Nanoparticle volume, mass, particle number concentration, and molarity are fundamental quantities for comparing nanoparticle formulations and designing experiments. These values can be calculated from particle dimensions, material density, and measured mass concentration.
This guide explains how nanoComposix performs these calculations for common particle geometries, including spheres, rods, plates, cubes, and core-shell nanoparticles. It also reviews direct and indirect methods for measuring nanoparticle concentration.
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How to Calculate Nanoparticle Volume
The first step is determining particle dimensions. At nanoComposix, particle dimensions are primarily measured by transmission electron microscopy (TEM). Measurements from many particles and multiple representative images can then be used to estimate average dimensions and calculate particle volume.
Spherical Nanoparticles
For a spherical nanoparticle:
where r is the particle radius.
Rod-Shaped Nanoparticles
For a cylindrical approximation of a rod-shaped nanoparticle:
where r is the rod radius and l is its length.
Plate-Shaped Nanoparticles
For a circular plate approximated as a cylinder:
where r is the nanoplate radius and h is its thickness.
Cube-Shaped Nanoparticles
For a cube:
where d is the cube edge length.
Measuring the Required Dimensions
TEM images can be analyzed using software such as ImageJ/Fiji to measure many particles and determine representative dimensions.
Some geometries require complementary measurements. Nanoplates, for example, often lie flat on a TEM grid, making their lateral dimensions easy to measure but their thickness difficult to determine directly. Atomic force microscopy (AFM), high-resolution scanning electron microscopy (SEM), or other complementary techniques may therefore be needed.
In some composite structures, nanoplate thickness can also become visible by TEM. For example, silica-shelled plates may orient on edge during grid preparation, allowing direct measurement of plate thickness.
How to Calculate Nanoparticle Mass
Once particle volume is known, the mass of an individual nanoparticle can be estimated by multiplying its volume by the material density:
where m is particle mass, V is particle volume, and ρ is material density.
For many dense metallic nanoparticles, bulk material density provides a useful approximation. Other materials can require an effective nanoparticle density because porosity, atomic structure, hydration, or degree of condensation can cause the nanoparticle to differ from the corresponding bulk material.
| Material | Density Used for Nanoparticle Calculations (g/cm3) | Comparison with Bulk Density |
|---|---|---|
| Gold | 19.32 | Same as bulk |
| Silver | 10.5 | Same as bulk |
| Platinum | 21.45 | Same as bulk |
| Silica | 2.2 | Lower than bulk silica (~2.65) |
| Magnetite (Fe3O4) | 5.24 | Same as bulk |
Effective Density of Silica Nanoparticles
Silica nanoparticles are commonly prepared through variations of the Stöber process, in which silane precursors hydrolyze and condense in the presence of a base. The resulting silica network can contain silanol groups (Si–OH) and may be less densely condensed than silica produced under high-temperature bulk-processing conditions.
Further condensation can convert neighboring silanol groups into Si–O–Si bonds while releasing water. The degree of condensation, particle porosity, synthesis conditions, and subsequent processing can therefore influence the effective density of a silica nanoparticle.
For concentration calculations involving nanoComposix solid silica nanospheres, we use an effective density of approximately 2.2 g/cm3. Published studies using techniques such as aerosol particle mass analysis have likewise demonstrated that nanoparticle effective density can differ from ideal bulk-material values.
How to Calculate the Mass of a Core-Shell Nanoparticle
For nanoparticles composed of more than one material, the mass of each region must be calculated separately. Gold nanoshells, for example, consist of a silica core surrounded by a gold shell.
The total particle mass is:
Core Mass
For a spherical core:
Shell Mass
The shell volume can be calculated by subtracting the core volume from the volume of the complete core-shell particle. For a spherical core-shell structure:
The shell thickness is the difference between the total particle radius and core radius:
The total particle mass is then obtained by adding the calculated core and shell masses. The same general approach can be adapted to other core-shell particle geometries.
How to Calculate Nanoparticle Number Concentration
Particle number concentration requires two pieces of information: the amount of nanoparticle material present in a known volume of solution and the calculated mass of an individual particle.
A rough estimate of material concentration can sometimes be obtained by assuming complete conversion of synthesis reagents into nanoparticles. For example, one could assume that all added gold precursor becomes elemental gold. This approach does not account for incomplete reaction yield or material lost during purification and processing.
Direct elemental analysis provides a more accurate measurement. At nanoComposix, ICP-MS is used to measure elemental concentration in purified nanoparticle dispersions.
Particle number concentration can then be calculated as:
where:
- N = nanoparticle number concentration
- MC = measured nanoparticle mass concentration
- m = calculated mass of an individual nanoparticle
If mass concentration is expressed in g/mL and individual particle mass in g/particle, the resulting number concentration is expressed in particles/mL.
Depending on nanoparticle material, size, and formulation, particle concentrations can span many orders of magnitude. Typical nanoComposix formulations fall broadly within approximately 109–1015 particles/mL.
Calculating the Concentration of Core-Shell Nanoparticles
The same principle can be applied to core-shell structures, but the measured elemental concentration should be paired with the calculated mass of that element within each particle.
For a gold nanoshell, for example, ICP-MS can determine the total mass of gold per unit volume. Dividing that value by the calculated gold-shell mass per particle provides an estimate of nanoshell particle concentration.
A similar approach can be used for other multicomponent structures, including gold/silver bimetallic nanoparticles.
How to Calculate Nanoparticle Molarity
Concentration in chemistry and biology is frequently expressed as molarity, or moles per liter. For nanoparticles, it is important to distinguish particle molarity from the molar concentration of the atoms or molecules that make up the nanoparticle.
Particle molarity is calculated from particle number concentration using Avogadro's constant:
where N is particle number concentration in particles/L.
Nanoparticle particle molarities are commonly in the nanomolar (nM) to picomolar (pM) range. As an example, a 40 nm gold nanosphere dispersion at 0.05 mg/mL contains approximately 7.7 × 1010 particles/mL, corresponding to a particle molarity of approximately 130 pM.
Other Ways to Measure Nanoparticle Concentration
Particle concentration can also be measured or estimated using methods that count individual particles or infer particle number from other measured properties. Each technique has different sample requirements and size limitations.
| Approach | How It Works & Key Considerations |
|---|---|
| Electrical Particle Counting | Instruments such as Spectradyne and qNano detect individual particles as they pass through a small aperture and alter electrical resistance. Particle counts can be converted to concentration using a known sample volume or flow rate. These measurements often require sufficiently conductive media, such as approximately 1× PBS, and historically have been most applicable to particles around 50 nm and larger. Exact limits depend on the instrument, particle, and formulation. |
| Nanoparticle Tracking Analysis | Instruments such as the Malvern NanoSight optically track individual particles undergoing Brownian motion. Particle trajectories are used to estimate hydrodynamic size and particle concentration. Particles around 30 nm and larger can be accessible for suitable strongly scattering materials, although practical detection limits depend strongly on composition, refractive index, concentration, and instrument configuration. |
| Microscopy-Based Counting | Particles can be deposited onto a surface and individually counted using large-area electron microscopy images. Accurate particle-number determination requires well-controlled sample volume, deposition, and representative imaging. |
| Optical Modeling | For particles with well-defined optical properties, numerical models can calculate extinction cross sections from particle geometry and composition. Measured UV-Visible extinction can then be related to particle concentration. This approach is particularly useful for plasmonic nanoparticles when the particle geometry is sufficiently well characterized. |
Estimating Concentration from Optical Properties
Determining particle number accurately can be challenging, particularly when particles are too small or too weakly scattering for direct counting techniques.
For optically active nanoparticles, calculated extinction, absorption, and scattering cross sections provide another route to estimating particle concentration from measured UV-Visible spectra. Hendel and colleagues demonstrated this approach for colloidal gold while also discussing its limitations.1
The Mie Theory Calculator can calculate optical cross sections for spherical and concentric core-shell particles and can be used to explore the relationship between particle geometry and optical extinction.
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Selected Literature
- Hendel, T.; Wuithschick, M.; Kettemann, F.; Birnbaum, A.; Rademann, K.; Polte, J. In Situ Determination of Colloidal Gold Concentrations with UV-Vis Spectroscopy: Limitations and Perspectives. Analytical Chemistry 2014, 86, 11115–11124.
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