CT – 4: Phase diagrams

Phase diagrams: definition and types, mapping a phase diagram, implicitly defined functions and their derivatives. Optimisation methods: the principle of the least-squares method, the weighting factor, Marquardt‘s algorithm

Dimensions of phase diagramsDimensions of phase diagrams

Two- or three- dimensional manifold of phase equilibria can be visualized as Two- or three- dimensional manifold of phase equilibria can be visualized as phase diagram („Higher-order“ phase diagram – usually 2- or 3-phase diagram („Higher-order“ phase diagram – usually 2- or 3-

dimensional sections)dimensional sections)

The largest dimension of phase diagram – number of free variables under the The largest dimension of phase diagram – number of free variables under the equilibrium conditions (diminishing it introducing xequilibrium conditions (diminishing it introducing x ii, H, Hmm etc.- recalculating etc.- recalculating

extensive variables to intensive ones)extensive variables to intensive ones)

Example:Example:

Diagrams in two dimensions:Diagrams in two dimensions:

Binary phase diagram: p = const.Binary phase diagram: p = const.

Ternary phase diagram: p, T = const. or p, xTernary phase diagram: p, T = const. or p, x ii = const. = const.

Quaternary phase diagram: p, T, xQuaternary phase diagram: p, T, x i i = const. or: p, x= const. or: p, xii, x, xjj = const. = const.

n – components phase diagram: (n-1) – intensive variables constantn – components phase diagram: (n-1) – intensive variables constant

Phase diagram and property diagramPhase diagram and property diagram

Example: Example: Functional dependence (property diagram): Functional dependence (property diagram):

only points on the curve have meaning only points on the curve have meaning

VVmm = V(T), phase = V(T), phase , xi, p=const. , xi, p=const.

(V(Vmm = (V = (Voo/T/Too)T for ideal gas))T for ideal gas)

Phase diagram of unary system: p = f(T, VPhase diagram of unary system: p = f(T, Vmm))

can be drawn at the same coordinates, but T and Vcan be drawn at the same coordinates, but T and Vm m are are

independent variables, pressure depends on the two independent variables, pressure depends on the two coordinates T and Vcoordinates T and Vm m and coordinate space shows the and coordinate space shows the

„ „state of the system“ („state“ diagram – phase diagram).state of the system“ („state“ diagram – phase diagram).

Lines in phase diagram can be interpreted as Lines in phase diagram can be interpreted as functional dependencefunctional dependence

Under condition of monovariant equilibrium of a set Under condition of monovariant equilibrium of a set of specified phases, one coordinate of phase of specified phases, one coordinate of phase diagram is an implicitly defined function of the diagram is an implicitly defined function of the other one (functional dependence)other one (functional dependence)

Types of phase diagramsTypes of phase diagrams

Fig.2.6 Three types (Schmalzried and Pelton – 1973): Fig.2.6 Three types (Schmalzried and Pelton – 1973):

- Axis - two intensive variables (a),(b)- Axis - two intensive variables (a),(b)

- Axis - one intensive variable and quotient of extensive variables (c),(d)- Axis - one intensive variable and quotient of extensive variables (c),(d)

- Axis – two quotients of extensive variables (e), (f)- Axis – two quotients of extensive variables (e), (f)

Types of phase diagramsTypes of phase diagrams

LFS - CT

Types of phase diagrams –cont.Types of phase diagrams –cont.

Plane phase diagrams exist for n-component system with (n-1) Plane phase diagrams exist for n-component system with (n-1) intensive variables kept constantintensive variables kept constant

Lines in phase diagrams represents monovariant equilibria (in Lines in phase diagrams represents monovariant equilibria (in Fig.2.6 two-phase equilibria)Fig.2.6 two-phase equilibria)

The intensive variables are the same for all phases in equilibrium, The intensive variables are the same for all phases in equilibrium, extensive variables differextensive variables differ

Coordinates, quotients of extensive variables, represent overall Coordinates, quotients of extensive variables, represent overall state variable of the whole system state variable of the whole system as well as as well as individual state individual state variables of the phasesvariables of the phases

Types of phase diagrams –contTypes of phase diagrams –cont..Variables of the individual phases are represented by pair of lines Variables of the individual phases are represented by pair of lines

belonging to the two coexisting phases (Fig.2.6, c, d, e, f)belonging to the two coexisting phases (Fig.2.6, c, d, e, f)

Corresponding points on these pairs of lines are connected by Corresponding points on these pairs of lines are connected by stright lines called „tie-lines“ (lines connecting the compositions of stright lines called „tie-lines“ (lines connecting the compositions of phases in equilibriumphases in equilibrium – isotherms in – isotherms in binary binary phase phase diagram,diagram,

tie-lines or tie-triangles in ternary diagram)tie-lines or tie-triangles in ternary diagram)

Intensive coordinate (constant along the „tie-lines“) may be parallel Intensive coordinate (constant along the „tie-lines“) may be parallel to the axis of the quotient of extensive variables. (Fig.2.6,c,d)to the axis of the quotient of extensive variables. (Fig.2.6,c,d)

The direction of „tie-lines“ is an important part of the information The direction of „tie-lines“ is an important part of the information

given by phase diagram (Fig.2.6,e,f)given by phase diagram (Fig.2.6,e,f)

Types of phase diagrams –cont.Types of phase diagrams –cont.

The monovariant equilibria meet in invariant equilibria The monovariant equilibria meet in invariant equilibria (where variable of the phases are represented by (where variable of the phases are represented by points – single point in the Fig.2.6.a,b, separate points points – single point in the Fig.2.6.a,b, separate points in other diagrams in other diagrams

– – in the type of Fig.2.6.c,d three points on a straight line in the type of Fig.2.6.c,d three points on a straight line

- in the type of Fig.2.6.e,f they form triangle)- in the type of Fig.2.6.e,f they form triangle)

Reading of binary phase diagramReading of binary phase diagramBinary phase equilibria, eutectic, peritectic, eutectoid, peritectoidBinary phase equilibria, eutectic, peritectic, eutectoid, peritectoidExample: Fe-C diagramExample: Fe-C diagram (Callister W.D.: Materials Science and Engineering, John Wiley 1999.)(Callister W.D.: Materials Science and Engineering, John Wiley 1999.)

Stable and metastable Fe-C phase diagramStable and metastable Fe-C phase diagram

(Callister W.D.: Materials Science(Callister W.D.: Materials Science

and Engineering, John Wiley and Engineering, John Wiley

19991999.).)

Rules for monovariant linesRules for monovariant lines

Thermodynamically possible

Thermodynamically impossible

Reading ternary phase diagramReading ternary phase diagram

Ternary phase equilibria, two-phase field, three-phase fieldTernary phase equilibria, two-phase field, three-phase field

Example: Fe-Cr-Ni Example: Fe-Cr-Ni phase phase diagramdiagram (Raynor G.V., Rivlin V.G. Phase (Raynor G.V., Rivlin V.G. Phase Equilibria in Iron Ternary Alloys. Inst. Equilibria in Iron Ternary Alloys. Inst. Of Metals 1988.) Of Metals 1988.)

Reading ternary phase diagramReading ternary phase diagram – cont – cont..

Al-Cr-C system: Hallstedt B.:Calphad XXXVIII Prague 2009 - Book of abstracts

Two-phase fields with tie-lines

Three-phase fields are tie-triangles

Reading ternary phase diagramReading ternary phase diagram

Liquidus projectionLiquidus projection: eutectic, peritectic, phase transformation: eutectic, peritectic, phase transformation

Example: FeExample: Fe -- Mo – W Mo – W Vrestal J. in Landolt-Börnstein Series 2009 in print

Phase transformations on liquidus surfacePhase transformations on liquidus surface

E (eutectic) U (phase transformation) P (peritectic)

Vertical sections of ternary systemsVertical sections of ternary systems(isopleths)(isopleths)

Tie-lines generally are not in the plane of section!Tie-lines generally are not in the plane of section!

The lines do not belong to monovariant equilibria -The lines do not belong to monovariant equilibria -

they show „zero-phase-fraction“ equilibria.they show „zero-phase-fraction“ equilibria.

Lines show boundaries between an n- and (n-1)-phase field Lines show boundaries between an n- and (n-1)-phase field (calculated from equilibrium condition for the n-phase (calculated from equilibrium condition for the n-phase equilibrium with additional condition that amount of one phase equilibrium with additional condition that amount of one phase is zero, although still present in equilibrium).is zero, although still present in equilibrium).

Mapping a phase diagramMapping a phase diagram

2 or 3 variables of the conditions are selected as axis variables 2 or 3 variables of the conditions are selected as axis variables with lower and upper limit and maximal step.with lower and upper limit and maximal step.

All additional conditions – kept constant throughout the whole All additional conditions – kept constant throughout the whole diagramdiagram

Start: „initial equilibrium“ for Newton-Raphson calculation (with all Start: „initial equilibrium“ for Newton-Raphson calculation (with all phases „entered“)phases „entered“)

All results of calculations are usually stored – any phase diagram All results of calculations are usually stored – any phase diagram may be displayed at the end of calculations (In Fig.2.6. may be displayed at the end of calculations (In Fig.2.6. diagrams b) – e) could be plotted from single mapping)diagrams b) – e) could be plotted from single mapping)

In non-isoplethal phase diagram calculations: In non-isoplethal phase diagram calculations: monovariant equilibrium is traced, keeping except one of the axis variable monovariant equilibrium is traced, keeping except one of the axis variable constant and increasing the only variable axis in steps until an invariant constant and increasing the only variable axis in steps until an invariant equilibrium is found or one of the selected limits of the axis exceeded.equilibrium is found or one of the selected limits of the axis exceeded.Procedure repeated for all monovariant equilibriaProcedure repeated for all monovariant equilibria

In isoplethal phase diagrams calculations:In isoplethal phase diagrams calculations:„„zero-phase-fraction“ line is traced by setting the appropriate conditions:zero-phase-fraction“ line is traced by setting the appropriate conditions:the set of stable phases is constituted by all phases appearing in both areas the set of stable phases is constituted by all phases appearing in both areas adjacent to the line, plus the phase appearing only in one area –adjacent to the line, plus the phase appearing only in one area –this phase is assigned the fixed amount of zero. this phase is assigned the fixed amount of zero. Procedure is continued until an axis limit is reachedProcedure is continued until an axis limit is reached

If in phase diagram a set of lines is completely separated from another set, If in phase diagram a set of lines is completely separated from another set, the mapping procedure must be continued with an additional starting the mapping procedure must be continued with an additional starting equilibrium leading to the other set (Example: Ag-Pd FCC miscibility gap)equilibrium leading to the other set (Example: Ag-Pd FCC miscibility gap)

Mapping a phase diagram – cont.Mapping a phase diagram – cont.

Mapping a phase diagram – cont.Mapping a phase diagram – cont.Example (in Thermocalc - Cu-Sn system):Example (in Thermocalc - Cu-Sn system):

set-axis-variable 1 x(Sn) 0 1 .025

s-a-v 2 t 200 1400 10

map

Example (in Thermocalc – Bi-Cu-Sn system at 1000 K):Example (in Thermocalc – Bi-Cu-Sn system at 1000 K):

set-axis-variable 1 x(Cu) 0 1 .025

s-a-v 2 x(Sn) 0 1 .025

map

--------------------------------------------------------„sorry, can not continue“ is warning in Thermocalc here, that Newton-Raphson procedure does not

converge – may be for bad starting values

Types of phase diagramsTypes of phase diagrams

LFS - CT

Implicitly defined functions and their Implicitly defined functions and their derivativesderivatives

Fig.2.6.b: T=T(Fig.2.6.b: T=T(MgMg) or ) or MgMg = = MgMg(T) - implicitly(T) - implicitly

Fig.2.6.c: T implicitly defined function of Fig.2.6.c: T implicitly defined function of

xxMg Mg MgCu2MgCu2 as well as the inverse function as well as the inverse function

xxMg Mg C15-MgCu2C15-MgCu2(T) for the boundary of two-phase field „C15-(T) for the boundary of two-phase field „C15-

MgMg22Cu“ against „C15-MgCuCu“ against „C15-MgCu22““

Derivatives of these implicitly defined functions are also Derivatives of these implicitly defined functions are also defineddefined

Gibbs-Konovalov rule:Gibbs-Konovalov rule:

(dT/dx(dT/dx))coexcoex = = [[(x(x - x- x) T () T (22GGmm / / xx22))] /] /

[(H[(H11 - H - H11

) ) ((11- x- x)) + (H + (H22 - H - H22

) ) xx ]]

OptimizationOptimization methods methods

Set of n-measurable values WSet of n-measurable values W i i depends on set of unknown coefficients depends on set of unknown coefficients CCj j via functions Fvia functions F i i with values of independent variables xwith values of independent variables xkiki::

WWii = F = Fii (C (Cjj, x, xkiki) i = 1,..., n, j = 1,..., m) i = 1,..., n, j = 1,..., m

(k distinguishes the various independent variables (T, x(k distinguishes the various independent variables (T, x ii...) belonging ...) belonging to measurement i, n to measurement i, n m) m)

WWii corresponds to measured value L corresponds to measured value L ii

Best set of CBest set of Cjj: sum of squares of the „errors“ : sum of squares of the „errors“

(calculated values by F(calculated values by F ii – experimental values L – experimental values L ii) )

must be minimalmust be minimal

„„Errors“ are multiplied by weighting factor pErrors“ are multiplied by weighting factor p ii

Weighting factorWeighting factor

Weighting factor pWeighting factor p ii is is defineddefined by „error equation“: by „error equation“:

(F(Fii(C(Cj,j,xxkiki) – L) – Lii ) . p ) . pii = = ii

Condition for the best values Cj:Condition for the best values Cj:

nni=1i=1 ii

22 = Min (with respect to the Cj) = Min (with respect to the Cj)

Weighting factor – contWeighting factor – cont..

nni=1i=1 ii

22 = Min (with respect to the Cj) = Min (with respect to the Cj)

For every coefficient CFor every coefficient Cjj, , mm equations can be derived: equations can be derived:

nni=1i=1 ii

..( ( ii / / CCjj) = 0 j = 1, ..., ) = 0 j = 1, ..., mm

Gauss: Gauss: i i expanded into a Taylor series with only linear expanded into a Taylor series with only linear terms:terms:

ii((CCjj, , xxkiki) ) iioo ( (CCjj

oo, , xxkiki) + ) + mml=1l=1

..( ( ii / / CCll) .) .CCll

CCj j are corrections to the coefficients Care corrections to the coefficients Cjj

Weighting factor – cont.Weighting factor – cont.

Calculation of corrections Calculation of corrections CCjj::

mml=1l=1((nn

i=1i=1((ii / /CCjj)()(ii / /CCll) ) CCl l = - = - nni=1i=1ii

..((ii / /CCjj) )

j = 1, ..., mj = 1, ..., m

set of set of mm such linear equations for such linear equations for mm unknowns unknowns CCjj::

„„Gaussian normal equations“Gaussian normal equations“

Measure of the fitMeasure of the fit

Fit between the resulting coefficients and the measured Fit between the resulting coefficients and the measured values can be defined by:values can be defined by:

Mean square error = Mean square error = nni=1i=1((ii

22/(n-m))/(n-m))

where:where:

n - measurable values, m – unknown coefficients Cn - measurable values, m – unknown coefficients Cjj, ,

i i -- defined as (Fdefined as (Fii(C(Cj,j,xxkiki) – L) – Lii ) . p ) . pii = = i i , p, pii weighting factor weighting factor

Weighting factor – cont.Weighting factor – cont.

In the optimisation procedures (PARROT etc.):In the optimisation procedures (PARROT etc.):

estimated uncertainities of measured quantities are introducedestimated uncertainities of measured quantities are introduced

Additionally Additionally introduced introduced dimensionless factor „weightdimensionless factor „weight“ is left to the “ is left to the responsibility of the responsibility of the useruser to assign weights that reflect the to assign weights that reflect the relative importance of data (default settings of all weights are relative importance of data (default settings of all weights are equal 1)equal 1)

First attempt to optimizationFirst attempt to optimization

Trial and error optimization (by hand):Trial and error optimization (by hand):

Influence of the change of parameter to theInfluence of the change of parameter to the

position of phase boundaries – starting valueposition of phase boundaries – starting value

for optimisation program (e.g.PARROT in TC).for optimisation program (e.g.PARROT in TC).

Example: Cr-Ti Laves phasesExample: Cr-Ti Laves phases

Cr-Ti-Laves phase C14Cr-Ti-Laves phase C14

PARAMETER G(LAVES_PHASE,CR:TI:CR;0) 298.15 +4*GHSERTI+8*GHSERCR -101605.-3.20*(T-T*ln(T)); 6000.0 N 93 ! PARAMETER G(LAVES_PHASE,CR:TI:CR,TI;0) 298.15 -50000.; 6000.0 N 93 !

Correct value = 3.20 Trial value = 2.90 - wrong

Pavlů J., Vřešťál J., Šob M.: Calphad 34 (2010) 215

MarquardtMarquardt’s’s algorithm algorithm

Combination of Newton-Raphson method with steepest-descent Combination of Newton-Raphson method with steepest-descent method for equations, nonlinear in coefficients (D.W.Marquardt method for equations, nonlinear in coefficients (D.W.Marquardt 1963)1963)

Calculation of corrections Calculation of corrections CCjj::

mml=1l=1((nn

i=1i=1((ii//CCjj)()(ii//CCll))CCl l = -= -nni=1i=1ii

..((ii//CCjj) )

j = 1, ..., mj = 1, ..., m

may not converge for may not converge for CCjj, then factor called Marquardt, then factor called Marquardt parameter parameter is added to the normalized matrix of equation above. For large is added to the normalized matrix of equation above. For large Marquardt parameter, the steepest descent method is used, for Marquardt parameter, the steepest descent method is used, for the small of it, the pure Newton-Raphson technique is in work.the small of it, the pure Newton-Raphson technique is in work.

Calculation of phase diagram by suspending Calculation of phase diagram by suspending one phaseone phase

Example: Example:

Metastable Fe-FeMetastable Fe-Fe33C diagramC diagram can be calculated can be calculated ( (withwith cementitecementite))

if graphiteif graphite (and diamond) are (and diamond) are suspended suspended

Calculation of metastable equilibriaCalculation of metastable equilibria

Extending this idea: we can calculate any Extending this idea: we can calculate any equilibrium for phaseequilibrium for phase,, or an assemblage of or an assemblage of phases, irrespective of whether these phases phases, irrespective of whether these phases represent the stable state of the systemrepresent the stable state of the system

Calculation of metastable equilibria – cont.Calculation of metastable equilibria – cont.

The same rules apply to the metastable equilibria as to the stable The same rules apply to the metastable equilibria as to the stable one.one.

The total Gibbs energy (at const.T,p) is higher then that of stable The total Gibbs energy (at const.T,p) is higher then that of stable equlibriumequlibrium (could be used for the check). (could be used for the check).

SimilaritySimilarity with stable equilibria with stable equilibria: it is still a minimum for the phases : it is still a minimum for the phases included in the calculation and the chemical potentials for all included in the calculation and the chemical potentials for all components are the same in all phases.components are the same in all phases.

Questions for learningQuestions for learning

1. Explain difference between phase diagram and property diagram1. Explain difference between phase diagram and property diagram

2. Find monovariant equilibria and invariant equilibria in binary and 2. Find monovariant equilibria and invariant equilibria in binary and ternary phase diagramsternary phase diagrams

3. Give the rules for extrapolation of monovariant lines in binary phase 3. Give the rules for extrapolation of monovariant lines in binary phase diagramsdiagrams

4. Give the rules for direction of monovariant lines in liquidus projection4. Give the rules for direction of monovariant lines in liquidus projection

at invariant pointsat invariant points

5. 5. Define weighting factor and explain its use during optimisation of Define weighting factor and explain its use during optimisation of themodynamic parametersthemodynamic parameters.. WWhathat is Marquardtis Marquardt‘s algorithm?‘s algorithm?