Chapter 1 Thermodynamics and Phase Diagrams

Transcription

1 Chapter Thermoynamics an Phase Diagrams Equilibrium Single component systems inary solutions Equilibrium in heterogeneous system inary phase iagrams The influence of interfaces on equilibrium Ternary equilibrium itional thermoynamic relationships for binary solutions Computation of phase iagrams The kinetics of phase transformations

2 Thermoynamics an Phase Diagrams Introuction - 由於我們需要對相圖 (phase iagrams) 和相變態 (phase transformation) 進行一些評價, 本章將探討一些基本的熱力學 (thermoynamics) 觀念 - 熱力學在物理冶金 (physical metallurgy) 的主要用途為預測合金是否在平衡狀態 對於相變態而言, 我們關心的是系統朝向平衡狀態的改變, 因此, 熱力學為一有效的工具 ; 然而, 系統的平衡狀態何時可以達到, 則非熱力學所能解答的, 這必須使用動力學才可完成 Metastable Diamon????? Stable raphite Diamon is forever!

3 Equilibrium Introuction - 對於相變態的探討, 我們首先處理在一個系統 (system) 內所發生的改變, 例如 : 一合金 (alloy) 可以是一個或多個相所組成的混合物 (mixture) - 對於相變態的探討, 主要是討論在一合金或一系統內, 一個或多個相如何改變成一個新的相或多相混合物 相變態的發生是因為系統的最初狀態相對於最終狀態為不穩定, 如何去測量相之穩定性? 這就是熱力學的功用了 Phase ( 相 ) component ( 成份 ) composition ( 組成 ) 的定義 ( 說明如下頁 ) - Phase portion of the system whose properties an composition are homogeneous an which is physically istinct from other parts of the system. - Component The ifferent element or chemical compoun which makes up the system. - Composition The relative amounts of each component. vogaro s number ( a ) The number of atoms or molecules within mol of material is given by vogaro s number an is

4 Equilibrium Phase Component Composition : a% + b% 4

5 Equilibrium Equilibrium system is sai to be in equilibrium when it is in the most stable state. t constant temperature an pressure, a close system (i.e. one of fixe mass an composition) will be in stable equilibrium if it has the lowest possible value of the ibbs free energy, or in mathematical terms 0 (.3) From the efinition of ibbs free energy, the highest stability will be that with the best compromise between low enthalpy an high entropy. H TS (.) H E P (.) - t low temperatures, soli phases are most stable since they have the strongest atomic boning an therefore the lowest internal energy (enthalpy). - t high temperatures, the -TS term ominates, an phases with more freeom of atom movement, liquis an gases, become most stable. 5

6 Equilibrium Stable, metastable, unstable re all the conitions stable, which satisfy = 0? slope = 0 - Stable equilibrium Configuration has the lowest free energy an satisfies = 0, is therefore a stable equilibrium. - Metastable equilibrium Configuration lies at a local minimum in free energy an therefore also satisfy = 0, but which oes not have the lowest possible value of. Such configurations are calle metastable equilibrium. Fig.. schematic variation of ibbs free energy with the arrangement of atoms. Configuration has the lowest free energy an is therefore the arrangement when the system is at stable equilibrium. Configuration is a metastable equilibrium. - Unstable The intermeiate states for which 0 are unstable. 6

7 Equilibrium Criterion for phase transformation - raphite an iamon at room temperature an pressure are stable an metastable equilibrium states. iven time, all iamon uner these conitions will transform to graphite. - ny transformation that results in a ecrease in ibbs free energy is possible. 0 (.4) : free energy of the initial state, : free energy of the final state. - The transformation nee not go irectly to the stable equilibrium state but can pass through a whole series of intermeiate metastable states. How fast oes a phase transformation occur? It is epenent on the free energy hump between the metastable an stable states. In general, higher humps or energy barriers lea to slower transformation rates. The stuy of transformation rates in physical chemistry belongs to the realm of kinetics. Intensive an extensive properties - Intensive properties are inepenent of the size of the system. T (temperature), P (pressure) - Extensive properties are irectly proportional to the quantity of material in the system. (volume), E (internal energy), H (enthalpy), S (entropy), (ibbs free energy) energy barrier 7

8 Single Component Systems ibbs free energy as a function of temperature Pressure effects The riving force for soliification 8

9 ibbs Free Energy as a Function of Temperature Introuction single component system coul be one containing a pure element or one type of molecule that oes not issociate over the range of temperature of interest. Specific heat The specific heat is the quantity of heat (in joules) require to raise the temperature of the substance by one egree Kelvin. Its empirical equation is illustrate as Fig.. (a) ariation of C P with temperature, C P tens to a limit of ~3R. C P a bt ct a, b, c: constants. 9

10 ibbs Free Energy as a Function of Temperature Enthalpy The variation of H with T can be obtaine from the variation of C P with T. H E P H E P P H ( Q P ) P P Q H P C p C p Q T P H T (.5) P Fig.. (b) ariation of enthalpy (H) with absolute temperature for a pure metal. Consequently H can be measure relative to any reference level which is usually one by efining H = 0 for a pure element in its most stable state at 98K (5C). The variation of H with T can then be calculate by integrating C P. H T 98 CPT (.6) 0

11 ibbs Free Energy as a Function of Temperature 相變態 Entropy The variation of entropy with temperature can be erive from the specific heat. Taking entropy at zero egrees Kelvin as zero, Fig..3 (c) ariation of entropy (S) with absolute temperature. P P P p P P P P C T T S T H C T H S P TS H T H H S T S T C T S P P (.7) T P T T C S 0 (.8)

12 ibbs Free Energy as a Function of Temperature Fig.. C P a bt ct C p H 98 CPT T H T P S T 0 S T C P T T CP T P

13 ibbs Free Energy as a Function of Temperature ibbs free energy From the two above figures, we can get the variation of with temperature. For a system of fixe mass an composition H TS ST P (.9) T P S This means that ecreases with increasing T at a rate given by -S. (.0) Fig..3 ariation of ibbs free energy with temperature. 3

14 ibbs Free Energy as a Function of Temperature ariation of H an with temperature for the soli an liqui phase of a pure metal - t all temperatures the liqui has a higher enthalpy than the soli. Therefore at low temperatures L > S. H = E + P - The liqui phase has a higher entropy than the soli phase an the ibbs free energy of the liqui phase therefore ecreases more rapily with increasing temperature than that of the soli. - For temperature up to T m the soli phase has the lowest free energy an is therefore the stable equilibrium phase, whereas above T m the liqui phase is the equilibrium state of the system. - Latent heat of melting (L) is require to convert soli into liqui. Fig..4 ariation of enthalpy (H) an free energy () with temperature for the soli an liqui phases of a pure metal. L is the latent heat of melting, T m the equilibrium melting temperature. 4

15 5 Pressure Effects 相變態 Clausius-Clapeyron equation (for single component system) If the two phases in equilibrium are an m m eq m m m m S S T P P T S S T S P T S P ) ( ) ( T S T S T P S T H S S T H H TS H TS H eq ) ( T H T P eq eq (.4) S T P eq (.3) (.)

16 Pressure Effects pplication of Clausius-Clapeyron equation t constant temperature the free energy of a phase increases with pressure such that ST P P T (.) (CC) (FCC) - P/T along -iron an -iron H H P T H 0 0 H T 0 - P/T along -iron an -iron H P T H eq eq H eq 0 0 H T eq 0 - P/T along -iron an liqui iron H (CC) Fig..5 Effect of pressure on the equilibrium phase iagram for pure iron. H P T L L H eq 0 0 H T eq 0 (HCP) 6

17 Driving Force for Soliification Driving force for soliification (for single component system) If a liqui metal is unercoole by ΔT below T m before it soliifies, soliification will be accompanie by a ecrease in free energy Δ (J/mol) as shown in Fig..6. This free energy ecrease provies the riving force for soliification. The free energies of the liqui an soli at a temperature T are given by L S H H L S TS TS L L S H TS S L S L S H H T S S (.5) Fig..6 Difference in free energy between liqui an soli close to the melting point. The curvature of the S an L lines has been ignore. 7

18 Driving Force for Soliification t the equilibrium melting temperature T m the free energies of soli an liqui are equal, i.e. 0 H T T m m S L T S 0 L S (entropy of fusion) (.6) For small unercooling (ΔT) the ifference in the specific heats of the liqui an soli can be ignore. H T L TS T m Richar s rule (for single component system) The entropy of fusion is a constant R (8.4 J/mol-K) for most metals. m L T L T m Tm T L Tm (.7) 8

19 inary Solutions The ibbs free energy of binary solutions Ieal solutions Chemical potential Regular solutions ctivity Real solutions Orere phases Intermeiate phases 9

20 ibbs Free Energy of inary Solutions Introuction In alloys, composition is also variable an to unerstan phase changes in alloys requires an appreciation of how the ibbs free energy of a given phase epens on compositions as well as temperature an pressure. ibbs free energy of binary solutions ssume that an have the same crystal structures in their pure states an can be mixe in any proportions to make a soli solution with the same crystal structure. Imagine that mol of homogeneous soli solution is mae by mixing together mol of an mole of., : mole fractions of an respectively in the alloy. (.8) 0

21 ibbs Free Energy of inary Solutions The mixing can be mae in two steps, as shown in Fig..7. () ring together mol of pure an mole of pure ; () llow the an atoms to mix together to make a homogeneous soli solution. fter step () the free energy of the system is given by (in Fig..8) (.9), : molar free energies of pure an pure at the temperature an pressure. Fig..8 ariation of (the free energy before mixing) with alloy composition ( or ). Fig..7 Free energy of mixing. -

22 ibbs Free Energy of inary Solutions The free energy of the system will not remain constant uring the mixing of the an atoms an after step () the free energy of the soli solution can be expresse as mix (.0) H H mix mix TS TS H mix H H T S S TS mix H mix = H H : heat absorbe or evolve uring step, S mix = S S : ifference in entropy between the mixe an unmixe states. mix H mix TS mix (.)

23 Ieal Solutions Ieal solutions - When H mix = 0, the resultant solution is sai to be ieal. H TS TS (.) - In statistical thermoynamics, entropy is quantitatively relate to raomness by the oltzmann equation, i.e. k: oltzmann s constant, : a measure of ranomness. - There are two contributions to the entropy of a soli solution- a thermal contribution S th an a configurational contribution S config. If there is no volume change or heat change uring mixing then the only contribution to S mix is the change in configurational entropy. efore mixing, the an atoms are hel separately in the system an there is only one istinguishable way in which the atoms can be arrange. - Stirling s approximation If mix S k S k 0 S mix S mix! mix mix mix 3 (.3)

24 ibbs Free Energy of inary Solutions - ssuming that an mix to form a substitutional soli solution an that all configurations of an atoms are equally probable, the number of istinguishable ways of arranging the atoms on the atom sites is : number of atoms, : number of atoms. Since we are ealing with mol of solution, a : vogaro s constant. config a a! a!! (.4) 4

25 S config config config config config config config ibbs Free Energy of inary Solutions ( a a a a R = a k: universal gas constant. a! a a a a )!! ( a a a a a a a a a a a R k config ak a )! S R ) (.5) mix ( a! a! S 0 mix 0 0 T mix S mix ) < 0 (.6) mix ( Fig..9 Free energy of mixing 5 for an ieal solution.

26 ibbs Free Energy of inary Solutions mix (.6) - s the temperature increases, an ecrease an free energy curves assume a greater curvature. The ecrease in an is ue to the thermal entropy of both components. ST P T P S Fig..8 Fig..9 Fig..0 The molar free energy (free energy per mole of solution) for an ieal soli 6 solution. combination of Fig..8 an.9.

27 Chemical Potential Chemical potential - If a small quantity of, n mol, is ae to a large amount of a phase at constant temperature an pressure, the size of the system will increase by n an therefore the total free energy will also increase by a small amount. If n is small enough will be proportional to the amount of ae. : partial molar free energy or chemical potential. - Definition of the chemical potential of is - For a binary solution at constant temperature an pressure the separate contributions can be summe: - For a binary solution at constant temperature an pressure the separate contributions can be summe, if T an P changes are also consiere: - If mol of the original phase containe mol an mole, the size of the system can be increase without altering its composition if an are ae in the correct proportions, i.e. such that n : n = :. ' n ' n ' n (T, P, n : constant) (.8) T, P, n n ' ST P n n 7 (.9) (.30)

28 Chemical Potential - When is known as a function of an, an can be obtaine by extrapolating the tangent to the curve to the sies of the molar free energy iagram as shown in Fig... - For an ieal, binary solution (J/mol) (.3) (.3) - - Fig.. The relationship between the free energy curve for a solution an the chemical potentials of the components. Fig.. The relationship between the free energy curve an chemical potentials for an ieal solution. 8

29 Regular Solutions Regular solutions - For a soli solution, usually mixing is enothermic (heat absorbe) or exothermic (heat evolve). - It is assume that the heat of mixing, ΔH mix, is only ue to the bon energies between ajacent atoms. For this assumption to be vali it is necessary that the volumes of pure an are equal an o not change uring mixing so that the interatomic istances an bon energies are inepenent of composition. Fig..3 The ifferent types of interatomic bon in a soli solution. Three types of interatomic bons are present: () - bons each with an energy, () - bons each with an energy, (3) - bons each with an energy. The internal energy of the solution E will epen on the number of bons of each type P, P an P such that E P P P 9

30 Regular Solutions efore mixing pure an contain only - an - bons respectively, the change in internal energy on mixing is given by H mix P (.33) (.34) (.34) 可以想像為在 - 間形成之鍵能增加為, 但卻少了 - 和 - 的可能鍵結, 因此減少了 /( + ) - When = 0 H mix = 0 ieal solution Fig..3 30

31 3 Regular Solutions 相變態 - How to calculate P? Suppose z is the number of bons per atom. Suppose mol of atom an mol of atom, = +. a is vogaro s number. The probability of - is ) ( The probability of - is ) ( The probability of - is ) ( The probability of - is ) ( The total number of bons is z z z z P (.35) mix z z P H z H mix (.36) (.37)

32 Regular Solutions - Real solutions that closely obey Equation.36 are known as regular solutions. H mix = (.36) - How to explain in Fig..4? H H H mix mix mix H mix If = 0, H mix If =, H mix (slope) (slope) Fig..4 The variation of H mix with composition for a regular solution. 3

33 33 Regular Solutions 相變態 The free energy change on mixing a regular solution mix mix mix R T S T H mix (.38) mix mix (.39) (.40) (.3)

34 Regular Solutions The effects of ΔH mix an T on Δ mix H mix mix z - Exothermic 0 0 H mix mix Enothermic 0 0 H mix mix or mix Fig..5 The effect of ΔH mix an T on Δ mix. 34

35 ctivity Definition of activity: a, a a a (.4) Remember those for an ieal solution The activity of a regular solution a a Definition of activity coefficient:, a a a (.43) a Fig..6 The relationship between molar free energy an activity. a a (.4) 35

36 Henry s law & Raoult s law For a ilute solution in, 0. ctivity - Henry s law (for solute) a a exp cons tant (.44) - Raoult s law (for solvent) a 0 a exp0 (.45) (clustering) (orere) Fig..7 The variation of activity with composition (a) a (b) a. Line : ieal solution (Raoult s 36 law). Line : ΔH mix < 0. Line 3: ΔH mix > 0.

37 ctivity - Explanation of Fig..7 Funamentals Line Line Line 3 H mix ( a ) H mix a H mix a H mix a - The actual meaning of activity an chemical potential ctivity an chemical potential are simply a measure of the tenency of an atom to leave a solution. If the activity or chemical potential is low the atoms are reluctant to leave the solution which means, for example, that the vapor pressure of the component in equilibrium with the solution will be relatively low. Fig..7 37

38 Real Solutions Real Solutions - The actual arrangement of atoms will be a compromise that gives the (a) lowest internal energy consistent with sufficient entropy, or ranomness, to achieve the minimum free energy. - In systems with < 0 the internal energy of the system is reuce by increasing the number of - bons, i.e. by orering the atoms as (b) shown in Fig..8(a). - If > 0 the internal energy can be reuce by increasing the number of - an - bons, i.e. by the clustering of the atoms into -rich an -rich groups, Fig..8(b). However, the egree of orering or clustering will ecrease as temperature increases ue to the increasing importance of entropy. - When the size ifference between the atoms is very large then (c) interstitial soli solutions are energetically most favorable, Fig..8(c). - In systems where there is strong chemical boning between the atoms there is a tenency for the formation of intermetallic phases. Fig..8 Schematic representation of soli solutions: (a) orere substitutional, (b) clustering, (c) ranom interstitial. 38

39 Orere Phases Short-range orer If the atoms in a substitutional soli solution are completely ranomly arrange each atom position is equivalent an the probability that any given site in the lattice will contain an atom will be equal to the fraction of atoms in the solution, similarly for the atoms. In such solutions P, the number of - bons, is given by P = z. If < 0 an the number of - bons is greater than z, the solution is sai to contain short-range orer (SRO). The egree of orering can be quantifie by efining a SRO parameter s such that P s P (max) P (max) : maximum number of bons possible, P (ranom) : number of bons for a ranom solution. P ( ranom) P ( ranom) 39

40 Orere Phases Example of short-range orer - Ranom - solution with a total of 00 atoms an = = 0.5, P 00. P s ( ranom) P P (max) P ( ranom) P z ( ranom) 4 00 (max) P 00 - Same alloy with short-range orer: P = 3, P (max) 00, P s P (max) P ( ranom) P ( ranom) (a) (b) Fig..9 (a) Ranom - solution with a total of 00 atoms an = = 0.5, P ~ 00, s =0. (b) Same alloy with short-range orer P = 3, P (max) ~ 00, s =

41 Materials Science an Engineering Structure of Crystalline Solis Share of toms in Unit Cells Corner: /8 Ege-center: /4 Face-center: / oy-center: 4

42 Orere Phases Long-range orer (superlattice) In solutions with compositions that are close to a simple ratio of : atoms another type of orer can be foun as Fig..8(a). This is known as long-range orer. The entropy of mixing of structures with long-range orer is extremely small an with increasing temperature the egree of orer ecreases until above some critical temperature there is no long-range orer at all. In many systems the orere phase is stable up to the melting temperature. Orere substitutional structures in the Cu-u system (b) Cu: (0,0,0), (/,/,0) 8 8 u: (0,/,/), (/,0,/) 4 (c) Cu: (/,/,0), (/,0,/), (0,/,/) 6 3 u: (0,0,0) (atoms) (atoms) 8 8 (atoms) (atoms) x z y Fig..8 (a) Fig..0 Orere substitutional structures in the Cu-u system; (a) high-temperature isorere structure. (b) Cuu superlattice. (c) Cu 3 u superlattice. 4

43 Orere Phases (a) L 0 (b) L (c) L 0 () D0 3 (e) D0 9 Fig.. The five common orere lattices, examples of which are: (a) L 0 : CuZn, FeCo, il, Fel, gmg; (b) L : Cu 3 u, u 3 Cu, i 3 Mn, i 3 Fe, i 3 l, Pt 3 Fe; (c) L 0 : Cuu, CoPt, FePt; () D0 3 : Fe 3 l, Fe 3 Si, Fe 3 e, Cu 3 l; (e) D0 9 : Mg 3 C, C 3 Mg, Ti 3 l, i 3 Sn. 43

44 Intermeiate Phases Intermeiate Phases Intermeiate phases are often base on an ieal atom ratio that results in a minimum ibbs free energy. When small composition eviations cause a rapi rise in the phase is referre to as an intermetallic compoun an is usually stoichiometric, i.e. has a formula m n where m an n are integers, Fig..3(a). In other structures fluctuations in composition can be tolerate by some atoms occupying wrong positions or by atom sites being left vacant, an in these cases the curvature of the is much less, Fig..3(b). Fig..3 Free energy curves for intermeiate phases: (a) for an intermeiate compoun with a very narrow stability range, (b) for an intermeiate phase with a wie stability range. 44

45 Equilibrium in Heterogeneous Systems Introuction - It is usually the case that an o not have the same crystal structure in their pure states at a given temperature. In such cases two free energy curves must be rawn, one for each structure. - It is clear from Fig..5(b) that -rich alloys will have the lowest free energy as a homogeneous phase an -rich alloys as phase. - For alloys with compositions near the cross-over in the curves the situation is not so straightforwar. In this case it can be shown that the total free energy can be minimize by the atoms separating into two phases.? Fig..5 (a) The molar free energy curve for the phase. (b) Molar free energy curves 45 for an phases.

46 Equilibrium in Heterogeneous Systems The molar free energy of a two-phase mixture (+) Phase : composition, free energy Phase : composition, free energy Phase mixture: composition a t first, raw the lines,,, be Then cbg ca, eg fc 0 cf c be bg bc a ac bc bg ac a bc ac Fig..6 The molar free energy of a two-phase mixture (+ ). eg g ab cf c ac ab eg cf ac ab ac bg eg bc ac 0 ab ac 0 Therefore, the length be represents the molar free energy of the phase mixture. 46

47 Equilibrium in Heterogeneous Systems How to get the minimum free energy e? - Consier alloy 0 in Fig..7(a). If the atoms are arrange as a homogeneous phase, the free energy will be lowest as, i.e. per mole. 0 - The system can lower its free energy if the atoms separate into two phases with compositions an for example. The free energy of the system will then be reuce to. - Equilibrium between two phases requires that the tangents to each curve at the equilibrium compositions lie on a common line, Fig..7(b). e 0 e Fig..7 (a) lloy 0 has a free energy as a mixture of +. (b) t equilibrium, alloy 0 has a minimum free energy e when it is a mixture of e + e. 0 e e ( ) ( ) 47

48 Equilibrium in Heterogeneous Systems ctivity v.s. composition for a binary system For heterogeneous equilibrium, each component must have the same chemical potential in the two phases. a (.46) a a a (.47) 0 cons tant a a cons tant ( Raoult' s) ( Henry' s) Henry s Roult s Roult s Henry s etween an e, an e an, where single phase are stable, the activities are proportional to the composition, for ieal solutions. Fig..8 The variation of a an a with composition for a binary system containing 48 two ieal solutions, an.

49 inary Phase Diagrams simple phase iagram Systems with a miscibility gap Orere alloys Simple eutectic systems Phase iagrams containing intermeiate phases The ibbs phase rule The effect of temperature on soli solubility Equilibrium vacancy concentration 49

50 Single Phase Diagram Single Phase Diagram The simplest case to start with is when an are completely miscible in both the soli an liqui states an both are ieal solutions. Conitions: T > T m() > T > T m() > T 3 () t a high temperature T, the liqui will be the stable phase for pure an pure, Fig..9(a). () t T m(), Fig..9(b),, an this correspons to point a on the - phase iagram, Fig..9(f). (3) t T, the free energy curves cross, Fig..9(c), an the common tangent construction inicates that alloys between an b are soli at equilibrium, between c an they are liqui, an between b an c equilibrium consists of a two-phase mixture (S+L) with compositions b an c. (4) t T m(), Fig..9(),, an this correspons to point on the - phase iagram, Fig..9(f). S S L L (5) t T 3, the soli will be the stable phase for pure an pure, Fig..9(e). 50

51 Single Phase Diagram Stable: L S+L Stable: L S S+L L L S+L S S S+L S Fig..9 The erivation of a simple phase iagram from the free energy curves for the liqui (L) an soli (S). 5

52 物理冶金 Miscibility ap ( 混合性間隙 ) Miscibility ap - The two-phase fiel, where an are both stable, constitutes an example of what is commonly calle a miscibility gap. - necessary conition for the formation of a miscibility gap in the soli state is that both components shoul crystallize in the same lattice form. - It is quite possible to consier the bounary of the miscibility gap as two solvus lines that meet at high temperatures to form a single bounary separating the Fig..7 ol-nickel phase iagram. two-phase fiel from the surrouning single-phase fiels. - The fact that gol-nickel alloys exhibit a miscibility gap woul generally be constructe as evience that the boning energy for a gol-nickel pair is larger than the average of the bon energies for gol-gol an nickel-nickel pairs because this effect is expecte in segregation. i-u > ½ ( i-i + u-u ) 5

53 物理冶金 Miscibility ap - -ray iffraction measurements show a small but efinite short-range orer to exist in the soli solutions above the miscibility gap. - The large ifference in size of the gol an nickel atoms amounts to about 5% an is at the Hume-Rothery limiting value for extensive solubility. When a ranom soli solution is forme of gol an nickel atoms, the fit between atoms is rather poor an the lattice is baly straine. One way that the strain energy associate with the misfit of gol an nickel atoms can be relieve is by causing the atoms to assume an orere arrangement. Less strain is prouce when gol an nickel atoms alternate in a crystal than when gol or nickel atoms cluster together. - still greater ecrease in the strain energy of the lattice is possible if it breaks own to form crystals of the gol-rich an nickel-rich phases. - Miscibility gap are not only foun in soli solutions, but also often in the liqui regions of phase iagrams. 53

54 Systems with a Miscibility ap Systems with a Miscibility ap ( 混合性間隙 ) Conitions: T > T > T 3 The free energy curves for a system in which the liqui phase is approximately ieal, but for the soli phase H mix > 0, i.e. the an atoms islike each other. Therefore, at low temperature (T 3 ) the free energy curve for the soli assumes a negative curvature in the mile, an the soli solution is most stable as a mixture of two phases an with composition e an f. The + region is known as a miscibility gap. The reason why all alloys shoul melt at temperatures below the melting points of both components can be qualitatively unerstoo since the atoms in the alloy repel each other making the isruption of the soli into a liqui phase possible at lower temperatures than in either pure or pure. 54

55 Systems with a Miscibility ap L +L L +L + S H mix H Fig..30 The erivation of a phase iagram where mix, Free energy v. 55 composition curves for (a) T, (b) T, an (c) T 3. L 0

56 Orere lloys Orere lloys H S mix 0 When, melting will be more ifficult in the alloys an a maximum melting point mixture may appear. This type of alloy also has a tenency to orer at low temperature as shown in Fig..3(c). soli liqui L L L + L + L H S mix 0 Fig..3 Phase iagram when. 56

57 Eutectic ( 共晶反應 ) Simple Eutectic Systems L + L S + S oth soli phases have the same structure S H mix If is much larger than zero the miscibility gap can exten into the liqui phase. In this case a simple eutectic phase iagram results as shown in Fig..3. S L Cooling S 57

58 Simple Eutectic Systems Fig..3 The erivation of a eutectic phase iagram where both soli phases have the same crystal structure. 58

59 Simple Eutectic Systems Each soli phase has a ifferent crystal structure Fig..33 The erivation of a eutectic phase iagram where each soli phase 59 has a ifferent crystal structure.

60 Phase Diagram Containing Intermeiate Phases Peritectic ( 包晶反應 ) L + S S Phase Diagram Containing Intermeiate Phases When stable intermeiate phase can form, extra free energy curves appear in the phase iagram. n example is shown in Fig..34, which also illustrates how a peritectic transformation is relate to the free energy curves. L Cooling S S 60

61 Phase Diagram Containing Intermeiate Phases Fig..34 The erivation of a complex phase iagram. 6

62 Phase Diagram Containing Intermeiate Phases Stable compositions of intermeiate phase It is epen on the free energy curves. + + Fig..35 Free energy iagram to illustrate that the range of composition over which a phase is stable epens on the free energies of the other phase in equilibrium. 6

63 Phase Diagram Containing Intermeiate Phases Stable compositions of intermeiate phase It is epen on the free energy curves. + + m n Stoichiometric composition 63

64 ibbs Phase Rule ibbs phase rule If a system containing C components an P phases is in equilibrium, the number of egrees of freeom F is given by P F C P: number of phase, F: egree of freeom, an intensive variable such as T, P,,, C: number of component. (.49) ibbs-duhen s Equation n n ni n i i Similarly, n n i n n 0 T, P, n j n n 3 n 3 n n 3 n n 3 n n n... 0 n n 3 n 3 n nii 0 nih i 0 n i Si 0 3 n 3... n n n

65 65 ibbs Phase Rule 相變態 How to erivate ibbs phase rule? - Total variables: CP + (: temperature & pressure) - t equilibrium Inepenent equations: C(P-) - ibbs-duhen s equation Inepenent equations: P P C C C C P P P (.48) P i P i i i i i i i - ariables ) ( ) ( C F P P C F P P C CP F

66 ibbs Phase Rule Examples to use ibbs phase rule - If pressure is maintaine constant, for a binary system (P + F = C + ) - Degree of one-phase region: + F = + F = Degree of two-phase region: + F = + F = Degree of three-phase region: 3 + F = + F =

67 Effect of Temperature on Soli Solubility Systems with a Miscibility ap -t T (Fig..36) - For a regular solution (.40) - is the ifference in free energy between pure in the stable -form an the unstable -form. For e e e (.5) - If the solubility is low e e e e exp (.5) Fig..36 Solubility of in. 67

68 e e H exp exp Effect of Temperature on Soli Solubility TS H TS H S R Q e exp (.53) S exp R Q H (.54) H is the ifference in enthalpy between the -form of an the -form in J/mol. is the change in energy when mol of with the -structure issolves in to make a ilute solution. Therefore Q is just the enthalpy change, or heat absorbe, when mol of with the -structure issolves in to make a ilute solution. 68

69 Equilibrium acancy Concentration Equilibrium acancy Concentration () The removal of atoms from their sites not only increases the internal energy of the metal, ue to broken bons aroun the vacancy, but also increases the ranomness or configurational entropy of the system. () For simplicity, we consier vacancies in a pure e metal, the problem of calculating is almost ientical to the calculation of mix for an atoms when H Fig..37 Equilibrium vacancy mix is positive. concentration. (3) ecause the equilibrium concentration of vacancies is small, the problem is simplifie because vacancy-vacancy interactions can be ignore, an the increase in enthalpy of the soli (H) is irectly proportional to the number of vacancies ae. H H : mole fraction of vacancies, ΔH : increase in enthalpy per mole of vacancies ae. 69

70 70 Equilibrium acancy Concentration 相變態 S T H (.55) (4) There is a small change in the thermal entropy of S per mole of vacancies ae ue to changes in the vibrational frequencies of the atoms aroun a vacancy. However, the largest contribution is ue to the increase in configurational entropy. The total entropy change is thus - The equilibrium concentration of vacancies is therefore e S T H S T H S T H S T H 0 R S S S T H

71 Equilibrium acancy Concentration - If << an at equilibrium H TS e e 0 e S R H exp exp (.56) e exp (.57) The first term on the right-han sie of Equation.56 is a constant ~3, inepenent of T, whereas the secon term increases rapily with increasing T. In practice H is of the orer of e per atom reaches a value of about at the melting point of the soli. e 7

72 Influence of Interfaces on Equilibrium Influence of Interfaces on Equilibrium () In real situations, surfaces, grain bounaries, interphase interfaces, islocations an other efects o exist an raise the free energies of the phases. Therefore, the minimum free energy of an alloy is not reache until all interfaces an islocations have been anneale out. () Interphase interfaces can become extremely important in the early stages of phase transformations. (3) If the phase is acte on by a pressure of atm, the phase is subjecte to an extra pressure ue to the curvature of the / interface, just as a soap bubble exerts an extra pressure on its contents. If is the / interfacial energy (J/m ) an the particles are spherical with a raius r W ( P P ) P P r 4 3 4r, r 3 P P 8r 4r P P P r r r Fig..38 The effect of interfacial energy on the solubility of small particles. 7

73 Influence of Interfaces on Equilibrium ibbs-thomson effect or capillarity effect ST P t constant temperature, P P r m( ) r m (.58) m : molar volume of the phase, : / interfacial energy, r: raius of spherical particle. r The concept of a pressure ifference is very useful for spherical liqui particles, but it is less convenient in solis. This is because finely isperse soli phases are often non-spherical. Fig..38 The effect of interfacial energy on the solubility of small particles. 73

74 Influence of Interfaces on Equilibrium Transfer of from large to a small particle () Consier a system containing two particles, one with a spherical interface of raius r, an the other with a planar interface (r = ), embee in an matrix as shown in Fig..39. () If the molar free energy ifference between the two particles is, the transfer of a small quantity (n mol) of from large to the small particle will increase the free energy of the system by a small amount () n given by. Fig..39 Transfer of n mol of from large to a small particle. (3) If the surface area of the large particle remains unchange, the increase in free energy will be ue to the increase in the interfacial area of the spherical particle. n (.59) n n 4 r 3 3 an 4r m r r n r 8 n 4r m n r m r m 74

75 Influence of Interfaces on Equilibrium Practical consequence of the ibbs-thomson effect - The solubility of in is sensitive to the size of the particles. - It can be seen that the concentration of solute in in equilibrium with crosse a curve interface ( r ) is greater than, the equilibrium concentration for a planar interface. - ssuming that phase is regular solution an the phase is almost pure, i.e. ~ exp (from.5) m - Similarly, r can be obtaine using ( ) in replace of (Compare Fig..36) r m This figure can explain r r exp the mechanism of ispersion harening. m r exp (.60) r If y 0 exp(y) = + y r r m (.6) Fig..38 The effect of interfacial energy on the solubility of small particles. 75

76 Influence of Interfaces on Equilibrium -Example = 00 mj/m = 0.00 J/m m = 0-5 m 3 R = 8.3 J/mol-K T = 500 K r r m r r 0 r 9 For r = nm r / = For r = 0 nm r / =. For r = 00 nm r / =.0 Quite large solubility ifferences can arise for particles in the range r = -00 nm. 76

77 Ternary Equilibrium ibbs triangle: composition - The composition of a ternary alloy can be inicate on an equilateral triangle (the ibbs triangle) whose corners represent 00%, or C as shown in Fig ll points on lines parallel to C contain the same percentage of, the lines parallel to C represent constant concentration, an lines parallel to constant C concentrations. - Total percentage must sum to 00%. (.6) C : percentage of component, : percentage of component, C : percentage of component C. Fig..40 The ibbs triangle. 77

78 Ternary Equilibrium Ternary phase iagram - Since most commercial alloys are base on at least three components, an unerstaning of ternary phase iagrams is of great practical importance. - The ibbs free energy of any phase can be represente by a vertical istance from the point in the ibbs triangle. If this is one for all possible compositions the points trace out the free energy surfaces for all the possible phases, as shown in Fig..4a. - The chemical potentials of, an C in any phase are given by the points where the tangential plane to the free energy surfaces intersects the, an C axes. - In Fig..4b, alloys with compositions in the vicinity of the intersection of the two curves consist of + L at equilibrium. The compositions of the two phases in equilibrium must be given by points connecte by a common tangential plane, for example s an l. These points can be marke on an isothermal section of the equilibrium phase iagram (Fig..4c). - The lines joining the compositions in equilibrium are known as tie-lines. y rolling the tangential plane over the two free energy surfaces a whole series of tie-lines will be generate, such as pr an qt, an the region covere by these tie-lines pqtr is a twophase region on the phase iagram. - The relative amounts of an L are simply given by the lever rule. 78

79 Ternary Equilibrium t higher temperature t lower temperature Fig..4 (a) Free energies of a liqui an three soli phases of a ternary system. (b) tangential plane construction to the free energy surfaces efines equilibrium between s an l in the ternary system. (c) Isothermal section through a ternary phase iagram obtaine in this way with a two-phase region (L + S) an various tie-lines. The amounts of l an s at point x are etermine by the lever rule. 79

80 Ternary Equilibrium - There are three-phase regions known as tie-triangles. The L++ triangle for example arises because the common tangential plane simultaneously touches the, an L surfaces. - If the temperature is lowere still further the L region shrinks to a point at which four phases are in equilibrium L+++. This is known as the eutectic points. Fig..4 Isothermal sections through Fig

81 Ternary Equilibrium Fig..44 The equilibrium soliification of alloy. 8

82 Ternary Equilibrium Fig..43 projection of the liquius surface of Fig..44 onto the ibbs triangle. Fig..45 vertical section between points, an in Fig

83 itional Thermoynamic Relationships Systems with a Miscibility ap From Fig..46, D FD HF F HF D FD F D FD D FD FD F The value of is, an the value of F is. F ( ) (.63) (.64) ( ) C D Fig..46 Evaluation of the change in chemical potential ue to a change in composition. ( ) (multiply ) H F E (.65) 83

84 itional Thermoynamic Relationships The erivation of (for a regular solution) For an ieal solution, = 0, (.67) 84

85 itional Thermoynamic Relationships The erivation of (for a regular solution) (.66) 85

86 86 itional Thermoynamic Relationships 相變態 The erivation of a (.68) (.69) a (.70) (.7) (.65)

87 Kinetics of Phase Transformations kinetics of Phase Transformations - The stuy of how fast processes occur belongs to the science of kinetics. - Driving force () If an are the free energies of the initial an final states = - - Thermal activation efore the free energy of the atom can ecrease from to, the atom must pass through a socalle transition or activate state with a free energy a above. Fig..47 Transformations from initial to final state through an activate state of higher free energy. s a result of ranom thermal motion of the atoms, the energy of any particular atom will vary with time an occasionally it may be sufficient for the atom to reach the activate state. This process is known as thermal activation. 87

88 Kinetics of Phase Transformations -Rate ccoring to kinetic theory, the probability of an atom reaching the activate state is given by exp(- a /kt). Rate exp( a / kt ) Fig..47 Δ a : activation free energy barrier, k: oltzmann s constant. - rrhenius rate equation a H a TS a a Rate exp( H / ) (.7) 88