Phase Diagrams And Phase Separation
Phase Diagrams and Phase SeparationBooksMF Ashby and DA Jones, Engineering Materials Vol2, PergamonP Haasen, Physical Metallurgy,G Strobl, The Physics of Polymers, SpringerIntroductionMixing two (or more) components together can leadto new properties:Metal alloys e.g. steel, bronze, brass .Polymers e.g. rubber toughened systems.Can either get complete mixing on theatomic/molecular level, or phase separation.Phase Diagrams allow us to map out what happensunder different conditions (specifically ofconcentration and temperature).Free Energy of MixingEntropy of MixingnA atoms of AnB atoms of BAM DonaldPhase Diagrams1
Total atoms N nA nBThen Smix k ln WN! k lnnA !nb!This can be rewritten in terms of concentrations ofthe two types of atoms:nA/N cAnB/N cBand using Stirling's approximationSmix/ kNSmix -Nk (cAln cA cBln cB)A0.5BThis is a parabolic curve.There is always a positive entropy gain on mixing(note the logarithms are negative) – so that entropicconsiderations alone will lead to a homogeneousmixture.The infinite slope at cA 0 and 1 means that it is veryhard to remove final few impurities from a mixture.AM DonaldPhase Diagrams2
This is the situation if no molecular interactions tolead to enthalpic contribution to the free energy (thiscorresponds to the athermal or ideal mixing case).Enthalpic ContributionAssume a coordination number Z.Within a mean field approximation there arenAA bonds of A-A type 1/2 NcAZcA 1/2 NZcA2nBB bonds of B-B type 1/2 NcBZcB 1/2 NZ(1cA)2and nAB bonds of A-B type NZcA(1-cA)where the factor 1/2 comes in to avoid double countingand cB (1-cA).If the bond energies are EAA, EBB and EAB respectively,then the energy of interaction is (writing cA as simplyc)1/2 NZ [cEAA (1-c)EBB c(1-c) (2EAB - EAA –EBB)]energy of 2 separate starting materials Umix 1/2 NZ c(1-c) (2EAB – EAA – EBB)AM DonaldPhase Diagrams3
The term (2EAB – EAA – EBB) determines whethermixing or demixing occurs.Define χkT Z/2 (2EAB – EAA – EBB)where χ is known as the interaction parameter, and isdimensionless.(The definition of χ in this way, including the kT term,is for historical reasons).Fmix/ kT c ln c (1-c) ln (1-c) χ kT c(1-c)Three cases to considerχ 0 athermal or ideal solution case.F mix/ kTÞFmix -TSmixFmix is always negative.AM DonaldPhase Diagrams4
χ 0 i.e. 2EAB EAA EBBχc(1-c)Fmix/ kTÞ-TSmixResultantFmix2 local minima resultχ 0Fmix/ kTÞχc(1-c)-TSmixF mixOne deep minimumAM DonaldPhase Diagrams5
These curves correspond to the case for one particularT.In order to plot out a phase diagram, we need to beable to see how temperature affects these curves.And we need to know whether or not these curvesimply phase separation.Finally we need to know, if phase separation occurs,how much of each phase is present.Lever Rule for Determining Proportions of PhasesImagine we have a system with 2 phases present(labelled 1 and 2), and two types of atoms A and B.In phase 1 concentration of A c1In phase 2 concentration of A c2and that there is x of phase 1 presentand (1-x) of phase 2 present.Total of N atoms, overall concentration of A c no of A atoms is Nc Nxc1 N(1-x) c2nmc1AM DonaldPhase Diagramsclc2c c2 n c1 c2 lÞx ÞLever Rule6
c c1 mc c2 nx and c1 c2l1 x c c1 mThus, when one wants to find the average of somequantity such as F, it is usually sufficient (neglectingsurface effects) to take a weighted average.Similarly 1-x Consider the free energy F as F(c)FSF1P Q Rc1c c2F2cF(c) F2 (F1 – F2) QR/PRBy similar triangles, Þ F(c ) SQ in the diagram.Therefore one can find free energy of any intermediatecomposition by drawing straight line between the freeenergies of the two constituent phases.Can now use this to interpret free energy curves.Consider cases of χ 0 or negative; free energy curvehad a single minimum.AM DonaldPhase Diagrams7
As pure components (A and B) F(c ) FFFBF1FAA1B1FnACBThis can be lowered by going to compositions A1,B1 togive free energy F1 etcAnd as A and B continue to dissolve more and more ofeach other, free energy continues to drop.Minimum energy for homogeneous single phase,energy Fni.e no phase separation occurs in this case.Composition overall determines the state of themixture.AM DonaldPhase Diagrams8
Minimum free energy will not be given (in general) bythe minimum in the free energy curve.e.g. starting with composition c, if this were to result incomposition corresponding to the minimum on thecurve cmin, would necessarily also have phase withcomposition c' present.c'ccminNecessary condition for homogeneous mixing to occuris ford2 f 0 everywheredc2However when χ is positive, this inequality does nothold, and the behaviour is very different.Start with homogeneous free energy F, for concentration c.AM DonaldPhase Diagrams9
FBFA1FAA3B1F1B3F3BcBAcAcThis can split into A1 and B1; energy drops to F1.Minimum in energy occurs at F3 when therepresentative points on the free energy curve arejoined by the lowest straight line: common tangentconstruction.In this case, have phase separation into compositionscA and cB, with phases α and β.c cA have α phasecA c cB have α βcB c have just β.Proportions of α and β given by Lever rule.For compositionsFor c cA A dissolves BFor cB c B dissolves AcA and cB define solubility limits.AM DonaldPhase Diagrams10
This common tangent construction can be extended toquite complicated situations, with several minima,which then give rise to quite complicated free energycurves and hence phase diagrams.For plotting a phase diagram we need to know howsolubility limits (as determined by the commontangent construction) vary with temperature.Have seen that if d2F/dc2 everywhere 0 have ahomogeneous solution.Phase separation occurs when free energy curve hasregions of negative curvature.This permits us to evaluate the limits of solubility interms of χ.For the symmetric case (i.e FA FB i.e terms involvingEAA and EBB are assumed equal)Fmix/ kT c ln c (1-c) ln (1-c) χc (1-c) per sited2 F11 2 kT( 2 χ)c 1 cdcCritical value when d2F/dc2 0Þc 1 12 1 χ2 2For a regular solution χ 1/T A/TAM DonaldPhase Diagrams11
Then limits of solubility vary with temperatureaccording toc 1 12T 1 A2 2i.e. as T raised the two compositions of phases α and βconverge, until at Tcrit A/2 there is no furtherseparation and a homogeneous mixture results.In general, critical value of χ 2.There is no solution for c for χ 2 so there is no phaseseparation.Also, for the symmetrical case, the common tangentconstruction reduces to the conditiondf/dc 0 (i.e. horizontal tangent)This equation defines the binodal or coexistencecurve.χ binodal 1cln()2c 1 1 cFor the regular solution case, with χ 1/Τ this allowsus to plot out how the binodal behaves.Some comments on this ApproachAM DonaldPhase Diagrams12
This approach of a 'symmetrical' AB mixture is verysimilar to the Ising model, where the 2 statescorrespond to opposite spins (although of course Acannot transform to B as spins can).The approach used has been a mean field theory.This will fail:Þ near a critical point: will give the wrong criticalpoint and the wrong critical exponents. Inpractice most of the phase diagram will be wellaway from this point, so this is not too severe.Þ for strongly negative χ. In this case there arestrong attractions between unlike atoms and theidea of random distributions breaks down. Thiscan lead to order-disorder transitions.In general it is difficult to calculate χ from 1stprinciples.Picture described here works best for liquid-liquidphase separation. Melting may complicate matters.For solid-solid transitions have additional problems,leading to very complicated phase diagrams, due toStrain energyOther intermediate compounds eg AB2.However for the regular solution model we now canconstruct a phase diagram – which contains all theessential physics.AM DonaldPhase Diagrams13
χ 2 , only a single minimum Þ homogeneous mixture0Free energy of mixingper site /kT-0.2-0.4chi 2.5chi 2.0chi 0chi onFree energy of mixingper site /kTχ 2 , two minima develop Þ phase separationT decreasingconcentrationAM DonaldPhase Diagrams14
Knowing the form of the free energy curves as afunction of T means that we can map out the limits ofsolubility etc.Fdf/dc 0concentrationPlotting out the locus of the binodal enables us to seethe shape of the phase 000.20.40.6Concentration of AAM DonaldPhase Diagrams150.81
The binodal or coexistence curve describes the limitsof solid solubility.Have seen condition for homogeneous mixingeverywhere is d2f/dc2 0 everywhere.The locus d2f/dc2 0 is called the spinodal curve.We need to distinguish how phase separation occursinside this curve from between binodal and spinodal.Spinodal Curve and Spinodal DecompositionIn the two phase region, two kinds of decompositioncan occur.1. Nucleation and Growth – which we have alreadytalked about, and is the most familiar.2. Spinodal DecompositionIn the nucleation and growth regime we know that anucleus of a critical size has to form before it isenergetically favourable for it to grow.Nucleation and growth corresponds to a metastableregion of the phase diagram.Spinodal decomposition occurs when any compositionfluctuation is unstable – the unstable region of thephase diagram.Section of free energy curve:AM DonaldPhase Diagrams16
CC1C2d 2f/ dc2 0limits of spinodaldecompositionStart at composition c, split into 2 slightly differentcompositions c1 and c2 to give net lowering of energy.Where the curvature is negative, any compositionfluctuation leads to a drop in F, and is thereforeunstable.This contrasts with region between spinodal andbinodal:C1d 2f/ dc2 0limits of spinodaldecompositionCβαC2common tangent constructionAM DonaldPhase Diagrams17
Net increase in F as compositions diverge – hencenucleation barrier, even though ultimately end up withoverall lowering of energy when split into phases αand β.Since any composition fluctuation is stable in thespinodal region, which wavelength dominates?Large fluctuations grow comparatively slowly becauseatoms have to diffuse over large distances.Small fluctuations are suppressed, because theyinvolve a lot of diffuse interfaces.However the fact that interfaces aren't sharp meansthat we have to think carefully about how to accountfor the interfacial energy.Local free energy density depends on the compositionat that point and in the vicinity, since this willdetermine the sharpness of the interface.Local free energy therefore can be written (1D case)AM DonaldPhase Diagrams18
c 2cF A ò f (c, , 2 ,.)dxdx dxwhere A is a constantExpand as Taylor series2 c 2c c 2cæ c öf (c, , 2 .) fo (c) k11 k12 k21 2 .è x ø x x x xDiscarding higher order terms, and noting that k11must be zero by symmetryé c 2 2c ùF A ò fo (c) k12 ( ) k21 2 dxêëdx x úûWhich can be shown to lead toæ c öF A ò [ fo (c) K]dxè x ø2Where K is the gradient energy coefficientK k12 k21 cThis theory is due to Cahn and Hilliard for metalalloys. An equivalent theory for magnetic domains isdue to Landau and Ginsburg.System will try to equilibrate by having a uniformchemical potential.AM DonaldPhase Diagrams19
General transport equation, flux of A, JA JA M ( µ µB ) x AOnsager relationM is an Onsager coefficientµA and µB are the chemical potentials of A and Brespectively.(µA - µB) is the free energy to remove 1 atom of A andreplace it by B atomµA µB æ c öfo (c) K( )2dc è x ø fo 2c 2K 2 c x 2 fo c 3x JA M 2 2 MK 3 c x xFor early times, c will not differ greatly from overallconcentration co. Can then make assumptions M, 2fo/ c2 and K are independent of c, i.e. linearise –linear Cahn-Hilliard theory.Now, continuity equation isAM DonaldPhase Diagrams20
c J t x 2 fo 2 c 4c M 2 2 2 MK 4 c x x[1]And if the term involving K, the gradient energy term,can be neglected, this equation is equivalent to astandard diffusion equation with an effective diffusioncoefficient Deff given by 2 foDeff M 2 cNote that this diffusion coefficient now containsthermodynamics, and is not simply a transportcoefficient.M contains transport information and is alwayspositive.But we know that within the spinodal regime 2fo/ c2 isalways negative, by definition.Thus Deff is negative – 'uphill diffusion'.All fluctuations tend to encourage concentrationfluctuations to increase, so that neighbouring regionsget more and more different.This is in contrast to most diffusion.AM DonaldPhase Diagrams21
The contrast between Spinodal Decomposition andNucleation and Growth.Nucleation and dal Decompositioncoarsening occurs, but by uphill diffusionNow return to the full equation involving the gradientenergy term.General solution to the diffusion equation isc( x,t) cos (qx) exp – (Dq 2t) coFor equation [1] this becomesAM DonaldPhase Diagrams22
2Kq 2c(x,t) co cos (qx) exp[ – (Deff q t(1 ) ]fo "2if the intitial composition is co, where fo" is the secondderivative of fo wrt composition.This form of the equation enables us to see whichwavelengths grow and which shrink.The amplitude of the composition fluctuation ofwavevector q changes with an amplification factor2Kq 2 2R(q) Deff (1 )qfo "Recall fo" is always negativeAmplification factor R(q) veqmax0-veWavevector qThe fastest growing wavelength is given byAM DonaldPhase Diagrams23
qmax AM DonaldPhase Diagrams12fo "K24
What does this Morphology look like?For nucleation and growth we are used to seeingdroplets of one phase nucleating in the other.For spinodal decomposition typically see (e.g. in anoptical microscope) a random, interconnectedstructure, with a characteristic lengthscale related toqmax.From such images, difficult to get a good measure ofthe average lengthscale.Easier to do this using scattering methods from whichqmax can be directly obtained.Scattering shows a broad Bragg-like peak, fromwhich qmax can be read off.The scattering radiation can be light, X-rays depending on actual lengthscale.Light scattering works well for polymers, becauselengthscales are large, but for atomic species X-raysneed to be used as much finer structures.This simple linearised theory will only work for earlytimes, during which time qmax will be constant.As time increases non-linear terms come into play,and the analysis gets more complicated.Domain size R(t); Interface width w(t)AM DonaldPhase Diagrams25
φxφ1coexEarlyR(t) w(t)Both increaseIntermediateR(t) w(t)R(t) increasesw(t) decreasesLate stagew(t) w( )constantφ2coexw( )At early times, linear theory OK; one lengthscale(corresponding to qmax)Intermediate times – analysis very complicated. Twolengthscales importantSize of domainsWidth of interfaceAM DonaldPhase Diagrams26
Late times – simple theory again with only onelengthscale (interfaces are sharp). Compositions havenow reached those of coexisting phases; coarseningcontinues to occur.During late stage growth, large droplets grow at theexpense of small ones.Morphology may cease to be interconnected, andstarts to resemble that of nucleation and growthregime.This phenomenon of coarsening also occurs in thenucleation and growth regime, and is known asOstwald ripening.Why do Small Droplets Shrink and Large OnesGrow?Imagine adding one atom from an ideal solution to aprecipitate particle, which thereby grows by r.AM DonaldPhase Diagrams27
(Same basic argument applies in both metastable andunstable regimes.) F 8πrγ γ rγ Fv4πr2 rsurface energychangebulk free energy entropy changechange-kT ln cAt equilibrium, F 0.Put Ω 4πr2 rThen0 2γ Ω FvΩ kT ln cr c exp( Fv Ω2γΩ)expkTrkT Fv Ωis the concentration inkTequilibrium with a flat surfaceNow c expcr c exp(2γΩ)rkTThe local concentration depends on particle size.Concentration highest just outside small particles,and the atoms will diffuse away to try to recover c .AM DonaldPhase Diagrams28
Hence small particles shrink, whilst overall volumefraction remains essentially constant (otherwise c would be changing).volumefractionin 2ndphasenuclnn(r)earlytimegrowth withOstwaldripeningtimen(r ) no/unit volumewith radius rn(r ) no/unitvolume withradius rlate timeCan work outhow particlesize scales with time by a heuristic argument.rAM DonaldPhase Diagrams29
Flux chemical potential gradient across interfacedc2γΩ cr c exp()rkTdr dc1 2 crrdrFlux also determines how radius grows or shrinks flux drdtdr 1 cdt r 2 rÞ r3 t orr t1/3Lifshitz-Slyozov lawThis is found to give good agreement with experimentwhen diffusion, rather than hydrodynamics, isdominating.AM DonaldPhase Diagrams30
Utilising Phase DiagramsPhase diagrams are inherently for equilibrium.In practice equilibrium may not always pertain, andone can end up with structures which are far fromequilibrium but are 'frozen in' because there isinsufficient thermal energy to allow furthertransformation to occur.Thus must always bear in mind kinetics as well asthermodynamics.Case Study I – Zone-refining SiliconSilicon chips are only possible because native silicon,which naturally contains sufficient impurities to messup any device, can be purified by a method known aszone-refining.Starting with co, when this is melted will be inequilibrium with a solid with concentration kco (k 1)level of impurities present.AM DonaldPhase Diagrams31
LiquidTL SSolidkco coconcentration ofimpuritiesThus the process of melting and solidification leads torejection of impurities into remaining liquid, whoseimpurity concentration goes up.By passing a heater over a solid bar, can effectivelypush impurities along.By doing this several times, end up with very pure Si(at one end).Case Study II – Hardening of aluminium by copperIn this case exactly what happens, depends onthermal history – e.g. cooling rate, ageing etc.AM DonaldPhase Diagrams32
At the so-called eutectic point Te, 3 phases are inequilibrium: L, α and θ.No degrees of freedom.Corresponds to the composition with the lowestmelting point.At Te, upon cooling, all remaining liquid solidifiesisothermally. 4% Cu in AL corresponds to Duralumins, designedwith Cu present to harden Al, which is typically afairly soft material.Above 500ûC 4% alloy is single phase α.AM DonaldPhase Diagrams33
Below 500ûC θ (CuAl2) phase starts to form.Proportion dependent on T and given by Lever Rule.However how the microstructure develops dependson rate of cooling.AM DonaldPhase Diagrams34
Cool slowly – low driving force for nucleating CuAl2precipitates.Few nuclei, but these then grow quite large.Large, well-separatedprecipitates, typically at grainboundaries, which do notprovide much barrier todislocation motionCool fast – large driving force for nucleation, andtherefore many nuclei form which then do not growmuch.Much finer structure more obstacles todislocation motionMaterial which has been fast cooled is much harder.However if you cool too fast, can kinetically frustratethis process, and Cu remains in supersaturated αsolid solution.AM DonaldPhase Diagrams35
Harden now by 'age-hardening' i.e. hold at low T(typically around 150ûC) for long time ( 100 hours).Fine precipitates will now form during this heattreatment, by a series of steps.1. Solid solution – Cu randomly distributed on lattice.2. Small 'Guinier-Preston' zones form – small discs ofCu form on 001 Al lattice planes.3. These zones coarsen and form θ" phase, which hascoherent lattice wrt Al lattice. This gives rise to'coherency strain' which strain field impedesdislocation motion. In this case size of latticemismatch and distances involved mean that latticeplanes can still bend to accommodate thismismatch.4. Further coarsening and formation of θ', which isincohere
P Haasen, Physical Metallurgy, G Strobl, The Physics of Polymers, Springer Introduction Mixing two (or more) components together can lead to new properties: Metal alloys e.g. steel, bronze, brass–. Polymers e.g. r
a time period. There are seven behavioral diagrams: Use Case Diagrams, Activity Diagrams, State Machine Diagrams, Communication Diagrams, Sequence Diagrams, Timing Diagrams and Interaction Overview Diagrams. UML 2 has introduced Composite Structure, Object, Timing and Interaction Overview diagrams.
node diagrams (N diagrams), node-link diagrams (NL diagrams) and node-link-groups diagrams (NLG diagrams). Each of these diagrams extends the previous one by making more explicit a characteristic of the input data. In N diagrams, a set of objects is depicted as points in a two or three dimensional space; see Figure1a. Clusters are typically .
wiring your motor p.06 9.1 nema three-phase wiring diagrams p.07 9.1.1 nema brake motor wiring diagrams p.08 9.2 iec three-phase wiring diagrams p.09 9.2.1 iec brake motor wiring diagrams p.10 9.3 nema farm duty wiring diagrams p.11 9.4 nema single-phase wiring diagrams p.12 9.5 iec single-phase wir
Class Diagrams etc. (e.g. Component Diagrams) Dynamic / Behavioral Modeling: capturing execution of the system Use Case Diagrams Sequence Diagrams State Chart Diagrams etc. (e.g. Activity Diagrams) UML Diagrams Models: high-level class relations Components: Class (rectangle) - Upper section: name of the class
2 GN 3 phase disc stack separator is a high-speed, stable, sealed, efficient, and automatic slag discharge 3 phase separation equipment. It is widely used for liquid, liquid, solid 3-phase separation or liquid-liquid separation. The 2-phase solid-liquid separation disc separator has only one liquid discharge port and one slag discharge port .
Use Case Diagrams Sequence Diagrams Questions Use Case Diagrams & Sequence Diagrams SE3A04–Tutorial JasonJaskolka Department of Computing and Software Faculty of Engineering McMaster University Hamilton, Ontario, Canada jaskolj@mcmaster.ca October14/15,2014 Jason Jaskolka
Lewis dot diagrams In grade 9, you learned how to represent the atom by using Bohr Diagrams. Since Bohr diagrams can be a bit cumbersome to draw, we can use a shorter, more convenient notation. Lewis Dot Diagrams (Electron Dot Diagrams) A Lewis dot diagram is a way to represent an element and its VALENCE electrons.
separation tank components are damaged, notify the freight carrier immediately. 2. After inspection, carefully remove the separation tank and air bleeder valve from the carton. 3. Assemble air bleeder valve on top of separation tank. 4. Mount separation tanks on a per
