"Spontaneous absorption of droplets into single pores of different radii”

G. Callegari 1

A. Neimark 2

K. Kornev 3

1. TRI/Princeton, Princeton, NJ, 08540, USA

2. Chem. Eng. Dept., Rutgers University, Piscataway, NJ, USA

3. Sch. of Materials Sc., Clemson University, Clemson, SC, USA

Outline

Fast spontaneous absorption of droplets by capillaries

Aplications: Droplet absorption/ spreading in porous materials

Introduction Spreading/wetting and absorption of droplets in porous materials

1- Absorption/Dewetting

2- Pure Fast absorption

thick, large pores, rough surface

thin, small pores, smooth surface

Inertial vs viscous effects

Introduction of Dynamic Contact Angle

Viscoelastic effects

Droplet absorption/ spreading in porous materials: Applications

Ink Jet Printing

Micro-chromatography

Granulation processAgglomeration of fine powders using liquids as binders

Industrial processes:Agriculture chemistryPharmaceuticalMineral processingFoodDetergency

Micro amounts of biological fluids for bio-components recognition

Spray Painting of porous materials

Absorption/Spreading of droplets into porous media

They all considered only viscous forces

Denesuk et al. (1993)Droplet absorption in thick porous material

Washburn

Denesuk et al., J. Coll.Int. Sc.., 158, 114, 1993

cs

oCA R

V

cos35.1

2

32

CAC 9

Marmur (1988)

Washburn

Marmur., J. Coll.Int. Sc.., 122, 209, 1988

Pinned contact line Absorbed dewetting, constant

Starov et al. (2002)Borhan et al. (1993)

Droplet absorption in thin porous material

Borhan et al., J. Coll.Int. Sc.., 158, 403, 1993 Starov et al., J. Coll.Int. Sc.., 252, 397, 2002

Considered competition between spreading and dewetting while absorbed

First fast spreading without absorption,

then dewetting absorption with constant : maximum radius

Dry Spreading + Aspired Dewetting

8x (dx/dt)/R= 2cos

Fv (Poiseuille) = FC

x t 1/2

Absorption/Spreading of droplets on thin porous materials

Hexadecane in PVA nanoweb

RLR is th e r a d iu s o f th e d r o p le t b a s e a n d L i s th e r a d iu s o f th e p r e c u r s o r a b s o rb e d b y th e m e m b ra n e .

W e t t in g - A b s o r p t io n E x p e r im e n t s

T im e d e p e n d e n c e o f th e d ro p le t b a s e r a d iu s , R /R o , a n d th e p r e c u r s o r le n g th , L /R o . I n i t ia l r a d iu s R o = 4 9 0 m

Ro = 490 m

dL/dt =kPc/[ ln(L/R)L]

Dynamics

kPc =1.6 10-4 dyn = r/(2ko)

r = 0.74 m

~ 10° H = 22 m

(R/R0)3 =A-B(L/R0)2

B= 4H/( R0)

A= 4 0 /( R03) A and B may depend

on t (through )

Volume conservation Polyvinyl alcohol

Pore size ~ 1-3 m Well described by existing models (Starov et al 2002)

125 fps

Change of dynamics means transition from spreading to absorption

Pure and fast absorption (thick materials)

0

0,2

0,4

0,6

0,8

1

1,2

1,4

1,6

1,8

2

0 50 100 150 200 250

height (mm)

time (ms)

Dry Spreading

Pure Absorption

2236

hahVdrop

dt

dha

dt

dVdrop

22

22 zazVabs pore

abs vadt

dza

dt

dV 22

dt

dV

Tdt

dVabsdrop

porevTdt

dh 2

Fast absorption in a capillary tube v=cte!!

1000 fps

Ink on a thick porous substrate (large pores, in the order of hundred of microns)

Vol = 7 mm3

10 mm

0.8 mm

τabs ~ 300 ms

Rabs ~ 0.4 cm

Rpore ~ 100 m

Re= v Rpore/~ 1!

Inertia is not negligible

τDenesuk ~ 1 ms

Kinetics of Droplet Absorption

Fast Spontaneous Absorption of Droplets by Capillaries

Linear Kinetics

Reynolds numbers Re = ρUD/ ~ 10 – 150 !

The time interval between pictures is 10 ms.

1000 fps

R = 375 m.

Bosanquet, Phil.Mag. 45, 525(1923)

Fast Spontaneous Absorption of Droplets by Capillaries

[zz”+z’2]+8zz’/R2= 2cos/R-gz

No inertia: Washburn eq.RVB

U cos2

[(z+cR)z”+z’2]+8zz’/R2= 2cos/R-gz Quere, Europhys. Lett. 39, 533(1997)

Added apparent mass

Experimental results can not yet fitted with the expression

All of them working in different regimes, invoque the effect of the dynamic contact angle

Quere

Berezkin et al. ..

Joos et al.

Siebold et al.

Hamraoui et al.

Zhmud et al.

Barraza et al.

when z=0

Meniscus Dynamics: Dynamic Contact Angle

Air pushing glycerol Ca=V/ = viscosity

Callegari, Hulin, Calvo, Contact Angle, Wettability and Adhesion, Vol. 4, K. L. Mittal (Ed.), VSP, Leiden (2006)

There is still a big question mark in partial wetting cases

Callegari, Hulin, Calvo, Contact Angle, Wettability and Adhesion, Vol. 4, K. L. Mittal (Ed.), VSP, Leiden (2006)

Blake et al

Huh, Scriven, Dussand, Rame, Garoff, Hocking, Cox, Voinov, Shikmurzaev, etcBlake

Theoretical contributions: Hydrodynamic and Molecular models (from 1971…)

Ca1/3 Tanner, Marmur et al and Cazabat et al.

Petrov and Sedev Hoffman’s d=(s3+3 Ca)1/3 (acc, 1% up to 140)

Wetting case: Droplet spreading

HoffmanIn capillaries

water

0

10

20

30

40

50

60

70

0,02 0,03 0,04 0,05 0,06

Bosanquet modif with prefactor=0.67

Dyn Cont Angle with Hoffmann=4.5

[(z+cR)z”+z’2]+8zz’/R2= 2Cos[(s3+3z’/)1/3]/R-gz

Fast Spontaneous Absorption of Droplets by Capillaries

0

1

2

3

4

5

6

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

x (cm)

t (s)

Two experiments in long horizontal capillaries

Viscous effects

BosanquetRVB

U cos2

Ss-lambda-DNAs. The time interval between pictures is 4 ms. Capillary radius in microns, from left to right: 250, 290, 375, 450, 580.

UB R-1/2 ?

DNA Racing1000 fps

Kornev, Callegari, Neimark, XXI ICTAM, FM4L_10140, ISBN 83-89687-01-1.

Viscoelastic Fluids

Kornev, Callegari, Amosova, Neimark, Abs Pap ACS 228: U495

Real Fluids vs Ideal FluidsViscous fluids

Shearing stress:

xy= dV/dx , - viscosity

xx= hydrostatic pressure

Visco-elastic fluids (Maxwellian Model)

xy= dV/dx ,

xx=- 2 (dV/dx) xy = - (dV/dx)2

= relaxation time

Kornev, Neimark JCIS, 262, 253(2003)

Balance of momentum:

A·[U2VE

+ XX ]= A· P + 2 · R A· /R

162cos2

R

RVE

U

20

40

60

80

100

120

0.1 0.2 0.3 0.4 0.5

UVE

cm/s

R, cm

UVE has the maximum at RVE

For R < RVE, the velocity decreases due to the Weissenberg effect.

For R > RVE, the velocity decreases because of reduction of the driving capillary pressure.

/4VE

R

Viscoelastic Fluids (Weissemberg effect)

PEO

0

10

20

30

40

50

60

70

0,02 0,03 0,04 0,05 0,06

0

10

20

30

40

50

60

70

0,02 0,03 0,04 0,05 0,06

0

10

20

30

40

50

60

70

0,02 0,03 0,04 0,05 0,06

DNA

Viscoelastic Fluids (Weissemberg effect)

0.1% ss DNA

0.1% ds DNA

water

0.02% PEO

0.05% PEO0.1% PEO

Visc = 1 cp, =65dyn/cm= 0.0023s (0.02% PEO)= 0.0055s (0.05% PEO) = 0.008s (0.1% PEO)

Visc = 1 cp, =65dyn/cm= 0.001s = 0.01s

In fast absorption in thick and rough substrates two mechanisms with different timescales were shown. The constant slope in the decrease of the height of the droplet in function of time goes against Washburn like kinetics in the porous material. Inertial term is important.

Summary and Conclusions

For simple liquids, it was shown that absorption velocity decreases with the capillary radius as predicted by Bosanquet. But the effect of the dynamic contact angle can not be neglected.

Droplet absorption experiments in glass capillary tubes of different diameters were performed. For the high velocity experiments conducted, Re is much larger than one and inertial effect prevails over viscous force. The velocity is found to be independent on time.

The spreading of droplets in thin smooth porous materials shows the “aspired dewetting” regime. The dynamic contact angle is constant in time. Experimental results agree with a simple model proposed.

For viscoelastic liquids, it was shown that the absorption velocity is a non-monotonous function of the capillary radius, with a well defined maximum. This important experimental result support the theoretical analysis previously done.

Summary and Conclusions

For small concentration of polymer in water, a simple maxwellian model seems to cuantitative explain the effect.

For larger concentrations even while the escential features are captured, the cuantitative agreement is not good. This is probable due to a cooperative effect in the interaction of the polymeric molecules.