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Yield Stress Fluids, Meeting #1
1. Introduction to yield stress fluidsunderstand definition/caveats of apparent yield stress fluids develop a feel for yield stress and viscosity values
2. Rheometry with yield stress fluidsidentify and avoid slip artifactscorrect for parallel plate artifactsrecognize LAOS response of yield stress fluids
Randy H. EwoldtJune 26, 2009
Part of the summer 2009 Reading Group: Yielding, Yield Stresses, ViscoelastoplasticityNon-Newtonian Fluids (NNF) Laboratory, led by Prof. Gareth McKinley
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References
• Barnes, H. A., "The yield stress - a review or `pi alpha nu tau alpha rho epsilon iota' -everything flows?," Journal of Non-Newtonian Fluid Mechanics 81(1-2), 133-178 (1999)
• Bonn, D. and M. M. Denn, "Yield Stress Fluids Slowly Yield to Analysis," Science 324(5933), 1401-1402 (2009)
• Brunn, P. O. and H. Asoud, "Analysis of shear rheometry of yield stress materials and apparent yield stress materials," Rheologica Acta 41(6), 524-531 (2002)
• Yoshimura, A. S. and R. K. Prudhomme, "Response of an elastic Bingham fluid to oscillatory shear," Rheologica Acta 26(5), 428-436 (1987)
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Yield Stress Fluids:
Apparently solid at low stress, σ < σYApparently fluid at high stress, σ > σY
e.g. paint, toothpaste, hand lotion, concrete, snail slime, nuclear waste sludge
Yield Stress Fluids:
Apparently solid at low stress, σ < σYApparently fluid at high stress, σ > σY
e.g. paint, toothpaste, hand lotion, concrete, snail slime, nuclear waste sludge
Idealized example:linear scale
Idealized example:log-log scale
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Toothpaste
Carbopol 980, polymer microgel (0.2wt%)
Tomato puree
Mayonnaise
Viscosity, σηγ
=
Apparent yield stress fluids:
1. Initially “high”viscosity
2. Dramatic viscosity drop over narrow range of stress
3. Flows
Apparent yield stress fluids:
1. Initially “high”viscosity
2. Dramatic viscosity drop over narrow range of stress
3. Flows
Detailed experiments
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Viscosity, η ≈ 108 Pa.sIg Nobel Award, Physics (2005) - Presented jointly to John Mainstone and Thomas Parnell of the University of Queensland,
Australia, for patiently conducting the so-called pitch drop experiment that began in the year 1927 — in which a glob of congealed black tar pitch has been slowly dripping through a funnel, at a rate of approximately one drop every nine years.
Started: 1927Drop #1: 1938Drop #8: 2000 (12 years between drops #7 and #8)Drop #9: ????
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An estimated 5 to 8 percent of all human-generated atmospheric CO2 worldwide comes from the concrete industry.
Concrete creep is caused by the rearrangement of particles at the nano-scale.
The basic building block of cement paste at the nano-scale -- calcium-silicate-hydrates, or C-S-H -- is granular in nature.
A more dense phase of C-S-H can be induced by additional smaller particles that fit into the spaces between the nano-granules of C-S-H, which greatly hinders the movement of the C-S-H granules over time.
The rate of creep is logarithmic, which means slowing creep increases durability exponentially.
A nano-indentation device allows them to measure in minutes creep properties that are usually measured in year-long creep experiments at the macroscopic scale.
Professor Franz-Josef Ulm M.I.T. Dept. of Civil and Env. Eng.
Vandamme and Ulm, PNAS, 2009
η ≈ 1014 Pa.s
η ≈ 1019 Pa.s
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Blood
Hematocritcontent
Olivine (a rock),various grain sizes
Plasticine(modelling material, like putty)
Apparent yield stress fluids?
Barnes, JNNFM, 1999
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Slime & simulant rheology: Flow
Carbopol (polymer microgel)Carbopol 940 in DIW, pH→7 with NaOH
Laponite (particulate gel)Laponite RD in DIW
Ewoldt et al., Soft Matter, 2007
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Magnetorheological Fluid
AR-G2 rotational rheometer, parallel plates with 600 grit sandpaper, h=500µm
D=40mm for ambient testD=20mm for magnetic field tests Using MRF Rheometer Cell from Murat Ocalan, Ph.D. Thesis (in progress), MIT
LORD Corp. MRF-132DGcarbonyl iron particles (1-20µm) in silicone oil
B
no field field-activated
Magnetic Field 0.462 T 0.221 T 0.100 T 0.046 T 0.030 T 0.012 T ambient
5 -21.4 10 Pa.Tα = ⋅
10-5 10-4 10-3 10-2 10-1 100 101 102 103100
101
102
103
104
105
Cor
rect
ed S
hear
Stre
ss, σ
R [P
a]
Rim Shear Rate [s-1]
(a)
Ewoldt, Ph.D. thesis, 2009
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Magnetorheological fluid
100 101 102 103 104 10510-1
100
101
102
103
104
105
106
107
108
(b)Magnetic Field
B=0.462 T B=0.221 T B=0.100 T B=0.046 T B=0.030 T B=0.012 T ambient
C
orre
cted
Vis
cosi
ty, η
[P
a.s]
Corrected Shear Stress, σR [Pa]
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Apparent Yield Stress FluidsSome quotes from Barnes, JNNFM, 1999
Yield stresses were usually only a ‘figment of peoples’ extrapolation
The existence of an essentially horizontal region in a double-logarithmic plot of stress versus strain rate is the most satisfactory criterion for the existence of a ‘yield stress’(quote from Evans [36])
[Barnes] is in substantial agreement with Evans in this view, especially if then term ‘yield stress’ is prefixed with the adjective ‘apparent’!
Nguyen and Boger [38] reviewed experimental methods of measuring the flow properties of yield stress fluids in 1992: … the especial value of which is the emphasis on possible sourcesof error in measurement.
[36] I. Evans, J. Rheol. 36 (7) (1992) 1313.[38] Q.D. Nguyen, D.V. Boger, Ann. Rev. Fluid Mech. 24 (1992) 47.
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Despite the practical importance, there is no reliable way at present to predict the onset of flow.
Perhaps the main difficulty with much of the literature on yield stress fluids is the prevailing presumption that the solid-liquid transition occurs at a single invariant stress. This assumption ignores the fact that the microstructure may adjust dynamically when flow begins.
Foams, emulsions, and Carbopol gels (such as hair gel) are probably closest to “ideal” yield stress materials, because they do not usually show measurable rejuvenation or aging; in these cases, a yield stress may be a material property
In the few direct comparisons between computed and experimental velocity fields for the falling sphere, the agreement is poor (14), probably because the yielding process in the experimental fluids is not adequately described by the models. Much fundamental work thus remains to be done.
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Examples of “ideal” yield stress fluids(negligible thixotropy)
• Nivea Lotion ~ 4 Pa• Gilette foamy shaving cream ~10 Pa• Aloe gel ~60 Pa• Nivea Cream, Toothpaste and Mayo ~200-300 Pa• Magnetorheological fluid 10 ~ 10,000 Pa
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Experimental Issue: Slip
161.000E-6 0.01000 100.0
shear rate (1/s)
1.000
10.00
100.0sh
ear s
tress
(Pa)
NiveaLotion-0001f
NiveaLotion-S02-h1050-0001f, Steady state flow stepNiveaLotion-S02-h450-0001f, Steady state flow stepNiveaLotion-S02-h700-0001f, Steady state flow stepNiveaLotion-S03-h1050-0001f, Steady state flow stepNiveaLotion-S03-h450-0001f, Steady state flow stepNiveaLotion-S03-h700-0001f, Steady state flow stepNiveaLotion-S04-h1050-0001f, Steady state flow step
Sandpaper required to avoid slip
FILLED)P/P w/ sandpaper, 3 different gaps
OPEN)P/P, NO sandpaper, 3 different gaps
Sandpaper from McMaster-CarrPart #) 47185A51Adhesive-backsandpaper, 8” dia. disc, 600 grit
Nivea Lotion
M
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Flow curves: different perspectives
1.000E-6 0.01000 100.0shear rate (1/s)
0.1000
1.000
10.00
100.0
1000
10000
1.000E5
1.000E6
1.000E7
visc
osity
(Pa.
s)
NiveaLotion-0001f
NiveaLotion-S02-h1050-0001f, Steady state flow stepNiveaLotion-S02-h450-0001f, Steady state flow stepNiveaLotion-S02-h700-0001f, Steady state flow stepNiveaLotion-S03-h1050-0001f, Steady state flow stepNiveaLotion-S03-h450-0001f, Steady state flow stepNiveaLotion-S03-h700-0001f, Steady state flow stepNiveaLotion-S04-h1050-0001f, Steady state flow step
1.000 10.00 100.0shear stress (Pa)
0.1000
1.000
10.00
100.0
1000
10000
1.000E5
1.000E6
1.000E7
visc
osity
(Pa.
s)
NiveaLotion-0001f
NiveaLotion-S02-h1050-0001f, Steady state flow stepNiveaLotion-S02-h450-0001f, Steady state flow stepNiveaLotion-S02-h700-0001f, Steady state flow stepNiveaLotion-S03-h1050-0001f, Steady state flow stepNiveaLotion-S03-h450-0001f, Steady state flow stepNiveaLotion-S03-h700-0001f, Steady state flow stepNiveaLotion-S04-h1050-0001f, Steady state flow step
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Parallel plate correction, steady flow
10-6 10-5 10-4 10-3 10-2 10-1 100 101 102 1031
10
100
OPEN: Apparent stress FILLED True stress
h=1050 µm h= 700 µm h= 450 µm
Sh
ear S
tress
, σ
[Pa]
Shear Rate [s-1]
Nivea LotionParallel plates with sandpaperD=40mm
M
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Parallel plate stress correction2
02 ( )
RM r r drπ σ= ∫
3
2( )AMRR
σπ
=
43
A
Y
σσ
→
3
ln32 lnR
R
M d MR d
σπ γ
= +
ln34 lnA A
RR
dd
σ σσγ
= +
for ( ) Yrσ σ=
M
( )r Hrσ =
4/3 ratio mentioned by Brunn and Asoud, Rheol. Acta, 2002
Torque balance
Linear assumption
Apparent rim stress
Error for perfect plastic
Correction formula
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Parallel plate stress correction, Nivea lotion steady flow
10-6 10-5 10-4 10-3 10-2 10-1 100 101 102 1031
10
100
OPEN: Apparent stress FILLED True stress
h=1050 µm h= 700 µm h= 450 µm
Sh
ear S
tress
, σ
[Pa]
Shear Rate [s-1]
Nivea LotionParallel plates with sandpaperD=40mm
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Roughened Cone – Works!
10-6 10-5 10-4 10-3 10-2 10-1 100 101 102 1031
10
100
S02, P/P, h= 1050µm (uncorrected, for reference) S02, P/P, h= 700µm S02, P/P, h= 450µm S04, P/P, h=1050µm S05, Cone
Shea
r Stre
ss, σ
[P
a]
Shear Rate [s-1]
Nivea Lotion on AR-G23 different samples to show repeatability (S02, S04, S05)
all geometries with sandpaperP/P) D=40mm, with plate-plate correctionCone) D=60mm, α=2o, no correction necessary
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Nivea lotion flow curve
10-3 10-2 10-1 100 101 102 103 10410-1
100
101
102
103
104
105
106
S02, P/P, h= 1050µm S02, P/P, h= 700µm S02, P/P, h= 450µm S04, P/P, h=1050µm S05, Cone
Vis
cosi
ty, η
[P
a.s]
Shear Stress [Pa]
Nivea Lotion3 different samples to show repeatability (S02, S04, S05)
all geometries with sandpaperP/P) D=40mm, with plate-plate correctionCone) D=60mm, α=2o, no correction necessary
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Creep tests, low stress
Cannot determine a steady flow viscosity below the yield stress
NEGATIVE strain-rates? Is this remnant elastic recovery from the previous flow test? I saw this also with the stress-ramp flow tests, from high-to-low stresses, that at sufficiently low stresses the strain-rate was negative (see Sample #1). Or maybe it’s instrument artifact?
0 100.0 200.0 300.0 400.0 500.0 600.0time (s)
-0.50000
0
0.50000
1.0000
1.5000
2.0000
2.5000
% s
train
NiveaLotion-S04-h1050-0002c
NiveaLotion-S04-h1050-0002c, CreepNiveaLotion-S04-h1050-0003c, Creep
OPEN) σ0 = 0.1 Pa
CLOSED) σ0 = 1 Pa
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1.000E-5 1.000E-3 0.1000 1.000 10.00 100.0shear rate (1/s)
1.000
10.00
100.0
shea
r stre
ss (P
a)
NiveaLotion-S05-ConeSP-0001f
NiveaLotion-S05-ConeSP-0001f, Steady state flow stepNiveaLotion-S06-ConeSP-0001f, Steady state flow stepNiveaLotion-S06-ConeSP-0005f, Steady state flow step
Repeatability with Rough Cone
S05-Run1, initial flow test
S06-Run1, initial flow testS06-Run5, flow after LAOS
Nivea Lotion
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Yield Stress ModelsSt
ress
, σ
/ Pa
( )( )
B
H-B
Casson
12 2
Carreau 0 1
Y Pm
Y
Y p
n
Kσ σ µ γ
σ σ γ
σ σ µ γ
σ γη λγ−
= +
= +
= +
= +
(Sisko is subset of Carreau)
Barnes, JNNFM, 1999 (hypothetical data)
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0.1000 1.000 10.00 100.0 1000ang. frequency (rad/s)
1.000
10.00
100.0
1000
G' (
Pa)
1.000
10.00
100.0
1000G
'' (Pa)
NiveaLotion-S06-ConeSP-0004o
NiveaLotion-S06-ConeSP-0004o, Frequency sweep step
Linear viscoelasticity
0.5% strain amplitude
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Large amplitude oscillatory shear
0.010000 0.10000 1.0000 10.000 100.00 1000.0 10000% strain
0.1000
1.000
10.00
100.0
1000
G' (
Pa)
0.1000
1.000
10.00
100.0
1000
G'' (P
a)
NiveaLotion-S06-ConeSP-0002o
NiveaLotion-S06-ConeSP-0002o, Strain sweep stepNiveaLotion-S06-ConeSP-0003o, Strain sweep step
“strain-controlled”LAOS on AR-G2
Run 2) 10 rad/sRun 3) 1 rad/s
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Nivea lotion LAOS
0.000 0.001 0.002
0.0
0.1
0.2
Stre
ss [P
a]
Strain [-]
Nivea Lotion, AR-G2ω=10 rad/s, 6cm 2deg cone with sandpaper, T=23 C
0.00 0.01 0.02-1.0
-0.5
0.0
0.5
1.0
1.5
Stre
ss [P
a]Strain [-]
Nivea Lotion, AR-G2ω=10 rad/s, 6cm 2deg cone with sandpaper, T=23 C
-0.1 0.0 0.1-10
-5
0
5
10
Stre
ss [P
a]
Strain [-]
Nivea Lotion, AR-G2ω=10 rad/s, 6cm 2deg cone with sandpaper, T=23 C
0.0 0.5 1.0 1.5 2.0-20
-10
0
10
20
Stre
ss [P
a]
Strain [-]
Nivea Lotion, AR-G2ω=10 rad/s, 6cm 2deg cone with sandpaper, T=23 C
-10 -5 0 5 10-60
-40
-20
0
20
40
60
Stre
ss [P
a]
Strain [-]
Nivea Lotion, AR-G2ω=10 rad/s, 6cm 2deg cone with sandpaper, T=23 C
0.010000 0.10000 1.0000 10.000 100.00 1000.0 10000% strain
0.1000
1.000
10.00
100.0
1000
G' (
Pa)
0.1000
1.000
10.00
100.0
1000
G'' (P
a)
NiveaLotion-S06-ConeSP-0002o
NiveaLotion-S06-ConeSP-0002o, Strain sweep step
γ0=0.13% γ0=1% γ0=10.6%
γ0=108%
γ0=1052%Needs inertia correction to
determine sample stressω = 10 rad/s
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Model responses to LAOS
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Pseudoplastic and yield stress fluid response to LAOS
( )( )1
2 20( ) 1
n
σ γ γη λγ−
= +
0( ) cost tγ γ ω ω=LAOS deformation
Purely viscous Carreau model
Response:
0 : yield stress1 : Newtonian
nn==
Ewoldt et al., submitted
0
( ) costy tγ ωγ
= =
1 0n = →
310
( ) sintx tγ ωγ
= =0
( ) costy tγ ωγ
= =
Pseudoplastic and yield stress fluid response to LAOS
( )( )1
2 20( ) 1
n
σ γ γη λγ−
= +
0( ) cost tγ γ ω ω=LAOS deformation
Purely viscous Carreau model
Response:
0 : yield stress1 : Newtonian
nn==
Ewoldt et al., submitted
max 0( ) at 10tσ σ λγ ω =
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Elastic Bingham model
Y pE
Y
E EYGσ
σ σ µ γ
γ
γ
γ γ
γ
<
= + =
=
Yoshimura & Prud’homme., Rheol. Acta, 1987
Elastic before yield
Bingham model after yield
Important things to keep in mind:
1) Not a full continuum model as represented here2) Elastic strain can be recovered when flow reverses3) Strain-induced yielding, therefore a well defined yield surface in plate-plate tests
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Elastic Bingham model - LAOS
Y pE
Y
E EYGσ
σ σ µ γ
γ
γ
γ γ
γ
<
= + =
=
Yoshimura & Prud’homme., Rheol. Acta, 1987
Elastic before yield
Bingham model after yield
00
maximum imposed strain~yield strainY
γγ
Γ =
0 maximum viscous stress~yield stress
p
Y
Nµ γ ωσ
=
35
Elastic Bingham model - LAOS
Y pE
Y
E EYGσ
σ σ µ γ
γ
γ
γ γ
γ
<
= + =
=
Yoshimura & Prud’homme., Rheol. Acta, 1987
Elastic before yield
Bingham model after yield
00
Y
γγ
Γ = 0p
Y
Nµ γ ωσ
=
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Elastic Bingham model - LAOS
Y pE
Y
E EYGσ
σ σ µ γ
γ
γ
γ γ
γ
<
= + =
=
Yoshimura & Prud’homme., Rheol. Acta, 1987
Elastic before yield
Bingham model after yield
00
Y
γγ
Γ = 0p
Y
Nµ γ ωσ
=
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Elastic Bingham model – Pipkin space
Y pE
Y
E EYGσ
σ σ µ γ
γ
γ
γ γ
γ
<
= + =
=
Yoshimura & Prud’homme., Rheol. Acta, 1987
Elastic before yield
Bingham model after yield
00
Y
γγ
Γ =
0p
Y
Nµ γ ωσ
=
0max 6.0Γ =
max 0.3N =
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LAOS plate-plate artifacts
Lissajous curves of (apparent) stress σ(t) vs. strain γ(t).
Maximum (apparent) stress shown above curve, σA/σY
1. Smoothes out sharp transitions, since a portion of the material is always in the linear strain regime
2. Over-estimates stress for shear-thinning fluids, by factor of 4/3 for a perfect plastic response
Homogeneous (e.g. cone/plate) Plate-plate response
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Elastic Bingham model - LAOS
Y pE
Y
E EYGσ
σ σ µ γ
γ
γ
γ γ
γ
<
= + =
=
00
Y
γγ
Γ =
model from: Yoshimura & Prud’homme., Rheol. Acta, 1987
0p
Y
Nµ γ ωσ
=
Elastic before yield
Bingham model after yield
00
Y
γγ
Γ =
0p
Y
Nµ γ ωσ
=
40
Elastic Bingham model - LAOS
Y pE
Y
E EYGσ
σ σ µ γ
γ
γ
γ γ
γ
<
= + =
=
00
Y
γγ
Γ =
model from: Yoshimura & Prud’homme., Rheol. Acta, 1987
0p
Y
Nµ γ ωσ
=
Elastic before yield
Bingham model after yield
00
Y
γγ
Γ =
0p
Y
Nµ γ ωσ
=
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( ) ( )( )0 ma
2
x
0 1
1 Perfect Plastic4 0.785 Newtonian
0 Purely Elast2 2
icd pp
dE GE γ σ
πγφ π→
′′ = = → = →
Perfect plasticdissipation ratio
42
Experiments
43
Shear-thinning xanthan gum (0.2wt%)
( )0 1
max
1 Perfect Plastic4 0.785 Newtonian
40 Purely Elastic
d
d pp
E GE
πγφ πσ
→′′ ≡ = → =
→
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Drilling fluid
( )0 1
max
1 Perfect Plastic4 0.785 Newtonian
40 Purely Elastic
d
d pp
E GE
πγφ πσ
→′′ ≡ = → =
→
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Nivea lotion LAOS
0.000 0.001 0.002
0.0
0.1
0.2
Stre
ss [P
a]
Strain [-]
Nivea Lotion, AR-G2ω=10 rad/s, 6cm 2deg cone with sandpaper, T=23 C
0.00 0.01 0.02-1.0
-0.5
0.0
0.5
1.0
1.5
Stre
ss [P
a]Strain [-]
Nivea Lotion, AR-G2ω=10 rad/s, 6cm 2deg cone with sandpaper, T=23 C
-0.1 0.0 0.1-10
-5
0
5
10
Stre
ss [P
a]
Strain [-]
Nivea Lotion, AR-G2ω=10 rad/s, 6cm 2deg cone with sandpaper, T=23 C
0.0 0.5 1.0 1.5 2.0-20
-10
0
10
20
Stre
ss [P
a]
Strain [-]
Nivea Lotion, AR-G2ω=10 rad/s, 6cm 2deg cone with sandpaper, T=23 C
-10 -5 0 5 10-60
-40
-20
0
20
40
60
Stre
ss [P
a]
Strain [-]
Nivea Lotion, AR-G2ω=10 rad/s, 6cm 2deg cone with sandpaper, T=23 C
0.010000 0.10000 1.0000 10.000 100.00 1000.0 10000% strain
0.1000
1.000
10.00
100.0
1000
G' (
Pa)
0.1000
1.000
10.00
100.0
1000
G'' (P
a)
NiveaLotion-S06-ConeSP-0002o
NiveaLotion-S06-ConeSP-0002o, Strain sweep step
γ0=0.13% γ0=1% γ0=10.6%
γ0=108%
γ0=1052%
Needs inertia correction
Outline revisited
1. Introduction to yield stress fluidsunderstand definition/caveats of apparent yield stress fluids develop a feel for yield stress and viscosity values
2. Rheometry with yield stress fluidsidentify and avoid slip artifactscorrect for parallel plate artifactsrecognize LAOS response of yield stress fluids
Randy H. EwoldtJune 26, 2009
Part of the summer 2009 Reading Group: Yielding, Yield Stresses, ViscoelastoplasticityNon-Newtonian Fluids (NNF) Laboratory, led by Prof. Gareth McKinley