Dust Rotational Dynamics in C-shocks: Rotational Disruption of Nanoparticles by Stochastic Mechanical Torques and Spinning Dust Emission

Polycyclic aromatic hydrocarbon (PAHs) and nanoparticles are expected to play an important role in many astrophysical processes due to its dominant total dust surface area, including gas heating, chemistry, star formation tracer, and anomalous microwave emission. In dense magnetized molecular clouds where low-velocity shocks are present, PAHs and nanoparticles are widely believed to originate from grain shattering due to grain-grain collisions. However, the question is whether these nanoparticles can survive in the dense and hot shocked regions, and how to constrain the size and abundance of these tiny particles via observations. In this paper, we show that, due to high gas temperature and supersonic drift of neutrals relative to charged grains, smallest nanoparticles can be spun-up to extremely fast rotation by stochastic atom bombardment such that centrifugal force can disrupt non-ideal nanoparticles of size below $\sim 2$ nm. The proposed disruption mechanism is shown to be more efficient than sputtering in affecting the lower cutoff of grain size distribution in C-shocks if nanoparticles have non-ideal tensile strength. We model spinning dust emission from spinning nanoparticles subject to supersonic drift and rotational disruption. We find that suprathermally rotating nanoparticles can emit strong microwave radiation and peak frequencies increasing with the drifting velocity. We suggest a new method to trace nanoparticles and shock velocities in dense regions using microwave emission from spinning dust.


INTRODUCTION
Polycyclic aromatic hydrocarbon (PAHs) and very small dust grains (hereafter referred to as nanoparticles) are expected to play an important role in gas heating via photoelectric effect (Weingartner & Draine 2001) and chemistry because they contribute the dominant surface area of dust grains (see Akimkin et al. 2013). In dense, low ionization molecular clouds, nanoparticles are believed to affect cloud dynamics and star formation such as ambipolar diffusion (Mestel & Spitzer 1956) due to its dominant charge carrier (see Zhao et al. 2016 and references therein).
Nanoparticles are expected to be depleted in dense regions due to grain coagulation and accretion of atoms on the grain surface (Draine 2003;Tibbs et al. 2016). At the same time, grain-grain collisions due to grain acceleration by magnetohydrodynamic (MHD) turbulence (Yan & Lazarian 2003; can shatter big grains to form nanoparticles. In a strong radiation field such as massive stars and supernovae, Hoang et al. (2018d) proposed that nanoparticles can be produced by rotational disruption of big grains by radiative torques. When a shock is passing through a dense cloud, nanoparticles can also be formed by shattering of big grains due to grain-grain collisions Jones et al. 1994).
Interstellar shocks are ubiquitous in the ISM. In the magnetized dense clouds with a low ionization fraction, neutral gas and neutral grains can drift relative to charged grains coupled to the magnetic field. When the shock speed is lower than the magnetosonic speed, physical parameters are continuous, for which the term C-type shocks are introduced (Draine 1980) (see also Ciolek et al. 2004). As a result, collisions between neutral small grains and charged large grains can shatter the grains and form nanoparticles, including PAHs, nanosilicates, and nanodiamonds (Jones et al. 1996;Guillet et al. 2011). The remaining question is (i) whether nanoparticles can survive passing the C-shock and (ii) how to constrain the size and abundance of this nanodust population.
The first issue is important for understanding the composition of interstellar nanoparticles, which can help to shed light on the exact carrier of AME (Hoang et al. 2016; see Dickinson et al. 2018 for a review). The second issue is important for quantitatively understanding of the effect of nanoparticles on the cloud dynamics and chemistry.
Due to the lack of UV/optical photons to trigger mid-IR emission, the potential new technique to probe PAHs and nanoparticles in shocked dense regions is to use microwave emission produced by rapidly spinning nanoparticles (Draine & Lazarian 1998;Hoang et al. 2010;Hoang et al. 2016;Hoang et al. 2018b). Spinning dust emissivity depends mostly on the rotation of nanoparticles, grain dipole moment, and its abundance and size distribution in which the smallest nanoparticles are the most important emitters (Draine & Lazarian 1998;Hoang et al. 2010;Hoang et al. 2016).
Rotational dynamics of interstellar dust grains is a fundamental astrophysical problem. The grain rotation controls a variety of astrophysical observations, both in emission and polarization. For instance, the rapid rotation of nanoparticles can emit electric dipole radiation which is an important galactic foreground contaminated to Cosmic Microwave Background (CMB) radiation. In particular, suprathermal rotation can help dust grains to be efficiently aligned with magnetic fields (Hoang & Lazarian 2016a; see Andersson et al. 2015 andLazarian et al. 2015 for recent reviews). Dust polarization induced by aligned grains is a powerful tracer of cosmic magnetic fields and also a major foreground for CMB polarization observations. In particular, extremely fast rotation of grains due to radiative torques in strong radiation fields can control the maximum cutoff of grain size distribution (Hoang et al. 2018d). Rotational dynamics of nanoparticles and dust grains has been well studied for the homogeneous diffuse interstellar medium (ISM), molecular clouds (Draine & Lazarian 1998;Hoang et al. 2010), and turbulent media (Hoang et al. 2011). To date, there is no study of grain rotation in interstellar shocks.
The size distribution of PAHs and nanoparticles in the shocked dense regions is poorly known due to the lack of observational constraints. In the diffuse ISM, the smallest size a ∼ 3.5Å is determined by thermal sublimation (Guhathakurta & Draine 1989;Hensley & Draine 2017). In very hot plasma, thermal sputtering takes over (Draine & Salpeter 1979). In the dense cold clouds, due to the lack of UV photons, both sublimation and thermal sputtering are not effective, such that one can expect a much smaller lower cutoff. Theoretically, in the grain-grain shattering model, the smallest size of nanoparticles is fixed to a min = 5Å without physical justification (Jones et al. 1996;Guillet et al. 2011). Micelotta et al. (2010) studied PAH destruction in shocks and found that PAHs are efficiently destroyed for shock velocities v s > 100 km s −1 . For lower shock velocities, PAHs and smallest nanoparticles are expected to survive the shock passage. As a result, constraining the lower size cutoff and abundance of nanoparticles is of great importance.
In the diffuse ISM, nanoparticles are found to rotate subthermally (i.e., rotational temperature lower than gas temperature) due to the dominance of rotational damping by electric dipole emission over the gas damping (Draine & Lazarian 1998;Hoang et al. 2010;Hoang & Lazarian 2016b). The situation is dramatically different in the dense regions where the gas collisional damping becomes dominant over electric dipole damping such that nanoparticles can achieve thermal rotation (Hoang et al. 2018b; see Section 4). In C-shocks where the gas can be heated to high temperature of T gas ∼ 3 × 10 3 K (see Section 2), nanoparticles at thermal equilibrium would rotate extremely fast, at frequencies ω rot 2π 2.5 × 10 10 where a is the particle size and a −7 = a/10 −7 cm (see Eq. 41). Therefore, nanoparticles of size a ≤ 0.5 nm will have ω rot /2π ≥ 1.4 × 10 11 Hz, just a factor of 2.6 lower than the critical limit for disruption of ω cri /2π ∼ 3.7×10 11 (0.5 nm/a)(S max /10 10 ergs cm −3 ) 1/2 Hz (see Section 4). We expect that supersonic neutral drift would spin-up charged nanoparticles to ω cri , resulting in their instantaneous disruption. This important effect will be quantified in the present paper. A high fraction of nanoparticles resulting from grain shattering combined with suprathermal rotation due to neutral drift are expected to induce strong spinning dust emission at microwave frequencies. On the other hand, the rotational disruption can decrease the abundance of smallest nanoparticles, resulting in a considerable reduction of spinning dust emission. We will perform a detailed modeling of spinning dust in the C-shock regions, accounting for this destruction effect.
The structure of the present paper is as follows. In Section 2, we describe the C-type shock model in dense magnetized clouds and compute the gas temperature as well as velocities of neutral, ion, and charged nanoparticles. In Section 3 we will study rotation dynamics of nanoparticles in the shocks where the supersonic drift between neutrals and charged grains is important. We will introduce the rotational disruption mechanism and calculate the minimum size of nanoparticles that can survive the shock passage in Section 4. In Section 5 we calculate spinning dust emissivity from nanoparticles in the shock regions. We will discuss the importance of rotational disruption and potential application of spinning dust for probing nanoparticles in shocks and shock tracing in Section 6. A short summary of our main results is presented in Section 7.
2. STRUCTURES OF C-SHOCK AND GRAIN DRIFTING VELOCITIES

Shock structure and physical parameters
Let us briefly describe the C-type shock in dense magnetized molecular clouds. In the shock reference frame, the ambient pre-shock gas flows into the shock such that their physical parameters change smoothly with the distance in the shock z. At the shock interface z = 0, the neutral and ion velocities are the same as the shock velocity considered in the shock reference frame. Due to the deceleration when colliding with the shock matter, neutrals and ions are slowed down until they move together with the shock front, i.e., v n = v i = 0. Due to the magnetic force by the magnetic field, ions and charged grains are coupled to the field and move slower than neutrals, resulting in v n > v i or drifting of neutral gas with respect to charged grains and ions.
We calculate the ion and neutral velocities for the different shock velocities using the one-dimensional plane-parallel Paris-Durham shock model (Flower & Pineau des Forêts 2015). Our initial elemental abundances in gas, grain cores, ice mantle, and PAH are the same as in the previous studies (Flower & Pineau des Forêts 2003;Lesaffre et al. 2013;Tram et al. 2018).
Different types of shock depend on the value of the shock's entrance speed relative to the entrance magnetosonic speed v m . C-shock is accordingly given when the shock speed is lower than v m , which is defined in charged fluid as v m = c 2 s + where ρ c is the total mass density of charged particles, including ions, charged dust grains (e.g., PAHs, nanosilicates, and big grains), c s and B/ √ 4πρ c are the sound speed and Alfvén speed of the charged fluid. Because of the low ionization, v m B/ √ 4πρ c = bv m1 , where v m1 is the magnetosonic speed obtained when the dimensionless magnetization parameter b is equal to unity. In our calculations, we find v m1 =19.2 km s −1 or v m1 =19.5 km s −1 for respective densities of n H = 10 4 cm −3 or n H = 10 5 cm −3 . The corresponding magnetic field scaling with the gas density is then (3) Table 1 presents the main physical parameters, including proton number density n H , gas temperature T gas , hydrogen ionization fraction x H , heavy element ionization fraction x M , the radiation strength χ relative to the average interstellar radiation field (ISRF, Mathis et al. 1983), molecular hydrogen fraction y, fraction of PAH in the shock x PAH . The magnetic field strength is evaluated for b = 2. Figure 1 shows the temperature structure of the Cshock for gas density n H = 10 4 cm −3 (upper panel) and n H = 10 6 cm −3 (lower panel). The value z = 0 is the location at the interface of the preshock and postshock medium. The positive value z corresponds to the location inside the shock, which is also the shocked gas. Shock compresses the gas and makes the gas temperature to first rise to the peak and then it decreases due to radiative cooling and infrared emission. The magnetic precursor forms an upstream of the discontinuity, where the charged and neutral fluids dynamically decouple ( Figure 2). The resulting friction between the two fluids heats up and accelerates the neutral fluid. Because of friction between the neutral and charged components, the kinetic energy dissipation is a much more gradual process and spreads over a much larger volume. For instance, the length L of the C-type shock can be up to 3 × 10 16 cm, or ∼ 0.01 pc with pre-shock density n H = 10 4 cm −3 . In a denser cloud (lower panel), however, the gas is swept and compressed by shock much stronger and earlier. This therefore makes the gas hotter and increasing much earlier than the one in less dense medium. Radiative cooling also occurs faster, narrowing the shock region to L ∼ 10 15 cm (see lower panel).

Drifting velocities
Figures 2 shows the velocity structure of neutral and ions, as well as the drift velocity v drift as a function of zthe distance from the point of interest to the shock transition position, assuming a gas density n H = 10 4 cm −3 . The dimensionless drifting parameter is defined as 1/2 is the thermal gas velocity.
The drift velocity v drift rises and reaches the maximum value at the middle of the shock and then it declines. The values s d increases rapidly with z, and then declines when the gas is heated to high temperature. Note the peak of s d does not coincide with the peak of v drift due to the effect of v th or T gas . Figure 3 shows the results for n H = 10 6 cm −3 . The apparent difference is higher gas temperatures due to collisional heating compared to Figure 2. The shock length is also narrower due to the fast radiative cooling. In the dense, low ionization regions, collisions with gas atoms and molecules play an important role in grain rotation dynamics. We first consider the damping and excitation by collisions of purely H thermal gas onto a spherical grain. This situation is the same as those derived in Jones & Spitzer (1967). The treatment below is also correct for molecular gas with n H = 2n(H 2 ).
The well-known damping process for a rotating grain is sticking collision with gas atoms, followed by thermal evaporation. Let consider a grain rotating along the zaxis with angular velocity ω z . The angular momentum carried away by an H atom from the grain surface is given by where r is the distance from the atom to the spinning axis z, I m = m H r 2 is the inertial moment of the hydrogen atom of mass m H , θ is the angle between the z-axis and the radius vector, and r = a sin θ is the projected distance to the center. In an interval of time, there are many atoms leaving the grain surface from the different location, θ. Then, assuming the isotropic distribution of θ the atoms leaving the grain, we can evaluate the mean angular momentum carried away per H atom. Thus, we can replace sin 2 θ =< sin 2 θ >= 2/3, which give rise to Using the collision rate of atomic gas, R coll = n H vπa 2 , one can derive the mean decrease of angular momentum of the grain per second is The mean velocity is defined by where Z = (m H /2πkT gas ) 3/2 is the normalization factor of the Boltzmann distribution of gas velocity. Thus, the gas damping rate is where I = 8πρa 5 /15 is the inertia moment of a spherical grain with ρ being the dust mass density, and the characteristic rotational damping time τ H reads Now, let us estimate the rotational excitation of grains due to gas random bombardment. 1 Each atom colliding with the grain surface at radius r induces an impulsive torque of r × m H v. The increase of the grain rotational energy per unit of time resulting from each impact can be described by Averaging over the grain surface, one obtains r 2 = a 2 /2 and Using the random walk theory for stochastic collisions, one can derive the total increase of grain rotational energy per unit of time as follows: Assuming that the evaporation provides the same amount of excitation, then, one obtains In the case of supersonic motion, v = v d v th , Equation (14) indicates the rate of energy increase for grain rotation.
In the thermal gas, the mean rate of energy increase is obtained by integrating over the velocity distribution, which results in where with C = (2πkT gas /m H ) −3/2 being the normalization constant. Finally, we obtain the excitation coefficient: where ω T = (2kT gas /I) 1/2 is the thermal angular velocity.

Ion collisions, plasma drag, and infrared emission
Realistic gas consists of atomic and molecular hydrogen as well as heavier elements. In addition to neutralgrain collisions, nanoparticles are also bombarded by ions, interact with passing ions (Draine & Lazarian 1998;Hoang et al. 2010).
Following Draine & Lazarian (1998) one takes the damping and excitation coefficients from purely hydrogen gas collisions and define dimensionless coefficients F and G with respect to these terms. The total damping and excitation rate from various interaction processes are given by where j = n, ion, p, IR denotes neutral-grain, ion-grain, plasma drag, and IR emission. Note F j = G j = 1 for grain collisions with purely atomic hydrogen gas. The diffusion coefficients from ion-grain collisions and plasma drag can be calculated assuming stationary grains, i.e., s d = 0, which is a valid assumption because the ions and charged grains are both coupled to the magnetic field. Moreover, in dense clouds, IR emission is subdominant.

Rotational damping and excitation in C-shocks
We move on to study the rotational dynamics of grains in C-shocks in the presence of supersonic neutral drift relative to charged grains and derive the damping and excitation coefficients.

Supersonic drift
For the supersonic drift of s d 1, one can easily obtain the total excitation coefficient as given by Equation (13): which corresponds to the dimensionless coefficient

Transonic drift
For the transonic case, the collisions consist of thermal random collisions and drifting velocity. We can adopt the diffusion coefficients derived in Roberge et al. (1995). Letxŷẑ be the reference frame fixed to the gas, such thatẑ-axis is directed along the magnetic field and the drift velocity v d lies inŷẑ plane with an angle α withẑ. In this paper, we consider a perpendicular shock such that α = 90 • . For the interest of grain rotation, we here consider only spherical grains. Nonspherical grains are important for grain alignment and dust polarization will be addressed in a future study.
For spherical grains, the dimensionless damping coefficient is given by where M 0 is a function of the drifting velocity, and M 0 = 1 for s d = 0, i.e., thermal collision and evaporation. The diffusion coefficients parallel and perpendicular to the drifting motion are where In the reference frame fixed to the ambient gas, the dimensionless diffusion coefficients become which results in the dimensionless excitation coefficient For s d 1, D T scales as s 3 d , such that G n returns to Equation (22).

Dynamical timescales in shocks
The most important dynamical timescale is the flow time of dust grains in the C-shock. The flow time of grains in the shock at location z can be estimated as where the drift velocity is a function of distance z. Assuming v drift ∼ 10 km s −1 for the shock length L = 10 15 cm, we can estimate the passage time as To understand the effect of rotational excitation by gas bombardment, we need to compare the spin-up time by stochastic collisions with the flow time. The characteristic timescale to spin up a grain from the rest to an angular momentum J: where Equation (14) has been used and v drift is the velocity of the neutral drift.
Assuming J = Iω T , the spin-up time by neutral gas drift is equal to By comparing Equations (33) with (35), it follows that the spin-up time by Gold stochastic torques is rather short compared to the time passing the shock structure. Therefore, nanoparticles can be rotating suprathermally in the substantial fraction of the shock. The spin-up timescale is larger for larger grains, such as classical grains of a = 0.1 µm because τ spin−up ∼ a −7 .
The rotational damping time is given by Equation 10, which corresponds to (36) which is long compared to the spin-up time (Eq. 35). Therefore, for supersonic drift, grains can be spun-up to suprathermal rotation before the damping by gas collisions.
Rapidly spinning nanoparticles emit electric dipole radiation, which results in electric dipole damping on a timescale where µ is the dipole moment andβ = β/0.4D with β being the dipole moment per structure of the grain (see Section 5), and the small contribution of dipole moment due to asymmetric charge distribution on the grain surface is ignored for numeric convenience (Draine & Lazarian 1998;Hoang et al. 2010).
Comparing τ ed with τ H , one can see that, for hot and dense shocked regions, the electric dipole damping rate is lower than the gas damping rate. Thus, we expect the excitation by gas collisions and neutral drifting is very efficient in spinning nanoparticles up to extremely fast rotation because electric dipole emission is not fast enough to remove the grain angular momentum.

Grain charge distribution
In the shock code used for computing the shock structure, calculations of grain charge distribution are not yet included. Therefore, to find the grain charge distribution, we take the physical parameters of the shock as input parameters for our charging code (Hoang et al. 2011;.
The physical parameters x e = n e /n H are assumed to be constant throughout the shock. The temperatures 0.0 0.5 3.56Å 3.56Å 3.56Å z = 10 15 cm z = 5.10 15 cm z = 10 16 cm T e , T i , T n , T d , number density of neutrals, ion species and ionization fraction (x H and x M ) at each location in the shock obtained from the shock model are used to find the collisional charging rate. Photoelectric emission is calculated as in Weingartner & Draine (2001) (see also , assuming the similar radiation spectrum as the ISRF, but with the strength χ shown in Table 1. 2 Figure 4 shows the charge distribution of PAHs at the different locations in the shock. For grains below 10Å, the grain charge is mostly between Z = −1 and Z = 0, with a considerable probability on Z = −1. The fraction of grains being on the negative charge states is increasing with increasing a, and it also varies with the distance in the shock. Since the gas-grain drifting is only relevant for charged grains, the high chance of being on Z = −1 indicates that the grain excitation by neutral drift is important.

Numerical results for damping and excitation coefficients
To account for the charge distribution of nanoparticles, we can write the net damping and excitation coefficients due to grain-neutral collisions as follows: 2 Note that the collisional charging time is comparable to the mean time between two collisions with electrons is 1/(nevπa 2 ) ∼ 1yr, which is much shorter than the flow time. Thus, we can find the charge distribution of nanoparticles in C-shocks. where the first terms for s d = 0 are calculated as in Draine & Lazarian (1998), and the second terms describe the effect of the charged grain drift in the shock which are given by Equation (30). Figure 5 shows the damping (upper panel) and excitation (lower panel) coefficients from the various interaction processes, assuming a supersonic drifting with s d = 4. The damping is dominated by neutral impact and plasma drag for small nanoparticles. As expected, the excitation by neutral-grain drift is dominant, while the excitation by infrared emission is negligible. Interestingly, F sd and G sd tend to increase with increasing the grain size a, while the coefficients of the other processes decrease. This can be understood in terms of grains having an increased fraction of the negative charge states (see Figure 4).

Rotational temperature and Rotation Rate
Let T rot be the rotational temperature of spinning nanoparticles, so that I ω 2 /2 = 3kT rot /2. Thus, using the rms angular velocity from Draine & Lazarian (1998), we obtain (40) where τ H and τ ed are the characteristic damping times due to gas collisions and electric dipole emission. For n H ≥ 10 4 cm −3 , one can see from Equations (36) and (37) that τ ed τ H . Figure 6 shows the rotational temperature relative to the neutral gas temperature, T rot /T gas , as a function of the grain size, at four different locations in the shock for three shock models. Nanoparticles rotate suprathermal velocities due to gas bombardment. The ratio T rot /T n increases with the grain size due to the increasing fraction of being on the negative charge states. The ratio T rot /T n also varies with the shock location and is larger inside the shock.
The rotation rate at the rotational temperature T rot is given by Hz, (41) whereρ = ρ/3 g cm −3 .

Rotational disruption
In this section, we introduce a new mechanism of dust destruction in shocks, namely rotational disruption.
Nanoparticles can be excited to extremely fast rotation in shocks as shown by Equation (41). When the rotation rate increases to some critical level such that the centrifugal force exceeds the maximum tensile strength of the material, nanoparticles are disrupted instantaneously. The critical angular velocity for the disruption is given by where S max is the maximum tensile strength of dust material and S max,10 = S max /10 10 ergs cm −3 is the tensile strength in units of 10 10 ergs cm −3 (Draine & Lazarian 1998;Hoang et al. 2018c). 3 The exact value of S max depends on the dust grain composition and structure. Compact grains can have 3 An alternative unit of the tensile strength is dyne/cm 2 , but in this paper we use the unit of ergs cm −3 for Smax. ω cri (S max = 10 9 erg cm −3 ) ω cri (S max = 10 10 erg cm −3 ) ω cri (S max = 10 11 erg cm −3 ) < ω > (d = 1.0x10 13 cm) < ω > (d = 6.0x10 14 cm) < ω > (d = 5.0x10 15 cm) < ω > (d = 2.0x10 16 cm) higher S max than porous grains. Ideal material without impurity, such as diamond, can have S max ≥ 10 11 ergs cm −3 . Burke & Silk (1974) suggested that S max ∼ 10 9 − 10 10 ergs cm −3 for polycrystalline bulk solid (see also Draine & Salpeter 1979). Composite grains as suggested by Mathis & Whiffen (1989) would have much lower strength. In Draine & Salpeter (1979), the value S max = 5 × 10 9 ergs cm −3 is taken for small graphite grains. Thus, in the following, we take S max = 10 10 ergs cm −3 as a typical value nanoparticles. Figure 7 shows the rms rotation rate ω 2 1/2 as a function of the grain size. The critical disruption rate is also shown for comparison. For instance, nanoparticles of size a < 0.8 nm can be disrupted by the centrifugal force at S max = 10 9 ergs cm −3 with shock density n H = 10 4 cm −3 and shock velocity v s = 20 km s −1 .

Disruption rotational temperature
From Equations (41) and (42) one can infer the rotational temperature for disruption: which follows that T rot ∼ 20000 K is required to destroy nanoparticles of a ≤ 0.5 nm, assuming the typical S max = 10 10 ergs cm −3 . For a lower tensile strength of S max = 10 9 ergs cm −3 , the disruption can occur at temperature T rot ∼ 2000 K for a ≤ 0.5 nm, i.e., smallest nanoparticles can be disrupted by thermal gas collisions without needing supersonic neutral drift. Using the calculations of the excitation by neutral drift in Section 3.4, we can evaluate the time required to spin-up nanoparticles to ω cri , i.e., disruption time:

Disruption time and critical drift velocity
where n 5 = n H /10 5 cm −3 . This disruption time is much lower than the flow time t flow for a typical drift velocity v = 10 km s −1 . Thus, rotational disruption is important in shock regions. Moreover, the disruption time scales as a 4 , thus, small nanoparticles tend to be disrupted faster than big grains.
For shock velocities v s < 50 km s −1 , the gas temperature is not high enough such that thermal rotation can disrupt nanoparticles. As a result, to disrupt grains, one requires t disr < τ H , which corresponds to v drift ≥ 9a −7 T gas 1000 K  (2) and (3) one can see that the drift velocity in C-shocks can easily satisfy this condition. Figure 8 shows the critical drift velocity above which nanoparticles are disrupted, as a function of the flow time. Several grain sizes and tensile strength are considered. The critical velocity decreases rapidly with increasing the flow time.

Disruption grain size
To calculate the smallest size a min that nanoparticles can withstand the rotational disruption, we compute ω 2 using the rotational temperature T rot as given by Equation (40) at each shock location for a grid of grain sizes from 0.35 − 10 nm and compare it with ω cri . Figure 9 shows the obtained minimum size a min as a function of the location in the shock for the different values of S max and three shock models. Ideal material grains can survive the shock passage (red line), while non-ideal material grains can destroyed. The disruption size is below 0.5 nm for the typical S max,10 = 1, but it can be increased to 2.0 nm for weaker materials (see blue, orange and green lines). The disruption size is largest for the shock model B (middle panel).

SPINNING DUST EMISSION FROM NANOPARTICLES IN C-SHOCKS
In this section, we present calculations of spinning dust emission from nanoparticles in the shocks. The minimum cutoff of the nanoparticle size distribution is self-consistently determined by rotational disruption. 5.1. Spinning dust model 5.1.1. Electric dipole moment and emission power The rotational emission mechanism is built upon the assumption that nanoparticles own non-zero electric dipole moments. PAH molecules can acquire intrinsic dipole moments due to polar bonds (see Draine & Lazarian 1998). The attachment of SiO and SiC molecules to the grain surface gives rise to the electric dipole moment for nanosilicates (Hoang et al. 2016).
Let N be the total number of atoms in a nanoparticle of effective size a that is defined as the radius of an equivalent sphere of the same volume. Assuming PAHs with a typical structure C:H=3 : 1 having mean mass per atom m ≈ 9.25 amu, one obtains N = 545a 3 −7 for the mass density ρ = 2 g cm −3 (Draine & Lazarian 1998). Assuming nanosilicate with a structure SiO 4 Mg 1.1 Fe 0.9 having m = 24.15 amu, one has N = 418a 3 −7 for ρ = 4 g cm −3 (Hoang et al. 2016).
Let β be the dipole moment per atom in the grain. Assuming that dipoles have a random orientation distribution, the intrinsic dipole moment of the grain can be estimated using the random walk formula: for PAHs, and µ 2 66.8(β/0.4D) 2 a 3 −7 D 2 for nanosilicates (Hoang et al. 2016).
The power emitted by a rotating dipole moment µ at angular velocity ω is given by the Larmor formula: where θ is the angle between ω and µ. Assuming an uniform distribution of the dipole orientation, θ, then, sin 2 θ is replaced by sin 2 θ = 2/3.

Angular momentum distribution function
In high-density conditions where collisional excitation dominates the rotation of nanoparticles (e.g., in shock regions), the grain angular velocity can be appropriately described by the Maxwellian distribution: where I is the moment of inertia of the spherical nanoparticle of mass density ρ, and T rot is the grain rotational temperature.

Size distribution: PAHs and nanosilicates
Following Li & Draine (2001), nanoparticles are assumed to follow a log-normal size distribution: where j = P AH, sil corresponds to PAHs and nanosilicate composition, a 0,j and σ j are the model parameters, and B j is a constant determined by where m X is the grain mass per atom X, b X = X H Y X with Y X being the fraction of X abundance contained in very small sizes and X H being the solar abundance of element X. In our studies, X =C for PAHs and X =Si for nanosilicates. In addition, m X = m C for PAHs, and m X = m(SiO 4 M g 1.1 F e 0.9 ) for nanosilicates of the adopted composition. The peak of the mass distribution a 3 dn j /d ln a occurs at a p = a 0,j e 3σ 2 j . Three parameters determine the size distribution of nanoparticles, including a 0,j , σ j , Y X .

Spinning dust emissivity and emission spectrum
Let j a ν (µ, T rot ) be the emissivity from a spinning nanoparticle of size a at location in the shock, where T rot in general is a function of the local conditions. Thus, where pdf (ν|ω) is the probability that the nanoparticle rotating at ω emits photons at observe frequency ν, and the relation ω = 2πν is assumed. The rotational emissivity per H nucleon is obtained by integrating over the grain size distribution (see Hoang et al. 2011): where dn/da = dn PAH,sil /da for spinning PAHs and nanosilicates, respectively. The minimum size a min at each location z in the Cshock is set equal to the disruption size (see Figure 9), which is a function of the shock velocity.

Emission spectrum
We use Equation (52) to calculate the spinning dust emissivity at various locations inside the shock. The emissivity is calculated assuming that dust composes of 90 % PAH and 10 % nanosilicate. Without considering the disruption effect, a min is taken to be equal to 3.56 Rotational emission spectrum of spinning nanoparticle for nH=10 4 cm −3 and vs=20 km s −1 computed at several positions in the shock without rotational disruption (upper panel) and with disruption with the maximum tensile strength Smax=10 9 erg cm −3 (lower panel). Dust is considered of 90% of PAH and 10% of silicate. Thermal dust emission from big grains is shown in dashed black line for comparison.
A. When the rotational disruption effect is taken into account, a min is determined as in Section 4. We fix the nanoparticle abundance throughout the shock, although the realistic abundance should vary in the shock due to grain shattering (Guillet et al. 2011). Figure 10 shows the emission spectrum of spinning dust as a function of frequency computed at different location z in the shock. The black dashed line shows the thermal dust emission from big grains (see Hoang et al. 2018b). When the rotational disruption is not taken into account, the rotational emissivity is much stronger than the thermal emission (see upper panel). When accounting for the rotational disruption, the rotational emissivity is reduced significantly due to the suppression of smallest grains, but spinning dust is still dominant over thermal dust at frequencies below ν < 100 GHz (lower panel).

Emission spectral flux
We now calculate the spectral flux of spinning dust emission from the shock region. Assuming the shocked regions being spherical, the spectral flux is calculated as follows: where D is the distance from the shocked region to the observer, V is the shock volume, and z i , z f are the initial and final distances of the shock. Figure 11 shows the spectral flux for the different shock velocities for three different shock models, assuming S max = 10 9 ergs cm −3 . For model A (upper panel), the spectra flux is increased significantly for v s = 25 km s −1 . The peak frequency occurs at ν ∼ 200 GHz, much larger than ν = 50 GHz for lower shock velocities. The emission flux for the shock model B (middle panel) is much larger than that for model A (upper panel) due to increased gas heating. However, increasing the gas density to n H = 10 6 cm −3 (model C, lower panel), the emission flux is significantly decrease due to increased rotational disruption. The peak frequency decreases due to the removal of smallest nanoparticles by rotational disruption and is lower than 100 GHz. Note that the high peak at ν ∼ 200 GHz at v s = 25 km s −1 in model A (upper panel) is completely destroyed due to the rotational disruption that remove smallest nanoparticles. Figure 12 shows the same results but for a higher tensile strength for which rotationla disruption is not quite efficient. In additional to the first peak at ν ∼ 40 GHz, there is an additional peak at high frequency of ν ∼ 300 − 500 GHz for the case shock velocity v s ≥ 15 km s −1 which originate from rotational excitation of nanoparticles by the shock. The presence of two peaks arises from the fact that the integration in Equation (53) accounts for the emissivity from all positions in shock region, however the small nanoparticles with high tensile strength are not disrupted by the suprathemal rotation at some positions inside the shocked region. Therefore, the sum of the spinning emissivity from small nanoparticles causes the second peak in the spectral flux. In addition, the second peak is much stronger than the first peak in model A, but it becomes weaker in model B and C. This originates from the fact that higher gas temperature increases the disruption efficiency, that transfer dust mass from smallest nanoparticles to larger one responsible for the first peak.  Figure 11. Spectral flux of spinning nanoparticles for three shock models A, B and C, with corresponding gas density nH = 10 4 cm −3 (top panel, zmax = 4 10 16 cm), nH = 10 5 cm −3 (middle panel, zmax = 6 10 16 cm), and nH = 10 6 cm −3 (bottom panel, zmax = 10 16 cm). The value Smax = 10 9 erg cm −3 , zi= 10 13 cm, and D=100 pc are considered.
6.1. Dust processing in shocks: comparisons of rotational disruption with other mechanisms In shock regions, shattering of large grains due to collisions between neutral and charged grains is suggested to produce PAHs and silicate nanoparticles, resulting in Same with Figure 11 but for Smax = 10 10 ergs cm −3 . The second, high frequency peaks are produced rotational excitation of nanoparticles by the shocks.
the modification of the original grain size distribution (Jones et al. 1996;Guillet et al. 2011). However, the lower cutoff of the resulting grain size distribution is arbitrarily chosen to be 5Å in the previous studies.
On the other hand, thermal sputtering is believed a dominant process to destroy smallest grains such as PAHs and nanoparticles in hot shocked regions. Let Y sp be the sputtering yield by a hydrogen atom bombarding on the dust grain at velocity v drift . The sputtering rate is given by which yields For the C-shock where the neutral drift velocity v drift < 50 km s −1 (see Figure 3), the sputtering yield is rather low, Y sp < 10 −3 (Draine 1995). Therefore, the destruction time for a grain size of a is given by where n 4 = n H /10 4 cm −3 . Comparing τ sp with τ disr (Eq. 45), we can see that the disruption time is much shorter than the sputtering time for a below a few nanometers.
In this paper, we identified a new mechanism of destruction for small grains in the shocks, namely rotational disruption by stochastic mechanical torques. We showed that nanoparticles are spun-up to extremely fast rotation due to high gas density, high temperature, and supersonic drift of neutrals relative to charged nanoparticles. As a result, nanoparticles are disrupted at some location in the shock when the rotation rate exceeds the maximum tensile strength. The high gas density and temperature enables gas collisional excitation to dominate over the electric dipole damping, making nanoparticles to rotate at thermal angular velocities. Supersonic drift then further drive grains to suprathermal rotation. The rotational disruption is quite efficient for non-ideal grains. For a typical material strength S max = 10 10 ergs cm −3 , nanoparticles of a < 1 nm (e.g., PAHs) are removed from the C-shock at v s = 30 km s −1 . We find that nanoparticles of ideal material of S max = 10 11 ergs cm −3 cannot be destroyed in C-shocks with v s ∼ 30 km s −1 . Therefore, nanodiamonds produced in the previous stage of high velocity shock by grain-grain collision (Tielens et al. 1987) would survive the shock and will be released in the ISM. Table 2 summarizes the destruction mechanisms in the shocks and their characteristic timescales. The grain-grain collision timescale is estimated as the mean time between two collisions, τ gg = 1/(πa 2 n gr v drift ) = 4ρa/(3n H m H v drift ), assuming the single size a distribution with the gas-to-dust mass ratio of 100. Moreover, thermal sublimation is obviously ineffective in dense regions with low radiation intensity considered in this paper. The rotational disruption appears to be the fastest mechanism to destroy nanoparticles in C-shocks. As a result, this mechanism would play an important role in Table 2. Grain destruction in C-shocks

Effect of material tensile strength on rotational disruption
The maximum tensile strength of dust grains and nanoparticles is very uncertain. Ideal material such as graphene has highest tensile strength of S max ∼ 1.3 × 10 12 ergs cm −3 . If ideal PAHs can be considered as a sheet of graphene originating from collisions of graphite grains. Thus, one expect ideal PAHs to have a high tensile strength. Nevertheless, in astrophysical conditions, energetic ion bombardment by cosmic rays can eject some carbon atoms (see Micelotta et al. 2011), resulting in some defects. As a result, the maximum strength of PAHs is significantly reduced, perhaps to S max ∼ 10 9 ergs cm −3 .
However, in shocked regions, nanoparticles can be heated to high temperature due to collisional heating. As a result, its maximum tensile strength is expected to be reduced (Idrissi et al. 2016). We show that nanoparticles with S max ≤ 10 9 ergs cm −3 and size a < 2 nm would be disrupted in C-shocks, while stronger material nanoparticles can survive the shock passage.

Implications for mid-IR emission from shock regions
Most observations of supernova remnants (SNRs) do not show mid-IR PAH emission features (λ = 3.3, 6.2, 7.7, 8.6, 11.3 µm) (Smith et al. 2009), except observations for N132D by Tappe et al. (2006). Since PAHs are expected to be abundant in shocks due to shattering of carbonaceous grains, this lack of PAH emission implies that PAH destruction is efficient in the shock regions (see Kaneda et al. 2011 for a review). On the other hand, observations shows the ubiquitous mid-IR emission features (e.g., 11 µm and 21 µm) which are produced by hot very small grains (a ∼ 5 − 50 nm) stochastically heated by UV photons (Andersen et al. 2011). Micelotta et al. (2010) studied destruction of PAH molecules in interstellar shocks with velocity v ∼ 50 − 200 km s −1 . The authors found that PAHs are destroyed by sputtering due to proton and electron bombardment for v > 100 km s −1 , but they can survive passing the shock at lower velocities.
In the light of this study, we suggest PAHs can be destroyed efficiently at low shock speeds of v < 50 km s −1 by rotational disruption, and this mechanism may play an important role in the lack of PAH emission from SNRs. Moreover, nanoparticles larger than ∼ 1 nm can survive the shock and be responsible for 9 and 21 µm emission features (Rho et al. 2018).

Tracing nanoparticles in shock regions with spinning dust
PAHs and nanoparticles are suggested to be abundant in shock regions due to grain-grain collisions (Jones et al. 1996;Guillet et al. 2011). However, to date, there is no direct observations to test this formation mechanism of PAHs/nanoparticles. By modeling microwave emission from spinning dust, we show that spinning dust is very strong in shock regions due to excitation by supersonic neutral drift. Even with abundance of Si and C in nanoparticles of only Y C = Y Si = 5%, the spectral flux of spinning dust is still dominant over thermal dust at frequencies below 100 GHz. The spinning dust emissivity is several orders of magnitude higher than thermal dust when the rotational disruption is disregarded. 4 Therefore, future radio observations with ALMA Band and ngVLA, would be used to probe nanoparti-cles and test the different dust destruction mechanisms in C-shocks. 6.5. Constraining the shock velocity in dense regions with spinning dust C-shocks are ubiquitous in the ISM, especially in the dense, magnetized molecular clouds due to the effect of jets from young stars. The popular technique to probe slow shocks is observing molecular emission lines (CO, SiO, and H 2 ). Here we suggest a new technique to trace shocks using continuum microwave emission from spinning dust. This technique is based on spinning dust mechanism which is most suitable in C-shock regions due to both highly collisional gas excitation, supersonic drift of neutral gas, and abundance of nanoparticles formed by grain-grain collisions. Figure 13 shows the maximum flux (peak flux) of spinning dust emission as a function of the shock velocity for the different shock models and three values of the tensile strengths (upper, middle, and lower panels). The peak flux increases rapidly with the shock velocity due to increased drift velocity and gas temperature. The peak flux increases by a factor of 2 when the shock velocity increases from v s = 15 km s −1 to 30 km s −1 . In particular for the model A (n H = 10 4 cm −3 , the peak flux increases significantly for v s ≥ 20 km s −1 . Figure 14 shows the peak frequency vs. shock velocity. The peak frequency increases with increasing v s for model A (n H = 10 4 cm −3 ) only, but it slightly changes with v s for the other models (B and C) due to the enhanced minimum size a min due to rotational disruption.

Grain alignment and tracing magnetic fields in shocks
Magnetic fields play an crucial role in regulating shock structures in dense clouds (Draine 1980). Dust polarization resulting from grain alignment is a powerful technique to trace magnetic fields. To date, a detailed study of grain alignment in shock regions is not yet available. We found that grains can be spun-up to suprathermal rotation of grains in shocks by Gold stochastic torques. Grains of irregular shape experience also regular mechanical torques and can be aligned with the magnetic field (Lazarian & Hoang 2007;Das & Weingartner 2016;Hoang et al. 2018a). If grains have iron inclusions which can greatly enhance its magnetic relaxation, grains can be aligned perfectly with the magnetic field (Hoang & Lazarian 2016a). Detailed studies of grain alignment and resulting dust polarization in shocks will be presented elsewhere in a near future.

SUMMARY
We study rotational dynamics of nanoparticles in Cshocks of dense molecular clouds and suggest a new S max = 10 11 erg cm −3 S max = 10 11 erg cm −3 S max = 10 11 erg cm −3 nH = 10 4 cm −3 nH = 10 5 cm −3 nH = 10 6 cm −3 Figure 13. Peak emission flux as a function of the shock velocity for three different shock models, nH = 10 4 cm −3 , nH = 10 5 cm −3 , and nH = 10 6 cm −3 . Upper, middle, and lower panels show the results for three values of the tensile strength Smax, respectively. The peak flux increases rapidly with increasing vs.
method to probe nanoparticles and shock velocities via spinning dust emission. Our principal results are summarized as follows: S max = 10 11 erg cm −3 S max = 10 11 erg cm −3 S max = 10 11 erg cm −3 nH = 10 4 cm −3 nH = 10 5 cm −3 nH = 10 6 cm −3 Figure 14. Same as Figure 13, but for the peak emission frequency. Peak frequency slightly changes with vs except for the model of nH = 10 4 cm −3 .
into account supersonic drift of neutral relative to charged nanoparticles. We find that nanoparticles can be rapidly spun-up to suprathermal rotation due to supersonic neutral drift, whereas electric dipole emission is rather slow to remove the grain angular momentum for the dense shocked regions.