F4280 Technologie depozice tenkých vrstev a
povrchových úprav 2. Gas Kinetics
Lenka Zajíčková
Přírodovědecká fakulta & CEITEC, Masarykova univerzita, Brno
lenkaz@physics.muni.cz
jarní semestr 2017 Central European Institute of Technology
BRNO I CZECH REPUBLIC
• 2.1 Vapors and Gases
• 2.2 Maxwell-Boltzmann Distribution
• 2.3 Ideal-Gas Law
• 2.4 Molecular Impingement Flux
• 2.5 Units of Measurement
• 2.6 Knudsen Equation
• 2.7 Mean Free Path
• 2.8 Transport Properties
F4280 Technologie depozice a povrchových úprav:		2.1 Vapors and Gases	Len	ka Zajíčková 3/16
p-V-T diagram				
The possible equilibrium states can be represented in pressure-volume-temperature (p-V-T) space for fixed amount of material (e.g. 1 mol = 6.02 x 1023):
A cut through the p-V-T surface below critical point for fixed T =^ relationship between p and Vm (molar volume)
► point a: highest V (lowest p) - vapor phase
► from point a to b\ reducing V ->> increasing p
► point b\ condensation begins
► from point b to c: V is decreasing at fixed p and T (b- c line is _L to the p-T plane, p is called saturation vapor pressure pv or just vapor pressure)
► point c: condensation completed
If V is abruptly decreased in b - c transition p would be pushed above the line b - c =>- non-equilibrium supersaturated vapors
F4280 Technologie depozice a povrchovych uprav:    2.1 Vapors and Gases
ka Zajíčková
4/16
p-V-T diagram
It is important to distinguish between the behaviors of vapors and gases:
Solid and liquid
Constant-temperature line
► vapors: can be condensed to liquid or solid by compression at fixed T =^ below critical point defined by pc, Vc and Tc
► gases: monotonical decrease of V upon compression =^ no distinction between the two phases
Surfaces "liquid-vapor", "solid-vapor" and "solid-liquid" are perpendicular to the p-T plane =^ their projection on that plane are lines.
p
(atm)
^—Equilibrium
^—Equilibrium
\
CRITICAL POINT
solid
vaporization
deposition
TRIPLE POINT
T (°C)
F4280 Technologie depozice a povrchových úprav:		2.1 Vapors and Gases	Len	ka Zajíčková 5/16
p-T diagram				
T fC)
► triple point: from triple line _L to p-T plane
► below T of triple point: liquid-phase region vanishes =^ condensation directly to the solid phase, vaporization in this region is sublimation
► pressure along borders of vapor region is vapor pressure pv
► pv increases exponentially with T up to pc that is well above 1 atm
=^ deposition of thin films is performed at p < pc, either p > pv (supersaturated vapors) or p < pv
F4280 Technologie depozice a povrchových úprav:    2.2 Maxwell-Boltzmann Distribution
Lenka Zajíčková
6/16
Maxwell-Boltzmann Distribution
Distribution of random velocities V in equilibrium state
m
f(V) = n
2irkB T
3/2      / mV2 exp -
2kBT
(1)
where kB = 1.38 x 10-23 m2 kg s-2 K_1 (or J K_1) is the Boltzmann constant, n, T and m are particle density, temperature and mass, respectively.
If the drift velocity is zero we do not need to distinguish between the velocity and random velocity, i.e. v = V.
Maxwell-Boltzmann distribution is isotropic =^ F(v) distribution of speeds v = \ v\ can be defined by integration of f(v) in spherical coordinates
47T f'iĹTľ
F(v)dv resulting in
o
f(v)v2 s\x\6d(j)d6dv
(2)
F4280 Technologie depozice a povrchových úprav:    2.2 Maxwell-Boltzmann Distribution
Lenka Zajíčková
7/16
Mean (Average) Speed, Molecular Impingement Flux
Vp = Most Probable Speed = Average Speed"
i
i -»_
vrms = Root-IVfeaii-Square Speed
i
' Ý Ý_
■> v
Velocity
Mean speed:
(v) = ^
1
'OO
av--  / F(v)vdV=X/^^(4) ľ! Jo V 7TA77
or
l/av =
"Ä/T
(5)
using molar mass M = uiNa in kg/mol and gas constant
R    kBNA    8.31 Jmol-^-1
where NA = 6.02 x 1023 mol-1 is Avogadro's number
Root-mean-square (rms) speed:
^rms —
3kBT urn
(6)
The most probable speed vl
dF{v) dv
= 0 => vp =
V=Vr
2kBT m
(7)
F4280 Technologie depozice a povrchových úprav:    2.3 Ideal-Gas Law
Lenka Zajíčková
8/16
Ideal-Gas Law
From the definition of pressure for ideal gas (not necessary to consider pressure tensor but only scalar pressure)
p=-mn{V2+V2+V2) = -mn{V2) = m[ V2f(V)d3V. (8) 3 y 3 Jv
The ideal-gas law is obtained by integration of (8) using Maxwell-Boltzmann distribution:
pV
p= nkBT     or = NkB (9)
where N is the number of particles.
Chemists are used to work in molar amounts:
► molar concentration nm = n/NA =^ p = rimRT
► number of moles A/m = N/A/a =^ p = NmRT/V
► molar volume l/„, = V/Na =>• p = RT/Vm
The ideal gas is obeyed if
► the volume of molecules in the gas is much smaller than the volume of the gas
► the cohesive forces between the molecules can be neglected.
Both assumptions are fulfilled for low n =^ always fulfilled for thin film deposition from the vapor phase (T > Troorn and p < patm), i.e. well away from the critical point (most materials pc > 1 atm or if not Tc < 25 °C
F4280 Technologie depozice a povrchových úprav:    2.3 Ideal-Gas Law
Energy Forms Stored by Molecules
Lenka Zajíčková
9/16
Molecules can store energy in various forms. Their energetic states are quantized (spacing between energy levels AE)
► electronic excitations - AEe is highest, transitions between different electronic states are possible only for extremely high T or collision with energetic particle
► vibrational excitations - energy levels correspond to different vibration modes of the molecule, AEV « 0.1 eV (1 eV = 11 600 K)
► rotational excitations - different rotational modes of the molecule, AEr « 0.01 eV
► translational energy - above performed description of molecular random motion
Et = 1 /2mV2, no details of inner molecule structure are considered, AEt negligible at ordinary T.
Ei
Rj"
v. 3
Energy level diagram
}
Rolaltonal
levels
energy levels
Eledrgnit
Energy
I
Fr.A z::_JLei1 c]«:Liu[l:ľ sLale
I_.--^^  Gro unci electronic stale
y~ty_e^__ iRutalJanal ]rvé.s
= = =        01 eV
Separation distance
From definition of absolute temperature - the mean thermal energy AT'/2 belongs to each translational degree of freedom and molecular translation energy is
=> equipartition theorem of classical statistical mechanics. Classical statistical treatment assumes very close quantized energy levels of molecules, i.e. approximated as a continuum. It is a good assumption for translational energy when T > 0 K.
► For atomic gases, Et is total kinetic energy content.
► For molecular gases, Er is added at ordinary T and Ev at very high 7~:
Molar heat capacity at constant volume Cy (for molecular gas) [J/(mol.K)] - increase of total kinetic energy for increasing T:
Cy     d£m     d(£t + Er + Ev)
A/A ~ dT ~ dT 1 ;
for atomic gases for small diatomic molecules at room T
(10)
- two rotational degrees of freedom are excited but vibrational ones are not
F4280 Technologie depozice a povrchových úprav:    2.3 Ideal-Gas Law
Lenka Zajíčková
11/16
Energy Content of Gas
The heat capacity of any gas is larger when measured at constant pressure cp - heat input is doing pdV work on the surroundings in addition to adding kinetic energy to the molecules:
Cp = Cv + R
(12)
We can write from thermodynamics
dUm\
dT   J y
Cy
(13)
where Um is internal energy per mol Um = HmA/A and dHm
ÔT
(14)
where /-/m is enthalpy per mol /-/m = Um + pV,
dHm
m
dT
dUm\     ,      (dVm + P
dT
dT
(15)
giving cp = Cy + R
F4280 Technologie depozice a povrchovych uprav:    2.4 Molecular Impingement Flux
ka Zajíčková
12/16
Molecular Impingement Flux - Knudsen Equatioi
The molecular impingement flux at a surface is a fundamental determinant of film deposition rate:
rOO      í' 7T f'2.71
V = n(vcosO)=       /   /    f(v)v2 cos OsinOd^dOdv
Jo   Jo Jo
(16)
Substituting Maxwell-Boltzmann distribution
fkBT\^2 1
(17)
F4280 Technologie depozice a povrchových úprav:    2.5 Units of Measurement	Lenka Zajíčková	13/16
		
2.5 Units of Measurement		
see Smith's book
F4280 Technologie depozice a povrchových úprav:    2.6 Knudsen Equation
ka Zajíčková
14/16
nudsen tquation
see Smith's book
F4280 Technologie depozice a povrchových úprav:    2.7 Mean Free Path
ka Zajíčková
15/16
2.7 Mean Free Pa
see Smith's book
F4280 Technologie depozice a povrchovych uprav:    2.8 Transport Properties	Lenka Zajíčková	16/16
		
2.8 Transport Properties		
see Smith's book