Contributions to Mineralogy and Petrology

, Volume 111, Issue 1, pp 74–86 | Cite as

Cation diffusion in aluminosilicate garnets: experimental determination in spessartine-almandine diffusion couples, evaluation of effective binary diffusion coefficients, and applications

  • Sumit Chakraborty
  • Jibamitra Ganguly


We present new experimental data on diffusion of divalent cations in almandine-spessartine diffusion couples in graphite capsules in the P-T range of 14–35 kb, 1100–1200° C. The tracer diffusion coefficients of the major divalent cations, viz. Fe, Mg and Mn, retrieved from the multicomponent diffusion profiles, have been combined with earlier data from our laboratory at 29–43 kb, 1300–1480° C (Loomis et al. 1985) to derive expressions of the P-T dependence of the diffusion coefficients at fO2 approximately corresponding to that defined by equilibrium in the system graphite-O2. We review the conditions, discussed earlier by Cooper, under which the flux of a component in a multicomponent system becomes proportional to its concentration gradient (Fickian diffusion), as if the entire solvent matrix behaves as a single component, and also suggest a method of incorporating the thermodynamic effect on diffusion in the same spirit. Regardless of the magnitude or sign of the off-diagonal terms of the D matrix, it is always possible to define an effective binary diffusion coefficient (EBDC) of a component in a semi-infinite multicomponent diffusion-couple experiment such that it has the property of the Fickian diffusion coefficient, provided that there is no inflection on the diffusion profiles. It is shown that the success of Elphick et al. in fitting the experimental diffusion profiles of all components over a limited concentration range by a single diffusion coefficient is due to fortuitous similarity of the EBDCs of the components (Fe, Mg, Mn and Ca) in their diffusion couple experiments. In common metapelitic garnets showing compositional zoning, the EBDCs of the divalent cations do not differ from each other by more than a factor of 2.5. However, the EBDC of a component changes from core to rim by a factor of 3 to 12, depending on the composition. We suggest a method of volume averaging of the EBDC which should prove useful in approximate calculations of diffusion flux during relaxation of compositional zoning. The EBDC of Mn is found to reduce essentially to DMnMn, the main diagonal term of the D matrix, and consequently can be calculated quite easily. Evaluation of EBDC of Fe, Mg and Mn in garnets from a prograde Barrovian sequence did not reveal any significant dependence on the extent of relaxation of garnet. The diffusion data have been applied to calculate the cooling rate of natural biotite-garnet diffusion couple from eastern Finland and diffusional modification of growth zoning in garnet in early Proterozoic Wopmay orogen, Canada. The results are in good agreement with geochronological and other independent constraints.


Orogen Divalent Cation Diffusion Couple Diffusion Profile Tracer Diffusion 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Symbols and abbreviations


Radius of a spherical garnet crystal


Back-scattered electron imaging


Column vector of (n-1) independent components


Diffusion coefficient matrix


An element of the diffusion matrix on the i th row and j th column


Tracer diffusion coefficient of component i


Effective interdiffusion coefficient (EIC) of various components in a multicomponent solution rich in the component i


Interdiffusion coefficient of components i and j in a binary solution


Effective binary diffusion coefficient of component i in a multicomponent solution


D i (EB) under condition of ideal thermodynamic mixing of the diffusing species


Thermodynamic component of Di(EB)


Interdiffusion coefficient at peak temperature T0 in the phase A


Pre-exponential factor in an Arrhenius relation


Effective binary diffusion coefficient between a solute and a multicomponent solvent matrix


Fixed edge composition model


Effective interdiffusion coefficient


Fugacity of component i


Hematite-magnetite oxygen fugacity buffer






Activation energy (enthalpy) of diffusion


Extent of relaxation defined as the difference between core and rim compositions normalized to the same difference in the initial zoning profile


Gas constant


Cooling rate

T0, TCh

Peak temperature and characteristic temperature, respectively




Variable edge composition model


Activation volume


Simple mixture interaction parameter between i and j


Effective simple mixture interaction parameter of a component i in a multicomponent solution


Margules interaction parameter between i and j


Mole fraction of component i


Activity coefficient of component i


A dimensionless variable =πD t/a2


Kronecker delta (i=j, δ ij =1; i≠j, δ ij =0)


Charge on the ion i


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Akella J, Vaidya SN, Kennedy GC (1969) Melting of sodium chloride at pressures to 65 kbar. Physical Rev 185:1135–1140Google Scholar
  2. Anovitz LMA, Chase CG (1990) Implications of post-thrusting extension and underplating for P-T-t paths in granulite terranes: a Grenville example. Geology 18:466–469Google Scholar
  3. Barrer RM, Bartholomew RF, Rees LVC (1963) Ion exchange in porous crystals. Part II. The relationship between self and exchange-diffusion coefficients. J Phys Chem Solids 24:309–317Google Scholar
  4. Boettcher AL, Windom KE, Bohlen SR, Luth RW (1981) Low friction, anhydrous, low-to high-temperature furnace assembly for piston cylinder apparatus. Rev Sci Instrum 52:1903–1904Google Scholar
  5. Brady JB (1975) Reference frames and diffusion coefficients. Am J Sci 275:954–983Google Scholar
  6. Chakraborty S, Ganguly J (1990) Compositional zoning and cation diffusion in aluminosilicate garnets. In: Ganguly J (ed) Diffusion, atomic ordering and mass transfer, advances in physical geochemistry, Vol 8. Springer, Berlin Heidelberg New York Tokyo, pp 120–175Google Scholar
  7. Clark SP Jr (1959) Effect of presure on the melting point of eight alkali halides. J Chem Phys 31:1526–1531Google Scholar
  8. Cohen LH, Klement W Jr, Kennedy GC (1966) Investigation of phase transitions at elevated temperatures and pressures by differential thermal analysis in pistoncylinder apparatus. J Phys Chem Solids 27:179–186Google Scholar
  9. Cooper AR (1968) The use and limitation of the concept of an effective binary diffusion coefficient for multicomponent diffusion, in Mass Transport in Oxides-Proc Symposium, US Dept of Commerce, NBS Sp Pub No 296Google Scholar
  10. Crank J (1975) The mathematics of diffusion, Oxford London, 414 ppGoogle Scholar
  11. Cullinan HT Jr (1965) Analysis of the flux equations of multicomponent diffusion. Ind Eng Chem Fundament 4:133–139Google Scholar
  12. Cygan RT, Lasaga AC (1985) Self diffusion of magnesium in garnet at 750° C to 900° C. Am J Sci 285:328–350Google Scholar
  13. Darken LS (1948) Diffusion, mobility and their interrelation through free energy in binary metallic systems. Am Inst Mining Metall Engineers Trans 175:184–201Google Scholar
  14. Dempster TJ (1985) Garnet zoning and metamorphism of the Barrovian type area, Scotland. Contrib Mineral Petrol 89:30–38Google Scholar
  15. Elphick SC, Ganguly J, Loomis TP (1981) Experimental study of Fe−Mg interdiffusion in aluminosilicate garnet. EOS (abstract). Tran Am Geophys Union 62:411Google Scholar
  16. Elphick SC, Ganguly J, Loomis TP (1985) Experimental determination of cation diffusivities in aluminosilicate garnets I. Experimental methods and interdiffusion data. Contrib Mineral Petrol 90:36–44Google Scholar
  17. England PC, Thompson AB (1984) Pressure-temperature-time paths of metamorphism I. Heat transfer during the evolution of regions of thickened continental crust. J Petrol 25:894–928Google Scholar
  18. Ferry JM, Spear FS (1978) Experimental calibration of the partitioning of Fe and Mg between biotite and garnet. Contrib Mineral Petrol 66:113–117Google Scholar
  19. Florence FP, Spear FS (1991) Effects of diffusional modification of garnet growth zoning on P-T path calculations. Contrib Mineral Petrol 107:487–500Google Scholar
  20. Freer R (1981) Diffusion in silicate minerals and glasses: a data digest and guide to literature. Contrib Mineral Petrol 76:440–454Google Scholar
  21. Ganguly J, Kennedy GC (1974) The energetics of natural garnet solid solution. I. Mixing of the aluminosilicate end-members. Contrib Mineral Petrol 48:137–148Google Scholar
  22. Ganguly J, Bhattacharya RN, Chakraborty S (1988) Convolution effect in the determination of compositional profiles and diffusion coefficients by microprobe step scans. Am Mineral 73:901–909Google Scholar
  23. Ganguly J, Saxena SK (1984) Mixing properties of aluminosilicate garnets: Constraints from natural and experimental data, and applications to geothermobarometry. Am Mineral 69:88–97Google Scholar
  24. Ganguly J, Ruiz J (1987) Time-temperature relation of mineral isochrons: a thermodynamic model, and illustrative applications for the Rb−Sr system. Earth Planet Sci Let 81:338–348Google Scholar
  25. Ganguly J, Saxena SK (1987) Mixtures and mineral reactions. Springer, Berlin Heidelberg New York TokyoGoogle Scholar
  26. Ganguly J, Chakraborty S, Rumble D III (1991) Constraint on the time scale of regional metamorphism from diffusion profiles in a natural garnet-garnet couple from Vermont. EOS (abstract) Am Geophys Union, Fall meetingGoogle Scholar
  27. Geiger CA, Newton RC, Kleppa OJ (1987) Enthalpy of mixing of synthetic almandine-grossular and almandine-pyrope garnets from high temperature solution calorimetry. Geochim Cosmochim Acta 51:1755–1763Google Scholar
  28. Guggenheim EA (1967) Thermodynamics. Elsevier, North Holland Amsterdam New YorkGoogle Scholar
  29. Hackler RT, Wood BJ (1989) Experimental determination of Fe and Mg exchange between garnet and olivine and estimation of Fe−Mg garnet mixing properties. Am Mineral 74:994–999Google Scholar
  30. Haselton HT, Newton RC (1980) Thermodynamics of pyrope-grossular garnets and their stabilities at high temperatures and pressures. J Geophys Res 85:6973–6982Google Scholar
  31. Hodges KV, Spear FS (1982) Geothermometry, geobarometry and the Al2SiO5 triple point at Mt. Moosilauke, New Hampshire. Am Mineral 67:1118–1134Google Scholar
  32. Hollister LS (1969) Contact metamorphism in the Kwoiek Area of British Columbia: An end member of the metamorphic process; Geol Soc Am Bull 80:2465–2494Google Scholar
  33. Jiang J, Lasaga AC (1990) The effect of post-growth thermal events on growth-zoned garnet: Implications for metamorphic P-T history calculations. Contrib Mineral Petrol 105:454–459Google Scholar
  34. Koziol AM (1990) Activity-composition relationships of binary Ca−Fe and Ca−Mn garnets determined by reversed, displaced equilibrium experiments. Am Mineral 75:319–327Google Scholar
  35. Koziol AM, Newton RC (1989) Grossular activity-composition relationships in ternary garnets determined by reversed displaced equilibrium experiments. Contrib Mineral Petrol 103:423–433Google Scholar
  36. Lasaga AC (1979) Multicomponent exchange and diffusion in silicates. Geochim Cosmochim Acta 43:455–469Google Scholar
  37. Lasaga AC (1983) Geospeedometry: An extension of geothermometry. In: Saxena SK (ed) Kinetics and equilibrium in mineral reactions, advances in physical geochemistry, Vol 3. Springer, Berlin Heidelberg New York Tokyo, pp 81–114Google Scholar
  38. Lasaga AC, Richardson SM, Holland HD (1977) The mathematics of cation diffusion and exchange between silicate minerals during retrograde metamorphism. In: Saxena SK, Bhattacharji SD (ed) Energetics of geologic processes. Springer, BAR, Berlin Heidelberg New York Tokyo, pp 353–388Google Scholar
  39. Lindstrom R, Vitanen M, Juhanoja J, Holtta P (1991) Geospeedometry of metamorphic rocks: examples in the Rantasalmi-Sulkava and Kiuruvesi areas, eastern Finland. Biotite-garnet diffusion couples. J Metamorphic Geol 9:181–190Google Scholar
  40. Loomis TP (1986) Metamorphism in metapelites: calculation of equilibrium assemblages and numerical simulations of the crystallization of garnet. J Met Geol 4:201–230Google Scholar
  41. Loomis TP, Ganguly J, Elphick SC (1985) Experimental determination of cation diffusivities in aluminosilicate garnets II. Multicomponent simulation and tracer diffusion coefficients. Contrib Mineral Petrol 90:45–51Google Scholar
  42. Luth R, Virgo D, Boyd FR, Wood BJ (1991) Ferric iron in mantle derived garnets: implications for thermobarometry and for the oxidation state of the mantle. Contrib Mineral Petrol 104:56–72Google Scholar
  43. Manning JR (1968) Diffusion kinetics for atoms in crystals. Van Nostrand, Princeton, USAGoogle Scholar
  44. McLellan E (1985) Metamorphic reactions in the kyanite and sillimanite zones of the Barrovian type area. J Petrol 26:789–818Google Scholar
  45. Morioka M (1981) Cation diffusion in olivine-II. Ni−Mg, Mn−Mg, Mg and Ca. Geochim Cosmochim Acta 45:1573–1580Google Scholar
  46. Morioka M, Nagasawa H (1990) Ionic diffusion in olivine. In: Ganguly J (ed) Diffusion, atomic ordering and mass transport, advances in physical geochemistry, Vol 8. Springer, Berlin, Heidelberg, New York, Tokyo, pp 176–197Google Scholar
  47. Muncill GE, Chamberlain CP (1988) Crustal cooling rates inferred from homogenization of metamorphic garnets. Earth Planet Sci Lett 87:390–396Google Scholar
  48. Onsager L (1945) Theories and problems of liquid diffusion. New York Acad Sci Ann 46:241–265Google Scholar
  49. Perchuk LL, Lavrent'eva IV (1983) Experimental investigation of exchange equilibria in the system cordierite-garnet-biotite. In: Saxena SK (ed) Kinetics and equilibrium in mineral reactions, advances in physical geochemistry Vol 3. Springer, Berlin Heidelberg New York Tokyo, pp 199–239Google Scholar
  50. Powenceby MI, Wall VJ, O'Neill HStC (1987) Fe−Mn partitioning between garnet and ilmenite: experimental calibration and applications. Contrib Mineral Petrol 97:116–126Google Scholar
  51. Press WH, Flannery BP, Teukolsky SA, Vetterling WT (1986) Numerical recipes — The art of scientific computing. Cambridge University Press, Cambridge, UKGoogle Scholar
  52. Robie RA, Hemingway BS, Fisher JR (1978) Thermodynamic properties of minerals and related substances at 298.15 K and 1 bar (105 pascals) pressure and at higher temperature. US Geol Survey Bull Vol 1452. Washington, USAGoogle Scholar
  53. Saxena SK, Fei Y (1987) Fluids at crustal pressures and temperatures I. Pure species. Contrib Mineral Petrol 95:370–375Google Scholar
  54. Shewmon PG (1963) Diffusion in solids. McGraw-Hill, New York, USAGoogle Scholar
  55. Smith D, Barron BR (1991) Pyroxene-garnet equilibration during cooling in the mantle. Am Mineral (in press)Google Scholar
  56. Spear FS, Silverstone J (1983) Quantitative P-Tpaths from zoned minerals: theory and tectonic applications. Contrib Mineral Petrol 83:348–357Google Scholar
  57. St-Onge MR (1987) Zoned poikiloblastic garnets: P-Tpaths and synmetamorphic uplift through 30 km of structural depth, Wopmay Orogen. Canada, J Petrol 28:1–21Google Scholar
  58. Tompson AFB, Gray WG (1986) A Second Order Approach for the modeling of Dispersive Transport in Porous Media I. Theoretical Development, Water Resources Res 22:591–599Google Scholar
  59. Toor HL (1964) Solution of the Linearized equations of Multicomponent mass transfer: II. Matrix Methods. J Am Inst Chem Eng 8:460–465Google Scholar
  60. Tracy RJ, Robinson P, Thompson AB (1976) Garnet composition and zoning in the determination of temperature and pressure of metamorphism, central Massachusetts. Am Mineral 61:762–775Google Scholar

Copyright information

© Springer-Verlag 1992

Authors and Affiliations

  • Sumit Chakraborty
    • 1
  • Jibamitra Ganguly
    • 2
  1. 1.Bayerisches GeoinstitutUniversität BayreuthBayreuthGermany
  2. 2.Department of GeosciencesUniversity of ArizonaTucsonUSA

Personalised recommendations