Modeling Strong Extensional Flows of Polymer Solutions and Melts

Antony N. Beris

CSCAMM WorkshopMultiscale Modeling and Simulation of Complex Fluids

April 19, 2007

Acknowledgements: Dr. Joydeep Mukherjee, Department of Chemical Engineering, University of Michigan

Art Metzner, 1927 - 2006Commemorative Symposium: May 14, 2007, Dept. of Chem. Engrg., Univ. of Delaware

Fiber Spinning of Semicrystalline Polymer Melts

Die swell

spinneret

Spinline

Necking

Take-up rolls

Disordered melt

Oriented semicrystallinemorphology

Semicrystalline Morphologies in Polymers

Spherulite[1]Shish-kebab[2] Fibrillar[3]

1 A. Keller, G.R. Lester and L.B. Morgan, Phil. Trans. Roy. Soc. (London) A247 1 (1954)

2 A. Keller and M.J. Machin, J. Macromol. Sci. Phys. B1 41 (1967)

3 D.C. Prevorsek, P.J. Harget, R.K. Sharma and A.C. Reimschuessel, J. Macromol. Sci. Phys. B8 127 (1973)

Most of the continuum models have a high degree of empiricism and lack capability to predict semicrystalline morphology.

Micro-Macro Hierarchical Approach[1]

Predictions:

Morphology

Macroscopic Model:

•Amorphous melt rheology and chain deformation

• Nonequilibrium thermodynamic potentials

Processing conditions

• Temperature

• Take-up roll speed

Microscopic Simulations

• Chain conformations and microscopic thermodynamic information under the influence of flow deformation for different semicrystalline morphologies using a microscopic model

1 Mukherjee, Wilson , Beris, Flow-induced non-equilibrium thermodynamics of lamellar semicrystalline polymers, J. Non-Newt. Fluid Mech., 120 (1-3): 225-240 (2004).

Outline• Objective: Examine/improve the internal consistency in micro-macro modeling following a hierarchical approach applied to a dense amorphous linear polymer system

• Microscopic modeling

Lattice subdivision-based Monte Carlo method

• Macroscopic modeling

Conformation tensor based models using Non-Equilibrium Thermodynamics

• Microscopic modeling under non-equilibrium conditions

•Conclusions

Microscopic Modeling: Lattice Models

• Originally developed by Flory and

Yoon[1]

• Has been found to be useful in the

study of dense semicrystalline polymers

• Various lattice based approaches:

Random walk models

Exact enumeration methods

Monte Carlo methods

Mean-field models

[1] Flory and Yoon, Polymer 18 (1977), pp. 509-513

Layer 12

.

Lattice-based Metropolis Monte Carlo Scheme

• A modular lattice subdivision approach.

• At any Monte Carlo step perform any ofMonte Carlo moves:

• Internal sublattice optimization

• Isomerization

• Accept the new state with a probability

Eη energetic penalty on tight folds non-dimensionalized with respect to the Boltzmann factor kBT

Orig

inal

sta

teA

fter a

Mon

te C

arlo

m

ove

( ) exp∝ − × tight foldsp E Nη

Tight folds

Initial Macroscopic Modeling of the Dense AmorphousPolymer Phase: Phan-Thien Tanner Model[2]

• Based on a network model

• Can also be described from bracket formulation of non-equilibrium

thermodynamics [3]

• The internal structure is described by the conformation tensor, ,the

second moment of the end-to-end n-segment chain distance.

c

6 …

R(n)1 2

3

4n

n-15c(n) = <R(n)R(n)>

____________________________________________________________________________________________________________________________________

[2] Phan-Thien N, Tanner R.I., J.Non-Newtonian Fluid Mech.,2(1977) 353-365

[3]A.N. Beris, B.J. Edwards, Thermodynamics of Flowing Systems, Oxford University Press, New York, 1994

Phan-Thien Tanner (PTT)Model

( )

( )*

*0

1( )

G 1

= − −λ

= −

* *c c 1c

σ c

( )**

0

1 1 exp (tr( ) 3)( )

= α −λλ

cc

Internal structure dependent relaxation time

Constitutive equationη - extensional viscosity λ - relaxation timeG - elastic modulus

0λ - zero shear rate relaxation time

~ 0.05 for PEα

A*- free energy of the system (non-dimensionalized by kBT)M- total number of entanglements in the system

( )( )1A M I2

**

*

−δ−

δ = c

c

Non-equilibrium thermodynamic information

( ) ( )( )** MA tr log det2⎡ ⎤= −⎣ ⎦

*c c

Hamiltonian Functional Formalism**Beris and Edwards, Thermodynamics of Flowing Systems, Oxford UP, 1994

• For any arbitrary functional F, its time evolution can be described as the sum of two contributions:– a reversible one, represented by a Poisson bracket:

• {F,H}

– an irreversible one, represented by a dissipative bracket:

• [F,H]

• The final dynamic equations are recovered through a direct comparison with the expression derived by differentiation by parts:

{ , } [ , ]dF F dF H F H dVdt dtδ

δ= + = ⋅∫ xx

Advantages of Hamiltonian Formalism (1)

• It only requires knowledge of the following:– A set of macroscopic variables, taken uniformly as volume

densities. The include, in addition to the equilibrium thermodynamic ones (the component mass density, ρi, for every component i, the entropy density si), the momentum density, ρv, and any additional structural parameter, again expressed as a density

– The total energy of the system or any suitable Lagrange transform of it, typically the total Helmholtz free energy, expressed as a functional of all other densities with the temperature substituting for the entropy density

– The Poisson bracket, {F,H}– The dissipation bracket, [F,H]

Advantages of Hamiltonian Formalism (2)

A set of macroscopic variables can easily be assumed depending on the physics that we want to incorporate to the problemThe total Helmholtz free energy can also easily be constructed as the sum of kinetic energy plus an extended thermodynamic free energy that typically includes an easily derived expression (in terms of the structural parameters) in addition to a standard equilibrium expressionThe Poisson bracket, {F,H} is rarely needed by itself: only when an equation is put together for the first time characteristic of the variables involved in this system; otherwise, its effect is probably already known from previous work: it corresponds to a standard reversible dynamics. For viscoelastic flows, this corresponds to the terms defining an upper convected derivativeThe dissipation bracket, [F,H] is the only one to contain major new information and is typically where our maximum ignorance lies. Barren any other information (say, by comparison against a microscopic theory) the main information that we can use is a linear irreversible thermodynamics expression: according to that, the dissipation bracket becomes a bilinear functional in terms of all the nonequilibrium Hamiltonian gradients with an additional nonlinear (in H) correction with respect to δF/δs (entropy correction) that can be easily calculated so that the conservation of the total energy is satisfied: [H,H] = 0.

Final Equations for Single Conformation Viscoelasticity

• For an isothermal system, we get the standard dynamics for a viscoelastic medium (together with the divergence-free velocity constraint):

DDt

Tspρ η= −∇ + ∆ +∇⋅v v T

( )D :Dt 2

T aξ ∂−∇ ⋅ − ⋅∇ = − ⋅ + ⋅ −

∂c v c c v γ c c γ Λ

c

2(1 )T aξ ∂= − ⋅

∂T c

c

Key Question: Consistency with the Underlying Microscopic Physics

• Approach:– Solve the Continuum Macroscopic Model for a well

defined flow to obtain:• Structural information: Conformation Tensor, c• Thermodynamic information: (Non-equilibrium) Free

Energy, A and Free Energy potentials, δA/δc– For the given flow, and the given potentials, solve

directly for the corresponding microscopic structure• Use a specially developed Non-Equilibrium Monte-Carlo

approach applied to a 3d Lattice model with full periodicity– Attest on the consistency through a direct

comparison of the microscopic/macroscopic results– Interpret results and suggest improvements to the

macroscopic model repeating the procedure if necessary

Uniaxial Extensional flow for PTTSolve the equations for uniaxial extensional flow at steady state

Diagonal Conformation tensor*xx

* * *xx yy

*zz

* *xx zz* *

*

**

1

c 0 00 c c 00 0 c

1 1c and c ,1 We 1 2We

where, We 0.5

( )We is the Weissenberg number −

⎡ ⎤⎢ ⎥= =⎢ ⎥⎢ ⎥⎣ ⎦

= =+ −

<

λ=

ε

c

c

x

y

z2ε

−

εNon-equilibrium thermodynamic potentials

* * ** *

* * *xx yy zz

A A M AWe ; MWec c 2 cδ δ δ

= = − =δ δ δ Macro-micro bridges

Notice that at equilibrium (We*~ 0), these potentials go to zero!

Non-equilibrium Thermodynamic Bias

Need to formulate the proper non-equilibrium thermodynamic guidance to drive the microscopic simulations

Ansatz:Accept the new microstate with a probability

**

*Axp :p e δ

< >

⎧ ⎫⎪ ⎪⎪ ⎪∝ ⎨ ⎬

⎪δ⎪ ⎪

⎪⎩ ⎭Non-equilibrium forcing term

RRc

R(n)

Evaluation of the Non-equilibrium Bias

2. Use it to evaluate the Non-equilibrium forcing term

** *

*

A : where1.( / )4n 8 5RRRR RR ,

cδ < >

< > < > =δ

R(n)

1. Evaluate the non-equilibrium thermodynamic potential from Macroscopic model (PTT) :

Ne : entanglement length (typically 100 lattice chain segments for polyethylene).For Ne segment chain, n = Ne , we have

* * ** *

* * *xx yy zz

A A M AWe ; MWec c 2 cδ δ δ

= = − =δ δ δ

x

y

z

e

2 2*x y* 2

z* 2all possible multisegment selections of length Ne

R RA 1: 1.845 We R2N

*

RR =

c

⎛ ⎞⎛ ⎞+δ< > × −⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟δ ⎝

∴⎝ ⎠⎠

∑

Consistency at low We* between Microscopic and PTT Models

Serious Qualitative Inconsistency at higher We*

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00

10

20

30

40

50

60

70

80

90

100

110

120

130

Czz

We*

PTT MC-data

Maximum limit

Inconsistencies in Czz at high We* between PTT and MC

Modified FENE-P/PTT Model

( )

( )

2

* *2

*0

1 L 3f ; f( ) L tr

G f 1

−= − − =

λ −

= −

* *c c 1c c

σ c

( )**

0

1 1 exp (tr( ) 3)( )

= α −λλ

cc

Internal structure dependent relaxation time

Constitutive equation

0

L - extensibility parameter λ - relaxation timeG - elastic modulus

0λ - zero shear rate relaxation time

~ 0.05 for PEα

( ) ( )( )1A M f I2

** *

*

−δ−

δ = c c

c

Non-equilibrium thermodynamic information

( ) ( )( )** 2 *f

MA (L 3)log(f ) log det Ma2⎡ ⎤= − − ≡⎣ ⎦

*c c

A*- free energy of the system (non-dimensionalized by kBT)M- total number of entanglements in the system

Uniaxial Extensional flow for FENE-P/PTTSolve the equations for uniaxial extensional flow at steady state

Diagonal Conformation tensor

ε

2ε

−

*xx

* * *xx yy

*zz

* *xx zz* *

*

**

1

c 0 00 c c 00 0 c

1 1c and c ,f We f 2We

where, We is unrestricted!

( )We is the Weissenberg number −

⎡ ⎤⎢ ⎥= =⎢ ⎥⎢ ⎥⎣ ⎦

= =+ −

λ=

ε

c

c

x

y

z

Same Non-equilibrium thermodynamic potentials as before! * * *

* ** * *xx yy zz

A A M AWe ; MWec c 2 cδ δ δ

= = − =δ δ δ Macro-micro bridges

This model is also compatible to the same microscopic simulations!

Comparison between FENE-P/PTT, PTT and MC

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00

10

20

30

40

50

60

70

80

90

100

110

120

130

Czz

We*

PTTMC-data FENE-P/PTT

Maximum limit

0 2 4 6 8 10-20

0

20

40

60

80

100

120

140

Free

Ene

rgy

We*

FENE-P/PTT PTT

Inconsistencies in the free energy

The total free energy difference between fully extended and fullyrandom state cannot be more thanlogW ≈L2

Bounded Free Energy FENE-PB/PTT Model

( )

( )

2

* *2

22*

0 * 2f

B L 3f ; f( ) L tr

LG B f 1 ; Ba L

−= − − =

λ −

⎛ ⎞= − = ⎜ ⎟+⎝ ⎠

* *c c 1c c

σ c

( )**

0

1 1 exp (tr( ) 3)( )

= α −λλ

cc

Internal structure dependent relaxation time

Constitutive equation

0

L - extensibility parameter λ - relaxation timeG - elastic modulus

0λ - zero shear rate relaxation time

~ 0.05 for PEα

( ) ( )( )1A M B f I2

** *

*

−δ−

δ = c c

c

Non-equilibrium thermodynamic information2 *

* f2 *

f

(L a )A M (L a )⎡ ⎤= ⎢ ⎥+⎣ ⎦

A*- free energy of the system (non-dimensionalized by kBT)M- total number of entanglements in the system

Uniaxial Extensional flow for FENE-PB/PTTSolve the equations for uniaxial extensional flow at steady state

Diagonal Conformation tensor

ε

2ε

−

*xx

* * *xx yy

*zz

* *xx zz* *

*

**

1

c 0 00 c c 00 0 c

B Bc and c ,Bf We Bf 2We

where, We is (asbefore) unrestricted!

( )We is the Weissenberg number −

⎡ ⎤⎢ ⎥= =⎢ ⎥⎢ ⎥⎣ ⎦

= =+ −

λ=

ε

c

c

x

y

z

Same Non-equilibrium thermodynamic potentials as before! * * *

* ** * *xx yy zz

A A M AWe ; MWec c 2 cδ δ δ

= = − =δ δ δ Macro-micro bridges

The microscopic simulations are also compatible to this model!

0 2 4 6 8 10-20

0

20

40

60

80

100

120

140

Free

Ene

rgy

We*

FENE-P/PTT PTT FENE-PB/PTT

Maximum limit for FENE-PB/PTT

FENE-PB/PTT has always a finite free energy

There is good agreement between FENE-PB/PTT and MC

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00

10

20

30

40

50

60

70

80

90

100

110

120

130

Czz

We*

PTT MC-data FENE-P/PTT FENE-PB/PTT

Maximum limit

There is also consistency in Czz between FENE-PB/PTT and MC

0 1 2 3 4 5 6 7 8 9 10

0

50

100

150

200

FENE-P/PTT FENE-PB/PTT

Trou

ton

ratio

We*

Extensional viscosity properties of the modified models are also realistic

Efficient 3-D lattice-subdivision based Monte Carlo simulations have

enabled us to check for microscopic consistency various continuum

viscoelastic models for dense linear polymer models.

Previous models have shown to be defective at high levels of

molecular extension. A new phenomenological model has been

developed (FENE-PB/PTT) that is consistent to finite levels on both the

molecular extension and the total free energy.

Preliminary results show that the new model captures well the sudden

changes accompanying the coil-to-stretch transition at high extensions.

Conclusions

Acknowledgements

Dr. Joydeep Mukherjee

Dr. Stevan Wilson

Randy Oswald

Brian Snyder

Funding:

• NSF GOALI (Materials processing and manufacturing division).

• Dupont-Nylon

Evaluating Microscopic Thermodynamic Information

A*(Eη,We*) can be evaluated incrementally close to static equilibrium as:

*

Equilibriu

* *We* * * * *

*0

* ** * * *

expanded* *

*expanded

m

A (E ,We )A (E ,We ) A (E ,We 0) dWe

We

A ANow we calculate from the relationship of A with A A : and theWe

fact that in the MC we calculate A

ηη η

∂= = +

∂

∂ ∂= −

∂ ∂

=

∫

cc

**i*

i microstate

* ** *expanded* * *

* * ** * *

* * * * * *

**

* *

Alog Z log exp : RRc

A AA :We We

A A A: : :We We We

A :We

−

⎛ ⎞⎛ ⎞δ− = − < >⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟δ⎝ ⎠⎝ ⎠

⎛ ⎞∂ ∂ ∂==> = + < > =⎜ ⎟∂ ∂ ∂⎝ ⎠

⎛ ⎞ ⎛ ⎞∂ ∂ ∂ ∂ ∂ ∂− < > + < > + < > =⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠∂ ∂

< >∂ ∂

∑

RRc

RR RR RRc c c

RRc