ii. rheometry

RheometryPart 2

Introduction to the Rheology of Complex Fluids

Dr. Aldo Acevedo - ERC SOPS 1

Rheometry

Making measurements of rheological material functions

To measure a material function, an experiment must be designed to produce the kinematics pescribed in th edefinition of the material function, then measure the stress components needed and calculate the material function.


Viscometer vs Rheometer

Viscometer – measures viscosityRheometer – measures rheological properties

A rheometer is a viscometer, but a viscometer is not a rheometer.


Experimental Methods/ Instruments Capillary viscometers

Cup Glass Extrusion rheometers

Rotational rheometers Parallel plates (disks) Cone-and-plate Couette Brookfield viscometers

Falling ball viscometers Extensional rheometers …


Rotational Rheometry

Rotational instruments makes it possible:1. To create within the sample the homogeneous

regime of deformation with strictly controlled kinematic and dynamic characteristics

2. Maintain assigned regime of flow for unlimited period of time

Different regimes of deformation:1. Constant angular velocity/frequency (constant

shear rate)2. Constant torque (constant stress)


Rotational Rheometry

Advantages:1. Small quantities of materials2. Smaller instrument sizes3. Preferred for samples which are sensitive to contractions and

expansions4. Longer residence times /testing times5. Multiple testing or complex testing protocols

Disadvantages:1. Lower maximum shear rates/stresses2. Lower shear rates (~10-3 s-1) limited by power drive and speed

control (reducing gears)3. High shear rates – heating of the sample (bad energy

dissipation), Weissenberg effect, flow instabilities 4. Wall slip and ruptures (detachment from wall)


Constant frequency of rotationTypical experimental results:

1. Low speed – monotonic dependence of T(t) until steady state flow is reached2. Increasing speed, during the transient stage, the shear stress maximum

(stress overshoot) appears.3. The stress overshoot becomes more pronounced, and although the steady

flwo is observed it is followed by a drop in torque (approach to unstable regime of deformation)

4. High speeds, steady flow is generally impossible.

A drop in torque is an indication of rupture in the sample or its detachment from the solid rotating or stationary surface. Dr. Aldo Acevedo - ERC SOPS 7

Constant torqueTypical experimental results:

1. Low torque – slow monotonic transition to the steady viscous flow2. Higher stresses - speed passes through a minimum and only then is steady

flow reached.3. At very high stresses – a steady flow is generally impossible due to a gradual

adhesive detachment of sample from the measuring surface or a cohesive rupture of sample.


Parallel Disks (Parallel Plates)

The upper plate is rotated at a constant angular velocity Ω, the velocity is:

zr

vv

0

0

0

01)(1

vzvv

rrrv

rzr

With this velocity field, and assuming incompressible flow, the continuity equation gives:


Parallel Disks (Parallel Plates)Assuming simple shear flow in θ-direction with gradient in z-direction (i.e. the velocity profile is linear in z)

Hzrv

Hzrvzv

rBzrAv

@0@0

)()(

Tvv )(

The boundary conditions:

Solving:

The rate-of-deformation tensor is then:



R

HR

Rr

RR

zrzr

T

zv

zv

zv

zv

rv

rv

rv

rv

vv

00

00000

00

0

00

)(

At the outer edge, we can write


Hr



Htrtd

Hrtdtt

tt

00

)(),0(

Assuming all curvature effects are negligible and unidirectional flow, viscosity can be calculated from:

The strain also depends on radial position:

R

Rr

R

Rrz

HR

21

0

21

0

21



zrzzz

z

rr

00

00

From the equation of motion (i.e. Cauchy-Euler), and assuming pressure does not vary with θ, then:

The strain also depends on radial position:

)(

0),(

rC

zzr

z

z

Unknown function

To measure shear stress, we must take measurements at specific values of r and evaluate viscosity at each position. 13


)()()(

)2)()((

)lever_arm)(stress(

0

21

0

rrr

rdrrT

dAT

z

R

Hzz

A

The viscosity at any value of r can be written as:

Although it is possible to measure stress, it is easier to measure the total torque required to turn the upper disk

drrTR

2

0

2

Rewritting in terms of viscosity, then:



3

0

33 )()(

2 RRR

RR

R

dRT

dd

Now to eliminate the integral, we differentiate both sides by the shear rate at the rim and using Leibnitz rule:

Now we need an expression of viscosity in terms of torque:

First, lets change variable from r to shear rate

dRTR

R0

332

0



RRR d

RTdRT

ln)2/ln(32)(33

To measure viscosity at the rim shear rate:• data at a variety of rim shear rates (rotational speeds) must be taken• torque must be differentiated •A correction must be applied to each data pair

Rearranging:

Warning – Since the strain varies with radius, not all material elements experience the same strain. The torque however, is a quantity measured from contributions at all r. For materials that are strain sensitive this gives results that represent a blurring of the material properties exhibited at each radius.



040

040

cos2)(

sin2)(

RHTRHT

SAOS material functions for parallel disk apparatus

It is also popular for SAOS where the results are:


Cone and PlateEliminates the radial dependence of shear rate (and strain).

Homogeneous flows produced only in the limit of small angles.

The velocity is:

rv

v

00

21 )(

0

CrCv

v

Assuming that single shearflow takes place in the Φ-direction with gradient in the (-rθ)-direction):

Thus,


Cone and Plate

02/2/0

rvv

The boundary conditions:

Applying BCs:

20

rv

The rate-of-deformation tensor:

The small cone angle.

zr

r

vrr

vr

r

vr

rv

rr

00

00000

0sin

sinsin

sin00

00

20

Cone and Plate

0

1

sinsin

vr

vr

Since θ is close to π/2, sin θ ~1 and:

Thus,

0

The strain is then:

00

00

)(),0(

ttdtdtt

tt


Cone and Plate

0

21

21

00 )(1 vr

The viscosity is thus:

Looking for an expression for the stress using torque:

))()((

)lever_arm)(stress(

2

0 0 2

rdrdrT

dAT

R

A

Since shear rate is constant through the flow domain, the viscosity and shear stress are constant, too.

Dr. Aldo Acevedo - ERC SOPS

22

Cone and Plate

30

0

21

23RT

Thus viscosity is:

In the limit of small angle, the cone-and-plate geometry produces constant shear rate, constant shear stress and homogeneous strain throughout the sample.

The uniformity of the flow is also an advantage with structure forming materials, such as liquid crystals, incompatible blends, and suspensions that are strain or rate sensitive.

Also, the first normal stress difference can be calculated from measurement of the axial thrust on the cone.

2

3

32

RT

Dr. Aldo Acevedo - ERC SOPS

23

Cone and Plate

atmPRrdrF 22

0 22

22

20

12

RF

The total thrust on the upper plate:

First Normal-stress coefficient in cone-and-plate

0300

0300

0

2cos3

2sin3

RTRTeR ti

SAOS for cone-and-plate


Couette (Cup-and-Bob)

zr

vv

0

0

1

1)(

kk

kRrkv

The velocity field is:

The velocity:

Shear rate:

r

r

kk

rv

0

21

0

21

1


Couette (Cup-and-Bob)

)2)()(()area)(lever_arm)(stress(

kRLkRTT

kRrr

322)1(

LkRkT

Torque:

Viscosity in Couette flow(bob turning):

Advantages: •Large contact area boosts the torque signal.

Disadvantages:•Limited to modest rotational speeds due to instabilities due to inertia or elasticity.


Commercial Rotational Rheometers

The biggest players: TA Instruments (originally Rheometrics

Scientific) Bohlin Paar Physica Haake (now part of Thermo Fisher) Reologica


The toppings…

Many other attachments or options may be used in rotational rheometers. These provide additional tests or independent measurements of data on the structure of fluids.

Magnetorheological cells Electrorheological cells Optical Attachments UV- and Photo- Curing accessories Dielectric Analysis


Capillary Flow

The velocity is:

123

20

00

xv

rv

rv zz

Rrrz

0

21

Assuming cylindrical coordinates:

The flow is unidirectional in which cylindrical surfaces slide past each other.

Near the walls, except in the θ-direction, this flow is simple shear flow.


Capillary Flow

Rrz rv

RRr

z

rvR

)(

zr

z

z

T

rv

rv

vv

00000

00

)(

Thus, is the shear at the wall


rvz


Capillary FlowThe viscosity for capillary flow is then:

R

Rrrz

Rr

z

Rrrz

rv

0

21

Now expressions for both the shear rate and stress in terms of experimental variables must be obtained.

The flow is assumed to be unidirectional and the fluid incompressible, thus, the continuity equation gives:

0

zvv z


Capillary Flow

01 22

rrrr

Assumption:• stresses and pressure are independent of θ-direction• the flow field does not vary with z (fully developed flow)• capillary is long, such that end effects are diminished• stress tensor is symmetric

•Thus, the θ-component of the equation of motion gives:

The equations of motion:

gzP

-

P P0


Capillary Flow - Stress

21

rC

r

Using the mathematical boundary condition that the stress is finite at the center (r=0). Thus, it equals zero.

The z-component:

The r-component:

Solving:

rr

rrr

rrrrz

zr

rr

rz

)(1

)(1),(

P

P


Capillary Flow - Stress

0rP

N2 is very small (negative) for polymers.

Less is known about tθθ. Thus, it seems reasonable to assume that this stress will be small or zero in a flow with assumed θ-symmetry.

Thus, the condition that both must be zero should be met easily by most materials.

Using the r-component and expressing it in terms of the normal stress coefficients:

rrN

rN

r

rrrrrrrr

22P

P


Capillary Flow- Stress

Rrr

LPP

RL

rz

20

Again, taking the stress as finite in the center, the integration constant must be zero.

Rearranging the z-component

Solving:

rCr

LPP

rrrrz

z

Lrz

rz

10

2

)(1)(

P

Shear stress in capillary flow


Capillary Flow – Shear Rate

Ra

La L

RPPRQ

1

2)(14 0

3

For Newtonian fluid, calculate the expression for the velocity directly:

The viscosity is then:

QR

LRPP

RQ

Rr

RQ

drdv

Rr

RQrv

L

R

R

R

z

z

42)(

4

4

12)(

30

0

21

3

3

2

2

Not so easily done for unknown material.However, it was observed that Q can be related to pressure drop.Dr. Aldo Acevedo - ERC SOPS

36

Weissenberg-Rabinowitsch expression:

Integrating by parts:

Applying a change in variables:


R

R

z

drrQ

rdrrvQ

0

2

0

)(2

Rr

Rrz rzrzR

R

dRQ

0

23

3


Differentiate with respect to tR and apply Leibnitz rule

Rearranging:


2

0

23

0

23

)(4)(4)(

)(4

RRrzrzrzR

RaR

rzrzrzRa

R

R

ddd

d

R

aaRR d

d

lnln3

41)(

rzrzrzR

a

R

dRQ

0

233 )(44

0

Weissenberg-Rabinowitsch correctionDr. Aldo Acevedo - ERC SOPS 38

Thus viscosity may be calculated by measurements of Q to obtain the shear rate and measurements of pressure drop to obtain stress, and the geometric constants R and L.

Capillary Flow – Viscosity

1

lnln34)(

R

a

a

RR d

d


Capillary FlowAdvantages:

1. Simple – experimentally and equipment set-up2. Inexpensive3. Higher shear rates

Disadvantages:1. May need multiple corrections:

End effects Wall slip Temperature

2. No good temperature control


Capillary Flow – Glass Viscometers


Extensional Rheometers Difficult to measure, difficult to construct. Usually “home-made” rheometers Common for solids, not for fluids


Filament Stretching Extensional Rheometers

Devices for measuring the extensional viscosity of moderately viscous non-Newtonian fluids

A cylindrical liquid bridge is initially formed between two circular end-plates. The plates are then moved apart in a prescribed manner such that the fluid sample is subjected to a strong extensional deformation.


Filament Stretching Extensional Rheometers

The kinematics closely approximate those of an ideal homogeneous uniaxial elongation.

The evolution in the tensile stress (measured mechanically) and the molecular conformation (measured optically) can be followed as functions of the rate of stretching and the total strain imposed.

Extensional flows are irrotational and extremely efficient at unraveling flexible macromolecules or orienting rigid molecules.

If it was possible to maintain the flow field, all molecules would eventually be fully extended and aligned.

McKinley and Sridhar, “Filament-Stretching Rheometry of Complex Fluids”, Annual Reviews of Fluid Mechanics, 34 375-415 (2002)


Instrument Design

The drive train accommodates the end plates, and the electronic control system imposes a predetermined velocity profile on one or both of the end plates.

The principal time-resolved measurements required are the force F(t) on one of the end plates and the filament diameter at the mid-plane.

The geometric dimensions and motor capacity of the motion-control system determine the range of experimental parameters accessible in a given device.


Operating Space

The maximum length, Lmax, and the maximum velocity, Vmax, bound the operating space.

An ideal uniaxial extensional flow is represented as a straight line on this diagram, with the slope equal to the imposed strain rate.

A given experiment will be limited by either the total travel available to the motor plates or by the maximum velocity the motors can sustain.

A characteristic value is the critical strain rate E* = Vmax/Lmax, where both limits are simultaneously achieved.

Operating Space

The operation space accessible for a given fluid may be constrained by instabilities associated with gravitational sagging, capillarity or elasticity.

The instabilities can arise from either the interfacial tension of the fluid or the intrinsic elasticity of the fluid column.


Flow

Initial aspect ratio Lo/Ro.

The diameter of the filament is axially uniform as desired for homogeneous elongation.

However, the no-slip condition at the endplates does cause a deviation from uniformity.

Thus, the diameter is usually measured at the middle of the filament.Dr. Aldo Acevedo - ERC SOPS 49

Flow

Initial aspect ratio Lo/Ro.

The diameter of the filament is axially uniform as desired for homogeneous elongation.

However, the no-slip condition at the endplates does cause a deviation from uniformity.

Thus, the diameter is usually measured at the middle of the filament.


Equations to Analyze Flow

The time-dependent total force needed to deform the sample can be measured by a load cell and related to the total stress as:

where, f(t) is the magnitude of the tensile force A(t) is the changing cross-sectional area

The normal stress difference is thus:

)()(tAtfPatmzz

)()(tAtf

rrzzrrzz



If the flow is homogeneous from start-up of steady elongation:

The elongational viscosity growth function can be calculated from a measurement of f(t) alone.

The steady elongational viscosity:

teAtA 00)(

00

00

0

0)(

Aef

Aetf

t

t

Usually not reached.

from the Hencky strain EQ 5.174



It is usually difficult to measure the length, thus the diameter at mid section is measure. However, these are not directly proportionally.

Ideal elongation of a cylinder -> p(t) = 2Lubrication theory (at short times) -> p(t) = 4/3

Experimentally a two-step procedure: Constant elongational rate based on the filament length is first imposed

and the mid filament diameter is measured. A calibration curve of Hencky strain based on length vs Hencky strain

based on mid-filament diameter is produced. The curve is then used in a second experiment to program the plate

separation that will result in exponentially decreasing diameter.

)(

0

0 )()(

tp

tDD

ltl


L-D Calibration Plot

)/ln(2

)/ln(

0

0

DD

ll

midD

L

Anna, etal “An interlaboratory comparison of measurements from filament-stretching rheometers using common test fluids”, Journal of Rheology 45(1) 83-114 (2001)


Controlled Filament Diameter Profiles


Elongational Viscosity

The unsteady extensional viscosity is obtained from:

Where the strain rate is obtained by fitting to the raw diameter data.

The Trouton ratio (or dimensionless extensional viscosity) is:

00

11330

)()()(

rrzz

0

0 )(

TrZero-shear steady shear viscosity

For Newtonian fluids Tr = 3.

The Trouton viscosity is defined as 3 times the z-s ss viscosity


Elongational Viscosity

Representative result


Pros and Cons

Advantages: The sample starts from a well defined initial rest state. Except near the ends, the strain of each material element

is the same.

Disadvantages The deformation near the ends is not homogeneous

uniaxial extension. At short times there is an induction period during which

a secondary flow occurs near the plates due to gravitational and surface tension forces.

Elongational rates calculated based on length differ from those calculated on radius.


Filament Evolution

The general evolution in the experiment typically exhibit three characteristic regimes:

A. Filament elongation • the radius decreases exponentially• At short times (early strains) there is a solvent-dominated peak

in the force followed by a steady decline due to the exponential decrease in the cross-sectional area.

• Intermediate times (or strains) the force begins to increase again owing to the strain hardening in the tensile stress. Since the area decreases, an increase in the force indicates that the stress is increasing faster that the exponential of the strain.

• At very large strains, a second maximm in the force may be observved after th eextensional stresses saturate and the extensional viscosity of the fluid recahes steady-state.


Filament Evolution

The general evolution in the experiment typically exhibit three characteristic regimes:B. Stress relaxation

• The radius remains almost constant.• This region is typically short, lasting only one or two fluid

relaxation times.• As elastic stresses decay, pressure and gravity stresses

dominate and filament breakup ensues

C. Filament break-up• The force decays and the radius decreases in similar manner


Haake CaBER IUses a high precision laser micrometer to accurately track the filament diameter as it thins. Aside from its resolution (around 10μm) the micrometer is also immune to large ambient light fluctuations and can resolve small filaments easily (a different issue from the resolution).

The plate motion is controlled by a linear drive motor. The fastest stretch time is of the order of 20 ms (depending on stretch distance) and the motor has a positional resolution of 20 μm.

Reference: Instruction Manual Haake CaBER IDr. Aldo Acevedo - ERC SOPS 61

References Faith Morrison, “Understanding Rheology,” Oxford

University Press (2001) Malkin, A.Y. & A.I. Isayev, “Rheology: Concepts,

Methods & Applications,” ChemTec Publishing, Toronto (2006)


ii. rheometry

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