# Dose calculations for size distributions of nano particles

Next to specifying a distribution for the diameter of the aerosol particle, the user may specify a distribution for the size of the nano particle. The size distribution is to be specified as a mass based distribution *P(v)* of the volume of the particle. *P**(**v**)**dv* specifies the fraction of the total mass that is taken up by particles with a volume between v and v + dv.

In ConsExpo nano, the volume distribution is assumed to be adequately described by a lognormal distribution, specified by the median particle volume and the arithmetic coefficient of variation (defined as the ratio of the standard deviation over the mean).

To specify the median volume, the user gives the shape and size of the median particle (e.g. the median particle diameter for a spherical particle, the diameter and height of the median cylindrical particle). ConsExpo nano constructs the volume distribution of the material from the median particle with the dispersion defined by coefficient of variation, assuming that the shape of the particle remains unchanged over the distribution. This means that for cylindrical particles the ratio of diameter to the height is assumed to be constant. Similarly, for nano sheets it is assumed that the ratio of the surface to the particle thickness is constant.

When considering a size distribution of the nano material, the following dose calculations remain unchanged:

- all aerosolbased dose metrics
- the mass of nano material

The calculation of the number of nano particles and the total surface area of the nano particles will change as follows:

*Number of nano particles*

The distribution of the volume of the nano material is given by the distribution function * **P**(**v**)*

From the distribution function, the fraction of the mass of material of nano material with consisting of particles with volume between v and v + dv follows as *P**(**v**)**dv*.

The mass dM of these particles is given by multiplication with the total mass M available for inhalation :

*dM=M×P**(**v**)**dv*

The number of particles dN in this mass dM is found by dividing through the mass of an individual particle with the volume v. This mass is given by ρ x v. Thus

*dN=**M×P**(**v**)**ρv**dv*

The total number of particles is found by integration of dN over the volume

*N=**M×P**(**v**)**ρv**dv*

Surface area of nano materials

The total surface area of nano material is determined by multiplying the number of particles of volume v with the surface area *σ(v)* of the individual particle. Thus the total surface area dS of particles with volume between v and v + dv is given by

*dS**=**σ**(**v**)**×**dN**=**σ**(**v**)**×**M**×**P**(**v**)**ρv**dv*

The total surface area S is found by integration over the volume

*S**=**M**ρ**×**σ**(**v**)×**P**(**v**)**v**dv*

To evaluate this integral, the surface area σ of the particle as a function of its volume needs to be given. This function depends on the nano material’s shape:

Spheres For spheres the relation between surface area and volume is given by:

*σ**v**v**= π**d**2**=π(**6v**π**)**2/3*

Cylinders For cylinders the volume is determined from the height h and diameter d:

*v=h×π**r**2**=h×**π**d**2**4*

As it is assumed that all particles have the same shape (i.e. the same ratio between h and d), h can be specified as *α**×**d*, where the ratio a is determined from the median particle as specified by the user.

With this, the volume of the particle is given by *v**=**π**×**α**×**d**3**4*

And the surface area of the particle as a function of v is given by

*σ**v**v**=**π**×**d**×h+**2**π**d**2**4**=**π**×**d**2**×**1**2**+**α**=**π**×**(**4**v**πa**)**2**3**×**1**2**+**α*

Sheets The shape of a sheet is determined by the surface area s and the thickness t. The ratio between these is considered constant. Therefore, t = βs^{1/2} with some parameter β that can be determined from the user input. With this the volume v of the particle becomes v = βs^{3/2}. The surface area as a function of v, finally is given by

*σ**v**v**= 2×s=2×(**v**β**)**2/3*