Introduc)ontoRheology

D.Vader,H.WyssWeitzlabgroupmee)ngtutorial

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Whatisrheology?

•  RheologyisthestudyoftheflowofmaBer:mainlyliquidsbutalsosoEsolidsorsolidsundercondi)onsinwhichtheyflowratherthandeformelas)cally.Itappliestosubstanceswhichhaveacomplexstructure,includingmuds,sludges,suspensions,polymers,manyfoods,bodilyfluids,andotherbiologicalmaterials.

Biopolymers

Emulsions

Foams

Cheese

Whatisrheology?

•  Thetermrheologywascoinedin1920s,andwasinspiredbyaGreekquota)on,"pantarei","everythingflows".

•  Inprac)ce,rheologyisprincipallyconcernedwithextendingthe"classical"disciplinesofelas)cityand(Newtonian)fluidmechanicstomaterialswhosemechanicalbehaviorcannotbedescribedwiththeclassicaltheories.

Basicconcepts

L + ΔL

F F

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

Responsetodeforma)on

Step strain

time

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Oscillatoryrheology

Elastic solid:

Viscous fluid:

Stress and strain are in phase

Stress and strain are out of phase

Viscoelastic material, use:

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LissajouplotsLissajou

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G'

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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

•  BUT…nowadays,feedbackloopsarefastenoughthatmostrheometerscanoperateOKinbothmodes

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,therebypreven<nglargedeforma<onsthatcouldthreaten<ssueintegrity.”(Stormetal.,2005)

stiffening weakening

Oscillatorystrainsweeps(collagengels)

LissajouplotsfromtheG2Rawdatatool

Lissajouplot,1%strain

Nonlinear Lissajou plot

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Nonlinear Lissajou plot

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RAW DATAσ' (elastic stress)

fit to σ'

2.4mg/mLcone‐plate

2.4mg/mLcone‐plate

MITLAOSMATLABpackage

Creep‐ringing

Creep‐ringing

•  Norman&Ryan’sworkhere(fibrin,jamming)•  GoodtutorialbyEwoldt&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)clesembeddedinamaterial

•  Mean‐squaredisplacementofpar)clesasafunc)onof)meprovidesmicroscopicinforma)ononlocalelas)candviscousmaterialproper)esasafunc)onoffrequency

•  MasonandWeitz,PRL,1995

Shortandlong)mescales

Short time scales: diffusive

r: position vector D: diffusion constant τ: lag time kT: thermal energy a: particle size η: viscosity

Long time scales: spring-like

K: effective spring-constant, linked to elastic properties

What about intermediate times?

GeneralizedStokes‐Einstein

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 microrheology Open symbols: G”

2-point microrheology calculates a mean-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.