1. Forced Fluid Imbibition
in a Powder-Packed Column
Jinwu Wang, Post Doctoral Associate
Sheldon Q. Shi, Assistant Professor
Department of Forest Products
Mississippi State University
2. Objectives
Develop 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
3. Explanation
When 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
4. Velocity Field
around the Moving Meniscus
Phys. Rev. Lett. (2007), Capillary Rise in Nanopores: Molecular Dynamics Evidence for the Lucas-Washburn Equation
5. Liquid Behaviors in Powders
A powder-packed column with radius R
air
Assume that a powder-packed
Liquid column consists of numerous capillary
tubes: a wicking-equivalent effective
capillary radius
The same governing equations as
those applied to a capillary tube
Capillary action
6. Free Body Diagram
r
Surface Tension
External vacuum
}Driving Forces
Poiseuille Viscous Force
Gravitation Force
Inertial Force
Z(t)
Dragging Forces
List of Variables:
volume = πr2z
g = gravity
r = radius of capillary tube
z = rising height, measured to the bottom of the meniscus, at time t ≥ 0
ρ = density of the surface of the liquid
γ = surface tension
θ = contact angle between the surface of the liquid and the wall of the tube
7. Explanation of the Forces
Surface Tension Force 2π r γ cos( θ )
Gravitational Force Fw = mg = ρπr 2 zg
Poiseuille Viscous Force Fdrag = 8πη z dz
dt
Vacuum Force πr2ΔP
Newton's Second Law of Motion
d (mv) d ⎛ 2 dz ⎞ ⎛ d 2 z ⎛ dz ⎞2 ⎞
∑ F=
dt dt ⎝ dt ⎠
2 ⎜
= ⎜πr zρ ⎟ = πr ρ z 2 + ⎜ ⎟ ⎟
⎜ dt ⎝ dt ⎠ ⎟
⎝ ⎠
8. Explanation of Differential Equation
Newton's Second Law of Motion:
Net Force = Surface Tension Force +Vacuum
- Poiseuitte Viscous Force - Gravitational Force
⎛ d 2 z ⎛ dz ⎞ 2 ⎞ dz
ρπr 2 ⎜ z 2 + ⎜ ⎟ ⎟ = 2πrγ cos(θ ) + πr 2 ΔP − 8πηz − ρgπr 2 z
⎜ dt ⎝ dt ⎠ ⎠ ⎟ dt
⎝
Dividing by πr2, the differential equation becomes:
⎛Zo = Z(0) =⎛0dz ⎞ 2 ⎞ 2
d 2z
ρ ⎜z +⎜ ⎟ ⎟ = γ cos(θ ) + ΔP − 8 ηz dz − ρgz
⎜ dt ⎝ dt ⎠ ⎟ r
2
⎝ ⎠ r 2 dt
Boundary Conditions:
z(0) = 0 and z’(z∞) = 0
9. The Effective Zone of Forces
The size of each zone
depends on the probe
Gravity Effective liquid properties and
Zone capillary structures
z Washburn Zone
z0 Inertial Force
2 8 dz
γ cosθ + ΔP − 2 ηz − ρgz = 0
r r dt
8η ⎛ ze ⎞ 2γ cos θ ΔP
t = 2 ⎜ z e ln
⎜ − z (t ) ⎟
⎟ ze = +
r ρg ⎝ z e − z (t ) ⎠ ρgr ρg
10. The Effect
of Capillary Radius on Wicking
Lucas-
Washburn equation:
1/ 2
⎛ γ r cos θ ⎞
z (t ) =⎜ ⎟
2
⎜ 2η ⎟ t
⎝ ⎠
Is valid when
Capillary diameter is small
At initial rising period
Viscous drag >> gravity force
Density is low, inertia is small
11. Column Wicking Diagram
Non-spontaneous inbibition
when the contact angle is larger than 900
by applying vacuum spontaneous inbibition
13. Rising Rate by Image Analysis
0s 2s 65 s 150 s 410 s 614 s 700 s
Imbibing was recorded by camera video
Scale was referenced with a caliper
Advancing front line vs. time processed by ImageJ image analysis
14. Observations
0.07
Hexane Replicate 1
0.06
Energy loss due to
Hexane Replicate 2
Rising Height (m)
0.05 Contact angle,
Methanol, Experimental partial wetting
0.04 (water)
Water, Experimental
0.03 Polar liquid
Hexane, theta = 0 swelling (methanol)
0.02
Methanol, theta = 0 Heat of wetting,
0.01
(water & methanol)
Water, theta = 0
0
0 100 200 300
Time (s)
γ η ρ
Assuming full wetting, i.e. contact angle is
mJ/m2 mPa.s g/cm3
zero. Rising rates: Water > Hexane > Methanol
Hexane 18.4 0.326 0.65 Experimental: Hexane > Methanol > Water
Water 72.8 1 1 Some energy is not used for rising in water and
Methanol 22.5 0.54 0.79 methanol imbibitions
15. Reproducibility & Vacuum: Hexane
0.07
Replicate 1
0.06
Replicate 2
0.05 Replicate 3
Replicate 4
0.04
Rising height (m)
replicate 5
0.03
Vacuum 453 Pa
0.02 Vacuum 1050 Pa
0.01 Vacuum 4700 Pa
Vacuum 5800 Pa
0
0 20 40 60 80 100
Time (s)
Reproducibility is good for hexane imbibitions
Rising rates increase with the vacuum
16. Reproducibility & Vacuum: Water
0.14
Replicate 1
0.12 Replicate 2
Replicate 3
0.1
Replicate 4
Rising Height (m)
0.08 Replicate 5
Replicate 6
0.06 Vacuum 2237 Pa
Vacuum 2362 Pa
0.04
Vacuum 2658 Pa
0.02 Vacuum 2856 Pa
0
0 100 200 300 400 500 600
Time (s)
Reproducibility for water is not as good as hexane imbibitions
Rising rates increase with the vacuum
17. Experimental Data: EG & Glycerol
0.1 0.06
0.09 Vacuum 2,914 Pa
0.08 0.05 Vacuum 26,319 Pa
Rising Height (m)
0.07 Vacuum 26,553 Pa
Rising Height (m)
0.04
0.06 vacuum 23,496 Pa
Vacuum 2353 Pa
0.05 0.03 Vacuum 22,668 Pa
Vacuum 2106 Pa
0.04
Vacuum 2053 Pa 0.02
0.03
0.02 Vacuum 2160 Pa
0.01 Vacuum 2266 Pa 0.01
0 Vacuum 2160 Pa
0
0 100 200 300 400
0 500 1000 1500
Time (s) Time (s)
γ η ρ
mJ/m2 mPa.s g/cm3 Ethylene glycol imbibes very slowly
Hexane 18.4 0.326 0.65 without external vacuum
Ethylene glycol 48 16.1 1.113 Glycerol cannot imbibe spontaneously
Glycerol 64 1420 1.261
18. Results and Discussion
Define the effective capillary radius with
hexane
The effect of polar liquids
Energy loss constant
Contact angle with water
Vacuum induced slip
19. Effective Capillary Radius from Hexane
2 8 dz
γ cosθ + ΔP − 2 ηz − ρgz = 0 Effective Capillary
Radius (r)
R2
r r dt Replicate 1 1.41E-06 1.00
⎛ ⎞ Replicate 2 1.41E-06 1.00
8η ze
t = ⎜ z e ln − z (t ) ⎟ Replicate 3 1.56E-06 0.98
r 2ρg ⎜
⎝ z e − z (t ) ⎟
⎠ Replicate 4 1.20E-06 1.00
Replicate 5 1.10E-06 0.99
2γ cos θ ΔP Average 1.34E-06
ze = + COV (%) 13.80
ρgr ρg
Quasi state ma=0 Average effective
No external vacuum, ΔP = 0 Capillary Radius
Full wetting, cos(θ) = 1
No swelling & release of heat of r = 1.34 × 10 −6
μm
wetting
20. Effect of Polar Liquid
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
liquid
π R 2 ρ m − (1 + δ v ) G m Gm, unit column mass of
rs = ⋅r the material (g/m)
πρ m R − G m
2
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.
22. Derivation of Energy Loss Constant
Quasi-state ma = 0; External vacuum ΔP = 0
C (J/m) R2
Deformable materials, r into rs
Rep. 1 5.59E-07 1.00
Energy loss is proportional to shrinkage and reverse
Rep. 2 4.88E-07 1.00
proportional to r2 by C
Rep. 3 5.57E-07 0.99
Fitting with methanol imbibition data, i.e. cos(θ) =0
Rep. 4 5.62E-07 0.98
2rs Cδ 8 dz Rep.5 4.45E-07 0.98
γ cos θ + ΔP − 2v − 2 ηz − ρgz = 0 Average 5.52E-07
r2 πr rs dt
Cov 9.6%
8η ⎛ ze ⎞
t = 2 ⎜ z e ln
⎜ − z (t ) ⎟
⎟
rs ρ g ⎝ z e − z (t ) ⎠
Average energy loss
2rsγ cos θ ΔP cδ constant
ze = + − 2v
ρgr 2 ρg πr ρg
−7
C = 5.52 × 10 J /m
23. Contact Angle with Water
2rs Cδ v 8 dz r = 1 . 34 × 10 − 6 μ m
γ cos θ + ΔP − 2 − 2 ηz − ρgz = 0
r 2
πr rs dt
C = 5 . 52 × 10 − 7 J / m
8η⎛ ze ⎞
t = 2 ⎜ z e ln − z (t ) ⎟ rs / r = 0 . 75
⎜
rs ρ g ⎝ z e − z (t ) ⎟
⎠ θ (°) R2
2rsγ cos θ ΔP cδ
ze = + − 2v Rep. 1 63 0.99
ρgr 2 ρg πr ρg
Rep. 2 57 0.99
Quasi-state ma = 0; External vacuum ΔP = 0 Rep. 3 65 0.97
Deformable materials, r into rs Rep. 4 48 0.93
Energy loss is proportional to shrinkage and reverse Rep. 5 53 0.84
proportional to r2 by C Rep. 6 64 0.94
Fitting with water imbibition data to calculate Average 58
cos(θ) COV (%) 12.6
The water contact angles calculated from the model (58°) is in agreement with the sessile drop
T. Nguyen and W. E. Johns, Wood Sci. Technol. 12, 63–74 (1978).
results (60°) from the literature V. R. Gray, For. Prod. J. 452–461 (Sept. 1962).
24. 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
26. r Force without Slip
2γ cosθ
capillary force: FSurfaceTesnsion =
R
Gravity: FGravity = mg
dz (t )
Z(t)
ηz (t )
8
viscous drag: Fviscous =
R2 dt
ESF-Exploratory Workshop Microfluidic: Rome, Sept. 28-30, 2007
27. Effect of Slip under Vacuum
dz (t )
ηz (t )
8
Fviscous =
(R + δ ) 2
dt
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).
28. Full Models
Vacuum Viscous Drag
Swelling
2 rs Cδ v 8 dz
γ cos θ + Δ P − − ηz − ρ gz = 0
r 2
πr 2
( rs + δ ) 2
dt
Surface Energy Slip gravity
Tension Loss Radius
8η ⎛ ze ⎞
t = ⎜ z e ln
⎜ − z (t ) ⎟
⎟ r = 1.34 × 10 −6 μm
( rs + δ ) ρ g ⎝
2
z e − z (t ) ⎠
C = 5.52×10−7 J / m
2rsγ cos θ ΔP cδ v
ze = + − 2 rs / r = 0 . 75
ρgr 2
ρg πr ρg
29. Slip Radius under Vacuum
1.8E-05
1.6E-05
y = 5E-10x + 2E-06
1.4E-05 R² = 0.898
1.2E-05
Slip Radius (m)
1.0E-05
8.0E-06
6.0E-06 Hexane
Methanol
4.0E-06 Water
2.0E-06 Ethylene Glycol
Glycerol
0.0E+00
0 5,000 10,000 15,000 20,000 25,000 30,000
Vacuum (Pa)
Assuming forced wetting under vacuum, cos(θ)=1
Slip radius is roughly proportional to vacuum
Contact angle and slip radius cannot be decoupled except for figuring out slip
radius with alternative methods
30. Conclusions
Rising rates of imbibitions can be measured precisely
with an image acquisition and analysis system
The effect of swelling and heat of wetting can be
calibrated by hexane and methanol
Contact angles for other polar and partial wetting liquids
can thus be measured reasonably
Vacuum 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.
31. Thank you for your attentions
Questions or Comments
?
