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Computational Thermodynamics 3 Outline Compound energy formalism, part 2 Associated liquid solution Interpolation to ternary systems Introduction to optimization Compound energy formalism, part 2 The green surface is represented by eq: surf 0 0 0 0 Gmref = yA yCGAC + yB yCGBC + yA yDGAD + yB yDGAD Compound energy formalism, part 2 The method for describing the Gibbs excess energy can, again, be best shown by using a two-sublattice system (A, B)1(C,D)1 before generalizing to a multi-component system. In this alloy A-C, A-D, B-C and B-D interactions are controlled by the Gibbs energy of the compounds AC, BC, AD and BD. Mixing on the sublattices controls A-B and C-D interactions and the simplest form of interaction is a regular solution format such that G =y y L xs m 1 1 0 A B A,B:* +y y L 2 2 0 C D *:C,D where L0 A, B:* and L0 *:C,D denote regular solution parameters for mixing on the sublattices irrespective of site occupation of the other sublattice. Compound energy formalism, part 2 A sub-regular model can be introduced by making the interactions compositionally dependent on the site occupation in the other sublattice. G =y y y L xs m 1 1 2 0 A B C A,B:C +y y y L 1 2 2 0 A C D A:C,D +y y y L 1 1 2 0 A B D A,B:D +y y y L 1 2 2 0 B C D B:C,D Compound energy formalism, part 2 Finally some site fraction dependence to these parameters can be added such that LA,B:C = å LA,B:C ( y - y n 1 A ) 1 n B n LA,B:D = å LA,B:D ( y - y ) LA:C,D = å LA:C,D ( y - y 2 n D LB:C,D = å LB:C,D ( y - y ) n 1 A 1 n B n n 2 C n n n 2 C 2 D ) n Compound energy formalism, part 2 Interstitial phases. These are predominant in steels and ferrous-based alloys, where elements such as C and N occupy the interstitial sites of the ferrite and austenite lattices. In this case the structure of the phase can be considered as consisting of two sublattices, one occupied by substitutional elements, such as Fe, Cr, Ni, Mn, etc., and the other occupied by the interstitial elements, such as C or N, and interstitial vacancies (Va). As the concentration of C, N,..., etc., increases the interstitial vacancies are filled until there is complete occupation. The occupation of the sublattices is shown below as (Fe, Cr, Ni, Mn… )u (Va, C, N...)v Compound energy formalism, part 2 In the case of an FCC_A1 structure, with u = v = 1, the state of complete occupation of interstitial carbon corresponds to a MC carbide with the NaCI lattice. For the HCP_A3 structure, with u = 1 and v = 0.5, complete occupation of the interstitial sites by carbon gives the M2C carbide with a hexagonal Fe2N-type structure. 1 2 0 2 0 1 0 0 Gm = yCr yVa GCr:Va + y1Fe yVa GFe:Va + yCr yC2 GCr:C + y1Fe yC2 GFe:C ( 1 1 2 2 +RT 1( yCr ln+ yCr + y1Fe ln+ y1Fe ) +1( yC2 ln+ yC2 + yVa ln+ yVa ) ) æ ö 1 1 2æ ö n 1 1 n n 1 1 n +y y y ç å LCr,Fe:Va ( yCr - yFe ) ÷ + yCr yFe yC ç å LCr,Fe:C ( yCr - yFe ) ÷ èn ø èn ø æ ö 1 2 2 æ ö 1 2 2 n 2 2 n n 2 2 n +yCr yC yVa çå LCr:C,Va ( yC - yVa ) ÷ + yFe yC yVa çå LFe:C,Va ( yC - yVa ) ÷ èn ø èn ø 1 1 2 Cr Fe Va Compound energy formalism, part 2 For the case of the FCC_A1, austenite phase in Cr-Fe-C, the Gibbs energy of the phase is represented by the formula (Cr, Fe)1 (C, Va)1 and its Gibbs energy given by the following equation 1 2 0 2 0 1 0 0 Gm = yCr yVa GCr:Va + y1Fe yVa GFe:Va + yCr yC2 GCr:C + y1Fe yC2 GFe:C ( 1 1 2 2 +RT 1( yCr ln yCr + y1Fe ln y1Fe ) +1( yC2 ln yC2 + yVa ln yVa ) ) æ ö 1 1 2æ ö n 1 1 n n 1 1 n +y y y ç å LCr,Fe:Va ( yCr - yFe ) ÷ + yCr yFe yC ç å LCr,Fe:C ( yCr - yFe ) ÷ èn ø èn ø æ ö 1 2 2 æ ö 1 2 2 n 2 2 n n 2 2 n +yCr yC yVa çå LCr:C,Va ( yC - yVa ) ÷ + yFe yC yVa çå LFe:C,Va ( yC - yVa ) ÷ èn ø èn ø 1 1 2 Cr Fe Va Compound energy formalism, part 2 The database PHASE SBSN % 2 .5 .5 ! CONSTITUENT SBSN :SB,SN : SB,SN : ! PARAMETER G(SBSN,SB:SB;0) 2.98150E+02 +GHSERSB#+3243.7917; 3.00000E+03 N REF0 ! PARA G(SBSN,SN:SB;0) 298.15 0; 6000 N ! PARAMETER G(SBSN,SB:SN;0) 2.98150E+02 +.5*GHSERSN#+.5*GHSERSB# -2951.1569.52608471*T; 6.00000E+03 N REF0 ! PARAMETER G(SBSN,SN:SN;0) 2.98150E+02 +GHSERSN#+4000-.043093938*T; 6.00000E+03 N REF0 ! PARAMETER G(SBSN,SB,SN:SB;0) 2.98150E+02 0.0 ; 3.00000E+03 N REF0 ! PARAMETER G(SBSN,SB:SB,SN;0) 2.98150E+02 0.0 ; 3.00000E+03 N REF0 ! PARAMETER G(SBSN,SN:SB,SN;0) 2.98150E+02 -573.04832; 3.00000E+03 N REF0 ! PARAMETER G(SBSN,SB,SN:SN;0) 2.98150E+02 -573.04832; 3.00000E+03 N REF0 ! Associated liquid solution In the systems with short range ordering (SRO), unlike atoms tend to stay together for a shorter or longer length of time. The term associate was introduced to denote an association between unlike atoms when the attractive forces between the atoms are not strong enough to form a stable chemical molecule. Fictitious constituents can be used in the same form of that equation for the surface Gibbs energy and thus creates additional degree of freedom. n srf Gm = å yiGi0 i=1 n srf Sm = -Rå yi lnyi i=1 Associated liquid solution The Gibbs energy of excess usually describes inteactions between an associate and pure elements. xs æ æ nö nö n n Gm = yA yAB çå LA,AB ( yA - yAB ) ÷ + yAB yB çå LAB,B ( yAB - yB ) ÷ èn ø èn ø Associated liquid solution SPECIES PBTE_L PB1TE1! PHASE LIQUID % 1 1.0 ! CONSTITUENT LIQUID :PB,PBTE_L,TE : ! PARAMETER G(LIQUID,PB;0) 2.98150E+02 -2977.961+93.949561*T -24.5242231*T*LN(T)-.00365895*T**2-2.4395E-07*T**3-6.019E-19*T**7; 6.00610E+02 Y -5677.958+146.176046*T-32.4913959*T*LN(T)+.00154613*T**2; 1.20000E+03 Y +9010.753+45.071937*T-18.9640637*T*LN(T)-.002882943*T**2+9.8144E-08*T**3 -2696755*T**(-1); 2.10000E+03 N REF0 ! PARAMETER G(LIQUID,PBTE_L;0) 2.98150E+02 -4.2572284E+04 -1.7419984E+00*T+GHSERPB#+GHSERTE#; 3.00000E+03 N REF0 ! PARAMETER G(LIQUID,TE;0) 2.98150E+02 -17554.731+685.877639*T -126.318*T*LN(T)+.2219435*T**2-9.42075E-05*T**3+827930*T**(-1); 6.26490E+02 Y -3165763.48+46756.357*T-7196.41*T*LN(T)+7.09775*T**2-.00130692833*T**3 +2.58051E+08*T**(-1); 7.22660E+02 Y +180326.959-1500.57909*T+202.743*T*LN(T)-.142016*T**2+1.6129733E-05*T**3 -24238450*T**(-1); 1.15000E+03 Y +6328.687+148.708299*T-32.5596*T*LN(T); 1.60000E+03 N REF0 ! Associated liquid solution PARAMETER G(LIQUID,PB,PBTE_L;0) 2.98150E+02 2.2173103E+04 -1.0995062E+01*T; 3.00000E+03 N REF0 ! PARAMETER G(LIQUID,PB,PBTE_L;1) 2.98150E+02 1.3818301E+03; 3.00000E+03 N REF0 ! PARAMETER G(LIQUID,PBTE_L,TE;0) 2.98150E+02 -4.2749684E+03+4.1391182E+00*T; 3.00000E+03 N REF0 ! Ternary systems The predominant method at the present time uses the equation developed by Muggianu. In this circumstance the excess energy in a multicomponent system, as given by the Equation: xs æ æ æ nö nö nö n n n Gm = xA xB çå LA,B ( x A - xB ) ÷ + x A xC çå LA,C ( x A - xC ) ÷ + xB xC çå LB,C ( xB - xC ) ÷ èn ø èn ø èn ø Besides that, there are also: Kohler and Toop interpolations. Ternary systems Ternary systems The ternary interaction parameter can be added: n-2 n-1 xs Gm = å å n åxx x L i j k ijk i=1 j=i+1 k= j+1 Higher-ordered interaction parameter can be also added: n-3 n-2 n-1 xs Gm = å å n å åxx x xL i=1 j=i+1 k= j+1 l=k+1 i j k l ijkl Ternary systems Database PARAMETER G(LIQUID,CU,SB,SN;0) 2.98150E+02 1.31473890E+05 -1.12335674E+01*T+0*T*LN(T) ; 3.00000E+03 N REF0 ! PARAMETER G(LIQUID,CU,SB,SN;1) 2.98150E+02 -1.72286975E+04 -2.00146866E+01*T+0*T*LN(T); 3.00000E+03 N REF0 ! PARAMETER G(LIQUID,CU,SB,SN;2) 2.98150E+02 -1.22041192E+04 -4.63210745E+01*T+0*T*LN(T) ; 3.00000E+03 N REF0 ! Ternary systems Let’s perform some calculation of ternary system. Ternary systems Ternary systems Ternary systems Ternary systems Try to change the picture by yourself Ternary systems Ternary systems Isothermal section at 500K. Ternary systems Ternary systems Isoplethal section Al05Mg05-Zn. This is not a phase diagram. Why? Introduction to optimization Next slides are based on Pandat User Guide available on www.computherm.com Introduction to optimization PanOptimizer Functions The major functions included in the PanOptimizer menu are shown in Figure Introduction to optimization Rough Search To perform the rough search, we should follow the steps as follow 1. Prepare the thermodynamic database file (.TDB) In order to do the optimization, we must first define the model parameters to be optimized in the thermodynamic database (.tdb). The model parameters to be optimized can be defined in Pandat workspace where all the built-in keywords are automatically highlighted. It can also be done through outside text editor such as “Notepad”. Each model parameter that needs to be optimized is defined by the keyword “OPTIMIZATION”. The format of defining a model parameter is: Optimization [parameter name][low bound][initial value][high bound] N ! Introduction to optimization A definition sample for liquid phase in binary Al-Zn system is given as follows Introduction to optimization 2. Prepare the experimental file (.POP) Keyword CREATE_ROUGH_EQUILIBRIUM is used to define phase equilibrium for rough search. When creating an equilibrium data set for rough search, the status of phases in equilibrium can only be FIXED. The keyword PHASE_POINT is used to define the state space of the phases in equilibrium including temperature, pressure and composition. No thermodynamic property is allowed to be optimized for multiple-phase equilibria. However, a special case is the single-phase equilibrium. The thermodynamic property data of the single-phase equilibrium can be used as experimental data to optimize the model parameters. Taking the binary Al-Zn system as an example, we will define a rough equilibrium between Fcc_A1 and Hcp_A3 at 500K. Note: For a phase described by a sublattice model, the initial value of an element or species should be specified using SET_START_VALUE. Introduction to optimization Introduction to optimization A complete POP file for rough search can be found at the Pandat installation directory Introduction to optimization 3. Load/compile experimental file 4. Rough Search There are four control areas in the control panel for rough search as shown in Figure Introduction to optimization Introduction to optimization There are two types of built-in optimization algorithms for Rough Search and they are Global Search and Local Search, as shown in area C. Global Search is to search a set of good initial values of the model parameters within the whole searching domain, while Local Search leads to a quick convergence to an optimal solution based on the given initial values of model parameters. The user can set the maximum number of function calls for both algorithms. Introduction to optimization Let’s again take the binary Al-Zn system as an example. The phase boundary and invariant reaction information as shown in Figure is used for the rough search and the result of rough search is also shown in the same Figure Introduction to optimization The calculated phase boundary compared with the experimental data after the rough search with Global Optimization Introduction to optimization By setting the maximum number of function evaluations to be 1000 and clicking Global Search, the program finally stops at the sum of squares of 2.91. Figure in previous slide shows the calculated phase boundaries after a global search with 1000 function evaluations. It can be seen that the optimized parameters give a very similar topology with the known Al-Zn phase diagram. By clicking Local Search, the sum of squares decreases from 2.91 to 0.059, which reproduces the phase diagram in better agreement with the experimental one as can be seen in Figure in next slide. Introduction to optimization The calculated phase boundary compared with the experimental data after the rough search with Local Optimization Introduction to optimization During rough search, user can check model parameters through Parameters. In addition, chemical potential differences between any two phases in equilibrium can be accessed by Experimental Data. They are reflected by the values in column “(Wtd. Residual)^2” as shown in the dialog window in Figure in next slide. Similarly, the weight factor of the equilibrium can be adjusted as needed during optimization procedure. The chemical potential values of each phase in equilibrium can be accessed by clicking the buttons in the first column. Introduction to optimization Introduction to optimization 5. Save/Open Optimization Results During optimization, the user may save and load optimization file through Save Optimization Results and Open Optimization Results. The optimization results file has the extension name of POR that can only be read by PanOptimizer. Intermediate optimization results can be saved and restored through these two operations. These functions are very useful especially when a user is optimizing model parameters for a complicated system. The user can always go back to a certain middle stage and restart from there. This will save user’s time by avoiding some repeated work. Introduction to optimization Introduction to optimization Normal Optimization In general, user should follow a few steps below to perform the model parameter optimization. 1. Prepare the thermodynamic database file (.TDB) Preparation of the fileis exactly the same as in case o rough optimization 2. Prepare the experimental file (.POP) Users need to provide their own experimental data file for optimization of model parameters. The most widely accepted format for experimental data file in the CALPHAD society is a POP file. Introduction to optimization PanOptimizer accepts most of the keywords in the POP format and adds a few special keywords. In a POP file, a phase can have four statuses: ENTERED, FIXED, DORMANT, and SUSPEND. The first two statuses were used most frequently. When phases are in the ENTERED status, PanOptimizer does not require user to input any initial values for calculation since the truly stable phase equilibria will be found automatically in this case with the built-in global optimization algorithm. On the other hand, for those phases in FIXED or DORMANT status, the initial values should be provided by the user. Example POP files are provided in the installation dictionary of Pandat. Introduction to optimization 3. Load thermodynamic database file 4. Load and compile experimental file Introduction to optimization 5. Perform Optimization Once the TDB file and the experimental POP file are loaded, user is ready to do the optimization. In the current version of PanOptimizer, the optimization is controlled through the optimization control panel as shown in Figure Introduction to optimization Histogram (A) Histogram is for displaying and tracing the history of the discrepancy between modelcalculated values and experimental data, which is characterized by the Sum of Squares displayed during the whole optimization procedure. The histogram plots the sum of squares vs. the number of function calls. The exact value of the sum of squares in the current step can be found at the up-right corner of this area. Introduction to optimization Free Bound/Unbound Variables (B) The user can choose the model parameters to be optimized in either the bounded or the unbounded mode. In the bounded mode, the low and high bounds defined in a TDB file will take into effects. Optimization (C) The goal of the optimization process is to obtain an optimal set of model parameters so that the model calculated results can best fit the given experimental measurements. The optimization process can be controlled by choosing 1 Iteration or Run. With 1 Iteration, the maximum number of function calls is set to be 2(N+2), where N is the number of model parameters to be optimized. The user can click Run mode several times until the optimal solution is found or the designated maximum number of function calls is reached. Introduction to optimization Optimization Results (D) During optimization, user can check model parameters through the Parameters button as shown in Figure. For each model parameter, user can change its low bound, upper bound and initial value in the dialog window as shown in Figure. User can Include or Exclude a certain parameter in the optimization through the “check box” in front of the parameter. If a set of optimized parameters is satisfactory, user can save this set of values as default ones through Set Default shown in Figure. Otherwise, user can reject this set of values and go back to previous default values through Get Default. User also can save the TDB file with the optimized model parameters through Save TDB. The standard deviation and relative standard deviation (RSD) of each parameter are computed during optimization. The parameter evolution during the last 100 iterations can be tracked by clicking the parameter name in the table as indicated in Figure. It should be noted that any changes on model parameters made by user will take into effect only after the “Apply” button is clicked. Introduction to optimization Here we take the binary Al-Zn system as an optimization example. The TDB and POP files are available at the installing directory of Pandat “/Pandat_Examples/PanOptimizer/”. In this example, there are totally 11 parameters to be optimized and all the initial values are set to be zero. The available experimental data include: Introduction to optimization Before the optimization, user may want to check the calculated results using the initial set of model parameters without optimization. Phase diagram calculation can be done through 2-D section calculation, and enthalpy calculation can be done through the 1-D line calculation. The parameters before optimization result in large discrepancies between the calculated values and the experimental measurements. Optimization of these model parameters (initially set as zero) is needed. Introduction to optimization The Figure shows the calculated Al-Zn phase diagram and enthalpy of mixing for liquid phase at 953K along with the given experimental data, respectively. Introduction to optimization By clicking Run twice (two rounds), and each run performs 50 function calls, the sum of squares decreases from 904896 to 3061 as shown in left Figure. After another two rounds of Run, the sum of squares decreases from 3061 to 2.8 as shown in a right Figure. Now, we can do the real time calculation using the instantly obtained optimized parameters. Introduction to optimization The two comparisons in Figure show excellent agreements between the calculated results and the measured data. Introduction to optimization Optimization Results (D) During optimization, user can check model parameters through the Parameters button as shown in Figure. For each model parameter, user can change its low bound, upper bound and initial value in the dialog window as shown in Figure. User can Include or Exclude a certain parameter in the optimization through the “check box” in front of the parameter. If a set of optimized parameters is satisfactory, user can save this set of values as default ones through Set Default shown in Figure. Otherwise, user can reject this set of values and go back to previous default values through Get Default. User also can save the TDB file with the optimized model parameters through Save TDB. The standard deviation and relative standard deviation (RSD) of each parameter are computed during optimization. The parameter evolution during the last 100 iterations can be tracked by clicking the parameter name in the table as indicated in Figure. It should be noted that any changes on model parameters made by user will take into effect only after the “Apply” button is clicked. Introduction to optimization Introduction to optimization Introduction to optimization 6. Save/Open Optimization Results During optimization, the user may save and load optimization file through Save Optimization Results and Open Optimization Results. The optimization results file has the extension name of POR that can only be read by PanOptimizer. Intermediate optimization results can be saved and restored through these two operations. These functions are very useful especially when a user is optimizing model parameters for a complicated system. The user can always go back to a certain middle stage and restart from there. This will save user’s time by avoiding some repeated work. Introduction to optimization