Introduction Tor He Ology 2

7/29/2019 Introduction Tor He Ology 2

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Whatisrheology?

• RheologyisthestudyoftheflowofmaBer:mainlyliquidsbutalsosoEsolidsorsolidsundercondi)onsin

whichtheyflowratherthandeformelas)cally.It

appliestosubstanceswhichhaveacomplexstructure,

includingmuds,sludges,suspensions,polymers,manyfoods,bodilyfluids,andotherbiologicalmaterials.

Biopolymers

Emulsions

Foams

Cheese



Whatisrheology?

• Thetermrheologywascoinedin1920s,andwasinspiredbyaGreekquota)on,"pantarei",

"everythingflows".

• Inprac)ce,rheologyisprincipallyconcernedwithextendingthe"classical"disciplinesof

elas)cityand(Newtonianfluidmechanicsto

materialswhosemechanicalbehaviorcannot

bedescribedwiththeclassicaltheories.



asicconcepts

L + ΔL

FF

F

F

Δx

L area A

h

area A



Simplemechanicalelements

Elastic solid: force (stress) proportional to strain

Viscous fluid: force (stress) proportional to strain rate

Viscoelastic material: time scales are important

Fast deformation: solid-like

Slow deformation: fluid-like



Oscillatoryrheology

Elastic solid:

Viscous fluid:

Stress and strain are in phase

Stress and strain are out of phase

Viscoelastic material, use:

-10x10 -3

-5

0

5

10

s t r a i n

12 10 8 6 4 2 0 time [s]

-10

-5

0

5

10

s t r e s s [ P a ] δ

strain stress



LissajouplotsLissajou

-10x10 -3

-5

0

5

10

s t r a i n r a

t e

[ 1 / s ]

12 10 8 6 4 2 0 time [s]

-10

-5

0

5

10

s t r e s

s [ P a ]

strain rate stress

-10x10 -3

-5

0

5

10

s t r a i n

12 10 8 6 4 2 0 time [s]

-10

-5

0

5

10

s t r e s s [ P a ] δ

strain stress

-10

-5

0

5

10

s t r e s s [ P a ]

-10x10 -3 -5 0 5 10 strain

G'

-10

-5

0

5

10

s t r e s s

[ P a ]

-10x10 -3 -5 0 5 10 strain rate [1/s]

G''/ ω



Strain‐controlvsstress‐control

Strain-controlled Stress-controlled

ARES Bohlin, AR-G2, Anton Paar



Strain‐controlvsstress‐control

• Strain‐controlledstatetypicallyconsideredbe2erdefined

• Stress‐controlledrheometershavebeBertorquesensi)vity

• Strain‐controlledrheometerscanprobehigherfrequencies

• UT…nowadays,feedbackloopsarefastenoughthatmostrheometerscanoperateOK

inbothmodes



Rheometergeometries

Cone-plate• uniform strain / strain-rate• fixed gap height

Plate-plate

• non-uniform strain• adjustable gap height• good for testing boundary effects like slip

Couette cell• good sensitivity for low-viscosity fluids



Linearviscoelas)city

strain amplitude γ0

storage modulus G’

loss modulus G”

Acquire data at constant frequency, increasing stress/strain



Typicalprotocol

• Limitsoflinearviscoelas)cregimeindesiredfrequencyrangeusingamplitudesweeps

=>yieldstress/strain,cri)calstress/strain

• Testfor)mestability,i.e)mesweepatconstainamplitudeandfrequency

• Frequencysweepatvariousstrain/stressamplitudeswithinlinearregime

• Studynon‐linearregime



Nonlinearrheology(ofbiopolymers

• “Unlikesimplepolymergels,manybiologicalmaterials—

includingbloodvessels,mesentery<ssue,lungparenchyma,

corneaandbloodclots—s<ffenastheyarestrained,thereby

preven<nglargedeforma<onsthatcouldthreaten<ssue

integrity.” (Stormetal.,2005

stiffening weakening



Oscillatorystrainsweeps(collagengels



LissajouplotsfromtheG2Rawdatatool

Lissajouplot,1%strain



Nonlinear Lissajou plot

-4

-2

0

2

4

s t r e s s [ P a

]

-40x10-3

-20 0 20 40strain



Nonlinear Lissajou plot

-4

-2

0

2

4

s t r e s s [ P a

]

-40x10-3

-20 0 20 40strain

RAW DATAσ' (elastic stress)fit to σ'



2.4mg/mLcone‐plate



MITLAOSMATLApackage



Creep‐ringing



Creep‐ringing

• Norman&Ryan’sworkhere(fibrin,jamming• Goodtutorialbywoldt&McKinley(MIT



Creep‐ringingresultsI:bulkproper)es



Morenonlinearrheology

• Stress/strainrampswithconstantrate• Pre‐stressmeasurements,i.e.smallstressoscilla)onsaroundaconstant(pre‐stress

• Pre‐strainmeasurements• TransientresponsesinLAOS(talktoStefan• Fourierdomainanalysis

• SRFS(talktoHans Linear behavior



Originofnonlinearbehavior

• Distribu)onoflength‐scales/inhomogenei)es

• Rearrangementofpar)cles/filaments• Non‐affinemo)on

• Howdowefindout?Observation at the microscopic scale:

• Microrheology• Microscopy



Microrheologybasics

• Generalidea:lookatthethermally‐drivenmo)onofmicron‐sizedpar)clesembeddedina

material

• Mean‐squaredisplacementofpar)clesasafunc)onof)meprovidesmicroscopic

informa)ononlocalelas)candviscousmaterial

proper)esasafunc)onoffrequency

• MasonandWeitz,PRL,1995



Shortandlong)mescales

Short time scales:diffusive

r: position vector

D: diffusion constant τ: lag time

kT: thermal energya: particle size

η: viscosity

Long time scales:spring-like

K: effective spring-

constant, linked toelastic properties

What about intermediate times?



GeneralizedStokes‐instein

Take Laplace transform of η( τ) numerically, to get η(s) – with s=iω.

From earlier, we know:

We can then get the generalized complex modulus, by analytically extending:

i.e.



2‐pointvs1‐pointmicrorheology

Black: bulk rheology

Red: 2-point microrheology

Blue: 1-point microrheologyOpen symbols: G”

2-point microrheology calculates amean-square displacement from

the correlated pair-wise motion of

particles, rather than the single-

particle MSD.



Otherconsidera)ons

• Non‐linearregimenon‐trivial,butmoreinteres)ng.

• Surfaceeffectscanbeimportant.• Imagingtofigureoutmechanisms.• Richnessofeffects,mechanisms,)me‐,length‐

andenergy‐scalespresentinsoEmaBer/

complexfluids.

• MoretoexploreonWeitzlabwebpage.

• MoreatComplexFluidsmee)ngs.

Introduction Tor He Ology 2

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