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End effect correction for orthogonal small strain oscillatory shear in a rotational shear rheometer

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Abstract

The orthogonal superposition (OSP) technique is advantageous for measuring structural dynamics in complex fluids subjected to a primary shear flow. This technique superimposes a small-amplitude oscillation orthogonal to a primary shear flow to measure the real and imaginary components of the complex shear modulus. The commercial availability of OSP geometries and bi-axial transducers is expected to increase its adoption as a more routine rheological technique. It is important to understand calibration procedures and the influence of intrinsic inhomogeneous flow fields, residual pumping flow effects, and boundary forces at the leading edges of the geometry components on experimental error and measurement uncertainty. In this work, we perform calibration measurements of viscosity standards on a commercial shear rheometer using a double-wall concentric cylinder geometry. Newtonian calibration fluids with viscosities that range from 0.01 to 331 Pa s are used to obtain the end-effect factors in primary steady shear and orthogonal oscillatory shear directions. The corrections needed for the viscosity measured in steady shear range from 16 to 21%; whereas for the orthogonal complex viscosity, the errors range from 19 to 25%. Computational fluid dynamics simulations are used to understand the relationship between the end-effect corrections, OSP flow cell, and the imposed shear flow fields. We show that approximate linear shear deformation profiles are attained, in the double gap, for both primary rotational shear and orthogonal oscillatory shear deformation, with only a slight deviation for the fluid in the vicinity of the bob ends. We also present information on the velocity, pressure, and shear rate distributions for fluid within the entire flow cell. The overestimation of the orthogonal viscosity is attributed to the pressure forces exerted on the bob end surfaces (9%) and a higher shear rate in the double gap that leads to higher viscous stresses on the bob cylindrical surfaces (8%). The Newtonian fluid field information provides a benchmark for future simulations involving non-Newtonian fluids. Additionally, the operational knowledge (i.e., consistent sample filling) and measurement window (i.e., viscosity and frequency) described within are critical for proper use of the instrument and measurement accuracy.

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Abbreviations

R 1 :

inside cup radius (m)

R 2 :

inside bob radius (m)

R 3 :

outside bob radius (m)

R 4 :

outside cup radius (m)

R b :

average radius of the bob (m)

R a :

difference between Rb and the average gap width (m), defined in Eq. 4

l :

inner cylinder height (m)

h :

bob effective length (m)

c L :

primary end-effect factor

c Lo :

orthogonal end-effect factor

K τ :

primary stress constant (Pa N−1 m−1)

K τo :

orthogonal stress constant (Pa N−1)

K γ :

primary strain constant (rad−1)

K γo :

orthogonal strain constant (m−1)

M :

torque (N m)

F ⊥ :

orthogonal oscillation force (N)

\( {G}_{\perp}^{\ast } \) :

orthogonal complex shear modulus (Pa)

fh :

fluid height (m)

d :

upper opening height (m)

D :

gap width (m)

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Acknowledgments

The authors thank Prof. Jan Vermant (ETH Zurich), Dr. Paul Salipante (NIST), and Dr. Fred Phelan (NIST) for their insights and expertise that greatly assisted this research. We also thank Dr. Siva Tholeti (COMSOL, Inc.) for the advices on simulation and the communications with TA Instruments Rheology team that sparked our initial interest in this research. Dr. Ryan Murphy and Dr. Paul Butler (NIST) are acknowledged for the assistance with the access of the rheometer and OSP geometry at NIST NCNR. We thank Dr. Paul Salipante (NIST) for the careful review of an early draft.

Greek letters

ω⊥ bob orthogonal oscillatory shear frequency (rad/s)

γ⊥ bob orthogonal oscillatory shear strain amplitude

τ shear stress (Pa)

\( \dot{\gamma} \)shear rate (s−1)

Ω cup angular velocity (rad/s)

η|| primary steady shear viscosity (Pa s)

\( {\eta}_{\perp}^{\ast } \)orthogonal complex viscosity (Pa s)

θ⊥ orthogonal oscillation displacement (m)

λs shear wavelength (m)

Funding

Funding from the National Institute of Standards and Technology (Award # 70NANB15H112) is gratefully acknowledged.

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Authors and Affiliations

  1. Materials Measurement Science Division, National Institute of Standards and Technology, Gaithersburg, MD, 20899, USA

    Ran Tao & Aaron M. Forster

  2. Department of Chemical Engineering, Texas Tech University, Lubbock, TX, 79409, USA

    Ran Tao

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  1. Ran Tao
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  2. Aaron M. Forster
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Correspondence to Ran Tao or Aaron M. Forster.

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Certain commercial equipment, instruments, or materials are identified in this paper in order to specify the experimental procedure adequately. Such identification is not intended to imply recommendation or endorsement by the National Institute of Standards and Technology, nor is it intended to imply that the materials or equipment identified are necessarily the best available for the purpose. This paper is an official contribution of the National Institute of Standards and Technology, US Department of Commerce; not subject to copyright in the USA.

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Movie S1

Surface plot of revolution 2D velocity (m/s) for orthogonal oscillatory motion of bob for a total of 2 cycles. (GIF 9238 kb)

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Tao, R., Forster, A.M. End effect correction for orthogonal small strain oscillatory shear in a rotational shear rheometer. Rheol Acta 59, 95–108 (2020). https://doi.org/10.1007/s00397-019-01185-5

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