Viscosity of magmatic liquids: A model

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Abstract

The viscosity of silicate melts controls magma transport dynamics, eruption style and rates of physicochemical processes (e.g., degassing, crystallization) in natural magmas. Thus a comprehensive viscosity model for magmatic liquids has long been a goal of earth scientists. Here we present a model that predicts the non-Arrhenian Newtonian viscosity of silicate melts as a function of T and melt composition, including the rheologically important volatile constituents H2O and F. Our model is based on > 1770 measurements of viscosity on multicomponent anhydrous and volatile-rich silicate melts. The non-Arrhenian T-dependence of viscosity is accounted for by the VFT equation [log η = A + B / (T(K)  C)]. The optimization assumes a common, high-T limit (A) for silicate melt viscosity and returns a value for this limit of − 4.55 (+ 0.2) (e.g., log η ~ 10 4.6 Pa s). All compositional dependence is ascribed to the parameters B and C and is accounted for by an additional 17 model coefficients. Our model is continuous in composition- and temperature-space and predicts the viscosity of natural volatile-bearing silicate melts (SiO2, Al2O3, TiO2, FeOtot, CaO, MgO, MnO, Na2O, K2O, P2O5, H2O, F2O 1) over fifteen log units of viscosity (10 1 1014 Pa s). The model for viscosity can also predict other transport properties including glass transition temperatures (Tg) and melt fragility (m). We show strong systematic decreases in Tg and m with increasing volatile content. This pattern has implications for predicting styles of volcanic eruption and understanding silicate melt structure. Our model transforms a quarter-century of experimental study of melt viscosities, into a parameterisation having a predictive capacity that makes it relevant to diverse fields of research including: volcanology, geophysics, petrology and material sciences.

Introduction

Viscosity is the single most important physical property governing the production, transport and eruption of magmas (Papale, 1999, Sparks, 2004, Dingwell, 2006). The viscosity of naturally-occurring silicate magmas can span more than 15 orders of magnitude (10 1–1014 Pa s) primarily in response to variations in temperature (T), melt composition and the proportions of suspended solids and/or exsolved fluid phases (Dingwell, 1996, Giordano et al., 2004a, Giordano et al., 2004b). Dissolved volatiles, such as H2O (Giordano et al., 2004a) and F (Giordano et al., 2004b), are of particular importance because small variations in their concentrations can generate large (> 105) and strongly nonlinear changes in melt viscosity that drastically affect magma transport, ascent and eruption dynamics. These effects may even dictate whether eruptions are effusive or explosive (Dingwell, 1996, Papale, 1999, Sparks, 2004). Other volatile components (CO2, Cl, Br, I, S) appear to be less important in terms of directly influencing viscosity (e.g., Dingwell and Hess, 1998, Bourgue and Richet, 2001, Zimova and Webb, 2006). Thus, the accurate prediction of silicate melt viscosity as a function of temperature and composition, including H2O and F, is of paramount importance for modelling and understanding magmatic and volcanic processes.

The task of creating such models has been hampered, to date, by the complexity of incorporating these volatile effects, together with multicomponent melt compositional effects, into the framework of a non-Arrhenian model (Baker, 1996, Hess and Dingwell, 1996, Russell et al., 2002, Giordano and Dingwell, 2003, Russell and Giordano, 2005, Giordano et al., 2006, Vetere et al., 2006, Hui and Zhang, 2007). The earliest models for predicting the viscosity of geologically-relevant silicate melts adopted a strictly Arrhenian temperature dependence (Shaw, 1972, Bottinga and Weill, 1972). These models were based on a relatively small number of high-T experiments (~ 130). They were, and remain, quite effective in reproducing silicate melt viscosities over a restricted range of compositions at high, generally near-liquidus, temperatures. Arrhenian models, however, inevitably fail when extrapolated to lower temperatures. This is a simple consequence of the now universal observation that silicate melts show pronounced non-Arrhenian temperature dependence (Dingwell et al., 1993, Richet and Bottinga, 1995).

Accordingly, the next generation of viscosity models for silicate melts adopted a non-Arrhenian T-dependence usually involving 3 (rather than 2) adjustable parameters. These models were, however, designed to span restricted ranges in melt composition and some are extremely effective in doing so. For example, several of these computational schemes were developed to model the effects of H2O on the viscosity of single specific melt composition (Baker, 1996, Hess and Dingwell, 1996, Giordano et al., 2004a, Vetere et al., 2006). There are several models for melt viscosity that incorporate a non-Arrhenian T-dependence and span wide ranges of melt composition (e.g., Giordano and Dingwell, 2003, Russell and Giordano, 2005, Giordano et al., 2006). However, they do not incorporate the effects of dissolved volatiles (especially H2O and F). The sole exception is the model published by Hui and Zhang (2007) which predicts melt viscosity as a function of temperature and composition and is applicable to hydrous silicate melts. Currently, this is the only viscosity model for multicomponent silicate melts that accounts for the effects of H2O.

Here we present a multicomponent chemical model for predicting the viscosity of naturally-occurring silicate melts based on experimental measurements of viscosity at T(K) on melts of known composition (Fig. 1) at atmospheric pressure (105 Pa). The model has the following attributes: i) it spans most of the compositional range found in naturally-occurring volcanic rocks, ii) it captures the effects of 10 major and minor oxide components and the volatile components H2O and F, iii) it is computationally continuous across the entire compositional and temperature spectrum of the database, iv) it is capable of accommodating both strong (near-Arrhenian T-dependence) and fragile (non-Arrhenian T-dependence) behaviour of silicate melts (e.g., Angell, 1985), and v) it reproduces observed relationships between melt composition and transport properties such as glass transition temperature (Tg) and fragility (m) (Angell, 1995, Plazek and Ngai, 1991) (see Section 5.2 for definition).

We anticipate this model finding widespread use in the petrological and volcanological sciences because of these traits. In particular, the low number (i.e. 18) of model parameters should facilitate incorporation into complex applications (e.g., volcanic eruption models) that involve dynamic changes in melt composition and for greater stability when the model is extrapolated outside the compositional range of the experimental database.

Section snippets

Experimental database

The primary dataset, against which the model is calibrated, comprises 1774 experimentally measured pairs of values of T(K) and log η on silicate melts of known composition. Our database includes measurements from many laboratories but most are from work in experimental labs from the past 20 years; sources of data are available as supporting material in the Appendix. Most measurements derive from high-T concentric cylinder experimentation (η ~ 10 1 to 105 Pa s) and from

Model optimization

The experimental database comprises only viscosity measurements that are coupled to melts for which there are complete and accurate compositions, including direct analysis of volatiles (e.g., spectroscopic or volumentric). We use the oxide mol% as a chemical basis and treat all iron as FeO even though there is evidence that Fe2O3 and FeO play distinct roles in governing melt structure (Mysen, 1988) and that the redox state of very Fe-rich melts does measurably influence viscosity (Dingwell, 1991

Model quality

There is excellent agreement between values of viscosity obtained by experiment and those predicted by the model (Fig. 3). The model has a root-mean-square-error (RMSE) of 0.40 log units and anhydrous and hydrous melts have an average misfit (∣observed  predicted∣) of ± 0.25 and ± 0.35 logunits, respectively. The data plotted in Fig. 3 show that the largest residuals derive from the low-T data where viscosity values exceed 108 Pa s (Table 3; Fig. 3A, B). Over this range of viscosity, the absolute

Model consequences

The quality of our model is demonstrated, in part, by how well it reproduces measured values of melt viscosity (Fig. 3). However, higher-level models have the power to extrapolate and have logical consequences that can serve as testable predictions (Greenwood, 1989). The model we have presented here can be tested against the expected behaviour of other transport properties with respect to composition (Fig. 7, Fig. 8).

Applications

Our multicomponent chemical model provides a means to predict the viscosity of volatile-enriched melts. It does so over the full range of compositions found in naturally-occurring silicate melts. A feature that sets this model apart from all others is that it allows for accurate, continuous prediction of melt properties as a function of temperature and melt composition. The effects of pressure on the silicate melt viscosity are not accounted for in this model. Natural systems feature constantly

Conclusions

We have presented a computational model that predicts, within inter-laboratory experimental error, viscosity and other melt properties (i.e., Tg and m) for most of the T-X space found in naturally-occurring silicate melts. The model accommodates the properties of fragile (non-Arrhenian) to strong (near-Arrhenian) melts equally well. The model also reproduces the T-log η relationships for melts not used for calibration purposes (see Appendix A) suggesting that the model can be extrapolated past

Acknowledgements

A portion of this research was completed when D. Giordano held a post-doctoral fellowship from the Dorothy Killam Trust administered by The University of British Columbia. JKR is supported by the Natural Sciences and Engineering Research Council of Canada. This work was supported by the 2005–2006 INGV-DPC project V3-2/UR17. Experimental viscometry activities of Dingwell´s group contributing to the completeness of the data set that was a prerequisite for this model have been supported by the

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