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Cation diffusion in olivine to 1400°C and 35 KB Misener, Donald James 1973-12-31

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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 a c c e p t t h i s t h e s i s as conforming t o t h e required standard  THE UNIVERSITY OF BRITISH COLUMBIA June, 1973  In p r e s e n t i n g an the  this  thesis in partial  advanced degree at the Library  University  s h a l l make i t f r e e l y  f u l f i l m e n t of the  of B r i t i s h Columbia, I agree  a v a i l a b l e f o r r e f e r e n c e and  I f u r t h e r agree t h a t p e r m i s s i o n f o r extensive for  s c h o l a r l y p u r p o s e s may  by h i s r e p r e s e n t a t i v e s .  be  g r a n t e d by  thesis for financial  written  permission.  gain  s h a l l not  Department The U n i v e r s i t y o f B r i t i s h V a n c o u v e r 8, C a n a d a  the  Head o f my  Columbia  be  that  thesis  Department  copying or  for  study.  copying of t h i s  I t i s understood that  of t h i s  requirements  or  publication  allowed without  my  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 Kilobars 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 where  N  2  (N ) 2  Kilocalories/mole  = 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)  w e r e  used to  determine the interdiffusion coefficient of Fe and Mg in olivine between 1200 and 1400° C. with increasing  The interdiffusion coefficient increases  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  i i  TABLE OF CONTENTS  iv  LIST OF FIGURES: PAPER #1  vi  LIST OF TABLES: PAPER #1  v i i  LIST OF FIGURES: PAPER #2  v i i  LIST OF TABLES: PAPER #2  v i i  ACKNOWLEDGEMENTS  vi 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  78  V  PAPER #2:  CATION DIFFUSION IN OLIVINE AND ITS  79  IMPORTANCE FOE CREEP IN THE UPPER MANTLE ABSTRACT  80  INTRODUCTION  81  THEORY  82  INTRODUCTION  82  RELATION BETWEEN CREEP AND DIFFUSION  82  RELATION BETWEEKF DIFFUSION COEFFICIENT AND  85  MELTING  TEMPERATURE  RESULTS OF DIFFUSION EXPERIMENTS  86  CALCULATIONS AND DISCUSSION  87  CONCLUSIONS  97  ACKNOWLEDGEMENTS  98  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 l g 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 Lnftversus cationic mole fraction (900- 32 1100°C) FIGURE 7 Dependence of Lnftupon crystallographic orientation  36  FIGURE 8A Ln D versus 1/T (°K ) at selected compositions  38  _1  FIGURE 8B Ln D versus 1/T (°K ) as a function of crystallographic 4.0 _1  orientation FIGURE 9 Ln Dgversus composition  42  FIGURE 10 AH* versus concentration  44  FIGURE 11 Ln D versus concentration as a bunction of pressure  47  FIGURE 12 Lnftversus pressure at selected concentrations  49  1  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 a x i s v 5 9 (1200-1300°C) FIGURE 17 Ln D versus composition for diffusion parallel to c axis 6 l (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(°K ) at selected compositions ri  6  7  (1200-1400°C) FIGURE 21 Ln D versus 1/T( K ) as a function of crystallographic 70 0  _1  orientation (1200-1400°C) FIGURE 22 Experimental determinations of Log o  versus 1/T( K ) 0  m  72  _1  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( K ) at selected compositions (900-1100°C): ?88 0  _1  FIGURE 3 Ln D versus pressure at selected compositions  90  FIGURE 4 Ln D versus 1/T(°K ) as a function of crystal lographic  92•..  _1  orientation (1200-1400°C)  LIST OF TABLES: PAPER #2 94  TABLE 1 Calculated values of the pre-exponential constant (Di^aarid the activation enthalpy ( A H * ) , (900-1100°C)  95  TABLE 2 Calculated values for the pre-exponential constant (D ) and  r  0  the activation enthalpy ( A H * ) , (1200-1400°C) TABLE 3 Calculated values of g^and g  y  TABLE 4 Experimental determinations of the flow laws in dunites and lherzolites  197 1Q0  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 investigation.  Mr. C. G. Hadidiacos assisted with the electron  microprobe analysis and the s k i l l 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 temperature creep are a l l 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 f i r s t 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 solidmedia 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 f i r s t 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. a l , 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 i t 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 i s tested with the activation parameters in the present investigation.  determined  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 N where  ^  Kcal./mole  2  = 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. diffusion coefficient increases with increasing with temperature.  The inter-  Fe content and  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. a l , 1973).  The temperature  and pressure dependence of the electrical conductivity of single crystals of olivine has been determined up to 1300°C, (Duba et. a l , 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 (800 CrJ-&2u§itC) and possibly to o  one of ionic conduction at high temperature (Shankland, 1969; Duba, 1972).  (T-1200°C),  At depths greater than 100  kilometers, the dominant conduction process seems likely to be ionic i f 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=a exp (-E/RT)  -(1)  0  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. a l , 1973) and thus when comparison of diffusion and conductivity are made care must be taken to assure that the experimental samples have a similar F e  +++  /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 [1973). 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 -(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 Co Si04, Borchardt and Schmalzried, +  2  1972, (2) F e  ++  N i i n Fo Fa  2  + +  93  In Mg Si0 , Naughton and Fujikawa, 1959, (3)  7  4  olivine, Clark, 1970, (4) N i  Olivine, Clark, 1970, (5) Mg Stamm, 1932, (6) Mg  ++  ++  + +  in Red Sea  in Mg Si0 powder, Jander and 2  4  in Mg Si04 from electrical conductivity, 2  Pluschkell and Engell, 1968.  T (°C) 900  10,00 11,00 12,0013,001,400  11 AG = decrease in free energy of component 1  where  y2 1 u  =  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 N  where  N  j=lfix  A  = Avogadro s number 1  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 a l l the diffusion forces. V. = m. . 6 p i + S 1  where  N " 6X  m. . 6yj+m. 6yv  j=2  A  A  6x  -(6)  N^ 6x V  A  = velocity of component i m.. = mobility of species i due to chemical IJ  potential gradient 6 y j 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 6 y i 1  where  N  A  -(7)  6 X  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 1  -1  N. A  1  7 6x  -(8)  12 Assuming conditions of constant pressure and temperature, the chemical potential may be written in the form U  where  i  =  %i  + R T l n  ( j)  "(9)  a  a^ = activity of component i u . = chemical potential at a standard state 0 1  R  The activity is related to the mole fraction a. = y.N. i i i where  -(10)  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) fij ~~6x 1  1  -(11)  1  Substituting the activity coefficient in (11) J. = =.-N. m. RT 6 A 1  1  {ln(y.)+ln(N.)}  = N.kT {1 + 61n(y )}{6N.} 61n(N^) 5^*"  -(12)  1  Writing Fick's f i r s t law in the form J. = -D.5N. l  -(13)  1 T ~ l  6x and equating (12) and (13) D. = m.kT[l + 51n(y.)] 61n(N^) 1  -(14a)  1  similarly D. = m,.kT[l + 61n(y.)] * 6dm (N^) Y Using the Gibbs-Duhem equation in the form  -(14b)  J  61n(y.)= 61n(y.) 61n(N*) 61n(N^)  -(15)  and substituting in (14) D. = m.  TTl  ~1  D. m. 3 3 In an ideal solution  -(16)  13 and equations [14) and (15) reduce to D. = m.kT i i  = D.* l  D. = m.kT 3 3  = D.* 3  -(17)  D^* is the self-diffusion coefficient of the i th component and  is the intrinsic coefficient of the i th component.  The f i r s t 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  1  D. = D.* 3  3  [1 + 61n(y.)] 61n(N^)  -(18a)  [1 + 61n(y.)] 61n(N^)  -(18b)  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. = 1  where  -(20)  N*7V  i m 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 = V ( N J M  2  1  - N ^ / C f i N ^ f i x )  "(21)  The equation necessary for the evaluation of the interdiffusion coefficient is  14  D(N ) 2  = (N^N")V (N *) {[1-Y*]/^ Y 6 x * ^ l - Y m  2  +Y  6x}  -(22)  where D(N *) = interdiffusion coefficient evaluated at 2  a distance on the concentration profile x = x* V (N *) m 2  molar volume of cations at N =N * auxiliary variable = N * -N i n i t i a l 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 f i r s t  derivatives were calculated using a five point f i r s t derivative f i l t e r (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.  A l l 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 T i O ^ C r ^  (O'Keefe and Ribble,. 1973),  and in the oxide-spinel system MgO-MgAl 0 (Whitney and Stubican, 2  4  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 V obtained by Fisher and Medaris, (1969) and Yoder and m  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) = D gxp (-AH*/RT)  -(23)  0  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)=Dexp(-AH*/RT)exp(-PAV*/RT) 0  where  P AV*  -(24)  = ambient hydrostatic pressure = 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) l i s t s 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  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  42.62  29.13  40.83  41.70  100.17  99.05  100.42  100.35  (WT%)  SI0  2  TOTAL  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  SOURCE  Dr. T. J. Shankland, M.I.T. Cambridge, Mass. Grown by flame fusion process.  FAYALITE  67.30  Dr. D. R. Wones, U.S.G.S. Sample from Rockport, Mass.  Fol  9.05  Dr. D. Virgo, Geophysical Laboratory, Carnegie Institute of Wash., D.C.  Sample from St. John's  Island, Red Sea, Egypt. Fo2  7.18  Dr. T. Richards, G.S.C. Vancouver, B. C. Locality unknown.  TABLE IB. Sample localities and sources. FeO weight percent.  Fe content given in  2.0 The ring was then f i l l e d 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 d i s t i l l e d 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 a l l  within 3° of arc of their original orientation.  ONE ATMOSPHERE EXPERIMENTS Two types of diffusion experiments were performed.  In the  f i r s t , crystallographically orientedediscs were placed in contact and wrapped in an inert metal f o i l .  Experiments at temperatures  over 1000°C were wrapped in Pt f o i l and those at lower temperatures in AgyQPd3Q. The f o i l wrapped diffusion couple was tightly wrapped in Pt wire and placed in a s i l i c a 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 s i l i c a glass  tube as in the other type of experiment. Platinum wound furnaces were employed in a l l high temperature experiments.  P l a t i n u m - 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, i t 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 l i f e of the s i l i c a glass tubes and  allowed experiments of up to two weeks. Immediately upon extraction from  the s i l i c a 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. a l , 1968; al, 1971).  Bell et.  A correction of +2 Kb. (Bell et. a l , 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 a l l 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. a l , 1967); Fujisawa, 1968).  Akimoto and  Thompson and Kushiro, (1972) have shown that  when graphite capsules are used at high pressure the resulting P 2 is in the range of wustite-magnetite. 0  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 micrometers, 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  f!  Graphite  = Talc  X  Fired Pyrophyllite  Steel Plug  Ceramic  Rh ). 1Q  Boron Nitride  = Pyrophyllite  Crushable  ceramic  24  V Scale I,  x thermal gradient A  \| \l \ \ \ \  x  X  X X  X X  X  X X  X X  X  X  X X  X  TiTTT7T  D . S . II  iH  TM X  900 T  1000 C C )  \ \ \ l \  X  X  x  x X X  X  X X  X  X X  x  \ii  \  X  X  7  0  X  X  II  X  i>:|i|l!|l:iii  x X  x  X  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. a l , 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 a l l 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 a l l 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 microprobe.  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. flourescence effect,  H  H  The  continuous  is also plotted.  28  0  l 5 5 150 260, 2^0 Dis8a n c e in m i c r o m e t e r s  .2,9  FIGURE 5.  Mg concentration profiles for experiment No. 14,  parallel to the C axis  — £  parallel to the B axis —J|| 950°C and 235 Hr.  ^ — £ —  , and experiment No. , Both experiments at  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 (1100 C) o  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. o  r\,  -  Values of D as a function of 1/T ( K  1  ) , at selected con-  centrations, are plotted in Figs. (8a) and (8b).  Regression  analysis of Ln, Iii versus 1/T results in values of D and AH* Q  which are shown in Figs. (9) and (10).  A general Arrhenius  relationship was formulated using these results ft = [1. 55± . 25-1.12 (N ) ] xlO'xexp [49. 85±4. 5+9. 05 (N ) ] -(25') 2  2  2  RT where  N  2  = cationic mole fraction Mg (0.1-Mg-0.8)  In a l l 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 NUMBER  TIME (SEC.)  TEMPERATURE (°C)  4  1.192xl0  14  8.46xl0  15  CRYSTAL ORIENTATION  900±5  C-AXIS  5  950±5  C-AXIS  8.46xl0  5  950±5  B-AXIS  1  3.56xl0  5  1000±5  C-AXIS  6  1.14xl0  6  1000±5  C-AXIS  59  3.64xl0  5  1000±5  B-AXIS  40  8. 95xl0  5  1100±5  C-AXIS  6  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 to B axis, —WM  ^ Hj—  ^  , Experiment No. 59, parallel Both experiments at 1000°C.  36  i  .0  .1  i—i—i—i—i—r—i—i—r .2  3  4  .5  .6  .7  .8  .9  Mg cation mole fraction  37 - u  FIGURE 8A.  and 60 —  Ln D versus 1/T  —  (°K)  _1  x 10  4  , Mg cation mole percent.  for D calculated at  Error bars indicate one  standard deviation calculated using regression analysis.  '38  T(°C)  9,00  10,00  11,00  2L u  22  85  75  8!0 1/T  (°K" )x10 1  4  3.9  FIGURE 8B.  Ln ft versus 1/T  ( K ) x 10 . 0  , parallel to B axis  _1  4  Parallel to C axis  4:0  9p0  T  ^ 15,00  lljOO  23  . 2BI 291  3d 3 U  8! 5  8.0 „  . 1/T (°K" ) x 1 0 1  7.'5 4  FIGURE 9.  Ln ft versus Mg cation mole f r a c t i o n . Q  indicate one standard deviation.  Error bars  42,  43  FIGURE 10.  AH* versus Mg cation mole fraction.  indicate one standard deviation.  Error bars  ,44  .0  I  .2  1  .4  1  .6  1~  .8  Mg cation mole fraction  1.0  45  TABLE  3  EXPERIMENT  TIME  TEMPERATURE  PRESSURE  CRYSTAL  NUMBER  (SEC)  (°C)  (KB) ±  ORIENTATION  19-P  5.11xl0  5  S00 900  10  C-AXIS  16-P  6.95xl0  5  900 900  25  C-AXIS  ±  10°  3-P  3.24xl0 3 00(1000  i-p  3.42xio  6-P 21-P  5  10  2  C-AXIS  10  C-AXIS  3.14xl0 11001100  10  C-AXIS  4.84xl0 11001100  15  B-AXIS  SGB-FA)  l-548xl0 11001100 6  25  C-AXIS  14-P  1.186xl0 110G1100  25  C-AXIS  18-P  4.07xl0 11001100  35  C-AXIS  TABLE 3.  5  KGQLIOO  5  5  6  5  Lists of the high pressure diffusion experiments  using Forsterite-Fayalite and Fayalite-MgO powder diffusion couples.  Time in s e c , temperature in ° C , and pressure in Kb.  46  FIGURE 11. Ln D versus Mg cation mole fraction at various pressures. (10 Kb.) A  Experiment 40 (1 A  atm.)-J|—J/^—,  Experiment 1-P  , Experiment 14-P (25 Kb.) - £  Experiment 18-P (35 Kb.) - A — A l l  ,  experiments at 1100  48  FIGURE 12.  Ln D versus pressure calculated at 10, 30, and 60  Mg cation mole percent. 1100°C  -0—.  Experiments at  900°C—fJ-B",  at  4.9  22  I  0  1—'•  5  I  10  1  15  Pressure  1  20  1  25  (Kb.)  1  30  1  35  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, Lnftdecreases 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 (dN /dX) in 2  this portion of the profile.  A similar, but less pronounced  effect, i s 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 TIME NUMBER (SEC.)  TEMP. (°C)±5  CRYSTAL ORIENTATION  DIFFUSION COUPLE  68  7.25xl0  5  1200  A-AXIS  Fo(l)-MG0  69  7.25xl0  5  1200  C-AXIS  Fo(l)-MG0  29  1.470xl0  1250  A-AXIS  Fo(l)-MG0  30  1.470xl0  1250  B-AXIS  Fo(l)-MG0  31  1.470xl0  1250  C-AXIS  Fo(l)-MG0  50  8,.80xl0  5  1300  B-AXIS  Fo(l)-MG0  51  8.80xl0  5  1300  C-AXIS  Fo(l)-MG0  52  1.210xl0  6  1300  C-AXIS  SYNTHETICFo(2)  26  9.46xl0  5  1350  A-AXIS  Fo(2)-MG0  27  9.46xl0  1350  C-AXIS  Fo(2)-MG0  72  3.41xl0  1400  A-AXIS  Fo(2)-MG0  73  3.41xl0  1400  B-AXIS  Fo(2)-MG0  74  3.41xl0  1400  C-AXIS  Fo(2)-MG0  71  3.41xl0  1400  C-AXIS  Fo(2)-MG0  TABLE 4.  6  5  6  5  5  5  5  5  List of the 1 atm. Experiments (Nitrogen furnace) using  Forsterite-MgO powder and synthetic Forsterite-FOgg olivine diffusion couples.  Time in s e c , temperature in °C.  54  FIGURE 14.  Fe concentration profiles for Experiment 29,  parallel to A axis A — E x p e r i m e n t B axis  30, parallel to  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.  29, parallel to A a x i s - A — * m r - ;  Experiment  Experiment 30, parallel to B  a x i s - J H — H — > Experiment 31, parallel to C axis  57  .0  .02  I  .04  1  .06  1  .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) experiments parallel to C axis.  ~%  All  59  Fe cation mole fraction  60  FIGURE 17.  Ln D versus Fe cation mole fraction.  Experiment 27 (1350°C) - A — ^ k r  > Experiment 74 (1400°C)  A l l experiments parallel to C axis.  61  I  I  I  .0  .02  .04  I .06 -  Fe cation mole fraction  62 o ,  FIGURE 18.  Fe concentration profile for synthetic forsterite-  FoggFay olivine diffusion anneal. 336 hr.)  Experiment 52 (1300°C) 1 atm.,  63  Distance in micrometers  64  FIGURE 19. 52.  Ln D versus Fe cation mole fraction for Experiment  Error bar indicates one standard deviation.  65  66  FIGURE 20.  Ln ft versus 1/T (°K ) x 10 forftcalculated at  cation mole fraction.  _1  4  Calculations for synthetic forsterite-  Fog^Fay olivine diffusion couple, 5 " " ' — Fe cation mole fraction.  A  , and 7  >  Error bars indicate one standard deviation.  (  67  12)00  T (°C) 13|00  6:5  14i00  6!0  1/T (°fC )xlO* 1  68  to the crystal powder couple. Values of D  Q  and AH* determined  from the profiles in Fig. ( 2 0 ) are in general agreement with the extrapolation of the lower temperature values and are plotted on Figs. ( 9 ) and ( 1 0 ) . 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. ( 2 2 ) along with the previously determined values of electrical conductivity in olivine. the experiments between calculations.  1200  and  Interdiffusion data from  1400°C  were used in the  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 ) a  m  =  D(T,P)KT  T  -(26)  69  FIGURE 21. Ln D versus 1/T ("IC ) x 10 for diffusion parallel 1  to aaaxis  - j ^  , b  4  axis'JJ—JJ-, and c axis  All calculations for Fe cation mole fraction = 0.07.  70  T(°C)  12|OQ  13,00  14^0  24  25  C ax is.  B axis  26  27  AaxisX  28 7!0  6.5  1/T (°K" )X10 1  6.0 4  FIGURE 22. Experimental determinations of Log C°K"^) x 10 . 4  , results of  versus 1/T 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) F o and Maruyama (1971), (6) F°82 (1967) .  o  n  v  i  n  e  i  92>  6 olivine Kobayashi  Mizutani and Kanamori  '72  10,00 T (°p)  i  7.5  1/T  12,00 1 3 , 0 0 1 4 , 0 0  r  7.0  (°K" )x10 1  4  •73  where  =  a.  m  independent determination of conductivity in the mantle  D(T,P) =  calculated value of the interdiffusion coefficient at T and P  T  =  absolute temperature  (°K)  substituting equation (24) in equation (26) °m  =  ( l ) o exp(-AH*/RT)exp(-PAV*/RT) -(27) K  D  thus Lna  = Ln(K )+Ln(D )-l_ 0_(AH*+PAV*) T RT  m  L  o  1  -(28)  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 |a (calculated) -a:l|<0.05l|a:i| m  m  m  -(29)  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. McDonald (1957).  Assuming the conductivity profile of  The two profiles calculated for Fog^Fag olivine;  the lower one using McDonald's estimate of a  m  the upper one using  the Eckhardt et. al estimate, bracket the geotherm of Ringwood and  Clark  774  FIGURE 23.  Temperature versus depth in the mantle.  geotherm, Ringwood and Clark, —  —.  Oceanic  Forsterite  stability f i e l d , 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 —^  ^—  , Fo  gi  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^  diffusion parallel to c axis.  A - . A l l calculations for  75  liquid  20QQ  15QO  U  o  'oceanic ^geotherm  1000  olivine/  spinel  5Qd  0  60  120  180  Depth  240  300  ( km)  360  76 and C l a r k (1966) below 150 Km.  Two two c a l c u l a t i o n s assuming  a AV* o f 5.50cm /mole y i e l d temperatures  a p p r o x i m a t e l y 450°C  h i g h e r a t a depth o f 300 Km. U n c e r t a i n t i e s s u r r o u n d i n g t h e importance o f t h e p r e s s u r e e f f e c t on t h e c o n d u c t i v i t y (Duba, 1972; Hughes, 1955), and u n c e r t a i n t i e s i n the c a l c u l a t e d a  ffl  p r o f i l e s from geomagnetic  and m a g n e t o t e l l u r i c measurements p r e c l u d e d e f i n i t e c o n c l u s i o n s as t o t h e v a l i d i t y o f t h e p r e s s u r e - i n c l u d e d v e r s u s p r e s s u r e excluded p r o f i l e s ;  however, p r e l i m i n a r y comments may be made.  The AV* used i n these c a l c u l a t i o n s was f o r s i n g l e c r y s t a l s and f o r a p o l y c r y s t a l l i n e a g g r e g r a t e p r e s s u r e e f f e c t s have been obs e r v e d t o be l e s s ( S c h u l t and Schober, B r a d l e y e t . a l , 1964).  1969; H a m i l t o n , 1965;  The p o l y c r y s t a l l i n e p r e s s u r e e f f e c t s may  be a c l o s e r a p p r o x i m a t i o n t o t h e mantle and thus t h e p r o f i l e s c a l c u l a t e d i n t h i s s t u d y u s i n g AV* = 5.50cm /mole may be t o o high.  The m a g n e t o t e l l u r i c s t u d i e s o f the e l e c t r o m a g n e t i c f i e l d  may d e f i n e a  m  v e r s u s depth i n t h e upper 400 Km. o f t h e mantle  more a c c u r a t e l y as they a r e r e c o r d i n g s h o r t e r p e r i o d v i b r a t i o n s thus o b t a i n i n g b e t t e r r e s o l u t i o n a t s h a l l o w depths. reasons, p r o f i l e  (A)  i n Figl  F o r these  (23) was assumed t o r e p r e s e n t t h e  b e s t f i t c a l c u l a t i o n based on t h e d a t a from t h i s s t u d y .  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±0.25-1.12N ) x 10" x exp (49.83±4.5+9.05N ) 2  2  9  RT  2)  where  900 - T - 1100,  and  N  2  0.10 - N  2  - 0.60  = cation mole fraction Mg  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 controlling 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. 1970;  (Goetze and Brace, 1972; Carter and Ave'Lallement,  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 T is the melting temperature at given pressure. m  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(a ) exp (-Q/RT) n  where  e  strain rate  a a  differential stresso(03-0^)  Q  activation energy for creep  -(1)  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. i t is 5. Assuming a relationship between diffusion and creep rate we may write: e = F(a ) D  -(2)  s  T)P  where D-p p = controlling diffusion coefficient T = absolute temperature, T>.5T  m  a = differential stress (in the range of tens to hundreds of bars) Experimental determinations of diffusion coefficents have shown I  T,P  D  where  Substituting  =  D  o P C-AH*/RT) exp (-PAV*/RT) -(3)" ex  D  = pre-expontential constant (10 tolO^)  AH*  = activation enthalpy for diffusion  AV*  = activation volume for diffusion  T  = absolute temperature  R  = gas constant  Q  (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. mantle.  Tectonic stresses are assumed values for the upper  84  u  c/) CO  O  experimental  limit  i  id  ui  tectonic stress  12  13  T  r  I  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) = D where  T  G  exp (-gT /T)  _( )  m  5  = melting temperature  m  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 where  and  T  =  S  =  q  g  v  g  q  = AH*/RT  g  v  = AV*7R(dT /dP)  mj0  m  -(6a) -(6b) -(6c)  = melting temperature at 1  m o  atmosphere dT /dP = pressure derivative of the melting m  temperature These relationships imply a simple relation between the activation rate parameters and the volume and enthalpy changes of melting.  In particular AH* W V *  = H^. T V m  m  If equations (6c), (6) and (5) are obeyed then gq/g =1-0 v  -07)  Equation (6c) also implies the validity of a parameter o f P* proposed by C.ietzc and -Brace (.1.372) ,• where-  86 •. t  P* proposed by Goetze and Brace (1972), where  where  p*  = p* = p*  P*  =  M  P  =  _ C8")  AH*/AV*  VM V  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/g  y  a n  d 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 c r i t i c a l 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 A H * , A H * and thence g /g , and P*. q v  The results are  summarized here in Figs. (2) to (4). Tables (1) and (2) present the results in numerical form appropriate to equation (6).  87  FIGURE 2  50  — ^ —  Ln D versus 1/T (°K. ) x 10  , and 60  — Q  for D calculated at  , Mg cation mole percent.  Error bars indicate one standard deviation calculated using regression analysis.  1  89  FIGURE 3  Ln ft versus pressure.  — A — c axis, 1000°C  Parallel to c axis, 1100°C  , parallel to b axis — | —  ^J-,  parallel to  , parallel to c axis, 900°C  Data calculated at 30 cation mole percent Mg.  90  ^  2  3  29 i 0  i  i  i  i  i  i  5  10  15  20  25  30  P r e s s u r e  (kb.)  r 35  91  FIGURE 4. to a axis  Ln D versus 1/T  (°K ) x 10  ;4fc— , b axis  1  4  for diffusion parallel , and c axis  All calculations for Fe cation mole fraction =  0.07.  92  T(°C) 12,00  13,00  14)00  24J  25  C a x is.  B axis  26  27  Aaxis^  28 7!0  6.5  6.0  1/T ( ° K " ) x 1 0 1  4  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 interdiffusion in the forsterite-fayalite system. in D and AH* are one standard deviation, o  The uncertainties  94  TABLE 1 CATIONIC MOLE FRACTION MG  TEMPERATURE RANGE (°C)  D + :&) (cV/sec)  AH* ± .<$AH* (Kcal/mole)  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 MOLE FRACTION MG  TEMPERATURE (°C)  PRESSURE RANGE  AV* ± "S"AV* cm m o l e  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  3  - 1  95  TABLE 2  DIFFUSION COUPLE  TEMPERATURE RANGE (°C)  D ± /D (xl0 ) (cm /Sec.)  AH* ± /AH* (Kcal/Mole)  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  MGO POWDER FORSTERITE CRYSTAL (100) AXIS 1200-1400  4.33 ±0.35  Q  D  +2  2  76.77 ±1.31  70.11 ±3.79  TABLE 2 Calculated values for the pre-expontential constant ( D ) and the activation enthalpy (AH*) as a function of temperature Q  and crystallographic orientation in the system MgO powderforsterite olivine single crystals. Values calculated at 0.07 cation mole fraction Fe.  '96 CALCULATIONS AND DISCUSSION  Values of g /g q  m a y  y  calculated from equation (6) as  D e  follows g /g = AH*(dT /dP)/AV*Tmo q  v  '  M  - ) (9  Results of these calculations are presented in table (3). The values of g /g , last column table (3) q v g /g  v  = 1-30 ± 0.5  (F°  g /g  y  = 1.16 ± 0.5  (Fo Fa )  g  q  Fa 10  93  9o) 7  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 COMPOSITION  AH Kcal mol"  AV _ cm mol  gO  Foj.^  50.8±4.0  5.5±1.0  17.±1.7  65.56±4.0  5.5±1.0  1 6 . U 1 . 6  F  A  q  gV  g g  V  q  13.±4.0  V  1.3±.5  9 0  Fo IFa  g 3  133934400  1.16±.5  ?  TABLE  3.  C a l c u l a t e d v a l u e s o f g^ and g^ a t  Fa., o l i v i n e  compositions.  FO^Q  F a  go  anc  *  F  ° 9 3  98 eratures ranged from 0.64T - T - 0.79T, • for the Fo^Fa., M M 93 7 calculations, 0.68T < T < 0.79T„. The values for the preM M W  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*(Fo Fa ) = 350 ± 100 Kb. (0-35 Kb.) 10  9()  P*(Fo Fa ) = 450 ± 100 Kb. (0-35 Kb.) 93  ?  Using the data of Davis and England (1964) for the melting of forsterite  P*(Eo J m 100 in  = 460 ± 10 Kb.  (0-50 Kb.)  The data of Akimoto et. al (1967) show a change in slope of (dT /dP) for fayalite at 1 atmosphere 200 ± 10 Kb. Pm^W dT where  m  /dP  (P = 1 atm.)  7.5°C/Kb.  At 50 Kb.  ^ loo>  430 ± 10 Kb.  dT /dP m  3.5°C/Kb.  Fa  (P = 50 Kb.)  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, i t appears that the melting parameters are related to the faster cation diffusion rates.  99 The validity of equations (2) and (4) depends on using the diffusion coefficient for the rate-limiting atom in the structure.  Table (4) l i s t s 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. Preliminary results, Goetze and Brace (1972), at low strainsrates (e = 10 ) and low stresses (a = 450 bars) indicate that the S  discrepancy is s t i l l present at small strain rates.  The  results from experiments with water present indicate a decrease in Q for creep;  however, i t is s t i l l 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 i s proportional to a  S  [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 dislocation climb is calculated from the data of Goetze and Kohlstedt (1973) to be D  d-C  * = 7.0 x 10"  14  cm /sec. 2  100  TABLE 4  MATERIAL  CONFINING PRESSURE (KB) (KB) 15.0]  A (SEC )  Q (KCAL/MOLE)  6.2 x 10  7979.9±7.5  Dunite 1 (dry)  15.0  1.2 x 1 0  Lherzolite 1 (wet)  15.0  3.2 x 10  Lherzolite 2 (dry)  15.0  1.0 x 10  Dunite 1 (wet)  N  2.4 ± 0.2  119.8±16.6  4.8 ± 0.4  7  79.8±9.4  2.3 ± 0.2  8  106.0±20.  10  5.0  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)a  n  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  cm /sec. 2  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  where  D  = D exp (-gT /T)  g  = 18.0  q  m  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 i n 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 approximately 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 and D,  pressure  i n a binary  may  be  described  the i n t e r d i f f u s i o n  dimensional components related  2,  system  at constant  i n terms o f a s i n g l e  coefficient.  diffusion 1 and  solid  parameter  I n t h e c a s e o f one-  the average v e l o c i t i e s  i n an a r b i t r a r y  to the molar fluxes  temperature  v^  co-ordinate  and  and v ^ o f system  by  v i VV ; W^VV J  where  V  fractions In  o f components  to use  regardless  V  -^  _ V  2  a  s  t  invariant  v^~^2  N^.  the d e f i n i t i o n  Thus  a  o f the choice ^  and N  1 and  systems i n v o l v i n g  ropriate  -ci*  / (  i s t h e m o l a r volume  m  and N„ a r e t h e m o l a r  n  1  2  2.  change  i n volume,  quantity w i l l  s  of co-ordinate P  e  i -  n  r  o  P  o  r  of  are  ' i t  o  n  a  l  t  0  i t i s app-  be  invariant  system. t  n  The  local  e  velocity  gradient i n  the i n t e r d i f f u s i o n  coefficient  becomes D  = N  N 1  Consider a n d N*.  Then  is  defined  by t h e  2  (v  - v  dN^/dx  a diffusion  N_  2  2  single  ^  +oo d  K  +  with  function  X = x  x  ;  -(2:).)  J J  2  m  initial  moi  fractions  o f t h e Matano  interface  +  xm 2- 2 N  N  V  t o Matano-Bolzmann valued  - N  ki /&  equation  m  a  2  2  couple  V  According  m  the co-ordinate  xm r  ) = V ( W J  2  m  the composition  of the a u x i l i a r y  o f t h e sample i s  variable  xiji  _ (4)  tl/2 The in  moi  rate  o f change  per unit  volume  i n t h e c o n c e n t r a t i o n o f each at a given  value  of  x  _  x m  component  i s equal  to the  103  negative divergence of the r e s p e c t i v e f l u x i n a co-ordinate system f i x e d w i t h r e s p e c t t o t h e Matano jL. ( N J St  = §_ (1-N )  V  m x=x  St V  CNo)  §-  «t  V m x =x  =  = -6J,  x,x  m  m  interface  v  J  m  m  -(5b)  «J?  §(* - x ) m  yv m  i n a c c o r d w i t h Sauer and F r e i s e (1962) the a u x i l i a r y Y  -(5a)  <5(x-x )  =  variable  N -N^  -(6)  9  i s i n t r o d u c e d as a measure o f c o m p o s i t i o n whereupon N  x  = (1-N*)Y + (1-N~) (1-Y)  -(7a)  N  2  = N*Y + N~ (1-Y)  -(7b)  S u b s t i t u t i n g equations  ( 4 ) , ( 5 a ) , and (5b) i n e q u a t i o n (7a)  and (7b) y i e l d s X_[(1-N^)6_(Y_) + ( 1 - N J 6 _(1-Y)] = S J 2t SA Vm SA Vm Sx  _(8a)  >L_fN^_(Y_)+N"S_ (1-Y)] 2t 6A Vm 6A Vm  -(8b)  1  =  SJ  C  M u l t i p l y i n g t h r o u g h e q u a t i o n (8a) by N and e q u a t i o n (8b) by ( l - N ^ ) and s u b t r a c t i n g c o r r e s p o n d i n g s i d e s , one has - A _ CN*-N:)6_CY_)=N:- S J . - C l - N J f i J 2t SA Vm Sx Sx 1  _(9a)  z  M u l t i p l y i n g t h r o u g h e q u a t i o n (8a) by N* and e q u a t i o n (8b) by (1 - N*) and s u b t r a c t i n g c o r r e s p o n d i n g s i d e s one has A _ (N^-N:)6_ (1-Y)=N*6J - ( 1 - N * ) 6 J 2t 6'A Vm Sx 6x 1  z  -(9b) z  E q u a t i o n (9a) i s m u l t i p l i e d t h r o u g h by dA=d* and i n t e g r a t e d tl/2 a t c o n s t a n t time between A=-°° and t h e p a r t i c u l a r v a l u e A-x* w i t h  * the mole f r a c t i o n N .  L i k e w i s e e q u a t i o n (9b) i s i n t e g r a t e d  * between A=x  and A=+«>.  A t l a r g e a b s o l u t e v a l u e s o f A , the  change i n c o m p o s i t i o n i s v e r y s m a l l .  Thus t h e l o c a l  inter-  104  diffusion coefficients are nearly constant and, therefore, values of N and Y as functions of X can be expressed in terms 2  of the error function. Consequently, at \=t°° i.e. x = -°° Y and 1 - Y tend quasi-exponentially to zero. gradient dN^/dX vanishes.  Further the  Therefore no movement of component 1  relative to component 2 occurs, i.e. v^-V2=0. thus  J /N - J /N = 0 or J N  hence  1_(N*-N~D [-A_V 2t Vm  2  1  +  —  Z  2  + /*YdA_] = 1_- [N~J* - ( l - N " ) J j -(10a) "°° Vm tl/2  A  *  *  -J N = 0 1  + 0 0  +  1  Z  *  +  Z  *  1_(N -N )[-A (1-Y ) - f ^ ( l - Y ) d X ] = l _ [ - N J ( l - N ) J ] 2t Vm* ' Vm ttjf.2 2  where  2  2  1+  2  -(10b)  2  and J are fluxes at X = X . Multiplying through *  *  equation (10a) by (1-Y ) and equation (10b) by Y and subtracting corresponding sides !_(N -N ) [1-Y ?  YdA + Y / Vm  ?  2t  (l-Y)dX] = _1_ [N J -(1-N )J ] -(11) Vm £i/2 Z  *  Substituting equation (2) for N =N in equation (11), letting 2  2  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 2t(dN /dx-  Vm  2  * where x i  s  -(12)  x Vm  * the distance at which N = N,,. 2  Since equation (12) involves only the value of dx i t is possible to use distance from an arbitrary plane of reference in order to calculate dN /dx and integrals in equation (12). There 2  is thus no need to determine explicitly the Matano interface. With the help of equation (12) one may obtain values forftas a function of N for the whole range of composition from a single 2  diffusion run.  105  BIBLIOGRAPHY  Akimoto, S., and H. Fujisawa, Olivine-spinel solid solution equilibria in the system Mg2Si0^-Fe2Si0^, J. Geophys. 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