Nanoparticle Agglomeration Calculator

Enter suspension and particle data

The form uses a 3-column layout on large screens, 2 columns on smaller screens, and 1 column on mobile screens.

Primary particle diameter (nm)

Single-particle diameter before clustering begins.

Initial concentration (particles/mL)

Used in the Smoluchowski population decay term.

Solids volume fraction, φ

Adds a simple crowding correction to collision frequency.

Temperature (°C)

Affects diffusion, electrostatics, and energy scaling.

Viscosity (mPa·s)

Higher viscosity lowers Brownian collision frequency.

Ionic strength (mM)

Controls double-layer compression through Debye length.

Zeta potential (mV)

A larger magnitude often improves electrostatic stability.

Hamaker constant (×10⁻²⁰ J)

Represents attractive van der Waals interaction strength.

Shear rate (s⁻¹)

Adds orthokinetic collision frequency from mixing or flow.

Sticking efficiency, α

Fraction of collisions that become permanent attachments.

Fractal dimension, Df

Converts cluster size into aggregate diameter growth.

Relative permittivity

Useful for non-water media or mixed solvents.

Minimum gap distance (nm)

Closest approach used for interaction-energy scanning.

Steric layer thickness (nm)

Optional soft barrier from adsorbed polymer or surfactant.

Process time (minutes)

Used to estimate concentration decay and size growth.

Reset

Example data table

This example uses the default inputs shown in the form.

Diameter	Ionic strength	Zeta potential	Shear rate	Time	Debye length	Aggregate diameter	Regime
80 nm	10 mM	-35 mV	150 s⁻¹	30 min	3.040 nm	2,378.011 nm	Rapid aggregate growth

Formula used

This tool combines Brownian and shear-induced collision kernels, then scales the total collision frequency using a simplified DLVO-style stability barrier. The Brownian kernel is: β_B = 8k_BT / 3μ. The shear kernel for equal spheres is: β_S = (4/3)γ(2a)³.

Ionic strength sets the Debye length with an aqueous approximation: λ_D(nm) ≈ 0.304 √[(ε_r/78.5)(T/298.15)] / √I, where I is in mol/L. Shorter Debye length compresses the electric double layer and reduces repulsive stabilization.

Interaction energy is estimated from van der Waals attraction, electrostatic repulsion, and an optional steric term. The total barrier is converted into a stability ratio: W ≈ exp(ΔV / k_BT). The effective collision kernel becomes: β_eff = α(β_B + β_S)(1 + 4φ) / W.

Particle number concentration follows the monodisperse Smoluchowski solution: N(t) = N₀ / [1 + β_effN₀t]. Mean particles per cluster are approximated by N₀/N(t), and aggregate diameter is estimated from a fractal scaling law: d_agg = d_p(N₀/N(t))^1/D_f.

These relations are practical engineering approximations. They are best for screening trends, comparing formulations, and identifying stability windows before deeper experimental work.

How to use this calculator

Enter the primary nanoparticle diameter and initial particle concentration.
Set the suspension conditions: volume fraction, temperature, and viscosity.
Provide ionic strength, zeta potential, Hamaker constant, and relative permittivity.
Add process-specific mixing information through the shear rate.
Choose a sticking efficiency and fractal dimension for your material system.
Set minimum gap distance and optional steric layer thickness.
Enter the process time, then click Calculate Agglomeration.
Read the summary metrics, examine both graphs, and export the report using CSV or PDF if needed.

Frequently asked questions

1) What does the stability ratio mean?

It estimates how strongly repulsive forces slow aggregation. A value near 1 suggests fast agglomeration. Larger values indicate collisions are less likely to stick because of an energy barrier.

2) Why does ionic strength matter so much?

Higher ionic strength compresses the electric double layer, reducing electrostatic repulsion. That usually lowers colloidal stability and can accelerate agglomeration, especially when van der Waals attraction is already strong.

3) What is the role of zeta potential?

Zeta potential is a practical indicator of electrostatic stabilization. Higher magnitude values, positive or negative, often increase repulsive interaction and raise the estimated energy barrier against agglomeration.

4) Why include both Brownian and shear kernels?

Nanoparticles collide from thermal motion and from velocity gradients in mixing or flow. Including both kernels helps the model capture still systems, stirred systems, and intermediate processing conditions.

5) What does fractal dimension change?

Fractal dimension translates cluster population growth into aggregate size. Lower values represent looser, more open aggregates, while higher values represent denser clusters with slower diameter expansion for the same number of particles.

6) When should I use steric thickness?

Use it when polymers, surfactants, or ligands create a protective shell around particles. The steric term can reduce close approach and increase the apparent barrier to agglomeration.

7) Is this suitable for concentrated slurries?

It can support quick screening, but concentrated systems may need richer hydrodynamic, many-body, and breakage models. Use the results as trend guidance rather than a final design basis.

8) Can I use this for non-aqueous dispersions?

Yes, but update viscosity and relative permittivity carefully. The Debye approximation and electrostatic term are simplified, so non-aqueous systems may still require calibration against experimental stability data.