Atomic Transport & Phase Transformations. Prof. Dr. G. Schmitz & PD Dr. Nikolay Zotov

Transcription

1 Atomic Transport & Phase Transformations Prof. Dr. G. Schmitz & PD Dr. Nikolay Zotov

2 Atomic Transport & Phase Transformations Lectures Part I Alloy Thermodynamics PD Dr. N. Zotov (5 weeks) Part II Diffusion Prof. Dr. G. Schmitz (7 weeks) Part III Solid State Reactions PD Dr. N. Zotov (3 weeks) Lecture Times: Mondays 9:15 11:15 2R4 No lectures 17.04; Tuesdays 9:45 11:15 2R H7!!! 2

3 Atomic Transport & Phase Transformations Practicals Part I Alloy Thermodynamics PD Dr. N. Zotov (5 weeks) Part II Diffusion Prof. Dr. G. Schmitz (7 weeks) Part III Solid State Reactions PD Dr. N. Zotov (3 weeks) Tuesdays 11:30 13:00 2P4 Practicals uploaded in advanced Problems solved writtenly at home Presentation of the problems by randomly selected students during the practicals Min 51% of all problems solved in order to be allowed to the written exam Randomly selected practicals checked by the examinators 3

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

5 Atomic Transport & Phase Transformations Part I Lecture Alloy Thermodynamics Short Description 1 Introduction; Review of classical thermodynamics 2 3 Phase equilibria, Classification of phase transitions 4 Thermodynamics of solutions I 5 Thermodynamics of solutions II 6 Binary Phase Diagrams I 7 Binary Phase Diagrams II 8 Binary Phase Diagrams III 9 Order Disorder Phase Transitions 5

6 Atomic Transport & Phase Transformations Lecture I-1 Outline Scope of Thermodynamics Importance of this course Brief Histoty of Thermodynamics Definitions Energy, Heat, Work 1 st Law of Thermodynamics Enthalpy Heat capacity 6

7 Thermodynamics Thermodynamics ist a part of physics, concerned with the preservation, transfer and conversion of energy bewteen different systems Thermodynamics is universal and applies to all type of phases!!! Phases with different bonding metals, semiconductors, isolators ceramics, polymers Phases in different aggregate state - Gasses, Liquids, Solids Phases with different degree of ordering - Crystalline solids, Amorphous materials 7

8 Importance of this course Phenomenological understanding of the behaviour of matter under different conditions Development of new materials Improvement of new materials Processing of materials Design and work of heat engines Chemical Engineering Thermodynamics Materials Engineering Earth Sciences Metallurgy 8

9 Importance of this course Design of chemical processing plants Oil refineries Production, transportation, recycling of energy & water Production of steel & alloys Safety regulations 9

10 Importance of this course Understanding and Prediction of Phase Equilibria at different conditions Understanding of Phase Diagrams Understanding of Phase Transformations 10

11 Brief History of Thermodynamics Fahrenheit (1715) measured Temperature (T) by the expansion of a fluid Celsius (1742) defined the melting T of ice and boiling T of water A. Celsius Lord Kelvin (19 century) introduced the notion of Absolute zero temperature W. Thompson (Lord Kelvin) 11

12 Brief History of Thermodynamics Carnot (1824) shows theoretically that heat is not completely convertible to work; work engines Sadi Carnot First Law of Thermodynamics (1850) Energy is related to heat and work W. Thompson (Lord Kelvin) Rudolf Clausius 12

13 Brief History of Thermodynamics R. Clausius (1850) defines a new thermodynamic property, the Enthropy R. Clausius Lord Kelvin, Max Planck and R. Clausius (middle of 19th century) formulate the 2nd Law of Thermodynamics Max Planck 13

14 Brief History of Thermodynamics L. Boltzmann (late 19th century) related Entropy to Order L. Boltzmann J. Gibbs ( end of 19th Century) formulated the Chemical Thermodynamics and introduces the Chemical Potential µ. W. Nernst (end of 19th Century) formulated the 3rd Law of Thermodynamics J. Gibbs W. Nernst 14

15 Definitions Species: Chemical Elements C, O Ions O 2- Molecules CO 2 Components: Species of fixed composition (Elements, Molecules) FeO 1-X 15

16 Definitions System: Continuous System: Ansamble of N species (microscopic definition) System, whose physical properties vary continuously in space. Types of systems (1): Isolated System: No energy and no matter can pass through the boundaries Open System: Both energy and matter may pass through the boundaries Energy Matter Closed System: Energy can pass through the boundaries Energy 16

17 Definitions Types of systems (2): Unary systems: Binary systems: contains one component (Sn) contains two components (Ag-Sn) Ternary systems: contains three components (Ag-Sn-Cu) This classification reflects the complexity of a system 17

18 Definitions Types of states: Equilibrium: (Stable) The properties of the system remain unchanged over a long period of time, if there are no external applied fields; Resistant against perturbations Metastable: Resistant against small perturbations Unstable: The properties of the system change easily under small perturbations 18

19 Definitions Process: Change of the state of the system(s) State A Process State B Types of processes (1): Quasi-static: Reversible: Process, which consists of a succession of equilibrium states, each of which differes only infinitesimaly from the previous state. Process, in which the initial state of the system could be restored. Path: Irreversible: Process, in which the initial state of the system could not be restored. The succession of states traveresed by the system from the initial to the final state. 19

20 Definitions Variables: Quantities, which characterize the state of the system. Types of variables (1): Intensive: Extensive: Variables, which do not depend on the size of the system. Examples: Temperature, Pressure Variables, which depend on the size of the system. Examples: Energy, Heat Capacity 20

21 Definitions Types of variables (2): State: Varaibles, which characterize given state of the system Examples: Temperature (T), Pressure (p), Energy (E) Volume (V) Process: Variables, which characterize the change of the state of the system(s) Examples: Work, Heat 21

22 Definitions Energy is the ability to do work Energy is the ability to cause changes Energy the quantity necessary to create a system in a given state Types of Energy (1): Potential Energy: Kinetic Energy: Energy stored in the bonds between species Examples: Nuclear - Energy of and between the protons and neutrons in the nuclei Chemical - Energy of the atoms in molecules or solids Energy of the motion of the species Internal Energy (U): The sum of the potential and the kinetic energies of the species constituting a given system (body) Teaching the concept of energy is a problem R.L. Coelho (2009) 22

23 Definitions Types of kinetic energy (motion): Translational Rotational Translational motion Rotational motion Vibrational Vibrational motion Units of Energy: SI Joule [1 J = 1 kg m 2 /s 2 ] E = mc 2 erg [1 erg = 1 g cm 2 /s 2 = 10-7 J] cal [1 cal = J] ev [1 ev = x J] J. Joule 23

24 Definitions Definition (1): Work: A process of transfering energy from or to a system, by changing the macroscopic state variables of the whole system Definition (2): Work: Energy transfered from the surrounding to a system by a generalized force acting on the system over a generalized distance. Types of work: W = -.d F Mechanical: W = -F. dx; W = - pdv Electrical: W = - q.df; q - charge, F - electrostatic potential Magnetic: W = - H.dB; H- magnetic field, B - magnetic induction Scalar product of 2 vectors: A x B x + A y B y + A z B z = A i B i. 24

25 Work Example: Stretching a thin rod F l o dl/2 W = -Fdl = - A o l o (F/A o ) (dl/l o ) = = -V o s de V o initial Volume s - applied stress (s > 0 tension) e - strain Work per unit volume: w = - sde 25

26 Definitions Temperature (T): A state variable, quantity, which is a measure of the average kinetic energy of the species in a given system. T ~ 2/3 <mv 2 >/k B Heat (Q): A process of transfering energy from the system to the surrounding or from the surrounding to the system, by changing the temperature of the system. Sign convention Work is negative, when the surrounding does work on the system and positive, when the system does work on the surrounding. Heat is positive, when the T of the system increases and is negative when the T of the system decreases. 26

27 1 st Law of Thermodynamics Mathematical Formulation: DU = Q W The change of the internal energy of a system is the sum of heat and work transfered to the system. Implications: # Although the work (W) and the Heat (Q) are NOT state variables, their sum is a state variable (the change of the internal energy); # Heat and Work have the same units as the Energy (SI: Joule); # The Work and the Heat are processes of energy transfer between the surrounding and the system Isolated system: DU = 0 # Conservation of energy for an isolated system the energy is constant, cannot be destroyed or created. # Reference state: DU = U 2 U 1 ; U 2 = U 1 + DU (Internal) energies are defined to within an (arbitrary) reference state. 27

28 Processes # W and Q are process variables; # Given state of the system can be reached by different processes Types of processes: Isochoric: V = const; δw = pdv = 0; DU = δq Isobaric: p = const; DU = δq pdv; (U 2 + pv 2 ) (U 1 + pv 1 ) = Q Isothermal: T = const Adiabatic: δq = 0 DU = δw 28

29 Enthalpy Mathematical definition: H = U + pv Implications: # U, p and V are extensive state variables the Enthalpy is also extensive state variable (function) # isobaric processes: DH = Q DH > 0 Endothermic processes DH < 0 Exothermic processes # Units: Joule Different types of enthalpies Enthalpy of melting (fusion) Enthalpy of formation Enthalpy of mixing Enthalpy of solution 29

30 Heat Capacity (1) Mathematical definition: C = Q/DT; C = δq/dt Specific Heat Capacity (c): Heat capacity per unit of volume (or per mole) Implications: # The heat Q is a process variable and depends on the path The heat capacity also depends (indirectly) on the process: # Constant-volume heat capacity C V = (δq/dt) V ; DU = Q = C V DT # Constant-pressure heat capacity C p = (δq/dt) p ; DH = Q = C p DT # The Enthalpy H and the Internal Energy U can be calculated from the heat capacities. 30

31 Heat Capacity (2) Relation between C p and C v : C p = (δq/dt) p = ( H/ T) p = ( U/ T) p + p ( V/ T) p ; C V = (δq/dt) V = ( U/ T) V ; Differentiating U with respect to T at constant p gives: C p C V = ( V/ T) p [p + ( U/ V) T ] Introducing the isobaric expansion coeficient a = 1/V ( V/ T) p C p C V = a[p + ( U/ V) T ] ; but ( U/ V) T = T( p/ T) V - p = V a T ( p/ T) V ; but ( V/ T) p = - ( V/ p) T ( p/ T) V ; C p C V = a 2 TV/ß(T) ß(T) = -V(dp/dV) T Isothermal compresibility; B = 1/ß Bulk Modulus 31

32 Heat Capacity (2) T = 300 K; V = 1 cm 3 = 1x10-6 m 3 ; Material a x 10-6 (1/K) B (Gpa) Corr (J/K) Moles Corr (J/mol.K) Al x10-3 Cu x10-3 Graphite x x10-6 Corr = a 2 TVB(T) Moles = Mass (g) / At. Mass (g/mol) = rv / At. Mas (g/mol) 32

33 Calculations of the Heat Capacity (1) Dulong & Petit (1819): At RT c V ~ 3R = 25 J/K.mol C p ~ C V 33

34 Calculations of the Heat Capacity (2) Einstein Model: # The system (solid) consists of N A harmonic oscilators vibrating with the same frequency n; # The average internal energy <U> of a harmonic oscilator is: <U o > = hn /[exp (hn/k B T) 1] # The internal energy <U> of a system of N A harmonic oscilators <U> = 3N A <U o > = 3N A hn /[exp (hn/k B T) 1] Differentiating with respect to T: 34

35 Calculations of the Heat Capacity (3) c V = ( <U>/ T) V = = 3N A k B (Q E /T) 2 exp(q E /T)/ [exp (Q E /T) 1] 2 ; = 3R (Q E /T) 2 exp(q E /T)/ [exp (Q E /T) 1] 2 ; T ~ RT c V ~ 3R T 0 c V 0 Q E = hn/k B Einstein Temperature 35

36 Debey Model: Calculations of the Heat Capacity (4) # Distribution of frequences of the harmonic oscilators g(n) ~ n 2, n < n D. Gaskel (2008) c V = 9R (T/Q D ) 3 x 4 e x / (e x 1) 2 dx; x = hn/k B T; Q D = hn D /k B. At low temperatures: c V ~ 9R (T/Q D ) 3. 36

37 T/Q D Gaskel (2008) 37

38 Calculations of the Heat Capacity (5) Vibrational Density of States (VDOS) Model: g(n) = δ(n n E ) g(n) ~ n 2 c V = 3R {x 2 e x / (e x 1) 2 } g(n) dn; Einstein model Debey Model x = hn/k B T Zotov (2002) 38

39 Measurements of the Heat Capacity # Calorimeters C P = (Q - Q loss )/ MDT - C calor /DT Q = P elec t exp = UI t exp ; DT = (T 1 T o ) 39

40 Measurements of the Heat Capacity Differential Scanning Calorimetry (DSC) Heat-Flux DSC Power-Compensated (Heat-Flow) DSC c p = 1/m ( H/ T) p = 1/m ( H/ t) / ( T/ t) = (1/m) Ṡ / ß c p Specific heat capacity (J/g.K) Ṡ Heat Flow to the sample (J/s) ß Heating rate (K/s) c P = K(1/m) Ṡ / ß 40

41 Measurements of the Heat Capacity Heat Flux DSC # Sample (S) and reference material (R) are heated with the same rate ß from a single heating source; # The difference in the temperatures DT of S and R is allowed to vary and is recorded. # The temperature difference DT is converted to Heat- Flow Ṡ = DT/R; R thermal resistance Lower hetaing/cooling rates possible Less sensitive to small changes of DH Moderate isothermal performance Baseline more flat Short-term noise lower 41

42 Measurements of the Heat Capacity Power-compensated (Heat-flow) DSC # Sample (S) and reference material (R) are heated with the same rate ß separately from two heating sources in two separate compartments; # The temperature difference DT = T S T R is kept constant; Different powers P are supplied to the S and R heaters to maintain the T of the S at the programmed value. # The difference in the electric powers DP ~ Ṡ. Higher heating/cooling rates possible More sensitive to small changes of DH Excelent isothermal performance Baseline less flat Short-term noise slightly higher 42

43 Measurements of the Heat Capacity Three measurements required for Experimental determination of C p by DSC: # Empty Pan # Reference Material # Sample c p (Sample) = (Ṡ S /Ṡ R ) (m R /m S ) c p (Ref) Ṡ S = Heat-Flow Sample Heat-flow Empty Pan Ṡ R = Heat-Flow Reference Heat-flow Empty Pan C p (J/g.K) 1,05 1,00 0,95 0,90 0,85 Sapphire 10 o C/min Weight 29.7 mg c P (Sample) = K(1/m) Ṡ (Sample) / ß c P (Ref) = K(1/m) Ṡ (Ref)/ ß 0, Temperature (oc) 43

44 Measurements of the Heat Capacity Epoxy Hitachi (1981) 44

45 Measurements of the Heat Capacity Perkin Elmer DSC Al pans o C 20 K/min Careful callibration O Neil (1966) 45

46 Measurements of the Heat Capacity O Neil (1966) C p = T /T 2 46

47 Recommended Literature (Part I & III) # E.J. Mittemeijer Fundamentals of Materials Science # H.-G. Lee Material Thermodynamics # D.A. Porter and K.E. Easterling Phase Transformations in Metals and Alloys # D. R. Gaskell Introduction to the Thermodynamics of Materials # A. Prince Alloy phase equilibria # R. DeHoff Thermodynamics in Material Science 47