On the stickiness of CO_{2} and H_{2}O ice particles
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On the stickiness of CO and H O ice particles
Sota Arakawa and Sebastiaan Krijt Division of Science, National Astronomical Observatory of Japan, Mitaka, Tokyo 181-8588, Japan College of Engineering, Mathematics and Physics Sciences, University of Exeter, Stocker Rd, Exeter EX4 4QL, UK
ABSTRACTLaboratory experiments revealed that CO ice particles stick less efficiently than H O ice particles,and there is an order of magnitude difference in the threshold velocity for sticking. However, thesurface energies and elastic moduli of CO and H O ices are comparable, and the reason why CO iceparticles were poorly sticky compared to H O ice particles was unclear. Here we investigate the effectsof viscoelastic dissipation on the threshold velocity for sticking of ice particles using the viscoelasticcontact model derived by Krijt et al. We find that the threshold velocity for sticking of CO iceparticles reported in experimental studies is comparable to that predicted for perfectly elastic spheres.In contrast, the threshold velocity for sticking of H O ice particles is an order of magnitude higherthan that predicted for perfectly elastic spheres. Therefore, we conclude that the large difference instickiness between CO and H O ice particles would mainly originate from the difference in the strengthof viscoelastic dissipation, which is controlled by the viscoelastic relaxation time.
Keywords: solid state: volatile — planets and satellites: formation — protoplanetary disks INTRODUCTIONPairwise collisional growth of dust aggregates is thefirst step of planet formation (e.g., Johansen et al. 2014).The stickiness and collisional behavior of silicate dustparticles/aggregates have been reported in a large num-ber of studies (e.g., Poppe et al. 2000; Blum & Wurm2008; Seizinger et al. 2013; Kimura et al. 2015; Gunkel-mann et al. 2016; Quadery et al. 2017; Planes et al.2020). Particles/aggregates composed of H O ice aregenerally found to be stickier (e.g., Shimaki & Arakawa2012; Gundlach & Blum 2015), although Kimura et al.(2020) claimed that H O ice particles might not be stick-ier than crystalline silicate particles. This difference inbehavior plays an important role in models of dust evo-lution and planetesimal formation in the inner a few auof circumstellar disks (e.g., Drążkowska & Alibert 2017).In the cooler outer region of circumstellar disks, notonly H O ice but also CO and/or CO ices are impor-tant constituents of icy dust particles (e.g., Öberg &Bergin 2020). The condensation temperatures of CO and CO ices are approximately
70 K and
20 K , respec-tively (see Okuzumi et al. 2016). Using the minimum
Corresponding author: Sota [email protected] mass solar nebula model (Hayashi 1981), Musiolik et al.(2016a) found that the location of the CO snow lineis at . from the Sun, which is close to the currentorbit of Saturn. Ali-Dib et al. (2014) also suggests thatUranus and Neptune might be formed near the CO snowline based on the high atmospheric C/H and low N/Hratios. Therefore, CO and CO ices may play a crucialrole in the planet formation.In addition, the stickiness of CO ice particles is ofgreat importance for understanding the dust growth andradial drift behavior in circumstellar disks (Pinilla et al.2017). Recent (sub)millimeter polarimetric observationsof circumstellar disks around young stars (e.g., Kataokaet al. 2017; Stephens et al. 2017) revealed the abun-dant presence of ∼ µ m -sized dust particles beyondthe H O snow line. In contrast, the classical theoryfor dust growth (e.g., Dominik & Tielens 1997; Wadaet al. 2009) suggests that H O ice particles can growinto significantly larger aggregates when turbulence ina circumstellar disk is moderate. To solve this discrep-ancy, Okuzumi & Tazaki (2019) proposed an idea thatthe low stickiness of CO ice particles reported by Musi-olik et al. (2016a,b) might be the key to explain the smallsize of dust particles observed in circumstellar disks.Laboratory experiments by Musiolik et al. (2016a,b)revealed that CO ice particles are less sticky comparedto H O ice particles. Pinilla et al. (2017) and Okuzumi a r X i v : . [ a s t r o - ph . E P ] F e b Arakawa & Krijt & Tazaki (2019) proposed that the large difference instickiness between H O and CO ice particles wouldoriginate from the difference in the dipole moment. Inother words, the low threshold velocity for sticking ofCO ice particles is due to the small surface free en-ergy of apolar CO ice. However, we note that theliterature value of the surface free energy of CO ice(
80 mJ m − ; Wood 1999) is comparable to that of H Oice (
100 mJ m − ; Israelachvili 2011). In addition, thevalues of elastic properties (i.e., the Young’s modulusand Poisson ratio) are also similar between two mate-rials (see Section 3). In the framework of Dominik &Tielens (1997), one would then expect the threshold ve-locity for sticking to be similar for H O and CO ices.In this study, we investigate another possibility to ex-plain the low threshold velocity for sticking of CO iceparticles compared to that of H O ice particles. Krijtet al. (2013) constructed a viscoelastic contact model,which is the advanced version of the contact theory forperfectly elastic spheres (e.g., Johnson et al. 1971; Wadaet al. 2007). The viscoelastic contact model of Krijtet al. (2013) takes into account a crack propagation atthe edge of the contact and an energy dissipation arisingfrom viscoelastic behavior beneath the contact. Apply-ing this model to water ice particles, Gundlach & Blum(2015) found that the threshold velocity for sticking isup to an order of magnitude higher than that predictedfrom the theory for perfectly elastic spheres. Therefore,we can potentially explain the large difference in stick-iness between H O and CO ice particles reported byMusiolik et al. (2016a,b) if CO ice particles follow moreclosely the contact theory for perfectly elastic adhesivespheres.The structure of this paper is as follows. In Section 2,we review the viscoelastic contact model derived by Krijtet al. (2013). In Section 3, we summarize the materialproperties of CO ice. In Section 4, we show the typicalresults for collisions between two viscoelastic spheres. InSection 5, we calculate the threshold velocity for stick-ing and compare our numerical results with experimen-tal data reported by Musiolik et al. (2016a,b). In Sec-tion 6, we evaluate the critical velocity for collisionalgrowth/fragmentation of dust aggregates. Implicationsof our results are discussed in Section 7, and we concludein Section 8. CONTACT MODELThe contact model used in this study is identical towhat Krijt et al. (2013) derived. In Section 2, we brieflysummarize their viscoelastic contact model.2.1.
Elastic strain energy stored in a contact
When two elastic spheres are pressed together, theywill deform locally and share a circular contact area withradius, a . The pressure distribution in the contact area, p ( r ) , is given as a function of the distance from the cen-ter of the contact, r , as follows (Muller et al. 1980): p ( r ) = E ∗ πR a − r + Rδ √ a − r , (1)where δ is the mutual approach, R is the reduced particleradius, and E ∗ is the elastic contact modulus. For acontact between two spheres with the same radius andmaterial, R and E ∗ are given by R = R / and E ∗ = E/ (cid:2) (cid:0) − ν (cid:1)(cid:3) , where R is the particle radius, E is the(relaxed) Young’s modulus, and ν is the Poisson ratio.Then the elastic strain energy stored in the contact, U E ,is given by (Muller et al. 1980) U E = 12 (cid:90) a d r πrp ( r ) w ( r )= E ∗ a R (cid:20) δ (cid:18) δRa − (cid:19) − a R (cid:18) δRa − (cid:19)(cid:21) , (2)where w ( r ) = δ − r / (2 R ) is the deformation of thesurface of spheres.2.2. Johnson–Kendall–Roberts theory
Johnson et al. (1971) introduced a surface energyterm, U S , to describe a contact between adhesive parti-cles: U S = − πa γ. (3)Assuming that the contact area changes quasistatically, ∂U S /∂a is given by ∂U S ∂a = − πaγ. (4)It is known that an equilibrium exists in the frame-work of Johnson–Kendall–Roberts theory (hereinafterreferred to as JKR theory; Johnson et al. 1971). If thereare no external forces, the contact radius at the equilib-rium is a eq = (cid:18) πγR E ∗ (cid:19) / , (5)and the mutual approach at the equilibrium is given by δ eq = a eq2 R − (cid:114) πγa eq E ∗ = (cid:18) π Rγ E ∗ (cid:19) / . (6)Johnson et al. (1971) assumed that there are no forcesacting outside the contact area for simplicity. This treat-ment works well when the Tabor parameter, µ , is suf-ficiently large, i.e., µ (cid:29) (Tabor 1977). The Tabor n the stickiness of CO and H O ice particles µ = 1 (cid:15) (cid:18) Rγ E ∗ (cid:19) / , (7)where (cid:15) is the range of action of the surface forces and γ is the surface energy. We set (cid:15) = 0 . in this study(Krijt et al. 2013). Using material properties of CO ice, we found that µ (cid:39) . for R = 60 µ m , and JKRtheory could be appliable for (sub)micron-sized CO iceparticles. 2.3. Viscoelastic crack velocity
If spheres are made of perfectly elastic materials, wecan use the surface energy term introduced by Johnsonet al. (1971). However, when the material is viscoelastic,the propagating cracks have non-zero velocities and wecan no longer use Equation (4) to calculate the surfaceenergy term. In this case, the energy released/absorbedwhen the crack is closed/opened, ∂U ∗ S /∂a , is ∂U ∗ S ∂a = − πaG eff , (8)where G eff is the apparent surface energy for bond-ing/cracking. The apparent surface energy, G eff , de-pends on the crack velocity, ˙ a . Here we introduce thenormalized apparent surface energy, β , and the normal-ized crack velocity, v : β ≡ G eff γ , (9) v ≡ σ T vis E ∗ γ ˙ a, (10)where σ = γ/(cid:15) is the attractive force acting across thecrack and T vis is the viscoelastic relaxation time (Green-wood 2004; Krijt et al. 2013). The normalized apparentsurface energy, β , is a function of k and v : β = β ( k, v ) , (11)where k is the ratio of relaxed to instantaneous elasticmoduli. In practice, k (cid:28) and we set k = 0 . asa fiducial value (Greenwood 2004). We note that Krijt The elastic strain distribution of viscoelastic media due to theinstant application of loads should be given by the instantaneous elastic modulus. In contrast, even if the loads remain constant,the strain distribution grows according to the creep, and the finalstrain distribution is given by the relaxed elastic modulus. There-fore we distinguish these two elastic moduli. The fact that theinstantaneous modulus is much larger than the relaxed modulus(i.e., k (cid:28) ) means that the stress relaxation is much faster thancreep (see Baney & Hui 1999; Greenwood 2004, and referencestherein). et al. (2013) set k = 0 . instead; however, this smalldifference in k hardly changes our numerical results (seeFigure 3). The normalized crack velocity is also a func-tion of k and β : v = v ( k, β ) . (12)Figure A1 shows the dependence of v on β for differentvalues of k (see Greenwood 2004). We describe how didwe calculate v ( k, β ) in Appendix A.2.4. Apparent surface energy in equilibrium
Krijt et al. (2013) assumed that the crack velocity, ˙ a ,and the apparent surface energy, G eff , adjust themselvesto satisfy the equilibrium contact condition: ∂∂a ( U E + U ∗ S ) = 0 , (13)which can be solved to give G eff = E ∗ πaR (cid:0) a − δR (cid:1) . (14)Then the apparent surface energy, G eff = G eff ( a, δ ) ,is given by Equation (14) and the crack velocity, ˙ a =˙ a ( a, δ ) , is given by Equation (12).2.5. Elastic and dissipative forces
The pressure distribution in the contact area is givenby Equation (1), and the integral over the contact areayields the elastic force between two particles, F E : F E = (cid:90) a d r πrp ( r )= 2 E ∗ R (cid:0) aδR − a (cid:1) . (15)When two viscoelastic particles collide and deform, asignificant amount of energy could be dissipated. Fol-lowing Krijt et al. (2013), we write the dissipative forceas follows: F D = (cid:90) a d r πrA ∂p ( r ) ∂δ ˙ δ = 2 AE ∗ a ˙ δ, (16)where A = T vis /ν (Brilliantov et al. 1996, 2007). Thedissipative force, F D , depends on a and ˙ δ , and it acts asa drag term.For a head-on collision of identical spheres, the timeevolution of the mutual approach is given by ¨ δ = − m ∗ ( F E + F D ) , (17)where m ∗ is the reduced mass, which is m ∗ = m / for a contact between two spheres with the same radius Arakawa & Krijt and material. The mass of each particle is given by m = 4 πρR / , where ρ = 1560 kg m − is the materialdensity of CO ice (Mazzoldi et al. 2008).Ignoring long-range forces, the moment of first con-tact is taken as t = 0 , and the initial condition for themutual approach is δ = 0 and ˙ δ = V in , where V in isthe collision velocity. The normalized crack velocity, v = v ( k, β ) , is defined within the range of k < β < /k (see Appendix A) and the apparent surface energy sat-isfies G eff = βγ > . Equation (14) is rewritten as G eff = E ∗ a / (cid:0) πR (cid:1) when δ = 0 , and it does not allow a = 0 as the initial condition. For our numerical integra-tions, we set β = (1 + ε ) k at t = 0 as Krijt et al. (2013)assumed. We use the small value of ε = 0 . . We endintegrations when β = (1 − ε ) /k , and two spheres willseparate immediately. These assumptions are justifiedas the contact evolves rapidly near a (cid:39) and δ hardlychanges (see Appendix B of Krijt et al. 2013). MATERIAL PROPERTIES OF CARBONDIOXIDE ICEIn Section 3, we discuss the elastic and adhesive mate-rial properties of CO and H O ices, needed for solvingEquation (17). Following Gundlach & Blum (2015), wetreat the main viscoelastic parameter T vis as a free pa-rameter that may depend on particle size.3.1. Surface free energy
The surface free energies of crystals are proportionalto their sublimation energies (e.g., Shuttleworth 1949;Benson & Claxton 1964). Using the crystal structureand the latent heat of sublimation of CO ice, Wood(1999) theoretically estimated the surface free energy ofCO ice as γ SV = 80 mJ m − . This value is widely usedin the studies of CO clouds in the martian atmosphere(e.g., Määttänen et al. 2005; Nachbar et al. 2016; Man-gan et al. 2017). Glandorf et al. (2002) also obtained anapproximate value of γ SV by using the Antonoff’s rule,and the surface free energy is estimated as
67 mJ m − ,which is in reasonable agreement with that obtained byWood (1999). For a contact between two spheres made of same ma-terial, the surface energy, γ , is (approximately) twice thesurface free energy (Johnson et al. 1971): γ = 2 γ SV . (18) Wood (1999) also tested the validity of the technique for esti-mating surface energy. Using the technique, they obtained thatthe surface energy of H O ice is
128 mJ m − for the prism face(and
120 mJ m − for the basal face). This theoretical estimateshows good agreement with the canonical value obtained fromexperiments (i.e.,
100 mJ m − ; Israelachvili 2011). For H O ice, the canonical value of γ SV is γ SV =100 mJ m − (Israelachvili 2011). We note, however,that the surface free energy of H O ice is still underdebate (see Section 5.2).3.2.
Young’s modulus and Poisson ratio
As the longitudinal and transversal velocities of sound, v lg and v ts , are related to the elastic properties, wecan calculate the Young’s modulus and Poisson ratiofrom the results of sound velocity measurements. Thesesound velocities, v lg and v ts , are given by (e.g., Han &Batzle 2004) v lg = (cid:115) ρ (cid:18) K + 43 G (cid:19) , (19) v ts = (cid:115) G ρ , (20)where K is the bulk modulus and G is the shear modulus.Both K and G can be rewritten by using E and ν asfollows: K = E − ν ) , (21) G = E ν ) . (22)Yamashita & Kato (1997) measured the longitudinaland transversal velocities of sound in CO ice and ob-tained v lg = 2900 m s − and v ts = 1650 m s − atthe temperature of
80 K (see Musiolik et al. 2016a).Then the Young’s modulus and Poisson ratio are E =10 . and ν = 0 . , respectively.For H O ice, the literature values of E = 7 GPa and ν = 0 . are widely used in previous studies (e.g., Do-minik & Tielens 1997; Wada et al. 2007; Gundlach &Blum 2015). STICKING, BOUNCING, AND DOUBLECOLLISIONSThe contact model reviewed in Section 2 can be usedto calculate the time evolution of the contact betweentwo colliding spheres. In Section 4, we show the typicalresults for collisions between two equal-sized spheres ofCO ice.We begin by setting R = 60 µ m and T vis = 10 − s ,and exploring a range of impact collision velocities V in .We found that there are three types of collision out-comes, namely, sticking collisions, bouncing collisions,and double collisions. Similar results are also reportedin Sections 3.1 and 3.2 of Krijt et al. (2013). n the stickiness of CO and H O ice particles
Sticking collision
The grey lines of Figure 1 show the evolution ofthe contact radius, a , the mutual approach, δ , andthe approaching velocity, ˙ δ , for a head-on collision at V in = 3 . − . The green stars mark the equilibriumpoint in JKR theory ( a = a eq , δ = δ eq , and ˙ δ = 0 ). Atthe start of the collision, t = 0 , the mutual approach is δ = 0 and the contact radius is given by a = (cid:20) πR E ∗ (1 + ε ) kγ (cid:21) / . (23)The contact radius initially grows very rapidly, as a in-creases to a (cid:39) . µ m with δ hardly changing. Krijtet al. (2013) described the details of the behavior ofthe viscoelastic contact, by comparing with that of JKRtheory (e.g., Wada et al. 2007).The most important difference between our viscoelas-tic contact model and JKR theory is whether the kineticenergy dissipates during contact or not. For the case of V in = 3 . − , the spheres cannot separate and in-stead oscillate back and forth. In δ – a and δ – ˙ δ planes,the contact spirals toward the equilibrium point of JKRtheory due to the dissipative effects when we use the vis-coelastic contact model. In the framework of JKR the-ory, in contrast, the oscillation would not be dampened.The dissipative effects increase the threshold velocity forsticking, V stick (see Section 5).4.2. Bouncing collision
Even if the dissipative effects work, collisions of twospheres will result in bouncing as the collision velocity isincreased. The red lines of Figure 1 show the evolutionof a , δ , and ˙ δ , for a head-on collision at V in = 4 . − .In this case, the contact radius finally becomes a (cid:39) ,and the mutual approach and the approaching velocityare δ > and ˙ δ < at the end of the contact. At thatpoint, the spheres separate and move away from eachother at a velocity V out (see Section 4.4).4.3. Double collision
There exists a narrow range of impact velocities forwhich we observe a “double collision”. This double colli-sion occurs as a result of energy dissipations and vis-coelastic cracking. The blue lines of Figure 1 showthe evolution of a , δ , and ˙ δ , for a head-on collision at V in = 4 .
05 cm s − . In this case, the mutual approachand approaching velocity are δ > and ˙ δ > at the endof the contact. As ˙ δ > , two spheres are expected torecollide after their separation. We therefore named thisoutcome as the “double collision”. We note that the col-lision velocity of the second collision is much lower than that of the first collision because of dissipative effects,and the second collision should result in sticking.4.4. Coefficient of restitution
We use the coefficient of restitution, e , to describecolllision outcomes. The definition of e is e ≡ − V out V in , (24)where V out is the approaching velocity at the end of thecontact. We set e = 0 for sticking collisions. For doublecollisions, negative values of the coefficient of restitutionwill be obtained from numerical calculations. We note,however, that the second collision may occur immedi-ately after the first collision and the final outcome ofthe collisional sequence is sticking. Then we can imaginethat the “observed” value of the coefficient of restitutionin laboratory experiments is e = 0 for double collisions.Figure 2 shows the variations of e with V in for R =60 µ m and T vis = 10 − s , a transition from sticking col-lisions to double collisions occurs at V in = 4 .
04 cm s − ,and a transition from double collisions to bouncing col-lisions occurs at V in = 4 .
14 cm s − . In this case, weobtain the threshold velocity for sticking as V stick =4 .
14 cm s − . As V stick depends on the particle radius andmaterial properties including γ SV and T vis , we can es-timate the relaxation time of viscoelastic particles fromliterature values of V stick which are experimentally deter-mined (e.g., Krijt et al. 2013; Gundlach & Blum 2015). THRESHOLD VELOCITY FOR STICKINGIn section 5, we calculate the threshold velocity forsticking using the viscoelastic contact model, and wealso compare our numerical results with experimentaldata reported by Musiolik et al. (2016a,b). We showthat V stick of both CO and H O ice particles observedin experiments are consistent with the theoretical pre-diction from the viscoelastic contact model. Especially, V stick of H O ice particles can be reproduced only whenwe consider the dissipative effects.5.1.
Carbon dioxide ice
Musiolik et al. (2016a) performed laboratory experi-ments of collisions of CO ice particles within a vacuumchamber at a temperature of
80 K . The collision veloc-ities are below . − , and the typical radius of theparticles is R = 60 µ m when we focus on the collisionsof small particles whose radii are less than µ m . The size distribution of the CO ice particles is shown in Figure4 of Musiolik et al. (2016a), and 80% of all particles are withinthe size range of µ m < R < µ m . Arakawa & Krijt a ( - m ) δ (10 -8 m) V in = 3.5 cm s -1 V in = 4.05 cm s -1 V in = 4.5 cm s -1 ★ -5 0 5 -1 0 1 2 d δ / d t ( c m s - ) δ (10 -8 m) V in = 3.5 cm s -1 V in = 4.05 cm s -1 V in = 4.5 cm s -1 -5 0 5 -1 0 1 2 ★ Figure 1.
Time evolution of the contact radius, a , the mutual approach, δ , and the approaching velocity, ˙ δ , for head-oncollisions. The green stars mark the equilibrium point in JKR theory ( a = a eq , δ = δ eq , and ˙ δ = 0 ). The left panel shows theevolution in δ – a plane, and the right panel is the evolution in δ – ˙ δ plane, respectively. The grey lines represent the evolutionarytrack for the case of V in = 3 . − , resulting in sticking. The blue lines are for the case of V in = 4 .
05 cm s − , resulting indouble collision in the collisional sequence. The red lines are for the case of V in = 4 . − , resulting in bouncing. e V in (cm s -1 )sticking collisiondouble collisionbouncing collision 0 0.25 0.5 0.75 1 Figure 2.
Variations of the coefficient of restitution, e ,with the collision velocity, V in , for head-on collisions of CO ice particles. Particle radius of R = 60 µ m and a relaxationtime of T vis = 10 − s are assumed here. The grey, blue, andred solid lines represent the values of e which are expected tobe observed as final outcomes, and the blue dashed line is thevalue of e obtained from numerical calculations (for doublecollisions). The grey point indicates the threshold velocityfor sticking, V stick = 4 .
14 cm s − . They found that the threshold velocity for sticking is V stick = (0 . ± .
02) m s − , although the uncertaintyis large.Figure 3 shows the dependence of V stick on T vis fordifferent values of k . As mentioned in Greenwood (2004)and Krijt et al. (2013), the evolution of contact radius isalmost independent of k except for the start and end ofthe contact. Then the collision outcomes hardly dependon the choice of k as long as we set k (cid:28) (see AppendixA).As shown in Figure 3, V stick hardly changes when T vis (cid:46) − s . In this case, V stick is almost identicalto that of JKR theory. According to Thornton & Ning(1998), in the framework of JKR theory, the threshold -2 -1 -12 -11 -10 -9 -8 V s ti c k ( m s - ) T vis (s) k = 0.1 k = 0.01 k = 0.00110 -2 -1 -12 -11 -10 -9 -8 ★ Figure 3.
Dependence of the threshold velocity for stick-ing, V stick , on the relaxation time, T vis , for different val-ues of k . The black dashed line represents the thresh-old velocity for sticking obtained from laboratory experi-ments and the yellow shaded region shows the uncertainty: V stick = (0 . ± .
02) m s − (see Musiolik et al. 2016a). Theblack star indicates the threshold velocity for sticking in-ferred from JKR theory, V stick , JKR (e.g., Thornton & Ning1998; Wada et al. 2007). The typical radius of CO ice par-ticles used in Musiolik et al. (2016a) is R = 60 µ m . velocity for sticking is given by V stick , JKR = (cid:18) . m ∗ (cid:19) / (cid:18) γ R E ∗ (cid:19) / = 1 . × − (cid:18) R µ m (cid:19) − / m s − . (25)The black star plotted in Figure 3 indicates the value of V stick , JKR , and it is clear that V stick → V stick , JKR for theshort- T vis limit, T vis → .In contrast, when T vis (cid:38) − s , the threshold velocityfor sticking is several times higher than that predictedfrom JKR theory. The increase of V stick with increasing n the stickiness of CO and H O ice particles T vis is also reported in previous studies (Krijt et al.2013; Gundlach & Blum 2015), and our results shownin Figure 3 are consistent with their results. Assumingthat k = 0 . , we can obtain the suitable range of T vis to reproduce V stick reported by Musiolik et al. (2016a)as follows: . × − s ≤ T vis ≤ . × − s; (26)though we do not reject the possibility that T vis (cid:28) − s and V stick is nearly identical to V stick , JKR .In numerical calculations, we assumed that CO iceparticles are spherical and the viscoelastic contact the-ory for spheres is appliable. We acknowledge, however,that CO ice particles used in Musiolik et al. (2016a) arenot spherical. Although Blum & Wurm (2000) suggestedthat irregular grains are slightly stickier than sphericalgrains, Musiolik et al. (2016a) mentioned that the effectof the irregular shape may be negligible.Musiolik et al. (2016a,b) did not report the surfaceroughness of ice grains, however, it might alter thethreshold velocity for sticking (e.g., Nagaashi et al.2018). Although our results for both CO and H O iceparticles are consistent with the cases for smooth par-ticles (see Sections 5.2 and 5.3), we need to assess theeffect of the surface roughness in future.It should also be noted that whether CO ice particleswere monolithic grains or aggregates is unknown. In thisstudy, however, we assume that CO ice particles whoseradii are R (cid:39) µ m are monolithic. This is because thecritical velocity for collisional growth/fragmentation, V frag , should be several times higher than .
04 m s − when CO ice particles are aggregates, even if themonomer grains of these aggregates behaved as perfectlyelastic spheres (see Section 7.1 for details).5.2. Water ice
Musiolik et al. (2016b) also performed laboratory ex-periments of collisions of pure H O ice particles (andmixtured ice particles of H O–CO ) within a vacuumchamber at a temperature of
80 K . The typical radiusof the particles is R = 90 µ m , and their experimentalresults suggest that V stick ∼ .
73 m s − for pure H Oice particles.Material properties of H O ice are reported in a largenumber of previous studies. We set E = 7 GPa , ν = 0 . , and ρ = 930 kg m − (Gundlach & Blum2015). The surface free energy of H O ice is still un-der debate. The canonical value of γ SV used in numer-ical simulations (e.g., Wada et al. 2013; Sirono & Ueno2017; Tatsuuma et al. 2019) is γ SV = 100 mJ m − (Is-raelachvili 2011). Measurements of the critical rollingfriction force of µ m -sized H O ice particles suggest that -3 -2 -1 -10 -9 -8 V s ti c k ( m s - ) T vis (s) γ SV = 190 mJ m -2 γ SV = 100 mJ m -2 γ SV = 20 mJ m -2 -3 -2 -1 -10 -9 -8 Figure 4.
Dependence of the threshold velocity for stick-ing, V stick , on the relaxation time, T vis , for different values of γ SV . The black dashed line represents the threshold velocityfor sticking obtained from laboratory experiments. The typi-cal radius of H O ice particles used in Musiolik et al. (2016b)is R = 90 µ m . γ SV = 190 mJ m − (Gundlach et al. 2011), although thevalue of γ SV depends on the assumed value of the criticalrolling displacement (see Krijt et al. 2014). Moreover,tensile strength measurements in a low-temperature en-vironment (Gundlach et al. 2018) suggest that γ SV =20 mJ m − at low temperatures below
150 K . Musio-lik & Wurm (2019) also reported that γ SV at
175 K isone to two orders of magnitude lower than the canonicalvalue based on their pull-off measurements of mm-sizedwater ice grains. Then we parameterize γ SV in our cal-culations.Figure 4 then shows the dependence of V stick on T vis for different values of γ SV . We found that we cannotexplain the reported value of V stick by using JKR theory,that is, the contact model for perfectly elastic adhesivespheres. Assuming that the range of the surface freeenergy is
20 mJ m − ≤ γ SV ≤
190 mJ m − , the requiredvalue of T vis is . × − s ≤ T vis ≤ . × − s , (27)and T vis of H O ice particles with R = 90 µ m maybe an order of magnitude larger than that of CO iceparticles with R = 60 µ m .For H O ice particles with R = 1 . µ m , Gundlach &Blum (2015) revealed that T vis = 1 × − s is plausibleto reproduce the value of V stick = (9 . ± .
3) m s − ob-tained from their experiments. In Section 5.3, we discussthe dependence of T vis on R .5.3. Relaxation time
The relaxation time is a fitted parameter in this studybecause we do not know how T vis relates to other fun-damental material properties. We note, however, thatthere is an empirical relation between T vis and R (Krijt Arakawa & Krijt -13 -12 -11 -10 -9 -8 -1 ✶ T v i s ( s ) R (10 -6 m)CO H O10 -13 -12 -11 -10 -9 -8 -1 Figure 5.
Dependence of the relaxation time, T vis , onthe particle radius, R , for both CO and H O ice particles.The blue and red points with error bars are the calculatedvalues of T vis for CO and H O ice particles in this study,respectively. The red star shows a reported value of T vis , H O from numerical calculations by Gundlach & Blum (2015).The dashed lines are the (empirical) fitting formulae of size-dependent T vis , CO and T vis , H O . et al. 2013; Gundlach & Blum 2015). Gundlach & Blum(2015) reported that the relaxation times obtained byKrijt et al. (2013) is consistent with a relation between T vis and R , that is, T vis ∝ R . .For H O ice particles with R = 1 . µ m , Gundlach &Blum (2015) revealed that T vis = 1 × − s . Therefore,the size-dependent relaxation time of H O ice particlesmay be given by (Gundlach & Blum 2015) T vis , H O = 1 × − (cid:18) R . µ m (cid:19) . s , (28)and this equation yields T vis , H O = 9 . × − s for H Oice particles with R = 90 µ m . This relation showsexcellent agreement with our numerical results (see Re-lation 27).Then we also apply the empirical relation to the size-dependent relaxation time of CO ice particles. FromRelation (26), we found that the following relation, T vis , CO = 1 × − (cid:18) R µ m (cid:19) . s , (29)is consistent with the experimental results for CO iceparticles with R = 60 µ m . Figure 5 shows the depen-dence of T vis on R for both CO and H O ice particles.5.4.
Size dependence of threshold velocity for sticking
Here we calculate V stick for (sub) µ m -sized CO iceparticles by using the size-dependent relaxation timederived in Section 5.3. We set γ SV = 80 mJ m − , E = 10 . , ν = 0 . , and ρ = 1650 kg m − (seeSection 3). The left panel of Figure 6 shows V stick as a functin of R . We also consider the dependence of T vis on V stick .The grey line represents the case of perfectly elastic con-tact model, i.e., V stick = V stick , JKR . The blue line repre-sents the standard model, i.e., T vis = T vis , CO (Equation29). We can find that the difference between these twomodels is within a factor of a few in V stick , and we can(roughly) evaluate V stick by using JKR theory, which iswidely used in previous studies (e.g., Dominik & Tielens1997; Wada et al. 2007).In contrast, the viscoelastic dissipation effects play agreat role when T vis is several times higher than thatwe assumed for CO ice particles. The red dashed lineis the threshold velocity for sticking, V stick , for the casewhen T vis = T vis , H O (Equation 28). As V stick is anorder of magnitude higher than V stick , JKR when we use T vis = T vis , H O , we can imagine that the large differenceof V stick between CO and H O ice particles (Musioliket al. 2016a,b) mainly originate from the large differenceof T vis between two materials.Pinilla et al. (2017) and Okuzumi & Tazaki (2019)mentioned that the low value of V stick for CO ice parti-cles is due to the small surface free energy of apolar CO ice. However, the literature value of γ SV = 80 mJ m − (Wood 1999) is comparable to that of H O ice, althoughfuture direct measurements of the surface free energy ofCO ice is essential. In addition, the values of elasticproperties, E and ν , are also similar between two ma-terials. Therefore, we proposed that the large differencein V stick between CO and H O ice particles is thoughtto originate from the large difference in T vis . CRITICAL VELOCITY FOR COLLISIONALFRAGMENTATION OF AGGREGATESHere we discuss the critical velocity for collisionalgrowth/fragmentation, V frag , of dust aggregates com-posed of µ m -sized monomer grains. The right panel ofFigure 6 shows the dependence of V frag on R . Here R is the radius of monomer grains. The cyan hatchedregion indicates the maximum collision velocity of dustaggregates in circumstellar disks with weak turbulence,i.e.,
25 m s − (cid:46) V col , max (cid:46)
50 m s − (e.g., Adachi et al.1976; Blum & Wurm 2008; Wada et al. 2013).It is empirically known that V frag is an order of mag-nitude larger than V stick and is almost independent ofthe number of constituent monomer grains (Dominik &Tielens 1997; Wada et al. 2009, 2013). Here we brieflyexplain the basic findings from numerical simulationsof collisions of dust aggregates. Based on JKR theory,the amount of energy dissipated in a bouncing collision, E stick , JKR , is given by (Thornton & Ning 1998; Wada n the stickiness of CO and H O ice particles -1 -2 -1 V s ti c k ( m s - ) R (10 -6 m)JKR T vis = T vis, CO T vis = T vis, H O -1 -2 -1 -2 -1 V fr a g ( m s - ) R (10 -6 m)JKR T vis = T vis, CO -2 -1 Figure 6.
Dependence of the threshold velocity for sticking (for collisions between monomer grains), V stick , and the criticalvelocity for collisional growth/fragmentation (for inter-aggregate collisions), V frag , as functions of the particle radius of monomergrains, R . The left panel shows V stick for different assumptions for the size-dependent relaxation time. The red dashed line is V stick for T vis = T vis , H O . The blue solid line is V stick for T vis = T vis , CO , and this is the standard model in this study. The greysolid line shows V stick , JKR as a lower limit of V stick . The right panel shows V frag for different assumptions for the size-dependentrelaxation time. The blue solid line is V frag for T vis = T vis , CO . The grey solid line also shows V frag , JKR as a lower limit of V frag .In our estimates, V frag is given by V frag = 10 V stick (Equation 33). et al. 2007) E stick , JKR = m ∗ V stick , JKR2 = 0 . F crit δ crit , (30)where F crit = 3 πγR/ is the maximum force needed toseparate two contact particles and δ crit = (9 / / δ eq isthe critical pulling length between the particles in con-tact. The energy necessary to break completely a con-tact in the equilibrium position, E break , JKR , is slightlylarger than E stick , JKR (e.g., Wada et al. 2007): E break , JKR = 1 . F crit δ crit , (31)and we usually use E break , JKR to interpret collision out-comes of dust aggregates.According to Wada et al. (2013), the critical velocityfor collisional growth/fragmentation of dust aggregateof perfectly elastic monomer grains, V frag , JKR , is empir-ically given by V frag , JKR = C (cid:114) E break , JKR m = 0 . C · V stick , JKR , (32)where C is a dimensionless constant: C (cid:39) for equal-sized collisions and C (cid:39) for different-sized collisions(Wada et al. 2009, 2013). Therefore, the scaling rela-tion between V frag , JKR and V stick , JKR is approximately Wada et al. (2009, 2013) numerically revealed that the value of C hardly depends on the size of aggregates when the numberof constituent monomer grains is in the range between and . We note, however, that the detailed reason why C hardlydepends on the size of aggregates is still unclear. given by V frag , JKR = 10 V stick , JKR . Although it is notclear that whether this relation between V frag and V stick is appliable for dust aggregates of viscoelastic monomergrains (e.g., Gunkelmann et al. 2016), we apply the fol-lowing assumption to evaluate the value of V frag : V frag = 10 V stick . (33)The right panel of Figure 6 suggests that V frag >V col , max when the radius of monomer grains is R (cid:28) . µ m , and V frag < V col , max for the case of R (cid:29) . µ m . In the context of dust growth in circumstel-lar disks, we usually assumed R = 0 . µ m in numericalcalculations (e.g., Okuzumi et al. 2012; Krijt et al. 2015;Homma & Nakamoto 2018). This assumption is at leastconsistent with the grain size in the surrounding enve-lope of proto-stellar objects inferred from near-infraredpolarimetry (e.g., Murakawa et al. 2008) and the size dis-tribution of interstellar dust grains (e.g., Mathis et al.1977; Weingartner & Draine 2001). For R = 0 . µ m ,dust aggregates of CO ice monomer grains can sticktogether without catastrophic fragmentation when thestrength of turbulence is weak. In this case, the max-imum size of dust aggregates is controlled not by frag-mentation but by radial drift (e.g., Okuzumi et al. 2012;Drążkowska & Alibert 2017), although bouncing and/orerosive collisions between particles with a high mass ra-tio might prevent dust aggregates from growing intolarger aggregates (e.g., Zsom et al. 2010; Krijt et al.2015). In Section 7.2, we discuss the possible mecha-nisms for altering the size of monomer grains. DISCUSSION7.1.
Morphology of ice particles used in experiments Arakawa & Krijt
We consider that CO ice particles used in Mu-siolik et al. (2016a) may be monolithic (see Section5.1). This is because the critical velocity for collisionalgrowth/fragmentation, V frag , is several times higherthan .
04 m s − when we assume that µ m -sized CO ice particles are dust aggregates. If the aggregate radiusis µ m , then the radius of monomer grains shouldbe smaller than the half of the aggregate radius, i.e., R (cid:46) µ m . In this case, V frag , JKR of dust aggregatescomposed of CO ice monomer grains with R (cid:46) µ m is V frag , JKR = 10 V stick , JKR = 2 . × − (cid:18) R µ m (cid:19) − / m s − . (34)Moreover, this large value of V frag , JKR gives the mini-mum estimate of V frag . Therefore, the estimated V frag isan order of magnitude higher than the threshold velocityreported by Musiolik et al. (2016a).We also note that the experimental setup of Musio-lik et al. (2016b) is identical to that of Musiolik et al.(2016a). The particle radius of CO and H O ices arevery similar: R = 60 µ m and µ m , respectively.Therefore, we conclude that both CO and H O ice par-ticles used in Musiolik et al. (2016a,b) are not aggregatesbut monolithic grains.7.2.
Size of monomer grains
The size of monomer grains is often taken to . µ m (e.g., Okuzumi et al. 2012); however, it is unclear towhat extent using a single and constant monomer size isappropriate. Here we discuss several possible scenariosthat can alter the size of monomer grains in circumstellardisks.Ros & Johansen (2013) proposed that condensationof H O vapor near the H O snow line might be a dom-inant particle growth mechanism when dust growth isprevented by bouncing and/or fragmentation. If con-densation of H O vapor controls the size of monomergrains, the physics of heterogeneous nucleation may playa crucial role. Laboratory experiments on heterogeneousnucleation by Iraci et al. (2010) revealed that the for-mation of a H O ice layer on a bare silicate surface re-quires a substantially high H O vapor pressure. ThenRos et al. (2019) showed that H O vapor may be de-posited predominantly on already ice-covered particlesand these icy particles can grow into cm -sized hugemonomer grains near the H O snow line. In this sce-nario, cm -sized huge monomer grains cannot agglomer-ate because V stick for cm -sized huge monomer grains istoo low even if they are covered by a H O ice mantle. Then icy planetesimals might be formed through grav-itational collapse of clumps of cm -sized icy grains (e.g.,Johansen et al. 2007; Bai & Stone 2010).This selective condensation process may be importantnot only near the H O snow line but also near the CO snow line. We note, however, that the formation processof the first CO ice layer may different from that of theH O ice layer. As CO ice might be formed via chemicalreaction of CO and OH on grain surfaces (e.g., Bosmanet al. 2018; Krijt et al. 2020), the size of monomer grainscovered by a CO ice mantle would be similar to thatcovered by a H O ice mantle near the CO snow line.Another possible mechanism for changing the size ofmonomer grains is evaporation and following reconden-sation of dust particles via flash-heating events (e.g.,Miura et al. 2010; Arakawa & Nakamoto 2016). Theflash-heating events in the early solar nebula are thoughtto be the plausible formation mechanisms of chondrulescontained within chondrites (e.g., Arakawa & Nakamoto2019). Recently, Fujiya et al. (2019) revealed that atleast some chondrite parent bodies were formed beyondthe CO snow line, based on C-isotope measurements oncarbonate minerals in carbonaceous chondrites. Thenthe flash-heating events might occur not only in theinner region of the solar nebula but also outside theCO snow line, and the following recondensation pro-cess would determine the size of monomer grains in theearly solar nebula.Based on the combination of dust evolution calcula-tions and synthetic polarimetric observations of a cir-cumstellar disk around a young star HL Tau, Okuzumi& Tazaki (2019) revealed that the plausible value of V frag is lower than − both inside the H O snow lineand outside the CO snow line, to explain the smalldust scale height (Pinte et al. 2016) and the observedaggregate radius of (cid:39) µ m (e.g., Kataoka et al. 2017;Stephens et al. 2017) simultaneously. This suggests thatthe size of monomer grains in the disk around HL Taumight be R (cid:38) µ m (see right panel of Figure 6),and some mechanisms for altering the size of monomergrains from that of interstellar dust grains are required.7.3. Impact of dust growth on the gas-phase abundanceof carbon monoxide in circumstellar disks
Understanding the astrochemistry of CO in circum-stellar disks is of great importance in the context of starand planet formation. This is because emission fromgas-phase CO and its isotopologues is widely used tostudy the structures of circumstellar disks, such as thedisk radius (e.g., Ansdell et al. 2018), the disk mass (e.g.,Ansdell et al. 2016), the temperature profile (e.g., Dulle- n the stickiness of CO and H O ice particles SUMMARYWe have investigated the reason for the low thresholdvelocity for sticking of CO ice particles compared tothat of H O ice particles. Using the viscoelastic contactmodel (Krijt et al. 2013), we succeeded in reproducingthe experimental results of collisions of CO and H Oice particles (Musiolik et al. 2016a,b). Our findings aresummarized as follows.1. For collisons between two viscoelastic spheres, wefound that there are three types of collision out-comes, namely, sticking collisions, bouncing col-lisions, and double collisions. We defined thethreshold velocity for sticking, V stick , as the tran-sition velocity from double collisions to bouncingcollisions (see Figures 1 and 2).2. In the viscoelastic contact model, the relaxationtime, T vis , is the key parameter to describe thestrength of viscoelastic effects (Krijt et al. 2013).We found that the relaxation time of CO ice par-ticles with the particle radius of R = 60 µ m is inthe range of . × − s ≤ T vis ≤ . × − s ,and V stick of CO ice particles is not so different from that predicted from JKR theory for perfectlyelastic spheres (see Figure 3).3. In contrast, we found that V stick of H O ice par-ticles is an order of magnitude higher than thatpredicted from JKR theory (see Figure 4). Therelaxation time of H O ice particles with the par-ticle radius of R = 90 µ m should be in the rangeof . × − s ≤ T vis ≤ . × − s , and thisvalue of T vis is an order of magnitude higher thanthat for CO ice particles. This relaxation timefor H O ice particles obtained from our numericalresults is consistent with the result of Gundlach &Blum (2015) when we use the empirical relationbetween T vis and R (see Figure 5).4. Therefore, we concluded that the large differencein stickiness between H O and CO ice particleswould mainly originate from the difference in thestrength of viscoelastic effects.5. We also evaluated the critical velocity for col-lisional growth/fragmentation, V frag , of dust ag-gregates composed of µ m -sized CO ice particles.Assuming that V frag is approximately given by V frag = 10 V stick and the radius of monomer grainsis R = 0 . µ m , we found that the maximum sizeof dust aggregates would be controlled not by frag-mentation but by radial drift even outside the CO snow line (see Figure 6).More broadly, our results highlight the importance ofadditional energy dissipation channels during collisionsof dust particles. Thus future studies on the (viscoelas-tic) material properties of ices, including H O, CO ,CO, CH , CH OH, and NH , are of great importanceto understand the physics and chemistry in circumstellardisks. We also need to study the interplay between dustgrowth and chemical evolution in circumstellar disks.ACKNOWLEDGMENTSWe would like to thank Stephen E. Wood for provid-ing information about the surface energy of CO ice.S.A. is very thankful to Kenji Furuya for fruitful discus-sions. S.A. is supported by JSPS KAKENHI Grant No.JP20J00598.APPENDIX2 Arakawa & Krijt A. DEPENDENCE OF APPARENT SURFACE ENERGY ON CRACK SPEEDGreenwood (2004) derived the apparent surface energy which depends on the crack velocity using the Maugis–Dugdale model of the surface force law (Dugdale 1960; Maugis 1992). The normalized apparent surface energy andcrack velocity, β and v , are given as functions of k and α , where α is the non-dimensional transit time (Greenwood2004). For the opening crack, β and v are given by β = 1 I ( k, α ) , (A1) v = − π αI ( k, α ) , (A2)and for the closing crack, β = [ I ( k, α )] I ( k, α ) , (A3) v = π αI ( k, α ) . (A4)Here I , I , and I are given by I ( k, α ) = k + (1 − k ) J ( α ) , (A5) I ( k, α ) = k + (1 − k ) J ( α ) , (A6) I ( k, α ) = 1 − (1 − k ) J ( α ) , (A7)and J , J , and J are functions of α : J ( α ) = 12 α (cid:90) d ξ exp [ − α (1 − ξ )] F ( ξ ) , (A8) J ( α ) = 12 α (cid:90) d ξ exp [ − α (1 − ξ )] A ( ξ ) , (A9) J ( α ) = 12 (cid:90) d ξ exp [ − α (1 − ξ )] ξ − / , (A10) F ( ξ ) = 2 (cid:112) ξ − (1 − ξ ) log (cid:12)(cid:12)(cid:12)(cid:12) √ ξ − √ ξ (cid:12)(cid:12)(cid:12)(cid:12) , (A11) A ( ξ ) = 2 (cid:112) ξ + (1 − ξ ) log (cid:12)(cid:12)(cid:12)(cid:12) √ ξ − √ ξ (cid:12)(cid:12)(cid:12)(cid:12) . (A12)As both β and v are the functions of α , we can regard α as an auxiliary variable. Then we obtained v = v ( k, β ) asshown in Figure A1. 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