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Ind. Eng. Chem. Res. 2004, 43, 2602-2604
Relationship between the Intrinsic Viscosity and Eyring-NRTL Viscosity Model Parameters Lawrence T. Novak† The Lubrizol Corporation, 29400 Lakeland Boulevard, Wickliffe, Ohio 44092-2298
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. This paper defines the mathematical relationship between the intrinsic viscosity and Eyring-NRTL viscosity model parameters. The mathematical relationship allows the extrapolation of viscosity data, over a wide range of concentrations, to the dilute solution intrinsic viscosity. For illustration purposes, intrinsic viscosities for several branched polyolefin-solvent systems were calculated from Eyring-NRTL parameters and compared to literature intrinsic viscosities for fractionated polyisobutylenes and oligomers over a Mn range of 224-1 500 000. Introduction Polymer-solvent mixture viscosity is a physical property that is often important in research, development, and engineering. The importance of polymer-solvent viscosity modeling is reflected by a large number of publications on this topic over the past 70 years. Selected publications1 of Flory contain key early publications relating intrinsic viscosity from dilute polymer solution studies to polymer coil size, coil swelling, and molecular weight. So, intrinsic viscosity is useful for dilute solution polymer characterization and for determination of the polymer molecular weight. In applied research and development, practical issues quite often result in the determination of polymersolvent mixture viscosities over a range of concentrations that cannot be graphically extrapolated to the dilute region for reasonable estimates of intrinsic viscosity. The purpose of this paper is to define the mathematical relationship between the intrinsic viscosity and Eyring-NRTL (nonrandom two-liquid) viscosity model2 parameters to allow the extrapolation of more concentrated polymer-solvent viscosity data to the dilute solution intrinsic viscosity of interest in polymer science. Novak2 proposed a multicomponent liquid mixture viscosity model to describe the mixture viscosity over the entire range of composition and over a range of temperature. On the basis of the Eyring3 and NRTL4 theories, this model was called the Eyring-NRTL model. The NRTL model is based on a local composition model representing the nonrandom distribution of molecules in a mixture at equilibrium. Considering polymersolvent systems as pseudobinary, viscosity data over the entire range of polymer composition were correlated by the following binary form of the Eyring-NRTL model. For a binary system, the Eyring-NRTL model for the mixture viscosity (ηm) was written as2
ln(ηmV ˜ m) ) x˜ 1 ln(η1V ˜ 1) + x˜ 2 ln(η2V ˜ 2) -
g˜ *e RT
(1)
† Tel.: (440) 347-5558. Fax: (440) 347-5948. E-mail: ltn@ lubrizol.com.
(
τ12G12 τ21G21 g˜ *e ) x˜ 1x˜ 2 + RT x˜ 1 + x˜ 2G21 x˜ 1G12 + x˜ 2
)
(2)
The equation variables used in this paper are defined in the Nomenclature section. The first two terms on the right-hand side of eq 1 can be thought of as the “ideal part”, and g˜ *e/RT can be thought of as the “nonideal part”. The NRTL model was used to predict the effect of composition and temperature on the molar excess Gibbs free energy of activation (g˜ *e) in eq 1. Parameters τ12 and τ21 are the composition-independent parameters referred to here as the NRTL binary interaction parameters for the activated state. In theory, the binary parameters determined from viscosity data should be the same as the binary parameters determined from phase equilibrium data. However, a relationship between the phase equilibrium NRTL parameters and the viscosity NRTL parameters has not been established. Because the above Eyring-NRTL model does not model the shear-rate-dependent viscosity, application is restricted to Newtonian and zero-shear viscosity applications. However, the Eyring-NRTL model could be extended to shear-rate-dependent viscosity applications using the approach of Song et al.5 Mathematical Relationship between the Intrinsic Viscosity and Eyring-NRTL Model Parameters If the mixture viscosity (ηm) in eq 1 is written in terms of the specific viscosity (ηsp) and then differentiated and evaluated at zero concentration of the polymer, the following equation is obtained for the intrinsic viscosity, when the volume change on mixing is zero.
{( ) (
[η] ) ln
)
}
η1 V ˜1 V ˜1 V ˜2 - 1 - (τ21 + τ12G12) (3) η2V ˜2 V ˜2 Mn
On the basis of the Eyring-NRTL model, the polymer dilute solution property of intrinsic viscosity is a function of pure-component viscosities, molar volumes, NRTL binary parameters, polymer number-average molecular weight, and temperature. Temperature de-
10.1021/ie040010z CCC: $27.50 © 2004 American Chemical Society Published on Web 04/10/2004
Ind. Eng. Chem. Res., Vol. 43, No. 10, 2004 2603
pendency is implicit in the pure-component viscosities (η), molar volumes (V), and NRTL binary interaction parameters4 (τij’s). The intrinsic viscosity has been shown to be a function of the polymer coil size and molecular weight for nearmonodisperse linear polymers.6
[η] ) Φ
( ) ( ) 〈r〉3/2 〈r2〉 )Φ Mn Mn
3/2
Mn1/2 ) KMn1/2
(4)
At high enough molecular weight, Φ is independent of the molecular weight and is called the “universal constant”. For polyisobutylene (PIB) of molecular weight greater than 50 000, Φ is independent of the molecular weight.6 The polymer coil size (〈r2〉1/2) can be written in terms of an unperturbed coil size (〈ro2〉1/2) and a chain expansion factor (R) quantifying the effect of coil expansion caused by a good solvent, relative to a Θ solvent, and temperature.
( ) 〈r2〉1/2
〈ro2〉1/2
)R
(5)
Combining eqs 3 and 4 produces a relationship between the polymer coil size, universal constant, and EyringNRTL viscosity model parameters for higher molecular weight near-monodisperse linear polymers.
{( ) (
〈r2〉3/2 ) ln
)
}
η1V ˜1 V ˜1 V ˜2 (6) - 1 - (τ21 + τ12G12) η2V ˜2 V ˜2 Φ
Equations 4 and 5 are classical polymer science equations that provide a basis for characterizing polymers in the dilute solution limit. When the intrinsic viscosity and Mn are known, the polymer coil size is determined. Equation 4 can also be used as a correlating equation to determine Mn from intrinsic viscosity measurements when K is known for the particular polymer-solvent system. Equations 3 and 6 provide a basis for extrapolation of mixture viscosity data, over a wide range of concentrations, to the dilute solution limit to determine the intrinsic viscosity and polymer coil size. Extrapolated mixture viscosity data over a range of Mn’s could be used to develop an intrinsic viscosity-Mn correlation. Discussion The intrinsic viscosity can be directly calculated from eq 3 when pure-component properties and binary parameters are known. Binary parameters are determined by correlating mixture viscosity data over a range of polymer concentrations.2 Inclusion of viscosity data at lower polymer concentrations should improve the estimate of the intrinsic viscosity from eq 3. Estimating unique binary parameters from the intrinsic viscosity is not possible unless pure-component properties and reasonable estimates are known for R12 and the concentrated solution binary interaction parameter (τ21), based on knowledge from similar polymersolvent systems. Under these conditions, the dilute solution binary interaction parameter (τ12) can be uniquely estimated. The binary form of the Eyring-NRTL model was used to correlate the Newtonian viscosity over a range of compositions (7-99.5 wt % polymer) and over a range of temperatures for several branched polyolefin-solvent systems.2 Additional literature data provided an op-
Figure 1. Intrinsic viscosity vs Mn for branched polyolefins: (b) fractionated PIB in benzene at 25 °C,10 (O) fractionated PIB in diisobutylene at 20 °C,9 ([) PIB in isooctane at 20 °C,7 (2) PIB in 500 Mn polybutene at 20 °C,8 and (4) branched polyolefin in n-heptane at 20 °C.2
portunity to determine Eyring-NRTL parameters for higher molecular weight PIB polymers.7,8 PIB-isooctane mixtures covered a 6-66 wt % PIB range,7 and PIBoligomer (500 Mn polybutene) mixtures covered a 0.719 wt % PIB range.8 Correlation of these sets of mixture viscosity data2,7,8 at Newtonian and zero-shear conditions produced respective sets of the three binary parameters (R12, τ12, and τ21). Intrinsic viscosities were then calculated from eq 3 by using respective pure-component properties and binary parameters obtained from concentrated mixture viscosity data.2,7,8 The solvent viscosity and density were obtained from the DIPPR databank correlations. PIB and oligomer density and viscosity were obtained from correlated and extrapolated BASF data. Branched polyolefin density and viscosity were obtained from correlated Lubrizol data. Intrinsic viscosities calculated from concentrated mixture viscosity data,2,7,8 using eq 3, are compared with measured intrinsic viscosities on fractionated PIBs9,10 in Figure 1. All intrinsic viscosities in Figure 1 are for 20 °C, except for ref 10. Use of the Eyring-NRTL model to extrapolate 25 °C intrinsic viscosities to 20 °C demonstrated that the effect of this temperature difference is only about 2%. Reference 10 appears to be the best reference data found on the intrinsic viscosity for PIB, because a Θ solvent for PIB was used and the fractionated PIB was characterized in terms of Mn and Mw/Mn. The trend lines in Figure 1 correspond to respective intrinsic viscosity-Mn correlations for the calculated7,8 and measured9,10 intrinsic viscosities. All trend lines based on calculated (eq 3) and measured intrinsic viscosities are linear, and the power on Mn is close to 1/ , as predicted by eq 4. These results suggest that eq 2 3 can be used to estimate the intrinsic viscosity by extrapolating the concentrated polymer-solvent mixture viscosity to the dilute regime. One should not expect a one-to-one agreement of all data in Figure 1 because there are differences in solvents, polydispersities (Mw/Mn), and methods used to determine Mn. Better solvents will shift the [η] vs Mn curves upward because of the effect of coil swelling (R). Higher polydispersities (Mw/Mn) will shift the [η] vs Mn curves to the left because Mv and Mw are higher moments than Mn. If one were to add a small amount
2604 Ind. Eng. Chem. Res., Vol. 43, No. 10, 2004
of lower molecular weight polymer to a higher molecular weight sample, Mn would be reduced more than the higher moments Mv and Mw and the intrinsic viscosity would be marginally reduced because it should correlate best to Mv for polydisperse systems.11
ηsp ) ηr - 1 ) specific viscosity [η] ) (ηsp/C)Cf0 ) intrinsic viscosity 〈r2〉1/2 ) root-mean-square distance from the beginning to the end of a polymer chain coil Φ ) universal constant11,12 ) 2.1 × 1021, for [η] in dL/g and 〈r2〉1/2 in cm
Conclusions
Subscripts
For the polymer-solvent systems studied, the Eyring-NRTL model provides a basis (eq 3) for extrapolation of viscosity data, over a wide range of concentrations, to the dilute solution intrinsic viscosity. This capability should be useful in extending viscosity data, from applied research and development studies, to the dilute solution regime for dilute solution polymer characterization and for determination of the polymer molecular weight.
m ) mixture 1 ) polymer 2 ) solvent o ) unperturbed polymer coil state
Acknowledgment The author acknowledges William Starr and Vadim Lvovich, The Lubrizol Corp. William Starr critically reviewed the manuscript, and Vadim Lvovich translated the study on PIB-isooctane viscosity reported in Usp. Khim.7 The author also thanks The Lubrizol Corp. for supporting this work and providing the approval for publication. Nomenclature C ) polymer concentration g˜ *e ) molar excess Gibbs free energy of activation Gij ) exp(-Rijτij) K ) proportionality constant in [η]-Mn correlation, eq 4 Mn ) polymer number-average molecular weight Mw ) polymer weight-average molecular weight Mv ) polymer viscosity-average molecular weight R ) gas constant T ) absolute temperature V ˜ ) molar volume x˜ ) mole fraction Greek Symbols R ) chain expansion factor, defined in eq 5 Rij, τij ) NRTL binary parameters η ) dynamic viscosity ηr ) ηm/η2 ) relative viscosity
Literature Cited (1) Mandelkern, L.; Mark, J. E.; Suter, U. W.; Yoon, D. Y. Selected Works of Paul J. Flory; Stanford University Press: Stanford, CA, 1985. (2) Novak, L. T. Modeling the Viscosity of Liquid Mixtures: Polymer-Solvent Systems. Ind. Eng. Chem. Res. 2003, 42, 1824. (3) Powell, R. E.; Roseveare, W. E.; Eyring, H. Diffusion, Thermal Conductivity, and Viscous Flow of Fluids. Ind. Eng. Chem. 1941, 33, 430. (4) Renon, H.; Prausnitz, J. M. Local Compositions in Thermodynamic Excess Functions for Liquid Mixtures. AIChE J. 1968, 14 (1), 135. (5) Song, Y.; Mathias, P. M.; Tremblay, D.; Chen, C. Liquid Viscosity Model for Polymer Solutions and Mixtures. Ind. Eng. Chem. 2003, 42, 2415. (6) Flory, P. J.; Fox, T. G. Treatment of Intrinsic Viscosities. J. Am. Chem. Soc. 1951, 73, 1904. (7) Tager, A. A.; Dreval, V. E. Newtonian viscosity of concentrated solutions of polymers. Usp. Khim. 1967, 36, 888. (8) Ueda, S.; Kataoka, T. Steady-Flow Viscous and Elastic Properties of Polyisobutylene Solutions in Low Molecular Weight Polybutene. J. Polym. Sci., Polym. Phys. Ed. 1973, 11, 1975. (9) Flory, P. J. Molecular Weights and Intrinsic Viscosities of Polyisobutylenes. J. Am. Chem. Soc. 1943, 65, 372. (10) Abe, F.; Einaga, Y.; Yamakawa, H. Intrinsic Viscosity of Oligo- and Polyisobutylenes. Treatments of Negative Intrinsic Viscosities. Macromolecules 1991, 24, 4423. (11) Flory, P. J. Principles of Polymer Chemistry; Cornell University Press: Ithaca, NY, 1953; p 312. (12) Newman, S.; Krigbaum, W. R.; Laugier, C.; Flory, P. J. Molecular Dimensions in Relation to Intrinsic Viscosities. J. Polym. Sci. 1954, 14, 451.
Received for review January 8, 2004 Revised manuscript received March 24, 2004 Accepted March 24, 2004 IE040010Z
