Atomic Transport & Phase Transformations Lecture 4

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1 tomic Transport & Phase Transformations Lecture 4 PD Dr. Nikolay Zotov zotov@imw.uni-stuttgart.de

2 Lecture I-4 Outline Classification of phase transitions Solutions - Definitions Types of solutions Experimental determination of solubilities Gibbs energy of solutions Enthalpy of Mixing Entropy of Mixing Ideal Solid Solution Model Substitutional Solutions - Hume-Rothery Rules Interstitial Solutions - Rules

3 Phase Transitions (Transformations) Definition: Phase transition is a conversion (transformation) of one phase into another phase (at a constant composition). Examples: Solid - Gas Solid - Liquid Liquid Solid Liquid - Gas Gas Liquid Sublimation Melting Solidification Evaporation Condensation

4 Phase Transitions (Transformations) Solid Solid Polymorphic transition Bismut

5 Phase Transitions Classification Classifications Structural (solid-solid) Thermodynamic Other examples of Phase transitions: # Conductor to insulator # Normal to - Superconductor

6 Phase Transitions Classification Solid Solid phase transitions Discontinuous Graphit Diamant # Chemical bond breaking and reconstruction (reconstructive) NaCl CsCl-Typ # No Group-Subgroup relations between parent hcp fcc, fcc hcp and product phase Continuous Displacive ß-Quarz α-quarz # Only small displacements of the atoms Order-Disorder ucu # Group Subroup relations between parent and product phase (Lecture 9)

7 Phase Transitions Structural Phase Transitions Reconstructive P 6 3 /mmc Fd3m Change of first coordination number (CN 4 3) No Group Subgroup relation 7

same, only small rotations of the SiO 4 tetrahedra No change of translational

8 Phase Transitions Structural Phase Transitions Displacive Space group Space group P P The first coordinations of Si and Oxygen remains the same, only small rotations of the SiO 4 tetrahedra No change of translational symmetry (both lattices are primitive) Change only of the point-group symmetry ( ) 8

9 Entropy Classification: Phase Transitions Thermodynamic Classifications (1) G ) Discontinuous Phase Transitions T C S # The entropy is discontinuous DS; S # There is a latent heat L = T c DS; # C p = - T( 2 G/ T 2 ) is finite for T T C and undefined at T C. T C T B) Continuous Phase Transitions G # The entropy is continuous; # No latent heat; # Singularities in C p and k T at T C. Discontinuities Divergences T C T C T

Phase Transitions Thermodynamic Classifications (2) Ehrenfest Classification: ) 1st Order Phase Transitions # The first derivatives of the Gibbs free energy V= ( G/ p) T and S = - ( G/ T) p are

10 Phase Transitions Thermodynamic Classifications (2) Ehrenfest Classification: ) 1st Order Phase Transitions # The first derivatives of the Gibbs free energy V= ( G/ p) T and S = - ( G/ T) p are discontinuous (exhibit jumps ). B) 2nd Order Phase Transitions #The first derivatives of the Gibbs free energy (V and S) are continuous; # The second derivatives C p = - T( 2 G/ T 2 ) p and ß T = - ( 2 G/ p 2 ) T /V are discontinuous (exhibit jumps ) at the transition temperature P. Ehrenfest

11 Phase Transitions Ehrenfest Classification 1. Order 2. Order Latent heat L = T C (S 1 S 2 ) No latent heat (L = 0) (Heat is released/absorbed at T = const) Co-existance of the two phase Overheating/Undercooling possible Discontinuous phase transition No coexistance of the two phase Overheating/Undercooling not possible Continuous phase transition Solid-Solid phase transitons No Group-Subgroup relations Group Subgroup relations mandatory

12 Phase Transitions Eherenfest Classifications Heat Capacities c p 1st Order 2nd Order c p T c T T c T Implications: C p is divergent at T c. C p has a jump at T c

13 Phase Tranisitons Examples: H 2 O J/mol Enthalpy J/mol.K Heat capacity 6006 J/mol

14 Phase Tranisitons Examples: La 0.7 Ca 0.3 MnO 3 Souza et al. (2005) # S is continuous # D C p discontinuous 2nd order magnetic phase transition

15 Phase Transitions Ehrenfest Classification 1. Order 2. Order Coexsistance of the two phase No coexistance of the two phase Spontaneous change of structural state in the whole macroscopic volume Formation of domains # change only of the point-group symmetry Number of domains = P HT / P LT Existence of Interface bewteen the two phases Growth of the product phase through movement of interfaces. Nucleation of the product phase in the parent phase (Part III)

16 Phase Transitions Ehrenfest Classification PbTiO 3 pervoskite SG PG Order HT P m -3 m m -3 m 48 LT P 4mm 4mm 8 Number of domains 48/8 = 6 Sicron et al. (1995)

17 Phase Transitions Ehrenfest Classification Examples 1. Order 2. Order Crystallization/Melting Sublimation Condensation Solid-Solid phase transformations Feroelectricity Ferromagnetism Normal-to-Superconducting state Solid-solid phase transformation Ce

Definitions: Solubility: The ability of a given component to dissolve into another component. Solvent: Solute: The component in greater amount, which dissolves the minor component.

18 Definitions: Solubility: The ability of a given component to dissolve into another component. Solvent: Solute: The component in greater amount, which dissolves the minor component. The component in lesser amount, which is dissolved. Solubility Limit: The maximum amount (concentration) of the solute, which can be dissolved in the solvent. Mechanical mixture: System consisting of two completely immissible components. solution is a homogeneous system it represents a single phase. System with C = 2 and P = 1

19 lcohol Methanol Ethanol Solubility in water at RT [g lc/100 ml water] Solubility of sugar in water Butanol 8.0 Hexanol 0.7 Solubility changes with T and pressure!!!

20 Types of solutions: Types of solutions: Gaseous Binary (C = 2) Liquid Solid (lloys) B Ternary (C = 3) Multicomponent ( C > 3) Substitutional C Interstitial

$methods, ICP, EDX SEM # Homogeneity Optical microscopy, Diffraction Variation of the Crystallization/Melting temperature of the solution with composition DSC, DT ( at ambient$

21 Determination of Solubilities Experimental methods: # Solubility of metals in metals Diffusion experiments (Part II) Quenching of liquid specimens # Composition Chemical methods, ICP, EDX SEM # Homogeneity Optical microscopy, Diffraction Variation of the Crystallization/Melting temperature of the solution with composition DSC, DT ( at ambient pressure)

22 Determination of Solubilities Melting is 1 st Order Phase transtion The Entalpy shows a jump. Heat capacity C p ~ dh/dt = Ŝ/ß ideally divergies at T m. In DSC/DT the C p peak broadens due to instrumental factors (heating rate; mass of the sample, shape of sample, time lag, etc.) The melting/crystallization temperaturs in DT/DSC are determined by the corresponding Onset Temperatures.

23 Determination of Solubilities Pure g (T m = o C) Liquidus Boettinger et al. Elsevier, 2007 Schematic change of DT/DSC peak with alloying DT curves at three different heating rates (5, 10, 20 K/min) Shift of onset temperatures, peak maximum with heating rate low heating/cooling rates for accurate results Boettinger et al. Elsevier, 2007

24 Gibbs Energy Mechanical mixture (C = 2): G = n G + n B G B ; N = n + n B ( number of moles); Molar Gibbs energy: G m = X G + X B G B ; X + X B = 1 Implications: X = n /(n + n B ); X B = n B / (n + n B ); G m = (1 X B ) G + X B G B ; The molar Gibbs energy of a mechanical mixture is a linear sum of the Gibbs energies

25 Gibbs Energy Solution (C = 2): G sol = n G + n B G B + DG mix ; DG mix - Gibbs energy of mixing Implications: Condition for the formation of a solution (1): DG mix < 0; but G = H TS Condition for the formation of a solution (2): DH mix < TDS mix DH mix DS mix - Enthalpy of Mixing (Heat of formation of the solution); - Entropy of Mixing (Entropy of formation of the solution).

26 Enthalpy of Mixing Water - Metanol Water - Ethanol Water - Propanol DH mix m = Q = C p DT Peteers et al. (1993)

27 Enthalpy of Mixing DH mix = H sol H H B ; H = U + pv ~ U Solid (Liquid) Solutions Neglect the kinetic energy of the species; U ~ Potential energy of the species U = ½ S E 2 (r ij ) + E 3 (r) + ; E 2 pair-wise potential energy between species E 3 tree-body potential energy U = ½ S E 2 (r ij ) + E many-body

28 Enthalpy of Mixing Potentials Van-der-Waals crystals/liquids (noble gases at low T): non-directional interactions Lenard-Jones potential E 2 LJ = 4e[(s/r) 12 - (s/r) 6 ];

29 Enthalpy of Mixing Potentials Ionic crystals/melts non-directional, electrostatic interactions E 2 ionic ~ C/r 12 e 2 /(4pe o r); e o dielectric constant of the vacuum

r 2 ); r c (cut-off) Cu a E many-body FS = - Si f(r i ); r Electron density of atom i; f(r) = r ½ ; r

30 Enthalpy of Mixing Potentials c Metals/metallic melts non-directional, delocalized electron-electron + electron-core interactions Metals Finnis-Sinclair (FS) potential E 2 FS = (r - c) 2 (c o + c 1 r + c 2 r 2 ); r c (cut-off) Cu a E many-body FS = - Si f(r i ); r Electron density of atom i; f(r) = r ½ ; r i = S j i 2 f(r ij ); f(r ij ) = (r ij -d) 2 ; r d = 0.93 ev/å c = 2.96 Å d = 4.05 Å Dai et al. (2006)

Covalent crystals/melts strongly-directional

E 2 SW = [B(s/r) 4 1]exp(s/(r-as)) E 3 SW =

31 Covalent crystals/melts strongly-directional interactions Stillinger-Weber (SW) potential Solutions Enthalpy of Mixing Potentials Si sp 3 hybridisation Covalent crystals SW potential E 2 SW = [B(s/r) 4 1]exp(s/(r-as)) E 3 SW = exp(s/(r ij as))exp(s/(r ik -as))(cos(q jik + 1/3) 2 ;

32 Enthalpy of Mixing H ~ U ~ ½ S { E 2 (r) + E 3 (r) + } ~ ½ S E 2 (r NN ) = ½ Nz e Bond ; N Number of atoms z - Coordination of the atoms e Bond = E 2 (r NN ) Bond energy between nearest-neighbours Substitutional solid-solution model # Components and B have the same coordination (z) # Bond Energies independent of composition U = ½ Nz e ; U B = ½ Nz e BB ; B B B B B B B B B B B B B B B B B B B B B B H sol ~ U sol = X 2 U + X B2 U B + Nz X X B e B ; X 2 ~ probability to find - bond X 2 B ~ probability to find B-B bond X X B ~ probability to find -B bond Random distribution

33 Enthalpy of Mixing DH mix m = X 2 U + X B2 U B + Nz X X B e B - X U - X B U B = = ½ Nz e X (1-X B ) + ½ Nz e BB X B (1-X ) + Nz X X B e B - ½ Nz X e - ½ Nz X B e BB = = ½ Nz X X B (2e B e e BB ) = ½ Nz X X B De ; De = 2e B e e BB. = WX X B ; W = ½ NzDe (N = N ) Implications: De > 0 DH mix > 0 De < 0 DH mix < 0

34 Ideal Solid Solution Model Ideal Solution Model: Solution for which the Enthalpy of mixing is zero (DH mix = 0). DG mix = -TDS mix ; Implications: De = 0 e B = ½ (e + e BB ); uniformity of the nearest-neighbhour interactions e B = e = e BB ; (not necessary condition)!! DV mix = ( DG mix / p) T = 0 V m id = X V o + X B V Bo ;

35 Entropy of Mixing DS mix = S sol n S n B S B ; DS mix = k B ln(w) Vibrational Entropy: W = W vib W elec W Conf ; DS mix = DS vib + DS Conf + DS Elec ; For a single component system (C = 1) without electronic degrees of freedom and defects, the entropy arrises from the vibrational degrees of freedom (Lecture 2) S (vib) = - ( G/ T) p ; S sol = n S + n B S B - ( DG mix / p) T ( p/ T) = n S + n B S B - DV mix ( p/ T) DS mix (vib) = -( p/ T) DV mix ; For the ideal solid-solution model DV mix = 0 DS mix id (vib) ~ 0

36 Entropy of Mixing # Substitutional solid solutions; # DS mix id (vib) ~ 0; # Neglecting electronic contributions to the entropy; # Only contribution from the different possible arrangements of the and B atoms in the lattice considered (configurational entropy) # ideal solid-solution model: similar forces between different types of atoms random distribution is the most probable: W ~ W conf = N!/ (X N)! (X B N)! ; DS mix = k B ln[n!/ (X N)! (X B N)! ]

37 DS mix = k B ln[n!/ (X N)! (X B N)! ] = Solutions Entropy of Mixing Sterling formula = = - k B N { X lnx + X B lnx B }; [= - k B N S X i ln(x i ) ] ; i = 1,2, C N = N ; ( 1 mole solution) Molar Entropy of mixing of the ideal solid solution model: DS id mix = - R{X lnx + X B lnx B } DS mix = X S + X B S B S i = - R ln(x i ); partial molar entropies of the components

38 Entropy of Mixing DS mix id > 0 Increase of Entropy by Mixing!!! Max X = X B = 0.5 DS id mix m = Rln2 = 5.76 J/mol.K

39 Gibbs energy of the ideal solution model G sol = n G + n B G B + DG mix ; T 2 > T 1 Molar Gibbs energy G sol m = X G + X B G B + (DH mix m TDS mix m ) Molar Gibbs energy of the ideal solution model: G m id = X G + X B G B + TR {X lnx + X B lnx B } DG mix = TR {X lnx + X B lnx B } < 0 The Gibbs energy of the ideal solution model decreases, compared to the mechanical mixture X G + X B G B. G m id has a minimum at X B = 1/{1 + exp[ (G B -G )/2RT]}; if G = G B ; min at X B = 0.5

40 Molar Gibbs energy of the ideal solution model: 1st Derivative: DG id m/ X B = RTln (X B /1-X B ) Solutions Gibbs energy of ideal solution model X B 0 DG id m/ X B Logarithmic divergence # ddition of small amount of solute (vacancies) decreases the Gibbs energy due to the configurational entropy 2nd Derivative: 2 DG id m/ X B2 = RT(1/X B 1/(1-X B ) = = RT/X B (1-X B ) Curvature k = 2 DG id m/ X B2 / [(1 + ( DG id m/ X B ) 2 ] 3/2 = RT/ {[X B (1 X B )]{1 + [RTln (X B /1-X B )] 2 } 3/2 } maximum curvature k max = 4RT at X B = ½

41 Ideal Solid Solution - Summary DS mix max = 5.76 J/mol.K DeHoff (2006)

Substitutional Solid Solutions Questions: when are formed ideal substitutional solid solutions, Hume-Rothery Rules: 1.

42 Substitutional Solid Solutions Questions: when are formed ideal substitutional solid solutions, Hume-Rothery Rules: * (R solute R solvent )/R Solvent < 15 %; R atomic radius 2. Crystal structures very similar (identical); 3. Solute and Solvent atoms typically have the same valence; 4. Small difference in the electronegativities : c c B 0. W. Hume-Rothery

43 Substitutional Solid Solutions Hume Rothery Rules Examples: Proprerty l Cu Radius (Å) % Structure fcc fcc Electonegativity Valence state +3 +2(+1) Limited solubility

Substitutional Solid Solutions Hume Rothery Rules Examples: Proprerty Ni Cu Radius (Å) 1.35 1.

44 Substitutional Solid Solutions Hume Rothery Rules Examples: Proprerty Ni Cu Radius (Å) % Structure fcc fcc Electonegativity Valence state +2 +2(+1) +2 Complete Solubility!!

45 Substitutional Solid Solutions Hume Rothery Rules Examples: Proprerty g Cu Radius (Å) % Structure fcc fcc Electonegativity Valence state +2(+1) +2(+1) Limited solubility (M) Solid solution with M as solvent

46 Darken Gurry Maps: Solutions Substitutional Solid Solutions [ (c c )/0.2] 2 + [ (r r )/0.075] 2 = 1 R. Ferro, Intermetallic Chemistry, Elsevier 2008

47 Darken Gurry Maps: Solutions Substitutional Solid Solutions

48 Interstitial Solid Solutions Criteria for formation of interstitial solid solutions # Number of Voids, coordination of interstitial atoms # Degree of distortion of the host (solvent) lattice Size of the Voids R V ~ 0.7 R atom ;

49 Interstitial Solid Solutions - bcc Size of the Voids TV: R V ~ 0.29 R atom ; OV: R V ~ R atom OV TV Distorted tetrahedral and octahedral voids

50 Interstitial Solid Solutions - fcc Size of the Voids TV: R V ~ R atom ; OV: R V ~ R atom TV OV undistorted

Size of the Voids (for ideal c/a ratio) TV: R V ~ 0.

51 Size of the Voids (for ideal c/a ratio) TV: R V ~ R atom ; Solutions Interstial Solid Solutions - hcp OV: R V ~ R atom

52 Interstitial Solid Solutions Type lattice Type of Void Cubic Tetrahedral Octahedral SC bcc Fcc hcp - 4 2

53 R Fe ~ 1.26 Å Size of Voids (Å) Lattice TV OV bcc Fcc Solutions Intersititial Solid Solutions Bond Distortion e B = (R I R V ) / R = R I / R - c; Total Distortion ~ ½ N V z V e B x 100 (%) Total Distortion Lattice TV OV bcc 7.7 % 8.1% R C = 0.77 Å; R N = 0.71 Å; R H = 0.46Å Fcc 6.2% 2.4% Smaller distrotion - Larger Solubility

Atomic Transport & Phase Transformations Lecture 4

Atomic Transport & Phase Transformations Lecture 4 PD Dr. Nikolay Zotov zotov@imw.uni-stuttgart.de Materials Science Colloquium Monday, the 24 th of April 2017 at 4:30 p.m., first presentation for the