# Rheometry

## Introduction

If a fluid and an outer cylinder are at rest and an inner cylinder revolves uniformly, a circular motion, communicated to the fluid, will be propagated by decreasing degrees through the fluid to the outer cylinder.

-Principia, Prop. LI, Cor. V.

Philosophiæ Naturalis Principia Mathematica, 1687

 Shearing stress = Force/Area = F/A = Newton/m2 Rate of shear = Change of velocity with distance = dv/dx (sec-1) $\frac{F}{A}=\eta \frac{dv}{dx}\,\!$ $\eta =\frac{\text{shear stress}}{\text{shear rate}}=\frac{\text{Newton }\times \text{ sec}}{\text{m}^{\text{2}}}=\text{Pascal-sec}\,\!$

## Couette viscometer

Couette viscometer is a viscometer in which the liquid whose viscosity is to be measured fills the space between two vertical coaxial cylinders, the inner one suspended by a torsion wire; the outer cylinder is rotated at a constant rate, and the resulting torque on the inner cylinder is measured by the twist of the wire. Also known as rotational viscometer.

In modern Couette viscometers the bob is either driven with a known stress and the resulting angular velocity measured or it is driven at a known angular velocity and the required stress measured. The first provides viscosity as a function of shear stress, the latter provides viscosity as a function of shear rate.

Morrison, Fig. 2-18
Morrison, Fig. 2-2

## Cone and plate rheometer

The liquid is placed on horizontal plate and a shallow cone placed into it. The angle between the surface of the cone and the plate is of the order of 1 degree—i.e. it is a very shallow cone. Typically the plate is rotated and the force on the cone measured. A well-known version of this instrument is the Weissenberg Rheogoniometer, in which the movement of the cone is resisted by a thin piece of metal which twists—known as a torsion bar. The known response of the torsion bar and the degree of twist give the shear stress, while the rotational speed and cone dimensions give the shear rate. In principle the Weissenberg Rheogoniometer is an absolute method of measurement providing it is accurately set up. Other instruments operating on this principle may be easier to use but require calibration with a known fluid. Cone and plate rheometers can also be operated in an oscillating mode to measure elastic properties, or in combined rotational and oscillating modes.

A cone and plate rheometer provides information on the mechanical response of a fluid material, in response to a shear deformation. For Newtonian fluids, the two are proportional: $\sigma = \eta\dot{\gamma}$. However, for viscoelastic material with non-Newtonian flow behaviors, the relation becomes : $\sigma = \eta\dot{\gamma} + \sigma_0$.

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 $\gamma = \gamma_0sin\omega t$, where $\omega$ is the frequency of deformation.

The resulting stress will be $\sigma = \sigma^{\prime}_0 sin\omega t + \sigma^{\prime\prime}_0 sin\omega t$.

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 $G^{\prime} = \frac{\sigma^{\prime}_0} {\gamma_0}$.

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

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

$tan \delta = \frac{G^{\prime\prime}}{G^{\prime}}$

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

For a Newtonian liquid, the velocity profile is parabolic.

$v(r)=-\frac{\Delta p}{4\eta L}\left( R^{2}-r^{2} \right)$

And the mass flow rate is:

$\omega =\frac{\pi \Delta pr^{4}}{8L\eta }$

Morrison, Fig. 2.2
Capillary viscosimeter

The capillary is calibrated by means of a visocosity standard:

$\eta _{1}=\eta _{2}\left( \frac{\rho _{1}t_{1}}{\rho _{2}t_{2}} \right)$

Morrison, Fig. 2.15

## Falling ball and rising bubble viscosimeters

Morrison, Fig. 2.16
Bubbles/spheres are easy to detect.

They sense the local viscosity.

Lots of interesting possibilities.

$\eta =\frac{2a^{2}g\left( \rho _{2}-\rho _{1} \right)t}{9d}\,\!$

## Bubble rise in complex fluids

The rate of rise of a bubble (or the sinking of a particle) depends, in part, on the velocity at the interface. For particles, the general assumption is that the velocity of the liquid at the particle surface is zero. The water-air interface is quite different; liquid motion can cause air motion.

$v=\frac{2\left( \rho _{1}-\rho _{2} \right)ga^{2}}{9\kappa \eta _{2}}$

$\kappa =\frac{3\eta _{1}+2\eta _{2}}{3\eta _{1}+3\eta _{2}}$

$\kappa \,\!$ is the Rybczynski-Hadamard factor.

As surfactant is adsorbed at the interface, the interface becomes more and more rigid.

Morrison, Fig. 14.6
Ratio of the observed velocity of ascent of a bubble to the calculated Stokes’ velocity in solutions of various concentrations of (a) polydimethylsiloxane in trimethylolpropane-heptanoate; (b) polydimethylsiloxane in mineral oil; (c) N-phenyl-1-1napthylamine in trimethylolpropane-heptanoate. Each figure shows the transition from the Hadamard to the Stokes regime.

## Creep of ice

Glaciers can act like a fluid. Looking at images of glaciers reminds people of common liquids spreading and even streamlines (lines of constant velocity).

Here is an example of ice flowing out of valleys into open spaces.

Here are lines of debris (rocks and dirt) which form into lines as the glacier flows downhill.

The movement of glaciers is called creep.

There are three dominating factors in glacial creep...

1. Creep is proportional to temperature. Not what you might think. Other than the obvious melting at the bottom of glaciers, high up in the mountains the temperature variation throughout the deep ice where the bottom of the glacier is heated by the Earth's heat causes a gradient of flow which causes stratification of debris in the ice.

2. Creep is proportional to stress (essentially proportional to the weight of overlying ice) The larger the glacier, the deeper the ice, the higher the weight and therefore the higher the flow.

3. There is a minimum stress, called the threshold stress, below which creep does not operate. If the glacier is not large/deep enough frictional forces overcome the weight of the ice and therefore no flow will occur.

[[1]]

Other cool notes:

"It is important to understand that the increase in flow rate is not related to present day air temperature, but to increased precipitation long ago."

"At the surface there is no stress, so the ice does not flow: at a certain depth the weight of ice is sufficient to cause flow. Between these two limits the ice is a brittle solid, being carried along on plastic ice beneath. Since the flow is uneven (greatest in the middle in valley glaciers) the solid, brittle ice is broken up by a series of cracks called crevasses."

 Griggs, D.T.; Coles, N.E. Creep of a single crystal of ice, SIPRE Technical Report 1954, 11, 24; Morrison, 2.4 The flow behavior of ice was first demonstrated by J.D. Forbes in 1842 by planting a line of stakes across the Mer de Glace glacier. On returning the following year he found that the linear array had moved into a parabolic curve. (Cunning ham, F.F. James David Forbes: Pioneer Scottish glaciologist; Scottish Academic Press: Edinburgh; 1990, p 329.)

## Flocculated suspensions

Morrison, Fig. 2.8a
Morrison, Fig. 2.8b

Flocculated dispersions form extended particle-particle contact structures. Sufficient shear stress to break the structure is easy to measure and is some indication of the particle-particle interaction. But it is difficult to formulate a rheological equation that describes the transition without detailed knowledge of the particle contact forces, the effect of re-arrangements, and the fluid flow between the particles.

The common method is to use empirical equations. Simple, but at great loss of understanding.

The more successful route is to envision properties (say with small motions, or as a function of concentration, etc.) that depend more simply on material properties and then perform the appropriate rheological test.

## 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."