The rotational disruption of porous dust aggregates from ab initio kinematic calculations

Stefan Reissl; Philipp Nguyen; Lucas M. Jordan; Ralf S. Klessen

doi:10.1051/0004-6361/202346068

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A&A, 692 (2024) A60

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Issue		A&A Volume 692, December 2024


Article Number		A60
Number of page(s)		17
Section		Interstellar and circumstellar matter
DOI		https://doi.org/10.1051/0004-6361/202346068
Published online		03 December 2024

A&A, 692, A60 (2024)

The rotational disruption of porous dust aggregates from ab initio kinematic calculations

Stefan Reissl¹^★, Philipp Nguyen¹, Lucas M. Jordan² and Ralf S. Klessen¹^,3

¹ Universität Heidelberg, Zentrum für Astronomie, Institut für Theoretische Astrophysik, Albert-Ueberle-Straße 2, 69120 Heidelberg, Germany
² Institut für Astronomie und Astrophysik, Universität Tübingen, Auf der Morgenstelle 10, 72076 Tübingen, Germany
³ Universität Heidelberg, Interdisziplinäres Zentrum für Wissenschaftliches Rechnen, Im Neuenheimer Feld 205, 69120 Heidelberg, Germany

^★ Corresponding author; reissl@uni-heidelberg.de

Received: 2 February 2023
Accepted: 3 October 2024

Abstract

Context. The size of dust grains in the interstellar medium follows a distribution where most of the dust mass is made up of smaller grains. However, the redistribution from larger grains towards smaller sizes, especially by means of rotational disruption, is still poorly understood.

Aims. We aim to study the dynamics of porous grain aggregates undergoing an accelerated rotation, namely, a spin-up process that rapidly increases the angular velocity of the aggregate. In particular, we aim to determine the deformation of the grains and the maximal angular velocity up to the rotational disruption event by caused by centrifugal forces.

Methods. We precalculated the porous grain aggregate by means of ballistic aggregation analogous to the interstellar dust as input for subsequent numerical simulations. We performed three-dimensional (3D) N-body simulations, mimicking the radiative torque spin-up process up to the point where the grain aggregates become rotationally disrupted.

Results. Our simulations results are in agreement with theoretical models predicting a characteristic angular velocity, ω_disr, on the order of 10⁸–10⁹ rad s⁻¹, where grains become rotationally disrupted. In contrast to theoretical predictions, we show that for large porous grain aggregates (⪆300 nm), the ω_disr values do not strictly decline. Instead, they reach a lower asymptotic value. Hence, such grains can withstand an accelerated rotation more efficiently up to a factor of 10 because the displacement of mass by centrifugal forces and the subsequent mechanical deformation supports the buildup of new connections within the aggregate. Furthermore, we report that the rapid rotation of grains deforms an ensemble with initially 50:50 prolate and oblate shapes, respectively, preferentially into oblate shapes. Finally, we present a best-fit formula to predict the average rotational disruption of an ensemble of porous dust aggregates dependent on the internal grain structure, total number of monomers, and applied material properties.

Key words: methods: laboratory: solid state / methods: numerical / ISM: abundances / dust, extinction

© The Authors 2024

Open Access article, published by EDP Sciences, under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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1 Introduction

Dust is a key component of the interstellar medium (ISM). It is important for regulating the properties of astrophysical objects across a wide range of scales: from the cooling of collapsing molecular clouds and subsequent star-formation down to the formation of planetary systems (Spitzer & Arny 1978; Dorschner & Henning 1995). However, the origin of dust, along with its initial physical properties and redistribution of grain sizes, is still a field of ongoing research (O’Donnell & Mathis 1997; Ormel et al. 2009; Birnstiel et al. 2010; Guillet et al. 2018; Draine & Hensley 2021a).

Dust composition and grain size distribution may be derived from the observed interstellar extinction curve and starlight polarization (Mathis et al. 1977; Draine & Lee 1984; Guillet et al. 2018; Draine & Hensley 2021a). Initially, the ISM is enriched by intermediate-mass stars at the asymptotic giant branch (AGB) as well as supernova ejecta both coming with a certain grain size distribution (Nozawa et al. 2007; Gail et al. 2009; Barlow et al. 2010; Matsuura 2011; Karovicova et al. 2013; Zhukovska et al. 2015; Bevan & Barlow 2016). Later, the grains grow in dense molecular clouds by the accretion of abundant elements and coagulation (Spitzer & Arny 1978; Chokshi et al. 1993; O’Donnell & Mathis 1997). Naturally, this process results in porous dust aggregates, rather than solid bodies (Ossenkopf 1993; Dominik & Tielens 1997; Wada et al. 2007). Dust destruction processes such as gas-grain sputtering or grain-grain collision (shattering) may redistribute the grown grains towards smaller sizes (Dwek & Scalo 1980; Tielens et al. 1994; Hirashita & Yan 2009).

More recently, the pioneering work presented by Hoang et al. (2019) offered a new dust destruction mechanism. Here, a radiation field causes radiative torques (RAT) acting on dust grains, which lead to an angular acceleration (see e.g., Lazarian & Hoang 2007a; Hoang et al. 2014). Given a sufficiently luminous environment, the grains would inevitably be disrupted by the emerging centrifugal force (Silsbee & Draine 2016; Hoang et al. 2019; Hoang 2020). The disruption process is usually quantified by the maximal tensile strength Smax, which is a measurement of how materials respond to stretching (Silsbee & Draine 2016; Tatsuuma & Kataoka 2021). For porous materials and dust grain analogs, the tensile strengths may be determined by numerical N-body simulations (Kataoka et al. 2013b; Seizinger et al. 2013b; Tatsuuma et al. 2019). However, simulations involving Smax end up missing the fact that it is not just the individual building blocks (so-called monomers) that undergo the stretching force between its neighbors and, in fact, it is the global centrifugal force acting on the entire aggregate. Therefore, up to this point, it has remained unclear whether the 𝒮_max parameter can accurately describe the internal processes of material displacement within the rotating aggregate.

The rotational disruption of fractal grain aggregates was already studied indirectly in Reissl et al. (2023, RMK22 hereafter) by evaluating 𝒮_max in the context of rapid grain rotation caused by a differential gas-dust velocity. However, in this paper, we aim to simulate the dynamics of rapidly rotating interstellar grains directly by taking the time evolution of the internal aggregate structure into account. The aim is to develop a model of the average disruption process of large ensembles of porous grains. This paper is structured as follows. In Sect. 2.1, we discuss the most likely composition of elements and minerals present in dust aggregates. An algorithm to mimic the growth of porous dust aggregates is outlined in Sect. 2.2, along with an introduction to the methods for quantifying the grain shape and porosity. In Sect. 2.3, we discuss the processes and forces acting between the monomers connected within the aggregates. The spin-up process of grains by means of RATs is outlined in Sect. 2.4. In Sect. 2.5, we discuss the numerical implementation of the set of equations that governs the internal aggregate dynamics under rapid rotation. In Sect. 3, we discuss our N-body simulation results. Finally, in Sect. 4, we summarize our findings.

Table 1

Ratio of different element abundances as observed in the ISM in comparison with the composition of the co-S material applied in this work.

2 Models and methods

In this section, we outline the construction of the applied dust aggregates, the considered physical effects on the surfaces between monomers, and the numerical aspects of the performed N-body simulations.

2.1 Dust grain composition

The exact composition of dust remains an open question, with a large variety of observed materials and sizes within the galaxy being similarly possible. Interstellar dust is usually modeled by grains with a spectrum of different sizes with silicate and carbonaceous components (Mathis et al. 1977; Weingartner & Draine 2001; Zhukovska et al. 2008; Voshchinnikov & Henning 2010; Guillet et al. 2018). Small spherical monomers condensate by depletion of the most abundant elements C, Mg, Si, Fe, and O from their immediate surrounding (see e.g., Kim et al. 2021). Elements such as Ti, Al, S Ca, and Ni are less abundant and may only contribute a few percent of the total dust mass (Hensley & Draine 2021). The abundance of molecules within the grains may be determined by observing the spectral dust absorption features. Silicate minerals from the olivine and pyroxene group are the most likely candidates to be present in order to account for the observed characteristic features (Mathis et al. 1977; Wada et al. 1999; Li & Draine 2001; Hensley & Draine 2021). However, it remains inconclusive whether these minerals form dust in a crystalline or amorphous structure and to what extend iron is present in its pure form (Draine & Li 2007; Rogantini et al. 2019; Do-Duy et al. 2020). Carbonaceous grains may consist of a regular graphite lattice or amorphous structures and be partly hydrogenated (Wada et al. 1999; Goto et al. 2003; Mennella 2006). Possibly, carbonaceous and silicate grains are not even separate dust populations; rather, they are baked into a single composite material (Draine & Hensley 2021a).

For the mimicking process, the optical properties of dust grains models based on the refractive indices of various materials have been developed and the data are publicly available¹ (see e.g., Draine & Li 2007; Jones 2012; Draine & Hensley 2021a). However, complementary well constrained models of the mechanical properties particularly for the composition of interstellar dust grain materials are still lacking. Commonly, grain analogs consisting exclusively of icy or pure quartz (SiO2) materials are utilized to simulate grain growth processes (e.g., Dominik & Tielens 1997; Wada et al. 2007; Seizinger et al. 2012). This is simply due to the fact that quartz is an easily available material on the market with well constrained properties by laboratory experiments (Kendall et al. 1987; Heim et al. 1999; Israelachvili 2011; Krijt et al. 2013).

In this paper, however, we explore the rotational disruption of three distinct grain materials labeled a-C, q-S, and co-S. Carbonaceous grains are represented by the mechanical parameters of amorphous carbon (a-C) and for compression silicate grains are considered built of pure quartz (q-S). In addition, we aim to approximate the mechanical properties of composite silicate (co-S) grains more precisely. Here, we assume the minerals forsterite (Mg₂SiO₄) and fayalite (Fe₂SiO₄) of the olivine series as well as enstatite (MgSiO₃) and ferrosilite (FeSiO₃) of the pyroxene series to be among the most likely ingredients of silicate grains (Petrovic 2001; Zolensky et al. 2006; Zhukovska et al. 2008; Gail et al. 2009; Takigawa & Tachibana 2012; Min et al. 2007; Kimura et al. 2015; Fogerty et al. 2016; Hoang et al. 2019; Escatllar et al. 2019; Kimura et al. 2020; Hensley & Draine 2021; Draine & Hensley 2021a). To get an approximation of the material mixture, we matched the abundance of individual elements within the considered minerals with the abundance of elements typical for the ISM (Min et al. 2007; Voshchinnikov & Henning 2010; Compiègne et al. 2011; Hensley & Draine 2021).

In Table 1, we present the relative abundances of elements within the ISM in comparison with the composition of our co-S model. For our best fit of the co-S model we get that each individual silicate monomer consist of a mixture of 31 % forsterite, 29 % fayalite, 20 % enstatite, and 20 % ferrosilite, respectively, and average the mechanical material properties accordingly. The abundance of elements within the co-S agrees well with the overall observations within the Milky Way but our co-S model shows a slight overabundance of Si. Naturally, the composition of co-S may locally be vastly different; for instance, in the vicinity of oxygen or carbon rich asymptotic giant branch (AGB) stars, where dust grains are newly formed (Zhukovska & Henning 2013). Thus, we remain agnostic concerning the actual composition of interstellar dust, however, we do consider our co-S model to be an improvement compared to simulations using pure quartz monomers.

2.2 Dust grain growth and aggregation

Dust grain aggregates may grow by ballistic hit-and-stick processes of monomers onto a grain’s surface. This process is usually called ballistic particle-cluster aggregation (BPCA, see e.g. Kozasa et al. 1992; Bertini et al. 2007). Newly formed grains may then grow to even larger aggregates by ballistic clustercluster aggregation (BCCA) (Ossenkopf 1993). Such grain-grain collisions result significantly compressed aggregates and, subsequently, gas pressure may compress an aggregate even more (see e.g., Dominik & Tielens 1997; Kataoka et al. 2013a; Michoulier & Gonzalez 2022, and references therein).

In our study, we modeled such compression effects by the ballistic aggregation with a migration (BAM) model introduced in Shen et al. (2008). Here, grains simply grow by means of ballistic aggregation (BA)² of monomers hitting the aggregate from random directions. In the BAM grain model, monomers may migrate along the surface once (BAM1) or twice (BAM2). Consequently, for BAM1, each monomer has at least two connections with the aggregate and, for BAM2, each monomer has at least three (Shen et al. 2008; Seizinger et al. 2013a).

Commonly, dust aggregation models utilize only a constant monomer size (see e.g., Kozasa et al. 1992; Shen et al. 2008; Wada et al. 2007; Bertini et al. 2007; Seizinger et al. 2012). However, it seems unlikely that a condensation process of elements in nature would lead to exactly one monomer size. In fact, numerous laboratory experiments clearly indicate that grown aggregates consist of monomers with a variable radius (Karasev et al. 2004; Chakrabarty et al. 2007; Slobodrian et al. 2011; Kandilian et al. 2015; Salameh et al. 2017; Paul et al. 2017; Baric et al. 2018; Kelesidis et al. 2018; Bauer et al. 2019; Wu et al. 2020; Zhang et al. 2020; Kim et al. 2021). In this study, we focus on a polydisperse system of monomers, where the exact distribution of the monomer radii within an aggregate may be approximated by a log-normal distribution (Köylü & Faeth 1994; Lehre et al. 2003; Slobodrian et al. 2011; Bescond et al. 2014; Kandilian et al. 2015; Liu et al. 2015; Bauer et al. 2019; Wu et al. 2020; Zhang et al. 2020).

We ran the BAM model with a Monte Carlo approach to create an ensemble of precalculated grain analogs resembling the observed parameters of dust in the circumstellar and interstellar medium. The radius of the i-th monomer a_mon,i is sampled from a range of a_mon,i ∈ [10 nm, 100 nm]. The log-normal distribution has a typical average of 〈a_mon〉 = 20 nm and a standard deviation of 1.65 nm. Successively, each newly sampled monomer was shot on a random trajectory into the simulation domain until it hit the aggregate. For simplicity, we assumed that each monomer sticks onto the surface when colliding. For BAM1 and BAM2, respectively, the i-th monomer migrates along the surface of the initially hit monomer in a random direction to establish additional connections. To create an aggregate in equilibrium, the monomer position X_i is corrected in such a way that each overlap between connected monomers agrees with the materialdependent equilibrium compression length, δ₀ (see Sect. 2.3 for details). The aggregation process is repeated until the dust aggregate reaches a certain volume of $V_{agg} = \frac{4 π}{3} \sum_{i=1}^{N_{mon}} a_{mon, i^{\cdot}}^{3}$ ${V_{{\rm{agg}}}} = {{4\pi } \over 3}\mathop {\mathop \sum \nolimits^ }\limits_{{N_{{\rm{mon}}}}}^{{\rm{i = 1}}} a_{{\rm{mon}},{{\rm{i}}^ \cdot }}^3$ (1)

Ballistic aggregates are usually quantified by the total number of monomers N_mon (see e.g., Wada et al. 2007; Shen et al. 2008; Seizinger et al. 2012). However, dust observations are tightly connected to the effective size of the grains (Mathis et al. 1977; Weingartner & Draine 2001). Hence, in our study, we opted to set controls on the effective radius by means of a biased sampling of the radius for each of the final three monomers to be added to the aggregate. The exact effective radius is determined by $a_{eff} = {(\frac{3 V_{agg}}{4 π})}^{\frac{1}{3}},$ ${a_{{\rm{eff}}}} = {\left( {{{3{V_{{\rm{agg}}}}} \over {4\pi }}} \right)^{{1 \over 3}}},$ (2)

instead of getting the grain size indirectly from $a_{eff} \approx N_{mon}^{1 / 3} 〈 a_{mon} 〉$ ${a_{{\rm{eff}}}} \approx N_{{\rm{mon}}}^{1/3}\,\langle {a_{{\rm{mon}}}}\rangle$ . Finally, we calculated the inertia tensor for each aggregate in order to determine the characteristic moments of inertia I_a1 > I_a2 > I_a3, along the unit vectors ${\hat{a}}_{1}, {\hat{a}}_{2},$ ${\hat a_1},{\hat a_2},$ and ${\hat{a}}_{3}$ ${\hat a_3}$ (see RMK22 for the exact procedure).

To connect the rotational disruption of the grain ensemble to distinct quantities associated with the internal structure of fluffy dust grains, we introduced the porosity, 𝒫, volume filling factor, ϕ, and fractal dimension, D_f, as well as the semi major axes, a < b < c, which are unique for each individual aggregate.

The porosity quantifies the empty space within an aggregate where 𝒫 = 0 is for a solid object and 𝒫 = 1 for a vacuum. Its value for an individual object depends on the definition of the surface that envelopes the aggregate. In our study, we followed the procedure of determining 𝒫 based on utilizing the moments of inertia of an aggregate as outlined in Shen et al. (2008). Here, the quantity $α_{i} = \frac{15}{8 π} \frac{I_{a i}}{ρ_{m a t} a_{eff}^{5}}$ ${\alpha _{\rm{i}}} = {{15} \over {8\pi }}{{{I_{a{\rm{i}}}}} \over {{\rho _{mat}}a_{{\rm{eff}}}^5}}$ (3)

is the ratio of the moment of inertia, I_ai, to that of a sphere with equivalent volume where the index i ∈ {1, 2, 3} denotes the three spatial directions. The corresponding semi major axes are $a = a_{eff} \sqrt{α_{2} + α_{3} - α_{1}},$ $a = {a_{{\rm{eff}}}}\sqrt {{\alpha _2} + {\alpha _3} - {\alpha _1}} ,$ (4) $b = a_{eff} \sqrt{α_{3} + α_{1} - α_{2}},$ $b = {a_{{\rm{eff}}}}\sqrt {{\alpha _3} + {\alpha _1} - {\alpha _2}} ,$ (5)

and $c = a_{eff} \sqrt{α_{1} + α_{2} - α_{3}},$ $c = {a_{{\rm{eff}}}}\sqrt {{\alpha _1} + {\alpha _2} - {\alpha _3}} ,$ (6)

respectively. These axes represent the length, width, and height of a spheroid with exactly the same moments of inertia I_a1, I_a2, and I_a3, respectively, as an individual irregularly shaped dust aggregate. Finally, the corresponding porosity of an aggregate may then be written as $P = 1 - \frac{a_{eff}^{3}}{a b c},$ ${\cal P} = 1 - {{a_{{\rm{eff}}}^3} \over {abc}},$ (7)

whereas the complementary quantity ϕ = 1 – 𝒫 is the volume filling factor (Shen et al. 2008).

The fractal dimension is a measure of the shape of an aggregate where D_f = 1 represents a one dimensional (1D) line and D_f = 3 is a compact sphere. We determined the fractal dimension, D_f, of each dust aggregate by the correlation function, $C χ^{D_{f} - 3} = \frac{n (χ)}{4 π χ^{2} l N_{mon}},$ $C{\chi ^{{D_{\rm{f}}} - 3}} = {{n(\chi )} \over {4\pi {\chi ^2}l{N_{{\rm{mon}}}}}},$ (8)

as outlined in Skorupski et al. (2014). Here, C is a scaling factor, χ is the distance from the center of mass, l is a length with l ≪ a_eff, and n(χ) is the number density of connected monomers within the shell [χ – l/2;χ + l/2]. Since the distribution of matter within each aggregate is of the essence for the rotational disruption, quantifying the shape of each aggregate by its fractal dimension, D_f, may potentially provide a valuable parameter in addition to the porosity, 𝒫, to quantify the rotational dynamics.

We created dust grains with effective radii in the range a_eff = 50 nm–550 nm in steps of 50 nm. For each a_eff, we repeated the MC dust growth process with 30 random seeds for the BA, BAM1, and BAM2 configurations and the a-C, q-S, and co-S materials. In total, we precalculated an ensemble of 2970 individual grains as input for our 3D N-body simulations.

We note that the grain growth by BA may heavily be impacted by grain charge (Matthews et al. 2012), high impact velocities (Dominik & Tielens 1997; Ormel et al. 2009), a preferential impact direction by means of grain alignment with the magnetic field (Lazarian & Hoang 2007a; Hoang 2022) or a gas-dust drift (RMK22). Hence, the resulting shapes in our grain ensemble may not be representative when compared for grain growth processes; for instance, in the vicinity of AGB stars (Zhukovska & Henning 2013) or in protostellar envelopes (Galametz et al. 2019). However, our grain model covers a sufficiently broad variety of grain shapes, allowing us to draw fitting conclusions about their rotational stability; noting, however, that the individual shapes are not all equally likely to exist in nature.

An exemplary selection of the grains is shown in Fig. 1. In Fig. 2, we present the characteristic quantities of the entire grain ensemble, as introduced above. The number of monomers, N_mon, as well as the number of connections, N_con, within each aggregate increase with effective radius, a_eff. By design, BA grains have generally less connections compared to BAM grains. Compared to the BAM grains with a fixed monomer size (Shen et al. 2008), the porosity, P, is not strictly increasing; rather, it stagnates for higher a_eff values because smaller monomers may easily migrate towards the center, making our aggregates overall more compact. The fractal dimension, D_f, shown in Fig. 2 increases slightly towards larger grains; however, the resulting grain BA, BAM1, and BAM2 grains of different sizes and shapes in this work are not well correlated with the fractal dimension, D_f. This is in contrast to the dust models of RMK22, where the grains are explicitly constructed to get an exact pre-determined D_f and are not to be compared with the BAM grains, as depicted in Fig. 1.

2.3 Inter-monomer contact effects

In this section, we introduce the forces and torques acting between the monomers within an aggregate in detail. In Fig. 3, we provide a schematic illustration of all considered monomer interactions. Two individual monomers in physical contact establish a common contact surface area and experience an attraction because of the van der Waals force. For some materials, stronger attractions such as Coulomb forces between charged monomers, dipole-dipole interaction within ices, or metallic binding between iron pallets may be relevant.

The attraction is quantified by the material dependent energy per surface area γ. Assuming monomers act like elastic spheres with a radius of a_mon,i and a_mon,j, respectively, their elastic deformation causes a repulsive force.

An analytical description of these forces was first presented in Johnson et al. (1971) with the so called JKR model (see also Hertz 1896), where the equilibrium radius of the contact surface is $r_{0} = {(\frac{9 π γ R^{2}}{E^{*}})}^{1 / 3}$ ${r_0} = {\left( {{{9\pi \gamma {R^2}} \over {{E^ * }}}} \right)^{1/3}}$ (9)

when no external force are acting between monomers; namely, the attractive and repulsive forces are in balance. Here, the quantity R = a_mon,ia_mon,j/(a_mon,i + a_mon,j) is the reduced monomer radius whereas the elastic parameter E* = E/(2 – 2ν²) is determined by the material specific constants of the Poisson ratio, ν, and Young’s modulus, E, respectively (we refer to Johnson 1987, for further details). Utilizing that a monomer contact breaks for the characteristic pulling force of $F_{C} = 3 π γ R,$ ${F_{\rm{C}}} = 3\pi \gamma R,$ (10)

as outlined in (Johnson et al. 1971), the normal force along the unit vector, $n_{c} = \frac{X_{i} - X_{j}}{| X_{i} - X_{j} |},$ ${n_{\rm{c}}} = {{{X_{\rm{i}}} - {X_{\rm{j}}}} \over {\left| {{X_{\rm{i}}} - {X_{\rm{j}}}} \right|}},$ (11)

may be written as $F_{N, ij} = 4 F_{C} [{(\frac{r}{r_{0}})}^{3} - {(\frac{r}{r_{0}})}^{3 / 2}] n_{c} .$ ${F_{{\rm{N}},{\rm{ij}}}} = 4{F_{\rm{C}}}\left[ {{{\left( {{r \over {{r_0}}}} \right)}^3} - {{\left( {{r \over {{r_0}}}} \right)}^{3/2}}} \right]{n_{\rm{c}}}.$ (12)

Consequently, for F_N,ij > 0 the i-th monomer is exerting a pushing force onto its j-th neighbour. Otherwise, for F_N,ij < 0, the j-th monomer is pulling on the i-th monomer.

The JKR model was eventually extended by Dominik & Tielens (1997), considering the mechanics of rolling (Dominik & Tielens 1995, 1997), sliding (Dominik & Tielens 1996, 1997), and twisting motions (Dominik & Tielens 1997) in between connected monomers. To track the relative motion of monomers, we used the formulation of the contact pointers n_i and n_j, as outlined in Dominik & Nübold (2002). These vectors point initially towards the centers of neighboring monomers when a new contact is established (see Fig. 3).

The force acting on the i-th monomer because of the sliding motion of the j-th monomer may be written as $F_{S, ij} = - 8 r_{0} G^{*} ζ \frac{(a_{mon, j} n_{j} - a_{mon, i} n_{i}) \cdot n_{c}}{| X_{i} - X_{j} |},$ ${F_{{\rm{S}},{\rm{ij}}}} = - 8{r_0}{G^ * }\zeta {{\left( {{a_{{\rm{mon}},{\rm{j}}}}{n_{\rm{j}}} - {a_{{\rm{mon}},{\rm{i}}}}{n_{\rm{i}}}} \right) \cdot {n_{\rm{c}}}} \over {\left| {{X_{\rm{i}}} - {X_{\rm{j}}}} \right|}},$ (13)

where $ζ = a_{mon, i} n_{i} - a_{monn, j} n_{j} - (a_{mon, i} n_{i} \cdot n_{c} - a_{mon, j} n_{j} \cdot n_{c}) \cdot n_{c}$ $\zeta = {a_{{\rm{mon,}},i}}{n_{\rm{i}}} - {a_{{\rm{monn,}},{\rm{j}}}}{n_{\rm{j}}} - \left( {{a_{{\rm{mon,}},{\rm{i}}}}{n_{\rm{i}}} \cdot {n_{\rm{c}}} - {a_{{\rm{mon,}},{\rm{j}}}}{n_{\rm{j}}} \cdot {n_{\rm{c}}}} \right) \cdot {n_{\rm{c}}}$ (14)

is the sliding displacement and $Γ_{S, ij} = - 8 r_{0} G^{*} a_{mon, i} n_{i} \times ζ$ ${{\bf{\Gamma }}_{{\rm{S}},{\rm{ij}}}} = - 8{r_0}{G^ * }{a_{{\rm{mon}},{\rm{i}}}}{n_{\rm{i}}} \times \zeta$ (15)

is the associated sliding torque (Johnson 1987; Dominik & Tielens 1996, 1997). Here, the material constant $G^{*} = G / (4 - 2 ν_{i}^{2})$ ${G^ * } = G/\left( {4 - 2v_{\rm{i}}^2} \right)$ depends on the shear modulus G = E/(2 + 2ν) (see e.g., Cardarelli 2008).

In contrast to sliding a rolling motion does not result in a force (see e.g., Dominik & Tielens 1997; Wada et al. 2007) but only in a torque of $Γ_{R, ij} = - 4 F_{C} n_{i} \times ξ,$ ${{\bf{\Gamma }}_{{\rm{R}},{\rm{ij}}}} = - 4{F_{\rm{C}}}{n_{\rm{i}}} \times \xi ,$ (16)

where the rolling displacement depends on the contact pointers via ξ = R(n_i + n_j).

The same for the twisting between monomers with a torque of $Γ_{T, ij} = - \frac{16}{3} G r_{0}^{3} ϕ;$ ${{\bf{\Gamma }}_{{\rm{T}},{\rm{ij}}}} = - {{16} \over 3}Gr_0^3\phi ;$ (17)

whereas the motion of the twisting follows the direction of the vector as $ϕ = n_{c} (t) \int_{0}^{t} (ω_{i} (t^{'}) - ω_{j} (t^{'})) \cdot n_{c} (t^{'}) d t^{'} .$ $\phi = {n_{\rm{c}}}(t)\mathop \smallint \limits_0^t \left( {{\omega _{\rm{i}}}\left( {t'} \right) - {\omega _{\rm{i}}}\left( {t'} \right)} \right){n_{\rm{c}}}\left( {t'} \right){\rm{d}}t'.$ (18)

Here, the angular velocities ω_i (t) and ω_j(t), respectively, describe the relative rotation of two monomers in contact (Dominik & Tielens 1997).

In this study we extend the monomer contact physics pioneered by Johnson et al. (1971), Dominik & Tielens (1995, 1996, 1997), and Wada et al. (2007) by introducing additional forces emerging from the accelerated rotation of the grain aggregate. In the notation of our grain model, the centrifugal force acting on each monomer may be written as $F_{cent,i} = - m_{mon, i} ω_{agg} \times (ω_{agg} \times X_{i}),$ ${F_{{\rm{cent,i}}}} = - {m_{mon,{\rm{i}}}}{\omega _{agg}} \times \left( {{\omega _{agg}} \times {X_{\rm{i}}}} \right),$ (19)

where $m_{mon,i} = (4 π / 3) ρ_{mat} a_{mon,i}^{3}$ ${m_{{\rm{mon,i}}}} = (4{\rm{\pi }}/3){\rho _{{\rm{mat}}}}a_{{\rm{mon,i}}}^3$ is the mass of the i-th monomer.

We emphasize that our numerical setup operates in a corotating coordinate system with its origin coinciding with the center of mass of the most massive fragment. Hence, the Coriolis effect acts as an additional fictive force on each individual monomers via $F_{cor, i} = - 2 m_{mon,i} ω_{agg} \times \frac{d X_{i}}{d t} .$ ${F_{{\rm{cor}},{\rm{i}}}} = - 2{m_{{\rm{mon,i}}}}{\omega _{{\rm{agg}}}} \times {{{\rm{d}}{X_{\rm{i}}}} \over {{\rm{d}}t}}.$ (20)

Furthermore, our aggregates are initially at rest and are gradually spun-up. Consequently, the acceleration of each monomer within the aggregate leads to an Euler force of $F_{eul,, i} = - m_{mon,i} \frac{d ω_{agg}}{d t} \times X_{i}$ ${F_{{\rm{eul,}},{\rm{i}}}} = - {m_{{\rm{mon,i}}}}{{{\rm{d}}{\omega _{{\rm{agg}}}}} \over {{\rm{d}}t}} \times {X_{\rm{i}}}$ (21)

acting on each monomer, where dω_agg/dt is the angular acceleration.

When monomers establish a new contact they start to oscillate around the equilibrium position driven by the surface attraction and monomer deformation. In nature, it is expected that the oscillation becomes dampened because the deformation dissipates energy. A weak damping force may be introduced based on the relative velocity of the monomers (Kataoka et al. 2013b; Seizinger et al. 2012) to artificially reduce oscillations. However, as shown in Seizinger et al. (2012), such a force may come with numerical instabilities. However, the damping model presented in Krijt et al. (2013) shows that the elastic dampening can significantly increase the dissipation of energy. In this study, we applied the viscoelastic damping force, $F_{D, ij} = \frac{2 E^{*}}{ν_{i}^{2}} \frac{d δ}{d t} r_{ij} T_{vis} n_{c},$ ${F_{{\rm{D}},ij}} = {{2{E^ * }} \over {v_{\rm{i}}^2}}{{{\rm{d}}\delta } \over {{\rm{d}}t}}{r_{ij}}{T_{vis}}{n_{\rm{c}}},$ (22)

as introduced by Seizinger et al. (2013a). Here, the dampening depends not on the relative velocity, but instead on the time evolution of the compression length $δ = a_{mon, i} + a_{min, j} - | X_{i} - X_{j} |;$ $\delta = {a_{{\rm{mon}},{\rm{i}}}} + {a_{{\rm{min}},{\rm{j}}}} - \left| {{X_i} - {X_j}} \right|;$ (23)

namely, the overlap between two spherical monomers with a distance of |X_i – X_j| apart from each other. The contact radius r_ij is related to the compression lengths δ via $\frac{δ}{δ_{0}} = 3 {(\frac{r}{r_{0}})}^{2} - 2 {(\frac{r}{r_{0}})}^{1 / 2} .$ ${\delta \over {{\delta _0}}} = 3{\left( {{r \over {{r_0}}}} \right)^2} - 2{\left( {{r \over {{r_0}}}} \right)^{1/2}}.$ (24)

Here, for an aggregate in equilibrium, the compression length is $δ_{0} = r_{0}^{2} / (3 R)$ ${\delta _0} = r_0^2/(3R)$ . At the exact moment of separation for two monomers, the compression length equals δ = –δ_C, where the critical length is δ_C = (9/16)^1/3δ₀ (Dominik & Tielens 1995).

The characteristic viscoelastic timescale T_vis is in the order of 1 ps – 10 ps (Krijt et al. 2013) but poorly constrained for specific grain materials. For smaller values of T_vis the material effectively behaves elastically and no damping is expected at all. For simplicity, we adopted a fixed value of T_vis = 5 ps for all considered grains, independent of material.

The full set of material parameters applied in our simulations is listed in Table 2. We emphasize that all parameters are the best estimates for ISM conditions, namely, when assuming a cold and dry environment. In general, these parameters (especially the surface energy γ), exhibit a wide range because they are highly sensitive to temperature (Schönecker et al. 2015; Bogdan et al. 2020), monomer size (Bauer et al. 2019), and humidity (Heim et al. 1999; Fuji et al. 1999; Kimura et al. 2015; Steinpilz et al. 2019; Bogdan et al. 2020). Subsequent studies on this matter are needed to complement the mechanical parameters utilized in this paper.

Fig. 1

Exemplary selection from the total ensemble of porous BA (top row), BAM1 (moddle row), and BAM2 (bottom row) aggregates for the effective grain radii a_eff = 200 nm (left column), a_eff = 350 nm (middle column), and a_eff = 500 nm (right column) with the corresponding numbers of monomers N_mon and neighbourhood connections N_con. Monomers are sampled to guarantee an exact effective radius a_eff. The radius a_out is associated to the smallest sphere enclosing the entire aggregate. Axes ${\hat{a}}_{1}, {\hat{a}}_{2},$ ${\hat a_1},{\hat a_2},$ and ${\hat{a}}_{3}$ ${\hat a_3}$ are defined by the grain’s moments of inertia I_a1 > I_a2 > I_a3, where ${\hat{a}}_{1}$ ${\hat a_1}$ is the designated axis of grain rotation.

Fig. 2

Distribution of the characteristic quantities of total number of monomers, N_mon (top left), number of connected neighbours, N_con (top right), porosity 𝒫 (bottom left), and fractal dimension, D_f (bottom right), for all BA (red), BAM1 (green), and BAM2 (blue) aggregates, respectively, dependent on effective radius, a_eff. Dots are the ensemble average over all shapes and materials while vertical bars represent the minima and maxima. Note: the data points have a small offset for better visibility.

Fig. 3

Schematic representation of the forces and torques acting in between the i-th and the j-th monomer at positions X_i and X_j, respectively. The common contact surface with radius r_ij is shaded in red. (a) Each individual monomer experiences external centrifugal F_cent, Coriolis F_cor, and Euler forces F_eul as a result of the aggregate’s accelerated rotation with angular velocity ω_agg. (b) The normal force F_N,ij acts normal to the contact surface of two monomers because of surface attraction and mechanical deformation with the contact pointers remaining anti-parallel n_i = –n_j. (c) The sliding force, F_S,ij, is parallel to the sliding displacement, ζ. Both F_S,ij and the corresponding sliding torque ΓS,ij work tangential to the contact surface. (d) The same is truefor the rolling of monomers within the aggregate: The rolling displacement ξ, the force F_R,ij, and the Γ_R,ij are tangential to the contact surface. (e) The twisting of monomers in contact is not associated with a net force. The resulting torque Γ_T,ij is parallel to the normal vector ϕ of the twisting. We note that the depicted quantities are not to scale since r_ij ≪ a_mon.

2.4 Radiative torque disruption (RATD)

In this section, we briefly outline the radiative torques (RAT) that lead to rapid grain rotation and potentially to the disruption by centrifugal forces. It has been well established that grains with an irregular shape acquire a certain amount of angular velocity when exposed to directed radiation (Dolginov & Silantev 1976; Draine & Weingartner 1996, 1997; Weingartner & Draine 2003; Lazarian & Hoang 2007a). Here, the gain of angular momentum over time follows $ω_{agg} (t) = ω_{RAT} [1 - \exp (- \frac{t}{τ_{drag}})],$ ${\omega _{{\rm{agg}}}}(t) = {\omega _{{\rm{RAT}}}}\left[ {1 - \exp \left( { - {t \over {{\tau _{{\rm{drag}}}}}}} \right)} \right],$ (25)

where τ_drag is the characteristic timescale of the rotational drag by means of gas collisions (Draine 1996) and photon emission (Draine & Lazarian 1998), along with $ω_{RAT} = \frac{Γ_{RAT} τ_{drag}}{I_{a 1}},$ ${\omega _{{\rm{RAT}}}} = {{{\Gamma _{{\rm{RAT}}}}{\tau _{{\rm{drag}}}}} \over {{I_{a1}}}},$ (26)

which is the terminal angular velocity. The torque, Γ_RAT, is characteristic for individual grains and depends on the spectrum of the radiation field as well as the shape of a particular grain and its material composition (see e.g., Hoang et al. 2014). A solution of Γ_RAT may be calculated by numerical approximations (Draine 1996; Draine & Flatau 2013; Herranen et al. 2019) or analytical toy models (Lazarian & Hoang 2007a). However, we emphasize that we do not evaluate ω_RAT explicitly within the scope of this paper; instead, we assume a maximal grain rotation to guarantee the disruption of all aggregates. For the corresponding angular acceleration follows then $\frac{d ω_{agg} (t)}{d t} = \frac{ω_{RAT}}{τ_{drag}} \exp (- \frac{t}{τ_{drag}}) .$ ${{{\rm{d}}{\omega _{{\rm{agg}}}}(t)} \over {{\rm{d}}t}} = {{{\omega _{{\rm{RAT}}}}} \over {{\tau _{{\rm{drag}}}}}}\exp \left( { - {t \over {{\tau _{{\rm{drag}}}}}}} \right).$ (27)

In principle, grains would also spin-up when exposed to a gaseous flow (Lazarian & Hoang 2007b; Das & Weingartner 2016; Hoang et al. 2018; RMK22) leading to a mechanical torque (MET) Γ_MET. However, the time evolution and acceleration of the angular velocity would still follow the same curves governed by Eqs. (25) and (27), respectively, but evaluated with Γ_MET instead of Γ_RAT.

A dust aggregate cannot simply be modeled as a rigid body. Internal relaxation processes such as Barnett relaxation (Purcell 1979; Lazarian & Roberge 1997), nuclear relaxation (Lazarian & Draine 1999), or inelastic relaxation (Purcell 1979; Lazarian & Efroimsky 1999) would dissipate the rotational energy. Subsequently, the dissipation would re-orient the direction of the angular velocity, ω_agg. For typical interstellar conditions the total relaxation time is much smaller than the drag time, τ_drag, (Weingartner & Draine 2003) and the grain axis ${\hat{a}}_{1}$ ${\hat a_1}$ becomes the most likely axis of grain rotation (compare Fig. 1). For simplicity we assume in this study that the angular velocity points always in the direction of ${\hat{a}}_{1}$ ${\hat a_1}$ , namely, $ω_{agg} (t) = {\hat{a}}_{1} ω_{agg} (t)$ ${\omega _{{\rm{agg}}}}(t) = {\hat a_1}{\omega _{{\rm{agg}}}}(t)$ for all of our 3D N-body simulations.

Potentially, the aggregate becomes ripped apart by centrifugal forces long before reaching the terminal angular velocity, ω_RAT, (Silsbee & Draine 2016; Hoang et al. 2019). A mathematical framework for the radiative torque disruption (RATD) of grains was presented in Hoang et al. (2019). Here, the critical angular velocity for grain suggested for RATD is: $ω_{S} = \frac{2}{a_{eff}} {(\frac{S_{max}}{ρ_{mat}})}^{1 / 2} .$ ${\omega _{\cal S}} = {2 \over {{a_{{\rm{eff}}}}}}{\left( {{{{{\cal S}_{{\rm{max}}}}} \over {{\rho _{{\rm{mat}}}}}}} \right)^{1/2}}.$ (28)

Any dust aggregate exceeding the rotational limit ω_agg > ω_𝒮 becomes inevitably destroyed. This theoretical upper limit is derived from the material density, ρ_mat, the effective radius, a_eff, and the maximal tensile strength, 𝒮_max, of the aggregate (see Hoang 2020, for further details). However, the tensile strength of porous dust aggregates is not well constrained and may be lower by several orders of magnitude compared to solid bodies because individual substructures are not fully connected. In Greenberg et al. (1995), a means to estimate the tensile strength of aggregates was provided by evaluating (see also Li & Greenberg 1997). $S_{\max} = \frac{3}{2} 〈 N_{con} 〉 \frac{ϕ E_{b}}{a_{mon}^{2} h} .$ ${{\cal S}_{\max }} = {3 \over 2}\langle {N_{{\rm{con}}}}\rangle {{\phi {E_{\rm{b}}}} \over {a_{{\rm{mon}}}^2h}}.$ (29)

Here, E_b is the binding energy, h is related to the overlap of monomers, ⟨N_con⟩ is the average number of connections between all monomers within the aggregate, and a_mon is the monomer size of a monodisperse aggregate.

Complementary N-body simulations by Seizinger et al. (2013b) suggest that the tensile strength is not directly related to the initial volume filling factor ϕ as assumed, for instance, by Greenberg et al. (1995) or Blum et al. (2006). Instead, the tensile strength is 𝒮_max ∝ ϕ^1.6 for static compaction aggregates and 𝒮_max ∝ ϕ^1.9, respectively, BAM aggregates. Comparable results are presented by Tatsuuma et al. (2019), where the relation 𝒮_max ∝ ϕ^1.8 was suggested in particular for icy and quartz BCCA aggregates and $S_{\max} \propto ϕ^{2 / (3 - D_{f})}$ ${{\cal S}_{\max }} \propto {\phi ^{2/\left( {3 - {D_{\rm{f}}}} \right)}}$ for arbitrary grain shapes with a fractal dimension, D_f. We emphasize that in these simulations the aggregates got merely stretched or compressed, respectively, upon breaking, but do not experience any effects associated with rotation at all.

Table 2

Material parameters of the constructed ballistic aggregates.

2.5 Numerical setup and code implementation

We implemented the physical effects (as outlined above) into a C++ code that combines the physics of Seizinger et al. (2012, 2013a) with the code presented in RMK22 for the growth of aggregates. The time evolution of the net force acting on each monomer, $m_{mon,, i} \frac{d v_{i}}{d t} = F_{cent, i} + F_{cor, i} + F_{eul, i} + \sum_{j = 1, i \neq j}^{N_{mon}} (F_{N, ij} + F_{S, ij}),$ ${m_{{\rm{mon,}},{\rm{i}}}}{{{\rm{d}}{\upsilon _{\rm{i}}}} \over {{\rm{d}}t}} = {F_{{\rm{cent,i}}}} + {F_{{\rm{cor}},{\rm{i}}}} + {F_{{\rm{eul}},{\rm{i}}}} + \mathop \sum \limits_{{\rm{j}} = 1,{\rm{i}} \ne {\rm{j}}}^{{N_{{\rm{mon}}}}} \left( {{F_{{\rm{N}},{\rm{ij}}}} + {F_{{\rm{S}},{\rm{ij}}}}} \right),$ (30)

and that of the corresponding torque $I_{mon, i} \frac{d ω_{i}}{d t} = \sum_{j = 1, i \neq j}^{N_{mon}} (Γ_{S, ij} + Γ_{R, ij} + Γ_{T, ij}),$ ${I_{{\rm{mon}},{\rm{i}}}}{{{\rm{d}}{\omega _{\rm{i}}}} \over {{\rm{d}}t}} = \mathop \sum \limits_{{\rm{j}} = 1,{\rm{i}} \ne {\rm{j}}}^{{N_{{\rm{mon}}}}} \left( {{{\bf{\Gamma }}_{{\rm{S}},{\rm{ij}}}} + {{\bf{\Gamma }}_{{\rm{R}},{\rm{ij}}}} + {{\bf{\Gamma }}_{{\rm{T}},{\rm{ij}}}}} \right),$ (31)

is calculated with second order accuracy by a symplectic leapfrog integration scheme. Here, the quantity 3i is the velocity of an individual monomer within the simulation domain and ω_i is the angular velocity caused by sliding, rolling, and twisting motions of its neighbors, whereas $I_{mon,i} = 2 / 5 m_{mon,i} a_{mon,i}^{2}$ ${I_{{\rm{mon,i}}}} = 2/5{m_{{\rm{mon,i}}}}a_{{\rm{mon,i}}}^2$ is the moment of inertia of a particular monomer.

To account for the dissipation of energy, we assume that sliding, rolling, and twisting operate in the elastic limit until the corresponding displacement reaches some characteristic critical limit. The theoretical limits are ζ_c = r₀(2 – ν)/(16π) for sliding and ϕ_c = 1/(16π) for twisting (compare Eqs. (14) and (18)). However, the critical limit of rolling ξ_crit (see Eq. (16)) is still highly debated. For silicate monomers of different sizes this limit is within the range of ξ_crit ∈ [0.2 nm, 3.2 nm] (Dominik & Tielens 1997; Heim et al. 1999; Paszun & Dominik 2008). In this study, we apply the conservative value of ξ_crit = 0.2 nm for the entire ensemble of aggregates independent of size and material. When the displacements because of the relative motion of monomers in contact exceed their critical values, energy is dissipated and our code modifies the contact pointers ni and nj to ensure that ∣ζ∣= ζ_c, |ϕ| = ϕ_c, and |ξ|= ξ_crit, respectively (see Dominik & Nübold 2002; Wada et al. 2007, for details).

Individual monomers operate on a characteristic time scale that may be written as (Wada et al. 2007): $t_{dis, i} = \frac{1}{6^{2 / 3}} \sqrt{\frac{m_{mon, i} r_{0}^{2}}{π γ R^{2}}} .$ ${t_{{\rm{dis}},{\rm{i}}}} = {1 \over {{6^{2/3}}}}\sqrt {{{{m_{{\rm{mon}},{\rm{i}}}}r_0^2} \over {\pi \,\gamma {R^2}}}} .$ (32)

Furthermore, a monomer was not allowed to move a distance larger than its own radius, limiting the time step further by t_vei,i = a_mon,i/v_i. The final time step for our leap-frog integration scheme is then the minimum of these characteristic times ∆t = 0.02 mini (t_dis,i, t_vel,i) of all the monomers as well as monomer connections within the simulation domain. The factor of ∆t is slightly smaller than the one suggested by Wada et al. (2007), which guarantees a conservation of energy below an error of 10^–3 . However, monomers may roll along the surface of the rapidly rotating aggregate and choosing a smaller time step allows for a more accurate spatial resolution of this kind of displacement of the monomer within the aggregate.

The rotational damping acts usually on short timescales, τ_drag, on the order of days up to several hundred years compared to typical astronomical processes in the ISM (see e.g., Weingartner & Draine 2003; Tazaki et al. 2017; Hoang 2022). A computational burden arises because the integration time, ∆t, is only a few ns. Consequently, some simulations may take up to 10¹⁸ time steps to terminate. Hence, a much smaller drag time τ_drag in the order of hours instead of years is selected to reduce the total run-time of the code. However, this does not impact the result of the rotational disruption simulations as long as the change in centrifugal forces per time step remains much smaller than the displacement processes for individual monomers to reach a new equilibrium position within the rotating aggregate. Even for a τ_drag of a few hours, it is still guaranteed that centrifugal forces are the dominant cause of grains disruption because the spin-up remains slow enough that the shear within the aggregate by Euler forces remains negligible. In detail, a τ_drag is selected such that the aggregate reaches an angular velocity whereby destruction is guaranteed within a few hours of the simulation time, but the centrifugal force remains always the dominant force exerted on each monomer; namely, |F_eul,i|≪ |F_cent,i|. The simulations setup starts at ω_agg = 0 rad s^–1 and follows the curve of Eq. (25) up to a given terminal angular velocity ω_RAT by default. Alternatively the code terminates for the condition N_con = 1; namely, only two connected monomers remain within the simulation domain.

For detecting monomer collisions with variable sizes and the tracking of contact breaking events, we used a loose octree data structure (see e.g., Ulrich 2000; Raschdorf & Kolonko 2009, for further details) to optimize the runtime of the code even more. A new contact was established when two moving monomers touch each other, namely, at δ ≤ 0. When individual monomers or even entire sub-structures of the initial aggregate would become unconnected, we only tracked the evolution and spin-up process of the most massive remaining fragment, while smaller fragments were allowed to leave the simulation domain. After each time step, the remaining connected fragments were identified with a recursive flood fill algorithm on an undirected graph that represents the monomer connection relations. For every thousandth time step, the affiliation of monomer to a certain fragment, the number of monomers, N_mon, remaining within the simulation domain, and the monomer positions, X_i, as well as the forces and connections in between monomers are recorded.

In order to quantify the total stress in between all connected monomers within the largest fragment we introduce the quantity of the total stress, $σ_{Σ} = \sum_{i = 1}^{N_{mon}} \sum_{j > i}^{N_{mon}} \frac{| F_{N, ij} |}{π r_{ij}^{2}},$ ${\sigma _\Sigma } = \mathop \sum \limits_{{\rm{i}} = 1}^{{N_{{\rm{mon}}}}} \mathop \sum \limits_{{\rm{j}} > {\rm{i}}}^{{N_{{\rm{mon}}}}} {{\left| {{F_{{\rm{N}},{\rm{ij}}}}} \right|} \over {\pi r_{{\rm{ij}}}^2}},$ (33)

where F_N,ij = 0 for non-connected monomers.

3 Results and discussion

We performed numerical N-body simulations for each individual precalculated grain aggregate with our numerical N-body setup upon rotational disruption. We note, that all grains are disrupted at an angular velocity of ω_disr before the overall spin-up process reaches the terminal angular velocity of ω_RAT = 3 ⋅ 10¹⁰ rad s^–1. Hence, we consider this exact value of ω_RAT as an upper limit in our simulation setup as well as all subsequent data analysis. We emphasize once again that the exact values of ω_RAT is of minor relevance for our N-body simulations as long as the disruption of each aggregate is guaranteed and the centrifugal force and the Euler force follow strictly the relation F_eul,i ≪ F_cent,i for the entire spin-up process as governed by Eqs. (25) and (27), respectively.

3.1 Time evolution of the rotational disruption process

A typical simulation result for a q-S BAM2 grain with an effective radius of a_eff = 350 nm is shown in Fig. 4. The evolution of the mass M(ω_agg) during the spin up process remains constant up to ω_agg ≈ 3 ⋅ 10⁸ rad s^–1. Once the grain spins even faster, contacts break and individual monomers as well as smaller fragments start to break from the aggregates surface. Simultaneously, monomers wander outwards driven by centrifugal forces. For a small period the total number connections exceeds even the initial value and drops then steadily. This features is most pronounced for BAM2 grains. Eventually, the simulations reach the condition of N_con = 1 and terminate. The breaking of contacts and the subsequent mass loss is continuous in nature while the angular velocity, ω_agg, increases by roughly one order of magnitude. To designate a characteristic angular velocity, ω_disr, of rotational disruption based on mass loss or broken connections would be arbitrary. In Fig. 4, we show also the time evolution of the total stress σ_Σ(ω_agg) of each aggregate. In contrast to the mass loss and the braking of contacts, stress σ_Σ(ω_agg) rises steadily but then drops abruptly. Hence, we define this unique feature in the time evolution of each aggregate to be associated with the angular velocity of ω_disr of rotational disruption. The value of ω_disr is close to but not identical with a mass loss of 25%.

Fig. 4

Evolution of the mass M(ω_agg) (top panel), number of connections N_con(ω_agg) (middle panel), and total stress σ_Σ(ω_agg) (bottom panel) within the largest fragment dependent on the increasing angular velocity ω_agg. The blue lines represent the exemplary ensemble of q-S BAM2 grains with a_eff = 350 nm. An arrow points to the peak value of the number of connections, N_con. Red dots indicate the characteristics angular velocity, ω_agg, where an individual aggregate has lost for the first time more than 25 % of its initial mass, initial number of connections, or reached its peak stress.

3.2 The fragmentation of rotating grains

In Fig. 5, we show snapshots of a typical spin-up process and rotational disruption event for one representative grain. The grain is identical to the BAM2 with a_eff = 500 nm depicted to the bottom right of Fig. 1. Once the grain has spun up to an angular velocity of ω_agg/ω_disr = 0.1, the aggregate starts to experience a stretching force and first connections break. However, all monomers are still connected to the same aggregate. Compared to its original configuration (depicted in Fig. 1), the aggregate changed its shape already, because monomers are re-arranged within the aggregate by means of rolling. The aggregate fragments for ω_agg/ω_disr = 0.8 and new connections are established as monomers move into a new equilibrium position driven by pulling forces and pushing forces between connected neighbors. We report that in this phase, small disconnected fragments may rarely reconnect to the most massive fragment. For an angular velocity of ω_agg/ω_disr = 1.0, larger fragments are separated from the most massive cluster. The remaining cluster goes through a phase of relaxation, where its total stress σ_Σ declines; simultaneously, the mass loss is still engaged in an ongoing process. We note that if the spin-up process of the grain were to stop at an angular velocity ω_agg/ω_disr ≤ 1.0 the aggregate would still fragment, but the mass loss would stop eventually as the remaining most massive fragment reaches a new equilibrium configuration.

Dust destruction processes such as shattering efficiently redistribute larger grains sizes towards smaller ones (Dwek & Scalo 1980; Tielens et al. 1994). In modeling this process, it is usually assumed that the new size distribution of the fragments follows a power-law (Hellyer 1970; Hirashita & Yan 2009; Kirchschlager et al. 2019). A similar power-law is assumed in models of the grain redistribution by means of RATD (e.g., Giang et al. 2020). However, the fragmentation process of rapidly rotating porous dust aggregates remains poorly constrained. In fact, our N-body simulation results so far indicate that the considered BAM aggregates almost completely break down into their individual building blocks. Hence, the resulting size distribution of the fragments is expected to be almost identical to the initial size distribution of the monomers. However, we note that our numerical setup does not track smaller fragments once they become separated from the most massive fragment.

Furthermore, each individual fragment will experience its own individual spin-up and drag (see Sect. 2.4). Smaller fragments may also eventually become too small for a RAT effective for a further spin-up upon the disruption point (Lazarian & Hoang 2007a). Such effects need to be taken into account in forthcoming studies to answer question about the resulting size distribution of an ensemble of rotationally disrupted grains conclusively.

3.3 Rotational deformation of grain shapes

In Fig. 6 we present the time evolution of the shapes of a-C grains with different sizes up to the breaking point. Initially, for ω_agg/ω_disr = 0.0 the ensemble of grains with an effective radius of a_eff = 200 nm is almost oblate and prolate shapes in equal parts. Larger grains with a_eff = 350 nm and a_eff = 500 nm are slightly more prolate where most of the axis ratios are c/b < 1.7. At the breaking point (i.e., ω_agg/ω_disr = 1.0), the a-C grains become deformed where oblate shapes are the most likely outcome with an axis ratio up to b/a ≤ 5.0 for BAM grains. The exception is the ensemble of small BA grains with a_eff = 200 nm where a prolate shape seems to be the more favorable configuration. We note that BA grains reach smaller axis ratios because they break more easily with a much shorter deformation phase compared to BAM1 and BAM2 grains. We report similar trends for the deformation of grains with q-S and co-S materials, respectively, where oblate grains are the most likely shape of rotational disruption.

Quantifying the grain shape by the fractal dimension, D_f, during the spin-up process reveals no clear trend. As depicted in Fig. 2 the fractal dimension is not well correlated with the grain shapes of the initial ensemble. Up to the braking point the variation of D_f becomes even larger. The same for the porosity 𝒫. At the beginning of the spin up process 𝒫 starts to increase slightly for BAM1 and BAM2 grains while the BA ones break without an increase in 𝒫. Close to the breaking point the porosity distribution has a large variation and becomes virtually identical for BAM1 and BAM2 grains. Eventually, the analysis of the fractal dimension, D_f, as well as the porosity, 𝒫, does no longer apply because the calculation of these quantities fails as the grain aggregates break down into its individual monomers (see Sect. 2.2).

The shape and porosity of interstellar dust is still a matter of debate. For example BA grain growth processes favor roundish grain aggregates with a fractal dimension of about D_f = 2.0, where the principle axes are a ≈ b ≈ c. Guillet et al. (2018) developed a grain model based on spheroids of amorphous silicate and amorphous carbon to reproduce both starlight polarization and polarized sub-millimeter emission. The best fit model suggest prolate grains with an axes ratio of b/a = 1/3 and a porosity of 𝒫 = 0.2. More recently, Draine & Hensley (2021b) suggest prolate grains with an axis ratio of b/a ≲ 0.4 or oblate grains with b/a ≳ 1.5 with a porosity of about 𝒫 = 0.4. Comparing the initial porosity of our ensemble grown by BAM (see Fig. 2) reveals that only the BAM2 ensemble would match the limitation in porosity with a value range of 𝒫 ≈ 0.4–0.6.

Elongated grains may be the result of hit and stick processes of BPCA with preferential direction; for instance, for magneto-hydrodynamic turbulence (Yan & Lazarian 2003) or for grain aggregates aligned with the magnetic field direction (Hoang 2022). However, the latter effect requires rapid grain rotation. Subsequently, the spin-up process would increase the relative velocity between the surface of the rotating aggregate and unconnected free impinging monomers. These impinging monomers may potentially destroy the entire aggregate in a catastrophic disruption event (Benz & Asphaug 1999; Morris & Burchell 2017; Schwartz et al. 2018). What we have found with our N-body simulations is that accelerated rotation preferentially deforms initially more roundish aggregates into oblate shapes.

For RATs, the axis ratio of the grains is linked to the radiation field via the maximal angular velocity, ω_RAT. The more extreme cases, as depicted in Fig. 6 with b/a = 4 – 5 would require the grains to be in close vicinity of a supernova or an active galactic nucleus (Hoang et al. 2019; Giang et al. 2020; Giang & Hoang 2021). We emphasize that the torques Γ_RAT and Γ_MET (see Sect. 2.4) are tightly connected to the grain shape (Lazarian & Hoang 2007a; Das & Weingartner 2016; RMK22). Consequently, the terminal angular velocity, ω_RAT, would be marginally increased during the deformation phase accelerating the grain rotation even more (see Eq. (27)). However, this effect would be continuous over a long time span and not considerably impact the internal grain dynamics as far as the final value of ω_disr is concerned. More severe is the change in ω_RAT during the fragmentation phase. Here, the RAT would decrease as the aggregate fragments and subsequently the rotation would slow down and the grain stabilizes by reaching a new equilibrium configuration. However, considering a dynamical spin-up process in our N-body simulation would not only require us to calculate the torque, Γ_RAT, but also the rotational drag timescale τ_drag for each time step by means of time consuming numerical approximate methods (Draine & Flatau 2013; RMK22). Potentially, both quantities (Γ_RAT and τ_drag) may be parameterized, based on the shape and material parameters of each individual BAM grain. At present, a dynamical spin-up goes beyond the scope of this paper.

Fig. 5

Three exemplary snapshots of a BAM2 grain aggregate with a_eff = 500 nm of a particular rotational disruption simulation with the corresponding number of connection N_con for the angular velocities of ω_agg/ω_disr = 0.1 (top row), ω_agg/ω_disr = 0.8 (middle row), andωagg/ωdisr = 1.0 (bottom row). We emphasize that the exemplary grain is identical to the one in the lower right corner of Fig. 1. Left column: Color-coded is the affiliation of the distinct connected fragments. Smaller fragments are arbitrarily colored whereas the most massive fragment is always depicted in blue. Right column: Monomers are colored according to the largest magnitude of the normal force F_N exerted from its connected neighbors, where F_N > 0 (green) represents pushing forces while F_N < 0 (red) are pulling forces, respectively.

Fig. 6

Evolution of grain shapes for a-C BA (red), BAM1 (green), and BAM2 (blue) grains with an effective radius of a_eff = 200 nm (left panel), a_eff = 350 nm (middle panel), and a_eff = 500 nm (right panel), respectively. Dots represent the initial grain shape i.e., ω_agg/ω_disr = 0 whereas crosses are the grain shape at disruption for ω_agg/ω_disr = 1. We note the clear tendency of the porous grains to become oblate in shape with increasing ω_agg.

3.4 Average rational disruption of the grain ensemble

In Fig. 7, we present the characteristic angular velocity ωdisr for the entire set of simulation results. The average magnitude of ω_disr is roughly in the range of about 5 ⋅ 10⁸–3 ⋅ 10⁹ rad s^–1 for different materials and grain sizes. This result agrees well with the predictive model presented in Hoang et al. (2019). However, the exact value of ω_disr depends on the exact material properties and the internal structure of the initial grain. While rapidly rotating solid bodies break into few fragments by driven by centrifugal forces, a porous material may fragment at lower angular velocity and into a larger number of pieces. The maximal tensile strength, 𝒮_max, is usually utilized to quantify the response of a material during stretching. However, what we find that utilizing the maximal tensile strength, 𝒮_max, as introduced in Eq. (29), to calculate the critical angular velocity, ωS, with Eq. (28) does not reproduce the average rotational disruption, ω_disr, resulting from our N-body simulations. We argue that 𝒮_max is insufficient to describe the dynamical behavior of rotating aggregates because in a stretching aggregate each monomers experience only the local forces in between its connected neighbors. In a rotating aggregate, however, each individual monomer experiences also the centrifugal force. The maximal tensile strength, 𝒮_max, is determined by stretching of granular materials performed along a distinct axis, whereas the centrifugal force is radial with respect to the rotation axis. Furthermore, considering only stretching acting on an aggregate does not lead to a large displacement of monomers compared to the aggregate’s scale.

It is important to note that our N-body simulations reveal that monomers are rolling outwards within the grain aggregates. This movement of N_mon monomers potentially allows us to establish up to $N_{mon} (N_{mon} - 1) \approx N_{mon}^{2}$ ${N_{{\rm{mon}}}}\left( {{N_{{\rm{mon}}}} - 1} \right) \approx N_{{\rm{mon}}}^2$ new connections. Naturally, some monomers are already connected and only a small fraction of all unconnected monomers is close enough within the aggregate to newly connect. Hence, the relation cannot be quadratic, but with a much smaller exponent α ≪ 2.

In summary, our simulations reveal that in order to model the maximal angular velocity ω_disr two effects have to be accounted for: (i) the number of potentially newly formed connection in the deformation phase and (ii) the number of of connections that need to be overcome to enter the deformation phase in the first place. As outlined above, the former may be represented by the potential number of potential partners to establish new connections, $N_{mon}^{α}$ $N_{{\rm{mon}}}^\alpha$ , where as the later depends on the initial number of connections and the volume filling factor; namely, (⟨N_con⟩ ϕ)^β, where β is an additional power-law exponent. Hence, we argue that the definition of the tensile strengths (see Eq. (29)) needs to be extended by the number of monomers, N_mon, to account for the outward movement of monomers along the surface. Consequently, the characteristic angular velocity of rotation disruption for a polydisperse grain aggregate then reads: $ω_{disr} = \frac{A}{a_{eff}} \sqrt{\frac{γ}{ρ_{mat} 〈 a_{mon} 〉}} {(〈 N_{con} 〉 N_{mon} ϕ)}^{α},$ ${\omega _{{\rm{disr}}}} = {A \over {{a_{{\rm{eff}}}}}}\sqrt {{\gamma \over {{\rho _{{\rm{mat}}}}\langle {a_{{\rm{mon}}}}\rangle }}} {\left( {\langle {N_{{\rm{con}}}}\rangle {N_{{\rm{mon}}}}\phi } \right)^\alpha },$ (34)

where ⟨a_mon ⟩ is the average monomer radius and A and α, respectively, are dimensionless fit parameters³ to match the simulation results. We emphasize that introducing a separate exponent to each of the individual quantities ⟨N_con⟩ and N_monϕ, respectively, does not improve the accuracy of the fit so that in the following we model the maximal angular velocity, ω_disr, only by two fit parameters: A and α. Best fit results of ω_disr are plotted in Fig. 7 and the corresponding fir parameters are listed in Table 3. The fit matches very well with the ensemble average of the different considered grain sizes. We emphasize that our best fits of the parameter α are smaller than the theoretical limit of 1/2 given, for instance, in Hoang et al. (2019). We argue that the smaller values of α are a result of the deformation the dust aggregates experience before the mass loss starts. This effect is not taken into account by Hoang et al. (2019), where the dust is assumed to be a briddle object where the disruption occurs nearly instantaneously.

In Fig. 8, we show a comparison of our best fit of ω_disr for co-S BAM2 grains with that calculated with different parameter- izations of the tensile strength, 𝒮_max, given in literature (see also Sect. 2.4). These parameterizations of 𝒮_max may not necessarily be evaluated with the parameters provided by our grain models. Hence, we scaled the result to match our results for ω_disr at an effective radius of a_eff = 100 nm. The comparison reveals that previous attempts of modeling ω_disr by a volume filling factor, ϕ, dependent tensile strength, 𝒮_max, cannot reproduce the asymptotic behavior of our simulation results towards larger grain radii, a_eff.

Finally, we emphasize that the presented results of rotational disruption are calculated for aggregates of carbonaceous and silicate monomers loosely connected by the van der Waals force. However, materials other than carbon or silicates may form much stronger bonds (Dominik & Tielens 1997). For instance, pure iron in the form of small pallets may be present in the interstellar dust that would create metallic bonds between monomers (Dominik & Tielens 1997; Draine & Hensley 2021a). The same for ices covering the surface of the grain aggregate where dipoledipole interactions may increase the resistance against rotational disruption.

Furthermore, high impact collisions of monomers or compressive stress acting on the aggregate may lead to sintering, an effect where the monomers fuse at the contact surface. Subsequently, sintering leads to a neck between monomers (Maeno & Ebinuma 1983; Blackford 2007; Sirono & Ueno 2017). This does strengthen the connection between monomers, but at the same time, it makes the aggregate a lot more brittle because the neck will not allow for rolling motions between monomers.

Moreover, an effect that may weaken monomer connections is dust heating. A higher dust temperature decreases the surface energy, γ, up to one order of magnitude (Bogdan et al. 2020). Any luminous environment with a radiation field strong enough to drive grain rotation up to ≈ 10⁹ rad s⁻¹ would inevitably heat the dust grains up to several hundreds of Kelvin. Thus, γ and subsequently ω_disr would decrease. Altogether, such effects need to be taken into consideration in forthcoming studies in order to complement our numerical setup of the rotational disruption of of porous dust aggregates.

Fig. 7

The angular velocity of disruption ω_disr dependent on grain size for the ensembles of co-S (top panel), q-S (middle panel), and aC (bottom panel) grain materials dependent on the effective radius a_eff. Color coded are the BA (red), BAM1 (green), and BAM2 (blue) grains. Vertical bars are the range between minimal and maximal values of ωdisr resulting from our N-body disruption simulations while solid lines represent the best fit model.

Table 3

Best-fit parameters of ω_disr based on our N-body disruption simulation results.

Fig. 8

Same as Fig. 7 but only for co-S BAM2 grains (blue) in comparison with the predictions of ω_agg based on the tensile strengths models presented in Greenberg et al. (1995) (G95), Seizinger et al. (2013b) (S13), and Tatsuuma et al. (2019) (T19), respectively (gray). The latter are re-scaled to match our simulation results for a_eff = 100 nm.

3.5 Impact of critical rolling displacement and viscous damping

The critical rolling displacement, ξ_crit, and the viscous damping time scale, T_vis, are the least constrained parameters in our N-body simulations. Laboratory data indicates a huge variation in these parameters of about one order of magnitude for silicates (Dominik & Tielens 1995; Heim et al. 1999; Krijt et al. 2013).

To quantify the impact ξ_crit and T_vis, respectively, on rational disruption, we repeated our N-body simulation for a subset of the precalculated grain aggregates, while varying ξ_crit values in the range 0.2 nm to 6.4 nm and T_vis from 1 ps to 9 ps. In Fig. 9, we present the exemplary results of the parameter test of a-C BA and BAM2 grains. Variations in the rolling displacement, ξ_crit, show little effect. The resulting angular velocity of rotational disruption, ω_disr, exhibits small fluctuations but no clear trend with an increase in ξ_crit. Hence, we attribute the fluctuation to numerical effects (see also Appendix A). In contrast to ξ_crit, an increase in T_vis leads to a clear increase of ω_disr. This effect is most evident for BAM2 grains with ω_disr being about 11 % higher for the largest applied viscous time of T_vis = 9 ps. Similar trends can be reported for the q-S and co-S materials. Further improvements in the predictive accuracy of our rational disruption simulations cannot be achieved based on forthcoming laboratory data, especially for the viscoelastic damping of oscillations within carbonaceous aggregates.

4 Summary

The aim of this paper is to study the rotational disruption of interstellar dust aggregate analogs. We precalculated an ensemble of porous dust aggregates by means of ballistic aggregation and migration (BAM). For BA, BAM1, and BAM2 grains, each monomer has at least one, two, or three connections, respectively, with its neighbors. Here, we modified the original BAM algorithm presented in Shen et al. (2008) to make it applicable to variable monomer sizes. We estimated the composition of the grain aggregates based on the abundance of elements in the ISM to approximate their mechanical properties. We performed numerical 3D N-body simulations to determine the characteristic angular velocity of rational disruption ω_disr for each aggregate individually. The numerical setup was based on the work of Dominik & Tielens (1997), Wada et al. (2007), Seizinger et al. (2012), and RMK22, respectively. We modified their setup by introducing additional forces associated with an accelerated rotation of an aggregate. A subsequent analysis of the disruption event allowed us to describe the average rotational disruption of porous grain ensembles. The findings of this study are summarized as follows:

Compared to the original BAM algorithm, considering only single a monomer size, we report that a grain growth by BAM with polydisperse monomers results in aggregates that are more compact and less porous as smaller monomer tend to migrate deeper into the aggregate. For the same reason, the porosity, 𝒫, becomes stagnated with an increasing effective radius, a_eff, with values of up to 𝒫 = 0.4–0.85 for BA; whereas BAM1 and BAM2 grains are less porous, with 𝒫 = 0.35–0.7 and 𝒫 = 0.35–0.5, respectively;
Our simulation results reveal that rotating porous dust aggregates are not disrupted by a single abrupt event characteristic for a brittle material, but, rather, by a continuous process where aggregates experience a continuous mass loss. Subsequently, the initial aggregate breaks ultimately down into fragments not larger than a few monomers;
The initial distribution of the pre-calculated polydisperse BAM grains have oblate and prolate shapes, respectively, almost in equal parts. However, under accelerated rotation, the grain aggregates enter a phase of deformation and, ultimately, the grain shapes become preferentially redistributed towards oblate shapes;
We introduced the quantity of the total stress, σ_Σ, as a measurement of the internal aggregate dynamics and time evolution. We report that σ_Σ increases up to a peak value characteristic for individual aggregates, while the rotation accelerates and subsequently starts to lose mass. This peak roughly coincides with a mass loss of 25% with respect to the initial grain mass. For higher angular velocities, σ_Σ drops sharply, while the mass loss continues. We utilized this peak in σ_Σ to define the breaking point on a physical basis and to finally determine the characteristic angular velocity, ωdisr, of rotational disruption;
In the deformation phase of BAM aggregates, individual monomers move along the grain surface driven by the centrifugal force and new connections may be established. Consequently, the additional connections stabilize the BAM aggregates against the disruptive centrifugal forces acting on the monomers;
Our N-body simulations reveal that the angular velocity ω_disr reaches an asymptotic limit towards larger grain sizes of a_eff ⪆ 300 nm. This finding is in contrast to previous attempts to describe the rational disruption of porous aggregates analytically based on the maximal tensile strength, where ω_disr would continuously decrease for an increasing a_eff.

Fig. 9

Same as Fig. 7 but only for a-C BA (dashed lines) and BAM2 (dotted lines) grains. The left panel shows the impact of critical rolling displacement ξ_crit ∈ [0.2 nm, 6.4 nm] on ω_disr, while the viscous dumping time T_vis remains constant. The right panel is for a constant ξ_crit but with T_vis ∈ [1 ps, 9 ps]. Red lines represent the default parameters of our N-body disruption simulations.

Acknowledgements

Special thanks goes to Wilhelm Kley for numerous fruitfull discussion about N-body simulations of aggregates. The authors thank Thiem Hoang, Cornelis P. Dullemond, and Bruce T. Draine for usefull insights into the topic of dust composition and dynamics. S.R., P.N., and R.S.K. acknowledge financial support from the Heidelberg cluster of excellence (EXC 2181 - 390900948) “STRUCTURES: A unifying approach to emergent phenomena in the physical world, mathematics, and complex data”, specifically via the exploratory project EP 4.4. S.R. and R.S.K. also thank for support from Deutsche Forschungsgemeinschaft (DFG) via the Collaborative Research Center (SFB 881, Project-ID 138713538) ’The Milky Way System’ (subprojects A01, A06, B01, B02, and B08). And we thanks for funding form the European Research Council in the ERC synergy grant “ECOGAL – Understanding our Galactic ecosystem: From the disk of the Milky Way to the formation sites of stars and planets” (project ID 855130). The project made use of computing resources provided by The Länd through bwHPC and by DFG through grant INST 35/1134-1 FUGG. Data are in part stored at SDS@hd supported by the Ministry of Science, Research and the Arts and by DFG through grant INST 35/1314-1 FUGG.

Appendix A Connection-breaking test

In this section, we provide a test scenario for the accuracy of our code. Due to the complexity of N-body simulations, it is not feasible to derive an analytical solution for an entire aggregate. However, the problem can be be accurately solved for two connected monomers with of equal size (i.e., a_mon,i = a_mon,j), where the reduced radius simply becomes R = a_mon,i/2. In this case, the centrifugal force (see Eq. 19) may be written as $F_{cent, i} = - m_{mon, i} a_{mon, i} ω_{agg}^{2} .$ ${F_{{\rm{cent}},{\rm{i}}}} = - {m_{{\rm{mon}},{\rm{i}}}}{a_{{\rm{mon}},{\rm{i}}}}\omega _{{\rm{agg}}}^2.$ (A.1)

By using F_cent = F_N > F_C (see Eqs. 10 and 12), which is the criterion of separation δ = –δ_C (see Eq. 24), the corresponding critical contact radius becomes r_C = (1/6)^2/3r₀. Putting r_C into Eq. (12) allows us to solve for the exact angular velocity, $ω_{ref} = \sqrt{\frac{5 π}{6} \frac{γ}{m_{mon, i}}},$ ${\omega _{{\rm{ref}}}} = \sqrt {{{5\pi } \over 6}{\gamma \over {{m_{{\rm{mon}},{\rm{i}}}}}}} ,$ (A.2)

where the two equally sized monomers become separated due to centrifugal forces. We use this quantity for reference to evaluate the accuracy of our code. A number of 80 benchmark runs are performed with randomly selected monomer radii a_mon,i ∈ [10 nm, 100 nm]. In Fig. A.1, we compare the resulting angular velocity ω_disr of our numerical setup introduced in Sect. 2.5 with Eq. (A.2). The resulting error (ω_ref – ω_disr)/ω_ref is below 1.8 % with a trend of underpredicting ω_ref. An error of 1.8 % is comparable to the range we observe for the parameter test of the rolling displacement ξ_crit as presented in Fig. 9. Hence, we assumed a numerical accuracy of about 1.8 % for our N-body simulations. Consequently, the uncertainties in our reported results of the angular velocity of rotational disruption, ω_ref, is most impacted by the lack of exact laboratory material parameters, rather than numerical errors.

Fig. A.1

Distribution of the angular velocity, ω_disr, resulting from a number of runs, N_run, of N-body simulations in compression with the corresponding exact analytical solution, ω_disr.

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¹

For more details, we refer to the website of Bruce Draine https://www.astro.princeton.edu/~draine/dust/dust.diel.html and the THEMIS model https://www.ias.u-psud.fr/themis/

²

BA and BPCA are used synonymous in literature.

³

Note that 〈N_con〉 = N_con/N_mon and thus Eq. (34) may also be written in an equivalent form with a factor (〈N_con〉 N_monϕ)^α = (N_conϕ)^α.

All Tables

Table 1

Ratio of different element abundances as observed in the ISM in comparison with the composition of the co-S material applied in this work.

In the text

Table 2

Material parameters of the constructed ballistic aggregates.

In the text

Table 3

Best-fit parameters of ω_disr based on our N-body disruption simulation results.

In the text

All Figures

Fig. 1

Exemplary selection from the total ensemble of porous BA (top row), BAM1 (moddle row), and BAM2 (bottom row) aggregates for the effective grain radii a_eff = 200 nm (left column), a_eff = 350 nm (middle column), and a_eff = 500 nm (right column) with the corresponding numbers of monomers N_mon and neighbourhood connections N_con. Monomers are sampled to guarantee an exact effective radius a_eff. The radius a_out is associated to the smallest sphere enclosing the entire aggregate. Axes ${\hat{a}}_{1}, {\hat{a}}_{2},$ ${\hat a_1},{\hat a_2},$ and ${\hat{a}}_{3}$ ${\hat a_3}$ are defined by the grain’s moments of inertia I_a1 > I_a2 > I_a3, where ${\hat{a}}_{1}$ ${\hat a_1}$ is the designated axis of grain rotation.

In the text

Fig. 2

Distribution of the characteristic quantities of total number of monomers, N_mon (top left), number of connected neighbours, N_con (top right), porosity 𝒫 (bottom left), and fractal dimension, D_f (bottom right), for all BA (red), BAM1 (green), and BAM2 (blue) aggregates, respectively, dependent on effective radius, a_eff. Dots are the ensemble average over all shapes and materials while vertical bars represent the minima and maxima. Note: the data points have a small offset for better visibility.

In the text

Fig. 3

Schematic representation of the forces and torques acting in between the i-th and the j-th monomer at positions X_i and X_j, respectively. The common contact surface with radius r_ij is shaded in red. (a) Each individual monomer experiences external centrifugal F_cent, Coriolis F_cor, and Euler forces F_eul as a result of the aggregate’s accelerated rotation with angular velocity ω_agg. (b) The normal force F_N,ij acts normal to the contact surface of two monomers because of surface attraction and mechanical deformation with the contact pointers remaining anti-parallel n_i = –n_j. (c) The sliding force, F_S,ij, is parallel to the sliding displacement, ζ. Both F_S,ij and the corresponding sliding torque ΓS,ij work tangential to the contact surface. (d) The same is truefor the rolling of monomers within the aggregate: The rolling displacement ξ, the force F_R,ij, and the Γ_R,ij are tangential to the contact surface. (e) The twisting of monomers in contact is not associated with a net force. The resulting torque Γ_T,ij is parallel to the normal vector ϕ of the twisting. We note that the depicted quantities are not to scale since r_ij ≪ a_mon.

In the text

Fig. 4

Evolution of the mass M(ω_agg) (top panel), number of connections N_con(ω_agg) (middle panel), and total stress σ_Σ(ω_agg) (bottom panel) within the largest fragment dependent on the increasing angular velocity ω_agg. The blue lines represent the exemplary ensemble of q-S BAM2 grains with a_eff = 350 nm. An arrow points to the peak value of the number of connections, N_con. Red dots indicate the characteristics angular velocity, ω_agg, where an individual aggregate has lost for the first time more than 25 % of its initial mass, initial number of connections, or reached its peak stress.

In the text

Fig. 5

Three exemplary snapshots of a BAM2 grain aggregate with a_eff = 500 nm of a particular rotational disruption simulation with the corresponding number of connection N_con for the angular velocities of ω_agg/ω_disr = 0.1 (top row), ω_agg/ω_disr = 0.8 (middle row), andωagg/ωdisr = 1.0 (bottom row). We emphasize that the exemplary grain is identical to the one in the lower right corner of Fig. 1. Left column: Color-coded is the affiliation of the distinct connected fragments. Smaller fragments are arbitrarily colored whereas the most massive fragment is always depicted in blue. Right column: Monomers are colored according to the largest magnitude of the normal force F_N exerted from its connected neighbors, where F_N > 0 (green) represents pushing forces while F_N < 0 (red) are pulling forces, respectively.

In the text

Fig. 6

Evolution of grain shapes for a-C BA (red), BAM1 (green), and BAM2 (blue) grains with an effective radius of a_eff = 200 nm (left panel), a_eff = 350 nm (middle panel), and a_eff = 500 nm (right panel), respectively. Dots represent the initial grain shape i.e., ω_agg/ω_disr = 0 whereas crosses are the grain shape at disruption for ω_agg/ω_disr = 1. We note the clear tendency of the porous grains to become oblate in shape with increasing ω_agg.

In the text

Fig. 7

The angular velocity of disruption ω_disr dependent on grain size for the ensembles of co-S (top panel), q-S (middle panel), and aC (bottom panel) grain materials dependent on the effective radius a_eff. Color coded are the BA (red), BAM1 (green), and BAM2 (blue) grains. Vertical bars are the range between minimal and maximal values of ωdisr resulting from our N-body disruption simulations while solid lines represent the best fit model.

In the text

	Fig. 8 Same as Fig. 7 but only for co-S BAM2 grains (blue) in comparison with the predictions of ω_agg based on the tensile strengths models presented in Greenberg et al. (1995) (G95), Seizinger et al. (2013b) (S13), and Tatsuuma et al. (2019) (T19), respectively (gray). The latter are re-scaled to match our simulation results for a_eff = 100 nm.
In the text

Fig. 9

Same as Fig. 7 but only for a-C BA (dashed lines) and BAM2 (dotted lines) grains. The left panel shows the impact of critical rolling displacement ξ_crit ∈ [0.2 nm, 6.4 nm] on ω_disr, while the viscous dumping time T_vis remains constant. The right panel is for a constant ξ_crit but with T_vis ∈ [1 ps, 9 ps]. Red lines represent the default parameters of our N-body disruption simulations.

In the text

	Fig. A.1 Distribution of the angular velocity, ω_disr, resulting from a number of runs, N_run, of N-body simulations in compression with the corresponding exact analytical solution, ω_disr.
In the text

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