The US Particle Accelerator School Vacuum Fundamentals

TheThe USUS ParticleParticle AcceleratorAccelerator SchoolSchool VacuumVacuum FundamentalsFundamentals

Lou Bertolini Lawrence Livermore National Laboratory June 10-14, 2002

USPAS June 2002 Vacuum Fundamentals Page 1 Your Instructor

• Lou Bertolini —Mechanical Engineer —Accelerator Technologies Engineering Group − E-mail: [email protected] − Phone: (925) 422-9532 − FAX: (925) 423-1767

USPAS June 2002 Vacuum Fundamentals Page 2 Vacuum is a highly interdisciplinary subject

Mechanical Fluid Engineering Mechanics Physics

Simulation Heat Sciences Transfer Vacuum Electronics Technology Chemistry

Surface Science Molecular Thermodynamics Mathematics Metallurgy

USPAS June 2002 Vacuum Fundamentals Page 3 Kinetic behavior of gas molecules

TheThe behaviorbehavior ofof aa collectioncollection ofof gasgas moleculesmolecules inin aa vesselvessel isis dependentdependent uponupon thethe pressure,pressure, temperaturetemperature andand compositioncomposition ofof thethe gas.gas.

Velocity of gas molecules. Kr He

Mean free path

Impingement rate

USPAS June 2002 Vacuum Fundamentals Page 4 Gas Laws

Charles’ Law volume vs temperature

Boyle’s Law pressure vs volume

Combined or General Gas Law pressure vs temperature vs volume

Avogadro’s Law volume vs amount

Ideal Gas Law TheseThese lawslaws applyapply toto allall pressure vs temperature moleculesmolecules andand atomsatoms vs volume vs amount regardlessregardless ofof theirtheir sizesize

USPAS June 2002 Vacuum Fundamentals Page 5 Charles’s Law

TheThe volume volume of of a a fixed fixed amount amount of of gas gas at at a a fixed fixed pressurepressure will will vary vary proportionally proportionally with with absolute absolute temperature.temperature.

10

8 V V ∝ T = constant T 6

4

    Volume (liters)  V1   V2    =   2  T1   T2 

0 0246810 Temperature (C)

USPAS June 2002 Vacuum Fundamentals Page 6 Boyle’s Law

ForFor aa fixedfixed amountamount ofof gasgas atat aa fixedfixed temperature,temperature, itsits pressurepressure isis inverselyinversely proportionalproportional toto itsits volume.volume.

10

8 1 V ∝ PV = constant P 6

4

= Volume (liters) P1V1 P2V2 2

0 0246810 Pressure (Torr)

USPAS June 2002 Vacuum Fundamentals Page 7 Finding volume of a vessel with Boyle’s Law

V = ? liters V1 = 1.5 liters TC1 2 TC2

P1 = 760 Torr

Vacuum Vessel 1 Vacuum Vessel 2 P2 = 20 Torr

USPAS June 2002 Vacuum Fundamentals Page 8 Combined Gas Law

ProvidesProvides relationshiprelationship betweenbetween pressure,pressure, volume,volume, andand temperaturetemperature forfor aa fixedfixed amountamount ofof gas.gas.

PV = constant T P V P V 1 1 = 2 2 T1 T2

USPAS June 2002 Vacuum Fundamentals Page 9 Example of combined gas law

P1 = 100 Torr P2 = 200 Torr V1 = 200 liters V2 = 80 liters T1 = 293 K T2 = ? K

P V P V 1 1 = 2 2 T1 T2 (()100 Torr (()200) liters )(()200 Torr (()80) liters ) = 293 K T2 = T2 234 K

USPAS June 2002 Vacuum Fundamentals Page 10 Avogadro’s Law

TheThe VolumeVolume occupiedoccupied byby anyany gas,gas, atat aa fixedfixed temperaturetemperature andand pressure,pressure, isis proportionalproportional toto thethe numbernumber ofof molesmoles ofof thatthat gas.gas.

V ∝ n

23 No = Avogadros’ Number = 6.02 x 10 particles = 1 mole

USPAS June 2002 Vacuum Fundamentals Page 11 Ideal Gas Law

ProvidesProvides relationshiprelationship betweenbetween pressure,pressure, volume,volume, Temperature,Temperature, andand amountamount ofof gas.gas.

PV = nRT

R = 0.08206 Atm-liter/ K-mole = 62.36 Torr-liter/K-mole

USPAS June 2002 Vacuum Fundamentals Page 12 Units in Gas Law calculations

GasGas lawlaw calculationscalculations maymay bebe performedperformed usingusing aa varietyvariety ofof pressurepressure unitsunits (Torr,(Torr, Bars,Bars, ATM,ATM, PSI,PSI, Pa,Pa, etc.)etc.) butbut thethe pressurepressure unitsunits mustmust remainremain consistentconsistent throughthrough thethe calculation.calculation.

V P  Torr  t = ln 1     St P2 Torr

TemperatureTemperature mustmust bebe inin absoluteabsolute unitsunits (K,R).(K,R).

USPAS June 2002 Vacuum Fundamentals Page 13 Pressure unit conversions

To convert from: Multiply by: To:

Atm Torr 760

Pascal Torr 7.5 x 10-3

mBar Torr 0.750

PSI Torr 51.7

USPAS June 2002 Vacuum Fundamentals Page 14 Maxwell’s Distribution Law

MaxwellMaxwell determineddetermined thatthat forfor aa largelarge populationpopulation ofof oneone typetype ofof molecule,molecule, therethere wouldwould bebe aa distributiondistribution ofof velocities.velocities. ThereThere isis notnot bebe oneone uniformuniform velocity.velocity.

3  2   -mv  π    m 2 2kT N = 4 N  v2e dv v π2 kT 

= where Nvdv the number of molecules found in the velocity interval between v and v + dv v = velocity of the gas molecules (m/sec) N = the total number of gas molecules k = Boltzmann's constant (1.38 x 10-23 J/K) m = mass of the molecule (kg) T = Temperature (K)

USPAS June 2002 Vacuum Fundamentals Page 15 A Typical Maxwell Velocity Distribution

= vrms root mean square velocity 1 1 V p  kT 2  T  2 V ≈ 1.7  ∝   avg  m   M  Vrms = vavg average velocity of population ≈ 0.98vrms = vp most probable velocity ≈ 0.82vrms

USPAS June 2002 Vacuum Fundamentals Page 16 Velocity of gas molecules

TheThe velocityvelocity ofof gasgas moleculesmolecules isis independentindependent ofof thethe pressurepressure ofof thethe gas,gas, andand dependsdepends onlyonly onon thethe molecularmolecular weightweight ofof thethe gasgas andand itsits absoluteabsolute temperature.temperature.

T V = 1.455 x 10 4 M

V = average velocity (cm/sec) T = absolute temperature (K) M = molecular weight of gas, (g/mole)

USPAS June 2002 Vacuum Fundamentals Page 17 Mean free path

TheThe meanmean freefree pathpath isis thethe averageaverage distancedistance thatthat aa gasgas moleculemolecule cancan traveltravel beforebefore collidingcolliding withwith anotheranother gasgas molecule.molecule. Mean Free Path is determined by: - Size of molecule - Pressure - Temperature

USPAS June 2002 Vacuum Fundamentals Page 18 Mean Free Path Equation

λ i kT = π 2 2 Pd i

λ cm i = mean free path of gas species “i” ( /sec ) k = Boltzmann’s constant (1.38 x 10-23 Joules/K) T = Temperature (K) P = Pressure (Pa)

di = gas species diameter (cm)

USPAS June 2002 Vacuum Fundamentals Page 19 Gas Flow

TheThe flowflow ofof gasesgases inin aa vacuumvacuum systemsystem isis divideddivided intointo threethree regimes.regimes. TheseThese regimesregimes areare defineddefined byby specificspecific valuevalue rangesranges ofof aa dimension-lessdimension-less parameterparameter knowknow asas thethe KnudsenKnudsen numbernumber.. λ K = a n a λ a = mean free path a = characteristic dimension of flow channel (typically a pipe radius)

USPAS June 2002 Vacuum Fundamentals Page 20 Gas Flow

λ Viscous Flow : Kn = a < 0.01 a Turbulent Laminar

Transition Flow : 0.01 < Kn < 1.0

Molecular Flow : Kn > 1.0

Molecular

USPAS June 2002 Vacuum Fundamentals Page 21 Viscous Flow

• Viscous Flow – Molecules travel in Process Chamber uniform motion toward lower pressure – Random motion of a Rough molecule is influenced Valve in the direction of the mass flow High P – Molecular motion Rough “against” mass flow Line unlikely Rough Pump

Low P

USPAS June 2002 Vacuum Fundamentals Page 22 Molecular Flow

Process Chamber • Molecular Flow Molecules move randomly in either direction - to or away from rough pump and vacuum pump – Oil molecules will

Backstream (move up Chamber the rough line) in this flow regime

Oil

Pump

USPAS June 2002 Vacuum Fundamentals Page 23 Gas Flux Density incident on a Surface (impingement rate) υ

π 1 / 2  kT  = N   2 m  PN where N = o RT υ = density of molecules per m3 1 / 2 PN  kT  = 103 o   RT  2πm 

liters/m3 conversion

USPAS June 2002 Vacuum Fundamentals Page 24 Gas Flux Density (Impingement rate)

TheThe numbernumber ofof gasgas moleculemolecule collisionscollisions withwith thethe innerinner surfacesurface ofof aa containercontainer isis givengiven byby:: P υ = 3.5 x 1022 MT

u = particle flux density (molecules/sec cm2) P = pressure (Torr) M = molecular weight of gas (g/mole) T = absolute temperature (K)

USPAS June 2002 Vacuum Fundamentals Page 25 Residence time

ResidenceResidence timetime isis thethe averageaverage amountamount ofof timetime aa moleculemolecule staysstays onon aa surface,surface, andand isis aa functionfunction ofof thethe molecularmolecular weightweight ofof thethe gasgas andand thethe temperaturetemperature ofof thethe surface.surface.

   Q   RT  t = toe

where tO = residence time (sec) Q = activation energy T = temperature (K)

USPAS June 2002 Vacuum Fundamentals Page 26 Vapor Pressure

Psat Pv < Psat

In this case, the gas above the In this case, there is not sufficient bulk bulk is considered to be saturated. material to support a saturation vapor pressure

• Vapor pressure is an important concept as it relates to cryopumps. • At thermal equilibrium the net flux of particles at the surface of bulk material is zero. • Each element has a specific saturation pressure for a given temperature. The saturation pressure is also referred to as the saturation vapor pressure, equilibrium vapor pressure, or just vapor pressure.

USPAS June 2002 Vacuum Fundamentals Page 27 Vapor Pressure Curve

USPAS June 2002 Vacuum Fundamentals Page 28 Pumping Speed

DefinedDefined asas aa measuremeasure ofof volumetricvolumetric displacementdisplacement (liters/sec,(liters/sec, cu.feet/minute,cu.feet/minute, cu.meters/minute)cu.meters/minute)

• It is the volume of gas flowing past a point per unit time.

• Pumping speed is independent of pressure.

• Pumping speed is an abstract concept used to describe the behavior of gas in a vacuum system. dV S = dt

USPAS June 2002 Vacuum Fundamentals Page 29 Throughput

DefinedDefined asas aa measuremeasure ofof gasgas flowflow raterate (Torr(Torr liters/sec) liters/sec)

• For constant pumping speed throughput varies with pressure: Qleak d(()PV ) Q = = SP Q dt outgas

• Under dynamic conditions: d(()PV ) = Q - S(()P - P ) Q dt o Pump

USPAS June 2002 Vacuum Fundamentals Page 30 Conductance

DefinedDefined asas aa measuremeasure ofof easeease withwith whichwhich abstractabstract volumesvolumes cancan passpass fromfrom oneone placeplace inin aa vacuumvacuum systemsystem toto another.another.

• Conductance is an abstract concept used to describe the behavior of gas in a vacuum system.

• Conductance is specific to a particular geometrical configuration.

• Conductance is specific to the actual gas species and temperature.

• When the mean free path of a gas species in a system is less than the dimensions of the system the conductance is pressure dependent.

USPAS June 2002 Vacuum Fundamentals Page 31 Conductance under Molecular Flow

Aperture T C = 3.64 A (liters/sec) M where A = the aperture area (cm2) T = Temperature (K) M = molecular weight (grams/mole)

Long Tubes (L > 10D) T D3 C = 3.81 (liters/sec) M L where D = pipe diameter (cm) L = pipe length (cm)

USPAS June 2002 Vacuum Fundamentals Page 32 Electrical Analogy (Ohm’s Law)

A vacuum system operating in the molecular flow regime can be thought of in terms of an electrical circuit.

Ve G =1/R I = GVe

I

C P Q = C(P1 –P2) 1 P2 Q

USPAS June 2002 Vacuum Fundamentals Page 33 Electrical Analogy (continued) Conductances in Series

G1 =1/R1 G2 =1/R2 G3 =1/R3 1 1 1 1 = + + Ge G1 G2 G3

C 1 = 1 + 1 + 1 C1 2 C3 CTotal C1 C2 C3

USPAS June 2002 Vacuum Fundamentals Page 34 Electrical Analogy (continued) Conductances in Parallel

G1 =1/R1 G =1/R 2 2 = + + Ge G1 G2 G3 G3 =1/R3

C1 = + + CTotal C1 C2 C3 C2

C3

USPAS June 2002 Vacuum Fundamentals Page 35 Dalton’s Law

TheThe partialpartial pressurespressures ofof gasesgases inin aa mixturemixture behavebehave independentlyindependently accordingaccording toto thethe idealideal gasgas laws.laws.

O P = P + P + P + P + . . . . . + P Total N2 O2 Ar C 2 n

QN QO Q QOC Q P = 2 + 2 + Ar + 2 . . . . . + n Total S S S S S N2 O2 Ar CO2 n

USPAS June 2002 Vacuum Fundamentals Page 36 Variable Pumping Speed

AllAll pumpspumps havehave bothboth highhigh andand lowlow applicableapplicable pressurepressure limits.limits.

140

• Base or blank-off pressure 120 (PB) is the minimum pressure a pump will achieve 100

80

• At base pressure pumping 60 speed is zero

40 Pump Speed (lps)

 P  20 S = S 1 - B  max 0   10 -5 0.0001 0.001 0.01 0.1 1 10 P Pressure (Torr)

USPAS June 2002 Vacuum Fundamentals Page 37 Adsorption and desorption (outgassing)

• Adsorption is the arrival of gas molecules on a surface —Adsorbed gas molecules exist as molecular layers and in some ways behave like a sheet of liquid —Rule of thumb – one monolayer consists of ~1015 molecules (atoms) per cm2

• Residence time is the amount of time a gas molecule stays on a surface

• Desorption is the departure of gas molecules from a surface —The rate of desorption is a function of the activation energy of the sorbent and the temperature of the surface

USPAS June 2002 Vacuum Fundamentals Page 38 Adsorbed gases affect material properties

• 1 monolayer of adsorbed gas influences bonding, wettability, and surface chemical reactions

• 1 to 10 monolayers of adsorbed gas affect lubrication and electrical conduction

• 1 to 200 monolayers of adsorbed gas change the absorption of light by a surface

• 200 to 2000 monolayers affect the visual color of surfaces

USPAS June 2002 Vacuum Fundamentals Page 39 Permeation is the transfer of a fluid through a solid

Atmosphere • Material combination (fluid & solid)

• Temperature

• Permeation thickness

• Area Vacuum • Pressure differential

USPAS June 2002 Vacuum Fundamentals Page 40