The paper forwards the conclusions of a survey performed on a 

mathematical model of the phase equilibrium from the ternary system 

Al-Cu-Si. The author presents the calculus of the statistic equation of 

the liquidus surface model from this diagram, the plotting and 

statistical-mathematical interpretation of the results obtained

Florentina A. Cziple

Directory of Open Access Journals

  141    Florentina A. Cziple The  Establishment,  Plotting  and  Statistic–Mathematical Interpretation of the Liquidus Surface from the Phase Equilibrium Diagram of the Ternary System Al-Cu-Si   The  paper  forwards  the  conclusions  of  a  survey  performed  on  a mathematical model of the phase equilibrium from the ternary system Al Cu Si.  The author presents the calculus of the statistic equation of the  liquidus  surface  model  from  this  diagram,  the  plotting  and statistical mathematical interpretation of the results obtained.   1. Introduction The  model  of  obtaining  a  liquidus  surface  through  thermodynamic mathematical computerised modelling consists of a thermodynamic analysis and a mathematical complex treatment of the pure components and of their interaction in the binary systems and in the ternary system. The phase equilibriums liquid solid  from  the  binary  diagram  Al Si  [1],  Al Cu  [6],  Cu Si  [7]  were  treated analytically, rigorously deducing he equations of the phase transformation curves delimiting  the  monophase  liquid  domain  form  the  biphase  liquid solid  domain based on the models of the regular un ideal solutions.  2. The computerised generation of the liquidus surface  In order to establish the liquidus curve that forms the perimeter of the liquidus surface  of  the  ternary  system  Al Cu Si,  we  mathematically  processed     through concatenation     the  following  functions  (Table  1)  inferred  from  the  phase equilibrium of the binary systems Si Al, Al Cu, Si Cu, valid in the specified sub intervals of the x variable (% atomic) [4]:  ANALELE UNIVERSITĂłII   “EFTIMIE MURGU” RE IłA  ANUL XIII, NR. 1, 2006, ISSN 1453 - 7397    142 Table 1     The mathematical functions of the fragments of liquidus curve from the binary diagrams Al Si, Si Cu, Cu Al Binary system Interval Equation  Coefficients [ ] 89 0¸ Î Al x   ( )( )LAlLAlxxT- × --=1 ln 987 . 1 189 , 73000 121002  Al Si ( ] 100 89¸ Î Al x ( )( )LAlLAlxxTln 987 . 1 189 , 71 3000 25702× -- -=   [ ] 17 0¸ Î Cu x  3 2 x d x c x b a y × + × + × + = y=Temp (K) x=atomic fraction Cu 9807 . 2407049536 . 5893 . 61254386 . 933- ==- ==dcba ( ] 32 17¸ Î Cu x 3 2 x d x c x b a y × + × + × + =  0285 . 60537855 . 64722222 . 23297285 . 582=- ===dcba ( ] 36 32¸ Î Cu x 3 2 x d x c x b a y × + × + × + =  6 . 49999902 . 51857143 . 178178327 . 21163- ==- ==dcba ( ] 52 36¸ Î Cu x 3 2 x d x c x b a y × + × + × + =   a =607.53922 b =350.58049 c =1289.9897 d = 128.99897 ( ] 60 52¸ Î Cu x 3 2 x d x c x b a y × + × + × + = a =10908.29 b = 57118.803 c =108281.59 d = 66600.059 ( ] 66 60¸ Î Cu x 3 2 x d x c x b a y × + × + × + = 367 . 8333378 . 146785564 . 84609008 . 17156- ==- ==dcba Al Cu ( ] 83 66¸ Î Cu x ( ) d x c b a y + × × + = cos  7903553 . 1183136 . 19765299 . 223147 . 1301- ====dcba   143 ( ] 100 83¸ Î Cu x 3 2 x d x c x b a y × + × + × + = 5882 . 974632 . 28631738 . 281586496 . 7916=- ==- =dcba [ ] 7 0¸ Î Cu x  4 3 2 x e x d x c x b a y × + × + × + × + =  y=Temp (K) x=atomic fraction Cu 1212 . 30877172 . 32965227 . 165872908 . 2628106 . 1682- ==- =- ==edcba ( ] 8 7 ¸ Î Cu x  2 x c x b a y × + × + =   a = 8649.5116 b =25778.253 c =   16990.83   ( ] 2 , 8 8¸ Î Cu x c bx ay+=1 4769895 . 15 0001547649 . 00010221998 . 0=- ==cba ( , 8 2 , 8 ¸ Î Cu x b ax y + =   500693==ba  Si Cu ( 100 6 , 8 ¸ Î Cu x cx e bay - × +=1 978334 . 177 . 10389138063 . 1376===cba  We  studied  the  binary  diagrams  Si Al,  Al Cu,  Cu Si,  determining  the  analytical equations and the optimised empirical statistical functions of the liquidus curves, with the experimental data from the literature [1],[6],[7]. In  order  to  establish  the  analytical  equations  of  the  liquidus  curves separating the liquid monophase domain from the bi phase domain where the solid component  has known thermal and thermodynamic properties, we applied three computation models for: the ideal liquid solution ( Cp=0 and  Cp≠0) and for the non ideal  solution  of  the  regular  type  [1],  [4].  The  empirical  equations  of  the liquidus curves were established for the cases where there were no thermal and thermodynamic data of the solid phases [4],[3],[6], [7].  The  selection  of  the  optimum  equations  of  the  equilibrium  curves  was performed with the help of the computer, based on the correlation coefficient r and on the estimated standard error S, by statistically mathematically processing the experimental data from the literature, using the statistic software CurveExpert The liquidus surface form the system Al Cu Si  modelled on the computer with the 3D software StudioMax, using the concatenation of functions (table 1), is presented in fig.1 .   144  Figure 1 The computerised generation of the liquidus surface from the ternary diagram Al Cu Si  The generation of the liquidus curved surface from the spatial diagram of phase equilibrium of the ternary system Al Cu Si was performed on a computer using  the  mathematical  instruments  that  utilise  sets  of  complex  mathematical functions which define the surface through a set of control points specified by the user (Fig.2).  These points resulted through the mathematical processing of the liquidus and  solidus  curves  of  the  binary  systems  Si Al,  Al Cu,  Cu Si,  with  the  graphic software 3D Studio Max. The  plotting  (Fig.2)  starts  from  the  Gibbs  triangle  with  the  concentration expressed in atomic %, and the axis perpendicular on the triangle plane represents the  temperature  in  Kelvin  degrees.  The  acknowledged  term  for  discretisation  is „mesh”.  The projection of isotherms of the liquidus surface (Fig.3) on the plane of the Gibbs triangle, obtained through the computerised slicing of the surface with horizontal  parallel  planes  offers  information  on  the  physical  shape  and  thermal characteristics of each point on these surfaces.  %Si %Al %Cu T (K)   145   Temp (K) %   Figure 2 The liquidus surface mathematically modelled, in the system Al Cu Si.     Figure 3 The projection of the liquidus surface on the plane of the Gibbs triangle and the delimitation of the ternary eutectic area.    146 The study of the mathematically modelled diagram under the temperature of 504oC (Fig.4) leads us to the conclusion that the ternary system Al Cu Si has four zones of composition where there may exist ternary eutectic, at the temperature of 476,4oC [4].         Figure 4 The projection on the Gibbs triangle of the isotherms form the zone of the ternary eutecticles. The area delimited by the isotherm of 476,4oC.    3. The statistical-mathematical equation of the surface  From the database of the mathematical model of the liquidus surface (Fig.2) we selected with the help of the computer 15467 points on the surface having the co ordinates : concentrations xAl,  xSi,  xCu, (atomic %) and  temperature (K). With these data [4], with the Statistica 5 software, we established the approximation equation (1) of the liquidus surface through non linear multiple regression  with the full cubic approximation model [5]. The specific graph offered is a  scatterplot with a 95% confidence interval, after a correlation analysis is run [4].  z y x z y z yz x z x y x y x z yz x y x z y x T× × × + - × ×+ - × × - - × × - × ×- × × - × × + × + × + × =369 , 2564 ) ( 135 , 2416) ( 364 , 1396 ) ( 968 , 1354 233 , 811939 , 249 962 , 409 557 , 1355 086 , 1491 959 , 918(1) where, T  temperature (K); x  at.% Al; y  at.% Si; z  at.% Cu    147 4. Conclusions The practical applicability of the equation of the statistical model (1) presented in this paper and of the plotting (Fig. 2, Fig. 3) is that it allows the approximation of the melting temperature of any alloy of the (Al Cu Si) type, provided we know its  chemical  composition  (at.%).  Furthermore  we  may  approximate  the  alloy’s composition knowing only the melting temperature and the concentration of one of the components. Through the thermodynamic mathematical analysis of the binary systems and determination of the functions of the separation curves of the phase domain from the equilibrium diagrams, we may represent the separation surface of the domains, in the 3D space of the ternary diagram between the components present in the binary systems.The theoretical calculus of the diagrams of phase equilibrium has the advantage of using a small amount of thermo dynamic data and the possibility of utilising computation techniques, with appropriate software. This  considerably  reduces  the  research  time  and  expenses.  It  is  necessary  to create  mathematical  methods,  software  and  even  special  languages  that  could simplify  the  computerised  calculus  of  the  phase  equilibrium  diagrams  and  in general the calculus in the domain of the thermodynamics of the physical chemical systems. [1],[2]. The theoretical thermodynamic computations of the equilibrium diagrams, based on the physical and chemical properties of substances and using the new technique, adequate mathematical and graphic software, allow [2]: The  tentative,  expected  construction  of  the  equilibrium  diagrams  which  haven’t been experimentally determined yet; The completion of the partially plotted equilibrium diagrams; The correction of the experimental equilibrium diagrams; The  qualitative  analysis  and  systematic  classification  of  the  possible  type  of equilibrium diagrams; The crystallisation process in alloys differs from that in pure metals  and  depends  on  the  number  and  nature  of  phases  separating  from  the melted  material  as  a  result  of  the  way  the  atoms  of  the  composing  elements interact. Being a mixture of different atoms, alloys solidify within a temperature range to be found in the thermal range between the liquidus and solidus surfaces. This  range  may  be  different  in value, depending  on the experiment  conditions. Possible causes of differences may be: researchers’ errors, different sensitivities of the experimental methods used for the diagram plotting, different purity of the initial materials (impurities can provoke the apparition of eutectic Each  theoretical  study  must  nevertheless  be  sustained  by  the  experimental confirmation through appropriate methods.     148 References [1]  Aloman,  A,  Theory  of  equilibrium  diagrams,  Romanian  Academy Publishing House, Bucharest, 1999, p.22, 64 66. 141, 591, 796 798, 843. [2]   Aloman,  A,  Thermodynamic mathematical  modelling  of  phase  binary equilibriums, Bren PH, Bucharest, 2001, p.32. [3]  Chang, A, Y. Yan,  Calculated phase diagrams of Aluminium alloys from binary  Al Cu  to  multicomponent  commercial  alloys,  Journal  Alloy Compounds, 2001, p.151,320 [4]  Cziple, F, Doctorship Dissertation, Timi oara, 2003 [5]  Muller, A, Mathematical description of solidification cooling curves of pure metals, Materials research, Vol. I, Nr.1, Porto Alegre, 1998, Brazilia, p.29 38 [6]  Murray, J.L,  Al Cu (Aluminium Cooper), Int.Met.Rev., 30(5), 1985, p.17 [7]  Olesinski  R.W,  Abbaschian,  G.J.   The  Cu Si  System,  Bulletin  of  Alloy Phase Diagram, v7, Nr.2, 1986, p170 178 Address: ·  Lect. dr.eng. Florentina Cziple, “Eftimie Murgu” University of Re iŃa, PiaŃa Traian Vuia, nr. 1 4, 320085, Re iŃa, cziple.chim@uem.ro    

The Establishment, Plotting and Statistic– Mathematical Interpretation of the Liquidus Surface from the Phase Equilibrium Diagram of the Ternary  System Al-Cu-Si

The Establishment, Plotting and Statistic– Mathematical Interpretation of the Liquidus Surface from the Phase Equilibrium Diagram of the Ternary System Al-Cu-Si

Abstract

Similar works

Full text

Available Versions

Directory of Open Access Journals