Herbert Kroll

New developments in two-feldspar thermometry

Abstract The thermodynamic model of the two-feldspar thermometer has been revised. From recent enthalpy and volume measurements in the (Na,Ca)- and (K,Ca)-feldspar binaries, new interaction parameters have been derived and previous ones have been updated. Entropy parameters have been fitted to the phase equilibrium data of Seck (1971) and Elkins and Grove (1990). The two data sets could be suitably combined into one. Ideal Ab, Or, and An activities have been expressed in terms of both the molecular mixing and Al-avoidance models. Two-feldspar pairs from high-grade metamorphic rocks that cooled slowly under dry conditions suffer from a distinct type of retrograde resetting. Whereas the original An content in both the plagioclase and the alkali feldspar is preserved because the intercrystalline Ca + Al ↔ (Na,K) + Si diffusion is sluggish, Na and K may be freely exchanged between phases. Mathematical reversal of the Na-K exchange at constant An yields the temperature at which the two feldspars originally coexisted. The shifts in Ab and Or contents obtained from the reversal reflect the relative plagioclase/alkali feldspar proportions observed in thin sections. Good agreement between calculated and measured ratios was found for feldspar pairs from Sri Lankan granulites. This observation represents a successful test of the reliability of the calculated Ab-Or shifts. In contrast to dry metamorphic rocks, similar application of chemical constraints is not indicated in the case of volcanic rocks. Then the two-feldspar thermometer delivers three, usually incongruent temperatures: T(Ab), T(Or), and T(An). From the abundance of temperatures, Fuhrman and Lindsley (1988) suggested adjusting compositions within assumed chemical uncertainties (e.g., ±2 mol%) so that congruent temperatures could be obtained. However, the result is not unique. Depending on minute variations in the starting compositions, the temperatures may vary by several tens of degrees. In addition, temperatures vary to a similar extent depending on the type of search algorithm. Therefore, we advise users to completely abandon this practice. Instead, a statistical procedure is suggested: Two-feldspar compositions are randomly generated according to Gaussian distributions with their means at the observed compositions and standard errors chosen according to the quality of the chemical analysis. This procedure returns normally distributed temperatures [T(Ab), T(Or), T(An)] together with means and standard deviations. From the overlap of the three Gaussian curves the question of equilibrium or non-equilibrium crystallization of feldspar pairs may be addressed.

Octahedral cation partitioning in Mg,Fe 2+ -olivine. Mössbauer spectroscopic study of synthetic (Mg 0.5 Fe 2+ 0.5 ) 2 SiO 4 (Fa 50 )

The high-temperature partitioning of Fe 2+ and Mg between the two non-equivalent octahedral M1 and M2 sites in synthetic olivine (Fa 50 ) was studied by Mossbauer spectroscopy. Powder samples have been equilibrated in annealing experiments performed under reducing oxygen fugacity at temperatures between 500°C and 800°C followed by rapid quenching in order to prevent redistribution of cations. M-site ordering with Fe 2+ preferring M1, Mg preferring M2 sites increases continuously with rising equilibrium temperature. K D values increase from 1.21 at 500°C to 1.48 at 750°C. The results are consistent with both room temperature as well as in situ high temperature single crystal X-ray diffraction experiments of Heinemann et al. (2003a and b).

Excess heat capacity and entropy of mixing in high structural state plagioclase

Abstract Low- and high-temperature heat capacities for a series of synthetic high structural state plagioclase crystals (Ab-An) were measured using both a relaxation calorimeter and a differential scanning calorimeter. The measurements were performed at temperatures between 5 and 800 K on milligram-sized polycrystalline samples that had been characterized in a previous study. The data show positive excess heat capacities of mixing at temperatures below 300 K with a maximum value of ~2 J/(mol·K). Below ~70 K, the excess heat capacities exceed two standard deviations and are thus significant. Above 300 K, the measurements indicate negative excess heat capacities with a maximum of ca. -1.5 J/(mol·K) at about 400 K, and do not exceed two standard deviations. The excess vibrational entropies of mixing are positive with an asymmetric variation. At T = 298.15 K, the largest deviation from ideal behavior occurs at Ab20An80 amounting to ΔSexvib = 2.8 ± 2.4 J/(mol·K). An asymmetric Margules mixing model was found to adequately describe the vibrational entropy-composition behavior, yielding WSvibAbAn = 16.4 J/(mol·K) and = WSvibAnAb 4.7 J/(mol·K).

Time and temperature variation of the intracrystalline Fe2+,Mg fractionation in Johnstown meteoritic orthopyroxene

The partitioning of Fe 2+ and Mg on the M1 and M2 sites of orthopyroxenes from the Johnstown meteoritic diogenite has been equilibrated between 1000°C and 700°C in ordering and disordering runs. The method of bivariate high order truncation analysis (Kroll et al., 1997) has been employed to refine the site occupancies from conventional X-ray intensity data. The Fe 2+ ,Mg distribution coefficient varies according to \[\mathrm{In\ K_{D}\ =\ 0.417\ (121)\ {-}\ 2540(136)/T[K].}\] From isothermal kinetic ordering and disordering experiments an exceptionally large activation energy was derived. Combining our data with those of Zema et al. (1997a) results in the Arrhenius equation for the rate constant: \[\mathrm{In\ k_{dis}[min^{{-}1}]\ =\ 41.4({\pm}0.9)\ {-}\ 97.8({\pm}1.9)[kcal/mol]\ /\ RT.}\] For the first time, non-linear continuous cooling experiments were performed in which the crystals were cooled from 850°C to 250°C at an average rate of 10°C/day. The Fe 2+ ,Mg distributions were determined after the crystals had reached 650°C, 550°C, 450°C, 350°C, and 250°C, respectively. Using the Mueller rate equation (Ganguly, 1982) and employing the temperature dependencies of K D and k dis as given above, the experimentally delineated ordering path is closely reproduced by the calculated path. However, due to the large activation energy, cooling rates calculated for the untreated crystals turn out to be physically unreasonable, i.e. some 10 −5 K/My. By contrast, Arrhenius parameters determined in the literature on orthopyroxenes with compositions similar to the Johnstown crystals produce physically reasonable rates of some hundred K/My. TEM studies do not show a significant difference between the microtextures of untreated and annealed samples. All orthopyroxenes studied contain clinopyroxene exsolution lamellae and abundant Guinier-Preston zones. At present, we can neither prove nor disprove the concept that the large activation energy of the Johnstown orthopyroxenes is related to their intricate exsolution microtexture.

Excess heat capacity and entropy of mixing in ternary series of high-structural-state feldspars

Low- and high-temperature heat capacities were measured for ternary series of synthetic, high structural state feldspars in the NaAlSi 3 O 8 –KAlSi 3 O 8 –CaAl 2 Si 2 O 8 system that had been characterised by X-ray diffraction and transmission light microscopy. The compositions of the phases lie in the NaAlSi 3 O 8 -rich part of the system where the stability is less limited by the ternary miscibility gap than in other compositional fields. The heat capacities of the end-members had been measured in previous studies using the same calorimeters as in this study (relaxation calorimeter and differential scanning calorimeter). Below 300 K, all samples showed strong positive excess heat capacities of mixing giving rise to excess entropies of mixing up to Δ S ex = 5.8 ± 1.6 J mol −1 K −1 at T = 298.15 K. Above 300 K, further contributions to the excess entropies of mixing do not appear to be significant. The non-ideal entropic mixing behaviour was described by a ternary asymmetric Margules model, resulting in the ternary interaction parameter W AbOrAn S = 93.9 J mol −1 K −1 .

Temperature dependence of Fe,Mg partitioning in Acapulco olivine

Abstract The temperature dependence of the intracrystalline Fe,Mg partitioning (KD) in two olivine crystals (Fa11) separated from the Acapulco meteorite was determined by single-crystal X-ray structure analysis. The independent atom model (IAM) was compared with a “bond model” which accounts for bond-induced charge accumulations on the Si-O bonds. Outliers in the set of structure amplitudes observed when using the IAM disappeared upon introducing the bond model. The crystals were equilibrated at 750, 650, and 500 °C. The refined site occupancies yield the relation ln(KD) = 0.345(60) - 204(53)/T, where T is in K, which is in qualitative agreement with earlier work. Comparison of these data with the higher temperature data of Artioli et al. (1995) suggests an unusual temperature variation of the Fe,Mg distribution within two temperature regimes. Below 880 °C, Fe tends to order onto the M1 site with increasing temperature whereas it concentrates on the M2 site above 880 °C. In principle, olivine may serve as a geospeedometer similar to orthopyroxene. However, at present its usefulness is restricted because (1) the relatively weak dependence of ln(KD) on temperature needs to be even more tightly constrained than presented here and (2) low-temperature extrapolation of the rate constants for the Fe,Mg site exchange, derived from interdiffusion coefficients, is uncertain.

Order and anti-order in olivine I: Structural response to temperature

The crystal structure of Fe0.48Mg0.52[SiO4] olivine from the Boseti volcano, Ethiopia, has been investigated by single-crystal X-ray diffractometry at temperatures between 20 °C and 900 °C. For temperatures up to 600 °C, data were collected on crystals equilibrated at 600 °C. These data can therefore be assumed to reflect structural changes that are exclusively caused by thermal effects, whereas data collected between 600 °C and 900 °C carry additional information about the Fe2+,Mg equilibrium distribution. The in situ experiments at elevated temperatures were complemented by ambient temperature data collections on quenched crystals in order to check for possible Fe2+ and Mg redistributions during quenching. Such effects were found absent in crystals quenched from 800 °C or below. The derived temperature dependence of the Fe2+,Mg site distribution is \batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} \[lnK_{D}\ =\ 0.4422({\pm}0.0070)\ {-}\ 140.0({\pm}6.5)/T\ (K)\] \end{document} according to which Fe2+ progressively anti-orders into the M1 “octahedral” site as temperature is raised. A reverse ordering reaction at ≈650 °C leading to a strong segregation of Fe2+ into the other “octahedral” site, M2, as reported by Redfern et al. (2000), could not be detected. Both the and mean bond distances continuously increase with temperature, exhibiting, however, a change in the increase rate at about 600 °C which conforms with an enrichment of the larger Fe2+ cation on the M1 site and its concomitant depletion on M2. In terms of bond lengths, the octahedral distortion of the M2 site is larger than that of M1. The opposite is true for the distortion defined in terms of the angles subtended at the cation site. Similar to the distances, the behaviour of the distortion parameters both of which increase above 600 °C reflects the Fe2+,Mg anti-order. The relative magnitudes as well as the variation with temperature of both bond length and angular distortions can be rationalized considering the different geometrical environments of the M1 and M2 sites. With respect to isotropic displacement parameters, U(M1)equiv is found larger than U(M2)equiv at all temperatures, also at variance with Redfern et al. (2000).

Isothermal annealing and continuous cooling experiments on synthetic orthopyroxenes: temperature and time evolution of the Fe, Mg distribution

Isothermal annealing and continuous cooling experiments have been performed on synthetic orthopyroxene crystals with compositions En81Fs19 and En49Fs51. Their intracrystalline Fe2+, Mg-partitioning was determined by X-ray structure analysis. En81Fs19 was equilibrated at 550°C, 675°C and 850°C, En49Fs51 at 550°C, 650°C, 675°C, 800°C, and 850°C. The Fe2+, Mg distribution coefficients vary with temperature according to \batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} \[\mathrm{In(K\_{D})}\ =\ 0.633(31)\ {-}\ 2625(31)/\mathrm{T[K]\ (En\_{81}Fs_{19})}\] \end{document} \batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} \[\mathrm{In(K\_{D})}\ =\ 0.594(52)\ {-}\ 2581(51)/\mathrm{T[K]\ (En\_{49}Fs_{51})},\] \end{document} suggesting that in this compositional range In(KD) values do not significantly depend on the Fs content. Rate constants were determined in ordering runs at 550°C: \batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} \[\mathrm{k}\_{\mathrm{dis}}\ =\ 0.00165(37)\ \mathrm{[h^{{-}1}]\ (En\_{81}Fs_{19})}\] \end{document} \batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} \[\mathrm{k}\_{\mathrm{dis}}\ =\ 0.0080(18)\ \mathrm{[h^{{-}1}]\ (En\_{49}Fs_{51})}.\] \end{document} The results closely agree with predictions from Arrhenius equations given by Ganguly & Tazzoli (1994) and Kroll et al. (1997). Non-linear continuous cooling experiments were performed on both orthopyroxenes to test the ability of the Mueller rate equation (Mueller, 1967, 1969) to correctly predict the evolution of ordering. The crystals were cooled from 850°C to 250°C at an average rate of 13°C/day. The frozen site occupancies could be fully reproduced by the calculated ordering paths when the Mueller equation was run with the temperature dependencies of KD given above and Arrhenius parameters for kdis, taken from Kroll et al. (1997) and slightly adjusted within their error limits. Site refinements were also performed under the assumption that an error of ±1% total Fe had occurred in the microanalysis. This resulted in In (KD) lines which almost coincide for En49Fs51, but are clearly different for En81Fs19. Consequently, the error of calculated cooling rates is negligible for En49Fs51, but becomes significant for En81Fs19.

Rate equations for non-convergent order-disorder processes - a review and application to orthopyroxene

Various rate equations for non-convergent order-disorder processes in minerals are currently in use in the literature. Their interrelations are investigated and their formal equivalences are worked out. Their performance in reproducing kinetic experiments and modelling ordering paths is compared using orthopyroxene as an example. Macroscopic thermodynamic consideration of the entropy production in irreversible order-disorder processes yields for the variation of the order parameter, Q, with time, t, \batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} \[\mathrm{\frac{d{\bar{Q}}}{dt}\ =\ \frac{k}{RT}\ \left[\frac{{\partial}G^{ord}}{{\partial}{\bar{Q}}}\right]_{p,T}\ (k\ =\ rate\ constant).}\] \end{document} [Equation (1)][1] is valid for quasi-static processes in the linear regime between driving force, ∂Gord/∂Q, and ordering rate, dQ/dt. Under these constraints the Ginzburg-Landau equation (Salje, 1988) which is based on microscopic arguments reduces to the same form. [Equation (1)][1] can be elaborated along different routes. (1) The Gibbs energy due to ordering, Gord, may be expressed in terms of classical or Landau formulations. From the reciprocal solution model a rather simple rate law is obtained in which the driving force relates to the difference between the logarithmic temporal and equilibrium distribution coefficients, KD, \batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} \[\mathrm{\frac{d{\bar{Q}}}{dt}\ =\ k\left[\frac{1}{2}\left(1nK\_{D}\ -\ 1nK^{E}\_{D}\right)\right]\ (E\ =\ equilibrium).}\] \end{document} (2) Reaction rate theory provides another path to rate equations. The popular Mueller (1967, 1969) equation can be developed along two lines yielding formally different, but numerically identical results, \batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} \[\mathrm{\frac{d{\bar{Q}}}{dt}\ =\ k_{dis}\left[-\ A{\cdot}\left({\bar{Q}}\ -\ {\bar{Q}}^{E}\right)-\ B{\cdot}\left({\bar{Q}}\ -\ {\bar{Q}}^{E}\right)^{2}\right]\left(A,\ B\ =\ constants\right)}\] \end{document} and \batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} \[\mathrm{\frac{d{\bar{Q}}}{dt}\ =\ k\_{dis}\left[-{\bar{X}}^{M2,E}\_{Fe}{\bar{X}}^{M1,E}\_{Mg}\left(1\ -\ exp\left({\Delta}G\_{exch}/RT\right)\right)\right]}\] \end{document} where XE denotes equilibrium site occupancies and ΔGexch is the reaction Gibbs energy for order-disorder. (3) [Equations (3)][2] and [(4)][3] yield new expressions when reformulated for processes close to equilibrium. From (4) one obtains a form that is similar to (2), \batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} \[\mathrm{\frac{d{\bar{Q}}}{dt}\ =\ k\_{dis}\left[-{\bar{X}}^{M2,E}\_{Fe}{\bar{X}}^{M1,E}\_{Mg}\left(1nK\_{D}\ -\ 1nK^{E}_{D}\right)\right],}\] \end{document} whereas (3) can be simplified by neglecting the quadratic term so that \batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} \[\mathrm{\frac{d{\bar{Q}}}{dt}\ =\ k_{dis}\left[-A\left({\bar{Q}}\ -\ {\bar{Q}}^{E}\right)\right].}\] \end{document} This expression also results (a) when a second order Landau potential is inserted into (1), or (b) when the formalism of Sha & Chappell (1996a) for two-site multi-cation ordering is rewritten for the usual case of two-cation ordering. It is thus seen that rate equations are closely interrelated although they are derived from different starting points and appear in quite different formulations in the literature. Inspection of the driving forces for ordering and disordering reveals that [equations (2)][4] and [(5)][5] predict disordering rates to be faster than ordering rates, whereas the opposite is predicted by [equations (3)][2] and [(4)][3], and no preference is expected from [equation (6)][6]. Testing these predictions on literature data of kinetic experiments fails, however, because the data scatter is too large to allow distinction between the predictions. Rate constants were calculated from [equations (2)][4][][2][][3][][5]–[(6)][6] using published kinetic data Q(t). The composition and temperature dependences of the rate constants were fitted to an Arrhenius equation. The activation energy was found to vary quadratically with composition when fitting rate constants derived from the Mueller related [equations (3)][2][][3][][5]–[(6)][6], whereas a linear variation resulted for rate constants obtained from (2). From the quality of fit of the Arrhenius equations no decision could be made on the most appropriate rate equation. This failure may be related to both the close formal correspondences between the rate equations and a lack of consistency in the experimental rate constants. However, when employing the rate equations in ordering path calculations to obtain cooling rates, differences turn up. They are largest for [equations (2)][4] and [(3)][2] amounting in the worst case to a factor of two in the rate of a slowly cooled compositionally intermediate orthopyroxene. [1]: #disp-formula-1 [2]: #disp-formula-3 [3]: #disp-formula-4 [4]: #disp-formula-2 [5]: #disp-formula-5 [6]: #disp-formula-6

The heat capacity of fayalite at high temperatures

Abstract The high-temperature heat capacity of fayalite was reinvestigated using drop and differential scanning calorimetry. The resulting data together with drop calorimetry data taken from the literature were analyzed yielding CP J/(mol·K) = -584.388 + 129 440·T−1 - 3.84956·107·T−2 + 4.10143·109·T−3 + 98.4368·ln(T). This new CP polynomial is recommended for calculating phase equilibria involving fayalite at mantle conditions. Using thermal expansion coefficient and isothermal bulk modulus data from the literature, the isochoric heat capacity was calculated resulting in CV J/(mol·K) = - 217.137 + 63 023.1·T−1 - 2.15863·107·T−2 + 2.23513·109·T−3 + 51.7620·ln(T).