Jinwu Wang, Post Doctoral AssociateSheldon Q. Shi, Assistant Professor

Forced Fluid Imbibitionin a Powder-Packed Column

Department of Forest ProductsMississippi State University

ObjectivesDevelop a tool to measure contact angles and surface energies for both– Spontaneous and – non-spontaneous imbibing liquids in powders

Current Problem– Spontaneous inbibition is not achieved in many

cases when the wetting angle is larger than 900

ExplanationWhen a rigid container is inserted into a fluid, the fluid will rise in the container to a height higher than the surrounding liquid

Capillary Tube Wedge Sponge

Professor John Pelesko and Anson Carter, Department of Mathematics, University of Delaware

Phys. Rev. Lett. (2007), Capillary Rise in Nanopores: Molecular Dynamics Evidence for the Lucas-Washburn Equation

Velocity Field around the Moving Meniscus

Liquid Behaviors in Powders

Assume that a powder-packed column consists of numerous capillary tubes: a wicking-equivalent effective capillary radius

The same governing equations as those applied to a capillary tube

Liquid

air

A powder-packed column with radius R

Capillary action

List of Variables:volume = πr2z g = gravityr = radius of capillary tubez = rising height, measured to the bottom of the meniscus, at time t ≥ 0ρ = density of the surface of the liquid γ = surface tensionθ = contact angle between the surface of the liquid and the wall of the tube

Free Body Diagram r

Z(t)

Poiseuille Viscous ForceGravitation Force Inertial Force

Surface TensionExternal vacuum }Driving Forces

Dragging Forces

Explanation of the Forces

Gravitational Force

Poiseuille Viscous Force

Vacuum Force

)cos(2 θγπr

zgrmgFw2ρπ==

dtdzzFdrag πη8=

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛+=⎟

⎠⎞

⎜⎝⎛==∑

2

2

222)(

dtdz

dtzdzr

dtdzzr

dtd

dtmvdF ρπρπ

Newton's Second Law of Motion

Surface Tension Force

πr2ΔP

Explanation of Differential Equation

Newton's Second Law of Motion:Net Force = Surface Tension Force +Vacuum

- Poiseuitte Viscous Force - Gravitational Force

Dividing by πr2, the differential equation becomes:

Zo = Z(0) = 0

gzdtdzz

rP

rdtdz

dtzdz ρηθγρ −−Δ+=⎟

⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛+ 2

2

2

2 8)cos(2

Boundary Conditions:z(0) = 0 and z’(z∞) = 0

zrgdtdzzPrr

dtdz

dtzdzr 22

2

2

22 8)cos(2 πρπηπθγπρπ −−Δ+=⎟

⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛+

The Effective Zone of Forces

The size of each zone depends on the probe liquid properties and capillary structures

z

z0Inertial Force

Washburn Zone

Gravity Effective Zone

08cos22 =−−Δ+ gz

dtdzz

rP

rρηθγ

gP

grze ρρ

θγ Δ+=

cos2⎟⎟⎠

⎞⎜⎜⎝

⎛−

−= )(

)(ln8

2 tztzz

zzgr

te

eeρ

η

The Effect of Capillary Radius on Wicking

Lucas-Washburn equation:

( ) trtz2/1

2

2cos

⎟⎟⎠

⎞⎜⎜⎝

⎛=

ηθγ

Is valid whenCapillary diameter is smallAt initial rising periodViscous drag >> gravity forceDensity is low, inertia is small

Column Wicking Diagram

Non-spontaneous inbibitionwhen the contact angle is larger than 900

by applying vacuum spontaneous inbibition

Experimental Setup

Vacuum Regulator

Vacuum Pump

Vacuum Gauge

Sample Liquid

Rising Rate by Image Analysis

0 s 150 s 2 s 410 s 614 s 700 s65 s Imbibing was recorded by camera videoScale was referenced with a caliperAdvancing front line vs. time processed by ImageJ image analysis

γmJ/m2

ηmPa.s

ρg/cm3

Hexane 18.4 0.326 0.65Water 72.8 1 1

Methanol 22.5 0.54 0.79

Assuming full wetting, i.e. contact angle is zero. Rising rates: Water > Hexane > Methanol

Experimental: Hexane > Methanol > WaterSome energy is not used for rising in water and

methanol imbibitions

Energy loss due to

Contact angle, partial wetting (water)

Polar liquid swelling (methanol)

Heat of wetting, (water & methanol)

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0 100 200 300

Ris

ing

Hei

ght (

m)

Time (s)

Hexane Replicate 1

Hexane Replicate 2

Methanol, Experimental

Water, Experimental

Hexane, theta = 0

Methanol, theta = 0

Water, theta = 0

Observations

Reproducibility & Vacuum: Hexane

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0 20 40 60 80 100

Ris

ing

heig

ht (m

)

Time (s)

Replicate 1

Replicate 2

Replicate 3

Replicate 4

replicate 5

Vacuum 453 Pa

Vacuum 1050 Pa

Vacuum 4700 Pa

Vacuum 5800 Pa

Reproducibility is good for hexane imbibitionsRising rates increase with the vacuum

Reproducibility & Vacuum: Water

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0 100 200 300 400 500 600

Ris

ing

Hei

ght (

m)

Time (s)

Replicate 1Replicate 2Replicate 3Replicate 4Replicate 5Replicate 6Vacuum 2237 PaVacuum 2362 PaVacuum 2658 PaVacuum 2856 Pa

Reproducibility for water is not as good as hexane imbibitionsRising rates increase with the vacuum

Experimental Data: EG & Glycerol

00.010.020.030.040.050.060.070.080.09

0.1

0 100 200 300 400

Ris

ing

Hei

ght (

m)

Time (s)

Vacuum 2353 PaVacuum 2106 PaVacuum 2053 PaVacuum 2160 PaVacuum 2266 PaVacuum 2160 Pa

0

0.01

0.02

0.03

0.04

0.05

0.06

0 500 1000 1500R

isin

g H

eigh

t (m

)Time (s)

Vacuum 2,914 PaVacuum 26,319 PaVacuum 26,553 Pavacuum 23,496 PaVacuum 22,668 Pa

γmJ/m2

ηmPa.s

ρg/cm3

Hexane 18.4 0.326 0.65Ethylene glycol 48 16.1 1.113

Glycerol 64 1420 1.261

Ethylene glycol imbibes very slowly without external vacuum

Glycerol cannot imbibe spontaneously

Results and DiscussionDefine the effective capillary radius with hexane The effect of polar liquidsEnergy loss constantContact angle with waterVacuum induced slip

Effective Capillary Radius from Hexane

08cos22 =−−Δ+ gz

dtdzz

rP

rρηθγ

⎟⎟⎠

⎞⎜⎜⎝

⎛−

−= )(

)(ln8

2 tztzz

zzgr

te

eeρ

η

gP

grze ρρ

θγ Δ+=

cos2

Quasi state ma=0 No external vacuum, ΔP = 0 Full wetting, cos(θ) = 1 No swelling & release of heat of wetting

Effective Capillary Radius (r) R2

Replicate 1 1.41E-06 1.00Replicate 2 1.41E-06 1.00Replicate 3 1.56E-06 0.98Replicate 4 1.20E-06 1.00Replicate 5 1.10E-06 0.99

Average 1.34E-06COV (%) 13.80

Average effective Capillary Radius

mr μ 1034.1 6−×=

Effect of Polar Liquid

rGR

GRrmm

mvms ⋅

−+−

= )1(2

2

πρδρπ

S.Q. Shi and D.J. Gardner, A new model to determine contact angles on swelling polymer particles by the column wicking method, Journal of Adhesion Science and Technology, 14 (2000) 301-314.

r, average capillary radius (m)rs, average capillary

radius after material swelling (m)R, inner radius of the

column tube (m) ρm, material density

(g/cm3 )δv, volume shrinkage

after absorbing probe liquidGm, unit column mass of

the material (g/m)

Characteristics of Packing tubes

Name Water Methanol Ethylene glycol GlycerolVolume Shrinkage (%) 15.0 13.8 17.4 20.0

Inner d (mm) 3.77 3.84 3.78 3.83G0 (tube weight) (g) 5.87 4.06 4.05 4.07

G1 (g) 6.56 4.63 4.58 4.64G2 (g) wet weight 7.35 5.03 5.29 5.14

Packing Length, mm 161.4 126.9 127.5 129.3wetting Length,mm 91.4 69.0 72.7 40.0

density (g/cm3) 0.38 0.39 0.37 0.38Gm (g/m) 4.27 4.53 4.19 4.38

wet (g/g wood) 2.01 1.25 2.34 2.85Wet(g/cm) 0.09 0.06 0.10 0.12

r/rs 0.75 0.80 0.78 0.71

Derivation of Energy Loss ConstantQuasi-state ma = 0; External vacuum ΔP = 0 Deformable materials, r into rs

Energy loss is proportional to shrinkage and reverse proportional to r2 by CFitting with methanol imbibition data, i.e. cos(θ) =0

08cos2222 =−−−Δ+ gz

dtdzz

rrCP

rr

s

vs ρηπδθγ

⎟⎟⎠

⎞⎜⎜⎝

⎛−

−= )(

)(ln8

2 tztzz

zzgr

te

ee

s ρη

grc

gP

grrz vs

e ρπδ

ρρθγ

22

cos2−

Δ+=

C (J/m) R2

Rep. 1 5.59E-07 1.00Rep. 2 4.88E-07 1.00Rep. 3 5.57E-07 0.99Rep. 4 5.62E-07 0.98Rep.5 4.45E-07 0.98

Average 5.52E-07Cov 9.6%

mJC / 1052.5 7−×=

Average energy loss constant

Contact Angle with Water

Quasi-state ma = 0; External vacuum ΔP = 0 Deformable materials, r into rs

Energy loss is proportional to shrinkage and reverse proportional to r2 by CFitting with water imbibition data to calculate cos(θ)

08cos2222 =−−−Δ+ gz

dtdzz

rrCP

rr

s

vs ρηπδθγ

⎟⎟⎠

⎞⎜⎜⎝

⎛−

−= )(

)(ln8

2 tztzz

zzgr

te

ee

s ρη

grc

gP

grrz vs

e ρπδ

ρρθγ

22

cos2−

Δ+=

θ (°) R2

Rep. 1 63 0.99Rep. 2 57 0.99Rep. 3 65 0.97Rep. 4 48 0.93Rep. 5 53 0.84Rep. 6 64 0.94

Average 58COV (%) 12.6

mJC / 1052.5 7−×=

mr μ 1034.1 6−×=

75.0/ =rrs

The water contact angles calculated from the model (58°) is in agreement with the sessile drop results (60°) from the literature T. Nguyen and W. E. Johns, Wood Sci. Technol. 12, 63–74 (1978).

V. R. Gray, For. Prod. J. 452–461 (Sept. 1962).

Effect of Vacuum

Under vacuum, the rise of the liquid proceeds much faster than predicted even with con(θ) = 1, clearly

indicating a slip radius δ in the interface

Slip under Vacuum

capillary force: RF nsionSurfaceTes

θγ cos2=

viscous drag: ( ) ( )dt

tdztzR

Fviscous η2

8=

ESF-Exploratory Workshop Microfluidic: Rome, Sept. 28-30, 2007

Force without Slipr

Z(t)

Gravity: mgFGravity =

( ) ( )dt

tdztzR

Fviscous ηδ 2)(

8+

=

Effect of Slip under Vacuum

D.I. Dimitrov, A. Milchev, and K. Binder, Capillary rise in nanopores: Molecular dynamics evidence for the Lucas-Washburn equation, Physical Review Letters, 99 (2007).

Full Models

⎟⎟⎠

⎞⎜⎜⎝

⎛−

−+= )(

)(ln

)(8

2 tztzz

zzgr

te

ee

s ρδη

grc

gP

grrz vs

e ρπδ

ρρθγ

22

cos2−

Δ+=

mJC / 1052.5 7−×=

mr μ 1034.1 6−×=

75.0/ =rrs

0)(

8cos2222 =−

+−−Δ+ gz

dtdzz

rrCP

rr

s

vs ρηδπ

δθγ

SurfaceTension

Vacuum

EnergyLoss

Viscous Drag

gravitySlip Radius

Swelling

Slip Radius under Vacuum

y = 5E-10x + 2E-06R² = 0.898

0.0E+00

2.0E-06

4.0E-06

6.0E-06

8.0E-06

1.0E-05

1.2E-05

1.4E-05

1.6E-05

1.8E-05

0 5,000 10,000 15,000 20,000 25,000 30,000

Slip

Rad

ius (

m)

Vacuum (Pa)

HexaneMethanolWaterEthylene GlycolGlycerol

Assuming forced wetting under vacuum, cos(θ)=1Slip radius is roughly proportional to vacuumContact angle and slip radius cannot be decoupled except for figuring out slip

radius with alternative methods

ConclusionsRising rates of imbibitions can be measured precisely with an image acquisition and analysis systemThe effect of swelling and heat of wetting can be calibrated by hexane and methanolContact angles for other polar and partial wetting liquids can thus be measured reasonablyVacuum induced slip; the slip and partial wetting were coupling together such that contact angle could not be measured separately in this investigation. Further investigation is needed to correlate the extent of slip and vacuum.

Thank you for your attentions

Questions or Comments

?