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Ind. Eng. Chem. Res. 2006, 45, 844-855
Model for Calculating the Viscosity of Non-Newtonian Aqueous Solutions of Poly(ethylene glycol) 6000 at 313.15 K and 0.1 MPa Raphael da C. Cruz,† Rosana J. Martins,†,‡ Manoel J. C. Esteves,† Ma´ rcio J. E. de M. Cardoso,*,† and Oswaldo E. Barcia† Laborato´ rio de Fı´sico-Quı´mica de Lı´quidos e Eletroquı´mica, Departamento de Fı´sico-Quı´mica, Instituto de Quı´mica, UniVersidade Federal do Rio de Janeiro, Centro de Tecnologia, Bloco A, sala 411, Cidade UniVersita´ ria, CEP 21949-900, Rio de Janeiro, RJ, Brazil, and Departamento de Fı´sico-Quı´mica, Instituto de Quı´mica, UniVersidade Federal Fluminense, Outeiro de Sa˜ o Joa˜ o Batista s/n°, CEP 24020-150, Nitero´ i, RJ, Brazil
In this work, a model for calculating the dynamic viscosity of polymer solutions was developed. The model is based on the Eyring absolute rate theory and on the solution theory of McMillan-Mayer. An equation of state is used for the calculation of the solution osmotic pressure and, thus, the excess molar McMillanMayer free energy. The final expression shows an explicit dependence between the viscosity of the polymer solution and the applied shear stress. The proposed model contains three terms. The first term describes the viscosity of an ideal polymer solution. The second term takes into account non-Newtonian behavior of the polymer solution. Finally, the third term represents the deviation from the thermodynamic ideal behavior. The whole model presented five adjustable parameters, with two of them also considered as being a function of the applied shear stress. To test the proposed model, we have measured experimental rheological data for poly(ethylene glycol) aqueous solutions (nominal molecular weight 6000 g/mol) for nine different polymer concentrations, at different shear rates, at 313.15 K and 0.1 MPa. The proposed model has been used for correlating the experimental viscosity data of these polymer solutions at different values of the applied shear stress and polymer concentration. It has been found that the agreement between the experimental and calculated values is within the experimental error. 1. Introduction The poly(ethylene glycol) (PEG) is a linear polymer that is soluble in water and in a large number of organic solvents and presents lower (LCST) and upper (UCST) critical solution temperatures.1-4 Because of its lubricity, good stability, and low toxicity, it is used in lubricants, electronics, cosmetics, and pharmaceuticals.5 PEG aqueous solutions mixed with other polymers or salts are frequently used as biomolecules partitioning mediums.6 Since PEG is compatible with biopolymers, it can replace them, providing an insight into the understanding of their behavior and biological functions.7 In addition, its capacity for inhibiting protein adsorption makes PEG micelles, PEG gels, and others aggregates of PEG good candidates for drug delivery purposes.8,9 Another important use of PEG is in the oil industry, as a polymeric flocculant agent for water-oil emulsions.10 There is a quite large number of recently published papers on the viscosity measurements of PEG aqueous and nonaqueous solutions, for different PEG molecular weights, with or without added salts, at different temperatures.11-22 Gonza´lez-Tello et al.11 have used a rotatory viscometer (Contraves Rheomat 180E/ R) and have found that, for the systems they have studied (concentrated aqueous solutions of PEG with molecular weights of 1000, 3350, and 8000 g/mol), the measured viscosity values are independent of the shear rate. Rahbari-Sisakht et al.12 mentioned the Gonza´lez-Tello et al.11 results as a justification for assuming Newtonian flow behavior in their measurements. Several other authors13-22 have performed their experiments by * To whom correspondence should be addressed. Tel.: +55-212562-7172. Fax: +55-21-2562-7265. E-mail:
[email protected]. † Universidade Federal do Rio de Janeiro. ‡ Universidade Federal Fluminense.
means of open-gravitational capillary viscometers. The calculation of the viscosity of a Newtonian polymer solution as a function of the polymer concentration has only recently been addressed in the literature.23-27 Schnell and Wolf23 have proposed a model based on the concept of energy dissipation. This model is developed by means of particle surface fractions. Their model has four adjustable parameters, and it has been used for correlating zero-shear viscosity data for two polymer-solvent systems (diethyl phthalate/poly(vinyl acetate) and diethyl phthalate/poly(methyl acrylate)). Song et al.24 have also proposed a model for calculating the viscosity of Newtonian polymeric solutions. Viscosity literature experimental data for solutions of PEG in 1,3-dioxolane and of polystyrene in styrene were used for adjusting the model parameters. In their model, Song and co-workers have used a modified Mark-Houwink equation28 to take into account the dependence of the viscosity of pure melted polymer on the temperature, and the Andrade equation29 to describe the temperature dependence of pure solvent viscosity. To take into account the effect of the polymer concentration in the viscosity of the polymeric solutions, the authors have used a modification of the mixing rule proposed by Kendall and Monroe.30 In their proposed modification, the mole fraction was changed to mass fraction and they have also added quadratic and antisymmetric cubic terms. Song et al.24 have also extended their model to take into account non-Newtonian behavior of polymer solution. The dependence of the solution viscosity on the shear rate was considered by means of the phenomenological model by Carreau and co-workers.31,32 Novak and co-workers25,26 proposed an Eyring-NRTL model in order to calculate the concentration dependence of the viscosity of a Newtonian polymer solution. The original form
10.1021/ie050876k CCC: $33.50 © 2006 American Chemical Society Published on Web 12/07/2005
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of the Eyring-NRTL model25 is component-based and is also applicable to solvent mixtures. In a subsequent work, a segmentbased model was developed.26 This segment-based EyringNRTL viscosity model has shown good agreement with experimental literature viscosity data for several polymer solvent systems. Sadeghi27 has recently proposed a segment-based EyringWilson model for correlating the viscosity of polymer solutions. For a binary polymer solution, the proposed model presents two binary interaction parameters. But for some systems, when the pure polymer viscosity is not avaible, it is necessary to take this property as an extra adjustable parameter of the model. The model has been applied to several binary polymer solutions showing a relatively good agreement with the experimental viscosity values. Comparatively, there are fewer papers in the literature on the experimental determination of the rheological flow behavior of PEG aqueous solutions.33-36 Powell and Schwarz33,34 have used both a Weissenberg Rheogoniometer and a Cannon-Fenske capillary viscometer (with variable pressure head) for determining the rheological curves of dilute and concentrated aqueous PEG solutions (molecular weight between 2.9 × 106 and 3.6 × 106 g/mol). They have observed the influence of the material degradation, due to aging, on their measurements. Ortiz et al.35 have studied the rheological bahavior of concentrated aqueous solutions of PEG with different molecular weights (from 2.8 × 105 to 4.3 × 106 g/mol) at 298 K. They have also studied mixed-solvent (50 wt % mixture of water and glycerine) PEG solutions. The authors performed both steady shear and oscillatory determinations by means of four different rheometers. The authors have also developed a modification of a previously proposed model37,38 for calculating the dimensionless viscosity of PEG solutions by means of two characteristic time parameters in order to take into account the polymer polydispersity. Kalashnikov36 has measured the rheological behavior of very dilute aqueous solutions of PEG (molecular weights: 9.1 × 104 to 1.4 × 107 g/mol). He has used a Zimm-Crothers type of rotational viscometer (for low shear rates, 0.1 to 10 s-1), a horizontal straight capillary viscometer (for intermediate shear rates, 50 to 500 s-1) and a commercially available OstwaldUbbelohde viscometer (for high shear rates, 1000 to 3000 s-1). Kalashnikov has also carefully determined the effects of the polymer degradation due to the time of the solution storage, the characteristics of the flow process, and the solution temperature. The author has concluded that some type of supramolecular structure must exist even at very low polymer concentrations. The viscosity of non-Newtonian solutions can be calculated by means of a continuum mechanistic approach, for example, the Oldroyd-B and (K)-BKZ equations.39 These types of models contain a large number of parameters which are difficult or, in some cases, impossible to be determined by fitting the experimental viscosity data.40 A second approach for the modeling of the viscosity of a non-Newtonian solution is to consider the molecular modeling. In the last 50 years, several dynamic theories have been proposed with this aim.40-46 These dynamic theories are based on the diffusion-convection Fokker-Planck equation or on the stochastic Langevin equation.40,46 Because of the closure condition that appears in theses models, different methodologies have been employed for breaking the equation hierarchy, leading to different types of constitutive theories.40 Examples of these models are the Rouse theory,41 the Zimm theory,42 the “Internal
Viscosity Models” (IVM),43-45 the finitely extensible (FENE) models, and the Doi-Edwards reptation theory.40,46 The first three models are employed exclusively to very dilute solutions.44 The FENE models have been applied up to medially dilute solution.40 The Doi-Edwards reptation theory40,46 gives good results in the treatment of entangled polymer solutions. The complexity of the Doi-Edwards reptation theory40,46 makes its use in practical rheological applications difficult. The modeling of non-Newtonian flow behavior is often made by means of phenomenological models, like those of Casson,32,47 Cross,32,48 Carreau,31,32 Yasuda,32,49 and Eyring.32,50 In these models, the adjusted parameters, for a given solute and solvent system, depend on the solution composition, temperature, and pressure. These models are also restricted, in general, to the range of shear rate to which the model parameters have been fitted. It should be stressed that, when using such a model, the region of its validity must be carefully observed. The viscosity of a polymer solution can be conveniently expressed by means of the so-called reduced viscosity,51-53 ηred ) (η - η1)/cmη1, where η and η1 are, respectively, the solution and the pure-solvent viscosities and cm is the polymer concentration in g/cm3. The reduced viscosity of a polymer solution can be calculated by means of different empirical relationships, for example, the Schulz and Blaschke and Huggins equations.15,52 The Schulz and Blaschke equation is given as follows,
{
ηred ) [η] 1 +
}
kSB(η - η1) η1
(1)
and the generalized Huggins equation reads
ηred ) [η] + k1[η]2cm + k2[η]3cm2
(2)
where [η] is the polymer intrinsic viscosity (in a given solvent),51-53 kSB is the Schulz and Blaschke constant, and k1 and k2 are the two parameters of the generalized Huggins equation. It should be stressed, as previously remarked by Flory,51 that, for large values of the polymer intrinsic viscosity, one has to utilize the extrapolated values, to zero shear stress, of the solution reduced viscosity, for each polymer concentration, for calculating the polymer intrinsic viscosity. This procedure is necessary to ensure that the polymer solution viscosity values, for each polymer concentration, and, thus, the determined polymer intrinsic viscosity are not dependent on the applied shear stress. It is interesting to note that, in the limit of sufficiently dilute polymer solution (i.e., η ) ηid), both the Schulz and Blaschke and the generalized Huggins equations would reduce to the following expression:
ηid ) η1 + η1[η]cm
(3)
This expression shows a linear dependence of the ideal solution viscosity on the polymer concentration. In this article, we present a theoretical model based on the Eyring absolute rate theory54-56 and on the McMillan-Mayer solution theory57 for calculating the rheological behavior of nonNewtonian polymeric solutions. The present model presents an explicit shear stress dependence of the viscosity of the solution. The parameters of the proposed model have been fitted by means of experimental rheological data measured for aqueous solutions of PEG of a nominal molecular weight of 6000 g/mol (PEG6000). The measurements were made at the temperature
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Figure 1. PEG6000 aqueous solution experimental viscosity data at 313.15 K and 0.1 MPa, η, as a function of the applied shear stress, τ, for different polymer molar concentrations: (9) 0.0154; (b) 0.0219; (2) 0.0280; (1) 0.0342. The solid lines represent the polynomial regressed values.
of 313.15 K, and at the pressure of 0.1 MPa, for different values of shear stresses, in the range from 1.92 up to 48.60 Pa.58 The organization of the rest of this article is as follows. In Section 2, details of the experimental procedures are given. In Sections 3 and 4 are presented the theoretical framework of the proposed model. Section 5 presents the model results. In Section 6, our conclusions are summarized. Appendix I shows how to calculate the contribution of the solute species to the viscosity of the solution. Finally, in Appendix II, a brief review of the McMillan-Mayer solution theory is presented. 2. Experimental Section In this work, the aqueous solutions of PEG6000 were prepared with nine different weight fractions, namely, 0.0916, 0.1300, 0.1650, 0.2000, 0.2544, 0.2847, 0.3115, 0.3347, and 0.3546. To convert the polymer weight fractions to polymer molar concentrations, necessary for modeling proposes, a volumetric study of this system was performed and is presented elsewhere.59 The molar concentrations obtained were, respectively, 0.0154, 0.0219, 0.0280, 0.0342, 0.0438, 0.0493, 0.0542, 0.0584, and 0.0621 mol/L. The solutions were prepared by mass using a digital balance (Chyo YMC, model JK-180, Kyoto, Japan), with an uncertainty of (0.1 mg, in airtight stopped bottles. The estimated error in mass fraction is
