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?