 − i xi

ln γifv = ln γicomb−fv − ln γicomb

65

 + fifreed,FH

(10)

New UNIFAC group interaction parameters have been regressed using the original Freed-FV model (Eqs. (1) and (7)) for aqueous and alcohol solutions using the “physical” R and Q parameters for water, OH and COO groups, shown in Table 1. The group parameters are provided in Table 2. The parameters are optimized based on experimental VLE data of polymer–water and alcohols as well as polystyrene–alcohol mixtures.

Table 2 New regressed main group interaction parameters (amn ) using the Freed-FV model [Eqs. (1) and (7)], suitable for aqueous solutions

CH2 ACH ACCH2 OH H2 O CCOO CH2 O COOH COO

CH2

ACH

ACCH2

OH

H2 O

CCOO

CH2 O

COOH

COO

– – – 357.79 792.22 – – – 160.80

– – – −43.15 – – – – –

– – – 92.58 – – – – –

1737.19 828.97 828.97 – 40.92 186.79 240.01 – 171.29

13281.28 – – −382.01 – 758.52 509.29 3991.59 –

– – – 34.71 225.93 – – – –

– – – 0.9634 −101.33 – – – –

– – – – −505.29 – – – –

2422.31 – – 199.99 – – – – –

The same parameters are then used also for the other FV models.

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Table 3 Average absolute percentage deviations (%AAD) between predicted and experimental activity coefficients for various polymer systems System

T (K)

Athermal polymer systems, finite dilution Athermal polymer systems, infinite dilution Non-aqueous polymer systems, finite dilution Non-aqueous polymer systems, infinite dilution Aqueous polymer systems, finite dilution Aqueous polymer systems, infinite dilution

293.2–458.2 298.2–373.2 293.2–373.2 298.2–502.6 298.2–423.2 298.2–423.2

Ns

31 43 80 129 13 5

Np

463 121 1200 348 142 41

%AAD

References

Freed-FV

UNIFAC-FV

Kannan-FV

Modified Freed-FV

5.7 15.2 23.8 39.9 29.1 74.0

7.5 21.1 25.9 34.2 66.2 81.7

8.8 9.3 27.1 43.8 16.3 29.8

10.2 9.3 26.5 43.6 16.9 29.5

[3,22,23] [3,22] [3,22,24–27] [3,22] [3,22,28,29] [3,22]

Ns indicates the number of systems and Np is the number of experimental points.

The new interaction parameters were obtained by minimizing the following objective function:  2  aexp − acorr i i OF = (13) exp ai i exp

where ai and aicorr are the experimental and correlated activity of solvent, respectively. Use of FV models requires accurate values of the molar volume for both polymers and solvents. Volumes of solvents have been taken from the DIPPR correlation [16] and for polymers from the Tait equation [17] except for PAA, PMAA, PPO and PVAL where the predictive GC-VOL method is used [18]. 4. Results and discussion 4.1. Comparison of FV models for size-asymmetric systems We have considered a very large database but for brevity the results are summarized in Table 3. The following points highlight the major observations:

Fig. 1. Predicted and experimental (䊉) water activities for the PVAc (50,000)–water system at 313.15 K using the UNIFAC-FV model (—), KannanFV model (- - -), original Freed-FV model (· · ·) and the modified Freed-FV model proposed in this work (– - –).

1. For athermal polymer solutions, the modified Freed-FV performs as satisfactorily as Freed-FV and Kannan-FV, but overall better than UNIFAC-FV. 2. For non-athermal non-aqueous polymer solutions, the modified Freed-FV performs similarly to the other models at finite concentrations but a bit worse at infinite dilution. 3. Modified Freed-FV and Kannan-FV are the best models for aqueous polymer solutions, and in this case the results are much better than Freed-FV and UNIFAC-FV, which fail especially at infinite dilution. Two typical results are shown in Figs. 1 and 2 for a finite concentration (PVAC–water) and an infinite dilution system (PEO–water). The results with all models shown in Table 3 and Figs. 1 and 2 are based on the “classical” approach with respect to R and Q, i.e. using the fitted Q parameters (for OH, water and COO) for the residual term and the physical parameters in the various combinatorial-FV terms. In summary, and in terms of the Freed-FV models, the results of Table 3 suggest using the original version for non-aqueous systems and the modified version for the aqueous ones.

Fig. 2. Predicted and experimental (䊉) water activity coefficients at infinite dilution for the PEO (400)–water system using the original UNIFAC-FV model (—), Kannan-FV model (- - -), original Freed-FV model (· · ·) and the modified Freed-FV model proposed in this work (– - –).

Table 4 Prediction and correlation results (% AAD) for aqueous and alcohol polymeric systems using the new regressed interaction parametersa,b Polymer

Solvent

T-range (K)

Ns

Np

AADc (%) Freed-FV New

PVAL PEO PPO PAA PMAA PVAC

Water Water Water Water Water Water

PEO PPO PIB PBMA PVAC PS

–

PVAC PBMA PEA PEMA PMA Overall

New

Old

New

Old

References

3 12 2 1 1 2

28 159 18 12 11 15

12.8 4.7 44.4 26.6 37.9 6.2

42.7 8.0 48.0 65.2 19.8 65.4

9.5 4.1 47.7 24.7 45.5 19.9

29.0 11.7 55.0 47.8 24.1 71.5

14.0 3.9 41.5 27.0 39.0 7.2

95.6 7.6 16.6 258.6 105.0 15.5

Fin Fin Fin Fin Fin Fin

c c c c c c

[22,28] [3,22,24,29] [3,25] [3] [3] [3,22]

21

243

11.0

21.8

11.5

23.0

10.6

35.7

Fin

c

–

Ethanol, 2-propanol Ethanol, 1-propanol Ethanol, 1-propanol, 2-propanol, 1-butanol, 2-butanol Ethanol, 1-propanol, 2-propanol, 1-butanol, 2-butanol Ethanol, 1-propanol, 2-propanol, 1-butanol, 2-butanol Ethanol, 1-propanol, 2-propanol, 1-butanol, 1-pentanol

298.2–303.2 303.15 293.2–353.2

3 2 6

33 13 180

1.5 20.8 25.8

6.0 54.1 61.1

3.0 28.3 30.1

3.3 35.0 38.0

2.0 25.2 28.4

11.4 81.3 74.5

Fin Fin Fin

c c c

[25,26] [25] [27]

293.2–353.2

6

175

19.3

22.2

23.1

16.9

22.8

25.5

fin

c

[27]

293.2–353.2

7

130

15.4

48.6

24.3

29.2

21.6

65.2

Fin

c

[3,27]

435.5–502.6

5

20

48.1

67.7

49.4

55.6

46.2

Inf

c

[22]

29

553

20.5

41.6

25.6

26.7

24.0

52.3

Fin, inf

c

–

Water Ethanol, 1-propanol, 2-propanol, 1- butanol, 2-butanol Ethanol, 1-propanol, 2-propanol, 1- butanol Ethanol, 1-propanol, 2-propanol, 1- butanol 1-Propanol 1-Propanol, 1-butanol Ethanol, 1-propanol,

328.2–423.7 334.7–398.2

6 17

50 51

24.7 10.1

76.1 58.8

30.3 12.5

80.3 37.2

25.2 10.8

35.7 90.1

Inf Inf

p p

[22] [22]

353.2–473.2

5

17

25.5

119.6

11.1

79.9

12.0

158.5

Inf

p

[22]

329.9–417.2

4

20

14.0

29.7

25.3

29.7

24.4

46.7

Inf

p

[22]

393.2 393.2–417.2 349.2–398.2

1 3 2

1 3 3

10.3 16.5 13.8

14.2 12.7 17.6

26.6 13.7 17.3

29.7 8.1 20.4

26.6 14.0 17.1

10.6 11.8 22.8

Inf Inf Inf

p p p

[22] [22] [22]

38

165

17.7

65.8

20.5

53.6

18.1

69.8

Inf

p

–

Overall PEO PEO

Kannan-FV

Notee

–

–

205

H.Reza. Radfarnia et al. / Fluid Phase Equilibria 257 (2007) 63–69

Overall

303.2–383.2 293.1–338.2 303.2–323.2 299.2 313.2 313.2

Old

Entropic-FV

Datad

a The molecular weights of polymers used in this work are: aqueous systems (finite dilution): PVAL (14,700, 6700, 64,444), PEO (296, 335, 600, 1000, 1500, 3000, 4237, 5000, 12,000, 20,000), PPO (400, 2000), PAA (50,000), PMAA (50,000), PVAC (50,000, 113,000). aqueous systems (infinite dilution): PEO (400, 600, 4000, 7500, 10,000, 400,0000). alcohol systems (finite dilution): PEO (300, 600, 1000), PPO (2000), PIB (500,000), PBMA (337,000, 337000, 337000, 337000, 337000), PVAC (167,000). alcohol systems (infinite dilution): PEO (200, 300, 600, 1000, 2000, 3600, 4000, 10,000, 10,700), PVAC (47,800, 83,400, 257,000), PBMA (8716, 35,000, 2,050,000), PEA (80,000), PEMA (40,000, 144,000), PMA (90,000, 381,000), PS (20,000). b All VLE results were predicted for the Kannan-FV/UNIFAC and Entropic-FV/UNIFAC models using the regressed interaction parameters of Freed-FV/UNIFAC model. c AAD (%) = (100/N)|a i,calc − ai,exp |/ai,exp . d fin: finite, inf: infinite. e c: correlated results for Freed-FV/UNIFAC model, p: predicted results for Freed-FV/UNIFAC model.

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and infinite dilution activity coefficients for the PPO–ethanol and PEO–ethanol solutions. An effective comparison is also provided using the results predicted by the original UNIFAC parameters. It can be seen that a significant improvement is obtained when the new parameters are used and for all models considered. The improvement is particularly pronounced at infinite dilution conditions. 5. Conclusion

Fig. 3. Calculated and experimental (䊉) ethanol activities for the PPO (Mn = 2000)–ethanol system at 303.15 K using the “newly” regressed and the “old” regressed interaction parameters with the original Freed-FV model: new parameters ( ), old parameters (—); Entropic-FV model: new parameters ), old (– – –), old parameters (- - -); Kannan-FV model: new parameters ( parameters (· · ·).

4.2. Aqueous and alcohol polymer solutions using the new interaction parameters The results are summarized in Table 4, while Figs. 3 and 4 show some typical results for the finite concentration activities

We have compared the performance of the “classical” (UNIFAC-FV, Entropic-FV) and “recent” free-volume (FV) models (Kannan-FV, Freed-FV) for a variety of size-asymmetric systems including polymer solutions. Two improvements have been developed. First, similar to the work of Kannan et al. [5], a modified free-volume expression has been developed in order to improve the performance of Freed-FV based models for aqueous polymer systems by subtracting the Freed Flory-Huggins hard-core combinatorial expression from the combinatorialfree-volume term of Freed-FV model proposed in a previous work. The results predicted by the modified Freed-FV model are superior to the original version of the model. On a second level, new UNIFAC parameters have been regressed for alcohol and aqueous mixtures using the physical (Bondi-derived) values for the van der Waals volumes and surface areas of the OH, COO and water groups. The new parameters have been estimated using the Freed-FV model but used in the other models as well. Much improved results have been obtained with all FV models considered for such complex solutions and at both finite concentrations and infinite dilution. List of symbols a activity rfv ratio of molar free molar volume of polymer to solvent rh ratio of molar hard-core volume (van der Waals volume) of polymer to solvent v molar volume vh hard-core volume x mole fraction Greek letters ␣ non-random parameter φfv molar free-volume fraction φh hard-core volume fraction γ molar based activity coefficient Subscripts i, j component

Fig. 4. Calculated and experimental ethanol activity coefficient at infinite dilution for the PPO (Mn = 200, 300, 1000, 2000)–ethanol system using the “newly” regressed and the “old” interaction parameters with the original Freed-FV model. Experimental (Mn-200: (), Mn-300: (), Mn-1000: (), Mn-2000: (䊉)) and predictions: new parameters: Mn-200 ( ), Mn-300 (– –), Mn-1000 ( ), Mn-2000 (– - –); old parameters: Mn-200 (—), Mn-300 (- - -), Mn-1000 (· · ·), Mn-2000 (- · - · -).

Superscripts vdW van der Waals comb combinatorial fv free-volume exp experimental predict prediction

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