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Cation diffusion in olivine to 1400°C and 35 KB Misener, Donald James 1973-03-04

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/72ti CATION DIFFUSION IN OLIVINE TO 1400°C AND 35 KB. BY DONALD JAMES MISENER B.A.SC, UNIVERSITY OF TORONTO, 1967 M.SC, UNIVERSITY OF BRITISH COLUMBIA, 1971 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY IN THE DEPARTMENT OF GEOLOGICAL- SCIENCES We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA June, 1973 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. the Library shall make it freely available for reference and study. Department The University of British Columbia Vancouver 8, Canada ABSTRACT Thirty "diffusion experiments were performed on crystalline sampl of Fe-Mg olivine. Interdiffusion coefficients of Fe and Mg have been determined between 900 and 1100°C and from 1 atm. to 35 Kilo-bars using diffusion couples of fayalite from Rockport, Mass. and Fog^Fag olivine from St. John Island, Red Sea, Egypt. The diffusion of cations is strongly dependent on olivine composition and crystallographic orientation. The diffusion coefficient varies with temperature and pressure according to an empirical Arrhenius relationship, with an activation enthalpy for diffusion of A H* = 49.83 + 9.05 (N2) Kilocalories/mole where N2 = Cation mole fraction Mg An average value of 5.50 cm /mole was calculated for the activation volume of diffusion. Diffusion couples of Red Sea olivine-MgO powder and couples of Fog3Fay olivine-synthetic forsterite (FO^QO) were used to determine the interdiffusion coefficient of Fe and Mg in olivine between 1200 and 1400° C. The interdiffusion coefficient increases with increasing Fe content and with temperature. Diffusion is faster parallel to the c, [001], axis than paralleltto either a, [100], or b, [010]. At 7 cation mole percent Fe in the olivine, the activation enthalpy [001] is 65.6 - 3.6 kcal/mole. Calculations of ionic electrical conductivity in olivine usingCresultsiofiathis investigation agree with observed conduct ivity measurements. The results indicate that at depths greater than 100 Km. in the mantle ionic conduction is the dominant mechanism of electrical conduction. Estimates of temperature versus depth are made using the derived donductivities in con-junction with conductivity-depth profiles calculated from published electromagnetic depth sounding results. Experimental and theoretical results of steady state creep studies suggest that the large scale deformation of the upper mantle is ultimately controlled by diffusion in the olivine lattice. Results of this investigation indicate that cationic diffusion is not the rate controlling process in the deformation of silicates. The investigation also indicates that the theories relating cation diffusion to melting in metals may be extended to include silicates. iv TABLE OF CONTENTS PAGE ABSTRACT ii TABLE OF CONTENTS iv LIST OF FIGURES: PAPER #1 vi LIST OF TABLES: PAPER #1 viLIST OF FIGURES: PAPER #2 vii LIST OF TABLES: PAPER #2 viACKNOWLEDGEMENTS v i i i INTRODUCTION TO THE THESIS AND PRELIMINARY 1 STATEMENT PAPER #1: CATION DIFFUSION IN OLIVINE TO 4 1400°C AND 35 KB. ABSTRACT 5 INTRODUCTION 6 THEORY ELECTRICAL CONDUCTION 6 INTERDIFFUSION COEFFICIENT 8 EXPERIMENTAL DETAILS AND PROCEDURES SAMPLE PREPARATION 17 1 ATMOSPHERE EXPERIMENTS 20 HIGH PRESSURE EXPERIMENTS 21 EXPERIMENTAL DATA AND ANALYSIS 26 GEOPHYSICAL IMPLICATIONS OF THE DIFFUSION DATA 68 CONCLUSIONS 77 ACKNOWLEDGEMENTS 8 V PAPER #2: CATION DIFFUSION IN OLIVINE AND ITS 79 IMPORTANCE FOE CREEP IN THE UPPER MANTLE ABSTRACT 80 INTRODUCTION 1 THEORY 2 INTRODUCTION 8RELATION BETWEEN CREEP AND DIFFUSION 82 RELATION BETWEEKF DIFFUSION COEFFICIENT AND 85 MELTING TEMPERATURE RESULTS OF DIFFUSION EXPERIMENTS 86 CALCULATIONS AND DISCUSSION 87 CONCLUSIONS 9ACKNOWLEDGEMENTS 8 APPENDIX A: DERIVATION OF THE INTERDIFFUSION 99 COEFFICIENT BIBLIOGRAPHY 100 VI LIST OF FIGURES: PAPER #1 PAGE FIGURE 1 Experimental determinations of Ln D versus 1/T ( K- «) 1Q FIGURE 2 Graphical representation of the quantities necessary for lg the calculation of the interdiffusion coefficient FIGURE 3 Illustration of the high pressure assembly 24 ~ FIGURE 4 Mg concentration profile for Experiment No. 6 28 FIGURE 5 Mg concentration profiles for Experiments No. 14 and 15 30' FIGURE 6 Calculation of Ln ft versus cationic mole fraction (900- 32 1100°C) FIGURE 7 Dependence of Ln ft upon crystallographic orientation 36 FIGURE 8A Ln D versus 1/T (°K_1) at selected compositions 38 FIGURE 8B Ln D versus 1/T (°K_1) as a function of crystallographic 4.0 orientation FIGURE 9 Ln Dgversus composition 42 FIGURE 10 AH* versus concentration 4 FIGURE 11 Ln D versus concentration as a bunction of pressure 47 FIGURE 12 Ln ft versus pressure at selected concentrations1 49 FIGURE 13 AV* versus composition 51 FIGURE 14 Fe concentration profiles parallel to a, b, and c axis 55 at 1250°C FIGURE 15 Ln D versus composition for the profiles in Fig. 14 Si? FIGURE 16 Ln D versus composition for diffusion parallel to c axisv59 (1200-1300°C) FIGURE 17 Ln D versus composition for diffusion parallel to c axis 6l (1350-1400°C) FIGURE 18 Fe concentration profile for Experiment 52 63 FIGURE 19 Ln D versus composition for the profile in Fig. 18 65 FIGURE 20 Ln D versus l/T(°Kri) at selected compositions 67 (1200-1400°C) FIGURE 21 Ln D versus 1/T(0K_1) as a function of crystallographic 70 orientation (1200-1400°C) FIGURE 22 Experimental determinations of Log om versus 1/T(0K_1) 72 FIGURE 23 Temperature versus depth in the mantle 75; LIST OF TABLES: PAPER #1 TABLE l.AElectron microprobe analysis of olivine samples 218 TABLE IB Sample localities and sources 21>9 TABLE 2 List of the one atmosphere diffusion experiments (900-110G°C) 34 TABLE 3 List of the high pressure diffusion experiments 45 TABLE 4 List of the one atmosphere diffusion experiments (1200-1400°C) 53 LIST OF FIGURES: PAPER #2 FIGURE 1 Experimental determinations of Strain rate versus stress 85 FIGURE 2 Ln D versus 1/T(0K_1) at selected compositions (900-1100°C): ?88 FIGURE 3 Ln D versus pressure at selected compositions 90 FIGURE 4 Ln D versus 1/T(°K_1) as a function of crystal lographic 92•.. orientation (1200-1400°C) LIST OF TABLES: PAPER #2 TABLE 1 Calculated values of the pre-exponential constant (Di^aarid 94 the activation enthalpy (AH*), (900-1100°C) TABLE 2 Calculated values for the pre-exponential constant (D0) and r95 the activation enthalpy (AH*), (1200-1400°C) TABLE 3 Calculated values of g^and gy 197 TABLE 4 Experimental determinations of the flow laws in dunites 1Q0 and lherzolites ACKNOWLEDGEMENTS I am especially indebted to Professor H. J. Greenwood for his continued interest and direction throughout the investigation. I am also indebted to Drs. H. S. Yoder Jr., F. J. Boyd, L. W. Finger, and in particular Dr. P. M. Bell of the Geophysical Laboratory, Carnegie Institution of Washington, who gave generously of their time on innumerable occasions to discuss various aspects of the in vestigation. Mr. C. G. Hadidiacos assisted with the electron microprobe analysis and the skill at machining of Mr. C. A. Batten made the high pressure experiments possible. I would also like to express my appreciation to the Carnegie Institution of Washington for financial and practical support. The work was made possible by a Pre-Doctoral Fellowship at the Geophysical Laboratory, financed jointly by the Carnegie Institution of Washington and the National Research Council of Canada Research, through Grant No. A-4222 awarded to Dr. H. J. Greenwood. 1 INTRODUCTION AND PRELIMINARY STATEMENT Atoms in a crystal lattice vibrate about their equilibrium position when the temperature is increased above absolute zero. When the thermal energy is sufficient, the atoms jump from one equilibrium site to another. This process is responsible for diffusion in solids. Temperatures and stresses within the mantle are large enough to activate the process of diffusion in silicates. Crystal growth and zonation, electrical conductivity, and high temp erature creep are all processes controlled by the diffusion of ions in the crystal lattice. In the present investigation, the olivine lattice is treated, as a aolid continuum and thus Fick's first and second law for diffusion may be used as a basis for the diffusion coefficient calculations. Using the continuum model, a classical thermodynamic approach may be used to analyse the effect of temperature and pressure on the diffusion of ions in the crystal lattice. The primary objective of the present investigation has been to determine the effect of temperature, pressure, crystallographic orientation and composition on the rate of cation diffusion in the Fe-Mg olivine solid solution. The diffusion experiments were performed in high temperature furnaces and high pressure solid-media apparatus at the Geophysical Laboratory, Carnegie Institution of Washington, D. C. The model 400 M.A.C. electron microprobe at the Geophysical Laboratory was used to analyse the diffusion profiles. The interdiffusion coefficient (L^) of Fe and Mg in olivine was calculated from the concentration profiles using the theory 2 of interdiffusion coefficients developed by Wagner (1962). A complete derivation of the equation used in the calculation of the interdiffusion coefficient is given in Appendix A. The experimental results and geophysical implications are presented in the form of two manuscripts to be presented for publication. The first paper outlines in detail the experimental procedures and results of the diffusion experiments. New estimates of ionic conductivity in the mantle are made using the present results and the theoretical Nernst-Einstein relation between ionic conduction and ionic diffusion. Estimates of temperature in the mantle are made using the experimentally determined values for the effect of temperature and pressure on the derived electrical conductivity in conjunction with conductivities estimated from magnetic (McDonald, 1957) and magneto-telluric (Eckhardt et. al, 1963) techniques. The second paper deals with the application of the results to steady st-Stoe deformation in silicates. Results of steady state creep experiments on metals and halides indicate that the effect of temperature and pressure on the creep rate is identical to their effect on the diffusion of the controlling vacancy diffusion in the creep pi^cess.. Thus to extrapolate results of creep experiments it is necessary to know the activation parameters for diffusion in the constituent minerals. Olivine is the major mineral in the upper mantle (Ringwood, 1966) and thus quantitative estimates for the diffusion of the various ions in olivine are necessary for extrapolation to mantle temperatures, pressures and stresses. The empirical relationship existing between melting temperature and diffusion rates in metals has been proposed by Rice and Nachtrieb (1959) as a method of estimating diffusion parameters 3 from melting data of olivine (Weertman, 1970; Goetze and Brace, 1972). This procedure is tested with the activation parameters determined in the present investigation. The activation parameters for creep and cation diffusion differ significantly, indicating that diffusion of cations in olivine is not the rate controlling mechanism in high temperature creep of dunites and lherzolites. 4 PAPER # 1 CATIONIC DIFFUSION IN OLIVINE TO 1400°C AND 35 KB. ABSTRACT Thirty diffusion experiments were performed on crystalline samples of Fe-Mg olivine. Interdiffusion coefficients of Fe and Mg have been determined between 900 and 1D0D£C and from 1 atm. to 35 Kb. Usin^ diffusion couples of Fayalite from Rockport, Mass. and FSg^Fag olivine from St. John Island, Red Sea, Egypt. The diffusion of cations is strongly dependent on olivine composition and crystallographic orientation. The diffusion coefficient varies with temperature and pressure according to an empirical Arrhenius relationship, with an activation enthalpy for diffusion of AH* = 49.83 + 9.05 N2 Kcal./mole where ^ = Mg cation mole fraction 3 An average value of 5.50 cm /mole was calculated for the activation volume of diffusion. Diffusion couples of Red Sea divine-MgO powder and couples of FQggFay o 1 ivine-synthetic forsterite (FO-^QQ) were used to determine the interdiffusion coefficient between 1200 and 1400°C. The inter-diffusion coefficient increases with increasing Fe content and with temperature. Diffusion is faster parallel to the c, [001], axis than parallel to either a, [100], or b, [010]. At 7 cation mole percent in the olivine, the [001] activation enthalpy is 65.6 4 3.6 Kcal./mole. Calculations of ionic electrical conductivity in olivine using results of this investigation agree with observed conductivity measurements. The results indicate that at depths greater than 100 Km. in the mantle ionic conduction is the dominant mechanism of electrical conduction. Estimates of temperature versus 6 depth are made using the derived conductivities in conjunction with conductivity-depth profiles calculated from published electromagnetic depth sounding results. INTRODUCTION Recent experimental determinations of electrical conductivity in olivine, (Duba, 1972; Duba and Nichols, 1973; Shankland, 1969), taken in conjunction with profiles of conductivity versus depth in the mantle, (McDonald, 1957), have been used to calculate temperatures in the mantle, (Duba et. al, 1973). The temperature and pressure dependence of the electrical conductivity of single crystals of olivine has been determined up to 1300°C, (Duba et. al, 1973) and up to 10 kb. (Hughes, 1955). Measurements of the electrical conductivity have been made on olivine crystals assumed to be of upper mantle composition (fayalite content = 10%). In the work reported here, the interdiffusion coefficients of Mg and Fe in olivine are reported and coupled with the above work to refine estimates of the conduction mechanism and geothermal gradients. HiHEORY: CONDUCTION The conduction mechanism has been found to change from impurity conduction at low temperature (T-800°C) to one of intrinsic semi-conduction (800oCrJ-&2u§itC) and possibly to one of ionic conduction at high temperature (T-1200°C), (Shankland, 1969; Duba, 1972). At depths greater than 100 kilometers, the dominant conduction process seems likely to be ionic if one assumes the temperature profile corresponding to the oceanic geotherm of Ringwood, (1966) and Clark and Ringwood, (1964). Ionic conduction involves the migration of defects (vacancies) and ions across a potential energy barrier separating the two sites. ,The pressure and temperature dependence of the process may be expressed as a=a0exp (-E/RT) -(1) where AE =°AH+PAV (Kcal/mole) R = gas constant T = absolute temperature a = conductivity (ficm) ~^ If the process of ionic electrical conduction and ionic diffusion are the same, the relationship between the electrical conductivity and the diffusion coefficient may be expressed by the Nernst-Eistein equation \ a = e ^ c D KT where ez = charge of the migrating species C = concentration of the migrating species K = Boltzmann's constant T = absolute temperature The few reported studies of cationic diffusion in olivine (Clark, 1971; Buening and Buseck, 1973; Misener, 1972) suggest that vacancy diffusion governs the rate of ionic migration in a chemical potential gradient, consequently, the processes may operate by the same mechanism and be strictly coupled. The interdiffusion coefficient of Fe and Mg has been determined in olivine using a diffusion couple technique (Misener, 1972). Interdiffusion coefficients were obtained for the olivine solid solution up to 1100°C and 35 Kb. Between 1200° and 1400° C, 8 diffusion couples of MgO powder and single crystals of forsterite were used. Two high-temperature runs are also reported for a crystal-crystal couple consisting of synthetic forsterite and 7%Fa olivine. The oxidation state of the Fe ions in the olivine changes the electrical conductivity by as much as a factor of 10^ (Duba  et. al, 1973) and thus when comparison of diffusion and conductivity are made care must be taken to assure that the experimental samples have a similar Fe+++/Fe++ ratio. The choice of Red Sea olivine crystals for the diffusion experiments permits reliable comparison with the work of Hughes (1955), Duba (1972), and Duba and Nichols Previous studies of diffusion rates of cations in olivine have been concerned chiefly with temperature dependence. Literature values along with the present results are shown in Fig. (1). The results of Naughton and Fujikawa (1959) and Jander and Stamm (1932) were obtained for rates in powdered samples while the other results are for studies on single crystals. THEORY: INTERDIFFUSION COEFFICIENT Multicomponent diffusion is most conveniently considered from a thermodynamic viewpoint. In general, diffusion occurs when a concentration gradient exists. This 'driving force' for diffusion may be expressed in terms of the chemical potential. Defining the decrease in free energy of component 1 accompanied by one gram-atom of material diffusing from site 1 to site 2 as a Taylor series expansion [1973). -(3) 9 FIGURE 1. Experimental determinations of Ln D versus 1/T (°K ^) . Results of the present study Percentages correspond to Mg cation mole percents. Results of other authors; (1) Co+* in Co2Si04, Borchardt and Schmalzried, 1972, (2) Fe++ In Mg2Si04, Naughton and Fujikawa, 1959, (3) Ni++in Fo93Fa7 olivine, Clark, 1970, (4) Ni++ in Red Sea Olivine, Clark, 1970, (5) Mg++ in Mg2Si04 powder, Jander and  Stamm, 1932, (6) Mg++ in Mg2Si04 from electrical conductivity, Pluschkell and Engell, 1968. T (°C) 900 10,00 11,00 12,0013,001,400 11 where AG = decrease in free energy of component 1 y2 u1 = chemical potential of component 1 at site 2 and site 1 then the diffusion force acting on one gram-atom of material may be approximated by F, = -5y -(4) and for one gram-atom of species i in a multicomponent system NA j=lfix where N = Avogadro1s number At &\ij = driving force due to species |. 6x The rate of flow of one atom of species i (i.e., velocity) will be proportional totibhe sum of all the diffusion forces. V. = m. . 6pi + S m. . 6yj+m. 6yv -(6) 1 N" 6X j = 2 6x N^V6x A A A where = velocity of component i m.. = mobility of species i due to chemical IJ potential gradient 6yj 6x m. ,6uv = mobilities and chemical potential gradient of vacancies Assuming that the vacancies are in equilibrium (6yv-»&) and 6x that no coupling exists between the components, (m„->()), then V. = -mi 6yi -(7) 1 NA 6X where m^^ = mobility of component i (mean drift velocity per unit chemical potential gradient) Thus the total flux of component i, of concentration N^, will be J. = -N. m. <5yi -(8) 1 -1 1 7 N. 6x A 12 Assuming conditions of constant pressure and temperature, the chemical potential may be written in the form Ui = %i+RTln(aj) "(9) where a^ = activity of component i u . = chemical potential at a standard state 01 R The activity is related to the mole fraction a. = y.N. -(10) iii where y^ = activity coefficient of species i = mole fraction of species i Differentiating (9) with respect to x and substituting in (8) J. = -N. m. RT 61n(ai) -(111 fij 1 ~~6x 1 Substituting the activity coefficient in (11) J. = =.-N. m. RT 6 {ln(y.)+ln(N.)} 1 1 A = N.kT {1 + 61n(y )}{6N.} -(12) 1 61n(N^) 5^*" Writing Fick's first law in the form J. = -D.5N. -(13) l 1T~l 6x and equating (12) and (13) D. = m.kT[l + 51n(y.)] -(14a) 1 1 61n(N^) similarly D. = m,.kT[l + 61n(y.)] -(14b) J * 6dm (N^) Y Using the Gibbs-Duhem equation in the form 61n(y.)= 61n(y.) -(15) 61n(N*) 61n(N^) and substituting in (14) D. = m. -(16) TTl ~1 D. m. 3 3 In an ideal solution 13 and equations [14) and (15) reduce to D. = m.kT = D.* -(17) ii l D. = m.kT = D.* 3 3 3 D^* is the self-diffusion coefficient of the i th component and is the intrinsic coefficient of the i th component. The first term on the right hand side of equation (14) is associated with the random jumping in an ideal solution, the other reflects the modification of these jumps due to nonideality. thus D. = D.* [1 + 61n(y.)] -(18a) 1 1 61n(N^) D. = D.* [1 + 61n(y.)] -(18b) 3 3 61n(N^) r Darken, (1948), using the above definition for the intrinsic diffusion coefficient, has defined the chemical, or binary diffusion coefficient, D.. D. . = (x.D.*+x.D.*)[1 + 6In(y-)] ~(19) l Sauer and Freise, (1962) and Wagner, (1969) have extended the concept of the binary diffusion coefficient and formulated a. the theory for the interdiffusion coefficient, D. Assuming that the velocity of a mobile component may be expressed in an arbitrary co-ordinate system. V. = -(20) 1 N*7V i m where J\ = Flux of the i th component I\L = mole fraction of the i th component V = molar volume m Wagner, (1969) defines the binary interdiffusion coefficient D as ft = VM(N2J1 - N^/CfiN^fix) "(21) The equation necessary for the evaluation of the interdiffusion coefficient is 14 D(N2) = (N^N")Vm(N2*) {[1-Y*]/^ Y 6x+Y*^l-Y 6x} -(22) where D(N2*) = interdiffusion coefficient evaluated at a distance on the concentration profile x = x* * V (N *) m 2 molar volume of cations at N =N * auxiliary variable = N -N initial concentrations of component 2 in the two diffusion couple members t time interval A complete derivation of equation (22) is given by Wagner, (1969). Fig. (2) illustrates the necessary integrals and derivatives for the calculation of the interdiffusion coefficient. The first derivatives were calculated using a five point first derivative filter (McCormick and Salvadori, 1965). The integral were calculated using Simpson's 3/8 Rule with a spacing distance of 5 micrometers on the concentration-distance profile. All comp utations were carried out on an I.B.M. Model 360-67 computer at the University of British Columbia. Equation (22) has been used successfully in determining interdiffusion coefficients in the systems MgO-Cr,^ (Greskovich  and Stubican, 1969), and TiO^Cr^ (O'Keefe and Ribble,. 1973), and in the oxide-spinel system MgO-MgAl204 (Whitney and Stubican, 1971) . There are certain advantages in using equation (22) for diffusion studies in silicate systems. It is not necessary to determine accurately the location of the Matano interface (Matano, 1933) or the original diffusion couple interface. Variation in the molar volume across the olivine solid solution 15 FIGURE 2. Graphical representation of the quantities necessary for the evaluation of the interdiffusion coefficient. 16 Distance (x) 17 can be included in the determination of D by using the values for Vm obtained by Fisher and Medaris, (1969) and Yoder and  Sahama (1957). An arbitrary reference point can be selected for analysis of the profile. The preceeding development was for diffusion at constant temperature and pressure. Numerous experiments on diffusion in oxides conform that D(T) = D0gxp (-AH*/RT) -(23) where AH* = enthalpy of activation for diffusion R = gas content T = absolute temperature Experimental determinations of D as a function of pressure (Lazarus and Nachtrieb, 1963) indicate that D(T,P)=D0exp(-AH*/RT)exp(-PAV*/RT) -(24) where P = ambient hydrostatic pressure AV* = volume of activation for diffusion EXPERIMENTAL DETAILS AND PROCEDURES SAMPLE PREPARATION Table (la) lists the electron microprobe analysis of the olivine samples used in this study and table (lb) lists the sources of the samples. Crystallographic orientation of the crystals used in the diffusion couples was achieved as follows. Crystals were mounted on an X-ray goniometer head and the Buerger precession method was used to align the crystals (Buerger, 1942). Orientation was accurate to ±1/2° of arc. The goniometer head was transferred to a mount and the oriented crystal lowered into a one inch diameter Bakelite ring. 18 TABLE IA OXIDE OLIVINE SAMPLE SYNTHETIC FAYALITE Fo 1 Fo2 (WT%) FEO 0.00 67.30 9.05 7.18 MNO 0.00 2.38 0.16 N.D. NIO 0.00 0.19 0.42 0.39 MGO 57.55 0.07 50.27 51.08 SI02 42.62 29.13 40.83 41.70 TOTAL 100.17 99.05 100.42 100.35 TABLE IA. Electron microprobe analysis of the olivine samples. N.D. = npt determined. Synthetic forsterite grown by flame fusion process (Shankland, 1969). Other samples were natural crystals. 19 TABLE IB OLIVINE SAMPLE FEO WT % SYNTHETIC FORSTERITE 0.00 FAYALITE 67.30 Fol 9.05 Fo2 7.18 SOURCE Dr. T. J. Shankland, M.I.T. Cambridge, Mass. Grown by flame fusion process. Dr. D. R. Wones, U.S.G.S. Sample from Rockport, Mass. Dr. D. Virgo, Geophysical Laboratory, Carnegie Institute of Wash., D.C. Sample from St. John's Island, Red Sea, Egypt. Dr. T. Richards, G.S.C. Vancouver, B. C. Locality unknown. TABLE IB. Sample localities and sources. Fe content given in FeO weight percent. 2.0 The ring was then filled with epoxy. The specimen was then polished on both sides, mounted on a glass slide, and cut into discs 0.070" thick with a 0.010" diamond blade. The individual platelets were ground and polished to O.SyA^Og (Buehler) and mounted on glass slides which were then mounted on a piece of clear plexiglass. A brass coring tool, 0.125" in diameter, was used with a slurry of 600 grade Carborundum SiC grains and distilled water to bore cylinders from the platelets. Individual cores were cleaned with alcohol and stored in an evacuated bell jar. Cores exhibiting cracks or scratches on the surface were rejected. Five cores were remounted on a gonometer head and precession photographs taken to confirm orientation. The cores were all within 3° of arc of their original orientation. ONE ATMOSPHERE EXPERIMENTS Two types of diffusion experiments were performed. In the first, crystallographically orientedediscs were placed in contact and wrapped in an inert metal foil. Experiments at temperatures over 1000°C were wrapped in Pt foil and those at lower temperatures in AgyQPd3Q. The foil wrapped diffusion couple was tightly wrapped in Pt wire and placed in a silica glass tube which was evacuated. In the second type of experiment, an oriented disc of olivine was pressed against a pellet of MgO powder, inserted into a Pt tube, followed by more MgO powder and the tube tightly crimped at both ends. The Pt tube was placed in a silica glass tube as in the other type of experiment. Platinum wound furnaces were employed in all high temperature experiments. Platinum - Platinum 90% Rhodium 10% were used to monitor temperature. Total temperature uncertainty due to position ing in the thermal gradient, thermocouple error and temperature controller error was - 5°C over the temperature range studied. At the completion of each experiment the samples were slowly cooled (over a period of two to three hours). If the tubes appeared cloudy (due to devitrification) or a 'pop' was not heard when the tubes were opened, it was assumed that the vacuum was not maintained and the experiment was rejected. In the case of the experiments between 1200 and 1400°C a nitrogen atmosphere was used. This prolonged the life of the silica glass tubes and allowed experiments of up to two weeks. Immediately upon extraction from the silica glass tube, the diffusion was impregnated with epoxy under vacuum and prepared for microprobe analysis. The original interface was oriented perpendicular to the polished surface. HIGH PRESSURE EXPERIMENTS The high pressure experiments were carried out on a 0.75" diameter single stage, solid media pressure apparatus developed and modified at the Geophysical Laboratory (Boyd and  England, 1960). Pressure was calculated by multiplying the recorded ram pressure by the ratio of ram area to piston area. Two factors may cause the 'true' sample pressure to deviate from the calculated value; 1) Internal friction generated between the piston and cylinder walls and friction between the constituent parts of the sample assembly. 2) Anvil effects due to the differences in strength of the various parts making up the high pressure assembly. 22 When applying force to advance the piston, gauge pressure is greater than sample pressure (Richardson et. al, 1968; Bell et. al, 1971). A correction of +2 Kb. (Bell et. al, 1971) was applied to the pressure by 'overpumping' a value equal to 2 Kb. on the sample. Richardson et. al (1968) state that anvil effects decrease as temperature increases and that at 1300°C frictional error would account for almost all discrepancy between 'observed' and 'true' pressure. No attempt was made to measure this effect, but an error allowance of - 2 Kb. is included. The experimental assembly is illustrated in Fig. (3). A graphite sample holder was used in order to minimize anvil effects on the sample by absorbing internally generated shear stresses. Resulting sample pressure was assumed to be hydrostatic. Recent results by Hays and Bell, (1973) on the albite-jadeite+ quartz equilibrium indicate excellent agreement between gas pressure media apparatus and solid media apparatus up to approximately 17 Kb. The graphite capsule also controls the ambient P and even for runs of several hundred hours duration olivine remains stable. Experiments using fayalite in graphite capsules have produced similar results (Akimoto et. al, 1967); Akimoto and  Fujisawa, 1968). Thompson and Kushiro, (1972) have shown that when graphite capsules are used at high pressure the resulting P02 is in the range of wustite-magnetite. Temperature was controlled using a solid state controller (Hadidiacos, (1972), with a Pt-Pt90%RhlO% thermocouple sensing element. The thermal gradient in the sample cavity was determined (Fig. 3) and over the diffusion zone, approximately 200 micro meters, temperature variation was approximately 2°C. The hydrostatic 2r3 FIGURE 3. Illustration of the sample assembly used in the high pressure experiments. Experimentally determined thermal gradient in the region of the sample and graphite sample holder are also shown. D.S. = Diffusion sample. Th. = Thermocouple (Pt-Pt Rh1Q). f! Graphite = Talc Boron Nitride X Fired Pyrophyllite = Pyrophyllite Steel Plug Crushable ceramic Ceramic 24 Scale thermal gradient A 0 iH 900 1000 T CC) \| \l \ \ \ \ \ \ \l \ \ii x x X X X X X X X X V I, II X X X X X X X X X X X 7TiTTT7T D.S. TM II X X X x x X X X X X X x X x x X X X X i>:|i|l!|l:iii \ K l\ <25 pressure effect on the thermocouple e.m.f. was calculated using the data of Getting and Kennedy, (1970) assuming a seal temperature of 20°C. Maximum hydrostatic pressure effect at 1100°C and 35Kb. was approximately 15°C and the controller was set to compensate for this effect. Thermocouple drift effects, although small at these temperatures (Mao et. al, 1971) were compensated for by increasing the temperature 1°C every 50 hours. Thermocouple contamination, diffusion effects at the junction, and the nonhydrostatic stress effects were not determined. The total uncertainty from all sources is estimated as follows. Controller error = ± 1° Gradient and positioning error = - 3 Hydrostatic error= - 3° Other errors = - 3° Total temperature uncertainty = - 10°C A uniform experimental procedure was adopted for all high pressure experiments. Initially, pressure was increased to within 5 Kb. of the desired pressure, temperature was increased to the desired value, then the pressure was increased to the final value. At the completion of each experiment, temperature and pressure were simultaneously lowered over a period of two to three hours. The graphite capsule containing the diffusion couple was impregnated with epoxy and prepared for electron microprobe analysis in the same manner as the one atmosphere experiments. 26 EXPERIMENTAL DATA AND ANALYSIS Experimental study of Fe-Mg interdiffusion in olivine involved the collection of compositional profiles produced by diffusion for different times, temperatures and pressures. Concentration profiles were obtained for each diffusion experiment using the M.A.C. model 400 electron microprobe at the Geophysical Laboratory. The concentration data were collected at intervals of 2, 5, 10, or 20 micrometers, depending on the rate of change of composition with distance. The microprobe was calibrated using standards of 7.42% and 31.14% Mg, 5.62%, 17.18%, and 31.86% Fe, 4.19%, Mn, and 19.10% Si. The raw data, in counts per second, were reduced to oxide weight per-cent at the time of analysis using a computer program developed by Finger and Hadidiacos (1972). The Bence-Albee correction procedure was used for the matrix corrections (Bence and Albee, 1968) with the coefficients given by Albee  and Ray (1970). The continuous flourescence effect (Reed and Long, 1963) may become important when measuring small amounts of one element 1 in phase 2 near a boundary with another phase 1 which is rich in the element. A calibration couple of forsterite-fayalite was prepared in a manner similar to that for the one atmosphere experiments and analyzed on the electron micro probe. The resulting curve is plotted in Fig. (4). The correction is of minor importance within 10 micrometers of the interface and is negligible further away. The couples used in the 1 atmosphere study of interdiffusion of Fe and Mg in the forsterite-fayalite system are listed in 27 FIGURE 4. Mg concentration profile for experiment No. 6 (1000°C) 311 Hr. , 1 atm.). Plotted points are corrected micro probe analysis along the diffusion zone. The continuous flourescence effect, H H is also plotted. 28 0 Dis8 l55 150 260, 2^0 ance in micrometers .2,9 FIGURE 5. Mg concentration profiles for experiment No. 14, parallel to the C axis —£ ^— , and experiment No. parallel to the B axis —J|| £— , Both experiments at 950°C and 235 Hr. FIGURE 6. Calculation of Ln D versus Mg cation mole fraction. Experiment 4 (900°C) — # , Experiment 14 (950°C) , Experiment 1 (1000°C) Experiment 6 (1000°C) , Experiment 40 (1100oC) Table (2). Figs. (4) and (5) are representative Mg concentration profiles of forsterite-fayalite couples at 950°C and 1000°C. Fig. (6) illustrates the results of calculations of D versus concentration using equation (22). The resulting profiles show a logarithimic dependence of the interdiffusion coefficient on concentration in the ragge (0.l^Mg/Mg+Fe^O.8). Fig. (7) shows the dependence of & upon crystallographic orientation. r\, o - 1 Values of D as a function of 1/T ( K ) , at selected con centrations, are plotted in Figs. (8a) and (8b). Regression analysis of Ln, Iii versus 1/T results in values of DQand AH* which are shown in Figs. (9) and (10). A general Arrhenius relationship was formulated using these results ft = [1. 55± . 25-1.12 (N2) ] xlO' 2xexp [49. 85±4. 5+9. 05 (N2) ] -(25') RT where N2 = cationic mole fraction Mg (0.1-Mg-0.8) In all figures using regression techniques, the standard deviation in the dependent variable is indicated by the large (I) error bars . •• The high pressure experiments are listed in table (3). Fig. (11) represents the results of calculations using equation (22) for the experiments at 1100°C, parallel to [001]. Ln. ^ shows a linear decrease with increasing mole percent Mg as in the one Atmosphere experiments. Fig. (12) shows the effect of hydrostatic pressure on the interdiffusion coefficient. Regression analysis of the data for 900°C and 1100°C yield values of the activation volume, AV*, plott in Fig. (13). The average value of AV* is approximately 3 5.50 cm /mole with a slight decrease in AV* between 1100°C and 900°C. Calculated values of AV* using experiments 3-P and 21-P 3, yield values consistent with the average AV* of 5.50 cm /mole. 34 TABLE 2 EXPERIMENT TIME TEMPERATURE CRYSTAL NUMBER (SEC.) (°C) ORIENTATION 4 1. 192xl06 900±5 C-AXIS 14 8. 46xl05 950±5 C-AXIS 15 8. 46xl05 950±5 B-AXIS 1 3. 56xl05 1000±5 C-AXIS 6 1. 14xl06 1000±5 C-AXIS 59 3. 64xl05 1000±5 B-AXIS 40 8. 95xl05 1100±5 C-AXIS TABLE 2 List of the one atmosphere diffusion experiments (900-1100°C). 35 FIGURE 7. Ln S versus Mg cation mole fraction, Experiment No. 1 parallel to C axis ^ ^ , Experiment No. 59, parallel to B axis, —WM Hj— Both experiments at 1000°C. 36 i i—i—i—i—i—r—i—i—r .0 .1 .2 3 4 .5 .6 .7 .8 .9 Mg cation mole fraction 37 - u FIGURE 8A. Ln D versus 1/T (°K)_1 x 104 for D calculated at and 60 —— , Mg cation mole percent. Error bars indicate one standard deviation calculated using regression analysis. '38 T(°C) 9,00 10,00 11,00 2L u 22 85 8!0 75 1/T (°K"1)x104 3.9 FIGURE 8B. Ln ft versus 1/T (0K_1) x 104. Parallel to C axis , parallel to B axis 4:0 9p0 T ^ 15,00 lljOO 23 . 2BI 291 3d 3U 8! 5 8.0 „ 7.'5 . 1/T (°K"1 )x104 FIGURE 9. Ln ftQ versus Mg cation mole fraction. Error bars indicate one standard deviation. 42, 43 FIGURE 10. AH* versus Mg cation mole fraction. Error bars indicate one standard deviation. ,44 I 1 1 1~ .0 .2 .4 .6 .8 1.0 Mg cation mole fraction 45 TABLE 3 EXPERIMENT TIME TEMPERATURE PRESSURE CRYSTAL NUMBER (SEC) (°C) ± 10° (KB) ± 2 ORIENTATION 19-P 5.11xl05 S00 900 10 C-AXIS 16-P 6.95xl05 900 900 25 C-AXIS 3-P 3.24xl05 3 00(1000 10 C-AXIS i-p 3.42xio5 KGQLIOO 10 C-AXIS 6-P 3.14xl05 11001100 10 C-AXIS 21-P 4.84xl05 11001100 15 B-AXIS SGB-FA) l-548xl06 11001100 25 C-AXIS 14-P 1.186xl06 110G1100 25 C-AXIS 18-P 4.07xl05 11001100 35 C-AXIS TABLE 3. Lists of the high pressure diffusion experiments using Forsterite-Fayalite and Fayalite-MgO powder diffusion couples. Time in sec, temperature in °C, and pressure in Kb. 46 FIGURE 11. Ln D versus Mg cation mole fraction at various pressures. Experiment 40 (1 atm.)-J|—J/^—, Experiment 1-P (10 Kb.) A A , Experiment 14-P (25 Kb.) -£ , Experiment 18-P (35 Kb.) -A—All experiments at 1100 48 FIGURE 12. Ln D versus pressure calculated at 10, 30, and 60 Mg cation mole percent. Experiments at 900°C—fJ-B", at 1100°C -0—. 4.9 22 I 1—'• I 1 1 1 1 1 0 5 10 15 20 25 30 35 Pressure (Kb.) 50 FIGURE 13. AV* versus concentration (Mg cation mole fraction). Error bars indicate one standard deviation. 51 n52 A diffusion couple of Mgo powder-fayalite crystal was run at 1100°C and 25 Kb. (experiment No. 12-P) to observe the effect of the crystal-powder interface on the diffusion coefficient. Calculated values of D agree with the crystal-crystal experiments within experimental error. The MgO powder-olivine single crystal experiments are listed in table (4). Fig. (14) shows profiles parallel to the a, b, and c crystallographic axis at 1250°C and one atmosphere. a. Fig. (15) represents calculations of D versus Mg concentration for the profiles in Fig. (14). As in the lower temperature exper iments, Ln ft decreases with increasing Mg concentration. The plot of Ln D versus concentration departs from linearity at Mg concentrations greater than 98%. This is probably due to large errors in the calculation of the derivative (dN2/dX) in this portion of the profile. A similar, but less pronounced effect, is observed when the Mg concentration is less than 92%. Figs. (16) and (17) illustrate analysis of the other MgO powder-single crystal experiments parallel to [001]. Figs. (18) and (19) show the concentration profile and LnD versus concentration calculation for a synthetic forsterite-Fog^Fay olivine diffusion couple at 1300°C. Fig. (20) illustrates the regression analysis of Ln D versus 1/T (°K~"'") for the interdiffusion of Fe-Mg in olivine between 1200 and 1400°C parallel to the C axis. The standard ^ as calculated using the regression analysis, is error in Ln D indicated by the large error bars. The slight increase in the a. values of D obtained from the crystal-crystal experiments, relative to crystal-powder experiments, may indicate a decrease in the contact resistance in the crystal-crystal couple relative 53 TABLE 4 EXPERIMENT NUMBER TIME (SEC.) TEMP. (°C)±5 CRYSTAL ORIENTATION DIFFUSION COUPLE 68 7.25xl05 1200 A-AXIS Fo(l)-MG0 69 7.25xl05 1200 C-AXIS Fo(l)-MG0 29 1.470xl06 1250 A-AXIS Fo(l)-MG0 30 1.470xl05 1250 B-AXIS Fo(l)-MG0 31 1.470xl06 1250 C-AXIS Fo(l)-MG0 50 8,.80xl05 1300 B-AXIS Fo(l)-MG0 51 8.80xl05 1300 C-AXIS Fo(l)-MG0 52 1.210xl06 1300 C-AXIS SYNTHETIC-Fo(2) 26 9.46xl05 1350 A-AXIS Fo(2)-MG0 27 9.46xl05 1350 C-AXIS Fo(2)-MG0 72 3.41xl05 1400 A-AXIS Fo(2)-MG0 73 3.41xl05 1400 B-AXIS Fo(2)-MG0 74 3.41xl05 1400 C-AXIS Fo(2)-MG0 71 3.41xl05 1400 C-AXIS Fo(2)-MG0 TABLE 4. List of the 1 atm. Experiments (Nitrogen furnace) using Forsterite-MgO powder and synthetic Forsterite-FOgg olivine diffusion couples. Time in sec, temperature in °C. 54 FIGURE 14. Fe concentration profiles for Experiment 29, parallel to A axis A—Experiment 30, parallel to B axis f/^~; Experiment 31, parallel to G axis . Experiments at 1 atm. 1250°C, and 408 hr. 55 56 FIGURE 15. Ln D versus Fe cation mole fraction. Experiment 29, parallel to A axis-A—*mr-; Experiment 30, parallel to B axis-JH—H— > Experiment 31, parallel to C axis 57 I 1 1 " .0 .02 .04 .06 .08 Fe cation mole fraction 58 a. FIGURE 16. Ln D versus Fe cation mole fraction. Experiment 69 (1200°C) J4~ , Experiment 31 (1250°C) —, Experiment 51 (1300°C) ~% All experiments parallel to C axis. 59 Fe cation mole fraction 60 FIGURE 17. Ln D versus Fe cation mole fraction. Experiment 27 (1350°C) -A—^kr > Experiment 74 (1400°C) All experiments parallel to C axis. 61 I I I I .0 .02 .04 .06 -Fe cation mole fraction 62 o , FIGURE 18. Fe concentration profile for synthetic forsterite-FoggFay olivine diffusion anneal. Experiment 52 (1300°C) 1 atm., 336 hr.) 63 Distance in micrometers 64 FIGURE 19. Ln D versus Fe cation mole fraction for Experiment 52. Error bar indicates one standard deviation. 65 66 FIGURE 20. Ln ft versus 1/T (°K_1) x 104 for ft calculated at cation mole fraction. Calculations for synthetic forsterite-Fog^Fay olivine diffusion couple, 5""'— A , and 7 > Fe cation mole fraction. Error bars indicate one standard deviation. 67 ( 12)00 T (°C) 13|00 14i00 6:5 6!0 1/T (°fC1)xlO* 68 to the crystal powder couple. Values of DQ and AH* determined from the profiles in Fig. (20) are in general agreement with the extrapolation of the lower temperature values and are plotted on Figs. (9) and (10). The dependence of u on crystallographic orientation between 1200 and 1400°C is illustrated in F£g. (21). Diffusion parallel to the c axis is approximately 4.5 times faster than diffusion parallel to the b axis at 1200°C. At 1400°C this ratio has decreased to a value of approximately 2.6. GEOPHYSICAL APPLICATION OF THE DIFFUSION COEFFICIENTS The interdiffusion coefficient has been used to calculate the conductivity as a function of temperature. This calculation depends on the assumption that D measured at low Fe concentrations approximates the self diffusion coefficient for vacancies. Results of this calculation, using equation (2) are shown in Fig. (22) along with the previously determined values of electrical conductivity in olivine. Interdiffusion data from the experiments between 1200 and 1400°C were used in the calculations. Agreement between calculated and experimentally determined values of the electrical conductivity and the enthalpy of activation for conductivity indicate that ionic migration is the dominant mechanism of electrical conduction in olivine when the temperature is greater than 1200°C. Estimates of temperatures in the mantle may also be made using the P-T dependence of D. From equation (2) am = D(T,P)KT -(26) T 69 FIGURE 21. Ln D versus 1/T ("IC1) x 104 for diffusion parallel to aaaxis -j^ , b axis'JJ—JJ-, and c axis All calculations for Fe cation mole fraction = 0.07. 70 24 25 26 27 28 12|OQ T(°C) C ax is. AaxisX 13,00 7!0 14^0 B axis 6.5 6.0 1/T (°K"1)X104 FIGURE 22. Experimental determinations of Log versus 1/T C°K"^) x 104. , results of calculated using equation 2 for Fe cation mole fraction of 0.03 and 0.09. Results from other authors; (1) Synthetic Forsterite, Shankland, (1969), (2) Red Sea Olivine, Hughes (1959), (3) and (4) San Carlos Olivine, Duba and Nichols (1973), (5) Fo92>6 olivine Kobayashi  and Maruyama (1971), (6) F°82 onvinei Mizutani and Kanamori (1967) . '72 10,00 T (°p) 12,00 13,0014,00 i r 7.5 7.0 1/T (°K"1)x104 •73 where a. = independent determination of m conductivity in the mantle D(T,P) = calculated value of the interdiffus ion coefficient at T and P T = absolute temperature (°K) substituting equation (24) in equation (26) °m = (Kl)Do exp(-AH*/RT)exp(-PAV*/RT) -(27) thus Lnam = Ln(KL)+Ln(Do)-l_10_(AH*+PAV*) -(28) T RT Using equation (28) with the conductivity data of McDonald (1957) and Eckhardt et. al (1963) and a starting temperature versus depth profile of Ringwood and Clark (1964) and Ringwood (1966) anditerative procedure was used to calculate a new temperature versus depth profile. Iteration of temperature in equation (28) was continued until The temperature profiles calculated using equation (28) are shown in Fig. (23) along with the olivine stability field and the oceanic geotherm of Ringwood (1966). The two points with the large error bars are estimates by Duba and Nichols (1973) based on electrical conductivity measurements on Red Sea olivine at 1 atmosphere and 8 Kb. Assuming the conductivity profile of McDonald (1957). The two profiles calculated for Fog^Fag olivine; the lower one using McDonald's estimate of am the upper one using the Eckhardt et. al estimate, bracket the geotherm of Ringwood and Clark |am (calculated) -am:l|<0.05l|am:i| -(29) 774 FIGURE 23. Temperature versus depth in the mantle. Oceanic geotherm, Ringwood and Clark, — —. Forsterite stability field, Akimoto and Fujisawa (1968), Davis and England (1964), . Experimentally determined values, Duba (1973) (j) . Calculated values using experimentally determined diffusion coefficients, this study; Fog^ olivine, AV* = 0.0, McDonald's (1957) conductivity estimate —^ ^— , Fogi olivine, AV* = 0.0 Eckhardt et. al (1963) conductivity estimate ~A a^- , F°g-i olivine, AV* = 5.50 cm /mole, McDonald's (1957) conductivity estimate Fog-^ olivine, AV* = 5.50 cm /mole, Eckhardt et. al (1963) conductivity estimate -4^ A- . All calculations for diffusion parallel to c axis. 75 20QQ 15QO U o 1000 5Qd liquid 'oceanic ^geotherm olivine/ spinel 0 60 120 180 240 300 360 Depth ( km) 76 and Clark (1966) below 150 Km. Two two calculations assuming a AV* of 5.50cm /mole yield temperatures approximately 450°C higher at a depth of 300 Km. Uncertainties surrounding the importance of the pressure effect on the conductivity (Duba, 1972; Hughes, 1955), and uncertainties in the calculated affl profiles from geomagnetic and magnetotelluric measurements preclude definite conclusions as to the validity of the pressure-included versus pressure-excluded profiles; however, preliminary comments may be made. The AV* used in these calculations was for single crystals and for a polycrystalline aggregrate pressure effects have been ob served to be less (Schult and Schober, 1969; Hamilton, 1965; Bradley et. al, 1964). The polycrystalline pressure effects may be a closer approximation to the mantle and thus the profiles calculated in this study using AV* = 5.50cm /mole may be too high. The magnetotelluric studies of the electromagnetic field may define am versus depth in the upper 400 Km. of the mantle more accurately as they are recording shorter period vibrations thus obtaining better resolution at shallow depths. For these reasons, profile (A) in Figl (23) was assumed to represent the best fit calculation based on the data from this study. 77 CONCLUSIONS 1) The interdiffusion coefficient for Fe and Mg in olivine decreases with increasing Mg cation mole fraction. The dependence of ft upon temperature and composition is given by D = (1.53±0.25-1.12N2) x 10"2 x exp (49.83±4.5+9.05N9) RT where 900 - T - 1100, 0.10 - N2 - 0.60 and N2 = cation mole fraction Mg 2) The interdiffusion coefficient is a function of the crystallographic orientation. In the temperature range (900 - T - 1100, D[001] > D[010]. In the temperature range 1200 - T - 1400, D[001] > D[000:} > D[010]. 3) The interdiffusion coefficient decreases with increasing pressure. A hydrostatic pressure of 35 Kilobars decreases the interdiffusion coefficient by approximately a factor of ten. 4) Ionic conductivities calculated from the results of the diffusion experiments between 1200 and 1400°C agree with the experimental determinations of electrical conductivity in olivine. Below a depth of 100 Kilonet-e'Es in the mantle ionic conduction is probably the dominant mechanism of electrical conduction. 78 ACKNOWLEDGEMENTS: PAPER #1 I wish to thank Dr. H. J. Greenwood of the University of British Columbia and Drs. H. S. Yoder Jr., F. R. Boyd, L. W. Finger, and P. M. Bell of the Geophysical Laboratory, Carnegie Institution of Washington for their interest in the investigation and their helpful discussion. I also wish to thank Mr. C. G. Hadidiacos for his help in the electron micro probe analysis. The work was supported by the Carnegie Institution of Washington and Grant A-4222 of the National Research Council Canada to H. J. Greenwood. 79 PAPER #2 CATION DIFFUSION IN OLIVINE AND ITS IMPORTANCE FOR CREEP IN THE UPPER MANTLE 80 ABSTRACT Calculations based on experimental data on diffusion of Fe and Mg in olivine and a corresponding state theory of creep indicate that cationic diffusion is not the rate con trolling step in either dislocation climb or creep. The melting behaviour of olivine appears to be closely related to the mobility of cations rather than of anions. INTRODUCTION Experimental determinations of the rheological behaviour of dunites and lherzolites have indicated possible flow mechanisms responsible for large scale deformation in the mantle. (Goetze and Brace, 1972; Carter and Ave'Lallement, 1970; Raleigh and Kirby, 1970). Extrapolation of these data to mantle conditions have been made assuming that the pressure effect on the creep is reflected by the pressure effect on the self-diffusion coefficient of the rate controlling species in the olivine lattice (Weertman, 1967; Goetze and Brace, 1972). Estimates of the pressure effect on the diffusion coefficient have been made using the relationship, derived empirically from studies on metals, that the logarithm of D is proportional to T /T, where Tm is the melting temperature at given pressure. In the present study, interdiffusion coefficients for Fe and Mg in olivine reported in detail in a companion paper (Misener, 1973), are applied to test this relationship and to compare the energetics of different flow mechanisms. The effects of temperature on the diffusion coefficient and creep ,81 rate are compared. This comparison indicates that cation diffusion in olivine is not the rate controlling process in upper mantle creep. THEORY INTRODUCTION The theory and calculations that follow are presented to test certain equations for extrapolation of experimental results, these are summarized briefly. Flow rates depend exponentially on temperature and pressure, as does diffusion. Comparison of their activation parameters permits assessment of the degree of correspondence between their fundamental mechanisms. Experimentally determined activation parameters can be related to experimentally determined melting curves and conclusions made regarding controls on the melting process. The equations are presented below with an outline of their interrelations. RELATION BETWEEN CREEP AND DIFFUSION The experimental studies of Carter and Ave'Lallement (1970); and Raleigh and Kirby (1970) have resulted in the following creep law for upper mantle material. A(an) exp (-Q/RT) -(1) where e strain rate a a differential stresso(03-0^) Q activation energy for creep 82 T = absolute temperature rj; = positive number (2-iij-5) Geotze and Brace (1972) have presented a compilation, fitting published creep data to equation (1). Their results are shown in Fig. (1). At differential stresses less than 1Kb. The exponent 'n* is 3 and above 1 Kb. it is 5. Assuming a relationship between diffusion and creep rate we may write: e = F(as) DT)P -(2) where D-p p = controlling diffusion coefficient T = absolute temperature, T>.5Tm a = differential stress (in the range of tens to hundreds of bars) Experimental determinations of diffusion coefficents have shown I DT,P = Do exP C-AH*/RT) exp (-PAV*/RT) -(3)" where DQ = pre-expontential constant (10 tolO^) AH* = activation enthalpy for diffusion AV* = activation volume for diffusion T = absolute temperature R = gas constant Substituting (3) in (2) and equating (2) and (1) results in Q = AH* + PAV* -(4) 83 FIGURE 1. Experimental determination of strain rate versus stress (from Geotze and Brace, 1972) for results at 100°C. Tectonic stresses are assumed values for the upper mantle. 84 u c/) CO O id i ui 12 13 tectonic stress experimental limit T I r LOG 6 3 (bars)4 85 RELATION BETWEEN DIFFUSION COEFFICIENT AND MELTING TEMPERATURE Sherby and Simnad (1961) have proposed an empirical relation between the diffusion coefficient and the melting temperature. D(T,P) = DG exp (-gTm/T) _(5) where Tm = melting temperature g = dimensionless^constanfct (18.0 for most metals) Weertman (1970) has shown that the constant g can be related to the thermodynamic rate parameters in the following way g = Sq = gv -(6a) where gq = AH*/RTmj0 -(6b) gv = AV*7R(dTm/dP) -(6c) and Tmo = melting temperature at 1 atmosphere dTm/dP = pressure derivative of the melting temperature These relationships imply a simple relation between the activation rate parameters and the volume and enthalpy changes of melting. In particular AH* = H^. WV* TmVm If equations (6c), (6) and (5) are obeyed then gq/gv =1-0 -07) Equation (6c) also implies the validity of a parameter of P* proposed by C.ietzc and -Brace (.1.372) ,• where-86 •. t P* proposed by Goetze and Brace (1972), where p* = p* = p* _ C8") where P* = AH*/AV* PM = VVM Consequently the validity of equation (6c) and equations (6) and (7) may be tested by comparing gq/gy- with 1.0 and P* with calculated values from other workers. The validity of extrapolations may then be judged. Both gq/gy and P* have been used to estimate the change of D with pressure and in conjunction with equation (2) to extrapolate flow laws to mantle conditions. Extrapolation of this or similar equations to mantle conditions is subject to large errors. It has been demonstrated (Griggs, 1967) Carter and  Ave'Lallement, 1970) that the amount of water present is critical in determining the flow law. The deformation mechanism may: change at the lower stresses assumed to ineimaintained in the upper mantle. The large uncertainties in the temperature estimates of the mantle cause or-derfofsagagrtiifisKiererjrGrs.ninht'he estimation of the diffusion coefficient. RESULTS Experimental determinations of interdiffusion coefficients, reported in a companion paper (Misener, 1973) , are used to evaluate AH*, AH* and thence g /g , and P*. The results are q v summarized here in Figs. (2) to (4). Tables (1) and (2) present the results in numerical form appropriate to equation (6). 87 FIGURE 2 Ln D versus 1/T (°K. ) x 10 for D calculated at 50 —^— , and 60 —Q , Mg cation mole percent. Error bars indicate one standard deviation calculated using regression analysis. 1 89 FIGURE 3 Ln ft versus pressure. Parallel to c axis, 1100°C —A— , parallel to b axis —| ^J-, parallel to c axis, 1000°C — , parallel to c axis, 900°C Data calculated at 30 cation mole percent Mg. 90 ^23 29 i i i i i i i r 0 5 10 15 20 25 30 35 Pressure (kb.) 91 FIGURE 4. Ln D versus 1/T (°K 1) x 104 for diffusion parallel to a axis ;4fc— , b axis , and c axis All calculations for Fe cation mole fraction = 0.07. 92 24J 25 26 27 28 12,00 T(°C) C ax is. Aaxis^ 13,00 14)00 B axis 7!0 6.5 6.0 1/T (°K"1)x104 93 TABLE 1 Calculated results for the pre-exponential constant (D ) and the activation enthalpy (AH*) as a function of temperature, pressure, and Mg cation mole fraction for inter diffusion in the forsterite-fayalite system. The uncertainties in D and AH* are one standard deviation, o 94 TABLE 1 CATIONIC TEMPERATURE D + :&) AH* ± .<$AH* MOLE FRACTION RANGE (°C) (cV/sec) (Kcal/mole) MG 0.0)0' 900-1100 1.49±0.19 50.75±4.13 0.20 900-1100 1.23±0.25 51.68±3.43 0.30 900-1100 1.15±0.27 52.45±3.08 0.40 900-1100 1.08±0.28 53.47±3.14 0.50 900-1100 0.98±0.22 54.36±3.55 0.60 900-1100 0.87±0.18 55.27±4.22 CATIONIC TEMPERATURE PRESSURE AV* ± "S"AV* MOLE FRACTION (°C) RANGE cm3 mole-1 MG 0.10 1100 1 atm.-35Kb. 6.69±0.85 0.20 1100 1 atm.-35Kb. 6.25±0.72 0.30 1100 1 atm.-35Kb. 6.28±0.79 0.40 1100 1 atm.-35Kb. 6.08±0.78 0.50 1100 1 atm.-35Kb. 5.88±0.79 0.60 1100 1 atm.-35Kb. 5.69±0.79 0.10 900 1 atm.-25Kb. 4.47±0.30 0.20 900 1 atm.-25 Kb. 4.60±0.64 0.30 900 1 atm.-25Kb. 4.58±0.94 0.40 900 1 atm.-25Kb. 4.61±1.27 0.50 900 1 atm.-25Kb. 4.60±1.50 0.60 900 1 atm.-35Kb. 4.58±1.80 95 TABLE 2 DIFFUSION TEMPERATURE DQ ± /DD AH* ± /AH* COUPLE RANGE (°C) (xl0+2) (Kcal/Mole) (cm2/Sec.) MGO POWDER FORSTERITE CRYSTAL IfOaipdAXTS DRYST200-1400 2.16 ±0.35 65.56 ±3.56 MGO POWDER FORSTERITE CRYSTAL (010) AXIS 1200-1400 2.23 ±0.05 76.77 ±1.31 MGO POWDER FORSTERITE CRYSTAL (100) AXIS 1200-1400 4.33 ±0.35 70.11 ±3.79 TABLE 2 Calculated values for the pre-expontential constant (DQ) and the activation enthalpy (AH*) as a function of temperature and crystallographic orientation in the system MgO powder-forsterite olivine single crystals. Values calculated at 0.07 cation mole fraction Fe. '96 CALCULATIONS AND DISCUSSION Values of gq/gy may De calculated from equation (6) as follows gq/gv = AH*(dTM/dP)/AV*Tmo ' -(9) Results of these calculations are presented in table (3). The values of g /g , last column table (3) q v gg/gv = 1-30 ± 0.5 (F°10Fa9o) gq/gy = 1.16 ± 0.5 (Fo93Fa7) May be compared with the theoretical value of 1.0 obtained in equation (6d). Results of calculations on other materials, Weertman (1970), range from 0.7 to 1.35. These results indicate that the realtionship derived by Sherby and Simnad (1961) may be extended to olivine where AH* and AV* are the activation parameters for cation diffusion. PARAGRAPH" A' Values for the parameters g^, g^, and P* are calculated for cation diffusion in olivine and compared with values obtained for other materials. At low solute concentrations the approximation is made that the interdiffusion coefficient (ft) is within an order of magnitude of the intrinsic lattice self-diffusion coefficient. The intrinsic diffusion, or the creation and motion of point defects, dominates over grain boundary and extrinsic (impurity) diffusion for most materials when the temperature is greater than about one half of the melting temperature. For the calculations using Fo^FagQ olivine diffusion parameters, the experimental temp-97 TABLE 3 OLIVINE AH AV _ gO gV g g COMPOSITION Kcal mol" cm mol q V q V Foj.^ 50.8±4.0 5.5±1.0 17.±1.7 13.±4.0 1.3±.5 FA90 Fog3 65.56±4.0 5.5±1.0 16.U1.6 133934400 1.16±.5 IFa? TABLE 3. Calculated values of g^ and g^ at FO^Q Fago anc* F°93 Fa., olivine compositions. 98 eratures ranged from 0.64TW- T - 0.79T, • for the Fo^Fa., M M 93 7 calculations, 0.68T < T < 0.79T„. The values for the pre-M M r _9 expontential constant, D , (^10 ) are within the range for intrinsic diffusion. The parameter P* calculated from equation (7a) using the values listed in Table (1) and Table (2) results in the following values P*(Fo10Fa9()) = 350 ± 100 Kb. (0-35 Kb.) P*(Fo93Fa? ) = 450 ± 100 Kb. (0-35 Kb.) Using the data of Davis and England (1964) for the melting of forsterite P*(EoinJ = 460 ± 10 Kb. (0-50 Kb.) m 100 The data of Akimoto et. al (1967) show a change in slope of (dT /dP) for fayalite at 1 atmosphere where where The average value of P* (Fa^^) is approximately 315 Kb. This value agrees with that of Goetze and Brace (1972), (P* = 300Kb.). The parameters, P* and P*, are assumed to be equal within experimental error. Goetze and Brace (1972), using an activation enthalpy of lOOKcal/mole reach the conclusion that equation (2) may not be valid for olivine. They obtain the' value of lOOKcal/mole from the activation enthalpy for creep and equate this to the anion enthalpy for diffusion in olivine. From the present calculations, it appears that the melting parameters are related to the faster cation diffusion rates. Pm^W dT /dP m At 50 Kb. ^Faloo> dT /dP m 200 ± 10 Kb. (P = 1 atm.) 7.5°C/Kb. 430 ± 10 Kb. (P = 50 Kb.) 3.5°C/Kb. 99 The validity of equations (2) and (4) depends on using the diffusion coefficient for the rate-limiting atom in the structure. Table (4) lists reported activation parameters for creep experiments on dunites and lherzolites. Comparison with tables (l),afi2), and (3) reveal that the activation energy for creep experiments is significantly greater than for cation diffusion. At confining pressures of 15 Kb. and stresses of the order of 1 Kb., the activation energy for the creep of dry material is approximately two to three times greater than the activation enthalpy for the diffusion of cations. Pre liminary results, Goetze and Brace (1972), at low strainsrates (e = 10 S) and low stresses (a = 450 bars) indicate that the discrepancy is still present at small strain rates. The results from experiments with water present indicate a decrease in Q for creep; however, it is still a factor of one and a half to two greater than AH* for cationic diffusion. Direct observation of dislocation climb in olivine by Goetze and Kohlstedt (1973) is in agreement with the above observations. The process of dislocation climb has been pro posed by Weertman (1970) £5 gQVern the flow of olivine in the region where e is proportional to aS [see Fig. (1)]. Goetze  and Kohlstedt (1973) obtain a value of 135 ± 30 Kcal/mole for the activation enthalpy of dislocation climb. This value agrees with the experimentally determined values for Q but is three times greater than AH* for cation diffusion. The diffusion coefficient of a vacancy or ion accompanying dis location climb is calculated from the data of Goetze and Kohlstedt (1973) to be Dd-C* = 7.0 x 10"14 cm2/sec. 100 TABLE 4 MATERIAL CONFINING A Q N PRESSURE (SEC ) (KCAL/MOLE) (KB) (KB) Dunite 1 15.0] 6.2 x 10 7979.9±7.5 2.4 ± 0.2 (wet) Dunite 1 15.0 1.2 x 1010 119.8±16.6 4.8 ± 0.4 (dry) Lherzolite 1 15.0 3.2 x 107 79.8±9.4 2.3 ± 0.2 (wet) Lherzolite 2 15.0 1.0 x 108 106.0±20. 5.0 (dry) TABLE 4. Experimental determinations of the flow law in dunites and lherzolites. The parameters A, Q, and N refer to the equation i = Aexp(-Q/RT)an Numbers after the samples refer to the authors: (1) Carter and  Ave'Lallement, 1970 (2) Raleigh and Kirby, 1970. 10.1 from the present work D (1400°C) = 5.10 x 10-11 cm2/sec. The cation diffusion coefficient is approximately 700 times larger than the rate of dislocation climb and thus cannot be the rate limiting step in dislocation climb. CONCLUSIONS 1) Experimental determinations of the activation enthalpy and activation volume for Fe-Mg interdiffusion inoolivine indicate that the relationship D = Dq exp (-gTm/T) where g = 18.0 is valid for cation diffusion. 2) Verification of the equation AH*/AV* = AH /AV m/ m indicates that the melting of olivine is related to the mobility of cations rather than of the anions in the crystal. 3) The activation enthalpy for cation diffusion in olivine is one-half to one-third the value for the activation enthalpy for high temperature creep and is one-third the value for diffusion-controlled dislocation climb in olivine. Experimentally determined values of cation diffusion coefficients are approx imately 700 times larger than for dislocation climb. It is concluded that the diffusion of cations in the olivine structure is not the rate limiting process for high temperature creep in the upper mantle. 102 APPENDIX A Diffusion in a binary solid system at constant temperature and pressure may be described in terms of a single parameter D, the interdiffusion coefficient. In the case of one-dimensional diffusion the average velocities v^ and v^ of components 1 and 2, in an arbitrary co-ordinate system are related to the molar fluxes and by vJi/(VV ; W^VV -ci* where V is the molar volume and Nn and N„ are the molar m 12 fractions of components 1 and 2. In systems involving a change in volume, it is app ropriate to use V-^_V2 as tni-s quantity will be invariant regardless of the choice of co-ordinate system. The velocity invariant v^~^2 ^e ProPor'tional t0 tne local gradient in N^. Thus the definition of the interdiffusion coefficient becomes D = N N (v - v ) = V(WJ - N JJ -(2:).) 1 2dN^/dx 2 m 2 ki2/& 2 Consider a diffusion couple with initial moi fractions N_ and N*. Then the co-ordinate x of the Matano interface 2 2 m is defined by the equation xm - +oo + r^ dK + ;xmN2-N2 V V m m According to Matano-Bolzmann the composition of the sample is a single valued function of the auxiliary variable X = x xiji _ (4) tl/2 The rate of change in the concentration of each component in moi per unit volume at a given value of x_xm is equal to the 103 negative divergence of the respective flux in a co-ordinate system fixed with respect to the Matano interface jL. (NJ = §_ (1-N ) = -6J, -(5a) St V St V <5(x-x ) m x=xm m x,xm v mJ §- CNo) = «J? -(5b) «t V §(* -x ) m x =xm yv- m in accord with Sauer and Freise (1962) the auxiliary variable Y = N9-N^ -(6) is introduced as a measure of composition whereupon Nx = (1-N*)Y + (1-N~) (1-Y) -(7a) N2 = N*Y + N~ (1-Y) -(7b) Substituting equations (4), (5a), and (5b) in equation (7a) and (7b) yields X_[(1-N^)6_(Y_) + (1-NJ6 _(1-Y)] = SJ _(8a) 2t SA Vm SA Vm Sx1 >L_fN^_(Y_)+N"S_ (1-Y)] = SJ -(8b) 2tC 6A Vm 6A Vm Multiplying through equation (8a) by N and equation (8b) by (l-N^) and subtracting corresponding sides, one has -A_ CN*-N:)6_CY_)=N:- SJ.-Cl-NJfiJ _(9a) 2t SA Vm Sx1 z Sx Multiplying through equation (8a) by N* and equation (8b) by (1 - N*) and subtracting corresponding sides one has A_ (N^-N:)6_ (1-Y)=N*6J -(1-N*)6J -(9b) 2t 6'A Vm Sx1 z 6xz Equation (9a) is multiplied through by dA=d* and integrated tl/2 at constant time between A=-°° and the particular value A-x* with * the mole fraction N . Likewise equation (9b) is integrated * between A=x and A=+«>. At large absolute values of A, the change in composition is very small. Thus the local inter-104 diffusion coefficients are nearly constant and, therefore, values of N2 and Y as functions of X can be expressed in terms of the error function. Consequently, at \=t°° i.e. x = -°° Y and 1 - Y tend quasi-exponentially to zero. Further the gradient dN^/dX vanishes. Therefore no movement of component 1 relative to component 2 occurs, i.e. v^-V2=0. thus J /N - J2/N2 = 0 or J N - J N = 0 hence 1_(N*-N~D [-A_V + /*YdA_] = 1_- [N~J* -(l-N")Jj -(10a) 2t 1 Z VmA "°° Vm tl/2 1 1 Z Z + — * * +00 + * + * 1_(N2-N2)[-A (1-Y ) -f^(l-Y)dX]=l_[-N2J1+(l-N2)J2] -(10b) 2t Vm* ' Vm ttjf.2 where and J are fluxes at X = X . Multiplying through * * equation (10a) by (1-Y ) and equation (10b) by Y and sub tracting corresponding sides !_(N?-N?) [1-Y YdA + Y / (l-Y)dX] = _1_ [N J -(1-N )J ] -(11) 2t Vm Vm £i/2 Z * Substituting equation (2) for N2=N2 in equation (11), letting dX=dx in accord with equation (4) and solving for D one tl/2 obtains ft(N*) = (<-N;)Vm(N*)[(l-Y*)/*Y dx + Y*£~l-Ydx -(12) 2t(dN2/dx- Vm x Vm * * where x is the distance at which N2 = N,,. Since equation (12) involves only the value of dx it is possible to use distance from an arbitrary plane of reference in order to calculate dN2/dx and integrals in equation (12). There is thus no need to determine explicitly the Matano interface. With the help of equation (12) one may obtain values for ft as a function of N2 for the whole range of composition from a single diffusion run. 105 BIBLIOGRAPHY Akimoto, S., and H. Fujisawa, Olivine-spinel solid solution equilibria in the system Mg2Si0^-Fe2Si0^, J. Geophys. Res., 73, 1467-1479, 1968. Akimoto, S.,E. Komada, and I. Kushiro, Effect of pressure on the melting of olivine and spinel polymorph of Fe2Si0^, J. Geophys. Res., 72, 679-686, 1967. Albee, A.L. and L. Ray, Correction factors for electron probe microanalysis of silicates, oxides, phosphates, carbonates, and sulfates, Anal. Chem., 42, 1408-1414, 1970. Bell, P.M., H. K. Mao, and J. L. England, A discussion of pressure distribution in modern solid-pressure-media apparatus, Carnegie Inst., Washington Year Book, 70, 277-281, 1971. Bence, A.E., and A. L. Albee, Empirical correction factors for the electron-microanalysis of silicates and oxides, J. Geol., 76, 382-403, 1968. Borchardt, V.G., and H. Schmalzried, Diffusion in orthosilicaten, Ber. Deut. Keram. Ges., 49, 5-9, 1972. Bowen, N. L. and J. F. Schairer, The system, Mg0-Fe0-Si02, Amer. J. Science, 29, 151-217, 1935. Boyd, F.R., and J. L. England, Apparatus for phase-equilibrium measurements at pressures up to 50 Kb. and temperatures up to 1750°C. J. Geophys. Res., 65, 741-748, 1960. Bradley, R.S., AnK. Jamil, and D. C. 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Bell, Albite-jadeite-quartz equilibrium: A hydrostatic determination, (abstract) Eos Trans. AGU, 54, 482T 1973. Hughes, H., The pressure effect on the electrical conductivity of olivine, J. Geophys. Res., 60, 187-191, 1955. 108 Jander, W., and W. Stamm, Der innere aufbrau fester anorganischer verbindungen bei hoheren temperature, Z. Anorg. Allg. Chem., 207, 289-307, 1932. Lazarus, D., and N. H. Nachtrieb, In solids under pressure, edited by W. Paul and D. M. Warschauer, McGraw-Hill, New York, pp-43, 1963. McDonald, K. L., Penetration of the geomagnetic secular field through the mantle with variable conductivity, J. Geophys. Res., 62, 117-141, 1959. Mao, H. K., P. M. Bell, and J. L. England, Tensional error and drift of thermocouple electromotive force in the single-stage, piston-cylinder apparatus, Carnegie Inst., Washington Year Book, 70, 281-287, 1971. Misener, D. J., Interdiffusion studies in the system Fe2Si0^-Mg^SiO^, Carnegie Inst. Washington Year Book, 71, 576-520, 1972. 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C. Gilbert, Kyanite-silliman-ite equilibrium between 700°C and 1300°C, Amer. J. Science, 266, 513-541, 1968. Ringwood, A. E., Mineralogy of the mantle, in Advances in Earth sciences, editor, P. M. Hurley, M.I.T. Press, Cambridge, Mass., pp-357, 1966. Sauer, F. arid V. Freiese, Diffusion in binaren gemischen mit volumanderung, Z. Electochem., 66, 353-363, 1962. Rice, S. A. and N. H. Nachtrieb, On the dynamocal theory of D+ diffusion in crystals 11; pressure dependence of the self diffusion constant, J. Chem. Phys., 31, 139-145, 1959. Schult, A., and M. Schober, Measurements of electrical conductivity of natural olivine at temperatures up to 950°C and pressures up to 40 Kilobars, Z. Geophys., 35, 105-112, 1969. Shankland, T. J., Transport properties of olivine, in the application of modern physics to the earth and planetary interiors, editor, S. K. Runcorn, pp-175, Interscience, New York, 1969. Sherby, 0. D., and P. M. Burke, Mechanical behaviour of crystalline solids at elevated temperatures, in Progress in materials science, editors, B. Chalmers and W. H. Rathey, 13, 325-390, 1967. Sherby, 0. D., and M. T. Simnad, Prediction of atomic mobility in iro metallic systems, Trans. Amer. Soc. Metals, 54, 227-242, 1961. Thompson, R. N., and I. Kushiro, Oxygen fugacity with graphite capsules in piston cylinder apparatus at high pressure, Carnegie Inst. Washington Year Book, 72, 615-616, 1972. Wagner, C, The evaluation of data obtained with diffusion couples of binary single phase and multiphase systems, Acta. Met., 17, 99-107, 1969. Weertman, J., The creep strength of the upper mantle, Rev. Geophys., 8, 145-168, 1970. Whitney, W. P., and V. S. Stubican, Interdiffusion in the system Mg0-MgAl204, J. Amer. Cer. Soc, 54, 349-352, 1971. Yoder, H. S. Jr., and Th. G. Sahama, Olivine, X-ray determinative curve, Amer. Mineral,, 42, 475-491, 1967. McCormick, J. M., and M. G. Salvadori, Numerical methods in FORTRAN, Prentice-Hall. Eaglewood Cliffs, New Jersey, pp38-43, 1965. 

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