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METALS AND MATERIALS International, Vol. 9, No. 3 (2003), pp. 247~253

Evaluation of Enthalpy and Entropy Changes for Binary Alloy Solidification Process 1,2, 1 M. Zheng * and G. Zhou

1 2

School of Material Science & Engineering, Xian Jiaotong University, Xian, 710049, P. R. China School of Chemistry & Material Science, Nanchang University, Nanchang, 330029, P. R. China

Analytical expressions of the entropy and enthalpy changes and apparent specific heat capacity of binary alloys in the solidification process are derived. The non-equilibrium lever rule is employed in the assessments of the relative concentration of binary alloys during solidification. The effect of the partial ordering of alloys in the liquid state is ignored and the alloy solidification process is divided into steps in the evaluations. Furthermore, experimental data from the available literature for Al-Cu alloys are employed to check the predictions of the current approaches, which indicates the reasonability of the current expressions. Keywords: non-equilibrium lever rule, binary alloy, solidification, entropy change, enthalpy change, specific heat capacity

1. INTRODUCTION The evaluation of entropy and enthalpy changes in alloys in their first order phase transition process (solidification) is of significance and undoubtedly beneficial to the investigations concerning the solidification process and quality control of alloys. To the best of our knowledge, expressions for such evaluations have not yet been proposed. Correspondingly, the assessment of the liberation of heat energy during the solidification of alloys has not yet been properly performed [1-4]. As to pure metals, empirical correlations among entropy and enthalpy changes and melting temperature have been established for their first order phase transition (solidification) [1,2]. The phase transition latent heat and entropy are due to the change in the free volume of alloys from their liquid to solid state [2,3]. Semi-empirical evaluations, such as the Miedema method and ab initio approach, etc., can only calculate solid and liquid formation enthalpy instead of the phase transition type [2,3]. In addition, the presence of a mushy zone in alloy solidification makes the evaluation troublesome. In this paper, we propose approaches to evaluating the entropy and enthalpy changes and apparent specific heat capacity of binary alloys in their first order phase transition process (solidification). In actual solidification processes [1,4,5], the diffusion coefficients in metallic liquids are 3-4 orders of magnitude *Corresponding author: [email protected]

greater than those in solid state, for example, the diffusion coefficient in solid Al-Cu alloy is 3×10−13 m2/sec while in liquid it is 3×10−9 m2/sec; for Al-Si alloys the corresponding values are 1×10−12 m2/sec and 3×10−9 m2/sec, respectively. Thus, the diffusion in the solidified part could be neglected while that in metallic liquid state be assumed to be approximately sufficient during solidification. As a consequence, the Scheil equation, i.e., the non-equilibrium lever rule, can be reasonably employed to evaluate the relative concentration during the alloy solidification process provided that the moving velocity of the solidification front be low enough.

2. HYPOTHETICAL STEPS IN ALLOY SOLIDIFICATION PROCESS In order to evaluate entropy and enthalpy changes and specific heat capacity of alloy solidification appropriately the actual solidification can be divided into the following hypothetical steps. Since in common solidification processes, there is not enough time for the diffusion in the solidified part to progress sufficiently though the diffusion in the alloy liquid could conduct completely due to the big difference in the diffusion coefficients of the two states. As a consequence [5], the actual solidification process could always continue until the eutectic temperature for the eutectic alloy system, even the initial solute concentration, is rather low, and the remaining alloy liquid at eutectic temperature has to be solidified completely at the eutectic temperature in the “eutectic reaction” manner.




M. Zheng and G. Zhou

According to solidification theory, the solid-liquid interface is in local equilibrium during the solidification process. Thus the Scheil equation holds approximately valid provided that both the solidus and liquidus lines are linear and the moving velocity of the solidification front is low enough. With the Scheil equation, the instant concentration in the solidifying part and the “equilibrium concentration” in the liquid as well as the fraction solid could be evaluated. The liquid concentration varies along the liquidus line as the solidification continues till eutectic temperature. The above discussion indicates that an actual solidification can be divided into a solid solution and the remaining eutectics, the former has gradually changed concentration due to the alloy liquids changing its concentration along the liquidus line. As a result, the physical quantities concerned with the solidification process can be evaluated as an integral of the corresponding parts. Furthermore, as to an instant solid solution solidification or eutectic reaction process, the apparent process (α) can be broken into two hypothetical steps (β) and (γ) as in the following equations [1]: (α): iA(L) + jB(L) → AiBj(S), (β): iA(L) + jB(L) → AiBj(L), (γ): AiBj(L) → AiBj(S),

N AB = 4N AN B ⎛ ⎛ 2 ε ⎞ ⎞ 0.5 ⎫ 2ε 1⎧ - exp ------- – 1 – 1 ⎬ ⁄ exp ⎛ -------⎞ – 1 , ZN --- ⎨ 1 + --------------2 ⎝ ⎠ 2⎩ RT N ⎝ ⎝ RT⎠ ⎠ ⎭ 2ε (3) ≈ ZNX AX B 1 – XA XB ⎛⎝ exp⎛⎝ -------⎞⎠ – 1⎞⎠ RT in which ε is the exchange energy, ε = UAB − 0.5 (UAA+UBB), XA=NA/N, XB=NB/N. If ε is negative, the attraction between unlike atoms is stronger; while, if ε is positive, the same kind of atoms attract more strongly than do different ones, thus leading to a tendency of separation for the system into A-rich and B-rich phases. However, where ε equals zero, this gives an ideal solution, and the atoms can arrange themselves randomly. In the light of Eq. 2, Eq. 1 can be further written as, F=F0(NA, NB)+0.5Z(NAUAA+NBUBB)+NAB(UAB−0.5(UAA+UBB) −TS (4) =F0(NA, NB)+0.5Z(NAUAA+NBUBB)+NABε-TS While the configuration entropy in liquid state can be written as, S = −RN(XlnX+(1−X)ln(1−X))+2/ ε /NX(1−X)(XlnX+(1−X) ln(1−X)) /T = − RN[XlnX+(1−X)ln(1−X)][1−2/ε /X(1−X)/(TR)] (5)

in which i and j represent the molar numbers of atoms A and B in the processes and S and L the solid and liquid states, respectively. The corresponding entropy and enthalpy changes can be evaluated according to the corresponding broken.

3. FREE ENERGY AND CONFIGURATION ENTROPY OF BINARY ALLOY IN LIQUID STATE In liquid state, the free energy of binary alloys is [1]

where R is the gas constant, and X = NB/N the molar concentration of atom B in the alloy. The second term in Eq. 5 accounts for the effect of exchange energy, which leads to ordering in some degree. In general, X · (1−X) is much less than 1, /ε/ is less than RT, as a first order approximation, the second term in Eq. 5 can be neglected. Thus, this leads to a first order approximated expression for the configuration entropy of alloy in liquid state, the ideal mixing one.

4. ENTROPY CHANGE OF BINARY ALLOYS IN FIRST ORDER PHASE TRANSITION

F=F0(NA, NB)+NAAUAA+NABUAB+NBBUBB − TS

(1)

2NAA+NAB=ZNA, 2NBB+NAB=ZNB, NA+NB=N

(2)

where, N, NA, NB, NAA, NBB and NAB express the total number of atoms, the number of atom A, the number of atom B, as well as the corresponding atomic pairs number, respectively. F0(NA, NB) expresses the free energy components related to the translation, rotation, and vibration degrees of freedom as for the liquid state, T and S represent the absolute temperature and the configuration entropy, respectively. U is the corresponding pair interaction energy between atoms. Z is the coordination number in the system. The number of A-B pairs in the non-regular liquid state can be determined by quasi-chemical approximation [1],

The configuration entropy change for binary alloy with an initial solute concentration C0 to solidify completely can be evaluated as follows. Since the position for each atom in solid state is highly ordered or well defined, the configuration entropy of alloy in solid state could be assumed to be zero as compared to that in liquid state [1]; According to the last paragraph, the partial ordering effect of the atom distribution in liquid state can be neglected; the configuration entropy for a binary alloy per mole with instant liquid concentration Cl is: Sc = – R [ Cl ln( C l) + (1 – C l )ln ( 1 – C l ) ]

(6)

Furthermore, ignoring the diffusion in the solidified part and assuming that the diffusion in liquid is approximately




Evaluation of Enthalpy and Entropy Changes for Binary Alloy Solidification Process

sufficient due to the reason stated in the previous paragraphs, the instant solid fraction, fs, can be derived with the non-equilibrium lever rule, i.e., the Scheil equation provided that the moving velocity of the solidification front is low enough [5], 1 ---------

2–k ---------

C k– 1 1 C k – 1 dC fs = 1 – ⎛⎝ -----l ⎞⎠ , dfs = – --------- ⎛⎝ -----l ⎞⎠ --------l – k 1 C0 C0 C0

(7)

Thus for an alloy with the initial solute concentration C0 and eutectic concentration Ce, the configuration entropy change due to solidification can be written as,

where ΔH1 is the partial enthalpy change due to the configuration change for solidification till eutectic temperature; ΔH2 is the partial configuration enthalpy change of the remaining eutectic to be solidified completely; ΔH3 is the partial phase transition induced enthalpy of the solidification process till eutectic temperature; ΔH4 is the partial phase transition enthalpy change of the remaining eutectic due to its complete solidification. The detailed expressions of the above terms can be obtained according to their corresponding thermodynamic definitions. fse

fse

ΔSc = ∫ S cdfs – R ( 1 – fse)[ C eln( Ce ) + (1 – C e) ln( 1 – C e )] (8)

ΔH 1 = ∫ TScdfs = 0

0

1 --------k–1

C fse is the solid fraction at eutectic temperature, fse = 1 – ⎛⎝ -----e⎞⎠ . C0 In order to obtain an analytical expression, Taylor expansion can be employed to expand ln(1−Cl) provided that Cl is smaller than 1, i.e., ln( 1 – C l) ≈ –⎛ C l + --- Cl + --- C l ⎞

⎝

1 2

2

1 3

3

(9)

⎠

fse

–R ∫ ( T0 + ml C l) ⋅ [ Cl ln(C l ) + ( 1 – C l )ln ( 1 – C l ) ]dfs

in which the linear liquidus line T = T0 + mlCl is employed with T0, the melting temperature of the pure component metal and ml, the slope of the liquidus line. Again, by expanding the relevant terms as done in evaluating ΔS, yields:

Substituting Eq. 9 for Eq. 8 thus yields, ⎧1 C ΔS c = RC 0 ⎨ ---⎛⎝ -----e⎞⎠ ⎩ k C0

k --------k–1

k--------

⎛ -----e⎞ ln( C e) – --- ln(C 0 ) – --------2 k– 1 C k ⎝ C 0⎠

1 k

2k – 1------------

k--------

2

k --------k–1

–1 + k--------2 k

3k – 2------------

4k – 3------------

(10)

Furthermore, the total entropy change, i.e. the configuration entropy change and the phase transition induced latent entropy change, can be approximately written as, ΔS = ΔS l + ΔSc

(11)

where ΔSl is the phase transition induced latent entropy change; it can be assumed approximately the same as that for pure metals due to the same resource, says, R [1-4].

5. ENTHALPY CHANGE OF BINARY ALLOYS IN FIRST ORDER PHASE TRANSITION According to the statements in the previous paragraphs the enthalpy change due to solidification can also be estimated with a procedure similar to those for evaluating entropy, (12)

2k – 1------------

2

C k–1 ml RC 0 - ln C e ⋅ ⎛ -----e⎞ – ln C0 + -------------⎝ 2k – 1 C 0⎠ 2k – 1------------

ml RC 0 ( k – 1 ) ⎛ C e⎞ k – 1 ----– ----------------------------–1 2 ⎝ C 0⎠ ( 2k – 1 ) 2

3k – 2------------

2⎧ 1 –ml RC0 ⎨ -------------

3k – 2------------

C k–1 C e⎞ k – 1 C0 ⎛ ----------------------- ⎛ -----e⎞ – 1 – –1 ⎝ ⎠ 2 ( 3k – 2 ) ⎝ C 0⎠ ⎩ 2k – 1 C 0

2

ΔH = ΔH 1 + ΔH 2 + ΔH 3 + ΔH 4

3k – 2------------

2

3 2 3 C 1 ⎫ C 1 C k–1 1 C C 1 – -----0 ------------- ⎛ -----e⎞ – --- +-----0 +-----0 ------------- +-----0------------- ⎬ 3 4k – 3 ⎝ C0⎠ k 4k 6 3k – 2 3 4k – 3 ⎭

1 ---------

C k–1 –R ⎛ -----e⎞ [ C eln( C e) + ( 1 – Ce) ln ( 1 – C e) ] ⎝ C 0⎠

2k – 1------------

k ---------

C k – 1 C C k – 1 C 1 ⎛ C e⎞ k – 1 1 - ------- ⎛ -----e⎞ – -----0 ⎛ -----e⎞ – -----0 -----------4k ⎝ C 0⎠ 6 3k – 2 ⎝ C 0⎠ k ⎝ C 0⎠

4k – 3------------

2 3 3 C 1 ⎫ C 1 C k–1 1 C C 1 – -----0 ------------- ⎛ -----e⎞ – --- + -----0 + -----0 ------------- + -----0 ------------- ⎬ 3 4k – 3 ⎝ C 0⎠ k 4k 6 3k – 2 3 4k – 3 ⎭

k ---------

⎧1 C k–1 C e⎞ k – 1 1 k – 1 ⎛ ----ΔH1 = RC0 T0⎨ --- ⎛⎝ -----e⎞⎠ ln( C e) – --- ln( C 0) – --------2 k k ⎝ C 0⎠ ⎩ k C0 k–1 – + --------2 k

1 C k–1 C C k–1 C 1 C k–1 – --- ⎛ -----e⎞ – -----0 ⎛ -----e⎞ – -----0 -------------⎛ -----e⎞ 4k ⎝ C0⎠ 6 3k – 2 ⎝ C 0⎠ k ⎝ C 0⎠

(13)

0

C0 – ------------------6 ( 4k – 3 )

4k – 3------------

3

Ce ⎞ k – 1 C0 ⎛ ----– 1 – ------------------⎝ C 0⎠ 3 ( 5k – 4 )

5k – 4------------

⎫ C e⎞ k – 1 ⎛ ----– 1 ⎬ (14) ⎝ C 0⎠ ⎭

ΔH 2 = – RTe (1 – fse )[ Ce ln( C e) + ( 1 – C e )ln(1 – C e) ] 1 ---------

C k–1 = – RTe⎛ -----e⎞ [ C eln( C e) + ( 1 – C e )ln(1 – C e) ] ⎝ C 0⎠

(15)




M. Zheng and G. Zhou fse

again the assumption ΔSl = R is employed. The approximate enthalpy change can be evaluated by assuming that the release of phase transition latent heat is uniform in the mushy zone, df’s=− dT/ΔT,

2–k ---------

Ce

1 C k – 1 dC ΔH3 = ∫ RTdfs = –R ∫ ( T0 + m lC l ) --------- ⎛⎝ -----l ⎞⎠ --------l k – 1 C0 C0 C0 0 1 ---------

k ---------

C k–1 C C k–1 = – RT 0 ⎛ -----e⎞ – 1 – -----0 m lR ⎛ -----e⎞ – 1 ⎝ C 0⎠ ⎝ C 0⎠ k C ΔH4 = RT e( 1 – fse) = RTe⎛⎝ -----e⎞⎠ C0

1 --------k–1

(16)

1

ΔH′ = ∫T (ΔS l + ΔS c )dfs′ 0

1

(17)

= R∫ T[ 1 – C0 ln (C 0 ) + ( 1 – C 0) ln ( 1 – C 0 )]dfs′ 0

Ts

6. APPARENT SPECIFIC HEAT CAPACITY

= –R ∫ T [1 – C 0ln ( C 0) + ( 1 – C0 ) ln( 1 – C0 ) ]dT ⁄ ( ΔT) Tl

During solidification, both the liberation of latent heat and true specific heat induced heat release contribute to the actual measurements for the heat analysis [4,6]. Thus, one needs to add both contributions so that comparable predictions can be made. Therefore, the apparent specific heat capacity can be written as dH Cp * = ------- + C p dT

Ts + Tl = R[ 1 – C0ln ( C0 ) + ( 1 – C 0 ) ln( 1 – C 0 ) ] ------------2

(22)

where Ts and Tl express solidius and liquidius temperatures, respectively. ΔT = Tl − Ts. The phase diagrams in [7] are employed to take the relevant values for some alloys; thus, the applications of the

(18)

in which Cp presents the true specific heat capacity of the mushy alloy in the freezing range. According to the previous paragraph, both ΔH1 and ΔH3 of Eqs. 13 and 16 contribute to the enthalpy change H of Eq. 18 in the mushy zone during freezing. Substituting Eqs. 13 and 16 into Eq. 18, yields: ⎧ T–T T–T T–T T–T ⎫ Cp* = RT⎨ 1 – ⎛⎝ ------------0⎞⎠ ln⎛⎝ ------------0⎞⎠ – ⎛⎝ 1 – ------------0⎞⎠ ln ⎛⎝ 1 –------------0⎞⎠ ⎬
m m m ml ⎭ l l l ⎩ 2–k ---------

m l C0 ⎞ 1 – k 1 1 ⎛ ------------------------------ + Cp 1 – k ⎝ T – T0⎠ ml C0

(19) Fig. 1. Configuration entropy change of Ag-Cu alloy solidification.

At a given fraction solid, fs, the representative value of the true specific heat capacity Cp of alloy in the mushy zone can be expressed as a linear addition of the solid part Cps and the liquid part Cpl, correspondingly [4], Cp = fsCps + (1−fs)Cpl

(20)

7. PREDICTIONS In a completely ideal process, one considers the phase transition process to be very slow such that the diffusions in both solid and liquid are all sufficient, i.e., an ideal equilibrium phase transition; the relevant entropy change can be calculated only according to the phase diagram, which is ΔS′ = ΔS l + ΔS c = R – R[ C 0 lnC 0 + ( 1 – C 0 )ln ( 1 – C 0 )] = R [ 1 – C 0 lnC 0 + ( 1 – C 0 )ln ( 1 – C 0 ) ]

(21)

Fig. 2. Enthalpy change of Ag-Cu alloy solidification.

Evaluation of Enthalpy and Entropy Changes for Binary Alloy Solidification Process




Fig. 6. Enthalpy change of Al-Mg alloy solidification. Fig. 3. Configuration entropy change of Al-Cu alloy solidification.

Fig. 4. Enthalpy change of Al-Cu alloy solidification.

Fig. 7. Configuration entropy change of Al-Si alloy solidification.

Fig. 5. Configuration entropy change of Al-Mg alloy solidification.

Fig. 8. Enthalpy change of Al-Si alloy solidification.

above approaches for evaluating entropy and enthalpy changes for these alloys could be performed, which are given in Figs. 1 through 8, respectively. The predictions for non-equilibrium solidifications are from Eqs. 10 and 12, while for equilibrium solidifications the predictions are

from Eqs. 21 and 22. For entropy changes, only the configuration parts are drawn since the phase transition induced latent part is assumed constant, says R [1-4]. The variations of the apparent specific heat capacities for





M. Zheng and G. Zhou Table 1. Specific heat capacity of Al-Cu alloys in liquid and solid states [6] Property

Al-2%Cu

Al-5%Cu

Al-5.5%Cu

1.156 1.278

1.056 1.253

1.052 1.249

o

Cps (KJ/Kg C) o Cpl (KJ/Kg C)

Al-Cu alloys with respect to temperature are shown in Figs. 9 through 11. Shown in Table 1 are the specific heat capacities for Al-Cu alloys in liquid and solid states, which, as well as the experimental data in Figs. 9 though 11, are taken from [6]. The predictions in Figs. 9 through 11 are from Eqs. 19 and 20.

8. CONCLUDING REMARKS Fig. 9. Cp* vs T for Al-2%Cu alloy.

By dividing the solidification process into steps, the expressions for evaluating the changes in entropy, enthalpy and specific heat capacity during solidification are performed analytically. The experimental data from the available literature indicate the reasonability of these evaluations.

ACKNOWLEDGMENT The Chinese High Technology Project Committee (863, 2002AA331110) is acknowledged for supporting this work.

REFERENCES

Fig. 10. Cp* vs T for Al-5%Cu alloy.

1. R. A. Swalin, Thermodynamics of Solids, 2nd edition, John Wiley & Sons, New York (1972). 2. F. Sommer, R. N. Singh, and V. Witusiewicz, J. Alloys & Compounds 325, 118 (2001). 3. K. Ohsaka and E. H. Trinh, Appl. Phys. Lett. 66, 3123 (1995). 4. D. G. Sharma, M. Krishnan, and C. Ravindran, Materials Characterization 44, 309 (2000). 5. M. C. Flemings, Solidification Processing, McGraw-Hill Book Co., New York (1974). 6. A. I. Veinik, Thermodynamics for the Foundryman, Maclaren & Sons Ltd., London (1968). 7. M. Hansen and K. Anderko, Constitution of Binary Alloys, McGraw-Hill Book Co., Inc., New York (1958).

APPENDIX Derivation for Eq. 10. Substituting Eq. 9 for 8 and using Eq. 6, yields: fse

ΔSc = ∫ Scdfs – R ( 1 – fse) [ C eln( C e) + ( 1 – C e) ln ( 1 – C e) ] 0

k--------

Fig. 11. Cp* vs T for Al-5.5%Cu alloy.

k--------

Ce ⎞ k – 1 1 1 C k–1 k – 1 ⎛ ----= RC0 --- ⎛ -----e⎞ ln (C e ) – RC0 --- ln ( C 0) – RC0 --------2 ⎝ ⎝ ⎠ k k C0 k C 0⎠




Evaluation of Enthalpy and Entropy Changes for Binary Alloy Solidification Process

k–1 – RC 0 +RC 0--------2 k 3

2k – 1------------

k ---------

2

4k – 3------------

2

3

C 1 C 1 C k – 1 1 C 0 C0 1 – --- + ----- + ----- ------------- + -----0 ------------– -----0 ------------- ⎛⎝ -----e⎞⎠ 3 4k – 3 C0 k 4k 6 3k – 2 3 4k – 3 – R (1 – fse)[ C eln( C e) + ( 1 – C e) ln (1 – C e )] k ---------

3k – 2------------

C e⎞ k – 1 C 0 ⎛ C e ⎞ k – 1 C 0 1 ⎛ C e ⎞ k – 1 1---⎛ ----– -----⎝ -----⎠ – ----- ------------- ⎝ ----- ⎠ ⎝ 4k C 0 6 3k – 2 C 0 k C 0⎠

k ---------

k ---------

2

3k – 2------------

2 3 C 1 ⎫ 1 C C 1 –--- + -----0 + -----0 ------------- + -----0 ------------- ⎬ k 4k 6 3k – 2 3 4k – 3 ⎭ 1 ---------

⎧1 C k– 1 C e⎞ k – 1 k – 1 1 k – 1 ⎛ ----= RC0 ⎨ --- ⎛⎝ -----e⎞⎠ ln ( Ce ) – --- ln( C o) – --------+ --------2 2 ⎝ k k C 0⎠ k ⎩ k C0

2k – 1------------

3

4k – 3------------

1 C k–1 C C k –1 C 1 C k –1 C 1 C k –1 – --- ⎛⎝ -----e⎞⎠ – -----0 ⎛⎝ -----e⎞⎠ – -----0 ------------- ⎛⎝ -----e⎞⎠ – -----0 -------------⎛⎝ -----e⎞⎠ 4k C0 6 3k – 2 C 0 3 4k – 3 C 0 k C0

C k–1 – R⎛ -----e⎞ [ C eln( C e) + ( 1 – C e) ln ( 1 – C e) ] ⎝ C 0⎠