Atomic Transport & Phase Transformations. PD Dr. Nikolay Zotov

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1 Atomic Transport & Phase Transformations PD Dr. Nikolay Zotov

2 Atomic Transport & Phase Transformations Part II Lecture Diffusion Short Description 1 Introduction; Non-equilibrium thermodynamics 2 Diffusion Equations 3 Solutions of the Diffusion Equations 4 Atomic Mechansims of Diffusion, Statistical Interpretation 5 6 Measuremens of Diffusion Coeffíents 7 8 Diffusion in Concentrated/Ordered Alloys 9 High Diffusion Pathways 2

3 Atomic Transport & Phase Transformations Lecture II-1 Outline Importance of Diffusion Homogeneous Heterogeneous Systems Irreversible Processes Fluxes Balance Equations Entropy Production 3

Importance of Diffusion Mixing of Gasses/Liquid

blood) Purification (filters) Solid Oxide Fuel

4 Importance of Diffusion Mixing of Gasses/Liquid (T. Graham 1833) Inert Gas Welding, Anestesiology, Scuba diving Transport through membranes Breathing (Oxygen diffusion in the blood) Purification (filters) Solid Oxide Fuel Cells Batteries Cell Phones, Tablets, Electic cars, Storage devices 4

5 Importance of Diffusion Steel-Hardening The Carbon (Nitrogen) atoms diffuse from a C(N 2 )-rich atmosphere into the steel through the surface. Result: A C(N)-rich surface layers is formed, which makes the steel surface harder. Nitride layer with enhanced hardness Fu et al. (2015) 5

Homogeneous and Heterogeneous Systems Homogeneous Systems: The state variables (T, P, X,S) are macroscopically everywhere the same (small fluctuations are neglected)

6 Homogeneous and Heterogeneous Systems Homogeneous Systems: The state variables (T, P, X,S) are macroscopically everywhere the same (small fluctuations are neglected) Equilibrium Thermodynamics Heterogeneous Sytems: The state variables vary in space and/or in time T(x,t), µ(x,t), S(x,t) Non-equilibrium Thermodynamics; Transport phenomena 6

7 Irreversible Processes G = H - TS Second law of Thermodynamics: ds/dt 0 Equilibrium reached when ds/dt = 0 7

8 Equilibrium Thermodynamics DS = DS ext + DS int Second Law of Equilibrium Thermodynamics (Homogeneous systems) S = S(U, V, X); Reversible (Eq.) processes DS int = 0 Irreversible processes DS int > 0 (Entropy production) 8

9 Equilibrium Thermodynamics Equilibrium Thermodynamics S = S(U,V,N) TdS = du + pdv - k µ k dn k, State variable k = 1,2,... C No Change of volume (δw = pdv ; No work) No convection TdS = du - k µ k dn k ; Tds = du - k µ k dc k ; s Entropy density [s] = J/m 3.K u Internal energy density [u] = J/m 3 ; c k = N k /V o ; V o constant (average) cell volume [c k ] = mol/m 3 T s/ t = u/ t - k µ k c k / t 9

10 Irreversible Processes Irreversible Thermodynamics (Heterogeneous system) # Local equilibrium The system is separated in small regions, which locally are homogeneous and the laws of equilibrium thermodynamics are still valid locally there (coarse graining), although the whole system is not in equilibrium. # The gradeints from cell to cell are not too large # Continuous description S, U, µ are continuous functions in space and time # Microscopic cells are open systems and entropy can flow from/or to the cells # T s/ t = u/ t - k µ k c k / t is still valid for small regions in local equilibrium 10

11 Fluxes q r q v q Quantity, which can flow (particles, heat, charges, energy, entropy, cars) Density of q ([r q ] = amount of q/m 3 ); r q = r q (x,y,z,t) Velocity of q ([v q ] =m/s) dq/dt Change of q in a given volume V with surface S S Balance Equation dq/dt = d/dt r q dv = r q / t dv = V r q / t dv = - r q v q. ds + Ř dv ; Divergence theorem [Ostrogradsky (1826)] Ř rate of creation/annihilation of q in the volume V; [Ř] = am. q/m 3.s 11

12 Fluxes r q / t dv = - div(r q v q ) dv + Ř dv r q / t = - div(r q v q ) + Ř ; J q = r q v q ; Flux of q ( amount of q/m 2.s) Rate (Balance) Equation r q / t = - div(j q ) + Ř ; div(a) = A/ x + A/ y + A/ z = grad.a 12

13 Balance Equation for the Entropy Irreversible Processes Entropy Change Entropy Flux Entropy Production s/ t = - div(j S ) + s; (1) 2nd Law s 0 T s/ t = u/ t - k µ k c k / t (2) 13

14 Mass Balance Equation m k Mass of component k, k = 1,2,...C r k Mass density of component k ([r k ] = kg/m 3 ) r = r i ; Mass density of the material Ệ k Rate of mass generation/annitilation ([Ệ k ] = kg/m 3.s ) dm k /dt = d/dt ( r k dv) = r k / t dv Mass Balance Equation r k / t dv = - r k v k.ds + Ệ dv r k / t = - div(r k v k ) + Ệ k ; 14

15 Mass Balance Equation Ĵ k = r k v k Mass Flux of component k (Diffusion); [Ĵ k ] = kg/m 2.s No net macroscopic mass flow of matter: Condition: Ĵ k = 0; The fluxes are not independent!! r k v k = <v> m Mass-density average velocity; ( J # k = Ĵ k - r k <v> m ; J # k 0; Coordinate system moving with the centre of mass) r k / t = - div(ĵ k ) + Ệ k ; 15

16 Mass Balance Equation r k / t = - div(ĵ k ) + Ệ k ; w k Mass fraction of component k w k = r k /r; r Total density; w i = 1 (rw k )/ t = - (Ĵ k ) + Ệ k ; c k Concentration of component k (number of species per unit volume) c k = rw k /M k N k /V o [c k ] = 1/m 3 ([c k ] = mol/m 3 ) V o Total Volume of the cell (constant); c i = A c k / t = - (1/M k )div(ĵ k ) + (1/M k )Ệ c k / t = - div(j k ) + E k ; (3) J k = r k v k /M k = c k v k Atomic flux; [J k ] = mol/m 2.s E k Atomic production/annihilation rate 16

17 Energy Balance Equation u/ t = - div(j U ) (Internal energy is conserved - 1st Law of thermodynamics) J U = J Q + J D J D = y k J k Flux of potential energy due to Diffusion of the components y k Scalar potential energy of component k in external field [y] = J F k = -grad(y k ) External force acting on component k; [F k ] = N = J/m J Q Heat flux [J/m 2.s] Electric Field E F el = ZeE; E = -grad(u) 17

18 Energy Balance Equation u/ t = - div(j U ) = - div[j q + y k J k ] = = - div(j Q ) - div(y k J k ) = - div(j Q ) { [J k.grad(y k ) + y k div(j k )]} = = - div(j Q ) + J k.f k - y k div(j k ) div(j Q ) + J k.f k - <y> div( J k ) u/ t = - div(j Q ) + J k.f k (4) 18

19 Entropy Balance T s/ t = u/ t - k µ k c k / t = No mass generation; No chemical reactions = {- div(j Q ) + J k.f k } { k µ k [- div(j k ) ] }= s/ t = - 1/T div(j Q ) + 1/T J k.f k + 1/T µ k div(j k ) (5) div(j Q /T) = 1/T div(j Q ) - 1/T 2 grad(t).j Q div[1/t µ k J k ] = 1/T µ k div(j k ) + J k.grad(µ k /T) s/ t = - div[ 1/TJ Q 1/T µ k J k ] + {- J k.grad(µ k /T) F k ) - 1/T 2 grad(t).j Q } (6) s/ t = - div(j S ) + s; (1) 19

20 Entropy Balance Entropy Flux J S = 1/T(J Q - µ k J k ) Heat-Entropy Flow J Q /T + Flux of Entropy due to Diffusion Entropy Production s = - 1/T 2 grad(t).j Q - J k.(grad(µ k /T) F k ); s = J Q.grad(1/T) - J k.(grad(µ k /T) F k ) > 0 Heat Conduction + Diffusion The Entropy production will stop when there is no more temperature gradient (grad(t) = 0) and no more Diffusion (J k = 0) 20

21 Entropy Production No external field s = - J k.grad(µ k /T) - 1/T 2 grad(t).j Q > 0 Diffusion under Isothermal conditions s = - 1/T J k.grad(µ k ) > 0 J k must be opposite in direction to grad(µ k )!!! No Diffusion s = - 1/T 2 grad(t).j Q > 0 J Q must be opposite in direction to grad(t)!!! 21

22 Entropy Production grad(µ k /T) = 1/Tgrad(µ k ) (µ k /T 2 )grad(t) Ts = - J k.[grad(µ k /T) F k ] - 1/Tgrad(T).J Q = -F k = grad(y k ) = - J S.grad(T) - J k.[grad(µ k ) + grad(y k )] = - J S.grad(T) - J k.grad(µ k +y k ) Ts = J k. F* k + J S.F T ; (5) each term has units J/m 3.s Conjugate fluxes and forces F T = -grad(t) Thermal force F* k = -grad(µ k + y k ) Thermodynamic Force acting on component k due to diffusion and/or presence of external field The thermodynmic force is the gradient of the chemical potential! Equilibrium is reached when all thermodynamic forces become zero 22

23 Chemical Gradient Complete Solid Solution Alloy P(A-rich) X B = X P G P Alloy Q (B-rich) X B = X Q G Q Tangent Construction µ AP, µ B P µ AQ, µ B Q x t = 0 Alloy P + Q is still not in equilibrium Overall mole fraction of B is X B R > X BP (X BR < X BQ ) t =, The system reaches equilibrium G R ; µ AR, µ B R µ AP > µ AR > µ AQ - At every x and t there is difference Dµ A (x) = µ AR + µ A / x 23

24 Chemical Gradient µ B µ B R x Diffusion of A species from left to right A Diffusion of B species from right to left B 24

25 Chemical Gradient Entropy Production Configurational Entropy (Ideal/Regular Solid Solution) DS mix = - R[(1-X B )ln(1-x B ) + X B ln(x B )] DS mix R > DS mix P Increse of the entropy DS [J mol -1 K -1 ] ,0 0,2 0,4 0,6 0,8 1,0 A B x B X P X R 25

26 Relations between Fluxes and Forces J k = f k (F); k = 1,2,...C or Q or external field At equilibrium all F = 0 and J k = f k (0) = 0 For small deviation from equilibrium (Taylor s expansion); Isotropic system! J k = m ( f k / F m )F m + m n ( 2 f k / F m F n )F m F n + Empirically for a large class of irreversible processes J k = m ( f k / F m )F m Linear Laws (Linear response theory) J k ~ L km F m ; L km ~ f k / F m (postulate) L km phenomenological coefficients [L km ] = 1/m 2.s.N Examples: Heat conductions, Diffusion, Electrical Conduction 26

27 Relations between Fluxes and Forces Ts = J k *. F k = k m L km F k. F m > 0 Properties of the Phenome nological Coefficients L kk > 0; L ii L kk > ¼ (L ik + L ki ) 2 Onsager s principle (for conjugate fluxes and forces) L ik = L ki ; L ik (L ii L kk ) ½ No mass flow condition 0 = k J k = k m L km F m = m F m ( k L km ) Lars Onsager k L km = 0 27

28 Heat Conduction Pure Heat Conduction J Q = lf T = - l grad(t); Fourier s law l thermal conductivity [J Q ] = J/m 2.s = W/m 2 [grad(t)] = K/m [l] = J/ K.m.s = W/K.m Entropy production s = - J Q.grad(T)/T 2 = = l [grad(t)] 2 /T 2 > 0 Jean-Baptise Fourier 28

29 Diffusion Energy Production No exteranal forces; No thermal gradient Ts = = k m L km F* k. F* m = k m L km grad(µ k ).grad(µ m ) L km = L kk δ km ; No interaction between the different species Ts = k L kk (grad(µ k )) 2 > 0; L kk > 0 29

30 Recommended Literature (Part II) # E.J. Mittemeijer Fundamentals of Materials Science # H. Mehrer Diffusion 30

Properties of Entropy

Properties of Entropy Due to its additivity, entropy is a homogeneous function of the extensive coordinates of the system: S(λU, λv, λn 1,, λn m ) = λ S (U, V, N 1,, N m ) This means we can write the entropy