Thermodynamics of Nonisothermal Polymer Flows: Experiment, Theory, and Simulation Brian J. Edwards Department of Chemical and Biomolecular Engineering

Thermodynamics of Nonisothermal Polymer Flows:Experiment, Theory, and Simulation

Brian J. Edwards

Department of Chemical and Biomolecular EngineeringUniversity of Tennessee-Knoxville

University of KentuckyLexington, KentuckyFebruary 18, 2009

Collaborators and Funding

Tudor Ionescu: Graduate student, UTK Vlasis Mavrantzas, Professor, University of Patras

Grant #41000-AC7, The Petroleum Research Fund, American Chemical Society

Outline

Part I: Introduction and Background• Introduction to Viscoelastic Fluids• Definition of the concept of Purely Entropic Elasticity• Objective

Part II: Experiment and Theory• Experimental Approach• Theoretical Approach

Part III: Molecular Simulations• Equilibrium Simulations• Nonequilibrium Simulations

Conclusions

Part I: Introduction and Background

The phenomenon described in this presentation is one manifestation of viscoelastic fluid mechanics

Viscoelastic fluids display complex non-Newtonian flow properties under the application of an external force:

» Pressure gradient

» Shear stress

» Extensional strain (stretching)

Paint (&) Crude oil Asphalt Cosmetics Biological fluids

• Blood• Protein solutions

Pulp and coal slurries

Toothpaste Grease Foodstuffs

• Ketchup• Dough• Salad dressing

Plastics• Polymer melts• Rubbers• Polymer solutions

Examples of Viscoelastic Fluids

The dynamics of an incompressible Newtonian fluid can be described completely with three equations:

The Cauchy momentum equation:

The divergence-free condition:

The Newtonian constitutive equation:

p

Dt

Dv

0 v

vv

Newtonian Fluid Dynamics

Newtonian Flow Equations Are Remarkably Robust:

Simple, low-molecular-weight, structureless fluids are well described in three dimensions: Laminar shear and extensional flows Turbulent pipe and channel flows Free-surface flows

• The simple, structureless fluid:

Viscoelastic Fluid Dynamics

A viscoelastic fluid has a complex internal microstructure Today’s topic: Polymer melts

A high-molecular-weight polymer is dissolved in a simple Newtonian fluid

At equilibrium, the polymer molecules assume their statistically most probable conformations, random coils:

Polymer solution

Viscoelastic Flow Behavior

These conformational rearrangements produce very bizarre “non-Newtonian” flow phenomena!

Viscoelastic fluids have very long relaxation times:

Viscoelastic fluid

Newtonian fluid

t Flow off

12


Viscoelastic fluids typically display shear-rate dependent viscosities:

Shear-thinning fluid

Newtonian fluid


Viscoelastic fluids develop very large normal stresses:

Example: Paint

22211

1

Viscoelastic fluid

Newtonian fluid

Nonisothermal Flows of VEs

Nonisothermal flow problems defined by a set of four PDE’s:

• 1) Equation of motion:

• 2) Equation of continuity:

Incompressible fluid:

• 3) Internal energy equation:

• 4) An appropriate constitutive equation: Upper-Convected Maxwell Model (UCMM)

gpDt

D v

v

t

0 v

vv :ˆ

pqDt

UD

vvG 1

The concept of Purely Entropic Elasticity

For simplicity, the internal energy of a viscoelastic liquid is considered as a unique function of temperature (i.e. not a function of deformation) [1,2]:

This let us define the constant volume heat capacity as:

For an incompressible fluid with PEE, the heat equation becomes:

PEE is always assumed in flow calculations!!!

TUU ˆˆ

T

TUcv d

)(ˆdˆ

1. Sarti, G.C. and N. Esposito, Journal of Non-Newtonian Fluid Mechanics, 1977. 3(1): p. 65-76. 2. Astarita, G. and G.C. Sarti, Journal of Non-Newtonian Fluid Mechanics, 1976. 1(1): p. 39-50.

v :ˆ qDt

DTcv

Implications of PEE

What happens to the energy equation if one does not assume PEE?• First, the internal energy is taken as a function of temperature and an

appropriate internal structural variable (conformation tensor):

• Next, the heat capacity is defined as:

• Then, the substantial time derivative of the internal energy becomes:

• The complete form of the heat equation becomes:

c,ˆˆ TUU

cT

Ucv

ˆˆ

cc

c

c

cv

t

U

Dt

DTc

Dt

DU

Dt

DTc

Dt

UD

VT

v

VT

v :ˆ

ˆ:ˆ

ˆˆ

,,

vqvt

U

Dt

DTc

VT

v

::ˆ

ˆ,

cc

c

Objective

Test the validity of PEE under a wide range of processing conditions using experimental measurements, theory and molecular simulation• Experimental approach

Solve the temperature equation numerically using a finite element modeling method (FEM)

Measure the temperature increase due to viscous heating, and compare the results to the FEM predictions

• Theoretical approach Identify all possible causes for the deviations from the FEM predictions

observed in the experimental measurements Use a theoretical model to propose a more accurate form of the

temperature equation and test it through the FEM analysis

• Molecular simulation approach Use a molecular simulation technique to evaluate the energy balances

under non-equilibrium conditions for compounds chemically similar to the ones used in the experiments

Part II: Experiment and Theory

Experimental Approach• Identify a flow situation in which high degrees of orientation are

developed Uniaxial elongational flow generated using the semi-hyperbolically converging

dies (Hencky dies) The analysis is not possible in capillary shear flow

• Find numerical solutions to the temperature equation at steady state using the PEE assumption for this particular flow situation

The solution to this equation will yield the spatial temperature distribution profiles inside the die channel

Compute the average temperature value for the exit axial cross-section of the die

• Under the same conditions used in the FEM calculations, measure the temperature increase due to viscous heating

vv :ˆ 2 TkTcv

Experimental Approach

The semi-hyperbolically converging die (Hencky die)• Proven to generate a uniaxial elongational flow field under special

conditions 2

00 lnln

eeH D

D

A

A

Bz

Azr

2

220

220

e

e

RR

RRLA

22

0

2

e

e

RR

RLB

Hencky 6 Die:

mmD 96.190

mmDe 9937.0

6H


Materials used in this study

Material Grade MI

(g/10min)

Density

(g/cm3)

Thermal conductivity

(Wm-1K-1)

MW PI

LDPE Exact 3139 7.5 0.901 0.3 56,950 1.99

HDPE Paxxon AB40003

0.3 0.943 0.5 105,200 9.74


Calculation of the steady-state spatial temperature distribution profiles• Used a FEM method to find numerical solutions to the temperature equation

• First, elongational viscosity measurements are needed in order to evaluate the viscous heating term:

• The elongational viscosity is identifiable with the “effective elongational viscosity” [1] which can be measured using the Hencky dies and the Advanced Capillary Extrusion Rheometer (ACER)

vv :ˆ 2 TkTcv

2

2

3: ezz v

Hefe

P

1. Feigl, K., F. Tanner, B.J. Edwards, and J.R. Collier, Journal of Non-Newtonian Fluid Mechanics, 2003. 115(2-3): p. 191-215.


Advanced Capillary Extrusion Rheometer (ACER 2000)

1exp

Hram

Lv


Effective elongational viscosity results• HDPE

1.00E+05

1.00E+06

1.00E+07

1 10 100Strain Rate (1/s)

Eff

eciv

e E

lon

gat

ion

al V

isco

sity

(P

a•s)

190ºC

210ºC

230ºC


Effective elongational viscosity results• LDPE

1.00E+05

1.00E+06

1.00E+07


Eff

eciv

e E

lon

gati

on

al

Vis

co

sit

y (

Pa•s

)

150ºC

170ºC

190ºC


FEM calculations

• The heat capacity is considered a function of temperature the tabulated values for generic polyethylene are used from [1]

• The thermal conductivity is considered isotropic, and taken as a constant with respect to temperature and position [1]

• The input velocity field corresponds to a uniaxial elongational flow field in cylindrical coordinates [2]

• The effective elongational viscosity is taken as a function of temperature [3], according to our own experimental measurements

1. Polymer Handbook. 1999, New York: Wiley Interscience.2. Feigl, K., F. Tanner, B.J. Edwards, and J.R. Collier, Journal of Non-Newtonian Fluid Mechanics, 2003. 115(2-3): p. 191-215.3. Dressler, M., B.J. Edwards, and H.C. Ottinger, Rheologica Acta, 1999. 38(2): p. 117-136.

22ˆ efv TkTc v

rvr 2

1 zvz

Tk

AT

B

00 exp


Sample FEM calculation results• HDPE, Tin = Twall = 190oC

12 s 110 s 150 s


Sample FEM calculation results• Axial temperature profiles

• HDPE, Tin = Twall = 190oC

0

2

4

6

8

10

12

14

16

0 5 10 15 20 25 30z (mm)

ΔT

= T

(r=

0,z)

-Tin

(K

)


Sample FEM calculation results• Radial temperature profiles

• HDPE, Tin = Twall = 190oC

0

2

4

6

8

10

12

14

16

-0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45r(mm)

ΔT

= T

(r,z

=2

5mm

)-T

in (

K)


Complete FEM calculation results• Average exit cross-section temperature increases with respect to the inlet

• HDPE

0

1

2

3

4

5

6

7

8

9

10


ΔT

=<

T(r

,z=

25

mm

)>-T

in (

K)

Tin=190ºC

Tin=210ºC

Tin=230ºC

R

R

exitrr

rrLzrTT

0

0

d

d,


Complete FEM calculation results• Average exit cross-section temperature increases with respect to the inlet

• LDPE

R

R

exitrr

rrLzrTT

0

0

d

d,

0

2

4

6

8

10

12

14


ΔT

=<

T(r

,z=

25m

m)>

-Tin

(K

)Tin=150ºC

Tin=170ºC

Tin=190ºC


Experimental design for the temperature measurements


Complete temperature measurement results• HDPE

0

2

4

6

8

10

12

14

16


ΔT

=<

T(r

,z=

25m

m)>

-Tin

(K

)Tin=190ºC Measured

Tin=210ºC Measured

Tin=230ºC Measured

Tin=190ºC FEMLAB

Tin=210ºC FEMLAB

Tin=230ºC FEMLAB

Theoretical Approach

Identify all the factors that may be responsible for the deviations observed at high strain rates

Key assumptions made for the derivation of the temperature equation used in the FEM analysis• Started with the general heat equation

• Assumption 1: Incompressible fluid

• Assumption 2: Flow is steady and

• Assumption 3: Fluid is Purely Entropic and• Obtained the temperature equation solved using FEM

vvq :ˆ

pDt

UD

0 v

0ˆ

t

UU

Dt

UD ˆˆ

v

TUU ˆˆ T

TUcv d

ˆdˆ

22ˆ efv TkTc v


Furthermore• As a consequence of Assumption 3, the heat capacity is a

function of temperature only• Assumption 4: the thermal conductivity is isotropic• Assumption 5: the velocity flow field corresponds to uniaxial

elongational stretching (with full-slip boundary conditions)

Identified Assumptions 3, 4, and 5 as possible candidates responsible for the deviations mentioned earlier

rvr 2

1 zvz


Elimination of Assumptions 4 and 5• Considered anisotropy into the thermal conductivity

Increased k|| by 20%

Decreased k┴ by 10%

• Axial temperature profile calculated for HDPE at Tin = 190oC and a strain rate of 34s-1

0

2

4

6

8

10

12

14

0 5 10 15 20 25z (mm)

ΔT

= T

(r=

0,z)

-Tin

(K

)

k_isotropic

k_anisotropic


Clearly, the PEE assumption seems to be the only remaining factor that is potentially responsible for the deviations observed at high strain rates

How do we eliminate it?• Start with the complete form of the temperature equation for an

incompressible fluid defined earlier

(*)

• First correction: introduce conformation information into the heat capacity [1,2]

• Second correction: introduce the second term on the left side of equation (*)

vqvt

U

Dt

DTc

VT

v

::ˆ

ˆ,

cc

c

1. Dressler, M., B.J. Edwards, and H.C. Ottinger, Rheologica Acta, 1999. 38(2): p. 117-136.2. Dressler, M., The Dynamical Theory of Non-Isothermal Polymeric Materials. 2000, ETH: Zurich.

2

2

00 trtr2

1

T

TKTcc

cc


Both corrections mentioned above require knowledge of the conformation tensor• We can use the UCMM to evaluate the conformation tensor

components inside the die channel

• In Cartesian coordinates, the diagonal components of the normalized conformation tensor work out to be:

TK

Tkcc B

1

1

1exp

1~~

11

HR

yyxx

t

L

zcc

12

12exp

12

2~1

2

HR

zz

t

L

zc


Relaxation time measurements• Complete results for HDPE and LDPE

0.001

0.01

0.1

1

0.0019 0.002 0.0021 0.0022 0.0023 0.0024 0.0025 0.0026 0.00271/T (K^-1)

λ (s

)

LDPE

HDPE

Exponential Fit LDPE

Exponential Fit HDPE


Conformation tensor predictions using the UCMM• HDPE, Tin = 190oC

0

1

0 5 10 15 20 25z(mm)

12 s

13

s

1

5.4

s

1

10

s

1

50

s

mm

z25

~tr/

~tr

c

c


Conformation tensor predictions using the UCMM• HDPE, all temperatures

0.0E+00

5.0E+04

1.0E+05


T = 190°CT = 210°CT = 230°C

mm

z25

~tr

c


Correlation between the conformation at the exit cross-section and the difference between the measured and calculated ΔT

0.0E+00

5.0E+04

1.0E+05


0

1

2

3

4

5

6

ΔT

_m

easu

red

-ΔT

_calc

ula

ted

(K

)

tr(c)(z=25mm) at Tin = 190°C



ΔT_measured-ΔT_calculated at Tin = 190°C

ΔT_measured-ΔT_calculated at Tin = 210°C

ΔT_measured-ΔTcalculated at Tin = 230°C

mm

z25

~tr

c


First correction: the conformation dependent heat capacity

• For example, the total heat capacity evaluated at the die axis for HDPE at Tin = 190oC

2

2

00 trtr2

1

T

TKTcc

cc

30

32

34

36

38

0 5 10 15 20 25 30

z (mm)

11

0

K

Jmol

cc

cco

nf

12 s110 s

115 s

123 s

134 s

150 s50 s-1

2 s-1


Second correction• Rearranging the complete form of the heat equation and making the

appropriate simplifications, we get:

• The axial gradient of czz is already known from the UCMM

• The derivative of the internal energy with respect to czz can also be evaluated using the UCMM [1]:

z

c

c

UvvTkTvc zz

zzzv

ˆ:ˆ 2

ctr2

1

T

TKTTKu

cc

tr

ˆˆtr

U

c

Uc

zzzz

T

TKTTK

c

U

zz

2

1ˆ

1. Dressler, M., The Dynamical Theory of Non-Isothermal Polymeric Materials. 2000, ETH: Zurich.


Examining the effect of introducing corrections 1 and 2 detailed above

• HDPE, Tin = 190oC

0

5

10

15

20


ΔT

=<T

(r,z

=25m

m)>

-Tin

(K

) Calculated w/ no correction

Calculated w/ Correction 1 UCMM

Calculated w/ Correction 2 UCMM

Calculated w/ Correction 2 Giesekus (β = 0.0065)

Measured

Part II: Summary

Provided experimental evidence that PEE is not universally valid

Verified a new form for the temperature equation by essentially eliminating the PEE assumption

Using the UCMM, two corrections have been made to the traditional temperature equation• 1) The conformational dependent heat capacity

Was found to have a significant decrease with increasing orientation Had a negligible effect on the calculated temperature profiles

• 2) The extra heat generation term Quantified the temperature profiles in agreement with the experimental

values

Part III: Molecular Simulations

Simulation Details• NEMC scheme developed by Mavrantzas and coworkers was used• Polydisperse linear alkane systems with average lengths of 24, 36, 50

and 78 carbon atoms were investigated• Temperature effects were also investigated (300K, 350K, 400K and

450K)• A uniaxial orienting field was applied

• Simulations were run at constant temperature and constant pressure P=1atm

200

02

0

00

α

xx

xx

xx

Molecular Simulations

Background• The conformation tensor is defined as the second moment of the end-to-

end vector R

• The normalized conformation tensor is:

• The overall chain spring constant is then defined as:

• The “orienting field” α:

RRc

cc ~

0

2

3

R

0

2

3

R

TkTkTK B

B

c~,,

,,c~

1

TB

TN

A

Tk

c


Thermodynamic Considerations• How do we test the validity of PEE under this framework?

• The steps involved in accomplishing this task include: Evaluate ΔA via thermodynamic integration

Evaluate ΔU directly from simulation

chchchchch N

ST

N

U

N

AT

N

AT

N

A

Ic ,,,, 0

Ic ,,,, TUTUN

Utottot

ch

c:

11

0,,0

0

TkN

MbdcTk

N

A

N

AB

AbT

Bchch


Potential Model Details• Siepmann-Karaborni-Smit (SKS) force field

bondednontorsionanglebond UUUUU

202

1 kU angle

3

0

cosk

kktorsion aU

0Rigid rrIJ

612

4rr

U bondednon

σε

interintrabondednon UUU

Equilibrium Simulations

The equilibrium mean-squared end-to-end distance• Used in the evaluation of the conformation tensor normalization factor

and the chain spring constant

• Can be evaluated for the entire molecular weight distribution interval

• Its molecular weight dependence can be fitted to a polynomial function proposed by Mavrantzas and Theodorou [1]

0

2R

0

2

3

R Tk

R

TkTK B

B 0

2

3

33

221

020

2

1111

XXXbX

RCX

1. Mavrantzas, V.G. and D.N. Theodorou, Macromolecules, 1998. 31(18): p. 6310-6332.


The equilibrium mean-squared end-to-end distance• All systems at T = 450K

0

2R

0

500

1000

1500

2000

2500

3000

0 20 40 60 80 100 120 140

X (#C Atoms)

C24

C36

C50

C78

Polynomial Fit

2

0

2Å

R


The equilibrium mean-squared end-to-end distance• The polynomial fitting constants

• For polyethylene, the measured characteristic ratio at T = 413K [2]

0

2R

Temperature α0 α1 α2 α3

450K 8.8427 -77.9066 521.951 -2141.85

400K 8.6677 -30.5968 -681.681 6030.121

350K 9.219 -9.298 -1573.36 13064.89

300K 11.9351 -183.756 2022.19 -9320.51

450K ref. [1] 9.1312 -75.1865 315.742 -500.3518

1. Mavrantzas, V.G. and D.N. Theodorou, Macromolecules, 1998. 31(18): p. 6310-6332.2. Fetters, L.J., W.W. Graessley, R. Krishnamoorti, and D.J. Lohse, Macromolecules, 1997. 30(17): p. 4973-4977.

4.08.7 C


The equilibrium mean-squared end-to-end distance• Polynomial fits, all temperatures

0

2R

0

500

1000

1500

2000

2500

3000

3500

0 20 40 60 80 100 120 140X (# C Atoms)

300K

350K

400K

450K

Ref. [1]

2

0

2Å

R

1. Mavrantzas, V.G. and D.N. Theodorou, Macromolecules, 1998. 31(18): p. 6310-6332.


The conformation tensor normalization factor μ• Usually taken as a constant with respect to temperature (PEE

assumption)• Gupta and Metzner [1] proposed the following for the temperature

dependence of μ

• This expression was used to fit our equilibrium simulation data with great success

1 BT

1. Gupta, R.K. and A.B. Metzner, Journal of Rheology, 1982. 26(2): p. 181-198.


Theoretical considerations for the behavior of μ with respect to temperature• If B= - 1, μ is a constant and K(T) is a linear function of temperature

The configurational part of the internal energy density of a fluid particle given by the UCMM vanishes

• If B< - 1, μ increases with temperature and decreases with temperature

The configurational part of the internal energy density of a fluid particle given by the UCMM may become important at high degrees of orientation

1BT TkTK B

0tr2

1

cT

TKTTKu

0

T

TKTTK

0

2R

1

0

2

3 BTR


The temperature exponent B

-1.7

-1.6

-1.5

-1.4

-1.3

-1.2

-1.1

-1

-0.9

0 20 40 60 80 100 120 140X (#C Atoms)

B

Temperature Exponent B

Extrapolation

Non-equilibrium Simulations

The applied “orienting field” α:

• The magnitude of αxx will uniquely describe the “strength” of the orienting field

Following the definition of α, the conformation tensor will also have a diagonal form

• Therefore, the trace of the conformation tensor may be used as a unique descriptor for the degree of orientation and extension developed in the simulations

200

02

0

00

α

xx

xx

xx

c~,,

,,c~

1

TB

TN

A

Tk

c ][,,

,,1~

bTchB

TbN

G

Tkc


Molecular weight dependence of the degree of orientation• All systems, T = 450K

0

2

4

6

8

10

12

14

16

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

24C

36C

50C

78C

xx

c~tr


Temperature dependence of the degree of orientation

• C36 system , all temperatures

0

1

2

3

4

5

6

7

8

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

300K

350K

400K

450K

xx

c~tr


Energy balances for the oriented systems• All systems, T = 450K

-40

-30

-20

-10

0

10

20

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

En

erg

y C

han

ge

(J/g

)

chNU /

chNA /

24C 36C 50C 78C

xx


Energy balances for the oriented systems• C36 system, all temperatures

-40

-30

-20

-10

0

10

20

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

En

erg

y C

han

ge

(J/g

)

chNU /

chNA /

xx

450K 400K 350K 300K

b)


Internal energy broken down into individual components• C24 system, T = 400K

-30

-25

-20

-15

-10

-5

0

5

0 0.2 0.4 0.6 0.8

En

erg

y C

han

ge

(J/g

)

xx

chtotal NU /

changle NU /

chtorsion NU /

chinter NU /

chintra NU /

ch

inter

ch

intra

ch

torsion

ch

angle

ch

total

N

U

N

U

N

U

N

U

N

U


The UCMM prediction for the change in Helmholtz free energy

cc ~detln2

13~tr

2

1TkTk

N

ABB

ch

0

5

10

15

20

25

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

UCMM

Integration

gJ

NA

ch/

/

xx


The conformational part of the heat capacity• The MW dependence, T = 450K

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8xx

11

KJm

olc co

nf

24C

36C

50C

78C


The conformational part of the heat capacity• The temperature dependence, C36 system

-0.35

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

450K400K350K300K

xx

11

KJm

olc co

nf

Part III: Summary

Equilibrium simulations• Revealed a non-linear dependence of K(T) with respect to

temperature• Improved agreement with experiment in terms of the

characteristic ratio C∞ and temperature exponent B

Non-equilibrium simulations• The changes in free energy and internal energy are of similar

magnitude• The examination of the individual components of the internal

energy provided two useful insights The elastic response of single chains is indeed purely entropic The inter-molecular contribution to the internal energy of an ensemble of

chains (missing in the isolated chain case) is very important and explains the trends observed during the experiments

Part IV: Published Research

“Structure Formation under Steady-State Isothermal Planar Elongational Flow of n-eicosane: A Comparison between Simulation and Experiment”[1]

• First, we examined the liquid structure predicted by simulation under equilibrium conditions

Simulation performed in the NVT ensemble (number of particles N, system volume V and temperature T are kept constant)

The state point was chosen the same as in the experiment case (T = 315K and ρ = 0.81 g/cc), and the experimental data were taken from literature (**)

1. Ionescu, T.C., et al.,. Physical Review Letters, 2006. 96(3). (*) A. Habenschuss and A.H. Narten, J. Chem. Phys., 92, 5692 (1990)

0

0.5

1

1.5

2

2.5

3

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16k (1/Å)

s(k)

Experimental (*)

Simulation

Simulated Elongated Structure

Next, we examined the structure when the flow field is turned on at steady-state in terms of the pair correlation function • The applied velocity gradient is of the form:

• Results shown at a reduced elongation rate =1.0 • The state point was the same as in the equilibrium case (T=315K and ρ = 0.81

g/cc)

000

00

00

u

2/12 )/( m

0

1

2

3

4

5

6

7

8

9

10

0 1 2 3 4 5 6 7 8 9 10 11 12r (Å)

g(r

)

Quiescent Melt

Elongated Structure

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 2 4 6 8 10 12 14 16 18r (Å)

g_i

nte

r(r)

Quiescent Melt

Elongated Structure


Same structural data, in terms of the static structure factor s(k)

0

0.5

1

1.5

2

2.5

3

0 2 4 6 8 10 12 14 16k (1/Å)

s(k)

Quiescent Melt

Elongated Structure

0

0.5

1

1.5

2

2.5

3

0 2 4 6 8 10 12 14 16k (1/Å)

s_in

ter(

k)

Quiescent Melt

Elongated Structure

Comparison with Experiment

The structure factor s(k) determined via x-ray diffraction from the n-eicosane crystalline sample• Identify two regions:

Inter-molecular region (k<6Å-1), where sharp Bragg peaks are present Intra-molecular region (k>6Å-1), where the agreement with simulation is excellent

0.00

0.50

1.00

1.50

2.00

2.50

3.00

0 2 4 6 8 10 12 14 16k (1/Å)

s(k

)

Liquid XRD (*)

Crystal XRD (this work)

0.00

0.50

1.00

1.50

2.00

2.50

3.00

0 2 4 6 8 10 12 14 16k (1/Å)

s(k

)


Crystal XRD

(*) A. Habenschuss and A.H. Narten, J. Chem. Phys., 92, 5692 (1990).

Conclusions and Directions for Future Research

Conclusions• We successfully combined experiment, theory and simulation to

investigate the nature of the free energy stored by polymer melts subjected to deformation

• First, it was shown that the Theory of Purely Entropic Elasticity is applicable to polymer melts only at low deformation rates

• Second, molecular theory (the UCMM) was used to propose a recipe for eliminating the PEE assumption with great results

• In the end, the Molecular Simulation study helped us elucidate the trends observed in the experimental part The simulated conformational dependent heat capacity was found in

good qualitative agreement with experiment

Conclusions and Directions for Future Research

Directions for Future Research• More polymers and processing conditions

Effects of molecular characteristics

• More accurate viscoelastic models

• Our work in Part IV already led to the development of a constant pressure version of the NEMD algorithm used

Longer chain systems and different flow situations also worth investigating

Acknowledgements

My advisors, Drs. Brian Edwards and David Keffer

Dr. Vlasis Mavrantzas

Dr. Simioan Petrovan

Doug Fielden

ORNL – Cheetah and UT SInRG Cluster

PRF, Grant No. 41000-AC7

Questions?


Same analysis performed for shear flow using a capillary die• LDPE, Tin = Twall = 170oC, D = 1mm, L = 25mm

1150 s 14500 s


Shear flow temperature profile in the measurement device

14500 s

Documents

Thermodynamics of Nonisothermal Polymer Flows: Experiment, Theory, and Simulation Brian J. Edwards Department of Chemical and Biomolecular Engineering