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1.2 Low Temperature Properties of Materials

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1.2 Low Temperature Properties of Materials 1.2 Low Temperature Properties of Materials Materials properties affect the performance of cryogenic systems. Properties of materials vary considerably with temperature Thermal Properties: Heat Capacity (internal energy), Thermal Expansion Transport Properties: Thermal conductivity, Electrical conductivity Mechanical Properties: Strength, modulus or compressibility, ductility, toughness Superconductivity Many of the materials properties have been recorded and models exist to understand and characterize their behavior Physical models Property data bases (Cryocomp®) NIST: www.cryogenics.nist.gov/MPropsMAY/material%20properties.htm What are the cryogenic engineering problems that involve materials? USPAS Cryogenics Short Course Boston, MA 6/14 to 6/18/2010 1 Cooldown of a solid component Cryogenics involves cooling things to low temperature. Therefore one needs to understand the process. If the mass and type of the object and its material are known, then the m Ti = 300 K heat content at the designated temperatures can be calculated by integrating 1st Law. ~ 0 dQ Tds= = dE+ pdv The heat removed from the component is equal to its change of T = 80 K m f internal energy, ⎛ Ti ⎞ EΔ m = ⎜ CdT⎟ ⎜ ∫ ⎟ ⎝ T f ⎠ Liquid nitrogen @ 77 K USPAS Cryogenics Short Course Boston, MA 6/14 to 6/18/2010 2 Heat Capacity of Solids C(T) General characteristics: The heat capacity is defined as the change in the heat content with temperature. The heat capacity at constant volume is, 0 300 ∂E ∂s T(K) Cv = and at constant pressure,CTp = ∂T v ∂T p 3rd Law: C 0 as T 0 rms of the heat capacity are oThese two f related through the following thermodynamic relation, 2 ∂v ⎞ ∂p ⎞ Tvβ 2 1 ∂v ⎞ 1 ∂v ⎞ CCT− = − = κ= − ⎟ β= − ⎟ p v ⎟ ⎟ v ∂p ⎟ v ∂T ∂T ⎠p∂v ⎠ T κ ⎠T ⎠ p Isothermal Volume Note: C –C is small except for p v compressibility expansivity gases, where ~ R = 8.31 J/mole K. USPAS Cryogenics Short Course Boston, MA 6/14 to 6/18/2010 3 Heat Capacity of Solids (Lattice Contribution) Lattice vibration (Phonon) excitations are the main contribution to the heat capacity of solids at all except the lowest temperatures. h Internal energy of a phonon gas is given by E = ωd ω D()() ω n ω ph 2π ∫ D(ω) is the density of states and depends on the choice of model n(ω) is the statistical distribution function 1 n()ω = h = Planck’s constant = 6.63 x 10-34 J.s hω k = Boltzmann’s constant = 1.38 x 10-23 J/K e 2kπ B T −1 B Debye Model for density of states Constant phonon velocity Maximum frequency = ωD Debye temperature: ΘD = hωD/2πkB USPAS Cryogenics Short Course Boston, MA 6/14 to 6/18/2010 4 Debye Internal Energy & Heat Capacity In Debye mdel the internal enoergy and heat capacity have simple forms -l 3 K ⎛ T ⎞ xD x3 E= 9 RT⎜ ⎟ dx ph ⎜ ⎟ ∫ x ⎝θD ⎠ 0 e −1 3 ⎛ T ⎞ xD x4 ex CR= 9 ⎜ ⎟ dx ph ⎜ ⎟ ∫ x 2 ⎝θD ⎠ 0 ()e −1 where x = hω/2πkBT and xD = ΘD/T Limits: T > Θ , (Dulong-Petit value) D CRph ≈ 3 3 ⎛ T ⎞ T << ΘD, ⎜ ⎟ CRph ≈ 234 ⎜ ⎟ ⎝θ D ⎠ 8.31 J/mole K (gas constant) Example R= N0 B k = USPAS Cryogenics Short Course Boston, MA 6/14 to 6/18/2010 5 Values of Debye Temperature (K) 1 2 Metals ΘD Non-metals & Compounds ΘD Ag 225 C (graphite) 2700 Al 428 C (diamond) 2028 Au 165 H2 (solid) 105-115 Cd 209 He (solid) 30 Cr 630 N2 (solid) 70 Cu 343 O2 (solid) 90 Fe 470 Si 630 Ga 320 SiO2 (quartz) 255 Hf 252 TiO2 450 Hg 71.9 In 108 The Debye temperature is normally determined Nb 275 by measurements of the specific heat at low Ni 450 temperature. For T << ΘD , 3 Pb 105 ⎛ T ⎞ ≈ 234RC ⎜ ⎟ Sn 200 ph ⎜ ⎟ ⎝θ D ⎠ Ti 420 1 Kittel, Introduction to Solid State Physics V 380 2 Timmerhaus and Flynn, Cryogenic Process Zn 327 Engineering Zr 291 USPAS Cryogenics Short Course Boston, MA 6/14 to 6/18/2010 6 Electronic Heat Capacity (Metals) T = 0 f(ε) The free electron model treats electrons as a gas of particles obeying Fermi-Dirac T > 0 statistics E=ε d ε( D ε) ( ) f ε e ∫ ε εφ Where 2 1 h2 N 3 f ()ε = ⎛ 2 ⎞ ε− εf ε f ≡ 2 ⎜3π ⎟ k T 8π m ⎝ V ⎠ e B +1 3 2 2 V⎛ 8π m⎞ 1 and D ε = ⎜ ⎟ ε 2 () 2 ⎜ 2 ⎟ 2π ⎝ h ⎠ 4 At low temperatures, T << εf/kB ~ 10 K 1 C≈π D2 ( ε) k2 = T γ T e 3 f B The electronic and phonon e heat thcontributions to capacity of copper are approximately equal at 4 K USPAS Cryogenics Short Course Boston, MA 6/14 to 6/18/2010 7 Summary: Specific Heat of Materials General characteristics: Specific heat decreases by ~ 10x between 300 K and LN2 temperature (77 K) Decreases by factor of ~ 1000x between RT and 4 K Temperature dependence C ~ constant near RT C ~ Tn, n ~ 3 for T < 100 K C ~ T for metals at T < 1 K USPAS Cryogenics Short Course Boston, MA 6/14 to 6/18/2010 8 Thermal Contraction All materials change dimension with temperature. The expansion coefficient is a measure of this effect. For most materials, the expansion coefficient > 0 T1 T~2 <300 T1 K D D-ΔD ΔL Linear expansion coefficient 1∂L ⎞ 1 α = ⎟ = β For isotropic L ∂T ⎠ p 3 materials Bulk expansivity (volume change): 1 ∂v ⎞ 1 ∂ρ ⎞ β = ⎟ = − ⎟ v ∂T ⎠ p ρ ∂T ⎠ p Expansivity caused by anharmonic terms in the lattice potential USPAS Cryogenics Short Course Boston, MA 6/14 to 6/18/2010 9 Temperature dependence to α and β Most materials contract when cooled The magnitude of the effect depends on materials Plastics > metals > glasses Coefficient (α) decreases with temperature (2006)From Ekin Thermodynamics: Tvβ 2 CC−p = v →0 κ T →0 ∂v ⎞ ∂s ⎞ ⎟ = − ⎟ → 0 ⎟ T →0 ∂T ⎠ p ∂p ⎠T (Third law: s 0) USPAS Cryogenics Short Course Boston, MA 6/14 to 6/18/2010 10 Expansion coefficient for materials USPAS Cryogenics Short Course Boston, MA 6/14 to 6/18/2010 11 Thermal contraction of supports The change in length of a support is determined by the TH temperature distribution along the support Tabulated ΔL/L values are for A L ΔL (greatly exaggerated) uniform temperature of m support m TL For non-uniform temperature TL L TH L LΔ dx = α() T dT Actual T(x) ∫∫ x 0 T() x Where T(x) is defined according to, T T H ⎛ x ⎞ H TL TH k T() dT= ⎜ k⎟ T() dT ∫L ∫ Tx ⎝ ⎠TL USPAS Cryogenics Short Course Boston, MA 6/14 to 6/18/2010 12 Thermal stress in a composite A2 A1 L Assumptions: Force balance Composite is stress free at T 0 FAFA1= 1σ 1= 2= σ 2 2 Materials remain elastic: σ = Eyε ΔL ΔL ε1 = − No slippage at boundary L 1 L composite Ends are free ΔL ΔL ε = − 2 L L ⎡ ⎤ composite 2 ⎢ ΔL − ΔL ⎥ L L σ=EE ε = ⎢ 1 2 ⎥ 1 1y 1y ⎛ 1 EA ⎞ ⎢ 1+ 1y 1 ⎥ ⎢⎜ EA⎟⎥ ⎣⎝ y2 2⎠⎦ USPAS Cryogenics Short Course Boston, MA 6/14 to 6/18/2010 13 Electrical conductivity of materials The electrical conductivity or resistivity of a material is defined in terms of Ohm’s Law: V = IR V L I R = dV/dI V A I Heat generation by electrical conduction Q = I2R Conductivity (σ) or resistivity (ρ) are material properties that depend on extrinsic variables: T, p, B (magnetic field) ρL L R == A σA USPAS Cryogenics Short Course Boston, MA 6/14 to 6/18/2010 14 Instrumentation leads An instrumentation lead usually carries current between room temperature TH and low temperature Material selection (pure metals vs. alloys) L A Lead length determined by application Optimizing the design m Leads are an important component and TL should be carefully designed Often one of the main heat loads to Instrumentation lead the system Poor design can also affect one’s ability to make a measurement. USPAS Cryogenics Short Course Boston, MA 6/14 to 6/18/2010 15 Cryogenic temperature sensors A temperature sensor is a device that has a measurable property that is sensitive to temperature. Measurement and control of temperature is an important component of cryogenic systems. Resistive sensors are frequently used for cryogenic temperature measurement V RT()= I T Knowing what temperature is being measured is often a challenge in cryogenics (more later) USPAS Cryogenics Short Course Boston, MA 6/14 to 6/18/2010 16 Electrical Conductivity in Metals Resistivity in metals develops from two electron scattering mechanisms (Matthiessen’s rule) Electron-phonon scattering, T > ΘD Scattering probability ~ mean square displacement due to thermal 2 motion of the lattice: <x > ~ kBTand ρ ~ T Electron-defect scattering (temperature independent), T << ΘD Scattering depends on concentration of defects ρ ~ ρ0 ~ constant; Residual Resistivity Ratio (RRR = ρ(273 K)/ρ0) indication of the purity of a metal Intermediate region, T ~ ΘD/3 ρ ~ (phonon density) x (scattering probability) ~ T5 Scattering process: “l “ is the mean free path l τ is the mean scattering time = l/v f USPAS Cryogenics Short Course Boston, MA 6/14 to 6/18/2010 17 Resistivity of Pure Metals (e.g. Copper) dR/dT ~ constant Residual Resistivity R ~ constant USPAS Cryogenics Short Course Boston, MA 6/14 to 6/18/2010 18 Kohler plot for Magneto-resistance The resistance of pure metals increases in a magnetic field due to more complex electron path and scattering (Why?) How to use a Kohler plot? Copper 1.
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