Author's personal copy126 D. Giordano et al. / Earth and Planetary Science Letters 271 (2008) 123–134Table 1Coefficients for calculation of VFT parameters B and C [A =−4.55 (±0.21) a ] from meltcompositions expressed as mol% oxidesOxides Values Oxides Valuesb 1 SiO 2+TiO 2 159.6 (7) c 1 SiO 2 2.75 (0.4)b 2 Al 2 O 3 −173.3 (22) c 2 TA c 15.7 (1.6)b 3 FeO(T)+ MnO+P 2 O 5 72.1 (14) c 3 FM d 8.3 (0.5)b 4 MgO 75.7 (13) c 4 CaO 10.2 (0.7)b 5 CaO −39.0 (9) c 5 NK e −12.3 (1.3)b 6 Na 2 O+V b −84.1 (13) c 6 ln(1 + V) −99.5 (4)b 7 V +ln(1 +H 2O) 141.5 (19) c 11 (Al 2O 3+FM+CaO0.30 (0.04)b 11 (SiO 2+ TiO 2)⁎(FM) −2.43 (0.3) − P 2O 5)⁎(NK+V)b 12 (SiO 2 + TA−0.91 (0.3)+ P 2 O 5 )⁎(NK+H 2 O)b 13 (Al 2 O 3 )⁎(NK) 17.6 (1.8)aNumbers in brackets indicate 95% confidence limits on values of model coefficients.b Sum of H 2 O+F 2 O − 1 .c Sum of TiO 2+Al 2O 3.dSum of FeO(T)+ MnO+MgO.e Sum of Na 2 O+K 2 O.the compositional controls on B, and iii) 7 coefficients (c i , c 11 ) for C. Weobtained the optimal solution by minimization of the function:"Χ 2 ¼ Xnh logg Bðx i Ai Þ½ðT i¼1 i Cðx i ÞŠi 2 #=r 2 iwhere the summation is over the entire n experiments and each dataarray includes observed values of log η i , T i , the melt composition (x i )and the experimental uncertainty on log η i (i.e., σ i ). The optimizationinvolves simultaneous solution of this system of non-linear equationsand the methods we used are discussed more fully in Russell et al.(2002) and Russell and Giordano (2005).The optimal values for the 18 model parameters and theircorresponding confidence limits are reported in Table 1; a demonstrationcalculation is provided for in Table 2. In our model, A is one of theparameters we solve for and the optimization returns an independentand robust estimate for this high-T limit to viscosity. The optimal value(A=−4.55±0.21; η ∞ ~10 −4.6 Pa s; Table 1) accords well with values(~10 5 Pa s) postulated for low-T glass-forming systems (Bottinga andWeill, 1972) and predicted by theory (Myuller, 1955; Eyring et al., 1982).We have also performed an additional set of numerical experiments tofurther test the concept that all silicate melts, regardless of composition,converge to a single value at high temperature (Vogel, 1921; Richet andBottinga, 1995; Russell et al., 2002; Giordano and Russell, 2007). Ourexperiments involved running the optimization code separately on thedatasets for volatile-free (N=932) and volatile-rich (N=842) meltswhich returned values for A of −4.36±0.3 and −4.11±0.45, respectively.The values of A for partitioned datasets are clearly within error of thevalue (−4.55±0.21) obtained for the larger optimization problem. Wetake this as a strong empirical endorsement of the “constant A concept”.ð4ÞAn equally valid interpretation, reflecting a numerical modellingperspective, is that, even the highest temperature measurements ofmelt viscosity, only weakly constrain the value of A. Thus, one of thestrengths of a large scale optimization, such as this, is that the optimalA must satisfy all of the experimental observations. The consequenceis that, as long as the model assumptions are consistent with the data,increasing the number of data in the optimization problem willproduce tighter confidence limits on the model parameters. The factthat confidence limits on A are tightest for the complete datasetprovides another level of verification for this model assumption.Some of the compositional coefficients (i.e., b 11 , b 12 , c 11 ) are smallrelative to the other coefficients. However, our analysis shows all valuesto be significant at the 95% confidence level and they cannot be droppedwithout reducing the quality of the fit (Table 1). Conversely, the totalnumber (17) of compositional coefficients (Eqs. (2) and (3); Table 1)results, in part, from a sequence of earlier optimizations that includedadditional compositional terms. Higher order models were consideredto be overfit and rejected unless they returned a significantly better fit tothe original data and all adjustable parameters were shown to besignificantly different from zero.4. Model assessment and comparison4.1. Model qualityThere is excellent agreement between values of viscosity obtained byexperiment and those predicted by the model (Fig. 3). The model has aroot-mean-square-error (RMSE) of 0.40 log units and anhydrous andhydrous melts have an average misfit (∣observed−predicted∣) of ±0.25and ±0.35 logunits, respectively. The data plotted in Fig. 3 show that thelargest residuals derive from the low-T data where viscosity valuesTable 2Sample calculation of viscosity (Pa s) using model coefficients (Table 1)Sample a wt.% wt.% N mol% B-terms Values C-terms ValuesSiO 2 62.40 61.23 62.38 b 1 10018.8 c 1 171.5TiO 2 0.55 0.54 0.41 b 2 −2043.2 c 2 191.8Al 2 O 3 20.01 19.63 11.79 b 3 6.69 c 3 40.27FeO(T) 0.03 0.03 0.03 b 4 363.2 c 4 99.2MnO 0.02 0.02 0.02 b 5 −379.1 c 5 −49.21MgO 3.22 3.16 4.80 b 6 −858.7 c 6 −204.5CaO 9.08 8.91 9.73 b 7 1253.4 c 11 85.3Na 2 O 3.52 3.45 3.41 b 11 −738.7K 2 O 0.93 0.91 0.59 b 12 −733.8 A (constant) b −4.55P 2 O 5 0.12 0.12 0.05 b 13 831.6 B (computed) b 7720H 2 O 2.00 2.00 6.80 C (computed) b 334a Iron-free andesite melt with 2.00 wt.% H 2O.b Log η=A+B/(T(K)−C); predicted value for this melt at 1273 K is 3.67 Pa s.Fig. 3. Results of VFT-based model for compositional dependence of silicate meltviscosity. Observed values of viscosity are compared against model values for(A) anhydrous (N = 932 experiments) and (B) volatile-rich (N=842 experiments)melts. Dashed lines indicate ~5% uncertainty on measured values of log η.
Author's personal copyD. Giordano et al. / Earth and Planetary Science Letters 271 (2008) 123–134127exceed 10 8 Pa s (Table 3; Fig. 3A, B). Over this range of viscosity, theabsolute values of residuals on volatile-rich melts are slightly larger thanthose on the corresponding volatile-free melts. However, the accuracy ofthe model at conditions relevant to volcanic processes is usually within5% relative error (Fig. 3).4.2. Model parameters B and CThe values of B and C calculated from melt compositions using ourchemical model (Fig. 4) vary from 4450 to 12,000 J mol − 1 , and −125 to668 K, respectively. Melts enriched in H 2 O and F show roughly thesame range of values for B (4600 to 12,000 J mol − 1 ) compared toanhydrous melts (4450 to 11,950 J mol − 1 ) but model values of Cdecrease substantially with increased volatile content (Fig. 4; opensymbols). The predicted values of C are substantially lower (−125 to550 K) for volatile-rich melts than predicted for anhydrous melts (190to 668 K). Values of C represent the temperature where the viscositybecomes singular and is related to the Kauzmann temperature, wherethe entropy difference between solid and liquid is zero. In principle,the values of C should always be greater than 0 K (cf. Fig. 4).Values of B and C for the 58 anhydrous melts (solid circles; Fig. 4)show a strongly negative linear relationship. The two largest outliersto this general trend are compositions SFB60 (wt.% SiO 2 =74.8) andSFB40 (wt.% SiO 2 =58.9). These outliers derive from the literature(Bouhifd et al., 2004; Appendix B), rather than from our own viscositymeasurements, and we cannot comment further on their validity. Thepredicted values of B and C for volatile-enriched melts (grey circles;Fig. 4) show a dramatic departure from the linear pattern establishedby anhydrous melts. Values of B for anhydrous melts decrease withincreasing proportions of network-modifier cations (i.e. Na, K, Ca, Mg)(Fig. 4). Increasing volatile content causes only a slight decrease in B(Fig. 4). The values of C are highest for melts having the highestcontents of network-modifiers. In contrast to B, the C parameter isstrongly dependent on volatile content. Relative to the anhydrouscompositions, values of C decrease strongly with increased volatilecontents (Fig. 4). This pattern suggests that, although water andfluorine are normally considered network modifiers, they play asomewhat different role in the structure of silicate melts relative toother network-modifying cations (see discussion below).4.3. Model comparisonsThere are presently a number of models for predicting viscosity ofmelts as a function of temperature and composition (Shaw, 1972;Bottinga and Weill, 1972; Persikov, 1991; Giordano and Dingwell,Table 3Statistical comparison of models for predicting silicate melt viscosityModel a Dataset Anhydrous Volatile-rich TotalGRD F-free data b N 932 590 1508χ 2 6549 1775 8325RMSE c 0.34 0.50 0.41SH χ 2 251,356 35,020 286,377RMSE 2.53 2.47 2.51HZ χ 2 24,096 2300 26,396RMSE 0.52 0.54 0.53HZ-data N 757 694 1451HZ RMSE 0.33 0.28 0.31GRD RMSE 0.91 0.71 0.82GRD Full dataset N 932 842 1774χ 2 6993 2232 9226RMSE 0.34 0.46 0.40aGRD: model for viscosity of volatile-rich melts presented here and calibrated on1774 experiments; HZ: model of Hui and Zhang (2007); SH: model of Shaw (1972).b All data in GRD database excluding F-bearing compositions for comparativepurposes.c RMSE is the root mean square error.Fig. 4. Variations in values of parameters B (kJ per mole) and C (K) predicted for volatilefree(solid) and volatile-rich melts by our model for melt viscosity. Predicted values of Band C vary from 4400 to 12,000 J mol − 1 and from −120 to 670 K, respectively. Relative todry melts, melts enriched in H 2 O and F show the same range of B values but lowervalues of C (see text). to 12,000 J mol − 1 and from −120 to 670 K, respectively. Relative todry melts, melts enriched in H 2 O and F show the same range of B values but lowervalues of C (see text).2003; Giordano et al., 2006; Hui and Zhang, 2007; Avramov, 2007).Below we compare our model to two of these; the most commonlyused Arrehnian model of Shaw (1972) and the most recent non-Arrhenian model of Hui and Zhang (2007). Both of these models aredesigned for natural hydrous melts. For comparative purposes wehave dropped all fluorine-bearing melts from our database as neitherof these previous models is calibrated for fluorine.4.3.1. The Shaw modelThe model of Shaw (1972) has an Arrhenian basis of the formlogη=logη ∞ +E a /T(K), where both log η ∞ and E a (i.e., the activationenergy) have compositional dependence, and it is calibrated with~130 experiments spanning a wide range of temperatures (~600 to1700 °C) and viscosities (~10 − 0.2 to 10 12 Pa s). The compositionaldependence is embedded in the term the author refers to as a characteristicslope (s) and which is calculated from melt compositionsusing a total of 5 model parameters.We have used the SH model to predict the viscosity of melts in ourdatabase (Table 3; Fig. 5). We fully expect it to fail because it has a strictlyArrhenian temperature dependence and cannot possibly cope with theslight to pronounced non-Arrhenian behaviour shown by most naturalmelts. Table 3 is a statistical analysis of how well the SH model reproducesour experimental database after removing the F-bearing melts. Themodel is applied to the anhydrous, hydrous and combined (anhydrous+hydrous) datasets. As expected, the SH model reproduces the data poorly(cf. values of χ 2 ) and RMSE values are 5–8 times higher than obtained byour model (GRD). The SH model does a good job of reproducing meltviscosity at high temperatures (Fig. 5A, B). This reflects the fact that mostsilicate melts show near-Arrhenian behaviour at liquidus conditions.However, it is also a testament to the robustness of this parameterization(Shaw, 1972). At lower temperatures the SH model systematically missesthe anhydrous dataset by always underestimating melt viscosity (Fig. 5A).This results from the Arrhenian model being linearly extrapolated tolower temperatures coupled to the fact that viscosity curves for all non-Arrhenian silicate melts are concave up (Fig. 2) and steepen withdecreasing temperature. Hydrous melts show a similar pattern exceptthat the systematic deviation is not as pronounced and there are somemelts for which the SH model overestimates melt viscosity (Fig. 5B).Hydrous melts tend to be more Arrhenian-like than their anhydrouscounterparts (see below) and, thus, the SH model can be applied tohydrous melts over larger temperature intervals.Using the Shaw (SH) model, we have also calculated values of logη ∞ for all anhydrous and hydrous melts in the viscosity database. Thisparameter in the SH model represents the high-T limit to melt