Molecular dynamics simulations of bubble nucleation in dark matter detectors

0.53 MB·

by  Philipp Denzel

Molecular dynamics simulations of bubble nucleation in dark matter detectors
Philipp Denzel,∗ Ju¨rg Diemand, and Raymond Ang´elil
Institute for Computational Science, University of Zurich, 8057 Zurich, Switzerland
(Dated: January 28, 2016)
Bubble chambers and droplet detectors used in dosimetry and dark matter particle search ex-
periments use a superheated metastable liquid in which nuclear recoils trigger bubble nucleation.
This process is described by the classical heat spike model of F. Seitz [Phys. Fluids (1958-1988)
1, 2 (1958)], which uses classical nucleation theory to estimate the amount and the localization of
the deposited energy required for bubble formation. Here we report on direct molecular dynamics
simulations of heat-spike-induced bubble formation. They allow us to test the nanoscale process
6 describedintheclassicalheatspikemodel. 40simulationswereperformed,eachcontainingabout20
1 millionatoms,whichinteractbyatruncatedforce-shiftedLennard-Jonespotential. Wefindthatthe
0 energy per length unit needed for bubble nucleation agrees quite well with theoretical predictions,
2 but the allowed spike length and the required total energy are about twice as large as predicted.

n This could be explained by the rapid energy diffusion measured in the simulation: contrary to the
a assumption in the classical model, we observe significantly faster heat diffusion than the bubble
J formation time scale. Finally we examine α-particle tracks, which are much longer than those of

7 neutronsandpotentialdarkmatterparticles. Empirically,αeventswererecentlyfoundtoresultin
2 louderacousticsignalsthanneutronevents. Thisdistinctioniscrucialforthebackgroundrejection
in dark matter searches. We show that a large number of individual bubbles can form along an α

] track, which explains the observed larger acoustic amplitudes.
h
p
-
p I. INTRODUCTION Thanks to increasing computing power, and highly
m scalableandefficientmoleculardynamicscodes,itisnow
Liquids can be heated to a temperature higher than possible to simulate phase transitions directly, within
o
c their boiling point, without yet going through a phase a realistic environment[18–20]. Homogeneous liquid-to-
. transition. The liquids then exist in a metastable, or su- vapor nucleation has recently been simulated by some
s
c perheatedstate,whichcanbelocallydisruptedtoinduce of us [21], and the resulting nucleation rates and bubble
i bubblenucleation. Forexample, thescatteringofcosmic properties were found to differ somewhat from classical
s
y radiation off the liquid nuclei is an energy deposition, nucleation theory. In this work we simulate a similar
h which can lead to bubble nucleation. liquid, just at a lower level of superheating to avoid ho-
p Inthe1950s,thisprocessmotivatedtheuseofsuchsu- mogeneous nucleation, and we add model heat spikes to
[ perheated liquids to detect particles in bubble chambers probetheirabilitytoinducebubbleformation. Thesim-
1 [1], the most notable being Gargamelle at CERN, which ulations presented here provide an atomistic description
v is credited with the discovery of weak neutral currents of heat-spike-induced bubble nucleation.
0 [2]. While these detectors have been used extensively in
9
the past, their applications today are limited mostly to
3
monitoring and dosimetry [3].
7 II. HEAT SPIKE MODEL
However, recent research projects, such as SIMPLE
0
. [4–6], PICASSO [7–9], COUPP [10, 11], and PICO [12]
1 The classic heat spike model underlies the theory of
employ bubble chambers (or superheated droplet detec-
0 the bubble detector’s functionality and induced bubble
tors—modifications of bubble chambers) in the hope of
6 nucleation, and was developed by Seitz in 1958 [22]. It
1 directly detecting dark matter in the form of weakly in-
has since been used to calibrate various detectors. How-
: teracting massive particles (WIMPs). While detection
v ever, its assumptions and predictions on the nanoscale
in bubble chambers used to consist of event picture tak-
i processofbubblenucleationremainuntested. Themodel
X ing, nowadays this is achieved by taking advantage of
describestheformationofbubblesinhighlylocalizedand
r the explosive nature of the phase transition. The ex-
a plosions are accompanied by an acoustic shock wave, hotregions. Theformationprocessofamacroscopicbub-
ble can be split into two parts: the formation of a mi-
which is recorded by piezoelectric transducers, located
croscopicprotobubbleornucleus,followedbythegrowth
on the chamber walls [13]. Superheated liquid detectors
phase into a macroscopic, gas-filled bubble. Because we
arecompetitiveandcomplementarytootherdarkmatter
wish to determine the deposited energy and track length
detection methods [14–16]. However there is still some
thresholds for stable bubble formation, in this paper our
discussiononseveralissues,onebeingthediscrimination
focus remains on the first part.
of WIMP signals from the α-particle background [17].
A spherical bubble in mechanical equilibrium experi-
ences a few competing influences: the outwards pressure
of the vapor, the pressure from the surface tension act-
∗ [email protected] ingtocollapsethebubble,andthefluidpressurepushing
2
inwards. This leads to the condition SIMPLE experiment can be approximated quite well us-
ing the following factors when converting from LJ to SI
2γ
R = , (1) units: σ = 0.501 nm, (cid:15) = 377.65k K = 0.0325 eV and
c ∆p m = 4.466 g/mol. At 9℃(the runBning temperature of
where R is the minimal critical radius, γ the surface SIMPLE) this results in an equilibrium pressure and a
c
tension,and∆p=p −p thepressuredifferencebetween surface tension which both agree to within 26 percent
v l
the (low) pressure pl in the superheated liquid and the between C2ClF5 [25] and our LJ fluid. The main dif-
vapor pressure inside the bubble p . ference is that our simulations have a higher amount of
v
Theenergyrequiredfortheformationofabubblewith superheat, a smaller critical radius (10.47σ = 5.26 nm
critical radius R is the sum of surface energy and the vs. around 40 nm in SIMPLE [6]) and a much smaller
c
energy of vaporization required energy deposition (Em = 2164(cid:15) = 70.42eV vs.
a few keV in SIMPLE). Simulating heat spikes at com-
4π parable thermodynamic conditions to those in the ex-
E =4πγR2+ R3n H , (2)
m c 3 c v s periments is therefore possible with the same interaction
potential, but will require significantly larger simulation
where n is the number of moles per unit volume at the
v volumes(whichiscomputationallyveryexpensive)toac-
equilibrium pressure and H is the heat of sublimation
s commodatetheformationoflargecriticalbubblesatcon-
per mole. This means that bubbles are formed if suffi-
stant ambient pressure.
cient energy is deposited along a track of critical length
To prepare the liquid in a superheated, metastable
l = ΛR or shorter. The classical heat spike model as-
c c
state, we use a procedure similar to that of Diemand
sumes that the critical length is equal to the diameter
et al. [21]: First19’652’000atomsareplacedonalattice
of the critical bubble, i.e., Λ = 2. The deposited energy
corresponding to a density of ρ=0.58mσ−3. The parti-
E is calculated using the mean energy deposited per
dep
cles are then randomly assigned velocities corresponding
unit distance, or linear energy transfer (hereafter LET):
to a temperature of T =0.95(cid:15)/k . A fixed time step of
B
(cid:112)
E dE ∆t=0.0025τ =0.0025 σ2m/(cid:15)wasusedthroughoutall
LET = m ≤ dep. (3)
c ΛR dx runs. A run of 10’000 time steps is initiated under NVT
c
integration (constant particle number, volume, and tem-
perature ensemble) at a temperature of T = 0.95(cid:15)/k .
B
These are stable liquid conditions, used to equilibrate
III. SIMULATION SETUP the atoms which were initially placed on a lattice. The
liquid is superheated by lowering the temperature from
T = 0.95(cid:15)/k to T = 0.855(cid:15)/k over a relatively short
We use the classical molecular dynamics code, the B B
period of 15’000 steps as an NVT ensemble.
Large-scale Atomic/Molecular Massively Parallel Simu-
Another NVT run of 15’000 steps is performed to sta-
lator (or LAMMPS) [23]. We use simulation boxes of
bilizethesystematthenewtemperatureT =0.855(cid:15)/k .
length L = 323.6σ with periodic boundary conditions. B
Thethermostatisthenturnedoff,andthesimulational-
The interaction between the particles is described by a
lowed to continue for 450’000 time steps with simple di-
truncated force-shifted Lennard-Jones (LJ) potential:
rect integration of the classical equations of motion (i.e.,
(cid:26) u (r)−(r−r )u(cid:48) (r )−u (r ) as a microcanonical or NVE ensemble). A typical sim-
u (r)= LJ c LJ c LJ c
TSF 0 if r >r , ulation spreads 256 MPI tasks over the same number of
c
(4) cores, and runs for between six and eight hours.
with the well-known Lennard-Jones potential u , and Bubbles are identified with the same method as in
LJ
a cutoff distance r = 2.5σ, where the potential and [18, 26]: The simulation volume is divided up into cubic
c
force smoothly approach zero. At our run tempera- cells of size (3σ)3. Cells with a density ρ<0.2mσ−3 are
ture (T = 0.855(cid:15)/k ) the truncated force-shifted LJ marked as vaporized regions and are recursively linked
B
fluid has a surface tension of γ = 0.0895(cid:15)σ−2, a la- withneighboringvaporcellsintoaconnectedvaporbub-
tent heat of H = 6.9(cid:15), and an equilibrium pressure of ble. As expected, these runs did not result in sponta-
s
P =0.0461(cid:15)σ−3 ((cid:39)p ; see [21, 24] for details). At the neously nucleated bubble formation due to the fact that
eq v
simulation density (ρ = 0.5792mσ−3) the liquid pres- thedensitywastooclosetothestableequilibriumvalue,
sure is p = 0.028(cid:15)σ−3, which results in a minimal crit- making the free energy barrier between the liquid and
l
ical bubble radius of R = 10.47σ and required energy the gaseous phase too high for spontaneous homogenous
c
deposition of E = 2164(cid:15), where ∼ 2041(cid:15) is used for bubble nucleation.
m
vaporization and ∼ 123(cid:15) goes into work against surface Following Seitz’s model, we mimic the nuclear recoil
tension. energy injection heat spike by assigning higher velocities
LJ fluids are often used to model noble gases. For- to all particles within a cylindrical region (see the upper
tunately our LJ fluid also has thermodynamic proper- panels of Fig. 1). Each simulation continues for 200’000
ties similar to those of the liquids used in dark matter to 500’000 time steps after the heat spike, in order to
search experiments. For example C ClF used in the determine whether or not a stable, growing bubble man-
2 5
3
agestoformoutoftheheatspike. Intotal40simulations Wetestedthestabilityofoursimulationsbyobserving
with different deposited energies and track lengths were the box pressure after the heat spike throughout all the
performed. runs and found a stable pressure of P = 0.0288 ±
init
0.002(cid:15)σ−3 averaged over 10’000 time steps. This tells us
that the simulation box size and number of atoms that
we use to model our fluid, are chosen reasonably and do
not effect the physics of the problem investigated.
time [ns]
0 7 15 22 30 37 44
65450
stage 1 stage 2 stage 3
20 33510
σ 3σ
of of
nits 15 14137nits
u u
n n
s iRc 4189 e i
u m
di u
FIG. 1. (Color online) Projected snapshots of run at 7.5τ, ra ol
37.5τ, 125τ, 500τ, 812.5τ, and 1250τ after a heat spike of v
5 524
12k(cid:15) along a length of 8R .
c nucleated bubble
collapsed bubble
0 0
0 200 400 600 800 1000 1200
time in units of τ
IV. SIMULATION RESULTS
FIG. 2. (Color online) Three-stage bubble growth plot of a
The bubble nucleation event can be split into three run with a heat spike of 5000(cid:15) (or 3000(cid:15)) along a track of
stages (see Fig. 2). 1Rc yielding a stable growing (collapsing) bubble.
1. Thefirststageisprecededbyaheatspike,whichre-
Figures 3 and 4 show the outcome of simulations with
sultsinanexplosionofthebubbleradius[27],with
a fixed cylinder radius of 2σ in a plot of the deposited
an initial expansion rate of about 0.40σ/τ and up
energy ∆E relative to the length of the cylinder l .
to 1.0σ/τ for simulations with large energy depo- c
sition. In all cases we observe an overshoot right
afterthisearlyexplosivephaseandthebubblevol-
ume decreases slightly. If the heat spike produces V. COMPARISON TO THEORY
a bubble with a radius of about 10σ to 15σ at the
end of the first stage, it generally is stable and will Seitz’smodelpredictsthattheenergythreshold∆E in
transition into the second growth stage. Bubbles our simulated fluid is 2164(cid:15)=70.3eV. From the energy
which are too small, however, are unstable, and inputs required to generate a growing bubble (Fig. 3)
recollapse completely. we find ∆E(Λ=2)=4166.1±125.0(cid:15)=135.4±4.1eV,
whichisaboutafactorof2higher. Thelengthscaleover
2. During the second stage we observe linear growth
which the energy must be deposited (ΛR ) is also un-
perturbed by small oscillation patterns. Since the c
derestimated by the theory: According to Seitz, the en-
frequency of the oscillations matches the speed of
ergycontributingtoonebubbleformationeventmustbe
soundinsidethemediumoverthelengthofthebox,
confined within a length of 2R ; i.e., his model assumes
the patterns can be explained by a pressure wave c
Λ = 2. The simulations however show that the critical
propagating through the simulation box.
energy can be spread over up to 4 critical lengths with-
3. In the third stage, the bubble grows linearly un- out losing bubble formation efficiency—a discrepancy of
til the system pressure increases due to the fi- a factor of 2. On tracks longer than Λ=4, the required
nitesimulationboxsize,whichunrealisticallyslows total energy for bubble formation is higher, since only a
the growth of large bubbles in the final stages of fractionofthisnowtoowidelyspreadenergycontributes
our simulations. Typical average expansion speeds to a single bubble formation event.
over the linear regime are around 0.0085σ/τ to For analysis it is also convenient to consider the rela-
0.0130σ/τ. These growth rates are in good agree- tionship between the LET [see Eq. (3)] and the cylinder
ment with Ang´elil et al. [24], where the bubble length (see Fig. 4). For Λ = 4 and all larger values
growthofhomogeneousnucleationwasinvestigated we find LET values which are consistent with a constant
under similar conditions. value of 69.31 ± 4.41(cid:15)/σ = 4.50 ± 0.29keV/µm. The
4
gies are almost equal, even though the number of atoms
in the cylinder differs by a factor of (cid:0) 2σ (cid:1)2 ≈ 7.22.
1000 0.744σ
growing bubble 1600 Formuchhighercylinderradii,therequiredenergymust
indeterminate be higher also, because heat diffusion is accelerated with
no stable bubble 1400
800 more atoms.
prediction: E=2164†
1200 Althoughthemodelmakessomequiteaccuratepredic-
tions (e.g. for the LET), it contains several unrealistic
V] 600 1000V] simplifying assumptions:
e e
[ [
E 800 E
∆ ∆ • Themicrobubblethermophysicalpropertiesareas-
400
600 sumedtobethesameasinaplanar,static,isother-
mal interface in equilibrium between bulk liquid
400
200 and bulk vapor. The inclusion of a more realis-
200 tic surface tension, specifically, its dependence on
bubble size [28, 29], should be addressed in future
0 0
0 1 2 3 4 5 6 7 8 926 27 2853 54 55 refinement of the model.
Λ
• Viscosity is not included in the model.
FIG. 3. (Color online) Energy plot of all runs with cylinder
radius rcyl =2σ and cylinder length lcyl =ΛRc. The critical • Microbubbles are assumed to be spherical and to
energy to nucleate a bubble lies around 4500(cid:15), and can be have a steplike vapor-liquid transition region. But
spread over up to a length of four critical radii, which is in
critical bubbles are nonspherical and the width of
both cases a factor of 2 greater than the theoretical predic-
their transition regions can be comparable to their
tions.
radii [24].
• A more realistic model would include the compli-
theoretical prediction for the LET is quite accurate, just
cated and evolving non-isothermal bubble temper-
somewhat larger: 89.80(cid:15)/σ =5.83keV/µm.
ature profiles due to the latent heat of transforma-
The LET plots show that the Seitz predictions for the
tionaswellascompressiveheatinthefluiddirectly
LETareaccurate,providedthattheLETissustainedfor
outside the bubble due to rapid bubble expansion.
at least four times the critical radius. Twice the critical
Significant non-isothermal effects are found in our
radius is insufficient. Even up to lengths of Λ = 26.88
heat spike simulations and also in homogeneous
and Λ=53.71 for α-particle runs, the results match the
bubble nucleation simulations; see [24].
predictions.
VI. DISSIPATION OF HEAT
growing bubble
25 indeterminate 25 Seitz’s model assumes that a critically-sized bubble
no stable bubble formsbeforethedepositedenergydiffusesoutofthecrit-
] ]
m prediction: LET =5.82keV/µm m icalvolume. Asweshallshow,thisisnotthecaseforour
µ µ
/20 20/
V V simulated liquid. To establish a criterion related to the
e e
k[ k[ dissipation of the deposited energy, we assume that the
r) crit∗15 15r) crit∗ edniuesrgRy0issmdeaplolesrittehdaunnRifoc.rmTlyheinvoaluspmheersihcaalllrbegeioenquoaflrtao-
Λ Λ
E / (10 10E / ( asimcyillainrdteoravotlruamckecoofvleernegdthby2Racreacnodil.craosiss scehcotsioenn sπuac2h,
∆ ∆ that 4πa3 is the mean molecular volume, where a is the
5 5 3
van der Waals radius. Then, the critical time τ for the
c
dissipation of heat in a spherical spike of radius r is de-
00 1 2 3 4 5 6 7 8 926 27 2853 54 550 termined by setting r =Rc:
Λ
R2
τ = c, (5)
FIG. 4. (Color online) LET plot of same runs as in Fig. c 4D
3. The LET matches the theoretical prediction, provided a
length of at least four critical radii. inwhichD isthediffusioncoefficientforheat. Thisisan
approximation,yetallthequantitiesaremostlyconstant
We have also investigated the dependence of the de- with variable pressure, so the values should be accurate
positedenergyonthecylinderradius. Forsmallcylinder to a factor of 2. With Eq. (5) one can assign an average
radii (0.744σ and 2σ) and small Λ, the required ener- minimumvelocitywithwhichthewallsofabubblemust
5
expand to achieve stability before the deposited heat is surements were much too noisy to deduce anything from
dissipated, them.
R
v = c (6)
c τc 2.6 Rc 0.63−
σ
Iovtaflisusoeuisnn,tdeSreieinsttztinhfigendtliosquctiohdmatvpsavocruenidst.haiCslmovmoeslpotcaairtiynfagwcttiothhreotlhiftee1r0saptleueersdes 22..42 Eρfitkin 0.5 units of
twhhainchvtshouendw.allCsoonfstehqeuebnutblyb,lethheavaevetroageexpvaenlodciitnyowrditehr †s of 2.0 0.4 me in
to form a bubble during a time around τc is subsonic unit1.8 0.3 olu
(apsoesnepehn[e2tr2iea]llafyot.rthmBeyocrerfiittdticienatglairlasa)nd.iuesxTpihsoeneekxniptneieactltiecddeectnaoeyrdgefyucaniynctsieiodxne- E in kin11..46 0.2 ritical v
c
to the simulation data, the diffusion time and veloc- 1.2 n
ivty=ca3n.5b4eσo/bτt=ain6e0d1:.0τ3cm=/s3.(4s1eeτF=ig.6.852).×T1h0e−s1e2vsalauneds 1.0 0.1 sity i
c n
can be compared to the diffusion time calculated with 0.80 20 40 60 80 1000.0 de
literature values for Lennard-Jones MD simulations, ac- time in units of τ
cording to Eq. (5): v = 96.02m/s. This value is
diss
in agreement with the simulations if one takes into ac- FIG. 5. (Color online) Evolution of kinetic energy and den-
count that the analysis assumes perfectly spherical heat sity within a spherical volume of radius R . By fitting the
c
spike bubbles, which is not the case in our runs (see Fig. data for the kinetic energy with an exponential decay func-
1). Seitz assumes that critical bubble formation occurs tionEkin(t)=A·exp(K·t)+btherelaxationtimeτc =K−1
within the dissipation time scale τ . We find that this is canbeobtained. Theresultsare(A,K,b)=(1.3195,7.4640e-
c
4, 1.1124) with squared diagonal elements of the covariance
clearly not the case: The kinetic energy to density com-
matrix (6.8294e-7, 3.7630e-19, 8.5889e-11) in units of ((cid:15),
parison makes it clear that the heat dissipation is sig-
0.0025·τ, (cid:15)).
nificantly faster than the bubble expansion: the average
kinetic energy in a sphere of critical radius disappears
before the bubble number density drops. The measured
initial expansion velocities imply that the heat dissipa-
tion is a factor of 3−10 times faster then the bubble VII. α-PARTICLE DISCRIMINATION
expansion.
On the other hand, Seitz compares his diffusion ve- Superheated liquids function as threshold detectors.
locities with acoustic velocities of the same media and All particles able to deposit more energy than this (tun-
concludes that the diffusion is subsonic. The contrary is able) detector threshold will lead to bubble nucleation
the case for our simulations. The expansion velocities of and an observable acoustic signal. A setup to search
simulated bubbles are 3 to 10 times slower and lie in the for WIMPs is also sensitive to α particles, because they
same range as the speed of sound for a monoatomic gas. are able to deposit higher energies.Tracks from α par-
InFig. 5,weshowtheevolutionofdensityandaverage ticles are far longer, and their discrimination is essen-
of the kinetic energy, which is proportional to tempera- tial for a convincing dark matter detection. Empirically
ture, averaged over a fixed volume of radius Rc centered it has been found that the longer α-particle tracks lead
on the midpoint of the heat spike, which was inserted at to larger acoustic signals compared to neutrons [11, 17]
time t=0. After a time of about 60 τ the density values (which have tracks similar to typical WIMPs), presum-
do not change significantly anymore inside this volume, ably because a large number of bubbles can form along
since the bubble exceeds the critical size after this time. a longer track. However the origin of the louder sig-
Aquickdimensionalanalysiswithvaluesfrom[30]letsus nal remains unclear. Currently there is some discussion
expect a thermal conductivity of 0.1 JK−1m−1s−1, and aroundrobustnessoftheα-particlediscrimination[6,17].
we get a heat flow of Qt = 3.4J/τc. The internal energy For this reason, we go beyond testing Seitz’s model, and
per atom is U = 3k T ≈ 10−21J within a time of τ , simulate far longer tracks (see Fig. (3) and (4)). Only
2 B c
which means that an energy of about 32’000 atoms can a small fraction of an α track can be simulated, because
be transported out of the critical volume. These dimen- the computation cost of simulation boxes of lengths in
sions lie well within the range of our simulation results. the µm range would be too large. For the α runs we
Itwouldalsobeveryinterestingtomeasuretheevolu- set the track length equal to the box size L. We also
tion of pressure inside the bubble volume after the heat ran additional large box α simulations with L = 647.1σ
spike, which then would give insights to the surface ten- and157’216’000atomsandfoundconsistentresults: Fig.
sion at the bubble boundary, i.e., the surface tension as (4) shows that even for long tracks with Λ = 53.71, the
a function of the bubble radius. Unfortunately, the mea- required LET is the same. α tracks above the required
6
LET produce several stable bubbles separated by only larger-than-expectedrequiredenergydepositionfor
4 to 5 critical radii. Rescaled to an experiment such as bubble formation.
SIMPLE (R ∼ 40 nm), this indicates a bubble density
c • In the bubble-chamber dark matter particle search
of 5 per micrometer and around 200 microbubbles on a
communitythereisstillsomediscussiononthedis-
typical α-particle track. The acoustic signal is generated
crimination of WIMP signals from the α-particle
mostlyinthelineargrowthphase[17]andalargenumber
background [17], which depends on exact energy
of linearly growing (and later merging) bubbles explains
predictions from the model of induced bubble nu-
the louder signal compared to the single bubble case.
cleation. By giving corrections to the energy pre-
dictions of the model, this paper and further work
couldhelpinthecorrectcalibrationofthedetectors
VIII. CONCLUSIONS and analysis of data, especially the discrimination
ofα-particlebackground,andgenerallygivesabet-
ter understanding of the measuring processes.
Wehavestudieddarkmatterinteractioninducedbub-
ble nucleation using direct molecular dynamics simula-
Large future simulations would be needed to explore
tions of the process. Our molecular dynamics simula-
more stable fluid critical bubbles as large as in the
tions of dark matter bubble detectors are the first at-
detectors. The rescaled thermodynamic properties of
tempt of an atomistic, computational description of the
our LJ fluid are comparable to some of the detector
heat-spike-induced bubble nucleation event. Our results
fluids; however to closely match one specific detector
qualitatively confirm the general framework assumed in
it would be worthwhile to simulate more specific and
the classical heat spike model by Seitz, but also show
more complicated fluids. Our simulations demonstrate
some interesting quantitative differences.
complications (bubble properties, non-isothermal ef-
fects, etc.) which are neglected in the classical heat
• In comparison to the model predictions, for sta- spike model by Seitz, which might provide an opportu-
ble bubbles to be successfully nucleated, we find nityforconstructingmorerealisticmodelsoftheprocess.
that the deposited energy must be approximately
twice as much, whereas the length over which it is
deposited can be twice as long. This means that
the model predictions actually are rather accurate ACKNOWLEDGMENTS
for the linear energy transfer prediction for tracks,
which are at least 4 critical radii long, twice the We thank Kyoko and Hidekazu Tanaka, Tom Girard,
length assumed in the model. The measured heat andtheSIMPLEteamforvaluablediscussionsandfeed-
diffusiontimescaleisshorterthanthetimeittakes back. We also thank the anonymous referees for useful
to form a critical size bubble, contrary to what comments and suggestions. J.D. and R.A. acknowledge
is assumed in the model, which could explain the support from the Swiss National Science Foundation.
[1] D.A.GlaserandD.C.Rahm,Phys.Rev.97,474(1955). [9] S. Archambault et al. (PICASSO Collaboration), Phys.
[2] F. Hasert et al. (Gargamelle Collaboration), Phys. Lett. Lett. B 711, 153 (2012).
B 46, 121 (1973). [10] E. Behnke et al. (COUPP), Phys. Rev. D 86, 052001
[3] R. E. Apfel, Nucl. Instrum. Methods 162, 603 (1979). (2012).
[4] M. Felizardo, T. Morlat, A. C. Fernandes, T. A. Girard, [11] E.Behnkeet al.(COUPP),Phys.Rev.D88,021101(R)
J. G. Marques, A. R. Ramos, M. Auguste, D. Boyer, (2013).
A. Cavaillou, C. Sudre, et al. (SIMPLE Collaboration), [12] C. Amole et al. (PICO Collaboration), Phys. Rev. Lett.
Phys. Rev. Lett. 105, 211301 (2010). 114, 231302 (2015).
[5] M. Felizardo, T. A. Girard, T. Morlat, A. C. Fernandes, [13] L.Hamel,L.Lessard,L.Rainville,V.Zacek, andB.Sur,
A. R. Ramos, J. G. Marques, A. Kling, J. Puibasset, Nucl. Instrum. Methods A 388, 91 (1997).
M. Auguste, D. Boyer, et al. (SIMPLE Collaboration), [14] E. Aprile et al. (XENON100 Collaboration), Astropart.
Phys. Rev. Lett. 108, 201302 (2012). Phys. 35, 573 (2012).
[6] M. Felizardo, T. A. Girard, T. Morlat, A. C. Fernandes, [15] T. C. I. Collaboration, Science 327, 1619 (2010).
A. R. Ramos, J. G. Marques, A. Kling, J. Puibasset, [16] G. Angloher et al. (CRESST-II Collaboration), Eur.
M. Auguste, D. Boyer, et al. (SIMPLE Collaboration), Phys. J. C 74, 3184 (2014).
Phys. Rev. D 89, 072013 (2014). [17] F.Aubinet al.(PICASSOCollaboration),NewJ.Phys.
[7] S. Archambault et al. (PICASSO Collaboration), Phys. 10, 103017 (2008).
Lett. B 682, 185 (2009). [18] J. Diemand, R. Ang´elil, K. K. Tanaka, and H. Tanaka,
[8] M. Barnab´e-Heider et al. (PICASSO Collaboration), J. Chem. Phys. 139, 074309 (2013).
Nucl. Instrum. Methods Phys. Res. A 555, 184 (2005). [19] R. Hol(cid:32)yst, M. Litniewski, and P. Garstecki, Phys. Rev.
7
E 82, 066309 (2010). 051604 (2010).
[20] R. Hol(cid:32)yst, M. Litniewski, and P. Garstecki, Phys. Rev. [27] We assume spherical bubbles when converting the ob-
E 85, 056303 (2012). servedbubblevolumetoaradius.However,immediately
[21] J. Diemand, R. Ang´elil, K. K. Tanaka, and H. Tanaka, after the heat spike, the bubbles are far from spherical
Phys. Rev. E 90, 052407 (2014). (see Fig. 1). Large, stable bubbles eventually do become
[22] F. Seitz, Physics of Fluids (1958-1988) 1, 2 (1958). spherical [24].
[23] S. Plimpton, J. Comput. Phys. 117, 1 (1995). [28] R. C. Tolman, J. Chem. Phys. 17, 333 (1949).
[24] R. Ang´elil, J. Diemand, K. K. Tanaka, and H. Tanaka, [29] K. K. Tanaka, H. Tanaka, R. Ang´elil, and J. Diemand,
Phys. Rev. E 90, 063301 (2014). Phys. Rev. E 92, 022401 (2015).
[25] M. Okada, T. Arima, M. Hattori, and K. Watanabe, J. [30] S.-H. Lee, D.-K. Park, and D.-B. Kang, Bull. Korean
Chem. Eng. Data 33, 399 (1988). Chem. Soc. 24, 178 (2003).
[26] H. Watanabe, M. Suzuki, and N. Ito, Phys. Rev. E 82,