doi: 10.1038/nmat4335 supplementary information potential … · 2015-08-20 · and the...

SUPPLEMENTARY INFORMATIONDOI: 10.1038/NMAT4335

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1

Supplementary Information Potential Dependent Dynamic Fracture of Nanoporous Gold

Shaofeng Sun1, Xiying Chen1, Nilesh Badwe1, Karl Sieradzki1*

1 Ira A. Fulton School of Engineering, Arizona State University, Tempe, Arizona 85281 *Corresponding author ([email protected])

Linear sweep voltammetry and chronocoulometry for the fabrication of crack-free NPG.

Figure S1 | Electrochemical protocols used for producing crack-free NPG. a, Potential dynamic scan of Ag-28 at% Au in 1 M HNO3. Scan rate 5 mV/s. b, Chronoamperomtery at 1.17 V (NHE), showing how the current density decays with time over ~ 4 days. The oscillations in the current are real and result from transport limitations associated with forming the monolithic NPG samples.

Figure S2 | Cyclic voltammetry of imbibed NPG samples in 1M HClO4; fully immersed in a beaker (red) and imbibed on the tensile device (black). The difference corresponds to small portions of the sample that were glued to the tensile stage.

a b

Potential-dependent dynamic fracture of nanoporous gold

© 2015 Macmillan Publishers Limited. All rights reserved

http://dx.doi.org/10.1038/nmat4335

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SUPPLEMENTARY INFORMATION DOI: 10.1038/NMAT4335 2

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Benchmark experiments for DIC measurements. Figure S3 | DIC benchmarks with the Phantom V12 Camera. a, the average DIC displacement versus that obtained from the Thorlabs stage and b, The standard deviation in the measurement. Data from 4 separate regions of the sample. The waviness in the average displacement resulted from low frequency building vibration over a time scale much longer than the duration of the dynamic fracture tests. The standard deviation generally increased with total displacement and for a 7.0 µm displacement the standard deviation was almost 0.2 µm.

Figure S4 | DIC benchmarks for the EO-5012M CMOS Monochrome camera showing the average displacement obtained from DIC versus the Thorlabs stage. The different colored points correspond to displacement steps of 0.1 (black), 0.2 (green), 0.5 (blue) and 1.0 (red) µm per step. The standard deviation was 0.02 µm. Evaluation of the longitudinal, shear and Raleigh wave velocities In an isotropic elastic solid, the longitudinal, cl, shear wave, cs, and Rayleigh, cR, velocities are given by1,

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DIC

ave

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a b

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cl =

E(1−υ)ρ(1+υ)(1− 2υ)

; cs =µρ

; cR ≅0.862 +1.14υ

1+υ⎛⎝⎜

⎞⎠⎟

cs ,

where E is Young’s modulus, µ is the shear modulus equal to E/2(1+ν), ρ is the density of 40 nm NPG) and ν is Poisson’s ratio. Using the values of E (2.5 GPa) and ν (0.19) and taking the density to be 0.28 the density of Au (19300 kgm-3) we obtain,

cl = 713 m/s; cs = 441 m/s; cR ≅ 400 m/s . Our measured values for E and ν are similar to what has been previously reported for NPG in tension2 and the longitudinal and shear wave velocities that we calculate from the measured elastic constants are in accord with recent non-contact laser-based ultrasonic measurements3. Evaluation of critical stress intensity factor The critical stress intensity factor at fracture, Kcrit, for the single edge-notch geometry of our samples was evaluated from4,

𝐾𝐾! = 𝑌𝑌𝑌𝑌 𝑎𝑎

𝑌𝑌 = 1.99− 0.41 !!+ 18.7 !

!

!− 38.48 !

!

!+ 53.85 !

!

! ,

where, a is the crack length, σ is the stress and w is the sample width.

Crack velocity data for imbibed samples at 0.7 and 1.0 V (NHE) Figure S5 | Crack velocity-crack length data for samples imbibed with 1M HClO4. a, 0.7 V and b,1.0 V.

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SUPPLEMENTARY INFORMATIONDOI: 10.1038/NMAT4335 2

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Benchmark experiments for DIC measurements. Figure S3 | DIC benchmarks with the Phantom V12 Camera. a, the average DIC displacement versus that obtained from the Thorlabs stage and b, The standard deviation in the measurement. Data from 4 separate regions of the sample. The waviness in the average displacement resulted from low frequency building vibration over a time scale much longer than the duration of the dynamic fracture tests. The standard deviation generally increased with total displacement and for a 7.0 µm displacement the standard deviation was almost 0.2 µm.

Figure S4 | DIC benchmarks for the EO-5012M CMOS Monochrome camera showing the average displacement obtained from DIC versus the Thorlabs stage. The different colored points correspond to displacement steps of 0.1 (black), 0.2 (green), 0.5 (blue) and 1.0 (red) µm per step. The standard deviation was 0.02 µm. Evaluation of the longitudinal, shear and Raleigh wave velocities In an isotropic elastic solid, the longitudinal, cl, shear wave, cs, and Rayleigh, cR, velocities are given by1,

0

1

2

3

4

5

6

0 1 2 3 4 5 6Displacement from Thorlabs stage (µm)

DIC

ave

rage

dis

plac

emen

t (µm

)

a b

3

cl =

E(1−υ)ρ(1+υ)(1− 2υ)

; cs =µρ

; cR ≅0.862 +1.14υ

1+υ⎛⎝⎜

⎞⎠⎟

cs ,

where E is Young’s modulus, µ is the shear modulus equal to E/2(1+ν), ρ is the density of 40 nm NPG) and ν is Poisson’s ratio. Using the values of E (2.5 GPa) and ν (0.19) and taking the density to be 0.28 the density of Au (19300 kgm-3) we obtain,

cl = 713 m/s; cs = 441 m/s; cR ≅ 400 m/s . Our measured values for E and ν are similar to what has been previously reported for NPG in tension2 and the longitudinal and shear wave velocities that we calculate from the measured elastic constants are in accord with recent non-contact laser-based ultrasonic measurements3. Evaluation of critical stress intensity factor The critical stress intensity factor at fracture, Kcrit, for the single edge-notch geometry of our samples was evaluated from4,

𝐾𝐾! = 𝑌𝑌𝑌𝑌 𝑎𝑎

𝑌𝑌 = 1.99− 0.41 !!+ 18.7 !

!

!− 38.48 !

!

!+ 53.85 !

!

! ,

where, a is the crack length, σ is the stress and w is the sample width.

Crack velocity data for imbibed samples at 0.7 and 1.0 V (NHE) Figure S5 | Crack velocity-crack length data for samples imbibed with 1M HClO4. a, 0.7 V and b,1.0 V.

0

50

100

150

500 1500 2500 3500

Vel

ocity

(m/s

)

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a b





Estimation of remaining stored elastic strain energy Figure 2b. We make a lower bound estimate by assuming that the sample is fully equilibrated with its boundaries and that the large majority of the remaining stored elastic energy is in the un-cracked region of the sample. For a crack 2.5 mm in length the remaining strain energy is ~ 74 Jm-2 × (1.5/3.2) ≈ 35 Jm-2. Estimation of plastic strain-rate at a crack velocities of ~ 102 m/s. The extent of the plastic zone, rp, at fracture initiation is ~ 50 µm. The average plastic strain rate

γ p can be estimated from the size of this zone and the crack velocity, v, from5 :

γ p = v / rp = 2 ×106 s-1. We note that strain rates become larger than average, the closer the

ligaments within the plastic zone are to the advancing crack tip. Dynamic stress intensity and energy release rate. We evaluated Kdyn from the particle velocity components near the moving crack edge from1:

u1 vKdyn

µD 2πr1+α s

2( ) cos θd / 2( )γ d

− 2α dα s

cos θs / 2( )γ s

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

u2 vα d Kdyn

µD 2πr1+α s

2( ) cos θd / 2( )γ d

− 2sin θs / 2( )

γ s

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

Here, Kdyn is the time-dependent mode I stress intensity factor, v is the crack speed, µ is the shear modulus and r, θ define the particle position with respect to the moving crack. The remaining parameters are defined as follows:

Kdyn for an unbounded sample was evaluated from (1),

The dynamic energy release rates were evaluated from (1),

Gdyn =

Kdyn2

EA(v),where A(v) =

v2α d

(1− v)cs2 D

.

D = 4α dα s − 1+α s2( )2

, α d = 1− v2 / cl2( ), α s = 1− v2 / cs

2( ), γ d = 1− v sinθ / cl( )2, γ s = 1− v sinθ / cs( )2

tanθd = α d tanθ , and tanθs = α s tanθ.

Kdyn = Kstatick(v), where k(v) = 1− v / cR( ) / 1− v / cl .

5

Estimation of dislocation density in 40 nm monolithic NPG. The mean separation between dislocations can be estimated from ρΑ

-1/2 where ρΑ is the dislocation density per unit area. Typically well-annealed parent phase will have a dislocation density between 105-108 cm-2 6,7,8. This range in density corresponds to a mean separation ranging from about 1 µm (108 cm-2) to 30 µm (105 cm-2). The fraction of gold ligaments containing any dislocations inherited from the parent phase (estimated from the ratio the ligament size to the mean separation) is 0.001 (105 cm-2) - 0.04 (108 cm-2). In addition to defining a dislocation density in terms of number per unit area one can also define the density in terms of the length of dislocation lines per unit volume, ρV. Nabarro has shown9 that these different measures of dislocation density are related (ρV ≈ 2ρΑ ), so our estimate of the fraction of ligaments containing dislocations is not significantly altered by how one chooses to define the measure of dislocation density. Relevance of Dynamic Fracture Results to SCC.

Since the dealloyed film thickness that forms in SCC, between successive discontinuous cracking events, is only of order a few hundred nanometers10, how does data such as that shown in Figure 2, 3 and S5 connect to dealloying induced SCC? The data show that under the experimental conditions used, cracks do not reach high speeds (~100 ms-1 or greater) until the crack has extended to ~2000 µm. In an unbounded sample, the crack tip equation of motion contains no acceleration term and the crack velocity only depends on the instantaneous value of the stress intensity, K [or alternatively 𝐺𝐺!"#"$% 𝑙𝑙 ] and Γ(v), the velocity dependent energy dissipated per unit crack advance1,11. This means that at the initiation of a fracture event, the crack immediately jumps to the velocity defined by the “instantaneous” K value, which can be arbitrarily large11. Figure. 2d shows that in monolithic 40 nm NPG at 0.5 V NHE, a crack attains a velocity of ~100 m/s at a K value of ~0.4 MPa-m1/2. For a 1 mm size crack in a silver-gold sample-undergoing SCC an applied stress of 7 MPa (𝐾𝐾 ≈ 𝜎𝜎!""# 𝜋𝜋𝜋𝜋) would be sufficient to generate such a high-speed crack within a very thin (~ 200 nm or less) dealloyed layer. Supplemental References

1. Freund, L.B., Dynamic Fracture Mechanics, Cambridge Univ. Press (1998).

2. Briot, N.J., Kennerknecht, T., Eberl, C., Balk, T.J. Mechanical properties of bulk single crystalline nanoporous gold investigated by millimetre-scale tension and compression testing. Philos. Mag. 94, 847-866 (2014). 3. Ahn, P., Balogun, O. Elastic characterization of nanoporous gold foams using laser based ultrasonics, Ultrasonics 54, 795-800 (2014). 4. Broek, D., Elementary Engineering Fracture Mechanics, Nordoff Publishing (1974).




SUPPLEMENTARY INFORMATIONDOI: 10.1038/NMAT4335 4

Estimation of remaining stored elastic strain energy Figure 2b. We make a lower bound estimate by assuming that the sample is fully equilibrated with its boundaries and that the large majority of the remaining stored elastic energy is in the un-cracked region of the sample. For a crack 2.5 mm in length the remaining strain energy is ~ 74 Jm-2 × (1.5/3.2) ≈ 35 Jm-2. Estimation of plastic strain-rate at a crack velocities of ~ 102 m/s. The extent of the plastic zone, rp, at fracture initiation is ~ 50 µm. The average plastic strain rate

γ p can be estimated from the size of this zone and the crack velocity, v, from5 :

γ p = v / rp = 2 ×106 s-1. We note that strain rates become larger than average, the closer the

ligaments within the plastic zone are to the advancing crack tip. Dynamic stress intensity and energy release rate. We evaluated Kdyn from the particle velocity components near the moving crack edge from1:

u1 vKdyn

µD 2πr1+α s

2( ) cos θd / 2( )γ d

− 2α dα s

cos θs / 2( )γ s

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

u2 vα d Kdyn

µD 2πr1+α s

2( ) cos θd / 2( )γ d

− 2sin θs / 2( )

γ s

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

Here, Kdyn is the time-dependent mode I stress intensity factor, v is the crack speed, µ is the shear modulus and r, θ define the particle position with respect to the moving crack. The remaining parameters are defined as follows:

Kdyn for an unbounded sample was evaluated from (1),

The dynamic energy release rates were evaluated from (1),

Gdyn =

Kdyn2

EA(v),where A(v) =

v2α d

(1− v)cs2 D

.

D = 4α dα s − 1+α s2( )2

, α d = 1− v2 / cl2( ), α s = 1− v2 / cs

2( ), γ d = 1− v sinθ / cl( )2, γ s = 1− v sinθ / cs( )2

tanθd = α d tanθ , and tanθs = α s tanθ.

Kdyn = Kstatick(v), where k(v) = 1− v / cR( ) / 1− v / cl .

5

Estimation of dislocation density in 40 nm monolithic NPG. The mean separation between dislocations can be estimated from ρΑ

-1/2 where ρΑ is the dislocation density per unit area. Typically well-annealed parent phase will have a dislocation density between 105-108 cm-2 6,7,8. This range in density corresponds to a mean separation ranging from about 1 µm (108 cm-2) to 30 µm (105 cm-2). The fraction of gold ligaments containing any dislocations inherited from the parent phase (estimated from the ratio the ligament size to the mean separation) is 0.001 (105 cm-2) - 0.04 (108 cm-2). In addition to defining a dislocation density in terms of number per unit area one can also define the density in terms of the length of dislocation lines per unit volume, ρV. Nabarro has shown9 that these different measures of dislocation density are related (ρV ≈ 2ρΑ ), so our estimate of the fraction of ligaments containing dislocations is not significantly altered by how one chooses to define the measure of dislocation density. Relevance of Dynamic Fracture Results to SCC.

Since the dealloyed film thickness that forms in SCC, between successive discontinuous cracking events, is only of order a few hundred nanometers10, how does data such as that shown in Figure 2, 3 and S5 connect to dealloying induced SCC? The data show that under the experimental conditions used, cracks do not reach high speeds (~100 ms-1 or greater) until the crack has extended to ~2000 µm. In an unbounded sample, the crack tip equation of motion contains no acceleration term and the crack velocity only depends on the instantaneous value of the stress intensity, K [or alternatively 𝐺𝐺!"#"$% 𝑙𝑙 ] and Γ(v), the velocity dependent energy dissipated per unit crack advance1,11. This means that at the initiation of a fracture event, the crack immediately jumps to the velocity defined by the “instantaneous” K value, which can be arbitrarily large11. Figure. 2d shows that in monolithic 40 nm NPG at 0.5 V NHE, a crack attains a velocity of ~100 m/s at a K value of ~0.4 MPa-m1/2. For a 1 mm size crack in a silver-gold sample-undergoing SCC an applied stress of 7 MPa (𝐾𝐾 ≈ 𝜎𝜎!""# 𝜋𝜋𝜋𝜋) would be sufficient to generate such a high-speed crack within a very thin (~ 200 nm or less) dealloyed layer. Supplemental References

1. Freund, L.B., Dynamic Fracture Mechanics, Cambridge Univ. Press (1998).

2. Briot, N.J., Kennerknecht, T., Eberl, C., Balk, T.J. Mechanical properties of bulk single crystalline nanoporous gold investigated by millimetre-scale tension and compression testing. Philos. Mag. 94, 847-866 (2014). 3. Ahn, P., Balogun, O. Elastic characterization of nanoporous gold foams using laser based ultrasonics, Ultrasonics 54, 795-800 (2014). 4. Broek, D., Elementary Engineering Fracture Mechanics, Nordoff Publishing (1974).





5. Freund, L.B., Hutchinson, J.W. High strain-rate crack growth in rate-dependent solids, J. Mech. Phys. Solids 33, 169-191 (1985). 6. McClintock, F.A. & Argon, A.S. Mechanical Behavior of Materials, Addison-Wesley (1966), pg.106. 7. Lee-S-W & Nix, W.D. Geometrical analysis of 3D dislocation dynamic simulations of FCC micro-pillar plasticity, Mat. Sci. Eng. A-Struct. 527, 1903-1910, (2010). 8. Rao, S.I. et al. Athermal mechanisms of size-dependent crystal flow gleaned from three-dimensional discrete dislocation simulations, Acta Mater. 56, 3245-3259 (2008). 9. Nabarro, FRN. Theory of Crystal Dislocations, Oxford Press (1967), p. 44.

10. Sieradzki, K., & Newman, R.C. Brittle behavior of ductile metals during stress-corrosion cracking, Philos. Mag. A51, 95-132 (1985). 11. Sharon, E & Fineberg, J. Confirming the continuum theory of dynamic brittle fracture for fast cracks, Nature, 307, pp. 333-335 (1999).



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