Chemical Geology 236 (2007) 65 – 74 www.elsevier.com/locate/chemgeo

Ti diffusion in quartz D.J. Cherniak ⁎, E.B. Watson, D.A. Wark Department of Earth and Environmental Sciences, Science Center 1W19, Rensselaer Polytechnic Institute, 110 8th St., Troy, NY 12180, USA Received 28 April 2006; received in revised form 4 September 2006; accepted 6 September 2006 Editor: P. Deines

Abstract We have measured Ti diffusion in quartz under dry 1-atm conditions. Experiments were performed using synthetic and natural quartz and a TiO2 powder source, with Ti profiles obtained by Rutherford Backscattering Spectrometry (RBS). Over the temperature range 700–1150 °C, the following Arrhenius relation was obtained for diffusion parallel to (001): DTi ¼ 7  10−8 expð−273F12kJ mol−1 =RT Þm2 sec−1 Similar diffusivities were obtained for both synthetic and natural quartz, and for a range of Ti source materials, including natural titanite powder and rutile–quartz single crystal diffusion couples. Although Ti diffusion appears slightly slower in the direction normal to c, this difference is not great. Using these diffusion parameters, calculations indicate that distances over which diffusional alteration of Ti concentrations in quartz could occur in a million years would be on order of 500 μm at 800 °C, and ∼ 15 μm at 600 °C. The Arrhenius relation above should be broadly applicable, and may find application in constraining metamorphic histories under conditions where the Ti-inquartz geothermometer can be reasonably applied. This relation can also be used to constrain peak temperatures and cooling histories from observations of exsolved rutile needles in quartz and the width of the Ti-depleted zone that develops around the growing needles. Finally, these findings may find utility in evaluating crystal residence times in magmatic systems when coupled with observations of the sharpness of zoning patterns in quartz. © 2006 Elsevier B.V. All rights reserved. Keywords: Quartz; Diffusion; Titanium; RBS

1. Introduction Quartz is one of the most abundant minerals in Earth's crust, and occurs as a primary constituent of many rock types. It can contain minor amounts of aluminum, alkalis, and transition elements. Quartz color, and in some cases luminescence, can be attributed to the ⁎ Corresponding author. Fax: +1 518 276 2012. E-mail address: [email protected] (D.J. Cherniak). 0009-2541/$ - see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.chemgeo.2006.09.001

presence of these elements and the defects they create. For example, tiny rutile needles or TiO2 in the colloidal state may be responsible for rose and blue quartz coloring, respectively (e.g., Deer et al., 1992). Quartz often reveals detailed records of growth history in fine scaled zoning in individual grains. Some of this zoning is observable by variations in cathodoluminescence intensity, and correlated with Ti concentrations. An experimentally calibrated geothermometer utilizing the temperature dependence of Ti solubility in

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quartz has recently been developed (Wark and Watson, in press). Given the broad potential application of the Ti in quartz geothermometer, knowledge of Ti diffusivities in quartz is clearly of considerable import, and preservation of zones of distinct Ti composition and the thermal information they may provide is dependent on the degree of intracrystalline Ti diffusion. In this work, we report Ti diffusivities for both synthetic and natural quartz under anhydrous, 1-atm conditions. We also investigate anisotropy of diffusion. Examples of application of the diffusion findings to both igneous and metamorphic systems are then presented, and their implications discussed. 2. Experimental procedure The quartz crystals used in this study included both hydrothermal synthetic quartz and natural quartz from Arkansas. Crystals were oriented by crystal habit and sectioned with a low-speed saw, polished to 0.3 μm alpha alumina, and finished with a chemical polish using colloidal silica. The polished specimens were then preannealed for 1 day at 1200 °C in a 1-atm furnace in order to heal damage that may have been produced during mechanical polishing. Procedures of polishing and preannealing to anneal damage produced by cold-working have been employed for some time in preparation of sample surfaces of minerals and other oxides for ion beam analysis (e.g., Reddy and Cooper, 1982). Final polishing with colloidal silica should remove any such damage, but the final annealing step was done as an added precaution. A range of sources were employed in the Ti diffusion experiments. One source of diffusant consisted of commercial TiO2 powder, 99.9% purity. For some of these experiments, the powder was pre-fired in a 1-atm furnace for a few hours at 1200 °C, for others the powder was merely dried overnight at low temperature in a drying oven. An experiment was also run using a source consisting of finely ground natural titanite (from Minas Gerais, Brazil). For the powder source experiments, the pieces of polished and pre-annealed quartz, typically a few mm on a side, were placed in Pt capsules with TiO2 sources, with source packed tightly around the quartz samples. An additional set of experiments was conducted using quartz–rutile diffusion couples. For these experiments, polished pieces of quartz and synthetic rutile (from Princeton Scientific Corporation) were placed with polished faces together and wrapped in Pt mesh. The wrapped couple was then set in a small Pt crucible and topped with several Pt plugs to weight the assembly. The prepared capsules were heated in 1-atm furnaces for times from 30 min to a few months, at temperatures

from 1150 to 648 °C. Quartz is metastable above ∼ 870 °C, but no signs of transformation were noted in optical examination of run products. Following completion of diffusion anneals, samples were quenched simply by removing them from the furnace and allowing them to cool in air. Samples were then separated from the source material and ultrasonically cleaned in alternate baths of distilled water and ethanol. A “zero time” experiment was also run with a synthetic quartz specimen and TiO2 powder source to investigate whether significant Ti uptake might occur in the stages of sample heat-up and quench during diffusion anneals, and to serve as means to highlight other possible problems with experimental protocol. The sample and source were prepared as described above, and sample brought up to run temperature (900 °C) and immediately quenched. Time series, with a series of experiments run for differing times but at the same temperature, were run at 900 °C for both un-fired and pre-fired TiO2 sources. 3. RBS analysis RBS has been used for depth profiling in many of our diffusion studies (e.g., Cherniak and Watson, 1992; Cherniak, 1995), and the experimental and analytical approach used here is similar to that employed in our previous work. For analysis of these experiments, an

Fig. 1. RBS spectra from Ti diffusion experiments. The spectra were taken with 4He+ ions of 2.5 MeV incident energy. Elemental edges for Si and O are indicated in the figure, along with the location of Ti. The “peak” in the spectrum for oxygen is due to non-Rutherford scattering from 16O at this energy. The inset figure shows the region of the spectrum containing the Ti profile and the contribution to the spectrum due to the presence of Fe in natural quartz. A spectrum from a Ti diffusion experiment run with synthetic quartz (open circles) is shown for comparison.

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incident 4He+ beam with energy of 2 or 2.5 MeV was employed. Spectra were converted to Ti concentration profiles using procedures outlined in publications cited above. The resultant profiles were fit with a model to determine the diffusion coefficient D. Uncertainties in concentration are primarily a function of counting statistics (i.e., proportional to N½, where N is the number of counts in a particular channel) in the RBS spectra. In the natural quartz, an additional factor included in the determination of uncertainties was the background signal from other elements heavier than Ti present in the quartz (an example spectrum is shown in Fig. 1), which is superimposed on the signal from the Ti diffusion profile. This is incorporated in the error calculation with uncertainty varying as 2(Nb + No)½ / No, where No is the total number of counts in the multichannel analyzer channel, and Nb is the number of counts in the background in that channel. Uncertainties in depth determination are primarily a function of the resolution of the detector used to detect backscattered particles, and the statistical energy spread of individual ions comprising the beam as they travel through the sample. Diffusion is modeled in these cases as simple onedimensional, concentration independent diffusion in a semi-infinite medium with a source reservoir maintained at constant concentration (i.e., a complementary error function solution, C(x,t) = Coerfc (x / (4Dt)1/2 )). The rationale for the use of this model has been addressed elsewhere (e.g., Cherniak and Watson, 1992, 1994) for experiments using fine-grained sources. In the case of the diffusion couple, Ti diffusion in the rutile is much more rapid than in the quartz (e.g., Akse and Whitehurst, 1978). Diffusivities are evaluated by plotting the inverse of the error function (i.e., erf− 1((Co − C(x,t)) / Co)) vs. depth (x) in the sample. A straight line of slope (4Dt)− 1/2 results if the data satisfy the conditions of the model. Co,

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the surface concentration of diffusant, is determined by iteratively varying its value until the intercept of the line converges on zero. This approach is taken because of its simplicity and straightforwardness given the experimental configuration, but direct non-linear fits of the profiles without inversion could have also been performed with similar results to extract diffusivities. In Fig. 2, a typical diffusion profile and inversion through the error function is shown. The uncertainties in concentration and depth from each data point were used in evaluating the uncertainties in the diffusivities determined from fits to the model. 4. Results The results for Ti diffusion anneals of quartz are presented in Table 1 and plotted in Fig. 3. For diffusion parallel to c in the synthetic quartz using the pre-fired TiO2 source, an activation energy 273 ± 12 kJ mol− 1 and pre-exponential factor 7.01 × 10− 8 m2 s− 1 (log Do = − 7.154 ± 0.525) are obtained by a least squares univariant regression fit. Diffusivities for the direction normal to c do not differ significantly from those parallel to c. Rates for diffusion for the natural quartz are similar, and diffusivities from experiments using the various diffusant powder sources are also comparable. However, for reasons not well understood, Ti concentrations in samples run with the un-fired TiO2 source show higher surface concentrations than do those from the pre-fired TiO2 source, although, as noted, Ti diffusivities themselves do not differ significantly. Diffusivities for experiments using the quartz–rutile couple yield the following Arrhenius parameters: activation energy 283 ± 26 kJ mol− 1 and pre-exponential factor 2.65 × 10− 8 m2 s− 1 (log Do = − 7.576 ± 1.084). These are slightly and apparently systematically lower, but the diffusion parameters fall within experimental

Fig. 2. A typical Ti concentration profile from a Ti diffusion experiment on quartz. The profile was measured by RBS. In (a) the diffusion data are plotted with a complementary error function curve. (b) The data are inverted through the error function. The slope of the line is equal to (4Dt)− 1/2.

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Table 1 Ti diffusion in quartz T (°C) Time (s)

D (m2 s− 1)

Synthetic quartz, parallel to c: TiQ-42 1200 1.80 × 103 2.33 × 10− 18 TiQ-29a 1151 1.80 × 103 7.12 × 10− 18 TiQ-41 1150 2.70 × 103 2.01 × 10− 18 TiQ-27a 1101 3.60 × 103 3.24 × 10− 18 TiQ-40 1102 7.20 × 103 2.44 × 10− 19 TiQ-12 1052 1.26 × 104 5.68 × 10− 19 TiQ-25a 1051 1.26 × 104 1.28 × 10− 18 TiQ-38 1050 1.62 × 104 1.41 × 10− 19 TiQ-11 1000 6.48 × 104 2.13 × 10− 19 TiQ-22a 999 1.80 × 104 4.78 × 10− 19 TiQ-37 1000 6.66 × 104 1.05 × 10− 19 TiQ-10 952 8.64 × 104 8.48 × 10− 20 TiQ-20 952 6.12 × 104 1.11 × 10− 19 TiQ-39 950 3.28 × 105 9.38 × 10− 21 TiQ-9 903 1.94 × 105 2.08 × 10− 20 TiQ-15 901 5.18 × 105 1.96 × 10− 20 TiQ-17 901 6.48 × 104 2.57 × 10− 20 TiQ-30a 903 1.66 × 105 2.73 × 10− 20 TiQ-35 900 1.46 × 105 4.55 × 10− 21 TiQ-36 900 1.46 × 105 7.61 × 10− 21 TiQ-13 852 2.63 × 105 1.13 × 10− 20 TiQ-23a 850 6.77 × 105 1.12 × 10− 20 TiQ-43 848 7.76 × 105 2.50 × 10− 21 TiQ-16 801 6.28 × 105 3.46 × 10− 21 TiQ-26a 801 5.17 × 105 3.75 × 10− 21 TiQ-44 803 2.59 × 106 5.48 × 10− 22 TiQ-28a 750 1.04 × 106 8.23 × 10− 22 TiQ-24a 700 2.76 × 106 4.72 × 10− 22 TiQ-19 648 7.33 × 106 8.98 × 10− 23

log D

±

Source⁎

− 17.63 − 17.15 − 17.70 − 17.49 − 18.61 − 18.25 − 17.89 − 18.85 − 18.67 − 18.32 − 18.97 − 19.07 − 18.95 − 20.03 − 19.68 − 19.71 − 19.59 − 19.56 − 20.34 − 20.12 − 19.95 − 19.95 − 20.60 − 20.46 − 20.43 − 21.26 − 21.08 − 21.33 − 22.05

0.36 0.15 0.40 0.12 0.41 0.05 0.34 0.42 0.05 0.49 0.36 0.04 0.33 0.47 0.08 0.05 0.22 0.17 0.50 0.50 0.12 0.45 0.45 0.07 0.15 0.47 0.42 0.45 0.50

Couple TiO2, a Couple TiO2, a Couple TiO2, u TiO2, a Couple TiO2, u TiO2, a Couple TiO2, u TiO2, a Couple TiO2, u TiO2, u TiO2, u TiO2, a Couple Titanite TiO2, u TiO2, a Couple TiO2, u TiO2, a Couple TiO2, a TiO2, a TiO2, u

the experimental approach. Diffusivities from time series at 900 °C for both the fired and un-fired TiO2 sources are quite similar for times ranging over more than an order of magnitude (Table 1, Fig. 4), and show good agreement with diffusivities determined from experiments using other Ti sources, suggesting that volume diffusion is the dominant contributor to the observed diffusion profiles. The zero-time experiment (not shown) displays little evidence of significant near-surface Ti during the heatup and quench phases of the anneal, offering further confirmation that measured Ti profiles are a consequence primarily of lattice diffusion. 5. Partitioning A relationship for Ti uptake in quartz as a function of temperature was determined in the experimental

Natural quartz, parallel to c: TiQ-32 903 2.48 × 105 1.24 × 10− 20 − 19.91 0.13 TiO2, a TiQ-33 1002 1.44 × 104 2.55 × 10− 19 − 18.59 0.29 TiO2, a Synthetic quartz, normal to c: TiQ-27b 1101 3.60 × 103 TiQ-25b 1051 1.26 × 104 TiQ-22b 999 1.80 × 104 TiQ-30b 903 1.66 × 105 TiQ-23b 850 6.77 × 105 TiQ-26b 801 5.17 × 105 TiQ-28b 750 1.04 × 106

7.12 × 10− 19 1.92 × 10− 19 2.28 × 10− 19 1.43 × 10− 20 2.21 × 10− 21 2.48 × 10− 21 1.83 × 10− 21

− 18.15 − 18.72 − 18.64 − 19.84 − 20.66 − 20.61 − 20.74

0.48 0.47 0.12 0.28 0.46 0.32 0.50

TiO2, a TiO2, a TiO2, a TiO2, a TiO2, a TiO2, a TiO2, a

⁎Sources: TiO2, a: TiO2 powder, heated at 1200 °C; TiO2, u: TiO2 powder, dried; couple: quartz–rutile couple; titanite: ground natural titanite.

uncertainties of those obtained for the TiO2 powder source. The “zero-time” experiment, as well as a time-series study at 900 °C, were performed in order to verify that the measured concentration profiles represent volume diffusion and are not a result of other phenomena such as surface reaction that may otherwise result in enhanced Ti yields in the near-surface region. The “zero-time” anneal also serves to highlight possible systematic problems in

Fig. 3. Arrhenius plot of Ti diffusion in quartz. Plotted are diffusion experiments on synthetic (white circles) and natural quartz (grey triangles) for diffusion parallel to c, and for diffusion normal to c (black circles) for synthetic quartz, all using the “pre-fired” TiO2 source. The line is a least-squares fit to the diffusion data for synthetic quartz for transport parallel to c. Arrhenius parameters extracted from the fit are: activation energy 273 ± 12 kJ mol− 1 and pre-exponential factor 7.01 × 10− 8 m2 s− 1 (log Do = −7.154 ± 0.525). Little anisotropy for Ti diffusion is evident, as diffusivities in quartz parallel and perpendicular to c are similar. Ti diffusion for both synthetic and natural quartz is also quite similar, indicating that differences in minor element content among quartz specimens have little effect on Ti diffusion. Also plotted are diffusion data extracted from synthetic quartz–rutile couples (black squares). These yield slightly lower diffusivities overall, but the Arrhenius parameters derived from the data (activation energy 283± 26 kJ mol− 1 and pre-exponential factor 2.65 × 10− 8 m2 s− 1 [log Do = −7.576± 1.084]) agree within experimental uncertainties with those obtained for the TiO2 powder source. The diffusivity for the experiment run with a natural titanite source (white diamond) is also consistent with the other diffusion findings, as are those run with the “un-fired” TiO2 powder source (grey circles).

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partitioning study of Wark and Watson (in press). Over the temperature range 600–1000 °C at 1 GPa pressure, they obtain the following relationship between temperature (in degrees K) and Ti content in quartz (XTi,qtz) : T ðKÞ ¼ −3765=½logðXTi;qtz =aTiO2 Þ−5:69 where the Ti content is in wt. ppm, and aTiO2 is the activity of TiO2 relative to that required for rutile saturation (aTiO2 = 1), assuming Henrian behavior. In Fig. 5, we plot values for Ti surface concentrations determined from our diffusion profiles from experiments run with the “pre-fired” TiO2 source and those from quartz–rutile crystal couples along with the partitioning relationship above. The Ti surface concentrations from the diffusion experiments display relatively good agreement with the partitioning values. 6. Diffusion in quartz Some of the extant diffusion data for quartz are shown in Fig. 6. Ti diffusion in quartz is faster than Si selfdiffusion (Cherniak, 2003), and somewhat slower than Al+ 3 (Pankrath and Flörke, 1994) and Ga+ 3 (Mizutani et al., 1982) diffusion. Ti most likely substitutes for Si in the quartz lattice, as do Al and Ga. Ti has a higher charge (+ 4 vs. +3) than both Al and Ga, which may contribute to its lower diffusivity with respect to these elements. The ionic radius of Ti+ 4 brackets the values for Al and Ga (0.42 Å for Ti, 0.39 and 0.47 for Al and Ga, respectively, in 4-fold coordination; Shannon, 1976). Ti diffusion is significantly slower than diffusion of the

Fig. 5. Ti surface concentrations and solubilities. Ti surface concentrations from diffusion experiments using the “pre-fired” TiO2 source (circles), and quartz–rutile couples (squares), are plotted against the line describing the temperature dependence of Ti solubility in quartz determined by Wark and Watson (in press). While there is a fair amount of scatter, the data do cluster around the line defining values expected from the solubility studies.

alkalis (Verhoogen, 1952; Frischat, 1970), not surprising given the typical rapid transport of alkali elements in most materials. Ti diffusion is faster than oxygen diffusion in quartz under dry (Dennis, 1984a) conditions, and slower than oxygen diffusion under CO2 present (Sharp et al., 1991) and hydrothermal (Dennis, 1984b) conditions; hence, the potential exists for diffusional decoupling of oxygen isotope ratios and Ti concentrations within quartz crystals. 7. Geological implications 7.1. Rutile needles in quartz — observations and modeling

Fig. 4. Time series at 900 °C for Ti diffusion in quartz, for experiments run with both the pre-fired (squares) and un-fired (circles) TiO2 sources. Measured diffusivities at this temperature show fairly consistent values for experiment durations differing by up to an order of magnitude, suggesting that the dominant process being measured is volume diffusion. Also shown are data from 900 °C experiments using titanite powder and rutile crystal sources, for which similar diffusivities are obtained at this temperature.

Rutile needles are commonly observed as inclusions in natural quartz crystals, and in many cases these appear to have been formed by exsolution of Ti from the quartz lattice. An example of rutile needles in quartz in a Grenville age metamorphosed volcaniclastic sediment from the Honey Brook Upland in southeast Pennsylvania is shown in Fig. 7 (Pyle, 2006). Fig. 7A shows a transmitted light image. The box in the image indicates the 200 μm area of the cathodoluminescence image shown in Fig. 7B. Randomly oriented needles of rutile (diameter b 1 μm) penetrate the polished section at the sites of dark “spots” in the CL image, which are interpreted as Ti-depletion zones that formed during rutile exsolution following peak metamorphism. Peak temperatures were estimated by Pyle (2006) to be 730– 740 °C, consistent with Ti contents in un-exsolved

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diffusion profile translated through it, the other with a moving interface that ‘consumed’ cylindrical shells to accommodate rutile growth. Good agreement was achieved with these two approaches. The solubility of Ti in quartz (the equilibrium condition at the rutile/ quartz interface) was calculated from the TitaniQ thermometer of Wark and Watson (in press). For most

Fig. 6. Summary of diffusion data for cations and anions in quartz. Ti diffusion in quartz is faster than Si self-diffusion, and somewhat slower than Al and Ga diffusion. Ti most likely substitutes for Si in the quartz lattice, as do Al and Ga. Ti has a higher charge (+ 4 vs. +3) than both Al and Ga. The ionic radius of Ti+ 4 brackets the values for Al and Ga (0.42 Å for Ti, 0.39 and 0.47 for Al and Ga, respectively, in 4-fold coordination; Shannon, 1976). Ti diffusion is significantly slower than diffusion of the alkalis, not surprising given the typical rapid transport of the alkalis in most materials. Sources for data: Si– Cherniak, 2003; Al– Pankrath and Flörke, 1994; Ga– Mizutani et al., 1982; O (parallel to c)– Dennis, 1984a,b; Sharp et al., 1991; Li, K– Verhoogen, 1952; Na– Frischat, 1970; Verhoogen, 1952.

quartz of roughly 86 ppm, which corresponds to a temperature of 733 °C using the TitaniQ geothermometer (Wark and Watson, in press). Growth of the needles can be understood quantitatively using the diffusion data reported in this study. Rutile growth causes a local depletion of Ti in the host quartz whose spatial extent is determined by the Ti diffusivity and whose limiting value at the interface between rutile and quartz is given by the Ti solubility (see Fig. 8). Knowledge of these two quantities– diffusivity and solubility of Ti in quartz–enables detailed modeling of rutile growth with cooling using standard numerical approaches. Because the habit of exsolved rutile is invariably needle-like, it is reasonable to assume that the needles grow radially by diffusive supply of Ti from the quartz in a cylindrical diffusion field. Accordingly, numerical simulations were made using the explicit finite-difference method implemented with volume elements consisting of concentric cylindrical shells. The main complication to the overall modeling problem–i.e., the moving boundary aspect of the rutile/quartz interface– was addressed using two different algorithms: one in which the rutile/quartz interface was fixed and the

Fig. 7. Rutile needles in a Grenville-age metamorphosed volcaniclastic sediment (sample Yhga1 of Pyle, 2006) from the Honey Brook Upland in southeast Pennsylvania. 7(A) is a transmitted light image of the quartz. The boxed area is the 200 μm region shown in a CL image in (B). Randomly oriented needles of quartz (diameter b 1 μm) penetrate the polished section at the sites of dark “spots” in the CL image, which are interpreted as Ti-depletion zones that formed during rutile exsolution following peak metamorphism. Peak temperatures were estimated by Pyle (2006) to be 730–740 °C, consistent with Ti contents in un-exsolved quartz of roughly 86 ppm, which corresponds with a temperature of 733 °C using the TitaniQ geothermometer (Wark et al., in press).

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Fig. 8. Schematic representation of the Ti diffusion field in quartz surrounding a single rutile needle that has grown progressively during cooling. See text for discussion.

simulations, the initial bulk Ti content of the quartz was taken as the solubility at the starting temperature of the simulation—i.e., rutile nucleation was assumed to occur immediately upon undercooling with an initial ‘seed’

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needle of 10 nm radius. A node spacing of ∼0.1– 0.15 μm was used near the interface where the diffusion profile was steep, but this was ‘relaxed’ to much larger values (up to ∼ 30 μm) at greater distances where the profile was shallow (see example profiles in Fig. 9A). The near-field node spacing was chosen by exploring progressively smaller values until the time-integrated transport across the interface did not change appreciably. To confirm mass conservation during the simulations, the final diffusion profiles were numerically integrated to obtain the total Ti atoms lost from quartz; these values were found to be equivalent within ∼ 0.5% to the total Ti in the newly formed rutile needles. Fig. 9B illustrates progressive rutile growth during cooling at three different rates from three different initial temperatures (800, 775 and 700 °C). An overall summary of the results of the numerical simulations is presented in Fig. 10, which portrays final rutile radius as a function of cooling rate for initial temperatures ranging from 650 to 800 °C. Also shown on the figure are smooth curves given by R¼

expð−37:179 þ 0:07882Ti −3:832  10−5 Ti2 Þ pffiffiffiffiffiffiffiffiffiffiffiffi dT =dt ;

which is a general fit to the numerical results that can be used for calculating the radius (R) of an exsolved rutile needle produced by cooling from an initial temperature (Ti, °C) at a given rate, dT/dt (°/Ma). For illustrative purposes, a few simulations were also run in which rutile nucleation was delayed during cooling by up to 100 °C in order to observe growth

Fig. 9. (A) Illustrative Ti concentration gradients in quartz adjacent to rutile needles produced by cooling from three initial temperatures at three different rates (shown on figure) assuming immediate nucleation upon undercooling. The horizontal axis is radial distance moving outward from the rutile/quartz interface. (B) Radial growth of rutile as a function of temperature for the three cooling scenarios described in (A). Note that the temperature at which growth effectively stops (‘closes’) depends upon the T–t path. See text and Fig. 8.

Fig. 10. Compilation of the results of rutile growth simulations (plotted points) in comparison with a ‘master equation’ generated by multivariate fitting of the ‘data’. Each smooth curve is an isotherm representing cooling from a specific initial temperature. The master equation should not be used appreciably beyond its range of ‘calibration’, and applies only to the case of immediate nucleation.

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behavior under circumstances of significant oversaturation of quartz in TiO2. The results are shown in Fig. 11, from which it is clear that delayed nucleation (‘undercooling’) can significantly reduce the final radius of the rutile needles. The details will depend upon the assumed initial conditions and the cooling rate, but undercooling by ∼ 100° under plausible conditions can reduce the needle radius by a factor of ∼ 4 during cooling at 10°/Ma. All simulations assume that the Ti diffusion fields in quartz associated with individual rutile crystals do not overlap. The validity of this assumption is determined by the rutile nucleation density; unfortunately, this is a difficult quantity to evaluate, and may depend upon the presence of defects and/or impurities in the quartz lattice. Note, however, that the cylindrical diffusion fields can overlap significantly without dramatically affecting the rutile growth rate because the steep part of

Fig. 12. This plot shows calculations for the “blurring” of Ti concentrations in quartz for regions or zones of different width. For conditions on the curves, concentration will change by 10% one-tenth of the way into the zone due to diffusional modification. For conditions above each curve, the effects on Ti composition due to diffusion will be greater than this; below the curves they will be smaller. For example, a 100 μm zone would experience a 10% compositional change 10 μm from its edge in about 1 Ma at 600 °C. See text for further details.

the diffusion ‘well’ is confined to a relatively small radius around the rutile (see Fig. 9A). For circumstances in which high nucleation density leads to spacing of rutile needles that is small relative to the diffusive length scale, rutile growth during cooling will lead to relatively uniform ‘draw-down’ of the host quartz Ti content. This behavior would be expected, for example, during the earlier stages of cooling from ultra-high-temperature (UHT) metamorphic conditions. 7.2. Preservation of Ti zoning in quartz

Fig. 11. Numerical results for rutile growth under circumstances of delayed nucleation, described in terms of undercooling by U degrees below the saturation temperature. In (A) the initial temperature is 750 °C (with Ti appropriate for saturation at that temperature) and the cooling rate is 10 °C/Ma in all cases. The system was allowed to undercool by up to 100 °C. In (B), rutile nucleation occurs at 725 °C in all cases, with varying amounts of undercooling from higher temperatures. See text for discussion.

Titanium shows spatial variation in quartz that may be expressed as variation in cathodoluminescence emission intensity (e.g., Wark and Spear, 2005). Given the broad potential for the use of Ti concentrations in geothermometry, it is useful to consider the thermal conditions under which Ti compositions of zones or regions of quartz grains may be modified through diffusion. Ideally, it may be possible to evaluate the degree of diffusive exchange by assessing the sharpness of zones visible in CL if they could be attributed to Ti, or by evaluating the steepness of concentration gradients bounding distinct regions through electron microprobe or ion microprobe step scans. We consider a simple model, with zones modeled as plane sheets of thickness l; adjacent planes have different concentrations of diffusant. Only diffusion normal to the planar interface is considered. A (somewhat arbitrary) criterion for compositional modification of zones is

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employed. Zones are considered to be “blurred” if a compositional change of 10% is attained at a distance 10% from edge of the zone; the dimensionless parameter Dt/l2 will be equal to 1.8 × 10− 3 when this condition occurs (e.g., Crank, 1975). Fig. 12 shows curves constraining the time–temperature conditions under which Ti zoning of 10 μm, 100 μm and 1 mm scales will be modified in quartz given the above criteria. For example, a 100 μm zone would experience a 10% compositional change 10 μm from its edge in about 1 Ma at 600 °C. For conditions above the curves, there will be changes in Ti concentration greater than the 10% change 10% into the zone described above, while for conditions below there will be less “blurring” of zones. 7.3. Applications to igneous systems— an example from the Bishop Tuff Ti zoning patterns in quartz, as noted above, can provide information on crystallization and growth histories. The observed sharpness of gradients between these zones, when coupled with the diffusion findings for Ti in quartz, may be used to place limits on crystal residence times in magmatic systems. An example where this has been applied is in interpretation and timing of igneous events leading to formation of the Bishop Tuff and collapse of the Long Valley caldera (Wark et al., in press). Quartz crystals from the Bishop Tuff show complex cathodoluminescence zoning (see Peppard et al., 2001), particularly in quartz from late-erupted pumices, which record FeTi oxide temperatures of ∼ 780 °C (Hildreth, 1979). Observed variations in CL intensity are correlated with Ti content, as measured by electron microprobe across boundaries separating dark-CL cores from brightCL rim zones (Wark and Spear, 2005). Assuming initial, infinitely steep Ti concentration gradients between core and rim, and using a diffusivity of 2 × 10− 21 m2 s− 1 (based on our new Arrhenius relation), Wark et al. (in press) calculated Ti-concentration profiles that would result from interdiffusion between core and rim at 780 °C for different periods of time. Measured profiles resemble those calculated for ∼ 100 yrs, suggesting that the bright-CL rims formed within 100 yrs of the 0.76 Ma eruption that produced the Bishop Tuff. 7.4. Applications of Ti diffusivities to metamorphic systems Quartz in some metamorphic rocks is zoned in CL and in Ti, as shown by Wark and Spear (2005). Observed CL and Ti zoning may reflect temperature conditions of

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quartz crystallization during metamorphic events, or they may be the result of diffusional modification on either prograde or retrograde paths. Depending on metamorphic grade, however, effective diffusion distances may be quite limited: At 500 °C, for example, diffusion will be effective over distances of only 0.05 μm, 2 μm and 5 μm in 0.001, 1.0, and 10 Ma, respectively. At 800 °C, however, these distances increase to 16, 500, and 1600 μm over the same time ranges. This suggests that the Ti content in the interiors of metamorphic quartz grains may record peak T conditions only if high metamorphic grades are reached, or if the quartz is fine grained. Acknowledgements This work was supported by grants EAR-0440228 and EAR-0337481 (to E.B. Watson) and EAR-0409622 (to D. Wark) from the National Science Foundation. References Akse, J.R., Whitehurst, H.B., 1978. Diffusion of titanium in slightly reduced rutile. J. Phys. Chem. Solids 39, 457–465. Cherniak, D.J., 1995. Sr and Nd diffusion in titanite. Chem. Geol. 125, 219–232. Cherniak, D.J., 2003. Silicon self-diffusion in single-crystal natural quartz and feldspar. Earth Planet. Sci. Lett. 214, 655–668. Cherniak, D.J., Watson, E.B., 1992. A study of strontium diffusion in Kfeldspar, Na–K feldspar and anorthite using Rutherford Backscattering Spectroscopy. Earth Planet. Sci. Lett. 113, 411–425. Cherniak, D.J., Watson, E.B., 1994. A study of strontium diffusion in plagioclase using Rutherford Backscattering Spectroscopy. Geochim. Cosmochim. Acta 58, 5179–5190. Crank, J., 1975. The Mathematics of Diffusion, 2nd ed. Oxford, 414 pp. Deer, W.A., Howie, R.A., Zussman, J., 1992. An Introduction to the Rock-Forming Minerals, 2nd ed. Longman, 696 pp. Dennis, P.F., 1984a. Oxygen self-diffusion in quartz under hydrothermal conditions. J. Geophys. Res., B 89, 4047–4057. Dennis, P.F., 1984b. Oxygen self diffusion in quartz. Sixth progress report of research supported by N.E.R.C., 1981–1984. Prog. Exp. Petrol. 6, 260–265. Frischat, G.H., 1970. Sodium diffusion in natural quartz crystals. J. Am. Ceram. Soc. 53, 357. Hildreth, W., 1979. The Bishop Tuff: evidence for the origin of compositional zonation in silicic magma chambers. Spec. Pap. Geol. Soc. Am. 43. Mizutani, S., Ohdomari, I., Miyazawa, T., Iwamori, T., Kimura, I., Yoneda, K., 1982. Diffusion of gallium in quartz and bulk-fused silica. J. Appl. Phys. 53, 1470–1473. Pankrath, R., Flörke, O.W., 1994. Kinetics of Al–Si exchange in low and high quartz: calculation of Al diffusion coefficients. Eur. J. Mineral. 6, 435–457. Peppard, B.T., Steele, I.M., Davis, A.M., Wallace, P.J., Anderson, A.T., 2001. Zoned quartz phenocrysts from the rhyolitic Bishop Tuff. Am. Mineral. 86, 1034–1052. Pyle, J.M., 2006. Temperature–time paths from phosphate accessory phase paragenesis in the Honey Brook Upland and associated cover sequence, SE Pennsylvania, USA. Lithos. 88, 201–232.

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Reddy, K.P.R., Cooper, A.R., 1982. Oxygen diffusion in sapphire. J. Am. Ceram. Soc. 65, 634–638. Shannon, R.D., 1976. Revised effective ionic radii and systematic studies of interatomic distances in halides and chalcogenides. Acta Crystallogr., A 32, 751–767. Sharp, Z.D., Giletti, B.J., Yoder jr., H.S., 1991. Oxygen diffusion rates in quartz exchanged with CO2. Earth Planet. Sci. Lett. 107, 339–348. Verhoogen, J., 1952. Ionic diffusion and electrical conductivity in quartz. Am. Mineral. 37, 637–655.

Wark, D.A., Spear, F.S., 2005. Ti in quartz: Cathodoluminescence and thermometry. Goldschmidt 2005 abstract. Wark, D.A., Watson. E.B., in press. TitaniQ: A Titanium-in-Quartz Geothermometer Contributions to Mineralogy and Petrology. (Published electronically — doi:10.1007/s00410-006-0132-1). Wark, D.A., Hildreth, W., Spear, F.S., Cherniak, D.J., Watson, E.B., in press. Pre-eruption recharge of the Bishop magma chamber. Submitted to Geology.