static and dynamic instabilities in a drop-rod system · a. antkowiak b. audoly c. josserand s....

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

http://www.memsnet.org

http://www.memsnet.org

Syms et al, Journal of MEMS (2003)


surfacemicro-machining

expensivelimited shapes

R. Syms, Journal of MEMS (1995)

rotated hinged joints

re-solidification

melting of solder (tin)


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


Instant Origami

12

L = 7 mmR = 1.5 mm

impacting speed U = 0.53 m/s

targetdrop


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


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


17

no encapsulation


18

encapsulation


19

no encapsulation


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:

static and dynamic instabilities in a drop-rod system · a. antkowiak b. audoly c. josserand s....

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