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

Effects of Polymer Concentration and Molecular Weight on the Dynamics of Visco-Elasto- Capillary Breakup Matthieu Verani Advisor: Prof. Gareth McKinley Mechanical Engineering Department January 30, 2004

Capillary Breakup Extensional Rheometer (CABER) Balance of Top and Bottom Cylinders: Diameter (D 0 ) = 6 mm capillary, viscous, and Initial height = 3 mm elastic forces Final height = 13.2 mm Uses ~90 µ L of sample Time to open: 50 ms H Λ = Initial aspect ratio = .5 D 0 Final aspect ratio = 2.2

Balance of Stresses If = F 0 total σ 2 dR = η − + τ − τ 3 ( ) ( ) S zz rr R R dt (capillary) (viscous) (elastic) Normal stress differences: ( ) [ ] ∑ i i τ = τ − τ = − G f A A p zz rr i i zz rr i η − η 1 ∑ = G 0 S elastic moduli: λ + ν 2 i i z

Initial conditions • At early times, the viscous response of the solvent is not negligible for dilute polymer solutions The initial value of the axial stretch is chosen to fit the curve in order to obtain a shape as good as possible. The polymeric stretch grows as t ( ) 0 λ = 3 A t A e z zz zz • Other initial conditions are: ( ) ( ) ≠ i 1 = = i t = = A t 0 1 A 0 1 and zz rr (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: η 0 = Oh ρσ R It can be seen as a Reynolds number: σ ρ vR − = = v 2 where the capillary velocity is Oh η η 0 0 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: λ = De ρ σ 3 R The ratio of these two numbers is then an elasto-capillary number: λσ De = η Oh R 0

Extensional Rheology: CABER experiments Progressive dilution decreases time to breakup: a) b) 1 1.0 8 ps 025 ps 025 6 ps 008 ps 008 4 ps 0025 ps 0025 0.8 ps 0008 ps 0008 2 ps 00025 ps 00025 ps 00008 ps 00008 0.1 ps 000025 ps 000025 0.6 8 R / Ro R / Ro 6 4 0.4 2 0.01 8 0.2 6 4 0.0 0 10 20 30 40 50 0 10 20 30 40 50 Re-scaled time [s] Re-scaled time [s] η η 1 [ ] M λ = υ − s w 3 1 η = Compared to Kuhn Chain formula: [ ] K M . with ζ υ w (3 ) N k T A B Concentration [wt.%] 0.025 0.008 0.0025 0.0008 0.00025 0.00008 0.000025 Ratio c/c* 0.273 0.0873 0.0273 0.00873 0.00273 0.000873 0.000273 Rel. Time Kuhn 3.14 3.14 3.14 3.14 3.14 3.14 3.14 Rel. Time CABER [s] 4.17 1.98 1.98 1.97 1.95 1.86 1.80

Shear Rheology: Cone and Plate Rheometer 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. 6 10 ' '' G , G [ P a ] 4 10 2 10 0 10 -2 10 G' prediction -4 G" prediction 10 o G' experiment x G" experiment -2 0 2 4 10 10 10 10 ω [ rad / s ] Concentration 0.025 0.008 0.0025 0.0008 [wt.%] λ [s] 5.02 1.85 3.97 1.49 F it

Governing equations: FENE-P model Entov & Hinch, JNNFM 1997 The radius decreases according to 1 ε � : axial strain-rate of the axisymmetric � = − ε � R R 2 extensional flow Axial deformation f � i i i = ε − − � A 2 A i ( A 1) zz zz zz λ i Radial deformation f � i i i = − ε − − � A A i ( A 1) − υ 1 Finite extensibility: L ∼ M rr rr rr λ w i 1 L = = f L λ , With relaxation times and FENE factors i i i ν i trA i − 1 2 L i

Numerical Simulations Decrease of the radius as a function of time. Inputs of the simulation: η η λ 0 , , , A 0 S Zimm zz PS025 0 10 PS008 PS0025 PS0008 − 1 Radius R mid (t)/R 1 10 − 2 10 − 3 10 0 5 10 15 Time t/ λ 1

Asymptotic Behaviors 1- Early viscous times: For a strong surface tension with no elastic stress… σ = η ε � 3 Stress balance: S R σ = − R R t 1 η 6 S 2- Middle elastic times: Elastic stress grows. Viscous stress drops with the strain-rate. Balance between capillary pressure and elastic stress. i i Assumption: >> > A 1 A zz rr ( ) f = i 2 << 1 A L The deformation is smaller than the finite extension limit: i zz i

Asymptotic Behaviors (2) The radius decreases exponentially: 1 t   R G t ( ) − 3 = ∑ ( ) = G e λ R t ( ) R 1 with G t   i 1 σ i   i 3- Late times limited by finite extension: Viscous stress is the difference of large numbers. The system of equations is very stiff. σ = ( ) ∑ i − i Balance capillary pressure/elastic stress: G f A A i i zz rr R i The FENE fluid is now behaving like a suspension of rigid rods, with an effective viscosity = ∑ 2 2 η λ * G L i i i 3 i The decrease of radius is then linear: σ ( ) ( ) = − R t t t b η * 6

Correspondence between numerics and asymptotes

Importance of Gravity: New Test Fluid (MV1) Competition between gravitational and viscous forces: Bo/Ca ρ gR ρ Bo gR ε = � 0 = 0 sag η η ε � Ca 0 0 λρ η ρ gR [ ] M gR × − 1.59 ≤ 5 = = ≤ 1.3 10 M 0.5 De 0 w 0 0.5 w sag η ζ υ + η (3 ) N k T (1 c [ ]) 0 A B ≤ M 765000 / g mol w = M 750000 / g mol w Experimental Results: Fluid PS 025 MV1 Ratio c/c* 0.273 0.50 Zero-shear viscosity [Pa.s] 49 53.5 Solvent viscosity [Pa.s] 45.5 48.5 Relaxation time Kuhn Chain [s] 3.14 0.78 Relaxation time by fitting with Zimm theory [s] 5.02 1.04 Relaxation time CABER [s] 4.17 1.09

Filament Thinning and Gravitational Sagging Newtonian Fluid: Glycerol t = 0.01s t = 0.06s t = 0.11s t = 0.18s New test fluid: MV1 t = 0.01s t = 9.00s t = 18.00s t = 27.00s Viscoelastic Fluid: PS 025 t = 34.00s t = 0.01s t = 11.71s t = 22.70s

Force Transducer: Experimental Setup Laptop 1mm 2Ro=6.35mm L(t) BNC-2110 Force Transducer Calibration: 3 Curve Fit: y = a+bx 10x10 with a = -12.025 ± 0.0031 mV b = 10.032 ± 0.012 mV/mg Voltage [mV] 8 6 4 Gain = 10 V/g 2 Maximal force =0.01 N = 1 g 0 0 200 400 600 800 1000 Mass [mg]

Force measured on the bottom plate of the CABER Force Balance: 2 R L = η επ 2 + πσ + ∆ τ π 2 − ρ π � F 3 R t ( ) R t ( ) R t ( ) g 0 0 N S p 2 Elongation: v − = 34 − ε � t R (0 ) R e ε = plate � − − ε = � 1 (0 ) 15.38 s and 0 L − − = × 2 F (0 ) 2.07 10 N V − − = × 4 F (0 ) 2.12 10 N σ Stress Relaxation: σ 2 dR ε + = − � 1 ε = − = (0 ) .158 s � Visco-Capillary part: η R dt 3 R S   σ t = πσ + − F 2 R (0 )   N η 6   S

Force measured on the bottom plate of the CABER (2) 2   0 πη 3 Elasto-Capillary part: 2 GA R 2 t − 2 = S zz i λ F R e 3   V i λ σ   2   0 3 GA R t   − 0 = πσ GA R F R zz i e λ 3   t − = R t ( ) R zz i e λ   3 σ i σ i   σ   2 3   0 GA R t − 0 2 = π F GA R zz i e λ   3 E zz i σ   e β − + = α t F F Measure of Azz0: exponential fit of the force data σ E Fluid PS 025 MV1 Fluid PS 025 MV1 F_exp [N] 4.64 10 − × 4.49 10 − × 4 4 49.7 8.09 0 A zz F_num [N] λ 4.25 10 − × 4.26 10 − × 4 4 [s] 4.83 1.82 F_an [N] 4.25 10 − × 4.25 10 − × 4 4

Experimental Results -3 2.0x10 MV1, sample 1 MV1, sample 2 1.5 Styrene PS 025 Force [N] 1.0 0.5 0 5 10 15 20 25 30 Time [s] 2 MV1, sample 1 -3 MV1, sample 2 10 8 Styrene 6 PS 025 Force [N] 4 2 -4 10 8 6 4 0 5 10 15 20 25 30 Time [s]

Comparison to the Simulations 20 20 18 18 16 16 14 14 12 12 H [mm] H [mm] 10 10 8 8 1 2 6 8 6 6 1 4 4 4 6 2 2 4 0 2 6 4 2 0 2 4 6 0 R [mm] 6 4 2 0 2 4 6 2 R [mm] R/Ro 0.1 R/Ro 8 0.1 6 6 4 4 Exp. data Exp. data 2 2 Num. simulation Num. simulation 0.01 0.01 8 6 6 0 2 4 6 8 10 12 14 0 5 10 15 t/lambda t/lambda -3 -3 10 10 8 MV1 Exp. PS 025 Exp. 7 6 6 Simu Azz=49.7 Simu Azz=8.09 5 4 Simu Azz=1 Simu Azz=1 4 Force [N] Force [N] 3 2 2 -4 10 8 6 -4 10 4 7 6 5 2 4 5 10 15 20 25 30 5 10 15 20 Time [s] Time [s]

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

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