Effects of Polymer Concentration and Molecular Weight on the - - PowerPoint PPT Presentation

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

H D Λ =

) ( ) 2 ( 3

rr zz S

dt dR R R τ τ η σ − + − =

(capillary) (viscous) (elastic)

Balance of Stresses

Normal stress differences:

[ ]

( )

i i p zz rr i i zz rr i

G f A A τ τ τ = − = −

∑

2

1

S i z

G i

ν

η η λ

+

− =

∑

elastic moduli: If

total

F =

Initial conditions

( )

3

z

t zz zz

A 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

1

i zz

A t

≠

= =

( )

1

i rr

A 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

η σ = v

3

De R λ ρ σ = R Oh ρσ η 0 =

2

vR Oh ρ η

− =

The ratio of these two numbers is then an elasto-capillary number:

De Oh R λσ η =

Extensional Rheology: CABER experiments

4 6 8

0.01

2 4 6 8

0.1

2 4 6 8

1 R / Ro 50 40 30 20 10 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 50 40 30 20 10 Re-scaled time [s]

ps 025 ps 008 ps 0025 ps 0008 ps 00025 ps 00008 ps 000025

b)

0.000273 0.000873 0.00273 0.00873 0.0273 0.0873 0.273

Ratio c/c*

3.14 3.14 3.14 3.14 3.14 3.14 3.14

• Rel. Time Kuhn

1.80 1.86 1.95 1.97 1.98 1.98 4.17

• Rel. Time CABER [s]

0.000025 0.00008 0.00025 0.0008 0.0025 0.008 0.025

Concentration [wt.%] Progressive dilution decreases time to breakup: Compared to Kuhn Chain formula:

[ ] 1 (3 )

s w A B

M N k T η η λ ζ υ =

with

3 1

[ ] .

w

K M

υ

η

−

=

Shear Rheology: Cone and Plate Rheometer

10

• 2

10 10

2

10

4

10

• 4

10

• 2

10 10

2

10

4

10

6

G' prediction G" prediction

• G' experiment

x 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.

[ / ] rad s ω

' ''

, [ ] G G P a

1.49 3.97 1.85 5.02 [s] 0.0008 0.0025 0.008 0.025 Concentration [wt.%]

F it

λ

Governing equations: FENE-P model

1 2 R R ε = −

• 2

( 1)

i i i i zz zz zz i

f A A A ε λ = − −

• (

1)

i i i i rr rr rr i

f A A A ε λ = − − −

• 2

1 1

i i i

f trA L = −

i

L L iν =

The radius decreases according to

ε : axial strain-rate of the axisymmetric

extensional flow Axial deformation Radial deformation With relaxation times and FENE factors ,

i

λ

Entov & Hinch, JNNFM 1997

1 w

L M

υ −

∼

Finite extensibility:

Numerical Simulations

Inputs of the simulation:

η

S

η

Zimm

λ

,

zz

A

, ,

5 10 15 10

−3

10

−2

10

−1

10 Time t/λ1 Radius Rmid(t)/R1 PS025 PS008 PS0025 PS0008

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

S

R σ η ε =

• 1

6

S

R R t σ η = −

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:

1

i i zz rr

A A >> >

2 i zz i

A L <<

( )

1

i

f =

Asymptotic Behaviors (2)

1 3 1 1

( ) ( ) R G t R t R σ   =    

( )

i

t i i

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 i i i zz rr i

G f A A R σ = −

∑

Balance capillary pressure/elastic stress: The FENE fluid is now behaving like a suspension of rigid rods, with an effective viscosity

2 *

2 3

i i i i

G L η λ = ∑

The decrease of radius is then linear:

( ) ( )

*

6

b

R t t t σ η = −

Correspondence between numerics and asymptotes

Importance of Gravity: New Test Fluid (MV1)

0.50 0.273 Ratio c/c* 1.09 4.17 Relaxation time CABER [s] 1.04 5.02 Relaxation time by fitting with Zimm theory [s] 0.78 3.14 Relaxation time Kuhn Chain [s] 48.5 45.5 Solvent viscosity [Pa.s] 53.5 49 Zero-shear viscosity [Pa.s] MV1 PS 025 Fluid gR Bo Ca ρ η ε =

• sag

gR ρ ε η =

• Experimental Results:

Competition between gravitational and viscous forces: Bo/Ca

[ ] 0.5 (3 ) (1 [ ])

w sag A B

gR M gR De N k T c λρ η ρ η ζ υ η = = ≤ +

1.59 5

1.3 10 0.5

w

M

−

× ≤ 765000 /

w

M g mol ≤ 750000 /

w

M 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

10x10

3

8 6 4 2 Voltage [mV] 1000 800 600 400 200 Mass [mg]

Curve Fit: y = a+bx with a = -12.025 ± 0.0031 mV b = 10.032 ± 0.012 mV/mg

Calibration:

Force measured on the bottom plate of the CABER

2 2 2

3 ( ) ( ) ( ) 2

N S p

R L F R t R t R t g η επ πσ τ π ρ π = + + ∆ −

• 1

(0 ) 15.38s ε

− −

=

• 34

(0 )

t

R R e

ε − − =

• Force Balance:

Elongation: Stress Relaxation:

plate

v L ε =

• and

2

(0 ) 2.07 10

V

F N

− −

= ×

4

(0 ) 2.12 10 F N

σ − −

= ×

Visco-Capillary part:

2 3

S

dR R dt R σ ε η = − =

• 1

(0 ) .158s ε

+ −

=

• 2

(0 ) 6

N S

t F R σ πσ η

+

  = −    

Force measured on the bottom plate of the CABER (2)

3

( )

t zz i i

GA R R t R e

λ

σ

−

  =    

2 3 2 2 3

2

t S zz i V i

GA R F R e

λ

πη λ σ

−

  =    

2 3 3 t zz i i

GA R F R e

λ σ

πσ σ

−

  =     Elasto-Capillary part:

2 3 2 3 t zz i E zz i

GA R F GA R e

λ

π σ

−

  =    

Measure of Azz0: exponential fit of the force data

t E

F F e β

σ

α

−

+ =

1.82 4.83 [s] 8.09 49.7 MV1 PS 025 Fluid

zz

A

λ

F_an [N] F_num [N] F_exp [N] MV1 PS 025 Fluid

4

4.64 10− ×

4

4.49 10− ×

4

4.25 10− ×

4

4.25 10− ×

4

4.25 10− ×

4

4.26 10− ×

Experimental Results

2.0x10

• 3

1.5 1.0 0.5

Force [N]

30 25 20 15 10 5

Time [s]

MV1, sample 1 MV1, sample 2 Styrene PS 025

4 6 8

10

• 4

2 4 6 8

10

• 3

2

Force [N]

30 25 20 15 10 5

Time [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/Ro

14 12 10 8 6 4 2

t/lambda

• Exp. data
• Num. simulation

R [mm] H [mm]

2 4 6 8

10

• 4

2 4 6 8

10

• 3

Force [N]

30 25 20 15 10 5

Time [s]

Exp.

Simu Azz=49.7 Simu Azz=1

PS 025

6 8

0.01

2 4 6 8

0.1

2 4 6 8

1

R/Ro

15 10 5

t/lambda

• Exp. data
• Num. simulation

R [mm] H [mm]

4 5 6 7

10

• 4

2 3 4 5 6 7

10

• 3

Force [N]

20 15 10 5

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?