Effects of Polymer Concentration and Molecular Weight on the Dynamics of Visco-Elasto- Capillary Breakup

Transcription

1 Effects of Polymer Concentraton and Molecular Weght on the Dynamcs of Vsco-Elasto- Capllary Breakup Mattheu Veran Advsor: Prof. Gareth McKnley Mechancal Engneerng Department January 3,

2 Capllary Breakup Extensonal Rheometer (CABER) Top and Bottom Cylnders: Dameter (D ) mm Intal heght 3 mm Fnal heght 13. mm Uses ~9 µl of sample Tme to open: 5 ms Intal aspect rato.5 Fnal aspect rato. Λ H D Balance of capllary, vscous, and elastc forces

3 Balance of Stresses If F total σ R dr 3η S( ) + ( τzz τrr) R dt (capllary) (vscous) (elastc) Normal stress dfferences: [ ] ( Gf A A ) τ τ τ p zz rr zz rr elastc modul: G η η S λz 1 + ν

4 Intal condtons At early tmes, the vscous response of the solvent s not neglgble for dlute polymer solutons The ntal value of the axal stretch s chosen to ft the curve n order to obtan a shape as good as possble. The polymerc stretch grows as zz () zz t 3 z A t A e λ Other ntal condtons are: Azz 1 ( t ) 1 and ( ) A t 1 rr (undeformed materal)

5 Ohnesorge and Deborah Numbers The Ohnesorge Number evaluates the mportance of vscous effects over nertal effects and, n our case, s defned by: Oh η ρσr It can be seen as a Reynolds number: Oh ρvr η where the capllary velocty s v σ η The Deborah Number s the dmensonless deformaton rate computed as the rato of the relaxaton tme of the flud by the characterstc tme of the experment. It can be defned as: De λ ρr 3 σ The rato of these two numbers s then an elasto-capllary number: De Oh λσ η R

6 Extensonal Rheology: CABER experments a) 1 b) 1. ps 5 ps ps 5 ps. ps 5 ps.1 ps 5. R / Ro Progressve dluton decreases tme to breakup: R / Ro. ps 5 ps ps 5 ps ps 5 ps ps Re-scaled tme [s] Compared to Kuhn Chan formula: [] η ηsm λ ζ (3 υ ) NkT w wth A B 3 Re-scaled tme [s] [ η] KM. w 3 υ 1 5 Concentraton [wt.%] Rato c/c* Rel. Tme Kuhn Rel. Tme CABER [s]

7 Shear Rheology: Cone and Plate Rheometer From the data gven by oscllatory shear flow wth a cone and plate rheometer, one obtans the storage modulus G' and the loss modulus G". Fttng these data yelds the relaxaton tme through Zmm theory. 1 ' G G Pa '', [ ] G' predcton G" predcton o G' experment x G" experment ω [ rad / s] Concentraton [wt.%] λ F t [s]

8 Governng equatons: FENE-P model The radus decreases accordng to Entov & Hnch, JNNFM 1997 R 1 ε R ε : axal stran-rate of the axsymmetrc extensonal flow Axal deformaton f A zz ε Azz ( Azz 1) λ Radal deformaton f A rr ε Arr ( Arr 1) λ Wth relaxaton tmes λ and FENE factors, f Fnte extensblty: 1 1 tra L L M υ L 1 w L ν

9 Numercal Smulatons Decrease of the radus as a functon of tme. Inputs of the smulaton: η, ηs λ Zmm,, A zz 1 PS5 PS PS5 PS Radus R md (t)/r Tme t/λ 1

10 Asymptotc Behavors 1- Early vscous tmes: For a strong surface tenson wth no elastc stress Stress balance: σ 3η S ε R R R σ t 1 η S - Mddle elastc tmes: Elastc stress grows. Vscous stress drops wth the stran-rate. Balance between capllary pressure and elastc stress. Assumpton: A zz >> 1 > rr The deformaton s smaller than the fnte extenson lmt: A A zz << L ( f 1)

11 Asymptotc Behavors () The radus decreases exponentally: Rt () 1 R 3 1Gt () R1 σ wth G t () Ge λ t 3- Late tmes lmted by fnte extenson: Vscous stress s the dfference of large numbers. The system of equatons s very stff. Balance capllary pressure/elastc stress: σ R ( ) G f A A zz rr The FENE flud s now behavng lke a suspenson of rgd rods, wth an effectve vscosty * η GλL 3 The decrease of radus s then lnear: R t σ t * b t η () ( )

12 Correspondence between numercs and asymptotes

13 Importance of Gravty: New Test Flud (MV1) Competton between gravtatonal and vscous forces: Bo/Ca Bo ρgr ρ gr ε sag Ca η ε η De sag λρgr [ η] M ρgr w η ζ(3 υ) NkT A B (1 + c[ η]) M w M w 75 g/ mol M w 75 g/ mol Expermental Results: Flud Rato c/c* Zero-shear vscosty [Pa.s] Solvent vscosty [Pa.s] Relaxaton tme Kuhn Chan [s] Relaxaton tme by fttng wth Zmm theory [s] Relaxaton tme CABER [s] PS MV

14 Flament Thnnng and Gravtatonal Saggng Newtonan Flud: Glycerol New test flud: MV1 t.1s t.s t.11s t.1s t.1s t 9.s t 1.s t 7.s Vscoelastc Flud: PS 5 t.1s t 11.71s t.7s t 3.s

15 Force Transducer: Expermental Setup 1mm Laptop Ro.35mm L(t) BNC-11 Force Transducer Calbraton: Gan 1 V/g Maxmal force.1 N 1 g Voltage [mv] 1x1 3 Curve Ft: y a+bx wth a -1.5 ±.31 mv b 1.3 ±.1 mv/mg 1 Mass [mg]

16 Force measured on the bottom plate of the CABER Force Balance: Elongaton: R L N 3 ηεπ S ( ) + πσ ( ) + τπ p ( ) ρ π F R t R t R t g ε v plate L ε ( ) 15.3s FV 1 ( ).7 1 and N R( ) R e 3 εt F σ ( ).1 1 N Stress Relaxaton: Vsco-Capllary part: dr ε Rdt σ R 3η S ε ( ).15s + 1 F N + σt πσ R( ) η S

17 Force measured on the bottom plate of the CABER () Elasto-Capllary part: GA t zz R 3λ Rt () R e σ 3 πη t S GAzz R 3 λ FV R e λ σ 3 GAzz R σ πσ t 3 λ F R e σ 3 GAzz R t 3 E π zz λ F GA R e σ Measure of Azz: exponental ft of the force data t F + F αe β σ E Flud PS 5 MV1 Flud PS 5 MV1 A zz λ [s] F_exp [N] F_num [N] F_an [N]

18 Expermental Results Force [N].x MV1, sample 1 MV1, sample Styrene PS Tme [s] 5 3 Force [N] MV1, sample 1 MV1, sample Styrene PS Tme [s] 5 3

19 Comparson to the Smulatons R/Ro Exp. data Num. smulaton H [mm] 1 R [mm] R/Ro Exp. data Num. smulaton H [mm] 1 R [mm] t/lambda t/lambda 15 Force [N] PS 5 15 Tme [s] Exp. Smu Azz9.7 Smu Azz1 5 3 Force [N] MV1 1 Tme [s] Exp. Smu Azz.9 Smu Azz1 15

20 Concluson and Future Work Expermental and numercal demonstraton of the concentraton dependence for relaxaton tmes. Breakdown of the neckng n three asymptotc behavors. Fabrcaton of a new vscoelastc flud. Measure of the force on the bottom plate of the CABER: measure of Azz. Smulaton of the evoluton of the force and comparson wth expermental data. slope 1

21 QUESTIONS?