Liquid and magma viscosity in the anorthite-forsterite-diopsidequartz system and implications for the viscosity-temperature paths of cooling magmas

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1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 112,, doi: /2006jb004812, 2007 Liquid and magma viscosity in the anorthite-forsterite-diopsidequartz system and implications for the viscosity-temperature paths of cooling magmas Jacqueline M. Getson 1 and Alan G. Whittington 1 Received 20 October 2006; revised 19 April 2007; accepted 31 July 2007; published 18 October [1] We measured the viscosity of 17 liquids in the systems anorthite-forsterite-quartz and anorthite-diopside-forsterite, representing analogs of dacitic, basaltic andesitic, and basaltic magmas. The three series lie in the anorthite liquidus fields and represent liquid lines of descent. The viscosity of evolving basaltic and basaltic andesite liquids changes little during cooling because the viscosity decrease associated with removal of polymerized anorthite component from the melt offsets the viscosity increase associated with cooling. A minimum liquidus viscosity in the basaltic system is encountered in the anorthite stability field. In contrast, dacitic liquids appear insensitive to the compositional changes during anorthite crystallization, and liquidus viscosity increases monotonically between anorthite and the anorthite-forsterite-quartz eutectic. Using the Einstein-Roscoe equation to calculate the rheological effect of crystals, we show that magma viscosity always increases during crystallization for closed systems and that crystallinity is the dominant control on the viscosity of all three magma series. We introduce the concept of viscosity paths, which may approach one of four end-members: (1) equilibrium crystallization (closed system) and (2) perfect fractional crystallization (open system) are both associated with changing residual liquid compositions; (3) supercooling without crystal growth maintains constant liquid composition, while (4) increasing crystallinity at constant liquid composition is only permissible for eutectic starting compositions. Because of the dependence of crystal-liquid segregation rates on liquid viscosity and crystal buoyancy, feedback relations exist between evolving liquid composition, crystallinity, and magma viscosity. Crystal-liquid segregation rates may increase or decrease during progressive cooling and crystallization of basaltic liquids. Citation: Getson, J. M., and A. G. Whittington (2007), Liquid and magma viscosity in the anorthite-forsterite-diopside-quartz system and implications for the viscosity-temperature paths of cooling magmas, J. Geophys. Res., 112,, doi: /2006jb Introduction [2] The viscosity of aluminosilicate liquids and magmas exerts a fundamental control over the rates and styles of many geological processes, including segregation of melt from source regions, the ascent and emplacement of magmas, mixing and mingling in magma chambers, and eruption style. Magma viscosity can vary by many orders of magnitude within a single environment, and is very sensitive to liquid composition, temperature, and crystal content. The viscosity of magma is dependent on its constituent phases: silicate liquid and entrained crystals. The most reliable way to determine silicate liquid viscosity is to measure it directly, at both superliquidus conditions and close to the glass transition, to fully account for non- 1 Department of Geological Sciences, University of Missouri-Columbia, Columbia, Missouri, USA. Copyright 2007 by the American Geophysical Union /07/2006JB004812$09.00 Arrhenian behavior. During cooling, progressive crystallization leads to changes in the bulk composition of the residual liquid, leading to potentially complex changes in magma viscosity. The bulk viscosity of magma also includes the effects of crystals, and bubbles if present. The physical effect of crystals, approximated as rigid spherical inclusions, can be assessed using the Einstein- Roscoe equation [e.g., Marsh, 1981]. The ratio of the mixture (melt + crystals) viscosity, h mix, to the melt viscosity, h melt,is h r ¼ h mix =h melt ¼ ð1 f=f 0 Þ 2:5 ð1þ where f is the crystallinity, and f 0 is the maximum packing fraction. A modification that takes into account melt trapped between solid particles and hence not available for shear was given by Spera [2000]: h r ¼ h mix =h melt ¼ ð1 f=y M Þ q ð2þ 1of19

2 Figure 1. Physical effect of crystals on magma viscosity, calculated using the Einstein-Roscoe equation [Marsh, 1981] and a modified form [Spera, 2000]. where q is a positive integer between 2 and 3, and y M depends on f according to y M ¼ 1 f½1 f 0 Š=f 0 ð3þ Both equations (1) and (2) suggest similar increases in magma viscosity increases with crystallinity, of 0.5 log units for a crystallinity of vol %, and 1 log unit for 40 vol % (Figure 1). At higher crystal contents, magmas first behave like Bingham fluids that possess a yield strength, and then rheology becomes highly non-newtonian, as the crystal mush approaches a semirigid crystalline network containing interstitial melt [e.g., Petford, 2003]. Experiments on magma viscosity during congruent crystallization of liquid Mg 3 Al 2 Si 3 O 12 and CaAl 2 Si 2 O 8 at low strain rates have confirmed the general utility of equation (1) up to 40 vol % crystallinity for a variety of crystal shapes [Lejeune and Richet, 1995; Lévesque, 1999]. The transition to non- Newtonian rheology arising from particle interaction is less abrupt for crystals having different shapes and showing a size distribution [Lévesque, 1999] than for spherical particles of nearly the same size [Lejeune and Richet, 1995]. Costa [2005] developed an expression for relative viscosity of a crystal mush that reproduces experimental data well over almost the entire range of solid fraction, which gives values close to those of equations (1) and (2) for solid fractions less than 0.5. [3] During cooling and crystallization, the bulk composition of the residual liquid phase within the magma undergoes incremental changes in chemical composition. This causes a change in both the residual liquid viscosity, and also the bulk viscosity of the magma. This chemical effect of crystallization on magma viscosity is typically ignored in petrologic models of igneous processes, in part because of the lack of a reliable predictive model for liquid viscosity as a function of temperature and bulk composition, although its potential importance has been stressed [McBirney and Murase, 1984]. [4] Previous measurements of the viscosity of crystallizing magma have been relatively few. While all have recognized the combined influence of crystal content and changing residual liquid viscosity, the latter has typically been calculated using models, in order to extract the physical effect of crystals from the data. Both Ryerson et al. [1988], studying picritic Kilauea Iki basalt between 1249 and 1149 C, and Ishibashi and Sato [2007], studying alkali olivine basalt from 1230 to 1140 C, found reasonable agreement with the predictions of the Einstein-Roscoe equation. Sato [2005], studying high-alumina basalt from Mount Fuji between 1230 and 1130 C, and Murase et al. [1985], studying Mount St. Helens dacite between 800 and 1225 C, both calculated liquid viscosity using the model of Shaw [1972] and inferred a larger effect of crystals than would be predicted from the Einstein-Roscoe equation. Sato [2005] ascribed this to entanglement of acicular plagioclase even at low crystallinities. The experiments of Murase et al. [1985] required a significant extrapolation of Arrhenian behavior to temperatures several hundred degrees below the liquidus, resulting in underestimation of residual liquid viscosity and overestimation of the physical effect of crystals. Other potential complicating factors in their measurements, such as non-newtonian behavior and the effect of strain rate on apparent viscosity, are well reviewed by Pinkerton and Stevenson [1992]. Bouhifd et al. [2004] studied the viscosity of alkali basalt during progressive crystallization between about 650 and 880 C, and directly measured the viscosity of several residual liquids. Their results indicate that increasing crystal content has a greater effect than changing residual liquid composition, at least at the relatively low temperatures for these experiments, and are also consistent with the Einstein-Roscoe equation. The employed low temperatures led to crystallization of metastable spinel and silica-poor pyroxene, so that the experiments do not track equilibrium crystallization. [5] Here we adopt an alternative approach to determine the chemical effect of changing residual liquid composition on the viscosity of crystallizing magma, by measuring the viscosity of liquid compositions along the liquid line of descent. Starting with any given bulk composition along this line, the crystal fraction can then be calculated from the phase diagram, and the magma viscosity can be calculated if the liquid viscosity is known as a function of composition and temperature along the liquidus. Phase equilibria for natural magmas are complex, and have several liquidus phases by the time the eutectic is reached. Therefore we chose to conduct a preliminary study in the CaO-MgO- Al 2 O 3 -SiO 2 (CMAS) system, a good first-order simplification of the major igneous rock types, where phase equilibria are well known and only one or two crystalline phases will be present over large temperature intervals. There are some caveats to this approach: We rely on the Einstein-Roscoe equation to calculate magma viscosity, which means that factors such as crystal size and shape distributions, and development of a yield stress or other non-newtonian behavior, cannot be explicitly considered. However, these 2of19

3 Table 1a. Compositions of Synthetic Glasses in the Anorthite-Diopside-Forsterite System BA81 BA58 BA52 BA42 BA36 BA24 BA11 BA0 Oxides, wt % SiO (0.21) a (0.18) (0.18) (0.13) (0.15) (0.21) (0.25) (0.14) Al 2 O (0.19) (0.10) (0.24) (0.13) (0.08) (0.06) 6.06 (0.07) 0.44 b (0.02) MgO 3.39 (0.03) 7.97 (0.07) 9.06 (0.09) (0.09) (0.05) (0.11) (0.09) (0.08) CaO (0.07) (0.12) (0.05) (0.03) (0.09) (0.11) (0.06) (0.17) Total (0.50) (0.47) (0.56) (0.38) (0.37) (0.49) (0.47) (0.41) Oxides, mol % SiO Al 2 O MgO CaO Normative Mineralogy, mol % An Fo Di Density, c gcm Molar mass, g NBO/T d Al/(Al+Si) SM e a Values in parentheses are the standard deviations of six spots measured on a JEOL 733 Superprobe at Washington University, St. Louis. b Alumina was not added to this sample, all calculations were performed assuming 0 wt % Al 2 O 3. c Density determined using Archimedes principle, with distilled water as the immersion liquid. d NBO/T = (2 (CaO + MgO-Al 2 O 3 ))/(SiO 2 +Al 2 O 3 +Al 2 O 3 ), from Mysen and Richet [2005]. e SM parameter = (CaO + MgO), based on the work by Giordano and Dingwell [2003]. factors are strictly physical in nature, and can be simply accounted for in the magma viscosity calculation if so desired, for example by applying the shape factor correction of Simha [1940]. [6] Previous experiments on basaltic magma viscosity during the crystallization of mafic phases [Ryerson et al., 1988; Bouhifd et al., 2004; Ishibashi and Sato, 2007] confirm that increasing crystal content and increasingly silicic residual liquid both produce viscosity increases, so that resolving the two effects can be difficult. Molten feldspars are highly polymerized, in contrast to liquids corresponding to mafic phenocryst phases such as olivine and pyroxene. While plagioclase can be the liquidus phase in high-alumina basalts, and in some lunar ferrobasalts [e.g., Grove, 1978], it is more commonly one of the later phases to appear during crystallization. However, its near ubiquity as a crystalline phase in common magmas underlines the importance of understanding both the physical and chemical effects of its crystallization on magma rheology. [7] We quantify the change in liquid viscosity due to removal of the polymerized feldspar component, and test the hypothesis that this effect may at least partially offset the physical effect of crystals, particularly for low to intermediate crystal volume fractions typical of erupted lavas. Experimental measurements are presented of the viscosity of 17 liquids in the anorthite-forsterite-quartz and anorthitediopside-forsterite systems, representing basaltic to dacitic liquidus compositions during cooling and plagioclase crystallization. Combined with equations for the effect of crystals on magma rheology, we calculate changes in the viscosity of bulk magmas (liquid + crystals), during cooling and progressive crystallization. The results indicate distinctly nonlinear changes in bulk magma viscosity with decreasing temperature, and different behavior for dacitic and basaltic magmas. 2. Sample Compositions [8] Seventeen CaO-MgO-Al 2 O 3 -SiO 2 glasses were chosen, forming three linear compositional series in the anorthite-diopside-forsterite-silica system (Tables 1a and 1b). The series correspond approximately to liquids of basalt, basaltic andesite, and dacite composition. All three compositional series have anorthite as one end-member, so that the liquids fall on the liquid line of descent for the most anorthite-rich composition in each series. [9] The eight samples of the basaltic analog series BA lie in the CaAl 2 Si 2 O 8 -Mg 2 SiO 4 -CaMgSi 2 O 6 system, along a line between anorthite and Di 90 Fo 10, passing through the An-Di-Fo ternary eutectic (Figure 2a). Although this is strictly a piercing point, as the true eutectic composition lies outside the plane due to diopside exhibiting minor solid solution with the Ca-Tschermak s end-member CaAl 2 SiO 6 [e.g., Morse, 1994], for simplicity, we will refer to this piercing point as the eutectic composition in subsequent discussion. On the magnesium-rich side of the eutectic point, this line is approximately coincident with the diopside-forsterite cotectic in the 1 atm phase diagram. These liquids therefore fall on the liquid line of descent for the anorthite-free end-member of this series. The compositions outside the anorthite stability field therefore allow us to investigate the effects of fractionating depolymerized components (pyroxene and olivine) from the melt. [10] The four samples of the basaltic andesite analog series SM lie in the CaAl 2 Si 2 O 8 -Mg 2 SiO 4 -SiO 2 system, along the anorthite-enstatite join. The least anorthite-rich 3of19

4 Table 1b. Compositions of Synthetic Glasses in the Anorthite-Forsterite-Quartz System SM66 SM52 SM44 SM36 SA55 SA47 SA42 SA31 SA23 Oxides, wt % SiO (0.18) (0.18) (0.15) (0.20) (0.32) (0.23) (0.20) (0.11) (0.15) Al 2 O (0.09) (0.12) (0.11) (0.11) (0.11) (0.04) (0.08) (0.15) (0.10) MgO 7.74 (0.09) (0.06) (0.08) (0.04) 3.74 (0.05) 4.76 (0.02) 5.59 (0.04) 7.34 (0.08) 9.20 (0.07) CaO (0.12) (0.03) (0.05) (0.08) (0.06) (0.07) (0.04) (0.04) (0.06) Total (0.48) (0.39) (0.39) (0.43) (0.54) (0.36) (0.36) (0.38) (0.38) Oxides, mol % SiO Al 2 O MgO CaO Normative Mineralogy, mol % An Fo Q Density, g cm Molar mass, g NBO/T Al/(Al+Si) SM composition lies on the anorthite-forsterite cotectic in the 1 atm phase diagram (Figure 2b). The five samples of the dacitic analog series SA lie in the same system, on a line between anorthite and the anorthite-enstatite-tridymite eutectic. These two series were chosen as analogs to calcalkaline arc magmas of basaltic andesite and dacite composition, based on whole rock analyses from Santa Maria and Santiaguito volcanoes in Guatemala [Avard et al., 2006]. These analyses were converted to CIPW norms, and the normative mineralogy then converted to molar fractions. Quartz, hypersthene, and anorthite account for an average of 89% of the normative mineral assemblage on a molar basis. All the Santiaguito samples fall in the anorthite stability field on the ternary diagram (Figure 2b). Samples collected from Santiaguito all contain abundant plagioclase feldspar as phenocrysts and groundmass microlites (Figure 3), with orthopyroxene as a subordinate phenocryst. We therefore expect the simplified compositions of the SA and SM series Figure 2. Liquid compositions synthesized for this study. Sample names consist of two letters to identify the series, and a number corresponding to the anorthite content in mol %. Compositions are given in Tables 1a and 1b. (a) CaAl 2 Si 2 O 8 -CaMgSi 2 O 6 -Mg 2 SiO 4 phase diagram (wt %) at 1 atm pressure, modified after Osborn and Tait [1952] with compositions of glass series BA. (b) CaAl 2 Si 2 O 8 - Mg 2 SiO 4 -SiO 2 phase diagram (wt %) at 1 atm pressure, modified after Anderson [1915], with compositions of glass series SM and SA. Grey field contains bulk rock compositions of basaltic andesites and dacites from Santa Maria and Santiaguito volcanoes, Guatemala, as projected into this system. 4of19

5 experiments reported here) because the viscometer is calibrated empirically. [13] Viscosity is defined as the ratio of applied stress to resulting shear strain rate. For the concentric cylinder geometry used here, stress s depends on torque t applied by the measuring head, rotor radius R b, and the effective length L of the rotor immersed in the melt, such that s ¼ t= 2pR 2 b L ð4þ Figure 3. Thin section of dacitic lava from Santiaguito dome, Guatemala, showing abundant plagioclase feldspar, predominantly as phenocrysts but also as groundmass microlites showing trachytic texture. Volumetrically minor orthopyroxene can be seen at top center. Crossed polars, field of view is 2 mm. to be reasonable rheological analogs for anhydrous calcalkaline magmas. 3. Experimental Methods [11] Synthetic glasses were produced from a combination of oxide and calcium carbonate powders. After weighing, these were mixed and ground under acetone in a shatterbox, decarbonated by slow heating to 1600 C in platinum crucibles in a muffle furnace, and quenched by pouring on to a thick copper plate. This was followed by repeated cycles of grinding, fusion and quenching, to ensure homogeneity. Optical and electron microscopy confirmed that the glasses were bubble- and crystal-free, and compositions were checked by electron microprobe (Tables 1a and 1b). [12] Viscosity was measured over two different temperature-viscosity ranges, using two different instruments. Hightemperature viscosity measurements were conducted at superliquidus conditions using concentric cylinder viscometry. The apparatus is a Theta Industries Rheotronic II 1600C Rotating Viscometer, using a Brookfield HBDV-III Ultra measuring head, which can measure over the range 1 to 10 5 Pa s, with an upper temperature limit of 1595 C. The outside cylinder, which contains the melt, remains stationary. A concentric cylindrical rotor is lowered into the melt, and is rotated by a motor through a calibrated spring within the measuring head. The melt exerts a viscous drag on the spindle, and the measuring head records the torque required to achieve a particular angular velocity, which can be varied between 0.1 and 250 revolutions per minute. A typical experimental run uses approximately 60 g of sample placed within an alumina crucible, 33 mm inside diameter by 65 mm high, which is centered on three alumina sample holder rods and lowered to a set position within the furnace. Great care is taken to ensure that the rotor is always lowered exactly the same distance into the melt (18 mm for all The effective length accounts for end effects arising from a finite cylinder length, which are minimized by keeping the rotor at least 10 mm from the bottom of the crucible. We used alumina rotors with a diameter of 6.25 mm. The angular strain rate dg/dt depends on the geometry of the two cylinders, R c being the inside diameter of the outer cylinder, and the angular velocity w: d g =dt ¼ 2wR 2 c = R2 c R2 b In practice, the viscometer head measures angular velocity and torque. The three variables relating to crucible geometry (R b, R c and L) were the same in every experiment we conducted. In order to account for end effects in calculating the effective length, a constant calibration factor was determined through experiments on an oil viscosity standard at room temperature. We then checked this calibration constant by comparing measured and reference viscosity values for National Institute of Standards and Technology (NIST) standard glass 717a at high temperatures, resulting in an adjustment to the constant of 0.4% or less than log units. [14] After calibration, we measured both NIST standard glasses 717a and 710a as unknowns, and compared the values to the certified reference values. Over the temperature range 850 to 1500 C and the viscosity range 6 to 55,000 Pa s, the average absolute deviation (AAD) between measured and certified values was log units for 717a (number of measurements n = 10) and log units for 710a (n = 19), with root mean square (RMS) deviations of log units for 717a and log units for 710a. For viscosities higher than 100 Pa s, the AAD is and the RMS is log units for 717a (n = 9) and the AAD is and the RMS is log units for 710a (n = 10). On the basis of repeat experiments using standard glass 710a, the reproducibility of the measurements is better than 0.02 log units. For comparison, the certified viscosity values are quoted with uncertainties of ±0.1 log unit (Figure 4), and the slight discrepancy between our measured values and the certified values for NIST 710a is within the certificate uncertainty. On the basis of the agreement with standard glasses, the experimental setup is certainly accurate to within 0.06 log units, and probably to within 0.02 log units. Relative uncertainties between samples are less than 0.02 log units. [15] During all concentric cylinder viscometer experiments, multiple shear rates were used at each temperature to verify Newtonian behavior, such that strain rate was proportional to shear stress over the range of investigated conditions. Every melt investigated was Newtonian, over a range of shear stresses between 440 and 1350 Pa, and shear ð5þ 5of19

6 alumina dissolution occurred, it had a negligible effect on measured viscosity. [16] Low-temperature viscosity measurements were conducted near the glass transition using parallel plate viscometry. The apparatus is a Theta Industries Rheotronic III 1000C Parallel Plate Viscometer, which can measure over the range Pa s. Cylindrical glass cores with polished parallel ends are placed between silica plates and subjected to a known uniaxial stress, s. The sample height is measured to ±0.1 mm using a transducer and the vertical strain rate de/dt is used to calculate the viscosity: h ¼ s=3 ðde=dtþ ð6þ The factor of three arises from the conversion of a linear strain rate to the volumetric shear viscosity. Platinum foil is placed between the glass cylinder and the silica plates to avoid adhesion, and the equation used is for the perfect slip condition where the sample remains cylindrical [e.g., Dingwell, 1995]. Visual inspection and comparison with NIST standards confirmed that the perfect slip condition is a valid for our experiments. Typical runs used cores 10 mm long and 6.5 mm in diameter, with a 1500g vertical load. The range of longitudinal strain rates was between to s 1, and most runs involved less than 20% shortening. The accuracy and precision of the parallel plate measurements is ±0.05 log units, based on agreement with recommended values for NIST standard glass 717a, and on duplicate experiments. Full details are given by Hellwig [2006]. Figure 4. Concentric cylinder viscometry data for NIST standard glasses 717a (soda lime silicate) and 710a (borosilicate). Dotted lines represent ±0.1 log units from the certified TVF equations. strain rates between 0.05 and 55 s 1. Dissolution of the alumina beaker and spindle into alumina-undersaturated melts can potentially lead to changes in the liquid composition. We conducted microprobe analyses of glass compositions on traverses away from crucible walls after the experiments were complete. Minor reaction was observed, being greatest in BA11, the least aluminous melt (BA0 could not be analyzed because it crystallized completely during quenching). Beyond a 2mm zone of significant contamination adjacent to the crucible wall, compositions returned to within 2 wt % Al 2 O 3 of the starting value (6.12 wt %) and were homogeneous. Dacitic liquids have a contamination zone less than 500 mm wide, beyond which they return to exactly the original composition. For this reason, all experiments concluded by returning to the original high temperature to verify no drift during measurement, and in no case was the drift greater than the measurement uncertainty. This confirms that where minor 4. Results [17] The results of viscosity measurements for the 17 liquid compositions using concentric cylinder and parallel plate viscometry are listed in Tables 2a 2c and 3a 3c, respectively. The data are plotted as a function of inverse temperature in Figure 5, for all three series. To allow interpolation of liquid viscosity to the intermediate temperatures at which each composition would be on the liquidus within its phase diagram, the data were fit to Tammann- Vogel-Fulcher equations of the form log h ¼ A þ B= ðt CÞ ð7þ where T is temperature in K and A, B, and C are adjustable parameters [Vogel, 1921]. If C = 0, this reduces to an Arrhenian (straight line) relationship between viscosity and inverse temperature. The parameters for each composition are listed in Table 4. For anorthite liquid, we used the Tammann-Vogel-Fulcher (TVF) equation derived by Russell and Giordano [2005] from a compilation of literature data. High-temperature data were not obtained for sample BA81 because of its high liquidus temperature, close to the operational limit of the viscometer. The low-temperature data alone do not allow three independent parameters to be well constrained, and the TVF parameters given for this composition were obtained by viscosity modeling (section 5.2). [18] Within each series, viscosity increases with increasing anorthite content, which corresponds to increasing bulk 6of19

7 Table 2a. High-Temperature Viscosity Data for the BA Series a BA58 BA52 BA42 BA36 BA24 BA11 BA0 T, K log h T, K log h T, K log h T, K log h T, K log h T, K log h T, K log h a All viscosities are reported in Pa s. polymerization of the liquid, represented by a decreasing ratio of nonbridging oxygens to tetrahedral cations (NBO/T) [Mysen, 1988]. At high temperatures, all samples in the SA series have very similar viscosities, and the least anorthiterich sample apparently has the highest viscosity. However, viscosity differences between samples are much greater at low than at high temperatures, and TVF curves for all liquids have intercepts on the viscosity axis between 6.5 and 4.9 log Pa s (Table 4). For a given anorthite content, the BA series has both the lowest bulk polymerization, and the lowest viscosity. [19] Departures from Arrhenian behavior can be quantified via the parameter F D = C/B [Russell and Giordano, 2005], which is one expression of the fragility of a melt. Fragile liquids show dramatic changes in melt structure and viscosity with changes in temperature, leading to substantially non-arrhenian behavior, while strong liquids undergo only small structural changes and have near-arrhenian behavior [Angell, 1991]. The BA series shows the greatest departure from Arrhenian behavior, and the SA series is closest to Arrhenian, but within each series the value of F D is always smallest at intermediate compositions, indicating that fragility does not correlate directly with NBO/T. [20] These results allow an important conclusion to be drawn: that the incremental removal of a polymerized component from an already depolymerized melt can result in an incremental decrease in melt viscosity. The variations in liquid viscosity are largest for the BA series, which also has the largest range in NBO/T values, and are smallest for the SA series, which has only a small range of NBO/T. However, the viscosity differences between liquids with similar NBO/T in the three systems emphasize that this parameter is not a complete description of the chemical and structural state of a melt. For example, the viscosities of theoretically fully polymerized melts vary substantially between albite, orthoclase and anorthite [e.g., Urbain et al., 1982], but NBO/T is nonetheless useful in interpreting the bulk physical properties of depolymerized melts. 5. Modeling Liquid Viscosity in the CaAl 2 Si 2 O 8 - Mg 2 SiO 4 -CaMgSi 2 O 6 System [21] The aim of this work is to examine the viscosity of liquids and magmas during cooling and crystallization. Before applying the results of the measurements described Table 2c. High-Temperature Viscosity Data for the SA Series Table 2b. High-Temperature Viscosity Data for the SM Series SM66 SM52 SM44 SM36 T, K log h T, K log h T, K log h T, K log h SA55 SA47 SA42 SA31 SA23 T, K log h T, K log h T, K log h T, K log h T, K log h of19

8 Table 3a. Low-Temperature Viscosity Data for the BA Series a BA81 BA58 BA52 BA42 BA36 BA24 BA11 BA0 T, K log h T, K log h T, K log h T, K log h T, K log h T, K log h T, K log h T, K log h (c) (c) (b) (b) (b) (c) (d) (b) (c) (d) (a) (b) (a) (b) (c) (c) (d) (d) (a) (c) (a) (b) (d) (b) (d) (c) (b) (b) (b) (c) (c) (c) (c) (d) (a) (c) (a) (b) (d) (b) (c) (c) (b) (b) (b) (c) (c) (c) (d) (d) (a) (c) (a) (c) (d) (b) (d) (c) (b) (b) (b) (b) (c) (c) (c) (d) (a) (c) (a) (c) (d) (b) (d) (c) (b) (c) (b) (b) (c) (c) (c) (d) (a) (c) (a) (c) (d) (b) (c) (c) (b) (c) (b) (b) (c) (c) (d) (d) (a) (a) (c) (b) (d) (c) (b) (b) (b) (c) (d) (a) (a) (b) (b) (c) (a) a All viscosities are reported in Pa s. The a, b, c, and d in parentheses denote experiments using different glass cylinders. above, it is pertinent to assess whether existing predictive models of anhydrous silicate liquid viscosity are adequate to allow calculation of liquid viscosities without requiring further experimental measurements. In section 5.1 we compare our experimental viscosity determinations with commonly used literature models, and find the models insufficiently precise. In section 5.2 we generate a viscosity model for compositions of the BA series, which can be combined with knowledge of the liquidus surface to calculate the liquidus viscosity at any point between anorthite and the ternary eutectic, and along the Di-Fo cotectic Comparison With Previous Viscosity Models [22] Three models for the viscosity of multicomponent silicate melts exist in the literature. Bottinga and Weill [1972] and Shaw [1972] adopted the approximation of Arrhenian behavior, limiting their applicability to the high (superliquidus) temperatures at which most measurements at the time were obtained. Bottinga and Weill [1972] did not give coefficients for temperatures less than 1200 C, preventing use of their model at lower temperatures. Although the Shaw [1972] model was not intended to be used for viscosities greater than 10 5 Pa s, it can easily be extrapolated to lower temperatures, and commonly has been in the literature. Giordano and Dingwell [2003] adopted a different approach based on the TVF equation (7), which allows for non-arrhenian behavior. From a data set of 19 natural multicomponent melts analyzed via concentric cylinder, parallel plate, and micropenetration methods, spanning a viscosity range of Pa s, Giordano and Dingwell [2003] developed equations where the three parameters of the TVF equation are each dependent on a structural parameter for that melt, calculated from its composition. Two parameterizations were produced, one as a function of NBO/T, and the other a function of the SM parameter, which is the mole fraction of structural modifier oxides. Although the SM parameter does not account for some of these oxides being in a charge-balancing role for tetrahedrally coordinated Al 3+ cations, it reproduced the experimental data set slightly better than the parameterization based on NBO/T. Giordano et al. [2006] revised the model to include an additional parameter reflecting the ratio of alkalis to alumina, producing lower viscosities for peralkaline melts and higher viscosities for peraluminous melts. [23] The models described above are compared with experimental data at 1300 C in Figure 6. We chose this temperature because it is within the range of liquidus temperatures for the samples investigated, and falls within the temperature range at which viscosity can be reliably interpolated using the TVF equations given in Table 4. Each series is plotted as a function of NBO/T, which reflects the anorthite content of each sample within a series. The Bottinga and Weill [1972] model reproduces the pattern of decreasing viscosity with increasing NBO/T for the BA and Table 3b. Low-Temperature Viscosity Data for the SM Series SM66 SM52 SM44 SM36 T, K log h T, K log h T, K log h T, K log h (a) (d) (a) (b) (a) (d) (b) (b) (b) (c) (a) (c) (a) (d) (b) (b) (b) (b) (b) (c) (a) (c) (a) (b) (b) (c) (b) (c) (a) (d) (a) (b) (b) (c) (b) (c) (b) (d) (a) (b) (b) (b) (c) (c) (a) (b) (b) (b) (c) (d) (a) (b) (d) (b) (c) (b) (a) (c) (a) (b) (a) (c) (d) (d) (b) (c) 8of19

9 Table 3c. Low-Temperature Viscosity Data for the SA Series SA55 SA47 SA42 SA31 SA23 T, K log h T, K log h T, K log h T, K log h T, K log h (c) (c) (c) (a) (c) (b) (c) (c) (c) (c) (c) (b) (a) (b) (b) (b) (c) (b) (c) (c) (c) (c) (c) (a) (b) (b) (b) (b) (b) (c) (c) (b) (a) (c) (b) (b) (c) (b) (a) (b) (b) (c) (a) (b) (c) (c) (b) (c) (a) (b) (b) (c) (a) (c) (c) (c) (b) (c) (b) (b) (c) (c) (b) (a) (c) (b) (b) (b) (c) (b) (b) (b) (c) (b) (c) (c) (c) (a) (a) (b) (b) (b) (a) (c) (b) (c) (c) (c) (b) (c) (c) (c) (b) (a) (c) (b) (b) (c) (c) (b) (b) (b) (c) (c) (a) (c) (b) (a) (c) (b) (a) (c) (b) (b) (a) (c) (a) SM series, and the calculated values are within 0.5 log units of the measured data at 1300 C for all three series. The Shaw [1972] model significantly underestimates the viscosity of most samples in each series, and also predicts a viscosity maximum in the BA series, which was not observed. The Giordano and Dingwell [2003] model using NBO/T as the composition-dependent parameter shows good agreement with the most anorthite-poor members of each series, but greatly overestimates the viscosity of anorthite-rich samples. The Giordano et al. [2006] model using SM as the composition-dependent parameter systematically underestimates liquid viscosities by about 0.5 log units at 1300 C (Figure 6d). The best model for predicting viscosity at liquidus temperatures in the simple systems considered here is that of Bottinga and Weill [1972]. This is not surprising, because it is based on a data set that is dominated by simple compositions containing only a few components, and only on measurements at superliquidus temperatures. We still find it insufficiently accurate for predicting the evolution of viscosity along the liquid line of descent because it overestimates the viscosity increase associated with basaltic liquids as they approach full polymerization. Therefore further measurements will be needed for similar analysis of other systems, and particularly for more complex systems with liquidus temperatures below 1200 C A Viscosity Model for the CaAl 2 Si 2 O 8 - Ca 0.9 Mg 1.1 Si 1.9 O 5.8 Pseudobinary [24] The ideal starting point for a model of evolving magma viscosity during cooling would include equations for the liquid composition, crystal fraction, and liquid viscosity as a function of temperature. For the basaltic andesite (SM) and dacite (SA) series, we have enough data to well constrain these parameters at several points from individual measurements over the anorthite stability field, which is the compositional range of interest. For the basaltic (BA) series we have data along the entire CaAl 2 Si 2 O 8 - Ca 0.9 Mg 1.1 Si 1.9 O 5.8 pseudobinary, enabling us to develop a simple predictive viscosity model based on variations of the TVF parameters along this join. [25] Russell and Giordano [2005] determined that viscosity data in the CaAl 2 Si 2 O 8 -NaAlSi 3 O 8 -CaMgSi 2 O 6 system could be fitted well using a constant A parameter of 5.06 log Pa s. Because the BA series shares two of these three components, we fitted TVF equations twice: first allowing A to vary and second assuming a constant A of We found that both models reproduce the experimental data well, with global root-mean-square deviations from the measured values of 0.07 log units for variable A fits and 0.08 log units for constant A fits over the 192 data for the BA series (Tables 4 and 5). [26] The next step is to find expressions for A, B, and C as a function of X An. All three parameters vary smoothly as a 9of19

10 where X is the mole fraction of anorthite, A 0, B 0,orC 0 and A 1, B 1,orC 1 describes the parameter value for each series end-members, and W A,B,C is the Margules parameter for each of the TVF parameters. For example, equation (9) describes TVF parameter A for the BA series: A ¼ 4:90ð1 X An Þ 4:71X An þ W A ½ð1 X An ÞX An Š ð9þ In this case, the value of A is 4.90 for BA0 and 4.71 for anorthite. The best fit values of the Margules parameters W A, W B and W C are listed in Table 6, and the resulting model TVF parameters as a function of composition are given in Table 7, along with the deviations from our measured data. The TVF parameters for BA42 were not used in fitting the Margules parameters, but the calculated TVF equation for BA42 still reproduces the experimental data with a root-mean-square error of only 0.12 log units. Overall, the model reproduces the entire data set for the BA series (n = 192) with a root-mean-square error of 0.10 log units. [27] This methodology was successfully employed to model liquid viscosity in the CaAl 2 Si 2 O 8 -NaAlSi 3 O 8 - CaMgSi 2 O 6 ternary system by Russell and Giordano [2005], who also constrained their B and C parameters by using TVF equations fitted to a common A value of Because A is the theoretical viscosity at infinite temperature, relaxation times given by the Maxwell relation h = G 1 t, where G 1 is the shear modulus at infinite frequency, should approach the periods of atomic vibrations and be independent of structural features [Richet, 1984; Dingwell and Webb, 1989; Russell et al., 2003]. While it is generally accepted that G 1 is approximately 10 GPa, and varies by no Figure 5. Measured liquid viscosity data against inverse temperature. Solid lines are best fit TVF equations, with parameters given in Table 4, and the anorthite TVF equation is from Russell and Giordano [2005]. (a) BA series, with TVF equation for sample BA81 calculated using the model parameters in Table 6 (see text). (b) SM series. (c) SA series. function of composition with the exception of sample BA42 (Figure 7). The TVF parameters exhibit near-parabolic curves, so we modeled their compositional variation using a simple symmetric mixing relationship of the form A; B; C ¼ A 0 ; B 0 ; C 0 ð1 X An ÞþA 1 ; B 1 ; C 1 X An þ W A;B;C ½ð1 X An ÞX An Š ð8þ Table 4. Parameters for TVF Equations, log h = A + B/(T C) Sample A, a Pa s B, PasK 1 C,K T 12, b K F D c RMS d An e BA81 f BA (0.20) 6567 (290) (12.9) BA (0.18) 6396 (249) (11.1) BA (0.19) 5470 (253) (11.6) BA (0.14) 6193 (193) (8.4) BA (0.15) 5489 (192) (8.9) BA (0.19) 5194 (242) (11.3) BA (0.08) 4372 (95) (4.6) SM (0.15) 6428 (221) (10.1) SM (0.09) 6909 (131) (5.8) SM (0.15) 7923 (248) (10.4) SM (0.15) 6566 (209) (9.1) SA (0.07) 6975 (103) (4.5) SA (0.20) 8479 (324) (12.9) SA (0.13) 8013 (202) (8.4) SA (0.13) 8092 (199) (8.0) SA (0.16) 7745 (239) (10.2) a Numbers in parentheses are absolute uncertainties. b T 12 indicates the temperature at which the viscosity is Pa s, approximating the glass transition temperature. c The fragility is calculated where F D = C/B [Russell and Giordano, 2005]. d RMS is the root-mean-square deviation between measured values and those calculated using the TVF equation. e Anorthite data from Russell and Giordano [2005, p. 5334, Table 1] with root-mean-square error of f High-temperature data could not be obtained for BA81. See text for details. 10 of 19

11 Figure 6. Comparison between experimentally determined viscosity and predicted viscosity based on literature models at 1300 C. (a) BA series, (b) SM series, and (c) SA series. Graphs a), b) and c) are plotted against NBO/T, which approximates bulk melt polymerization. Solid circles are experimentally constrained points, which were interpolated to 1300 C using the TVF equations of Table 4. Other lines connect points calculated using the work by Bottinga and Weill [1972] (short dashes); the work by Shaw [1972] (long dashes); and the NBO/T model of Giordano and Dingwell [2003] (solid lines). (d) Viscosity of all three series at 1300 C as a function of the structure modifier parameter, compared with values predicted using the Giordano et al. [2006] model. Inset shows the SA series. more than 1 order of magnitude between different silicate liquids [Dingwell and Webb, 1989], it can be argued that silicate melts of such fundamentally different structure as anorthite and diopside-forsterite may be expected to have different values of A. Our TVF equations using constant A reproduced all the experimental data with a root-meansquare error of 0.08 (Table 5), compared with 0.10 for the modeled TVF equations incorporating a mixing term into the expression for A (Table 7), which does not conclusively resolve the question. Further experiments with more mafic compositions may reveal the true nature of A, if the experimental difficulties associated with quenching such depolymerized liquids can be overcome. [28] The rationale for using a mixing equation to describe the compositional variation of the TVF parameters is based in configurational entropy theory [Adam and Gibbs, 1965; Richet, 1984], whereby liquid viscosity is inversely proportional to its configurational entropy (S conf ) at any given temperature: log h ¼ A e þ B e =TS conf ð10þ where A e is the viscosity intercept at infinite temperature, and B e is approximately a constant proportional to the energy barriers hindering cooperative rearrangements of the structure. The configurational entropy is related to the configurational heat capacity C conf P, given by the difference in heat capacities between the liquid and glass, by S conf ðtþ ¼ S conf T g þ Z C conf P =T dt ð11þ 11 of 19

12 Table 5. TVF Parameters for the BA Series Calculated Using a Constant Value of A and Comparison With the Measured Viscosity Data Sample X (An) A s B s C s n a RMS b An c BA81 d BA BA BA BA BA BA BA a Number of experimental measurements (Tables 2 and 3). b RMS is the root-mean-square deviation between measured values and those calculated using the TVF equation. c Anorthite values from Russell and Giordano [2005, p. 5340]. d Data for BA81 were not used in constructing the model. where C conf P is integrated from T g to T. The TVF equation (7) and the Adam-Gibbs equation (10) have different forms, so that there is no direct equivalence between the B (TVF) and B e (Adam-Gibbs) parameters, for example. However, because both reproduce viscosity data well, the thermodynamic effects of mixing that can only be properly interpreted using equation (10) are actually well accounted for by adopting a mixing approach to the empirically fitted parameters of equation (7). [29] When considering binary or ternary systems, the entropy of mixing contributes to the overall configurational entropy, resulting in lower viscosities for intermediate liquids than for end-members [Neuville and Richet, 1991]. At low temperatures near the glass transition, where the entropy of mixing contribution is a significant fraction of the total configurational entropy, this results in particularly noticeable viscosity decreases for intermediate compositions. At high temperatures, the entropy of mixing is now a small fraction of the total configurational entropy, and viscosity is typically quasi-linear as a function of composition. If mixing is near ideal, one would expect nearparabolic trends in the TVF parameters B and C as a function of composition along a binary or pseudobinary join. [30] This explanation works very well for simple systems such as MgSiO 3 -CaSiO 3 and Mg 3 Al 2 Si 3 O 12 -Ca 3 Al 2 Si 3 O 12 investigated by Neuville and Richet [1991], where ideal mixing of Ca and Mg is observed in the crystalline equivalent solid solutions. In the CaAl 2 Si 2 O 8 -NaAlSi 3 O 8 - CaMgSi 2 O 6 and CaAl 2 Si 2 O 8 -(CaMgSi 2 O 6 -Mg 2 SiO 4 ) systems, mixing occurs between end-members of substantially different composition, structure and viscosity behavior. However, Hummel and Arndt [1985] and Tauber and Arndt [1987] found a two-lattice entropy model to be adequate for calculating the viscosities of CaAl 2 Si 2 O 8 -NaAlSi 3 O 8 and CaAl 2 Si 2 O 8 -CaMgSi 2 O 6 liquids. Our results, and those of Russell and Giordano [2005], suggest that symmetric solution models for TVF parameters also provide a reasonable approximation to the viscosity behavior of ternary systems. 6. Viscosity Along the Liquid Line of Descent [31] The viscosity data presented in section 4 are for three compositional series that each form a liquid line of descent Figure 7. TVF parameters A, B, and C as a function of molar anorthite content for the BA series. Curves are mixing lines as discussed in the text. Parameters for sample BA42 were not used in fitting the curves. 12 of 19

13 Table 6. Parameters for a Mixing Model for the Viscosity of Liquids in the BA Series BA0 An W a A, Pas (0.16) B, PasK (237) C, K (11.1) a A, B, C = A 0, B 0, C 0 (1 X An )+A 1, B 1, C 1 X An + W A,B,C [(1 X An )X An ]. Absolute uncertainties are in parentheses. within the anorthite stability field of their respective phase diagrams. The viscosity along the liquid line of descent can therefore be calculated by plotting the viscosity of each composition for the temperature at which it is on the liquidus. For the SM and SA series, we simply plot the available data and draw a smooth line joining the liquidus viscosity points. For the BA series, having developed an empirical model to predict viscosity as a function of composition and temperature in section 5, we can predict the viscosity along the liquid line of descent for any X An,as long as the liquidus composition is known as a function of temperature. We estimated the relationship between liquidus composition and temperature for the BA series from the phase diagram of Osborn and Tait [1952] and fit the following equations for the silicic and mafic sides of the eutectic point (1270 C), respectively: T liquidus ð C Þ ¼ 0:0472 ðx An Þ 2 þ 10:848ðX An Þ þ 940:77; 100 X An 35:5 ð12aþ T liquidus ð CÞ ¼ :295ðX An Þ; 35:5 X An 0 ð12bþ Given the graphical method of interpolating liquidus temperature, higher-order polynomials are neither necessary nor justified. Combining equation (12) with the model developed in section 5 gives expressions for liquidus viscosity at any temperature along the pseudobinary. Calculated liquid viscosities along the liquid line of descent are shown for each series in Figure 8. Two branches are shown for the BA series, one originating from each endmember, and converging at the ternary eutectic point. Liquids on the diopside-forsterite side of the eutectic in the BA series show a relatively rapid viscosity increase along the liquidus surface, consistent with the combined effects of decreasing temperature and increasing anorthite content. The most striking features of Figure 8 are the initial decrease and subsequent increase in liquidus viscosity of the BA series, and the small overall increase in viscosity during a temperature decrease of 250 C for both the BA and SM series. [32] Within the SM series, the cotectic liquid (SM36) has a viscosity of 18 Pa s at 1320 C, and the available data indicate a slight liquidus viscosity decrease approaching the cotectic composition. In contrast to these systems, the variation in liquid viscosity within the SA series is very small, and the liquidus viscosity increases smoothly to almost 1000 Pa s at the An-En-Q eutectic temperature of 1240 C (SA23). We note that this implies a viscosity increase of a factor of 50 for liquids between the cotectic and the eutectic, over a temperature interval of 100 C. Because SA23 is more viscous than SM36, and the two have subparallel viscosity curves separated by 1 log unit, this increase can be ascribed mostly to compositional effects, with the temperature difference accounting for only a factor of 5 between the two points. [33] Liquidus viscosity can be continuously calculated for the BA series using equation (12) and the mixing model for liquid viscosity as a function of composition. The viscosity of liquid anorthite is 5.0 Pa s at the anorthite melting point of 1553 C. With decreasing anorthite content, liquidus viscosity decreases slightly to reach a minimum value of 3.5 Pa s at a liquidus temperature of 1498 C and a composition of 77.5 mole% anorthite, then ascends again until the eutectic temperature of 1270 C, where the viscosity of the An-Di-Fo eutectic liquid is 20.4 Pa s (Figure 9). On the mafic side of the eutectic, both the liquidus temperature and viscosity change more linearly with composition. The viscosity decreases dramatically due to the rapid change in melt structure approaching the mafic end-member, while the temperature increases relatively slowly along the diopside-forsterite cotectic. [34] Another important conclusion can be drawn from these results: that changing melt composition during crystallization can have effects on the liquid viscosity of similar magnitude to the accompanying temperature changes. Over the temperature-composition ranges considered here, temperature effects dominate in most cases, while compositional effects are generally expected to be more important in the early stages of crystallization where the liquidus surface has a lower dt/dx slope. 7. Magma Viscosity During Cooling and Crystallization 7.1. Magma Viscosity Along the Liquid Line of Descent [35] Having calculated the changing liquidus viscosity for each of the three series, the next consideration is how the viscosity of the bulk magma will change during progressive cooling and crystallization. Using a geochemical analogy, the liquidus viscosity is one end-member of magma viscosity, equivalent to perfect fractional crystallization, such that Table 7. TVF Parameters for the BA Series Calculated Using the Mixing Model (Table 6), and Comparison With the Measured Viscosity Data Sample X (An) A s B s C s n a RMS b An c BA81 d BA BA BA BA BA BA BA a Number of experimental measurements (Tables 2a 2c and 3a 3c). b RMS is the root-mean-square deviation between measured values and those calculated using the TVF equation. c Anorthite values from Russell and Giordano [2005, p. 5334]. d Data for BA81 were not used in constructing the model. 13 of 19

14 Figure 8. Viscosity of liquids along the liquid line of descent for (a) BA series, (b) SM series, and (c) SA series. (top) Molar anorthite content of the liquid and (bottom) log viscosity as a function of temperature. The viscosity of liquids, calculated from the TVF equations of Table 4, are shown as thin solid, dashed and dotted lines. Bold lines mark the intersection of the viscosity curves of individual samples with the temperature at which each composition is on the liquidus. crystals are removed as soon as they form. The other, more geologically reasonable, end-member scenario is for equilibrium crystallization, where all crystals are retained in the magma, which behaves as a closed system. [36] Making the closed system assumption and ignoring potential complexities of heterogeneous crystal size and shape distributions, magma viscosity is easily calculated by combining liquidus viscosities (section 6) with an expression for the physical effects of crystals on magma viscosity. For simplicity we use equation (1), with f 0 = 60%, and calculate the bulk viscosity only for crystal volume contents less than 55%. Crystal volumes were calculated from the phase diagram using the lever rule, combined with liquid volumes at each temperature calculated using the model of Lange [1997]. Molar volumes of crystalline phases were calculated using room temperature volumes of cm 3 for anorthite, cm 3 for diopside and cm 3 for forsterite [Smyth and McCormick, 1995], and thermal expansivity coefficients of K 1 for anorthite, K 1 for diopside, and K 1 for forsterite [Fei, 1995]. [37] Crystal volume fraction and bulk magma viscosity are plotted as a function of temperature in Figure 10, for bulk compositions corresponding to each of the initial liquids. The viscosity of bulk magma within the SA series always increases as crystal content increases and temperature decreases. The same is true for the SM and BA series, even though these do not show a monotonic increase in liquidus viscosity during cooling. Additional tests were conducted using the viscosity model for the BA series developed in section 6, which confirm that even over those temperature intervals where liquidus viscosity decreases, magma viscosity always increases for any initial starting liquid composition. [38] These results demonstrate that, for the synthetic systems considered here, magma viscosity will always increase during progressive crystallization in a closed system, even if the viscosity of the residual liquid may decrease. In other words, the physical effect of crystals will always outweigh the chemical effect of removing components from the liquid to form those crystals. Most mafic magmas have mafic phases on their liquidus, corresponding to the anorthite-poor branch of the BA series for example, so it is unlikely that natural magmas would ever undergo viscosity decreases during closed system cooling and crystallization. However, our results indicate that they may Figure 9. Liquidus temperature and liquidus viscosity for the BA series, as a function of anorthite content. See text for details. 14 of 19

15 Figure 10. Magma viscosity during closed system crystallization (all crystals retained) for (a) BA series, (b) SM series, and (c) SA series. (top) Calculated crystal volume fraction at each temperature. (bottom) Magma viscosity for each bulk composition. Note the difference in viscosity scales for the different series. Solid lines in Figure 10a are for bulk compositions in the diopside + forsterite field of the BA series. Curves are labeled with mol % An of the bulk composition. undergo smaller viscosity increases than would be predicted from simple application of the Einstein-Roscoe equation Viscosity Paths [39] We previously noted that the viscosity along the liquid line of descent would also represent magma viscosity if no crystals were present, i.e., for instantaneous and complete crystal removal (perfect fractional crystallization). Conversely, the magma viscosity calculated in section 7.1, assuming a closed system, represents zero crystal removal from the magma (equilibrium crystallization). While the common occurrence of phenocrysts in plutons and lavas suggests that the latter is more reasonable than the former, many magmas are likely to undergo partial crystal removal during their ascent and cooling history, which will result in an intermediate magma viscosity-temperature curve, or viscosity path. Four end-member scenarios are illustrated schematically in Figure 11, with Figures 11a to 11b representing perfect fractional crystallization and Figures 11a to 11c representing equilibrium (closed system) crystallization. The change in liquid composition is the same in both cases. A third scenario is for the magma to remain in state shown in Figure 11a, which would correspond to rapid cooling relative to crystal nucleation rates, and quenching to a glass, such that no crystals are formed and the liquid composition remains constant throughout cooling. [40] A fourth end-member applies to the crystallization of eutectic liquids, where the liquid composition remains the same throughout crystallization. This corresponds to transformation from state shown in Figure 11a to state shown in Figure 11d. For noneutectic compositions, the application of the Einstein-Roscoe equation without also accounting for changing residual liquid composition implicitly violates conservation of mass for a closed system. This scenario could also apply to open systems, for example during cumulate formation where phenocrysts are brought into the system from outside. [41] Example viscosity paths are shown in Figure 12, starting with a given bulk composition at its liquidus and Figure 11. Phase evolution of a silicate magma during cooling. (a) Initial liquidus composition, which may remain metastable during undercooling, and would ultimately form a glass; (b) evolved liquid with zero crystal retention (perfect fractional crystallization); (c) evolved liquid with full crystal retention (equilibrium crystallization); and (d) initial liquid composition with crystals also present. In a closed system this scenario is only permissible for eutectic liquid compositions, but under open system conditions it could apply to cumulate formation. 15 of 19

16 Figure 12. End-member paths for the evolution of magma viscosity during cooling and crystallization for (a) BA series, starting from composition BA58; (b) SM series, starting from composition SM52; (c) SA series, starting from composition SA47. Thin solid line mu represents the initial liquid composition (metastable undercooling, see Figure 11a); thick solid line pfc represents the liquid line of descent (perfect fractional crystallization, see Figure 11b); thick dashed line ec represents evolving liquid with full crystal retention (equilibrium crystallization, see Figure 11c); dotted line ilxt represents the initial liquid with crystal retention (impossible for noneutectic compositions; see text and Figure 11d). The shaded area therefore indicates the range of physically possible magma viscosity values. calculating the magma viscosity for each additional composition for which data are available before the eutectic. We start with composition BA58, with a liquidus viscosity of 4.6 Pa s at 1410 C (Figure 12a). On cooling to the eutectic temperature of 1270 C, the smallest viscosity increase is calculated for the perfect fractionation path along the liquid line of descent, reaching 21 Pa s at 1270 C. The viscosity of supercooled BA58 liquid at 1270 C is somewhat higher, at 50 Pa s, while equilibrium crystallization leads to a magma viscosity of 182 Pa s, including the effect of a 29% crystal volume fraction. The impossible scenario based on supercooled BA58 liquid containing the same crystal fraction has the highest calculated viscosity of 363 Pa s. The trends are quasi-linear, and the range of viscosities arising for different paths is 21 to 182 Pa s, or an increase of between 5 and 40 times from the original liquidus temperature. For the SM and SA series, we evaluate the evolution of compositions equivalent to whole rock basaltic andesite and dacite compositions from Santa Maria and Santiaguito. The different paths calculated for the SM series (Figure 12b) are qualitatively similar to those for the BA series. The pattern is somewhat different for the dacitic SA series, because the changing liquid composition has almost no effect on viscosity at the temperatures considered (Figure 12c). [42] In summary, the lowest possible final viscosity corresponds to the liquid line of descent, due to the chemical effects of removing the anorthite component from the melt. Next lowest is the supercooled starting liquid, which is close to the liquid line of descent for the most silicic magma considered here. Closed system magma viscosity increases linearly with decreasing temperature for the BA and SA series, and is close to linear for the SM series, although data for more compositions would be required to evaluate the SM series in detail. These graphs allow the important distinction to be drawn between the chemical and physical effects of crystallization on magma viscosity. The chemical effects due to changing residual liquid composition are negligible for the dacitic analog SA series, but significant for both the basalt and basaltic andesite analogs. The physical effects of crystals depend on the distance traveled from the original composition on the phase diagram, which controls crystal volume fraction produced under equilibrium conditions, and of course on the degree of crystal nucleation and retention, which reflect how closely the system follows the equilibrium scenario. [43] For the BA series, the model developed in section 5.2 allows for magma viscosity to be calculated at any temperature, for starting compositions on both sides of the eutectic (Figure 13). Lines tracking equilibrium crystallization with complete crystal retention are steep, but become less steep for starting compositions closer to the eutectic composition. Lines tracking evolving magma viscosity at constant crystal content run parallel to the liquidus viscosity (f = 0), and are necessarily less steep than equilibrium crystallization lines, although the two become more similar as the eutectic composition is approached. Figure 13 also allows an appreciation of the effects of crystal-liquid segregation. For example, magma ponding and cumulate formation may decrease the crystallinity of the magma, potentially resulting in an overall viscosity decrease even as the liquid continues to cool and evolve toward the eutectic. This effect will be stronger on magmas on the anorthite side of the eutectic, 16 of 19