PDV-based estimation of ejecta particles’ mass-velocity

function from shock-loaded tin experiment

Jean Eloi Franzkowiak, G. Prudhomme, Patrick Mercier, S. Lauriot, E.
Dubreuil, Laurent Berthe

To cite this version:

Jean Eloi Franzkowiak, G. Prudhomme, Patrick Mercier, S. Lauriot, E. Dubreuil, et al.. PDV-based
estimation of ejecta particles’ mass-velocity function from shock-loaded tin experiment. Review of
Scientific Instruments, 2018, 89, pp.033901 (2018). �10.1063/1.4997365�. �hal-02163209�

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PDV-based estimation of ejecta particles’ mass-velocity function
from shock-loaded tin experiment
J.-E. Franzkowiak,1,2,a) G. Prudhomme,1 P. Mercier,1 S. Lauriot,1 E. Dubreuil,1 and L. Berthe2
1 CEA, DAM, DIF, F-91297 Arpajon, France
2 PIMM, UMR 8006 CNRS-Arts et Métiers ParisTech, 151 bd de l’Hôpital, F-75013 Paris, France

A metallic tin plate with a given surface finish of wavelength λ ' 6 0 µm and amplitude h ' 8 µm
is explosively driven by an electro-detonator with a shock-induced breakout pressure PSB = 28 GPa
(unsupported). The resulting dynamic fragmentation process, the so-called “micro-jetting,” is the
creation of high-speed jets of matter moving faster than the bulk metallic surface. Hydrodynamic
instabilities result in the fragmentation of these jets into micron-sized metallic particles constitut-
ing a self-expanding cloud of droplets, whose areal mass, velocity, and particle size distributions are
unknown. Lithium-niobate-piezoelectric sensor measured areal mass and Photonic Doppler Velocime-
try (PDV) was used to get a time-velocity spectrogram of the cloud. In this article, we present both
experimental mass and velocity results and we relate the integrated areal mass of the cloud to the PDV
power spectral density with the assumption of a power law particle size distribution. Two models of
PDV spectrograms are described. The first one accounts for the speckle statistics of the spectrum and
the second one describes an average spectrum for which speckle fluctuations are removed. Finally, the
second model is used for a maximum likelihood estimation of the cloud’s parameters from PDV data.
The estimated integrated areal mass from PDV data is found to agree well with piezoelectric results.
We highlight the relevance of analyzing PDV data and correlating different diagnostics to retrieve the
physical properties of ejecta particles.

I. INTRODUCTION beam (λ = 1.55 µm, i.e., no frequency up-shifting). For a sin-

gle moving object, the interference results in a time-beating
The fragmentation of a roughened metallic plate under
intensity I(t) on the photodetector whose beat frequency ∆f
shock-loading has been extensively investigated in the last
can be related to V using the formula V = λ2 ∆f ,
decade, theoretically1–6 and experimentally.7–13 Many physi-
cal mechanisms have to be taken into account: spalling, micro- I(t) = αs Is + αr Ir + 2 αr αs Is Ir cos(2π∆ft + φ).
p
(1)
spalling, melting, and micro-jetting being the main ones. The
increasing interest to characterize these surface destruction I s and I r are, respectively, the beam intensities of the backscat-
products has been leading to many developments, such as tered signal and of the reference signal. α r and α s are the
X-ray shadowgraphy, piezoelectric pins, Asay foils,14 Mie- coupling parameters of the PDV setup and φ is the phase
sizing diagnostic, Photonic Doppler Velocimetry (PDV),15 and difference between reference and backscattered waves.
holography. The purpose is to determine the areal mass of For a large number of particles with different velocities,
ejecta, the particle sizes, and their velocities. In this article, the backscattered electric field E s (t) is the sum of each con-
we focus on the fastest ejecta particles obtained from shock- tribution with its given amplitude E j , frequency ωj , and phase
wave loading at a metal vacuum tin interface with a particular φj ,
surface finish (PSB = 28 GPa). We describe a PDV simula- X f g
Es (t) = Ej exp i(ωj t + φj ) . (2)
tion model to analyze time-velocity spectrograms of ejecta
j
clouds. Finally, we show that experimental PDV results can be
related to simultaneous piezoelectric measurements through Assuming a random particle arrangement (i.e., uncorre-
numerical investigations of PDV spectra and mass-velocity lated positions and phases φj uniformly distributed between 0
distributions, in line with experimental results. and 2π), the resulting intensity Is (t) ∝ |Es (t)| 2 has a speckle
A standard PDV setup,15 presented in Fig. 1, has been structure16 in both time and frequency (the amplitude and
implemented in our experiment. It consists in mixing two dif- the phase are treated as random variables). The Power Spec-
ferent frequencies to get a beat frequency ∆f related to the tral Density (PSD), or the so-called PDV spectrogram, is the
velocity V of the object. Moving particles are illuminated with squared modulus of the Short-Term Fourier Transform (STFT)
a single mode continuous-wave fiber laser at λ = 1.55 µm using of I(t) given by Eq. (1). Recently, Andriyash et al.17 studied
a fiber collimator (PDV probe). Doppler-shifted wavelets are the influence of the multiple scattering of light on PDV spectra,
collected in the same collimator and mixed with a reference based on a two-flux solution of the Radiative Transfer Equation
(RTE) in the ejecta cloud. They applied this model to analyze
the properties of densely packed debris (∼30 µm size) with
a) Email: jean-eloi.franzkowiak@cea.fr relatively slow velocity gradients (between 100 and 500 m/s).
 fastest particle). lext (t, z(t)) is the extinction length of a slab of
particles between z(t) and z(t) + δz(t), depending on the particle
sizes in the slab and on the volume fraction in particles. Re is
the real part.
The efficiency η(r, z) depends on both particle’s position
and particle’s characteristics [size, shape, complex refractive
index, and backscattering efficiency Qbs (d p ) defined by the
radar hypothesis in Ref. 22]. In the following, only spherical
particles will be considered.
In Eq. (3), Einc (rj , zj (t)) is the gaussian distribution of the
FIG. 1. Standard PDV setup. 1: Laser (beam intensity I 0 ), 2: optical circula-
tor, 3: optical collimator, 4: moving particles, 5: reference (beam intensity I r ),
irradiance received by each particle at position (r j , zj (t)) (see
6: optical coupler, 7: photodetector and digitizer. Fig. 2). The negative exponential accounts for the attenuation
by scattering (in the forward and backward directions). Some
assumptions must be made to write Eq. (3):
In this article, we study the properties of ejecta clouds with
typical particle sizes between 1 and 10 µm and velocity gradi- • A first order solution of light multiple scattering in the
ents around 1500 m/s. We develop two complementary models ejecta cloud. A scatter-induced attenuation of the beam
of PDV spectra for sparse ejecta cloud using an assumption is taken into account and multiply scattered photons are
on the ejecta size distribution (based on holographic mea- neglected, which is valid for dilute systems.
surements18 ). We show that the integrated areal mass-velocity • Velocities and particle sizes are uncorrelated. This may
distribution estimated from PDV data is in line with piezo- only be true23 for the high-velocity region of the spec-
electric experimental results, making possible to solve the trum (region visible to PDV). For the estimation of the
inverse problem of determining the areal mass of ejecta with optical extinction in each slab, a mean particle diameter
a non-invasive optical diagnostic. dp and a mean extinction efficiency Qext will be inferred
from the size distribution. Based upon previous results
obtained by Sorenson et al.,18 a power law dependence
II. SIMULATION OF PDV SPECTRA with a specific exponent γ d is assumed,
In Secs. II A and II B, two models of PDV spectra for −γ
fd (dp ) ∝ dp d . (4)
high-speed ejecta particles are presented.
We define an effective volume Vp and the corresponding
A. A general simulation of PDV spectrograms diameter dp , determined from f d ,
The physical properties of ejecta cloud and their influence  !  d max ! −1
on PDV spectra will be detailed. If the particles are dragging d max
π 3
Vp = fd (dp ) dp ddp · fd (dp )ddp , (5)
in a gas, the PDV spectrum is time-dependent.19–21 Metallic d min 6 d min
particles ejected in a vacuum will be considered in this arti-
cle, leading to a statistical time-invariant spectrum if the PDV 1
probe’s efficiency does not depend on the viewing distance. In  (1 − γ )(d 4−γd − d 4−γd )  − 3
d max min 
its most general description, the digitized signal U s (t) (in volts) dp =  1−γd 1−γ 
. (6)
 (4 − γd )(d max − d min d ) 
supplied by a collection of N spherical particles depends on
many parameters: the incoming gaussian irradiance Einc (r, z) d min and d max (µm) are the minimum and maximum par-
(W/m2 ), the probe’s efficiency η(r, z), the system’s response ticle diameters. An effective particle extinction efficiency Qext
Rsys (V/W), the coupling parameters and the reference inten-
sity [respectively, α r , α s , I r (W)], the volume fractions inside
the cloud (volume of matter divided by occupied volume),
directly related to the size-velocity distribution, and finally the
phase φj and the velocity V j (m/s) of each particle. The noise-
less expression of the raw PDV signal, corresponding to the
“AC” part of the signal (digitizers are usually AC coupled so
DC terms can be safely ignored), is, for a system of N particles,
X N q
Us (t) = 2Rsys αs αr Ir × Re  η(rj , zj (t))Einc (rj , zj (t))
p
 j=1
"  zj (t) # #
1 
× exp −2 dz . exp 2ik0 Vj t + φj ,
zmin (t) lext (t, z(t))
(3)
where k 0 (1/m) is the wave vector at λ = 1.55 µm and zmin (t) FIG. 2. Coordinates (r p , zp ) of a particle in 3D-space with respect to the PDV
corresponds to the head of the cloud at time t (position of the probe.
is defined similarly, entrance pupil with the back-propagated gaussian mode of the
 d max !  d max ! −1 fiber G0 (r) in the same plane,
Qext = Qext (dp )fd (dp )ddp · fd (dp )ddp . (7)  2
1
d min d min η(rp , zp ) = Es∗ (r, z = 0)G0 (r)dr × . (11)
pupil Einc p , zp )
(r
In Eqs. (5) and (7), f d is the particle size distribution and
Qext (d p ) is the extinction efficiency of a spherical particle cal- For each distance z between the probe and the particle, a
culated using the Mie theory of scattering. The cloud being mean coupling efficiency η̄(z) is obtained [η(r p , zp ) is averaged
inhomogeneous along z due to the velocity gradient, we decide over a disk of radius ΦP /2 located at a distance zp from the
to split it into Ns slabs of identical volume Vs (t). In each par- probe with ΦP the width of the beam],
ticle slab, the volume fraction f vol (z(t), t) is estimated at each η(r, z)Einc (r, z) = η(z)Eu (z), (12)
time t. We make the assumption that in each slab, the num-
ber of particles is large (N(z, z + δz)  1) and the volume where Eu (z) (W/m2 ) is the equivalent uniform irradiance.
of matter in a slab is approximated by N(z, z + δz)Vp . As the Given a discrete number of particles whose positions and
cloud expands in a vacuum, the mesh evolves similarly such velocities are known at t = 0 in the mesh, we can construct
that each slab of particles keeps the same amount of matter, the raw PDV signal at each time step and obtain the PDV
spectrum Φ(V, t) by computing a conventional STFT.
N(z(t), z(t) + δz(t))Vp An example of such a PDV spectrogram Φ(V, t) is shown
fvol (z(t), t) = . (8) in Fig. 3. We observe the typical time-varying speckle fluctu-
Vs (t)
ations coming from the interferences between the Doppler-
The extinction length in Eq. (3), using the first order
shifted wavelets and the reference beam. For neighboring
solution of light scattering, is written as
particles having close velocities, their velocity traces are not
2dp discriminated from the Fourier analysis and a single particle
lext (t, z(t)) = . (9) velocity cannot be extracted in the spectrogram. On the left,
3fvol (z(t), t)Qext
we can observe the evolution of the coherent irradiance inside
To get an estimation of the collection efficiency η(r, z), the cloud versus the penetration distance z.
i.e., the optical power coupled in the system, we detail the
calculation for a fixed single particle of coordinates (r p , zp ). B. Analytical model of PDV spectrograms
Using the radar hypothesis, for a particle in the far field of the
entrance pupil of the probe (lens), the scattered electric field E s We add new assumptions to get an analytical description
impinging on the lens is given by the following relationship: of a PDV spectrum for an N-particle system expanding in a vac-
uum. In the case of linear Richtmyer-Meshkov instabilities (for
dp q kh < 1) and based on recent molecular dynamics results,5,23–25
Es (r, z = 0) = Einc (rp , zp )Qbs (dp ) exp (−ik0 r) (10)
4r the integrated mass-velocity distribution can be very well
q approximated by an exponential function. Consequently, an
with r = (x − xp )2 + (y − yp )2 + zp2 . exponential distribution of velocities will be assumed,
The optical power coupling efficiency η(r p , zp ) is the (V −V min )
scalar product of the field scattered by the particle in the fv (V ) = J0 e−γv V max , (13)

FIG. 3. Noiseless simulated PDV spectrogram Φ(V, t) (right) and evolution of the irradiance Einc (z) inside the cloud (left) from Eq. (3) for a polydisperse
expanding cloud in a vacuum at time t = 500 ns. z = 0 is the head of the cloud (fastest particles) and L cloud = 725 µm. Velocities are exponentially distributed
with γ v = 18 [cf. Eq. (13)], and particle sizes follow a power law distribution with characteristic exponent γ d = 4. d min = 1 µm, d max = 10 µm, V min = 2200 m/s,
Φ2
V max = 3650 m/s, and M s = 2 mg/cm2 . Collimated optical probe (ω 0 = 150 µm, zf = 0), φ P = 300 µm, I0 = Einc (r, 0)π 4P = 400 mW. [Short-Time Fourier
Transform (STFT) parameters: sliding window F w : Hanning, width T w = 50 ns, time step δT w = 10%, zero-padding Z p = 4.]
where J 0 is a multiplicative factor. The integration of f v For a rectangular window of width T w and an exponen-
between V min and V max is equal to the total number of particle tial velocity distribution f v (V ) [Eq. (13)], we finally get for
N tot , each V
V max  γv ∆V
  +∞ γv (Ṽ −V min )
Ntot = J0 1 − e− V max . (14) 2Tw
hΦ(V , t)it∞ =  · βMs · e− 2·V max
p
γv λ −∞
∆V = V max V min . N tot is related to the total areal mass
 3Q V
of the cloud M s (kg/m2 ) through the relationship V max
 Ṽ −V min
ext p
· exp − · β · Ms · e−γv V max
 dp πφ 2 γv
4 Ntot Vp ρ

P
= Ms . (15)
πΦ2E ! 2
  · sinc 2Tw (V − Ṽ ) d Ṽ .
 
−γv V∆V
−e max (21)
ρ is the density of tin (kg m 3 ) and πΦ2E /4 is the surface   λ
of ejection (m2 ). Finally, f v is written as h.it∞ is the time average. If we average an infinite num-
γv πΦ2E 1 (V −V min ) ber of independent speckle realizations, the fluctuations are
fv (V ) = Ms · f g · e−γv V max . removed and the PSD is hΦ(V , t)it∞ , the parameter of the
4 Vp ρV max ∆V
1 − exp −γv V max speckle statistics. When the sliding time step of the Fourier
(16) window is equal to its width, adjacent points in the PDV spec-
trogram are independent and time-averaging is equivalent to
We add two points in order to precise the model: averaging over different particle cloud realizations (ensem-
• The first point deals with the self-expansion of the cloud ble average).  (W) is a multiplicative constant accounting
that links the coordinates z of the particles to their for the collection efficiency of the probe ( = η · Eu ) and
velocities (no dragging), i.e., ∆V = V max V min . The transverse size of the cloud φE is
equal to the width of the beam φP . M s is the total areal mass
N(z)δz = N(V )δV . (17) of the cloud (kg/m2 ), sin c(x) = sin(πx)/πx, and β (m s/kg) is
given by the following relationship:
• The second point is focused on replacing the backscatter-
ing efficiency of each particle with an average efficiency γv πΦ2P 1
β= · g. (22)
Qbs ,
f
4 Vp ρV max 1 − exp −γv V∆V
max
 d max !  d max ! −1
hΦ(V , t)it∞ can be summed to recover the total optical
Qbs = Qbs (dp )fd (dp )ddp · fd (dp )ddp .
d min d min intensity (W). Equation (21) gives an estimate of the average
(18) PSD for a self-expanding particle cloud in a vacuum, whose
size and velocity distributions have, respectively, a power law
As the cloud rapidly self-expands in a vacuum, the first and an exponential dependence. Since the constant  does not
point is meaningful. The second point is only valid if the sta- depend on the properties of the cloud, it will be held constant.
tistical properties of the cloud are invariant by translation in This will not have any consequence for a relative comparison
the x and y directions and if the total number of particles in between different PDV spectra.
each slab is large (N  1). In Fig. 4, we can observe the time-averaged PSD (spectro-
The average PSD, P(V ) = hΦ(V , t)it∞ (W), between veloc- gram of Fig. 3) and the PSD obtained from Eq. (21). The red
ities V and V + δV, is proportional to the average intensity curve would be obtained if an infinite number of independent
scattered in the velocity range [V, V + δV ]. Position being realizations (see Fig. 3) were averaged.
related to velocity, the spatial integral in the exponential argu- With these two models, we get access to a fluctuat-
ment [Eq. (3)] can be replaced with an integral over velocities. ing PDV spectrogram (model 1) and to what is known as
For a collimated beam, Eu (z) = Eu . Introducing the velocity an “average PDV spectrogram” for which the characteristic
distribution f v , the average PSD hΦ(V , t)it∞ is speckle fluctuations are removed (model 2). Before draw-
 +∞ q ing a direct comparison with experimental results, the way
2
hΦ(V , t)it∞ = P(V ) = fv (V ) · η · Eu in which the parametric dependencies of the ejecta cloud
p
−∞ modify the shape of the spectrum has to be considered
 3Q V  V max  2 [Eq. (21)].
ext p
× exp −
 fv (V )dV  dV .

 dp πφ2 V 
P C. PDV spectrum and parametric dependencies
(19)
The influence of ejecta parametric dependencies on the
To take into account the Fourier window F w of finite width PDV spectrum will be briefly discussed, based on the calcu-
T w , the upper definition of hΦ(V√, t)it∞ must be modified and lation of the average spectrogram hΦ(V , t)it∞ from Eq. (21)
a convolution between F w and P(V ) must be computed for (model 2). We recall that this analysis is only valid for a
each velocity V, self-expanding cloud in a vacuum. The integrated areal mass-
2
velocity M(V ) is the cumulative areal mass (mg cm 2 ) between
hΦ(V , t)it∞ = P(V ) ⊗ Fw (V ) .
p
(20) V and V max with M(V min ) = M s .
 FIG. 4. Comparison between hΦ(V, t)it
averaged over 500 ns, obtained from
model 1, and the PDV spectrum
hΦ(V , t)it∞ obtained from model 2
[Eq. (21)] for a polydisperse expanding
cloud in a vacuum. The cloud’s proper-
ties are described in the caption of Fig. 3.
The red curves correspond to Eq. (21)
and the blue ones correspond to Eq. (3).
On the right are plotted the spatial evo-
lutions of the volume fractions inside
the cloud for the 2 models (total length
of the cloud L cloud = 725 µm) at time
t = 500 ns. z = 0 is the head of the cloud
(fastest particles).

1. Influence of the areal mass Ms A change in velocity distribution drastically impacts both
slopes and amplitude of the spectrum hΦ(V , t)it∞ . For large
The properties of the cloud remain unchanged (size and
values of γ v (see green curves in Fig. 6), the velocity distribu-
velocity distributions), except its areal mass M s (mg/cm2 ).
tion is highly peaked at low velocities and the few high-speed
The impact of increasing the areal mass on the mean PDV
particles (near 3650 m/s) are not detected. The location of the
spectrum is shown in Fig. 5. The maxima of the spectrum are
maximum always depends on a compromise between extinc-
shifted to higher velocities. It arises from the fact that particle
tion and scattering of light. For small values of γ v , particle
volume fractions are increasingly densified at high velocities.
densities are increased in the high-velocity region and the
Two phenomena must be taken into account: the first one is
trade-off between attenuation and scattering appears for higher
the increased attenuation by scattering due to increased particle
velocities (around 2750 m/s for γ v = 18).
densities and the second one is the increase of the mean col-
lected intensity (more particles). The penetration of the beam
is large for a small areal mass (see green curves in Fig. 5) 3. Influence of the size distribution
and reduced for a larger one (red curves): the PSD drasti- We study the influence of the exponent γ d on the PDV
cally falls below V = 2500 m/s. hΦ(V , t)it∞ remains large at spectrum,
high velocities, where the beam has not yet been impacted by −γ
attenuation. fd (dp ) = dp d .
In Fig. 7, we can see the impact of increasing γ d . For
2. Influence of the velocity distribution
γ d = 2, the average particle diameter is larger than for a
All properties are still unchanged except the characteristic highly peaked size distribution obtained with γ d = 5.5. As the
exponent γ v of the velocity distribution. We recall the form of areal mass density is kept constant (3 mg/cm2 ), the number of
the velocity distribution f v , particles contributing to the amplitude of the spectrum grows
(V −V min )
with increasing γ d . Regarding the smallest particles (γ d = 6),
fv (V ) ∝ e−γv V max . we can also notice that the optical attenuation of light is more

FIG. 5. Influence of the areal mass M s

(mg/cm2 ) on the average PDV spectrum
hΦ(V , t)it∞ (W) and on M(V ) (mg/cm2 )
inside the cloud. γ d = 4, γ v = 18, V min
= 2200 m/s, V max = 3650 m/s, d min
= 1 µm, and d max = 10 µm.
 FIG. 6. Influence of velocity distribu-
tion fv on the average PDV spectrum
hΦ(V , t)it∞ (W) and on M(V ) (mg/cm2 )
inside the cloud. γ d = 4, M s = 6 mg/cm2 ,
V min = 2200 m/s, V max = 3650 m/s,
d min = 1 µm, and d max = 10 µm.

FIG. 7. Influence of the size distribu-

tion f d on the average PDV spec-
trum hΦ(V , t)it∞ (W) and on M(V )
(mg/cm2 ) inside the cloud. γ v = 18,
M s = 3 mg/cm2 , V min = 2200 m/s,
V max = 3650 m/s, d min = 1 µm, and
d max = 10 µm.

severe and the position of the maximum is shifted to high The unsupported peak pressure at the free surface is approx-
velocities. imately 28 GPa. The signals were recorded with a sampling
This parametric study is of great interest to understand the frequency of 50 GS/s (20 ps time step) during several µs. We
influence of the microphysical properties of the cloud on the do not focus on the ejection mechanism which has been exten-
PDV spectrum. We are aware that a real PDV spectrum is inher- sively studied, and this way experimental conditions will be
ently noisy due to the speckle statistics (Fig. 3) and average kept constant (shockwave pressure, plate thickness, and sur-
PDV spectra shown in Figs. 5–7 will never be obtained in any face finish). PDV probes are gradient index (GRIN) lenses with
ejecta experiment. However, in the case of a self-expanding an output beam diameter of 100 µm (purchased from IDIL
ejecta in a vacuum, a relevant time-averaging can be done to Fibres Optiques, Inc., reference COCOM02472). Despite the
mitigate speckle fluctuations and an average model of the PDV fact that the coupling efficiency of this probe has not been
spectrum can be used to find the best fit and try to resolve precisely defined, our analysis will not be limited since we are
the inverse problem of determining ejecta’s properties with interested in relative comparisons between PDV spectrograms.
non-invasive PDV diagnostic. The optical power delivered by the PDV probe is set to 300 mW
at λ = 1.55 µm. The use of piezoelectric probes for areal mass
measurements has been demonstrated26–28 and we used a stan-
III. EXPERIMENT AND COMPARISONS
dard lithium-niobate (LN) sensor (purchased from Dynasen,
The experimental setup is described below. An electro- Inc.).
detonator is used to generate a shockwave in a 1 mm-thick
A. PDV spectrogram results
tin plate. In the first experiment, the free surface of the plate
is flat (e.g., diamond turned). For the second one, the surface To get an estimate of the free-surface velocity, a first exper-
was finished to get 2D periodic triangular patterns of width iment was performed on a flat tin surface of width 1 mm
60 µm and height 8 µm. The distance between the probes (the setup is described in Fig. 8). Two frontal PDV probes
and the plate is set to 7 mm. The lateral size of the plate is are located 1.75 mm off center at 7 mm from the surface.
4 mm. For the second experiment, the shockwave interaction The shock is not perfectly plane at 1.75 mm and a correction
with the perturbed surface creates metal sheets which fragment must be made since PDV measures the line-of-sight velocity
into high-speed micron-sized metallic particles in a vacuum. vector V z . From a hydrodynamic simulation using CEA
 FIG. 10. Second experiment: Experimental PDV spectrogram Φexp (V, t) for
a shockwave loaded 1 mm-thick tin plate experiment (60 × 8 µm grooves) at
PSB ' 28 GPa, the configuration being described in Fig. 8. Fourier window:
Rectangular, Tw = 50 ns, δTw = 100%, no zero-padding. The black dashed
FIG. 8. Experimental setup. box defines the limits of our analysis: time ranges from t = 1.95 µs to t = 2.9
µs and velocities from V min = 2281 m/s to V max = 3676 m/s. hBΦexp i is the
average background noise of the PDV spectrogram.
Hesione code, an angle of θ ' 6◦ is found between the free-
surface velocity vector V~fs and the z-axis such that V fs ' V z /
cos θ = 2013 m/s (see Fig. 9). This free-surface velocity will be t ' 3.8 µs, the fastest particles hit the probe, corresponding
our reference for the following analysis on piezoelectric and to a time of flight of approximately 2 µs for a particle mov-
PDV data. ing at 3500 m/s. Within the first microsecond, we assume
In a second experiment, simultaneous frontal PDV and that the statistics of the cloud is unchanged (the collection
LN piezoelectric probes have been implemented (Fig. 8). The efficiency of the probe is assumed to be constant) and the
average background noise hBΦexp i is estimated in a region of the data Φexp (V, t) are averaged on a finite interval, between
spectrum Φexp (Fig. 10) without signal (V ≥ 5000 m/s). Since t = 1.95 µs and t = 2.9 µs, to get hΦexp (V )it .
the probe’s collection efficiency is not constant with z (diverg-
ing beam), both the fastest and the slowest particles appear B. Piezoelectric pin results
at later time, the high optical energy density near the probe
increasing the penetration of the beam in the cloud. Around The piezoelectric pin used in this experiment was
a Lithium-Niobate (LN) y + 36◦ -cut crystal of diameter
1.27 mm. The compression of the crystal, induced by accu-
mulation of particles and in the absence of any applied electric
field, leads to the creation of a charge density Di given by

Di = dij σj , (23)

where d ij is the piezoelectric sensitivity and σ j is the applied

stress. It will be assumed that the stress is applied in the z-
direction (i = j for uniaxial compression), that all particles
are ejected instantaneously at shock breakout (t 0 ), and that
particle collisions are inelastic. The sensitivity of the LN-pin
is assumed to be d ii = 24 pC/N and the fail pressure is around
0.6 GPa. The time-dependent voltage response U(t) across an
impedance R (50 Ω) shown in Fig. 11 is linked to the time-
dependent applied stress σ(t) (GPa),
t
1 U(t)
σ(t) ' dt. (24)
dii A t0 R
The integrated areal mass M a (t) is calculated as follows:
FIG. 9. First experiment: Velocity of an explosively driven 1 mm-thick flat t
σ(t) 0
tin surface measured by two frontal PDV probes located at 1.75 mm off center. Ma (t) = dt , (25)
The mean velocity is V fs ' 2013 m/s (with an angle correction of 6◦ ). t0 Vp (t)
 PDV spectrogram (V j , t i ) in the absence of signal, the probabil-
ity distribution of background noise BΦexp (Vj , ti ) (experimental
results) is well approximated by a negative exponential,

BΦexp (Vj , ti )
" #
1
p(BΦexp (Vj , ti )) = exp − . (26)
hBΦexp i hBΦexp i

If the background noise were absent (hBΦexp i = 0), the dis-

tribution of independent values Φexp (V, t i ) (experimental PDV
spectrogram) at different times t i (i = 1, . . ., N) (N being
the number of independent realizations from the PDV spec-
trogram) would follow a speckle statistics. The probability
density function (PDF) would be
" #
1 Φexp (V , ti )
p(Φexp (V , ti )) = exp − . (27)
hΦ(V , t)it∞ hΦ(V , t)it∞

To ensure the independence between Φ(V, t i ) and Φ(V, t j ) at

FIG. 11. LN-pin voltage U(t) (V), located at 7 mm from the sample.
constant velocity, the Fourier analysis on the raw PDV signal
is made with a rectangular window (width T4 = 50 ns, sliding
where V p (t) (m/s) are particle velocities. The expression of
time step δT4 = T4 , and no zero-padding). However, as back-
M a (t) as a function of velocity V is called M(V ). At V = V fs ,
ground noise is present, we must correct Eq. (27) to take into
M(V = V fs ) = M s , the total areal mass of ejecta.
account both the signal Φexp (V, t i ) and average noise hBΦexp i.
In Fig. 12, both pressure on the pin and integrated areal
When background noise is present, the PDF p(Φexp (V, t i )) is
mass versus velocity are depicted. With V fs ' 2013 m/s (first
the PDF of the sum of three random variables:
experiment), the total areal mass is M(V fs ) = M s ' 7.3 mg/cm2 .
• X = Φexp (V, t i ) whose PDF is given by Eq. (27),
C. Maximum likelihood estimation • Y = BΦexp (Vj , ti ) whose PDF is given by Eq. (26),
√ √
• and Z = 2 · X · Y cos(Θ), where Θ √ follows√ an uni-
To make comparison between experiment and simula-
form distribution between 0 and 2π, X and Y being
tions, we make the assumption of a power-law size distribution
described by two independent Rayleigh distributions.
with exponent18,29 γ d = 5.6. Model 2 described by Eq. (21) is
parametric and will be used for simulating average PDV spec- We get,
tra hΦ(V , t)it∞ with the given cloud’s characteristics (γ d , γ v ,
etc.). To the extent that a simple visual comparison is not rel- 1
p(Φexp (V , ti )) =
evant, we perform a Maximum Likelihood (ML) estimation (hΦ(V , t)it∞ + hBΦexp i)
of the parameters with our model. The purpose is to estimate "
Φexp (V , ti )
#
the mass-velocity function M(V ) from PDV data. The model × exp − . (28)
(hΦ(V , t)it∞ + hBΦexp i)
has been reduced to four parameters, two of whom with a real
interest: the integrated areal mass density M(V min ) and the
coefficient from the velocity distribution γ v . The third param- For the experimental time-averaged PDV spectrum
eter is a multiplicative coefficient κ and the fourth parameter is hΦexp (V , t)it , the PDF becomes Gamma distributed30 (PDF of
the background noise amplitude hBΦ i. For a given point of the the mean of N independent speckle realizations) and Eq. (28)

FIG. 12. Cumulative areal mass M(V )

(mg/cm2 ) and applied stress σ (GPa) on
the piezoelectric probe versus reduced
velocity V /V fs with V fs = 2013 m/s,
determined from the first experiment.
 The first three parameters of ζ = [M(V min ), γ v , κ, hBΦ i]
are hidden in hΦ(V , t)it∞ = hΦ(V , t, M(Vmin ), γv , κ)it∞ . Γ is the
Gamma function. M(V min ) will be denoted as M Vmin . The goal
is to find the maximum of the likelihood function defined by

L : (κ, γv , MVmin , hBΦ i)

NV
Y N N hΦexp (Vk , t)iN−1t
7→
k=1
Γ(N)(hΦ(V k , t, κ, γv , MVmin )it∞ + hBΦ i)
N

" #
N hΦexp (Vk , t)it
× exp − . (30)
h(Φ(Vk , t, κ, γv , MVmin )it∞ + hBΦ i)

N V = 101 is the number of points extracted from the

spectrogram, between V k =1 = V min = 2281.3 m/s and V k =101
= V max = 3676 m/s, and N = 20 is the number of independent
temporal PDV slices. A Raphson-Newton algorithm is used
to minimize the negative log-likelihood function. The set of
FIG. 13. Comparison between the normalized experimental time-averaged
parameters, which is the most likely to have generated experi-
PDV data hΦexp (V , t)it (blue) and the PDV spectrum hΦ(V )it∞ obtained from
the ML estimation: ζ o = [M(V = 2281 m/s) = 2.05 mg/cm2 , γ v = 23.28,
mental PDV data, is ζ o = [M Vmin , γ v , κ, hBΦ i] = [2.05 mg/cm2 ,
κ = 6.6 × 10 3 , hBΦ i = 2.8 × 10 3 ] (red), for V between 2281 m/s and 3676 m/s. 23.28, 6.6 × 10 3 , 2.8 × 10 3 ].
Fourier parameters: Rectangular window (Tw = 50 ns, δTw = Tw without zero- In Fig. 13, both experimental time-averaged PDV spectro-
padding). 20 PDV independent slices between 1.95 and 2.9 µs were used to gram hΦexp (V , t)it and average PDV spectrogram hΦ(V , t)it∞
average the spectrum. The black dashed line is the estimated background noise
amplitude hBΦ i. The error bars (± 2 σ̂(V ), standard deviation) are estimated obtained from the ML estimation are depicted. The shape of
from the experimental PDV spectrogram. the PDV spectrogram is well reproduced by our model, taking
into account both increasing PSD related to particle densities
becomes in the cloud and decreasing PSD due to the scatter-induced
attenuation in the low velocity region. For each velocity, the
N N hΦexp (V , t)iN−1
t blue error bars are estimated from√ the experimental PDV
p(hΦexp (V , t)it ) =
Γ(N)(hΦ(V , t)it∞ + hBΦexp i)N spectrogram, σ̂(V ) = (hΦexp (V , t)it / N) on N = 20 uncorre-
" # lated points. Around V = 2600 m/s, the strong peak could be
hΦexp (V , t)it
× exp −N . (29) explained by local density fluctuations in the particle cloud,
(hΦ(V , t)it∞ + hBΦexp i) the standard deviation being underestimated by the Gamma

FIG. 14. Top left, top right, and bot-

tom left: Projections of the constant
likelihood function L(ζ ) such that the
equality of Eq. (31) is verified (joint
confidence region at 95%). M(V min )
has units of mg/cm2 . Scatter plot: ML
estimation. Bottom right: Comparison
between experimental results and ML
estimation of the mass-velocity distribu-
tion M(V ) (mg/cm2 ) for V /V fs between
1.11 and 1.8. The error bar at ±2σ
and V min /V fs = 1.13 is calculated using
the likelihood ratio method (boundaries
of L).
density function of Eq. (29). The joint confidence region for estimation from PDV data being limited to V min = 2281 m/s,
the set of parameters [M(V ), γ v , κ, hBΦ i] is a four-dimensional an extrapolation of M(V ) to smaller velocities is possible
volume in the space of parameters and is estimated with the but uncertain (no information gain). In our case, kh = 0.42
L(ζo )
(k = 2π/60 µm 1 and h = 4 µm) and an extrapolation using the

Likelihood Ratio (LR) method, 2 log L(ζ ) being approxi-
mated as a χ24 (1−α) with α = 0.05 (confidence region of 95%), exponential model of the mass-velocity seems therefore rele-
( ! ) vant. A total areal mass of M s = 11.2 ± 3.4 mg/cm2 is obtained
L(ζo ) from our PDV-based analysis, compared to LN-pin result
ζ : 2 log ≤ χ4 (1 − α) .
2
(31)
L(ζ) M s = 7.3 mg/cm2 (see Fig. 15).
In Fig. 14, three projections of a constant likeli-
hood function are provided to illustrate the joint con- IV. DISCUSSION
fidence region forf the set of g parameters ζ, such that
L(ζ) = L(ζo ) exp − 21 χ24 (1 − α) . κ and hBΦ i have a small Section IV discusses how the estimate is modified when
impact on the estimation. On the contrary, the main uncer- various assumptions are violated and how combining different
tainty on MVmin comes from the coupling of crucial importance diagnostics may improve the analysis.
between MVmin and γ v . The integrated mass-velocity func-
tion M(V ) (bottom right) is deduced from the two parameters A. Random particle arrangement
M Vmin and γ v using In this article, a random particle arrangement is assumed
" #
(V − Vmin ) and fully justified by the speckle nature of the PDV spectro-
M(V ) = MVmin exp −γv . (32)
Vmax gram, hΦ(V , t)it∞ being described by the negative exponential
distribution of a random phasor. The following relationship
A good agreement is obtained between the ML estimation
can be used to test whether or not experimental results follow
and the experimental results [MVmin = 2.05 ± 0.36 mg/cm2 for
this distribution:
the ML estimation and M(V min ' 2.48 mg/cm2 ) for experimen-
tal LN-pin data]. It appears that for V /V fs > 1.3, the LN-pin hΦNexp (V , t)it = hΦexp (V , t)iNt N!. (33)
does not detect the presence of high-speed ejecta particles. This relation has been verified up to N = 4 for the experimental
In this case, an estimate from PDV data is relevant. Further results shown in Fig. 13.
comparisons between model 2 [Eq. (21)] and experimental
PDV data, for which a time-averaging of the spectrum can be
B. Particle shapes
done over a larger interval, are of great interest. If the number
of points used to average the spectrum is large, the proba- The average extinction efficiency Qext is determined using
bility density function of the averaged spectrum will be well the Mie theory only valid for spherical particles. In practice,
approximated by a normal distribution and the ML estimate particles are not perfectly spherical and ellipsoidal shapes may
will converge with which it is obtained from a non-linear least be present in the ejecta. An extended Mie scattering theory
square minimization. Furthermore, since the joint confidence or any electromagnetic solver can be used to study how the
intervals will be reduced, the estimation will be improved. Our analysis might be affected. For randomly oriented ellipsoidal
particles with respect to the PDV beam, the extinction effi-
ciency will not be drastically modified for aspect ratios (the
ratio between the length and the width of the particle) between
1 (sphere) and 2. We believe that assuming particles’ sphericity
is reasonable and introducing ellipsoidal shapes will unneces-
sarily increase the complexity of the problem for a first order
theory of scattering in the ejecta cloud.

C. Particle sizes
In this article, a power law dependence with exponent
γ d = 5.6 has been assumed, between d min = 1 µm and d max
= 10 µm. Since most of the mass is present in the region near
d min , a precise knowledge of the lower bound of the distribution
is of paramount importance. Since it is likely that submicron
particles are present in the ejecta, the impact of lowering d min
down to 0.3 µm must be studied. We also analyze how the
areal mass estimate is modified when the critical exponent is
lowered down to γ d = 3.4.
The impact of lowering d min is shown in Fig. 16. Down to
FIG. 15. Comparison between the ML estimate M(V ) (mg cm 2 ) performed 0.5 µm, the estimated areal mass M s decreases almost linearly.
on experimental PDV data and experimental M(V ) measured with a piezo-
electric LN-pin, for V /V fs between 1 and 1.8. The error bar at ±2σ and
For d min = 0.3 µm, M s increases up to 8 mg/cm2 . In this region,
V /V fs = 1 is estimated from the uncertainty determined at V /V fs = 1.13 using extinction cross sections are so small that their contribution
the likelihood ratio method (boundaries of L). to the extinction of light is reduced. On the contrary, their
 for σ, we get 21.2, and 22.2 and 28.4 mg/cm2 . The increased
estimate is justified by the reduced number of small particles
(tail of the log-normal distribution) in the cloud.

D. Multiple scattering
The first order solution presented in this paper does not
take into account multiply scattered photons. For the high-
velocity region of the spectrum, they may contribute17 at
second order in the formation of the Doppler signal. The
discrepancy between our model and experimental results
observed around 2600 m/s in Fig. 13 may also be explained
by the multiple scattering of light in the ejecta cloud. After the
first scattering event, the photon can be preferentially scattered
in the forward direction and will not acquire any Doppler shift
since incoming and scattered directions will be identical. If this
event takes place in the high-velocity region of the cloud above
FIG. 16. Influence of lowering d min on the estimated areal mass M s from 2600 m/s, a second scattering may eventually occur in the back
PDV data (power law particle size distribution with exponent γ d = 5.6). direction around 2600 m/s, increasing the PSD in this region
with respect to the amplitude obtained using a first order the-
ory of scattering. Further investigations are deemed necessary
contribution to the total mass of the cloud is not negligible. The to validate this assumption. The general method described in
evolution of the estimated areal mass M s versus γ d is shown this paper can be applied to other models of average PDV
in Fig. 17. The decrease is mostly attributed to an increased spectrum hΦ(V , t)it∞ for which more physics is taken into
average particle diameter in the ejecta cloud. account.
The analysis will drastically be improved by a precise
knowledge of the size distribution. Indeed, the inversion from E. Combining different diagnostics
PDV data is severely under-constrained and the data will never Additional diagnostics are needed to improve the inver-
be inverted if a particle size distribution is not assumed (the sion. An advantage of PDV inversion is that the retrieved
consistency of the ML estimator would be lost). Further exper- areal mass depends neither on the time of shock breakout
imental investigations are needed to relax this strong assump- nor on the distance between the PDV probe and the cloud
tion. A log-normal distribution may also be assumed for the (for a collimated beam). Asay foil, X-ray absorption and
size distribution:31 piezoeletric pins diagnostics can be used to measure the mass-
(log dp −log µ)2 velocity distribution in the ejecta cloud. In this case, the mass-
1 −
fd (dp ) ∝ √ e 2σ 2 (34) velocity function can be related to the velocity distribution
dp σ 2π
M(V ) ∝ ∫ VVmax fv (V )dV . Different particle size distributions can
For µ = 2 and σ = 0.1, 0.2, and 0.5, M s are, respectively, equal accordingly be tested to find out which one is likely to explain
to 13.9, 14.3, and 19.3 mg/cm2 . For µ = 3 and the same values experimental PDV results. Further experiments using Asay foil
diagnostic will be very relevant and will certainly give more
information on the shape of the mass-velocity distribution (the
assumption of an exponential function will be relaxed). Small
Asay foils32 may be implemented nearest to a PDV probe such
that the two diagnostics will look at the same region of the
ejecta. As already discussed in Sec. IV C, any particle sizing
diagnostic will also drastically improve the analysis.

V. CONCLUSION
In this article, we have developed two models to study
the way how the properties of an ejecta cloud expanding in
a vacuum influence the shape of the time-velocity PDV spec-
trogram. The first one accounts for the speckle fluctuations
observed on PDV data and the second one describes an average
spectrum which would be obtained from averaging an infi-
nite number of realizations. This model [Eq. (21)] is used to
study the influence of the areal mass, the size and the velocity
FIG. 17. Influence of lowering γ d on the estimated areal mass M s from PDV
distributions. A comparison between the time-averaged PDV
data (power law particle size distribution with exponent γ d ). d min = 0.5 µm spectrum measured on a shock-loaded tin plate experiment
and d max = 10 µm. and our model is presented, showing promising comparisons
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