An introduction to solid solutions and disordered systems MSS. MSSC2016 Torino

Transcription

1 An introduction to solid solutions and disordered systems MSSC2016 Torino 1 Institut des Sciences de la Terre de Paris (UMR 7193) UPMC - Sorbonne Universités, France September 6, 2016 this talk is dedicated to Roberto Orlando

2 Goal of these presentations give general idea to numerically study disordered systems present unfamiliar application of the symmetry introduce the techniques implemented in CRYSTAL show few examples CONFCNT CONFRAND/RUNCONFS Comparing computation and experiment is not straightforward The same ideas or techniques are valid to study defects.

3 Non ideal crystalline systems Using space group supposes that : translational symmetry is strictly obeyed! Some crystalline systems cannot be strictly periodic at the atomic scale! Disordered materials Solid solutions Magnetic spin distribution

4 Disordered systems Different atoms occupy positions that are equivalent at some temperature Ex : Gehlenite Ca 2 Al 2 SiO 7 Space group : P 42 1 m, a= Å, c= Å atom x y z occ Ca AlT AlT SiT O O O XRD cannot resolve atomic distribution over different positions (sites) Atoms are very similar (Si, Al) Statistical disorder Dynamical disorder

5 Textbook example of Disordered system Normal to inverse spinel A IV B VI 2 O 4 : AB 2 O 4 B (A,B) O 4 Symmetry remains cubic statistical occupation of the octahedral site! In general disordered over A and B sites : A 1 x B x A x B 2 x O 4. Non-converging disorder!

6 Solid solutions Crystalline materials displaying large chemical variations without apparent discontinuity. Isostructural materials : Alloys Simple oxides : Al 2 O 3 -Cr 2 O 3 Complex silicates : NaAlSi 3 O 8 KAlSi 3 O 8 Chemical variations are described on the basis of end-members that are considered as pure compounds. End-members may be stable or not : ZnS (Zincblende) -FeS (NiAs). The solution can be either continuous or partial between end-members.

7 Macroscopic properties depend on composition/disorder Gehlenite Ackermanite Melilite Ca 2 Al 2 SiO 7 Ca 2 MgSi 2 O Volume/formula (A 3 ) 303 Δ H sol /formula (kcal) Ge Mol. per cent Ca 2MgSi 2O 7 Ak Ge Mol. per cent Ca 2MgSi 2O 7 Ak Charlu et al, 1981, geochim. Cosmochim. Acta, 45, p

8 Macroscopic properties depend on composition/order ZnS-FeS system Very large exces volume. The apparent volume of the FeS end member is almost 25% smaller than the volume of its stable form. (Zn,Fe)S is cubic, but the stable form of FeS is hexagonal (NiAs).

9 Macroscopic properties Energy Observed atomic distribution Volume are Boltzmann average values over all the possible states of the system at a given temperature. ET x = s ɛs exp k b T ɛ s ɛs s exp k b T Difficulties : What are the states? How many are they?

10 What are the states? At a given composition or (α β)spin state configuration 2 atomic species (Red,Blue) on 4 positions 4R 3R1B

11 How many are they? The number of configurations is composition dependent. Composition N S = 16 Configurations 0/4 1 1/3 4 2/2 6 3/1 4 4/0 1

12 How many are they? The number of configuration is dependent on the size of the cell x=1/4

13 How many are they? The number of configuration is dependent on the size of the cell x=1/4 The number of configurations can be very large! The computational cost increases rapidly as the size of the considered cell enlarges and as the symmetry reduces

14 Cell Supercell, why? Conventional cells are usually too small because the number of accessible compositions is quite reduced they do not account for a large number of configurations

15 A huge, but finite, number of configurations A site of multiplicity D is occupied by R atoms R D configurations exists. R = {r 1,..., r R } D = {d 1,..., d D } S = R D = {s 1,..., s R D }

16 A huge, but finite, number of configurations A site of multiplicity D is occupied by R atoms R D configurations exists. R = {r 1,..., r R } S = R D = {s 1,..., s R D } D = {d 1,..., d D } G G partitions S into equivalence classes. S = i=1, (S) (S) depends on D, R and G. ω i

17 Action of the symmetry Two configurations s i and s j are equivalent : s i s j g G; gs i = s j , σ v1, σ v2 σ d1, σ d2

18 Action of the symmetry Two configurations s i and s j are equivalent : s i s j g G; gs i = s j , σ v1, σ v2 σ d1, σ d2 4 σ vi

19 Symmetry-equivalent classes (SICs)? Comp. Multiplicity Representatives G s 0/ / C 4 v 1 2 2/ / /0 1 8 G / G s

20 Polya s theorem The number of classes due to the action of G on R D = S is 1 (S) = 1 R cycg G g G cyc g : number of cycles of d D describing the action of g on D (see Cayley s theorem) C 4v acting on four equivalent positions 1,2,3,4 E σ v σ v σ d σ d cyc g (S) = 1 8 ( ) = 6 Polya s th. does not say how to find a representative of each class! 1 D sites occupied by R different species

21 Lexicographical order / CONFCNT Permutations Words (0,0,0,0) (0,0,0,1) (0,0,1,1) (0,1,0,1) (0,1,1,1) (1,1,1,1) (0,0,1,0) (0,1,0,0) (0,1,1,0) (1,0,0,0) (1,0,0,1) (1,0,1,0) (1,0,1,1) C 4v acting on vertices of a square (1,1,0,0) (1,1,0,1) (1,1,1,0)

22 Class factorization ET x = ω M ω exp ɛsω ω M ω exp ɛsω k b T k b T ɛ s ω

23 Application to garnets (Ia 3d) At/site Mult. x y z Occ Si Al Ca Fe Mn O Novak G A, Gibbs G V, American Mineralogist 56 (1971) Primitive Cell Conventional Cell R N R 8 N R 12 N R 16 N R

24 Strategies to face large number of SICs (S) Cluster Expansion Monte Carlo sampling

25 Cluster expansion approximation Any scalar function E : Empty E = J 0 + Point i σ ij i + Pair i>j σ iσ j J i,j + triangle MB i>j>k σ iσ j σ k J i,j,k + spin variable on site i Extension of Ising Hamiltonian interaction coefficient

26 Monte Carlo sampling over configurations Prob(s) = 1 S Direct sampling S s ω

27 Monte Carlo sampling over configurations Prob(s) = 1 S Direct sampling S s Prob(ω) = ω S ω = G (S) G s ; ω i = S ; G s G i Large classes, low symmetry confs., are more frequently found! ω

28 MC Sampling the configurational space Parent structure index=n supercell G G = n G As n increases The multiplicity of the low symmetry SICs increases Larger the cell, lower the probability to reach a symmetric SIC If the most stable configuration is missing, statistical results might be biased.

29 Need of a sampling over the SICs Conjecture The most stable and the less stable configurations would have some symmetry. The most stable and the less stable configurations would have to search among the most symmetric configuration. (The most stable and the less stable configuration would have a low multiplicity)

30 Symmetry-adapted MC over SICs (S) = 1 G g G R cycg = 1 G c C c R cycgc (S) G c R cycgc c C = 1

31 Symmetry-adapted MC over SICs (S) = 1 G g G R cycg = 1 G c C c R cycgc (S) G c R cycgc c C = 1 Select a conjugacy class draw g c c s gc

32 Symmetry-adapted MC over SICs Select a conjugacy class draw g c c s gc ω

33 Symmetry-adapted MC over SICs Select a conjugacy class draw g c c s gc Prob(ω) = ω S gc S gc ω Repeating the process, SICs are obtained with the same probability 1 (S)

34 Efficiency of Monte Carlo sampling in CRYSTAL Supercell of calcite 120 atoms 144 symmetry operations Carbonate case study : average number of tries (< t >) required to find the full set of SICs. Studied compositions : α = 4 24, 6 24, 7 24, 8 24, 9 24, 10 24, 11 24,

35 Tuning Monte Carlo in CRYSTAL 0.04 (a) Supercell of calcite 120 atoms 144 symmetry operations (b) Carbonate case study : frequency of orbit finding ν hit as a function of the class length (ω). Two compositions have been considered : (a) 4/24 and (b) 12/24.

36 Spinel example Conventional cell 16 octahedral sites 192 symmetry operations Spinel case study (Mg(Fe aal 1 a) 2O 4 Volume of SIC representatives versus energy for 3 compositions : (a) a=0.25, (b) a=0.50, (c) a=0.75. Symbols refer to multiplicities : red dot: 192, blue up triangle : 96, green down triangle: 48, blue diamond: 24, green square: 12 and 6.

37 Garnet case : Grossular-Katoite Katoite : Ca 3 Al 2 (O 4 H 4 ) 3 O 12 Grossular : Ca 3 Al2Si 3 O 12 Ca 3 Al 2 (O 4 H 4 ) 3 n (SiO 4) n (n = 0,..., 12) 4 4 Primitive cell : 12 involved positions / B3LYP Hamiltonian / 136 SICs

38 Experimental/calculated volume V - (Å 3 ) V - (Å 3 ) Kat Gro n Calculated (black symbols) and experimental (red symbols) volume and excess volume versus composition (n)

39 Need of a split atom model? δ-o (A ) n Tetrahedral δ-o distance versus composition n. Full circles and squares : average δ-o values for SiO 4 and H 4O 4 tetrahedra, respectively. Error-bars range from the minimum to the maximum δ-o distance in each tetrahedron. The gray stripe : average δ-o distances weighted over the SiO 4 and the H 4O 4 tetrahedra. Red asterisks : experimental data.

40 Disorder in gehlenite (Ca 2 Al[AlSi]O 7 ) Supercell 2x2x1 16 involved positions 32 symmetry operators 440 SICs 66 symmetric SICs ΔE (mh/cell) x32 x16 x8 x4 x Volume (A 3 ) In the supercell, formally equivalent tetrahedral positions occupied by Si or Al define 8 T a 2T b 2 groups. Three different groups can be distinguished : Al Si, Si Si and Si Si. The different strips are related to the number of groups of each type.

41 Testing x-ray diffraction pattern A A c B C d B c d 3 0 0

42 Few points of thermodynamics E x T = Z P ω ω ω Mω exp ɛsω k b T ɛsω ɛsω Mω exp ( k b T ɛsω ( ) k b T Mω exp ɛsω k b T Z = ω M ω exp = ) ET x = ω P m > P n m more stable than n P ω ɛ sω P m > P n ln P m > ln P n Then ln P m = E m k b T + ln M ω ln Z ln P n = E n k b T + ln M ω ln Z E m K b T ln M ωm < E n K b T ln M ωn

43 Few points of thermodynamics Maximal accessible entropy S max (x, N) = 1 N K N! b [Nx]![N(1 x)]! Only when N + S max (x) = K b [x ln x + (1 x) ln(1 x)]

44 CRYSTAL today Efficient numeration limited to 2 colors (but general) CONFCNT : Mustapha et al. (2013) On the use of symmetry in configurational analysis for the simulation of disordered solids. Journal of Physics-Condensed Matter, 25, Possible uniform random generation of the SICs CONFRAND : D Arco et al.(2013) Symmetry and random sampling of symmetry independent configurations for the simulation of disordered solids. Journal of Physics-Condensed Matter, 25, with RUNCONF// CONFNEIGH

45 Workers/actors Sami Mustapha Valentina Lacivita Marco de la Pierre Matteo Ferrabone Yves Noel