static and dynamic instabilities in a drop-rod system · a. antkowiak b. audoly c. josserand s....
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A. AntkowiakB. AudolyC. JosserandS. NeukirchM. Rivetti
Static and dynamic instabilities in a drop-rod system
CNRSUniv. P. & M. Curie (Paris 6)
Paris - France1
PhD work
1 - Motivations
2
2 - Experiments
3 - Model
Nano-structures fabrication
www.memsnet.org
Opticallithography
planar only3
Syms et al, Journal of MEMS (2003)
Nano-structures fabrication
surfacemicro-machining
expensivelimited shapes
R. Syms, Journal of MEMS (1995)
rotated hinged joints
re-solidification
melting of solder (tin)
Nano-structures fabrication
5
Micro-Origamis
Dahlmann, Electron Lett. (2000)
3D electrical components (here an inductor)assembled by surface tension
inductor has to beaway from (metallic) substrate to prevent magnetic field loss
6
Micro-Origamis
Linderman et al, Sens. Actuators (2002)
microfan with polysilicon180 rpmmicro-fluidic systems
folding by surface tension of Pb:Sn solder spheres
7
Py et al, Phys. Rev. Lett. (2007)
Micro-Origamis
8
evaporationwater dropon elastic surface
Guo et al, PNAS (2009)
Micro-Origamis
3D photovoltaic devices
9
microsolaroven
1 - Motivations
10
2 - Experiments
3 - Model
Antkowiak et al, PNAS (2011)
Instant Origami
11
L = 7 mmR = 1.5 mm
impacting speed U = 0.53 m/s
targetdrop
encapsulation in 20 ms
total video length 133 ms
Antkowiak et al, PNAS (2011)
Instant Origami
12
L = 7 mmR = 1.5 mm
impacting speed U = 0.53 m/s
targetdrop
Antkowiak et al, PNAS (2011)
Instant Origami: final shape selection
13
L = 10 mmR = 1.5 mm
Ua = 0.68 m/s
same targetsame drop
Ua = 0.92 m/s
Antkowiak et al, PNAS (2011)
Instant Origami: final shape selection
14
L = 10 mmR = 1.5 mm
different impacting speedsUa = 0.68 m/sUb = 0.92 m/s
same targetsame drop
Instant Origami: 1-D setup
We =
�ρRU2
γWeber number
kinetic energysurface energy
�eg =
�EI
µg
�1/3gravito-elastic length
~ 3.6 mm
bending rigidityweight per length
15
R ∼ 1.5mm
encapsulation
Instant Origami: 1-D setup
17
no encapsulation
Instant Origami: 1-D setup
18
encapsulation
Instant Origami: 1-D setup
19
no encapsulation
Instant Origami: 1-D setup
Instant Origami: Exp. results
1 - Motivations
21
2 - Experiments
3 - Model
Variational approach
22
Eel =1
2
� L
0EI κ2(s) ds
Eγ
Eg
surface energy
gravitational energy
+ constraints
bending energy
Elasto-capillary length
23
(fully wetting conditions)
Lec =
�EI
γw
2R>L ec
2R < Lec
rigid cylinder(depth w)
Elasto-capillary wrapping
24
α α
Eel + Eγ
π
α
Eel + Eγ
π
2R < Lec
2R > Lec
E = Eel + Eγ
= 4γwRα
�L2ec
(2R)2− 1
�
Weight as energy barrier
25
α α
Eel + Eγ
π
2R > Lec
απ
Eel + Eγ + Eg
g
g = 0
g > 0
Discrete `two elements’ model
26
g
!" #"
$
!$%
s ∈ (−D;D) circular arcs ∈ (D;L) straight (heavy) beam
drop : no weight (circular interface)
(depth w)
Contact line pinning
27
θ < θ∗ θ = θ∗
θ θD D
L L
pinning: D fixed advancing: D increases
θ∗θ∗
Bending energy
28
g
!" #"
$
!$%
Eel =1
2
� D
−DEI κ2(s) ds = EI κ2 D
Gravitational energy
29
g
!" #"
$
!$%
Egρgwh
=1
κ
�2D − sin(2κD)
κ
�+1− cos(2κD)
κ(L−D)+
1
2(L−D)2 sin(2κD)
Surface energy
30
Eγ = 0.87
��
∂Vγ dA+ 2γslDw + γsv(L+ L� − 2D)w
-D 0D
L
-L’
Surface energy
31
Eγ = 0.87
��
∂Vγ dA+ 2γslDw + γsv(L+ L� − 2D)w
-D 0D
L
-L’
Eγ = 0.87 γ (2αRw + 2Ac)− 2γDw cos θY + γsv(L+ L�)w
cos θY =γsv − γsl
γ
with
Ac
R
(depth w)
Surface correction factor
32
×0.87
Surface Evolver
cylinder + caps
Constraints
(1/κ) sinβ = R sinα
κD = β
V0
w=
1
κ2
�β − sin 2β
2
�+R2
�α− sin 2α
2
�= Ac
kinematics
conservedvolume
Eel + Eg + Eγ = E(α,β, R,κ, D) 5 variables
contact line
pinning
D = D∗
θ = θ∗either
or 4 constraints
Eel + Eg + Eγ = E(α,β, R,κ, D)
Results
34
13
12
0
U
maximal spreading
−D∗
t = 0 Ekin
total mechanical energy
-D 0D
L
-L’
D∗(U)
for given:- total length L=La- impact speed U
Results
35
!6
!3
!2
"
12
13
13.5
13.40
0
same impact speed Ulonger length: Lb > La
final stateis stillencapsulated
Results
36final state is open
!6
!3
!2
"
12
1313.5
13.40
0
same impact speed Ulonger length: Lc > Lb > La
13
12
0
Results
37
final state is flat
same impact speed Ulonger length: Ld > Lc > Lb > La
Results
38
!" #" $" %"
#&'
$&'
%&'
'&'
(a) (b)(c)
(d)θY = 110◦
θ∗ = 150◦
Lec = 0.55 mm
Leg = 3.6 mm
withexperimentallymeasured: