Advertisement

Contributions to Mineralogy and Petrology

, Volume 161, Issue 5, pp 683–702 | Cite as

Uphill diffusion, zero-flux planes and transient chemical solitary waves in garnet

  • D. VielzeufEmail author
  • A. Saúl
Original Paper

Abstract

Diffusion profiles in minerals are increasingly used to determine the duration of geological events. For this purpose, the distinction between growth and diffusion zoning is critical; it requires the understanding of complex features associated with multicomponent diffusion. Seed-overgrowth interdiffusion experiments carried out in the range 1,050–1,250°C at 1.3 GPa have been designed to quantify and better understand Fe–Mg–Ca interdiffusion in garnet. Some of the diffusion profiles measured by analytical transmission electron microscope show characteristic features of multicomponent diffusion such as uphill diffusion, chemical solitary waves, zero-flux planes and complex diffusion paths. We implemented three different methods to calculate the interdiffusion coefficients of the D matrix from the experimental penetration curves and determined that with Ca as the dependent component, the crossed coefficients of the D matrix are negative. Experiments and numerical simulations indicate that: (1) uphill diffusion in garnet can be observed indifferently on the three components Fe, Mg and Ca, (2) it takes the form of complementary depletion/repletion waves and (3) chemical waves occur preferentially on initially flat concentration profiles. Derived D matrices are used to simulate the fate of chemical waves in time, in finite crystals. These examples show that the flow of atoms in multicomponent systems is not necessarily unidirectional for all components; it can change both in space along the diffusion profile and in time. Moving zero-flux planes in finite crystals are transitory features that allow flux reversals of atoms in the diffusion zone. Interdiffusion coefficients of the D matrices are also analyzed in terms of eigenvalues and eigenvectors. This analysis and the experimental results show that depending on the composition of the diffusion couple, (1) the shape of chemical waves and diffusion paths changes; (2) the width of the diffusion zone for each component may or may not be identical; and (3) the width of diffusion calculated at a given D and duration may greatly vary. D matrices were retrieved from thirteen sets of diffusion profiles. Data were cast in Arrhenius relations. Linear regressions of the data yield activation energies equal to 368, 148, 394, 152 kJ mol−1 at 1 bar and frequency factors Do equal to 2.37 × 10−6, −4.46 × 10−16, −1.31 × 10−5, 9.85 × 10−15 m2 s−1 for \( \tilde{D}_{FeFe}^{Ca} \), \( \tilde{D}_{FeMg}^{Ca} \), \( \tilde{D}_{MgFe}^{Ca} \), \( \tilde{D}_{MgMg}^{Ca} \), respectively. These values can be used to calculate interdiffusion coefficients in Fe–Mg–Ca garnets and determine the duration of geological events in high temperature metamorphic or magmatic garnets.

Keywords

Garnet Multicomponent diffusion Fe–Mg–Ca interdiffusion Uphill diffusion Chemical solitary wave Zero-flux plane 

Notes

Acknowledgments

This work was supported by the French Centre National de la Recherche Scientifique—Institut National des Sciences de l’Univers through grants DyETI 2005 to D. V. We performed the ATEM analyses at the French Earth Science TEM facility (Lille and Marseille). We thank A. Addad for his supervision during the analytical sessions at Lille. Part of this work was completed while D. V. was at Caltech as part of a CNRS/Caltech exchange; E. M. Stolper is gratefully acknowledged for his recurrent support. We are grateful to H. Kawabata and H. Raimbourg for providing the analytical data shown in Fig. 11. This paper benefited from discussions with A. Lupulescu, J. P. Monchoux and F. Costa, comments of an early version of the manuscript by J. Ganguly and P. Wynblatt, and careful reviews and constructive comments by J. E. Morral and S. Chakraborty. We thank F. Poitrasson for efficient editorial handling. Matlab scripts developed in this study are available upon request.

References

  1. Bejina F, Jaoul O, Liebermann RC (2003) Diffusion in minerals at high pressure: a review. Phys Earth Planet Inter 139(1–2):3–20CrossRefGoogle Scholar
  2. Boltzmann L (1894) Zur integration der Diffusionsgleichung bei variabeln Diffusionscoefficienten. Annalen der Physik 53:959–961CrossRefGoogle Scholar
  3. Bouchet R, Mevrel R (2002) A numerical inverse method for calculating the interdiffusion coefficients along a diffusion path in ternary systems. Acta Mater 50(19):4887–4900CrossRefGoogle Scholar
  4. Carlson WD (2006) Rates of Fe, Mg, Mn, and Ca diffusion in garnet. Am Mineral 91(1):1–11CrossRefGoogle Scholar
  5. Chakraborty S, Ganguly J (1991) Compositional zoning and cation diffusion in garnets. In: Ganguly J (ed) Diffusion, atomic ordering and mass transport, vol 8. Springer, Berlin, pp 120–175Google Scholar
  6. Chakraborty S, Ganguly J (1992) Cation diffusion in aluminosilcate garnets—Experimental determinations in spessartine-almandine diffusion couples, evaluation of effective binary diffusion coefficients and applications. Contrib Mineral Petrol 111(1):74–86CrossRefGoogle Scholar
  7. Chakraborty S, Dingwell DB, Rubie DC (1995a) Multicomponent diffusion in ternary silicate melts in the system K2O-Al2O3-SiO2.1. Experimental measurements. Geochim Cosmochim Acta 59(2):255–264CrossRefGoogle Scholar
  8. Chakraborty S, Dingwell DB, Rubie DC (1995b) Multicomponent diffusion in ternary silicate melts in the system K2O-Al2O3-SiO2 2 Mechanisms, systematics, and geological applications. Geochim Cosmochim Acta 59(2):265–277CrossRefGoogle Scholar
  9. Clemens JD, Wall VJ (1988) Controls on the mineralogy of S-type volcanic and plutonic rocks. Lithos 21(1):53–66CrossRefGoogle Scholar
  10. Dachs E, Proyer A (2002) Constraints on the duration of high-pressure metamorphism in the Tauern Window from diffusion modelling of discontinuous growth zones in eclogite garnet. J Metamorph Geol 20(8):769–780CrossRefGoogle Scholar
  11. Darken LS (1948) Diffusion, mobility and their interrelation through free energy in binary metallic systems. Trans AIME (Am Inst Min Metall Eng) 175:184–201Google Scholar
  12. Darken LS (1949) Diffusion of carbon in austenite with a discontinuity in composition. Trans AIME (Am Inst Min Metall Eng) 180:430–438Google Scholar
  13. Day KM, Ram-Mohan LR, Dayananda MA (2005) Determination and assessment of ternary interdiffusion coefficients from individual diffusion couples. J Phase Equilib Diffus 26(6):579–590Google Scholar
  14. Dayananda MA, Kim CW (1979) Zero-flux planes and flux reversals in Cu-Ni-Zn diffusion couples. Metall Trans a-Phys Metall Mater Sci 10(9):1333–1339CrossRefGoogle Scholar
  15. Dayananda MA, Sohn YH (1999) A new analysis for the determination of ternary interdiffusion coefficients from a single diffusion couple. Metall Mater Trans a-Phys Metall Mater Sci 30(3):535–543CrossRefGoogle Scholar
  16. den Broeder FJA (1969) A general simplification and improvement of the Matano-Boltzmann method in the determination of the interdiffusion coefficients in binary systems. Scripta Metall 3:321–326CrossRefGoogle Scholar
  17. Emmanuel S, Cortis A, Berkowitz B (2004) Diffusion in multicomponent systems: a free energy approach. Chem Phys 302(1–3):21–30CrossRefGoogle Scholar
  18. Faryad SW, Chakraborty S (2005) Duration of Eo-Alpine metamorphic events obtained from multicomponent diffusion modeling of garnet: a case study from the Eastern Alps. Contrib Mineral Petrol 150(3):306–318CrossRefGoogle Scholar
  19. Ganguly J (2002) Diffusion kinetics in minerals: principles and applications to tectonic-metamorphic processes. EMU Notes Mineral 4:271–309Google Scholar
  20. Gilbert JS, Rogers NW (1989) The significance of garnet in the Permo-Carboniferous volcanic rocks of the Pyrénées. J Geol Soc 146:477–490CrossRefGoogle Scholar
  21. Glicksman ME, Lupulescu AO (2003) Dynamics of multicomponent diffusion with zero flux planes. Acta Mater 51(4):1181–1193CrossRefGoogle Scholar
  22. Gupta PK, Cooper AR (1971) The [D] matrix for multicomponent diffusion. Physica 54:39–59CrossRefGoogle Scholar
  23. Harangi S, Downes H, Kosa L, Szabo C, Thirlwall MF, Mason PRD, Mattey D (2001) Almandine garnet in calc-alkaline volcanic rocks of the northern Pannonian Basin (eastern-central Europe): Geochemistry, petrogenesis and geodynamic implications. J Petrol 42(10):1813–1843CrossRefGoogle Scholar
  24. Kailasam SK, Glicksman ME (1999) A simplified method for calculating the diffusivity matrix in ternary alloys. Acta Mater 47(3):905–913CrossRefGoogle Scholar
  25. Kawabata H, Takafuji N (2005) Origin of garnet crystals in calc-alkaline volcanic rocks from the Setouchi volcanic belt, Japan. Mineral Mag 69(6):951–971CrossRefGoogle Scholar
  26. Kirkaldy JS (1957) Diffusion in multicomponent metallic systems. Can J Phy 35:435–440Google Scholar
  27. Kirkaldy JS, Young DJ (1987) Diffusion in the condensed state. vol. The Institute of Metals, London, p 527Google Scholar
  28. Kirkaldy JS, Zia-Ul-Haq L, Brown LC (1963) Diffusion in ternary substitutional systems. Trans Am Soc Metals 56:834–837Google Scholar
  29. Lesher CE (1994) Kinetics of Sr and Nd exchange in silicate liquids–Theory, experiments, and applications to uphill diffusion, isotopic equilibration, and irreversible mixing of magmas. J Geophys Res-Solid Earth 99(B5):9585–9604CrossRefGoogle Scholar
  30. Liang Y, Richter FM, Watson EB (1996) Diffusion in silicate melts.2. Multicomponent diffusion in CaO-Al2O3-SiO2 at 1500 C and 1 GPa. Geochim Cosmochim Acta 60(24):5021–5035CrossRefGoogle Scholar
  31. Loomis TP (1978) Multicomponent diffusion in garnet.2. Comparison of models with natural data. Am J Sci 278(8):1119–1137CrossRefGoogle Scholar
  32. Matano C (1933) On the relation between diffusion coefficients and concentration of solid metals (the nickel-copper system). Jpn J Phys 8:109–113Google Scholar
  33. Milman-Barris MS, Beckett JR, Baker MB, Hofmann AE, Morgan Z, Crowley MR, Vielzeuf D, Stolper E (2008) Zoning of phosphorus in igneous olivine. Contrib Mineral Petrol 155(6):739–765CrossRefGoogle Scholar
  34. Nishiyama T (1998) Uphill diffusion and a new nonlinear diffusion equation in ternary non-electrolyte system. Phys Earth Planet Inter 107(1–3):33–51CrossRefGoogle Scholar
  35. Onsager L (1945) Theories and problems of liquid diffusion. Ann N Y Acad Sci 46:241–265CrossRefGoogle Scholar
  36. Perchuk AL, Burchard M, Schertl HP, Maresch WV, Gerya TV, Bernhardt HJ, Vidal O (2009) Diffusion of divalent cations in garnet: multi-couple experiments. Contrib Mineral Petrol 157(5):573–592CrossRefGoogle Scholar
  37. Philibert J (1991) Atom movements—diffusion and mass transport in solids. vol, Les Ulis, p 577Google Scholar
  38. Raimbourg H, Goffe B, Jolivet L (2007) Garnet reequilibration and growth in the eclogite facies and geodynamical evolution near peak metamorphic conditions. Contrib Mineral Petrol 153(1):1CrossRefGoogle Scholar
  39. Sabatier JP, Vignes A (1967) Study of diffusion phenomena in ternary Fe-Ni-Co system. Mém Sci Rev Métall 64:225–240Google Scholar
  40. Sauer F, Freise V (1962) Diffusion in binären Gemischen mit Volumenänderung. EZeitschrift für Elektrochemie 66:353–363Google Scholar
  41. Schwind M, Helander T, Agren J (2001) On zigzag shaped diffusion paths in multi-phase diffusion couples. Scripta Mater 44(3):415–421CrossRefGoogle Scholar
  42. Sohn YH, Dayananda MA (2000) A double-serpentine diffusion path for a ternary diffusion couple. Acta Mater 48(7):1427–1433CrossRefGoogle Scholar
  43. Stalker MK, Morral JE (1990) Classification of concentration profiles in quaternary diffusion couples. Acta Metallurgica Et Materialia 38(3):439–447CrossRefGoogle Scholar
  44. Stalker MK, Morral JE, Romig AD (1992) Application of the square root diffusivity to diffusion in Ni-Cr-Al-Mo alloys. Metall Trans a-Phys Metall Mater Sci 23(12):3245–3249CrossRefGoogle Scholar
  45. Thompson MS, Morral JE (1986) The effect of composition on interdiffusion in ternary alloys. Acta Metall 34(2):339–346CrossRefGoogle Scholar
  46. Thompson MS, Morral JE, Romig AD (1990) Applications of the square root diffusivity to diffusion in Ni-Al-Cr alloys. Metall Trans a-Phys Metall Mater Sci 21(10):2679–2685CrossRefGoogle Scholar
  47. Vielzeuf D, Holloway JR (1988) Experimental determination of the fluid-absent melting relations in the pelitic system—Consequences for crustal differentiation. Contrib Mineral Petrol 98(3):257–276CrossRefGoogle Scholar
  48. Vielzeuf D, Veschambre M, Brunet F (2005) Oxygen isotope heterogeneities and diffusion profile in composite metamorphic-magmatic garnets from the Pyrenees. Am Mineral 90(2–3):463–472CrossRefGoogle Scholar
  49. Vielzeuf D, Baronnet A, Perchuk AL, Laporte D, Baker MB (2007) Calcium diffusivity in alumino-silicate garnets: an experimental and ATEM study. Contrib Mineral Petrol 154(2):153–170CrossRefGoogle Scholar
  50. Vignes A, Sabatier JP (1969) Ternary diffusion in Fe-Ni-Co alloys. Trans AIME 245:1795–1802Google Scholar
  51. Wakabayashi H, Oishi Y (1978) Liquid-state diffusion of Na2O-CaO-SiO2 system. J Chem Phys 68(5):2046–2052CrossRefGoogle Scholar
  52. Wolf MB, London D (1994) Apatite dissolution into peraluminous haplogranitic melts—an experimental study of solubilities and mechanisms. Geochim Cosmochim Acta 58(19):4127–4145CrossRefGoogle Scholar
  53. Wu K, Morral JE, Wang Y (2006) Horns on diffusion paths in multiphase diffusion couples. Acta Mater 54(20):5501–5507CrossRefGoogle Scholar
  54. Zhao J, Garay JE, Anselmi-Tamburini U, Munir ZA (2007) Directional electromigration-enhanced interdiffusion in the Cu-Ni system. J Appl Phy 102(11), article no. 114902, 7 pagesGoogle Scholar
  55. Ziebold TO, Ogilvie RE (1967) Ternary diffusion in copper-gold-silver alloys. Trans AIME 239:942–953Google Scholar

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Centre Interdisciplinaire de Nanoscience de MarseilleCNRS, Aix-Marseille UniversityMarseilleFrance

Personalised recommendations