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 Flow Behavior
Viscoelastic fluids typically display shear-rate dependent viscosities:
Shear-thinning fluid
Newtonian fluid
Viscoelastic Flow Behavior
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
Experimental Approach
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
Experimental Approach
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.
Experimental Approach
Advanced Capillary Extrusion Rheometer (ACER 2000)
1exp
Hram
Lv
Experimental Approach
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
Experimental Approach
Effective elongational viscosity results• LDPE
1.00E+05
1.00E+06
1.00E+07
1 10 100Strain Rate (1/s)
Eff
eciv
e E
lon
gati
on
al
Vis
co
sit
y (
Pa•s
)
150ºC
170ºC
190ºC
Experimental Approach
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
Experimental Approach
Sample FEM calculation results• HDPE, Tin = Twall = 190oC
12 s 110 s 150 s
Experimental Approach
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
)
Experimental Approach
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)
Experimental Approach
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
1 10 100Strain Rate (1/s)
ΔT
=<
T(r
,z=
25
mm
)>-T
in (
K)
Tin=190ºC
Tin=210ºC
Tin=230ºC
R
R
exitrr
rrLzrTT
0
0
d
d,
Experimental Approach
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
1 10 100Strain Rate (1/s)
ΔT
=<
T(r
,z=
25m
m)>
-Tin
(K
)Tin=150ºC
Tin=170ºC
Tin=190ºC
Experimental Approach
Experimental design for the temperature measurements
Experimental Approach
Complete temperature measurement results• HDPE
0
2
4
6
8
10
12
14
16
1 10 100Strain Rate (1/s)
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
Theoretical Approach
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
Theoretical Approach
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
Theoretical Approach
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
Theoretical Approach
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
Theoretical Approach
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
Theoretical Approach
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
Theoretical Approach
Conformation tensor predictions using the UCMM• HDPE, all temperatures
0.0E+00
5.0E+04
1.0E+05
1 10 100Strain Rate (1/s)
T = 190°CT = 210°CT = 230°C
mm
z25
~tr
c
Theoretical Approach
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
1 10 100Strain Rate (1/s)
0
1
2
3
4
5
6
ΔT
_m
easu
red
-ΔT
_calc
ula
ted
(K
)
tr(c)(z=25mm) at Tin = 190°C
tr(c)(z=25mm) at Tin = 210°C
tr(c)(z=25mm) at Tin = 230°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
Theoretical Approach
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
Theoretical Approach
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.
Theoretical Approach
Examining the effect of introducing corrections 1 and 2 detailed above
• HDPE, Tin = 190oC
0
5
10
15
20
1 10 100Strain Rate (1/s)
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
Molecular Simulations
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
Molecular Simulations
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.
Equilibrium Simulations
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
Equilibrium Simulations
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
Equilibrium Simulations
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.
Equilibrium Simulations
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.
Equilibrium Simulations
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
Equilibrium Simulations
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
Non-equilibrium Simulations
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
Non-equilibrium Simulations
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
Non-equilibrium Simulations
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
Non-equilibrium Simulations
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)
Non-equilibrium Simulations
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
Non-equilibrium Simulations
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
Non-equilibrium Simulations
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
Non-equilibrium Simulations
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
Simulated 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
)
Simulated Elongated Structure
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?
Experimental Approach
Same analysis performed for shear flow using a capillary die• LDPE, Tin = Twall = 170oC, D = 1mm, L = 25mm
1150 s 14500 s
Experimental Approach
Shear flow temperature profile in the measurement device
14500 s