Thermodynamic Modeling of the Rheological Behavior of PEG Aqueous Solutions as a Function of the Solute Molecular Weight and Shear Stress, at 298.15 K and 0.1 MPa


  • Raphael da C. Cruz Fluminense Federal University
  • Márcio J.E. de M. Cardoso Federal University of Rio de Janeiro
  • Oswaldo E. Barcia Federal University of Rio de Janeiro



Viscosity, rheology, thermodynamics, modeling, polymer solutions, polyethylene glycol


This work presents a study of the influence of the molecular weight on the thermodynamic modeling of the viscosity of non-newtonian polymer solutions. The employed model is based on the absolute rate theory of Eyring and on the solution theory of McMillan-Mayer. The Soave-Redlich-Kwong equation of state was adopted for the calculation of the excess molar McMillan-Mayer free energy derived from the osmotic pressure of the solution. The model presents parameters that take account separately the different possibilities of interaction in the macromolecular environment. As the tertiary structure of a polymer molecule can be affected by applied shear stress, only the parameters related with the intramolecular interactions are dependent of the shear stress. The experimental rheological curves for different molecular weights of polyethylene glycol aqueous solutions have been measured at several concentrations, within the whole polymer solubility range, at 298.15 K and 0.1 MPa. The dependence on the molecular weight for all parameters of the model was analyzed and characterized. The dependence of the shear sensitive parameters on the shear stress was also studied.

Author Biographies

Raphael da C. Cruz, Fluminense Federal University

Department of Physical Chemistry, Institute of Chemistry

Márcio J.E. de M. Cardoso, Federal University of Rio de Janeiro

Department of Physical Chemistry,
Institute of Chemistry

Oswaldo E. Barcia, Federal University of Rio de Janeiro

Department of Physical Chemistry,
Institute of Chemistry


[1] Coleman BD, Noll W. Foundations of Linear Viscoelasticity. Rev Mod Phys 1961; 33: 239-49.
[2] Rivlin RS, Sawyers KN. Nonlinear Continucm Mechanics of Viscoelastic Fluids. Annu Rev Fluid Mech 1971; 3: 117-46.
[3] Feigl K, Öttinger HC. The equivalence of the class of Rivlin– Sawyers equations and a class of stochastic models for polymer stress. J Math Phys 2001; 42: 796-17.
[4] Leonov AI, Padovan J. On a kinetic formulation of elastoviscoplasticity. Int J Engng Sci 1996; 34: 1033-46.
[5] Wineman A. Nonlinear Viscoelastic Solids – A Review. Math Mech Solids 2009; 14: 300-66.
[6] Bernstein B, Kearsley EA, Zapas LJ. A Study of Stress Relaxation with Finite Strain. Trans Soc Rheol 1963; 7: 391- 10.
[7] Kaye A. Note No. 134 – Non-Newtonian Flow in Incompressible Fluids. Cranfield: College of Aeronautics 1962.
[8] Kaye A. A Bouncing Liquid Stream. Nature 1963; 197: 1001- 1002.
[9] Leal LG, Oberhauser JP. Non-Newtonian fluid mechanics for polymeric liquids: A status report. Korea-Australia Rheol J 2000; 12: 1-25.
[10] Rouse PE. A Theory of the Linear Viscoelastic Properties of Dilute Solutions of Coling Polymers. J Chem Phys 1953; 21: 1272-80.
[11] Zimm BH. Dynamics of Polymer Molecules in Dilute Solution: Viscoelasticity, Flow Birefringence and Dielectric Loss. J Chem Phys 1956; 24: 269-78.
[12] Cerf RJ. La Macromolecule en Chaine dans un Champ Hydrodynamique – Theorie Generale – Proprietes DynamoOptiques. Polym Sci 1957; 23: 125-50.
[13] Zimmerman RD, Williams MC. Evaluation of Internal Viscosity Models. Trans Soc Rheol 1973; 17: 23-46.
[14] Bazúa ER, Williams MC. Rheological Properties of Internal Viscosity Models with Stress Symmetry. J Polym Sci Polym Phys 1974; 12: 825-48.
[15] Doi M, Edwards SF. The Theory of Polymer Dynamics. New York: Oxford University Press 2001.
[16] Carreau PJ, De Kee DCR, Chhabra RP. Rheology of Polymeric Systems: Principles and Applications. Munich: Carl Hanser Verlag 1997.
[17] Bird RB, Stewart WE, Lightfoot EN. Transport Phenomena. 3rd press. New York: John Wiley & Sons, Inc. 1963.
[18] Eyring HJ. Viscosity, Plasticity, and Diffusion as Examples of Absolute Reaction Rates. Chem Phys 1936; 4: 283-91.
[19] Kincaid JF, Eyring H, Stearn AE. The Theory of Absolute Reaction Rates and its Applications to Viscosity and Diffusion in the Liquid State. Chem Rev 1941; 28: 301-65.
[20] Glasstone S, Laidler KJ, Eyring H. The Theory of Rate Process. New York: McGraw-Hill 1944.
[21] Bouchbinder E, Langer JS. Nonequilibrium thermodynamics of driven amorphous materials. III. Shear-transformationzone plasticity. Phys Rev E 2009; 80: 0311331-0311338.
[22] Macosko CW. Rheology: Principles, Measurements and Applications. New York: Wiley-VCH, Inc. 1994.
[23] Bohdaneck M, Ková , J. Viscosity of Polymer Solutions. Amsterdam: Elsevier Scientific Publishing Company 1982.
[24] Ninni L, Burd H, Fung WH, Meirelles AJA. Kinematic Viscosities of Poly(ethylene glycol) Aqueous Solutions. J Chem Eng Data 2003; 48: 324-29.
[25] González-Tello P, Camacho F, Blázquez. Density and Viscosity of Concentrated Aqueous Solutions of Polyethylene Glycol. J Chem Eng Data 1994; 39: 611-14.
[26] Cruz RC, Martins RJ, Esteves MJC, Cardoso MJEM, Barcia OE. Model for Calculating the Viscosity of Non-Newtonian Aqueous Solutions of Poly(ethylene glycol) 6000 at 313.15 K and 0.1 MPa. Ind Eng Chem Res 2006; 45: 844-55.
[27] McMillan Jr. WG, Mayer JE. The Statistical Thermodynamics of Multicomponent Systems. J Chem Phys 1945; 13: 276- 305.
[28] Kozak JJ, Knight WS, Kauzmann W. Solute-Solute Interactions in Aqueous Solutions. J Chem Phys 1968; 48: 675-90.
[29] Franks F, Pedley M, Reid DS. Solute Interactions in Dilute Aqueous Solutions, Part 1. – Microcalorimetric Study of the Hydrophobic Interaction. J Chem Soc Faraday Trans 1 1976; 72: 359-67.
[30] Cardoso MJEM, Medeiros JL. A thermodynamic framework for solutions based on the osmotic equilibrium concept – 1. General formulation. Pure Appl Chem 1994; 66: 383-86.
[31] Cruz RC, Esteves MJC, Teixeira RGD, Cardoso MJEM, Barcia OE. Calculation of the Osmotic Pressure and Theta Temperature of Polymer Solutions Through Cubic Equations of State and the McMillan–Mayer Solution Theory Framework. J Macromol Sci Phys 2010; 49: 1083-93.
[32] Cruz RC. Modelagem do Comportamento Reológico de Soluções Aquosas de Polietilenoglicol de Diferentes Massas Moleculares a 298,15 K e 0.1 MPa (Ph.D. Thesis). Rio de Janeiro: Federal University of Rio de Janeiro 2007.
[33] Cruz RC, Martins RJ, Cardoso MJEM, Barcia OE. Volumetric Study of Aqueous Solutions of Polyethylene Glycol as a Function of the Polymer Molar Mass in the Temperature Range 283.15 to 313.15 K and 0.1 MPa. J Solution Chem 2009; 38: 957-81.
[34] Kirini S, Klofutar C. Viscosity of aqueous solutions of poly(ethylene glycol)s at 298.15 K. Fluid Phase Equilib 1999; 155: 311-25.
[35] Benchabane A, Bekkour K. Rheological properties of carboxymethyl cellulose (CMC) solutions. Colloid Polym Sci 2008; 286: 1173-80.
[36] Ebagninin KW, Benchabane A, Bekkour K. Rheological characterization of poly(ethylene oxide) solutions of different molecular weights. J Colloid Interf Sci 2009; 336: 360-67.
[37] Schlichting H, Gersten K. Boundary Layer Theory. 8th ed. Heidelberg: Springer-Verlag 2000.
[38] Cruz RC. Transições em Baixo Cisalhamento e Evolução da Camada-Limite (Proceedings of the 12th Meeting of the Brazilian Society of Chemistry - Regional Rio de Janeiro). Rio de Janeiro: Brazilian Society of Chemistry 2009.
[39] Cruz RC, Vaz Jr CA. Transitions in low shear by development of the boundary layer (Proceedings of the 5 Brazilian Congress on Rheology). Rio de Janeiro: Brazilian Society of Rheology 2010.
[40] Wagner NJ, Brady JF. Shear thickening in colloidal dispersions. Phys Today 2009; 62: 27-32.
[41] Hoffman RL. Discontinuous and Dilatant Viscosity Behavor in Concentrated Suspensions. II. Theory and Experimental Tests. J Colloid Interface Sci 1972; 46: 491-506.
[42] Boersma WH, Laven J, Stein HN. Shear thickening (dilatancy) in concentrated dispersions. AIChE J 1990; 36: 321-32.
[43] Hoffman RL. Explanations for the cause of shear thickening in concentrated colloidal suspensions. J Rheol 1998; 42: 111-23.
[44] Bender JW, Wagner NJ. Optical Measurement of the Contributions of Colloidal Forces to the Rheology of Concentrated Suspensions. J Colloid Interface Sci 1995; 172: 171-84.
[45] Maranzano BJ, Wagner NJ. The effects of particle-size on reversible shear thickening of concentrated colloidal dispersions. J Chem Phys 2001; 114: 10514-27.
[46] Ma T, Wang S. Boundary-layer and interior separations in the Taylor–Couette–Poiseuille flow. J Math Phys 2009; 50: 03310101-03310129.
[47] Gebrüder Haake GmbH Rheometer RheoStress® RS150 – Sensor systems. Karlsruhe: Gebrüder Haake GmbH 1998.
[48] ThermoHaake GmbH Rheometer RheoStress® 1 – Instruction Manual. Karlsruhe: ThermoHaake GmbH 2002.
[49] Whorlow RW. Rheological Techniques. Sussex: Ellis Horwood 1980.
[50] Poling BE, Prausnitz JM, O’Connell JP. The Properties of Gases and Liquids. 5th ed. New York: McGraw-Hill 2001.
[51] Kalashnikov VN. Shear-rate dependent viscosity of dilute polymer solutions. J Rheol 1994; 38: 1385-403.
[52] Nelder JA, Mead RA. Simplex Method for Function Minimization. Computer J 1965; 7: 308-13.
[53] Himmelblau DM. Applied Nonlinear Programming. New York: McGraw-Hill 1972.
[54] Huggins ML. The Viscosity of Dilute Solutions of Long-Chain Molecules. IV. Dependence on Concentration. J Am Chem Soc 1942; 64: 2716-18.
[55] Huggins ML. Physical Chemistry of High Polymers. New York: John Wiley & Sons, Inc. 1958.
[56] Schulz GV, Blaschke FJ. Eine Gleichung zur Berechnung der Viscositäts-zahl für sehr kleine Konzentrationen,
[Molekulargewichtsbestimmungen an makromolekularen Stoffen, IX]. J Prakt Chem 1941; 158: 130-35.
[57] Cline AK, Renka RL. A Storage-Efficient Method for Construction of a Thiessen Triangulation. Rocky Mt J Math 1984; 14: 119-39.
[58] ThermoHaake GmbH Test certificate. Karlsruhe: ThermoHaake GmbH 1997.
[59] Bird RB, Stewart WE, Lightfoot EN. Fenômenos de Transporte. 2nd ed.. Rio de Janeiro: LTC 2004.
[60] Song L, Chen Y, Evans JW. Measurements of the thermal conductivity of poly(ethylene oxide)-lithium salt electrolytes. J Electrochem Soc 1997; 144: 3797-800.
[61] Griskey RG. Transport Phenomena and Unit Operations: A Combined Approach. New York: John Wiley & Sons, Inc. 2002.
[62] Roscoe R. The end correction for rotation viscometers. Br J Appl Phys 1962; 13: 362-66.
[63] Schramm G. Reologia e Reometria – Fundamentos Teóricos e Práticos. São Paulo: Artilber 2006.
[64] Pahl M, Gleißle W, Laun HM. Praktische Rheologie der Kunststoffe und Elastomere. Düsseldorf: VDI-Verlag 1995.
[65] Wein O, Vecer M, Havlica J. End effects in rotational viscometry I. No-slip shear-thinning samples in the Z40 DIN sensor. Rheol Acta 2007; 46: 765-72.
[66] Wein O, Vee M, Tovigreko VV. AWS rotational viscometry of polysaccharide solutions using a novel KK sensor. J Non-newtonian Fluid Mech 2006; 139: 135-52.
[67] Vee M. Experimentální Studium Zdánlivého Skluzu P i St n (Ph.D. Thesis). Prague: Institute of Chemical Process Fundamentals 2004.
[68] Wein O. Research Report ICPF No. 2/2005 – Edge Effects In Rotational Viscometry. III. ZZ and KK sensors, Total slip pseudosimilarity. Prague: Institute of Chemical Process Fundamentals 2005.
[69] Mooney M. Explicit Formulas for Slip and Fluidity. J Rheol 1931; 2: 210-22.
[70] Yoshimura AS, Prud’homme RK. Wall Slip Effects on Dynamic Oscillatory Measurements. J Rheol 1988; 32: 53- 67.
[71] Kilja ski T. A method for correction of the wall-slip effect in a Couette rheometer. Rheol Acta 1989; 28: 61-64.
[72] Yeow YL, Choon B, Karniawan L, Santoso L. Obtaining the shear rate function and the slip velocity function from Couette viscometry data. J Non-newtonian Fluid Mech 2004; 124: 43- 49.
[73] Tarantola A. Inverse Problem Theory and Methods for Model Parameter Estimation. Philadelphia: Soc Ind Appl Math 2005.
[74] Ancey C. Solving the Couette inverse problem using a wavelet-vaguelette decomposition. J Rheol 2005; 49: 441-60.
[75] Wein O, Tovchigrechko VV. Rotational viscometry under presence of apparent wall slip. J Rheol 1992; 36: 821-44.
[76] Wein O. Private communication 2010.






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