completion talk 02

Effect of surface stress change on the stiffness ofcantilever plates

Michael LachutSupervisor: Prof. John E. Sader

Department of Mathematics and StatisticsUniversity of Melbourne

June 25, 2013

Michael Lachut Supervisor: Prof. John E. Sader (Department of Mathematics and Statistics University of Melbourne)Effect of surface stress change on the stiffness of cantilever platesJune 25, 2013 1 / 27

Micromechanical devices

a b

Human Hair


Resonant frequency

Consider the resonant frequency of a resonator

ω =1

2π

√

k

m

Spring constant k assumed constant

The resonators mass m changes due to molecular absorption


Resonant frequency

Consider the resonant frequency of a resonator

ω =1

2π

√

k

m

Spring constant k assumed constant

The resonators mass m changes due to molecular absorption

BUT!!

Miniaturization to the micro- and nanoscale enhances surface effects

Particulary, surface stress shown to affect cantilever stiffness

Mass measurements potentially ambiguous


Axial-force model

1x

2x3x

−

sσ

+sσ


Lagowski model

ω = ω0

[

1− γ(

L2

h3

)

σ]1/2

, where γ = (1− ν2)/E

Lagowski et. al, Appl. Phys. Lett. 26 (1975)


Invalidity of the axial-force model

σ δ(x + h/2)s

−

3

3x

σ (b)h1x

σ δ(x − h/2)s

+

3

Gurtin et. al, Appl. Phys. Lett. 29 (1976)

T = b

∫ h/2

−h/2σ(b) + σ(x3) dx3= 0

where σ(x3) = σ+s δ(x3 − h/2) + σ−s δ(x3 + h/2)


Invalidity of the axial-force model

σ δ(x + h/2)s

−

3

3x

σ (b)h1x

σ δ(x − h/2)s

+

3

Gurtin et. al, Appl. Phys. Lett. 29 (1976)

T = b

∫ h/2

−h/2σ(b) + σ(x3) dx3 = 0

where σ(x3) = σ+s δ(x3 − h/2) + σ−s δ(x3 + h/2)


Recent references still using the axial-force model

G. Y. Chen et al., J. Appl. Phys., 77, 3618, (1995)

Y. Zhang et al., J. Appl. Phys. D, 37, 2140, (2004)

A. W. McFarland et al., Appl. Phys. Lett., 87, 053505, (2005)

J. Dorignac et al., Phys. Rev. Lett., 97, 186105, (2006)

K. S. Hwang et al., Phys. Rev. Lett., 89, 173905, (2006)

G. F. Wang et al., Appl. Phys. Lett., 90, 231904, (2007)

S. Zaitsev et al., Sens. Actu. A, 179, 237, (2012)

Y. Zhang, Sens. Actu. A, 194, 169, (2013)


Surface stress load on an unrestrained plate

FREE PLATE

Before Surface Stress Load



FREE PLATE

σsT

After Surface Stress Load



FREE PLATE

σsT

u1(x1, x2) = −

(1−ν)σTs

Eh x1

u2(x1, x2) = −

(1−ν)σTs

Eh x2


Problem decomposition

Decompose into two Sub-Problems

T

sσ

T

sσ

u = σ x22

Free plate has no stiffness effect

Stiffness of original problem given by that of the clamp loadedproblem


Scaling analysis of clamp loaded problem

σ N = 0 ij

x < o(b)1

N = 0 ij

x > o(b)1



σ N = 0 ij

x < o(b)1

N = 0 ij

x > o(b)1

Consider the thin plate equation

D0∂2

∂xi ∂xi

(

∂2w

∂xj ∂xj

)

−Nij∂2w

∂xi ∂xj= q



σ N = 0 ij

x < o(b)1

N = 0 ij

x > o(b)1


D0∂2

∂xi ∂xi

(

∂2w

∂xj ∂xj

)

−Nij∂2w

∂xi ∂xj= q

<<If σ << O(h / b)2



σ N = 0 ij

x < o(b)1

N = 0 ij

x > o(b)1


D0∂2

∂xi ∂xi

(

∂2w

∂xj ∂xj

)

−Nij∂2w

∂xi ∂xj= q

>>If σ << O(h / b)2



σ N = 0 ij

x < o(b)1

N = 0 ij

x > o(b)1


D0∂2

∂xi ∂xi

(

∂2w

∂xj ∂xj

)

−Nij∂2w

∂xi ∂xj= q

Relative change in effective rigidity (for x1 < O[b])

Deff

D0− 1 ∼ O

(

σ̄

(

b

h

)2)



σ N = 0 ij

x < o(b)1

N = 0 ij

x > o(b)1


D0∂2

∂xi ∂xi

(

∂2w

∂xj ∂xj

)

−Nij∂2w

∂xi ∂xj= q

Relative change in effective rigidity (for x1 < o[b])

Deff

D0− 1 ∼ O

(

σ̄

(

b

L

)(

b

h

)2)



σ N = 0 ij

x < o(b)1

N = 0 ij

x > o(b)1


D0∂2

∂xi ∂xi

(

∂2w

∂xj ∂xj

)

−Nij∂2w

∂xi ∂xj= q

Relative change in effective rigidity (for x1 < o[b])

Deff

D0− 1 ∼ O

(

σ̄

(

b

L

)(

b

h

)2)

Relative frequency shift

∆ω

ω0= φω(ν)σ̄

(

b

L

)(

b

h

)2


Results for ∆ω/ω0 when L/b = 25/3

−2 −1 0 1 2

−0.004

−0.002

0

0.002

0.004

Increasing Poisson's Ratio ν

0

∆ωω

(b / h)σ2

L / b = 25/3


Final result for the relative change in effective stiffness

Relative change in resonant frequency

∆ω

ω0= −0.042ν σ̄

(

b

L

)(

b

h

)2


Final result for the relative change in effective stiffness

Relative change in resonant frequency

∆ω

ω0= −0.042ν σ̄

(

b

L

)(

b

h

)2

Relative change in Effective Spring Constant

∆keffk0

= −0.063ν σ̄

(

b

L

)(

b

h

)2


Problem decomposition for V-shaped cantilevers

Decompose into two Sub-Problems

FREE

CLAMP LOADED

ORIGINAL

σsT

σsT

u=

σx 2

2

c

d

d

L


V-shape cantilever stiffness vs. Rectangular cantilever

stiffness

V-shape

Decreasing Poisson’s Ratio

Rectangle

Increasing Poisson’s Ratio

0.002

0.001

0

No

rma

lize

d F

req

ue

ncy

Sh

ift

0.0015

0.0005

−0.0005

0.05 0.1 0.15 0.2 0.25 0.3

d / c


Physical Mechanisms

(a) (b)


Practical implications

P. Li, Z. You and T.Cui, Appl. Phys. Lett. 101, 093111, (2012)

Cantilever σTs

Device −1

Si3 N4

V-shape

Silicon Nitride Devices

97, 0.8

(µm)

b,h)(L,

499,

+−

+− +−

∆ω / ω0

(Nm )

0.01

2.1x

Dimensions

180,18,0.8

c,d,h)(L,

180, +−

Rectangular 60

)(

Karabalin et. al, Phys. Rev. Lett. 108, (2012)

1 10−3

Stress Effect: ∆ω / ω = − 0.042νσ(b / L)(b / h) b / h >> 102

Geometric Effect: ∆ω / ω = (1+2ν) / (1−ν)σ b / h ~ 3



Cantilever σTs

Device −1

Si3 N4

V-shape

Silicon Nitride Devices

97, 0.8

(µm)

b,h)(L,

499,

+−

+− +−

∆ω / ω0

(Nm )

0.01

2.1x

Dimensions

180,18,0.8

c,d,h)(L,

180, +−

Rectangular 60

)(


1 10−3

Stress Effect: ∆ω / ω = − 0.042νσ(b / L)(b / h) b / h >> 102

Geometric Effect: ∆ω / ω = (1+2ν) / (1−ν)σ b / h ~ 3

Observed frequency shifts remain unexplained



Introduction to buckling

Column



Lb

What load will buckle a

cantilever plate?

T

sσ


Scaling analysis for the critical strain load

Consider again the thin plate equation

D0∂2

∂xi ∂xi

(

∂2w

∂xj ∂xj

)

−Nij∂2w

∂xi ∂xj= q




D0∂2

∂xi ∂xi

(

∂2w

∂xj ∂xj

)

−Nij∂2w

∂xi ∂xj= q

σ

x < o(b)1 x > o(b)1

Balance

x < o(b)1




D0∂2

∂xi ∂xi

(

∂2w

∂xj ∂xj

)

−Nij∂2w

∂xi ∂xj= q

σ

x < o(b)1 x > o(b)1

Balance

x < o(b)1 x > o(b)1




D0∂2

∂xi ∂xi

(

∂2w

∂xj ∂xj

)

−Nij∂2w

∂xi ∂xj= q

σ

x < o(b)1 x > o(b)1

Balance

x < o(b)1 x > o(b)1

N = 0 ijk 0




D0∂2

∂xi ∂xi

(

∂2w

∂xj ∂xj

)

−Nij∂2w

∂xi ∂xj= q

σ N = 0 ij

x < o(b)1 x > o(b)1

Balance

D 0

x < o(b)1 x > o(b)1

N = 0 ijk 0




D0∂2

∂xi ∂xi

(

∂2w

∂xj ∂xj

)

−Nij∂2w

∂xi ∂xj= q

σ N = 0 ij

x < o(b)1 x > o(b)1

Balance

D 0

x < o(b)1

k 0




D0∂2

∂xi ∂xi

(

∂2w

∂xj ∂xj

)

−Nij∂2w

∂xi ∂xj= q

σ N = 0 ij

x < o(b)1 x > o(b)1

Balance

D 0

x < o(b)1

k 0

Critical strain load scales by

σ̄cr ∼ O

(

[

h

b

]2)




D0∂2

∂xi ∂xi

(

∂2w

∂xj ∂xj

)

−Nij∂2w

∂xi ∂xj= q

σ N = 0 ij

x < o(b)1 x > o(b)1

Balance

D 0

x < o(b)1

k 0

Critical strain load scales by

σ̄cr ∼ O

(

[

h

b

]2)

General form of the critical strain load

σ̄cr = ψ(ν)

(

h

b

)2


Independence of aspect ratio (L/b)

b / h = 480

−0.4

−0.2

−0.6

−0.8

−1.0−2 2−1 10

σ (x 10 ) −2

Increasing L / b

kk0

−1

L / b = 2, 4, 8, 16


Dependence on thickness-to-width ratio (b/h)

0 0.01 0.02 0.03 0.04 0.05−60

−40

−20

0

20

40

60

80

Increasing νσ

cr(b / h)2

h / b

ν = 0, 0.25, 0.49


Expressions for the positive and negative critical strain

loads

Positive strain load

σ̄(+)cr = 63.91

(

1− 0.92ν + 0.63ν2)

(

h

b

)2


Expressions for the positive and negative critical strain

loads


σ̄(+)cr = 63.91

(

1− 0.92ν + 0.63ν2)

(

h

b

)2

Negative strain load

σ̄(−)cr = −38.49

(

1 + 0.17ν + 1.95ν2)

(

h

b

)2


Buckled mode shapes

−0.15

0.5

1.0 −0.5

0

0.5

−0.1

−0.05

0

0.05

x

b

W

1

1

−0.5

0.5−1.0

−0.5

0

2

3

4

0

W

0.5

1.0 −0.5

0

0.5−0.15

−0.1

−0.05

0

0.05

W

1

−0.5

0.5−1.0

−0.5

0

2

3

4

0

W

Positive strain loads (σ > 0) Negative strain loads (σ < 0)

x

b1

x

b1

x

b1

x

b2 x

b2

x

b2 x

b2


Mechanical pressure compared to buckled mode shapes

0 0.2 0.4 0.6 0.8 1

−0.4

−0.2

0

0.2

0.4

-40

-30 0

5

510

10

-40

-20 -10 -2.5

2.5

2.5

Region 2

Region 1

Normalized Mechanical Pressure: P

(a)

0 0.2 0.4 0.6 0.8 1

-0.2

0

0.2

0.41.5

1.5-0.4

10.5

0

10.5

0

-2 -4 -6 -8 -10 -12ν = 0.25

Buckled Mode Shape: σ > 0

(b)

0 0.2 0.4 0.6 0.8 1

-0.2

0

0.2

0.4

-0.4

-2

01

-2 -4 -6 -8 -10

-12

2

3

-2

-14

-14

(c)

Region 1

10 −2)(×

Buckled Mode Shape: σ < 0 10 −2)(× 10 −2)(×

ν = 0.25

x / b2

x / b2 x / b2

x / b1

x / b1 x / b1



Mechanical pressure compared to buckled mode shapes

0 0.2 0.4 0.6 0.8 1

−0.4

−0.2

0

0.2

0.4

40

30 0

−5

−5−10

−10

40

20 10 2.5

−2.5

−2.5

Region 2

Region 1

Normalized Mechanical Pressure: P

(a)

0 0.2 0.4 0.6 0.8 1

-0.2

0

0.2

0.41.5

1.5-0.4

10.5

0

10.5

0

-2 -4 -6 -8 -10 -12ν = 0.25

Buckled Mode Shape: σ > 0

(b)

0 0.2 0.4 0.6 0.8 1

-0.2

0

0.2

0.4

-0.4

-2

01

-2 -4 -6 -8 -10

-12

2

3

-2

-14

-14

(c)

Region 1

10 −2)(×

Buckled Mode Shape: σ < 0 10 −2)(× 10 −2)(×

ν = 0.25

x / b2

x / b2 x / b2

x / b1

x / b1 x / b1

strain loadNegative



Cantilever σTs > 0(< 0) ∆T > 0 (< 0)

Material (Nm )− 1 (K)

Si3 N4 98.7 (−78.5) 4010 (unphysical)

Graphene 0.0324 (−0.0276) 8.9 (−7.6)

Silicon Nitride: × 12 × 0.09 µm 3× b ×h)(L× ×

Graphene (2-layer): × 0.8× 0.0006 µm3

30

3.2 × b ×h)(L× ×



Cantilever σTs > 0(< 0) ∆T > 0 (< 0)


Si3 N4 98.7 (−78.5) 4010 (unphysical)

Graphene 0.0324 (−0.0276) 8.9 (−7.6)

Graphene cantilever stability is marginal!!



30

3.2 × b ×h)(L× ×



Li et. al, Appl. Phys. Lett. 101, (2012)

Cantilever σTs > 0(< 0) ∆T > 0 (< 0)


Si3 N4 98.7 (−78.5) 4010 (unphysical)

Graphene 0.0324 (−0.0276) 8.9 (−7.6)



30

3.2 × b ×h)(L× ×


Conclusion

Positive/negative surface stress loads induce a negative/positivechange in stiffness

Practical V-shaped cantilevers more sensitive to surface stress changes

Observed frequency shifts not due to surface stress

Buckling driven by compressive in-plane stresses

Novel graphene cantilevers buckle under surface stress and thermalloads


Publications

M. J. Lachut and J. E. Sader, Phys. Rev. Lett. 99, 206102 (2007)

M. J. Lachut and J. E. Sader, Appl. Phys. Lett. 95, 193505 (2009)

M. J. Lachut and J. E. Sader, Phys. Rev. B. 85, 085440 (2012)

M. J. Lachut and J. E. Sader, J. Appl. Phys. 113, 024501 (2013)


Acknowledgements

Thanks to:

Family and Friends for their emotional support

Esteemed colleagues for their insightful comments

Confirmation panel for their support, and

Supervisor Prof. John E. Sader for being an inspirational mentor


completion talk 02

Technology

eect of surface stress

h2michael lachut supervisor

obmichael lachut supervisor

s x3 h2x3bx1hs x3

stiness of original

x3 dx3

axialforce modelx3 s

h2 gurtin et