Fluid dynamics of droplet impact on
porous membranes.
Molecular diagnostics
Biomicrofluidics 2 044101 (2008)
Frits Dijksman and Anke Pierik
Philips Research Europe Eindhoven
Workshop Contact Line Instabilities Lorentz Centre Leiden University
January 4-8, 2010
Trends in Molecular Diagnostics
• Multiple parameter testing on molecular level
(DNA, RNA, proteins, enzymes, hormones, cells,…)
• Point of Care Rapid Diagnostic Testing
– Sensitive
– Fast
– Integrated
– 24h/7d access
– Easy to use
– High multiplex grade
Use of microarrays
CONFIDENTIAL
Dijksman/Pierik, Workshop Lorentz Centre January 4-8 , 2010
2
Microarrays
•Microarray: substrate containing multiple spots (DNA-fragments,
proteins, …) that can each specifically capture a single target
Advantages of microarrays
•Highly parallel, especially important for complex diseases
 low price per result
•Small sample volumes can be used
•More sensitive due to miniaturization
•Shorter reaction times due to shorter diffusion distances
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Dijksman/Pierik, Workshop Lorentz Centre January 4-8 , 2010
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Use of microarrays in biomedical devices
Developing very sensitive
detection techniques for
biomedical devices
Microarray containing ink jet
printed capture probes
Sample
Integrated systems
Cartridge
Measurement
equipment
Sample preparation and
detection equipment
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Dijksman/Pierik, Workshop Lorentz Centre January 4-8 , 2010
Example of microarrays: DNA arrays
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Dijksman/Pierik, Workshop Lorentz Centre January 4-8 , 2010
Applications of microarrays
• Genotyping
– Identification: food, diagnostics
• Gene / protein expression
– Personalized prognosis, patient stratification
• Pharmacogenomics
– Personalized medicine
CONFIDENTIAL
Dijksman/Pierik, Workshop Lorentz Centre January 4-8 , 2010
Introduction on micro arrays
Microarray (5 by 6 mm)
with capture probe spots
Before
hybridisation
Today:
Dot size 150-200 m,
Permeation depth 100 m
Dot pitch 254-400 m
Number of dots: 100-400
CONFIDENTIAL
After
hybridisation
Near future:
Dot size 50-100 m,
Permeation depth 50 m
Dot pitch 100-200 m
Number of dots: > 1000
Dijksman/Pierik, Workshop Lorentz Centre January 4-8 , 2010
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Manufacturing process of micro arrays
3D distribution of probes depends on:
Nylon membranes, neutral or
positively charged
•Volume of droplet Vdroplet
•Total volume of dot Vtotal
Average pore size: 450 nm
Thickness of membranes: 150 m
•Velocity of droplet vdroplet
•Print frequency
•(Auto) diffusion of capture probes
•Surface properties membrane (charge)
and properties solution and oligo-(dye)
Cross-section Nytran membrane
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Flow
through
direction
Membrane with printed
spots containing
oligonucleotide probes
Unit operations dot formation / capture probes
Velocity print head
Nozzle
front
Droplet formation
Evaporation of free flying droplet
Slowing down due to air friction
Evaporation at surface
membrane
Spreading on surface of
membrane
Print head
Air
Surface of
substrate
Penetration into porous
structure of membrane
Diffusion of capture
probes towards inner
surface membrane
Substrate
Evaporation from inner
structure of membrane
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Dijksman/Pierik, Workshop Lorentz Centre January 4-8 , 2010
Timing of:
•Droplet formation
•Flight
•Evaporation
•Spreading
•Penetration
•Diffusion
•Drying
9
Issue: print heads
Single nozzle pipette
Droplet volume: 100-150 pl
Droplet speed: 1-3 m/s
Max frequency: 1 kHz
Volume: 25 l
Linear array 16 nozzle
print head.
Droplet volume: 10 pl
Droplet speed: 5-10 m/s
Max frequency: 4-6 kHz
Cartridge volume: 1.5 ml
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Dijksman/Pierik, Workshop Lorentz Centre January 4-8 , 2010
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Issue: printers
Single nozzle
printer:
large droplets
Total QC
A. Pierik et al, J. Biotechnology Vol 3, 12, 2008
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Dijksman/Pierik, Workshop Lorentz Centre January 4-8 , 2010
High speed
multi-fluid printer:
small droplets
11
Main issue
Wet dot size 500-1000 pl
Dimatix:
Droplet volume 10 pl
50-100 droplets, 4 kHz
Microdrop:
Droplet volume 120-150 pl
7-10 droplets, 100 Hz
Are dot dimensions dependent on droplet
size, jetting frequency and number?
Is the capture probe distribution
dependent on droplet size and number?
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Dijksman/Pierik, Workshop Lorentz Centre January 4-8 , 2010
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Dot spreading upon impact
Nozzle
front
Droplet formation
Evaporation of free flying droplet
Slowing down due to air friction
Evaporation at surface
membrane
Spreading on surface of
membrane
Print head
Air
Surface of
substrate
Penetration into porous
structure of membrane
Diffusion of capture
probes towards inner
surface membrane
Substrate
Evaporation from inner
structure of membrane
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Dijksman/Pierik, Workshop Lorentz Centre January 4-8 , 2010
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Results Christophe Le Clerc and Michel Bruyninckx
Impact of a water droplet with
R = 42.3 m (317 pl)
and velocity 5.1 m/s.
The delay between each droplet
is 3 sec.
C. Le Clerc and D.B. van Dam
Phys. Fluids 16, 3403 (2004)
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Dijksman/Pierik, Workshop Lorentz Centre January 4-8 , 2010
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Results Christophe Le Clerc and Michel Bruyninckx
Impact of a water droplet with
R = 33 m (150 pl)
and velocity 11.4 m/s.
The delay between each droplet
is 0.25 sec.
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Dijksman/Pierik, Workshop Lorentz Centre January 4-8 , 2010
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Impacting droplets in a pixel.
Movie made by Thijs van der Zanden Eindhoven University of Technology
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Dijksman/Pierik, Workshop Lorentz Centre January 4-8 , 2010
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Results Bayer and Magarides (2004)
Impact of a water droplet with
R = 0.69 mm (1.38 l)
and velocity  0.5 m/s.
Ilker S. Bayer and
Constantine M. Magarides at
XXI ICTAM 15-21 August 2004
Warsaw Poland
Wetting
surface
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Neutral
surface
Anti-wetting
surface
Dijksman/Pierik, Workshop Lorentz Centre January 4-8 , 2010
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Computational results Pasindideh et al 1996
Impact of a water
droplet with R = 1mm
and velocity = 1 m/s.
M. Pasandideh-Fard, Y.M. Qiao, S. Chandra and J.
Mostaghimi,
“Capillary effects during droplet impact on a solid surface:
Physics of Fluids 8 (3), March 1996, 650-659.
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Dijksman/Pierik, Workshop Lorentz Centre January 4-8 , 2010
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Static wetting
E
Dome
h
Surface of
substrate
Rd
a
Hütte
Mathematische Formeln
und Tafeln
I Szabo
Berlin 1959
1
1
h(3a 2  h 2 )  h 2 (3Rd  h)
6
3
O  2Rd h (this is the surface area of the curved part of the dome)
V 
cos  
Rd  3
CONFIDENTIAL
Rd -h
,
Rd
sin  
a
Rd
3V
 (2  3 cos   cos 3  )
Dijksman/Pierik, Workshop Lorentz Centre January 4-8 , 2010
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Definition of dynamic contact angle
Jiang, Oh and Slattery formula
U: velocity of moving contact line
Ca 
U

cos( E )  cos( D )
 tanh( 4.96 Ca 0.702 )
cos( E )  1
Terence D. Blake and Kenneth J. Ruschak,
“Wetting: static and dynamic contact lines”,
Liquid Film Coating edited by Kistler and Schweitzer,
Chapman & Hall 1997
CONFIDENTIAL
Dijksman/Pierik, Workshop Lorentz Centre January 4-8 , 2010
20
Dimensionless numbers
We =
2ρRv 2
Dimatix,
16 nozzle 10 pl
print head
γ
, Re =
η
η
, Oh =
, K =We Oh
2Rγρ
-2 / 5
Yarin no-splashing criterion on wet surface:
v < Vos = 18( /)1/4(/)1/8f 3/8
Volume Radius Velocity
[pl]
[m]
[m/s]
10
R = 13.4
6
143
R = 32.4
3
Microdrop single nozzle
143 pl print head
CONFIDENTIAL
2ρvR
We
Re
Oh
19
12
160
198
0.03
0.02
K <650
dry
81
59
Vos
Ca
wet U=3/4 v
6
0.09
1.5
0.045
A.L. Yarin: Annual Review Fluid Mechanics
38 (2006)
A.L. Yarin and D.A. Weiss, J. Fluid Mech. 1995
Dijksman/Pierik, Workshop Lorentz Centre January 4-8 , 2010
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Energy consideration to calculate max. spreading
We 
2 Rv 2

,
Re 
2 vR

fluid vapour surface 2Rd h
fs 

fluid solid surface
a 2
Loss surface
energy air solid
interface
1 4 3 2
 R v  4R 2  Rs2 ( f s   SL   SO )
2 3
Kinetic
energy
droplet
CONFIDENTIAL
Surface
energy
droplet
Surface
energy
dome dot
Dijksman/Pierik, Workshop Lorentz Centre January 4-8 , 2010
Gain surface
energy fluid
solid interface
22
Energy consideration to calculate max. spreading
With Young' s relation :
1 4 3 2
 R v   4R 2  Rs2 ( f s   cos  E )
2 3
Rs
We  12
Energy considerat ion :

(?????)
R
3( f s  cos  E )
Rs
We  12
Pasandideh - Fard et al model :

We
R
3(1  cos  D )  4
Re
26 < We < 642,
213 < Re < 35339
M. Pasandideh-Fard, Y.M. Qiao, S. Chandra and J. Mostaghimi,
“Capillary effects during droplet impact on a solid surface:
Physics of Fluids 8 (3), March 1996, 650-659.
CONFIDENTIAL
Dijksman/Pierik, Workshop Lorentz Centre January 4-8 , 2010
23
Pancake model
R

v
Rs

hs
SO
Situation before
impacting of droplet
CONFIDENTIAL
Substrate
SL
Dijksman/Pierik, Workshop Lorentz Centre January 4-8 , 2010
Situation after
impacting of droplet
24
Pancake model with dynamic contact angle
for U = ¾ v
Incoming kinetic energy  excess surface energy
Espreading   (-Rs2 cos D  Rs2  2Rs hs - 4R 2 )
1 4 3 2
Ekinetic   R v
2 3
Rs 3
Rs
E1
8
( ) 

0
2
R
 (1  cos  D )R R 3(1  cos  D )
Hütte
Mathematische
Formeln
und Tafeln
I Szabo
Berlin 1959
E1  Ekinetic   4R 2 (being the total energy of the droplet)
CONFIDENTIAL
Dijksman/Pierik, Workshop Lorentz Centre January 4-8 , 2010
25
Results of the pancake model with dynamic
contact angle compared to other models
Dynamic spreading of droplets impacting on a solid surface.
Droplet volume 150 pl, surface tension 50 mN/m, equilibrium
contact angle 40 degrees, density 1000 kg/m^3, viscosity 1 mPas.
Dymanic spreading Rs/R
14.0
12.0
10.0
Static
Energy consideration
Pasandideh-Fard
Dynamic
8.0
6.0
4.0
2.0
0.0
0
1
2
3
4
5
6
7
8
9
10
Droplet speed in m/s
CONFIDENTIAL
Dijksman/Pierik, Workshop Lorentz Centre January 4-8 , 2010
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Results of the pancake model with dynamic
contact angle compared to other models
Dynamic spreading of droplets impacting on a solid surface. Droplet volume 10
pl, surface tension 50 mN/m, equilibrium contact angle 40 degrees, density
1000 kg/m^3, viscosity 1 mPas.
Dymanic spreading Rs/R
9.0
8.0
7.0
6.0
Static
5.0
Energy consideration
4.0
Pasandideh-Fard
Dynamic
3.0
2.0
1.0
0.0
0
1
2
3
4
5
6
7
8
9
10
Droplet speed in m/s
CONFIDENTIAL
Dijksman/Pierik, Workshop Lorentz Centre January 4-8 , 2010
27
Timing of spreading
• In order to calculate the timing of the spreading event we consider the
following model:
M.J. de Ruijter,
Axis of
J. de Coninck and
symmetry
Velocity of
Pancake of
G. Oshanin,
contact line
Velocity
expanding dot
profile
Lamgmuir 15 (1999)
Substrate
Velocity of centre of gravity of
dot, pointing in negative direction
Generalised co-ordinates:
position and velocity of centre of gravity of expanding dot.
CONFIDENTIAL
Dijksman/Pierik, Workshop Lorentz Centre January 4-8 , 2010
28
Velocity field expressed in terms of generalised
co-ordinates and definition of latent energy or
Lagrangian L
1 24Z centre
2
vr 
r
(
z
 2hs z )
3
2 5hs
24Z centre 1 3
2
vz  
(
z

h
z
)
s
3
5hs
3
hs Rs
1
T     2rdrdz (vr2  vz2 ), U  excess surface energy
2
0 0
L  T U
CONFIDENTIAL
Dijksman/Pierik, Workshop Lorentz Centre January 4-8 , 2010
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Lagrange’s equation
• Joseph-Louis, comte de Lagrange (17361813).
• Opus magnus: Mécanique Analytique (1787)
d
L
L

0
dt Z centre Z centre
Results in an equation of motion.
 *2
Z
0
.
512

1
*3
Z* ( *3  4.827428)  0.768 *4 
[

(
1

cos

)

6
Z
]0
D
2 3
*2
Z
Z
 R Z
With initial conditions we solved this equation numerically.
CONFIDENTIAL
Dijksman/Pierik, Workshop Lorentz Centre January 4-8 , 2010
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Timing of spreading according to Lagrange
Time evolution of spreading radius and velocity and position of
centre of gravity of a spreading dot after impacting of a 150 pl dot.
Droplet velocity 3 m/s, surface tension 0.05 N/m, equilibrium
contact angle 40 degrees and viscosity 1 mPas.
3.00
0.80
Position of centre
of gravity
0.60
0.40
2.00
0.20
1.50
0.00
-0.20
1.00
-0.40
Velocity of centre
of gravity
0.50
-0.60
0.00
-0.80
0
First touch of
droplet with
substrate
CONFIDENTIAL
Dimensionless
position and velocity
of centre of gravity
Dimensionless
spreading number
Rs/R
2.50
Spreading radius
0.5
1
1.5
2
2.5
3
Dimensionless time vt/R
Dijksman/Pierik, Workshop Lorentz Centre January 4-8 , 2010
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Comparison with experimental results of Le Clerc
End of
dynamic
spreading
process
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Dijksman/Pierik, Workshop Lorentz Centre January 4-8 , 2010
Spreading
ratio
=Rs/R
32
Permeation model according to De Gennes
Hole
Washburn equation (1921)
Rpore
Unit cell
x
Hmembrane
L
CONFIDENTIAL
L
1  cos  E
x
R pore t ,
2 
2
x2
t
 cos  E R pore
Dijksman/Pierik, Workshop Lorentz Centre January 4-8 , 2010
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Diffusion of capture probes (1)
• Diffusion time given by:
td 
2
R pore
DAB
• We have three cases (tp : permeation time):
• t d  t p : capture probes flows along with the fluid, dot distribution
in membrane = capture probes distribution, evaporation can
influence capture probe distribution
• t d  t p : capture probes caught in upper part of the membrane,
dot distribution not equal capture probes distribution
•
t d  t p : capture probe distribution equal to fluid distribution,
evaporation does not change capture probe distribution
CONFIDENTIAL
Dijksman/Pierik, Workshop Lorentz Centre January 4-8 , 2010
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Diffusion of capture probes (2)
•
Diffusion coefficient literature:
•
R.B. Bird. W.E. Stewart and E.N. Lightfoot, “Transport Phenomena”, Second Edition, John Wiley
& Sons, 2002.
W. Eimer, J.R. Williamson, S.G. Boxer and R. Pecora, “Characterisation of the Overall and
Internal Dynamics of short Oligonucleotides by Depolarized Dynamic Light Scattering and NMR
Relaxation Measurements, Biochemistry 1990, 29, 799-811.
W. Eimer and R. Pecora, “Rotational and translational diffusion of short rodlike molecules in
solution: Oligonucleotides”, J. Chem. Phys. 94 (3), 1 febrauray 1991, 2324-2329.
R. Pecora, “DNA: A Model Compound for Solution Studies of Macromolecules”, Science, Vol.
251, 22 February 1991, 893-898.
D. Brune and S. Kim, “Predicting protein diffusion coefficients”, Proc. Natl. Acad. USA, Vol. 90, pp
3835-3839, May 1993.
G. F. Bonifacio, T. Brown, G.L. Conn and A,N. Lane, “Comparison of the Electrophoretic and
Hydrodynamic Properties of DNA and RNA Oligonucleotide Duplexes”, Biophysical Journal
Volume 73 September 1997 1532-1538.
Hongmei Jian, Alexander V. Vologodskii and Tamar Schlick, “A Combined Wormlike-Chain and
Bead Model for Dynamic Simulations of Long Linear DNA”, Journal of Computational Physics
136, 168-179 (1997).
Lenigk et al, "Plastic biochannel hybridization devices: a new concept for microfluidic DNA
arrays", Anal. Biochem. 311 (2002) 40-49
•
•
•
•
•
•
•
CONFIDENTIAL
Dijksman/Pierik, Workshop Lorentz Centre January 4-8 , 2010
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Diffusion of capture probes (3)
Translational diffusion coefficients of short and long
oligonucleotides as function of the number of base pairs
1.00E-09
1
10
100
1000
10000
Diffusion coefficient
Our case
1.00E-10
1.00E-11
y = 7E-10x-0.6421
R2 = 0.9938
1.00E-12
Number of base pairs
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Dijksman/Pierik, Workshop Lorentz Centre January 4-8 , 2010
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3 D dot shape 7 143 pl droplets
Droplet 7
Thickness
membrane
Penetration
depth per
droplet:
xpenetration
Droplet 1
Total penetration
depth fluid column in
membrane core: x0
CONFIDENTIAL
Wetting radius of
dry substrate
membrane: Rs
Dijksman/Pierik, Workshop Lorentz Centre January 4-8 , 2010
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3 D dot shape 100 10 pl droplets
Fluid on surface
Penetration depth
outer ring core: x2
Static wetting
radius of Vexcess
on membrane: a
Thickness
membrane
Droplet 3
Penetration depth
outer ring core: x1
Droplet 2
Droplet 1
Fluid on surface
Penetration depth
inner core: x0
Fluid distribution in membrane
during printing
Thickness
membrane
Fluid distribution in membrane
after printing
CONFIDENTIAL
Dijksman/Pierik, Workshop Lorentz Centre January 4-8 , 2010
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Timing results
CONFIDENTIAL
Droplet size
10 pl
143 pl
Spreading time of one droplet on dry substrate
7.17 s
31.9 s
Penetration time of one droplet on virgin
substrate
8 s
57.66 s
Penetration depth first droplet
6.2 m
16.6 m
Penetration time last droplet
Stays at surface
0.792 ms
Total process time for 1000 pl
100 droplets at 4 kHz
25 ms
(per droplet 0.25 ms)
7 droplets at 100 Hz
70 ms
(per droplet 10 ms)
Final spreading (diameter)
125 m
153 m
Final permeation depth
150 m
116 m
Total drying time
40 s
34 s
Diffusion time of capture probes in water
0.625 ms
0.625 ms
Dijksman/Pierik, Workshop Lorentz Centre January 4-8 , 2010
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Conclusions
• There is quite some difference between high frequency spitting of 10 pl
droplets and low frequency jetting of large droplets.
• High frequency jetting of small droplets leads to a more confined spot of
which the capture probes are more concentrated at the surface
compared to low frequency jetting of large droplets
• The pancake model gives good correspondence with experimental
results. Lagrange’s equation (and method) leads to a correct estimate of
the timing of the spreading event
• Timing of spreading is important in relation to other phenomena taking
place on and in the membrane
• Diffusion time of capture probes interfere with processes associated to
printing, such as spreading, permeation and jetting frequency
CONFIDENTIAL
Dijksman/Pierik, Workshop Lorentz Centre January 4-8 , 2010
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