Rheometry CM4650 4/27/2018

Home , Circulation (physics), Rheometer, Rheometry

Capillary Rheometer piston CM4650 F entrance region Polymer Rheology r A barrel z

2Rb polymer melt L 2R reservoir B

exit region

Professor Faith A. Morrison Department of Chemical Engineering Michigan Technological University 1 © Faith A. Morrison, Michigan Tech U.

Rheometry (Chapter 10)

measurement

All the comparisons we have discussed require that we somehow measure the material functions on actual fluids.

2 © Faith A. Morrison, Michigan Tech U.

1 Rheometry CM4650 4/27/2018

Rheological Measurements (Rheometry) – Chapter 10 Tactic: Divide the Problem in half Modeling Calculations Experiments

Dream up models RHEOMETRY

Calculate model Build experimental predictions for Standard Flows apparatuses that stresses in standard allow measurements flows in standard flows

Calculate material Calculate material functions from Compare functions from model stresses measured stresses

Pass judgment

on models U. A. Morrison,© Tech MichiganFaith

Collect models and their report 3 cards for future use

Chapter 10: Rheometry

Capillary Rheometer piston CM4650 F entrance region Polymer Rheology r A barrel z

2Rb polymer melt L 2R reservoir B

exit region

Professor Faith A. Morrison Department of Chemical Engineering Michigan Technological University 4 © Faith A. Morrison, Michigan Tech U.

2 Rheometry CM4650 4/27/2018

Rheological Measurements (Rheometry) – Chapter 10 Tactic: Divide the Problem in half Modeling Calculations Experiments

Dream up models RHEOMETRY

Calculate model Build experimental predictions for Standard Flows apparatuses that stresses in standard allow measurements in standard flows As withflows Advanced

Rheology,Calculate material there is way Calculate material functions from Compare functions from too muchmodel stresseshere to cover in measured stresses the time we have. Pass judgment

on models U. A. Morrison,© Tech MichiganFaith (we will be taking a tour) Collect models and their report 5 cards for future use

Rheological Measurements (Rheometry) – Chapter 10

Thumbnail of Rheometry  Need to create the flow  Measure the stresses accurately

Shear (not too hard) Elongation (hard) • Capillary (Weissenberg- • Capillary entrance losses – Rabinowich and slip corrections, many assumptions good for high ; Ψ from die swell, • Filament stretching – hard to Ψ) produce; research only • Torsional flow • Counter-rotating rolls – pretty • Parallel plate (correction accessible needed; low ; Ψ, Ψ) • Cone and plate (low ; can do Ψ,Ψ • Couette (cup and bob; low ; Ψ, Ψ) © Faith A. Morrison, Michigan Tech U. Tech Michigan A. Morrison, © Faith

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Rheological Measurements (Rheometry) – Chapter 10

Standard Flows Summary

Choose velocity field: Symmetry of flow alone implies:

  con 11 12 0  0   v (x ) x ≡ H 1 2 2 0     0  21 22 x 0 1  0 0    33123 x3 2 11 0 0  ≡      0 22 0  x1, x2 2  0 0    33123

To measure the stresses we need for material functions, we must produce the defined flows 7 © Faith A. Morrison, Michigan Tech U.

Rheological Measurements (Rheometry) – Chapter 10

Simple Shear flow (Drag) (z-plane ≡ 0 section) y 0 x

Challenges: •Sample loading •Maintain parallelism •Producing linear motion •Stress measurement (Edge effects) •Signal strength

From the McGill website (2006): Hee Eon Park, first-year postdoc in Chemical Engineering, works on a high-pressure sliding plate rheometer, the only instrument of its kind in the world.

J. M. Dealy and S. S. Soong “A Parallel Plate Melt Rheometer Incorporating a Shear Stress Transducer,”J. Rheol. 28, 8 355 (1984) © Faith A. Morrison, Michigan Tech U.

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Rheological Measurements (Rheometry) – Chapter 10

Although we stipulated simple, homogeneous shear flow be produced throughout the flow domain, can we, perhaps, relax that requirement?

≡ 0 0

9 © Faith A. Morrison, Michigan Tech U.

Rheological Measurements (Rheometry) – Chapter 10

Viscometric flow: motions that are locally equivalent to steady simple shearing motion at every particle

•globally steady with respect to some frame of reference •streamlines that are straight, circular, or helical •each flow can be visualized as the relative motion of a sheaf of material surfaces (slip surfaces) •each slip surface moves without changing shape during the motion •every particle lies on a material surface that moves without stretching (inextensible slip surfaces)

Viscometric Flows: 1. Steady tube flow 2. Steady tangential annular flow 3. Steady torsional flow (parallel plate flow) 4. Steady cone-and-plate flow (small cone angle) 5. Steady helical flow

Wan-Lee Yin, Allen C. Pipkin, “Kinematics of viscometric flow,” Archive for Rational Mechanics and Analysis, 37(2) 111-135, 1970 R. B. Bird, R. Armstrong, O. Hassager, Dynamics of Polymeric Liquids, 2nd edition, Wiley (1986), section 3.7. 10 © Faith A. Morrison, Michigan Tech U.

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Rheological Measurements (Rheometry) – Chapter 10 Viscometric Flows: Experimental Shear Geometries (viscometric flows) 1. Steady tube flow 2. Steady tangential annular flow 3. Steady torsional pp flow (z-plane 4. Steady cone-and-plate flow y section)   5. Steady helical flow x  H r  (z-plane r section) y piston F (-plane (plane x section) section) A barrel

2Rb polymer melt reservoir  B  (-plane (-plane section) (z-plane section) (z-plane section) section)

 r Q

11 © Faith A. Morrison, Michigan Tech U.

Rheological Measurements (Rheometry) – Chapter 10

Types of Shear Rheometry

Mechanical:

•Mechanically produce linear drag flow; Measure (shear strain transducer): 1. planar Couette Shear stress on a surface

•Mechanically produce torsional drag flow; Measure: (strain-gauge; force rebalance) 1. cone and plate; Torque to rotate surfaces 2. parallel plate; Back out material functions 3. circular Couette

•Produce pressure-driven flow through conduit Measure: 1. capillary flow Pressure drop/flow rate 2. slit flow Back out material functions

12 © Faith A. Morrison, Michigan Tech U.

6 Rheometry CM4650 4/27/2018

Shear Rheometry– Capillary Flow Capillary Rheometer piston F entrance region

r A barrel z

2Rb polymer melt L 2R reservoir B

exit region

Q 13 © Faith A. Morrison, Michigan Tech U.

Shear Rheometry– Capillary Flow Capillary Rheometer piston The basics F entrance region

A barrel r z 2Rb

polymer melt 2R well-developed flow reservoir B

exit region

Q 14 © Faith A. Morrison, Michigan Tech U.

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Shear Rheometry– Capillary Flow

Exercise:

• What is the shear stress in capillary flow, for a fluid with r unknown constitutive equation? •What is the shear rate in capillary z flow?

15 © Faith A. Morrison, Michigan Tech U.

Shear Rheometry– Capillary Flow

To calculate shear rate, shear stress, look at EOM: P  p  gz  v    v v  P     t 

steady Assume: unidirectional state •Incompressible fluid •no -dependence •long tube  P   1 r   0    rr    •symmetric stress tensor    r   r r r  •Isothermal      1 r 2  •Viscosity independent of pressure 0   0    r       r 2 r     P   1 r  0    rz   rz      z rz  r r rz 16 © Faith A. Morrison, Michigan Tech U.

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Shear Rheometry– Capillary Flow Shear stress in capillary flow, for a fluid with unknown constitutive equation Shear stress in capillary flow: P  P r r   0 L  rz R  P  2L R   constant  z  (varies with position, i.e. inhomogeneous flow)

What was the shear stress in drag flow?

17 © Faith A. Morrison, Michigan Tech U.

Shear Rheometry– Capillary Flow Viscosity from capillary flow – inhomogeneous shear flow

x2 R r

x Shear coordinate system 1 near wall:

eˆ1  eˆz      It is not the 21 rz rR R same shear eˆ2  eˆr v v rate   z   z     0 rR R everywhere, eˆ3  eˆ (r) r but if we   focus on the   21  R wall shear stress wall we can still get ( )    R 0 R wall shear rate 18 © Faith A. Morrison, Michigan Tech U.

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Shear Rheometry– Capillary Flow

Viscosity from Wall Stress/Shear rate Note: we are assuming no-slip at the wall

Wall shear stress in capillary flow: P  P r PR   0 L  rz rR 2L 2L rR  P    constant  z 

What is shear rate at the wall in capillary flow?

If is known, this is easy to calculate.

19 © Faith A. Morrison, Michigan Tech U.

Shear Rheometry– Capillary Flow If is known, is easy to calculate.

Velocity Fields, Flow in a Capillary

2 2Q   r   Newtonian fluid: vz (r)  2 1    R   R  

Power-law GNF fluid:

1 1 1  1  1 P  P  n  1   r  n n 0 L    vz (r)  R     1    2mL  1 n 1  R    

20 © Faith A. Morrison, Michigan Tech U.

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Shear Rheometry– Capillary Flow If is known, is easy to calculate.

Wall shear-rate for a Newtonian fluid

Hagen-Poiseuille: PR4 Q  8L 44Q 4Q 1 PR a   R3 R3  2L

slope = 1/

PR   R 2L

21 © Faith A. Morrison, Michigan Tech U.

Shear Rheometry– Capillary Flow If is known, is easy to calculate.

Wall shear-rate for a Power-law GNF

PL-GNF flow rate: 10

1 3 44Q  PR  n 1 nR   Q    1 n a  2L  m 1 3n R3

 1/n  1  4m   1/ n  3  intercept = 

slope = 1/n

0.1 0.1 1 10 PR   R 2L 22 © Faith A. Morrison, Michigan Tech U.

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Shear Rheometry– Capillary Flow If is known, is easy to calculate.

For an unknown, non-Newtonian fluid, is not known, and we need to take special steps to determine the wall shear rate

The wall shear rate is generally greater than for a Newtonian fluid.

vz(r)   ,R,nonNewtonian Newtonian velocity profile

non-Newtonian velocity profile

,R,Newtonian r

23 © Faith A. Morrison, Michigan Tech U.

Shear Rheometry– Capillary Flow If is not known, we must take extra steps to determine .

For a General non-Newtonian fluid

Q  ?

Something wall shear-rate-ish ?

Something wall shear-stress-ish

24 © Faith A. Morrison, Michigan Tech U.

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Shear Rheometry– Capillary Flow If is not known, we must take extra steps to determine .

Weissenberg-Rabinowitsch correction

4 1 ln 3 10 4 ln slope is a

44Q function of R a  R3

dlnln slope  a dlnlog R

0.1 0.1 1 10 PR 25  R  2L © Faith A. Morrison, Michigan Tech U.

Shear Rheometry– Capillary Flow Other corrections for capillary flow

Capillary flow

Assumptions: •Steady…………………………•No intermittent flow allowed • symmetry……………………•No spiraling flow allowed •Unidirectional …………………•Check end effects •Incompressible………………. •Avoid high absolute pressures •Constant pressure gradient…•Check end effects •No slip…………………………•Check wall slip

Methods have been devised to account for •Slip •End effects

26 © Faith A. Morrison, Michigan Tech U.

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Shear Rheometry– Capillary Flow Other corrections for capillary flow Slip at the wall - Mooney analysis

Slip at the wall reduces the shear rate near the wall.

v1(x2 ) v1(x2 )

no slip v slip x x2 1,slip 2 dv dv  1  1 , smaller x1 x1   , dx2 dx2

dv1 dv1 dx2 dx2

27 © Faith A. Morrison, Michigan Tech U.

Shear Rheometry– Capillary Flow Other corrections for capillary flow Slip at the wall - Mooney analysis Q Slip at the wall reduces the shear vz,av  2 rate near the wall. R vz,true  vz,measured  vz,slip 44vz,av 44Q ,   R R3 a

4 4, , , 4 1 , 4 , , At constant wall shear stress, take data in capillaries of various R 4v 4Q z,av  measured slope intercept R R3

The Mooney correction is a correction to the apparent shear rate 28 © Faith A. Morrison, Michigan Tech U.

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Shear Rheometry– Capillary Flow Other corrections for capillary flow Slip at the wall - Mooney analysis 800

pR, MPa 2L Note: each line is 600 0.35 at constant wall 4Q 0.3 shear stress 3 0.25  R 4vz,slip (data may need 0.2 to be interpolated -1 0.14 to meet this (s ) 400 0.1 requirement). 0.05 0.01

200 Turns out there is slip sometimes!

0 01234 1/R (1/mm)

Figure 10.10, p. 396 Ramamurthy, LLDPE 29 © Faith A. Morrison, Michigan Tech U.

Shear Rheometry– Capillary Flow Other corrections for capillary flow Entrance and exit effects - Bagley correction

The pressure gradient is not entrance region accurately represented by the raw pressure drop: r Preservoir  Patm z L 2R well-developed flow P(z)

p exit region  L raw

p  L corrected

z=0 z=L z

30 © Faith A. Morrison, Michigan Tech U.

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Shear Rheometry– Capillary Flow Other corrections for capillary flow Entrance and exit effects - Bagley correction

entrance region PR L  R   P  2 R r 2L R z

2R well-developed flow Constant at constant Q Run for different

capillaries exit region

This is the P result when the 2 R end effects are negligible.

L The Bagley correction is a correction R to the wall shear stress 31 © Faith A. Morrison, Michigan Tech U.

Shear Rheometry– Capillary Flow Other corrections for capillary flow Bagley Plot 1200

Δ 1  (s ) effects 1000 a 250 120 800 90 60 Turns out that end 40 effects are visible 600 sometimes!

Pressure drop drop (psi) Pressure 400

200 Δ 0 -10 0 10 20 30 40 L/R e(250, s -1 )

Figure 10.8, p. 394 Bagley, PE 32 © Faith A. Morrison, Michigan Tech U.

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Shear Rheometry– Capillary Flow Other corrections for capillary flow

1200 y = 32.705x + 163.53 1 a (s ) R2 = 0.9987 1000 250 y = 22.98x + 107.72 120 2 90 R = 0.9997 800 60 y = 20.172x + 85.311 40 R2 = 0.9998 600 y = 16.371x + 66.018 R2 = 0.9998 y = 13.502x + 36.81 400 2

Steady State Pressure, Pressure, State Steady R = 1

200 The intercepts are equal to the entrance/exit pressure 0 losses; these must be 0 10203040subtracted from the data to get true pressure versus L/R L/R.

Figure 10.8, p. 394 Bagley, PE 33 © Faith A. Morrison, Michigan Tech U.

Shear Rheometry– Capillary Flow Other corrections for capillary flow 1200 y = 32.705x + 163.53 2 1 R = 0.9987  (s ) 1000 a y = 22.98x + 107.72 250 2 120 R = 0.9997 800 90 y = 20.172x + 85.311 60 R2 = 0.9998 40 600 y = 16.371x + 66.018 R2 = 0.9998 y = 13.502x + 36.81 400 2

Steady State Pressure, Pressure, State Steady R = 1

200

Figure 10.8, p. 394 Bagley, PE 0 The slopes are equal to 0 10203040twice the shear stress for the various apparent shear rates L/R pR  L   rz   p  2 rz   34 2L  R  © Faith A. Morrison, Michigan Tech U.

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Shear Rheometry– Capillary Flow Other corrections for capillary flow The data so far:   a Pent  R  R gammdotA deltPent slope sh stress sh stress (1/s) psi psi psi Pa 250 163.53 32.705 16.3525 1.1275E+05 120 107.72 22.98 11.49 7.9220E+04 90 85.311 20.172 10.086 6.9540E+04 60 66.018 16.371 8.1855 5.6437E+04 40 36.81 13.502 6.751 4.6546E+04

Now, turn apparent shear rate into wall shear rate (correct for non- parabolic velocity profile).

Figure 10.8, p. 394 Bagley, PE 35 © Faith A. Morrison, Michigan Tech U.

Shear Rheometry– Capillary Flow Weissenberg-Rabinowitsch correction

4 1 ln 3 4 ln 10 slope is a 4Q function of    4 R a 3  R Sometimes the WR correction varies from point-to-point; 1 sometimes it is a constant that applies to d ln all data points (PL slope  lna region). d log R ln

0.1 0.1 1 10 PR   R 2L 36 © Faith A. Morrison, Michigan Tech U.

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Shear Rheometry– Capillary Flow Weissenberg-Rabinowitsch correction 7 ln 2.0677 ln y = 2.0677x - 18.537 6 R2 = 0.9998 This slope is usually >1

ln(apparent shear rate, 1/s) shear ln(apparent 4

3 10 10.4 10.8 11.2 11.6 12 ln(shear stress, Pa) Figure 10.8, p. 394 Bagley, PE 37 © Faith A. Morrison, Michigan Tech U.

Shear Rheometry– Capillary Flow

The data corrected for entrance/exit and non-parabolic velocity profile:

 P P a ent ent  R R gammdotA deltPent deltPent sh stress ln(sh st) ln(gda) WR gam-dotR viscosity (1/s) psi Pa Pa correction 1/s Pa s 250 163.53 1.1275E+06 1.1275E+05 11.63289389 5.521460918 2.0677 316.73125 3.5597E+02 120 107.72 7.4270E+05 7.9220E+04 11.2799902 4.787491743 2.0677 152.031 5.2108E+02 90 85.311 5.8820E+05 6.9540E+04 11.14966143 4.49980967 2.0677 114.02325 6.0988E+02 60 66.018 4.5518E+05 5.6437E+04 10.9408774 4.094344562 2.0677 76.0155 7.4244E+02 40 36.81 2.5380E+05 4.6546E+04 10.74820375 3.688879454 2.0677 50.677 9.1849E+02

Now, plot viscosity versus wall- shear-rate

Figure 10.8, p. 394 Bagley, PE 38 © Faith A. Morrison, Michigan Tech U.

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Shear Rheometry– Capillary Flow

Viscosity of polyethylene from Bagley’s data 1.0E+04

1.0E+03 Viscosity, Pa s

1.0E+02 10 100 1000   R,Shear rate, 1/s 39 Figure 10.8, p. 394 Bagley, PE © Faith A. Morrison, Michigan Tech U.

Shear Rheometry– Capillary Flow Viscosity from Capillary Experiments, Summary:

1. Take data of pressure-drop versus raw data: P(Q) flow rate for capillaries of various lengths; perform Bagley correction final data: on p (entrance pressure losses) 2. If possible, also take data for capillaries of different radii; perform Mooney correction on Q (slip) 1.0E+04    R 3. Perform the Weissenberg-  R Rabinowitsch correction (obtain correct wall shear rate) 1.0E+03 4. Plot true viscosity versus true wall Viscosity, Pa s Pa Viscosity, shear rate 5. Calculate power-law m, n from fit to final data (if appropriate) 1.0E+02 10 100 1000   R Shear rate, 1/s ,

40 © Faith A. Morrison, Michigan Tech U.

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Shear Rheometry– Capillary Flow What about the shear normal stresses, , from capillary data?

Extrudate swell -relation to N1 is model dependent (see discussion in Macosko, p254)

6  D   Assuming unconstrained N 2  8 2  e 1 1 R    recovery after steady shear,  2R   K-BKZ model with one relaxation time De  Extrudate diameter

Not a great method; can perhaps be used to index materials

 ? (cannot obtain from capillary flow, but…)

Macosko, Rheology: Principles, Measurements, 41 and Applications, VCH 1994. © Faith A. Morrison, Michigan Tech U.

Shear Rheometry

We can obtain   from slit-flow data: Hole Pressure-Error

Pressure transducers mounted in an access channel (hole) do not measure the same pressure as those that are “flush-mounted”:

Hou, Tong, deVargas, Rheol. Acta 1977, 16, 544

ph  p flush  phole

 d lnp  Slot transverse to flow:  h  N1  2ph    d ln w   d lnp  Slot parallel to flow:  h  N2  ph    d ln w   d lnp  Circular hole:  h  N1  N2  3ph    d ln w 

(We can of course obtain  also from slit- flow data; the equations are analogous to Lodge, in Rheological Measurement, Collyer, Clegg, eds. the capillary flow equations) Elsevier, 1988 Macosko, Rheology: Principles, Measurements, and 42 Applications, VCH 1994. © Faith A. Morrison, Michigan Tech U.

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Shear Rheometry– Capillary Flow

Limits on Measurements: Flow instabilities in rheology capillary flow 10,000

LPE-1 4Q -1 LPE-2 3 (s )  R LPE-3 LPE-4 1,000 Flow driven by Spurt flow constant pressure 100

10 1.E+05 1.E+06 1.E+07 PR , dynes / cm2 2L Figure 6.9, p. 176 Pomar et al. LLDPE Figure 6.10, p. 177 Blyler and Hart; PE 43 © Faith A. Morrison, Michigan Tech U.

Limits on Measurements: Flow instabilities in rheology

4Q 8 v  R3 D Flow driven by constant piston speed ( constant Q)

Slip-Stick Flow

44 D. Kalika and M. Denn, J. Rheol. 31, 815 (1987) © Faith A. Morrison, Michigan Tech U.

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Shear Rheometry– Torsional Flow

 Torsional Parallel Plates  Viscometric flow

H  0    r v  v (r, z)  0   rz

(-plane section)

45 © Faith A. Morrison, Michigan Tech U.

Shear Rheometry– Torsional Flow

 0  To calculate shear rate:   v   A(r)z  B(r)  0  v  A(r)z  B(r)  rz rz v  (due to boundary conditions)  H v v  0 0    0  0   r r  v v v    r  r 0 z   v 0   0  0   z rz H 0 0 r  ?

? (-plane section)

46 © Faith A. Morrison, Michigan Tech U.

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Shear Rheometry– Torsional Flow  0     rz Ω H Result: v   H  r   0 (-plane  rz section)

To calculate shear stress, look at EOM:  0 0   rr  P  p  gz    0  z   0     v   z zz rz   v v  P     t  (viscometric flow)

steady neglect  1 r   state inertia  rr    Assume: r r r •Form of velocity P   0  r    •no -dependence      0   0    z  •symmetric stress tensor  z  •neglect inertia 0  P   rz  z rz   •no slip     zz  •isothermal  z rz 47 © Faith A. Morrison, Michigan Tech U.

Shear Rheometry– Torsional Flow  Result: z  0 z

 z  f (r)

The experimentally measurable variable is the torque to turn the plate: T  R nˆ  dS surface S 2 R T  reˆ eˆ  rdrd r z zH 00 R T  2  r 2dr z z zH 0

Following Rabinowitsch, replace stress with viscosity, r with shear rate, and differentiate. 48 © Faith A. Morrison, Michigan Tech U.

24 Rheometry CM4650 4/27/2018

Shear Rheometry– Torsional Flow

Torsional Parallel-Plate Flow - Viscosity

 cross-sectional Measureables: view:  Torque  to turn plate Rate of angular rotation 

z Note: shear rate H r experienced by fluid R elements depends on their r Ω position. (consider effect on complex fluids)

ln /2 By carrying out a Rabinowitsch-like 3 calculation, we can obtain the stress at 2 ln the rim (r=R).

| Correction required 49 © Faith A. Morrison, Michigan Tech U.

Shear Rheometry– Torsional Flow

Torsional Parallel-Plate Flow - correction

10 slopeslope is a function of R T function of 2 R3

d log T / 2 R3 slope  / sloped log  R

0.1 0.1 1 10 RΩ   R H /2 ln /2 3 50 ln © Faith A. Morrison, Michigan Tech U.

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Shear Rheometry– Torsional Flow

Torsional Parallel-Plate Flow – Viscosity – Approximate method

ln /2 =1 for Newtonian 3 2 ln ≡

ln /2 For many materials: 1.4 ln ( ) a ( a )  2%

  a (r  0.76R) Giesekus and Langer Rheol. Acta, 16, 1 1977

. . No correction required . (but making a material assumption)

51 Macosko, Rheology: Principles, Measurements, and Applications, VCH 1994, p220 © Faith A. Morrison, Michigan Tech U.

Shear Rheometry– Torsional Flow

Torsional Parallel-Plate Flow – Normal Stresses

Similar tactics, logic (see Macosko, p221)

ln 2 ln

(Not a direct material function)

52 Macosko, Rheology: Principles, Measurements, and Applications, VCH 1994, p220 © Faith A. Morrison, Michigan Tech U.

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Shear Rheometry– Torsional Flow

 Torsional Cone and Plate  (spherical coordinates)

0 r  0    v   0  v (r, ) (plane   r section)

53 © Faith A. Morrison, Michigan Tech U.

Shear Rheometry– Torsional Flow

 0  To calculate shear rate, :   v  0 v  A(r )  B    A(r )  B r   r   (due to boundary conditions) v     0  2 

v  0 0 r     00r r  v    0 0 sin    r  sinsin  v v    00sin    r r r r  sin 0 sin  r sin  0  ? sin  ?  r 0

(plane section)

54 © Faith A. Morrison, Michigan Tech U.

27 Rheometry CM4650 4/27/2018

Shear Rheometry– Torsional Flow   0      v   0  constant  Result:   r 0 0  r    2   0 r (plane section)

Note: The shear rate is a constant.  0 0   rr  The extra stresses  are only a function of the    0    ij   shear rate, thus the ij are constant as well. 0      r (viscometric flow)

constant Assume: Result:  ij  •Form of velocity •no -dependence •no slip Torsional cone and plate is a •isothermal homogeneous shear flow.

55 © Faith A. Morrison, Michigan Tech U.

Shear Rheometry– Torsional Flow

Result:

The experimentally measurable variable is the torque to turn the cone: T  R nˆ  dS surface S 2 R

T  reˆr  eˆ   rdrd    2 00 3 2R  T  Tz  3 For an arbitrary fluid, we are able to relate the torque and the shear stress.

56 © Faith A. Morrison, Michigan Tech U.

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Shear Rheometry– Torsional Flow

Torsional Cone-and-Plate Flow - Viscosity

  (-plane section) Measureables: Torque to turn cone  Rate of angular rotation Ω polymer melt r  The introduction of the cone Ω means that shear rate is independent of r. Θ R

Since shear rate is constant 3 everywhere, so is extra stress, and we can calculate stress from torque. 2

3Θ No corrections needed in cone-and-plate 2 Ω (and no material assumptions) 57 © Faith A. Morrison, Michigan Tech U.

Shear Rheometry– Torsional Flow   To calculate normal stresses, look at EOM:  P  p  gz r 0

(plane  v  section)   v v  P     t   rr 0 0  Although the stress is   steady neglect constant, there are    0    state inertia some non-zero terms in   the divergence of the  0    Also, the stress in the r r pressure is coordinate system (viscometric flow) not constant

 1 r 2     rr     Assume:  r 2 r r  •Form of velocity 0   P       r   cot •no -dependence 1 P  1  sin   0      •symmetric stress tensor r   r sin  r      •neglect inertia 0  0    r r   •no slip 1  sin  cot    •isothermal  r sin  r r 58 © Faith A. Morrison, Michigan Tech U.

29 Rheometry CM4650 4/27/2018

Shear Rheometry– Torsional Flow   On the bottom plate, sin=1, cos=0:   2    r 0  rr     0   P   r r  (plane    r    section) 1 P 0   r     0  0  0     r  r    0   r P 2   0    rr    r r r (P  ) 2   0     rr    (valid to insert, since extra stress is constant) r r r Ψ (by definition) Ψ

Π Ψ 2Ψ ln

59 © Faith A. Morrison, Michigan Tech U.

Shear Rheometry– Torsional Flow Π Result: Ψ 2Ψ ln

The experimentally measurable variable is the fluid

thrust on the plate minus the thrust of Patm: 2 N  Fz R Patm

F  nˆ  dS surface S 2 R

F   eˆ   rdrd    2 00 R

F  Fz  2   rdr    2 0

60 © Faith A. Morrison, Michigan Tech U.

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Shear Rheometry– Torsional Flow

Integrate: Boundary condition:

Π Ψ 2Ψ ln Π Π Ψ 2Ψ ln Ψ

Directly from Π Ψ 2Ψ ln Ψ definition of 

2 N  Fz R Patm R 2 N  2   rdr  R Patm    2 0 . . .

61 © Faith A. Morrison, Michigan Tech U.

Shear Rheometry– Torsional Flow

Torsional Cone-and-Plate Flow – 1st Normal Stress

  Measureables: Normal thrust F 

 r 0 R   2 (plane N  2   rdr R patm       section)  0 2 

The total upward thrust of the cone can be 2Θ related directly to the Ψ (see also DPL pp522) first normal stress Ω coefficient.

62 © Faith A. Morrison, Michigan Tech U.

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Shear Rheometry– Torsional Flow

Torsional Cone-and-Plate Flow – 2nd Normal Stress

If we obtain  as a function of r / R, we Π Ψ 2Ψ ln Ψ can also obtain 2.

•MEMS used to manufacture sensors at different radial positions

RheoSense Incorporated (www.rheosense.com)

•S. G. Baek and J. J. Magda, J. Rheology, 47(5), 1249-1260 (2003) 63 •J. Magda et al. Proc. XIV International Congress on Rheology, Seoul, 2004. © Faith A. Morrison, Michigan Tech U.

Shear Rheometry– Torsional Flow RheoSense Incorporated Comparison with other instruments

64 S. G. Baek and J. J. Magda, J. Rheology, 47(5), 1249-1260 (2003) © Faith A. Morrison, Michigan Tech U.

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Shear Rheometry– Torsional Flow Treated in detail in Macosko, pp 188-205 Couette Flow (1890)

•Tangential annular flow •Cup and Bob geometry

 (-plane fluid section)

cup

bob

air pocket

Assume: •Form of velocity •no -dependence •symmetric stress tensor •Neglect gravity •Neglect end effects •no slip •isothermal

Macosko, Rheology: Principles, 65 Measurements, and Applications, VCH 1994. © Faith A. Morrison, Michigan Tech U.

Shear Rheometry– Torsional Flow Couette Flow

Assume:  •Form of velocity •no -dependence T 1 •symmetric stress tensor   •Neglect gravity 2R2 L 3 •Neglect end effects •no slip R •isothermal   inner Router

As with many measurement systems, the assumptions made in the analysis do not BUT •Generates a lot of signal always hold: •End effects are not negligible •Can go to high shear rates •Wall slip occurs with many systems •Is widely available •Inertia is not always negligible •Is well understood •Secondary flows occur (cup turning is more stable than bob turning to inertial instabilities; there are elastic instabilities; there are viscous heating instabilities) •Alignment is important •Viscous heating occurs

•Methods for measuring 1 are error prone •Cannot measure 2

Macosko, Rheology: Principles, 66 Measurements, and Applications, VCH 1994. © Faith A. Morrison, Michigan Tech U.

33 Rheometry CM4650 4/27/2018

Shear Rheometry– Torsional Flow For the PP and CP geometries, we can also calculate G’, G”:

G() 2HT sin  0 Parallel plate  ()   4  R 0 Amplitude of oscillation G() 2HT cos  0  ()   4  R 0

3 T sin Cone and plate  0 0  ()  3 2R 0 3 T cos  0 0  ()  3 2R 0 Amplitude of oscillation

•A typical diameter is between 8 and 25mm; 30-40mm are also used •To increase accuracy, larger plates (R larger) are used for less viscous materials to generate more torque. •Amplitude may also be increased to increase torque •A complete analysis of SAOS in the Couette geometry is given in Sections 8.4.2-3 67 © Faith A. Morrison, Michigan Tech U.

Shear Rheometry– Torsional Flow Limits on Measurements: Flow instabilities in rheology

Cone and plate/Parallel plate flow

High (steady shear) or (SAOS) cause these instabilities to be observed.

68 Figures 6.7 and 6.8, p. 175 Hutton; PDMS © Faith A. Morrison, Michigan Tech U.

34 Rheometry CM4650 4/27/2018

Shear Rheometry– Torsional Flow

Taylor-Couette flow

1923 GI Taylor; inertial instability 1990 Ron Larson, Eric Shaqfeh, Susan Muller; elastic instability

•GI Taylor "Stability of a viscous liquid contained between two rotating cylinders," Phil. Trans. R. Soc. Lond. A 223, 289 (1923) •Larson, Shaqfeh, Muller, “A purely elastic instability in Taylor- 69 Couette flow,”J. Fluid Mech, 218, 573 (1990) © Faith A. Morrison, Michigan Tech U.

Torsional Shear Flow: Parallel-plate and Cone-and-plate

Why do we need more Instabilities in than one method of torsional flows measuring viscosity? log

Torsional flows Capillary flow •At low deformation rates, torques & pressures become low •At deformation high rates, torques & pressures become high; flow instabilities set in Low signal in capillary flow The choice is determined by experimental issues (signal, noise, instrument limitations. log

70 © Faith A. Morrison, Michigan Tech U.

35 Rheometry CM4650 4/27/2018

Shear measurement Material Function Calculations © Faith A. Morrison, Michigan Tech U.

Morrison, UR, Table 10.2 See also Macosko, Part II

Shear measurements Pros and Cons

Morrison, UR, Table 10.3 See also Macosko, Part II 72 © Faith A. Morrison, Michigan Tech U.

36 Rheometry CM4650 4/27/2018

Stress/Strain Driven Drag Pressure-driven Shear Optical Multipurpose rheometers

(diffusive wave spectroscopy; G’G”)

(plus attachments on multipurpose rheometer)

(capillary, slit flow; melts)

(Slit flow; microfluidics)

Interfacial Rheology

(drag flow on interface)

73 © Faith A. Morrison, Michigan Tech U.

Elongational Flow Measurements

fluid

74 © Faith A. Morrison, Michigan Tech U.

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Elongational Rheometry

Experimental Elongational Geometries

x1 air-bed to support sample fluid x3

to to+tto+2t

x3 R(t) h(t) x1 h(to)

thin, lubricating layer on each plate R(to)

Elongational Rheometry Uniaxial Extension

tensile force f (t)     zz rr A(t) load measures cell force time-dependent f (t) cross-sectional area

For homogeneous flow: z fluid sample r

38 Rheometry CM4650 4/27/2018

Elongational Rheometry

Experimental Difficulties in Elongational Flow

experimental ideal elongational challenges deformation end effects

inhomogeneities

initial

initial final

effect of gravity, drafts, surface tension

Elongational Rheometry

Filament Stretching Rheometer (FiSER)

Tirtaatmadja and Sridhar, J. Rheol., 37, 1081-1102 (1993)

•Optically monitor the midpoint size •Very susceptible to environment •End Effects

McKinley, et al., 15th Annual Meeting of the International Polymer Processing Society, June 1999.

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Elongational Rheometry

Filament Stretching Rheometer (Design based on Tirtaatmadja and Sridhar)

“The test sample (a) undergoing investigation is placed between two parallel, circular discs (b) and

(c) with diameter 2R0=9 mm. The upper disc is attached to a movable sled (d), while the lower disc is in contact with a weight cell (e). The upper sled is driven by a motor (f), which also drives a mid-sled placed between the upper sled and the weight cell; two timing belts (g) are used for transferring momentum from the motor to the sleds. The two toothed wheels (h), driving the timing belts have a 1:2 diameter ratio, ensuring that the mid-sled always drives at half the speed of the upper sled. This means that if the mid-sled is placed in the middle between the upper and the lower disc at the beginning of an experiment, it will always stay midway between the discs. On the mid-sled, a laser (i) is placed for measuring the diameter of the mid-filament at all times.

Bach, Rasmussen, Longin, Hassager, JNNFM 108, 163 (2002)

•Steady and startup flow •Recovery •Good for melts

40 Rheometry CM4650 4/27/2018

Achieving commanded strain requires great care.

Use of the video camera (although tedious) is recommended in order to get correct strain rate.

Elongational Rheometry

Sentmanat Extension Rheometer (2005)

•Originally developed for rubbers, good for melts •Measures elongational viscosity, startup, other material functions •Two counter-rotating drums •Easy to load; reproducible

www.xpansioninstruments.com

41 Rheometry CM4650 4/27/2018

Elongational Rheometry

Sentmanat et al., J. Rheol., 49(3) 585 (2005)

Comparison on different host instruments

Comparison with other instruments (literature)

Elongational Rheometry CaBER Extensional Rheometer •Polymer solutions •Works on the principle of capillary filament break up •Cambridge Polymer Group and HAAKE For more on theory see: campoly.com/notes/007.pdf

Brochure: www.thermo.com/com/cda/product/detail/1,,17848,00.html

Operation •Impose a rapid step elongation •form a fluid filament, which continues to deform •flow driven by surface tension •also affected by viscosity, elasticity, and mass transfer •measure midpoint diameter as a function of time •Use force balance on filament to back out an apparent elongational viscosity

42 Rheometry CM4650 4/27/2018

Elongational Rheometry

Capillary breakup experiments

Comments •Must know surface tension •Transient agreement is poor •Steady state agreement is acceptable •Be aware of effect modeling assumptions on reported results Filament stretching apparatus

Elongational Rheometry

Elongational Viscosity via Contraction Flow: Cogswell/Binding Analysis

Fluid elements along the centerline undergo funnel-flow considerable region elongational flow corner vortex z R(z) y By making strong assumptions about the flow we can relate the pressure drop across the contraction to an Ro elongational viscosity

43 Rheometry CM4650 4/27/2018

Elongational Rheometry

This flow is produced in some capillary rheometers:

P  P P P reservoir atm  entrance  true entrance region L L L r We can use this z “discarded” measurement to rank 2R well-developed flow P(z) elongational properties

p  exit region L raw

p  L corrected

z=0 z=L z

Elongational Rheometry Elongational properties may be inferred from shear capillary flow data. Bagley Plot 1200

Δ 1  (s ) effects 1000 a 250 120 800 90 60 40 600

Pressure drop drop (psi) Pressure 400

200 Δ 0 -10 0 10 20 30 40 L/R e(250, s -1 )

44 Rheometry CM4650 4/27/2018

Elongational Rheometry Elongational properties may be inferred from shear capillary flow data. 1200 y = 32.705x + 163.53 1 a (s ) R2 = 0.9987 1000 250 y = 22.98x + 107.72 120 2 90 R = 0.9997 800 60 y = 20.172x + 85.311 40 R2 = 0.9998 600 y = 16.371x + 66.018 R2 = 0.9998 y = 13.502x + 36.81 400 2

Steady State Pressure, Pressure, State Steady R = 1

200 The intercepts are equal to the entrance/exit pressure 0 losses; these are obtained 0 10203040as a function of apparent shear rate L/R

Elongational Rheometry Elongational properties may be inferred from shear capillary flow data. Assumptions for the Cogswell Analysis • incompressible fluid • funnel-shaped flow; no-slip on funnel surface • unidirectional flow in the funnel region • well developed flow upstream and downstream • -symmetry z y • pressure drops due to shear and elongation R(z) may be calculated separately and summed to give the total entrance pressure-loss • neglect Weissenberg-Rabinowitsch correction

• shear stress is related to shear-rate through a Ro power-law • elongational viscosity is constant • shape of the funnel is determined by the minimum generated pressure drop • no effect of elasticity (shear normal stresses neglected) • neglect inertia ̅ constant

F. N. Cogswell, Polym. Eng. Sci. (1972) 12, 64-73. F. N. Cogswell, Trans. Soc. Rheol. (1972) 9016, 383-403. © Faith A. Morrison, Michigan Tech U.

45 Rheometry CM4650 4/27/2018

Elongational Rheometry Elongational properties may be inferred from shear capillary flow data. Cogswell Analysis

elongation rate 2 4

elongation normal 3 stress     p (n 1) 11 22 8 ent

elongation 9 1 Δ viscosity ̅ 32

Elongational Rheometry Elongational properties may be inferred from shear capillary flow data.

Cogswell Analysis – using Excel From shear (power law): (Bagley’s data)

̅ 2 4 3 pent pent  R RAW DATA RAW DATA Cogswell CogswellTrouton gammdotA deltPent(psi) deltPent(Pa) sh stress(Pa) N1(Pa) e_rate elongvisc 3*shearVisc 250 163.53 1.13E+06 1.13E+05 -6.27E+05 2.25E+01 2.79E+04 1.55E+03 120 107.72 7.43E+05 7.92E+04 -4.13E+05 1.15E+01 3.59E+04 2.27E+03 90 85.311 5.88E+05 6.95E+04 -3.27E+05 9.56E+00 3.42E+04 2.65E+03 60 66.018 4.55E+05 5.64E+04 -2.53E+05 6.69E+00 3.79E+04 3.23E+03 40 36.81 2.54E+05 4.65E+04 -1.41E+05 6.59E+00 2.14E+04 4.00E+03

3 Results in one data point for 11  22   pent (n 1) elongational viscosity for each 8 entrance pressure loss (i.e. each apparent shear rate) 92 © Faith A. Morrison, Michigan Tech U.

46 Rheometry CM4650 4/27/2018

Elongational Rheometry Elongational properties may be inferred from shear capillary flow data. Assumptions for the Binding Analysis • incompressible fluid • funnel-shaped flow; no-slip on funnel surface • unidirectional flow in the funnel region •well developed flow upstream and downstream •  -symmetry z • shear viscosity is related to shear-rate through a R(z) y power-law • elongational viscosity is given by a power law • shape of the funnel is determined by the minimum R work to drive flow o • no effect of elasticity (shear normal stresses neglected) • the quantities  dR dz 2 and d 2 R dz 2 , related to the shape of the funnel, are neglected; implies that the radial velocity is neglected when calculating the rate of deformation • neglect energy required to maintain the corner ̅ circulation • neglect inertia

Elongational Rheometry Elongational properties may be inferred from shear capillary flow data. Binding Analysis

l, elongational prefactor

1 (1t) 2m(1 t)2 lt(3n 1)ntI  p  nt  t(n1) (1t) 13t(n1) (1t) ent 2 2   Ro  3t (1 n)  m 

t1 1  3n 1 I  2    11 n  d nt  n 0   (3n 1) Q   Ro 3 nRo

elongation viscosity ̅

47 Rheometry CM4650 4/27/2018

Elongational Rheometry Elongational properties may be inferred from shear capillary flow data. Binding Analysis

Evaluation Procedure 1. Shear power-law parameter n must be known; must have data for Δversus 2. Guess ,

3. Evaluate by numerical integration over 4. Using Solver, find the best values of t and l that are

consistent with the Δ versus data

Note: There is a non-iterative solution method described in the text; The method using Solver is preferable, since it uses all the data in finding optimal values of and . Results in values of , for a model (power-law)

Elongational Rheometry Elongational properties may be inferred from shear capillary flow data. Binding Analysis – using Excel Solver

Evaluate integral t1 1  3n 1 numerically I  2    11 n  d nt  n 0  

phi f(phi) areas 00 1 0.005 0.023746502 5.93663E-05 area  (b  b )h 0.01 0.047492829 0.000178098 2 1 2 0.015 0.071238512 0.000296828 0.02 0.094982739 0.000415553 0.025 0.118724352 0.000534268 0.03 0.142461832 0.000652965 0.035 0.166193303 0.000771638 Summing: 0.04 0.189916517 0.000890275 Int= 1.36055 0.045 0.213628861 0.001008863 0.05 0.237327345 0.001127391 0.055 0.261008606 0.00124584 0.06 0.2846689 0.001364194 0.065 0.308304107 0.001482433 96 . . . © Faith A. Morrison, Michigan Tech U.

48 Rheometry CM4650 4/27/2018

Elongational Rheometry Elongational properties may be inferred from shear capillary flow data.

Binding Analysis – using Excel Solver

Optimize t, l using Solver

By varying these cells: predicted  actual 2 2 t_guess= 1.2477157 actual l_guess= 11991.60895 ******* SOLVER SOLUTION ******** predicted exptal DeltaPent DeltaPent difference 1.26E+06 1.13E+06 1.35E-02 6.88E+05 7.43E+05 5.51E-03 5.43E+05 5.88E+05 6.02E-03 3.89E+05 4.55E+05 2.14E-02 2.78E+05 2.54E+05 9.28E-03 target cell 5.57E-02 Sum of the differences: Minimize this cell

Elongational Rheometry Elongational properties may be inferred Example from shear capillary flow data. calculation 1.E+05 ̅ Cogswell ̅ Binding from Bagley’s Data This Binding curve was 1.E+04 calculated using the Solver solution procedure in the text

3 1.E+03

-0.5165 1.E+02 shear viscosity y = 6982.5x Cogswell elong visc R2 = 0.9998 Trouton prediction Binding elong visc Binding Solver

shear or elongatinal viscosity (Pa s) 1.E+01 Power (shear viscosity)

Bagley's data from Figure 10.8 Understanding Rheology Morrison; assumed contraction was 12.5:1 1.E+00 1.E+00 1.E+01 1.E+02 1.E+03

49 Rheometry CM4650 4/27/2018

Elongational (Pseudo) Rheometry

Rheotens (Goettfert) from their brochure: "Rheotens test is a rather complicated function of the characteristics of the polymer, dimensions of the capillary, length of the spin line and of the extrusion history"

www.goettfert.com/downloads/Rheotens_eng.pdf

•Does not measure material functions without constitutive model •small changes in material properties are reflected in curves •easy to use •excellent reproducibility •models fiber spinning, film casting •widespread application

Elongational (Pseudo) Rheometry Raw data

An elongational viscosity may be extracted from a “grand master curve” under some conditions

"The rheology of the rheotens test,“ M.H. Wagner, A. Bernnat, and V. Schulze, J. Rheol. 42, 917 (1998)

Grand master curve vs. draw ratio V  v vdie exit b  shift factor

Draw resonance

50 Rheometry CM4650 4/27/2018

Elongational Rheometry

Elongational measurements Pros and Cons

Elongational Rheometry

Extensional Measurement of (dual drum windup) elongational viscosity is still a labor of love. (filament stretching)

(capillary breakup)

(drum windup)

51 Rheometry CM4650 4/27/2018

Newer emphasis:

C. Clasen and G. H. McKinley, "Gap-dependent microrheometry of complex liquids," JNNFM, 124(1-3), 1-10 (2004) 103 © Faith A. Morrison, Michigan Tech U.

http://www.ksvnima.com/file/ksv-nima-isrbrochure.pdf

KSV NIMA Interfacial Shear Rheometer •a probe floats on an interface; •is driven magnetically; •material functions are inferred.

52 Rheometry CM4650 4/27/2018

Strong multiple scattering of light + model = rheological material functions

•D.J. Pine, D.A. Weitz, P.M. Chaikin, and E. Herbolzheimer,“Diffusing- Wave Spectroscopy,” Phys. Rev. Lett. 60, 1134-1137 (1988). •Bicout, D., and Maynard, R., “Diffusing wave spectroscopy in inhomogeneous flows,” J. Phys. I 4, 387–411. 1993 106 © Faith A. Morrison, Michigan Tech U.

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Flow Birefringence - a non-invasive way to measure stresses

no net force, isotropic chain, isotropic polarization For many polymers, n stress and refractive- index tensors are coaxial (same principal axes):

n n  C  BI

Stress-Optical Law force applied, anisotropic chain, anisotropic polarization = birefringent

Large-Amplitude Oscillatory Shear A window into nonlinear viscoelasticity

Gareth McKinley, Plenary, International Congress on Rheology, Lisbon, August, 2012 108 http://web.mit.edu/nnf/ICR2012/ICR_LAOS_McKinley_For%20Distribution.pdf © Faith A. Morrison, Michigan Tech U.

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Summary

SHEAR •Shear measurements are readily made •Choice of shear geometry is driven by fluid properties, shear rates •Care must be taken with automated instruments (nonlinear response, instrument inertia, resonance, motor dynamics, modeling assumptions)

•Microrheometry

ELONGATION

•Elongational properties are still not routine •Newer instruments (Sentmanat,CaBER) have improved the possibility of routine elongational flow measurements •Some measurements are best left to the researchers dedicated to them due to complexity (FiSER) •Industries that rely on elongational flow properties (fiber spinning, foods) have developed their own ranking tests

Done with Rheometry.

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Course Summary

1. Introduction CM4650 2. Math review Polymer Rheology 3. Newtonian Fluids 4. Standard Flows 5. Material Functions 6. Experimental Data 7. Generalized Newtonian Fluids 8. Memory: Linear Viscoelastic Models 9. Advanced Constitutive Models 10. Rheometry

Professor Faith A. Morrison Department of Chemical Engineering Michigan Technological University 111 © Faith A. Morrison, Michigan Tech U.