Collisions of CO$_2$ Ice Grains in Planet Formation

2 MB·

by  Grzegorz Musiolik

Draft version January 20, 2016
PreprinttypesetusingLATEXstyleemulateapjv.5/2/11
COLLISIONS OF CO ICE GRAINS IN PLANET FORMATION
2
Grzegorz Musiolik∗, Jens Teiser, Tim Jankowski, Gerhard Wurm
Fakult¨atfu¨rPhysik,Universit¨atDuisburg-Essen,Lotharstr. 1,47048Duisburg,Germany
Draft version January 20, 2016
ABSTRACT
Inprotoplanetarydisks,CO issolidicebeyonditssnowlineat∼10AU. Duetoitshighabundance,
2
itcontributesheavilytothecollisionalevolutioninthisregionofthedisk. Forthefirsttime,wecarried
out laboratory collision experiments with CO ice particles and a CO -covered wall at a temperature
2 2
6 of 80 K. Collision velocities varied between 0 - 2.5 m/s. Particle sizes were on the order of ∼ 100 µm.
1 Wefindathresholdvelocitybetweenthestickingandthebouncingregimeat0.04m/s. Particleswith
0 greatervelocitiesbutbelow1m/sbounceoffthewall. Foryetgreatervelocities,fragmentationoccurs.
2 Wegiveanalyticalmodelsforthecoefficientsofrestitutionandfragmentationstrengthconsistentwith

n the experimental data. Set in context, our data show that CO2 ice and silicate dust resemble each
a otherinthecollisionalbehavior. Comparedtowatericethestickingvelocityisanorderofmagnitude
J smaller. One immediate consequence as example is that water ice particles mantled by CO ice lose
2

9 any ”sticking advantage.” In this case, preferential planetesimal growth attributed to the sticking
1 properties of water ice will be limited to the region between the H2O ice line and the CO2 ice line.

]
P 1. INTRODUCTION calculationmethods,theCO andH Osnowlinesmight
2 2
E shift,but,ingeneral,thevaluesareconsistentwiththose
The role of silicate dust and water ice in planetesi-
h. mal formation by collisional growth has been studied from other authors like O¨berg et al. (2011) for example.
p for many years (Johansen et al. 2014; Blum & Wurm Essentially no data exist to date on the collisional be-
- 2008; Dominik et al. 2007). The role of CO2 ice and haviorofCO2 icegrains. Wenotethatthereareinvesti-
o CO ice has not been touched upon yet in the frame- gationsrelatedtoCO2 snow. Thesehaveapplicationson
r work of collisional evolution, although carbon oxides are Earth in mind, e.g. Lin et al. (2014) explain an agglom-
t
s highly abundant. For comet Churyumov-Gerasimenko, eration mechanism of 100 µm sized CO2 snow inside a
a Rosetta recently found significant CO gas fractions in tube and describe its dependency on jet flows with vor-
[ 2
the coma composition (H¨assig et al. 2015). Pontoppidan tices. This cannot be applied to astrophysical environ-
1 et al. (2014) and O¨berg et al. (2011) estimate the num- ments directly, though. Quantitative experimental stud-
v ber ratios of H O:CO :CO and give values of 100:38:31 iessimilartothosemadeforplanetesimalformationwith
4 2 2 silicates (Blum & Wurm 2008; Wurm et al. 2005; Teiser
in cloud core environments and 100:13:13 toward proto-
5 stars. et al. 2011; Gu¨ttler et al. 2010) or H2O ice (Aumatell &
8 Wurm2011;Aumatell&Wurm2014;Gundlach&Blum
In the general scheme of planetesimal formation, the
4 2015) are missing so far.
icesofthesevolatilesplayarolebeyondtheirsnowlines.
0 Inthecontextofplanetesimal(orcomet)formation, it
Ali-Dibetal.(2014)suggestthattheformationofUranus
. is important to know threshold velocities between stick-
1 and Neptune might be tight to the CO ice line. For TW
ing and bouncing as a function of grain size for all rele-
0 Hya, observations by ALMA imply the existence of a
6 CO ice line at ∼30 AU (Qi et al. 2013). Further inward vant condensed species. It is also important to have an
1 though is the CO ice line. Its position can be calcu- estimate of how much energy is dissipated in a bouncing
v: lated using the su2blimation pressure curve and a disk collision. Last but not least, the onset and strength of
fragmentation are also fundamental properties to know.
i model. We fitted the experimental sublimation pressure
X data from Bryson et al. (1974) with a modified sublima- Wettlaufer (2010) e.g. considers collisional fusion of ice,
where dissipation might lead to local heating and sinter-
r tionpressure curve fromSpan& Wagner(1996), leading
a to ing, which might be important at higher temperatures.
Wurmetal.(2001)considerreaccretionofparticlesafter
(cid:18)p (cid:19) T (cid:34) (cid:18) T (cid:19)l1 (cid:18) T (cid:19)l2(cid:35) acollisionbygasdrag. Currently,thisisalsodetailedby
ln psub = Tc k1 1− T +k2 1− T Jankowski et al. (2015) and the rebound speed would be
c c c important then.
(1) It should be obvious, but we would like to stress that
with Tc = 216.592 K, pc = 0.51795 MPa, k1 = CO2 and H2O are quite different. Besides the general
−12.3081445, k2 = −5.3061778, l1 = 0.8472, l2 = common feature of volatility that kind of labels both
3.47023. Usingtheminimummasssolarnebula(MMSN) solidphasesasice,itisapriorinotpossibletoequalthe
disk properties by Hayashi et al. (1985) and setting collisionalbehaviorofCO -andH O-icewithoutconsid-
2 2
psub = pdisk, the CO2 snow line can be calculated to ering the microphysics of collisional particle interaction.
beat9.3AU.Withthesamemethod,theH2Osnowline IncontrasttoH2O,CO2 moleculesdonothaveanyelec-
is calculated to be at 2 AU. Using different models and tricaldipolemoment(Morrison&Hay1979). Thisisim-
portant for cohesion and collisional sticking. As shown
∗ E-Mail: [email protected] in this paper, the term ice is not a synonym for sticky.
2
Therefore, a possible zoned setting of a protoplanetary zone 2 being the one supposed to favor rapid formation
disk might look like fig. 1. The idea underlying this pic- of larger bodies.
tureisanonion-likegrainstructure. Themostrefractory Not considered so far is how far zone 2 extends. Fur-
material is in the center; the most volatile ice is on the ther outward in a disk, other ice lines exist. As outlined
surface. This picture can be found throughout the liter- above, CO is very abundant. We consider it to be the
2
ature (Sirono 2013; Fayolle et al. 2011). As shown later, next major ice line. In the picture of mantled grains,
growth of particles might preferably happen in zone 2, colliding grains beyond this line will no longer be domi-
where H O ice dominates the collisional behavior. nated by water-ice properties. It is therefore important
2
In a complex scenario of sublimation and recondensa- to know how the picture changes for CO particles.
2
tion,especiallyinthevicinityofthedifferenticelines,the Inthispaper,wequantifycollisionaloutcomesbetween
picture of layered grains might be somewhat simplified. ∼100µmsizedCO particlesandaCO coveredwallat
2 2
However, it will readily show the potentially wide im- atemperatureof80Kandambientpressureof0.5mbar.
plications of the CO collisional behavior. Mostly mod- The collision velocities reach from 0 to 2.5 m/s. We de-
2
erated by surface forces, the material of the outermost termine the impact velocity thresholds between sticking,
layer counts at least for major species like CO that can bouncing, and fragmentation. Furthermore, we study
2
can build sufficient mantle thickness. the fragment size distribution when critical fragmenta-
The composition of the mantle determines the range tion occurs. We present an analytical model for frag-
of velocities and particle masses / sizes for which hit- mentation and restitution that describes the measured
and-stick collisions occur as grains collide (Geretshauser collisional behavior, which we finally put in context of
et al. 2011). For silicates in the warm region (zone 1 planetesimal formation.
in fig. 1), growth via collisions easily works to millime-
ter size. However, from this size on, particles tend to 2. EXPERIMENTAL SETUP
bounce off each other rather than stick together. This
The setup of the experiment is shown in fig. 2. The
is known as the bouncing barrier (Zsom et al. 2010) and
experiment consists of a vacuum chamber, a liquid ni-
has also been shown to be a robust finding in experi-
trogen cooled cryostat with a connected copper plate 15
ments(Kellingetal.2014). Iflargegrainsareintroduced
cm below, a CO injecting nozzle, and a motor with a
by some way collisional growth can still proceed (Wind- 2
cogwheel.
marketal.2012;Wurmetal.2005;Teiser&Wurm2009;
Asfirststep,thevacuumchamberisfloodedwithCO
Deckers & Teiser 2014). This is based on collisions with 2
gas. Since CO gas is deposited on surfaces below 195
fragmentation where part of a smaller body is added to 2
K for atmospheric pressures, a 2 mm thick, solid CO
the larger one. 2
ice layer grows on the walls of the cryostate and on the
Inthecoolerregionsofthedisk(zone2infig. 1)water
copper plate within several minutes. In a second step,
ice is the next solid to dominate the collisional outcome.
the vacuum chamber is sealed and evacuated to an am-
Duetotheincreasedsurfaceenergyofthewatericewith
bient pressure of 0.5 mbar. Using the motor with the
its polar molecules, bindings are much stronger and par-
gearwheel, which can be slided and rotated along the
ticles can grow larger. This has been seen in numerical
chamber from the outside, CO ice can then be beveled
simulations (Dominik & Tielens 1997; Wada et al. 2009) 2
offthecryostateandcollideatspeedsbelow2.5m/swith
and laboratory experiments (Gundlach & Blum 2015;
the CO layer on the copper plane. Low-velocity colli-
Aumatell & Wurm 2014). Okuzumi et al. (2012) even 2
sions at ∼0.05 m/s can be observed due to rebound and
considered this to lead to very large planetesimal size
gas-grain coupling of the aggregates. The collisions are
porous ice aggregates. Follow up work by Kataoka et al.
imagedusingahigh-speedcamerawithamicroscopelens
(2013) also supports this view. In total, zone 2 seems
recording at 1250 frames per second. The spatial resolu-
very favorable for collisional planetesimal-size formation
tion is limited to 10 µm. Sample images from the data
due to the properties of water ice.
set are shown in fig. 3.
We note that other concepts of planetesimal forma-
Intotal,weperformedaround20experimentsandeval-
tion are also intensively studied currently. To span the
uated 96 collision events therein.
gap from sub-meter particles to planetesimal gravito-
turbulent concentration, concentration in baroclinic in-
3. RESULTS AND DISCUSSION
stabilitiesorstreaminginstabiliiesareconsideredtocon-
centratesolidswithinthedisktoapointofgravitational For further analysis, we consider the particles to be
collapse (Chiang & Youdin 2010; Dittrich et al. 2013; spheres with a density of 1560 kg/m3 (Mazzoldi et al.
Johansen et al. 2014). Both kinds of formation mecha- 2008). We take the radius of a sphere with the same
nisms – collisional growth and gravitational instabilities cross section as the observed particle. Fig. 4 shows the
– might also go hand in hand. More efficient sticking distribution of the particle radii used for this study.
might provide the necessary seeds of certain sizes for in- Irregular grains might behave differently in individual
stabilities more easily (Johansen et al. 2014) collisionscomparedtosphericalparticles. Blum&Wurm
Todate,alsopartlyduetotheincreasedmassreservoir (2000)andPoppeetal.(2000)suggestirregulargrainsof
ofsolids,giant-planetcoresaremostoftensupposedtobe samesizetobeslightlystickier. However,itismostlythe
formed beyond the water snow line and formation mod- size that determines the threshold velocity for sticking.
els are typically set in this regime (Pollack et al. 1996; We therefore consider our collisions to be representative
Helled et al. 2014). Close-in giant planets are thought of collisions of solid ∼100 µm CO grains. This holds for
2
to be migrated there afterward (Baruteau et al. 2013; the sticking and bouncing collisions. It should be noted
Kley & Nelson 2012; Dittkrist et al. 2014). In any case, thatwecouldnotunambiguouslydetermineifthegrains
the difference between zone 1 and zone 2 is evident with are monolithic or aggregates and the scaling to smaller
3
0.4 AU 2.0 AU 9.3 AU
zone 1 (dust)
star
zone 2 (dust + H O)
2
zone 3 (dust + H O + CO )
2 2
Fig. 1.— Initialsettingofaprotoplanetarydisk. Whilstinthefirstzoneonlysilicateparticlesexist,inthesecondzonebeyondtheH2O
snowlinethesilicateparticlesmightbecoveredbywaterice. TheCO2 snowlineindicatesthebeginningofthethirdzone,wheresilicate
particlesarecoatedwithtwolayers: oneinnerH2OlayerandanouterCO2 layer. Notethatforbettervisualizationthecoatingsaresolid
andcolored,green: silicates;blue: H2O;red: CO2. Whilewithinthefirst,silicate-dominatedzoneaswellasinthethird,CO2-dominated
zoneparticlescannotgrowquicklytolargersizes,H2Ocoatedparticlesinthesecondzonemightgrowtolargersizesmoreefficiently.
cryostate with LN
vacuum pump
copper plate
motor with gearwheel
motor steering
CO2 in
CO2 out
CO2 nozzle
pressure meter
CO2
ice-aggregates
highspeed camera
Fig. 2.— Experimentalsetup. Withinavacuumchamber,aliquid-nitrogen-cooledcryostatcoolsthetopflange. Thechamberisflooded
withgaseousCO2 whichisdepositedatthecryostatandathermallycoupledcopperplatebelowthecryostate,forminganapproximately
2mmthicklayerofsolidCO2. Usingacogwheelplacedonamotor,iceaggregatescanbebeveledoffthecryostatandtheparticlescollide
withtheCO2 layeratthecopperplate.
sizesshouldonlybeconsideredasanorderofmagnitude connectionsaresolid,theymightnot. Duetotheforma-
estimate for these first experiments. For fragmentation tion mechanism of scraping grains from a solid surface,
athigherspeed,theoutcomeisinfluencedbytheinternal we do not know this a priori. However, the way the par-
binding. If the larger grains (see fig. 3 right) are aggre- ticles fragment into a large number of smaller units at
gatesboundbysurfaceforces,theyfragmenteasily. Ifall low velocities indicates an aggregate structure. Compar-
4
Sticking Bouncing Fragmentation foreconsiderthevelocitydependenceofthecoefficientof
restitution to be representative of the average-size parti-
cles.
The CO particles show the same well-known qualita-
2
0 tive behavior as silicate particles. At low velocities, the
CO particles ”hit & stick.” If the speed is greater than
2
a sticking velocity v , the aggregates bounce off the
stick
surface in inelastic collisions. For these collisions, nei-
ther the surface is eroded nor do the particles fragment.
Onlyatagreatervelocitythanv , doesfragmentation
frag
start. In our experiment, we were not able to see mass
1 transfer during fragmenting collisions.
.6
Themotionofaparticlecanbeapproximatedaslinear
in a time frame of ±5 ms around the collision, like fig. 5
shows exemplarily for two bouncing particles. By means
]
m 
m 1.5  
3 [   
.2 e 1 
t
tim na 0.5 
e i 0
[m rd
s 1 mm o 0 1 2 4 5 6 7
] o
c
Fig. 3.— Sampleimagesfromthedataset. CO2 particlescom- time [ms]
ing from the top are impinging the CO2 ice layer at the bottom. ]
Aggregateswithsurfaceequivalentradiigreaterthan500µmwere m 
notstudied. Theexamplesshowtimesequencesofthetypicalcol- m 0.3  
[ 
lisionaloutcomesobserved: sticking,bouncing,fragmentation
e 0.2 
t
a 
150 n 0.1 
i
rd 0 
o 0 1 2 3 4 5 6 7 8
o
s 100 c
t
n time [ms]
u
o Fig. 5.— Two dimensional motion of a bouncing particle with
c 50 overplotted linear fit functions (circles for camera x-coordinate,
squaresforcameray-coordinate)beforethecollision(left)andafter
the collision (right). On top an example for higher velocities at
about1m/sisshown. Belowwegiveanotherexampleforvelocities
around0.05m/s.
0
0 100 200 300 400 500
of linear fits, we were able to determine the velocities of
radius [μm] theparticlebeforeandafteracollisionv , v andv ,
i,x i,y o,x
v alongthex(horizontal)andy(vertical)coordinates,
Fig. 4.— Size distribution of the CO2 particles. 690 particles reos,ypectively. Wecalculatethecoefficientofrestitution(cid:15),
wereanalyzed. Theaveragesizeis71µm. Thebinsizeof20µmis
chosenwithrespecttothespatialresolutionoftheopticalsystem which we define as the ratio
v (cid:113) (cid:46)(cid:113)
ignrgainenseorfg∼ie1s00beµlomwinsudpipaomrtestetrhaisndviaegwgrtehgaattewsethperroedouf.ce (cid:15)(vo,vi)= voi = v2o,x+v2o,y v2i,x+v2i,y. (2)
The collision behavior of dust depends mainly on the Moreover, we examined the fragmentation strength µ,
initialcollisionvelocitiesandthemassesofthegrainsand which is the ratio of the largest fragment’s mass m and
can be divided into several types (Gu¨ttler et al. 2010). the original aggregate mass m after a fragmentation
0
80% of all particles are in the size range between 40 and
m
120 µm. The average size of the particles is 71 µm. In µ(m,m )= . (3)
terms of mass of individual grains and therefore energy 0 m0
(R3) this is a variation in particle energy by a factor of Fig. 6 shows both quantities for all 96 observed col-
about 1203/403 = 27. The collision velocity varies be- lisions. The coefficient of restitution was evaluated for
tween 0.05 m/s and 2.5 m/s, which in terms of energy all grain sizes from fig. 4. As shown in fig. 7 the larger
(v2)isafactor2500whichismuchlargerthantheenergy grains collide at greater velocities due to diminished gas
variationbymass. Also,fig. 7showsanextremedivision drag. Where both size ranges are present, the coefficient
for the coefficient of restitution for particles smaller and ofrestitutionshowsnodifferencewithinthestudiedtotal
larger than 150 µm that show no difference. We there- range.
5
1.0 1.0
thiswork
Higaetal.(1996,1998)
0.8 0.8
Thornton&Ning(1998)
S
F 0.6 0.6 Krijtetal.(2013),eA=eJKR
, R
O
C
R 0.4 0.4
O
C
0.2 0.2
0.0 0.0
0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0
initial velocity [m/s] initial velocity [m/s]
Fig. 6.— Coefficientofrestitution(COR;bluesqaures)andthe Fig. 8.— Comparison of different models for the coefficient of
fragmentation strength (FS; red dots) of 96 CO2 collisions. The restitution.
dataarefittedusingthemodelsforrestitution(solidblueline;eq.
(27))andforfragmentation(dashedredline;eq. (16)).
between the other models shown in the fragmentation
regime.
1.0
Regarding the fragmentation strength, Krijt & Kama
<150μm
(2014) investigated a fragmentation model recently.
>150μm
0.8 Their work mainly deals with the smallest fragment af-
ter collisions. Nevertheless, they give an expression for
the largest spherical fragment from which we can derive
0.6
a fragmentation strength, like fig. 9 shows. The curve
R
O
C
0.4
1.0
0.2 0.8
0.0 0.6
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 S
F
initial velocity [m/s] 0.4
Fig. 7.— Coefficient of restitution for grains of different sizes.
Theseparationisdoneinrespecttotheparticlesradii.
0.2
thiswork
The coefficient of restitution peaks at an impact ve- Krijtetal.(2014)
0.0
locity of about 0.25 m/s and decreases continuously for
0.5 1.0 1.5 2.0
increasing velocities. The data are fitted with the model
functions for the coefficient of restitution (eq. (27)) and initial velocity [m/s]
for the fragmentation strength (eq. (16)) as described in Fig. 9.— Comparison between the fragmentation model in this
detail in the next section. We also fitted other restitu- workandKrijt&Kama(2014).
tionfunctionsfor(cid:15),e.g. byHigaetal.(1996);Higaetal.
(1998), Andrews (1930), Borderies et al. (1984), Thorn- fitted from Krijt & Kama (2014) fits as well to the data
ton & Ning (1998), Krijt et al. (2013), and a number for the fragmentation events. However, it does not have
of other models taken from Antonyuk et al. (2010). For a threshold for a transition between bouncing and frag-
none did we get as well matching fits. In fact, most of menting,whichisclearlyvisibleinthedata. Beyondthis
these models were originally used to describe restitution model, we are not aware of any other analytical models
behaviorintheinelasticregimeanddonottreatcohesion for the fragmentation strength depending on the impact
effects for velocities near the sticking limit. For adhesive velocity, which we could compare to eq. (16).
models, fig. 8 shows the fitted curves for the coefficient Wedoseestickingatlowcollisionvelocitiesandatran-
of restitution. sition to bouncing. However, under gravity, sticking at
For velocities around the sticking-bouncing transition, very low speeds and bouncing with a very small coef-
alloftheshownmodelsdescribethebehaviorofthecoef- ficient of restitution cannot be disentangled. This spa-
ficient of restitution well. For high velocities in the frag- tial and temporal resolution limit is reached somewhat
mentation regime, Thornton & Ning (1998) and Krijt below 5 cm/s collision velocity. Therefore, to specify
et al. (2013) predict a slope of -1/4, which is too flat. a single sticking velocity v , we do not consider the
stick
Comparedtothis,theempiricalfunctionfromHigaetal. sticking collisions but use our model fit function for the
(1996); Higa et al. (1998) decreases too fast. The model coefficient of restitution. We determine the velocity for
which we present in this work contains the observed co- (cid:15) = 0 and find v = (0.04±0.02) m/s. This sticking
stick
hesive behavior at low velocities and strikes a balance threshold is valid for particles with radii below 150 µm
6
only. As larger particles do also have larger velocities, According to Dominik & Tielens (1997) fragmentation
no statement on their sticking velocity can be made, as for an aggregate occurs at an energy of
fig. 7shows. AccordingtoDominik&Tielens(1997)the
E =3nE (6)
sticking velocity can be calculated as frag br
with E as energy to break contact that is 4.5 times the
br
v = 1.07 · 1 ·γ5/6 (4) sticking energy Estick, or
stick ρ1/2E1/3 R5/6
py E /E =13.5·n. (7)
frag stick
with the surface energy γ, the reduced radius R and the In our case, it is
particle mass density ρ. The elastic properties enter via
E =(1.38±1)·10−12J (8)
E , defined as E = E /(2(1−ν2)). Here, E is the stick
py py y p y
Young’s modulus and ν is the Poisson’s ratio. and we get
p
The first term (density, elasticity) is known for solid E /E =351 (9)
frag stick
CO . The second term shows an almost inversely linear
2
for the onset of CO -aggregate fragmentation. This im-
dependence on particle size. Even without knowing the 2
pliesanumberofcontactsof26,whichisconsistentwith
third term (surface energy), the size dependence allows
the fragment count of catastrophic collisions that we ob-
a comparison of our CO data to data on silicate col-
2
served at higher impact speeds. In detail, the fragmen-
lisions and water-ice collisions. Experiments by Poppe
tationenergydependsonthesizeofanindividualaggre-
et al. (2000) and Gundlach & Blum (2015) determined
the sticking velocity of ∼ 1µm grains in collisions with gateinacollisionanditsnumberofcontacts. Wecannot
quantify this in a collision at the onset of fragmentation
walls. Poppeetal.(2000)foundv ≈1.2m/sforsilica
stick
whichisnotcompletelycatastrophic. Thereforethecon-
spheres with a 0.6µm radius colliding with a wall . For
siderationsgivenbeforearemoreofaqualitativesupport
1.5µmradiuswatericespheresGundlach&Blum(2015)
of the aggregate nature of larger particles.
found v ≈ 9.6m/s. They found roughly a factor of
stick
For very large aggregates, we analyzed the mass dis-
10 increase in sticking velocity compared to silicates of
tribution n(m) of the fragments of 12 collisions, where
similar size. Using the size dependence in eq. 4, we get
n(m)dm means that n fragments are found in the mass
v ≈1.2m/s for 1.5µm radius CO grains. This does
stick 2
range [m;m+dm]. We did not count the largest frag-
not match the values for water ice but is rather close
ment and binned the data into 10 masses. The range
to the value for silica of a similar grain size. This is
of impact velocity varied between 0.6 and 2.2 m/s. The
especially true considering that the CO grains are not
2
mass distribution is shown in fig. 10. This mass density
spherical as there seems to be a tendency that irregular
grains have somewhat higher sticking velocities in com-
parison to spheres (Poppe et al. 2000).
To allow an analytical treatment of CO collisions for 50
2
differentparametersinthefuture,weextractthesurface
energy from our experiments as the single most impor-
tant parameter. Using the longitudinal and transversal
s
velocities vlg = 2900 m/s and vts = 1650 m/s of sound t 10
n
in solid carbon dioxide for 80 K (Yamashita & Kato u
1997), we can calculate Young’s modulus Ey = v2lgρ ≈ co 5
13.12 GPa and the bulk modulus G = v2ρ ≈ 4.25 GPa.
ts
Finally, the Poisson ratio is given by ν =E/(2G)−1≈
p
0.544. With an average particle radius of 60 µm (only
particles below 150 microns are relevant for sticking), 1
this gives a surface energy of γ = 0.17+0.26 J/m2. The 0.05 0.10 0.20 0.50
−0.13
error in γ results from the fit of the sticking velocity
mass fraction (m/m0)
and the standard deviation from the size distribution.
From the fragmentation model of Krijt & Kama (2014) Fig. 10.— Numbermassdistributionoffragmentsafteracatas-
trophicfragmentation. m0isthemassofawholeaggregatebefore
we also get the surface energy as a fit parameter to
itsfragmentation.
γ = (0.325±0.088)J/m2, which is consistent with the
K
valuebasedonthestickingvelocity. Comparedtothis,?
function follows a power law
estimatesansurfaceenergyforCO between0.08−0.097
2
J/m2. Given the uncertainties of the particle contacts, n∝m−1.2±0.1. (10)
one should keep in mind that the slightly larger value in
our experiment are effective values for the specific given This falls into the range found for mass density dis-
setting. Within these limits the results are also consis- tributions for silicate aggregate collisions where power
tent with the value by ?. laws with exponents between -1.07 and -1.36 are found
At v =(0.75±0.05) m/s we find the onset of frag- (Gu¨ttleretal.2010;Deckers&Teiser2014). Inthiscon-
frag
mentationasseeninfig. 3.Theaveragekineticenergyat text, another way to characterize fragmentation is to es-
the onset of fragmentation is timatethecriticalfragmentationenergy(µ=0.5). With
v =(2.19±0.05) m/s we get
crit
Efrag = 21(cid:18)4πρ3r3m(cid:19)v2frag =(5.25±0.7)·10−10J. (5) Ecrit =(4.47±0.2)·10−9J. (11)
7
4. TWO NEW MODELS FOR FRAGMENTATION Fitting this to our data results in c =(−0.073±0.027)
1
AND RESTITUTION s2/m2 andc =−1.49±0.1,whichgivesourmodelfunc-
2
In this section, we develop our own models for the ve- tion plotted in fig. 6.
locity dependence of fragmentation strength and coeffi- For the limiting case Tf+χ (cid:29) V, we furthermore get
cient of restitution. a proportionality of the ratio between the volume of the
smallest fragment and the whole aggregate and the con-
4.1. Fragmentation model tact energy V to
Our fragmentation model is based on an energy bal-
V
ance. The energy of the aggregate with mass M impact- sf ∝V−1/(b+1). (17)
ing with velocity vi has to balance the kinetic energy of Vagg
all fragments after the collision T, the energy to break
f
thecontactsandaccountfordissipatedenergyχ. Weas- We note that other authors like Beitz et al. (2011)
sumethateachoftheNfragments(subaggregates)stuck or Deckers & Teiser (2016) describe the fragmentation
to the largest fragment with an energy of V. The energy strengthof(dust)aggregatesbyasimplepowerlaw. Cer-
balance then reads tainlyourdatacouldalsobefittedbyapowerlaw. How-
M ever, there is no physical meaning in this power law but
2 v2i =NVΘ(vi−vfrag)+Tf+χ (12) to provide an analytic equation and the spread of data
forallciteddataandoursissolargethatthereisnoclear
Here, vfrag is the threshold velocity for fragmentation. discriminator between these functions. A striking differ-
TheHeavisidefunctionΘimplicatesthatdisintegrations ence is that due to the Heaviside function our model has
donothappenforvi <vfrag. Tf isthetotalkineticenergy a limited upper value for fragmentation strength some-
of all fragments after the collision. We assume that χ what below 1 if fragmentation occurs.
does not depend on vi for vi > vfrag. As seen before, Krijt et al. (2013) used similar assumptions to derive
the size distribution of the fragments follows a power the smallest size of the fragment, namely, a power law
law n ∝ a·Vb. Here, V is the fragment volume, and size distribution, and the dissipation of energy by creat-
theconstantsaandbdependonthematerialproperties ing fragments though their approach is based on surface
(Gu¨ttleretal.2010;Geretshauseretal.2011). Therefore, energyofthenewparticlesurfacecreated. Ifthesmallest
we can express the fragment number in terms of volume size would be fixed e.g. due to monomers in an aggre-
and get gate and the sticking energy would be size independent
(cid:90) (s. factor NV from eq. (12)), this should result in the
M Vfrag
v2 =VΘ(v −v ) dVaVb+T +χ. (13) same fragmentation strength model.
2 i i frag f
Vsf
with Vagg = Vl+Vfrag, where Vagg is the volume of the 4.2. Extension of the fragmentation model
totalaggregate, V isthevolumeofthelargestfragment,
l The simple fragmentation model has a cutoff in frag-
V the volume of the smallest fragment, and V is the
sf frag mentation strength somewhat below 1 at v . We con-
volume of all fragments but without the largest. Inte- frag
siderthissufficientforourdatasetandotherdatasetson
grated and solved for V , this yields
frag first order (e.g. also fig. 10 in Deckers & Teiser (2014)
V =(cid:32)(b+1)(cid:0)M2 v2i −Tf−χ(cid:1) +Vb+1(cid:33)1/(b+1) sshitoiowns. aIncutthoiffs)s.ecHtioown,evweer,tihterdeofoesreneoxttetnredatthtihsemtoradne-l
frag aV sf slightly. We consider the dissipation χ(vi) to depend on
the initial velocity v.
i
×Θ(vi−vfrag). (14) We mainly attribute the dissipation within the ag-
gregate to phononic interaction. To approximate this
for b<−1.
interaction, we relate it to the harmonic oscillator so
The fragmentation strength µ(v) can then be de-
i that the phonons can take the discrete energy levels
scribed by
E = ¯hω(n + 1/2) only. Furthermore, the dispersion
n
V V −V relation should be linear ω ∝ p with the momentum p
µ(vi):= V l = aggV frag of the phonons. With these two assumptions, the mo-
agg agg mentum transfer ∆p to phonons will only be possible at
1 (cid:32)(b+1)(cid:0)Mv2−T −χ(cid:1) (cid:33)1/(b+1) the multiple n of a discrete velocity vR,
=1− 2 i f +Vb+1
V aV sf (cid:88)
agg
∆p(v)∝ δ(v −nv ). (18)
i i R
×Θ(v −v ), (15)
i frag n
Substituting c = M(b + 1)/(cid:0)2aVVb+1(cid:1) and c =
1 agg 2 To describe the macroscopic body, we smooth the delta
(cid:0)(b+1)(Tf+χ)/aV −Vsbf+1(cid:1)/Vabg+g1, this simplifies to distributions δ(ii−nvR) with gaussian curves
µ(vi)=1−(cid:2)c1v2i −c2(cid:3)1/(b+1)Θ(vi−vfrag) (16) (cid:88)δ(v −nv )−→(cid:88)exp(cid:0)−(v −nv )2(cid:1). (19)
i R i R
where we set b = −1.2 according to the measured n n
massdistributionandsetthefragmentationthresholdto
v = 0.75 m/s as determined from the measurements. and look at the limiting case v →0. This provides the
frag R
8
dissipative term χ(v) as 1.0
i
v(cid:90)i 0.8
χ(v)∝ lim ∆p(v∗)dv∗
i vR→0 i i
0 0.6
v(cid:90)i OR
∝ exp(cid:0)−v∗i2(cid:1)dv∗i ∝erf(vi) (20) C 0.4
0
or with a constant χ 0.2
0
χ(v)=χ erf(v). (21)
i 0 i 0.0
0.05 0.10 0.50 1
Forthefragmentationstrength,thisextensionleadsto
initial velocity [m/s]
µ(v)=1−(cid:2)ζ v2−ζ erf(v)−ζ (cid:3)1/(b+1)Θ(v −v )
i 1 i 2 i 3 i frag Fig. 12.— Coefficient of restitution on a logarithmic scale. An
(22)
ellipticalenvelopeisoverplottedaroundtheaxisofsymmetry.
with the constants ζ = c = M(b + 1)/(cid:0)2aVVb+1(cid:1),
1 1 agg
ζ2 = (cid:0)(b+1)χ0/aV −Vsbf+1(cid:1)/Vabg+g1, and ζ3 = vi → v˜i = ln(vi/vc). Forces, which are responsible for
(cid:0)(b+1)T/aV −Vb+1(cid:1)/Vb+1. Fig. 11 shows a com- the cohesion, seem to have a similar nature to forces be-
f sf agg tween contacts of grains, which have to be overcome for
parision of the models for phononic and non-phononic
plastic deformation on a multiplicative scale.
dissipation.
The enclosing envelope G (v) can be parameterized
(cid:15) i
by
1.0 (cid:115)
(cid:18)v (cid:19)2
G (v)=δ + δ +δ ln i (23)
(cid:15) i 1 2 3 δ
0.8 4
with δ = −0.5,δ = 2.19,δ = −0.55,δ = 0.24 m/s
1 2 3 4
0.6 for the upper boundary and δ1 = −0.5,δ2 = 1.2,δ3 =
S −1.7,δ4 =0.24m/sforthelowerboundary. δ4 describes
F the critical velocity v :=δ .
0.4 Once this symmetrcy app4lies, the derivative (cid:15)(cid:48)(v˜) has
O O i
tobepointsymmetricalinv˜. Here,weapproximatethis
Krijtetal.(2014) i
0.2 byalinearsymmetry(cid:15)(cid:48)(v˜i)/(cid:15)(v˜i)=a1v˜i withaconstant
χ=χ(vi) a . This might also be written as
1
χ=const.
0.0 (cid:15)(cid:48)(v˜) d
0.5 1.0 1.5 2.0 i = ln((cid:15)(v˜))=a v˜. (24)
(cid:15)(v˜) dv˜ i 1 i
initial velocity [m/s] i i
Integrating and solving for (cid:15)(v˜), this gives
Fig. 11.—Comparisonoffragmentationmodelsforphononicand i
non-phononicdissipation. Thecircleddatapointswerenotusedfor (cid:16) (cid:17)
thefit. (cid:15)(v˜)=exp a v˜2+a (25)
i 1 i 2
For the constants from the fit, we get ζ = (0.01± with a second constant a . Now we can transform back
1 2
0.036) s2/m2, ζ = 2.32 ± 1.01 and ζ = −3.5 ± 0.9. to a term depending on v. This finally leads to
2 3 i
Tbehheabveionrefifotrovfith→em0.odHeelrwe,itwhephgoetnoµni→cd1issfiopravtiforangi→sth0e. (cid:15)(v)=A·exp(cid:32)a (cid:18)ln(cid:18)vi(cid:19)(cid:19)2(cid:33) (26)
Wenotethatthismodeliscapableofhandlingthesteep i 1 v
c
transitionsinDeckers&Teiser(2014). Italsoagreeswell
with better resolved transition data (Deckers and Teiser with A=exp(a ).
2
personal communication). In the experiment we point out a sticking velocity
For a simple consideration of the fragmentation v , which eq. (26) does not consider so far. We
stick
strength without smooth transitions at v , the depen- can fulfill this condition with a linear transformation
frag
dence on velocity can be neglected in the extended ad- v →v −v andmultiplicationbyΘ(v −v ),sothat
i i stick i stick
dend ζ erf(v). As a result, the data can then be fitted (cid:15)(v)=0 ∀ v <v . In total, we gain a function for
2 i i i stick
with two parameters only, like shown in fig. 6. the coefficient of restitution
4.3. Restitution model (cid:15)(v)=A·exp(cid:32)a (cid:18)ln(cid:18)vi−vstick(cid:19)(cid:19)2(cid:33)Θ(v −v )
i 1 v i stick
As a starting point, we take a look at the coefficient c
of restitution on a logarithmic scale, as fig. 12 shows. (27)
Overplotting an elliptical envelope centered at a veloc- Strictly speaking, this transformation breaks the as-
ity v suggests that the shape of the data implies the sumedsymmetrytosomeextent. Thecoefficientofresti-
c
existence of a line symmetry as soon as we transform tutiondecreasesmoreintheneighborhoodofthesticking
9
velocity than for high velocities, which can be observed CO was already mentioned above. How CO collisions
in fig. 12. proceed is unknown. Considering the dipole moment as
For our data, we get A = 0.67±0.04,v = (0.189± a crucial factor for the collisional outcome, CO should
c
0.025) m/s, a =−0.36±0.04, and v =(0.04±0.02) behave similar to CO ice since the dipole moment of
1 stick 2
m/s. Thedependencyineq. (27)hasauniversalnature. CO of 0.112 D is very small compared to the dipole mo-
For v =0, it gives the lognormal-distribution, which ment of H O of 1.84 D (Scuseria et al. 1991). There are
stick 2
isusedwidely,forexample,infittingdistributionsofpar- also ”high temperature” ice lines. Lodders (2004) e.g.
ticlesizes(Raabe1971),describingthemagnetizationbe- propose the existence of a tar line inside of the water ice
havior of nm-Al-particles (Buhrman & Granqvist 1976) line to explain compositional details of Jupiter.
or even biological systems (Koch 1966). For a = −1 The similarity to silicate properties does not necessar-
1
and v = 0, we obtain the empirical equation from ily prevent formation of planetesimals in zone 3 as there
stick
Higa et al. (1996); Higa et al. (1998), where are terrestrial planets supposed to be formed in zone 1.
However, formation timescales and final size and com-
(cid:15)(v)=A·(cid:18)vi(cid:19)−ln(vvci) =A·exp(cid:32)−(cid:18)ln(cid:18)vi(cid:19)(cid:19)2(cid:33) pploasniteito)nmoifgthhtebfeordmiffinergenotb.jects (comet, planetesimal, or
i v v
c c
(28) 6. CONCLUSION
wasusedtodescribethecollisionbehaviorofcentimeter- We measured the threshold impact speeds between
sized H2O-ice spheres in the inelastic region. sticking and bouncing and between bouncing and frag-
mentation of ∼ 100 µm sized CO -ice aggregates. We
2
5. CONSEQUENCES FOR PLANET AND COMET obtain a sticking threshold of vstick = (0.04 ± 0.02)
FORMATION m/s. Scaled in size, this is similar to silicates (Poppe
et al. 2000) but a factor of 10 less than for water ice
IfgrainsoutsidetheCO icelinehaveonionlikeshells,
2 (Gundlach & Blum 2015). We derive a surface energy
theouterCO shellisthickenoughtodominatethestick-
2 of γ = 0.17+0.26 J/m2 for CO ice. After the bouncing
ingproperties. Certainly,fragmentationofshell-likebod- −0.13 2
regimefragmentationsetsinatv =(0.75±0.05)m/s.
ies might be different (Geretshauser et al. 2011). How- frag
Criticalfragmentationoccursatv =(2.19±0.05)m/s.
ever, for the low-energy collisions considered here, we crit
The size distribution of fragments except the largest one
regardthepureCO collisionsaswell-suitedanalog. We
2 follows a power law n ∝ m−1.2±0.1, consistent with ag-
picture a possible simple result of our studies in fig. 1.
gregatesofsilicategrains(Gu¨ttleretal.2010). Thecriti-
If no further recycling and phase transitions occur,
calfragmentationenergyisalsocomparabletotheenergy
therewillbeatorusofseveralAU(zone2)betweenwater
deduced from tensile strength measurements of 100 µm
icelineandCO icelinewhereplanetesimalsformprefer-
2
basalt-,andpalagoniteaggregates(deBeuleetal.2015).
entially. Farther in and farther out, growth will proceed
This all implies that CO ice behaves like silicates in
slower or get stuck, e.g. at bouncing barriers before in- 2
collisions.
stabilities can kick in. Another possibility is that larger
We further developed a model for fragmentation (eq.
ice structures could form in upper (warmer) disk layers,
(16)) and restitution (eq. (27)). Within this model, the
where H O ice in contrast to CO ice is still present.
2 2
fragmentation strength depends mainly on the size dis-
Those aggregates could cross the CO snow line due to
2
tribution of the fragments, which has to be known from
vertical settling resulting in sweep up of smaller CO
2
empirical measurements. We obtain a similar equation
coated particles and finally runaway growth of a limited
forthecoefficientofrestitutionasHigaetal.(1996);Higa
number of aggregates as, e.g. described for ”lucky win-
et al. (1998) used for centimeter-sized H O-ice spheres.
ners” in Windmark et al. (2012). 2
To conclude, the term ice in planetesimal formation is
Other scenarios are possible as well. Gravitational in-
often used with the tacit understanding that it is water
stability, for example, can be partly sensitive to certain
ice. Water ice is more sticky due to the dipole interac-
sizes. Therefore,itmightbebeneficialifgrowthdoesnot
tions. It is tempting to assume that beyond the water
proceed to large structures. In this case, zone 1 or zone
snow line the stickiness increases for all kinds of parti-
3 might be favored for planetesimal formation. Also, in
cles and aggregates. As shown here, this is not the case.
the vicinity of the ice lines and especially in turbulent
Collisions of (pure) CO ice grains are much more com-
disks phase transitions might be frequent with sublima- 2
parable to the collisions of silicate grains. If CO cov-
tion and recondensation changing the structure of the 2
ers water-ice grains beyond 10 AU, collisions farther out
solids constantly. This might also be beneficial to form
will be less sticky then farther inward. This restricts a
largerbodies(Ros&Johansen2013). Nevertheless,CO
2
water-ice-dominated preferential planetesimal formation
collisions would dominate the further evolution also in
to a torous of a few AU between the water ice line and
this picture at later times.
the CO ice line.
This shows some potential outcomes, which require 2
more complex modeling, but put emphasize on the im-
7. ACKNOWLEDGEMENTS
portance of CO collisions. It is crucial to note that ices
2
canbehaveverydifferently. WeonlyconsiderCO icein This work is supported by the DFG under the grant
2
this work but the concept of ice lines does not end here. number WU321/12-1 and TE890/1-1.
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