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CONFIGURATIONAL THERMODYNAMICS

CONFIGURATIONAL THERMODYNAMICS. Rafał Kozubski. Institute of Physics Jagellonian University Krakow, Poland. FOUNDATIONS OF THERMODYNAMICS. Thermodynamic equilibrium 2. Basic parameters and functions: Internal energy U : sum of all types of energies of atoms/molecules

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CONFIGURATIONAL THERMODYNAMICS

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  1. CONFIGURATIONAL THERMODYNAMICS Rafał Kozubski Institute of Physics Jagellonian University Krakow, Poland

  2. FOUNDATIONS OF THERMODYNAMICS • Thermodynamic equilibrium • 2. Basic parameters and functions: • Internal energy U: sum of all types of energies of atoms/molecules building a macroscopic body. • Heat Q: energy transferred between systems being in thermal contact •  Temperature T: parameter, whose value is equal in bodies being • in thermal equilibrium

  3. Irreversible evolution of systems towards equilibrium – experimental observation • General concept of entropy: • Entropy S is an extensive andadditive function of a state of a system: • SYSTEM 1: entropy S1 • SYSTEM 2: entropy S2 • SYSTEM composed of SYSTEM 1 AND SYSTEM 2: • entropy S = S1 + S2 • Entropy S increases in any spontaneous evolution of an isolated system. • Consequence: Maximum possible value of entropy is attributed to an equilibrium state of an isolated system

  4. In general S is a function of system internal energy U and all remaining parameters {xi} The following relations are proven: shows features of temperature – i.e. is equal in systems being in thermal equlibrium 1. 2. As: the I-st principle of thermodynamics requires that: , where DQ denotes heat transfer

  5. Isolated supersystem: SYSTEM + large surrounding in thermal equilibrium S – entropy of a SYSTEM Surrounding (very large) SYSTEM Entropy production Seli It MUST hold: Entropy flux Infinitesimal change of S For the whole process: Hence: II-nd principle of thermodynamics

  6. The II-nd principle of thermodamics leads to the formulation of equilibrium conditions in particular circumstances. Implementing theI-st principle of thermodynamics: one obtains: , where Wel denotes work performed ON the system hence:

  7. , adiabiatic-isochoric process (V=0): Equilibrium is determined by minimum internal energy U ________________________________________________________ , adiabatic-isobaric process (p=0): Equilibrium is determined by minimum enthalpyH ________________________________________________________ ,system in thermal equilibrium with surrounding, V=0: Equlibrium is determined by minimum Helmholtz free energyF ________________________________________________________ , system in thermal equilibrium with surrounding, p=0: Equilibrium is determined by minimum Gibbs free enthalpy G

  8. USEFUL RELATIONSHIPS: Chemical potentiali of an i-th component:

  9. BASIC IDEAS OF STATISTICAL THERMODYNAMICS Objects:systems composed of many particles Natural application: macroscopic bodies composed of N  1023 atoms/molecules Basic notions: Microscopic state of a macroscopic body: a state given by particular states of all atoms/molecules  Macroscopic state of a macroscopic body: a state directly observed (shape, hardness, roughness, colour etc.) Basic postulate: A macroscopic state may be realised by microscopic states{si} appearing with a probability P(si) andcharacterised by energies E(si)Observable parameters of macroscopic states are averages over corresponding parameters of microscopic states. E.g.

  10. QUANTITATIVE APPROACH: STATISTICAL INTERPRETATION OF ENTROPY Ludwig Boltzmann born 20 February 1844 in Vienna, Austriadied 5 October 1906 in Duino near Trieste

  11. ISOLATED SYSTEM: U = const Macroscopic states of an isolated system are realised EXCLUSIVELY by microscopic states{si} with E(si) = U. All the microscopic states {si} appear with THE SAME probability where p is the number of microscopic states{si} ENTROPY S of the isolated system is identified with: Consequence:macroscopic states of the isolated system evolve PRACTICALLY in the way that p increases: S . Reverse processes are NEGLIGABLY probable

  12. Surrounding (very large) SYSTEM SYSTEM IN THERMAL EQUILIBRIUM WITH SURROUNDING STANDARD ASSUMPTIONS: The system is much smaller than the surrounding  The whole supersystem (system + surrounding) is isolated - hence its energy U0is constant and all microscopic states are equiprobable  The system is in thermal equilibrium with the surrounding - hence its temperature equals T0 Question: What is the probability P(si) that the system is in a microscopic state Q with the energy E(si) ? NOTE ! The question makes sense, because the system is not isolated !!

  13. p – number of possible microstates of the supersystem, for which the system is in the microstate si P(si) p EntropyS0of the surrounding: But: and thus: partition function where It is proven that:

  14. PROOF: Let us assume: then: finally: -S U

  15. The theory is consistent if we assume that: For an isolated system: hence

  16. Interpretation of temperature: a quantity T proportional to an average kinetic energy of one atom/molecule - a measure of chaotic motion of atoms/molecules Factors controlling the generation of a concrete macroscopic state of a macroscopic body: (I) Tendency for attaining possibly lowest value of internal energy U, (II) Probability for reaching such energy – probability of appearanceP(s) of microscopic states corresponding to this energy. Importance of the factor (II) increases with increasing temperature, because CONSEQUENCE: The observed equilibriummacroscopic state is a compromise between both factors. Quantitatively it corresponds to a minimum value of a function F = E + T  kB <ln(P)>  free energy

  17. A B A - B Thermodynamics of solid solutions Solid solution: A solid multicomponent phase existing for a finite range of the compoment concentrations. Configurational free energy of a solid solution: part of the free energy which depends exclusively on the configuration of atoms over the crystalline lattice configurational entropy of mixing configurational free energy of mixing configurational internal energy of mixing

  18. THEORETICAL JUSTIFICATION FOR THE SEPARATION OF NON-CONFIGURATIONAL TERMS OF THE FREE ENERGY:

  19. IDEAL SOLUTION Micoscopic states: distinct configurations of A and B atoms over crystalline lattice sites of the solid solution As Du=0 all the configurations appear with the same probability p– number of distinct configurations of A and B atoms over the crystalline lattice of the solid solution therefore valid for large N

  20. REGULAR SOLUTION internal energy of a solution internal energy of not-mixed elements THERMODYNAMICS: COMMENT: Such grouping of energies is reasonable due to different time scales (mentioned earlier) configurational energy energy of other degrees of freedom related to particular configuration Consequence:

  21. The partition function Z may thus be written down as: where “average” configurational energy accounting for averaging over non-configurational degrees of freedom assigned to a particular configuration Modelling of the (average) configurational energy First approximation: pair-interactions of nearest-neighbouring (nn) atoms – the Ising model: ,where Nij is a number of i-j nn pairs, Vijis the i-j nn pair interaction energy

  22. 1 1 1 1 1 1 1 1 2 TWO-COMPONENT (BINARY) A-B SYSTEM Description of the atomic configuration of the system: N – total number of atoms (both A- and B-type), C – concentration of B-type atoms C = NB/N Z – co-ordination number: number of nn lattice sites surrounding a lattice site; in bcc structure Z = 8 Numbers of A- and B-type atoms occupying 1- and 2-type sublattice sites bcc-type lattice with two sublattices consisting of N(1) and N(2) lattice sites, respectively Fundamental relationships: CONCLUSION: are independent variables !! and

  23. ATOMIC (CHEMICAL) ORDER Long-range order (LRO): Differentiation „” of probalilities of particular sublattice sites being occupied by particular atoms. Diffraction: superstructure peaks Related variables: LRO parameter: presence of LRO absence of LRO Short-range order (SRO): Tendency for A-(B-) atoms to be preferencially surrounded by B- or A-atoms (correlation functions). Diffraction: diffuse scattering, background modulation between Bragg peaks Related variables: P(NAA) = 2NAA/ (ZxN) SRO parameter:  = P(NAA)-(1-c)2 NOTE:  and  are totally independent !

  24. A B A - B CONFIGURATIONAL ENERGY OF MIXING EM CONCLUSION: the configurational energy of mixing depends exclusively on NAA – as a configurational variable and on W – as an energetic variable

  25. variables, variables variables, variables GENERALIZATION: Within pair-interaction approximation: n-component systems, structures with m sublattices + pair-interactions in r co-ordination zones + Beyond pair-interaction approximation: many-body interactions numbers of clusters with particular atomic configurations

  26. CONFIGURATIONAL FREE ENERGY OF MIXING constant factor Within the nn pair-interaction approximation: number of ALL configurations showing similar values of NAA Basic thermodynamical approximation: The partition function is approximated by its maximum term

  27. Hence: Where the functional of the configurational entropy of mixing: Conclusion: the equilibrium value of the parameter NAA at temperature T minimises the free energy functional F BASIC DIFFICULTY AND THE PRINCIPAL PROBLEM OF CONFIGURATIONAL THERMODYNAMICS: It is impossible to exactly evaluate the number g(NAA). (The exact solution (by Onsager) exists only for 1- and 2-dimensional lattice) The same problem appears when working with many-body potentials – no exact evaluation of g({Nijk…}). Formulation of appropriate approximation methods for the evaluation of g is one of the main tasks of the configurational thermodynamics.

  28. CONCEPT OF CLUSTER VARIATION (CVM) R. Kikuchi, Phys.Rev. 81, 988, (1951). ijkl pi Nii ijk Complete description of an atomic configuration of a crystal: information on the occupation of EACH lattice site – unfeasible ! – but necessary for accurate determination of the free energy General assumption of the CVM: The atomic configuration of a crystal is given in terms of cluster variables {ijk… }: the probabilities that finite clusters of the lattice sites appear in particular configurations (feasible to be given explicitly). In an effective analysis clusters up to an arbitrarily chosen biggest one are considered. The bigger is the largest cluster, the more accurate is the description. Asymptotically, the exact description is achieved if the entire crystal is taken as the biggest cluster. The effective procedure consists of the minimisation the F functional with respect to{ijk… }: in nn pair approximation Methods are developed for finding g({ijk…})

  29. “0th” (Bragg-Williams) approximation: B.J. Bragg, E.J. Williams, Proc.Roy.Soc., A151, 540, (1935); A152, 231, (1935) The biggest cluster:a single lattice site Cluster variables:pA1, pA2, pB1, pB2  Basic approximation: The approximation consists of the negligence pf pair-correlations and relates NAA to  - which, as was shown, is not true ! The approximation yields:

  30. The approximation yields: and: Stirling formula yields: g() is a DECREASING function of 

  31. decreasingfunction of  W>0: increasing function of  W<0: decreasing function of  CONCLUSION: If W > 0, F = F(=0) – no atomic ordering at any temperature If W < 0, F = F(=1) at T=0 K – atomic ordering within a finite range of temperatures.

  32. What happens when W < 0 ? F T1 T2 T3 T4  T5 T7 T6 TC  T T1>T2>T3>T4>T5>T6>T7 B2 superstructure: At T T3Fmin = Fmin(=0) At T  T4Fmin = Fmin(>0) Only one single minimum of F appears ! Consequently: TCevaluated from the equations: Continuous “order-disorder” phase transition at T = TC

  33. TT2: Fmin = Fmin(=0) T<T2: F(1)min (=0), F(2)min(>0); F(2)min(>0) > F(1)min (=0) T=Tt: F(2)min(>0) = F(1)min (=0) T < Tt: F(2)min(>0) < F(1)min (=0) T  TC:Fmin = Fmin(>0) F T1 T2 T3 T4  L12 superstructure: T1>T2>T3>T4 two minima of F() Tt Tt Ordered (>0) and disordered (=0) phasescoexistas long as Fshows two minima Discontinuous “order-disorder” phase transition at T = Tt

  34. What happens when W > 0 ?  = 0 T1 < T2 < T3

  35. Phase 1: NA(1), NB(1) N(1)= NA(1)+ NB(1) NB(1)=c(1)N(1) f1 – free energy per one molecule Phase 2: NA(2), NB(2) N(2)= NA(2)+ NB(2) NB(2)=c(2)N(2) f2 – free energy per one molecule Interpretation: two-phase equilibrium: lever rule Entire system: A-B: N=N(1)+N(2) NA=NA(1)+NA(2) NB=NB(1)+NB(2) NB=C0N Hence:

  36. DECOMPOSITION OF A BINARY SOLUTION A-B INTO TWO PHASES: c0 – concentration of B-atoms in the homogeneous solution (before the decomposition c1, c2 – concentrations of B-atoms in the two phases, into which the solution decomposes Decomposition DECREASES the free energy of the system Decomposition INCREASES the free energy of the system Decomposition into c1’ and c2’ INCREASESF, but the continuation to c1 and c2 finally DECREASES the free energy of the system

  37. SPONTANEOUS AND ACTIVATED DECOMPOSITION activated decomposition: nucleation & growth activated decomposition: nucleation & growth Spinodal (barrier-less) decomposition homogeneous solution homogeneous solution T = T2 The solution withc1eq < c0 < c2eqdecomposes into two phases with concentrations equal toc1eqandc2eq

  38. MISCIBILITY GAP AND SPINODAL homogeneous solution spinodal decomposition nucleation & growth

  39. PHASES IN EQUILIBRIUM USEFUL RELATIONSHIPS: Chemical potentiali of an i-th component:

  40. SUBJECT OF INTEREST: System composed of „i” components and „” phases: • Assumptions: • As a whole, the system is closed, component particles are free • to migrate (diffuse) from phase to phase. • Isothermai-isobaric process is considered: T = const, p = const Particular process: dni i-type particles move from -phase to -phase: • Conclusions: • i-type particles move spontaneously from -phase to -phase (dG0) • if min < mim. • The system is in equilibrium with respect to particle diffusion from • phase to phase provided chemical potentials of particular • components are equal in all phases.

  41. It holds: Differentiation of both sides with respect to l and substitution  = 1 yields: Attention: The above reasoning IS NOT CORRECT for the Helmholtz free energy F: V is , however, NOT an intensive parameter and may be ni-dependent !!

  42. PRACRICAL METHOD FOR THE EVALUATION OF CHEMICAL POTENTIALS 1. Binary system A-B

  43. GRAPHIC INTERPRETATION OF PHASE EQUILIBRIUM A tangent common to g(c) curves corresponding to phases 1 and 2 means that the chemical potentials of A- and B-components are respectively equal in both phases if their compositions are given by C(1) and C(2). Conclusion: At particular temperature and pressure T, P the phases1and 2 may coexist provided the respective concentrations equalC(1)and C(2).

  44. EQUILIBRIUM OF AN OPEN SYSTEM I-st principle of thermodynamics: Surrounding TO, O System T0, 0 II-nd principle of thermodynamics: Hence for T = TO,  = O,Wel =0 (V=const): Attention: Minimisation of is equivalent to conditional minimisation of F at i = const - Grand potential

  45. PHASE EQUILIBRIUM AS A DISCONTINUOUS PHASE TRANSITION IN THE FIELD OF CHEMICAL POTENTIALS  functional: Binary system A-B:

  46. THEORY OF SPINODAL DECOMPOSITION Free energy of an inhomogeneous system CONTINUOUS MEDIUM approach: c – average concentration free energy per one atom free energy per one atom for a homogeneous system

  47. Cubic crystal: inversion:xj - xi: rotation: xixj:

  48. but: hence and: For systems which are homogeneous at any temperature there must hold:

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