Coiling rigid microfibers inside liquid droplets

Paul GrandgeorgeArnaud Antkowiak

Sébastien Neukirch

1

Sorbonne Université, UPMC Paris 06, CNRS, Institut Jean Le Rond d'Alembert, UMR 7190, Paris

Liquid wires

Nephila golden orb weaver

Biological inspiration

Interesting properties of its capture silk

2

Liquid wires?

1 mm

Nephila spider capture silk thread coated with small water droplets. Throughout the compression, the fiber does not sag,

it remains under tension. Video credit: Hervé Elettro

3

Nephila capture silkIts highly compressible property

Real TimeH. Elettro et al.

PNAS - 2016

50μm

4

During the compression, the thread spools inside the water droplets.

Nephila capture silkA closer look inside the droplets

5

Elastocapillary in-drop spoolingThe movie

Silicone oil droplet on a RTV (silicone polymer) fiber. The system is immersed in a water bath.

Fiber: radius a≈35 μm, Young's modulus E≈1 Mpa. Droplet : radius R=1.5 mm, Δ𝛾≈40 mN/m.

3 mmReal Time

6

VA = PL1�sv + PL2�sl + L21

2

EI

R2+ P�l�sv

VB = PL1�sv + PL2�sl + L21

2

EI

R2+ P�l�sl +�l

1

2

EI

R2

Elastocapillary in-drop spoolingThe energetic approach

Solid-vapor interface energy

Solid-vapor interface energy

Solid-liquid interface energy

Bendingenergy

Solid-liquid interface energy

Bendingenergy

7

Circular cross section fiber of

radius : P = 2⇡a

I =⇡a4

4

a

Typical values (spider silk):

If it is thin and elastic enough, a thread can coil inside a droplet!

R ⇠ 100µm

E ⇠ 10 MPa

⇠ 10µm

VA = PL1�sv + PL2�sl + L21

2

EI

R2+ P�l�sv

VB = PL1�sv + PL2�sl + L21

2

EI

R2+ P�l�sl +�l

1

2

EI

R2

�� ⇠ 50 mN/m

Elastocapillary in-drop spoolingThe energetic approach

T = ��V

�l=

✓P�� � 1

2

EI

R2

◆

�V = �l

✓1

2

EI

R2� P��

◆

T > 0 ?

2⇡a� � ⇡a4E

8R2> 0

a <

✓16�

E

◆1/3

R2/3

8

Elastocapillary in-drop spoolingThe pre-movie II

2 mm

Bear thermoplastic polyurethane microfiber with no droplet sitting on it. Fiber: radius a=3.3 μm, Young's modulus E=20 Mpa.

Real Time

9

Elastocapillary in-drop spooling

Silicone oil droplets on said thermoplastic polyurethane microfiber. An artificial ultra compressible/extensible device.

Fiber: radius a=3.3 μm, Young's modulus E=20 Mpa. Final droplet : radius R=106 μm, 𝛾=21 mN/m.

The movie II

2 mmReal Time

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The coiling fibers

Phase diagram of coiling and non-coiling fibers Fiber: Considered Young's modulus E = 70 MPa.

Droplet : 𝛾≈21 mN/m.

Phase diagram

2⇡a� � ⇡a4E

8R2> 0

Capillary Elastic- > 0

11

Elastocapillary in-drop spooling

Big silicone oil droplet on a thermoplastic polyurethane microfiber. Gravity kicks in.

Fiber: radius a=3.4 μm, Young's modulus E=20 Mpa. Droplet : radius R=430 μm, 𝛾=20 mN/m.

Gravity

2 mmReal Time

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Elastocapillary in-drop spoolingGravity

P�� � 1

2

EI

R2>

mg

2

2⇡a� � ⇡a4E

8R2>

4

3

⇡⇢gR3

2

P = 2⇡a

I =⇡a4

4� < ⇡/2

13

The coilable fibers

Phase diagram of coilable and non-coilable fibers Fiber: Considered Young's modulus E = 70 MPa.

Droplet : 𝛾=21 mN/m.

Phase diagram

2⇡a� � ⇡a4E

8R2>

4

3

⇡⇢gR3

2

2⇡a� � ⇡a4E

8R2> 0

14

The coiling fibers

Phase diagram of coiling and non-coiling fibers with gravity. Fiber: Considered Young's modulus E = 70 MPa.

Droplet : 𝛾=21 mN/m, 𝜌g = 9600 N/m3.

Phase diagram

2⇡a� � ⇡a4E

8R2>

4

3

⇡⇢gR3

2

amax

= c(⇢g, �) · E�3/7

Capillary Elastic- > 1/2 Weight

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The coiling fibersPhase diagram

Phase diagram of coilable and non-coilable fibers. Droplet : 𝛾=21 mN/m, 𝜌g = 9600 N/m3.

amax

= c(⇢g, �) · E�3/7

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Biggest coilable fibreGravity

P = 2⇡a

I =⇡a4

4

P�� � 1

2

EI

R2>

mg

2

2⇡a� � ⇡a4E

8R2>

4

3

⇡⇢gR3

2

� < ⇡/2

17

Biggest coilable fibreThe tutor

I =⇡a4

4

P = 2⇡a+ Psoft

' Psoft

Psoft

� � ⇡a4E/4 + Esoft

Isoft

2R2

>4

3

⇡⇢gR3

2

P�� � 1

2

EI

R2>

mg

2

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Biggest coilable fibreThe tutor

Phase diagram of coiling and non-coiling fibers with gravity. Rigid fiber: Considered Young's modulus E = 70 MPa.

Soft fiber : Young's modulus Esoft=200 kPa, width w = 180 μm, height h = 50 μm. Droplet : 𝛾=21 mN/m, 𝜌g = 9600 N/m3.

Psoft

� � ⇡a4E/4 + Esoft

Isoft

2R2

>4

3

⇡⇢gR3

2

amax

= C(⇢g, Psoft

�, Esoft

Isoft

) · E�1/4

19

The coiling fibersPhase diagram

Phase diagram of coilable and non-coilable fibers. Droplet : 𝛾=21 mN/m.

am

a

x =c · E �

3/7

amax

= C(⇢g, Psoft

�, Esoft

Isoft

) · E�1/4

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Coiling the uncoilableThe tutor

Phase diagram of coilable and non-coilable fibers with tutor. Soft fiber : Young's modulus Esoft=200 kPa, width w = 180μm, height h = 50μm.

Droplet : 𝛾=21 mN/m, 𝜌g = 9600 N/m3.

am

a

x = C · E �1/

4

amax

= C(⇢g, Psoft

�, Esoft

Isoft

) · E�1/4

am

a

x =c · E �

3/7

21

2 mm

Coiling the uncoilableUncoilable

Silicone oil droplet on a poly-actif acid (PLA) microfiber. Fiber: radius a=4.1 μm, Young's modulus E=2.7 GPa.

Droplet : radius R=256 μm, Δ𝛾≈20 mN/m.

Real Time

22

Coiling the uncoilable

Silicone oil droplet on a composite poly-actif acid (PLA) microfiber + polyvinyl siloxane (PVS) tutor fiber.

PLA fiber: radius a=4.1 μm, Young's modulus E=2.7 GPa. PVS fiber: width w=100 μm, height h = 35 μm, Young's modulus E=200 kPa,

Droplet: radius R=430 μm, Δ𝛾≈20 mN/m.

The coilabe composite fiber

2 mmReal Time

23200 μm

Coiling the uncoilableCoiled!

24

Thank you!

0.5 mmReal Time