Nanoparticle volume, mass and concentration are fundamental nanoparticle characteristics. In this module, we describe how we calculate these parameters for both solid particles and core/shell particle geometries.

## How to Calculate the Volume of a Nanoparticle

The volume of a nanoparticle is determined by first measuring its dimensions. At nanoComposix we primarily use a transmission electron microscope (TEM) to measure particle dimensions, allowing the volume to be calculated.

For spherical nanoparticles, the volume is: *V=4/3*𝜋*r ^{3}*, where

*r*is the radius of the sphere

For rod shaped nanoparticles, the volume is:*V=*𝜋*r ^{2}l*, where r is the radius of the rod and

*l*is the length

For plate shaped nanoparticles, the volume is *V=*𝜋*r ^{2}h* , where

*r*is the radius of the nanoplate and

*h*is the thickness.

For cube shaped nanoparticles, the volume is : *V=d ^{3}*, where d is the diameter of the cube.

To obtain these dimensions, TEM images are analyzed with a program such as ImageJ/Fiji to measure many particles from multiple TEM grids. The measurements are averaged and substituted in the formulas above. Sometimes, all of the needed dimensions cannot be obtained with TEM alone. For example, nanoplates typically sit flat on the TEM grid so it is not possible to measure the thickness directly with TEM, and complimentary measurement techniques, such as atomic force microscopy (AFM) or high-resolution scanning electron microscopy (SEM) may be needed to measure the plate thickness. Another method of measuring plate thickness is to measure the plate in composite particles. For example, when silica shelled, the nanoplates will often be rotated on edge when dried onto a TEM grid and a direct TEM measurement of the thickness can be made.

## How to Calculate the Mass of a Nanoparticle

Once the nanoparticle volume has been calculated the mass can be determined simply by multiplying the volume by the material density (ρ): *m=Vρ*. In most cases the density of nanomaterials is the same as the bulk density, but for some materials the atomic structure is different than the bulk and a corrected density must be used. The mass calculation is also adjusted for nanoparticles made of multiple materials, such as core/shell nanoparticles.

Material | Nanoparticle Density (g/cm^{3}) |
Bulk Density (g/cm^{3}) |
---|---|---|

Gold |
19.32 | Same |

Silver |
10.5 | Same |

Platinum |
21.45 | Same |

Silica |
2.2 | 2.65 |

Magnetite (Fe_{3}O_{4}) |
5.24 | Same |

### Effective Density of Silica Nanoparticles

Silica nanoparticles are typically prepared using the Stober method, in which silane precursors are condensed in the presence of a base. Depending on the fabrication, environment and storage conditions, the degree to which the silica is condensed varies. Initially, there will be many -OH groups within the silica particle; the number of hydroxyl groups can be reduced by heating, which converts two -OH bonds into a Si-O-Si bond while releasing a water molecule. This condensation process leads to the silica becoming less porous and more dense, but is still typically a lower density than bulk silica prepared at high temperatures.

We use an effective density of 2.05 g/cm^{3} for our silica nanoparticles, which is similar to other reported values in the literature measured by other techniques such as an aerosol particle mass analyzer (Kimoto 2014, Kimoto 2017)

## How to Calculate the Mass of a Core-Shell Nanoparticle

When a nanoparticle is made from more than one material, separate calculations must be made to determine the particle mass. For example, gold nanoshells consist of a silica core surrounded by a thin gold shell, and the mass of the core and the mass of the shell must be calculated separately to determine the total mass of the particle. In this example, the total particle mass is calculated by

*m _{total} = m_{core} + m_{shell} = V_{core}ρ_{core} + V_{shell}ρ_{shell}*

The mass of the core is the volume multiplied by the density of the core. For a spherical core particle the mass is given by

*m _{core}* =

*4/3*𝜋

*r*

_{core}^{3}ρ_{core}The mass of the shell is the volume of the shell multiplied by the density of the shell. In some cases, it may be easiest to calculate the shell volume by measuring the total particle volume and subtracting the volume of the core. For example, for gold nanoshells we measure the diameter of the core silica nanoparticles first, and then measure the final diameter of the gold nanoshells. The thickness of the gold nanoshell is determined to by subtracting the total radius of the gold nanoshell from the radius of the core. For spherical core/shell particles like these, the mass of the shell is given by

*m _{shell} = 4/3𝜋(r_{total}^{3 }– r_{core}^{3})ρ_{shell}*

The total mass of the particle is then the sum of the mass of the core and the mass of the shell. The formulas above are adjusted to account for other particle geometries.

## How to Calculate Nanoparticle Concentration

To calculate nanoparticle concentration you must first determine the total mass of the element of interest in nanoparticle form in the solution. A rough approximation can be obtained by assuming that all of the initial reagents were converted into nanoparticle form (for example, all added gold chloride is reduced to elemental gold), but does not account for lower reaction yield or processing losses, and analytical methods to directly measure elemental concentration will provide more accurate results.

At nanoComposix we use ICP-MS to directly measure the elemental concentration in our final purified nanoparticle solution. Using this concentration we can calculate the number concentration of the nanoparticles by dividing by the total mass in solution by the mass of a single nanoparticle:

*N = M _{C }/ m*

where *M _{C}* is the mass concentration of the measured element and

*m*is the mass of an individual nanoparticle. If the total mass concentration is expressed in units of g/mL, and the particle mass has units of g/particle, the calculated concentration has units of particles/mL. For typical nanoparticle formulations we provide, this concentration is in the range of 10

^{9}to 10

^{15}particles/mL, depending on the material and product.

## How to Calculate Concentrations of Core/Shell Nanoparticles

When calculating the concentration of nanoshells, the total mass of gold per mL is determined by ICP-MS and then divided by the mass of gold in a nanoshell to yield the total particles/mL. For other core/shell geometries (such as gold/silver bimetallic particles), a similar strategy is used.

## Nanoparticle Concentration in Molar Concentration

In chemistry and biology, concentration is often expressed as molarity, which refers to the number of moles of a substance per liter. In some cases, it is useful to perform calculations using the *particle* molarity, which is different than the molar concentration of the elements making up the nanoparticles. The particle molarity is calculated as

*M = N / 6.02 × 10 ^{23} *

where N is the number concentration of the nanoparticles in units of particles/L and the denominator is Avogadro’s number.

Typical molar concentrations of nanoparticles are in the nanomolar (nM) to picomolar (pM) concentration range. For example, our NanoXact 40 nm-diameter gold nanospheres have a total elemental gold concentration of 0.05 mg/mL, which corresponds to a particle number concentration of 8.1 × 10^{10} particles/mL and a particle molarity of 130 pM.

## Other Direct and Indirect Measurements of Nanoparticle Concentration

There are other methods of directly measuring nanoparticle concentration in solution. There are a number of instruments that count particles by monitoring the passage of particles through a small orifice. When the nanoparticle passes through the orifice, a portion of the solution is displaced which changes the electrical resistance. With a known flow rate, each electrical pulse can be counted and a particle concentration can be measured. Two such instruments are the Spectrodyne and the qNano. Typically, particles are required to be dispersed in a 1× PBS buffer or a solution with a similar level of salt concentration in order to make a measurement. Also, the lower size limit using this technique is approximately 50 nm and requires that particles be stable in high salt environments. Another method of measuring particle concentration is to optically count the number of particles in a small volume of solution. The Malvern NanoSight visually tracks individual nanoparticles and calculates their size based on diffusion. Particles as small as 30 nm can be measured with this instrument and there is no salt requirement for solution. However, the particles will drift in and out of the focal plane so the instrument must be calibrated with particle number standards first to obtain accurate particle number concentration measurements. One of the most accurate methods of counting particles is to dry the particles onto a surface and individually count each one. Large area images captured with a scanning electron microscope can be used to count particles. This technique relies on very accurate solution volumes to be applied to the sample during preparation.

Overall, it is surprisingly difficult to accurately determine particle number. An alternative method to the analytical solutions presented above is to use numerical models to predict the optical properties of a particle with a particular geometry. Calculation of the extinction, absorption and scattering cross sections can be used to predict the particle concentration based on the measured extinction in solution using a UV-visible spectrophotometer (Hendel 2014). Our Mie Theory particle calculator can be used to calculate cross sections of spherical and core/shell spherical particles and has been shown to agree well with analytical measurements.

## References

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." *Anal. Chem.* **2014**, *86(22)*, 11115-11124.