Problem no. 3 "Plasma"

Measurements had to be done first. As an example for the results of our measurements we want to show you two diagrams. The measurements were quite difficult, because there were lots of fluctuations. So the results are only average values.
This diagram shows the dependence of the current from the horizontal distance of the electrodes.

Diagram: Dependence of the current from the horizontal distance of the electrodes experimental data:
  • sharp electrodes
  • on the same level in the flame
  • voltage: 150 V, DC

This diagram shows another example of our measurements.

Diagram: Dependence of the current from the vertical distance of the electrodes experimental data:
  • sharp electrodes
  • above each other
  • intersection about 4 mm
  • voltage: 150 V, DC

After our measurements we drew the following conclusions:

Now we want to show you, what we found out about the influence of these parameters.
Let's start with the gas flow:
For our research on gas flow, we "recycled" a theory and measurement results from the previous year. For those who can't remember: In the 12th IYPT the problem no. 13 "Gas flow" concerned the gas flow in the flame of a candle. So we now show you what we found out one year ago:

This sketch shows you, how we measured the gas flow:

Sketch of the Gas Flow measurements

left mass: 2g; right mass: 1g; connected with a circular Al - plate (in the flame: Fe - plate); Temperature measurement with Pt-100 or thermocouple

The streaming force FStr is an upward force on the right mass which produces an additional downward force on the electronic scales. This causes the scales to show a higher weight, so FStr can be calculated using this "mass difference".
With the streaming force we are able to calculate the velocity of the gas flow using this formula:

Formula for the streaming velocity Cw = 1,1 (for circular plates)
A = r2.p (area of the circular plate)
FStr ... has been measured
rho ... density of air: calculated (see following)

Calculation of air density: using the ideal gas equation

ideal gas equation
ppressure [Pa]
Vvolume [m3]
nnumber of moles
Rgas constant (= 8,31) [ J / (K . mol) ]
Ttemperature [K]
Mmolecular weight
rhodensity [kg/m3]

calculation of the molecular weight M:
values from literature: air at 0C, 1 bar: r = 1,293 [kg/m3]

calculation of the molecular weight M (rho [kg/m3] --> M [kg])

This diagram finally shows the result of the gas flow measurements: Velocity of the flow versus the vertical distance from the flame:

Speed of Gas Flow versus vertical distance from the centre of the flame

Furthermore we considered some theory about plasma. Useful statements can only be given about a balance plasma. For a balance plasma, 3 distributions have to be fulfilled:

The following formula is the Maxwell distribution. It indicates how many molecules have a distinct velocity. It is possible to calculate the fraction of molecules for each velocity v.
Maxwell distribution
Nnumber of molecules
kBoltzmann constant (approx. 1,38.10-23 [J/K])
mMmass of one molecule
TTemperature [K]

We wanted to find out, how many molecules are ionised. In order to calculate the minimum velocity for ionisation through hits, we set the ionisation energy equal to the kinetic energy of a molecule. Ionisation energy
After having calculated the minimum velocity, we used the Maxwell distribution to calculate the fraction x of molecules which are faster than the minimum speed. Maxwell distribution
It is possible to calculate this for each kind of molecules in the plasma. Here are two results for example:
results: e.g. for O2: x = approx. 3.10-64; for N2: x = approx. 9.10-91
As you can see the fraction of ionised molecules is so small that it is possible to neglect it.


The Saha distribution:

The Saha equation indicates the ionisation balance between the electrons and the ions versus neutral particles:
Ionisatoin balance

Saha distribution
memass of the electron
hPlanck constant
kBoltzmann constant
Ttemperature
Eiionisation energy

Also this equation proofs that there are actually no ions in the plasma.


Influence of the electric field:

The electric field causes electrons to drift through the plasma with a current density j:
formula
Moreover we used some formulas which we found in literature. Formulas from literature

Nenumber of electrons
eelementary charge
vddrifting speed
Eelectric field strength
memass of the electron
taverage time between 2 hits
sconductivity
mobility of the electrons
lambdaefree distance between 2 hits
veaverage speed of the electrons (Maxwell distribution)
kBoltzmann constant
Ttemperature

Using the shown formula, we can proof that the conductivity is proportional to one divided by squareroot of the mass.

formula

By taking an element with an average mass from the periodic system of the elements we can show that the share of ions of the current is smaller than 0,4 %.
So we can restrict ourselves to the consideration of drifting electrons only.


The Richardson-Dushman equation indicates, how many electrons are emitted by the electrodes.

Richardson-Dushman equation
WAwork function
kBoltzmann constant
TTemperature
hPlanck constant
mrest mass of the electron

The Richardson-Dushman equation shows only the influence of the temperature and does not consider the point effect!

The electric field also causes the electrodes to emit electrons, as it is indicated by the Fowler-Nordheim equation.

Fowler-Nordheim equation
Eelectric field strength
Fforce
Qcharge
ecelementary charge
jcurrent density

It shows that the current density is proportional to the electric field strength squared times Euler's number to the minus one divided by the field strength.
To determine the electric field strength we made a computer simulation. The following plots shall give you an idea how the field depends on the shape and arrangement of the electrodes.

Click on a thumbnail to enlarge...

Click to enlarge thumbnailField between a point electrode and a spherical electrode.
Click to enlarge thumbnailField on the surface of a spherical electrode.
Click to enlarge thumbnailPotential distribution on the surface of two needel electrodes.
Click to enlarge thumbnailField between two spherical electrodes (quarter of the symmetry)

In conclusion the following things have to be said: The connection between our experiments and our theory is not possible. The current is determined by so many parameters which correlate so much that it is not possible to measure the dependence of the current from one single parameter. Therefore there can be no real connection between theory and experiments. Moreover, the phenomenon is much too complex so that there can be no final solution formula which describes the behaviour of the current.