IV. Phase Diagrams 6. Equilibrium Phase Diagrams · IV. Phase Diagrams 6. Equilibrium Phase...

56 Thermodynamics and Kinetics of Solids ________________________________________________________________________________________________________________________

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IV. Phase Diagrams

6. Equilibrium Phase Diagrams Phase: Physically homogeneous region in thermodynamic equilibrium which is limited by a surface that allows to separate it mechanically from other parts of the sample. Phases may be distinguished by their different physical character (structure etc.). Component: Independently variable chemical constituent (e.g. 2 in the case of Al2O3 + SiO2). Degree of Freedom: Number of intensive variables (e.g. temperature, pressure, concentration) which have to be fixed in order to define the system completely. Phase rule (Gibbs) f = c – p + 2 (6.1) 6.1. Monocomponent Systems f = 3 – p p = 1 : f = 2 (T and p are variable) p = 2 : f = 1 (boarder lines IA, IB and IC in Fig. 6.1) p = 3 : f = 0 (point I in Fig. 6.1) I: Invariant or Triple Point Boarder Lines: Univariant equilibria Areas: Bivariant equilibria AI: Sublimation curve (solid state + vapour coexist) IC: Evaporation curve (liquid + vapour coexist)

Upper End: Critical Point (gas may not be liquified

above this temperature) IB: Melting Curve (solid state and liquid coexist) (variation of the melting point of the solid phase

with pressure change) Metastability. A metastable phase has always a higher vapour pressure than another (more stable) phase at the same temperature. A metastable phase is prepared by a specific procedure. Example: Supercooled water may be only prepared by decreasing the temperature of water, but not by increasing the temperature of ice.

T Fig. 6.1. Equilibrium - pressure – temperature diagram of a monocomponent system (general)

Fig. 6.2 Pressure-temperature phase diagram of water.

Fig. 6.3 Polymorphic phases of SiO2 (allotropism)

Thermodynamics and Kinetics of Solids 57 ________________________________________________________________________________________________________________________

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SiO 2: As in the case of many other ceramic crystals: high melting point, low vapour pressure (Figs. 6.3 and 6.4). Most s table state: Minimum of Gibbs energy G = H – TS. The entropy and enthalpy increase with increasing temperature, but the Gibbs energy decreases (Figs. 6.5 and 6.6).

Fig. 6.4. Pressure-temperature diagram of SiO2.

Fig. 6.8 Stable and metastable phases of ZrO2 as a function of temperature. M → T at ̃ 1000 °C, large volume change (˜ 7%). T → C at ̃ 2300 °C, melting point: 2715 °C

TiO 2, ZrO 2

Fig. 6.7. Stable and metastable phases of TiO2 as a function of temperature.

Fig. 6.5 Gibbs energy, enthalpy and TS of SiO2 as a function of

temperature (schematic).

Carbon

Fig. 6.9. Phase diagram of carbon as a function of pressure and temperature Graphite → Diamond: > 4000 °C, p ≈ 150 kbar

Fig. 6.6 Temperature at which a transition from the α to the β

structure occurs (SiO2) in view of the minimum of the Gibbs

energy.


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6.2. Two-Component Systems (Binary Systems) 3 variables: T, p, x p-T-diagrams for the pure components A and B: In most cases determination of the phase relations for p = 1 atm (Fig. 6.10). In this case, the variables are T, x (Fig. 6.11). ME: Liquidus-Line (no crystals exist above this temperature) FG: Solidus-Line (no liquid exists below this temperature) Phase rule: p = 1 : f = 2 (T, x) p = 2 : f = 1 (T or x) Isothermal line through the 2-phase-region: Tie-Line („conode“). This line connects conjugated phases. Tie-

Line A3-M3 (Fig. 6.11): Liquid consists of 62 % B and 38% A. Binary Eutecticum We consider the path X in Fig. 6.11. The addition of component B to A decreases the freezing point → downward benditure of the liquidus line. More crystals are being formed with decreasing temperature which become separated from the melt. The melt becomes more rich in component B with decreasing temperature. At E the last remaining liquid crystallises. The reaction of separation of a liquid into 2 crystalline phases is called “eutectic reaction”; E is called "eutectic point": melt ⇔ crystal A + crystal B The reaction is an isothermal 3-phase reaction. At the eutectic ratio, a characteristic microstructure is being formed. This is frequently a lamellar structure. Both phases grow simultaneously from the supercooled melt. For example, first a nucleus of phase A is formed

Fig. 6.12 Indication of the composition at S: The distance A-S and S-B correspond to the amount of B and A, respectively. The ratios may be given in mass-% or atom mol %.

Fig. 6.13 Liquidus lines ME, NE: Solubility limits of A, B in the melt.

Fig. 6.11 Binary phase diagram A-B at constant pressure, e.g. 1

atm.

Fig. 6.10 T, p, x-phase diagram of a binary system.


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which starts to grow (Fig. 6.16). Atoms A are taken from the melt which becomes more rich in B and phase B crystallizes. The region around B becomes more rich in A which results in the formation of phase A again. With the simultaneous growth of the lamellos in the melt, a counter diffusion of A and B atoms along the preceeding interface solid-melt takes place.

The velocity depends on the degree of supercooling and the diffusion rates. Practical application: Control of the electrical, magnetic, optical, mechanical and thermal properties by controlled microstructure. 6.3. Binary Systems with Intermediate Compounds Classification with regard to the melting behaviour. AB2 (Fig. 6.17) melts congruently since the compound melts directly into a liquid of the same chemical composition. Congruently melting compounds separate a systems into separate smaller binary systems (AB2 – A und AB2 – B; both are simple eutectic systems). Fig. 6.18 shows the reactions uponcooling of a sample at

constant composition (35 w/o Al2O3, 65 w/o MnO): 1520+ °C: 23 units of melt + 10 units of crystalls (MnO · Al2O3) 1520– °C: 23 units of eutectic crystals + 10 units of crystals (MnO · Al2O3)

Fig. 6.18 Reactions upon cooling of a sample with constant composition (35 w/o Al2O3, 65 w/o MnO)

Fig.. 6.14 Microscopic description of the phases and liquid formed upon cooling at 30% B.

Fig. 6.15 Formation of a lamella structure upon cooling a eutectic mixture.

Fig. 6.16 Microscopic description of the diffusion of A and B atoms along the interface solid-liquid.

Fig. 6.17 Formation of an intermediate phase AB2 which melts congruently and separates the phase diagram into two separate smaller binary systems.


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Fig. 6.19: Microstructure at the composition x. Incongruently melt ing compounds do not melt into one liquid of the same composition but melt into another liquid and form another crystalline phase. Fig. 6.20: 3 phases are in equilibrium with each other along the line PN:

Cr2O3 · V2O5 ⇔810°C

Cr2O3 + melt. This reaction of incongruent melting is being called peritectic reaction. P: Peritectic Point. Temperature decrease at the composition x: Liquidus line → Cr2O3; 810 °C → Cr2O3 + melt → Cr2O3 · V2O5. The composition of the melt has not changed; however, the amount of melt has been reduced. The disappearance of Cr2O3 is called resorption. Rapid Quenching: Microstructure with only partially reaction (Fig. 6.21). Formation of a central nucleus.

Decomposition: - Formation of 2 other crystalline phases (Fig. 6.22) - Phase transformation (Fig. 6.23) 6.4. Solid Solutions. Cooling along x (Fig. 6.24) → increasing amount of solid solution and change in the composition. Counter Diffusion of Ni and Mg has to occur. Because of the limited velocity, a concentration gradient is being formed. Solid solutions may also have maxima and minima (Fig. 6.26). I: Indifferent or congruent point.

Fig. 6.22 Formation of an intermediate phase between temperatures T 1 and T2.

Fig. 6.23 Phase diagram with phase transformations of the intermediate compound A2B.

Fig.. 6.19 Microstructure upon cooling of a binary eutectic

systems.

Fig. 6.20 Binary phase diagram of a peridectic system.

Fig. 6.21 Microstructure with only partial reaction; Formation of a nucleus


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A miscability gap may occur and may extend up to the liquid phase and may form an eutectic system (Fig. 6.27).

Fig. 6.25 Solidification of the melt upon passing through the regime between the liquidus and solidus line.

Fig. 6.24 Binary phase diagram of a solid solution system with liquidus and solidus line.

Fig. 6.27 Growth of a miscability gap of a solid solution system A-B. Eventually a two-component system of immiscible components A and B is being formed.

Fig. 6.26 At I: Tangent of the liquidus curve = Tangent of the

solidus curve = Isothermal line.


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Fig. 6.29 shows a liquid immiscibility (example: ether and water). Cooling along x below MRT L1 → A + L2 (liquid → crystal + liquid) (Monotectic reaction) M: Monotectic point MRT has to be isothermal (phase rule)

Examples of phase diagrams with solid solutions are shown in Fig. 6.28

Fig. 6.28 Typical phase diagrams with solid solutions of the A and B phase

Fig. 6.29 Phase diagram of an immiscibility of two liquids (ether and water)


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6.5. Ternary Systems. f = 3 – p + 2. Invariant point: 5 phases coexist: p = const: 4 phases. Representation of ternary systems: see Fig. 6.31. Liquidus: Surface with hills and valleys. Sides: binary systems with binary eutectic points f, g, h. Eutectic point: E (invariant point, since 4 phases coexist.) The ternary phase diagram consists of „volumes“ or „spaces“: - Liquid-Space - Spaces of primary crystallisation (A + liquid, B +

liquid, C + liquid; Fig. 6.32) - spaces of binary crystallisation (A + B + liquid, A + C

+ liquid, B + C + liquid; Fig. 6.33)

Fig. 6.30 shows a spinodal decomposition: The microstructure shows diffuse boundary layers between the phases.

Fig. 6.30 Spinodal Decomposition

Fig. 6.31 Separation of a ternary phase diagram into sections of primary and secondary crystallisation

Fig.. 6.33 Description of the composition of a ternary phase triangle according to the intersections along binary legs.

Fig. 6.32 Description of the position in a ternary phase triangle. The distance to the side lines corresponds to the amount of the indicated components.


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Presentation of the composition: - The lengths of the lines perpendicular to the binary legs

of the triangle composed of the components A, B and C are proportional to their concentrations (Fig. 6.32).

- Lines parallel to the legs of the triangle. The intersections with the binary legs provide the relative concentrations. This method is also applicable for non-equilateral triangles (Fig. 6.33).

Representation of the changes during cooling of a melt by application of the planar projection of the liquidus plane (Fig. 6.34). Temperature scale: Series of isothermal lines along the liquidus planes. Projection of these isotherms onto the basis plane. The cooling path of each melt runs down to the eutectic point E (Fig. 6.35). The areas BHEF, AFEG and CHEG are called „primary phase fields“ of B, A and C.

In the case of a melt with the composition in the primary phase field B, the first crystal being formed is B. Along the boarder lines HE, GE and FE, 2 crystalline phases are in equilibrium with the melt. A straight line which connects the compositions of the primary crystals of 2 areas with common boarder line is called Alkemade-line. In the case shown in the Fig. 6.36, an intermediate compound BC with a primary phase field bc is being formed. The line A - BC is a line connecting the compositions of the primary crystals of two areas with common boarder line (ED). A - BC is an Alkamade-line. B - C is not an Alkamade-line. Alkemade-Theorem: The intersection of a boarder line between two liquidus planes with the corresponding

Fig. 6.34 3-dimensional representation of a ternary phase diagram with liquidus planes, isothermal lines along the planes and eutectic point E

Fig. 6.35 Projection of the liquidus plane and isotherms of the liquidus plane onto the basis plane.

Fig. 6.36 Boarder lines of the liquidus planes and Alkemade-

line of a ternary system with a binary compound BC.


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Alkemade-line represents a temperature maximum along the boarder line and a temperature minimum along the Alkemade line. 3 boarder lines come together at a common intersection at E and D (= eutectic points). Compositional Triangles. Alkemade-lines separate a ternary diagram into compositional triangles. 3 Alkemaade lines generate a compositional triangle. Alkemade lines never cross each other. Upon cooling of a melt, a combination of crystals with a composition of the corners of the compositional triangles is being formed. Example: The composition Y is within the triangle A-BC-C in Fig. 6.37. The relative amounts are given by the intersections of the lines parallel to the legs of the compositional triangles. Eutectic point: D (30% A, 48% BC, 22% C). A - BC represents a true binary system. If the Alkemade line does not cross the boarder line, there will be no a true binary system being formed by the Alkemade line. Isothermal Cuts Fig. 6.38 shows the isotherm of Fig. 6.37. Fig. 6.39 shows the isothermal cut at 700 °C. The areas with the straight lines represent two-phase regions. Fig. 6.40: 600 °C, Fig. 6.41: 400 °C, Fig. 6.42: 300 °C

Fig. 6.39 Isothermal cut of the ternary system at 700 °C.

Fig. 6.38 Isotherms of the ternary phase diagram shown in Fig. 6.37.

Fig. 6.40 Isothermal cut of the ternary system at 600 °C.

Fig. 6.37 Representation of the composition Y within the 3-phase equilibrium A-BC-B by drawing lines parallel to the legs of the ternary region.


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Peritectic and Eutectic Reactions during Cooling. Fig. 6.43: Before Solidification of the melt Y both a peritectic and a eutectic reaction occur. The following path is observed: i) Crystals B, C and AB are being formed ii) Cooling path: Cooling to the composition where the

crystals A, C and AB are in equilibrium with each other.

Y is within the primary phase field B; accordingly B is formed as the first crystal. The composition of the melt changes along the line YG. B and C are simultaneously formed while the composition changes from G → P. At the temperature below that of P, a peritectic reaction occurs, i.e. the melt reacts with one of the crystals in order to form two different crystalline phases After the peritectic reaction is completed, further cooling results in the change of the composition along the boarder line from P to E. Phase-Transformations. Formation of polymorphic crystalline phases. Example of representation in ternary systems: Fig. 6.44. Since the phase transformation occurs (in equilibrium) at a defined temperature, the boarder line between two polymorphes is in agreement with the corresponding isotherm. E.g. the boarder line between Cristobalit and Tridymit agrees with the 1470 °C-isotherm.

Formation of Intermediate Ternary Compounds. Fig. 6.45: Formation of a congruently melting compound ABC. Representation of the compound within the primary phase field. ABC forms 3 ternary subsystems: A-ABC-B, A-ABC-C and B-ABC-C. In case that ABC melts incongruently, the primary phase field of this compound is separated from the position which represents the compound. The ternary subsystems are not true ternary systems Ternary Solid Solutions. Fig. 6.46: Complete solid solubility between the components A and B under formation of the phase α. Limited solubility of C in A and C in B. The eutectic points of the binary systems B-C and A-C occur at E1 and E2, which are connected to each other by the 3 phase boundary E1E2.

Fig. 6.42 Isothermal cut at 300 °C.

Fig. 6.43 Cooling path and solidification of a melt Y with

peritectic and eutectic reactions.

Fig. 6.41 Isothermal cut at 400 °C

Fig. 6.44 Representation of polymorphic phases. The phase transformations between the various structures of SiO2 correspond to the isotherms of the corresponding transformation temperatures.


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Example: System MgO-Al 2O 3-SiO 2 Fig. 6.47: A large number of commercial products is based on this diagram, especially electrical insulators. Binary leg MgO-SiO2: Periclas (MgO), Forsterit (2MgO·SiO2), Eustatit (MgO·SiO2) = Magnesiummetasi-licate, which is called Eustatit at low temperatures and is transformed in its high temperature structure Protoenstatit at 1040 °C. Ternary Compound: Cordierit (2MgO·2Al2O3·5SiO2) which shows good thermal shock resistance and high electrical resistance. Starting materials for the preparation of these materials: Clay (Al2O3·2SiO2·2H2O) and talcum (3MgO·4SiO2·H2O). Fig. 6.46 Ternary phase diagram of a system with solid solubility between AB under formation of the phase α with limited solubility of C in A and C in B. The binary systems B-C and A-C show eutectic points at E1 and E2.

Fig. 6.45 Formation of a congruently melting compound ABC. This compound is formed within its primary phase field.


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Isothermal Ternary Cut (Gibbs-Triangle)

Fig. 6.47 Projection of the ternary phase diagram SiO2, MgO and Al2O3

Fig. 6.48 Representation of the single phase, 2-phase and 3-phase regions of a ternary system ABC.

Lines of equal activity


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6.6. Non-equilibrium Phases and Reactions Fig. 6.49: Heating at X0; A and B are powders with a microstructure shown in Fig. 6.50. T → T2: Melt with a composition X1 and solid body A (in equilibrium). First liquid phase eutectic composition. The part of the melt which is in contact with A, will dissolve further amounts of A. This holds similarly also for B. The simultaneous solution of A at a surface of the melt and of B at the other surface generates gradients of the composition in the melt → Counter diffusion of A and B. The melts will not reach saturation concentration. The process of solution will be continued until all B is consumed. Fig. 6.52: Suppression of the formation of an intermediate compound. Example: Suppression of the formation of mullite SiO2-Al2O3-system.

Fig. 6.51 Formation of a melt in-between phases A and B with diffusion of the two components in opposite direction for

equilibration.

Fig. 6.52 Suppression of the formation of the intermediate

phase AB. The system takes up the phase diagram of a simple

binary system with an eutectic point.

Fig. 6.50 Starting distribution of grains of A and B when

heating at the composition X0 in Fig. 6.49.

Fig. 6.49 Binary phase diagram with heating of a mixture of powders of A and B as shown in Fig. 6.50 at the composition X0.

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