T C – P R I S M A User's Guide and Examples - Thermo
Contents
1. 1 Temperature profile x BINARY CONDITIONS PL 1 Temperature profile cu4ti x A 370 MATRIX PHASE PRECIPITATE PHASE 365 Phase name fee Phase name cu4ti Composition Element mole fraction Composition Element nl 360 cu 0 981 cu ti 1 9e 2 oon Diffusivity Prefactor 2 5e 19 Solvus Line Prefactor g 350 Activation Energy 0 Activation Energy 2 Volume 0 8e 5 Volume q i 345 All m7 r System info Calculation info am pacman Matrix 0 981 cu 1 9e 2 ti Precipitate 1 0 8 cu 0 2 ti Temperature 350 C Simulation time ie4 s ll Matrix phase fcc Precipitate phase cu4ti 335 330 O0 1000 2000 3000 4000 5000 6000 7000 8000 3000 10000 Time s x 27 93296089 y 346 1849711 Simulation time 0 seconds Abort Continue LD _ _ _______ _ __ __ ____ ___ ___ ii A With the BINARY tab one can provide all necessary information including element phase name any name can be used phase diagram solvus line kinetics diffusivity volume of different phases interfacial energy etc More than one tab for precipitate phases can be added by clicking 3 2 Set Simulation Condition After your system is defined in either way by using database or user input binary data you can click and activate the CONDITIONS tab Here materials chemistry thermal profile temperature and time nucleation p2. The maximum value allowed for relative radius change in one time step This parameter limits the time step according to the following relation which is controlled by the particle growth At lt P x r dr dt for r gt rg where fat is a cut off subcritical size defined by the next parameter The growth rates of supercritical particles with r gt 7 are always bounded and there is a size class and the corresponding growth rate that controls the time step The subcritical particles with r lt 1 however has a mathematical singularity negative infinity in growth rate as r approaches 0 This means that the time step can become extremely small if applying the above criterion to very small subcritical particles In open literature several researchers have tried mathematical transformation to avoid this singularity Unfortunately the transformation also complicates the formulation of the models TC PRISMA implementation uses a simple approach to deal with this issue by defining a cut off size rgt All the particles with r lt rat may disappear within one time step ragt is determined by the next input parameter Maximum relative volume fraction of subcritical particles allowed to dissolve in one time step Prat This input parameter represents the portion of the volume fraction that can be ignored when determining the time step It is used to calculate the cut off subcritical size Tgt for the above time step control that al
3. 3 1 1 CALPHAD Database CalculatiON oooonocccnnnnnnnconnnonnnnnnnnnononannnnnnnnnnonnnnnrnnnnnnnnonnnnrnnnnnnnnronananennss 9 3 1 2 User Defined Binary Calculation ccccccsscccccsssccccessececeesececeeeceeeeeseceseeececeeeceesensccetsusesesseness 10 3 2 SEU SMM Ula ENON CONAN rica 10 Ads A A 10 EA AA A A O ERE OM 11 33 Ser Nucia on Pro DE CCS senadores 11 34 Set Physical care 2 amp 1 rn iia 12 A enn teen E ee 12 3 6 Open and Save Workspace File serial 12 sl E E E E E E E ects E E acre A E E O E A 13 3 8 Add a New Database Package cccccccsssccccesseccceeneccceesseceesusececeuececsuaeceeeeesecesseecesseaecessegecetsageeeetas 13 3 9 Post Processing of Multi modal Size DistributiONS occccoooccnncnnocnnnnnncnnnonanonnnnnnonnnonarnnnonaronnonanonoss 14 Bye Be MOU e o E EE AA 14 4 Theoretical Models and Numerical Methods cccccccssssssseeceececcaeeseeecceeeesuaesseeeceeeessaaaaeeeceeeesaaganeses 15 PL VOT A E o o OO Om E 15 12 NUCE O Nas 16 4 2 HOMO Seneous Nucia lee btojeaa 16 4 2 2 Heterogeneous Nucleation semis 17 4 2 3 Nucleation during a Non Isothermal Process csccccccssssseccccceessecccccuensecccssuaeececseuaneeeeesagaess 19 A GOWN RS 19 AG ACO AUSSI VIVE A A a Raa I hea 21 AS CONUNMIAE QUAN ada 21 4o Mass CONS ALON ds 22 4 7 Numerical Method idas iaa as 22 AS BINALY ADOrOXIMa ON ae da is 24 4 9 Estimation of Coherent Interfacial Energy cccccccs
4. 831 6 References poe MD Lodo Lan Oy USGS b l l l l al ON UI a es 0 T O0 N sess m E eee e e VD Me Ll 13 Langer J Schwartz A Phys Rev 1980 21 948 Wagner R Kampmann R Homogeneous Second Phase Precipitation in Haasen P Ed Mater Sci Technol Compr Treat Weinheim Wiley VCH 1991 Kashchiev D Nucleation Butterworth Heinemann 2000 Russell KC Adv Colloid Interface Sci 1980 13 205 Svoboda J Fischer FD Fratzl P Kozeschnik E Mater Sci Eng 2004 385 166 Feder J Russell KC Lothe J Pound GM Adv Phys 1966 15 111 Cahn JW Acta Met 1956 4 449 Jou HJ Voorhees PW Olson GB Computer Simulations for the Prediction of Microstructure property Variation in Aeroturbine Disks in Green KA Pollock TM Harada H Howson TE Reed RC Schirra JJ Walston S Eds Seven Springs PA TMS 2004 Chen Q Jeppsson J Agren J Acta Mater 2008 56 1890 Chen MK Voorhees PW Model Simul Mater Sci Eng 1993 1 591 Aaron HB Fainstein D Kotler R J Appl Phys 1970 41 4404 Kampmann R Wagner R Kinetics of Precipitation in Metastable Binary Alloys Theory and Application to Cu 1 9 at Ti and Ni 14 at Al in Haasen P Gerold R Wagner R Ashby MF Eds Decompos Alloys Early Stages Oxford Pergamon Press 1984 Becker R Ann Phys 1938 424 128
5. X Atorevious logU1 J2 I where nucleation rate J4 and J2 occurs at the beginning and end of At orevious e Maximum relative change of critical radius P During the nucleation under high supersaturation the critical radius can vary dramatically Hence this input parameter can be used to place a constraint on how fast the critical radium can vary and thus put a limit on time step At lt Pre X Atarevions Ia 2 Te1 e Minimum radius for a nucleus to be considered as a particle Pymin The cut off lower limit of precipitate radius Particles with radius smaller than the value specified for this parameter are discarded In reality the particle cannot be smaller than an atom hence there is no reason to keep track of particles of unphysical sizes e Maximum time step during heating stages Phemax The upper limit of the time step that has been enforced in the heating stages The current algorithm may over estimate the subsequent time increment when temperature is increased It is thus required to reduce this value when the calculation terminates unexpectedly during or after a heating stage The default values for the above numerical control parameters are given in Table 1 Table 1 Default value for numerical parameters that controls the size grid distribution and time step Pate Pur Pe Par Pr Prat Pec Progs Pre Prmin 0 1 0 01 4 8 Binary Approximation Many investigations on precipit
6. and it takes time when possible a less rigorous but simple and efficient model is preferred in many applications The simplified model is based on the advanced model but avoids the difficulty to find the operating tie line and uses simply the tie line across the bulk composition All models treat a spherical particle of stoichiometric composition or with negligible atomic diffusivity growing under the local equilibrium condition Other precipitate shapes and interface conditions such as para equilibrium and finite interface mobility can be considered easily within the same approach which is due to be available in the next version of TC PRISMA The governing equations for the advanced and simplified models are given below For the binary dilute solution model please see Section 4 8 According to the CJA model the interface velocity U can be obtained together with interface concentrations by numerically solving 2n 1 equations comprising of the flux balance equations for n 1 independent components and the local equilibrium conditions for all n components a alp pla _ calB calbyg Fi Ph 11 Gil al p Bla 20V li H t T 12 ole where cP and Cc P are the volume concentrations of component 1 at the interface in the precipitate and matrix respectively M is the corresponding atomic mobility in the matrix u and f f 7 Le P are the chemical potentials in the matrix of the mean field concentration and at
7. avoid undesirable precipitations during heat treatments of alloys such as aging and tempering A summary for the input and output of TC PRISMA is shown in Figure 1 Input Output e Alloy composition e Particle size distribution e Temperature e Number density e Simulation time e Average particle radius e Thermodynamic data Volume fraction e Kinetic data e Matrix composition e Property data Precipitate composition e Nucleation sites e Nucleation rate e Initial microstructure e Critical radius e TTP diagrams Figure 1 Input and output of TC PRISMA TC PRISMA relies on CALPHAD based software tools and databases to provide necessary bulk thermodynamic and kinetic data for phases in multi component systems The CALPHAD approach has been developed for more than 50 years and is now routinely applied to design new alloys and optimize existing materials within various metal industries such as steels and alloys of nickel titanium aluminum and magnesium The power of this approach is due to the adopted methodology where free energy and atomic mobility of each phase in a multicomponent system can be modeled hierarchically from lower order systems and model parameters are evaluated in a consistent way by considering both experimental data and ab initio calculation results TC PRISMA is directly integrated with Thermo Calc and DICTRA which are CALPHAD based computer programs for calculating phase equilibrium and diffusion controlled p
8. barrier and dissolve in the matrix The Zeldovich factor Z is a measure of such probability and is related solely to the thermodynamics of the nucleation process B Fa 4 2N r NAT A x m i s 3 F where N is the Avogadro number and r the critical radius is given by s 20V r AGO 5 The atomic or molecular attachment rate B reflects the kinetics of mass transport in the nucleation process and is given by 5 aa xr da a a o i f where a is the lattice parameter X i and X P are the mole fractions of element 7 at the interface in the precipitate and matrix respectively D is the corresponding diffusion coefficient in the matrix The incubation time for an isothermal reaction is l a 7 where 0 differs from one derivation to another The value 2 from Feder et al 6 is adopted T 4 2 2 Heterogeneous Nucleation All equations remain the same for the calculation of heterogeneous nucleation rate within the framework of CNT but the nucleation energy and available nucleation site are different To a first approximation the nucleation energy is calculated by assuming an effective interfacial energy for each heterogeneous nucleation site In future versions of TC PRISMA more elaborate treatment including grain boundary energy as a parameter will be implemented 1 Z A Figure 4 Tetrakaidecahedron approximation of grains S The number of different nucleation site is dependent
9. condition In a non isothermal process the exact calculation of the incubation time requires a solution to the Fokker Plank equation In TC PRISMA an approximation approach has been employed to deal with the transient nucleation which gives the incubation time as an integral form of past thermal history 8 a 1 p t dt 4 OZ T 10 where T is the incubation time 8 is the impingement rate for solute atoms to the critical cluster as defined in Eq 6 and Z is the Zeldovich factor previously defined in Eq 4 but now as a function of T derived from temperature change The starting point of the integral t O is either the starting time if there is an initial nucleation driving force or the latest time when the nucleation driving force is vanished 4 3 Growth Three different growth rate models have been implemented in TC PRISMA and they are called advanced simplified and binary dilute solution models respectively The advanced model is proposed by Chen Jeppsson and Agren CJA 9 and calculates the velocity of a moving phase interface in multicomponent systems by identifying the operating tie line from the solution of flux balance equations This model can treat both high supersaturation and cross diffusion rigorously Spontaneous transitions between different modes LE and NPLE of phase transformation can be captured without any ad hoc treatment Since it is not always possible to solve the flux balance equations
10. distribution has For the definition used see http en wikipedia org wiki Kurtosis Minimum Separation Limit This is a rough method to prevent that noise or wiggles are interpreted as different distributions If a local minima is encountered above this absolute value the following peak is not interpreted as a new distribution Minimum Distribution Peak Any separated peak below this value is removed 4 Theoretical Models and Numerical Methods 4 1 Overview Based on Langer Schwartz theory 1 TC PRISIMA adopts Kampmann Wagner numerical KWN method 2 to simulate the concomitant nucleation growth and coarsening of precipitates in multicomponent and multiphase alloy systems The KWN method is an extension of the original Langer Schwartz LS approach 1 and its modified MLS form 2 where the temporal evolution of the mean radius and particle density over the whole course of precipitation are predicted by solving a set of rate equations derived with certain assumptions for the rates of nucleation and growth as well as the function of particle size distribution PSD The MLS approach differs from the LS with respect to the Gibbs Thomson equations used for calculating equilibrium solubilities of small particles The former applies the exact exponential form whereas the latter takes the convenient linearized version Instead of assuming a PSD function a prior and working with rate equations for determining only mean radius and particle
11. on the shape and size of grains in the matrix Assuming all grains are equally large tetrakaidecahedra with the distance between one pair of square faces as H and that between the other two pairs as D see Figure 2 we obtain the densities p P and P for the grain boundary area edge length and grain corner number respectively _ 6Vv1 2A 1 24 y P2 4A 8 1 2 i ANNA p2 8 2 A ARO py D 8 3 where A H D is the aspect ratio defining the degree of elongation of the grains By default the aspect ratio is set to 1 and the densities are then the same as obtained by Cahn 7 Once the densities are known if taking boundary thickness as one atomic layer the available nucleation sites can be calculated by 1 3 N o Za i 2 1 0 9 1 m where V is the molar volume of the matrix phase and N is the Avogadro number For a crystalline material given a dislocation density p the number of nucleation sites at the dislocations N can be calculated with the same form 1 3 N al te 9 2 m 4 2 3 Nucleation during a Non Isothermal Process Under a non isothermal conditions temperature dependency of key parameters such as nucleation driving force solute diffusivities and solute concentrations etc have been taken into account and are updated automatically during a simulation Another important parameter that depends on thermal history is the incubation time defined by Eq 7 for an isothermal
12. pressing the Thermal Profile button which brings up the Thermal Profile window Here the time and temperature coordinates of thermal profile points can be entered A minimum of two points 1s required to set up a thermal profile 3 2 2 TTT Diagram File Options Data Formatfigure Help Cvs aa 9 Calphad Database Calculation User Defined Binary Calculation SYSTEM CONDITIONS min max Step Alloy composition mole Temperature 200 700 20 c y cu 98 1 Simulation time jg s TI 1 9 Phase fraction 9 001 Enable Nucleation Properties gar K mm r E System info Calculation info DICINCICI TOTS TO ee Min 200 Max 700 Temperature step 20 Fraction 0 001 Matrix phase FCC Al Precipitate phase CU4TI LE m Simulation time 0 seconds Abort Continue Everything is the same as for a single thermal profile calculation except that one has to specify a temperature range and a phase fraction for the precipitate phase 3 3 Set Nucleation Properties FA Nucleation Set nudeation properties E Nudeation site type IB Automatic calculation of nucleation sites By pressing the Properties button on the CONDITIONS tab one gets a pop up Nucleation window where one specifies the type of nucleation site and the available number of it For grain boundaries edges and corners
13. the grain size and aspect ratio of the matrix phase have to be given For dislocations the dislocation density has to be specified Unchecking Automatic calculation of nucleation sites one can enter a specific value for the number of nucleation site instead of accepting the automatically calculated value 3 4 Set Physical Parameters 7 ad Set Parameters t parameters Inter facial Energy Bulk H Phase Boundary Mobility Grain boundaries Phase Energy Addition Mobility Enhancement Factor Edge Volume Grain corners Dislocations gt Use estimated value with Prefactor 5 Use data from the property database Close Click Set Parameters in the Data menu one can get a pop up Set Parameters window where one can set the interfacial energy volume interface mobility for considering deviation from local equilibrium due to interface friction mobility enhancement factor for considering grain boundary diffusion etc or phase energy addition to the precipitate phase s for altering phase diagram 3 5 Open and Save Script File Format figure Hel Na Mths ms TC PRISMA comes with a number of example scripts that are placed under My Documents Thermo Calc TC PRISMA examples Opening a script file with extension PRM will give a complete setup of a simulation Any user defined simulation can be save in a script by clicking the SAVE button in the top or through the File menu Click Rei
14. Edit Database List window one can select a database and add it to the list of either thermodynamic or kinetic or property databases by clicking Apply Close this window and reopen the Edit Database Package window now one can enter a new package name in the available blank box and then in the same row choose the right database from each category of databases Make sure that everything is correct before clicking Apply A new database package is then generated and stored available for future uses E Edit Database Pacino ta g Package name Thermodynamic database Kinetic database Property database C STEEL TCFE6 MOBFE1 NONE CUTI S50L4 MoB2 NONE ALSC USER C Users ging Document USER C Users ging Document NONE NIALCR USER C Users ging Document x USER C Users ging Document NONE USER C Wsers ging Document USER C Wsers ging Document USER C Users ging Document One can also add a new database package by directly editing the TKDBPACK PRP file One must make sure that each database in the new package is available in the TKDBLIST PRP file If not please add it to the corresponding category in this file Be careful that the specified databases should be actually available on your computer Corrupted PRP file may cause the TC PRISMA software not able to start Therefore 1t is always a good idea to keep a copy of the original files 3 9 Post Processing of Multi modal Size Distributio
15. TC PRISMA User s Guide and Examples Version 2 0 Thermo Calc Software AB Notra Stationsgatan 93 Plan 5 SE 11364 Stockholm Sweden Thermo Calc Software AB 2011 2013 Copyright The Thermo Calc and DICTRA software are exclusive copyright properties of the STT Foundation Foundation of Computational Thermodynamics Stockholm Sweden All rights are reserved worldwide The TC PRSIMA software is an exclusive copyright property of Thermo Calc Software AB Stockholm Sweden All rights are reserved worldwide Thermo Calc Software AB has exclusive rights for further developing and marketing all kinds of versions of Thermo Calc DICTRA and TC PRISMA softwate database interface packages worldwide This TC PRISMA User s Guide and Examples as well as all other related documentation is the copyright property of Thermo Calc Software AB It is absolutely forbidden to make any illegal copies of the software databases interfaces and their manuals User s Guide and Examples Book and other technical publications Reference Book and Technical Information Any unauthorized duplication of such copyrighted products is a violation of international copyright law Individuals or organizations companies research companies governmental institutes and universities that make or permit to make unauthorized copies may be subject to prosecution The utilization of the Thermo Calc DICTRA and TC PRISMA software database interface packages an
16. ation modeling were based on assumptions of stationary growth rate and dilute solution in matrix to calculate nucleation and growth rates in binary or pseudo binary systems It offers a first approximation to the precipitation problem with all necessary model parameters calculated analytically if values of the phase boundary volume and diffusivity are known from any sources With dilute solution assumption the driving force of nucleation is given by B_ ya ya AG X TAs pr In 20 l X7 X and the Gibbs Thomson equation becomes 1 X 20v YAP x ex e m f 21 A JT T 21 The growth dissolution rate of a particle in the steady state approximation is represented by Xe Dp 0S aa s Xi 7 22 In addition to the formula given before for the calculation of the attachment rate and incubation time Eq 6 7 other ways were also widely used in many publications According to Russell 4 the attachment rate in a binary system is written as 4m X D fo 23 a It should be noted that Eq 6 for multicomponent systems does not reduce to this representation for binary systems The incubation time was first approximated by Kampmann and Wagner 12 as 1 X X e A 24 2D X X where c is an adjustable parameter for each particular alloy system and ranges from 1 4 to 3 5 Another formula suggested by Wagner and Kampmann 2 is 1 T 225 25 Although these descriptions are not consistent with more rigorous theoretical d
17. be compared with experimental data collected from R Wagner and R Kampmann Homogeneous second phase precipitation in P Haasen editor Materials Science and Technology A comprehensive treatment Weinheim Wiley VCH 1991 pp 213 303 The measurements are in minutes and nanometers respectively 5 2 1 Use binary dilute solution model 5 2 2 Use a database package 5 2 3 TTT calculation 5 3 Fe C This example simulates the kinetics of precipitation of carbides from a Bcc Fe solution phase 5 3 1 Use binary dilute solution model two carbides 5 3 2 Use binary dilute solution model TTT calculation 5 3 3 Use a database package 5 4 Fe Cr C This example simulates the kinetics of precipitation of both stable and metastable carbides from ferrite phase It demonstrates that metastable carbides may first emerge and then disappear and the stable phase will prevail 5 5 Ni Al Cr This example simulates the kinetics of precipitation of y phase from y phase The simulation results can be compared with experimental data collected from C K Sudbrack et al Acta Mater 56 2008 448 463 The measurements are in hours and nanometers respectively 5 6 Non isothermal The two examples simulate the kinetics of precipitation of y phase from y phase in Ni 8A1 8Cr and Ni 10A1 10Cr at alloys during continuous cooling The simulation results can be compared with experimental results from T Rojhirunsakool et al J Mater Sci 48 2013 825
18. ction time The steady state nucleation rate J can be expressed by x AG J ZB N exp S P N exp kT 2 where Zis the Zeldovich factor is the rate at which atoms or molecules are attached to the critical nucleus N is the number of available nucleation sites of a particular type per unit volume AG is the Gibbs energy of formation of a critical nucleus amp is Boltzmann s constant and T is absolute temperature In the case of homogeneous nucleation each atom in the mother phase is a potential nucleation site The Gibbs energy of formation of a critical nucleus is expressed as 1620 where is the interfacial energy AG is the molar Gibbs energy change for the formation of the P precipitate of the critical composition X i from the A matrix i e the maximum driving force for the gt f phase transformation ve is the molar volume of the p precipitate phase The maximum driving force AG and the critical composition Xf can be obtained by finding the parallel tangent lines or surfaces passing through the alloy composition of the matrix see Figure 2 which is a routine calculation in Thermo Calc Nucleation is a stochastic process and can be considered formally as a diffusion process or Brownian movement in the particle size space Supercritical nuclei with radius slightly larger than the critical radius have a probability of passing back across the free energy
19. d their manuals and other technical information are extensively and permanently governed by the Thermo Calc Software AB END USER LICENSE AGREEMENT EULA which is connected with the software Disclaimers Thermo Calc Software AB and the STT Foundation reserve the rights to further developments of the Thermo Calc DICTRA and TC PRISMA software and related software database interface products and to revisions of their manuals and other publications with no obligation to notify any individual or organization of such developments and revisions In no event shall Thermo Calc Software AB and the STT Foundation be liable to any loss of profit or any other commercial damage including but not limited to special consequential or other damage Please visit the Thermo Calc Software web site www thermocalc se for any modification and or improvement that have been incorporated into the programs or for any amendment that have made to the contents of the various User s Guides and to the FAQ lists and other technical information publications Table of Contents A e ee eee eee ee eee ee eee 2 A E E ctewunanneecaee staan 2 Se NTI CUNO reas o y E ows atcnieere ts ee epost POE EIA 5 DA ES HOM AI SIO A Pe ErEA nn a 6 A O O 50 TE EE E E E E 7 A A 7 2 2 Make a New SIMUIS ON ricas 7 2 3 Work with Different Types Of Files ooocccccoocccnconocnnnnnanonnnnanonnnonaronnonanonnnnnncnnnonanonnonononnnonacnnnonanonoss 8 Serapio User INTEL e 9 NAO tas 9
20. density the KWN method extends the LS and MLS approaches by discretizing the PSD and solving the continuity equation of the PSD directly Therefore the time evolution of the PSD and its th moment 0 th number density 7 th mean radius 3 th volume fraction can be obtained altogether during the simulation The key elements of the KWN method are the models for nucleation and growth under the mean field mass balance condition and the numerical algorithm for solving the continuity equation of the PSD Figure 1 Coarsening will come out naturally without any ad hoc treatment Mass Balance Particle Size Distribution Nucleation Kinetics Continuity Equation Figure 3 The key elements of the KWN method and their relations as implemented in TC PRISMA TC PRSIMA is integrated with Thermo Calc and DICTRA in order to obtain directly all necessary thermodynamic and kinetic information required in the KWN method which has mostly so far been applied for binary or pseudo binary systems where the nucleation and growth rates can be approximated quite easily For industry relevant multicomponent alloys thermodynamic and kinetic databases and calculation tools have to be used in order to obtain various quantities in the multicomponent models for nucleation and growth such as the driving forces for the formation of embryos and their compositions the atomic mobilities or diffusivities in the matrix the operating interface composition
21. e evolution of the PSD 4 5 Continuity Equation Let f r be the PSD of a precipitate phase N the number of particles per unit volume r the mean radius and the particle volume fraction is expressed as N f ar f De IN rf r dr p Zar far l 16 The time evolution of f r follows the continuity equation 1 nc NL 0 17 where U r is the growth rate of a particle of size r and J r is the distributed nucleation rate which is defined by J 0 i ar 18 where J is the nucleation rate 4 6 Mass Conservation The matrix concentration is updated at each time step according to the law of mass conservation If the alloy concentration i e initial mole fraction of component i in the matrix phase is X the new concentration X can be obtained from the following mass balance equation Ep pt 2 x f A 2 td 19 where X r t is the mole fraction of element 1 at the interface in the precipitate phase p of particle size r at time t f r t U r f and V are the PSD function growth rate and molar volume of the precipitate phase p respectively t is the time at time step j Apparently the new matrix concentration and thus the updated supersaturation nucleation rate and interface velocity are all functions of the PSD function This will inevitably lead to the complex non linear behavior of and great difficulty in solving the continuity equation 4 7 Numerical Method Since it is i
22. energy of solution in a multicomponent system involving the two phases considered In the actual calculation the coefficients taken distinguish between only bcc and fcc structure for the 110 and 111 lattice plane respectively It should be emphasized that the method is merely a simple approximation and one should consider it fortuitous if the estimated value gives excellent results for precipitation modeling In most situations one may need to do some adjustment and calibration work on the basis of the estimation This can be justified by the fact that interfacial energy is dependent on so many factors such as temperature incoherency orientation and curvature etc which have been ignored in the estimation Since the method is taking advantage of the existing databases the uncertainties of assessed thermodynamic data should also be kept in mind in the cautious use of the estimated value for the interfacial energy 5 Examples 5 1 Al Sc This example simulates the kinetics of precipitation of Al3Sc from an Fcc Al solution phase The simulation results can be compared with experimental data collected from E A Marquis and D N Seidman Acta Mater 49 2001 1909 1919 and G M Novotny and A J Ardell Mater Sci amp Eng A318 2001 144 154 The measurements are in seconds and nanometers respectively 5 2 Cu Ti This example simulates the kinetics of precipitation of Cu4Ti from an Fee Cu solution phase Simulation results can
23. er of magnitude in size space The number determines a default ratio between two adjacent grid points When there is a need to create new grid points such as nucleating at a new radius not covered by the current range of PSD this default ratio is used to add these new radius grid points A larger value of this parameter enforces a finer grid to allow better numerical accuracy However this also comes with performance penalty since finer grid in the size space often requires smaller time step to resolve the calculations Maximum number of grid points over one order of magnitude in r Phr The maximum allowed number of grid points in size space This parameter determines a lower bound limitation for the ratio of every two next nearest grid points in order to maintain adequate computational efficiency When a ratio of two next nearest grid points is less than this limit the middle grid point will be removed and the corresponding size class merged with the two neighbouring ones Minimum number of grid points over one order of magnitude in r PP The minimum allowed number of grid points in size space This parameter determines an upper bound limitation for the ratio of every two adjacent grid points in order to maintain proper numerical accuracy When a ratio of two adjacent grid points exceeds this limit a new grid point is then inserted between the two adjacent grids to keep the required resolution Maximum relative radius change P
24. erivations they are offered as options to choose in the specialized BINARY module due to their acceptance in many investigations 4 9 Estimation of Coherent Interfacial Energy Interfacial energy is probably the most important parameter in the calculation of rates of nucleation growth dissolution and coarsening Yet this parameter is also a physical property that is least known to everyone so far As mentioned in the introduction it is hoped that in future interfacial energy can also be modeled in the spirit of CALPHAD method and interfacial energy databases can be developed by considering various kinds of experimental data with help of computational tools such as TC PRISMA Before this day comes we have to provide a value for this important parameter to conduct precipitation simulations Since the value of interfacial energy can vary dramatically mostly between 0 01 to 2 0 J m it would be a great help to many users of TC PRISMA to have a mean to obtain a rough number to start with Therefore an extended Becker s model 13 has been implemented in TC PRISMA to estimate coherent interfacial energy by using thermodynamic data from existing thermodynamic databases nz O AE 26 N AS where o is the coherent interfacial energy n is the number of atoms per unit area at the interface z is the number of cross bonds per atom at the interface z is the coordination number of an atom within the bulk crystal lattice and AE is the
25. hase transformation in multicomponent systems and have a wide spectrum of accompanying thermodynamic and kinetic databases With Thermo Calc and DICTRA and accompanying databases almost all fundamental phase equilibrium and phase transformation information such as driving forces for nucleation and growth operating tie line under local equilibrium or para equilibrium condition deviation from local equilibrium at interface due to interface friction atomic mobilities or diffusivities in the matrix phase etc can be calculated without unnecessary and inaccurate approximations In addition to bulk thermodynamic and kinetic data a few other physical properties such as interfacial energy and volume are needed in precipitation models implemented in TC PRISMA These additional physical parameters can be obtained by experiments or other estimation models or first principles calculations As a matter of fact volume data for steels and nickel based alloys has already been assessed and included in TCFE6 TCNI5 and TCAL2 databases It is hoped that in future interfacial energy can also be modeled in the spirit of CALPHAD method and the model parameters can be assessed by considering various kinds of experimental data with help of TC PRISMA In this document we show how to operate the TC PRISMA software and describe various theoretical models and numerical algorithms that has been implemented in the software Chapter 2 gives a shott instruction on how t
26. lows a maximum relative radius changes for all particles f fr3dr f fredr Prat Relative radius change for avoiding class collision P For the supercritical particles the growth rate is non linear usually it first increases with r and then decreases after a certain size In the region s with growth rate decreasing with r it is possible that the smaller size grid point can catch up with the larger size grid if the time step is not controlled To prevent this from happening an additional user input parameter Pec can be used to set a limit on time step according to the following relation At lt P X N41 r dr dt dr dt for Tia gt i414 and dr dat lt dr dt Maximum overall volume change Py This input parameter defines the maximum absolute not ratio change of the volume fraction allowed during one time step This parameter is also used in controlling allowable variation in volume fraction due to the newly created particles within one time step That is At lt P 10 Anr J 3 where Terr and J are effective radius and nucleation rate respectively e Maximum relative change of nucleation rate in logarithmic scale Pigg This input parameter ensures accuracy for the evolution of effective nucleation rate It sets a limit on time step so that the relative change of nucleation rate does not exceed the specified value based on the information of previous step That is At lt Pigg
27. matrix composition e Set Nucleation Properties Click Properties button to get the Nucleation window select nucleation site and enter necessary microstructure information in the case a heterogeneous nucleation site is selected e Set Parameters Enter interfacial energy volume data etc if they are not available from the databases used The Set Parameters window can be accessed by clicking Data gt Set Parameters Simulate Plot results Define System Set Simulation Condition Set Nucleation Properties l Set Parameters Plot Results Figure 2 How to start a simulation from scratch 2 3 Work with Different Types of Files TC PRISMA has following types of files e PRP file Database list file or database package configuration file Be careful when trying to modify these files Make sure that added databases are actually available on your computer Corrupted PRP file may cause the TC PRISMA software not able to start e PRM file Macro file PRI file Workspace file A binary file storing everything It can be loaded for reviewing simulation results e PRD file Result data file 3 Graphic User Interface TC PRISMA offers a user friendly and intuitive Java based Graphic User Interface GUI With only a few clicks and inputs a simulation can be setup and performed After the simulation is completed the results can directly be plotted and compared with pre compiled experimental data The obtained figu
28. mpossible to have a general close form solution the complex non linear continuity equation of the PSD function has to be solved numerically Different numerical methods can be found in literature to solve this equation such as finite difference and method of characteristics In all approaches the continuous PSD is discretized into a sufficiently large number of size classes and then integrated discretely in the time space TC PRISMA utilizes the method of characteristics where the number of particles in each size class is kept constant unless particles of the same size have been nucleated An adaptive geometric grid allocation method has been used for particle size discretization since from nucleation to coarsening the particle size can span several orders of magnitude In this approach the new size grids are allocated evenly in a logarithmic scale and the number of grid points over one order of magnitude is kept almost constant during the whole process by class merging and splitting The time step is controlled by several adjustable numerical parameters based on mechanistic quantities Summarised below are all numerical parameters used in this approach to control either the size grid distribution or time steps e Maximum time step fraction PH as fraction of the simulation time Maximum time step allowed for time integration Number of grid points over one order of magnitude in rt Pyy Default number of grid points for every ord
29. nitiate before opening a new script 3 6 Open and Save Workspace File The TC PRISMA scripts only contain setup for a simulation The simulation data is saved in TC PRISMA workspaces These are files with the file extension PRI As a default all the current simulation data is saved in a workspace called TKDATA PRI From the File menu one can open and save workspaces e Open Workspace This will open an old workspace and load the data stored in this workspace Click Reinitiate before opening a new workspace e Save Workspace This enables the user to rename the workspace Very important This will RESET the workspace and clear all data from it Save Workspace must be performed BEFORE doing any simulations 3 7 Reinitiate A TC PRISMA File Options Data Format fiqu ole ta E At the top left of the GUI is the Reinitiate button This action can also be found in the File menu Clicking this button will terminate the current simulation and restart the program It is necessary to do this before continuing for a new simulation 3 8 Add a New Database Package To add a new database package one can select Edit Database Package from the Data menu Check first if the databases to be included in the new package are actually available in the drop down list in each corresponding category by clicking at the right side of any box If not close this window and then select Edit Database List from the Data menu From the pop up
30. ns When doing non isothermal simulations it is common that particles grow in different generations This will result in multi modal size distributions To correctly estimate the properties of these different generations of particles it is possible in TC PRISMA to separate the peaks of multi modal distributions into separate functions This functionality is activated when checking Separate multi modal size distributions When turned on the size distribution will be evaluated at the given time steps and checked for multi modal peaks These will be separated and used to calculate the specified property It can be applied on the following plot properties e Size Distribution e Mean Radius e Volume Fraction e Number Density 3 9 1 Input Fields Number of Evaluation Points Since the evaluation of multi modality at each time step is costly the user can in this field specify how many evaluation points should be used The points are geometrically distributed over the time scale where nucleated particles are found Excess Kurtosis for Binomial Split This number specifies the sensitivity when the program evaluates whether a curve is bimodal or not The evaluation process will calculate the excess kurtosis of the given size distribution and compare it with the input value If the value is lower than the given value the distribution will be split The excess kurtosis is a statistical measure that measures how peaked and how heavy tail a
31. o perform a TC PRISMA simulation right away The Graphic User Interface GUD is described in Chapter 3 and various windows are introduced one by one following the main procedures of a fresh simulation Chapter 4 covers the precipitation models and numerical method Detailed description of physical and numerical parameters can be found in this chapter Simulation examples are summarized in Chapter 5 1 1 Version History e TC PRISMA version 1 0 and 1 1 treat isothermal precipitation process e TC PRISMA version 2 0 enables arbitrary temperature time profiles especially continuous cooling conditions For the convenience of users interfacial energy between matrix and precipitate phases can be estimated in this version by using an approximation model based on Becker s bond energy approach 2 Quick Start Once TC PRISMA is installed it would be good to start by running a few examples first and confirm that the installation is successful Afterwards you can try to perform simulations for your own alloys of interest Some examples may not work due to that commercial databases are used in these examples and the access to these databases requires a valid license which unfortunately cannot be found on your computer Examples 5 1 5 2 1 5 3 1 5 3 2 5 5 and 5 6 need no commercial databases and should always be possible to run 2 1 Run Examples Examples can be found by clicking File gt Open Script or the open file icon A macro file with e
32. res can be edited printed and saved into different graphic format We now give an introduction to all major features of the GUL It is believed that most of the menus buttons and boxes are self evident and here we focus only on a few most important ones 3 1 Define System The startup window allows one to choose between a simulation with CALPHAD database input and one with user input binary data Depending on this choice either SYSTEM tab or BINARY tab will show up and allow you to define your system accordingly 3 1 1 CALPHAD Database Calculation DATABASE PACKAGE TC STEEL ELEMENTS Add elements PHASES Matrix Add phases System info Calculation info Thermodynamic database TCFE6 ase MOBFE1 Continue Simulate With the SYSTEM tab one can e Select a pre defined database package from the drop down list of available packages A package consists of three compatible databases one for thermodynamic data one for kinetic data and the other for properties such as interfacial energy The user can define a new set of database packages see Section 3 8 e Add elements e Choose a matrix phase e Add precipitate phase s Maximum 5 different phases are allowed 3 1 2 User Defined Binary Calculation Memu o File Options Data Format figure Help v k gt a Calphad Database Calculation User Defined Binary Calculation
33. roperties can be specified Two types of calculations are allowed one for a single thermal profile simulation and the other for a TTT calculation When all are set a simulation can be started by pressing the Simulate button 3 2 1 Single Thermal Profile C PRISN File Options Data Format figure Help Cds Ya 9 Calphad Database Calculation User Defined Binary Calculation 1 Temperature profile x SYSTEM CONDITIONS 1 Temperature profile 9 Multi partide TIT diagram 1200 Thermal Profile oct ea esac AL 10 1100 CR 10 E gt E 1000 seconds mole Enable Nucleation gt a 800 3 E o 2 700 2 600 x 500 System info Calculation info A SEF iaar SLO SENSO aaaea O CS Ws Gs aaa A tdb 400 Property database NONE Elements 10 AL 10 CR 80 NI pe i i mes 300 seettarson tise 1300 0151 J o 500 1000 1500 2000 2500 3000 3500 Matrix phase FCC Al J Time s Precipitate phase GAMMA PRIME 3 Simulation time 0 seconds Abort Continue Here one needs to specify heat treatment schedule and alloy compositions One needs also to set the nucleation properties see Section 3 3 by pressing the Properties button and perhaps some physical parameters see Section 3 4 if for example there is no interfacial energy or volume data available in the database package Heat treatment schedule can be defined by
34. s under local equilibrium or para equilibrium condition the Gibbs Thomson effect and the deviation from local equilibrium due to interface friction etc With Thermo Cale and DICTRA and accompanying databases all these properties and effects can be calculated without unnecessary and inaccurate approximations In the following sections various models and numerical methods implemented in TC PRISMA are introduced 4 2 Nucleation Precipitation process starts from the nucleation of clusters that can be considered as embryos of new phases with distinctive structures or compositions In a perfect single crystal nucleation happens homogeneously In an imperfect crystal or polycrystalline materials nucleation tends to occur heterogeneously due to the presence of dislocations grain boundaries edges and corners These imperfections or defects reduce the nucleation barrier and facilitate nucleation However if supersaturation or driving force is very large homogeneous nucleation is also possible since all sites including those inside a grain can be activated 4 2 1 Homogeneous Nucleation The classic nucleation theory CNT 3 4 has been extended for the purpose of modeling nucleation in multicomponent alloy systems The time dependent nucleation rate J t is given by I t Js p z 1 where J is the steady state nucleation rate 7 is the incubation time for establishing steady state nucleation conditions and f is the isothermal rea
35. scccccsseccccssccceesececeesececceeeceeeeesccesseecesseeceesenes 25 SAM DSi lanar amen taunt E RETEA 27 ES sitesntc hone secs aueoay voscres vec a cescieg A 27 A A N 27 adas 27 A aeha cto ce ated uacen te acca isiaage ce eeee cee sea lo raee aes ugee aeesieetamecuiaeeueneuenereeoimeceane eceeeneetes 27 O A O 27 A O 28 a O O A AI 29 1 Introduction The TC PRISMA software is a general computational tool for simulating kinetics of diffusion controlled multi particle precipitation process in multi component and multi phase alloy systems Precipitation formation of particles of a second phase or second phases from a supersaturated solid solution matrix phase is a solid state phase transformation process that has been exploited to improve the strength and toughness of various structural alloys for centuries This process is thermochemically driven and fully governed by system bulk and interface thermodynamics and kinetics Typically a precipitation process has three distinctive stages nucleation growth and coarsening However under certain conditions they could happen also at the same time With TC PRISMA the kinetics of concurrent nucleation growth and coarsening can be simulated by calculating the evolution of the probability distribution of the particle number densities usually called particle size distribution PSD The simulation results can be used to understand and guide how to obtain desirable precipitates with certain PSD or to
36. system if the particle sizes are not uniform Solutes at the interface in the matrix near a particle of a radius smaller than critical radius will have a higher chemical potential than that corresponding to the mean concentration of the matrix As a result the solutes will diffuse from the precipitate matrix interface to the inside of the matrix and cause dissolution of the particle Conversely particles with radius larger than the critical size will have lower interface potentials and the solutes will diffuse to the interface and cause growth of the particles Since it is not possible to find a closed form analytic solution for the problem of diffusion controlled spherical particle dissolution 11 we simply apply Eq 11 14 with the absolute value of 2 to calculate the interface velocity for particles of all sizes As can be easily seen if r lt r then the Gibbs Thomson equation Eq 12 gives u oS LU and a negative velocity will result from Eq 11 for particles having r lt r which means that they will shrink Results for particles having r gt r will be obtained vice versa In all situations when the absolute values of 2 are very small the steady state solution for either growth or dissolution will be recovered In conclusion we have essentially treated dissolution as the reverse of growth 11 and the coarsening shall come out naturally either together with nucleation and growth or as a dominant process finally in the course of th
37. the interface respectively u is the chemical potential at the interface in the precipitate In the above local equilibrium condition the multicomponent Gibbs Thomson effect has been taken into account by adding a curvature induced pressure term to the Gibbs energy of the precipitate phase The introduced effective diffusion distance factor E for each independent component is given by Q aa 13 2A Sen Grae where Q O is the so called dimensionless supersaturation for individual component EA i and A is obtained via the following equation 24 243 V7 expla berfe 2 14 Combining Eq 11 and 12 the simplified model can be derived K 20V 15 r r gt 7 XP Pm E where K SY IZ and AG is the driving force The interface compositions XP r M de X r r and X Pr are simply taken from the tie line across the matrix composition The non zero volume correction to the velocity according to Chen and Voorhees 10 has been taken into account and it follows p v I rJ4aN lt r gt where U is the corrected velocity lt r gt is the mean radius and N is the number density 4 4 Coarsening Physically speaking coarsening or Ostwald ripening where big particles grow and small particles shrink is a process driven by lowering the total surface energy of the system From a thermodynamic point of view the Gibbs Thomson effect leads to inhomogeneous chemical potentials in the
38. xtension PRM for each example in the corresponding folder can be loaded for setting up the simulation task If everything is OK the Simulate button at the lower right corner will then be activated Click the button and the simulation will start For some examples experimental data is collected from publications and can be compared against the simulation results by clicking Format figure gt Add experiments Time and length units can be changed by clicking Format figure gt Time units and Format figure gt length units respectively in order to make proper comparisons A description of all examples is given in Chapter 5 Examples 2 2 Make a New Simulation To perform a simulation for your own alloy you can try by copying an example script file to a new folder and making necessary modifications to the script or alternatively by starting from scratch and going through the following steps see Figure 2 When starting from scratch it is advised to save your setup to a script file and during the save script operation select always a different folder or if necessaty make a new folder in order to avoid existing files being overwritten or output files being mixed up when running several simulations at the same time e Define System Choose a database package select elements select matrix phase and select precipitate phase s e Set Simulation Conditions Activate the CONDITIONS panel enter thermal profile and enter
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