Ideal Mixing Rules for the Viscosity of Complex Polymer−Solvent

Ind. Eng. Chem. Res. 2006, 45, 7293-7300

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Ideal Mixing Rules for the Viscosity of Complex Polymer-Solvent Mixtures: Assessment of Segment-Fraction Approximations Stephan Machefer* and Klaus Schnitzlein Department of Chemical Reaction Engineering, Brandenburg Technical UniVersity, Cottbus, Burger Chaussee 2, 03046 Cottbus, Germany

To predict viscosities for complex polymer-solvent mixtures according to Eyrings kinetic theory of viscous flow, a suitable segmentation approach has to be applied. Reasonable segmentation approaches are elaborated and discussed in detail. Considering viscous flow as surface phenomenon, a new segmentation approach is developed based on molecular surfaces of each species in the mixture. For six binary polymer-solvent mixtures of different complexity, mixture viscosities were experimentally obtained and compared to model predictions. For all mixtures, the best agreement between model predictions and experimental data is obtained by using surface-based weighing fractions. Moreover, theoretical evidences for choosing surface-based weighing fractions are provided. Introduction Modeling of polymer-solvent mixture viscosities is one of the current challenges in physical property estimation of liquid mixtures. So far, models exist predominantly for liquid mixtures of species of low molecular weight. A compilation of viscosity models based on very different approaches is presented in excellent reviews by Monnery et al.1 and Poling et al.2 The most powerful and therefore widely used models are those based on the reaction rate theory developed by Eyring and co-workers more than six decades ago.3 For molecules of approximately equal size, a mixing rule based on mole fractions was derived. With these models, viscosities even for strongly nonideal mixtures can be predicted with high accuracy.4 However, when macromolecules or polymers are involved, these models lead to large errors requiring parameter adjustments in order to obtain a sufficient agreement with experimental data.5 As has been found very early from experimental investigations, viscous flow of polymers should be regarded rather as a sequence of movements of their segments6 instead as of one single stiff flow unit. In some recent approaches, the polymers are subdivided in a number of segments transforming the molecular mixture into a mixture of segments. This way, mixing rules are obtained based on segment fractions instead of mole fractions.7,8 However, the appropriate definition of segments or segment fractions, respectively, is far from being unambiguous, especially when the polymer structure gets more complex and irregular. Therefore, it is the objective of this study to elaborate and to assess reasonable segmentation approaches for heterogeneous polymers. Model predictions are compared to experimental mixture viscosities for different binary polymer-solvent mixtures exhibiting Newtonian flow behavior. Mixing Rules for Polymer-Solvent Mixtures According to the Eyring kinetic theory, viscous flow can be regarded as an activated process. An activation energy ∆F has * To whom correspondence should be adressed. E-mail: [email protected]. Phone: +49 355 691178. Fax: +49 355 691110.

Figure 1. Viscous flow according to Eyrings association. The activation energy ∆F has to be exerted in order displace a flow layer.

to be exerted in order to move the sliding layer from one energy minimum to the next by one flow unit (see Figure 1). The preexponential factor is derived from the flow area a of each flow unit and the distances between two flow units perpendicular λf and parallel λv to the viscous force f :9

η)

( ) { } hλv 2

λf a

exp

∆F RT

(1)

For species with low molecular weight, these geometrical values are similar and can be approximated by the molecular size (e.g., the van der Waals diameter). To extend eq 1 to mixtures, averaged values for the pre-exponential factor and

10.1021/ie060429k CCC: $33.50 © 2006 American Chemical Society Published on Web 09/08/2006

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fractions ξi, i ) 1, ..., N with ti representing the number of segments of species Ai:

the activation energy are considered:

η j)

(){} hλv

∆F

exp

2

(2)

RT

λf a

Generally, it is assumed that the averaged activation energy can be calculated in terms of pure component values ∆Fi using a mixing rule with mole fractions xi as fractional weights. By introducing a binary excess term ∆FEij nonideal effects can be taken into account, too: N

∆F )

∑ i)1

N

xi∆Fi +

N

∆FEij ∑ ∑ i ) 1 j)i+1

(3)

Thus, the viscosity of the mixture can be expressed as a function of the viscosities of the pure components: N

ln (η jΦ h)) with

N

N

xi ln (ηiΦi) + ∑ ∑ ∑ i)1 i)1 j)i+1

( )

λf2a Φ h ) hλv

∆FEij

( )

λif2ai and Φi ) hλvi

RT

(4)

(5)

(6)

eq 4 can further be simplified, yielding a mixing rule for mixture viscosity in terms of molar fractional weights: N

ln η j)

N

N

∑i xi ln ηi + ∑ ∑ i)1 j)i+1

∆FEij RT

(7)

Generally, based on a dimensional analysis, the preexponential factor Φi is interpreted as molar volume of species Ai.3 Moreover, assuming equal molar volumes of all species, the assumption eq 6 can be satisfied. Then, the well-known ideal mixing rule with linear relationship for the viscosity of the mixture is derived provided that nonideal effects can be entirely neglected: N

ln η j)

∑i xi ln ηi

xi

)

N

tj

N

∑j xjtj ∑ xjt j

(9)

i

For monodisperse homopolymers, this approach has been applied successfully.7 It should be mentioned that the estimation of the number of flow segments or segment ratios for all species in the mixture is somewhat arbitrary. However, different molecular properties can be considered in order to represent the segment ratios tj/ti: (1) Number of Repeating Structures. Intuitively, segment ratios can be represented by the ratio of the numbers of repeating structures gj/gi or monomers, respectively. Then the weighing functions are represented as

ξi(gi) )

xi gj

N

(10)

i

N

∑i xi ln Φi

xiti

∑j xjg

Provided that

ln Φ h )

ξi )

(8)

Obviously, eq 8 does not hold for species with significantly different molar volumes. This is particularly true for polymersolvent mixtures. Thus, considerable deviations between predictions and experimental data are reported.6 As a remedy, a segmentation approach is widely proposed. It has been observed experimentally that ∆F becomes independent from molecule size when the polymer chain becomes sufficiently large. This was regarded as an indication that viscous flow of polymer molecules cannot be represented by the movement of one single molecule. Segments of equal size, with a corresponding activation energy,6 should be regarded as characteristic elements of viscous flow instead. Consequently, mole fractions are to be replaced by segment-based weighing

The solvent is regarded as one flow unit,7 presupposing that solvent molecule and polymer-substructure are similar segments. It is obvious, that in cases where either the polymer segments are large in size or associated solvents are used, this approach is questionable. Furthermore, in cases of heteropolymers a logical and consistent segment definition is not possible. (2) Size of Molecular Volumes. Taking into account the different molecular volumes of the species in a polymer-solvent mixture, segment ratios can be represented by the ratio of their respective molecular volumes.4 Here, it will be assumed that molecular volumes can be substituted by normalized van der Waals (VdW) volumes. Provided that the chemical composition of each species is known, the respective VdW volume for any species can be calculated in terms of group contributions ∆Vk.4 Appropriate group contributions were published by Bondi:10 L

V/i )

∑k νik∆Vk

(11)

The same values in a normalized form ∆Vk are used within the original UNIFAC group contribution method to calculate combinatorial contributions to the Gibbs excess free energy.11 Modifications of the original UNIFAC-method should not be taken as a source, since their ∆Vk values are often included in the global group correlation procedure, thereby partly loosing their original geometric information. Thus, the weighing fractions become

xi

ξi(V/i ) ) N

V/j

(12)

∑j xj

V/i

(3) Size of Macroscopic Volumes. Vacant or free volume effects are known to become more significant especially when polymers are involved. To account for these effects, segment ratios can be represented in terms of macroscopic

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volumes for each species:

Vi )

Mi FiNA

(13)

thus identifying the resulting weighing functions as macroscopic volume fractions:

ξi(Vi) )

xi Vj

N

(14)

∑j xjV

i

These volume fractions are temperature dependent. (4) Size of the Molecular Surfaces. When viscous flow is considered as surface phenomenon, the viscous contribution of each flow unit should be related to its contact area. Therefore, a new approach is proposed here, representing segment ratios by the ratio of the respective molecular surfaces. Again, it will be assumed that molecular surfaces can be substituted by normalized VdW surfaces, which can be calculated in terms of group contributions ∆sk.4 Appropriate group contributions were published by Bondi:10 L

s/i )

∑k νik∆sk

(15)

Figure 2. Molar weight distributions of polyether-polyols used. Table 1, Pure Liquid Viscosity Correlations for the Polyols and the Solvents (least-squares fit in ln η) Eyring (eq 1)

P1 P2 P3 S1 S2 a

Thus, the weighing functions can be identified as molecular surface fractions:

xi

ξi(s/i ) ) N

s/j

(16)

∑j xj / si

For practical applications, it is important to determine which weighing function is best suited for predicting viscosities of polymer-solvent mixtures. In principle, eq 6 can be used as assessment criterion. For example, its validity has to be checked for every polymer-solvent mixture under consideration over the entire composition range. Numerical errors, introduced by fitting experimental data for pure component and mixture viscosities to the Eyring eq 1 may however disguise the real effects. Therefore, it is more appropriate to assess these approaches by comparing model predictions to experimental data obtained for polymers and solvents with different molecular properties. Experimental Section Polymer-Solvent Systems under Investigation. Three polyether-polyols (P1, P2, and P3) produced by an alkoxylation polymerization process were chosen as representative polymers. Due to the different compositions of one or more starters with one or more propagators, multiple reaction pathways are possible leading to polydisperse heteropolymers. Polyols P1 and P2 are produced from the same starters and propagators but with different compositions. As shown in Figure 2, both exhibit a broad and polydisperse molecular weight distribution (MWD) covering nearly 2 orders of magnitude concerning molecular weight. The MWD of P1 is shifted to lower molar masses and is more condensed, thereby approaching a bimodal character. Consequently, P1 and P2, although chemically similar, differ considerably in their polydispersity. Using these two polyols,

modified Eyring (eq 17)

1/Φ

∆F

δη [%]

3.397E-08 1.759E-08 6.187E-11 1.419E-03 8.620E-03

64508.7 68123.5 81455.7 15988.5 10278.0

11.1 10.3 11.3 4.29 0.34

1/Φ′

∆F′

C

7.267E-03 14117.6 -177.02 6.764E-03 15255.2 -175.15 3.010E-02 9563.0 -210.13 3.103E-02 4089.3 -151.61 1.315E-02 8442.0 -25.51

δη [%]a 0.93 0.03 1.64 0.25 0.26

δη ) ∑1/N‚|(ηcalc - ηexp)/ηexp|‚100.

the ability of the different methods to compensate for polydispersity can be elaborated. On the other hand, P3 is produced from chemically different starters and propagators, exhibiting a broad but condensed unimodal MWD. Using a chemically different polyol, the influence of different intermolecular interactions between solvent and polymer can be elaborated too. Due to their hydrophilic property, the polyols are miscible in polar solvents. Therefore, water (S1) and methanol (S2) have been chosen as solvents. Both are forming hydrogen bonds with the polar hydroxy groups of the polyols. On the other hand, both solvents can be regarded as fundamentally different concerning their chemical structure, one containing an alkyl, hence hydrophobic, group and the other (water) not. This chemical difference manifests itself in the strong hydrogen-bonded associated structures that water is able to form. In fact, the cohesive energy density of water is 2.6 times that of methanol.12 Pure Component Viscosities. Pure component viscosities for P1 and P2 were measured with a high precision falling sphere viscometer in the temperature range of 25-90 °C with a relative standard deviation well below 1%. Viscosity data for P3, S1, and S2 were taken either from the manufacturer’s database13 or from the literature.14,15 These pure component viscosity data have been fitted to eq 1 by a least-squares analysis adjusting the two physically meaningful Eyring parameters 1/Φ and ∆F. The results are summarized in Table 1, showing deviations of 4% for water and of about 10% for the polyols under investigation. Since water is known to exhibit a unique behavior in most of its physical properties,12 this may be an explanation for the higher deviations observed in comparison to methanol. The deviations of the polyols can be simply explained by the fact that their viscosity values cover nearly 3 orders of magnitude over the range of measured temperatures (25-90 °C). In view of assessing different mixing rules, a higher accuracy for predicting the pure viscosities is desirable. From a math-

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Table 2. Experimental Conditions Regarding Temperature Range and Composition for the Investigated Polyol-Solvent Mixture Viscosities experimental system

T range (°C)

wS range

P1/S1 P2/S1 P3/S1 P2/S2 P3/S2

25-95 25-95 25-70 25-60 25-60

0-1 0-1 0-0.35 0-0.15 0-0.15

typically occur can be ignored since each component is further subdivided into its functional groups and, therefore, neglecting sterism. It is strongly recommended to check the validity of the aforementioned components by performing elemental mass balance calculations. The molecular properties of each component can easily be determined by considering respective functional group contributions; then for every species Ai in the mixture, respective molecular properties are obtained in terms of properties for each component:

ematical point of view, eq 1 is identical with the two-parameter Antoine expression for the vapor-pressure curve. Actually, it has been found that ∆F is correlated to the heat of evaporation.9 Therefore, a third model parameter is introduced yielding a modified Eyring equation similar to the three-parameter Antoine expression:16

{

∆F' 1 exp η) Φ' R(C + T)

}

∑j xijMj

(19)

Qi

Vi )

∑i xij Vj

Qi

wij/Mj Qi

Mi )

(20)

(17)

Thus a significantly better representation of the pure viscosities could be achieved as shown in Table 1. Pure component viscosities for polyols and solvents can now be calculated with a mean relative error of less than 1% in nearly all cases. However, the modified Eyring parameters completely loose their physical interpretability. While for low and high molecular species the values for 1/Φ and ∆F differ by several orders of magnitude, this distinction vanishes for 1/Φ′ and ∆F′. Mixture Viscosities. To cover the full composition range (2.5-90%) within the temperature range of interest, viscosities of binary polymer-solvent mixtures P1/S1 and P2/S1 have been measured with the same high precision falling sphere viscometers. Viscosity data for the mixtures P3/S1 and P2/S2, P3/S2 were taken from the manufacturer’s database.13 The respective experimental conditions are compiled in Table 2. Using methanol as solvent, the composition as well as the temperature range is restricted due to the high volatility of methanol. Definition of Polyol-Pseudocomponents. To obtain the molecular based segmentation ratios, the molecular structure of all species in the mixture must be known. For the solvent, this generally constitutes an easy task while the molecular structure for the polymers is unknown in detail. As a remedy, the molecular structure of any polymer species is reconstructed based on the experimentally determined molecular weight distribution interpreted in terms of the chemistry involved in the polymerization process. Instead of considering the polymer as a mixture of polymers with different degrees of polymerization, the polyols are treated as one single pseudo-species consisting of different components. Thus, every peak in the MWD is regarded to represent a single component with a respective molecular mass. The mass fraction of each component can be calculated either in terms of peak height or peak area. Accordingly, the mole fraction of component Aij is obtained as

xij )

Qi

V/i )

∑j ∑k νijk∆Vk) Qi

s/i

)

L

xij(

(21)

L

∑j xij(∑k νijk∆sk)

(22)

Exemplarily, the overall procedure is illustrated in Figure 3 for the calculation of the molecular surface for polymer P1. In a first step, five separate peaks are identified from the MWD and attributed to five subcomponents (A1j, j ) 1, ..., 5). Knowing the overall reaction scheme (epoxylation of alcohol), every subcomponent is identified as the most probable reaction product matching the respective molar mass M1j. Mass and mole fractions for each subcomponent (w1j and x1j) are then calculated in terms of peak heights. Since the chemical structure for each subcomponent is known, six different functional groups can be identified to represent every subcomponent A1j, the frequency of which is denoted by ν1jk. The normalized VdW surface of every subcomponent is then calculated using tabulated values for the respective group contributions11 according to eq 22. Accordingly, molecular surfaces for polymers P2 and P3 are obtained by considering eight or six subcomponents, respectively. Data for the molecular surfaces of the solvents were also taken from the VdW parameter tables.11 Values for molecular properties of all species are summarized in Table 3. As can be seen, the molecular volumes V/i as well as the molecular surfaces s/i for polymers P1 and P2 are quite similar, despite the fact that the respective Eyring parameters are considerably different (see Table 1). Furthermore, it is noted that the ratio VP/VS is nearly the same as the ratio V/P/V/S for all polyol-solvent mixtures, although the absolute values for Vi and V/i differ considerably. Hence, very similar weight fractions ξ(V/i ) and ξ(Vi) are to be expected. Equally, the use of molecular surface fractions may be a reasonable approximation for the macroscopic viscous flow surface.

(18)

∑k wik/Mk with Qi being the total number of components for species Ai or the degree of polydispersity, respectively. The chemical structure of each component can be obtained by considering the most probable product based on the polymerization reaction mechanism. Multiple isomeric structures that

Results For all polymer-solvent systems mixtures viscosities were calculated in terms of segment fractions introduced above and compared to experimental data. Table 4 summarizes the results of the different weight fractions for all systems in terms of average deviations for all temperatures investigated. Additionally, molar and mass fractions were used for comparison purposes. Mass fractions have been widely used for systems in

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Figure 3. Sketch of the overall procedure to determine molecular properties for polymer P1 (see text). Table 3. VdW Properties for All Speciesa Ai

Qi

Mi

s/i

V/i

Vi (30 °C)

Vi (90 °C)

P1 P2 P3 S1 S2

5 8 6 1 1

488.2 511.3 339.7 18.0 32.0

19.69 19.87 13.14 1.40 2.05

20.40 20.53 15.46 0.92 1.90

48.58 50.19 34.60 1.977 4.47

50.54 52.37 36.34 2.044 4.86

a

The values for Vi have been normalized in the same way as for V/i .

Table 4. Comparison of Ideal Mixing Rules Based on Different Weighting Functions with Binary Viscosity Data for Different Polyol-Solvent Mixtures ∆ln ηa system

xi

wi

ξ(Vi)

ξ(V/i )

ξ(s/i )

P1/S1 P2/S1 P3/S1 P2/S2 P3/S2

-2.532 -2.777 -3.394 -2.843 -1.723

0.658 0.874 1.056 1.887 1.734

0.549 0.743 0.947 1.548 1.471

0.451 0.57 0.923 1.504 1.515

-0.035 -0.002 0.141 1.379 1.283

a

∆ln η ) ∑1/N(ln ηcalc - ln ηexp).

which polymers are involved, because in this case obviously the original Eyring approach (eq 8) fails. Theoretical justification for this replacement, however, does not exist.5 Figure 4 shows the comparisons for all binary polymer-water mixtures. As expected, using mole fractions xi as weighing fractions mixture viscosities are predicted differing from the experimental values by orders of magnitude. Using mass fractions a significantly better agreement is obtained, but mixture viscosities are overestimated by a factor of 2-6. Taking into account molecular properties, a much better agreement between model predictions and experimental data can be obtained. Replacing molecular volume based weighing fractions ξ(Vi) by VdW volume-based weighing fractions ξ(V/i ) yields, only a slight improvement of model predictions, indicating that free volume effects are only of minor importance when using associating solvents. The best agreement between model predictions and experimental data is achieved using VdW surface based weighing fractions ξ(s/i ). The good agreement for all polymers can be regarded as a hint that with this approach, polydispersity as well as different intermolecular interactions between solvent and

Figure 4. Binary viscosities for different polyol-water mixtures as a function of temperature (symbols, experiments; lines, model predictions).

polymer can be well accounted for. The average deviation is calculated to be within the range of 5-10% with respect to η. This accuracy is regarded to be acceptable for engineering and design calculations. The deviation encountered for P3/S1, especially at higher temperatures and high mass fraction of water may indicate that nonlinear effects will increasingly become

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Figure 6. Typical structure of polyol compared to the solvent molecules. The dotted hull marks the VdW volume or surface. Figure 5. Binary viscosities for different polyol-methanol mixtures as a function of temperature (symbols, experiments; lines, model predictions).

significant. In contrast to P1 and P2, the fraction of hydrophobic groups is larger and the amount of hydroxy groups less in P3. Using methanol as solvent, considerably larger deviations are obtained (Figure 5). Nevertheless, again the best agreement is achieved using surface-based segment fractions. The larger deviation may indicate that in this case nonlinear effects become important. Therefore, the assumption ∆FEij ≈ 0 seems to be only justified for the polyol mixtures with water. Since surface fractions lead to a close representation of viscosities for polyol-water mixtures, it can be concluded that nonlinear effects are of minor importance for these mixtures. This is not surprising since in these mixtures hydrogen bonds are the all-dominant intermolecular interactions. It is a fact that the polyols chemically consist of alkyl and ether groups, but these typically constitute the core of the polymer which is sterically difficult to access. The outer hull is nearly exclusively taken up by hydroxyl groups. For polymer-methanol mixtures, however, mechanisms involving repulsive van der Waals forces for the alkyl part of methanol may be considered. Alternatively, these deviations can be explained in terms of so-called “solvent-accessible surface”17 of the polymer with respect to the solvent used. Since the VdW surface constitutes a rather rough representation with lots of gussets and other nonaccessible space, not all of the VdWsurface is actually in contact with the solvent. For illustration purposes, the molecular structures of a typical branched polyol and both solvents are sketched in Figure 6. Considering the different molecule sizes for water and methanol, it can be concluded that the reduction in the solvent accessible surface for each polyol should be more pronounced for systems that are using methanol as solvent rather than for systems using water. Thus it is expected that the accurary in the predictions for systems P1/S2 and P3/S2 can be improved. However, according to a rough estimate, even for a reduction in surface of 10%, the mean absolute deviations will only decrease by approximately 10%. Therefore, this reasoning cannot be regarded as sufficient to explain the observed deviations entirely.

Discussion Interpreting viscous flow as a surface-specific phenomenon allows viscosities for six different polymer-solvent mixtures to be well-represented over a large range of temperatures and compositions. Since the experimental study is restricted to the mixtures used, the question of general validity arises. The assumption that viscosity actually is a surface-specific phenomenon, however, cannot be proved rigorously, although there are some evidences and considerations that support this view. Viscosity as inner friction of fluids may be expected to depend on the inner surface of a liquid since friction depends on the area attractive forces act on. This is particular by evident for mixtures containing molecules of very differing sizes where surface contributions may overcome excess energy contributions. Further support for this view can be obtained from the definition of viscosity according to Eyring. Viscous flow (cf. Figure 1) f is regarded to be the force acting on one molecule of surface area a moving it with velocity u through the liquid. The displacement dz corresponds to a finite flow unit displacement about λf in terms of Figure 1):

η)

df dz da du

(23)

On a microscopic level, viscosity is regarded as an phenomenon based on friction between molecules or layers of molecules. Consequently, the viscosity of a liquid is related to its inner surface. In a branched macromolecule, the outer shell represents the surface exposed to viscous flow. The fractional weight of the linear mixing rule should therefore be able to take this into account. This is shown subsequently by comparing VdW surface to VdW volume-derived fractions. When subdividing a polymer into segments of equal size, every segment approximately contributes the same volume, since overlapping volumes are relatively small. This is different in case of segment surfaces. Overlapping reduces the segment surface significantly. For example, segments at the end of a polymer contribute a much larger fraction to the overall surface than segments in the core. In other words, functional VdW end groups dominate the surface

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Acknowledgment This work has been supported financially by BASF AG, Polymer Research Division, and by BASF Schwarzheide GmbH, Polyurethane Chemical Development. Financial support as well as cooperative knowledge exchange are greatly acknowledged, especially the contribution from Dr. Thomas Ostrowski (BASF AG Ludwighafen) and from Sascha Bergmann and Dr. Olaf Otto (both at BASF Schwarzheide GmbH). Nomenclature

Figure 7. Dependency of ∆sk/∆Vk for different C-atoms in a polymer chain.

contribution to the overall surface. The volume contribution ∆Vk to the overall volume V/i , however, does not depend that much on the position of the segment in the polymer. This is shown in Figure 7, where the ratio ∆sk/∆Vk is plotted as a function of covalent bonds to other groups except hydrogen atoms. For all groups under consideration, the ratio decreases from primary to ternary carbon atoms. A quarternary sp3-hybrid carbon group does not even contribute surface at all (∆sk ) 0). Thus, fractional weighting on a VdW group contribution surface basis (ξ(s/i )) is a suitable fractional expression that generally weights the influence of the outer shell of a polymer more than the contribution of inner molecular parts. Molecular surface fractioning is therefore more suitable than volume fractioning as basis for the calculation of the viscosity of liquid viscosities with mixing rules. Another hint for the general validity of our results comes from observations made regarding the correlation between viscosity and polymer chain length. In polymer solutions, viscous flow is traditionally regarded as sequential flow of chain segments. It has been found that viscosity is mainly determined by the number of vacancies or, alternatively, the free volume allowing for segment dislocations. In very early studies, a direct relation between viscosity and free volume fraction yfv has been found for many chain molecules:6

η ∝ (yfv)-1

(24)

When regarding the free volume as gas phase, creating a gasliquid interface, ∆F may be expressed as18

∆F ) σ da

(25)

with σ as the mean surface tension. Hence, it can be concluded that the Eyring parameters Φ-1 ∝ a-1 and ∆F ∝ σ, respectively, are rather surface specific. Conclusions Based on the concept of segmental viscous flow of polymer molecules it was shown experimentally that viscosities of polymer-solvent mixtures can reliably be calculated using an ideal linear mixing rule, provided surface-based weighing functions are applied. Though some theoretical considerations are presented that suggest the use of surface fractions, further studies are needed to support this new approach. Provided that segments for the polymer and the solvent can be obtained rigorosly (e.g., by means of an equation of state), a comparison of the obtained segment ratios with VdW surface and VdW volume ratios may yield further insights.

Ai ) species or subspecies i C [K] ) modified Eyring parameter ∆F [J‚mol-1] ) activation energy of viscous flow (Eyring) ∆F′ [J‚mol-1] ) modified Eyring parameter L [-] ) number of functional groups M [kg‚mol-1] ) molar mass N [-] ) number of species NA [mol-1] ) Avogadro’s number Pi ) polyol (pseudo-) species A Qi [-] ) number of subcomponents in Pi R [J‚mol-1‚K-1] ) universal gas constant Si ) solvent species i T [K] ) temperature a [m2] ) area of viscous flow f [kg‚m‚s-2] ) force vector gi [-] ) number of repeating units in species i ti [-] ) number of viscous flow segments in species i s/i [m2‚m-2] ) normalized molecular VdW surface of species i ∆sk [m2‚m-2] ) normalized VdW surface contribution of functional group k u [m‚s-1] ) viscous flow velocity vector Vi [m3‚mol-1] ) macroscopic specific volume of species i V/i [m3‚m-3] ) normalized molecular VdW volume of species i ∆Vk [m3‚m-3] ) normalized VdW volume contribution of functional group k wi [kg‚kg-1] ) mass fraction of species i xi [mol‚mol-1] ) mole fraction of species i yfv [m3‚m-3] ) fraction of void or free volume z [m] ) flow coordinate Greek Symbols ∆x [var.] ) absolute deviation in property x Φ [kg‚m-1‚s-1] ) preexponential factor (Eyring) Φ′ [kg‚m-1‚s-1] ) modified Eyring parameter δ [var.] ) relative deviation in property x η [kg‚m-1‚s-1] ) viscosity λ [m] ) size of viscous flow element νi [-] ) number of functional groups in species i ξ [-] ) weighting fraction F [kg‚m-3] ) density σ [kg‚s-2] ) surface tension Subscripts f ) parallel to f i, j ) species or subspecies k ) functional group v ) vertical to f Superscripts / ) molecular property E ) excess property

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AbbreViations MWD ) molecular weight distribution UNIFAC ) UNIQUAC functional group activity coefficient UNIQUAC ) universal quasi-chemical VdW ) van der Waals Literature Cited (1) Monnery, W. D.; Svrcek, W. Y.; Mehrotra, A. K. Viscosity: a critical review of practical predictive and correlative methods. Can. J. Chem. Eng. 1995, 73, 3-40. (2) Poling, B. E.; Prausnitz, J. M.; O’Connel, J. P. The Properties of Gases and Liquids; McGraw-Hill: New York, 2001. (3) Kincaid, J. F.; Eyring, H.; Stearn, A. E. The theory of absolute reaction rates and its application to viscosity and diffusion in the liquid state. Chem. ReV. 1941, 28, 301-365. (4) W. Cao, W.; Knudsen, K.; Fredenslund, A.; Rasmussen, P. Group contribution viscosity predictions of liquid mixtures using UNIFAC-VLEparameters. Ind. Eng. Chem. Res. 1993, 32, 2088-2092. (5) Song, Y.; Mathias, P. M.; Trembley, D.; Chen, C. C. Liquid viscosity model for polymer solutions and their mixtures. Ind. Eng. Chem. Res. 2003, 42, 2415-2422. (6) Kauzmann, W.; Eyring, H. The viscous flow of large molecules. J. Am. Chem. Soc. 1940, 62, 3113-3125. (7) Novak, L. T.; Chen, C. C.; Song, Y. Segment-based Eyring-NRTL viscosity model for mixtures containing polymers. Ind. Eng. Chem. Res. 2004, 43, 6231-6237. (8) Novak L. T. Modeling the viscosity of liquid mixtures: polymersolvent systems. Ind. Eng. Chem. Res. 2003, 42, 1824-1826.

(9) Powell, R. E.; Roseveare, W. E.; Eyring, H. Diffusion, thermal conductivity, and viscous flow of liquids. Ind. Eng. Chem. 1941, 33, 430435. (10) Bondi, A. Physical Properties of Molecular Liquids, Crystals and Glasses; Wiley: New York, 1968. (11) Magnussen, T.; Rasmussen, P.; Fredenslund, A. UNIFAC parameter table for prediction of liquid-liquid equilibria. Ind. Eng. Chem. Proc. Des. DeV. 1981, 20, 331-339. (12) Chaplin, M. F. Water structure and behavior. Retrieved August 1, 2005, from http://www.lsbu.ac.uk/water/, 2005. (13) BASF. Internal Chemical Property Database; BASF, Inc.: 2004. (14) Wu, C. Engineering fundamentals. Retrieved July 7, 2005, from http://www.efunda.com/materials/, 2005. (15) Lide, D. R. CRC Handbook of Chemistry and Physics; CRC Press: Boca Raton, FL, 1996. (16) Goletz, E.; D. Tassios, D. An Antoine type equation for liquid viscosity dependency to temperatature. Ind. Eng. Chem. Proc. Des. DeV. 1977, 16, 75-79. (17) Connolly, M. L. Molecular surfaces: a review. Retrieved December 12, 2005, from http://www.netsci.org/Science/Compchem/feature14.html, 1996. (18) Vogel, H. Gehrtsen Physik; Springer: Berlin, 1999.

ReceiVed for reView April 6, 2006 ReVised manuscript receiVed July 13, 2006 Accepted August 1, 2006 IE060429K