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  ISSN 0031918X, The Physics of Metals and Metallography, 2013, Vol. 114, No. 12, pp. 1053–1060. © Pleiades Publishing, Ltd., 2013. 1053 1 INTRODUCTIONIt is important to know the value of the Gibbsenergy of mixing associated with ternary and multicomponent systems. Accordingly, it is expected that aconsistent thermodynamic model of the Gibbs energy of mixing associated with ternary and multicomponent systems should give quantitative predictions for the Gibbs energy of mixing observed in the alloy phases. There are various methods of processing thermodynamical data and performing thermodynamiccalculations in the case of multicomponent systems.To all appearance, the method CALPHAD is still themost widely used one today. One of its main featuresconsists in that it allows one to compile a thermodynamic description of multicomponent systems, basedon the descriptions of the systems of lower order.Moreover, the calculations can be carried out even inthe case when the data on corresponding subsystems isincomplete. In such a case, in order to minimize theinaccuracy of calculations one can attempt to estimatedeficient thermodynamic parameters via attractioninto consideration of one or another theoreticalmodel. Thus, for instance, in the works [1–5], for performance of calculations of the solubility of carbonitrides in multicomponent steels the parameters of interaction of the elements in the nonmetal and themetal sublattice of the phases with a bcc structure wereestimated in terms of the model that had been proposed in [6] and improved in [7]. However, in case of  1 The article is published in the srcinal. acute deficiency of thermodynamic data, especially, when of all corresponding subsystems there are available data only of binary ones, satisfactory results of thermodynamic calculations can be obtained via other methods, such as Chou’s general model [8]. Chou’sgeneral solution model breaks the boundary betweensymmetric and asymmetric models and simplifies various kinds of models to one. Based on these facts Chouet al. have proposed a general solution model that canovercome all these defects mentioned above. As corrected and accurate model, this general solutionmodel has been successfully applied by many researchers to various kinds of scientific and technological topics including calculation of thermodynamicproperties [9–14], construction of phase diagrams[15–18]. Recently, much attention has been paid tothe high order systems to fulfill the requirements of  both research and practical applications [8, 19–23]. When a binary system is compared with a higher system which have may difficulties in measurements,data of high order systems are very scarce in literature. All srcinal traditional geometrical models were only designed for ternary system only and extension of  binary information into ternary or higher order systems has been studied for some time long ago. In thepresent study, taking into consideration this case, we will stress the general form and application of thisChou’s general solution model to a penternary or higher order system such as six components system. Although Ni–Cr–Co–Al–Mo–Ti six componentssystem is difficult to study experimentally, due to its  An Analytical Approach for Thermodynamic Properties of the Six Components Systems Ni–Cr–Co–Al–Mo–Ti and Its Subsystems 1 H. Arslan a , *, A. Dogan a , and T. Dogan b a Kahramanmaras Sutcuimam University, Science and Art Faculty, Department of Physics,  Avsar Campus 46100 Kahramanmaras, Turkey b Cukurova University Engineering Faculty, Balcah, Adana, Turkey*email: hseyin_arslan@yahoo.com Received @@@@  Abstract —In the present study, the results of some thermodynamic prediction methods were applied to theNi–Cr–Co–Al–Mo–Ti system of six components. The Chou’s general solution model and the traditionalmodels of Kohler and Muggianu were included in the calculation for the comparison and discussion. Theexcess Gibbs energy dependences on composition for two investigated cross sections at 2000 K, were obtainedaccording to the applied models. The comparison between the results of the three models shows good mutualagreement. Keywords : geometrical model, Redlich–Kister coefficients, excess gibbs energy, thermodynamic model DOI: 10.1134/S0031918X13220018 STRUCTURE, PHASE TRANSFORMATIONS, AND DIFFUSION  1054 THE PHYSICS OF METALS AND METALLOGRAPHY Vol. 114 No. 12 2013  ARSLAN et al. high melting point, it will be significant to study theoretically the excess energies of mixing in the liquidphase of such a system. Fundamental Procedure  for the Solution Model Analysis In order to express the excess Gibbs energy of mixing of a binary system, the Redlich–Kister (R–K)type polynomial is always used in the following form:(1) where called the Redlich–Kister, is a parameter independent of composition and only relies on temperature. G  E   is the excess Gibbs energy of mixing and  X  i  ,  X   j   indicates the mole fractions of component i   and  j   in the i  –  j   binary system, respectively. Equation (1)can be reduced to the regular solution model if the value of n is 0 or subregular solution model if n is 1. ntakes a higher value when the binary system is out of these two solution models. When the general solution model is extended to asix components system, the excess Gibbs energy of mixing can be expressed as;(2) where W  ij   represents the weight probability of eachcorresponding binary composition point and can becalculated from;(3)Here  x  i   corresponds to the mole fractions of components in the alloy system. Using Eqs. (1)–(3), theexcess Gibbs energy is G  ij E   X  i   X   j   A ij k   X  i   X   j  – ( ) k  , k   0 = n ∑ =  A ij k  , G  E  W  12 G  12 E  W  13 G  13 E  W  14 G  14 E  W  15 G  15 E  +++= + W  16 G  16  W  23 G  23 E  W  24 G  24 E  W  25 G  25 E  W  26 G  26 E  + + + + + W  34 G  34 E  W  35 G  35 E  W  36 G  36 E  W  45 G  45 E  + + + + W  46 G  46 E  W  56 G  56 E  , + W  ij   x  i   x   j  '  X  i   X   j   . = G  E   x  1  x  2  A 12 k  2  X  1 12 ( )  1 – ( ) k k   0 = n ∑ = +  x  1  x  3  A 13 k  2  X  1 13 ( )  1 – ( ) k   x  1  x  4  A 14 k  2  X  1 14 ( )  1 – ( ) k k   0 = n ∑ + k   0 = n ∑ +  x  1  x  5  A 15 k  2  X  1 15 ( )  1 – ( ) k   x  1  x  6  A 16 k  2  X  1 16 ( )  1 – ( ) k k   0 = n ∑ + k   0 = n ∑ +  x  2  x  3  A 23 k  2  X  2 23 ( )  1 – ( ) k k   0 = n ∑ (4)Note that,  X  i  ,  X   j   values are different for differentmodels and These are defined in Kohler model [24] as(5)and for Muggianu model [25, 26](6)In general solution model,  X  ij   in Eq. (4), that is therelationship between compositions of components in amulticomponent system and the selected compositions of i and  j   in the ij   binary system, can be expressedin the form;(7)Here the coefficient represents the similarity coefficient of component k   to component i   in ij   system, and is defined as(8) where n ( ij  ,  ik  ) is the function, called “deviation sum of squares”, related to the excess Gibbs energy of   ij and ik   binaries, and is given by (9)Hence substituting Eq. (9) into Eq. (8), the similarity coefficients can be obtained. It should be pointed +  x  2  x  4  A 24 k  2  X  2 24 ( )  1 – ( ) k k   0 = n ∑ +  x  2  x  5  A 25 k  2  X  2 25 ( )  1 – ( ) k   x  2  x  6  A 26 k  2  X  2 26 ( )  1 – ( ) k k   0 = n ∑ + k   0 = n ∑ +  x  3  x  4  A 34 k  2  X  3 34 ( )  1 – ( ) k k   0 = n ∑ +  x  3  x  5  A 35 k  2  X  3 35 ( )  1 – ( ) k k   0 = n ∑  +  x  3  x  6  A 36 k  2  X  3 36 ( )  1 – ( ) k k   0 = n ∑ +  x  4  x  5  A 45 k  2  X  4 45 ( )  1 – ( ) k k   0 = n ∑ +  x  4  x  6  A 46 k  2  X  4 46 ( )  1 – ( ) k   x  5  x  6  A 56 k  2  X  5 56 ( )  1 – ( ) k  . k   0 = n ∑ + k   0 = n ∑  X  i ij  ( )  x  i   x  i   x   j  +  =  X  i ij  ( )  x  i  1  x  i   x   j  + ( ) – 2 . +=  X  i ij  ( )  x  i   x  k  ξ i ij  ( ) k  . k   1 = k i j  ,≠ m ∑ += ξ i ij  ( ) k  ξ i ij  ( ) k  1  n ji jk  ,( ) n ij ik  ,( )  + 1 – , = n ij ik  ,( )  G  ij E  G  ik E  – ( ) 2  X  i  . d  01 ∫  =  THE PHYSICS OF METALS AND METALLOGRAPHY Vol. 114 No. 12 2013  AN ANALYTICAL APPROACH FOR THERMODYNAMIC PROPERTIES1055 out that the parameters appearing in above equationsprove the following relation(10)This relation can be proved easily via Eq. (1). All problems appearing in the aforementioned traditional models will disappear when the general solution model extended to the six components is also usedin the present study. For example, it is easy to prove if all binaries in a multicomponent system are ideal.Then the multicomponent system will also be ideal. If  binary systems are ideal, i.e., = 0, after substitutingit into Eq. (2), G  E   from Eq. (1) will be equal to zero.Similarly, if all binaries are regular solutions, i.e., =constant. Note that the excess Gibbs energy of mixingcan be expressed as G  E   = for a binary regular solution model, combining this with Eqs. (2) and (3),the equation becomes(11) Which is the expression of a regular solution for amulticomponent system. Now for example if two components in a multicomponent system are identical, weprove that this model can reduce to a lower order model.On the other hand, the excess Gibbs energy of mixing of a penternary system can be written as;(12)In case the component 5 is identical to component 4,the following relations can considering Eq. (1), as: and (13)and n (14, 12) = n (15, 12) and n (41, 42) = n (51, 52).(14)Substituting Eq. (14) into Eq. (8), one finds(15)From n (14, 13) = n (15, 13) and n (41, 43) = n (51, 53),(16)  A ij k  1 – ( ) k   A  ji k  . = G  ij E   A ij  0  A ij  0  X  i   X   j  , G  E   A ij  0  X  i ij  ( )  X   j ij  ( ) . i j  ,  1 = n ∑ = G  E   x  1  x  2  X  1 12 ( )  X  2 12 ( )  G  12 E   x  1  x  3  X  1 13 ( )  X  3 13 ( )  G  13 E   x  1  x  4  X  1 14 ( )  X  4 14 ( )  G  14 E  + += +  x  1  x  5  X  1 15 ( )  X  5 15 ( )  G  15 E   x  2  x  3  X  2 23 ( )  X  3 23 ( )  G  23 E   x  2  x  4  X  2 24 ( )  X  4 24 ( )  G  24 E  + + +  x  2  x  5  X  2 25 ( )  X  5 25 ( )  G  25 E   x  3  x  4  X  3 34 ( )  X  4 34 ( )  G  34 E  + +  x  3  x  5  X  3 35 ( )  X  5 35 ( )  G  35 E   x  4  x  5  X  4 45 ( )  X  5 45 ( )  G  45 E  . + G  45 E  0 , =  G  14 E  G  15 E  , =  G  24 E  G  25 E  =  G  34 E  G  35 E  = ξ 1 14 ( ) 2 ( ) ξ 1 15 ( ) 2 ( ) . = ξ 1 14 ( ) 3 ( ) ξ 1 15 ( ) 3 ( ) = and from n (14, 15) = n (15, 14) and n (41, 45) = n (51, 54),(17)Substituting Eqs. (15)–(17) into Eq. (7) gives rise tothe following expressions(18)(19)Therefore,(20)On the other hand, similarly, the following relations can be obtained  X  2(24)  =  X  2(25)  and  X  3(34)  =  X  3(35) .(21)Substituting Eqs. (13), (20) and (21) into Eq. (12), onemay obtain the following formula:(22) When symbol is used to represent  x  4  +  x  5 , one cansee that this equation is the expression of the excessGibbs energy of mixing for a quaternary system. In other  words, if component 5 is identical to component 4, thispenternary model will reduce to a quaternary one.Moreover, it is seen clearly that a quaternary model will similarly reduce to a ternary one.RESULTS AND DISCUSSION Although there are no reliable and completed thermodynamic experimental data in multicomponentsystems, we do have some reliable ternary data. Theternary system Cu–Mg–Ni has been studied andreported by reference [27], possessing a series of reliable data in a whole concentration range. The Chou’sgeneral solution model has been used to calculate thissystem, and a good agreement between calculated andexperiment results has been reached [23]. In thepresent article, we stressed to obtain the general formand application of Chou’s general solution model to apenternary or higher order system. In this section, anexample of calculating a six component system will betreated. It has been seen that this model has advantages in almost some aspects in comparison with thetraditional models such as Kohler and Muggianumodel. The advantages are given in detail inEqs.(11)–(22). Having in mind the problems of directexperimentation, it is obvious that six componentssystem Ni–Cr–Co–Al–Mo–Ti is difficult to study  ξ 1 14 ( ) 5 ( ) ξ 1 15 ( ) 4 ( ) . =  X  1 14 ( )  x  1  x  2 ξ 1 14 ( ) 2 ( )  x  3 ξ 1 14 ( ) 3 ( )  x  5 ξ 1 14 ( ) 5 ( ) , + + +=  X  1 15 ( )  x  1  x  2 ξ 1 15 ( ) 2 ( )  x  3 ξ 1 15 ( ) 3 ( )  x  4 ξ 1 15 ( ) 4 ( ) . + + +=  X  1 14 ( )  X  1 15 ( ) . = G  E   x  1  x  2  X  1 12 ( )  X  2 12 ( )  G  12 E   x  1  x  3  X  1 13 ( )  X  3 13 ( )  G  13 E  += +  x  1  x  4  x  5 + ( )  X  1 14 ( )  X  4 14 ( )  G  14 E   x  2  x  3  X  2 23 ( )  X  3 23 ( )  + +  x  2  x  4  x  5 + ( )  X  2 24 ( )  X  4 24 ( )  G  24 E   x  3  x  4  x  5 + ( )  X  3 34 ( )  X  4 34 ( )  G  34 E  . +  x  4 /  1056 THE PHYSICS OF METALS AND METALLOGRAPHY Vol. 114 No. 12 2013  ARSLAN et al. experimentally due to its high melting point and thelarge amount of work involved. There is a need for application of theoretical calculations. Six components system Ni–Cr–Co–Al–Mo–Ti contain fifteen binary systems, and the information of all these binary systems should be known before using the geometricmodel. Six components system Ni–Cr–Co–Al–Mo–Ti is not studied yet in the literature as far as weknow. For the sake of simplicity, instead of the elements Ni, Cr, Co, Al, Mo and Ti, we have used thenumbers 1, 2, 3, 4, 5 and 6, respectively. Based onTable 1, a total of 120 n  and 60 ξ  have been calculated viaEqs. (8) and (9). The only ξ  data are listed in Table 2.Substituting the data in Tables 1 and (2) into Eq. (7),fifteen binary compositions are obtained. When the values of in Table 1 and similarity coefficients inTable 2 are substituted into in Eq. (2), the numerical values of the Gibbs energy of mixing for the six components system mentioned above are obtained and theresults are presented in Figs. 1 and 2. The figures showonly results of excess Gibbs energy of mixing in the six components system Ni–Cr–Co–Al–Mo–Ti corresponding to the ratios  X   A1 /  X  Cr   = 1,  X  Mo /  X  Cr   = 1,  X  Ti /  X  Cr   = 1, r   =  X  Co /  X  Cr   and  X  Co /  X  Cr   = 1,  X  Mo /  X  Cr   = 1,  X  Ti /  X  Cr   = 1, r   =  X   Al /  X  Cr   at 2000 K, respectively. Inorder to compare numerical values obtained from thegeneral solution model with the results obtained for Kohler and Muggianu model, we also calculated theexcess Gibbs energies for these models as a function of the composition and the obtained results were plottedin Figs. 3 and 4. Since a large number of points on thesame figure result in overlapping of these points, for the sake of simplicity, the results calculated from thesemodels are again listed in Table 3. On the other hand,in order to compare between the results of the Chou’smodel and those of the two traditional models (Kohler and Muggianu model) the excess Gibbs energies for these models as a function of the composition wereplotted in Fig. 5 for the ratio r = 0, for example, i.e., without having Co component. It is seen from the figure that the excess Gibbs energies for these models as  A ij k  Table 1. RedlichKister coefficients for fifteen liquid phases of the Ni–Cr–Co–Al–Mo–Ti System at 2000 K System i  –  j  ReferencesNi–Cr (1–2)–836800[28]Ni–Co (1–3)334700[28]Ni–Al (1–4)–147728.5 – 134.178 T   + 0.13924T2 –2.7313 ×  l0 –5   T  3 –55647.5 – 3.972T0[29]Ni–Mo (1–5)–17.73 + 13.807 T  00[30]Ni–Ti (1–6)–160526.9 + 44.15253 T  –88542.4 + 44.63193 T  0[32]Cr–Co (2–3)–836800[28]Cr–Al (2–4)–4644200[29]Cr–Mo (2–5)19037 – 8.58 T  6485 – 2.72 T  0[31]Cr–Ti (2–6)525015000[33]Co–Al (3–4)–281347 + 118.003 T  174264 + 0.379 T   + 0.03612 T  2 0[30]Co–Mo (3–5)251000[30]Co–Ti (3–6)–119780 + 15.06 T  –21630[34] Al–Mo (4–5)–4602400[31] Al–Ti (4–6)–118048 + 41.972 T  –23613 +19.704 T  34757 – 13.844 T  [35]Mo–Ti (5–6)5487 – 9.727 T  00[36]  A ij  0 T  ( )  A ij  1 T  ( )  A ij  2 T  ( ) –2000–4000–6000–8000–100000.80.60.40.201.0 –120000    E   x   c   e   s   s   G   i   b   b   s   e   n   e   r   g   y    (   J    /   m   o   l    )  r   = 0 r   = 0.5 r   = 1 r   = 2Mole fraction of Ni Fig. 1.  Excess Gibbs energy of mixing for the Ni–Cr–Co– Al–Mo–Ti system with six components in Chou’s model( T   = 2000 K,  X   Al /  X  Cr   = 1,  X  Mo /  X  Cr   = 1,  X  Ti /  X  Cr = 1 and r  =  X  Co /  X  Cr  ).  THE PHYSICS OF METALS AND METALLOGRAPHY Vol. 114 No. 12 2013  AN ANALYTICAL APPROACH FOR THERMODYNAMIC PROPERTIES1057 Table 2. Similarity Coefficients for the Ni–Cr–Co–Al–Mo–Ti System at 2000 K 10.7875770.3998150.9564710.50.3285190.9846060.3978640.7761370.2448530.839790.1815140.704430.4413630.7244390.9215760.4045760.9494990.2824880.66592500.0833130.4704290.0273400.4832340.0904210.9999250.92549960.6114140.9947320.5043440.4972440.0299950.0202020.6403730.0299370.6198340.5223650.9072460.5388810.0122020.9953180.8775010.9690340.96604980.4231460.6894960.5972120.6551880.3363370.450290.4046190.0452370.7184090.3340150.00007610.8609250.1657240.5958550.125368 ξ 1 12 ( ) 3 ( ) ξ 1 12 ( ) 4 ( ) ξ 1 12 ( ) 4 ( ) ξ 1 12 ( ) 6 ( ) ξ 1 13 ( ) 2 ( ) ξ 1 13 ( ) 4 ( ) ξ 1 13 ( ) 5 ( ) ξ 1 13 ( ) 6 ( ) ξ 1 14 ( ) 2 ( ) ξ 1 14 ( ) 3 ( ) ξ 1 14 ( ) 5 ( ) ξ 1 14 ( ) 6 ( ) ξ 1 15 ( ) 2 ( ) ξ 1 15 ( ) 3 ( ) ξ 1 15 ( ) 4 ( ) ξ 1 15 ( ) 6 ( ) ξ 1 16 ( ) 2 ( ) ξ 1 16 ( ) 3 ( ) ξ 1 16 ( ) 4 ( ) ξ 1 16 ( ) 5 ( ) ξ 2 23 ( ) 1 ( ) ξ 2 23 ( ) 4 ( ) ξ 2 23 ( ) 5 ( ) ξ 2 23 ( ) 6 ( ) ξ 2 24 ( ) 1 ( ) ξ 2 24 ( ) 3 ( ) ξ 2 24 ( ) 5 ( ) ξ 2 24 ( ) 6 ( ) ξ 2 25 ( ) 1 ( ) ξ 2 25 ( ) 3 ( ) ξ 2 24 ( ) 4 ( ) ξ 2 25 ( ) 6 ( ) ξ 2 26 ( ) 1 ( ) ξ 2 26 ( ) 3 ( ) ξ 2 26 ( ) 4 ( ) ξ 2 26 ( ) 5 ( ) ξ 3 34 ( ) 1 ( ) ξ 3 34 ( ) 2 ( ) ξ 3 34 ( ) 5 ( ) ξ 3 34 ( ) 6 ( ) ξ 3 35 ( ) 1 ( ) ξ 3 35 ( ) 2 ( ) ξ 3 35 ( ) 4 ( ) ξ 3 35 ( ) 6 ( ) ξ 3 36 ( ) 1 ( ) ξ 3 36 ( ) 2 ( ) ξ 3 36 ( ) 4 ( ) ξ 3 36 ( ) 5 ( ) ξ 4 45 ( ) 1 ( ) ξ 4 45 ( ) 2 ( ) ξ 4 45 ( ) 3 ( ) ξ 4 45 ( ) 6 ( ) ξ 4 46 ( ) 1 ( ) ξ 4 46 ( ) 2 ( ) ξ 4 46 ( ) 3 ( ) ξ 4 46 ( ) 5 ( ) ξ 5 56 ( ) 1 ( ) ξ 5 56 ( ) 2 ( ) ξ 5 56 ( ) 3 ( ) ξ 5 56 ( ) 4 ( ) Table 3. Excess Gibbs energy (J/mol) associated with the Ni–Cr–Co–Al–Mo–Ti System at 2000 K calculated by different predicting methods (  X   Al /  X  Cr   = 1,  X  Mo /  X  Cr   = 1,  X  Ti /  X  Cr   = 1, r   =  X  Co /  X  Cr  Present study (J/mol)Kohler model (J/mol)Muggianu model (J/mol) Ni r   = 0 r   = 0.5 r = l r   = 2 r   = 0 r   = 0.5 r = l r   = 2 r   = 0 r   = 0.5 r = l r   = 20–8538.11–9744.22–10321.2–10540.3–2457.19–6631.7–10970.8–16645.1–2457.19–8256.69–10970.8–12410.80.2–11025.1–11119.7–10873.1–10120.3–7501.13–9586.08–11880.5–14762–6831.69–9883.43–11201.9–11604.20.4–11414.8–10833–10116.9–8856.3–11492.03–11879.4–12528–13161.9–9716.28–10715.4–10944.7–10463.40.6–9707.19–8884.13–8052.64–6748.23–11980.38–11364.3–11016.6–10335.8–10167.89–9908.4–9474.81–8484.780.8–5902.25–5273.14–4680.35–3796.13–8245.76–7475.65–6894.79–5982.06–7243.48–6618.02–6068–5164.831000000000000
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