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Physics of Thin Films

PES 449 / PHYS 549


Kinetic Theory of Gasses

Ohring: Chapter 2, sections 1, 2


Pressure and Vacuum

Many thin film processes involve vacuum.

"vacuum" = lower molecular density than in our atmosphere

results in a lower pressure of gas - so typically measure this

MANY different units are commonly used.

mbar
Pascals (N/m2)
atmospheres
Torr (mm Hg)
microns (µm Hg)
psi (lb/in2)
dyne/cm2

molecules / m3

1 mbar =

1
100
9.87x10-4
0.75
750
0.0145
1000
2.65x1022

1 Pa =

0.01
1
9.87x10-6
7.5x10-3
7.5
1.45x10-4
10
2.65x1020

1 atm =

1010
10,100
1
760
7.6x105
14.69
1.01x106
2.69x1025

1 Torr =

1.333
133.3
1.31x10-3
1
1000
0.0193
1333
3.53x1022

1 µm =

1.33x10-3
0.133
1.31x10-6
0.001
1
1.93x10-5
1.333
3.53x1019

1 psi =

68.94
6.89x103
0.068
51.71
5.17x104
1
6.89x104
1.83x1024

1 dyne/cm2 =

0.001
0.10
9.87x10-7
7.50x10-4
0.75
1.45x10-5
1
2.65x1019

1 molecule/m3 =

3.77x10-23
3.77x10-21
3.72x10-26
2.83x10-23
2.83x10-20
5.47x10-25
3.77x10-20
1

Pressure Conversion Calculator

enter exponents as "e-7"
for example: 5.3x10-7 should be entered 5.3e-7

-- still has some trouble displaying very small or very large numbers --

Pressure Conversion Calculator

mbar

Pascals

atmospheres

Torr

microns
psi
dyne/cm2
molecules/m3

 Composition of gas in vacuum chamber is very different from atmosphere

pumps remove certain gasses preferentially

Component
Volume % in dry air
Volume % in ion pumped chamber at 2x10-9 torr
N2
78 %
trace
O2
21 %
trace
Ar
0.93 %
trace
CO2
0.03 %
3 %
CH4
trace
3 %
H2O
trace
5 %
CO
trace
6 %
H2
trace
78 %


Ideal Gas Law

much of vacuum technology can be understood from the ideal gas law

more correctly: the equation of state of an ideal gas

PV = NkT

where

  • P = absolute pressure
  • V = volume
  • N = number of gas molecules
  • k = Boltzmann's constant
  • T = gas temperature (in K)

 
Kinetic Theory of Gasses - Gas Flow

Assumptions:
  • Gasses are composed of a very large number of very small particles.
    • "very small" => very small compared to the distance between particles
  • Particles are always moving rapidly in a straight line.
  • Particles exert no forces except during collisions.

Freeze other molecules and examine motion of one molecule:

gas molecule obeying kinetic theory

What is the distribution of velocities ?

determine most properties from this

Maxwell velocity distribution

Maxwell velocity distribution

higher T: shifts curve to right; broadens and lowers it

lighter mass: shifts curve to right; broadens and lowers it

See Figure 2-1 in Ohring.

 

How fast are the molecules moving ?

equation for rms velocity

 

k = Boltzmann's constant

T = temperature of the gas (K)

m = mass of the molecule

 

Not surprising:

The hotter it is, the faster they move.

The lighter they are, the faster they move.

 

At room temperature:
Molecule
vrms (m/sec)
vrms (miles/hour)
H2
1700
3790
N2
450
1000
Ar
380
850

How far does a molecule travel before it collides with another molecule ?

equation of mean free path
l = mean free path

d = diameter of a molecule

n = number per unit volume

For air at room temperature, the mean free path can be expressed as:

equation of mean free path - practical units

P = pressure in torr

l will be in cm.

Pressure
Mean Free Path
1 atm
6.7 x 10-6 cm
1 torr
5 x 10-3 cm
1 millitorr
5 cm
10-6 torr
50 m
10-9 torr
50 km

Gas Flow:

three regimes:
  • viscous flow
    • mean free path << size of the system (D)
    • gas - gas collisions dominate
    • molecules "drag" one another along in the flow
    • when D(cm) P (Torr) > 0.5
      • for air at room temperature
  • intermediate (transition) flow
    • mean free path comparable to size of system (D)
    • complicated flow
  • molecular flow
    • mean free path >> size of system
    • gas - wall collisions dominate
    • molecules move independently of one another
    • when D(cm) P (Torr) < 0.005
      • for air at room temperature

    This information is summarized in Ohring Figure 2-3.


Kinetic Theory of Gasses - Interactions with surface

How many gas molecules collide with a surface each second ?
F = 0.25 n vrms

F = collision rate of gas molecules

n = number of molecules per unit volume

vrms = average velocity of a gas molecule

In terms of things we can directly measure:

collisions with surface

F will be in molecules/ cm2 - sec

P is the pressure in torr

M is the molecular weight of the gas molecule

T is the temperature in K

For example:

Nitrogen (N2) has a molecular weight M = 28. If we have a chamber with nitrogen at room temperature (293 K) and a pressure of 1 x 10-7 torr:

F = 3.88 x 1013 molecules/cm2 - sec

How long does it take to form a single complete layer of gas on a surface ?

equation of time to form monolayer
tm = time to form a monolayer (in seconds)

n = number of molecules per unit volume

vrms = average velocity of the molecules

d = diameter of a molecule

For air at room temperature, we can express this as:

tm = 1.86 x 10-6 / P

where P is the pressure in torr.

pressure
tm
1 atm
2 x 10-9 sec
10-6 torr
2 seconds
10-9 torr
31 minutes


Vapor Pressure

in equilibrium, a certain pressure ot atoms (vapor pressure) will exist above solid surfaces

vapor pressure

Do not make high vacuum chambers out of Zinc. If you heat it to 200 C (476 K) the vapor pressure of Zn is 6 x 10-6 torr.

 


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