introduction tor he ology 2
TRANSCRIPT
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
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Whatisrheology?
• Thetermrheologywascoinedin1920s,andwasinspiredbyaGreekquota)on,"pantarei",
"everythingflows".
• Inprac)ce,rheologyisprincipallyconcernedwithextendingthe"classical"disciplinesof
elas)cityand(Newtonianfluidmechanicsto
materialswhosemechanicalbehaviorcannot
bedescribedwiththeclassicaltheories.
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asicconcepts
L + ΔL
FF
F
F
Δx
L area A
h
area A
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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
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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:
-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
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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
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Strain‐controlvsstress‐control
Strain-controlled Stress-controlled
ARES Bohlin, AR-G2, Anton Paar
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Strain‐controlvsstress‐control
• Strain‐controlledstatetypicallyconsideredbe2erdefined
• Stress‐controlledrheometershavebeBertorquesensi)vity
• Strain‐controlledrheometerscanprobehigherfrequencies
• UT…nowadays,feedbackloopsarefastenoughthatmostrheometerscanoperateOK
inbothmodes
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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
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Linearviscoelas)city
strain amplitude γ0
storage modulus G’
loss modulus G”
Acquire data at constant frequency, increasing stress/strain
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Typicalprotocol
• Limitsoflinearviscoelas)cregimeindesiredfrequencyrangeusingamplitudesweeps
=>yieldstress/strain,cri)calstress/strain
• Testfor)mestability,i.e)mesweepatconstainamplitudeandfrequency
• Frequencysweepatvariousstrain/stressamplitudeswithinlinearregime
• Studynon‐linearregime
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Nonlinearrheology(ofbiopolymers
• “Unlikesimplepolymergels,manybiologicalmaterials—
includingbloodvessels,mesentery<ssue,lungparenchyma,
corneaandbloodclots—s<ffenastheyarestrained,thereby
preven<nglargedeforma<onsthatcouldthreaten<ssue
integrity.” (Stormetal.,2005
stiffening weakening
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Oscillatorystrainsweeps(collagengels
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LissajouplotsfromtheG2Rawdatatool
Lissajouplot,1%strain
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Nonlinear Lissajou plot
-4
-2
0
2
4
s t r e s s [ P a
]
-40x10-3
-20 0 20 40strain
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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'
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2.4mg/mLcone‐plate
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2.4mg/mLcone‐plate
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MITLAOSMATLApackage
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Creep‐ringing
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Creep‐ringing
• Norman&Ryan’sworkhere(fibrin,jamming• Goodtutorialbywoldt&McKinley(MIT
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Creep‐ringingresultsI:bulkproper)es
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Morenonlinearrheology
• Stress/strainrampswithconstantrate• Pre‐stressmeasurements,i.e.smallstressoscilla)onsaroundaconstant(pre‐stress
• Pre‐strainmeasurements• TransientresponsesinLAOS(talktoStefan• Fourierdomainanalysis
• SRFS(talktoHans Linear behavior
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Originofnonlinearbehavior
• Distribu)onoflength‐scales/inhomogenei)es
• Rearrangementofpar)cles/filaments• Non‐affinemo)on
• Howdowefindout?Observation at the microscopic scale:
• Microrheology• Microscopy
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Microrheologybasics
• Generalidea:lookatthethermally‐drivenmo)onofmicron‐sizedpar)clesembeddedina
material
• Mean‐squaredisplacementofpar)clesasafunc)onof)meprovidesmicroscopic
informa)ononlocalelas)candviscousmaterial
proper)esasafunc)onoffrequency
• MasonandWeitz,PRL,1995
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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?
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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.
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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.
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Otherconsidera)ons
• Non‐linearregimenon‐trivial,butmoreinteres)ng.
• Surfaceeffectscanbeimportant.• Imagingtofigureoutmechanisms.• Richnessofeffects,mechanisms,)me‐,length‐
andenergy‐scalespresentinsoEmaBer/
complexfluids.
• MoretoexploreonWeitzlabwebpage.
• MoreatComplexFluidsmee)ngs.