ACCRETION IN PROTOPLANETARY DISKS BY COLLISIONAL FUSION

The origin of the solar system and the formation of solar-type stars are wed through contemporary studies of primitive solar nebula from a wide range of perspectives including observational and theoretical astrophysics, nucleosynthesis, planetary dynamics, accretion physics, and meteoritics (Whipple 1964; Adachi et al. 1976; Weidenschilling 1977; Lissauer 1993; Cuzzi & Weidenschilling 2006; Quitte et al. 2007; Scott 2007; Dullemond et al. 2007; Blum & Wurm 2008; Armitage 2010). The general scenario (Armitage 2010) includes a sequence of stages: (1) the initial collapse of interstellar gas nucleating the central protostar (∼0.1 Myr); (2) slow mass accretion onto the star and primary planetesimal formation around the evolving accretion disk (∼Myr) followed by (3) a phase (∼Myr) during which the accretion rate drops significantly allowing the photoevaporative wind to "sever" the disk into two portions at a radius that depends on the ratio of the stellar accretion rate to the mass-loss rate due to photoevaporation, and finally (4) a rapid clearing phase (∼0.1 Myr) during which the inner disk is accreted onto the central star and the lightest elements of outer disk are removed by direct exposure to photoevaporative UV flux. Radiometric dating reveals that calcium–aluminum-rich inclusions (CAIs) within carbonaceous chondrite meteorites are about 4.57 Gyr old, thereby constituting the first planetary materials (see, e.g., Scott 2007). Several Myr later, well after protoplanetesimals had already been processed, chondrules within chondrites were still forming (Scott 2007).

Whereas very small particles move ostensibly as does the slightly sub-Keplerian nebular gas, planet-sized objects couple very weakly to the gas. Therefore, the buildup of protoplanets reaches an impasse at the meter scale due to the exchange of angular momentum with the gas; this relative motion is responsible for a drag that causes them to drift inward and they are lost to the central star on timescales much less than the CAI to chondrule interval (Blum & Wurm 2008; Weidenschilling 2008; Armitage 2010). The competition between gravitational settling and turbulence can lead to a disk midplane enhanced by solid matter and thus the same basic impasse is operative due to solid–solid angular momentum exchange which dominates the net drag effect (e.g., Cuzzi et al. 1993, and references therein). Therefore, an outstanding problem in cosmogony is to understand how, when objects can agglomerate to the critical ∼meter scale by known low collisional velocity accretion processes (Dominik et al. 2007; Blum & Wurm 2008; Güttler et al. 2010), they are not then rapidly lost into the central star? Hence, it is sought to reveal the basic mechanisms through which matter can accrete sufficiently rapidly to slow their radial motion and thwart their demise.

1.2.1. Gravitational Collapse

1.2.2. Sequential Aggregation of Particles

Outside the framework of the Minimum Mass Solar Nebula (MMN), it is a challenging exercise to calculate the relevant phase fronts in an evolving accretion disk (Stevenson & Lunine 1988; Lecar et al. 2006; Ciesla & Cuzzi 2006; Garaud & Lin 2007; Armitage 2010). For example, depending on, inter alia, turbulent mixing, the gas density, the nebular optical depth, and the epoch of evolution, the snowline can vary many AU with a temperature ∼170 ± 20 K, moving inward as the disk evolves. Thus, separate from the feasibility of computing it, knowledge of the entire thermodynamic history of a particle undergoing a two-body collision is highly model-dependent. However, it is important to put the situation discussed here in thermodynamic context for particle collisions. For a given material the basic kinetics of growth at a fixed temperature in a nebula can strongly influence its bulk and surface properties and hence collisional physics. Based solely on equilibrium considerations, ice alone exercises an enormous area of its phase diagram in a disk. Regardless of the material, it is well known that a host of basic equilibrium and disequilibrium phenomena will influence the bulk and interfacial properties (Dash et al. 2006). Therefore, depending on the detailed history of a particle it may, over the long times associated with disk evolution, have annealed to obtain normal compact laboratory properties (Dash et al. 2006), or it may have a highly porous structure (Blum & Wurm 2008; Güttler et al. 2010). These issues are discussed in Section 3.3, but we note that, in the absence of fracture, the mechanism described here provides a wide range of high sticking probability collisions for a very broad range of particle densities and thermodynamic conditions. Hence, this new mechanism acts in concert with other processes of particle destruction and sticking (Güttler et al. 2010).

In this paper, a new thermodynamic mechanism of mass agglomeration, termed collisional fusion, is described. The mechanism involves the confluence of collisional energetics, intermolecular interactions, and structural surface and bulk phase transitions. While the stresses in solid/solid collisions have been embodied in a range of planetesimal studies through examination of, among other things, the restitution coefficient (Chokshi et al. 1993; Bridges et al. 1996; Higa et al. 1998; Güttler et al. 2010), their influence on the bulk and interfacial phase behavior has heretofore received little attention. Due to a combination of accessibility and the dual role of relevance to both the formation of the gas giants and the dynamics of Saturn's rings, there have been a number of experiments on the collisions of ice particles under various conditions (e.g., Bridges et al. 1984, 1996; Higa et al. 1998; Dominik et al. 2007; Blum & Wurm 2008), but none at the very high pressures and low temperatures where this new mechanism is operative. Depending on the temperature and pressure, many materials can take on multiple equilibrium crystal structures (polymorphs) and moreover, under other conditions the crystalline order of one of the polymorphs can be lost and the system becomes amorphous such as the case with glass. Water substance exhibits a rich polymorphism and amorphism in its phase behavior. Here, classical collision physics is modified by an extension of the idea of damage-assisted interfacial melting (Dash et al. 2006) to the very high pressures that drive the fusion of matter by momentary liquefaction/amorph-polymorphization and freezing/annealing. The theory is quantitatively borne out in solid ice (Ih and Ic) down to T∼ 150 K, low density amorphous ice (LDA), and high density amorphous ice (HDA) at lower temperatures. Moreover, the process occurs in silicon, which melts at ∼1690 K, and where the range of collisional fusion speeds is ΔV_c∼ 100–1000 m s⁻¹, as seen in the recent molecular dynamics simulations of Suri & Dumitricaˇ (2008). Finally, bodies of inhomogenous composition can fuse in this manner. For example, a body with an icy mantle of sufficient thickness can fuse with another or other multicomponent materials, such as olivine, with similar melting temperatures can act in the same manner so long as one has access to the relevant phase diagrams. Hence, the approach provides a framework for collisions from the inner to the outer nebula. For clarity and brevity the focus here is on the latter, where it is found that perfect sticking is extended to a range of collisional speeds ΔV_c∼ 1–100 m s⁻¹ which encompass both typical turbulent rms speeds and the velocity differences between boulder-sized and small grains ∼1–50 m s⁻¹ (Johansen et al. 2007).

The development of the paper is as follows. In Section 2, the basic physical processes at play and an outline of the logic structure of the mechanism of collisional fusion is summarized. The detailed treatment of the thermodynamic state theory of collisional fusion is described in Section 3 after which in Section 4 the ideas are used to address the bottleneck problem by calculating the fate of drifters beginning from several places in nebulae experiencing a wide range of turbulent enhancement/suppression of midplane solids. A guide to the variables and symbols used is provided in Table 1. Conclusions are drawn in Section 5.

Table 1. List of Symbols and Variables

Symbol/Variable	Definition
T, Tm	Temperature and bulk melting temperature
P, Pm	Pressure and bulk melting pressure
μq	Chemical potential of quantity q
qm	Latent heat of fusion or amorphization
ρs	Mass density of the solid phase
ρℓ	Mass density of the liquid or high density phase
Vc	Particle collisonal velocity
VL (VU)	Lower (upper) bound for collisional fusion
ΔVc = VU − VL	Range of particle collisonal velocity for fusion
Vth ≪ VL	Lower threshold of particle collisonal velocity for sticking
Uc	Collisional energy
ξ	Fraction of collisional energy converted to damage
$u_{\cal D}$	Damage-induced energy density
d	Damage-induced interfacial film thickness
${\cal A}_H$	Hamaker constant divided by 6π
E	Effective Young's modulus
ν	Poisson's ratio
$\cal M$	Effective mass of two-body collision
$\cal R$	Effective radius of two-body collision
Pc	Collisonal pressure
rc	Collisonal contact radius
τc	Collision time
τf	Freezing/annealing time for damage- induced disorder
ks	Thermal conductivity of solid
r(t)	Drifting particle radius as a function of time t
ro = r(t = 0)	Drifting particle initial radius
m(t)	Drifting particle mass as a function of time t
h	Specific angular momentum of drifting particle
ℓ ≡ hm	Angular momentum of drifting particle
R(t)	Nebular radial position as a function of time t. R(t = 0) = Ro.
ΔR(t) ≡ Ro − R(t)	Relative nebular radial position as a function of time t
M	Mass of central star
Ω(R)	Orbital frequency =2 × 10−7 R−3/2 with R in AU
G	Universal gravitational constant
$v_K\equiv \sqrt{GM/R}$	Keplerian velocity
α	Global turbulence intensity parameter
σp(σg)	Particle (gas) surface mass density in g cm−2
ρg(R) = 1.36 × 10−9 R−11/4	Midplane MMN gas density in g cm−3
ρp(R, α) = Enρg(R)	Midplane particle density in g cm−3
H, hp	Gas and particle vertical scale heights
St = St(R, r)	Particle Stokes number
En(R) ≡ ρp/ρg = 10−2 H/hp	Midplane enhancement factor (Equation (8))
T(R) = T0(R/R0)−1/2	Vertically averaged radial nebular temperature; $T_0=280\,\rm K$, R0 = 1 AU.
LDA	Low density amorphous Ice
HDA	High density amorphous Ice

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Aggregation of snow on the ground or dustballs on a floor is facilitated by weak interparticle collisional speeds and attractive intermolecular forces such as van der Waals interactions. If the temperature does not rise above freezing snow crystals will fuse together (sinter) and then densify through annealing/coarsening driven by the tendency to reduce surface energy. Such processes are driven by either surface or volume diffusion or the transport of mobile surface films (Dash et al. 2006) and they underlie the fact snow drifts soon form a crust on their surfaces or that one can form a solid metal object beginning with a powder (Herring 1951). Hence, the agglomeration and densification of matter is facilitated by time and very low collisional speeds where weak attractive interactions dominate. However, when we toss an ice cube against the wall it will bounce, falling to the floor and often shattering. Throwing it at a higher velocity results in shattering upon collision with the wall. The theory developed quantitatively here considers the situation in which the wall is made of ice, and an ice ball is thrown at it some 10 s of m s⁻¹. Upon contact, the interfacial pressure rises dramatically and some of the collisional energy momentary liquefies or disorders a thin region shared by the two surfaces. In a nebula, the ambient temperatures are sufficiently low relative to the melting points of materials under consideration that any liquid (or disordered material) will rapidly freeze (or anneal) thereby fusing the particles so long as their stored elasticity is not so large that rebound occurs first. Thus, there is a low-speed fusion and annealing of molecules and small particles, a higher speed bouncing and then an even higher speed–reentrant–fusion. This basic process is not specific to ice, but ice is particularly relevant to nebulae and holds common terrestrial experience. The process of high-speed collisional fusion simply depends on the phase diagram of the material and the place (and hence thermodynamic conditions) in a nebula where a collision occurs. It can operate, mutatis mutandis, in the inner or outer nebular regions and in all manner of materials. The bottleneck problem is addressed by analyzing a meter-sized midplane object rapidly spiraling into the central star and colliding with a range of smaller particles down to a fraction of a centimeter. The growth rate of the inspiraling object is calculated using the theory of collisional fusion and a disk model that includes a parameterization of turbulence to estimate the midplane particle density. It is found that the inspiraling object grows sufficiently rapidly by this mechanism to settle into a stable Keplerian orbit.

This basic mechanism is not specific to ice, but to make concrete predictions the theory is demonstrated with calculations that focus on ice in two nebular regions beyond the snowline. The logic sequence is as follows. (1) Upon collision the interfacial pressure rises dramatically. The associated shift in the melting transition, or structural phase transition to an amorph/polymorph is determined from the phase diagram of Strässle et al. (2007) shown in Figure 1. (2) The fraction of the collisional energy/area inducing damage and hence the associated interfacial liquidity/disorder is determined from Equation (3) and shown in Figure 2. (3) The laboratory experiments of Higa et al. (1998) are used to constrain the degree of interfacial damage, ξ, as shown in Figure 3 below. These are extended to the astrophysical regime by harnessing the low-temperature high-pressure phase behavior determined from the laboratory experiments of Mishima (1996) and Strässle et al. (2007). (4) If the time to refreeze (anneal) the liquid (amorph/polymorph) so produced is of order or less than the collision time then the particles fuse together. There is a critical velocity beyond which the collision time is too rapid and fusion fails. Thus, agglomeration is reentrant with collision speed V_c. Fusion occurs when V_U ⩾ V_c ⩾ V_L≫ V_th and for ice a new high-speed window of perfect sticking is found; ΔV_c = V_U − V_L∼ 1–100 m s⁻¹. This range is extended to larger V_c for higher melting temperature materials. Therefore, the degree of damage-induced disorder, and the temperature and particle-size-dependent collision versus freezing times, combine to determine the new collisional fusion window ΔV_c shown in Figure 4.

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The ubiquity of gravitational and intermolecular forces is responsible for their different, but omnipresent, influence on all macroscopic bodies. They are both power-law forces and both influence the structure and properties of materials. However, due to their vastly different strengths and ranges of interaction they influence matter on disparate length scales. While we have a more visceral intuition for the fact that the gravitational field layers the structure of planetary and stellar atmospheres and interiors, there is a much wider range of phenomena wherein intermolecular forces between media influence their behavior (e.g., Amit 1978; Dietrich 1988). For example, liquids less polarizable than the substrates upon which they stand will "wet" the surface by spreading into a thin film.

When studying the phases of matter one appeals to bulk phase diagrams. However, the presence of external fields (e.g., electric, magnetic, gravitational, intermolecular) can shift the region of phase stability substantially such as exists in the interior of a star due to gravitationally enhanced pressure. So it is that the presence of intermolecular force fields across a surface shift the equilibrium states of matter from bulk coexistence and can stabilize the liquid phase in the solid region of the bulk phase diagram (Dash et al. 2006). This so-called interfacial premelting occurs in all classes of materials; metals, rare gases, semiconductors, quantum solids, and molecular solids, including ice wherein there are a host of astrophysical and geophysical consequences (Dash et al. 2006). For the present development, while I will often refer to liquid, it should be viewed as a synonym for the available high density phase which can be amorphous or another polymorph of the solid as discussed in Section 3.3.

The concept of damage-assisted interfacial melting was developed by Dash et al. (2001) to describe mass and charge transfer in the ice/ice collision experiments of Mason & Dash (2000). This work is reviewed in Dash et al. (2006) and the basic mechanism has recently been adopted in a study of lightning in protoplanetary disks by Muranushi (2010). The Dash et al. (2001) theory focused on atmospheric ice rather than the very high pressures and very low temperatures that are encountered under nebular conditions, which substantially complicate the situation due to the detailed phase behavior of ice.

Consider the surface between a solid (s) such as ice and the gas phase at a given temperature T and pressure P. When one considers the field energy/molecule due to intermolecular attractions (polarization forces) across such a surface, there is a shift of bulk phase coexistence Δμ = μ_s(T, P) − μ_ℓ(T, P) describing the formation of a stable interfacial disordered or liquid film of thickness d whose presence lowers the chemical potential of the wetted or disordered surface ${\mu }_{\cal I}(d)<0$. Hence, this disordered equilibrium layer has a chemical potential μ_f that is lower than the bulk liquid μ_ℓ; $\mu _f (T,P,d) = \mu _{\ell }(T,P) - |{\mu }_{\cal I}(d)| = \mu _{s}(T,P)$. Furthermore, a variety of forms of disorder lower μ_f; surface roughness, polycrystallinity, excess strain produced intrinsic or surface-induced dislocations or Frank–Read sources, impurities or kinetic effects (Dash et al. 2006). Such effects can be responsible for the persistence of disorder in at least a few molecular layers to ultra high vacuum temperatures. Importantly, all such forms of disorder can be present on the surface or within the bulk of nebular solids be they ice particles, growing from water vapor near the snowline, or warmer silicates in the inner nebula. Although it is presently unfeasible to understand the detailed thermodynamic evolution of individual particles, the sign of the effect of disorder is the same for all. Moreover, during interparticle collisions with speed V_c the increased stress increases the density of disorder thereby further decreasing μ_f relative to the bulk liquid. All of these effects are embodied as damage-induced disorder, ${\mu }_{\cal D}(V_c)<0$, and hence the combined influence of intermolecular forces, preexisting and damage-induced disorder is to reduce the chemical potential of the interfacial liquid as $\mu _f (T,P,d, V_c) = \mu _{\ell }(T,P) - |{\mu }_{\cal I}(d)| - |{\mu }_{\cal D}(V_c)|$.

Expanding Δμ about a point at coexistence, T_m, P_m yields

$$\begin{equation} |{\mu }_{\cal I}(d)| + |{\mu }_{\cal D}(V_c)| = q_m \left(\frac{T_m - T}{T_m}\right) + \left(\frac{\rho _{\ell } - \rho _{s}}{\rho _{\ell } \rho _{s}}\right) (P_m - P), \end{equation} \tag{ 1 }$$

where q_m is the latent heat of fusion, ρ_s (ρ_ℓ) is the density of the solid phase (liquid or high density phase), and for the usual form of nonretarded polarization forces $|{\mu }_{\cal I}(d)| = \frac{|{\cal A}_H|}{ \rho _{\ell } d^3}$, where ${\cal A}_H$ is the Hamaker constant divided by 6π. ${\cal A}_H$ describes the strength of the polarization forces and is negative when the film is present (Dash et al. 2006). Whence, a general thermodynamic relationship between the film thickness, the strength of the intermolecular forces, and the damage-induced energy density $u_{\cal D} \equiv \rho _{\ell } |{\mu }_{\cal D}(V_c)|$ is found as

$$\begin{equation} d = \left[\frac{|{\cal A}_H|}{{\rho _{\ell }} \frac{q_m}{T_m}(T_m - T) + \left(\frac{\rho _{\ell } - \rho _{s}}{\rho _{s}}\right) (P_m - P) - u_{\cal D} }\right]^{1/3}. \end{equation} \tag{ 2 }$$

Consider, for example, that two particles of ice at fixed temperature T collide. The bulk sound speed in ice is ∼3 × 10⁵ cm s⁻¹ and the thermal diffusivity is ∼5 × 10⁻³ cm² s⁻¹ and thus only the interfacial region between the particles will relax thermally and mechanically and their remaining volume will dissipate the collision energy adiabatically. The rapid pressure buildup during the collision P_c drives the interfacial region toward equilibrium pressure at solid–liquid coexistence; P_m(T). Whence, both T → T_m⁻ and P → P_m⁻. Simultaneously, a fraction ξ of the collisional energy is converted into disorder-enhanced interfacial damage $u_{\cal D}$ (some is lost through other channels such as, e.g., surface waves and stored in other degrees of freedom) which lowers the chemical potential of the liquid phase. Thus, the damage energy intervenes to further shift the onset of interfacial liquidity to values away from bulk coexistence. Due to the rapid relaxation of the phonon modes in the interfacial region, thermodynamic state theory argues that, on the timescale of the collision, the damage associated with the collisional energy density is deposited uniformly over the maximum collisional area πr_c² to a depth d as $u_{\cal D} = \frac{\xi U_c}{\pi {r_c}^2 d}$, where $U_c = \frac{1}{2} {\cal M} {V_c}^2$ is the kinetic energy of the collision with ${\cal M} = {m_1 m_2}/({m_1 + m_2})$ as the effective two-body mass.

Analysis of Equation (2) for the range of astrophysically relevant particle sizes and collision energies shown in Figures 2 and 3 reveals that it is near the divergent limit, where the denominator vanishes, and hence

$$\begin{equation} d = \frac{\xi U_c}{\pi {r_c}^2 \left[{\rho _{\ell }} \frac{q_m}{T_m}(T_m - T) + \left(\frac{\rho _{\ell } - \rho _{s}}{\rho _{s}}\right) (P_m - P) \right]}. \end{equation} \tag{ 3 }$$

Equation (3) shows that the detailed nature of the interactions responsible for the film become unimportant, but the form of the result depends on their existence in the first place. Thus, for example, for other power-law interactions driving interfacial melting the exponent in Equation (2) changes. Moreover, for exponentially decaying forces, such as for example are present due to screening in metals from conduction electrons or intrinsic or impurity produced ions in ice, the relation can be logarithmic (Dash et al. 2006). However, regardless of the nature of the intermolecular interactions, the film thickness diverges by the same competition of effects in the denominator of Equation (2) and hence Equation (3) is the same for all. Therefore, while small changes in the nature of the attractive interactions between small particles, be they polarization forces or of some other form (Wang et al. 2005), may influence the rate of growth of small agglomerations (Dominik et al. 2007; Blum & Wurm 2008; Güttler et al. 2010) below the Chokshi et al. (1993) limit (see Figure 3), they will have little effect here.

The inelastic loss of energy into damage is treated as an inefficient Hertzian interaction. Experiments show the Hertzian formalism to be accurate for the losses of up to 40% of the incident kinetic energy (Gugan 2000). Thus, the maximum value of the contact pressure P ≡ ξP_c and contact radius r_c in Equation (3) are determined from

$$\begin{equation} P_c = \frac{3}{2 \pi } \left(\frac{4}{3}\right)^{4/5} \left(\frac{5 {\cal M} {V_c}^2 E^4}{4 {\cal R}^3}\right)^{1/5} {\mbox{and}} \end{equation} \tag{ 4 }$$

$$\begin{equation} r_c = \left(\frac{15 {\cal M} {\cal R}^2 {V_c}^2}{16 E}\right)^{1/5}, \end{equation} \tag{ 5 }$$

where ${\cal R} = {r_1 r_2}/({r_1 + r_2})$ is the effective two-particle radius and E is the effective Young's modulus which, for particles of the same material and Young's modulus $\tilde{E}$ and Poisson's ratio ν, is $E^{-1} = 2 (1 - \nu ^2)/{\tilde{E}}$; for ice E = 5.3 GPa. Figure 2 shows that the interfacial liquid increases rapidly with V_c and that this depends on the fraction ξ of the collisional energy channeled into disorder-enhanced damage.

Particle fusion requires that a sufficient number of new bonds must be formed in the melting/freezing (amorph-polymorphization/annealing) process to overcome the stored elastic energy driving rebound and the liquid must freeze before separation. The combination of the degree of damage and these conditions determine V_L and V_U. Collisions can clearly create sufficient interfacial liquidity (Figure 2) for fusion. The interfacial liquid refreezes on a timescale τ_f commensurate with the contact time τ_c and hence provides a velocity range for a given ${\cal R}$ in which fusion can occur. The contact time $\tau _c = 2.87\left({{\cal M}^2}/{{\cal R} V E^2 \xi ^5}\right)^{1/5}$ is increased from the Hertzian value by the loss factor ξ but is still dominated by the properties of the solid.

In the limit that a thin annular region of liquid surrounding a sphere of solid is much thinner than the radius of the sphere, the timescale for freezing of the former by heat conduction through the cold solid is ρ_sq_mr₁d/6k_sΔT, where k_s is the thermal conductivity of the solid and ΔT is the pressure-corrected deviation of the melt fluid from the precollision bulk freezing temperature (Carslaw & Jaeger 1959). Under conditions, for example, at 150 K the liquid formed is highly supercooled and thus likely freezes much more rapidly than this estimate which is very conservative. More importantly, in laboratory experiments on freezing greater volumes (∼30 mm³ as opposed to the maximum volume here for the collisions at 150 K of ∼20 mm³) with ΔT of only approximately 30 K is of order 10 ms (Spannuth et al. 2007). Moreover, in the timescale estimate above the entire sphere is considered to be covered by the film so this estimate is an upper bound. Nonetheless, for parsimony of development it is stated that the time it takes for the liquid formed by the damage to freeze is half that above due to heat loss from both sides of the interface; τ_f = ρ_sq_mr₁d/12k_sΔT. For centimeter-sized particles colliding with a meter scale drifter at V_c ∼ 10 m s⁻¹ this gives a range of τ_f for T = 105–150 K of 1–10 ms. Finally, as T decreases more energy is required for a given effective mass to create the same amount of liquid but τ_f decreases and hence the associated V_U increases. While these are rough estimates, a more refined analysis is not warranted by existing experimental data which do not cover the appropriate range of conditions. The obvious refinements lead to decreases in τ_f and hence increase in both V_L and V_U.

There is a material-specific interpretation of the contributions to τ_f and τ_c with a particular emphasis on the energetically favorable structure and hence the potential volume of fused material, dπr_c², which eventually drops below a value that provides sufficient bonding. While the collision time τ_c is interpreted as that of a weakly inelastic Hertzian process, in reality there is a continuous conversion of inelastic energy into the formation of interfacial melting throughout dπr_c². However, (1) this conversion is extremely rapid since it is controlled by the phonon speed of the solid and (2) most of the elastic interactions are carried by the solid, except under conditions not realized in practice as discussed below. Thus, while being mindful of our degree of ignorance, it is understood that both bounds are likely functions of the deviation of a collision from centrality, the surface history, and angular momentum of the particles, among other unknown factors. Finally, for ice there is an interesting but as yet unexplained size dependence of the maximum shear and tensile stresses (Higa et al. 1998) that may influence the slope of the upper curves in Figure 3.

3.2.1. Collisions at the Divergent Limit

As described in Section 1.3 any calculation of the precise thermodynamic evolution of a given particle, and hence that of any particular two-body collision, is highly model-dependent. Here, the relevant phase behavior is described and the implications for other materials of particular relevance for the inner nebula are developed.

At temperatures as low as 150 K and high collisional pressures, ices Ih and Ic transform to supercooled water (likely rapidly through metastable ices II, III, or IX) (Mishima 1996; Strässle et al. 2007) whereas at lower temperatures LDA undergoes amorphization to HDA in the same modality as has recently been studied at high temperatures in H-passivated Si spheres (Suri & Dumitricaˇ 2008). Rapidly deposited vapor may form as porous LDA but it will have ample time to anneal both thermally and due to cosmic irradiation (Palumbo 2005) and, unless continuous rapid growth continues (which will allow the disorder to persist), such a particle will be either ice Ih or Ic upon collision. Under the GPa pressures induced by typical high-speed collisions these ices will rapidly pass through the stable region of ice IX to transform into supercooled water (Mishima 1996; Strässle et al. 2007). One can envision growth histories and outward drift trajectories to regions substantially lower than 150 K, (Stevenson & Lunine 1988; Lecar et al. 2006; Ciesla & Cuzzi 2006; Garaud & Lin 2007) and thus collisions between LDA ice, that can undergo an amorphization transition to HDA ice, or annealed crystalline ice, can collisionally melt by a phonon softening mechanism (Strässle et al. 2007) represented by the phase diagram shown in Figure 1. Regardless, the essential process discussed here is the same whether ice persists as LDA and then undergoes collisionally induced high-pressure polyamorphization to HDA or anneals to Ih or Ic and collisions induce the formation of supercooled liquid (Mishima 1996; Strässle et al. 2007).

The molecular dynamics simulations of Suri & Dumitricaˇ (2008) provide an excellent example of damage-induced polyamorphism and fusion in Si which experimentally melts at ∼1690 K. They find that in high-speed and hence high-pressure collisions the β-tin phase forms in an interfacial region followed by picosecond annealing to the a-Si phase. While the particles in their study are sufficiently small that were they in the inner nebula they would be strongly coupled to the gas phase, their work demonstrates the basic effect of low-speed rebound (V_c∼ 900 m s⁻¹) and high-speed (V_c∼ 1640 m s⁻¹) damage-induced fusion. Clearly too, perhaps using simulations such as those on silicon (Suri & Dumitricaˇ 2008), it is important to microscopically examine the role of the depth of disorder/liquidity necessary to fuse particles. While here material with laboratory determined properties is used, it is understood that agglomeration of micron scale particles can create highly porous "pre-planetesimals" in experiments and simulations (Blum & Wurm 2008; Güttler et al. 2010). However, one must appreciate that over nebular timescales annealing can densify, and rapid growth kinetically roughen the surfaces of, all classes of crystalline material (Dash et al. 2001, 2006). Note too that while porous materials may be more fragile, the higher roughness and porosity of non-annealed particles can be treated in this framework and the results may be qualitatively the same, but the preexisting disorder enhances the interfacial effects leading to fusion. Thus, when wholesale fracture or spalling does not intervene, the mechanism described here provides a conservative range of collisional fusion.