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Matthieu Verani
Advisor: Prof. Gareth McKinley
Effects of Polymer Concentration and Molecular Weight on the Dynamics of Visco-Elasto-
Capillary Breakup
Mechanical Engineering Department
January 30, 2004
Capillary Breakup Extensional Rheometer (CABER)
Balance of
capillary, viscous, and elastic forces
Top and Bottom Cylinders: Diameter (D0) = 6 mm
Initial height = 3 mm
Final height = 13.2 mm
Time to open: 50 ms
Initial aspect ratio = .5
Final aspect ratio = 2.2
Uses ~90 µL of sample
0
HD
Λ =
)()2(3 rrzzSdtdR
RRττησ −+−=
(capillary) (viscous) (elastic)
Balance of Stresses
Normal stress differences:
[ ] ( )i ip zz rr i i zz rr
iG f A Aτ τ τ= − = −∑
02
1S
iz
Gi ν
η ηλ +
−= ∑elastic moduli:
If 0totalF =
Initial conditions
( ) 0 3 zt
zz zzA t A e λ=
The initial value of the axial stretch is chosen to fit the curve in order to obtain a shape as good as possible.
• At early times, the viscous response of the solvent is not negligible for dilute polymer solutions
The polymeric stretch grows as
• Other initial conditions are:
( )1 0 1izzA t≠ = = ( )0 1i
rrA t = =and
(undeformed material)
Ohnesorge and Deborah Numbers
The Ohnesorge Number evaluates the importance of viscous effects over inertial effects and, in our case, is defined by:
The Deborah Number is the dimensionless deformation rate computed as the ratio of the relaxation time of the fluid by the characteristic time of the experiment. It can be defined as:
It can be seen as a Reynolds number:
where the capillary velocity is 0η
σ=v
3De
Rλ
ρ σ=
ROh
ρση 0=
2
0
vROh ρη
− =
The ratio of these two numbers is then an elasto-capillary number:
0
DeOh R
λση
=
Extensional Rheology: CABER experiments
4
68
0.01
2
4
68
0.1
2
4
68
1
R /
Ro
50403020100 Re-scaled time [s]
ps 025 ps 008 ps 0025 ps 0008 ps 00025 ps 00008 ps 000025
a) 1.0
0.8
0.6
0.4
0.2
0.0
R /
Ro
50403020100 Re-scaled time [s]
ps 025 ps 008 ps 0025 ps 0008 ps 00025 ps 00008 ps 000025
b)
0.0002730.0008730.002730.008730.02730.08730.273Ratio c/c*
3.143.143.143.143.143.143.14Rel. Time Kuhn
1.801.861.951.971.981.984.17Rel. Time CABER [s]
0.0000250.000080.000250.00080.00250.0080.025Concentration [wt.%]
Progressive dilution decreases time to breakup:
Compared to Kuhn Chain formula:[ ]1
(3 )s w
A B
MN k Tηηλ
ζ υ= with 3 1[ ] . wK M υη −=
Shear Rheology: Cone and Plate Rheometer
10-2
100
102
104
10-4
10-2
100
102
104
106
G' predictionG" predictiono G' experimentx G" experiment
From the data given by oscillatory shear flow with a cone and plate rheometer, one obtains the storage modulus G' and the loss modulus G". Fitting these data yields the relaxation time through Zimm theory.
[ / ]ra d sω
' '', [ ]G G P a
1.493.971.855.02[s]
0.00080.00250.0080.025Concentration [wt.%]
F i tλ
Governing equations: FENE-P model
12
R Rε= −
2 ( 1)i i iizz zz zz
i
fA A Aελ
= − −
( 1)i i iirr rr rr
i
fA A Aελ
= − − −
2
1
1i i
i
ftrA
L
=−
iLLiν
=
The radius decreases according to
ε : axial strain-rate of the axisymmetric extensional flow
Axial deformation
Radial deformation
With relaxation times and FENE factors ,iλ
Entov & Hinch, JNNFM 1997
1wL M υ−∼Finite extensibility:
Numerical Simulations
Inputs of the simulation:
0η Sη Zimmλ, 0zzA, ,
0 5 10 1510
−3
10−2
10−1
100
Time t/λ1
Rad
ius
Rm
id(t
)/R
1
PS025PS008PS0025PS0008
Decrease of the radius as a function of time.
Asymptotic Behaviors
1- Early viscous times:
For a strong surface tension with no elastic stress…
3 SRσ η ε=
1 6 S
R R tση
= −
Stress balance:
2- Middle elastic times:
Elastic stress grows. Viscous stress drops with the strain-rate. Balance between capillary pressure and elastic stress.
Assumption:
The deformation is smaller than the finite extension limit:
1i izz rrA A>> >
2izz iA L<< ( )1if =
Asymptotic Behaviors (2)
13
11
( )( ) R G tR t Rσ
=
( ) i
t
ii
G t G e λ−
=∑
The radius decreases exponentially:
with
3- Late times limited by finite extension:
Viscous stress is the difference of large numbers. The system of equations is very stiff.
( )i ii i zz rr
iG f A A
Rσ = −∑Balance capillary pressure/elastic stress:
The FENE fluid is now behaving like a suspension of rigid rods, with an effective viscosity
2* 23 i i i
iG Lη λ= ∑
The decrease of radius is then linear:
( ) ( )*6 bR t t tση
= −
Correspondence between numerics and asymptotes
Importance of Gravity: New Test Fluid (MV1)
0.500.273Ratio c/c*
1.094.17Relaxation time CABER [s]
1.045.02Relaxation time by fitting with Zimm theory [s]
0.783.14Relaxation time Kuhn Chain [s]
48.545.5Solvent viscosity [Pa.s]
53.549Zero-shear viscosity [Pa.s]
MV1PS 025Fluid
0
0
gRBoCa
ρη ε
= 0
0sag
gRρεη
=
Experimental Results:
Competition between gravitational and viscous forces: Bo/Ca
0 0
0
[ ] 0.5(3 ) (1 [ ])
wsag
A B
gR M gRDeN k T c
λρ η ρη ζ υ η
= = ≤+
1.5951.3 10 0.5wM−× ≤
765000 /wM g mol≤
750000 /wM g mol=
Viscoelastic Fluid: PS 025
t = 0.01s t = 11.71s t = 22.70s t = 34.00s
Newtonian Fluid: Glycerol
t = 0.01s t = 0.06s t = 0.11s t = 0.18s
Filament Thinning and Gravitational Sagging
New test fluid: MV1
t = 0.01s t = 9.00s t = 18.00s t = 27.00s
Force Transducer: Experimental Setup
2Ro=6.35mm L(t)
Laptop
BNC-2110
Force Transducer
1mm
Gain = 10 V/g
Maximal force =0.01 N = 1 g
10x103
8
6
4
2
0
Vol
tage
[mV
]
10008006004002000
Mass [mg]
Curve Fit: y = a+bxwith a = -12.025 ± 0.0031 mV b = 10.032 ± 0.012 mV/mg
Calibration:
Force measured on the bottom plate of the CABER
22 2 0 03 ( ) ( ) ( )
2N S pR LF R t R t R t gη επ πσ τ π ρ π= + + ∆ −
1(0 ) 15.38sε − −=3
40(0 ) tR R e ε−− =
Force Balance:
Elongation:
Stress Relaxation:
platevL
ε = and
2(0 ) 2.07 10VF N− −= ×4(0 ) 2.12 10F Nσ
− −= ×
Visco-Capillary part:2
3 S
dRR dt R
σεη
= − = 1(0 ) .158sε + −=
2 (0 )6N
S
tF R σπση
+ = −
Force measured on the bottom plate of the CABER (2)
03( )
tzz i
iGA RR t R e λ
σ−
=
20 3 22 32 t
S zz iV i
GA RF R e λπηλ σ
− =
2
0 33
tzz i
iGA RF R e λ
σ πσσ
− =
Elasto-Capillary part:
2 300 2 3
tzz i
E zz iGA RF GA R e λπ
σ−
=
Measure of Azz0: exponential fit of the force data tEF F e β
σ α −+ =
1.824.83[s]
8.0949.7
MV1PS 025Fluid
0zzA
λF_an [N]
F_num [N]
F_exp [N]
MV1PS 025Fluid
44.64 10−× 44.49 10−×
44.25 10−×
44.25 10−× 44.25 10−×
44.26 10−×
Experimental Results2.0x10
-3
1.5
1.0
0.5
For
ce [N
]
302520151050Time [s]
MV1, sample 1 MV1, sample 2 Styrene PS 025
4
6
810
-4
2
4
6
810
-3
2
For
ce [N
]
302520151050Time [s]
MV1, sample 1 MV1, sample 2 Styrene PS 025
Comparison to the Simulations
6
0.01
2
4
6
0.1
2
4
6
1
2
R/R
o
14121086420
t/lambda
Exp. data Num. simulation
6 4 2 0 2 4 60
2
4
6
8
10
12
14
16
18
20
R [mm]
H [m
m]
2
4
6
810
-4
2
4
6
810
-3
For
ce [N
]
30252015105
Time [s]
Exp. Simu Azz=49.7 Simu Azz=1
PS 025
68
0.01
2
4
68
0.1
2
4
68
1
R/R
o
151050
t/lambda
Exp. data Num. simulation
6 4 2 0 2 4 60
2
4
6
8
10
12
14
16
18
20
R [mm]
H [m
m]
4
567
10-4
2
3
4
567
10-3
For
ce [N
]
2015105
Time [s]
Exp. Simu Azz=8.09 Simu Azz=1
MV1
Conclusion and Future Work
•Experimental and numerical demonstration of the concentration dependence for relaxation times.
•Breakdown of the necking in three asymptotic behaviors.
•Fabrication of a new viscoelastic fluid.
•Measure of the force on the bottom plate of the CABER: measure of Azz0.
•Simulation of the evolution of the force and comparison with experimental data.
slope = 1
QUESTIONS?
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