Dust Grain Size Redistribution

• Redistribution of Dust Grain Sizes is the process of altering the grain size spectrum in astrophysical environments through accretion, shattering, coagulation, and destruction.
• Modeling employs shift operators and discrete or continuous bin methods to capture grain growth dynamics, revealing sensitivity to initial size distributions and accretion efficiency.
• These processes critically impact star formation, extinction curves, and critical metallicity, thereby influencing dust enrichment and galaxy evolution.

Redistribution of Dust Grain Sizes

Dust in astrophysical environments is subject to dynamic and continuous redistribution in grain size due to a combination of physical processes, including accretion, shattering, coagulation, and destruction mechanisms. The grain size distribution (GSD)—the number of grains per unit volume as a function of grain radius—is a fundamental parameter controlling the efficiency of dust evolution, regulating processes such as metal accretion, molecular formation, thermal emission, and radiative transfer. Redistribution refers specifically to changes in the grain size spectrum, encompassing both the vertical (mass-weighted abundance in different size bins) and horizontal (existence/nonexistence of certain sizes) migration of dust as dictated by the physics in the relevant environment.

1. Analytical Description of Grain Growth and Redistribution

Dust mass growth by metallic accretion in the interstellar medium is regulated by the surface-to-volume ratio of dust grains. If the initial dust GSD is $n(a, 0)$, then as grains accrete metals homogeneously from the gas phase, each grain’s radius increases by $A(t)$, so the GSD at time $t$ becomes

$n(a, t) = n(a – A(t), 0)$

for $a > A(t)$ and $n(a, t) = 0$ otherwise. This "shift-operator" formalism is exact under the idealization of uniform growth by accretion (Hirashita et al., 2011).

The total dust mass, proportional to the third moment of the radius distribution ($\langle a^3 \rangle$), evolves as a polynomial in $A(t)$: $$\frac{\langle a^3 \rangle(t)}{\langle a^3 \rangle_0} = 1 + \frac{3\langle a^2 \rangle_0}{\langle a^3 \rangle_0}A(t) + \frac{3\langle a \rangle_0}{\langle a^3 \rangle_0}A(t)^2 + \frac{A(t)^3}{\langle a^3 \rangle_0}$$ The rate of redistribution, and thus grain growth, is highly sensitive to $\langle a^2 \rangle_0/\langle a^3 \rangle_0$; this ratio is large for distributions biased toward small grains. Steep initial size distributions ($n(a)\propto a^{-r}$ with $r\gtrsim3.5$) or distributions with small $a_\mathrm{min}$ lead to rapid mass growth under accretion, while a sharply peaked distribution at large $a$ suppresses mass growth due to the low available surface area.

2. Sensitivity to Initial Grain Size Distributions

Redistribution is shaped by both the input GSD from stellar sources (e.g., supernovae, AGB stars) and subsequent interstellar medium processing. Supernovae often inject a GSD biased toward large grains, as small grains are selectively destroyed by sputtering behind the reverse shock (Kuo et al., 2012, Temim et al., 2013, Zhao et al., 15 Sep 2025). This initial injection delays efficient dust growth by accretion because the surface-to-volume ratio is low. However, secondary processing—such as shattering induced by grain–grain collisions in turbulent or shocked ISM—populates smaller size bins, increasing the attainable dust mass.

In modeling the evolution of galaxies and quasars, the inclusion of shattering sharply lowers the critical metallicity ($Z_\mathrm{cr}$) for runaway grain growth, as accretion on small grains with high surface area becomes effective. Without adequate small grains, efficient growth occurs only at super-solar $Z_\mathrm{cr}$, while with shattering (and thus small grains), $Z_\mathrm{cr}$ can fall below $0.5\,Z_\odot$ (Hirashita et al., 2011, Kuo et al., 2012).

A summary of the outcomes under different initial GSDs is shown below.

Initial GSD Type	Accretion Efficiency	Critical Metallicity ($Z_\mathrm{cr}$)
Delta function at $a=0.1\,\mu$m	Low	$\sim$ 2–3$\,Z_\odot$
Power law, steep ($r\gtrsim3.5$)	High	$\sim$ 0.2–0.3$\,Z_\odot$
SN-ejecta w/o shattering	Low	High
SN-ejecta + shattering	High	Low

This sensitivity underlies galaxy–to–galaxy variations and the evolutionary progression of the dust–to–gas ratio and extinction properties.

3. Processes Governing Redistribution

Redistribution of grain sizes occurs through several interrelated mechanisms, each with characteristic dependencies on local ISM conditions:

• Accretion: Boosts grain radius as gas-phase metals condense onto pre-existing grains, most effective for small grains due to their high $S/V$.
• Shattering: Grain–grain collisions in the diffuse ISM fragment large grains, populating small sizes and enhancing surface area.
• Coagulation: In dense phases, small grains stick to one another, progressively forming larger aggregates; coagulation competes with accretion to shape the GSD.
• Destruction (Sputtering, SN Shocks): Destroys both large and small grains, typically more rapidly for the latter; the net result is a shift in GSD toward larger sizes, as seen in SNRs (Zhao et al., 15 Sep 2025).

These mechanisms are often encoded in simulations via discrete or continuous distribution functions, updating the GSD as a function of time using deterministic or stochastic evolution equations.

4. Consequences for Observational Diagnostics

Observational evidence for redistribution is seen in extinction curve variations and in the gas-phase metal and molecular budgets. Notably, the total surface area per unit volume, and hence the depletion of gas-phase species by dust, is directly controlled by the GSD. For example, redistribution toward larger grains (by coagulation or shock processing) reduces UV extinction (higher $R_V$; i.e., flatter extinction curve), as directly observed in SNRs, where $$\Delta R_V = R_{V,\mathrm{SNR}} - R_{V,\mathrm{ISM}}$$ correlates with SNR radius and progression through the Sedov phase (Zhao et al., 15 Sep 2025).

In AGB outflows and chemically significant regions, the total rate of molecular ice formation and gas-phase depletion is a strong function of the average grain surface area as set by the GSD (Sande et al., 2020). Models that depart from the canonical MRN assumptions reveal that enhancements in ultrasmall grain abundance drive efficient depletions and changes in observed spectral signatures.

In galaxies, the GSD shape and its evolution over cosmic time control features such as the 2175 Å bump and the far-UV slope, allowing for the inversion of observed extinction curves to diagnose the underlying GSD (Huang et al., 2020, Chang et al., 2022).

5. Modeling Approaches and Integration into Simulations

Simulations employ either fully resolved continuous GSDs (e.g., 32–64 bin logarithmic discretizations (Aoyama et al., 2019, Huang et al., 2020)) or efficient two-size approximations (tracking only "small" and "large" mass fractions) (Hou et al., 2019). These are coupled with multiphase ISM prescriptions (separating dense/cold and diffuse/warm media) to weight which processes operate where.

Redistribution is typically encoded via integro-differential equations for the number/mass in each size bin, with accretion modeled as a uniform shift, shattering as a transfer of mass from larger bins into smaller via collision kernels, and coagulation as a merger process moving mass upward in size. Piecewise linear methods and subcycling are required to ensure accurate integration given the disparate timescales (from $10^5$ yr for shattering or coagulation up to Gyr for galactic evolution) (McKinnon et al., 2018, Huang et al., 2020).

Empirical recipes facilitate integration into analytic and semi-analytic galaxy evolution codes by encoding the effect of the GSD on the dust mass as a shift parameter $\beta$, which is then used in the galaxy’s dust enrichment equations (Hirashita et al., 2011).

6. Astrophysical Implications and Environment Dependence

The redistribution of dust grain sizes underpins a wide range of astrophysical phenomena:

• Star Formation and GMC Thermodynamics: Redistribution toward larger grains via coagulation in dense GMCs decreases dust opacity, resulting in enhanced radiation penetration, increased gas temperatures, and suppressed star formation efficiency, especially for low-mass stars. The SFE can decrease by an order of magnitude when the maximum grain size increases by an order of magnitude (Soliman et al., 12 Jul 2024).
• Survival and Rapid Enrichment in SNRs: Preferential destruction of small grains by shocks, and subsequent enrichment of the ISM with large grains, enhances dust resilience and is critical to rapid dust enrichment up to high redshifts, as observed in galaxies at $z \sim 8$ (Zhao et al., 15 Sep 2025).
• Critical Metallicity and Feedback: Redistribution controls the critical metallicity for nonlinear grain growth—a lower $Z_\mathrm{cr}$ accelerates dust mass buildup and allows for efficient ISM cooling and molecular chemistry at earlier epochs (Hirashita et al., 2011).
• Planetesimal Formation: In protoplanetary disks, only grains above a certain size participate in streaming instabilities and the formation of planetesimals, producing a size-selective mechanism that sets a lower grain size cutoff for inclusion in asteroids or comets (Rucska et al., 2023).

7. Integration with Broader Cosmological Evolution

Over cosmological timescales, the combined action of stellar dust production, ISM processing (accretion, shattering, coagulation), and feedback from star formation and ISM phase transitions drives the system toward quasi-universal GSDs such as the MRN distribution $n(a)\propto a^{-3.5}$, modulated by environmental parameters like metallicity, dense gas fraction, and the star formation history (Aoyama et al., 2019, Huang et al., 2020).

The balance among processes ensures that, in typical late-type galaxies, the GSD at $z\sim1$ to 0 approaches a steady state with small grains efficiently produced and cycled into larger ones by ISM processing, reproducing both observed extinction curves and dust mass–metallicity relations (Huang et al., 2020, Chang et al., 2022), while allowing for significant variations in extreme environments.

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The redistribution of dust grain sizes is thus a physically robust and observationally manifest process, shaped by a hierarchy of mechanisms whose efficiencies are set by local ISM conditions, initial injection from stellar sources, and subsequent interstellar processing. These physical principles underpin models of dust enrichment, extinction, molecular formation, and star formation regulation across the universe.