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INTRODUCTION

SOLID STATE THEORY

PARTICULATE MODEL 1-D THERMAL MODEL

Low Temperature Thermal Properties of Lunar Soil

R. Woods-Robinson¹, D. A. Paige¹

¹Department of Earth, Planetary and Space Sciences, UCLA

From Apollo laboratory tests, thermal and material properties of lunar regolith soils are well understood for temperatures down to approximately 100K (Fig. 1). Recent Diviner measurements reveal polar areas in permanent shadow where temperatures may be as low as 20K (Fig. 2). These

Figure 2. LRO Diviner thermal image of the south polar region of the moon showing brightness temperatures as low as ~30K.

Figure 5: A visual of the Debye relation k  =  CV  v  l    with (a) mean free path7, (b) specific heat7 and (c) thermal conductivity8 as functions of temperature for crystalline (blue) and amorphous (red) SiO2.

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ThermalConductivityHWê

mKL(a) Mean free path l(T) (b) Specific heat CV(T) (c) Thermal conductivity k(T)

Thermal Properties of “Generic” Model: Solid SiO2

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Phonon

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Ø  Heat Capacity CV = energy storage capability in phonon vibrational modes (specific heat = per unit mass) Ø  Thermal conductivity (TC) k = ability of phonons to propagate through material Ø  Mean free path (mfp) l  = inverse of scattering probability Ø  Sound velocity v  = speed of phonon propagation

10084 12001 14163 15020* 68501 Agglutinate 52% 45.7% 55.2% 36.7% Basalt 25.1 2.8 5.7 0.3 Breccia 8.3 31 1 39.8 Pyroxene 61.8 2.6 19.5 2.9 Plagioclase 2.3 15.7 8 9.3 14.8 Opaques 1.1 9.5 0.2 0.1 Glass 6.6 12.9 10 8.5 5.5

Dominant Modal Mineralogy of Lunar Soil

THERMAL CONDUCTIVITY

SPECIFIC HEAT

DISCUSSION

Figure 1. Constituent minerals in the lunar regolith soils used in this study1

!

1. Solid state: Consider both specific heat ! and TC !

as functions of their solid constituents for ! → (!,!): a. Specific heat9: !!"#$ ∝ !!!!!!!"#$%! b. TC10: !!"" = !! + !! ! !! ∝ ! !!!!!!!"#$%!

(Note !!: volume content, homogeneous material: !!!!! → !!"#$%)

2. Two-phase model: due to incomplete data, simplify solid components (to first order) as linear combination of the two “generic” solids, crystalline and amorphous SiO2, such that

!!"#$% = !!!" + (!− !)!!"

3. Amorphous assumption: assume amorphous

component dominates (! = !, !!"#$% → !!") because of small particle size and glass-like TC behavior of feldspar crystals

4. Damping: introduce damping coefficients or functions where necessary ! dC, !(!), !(!)

5. Application: Apply this model to lunar soil (for TC, first to particulate basalt lunar simulant then to lunar soil) to solve for damping functions

6. Crystalline sensitivity: Adjust x to observe potential effects of crystallinity temperatures are lower than any measured

data. Therefore, existing models based off fits to this data may be inaccurate (Fig. 3).  

In this study we investigate solid state theory and analog materials to derive a physical-based model of specific heat and thermal conductivity (TC) in lunar soils in the range 20-100K.

REFERENCES

Figure 6: Revised estimate of the specific heat of lunar soil. Estimate is valid in range 10-400K."

Figure 7: Sensitivity analysis to various generic crystalline components (x) (left) compared to specific heat of a basaltic meteorite11 and its various

mineral components (right)"

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ksoil (T,ρ, p, x) = a(ρ)T3 + b(ρ)(1− p) xkcr (T )+ (1− x)kam (T )[ ]

Radiative component kr" Conductive component kc"

Bulk Density (ρ) dependent damping functions" Crystalline TC" Amorphous TC"Porosity"

Crystalline volume fraction"

csoil (T, x) = dC xccr (T )+ (1− x)cam (T )[ ]Damping!

coefficient"Crystalline

volume fraction"Crystalline c" Amorphous c"

Figure 10: Sensitivity analysis to (a) various porosities and (b) various generic crystalline components x compared to © thermal conductivity of various meteorites12 (right)"

Figure 9: New TC fit at ρ = 1300 kg/m3, x = 0 and p = 1 – ρ/ρbasalt"plotted against TC data from five different Apollo missions"

Lunar Soil, ρ = 1300 kg/m3

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Apollo 11 soil 10084 Apollo 12 soil 12001 Apollo 14 soil 14163* Apollo 15 soil 15013 Apollo 16 soil 68501

Basalt fit, ρ = 1300 kg/m3 Vasavada’s fit to Apollo

12 soil, ρ = 1300 kg/m3

Figure 8: The thermal conductivity of particulate basalt12 fit to Watson’s equation (dotted) as in original publication and fit to the

above equation (solid) at x = 0 and p = 1 – ρ/ρbasalt"

Lunar Simulant: Particulate Basalt

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1500 kg/m3, old, new 1300 kg/m3, old, new 1130 kg/m3, old, new 980 kg/m3, old, new 880 kg/m3, old, new Apollo 12 soil, ρ = 1300 kg/m3

Vasavada’s fit to Apollo 12 soil, ρ = 1300 kg/m3

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Temperature HKL

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Specific Heat of Lunar Soil

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Apollo 11 soil 10084 Apollo 14 soil 14163 Apollo 15 soil 15013 Apollo 16 soil 68501 Breccia rock 10046 (Morrison & Norton 1970)

New fit to soil, x = 0 Old fit (Ledlow 1997)

x = 0 x = 0.05 x = 0.1 x = 0.5 x = 1 Apollo 11 soil 10084 Old fit (Ledlow 1997)

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10084 12001 14163* 15013 6850

Vasavada’s fit to Apollo 12 soil, ρ = 1300 kg/m3

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x = 0 (Basalt fit) x = 0.001 x = 0.002 x = 0.005 x = 0.01 x = 0.02 Apollo 11 soil 10084 Apollo 12 soil 12001 Vasavada’s fit to Apollo 12 soil, ρ = 1300 kg/m3

(b)" (c)"

Sensitivity Analyses

Sensitivity Analysis

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a) Goal: Assess the effects of new thermal properties calculated surface and subsurface soil temperatures b) Assumptions: Standard 2-layer density (ρ=1300 depths 0-2 cm, ρ=1800 below)5

c) Runs: 2 standard cases: i) Equator at noon ii) Simulated polar conditions à same as (i) but w/ solar constant 0.25 W/m2 d) Results: Equatorial temps not affected, polar surface temps show significantly enhanced diurnal variability w/ shallower depth penetration

Figure 11: Comparison of the old and new fits input into the Diviner two-layer depth-temperature thermal model."

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Figure 4: TC in (a) crystalline, (b) impure crystalline &

(c) amorphous solids7

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General Implications • Introduces new approach to estimate thermal parameters • Consistent w/ old models at high temps, but at low temps suggest higher

specific heat values (2x at 20K) & lower TC values (5x at 20K) • Lab measurements suggest various Apollo samples do indeed have

different thermal properties ! observe w/ Diviner (see below) • TC fits best for Apollo 11 & 12 soils, others need further analysis • Specific heat slightly higher than constituent crystals ! additional

vibrational modes in particulates? Validity of the Solid Two-Phase Model • Specific Heat: accurate trend to first order, may differ by material • Thermal Conductivity (Fig. 5c)

• Generic amorphous component kam " Relatively good estimator " Accurate if indeed amorphous effects dominate

• Generic crystalline component kcr " Less accurate: peak amplitude & location vary " If high crystallinity, need more thorough model

• kcr & kam orders different at low temps ! constituent crystalline variation most significant at low temps, where there is no data

• Non-ideal particles: more complex than linear combination? Density, Porosity & Particle Size • kc increases w/ bulk density in range of interest ! consistent w/ literature • Porosity “damping” may affect both kc & kr, needs investigation • Range in particle size distribution could account for TC variations;

suggested that increasing grain size ! kr decreases, kc increases Uncertainties • Statistical: only one source of TC data used in calibrations • Application of basalt data to lunar soil? • Differing methodologies in data used, accuracy debated in literature • Different fitting methods possible, unclear which is “best” fit Future Work ! Thorough mineralogy analysis, incorporate dominant modes ! Decoupling & incorporation of particle size, stress, emissivity, etc. ! Synthesis of other TC models and particulate data from the literature ! TC variation observations from Diviner diurnal temperature variations ! Low temperature measurements to confirm low temperature behavior of

particulate materials

1 Meyer, C. (2005). NASA STI/Recon Technical Report N, 6, 11039. 2 Robie, R., Hemingway, B., & Wilson, W. (1970). Science, 0–1. 3 Ledlow et al. (1992). The Astrophysical Journal, 384, 640-655. 4 Cremers, C., & Hsia, H. (1974). Proceedings of the Fifth Lunar Conference, 3, 2703–2708. 5 Vasavada, A. R. et al. (1999). Icarus, 141(2), 179-193. 6 Watson, K. (1964). Unpublished PhD Thesis. California Institute of Technology. 7 Cahill, D., & Pohl, R. (1988). Annual Review of Physical Chemistry, 39(1), 93–121. 8 Touloukian, Y.S. & Ho, C.Y. (1970). Thermophysical properties of matter-The TPRC Data Series, 1. 9 De Vries, D. (1963). Physics of plant environment. 10 Merrill, R. B. (1969). National Aeronautics and Space Administration. 11 Opeil et al. (2012). Meteoritics & Planetary Science, 47(3), 319-329. 12 Fountain, J & West, E. (1970). Journal of Geophysical Research, 75(20), 4063–4069.

§  Best fit w/ x = 0, a(ρ) = 1.3 × 10-11, b(ρ) = 9.9 × 10-4 + 9.2 × 10-10 ρ2, p = 1 – ρ/2800"

§  Range: T = 10-400 K, ρ = 1000-2000 kg/m3 §  Degree of uncertainty is uncertain

100 101 10210-610-510-410-310-210-1100

Temperature HKL

SpecificHeatHJê

gKL

Amorphous SiO2 Crystalline SiO2

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ii)  

§  Varying x à defines uncertainty

§  Meteorites à suggest composition not the full story

§  Best fit w/ x = 0, dC = 1.04 §  Piece-wise function: new

model ~10-200K, old model ~100K+

!Solid Lunar Ex. CV k

Pure Crystalline

Pyroxene, breccia, feldspar

• Very low temp: T3 (Debye) • Very high temp: constant (Dulong-Petit) • Between: roughly linear

• Umklapp scattering ! drop in mfp, peak in TC • Peak location and amplitude differ by material (usually <100K) • Exceptions: optical-acoustic coupling ! amorphous-like TC (ex: feldspar)

Impure Crystalline

Basalt, others

• Approx. same trend as pure crystalline

• Amplitude decr. with % impurity (higher scattering probability) • Peak in same location as that of pure material

Amorphous Glasses

• Similar trend above 100K, variation due to additional localized modes

• Mfp orders lower than in crystalline ! TC orders lower • Monotonically increases • Same general trend for all glasses. The “plateau”, TCs w/in an order of magnitude

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Temperature HKL

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Specific Heat: Old Fits

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Thermal Conductivity: Old Fits

Figure 3: Specific heat measurements2 from Apollo missions fit the current Diviner model3, and to the best third order polynomial fits (left). Thermal conductivity

measurements of regolith soil4 fit to Watson’s equation5,6 k = B+AT3 (right). Note: TC of rocks is ~ an order of magnitude higher and thus isn’t shown for comparison

, Apollo 11 soil 10084 ,! Apollo 12 soil 12001 , Apollo 14 soil 14163* , Apollo 15 soil 15013 , Apollo 16 soil 68501

Vasavada’s fit to Apollo 12 soil, ρ = 1300 kg/m3

o Apollo 11 breccia 10021 + Apollo 11 basalt 10057

, Apollo 11 soil 10084 , Apollo 14 soil 14163 , Apollo 15 soil 15013 , Apollo 16 soil 68501

Ledlow 1997 fit

•  Phonon = quanta of elastic energy •  Debye’s wave theory: k  =  CV  v  l