Science, Faculty of
Earth, Ocean and Atmospheric Sciences, Department of
DSpace
UBCV
Li, Mu
Master of Science - MSc
University of British Columbia
A model for the viscosity (η) of melts in the system CaMgSi₂O₆-CaAl₂Si₂O₈-NaAlSi₃O₈ (Di-An-Ab) based on data compiled from the literature is presented. The model is calibrated on 560 measurements of melt viscosity at 1 bar pressure on 40 individual melt compositions and 303 measurements obtained at high pressures on 18 melt compositions. The model uses the Tammann-Vogel- Fulcher equation for temperature (T)- dependence and accounts for variations in melt composition (X) and pressure (P). At ambient pressures, the model spans 15 orders of magnitude of η, a T range of 933-2450 K, and reproduces the datasets well (RMSE = 0.26). The high-P model is calibrated over a T range of 970-2470 K and P range of 50 to 13100 MPa and reproduces the high-pressure data sets to within (larger) experimental uncertainties (RMSE = 0.35). The T-P-X model predicts the glass transition temperatures (Tg), melt fragilities (m) and activation energies (Ea) across the ternary with increasing P. Used in conjunction with chemical proxies for melt structural organization (i.e. SM and NBO/T), the model informs on changes in melt polymerization as functions of T, X, and P. Di-An-Ab melts with SM values > 25 have positive pressure coefficients and show a minimal to substantial increase in viscosity with increasing P. Melts with SM less than 25 show negative P-coefficients indicating a decrease in viscosity with increasing P. The P-effect might be correlated with the degree of polymerization and explained by the increasing coordination numbers of silicate and aluminum cations at elevated pressures. Model values of Tg for Di-An-Ab melts range between 930 K and 1130 K at ambient conditions; whilst at elevated pressures, Tg values are depressed for highly polymerized melts by as much as 150K at 5000 MPa but are higher for more depolymerized melts (i.e. SM > 25). A comparison of the liquidus surface topology for the ternary system to the corresponding isokoms of viscosity shows the relative effects of X and T on melt viscosity to be highly varied.
Temperature-Pressure-Composition Model for Melt Viscosity inthe Di-An-Ab SystembyMu LiB.Sc., The University of Waterloo, 2018B.Sc., China University of Geosciences (Beijing), 2018A THESIS SUBMITTED IN PARTIAL FULFILLMENTOF THE REQUIREMENTS FOR THE DEGREE OFMASTER OF SCIENCEinTHE FACULTY OF GRADUATE AND POSTDOCTORAL STUDIES(Geological Sciences)The University of British Columbia(Vancouver)August 2020c©Mu Li, 2020The following individuals certify that they have read, and recommend to the Faculty of Graduateand Postdoctoral Studies for acceptance, the thesis entitled:Temperature-Pressure-Composition Model for Melt Viscosity in the Di-An-Ab Systemsubmitted by Mu Li in partial fulfillment of the requirements for the degree of MASTER OFSCIENCE in Geological Sciences.Examining Committee:James Kelly Russell; Earth, Ocean and Atmospheric SciencesSupervisorLori Kennedy; Earth, Ocean and Atmospheric SciencesSupervisory Committee MemberJames Scoates; Earth, Ocean and Atmospheric SciencesSupervisory Committee MemberLee Groat; Earth, Ocean and Atmospheric SciencesExternal ExamineriiAbstractA model for the viscosity (η) of melts in the system CaMgSi2O6-CaAl2Si2O8-NaAlSi3O8(Di-An-Ab) based on data compiled from the literature is presented. The model is calibrated on560 measurements of melt viscosity at 1 bar pressure on 40 individual melt compositions and 303measurements obtained at high pressures on 18 melt compositions. The model uses the Tammann-Vogel- Fulcher equation for temperature (T)- dependence and accounts for variations in melt com-position (X) and pressure (P). At ambient pressures, the model spans 15 orders of magnitude ofη , a T range of 933-2450 K, and reproduces the datasets well (RMSE = 0.26). The high-P modelis calibrated over a T range of 970-2470 K and P range of 50 to 13100 MPa and reproduces thehigh-pressure data sets to within (larger) experimental uncertainties (RMSE = 0.35). The T-P-Xmodel predicts the glass transition temperatures (Tg), melt fragilities (m) and activation energies(Ea) across the ternary with increasing P. Used in conjunction with chemical proxies for melt struc-tural organization (i.e. SM and NBO/T), the model informs on changes in melt polymerization asfunctions of T, X, and P. Di-An-Ab melts with SM values > 25 have positive pressure coefficientsand show a minimal to substantial increase in viscosity with increasing P. Melts with SM less than25 show negative P-coefficients indicating a decrease in viscosity with increasing P. The P-effectmight be correlated with the degree of polymerization and explained by the increasing coordinationnumbers of silicate and aluminum cations at elevated pressures. Model values of Tg for Di-An-Abmelts range between 930 K and 1130 K at ambient conditions; whilst at elevated pressures, Tgvalues are depressed for highly polymerized melts by as much as 150K at 5000 MPa but are higherfor more depolymerized melts (i.e. SM > 25). A comparison of the liquidus surface topology forthe ternary system to the corresponding isokoms of viscosity shows the relative effects of X and Ton melt viscosity to be highly varied.iiiLay summarySilicate melt are an essential component for almost all igneous processes. Laboratory ex-periments cannot fully describe the properties of magmas deep with Earth. Models are designedto aid in developing the understanding of the structure and dynamics of silicate melts under theP-T conditions of the Earth’s crust and mantle. In particular, the viscosity of magmatic meltsgoverns virtually all geological processes, including rates of melt extraction and magma ascent,the viability of magmatic differentiation processes. The diopside-anorthite-albite ternary systemis a simple chemical system that serves as an analogue for dacitic, basaltic and andesitic mag-mas. Understanding this system, where there are many data, provides insight into the rheology ofthese natural magmatic systems. The new predictive viscosity model presented here, provides thatunderstanding.ivPrefaceThis thesis comprises an introduction, a summary and seven research chapters. The lattercomprises a draft manuscript, titled ’Temperature-Pressure-Composition Model for Melt Viscosityin the Di-An-Ab System’, that has been accepted, subject to revisions for publication in ChemicalGeology. Authors: M. Li, J.K. Russell, D. Giordano.The core of this thesis is a predictive viscosity model for silicate melts in the diopside-anorthite-albite system which I built during my time at UBC. I am the lead author for each chapterand the manuscript and subsequently received revisions and comments from my supervisor KellyRussell and my co-author, D. Giordano.Chapter 2 is the data assessment for the 105 Pa and high-P viscosity database I used tocalibrate the model. This database was compiled from many published papers in the scientificliterature. Chapter 3, 4, 5 present an oxide-based, multicomponent 1 bar viscosity model forsilicate melts as functions of temperature, pressure and melt composition. Chapter 6, 7 summarizethe effects of pressure on the viscosity of natural and model silicate melts and presented a modelaccounting for the effects of pressure, temperature and composition.I built algorithms following the studies of Kelly Russell, who helped develop and refine thecode for viscosity modelling.vTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xList of Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiList of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Data assessment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.1 Low-pressure (1 bar) data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Pressure data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 Temperature-dependence of viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . 163.1 T-dependence of viscosity model analyses . . . . . . . . . . . . . . . . . . . . . . 163.2 Baseline VFT functions for individual melts . . . . . . . . . . . . . . . . . . . . . 173.2.1 Quality of fits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.2.2 Compositional parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 193.2.3 Other melt properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 VFT with a constant Av f t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.1 The “constant A” concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.2 VFT functions fitting to a constant Av f t . . . . . . . . . . . . . . . . . . . . . . . . 254.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27vi5 Multicomponent 1 bar model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315.1 Linear model-A first approximation . . . . . . . . . . . . . . . . . . . . . . . . . 315.1.1 Quality of fits: linear model . . . . . . . . . . . . . . . . . . . . . . . . . 345.2 A non-linear- optimal model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355.2.1 Quality of fits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375.2.2 Compositional dependence of Bv f t and Cv f t . . . . . . . . . . . . . . . . . 405.2.3 Model values of Tg and melt fragility m . . . . . . . . . . . . . . . . . . . 446 1 bar model implication and application . . . . . . . . . . . . . . . . . . . . . . . . . 486.1 Model prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 486.2 Model extrapolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 506.3 Isotherms and Isokoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 526.4 Correlation of melt rheology and phase relationship in Di-An-Ab system . . . . . . 567 Pressure-dependence of viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 597.1 P-effects on viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 597.2 Modelling P-T-X dependence of viscosity . . . . . . . . . . . . . . . . . . . . . . 617.3 A model for Di-An-Ab melts at P . . . . . . . . . . . . . . . . . . . . . . . . . . 647.3.1 Quality of fits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 647.3.2 P-X dependence of Bp . . . . . . . . . . . . . . . . . . . . . . . . . . . . 677.3.3 Di-An-Ab melt viscosity at P . . . . . . . . . . . . . . . . . . . . . . . . . 728 P-T-X model implication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 758.1 Tg and m at P . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 758.2 Activation energy at P . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 788.3 Activation volume of viscous flow (Va) . . . . . . . . . . . . . . . . . . . . . . . . 808.4 Correlation of melt rheology and phase relationship at P . . . . . . . . . . . . . . . 829 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 849.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 849.2 Geological applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87viiA Viscosity Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94A.1 Adam-Gibbs (AG) equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94A.2 Voegel–Fulcher–Tammann (VFT) equation . . . . . . . . . . . . . . . . . . . . . . 95A.3 Avramov (AV) equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95B Experimental Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97viiiList of Tables2.1 Ideal weight percentages of oxides for Di, An, and Ab melts calculated frommolar mass and numbers of oxides. . . . . . . . . . . . . . . . . . . . . . . . . 123.1 Model parameters (Av f t ,Bv f t ,Cv f t) and calculated melt properties Tg, steepnessindex m from VFT equation individual fitting to 40 melt compositions. . . . . . 184.1 Model parameters (Bv f t , Cv f t), Tg, m, and RMSE solved from fitting the VFTfunction with a constant Av f t = -4.55 Pa s to 40 melt compositions. . . . . . . . 305.1 Coefficients for Bv f t and Cv f t by solving the linear model. Av f t is -3.80 Pa s. . . 335.2 Coefficients for the 13-parameter multicomponent 1 bar model for Di-An-Abmelts. Confidence limits (1σ ) on model parameters are also reported in brack-ets. Data were fitted to the multicomponent equation with a constant value forAv f t (-4.73 ± 0.29 Pa s). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365.3 Model parameters Bv f t ,Cv f t predicted for each melt in the system Di-An-Ab (cf.Table 3.1) using the multicomponent 1 bar model (Table 5.2) with a constanthigh-T limit to viscosity (i.e. Av f t = -4.73 ± 0.29 Pa s). Also reported are thencalculated Tg, steepness index (m), RMSE as the quality of fit for individualmelt compositions, and calculated liquidus temperatures (TL) from Weill et al.(1980). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397.1 Coefficients for the pressure bootstrap (Eq. 7.4) in the multicomponent 1 barmodel for Di-An-Ab melts. Data were fitted to the high-P database with a con-stant value for Av f t (-4.73 ± 0.29 Pa s). . . . . . . . . . . . . . . . . . . . . . . 667.2 Summary of high-P data sets for Di-An-Ab melts and including the number ofdata (N) and sources1. These data used to extend the multicomponent 1 barmodel to higher pressures by expanding the term for Bv f t in the VFT functionas: BP =B1bar + [bp1(SM- bp4) + bp2 (SM- bp4)2]*(P-0.001) + bp3 (SM- bp4)(SAC). Also included are model values of BP for individual melt compositionsand calculated values of ∆BP (BP - B1bar), Tg, m, and RMSE . . . . . . . . . . . 69B.1 Summary of experimental methods used by different works for the collectedviscosity data in the Di-An-Ab system. . . . . . . . . . . . . . . . . . . . . . . 98ixList of Figures1.1 Liquidus surface in the system Di-An-Ab . . . . . . . . . . . . . . . . . . . . 21.2 Fragility (or Angell) plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Alternative models for melt structure . . . . . . . . . . . . . . . . . . . . . . . 41.4 Methods of determining viscosity . . . . . . . . . . . . . . . . . . . . . . . . 41.5 Flow diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.1 XAl2O3 - 0.25 XAn against XNa2O. . . . . . . . . . . . . . . . . . . . . . . . . . 82.2 ± 0.5 wt.% uncertainty limits of five oxides for three end-members . . . . . . . 82.3 NBO/T distribution for Di, An, Ab melts by Monte Carlo Simulation . . . . . . 92.4 Calculated NBO/T values for melts in the original data sets . . . . . . . . . . . 102.5 Final 1 bar data sets for calibration . . . . . . . . . . . . . . . . . . . . . . . . 132.6 Final high-P data for calibration . . . . . . . . . . . . . . . . . . . . . . . . . 143.1 Results for VFT individual fits. . . . . . . . . . . . . . . . . . . . . . . . . . . 204.1 Quality of fitting VFT with a constant Av f t . . . . . . . . . . . . . . . . . . . . 264.2 Model Bv f t and Cv f t by fitting VFT functions with a constant Av f t . . . . . . . . 274.3 VFT-constant A model parameters variations . . . . . . . . . . . . . . . . . . 295.1 Quality of the linear model: A first approximation . . . . . . . . . . . . . . . . 345.2 Model Bv f t and Cv f t solved by the linear model. . . . . . . . . . . . . . . . . . 355.3 Quality of the 13-parameter multicomponent 1 bar model. . . . . . . . . . . . 385.4 Model predictions for individual melt groups . . . . . . . . . . . . . . . . . . 385.5 Model Bv f t : Cv f t by the multicomponent model . . . . . . . . . . . . . . . . . 405.6 Model Bv f t and Cv f t plotted against mole fractions for end-members . . . . . . 415.7 Model Bv f t and Cv f t plotted against NBO/T and SM and their contoured ternaryplots. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425.8 Contoured ternary diagrams for Tg and m, and Tg and m against SM. . . . . . . 445.9 Model Tg and m comparisons . . . . . . . . . . . . . . . . . . . . . . . . . . . 455.10 Model predictions for melts with insufficient data . . . . . . . . . . . . . . . . 465.11 Doremus ratio against the steepness index and SM . . . . . . . . . . . . . . . 476.1 Model log η vs. 10000/T (K) for specific melt compositions . . . . . . . . . . 49x6.2 Model predictions beyond the calibration . . . . . . . . . . . . . . . . . . . . 506.3 Model predictions for NMCAS melts . . . . . . . . . . . . . . . . . . . . . . 516.4 Isothermals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 536.5 Isokoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 556.6 Liquidus surface of the Di-An-Ab ternary system . . . . . . . . . . . . . . . . 577.1 Variations of VFT parameter BP against P and SM . . . . . . . . . . . . . . . . 637.2 Quality of fits for the T-P-X viscosity model . . . . . . . . . . . . . . . . . . . 657.3 Contoured ternary diagrams for SM and SAC . . . . . . . . . . . . . . . . . . 677.4 P- effects on the VFT model parameter BP . . . . . . . . . . . . . . . . . . . . 687.5 Model values of melt viscosity at pressures related to geological settings . . . . 737.6 Model values of ∆Bp and melt viscosity plotted against mole fraction of end-members . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 748.1 Contoured ternary diagrams for Tg and ∆Tg at pressures . . . . . . . . . . . . . 768.2 Contoured ternary diagrams for m and ∆m at pressures . . . . . . . . . . . . . 778.3 Contoured ternary diagrams for Ea at pressures . . . . . . . . . . . . . . . . . 798.4 Contoured ternary diagrams for Va . . . . . . . . . . . . . . . . . . . . . . . . 818.5 Viscosity at liquidus surface at 2000MPa . . . . . . . . . . . . . . . . . . . . . 83xiList of AbbreviationsAn anorthiteAb albiteAG Adam- Gibbs equationAV Avramov equationAAMA solid phase apparatuses of anvil and multi-anvilBB beam-bending techniqueCBS: counter-balanced sphere methodCC concentric-cylinder viscometerCRN Continuous Random Network modelDi diopsideFE fiber elongation methodIHPV internally heated pressure vesselMRN Modified Random Network modelMC MgO+CaO (mole percent)MP micropenetration techniqueNMCAS Na2O-MgO-CaO-Al2O3-SiO2NMR nuclear magnetic resonance spectroscopyOS oscillation viscometerPCA solid-media, piston-cylinder apparatusPPV parallel plate viscometerRC rotating cylinder apparatusRV rotational viscometerRVHGP radiation viscosimeter of high gas pressureSA SiO2 + Al2O3 (mole percent)SAC SiO2 + Al2O3-CaO (mole percent)SFS/FS sinking (falling) and floating sphere methodVFT Tammann- Vogel- Fulcher equationxiiList of SymbolsAe parameter A in Adam-Gibbs equationAv f t parameter A in Vogel-Fulcher-Tammann equationAAV parameter A in Avramov equationAl aluminumAl2O3 aluminum oxideBe parameter B in Adam-Gibbs equationBv f t parameter B in Vogel-Fulcher-Tammann equationBAV parameter B in Avramov equationCv f t parameter C in Vogel-Fulcher-Tammann equationCaO calcium oxideEa activation energy (kJ/mol)Fe ironG∞ bulk shear modulus; 1010 PaH enthalpy (kJ mol−1)k volume expansion coefficientko volume expansion coefficient at no pressurem fragilityMgO magnesium oxideNa2O sodium oxideNBO/T non-bridging oxygen per tetrahedrally coordinated cationP pressure (bar or MPa)Q cooling/ heating rateRD Doremus’ ratioRgas universal gas constant (cm3 MPa/K mol)Scon f configurational entropy (J K−1 mol−1)SM structual modifier; = MgO + CaO + Na2OSiO2 silicon dioxideTaAV parameter T in Avramov equation; aAV = 2Cp/RZTAG parameter T in Adam-Gibbs equationT temperature (◦C or K)Ti titaniumTg glass transition temperature (◦C or K)TL liquidus temperature (◦C or K)χ2 chi square functionη viscosity (Pa s)xDi concentration of Di (mol fraction)τ relaxation timescale of melt (s); =η /G∞η viscosity (Pa s)Va activation volume (cm3/mol)∆Cp heat capacity (J/K)σ standard deviationZ coordination numberxiiiAcknowledgementsFirst and foremost, I would like to thank my supervisor, Kelly Russell for your time, support,guidance, enthusiasm and insights. Your passion for new ideas and volcanology are inspiring.Thank you for bearing with my poor spoken and written English. My success can certainly beattributed to your patience and encouragement. It has been a privilege to study and work with you.I would like to thank my supervisory and examining committees (Lori Kennedy, JamesScoates, Lee Groat) and a big thank you to my coauthor Daniele Giordano.Additionally, I would like to thank my lab mates, VPL alumni, fellow grad students andfriends over the past few years, particularly: Alex Wilson, Amy Ryan, Ryan Kroner, David Sasse,Sophie Leither, Martin Harris, Katherine Landoni, Annie Borch, Elaine Lu, Xueying Yi, Sang Yi,Kerry Luo. You have helped me get to this point and keep me excited.Special thank you to my undergraduate supervisor Dr Changcheng Li, who is patient withme and had many excellent discussions with me. And I would like to thank Dr Hejiu Hui who gaveme many suggestions and support.Finally, I would like to express my sincere gratitude to my roommate, friends and family forsupporting and loving me.This study was supported by the Natural Sciences and Engineering Research Council ofCanada (NSERC) Discovery Grants program (RGPIN-2018-03841; JKR).xivChapter 1: IntroductionThe viscosity of magmatic melts governs many geological processes, including rates of meltextraction and magma ascent, the viability of magmatic differentiation processes (e.g. magmamixing, assimilation), as well as, the styles and intensity of volcanic eruptions. Silicate melt vis-cosity varies strongly with temperature (T), melt composition (X) and pressure (P) and predictivemodels that account for these variations are critical tools for providing quantitative understandingfor a wide range of geological processes. However, such models depend on an abundance of high-quality experimental measurements of melt viscosity in the T-X-P-space relevant to volcanic andmagmatic systems (Giordano et al., 2008; Hess and Dingwell, 1996; Russell and Giordano, 2005).The chemical system CaMgSi2O6-CaAl2Si2O8-NaAlSi3O8 (i.e. Di-An-Ab) is an essentialgeological system relevant to petrologists, geochemists, mineralogists, and geophysicists. It hasbeen used extensively and widely as an analogue for understanding the crystallization and differ-entiation processes in magmatic systems (Bowen, 1915; Kushiro, 1978), trace element partitioningbetween melts and crystals (Mysen, 2004), thermodynamic modelling of magmatic systems (Burn-ham and Nekvasil, 1986; Weill et al., 1980), and physical and geophysical properties of materialsat crustal and mantle pressures (Brearley et al., 1986; Taniguchi, 1992).Furthermore, melts in the Di-An-Ab system have been studied extensively for their viscosity(η) at elevated T and P. The system has also been viewed as a useful analogue for the rheologi-cal properties of terrestrial melts. Di-An-Ab melts vary between fully polymerized (Ab) to moredepolymerized (Di) melts similar to variations between natural rhyolites and basalts, respectively(Bowen, 1915; Kushiro, 1978). In many ways the system serves as a proxy for the phonolitic,dacitic, picritic and andesitic magmas (Getson and Whittington, 2007). For example, this sys-tem can be used to illustrate the crystallization processes for those magmas (Bowen, 1915), orto study the evolution of magma viscosity during cooling and crystallization (Getson and Whit-tington, 2007) although some properties differ by several orders of magnitude (e.g., Giordano,2019 and reference therein). This system also allows comparisons of phase equilibria relationships1(Fig. 1.1; Weill et al., 1980) to its rheological properties. For example, models for the T-P-Xdependence of silicate liquid viscosities can be combined with phase equilibrium models for theDi-An-Ab system to compute melt viscosity along the liquidus surfaces to illustrate rheologicalchanges attending crystallization.AbDiAn13501300125012001200 150014501250130013501400Figure 1.1: Liquidus surface and cotectic (in red) for the system Di-An-Ab after (Weill et al., 1980)contoured for liquidus temperature (◦C).As part of the NMCAS (Na2O-MgO-CaO-Al2O3-SiO2) chemical system, Di-An-Ab meltsalso have relevance to the glass sciences and industry (Solvang et al., 2005). Di-An-Ab meltsspan a wide range of melt fragilities (Angell, 1985, 1991) (Fig. 1.2) which are plotted againstreduced temperature (Tg/ T) where Tg is the glass transition temperature. Di and An melts showsimilar behaviors corresponding to moderately fragile melts having non-Arrhenian temperaturedependence. Conversely, Ab is a strong melt shows Arrhenian behavior. It serves as a proxy forrhyolitic melts.The strong to fragile behaviors reflect the melt structure of the silicate melts. Structures forstrong melts (e.g. Ab with 75 mol.% SiO2 content) are more ordered compared to fragile meltstructures (e.g. Di with 50 mol.% SiO2 content) are more disordered. Perturbations in the struc-ture order are reflected in the thermodynamic configurational properities such as configurationalentropy (Scon f ) and leads to changes in the transport properties (i.e. viscosity).Silicate melts can be more or less polymerized. The degree of polymerization can be quan-titively described by two pseudo-structure parameters: NBO/T and SM. The parameter NBO/T as2s gf � 12 10 8 6 4 2 0 -2-40 12 10 8 6 4 2 0 -20.2 0.6 0.8 1.0 AbAnDiFigure 1.2: Di, An, Ab fragility lines on the Fragility (or Angell) plots: logarithm of viscosity (logη) vs. reduced temperature ( Tg/ T) plot for 20 natural glass forming liquids.the number of non-bridging oxygen per tetrahedrally coordinated cation, which is calculated frommelt compositions as advocated by Mysen et al. (1982) and Mysen (1986) serves as a chemicalproxy for the structural organization of the melt and the degree of structure polymerization. Theoriginal definition of NBO/T is based on the Continuous Random Network (CRN) model for themelt structure (Fig. 1.3A) from Zachariasen (1932). It is formed by an extended 3-D intercon-nected network where cations such as Si4+, Al3+, IV Fe3+, P5+, Ti4+ (sometimes) serve as thenetwork-forming cations, other cations such as Na, Ca, Mg provide charge-balancing of Al3+ andFe3+ tetrahedra. All excess cations, and cations which are not stable if tetrahedrally coordinated,act as network modifiers (e.g., K, Ca, Mg, V IFe2+; (Mysen, 1986; Mysen et al., 1982).SM is the “structual modifier”, proposed by Giordano and Dingwell (2003), based on theModified Random Network (MRN) model for melt structure (Fig. 1.3B) (Greaves, 1985). Theparameter SM deos not consider the effect of stabilizing the charge, and is simply an additivesummation of MgO, CaO, and Na2O. For Di-An-Ab system, SM ranges from 12.5 for Ab melt to50 for Di melt, whereas, NBO/T ranges from 0 for An-Ab melts and 2 for Di melts.Lastly, the 1 atmosphere viscosity data available for Di-An-Ab system is reasonably abun-dant (Table 3.1; Fig. 2.5) and spans the full range of temperatures relevant to igneous petrologyand volcanology (933-2450 K). The most common measuring techniques employed at atmosphericpressure and high temperatures, where viscosity is relatively low (i.e. 10−2 to 105.5 Pa s), are rota-3(A) (B)Figure 1.3: (A) Continuous Random-Network (CRN) model by Zachariasen (1932); (B) Modifiedrandom network (MRN) model Greaves (1985). Perculation channels highlighted in blue.tional viscometry (i.e. concentric cylinder Fig. 1.4C) and falling sphere viscometry (Fig. 1.4A; B).At lower temperatures approaching glass transition temperatures (Tg), corresponding to viscositiesof ∼ 108 to 1013 Pa s (lower-T), the most commonly employed techniques are parallel plate andmicropenetration viscometry (Fig. 1.4H). Experiments designed to retrieve measurements of melt' ··-.-: i !LI/�l.- .... •-e) 1-.----------.---,. M ±L1p' ,,,----� ..... (b) (c)� (T) ·�:.di__ -··· -· .... .. :. -�-- .. (T) p (f) (g) (d) p l (h) (i)Figure L Methods of determining viscosity: (a) faJling sphere, (b) counterbalanced sphen, (c) Couette viscometer, (d) sink point measurement, (e) free fiber elongtion. (f) loaded fiber elongation, (g) beam bending. (h) micropenetration. (i) torion of a cylinder. (Redrawn from Zarzycki, 1991.)Figure 1.4: (a) falling sphere. (b) count rbala ced sphere (c) C u ttte (rotational) viscometer (d)sink point measurement (e) free fiber elongtion (f) loaded fiber elongation (g) beam bending (h)micropenetration (i) tortion of a cyliner. Zarzycki (1991).viscosity at elevated T and P relevant to magmatic and volcanic environments are technically chal-lenging and time-consuming. They demand substantial and careful experimental effort and, thus,there are substantially fewer reliable determinations of melt viscosity at high-P than at atmosphericpressures. Therefore, there are only a few liquid viscosity models which account for P-dependence.High-P measurements made near Tg are obtained by subjecting melts within externally heated pres-sure vessels to pure or compressive shear stresses (e.g. Paterson apparata, Paterson and Olgaard,42000). Unfortunately, the gas apparatus can only achieve a confining pressure of 500 MPa (Pa-terson and Olgaard, 2000). At higher temperatures, where viscosities are ∼ 10−4 Pa s to 104.5 Pas, high-P measurements can be made using falling sphere methods but can have experimental un-certainties up to 20% (Behrens and Schulze, 2003; Brearley et al., 1986; Giordano, 2019; Liebskeet al., 2005; Suzuki et al., 2005; Vetere et al., 2006). For example, at viscosities >104.5 Pa s typicalof SiO2-rich melts such as albite and haplogranite, the sinking distance of the falling sphere canbe close to the size of the sphere resulting in a rise in experimental uncertainties or potential errors(Kushiro, 1978; Persikov and Bukhtiyarov, 1999).High-P measurements on low viscosity (< 1 Pa s) melts (i.e. high-T, ultramafic melts) havebeen using real-time movies of Pt spheres falling through the melt such as “in-situ” X-rays analyses(Brearley et al., 1986; Kushiro, 1978; Mori et al., 2000; Poe et al., 2006; Taniguchi, 1992). Ardiaet al. (2008) used “ex-situ” post-run X-ray analysis of the samples to increase the precision onmeasurements of distance travelled by the sphere. High-P measurements of viscosity have alsobeen made using a centrifuge to increase the centripetal acceleration and settling speed of themetal sphere through the melt (Ardia et al., 2008; Dorfman et al., 1996). Centrifuge-assisted fallingsphere techniques allow for experiments over a broader range of conditions where viscosities riseto ∼ 107 Pa s. Methods used for the experimental data I collected from literature are summerizedin Appendix B. (Ardia et al., 2008; Dorfman et al., 1996).The high-P viscosity data for Di-An-Ab melts collected here suggest that the viscositiesof Di melts have positive P-dependence (Brearley et al., 1986; Scarfe, 1987; Taniguchi, 1992),whilst Ab melts exhibit negative effects of P (Behrens and Schulze, 2003; Kushiro, 1978; Moriet al., 2000). The same datasets suggest that the viscosities of An melts are nearly independent ofpressure (Behrens and Schulze, 2003; Del Gaudio and Behrens, 2009; Taniguchi, 1992).In this study, I collected and assessed the available experimental data from literature with thegoal of developing a chemical model for the T-X dependence of melt viscosity in the Di-An-Absystem. The resulting model can reproduce the original 1 bar data to within experimental errorand predicts the rheological behaviour for Di-An-Ab melts smoothly and continuously across the5ternary system. The model also has the capacity to predict ancillary melt properties, including theglass transition temperature (Tg) and melt fragility (m). The model is then extended to predict meltviscosity at high-P based on the available (sparse) experimental data. My approach is to coupleadditional P-dependent terms (i.e. adjustable parameters) to the ambient model yielding a singleT-P-X model for melt viscosity. Here, I present a flow diagram that summarizes the steps to betaken in this study (Fig. 1.5).Figure 1.5: Flow diagram for the steps to taken throughout this research program.6Chapter 2: Data assessment2.1 Low-pressure (1 bar) dataThe primary viscosity data set is for anhydrous melts in the Di-An-Ab ternary system mea-sured at 1 atm confining pressure. Measurements span the full compositional range of the systemand cover a wide range of temperature (◦C). The original compilation considered 680 viscositymeasurements deriving from experiments on 54 different melt compositions (Fig. 2.5A). For eachmelt composition, the number of measurements varies. Measurements span the full compositionalrange of the system and cover a wide range of temperature (933 – 2450 K). The majority of dataare for the end-members and binary melt compositions; there are only eight ternary melt compo-sitions (Fig. 2.5A). Melt compositions are reported as five oxide components SiO2, Al2O3, CaO,MgO, Na2O (reported wt.% or mol.%), and I restricted the data sets to measurements made onmelts strictly within the Di-An-Ab ternary system. Not all of the experimental data reported forthe ternary system were used for model calibration purposes.Firstly, I did not consider data where there was no independent analysis of the nominal meltcompositions, or where there were large discrepancies between measured and nominal compo-sitions (e.g. Kozu and Kani (1935)). In some cases, there were multiple sets of viscosity mea-surements on the same melt composition for overlapping ranges of temperature. Where there weresignificant (∆log η > 0.5 Pa s) discrepancies, I elected to use the more recent datasets (e.g., An100;Sipp et al. (2001) vs. Hummel and Arndt (1985)). I also required the sum of end-member com-ponents (i.e. XDi, XAn and XAb), calculated from the oxides, to equal 100 ± 4 mol.%, where XDi= 4XMgO; XAn = 4XCaO - XDi; XAb = 8XNa2O. XAb can also be calculated as 8XAl2O3 – 2XAn, andrejected data where the two calculations of XAb differed by > 5 mol.% (Fig. 2.1). The six meltcompositions excluded by this criterion are denoted as open circles (Fig. 2.1): “Di” Urbain et al.(1982); “Di” Behrens and Schulze (2003); “Naexc” Romano et al. (2001); “An50Di50” Del Gaudio7and Behrens (2009); “Di13.3An3.3Ab83.4” Tauber and Arndt (1987); “Di83An14” Tauber and Arndt(1987).0 0.02 0.04 0.06 0.08 0.1 0.12X (Na2O)-0.0200.020.040.060.080.10.12X(Al 2O3) - 0.25 X AnFigure 2.1: Comparison of calculated values mole fractions of Na2O to values of [XAl2O3 - 0.25XAn] used to validate data sets; dashed lines denote acceptable uncertainty limits (' 5 mol.% XAb).Open circles denote excluded data (see text).wt% wt% wt%Figure 2.2: Start and end values for the ± 0.5 wt.% uncertainty limits of five oxides for threeend-members.I also use the calculated value of NBO/T for each melt as a criterion to check the accuracyof chemical measurements. The parameter NBO/T serves as a chemical proxy for the structuralorganization of the melt and the degree of structure polymerization. For some cases, NBO/Tvalues calculated from the measured oxide concentrations differ significantly from NBO/T valuesfor the nominal compositions. I rejected data where the differences in model vs calculated NBO/Tvalues (∆NBO/T) > 0.04 representing an estimated 2σ uncertainty limits on NBO/T derived froma Monte Carlo simulation (N=1000) of average analytical errors on the oxides (Fig. 2.3; 2.4).8To quantify the uncertainty on NBO/T for every melt, values of oxide concentrations wereassigned uncertainties of 0.5 wt.%. I randomly select values for five oxides within the 0.5 wt.%uncertainty intervals for one thousand times (Fig. 2.2). Values for every oxide are random variablesand follow normal distributions, respectively. NBO/T can be calculated as:NBO/T =2∗ (2SiO2+3Al2O3+MgO+CaO+Na2O−4SiO2+8Al2O3)SiO2+2Al2O3+2Na2O(2.1)Distributions of NBO/T derived by this simulation for end-members are illustrated in Figure2.3. The ideal NBO/T values for Di, An, Ab are 2.00, 0.00, and 0.00, respectively. The 2σ (95%)ranges highlighted in orange for three end-members does not vary considerable (± 0.03; ± 0.03;± 0.05). For example, there is less than 5% possibility for “Di” melt has an NBO/T value lessthan 1.94 or exceeded 2.05. Since chemical compositions were measured by probe, uncertaintieswithin the oxide concentrations (wt.%) are identical; therefore, the uncertainty for NBO/T shouldbe identical. To improve the confidence in the experimental data I used in calibration, I decided tochoose the average value of the 2σ ranges calculated for all melts in calibration as the acceptableuncertainty limit for NBO/T of ± 0.04.Figure 2.3: NBO/T distribution calculated for Di, An, Ab melts by Monte Carlo Simulation (N=1000) with normal 0.5 wt.% error in the chemical measurements. Orange area is two standarddeviations 2σ of the fitted normal distribution (same mean and standard deviation obtained fromcalculated NBO/T values.9Therefore, the acceptable NBO/T range is ± 0.04 with respect to the ideal values for meltcompositions. Variations of NBO/T along binary joints are summarized in Figure 2.4. If NBO/Tcalculated with the measured melt concentrations lies outside the interval, I then classify it as adifferent melt composition and drop it from the calibration. For example, ideal Diopside has anNBO/T of 2.0 whilst the composition of “Diopside” melt from Urbain et al. (1982) has an NBO/Tof 1.89 forcing me to exclude these data (i.e. ∆NBO/T ∼ 0.11; Fig. 2.4A). I reserved these “re-jected” data, and other data outside of the ternary, for independent testing the capacity of the modelfor extrapolation in the more general NMCAS (Na2O-MgO-CaO-Al2O3-SiO2) system. In Figure0 0.2 0.4 0.6 0.8 100.511.52NBO/TDiAnData usedData dropped0 0.2 0.4 0.6 0.8 100.511.52NBO/TDiAb0 0.2 0.4 0.6 0.8 1-0.04-0.0200.020.040.06NBO/TAnAb0 0.2 0.4 0.6 0.8 100.511.52NBO/TDiAn + AbTernary meltsFigure 2.4: Calculated NBO/T for melts in the original data sets vs. mol fractions for end-members.(A) Di-An melts vs. XDi. (B) Di-Ab melts vs. XDi. (C) Ab-An melts vs. mol fraction of An (XAn).(D) Ternary melts vs. XDi. Solid lines denote ideal NBO/T values for binary and ternary melts.Dashed lines indicate ± 0.04 uncertainty limits.102.4A, I show four compositions along Di-An joint whose calculated NBO/T lying outside the ac-ceptable range: An melt, An50Di50 (Del Gaudio and Behrens, 2009), Di (Behrens and Schulze,2003), and Di (Urbain et al., 1982). Di-Ab, An-Ab and ternary melts lie inside the NBO/T uncer-tainty limit.11Table 2.1: Ideal weight percentages of oxides for Di, An, and Ab melts calculated from molar mass and numbers of oxides.Di An AbName Molar mass (g/mol) No. of Oxide wt % No. of Oxide wt % No. of Oxide wt %SiO2 60.08 2 120.16 55.49 2 120.16 43.19 6 360.48 68.74Al2O3 101.96 0 0 0 1 101.96 36.65 1 101.96 9.44MgO 40.30 1 40.30 18.61 0 0 0 0 0 0CaO 56.08 1 56.08 25.89 1 56.08 20.16 0 0 0Na2O 61.98 0 0 0 0 0 0 1 61.98 1.82SUM 216.54 278.19 524.421 wt %: weight per cent12The uncertainties on viscosity measurements should not exceed ±0.5 log units. Some meltshave multiple sets of viscosity measurements over the same range of temperatures. If there arediscrepancies between values, the more recent data were accepted. The more recent data trendto be more precise and the melt compositions to be better characterized, (i.e. An100; Sipp etal., 2001 vs. Hummel and Arndt, 1985). The low-T measurements for Ab and Ab80An20 meltfrom Cranmer and Uhlmann (1981) were not used, because they are systematically low comparedto the data given by Hummel and Arndt (1985). Measurements for Ab50An50, Ab25An75 meltsof Cranmer and Ulhmann, (1982) were not used either because they are identical to the data ofHummel and Arndt, (1985). Partial crystallization should not occur in the experiments during themeasurement.These steps for assessing the 1 bar data reduced the ambient data sets to 560 viscosity mea-surements performed on 40 individual melt compositions (Fig. 2.5; Table 3.1). The data are evenlydistributed along the binaries but are relatively sparse within the ternary and span a temperaturerange of 933 to 2450 K and values of viscosity from 10−1.5 to 1014.5 Pa s (Fig. 2.5).7067752010 9 7 9 6 5 8 1 1 118 6 6 5 29 8 6 10 5 410 91914191419 6 6 9 8 9 9 7 8 8AbDiAn(a)4 6 8 10 1210000/T(K)051015log [ (Pas)]No. of MeltsNo. of Data40560End-membersDi-AbDi-AnAn-AbTeranry melts(b)Figure 2.5: Final 1 bar data sets for calibration: (a) Ternary diagram Di-An-Ab shows the rangeof melt compositions in the ambient data sets and the total number of experiments at each meltcomposition. Black circles are data used (N = 40). And the yellow circles are data dropped fromthe datasets (N = 15). (b) log η vs 10000/T for the 1 bar data sets used for calibration.132.2 Pressure dataThe original high-P data sets contain 314 viscosity measurements on 21 anhydrous meltcompositions within the Di-An-Ab system. P spans a range of 50 to 13100 MPa. Most high-P measurements are for the Di-An-Ab end-members (N = 48, 18, 47, respectively) and on thebinaries Di-Ab (N = 141) and Di-An (N = 60). There are no high-P data for An-Ab melts nor forternary melts. This study does not focus on the effect of water content; therefore, high-P data setsonly include anhydrous melt compositions containing < 0.05 wt.% H2O content. Similar criteriaAbDiAn718182624167208211821221024209554Figure 2.6: Ternary diagram Di-An-Ab shows the range of melt compositions in the high-P datasets and the total number of experiments at each melt composition.as above were used to filter the compiled high-P data. I required independent analyses of thenominal melt compositions. The compositional range was restricted to within the ternary Di-An-Ab system, although I used a higher tolerance level for the high-P datasets due to the scarcityof data, for example, the NBO/T uncertainty limit was relaxed. In one case I elected to dropviscosity measurements where there were significant discrepancies between workers, for instance,measurements for “Di” melt at 4000 and 5000 MPa from Persikov and Bukhtiyarov (1999) vsTaniguchi (1992), and Brearley et al. (1986). Generally, where I observed substantial differencesin viscosity values for the same melt at equivalent T-P conditions, I again used the more recent data(e.g. Di at 1600◦C, 500 MPa; Taniguchi, 1992 vs. Brearley et al., 1986). After those assessmentprocesses, the high-P data sets were reduced to 303 measurements in total on 18 melt compositions14spanning a pressure range of 50 to 13100 MPa and a temperature range of 933 K to 2500 K (Fig.2.6; Table 7.2).15Chapter 3: Temperature-dependence of viscosity3.1 T-dependence of viscosity model analysesThe purpose of this study is to describe the melt viscosity of silicate melts in the syntheticDi-An-Ab system as functions of composition (X) (consisting of oxides of Si, Al, Mg, Ca, and Na),temperature (T), and pressure (P). Viscosity generally varies by more than 12 orders of magnitudebetween superliquidus conditions and glass transition range to account for the non-Arrhenian be-haviour. When putting aside P first, early models studying the T-dependence of the viscosity ofsilicate melts were restricted to a narrow temperature region, where viscosity is usually between0.1 to 104 Pa s. Therefore, they adopted the Arrhenian equation for T- dependence (Bottinga andWeill, 1972; Shaw, 1972). Those models are able to reproduce viscosity data over a range oftemperatures of 1500 K to 2500 K. The Arrhenius-type equation can be simplified asln η = ln A+BT(3.1)where T is the absolute temperature (K). A, B are adjustable parameters. A is the high-T limit ofviscosity.Measurements made at lower temperatures and approaching the glass transition temperature(Tg) show that many silicate melts have an increasing non-Arrhenian T-dependence and an increasein activation energy (Ea). This is illustrated by increasing curvature in the Arrhenian temperatureplot as log η vs reciprocal temperature. The non-Arrhenian behaviours in viscosity result from thetemperature-induced changes in the melt structure.Three non-Arrhenian, tri-parametrical T-η equations are extensively used to predict the vis-cosity of silicate melts over the whole T- range: Adam-Gibbs (AG) equation (Adam and Gibbs,1965; Richet, 1984); the Voegel–Fulcher–Tammann (VFT) equation (Fulcher, 1925; Tammann16and Hesse, 1926; Vogel, 1921); and the Avramov (AV) equation (Avramov, 1998). A summary ofviscosity models is reported in Appendix A.The VFT equation is proposed on the purely empirical basis, which can be derived from the“free volume” theory (Williams et al., 1955). It describes the T-dependence of viscosity for meltsvery well. Because of its simplicity and pervasiveness, I have adopted the VFT equation for the T-dependence of viscosity for this Di-An-Ab system.logη = Av f t +Bv f tT −Tv f t (3.2)where T is the absolute temperature (K). Av f t , Bv f t , and Cv f t are adjustable parameters (AppendixA)3.2 Baseline VFT functions for individual meltsMy initial approach is to fit the VFT function (Eq. 3.2) to the T (K): log η data sets forindividual melt compositions. Each melt composition is fitted for unique values of Av f t , Bv f t andCv f t . The fits to data sets for individual melts are solved simultaneously for individual values ofAv f t , Bv f t and Cv f t by minimization of the χ2 function:χ2 =n∑j=1[logηi, j−A j +[ B jTi, j−C j ]]2 (3.3)where j denotes each of the N melt compositions (i.e. jth data set) having m j paired measurementsof Ti j(K)- log ηi j. The total number of adjustable parameters is, therefore, 3N melt compositions(Table 3.1).17Table 3.1: Model parameters (Av f t ,Bv f t ,Cv f t) and calculated melt properties Tg, steepness index mfrom VFT equation individual fitting to 40 melt compositions.Label1 N of data set N of Exp Sources2 A B C Tg (K) m 5 RMSE3Di 1 70 a;b;c;d;e;f;g -4.58 4492 723 994 61 0.11An 2 67 h; q; f;a;i;j; -4.71 5536 798 1129 57 0.14Ab 3 75 e;f;k;l;i;j -5.08 11783 396 1085 27 0.14Di50Ab50 4 20 e;m -3.37 4945 619 940 45 0.03Di25Ab75 5 10 e -3.88 7505 462 934 31 0.00Di60Ab40 6 9 m -1.51 3518 682 943 49 0.03Di69.3Ab30.7 7 7 m -1.99 3658 683 945 51 0.01Di75Ab25 8 9 e -2.32 2349 868 1032 90 0.01Di84.2Ab15.8 9 6 m -2.69 3734 707 962 56 0.01Di95Ab5 10 5 e -4.64 6510 332 723 31 0.00Di30Ab70 11 8 m -4.83 7312 504 938 36 0.03Ab80Di20 12 1 r 1.79 0 450 450 \4 0.00Ab57Di43 13 1 r 1.30 0 450 450 \ 0.00Ab30Di70 14 1 r 0.30 0 450 450 \ 0.00Ab80An20 15 18 o;j -4.16 8371 555 1074 33 0.08An30Ab70 16 6 o -6.97 10910 492 1067 35 0.08An34Ab62 17 6 o -4.74 7858 597 1066 38 0.04An40Ab56 18 5 o -4.33 7129 635 1072 40 0.03An50Ab50 19 29 o;j;n -3.70 6048 691 1076 44 0.09An55Ab45 20 8 o -10.13 13262 474 1074 40 0.05An61Ab37 21 6 o -5.17 7263 660 1083 44 0.10An75Ab25 22 10 o;j -4.24 5972 724 1092 48 0.03An80Ab18 23 5 o -13.08 15072 497 1098 46 0.07An87Ab9 24 4 o -3.52 5486 751 1105 48 0.17Di50An50 25 10 q -2.41 2116 976 1123 110 0.01Di58An42 26 9 q -2.56 2273 934 1090 102 0.01Di20An80 27 19 a; q -4.27 4792 794 1089 60 0.11Di30An70 28 14 p;q -5.63 6521 701 1071 51 0.15Di40An60 29 19 a;q -4.03 4440 766 1043 60 0.12Di64An36 30 14 a -4.31 4435 749 1021 61 0.12Di80An20 31 19 a; q -4.67 4658 730 1009 60 0.14Di90An10 32 6 q -3.67 3930 640 890 56 0.00Di18.5An49.8Ab31.7 33 6 m -2.07 4351 705 1015 46 0.01Di21.2An8.8Ab70 34 9 m -3.39 6286 556 964 36 0.04Di28.2An15.5Ab56.3 35 8 m -2.61 4930 624 962 42 0.01Di35.9An21.7Ab42.2 36 9 m -2.70 4762 643 967 44 0.03Di38.3An5.9Ab55.8 37 9 m -1.21 3658 675 952 45 0.03Di43.8An27.6Ab28.6 38 7 m -5.09 6420 599 975 44 0.02Di51.7An32Ab16.3 39 8 m -7.86 8703 544 982 45 0.03Di55.1An14.5Ab30.4 40 8 m -3.93 5119 641 963 48 0.011Labels are reported in the source paper, whose subscripts are nominal compositions (mol2Sources: (a) Taniguchi, (1992); (b) Brearley et al. (1986); (c) Kirkpatrick (1974); (d) Licko and Danek(1986); (e) Scarfe and Cronin (1986); (f) Sipp et al. (2001); (g) Neuville and Richet (1991); (h) Sykes et al.(1993); (i) Urbain et al., (1982); (j) Cranmer and Ulhmann, (1982); (k) Stein and Spera (1993); (l) Persikov,(1991); (m) Tauber, (1987); (n) N’DALA et al. (1984); (o) Hummel and Arndt, (1985); (p) Tauber andArndt, (1987); (q) Scarfe et al. (1983); (r) Persikov and Bukhtiyarov (1999);3RMSE is root mean square error4non-feasible solutions for melt compositions having less than 3 experiments.5 Steepness index (m) as a proxy for melt fragility.183.2.1 Quality of fitsThe individual fits for each composition have a set of unique values of Av f t , Bv f t , and Cv f t ,which is summarized in Table 3.1. The accuracy of the VFT fits is illustrated in Figure 3.1. Iprovide the root mean square errors (RMSE), as a relative measure for the quality of fitting. Forindividual fitting the VFT equation, RMSE are always less than 0.15 log units. Majority of mea-surements are reproduced perfectly to within ±0.25 log units. The quality of individual fits iscontrolled by the number and the distribution of viscosity measurements. Where there are fewdata, some individual fits have irrational values (i.e. Av f t> 0). Thus, there should be at least threemeasurements at different temperatures for one melt composition in order to be solved. The moreextensive the range of T, the more reliable the 3-parameter fit.3.2.2 Compositional parametersParameter values should obey both numerical and physical regulations for silicate meltsproperties. Bv f t (units of K) is proportional to the activation energy (Ea) associated with the viscousflow, which represents the energy barrier for structure rearrangements. Hence, polymerized melts(i.e. albite), which mainly contain tetrahedral Si-O and Al-O bonds, require higher energy forstructure rearrangements. Values for Bv f t should be above zero in order that viscosity decreaseswith increasing temperature. Av f t (units of Pa s) is the high- T limiting viscosity, which shouldbe below zero. Cv f t is the temperature (K) at which viscosity becomes infinite, which is alsoconstrained to be positive in value (Angell, 1991; Bottinga and Richet, 1995; Richet, 1995; Rössleret al., 1998; Russell and Giordano, 2005).For three end members, parameter Av f t are -4.58 (Di), -4.71 (An), -5.08 (Ab) Pa s, respec-tively. The maximum Bv f t is ascribed to Ab melt (11783), which is the most polymerized melt.The minimum Bv f t is attributed to Di melt (4492), which is the least polymerized melt. And themaximum, and minimum Cv f t belong to An melt (798) and Ab melt (396), respectively. The datafor end-members are the most abundant; therefore, the values for parameters are best constrainedand should be most reliable.19Figure 3.1: Results of fitting VFT function to individual data sets. (A) Values of log η modelledby individual VFT fits compared to original data. Dashed lines are ± 0.25 log units of viscosity.(B) Optimized values of parameter Av f t vs. Bv f t (Table 3.1). (C) Optimized values of parameterAv f t vs. Cv f t . (D) Optimized values of parameter Bv f t vs. Cv f t . Lines denote the suggested areadefined by the three end-members. Non-feasible solutions are whose Av f t above zero.The area defined by three end-members does not necessarily enclose the values for binarymelts or ternary melts, whose Av f t (-14 – 0), Bv f t (2000 – 16000), and Cv f t (300 – 1000) definea much larger space (Fig. 3.1). Values of parameters for binary and ternary melts are mixed, andvariations of parameters along binary joins do not have patterns. The maximum, and minimumvalues of Bv f t for binary melts belong to An80Ab18 melt (15072), and Di58An42 melt (2273). Themaximum, and minimum values of Cv f t are ascribed to Di95Ab5 melt (332), and Di58An42 melt(934). Parameters values are primarily controlled by the quality of data. There are some unrealisticsets of values, and they result from two main causes: i), experimental uncertainties in the compiled20data, and ii) insufficient measurements. In the latter case, the measurements do not span the entire Trange; therefore, are not able to define the whole T-effect on viscosity. For example, there are eitherno low-T measurements below 1150 K nor high-T measurements above 2000 K (e.g. for Di10An89melt, measurements are localized for temperatures from 1100 K to 1153 K). Thus, the data are notable to define the non-Arrhenian curvature of the T-dependence. As a result, parameters dependingon a limited data set yield illogical values.3.2.3 Other melt propertiesParameterizing the VFT equation can help us investigate melt properties, and other derivativemelt properties of silicate melts, such as the melt fragility and glass transition temperature (Tg).Those parameters represent derivative properties of the viscosity model, which are themselvesdependent on melt composition.Glass transition temperatureTg is the temperature where we find a transition from the relaxed liquid state to the unrelaxedglassy state occurring (Adam and Gibbs, 1965). This temperature region also is close to the regionof metastable unrelaxed liquid (Angell, 1985, 1991; Ardia et al., 2008; Scholze and Kreidl, 1986).Viscosity is typical 1012 Pa s at Tg. I take Tg as the temperature at which melt viscosity reaches avalue of 1012 Pa s (Bottinga and Richet, 1995; Sipp et al., 2001; Webb and Knoche, 1996) and iscalculated from model values of Av f t , Bv f t and Cv f t as:Tg =Bv f t12−Av f t +Cv f t (3.4)For individual fits, values of Tg range from 890 K to 1129 K. The minimum and maximum valueof Tg belong to Di90An10 melt and An melt, respectively. The predicted values of Tg for the end-member melt compositions (Di, An, Ab) are 994 K, 1129 K, and 1085 K. These values agree wellwith the calorimetric measurements of Tg for the end members Di (1003–1009 K), An (1135–1143K), and Ab (978–1076 K) based on a cooling/heating rate of 5 K/min (Webb and Knoche, 1996).21The model value for An is slightly lower, which might be because our compositions are differentfrom those measured by Webb and Knoche (1996).Melt fragilityFragility measures how rapid the flow properties change with temperature as they approachthe Tg. Fragility (m) distinguishes silicate melts into two broad groups, each having extreme char-acteristics: strong and f ragile. Strong melts typically show Arrhenian behaviour, and resistantfirmly to structural changes, which implies that melt structure and rheological properties of meltsare not sensitive to the temperature variations. Fragile melts show more significant deviations fromthe Arrhenian behaviour. Melt properties change much faster with temperature compared to a re-sponding to strong melt. Fragile melts easily assume a large variety of configurational states inresponse to thermal perturbations (Ardia et al., 2008; Angell, 1985; 1991). Both viscosity andrelaxation of thermodynamic quantity (e.g. Enthalpy H) can be the measures of fragility (Brüningand Sutton, 1996; Moynihan, 1993).One method for ranking fragility is the steepness index (m), which is the derivative of log ηwith respect to the reduced T at Tg (Böhmer and Angell, 1992; Plazek and Ngai, 1991; Rossler etal., 1998):m =∂ (logη)∂ (Tg/T )∣∣∣∣T=Tg=Bv f tTg(1−Cv f t/Tg)2 (3.5)where Bv f t , Cv f t , and Tg are unique for each melt composition.Low values of m correspond to strong liquids, whereas high m values correspond to fragileliquids. The Di-An-Ab ternary system covers a wide range of model values of fragility, from astrong Ab melt (27) to a f ragile Di melt (61).To correlate fragility with more melt properties, I used the Doremus fragility parameter(Doremus, 2002). Doremus suggested that practically all melts deviate from the Arrhenian be-haviour, e.g. the activation energy changes from a high Ea at low temperatures to a low Ea at hightemperatures. Therefore, Doremus introduced RD, which is based on the changes that occur in the22activation energy (Ea) that are unambiguously characterized by the ratio between two values ofEa at low and high temperatures. RD is yielded as RD =EaHighEaLow. The higher RD, the more fragileare the liquids, Doremus’ fragility ratios range from 1.33 for germania to 7.26 for diopside melts(Doremus, 2002).Asymptotically both at low and high temperatures the Ea of viscosity becomes constant, e.g.independent of temperature. However, this method depends on an inflection point for the log η vs.1/T relation to define the onset temperatures for the two Arrhenian regions. Experimental data forsilicate melts commonly suggest a smooth transition of viscosity from low T to high T. And the twonear-Arrhenian regions for each homogeneous melt can be observed in the Arrhenius plots withT approaching Tg. The high-T Arrhenian region is near and above the liquidus whilst the low-TArrhenian is located close to Tg. The high-T Arrhenian region characteristic by liquidus observedin completely liquid silicates may be attributed to the classical mechanism of viscous flow occurredin monatomic liquids (Glasstone et al., 1941). At high temperatures, above the liquidus, viscousflow happens by jumping of atoms from one equilibrium position to another under the appliedshear stress. The height of the potential barrier is T-independent. Molecular dynamic simulationsby Karki et al. (2013) also confirmed the Arrhenian T-dependence of viscosity at temperatures wellabove the liquidus.Activation energyThe activation energy (Ea) with the unit of J/mol, implies the height of a potential barrierthat has to be overcome for viscous flow to occur and records the T-sensitivity of the structuralrelaxation times for the molecular reorientation (Glasstone et al., 1941; Putnis, 1992). Therefore,Ea for polymerized Ab melt at low-T and high-T changes only slightly because Ab melt structureis more resistant to temperature disturbances. Conversely, Ea for depolymerized Di melt increasessignificantly with T approaching Tg. Depolymerized (i.e. fragile) melts show a large variety of theconfigurational states. The definition of activation energy (Ea) derives from the Arrhenian equation23(Glasstone et al., 1941):η = ηo exp (EaR T) (3.6)This expression is only valid for a limited range of temperature (i.e. higher temperatures). If Ea isT independent, data should give a straight line when plotted in Arrhenian coordinates. However,experimental data commonly show a non-Arrhenius T-dependence for many silicate melts (Fig.2.5B). Therefore, Ea depends on T as Ea = Ea(T).The slope of data plotted in Arrhenian diagrams (e.g. ∂ log η/∂ 1/T) is proportional to Ea.The expression for Ea can yield from Eq. 3.6:Ea = Rgas∂ lnη∂1/T= 2.303 Rgas∂ logη∂1/T(3.7)where Rgas is the gas constant (8.314 J mol−1K−1), log η is the viscosity at corresponding T foreach melt composition.24Chapter 4: VFT with a constant Av f t4.1 The “constant A” conceptThis study accepted the assumption that the pre-exponential parameter Av f t (Eq. 3.2) forsilicate melts is a single, but unknown constant (Angell, 1991; Angell et al., 2000; Myuller, 1955;Russell et al., 2003, 2002). Av f t represents a high-T limit to silicate melt viscosity. This high-Tlimit is suggested when considering the timescales of relaxation processes in melts (Angell, 1991;Richet and Bottinga, 1995; Angell et al., 2000). The Maxwell relationship (τ = η/G∞) constrainsthe lower limits to melt viscosity, where G∞ is the bulk shear modulus at the infinite frequency,approaching the periods of atomic vibrations (Richet, 1984; Dingwell and Webb, 1989; Russellet al., 2003). It is generally accepted that G∞ is approximately 1010 Pa. The value of relaxationtimescale (τ) reflects the periods of atomic vibration, and is limited by the quasi-lattice vibrationperiod (10−14 s) (Angell, 1991; Angell et al., 2000; Dingwell and Webb, 1989; Toplis, 1998).Thus, the lower limiting value to viscosity (η∞) should approximate 10−4±1 Pa s (Dingwelland Webb, 1989; Angell, 1991; Toplis, 1998; Angell et al., 2000). In studies of low-T glass-forming systems (Angell 1991, Angell et al., 2000), data strongly suggest that viscosities for bothstrong and fragile melts converge to a common value (10−5 Pa s) at temperatures well above Tg.Av f t derived from Russell et al. (2002) is −4.3±0.74 Pa s where they fitted VFT equationsimultaneously to data sets for Di, An, and Ab. This value is also consistent with the values ofAv f t solved by fitting VFT independently for three end-members: Di (-4.58), An (-4.71), and Ab(-5.08).4.2 VFT functions fitting to a constant Av f tFollowing previous work, I assume that all melts converge to a common high-T limit (i.e.Av f t) and that Bv f t and Cv f t are dependent on melt composition. Thus, each melt compositionshares a single, common value of Av f t but has unique values of Bv f t and Cv f t . The fits to data sets25for individual melts are solved simultaneously for a common Av f t and individual values of Bv f tand Cv f t by minimization of the χ2 function:χ2 =n∑j=1[log ηi, j−Av f t +[ B jTi, j−C j ]]2 (4.1)where j denotes each of the N melt compositions (i.e. jth data set) having m j paired measurementsof Ti j(K)- log ηi j. There is one constant value of Av f t .The original data are well-reproduced (±0.25 log units) by this fitting (Fig. 4.1); the averageroot mean square error (RMSE) for each melt composition is 0.06 log units (Table 4.1), and theresiduals are randomly distributed against measured values of log η . This optimization provides arobust fit to the original viscosity data and unique values of Bv f t and Cv f t for VFT functions (Table4.1) that model the T-dependence of individual melts well. The quality of these fits (e.g., Fig. 4.1)serves as a baseline for the compositional-dependent model to follow.0 5 10 15log [Measured]051015log [Model]Constant A= -4.55 (a)-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 log [Individual Fits]-0.5-0.4-0.3-0.2-0.100.10.20.30.40.5 log [VFT Constant A] (B)(b)Figure 4.1: Results of fitting VFT with a constant Av f t to individual data sets. (a) Values of log ηmodelled by model VFT functions compared to original data. Dashed lines are ± 0.25 log units ofviscosity. (b) ∆log η individual fits (model observed) vs ∆ log η by fitting VFT functions with aconstant Av f t .The optimal value of Av f t is -4.55 Pa s which is in good agreement with the lower limitingvalue to viscosity for melts of 10−4±1 Pa s suggested by previous studies (Myuller, 1955; Angell,1991; Getson and Whittington, 2007; Russell. et al., 2003, among references). Values of Bv f t and26Cv f t vary with the melt composition as exemplified by the end-members Di (4463 / 725), An (5341/ 806) and Ab (10764 / 436) (Fig. 4.2). The three end members joined by lines define a field whichillustrates the magnitude of non-linear variations in Bv f t and Cv f t . The maximum and minimumvalues of Bv f t correspond to Ab and Di respectively; the maximum value of Cv f t belongs to An, andthe minimum value is for the melt Di75Ab25. Although Bv f t : Cv f t values for the join An: Ab arealmost linear, the optimal values of Bv f t and Cv f t for the other solid solutions are highly scattered.Di-An melts show a concave down curve. Much of the scatter observed in Figure 4.2 derives fromdata sets for individual melts that have only high-T or high low-T data. The consequence of thislimited temperature distribution of viscosity data is that values of Bv f t or Cv f t (or both) are onlyloosely constrained (Russell et al., 2003, 2002).300 400 500 600 700 800 900C [Model]4000500060007000800090001000011000B [Model]DiAnAb End MembersDp-AbDp-AnAn-AbTeranryFigure 4.2: Optimized values of parameter Bv f t vs. Cv f t . Lines denote the suggested area definedby three end-members.4.3 DiscussionBy assuming a common value for Av f t , I reduced the number of parameters significantlyfrom 3N to 2N + 1 and the quality of fits does not deteriorate substantially. Majority of discrep-ancies between observed and the modelled values of log η are ± 0.2 log unit. Also, by requiringAv f t to be a constant for all melts, Bv f t and Cv f t become better-constrained. Variations in Bv f t andCv f t with melt composition provide clearer insights into their compositional dependences.The suggested straight boundaries for the ternary area show how the parameters vary if thecompositional dependence is purely linear. Model Bv f t : Cv f t lie outside the area, showing concave27patterns along the binary joins (Fig. 4.2), which implies rather than linear mixing of end members,there are non-linearly interactions between molecules. Generally, larger discrepancies to the linearlines indicate larger non-linearly components in the compositional dependence. Along the Di-Anjoin, parameters show a concave curve. Bv f t increases faster than Cv f t with a small addition of Ancontent till a peak for Bv f t . After the inflection point, Cv f t becomes more sensitive to the additionof An content. Ab-An join also shows a concave curve, which curvature is smaller compared toDi-An join. Di-Ab melts lie outside the space, especially the melts with higher Ab content (Fig.4.2).I also illustrated how model parameters vary with melt compositions by plotting Bv f t andCv f t against corresponding melt structural organizations (i.e. SM and NBO/T) (Fig. 4.3). NBO/T(Eq. 2.1) is not a good proxy for An-Ab melts since its value are 0.0 for both. SM parameter, whichis the mole fraction of “structural modifier” oxides, can be calculated as XMgO +XCaO +XNa2O(Giordano and Dingwell, 2003). Variations in Bv f t shows a concave up curve along the Di-Ab join,and nearly linearly pattern for Di-An join against both variables. An-Ab melts show a concave updecreasing pattern between Bv f t and SM (Fig. 4.3C). Cv f t is corresponding the variations in Bv f t ,which shows a concave up curve along Di-An join (Fig. 4.3B; D). Ternary melts lie within thespace defined by binaries. Therefore, there might be more significant non-linearity distributions tothe parameters for Di-Ab and Di-An melts compared to that for Ab-An melts.The parameters from fitting the VFT function with a constant Av f t and the quality of thosefits provide a base against which multicomponent model can be compared.28-0.5 0 0.5 1 1.5 2 2.5NBO/T4000500060007000800090001000011000B [VFT Constant A]AnAbDiEnd MembersDi-AbDi-AnAn-AbTeranry(a)-0.5 0 0.5 1 1.5 2 2.5NBO/T400500600700800900C [VFT Constant A]AbAnDiEnd MembersDi-AbDi-AnAn-AbTeranry(b)10 15 20 25 30 35 40 45 50 55SM4000500060007000800090001000011000B [VFT Constant A]AnAbDiEnd MembersDi-AbDi-AnAn-AbTeranry(c)10 15 20 25 30 35 40 45 50 55SM300400500600700800900C [VFT Constant A]AnAbDiEnd MembersDi-AbDi-AnAn-AbTeranry(d)Figure 4.3: Model parameters plotted against melt composition variables NBO/T, SM: (a) Bv f t vs.NBO/T. (b) Cv f t vs. NBO/T. (c) Bv f t vs. SM. (d) Cv f t vs. SM.29Table 4.1: Model parameters (Bv f t , Cv f t), Tg, m, and RMSE solved from fitting the VFT functionwith a constant Av f t = -4.55 Pa s to 40 melt compositions.Label Bv f t Cv f t Tg (K) m RMSE1 4463 725 994 61 0.112 5341 806 1129 58 0.153 10764 436 1086 28 0.144 6510 542 935 39 0.145 9139 343 895 27 0.016 5801 590 941 44 0.067 5429 616 944 48 0.038 7110 303 732 28 0.039 5022 657 960 52 0.0110 6279 357 736 32 0.0011 7026 514 939 37 0.0312 1000 400 805 22 0.0013 10000 400 598 17 0.0014 10000 400 532 17 0.0015 9069 524 1072 32 0.0816 7946 588 1068 37 0.0817 7655 604 1066 38 0.0418 7364 627 1072 40 0.0319 7413 621 1069 40 0.1120 6745 668 1076 44 0.0721 6667 680 1083 45 0.1122 6434 702 1091 46 0.0423 6028 734 1098 50 0.0724 6364 720 1105 48 0.1725 5942 534 893 41 0.0426 5861 525 879 41 0.0427 5137 778 1088 58 0.1128 5138 761 1072 57 0.1829 5069 737 1043 56 0.1330 4687 737 1020 60 0.1231 4530 736 1010 61 0.1432 6005 393 756 34 0.0033 6333 631 1014 44 0.0234 7451 514 964 35 0.0435 6601 562 961 40 0.0236 6326 584 967 42 0.0437 6257 572 950 42 0.0438 5957 615 975 45 0.0239 5688 639 983 47 0.0440 5609 624 963 47 0.0230Chapter 5: Multicomponent 1 bar model5.1 Linear model-A first approximationThe purpose of this chapter is to build a chemical model for the T-X-η relation of silicatemelts. The chemical model is able to reproduce the data equally well compared to the baseline VFTfits with a constant Av f t . Previous viscosity models have generally expressed the chemical effectby a combination of oxides, or molecules, or melt structural organization (i.e. SM, NBO/T) anda range of adjustable parameters (Shaw, 1972; Bottinga and Weill, 1972; Giordano and Dingwell,2003; Russell and Giordano, 2005; Giordano et al., 2006, among references).Bottinga and Weill (1972) and Shaw (1972) adopted the Arrhenian equation (Eq. 3.1), limit-ing their applicability to the superliquidus temperatures. Bottinga and Weill (1972) did not providecoefficients for T less than 1200◦C; therefore, their model can not be used at lower temperatures.Shaw model is suggested to be used for viscosities lower than 105 Pa s. It can also easily beextrapolated to lower temperatures. Giordano and Dingwell (2003; GD model) adopted the VFTequation (Eq. 3.2), which allows for non-Arrhenian behaviour. They fitted VFT function to a dataset of 19 natural multicomponent melts, spanning a viscosity range of 100 to 1012 Pa s. GD modeldeveloped two equations where the three parameters of the VFT equation are each dependent on astructural parameter: NBO/T and SM, respectively.For modelling viscosity for Di-An-Ab melts, Russell and Giordano, (2005; RG model) ex-pressed Bv f t and Cv f t by the abundances of Di, Ab, An molecules, three quadratic terms, and 13adjustable parameters. RG model reproduced the data in Di-An-Ab ternary system well with a min-imal number of parameters. Also, their model is able to correlate model parameters with derivativemelt properties, like Tg, and melt fragility. Giordano and Russell (2007) also modelled the T-, X-dependence of viscosity across MgO-CaO-Al2O3-SiO2 (MCAS) system, which overlaps Di-An.GR model expresses the chemical effect by a combination of four oxides: MgO-CaO-Al2O3-SiO2,two cross-terms: SiO2×(MgO + CaO), and Al2O3×(MgO + CaO), and 11 adjustable parameters.31Since they calibrated the model on a much more extensive database, they were able to reproducethe data for Di-An melts very well. Two cross-terms in Bv f t are responsible for the nonlinearity inthe compositional effect. They suggested the non-linear contributions in Bv f t is smaller than thelinear contributions (Giordano and Russell, 2007).I have expressed the compositional dependence of the terms Bv f t and Cv f t in terms of themole.% abundance of oxides: MgO, CaO, Al2O3, SiO2, and Na2O. The choice of an oxide basis,over a basis of mineral end members Di-An-Ab (i.e. Russell and Giordano, 2005), improves thepotential for extrapolation of the model to beyond the ternary system used for calibration.I first fitted the 1 bar data sets to a purely linear VFT model. The parameters Bv f t and Cv f tare simply modelled as a linear-mixing of five oxides. The idea of this approximation is to studyhow significant non-linear interactions are. The parameters Bv f t and Cv f t are expanded as follows:Bv f t =5∑i=1biXi; Cv f t =5∑i=1ciXi (5.1)where subscripts i refer to five oxides: SiO2, Al2O3, MgO, CaO, and Na2O. And this formularepresents a purely linear compositional dependence of melt viscosity. There are 11 parameters intotal: Av f t ,b1,b2,b3,b4,b5,c1,c2,c3,c4, and c5. The values for 11 parameters are given in Table5.1. The linear model could reproduce viscosities for An-Ab melts but not Di-An and Di-Ab melts,suggesting the non-ideal interactions between An, Ab and Di are not negligible. Na2O shows thelargest contribution in Bv f t whilst Al2O3 has the smallest effect.32Table 5.1: Coefficients for Bv f t and Cv f t by solving the linear model. Av f t is -3.80 Pa s.Name B coefficients C coefficientsSiO2 255.83 -11.64Al2O3 -53.39 39.29MgO -91.31 36.12CaO -280.93 17.59Na2O -573.82 47.79335.1.1 Quality of fits: linear modelThe accuracy of this linear model is illustrated in Figure 5.1. This linear model can reproducemost high-T measurements to within 0.5 log units, where the T-dependence is more Arrhenian. Thedeviations at lower T and higher values of viscosity become considerably larger, especially alongthe Di/An- Ab binaries, and ternary. The residuals between model and observed viscosities valuesare substantial at lower temperatures (e.g. Ab melt and An50Di50 melt; Fig. 5.1B). Those misfitssuggest that there are higher-order interactions between the oxides.-4 -2 0 2 4 6 8 10 12 14 16log [Measured]-4-20246810121416log [Model]A = -3.8049(A)(a)-4 -3 -2 -1 0 1 2 log [VFT Constant A]-4-3-2-1012 log [Linear model](B)Ab100(b)Figure 5.1: Results of the linear model. (a) log η modelled by the linear model compared tooriginal data. (b) Residuals (model observed) from the linear model compared to residuals due tofitting VFT with a constant Av f t . Dashed lines are ± 0.5 log units of viscosityThe optimal value for A is -3.80 Pa s, which is lower than the high-T limit to the viscosity of-4.3± 0.74 Pa s provided by theoretical considerations (Russell et al., 2002; Russell and Giordano,2005; Angell, 1985; Toplis, 1998; Russell et al., 2003). Values for VFT parameters are calculatedby the coefficients listed in Table 5.1. The Bv f t and Cv f t parameters show similar variations withthat by VFT with a constant Av f t model (Fig. 4.2). Bv f t and Cv f t lie within the area bounded bythree end members, and the parameters vary linearly along each join. The maximum and minimumof Bv f t are assigned to Ab and Di melts, respectively. An and Ab melts have the maximum andminimum of Cv f t values. Since the linear model does not reproduce the data well, it suggests thecompositional dependence of viscosity contains significant non-linear contributions.34200 300 400 500 600 700 800 900C [Model]3000400050006000700080009000100001100012000B [Model]DiAnAbA = -3.8049End MembersDi-AbDi-AnAn-AbTernaryFigure 5.2: Values of parameters Bv f t vs. Cv f t . Lines denote the suggested area defined by threeend-members.5.2 A non-linear- optimal modelI have developed a non-linear multicomponent model using a 13-parameter oxide basis.Previous tests reduced the number of adjustable parameters from a 21-parameter chemical model,which includes ten quadratic terms to account for the nonlinearity in the parameter Bv f t (the com-binations of each two oxides). This model uses the minimum number of quadratic terms whichare needed to reproduce the viscosity data sets. Additional terms did not considerably improve thequality of fitting. The VFT coefficient Bv f t is expanded as follows:Bv f t =5∑i=1biXi+3∑j=1bb jXYj (5.2)where bi and bb j are adjustable parameters related to the five oxides (xi, mol.%) and XYj are thethree compositional cross-terms: SiO2×(MgO+CaO), Al2O3× (MgO+CaO), and Na2O×(SiO2+Al2O3),respectively. The compositional dependence of the VFT coefficient Cv f t is modelled as:Cv f t =4∑k=1ckXk (5.3)where the summation is over all of the oxides, excluding SiO2, and ck are the adjustable parameters(Table 5.2) with 1σ uncertainties on model parameters.35Table 5.2: Coefficients for the 13-parameter multicomponent 1 bar model for Di-An-Ab melts. Confi-dence limits (1σ ) on model parameters are also reported in brackets. Data were fitted to the multicom-ponent equation with a constant value for Av f t (-4.73 ± 0.29 Pa s).Linear Terms SiO2 Al2O3 MgO CaO Na2Obi 186.90 (33.6) -225.08 (439.4) 65.19 (439.6) 520.83 (448.3) 842.06 (597.6)ci 41.43 (28.5) 37.74 (28.5) -9.86 (28.6) -10.58 (28.5)B Cross-Terms1 SiO2∗MC Al2O3∗MC Na2O ∗SA-7.61 (1.9) -2.48 (0.4) -9.43 (5.4)1 SA: SiO2 + Al2O3 ; MC: MgO + CaO.36The optimal solution was obtained by χ2 minimization of the function:χ2 =n∑i=1[logηi− (A+∑5j=1 b jx j,i+∑3k=1 bbkXYk,iTi−∑4m=1 cmxm,i)]2 (5.4)calibrated against 560 (i.e. N) viscosity measurements with known log η , T(K) and compositionfor each data. The optimal 13 model coefficients and their confident limits (95%) are listed in Table5.2. The compositional cross-terms for Bv f t involve a combined term for the structural modifyingcations (MC = MgO+CaO) and network forming cations (SA = SiO2+Al2O3). The addition ofthese terms allows for non-ideal interactions between components (Giordano and Russell, 2007;Giordano et al., 2008); I found that additional quadratic terms to those employed here did notimprove the quality of fits significantly.5.2.1 Quality of fitsThe model reproduces the data well (Fig. 5.3), and the average RMSE for individual meltcompositions is 0.26 log units (Table 5.3). The model reproduces the majority of data to ± 0.5log units; deviations are largest for Di-Ab melts at viscosities >107 Pa s (Fig. 5.4D). The qualityof fit for this T-X model for melt viscosity is excellent when compared to the baseline VFT fits(Fig. 4.1A; Table 4.1) given the reduction in adjustable parameters from 81 to 13 (Table 5.3).Furthermore, the present 13-parameter, oxide-based model represents a marginal improvementover that developed in Russell and Giordano (2005) and potentially allows for extrapolation outsidethe Di-An-Ab system to the wider NMCAS system.The best-estimate value for A is -4.73 ± 0.29 Pa s which is only slightly different from thevalue obtained by Giordano and Russell, (2007) (-4.67) investigating the MCAS system, and itagrees well with theory (i.e. 10−4±1 Pa s; Myuller, 1955, Angell, 1991).37-4 -2 0 2 4 6 8 10 12 14 16log [Measured]-4-202468101214log [Model]A = -4.7318(A)(a)-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 log [VFT Constant A]-1-0.8-0.6-0.4-0.200.20.40.60.81 log [13-parameter](B)(b)Figure 5.3: Quality of the 13-parameter multicomponent model: (a) Values of log η predicted bythe multicomponent 1 bar model compared to original data; dashed lines denote ± 0.5 log units ofviscosity. (b) Residuals (model observed) due to 13-parameter model compared to residuals due tofitting VFT with a constant Av f t . Dashed lines denote ± 0.5 log units of viscosity.Figure 5.4: Values of log η predicted by the multicomponent model compared to original data for:(A)-(C) Di, An, and Ab melts, respectively; (D)-(F) Di-Ab, Di-An, and An-Ab melts, respectively;(G) ternary melts. Dashed lines denote ± 0.5 log units of viscosity.38Table 5.3: Model parameters Bv f t ,Cv f t predicted for each meltin the system Di-An-Ab (cf. Table 3.1) using the multicom-ponent 1 bar model (Table 5.2) with a constant high-T limitto viscosity (i.e. Av f t = -4.73 ± 0.29 Pa s). Also reported arethen calculated Tg, steepness index (m), RMSE as the qualityof fit for individual melt compositions, and calculated liquidustemperatures (TL) from Weill et al. (1980).Label B C Tg (K) m RMSE TL(K)Di 4969 697 994 56 0.18 1665An 5673 789 1128 56 0.16 1825Ab 11419 386 1068 26 0.25 1394Di50Ab50 6532 541 932 40 0.22 1545Di25Ab75 8560 463 975 32 0.13 1473Di60Ab40 5917 574 928 44 0.43 1566Di69.3Ab30.7 5634 589 926 46 0.62 1586Di75Ab25 5335 619 938 49 0.31 1601Di84.2Ab15.8 5026 645 946 53 0.58 1624Di95Ab5 4976 681 979 55 0.09 1651Di30Ab70 8206 469 959 33 0.81 1487Ab80Di20 8244 474 966 33 0.18 1485Ab57Di43 6535 541 932 40 0.26 1545Ab30Di70 5357 617 937 49 0.40 1600Ab80An20 9869 466 1056 30 0.19 1606An30Ab70 9329 490 1048 31 0.28 1660An34Ab62 9022 514 1053 33 0.20 1678An40Ab56 8581 543 1056 34 0.26 1698An50Ab50 7871 589 1059 38 0.15 1725An55Ab45 7555 617 1069 40 0.16 1733An61Ab37 7211 647 1078 42 0.13 1749An75Ab25 6716 682 1083 45 0.16 1775An80Ab18 6285 726 1101 49 0.15 1794An87Ab9 5955 755 1111 52 0.21 1812Di50An50 4933 743 1038 59 0.17 1638Di58An42 4887 736 1028 59 0.19 1584Di20An80 5284 771 1086 58 0.14 1776Di30An70 5136 761 1068 58 0.25 1736Di40An60 5019 752 1052 59 0.29 1694Di64An36 4865 730 1021 59 0.13 1548Di80An20 4862 715 1006 58 0.17 1604Di90An10 4900 706 999 57 0.22 1637Di18.5An 9.8Ab31.7 6526 632 1022 44 0.31 1686Di21.2An8.8Ab70 8518 474 983 32 0.70 1464Di28.2An15.5Ab56.3 7514 521 971 36 0.48 1489Di35.9An21.7Ab42.2 6510 583 972 42 0.20 1504Di38.3An5.9Ab55.8 7099 526 950 37 0.21 1516Di43.8An27.6Ab28.6 5850 626 976 47 0.08 1526Di51.7An32Ab16.3 5440 658 983 51 0.17 1537Di55.1An14.5Ab30.4 5690 618 958 47 0.19 1552395.2.2 Compositional dependence of Bv f t and Cv f tThe compositional-dependent model values of Bv f t and Cv f t for all 40 melt compositions arereported in Table 5.3 and illustrated in Figure 5.5. The range of Bv f t (4862–11419 K), and Cv f t(386-789 K) is smaller and better constrained than the range found for the baseline VFT constantA fits (Bv f t : 4463- 10760; Cv f t : 303-806; Fig. 4.2). The model incorporates the effects of, both,temperature and composition, which provides a strong coupling between datasets for differentmelt compositions. The consequence is well-constrained estimates of Bv f t and Cv f t for all melts asevidenced by substantially improved correlations between B- C pairs for melts along the binariesand within the ternary system (Fig. 5.5). This is true even for melts that have only a few viscositymeasurements over narrow ranges of temperature (i.e. Di90An10; N=6).400 500 600 700 800C [Model]4000600080001000012000B [Model]A = -4.73DiAnAbFigure 5.5: Model values of Bv f t vs. Cv f t (Table 5.3); symbols as in Figure 2.5B. Dashed lines aremodel values of Bv f t and Cv f t from Russell and Giordano (2005).Values of Bv f t are the maximum value for Ab melt (11419) and at a minimum along the Di-An join (Di80An20; 4862). An melt has the maximum value of C (789 K), and Ab has the lowestvalue (386 K). Bv f t decreases non-linearly with increasing Di content along Ab-Di join and showsa concave up relation (Fig. 5.5) whilst Bv f t and Cv f t vary nearly linearly along the Ab-An join.Along the Di-An join, values of Bv f t show a concave upwards trend with increasing An content.With small additions of An, values of Bv f t initially decrease from 4969 to a minimum (4862) at 80mol.% Di. After the minima, Bv f t increases to 5673 for An melt. This implies small additions of40An to Di-rich melts perturb the melt structure significantly. The values of Cv f t increase from Abtowards An-Di melts and decrease from An-rich to Di-rich melts.Importantly, the optimal values of Bv f t and Cv f t for melts on the binary joins (symbols) fallon or close to the model lines (solid lines; Fig. 5.5; Fig. 5.6) which are themselves smooth andcontinuous. Additionally, the model lines for the binaries define a field which encloses the modelvalues of B: C for all ternary melt compositions (black symbols, Fig. 5.5).Figure 5.6: Values of Bv f t and Cv f t solved by the multicomponent 1 bar model for melts in systemDi-An-Ab plotted against compositional parameters: (A)-(B) Bv f t and Cv f t vs. XDi; (C)-(D) Bv f tand Cv f t vs. XAb. (E)-(F) Bv f t and Cv f t vs. XAn.I have explored the relationships between model values of Bv f t and Cv f t , melt composition,and the pseudo-structural parameters SM and NBO/T (Fig. 5.7A-D). NBO/T can serve as a chem-ical proxy for the structural organization of the melt and the degree of structure polymerization.41However, values of NBO/T in the Di-An-Ab system vary from 0 (An, Ab melts) to 2 (Di melt).Similarly, the SM parameter (MgO+CaO+Na2O, mol.%; Giordano et al., 2006) is an alternativefirst-order, chemical proxy that can track relative changes in the structural configuration of silicatemelts. The SM parameter has the benefit over NBO/T that the range of values in the Di-An-Absystem is extended and the end-members have unique values: Ab (12.5), An (25), and Di (50) (Fig.5.7A-D).0 0.5 1 1.5 2NBO/T600080001000012000B [Model]DiAbAn(a)0 0.5 1 1.5 2NBO/T300400500600700800C [Model]AnDiAb(b)10 20 30 40 50SM4000600080001000012000B [Model]DiAnAb(c) (d)AbDiAn500055006000650070008000900010000(e)AbDiAn450750700650600550500(f)Figure 5.7: Model Bv f t and Cv f t solved by the multicomponent 1 bar model for Di-An-Ab meltsplotted against compositional dependent parameters: (a) Bv f t vs. NBO/T. (b) Cv f t vs. NBO/T.(c) Bv f t vs. SM. (d) Cv f t vs. SM. (e) Bv f t contoured ternary diagram. (f) Cv f t contoured ternarydiagram.42The model values of Bv f t and Cv f t are strongly correlated to melt structure as representedby the chemical proxies SM and NBO/T (Fig. 5.7A-D). The most polymerized melt (Ab; NBO/T= 0; SM = 12.5) has the maximum value of Bv f t . Increasing Di contents along, both, the Ab-Di, and An-Di binaries cause a marked, smooth decrease in the Bv f t parameter. The values of Bv f tchange almost linearly along the Ab-An joins and non-linearly along the Di-Ab join (Fig. 5.7A;C).The peak value of Cv f t corresponds to An melt (NBO/T = 0; SM = 25) and the minimum valuecorresponds to Ab (NBO/T = 0; SM = 12.5). With increasing Ab-content along An-Ab join, valuesof Cv f t decrease by 370, whilst values decrease by 81 from An to Di (Fig. 5.7B;D). CommonlyCv f t values increase as the degree of polymerization decreases, but that appears not to be the casefor the melts of the An-Di join.The decrease in Bv f t values observed for these three binary systems (Fig. 5.7A; C) is par-alleled by a continuous increase in SM implying a decrease in the degree of melt polymerization.The largest change in Bv f t values occurs between Ab and Di, which is also the largest change inSM parameter (∆∼37). Notably, the decrease in Bv f t for Ab-An melts is almost as large althoughthe change in SM is significantly smaller (∆ ∼12; Fig. 5.7A;C). Model values of Cv f t along thebinary joins are linearly correlated to SM (Fig. 5.7B;D). The largest change in Cv f t is between Aband Di melts, which again coincides with the greatest change in SM.I have used this melt viscosity model to contour the ternary compositional space for values ofBv f t (Fig. 5.7E) and Cv f t (Fig. 5.7F). Values of Bv f t show a nearly linear increase with increasingAb-content; slight non-linear effects are only observed at high Di-contents despite the higher-order, cross-terms in Bv f t (Eq. 5.2). The expansion of Cv f t (Eq. 5.3) has only linear terms andtherefore the model contours of Cv f t are linear across the Di-An-Ab ternary, increase linearlyfrom a minimum at Ab (386K) towards Di-An melts, and have a maximum at An (1128K; Fig.5.7F). The change in values of Cv f t across the binaries varies substantially: Di-An (∆92K), Di-Ab(∆311K) and Ab-An (∆403K).435.2.3 Model values of Tg and melt fragility mThe 13-parameter multicomponent model for melt viscosity provides an opportunity to in-vestigate Tg (Fig. 5.8A; C) and fragility (Fig. 5.8B; D).AbDiAn950100010501100(a)AbDiAn455055353040(b)10 20 30 40 50SM9009501000105011001150T g (K)AnAbDi(c)10 20 30 40 50SM2030405060Steepness index (m)AnDiAb(d)Figure 5.8: Values of Tg and m solved by the multicomponent 1 bar model for Di-An-Ab melts:(a) Tg contoured ternary diagram. (b) m contoured ternary diagram. (c) Tg vs. SM. (d) m vs. SM.Model values of Tg are contoured for the Di-An-Ab system in Figure 5.8A. The highest valueof Tg is ascribed to An melt (1128 K), and Tg decreases by 50-125 K from An towards the Ab-Dijoin. Values of Tg for Ab and Di are 1068 K and 994 K, respectively, and the minimum occursalong the Di-Ab join at ∼60 mol. % Di. Values of Tg vary non-linearly between end-memberson all three binaries (Fig. 5.8A), and the Di-Ab join shows the strongest nonlinearity. I plottedthe model values of Tg for 19 individual melts against the corresponding values measured directlyby calorimetry at a cooling/ heating rate of 5 K/min (Webb and Knoche, 1996). The agreement44is excellent between these measurements of Tg and model values calculated at η =1012 Pa s (Fig.5.9A). Measured calorimetry Tg sensitively depends on the thermodynamic history (cooling/ heat-ing rate Q). Increasing cooling/heating rate will result in a faster characteristic relaxation timescale(i.e. ∆τ = ∆T/Q) and the glass transition will shift to higher T region when performed at a morerapid rate, therefore, a higher value of measured Tg (Stebbins et al., 1995). The higher measuredTg matches rheological Tg’s corresponding to the faster characteristic relaxation timescales foundat lower viscosities.950 1000 1050 1100 1150Tg (K) Calorimetric9501000105011001150Tg (K) Model< 1012> 1012-Q +Q(a)20 30 40 50 60m- VFT Constant A2030405060m- ModelDi75Ab25Di50An50Di95Ab05Di90An10Di58An42L(b)Figure 5.9: (a) Model values of Tg taken as the temperature where η = 1012 Pa s compared to mea-sured values of calorimetric Tg for Di-An-Ab melts (Webb and Knoche, 1996). Arrows indicate therelative effects on operational or measured values of Tg by adopting higher or lower critical valuesof η or using faster or slower calorimetric scan rates. (b) m values solved by the multicomponentmodel (Table 5.3) compared to m values solved by fitting VFT with a constant Av f t (Table 4.1).The solid line is the 1:1 line. Dashed lines denote 5% uncertainty limitsMelts in Di-An-Ab systems cover a wide range of fragility from strong (Ab; m 25-30) tofragile (Di and An; m 55-60). The fragility index increases almost linearly from the lowest value atAb melt towards melts along the Di-An join and contours of m are practically parallel to the Di-Anboundary. The maximum value of m is ascribed to a melt composition on the Di-An join at ∼50mol.% Di (Fig. 5.8B; D).I plotted model values of m by the multicomponent 1 bar model against the model valuesfrom the VFT-constant A model (Fig. 5.9B). Majority of values are in good agreement within5% uncertainties except for five compositions (labelled on Fig. 5.9B). The reason for the diver-45gence is that those five compositions only have viscosity data at high temperatures; therefore, theT-dependence is poorly defined (Fig. 5.10). In Fig. 5.10, model lines calculated from the mul-ticomponent model (solid lines) are compared to that from the VFT- constant A model (dashedlines). Model lines diverge at low temperatures. VFT-constant A model was mainly parameterizeddepended on the quality of data (i.e. distribution) without considering compositional dependence.The values predicted by the multicomponent model are more consistent with respect to chemicalcomposition. The intermediate melts show intermediate T-dependence of viscosity.4 6 8 10 1210000/T(K)-2024681012log (Pa s)Di75Ab25Di95Ab05(a)4 6 8 10 1210000/ T(K)024681012log (Pa s)Di58An42Di50An50Di90An10(b)Figure 5.10: (a) Model log η against 10000/T (K) for two Di-Ab melts from Scarfe and Cronin(1986) compared to original data. (b) Model lines for three Di-An melts from Scarfe et al. (1983)compared to original data. Solid lines are model lines from the multicomponent model. Dashedlines are model lines from VFT- constant A model.Another quantity to measure fragility is the Doremus ratio of fragility (RD), which is intro-duced in Chapter 3. RD, which is based on the changes that occur in the activation energy (Ea) thatare unambiguously characterized by the ratio between two values of Ea at low and high temper-atures. I chose the liquidus and Tg as the high and low characteristic temperatures for these tworegions respectively. Therefore, RD can be calculated as RD =Ea@TgEa@TL.The model values of Ea derive from the derivative of log η with respect to reciprocal Tcalculated as:Ea = 2.303 R∂ logη∂1/T= 2.303 RB(CT −1)2(5.5)where Bv f t and Cv f t are calculated by multicomponent 1 bar model (Eq. 5.2; 5.3)46I have used the models of Weill et al., (1980) to calculate the topology of the liquidus surfacefor the Di-An-Ab system (See Chapter 6 for the calculation). Ab melt has the lowest RD (1.28)whilst Di and An melts have similar, larger ratios (3.79; 3.57). The model value of RD for Di melt(3.79) is lower than the suggested value by Doremus (2002). This deviation might be explainedby the choices of two single temperatures (e.g. liquidus and Tg) rather than using two Arrhenianregions with constant Ea values. The higher ratio, the more fragile are the melts. The maximumvalue of RD belongs to the melt along Di-An join. I then plotted the RD against the steepness index(m). These two quantities are indeed proportional (Fig. 5.11A) for Di-Ab, An-Ab and ternarymelts. For Di-An melt, the correlation is not as clear as other melts because of the turn-over in maround 50 mol.% Di (Fig. 5.8D) whilst RD has a minimum value at SM around 40.1 1.5 2 2.5 3 3.5 4RD2530354045505560Steepness index m(a)10 20 30 40 50SM11.522.533.54R D(b)Figure 5.11: (a) RD against the steepness index (m). symbols are the same as above figures. (b) RDagainst SM.47Chapter 6: 1 bar model implication and application6.1 Model predictionThis model captures the full rheological behaviour of melts in the system Di-An-Ab. Thismodel can explicitly predict the T-dependence (Arrhenian and non-Arrhenian) of melt viscosityacross the full compositional range of the Di-An-Ab ternary system. I illustrate the ability of thismodel by plotting model lines for a variety of melt compositions across the temperature range900 to 2500 K (Fig. 6.1). Model lines are plotted for individual melt compositions at 10 mol.%increments along each of the binaries (Fig. 6.1A–C) and ternary compositions at 10 mol.% incre-ments (Fig. 6.1D–F). Along Di-An join, melts show fragile behaviour. The model lines are almostparallel, which indicates a similar T-dependence of viscosity for Di-An melts. At low T region,viscosity varies more than seven orders of magnitude along the join, whilst that varies less than 1log unit at high T.Both Ab-Di and An-Ab binaries show a crossover as a function of temperature becausemelt rheology spans from strong (Ab) to fragile (An/ Di). The crossover is weaker for Di-Abjoin. Intermediate melts lie sequentially between two end members. At high T > 1500 K, Ab andAb-rich melts have higher values of viscosity. At low T < 1200 K, An and An-rich melts havethe highest values of viscosity. This crossover was also suggested by the natural melts betweenrhyolite (Ab) and basalt (An; Giordano and Dingwell, 2004). Compared to Di-An melts, viscosityshows a more pronounced variation along Di-Ab join at high T.Model lines for ternary melt compositions having equal proportions of An: Di (Fig. 6.1D),An: Ab (Fig. 6.1E), and Di: Ab (Fig. 6.1F) were also plotted. Curves are calculated for meltsat intervals of 10 mol.% of x. For melts having equal proportions of An and Ab, melt viscositygradually decreases with increasing Di content (Fig. 6.1E). At T < 1000 K (viscosity > 1010 Pa s)where there is a minor crossover involving Di-rich melts. For melts having equal Di-Ab contents(Fig. 6.1F), model lines at high T are identical. Regardless of the An content, these melts show484 6 8 1010000/T (K)0510log (Pa s) Di-An4 6 8 1010000/T (K)0510log (Pa s) Di-Ab4 6 8 1010000/T (K)0510log (Pa s) An-Ab4 6 8 1010000/T (K)0510log (Pa s) Di xAnxAb1-2x4 6 8 1010000/T (K)0510log (Pa s)Di 1-2xAnxAbx4 6 8 1010000/T (K)0510log (Pa s)DixAn1-2xAbx(B)(C) (D)(E) (F)(A)DiAbAnAbAbDiDiAnAnFigure 6.1: Model lines showing T- dependence of melt viscosity in system Di-An-Ab for specificmelt compositions. (A) binary An-Di at intervals of XDi of 0.1; (B) binary Ab-Di at intervals ofXDi of 0.1; (C) binary Ab-An at intervals of XAn of 0.1; (D) Dix-Anx-Ab1−2x; (E) Di1−2x-Anx-Abx;and (F) Dix-An1−2x-Abx at intervals of x = 0.1essentially uniform values of Arrhenian-like viscosity over the temperature interval 1500 to 2500K. Below this temperature, the melts show strongly non-Arrhenian behaviour, and the effect ofAn-content is to increase viscosity at a constant temperature. At 1150 K, the values of viscosityacross this system vary by at least 6 orders of magnitude. The pattern that emerges is very similarto that shown in Figure 6.1A for An-Di melts; this reflects the fact that the viscosity of Di and Anmelts has similar temperature dependencies (Fig. 6.1A).The model is able to predict the data beyond the original calibration data set. I selectedseveral melt compositions which were not used for calibration and plotted the model lines against49the experimental data (Fig. 6.2). Compositions from Kozu and Kani (1935) (e.g. Di20An20Ab60,Di20An60Ab20, Di20An40Ab40) were dropped from calibration because there were no indepen-dent analyses on the nominal compositions. The predictions are excellent. The compositionswhich only have high-T data are also well-described (Fig. 6.2). Composition from Tauber (1987)(Di13.3An3.3Ab83.4) was dropped because of large Ab discrepancy between the nominal and mea-sured composition of the melt. The comparison between model and data is within the uncertaintiesof the viscosity measurements.2 4 6 8 10 1210000/T(K)051015log (Pa s)Di 20An40Ab40Di 20An60Ab20Di 13.3An3.3Ab83.4Di 20An20Ab60Figure 6.2: Predictions of viscosity for melt compositions not used for calibration compared to ex-perimental data. (A) Di20An20Ab60 (B) Di20An60Ab20 (C) Di20An40Ab40 (D) Di13.3 An3.3Ab83.46.2 Model extrapolationThe oxide-based model has the potential to extrapolate beyond the simple system. The modelprovided by Giordano and Russell (2007) for the MgO–CaO–Al2O3–SiO2 (MCAS) system has asimilar parameterization with this model except for the Na2O term. I use my model to calculatethe viscosities for melt compositions in the CAS, MCAS, and NAS systems provided by Toplisand Dingwell (2004), Solvang et al. (2004) and Solvang et al. (2002), and compare to the reportedmeasured values (Fig. 6.3). Measured viscosities for CAS melts with 50, 67, and 75 mol.% SiO2content are from Toplis and Dingwell (2004). They measured the viscosity for CAS melts with 67,and 75 mol.% SiO2 content at a temperatures range 1400 to 1700◦C, and my model can reproducetheir viscosities to within experimental error± 0.5 log units (Fig. 6.3B). Deviations between modeland observed viscosities become larger at lower T for CAS melts with 50 mol.% SiO2 content (Fig.506.3B). There is no low-T data for CAS melts with a higher content of SiO2. Solvang et al. (2004)measured viscosities for CAS melts with lower mol.% SiO2 (30- 50 mol.%). Model values deviatesignificantly from those observed values (Fig. 6.3A). The same result showed in Figure 6.3D,where the model reproduces high-T viscosities for NAS melts with higher contents of SiO2. Thismodel could not reproduce viscosities for MCAS melts (Fig. 6.3C). This result implies that meltstructures change significantly with variations of individual structure modifiers (Na, Mg, Ca) andformers (i.e. Si) even SM values are similar.0 5 10 15log Measured [Pa s]051015log Model [Pa s]9 CAS melts withSiO2< 50 mol%Wo (CaSiO3)Geh (Ca2Al2SiO7)Solvang (2002)(a)0 5 10 15log Measured [Pa s]051015log Model [Pa s]CAS melts: 75mol% SiO 2: 61 to CAS melts 50mol% SiO 2: 44Toplis & Dingwell (2004)(b)-20 -10 0 10 20 30log Measured [Pa s]-20-100102030log Model [Pa s]MCAS18MCAS17MCAS15MCAS16Solvang (2002)(c)0 5 10 15log Measured [Pa s]051015log Model [Pa s]NAS 82 mol% SiO2: 50.to NAS 50 mol% SiO 2: 49Toplis et al. (1997)(d)Figure 6.3: Model predictions by the multicomponent model for NMCAS melts (Solvang et al.,2002; Toplis and Dingwell, 2004) (a) CAS melts with low SiO2-content. (b) CAS melts with highSiO2-content. (c) MCAS melts. (d) NAS melts.516.3 Isotherms and IsokomsThis viscosity model for the Di-An-Ab system is able to predict, both, Arrhenian and non-Arrhenian T-dependent behaviour of melts across the entire Di-An-Ab ternary system. In Figure6.4, I illustrate the variations in isothermal melt viscosity predicted by the multicomponent 1 barmodel at temperatures of 800◦C, 1000◦C, 1200◦C and 1600◦C. The isothermal viscosity curvesare plotted against the SM parameter which I use as a proxy for the degree of melt polymerizationand to track changes in melt structural properties.The compositional-driven variations in melt viscosity are greatest at low temperature (Fig.6.4). The 800◦C isotherm (Fig. 6.4A) corresponds to low-T viscosity data close to the glass tran-sition temperature interval. This temperature is also close to the eruptive temperatures of the manysilicic melts which are, themselves, commonly close to the melts’ anhydrous Tg’s, and it is perti-nent to post-eruptive processes such as welding and rheomorphic flow of ignimbrites. At 800◦C,viscosity spans 8 orders of magnitude (107 to 1015 Pa s) and the effects of melt composition areaccentuated. Interestingly, at 800◦C the highest viscosity in the system occurs unexpectedly atintermediate values of SM coinciding with An melt. The lowest viscosity coincides with higherSM values associated with ∼Di70Ab30 as opposed to the most depolymerized melt Di (SM=50).I have used the model of Giordano et al. (2006) to calculate the viscosity of natural melts at eachtemperature over the same range of SM values (Fig. 6.4; dashed line). Giordano et al. (2006) re-vised the model by Giordano and Dingwell (2003), which is a function of SM parameter to includean additional parameter reflecting the ratio of alkalis to alumina, producing lower viscosities forperalkaline melts and higher viscosities for peraluminous melts. The natural melts most closelyapproximate the viscosity of melts along the Ab-Di join. The substantial differences in valuesbetween the natural melt trend and the 800◦C isotherm for the Di-An-Ab system suggest largefundamental differences in melt structure even at similar SM values. In particular, melts alongthe Ab-An binary show a pronounced increase in viscosity with SM that is virtually perpendicularto the natural melt trend over the same range of SM values (12.5-25). Similarly, at SM valuesfrom 40 to 55, natural melt viscosity decreases continuously whilst the viscosity of Ab-Di melts5220 30 40 50SM6810121416log (Pa s)DiAnAbT = 800 oC(a)10 20 30 40 50SM246810log (Pa s)DiAnAb T = 1000 oC(b)10 20 30 40 50SM0123456log (Pa s)DiAnAb T = 1200 oC(c) (d)Figure 6.4: Isothermal log η at 800, 1000, 1200, 1600 ◦C against SM. (a) 800◦C. (b) 1000◦C.(c) 1200◦C. (d) 1600◦C. Dashed lines denote model viscosity values for natural multicomponentmelts after Giordano et al., (2006).increases. At low temperatures, near Tg, it is clear that the linkages between chemical compositionand the structural organization of melts in simple systems such as Di-An-Ab are significantly morecomplex than in natural multicomponent melts.The temperature range of 1000- 1200◦C is pertinent to many volcanic and magmatic pro-cesses (Fig. 6.4B, C). At these temperatures, well above the glass transition temperature interval,the range of viscosities in the Di-An-Ab system is reduced, and the effects of melt compositionbegin to diminish. At 1000◦C, viscosity values span < 5 orders of magnitude (103.5 to 108 Pas) and the differences in viscosity along the binaries is reduced substantially. Furthermore, thehighest viscosity now corresponds to the lowest SM value (Ab). An melts remain anomalouslyhigh relative to the log η : SM trend for Ab-Di melts indicating a significant difference in struc-53tural properties. At 1200◦C the log η : SM curves for all 3 binaries have collapsed substantiallyto form a single trend which parallels and is virtually the same as the trend predicted for naturalmelts (Fig. 6.4C; Giordano et al. 2006). At higher temperatures, the structural organization ofmelts attributable to composition begin to diminish and the differences observed between simpleand multicomponent natural melts at lower T (Fig. 6.4A) become negligible.At T > 1200◦C, the structural properties of melts (and viscosity) are dominated by topolog-ical contributions to the configurational entropy of the melt (Fig. 6.4D). The topological contri-bution is an expression of the increasing disorder in the melt network related to the average Si-Obond distances and Si-O-Si bond lengths (Stebbins et al., 2008; Giordano et al., 2008). At lowertemperatures, the topological contribution is subordinate to the chemical contributions to entropydue to mixing of different components in similar structural positions. At lower temperatures, es-pecially approaching the glass transition temperature interval, the contribution to entropy due tomixing dominates over the topological contribution and pronounced changes in viscosity can ariseover small changes in SM (Fig. 6.4A; Ab-An). The degree of order within the silicate network(i.e. polymerization) also dictates how temperature affects the relative topological and chemicalcontributions. More polymerized melts (SM: 10-25), such as Ab: An melts, are most affected byhigher temperatures indicating a rapid increase in the topological contribution to entropy. Con-versely, depolymerized melts such as Di (SM = 50) show less dramatic variations in viscosity withincreasing temperature suggesting that the chemical contribution to entropy is small relative to thetopological contribution even at low temperatures close to Tg.I also plotted model isokoms of viscosity at 103, 107 and 1012 Pa s showing changes in Twith melt composition represented by SM (Fig. 6.5). These three isokom values are somewhatrepresentative of: i) the high-T viscosity for some natural magmas (Di as a proxy for basalt; ii) theviscosity of melts close to fragmentation conditions, and iii) the viscosity at the glass transitiontemperatures, respectively. The lowest temperature associated with each isokom occurs along theAb-Di join and the minimum temperature shifts to higher Ab contents with increasing viscosity.This minimum along the Ab-Di join was also observed in the isothermal plots (cf. Fig. 6.5). The54highest temperature for each isokom changes from Ab at 103 and 107 Pa s to An melt close to theTg (i.e. 1012 Pa s). The range in temperatures associated with these isokoms is most extreme atlower viscosity (i.e. 103 Pa s; ∆∼575◦C) and least close to the Tg (i.e. 1012 Pa s; ∆∼225◦C) (Fig.6.5). The low viscosity isokom (103 Pa s) exhibits a cross-over between the Ab-Di join and theAb-An-Di joins. Isokom temperatures are highest for Ab-Di melts over the range of SM values12.5-25 and lowest at higher SM values. At higher viscosities, Ab-An and An-Di isokoms alwayshave higher (or equal) temperatures than Ab-Di melts.Figure 6.5: Isokom temperatures for η values of 103, 107, 1012 Pa s. (symbols as in previousfigures).556.4 Correlation of melt rheology and phase relationship in Di-An-Ab systemThe multicomponent 1 bar model can be used to calculate melt viscosities for liquidus condi-tions in the Di-An-Ab system (Fig. 6.6). I intent is to explore the connections between the liquidusphase relationships and the melt rheology in this simple system.The topology of the liquidus surface for the Di-An-Ab system was calculated after the mod-els of Weill et al. (1980). For pyroxene saturation surface:Tliqupyro = 1085.6+575.2∗ (x)+72.5∗ (y)−544.7∗ (x2)−848.4∗ (y2)+276.8∗ (x3)+ ...1846.1∗ x∗ (y3);(6.1)for plagioclase saturation surface:Tliquplagioclase = 1121.2+788.4∗ x+1551.6∗ y−7481.8∗ x2−6640.0∗ x∗ y−3044.6∗ y2+ ...15769.2∗ x3+37941.1∗ x2 ∗ y+9212.1∗ x∗ y2+3080.4∗ y3−7823.1∗ x4− ...74614.5∗ x3 ∗ y−32077.7∗ x2 ∗ y2−3025.0∗ x∗ y3−1156.5∗ y4+32215.8∗ x4 ∗ y+48865∗ x3 ∗ y2;(6.2)where x is the mole fraction of Di, y is the mole fraction of An. I then contoured the diagram for thetemperatures of the pyroxene and plagioclase saturation surfaces and the cotectic intersection (Fig.6.6A). The highest liquidus temperature is for An melt (1553◦C), followed by Di melt (1393◦C)and then Ab melt (1121◦C) (Fig. 6.6A); the cotectic temperatures range from ∼1300◦C on theDi-An join to ∼1190◦C on the Ab: Di join (Fig. 6.6A).I have then used the model liquidus temperatures across the ternary (Fig. 6.6A) to calculatethe 1 bar values of melt viscosity and contoured the ternary system for log η (Fig. 6.6B). Thehighest viscosity (106 Pa s) is for Ab melt which is also the lowest liquidus temperature (1121◦C).56AbDiAn1.81.621.82.22.42.62.42.222.62.82.8 333.23.23.43.43.63.73.61.4AbDiAn0.50.61.51.21.20.80.80.6223345 461.5AbDiAn13501300125012001200130015001400145012501350AbDiAn1.51.20.80.80.62233461.541.20.6(A)(B)(C)(D)Figure 6.6: (A) Liquidus surface and cotectic (in red) for the system Di-An-Ab after (Weill et al.,1980) contoured for liquidus temperature (◦C). (B) isokoms of log η (Pa s) calculated at 1 bar liq-uidus temperatures using the viscosity model presented here (Table 5.3). (C) combined isothermsand isokoms as shown in panel A and B delineating where melt viscosity is dominated by tem-perature (pseudo-parallel contours) versus composition (high angle contour intersections). (D)contoured RD.An and Di melts have higher liquidus temperatures (i.e. 1553, and 1393◦C) and have viscositiesof 100.7 Pa s and 100.4 Pa s, respectively. The viscosity of pyroxene saturated melts increases withincreasing Ab-An content until intersecting the cotectic where melts ultimately reach a viscosityof ∼106 Pa s. Plagioclase saturated melts increase significantly in viscosity with increasing Ab-content but melts moving towards the cotectic undergo only slight changes in viscosity even withdecreasing temperature (Fig. 6.6B).57I have plotted the contoured surfaces for liquidus temperatures (solid lines) and 1 bar meltviscosity (dashed lines) together in Figure 6.6C. The cotectic divides the ternary space into tworegimes (pyroxene vs plagioclase saturation) that show different relationships between the twocontoured surfaces. Pyroxene saturated melts have isokoms of viscosity that are nearly parallelto the isotherms indicating that the viscosity of these melts is mainly controlled by temperature(rather than melt composition). In contrast, the isokoms for plagioclase saturated melts intersectthe isotherms at high angles (nearly perpendicular) (Fig. 6.6C). For plagioclase melts, variations inmelt composition dominate liquidus viscosity values over the effects of temperature. For example,melts that are driven to the cotectic from high-An plagioclase will undergo little change in viscositydespite the decrease in temperature. Melts that are forced away from the cotectic towards Ab willsuffer a substantial rise in viscosity even for a lower drop in liquidus temperature.In Figure 6.6D, I illustrated the modelled values of RD, which is used as a proxy for meltfragility. Values of RD represents the changes in Ea at liquidus and Tg, respectively. Contouredlines intersect the isotherms at high angles (nearly perpendicular) for two regimes (Fig. 6.6A). Theminimum value of RD belongs to polymerized Ab melts. RD increases almost linearly from Ab toDi-An binary. Along Di-An join, RD shows a complex variation with Di content (i.e. SM). Theminimum value of RD along this join belongs to the cotectic composition, and the maximum valuebelongs to melt with ∼Di70.58Chapter 7: Pressure-dependence of viscosity7.1 P-effects on viscosityPressure is an important thermodynamic parameter of magmatic systems. The effects of P onthe viscosity of natural magmatic and silicate melts has been largely studied (Allwardt et al., 2007;Brearley et al., 1986; Karki and Stixrude, 2010; Kushiro, 1980; Liebske et al., 2005; McMillanand Wilding, 2009; Mysen et al., 1982; Persikov et al., 1989; Persikov, 1991, 1998; Persikov andBukhtiyarov, 2004; Persikov et al., 2018; Reid et al., 2003; Scarfe, 1987; Suzuki et al., 2005; Waff,1975; Wolf, 1995). Studies suggested that melt viscosity varies exponentially with increasing P(Brush, 1962; Liebske et al., 2003, 2005; Persikov, 1991, 1998; Scarfe, 1987; Wolf, 1995), whichcan be expressed in the Arrhenian equation as:η = ηo exp (Ea+Va PRGas T) (7.1)where Va is the activation volume (cm3/mol). Other notations are the same as in Eq. 3.6.The nature of P-dependence of viscosity depends on melt composition, which has been re-lated to the degree of polymerization of the melt in general. Viscosities of anhydrous rhyolite melt(e.g., Scarfe, 1987), jadeite melt (Kushiro, 1976), dacite melt (Tinker et al., 2004), andesite melt(Kushiro et al., 1976), and some basalt melts (e.g., Scarfe, 1987; Persikov et al., 2018) decreasewith increasing P in the first several GPa, whereas those of Di melt (e.g., Scarfe et al., 1979),haplokimberlite melt (Persikov et al., 2018), and peridotite melt (Liebske et al., 2005), all highlydepolymerized, increase with increasing P under the P-T conditions related to crust and mantle.Persikov et al. (2018) show at isothermal conditions (1800◦C), the viscosity of haplokimberlitemelts increases exponentially by about an order of magnitude as pressure increases from 100 MPato 7.5 GPa but the P- dependence is negative for basaltic melts, with the minimum at ∼5.5 GPa.59The negative effects of P on melt viscosity may be also related to the increasing coordinationnumbers of network formers Si4+ and Al3+ (e.g. Waff, 1975; Kushiro, 1980; Wolf and McMillan,1995; Persikov and Bukhtiyarov, 2004; Allwardt et al., 2007). This means that Al and Si trans-fer from network-former to modifier cations following an increase in depolymerization degree ofmelts. Higher coordinated Si-O and Al-O bonds (five- and six-fold coordinated) are longer andweaker than SiO4 and AlO2 tetrahedral bonds, therefore, less energy is required to break bonds(i.e. lower viscosity). Studies using NMR spectroscopy found that higher coordinated structureformers Si and Al are formed during compression (Farber and Williams, 1996; Stein and Spera,1993; Yarger et al., 1995). Another idea is that increasing P induces a decrease in the T-O-T angle,weakening the bridging bonds, causing a decrease in viscosities (Scarfe et al., 1987). Poe et al.,(1997) observed that below 10 GPa, the average Si-O-Si angle between tetrahedra decreases withpressure, but not depolymerizing the melt. Pressure might act on the ring structure of polymerizedmelts, weakening the T-O-T bonds.Also, the nature of P-dependence of viscosity for some melts can change with further in-creases in P. The effects of P on peridotite melts (Liebske et al., 2005), for example, are positive upto ∼8.5 GPa but then become negative between ∼8.5 and 13 GPa. Funakoshi et al. (2002) showsP-dependence for albite melts changes from negative to positive with higher P. Therefore, theremight be a viscosity maximum or minimum with increasing P (Persikov et al., 1989; Persikov,1998; Reid et al., 2003; Persikov and Bukhtiyarov, 2004; Liebske et al., 2005; Persikov et al.,2018).In the low T region (i.e. high viscosity), only a few studies have been conducted on theP-dependence of the viscosity of silicate melts over a pressure range up to 0.4 GPa. The viscosityof melts on the Ab-Di binary shows a change from negative for albite to positive for diopside(Behrens and Schulze, 2003; Schulze et al., 1999). The P-dependence for haploandesite in thehigh viscosity range changes from negative for anhydrous melt to positive for hydrous melt with1.06 wt.% water or up (Liebske et al., 2003). This is consistent with P-dependence for andesitemelts in the low viscosity range in general (Kushiro et al., 1976; Kushiro, 1978).60Various viscosity models can model the P dependence, for instance, Avramov et al. (2005)handled the positive to negative changes in P-effects as follows:η = η∞ exp [2.3(13.5− lg η∞)(TrT )α(1+PΠ)β ] (7.2)where β = 2κoVmZR , and Π= ακoVmCpΠ. κ is the volume expansion coefficient, Π is a constant and κois κ at no pressure. It is reasonable to assume that κ will be inversely proportional to P (Avramov etal., 2005). The value of κo (i.e. β ) could be positive as well as negative. Many glasses are knownto have temperature-independent density. Therefore, there are certain compositions for whichviscosity does not increase with pressure (e.g. anhydrous andesitic melts viscosity is independentof pressure).7.2 Modelling P-T-X dependence of viscosityThe P-dependence of melt viscosity has been modelled for some natural and synthetic sil-icate melts, such as peridotite liquid (Liebske et al., 2005), rhyolitic melts (Ardia et al., 2008),hydrous rhyolitic melts (Hui et al., 2009), haplokimberlite and basaltic melts (Persikov et al.,2018). The pressure effects on natural and synthetic melts has been modelled using VFT functionwith an expansion of the parameter Bv f t to account for pressure dependence (Ardia et al., 2008;Hui et al., 2009; Liebske et al., 2005; Romine and Whittington, 2015). Following these studies,and guided by the available experimental data, I have also treated parameter Bv f t as P-dependent,and introduced a new VFT parameter denoted as BP.My approach is to bootstrap the P-dependent model to the 1 bar chemical model. On thatbasis, I accept the 1 bar model values for each melt (i.e., A1bar, B1bar, and C1bar) and then solvefor additional coefficients to account for the deviations between the optimal values of B1bar and BPat pressurized conditions. The advantages of bootstrapping the pressure model to the 1 bar modelare that the integrity of the 1 bar model is maintained and the same model can be extrapolated to61higher P. This approach is also partly predicated on the fact that the ambient data set is much moredensely populated relative to the high P data.My initial strategy is to solve directly for an optimal value of ∆BP (i.e. BP – B1bar) for eachhigh-P viscosity experiment as:∆BP = (logη−A1bar)(T −C1bar)−B1bar (7.3)The optimal values of BP from each high-P experiment are explicitly illustrated as a function of P(Fig. 7.1A) and composition (i.e. SM, Fig. 7.1B). The data are sparse, and the variance is high dueto the larger experimental uncertainties associated with high-P experimentation, however, there isa general pattern. The available data suggest that the magnitude and nature of the P-effect on Bpvaries with melt composition (Fig. 7.1A). Di and Di-rich melts show positive P effects (∼+20%)whilst Ab and Ab-rich melts (e.g. Ab, Ab80Di20) shows significant negative P effects (∼-40%)similar to SiO2-rich rhyolitic melts (e.g., Ardia et al., 2008; Hui et al., 2009). Conversely, Anmelts appear to be nearly independent of P. Melts along the binaries show intermediate P-effects.The pressure effects on BP for most melt compositions are nearly linear. Ab shows a slightlynon-linear monotonic decrease in BP from ∼11420 at the ambient condition to ∼6750 at 7000MPa (∆-4670) with the exception of a few inconsistent data at P < 500 MPa (Fig. 7.1A). Data forDi melts are sparse but generally BP value increase from 4969 at 1 bar to 6082 at P of 104MPa(∆+1100). There are only a few high-P datasets for An-bearing melts and most are at P < 2000MPa where the effects of pressure are negligible to slightly positive (Fig. 7.1A).The role of composition on the P-dependence of the melts is summarized using the SMparameter in Figures 7.1(B, D). For Ab melt and Ab-rich melts in the range of SM values of12.5 (Ab) – 32 (∼Ab50Di50), BP decreases with increasing pressure. For melts with SM > 32,BP increases with increasing pressure. These relationships are best illustrated by the values of∆BP in Figure 7.1D. ∆BP is not linearly dependent on SM. The most pronounced effects of meltcomposition on the P-dependence of viscosity coincides with SM values of 12.5 (Ab) to 25 (An)620 50 100 150Pressure (MPa)4000600080001000012000B P VFTDiAnAbDi-AnDi-Ab10e2(a)10 20 30 40 50SM-6000-4000-200002000 Bp (B at P - B 1bar)<=200MPa(200-500](500-1000](1000-2000](2000-5000](5000-9000]>9000No P effect(b)0 50 100 150Pressure (MPa)-4000-200002000 Bp (B at P - B 1bar)10e2(c)10 20 30 40 50SM-6000-4000-200002000 Bp (B at P - B 1bar)AnAbDi(d)Figure 7.1: Variations of VFT parameter BP as functions of P and SM. (a) Optimal values ofthe VFT parameter BP from refitting 1 bar model to high-P data sets for each melt compositionwith constant A1bar and C1bar (see text). (b) The same data as in (A) plotted against SM showingcovariation of ∆BP as BP - B1bar with P. Colours denote P intervals in the range of 50 to 13100MPa. (c) Predicted values of ∆BP (solid lines) compared to the original data for end-members(Black circles denote Di melts; Grey circles denote An melts; Open circles denote Ab melts). (d)Model ∆BP against SM.at SM values. The SM parameter for An appears to serve as a divide between melts where thepressure term has a strong compositional dependence (low SM) and melts where there is a weakercompositional control on the pressure effect (Fig. 7.1C). The magnitude of P-effect parallels thisdivision in that it is largest for melts where SM < 25 (Fig. 7.1D).Values of BP for melts with SM values 25-32 are nearly independent of pressure and meltviscosity, itself, is not affected (or only weakly) by pressure. This observation is consistent withthe measurements from Taniguchi (1992) and del Gaudio and Behrens (2009), where viscosity63measurements for An melt at 1650◦C and 1800◦C are within error (∆η< 0.5 log units) up to 1000MPa pressure. del Gaudio and Behrens (2009) suggested a slightly negative P effect on the viscos-ity for An melt at T< 1000◦C and P < 400 MPa. Lastly, all experiments on An-Di melts (i.e. SM =25 - 50) SM show slightly positive P effects on melt viscosity.7.3 A model for Di-An-Ab melts at PBased on the patterns observed in BP, I have parameterized the effects of P on melt viscos-ity using the SM parameter and an additional term of combined network-forming oxides. Thisparameterization of the P-dependent VFT term BP comprises four adjustable parameters (bpi):BP = B1bar +[bp1(SM−bp4)+bp2(SM−bp4)2](P−0.001)+bp3(SM−bp4)(SAC) (7.4)where B1bar is the calculated value of Bv f t from the 1 bar chemical model for viscosity (Table 5.3),P is pressure in kb, and SAC is SiO2+Al2O3-CaO (mol.%). The parameter bp4 marks a crossovervalue in SM where the pressure effect on melt viscosity changes from negative to zero to positive.Unit of kb is used for the simplicity of coefficients and directly correlates with 1 bar coefficients.Optimal values of coefficients were solved by non-linear regression of equations for 303high-P viscosity measurements on 18 melt compositions by minimizing:χ2 =n∑i=1[logηi− (A1bar +∑5j=1 b jx j,i+∑3k=1 bbkXYk,i+BPTi−∑4m=1 cmxm,i)]2 (7.5)7.3.1 Quality of fitsThe model reproduces the majority of high-P data to within 1 log unit with an average RMSEequal 0.35 log units (Fig. 7.2, Table 7.2). The accuracy is less than for the 1 bar model (cf. Fig.5.3) but this mainly reflects the inherent variance in the high-P data sets and the paucity of data.For example, the three end-member melts (Di-An-Ab) collectively comprise a total of 66 viscositymeasurements spanning the full range of T and P values characteristic of volcanic and deep mantle64environments. For some melts, there is only a single measurement at a single T at elevated P. Allexperimental data are treated as equal weight, despite the reported amount of impurities.-2 0 2 4 6 8 10 12 14log [P-measured]-202468101214log [P-model](a)0 20 40 60 80 100 120Pressure (MPa)-2-1012 log [Pa s]10e2(b)Figure 7.2: (a) Model values of melt viscosity compared to original high-P data. Dashed linesdenote ± 1 log units of viscosity. (b) ∆log η (model – measured) against pressure in MPa.Most importantly, the model is able to reproduce the patterns of P-dependence for viscosityacross the range of compositions and pressures in the high-P database. This is exemplified bycomparison of values of ∆BP for the individual datasets (Eq. 7.3) to the model lines of ∆BPcalculated as a function of P and X (Eq. 7.4) (Fig. 7.1C, D). The model lines agree with the datato within experimental uncertainty which, as evident in the end member datasets (Fig. 7.1C), issignificantly greater than for 1 bar measurements of viscosity. More data and new refined methodsof measurement are required to make significant improvements.65Table 7.1: Coefficients for the pressurebootstrap (Eq. 7.4) in the multicomponent1 bar model for Di-An-Ab melts. Datawere fitted to the high-P database with aconstant value for Av f t (-4.73 ± 0.29 Pas).bp1 bp2 bp3 bp4bpi 2.8195 -0.1011 0.5531 25.34667.3.2 P-X dependence of BpThis viscosity model is valid for pressures > 50 MPa and I suggest using only the 1 bar 13-parameter viscosity model (Table 5.2) for predicting melt viscosity at P < 50 MPa. The P-model(Table 7.2) is continuous in temperature and composition and predicts the P-dependence (positiveto negative) of Bp across the entire ternary system (Fig. 7.1C, D). In Figure 7.3, SM varies linearlyfrom 12.5 (Ab) to 50 (Di) whereas SAC varies from 87.5 (Ab) to 12.5 (Di). The model predictszero pressure effect for melts where SM= 25.34 (i.e. bp4) which is consistent with the observationthat An (SM = 25) shows a near-zero P-dependence (Taniguchi 1992; del Gaudio and Behrens2009). It also predicts negative P-dependence of viscosity for Ab and Ab-rich melts with SM <25.34, and positive P-dependence for An- and Di-rich melts (SM> 25.34).AbDiAn45403530252015(a)AbDiAn405060708030(b)Figure 7.3: Contoured ternary diagrams for (a) SM (b) SAC (see text)I have contoured the Di-An-Ab ternary system for model values of ∆BP (i.e. BP– B1bar) as ameans of explicitly illustrating the effects of composition and pressure (200, 1000, 2000 and 5000MPa) on melt viscosity (Fig. 7.4). Where ∆BP is zero, there is no effect of P (i.e. VFT term B =B1bar). Positive numbers indicate a rise in viscosity with pressure whilst negative numbers indicatepressure reduces viscosity. The largest magnitude of ∆BP (negative) coincides with the Ab apex(-500 to -2000) indicating a relatively substantial reduction in viscosity with P. The largest positivevalues of ∆BP, indicating an increase in viscosity with P, are focused near the Di apex (∼Di75).67As P increases from 200 – 5000 MPa, these effects are enhanced. The crossover from positive tonegative P effects (0 contour line) occurs along a tie line from An to ∼Ab75-Di25.Most of the contours are linear indicating that ∆BP is linearly dependent on composition at allpressures. The exception is for more depolymerized Di-rich melts (SM > 35) which are associatedwith more complex contours of ∆BP indicating a nonlinear dependence on composition. Notably,as pressure rises this field for nonlinear compositional dependence (i.e. complex contours) shrinks(Fig. 7.4).200 MPaAbDiAn3753503000-500200-200(a)1000 MPaAbDiAn5004002005400-200-400-600-800(b)2000 MPaAbDiAn7000650500200-500-1000725(c)5000 MPaAbDiAn1250013001000500-2000 -1000(d)Figure 7.4: Contoured ternary diagrams for melts in the Di-An-Ab system to show the P-effects onthe model parameter BP expressed as ∆BP (BP– B1bar) at pressures of (a) 200 MPa (b) 1000 MPa(c) 2000 MPa (d) 5000 MPa68Table 7.2: Summary of high-P data sets for Di-An-Ab melts and including the number of data(N) and sources1. These data used to extend the multicomponent 1 bar model to higher pressuresby expanding the term for Bv f t in the VFT function as: BP =B1bar + [bp1(SM- bp4) + bp2 (SM-bp4)2]*(P-0.001) + bp3 (SM- bp4) (SAC). Also included are model values of BP for individual meltcompositions and calculated values of ∆BP (BP - B1bar), Tg, m, and RMSELabel N Source P (MPa) B1bar C1bar BP ∆BP Tg m RMSEDi 5 c 500 4969 697 5351 381 1017 53 0.294 e 1000 4894 705 5318 424 1023 54 0.183 d 1500 4971 697 5433 462 1021 53 0.061 c;d 2000 4969 697 5472 503 1024 52 0.011 i 3500 4969 697 5593 624 1031 52 0.072 i 7000 4969 697 5876 906 1048 50 0.011 i 8000 4969 697 5956 987 1053 49 0.481 i 8430 4969 697 5991 1022 1055 49 0.621 i 9500 4969 697 6078 1108 1060 49 0.001 i 13100 4969 697 6368 1399 1077 47 0.41An 4 b 50 5991 743 5954 -37 1099 52 0.843 b 100 5991 743 5952 -39 1099 52 1.064 b 200 5991 743 5949 -42 1099 52 0.744 b 300 5991 743 5945 -46 1098 52 0.823 b 400 5991 743 5941 -49 1098 52 0.762 c 500 5673 789 5659 -14 1127 56 0.091 c 750 5673 789 5657 -17 1127 56 0.012 c 1000 5673 789 5654 -19 1127 56 0.031 c 1500 5673 789 5650 -24 1127 56 0.131 c 2000 5673 789 5645 -29 1126 56 0.11Ab 3 a 100 11587 366 10921 -666 1019 26 0.153 a 150 11587 366 10895 -692 1018 26 0.166 a 200 11587 366 10869 -718 1016 26 0.063 a 300 11587 366 10817 -771 1013 26 0.063 a 400 11587 366 10764 -823 1010 26 0.071 e 400 11416 386 10584 -833 1018 27 0.531 f;h 500 11419 385 10534 -886 1015 27 0.461 f;h 850 11419 385 10349 -1071 1004 27 0.481 d 1000 11419 385 10269 -1150 999 27 0.142 f;h 1200 11419 385 10163 -1256 993 27 0.523 f;h 1500 11419 385 10005 -1414 983 28 0.321 f;h 2000 11419 385 9741 -1679 968 28 0.422 i 2500 11419 386 9475 -1943 952 28 0.261 g;h 2620 11419 386 9412 -2007 948 28 0.431 f 3000 11419 385 9212 -2207 936 28 0.251 g;h 3300 11419 386 9052 -2366 927 29 0.201 f 4000 11419 385 8683 -2736 904 29 0.191 g;h 4200 11419 386 8576 -2842 898 29 0.041 f 4500 11419 385 8419 -3000 889 30 0.351 g;h 4600 11419 386 8365 -3054 886 30 0.101 f 5000 11419 385 8155 -3265 873 30 0.361 g;h 5100 11419 386 8101 -3318 870 30 0.121 g;h 5180 11419 386 8058 -3361 867 30 0.081 g;h 5320 11419 386 7984 -3435 863 30 0.051 f 6000 11419 385 7626 -3793 841 31 0.001 f 6500 11419 385 7361 -4058 825 31 0.011 f 7000 11419 385 7097 -4322 810 32 0.22An50Di50 4 b 50 5315 706 5580 264 1039 52 0.345 b 100 5315 706 5589 274 1040 52 0.337 b 200 5315 706 5609 294 1041 52 0.40Continued on next page69Table 7.2 – Continued from the previous pageLabel N Source P (MPa) B1bar C1bar BP ∆BP Tg m RMSE4 b 300 5315 706 5629 313 1042 52 0.194 b 400 5315 706 5648 333 1043 52 0.31An75Di25 3 b 50 5619 723 5790 171 1069 52 0.314 b 100 5619 723 5798 179 1070 52 0.193 b 200 5619 723 5813 194 1071 52 0.233 b 300 5619 723 5828 209 1072 51 0.103 b 400 5619 723 5843 224 1073 51 0.204 b 50 5217 686 5537 320 1017 51 0.744 b 100 5217 686 5546 329 1017 51 0.415 b 200 5217 686 5563 346 1019 51 0.554 b 300 5217 686 5581 364 1020 51 0.563 b 400 5217 686 5598 381 1021 51 0.47Di64An36 1 c 500 4865 730 5256 391 1044 56 0.433 c 1000 4865 730 5353 488 1050 55 0.352 c 1500 4865 730 5450 585 1056 54 0.331 c 2000 4865 730 5547 682 1062 54 0.20Di94 3 a 50 5482 640 5814 331 987 48 0.013 a 100 5482 640 5818 336 987 48 0.079 a 200 5482 640 5826 344 988 47 0.306 b 300 5262 660 5623 361 996 50 0.333 b 400 5262 660 5631 369 996 50 0.65Ab50Di50 2 d 500 6587 538 6841 254 947 39 0.112 d 1000 6587 538 6908 321 951 39 0.082 d 1500 6587 538 6974 388 955 38 0.172 d 2000 6587 538 7041 454 959 38 0.401 d 2500 6587 538 7108 521 963 38 0.46Ab75Di25 2 d 500 8204 475 8093 -111 959 33 0.233 d 1000 8204 475 8062 -142 957 33 0.112 d 1500 8204 475 8030 -174 955 33 0.011 d 2000 8204 475 7999 -205 953 33 0.16Ab74Di26 4 a 100 8644 443 8533 -110 953 31 0.066 a 200 8644 443 8525 -119 952 31 0.037 a 300 8644 443 8517 -127 952 31 0.084 a 400 8644 443 8509 -135 951 31 0.18Ab42Di58 4 a 100 6462 520 6719 258 922 38 0.976 a 200 6462 520 6736 274 923 38 0.824 a 300 6462 520 6753 291 924 38 1.034 a 400 6462 520 6769 308 925 38 1.12Ab24Di76 4 a 100 5749 567 6102 353 931 43 0.929 a 200 5749 567 6122 373 932 43 0.954 a 300 5749 567 6141 392 934 43 1.084 a 400 5749 567 6161 412 935 42 1.24Ab15Di85 5 a 100 5545 596 5913 369 950 45 0.768 a 200 5545 596 5931 386 951 45 0.765 a 300 5545 596 5948 404 952 45 0.914 a 400 5545 596 5966 421 953 45 1.00Ab25Di75 4 d 500 5335 619 5776 441 964 47 0.132 d 1000 5335 619 5874 538 970 46 0.212 d 1500 5335 619 5971 636 976 46 0.341 d 2000 5335 619 6069 733 982 45 0.231 d 2500 5335 619 6166 831 988 45 0.16Ab80Di20 1 e 500 8244 474 8120 -124 959 33 0.251 e 1000 8244 474 8085 -159 957 33 0.121 e 1500 8244 474 8050 -194 955 33 0.121 e 4000 8244 474 7875 -369 944 34 0.581 e 5000 8244 474 7805 -439 940 34 0.74Ab57Di43 1 e 500 6535 541 6788 254 947 39 0.30Continued on next page70Table 7.2 – Continued from the previous pageLabel N Source P (MPa) B1bar C1bar BP ∆BP Tg m RMSE1 e 1000 6535 541 6855 320 951 39 0.131 e 1500 6535 541 6922 387 955 39 0.181 e 2000 6535 541 6989 454 959 38 0.331 e 5000 6535 541 7389 854 983 37 0.58Ab30Di70 1 e 500 5357 617 5798 441 964 47 0.341 e 1000 5357 617 5895 538 970 46 0.261 e 1500 5357 617 5993 636 975 46 0.501 e 2000 5357 617 6090 733 981 45 0.341(a) Behrens and Schulze (2003); (b) Del Gaudio and Behrens, (2009); (c) Taniguchi (1992); (d) Brear-ley et al., (1986); (e) Persikov and Bukhtiyarov, (1999); (f) Mori et al., (2000); (g) Suzuki et al., (2002);(h) Kushiro, (1978); (i) Reid et al., (2003).2Values of model coefficient are: bp1 (2.8195), bp2 (-0.1011), bp3 (0.5531), bp4 (25.34). SAC = SiO2 +Al2O3 - CaO (mol %).717.3.3 Di-An-Ab melt viscosity at PI have calculated melt viscosities for the three binaries at three different P (Fig. 7.5) toillustrate the utility of our model for predicting viscosity continuously across T-P-X space for Di-An-Ab melts. I chose pressures of 200, 2000 and 5000 MPa representative of shallow crustal (orvolcanic), lower crust and mantle, and deep mantle conditions, respectively. The viscosity curvesare shown for melt compositions in each binary at mole fraction increments of 0.1 (Fig. 7.5).The effects of 200 MPa pressure on melt viscosity is minimal (≤1 log unit) for all threebinary systems. This is evident by comparing the model curves to the 1 bar model lines for the twoend member compositions (heavy blue lines; Fig. 7.5). The shapes of the T-dependent curves andoverall pattern are very similar to the 1 bar viscosity curves (Fig. 7.5 A, D, G).At higher P and lower T, there are significant changes in viscosity across the binaries. TheAb-Di features a weak crossover at 200 MPa where, at low T, Di melts have higher viscosities thanAb melts (Fig. 7.5A). As P rises to 2000 MPa, this crossover becomes more pronounced over awider range of melt compositions and occurs at higher temperatures (> 1000◦C; Fig. 7.5B). At5000 MPa, Ab has the lowest viscosity over a wide range of temperatures relative to other Di-Abmelts. At high T, all melts have very similar viscosity (Fig. 7.5C). An-Ab melts show a similartrend with increasing P. At low P, there is a crossover in viscosity at lower temperatures whereAn and An-rich melts have higher viscosities than Ab melts (Fig. 7.5D). Increasing P causes thiscrossover to migrate to higher T and to include most An-bearing melts (Fig. 7.5E, F). At 5000MPa, Ab melts have the lowest viscosity except at highest T where all melts have similar viscosity(< 1 log unit variation; Fig. 7.5F).An-Di melts show a different pattern with increasing P. Firstly, there is no crossover in meltviscosity with temperature nor pressure (Fig. 7.5G, H, I). An melts always have the highest vis-cosity relative to Di. The effect of increasing P is to reduce the isothermal differences in viscosityof melts across the binary.724 5 6 7 8 9 10 1110000/T(K)051015log [Pa s]5000 MPaDiAn4 5 6 7 8 9 10 1110000/T(K)051015log [Pa s]DiAn2000 MPa5 6 7 8 9 1010000/T(K)051015log [Pa s]DiAn200 MPa4 5 6 7 8 9 10 1110000/T(K)051015log [Pa s]5000 MPaAnAb4 5 6 7 8 9 10 1110000/T(K)051015log [Pa s]AnAb2000 MPa4 5 6 7 8 9 10 1110000/T(K)051015log [Pa s]Ab200 MPaAn4 5 6 7 8 9 10 1110000/T(K)051015log [Pa s]AbDi5000 MPa4 5 6 7 8 9 10 1110000/T(K)051015log [Pa s]AbDi2000 MPa4 5 6 7 8 9 10 1110000/T(K)051015log [Pa s]DiAb200 MPa(I)(H)(G)(F)(E)(D)(A) (B) (C)Shallow (Volcanic) Lower Crust-Upper Mantle Deep MantleFigure 7.5: Model values of melt viscosity plotted as log η vs. 10000/T(K) at pressures of 200,2000 and 5000 MPa for melts in the binary systems: (A-C) Ab-Di, (D-F) Ab-An, and (G-I) An-Di.Solid black lines are for end-members and dashed lines are mole fraction increments of 0.1. Solidblue lines (A, D, G) denote the 1 bar viscosity values for binary end-members.Overall, the effects of P are small, compared to the effects of T, but P can cause interestingcrossovers in viscosity and can reduce the differences between melts of different compositions.I then plotted the model value of ∆Bp and viscosity at 1250◦C along three binaries composi-tions (Fig. 7.6). P-effect on viscosity is near zero for An melt as ∆Bp is zero. Along Di-An binary,the largest P-effect on viscosity comes from the melt with Di mol.% around 60 (Fig. 7.6B). Thepositive to negative P-dependence along the Di-Ab join is well-described in Fig. 7.5C; D. Underthe same degree of pressure variation,viscosity varies by 5 order of magnitude for Ab melt whereasviscosity for Di melt varies by 1 order of magnitude (Fig. 7.6D). Di34Ab66 has zero P-effect on730 20 40 60 80 100Di mol.%1.522.533.544.5log @ T =1250oCAn Di0 20 40 60 80 100Ab mol.%-6000-5000-4000-3000-2000-10000 B pAnAb0 20 40 60 80 100Di mol.%-5000-4000-3000-2000-10000100020003000 B p An Di0 20 40 60 80 100Di mol.%-6000-4000-2000020004000 BpAbDiFigure 7.6: Model values of ∆Bp and melt viscosity plotted against mole fraction of end-membersfor three binaries: (A, B) An-Di, (C-D) Ab-Di, and (E-F) An-Ab.viscosity. Because the P-effect on viscosity is fairly small, it is reasonable to say that P effects forcompositions around this inflection point are negligible. Degree of P effects along Ab-An binarydecreases monotonically from Ab to An melt (Fig. 7.6E; F).74Chapter 8: P-T-X model implication8.1 Tg and m at PThis model can investigate the effects of P on Tg and melt fragility m. I have contoured theternary system for calculated values of Tg at pressures of 200, 2000 and 5000 MPa (Fig. 8.1A, C,E). Contours for Tg are nonlinear and at 200 MPa are sub-parallel and concave towards the Ab-Dijoin (∆150). In contrast, at 5000 MPa the contours are non-linear, sub-parallel and concave towardsthe Di-An join (∆200). Values of Tg at elevated P parallel the P effects on melt viscosity; increasedP causes an increase in Tg values of Di and does not affect that of An melts. Tg values for Abmelts decrease with elevated pressure. The changes in model Tg values (∆Tg) due to increased Pare explicitly shown and contoured in Figure 8.1B, D, F. The zero contour is the tie line from An to∼Ab75-Di25 coinciding with ∆BP values of 0 where pressure has no effect on viscosity nor Tg. Thetie line separates the Di-rich side of the ternary, where Tg values increase (positive values) due topressure, from more Ab-rich melts where Tg decreases (negative values) with increasing pressure.The contour patterns are essentially parallel to the pattern of contoured values of ∆BP (Fig. 6.3).The peak increases in Tg values for the Di75Ab25 melt with pressures of 200, 2000, and 5000 MPaare ∼25, 45, and 80, respectively (Fig. 8.1B, D, F). The largest depressions in Tg values are for Abmelt and, for the same pressures, the decreases are ∼ -35, -80 and -150, respectively.Previous workers have documented pressure-driven drops in Tg values for Ab melt of 2.5Kto 5 K per 100 MPa implying a decrease of 125-250 K over 5000 MPa (Rosenhauer et al. 1979;Behrens and Schulze, 2003). If I adjust the estimate of Tg to a viscosity of 1011.8 Pa s to match thequench rate reported by Rosenhauer et al. (1979), our model suggests a P-dependent decrease ofTg of -3.1 K/100Mpa (-150K at 5000 MPa). Considering the diversity and sparseness of our high-Pdatasets, my model values agree well with Rosenhauer et al. (1979) observed Tg depression with P.In contrast with Tg, P has a limited effect on the steepness index (m). I have contoured theternary system for values of m and the differences in steepness index due to pressure (i.e. m at75AbDiAn7870500-50-100-150AbDiAn9009501000105011001075AbDiAn42.54030-60 -40 -20 020-80AbDiAn10009751100105010251075AbDiAn22.520100-10-20-30AbDiAn95010501075 110010251000975(A)(F)(E)(D)(C)(B)200 MPa2000 MPa5000 MPaFigure 8.1: Ternary diagrams for melts in the Di-An-Ab system contoured to show model valuesof Tg as a function of composition and at pressures of (A-B) 200, (C-D) 2000 and (E-F) 5000MPa. The right column of ternary diagrams is contoured for ∆Tg showing explicitly the differencebetween Tg at 1 bar versus at elevated pressures. (i.e. Tg – Tg1bar).P – m1bar) (Fig. 8.2). The most fragile melts at 200 MPa are along the Di-An join; at higherpressures of 2000 and 5000 MPa, An becomes the most fragile melt. The strongest melt (lowestm) at elevated P is Ab. Contours of m are sub-parallel to the Di-An join but change substantiallywith pressure. At 200 MPa contours are concave up to Di-An binary, at 2000 MPa the contours arenearly linear, and at 5000 MPa the contours are concave up to the Ab apex. m values at 1 bar forthis system range from ∼25 to 60 and changes in m with P up to 5000 MPa change only slightly(Fig. 8.2) despite the substantial changes in melt viscosity (i.e. Fig. 7.5). More polymerizedmelts (i.e. Ab-rich) become slightly more fragile with P whilst more depolymerized melts (i.e. Di-76AbDiAn-8-6-4-2024AbDiAn3235404550AbDiAn-5-4-3-2-1012AbDiAn404535305055AbDiAn0.50-0.5-1-2-2.5AbDiAn303540455055Figure 12. [Li et al., 2020; Chemical Geology](A)(F)(E)(D)(C)(B)200 MPa 2000 MPa5000 MPaFigure 8.2: Ternary diagrams for melts in the Di-An-Ab system contoured to show model valuesof steepness index (m) as a function of composition and at pressures of (A-B) 200, (C-D) 2000 and(E-F) 5000 MPa. The right column of ternary diagrams is contoured for ∆m showing explicitly thedifference between m at 1 bar versus at elevated pressures. (i.e. m at P – m1bar).rich) become less fragile (Fig. 8.2). The magnitude of changes in fragility (positive and negative)increases regularly with increasing pressure from ∼5% at 200 MPa to ∼15% at 5000 MPa (Fig.8.2B vs. Fig. 8.2F). The enhancement in fragility for polymerized melts is associated with negativeP-effect and may be supported by the molecular dynamics ionic model for pure silicate by Barratet al. (1997). They conclude that fragility of pure silicate increases with increasing pressure.778.2 Activation energy at PExperimental data suggest Ea is T and P dependent. Therefore, I contoured Ea (Eq. 5.5)for the ternary systems at four temperatures 800◦C, 1000◦C, 1200◦C and 1600◦C and three Pconditions 1 bar, 2000 MPa, 5000 MPa (Fig. 8.3).The T effect on Ea is prominent compared to the effect of P. Ea broadens at low temperatures(e.g. 800◦C) changing from the lowest value of 500 for melt along Di-Ab join with 20 mol.% Dito the maximum value of 1400 for An melt. At 1600◦C, Ea changes from 225 to 335 for Di80Ab20and Ab melt, respectively. Values of Ea vary by ∼ ∆900 at 800◦C, and by ∆100 at 1600◦C fromhighly polymerized melts (Ab and Ab-rich) to depolymerized melts (i.e. An). With increasing T,Ea becomes less sensitive to melt composition. Ea is proportional to the slope of ∂ logη/∂1/T ,thus, the magnitude of changes in Ea from low T to high T is an indication of the degree of non-Arrhenian behaviour (i.e. deviation from the Arrhenius- type plot). Polymerized melts like Ab meltare rigid to T changes and remains Arrhenian for the investigated T range. Ea for Di melt dropsfrom 700 to 240 with T increases from 800◦C to 1600◦C, and Ea for An melt drops from 1250to 320 with increasing T. At 1600◦C, Ea changes slightly with melt compositions since the linesin Arrhenius plot for all melts are essentially parallel to each other, thus, the slopes only changeby few per cent. This observation suggests that viscosity for all melts are converging to a narrowrange at high T. Therefore, at infinite temperature, viscosity has a constant value for silicate melts.The minimum value for Ea does not necessarily correlate with the degree of polymerization norfragility (Angell 1985; 1991) since Ea only demonstrates the tangent of lines at one P: T condition.And the contours vary with temperatures we chose to calculate.The model values of Ea do not change significantly at pressures up to 2000 MPa. Ea for Abmelt at 800◦C, decreases from 530 (1 bar) to 450 at 2000 MPa, whilst Ea for Di melt increasesfrom 700 (1 bar) to 850. Ea for An melt is independent to P. At 2000 MPa, Ea is still dependent onmelt composition at low T and shows a range of 450-1400. The minimum Ea belongs to Ab meltinstead of Ab80Di20 melt at 1 bar. The maximum Ea at 800◦C is still ascribed to An melt. The782000MPa [binary] bar 800oCAbDiAn525550600650700750800900100012001 bar 1000oCAbDiAn3807004004254504755005255505756006501 bar 1200oCAbDiAn310 325350375400425450 4751 bar 1600oCAbDiAn230235 240245250260270280300320330Figure 8.3: Ternary diagrams for melts in the Di-An-Ab system contoured to show model valuesof Ea as a function of T: 800, 1000, 1200, 1600 ◦C and P: 1 bar; 2000MPa, 5000 MPa.contour lines at 2000 MPa, 800◦C are almost linear showing Ea increases linearly along binaryjoins from Ab melt. At 1600◦C, Ea for Di increases from 240 (1 bar) to 265 at 2000 MPa, and Eafor Ab melts decreases from 335 to 292. Ea for An remains the same as 320. The effects of P aremore pronounced at lower temperatures.At 5000 MPa and 800◦C, contour lines for Ea rotate relative to that for 1 bar, which are stillnearly linear for Ab-rich melts. Minimum Ea comes from Ab melt ∼ 400 whilst the maximum79value still comes from An melt (1400) since An melt is independent to P. Ea for Di melt increasesto 900. Ea changes by ∆1000 whilst Ea changes by ∆900 at 1 bar.These observations show the effects of P on Ea are similar to those on melt viscosity. ForAn melt, Ea at 800◦C and 1600◦C remain 1400 and 320 respectively at all P conditions. For Abmelt, Ea at 800◦C decreases from 530 at 1 bar to 450 at 2000 MPa then to 400 at 5000 MPa. Eaat 1600◦C decreases from 335 at 1 bar to 292 at 2000 MPa to 250 at 5000 MPa. For Di melt, Eaat 800◦C increases 750 at 1 bar to 850 at 2000 MPa to 900 at 5000 MPa. Ea at 1600◦C increasesfrom 240 to 265 to 270 with P elevated from 1 bar to 2000 MPa then to 5000 MPa.8.3 Activation volume of viscous flow (Va)The P-effect on viscosity yields information for the activation volume of viscous flow (Va).By inserting Eq. (5.2) into Eq. (7.1), the expression for Va becomes:Va = 2.303 RGas T∂ log η∂P= 2.303 RGas Tbp1(SM−bp4)+bp2(SM−bp4)2T −C (8.1)where Va is the activation volume of viscous flow. RGas is 8.314 cm3 MPa/K mol. The slope oflines ∂ logη/∂P, is proportional to Va which represents the average value over the whole P rangeat each T. Va mirrors the volume requirement for local motion. In accordance with transition statetheory (Glasstone et al., 1941), Va is defined as the difference between the volume occupied bya molecule, or structural atomic cluster, in activated versus non-activated states (Eyring, 1935;Putnis, 1992). The values of Va provide information on the pressure sensitivity of the structuralrelaxation times necessary for molecular reorientations or on the molecular size of the relaxingunits. Larger molecules or clusters need more space to reorient than smaller one.I contoured Va at four temperatures: 800, 1000, 1200, and 1600◦C (Fig. 8.4). Value of Vaimplies the P-dependence of melt viscosity. Thus, the zero tie-line is the same with that on Figure7.4B, and indicates zero effect of P for melts with SM' 25.34 (i.e. bp4). Ab and Ab-rich meltshave negative Va values which increase slightly with temperature (average value of -13 cm3/mol).80AbDiAn684-8 -6 -4 -20-12-14 -10102(a)AbDiAn8420-2-4-6-10 -8-127667(b)AbDiAn7420-2-4-6-8-1066-12(c)AbDiAn6420-2-4-6-8-1055(d)Figure 8.4: Activation volume Va at: (a) 800 ◦C (b) 1000 ◦C (c) 1200 ◦C (d) 1600 ◦CIn particular, the observed negative activation volumes of viscous flow imply that a structuralcollapse takes place before viscous flow can be initiated (Brearley et al., 1986). Di- bearing meltsshow a quite small change in Va with decreasing temperature (average value of 5 cm3/mol). Themagnitude of Va indicates the degree of P-dependence of melt viscosity. Therefore, viscosities forAb and Ab-rich melts are more sensitive to pressure versus Di melts. P-dependence for Di-An-Abmelts does not vary significantly with temperature. Behrens and Schulze (2003) calculated Va asEq. 8.1 using linear regressions of log η vs. pressure at constant temperatures. Their model valuesof Va for Ab melts with an average value of -21.0± 3.0 cm3/mol. Kushiro (1978) got an average Vavalue of –13 cm3/mol while assuming an Arrhenian P- dependence in the range 0–2 GPa. Brearleyand Montana (1989) got a value of –14 cm3/mol for Ab melts at 1673 K. Persikov and Bukhtiyarov(1999) and Mori et al. (2000) indicated that Va values for Ab melts do not vary significantly in the81temperature range 1673–2173 K and in the pressure range 0–7 GPa. Deviations between thosevalues and our model values might result from Ab stoichiometry. Ardia et al., (2008) calculatedthe values of Va for rhyolite melts, which were always negative changing from -20 to -14 cm3/molat 750 to 1350◦C, respectively.Behrens and Schulze (2003) obtained values of Va for Di melts from 52.0 ± 2.7 cm3/mol at1020 K to 24.8± 6.6 cm3/mol at 1057 K, which are significantly larger than our model values. Anaverage Va value of 12 cm3/mol at 2123 K can be derived from the data provided by Scarfe et al.(1979), Brearley et al., (1986), and Taniguchi, (1992). Viscosity data for “Di” from Behrens andSchulze (2003) are indeed fitted poorly by our model. The magnitude of P-dependence for “Di”they suggested does not coincide with the effect of P observed by Reid et al., (2003), Taniguchi,(1992), and Brearley et al., (1986), who observed a minimal variation of viscosity with pressurefor Di melts in the P range 0–2 GPa at temperatures 1823– 2073 K (Scarfe et al., 1979; Brearleyet al., 1986; Taniguchi, 1992). For highly depolymerized systems and pressures up to 8 GPa, Va ispositive and considered to be constant (Liebske et al., 2005).Based on the experimental data, the effect of P on the melt viscosity of Di melt is smallerthan that of Ab melt. Therefore, the magnitude of Va for Di melts should be lower than that for Abmelts. Although model values scatter widely, we can conclude clearly that the pressure effect onviscosity for depolymerized Di melts is positive.8.4 Correlation of melt rheology and phase relationship at PThis T-P-X model can be used to calculate melt viscosities for liquidus conditions in the Di-An-Ab system at pressurized conditions (Fig. 8.5). I have tried to contour the ternary system forisokoms of liquidus viscosity at elevated P conditions. The viscosity does not vary significantly upto 2000 MPa (Fig. 8.5). The 2000 MPa isokoms of viscosity show a marked reduction in the rangeof melt viscosity values across the ternary at 1 bar liquidus temperatures (Fig. 6.6B). At 1 barviscosity varies by 7 orders of magnitude whilst at 2000 MPa viscosity varies only by 3 orders ofmagnitude (Fig. 8.5). This reflects the fact that at 2000 MPa there is an increase in the viscosity of82AbDiAn0.84332.52.5221.61.21.21.61.0Figure 8.5: isokoms of log η (Pa s) at 2000MPa calculated at 1 bar liquidus temperatures using theT-P-X viscosity model presented hereDi-rich melts and a decrease in viscosity for Ab-rich melts. Interestingly, isokoms for plagioclasesaturated melts have changed significantly and are essentially parallel (rather than perpendicular)to the 1 bar liquidus isotherms (Fig. 6.6A).83Chapter 9: Conclusion9.1 SummaryI have presented a chemical model for the prediction of melt viscosity in the Di-An-Absystem as functions of temperature, composition, and pressure. I adopted the VFT function tocapture the non-Arrhenian T-dependence of viscosity which features three adjustable parameters:Av f t , Bv f t , and Cv f t . The models assume that all silicate melts have a constant viscosity value athigh temperature. All compositional dependence is therefore embedded in the parameters Bv f t , andCv f t and, for the 1 bar viscosity model, I use 12 adjustable parameters to model their compositionaldependence. The non-linear terms in Bv f t account for possible non-ideal interactions between thestructure formers and the structure modifier cations. Optimal values for Av f t (-4.73 ± 0.29 Pas) and the 12 coefficients for the Bv f t , and Cv f t parameters are solved by fitting the model to560 measurements on different melt compositions. The high-P data sets suggest that the effectsof P on melt viscosity are coupled to melt composition related to the degree of polymerization.Thus, to account for the effects of P on melt viscosity, I expanded the VFT-parameter Bv f t as:Bp = [bp1(SM− bp4)+ bp2(SM− bp4)2](P− 1/1000)+ bp3(SM− bp4)(SAC) where the bpi’sare additional adjustable parameters. The optimal parameters for the pressure model (i.e. bpi’s) were solved for independently by adopting the ambient model values for Av f t , Cv f t and Bv f t(at 1 bar) for each high-pressure melt composition. This P-model is valid for predictions of meltviscosity in the Di-An-Ab system for pressures > 50 MPa and up to and over 13 GPa. The model iscontinuous in T-P-X space and is capable of reproducing the pronounced nonlinearities within theoriginal data sets including crossing VFT functions for different melt compositions and nonlinearchanges in Tg with composition and pressure.One issue raised by our T-P-X model is its discordance with the complex pressure depen-dence of viscosity observed for depolymerzied melts such as Di melts (e.g. Taniguchi, 1992; Per-sikov et al., 2018). In our model, there is a montonic crossover in the effects of P at all P conditions84at SM = 25.34 (i.e. bp4). The potential changes in the nature of P effects with increasing P werenot considered in this model which may explain the large deviations (∼1 log unit) for Di melts at0-300 MPa (Behrens and Schulze, 2003). However, my model agrees with other results showingthat the viscosity for Di melts will increase by around 1 log unit up to 2GPa (2000MPa; Taniguchi,1992; Brealey et al., 1986; Scarfe et al., 1987). The observed variations in P-effects in differentP ranges may be partially or completely due to the uncertanities in the measurements. Furtherhigh-P datasets will be necessary to validate this argument. Another limitation of our model isthat the T-P-X model can not be directly used for ambient conditions since there is an additionalterm introduced by the pressure bootstrap on the 13-parameter model which will not become zeroat ambient conditions (i.e. the term with the adjustable parameter bp3). Adding this term willimprove the quality of fitting. But for atmospheric conditions, I suggest to use the 13-parametermodel to calculate viscosity for Di-An-Ab melts and use the T-P-X model for P conditions > 50MPa.9.2 Geological applicationsIt was noted that melt compositions in the Di-An-Ab system may be used as analogs fornatural melts from rhyolitic (Ab) to basaltic (An-Di) compositions. With the understanding fromthis T-P-X model, it is clear that the effects of P on viscosity can be correlated generally with thedegree of polymerization. It also should be noted that these considerations will apply to anhydrousmelts or nearly dry erupted melts (i.e. lunar compositions).The value for the crossover in P-dependence may vary when extrapolating this subsystem tonatural melts. However, for polymerized melts with lower values of SM or NBO/T, that are SiO2-saturated basalts, andesites and more silicic compositions, such as rhyolite, will show a negative P-dependence of viscosity. Conversely, strongly depolymerized melts such as SiO2- undersaturatedbasalts, haplokimberlite, picrites, komatiites will exhibit a positive P-dependence, in a similarfashion to the Di or Di-rich melts here.85Because these differences in the P-dependence of viscosity, polymerized melts are less vis-cous and more mobile (i.e. fluid) at lower crustal, upper mantle conditions corresponding to theirsource depths. Less viscous behavior with rising P will enhance the crystal-melt separation, ag-gregation and flow of melts in the source region (Brearley et al., 1986; Fujii, 1977; Kushiro, 1976;Scarfe and Cronin, 1986; Scarfe et al., 1979). Crystal settling and fractionation will be more ef-fective at high P for polymerized melts. Depolymerized melts will tend to behave less mobile attheir source regions. Melt separation from the upper mantle will more or less inhibited due to theirhigher viscosities. Brearley and their coworkers stated that the inhibited melt separation may bean explanation for the low abundance of these melts at the earth’s surface. However, during ascentthese melts will have decreasing viscosity and become more mobile.Rhyolitic melts are most abundant in the lower crust, where the effects of P are minimal.However, picrites, komatiites source in the deeper mantle, where the effects of P are not negligible.Komatiites may be generated at 5-7 GPa (Takahashi and Scarfe, 1985). If komatiites exhibit asimilar P-dependent viscosity as Di melts, they will be significantly more viscous at these depthswhich will control the melt separation from mantle and ascent to the earth’s surface.With this basic understanding of anhydrous melts in the Di-An-Ab system, it is interesting tostudy how water will effect the P-dependences. H2O and F in rhyolite melts decreases the viscosityand preserves the negative P-dependence (Dingwell and Mysen, 1985). 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Cambridge university press.93Appendix A: Viscosity ModelsIn this appendix, I provided a short discussion on the three extensively-used, non-Arrhenian,tri-parametrical T-η equations for predicting melt viscosity of silicate melts over the whole T-range: Adam-Gibbs (AG) equation (Adam and Gibbs, 1965; Richet, 1984); the Voegel–Fulcher–Tammann(VFT) equation (Vogel, 1921; Fulcher, 1925; Tammann and Hesse, 1926); and the Avramov (AV)equation (Avramov, 1998).A.1 Adam-Gibbs (AG) equationThe AG equation is expressed in the frame of Adam-Gibbs theory which is the only theorythat addresses T, P, X and even time dependence of the viscosity together:η = Ae exp[BeT Scon f] (A.1)where Ae is the pre-exponential factor independent of temperature, Be is an energy term, T is thetemperature (K), and Scon f is the configurational entropy of the silicate melt at T, which is T, P-dependent. AG equation is based on the configurational entropy theory. Configurational entropy isexpected to increase with rising temperature, supporting easier structural relaxations, and leadingto a decrease in viscosity. Assuming Scon f is described by:Scon f = Scon f (Tg)+∫ TTg(∆CpT)dT (A.2)and substituted into Eq. A.1 yields:lnη = ln Ae+[BeT [Scon f (Tg)+∫ TTg(∆CpT )dT ]] (A.3)where Tg is the glass transition temperature. ∆Cp is the configurational heat capacity of the liquid,which can be calculated as ∆Cp= ∆Cp(T ) - ∆Cp(Tg). For end-members, I can use calorimetrical94values for the residual entropy and Tg. This equation becomes two parameters: Ae, Be. Becauseof lacking thermodynamic data for other melts in the calibration datasets, I also need to considerScon f (Tg), and Tg as the adjustable parameters. If I introduce an ideal temperature T2, where Scon fvanishes, this equation includes three adjustable parameters: Ae, Be, and T2.If the Ccon fp is assumed to be constant, the AG equation can be transformed into the followingequation:lnη = ln Ae+[B′eT ln(T/TAG)) (A.4)where B′e = Be / Ccon fp , TAG = Tg exp (−Scon fg /Ccon fp ).A.2 Voegel–Fulcher–Tammann (VFT) equationThe VFT equation is given in Chapter 3 (Eq. 3.2). The parameters by fitting VFT functionto the viscosity data are listed in Table 3.1. Av f t is the pre-exponential factor representing thehigh-T limits to viscosity. The term Bv f t is the pseudo-activation energy for viscous flow and isan expression of the resistance to the structural rearrangement of the melt during shear (Richet,1984; Bottinga and Richet, 1995). Values of Bv f t are also required to be greater than zero andmainly > 4000 for natural melts. The parameter Cv f t(K) is the VFT temperature where viscosityrises to infinity and is also constrained to be positive in value. As the value of Cv f t approacheszero melts tend to become increasingly Arrhenian in behaviour. The VFT equation was proposedpurely on the empirical basis independently by Vogel (1921), Fulcher (1925) and Tammann andHesse, (1926). Williams and co-workers (Williams et al., 1955) showed that the VFT equationcould be derived from the “free volume theory”.A.3 Avramov (AV) equationAvramov (AV) equation was developed based on the assumption that Ea is temperature-dependent with the variation linked to the total entropy:logη = AAV +(BAV/T aAV ); (A.5)95where AAV , BAV and aAV = 2Cp/RZ are the model parameters (Z ∼ coordination number). Sev-eral other formulations exist that can be reconciled with these three model equations, for instance,most recent GRD model (Giordano et al., 2008) used to describe the viscosity of anhydrous andvolatile-bearing melts where B, C and Scon f parameters accommodate the effects of melt compo-sition. Literature found there are correlations between different parameters of these models. Thecorrelations between log Av f t and log AAG or between Tv f t and TAG are strong and almost linear(Kondratiev and Khvan, 2016; Russell et al., 2003, 2002).96Appendix B: Experimental MethodsViscosity varies over 16 orders of magnitude with T, X, P in the synthetic and natural meltsystem. One single method is not able to fully characterize the various behavior. Therefore, a largevariety of devices for the measurements of viscosity have been presented in literature. For low Tranges (i.e. high viscosities ∼108 to 1013 Pa s), the commonly used methods are micropenetrationand parallel plate techniques. For high T ranges where viscosity is low (∼< 105.5 Pa s), exper-imentalists can use the falling-sphere method and rotational viscometer (i.e. concentric-cylinderviscometer). Other commonly used methods include fiber-enlongation, beam-bending techniques.For the sake of brevity, I will not report the detailed procedure of each experimental device here. Isummarized the devices used in the relevant reference papers in Table B.1.97Table B.1: Summary of experimental methods used by different works for the collected viscosity data in the Di-An-Ab system.Sources Composition T(K) P (MPa) Methods 1Kirkpatrick 1974 Di 1033-1800 0.1 RC; BBScarfe and Cronin, 1986 Di-Ab 1423-1873 0.1 CCSipp et al., 2001 Di, Ab, An 960-2067 0.1 MPLicko and Danek, 1985 Di 1675-1915 0.1 OSSykes et al., 1992 Di 1673-1773 0.1 BB Brookfield LVTD and HBTD RVNeuville and Richet 1991 Di 987-1057 0.1 Creep apparatusCranmer and Ulhmann, 1982 An-Ab 1073-1873 0.1 BB; CCTauber 1987 Di-An-Ab 989-1124 0.1 MPStein and Spera, 1993 Ab 1273-1573 0.1 CCHummel and Arndt, 1985 Ab-An 1073-1253 0.1 MPUrbain et al., 1982 Di, An, Ab 996-2249 0.1 CCScarfe et al., 1983; Di-An 1473-1898 0.1 CCScarfe et al., 1987 Di 1913 0.1-1500 FS in AAMABehrens and Schulze, 2003 Ab-Di 975-1048 0.1 -400 PPV IHPVDel Gaudio and Behrens, 2009 An-Di 624-1158 0.1- 400 PPV IHPVTaniguchi, 1992 An-Di 1673-2073 0.1-2000 FE for low T,1 atmosphere; CBS for high T, 1 atmosphere; SFS for high PBrearley et al., 1986 Ab-Di 1773-1873 500-2500 FS PCAPersikov and Bukhtiyarov, 1999 Ab-Di 1673-2173 0.1-4000 FS AAMA; RVHGP for H2O and ArMori et al., 2000 Ab 2000 3000- 7000 FS kawai AAMASuzuki et al., 2002 Ab 1873-1973 3000-5320 FS AAMA at SPring-8Kushiro 1978 Ab 1673 0- 2500 In-situ FS with X-ray in PCAReid et al., 2003 Di 2273-2473 3000-13100 In-situ FS AAMA at SPring-81RC: rotating cylinder apparatus; BB: beam-bending technique; CC: concentric-cylinder viscometer; RV: rotational viscometer; MP: micropen-etration technique; OS: Oscillation viscometer; PPV: parallel plate viscometer; IHPV: internally heated pressure vessel; FE: fiber elongationmethod; CBS: counter-balanced sphere method; SFS/FS: sinking (falling) and floating sphere method; PCA: solid-media, piston-cylinder appa-ratus; AAMA: solid phase apparatuses of anvil and multi-anvil; RVHGP: radiation viscosimeter of high gas pressure.98
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Geological Sciences
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Temperature-pressure-composition model for melt viscosity in the Di-An-Ab system
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