Modified Increment Method Research Articles

Calibration of oxygen isotope fractionation and calcite‐corundum thermometry in emery at Naxos, Greece

AbstractCorundum (Crn), including sapphire, occurs in emery pods surrounded by marble on the island of Naxos, Greece. The emery formed from bauxite deposited in karst that was metamorphosed to 400–700°C at 20–15 Ma. Many of these rocks initially appeared well suited for refractory accessory mineral (RAM) thermometry, which uses oxygen isotope fractionation between a RAM – corundum – and a modally dominant phase with faster diffusion of oxygen – calcite (Cc) – to determine peak metamorphic temperatures. However, previous attempts at oxygen isotope thermometry were confounded by highly variable fractionations (Δ18O) measured at mm‐scale and the uncertain calibration of Δ18O(Cc‐Crn) versus temperature. Secondary ion mass spectrometry (SIMS) permits in situ analysis of δ18O in corundum and calcite at the 10‐μm scale in adjacent grains where textures suggest peak metamorphic equilibrium was attained. SIMS analyses of adjacent mineral pairs in eight rocks yield values of Δ(Cc‐Crn) that systematically decrease from 7.2 to 2.9‰ at higher metamorphic grade. Pairing these data with independent temperature estimates from mineral isograds yields an empirical calibration of 1,000 lnα(Cc‐Crn) = 2.72 ± 0.3 × 106/T2 (T in K). The new fractionations (2.7‰ at 1,000 K) are significantly smaller than those calculated from the modified increment method (6.5‰ at 1,000 K; Zheng, Geochimica et Cosmochimica Acta, 1991, 55:2299–2307; Zheng, Mineral Mag, 1994, 58A:1000–1001), which yield unreasonably high temperatures of 630 to 1,140°C when applied to the new Naxos data. The new calibration of Δ(Cc‐Crn) can be combined with published fractionations to calculate A‐factors for corundum versus a range of 14 other minerals. These new fractionation factors can be used for thermometry or to constrain the genesis of corundum. A compilation of gem corundum δ18O values shows that many igneous sapphires, including important deposits of basalt‐associated sapphire, are mildly elevated in δ18O relative to the calculated range in equilibrium with mantle values (4.4–5.7‰) and formed from evolved magmas.

Forming limit evaluation by considering through-thickness normal stress: Theory and modeling

For hydroforming process and incremental forming, the plane stress assumption would be invalid to predict material formability. To consider the influence of normal stress on the forming limit strains, the instability perturbation approach proposed by Hu et al. is extended with normal stress. The M–K model with normal stress is chosen to compare with perturbation approach by implementing Hill’48 and Yld2000-2d yield criteria and swift and modified voce hardening laws. To guarantee the convergence of solving process in M–K model with normal stress, the modified increment method is implemented into it. Through comparing the experimental forming limit strains of AA5754-O under plane stress state and the corresponding theoretical values predicted by perturbation approach and M–K model, the suitable yield criterion and hardening law for these two methods are determined. The influences of through-thickness normal stress on the predicted forming limit strains under different stress states are investigated. The results show that forming limit curves (FLCs) in both forms of traditional Forming Limit Diagram (FLD) and equivalent plastic strain (EPS) based FLD (epFLD) enhance with increasing through-thickness normal stress under linear and nonlinear strain paths. The sensitivity of FLD to normal stress is related to the hardening law and the range of strain path in FLD is determined by the yield criterion.

New robust algorithms for Marciniak–Kuczynski model to calculate the forming limit diagrams

To solve the non-convergence of Newton–Raphson (N–R) method in the framework of Marciniak–Kuczynski (M–K) model for predicting sheet metal's forming limit, the relationship between the internal stress variables in the groove of sheet metal is investigated. The number of unknown variables in N–R method reduces to two from three. And the N–R method is changed as modified N–R method by considering this relationship. Since the convergences of N–R method and modified N–R method are not promised, an increment method for M–K model is developed, which can unconditionally guarantee its convergence. To improve the computational efficiency, a modified increment method is established by combining modified N–R method and increment method. For N–R method and modified N–R method, the accuracy of solutions is depended on the artificial defined errors for those equilibrium equations, while for increment method and modified increment method, the accuracy of solutions can be easily promised. To verify the validity of these methods, the material AA6111-T3 is adopted. It's found that increment method is convergent, whether combining the method with Hill’48 or Yld2000-2d yield criterion. The convergences of N–R method and modified N–R method are affected by strain path, necking angle and so on. The modified increment method is unconditionally convergent, and the computation efficiency is almost equal to N–R method. The modified increment method is recognized as the best method by considering from computation efficiency and convergence. The M–K model is also applied to Ti–6Al–4V, which shows that no matter for Hill’48 or Yld2000-2d yield criterion, the M–K model can predict the forming limit well under negative strain path, while overestimates the forming limit strains under positive strain path.

Response to the Comment by J. Horita and R.N. Clayton on “The studies of oxygen isotope fractionation between calcium carbonates and water at low temperatures”

The apparent inconsistency in calcite–water fractionation does occur between the arithmetic combination of Zhou and Zheng [Zhou G.-T., and Zheng Y.-F. (2003) An experimental study of oxygen isotope fractionation between inorganically precipitated aragonite and water at low temperatures. Geochim. Cosmochim. Acta 67, 387–399] and the experimental determination of Zhou and Zheng [Zhou G.-T., and Zheng Y.-F. (2005) Effect of polymorphic transition on oxygen isotope fractionation between aragonite, calcite and water: a low-temperature experimental study. Am. Mineral 90, 1121–1130]. To resolve this issue is to acknowledge whether or not the isotope salt effect of dissolved minerals would occur on oxygen isotope exchange between water and the minerals of interest. The question is whether or not a term of mineral–water interaction should be taken into account when calculating mineral–water 10 3ln α factors by an arithmetic combination between theoretical 10 3ln β factors for mineral and water, respectively. The hydrothermal experiments of Hu and Clayton [Hu G.-X., and Clayton R.N. (2003) Oxygen isotope salt effects at high pressure and high temperature, and the calibration of oxygen isotope geothermometers. Geochim. Cosmochim. Acta 67, 3227–3246] demonstrate the absence of isotope salt effect on the oxygen isotope fractionation between calcite and water, and this abnormal behavior reasonably explains the so-called inconsistency in the calcite–water fractionations of Zhou and Zheng (2003, 2005). We argue that the mineral–water correction is still necessary for calculation of fractionations in mineral–water systems. New experimental data for oxygen isotope fractionations involving dolomite and cerussite are consistent with the calculations of Zheng [Zheng Y.-F. (1999a) Oxygen isotope fractionation in carbonate and sulfate minerals. Geochem. J. 33, 109–126], but also shed light on the assumptions used in modifying the increment method. We argue that the modified increment method has developed into a theoretical mean of predictive power for calculation of oxygen isotope fractionation factors for crystalline minerals of geochemical interest.

Comment on the studies of oxygen isotope fractionation between calcium carbonates and water at low temperatures by Zhou and Zheng (2003; 2005)

Experimental and theoretical aspects of oxygen isotope fractionation in the system calcite–water at low temperatures were critically examined. Contrary to the claim made by Zhou and Zheng [Zhou G.-T., and Zheng Y.-F. (2003) An experimental study of oxygen isotope fractionation between inorganically precipitated aragonite and water at low temperatures. Geochim. Cosmochim. Acta 67, 387–399], there is excellent agreement between fractionation factors that were experimentally determined by means of slow, inorganic precipitation of calcite from solutions and those obtained largely from theoretical, statistical–mechanical calculations of the reduced partition function ratios. This agreement strongly suggests that calcite was precipitated from a solution very close to isotopic equilibrium. However, recently Zhou and Zheng [Zhou G.-T., and Zheng Y.-F. (2005) Effect of polymorphic transition on oxygen isotope fractionation between aragonite, calcite and water: a low-temperature experimental study. Am. Miner. 90, 1121–1130] presented, without any explanation, conclusions on these major aspects that contradict the previous statements of Zhou and Zheng (2003). The apparent discrepancy in calcite–water oxygen isotope fractionation between experimental and theoretical studies discussed by Zhou and Zheng (2003) originates from the “mineral–water interaction” term in the modified increment method, which was developed by one of the authors (Y.-F. Zheng). We call for evidence for the theoretical nature of the modified increment method, which has never been presented in any of Zheng’s papers. Without such evidence, great caution must be exercised in using fractionation factors derived from the modified increment method.

Oxygen-Isotope Fractionation between Aluminum-Hydroxide Phases and Water at <60°C: Results of Decade-Long Synthesis Experiments

AbstractOxygen-isotope data were obtained for synthetic aluminum-hydroxide phases precipitated over 65–125 mo and have been compared to results from similar experiments conducted for 3–56 mo. The Al(OH)3 polymorphs, gibbsite, nordstrandite, and bayerite, were synthesized, but gibbsite was dominant in most samples, and commonly the only phase present. Using pure gibbsite samples, the following oxygen-isotope fractionation factors, αgibbsite−H2O, were obtained: 1.0167 ± 0.0003 (9 ± 1°C), 1.0147 ± 0.0007 (24 ± 2°C), 1.0120 ± 0.0003 (51 ± 2°C). These values, and the associated equation for an oxygen-isotope geothermometer for the interval 0–60°C 103ln αgibbsite−H2O=2.04×106/T2−3.61×103/T+3.65 (T in K), are not significantly different from those obtained from experiments of much shorter duration. These results, and the good agreement with αgibbsite−H2O values obtained for well-constrained natural systems, suggest that the experimentally determined fractionation factors describe equilibrium conditions for gibbsite that has precipitated directly from solution.As also proposed by others using a modified-increment calculation, our synthesis experiments suggest that αAl(OH)3−H2O is polymorph-dependent at low temperatures and that a significant temperature-dependent trend exists in the values of αAl(OH)3−H2O. However, previously calculated fractionation factors obtained using the modified-increment method are higher than those obtained from the experiments, with this discrepancy becoming larger as temperature decreases.

Oxygen isotope fractionation between hydroxide minerals and water

The modified increment method has been applied to the calculation of oxygen isotope fractionation factors for hydroxide minerals. The results suggest the following sequence of 18O-enrichment in the common hydroxides: limonite > gibbsite > goethite > brucite > diaspore. The hydroxides are significantly enriched in 18O relative to the corresponding oxides. The sequence of 18O-enrichment in the hydroxides and oxides of trivalent cations is as follows: M(OH)3 > MO(OH) > M2O3. There are also considerable fractionations within the polymorphos of Al(OH)3. The internally consistent fractionation factors for hydroxide–water systems are obtained for the temperature range of 0 to 1200 °C, which are comparable with the data derived from synthesis experiments and natural samples at surficial temperatures. Temperature dependence of oxygen isotope fractionations between goethite, gibbsite, boehmite and diaspore and water are significant enough for the purpose of geothermometry. Thus the hydroxide–water pairs hold great promise of serving as reliable paleothermometers in surficial geological environments.

Theoretical calculation of oxygen isotopic fractionation of tenorite

Tenorite is an oxide of cupper which has been well documented on its structure, force constants and spectrum. Based on the isotopic shifts of its spectrum calculated from the force constants of tenorite, its reduced partition function ratios were calculated, from which an oxygen isotopic fractionation curve between tenorite and water was obtained: 103lnα = 2.51X 2-14.87X + 6.31 (X = 103/T) Oxygen isotopic fractionation of tenorite was also calculated with the modified increment method of Zheng (1991), and another equation was obtained: 103lnα= 2.89 X3 - 13.10X + 3.92 (X = 103/T) Calculated results of the two different models were fitted and possible deviation involved in the calculation was also discussed.

Oxygen isotope fractionation in zinc oxides and implications for zinc mineralization in the Sterling Hill deposit, USA

Oxygen isotope fractionation in the zinc oxides has been calculated by means of the modified increment method. The results suggest that zincite is slightly enriched in 18O relative to the franklinite of the spinel-type structure but considerably depleted in 18O relative to the franklinite of the inverse spinel-type structure. The zinc oxides are significantly depleted in 18O relative to water under hydrothermal and metamorphic conditions. The oxygen isotope analyses of mineral pairs including the zinc oxides and the common gangue minerals such as calcite and quartz can constitute a sensitive isotope geothermometer. Application of oxygen isotope geothermometry to natural assemblages is attempted for the calcite-zinc ore mineral pairs from the Sterling Hill deposit in USA. The results indicate that the temperature of the zinc mineralization may be in the range from 410° to 630 °C and thus lower than the metamorphic temperatures of granulite facies. A metamorphic fluid could have been involved in the formation of the zinc ore minerals. Franklinite would structurally be an inverse spinel in the infancy of its formation, and thus could have originally evolved from Zn2 + substitution to Fe2 + of magnetite at the high temperatures.

Oxygen isotope fractionation in magnetites: structural effect and oxygen inheritance

The modified increment method is applied to the calculation of oxygen isotope fractionation in different structures of magnetite. The results show that, at isotopic equilibrium, the magnetite of a spinel-type structure has the very similar behavior of oxygen isotope partitioning to hematite, but it is considerably depleted in 18O relative to the magnetite of an inverse spinel-type structure. This provides a resolution to the systematic discrepancy in the quartz-magnetite oxygen isotope fractionation between previous theoretical calculations of the author and the experimental determinations of H. Chiba and coworkers. An oxygen isotope inheritance is suggested in which the magnetites used in the experiments could have inherited the oxygen isotope feature of a precursor mineral hematite. The oxygen isotope inheritance may be common in the formation of secondary minerals in laboratory and nature. Most magnetite would be of the spinel-type structure in the infancy of its crystallization. The oxygen isotope study on magnetite may provide insight into the mechanism of mineralogical reactions during magnetite formation under laboratorial or natural conditions.

Oxygen isotope fractionation in TiO2 polymorphs and application to geothermometry of eclogites

Oxygen isotope fractionation in TiO2 polymorphs has been calculated by the modified increment method. The results suggest that rutile is enriched in18O relative to brookite but depleted in18O relative to anatase. Due to the same crystal structure, oxygen isotope partitioning in the TiO2 polymorphs is determined by the cation-oxygen interatomic distances. The theoretical calibrations involving rutile are in fair agreement with known experimental measurements and empirical estimates. Application of the theoretical quartz-rutile calibration to geothermometry of natural eclogite assemblages indicates the preservation of isotopic equilibrium at high temperatures. The isotopic temperatures calculated are only slightly lower than the non-isotopic temperatures, indicating the slow rates of exchange for oxygen diffusion in rutile. The kinetics of exchange for oxygen diffusion in rutile is accordingly estimated by reconciling the differences between the isotopic and the non-isotopic temperatures. The rates of exchange for oxygen diffusion in rutile should be smaller than those for hornblende, but may be equal to or greater than those for diopside.

Oxygen Isotope Fractionation in Metal Monoxides

Studies of the equilibrium oxygen isotope properties of solid minerals are essential to quantitative application of oxygen isotope data to geological thermometry and geochemical tracing. Metal monoxides appear to have been overlooked in this respect. Neither theoretical calculations nor experimental measurements nor empirical estimates have been reported for oxygen isotope fractionation in them. Oxygen isotope fractionation in metal multi-oxide minerals has been systematically calculated by Zheng (1991) using the modified increment method. The results obtained are in good agreement with known experimental and/or empirical data. The lowtemperature validity of the theoretical calibrations has been confirmed by the synthetic experiments of Bird et al. (1993). The goal of this study is to extend the theoretical calculation to metal monoxides.

Calculation of oxygen isotope fractionation in hydroxyl-bearing silicates

The modified increment method has been applied to the calculation of oxygen isotope fractionation factors for hydroxyl-bearing silicate minerals. The order of 18O enrichment obtained in common rock-forming minerals is: pyrophyllite > kaolinite > tourmaline ⩾ talc > prehnite ⩾ topaz > illite > phengite > lepidolite ⩾ muscovite ⩾ staurolite > epidote > glaucophane > serpentine ⩾ chlorite > tremolite > hornblende > phlogopite ⩾ biotite > humite > norbergite > ilvaite. Hydroxyl-bearing silicates are enriched in 18O relative to hydroxyl groups but depleted in 18O relative to anhydrous counterparts. Three sets of self-consistent fractionation factors: between quartz and the hydroxyl-bearing silicate minerals, between calcite and the silicate minerals, and between the silicate minerals and water, have been calculated for a temperature range of 0–1200°C. The fractionation factors calculated for mineral pairs are applicable to isotopic geothermometry in igneous, metamorphic and sedimentary petrology. They can be used as a test of isotopic equilibrium or disequilibrium in natural mineral assemblages over all temperature ranges of geological interest. The difference in oxygen isotope composition between the hydroxyl-bearing mineral and the OH group is quantitatively demonstrated to be temperature dependent and, therefore, can be used as a single-mineral geothermometer.

Oxygen isotope fractionation in SiO2 and Al2SiO5 polymorphs: effect of crystal structure

Oxygen isotope fractionation factors for SiO 2 and Al 2 SiO 5 polymorphs have been calculated by means of the modified increment method. The results suggest that the 18 O is depleted in the high pressure forms (i.e. the more dense phases). The effect of crystal structure on isotopic partitioning in polymorphic phases is important, because the cation-oxygen interatomic distances and coordination numbers can change due to compression or thermal expansion in geological processes

Calculation of oxygen isotope fractionation in anhydrous silicate minerals

Thermodynamic oxygen isotope factors for anhydrous silicate minerals have been calculated by means of the modified increment method. The obtained order of 18O enrichment in common rockforming minerals is quartz > albite ≥ K-feldspar > sillimanite ≥ leucite > andalusite > Jadeite > kyanite > anorthite ≥ cordierite > diopside > wollastonite > zircon ≈ garnet > olivine. Two sets of self-consistent fractionation factors between quartz and the silicate minerals and between the silicate minerals and water have been respectively obtained for a temperature range of 0 to 1200°C. The present results on the quartz-mineral systems are in good agreement with high-temperature data derived from semi-empirical calibrations and calcite-exchange experiments, demonstrating that the present calculated fractionation factors for mineral pairs are applicable to isotopic geothermometry and as a test of disequilibrium in natural mineral assemblages over all temperature ranges of geological interest. The silicate-water fractionations obtained from the present calculations also match those from hydrothermal experiments (except 3-isotope exchange results). The present calculations provide a potential insight into the quantitative dependence of oxygen isotope fractionation upon chemical composition and structural state of minerals. Minerals may be depleted in 18O with increasing pressure as a result of the change in crystal structures.

Oxygen isotope fractionation in wolframite

Thermodynamic oxygen isotope factors for wolframite are calculated using the modified increment method and assuming a predominant contribution of the 18 O-increment in complex anion to the I- 18 O index of complex mineral. The oxygen isotope fractionation factors obtained between quartz, wolframite and water are : 10 3 ln α quartz-wolframite =0.38×10 6 /T 2 +4.76×10 3 /T-2.48, 10 3 ln α wolfr a mite-water =3.86×10 6 /T 2 -8.54×10 3 /T + 1.44 for the temperature range from 0 to 1200°C. No difference in fractionation between ferberite and huebnerite is observed, indicating that the substitution of Fe for Mn does not significantly affect the 18 O/ 16 O partitioning in the wolframite series. When the present calibration for the quartz-wolframite system is applied to isotopic geothermometry in hydrothermal tungsten mineralizations, the temperatures calculated are well comparable to results from other geothermometric methods

Oxygen isotope fractionation in hematite and magnetite: A theoretical calculation and application to geothermometry of metamorphic iron-formations

Thermodynamic oxygen isotope factors for hematite and magnetite are calculated using the modified increment method and incorporating the theoretical and experimental reduced partition function ratios determined for quartz and water in the temperature range 0 to 1200 o C. The obtained oxygen isotope fractionation factors between hematite, magnetite and water are expressed as 10 3 ln α Fe2O3-H2O =2.69×10 6 /T 2 -12.82×10 3 /T+3.78, 10 3 ln α Fe3 O 4- H 2 O=3.02×10 6 /T 2 -12.00×10 3 /T+3.31, and those between quartz and hematite or magnetite are expressed as 10 3 ln α Si O 2-FeO3 =1.55×10 6 /T 2 +9.05×10 3 /T-4.82, 10 3 ln α SiO2-Fe3O4 =1.22×10 6 /T 2 +8.22×10 3 /T-4.35. The present results are well in agreement with existing theoretical, experimental and empirical calibrations involving magnetite and hematite. A significant oxygen isotope fractionation is obtained between hematite and magnetite, whith hematite being depleted in 18 O relative to magnetite. This matches with the observation from Precambrian iron-formation in the Hamersley Range, Australia. The rule of effect of oxidation state on the oxygen isotope fractionation seems not to work for the two iron oxides. Previous experimental calibration of magnetite-water system by reduction of hematite to magnetite is concordant with the present result for hematite-water system. This is attributed to the quantitative redox of iron in the experiments, that can in turn explain the nearly identical isotope composition of oxygen in coexisting hematite and magnetite pairs from the banded iron-formation in the Iron Quadrangle, Brazil

Calculation of oxygen isotope fractionation in metal oxides

A theoretical model calculating the thermodynamic isotope factors of oxygen in metal oxides of geochemical interest is presented for temperatures between 0 and 1200°C on the basis of the modified increment method. The following order of 18O-enrichment is obtained: cassiterite > rutile > magnetite > ilmenite > ulvöspinel > hematite > uraninite > chromite > corundum > spinel. The results are expressed as 10 3 In α = A × 10 6/ T 2 + B × 10 3/ T + C, which compare well with existing theoretical, experimental, and empirical calibrations on quartz-oxides and oxides-water systems. The present calculations enable not only estimation of the oxygen isotope composition of fluids from which the metal oxides were precipitated when independent temperature information is available, but can also be applied as oxygen isotope geothermometers to igneous and metamorphic rocks.