Compression of microspheres: Theory and application

Compression of microspheres:
Theory and application
2 µm
Jonas Paul, Marc Ziener, Stefan Romeis and Wolfgang Peukert
University of Erlangen-Nuremberg, Institute of Particle Technology
wolfgang.peukert@lfg.fau.de

Experimental Setup
Custom built SEM supported micromanipulator :
(J. Paul et al., Rev. Sci. Instruments 2012)
Characteristics:
• Tunable spring constant
• Precise determination of spring
constant
• Force resolution < 50nN
• Deformation resolution < 5nm
• Lateral scan field 80µm x 80µm
Technical data:
LVDT
Electronic noise: 250nV pp @10kHz
Sensitivity: 1.8·10 -7 V/nm
Displacement resolution: ∆z = 1,4nm pp @10kHz
J. Paul PiKo-Workshop Siegen 2012
Force resolution: ∆F = ∆z·k > 50nN pp
(35N/m < k < 15000N/m) @10kHz
Strain gauge resolution: 5nm pp @10kHz
Loadingrate: 50 – 5000µN/s
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Obtained Data
Representative force-deformation plot of a silica sphere (x = 500nm):
Information on:
• Young’s modulus
• Elastic range
• Yield strength
• Hardness
• Crack initialization
• Elastic energy
• Dissipated energy
• Elastic-plastic loading index
Force [µN]
1200
1000
800
600
400
200
Relative deformation
0.00 0.18 0.37 0.55 0.73
SiO 2
(x=500nm)
elastic-plastic
loading
elastic loading
0
0 100 200 300 400
Deformation [nm]
elastic
unloading
J. Paul PiKo-Workshop Siegen 2012
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Young’s modulus
Hertzian contact theory: 4 3 ∗
/
F
: loading force
∗ : reduced Young’s modulus
: particle radius
: contact deformation
diamond
∗
1 ²
1 ²
R
measured variable:
silicon
single contact deformation and unknown
J. Paul PiKo-Workshop Siegen 2012
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Young’s modulus
Determination of single contact deformations and :
∗
silicon-silica contact:
∗
diamond-silica contact:
/
∗
1
/
∗
4 3 1
∗
1
∗
/ 1 /
/
Example: Stöber silica (R = 250nm)
∗
Q = 0.500
∗
Q = 0.530
27.7 1.9
25.9 1.8
J. Paul PiKo-Workshop Siegen 2012
Q = δ 1
/δ t
0.8
0.7
0.6
0.5
PS
fused
silica
diamond
0.1 1 10 100 1000
E * particle [GPa] Si (100)
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Young’s modulus
Comparison of different contact theories:
Hertz , , , ∗
0.1 ∗ 27.7 1.9 non-adhesive
Tatara , , , ∗ ,
0.6 ∗ 22.7 1.7 non-adhesive
JKR , , , ∗ ,
0.1 ∗ 27.5 1.9 adhesive
DMT , , , ∗ ,
0.1 ∗ 27.4 1.9 adhesive
Hertz/JKR/DMT: similar ∗
Weak adhesive forces compared
to applied load
Tatara: decreased ∗ compared to Hertz
/ 0.1
J. Paul PiKo-Workshop Siegen 2012
Force [µN]
800
600
400
200
Rel. deformation
0.00 0.21 0.42 0.63 0.84
Hertz
Tatara
JKR
DMT
SiO 2
(R = 250nm)
0
0 100 200 300 400
Deformation [nm]
6

Yield strength
Yield criteria:
max , ,
1/2 ² ² ²
stress/ρ 0
1.0
0.8
0.6
0.4
ν = 0.17
σ z /ρ 0
σ r/θ /ρ 0
τ/ρ 0
τ max
( : principal stress; : maximum shear stress)
0.2
| / |
Hertzian theory:
• Calculation of principle stresses
• Hertzian pressure: ∗ /
Example: Stöber silica (R = 250nm)
2∙0.34 3.9
J. Paul PiKo-Workshop Siegen 2012
Force [µN]
0.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0
800
600
400
200
z/a
Rel. deformation
0.00 0.21 0.42 0.63 0.84
SiO 2
(R = 250nm)
yield point
Hertz
0
0 100 200 300 400
Deformation [nm]
7

Hardness
Abbott & Firestone:
Contact area & 2
Hardness / &
& 2.8 0.1
• Simplification of contact mechanics
• No volume conservation
• Discontinuities at yield point
• Fixed relation of yield strength and hardness
• Conservative
CEB:
Contact area 2
Hardness /
( 0.454 0.41)
5.3 0.2
• Volume conservation
• Small plastic deformation
• Discontinuities at yield point
• Fixed relation of yield strength and hardness
in fully plastic regime
Force [µN]
J. Paul PiKo-Workshop Siegen 2012
Count
800
600
400
200
20
16
12
0.00 0.21 0.42 0.63 0.84
0
0 100 200 300 400
8
4
SiO 2
(R = 250nm)
yield point
Rel. deformation
Hertz
A&F/CEB
Deformation [nm]
Abbott&Firestone
SiO 2
(R = 250nm)
0
2 3 4 5 6
Hardness [GPa]
CEB
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Unloading
Young’s modulus:
Fully elastic unloading process
Unknown particle shape
Unknown correlation to ∗
Residual stresses
L.-Y. Li et al. (2002):
4 3 ∗ /
4 3
∗
/
Uniform pressure distribution:
̅
16
3 ∗ 16
3² ∗
Average true stress/strain:
̅
/
Red. E-Modul (GPa)
70
60
50
40
30
20
Sphere Cap Cylinder
10
0.0 0.1 0.2 0.3 0.4
Rel. deformation
Hertz
Cyl-fit (uniform pressure) Cap-fit (uniform pressur
Cyl-fit (true stress/strain) Li-fit
J. Paul PiKo-Workshop Siegen 2012
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Calculation of Energy
Calculation of:
• Expended energy: d
• Elastic energy:
• Elastic plastic loading index:
1
Energy / volume [mJ/µm³]
6 SiO 2
:
5
4
3
2
1
0
x = 500nm
x = 1000nm
0.2 0.4 0.6 0.8 1.0
Max. relative deformation
Rel. force
1.0 Polystyrene (EPL = 0.67)
Silica (EPL = 0.87)
0.8
0.6
0.4
0.2
0.0
0.0 0.2 0.4 0.6
Rel. deformation
J. Paul PiKo-Workshop Siegen 2012
Elastic-plastic index
1.0
0.8
0.6
SiO 2
:
x = 500nm
x = 1000nm
0.4
0.2 0.4 0.6 0.8 1.0
Max. relative deformation
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Thank you for your attention!
For financial support:
DFG (SPP 1486/2)
For excellent cooperation:
All colleagues at LFG Erlangen
Anfatec AG for continuous support
J. Paul PiKo-Workshop Siegen 2012
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