Nucleation Discussion¶

Nucleation, or new particle formation (NPF), is the gas-to-particle conversion process that creates new aerosol particles directly from vapor molecules. Unlike condensation, which grows existing particles, nucleation generates new particles by assembling molecular clusters that become thermodynamically stable once they exceed a critical size. Nucleation controls aerosol number concentrations in many environments, seeds the growth that produces cloud condensation nuclei, and is the starting point of the NPF-to-cloud-droplet size range.

This follows Chapter 11 of Seinfeld, J. H., & Pandis, S. N. (2016). Atmospheric Chemistry and Physics: From Air Pollution to Climate Change (3^rd ed.). Wiley.

Homogeneous Nucleation (Classical Nucleation Theory)¶

Classical nucleation theory (CNT) describes single-species (homomolecular) homogeneous nucleation: clusters of a vapor form spontaneously in a supersaturated gas without pre-existing surfaces. The competition is between the free-energy cost of creating new surface and the free-energy gain of moving molecules from a supersaturated vapor into the condensed phase.

Saturation Ratio¶

Equation 1: Saturation Ratio

S = pᵢ / pᵢ^sat

Where:

S: Saturation ratio (dimensionless).
pᵢ: Partial pressure of the nucleating species i in the gas phase.
pᵢ^sat: Saturation vapor pressure of pure species i over a flat surface.

Description:

Nucleation requires supersaturation (S > 1). The larger the saturation ratio, the smaller the critical cluster and the faster the nucleation rate. For S ≤ 1, cluster formation is always uphill in free energy and no nucleation occurs.

Gibbs Free Energy of Cluster Formation¶

Equation 2: Free Energy of an r-Sized Cluster

ΔG® = 4 × π × r² × σ − (4 × π × r³ / (3 × v₁)) × k_B × T × ln S

Where:

ΔG®: Gibbs free energy change to form a spherical cluster of radius r.
r: Cluster radius.
σ: Surface tension of the cluster (bulk-liquid value in CNT).
v₁: Volume of one molecule in the condensed phase (v₁ = molar massᵢ / (ρ × N_A)).
k_B: Boltzmann constant.
T: Temperature.
ρ: Density of the condensed phase.
N_A: Avogadro's number.

Description:

The first term is the surface-energy penalty, which grows as r². The second term is the volume free-energy gain from transferring molecules out of the supersaturated vapor, which grows as r³. For S > 1 the sum passes through a maximum at the critical radius: clusters smaller than the critical size tend to evaporate, and clusters larger than it tend to grow.

Critical Radius¶

Equation 3: Critical Cluster Radius (Kelvin Relation)

r* = (2 × σ × v₁) / (k_B × T × ln S)

Where:

r*: Critical cluster radius.

Description:

The critical radius is where dΔG/dr = 0. It is the same Kelvin relation that governs the equilibrium vapor pressure over curved surfaces in condensation (see the condensation equations). A cluster of radius r* is in unstable equilibrium with the vapor: any fluctuation pushes it toward growth or evaporation.

Nucleation Barrier¶

Equation 4: Free-Energy Barrier

ΔG* = (16 × π × σ³ × v₁²) / [3 × (k_B × T × ln S)²] = (4/3) × π × σ × (r*)²

Where:

ΔG*: Free-energy barrier at the critical radius.

Description:

The barrier height controls the nucleation rate exponentially. Because ΔG* scales as σ³ / (ln S)², nucleation rates are extraordinarily sensitive to both surface tension and saturation ratio: small changes in either can shift the rate by many orders of magnitude. This sensitivity is the main practical difficulty in applying CNT quantitatively.

Nucleation Rate¶

Equation 5: Classical Homogeneous Nucleation Rate

J = z × β* × N₁ × exp( −ΔG* / (k_B × T) )

Where:

J: Nucleation rate (new stable clusters per unit volume per unit time).
z: Zeldovich non-equilibrium factor (typically ~0.01-1), accounting for the fact that some clusters at the barrier top still evaporate.
β*: Rate at which vapor molecules collide with the critical cluster (condensation flux onto the critical cluster).
N₁: Number concentration of vapor monomers.

Description:

The prefactor z × β* × N₁ is kinetic: how often clusters at the critical size are hit by another monomer, weighted by the population of monomers available. The exponential is thermodynamic: the probability of fluctuating over the barrier. A commonly used closed form for a single species (Seinfeld & Pandis, 2016, Eq. 11.47) is:

J = ( (2 × σ) / (π × m₁) )^(½) × v₁ × N₁² × exp( −ΔG* / (k_B × T) )

where m₁ is the molecular mass of the nucleating species.

Binary and Multicomponent Nucleation¶

Atmospheric nucleation is rarely a single-species process. The most studied system is binary sulfuric acid-water nucleation, where the critical cluster contains both species and the free-energy surface is two-dimensional (a saddle point rather than a simple maximum).

Equation 6: Binary Cluster Free Energy (Conceptual Form)

ΔG(n₁, n₂) = n₁ × (μ₁,liquid − μ₁,gas) + n₂ × (μ₂,liquid − μ₂,gas) + 4 × π × r² × σ(x)

Where:

n₁, n₂: Number of molecules of species 1 and 2 in the cluster.
μᵢ,liquid − μᵢ,gas: Chemical potential difference for species i between the cluster liquid and the gas phase.
σ(x): Composition-dependent surface tension at cluster mole fraction x.

Description:

The critical cluster sits at the saddle point of ΔG(n₁, n₂), and the nucleation rate follows the lowest free-energy path through that saddle. In practice, binary H₂SO₄-H₂O nucleation rates are evaluated with fitted parameterizations (for example Vehkamäki et al., 2002) rather than by direct evaluation of CNT, because CNT's bulk-property assumptions fail for clusters of a few molecules. Ternary systems (adding ammonia or amines) and ion-induced nucleation further lower the effective barrier and are handled by dedicated parameterizations or cluster-dynamics models.

Empirical Nucleation Parameterizations¶

Because CNT is quantitatively unreliable for atmospheric systems, boundary-layer NPF is often represented with empirical rate laws fitted to observations. Two standard forms relate the nucleation rate to the sulfuric acid vapor concentration:

Equation 7: Activation-Type Nucleation

J = A × [H₂SO₄]

Equation 8: Kinetic-Type Nucleation

J = K × [H₂SO₄]²

Where:

C (or [H₂SO₄]): precursor gas number concentration [#/m³].
A: Activation coefficient [s⁻¹], fitted for a site and set of conditions.
K: Kinetic coefficient [m³/s], fitted for a site and set of conditions.
J: Potential formation-event rate [#/m³/s].

Description:

The activation form assumes existing thermodynamically stable clusters are activated by a single sulfuric acid molecule; the kinetic form assumes the rate-limiting step is a collision between two sulfuric-acid-containing molecules or clusters. The activation interpretation follows the cluster-activation context of Kulmala, Lehtinen, and Laaksonen (2006); both forms are empirical parameterizations, rather than replacements for CNT or a general multicomponent mechanism. Their coefficients must use units consistent with the chosen concentration units: a value reported in cm³/s must be converted before use with C in #/m³. The broader nucleation framework and CNT limitations follow Seinfeld and Pandis (2016).

Scalar Potential-Rate Boundary¶

The bounded rate evaluator computes only potential event rates. It is used by the immutable P4 construction boundary; it is not itself a particle-source transaction. It first converts precursor mass concentration to number concentration:

Equation 9: Precursor Number Concentration

\[ C = \frac{c_m}{M} N_A \]

where \(c_m\) is precursor mass concentration [kg/m³], \(M\) is precursor molar mass [kg/mol], and \(N_A\) is Avogadro's constant [#/mol]. Thus \(C\) has units [#/m³]. The resulting configured rate equations, including a dimensionless survival factor \(f_{\mathrm{surv}}\), are

\[ J_{\mathrm{activation}} = f_{\mathrm{surv}} A C \qquad\text{and}\qquad J_{\mathrm{kinetic}} = f_{\mathrm{surv}} K C^2. \]

The unit check is intentional: \(A C\) and \(K C^2\) both yield [#/m³/s]. If a published kinetic coefficient is expressed in cm³/s, convert it before use with concentration in #/m³:

\[ K_{\mathrm{m^3/s}} = 10^{-6} K_{\mathrm{cm^3/s}}. \]

The survival factor is supplied configuration, not a quantity inferred from the state. It is dimensionless and represents a caller's modelling choice about the relation between the formation size and another size of interest.

flowchart TD
    I[Validate basic mass, molar-mass, temperature, and saturation inputs] --> C[Convert c_m with M and N_A to C #/m³]
    C --> ZC{Zero coefficient, C, or survival?}
    ZC -->|yes| Z[Return zero potential rate]
    ZC -->|no| G{C and T inside inclusive intervals?}
    S[Configured saturation interval] --> SG{Saturation gate}
    G -->|yes| SG
    SG -->|below lower bound| Z
    SG -->|within interval| R[Evaluate A C or K C²]
    F[Configured survival factor] --> R
    R --> P[Potential event rate J #/m³/s]
    G -->|no| X[Reject evaluation]
    SG -->|above upper bound| X
    P --> B[No particle source or inventory mutation]

Basic input validation, including saturation presence and form, and mass-to- number conversion occur before every zero return. A valid zero coefficient, concentration, or survival factor then returns exactly zero before closed-domain membership is checked. Validity intervals are closed: their lower and upper endpoints are accepted for nonzero calculations. Saturation is optional; when it is configured, saturation below its lower bound is a deliberate zero-rate gate, while saturation above its upper bound is rejected. This asymmetric behavior prevents an unsupported high-saturation extrapolation from being silently treated as a valid rate.

Formation-size metadata and injection composition may accompany the rate configuration so that a later source process can state its intended physical representation. They do not alter either equation. In particular, rate evaluation neither creates particles nor gas/particle inventories, chooses slots, depletes precursor vapor, or applies a timestep. P4 is the bounded, CPU-only public construction API for immutable activation and kinetic strategies, their builders, factory, and source-selection configuration through particula.dynamics.nucleation and particula.dynamics; it is not a runnable or GPU capability.

Survival to Detectable and Model-Resolved Sizes¶

Freshly nucleated clusters must grow through the smallest sizes, where coagulational scavenging by pre-existing particles is fastest, before they matter for the resolved aerosol population. The apparent formation rate at a larger diameter d is related to the "true" nucleation rate at diameter d* by a survival probability.

Equation 10: Kerminen-Kulmala Survival Relation

J_d = J_d* × exp( γ × (1/d − 1/d*) × CS' / GR )

Where:

J_d: Apparent particle formation rate at diameter d.
J_d*: Nucleation rate at the initial cluster diameter d*.
γ: Proportionality constant expressed in units compatible with the selected diameter, condensation-sink, and growth-rate units. For SI substitution, use meters for d and d* and choose γ so that γ × CS' / GR has units of meters; do not mix the original fitted-unit coefficient with SI inputs without converting it.
CS': Condensation sink of the pre-existing particle population (scavenging strength), in the units used by the selected γ.
GR: Growth rate of the freshly formed particles, in units compatible with γ and CS'.

Description:

The exponential expresses the competition between growth (GR, escape to safety at larger sizes) and coagulational loss to the existing aerosol surface (CS'). High pre-existing surface area suppresses observable NPF even when the nucleation rate itself is large. In a simulation, this relation is a consistency check: if the model injects particles at a size larger than the true cluster size, the injection rate should be the survival-corrected J_d, not the raw J_d*. P4 does not evaluate this relation; its survival factor is an externally selected value whose scientific justification remains the caller's responsibility.

For supported execution, see the CPU nucleation example and CPU Nucleation Strategy System. The hand-built custom single-species notebook is illustrative only and is not the supported API.

Nucleation as a Source Term in Aerosol Dynamics¶

In an aerosol dynamics model, nucleation enters the population balance as a source of new particles at the smallest resolved size:

Equation 11: Number Source Term

dN/dt |_nucleation = J

Equation 12: Coupled Gas Depletion

dCᵢ/dt |_nucleation = − J × n*ᵢ × (molar massᵢ / N_A)

Where:

dN/dt |_nucleation: Rate of new particle number production per unit volume.
dCᵢ/dt |_nucleation: Rate of change of gas-phase mass concentration of species i due to nucleation.
n*ᵢ: Number of molecules of species i in a freshly formed particle at the injection size.

Description:

Each nucleation event moves a small but nonzero mass of each participating vapor into the particle phase, so number production and gas depletion must be applied together to conserve mass. Numerical treatment differs by representation:

Binned/sectional: add J × Δt particles (and the corresponding mass) to the smallest bin each timestep.
Particle-resolved with fixed-capacity slots: activate inactive particle slots with the injection-size mass and composition. Because nucleation rates can be large, one computational particle typically represents many real particles via its concentration/weighting factor; the number of slots activated per step and the weight assigned to each is a resolution decision. When inactive slots run out, a resampling or volume-scaling policy is required.
Stiffness coupling: freshly nucleated particles sit at the fast-equilibration end of the condensation stiffness range, so the nucleation source interacts directly with the time-integration scheme chosen for condensation.

Fixed-capacity primitive boundary¶

Particula ships bounded CPU slot-exhaustion primitives for planning, resampling, and representative-volume scaling, plus direct Warp primitives for fixed-shape resampling and representative-volume scaling. Their ownership, planning, and mutation boundaries are documented in the Fixed-Capacity Slot Exhaustion Primitives.

These are slot-management primitives, not a nucleation process: they do not construct a particle source or deplete gas. The supported process boundary is the CPU-only, single-box particula.dynamics.Nucleation runnable, configured with NucleationCommitConfig. It adapts the legacy Aerosol backing containers and mutates the backing particle and partitioning-gas data by identity; it does not provide GPU execution.

Implementation boundary:

The CPU-only P4 activation and kinetic potential-rate strategies, builders, source-selection metadata, and factory are supported public construction APIs. They remain immutable configuration and rate-evaluation boundaries only.
The concrete CPU P2 particle-source planner and P3 finalization transaction remain in particula.dynamics.nucleation.particle_source. They are not re-exported through particula.dynamics.nucleation or particula.dynamics. P2 returns inventory-limited source demand and diagnostics without mutation; P3 owns the bounded particle/gas transaction.
Nucleation applies equal sequential substeps and re-reads the current gas state before each rate calculation, so completed substeps can change later rates. Atomicity is per attempted P3 substep only: a completed earlier substep is not rolled back if a later substep fails. P2/P3 helpers remain concrete-only, and GPU support remains deferred.

Shipped CPU source transaction¶

The layers deliberately separate rate selection from irreversible state updates:

flowchart LR
    P4[P4: immutable strategy and source selection] --> R[Potential rate J in #/m³/s]
    R --> P2[P2: nonmutating demand finalization]
    P2 --> A[Shared inventory-admitted events and limiting-species diagnostics]
    A --> P3[P3: staged fixed-capacity commit]
    P3 --> T[Particle/gas transaction and finalized diagnostics]
    T --> P5[P5: CPU-only one-box runnable]
    P5 --> G[Re-read current gas for the next equal substep]

For each P5 substep, the current potential rate is multiplied by the equal substep duration to obtain a potential event count [#/m³]. Concrete-only P2 uses the per-event species mass

\[ m_{\mathrm{event},i}=n_i\frac{M_i}{N_A}\quad[\mathrm{kg/event}] \]

and admits one shared count across participating species, limited by the tightest gas inventory, before any mutation. Concrete-only P3 stages slot activation, resampling, and representative-volume scaling on private particle state, then transfers each represented event's species mass from partitioning gas to particle mass in one commit. Its bookkeeping uses concentration-weighted particle inventory,

\[ I_{p,i}=\sum_j c_j m_{j,i}\quad[\mathrm{kg/m^3}], \]

so every box/species ledger is checked as particle plus gas with rtol=1e-12 and atol=1e-30.

Diagnostics distinguish potential, admitted, gas-limited, represented, and reduced events, identify limiting species, and report requested/activated/ released slots and selected exhaustion policy and scale. Resampling is the first exhausted-capacity policy; representative-volume scaling is its fallback. For scaled rows the reference total is scale * pre_total; the supported unscaled example uses requested_scale == minimum_scale == 1.0 and directly asserts total-mass conservation. P2 is nonmutating, P3 preflight rejection is atomic, and P5 rollback scope is one attempted substep rather than a complete call.

P5 is the supported public Nucleation runnable. It is CPU-only and accepts exactly one legacy Aerosol box, retaining its backing particle and partitioning-gas containers by identity. It performs equal sequential substeps, recomputing the rate from the gas left by each successful commit. A successful earlier substep therefore remains visible if a later P2/P3 attempt fails; P5 deliberately offers no whole-call rollback.

Direct-Warp P1--P5 correspondence¶

The bounded direct-Warp step is distinct from the CPU runnable. It computes C = c N_A / M [#/m³], activation J = S * A * C with A [s⁻¹], or kinetic J = S * K * C² with K [m³/s]. Its potential demand is E_pot = J * dt [#/m³]. P2 admits one shared inventory-limited event demand before the P5 particle/gas transfer. In these rate laws, S is the configured, dimensionless survival factor (not the saturation ratio); c is [kg/m³] and M is [kg/mol].

Before P3 stages discrete events, accepted_demand * volume must be finite, nonnegative, integral, and representable as an int32 event count. This representability rejection can occur after P2 has written its documented sidecars; particle/gas preservation and the existing rollback limits are unchanged.

Direct P1 accepts finite nonnegative duration, coefficient, and survival; inclusive configured precursor bounds; positive temperature; optional configured saturation bounds; positive formation diameter [m]; and nonnegative integer molecule counts with at least one positive entry. Temperature is either a positive scalar or same-device wp.float64 (B,); configured saturation is same-device wp.float64 (B, S) or comes from a validated environment.

The bounded public ordering is P1 read-only domain/schema preflight, read-only validation of the public P4 controls and buffers, P2 inventory admission, P3 fixed-slot staging, participating-molecule eligibility validation, P4 resampling-first/scaling-fallback resolution, and P5 handoff validation followed by the fused commit when final counts are nonzero. An empty box dimension returns after the public P4 validation. A configured zero-work or no-admission result still has its applicable planning, staging, and P4 diagnostic phases, but P5 observes zero final counts and launches no particle/gas writer; it is a successful no-op, not an error.

Invalid rejection during the read-only public checks preserves particles and gas. P2--P4 may have changed documented sidecars before a later rejection, and rollback is not promised after an entered exhaustion primitive or a P5 launch.

For an unscaled P5 row, concentration-weighted particle plus gas inventory is conserved per box/species at rtol=1e-12, atol=1e-30. A scaled row compares against s * initial_particle + initial_gas; P5 source transfer still balances the gas removal. This is direct-kernel conservation evidence, not CPU parity.

The direct step excludes hidden transfer/synchronization, CPU fallback, resize/compaction, GPU Runnable/scheduling/backend selection, E6-F9 orchestration, expanded physics, graph capture, autodiff, and performance guarantees. See the direct example and feature contract.

Variable Descriptions¶

Understanding the Parameters:

Saturation Ratio (S):
The single most important control on homogeneous nucleation; rates change by orders of magnitude over small changes in S.
Set by the gas-phase concentration and the temperature-dependent saturation vapor pressure, so accurate vapor pressures are prerequisites for any nucleation calculation.
Surface Tension (σ):
Enters the barrier as σ³; the dominant uncertainty in CNT.
Bulk surface tension is a poor approximation for clusters of a few molecules (the main criticism of CNT); composition dependence matters for multicomponent clusters.
Molecular Volume (v₁):
Condensed-phase volume per molecule; links the molar mass and liquid density of the nucleating species.
Critical Radius (r*):
Typically ~0.5-2 nm for atmospheric conditions; clusters at this size contain only tens to hundreds of molecules.
The same Kelvin physics that penalizes condensation onto the smallest particles.
Zeldovich Factor (z):
Corrects the equilibrium cluster distribution for the fact that the barrier crossing is a diffusive process in cluster-size space.
Condensation Sink (CS'):
Integral measure of how quickly vapors and small clusters are scavenged by the pre-existing particle population.
Couples nucleation to the rest of the aerosol: more pre-existing surface means less survival of fresh clusters.
Growth Rate (GR):
Diameter growth rate of freshly formed particles, set by condensation of available vapors.
Together with CS', controls what fraction of nucleated clusters survive to climate- and health-relevant sizes.
Formation-Size Molecule Count (n*ᵢ):
Composition of the particle injected into the model at the formation size; needed for mass conservation between the gas and particle phases.

Applications and Implications:

Aerosol Number Budgets: Nucleation is the dominant source of particle number in many clean and moderately polluted environments, feeding the accumulation mode through subsequent growth.
Cloud Condensation Nuclei: A substantial fraction of CCN originate as nucleated particles that grew by condensation and coagulation, so NPF connects gas-phase chemistry to cloud microphysics.
Chamber and Flow-Tube Experiments: Prescribed-precursor experiments (the multi-box and parcel use cases) often begin with a nucleation burst; simulating them requires a particle source, not just growth of an initial population.

Assumptions and Limitations:

Capillarity Approximation: CNT treats molecular clusters as spherical droplets with bulk liquid density and surface tension. For critical clusters of tens of molecules this is a strong assumption and the main source of CNT's quantitative error.
Steady-State Cluster Distribution: The classical rate assumes the sub-critical cluster population is in steady state with the vapor; rapid changes in vapor concentration or temperature violate this.
Parameterization Validity Ranges: Empirical forms (Equations 7-8) and fitted parameterizations (for example Vehkamäki et al., 2002) are only valid within the temperature, humidity, and concentration ranges of the underlying data; extrapolation can produce unphysical rates.
Bounded Scalar Evaluation: The P4 rate strategies accept scalar precursor state only and enforce configured, inclusive concentration and temperature intervals. Optional saturation below the lower interval is treated as no event; saturation above its upper interval is not extrapolated. A zero potential rate is not evidence that nucleation is physically absent outside a parameterization's domain.
Bounded conservation step: Without representative-volume scaling, the shipped P5 source transaction conserves supported partitioning-gas and particle inventory directly. For a row selected for representative-volume scaling, accounting instead compares the post-step inventory to the selected scale times its pre-step inventory, with the finalized source-mass transfer included in that transaction reference. This CPU-only, single-box, fixed-capacity work is not a general multiphysics source.
Injection-Size Convention: Models inject particles at a chosen formation size, not at the true critical size. The nucleation rate must be survival-corrected (Equation 10) to be consistent with that choice.

Further Considerations:

Ion-Induced and Heterogeneous Pathways: Ions and pre-existing surfaces lower the nucleation barrier; these pathways need separate rate expressions.
Cluster Dynamics Models: Explicit cluster-population models (for example ACDC-type birth-death schemes) replace the CNT barrier picture with molecule-by-molecule kinetics and are the current standard for sulfuric acid-base systems.
Differentiability: Nucleation rate expressions are smooth functions of gas concentrations and temperature, but the act of activating discrete particle slots is a discrete event. For gradient-based optimization, a binned or expected-value (mean-field) source term is differentiable, while discrete slot activation requires the same surrogate treatment as stochastic coagulation.

Conclusion¶

Nucleation converts supersaturated vapor into new particles through a barrier-crossing process whose rate is exponentially sensitive to saturation ratio and surface tension. The bounded P4/P5 CPU implementation is narrower: it evaluates configured empirical potential rates and commits a one-box, fixed-capacity, gas-conserving source transaction. Its configured survival factor provides no independently calculated growth or scavenging physics. Vehkamäki et al. (2002) is scientific context only, not implemented physics.

References¶

Seinfeld, J. H., & Pandis, S. N. (2016). Atmospheric Chemistry and Physics: From Air Pollution to Climate Change (3^rd ed.), Chapter 11: Nucleation. Wiley.
Vehkamäki, H., Kulmala, M., Napari, I., Lehtinen, K. E. J., Timmreck, C., Noppel, M., & Laaksonen, A. (2002). An improved parameterization for sulfuric acid-water nucleation rates for tropospheric and stratospheric conditions. Journal of Geophysical Research: Atmospheres, 107(D22), 4622. DOI: 10.1029/2002JD002184
Kerminen, V.-M., & Kulmala, M. (2002). Analytical formulae connecting the "real" and the "apparent" nucleation rate and the nuclei number concentration for atmospheric nucleation events. Journal of Aerosol Science, 33(4), 609-622. DOI: 10.1016/S0021-8502(01)00194-X
Kulmala, M., Lehtinen, K. E. J., & Laaksonen, A. (2006). Cluster activation theory as an explanation of the linear dependence between formation rate of 3 nm particles and sulphuric acid concentration. Atmospheric Chemistry and Physics, 6, 787-793. DOI: 10.5194/acp-6-787-2006
Zhang, R., Khalizov, A., Wang, L., Hu, M., & Xu, W. (2012). Nucleation and growth of nanoparticles in the atmosphere. Chemical Reviews, 112(3), 1957-2011. DOI: 10.1021/cr2001756