The Specific Heat of Astro-materials: Review of Theoretical Concepts, Materials, and Techniques

Abstract

We provide detailed background, theoretical and practical, on the specific heat of minerals and mixtures thereof, 'astro-materials,' as well as background information on common minerals and other relevant solid substances found on the surfaces of solar system bodies. Furthermore, we demonstrate how to use specific heat and composition data for lunar samples and meteorites as well as a new database of endmember mineral heat capacities (the result of an extensive literature review) to construct reference models for the isobaric specific heat c _P as a function of temperature for common solar system materials. Using a (generally linear) mixing model for the specific heat of minerals allows extrapolation of the available data to very low and very high temperatures, such that models cover the temperature range between 10 K and 1000 K at least (and pressures from zero up to several kbars). We describe a procedure to estimate c _P (T) for virtually any solid solar system material with a known mineral composition, e.g., model specific heat as a function of temperature for a number of typical meteorite classes with known mineralogical compositions. We present, as examples, the c _P (T) curves of a number of well-described laboratory regolith analogs, as well as for planetary ices and 'tholins' in the outer solar system. Part II will review and present the heat capacity database for minerals and compounds and part III is going to cover applications, standard reference compositions, c _P (T) curves, and a comparison with new and literature experimental data.

Supplementary information: The online version contains supplementary material available at 10.1007/s10765-022-03046-5.

Keywords: Meteorites; Minerals; Rocks; Solid matter; Specific heat; Thermophysical properties.

Fig. 1

Example c_P curves, (magnetic) transition peaks in some iron oxides, quartz with the λ transition (α–β) at 843 K, fayalite with its low-temperature magnetic transition, forsterite and anorthite with no anomalies. Note that magnetite has a small broad Verwey peak at ~ 124 K which here shows only as a ‘bump.’ Akaganéite here is β-FeOOH⋅0.65H₂O and the ferrihydrite is 2-line

Fig. 2

Plotting c_P/T versus T² for low temperatures, less than about 15 K, gives straight lines for most solids; the slope is ∝ 1/θ_D³, and extrapolation to 0 K gives directly γ, the electronic heat capacity term, while for Debye solids it is zero. Low-temperature anomalies (e.g., Schottky) also show up clearly. Smoothed c_P data of our database have been used

Fig. 3

The C_P (upper panel, a) and c_P (lower panel, b) of olivines, after [95]. Note the X-point at ~ 125 K, where all compositions have about the same mass-based specific heat, which is not the case in the molar C_P. This is a quite natural effect of the vastly different formula weights of fayalite (203.778) and forsterite (140.693). Parameter in legend: x_Fo, mole fraction forsterite (w_Fo = x_Fo × 140.693/(203.778 − 63.085 × x_Fo)). Higher-resolution data around the transition peaks not shown for clarity

Fig. 4

Bronzite [92], data and ideal c_P calculated for ideal composition

Fig. 5

Bronzite [92], low T, data, and ideal c_P calculated for the ideal composition. The Fs transition at 38 K and the Schottky peak of bronzite near 12 K do not scale linearly

Fig. 6

After [127] Schematic form of the principal thermodynamic parameters through a phase transformation at T_c. Column I = first order; column II = second order; column III = λ transformation with a small first-order break at T_c; column IV = λ transformation with no first-order break. G = free energy, H = enthalpy, S = entropy, η_l = long-range order parameter, η_s = short-range order parameter describing precursor ordering above T_c; C_P = specific heat; I_k = integrated intensity of a superlattice reflection. D = disordered state, O = ordered state. LRO = long-range order, SRO = short-range order. Volume is not shown, but must be continuous or discontinuous in some manner analogous to H and S

Fig. 7

Values of θ_D (T) for representative minerals. room temperature elastic values θ_D are shown by circles at 300 K; they are assumed to apply, approximately, at low temperatures, T → 0, as well. After [61]

Fig. 8

Compositional phase diagram of the different minerals that constitute the feldspar solid solution. Ternary phase diagram of the feldspars (at 900 °C). Miscibility gap line after (Benisek, Dachs et al. 2010c)

Fig. 9

Excess heat capacity (ΔCp) of the ordering transition in anorthite, which was defined as C_P Monte Somma—CpAn100 (solid circles) and C_P Pasmeda –C_P An100 (open symbols), respectively. An100 is a synthetic anorthite crystallized at 1573 K, Monte Somma is a volcanic anorthite (An98), and Pasmeda is a metamorphic anorthite (An100). From [243]. The average C_P (without the peak) at 510 K is ~ 272 J·mol⁻¹·K⁻¹

Fig. 10

Pyroxene quadrilateral. Note that the old name ‘hypersthene’ is an orthopyroxene with ~ 30 % En and rest Fs. Macke (2018 priv. comm.) takes diopside for hypersthene, which is incorrect. Bronzite is a member of the pyroxene group of minerals, belonging with enstatite and hypersthene to the orthorhombic series of the group. Rather than a distinct species, it is really a ferriferous (12 to 30 % iron(II) oxide) variety of enstatite. The augites (Di–Hed) are monoclinic: ‘clinopyroxenes.’ In natural orthopyroxenes, a small amount of Ca (< 2%) is always present in the structure

Fig. 11

Amphibole quadrilateral. The orthorhombic anthophyllites (low Ca, ≤ 3.8 at.%) extend up to ~ 30 at.% Fe, the monoclinic cummingtonite–grunerite (low Ca, ≤ 4.5 at.%) series from ~ 30 at.% to 100 at.% Fe. The calcium content of the actinolite series is centered around 2/7 ≈ 29 at.% (referred to total metal cations)

Fig. 12

Specific heat of FeNi alloys [261] together with the curves for pure Fe and pure Ni. In alloys, the iron α → γ transition at ~ 1190 K seems to vanish and the amplitude and position of the magnetic Ni transition varies systematically with composition. The black dots are the (digitized) data of [262], they seem systematically off

Fig. 13

Bounds on crystal water CP contribution. It appears that the curve given by Gurevich 2007 is a good estimate for crystal water, we use it as our default

Fig. 14

Heat capacity behaviour of confined H₂O in armenite and epididymite as well as for hemimorphite [289] and analcime [286] at 0 K < T < 300 K. The squares with e + symbol are the C_P of ice [290], the squares with the symbol the C_P of super-cooled liquid water [291] and the circles with the symbol the C_P of ideal H₂O gas [292]. (from [287], their Fig. 9)

Fig. 15

Schematic C_P curve of a glass, T_G is the glass transition temperature, T_M the melting point. After [181]. The heat capacity of the liquid, or the glass for T > T_G, is always greater than the heat capacity of the solid [89]

Fig. 16

Overview: c_p(T) of common solar system ices. The curious dip for methanol at 157.34 K is real, it is the α/β conversion ‘from crystal II to crystal I’ just before melting

Fig. 17

Overview, specific heat of some model tholins. The range of specific heat at low temperatures is about one order of magnitude. The large λ peak at ~ 20 K is due to methane, the small anomaly near 35 K due to nitrogen. For comparison with common silicates, our lunar regolith curve is given

Fig. 18

The c_P of tholins, model 1. The blue and dark yellow curves indicate the likely range, the black curve is just the average of the blue and red ones

Fig. 19

The c_P(T) of basic tholins, model 2

Fig. 20

Tholin model 3, specific heat of solid ammonia dihydrate, data of [316]

Fig. 21

Calculated c_P(T) of DI regolith simulants. For comparison, the standard lunar c_P curve is given [317]. Note the ‘theoretical’ fayalite peak at ~ 60 K and the ‘theoretical’ magnetite peak at 840 K; the fayalite peak is expected to be smeared out in natural samples with the same mean fayalite content but from a range of olivine compositions. For comparison, the standard lunar c_P curve is given [317]

Fig. 22

The specific heat model for lunar regolith. In the lower-temperature regime (≤ 350 K), a fit from [318] based on Apollo data is used. At higher temperatures (> 350 K), a model by [325] is used. Melting temperature is 1500 K. [Reprinted from Schreiner et al. [324] with permission from Elsevier]

Fig. 23

Synthetic lunar c_P curve, 0 K to 1000 K, bold black line. Apollo data with error bars, separately for each of the 9 sample, are plotted with symbols as indicated in the legend; pure anorthite (An, green dashed) and analytical curve of Biele et al., 2018 [317] (red dotted line) for comparison

Fig. 24

Enlarged portion of Fig. 23 showing all the well-known Apollo data points (numerical values: see Online Appendix, chapter 4.1

Fig. 25

Enlarged part of Fig. 23, cryogenic temperatures. Added are the specific heats of the two samples measured at LHe temperatures [326] which are ~ 2 orders of magnitude larger than expected. This cannot be explained by a glass excess c_P (factor 2 ‒3 only), maybe it indicates a Schottky anomaly in the liquid Helium temperature range or it is due to experimental errors

Fig. 26

Relative deviations of modeled ‘synthetic’ c_P curve to Apollo data

Fig. 27

Comparison of c_P(T) curves for the Moon (‘basaltic,’ reasonable also for S-type asteroids) and for 4 different phyllosilicates or clay minerals. Talc decomposes for T > 750 K to 800 K, this is why the curve ends at 800 K