3. Theory and Simulations
3.1. Vapour pressure and evaporation rate
A characteristic quantity for the evaporation of substances is their vapour pressure. Being kept in an enclosure with a certain range of pressures and temperatures all substances can exist in an equilibrium with both a gas and a condensed phase, so either liquid or solid[40].
Molecules in this case change between those two phases by either evaporation or subli-mation from the condensed phase to the gas or condensation from the gas to the condensed phase. The equilibrium is characterized by the same rates of evaporation and condensa-tion. The partial pressure that is exerted by the gas onto the walls of a container in such an equilibrium is called the vapour pressure. It depends on the actual substance and on the temperature in the container.
If the temperature rises, the vapour pressure also rises, which leads to a bigger part of the substance going into the gas phase. In this way the equilibrium could stop existing if the vapour pressure is so high that it requires all molecules of the substance to be part of the gas phase. On the other hand if e.g. a gas liquid equilibrium is kept in a container and one adds more of the respective substance while not changing the temperature of the system, the additional substance would only add to the condensed phase. The described
behaviour can only be observed within a temperature limit. Above this critical temperature a substance will not condense at any pressure and a two phase system is not possible (at very high pressures the liquid and gas phase become indistinguishable). This is for example the case for oxygen or nitrogen at room temperature[41].
3.1.1. The lead vapour pressure
The vapour pressure of a substance depends on its temperature. An approximate way to derive the vapour pressure for different temperatures is the Clausius Clapeyron equation.
In its integrated form ([42]) it gives the relation between two parameter combinations of vapour pressure and temperature,pv,1atT1andpv,2atT2, using the enthalpy of vaporization
∆Hvapand the gas constantRgas:
lnpv,2
pv,1 =∆Hvap Rgas
1 T1− 1
T2
. (3.1)
This equation assumes the gas to be an ideal gas and the substance dependent enthalpy of vaporization to be constant over the temperature range ofT1toT2. For the vapour pressure of liquid lead an empirical relation, given in[43](originally from[44]), can be used:
log10(pv [Pa]) =5.006+APb+ BPb
T[K] , (3.2)
wherepv is the lead vapour pressure in Pa andAPb andBPb are measured coefficients.
These are: APb =4.911 and BPb = −9701. The temperatureT can be chosen from the melting point of lead atT =327.5 °C to at leastT =1200 °C and the results will have a 5 % accuracy or better compared to measured values. Equation (3.2) with different coefficients and if needed also higher order terms, can be used for a large range of metallic elements and substances. Figure 3.1 shows the lead vapour pressure calculated from equation (3.2) together with 3 values from[43]for a large range of temperatures.
3.1.2. Evaporation
Evaporation is the process of particles leaving a condensed phase and becoming part of a vapour. When the same amount of particles condensate back into the liquid or solid, the two phases are in equilibrium. The molecular theory of gases gives a relation between the flux density of ideal gas particles impinging on a surface, J, their average velocity ¯v and the density of particlesn:
3.1. Vapour pressure and evaporation rate
300 400 500 600 700 800 900 1,000
10−9 10−8 10−7 10−6 10−5 10−4 10−3 10−2 10−1 100 101
T [°C]
pv[mbar]
Eq. (3.2)
Literature values from[43]
Figure 3.1.:Vapour pressure of lead following from equation (3.2), where the values of Pa where converted to mbar and the temperature K to °C. The plot also shows three literature values from[43].
J =nv¯
4 . (3.3)
From the Maxwell-Boltzmann distribution of speeds within the gas at a temperatureT it follows:
v¯= v t8kBT
πm . (3.4)
Heremis the mass of the gas particles andkBthe Boltzmann constant. As the gas density can also be linked to the pressurep at temperatureT, using:
p =n kBT , (3.5)
equation (3.3) can be brought into the form:
J = p
p2πm kBT . (3.6)
If one considers a liquid in a container that is in equilibrium with its vapour phase
(p=pv at temperatureT), equation (3.6) can be used to calculate the amount of particles hitting the surface of the liquid phase. A fraction of these molecules will enter the liquid phase again, making them the condensation flux densityJcond:
Jcond=αJ . (3.7)
The coefficientαgives the fraction of particles, that condensate when they are hitting the liquid phase. If the container is closed, the rate of particles exiting the liquid, Jevap, and condensing on it is the same, soJevap=Jcond. Now one can assume an opening in the container, reducing the surrounding pressure top<pv. This leads to a reduction in the flux of particles onto the liquid phase, while the amount of particles evaporating stays the same. Hence the net evaporation rate per surface areaJevap,netbecomes:
Jevap,net= α(pv−p)
p2πm kBT . (3.8)
This is the Hertz-Knudsen equation for evaporation, that allows calculating the rate of particles evaporating from a liquid substance at a certain temperature, if its vapour pressure and its evaporation coefficientαis known. From the origin of the equation the different values can be resumed:
• Jevap,net: The evaporation rate of the substance per surface area in at s−1m−2.
• pv: Vapour pressure of the concerned substance in Pa at the temperatureT.
• αEvaporation coefficient of the substance. A unit less number.
• p: Partial pressure of the vapour from the concerned substance in Pa. The presence of another gas is not considered, i.e. this is not a background pressure from the atmosphere.
• m: The atomic mass in kg.
• T: The temperature of the vapour and of the liquid in K.
It is used for applications where vacuum evaporation takes place and gives usable results for the evaporation of metals when the right evaporation coefficient is chosen[45]. But it needs to be taken into account that it is an interpolation from an assumed equilibrium and does not include a theory of how the evaporation mechanism actually works. To fit the results to measured values the evaporation coefficient is adapted for different substances.