Atomic Transport & Phase Transformations Lecture 4

Size: px

Start display at page:

Download "Atomic Transport & Phase Transformations Lecture 4"

Camilla Greene
5 years ago
Views:

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

2 Materials Science Colloquium Monday, the 24 th of April 2017 at 4:30 p.m., first presentation for the Summer Semester 2017 in our the Materials Science Colloquium will take place Prof. Radovan Černý, Department of Quantum Matter Physics, Laboratory of Crystallography, University of Geneva, Switzerland with the topic: Metal hydrides as solid electrolytes for batteries. We look forward to your attendance. Kind regards, Guido Schmitz

3 Small corrections Lecture 2, Slide 20 S = k B ln (W) = k B ln (M!/ m 1!m 2! m r!) = k B [ln M! ln(m i!)] Sterling s approximation ln(x!) for large number of species xln(x) x = k B {Mln(M) M [m i ln(m i ) m i ]} = = k B { Mln(M) M m i ln(m i ) + M} = = k B { m i [ ln(m) ln(m i )]} 3

4 Lecture 2, Slide 20 Murnaghan EOS V(p) = V o [1 + p(b /B)] 1/B B first derivative of B with respect to p B = B o + B p

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

Phase Transitions (Transformations) Definition: Phase transition is a conversion (transformation) of one phase into another phase (at a constant composition).

6 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 Solid Solid Sublimation Melting Solidification Evaporation Condensation Polymorphic transition

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

8 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 AuCu # Group Subroup relations between parent and product phase (Lecture 9)

9 Phase Transitions Structural Phase Transitions Reconstructive P 6 3 /mmc Fd3m Change of first coordination number (CN 4 3) No Group Subgroup relation 9

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

10 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 ( ) 10

11 Entropy Classification: Phase Transitions Thermodynamic Classifications (1) G A) Discontinuous Phase Transitions T C S # The entropy is discontinuous S; S # There is a latent heat L = T c S; # 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: A) 1st Order Phase Transitions # The first derivatives of the Gibbs free energy V= ( G/ p) T and S = - ( G/ T) p are

12 Phase Transitions Thermodynamic Classifications (2) Ehrenfest Classification: A) 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

13 Phase Transitions Ehrenfest Classification 1. Order 2. Order Latent heat L = T C (S 1 S 2 ) (Heat is released/absorbed at T = const) Co-existance of the two phase Overheating/Undercooling possible Discontinuous phase transition No latent heat 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

14 Phase Transitions Eherenfest Classifications Heat Capacities 1st Order 2nd Order c p c p T c T T c T Implications: C p is divergent at T c. C p = -T( 2 G/ T 2 ) p ~ Curvature of G

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

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

17 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)

18 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.

19 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. Implications: A solution is a homogeneous system it represents a single phase.

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

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

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

22 Determination of Solubilities Experimental methods: # Solubility of metals in metals Diffusion experiments 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, DTA ( at ambient pressure)

23 Determination of Solubilities Melting is 1 st Order Phase transtion The Entalpy shows a jump. Heat capacity C p ~ dh/dt ideally divergies at T m. In DSC/DTA 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 DTA/DSC are determined by the corresponding Onset Temperatures.

24 Determination of Solubilities Pure Ag (T m = o C) Boettinger et al. Elsevier, 2007 Schematic change of DTA/DSC peak with alloying DTA 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

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

26 Gibbs Energy Solution: G sol = n A G A + n B G B + G mix ; G mix - Gibbs energy of mixing Implications: Condition for the formation of a solution (1): G mix < 0; but G = H TS Condition for the formation of a solution (2): H mix < T S mix H mix - Enthalpy of Mixing (Heat of formation of the solution); S mix - Entropy of Mixing (Entropy of formation of the solution).

27 Enthalpy of Mixing Water - Metanol Water - Ethanol Water - Propanol H mix m = Q = C p T C p = n W C P W + n A C pa ; Peteers et al. (1993)

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

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

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

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

31 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 = - i f( i ); Electron density of atom i; f( ) = ½ ; i = j i A 2 (r ij ); (r ij ) = (r ij -d) 2 ; r d A = 0.93 ev/å c = 2.96 Å d = 4.05 Å Dai et al. (2006)

32 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( /r) 4 1]exp( /(r-a ) E 3 SW = exp( /(r ij a )exp( /(r ik -a )(cos( jik + 1/3) 2 ;

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

34 Enthalpy of Mixing H mix m = X 2 A U A + X 2 B U B + Nz X A X B AB - X A U A - X B U B = = ½ Nz X A (1-X B ) + ½ Nz BB X B (1-X A ) + Nz X A X B AB - ½ Nz - ½ Nz BB = = ½ Nz X A X B (2 AB AA BB ) = ½ Nz X A X B ; = 2 AB AA BB. = X A X B ; = ½ Nz (N = N A ) Implications: > 0 H mix > 0 < 0 H mix < 0

35 Ideal Solid Solution Model Ideal Solution Model: Solution for which the Enthalpy of mixing is zero ( H mix = 0). G mix = -T S mix ; Implications: = 0 AB = ½ ( AA + BB ); uniformity of the nearest-neighbhour interactions AB = AA = BB ; (not necessary condition)!! V mix = ( G mix / p) T = 0 V m id = X A V Ao + X B V B o ;

36 Entropy of Mixing S mix = S sol n A S A n B S B ; The pure components are taken in their ground state (S A = S B = 0) S mix = S sol = k B ln(w sol ) Vibrational Entropy: W sol = W vib W elec W Conf ; S mix = S vib + S Conf + S Elec ; For a single component system (C = 1) without electronic degrees of freedom and defects, the entropy arrises from the vibrational degrees of freedom S (vib) = - ( G/ T) p ; S sol = n A S A + n B S B - ( G mix / p) T ( p/ T) = n A S A + n B S B - V mix ( p/ T) S mix (vib) ~ -( p/ T) V mix ; For the ideal solid-solution model V mix = 0 S mix id (vib) ~ 0

37 Entropy of Mixing # Substitutional solid solutions; # S mix id (vib) ~ 0; # Neglecting electronic contributions to the entropy; # Only contribution from the different possible arrangements of the A 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 sol (conf) = N!/ (X A N)! (X B N)! ; S mix = k B ln[n!/ (X A N)! (X B N)! ]

38 S [J mol -1 K -1 ] S mix = k B ln[n!/ (X A N)! (X B N)! ] = Solutions Entropy of Mixing Sterling formula = - k B N { X A lnx A + X B lnx B } N = N A ; ( 1 mole solution) Molar Entropy of mixing of the ideal solid solution model: S mix id = - R{X A lnx A + X B lnx B } S mix = X A S A + X B S B S i = - R ln(x i ); partial molar entropies of the components S mix max = 5.76 J/mol.K ,0 0,2 0,4 0,6 0,8 1,0 A x B B

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

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

41 Gibbs energy of ideal solution model Molar Gibbs energy of the ideal solution model: 1st Derivative: G id m / X B = RTln (X B /1-X B ) X B 0 G id m / X B Logarithmic divergence # Addition of small amount of solute always decreases the Gibbs energy due to the configurational entropy 2nd Derivative: 2 G id m / X 2 B = RT(1/X B 1/(1-X B ) = = RT/X B (1-X B ) Curvature = 2 G id m / X 2 B / [(1 + ( G 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 max = 4RT at X B = ½

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

Substitutional Solid Solutions Question: when are formed ideal substitutional solid solutions, what is the solubility limit? Hume-Rothery Rules: 1.

43 Substitutional Solid Solutions Question: when are formed ideal substitutional solid solutions, what is the solubility limit? 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 : A B 0. W. Hume-Rothery

44 Substitutional Solid Solutions Hume Rothery Rules Examples: Proprerty Al 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.

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

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

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

48 Darken Gurry Maps: Solutions Substitutional Solid Solutions

49 Interstitial Solid Solutions Size of the Voids R V ~ 0.7 R atom ;

50 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

51 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.

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

53 Interstitial Solid Solutions Type lattice Type of Void Cubic Tetrahedral Octahedral SC bcc Fcc hcp Criteria for formation of interstitial solid solutions # Number of Voids, coordination of interstitial atoms # Degree of distortion of the host (solvent) lattice

R Fe ~ 1.26 Å Size of Voids (Å) Lattice TV OV bcc 0.37 0.20 Fcc 0.28 0.

54 R Fe ~ 1.26 Å Size of Voids (Å) Lattice TV OV bcc Fcc Solutions Intersititial Solid Solutions Bond Distortion B = (R I R V ) / R A = R I / R A - Total Distortion ~ N V (z V /2) 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 tomic Transport & Phase Transformations Lecture 4 PD Dr. Nikolay Zotov zotov@imw.uni-stuttgart.de Lecture I-4 Outline Classification of phase transitions Solutions - Definitions Types of solutions Experimental