Yield Stress Fluids, Meeting #1

Home , Stress (mechanics), Yield surface

1. Introduction to yield stress fluids understand definition/caveats of apparent yield stress fluids develop a feel for yield stress and viscosity values 2. Rheometry with yield stress fluids identify and avoid slip artifacts correct for parallel plate artifacts recognize LAOS response of yield stress fluids

Randy H. Ewoldt June 26, 2009

Part of the summer 2009 Reading Group: Yielding, Yield Stresses, Viscoelastoplasticity Non-Newtonian Fluids (NNF) Laboratory, led by Prof. Gareth McKinley 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)

2 YieldYield Stress Stress Fluids: Fluids: Idealized example: Apparently solid at low stress, σ < σY Apparently solid at low stress, σ < σY linear scale Apparently fluid at high stress, σ > σY Apparently fluid at high stress, σ > σY e.g.e.g. paint, paint, toothpaste, toothpaste, hand hand lotion, lotion, concrete,concrete, snail snail slime, slime, nuclear nuclear waste waste sludge sludge

Idealized example: log-log scale

3 Detailed experiments

ApparentApparent yield yield stress stress fluids:fluids: Tomato puree 1.1. Initially Initially “high” “high” viscosityviscosity Toothpaste 2.2. Dramatic Dramatic viscosity viscosity dropdrop over over narrow narrow σ range of stress Viscosity, η = range of stress γ 3.3. Flows Flows

Mayonnaise

Carbopol 980, polymer microgel (0.2wt%)

4 Viscosity, η ≈ 108 Pa.s

Ig 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: 1927 Drop #1: 1938 Drop #8: 2000 (12 years between drops #7 and #8) Drop #9: ????

5 6 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 Professor Franz-Josef Ulm granular in nature. M.I.T. Dept. of Civil and Env. Eng.

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 η≈1014 Pa.s hinders the movement of the C-S-H granules over time.

The rate of creep is logarithmic, which means slowing creep increases durability exponentially.

η≈1019 Pa.s 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.

7 Vandamme and Ulm, PNAS, 2009 Apparent yield stress fluids?

Blood

Hematocrit content

Olivine (a rock), various grain sizes

Plasticine (modelling material, like putty)

8 Barnes, JNNFM, 1999 Slime & simulant rheology: Flow

Carbopol (polymer microgel) Laponite (particulate gel) Carbopol 940 in DIW, pH→7 with NaOH Laponite RD in DIW 9 Ewoldt et al., Soft Matter, 2007 Magnetorheological Fluid

LORD Corp. MRF-132DG carbonyl iron particles (1-20µm) in silicone oil B

α =⋅1.4 105-2 Pa.T no field field-activated

105 (a) 4

[Pa] 10 R σ Magnetic Field 3 10 0.462 T 0.221 T AR-G2 rotational rheometer, 2 0.100 T 10 parallel plates with 600 grit sandpaper, h=500µm 0.046 T 0.030 T D=40mm for ambient test 1 10 0.012 T D=20mm for magnetic field tests ambient Using MRF Rheometer Cell from Corrected Shear Stress, Stress, Shear Corrected 100 Murat Ocalan, Ph.D. Thesis (in progress), MIT 10-5 10-4 10-3 10-2 10-1 100 101 102 103 Rim Shear Rate [s-1] 10 Ewoldt, Ph.D. thesis, 2009 Magnetorheological fluid

108

7 Magnetic Field 10 (b) B=0.462 T B=0.221 T 6 10 B=0.100 T

[Pa.s] B=0.046 T 5 η 10 B=0.030 T B=0.012 T 104 ambient

103 102 101 100 Corrected Viscosity, 10-1 100 101 102 103 104 105 Corrected Shear Stress, σ [Pa] R

11 Apparent Yield Stress Fluids

Some 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 sources of error in measurement.

[36] I. Evans, J. Rheol. 36 (7) (1992) 1313. 12 [38] Q.D. Nguyen, D.V. Boger, Ann. Rev. Fluid Mech. 24 (1992) 47. 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. 13 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

14 Experimental Issue: Slip

15 Sandpaper required to avoid slip

NiveaNiveaLotion-0001f Lotion

100.0 NiveaLotion-S02-h1050-0001f, Steady state flow step NiveaLotion-S02-h450-0001f, Steady state flow step NiveaLotion-S02-h700-0001f, Steady state flow step NiveaLotion-S03-h1050-0001f, Steady state flow step NiveaLotion-S03-h450-0001f, Steady state flow step NiveaLotion-S03-h700-0001f, Steady state flow step NiveaLotion-S04-h1050-0001f, Steady state flow step

FILLED) P/P w/ sandpaper, 3 different gaps 10.00 M shear stress (Pa) OPEN) P/P, NO sandpaper, 3 different gaps

Sandpaper from 1.000 McMaster-Carr 1.000E-6 0.01000 100.0 Part #) 47185A51 Adhesive-back shear rate (1/s) 16 sandpaper, 8” dia. disc, 600 grit Flow curves: different perspectives

NiveaLotion-0001f NiveaLotion-0001f

NiveaLotion-S02-h1050-0001f, Steady state flow step 1.000E7 NiveaLotion-S02-h1050-0001f, Steady state flow step 1.000E7 NiveaLotion-S02-h450-0001f, Steady state flow step NiveaLotion-S02-h450-0001f, Steady state flow step NiveaLotion-S02-h700-0001f, Steady state flow step NiveaLotion-S02-h700-0001f, Steady state flow step NiveaLotion-S03-h1050-0001f, Steady state flow step NiveaLotion-S03-h1050-0001f, Steady state flow step NiveaLotion-S03-h450-0001f, Steady state flow step 1.000E6 NiveaLotion-S03-h450-0001f, Steady state flow step 1.000E6 NiveaLotion-S03-h700-0001f, Steady state flow step NiveaLotion-S03-h700-0001f, Steady state flow step NiveaLotion-S04-h1050-0001f, Steady state flow step NiveaLotion-S04-h1050-0001f, Steady state flow step 1.000E5 1.000E5 10000 10000 1000 1000 100.0 100.0 viscosity (Pa.s) viscosity (Pa.s) 10.00 10.00 1.000 1.000 0.1000 0.1000 1.000 10.00 100.0 1.000E-6 0.01000 100.0 shear stress (Pa) shear rate (1/s)

17 Parallel plate correction, steady flow

100 Nivea Lotion Parallel plates with sandpaper D=40mm OPEN: Apparent stress FILLED True stress [Pa] h=1050 µm σ h= 700 µm 10 h= 450 µm

M Shear Stress, Shear 1 10-6 10-5 10-4 10-3 10-2 10-1 100 101 102 103 -1 Shear Rate [s ] 18 Parallel plate stress correction M R Torque balance M = 2()πσrrdr2 ∫0

Linear assumption σ ()rHr=

2M Apparent rim stress σ ()R = Correction formula A π R3 M  dMln  σ R =+3 3  2lnπ Rdγ Error for perfect plastic for σ (r ) = σ  R  Y σ

σ A 4 → σ A  d lnσ A  R =+3  σY 3 4ln d γ R 

19 4/3 ratio mentioned by Brunn and Asoud, Rheol. Acta, 2002 Parallel plate stress correction, Nivea lotion steady flow

100 Nivea Lotion Parallel plates with sandpaper D=40mm OPEN: Apparent stress FILLED True stress [Pa] h=1050 µm σ h= 700 µm 10 h= 450 µm

Shear Stress, Shear 1 10-6 10-5 10-4 10-3 10-2 10-1 100 101 102 103 -1 Shear Rate [s ] 20 Roughened Cone – Works!

100 Nivea Lotion on AR-G2 3 different samples to show repeatability (S02, S04, S05)

all geometries with sandpaper P/P) D=40mm, with plate-plate correction o Cone) D=60mm, α=2 , no correction necessary [Pa] σ 10

S02, P/P, h= 1050µm (uncorrected, for reference) S02, P/P, h= 700µm S02, P/P, h= 450µm S04, P/P, h=1050µm

Shear Stress, Shear S05, Cone 1 10-6 10-5 10-4 10-3 10-2 10-1 100 101 102 103 -1 Shear Rate [s ] 21 Nivea lotion flow curve

6 10 Nivea Lotion 3 different samples to show repeatability (S02, S04, S05) 5 S02, P/P, h= 1050µm 10 S02, P/P, h= 700µm all geometries with sandpaper S02, P/P, h= 450µm P/P) D=40mm, 4 S04, P/P, h=1050µm with plate-plate correction o 10 S05, Cone Cone) D=60mm, α=2 , no correction necessary 3

[Pa.s] 10 η

102 101

Viscosity, Viscosity, 10 10-1 10-3 10-2 10-1 100 101 102 103 104 Shear Stress [Pa] 22 Creep tests, low stress

NiveaLotion-S04-h1050-0002c 2.5000 Cannot determine a steady flow viscosity below the yield stress 2.0000 NiveaLotion-S04-h1050-0002c, Creep CLOSED)NiveaLotion-S04-h1050-0003c, σ = 1Creep Pa 0 NEGATIVE strain-rates? Is this 1.5000 remnant elastic recovery from the previous flow test? I saw 1.0000 this also with the stress-ramp

% strain flow tests, from high-to-low 0.50000 stresses, that at sufficiently low stresses the strain-rate was 0 negative (see Sample #1). Or OPEN) σ0 = 0.1 Pa maybe it’s instrument artifact? -0.50000 0 100.0 200.0 300.0 400.0 500.0 600.0 time (s)

23 Repeatability with Rough Cone

NiveaLotion-S05-ConeSP-0001f 100.0 Nivea Lotion

NiveaLotion-S05-ConeSP-0001f, Steady state flow step NiveaLotion-S06-ConeSP-0001f, Steady state flow step NiveaLotion-S06-ConeSP-0005f, Steady state flow step

10.00 shear stress (Pa) S05-Run1, initial flow test

S06-Run1, initial flow test S06-Run5, flow after LAOS

1.000 1.000E-5 1.000E-3 0.1000 1.000 10.00 100.0 shear rate (1/s) 24 Yield Stress Models

σ B = σµγYP+ σσ γm H-B =+Y K

/ Pa σ Casson =+σµγYp σ σγηλγn−1 2 2 Carreau=+ 0 ()1 () Stress, (Sisko is subset of Carreau)

Barnes, JNNFM, 1999 (hypothetical data) Linear viscoelasticity

NiveaLotion-S06-ConeSP-0004o 1000 1000

NiveaLotion-S06-ConeSP-0004o, Frequency sweep step

100.0 100.0 G'' (Pa) G' (Pa) 10.00 10.00

0.5% strain amplitude

1.000 1.000 0.1000 1.000 10.00 100.0 1000 ang. frequency (rad/s) 26 Large amplitude oscillatory shear

NiveaLotion-S06-ConeSP-0002o

1000 1000

NiveaLotion-S06-ConeSP-0002o, Strain sweep step NiveaLotion-S06-ConeSP-0003o, Strain sweep step

100.0 100.0 G'' (Pa)

10.00 10.00 G' (Pa)

“strain-controlled” LAOS on AR-G2

1.000 Run 2) 10 rad/s 1.000 Run 3) 1 rad/s

0.1000 0.1000 0.010000 0.10000 1.0000 10.000 100.00 1000.0 10000 % strain 27 Nivea lotion LAOS

Nivea Lotion, AR-G2γ =0.13% Nivea Lotion, AR-G2 γ =1% Nivea Lotion, AR-G2 0 0 γ =10.6% ω=10 rad/s, 6cm 2deg cone with sandpaper, T=23 C ω=10 rad/s, 6cm 2deg cone with sandpaper, T=23 C ω=10 rad/s, 6cm 2deg cone0 with sandpaper, T=23 C 1.5 10

1.0 0.2 5 0.5

0.1 0.0

Stress [Pa] -5 Stress [Pa] Stress [Pa] 0.0 -0.5

-1.0 -10 0.000 0.001 0.002 0.00 0.01 0.02 -0.1 0.0 0.1 Strain [-] Strain [-] Strain [-]

Nivea Lotion, AR-G2γ =108% 0 ω=10 rad/s, 6cm 2deg cone with sandpaper, T=23 C NiveaLotion-S06-ConeSP-0002o 20 1000 1000 10

100.0 100.0 -10 Stress [Pa] Stress

-20

G'' (Pa) 0.0 0.5 1.0 1.5 2.0 10.00 10.00 Strain [-] G' (Pa)

Nivea Lotion, AR-G2γ =1052%

ω=10 rad/s, 6cm 2deg cone0 with sandpaper, T=23 C 60 Needs inertia 1.000 1.000 40 NiveaLotion-S06-ConeSP-0002o, Strain sweep step 20 correction to determine ω = 10 rad/s 0 -20 sample stress

0.1000 0.1000 Stress [Pa] 0.010000 0.10000 1.0000 10.000 100.00 1000.0 10000 -40 % strain -60 28 -10 -5 0 5 10 Strain [-] Model responses to LAOS

29 Pseudoplastic and yield stress fluid response to LAOS

n−1 2 2 n = 0 : yield stress Purely viscous Carreau model σγ() γη=+ λγ0 () 1 () n =1 : Newtonian

LAOS deformation γ()tt= γω0 ω cos Response:

n =→10

γ()t yt==cosω γ0 30 Ewoldt et al., submitted Pseudoplastic and yield stress fluid response to LAOS

n−1 2 2 n = 0 : yield stress Purely viscous Carreau model σγ() γη=+ λγ0 () 1 () n =1 : Newtonian

LAOS deformation γ()tt= γω0 ω cos Response: σ (t )σλγωmax at 0 = 10

γ ()t γ ()t xt==sinω yt==cosω γ0 γ 0 31 Ewoldt et al., submitted 32 Elastic Bingham model

E E σ = Gγ < γ Y Elastic before yield σσ=+ µγ E Ypγγ = γ Y Bingham model after yield

Important things to keep in mind: 1) Not a full continuum model as represented here 2) Elastic strain can be recovered when flow reverses 3) Strain-induced yielding, therefore a well defined yield surface in plate-plate tests

33 Yoshimura & Prud’homme., Rheol. Acta, 1987 Elastic Bingham model - LAOS

E E σ = Gγ < γ Y Elastic before yield σσ=+ µγ E Ypγγ = γ Y Bingham model after yield

γ 0 maximum imposed strain Γ=0 ~ γ Y yield strain

µ γω maximum viscous stress N = p 0 ~ σY yield stress

34 Yoshimura & Prud’homme., Rheol. Acta, 1987 Elastic Bingham model - LAOS

E E σ = Gγ < γ Y Elastic before yield σσ=+ µγ E Ypγγ = γ Y Bingham model after yield

γ µ γω Γ= 0 N = p 0 0 γ Y σY

35 Yoshimura & Prud’homme., Rheol. Acta, 1987 Elastic Bingham model - LAOS

E E σ = Gγ < γ Y Elastic before yield σσ=+ µγ E Ypγγ = γ Y Bingham model after yield

γ µ γω Γ= 0 N = p 0 0 γ Y σY

36 Yoshimura & Prud’homme., Rheol. Acta, 1987 Elastic Bingham model – Pipkin space

E E σ = Gγ < γ Y Elastic before yield σσ=+ µγ E Ypγγ = γ Y Bingham model after yield

γ 0 Γ=0 γ Y

Γ=0max 6.0

Nmax = 0.3 µ γω N = p 0 37 σ Yoshimura & Prud’homme., Rheol. Acta, 1987 Y LAOS plate-plate artifacts

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

Lissajous curves of (apparent) stress σ(t) vs. strain γ(t).

Maximum (apparent) stress shown above curve, σA/σY

38 Elastic Bingham model - LAOS

E E σ = Gγ < γ Y Elastic before yield σσ=+ µγ E Ypγγ = γ Y Bingham model after yield

γ Γ= 0 γ 0 Γ = 0 γ Y 0 γ Y

µ pγω0 µ γω N = N = p 0 σY σ Y 39 model from: Yoshimura & Prud’homme., Rheol. Acta, 1987 Elastic Bingham model - LAOS

E E σ = Gγ < γ Y Elastic before yield σσ=+ µγ E Ypγγ = γ Y Bingham model after yield

γ Γ= 0 γ 0 Γ = 0 γ Y 0 γ Y

µ pγω0 µ γω N = N = p 0 σY σ Y 40 model from: Yoshimura & Prud’homme., Rheol. Acta, 1987 φπ →1 Perfect Plastic 2  Perfect plastic EGd πγ 01′′  == →=4 0.785 Newtonian dissipation ratio E 22γσ ()d pp ()()0max   → 0 Purely Elastic 41 Experiments

42 Shear-thinning xanthan gum (0.2wt%)

φπ →1 Perfect Plastic EGπγ ′′  ≡=d 01  →=4 0.785 Newtonian E 4 ()d pp σmax   → 0Purely Elastic

43 Drilling fluid

φπ →1 Perfect Plastic EGπγ ′′  ≡=d 01  →=4 0.785 Newtonian E 4 ()d pp σmax   → 0Purely Elastic

44 Nivea lotion LAOS

1.0 0.2 5 0.5

0.1 0.0

Stress [Pa] -5 Stress [Pa] Stress [Pa] 0.0 -0.5

-1.0 -10 0.000 0.001 0.002 0.00 0.01 0.02 -0.1 0.0 0.1 Strain [-] Strain [-] Strain [-]

Nivea Lotion, AR-G2γ =108% 0 ω=10 rad/s, 6cm 2deg cone with sandpaper, T=23 C NiveaLotion-S06-ConeSP-0002o 20 1000 1000 10

100.0 100.0 -10 Stress [Pa] Stress

-20

G'' (Pa) 0.0 0.5 1.0 1.5 2.0 10.00 10.00 Strain [-] G' (Pa)

Nivea Lotion, AR-G2γ =1052%

ω=10 rad/s, 6cm 2deg cone0 with sandpaper, T=23 C 60

40 1.000 1.000 Needs inertia NiveaLotion-S06-ConeSP-0002o, Strain sweep step 20 correction

-20

0.1000 0.1000 Stress [Pa] 0.010000 0.10000 1.0000 10.000 100.00 1000.0 10000 -40 % strain -60 45 -10 -5 0 5 10 Strain [-] Outline revisited

Randy H. Ewoldt June 26, 2009

Part of the summer 2009 Reading Group: Yielding, Yield Stresses, Viscoelastoplasticity Non-Newtonian Fluids (NNF) Laboratory, led by Prof. Gareth McKinley