Ind. Eng. Chem. Res. 2004, 43, 6231-6237
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Segment-Based Eyring-NRTL Viscosity Model for Mixtures Containing Polymers Lawrence T. Novak,*,† Chau-Chyun Chen,‡ and Yuhua Song‡ The Lubrizol Corporation, 29400 Lakeland Boulevard, Wickliffe, Ohio 44092-2298, and Aspen Technology, Inc., Cambridge, Massachusetts 02139
A multicomponent liquid mixture viscosity model based on the Eyring and nonrandom twoliquid (NRTL) theories was recently presented and evaluated over the entire concentration range using data from several polymer-solvent systems. In this paper, the original component-based Eyring-NRTL viscosity model is transformed to a segment-based Eyring-NRTL viscosity model for polymer-solvent and polymer-polymer systems. The segment-based approach provides a more physically realistic model for large molecules when diffusion and flow is viewed to occur by a sequence of individual segment jumps into vacancies rather than jumps of the entire large molecule. Evaluation of a segment-based Eyring-NRTL model with polyisobutylene-isooctane mixture viscosity data demonstrates an improvement in the correlation capability over the Mn range of 900-1 200 000. Additional evaluations with polystyrene in styrene, poly(ethylene glycol) in oxolane (THF), poly(dimethylsiloxane)s in pentamer, and poly(dimethylsiloxane) blends provide further support for using the segment-based Eyring-NRTL viscosity model to correlate the viscosity of mixtures containing polymers. For several cases evaluated, the segment-based Eyring-NRTL viscosity model is also found to provide some predictive capability because the binary parameters are essentially independent of polymer Mn. Introduction Polymer-solvent and polymer-polymer mixture viscosity is an important physical property in polymer research, development, and engineering. Novak proposed a multicomponent model to describe the liquid mixture viscosity over the entire range of compositions and over a range of temperatures.1 On the basis of the Eyring2 and NRTL3 formulations, this model was called the Eyring-NRTL model. The Eyring viscosity model2 is based on absolute rate theory, and the NRTL model3 is based on the local composition concept for binary interactions at equilibrium. The original Eyring-NRTL model1 is called a component-based model because diffusion and flow are viewed to occur, according to the Eyring model, by a sequence of individual molecular, or component, jumps of solvent and polymer molecules. For the polymer-solvent systems studied, the Eyring-NRTL model provided a significant improvement over the conventional liquid mixture viscosity model based on Eyring’s model with ideal mixing.1 By considering polymer-solvent systems as binary systems, mixture viscosity data over the entire range of polymer compositions were correlated by the following binary form of the component-based Eyring-NRTL model.1 Original Component-Based Eyring-NRTL Viscosity Model (Binary Form)
ln(ηmV ˜ m) ) x˜ 1 ln(η1V ˜ 1) + x˜ 2 ln(η2V ˜ 2) ˜ 1 + x˜ 2V ˜2 V ˜ m ) x˜ 1V
g˜ *e RT
(1)
(
)
G′12τ′12 G′21τ′21 g˜ *e + ) x˜ 1x˜ 2 RT x˜ 1 + x˜ 2G′21 x˜ 1G′12 + x˜ 2
(3)
where η and V ˜ are the viscosity and molar volume, respectively, of the mixture (m) and binary components (1 and 2). The mixture composition is defined in terms of the component mole fraction (x˜ ). Mathematically, eq 3 is the same as the binary form of the NRTL model3 for molar excess Gibbs free energy (g˜ e). However, the excess term in the viscosity model is for the activated state molar excess Gibbs free energy (g˜ *e). The first two terms on the right-hand side of eq 1 can be thought of as the ideal term, and g˜ *e/RT can be thought of as an excess term. The ideal term is the Eyring model under ideal mixing conditions. The use of -g˜ *e/RT instead of +g˜ *e/RT in eq 1 is not consistent with the thermodynamic definition for an excess property. However, other investigators4-9 using local composition models for g˜ *e have used -g˜ *e/RT as a correction term to fit mixture viscosity data for nonpolymer liquid mixtures. For small molecules, the use of g˜ *e/RT in eq 1 can be rationalized because intermolecular attractive forces and increased nonrandomness would be expected to increase the intermolecular friction and viscosity. It should be noted that, with large polymer molecules, other factors such as polymer chain entanglement and polymer coil swelling may become equally significant in determining the intermolecular friction and viscosity. These effects are not accounted for in the Eyring-NRTL model.
(2)
* To whom correspondence should be addressed. Tel.: (440) 347-5558. Fax: (440) 347-5948. E-mail:
[email protected]. † The Lubrizol Corp. ‡ Aspen Technology, Inc.
Segment-Based Eyring-NRTL Viscosity Model (Binary Form) A segment-based model should be more physically realistic for large molecules when diffusion and flow are viewed to occur by a sequence of individual segment
10.1021/ie0401152 CCC: $27.50 © 2004 American Chemical Society Published on Web 08/12/2004
6232 Ind. Eng. Chem. Res., Vol. 43, No. 19, 2004
jumps into vacancies rather than large molecule jumps into large vacancies. It has been found that the ratio of heats of activation for viscous flow relative to heats of vaporization decreases as molecules of normal paraffin get larger, suggesting that vacancies for jumps are smaller than the size of larger molecules.10 Although the appropriate size of a segment for diffusion and flow of large molecules remains an open question, this paper will follow a segment-based approach similar to the Chen polymer NRTL model that was successfully used for polymer solution-phase equilibria.11 In the following development, the segment approach will be applied to both the Eyring rate-based model and the NRTL formulation. When the Chen model is followed,11 segments are defined as the repeat units of polymers and oligomers and solvents are normally treated as components. On occasion, oligomers of lower molecular weight might be treated as components. Application of the segment concept to homopolymers results in the transformation of eqs 1-3 into the following binary form of a segmentbased Eyring-NRTL model for polymer-solvent and polymer-polymer binaries.
ln(ηmVm) ) x1 ln(η1V1) + x2 ln(η2V2) + Vm ) x1V1 + x2V2
(
G21τ21 G12τ12 g*e ) x1x2 + RT x1 + x2G21 x1G12 + x2 x1 )
r1x˜ 1 r1x˜ 1 + r2x˜ 2
g*e RT
(4) (5)
)
(6)
(7)
G12 ) exp(-R12τ12)
(8)
G21 ) exp(-R12τ21)
(9)
N g˜ *e g*e ) RT Ns RT
(10)
The composition (x) and volume (V) variables in eqs 4-9 are now defined on the basis of segment moles instead of component moles. If component “1” is the polymer and component “2” is the solvent, r1 is the average number of segments in the polymer component and r2 ) 1 for the solvent when the solvent is viewed as a component. The polymer segment molar volume is now used because it is a unit for diffusion and flow in this segment-based Eyring model. When the solvent is viewed as a component, the segment molar volume for the solvent is identical with the solvent molar volume. For solvents that are large molecules, it will be more physically correct to also view the solvent as composed of segments. Viscosities η1 and η2 correspond to the polymer and solvent component viscosities, respectively. The sign of the excess term (g*e/RT) in eq 4 is now consistent with the thermodynamic definition for an excess property. However, g*e is defined in terms of moles of segments, and g˜ *e is defined in terms of component moles. The relationship between these two excess properties is provided by eq 10, where N is the total number of polymer and solvent moles and Ns is
the total number of polymer segment and solvent moles in the mixture. The excess term in the Eyring-NRTL viscosity model, like the NRTL excess Gibbs energy expression, has some semiempirical basis, and the model parameters carry plausible physical significance resulting from the local composition concept. The segment version of the NRTL model assumes that the liquid has a lattice structure that can be described as cells with central species (segments or components) that are surrounded by various species in the mixture. The distribution of the species around these central species is determined by energies of interaction for the activated state. The nonrandomness factors, R’s, correspond to the coordination numbers of the liquid lattice structures, while the binary interaction parameters, τ’s, correspond to the differences of interaction energies between two different species and between two identical species. For a binary mixture with τ21 < 0, the two species are attracted because the energy state of the 2-1 mixture is lower than the energy state of the 1-1 mixture. In this case, the local composition of species “2” relative to species “1” (in a local cell with “1” in the center) will be larger than the ratio of bulk compositions of “2” relative to “1”. For τ21 > 0, the opposite occurs. For τ21 ) 0, the ratio of local compositions (in the cell with “1” in the center) is the same as the ratio of bulk compositions. The use of a local composition model, such as NRTL, to model the excess term is a reasonable approach because intermolecular friction and viscosity should be affected by nearest neighbors. Parameters R12, τ12, and τ21 are the composition-independent parameters referred to here as the segment-based NRTL binary parameters for the activated state. Temperature dependency in the ideal term and excess term is embodied in the formulations used for the component viscosity (η), segment molar volume (V), and NRTL binary interaction parameters3 (τij’s). In practice, the NRTL parameters in the Eyring-NRTL viscosity models are determined by fitting experimental mixture viscosity data. The NRTL term in the segment-based viscosity model refers to an “energy barrier” associated with a “segment jump”, whereas the NRTL activity coefficient model refers to molecular interactions in the liquid phase. Clearly, there is an open question regarding the relationship between NRTL parameters obtained by fitting viscosity data and those obtained by fitting phase equilibrium data. Systematic studies of mixture viscosity data with the Eyring-NRTL model and mixture phase equilibria data with the NRTL model are needed to answer this question. Because the component- and segment-based EyringNRTL models do not model shear-rate-dependent viscosity, application is restricted to Newtonian and zero-shear viscosity applications. However, binary Eyring-NRTL models could be extended to shearrate-dependent viscosity applications using the approach of Song et al.13 Simplified Segment-Based Eyring-NRTL Viscosity Model (General Form) The segment-based NRTL model for the excess term (g˜ e/RT) has been generalized to multicomponent mixtures.12 The proposed generalized model for homogeneous multicomponent mixtures containing solvents,
Ind. Eng. Chem. Res., Vol. 43, No. 19, 2004 6233 Table 1. Pure-Component Properties of Mixtures Studied componenta
Mn
2-propanol25 acetone25 methanol40 water40 styrene40 PS40 n-heptane0 branched polyolefin0 branched polyolefin0 isooctane20 PIB20 PIB20 PIB20 PIB20 PIB20 PIB20 THF30 PEG30 PEG30 pentamer30 PDMS30 PDMS30 PDMS30 PDMS30 PDMS30
60.1 58.1 20.0 18.0 104.2 171000 100.2 2282 2917 114.2 900 20000 110000 240000 650000 1200000 72.1 192 408 383 1900 12000 44000 76000 112000
density, kg/m3
viscosity, Pa‚s × 1000
data source
780.4 784.4 775.0 987.3 887.0
2.036 0.3025 0.4467 0.6711 0.5736 1.0 × 109 0.506 1.07 × 107 1.67 × 107 0.51 3.18 × 103 4.20 × 105 3.22 × 108 4.10 × 109 9.0 × 109 1.0 × 1012 0.442 37.68 71.78 1.65 27 8.3 × 102 2.3 × 104 2.4 × 105 8.2 × 105
14 14 Aspen-Plus Rel 11.1 Aspen-Plus Rel 11.1 Aspen-Plus Rel 11.1 b Aspen-Plus Rel 10.2 1 1 Aspen-Plus Rel 11.1 b and c b and c b and c b and c b and c b and c 18 18 18 19 19 19 19 19 19
693 892 892 695 887 914 915 915 915 915 876.4 1117 1118 866 949.7 965.3 965.3 965.3 965.3
a Superscript on components corresponds to temperature (°C). b Unavailable pure polymer viscosity was treated as an adjustable parameter in the model and extrapolated from mixture viscosity data. c Correlated and extrapolated density data from BASF.
( )
oligomers, homopolymers, and copolymers is
ln ηm )
∑i xi ln ηi + ∑i xi xi )
∑I ri,I x˜ I
∑j xjGjiτji
∑j ∑J rj,J x˜ J
ηi )
∑I ri,IηI ∑I ri,I
∑j xjGji
modeling and simulation software that already makes use of the segment-based NRTL model for phase equilibria.
(11)
(12)
(13)
Gij ) exp(-Rijτij)
(14)
Gji ) exp(-Rijτji)
(15)
Lower case indexes (i,j) refer to segments, such as repeating units in oligomers and polymers. Upper case indexes (I,J) refer to components, such as solvents, oligomers, and polymers. Because the segment-based extension of the EyringNRTL viscosity model is based on the concept of similarsized entities jumping into vacancies, it is reasonable to assume that the volume dependency in eq 4 under these conditions would not be significant and would essentially mathematically cancel out of eq 4. Under these conditions, the volume can be ignored, resulting in the above simplified general form (eqs 11-15) that can be efficiently implemented and used in process
Application of the General Form The above general form can mathematically accommodate a broad range of homogeneous liquid viscosity applications as discussed in specific cases below. Pure Components. For component I, such as solvent or homopolymer, the excess term is zero because there is only one segment (xi ) 1) and eqs 11-13 reduce to ηm ) ηI, the component viscosity. For a copolymer component I, the eq 13 mixing rule reduces to ηi ) ηI, the copolymer viscosity. Assuming that no interaction exists between different segments within the same copolymer,11,12 the excess term is zero and ηm ) ηI, the copolymer viscosity. Nonpolymer Mixtures. For nonpolymer mixtures, each segment corresponds to a different component in the mixture and the number of segments in each component is unity, by definition. Under these conditions, eqs 11-15 are directly applicable and reduce to a component-based general form. Binary Polymer-Solvent Mixtures. For an oligomer, homopolymer, or copolymer in a single solvent, eqs 11-15 reduce to the simplified, volume-free, version of eqs 4-9. Single Polymer-Multicomponent Solvent Mixtures. For a single oligomer, homopolymer, or copolymer in a multicomponent mixture of solvents, eqs 1115 are directly applicable. Polymer Mixtures. Blends of oligomers, homopolymers, and copolymers require special treatment because the eq 13 mixing rule may be inadequate when different polymers in the blend have common segments but different microstructure and different viscosities. Under these conditions, respective oligomers, homopolymers, and copolymers in a mixture can be considered to be
6234 Ind. Eng. Chem. Res., Vol. 43, No. 19, 2004 Table 2. Component-Based Eyring-NRTL Viscosity Model Binary Parameters and AAD % from Experiment component 1
component 2
polymer Mn
AAD %
R12
τ12 at 20 °C
τ21 at 20 °C
2-propanol branched polyolefin branched polyolefin PIB PIB
acetone n-heptane n-heptane isooctane isooctane
NA 2282 2917 900 20000
0.5 13.9 18.0 26.6 40.3
0.20 0.20 0.20 0.20 0.20
2.9228 -11.6213 -13.1246 -8.8204 -14.3223
-0.2696 1.2278 4.0914 0.0 0.0
Figure 1. Mixture viscosity of 2-propanol in acetone14 at 25 °C.
Figure 3. Mixture viscosity of methanol in water15 at 40 °C.
Figure 4. Mixture viscosity of PS in styrene16 at 40 °C. Figure 2. Mixture viscosity of 2917 Mn branched polyolefin in n-heptane1 at 0 °C.
different and defined as unique components. By association of a unique component viscosity and unique segments for each different oligomer, homopolymer, and copolymer, eqs 11-15 can be used to calculate the ideal term and excess term contributions to the mixture viscosity. For example, a binary homopolymer blend reduces to the simplified, volume-free, version of eqs 4-9. Results and Discussion Several examples are presented here to illustrate the capability and limitations of the original componentbased Eyring-NRTL viscosity model.1 Corresponding component properties used in this work and data sources are listed in Table 1. In some cases, the pure polymer viscosity was not available. In these cases, the polymer viscosity was treated as an adjustable parameter and “extrapolated” with the model, as indicated in Table 1. Figures 1 and 2 demonstrate the ability of eqs 1-3 to correlate mixture viscosity data for 2-propanol in acetone14 and a low molecular weight branched polyolefin in n-heptane.1 The straight line in Figure 1 refers to the conventional model, corresponding to the ideal component-based Eyring mixture viscosity model for similar molecules.1 In the ideal case, the excess term is zero in eq 1 and the molar volumes cancel.
The corresponding binary parameters and average absolute percent deviation (AAD) from experiment are listed in Table 2 for the component-based model. A comparison of the two branched polyolefin polymers in Table 2 suggests that the component-based binary interaction parameters are not independent of the molecular weight. From the standpoint of predictability, it would be desirable for binary parameters to be independent of polymer Mn. An attempt was made to correlate the viscosity data of mixtures of polyisobutylene (PIB) in isooctane,17 over a Mn range of 900-1 200 000, using the componentbased Eyring-NRTL model. Table 2 illustrates that AAD increases significantly with Mn. It was not possible with the component-based model to fit any of the PIBisooctane viscosity data above 20 000 Mn because of further increases in AAD with polymer Mn. Examples are presented here in Figures 3-8 to illustrate the capability of the segment-based EyringNRTL viscosity model (eqs 4-9). The corresponding segment-based binary parameters are listed in Table 3 along with AAD. All polymers and oligomers were modeled by segments, and all solvents, except pentamer oligomer, were modeled as components. Figure 3 illustrates the capability of the segmentbased Eyring-NRTL viscosity model for correlating methanol-water mixture viscosity data.15 In this case, the segment-based model reduces to the component-
Ind. Eng. Chem. Res., Vol. 43, No. 19, 2004 6235 Table 3. Segment-Based Eyring-NRTL Viscosity Model Binary Parameters and AAD % from Experiment component 1
component 2
polymer Mn
AAD %
R12
τ12 at 20 °C
τ21 at 20 °C
methanol PS branched polyolefin branched polyolefin PIB PIB PIB PIB PIB PIB PEG PEG PDMS PDMS PDMS PDMS PDMSa PDMSa PDMSa
water styrene n-heptane n-heptane isooctane isooctane isooctane isooctane isooctane isooctane THF THF pentamer pentamer pentamer pentamer PDMS PDMS PDMS
NA 171 000 2 282 2 917 900 20 000 110 000 240 000 650 000 1 200 000 192 408 12 000 44 000 76 000 112 000 1 900 44 000 76 000
0.8 13.1 19.0 22.8 19.7 22.5 18.3 18.3 20.2 19.8 0.2 1.5 11.3 5.6 5.1 5.1 4.2% 2.4% 1.9%
0.220 0.050 0.200 0.200 0.050 0.050 0.030 0.015 0.017 0.015 0.200 0.200 0.200 0.200 0.200 0.200 0.070 0.070 0.070
-0.4403 0.4896 0.1721 2.5885 -4.2077 -7.0877 -13.1689 -20.0567 7.8338 5.2191 1.2712 1.2793 3.5121 3.4693 3.4569 3.4742 1.5111 1.1513 0.2629
6.5139 67.6100 -11.3197 -12.0833 76.1142 51.5218 49.3414 55.9865 189.0267 182.3186 -3.9293 -4.2630 -5.3788 -5.9015 -6.0883 -5.8283 15.9432 0.5462 0.1694
a
PDMS with 112 000 Mn.
Figure 5. Mixture viscosity of PIB in isooctane17 at 20 °C.
Figure 7. Mixture viscosity of PDMS in pentamer19 at 30 °C.
Figure 6. Mixture viscosity of PEG in THF18 at 30 °C.
Figure 8. Mixture viscosity of PDMS blends20 at 30 °C.
based case because the segments are defined as molecular species, or components. The polymer-solvent and polymer-polymer systems evaluated include polystyrene (PS) in styrene16 (Figure 4), PIB in isooctane17 (Figure 5), poly(ethylene glycol) (PEG) in tetrahydrofuran (THF)18 (Figure 6), poly(dimethylsiloxane) (PDMS) in pentamer19 (Figure 7), and PDMS-PDMS20 (Figure 8). All PDMS polymers and the pentamer oligomer are methyl-terminated. The pentamer is dodecamethylpentasiloxane. For PS in styrene, the simplified segment-based model (eqs 1115) was used. It is evident that the segment-based Eyring-NRTL viscosity model is capable of correlating mixture viscos-
ity data for all of the evaluated mixtures containing polymers. Although the AADs for branched polyolefin polymers increased from the component-based approach, the AADs for PIB-isooctane are now lower and somewhat uniform over the range of Mn studied. The segment-based binary parameters are not exactly independent of polymer Mn for the branched polyolefin and PIB. However, in some cases the segment-based binary parameters are essentially independent of polymer Mn. For PEG in THF, the segment-based binary parameters are essentially independent of Mn over the range studied. Also, for 12 000 to 112 000 Mn PDMS in pentamer, the segment-based binary parameters are essentially independent of Mn.
6236 Ind. Eng. Chem. Res., Vol. 43, No. 19, 2004
Nomenclature
Figure 9. Dimensionless deviations from the ideal term.
Further systematic viscosity studies of mixtures containing polymers should be valuable for developing guidelines for viscosity predictability. Although these studies are beyond the scope of this work, several examples in this work illustrate the potential for developing guidelines for viscosity predictability using the segment-based Eyring-NRTL viscosity model. The excess term in eq 4 can be thought of as a correction accounting for deviations from the ideal term in the segment-based Eyring-NRTL viscosity model. Two examples of deviations from the ideal term are illustrated in Figure 9. Positive deviations are observed for PIB-isooctane mixtures, and negative deviations are observed for branched polyolefin-heptane mixtures. It is evident that, while the NRTL term captures the deviations, such deviations cannot be accounted for with the segment-based configurational entropy term that is shown as the dotted line in Figure 9. In summary, although local composition models may have the potential for modeling local transport phenomena like intermolecular friction, the application of the NRTL model to local transport phenomena in mixtures of polymers is largely semiempirical and incomplete because the Eyring-NRTL model does not account for the effects of polymer chain entanglement and polymer coil swelling. As a result of these omissions, good correlations and predictability in some cases may be limited. Conclusions The segment-based Eyring-NRTL viscosity model is an improvement over the component-based EyringNRTL viscosity model for correlating and predicting the viscosity of mixtures containing polymers. It was found capable of correlating mixture viscosity data for all of the polymer-solvent and polymer-polymer systems evaluated in this work. It also succeeded in correlating PIB-isooctane mixture viscosity data, over the entire 900-1 200 000 PIB Mn range, when the componentbased Eyring-NRTL model failed. For several cases evaluated, the segment-based Eyring-NRTL viscosity model was also found to provide some predictive capability because the binary parameters were essentially independent of polymer Mn. Acknowledgment The authors acknowledge The Lubrizol Corp. and Aspen Technology Inc. for supporting this work and providing the approval for publication.
D AAD ) (100/ND)∑N 1 |viscositycalc - viscosityexp|/viscosityexp, ND ) number of datapoints g˜ *e ) component-based molar excess Gibbs free energy of activation for flow g*e ) segment-based molar excess Gibbs free energy of activation for flow Mn ) polymer number-average molecular weight N ) total number of component moles Ns ) total number of segment moles ri,I ) average number of segments i in component I R ) gas constant T ) absolute temperature V ˜ I ) component-based molar volume, volume/moles of component I Vi ) segment-based molar volume, volume/moles of segment i x˜ I ) component-based mole fraction, moles of component I/total moles of components xi ) segment-based mole fraction, moles of segment i/total moles of segments
Greek Symbols R and τ ) NRTL binary parameters η ) dynamic viscosity Subscripts m ) mixture 1 ) polymer 2 ) solvent i, j ) indexes for segments, such as repeating units in oligomers and polymers I, J ) indexes for components, such as solvents, oligomers, and polymers Superscript ′ ) component-based
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Received for review April 16, 2004 Revised manuscript received June 28, 2004 Accepted July 2, 2004 IE0401152
