# Rheometry

## Contents

## Introduction

## Couette viscometer

## Cone and plate rheometer

A cone and plate rheometer provides information on the mechanical response of a fluid material, in response to a shear deformation. The fluid is placed between the flat bottom plate and the shallow, conical top plate as illustrated on the image. Typically the top plate is rotated (exerts shear) and the resulting strain is measured. For Newtonian fluids, the two are proportional: <math>\sigma = \eta\dot{\gamma}</math>. However, for viscoelastic material with non-Newtonian flow behaviors, the relation becomes : <math>\sigma = \eta\dot{\gamma} + \sigma_0</math>.

There are three popular types of measurements that are possible on a cone and plate rheometer:

1) Stress relaxation measurements: the strain is held constant and the decay of stress is monitored as a function of time.

2) Creep measurements: the stress is held constant and the increase in strain is monitored

3) Dynamic measurements : the applied strain is a function of time, usually an oscillatory strain, and the stress is monitored.

In the case of oscillatory strain <math>\gamma = \gamma_0sin\omega t</math>, where <math>\omega</math> is the frequency of deformation.

The resulting stress will be <math>\sigma = \sigma^{\prime}_0 sin\omega t + \sigma^{\prime\prime}_0 sin\omega t</math>.

The equation suggests that for a viscoleastic system, a component of stress will be in phase with the strain (elastic component), and a component will be out of phase (viscous component).

The in-phase, elastic modulus is defined as <math>G^{\prime} = \frac{\sigma^{\prime}_0} {\gamma_0}</math>.

The out-of-phase, viscous modulus is defined as: <math>G^{\prime\prime} = \frac{\sigma^{\prime\prime}_0} {\gamma_0}</math>.

Finally, the ratio of the two is called the loss tangent and is a measure of the energy loss per cycle:

<math>tan \delta = \frac{G^{\prime\prime}}{G^{\prime}}</math>

Images from: http://www.anton-paar.com/ and http://www.dactyl.com/scratchpad/pps/rheology.html

Text adapted from: http://people.seas.harvard.edu/~hwyss/files/Wyss_GIT_Lab_J_2007.pdf and Hamley I., Introduction to Soft Matter, England 2007

## Poiseuille flow

## Falling ball and rising bubble viscosimeters

## Bubble rise in complex fluids

## Creep of ice

## Flocculated suspensions

## Size limitations in viscosity measurements

There was a question in class on the lower limit of samples where rheological properties may be measured. The Suhling lab at King's College recently announced in JACS that they can observe the viscosity in human ovarian carcinoma cell line SK-OV-3 using a fluorescent probe called a molecular rotor. Essentially, in a medium of high viscosity, the rotational Brownian motion of the the rotor as a whole is slowed down. This slowing increases linearly and can be used to accurately determine the viscosity of the cytoplasm of a cell. Their technique uses fluorescence lifetime imaging, and allows them to analyze fluorescent decays as a function of viscosity in spatially resolved manner; this can show valuable information on the inhomogeneity of the intracellular viscosity.

The case for measuring intracellular viscosity is made in their introduction, reproduced below:

"Viscosity is one of the major parameters determining the diffusion rate of species in condensed media. In biosystems, changes in viscosity have been linked to disease and malfunction at the cellular level.1 These perturbations are caused by changes in mobility of chemicals within the cell, inﬂuencing fundamental processes such as signaling and transport and the efﬁciency of bimolecular processes governed by diffusion of short-lived intermediates, such as the diffusion of reactive oxygen species during an oxidative stress attack. While methods to measure the bulk macroscopic viscosity are well developed, imaging local microscopic viscosity remains a challenge, and viscosity maps of microscopic objects, such as single cells, are actively sought after.2–6 We report a new approach to image local microviscosity using the ﬂuorescence lifetime of a molecular rotor.literature data for similar compounds."

Paper, Supplemental, Newsy Article