Models for Polymer Solutions

Computer Aided Property Estimation for Process and Product Design G.M. Kontogeorgis and R. Gani (Editors) © 2004 Elsevier B.V. All rights reserved.

143

Chapter 7: Models for Polymer Solutions Georgios M. Kontogeorgis

7.1 INTRODUCTION - AREAS OF APPLICATION Knowledge of phase equilibria of polymer systems (solutions, blends, ...) is of interest to the design of a variety of processes related to polymers. Some examples are shown in the table below. Table 1. Applications of polymer thermodynamics. Properties/ Phase Equilibria Involved Polymer-solvent VLE Polymer-solvent LLE, often mixed solvents Polymer-solvent VLE, solvent activities Emissions from paint production Polymer recycling via physico-chemical Polymer-solvent LLE methods (selective dissolution) Product design Systems with co-polymers and polymer blends Design of flexible polymer pipes carrying Gas solubilities and diffusivities in subsea oil polymers Finding alternative plasticizers to PVC Compatibility (miscibility) of PVC plasticizers Deposition of polymer thin films using rapid Polymer-SCF (SGE) expansion from supercritical solution (RESS) Separation of proteins via aqueous two-phase LLE of polymer-water-protein often in systems presence of electrolytes Application Solvent devolatilization after polymerisation Selection of solvents for paints and coatings

This table shows a variety of systems and types of phase equilibria, which are of interest in the many practical situations where polymer thermodynamics plays a key role. For this reason, many different models have been developed for polymer systems and often the situation may seem rather confusing to the practising engineer. Polymer solutions and blends are complex systems: frequent existence of liquid-liquid equilibria (UCST, LCST, closed loop, etc.), the significant effect of temperature and polymer molecular weight including polydispersity in phase equilibria, free-volume effects and other factors may cause difficulties. The choice of a suitable model will depend on the actual problem and depends, specifically on:

144 -

type of mixture (solution or blend, binary or multicomponent,...) type of phase equilibria (VLE, LLE, SLLE, gas solubility,...) conditions (temperature, pressure, concentration) type of calculations (accuracy, speed, yes/no answer or complete design,...)

This chapter focuses mostly on simple activity coefficient models for polymers, which can be applied to a wide range of applications. These are group-contribution based (UNITAC) models, which account for some special effects in polymer systems such as free-volume differences. Some few recommendations on the vast but a bit confusing literature on equations of state for polymers will be also provided at the end of the chapter. Since most models often perform better for VLE than for LLE, indirect techniques are widely applied e.g. for solvent selection. These are briefly summarized in the next section.

7.2 CHOICE OF SOLVENTS A summary of some rules of thumb for predicting polymer-solvent miscibility, with focus on the screening of solvents for polymers, is presented here. These rules are based on wellknown concepts of thermodynamics (activity coefficients, solubility parameters) and some specific ones to polymers (Flory-Huggins parameter). Then, a brief discussion of some of the concepts involved is included. It can be roughly said that a chemical (1) will be a good solvent for a specific polymer (2), or in other words the two compounds will be miscible if one (or more) of the following 'rules of thumb' are valid^'^: i.

If the polymer and the solvent have 'similar hydrogen bonding degrees:

leal

-69 <1.J

ii.

(1)

\cm^ J

If the polymer and the solvent have very different hydrogen bonding degrees: ^<3,^ - 3,2 y+{^^ - ^,2 f+(^*,

- ^*2)' ^ R

(2)

where R is the Hansen-solubility parameter sphere radius. iii.

Q.^ < 6 (the lower the infinite dilution activity coefficient of the solvent, the greater the solvency of a chemical). Values of the infinite dilution activity coefficient above 10 indicate non-solvency. In the intermediate region, it is difficult to conclude if the specific chemical is a solvent or a non-solvent.

145 iv.

Zu ^ 0,5 (the lower the Flory-Huggins parameter value, the greater the miscibility, or, in other words, the greater the solvent's capacity of a specific chemical). Values much above 0.5 indicate non-solvency.

7.2.1 The Rules of Thumb Based on Solubility Parameters They are widely used. The starting point (in their derivation / understanding) is the equation for the Gibbs Free-Energy of mixing: AG""'' =AH-TAS

(3)

A negative value implies that a solvent/polymer system forms a homogeneous solution ie. the two components are miscible. Since the contribution of the entropic term ( - TAS) is always negative, it is the heat of mixing term that determines the sign of the Gibbs energy. The heat of mixing can be estimated from various theories e.g. the Hildebrand regular solution theory for non-polars systems, which is based on the concept of the solubility parameter. For a binary solvent(l)/polymer(2) system, according to the regular solution theory: AR = ^,(p,V{S,-S,f

(4)

where ^^is the so-called volume fraction of component i. This is defined via the mole fractions Xj and the molar volumes Vi, as (binary systems): % =

"'^' ,,

(5)

According to Eq. 4, the heat of mixing is always positive. For some systems with specific interactions (hydrogen bonding) the heat of mixing can be negative and Eq. 4 does not hold. Thus, the regular solution theory is strictly valid for non-polar/slightly polar systems, without any specific interactions. According to Eqs. 3 and 4, if solvent and polymer have the same solubility parameters, the heat of mixing is zero and they are thus miscible at all proportions. The lower the solubility parameter difference the larger the tendency to be miscible. Many empirical rules of thumb have been proposed based on this observation. Seymour^ suggests that if the difference of solubility parameters is below 1.8 (cal/cm^)^'^^, Eq. 1, then polymer and solvent are miscible. Similar rules can be applied for mixed solvent - polymer systems, which are very important in many practical applications, e.g. in the paints and coatings industry and for the separation of biomolecules using aqueous two-phase systems. The solublity parameter of a mixed solvent is given by the equation: 8 = Z

(6)

146 Barton^""^ provides empirical methods based on solubility parameters for ternary solvent systems. Charles Hansen introduced the concept of three-dimensional solubility parameters, which offer an extension of the regular solution theory to polar and hydrogen bonding systems. Hansen observed that when the solubility parameter increments of the solvents and polymers are plotted in three-dimensional plots, then the 'good' solvents lie approximately within a sphere of radius R (with the polymer being in the centre). This can be mathematically expressed as: ^<3,r - ^,2 f + {^pl - S,2 f + (^M - ^,2 y ^R

(2)

where subscript 1 denotes the solvents and subscript 2 the polymer. The quantity under the square root is the distance between the solvent and the polymer. Hansen found empirically that a universal value 4 should be added as a factor in the dispersion term to approximately attain the shape of a sphere. This universal factor has been confirmed by many experiments. Several other two-dimensional plots have been proposed, which employ all three contributions e.g. 5^-5^,df^-5^,5^-5^ or even combined plots such as the use of 5^-5^,5^

-Jd\

+ (^^ plots suggested by van Krevelen^. With few exceptions good

solvents lie within the circle of radius R, which mathematically can be expressed as:

^{8,,-5j^{5,,~5,,J


(7)

The justification for this plot lies in the fact that, of the three solubility parameter increments, the dispersion one varies the least and, via this average way, it can be treated together with the polar increment. The hydrogen bonding increment is very important and it is thus accounted for separately in Eq. 7. The Hansen method is very valuable. It has found widespread use particularly in the paints and coatings industry, where the choice of solvents to meet economical, ecological and safety constraints is of critical importance^. It can explain some cases in which polymer and solvent solubility parameters are almost perfectly matched and yet the polymer won't dissolve. The Hansen method can also predict cases where two non-solvents can be mixed to form a solvent. Still, the method is approximate, it lacks the generality of a full thermodynamic model for assessing miscibility and requires some experimental measurements. The determination of R is typically based on visual observation of solubility (or not) of 0.5 g polymer in 5 cm"^ solvent at room temperature. Given the concentration and the temperature dependence of phase boundaries, such determination may seem a bit arbitrary. Still the method works out pretty well in practice, probably because the liquid-liquid boundaries for most polymer/solvent systems are fairly 'flat'. A recent review of the Hansen method with extensive tables of solubility parameters is available^.

147 7.2.2 The Rule of Thumb Based on the Infinite Dilution Activity Coefficient Since in several practical cases concerning polymer/solvent systems, the 'solvent' is only present in very small (trace) amounts, the so-called infinite dilution activity coefficients are of importance. On a molar and weight basis, they are defined as follows: Y7 = lm,,_^o 7r

nr = lim

'xj^

(8)

The weight-based infinite dilution activity coefficient, Q^^, which can be determined experimentally from chromatography, is a very useful quantity for determining good solvents. Low values (typically below 6) indicate good solvents, while high values (typically above 10) indicate poor solvents according to rules of thumb discussed by several investigators''"^. The derivation of this rule of thumb is based on the Flory-Huggins model, discussed in the next section, 7.2.3. This method for solvent selection is particularly useful because it avoids the need for direct liquid-liquid measurements and it makes use of the existing databases of solvent infinite dilution activity coefficients, which is quite large (e.g. the DECHEMA and DIPPR databases). Moreover, in the absence of experimental data, existing thermodynamic models (such as the Flory-Huggins, the Entropic-FV and the UNIFAC-FV discussed later, section 7.3) can be used to predict the infinite dilution activity coefficient. Since, in the typical case today, existing models perform much better for VLE and activity coefficient calculations than directly for LLE calculations, this method is quite valuable and successful, as shown by sample results in Table 2. The thermodynamic models UNIFAC-FV (U-FV), Entropic-FV (E-FV), UNIFAC and GCLF, shown in Table 2, are used for obtaining the infinite dilution activity coefficient for several PVC systems^^. We can see that, with few exceptions, these models in combination with the rule of thumb mentioned above can identify which chemicals are good solvents for PVC and which are not. Similar results have been presented elsewhere^^'^ for selecting solvents for paint polymers such as PBMA and PMMA. This rule of thumb makes use of either experimental or predicted, by a model, infinite dilution activity coefficients. However, the results depend not only on the accuracy of the model, but also on the rule of thumb, which in turns depends on the assumptions of the FloryHuggins approach. A thermodynamically more correct method is to employ the activity concentration (aw) diagram, as shown in figures 1 and 2. Results for the PVC systems are shown in Table 2. The two plots have been generated with UNIFAC-FV. The maximum indicates phase split, while a monotonic increase of activity with concentration indicates a single liquid phase (homogeneous solutions).

148 Table 2. Observed and predicted weight fraction activity coefficients at infinite dilution and ai(w), activity-weight fraction, for different PVC (Mw=50 000) - solvent systems at 298 K. Nr.

Chemicals

Experimental

(E-FV)

(U-FV)

(GCLF)

UNIFAC

s/ns Hansen 92

Fred^75

Hansen '91

Qi"

ai(w)

Qi""

ai(w)

Qi"'

ai(w)

Qi""

ai(w)

S

5.08

S

3.08

S

4.54

s

2.45

S

Ns

2.98

8

4.44

S

4.54

%

2.U

S

S

3.7*-)

S

4.12

4.5

s

2.64

-

s -

-

-

5.67

s s

6.46

s

9.47

1.02

3.78

s

19.08

L34

5.76

.s

14.5

40.62

1.18

8.15

S

40.62

1.18 1 1.05

1

Monochlorobenzene

2

Chloroform

3

Dichloromethane

4

Nitroethane

Ns

-

5

Ethyl acetate

S

6.15

6

1,4-Dioxane

S

12.2

s s

7

Isopropanol

Ns

48.41

1.17

8

Methanol

Ns

70.72

1.23

94.5

1.4

9.48

s

61.81

9

Benzyl alcohol

Ns

23.2

S

19.97

1.05

10.14

s

15.42

10 Tetrahydrofurane

S

-

-

-

-

6.67

-

7.32

11 Toluene

S

2.54

S

19.9

1.72

9,16

s

2.87

s

12 Benzene

S

4.64

.S

4.47

s

8,23

s

3

s

13 Acetone

Ns

6.44

S

9.5

s

8,45

s

6.02

s

14 MEK (a)

S

5.09

S

6.63

s

9,35

s

15 MEK (b)

s s

5.35

7.51

s

9.72

s

4.93

s s

6

s

11.78

1.05

-

-

1.04

4.49

s 1.01

-

16 Methyl Isobutyl Ketone (a) 17 Methyl Isobutyl Ketone (b) 18 Di-n-Propyl Ether

s

4.72

s

6.21

s

12.3

Ns

13.58

1.03

17.1

1.13

23.49

1.31

9.52

19

Ns

18.89"'"

L 2 4 ~ ^' 1 6 ' 2 7 '

1.08 "' "29.89

T43

ir.57"

n-Nonane

1.22

s 1 1.01

20 n-Octane

Ns

17-72

1.2

15.72

1.06

27.34

l.,36

10.72

1.01

21 n-Pentane

Ns

15.98

1.2

15.88

1.02

22.56

1.22

8.71

s

22 n-Heptane

Ns

16.72

1.16

15.31

1.05

25.24

1.3

9.96

s

23 o-Xylene

S

(1,26

1.17

5.J6

Jv

10.2

1.02

3.85

s

s

2.06

S

L58

s

6.22

s

2.01

s

Ns

3.67

S

8.65

1.02

12.78

1.08

4.08

s

26 Ethylene di-chloride

S

4.19

S

4.32

s

5.22

s

3.22

s

27 Amyl acetate

6.43

s

8.41

24

1,2-dichlorobenzene

25 Butyl acrylate

s

6.42

S

1.01

22

s

28 Carbon disulfide

Ns

-

-

-

4.6

s

7.14

s

29 Vinyl chloride

Ns

2.17

3.93

s

3.8

s

-

-

30 Acetonitrile

Ns

-

s -

-

-

-

-

10.51

s

4

6

3

4

12

12

5

11

Wrong answers

26 26 28 28 26 26 1 Total 26 26 The dark colour indicates the cases where the rule of thumb can be applied, while the light colour indicates cases where 6
149

o

~^—'—\—'—\—'—r 0.20

0.40

0.60

0.80

1 0.20

Solvent weight fraction

'

I 0.40

'

\

'

0.60

r 0.80

Solvent weight fraction

Figure 1. Solvent-activity diagram of n- Figure 2. Solvent-activity diagram of heptane and PVAC at 300 K with UNIFAC- chloroform and PBMA at 300 K with UNIFAC-FV. Monotonically increasing line FV. The maximum indicates a phase spHt. indicates solubility at all concentrations.

7.2.3 The Rule of Thumb Based on the Flory-Huggins Model The Flory-Huggins (FH) model for the activity coefficient, proposed in the early 40's by Flory and Huggins^^, is a famous Gibbs free energy expression for polymer solutions. For binary solvent-polymer solutions and assuming that the parameter of the model, the so-called FH interaction parameter Xn i^ constant, the activity coefficient is given by the equation

(9)

= l n ^ + 1 — \p2+Xn9l where (p, can be volume or segment fractions and r is the ratio of the polymer volume to the solvent volume V2/V1 (approximately equal to the degree of polymerization). Using standard thermodynamics and Eq. 9, it can be found that for high molecular weight polymer-solvent systems, the polymer critical concentration is close to zero and the interaction parameter has a value equal to 0.5. Thus, a good solvent (polymer soluble in the solvent at all proportions) is obtained if Xu - ^-^' while values greater than 0.5 indicate poor solvency. Since the Flory-Huggins model is only an approximate representation of the physical picture and particularly the FH parameter is often not a constant at all, this empirical

150 rule is certainly subject to some uncertainty. Nevertheless, it has found widespread use and its conclusions are often in good agreement with experiment. This can be demonstrated by a socially important example, the choice of suitable (miscible) plasticizers with PVC^^. Typical results are shown in Tables 3 and 4 and figure 3. The infinite dilution activity coefficients calculated by E-FV and U-FV are also shown in these Tables. Table 3. Classification of the solvent power of various plasticizers from different calculation (thermodynamic) methods and experiments (dilute solution viscosity, apparent melting temperature, equilibrium swelling). Plasticizer X

a

App. DiluteQi* (298K) soln. Melting (E-FV) viscosity temperature

a,*

Qi"(298K) (U-FV)

Equilibrium swelling at350K

7.46

DOS

6.43

6.68 7.35

(350K) (E-FV)

DOS

0.62 0.8

DOA

DOS

3.85

DOA

0.48 1.4

DOS

DOA

2.30

13.7

DOA

4.03

DOP

0.05 2.4

DOP

DOP

2.31

4.22

DOP

4.31

BBP

0.17 2.6

BBP

BBP

2568

BBP

DBP

(350K) (U-FV)

3.87 19.11

3.42

2.12

The a and x values are taken from Bigg^^ The a values are simply (l-x)/MW. (MW is the molecular weight).

Table 4. Classification of the solvent power of various phthalates from different calculation (thermodynamic) methods. Plasticizer

Dihexyl phthalate

Mw X(323K) a Qi"' (323K) (g/mol) (E-FV) 3.75 0.98 DDP 446.7 0.56 2.27 3.31 BMP 194.20 0.56 DEP 222.2 2.61 2.89 0.42 2.77 DHP 334.5 3.38

Dioctyl phthalate

DOP

390.9

Dibutyl phthalate

DBP

278.3

Didecyl phthalate Dimethyl phthalate Diethyl phthalate

0.01

2.53

3.14

The a and % values are taken from Doty and Zable'"^.

Qi"(323K) (U-FV) 4.64 6.96 4.82

3.98 3.70

151 0.6^ 0.5-

DMP

DBF

DEP

DHP

DOP

DDP

4

i\,.

3.5

0.40.3-

3 1

#'"'"^

""""•#

2.5

\

0.2-

2

4

\

0.1 -

a 3^ 'w'

1.5

0-

1

-0.1 -

0.5

•

1

-0.2194

222

1 278

1

334

1

391

1

447

Molecular weight [g/mol] ^

Doty and Zable (at 326K)

A

A n a g n o s t o p o u l o s (360-403K) — » - " " A l p h a values by Bigg

.,....:%,...„., Doty and Zable (at 349K)

Figure 3 Dependency of the experimentally determined FH interaction parameter values and Big^ ;'s alpha values on molecular weight of phthalates. The x values are taken from Barton'^, There are several, still rather obscure issues about the Flory-Huggins model, which we summarize here together with some recent developments: 1. There is no single rigorous widely-accepted extension of the FH model to multicomponent systems. Several extensions have been proposed, but (at least) one Xi2-value is required per binary. 2. It has been shown that, unfortimately, the FH parameter is typically not a constant and should be estimated from experimental data. Usually it varies with both temperature and concentration, which renders the FH model useful only for describing experimental data. It cannot be used for predicting phase equilibria for systems for which no data is available. Moreover, when fitted to the critical solution temperature, the FH model cannot yield a good representation of the whole shape of the miscibility curve with a single parameter. 3. Accurate representation of miscibility curves is possible with the FH model using suitable (rather complex) equations for the temperature and the concentrationdependence of the FH-parameter^^"^^. 4. In some cases, a reasonable value of the FH parameter can be estimated using solubility parameters via the equation: X,2=Xs+X/,=0.35 + - ^ ( 5 , - 5 , ) '

(10)

152 Eq. 10, without the empirical 0.35 factor, is derived from the regular solution theory. The constant 0.35 is added for correcting for the deficiencies of the FH combinatorial term. These deficiencies become evident when comparing experimental data for athermal polymer and other asymmetric solutions to the results obtained with the FH model. A consistent underestimation of the data is observed, as discussed extensively in the literature^^, which is often attributed to the inability of the FH model in accounting for the free-volume differences between polymers and solvents or between compounds differing significantly in size such as n-alkanes with very different chain lengths. The term, which contains the "0.35 factor", corrects in an empirical way for these free-volume effects. However, and although satisfactory results are obtained in some cases, we cannot generally recommend Eq. 10 for estimating the FH parameter. Moreover, for many non-polar systems with compounds having similar solubility parameters, the empirical factor 0.35 should be dropped. Recently, Lindvig et al}^ proposed an extension of the Flory-Huggins equation using the Hansen solubility parameters for estimating activity coefficients of complex polymer solutions. 2 Inyi : l n ^ + l- 9l + X1292 ^1

(11) Xi2=0.6

RT

i(5dl -^62?

+0.25(5pi - 5 p 2 F +0.25(5^ - 6 ^ 2 ) ^

In order to achieve that, Lindvig et al.^^, as shown in Eq. 11, have employed a universal correction parameter, which has been estimated from a large number of polymer-solvent VLE data. Very good results are obtained, especially when the volume-based combinatorial term of FH is employed, as summarized in Table 5. Table 5. Average absolute deviations between experimental and calculated activity coefficients of paint-related polymer solutions using the Flory-Huggins/Hansen method and three group contribution models. From Lindvig et al}^. The second column presents the systems used for optimization of the universal parameter (358 points of solutions containing acrylates and acetates). The last two columns show predictions for two epoxy resins. Model FH/Hansen, Volume (Eq.ll) FH/Hansen Segment FH/Hansen Free-Volume Entropic-FV UNIFAC-FV GC-Flory

% AAD (systems in database) 22

% AAD Araldit 488

% AAD Eponol-55

31

28

25

—

—

26

—

—

35 39 18

34 119 29

30 62 37

153 6. Based on the Flory-Huggins model, several techniques have been proposed for interpreting and for correlating experimental data for polymer systems e.g. the socalled Schultz-Flory (SF) plot. Schultz and Flory^^ have developed, starting from the Flory-Huggins model, the following expression, which relates the critical solution temperature (CST), with the theta temperature and the polymer molecular weight: 1 CST

1 0

ui^'J_ J_'^

(12)

V

where \\f\= — Xs is the entropic parameter of the FH model (Eq.lO) and r is the ratio of molar volumes of the polymer to the solvent. This parameter is evidently dependent on the polymer's molecular weight. The SF plot can be used for correlating data of critical solution temperatures for the same polymer/solvent system, but at different polymer molecular weights. This can be done, as anticipated from Eq.l2 because the plot of 1/CST against the quantity in parentheses in Eq. 12 is linear. The SF plot can also be used for predicting CST for the same system but at different molecular weights than those used for correlation as well as for calculating the theta temperature and the entropic part of the FH parameter. It can be used for correlating CST/molecular weight data for both the UCST and LCST areas. Apparently different coefficients are needed.

7.3 THE FREE-VOLUME ACTIVITY COEFFICIENT MODELS 7.3.1 The Free-Volume Concept The Flory-Huggins model provides a first approximation for polymer solutions. Both the combinatorial and the energetic terms need substantial improvement. Many authors have replaced the random van-Laar energetic term by a non-random local-composition term such as those of the UNIQUAC, NRTL and UNIFAC models. The combinatorial term should be extended/modified to account for the free-volume differences between solvents and polymers. The improvement of the energetic term of FH equation is important. Local-composition terms like those appearing in NRTL, UNIQUAC and UNIFAC provide a flexibility, which cannot be accounted for by the single-parameter van Laar term of Flory-Huggins. However, the highly pronounced free-volume effects should always be accounted for in polymer solutions. The concept of free-volume (FV) is rather loose, but still very important. Elbro^ demonstrated, using a simple definition for the free-volume (Eq.l3), that the FV percentages of solvents and polymers are different. In the typical case, the FV percentage of solvents is greater (40-50%) than that of polymers (30-40%). There are two exceptions to this rule; water and PDMS: water has lower free-volume than other solvents and closer to that of most of the polymers, while PDMS has quite a higher free-volume percentage, closer to that of most

154 solvents. LCST is, as expected, related to the free-volume differences between polymers and solvents. As shown by Elbro^ the larger the free volume differences the lower the LCST value (the larger the area of immiscibility). For this reason, PDMS solutions have a LCST, which are located at very high temperatures. Many mathematical expressions have been proposed for the FV. One of the simplest and successful equations is^^'^^:

v^^v-v'=v-v^

(13)

originally proposed by Bondi^^ and later adopted by Elbro et al.^^ and Kontogeorgis et al.^^ in the so-called Entropic-FV model (described in 7.3.3). According to this equation, FV is just the 'empty' volume available to the molecule when the molecules' own (hard-core or closed-packed V*) volume is substracted. The free-volume is not the only concept, which is loosely defined in this discussion. Even the hard-core volume is a quantity difficult to define and various approximations are available. Elbro et al?^ suggested using V*=Vw, i.e. equal to the van der Waals volume (Vw), which is obtained from the group increments of Bondi and is tabulated for almost all existing groups in the UNIFAC tables. Other investigators^^'^"^ interpreted somewhat differently the physical meaning of the hard-core volume in the development of improved free-volume expressions for polymer solutions, which employ Eq. 13 as basis, but with V* values higher than Vw Table 6 shows that, due to the closed packed structure of molecules, a higher value of the hard-core volume would have been expected e.g. around 1.2-1.3 Vw. Indeed, investigations for athermal polymer systems (without any energetic interactions) demonstrate that the optimum resuhs with Entropic-FV (discussed below) and for both the solvent and polymer activities are obtained when V*=1.2Vw (Fig.4). This observation regarding the magnitude of hard-core volume related to Vw has helped not only in developments of the Entropic-FV model, but as shown in Figure 5, also in understanding problems of hard-core volume theories such as the one proposed by Guggenheim. This particular hard-core volume theory has been often used in models for estimating diffusion coefficients for polymeric systems. Table 6. Values of Packing Density and of the ratio fluids, as well as for various fluid families^^. Structure/Compound Open-packed cubic structure of spheres Closed-packed cubic structure of spheres Open-packed arays of infinite cylinders Close-packed arays of infinite cylinders Polyethylene Most organic compounds Random densely packed mixture of spheres with log normal size distribution

VVV^^VQA^W

Packing Density (po)

for various packing of

V*A^w=VoA^w

0.52

1.92

0.74

J 1.35

0.78 0.90 0.76 0.7....0.78 up to 0,8

1.27 1.11 1.31 1.43 ....1.28 down to 1.25

155

alkanes in PE alkanes in PIB

-alkanes in PDMS, PS, R/AC -n-short chain in long chain alkanes

1.05

1.1

1.15

1.2

1.25

1.3

Value of a-constant

1.35

1.4

1.45

. branched,cyclic short chain in long chain alkanes

Figure 4. Percentage deviation between experimental and calculated solvent infinite dilution activity coefficients, versus the a-parameter in the free-volume expression of the Entropic-FV model (Vf=V-aVw). From Kouskoumvekaki etalP.

Figure 5. Plot of the ratio V /Vw calculated from Guggenheim's hard-core volume equation (V^"^ = 0.286\4) as a function of the van der Waals volume Vw for n-alkanes. The critical volume (Vc) is obtained from two different sets of experimental data, those by Teja and those by Steele. Only those data by Teja have been verified by independent investigations based on molecular simulation. The plot shows that, using these "correct" data, the vVVw plot does not follow the physically expected trend. (Modified from Kontogeorgis et al}"^). The original UNIFAC model does not account for the free-volume differences between solvents and polymers and, as a consequence of that, it highly underestimates the solvent activities in polymer solutions,21,22,25 On the other hand, the various modified UNIFAC

156 versions (Lyngby and Dortmund, see chapter 4), which use exponential segment fractions, are also inadequate for polymer solutions. Although, their combinatorial terms are more satisfactory for alkane systems, they fail completely for polymer-solvent systems and as shown they significantly and systematically overestimate the solvent activities. Although these UNIFAC models are not adequate for polymer solutions, the problem seems, however, to lie more in the combinatorial term rather than the residual (energetic) term. In other words, improvements are required especially for describing the free-volume effects, which are dominant in polymer solutions. 7.3.2 The UNIFAC - FV model Various modifications - extensions of the classical UNIFAC approach to polymers have been proposed. All these approaches attempt to include the FV effects, which are neglected in the UNIFAC combinatorial term. All of them employ the energetic (residual) term of UNIFAC. The most well-known is the UNIFAC-FV model by Oishi and Prausnitz^^: Iny,. =lny^''"'+lnyf'+lny;'

(14)

The combinatorial and residual terms are obtained from original UNIFAC. An additional term is added for the free-volume effects. An approximation but at the same time an interesting feature of UNIFAC-FV, and other models of this type, is that the same UNIFAC group-interaction parameters - i.e. those of original UNIFAC- are used. No parameter reestimation is performed. The FV term used in UNIFAC-FV has a theoretical origin and is based on the Flory equation of state: Iny^" -3c,. In

\Vi

-1

Vi

1

(15)

V^i J where the reduced volumes are defined as: Vi =-

bV,^ (16)

Vm =

b{wy,^^+w,V,^^)

In Eq.l6, the volumes Vi and the van der Waals volumes are all expressed in cm /mol. Wi is the weight fraction. In the UNIFAC-FV model as suggested by Oishi and Prausnitz^^ the parameters ci (3ci is the number of external degrees of freedom) and b are set to constant values for all polymers and solvents (ci=l.l and b=1.28). The performance is rather satisfactory, as shown by many investigators, for a large variety of polymer-solvent systems. Some researchers have suggested that, in some cases, better agreement is obtained when these parameters are fitted to experimental data^^.

157 Originally, the UNIFAC-FV model was developed for solvent activities in polymers. It could be expected that the model (Eqs.14-16) is also valid for estimating polymer activities. However, such an application of UNIFAC-FV is rather problematic^^. It has been shown^^ that the performance of UNIFAC-FV in predicting the activities of heavy alkanes in shorter ones is not very good. Such problems limit the applicability of UNIFAC-FV to cases where the polymer activity is also of importance such as liquid-liquid equilibria for polymer solutions. Indeed, to our knowledge, UNIFAC-FV has not been applied to polymer-solvent LLE. 73.3 The Entropic-FV model A similar but somewhat simpler approach to UNIFAC-FV, which can be readily extended to multicomponent systems and liquid-liquid equilibria, is the so-called Entropic-FV model proposed by Elbro et aL^^ and Kontogeorgis et al?^\ Iny/ = lnyf°'"*-'^+ln yres ^

fv oomb-fv ^^%_^ '

1 - ^

Xi

Xi

(17) XiVi,fv

m^

X,>{Vi-V^i)

9/

i Inyf^ -» UNIFAC

j

^M^v~) (chapA)

As can been seen from Eq. 17, the free-volume definition given by Eq.l3, is employed. The combinatorial term of Eq. 17 is very similar to that of Flory-Huggins. However, instead of volume or segment fractions, free-volume fractions are used. In this way, both combinatorial and free-volume effects are combined into a single expression. The combinatorial - FV expression of the Entropic-FV model is derived from Statistical Mechanics, using a suitable form of the generalised van der Waals partition function. The residual term of Entropic-FV is taken by the so-called 'new or linear UNIFAC model, which uses a linear-dependent parameter table^^: ^r.n=^mn,i+^^n,20'-To)

(18)

This parameter table has been developed using the combinatorial term of the original UNIFAC model. As with UNIFAC-FV, no parameter re-estimation has been performed. The same group parameters are used in the "linear-UNIFAC" and in the Entropic-FV models. A common feature for both UNIFAC-FV and Entropic-FV is that they require the volumes of solvents and polymers (at the different temperatures where application is required). This can be a problem in those cases where the densities are not available experimentally and have to be estimated using a predictive group-contribution or other

158 method, e.g. GCVOL ' or van Krevelen methods. These two estimation methods perform quite well and often similarly even for low molecular weight compounds or oligomers such as plasticizers, as shown in figure 6 for the family of phthalates. Both UNIFAC-FV and Entropic-FV, especially the former, are rather sensitive to the density values used for the calculations of solvent activities.

1.1 g 1.05 CM

%

1

f <

0.95

^

0.9

I

0.85

A

% 1

>

0.8

1 1—-—1 DMP

DEP

DBF

DHP

DOP

DDP

Phthalates ^ GC-VOL m. van Krevelen k Experinnental

Figure 6. Volumes of different phthalates calculated by GC-VOL and van Krevelen, compared to the experimental volumes taken from Ellington^^ at 293.15 K (From Tihic^°)

7.3.4 Results and Discussion The UNIFAC-FV and Entropic-FV models have been widely applied to polymer solutions and some typical applications are shown in tables 7-9, figures 7-12 and discussed in this section, and figures 1-2 and tables 2-4 in the previous sections. Some of the most recent applications are included, while attention is paid to the advantages and shortcomings of the models. In some cases, comparisons with two group-contribution equations of state (GCFlory and GCLF) are presented. Vapor-Liquid Equilibria Both models have been extensively applied to vapour-liquid equilibria (VLE) - solvent activities in polymers and other size-asymmetric systems, including infinite dilution 22,27,33 VLE for co-polymer/solvent systems , solvent conditions for binary polymer solutions' activities in dendrimer solutions , VLE for a large variety of polar and hydrogen-bonding systems ' , VLE for paint-related polymer solutions including commercial epoxy resins ' , and recently also for VLE of ternary polymer-mixed solvent systems^ ^

159 Table 7 shows the performance of,the models for infinite dilution activity coefficients for some polyisoprene (PIP) solutions, while comparisons for complex systems (from a recent comparative study, Lindvig et al.^^) are shown in Table 8. Finally, results for some ternary polymer-mixed solvent systems^^ are shown in Table 9. Table 7. Prediction of infinite dilution activity coefficients for PIP systems with three predictive group contribution models. Experimental values and calculations are at 328.2 K

PIP systems +acetonitrile +acetic acid +cyclohexanone +acetone +MEK +benzene +1,2 dichloroethane +CCI4 +l,4dioxane +tetrahydrofurane +ethylacetate +n-hexane +chloroform

Exper. value 68.6 37.9 7.32 17.3 11.4 4.37 4.25 1.77 6.08 4.38 7.47 6.36 2.13

GC-Flory Ng 50.0 (32%) Ng 10.5 (39%) 7.5 (35%) 2.8 (37%) 6.6 (55%) 1.6(11% Ng Ng 4.4 (41%) 3.8 (39%) 2.6 (24 %)

Entropic-FV 47.7(31%) 33.5 (12%) 5.4 (27%) 15.9 (8%) 12.1 (6%) 4.5 (2.5%) 5.5 (29%) 2.1(20%) 6.3 (4%) 4.9 (14%) 7.3 (2 %) 5.1 (20%) 3.00(41%)

UNIFAC-FV 52.3 (24%) 17.7 (53%) 4.6 (38%) 13.4(23%) 10.1 (12%) 4.4 (0 %) 6.5 (54%) 1.8(0%) 5.9 (2%) 3.9 (10%) 6.6(11%) 4.6 (27%) 2.6 (20%)

ng: no groups available Table 8. Percentage deviation between calculated and experimental solvent activity coefficients from various thermodynamic models^^

FdarsoK^rts

NTHxIarsohals PCLMVHR

FO. FDVB

FBIE FB3 FM*\ FPCTE

FRD FVA) RA/E A^.%cla/.

ffV 24 169 53 53 183 462 24 35 7.4

UFV 9.9 172 25 63 183 466 23 40 Q8

QCF UNFA: 14.1 224 ***** 178 416 31 44 11.3 25 224 433 47.7 72 1.0 63 131 56 96

137

141

72

163

EV

h^clogen bcrdrg achats

UFV QCF UNR^C B ^

***** ***** ***** ***** ***** *****

UFV QCF UNFA:

***** ***** ***** ***** ***** ***** ***** ***** ***** ***** 628 358 121 450

***** ***** ***** *****

***** ***** ***** *****

***** ***** ***** *****

***** ***** ***** *****

***** ***** ***** *****

28 64

1.4 86

154 458 76

es2

3248

2^2 76

337 88

51.1

4^.3

34.7

36 366 222

218 190

31 240

11.7

11.0

134 96

***** ***** *****

*****

21.7

51.5

27.9

35

31

28

23

73 ^ 42

160 Table 9a: Average logarithmic deviations (xlOO) between experimental and predicted vapor phase mole fractions for some ternary polymer- mixed solvent systems. The color indications divide the deviations into the following groups: Grey: less than 20 %. Light grey: 20 - 50 %. Dark grey: above 50 %. Polymer

PS, r = 373.15 K PS, r - 3 9 3 . 1 5 K PS, r = 413.15 K

Solvent

Chloroform Carbontetrachloride Styrene Ethylbenzene Styrene Ethylbenzene St}Tene Ethylbenzene

SAFT EFV/UQ

FH

Pa-Ve

EFV

UFV

GCLF

FH/Ha

[45

37

i f

22 36

23 37

33 37

Table 9b: Average logarithmic deviations (xlOO) between experimental and predicted pressures for some ternary polymer-mixed solvent systems. Color indications as in Table 9a. SystOTL

PMVlA-butanQne-toluerie PS-benzene-toluene PS-toluene-ethylbenzme PS-toliene-c^lohexarK PS-chlQFofom>carbQiletradiloride PSnStyrai^^th^bQTz^^, r=373.15K PS-styrme-etii^bmzme, T=393.15 K PS^styr^ie-dir^bamie, r=413.15K

SAFT EFV/Uq 16 4 7 \\ 12 2 14 16 32 32 32

FH 15 12

P^-Ve 16 7

14 11

8 4

EFV 16 S 2 2 17 31 30 28

UFV GCLF FHffi 36 15 17 14 14 7 8 2 I 2 2 U 19 5 52 34 31 33 35 30 33 34 26 32

Figures 7 and 8 also show recent results^^, demonstrating the performance of the models for dendrimer solutions, including the sensitivity of the calculations to the density value employed.

161

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

weight fraction of solvent

Figure 7. Experimental and predicted activities of methanol in the dendrimer PANAM-G2 with the UNIFAC-FV and the Entropic-FV models' Results are shown using experimental and predicted densities (Reprinted with permission).

"- ENTROPIC-FV (d:precl) — UNIFAC-FV (d:pred)| A

EXPERIMENTAL DATA ENTROPIC-FV (d=1) -----UNIFAC-FV (d=1)

0.02

0.04

0.06

0.08

0.1

0.12

weight fraction of solvent

Figure 8. Experimental and predicted activities of acetone in the dendrimer A4 with the ,35 UNIFAC-FV and Entropic-FV models . Results are shown using experimental and predicted densities (Reprinted with permission). Overall, we can conclude that the Entropic-FV and UNIFAC-FV models, especially the former, provide satisfactory predictions of solvent activities, even at infinite dilution, for complex polar and hydrogen bonding systems including solutions of interest to paints and coatings, and rather satisfactory predictions when mixed solvents are present. Some more specific comments can be made from comparative investigations for different types of systems.

162 Athermal systems The articles cited in this section include investigations, which compare the performance of the UNIFAC-FV, Entropic-FV and several more recent free-volume equations for athermal systems. In such systems the energetic effects are zero or very small and they can be thus used for testing the combinatorial and free-volume terms of the models. Although the database used in the various investigations is not always the same, it typically consists of solutions having components differing significantly in size but which do not exhibit energetic interactions. Examples of these nearly athermal systems are solutions of polyethylene and polyisobutylene with alkanes (only solvent activities are available), alkane solutions (where both the activity of light and heavy-chain alkanes are available), polystyrene/ethylbenzene, polyvinyl acetate/vinyl acetate as well as "pseudo" experimental data for polymer activities generated with molecular simulation techniques. In their recent review, Pappa et alP, considered over 200 experimental datapoints for athermal polymer solutions at intermediate concentrations and about 100 points at infinite dilution and compared the Entropic-FV and Zhong-Masuoka^^ models. They found that the Entropic-FV formula yields lower error than the Zhong term, though the latter does not contain any volume terms (9% vs. 16%). Other literature comparisons'^^ also agree that the free-volume models with the volume-containing terms perform better than those models requiring no volume information. Thus, Entropic-FV, Flory-FV and related models provide a good basis for building a full thermodynamic model for polymers. UNIFAC-FV seems to offer no advantage over the simpler approaches and seem to be more sensitive^ ^'^^'^'^ to the volume values employed compared to simpler free-volume equations. Despite the overall successful performance of Entropic-FV and UNIFAC-FV models for a large number of systems and types of phase equilibria, it has been shown over the last years by a number of researchers^^''^^"'^'^, that the combinatorial/fi'ee-volume terms of both the Entropic-FV and UNIFAC-FV models have a number of deficiencies: i.

ii.

iii.

The solvent activities in athermal polymer solutions are systematically underestimated by, often, 10% (in the case of Entropic-FV) or more (for UNIFACFV). For athermal systems, the residual term is zero. Such an underestimation cannot be entirely attributed to the small interaction effects present in such systems. The activities of heavy alkanes in short-chain ones, available from SLE measurements, are in significant error, especially as the size difference increases. Due to the lack of experimental data on polymer activities, such SLE data can help test the models' applicability for the activities of heavy-compounds, The performance of the models is rather sensitive to the values used for the polymer density.

Numerous investigations and developments of new combinatorial/free-volume terms have been reported over the last 5 years. The general conclusions that can be drawn are: i.

The activities of alkane solvents in either alkane or athermal polymer (PE, PIB) solutions are very satisfactorily predicted (much better than with the Entropic-FV formula) by some more recent modified free-volume equations e.g. Chain-FV, pFV and R-UNIFAC. However, these models carmot be extended to multicomponent

163

ii. iii.

iv.

systems. This is a serious limitation for multicomponent systems. The Flory-FV and a recently developed model^^ do not suffer from this limitation. Volume-based models perform better than those not including volume-containing terms. The UNIFAC-FV expression, the first free-volume equation proposed, which is derived from the theory of Flory, is not as successful for athermal systems compared to more recent simpler equations. This may be due to the values of the parameters b and c employed in this model. Fitting these parameters may improve the performance of the UNIFAC-FV term. The results with this model seem particularly sensitive to the density values employed. All models perform clearly less satisfactorily for the activities of heavy alkanes in short-chain ones, especially as the size-asymmetry increases. Models without freevolume corrections such as UNIFAC, and Flory-Huggins are particularly poor in these cases. Unfortunately, such activity coefficient measurements, which could bee used for testing the performance of the models for the activities of polymers, are scarce. Direct measurements for polymer activities have not been reported. Molecular simulation studies can offer help in this direction'^^

Non-polar and slightly polar systems Numerous results (predictions and correlations) are available for such systems^^'^"^'"^^""^"^. Many models perform satisfactory even when pure predictions are considered. In a recent comparison, Pappa et al.^^ showed that Entropic-FV performs better than the Zhong-Masuoka model (11% vs. 20%). In the review by Lee and Danner'^'^, GCLF (Group-Contribution Lattice Fluid Equation of State) and Entropic-FV perform similarly for non-polar systems (15%), but GCLF appears to perform better for the weakly polar ones. This is attributed to problems of the Entropic-FV model for systems containing polyacrylates and polymethacrylates with acetates. UNIFAC-FV has an average error of 23% for these types of systems and GC-Flory of 20%. Water-soluble polymers and other hydrogen-bonding systems Predictions have been provided for some hydrogen bonding systems with a number of models. Pappa et alP report an average deviation of 26%) (in VLE) with both Entropic-FV and Zhong-Masuoka models, which is higher than the deviations observed for non-polar and polar systems. Lee and Banner's"^"^ comparison revealed that Entropic-FV is the best model for strongly polar solvents (23%), followed by GCLF (28%) and GC-Flory (31%). UNIFACFV does not seem to be very successful for such complex systems (mean deviation 65%). In some recent investigations^^'^^, several of these well-known group contribution models (Entropic-FV, UNIFAC-FV, GC-Flory) have been tested for VLE of paint-related systems. These are systems of polyacetates, polyacrylates, polymethacrylates, epoxies and a variety of solvents (non-polar, polar, hydrogen bonding, water). The performance of the models is overall similar with the Entropic-FV and GC-Flory being overall better than UNIFAC-FV in most situations, in agreement to the investigations reported earlier. Some results are shown in Table 8.

164 Co-polymer solutions Comparisons for co-polymer systems are not extensive, although several VLE data for solvent/co-polymers are available. Bogdanic and Fredenslund^"^, Pappa et al^^ and Lee and Danner"^"^ have presented comparisons for such systems, using the models Entropic-FV, GCFlory, Zhong-Masuoka, UNIFAC-FV and GCLF. In their comparison, Pappa et alP found that both Entropic-FV and Zhong-Masuoka models perform similarly for these systems with a deviation around 20%. Similar overall performance for the Entropic-FV, GC-Flory and UNIFAC-FV models was observed by Bogdanic and Fredenslund^"^, though the various models perform different for specific co-polymer systems. For example, the Entropic-FV model has problems in the presence of chloro-groups. GCLF is also shown to be quite successful for a number of co-polymer solutions in mostly non-polar/slightly polar solvents. Polymer-mixed solvent systems In a recent investigation^^, a database for ternary VLE systems (polymer-mixed solvents) has been compiled and used for evaluating the performance of several group contribution models (Entropic-FV, UNIFAC-FV, and GCLF). The performance of these predictive models, though inferior to the binary systems, can be considered quite satisfactory, considering also the experimental uncertainties involved in these measurements. The experimental measurements of solvent activities in mixed solvent/polymer systems are not easy and may be often associated with significant errors (J.M.Prausnitz, 2000. Personal Communication). Liquid-Liquid and Solid-Liquid Equilibria The application of free-volume models to liquid-liquid and solid-liquid equilibria of polymer solutions is much more limited compared to VLE, and only Entropic-FV has been widely used in such cases, and for both polymer solutions and blends. Entropic-FV has been applied to SLE of asymmetric alkane solutions"^^, LLE for binary polymer solutions"^^ and polymer blends"^^, SLLE for semicrystalline polymer/solvents'^^, as well as for LLE for ternary polymer-solvent-solvent (and solvent-antisolvent) systems'^ . Figures 9-12 show typical LLE results for binary and ternary polymer-solvent solutions with Entropic-FV. Entropic-FV can be readily applied to liquid-liquid and solid-liquid equilibria and can predict all types of phase diagrams present in polymeric systems (UCST, LCST, hourglass-type) e.g. the results in figure 9 for PS/acetone. However, the results are of qualitative than of quantitative value in most cases. A difference of 10-30 degrees should be expected in the predictions. The performance of Entropic-FV seems rather system-specific, e.g. for polyethylene/octylphenols the difference in UCST is 5-10 "^C while for polyethylene/octanol it is approximately 40 °C. In the cases of polymer blends"^^, where freevolume effects are not very important, the model deviates substantially from experimental data, although it can predict the UCST-type behavior. Compared to the other free-volume models, Entropic-FV may be considered as the most successful and widely used extension of UNIFAC to polymers. Besides binary polymer-solvent LLE, Entropic-FV has been also applied to ternary polymer-solvent LLE"^^, and compared to the Holten-Andersen et al. equation of state (a

165 previous version of the GC-Flory EoS). The polymer/mixed solvent systems considered include both two solvent-solvent and solvent-anti-solvent systems. The comparison was limited to eight PS-two solvent systems (benzene/acetone, benzene/methanol, methylcyclohexane/acetone, toluene/acetone, MEK/acetone, ethyl acetate/acetone, NNDMF/cyclohexane) and one PMMA system (with chlorobutane/butanol-2) for which full data are available. Qualitatively good results are obtained with both models, especially Entropic-FV (despite the fact that all group interaction parameters were based on low pressure VLE of non-polymeric systems). Typical results are shown in figures 11 and 12. Finally, in one of the very few works reported on the prediction of solid-liquid-liquid equilibria"^^ for polymer solutions, the Entropic-FV and UNIFAC models have been shown to yield similar results for SLLE.

600.00

^^500.00 \

n D n n •

n

' ooooooooo

o

0) 400.00 g 300.00 0)

• n • •

200.00

•

CU nnnnn 4800 OOOOO 10300 AAAAA 19800

100.00 U.UU

exp. data exp. data exp. data

~\ I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I r

0.00

0.20

0.40

0.60

0.80

Polymer weight fraction Figure 9. Correlation of PS/acetone LLE with Entropic/FV model (using GCVOL for the density of the polymer/^ The numbers (4800, 10300, 19800) correspond to the molecular weight of the polymer. (Reprinted with permission)

166 320.00 -1

280.00 d

^240.00 H U^ 200.00

OOOOO Experimental data Entropic-FV new UNIFAC original UNIFAC

160.00

120.00

I I I M I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I M I I I I I I I I I I I I I I I I M I I I I I

0

30000 60000 90000 120000 Polymer m o l e c u l a r weight

Figure 10. PS/cyclohexane LLE prediction with various predictive group contribution models"^^ (Reprinted with permission). benzene

\\ '^\

nnethanol

Entropic-FV model HA model OOOOO experimental data

PS

Figure 11. Ternary LLE for PS(300000)/benzene/methanol at T=298.15 K^^ (Reprinted with permission)

167 methyl cyclohexane

ooooo experimental data Entropic-FV model HA model

acetone

PS

Figure 12. Ternary LLE for PS(300000)/methyl cyclohexane/acetone, T-298.15 K!^\ (Reprinted with permission) 7.3.5 The Entropic-FV/UNIQUAC model Both UNIFAC-FV and Entropic-FV are group contribution models. This renders the models truly predictive, but at the same time with very little flexibility if the performance of the models for specific cases is not satisfactory. An interesting alternative approach is to employ the UNIQUAC expression for the residual term. This Entropic-FV/UNIQUAC model has been originally suggested by Elbro et a/.^'^^'^^ and has shown to give very good results for polymer solutions if the parameters are obtained from VLE data between the solvent and the low molecular weight monomer (or the polymer's repeating unit). The Entropic-FV/UNIQUAC model has been recently further developed and extended independently by two research groups'^^"^\ Both VLE and LLE equilibria are considered but the emphasis is given to LLE. Very satisfactory results are obtained as can be seen for two typical systems in figures 13 and 14. It has been demonstrated that the EntropicFV/UNIQUAC approach can correlate both UCST/LCST and closed loop behavior^^'^^ and even show the pressure dependency of critical solution temperatures (UCST and LCST)^\ 7.3.6 Extension to Semi-crystalline Polymers and Swelling When highly crystalline or cross-linked polymers are considered, e.g paints after drying, rubbers, polyolefms, the effects of cross-linking and crystallinity should be considered because they affect the solubility. Cross-linldng and crystallinity are often visualized as 'similar' (in some sense) phenomena and are described with the same theories: crystalline regions are assumed to act as 'physical or giant cross-links'.

168 530

0.00

0.02

0.04

0.06

0.08

Weight fraction of polymer

Figure 13. Correlation of LLE for PBMA/MEK system^^ correlation

Exp.data (Mw=200000 g/mol);

410-

400-

370-

O.CO

0.05

010

015

0.20

Weight fraction of polymer

Figure 14. Correlation and prediction of LLE for HDPE/1-dodecanol system"^" A Exp.data (Mw=60700 g/mol), correlation o Exp.data (Mw=77800 g/mol), prediction • Exp.data (Mw=21300 g/mol), prediction • Exp.data (Mw=6800 g/mol), prediction

169 Crystalline and cross-linked polymers do not dissolve (with a few exceptions) in solvents but only swell. Swelling equilibria is thus important. To account for the crystalline/cross-linking effect, an additional factor (elastic term) is typically required in thermodynamic models. Two popular theories to account for this effect are the Flory-Rechner^^: lnaf=£^(pf

(19)

where: pg is the density of the amorphous polymer Vi is the molar volume of the solvent Mc is the molecular weight between cross-links and the Michaels-Hausslein^^ equation: 1

Inaf1 = I

R

AHip,

T

T„

•
_

[3/(2%)-l]

(20)

where: Tm is the melting point temperature f is the fraction of elastically effective chains in amorphous regions As observed from these equations, both theories introduce at least one extra parameter, which needs to be determined from experimental data: the molecular weight between crosslinks Mc and the fraction of elastically effective chains f. They have been combined with free-volume models^'^"^'^^ and they have been applied to semicrystalline polymer/solvent systems. The results are satisfactory but they are not predictive: the Mc and f parameters should be estimated from experimental data. However, the swelling of cross-linked polymers can be estimated with such equations. 7.3.7 Extension of Free-Volume Models to Gas Solubilities in Elastomers Thorlaksen et al.^^ have recently combined the Entropic-FV term with Hildebrand's regular solution theory and developed a model for estimating gas solubilities in elastomers. The socalled Hildebrand - Entropic FV model is given by the equation: lnri=lnrf

+lnrf^^^

lnr^-^'=ln^^^l-^^

(21)

(22)

170 where : S] = solvent solubility parameter, 82= gas solubility parameter X2 = gas mole fraction in liquid/polymer. O2 is the 'apparent' volume fraction of solvent, given by:

and 02^^ is the 'free-volume' fraction given by:

V^ is a hypothetical liquid volume of the (gaseous) solute.

— = —--exp

''

(23)

f

/ 2 is the fugacity of the gas and f[ is the fugacity of the hypothetical liquid, which can be estimated from the equation: ln^

= 3.54811 - ^' ^"^^"^^ + 7.60151 • T^. - 0.87466 - 7/ + 0J0971 - T/

•'- c

(24)

r

Finally, the gas solubility in the polymer is estimated from the equation: — • exp\ X

/

2

(25)

\

Calculations showed that the hypothetical gas "liquid" volumes are largely independent to the polymer used, and moreover, for many gases (H2O, O2, N2, CO2 and C2H2) these are related to the critical volume of the gas by the equation: V2^=1.776V^-86.017 Very satisfactory results are obtained as shown in Table 10 and Figure 15.

(26)

171 Table 10. Summary of the performance of the models tested at T = 298 K; P = 101.3 kPa Errors associated with models for predicting gas solubilities in polymers^^.

Hildebrand-

Hildebrand-

Entropic FV

EntropicFV-

-1

2

Hildebrand/ Polymer

PIP

PIB

Gas

Michaels/Bixler Tseng/Lloyd

14.7

73

3.9

-7.9

-4.6

O2

-16.1

-4

14

10.8

11.8

Ar

-32.5

-23

-

29.4

-22.2

CO7

-3.2

13

4.5

8.7

4.6

N2

-2.5

-

6.8

3.1

5.0

O2

-6.1

-

-1.7

-8.3

1.7

-

-

-

CO2

32.8

-

-1.9

41.1

35.2

N2

22.3

-

8.1

8.1

12.6

N2

Ar PBD

PDMB

PCP

O2

14.9

-

-6

8.7

10.8

Ar

12.1

-

-

111.1

24.0

CO9

-9.7

-

-4.6

0.4

-4.0 -3.1 -15.9

N2

-

-23

-7.5

O2 Ar

-

-32

-16.8

-

-

-

-

CO2

-

-24

2.3

-2.2

N2

58.1

-

49

-7.0

-4.2

O2

43.7

-

60

-1.4

-1.4

Ar AAD:

Scott

-

-

-

-

CO9

8.8

-

27

-13.3

-17.1

|f|;|

19.8

28

18

16.8

10.6

Hildebrand Entropic-FV-1: The liquid volume of the gas is determined from its relationship with the critical volume, Eq. (26). Hildebrand Entropic-F V-2 : The average hypothetical liquid volume of a gas is used

172 Solubility Coefficients of atmospheric gases in PIP as a function of temperature. Model: Hildebrand Entr. FV 10

O r-l

•

^

o U

1—1

o

CD

01

3.05

3.10

3.15

3.20

3.25

3.30

3.35

3.40

Temperature' (xlO ), [K' ] ^

1

1.

.

1

~ Cai bui I dioxide Oxygen A Carbon dioxide, exp. • Nitrogen, exp.

— - — - Nitrogen Argon • Argon, exp • Oxygen, exp.

Figure 15. The solubiHty of several gases in PIP as a function of temperature predicted by the Hildebrand Entropic-FV model. 7.4 EQUATIONS OF STATE FOR POLYMERS Many equations of state (EoS) have been proposed for polymers, for a recent review see Kontogeorgis^^. Both cubic equations of state (vdW, SRK, PR, SWP) and non-cubic ones esp. GCLF and SAFT have been applied with success. In many of the cubic EoS, mixing rules based on the EoS/G^ principle have been employed using one of the previously described activity coefficient models (FH, E-FV, U-FV) in the mixing rules. In this case, the behavior of the activity coefficient model at low pressures is reproduced. In other cases (cubic EoS, SAFT) a single interaction parameter is used or group-based parameters (GCLF) are employed.

173 Numerous applications of these EoS have been reported: low and high-pressure VLB, Henry's law constants and LLE for systems including polymers, co-polymers and blends. Examples of what is possible with such EoS are shown in figures 16-20: LLE for polymer solutions with a cubic EoS^^ and VLE and LLE for binary and ternary systems with a recently developed modified SALT EoS^^"^^ This is still a very active open area of research and is difficult to recommend a specific approach. A serious problem with all EoS for polymers which, in our view, has not been adequately addressed as yet is the way to get the EoS parameters for polymers. Methods employed for low molecular weight compounds (see chapters 5 and 6) cannot be used since critical properties and vapour pressure data are not available (have no meaning) for polymers. Numerous indirect methods^^ have been employed using volumetric data and additional information, often phase equilibria data for mixtures of polymers with low molecular weight compounds. Such methods may be necessary since use of volumetric data alone do not seem to provide polymer EoS parameters useful for phase equilibrium calculations. Use of phase equilibria data, on the other hand, may render the parameters of pure polymers sensitive to the type of information employed. A thorough investigation on methods to obtain meaningful polymer parameters for equations of state will significantly improve and enhance the applicability of equations of state for polymers. A first effort towards this direction has been recently reported for the simplified PC-SAFT equation of state^^. PS with cyclohexane (correlation) 310

270 0.00

0,20

0,40

0,60

0,80

1,00

Polymer weight fraction Figure 16. Correlated UCST phase diagrams with the van der Waals equation of state for PS/cyclohexane at various molecular weights^^. A single interaction parameter is used (per molecular weight).

174 PnBMA(11600) with n - p e n t a n e

320

"T

I

I

I

I

r~

-I

1

1

1

1

T-

-|

1

1

1

r-

1«=0.17583 (pred.) li,=0.161478 (correl.) nnnnn Experimental data

w 300 h Q-

240 _i 0,00

I

I

I

I

0,20

I

\

I

_j

'

0,40

I

I

I

0,60

I

I

I

I

0,80

1,00

Polymer weight fraction Figure 17. Predicted and correlated UCST phase diagrams with the van der Waals equation of state for PBMA(11600)/n-pentane^^. A single interaction parameter is used. The predicted interaction parameter is based on a simple correlation with the solvent's molecular weight.

50

MOH

- T = 323.2 K T = 313.2K T = 303.2 K

30 20

10

0.05

0.1

0.15

0.2

weight fraction acetone

Figure 18. VLB prediction with the PC-SAFT equation of state for PVAC(170000)-acetone at three temperatures (the interaction parameter is set equal to zero). The PC-SAFT developed by von Solms et ai^^ is employed.

175 550 500 450 kn = 0 . 0 0 6 3

^400 H 350

L

^ft-

"-•'=-^—•-

300 250

• MW 37 0 0 0 • MW 6 7 0 0 0 0

Saeki et al. Macromolecules 6, 246 (1973).

200

0.05

0.1

0.15

0.2

0.25

w e i g h t fraction p o l y m e r

Figure 19. LLE with the PC-SAFT equation of state for PS-methylcyclohexane using a single interaction parameter. Polystyrene(l)

LOO-

90-

80

70

60

\

50-

ki3 = 0.006

ki2 = - 0.005

/

40

30 •

20

10

0

•

°l

/ /

7 t /

0

\

(

3'' 10

fo 20

Methylcyclohexane(3)

1

1

1

1

30

40

50

60

70

\ 90

100

Acetone(2)

Figure 20. Ternary LLE with the PC-SAFT equation of state for the ternary system PS(300000)-acetone-methylcyclohexane. The binary parameters are regressed from the binary systems.

176 LIST OF ABBREVIATIONS BR CST DBP DDP DEP DHP DMP DOA DOP DOS EoS Exper./exp. EFV FH FV GC GC-Flory GCF GCLF GCVOL HA Ha HDPE LCST LLE MEK MW NRTL Pa-Ve PBD PDMB PBMA PDMS PE PEO PIB PIP PMMA PR Pred. PS PVAC PVC

Butadiene Rubber Critical Solution Temperature dibutyl phthalate didecyl phthalate diethyl phthalate dihexyl phthalate dimethyl phthalate dioctyl adipate dioctyl phthalate dio(2-ethylhexyl) sebacate Equation of State Experimental Entropic-FV Flory-Huggins (model/equation/interaction parameter) free-volume Group Contribution (method/principle) Group Contribution Flory Equation of State Group Contribution Flory Equation of State Group Contribution Lattice Fluid Group Contribution Volume (method for estimating the density) Holten-Andersen equation of state Hansen solubility parameters High Density polyethylene Lower Critical Solution Temperature Liquid-Liquid Equilibria methyl ethyl ketone molecular weight non-random two liquid activity coefficient model Panayiotou-Vera equation of state Polybutadiene Polydimethylbutadiene Polybutyl methacrylate Polydimethylsiloxane Polyethylene Polyethylene oxide Polyisobutylene Polyisoprene Polymethyl methacrylate Peng-Robinsoon equation of state predicted Polystyrene Polyvinyl acetate Polyvinyl chloride

177 SCF SF SGE SRK S WP SLE SLLE UCST U-FV UNIFAC UQ vdW vdWlf VLE VOC

supercritical fluid Schultz-Flory plot solid-gas equilibria Soave Redlich Kwong equation of state Sako Wu Prausnitz equation of state Solid-liquid equilibria Solid-liquid-liquid equilibria Upper Critical Solution Temperature UNIFAC-FV Universal Functional Activity Coefficient Uniquac van der Waals equation of state van der Waals one fluid (mixing rules) Vapor-Liquid Equilibria Volatile Organic Content

REFERENCES 1. R. B. Seymour, Plastics vs. Corrosives, SPE Monograph Series, Wiley, 1982 2. Van Krevelen, Properties of polymers. Their correlation with chemical structure; their numerical estimation and prediction from additive group contributions, Elsevier, 1990. 3. A.F.M. Barton, Handbook of solubility parameters and other cohesion parameters, CRC Press, 1983. 4. A.F.M. Barton, CRC Handbook of polymer-liquid interaction parameters and solubility parameters, CRC Press, Boca Barton, FL, 1990. 5. J. Bentley and G.P.A. Turner, Introduction to Paint Chemistry and Principles of Paint Technology, 4* edition, Chapman and Hall, 1998. 6. CM. Hansen, Hansen Solubility Parameters. A User's Handbook, CRC Press, 2000. 7. J. Holten-Andersen and K. Eng, Progress in Organic Coatings, 16 (1988) 77. 8. H.S. Elbro, Phase Equilibria of polymer solutions - with special emphasis on free volumes, Ph.D Thesis. Department of Chemical Engineering, Technical University of Denmark, 1992. 9. J. Klein and H.E. Jeberien, Makromol. Chem., 181 (1980) 1237. 10. A. Tihic, Investigation of the miscibility of plasticizers in PVC, B. Sc. Thesis. Department of Chemical Engineering, Technical University of Denmark, 2003. 11. Th. Lindvig, M.L. Michelsen and G.M. Kontogeorgis, AIChE J., 47(11) (2001) 2573. 12. P.J. Flory, J. Chem. Phys, 9 (1941) 660. 13. D. C. H. Bigg, J. Appl. Polym. ScL, 19 (1975) 3119. 14. P.M. Doty and H.S. Zable, J. Polymer Sci., 1 (1946) 90. 15. C. Qian, S.J. Mumby and B.E. Eichinger, Macromolecules, 24 (1991) 1655. 16. Y.C. Bae, J.J. Shim, D.S. Soane and J.M. Prausnitz, J. Appl. Polym. Science, 47 (1993)1193. 17. J.M. Prausnitz, R.N. Lichtenthaler and E.G.D. Azevedo, Molecular thermodynamics of Fluid Phase Equilibria. Prentice-Hall International. 3^^^ Edition, 1999.

178 18. Th. Lindvig, M.L. Michelsen and G.M. Kontogeorgis, Fluid Phase Equilibria, 203 (2002) 247. 19. A.R. Schultz and P.J. Flory, J. Amer. Chem. Soc, 75 (1953) 496. 20. A. Bondi, Physical Properties of Molecular Crystals, Liquids and Glasses; John Wiley and Sons: New York, 1968 21. H.S. Elbro, Aa. Fredenslund and P. Rasmussen, Macromolecules, 23 (1990) 4707. 22. G.M. Kontogeorgis, Aa. Fredenslund and D.P. Tassios, Ind. Eng. Chem. Res., 32 (1993)362. 23.1. Kouskoumvekaki, M.L. Michelsen and G.M. Kontogeorgis, Fluid Phase Equilibria, 202(2) (2002) 325. 24. G.M. Kontogeorgis, LA. Kouskoumvekaki and M.L. Michelsen, Ind. Eng. Chem. Res., 41(18) (2002) 4848. 25. T. Oishi and J.M. Prausnitz, Ind. Eng. Chem. Process Des. Dev., 17(3) (1978) 333. 26. G.M. Kontogeorgis, Ph. Coutsikos, D.P. Tassios and Aa. Fredenslund, Fluid Phase Equilibria, 92 (1994) 35. 27. J.R. Fried, J.S. Jiang and E. Yeh, Comput. Polymer Science, 2 (1992) 95. 28. L.A. Belfiore, A.A. Patwardhan and T.G. Lenz, Ind. Eng. Chem. Res., 27 (1988) 284294. 29. H.K. Hansen, B. Coto and B. Kuhlmann, UNIFAC with lineary temperaturedependent group-interaction parameters, IVC-SEP Internal Report 9212, 1992. 30. H. S. Elbro, Aa. Fredenslund and P. Rasmussen, Ind. Eng. Chem. Res., 30 (1991) 2576. 31. E.C. Ihmels and J. Gmehling, Ind. Eng. Chem. Res., 42(2) (2003) 408-412. 32. J.J. Ellington, J. Chem. Eng. Data, 44 (1999) 1414. 33. G.D. Pappa, E.C. Voutsas and D.P. Tassios, Ind. Eng. Chem. Res., 38 (1999) 4975. 34. G. Bogdanic and Aa. Fredenslund, Ind. Eng. Chem. Res., 34 (1995) 324. 35.1. Kouksoumvekaki, R. Giesen, M.L. Michelsen and G.M. Kontogeorgis, Ind. Eng. Chem. Res., 41(19) (2002) 4848. 36. G.M. Kontogeorgis, Aa. Fredenslund, I.G. Economou and D.P. Tassios, AIChE J., 40 (1994)1711. 37. Th. Lindvig, L.L. Hestkjaer, A.F. Hansen, M.L. Michelsen and G.M. Kontogeorgis, Fluid Phase Equilibria, 663 (2002) 194. 38. Th. Lindvig, I. G. Economou, R.P. Danner, M.L. Michelsen and G. M. Kontogeorgis, Modelling of multicomponent vapor-liquid equilibria for polymer-solvent systems, Fluid Phase Equilibria (in press). 39. C. Zhong, Y. Sato, H. Masuoka and X. Chen, Fluid Phase Equilibria, 123 (1996) 97. 40. E.N. Polyzou, P.M. Vlamos, G.M. Dimakos, I.V. Yakoumis and G.M. Kontogeorgis, Ind. Eng. Chem. Res,, 38 (1999) 316-323. 41. E.C.Voutsas, N. S. Kalospiros and D. P. Tassios, Fluid Phase Equilibria, 109 (1995) 1. 42. G.M. Kontogeorgis, G.I. Nikolopoulos, D.P. Tassios, D.P and Aa. Fredenslund, Fluid Phase Equilibria, 127 (1997) 103. 43. G.M. Kontogeorgis, E.C. Voutsas and D.P. Tassios, Chem. Eng. Sci., 51 (1996) 3247. 44. B.-C Lee and R.P. Danner, AIChE J., 42 (1996) 837.

179 45. G.M. Kontogeorgis, A. Saraiva, Aa. Fredenslund and D.P. Tassios, Ind. Eng. Chem. Res., 34 (1995) 1823, 46. V.I. Harismiadis, A.R.D. van Bergen, A. Saraiva, G.M. Kontogeorgis, Aa. Fredenslund, Aa. and D.P. Tassios, AIChE J., 42 (1996) 3170. 47. V.I. Harismiadis and D.P. Tassios, Ind. Eng. Chem. Res., 35 (1996) 4667. 48. G.D. Pappa, G.M. Kontogeorgis and D.P. Tassios, Ind. Eng. Chem. Res., 36 (1997) 5461. 49. G. Bogdanic and J. Vidal, Fluid Phase Equilibria, 173 (2000) 241-252. 50. G. Bogdanic, Fluid Phase Equilibria, 4791 (2001) 1-9. 51. G.D. Pappa, E.G. Voutsas and D.P. Tassios, Ind. Eng. Chem. Res., 40 (2001) 4654. 52. P. J. Flory and J. Rechner, J. Chem. Phys., 11 (1943) 521. 53. M. J. Michaels and R.W. Hausslein, J. Polym. Sci. C , 10 (1965) 61. 54. J. S. Yoo, S. J. Kim and J. S. Choi, J. Chem. Eng. Data, 44 (1999) 16. 55. S. J. Doong and W. S. W. Ho, Ind. Eng. Chem. Res., 30 (1991) 1351. 56. P. Thorlaksen, J. Abildskov and G.M. Kontogeorgis, Fluid Phase Equilibria, 211 (2003) 17. 57. G. M. Kontogeorgis, Ch. 16: Thermodynamics of polymer solutions, in Handbook of Surface and Colloid Chemistry, 2"^ ed., CRC Press, 2003. 58. V.I. Harismiadis, G.M. Kontogeorgis, A. Saraiva, Aa. Fredenslund and D.P. Tassios, Fluid Phase Equilibria, 100 (1994) 63-102. 59. N. V. Solms, M. L. Michelsen and G. M. Kontogeorgis, Ind. Eng. Chem. Res., 42(5) (2003) 1098. 60.1.A.Kouskoumvekaki, N.v.Sokns, M.L.Michelsen and G.M.Kontogeorgis, Fluid Phase Equilibria, 215(1) (2004) 71-78. 61. Th.Lindvig, M.L.Michelsen and G.M. Kontogeorgis, Ind. Eng. Chem. Res, 43(4) (2004) 1125-1132. 62.1. A. Kouskoumvekaki, N. v. Solms, Th. Lindvig, M. L. Michelsen and G. M. Kontogeorgis, A novel method for estimating pure-component parameters for polymers: Application to the PC-SAFT equation of state, Ind. Eng. Chem. Res. (submitted).