Point defects and diffusion

1. Point defects and diffusion

2. Isovalent substitute atom Point defects in ionic solids Anti-Schottky defect: anion- cation-vacancy plus interstial pair. Schottky defect: anion-cation- vacancy pair. Frenkel defect: anion vacancy- interstitial cation pair. e- e- e- F-center: anion vacancy with excess electron in alkalihalides. M-center: two anion vacancies with one excess electron each.

3. Thermodynamics of point defects hf T=const. - Free energy of formation for n Schottky defect pairs in NaCl: G G G0: free energy of the perfect crystal Dh: enthalpy of formation of a defect Ds : entropy change n: number of defect pairs G0 Gmin -TDs n neq Consequence of the thermodynamic analyses: Above 0K, point defects will always be present in a crystal that is in thermodynamic equilibrium. The number of point defects will increase with increasing temperature.

4. Point defects and diffusion I Exchange mechanism Ring rotation mechanicsm Vacancy mechanism Interstitial mechanism The jump of an atom from one site to an adjacent empty side needs the "cooperation" of the neighboring atoms, e.g. the green atoms have to "open" a gap (through termal vibrations) exactly when the red atom tries to jump. that the jumping atom has enough energy to do the movement (activation energy EA ) The number of atoms which have enough energy and the number of ideal geometries increase with increasing temperature.

5. C x xi d2C dx2 > 0 d2C dx2 < 0 Point defects and diffusion II dC dx C x The diffusion rate J in direction x is proportional to a property of the system diffusant/diffusor ( diffusion constant D) and the concentration gradient along x. For a constant gradient the flux (e.g. migration of particles accross a unit area in unit time) is given by Fick's first law as: In most cases, however, the concentration gradient is not constant, but changes with time. Fick's second law describes diffusion with non steady-state concentration gradients: Qualitatively, the equations indicates, that in regions where the concentration gradient is convex, the flux (and the concentration) will decrease with time, for concave gradients it will increase. The order of magnitude of the diffusion distance can be calculated by the following thumb rule:

6. Fo100 Fo82 1 mm c* Point defects and diffusion III t In most cases the diffusion of one species above is not independent of the diffusion of a second species. For example the speed with which green atoms above can move into the red phase depend on the speed with which red atoms move to the left = interdiffusion. The arrangement above, often used to measure interdiffusion, is called a diffusion couple T = 1200 °C, 254h DEMP:9 x 10-13 DTEM:6 x 10-13 norm. conc. EMP TEM p1 TEM p2 distance mm T = 980 °C, 336h TEM EMP norm. conc. Determination of a diffusion profile in a olivine diffusion couple (Meissner et al., 1998) distance mm

7. Point defects and diffusion IV ln D 1/T The diffusion constant increases with temperature. D diffusion constant D0 preexponential factor EA activation energy K Bolzmann constant The dependency is shown in a lnD vs. 1/T (= inverse temperature) plot called Arrhenius diagram. Units of D: m2s-1 Self- and interdiffusion coefficient in garnets (Schwandt et al., 1996)

8. Solid solutions

9. Solid solutions Elements present in a pure phase with an ideal structure can be replaced by other elements. Such a replacement is called a substitution. A solid solution is a single phase which exists over a range in chemical compositions. Pure phases are calle endmembers, while partially substituted phases are intermediate members. Endmember 100% red atoms 0%greenatoms Solid solution 89% red atoms 11%greenatoms Endmember 0% red atoms 100%greenatoms Almost all minerals are able to tolerate variations in their chemistry (some more than others). Chemical variation greatly affects the stability and the behaviour of the mineral. Therefore it is crucial to understand:- the factors controlling the extent of solid solution tolerated by a mineral- the variation in enthalpy and entropy as a function of chemical composition- different types of phase transition that can occur in solid solutions

10. Olivine solid solution • Following rules apply for substitutions: - conservation of charge neutrality • - similar size of substituting ion • - preference for the same coordination of the substituting ion • - similar electronic configuration Olivin solid solution: Forsterite Fayallite Mg2SiO4 VI Mg2+ => VI Fe2+ Fe2SiO4 Magnesium in octahedral coordination can be replaced by Fe and Mn Cation radii: Mg2+ : 0.72Å vs. Fe2+: 0.78Å Vector notation for the substitution: Mg-1 Fe All intermediate compositions between pure Mg-olivine (forsterite) and pure Fe-olivine (fayallite) are possible => complete solid solution

11. Structural sites and substitutions Cation (anion) positions which are related by symmetry are called sites. Positions with the same polyhedral environment may not represent the same site. In the olivine structure f.ex. there are two octahedral cation positions, which are not related by symmetry, and represent,therefore, two different sites called M1 and M2. The two sites are not regular octahedra but are slightly distorted. The distorsion is not equal for both sites. M1 - site M2- site T-site av. M-O bond lengths in M1: 2.101 av. M-O bond lengths in M2: 2.135 Polyhedral representation of a Complete unit cell fragment of the olivine structure of the olivine structure viewed down the a-axis. In forsterite all octahedra are filled by magnesium, but can be replaced by other cations like Fe, Mn, Ca.Whereas Fe has a weak preference for M1, Mn and Ca enter preferentially the M2 site. The latter two are larger than Mg.

12. 1400 1200 1000 800 600 400 200 0.1 0.2 0.3 0.4 T ( C) Limited solid solution Calcite-dolomite solid solution: Calcite Dolomite Ca2(CO3)2 VI Ca2+ => VI Mg2+ CaMg(CO3)2 Cation radii: Ca2+ : 1.0 Å vs. Mg2+: 0.72Å !! large difference !! Vector notation for the substitution: Ca-1 Mg Large difference in size => only limited substitution possible => limited solid solution Composition range which is not possible: solution gap, miscibility gap The size of the miscibility gap is temperature dependent: • [Mg] • XMg : molar fraction = • [Mg] + [Ca] Miscibility gap The higher the temperature, the smaller the solutiongap. The diagram on the right is valid if the carbonates are heated in a pure CO2 atmosphere. 0.0 0.5 Calcite Dolomite XMg

13. Exsolution The system NaCl-KCl has, due to the difference in cation radii (1.0 vs 1.4Å) a miscibility gap. Above 410°C a crystal with a composition Na0.75K0.25Cl will be homogeneous, below this temperature, e.g. when cooling the crystal below the solvus, potassium rich exsolution lamellae will form, which grow and become richer in K with further cooling, whereas the matrix will become Na richer. Microstructures T=590°C composition of crystal: M0 T=410°C composition of matrix (99%) : M1 exsolution lamellae (1%): Ex1 M0 T=350°C composition of matrix (86%) : M2 Exsol. lamellae (14%): Ex2 Ex1 99% M1 100% Ex2 M2 Ex3 14% 86% T=200°C composition of matrix (76%) : M3 Exsol. lamellae (24%): Ex3 M3 24% 76 % Halite Sylvite

14. Cummingtonite lamellae formed within a glaucophane host seen in the TEM. The lamellae form in two symetrically related orientations. Exsolution in orthoamphiboles Between amphibole endmembers there are often miscibility gaps, f.ex. between cummingtonite and glaucophane. Cooling of a glaucophane (endmember: Na2Mg3Al2Si8O22(OH)2) with excess Mg and Ca will lead to exsolution of cummingtonite lamellae. The exsolution lamellae are very fine and bearly visible in the optical microscope

15. Coupled substitutions Albite-Anorthite solid solution Albite NaAlSi3O8 Anorthite CaAlSi3O8 IX Na+ => IXCa2+ Cation radii: Ca2+ : 1.18 Å vs. Na+: 1.24 Å ok! but violation of charge neutrality!! conservation of charge neutrality => second exchange Albite Anothite NaAlSi3O8 IX Na+ => IX Ca2+ CaAl(AlSi2)O8 IV Si4+ => IV Al3+ Two or more substitution at once: coupled substitutions Vector notation for the substitution: Na-1Si-1 Ca AlIV charge balance: Na1+ => Ca2+ + Si4+ => Al3+ : 1 + 4 =2+3 ok!! Vectors are often named after the most prominent occurence. The above vector is called plagioclase vector. Tschermak’s exchange: Diopside “Cats”-module CaMgSi2O6 VI Mg2+ => VI Al3+ CaAl(AlSi)O IV Si4+ => IV Al3+ AlIVAlVI Mg-1Si-1 Tschermak’s exchange

16. K1+ => K1+ => Pb2+ Vector notation: K-21+Pb2+ 1 + 1 = 2+0 Omission solid solutions Substitutions where charge balance is maintained by substituting only a part of the ions taken out is called omission substitution. Such a substitution will leave some sites empty, which are filled in the endmember. Perspective views of the kalifeldspar structure down the c-axis. potassium atoms potassium vacancy lead cation Pb2+ silicium tetrahedra aluminium tetrahedra Lead containing kalifeldspar has a green-blue color and is called amazonite.

17. Substoichiometry Omission substitution through oxidation Iron layer in pyrrhotite Fe1-xS Iron layer in troilite FeS metallic Fe Ferrous iron (Fe3+ ) Ferric iron (Fe2+ ) Oxidation reaction: 3Fe2+ => 2Fe 3+ + + Fe0 (metallic) ( = empty site, vacancy) Vector notation: Fe2+-3 Fe2+2 The stoichiometry of pyrrhotite can hardly be expressed by integral numbers, such phases are said to be substoichiometric The introduction of trivalent iron occurs often already during the growth of the crystal. If the oxidation occurs later, the reduced iron has to be expelled from the structure.

18. Interstitial solid solution Substituting ions can also go into interstitial sites (interstitial substitution), not occupied in the pure phase. Example: In the ring silicate beryl (Be3Al2Si6O18) (Fig. 2.27) alcali cations can lodge into the channels formed by silicate rings, which are empty in the pure phase. Charge balance is reestablished by replacing one silicon by aluminum: Exchange vector: Si-1 -1 Al Na Beryl structure seen along the open channels = c-axis Instead of alkali cations, neutral noble gas atoms or water molecules can occupy the channels, which require no charge balancing. Silica tetrahedra tetrahedral aluminum Beryllium tetrahedra alkali cation (e.g. Na+) Exchange vector: -1 (H2O) octahedral aluminum water molecule

19. Order-disorder I The order in a structure with two cations sharing the same site can be described by an order parameter Q. Disordered substitution Ordered substitution site: potential position for an atom Site  Atom A Site  Atom B Probability of A on : Probability of B on  1 - p Four possibilities: atom A on site  atom B on site  atom A on site  atom B on site 

20. Order-disorder II • Complete order : p = 1 • Complete disorder: p = 0.5 • Normalizing: long range order Q = 2p – 1 • Another way to define Q is: where the terms Aa is the fraction of atoms A on site a. For a phase with stoichiometry AmBn , Q is given as: and Disordering occurs not suddenly, but over a certain temperature range below the critical temperature. In the disordered structure it is impossible to predict what atom (green or red) occupies a certain site. Results from bulk property measurements of a disordered crystals are the same as the results from a hypothetical crystal with each site filled with half a green and half a red atom. Disordering is thus accompanied by a change in symmetry.For the example shown on the left, the symmetry would change from face-centered cubic to primitiv cubic.

21. Order-disorder III Long - vs. short range order bad long, bad short r. o. good long, good short r. o. bad long, good short r. o. boundary between perfectly ordered areas = anti-phase boundaries. The domains are related by a translation that is a fraction of the lattice translation of the ordered structure.

22. Ordering vs. exsolution Exsolution Cation ordering

23. Basic elements of the pyroxene structure The pyroxene structure is characterized by single tetrahedral chains (right). Two of the them sandwich an octahedral chain (top)

24. Ortho- vs. clinopyroxenes C2/c e.g. diopside CaMgSi2O6 Pbca e.g. enstatite Mg2Si2O6

25. Octahedral sites in pyroxenes

26. Pyroxene quadrilateral

27. Ordering in OPX I I-beam representation of the orthopyroxene structure. view is parallel to the silicon tetrahedra. Octahedral strip of the OPX structure.

28. M1 M2 Low temperature Mg Fe2+ Intermediate temperature Mg1-xFe2+x MgxFe2+1-x “Infinite” temperature Mg0.5Fe2+0.5 Mg0.5Fe2+0.5 Ordering in OPX II Fe2+ is slightly larger than Mg and prefers to sit on the larger M2 site (i.e. the crystal has a lower enthalpy when Fe2+ is sitting on M2). Example: For a composition (Mg0.5Fe2+0.5)SiO3 We can measure “x” experimentally and use it to determine what the cooling rate and effective equilibration rate of the mineral was (geospeedometry).

29. Ordering in OPX III