advances.sciencemag.org/cgi/content/full/3/6/e1602705/DC1

Supplementary Materials for

Dynamic fracture of tantalum under extreme tensile stress

Bruno Albertazzi, Norimasa Ozaki, Vasily Zhakhovsky, Anatoly Faenov, Hideaki Habara,

Marion Harmand, Nicholas Hartley, Denis Ilnitsky, Nail Inogamov, Yuichi Inubushi, Tetsuya Ishikawa,

Tetsuo Katayama, Takahisa Koyama, Michel Koenig, Andrew Krygier, Takeshi Matsuoka,

Satoshi Matsuyama, Emma McBride, Kirill Petrovich Migdal, Guillaume Morard, Haruhiko Ohashi,

Takuo Okuchi, Tatiana Pikuz, Narangoo Purevjav, Osami Sakata, Yasuhisa Sano, Tomoko Sato,

Toshimori Sekine, Yusuke Seto, Kenjiro Takahashi, Kazuo Tanaka, Yoshinori Tange, Tadashi Togashi,

Kensuke Tono, Yuhei Umeda, Tommaso Vinci, Makina Yabashi, Toshinori Yabuuchi, Kazuto Yamauchi,

Hirokatsu Yumoto, Ryosuke Kodama

Published 2 June 2017, Sci. Adv. 3, e1602705 (2017)

DOI: 10.1126/sciadv.1602705

The PDF file includes:

fig. S1. Single-shot x-ray spectra of Ta plasma irradiated by the high-power

optical laser.

fig. S2. Pulse shape of the optical beam at target center chamber.

fig. S3. Velocity of the Lagrangian particle provided by hydrodynamic (MULTI)

modeling.

fig. S4. Comparison between the new EAM Ta potential and experimental shock

Hugoniot curve.

fig. S5. Shadowgraph of mass distribution after nucleation of first voids in the

MD-simulated Ta sample.

fig. S6. MD simulation of the experiment.

fig. S7. Determination of the strain rate from the flow velocity profile just before

nucleation of the first voids in the MD simulation.

fig. S8. Determination of the strain rate from the flow velocity profile just before

nucleation of the first voids in the MD simulation.

fig. S9. Direct comparison between experimental profiles and x-ray profiles

derived from the MD simulation.

fig. S10. Experimental determination of the position of the different peaks at t =

1725 ps using the Gaussian method.

fig. S11. Experimental determination of the position of the different peaks at t =

2125 ps using the Gaussian method.

fig. S12. Experimental determination of the position of the different peaks at t =

2125 ps using the Lorentzian method.

fig. S13. Spall strength versus strain rate for tantalum.

table S1. Pressure and density retrieved from the experimental results at t = 1725

ps displayed in fig. S10.

table S2. Pressure and density retrieved from the experimental results at t = 2125

ps displayed in fig. S11.

References (34–51)

Other Supplementary Material for this manuscript includes the following:

(available at advances.sciencemag.org/cgi/content/full/3/6/e1602705/DC1)

movie S1 (.avi format). Evolution of density and pressure in the sample given by

MD simulation.

movie S2 (.avi format). Experimental data obtained at SACLA.

Reproducibility of the laser-matter interaction during the experiment

To characterize the laser-matter interaction, we implemented a 1 dimensional focusing

spectrometer with a spatial resolution (FSSR) using a spherical crystal (34). The X-ray

spectrum (Bremstrahlung radiation) produced by the Ta plasma was recorded by a vacuum

compatible ANDOR CCD in a single shot mode. The spectral distribution Ibs(E) depends on

the electron temperature as 𝐼𝑏𝑠(𝐸) = 𝐴𝑒𝑥𝑝(−

𝐸

𝑘𝑇𝑒) which enables the electron temperature to be

calculated by: 𝑑[ln(𝐼𝑏𝑠(𝐸))]

𝑑𝑡= −

1

𝑘𝑇𝑒.

fig. S1. Single-shot x-ray spectra of Ta plasma irradiated by the high-power optical

laser. (A) shows the highest intensity spectra and (B) the lowest. The maximum variation of

the x-ray intensity is ~ 6.4 % which implies a maximum variation of the electron temperature

of ~ 5%.

The accuracy in the determination of the electron temperature using the bremsstrahlung

radiation coming from the Ta plasma is on the order of 10%. The intensity variation of x-ray

spectra for 6 shots is of the order of 6.4 %, which implies a variation of the electron

temperature of ~ 5 % (see fig. S1). At nano and sub-nanosecond duration laser pulse with

relatively low laser intensities, the electron temperature follows the scaling laws Te ~ Ilaser2/3.

In that case, the shot-to-shot fluctuations of the laser intensities is approximately 2.3%. Thus

the laser-matter interaction is extremely stable and can launch similar shock wave inside our

sample for each shot (see Ref (35) for more details).

Determination of t = 0 in the experiment

Figure S2 shows the optical pulse with the time t = 0 defined as it has been explained in the

Materials and Methods section of the main text.

fig. S2. Pulse shape of the optical beam at target center chamber. (A) Full measurement

(B). Experimental determination of t = 0.

The lines broadening of the Au sample at t = 50 ps [see Fig.2.b of ref (33)] is due to sample

heating. It is possible to evaluate a laser intensity below 1011 W.cm-2 for such expansion,

which correspond to ~ 2.2.1011 W.cm-2. This point is defined as time t = + 50 ps both in

simulation and experiment. Figure S2 present the laser pulse shape where t = 50 ps is defined

with an accuracy of ± 25 ps, taking into account uncertainties. Time t = 0 is then shifted from

this value of 50 ± 25 ps.

Molecular dynamics simulation

To take into account the material response at the high-strain-rate stretching of tantalum

leading to spallation of the sample, a large scale Molecular Dynamics (MD) simulation has

been performed with the aim of planning the measurement timings in our experiments and

comparing directly simulation results with experimental results. The initial high power laser-

target interaction is modeled with the one dimensional (1D) Lagrangian radiation hydrocode

MULTI (36) coupled with SESAME equation of state (37). The pulse shape of the optical

laser beam measured in the experiment as well as the evaluated laser intensity are used as

input parameters in the hydrocode modeling. The velocity of the Lagrangian particle (LP) at

an initial position of 1050 nm from the front side of the target is used as a left boundary

condition in the MD simulation, see fig. S3. Such position of LP is large enough to escape

heating from the hot frontal layer. On the other hand, it is far enough from the rear-side

surface of the sample in order to give a necessary time for development of the material

fracture before the arrival of a rarefaction wave from the right boundary to the LP.

The initial length of the MD sample is Lx = 3950 nm, while the cross-section dimensions Ly

and Lz are (20.1 x 20.1) nm2 with periodic boundary conditions imposed on the y- and z-axes.

The left boundary condition was implemented as a piston moving along the x-axis to the right

with the prescribed velocity obtained from MULTI. A free boundary condition with vacuum

was set at the right side of the MD sample. It was found that pressure profiles obtained in the

MD simulation with such a left boundary condition are almost identical to those provided by

MULTI soon after passing that Lagrangian particle. The main difference between the

evolution in the MD simulation and the hydrocode MULTI appears after the reflection of the

shock wave from the free right boundary, where the tensile stress (negative pressure) is

formed in MD, while the pressure remains zero in MULTI modeling.

fig. S3. Velocity of the Lagrangian particle provided by hydrodynamic (MULTI)

modeling. It is used as a left boundary condition in MD simulation.

Embedded Atom Model (EAM) potential for Ta

With the aim of developing a potential capable of correctly reproducing the response of

tantalum to deformation over a wide range of compression/stretching, the stress-matching

method (38) was used. The fitting database is built of the stress tensor components 𝜎𝛼𝛽(𝑉) ≡

−𝑃𝛼𝛽(𝑉) calculated by DFT (Density Functional Theory) in a cold crystal lattice under

continuous hydrostatic and uniaxial deformations. Experimental elastic constants, equilibrium

density and the cohesive energy of Ta are also included in the database. The fitting procedure

also involves constraints such as monotonic behavior of 𝑃𝛼𝛽(𝑉), including requiring an

increase of the sound speed with compression, and an absence of solid-solid transition from

the stable bcc phase of tantalum.

To obtain the first-principles cold pressure curves of tantalum, DFT calculations using the

Vienna ab initio simulation package (VASP) (39) were performed. Electron wave functions of

crystals containing two atoms in a bcc-type cell were calculated with a Projector Augmented

Wave (PAW) pseudopotential (40, 41) and the Perdew-Burke-Ernzerhof functional (42).

These highly accurate DFT calculations, with the energy cutoff 500 eV and number of k-

points 21x21x21 generated according to the Monkhorst-Pack scheme for sampling the

Brillouin zone (43) were performed for the valence band 5p65d36s2 and the number of

unoccupied levels being 20. To calculate the uniaxial pressure components, a series of

stepwise static calculations with relaxation of atom positions were performed for normal

strains along the [100], [110], and [111] directions, respectively. The equilibrium bcc-cell of

crystal at P = 0 was found to have a size of 0.331 nm, consistent with the experimental value.

The high-order rational functions were used to represent the EAM potential consisting of a

pairwise energy, charge density and embedding energy. Fitting of potential coefficients was

performed by minimization of a target function with the use of a downhill simplex algorithm

(44) combined with random walk in a multidimensional space of the fitting coefficients.

The new EAM potential used in the present work shows excellent agreement with the

experimental shock Hugoniot curve of Ta presented in fig. S4.

fig. S4. Comparison between the new EAM Ta potential and experimental shock

Hugoniot curve. Dashed line shows cold pressure from DFT while solid line corresponds to

that calculated with the new EAM potential. Crosses and triangles show the shock Hugoniot

obtained from the MD simulation of Ta single crystals with different orientations, and open

circles present many experimental data taken from the shock-wave database (45).

Result of MD simulation

Typical results given by the MD simulation are displayed in figs. S5 to S8. The spallation

starts at approximately t =1798 ps with a spall strength (negative pressure) of about -17.5 GPa

(fig. S5, S6.A and S7). Figure S5 shows the shadowgraphs of mass distribution when the first

voids are nucleated and grow up to the cross-section dimensions of the simulated sample.

At later times the rightmost void is nucleated at the time of 1831 ps as shown on fig. S6.B and

S8. This void generates the sole spall shock, which is able to reach the right boundary of the

sample. Other shock waves generated before cannot propagate through this void when it

reaches the sample size. At a time t = 2001 ps, the thickness of the spalled layer is ~ 1 µm

(see fig. S6. C). Inside the spalled layer, a shock wave is generated with maximum pressure of

~ 10 GPa at t = 2124 ps after the beginning of the interaction. This value is in excellent

agreement with experimental results at t = 2125 ps (see Table 1 for example) where a strong

peak is observed with a corresponding pressure of 9.01 GPa as the 10 GPa pressure zone in

the MD simulation is large. On the other side, the maximum pressure obtained in the spalled

layer in the MD simulation (see fig. S6.E), is on the order of 15 GPa, but as the pressure zone

is small, it was not detected in the experiment.

fig. S5. Shadowgraph of mass distribution after nucleation of first voids in the MD-

simulated Ta sample. It takes about 25 ps for voids to grow in size up to the cross-section

dimensions of 20.1x20.1 nm2.

fig. S6. MD simulation of the experiment. Pressure and density given by the MD simuation

during the dynamic fracture of the Ta sample (A) when first two voids are nucleated at t =

1802 ps, (B) the rightmost void is formed at t = 1845 ps, (C) at a time of t = 2001 ps, (D)

when the spall shock wave approaches the right free boundary at t = 2124 ps and (E) at a

time t = 2226 ps. See movie 1 in the Supplementary Materials.

Determination of the strain rate just before the spallation occurs in the MD simulation

The strain rate in the MD simulation is determined from the mass velocity profile u(x)

obtained in the simulation. The expression of the strain rate is given by:

휀̇ =𝑑𝑢(𝑥)

𝑑𝑥

The MD simulation gives strain rates of 3.5.108 s-1 and 4.5.108 s-1 just before nucleation of

the first two voids under tensile stresses of -17.5 GPa and -18 GPa, respectively (see fig. S7).

Figure S8 shows nucleation of the rightmost void under tensile stress of -15.8 GPa at the

lower strain rate of 2.108 s-1 later time. This void will result in generation of the spall shock

shown on fig. S6 C. The shock waves generated earlier from other voids cannot propagate

through this void to the right boundary of sample.

fig. S7. Determination of the strain rate from the flow velocity profile just before

nucleation of the first voids in the MD simulation.

fig. S8. Determination of the strain rate from the flow velocity profile just before

nucleation of the first voids in the MD simulation.

The obtained density profiles 𝜌(𝑥, 𝑡) coming from the MD simulation are integrated to

estimate the corresponding x-ray diffraction signal from a polycrystalline sample with

randomly oriented grains via the formula:

𝑆(𝜃′) = ∫ ∫ 𝛿(𝜌(𝑥) − 𝜌′) 𝐷(𝜃, 𝑥) 𝑓(𝜃 − 𝜃′) cos(𝜃) 𝑑𝜃 𝑑𝑥, (1.1)

where 𝑥 is a distance from the rear-side boundary, 𝐷(𝜃, 𝑥) = exp{−[1

sin(200)+

1

sin(2𝜃−200)]𝑘∫ 𝜌(𝑥)𝑑𝑥} is an attenuation factor with the mass attenuation coefficient 𝑘 =

237.9 cm2/g , and a widening function 𝑓(𝜃 − 𝜃′) is used instead of 𝛿(𝜃 − 𝜃′) in order to fit

an experimental signal from the unshocked material. The density 𝜌′ corresponds to the Bragg

angle 𝜃′. The calculated signals 𝑆(𝜃′) are shown on Fig. 3.(A) of the main paper and an

additional comparison is shown in fig. S9.

fig. S9. Direct comparison between experimental profiles and x-ray profiles derived

from the MD simulation. at (A) t = 1925 ps, (B) t = 2125 ps and (C) t = 2425 ps.

Experimental determination of the pressure and density

Gaussian methods

This section is devoted to the method used in order to determine (i) the position of the

different peaks present in the experimental results, (ii) the density and (iii) the pressure related

to these peaks. Figure S10 shows an example of the data analysis at t = 1725 ps.

fig. S10. Experimental determination of the position of the different peaks at t = 1725 ps

using the Gaussian method.

The position of the maximum of the two main peaks is determined by fitting two gaussian to

the experimental data (see fig. S10). The form of the two Gaussian is given by:

𝑓(2𝜃) = 420 exp {(2𝜃 − 42.42

0.55)2

}

and

𝑓(2𝜃) = 730 𝑒𝑥𝑝 {(2𝜃 − 43.23

0.55)2

}

where 2θ = 42.42° for the first peak and 2θ = 43.23° for the second one with a fitting error of

the order of ± 0.01° for the position of the peak. We should note as well that this method

implies uncertainties on the FWHM of the Gaussian which is not a well known parameter.

The 2θ angle measured in the experiment is related to the lattice spacing d of the bcc (002)

plane of Ta via the Bragg’s law (nλ=2dsin(θ) with λ being the x-ray wavelength, d the spacing

of the (002) plane, n = 1 in that case and θ the angle between the x-ray beam and the lattice

plane of the target). The X-ray spectrum has been recorded for each shot allowing to have an

extremely small uncertainties on λ and as a consequence on the lattice spacing d. A small

remarks is that the slit, present in the XFEL beam causes diffraction patterns in the X-ray

beam profile which are not strong enough to influence our experimental data.

The density can be evaluated using the following formula 𝜌 = 𝑛𝐴/(𝑁𝑜𝑎3) where 𝑛 is the

number of atoms per unit cell, 𝐴 is the molar mass of the material, 𝑁𝑜 Avogadro’s number

and 𝑎 = √(ℎ2 + 𝑘2 + 𝑙2)𝑑2 the mesh parameter for a bcc lattice where h, k and l are the

Miller indices. We can use the isothermal equation of state (EOS) in solid state physics

developed by Birch and Murnagham. The third order Birch-Murnagham EOS (28–30) is

given by:

𝑃 = 3

2𝐵0𝑇 [(

𝑉

𝑉0)−7/3

− (𝑉

𝑉0)−5/3

] {1 −3

4(4 − 𝐵′) [(

𝑉

𝑉0)−2/3

− 1]} (1)

Where 𝐵0𝑇 is the bulk modulus, 𝐵′ the pressure derivative of the bulk modulus and 𝑉0 the

initial volume at zero pressure and room temperature. Table 1 summarizes the pressure

obtained in the experiment at t = 1725 ps after the beginning of the interaction using equation

(1). The bulk modulus 𝐵0𝑇 is taken to be 196.3 GPa (46) and 𝐵′~ 3.52 (47) as it has been

observed in experiment, which is also in good agreement with previous study (48).

table S1. Pressure and density retrieved from the experimental results at t = 1725 ps

displayed in fig. S10.

The analysis described above can be also applied to later data, i.e. when we observed the spall

shock. The highest compression achieved due to the spall shock occurs at t = 2125 ps. In the

same way as in fig. S10, one can construct two main peaks as can be seen in fig. S11.

fig. S11. Experimental determination of the position of the different peaks at t = 2125 ps

using the Gaussian method.

The position of the two peaks is respectively: 2θ = 43.59° and 2θ = 44.63° (error of ± 0.01°).

Using a similar method as described above, we can evaluate the pressure and the density for

both peaks. It is summarized in table 2.

table S2. Pressure and density retrieved from the experimental results at t = 2125 ps

displayed in fig. S11.

Lorentzian method

The Gaussian method is a simple method to get only the position of the peak. However,

another method could be used with a Lorentz function which gives the position of the peak, as

well as the FWHM. A typical example is presented below:

fig. S12. Experimental determination of the position of the different peaks at t = 2125 ps

using the Lorentzian method.

The position of the maximum of the two main peaks is determined, by fitting two lorentzian

to the experimental data (see fig. S12). The form of the two Lorentzian is given by:

𝑓(2𝜃) = 72

{

2𝜋/0.8326

1 + (2𝜃 − 43.590.8326/2

)2

}

𝑓(2𝜃) = 41

{

2𝜋/1.199

1 + (2𝜃 − 44.631.199/2

)2

}

As can be seen in the above equation, the position of the two peaks are respectively : 2θ =

43.59° and 2θ = 44.63° indicating that the Gaussian and the Lorentzian method give the same

pressure and density.

The spall strength limit of Tantalum

The fig. S13 displays the spall strength as a function of the strain rate. Our experimental study

allows to obtain data at a strain rate of 2-3.5.108 s-1. The dependence of the spall strength to

the strain rate for Ta is 𝜎𝑠𝑝~휀̇0.2344 for strain rate 휀̇ < 106𝑠−1 while for a strain rate 휀̇ >

106𝑠−1, it becomes 𝜎𝑠𝑝~휀̇0.1273.

fig. S13. Spall strength versus strain rate for tantalum. Quasi static loading (purple round),

laser shock (black stars) and MD simulation (purple triangle) data are taken from Ref (49).

The blue square is taken from Ref (19). The red and green triangle data are taken from Ref

(50). The orange triangle data are taken from Ref (51) and blue cross from Ref (22).