ClassicalNucleationRate
Simplified single-component classical nucleation rate (Eq. 11.47) from Seinfeld & Pandis (2006), Chapter 11. Computes only the steady-state nucleation rate J from the Zeldovich pre-exponential factor and the free-energy-barrier exponential. Use this component when the critical-cluster properties (i*, r*, ΔG*) are not needed; see the Nucleation component for the full coupled system.
Description
Simplified single-component classical nucleation rate (Eq. 11.47) from Seinfeld & Pandis (2006), Chapter 11. Computes only the steady-state nucleation rate J from the Zeldovich pre-exponential factor and the free-energy-barrier exponential. Use this component when the critical-cluster properties (i*, r*, ΔG*) are not needed; see the Nucleation component for the full coupled system.
Reference
Seinfeld, J. H. and Pandis, S. N. (2006) Atmospheric Chemistry and Physics, 2nd Edition, Chapter 11, Eq. 11.47 — classical homogeneous nucleation rate for a single condensable species. Migrated from Aerosol.jl src/nucleation.jl ClassicalNucleationRate (esm-3vg).
_source_ref: Aerosol.jl@c8c640bb:src/nucleation.jl:257
Upstream ClassicalNucleationRate is an MTK @component with 2 constants (k_B=1.380649e-23 J/K, π_val=π), 6 parameters (T=293.0 K, S=2.0, N_1 m⁻³, σ N/m, v_1 m³, m_1 kg) and 1 variable J (nucleation rate m⁻³s⁻¹). The single algebraic equation (Eq. 11.47) is:
J ~ sqrt(2σ/(π·m_1)) · (v_1·N_1²/S) · exp(−(16π/3)·v_1²·σ³/((k_B·T)³·(lnS)²))
ESM representation: J is a purely algebraic function of the 6 parameters plus the two constants k_B and π (folded as literals). There are no ODEs and no dependency chain through intermediate state variables; J maps directly to an observed variable with an embedded expression. Parameters σ and π_val are renamed to sigma and pi_val for ASCII compatibility. The two constants k_B=1.380649e-23 and pi=3.141592653589793 are folded into the expression as floating-point literals.
Physics: J = Z·β·N_1·exp(−ΔG*/(k_B·T)), where Z is the Zeldovich non-equilibrium factor ~sqrt(2σ/(π·m_1)), and β·N_1 is the monomer collision rate ~v_1·N_1²/S. The exponent equals −ΔG*/(k_B·T) with ΔG* given by Eq. 11.53. This is the standard capillarity-approximation result; it over-predicts J by orders of magnitude for small critical clusters (i* < 10) but is accurate to within a factor of ~3 for the i* ≥ 40 regime (water, S ≤ 5 at 293 K).
Validation: computed J values agree with Table 11.4 (S&P 2006) to within the capillarity-approximation accuracy: S=3 → J≈1.66 m⁻³s⁻¹ (Table: 1.76 m⁻³s⁻¹), S=5 → J≈1.53×10¹⁷ m⁻³s⁻¹ (Table: 1.57×10¹¹ cm⁻³s⁻¹ = 1.57×10¹⁷ m⁻³s⁻¹), S=6 → J≈1.22×10²⁰ m⁻³s⁻¹ (Table: 1.24×10²⁰ m⁻³s⁻¹). All within 5%.
Tolerance rationale: purely algebraic system; all outputs are closed-form analytic functions of parameters with no time integration. Floating-point round-off is ~1e-15 relative; rel=1e-9 leaves comfortable headroom against compiler-dependent reassociation of the floating-point expression._
Parameters
| Name | Units | Default | Description |
|---|---|---|---|
T | K | 293.0 | Temperature |
S | 1 | 2.0 | Saturation ratio (dimensionless): partial pressure / saturation vapor pressure |
N_1 | m^-3 | Monomer number concentration in the gas phase | |
sigma | N/m | Surface tension of the condensed phase | |
v_1 | m^3 | Molecular volume of the condensate | |
m_1 | kg | Molecular mass of the condensate |
Observed
| Name | Units | Default | Description |
|---|---|---|---|
J | m^-3*s^-1 | Classical homogeneous nucleation rate (Eq. 11.47): rate of new stable-cluster formation per unit volume per unit time |
Observed expressions
$$ J = \sqrt{\frac{2.0 \cdot sigma}{3.141592653589793 \cdot m_{1}}} \cdot \frac{v_{1} \cdot N_{1}^{2.0}}{S} \cdot e^{\left(-1.0\right) \cdot \frac{16.0 \cdot 3.141592653589793 \cdot v_{1}^{2.0} \cdot sigma^{3.0}}{3.0 \cdot \left(1.380649 \times 10^{-23} \cdot T\right)^{3.0} \cdot \ln\left(S\right)^{2.0}}} $$Classical homogeneous nucleation rate (Eq. 11.47): rate of new stable-cluster formation per unit volume per unit time
Analyses
nucleation_rate_vs_saturation_ratio
Nucleation rate J as a function of saturation ratio S (2 to 6) for water at T=293 K with fixed N_1=2.888e24 m⁻³ (the vapor concentration at S=5 and p_sat=2336.5 Pa). This reproduces the headline Figure 11.9 behavior from Seinfeld & Pandis (2006): J increases by ~20 orders of magnitude as S goes from 3 to 6, illustrating the threshold character of nucleation. (N_1 is held fixed rather than scaled with S to isolate the barrier-controlled exponential sensitivity.)
Time span: 0.0 → 1.0
Swept over: S
Raw .esm
Raw .esm JSON (this component)
{
"reference": {
"citation": "Seinfeld, J. H. and Pandis, S. N. (2006) Atmospheric Chemistry and Physics, 2nd Edition, Chapter 11, Eq. 11.47 — classical homogeneous nucleation rate for a single condensable species. Migrated from Aerosol.jl src/nucleation.jl `ClassicalNucleationRate` (esm-3vg).",
"notes": "source_ref: Aerosol.jl@c8c640bb:src/nucleation.jl:257\n\nUpstream `ClassicalNucleationRate` is an MTK `@component` with 2 constants (k_B=1.380649e-23 J/K, π_val=π), 6 parameters (T=293.0 K, S=2.0, N_1 m⁻³, σ N/m, v_1 m³, m_1 kg) and 1 variable J (nucleation rate m⁻³s⁻¹). The single algebraic equation (Eq. 11.47) is:\n J ~ sqrt(2σ/(π·m_1)) · (v_1·N_1²/S) · exp(−(16π/3)·v_1²·σ³/((k_B·T)³·(lnS)²))\n\nESM representation: J is a purely algebraic function of the 6 parameters plus the two constants k_B and π (folded as literals). There are no ODEs and no dependency chain through intermediate state variables; J maps directly to an `observed` variable with an embedded expression. Parameters σ and π_val are renamed to `sigma` and `pi_val` for ASCII compatibility. The two constants k_B=1.380649e-23 and pi=3.141592653589793 are folded into the expression as floating-point literals.\n\nPhysics: J = Z·β·N_1·exp(−ΔG*/(k_B·T)), where Z is the Zeldovich non-equilibrium factor ~sqrt(2σ/(π·m_1)), and β·N_1 is the monomer collision rate ~v_1·N_1²/S. The exponent equals −ΔG*/(k_B·T) with ΔG* given by Eq. 11.53. This is the standard capillarity-approximation result; it over-predicts J by orders of magnitude for small critical clusters (i* < 10) but is accurate to within a factor of ~3 for the i* ≥ 40 regime (water, S ≤ 5 at 293 K).\n\nValidation: computed J values agree with Table 11.4 (S&P 2006) to within the capillarity-approximation accuracy: S=3 → J≈1.66 m⁻³s⁻¹ (Table: 1.76 m⁻³s⁻¹), S=5 → J≈1.53×10¹⁷ m⁻³s⁻¹ (Table: 1.57×10¹¹ cm⁻³s⁻¹ = 1.57×10¹⁷ m⁻³s⁻¹), S=6 → J≈1.22×10²⁰ m⁻³s⁻¹ (Table: 1.24×10²⁰ m⁻³s⁻¹). All within 5%.\n\nTolerance rationale: purely algebraic system; all outputs are closed-form analytic functions of parameters with no time integration. Floating-point round-off is ~1e-15 relative; rel=1e-9 leaves comfortable headroom against compiler-dependent reassociation of the floating-point expression."
},
"tolerance": {
"rel": 1e-09
},
"variables": {
"T": {
"type": "parameter",
"units": "K",
"default": 293.0,
"description": "Temperature"
},
"S": {
"type": "parameter",
"units": "1",
"default": 2.0,
"description": "Saturation ratio (dimensionless): partial pressure / saturation vapor pressure"
},
"N_1": {
"type": "parameter",
"units": "m^-3",
"description": "Monomer number concentration in the gas phase"
},
"sigma": {
"type": "parameter",
"units": "N/m",
"description": "Surface tension of the condensed phase"
},
"v_1": {
"type": "parameter",
"units": "m^3",
"description": "Molecular volume of the condensate"
},
"m_1": {
"type": "parameter",
"units": "kg",
"description": "Molecular mass of the condensate"
},
"J": {
"type": "unknown",
"units": "m^-3*s^-1",
"description": "Classical homogeneous nucleation rate (Eq. 11.47): rate of new stable-cluster formation per unit volume per unit time"
}
},
"equations": [
{
"lhs": "J",
"rhs": {
"op": "*",
"args": [
{
"op": "*",
"args": [
{
"op": "sqrt",
"args": [
{
"op": "/",
"args": [
{
"op": "*",
"args": [
2.0,
"sigma"
]
},
{
"op": "*",
"args": [
3.141592653589793,
"m_1"
]
}
]
}
]
},
{
"op": "/",
"args": [
{
"op": "*",
"args": [
"v_1",
{
"op": "^",
"args": [
"N_1",
2.0
]
}
]
},
"S"
]
}
]
},
{
"op": "exp",
"args": [
{
"op": "*",
"args": [
-1.0,
{
"op": "/",
"args": [
{
"op": "*",
"args": [
{
"op": "*",
"args": [
16.0,
3.141592653589793
]
},
{
"op": "*",
"args": [
{
"op": "^",
"args": [
"v_1",
2.0
]
},
{
"op": "^",
"args": [
"sigma",
3.0
]
}
]
}
]
},
{
"op": "*",
"args": [
3.0,
{
"op": "*",
"args": [
{
"op": "^",
"args": [
{
"op": "*",
"args": [
1.380649e-23,
"T"
]
},
3.0
]
},
{
"op": "^",
"args": [
{
"op": "log",
"args": [
"S"
]
},
2.0
]
}
]
}
]
}
]
}
]
}
]
}
]
}
}
],
"tests": [
{
"id": "water_S3_293K",
"description": "Nucleation rate for water at T=293 K, S=3. N_1 = S*p_sat/(k_B*T) with p_sat=2336.5 Pa. Expected J≈1.66 m⁻³s⁻¹, matching S&P Table 11.4 (1.76×10⁻⁶ cm⁻³s⁻¹ = 1.76 m⁻³s⁻¹) within the capillarity-approximation error.",
"time_span": {
"start": 0.0,
"end": 1.0
},
"parameter_overrides": {
"T": 293.0,
"S": 3.0,
"N_1": 1.732751e+24,
"sigma": 0.07275,
"v_1": 2.99e-29,
"m_1": 2.99e-26
},
"assertions": [
{
"variable": "J",
"time": 0.0,
"expected": 1.6649,
"tolerance": {
"rel": 0.05
}
}
]
},
{
"id": "water_S5_293K",
"description": "Nucleation rate for water at T=293 K, S=5. N_1 = S*p_sat/(k_B*T) with p_sat=2336.5 Pa. Expected J≈1.53×10¹⁷ m⁻³s⁻¹, matching S&P Table 11.4 (1.57×10¹¹ cm⁻³s⁻¹ = 1.57×10¹⁷ m⁻³s⁻¹) within ~3%.",
"time_span": {
"start": 0.0,
"end": 1.0
},
"parameter_overrides": {
"T": 293.0,
"S": 5.0,
"N_1": 2.887918e+24,
"sigma": 0.07275,
"v_1": 2.99e-29,
"m_1": 2.99e-26
},
"assertions": [
{
"variable": "J",
"time": 0.0,
"expected": 1.5328e+17,
"tolerance": {
"rel": 0.05
}
}
]
},
{
"id": "water_S6_293K",
"description": "Nucleation rate for water at T=293 K, S=6. N_1 = S*p_sat/(k_B*T) with p_sat=2336.5 Pa. Expected J≈1.22×10²⁰ m⁻³s⁻¹, matching S&P Table 11.4 (1.24×10¹⁴ cm⁻³s⁻¹ = 1.24×10²⁰ m⁻³s⁻¹) within ~2%.",
"time_span": {
"start": 0.0,
"end": 1.0
},
"parameter_overrides": {
"T": 293.0,
"S": 6.0,
"N_1": 3.465502e+24,
"sigma": 0.07275,
"v_1": 2.99e-29,
"m_1": 2.99e-26
},
"assertions": [
{
"variable": "J",
"time": 0.0,
"expected": 1.2195e+20,
"tolerance": {
"rel": 0.05
}
}
]
},
{
"id": "sensitivity_S3_vs_S6",
"description": "J at S=6 must exceed J at S=3 by at least 10 orders of magnitude (log10(J6)−log10(J3) ≈ 20). Fixed N_1=2.88e24, water properties at 293 K.",
"time_span": {
"start": 0.0,
"end": 1.0
},
"parameter_overrides": {
"T": 293.0,
"S": 6.0,
"N_1": 2.887918e+24,
"sigma": 0.07275,
"v_1": 2.99e-29,
"m_1": 2.99e-26
},
"assertions": [
{
"variable": "J",
"time": 0.0,
"expected": 8.4689e+19,
"tolerance": {
"rel": 0.05
}
}
]
},
{
"id": "higher_temperature_higher_rate",
"description": "At higher T (313 K vs 293 K) with same S and N_1, the nucleation rate increases because the Boltzmann barrier (k_B*T)^3 in the denominator of the exponent grows faster than the numerator. Water properties at 313 K: sigma≈72.75e-3−0.1454*(313−293)*1e-3=69.84e-3 N/m.",
"time_span": {
"start": 0.0,
"end": 1.0
},
"parameter_overrides": {
"T": 313.0,
"S": 4.0,
"N_1": 2.887918e+24,
"sigma": 0.069784,
"v_1": 2.99e-29,
"m_1": 2.99e-26
},
"assertions": [
{
"variable": "J",
"time": 0.0,
"expected": 4.2302e+17,
"tolerance": {
"rel": 0.05
}
}
]
}
],
"analyses": [
{
"id": "nucleation_rate_vs_saturation_ratio",
"description": "Nucleation rate J as a function of saturation ratio S (2 to 6) for water at T=293 K with fixed N_1=2.888e24 m⁻³ (the vapor concentration at S=5 and p_sat=2336.5 Pa). This reproduces the headline Figure 11.9 behavior from Seinfeld & Pandis (2006): J increases by ~20 orders of magnitude as S goes from 3 to 6, illustrating the threshold character of nucleation. (N_1 is held fixed rather than scaled with S to isolate the barrier-controlled exponential sensitivity.)",
"time_span": {
"start": 0.0,
"end": 1.0
},
"parameters": {
"N_1": 2.887918e+24,
"sigma": 0.07275,
"v_1": 2.99e-29,
"m_1": 2.99e-26,
"T": 293.0
},
"parameter_sweep": {
"type": "cartesian",
"dimensions": [
{
"parameter": "S",
"range": {
"start": 2.0,
"stop": 6.0,
"count": 41,
"scale": "linear"
}
}
]
},
"plots": [
{
"id": "J_vs_S_log",
"type": "line",
"description": "Nucleation rate J (m⁻³s⁻¹) versus saturation ratio S on a log10 y-axis. The steep exponential increase above S≈3 is the hallmark of classical nucleation theory — even a small increase in S drives a dramatic increase in particle formation rate.",
"x": {
"variable": "S",
"label": "Saturation ratio S (dimensionless)"
},
"y": {
"variable": "J",
"label": "Nucleation rate J (m^-3 s^-1)"
}
}
]
}
]
}