1
Vacuum Technology
Page 1
Kinetic Theory of Gas
Vacuum Technology
Page 2
Vacuum Basics
Gas Volume % Pressure (Pa)
N
278 79,117
O
221 21,233
CO
2
0.033 33.4
Ar 0.934 946.4
Atmospheric Pressure = 101,323.2 Pa (760 torr)
(133Pa = 1 torr)
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Kinetic Theory of Gas - Thin Film Materials Processing - Lecture Slides, Slides of Material Engineering
These are the Lecture Slides of Thin Film Materials Processing which includes Vaporization, Vapor Pressure Curves, Thermal Desorption, Molecular Binding Energy, First Order Desorption, Desorption Rate, Real Surfaces, Diffusion of Gas Particles etc. Key important points are: Kinetic Theory of Gas, Vacuum Basics, Atmospheric Pressure, Gas Properties, Velocity Distribution, Maxwell Boltzmann Distribution, Mass Dependencies, Average Particle Velocity, Peak Velocity
Typology: Slides
2012/2013
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Page 1
Kinetic Theory of Gas
Vacuum Technology
Page 2
Vacuum Basics
Gas Volume % Pressure (Pa)
N 2 78 79,
O 2 21 21,
CO 2 0.033 33.
Ar 0.934 946.
Atmospheric Pressure = 101,323.2 Pa (760 torr) (133Pa = 1 torr)
Page 3
Vacuum Basics
Vacuum Pressure Range
(Pa)
Low 10
5
> P > 3.3x
3
Medium 3.3x
3
> P >
High 10
> P > 10
Very High 10
> P > 10
Ultra High 10
>P>
Extreme Ultra-
high
> P
Vacuum Technology
Page 4
Kinetic Picture of an Ideal Gas
- Volume of gas contains a large number of molecules
- Adjacent molecules are separated by distances that are large relative to the individual diameters
- Molecules are in a constant state of motion
- All directions of motion are possible (3-dimensions)
- All speeds are possible (though not equally probable)
- Molecules exert no force on each other except when they collide
- Collisions are elastic (velocity changes and energy is conserved)
Page 7
Temperature/Mass Dependencies
- Temperature Dependence • Molecular Mass Dependence
Vacuum Technology
Page 8
- Average particle velocity (Maxwell-Boltzmann)
- ↑Temperature, ↓ mass -- ↑ average particle velocity
Basic Expressions from Maxwell Boltzmann Distribution
m massofparticle
T Temperature
K Boltzman'sConstant
averagevelocity
1
where
m
KT
Page 9
Basic Expressions from Maxwell Boltzmann Distribution
- Peak Velocity (set first derivative of distribution = 0)
- Root Mean Square Velocity
- Maxwell-Boltzmann Statistics
- v (^) avg = 1.128v (^) p and v (^) rms = 1.225v (^) p
m
kT v (^) p
1 / 2
m
kT
v rms
Vacuum Technology
Page 10
Maxwell-Boltzmann Velocities
0
0 200 400 600 800 1000 1200 Velocity (m/s)
dn/dV
dN/dV peak average RMS
Page 13
For molecules traveling with velocity{Vx}, the distance they can travel in time interval ∆ t is: {Vx} ∆ t If they move towards the wall of area A and the number density is n (=N/V), the number of molecules that strike the wall in time ∆ t is: n•A{Vx} ∆ t, but half of the molecules move towards the surface, half away from the surface: (1/2)n•A{Vx} ∆ t When a molecule collides with the surface, it’s momentum changes from mV (^) x to -mVx (total 2mVx) (m=MW/N (^) A), hence the total momentum change is: = [(number of collisions)] • (momentum change per collision) = [(1/2)n•A{Vx} ∆ t] • (2m{Vx}) = n•m•A{Vx^2 } ∆ t
Pressure and Molecular Velocity
Vacuum Technology
Page 14
Since force is the rate of change of momentum: f = n•m•A{Vx^2 }
Pressure is the force per unit area: P = n•m•{Vx^2 }
Generalizing: {V^2 }= {Vx^2 } + {V (^) y^2 } + {Vz^2 } = 3 {Vx^2 }, P = (1/3)n•m{V^2 }
Generally VRMS is used here
1 atm = 1013 mbar = 1.013 bar = 760 mmHg 1 atm = 760 torr = 101,325 Pa = 101,325 Nm -
Pressure and Molecular Velocity
3
m
kT v (^) rms
P=nkT (where n=N/V)
Dr. Philip D. Rack Page 15
The time a molecule spends between collisions is 1/ Z.
A molecule of diameter “do ” sweeps out a collision cylinder of cross-sectional area: σ = πd 02 , and length {V}∆t, during period ∆t. For two colliding objects we must really take into account their relative speeds (not one fixed, one moving). The collision frequency Z (per unit time) per molecule is = √ 2 • σ{V}•n
Collision Frequency
Vacuum Technology
Page 16
↑ Pressure (↑ particle density) -- ↓ mean free path
The Mean Free Path
Mean free path (l) - average distance a particle travels before it collides with another particle:
n gas particle density
moleculardiameter
:
2
1 (^122)
=
=
=
o
o
d
where
π d n
λ
( )
- 6 ( ) P Pa
λ mm =
(for air at room temperature)
Page 19
Monolayer Formation Times
The inverse of the Gas impingement rate (or flux) is related to the Monolayer coverage time (t (^) c). If a surface has ~ 10 15 sites/cm^2
At 300K and 1 atm, if every Nitrogen molecule that strikes the surface remains absorbed, a complete monolayer is formed in about t = 3 ns. If P = 10 -3^ torr (1.3 x 10 -6^ atm), t = 3x10 -3^ s If P = 10 -6^ torr (1.3 x 10 -9^ atm), t = 3 s If P = 10 -9^ torr (1.3 x 10 -12^ atm), t = 3000 s or 50 minutes
t (^) c = 10^15 /s Γ, where S is the sticking coefficient Γ is the particle flux
Requirement for Experiment in Vacuum: Clean surface quickly becomes contaminated through molecular collision, ∴ p must be less than about 10 -12^ atm (7.67x10 -5^ torr). 10 -10^ to 10 -11^ torr (UHV-ultra high vacuum) is the lowest pressure routinely available in a vacuum chamber.
Vacuum Technology
Page 20
Page 21
Boyle’s Law (1622)
• P∝1/V (T and N constant)
P
V
Vacuum Technology
Page 22
Amontons’ Law (1703)
• P∝T (N and V constant)
T
P
Page 25
Avagadro’s Law (1811)
• P∝N (T and V constant)
N
P
Vacuum Technology
Page 26
Low Pressure Properties of Air
Page 27
Gas Transport Phenomena
- Viscosity -- due to momentum transfer via molecular collisions (development of a force due to motion in a fluid)
thispositionbetwen thetwosurfaces
rateofchangeofthegasvelocityat dy
dU
coefficientofviscosity
A surfaceareain x-zplane
forcein x-direction
:
xz
=
=
=
=
=
η
η
x
xz
x
F
where
dy
dU A
F
z
y
x
Moving Surface U 1
U
Fixed Surface
Axz
2
1
U 1 < U 2
Vacuum Technology
Page 28
Gas Transport Phenomena
- Viscosity
- Kinetic Theory
- More Rigorous Treatment
η nm νλ 3
η = 0. 4999 nm νλ
(wheny )
(^322)
(^12) λ π
η = ≥ d o
mkT
Viscosity ∝ (mT) 1/2^ and d (^) o^2 and independent of P (only true for y ≥ λ)
Page 31
Gas Transport Phenomena
- Heat Flow (y ≥ λ)
z
y
x
Hot Surface (T 2 )
T
Cold Surface (T 1 )
Axz
2
1
dy temperaturegradient T 1 < T 2
dT
specificheatatconstant volume
K heatconductivity c
H heatflow
:
v
=
=
= =
=
=
c v
where
dy H AKdT
η
Heat Flow ∝ (mT) 1/2^ and d (^) o^2 and independent of P (only true for y ≥ λ)
Vacuum Technology
Page 32
Gas Transport Phenomena
- Heat Flow (y ≥ λ) more detailed analysis of K (cf slide #31) - Simplified - Detailed
K = η c v
specificheatatcostant vo lume
specificheatatcostantpressure
c
c
:
( 9 5 ) 4
1
v
P
=
=
=
= −
v
p
v
c
c
where
K c
γ
γ η
333 (triatomic molecule)
667 (monatomicmolecule)
4 (diatomicmolecule) =
=
= γ
γ
γ
Page 33
Gas Transport Phenomena
- Heat Flow (λ >> y)
andtransferenergy)
(howeffectivethemoleculesabsorb
free-molecularheatconductivity
absorbenergy)
(howeffectivethesurfacestransferand
accomodationcoefficient
heatflow
:
( )
0
0 2 1
Λ =
=
=
= Λ −
α
α
E
where
E PT T
Heat Flow ∝ Pressure
Vacuum Technology
Page 34
Gas Transport Phenomena
d λ < d
d λ > d
Heat Flow controlled by particle-particle collisions
Heat Flow controlled by particle-wall collisions
Page 37
Gas Transport Phenomena
- Diffusion
xy
z
2N molecules
x-y plane at t=0, z=
z= -dz
+dz
betweenzandz dz
numberofmoleculeslocated
:
( )
( 4 ) (^12)
2
=
= −
dn
where
e Dt dn Ndz z Dt π
∞
=
=
locatedbetweenz and
fractionofmoleculesthatare
:
2 ( )
0
(^12) 0
f
where
Dt
f erfc z
moleculeshavediffusedafteratime t
minimumdistancethat10%ofthe
:
- 32 ( )
0
0 12
=
z
where
z Dt
Vacuum Technology
Page 38
Gas Transport Phenomena
d λ < d
d λ > d
Diffusion controlled by particle-particle collisions
Diffusion controlled by particle-wall collisions