Boltzmann load distribution

The Boltzmann charge distribution describes the resulting electrical bipolar charge equilibrium of gas-borne particles ( aerosol ) within a charge carrier cloud that contains positive and negative ions (as well as electrons ). The charge distribution corresponds to the probability of how high the charge state of a particle fraction is after a sufficiently long time in the charge cloud.

Charge distribution

The movement of gas-borne particles is determined by constant collision with other particles and with gas molecules . If at least one of the collision partners is electrically charged, an additional charge exchange takes place in the event of a collision, which depends on the charge level of the collision partner. The charge state of particles is strongly dependent on their size and surface properties. The charge distribution of a collective of particles, which occurs after a corresponding dwell time in a bipolar charge cloud, can be described with the Boltzmann charge distribution.

Boltzmann charge distribution, d _P  <20 nm

Boltzmann charge distribution, d _P  > 20 nm

The  following applies for particle diameters d _P <20 nm

${\displaystyle f(d_{\text{P}},n)={\frac {x(d_{\text{P}},n)}{\sum \limits _{n=-\infty }^{\infty }x(d_{\text{P}},n)}}}$

For particle diameters d _P  > 20 nm applies

${\displaystyle f(d_{\text{P}},n)={\frac {e}{\sqrt {4\cdot \pi ^{2}\cdot \varepsilon _{0}\cdot d_{\text{P}}\cdot k_{\mathrm {B} }\cdot T}}}\cdot x(d_{\text{P}},n)}$

With

${\displaystyle x(d_{\text{P}},n)=\exp \left(-{\frac {n^{2}\cdot e^{2}}{4\cdot \pi \cdot \varepsilon _{0}\cdot d_{\text{P}}\cdot k_{\mathrm {B} }\cdot T}}\right)}$

Here d _{P is} the particle diameter, n the number of charges (integer multiple of the elementary charge ), e the elementary charge 1.602 · 10 ⁻¹⁹  C, ε ₀ the permittivity of the vacuum 8.854 · 10 ⁻¹²  As / Vm, k _B the Boltzmann constant 1.308 · 10 ⁻²³  J / K and T is the gas temperature.

Assuming that the same number of positive and negative charge carriers are present, the resulting total charge is zero, which means that the particle collective has a neutral effect on the outside. Within the collective, the majority of the particles are also neutralized, especially as the particle size decreases. For d _P  >> 20 nm, the proportion of multiply charged particles increases.

In aerosol measurement technology, particles are often charged in a bipolar manner by ionizing the carrier gas, for example by a weakly radioactive source. Knowing the Boltzmann charge distribution and the electrical particle mobility distribution, one can then use a corresponding inversion algorithm to infer the particle size distribution.

See also

• Diffusion charging
• Particle charge

literature

• William C. Hinds: Aerosol Technology . John Wiley & Sons, New York 1982, ISBN 0-471-08726-2 .

Individual evidence

1. ^ William C. Hinds: Aerosol Technology . John Wiley & Sons, New York 1982, ISBN 0-471-08726-2 .