Rheologica Acta

, Volume 57, Issue 5, pp 415–428 | Cite as

Simultaneous and continuous measurement of shear elasticity and viscosity of liquids at multiple discrete frequencies

  • Tobias Brack
  • Sreenath Bolisetty
  • Jurg Dual
Original Contribution


This paper presents the extension of the classical torsional resonance viscometer towards the characterization of linear viscoelastic fluids. By simultaneously tracking multiple resonance frequencies, shear viscosity and elasticity can be monitored at multiple discrete frequencies at the same time. The proposed method is applied to the well-established sensor design of a simple rod-like structure, hence enabling a robust and versatile measurement system that can be used in different applications, such as in-line measurement or hand-held devices. In order to simultaneously control different resonance frequencies, multiple independent phase-locked loops are used, of which each subsystem is responsible for one natural vibration. An analytical description of the relationship between measurement parameters and physical fluid parameters is presented that is valid both for Newtonian and linear viscoelastic fluids. Consequently, calibration of the sensor is possible using Newtonian fluids only. To demonstrate the sensor, viscoelastic polymer and surfactant solutions are investigated at five frequencies between 2 kHz and 20 kHz. Additionally, the test fluids are characterized by means of a classical rotational rheometer (low frequency range) and diffusing-wave spectroscopy (high frequency range). The comparison of all methods shows very good agreement and validates the application of the sensor as resonant rheometer for viscoelastic fluids.

Graphical Abstract

Symbolization of the continuous measurement of the complex viscosity of fluids using multi-freuqency resonance tracking 172 × 194mm (300 × 300 DPI) page 1 of 17 Rheologica


Forced oscillations Polymer solution Viscoelastic fluid Xanthan gum solution Rheometry 



The authors would like to acknowledge Mathias Reufer from LS Instruments for his expert advice and competent measurements as well as the Laboratory of Food and Soft Materials at ETH Zurich with Prof. Raffaele Mezzenga for the fruitful collaboration.


  1. ASTM International (2010) ASTM B135-10 Standard specification for seamless brass tube.
  2. Bircher B A, Duempelmann L, Renggli K, Lang H P, Gerber C, Bruns N, Braun T (2013) Real-time viscosity and mass density sensors requiring microliter sample volume based on nanomechanical resonators. Anal Chem 85(18):8676–8683. CrossRefGoogle Scholar
  3. Blom C, Mellema J (1984) Torsion pendula with electromagnetic drive and detection system for measuring the complex shear modulus of liquids in the frequency range 80-2500 Hz. Rheol Acta 23:98–105. CrossRefGoogle Scholar
  4. Brack T (2017) Multi-frequency phase control of a torsional oscillator for applications in dynamic fluid sensing. PhD thesis, ETH ZürichGoogle Scholar
  5. Brack T, Kern D, Dual J (2016) Dynamics and stability of phase controlled oscillators. J Dyn Syst Measur Control 138(7):071007. CrossRefGoogle Scholar
  6. Brack T, Vujanic R, Dual J (2018) Simultaneous phase control of multiple frequencies of multi-degree-of-freedom systems. J Vib Control 24(2):438–450. CrossRefGoogle Scholar
  7. Choppe E, Puaud F, Nicolai T, Benyahia L (2010) Rheology of xanthan solutions as a function of temperature, concentration and ionic strength. Carbohydr Polym 82(4):1228–1235. CrossRefGoogle Scholar
  8. Cooper E, Johnson P, Donald A (1991) Probe diffusion in polymer solutions in the dilute/semi-dilute crossover regime: 1. poly(ethylene oxide). Polymer 32(15):2815–2822. CrossRefGoogle Scholar
  9. Dasgupta B R, Tee S Y, Crocker J C, Frisken B J, Weitz D A (2002) Microrheology of polyethylene oxide using diffusing wave spectroscopy and single scattering. Phys Rev E 65:051505. CrossRefGoogle Scholar
  10. Devanand K, Selser J C (1991) Asymptotic behavior and long-range interactions in aqueous solutions of poly(ethylene oxide). Macromolecules 24(22):5943–5947. CrossRefGoogle Scholar
  11. Dinser S M (2009) Resonator based superposition rheometry. PhD thesis, ETH ZürichGoogle Scholar
  12. Dontula P, Macosko C W, Scriven L E (1998) Model elastic liquids with water-soluble polymers. AIChE J 44(6):1247–1255. CrossRefGoogle Scholar
  13. Dual J (1989) Experimental methods in wave propagation in solids and4 dynamic viscometry. PhD thesis, ETH ZürichGoogle Scholar
  14. Dual J, O’Reilly O (1993) Resonant torsional vibrations: an application to dynamic viscometry. Arch Appl Mech 63(7):437–451. Google Scholar
  15. Dual J, Sayir M, Goodbread J (1990) Viscometer. Patent, US 4920787Google Scholar
  16. Dufour I, Maali A, Amarouchene Y, Ayela C, Caillard B, Darwiche A, Guirardel M, Kellay H, Lemaire E, Mathieu F, Pellet C, Saya D, Youssry M, Nicu L, Colin A (2012) The microcantilever: a versatile tool for measuring the rheological properties of complex fluids. J Sensors 2012:9. CrossRefGoogle Scholar
  17. Ebagninin K W, Benchabane A, Bekkour K (2009) Rheological characterization of poly(ethylene oxide) solutions of different molecular weights. J Colloid Interface Sci 336(1):360–367. CrossRefGoogle Scholar
  18. Ferry JD (1980) Viscoelastic properties of polymers. WileyGoogle Scholar
  19. Fritz G, Pechhold W, Willenbacher N, Wagner N J (2003) Characterizing complex fluids with high frequency rheology using torsional resonators at multiple frequencies. J Rheol 47(2):303–319. CrossRefGoogle Scholar
  20. Ghatkesar M K, Rakhmatullina E, Lang H P, Gerber C, Hegner M, Braun T (2008) Multi-parameter microcantilever sensor for comprehensive characterization of Newtonian fluids. Sens Actuators B 135(1):133–138. CrossRefGoogle Scholar
  21. Gökçek C (2003) Tracking the resonance frequency of a series RLC circuit using a phase locked loop. In: Proceedings of 2003 IEEE conference on control applications, pp 609–613.
  22. Hackley V A, Ferraris C (2001) Guide to rheological nomenclature for liquid-based particle systems. NIST Special Publications, pp 946Google Scholar
  23. Häusler K, Reinhart W, Schaller P, Dual J, Goodbread J, Sayir M (1996) A newly designed oscillating viscometer for blood viscosity measurements. Biorheology 33(4-5):397–404CrossRefGoogle Scholar
  24. Kern D, Brack T, Seemann W (2012) Resonance tracking of continua using self-sensing actuators. J Dyn Syst Measur Control 134(5):051004. CrossRefGoogle Scholar
  25. Konigsberg D, Nicholson T, Halley P, Kealy T, Bhattacharjee P (2013) Online process rheometry using oscillatory squeeze flow. Appl Rheol 23:3. Google Scholar
  26. Kutin J, Smrečnik A, Bajsić I (2003) Phase-locking control of the Coriolis meter’s resonance frequency based on virtual instrumentation. Sens Actuators Phys 104(1):86–93. CrossRefGoogle Scholar
  27. Landau LD, Lifshitz EM (1987) Fluid mechanics. Pergamon PressGoogle Scholar
  28. Langdon R M (1985) Resonator sensors - a review. J Phys Sci Instrum 18:2. CrossRefGoogle Scholar
  29. Larson R (1999) The structure and rheology of complex fluids. Topics in chemical engineering. Oxford University PressGoogle Scholar
  30. Martiel I, Sagalowicz L, Mezzenga R (2014) Viscoelasticity and interface bending properties of lecithin reverse wormlike micelles studied by diffusive wave spectroscopy in hydrophobic environment. Langmuir 30(35):10751–10759. CrossRefGoogle Scholar
  31. Mason W P, Baker W O, McSkimin H J, Heiss J H (1949) Measurement of shear elasticity and viscosity of liquids at ultrasonic frequencies. Phys Rev 75:936–946. CrossRefGoogle Scholar
  32. Meirovitch L (2001) Fundamentals of vibrations. McGraw-Hill Higher EducationGoogle Scholar
  33. Mezger T (2011) The rheology handbook: for users of rotational and oscillatory rheometers. European coatings tech files. Vincentz NetworkGoogle Scholar
  34. Mezzenga R, Schurtenberger P, Burbidge A, Michel M (2005) Understanding foods as soft materials. Nat Mater 4:729–740. CrossRefGoogle Scholar
  35. Milas M, Rinaudo M, Knipper M, Schuppiser J L (1990) Flow and viscoelastic properties of xanthan gum solutions. Macromolecules 23(9):2506–2511. CrossRefGoogle Scholar
  36. Nakken T, Nyland G H, Knudsen K D, Mikkelsen A, Elgsaeter A (1994) A new torsional rod instrument for high frequency dynamic viscoelastic measurements. J Non-Newtonian Fluid Mech 52(2): 217–232. CrossRefGoogle Scholar
  37. Oosterbroek M, Waterman H, Wiseall S, Altena E, Mellema J, Kip G (1980) Automatic apparatus, based upon a nickel-tube resonator, for measuring the complex shear modulus of liquids in the kHz range. Rheol Acta 19:497–506. CrossRefGoogle Scholar
  38. Padmanabhan M, Bhattacharya M (1994) In-line measurement of rheological properties of polymer melts. Rheol Acta 33(1):71–77. CrossRefGoogle Scholar
  39. Pipe C J, Majmudar T S, McKinley G H (2008) High shear rate viscometry. Rheol Acta 47(5):621–642. CrossRefGoogle Scholar
  40. Romoscanu A, Sayir M, Häusler K, Burbidge A (2003) High frequency parallel plate probe for the measurement of the complex viscosity of liquids. Rheol Acta 42:462–476. CrossRefGoogle Scholar
  41. Rüst P, Cereghetti D, Dual J (2013) A micro-liter viscosity and density sensor for the rheological characterization of DNA solutions in the kilo-hertz range. Lab Chip 13:4794–4799. CrossRefGoogle Scholar
  42. Schrag J L, Johnson R M (1971) Application of the birnboim multiple lumped resonator principle to viscoelastic measurements of dilute macromolecular solutions. Rev Sci Instrum 42(2):224–232. CrossRefGoogle Scholar
  43. Stokich T M, Radtke D R, White C C, Schrag J L (1994) An instrument for precise measurement of viscoelastic properties of low viscosity dilute macromolecular solutions at frequencies from 20 to 500 kHz. J Rheol 38(4):1195–1210. CrossRefGoogle Scholar
  44. Waigh T A (2005) Microrheology of complex fluids. Rep Prog Phys 68(3):685. CrossRefGoogle Scholar
  45. Wang Y Z, Xiong X M, Zhang J X (2008) New method of forced-resonance measurement for the concentrated and large-viscous liquid in the low frequency range by torsion resonator. J Rheol 52(4):999–1011. CrossRefGoogle Scholar
  46. Wang Y Z, Wang G H, Xiong X M, Wang B, Zhang L M, Zhang J X (2010) Viscoelastic measurement of complex fluids using forced oscillating torsion resonator with continuously varying frequency capability. Rheol Acta 49:1117–1126. CrossRefGoogle Scholar
  47. Wattinger T (2014) Modeling and experimental study of a flexural vibration sensor for density measurements. PhD thesis, ETH ZürichGoogle Scholar
  48. Willenbacher N, Pechhold W (2000) Torsional resonance oscillation - new prospects for an old technique. In: Proceedings of the international congress of rheology, vol 3Google Scholar
  49. Zhong L, Oostrom M, Truex M, Vermeul V, Szecsody J (2013) Rheological behavior of xanthan gum solution related to shear thinning fluid delivery for subsurface remediation. J Hazard Mater 244–245:160–170. CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institute for Mechanical SystemsETH ZurichZurichSwitzerland
  2. 2.Laboratory of Food and Soft MaterialsETH ZurichZurichSwitzerland

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