Miscibility in Binary Polymer Blends: Correlation and Prediction

Feb 6, 2004 - Laboratory of Thermodynamics & Transport Phenomena, School of Chemical Engineering, National Technical University of Athens, 9 Heroon ...
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Ind. Eng. Chem. Res. 2004, 43, 1312-1321

Miscibility in Binary Polymer Blends: Correlation and Prediction Epaminondas C. Voutsas,* Georgia D. Pappa, Christos J. Boukouvalas, Kostis Magoulas, and Dimitrios P. Tassios Laboratory of Thermodynamics & Transport Phenomena, School of Chemical Engineering, National Technical University of Athens, 9 Heroon Polytechniou Str., Zografou 15780, Athens, Greece

Correlation and prediction of miscibility in binary blends of homopolymers is investigated using an activity coefficient model, the entropic free-volume/UNIFAC (EFV/UNIFAC), and two equations of state: the Peng-Robinson (PR) and the Sanchez-Lacombe (SL) ones. Satisfactory correlation results are obtained with all models but their quality depends on whether T-dependent or T-independent interaction parameters (IPs) are used. Satisfactory prediction of the polymers molecular weight (MW) effect of blend miscibility is possible only with the EFV/UNIFAC and SL models, while the latter predicts successfully the MW effect in the single available in the literature case for homopolymers, the blend poly(n-pentyl methacrylate)/polystyrene, where UCST and LCST as well as hourglass behavior is observed. The SL model provides also satisfactory prediction of the pressure effect on the critical solution temperature (CST) but less so on the corresponding composition. It is, thus, the most successful of the three models considered here. Finally, an empirical scheme for the prediction of UCST using the PR EoS is presented. 1. Introduction In recent years the use of polymer blends is constantly increasing due to the properties they exhibit, which make them suitable for a variety of applications. Their production as well as their successful use requires that a number of their properties are known. Among them blend miscibility is of major importance since it affects the physical properties of the blend and consequently determines the field of its applications and uses. The kinds of factors that effect polymer-polymer miscibility are as follows: (i) entropy of mixing, (ii) dispersion forces, (iii) specific (Lewis acid-base or electrostatic) interactions, and (iv) free-volume differences. The dispersion forces lead always to positive heats of mixing and they are the main reason for phase separation in blends at low temperatures. Their effect weakens with increasing temperature and, at some point, they are overcome by the entropy of mixing, leading to the appearance of the UCST. This is the case for systems such as poly(methyl styrene) (PMS)/polystyrene (PS), poly(1-butene) (PB)/PS, poly(isoprene) (PIP)/PS, and poly(a-methyl styrene) (PaMS)/PS. The specific interactions can be sufficiently strong, leading to negative heats of mixing and polymer miscibility. For example, Fowkes et al.1 determined through spectroscopic measurements the enthalpy of acid-base interactions for miscible blends of poly(methyl methacrylate) (PMMA) with chlorinated poly(vinyl chloride) (CPVC), poly(vinyl fluoride) (PVF), and poly(vinylidene fluoride) (PVF2). They found values of -1.9, -6.8, and -2.9 kcal/mol, respectively. Other examples of miscible blends due to such interactions include poly(2,6-dimethyl-1,4-phenylene oxide) (PPO) with PS, poly(vinyl chloride) (PVC) with poly(-caprolactone) (PCL), and PVC with PMMA. The strength of these interactions and, consequently, their effect on polymer miscibility is not easy to predict. * To whom correspondence should be addressed. Tel.: +30 2107723137.Fax: +302107723155.E-mail: evoutsas@chemeng. ntua.gr.

For example, Kim and Paul2 report that polycarbonate (PC) is not miscible with PS, but tetramethyl bisphenolA carbonate (TMPC) is. They also demonstrate how this can be explained using charge distributions for the three polymers. Furthermore, Brinke and Karasz3 demonstrate that for polymer miscibility not only does one of them have to be basic and the other acidic but also their solubility parameters should not differ too much. The Lewis acid-base interactions are usually of a highly directional nature, leading to loss of rotational freedom for the polymer segments and, consequently, to a loss of entropy. This loss of entropy is significant for mixtures of small molecules but it is rather small for polymer blends. Thus, even though this tends to favor demixing, it is not sufficient to lead to this in the typical case. Free-volume (FV) differences between the polymers lead in the typical case to negative volume of mixing favoring demixing. Even though the FV differences are smaller here than for polymer/solvent systems, they can be sufficient to cause phase splitting. This effect, referred to also as EoS effect, becomes more important at higher temperatures, where the intermolecular interactions become weaker, leading to a LCST. Thus, at low temperatures and in the absence of specific interactions, polymer blends are not miscible due to dispersion forces. As the temperature increases, their effect diminishes and mixing occurs as a result of the entropy of mixing until the FV effects become important and immiscibility occurs, not necessarily at temperatures that can always be reached experimentally. One case, however, is available in the literature, the system poly(n-pentyl methacrylate) (PnPMA)/PS. In the presence of specific interactions, the system is miscible at low temperatures, but as the temperature increases, they become weaker, leading to immiscibility as a result of the FV effect. It is apparent that in the typical case phase separation should be expected and consequently the vast majority of the binary polymer blends are incompatible over much of the concentration range.

10.1021/ie0306269 CCC: $27.50 © 2004 American Chemical Society Published on Web 02/06/2004

Ind. Eng. Chem. Res., Vol. 43, No. 5, 2004 1313

Experimental measurement of polymer blend miscibility is far more difficult than that of polymer/solvent solutions. As a result, substantial work is available in the literature, from modifications of the original FloryHuggins (FH) expression to cubic and more complex EoS, for the correlation and prediction of blend miscibility. For example, Harismiadis et al.4 used the van der Waals (vdW) EoS with a T-independent interaction parameter for the correlation of the UCST and Tdependent one for the LCST; they also presented an empirical expression for the estimation of UCST with this EoS. Chang et al.5 used the Flory-Huggins model with interaction parameters functions of temperature and polymer volume fractions for the correlation of PS/ polybutadiene (PBD) and PS/hydrogenated PBD, while Kressler et al.6 used T-dependent interaction parameters with the Flory-Huggins, Flory-Orwoll-Vrij and Dee and Walsh models to correlate the miscibility of the PS/PMMA blend. Rostami and Walsh7 have used the Flory-Orwoll-Vrij EoS for the system PBD/PS while Prausintz and co-workers have used the perturbed hardsphere-chain (PHSC) equation of state again for the system PBD/PS8 and a simplified version of it for the PMS/PS and poly(vinyl methyl ether) (PVME)/PS systems.9 Finally, the Sanchez-Lacombe theory has been used by Kim and Paul2 for the PS/TMPC system, Chen et al.10 for the poly(ethylene oxide (PEO)/poly(vinyl acetate) (PVAc) system, and Economou11 for highdensity polyethylene (HDPE)/branched polyolefin systems. In this paper we consider three models: the EFV/ UNIFAC, the PR EoS, and the SL EoS and examine their performance in (i) the correlation of liquid-liquid equilibria (LLE) in polymer blends, (ii) the prediction of the effect of polymer molecular weight (MW) using interaction parameters (IP) obtained from one pair of MWs, and (iii) the prediction of the effect of system pressure on miscibility using IP obtained from lowpressure miscibility data. Moreover, the possibility of predicting blend miscibility is examined. All experimental data used here correspond to nearly monodispersed polymers, having polydispersity indices typically of less than 1.1, which is not expected to have a significant influence on blend miscibility. 2. Models

fraction of the component i. The free volume (Vfv i ) is assumed to be equal to w Vfv i ) Vi - Vi

(3)

where Vi is the molar volume and Vw i is the van der Waals volume of the component i as calculated by the Bondi method.13 Molar volumes, if not differently noted, are calculated using the Tait correlation with the parameters available in the publication by Rodgers.14 The residual part of the model is that of UNIFAC, which is calculated as the difference between the residual activity coefficient of group k (polymer segment) in the solution (Γk) and the corresponding activity coefficient in a reference solution containing only molecules of type i (Γ(i) k ):

ln γres i ) with

(

∑k v(i)k (ln Γk - ln Γ(i)k )

ln Γk ) Qk 1 - ln(

∑ m

ΘmΨmk) -

ΘmΨmk

(4)

)

∑ m ∑n ΘnΨnk

(5)

ν(i) k is the number of groups of type k in molecule i, Qk is the area parameter of group k and Θm is the area fraction of group m, calculated as

Θm )

QmXm

∑n

(6)

QnXn

where the mole fraction of group m in the mixture (Xm) is given by

Xm )

∑j v(j)m xj ∑j ∑n

(7)

v(j) n xj

2.1. The EFV/UNIFAC Model. In the various versions of the UNIFAC model activity coefficients (γi) are calculated as the sum of two terms: the combinatorial term, which accounts for the differences in the shape and size of the molecules, and the residual term, which mainly accounts for the energetic interactions between them. In this work the modification of Elbro et al.12 for the combinatorial term has been used. The expression is similar to the Flory-Huggins one, but free-volume (FV) fractions are used instead of volume fractions. Thus, both combinatorial and free-volume effects are included in a single combinatorial-FV term:

where Rmn is the interaction parameter (IP) between polymer segments m and n. The interaction parameters were treated either as temperature-independent or as linearly temperaturedependent according to the expression

ln γi ) ln γcomb-fv + ln γres i i

Rmn ) Rmn,0 + Rmn,1T

) ln ln γcomb-fv i

φfv φfv i i +1, φfv i ) xi xi

(1) xiVfv i

∑

xiVfv i

(2)

where xi is the mole fraction and φfv i is the free-volume

Finally, the group interaction parameter (Ψmn) is given by

(

Ψmn ) exp -

)

Rmn T

(8)

(9)

Interaction parameters between the polymer segments were estimated by fitting one set of experimental data per blend. These parameters were then used to predict either the effect of the polymers MWs or the effect of pressure on blend miscibility.

1314 Ind. Eng. Chem. Res., Vol. 43, No. 5, 2004 Table 1. Pure Polymer Parameters for the PR EoS polymera

a/MWd

b/MWe

T range (K)

PS PSc PBD PBDc PMMA PIP PaMeS hhPP sPIP sPBD sPIP 3-4

449843 466073 375411 473045 312502 558158 309599 674653 417428 417024 555038

0.8484 0.8504 0.9550 0.9919 0.8383 1.0706 0.8451 1.1348 0.8742 0.8695 0.7837

388-466 388-453 277-326 277-318 387-430 406-519 412-469 443-566 400-450 400-450 400-450

Table 2. Polymers Abbreviations

P range exp. data (bar) % ∆V b from 1 1-2000 1 1-2835 1 1 1 1 1 1 1

0.016 1.2 0.010 1.6 0.008 0.332 0.005 0.691 0.100 0.117 0.003

abbreviation

polymer

hhPP PaMeS PBD PEMA PEO PES PIP PMMA PnBMA PnHMA PnPMA PS sPBD sPIP sPIP 3-4

head-to-head polypropylene poly(a-methyl styrene) polybutadiene poly(ethyl methacrylate) poly(ethylene oxide) poly(ether sulfone) polyisoprene poly(methyl methacrylate) poly(n-butyl methacrylate) poly(n-hexyl methacrylate) poly(n-pentyl methacrylate) polystyrene saturated polybutadiene saturated poly(isoprene) saturated poly(isoprene) 3-4

Tait Tait Tait Tait Tait GCVOL Tait Tait GCVOL GCVOL GCVOL

a For abbreviations see Table 2. b % ∆V ) abs(V experimental Vcalculated)/Vexperimental × 100. c a and b values used for the prediction of the pressure effect on LLE. d a in cm6bar/mol2. e b in cm3/mol.

2.2. Peng-Robinson EoS. The expression for the Peng-Robinson EoS is15

RT a P) V - b V(V + b) + b(V - b)

polymera

(10)

Its application in polymer blends requires knowledge of the co-volume parameter (b) and the energetic parameter (a) for the pure polymers. For their estimation the method proposed by Louli and Tassios16 is followed: a single set of a/MW and b/MW parameters, independent of the polymer MW, is obtained for each polymer by fitting experimental PVT data. These were obtained by using either the Tait equation in the temperature range of its application with the parameter values proposed by Rodgers14 or the CGVOL17 predictive method for polymers for which the Tait equation parameters were not available. The obtained a/MW and b/MW parameters for the polymers examined in this work are shown in Table 1. The vdW one-fluid mixing rules were used to estimate the mixture EoS parameters:

amix )

∑i ∑j xixjaij

(11)

bmix )

∑i ∑j xixjbij

(12)

For the cross energy (aij) and cross co-volume (bij) parameters the geometric and arithmetic combining rules were employed, respectively:

aij ) xaiaj(1 - kij)

(13)

1 bij ) (bi + bj) 2

(14)

2.3. Sanchez-Lacombe EoS. The Sanchez-Lacombe (SL) model18,19 is a lattice-fluid (LF) equation of state developed to describe the thermodynamic properties of both small molecules and polymers. The SL EoS is given by the expression

[

(

˜ +T ˜ ln (1 - F˜ ) + 1 F˜ 2 + P

1 F˜ ) 0 r

)]

Table 3. Pure Polymer Parameters for the SL EoS along with the % Error in Molar Volume (% ∆V)

(15)

where F˜ , T ˜, P ˜ , and r are the reduced density, reduced temperature, reduced pressure, and the number of lattice sites occupied by one molecule, respectively.

PSb PBDb PBDc PMMAb PMMAd PEOb PESe PnPMAe,f

T range (K)

P range (bar)

T* (K)

P* (bar)

F* (kg/m3)

391-469 278-328 278-328 340-520 340-520 361-497 533-623 413-513

1-2000 1-2000 1-2000 1-2000 1-2000 1-2000 1-2000 1-2000

688.0 462.0 680.5 668.0 706.4 656.0 887.5 692.5

3715.0 4402.0 4195.1 5169.0 5066.5 4922.0 6393.5 5032.4

1119.9 1015.1 946.8 1281.2 1259.5 1177.6 1447.5 1103.4

% ∆V 0.33 0.73 1.10 0.46 0.60 0.19 0.16

a For abbreviations see Table 2. b Parameters reported by Rodgers. c Parameters estimated in this work by considering the LLE data for the PS(1900)/PBD(2350) along with the PVT experimental data. d Parameters estimated in this work by considering the LLE data6 for the PS(1250)/PMMA(6350) along with the PVT experimental data. e Parameters estimated in this work using PVT experimental data. f For PnPMA “pseudo” experimental PVT data were generated by interpolation using experimental data for PMMA, PEMA, PnBMA, and PnHMA and fitted to the Tait equation.

The reduced parameters are defined from characteristic fluid parameters:

T ˜ )

RT P v* F rv* T ) ; P ˜ ) ) P ; F˜ ) )F T* * P* * F* M

(16)

Thus, a pure polymer is completely characterized with the SL EoS by three equation of state parameters (T*, P*, and F*) or equivalently by three molecular parameters: characteristic energy, *, characteristic volume, v*, and the number of lattice sites occupied by one molecule, r. In eq 16, R is the gas constant and M is the molecular weight. The three parameters are usually estimated by utilizing pure polymer PVT experimental data14 and they are presented for the polymers of this study in Table 3. For a mixture it is assumed that if an i molecule occupies roi sites in the pure state, then it will occupy ri sites in the mixture so as riv* ) roi vi*, where v* is the mixture characteristic volume and vi* is the ith component characteristic volume. The number of sites occupied by a molecule, r, is given by the expression

r)

∑ xiri )

1 ) φi ri

∑

1 φoi

(17)

∑ ro i

where xi is the mole fraction of component i and φoi and

Ind. Eng. Chem. Res., Vol. 43, No. 5, 2004 1315

φi are segment fractions defined as

mi mi Fi*vi* Fi* and φi ) φoi ) mi mi Fi*vi* Fi*

∑

(18)

∑

where mi is the mass fraction of the component i. The mixture characteristic density, F*, volume, v*, and energy, *, are calculated by the following expressions, respectively:

1 mi Fi*

F* )

(19)

∑ v* ) * )

∑ φoi vi*

(20)

∑∑ φiφjij*

A binary interaction parameter, kij, is used to determine ij*:

ij* ) kijxi*j*

(21)

In the case of a binary mixture (i-j) the chemical potential of the component i is given by the expression

( )

[ (

)]

ri vi * µi ) ln φi + 1 - φj + roi F˜ Xij + 1 λ φ2 + RT rj vj* ij j P ˜i - F˜ F˜ 1 + roi - 1 ln(1 - F˜ ) + ln F˜ (22) roi Ti* T F˜

[(

)

]

where

i* + j* - 2ij* RT

(23)

Ti* - Tj* + (φi - φj)Xij ) -λji T

(24)

Xij ) λij )

The expression for µj is easily obtained by interchanging the indices i and j. Temperature-independent kij parameters have been determined by fitting the mixture critical temperature. The condition for the critical point is

( ) ∂µi ∂φi

and

) 0 (spinodal condition)

(25)

T,P

( ) ∂2 µ i

∂φi2

)0

(26)

T,P

When necessary, temperature-dependent kij parameters were determined by fitting binary polymer blend LLE data for one pair of molecular weights. 3. Results 3.1. EFV/UNIFAC. Typical correlation and MW effect prediction results with EFV/UNIFAC are shown in

Figure 1. Correlation and MW effect prediction results for the PS/PBD blend with the EFV/UNIFAC model. Comparison between T-independent and T-dependent IPs. (0) experimental data PS(1900)/PBD(2350); (O) experimental data PS(3300)/PBD(2350); (thin solid line) correlation using T-independent IPs: a12 ) 6.01539 (K), a21 ) 6.09855 (K); (- - -) prediction using T-independent IPs; (solid thick line) correlation using T-dependent IPs: a12 ) -49.9476 + 0.22767*T (K), a21 ) 85.0441 - 0.28193*T (K); (dashed thick line) prediction using T-dependent IPs.

Figure 1 for the system PS/PBD, where UCST behavior is observed. The experimental data for the PS(1900)/ PBD(2350) pair were used to obtain the interaction parameters of the model. The quality of both correlation and prediction depends on the temperature range employed for correlation. Thus, successful description of the whole concentration and temperature range of the experimental data requires T-dependent IPs, which, however, lead to poor prediction of the MW effect. On the other hand, T-independent IPs, which are based on correlation of only the vicinity of the critical point, lead to satisfactory prediction of the MW effect on the critical temperature but not on the whole phase envelope. We consider next the system PEO/PES, which is completely miscible at low temperatures due to specific interactions between the polymers but partially miscible at higher temperatures, where these interactions weaken and free-volume effects dominate.20 Use of T-independent IPs provides successful correlation only for the temperature and concentration region close to the critical ones, while use of T-dependent ones (Figure 2) leads to improved correlation for the whole phase envelope. In both cases the effect of the MW on LLE is somewhat overpredicted but the results can be considered satisfactory given that the MW of PEO increases by a factor of 50 as shown in Figure 2. Figure 3 presents results for the system PS/PnPMA, the only homopolymer mixture where both upper and lower CSTs have been observed.21 Molar volumes of PnPMA were obtained using the Tait equation as explained in Table 3. The LLE experimental data for the PS(7570)/PnPMA(6610) pair were used to obtain the IPs of the model. Quadratic interaction parameters with respect to temperature were required in order to simultaneously fit the UCST and LCST phase envelope. Correlation results are excellent but the effect of the

1316 Ind. Eng. Chem. Res., Vol. 43, No. 5, 2004

Figure 2. Correlation and MW effect prediction results for the PEO/PES blend with the EFV/UNIFAC model with linear Tdependent IPs: a12 ) 0.12772 + 0.45174*T (K), a21 ) -28.002 0.29169*T (K). (b) Experimental data PEO(4000)/PES(20000); (2) experimental data PEO(20000)/PES(20000); (9) experimental data PEO(200000)/PES(20000); (thick line) correlation PEO(4000)/PES(20000); (-) prediction PEO(20000)/PES(20000); (- - -) prediction PEO(200000)/PES(20000).

Figure 3. Correlation and MW effect prediction results for the PnPMA/PS blend with the EFV/UNIFAC model using T-dependent IPs: a12 ) -24.14 - 0.03336*T - 0.00019*T2 (K), a21 ) 51.66 0.07967*T + 0.000363*T2 (K). (b) Experimental data PnPMA(7570)/PS(6610); (9) experimental data PnPMA(7570)/PS(6835); (0) experimental data PnPMA(7570)/PS(6960); (2) experimental data PnPMA(7570)/PS(7110); (thick line) correlation PnPMA(7570)/PS(6610); (-) prediction PnPMA(7570)/PS(6835); (- - -) prediction PnPMA(7570)/PS(6960); (- - -) prediction PnPMA(7570)/PS(7110).

polymers MWs on LLE is significantly underpredicted, especially for the LCST. This is because the model considers the variation of pure polymer free volume with temperature, but it underestimates the volume of mixing. For the same

Figure 4. Correlation and MW effect prediction results for the PS/PBD blend with the SL EoS. Comparison between T-independent and T-dependent IPs. (0) Experimental data PS(1900)/PBD(2350); (O) experimental data PS(3300)/PBD(2350); (-) correlation using T-independent kij ) 0.99464; (- - -) prediction using Tindependent kij; (solid thick line) correlation using T-dependent kij ) 0.9850867 + 0.0000242*T; (dashed thick line) prediction using T-dependent kij.

reason it fails to predict the pressure effect on blend miscibility using IPs obtained at low pressures. 3.2. Peng-Robinson EoS. Correlation of UCST using the PR EoS is possible with a T-independent kij but for improved results a T-dependent kij is needed. For LCST a T-dependent kij is required since it is not possible to obtain correlation with a T-independent one. Prediction of the MW effect with the PR EoS using the same kij is not possible. This is not surprising since the interaction parameters reflect here interactions between molecules and not segments as is the case for the SL EoS shown next. What is surprising, however, is that the PR EoS performs poorly in the prediction of the system pressure effect on blend miscibility, which suggests poor prediction of the volume of mixing for the blend. 3.3. Sanchez-Lacombe EoS. Typical UCST correlation results with the SL EoS are presented in Figure 4 for the system PS(1900)/PBD(2350). In the same figure prediction of the MW effect on UCST using the kij determined from the correlation of the low MW LLE data is also presented. Use again of T-dependent kij, which provides a successful correlation of the whole concentration range, leads to poor prediction of the MW effect. On the other hand, a T-independent kij that is based on correlating only the UCST leads to satisfactory prediction of the MW effect on the CSTs. A similar behavior has been observed for other blends. The limitation of the SL EoS in describing the full concentration range with a kij determined from the critical point only is also encountered with other EoSs such as the perturbed hard-sphere chain one.8 It must be noted here that pure polymer EoS parameters obtained from PVT datasTable 3sdo not always lead to successful LLE calculations. For example, with use of the parameters reported by Rodgers14 for PS and PBD (Table 3), LLE correlation for the PS/PBD binary

Ind. Eng. Chem. Res., Vol. 43, No. 5, 2004 1317

Figure 5. SL predictions of the critical temperature in PBD(2350) blends with PS of different MWs at 1 atm. (b) Experimental points; (-) prediction with kij ) 0.99464.

of Figure 4 was not possible. This problem with the SL EoS and other noncubic equations of state has also been observed by other researchers.22-24 In this work, we reestimated some pure polymer parameters using LLE data for one set of MWs along with the PVT data. Notice however that, as shown in Table 3, the quality of PVT description is not sacrificed with this approach. Furthermore, pure polymer parameters determined by utilizing LLE data from one blend can be used for LLE calculation in other blends containing this polymer. For example, the PMMA parameters marked with superscript d in Table 3 gave also successful LCST correlation of the experimental data for the PMMA(130000)/PEO(300000) system.25 Figure 5 presents UCST predictions for mixtures of PBD(2350) with PS of various MWs, using the kij obtained from the critical temperature of the PS(1900)/ PBD(2350) blend. Excellent agreement with the observed values for low MWs is obtained, while for higher MWs the dependence of the CST on MW weakens, which is in agreement with the findings of Economou11 for the high-density polyethylene/poly(ethylene-alt-propylene) system using the same model. Even though the extrapolation of polymer parameters to such high temperatures may cast some doubt on these results, it appears thatsaccording to the SL modelsthe increase in dispersion interactions with an increase in MW is overcome by the temperature effect on them at very high temperatures. On the other hand, prediction of the critical weight fraction value is not as satisfactory as demonstrated by the results for the same system in Figure 6. This is in agreement with Chen et al.10 who used a MW-dependent ij (or kij) in order to reproduce satisfactorily the critical temperature and concentration for mixtures of polymers with different MWs. Correlation for the system PEO/PES is examined next using T-independent and T-dependent kij. T-dependent kij is needed for the description of the full phase envelope (Figure 7), which is in agreement with the findings of Sanchez and Balazs26 for the same EoS and for systems where specific interactions are present. In the same figure prediction of the MW effect on LCST are shown

Figure 6. SL predictions of the critical PS weight fraction of PBD(2350) blends with PS of different MWs as a function of PS MW at 1 atm. (b) Experimental points; (-) prediction with kij ) 0.99464.

Figure 7. Correlation and MW effect prediction results for the PEO/PES blend with the SL EoS using T-dependent kij ) 1.0137275 - 0.000011541*T. (b) Experimental data PEO(4000)/ PES(20000); (2) experimental data PEO(20000)/PES(20000); (9) experimental data PEO(200000)/PES(20000); (thick line) correlation PEO(4000)/PES(20000); (-) prediction PEO(20000)/PES(20000); (- - -) prediction PEO(200000)/PES(20000).

using kij determined for the pair with the lowest MWs. Comparison of the results obtained with T-dependent and T-independent parameters suggests that the former provides better prediction of the LCST and the latter better description of the composition effect. The use of two parameters in systems where specific interactions are present appears, however, to be preferred. Thus, Sanchez and Balasz26 have predicted LCST behavior for the pair PS(5.93 × 105)/PVME(1.1 × 106) using data for the pair PS(2.3 × 105)/PVME(3.89 × 105) with a two-interaction-parameter version of the

1318 Ind. Eng. Chem. Res., Vol. 43, No. 5, 2004

Figure 8. Correlation and MW effect prediction results for the PnPMA/PS blend with the SL EoS using T-dependent kij: 0.9980058 + 0.000012174 T - 2 × 10-8 T2. (b) Experimental data PnPMA(7570)/PS(6610); (9) experimental data PnPMA(7570)/PS(6835); (0) experimental data PnPMA(7570)/PS(6960); (2) experimental data PnPMA(7570)/PS(7110); (thick line) correlation PnPMA(7570)/PS(6610); (-) prediction PnPMA(7570)/PS(6835); (- - -) prediction PnPMA(7570)/PS(6960); (- - -) prediction PnPMA(7570)/PS(7110).

SL EoS that they developed. Their functional form for the interactions is, however, different from ours. Figure 8 presents correlation of LLE for the PS(7570)PnPMA(6610) system and prediction results for three other pairs of MW for the same system with the SL EoS. The PnPMA parameters for the SL EoS were determined as explained in Table 3. Excellent CST predictions for the whole range of PS MWs considereds involving both UCSTs and LCSTssare obtained as also shown in Figure 9 but the width of the phase envelopes is underpredicted in the case of the LCST behavior. Finally, the hourglass behavior is only qualitatively predicted. The kij values determined for the SL EoS were in all cases very close to unity, even for the polar systems such as the PEO/PES one. It must be noted, however, that LLE calculations with the SL EoS are very sensitive to the kij value. For example, with a change in the kij for the PS(1520)/PBD(2350) system at 1 atm from 0.994613 to 0.993618 (0.1% change), the predicted CST changes by 16.5%. On the other hand, LCST is not as sensitive to kij as the USCT, which can be realized by the following example for the system PEO(4000)/PES(20000) at 1 atm: a 0.1% change in the kij changes the predicted CST by 9%. This behavior is attributed to the fact that LCST is dominated by the equation of state effects, which are less sensitive to the kij value than the energetic ones that dominate the UCST behavior.19 In general, the kij for the SL EoS should be given with five important decimal digits. Except from the set of mixing rules presented in section 2.3, other sets proposed in the literature were also examined.28,29 No significant differences were observed in the results obtained from the different sets of mixing rules. Figure 10 presents prediction of the pressure effect on the UCST of the system PS(1520)/PBD(2350) for two

Figure 9. Variation of LCST and UCST with the PS MW for 50/ 50 (wt/wt) PS/PnPMA(7570) blends. Interaction parameters were obtained by fitting the UCST and LCST for the pair PS(6610)/ PnPMA(7570). (2) Experimental UCST data; (O) experimental LCST data; (-) experimental data, best fit line; (- - -) prediction with SL EoS.

Figure 10. Prediction of the critical temperature of the PS(1520)/ PBD(2350) blend as a function of pressure with the SL EoS. (b) Experimental points; (-) prediction with T-independent kij ) 0.99464 fitted to PS(1900)/PBD(2350) at 1 atm; (- - -) prediction with T-independent kij ) 0.994613 fitted to PS(1520)/PBD(2350) at 1 atm.

cases. In the first one, the kij value is obtained by fitting the atmospheric pressure UCST for the same system; in the second, the kij value is obtained from the UCST of a pair with different PS MW, equal to 1900, again at 1 atm. The increase in UCST with pressure suggests that, as pressure increases, pure polymer free-volume decreases as a result of better packing, leading to reduced miscibility. The pressure dependence of UCST is satisfactorily predicted by the SL EoS, which gives a pressure coefficient (dTc/dP) equal to 0.017 K/atm, with

Ind. Eng. Chem. Res., Vol. 43, No. 5, 2004 1319

Figure 11. SL predictions of the critical temperature of PEO(200000)/PES(2000) blend as a function of pressure. (b) Experimental points; (-) prediction with T-dependent kij ) 1.0137275 0.000011541*T fitted to PEO(4000)/ PES(20000) at 1 atm; (- - ) prediction with T-independent kij ) 1.0096 fitted to PEO(4000)/ PES(20000) at 1 atm.

the kij obtained by fitting the atmospheric pressure UCST for the PS(1520)/PBD(2350), close to the experimental value that was estimated from linear regression to be 0.015 K/atm. For the same system Rostami and Walsh7 used the Flory-Orwoll-Vrij equation of state to predict the pressure effect on UCST. To this purpose they determined, however, two binary parameters: one, common for all molecular weights, by fitting experimental enthalpies of mixing, and another, varying with the molecular weights of both polymers, by fitting the maximum cloud-point temperature at atmospheric pressure. Figure 11 presents prediction results of the pressure effect on LCST for the system PEO/PES. The LCST increases with pressure for the same reason as before. The SL EoS overpredicts the pressure dependence of LCST, giving a pressure coefficient equal to 0.14 K/atm, with the kij obtained by fitting the atmospheric pressure UCST for the PEO(200000)/PES(20000), while the experimentally observed is about 0.046 K/atm.7 The results presented here as well as those reported by Rudolf and Cantow30 demonstrate that the SL EoS with a temperature-independent binary interaction parameter can successfully predict the effect of pressure on blend miscibility. Similar results were also presented by Rudolf and Cantow30 with the Prigogine-FloryPaterson EoS and by Hino and Prausnitz31 with the perturbed hard-sphere-chain EoS.

4. Prediction of UCST for Blends The approach used for the PR EoS is similar to the one developed by Harismiadis et al. for the prediction of UCST using the van der Waals EoS.4 Here, however,

Figure 12. Optimum interaction parameter for a number of various polymer blends vs the ζ parameter (eq 27) and its fit with a linear equation.

the kij values obtained from UCST data are plotted vs the quantity

a1 a2 | 2 - 2| b1 b2 ζ) a12

(27)

b122 where a12 and b12 are given by eqs 13 and 14, respectively. The obtained plot is presented in Figure 12. As shown, the kij values (with the exception of three points) can be represented satisfactorily by a linear equation (kij ) A*ζ + Β), which should be used for ζ > 0.1 as suggested by the two low ζ value systems and the fact that for a polymer mixed with itself (ζ ) 0), kij must be equal to zero. This line is used to determine the kij values and consequently to predict the UCST. The predicted vs the experimental UCST values for the systems included in Figure 12sand for ζ > 0.1sare shown in Figure 13. With the exception of the two systems for which the kij values could not be represented by the aforementioned line, the predicted UCST values are within 50 K from the experimental ones. Moreover, very satisfactory USCT predictions are obtained for systems not included in Figure 12. For the prediction of LCSTs of binary blends Rodgers et al.32 have proposed the following method based on the Sanchez-Lacombe-Balasz (SLB) modification of the SL EoS. First, using heats of mixing data for mixtures of low MW compounds, they have developed a group-contribution method of the prediction of the heats of mixing (∆Hm) using their own modification of the Guggenheim quasichemical (MGQ) model. Using this model, they predict the heats of mixing for a polymer blend andsfrom thissthe two interaction parameters involved in the SLB model. Using IP values they have developed for a limited number of groups, they obtained satisfactory LCST predictions for blends such as PVME/PS, PMMA/PVC, and PMMA/PEO. This

1320 Ind. Eng. Chem. Res., Vol. 43, No. 5, 2004

Figure 13. Predicted vs experimental UCST for various polymer blends using the linear dependency of kij on ζ shown in Figure 12. (The asterisk indicates systems that were not included in the fit of Figure 12).

approach can of course be used only if the group interaction parameters are available, or if they can be determined from appropriate heats of mixing data. 5. Conclusions Three models, the EFV/UNIFAC, the PR EoS, and the SL EoS, are considered for the correlation of binary blends miscibility and the prediction of polymers MW and system pressure effect on miscibility. Excellent correlation results are obtained with all three models. Satisfactory predictions of the MW effect on polymer miscibility of both the UCST and LCST type, from data of one set of MWs, are possible with the EFV/UNIFAC and SL models but not with the PR EoS. Furthermore, the SL EoS is able to predict the MW effect for the single available in the literature case of homopolymer bends (PnPMA/PS), where USCT, LCST, and hourglass exist. The SL EoS provides also satisfactory prediction of the effect of pressure on blend miscibility. It is, thus, the most successful of the three models considered here. Prediction of the UCST for blends with the PR EoS is possible using the empirical method presented here, while for LCST the method of Rodgers et al.32 can be used, but for a limited number of blends. Literature Cited (1) Fowkes, F. M.; Tischler, D. O.; Wolfe J. A.; Lannigan, L. A.; Ademu-John, C. M.; Halliwell, M. J. Acid-Base Complexes of Polymers. J. Polym. Sci., Part A: Polym. Chem. 1984, 22 (3), 547. (2) Kim, C. K.; Paul, D. R. Interaction Parameters for Blends Containing Polycarbonates: 1. Tetramethyl Bisphenol-A Polycarbonate/Polystyrene. Polymer 1992, 33 (8), 1630. (3) ten Brinke, G.; Karasz, F. E. Lower Critical Solutions Behavior in Polymer Blends: Compressibility and DirectionalSpecific Interactions. Macromolecules 1984, 17, 815. (4) Harismiadis, V. I.; van Bergen, A. R. D.; Saraiva, A.; Kontogeorgis, G. M.; Fredenslund, A.; Tassios, D. P. Miscibility of Polymer Blends with Engineering Models. AIChE J. 1996, 42 (11), 3170.

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Received for review July 31, 2003 Revised manuscript received October 24, 2003 Accepted October 28, 2003 IE0306269