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2.341J Lecture 6: Extensional Rheometry:From Entangled Melts to Dilute Solutions

Professor Gareth H. McKinley Dept. of Mechanical Engineering, M.I.T.

Cambridge, MA 01239

http://web.mit.edu/nnf

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The Role of Fluid Rheology

• “Slimy”y • “Sticky”

• Other manifestations: ‘stringy’, ‘tacky’, ‘stranding’, ‘ropiness’, ‘pituity’,‘long’ vs. ‘short’ texture...

”

3 0

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Kinematics of Deformation

• As we have seen, there are three major classes of extensional flow:

vx = − 12

ε0(1+ b)x

v y = − 12 ε0(1− b)y

vz = ε0z

Simple Shear

Simple Shear-Free FlowFiber-spinning Thermoforming Calendering/rolling

b = flow type parameter

Fiber-spinning

vx = γ y

Fred Trouton (one paper on Rheology; 110 years ago)

doi:10.1016/S0377-0257(06)00214-X

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Bird, Armstrong & Hassager, (1987)© John Wiley and Sons. All rights reserved. This content is excluded from our Creative Commons license. For more information, see https://ocw.mit.edu/help/faq-fair-use/.

This image is in the public domain.

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Transient Extensional Rheometry

• The extensional viscosity is best considered as a transient material function: follow evolution from equilibrium to steady state (or break-up!)

106

105 Zero-Shear Rate Viscosity [Pa.s]104

103

102

101

100

10-1

10-2

Meissner Apparatus (RME)

SER Fixture

Opposed Jet DevicesContraction Flows (microfluidic)

Filament Stretching Rheometers(FISER)

Capillary Thinning & Breakup Rheometers

Melts

Dilute Solutions

McKinley & Sridhar, Sentmanat, Wang & McKinley, J. Rheol. May 2005 Ann. Reviews of Fluid Mech, 2002

20 μm

10-3

McKinley, Ann. Rheol. Reviews 2005

Pipe & McKinley, Rheol. Acta (AERC 2007)

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Importance of Extensional Rheology

2 cm

(d) (e)

20 μm

(a) (b)

(c)

• Dominates many processing operations• Effects are frequently transient in nature…

(f)

6.35

mm

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(e)�

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777

Nonlinear Extensional Viscosity

• Extensional flows are “strong flows” which result in extensive moleculardeformation, microstructural alignment and high tensile stresses

  Applications: fiber-spinning, blow-molding, sheet-molding, extrusion, coating

Strain Hardening

ηE+ ( ˙ ε 0 ,t) = τzz (t) −τrr (t)[ ] ˙ ε 0

limt→∞

ηE+ ( ˙ ε 0,t )→ηE( ˙ ε 0 )

Tension-Thickening

Transient Response Steady-State Response ?

Molecular alignment Increasing molecular increases with Hencky strain: alignment at higher strain rates

t Lε(t) = ∫ (t)

ε̇ (t ′ )dt ′ = ln−∞ L0

˙ ε 0t λ ˙ ε 0

Trouton (1906): ηE = 3μ

Ext

ensi

onal

Vis

cosi

ty

Ext

ensi

onal

Vis

cosi

ty

‘stretch’

‘coil’

Newtonian

Solution

Melt

De = λε0DeborahNumber

b = 0

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Tensile Stress Surface

• The evolution of the extensional viscosity can be visualized as a function oftime and strain rate.

Des = ε × τ s1

t s[ ]

Tr(De,t) =ηE

+

η0

Stretching of branched species for Des > 1

Constant velocity stretching

Necking

Increasing strain rate

Increasing viscosity (strain hardening)

Time of experiment

ηE (ε0, t) ⇔ Tr(De, ε)

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Polymer Melts

• SER Universal Testing Platform: specifically designedso that it can be easily accommodated onto a numberof commercially available torsional rheometers

• TA Instruments version;EVF = Extensional Viscosity Fixture

• Can be housed within the host system’s environmentalchamber for controlled temperature experiments.

  Requires only 5-200mg of material   Can be used up to temperatures of 250°C

  Easily detachable for fixture changeover/clean-up

Validation Experiments: LDPE (BASF Lupolen® 1840H) (Sentmanat, Wang & McKinley; JoR Mar/Apr (2005)  Mn = 17,000; Mw = 243,000; Mw/ Mn = 14.3 CH3/1000C = 23 Very similar to the IUPAC A reference material  Same polymer as that used by

Münstedt et al., Rheol. Acta 37, 21-29 (1998) ‘Münstedt rheometer’ (end separation method)

Sentmanat, Rheol. Acta (2004)

1.5”

1.5”

SER Principle of Operation

• “Constant Sample Length” Extensional Rheometer• Ends of sample affixed to windup drums, such that

for a constant drum rotation:

10

. ε0 = 2ΩR L

• Resulting torque on transducer (attached to housing)

T = 2(F + Ffriction )RSER Fixture with ARES Rheometer

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Comparison of LDPE Stress Growth Curves

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• Good agreement with LVE response at short times (t ≥ 0.01 s)• Increasing strain-hardening and sample rupture at high rates

• The results with the SER (red curves) show excellent agreementwith literature results from Münstedt et al. (black symbols & lines)

ηE+ = 3η+ (t)

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The Role of Chain-Branching

• Extensional stress growth is astrong function of the level ofmolecular branching.

• Branch points act as ‘crosslinks’that efficiently elongate chainsand transmit stress…

  Provided they are long enough to be entangled

H.M. Laun, Int. Cong. Rheol. 1980

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Results for Simple Fluids

• Newtonian Fluids • Ideal Elastic Fluids   McKinley & Tripathi, J Rheol., 2000   Entov & Hinch, JNNFM, 1997

RmidR0

= 0.0709σ

ηsR0tc − t( ) Rmid

R0=

GR0σ

⎛ ⎝

⎞ ⎠

1/ 3

exp −t3λ1

⎛ ⎝ ⎜

⎞ ⎠ ⎟

2R0 = 6 mm

tcap =η0R0σ

Poly

styr

ene

Oil

LaserMicrometer

xSM

-1 (500 ppm 2

106

~ 8 sec

g/mol.

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Dilute Polymer Solutions

• Increasing molecular weight delays elasto-capillary breakup• Measured time-scale agrees quantitatively with Zimm relaxation time

• Important in many biological processes (saliva, spinning of spider silk)

0.01

0.1

1

0 5 10 15 20 25 30 35 40

SM-1 FluidSM-2 FluidSM-3 FluidOldroyd-B

D/D

0

t/(η0R

0/σ)

−1 (3λZimm )

Finite Extensibility

100

101

102

103

106 107

λ z [se

c]

Mw

[g/mol]

100

101

102

103

106 107

η p [P

a.s]

Mw

[g/mol]

103

102

101

100

λ z [s

ec], η p

[Pa.

s]

Mw [g/mol]106

SM1

SM2

SM3

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8�

15�

Flow Through Porous Media�

• Solutions of flexible polymers and self-assembling wormlike surfactants arecommonly used in enhanced oil recovery (EOR) and reservoir fracturing operations.

Kefi et al. (2004) Oilfield Review�

Müller & Sanz in Kausch & Nguyen, 1997�

Dim

ensi

onle

ss�

Pres

sure

dro

p�

16�

Extensional Viscosity of Wormlike Surfactants�

• Solutions of EHAC (Schlumberger VES“clearfrac”,) in brine

Yesilata, Clasen & McKinley, JNNFM 133(2) 2006�Kim, Pipe, McKinley; Kor-Aust Rheol. J, 2010�

Dmid (t) ⇒ �ε(t) = − 2

Dmid

dDmid

dt⇒ ηE,apparent (t)

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Drag Reduction and Jet Breakup

• Extensional effects from polymer additives can dramatically reduce theextent of turbulent dissipation in high Reynolds number flows

• Applications include:  Wake reduction: sailing, submarines, high-speed swimming (dolphins)

  Flow-rate enhancement: storm drains, firehoses...

Union-Carbide“Rapid Water”!!

From W.R. Schowalter “Non-Newtonian Fluid Mechanics”( Pergamon) 1978

Commercial Interest in Jet (?) Breakup

New York Times, Dec 4th, 1984

“..if a few pieces of spaghetti are withdrawn gently there is little resistance.But if you jerk it out, it pulls on more than you have hold of. The resistance Is much higher.” (strain-hardening)

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Break Up of Low Viscosity Liquids

• 0.1 wt% PEO (2x106 g/mol) in water  Standard test fluid;   Plate diameter R0 = 3mm; Aspect Ratio = 2.3

η0 ≈ 1.10 ×10−3

Pa.s tR ≈ρR0

3

σ ≈ 0.020s

Rayleigh Time Scale

Rodd, Cooper-White, McKinley; Applied Rheol. 15(1), 12-27; 2005

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Low Viscosity Fluids

• Critical to Inkjet Printing  100-1000 drops per second   Drop volume 2-10 picoliter   Eliminate formation of spray and “satellite

droplets” • Identical viscosities and surface tensions

  One contains Poly(ethylene oxide) (PEO) Mw = 1,000,000 g/mol, c = 100 ppm

λDe0 =

ρR30 σ

• Viscous effects are never important compared to inertial, capillary, elastic effects!!

  Inviscid Viscoelastic Fluids….

~ O(1)Elastic effects Inertio-capillary effects

Oh =η0ρσR0

<<1Viscous effects Inertio-capillary effects

V. Tirtaatmadja, J.J. Cooper-White et al., Phys Fluids 18 (2006)

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• “I measure the properties in shear but it doesn’t helpme differentiate between different product grades that

spray (orĀ dispense, jet, spinĀ ) well”

ε̇,√3γ̇ [s−1]

ηand

(1/3)η

E

ε̇,√3γ̇ [s−1]

ηand

(1/3)η

E

ε̇,√3γ̇ [s−1]

ηand

(1/3)η

E

ε̇,√3γ̇ [s−1]

ηand

(1/3)η

E

η0 = 0.1Pa.sηs = 0.03 Pa.sL = 3.8τ = 0.01 s

ε̇,√3γ̇ [s−1]

ηand

(1/3)η

E

1

3ηE (at ε = ∞) =η0 +

2

3ηpL

2

η0 = 0.1Pa.sηs = 0.03 Pa.sL = 3.8τ = 0.01 s

Measure Shear & Extensional Viscosity of a typical commercial paint

[Pa.

s]

Capillary Thinning Experiments

The limit of infinite dilution: single molecule experiments

• Label DNA as a model “supersized” single flexible molecule

The Cross Slot Apparatus: A free-standing stagnation point : Perkins, Smith Chu, Science 1997

22

De Gennes (Perspective): “molecular individualism”

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Polymer Additives and Extensional Rheology�

• Large axial stresses (“Streamline tension”)• Transient extensional stretching of chains and tensile stress growth...

∇∇v =�ε0

2

⎛ −1 0 0 ⎞⎜ 0 −1 0 ⎟⎜ ⎟⎝ 0 0 +2⎠

Smith, Chu, Larson, Science 1997�Doyle, Shaqfeh, Spiegelberg, McKinley, JNNFM 1997�

O(L2 )~Mw �

100

101

102

103

0 1 2 3 4 5 6 7

Trou

ton

Ratio

Hencky Strain

SM-1 Fluid

Tr = 3(a)(b)

(c)

(d)

Toronto De = 12Monash De = 14M.I.T. De = 17

FENE-PM Theory�

(ext

ensi

onal

vis

cosi

ty)/

(she

ar v

isco

sity

)�

De = λ ˙ 1ε 0

Anna et al. J. Rheol. 2001�

Increasing �Flow strength�

Summary�

•  A number of well-characterized instruments now exist for performingmeasurements of transient extensional rheometry for fluids spanning range fromdilute solution to the melt�  ‘constant volume’ devices: e.g. FISER, CABER, Münstedt Rheometer �  ‘constant length’ devices: e.g. EVF, SER, RME = Meissner Rheometer

•  Understanding the kinematics imposed by the instrument and the dynamics offilament evolution is essential in order to extract the true material functions

•  Challenges still remain:�  Theory for filament deformation and rupture at very high strains �  Measurements for ‘weakly elastic’ materials “non-spinnable materials” �  Understanding and exploiting extensional viscosity on the microscale:

L. Mahadevan, Harvard� R. Cohn, U. Louisville�N. Kojic et al.�

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2.341J / 10.531J Macromolecular HydrodynamicSpring 2016

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