There Is Way More to This Than We Can Cover

Advanced CM4650 4/23/2018

Chapter 9: Advanced Constitutive Models

CM4650 Polymer Rheology

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

Chapter 9: Advanced Constitutive Models

CM4650 Polymer Rheology WARNING: There is way more to this than we can

Professor Faith A. Morrisoncover Department of Chemical Engineering Michigan Technological University 2 © Faith A. Morrison, Michigan Tech U.

1 Advanced CM4650 4/23/2018

Advanced Constitutive Modeling – Chapter 9

Fluids with Memory – Chapter 8

We have learned that the GLVE Summary: Generalized Linear-Viscoelastic Constitutive is a good model for: Equations

• Deformations at low rates PRO: •A first constitutive equation with memory • Deformation to low strains •Can match SAOS, step-strain data very well •Captures start-up/cessation effects (linear regime) •Simple to calculate with •Can be used to calculate the LVE spectrum

•Fails to predict shear normal stresses CON: The GLVE does not work for the •Fails to predict shear-thinning/thickening nonlinear regime, due to the problem •Only valid at small strains, small rates of lack of frame invariance. GLVE: •Not frame-invariant

Fluids with Memory – Chapter 8 Shear flow in a rotating frame of reference

y fluid x x, y, z Turntable  problem GLVE:

x, y, z 3 © Faith A. Morrison, Michigan Tech U.

Advanced Constitutive Modeling – Chapter 9

Fluids with Memory – Chapter 8 Strain-related issues? Shear flow in a rotating frame of reference

y fluid ⇒ perhaps the problem with the GLVE x

model is associated with how strain is x, y, z

mathematically described. 

GLVE:

x, y, z The question then becomes, how is strain mathematically described in the GLVE Model?

′

strain rate

4 © Faith A. Morrison, Michigan Tech U.

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Advanced Constitutive Modeling – Chapter 9

What is the strain measure that is used in the GLVE model?

′

strain rate

(use integration by parts; see hand calculations)

5 © Faith A. Morrison, Michigan Tech U.

Advanced Constitutive Modeling – Chapter 9 The infinitesimal Generalized Linear- strain tensor is the Viscoelastic Model: strain measure of the GLVE (strain version) ′ , ≡ ′ memory function

Infinitesimal Strain Tensor Turns out: It is the use of the infinitesimal strain , ′′ tensor as the strain measure that causes the frame-variance in the GLVE model.

6 © Faith A. Morrison, Michigan Tech U.

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Advanced Constitutive Modeling – Chapter 9

We have seen the infinitesimal strain tensor before: when we first defined strain (when we discussed material functions).

Infinitesimal strain tensor ≡

Displacement When formally ut(,)()()ref t rt rt ref developed, , is function related to the displacement function,  x (t)  1  , ′ . Particle r(t)   x2 (t) tracking vector  x (t)  3 123

7 © Faith A. Morrison, Michigan Tech U.

What is strain? Summary (from Ch5) Strain is our measure of deformation (change of shape)

For shear flow (steady or unsteady): Strain Strain is the , accumulates as integral of strain the flow rate progresses

The time The strain rate is derivative of the rate of strain is the strain instantaneous Deformation rate rate shape change

Applying this to each Infinitesimal component of and strain tensor ≡ generalizing to all flows:

8 © Faith A. Morrison, Michigan Tech U.

4 Advanced CM4650 4/23/2018

Advanced Constitutive Modeling – Chapter 9

Fluids with Memory – Chapter 8 Shear flow in a rotating frame of reference

In addition to the turntable y fluid example, another “flow” we x can use to test the GLVE x, y, z Turntable model is rigid body  problem rotation (no strain). GLVE:

x, y, z

Counter clockwise rigid body rotation (no strain).

No strain should  produce no stress. 9 © Faith A. Morrison, Michigan Tech U.

Advanced Constitutive Modeling – Chapter 9 No stress is generated when a fluid is rotated CCW through (from position at time to position at time ′, what does the GLVE predict? (Warning: later, we are going to consider CCW rotation from ′ to through an angle ; see Table 9.3) t t'



y y P(t') r' r cos P(t)  r r  r sin   x x

• calculate the infinitesimal strain tensor for rigid body rotation • use the strain-evident version of the GLVE

10 (note: we need , in the GLVE) © Faith A. Morrison, Michigan Tech U.

5 Advanced CM4650 4/23/2018

Advanced Constitutive Modeling – Chapter 9

What does the GLVE Predict for CCW Rigid-Body Rotation around the z-axis from to ?

′ ′ ′ ′

, ′

(see book for details)

11 © Faith A. Morrison, Michigan Tech U.

Advanced Constitutive Modeling – Chapter 9

What does the GLVE Predict for CCW Rigid-Body Rotation around the z-axis from to ? From geometry ̅ sin ̅ cos From trigonometry ′ ̅ sin ̅ sin cos sin cos cos sin ′ ̅ cos ̅ cos cos sin sin cos sin ′ From definitions of and cos sin ̅ ̅ cos sin 0 , ′ 12 © Faith A. Morrison, Michigan Tech U.

6 Advanced CM4650 4/23/2018

Advanced Constitutive Modeling – Chapter 9

GLVE Prediction for CCW Rigid-Body Rotation around the z-axis from to ′:

2 cos 1 0 0 02cos 1 0 ′ 000

13 © Faith A. Morrison, Michigan Tech U.

Advanced Constitutive Modeling – Chapter 9

GLVE Prediction for CCW Rigid-Body Rotation around the z-axis from to ′: WRONG

2 cos 1 0 0 02cos 1 0 ′ 000

Stress depends on angle of rotation! Why does GLVE make this erroneous prediction?

 (t,t)  u(t,t)  u(t,t) T u(t,t)  r(t)  r(t)

Because this vector, while accounting for deformation, also accounts for changes in orientation. 14 © Faith A. Morrison, Michigan Tech U.

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Advanced Constitutive Modeling – Chapter 9

 (t,t)  u(t,t)  u(t,t) T Accounts for changes in shape and orientation. u(t,t)  r(t)  r(t)

15 © Faith A. Morrison, Michigan Tech U.

Advanced Constitutive Modeling – Chapter 9

We desire a strain tensor that accurately captures large-strain deformation without being affected by translation and rotation.

time=t’ Consider: What change does the deformation time=t cause in the vector  that separates two P Q dr very nearby material elements? P Q

fixed coordinate system (xyz) 16 © Faith A. Morrison, Michigan Tech U.

8 Advanced CM4650 4/23/2018

Advanced Constitutive Modeling – Chapter 9 Define change-of-shape tensors that rely on How does dr map to dr’ along a particle path? relative location of two nearby particles

particle label (reference time t) Particle position at t’ ′ location at time t’ of the particle labeled r is a function of past position,

 x d r dr   r   y  f (r)    z r r xyz dx   df  dy  ?  dz   xyz

17 © Faith A. Morrison, Michigan Tech U.

Advanced Constitutive Modeling – Chapter 9 We can relate ′ to using the chain rule.

′ ′ ,, ′

? ? ?

(see text)

′ ′ ′ 18 © Faith A. Morrison, Michigan Tech U.

9 Advanced CM4650 4/23/2018

Advanced Constitutive Modeling – Chapter 9  x' y' z'  Combine answers from 3 directions:    x x x  x' y' z' dx dy dz  dx dy dz   xyz xyz  y y y   x' y' z'        z z z xyz dr  dr  F

 x' y' z'  Deformation-gradient   tensor  x x x  x' y' z' r' r ' F(t,t)      i eˆ eˆ   p i y y y r rp  x' y' z'    In Einstein notation:  z z z xyz ′ ′̂ , , ′ 19 © Faith A. Morrison, Michigan Tech U.

Advanced Constitutive Modeling – Chapter 9

We can calculate as follows:

1 Define: F  F  I Then use: dr  dr  F  ?

⋅ ⋅ ⋅⋅ ⋅ 20 © Faith A. Morrison, Michigan Tech U.

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Advanced Constitutive Modeling – Chapter 9

Deformation-gradient  x' y' z'  tensor    x x x  x' y' z' r' r ' dr  dr  F F(t,t)      i eˆ eˆ   p i y y y r rp  x' y' z'     z z z xyz

Inverse deformation-  x y z  gradient tensor   x' x' x' 1    1 x y z r r dr  dr  F F (t',t)      m eˆ eˆ   j m y' y' y' r' rj '  x y z        z' z' z' xyz These strain measures get rid of 21 the translation problem. © Faith A. Morrison, Michigan Tech U.

Advanced Constitutive Modeling – Chapter 9

We desire a strain tensor that accurately captures large-strain deformation without being affected by translation and rotation.

u These strain measures include translation,  deformation and orientation F These strain measures include F1 deformation and orientation

We can separate the deformation and orientation information in F 1 and F using a technique called polar decomposition.

(en.wikipedia.org/wiki/Polar_decomposition) 22 © Faith A. Morrison, Michigan Tech U.

11 Advanced CM4650 4/23/2018

Advanced Constitutive Modeling – Chapter 9

Polar Decomposition Theorem (en.wikipedia.org/wiki/Polar_decomposition)

Any tensor for which an inverse exists has two unique decompositions: ⋅

⋅ Pure rotation tensor

/ ⋅ Right stretch tensor Orthogonal tensor

/ , symmetric, ⋅ Left stretch tensor nonsingular tensors

⋅ ⋅ / ⋅

23 © Faith A. Morrison, Michigan Tech U.

Advanced Constitutive Modeling – Polar Decomposition

EXAMPLE: Calculate the right stretch tensor and rotation tensor for a given tensor. Calculate the angle through which rotates the vector u.

1 0 2 1     A  0 3 2 u  2     2 0 0 1  xyz  xyz

⋅ u w ⋅ ⋅ ⋅ A U Pure All the stretch; R v rotation some of the rotation 24 © Faith A. Morrison, Michigan Tech U.

12 Advanced CM4650 4/23/2018

Advanced Constitutive Modeling – Polar Decomposition

We have partially isolated the effect of rotation through polar decomposition.

pure rotation tensor left stretch tensor A  R U  V  R

right stretch tensor original (strain) tensor

We can further isolate stretch from rotation by considering the eigenvectors of and .

25 © Faith A. Morrison, Michigan Tech U.

Advanced Constitutive Modeling – Polar Decomposition ⋅ ⋅

U ˆk  kˆk

eigenvectors V ˆj  jˆj eigenvalues

Physical Interpretation

ˆ ⋅ i PATH I ⋅ ⋅ ⋅ ⋅ U R ⋅ (stretch first; R then rotate, or ˆ Pure i the reverse) Pure stretch rotation ˆ ˆ V i  i PATH II 26 © Faith A. Morrison, Michigan Tech U.

13 Advanced CM4650 4/23/2018

Advanced Constitutive Modeling – Polar Decomposition

U ˆk  kˆk

eigenvectors V ˆj  jˆj eigenvalues

Physical Interpretation

ˆ i PATH I

U R

(stretch first; R then rotate, or ˆ Pure i the reverse) Pure stretch Omitting the rotation ˆ ˆ V i  i details, the idea is: PATH II

• , are plausible strain measures (translation has been eliminated, but they contain rotation) • Eliminate rotation by decomposing , into pure stretch tensors using polar decomposition ⋅ / ⋅/

27 © Faith A. Morrison, Michigan Tech U.

Advanced Constitutive Modeling – Polar Decomposition ⋅ ⋅ Finite Strain Tensors

⋅ / ⋅ ⋅ ⋅/ proposed ⋅ ⋅ deformation tensors; contain stretch and ⋅ ⋅ rotation; and are symmetric ⋅ ⋅

proposed deformation Cauchy tensor ≡⋅ tensors; contain stretch of eigenvectors, BUT NO ROTATION Finger tensor ≡ ⋅

28 © Faith A. Morrison, Michigan Tech U.

14 Advanced CM4650 4/23/2018

Advanced Constitutive Modeling – Finite Strain Tensors

Now we can construct new constitutive equations using the new strain measures:

Replace: , with: ′, (we use the negative so that at small strains we Finite Strain Hooke’s Law of elastic solids: recover , , like in the GLVE) , 0

Finite Strain Maxwell Model:

′, ′ 29 © Faith A. Morrison, Michigan Tech U.

Advanced Constitutive Modeling – Chapter 9

Now we can construct new constitutive equations using the new strain measures:

Replace: , with: ′,

Finite Strain Hooke’s Law of elastic solids: Time to take these , 0 out for a spin Finite Strain Maxwell Model:

′, ′ 30 © Faith A. Morrison, Michigan Tech U.

15 Advanced CM4650 4/23/2018

Advanced Constitutive Modeling – Chapter 9

EXAMPLE: Calculate stress predicted in rigid-body rotation (around through a counter-clockwise angle ) by a finite-strain Hooke’s law.

, 0

(this didn’t work when the infinitesimal strain tensor , was used)

31 © Faith A. Morrison, Michigan Tech U.

Advanced Constitutive Modeling – Chapter 9

EXAMPLE: Calculate stress predicted in rigid-body rotation (around through a counter-clockwise angle ) by a finite-strain Hooke’s law.

, 0

Usual solution steps: 1. Begin with kinematics of the flow 2. Calculate the needed tensor elements ( before, now) 3. Calculate the stress 4. Calculate functions that rely on stress (material functions)

32 © Faith A. Morrison, Michigan Tech U.

16 Advanced CM4650 4/23/2018

Advanced Constitutive Modeling – Chapter 9

EXAMPLE: Calculate stress predicted in rigid-body rotation (around through a counter-clockwise angle ) by a finite-strain Hooke’s law.

, 0 Usually, start with Usual solution steps: , ,→ … 1. Begin with kinematics of the flow 2. Calculate the needed tensor elements ( before, now) 3. Calculate the stress 4. Calculate functions that rely on stress (material functions)

33 © Faith A. Morrison, Michigan Tech U.

Advanced Constitutive Modeling – Chapter 9

Our old constitutive equations were -based:

′ ′ .

And our “recipe cards” were, therefore, -based … 34 © Faith A. Morrison, Michigan Tech U.

17 Advanced CM4650 4/23/2018

Steady Shear Flow Material Functions Traditional “recipe card” Imposed Kinematics: 0, ≡ 0 0

0 0 constant

Material Stress Response: ̃

̃ ,

0 0

Material Functions: First normal-stress coefficient Ψ ≡ ̃ Viscosity ≡ Second normal- stress coefficient Ψ ≡

35 © Faith A. Morrison, Michigan Tech U.

Advanced Constitutive Modeling – Chapter 9

Our NEW constitutive equations are strain-based, , , , , etc.:

0, , 0 , , , ′ , ′ ′ ′ . .

Our recipe cards must now be deformation-based, , …

36 © Faith A. Morrison, Michigan Tech U.

18 Advanced CM4650 4/23/2018

Advanced Constitutive Modeling – Chapter 9

What is the Finger Tensor ,in CCW Rigid Body Rotation from ′ to through an angle ?

′ t t'



y y P(t') r' r cos P(t)  r r ′ ′  r sin   x x

37 © Faith A. Morrison, Michigan Tech U.

Advanced Constitutive Modeling – Chapter 9

Strain Tensor Prediction for CCW Rigid-Body Rotation around the z-axis from ′ to : ̅ cos ′ ̅ sin From geometry From trigonometry ̅ cos ̅ cos cos sin sin cos sin

̅ sin ̅ sin cos sin cos ′cos′sin ′

From definition: , ′ …

38 © Faith A. Morrison, Michigan Tech U.

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Advanced Constitutive Modeling – Chapter 9

Strain Tensor Prediction for CCW Rigid-Body Rotation around the z-axis from ′ to :

cos sin 0 , sin cos 0 ′ 001

(matches answer in Table 9.3)

NOTE: caption definition of is in error , ⋅

39 © Faith A. Morrison, Michigan Tech U.

CCW Rigid Body Rotation “Material Functions” Strain-centered “recipe card” Imposed Kinematics:

0 (in a coordinate system with origin within the fluid)

′ cos sin ′ ′ ′ cos ′ sin ′ ′

cos sin 0 , sin cos 0 , ̳ ′ 001

(no deformation ⇒ no stress)

40 © Faith A. Morrison, Michigan Tech U.

20 Advanced CM4650 4/23/2018

Advanced Constitutive Modeling – Chapter 9

EXAMPLE: Calculate stress predicted in shear by a finite-strain Hooke’s law. Compare with experimental results.

, 0

41 © Faith A. Morrison, Michigan Tech U.

Steady Shear Flow Material Functions Traditional “recipe card” Imposed Kinematics: 0, ≡ 0 0

0 0 constant

Material Stress Response: ̃

̃ ,

0 0

Material Functions: First normal-stress coefficient Ψ ≡ ̃ Viscosity ≡ Second normal- stress coefficient Ψ ≡ 42 © Faith A. Morrison, Michigan Tech U.

21 Advanced CM4650 4/23/2018

Steady Shear Flow Material Functions Strain-centered “recipe card” Imposed Kinematics: 0, ≡ 0 0 constant 0 0 ′ ′ ′ ′ ′ ′ 100 1 0 , 10 , 10 001 001

Material Functions: First normal-stress coefficient Ψ ≡ ̃ Viscosity ≡ Second normal- stress coefficient Ψ ≡ 43 © Faith A. Morrison, Michigan Tech U.

Advanced Constitutive Modeling – Chapter 9

EXAMPLE: Calculate stress predicted in shear by a finite-strain Hooke’s law. Compare with experimental results.

1 0 From shear kinematics: , 10 001

′,

, 0

1 0 10 (recall sign 001 convention on stress)

44 © Faith A. Morrison, Michigan Tech U.

22 Advanced CM4650 4/23/2018

Advanced Constitutive Modeling – Chapter 9

EXAMPLE: Calculate stress predicted in shear by a finite-strain Hooke’s law. Compare with experimental results.

60 NOTE: for the first time we have predicted 40 nonzero normal 1122 stresses in 20 shear.

0 -0.4 -0.2 0 0.2 0.4

Stress (kPa)Stress -20 21

-40

Solid lines, G = 160 kPa -60

Figure 9.6, p. 325 DeGroot; 45 solid rubber © Faith A. Morrison, Michigan Tech U.

Advanced Constitutive Modeling – Chapter 9 Table 9.3 has strain tensors for standard flows

(Note there is a typo in the definition of in the caption of Table 9.3; there is says from to ′, which is backwards. )

This is correct is the angle from to in ccw rotation around ̂ 46 © Faith A. Morrison, Michigan Tech U.

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Advanced Constitutive Modeling – Chapter 9

Now, let’s fix the Maxwell model.

Integral GLVE model ′ (rate version):

, ′ Integral GLVE model (strain version): ≡ ′

Integral Maxwell model (strain version): , ′

47 © Faith A. Morrison, Michigan Tech U.

Advanced Constitutive Modeling – Chapter 9

Lodge model

Integral Maxwell model , ′ (strain version):

substitute (-Finger tensor) for , infinitesimal strain tensor

Lodge Model: What , ′ does it predict? A finite-strain, viscoelastic constitutive equation

48 © Faith A. Morrison, Michigan Tech U.

24 Advanced CM4650 4/23/2018

Advanced Constitutive Modeling – Chapter 9

EXAMPLE: Calculate the material functions of steady shear flow for the Lodge model.

Lodge Model: , ′

49 © Faith A. Morrison, Michigan Tech U.

Advanced Constitutive Modeling – Chapter 9

EXAMPLE: Calculate the material functions of steady shear flow for the Lodge model.

Lodge Model: , ′

You try.

50 © Faith A. Morrison, Michigan Tech U.

25 Advanced CM4650 4/23/2018

Steady Shear Flow Material Functions Strain-centered “recipe card” Imposed Kinematics: 0, ≡ 0 0 constant 0 0 ′ ′ ′ ′ ′ ′ 100 1 0 , 10 , 10 001 001

Material Functions: First normal-stress coefficient Ψ ≡ ̃ Viscosity ≡ Second normal- stress coefficient Ψ ≡ 51 © Faith A. Morrison, Michigan Tech U.

Advanced Constitutive Modeling – Chapter 9

Lodge model , ′ 7 1.E+0710 =1 s 4 Go=4x10 Pa

1.E+06106

̅ Ψ

material function material 5 Lodge Model 1.E+0510 Report card: • does not shear thin 1.E+044 • Ψ is not zero! 10 0.001 0.01 0.1 1 10 100 1000 • Ψ 0 or , 52 © Faith A. Morrison, Michigan Tech U.

26 Advanced CM4650 4/23/2018

Start-up of Steady Shear Flow Material Functions Needs to become a Strain-centric “recipe card” Imposed Kinematics: 0, ≡ 0 0 00 0 0 0

Material Stress Response: ̃ , , , , , ,

0 0

Material Functions: First normal-stress growth coefficient Ψ , ≡ Shear stress ̃ growth , ≡ Second normal-stress function growth coefficient Ψ , ≡

53 © Faith A. Morrison, Michigan Tech U.

Cessation of Steady Shear Flow Material Functions Needs to become a Strain-centric “recipe card” Imposed Kinematics: 0, ≡ 0 0 0 0 0 00

Material Stress Response: ̃ , , , , , ,

0 0

Material Functions: First normal-stress Ψ, ≡ decay coefficient Shear stress ̃ decay , ≡ Second normal-stress function decay coefficient Ψ , ≡ 54 © Faith A. Morrison, Michigan Tech U.

27 Advanced CM4650 4/23/2018

Advanced Constitutive Modeling – Chapter 9

EXAMPLE: Does the Lodge model pass the test of objectivity posed by the turntable example? (remember, the GLVE failed this test)

y fluid x

x, y, z



x, y, z 55 © Faith A. Morrison, Michigan Tech U.

Advanced Constitutive Modeling – Chapter 9

y fluid Turntable Example: Lodge Model x

x, y, z  , ′ x, y, z  x y z     x' x' x'   1 r rm x y z F (t',t)   eˆjeˆm    r' rj'  y' y' y'  x y z     z' z' z'xyz ̅ ̅ ′′ ′ ̅ ̅′

56 © Faith A. Morrison, Michigan Tech U.

28 Advanced CM4650 4/23/2018

Advanced Constitutive Modeling – Chapter 9

Deformation in shear flow (strain)

 x (t )  1 ref  u  (t ,t)  1 r(tref )   x2 (tref ) 21 ref Shear strain x2  x (t )  3 ref 123

Displacement u , ≡ 0 function 0 57 © Faith A. Morrison, Michigan Tech U.

Advanced Constitutive Modeling – Chapter 9 Turntable Example Lodge Model: , ′

 2  1   0 C1    1 0    0 0 1  xyz Lodge prediction: rotating frame  2  (tt') 1   0 t      0 e    1 0 dt  2     0 0 1  xyz

58 © Faith A. Morrison, Michigan Tech U.

29 Advanced CM4650 4/23/2018

Advanced Constitutive Modeling – Chapter 9 Lodge turntable - from stationary frame

 x x  (y  y ) SC  CS  CC  (x  x ) SS  CC  CS     0 0 0  r   y   y0  (y  y0)CC  SS  SC  (x  x0)  CS  SC  SS      z z  xyz  xyz S  sin t S  sin t Now, calculate and . C  cost C  cost

 0(t ′t)  x y z     x' x' x'   1 r rm x y z F (t',t)   eˆjeˆm    r' rj'  y' y' y' T  x y z  1 1 1   C  F  F  z' z' z'xyz 59 © Faith A. Morrison, Michigan Tech U.

Advanced Constitutive Modeling – Chapter 9

http://pages.mtu.edu/~fmorriso/cm4650/Lodge_turntable.pdf

Result:  2 2 2 2 2   1 2CS  C  (C  S )  SC 0 C1(t,t)  (C2  S2)  SC 2 1 2CS  S2 2 0    0 0 1  xyz

Lodge Model prediction in stationary frame:  2 2 2 2 2  (tt') 1 2CS  C  (C  S )  SC 0 t      0 e  (C2  S2)  SC 2 1 2CS  S2 2 0 dt  2     0 0 1  xyz S  sin t C  cost To compare to previous result, S  sin t C  cost must consider shear coordinate system, e.g.    0(t′ t) 60 © Faith A. Morrison, Michigan Tech U.

30 Advanced CM4650 4/23/2018

Advanced Constitutive Modeling – Chapter 9

Lodge prediction: stationary frame, t=0 2 (tt') 1   0 t    0  IDENTICAL    e   1 0 dt  2     0 0 1  xyz

Lodge prediction: rotating frame  2  (tt') 1   0 t      0 e    1 0 dt  2     0 0 1  xyz

61 © Faith A. Morrison, Michigan Tech U.

Advanced Constitutive Modeling – Chapter 9

Lodge Model (Maxwell with Finger strain tensor) passes test of objectivity! 

What is the differential form of the Lodge model?

Lodge Model: , ′ ? ?

(see discussion in text…) 62 © Faith A. Morrison, Michigan Tech U.

31 Advanced CM4650 4/23/2018

Advanced Constitutive Modeling – Chapter 9

Differential Lodge Equation (Upper-Convected Maxwell Model)

⋅ ⋅ ⋅

Upper-Convected Maxwell Model

≡ ⋅ ⋅ Looks like the Maxwell model, but with a new type of time derivative. We call it the upper-convected time derivative. ≡ ⋅ 63 © Faith A. Morrison, Michigan Tech U.

Advanced Constitutive Modeling – Chapter 9

The Upper-Convected time derivative can be understood to be the time derivative calculated in a coordinate system that is translating and deforming with the fluid (see section 9.3).

y, x 2 material grid at time t'

x 2 same material grid at time t S Q P Q

P x 1 S x, x 1

-11

upper-convected time derivative ≡ ⋅ ⋅ 64 © Faith A. Morrison, Michigan Tech U.

32 Advanced CM4650 4/23/2018

Advanced Constitutive Modeling – Chapter 9

Other Convected Derivatives

Upper-convected time derivative ≡ ⋅ ⋅

Lower-convected time derivative ≡ ⋅ ⋅

Corotational time derivative 1 ≡ ⋅ ⋅ 2 ≡

The vorticity vector 65 © Faith A. Morrison, Michigan Tech U.

Advanced Constitutive Modeling – Chapter 9

Lodge Model: (upper-convected Maxwell) , ′

Cauchy-Maxwell Model: (lower-convected Maxwell) , ′ ′

Lodge Rubberlike Liquid ′ , ′ Model:

66 © Faith A. Morrison, Michigan Tech U.

33 Advanced CM4650 4/23/2018

Advanced Constitutive Modeling – Chapter 9

Lodge Model: (upper-convected Maxwell) , ′ (fix Maxwell with the Finger tensor)

Cauchy-Maxwell Model: (lower-convected Maxwell) , ′ ′ (fix Maxwell with the Cauchy tensor)

Lodge Rubberlike Liquid ′ , ′ Model:

(fix GLVE with the Finger tensor)

67 © Faith A. Morrison, Michigan Tech U.

Advanced Constitutive Modeling – Chapter 9

Lodge Model: (upper-convected Maxwell) , ′ (fix Maxwell with the Finger tensor) Time to take Cauchy-Maxwell Model: (lower-convected Maxwell) , ′ ′ (fix Maxwell withthem the Cauchy tensor) out for a

spin! Lodge Rubberlike Liquid ′ , ′ Model:

(fix GLVE with the Finger tensor)

68 © Faith A. Morrison, Michigan Tech U.

34 Advanced CM4650 4/23/2018

Advanced Constitutive Modeling – Appendix D

Lodge Equation (UCM)

69 © Faith A. Morrison, Michigan Tech U.

Advanced Constitutive Modeling – Appendix D

Cauchy- Maxwell Equation (LCM)

70 © Faith A. Morrison, Michigan Tech U.

35 Advanced CM4650 4/23/2018

Advanced Constitutive Modeling – Chapter 9 SUMMARY: Approaches to finite-strain constitutive equations replace with ,, or other, Non-objective time derivative objective, time derivatives differential Maxwell model

equivalent integral Maxwell , ′ model

Non-objective strain measure

replace with ,, or other, objective, strain measures 71 © Faith A. Morrison, Michigan Tech U.

Advanced Constitutive Modeling – Chapter 9 Methods of Improving Constitutive Equations

Advanced Constitutive Modeling – Chapter 9 We have seen two techniques SUMMARY: Approaches to finite-strain to generate finite-strain constitutive equations replace with ,, constitutive equations from the or other, Non-objective time derivative objective, time Maxwell equation. derivatives differential Maxwell model • Fix the time derivative

equivalent integral Maxwell • Fix the strain measure model

Non-objective strain measure

replace with ,, or other, objective, strain measures

We can also change the form of the basic equation.

•linear modifications

•non-linear modifications 72 © Faith A. Morrison, Michigan Tech U.

36 Advanced CM4650 4/23/2018

Advanced Constitutive Modeling – Chapter 9 We can also change the form of the basic equation. Maxwell Model - Mechanical Analog

Jeffreys Model - Mechanical Analog

73 © Faith A. Morrison, Michigan Tech U.

Advanced Constitutive Modeling – Chapter 9 We can also change the Other Constitutive Approaches form of the basic equation.

Simple Maxwell Model, shear flow only

Upper-Convected Maxwell Model, general flow

retardation time

Simple Jeffreys Model, shear flow only

Upper-Convected Jeffreys Model, general flow

(Oldroyd B Fluid) 74 © Faith A. Morrison, Michigan Tech U.

37 Advanced CM4650 4/23/2018

Advanced Constitutive Modeling – Chapter 9 We can also change the form of the basic equation. Non-linear modifications of the Maxwell Model

White-Metzner Model (brute force shear thinning)

Oldroyd 8-Constant Model: comprehensive continuum mechanics

1 1 1 ⋅ ⋅ : ̳ 2 2 2 1 ⋅ : ̳ 2

The Oldroyd 8-constant contains many UCM other constitutive equations as special cases. UCM  terms UCJ 75 © Faith A. Morrison, Michigan Tech U.

Advanced Constitutive Modeling – Chapter 9 We can also change the form of the basic equation.

The Oldroyd 8-Constant model contains all terms linear in stress tensor and at most quadratic in rate-of-deformation tensor that are also consistent with frame invariance. 1 1 1 ⋅ ⋅ : ̳ 2 2 2 1 ⋅ : ̳ 2

Giesekus Model :

quadratic The only way to choose among in stress these nonlinear models is to compare predictions. 76 © Faith A. Morrison, Michigan Tech U.

38 Advanced CM4650 4/23/2018

Advanced Constitutive Modeling – Appendix D We can also change the form of the basic equation. White- Metzner

77 © Faith A. Morrison, Michigan Tech U.

Advanced Constitutive Modeling – Appendix D

Oldroyd B (Convected Jeffreys)

We can also change the 78

form of the basic equation. U. Tech Michigan Morrison, A. © Faith

39 Advanced CM4650 4/23/2018

Advanced Constitutive Modeling – Chapter 9 We can also change the form of the basic equation. We can also modify integral models to add non-linearity and thus produce new constitutive equations.

Factorized Rivlin-Sawyers Model t  (t)   M (t  t)  (I , I )C   (I , I )C 1 dt  2 1 2 1 1 2  Factorized K-BKZ Model t  U U 1   (t)    M (t  t)2 C  2 C  dt   I2 I1  I1, I2 are the invariants of the Finger or Cauchy strain tensors (these are related). Again, the only way to choose among these nonlinear models is to compare predictions (see R. G. Larson, Constitutive Equations for Polymer Melts). 79 © Faith A. Morrison, Michigan Tech U.

Advanced Constitutive Modeling – Appendix D We can also change the form of the basic equation. Factorized Rivlin- Sawyers

80 © Faith A. Morrison, Michigan Tech U.

40 Advanced CM4650 4/23/2018

Advanced Constitutive Modeling – Chapter 9

Choosing Constitutive Equations

We have fixed all the obvious flaws in our constitutive equations, and now we have too many choices!

We could make predictions and compare with experimental data, but some of the models (Rivlin Sawyer, K-BKZ) have undefined functions that must be specified.

How to proceed? We need some guidance.

All along we have taken a continuum-mechanics approach. We have run that course all the way through. Now we must go back and seek some insight from molecular ideas of relaxation and polymer dynamics.

81 © Faith A. Morrison, Michigan Tech U.

Advanced Constitutive Modeling – Chapter 9 Some of what we have learned from Continuum Modeling

•We can model linear viscoelasticity. The GMM does a good job; there is no reason to play around with springs and dashpots to improve linear viscoelasticity •We can model shear normal stresses. The kind of deformation described by the Finger tensor (affine motion) gives a first normal stress difference and zero second-normal stress; the kind of deformation described by the Cauchy tensor gives both stress differences, but too much . •We can model shear thinning. But only by brute force (GNF, White-Metzner) •We can model elongational flows. But we predict singularities that do not appear to be present. •Frame-Invariance is important. Calculations outside the linear viscoelastic regime are incorrect if the equations are not properly frame invariant. •Thinking in terms of strain is an advantage. When we think only in terms of rate, we can only model Newtonian fluids. •Looking for contradictions when stretching a model to its limits is productive. •Continuum models do not give molecular insight. We can fit continuum models and obtain material functions (viscosity, relaxation times) but we cannot predict these functions for new, related materials 82 © Faith A. Morrison, Michigan Tech U.

41 Advanced CM4650 4/23/2018

Advanced Constitutive Modeling – Chapter 9

It’s time for a new approach.

Molecular Constitutive Modeling

•Begin with a picture (model) of the kind of material that interests you •Derive how stress is produced by deformation of that picture •Write the stress as a function of deformation (constitutive equation)

83 © Faith A. Morrison, Michigan Tech U.

second half Molecular Chapter 9: Advanced Constitutive Models

CM4650 Polymer Rheology

Professor Faith A. Morrison Department of Chemical Engineering Michigan Technological University 84 © Faith A. Morrison, Michigan Tech U.

42 Advanced CM4650 4/23/2018

Advanced Constitutive Modeling – Molecular modeling

At the beginning of the course we started with materials . . .

Chapter 3: Newtonian Fluid Mechanics Polymer Rheology

Molecular Forces (contact) – this is the tough one

choose a surface stress   through P f   at P  dS   on dS the force on P that surface

We need an expression for the state of stress at an arbitrary point P in a flow.

85 © Faith A. Morrison, Michigan Tech U.

Advanced Constitutive Modeling – Molecular modeling

At the beginning of the course we started with materials . . .

Molecular Forces (continued)

Think back to the molecular picture from chemistry:

At that time we wanted to avoid specifying much about our materials. The specifics of these forces, connections, and interactions must be captured by the molecular forces term that we seek.

86 © Faith A. Morrison, Michigan Tech U.

43 Advanced CM4650 4/23/2018

Advanced Constitutive Modeling – Molecular modeling

At the beginning of the course . . .we turned to continuum mechanics. Molecular Forces (continued)

•We will concentrate on expressing the molecular forces mathematically; •We leave to later the task of relating the resulting mathematical expression to experimental observations.

First, choose a surface: nˆ •arbitrary shape •small f dS  stress   at P dS  f What is f ?   on dS 

87 © Faith A. Morrison, Michigan Tech U.

Advanced Constitutive Modeling – Molecular modeling

At the beginning of the course . . .we turned to continuum mechanics. Molecular Forces (continued)

Assembling the force vector:

f  dS nˆ 11eˆ1eˆ1  21eˆ2eˆ1  31eˆ3eˆ1   eˆeˆ   eˆ eˆ   eˆ eˆ We swept all 12 1 2 22 2 2 32 3 2 molecular contact  13eˆ1eˆ3  23eˆ2eˆ3  33eˆ3eˆ3 forces into the 3 3  dS nˆ   pmeˆpeˆm stress tensor. pm1 1

 dS nˆ  pmeˆpeˆm f  dS nˆ  Total stress tensor Now, we seek to (molecular stresses) calculate molecular contact forces directly from a

molecular picture. 88 © Faith A. Morrison, Michigan Tech U.

44 Advanced CM4650 4/23/2018

Two Approaches to Stress‐ Deformation Modeling:

Continuum Molecular Modeling Modeling

• This is the ultimate smoothed over • We picture elements that can be model—matter is a continuum field modeled and use models to predict that abstracts all the molecular observable behavior structure to averaged properties • Modeling efforts identify features • Worked for Newtonian fluids! that are common and demonstrate • (density), (viscosity), (heat links to observable behavior capacity), (thermal conductivity), etc.

89 © Faith A. Morrison, Michigan Tech U.

Advanced Molecular Modeling in Rheology (Chapter 9, 2nd half)

Restart:

Molecular Modeling?

• Start with a specific material (known chemistry, structure) • Postulate a dominant physics that produces an observed behavior • See if it’s TRUE • Use model to address engineering, technology, end‐use questions

4/23/2018 90 © Faith A. Morrison, Michigan Tech U.

45 Advanced CM4650 4/23/2018

Advanced Constitutive Modeling – Molecular modeling

Long-Chain Polymer Constitutive Modeling

molecular tension ~ force on arbitrary f  dA nˆ ( ) stress tensor surface

We now attempt to calculate molecular forces by considering molecular models.

Polymer Dynamics end-to-end R vector, R

Long-chain polymers may be modeled as random walks. 91 © Faith A. Morrison, Michigan Tech U.

Advanced Constitutive Modeling – Molecular modeling

Long-Chain Polymer Constitutive Modeling

molecular tension ~ force on arbitrary f  dA nˆ ( ) stress tensor surface

We now attempt to calculateWARNING : molecular forces by considering molecularThere models. is way more to this than we can Polymer Dynamics end-to-end R cover; we’revector, R taking a

Long-chaintour polymersonly may be modeled as random walks. 92 © Faith A. Morrison, Michigan Tech U.

46 Advanced CM4650 4/23/2018

Advanced Constitutive Modeling – Molecular modeling

Polymer coil responds to deformation

A polymer chain adopts the most random configuration at equilibrium. end-to-end R vector, R

When deformed, the chain tries to recover that most random configuration, giving rise to a spring-like spring of equilibrium length restoring force. and orientation R

We will model the chain dynamics with a random walk. 93 © Faith A. Morrison, Michigan Tech U.

Advanced Constitutive Modeling – Molecular modeling

Gaussian Springs (random walk)

Equilibrium configuration distribution 3 2 function - probability a walk of N steps     RR  0 (R)    e of length a has end-to-end distance R    3   2Na 2 From an entropy calculation of the work needed to 3kT extend a random walk, we can calculate the force f  R needed to deform a the polymer coil Na 2

If we can relate this force, the force to extend the spring, to the force on an arbitrary surface, we can predict rheological properties

molecular tension ~ force on arbitrary f  dA nˆ  stress tensor surface 94 © Faith A. Morrison, Michigan Tech U.

47 Advanced CM4650 4/23/2018

Advanced Constitutive Modeling – Molecular modeling

Molecular force generated by deforming chain

Force on surface ~ Tension dA due to chains f  force on dA   of ETE R

Probability Probability Force exerted chain of ETE R chain has ETE by chain w/ crosses surface R ETE R dA

1 nˆ  R  3 3kT f  R (see next slide) Na 2  (R)dR1dR2dR3

= number of polymer 95 chains per unit volume © Faith A. Morrison, Michigan Tech U.

Advanced Constitutive Modeling – Molecular modeling

Probability chain of ETE R crosses surface dA dA b nˆ intersection I puta that in with dA 1 because this c 1 3  does not print a b to PDF right d 1 fam1 2012 1 3 1 3 nˆ  R nˆ

1 1      R Probability  3  3  nˆ  R    chain of ETE R    crosses surface  3  1/3 dA d c

96 = volume per polymer chain © Faith A. Morrison, Michigan Tech U.

48 Advanced CM4650 4/23/2018

Advanced Constitutive Modeling – Molecular modeling

Molecular force generated by deforming chain

1 ~ 3kT 3 f  nˆ  R  R Na 2 R  R  R  R (R)dR dR dR  1 2 3

BUT, from before . . . ~ molecular tension f  dA nˆ  force on arbitrary surface in terms of 

Comparing these two 2 we conclude, 3kT     R  R (dA  3 ) Na 2 Molecular force generated by deforming chain 97 © Faith A. Morrison, Michigan Tech U.

Advanced Constitutive Modeling – Molecular modeling

How can we convert this equation, 3kT    R  R Na 2 Molecular stress in a fluid generated by a deforming chain

which relates molecular ETE vector and stress, into a constitutive equation, which relates stress and deformation?

We need an idea that connects ETE vector motion to macroscopic deformation of a polymer network or melt.

49 Advanced CM4650 4/23/2018

Advanced Constitutive Modeling – Molecular modeling

Elastic (Crosslinked) Solid

Between every two crosslinks there is a polymer strand that follows a random walk of N steps of length a.

R 2 ETE = end-to-end vector R

Distribution of ETE vectors

Advanced Constitutive Modeling – Molecular modeling Affine motion

How can we relate changes in end-to-end vector to macroscopic deformation?

AN ANSWER: affine-motion assumption: the macroscopic dimension changes are proportional to the microscopic dimension changes

before after

There is no internal slippage of polymer chains: deformation with length scale.

50 Advanced CM4650 4/23/2018

Advanced Constitutive Modeling – Molecular modeling

Consider a general elongational deformation:

1 0 0  Inverse deformation 1   gradient tensor, F   0 2 0   0 0    3 123

For affine motion we can relate the components of the initial and final ETE vectors as, “ETE”=“end-to-end”

ETE after   R   1 1  R1 R2 R3 1  2  3  R(t)  2 R2  R1 R2 R3   R   3 3 123 ETE before

Advanced Constitutive Modeling – Molecular modeling

We are attempting to calculate the stress tensor with this equation: 3kT    R  R Na 2

R  R  R  R (R)dR dR dR  1 2 3

  R   1 1  But, where do R(t)  2 R2  we get this?   R   3 3 123 Configuration distribution function

51 Advanced CM4650 4/23/2018

Advanced Constitutive Modeling – Molecular modeling

Probability chain has ETE between R and R+dR:  (R)dR1dR2dR3

Configuration distribution function

3 Equilibrium configuration distribution     2 RR function:  0 (R)    e    3   2Na2 But, if the deformation is affine, then the number of ETE vectors between R and R+dR at time t is equal to the number of vectors with ETE between R’ and R’+dR’ at t’

Advanced Constitutive Modeling – Molecular modeling

Now we are ready to calculate the stress tensor. 3kT    R  R Na 2

R  R  R  R (R)dR1dR2dR3 R  R  i i    R  i  1 1  R(t)  2 R2  3      2 RR  (R)  (R )    e   R  0  3 3 123   

2 (much algebra Final solution: ˆ ˆ omitted; solved in    kTi eiei Problem 9.57) 104 © Faith A. Morrison, Michigan Tech U.

52 Advanced CM4650 4/23/2018

Advanced Constitutive Modeling – Molecular modeling 2 0 0   1  2 2 Final solution for stress:    kTi eˆieˆi   kT 0 2 0   0 0 2   3 123

Compare this solution with the Finger strain tensor for this flow.

2 0 0   1  1 1 T 1 2 C (t,t)  F  F   0 2 0   0 0 2   3 123 Affine motion

Since the Finger tensor for 1 any deformation may be    kTC written in diagonal form (symmetric tensor) our Which is the same as the finite-strain derivation is valid for all Hooke’s law discussed earlier, with . deformations.

Advanced Constitutive Modeling – Molecular modeling

What about polymer melts? Non permanent crosslinks Green-Tobolsky Temporary Network Model The model: • junction points per unit volume = constant •ETE vectors have finite lifetimes •when old junctions die, new ones are born •newly born ETE vectors adopt the equilibrium distribution Use a statistical approach:

Probability per unit Probability that strand time that strand dies 1 retains same ETE from t’ to and is reborn at   P t (survival probability) t ,t equilibrium 

53 Advanced CM4650 4/23/2018

Advanced Constitutive Modeling – Molecular modeling

What is the probability that a strand retains the same ETE vector between ’ and ’ Δ?

Probability that strand Probability that retains same ETE from ’ strand does not die P  t ,tt to (survival probability) over interval Δ

 1  Pt,tt  Pt,t 1 t    dP 1 t,t   P dt  t,t t ln P    C t,t  1 tt   Pt,t  e 107 © Faith A. Morrison, Michigan Tech U.

Advanced Constitutive Modeling – Molecular modeling Affine The contribution to the stress tensor of the motion individual strands can be calculated from,

Stress at t from Probability that Probability Stress generated by strands born strand is born that a strand an affinely deforming between ’ and = between ’ and survives from strand between ’ ’ ’ ’ ’ ’ and

tt     1   1 d   dte  GC (t,t)   

54 Advanced CM4650 4/23/2018

Advanced Constitutive Modeling – Molecular modeling

Oh no, back where we started! , ′ NO! Green-Tobolsky temporary network mode (Lodge model)

We now know that affine motion of strands with equal birth and death rates gives a model with no shear-thinning, no second-normal stress difference.

To model shear-thinning, N2, etc., therefore, we must add something else to our physical picture, e.g.,

•Anisotropic drag •nonaffine motion of various types

Advanced Constitutive Modeling – Molecular modeling

Anisotropic drag - Giesekus In a system undergoing deformation, the surroundings of a given molecule will be anisotropic; this will result in the drag on any given molecule being anisotropic too.

Starting with the dumbbell model (gives UCM), replace with an anisotropic mobility tensor /. Assume also that the anisotropy in is proportional to the anisotropy in .  B  I   G Giesekus Model :

55 Advanced CM4650 4/23/2018

Advanced Constitutive Modeling – Molecular modeling

Constitutive equations incorporating non-affine motion include:

Gordon and Schowalter: “strands of polymer slip with respect to the deformation of the macroscopic continuum”; see Larson, p130 (this model has problems in step-shear strains) strand slippage Non-Affine motion ≡ ⋅ ⋅ ⋅ ⋅ 2

•Phan-Thien/Tanner •Johnson-Segalman

Larson: uses non-affine motion that is a generalization of the motion in the Doi Edwards model; see Larson, Chapter 5

Wagner: uses irreversible non-affine motion; see Larson, Chapter 5

see Larson, Constitutive Equations for Polymer Melts, Butterworths, 1988

Advanced Constitutive Modeling – Molecular modeling

Reptation Theory (de Gennes)

Retraction (Doi-Edwards) Non-Affine motion

56 Advanced CM4650 4/23/2018

Advanced Constitutive Modeling – Molecular modeling

Step shear strain - strain dependence

10,000

G(t), Pa

1,000 <1.87 3.34 5.22 6.68 100 10 13.4 18.7 25.4

Advanced Constitutive Modeling – Molecular modeling

Step shear strain - Damping Function 10

100000 1

After retraction the relaxation is governed by the memory 10000 0.1 function ’ damping function, h

0.01 1000 0.1 1 10 100 strain

Depending on The Doi-Edwards the strain, a model does a good job 100 different

shifted G(t), Pa G(t), shifted of predicting the amount of damping function, stress is (see Larson p108) relaxed during 10 retraction Retraction time

57 Advanced CM4650 4/23/2018

Advanced Constitutive Modeling – Molecular modeling Non-Affine motion Predicts a strain Doi-Edwards Model measure

t     M (t  t)Q(t,t) dt 

 1 1  1 2  uˆ F uˆ F  Q(t,t)  5 sin dd   1 2  4 00  uˆ F   

tt Predicts a  0 Gi  8G 1 memory M (t  t)  e i N    Gi  2 2 i 2 function i odd i  i i

(Factorized K-BKZ type) Predicts a relaxation uˆ  unit vector that gives time orientation of strands at distribution time t’

M. Doi and S. Edwards J. Chem Soc. Faraday Trans II 74, 1818 115 (1978); ibid 74 560, 918 (1978); ibid 75, 32 (1979); ibid 75, 38 (1979) © Faith A. Morrison, Michigan Tech U.

Advanced Constitutive Modeling – Molecular modeling Ψ Doi-Edwards Model Steady Shear Ψ, SAOS

M. Doi and S. Edwards J. Chem Soc. Faraday Trans II 75, 38 (1979) Ψ Ψ, ∗

′

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Advanced Constitutive Modeling – Molecular modeling

Doi-Edwards Model Shear Start Up ,∞ M. Doi and S. Edwards J. Chem Soc. Faraday Trans II 75, 38 (1979)

Advanced Constitutive Modeling – Molecular modeling

Doi-Edwards Model Elongation Steady Elongation growth stress Elongation Startup difference M. Doi and S. Edwards J. Chem Soc. Faraday Trans II 75, 38 (1979)

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Advanced Constitutive Modeling 10,000

– Molecular modeling G(t), Pa

1,000 <1.87 3.34 Doi-Edwards Model 5.22 6.68 100 10 13.4 Large-Amplitude Step Shear 18.7 25.4

10 M. Doi and S. Edwards J. Chem Soc. Faraday Trans II 74, 1802 (1979) 1

0 1 10 100 1000 10000 time, s

0.1 damping function, h function, damping

0.01 0.1 1 10 100 strain Figure 6.58, p. 213 Einaga et al.; PS soln

Advanced Constitutive Modeling – Molecular modeling Non-Affine motion Doi-Edwards Model

Correctly predicts: •Ratio of •shape of start-up curves •shape of (nonlinear step strain, damping function) •Predicts Tentatively !!!! •shear thinning of ,Ψ •tension-thinning elongational viscosity conclude: shear thinning is an issue of Fails to predict: non-affine . motion • •shape of shear thinning of ,Ψ •reversing flows •Elongational strain hardening (branched polymers)

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Advanced Constitutive Modeling – Molecular modeling

Advanced Models Long-chain branched polymers

Pom-Pom Model (McLeish and Larson, JOR 42 81, 1998) Extended Pom-Pom (Verbeeten, Peters, and Baaijens, JOR 45 823, 2001)

•Single backbone with multiple branches •Backbone can readily be stretched in an extensional flow, producing strain hardening •In shear startup, backbone stretches only temporarily, and eventually collapses, producing strain softening •Based on reptation ideas; two decoupled equations, one for orientation, one for stretch; separate relaxation times for orientation and stretch) 121 © Faith A. Morrison, Michigan Tech U.

Advanced Constitutive Modeling – Molecular modeling

Extended Pom-Pom (Verbeeten, Peters, and Baaijens, JOR 45 823, 2001)

Predicts elongational strain hardening

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Advanced Constitutive Modeling – Molecular modeling

Extended Pom-Pom (Verbeeten, Peters, and Baaijens, JOR 45 823, 2001)

Advanced Constitutive Modeling – Molecular modeling

What about polymer solutions?

•Dilute solutions: chains do not interact Elastic Dumbbell Model •collisions with solvent molecules are W. Kuhn, 1934 modeled stochastically •calculate (R) by a statistical-mechanics solution to the Langevin equation Random force (ensemble averaging) models random collisions

R Drag on beads R models friction

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Advanced Constitutive Modeling – Molecular modeling

Elastic Dumbbell Model

Continuum modeling Momentum balance on a control volume (Navier-Stokes Equation)  v    v v  p  2 v   g  t  Inertia = surface + body

Mixed Continuum/Stochastic modeling (Langevin Equation) Momentum balance on a discrete body (mass m, velocity u) In a fluid continuum (velocity field v) Construct an ensemble of  du  dumbbells and m    u  R v  4kT 2 R  A  dt  seek the probability of a Inertia = drag + spring + random (Brownian) given ETE at t

Advanced Constitutive Modeling – Molecular modeling Construct an Elastic Dumbbell Model ensemble of dumbbells and Langevin Equation seek the probability of a  du  given ETE at t m    u  R v  4kT 2 R  A  dt 

To solve, (see Larson pp41-45). Consider an ensemble of dumbbells and seek the probability  that a dumbbell has an ETE R at a given time t. The equation for  is the Smoluchowski equation:

   4kT 2 2kT     R v  R    0 t R    R 

3kT We can calculate stress from:    R  R (R)dR dR dR Na 2  1 2 3

If we multiply the Smoluchowski equation by R  R and integrate over R space, we obtain an expression for  (i.e. the constitutive equation for this model) 126 © Faith A. Morrison, Michigan Tech U.

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Advanced Constitutive Modeling – Molecular modeling

Integration yields: see Larson, Constitutive Equations for Polymer Melts, Butterworths, 1988

Elastic dumbbell model Upper-Convected Maxwell Model! (same as temporary netrowk model) number of dumbbells/volume Two different models G kT give the same  bead friction factor constitutive equation   (because stress only 8kT 2 depends on the second moment of , 3  2  not on details of  2Na 2 from random walk

Advanced Constitutive Modeling – Molecular modeling

Elastic Dumbbell Model for Dilute Polymer Solutions

Polymer contribution

Solvent contribution

   Dumbbell Model p s (Oldroyd B) See problem 9.49

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Advanced Constitutive Modeling – Molecular modeling

Rouse Model •Multimodal bead-spring model •Springs represent different sub-molecules •Drag localized on beads (Stokes) N+1beads •No hydrodynamic interaction N springs

Advanced Constitutive Modeling – Molecular modeling

see Larson, Constitutive Equations for Rouse Model Polymer Melts, Butterworths, 1988

•Rouse wrote the Langevin equation for each spring. Each spring’s equation is coupled to its neighbor springs which produces a matrix of equations to solve. Langevin Equation

 du  m    u  R v  4kT 2 R  A  dt 

•Rouse found a way to diagonalize the matrix of the averaged Langevin equations;~ this allowed him to find a Smoluchowski equation for each transformed “mode” R i of the Rouse chain ~ •Each Smoluchowski equation gives a UCM for each of the modes R i

N Rouse Model for     i G  kT polymer solutions i1  (multi-mode UCM)        GI i 2 2 i i 16kT sin (i 2(N 1))

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Advanced Constitutive Modeling – Molecular modeling

see Larson, Constitutive Equations for Zimm Model Polymer Melts, Butterworths, 1988 •Multimodal bead-spring model •Springs represent different sub-molecules •Drag localized on beads (Stokes) •Dominant hydrodynamic interaction N+1beads N springs Rouse: solvent velocity near one bead is unaffected by motion of other beads (no hydrodynamic interaction)

Zimm: dominant hydrodynamic interaction) R

Advanced Constitutive Modeling – Molecular modeling (Mewis and Wagner, Colloidal Suspension What about suspensions? Rheology, Cambridge… 2012)

uniform flow

Dilute solution Einstein relation

 m 1 2.5 Stokes flow …

Increasing Concentrated complexity; suspensions solve NS Stokesian dynamics

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Advanced Constitutive Modeling – Suspensions Brady and Bossis, Ann. Rev. Fluid Mech, 20 111 1988 Wagner and Brady, Phys. Today Stokesian Dynamics 2009, p27

Langevin Equation for Dumbbells  du  m    u  R v  4kT 2 R  A  dt  Inertia = drag + spring + random (Brownian)

Another Langevin Equation Stokesian Dynamics for Concentrated Suspensions

dU M   F  F  F dt hydrodynamic particle Brownian

Hydrodynamic = everything the suspending fluid is doing (including drag) Particle = interparticle forces, gravity (including spring forces) Brownian = random thermal events

Advanced Constitutive Modeling – Suspensions

Stokesian Dynamics

Brady and Bossis, Ann. Rev. Fluid Mech, 20 111 1988

Spanning clusters increase viscosity

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Advanced Constitutive Modeling – Molecular modeling Summary

Molecular models may lead to familiar constitutive equations

•Rubber-elasticity theory = Finite-strain Hooke’s law model •Green-Tobolsky temporary network theory = Lodge equation (UCM) •Reptation theory = K-BKZ type equation •Elastic dumbbell model for polymer solutions = Oldroyd B equation

Model parameters have greater meaning when connected to a molecular model •

•, specified by model

Molecular models are essential to narrowing down As always, the the choices available in the continuum-based proof is in the see models (e.g. K-BKZ, Rivlin-Sawyers, etc.) prediction. Larson, esp. Ch 7

Modeling may lead directly to information sought (without ever calculating the stress tensor)

Advanced Constitutive Modeling – Molecular modeling Summary

Molecular models may lead to familiar constitutive equations

Caution: correct stress predictions do not imply that the molecular model is correct Stress is proportional to the second moment of , but different functions may have the same second moments.

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Advanced Constitutive Modeling – Molecular modeling Summary

Materials Discussed •Elastic solids •Linear polymer melts with affine motion (temporary network) •Linear polymer melts with anisotropic drag •Linear polymer melts with various types of non-affine motion •Chain slip •Reptation •Branched melts (pom-pom) •Polymer solutions •Suspensions

Resources R. G. Larson, Constitutive Equations for Polymer Melts R. G. Larson, The Structure and Rheology of Complex Fljuids J. Mewis and N. Wagner, Colloidal Suspension Rheology

Done with Advanced Constitutive Models

Let’s move on to Rheometry

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Chapter 10: Rheometry

CapillaryCapillary Rheometer 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 139 © Faith A. Morrison, Michigan Tech U.