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eISSN: 2576-4543

Physics & Astronomy International Journal

Research Article Volume 2 Issue 3

Study on the melting of interstitial alloy AB with FCC structure under pressure

Nguyen Quang Hoc, Bui Duc Tinh, Dinh Quang Vinh

Hanoi National University of Education, Vietnam

Correspondence: Bui Duc Tinh, Hanoi National University of Education, 136 Xuan Thuy Street, Cau Giay District, Hanoi, Vietnam, Tel 8498 6409 706

Received: November 02, 2017 | Published: June 15, 2018

Citation: Hoc NQ, Tinh BD, Vinh DQ. Study on the melting of substitution alloy AB with interstitial atom C and FCC structure under pressure. Phys Astron Int J. 2018;2(3):231-235. DOI: 10.15406/paij.2018.02.00091

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Abstract

From the model of interstitial alloy AB with FCC structure and the condition of absolute stability for crystalline state we derive analytic expression for the temperature of absolute stability for crystalline state, the melting temperature and the equation of melting curve of this alloy by the way of applying the statistical moment method. The obtained results allow us to determine the melting temperature of alloy AB ant zero pressure and under pressure. In limit cases, we obtain the melting theory of main metal A with FCC structure. The theoretical results are numerically applied for alloys AuSi and AgSi.

Keywords: interstitial alloy, absolute stability of crystalline state, statistical moment method

Introduction

Alloys in general and interstitial alloys in particular are typical materials in material technology and science. Study on interstitial alloys pays particular attention to many researchers. The melting temperature (MT) of materials under pressure is a very important problem of solid state physics and material science.1–3 Theoretically in order to determine the MT of crystal it is necessary to apply the equilibrium condition of solid phase and liquid phase. By this way, there are some methods such as the self–consistent phonon field method and the one–particle distribution function method. The obtained results from these methods are in not good agreement with experiments and are limited at low pressures.

In aid of the statistical moment method (SMM), Tang & Hung4,5 show that we absolutely only use the solid phase of crystal to determine the MT. The obtained results from the SMM are better than that from other methods in comparison with experiments.

Content of research

Analytic result

In the model of AB interstitial alloy with the face–centured cubic (FCC) structure, the A atoms with large size stay in the peaks and the face centers of cubic unit cell and the C interstitial atoms with smaller size stay in the body center. In6,7 we derived the analytic expressions of the nearest neighbor distance, the cohesive energy and the alloy parameters for atoms B, A, A1 (the atom A in the face centers) and A2 (the atom A in the peaks).

The equation of state of the AB interstitial aloy with FCC structure at temperature T and pressure P is described by

Pv= r 1 ( 1 6 u 0 r 1 +θxcthx 1 2k k r 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGqb GaamODaiabg2da9iabgkHiTiaadkhajuaGdaWgaaqcKfaG=haajugW aiaaigdaaSqabaqcfa4aaeWaaOqaaKqbaoaalaaabaqcLbsacaaIXa aajuaGbaqcLbsacaaI2aaaaKqbaoaalaaakeaajugibiabgkGi2kaa dwhajuaGdaWgaaqcKfaG=haajugWaiaaicdaaSqabaaakeaajugibi abgkGi2kaadkhajuaGdaWgaaqcKfaG=haajugWaiaaigdaaSqabaaa aKqzGeGaey4kaSIaeqiUdeNaamiEaiGacogacaGG0bGaaiiAaiaadI hajuaGdaWcaaGcbaqcLbsacaaIXaaakeaajugibiaaikdacaWGRbaa aKqbaoaalaaakeaajugibiabgkGi2kaadUgaaOqaaKqzGeGaeyOaIy RaamOCaKqbaoaaBaaajqwaa+FaaKqzadGaaGymaaqcbasabaaaaaGc caGLOaGaayzkaaaaaa@6CC2@  (1)

At 0K and pressure P, this equation has the form

Pv= r 1 ( u 0 r 1 + ω 0 4k k r 1 ). MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGqb GaamODaiabg2da9iabgkHiTiaadkhajuaGdaWgaaqcbasaaKqzadGa aGymaaWcbeaajuaGdaqadaGcbaqcfa4aaSaaaOqaaKqzGeGaeyOaIy RaamyDaKqbaoaaBaaajeaibaqcLbmacaaIWaaaleqaaaGcbaqcLbsa cqGHciITcaWGYbqcfa4aaSbaaKqaGeaajugWaiaaigdaaSqabaaaaK qzGeGaey4kaSscfa4aaSaaaOqaaKqzGeGaeS4dHGMaeqyYdCxcfa4a aSbaaKqaGeaajugWaiaaicdaaSqabaaakeaajugibiaaisdacaWGRb aaaKqbaoaalaaakeaajugibiabgkGi2kaadUgaaOqaaKqzGeGaeyOa IyRaamOCaSWaaSbaaKqaGeaajugWaiaaigdaaKqaGeqaaaaaaOGaay jkaiaawMcaaKqzGeGaaiOlaaaa@613B@  (2)

Knowing the form of the interaction potential φ i0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbiqaaqhbjugibi abeA8aQTWaaSbaaKqaGeaajugWaiaadMgacaaIWaaajeaibeaaaaa@3C13@ ,equation (1) allows os to determine the nearest neighbor distance r 1X ( P,0 )( X=B,A, A 1 , A 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGYb qcfa4aaSbaaKqaGeaajugWaiaaigdacaWGybaaleqaaKqbaoaabmaa keaajugibiaadcfacaGGSaGaaGimaaGccaGLOaGaayzkaaqcfa4aae WaaOqaaKqzGeGaamiwaiabg2da9iaadkeacaGGSaGaamyqaiaacYca caWGbbWcdaWgaaqcbasaaKqzadGaaGymaaqcbasabaqcLbsacaGGSa GaamyqaSWaaSbaaKqaGeaajugWaiaaikdaaKqaGeqaaaGccaGLOaGa ayzkaaaaaa@4F47@ at 0K and pressure P. Knowing r 1X ( P,0 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGYb WcdaWgaaqcbasaaKqzadGaaGymaiaadIfaaKqaGeqaaKqbaoaabmaa keaajugibiaadcfacaGGSaGaaGimaaGccaGLOaGaayzkaaaaaa@3FBB@ we can determine the parameters k X (P,0), γ 1X (P,0), γ 2X (P,0), γ X (P,0) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGRb WcdaWgaaqcbasaaKqzadGaamiwaaqcbasabaqcLbsacaGGOaGaamiu aiaacYcacaaIWaGaaiykaiaacYcacqaHZoWzlmaaBaaajeaibaqcLb macaaIXaGaamiwaaqcbasabaqcLbsacaGGOaGaamiuaiaacYcacaaI WaGaaiykaiaacYcacqaHZoWzlmaaBaaajeaibaqcLbmacaaIYaGaam iwaaqcbasabaqcLbsacaGGOaGaamiuaiaacYcacaaIWaGaaiykaiaa cYcacqaHZoWzlmaaBaaajeaibaqcLbmacaWGybaajeaibeaajugibi aacIcacaWGqbGaaiilaiaaicdacaGGPaaaaa@5AB9@ at 0K and pressure P for each case of X. The displacement y 0X (P,T) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWG5b qcfa4aaSbaaKqaGeaajugWaiaaicdacaWGybaaleqaaKqzGeGaaiik aiaadcfacaGGSaGaamivaiaacMcaaaa@3F72@ of atoms from the equilibrium position at temperature T and pressure P is determined.6,7 From that, we can calculate the neares neighbor distance r 1X ( P,T ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGYb WcdaWgaaqcKfaG=haajugWaiaaigdacaWGybaajqwaa+Fabaqcfa4a aeWaaKaaGfaajugibiaadcfacaGGSaGaamivaaqcaaMaayjkaiaawM caaaaa@441E@ at temperature T and pressure P as follows

r 1C (P,T)= r 1C (P,0)+ y A 1 (P,T), r 1A (P,T)= r 1A (P,0)+ y A (P,T), MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGYb qcfa4aaSbaaKqaGeaajugWaiaaigdacaWGdbaaleqaaKqzGeGaaiik aiaadcfacaGGSaGaamivaiaacMcacqGH9aqpcaWGYbWcdaWgaaqcba saaKqzadGaaGymaiaadoeaaKqaGeqaaKqzGeGaaiikaiaadcfacaGG SaGaaGimaiaacMcacqGHRaWkcaWG5bqcfa4aaSbaaKqaGeaajugWai aadgealmaaBaaajiaibaqcLbmacaaIXaaajiaibeaaaSqabaqcLbsa caGGOaGaamiuaiaacYcacaWGubGaaiykaiaacYcacaWGYbWcdaWgaa qcbasaaKqzadGaaGymaiaadgeaaKqaGeqaaKqzGeGaaiikaiaadcfa caGGSaGaamivaiaacMcacqGH9aqpcaWGYbWcdaWgaaqcbasaaKqzad GaaGymaiaadgeaaKqaGeqaaKqzGeGaaiikaiaadcfacaGGSaGaaGim aiaacMcacqGHRaWkcaWG5bWcdaWgaaqcbasaaKqzadGaamyqaaqcba sabaqcLbsacaGGOaGaamiuaiaacYcacaWGubGaaiykaiaacYcaaaa@6FCE@

r 1 A 1 (P,T) r 1C (P,T), r 1 A 2 (P,T)= r 1 A 2 (P,0)+ y C (P,T). MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGYb WcdaWgaaqcbasaaKqzadGaaGymaiaadgealmaaBaaajiaibaqcLbma caaIXaaajiaibeaaaKqaGeqaaKqzGeGaaiikaiaadcfacaGGSaGaam ivaiaacMcacqGHijYUcaWGYbWcdaWgaaqcbasaaKqzadGaaGymaiaa doeaaKqaGeqaaKqzGeGaaiikaiaadcfacaGGSaGaamivaiaacMcaca GGSaGaamOCaKqbaoaaBaaajeaibaqcLbmacaaIXaGaamyqaSWaaSba aKGaGeaajugWaiaaikdaaKGaGeqaaaWcbeaajugibiaacIcacaWGqb GaaiilaiaadsfacaGGPaGaeyypa0JaamOCaSWaaSbaaKqaGeaajugW aiaaigdacaWGbbWcdaWgaaqccasaaKqzadGaaGOmaaqccasabaaaje aibeaajugibiaacIcacaWGqbGaaiilaiaaicdacaGGPaGaey4kaSIa amyEaKqbaoaaBaaajeaibaqcLbmacaWGdbaaleqaaKqzGeGaaiikai aadcfacaGGSaGaamivaiaacMcacaGGUaaaaa@6CD8@  (3)

The mean neares neighbor distance between two atoms in AB interstitial alloy with FCC structure is approximately determined by

r 1A ( P,T ) ¯ = r 1A ( P,0 ) ¯ + y( P,T ) ¯ , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aa0aaaO qaaKqzGeGaamOCaSWaaSbaaKqaGeaajugWaiaaigdacaWGbbaajeai beaajuaGdaqadaGcbaqcLbsacaWGqbGaaiilaiaadsfaaOGaayjkai aawMcaaaaajugibiabg2da9Kqbaoaanaaakeaajugibiaadkhalmaa BaaajeaibaqcLbmacaaIXaGaamyqaaqcbasabaqcfa4aaeWaaOqaaK qzGeGaamiuaiaacYcacaaIWaaakiaawIcacaGLPaaaaaqcLbsacqGH RaWkjuaGdaqdaaGcbaqcLbsacaWG5bqcfa4aaeWaaOqaaKqzGeGaam iuaiaacYcacaWGubaakiaawIcacaGLPaaaaaqcLbsacaGGSaaaaa@5656@

r 1A (P,0) ¯ =( 1 c B ) r 1A (P,0)+ c B r 1A (P,0), r 1A (P,0)= 2 r 1B (P,0), MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aa0aaaO qaaKqzGeGaamOCaSWaaSbaaKqaGeaajugWaiaaigdacaWGbbaajeai beaajugibiaacIcacaWGqbGaaiilaiaaicdacaGGPaaaaiabg2da9K qbaoaabmaakeaajugibiaaigdacqGHsislcaWGJbWcdaWgaaqcbasa aKqzadGaamOqaaqcbasabaaakiaawIcacaGLPaaajugibiaadkhalm aaBaaajeaibaqcLbmacaaIXaGaamyqaaqcbasabaqcLbsacaGGOaGa amiuaiaacYcacaaIWaGaaiykaiabgUcaRiaadogalmaaBaaajeaiba qcLbmacaWGcbaajeaibeaajugibiqadkhagaqbaSWaaSbaaKqaGeaa jugWaiaaigdacaWGbbaajeaibeaajugibiaacIcacaWGqbGaaiilai aaicdacaGGPaGaaiilaiqadkhagaqbaSWaaSbaaKqaGeaajugWaiaa igdacaWGbbaajeaibeaajugibiaacIcacaWGqbGaaiilaiaaicdaca GGPaGaeyypa0tcfa4aaOaaaOqaaKqzGeGaaGOmaaWcbeaajugibiaa dkhalmaaBaaajeaibaqcLbmacaaIXaGaamOqaaqcbasabaqcLbsaca GGOaGaamiuaiaacYcacaaIWaGaaiykaiaacYcaaaa@73DD@

y( P,T ) ¯ =( 115 c B ) y A ( P,T )+ c B y B ( P,T )+6 c B y A 1 ( P,T )+8 c B y A 2 ( P,T ). MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aa0aaaO qaaKqzGeGaamyEaKqbaoaabmaakeaajugibiaadcfacaGGSaGaamiv aaGccaGLOaGaayzkaaaaaKqzGeGaeyypa0tcfa4aaeWaaOqaaKqzGe GaaGymaiabgkHiTiaaigdacaaI1aGaam4yaSWaaSbaaKqaGeaajugW aiaadkeaaKqaGeqaaaGccaGLOaGaayzkaaqcLbsacaWG5bWcdaWgaa qcbasaaKqzadGaamyqaaqcbasabaqcfa4aaeWaaOqaaKqzGeGaamiu aiaacYcacaWGubaakiaawIcacaGLPaaajugibiabgUcaRiaadogaju aGdaWgaaqcbasaaKqzadGaamOqaaWcbeaajugibiaadMhalmaaBaaa jeaibaqcLbmacaWGcbaajeaibeaajuaGdaqadaGcbaqcLbsacaWGqb GaaiilaiaadsfaaOGaayjkaiaawMcaaKqzGeGaey4kaSIaaGOnaiaa dogalmaaBaaajeaibaqcLbmacaWGcbaajeaibeaajugibiaadMhalm aaBaaajeaibaqcLbmacaWGbbWcdaWgaaqccasaaKqzadGaaGymaaqc casabaaajeaibeaajuaGdaqadaGcbaqcLbsacaWGqbGaaiilaiaads faaOGaayjkaiaawMcaaKqzGeGaey4kaSIaaGioaiaadogalmaaBaaa jeaibaqcLbmacaWGcbaajeaibeaajugibiaadMhalmaaBaaajeaiba qcLbmacaWGbbWcdaWgaaqccasaaKqzadGaaGOmaaqccasabaaajeai beaajuaGdaqadaGcbaqcLbsacaWGqbGaaiilaiaadsfaaOGaayjkai aawMcaaKqzGeGaaiOlaaaa@82DF@  (4)

The free energy of AB interstitial alloy with FCC structure and concetration condition c B << c A MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGJb WcdaWgaaqcbasaaKqzadGaamOqaaqcbasabaqcLbsacqGH8aapcqGH 8aapcaWGJbqcfa4aaSbaaKqaGeaajugWaiaadgeaaKqaGeqaaaaa@4058@ has the form

ψ AB =( 115 c B ) ψ A + c B ψ B +6 c B ψ A 1 +8 c B ψ A 2 T S c . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHip qElmaaBaaajeaibaqcLbmacaWGbbGaamOqaaqcbasabaqcLbsacqGH 9aqpjuaGdaqadaGcbaqcLbsacaaIXaGaeyOeI0IaaGymaiaaiwdaca WGJbqcfa4aaSbaaKqaGeaajugWaiaadkeaaSqabaaakiaawIcacaGL PaaajugibiabeI8a5TWaaSbaaKqaGeaajugWaiaadgeaaKqaGeqaaK qzGeGaey4kaSIaam4yaSWaaSbaaKqaGeaajugWaiaadkeaaKqaGeqa aKqzGeGaeqiYdKxcfa4aaSbaaKqaGeaajugWaiaadkeaaSqabaqcLb sacqGHRaWkcaaI2aGaam4yaSWaaSbaaKqaGeaajugWaiaadkeaaKqa GeqaaKqzGeGaeqiYdK3cdaWgaaqcbasaaKqzadGaamyqaSWaaSbaaK GaGeaajugWaiaaigdaaKGaGeqaaaqcbasabaqcLbsacqGHRaWkcaaI 4aGaam4yaSWaaSbaaKqaGeaajugWaiaadkeaaKqaGeqaaKqzGeGaeq iYdK3cdaWgaaqcbasaaKqzadGaamyqaSWaaSbaaKGaGeaajugWaiaa ikdaaKGaGeqaaaqcbasabaqcLbsacqGHsislcaWGubGaam4uaKqbao aaBaaajeaibaqcLbmacaWGJbaaleqaaKqzGeGaaiOlaaaa@7694@

ψ X U 0X + ψ 0X +3N{ θ 2 k X 2 [ γ 2X X X 2 2 γ 1X 3 ( 1+ X X 2 ) ]+ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHip qEjuaGdaWgaaqcKfaG=haajugWaiaadIfaaKqaGfqaaKqzGeGaeyis ISRaamyvaKqbaoaaBaaajqwaa+FaaKqzadGaaGimaiaadIfaaKqaGf qaaKqzGeGaey4kaSIaeqiYdK3cdaWgaaqcKfaG=haajugWaiaaicda caWGybaajqwaa+FabaqcLbsacqGHRaWkcaaIZaGaamOtaKqbaoaace aajaaybaqcfa4aaSaaaKaaGfaajugibiabeI7aXTWaaWbaaKazba4= beqaaKqzadGaaGOmaaaaaKaaGfaajugibiaadUgalmaaDaaajqwaa+ FaaKqzadGaamiwaaqcKfaG=haajugWaiaaikdaaaaaaKqbaoaadmaa jaaybaqcLbsacqaHZoWzlmaaBaaajqwaa+FaaKqzadGaaGOmaiaadI faaKazba4=beaajugibiaadIfalmaaDaaajqwaa+FaaKqzadGaamiw aaqcKfaG=haajugWaiaaikdaaaqcLbsacqGHsisljuaGdaWcaaqcaa waaKqzGeGaaGOmaiabeo7aNLqbaoaaBaaajqwaa+FaaKqzadGaaGym aiaadIfaaKqaGfqaaaqcaawaaKqzGeGaaG4maaaajuaGdaqadaqcaa waaKqzGeGaaGymaiabgUcaRKqbaoaalaaajaaybaqcLbsacaWGybqc fa4aaSbaaKazba4=baqcLbmacaWGybaajeaybeaaaKaaGfaajugibi aaikdaaaaajaaycaGLOaGaayzkaaaacaGLBbGaayzxaaqcLbsacqGH RaWkaKaaGjaawUhaaaaa@990A@

+ 2 θ 3 k X 4 [ 4 3 γ 2X 2 X X ( 1+ X X 2 )2( γ 1X 2 +2 γ 1X γ 2X )( 1+ X X 2 )( 1+ X X ) ] }, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaiGaaO qaaKqzGeGaey4kaSscfa4aaSaaaOqaaKqzGeGaaGOmaiabeI7aXTWa aWbaaKqaGeqabaqcLbmacaaIZaaaaaGcbaqcLbsacaWGRbWcdaqhaa qcbasaaKqzadGaamiwaaqcbasaaKqzadGaaGinaaaaaaqcfa4aamWa aOqaaKqbaoaalaaakeaajugibiaaisdaaOqaaKqzGeGaaG4maaaacq aHZoWzlmaaDaaajeaibaqcLbmacaaIYaGaamiwaaqcbasaaKqzadGa aGOmaaaajugibiaadIfalmaaBaaajeaibaqcLbmacaWGybaajeaibe aajuaGdaqadaGcbaqcLbsacaaIXaGaey4kaSscfa4aaSaaaOqaaKqz GeGaamiwaSWaaSbaaKqaGeaajugWaiaadIfaaKqaGeqaaaGcbaqcLb sacaaIYaaaaaGccaGLOaGaayzkaaqcLbsacqGHsislcaaIYaqcfa4a aeWaaOqaaKqzGeGaeq4SdC2cdaqhaaqcbasaaKqzadGaaGymaiaadI faaKqaGeaajugWaiaaikdaaaqcLbsacqGHRaWkcaaIYaGaeq4SdC2c daWgaaqcbasaaKqzadGaaGymaiaadIfaaKqaGeqaaKqzGeGaeq4SdC wcfa4aaSbaaKqaGeaajugWaiaaikdacaWGybaaleqaaaGccaGLOaGa ayzkaaqcfa4aaeWaaOqaaKqzGeGaaGymaiabgUcaRKqbaoaalaaake aajugibiaadIfajuaGdaWgaaqcbasaaKqzadGaamiwaaWcbeaaaOqa aKqzGeGaaGOmaaaaaOGaayjkaiaawMcaaKqbaoaabmaakeaajugibi aaigdacqGHRaWkcaWGybqcfa4aaSbaaKqaGeaajugWaiaadIfaaSqa baaakiaawIcacaGLPaaaaiaawUfacaGLDbaaaiaaw2haaKqzGeGaai ilaaaa@8CCE@

ψ 0X =3Nθ[ x X +ln(1 e 2 x X ) ], X X x X coth x X , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHip qEjuaGdaWgaaqcbasaaKqzadGaaGimaiaadIfaaSqabaqcLbsacqGH 9aqpcaaIZaGaamOtaiabeI7aXLqbaoaadmaakeaajugibiaadIhalm aaBaaajeaibaqcLbmacaWGybaajeaibeaajugibiabgUcaRiGacYga caGGUbGaaiikaiaaigdacqGHsislcaWGLbqcfa4aaWbaaSqabKqaGe aajugWaiabgkHiTiaaikdacaWG4bWcdaWgaaqccasaaKqzadGaamiw aaqccasabaaaaKqzGeGaaiykaaGccaGLBbGaayzxaaqcLbsacaGGSa GaamiwaSWaaSbaaKqaGeaajugWaiaadIfaaKqaGeqaaKqzGeGaeyyy IORaamiEaSWaaSbaaKqaGeaajugWaiaadIfaaKqaGeqaaKqzGeGaci 4yaiaac+gacaGG0bGaaiiAaiaadIhalmaaBaaajeaibaqcLbmacaWG ybaajeaibeaajugibiaacYcaaaa@6A34@  (5)

where ψ A MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHip qElmaaBaaajeaibaqcLbmacaWGbbaajeaibeaaaaa@3AC7@ is the free energy of A atom in A pure metal, ψ B MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHip qElmaaBaaajeaibaqcLbmacaWGcbaajeaibeaaaaa@3AC8@ is the free energy of B atom in interstitial alloy, ψ A 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHip qElmaaBaaajeaibaqcLbmacaWGbbWcdaWgaaqccasaaKqzadGaaGym aaqccasabaaajeaibeaaaaa@3D32@  and ψ A 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHip qElmaaBaaajeaibaqcLbmacaWGbbWcdaWgaaqccasaaKqzadGaaGOm aaqccasabaaajeaibeaaaaa@3D33@ respectively are free energy of A1 and A2 atoms and Sc is the configuration entropy of AC interstitial alloy.

The pressure is calculated by

P= ( ψ V ) T = a 3V ( ψ a ) T . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqi=KI8=fYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadcfacq GH9aqpcqGHsisllmaabmaakeaalmaalaaakeaajugibiabgkGi2kab eI8a5bGcbaqcLbsacqGHciITcaWGwbaaaaGccaGLOaGaayzkaaadda WgaaqcbasaaKqzadGaamivaaqcbasabaqcLbsacqGH9aqpcqGHsisl lmaalaaakeaajugibiaadggaaOqaaKqzGeGaaG4maiaadAfaaaWcda qadaGcbaWcdaWcaaGcbaqcLbsacqGHciITcqaHipqEaOqaaKqzGeGa eyOaIyRaamyyaaaaaOGaayjkaiaawMcaaSWaaSbaaKqaGeaajugWai aadsfaaSqabaqcLbsacaGGUaaaaa@5669@  (6)

Setting

γ G T = a AB 6 [ 1 k A k A a A ( 115 c B ) x A coth x A + 1 k B k B a B c B x B coth x B + MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqi=KI8=fYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabeo7aNX Waa0baaKqaGeaajugWaiaadEeaaKqaGeaajugWaiaadsfaaaqcLbsa cqGH9aqpcqGHsisllmaalaaakeaajugibiaadggalmaaBaaajeaiba qcLbmacaWGbbGaamOqaaWcbeaaaOqaaKqzGeGaaGOnaaaalmaadeaa keaalmaalaaakeaajugibiaaigdaaOqaaKqzGeGaam4AaSWaaSbaaK qaGeaajugWaiaadgeaaSqabaaaaaGccaGLBbaalmaalaaakeaajugi biabgkGi2kaadUgammaaBaaajeaibaqcLbmacaWGbbaajeaibeaaaO qaaKqzGeGaeyOaIyRaamyyaSWaaSbaaKqaGeaajugWaiaadgeaaSqa baaaamaabmaakeaajugibiaaigdacqGHsislcaaIXaGaaGynaiaado galmaaBaaajeaibaqcLbmacaWGcbaaleqaaaGccaGLOaGaayzkaaqc LbsacaWG4bWcdaWgaaqcbasaaKqzadGaamyqaaWcbeaajugibiGaco gacaGGVbGaaiiDaiaacIgacaWG4bWcdaWgaaqcbasaaKqzadGaamyq aaWcbeaajugibiabgUcaRSWaaSaaaOqaaKqzGeGaaGymaaGcbaqcLb sacaWGRbWcdaWgaaqcbasaaKqzadGaamOqaaWcbeaaaaWaaSaaaOqa aKqzGeGaeyOaIyRaam4AaWWaaSbaaKqaGeaajugWaiaadkeaaKqaGe qaaaGcbaqcLbsacqGHciITcaWGHbaddaWgaaqcbasaaKqzadGaamOq aaqcbasabaaaaKqzGeGaam4yaSWaaSbaaKqaGeaajugWaiaadkeaaS qabaqcLbsacaWG4baddaWgaaqcbasaaKqzadGaamOqaaqcbasabaqc LbsaciGGJbGaai4BaiaacshacaGGObGaamiEaSWaaSbaaKqaGeaaju gWaiaadkeaaSqabaqcLbsacqGHRaWkaaa@8C9E@
+ 1 k A 1 k A 1 a A 1 6 c B x A 1 coth x A 1 + 1 k A 2 k A 2 a A 2 8 c B x A 2 coth x A 2 ]. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqi=KI8=fYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabgUcaRS WaaSaaaKaaGfaajugibiaaigdaaKaaGfaajugibiaadUgalmaaBaaa jqwaa+FaaKqzadGaamyqaWWaaSbaaKazca4=baqcLbmacaaIXaaajq Maa+FabaaajeaybeaaaaWcdaWcaaqcaawaaKqzGeGaeyOaIyRaam4A aSWaaSbaaKazba4=baqcLbmacaWGbbaddaWgaaqcKjaG=haajugWai aaigdaaKazca4=beaaaKqaGfqaaaqcaawaaKqzGeGaeyOaIyRaamyy aSWaaSbaaKazba4=baqcLbmacaWGbbaddaWgaaqcKjaG=haajugWai aaigdaaKazca4=beaaaKqaGfqaaaaajugibiaaiAdacaWGJbWcdaWg aaqcKfaG=haajugWaiaadkeaaKqaGfqaaKqzGeGaamiEaSWaaSbaaK azba4=baqcLbmacaWGbbaddaWgaaqcKjaG=haajugWaiaaigdaaKaz ca4=beaaaKqaGfqaaKqzGeGaci4yaiaac+gacaGG0bGaaiiAaiaadI hammaaBaaajqwaa+FaaKqzadGaamyqaWWaaSbaaKazca4=baqcLbma caaIXaaajqMaa+Fabaaajqwaa+FabaqcLbsacqGHRaWklmaalaaaja aybaqcLbsacaaIXaaajaaybaqcLbsacaWGRbaddaWgaaqcKfaG=haa jugWaiaadgeammaaBaaajqMaa+FaaKqzadGaaGOmaaqcKjaG=hqaaa qcKfaG=hqaaaaalmaadiaajaaybaWcdaWcaaqcaawaaKqzGeGaeyOa IyRaam4AaWWaaSbaaKazba4=baqcLbmacaWGbbaddaWgaaqcKjaG=h aajugWaiaaikdaaKazca4=beaaaKazba4=beaaaKaaGfaajugibiab gkGi2kaadggalmaaBaaajqwaa+FaaKqzadGaamyqaWWaaSbaaKazca 4=baqcLbmacaaIYaaajqMaa+FabaaajeaybeaaaaqcLbsacaaI4aGa am4yaWWaaSbaaKazba4=baqcLbmacaWGcbaajqwaa+FabaqcLbsaca WG4bWcdaWgaaqcKfaG=haajugWaiaadgeammaaBaaajqMaa+FaaKqz adGaaGOmaaqcKjaG=hqaaaqcbawabaqcLbsaciGGJbGaai4Baiaacs hacaGGObGaamiEaWWaaSbaaKazba4=baqcLbmacaWGbbaddaWgaaqc KjaG=haajugWaiaaikdaaKazca4=beaaaKazba4=beaaaKaaGjaaw2 faaKqzGeGaaiOlaaaa@DB3C@  (7)

Here, γ G T MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqi=KI8=fYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabeo7aNX Waa0baaKqaGeaajugWaiaadEeaaKqaGeaajugWaiaadsfaaaaaaa@3C6E@ is the Grüneisen parameter of ABalloy. Then,

P= a AB 6 V AB [ ( 115 c B ) U 0A a A + c B U 0B a B +6 c B U 0 A 1 a A 1 +8 c B U 0 A 2 a A 2 ]+ 3 γ G T .θ V AB . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqi=KI8=fYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadcfacq GH9aqpcqGHsisllmaalaaajaaybaqcLbsacaWGHbWcdaWgaaqcKfaG =haajugWaiaadgeacaWGcbaajeaybeaaaKaaGfaajugibiaaiAdaca WGwbWcdaWgaaqcKfaG=haajugWaiaadgeacaWGcbaajeaybeaaaaWc daWadaqcaawaaSWaaeWaaKaaGfaajugibiaaigdacqGHsislcaaIXa GaaGynaiaadogammaaBaaajqwaa+FaaKqzadGaamOqaaqcKfaG=hqa aaqcaaMaayjkaiaawMcaaSWaaSaaaKaaGfaajugibiabgkGi2kaadw falmaaBaaajqwaa+FaaKqzadGaaGimaiaadgeaaKqaGfqaaaqcaawa aKqzGeGaeyOaIyRaamyyaSWaaSbaaKazba4=baqcLbmacaWGbbaaje aybeaaaaqcLbsacqGHRaWkcaWGJbWcdaWgaaqcKfaG=haajugWaiaa dkeaaKqaGfqaaSWaaSaaaKaaGfaajugibiabgkGi2kaadwfalmaaBa aajqwaa+FaaKqzadGaaGimaiaadkeaaKqaGfqaaaqcaawaaKqzGeGa eyOaIyRaamyyaSWaaSbaaKazba4=baqcLbmacaWGcbaajqwaa+Faba aaaKqzGeGaey4kaSIaaGOnaiaadogalmaaBaaajqwaa+FaaKqzadGa amOqaaqcbawabaWcdaWcaaqcaawaaKqzGeGaeyOaIyRaamyvaSWaaS baaKazba4=baqcLbmacaaIWaGaamyqaSWaaSbaaKazca4=baqcLbma caaIXaaajqMaa+FabaaajeaybeaaaKaaGfaajugibiabgkGi2kaadg galmaaBaaajeaybaqcLbmacaWGbbWcdaWgaaqccawaaKqzGeGaaGym aaqccawabaaajeaybeaaaaqcLbsacqGHRaWkcaaI4aGaam4yaSWaaS baaKazba4=baqcLbmacaWGcbaajeaybeaalmaalaaajaaybaqcLbsa cqGHciITcaWGvbWcdaWgaaqcKfaG=haajugWaiaaicdacaWGbbWcda WgaaqcKjaG=haajugWaiaaikdaaKazca4=beaaaKqaGfqaaaqcaawa aKqzGeGaeyOaIyRaamyyaSWaaSbaaKazba4=baqcLbmacaWGbbWcda WgaaqcKjaG=haajugWaiaaikdaaKazca4=beaaaKqaGfqaaaaaaKaa GjaawUfacaGLDbaajugibiabgUcaRSWaaSaaaKaaGfaajugibiaaio dacqaHZoWzlmaaDaaajqwaa+FaaKqzadGaam4raaqcKfaG=haajugW aiaadsfaaaqcLbsacaGGUaGaeqiUdehajaaybaqcLbsacaWGwbWcda WgaaqcKfaG=haajugWaiaadgeacaWGcbaajqwaa+FabaaaaKqzGeGa aiOlaaaa@DCAE@ (8)

From the condition of absolute stability limit

( P V AB ) T =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqi=KI8=fYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaalmaabmaakeaalm aalaaakeaajugibiabgkGi2kaadcfaaOqaaKqzGeGaeyOaIyRaamOv aWWaaSbaaKqaGeaajugWaiaadgeacaWGcbaajeaibeaaaaaakiaawI cacaGLPaaammaaBaaajeaibaqcLbmacaWGubaajeaibeaajugibiab g2da9iaaicdaaaa@4539@  hay ( P a AB ) T =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqi=KI8=fYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaalmaabmaakeaalm aalaaakeaajugibiabgkGi2kaadcfaaOqaaKqzGeGaeyOaIyRaamyy aWWaaSbaaKqaGeaajugWaiaadgeacaWGcbaajeaibeaaaaaakiaawI cacaGLPaaammaaBaaajeaibaqcLbmacaWGubaajeaibeaajugibiab g2da9iaaicdaaaa@4544@ (9)

we can derive the absolute stability temperature for crystalline state in the form

T s = T S 1 M S 1 ,T S 1 =2P V AB + a AB 2 6 [ ( 115 c B ) 2 U 0A a A 2 + c B 2 U 0B a B 2 +6 c B 2 U 0 A 1 a A 1 2 +8 c B 2 U 0 A 2 a A 2 2 ] ( 115 c B )[ a AB 2 k A ( k A a A ) 2 a AB 2 k A a A 2 ] ω A a AB 4 k A c B [ a AB 2 k B ( k B a B ) 2 a AB 2 k B a B 2 ] ω B a AB 4 k B MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaajugibi aadsfajuaGdaWgaaqcbasaaKqzadGaam4CaaWcbeaajugibiabg2da 9KqbaoaalaaakeaajugibiaadsfacaWGtbWcdaWgaaqcbasaaKqzad GaaGymaaqcbasabaaakeaajugibiaad2eacaWGtbWcdaWgaaqcbasa aKqzadGaaGymaaqcbasabaaaaKqzGeGaaiilaiaadsfacaWGtbWcda WgaaqcbasaaKqzadGaaGymaaqcbasabaqcLbsacqGH9aqpcaaIYaGa amiuaiaadAfalmaaBaaajeaibaqcLbmacaWGbbGaamOqaaqcbasaba qcLbsacqGHRaWkjuaGdaWcaaGcbaqcLbsacaWGHbqcfa4aa0baaSqa aKqzGeGaamyqaiaadkeaaKqaGeaajugWaiaaikdaaaaakeaajugibi aaiAdaaaqcfa4aamWaaOqaaKqbaoaabmaakeaajugibiaaigdacqGH sislcaaIXaGaaGynaiaadogalmaaBaaajeaibaqcLbmacaWGcbaaje aibeaaaOGaayjkaiaawMcaaKqbaoaalaaakeaajugibiabgkGi2UWa aWbaaKqaGeqabaqcLbmacaaIYaaaaKqzGeGaamyvaKqbaoaaBaaaje aibaqcLbmacaaIWaGaamyqaaWcbeaaaOqaaKqzGeGaeyOaIyRaamyy aSWaa0baaKqaGeaajugWaiaadgeaaKqaGeaajugWaiaaikdaaaaaaK qzGeGaey4kaSIaam4yaSWaaSbaaKqaGeaajugWaiaadkeaaKqaGeqa aKqbaoaalaaakeaajugibiabgkGi2UWaaWbaaKqaGeqabaqcLbmaca aIYaaaaKqzGeGaamyvaSWaaSbaaKqaGeaajugWaiaaicdacaWGcbaa jeaibeaaaOqaaKqzGeGaeyOaIyRaamyyaSWaa0baaKqaGeaajugWai aadkeaaKqaGeaajugWaiaaikdaaaaaaKqzGeGaey4kaSIaaGOnaiaa dogalmaaBaaajeaibaqcLbmacaWGcbaajeaibeaajuaGdaWcaaGcba qcLbsacqGHciITjuaGdaahaaWcbeqcbasaaKqzadGaaGOmaaaajugi biaadwfalmaaBaaajeaibaqcLbmacaaIWaGaamyqaSWaaSbaaKGaaf aajugWaiaaigdaaKGaafqaaaqcbasabaaakeaajugibiabgkGi2kaa dggalmaaDaaajeaibaqcLbmacaWGbbWcdaWgaaqccasaaKqzadGaaG ymaaqccasabaaajeaibaqcLbmacaaIYaaaaaaajugibiabgUcaRiaa iIdacaWGJbWcdaWgaaqcbasaaKqzadGaamOqaaqcbasabaqcfa4aaS aaaOqaaKqzGeGaeyOaIy7cdaahaaqcbasabeaajugWaiaaikdaaaqc LbsacaWGvbqcfa4aaSbaaSqaaKqzadGaaGimaiaadgealmaaBaaaji aibaqcLbmacaaIYaaajiaibeaaaSqabaaakeaajugibiabgkGi2kaa dggalmaaDaaajeaibaqcLbmacaWGbbWcdaWgaaqccauaaKqzadGaaG OmaaqccauabaaajeaibaqcLbmacaaIYaaaaaaaaOGaay5waiaaw2fa aKqzGeGaeyOeI0cakeaajugibiabgkHiTKqbaoaabmaakeaajugibi aaigdacqGHsislcaaIXaGaaGynaiaadogalmaaBaaajeaibaqcLbma caWGcbaajeaibeaaaOGaayjkaiaawMcaaKqbaoaadmaakeaajuaGda WcaaGcbaqcLbsacaWGHbWcdaWgaaqcbasaaKqzadGaamyqaiaadkea aKqaGeqaaaGcbaqcLbsacaaIYaGaam4AaKqbaoaaBaaajeaibaqcLb macaWGbbaaleqaaaaajuaGdaqadaGcbaqcfa4aaSaaaOqaaKqzGeGa eyOaIyRaam4AaSWaaSbaaKqaGeaajugWaiaadgeaaKqaGeqaaaGcba qcLbsacqGHciITcaWGHbWcdaWgaaqcbasaaKqzadGaamyqaaqcbasa baaaaaGccaGLOaGaayzkaaWcdaahaaqcbasabeaajugWaiaaikdaaa qcLbsacqGHsislcaWGHbWcdaWgaaqcbasaaKqzadGaamyqaiaadkea aKqaGeqaaKqbaoaalaaakeaajugibiabgkGi2UWaaWbaaKqaGeqaba qcLbmacaaIYaaaaKqzGeGaam4AaKqbaoaaBaaajeaibaqcLbmacaWG bbaaleqaaaGcbaqcLbsacqGHciITcaWGHbWcdaqhaaqcbasaaKqzad GaamyqaaqcbasaaKqzadGaaGOmaaaaaaaakiaawUfacaGLDbaajuaG daWcaaGcbaqcLbsacqWIpecAcqaHjpWDjuaGdaWgaaqcbasaaKqzad GaamyqaaWcbeaajugibiaadggalmaaBaaajeaibaqcLbmacaWGbbGa amOqaaqcbasabaaakeaajugibiaaisdacaWGRbWcdaWgaaqcbasaaK qzadGaamyqaaqcbasabaaaaKqzGeGaeyOeI0Iaam4yaKqbaoaaBaaa jeaibaqcLbmacaWGcbaaleqaaKqbaoaadmaakeaajuaGdaWcaaGcba qcLbsacaWGHbWcdaWgaaqcbasaaKqzadGaamyqaiaadkeaaKqaGeqa aaGcbaqcLbsacaaIYaGaam4AaSWaaSbaaKqaGeaajugWaiaadkeaaK qaGeqaaaaajuaGdaqadaGcbaqcfa4aaSaaaOqaaKqzGeGaeyOaIyRa am4AaKqbaoaaBaaajeaibaqcLbmacaWGcbaaleqaaaGcbaqcLbsacq GHciITcaWGHbWcdaWgaaqcbasaaKqzadGaamOqaaqcbasabaaaaaGc caGLOaGaayzkaaWcdaahaaqcbasabeaajugWaiaaikdaaaqcLbsacq GHsislcaWGHbWcdaWgaaqcbasaaKqzadGaamyqaiaadkeaaKqaGeqa aKqbaoaalaaakeaajugibiabgkGi2UWaaWbaaKqaGeqabaqcLbmaca aIYaaaaKqzGeGaam4AaKqbaoaaBaaajeaibaqcLbmacaWGcbaaleqa aaGcbaqcLbsacqGHciITcaWGHbWcdaqhaaqcbasaaKqzadGaamOqaa qcbasaaKqzadGaaGOmaaaaaaaakiaawUfacaGLDbaajuaGdaWcaaGc baqcLbsacqWIpecAcqaHjpWDlmaaBaaajeaibaqcLbmacaWGcbaaje aibeaajugibiaadggalmaaBaaajeaibaqcLbmacaWGbbGaamOqaaqc basabaaakeaajugibiaaisdacaWGRbqcfa4aaSbaaKqaGeaajugWai aadkeaaSqabaaaaKqzGeGaeyOeI0caaaa@6011@

6 c B [ a AB 2 k A 1 ( k A 1 a A 1 ) 2 a AB 2 k A 1 a A 1 2 ] ω A 1 a AB 4 k A 1 8 c B [ a AB 2 k A 2 ( k A 2 a A 2 ) 2 a AB 2 k A 2 a A 2 2 ] ω A 2 a AB 4 k A 2 , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqGHsi slcaaI2aGaam4yaSWaaSbaaKazba4=baqcLbmacaWGcbaajqwaa+Fa baqcfa4aamWaaKaaGfaajuaGdaWcaaqcaawaaKqzGeGaamyyaKqbao aaBaaajqwaa+FaaKqzadGaamyqaiaadkeaaKqaGfqaaaqcaawaaKqz GeGaaGOmaiaadUgalmaaBaaajqwaa+FaaKqzadGaamyqaSWaaSbaaK GaGeaajugWaiaaigdaaKGaGeqaaaqcKfaG=hqaaaaajuaGdaqadaqc aawaaKqbaoaalaaajaaybaqcLbsacqGHciITcaWGRbWcdaWgaaqcKf aG=haajugWaiaadgealmaaBaaajiaibaqcLbmacaaIXaaajiaibeaa aKazba4=beaaaKaaGfaajugibiabgkGi2kaadggalmaaBaaajqwaa+ FaaKqzadGaamyqaSWaaSbaaKGaGeaajugWaiaaigdaaKGaGeqaaaqc KfaG=hqaaaaaaKaaGjaawIcacaGLPaaalmaaCaaajqwaa+Fabeaaju gWaiaaikdaaaqcLbsacqGHsislcaWGHbWcdaWgaaqcbasaaKqzadGa amyqaiaadkeaaKqaGeqaaKqbaoaalaaajaaybaqcLbsacqGHciITlm aaCaaajqwaa+FabeaajugWaiaaikdaaaqcLbsacaWGRbWcdaWgaaqc KfaG=haajugWaiaadgealmaaBaaajiaibaqcLbmacaaIXaaajiaibe aaaKazba4=beaaaKaaGfaajugibiabgkGi2kaadggalmaaDaaajqwa a+FaaKqzadGaamyqaSWaaSbaaKGaGeaajugWaiaaigdaaKGaGeqaaa qcKfaG=haajugWaiaaikdaaaaaaaqcaaMaay5waiaaw2faaKqbaoaa laaajaaybaqcLbsacqWIpecAcqaHjpWDjuaGdaWgaaqcKfaG=haaju gWaiaadgealmaaBaaajiaibaqcLbmacaaIXaaajiaibeaaaKqaGfqa aKqzGeGaamyyaSWaaSbaaKazba4=baqcLbmacaWGbbGaamOqaaqcKf aG=hqaaaqcaawaaKqzGeGaaGinaiaadUgalmaaBaaajqwaa+FaaKqz adGaamyqaSWaaSbaaKGaGeaajugWaiaaigdaaKGaGeqaaaqcKfaG=h qaaaaajugibiabgkHiTiaaiIdacaWGJbWcdaWgaaqcKfaG=haajugW aiaadkeaaKazba4=beaajuaGdaWadaqcaawaaKqbaoaalaaajaayba qcLbsacaWGHbWcdaWgaaqcKfaG=haajugWaiaadgeacaWGcbaajqwa a+FabaaajaaybaqcLbsacaaIYaGaam4AaKqbaoaaBaaajqwaa+FaaK qzadGaamyqaSWaaSbaaKGaGeaajugWaiaaikdaaKGaGeqaaaqcbawa baaaaKqbaoaabmaajaaybaqcfa4aaSaaaKaaGfaajugibiabgkGi2k aadUgalmaaBaaajqwaa+FaaKqzadGaamyqaSWaaSbaaKGaGeaajugW aiaaikdaaKGaGeqaaaqcKfaG=hqaaaqcaawaaKqzGeGaeyOaIyRaam yyaSWaaSbaaKazba4=baqcLbmacaWGbbWcdaWgaaqccasaaKqzadGa aGOmaaqccasabaaajqwaa+FabaaaaaqcaaMaayjkaiaawMcaaKqbao aaCaaajeaybeqcKfaG=haajugWaiaaikdaaaqcLbsacqGHsislcaWG HbWcdaWgaaqcKfaG=haajugWaiaadgeacaWGcbaajqwaa+Fabaqcfa 4aaSaaaKaaGfaajugibiabgkGi2UWaaWbaaKazba4=beqaaKqzadGa aGOmaaaajugibiaadUgalmaaBaaajqwaa+FaaKqzadGaamyqaSWaaS baaKGaGeaajugWaiaaikdaaKGaGeqaaaqcKfaG=hqaaaqcaawaaKqz GeGaeyOaIyRaamyyaSWaa0baaKazba4=baqcLbmacaWGbbWcdaWgaa qccasaaKqzadGaaGOmaaqccasabaaajqwaa+FaaKqzadGaaGOmaaaa aaaajaaycaGLBbGaayzxaaqcfa4aaSaaaKaaGfaajugibiabl+qiOj abeM8a3TWaaSbaaKazba4=baqcLbmacaWGbbWcdaWgaaqccasaaKqz adGaaGOmaaqccasabaaajqwaa+FabaqcLbsacaWGHbWcdaWgaaqcKf aG=haajugWaiaadgeacaWGcbaajqwaa+FabaaajaaybaqcLbsacaaI 0aGaam4AaSWaaSbaaKazba4=baqcLbmacaWGbbWcdaWgaaqccasaaK qzadGaaGOmaaqccasabaaajqwaa+FabaaaaKqzGeGaaiilaaaa@407B@

M S 1 =( 115 c B ) a AB 2 k Bo 4 k A 2 ( k A a A ) 2 + c B a AB 2 k Bo 4 k B 2 ( k B a B ) 2 +6 c B a AB 2 k Bo 4 k A 1 2 ( k A 1 a A 1 ) 2 +8 c B a AB 2 k Bo 4 k A 2 2 ( k A 2 a A 2 ) 2 , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGnb Gaam4uaSWaaSbaaKazba4=baqcLbmacaaIXaaajqwaa+FabaqcLbsa cqGH9aqpjuaGdaqadaqcaawaaKqzGeGaaGymaiabgkHiTiaaigdaca aI1aGaam4yaKqbaoaaBaaajqwaa+FaaKqzadGaamOqaaqcbawabaaa jaaycaGLOaGaayzkaaqcfa4aaSaaaKaaGfaajugibiaadggalmaaDa aajqwaa+FaaKqzadGaamyqaiaadkeaaKazba4=baqcLbmacaaIYaaa aKqzGeGaam4AaKqbaoaaBaaajqwaa+FaaKqzadGaamOqaiaad+gaaK qaGfqaaaqcaawaaKqzGeGaaGinaiaadUgalmaaDaaajqwaa+FaaKqz adGaamyqaaqcKfaG=haajugWaiaaikdaaaaaaKqbaoaabmaajaayba qcfa4aaSaaaKaaGfaajugibiabgkGi2kaadUgalmaaBaaajqwaa+Fa aKqzadGaamyqaaqcKfaG=hqaaaqcaawaaKqzGeGaeyOaIyRaamyyaS WaaSbaaKazba4=baqcLbmacaWGbbaajqwaa+FabaaaaaqcaaMaayjk aiaawMcaaSWaaWbaaKazba4=beqaaKqzadGaaGOmaaaajugibiabgU caRiaadogalmaaBaaajqwaa+FaaKqzadGaamOqaaqcKfaG=hqaaKqb aoaalaaajaaybaqcLbsacaWGHbWcdaqhaaqcKfaG=haajugWaiaadg eacaWGcbaajqwaa+FaaKqzadGaaGOmaaaajugibiaadUgalmaaBaaa jqwaa+FaaKqzadGaamOqaiaad+gaaKazba4=beaaaKaaGfaajugibi aaisdacaWGRbWcdaqhaaqcKfaG=haajugWaiaadkeaaKazba4=baqc LbmacaaIYaaaaaaajuaGdaqadaqcaawaaKqbaoaalaaajaaybaqcLb sacqGHciITcaWGRbWcdaWgaaqcKfaG=haajugWaiaadkeaaKazba4= beaaaKaaGfaajugibiabgkGi2kaadggalmaaBaaajqwaa+FaaKqzad GaamOqaaqcKfaG=hqaaaaaaKaaGjaawIcacaGLPaaalmaaCaaajqwa a+FabeaajugWaiaaikdaaaqcLbsacqGHRaWkcaaI2aGaam4yaSWaaS baaKazba4=baqcLbmacaWGcbaajqwaa+Fabaqcfa4aaSaaaKaaGfaa jugibiaadggalmaaDaaajqwaa+FaaKqzadGaamyqaiaadkeaaKazba 4=baqcLbmacaaIYaaaaKqzGeGaam4AaKqbaoaaBaaajqwaa+FaaKqz adGaamOqaiaad+gaaKqaGfqaaaqcaawaaKqzGeGaaGinaiaadUgalm aaDaaajqwaa+FaaKqzadGaamyqaSWaaSbaaKGaGeaajugWaiaaigda aKGaGeqaaaqcKfaG=haajugWaiaaikdaaaaaaKqbaoaabmaajaayba qcfa4aaSaaaKaaGfaajugibiabgkGi2kaadUgalmaaBaaajqwaa+Fa aKqzadGaamyqaSWaaSbaaKGaGeaajugWaiaaigdaaKGaGeqaaaqcKf aG=hqaaaqcaawaaKqzGeGaeyOaIyRaamyyaSWaaSbaaKazba4=baqc LbmacaWGbbWcdaWgaaqccasaaKqzadGaaGymaaqccasabaaajqwaa+ FabaaaaaqcaaMaayjkaiaawMcaaSWaaWbaaKazba4=beqaaKqzadGa aGOmaaaajugibiabgUcaRiaaiIdacaWGJbWcdaWgaaqcKfaG=haaju gWaiaadkeaaKazba4=beaajuaGdaWcaaqcaawaaKqzGeGaamyyaSWa a0baaKazba4=baqcLbmacaWGbbGaamOqaaqcKfaG=haajugWaiaaik daaaqcLbsacaWGRbWcdaWgaaqcKfaG=haajugWaiaadkeacaWGVbaa jqwaa+FabaaajaaybaqcLbsacaaI0aGaam4AaSWaa0baaKazba4=ba qcLbmacaWGbbWcdaWgaaqccasaaKqzadGaaGOmaaqccasabaaajqwa a+FaaKqzadGaaGOmaaaaaaqcfa4aaeWaaKaaGfaajuaGdaWcaaqcaa waaKqzGeGaeyOaIyRaam4AaSWaaSbaaKazba4=baqcLbmacaWGbbWc daWgaaqccasaaKqzadGaaGOmaaqccasabaaajqwaa+Fabaaajaayba qcLbsacqGHciITcaWGHbWcdaWgaaqcKfaG=haajugWaiaadgealmaa BaaajiaibaqcLbmacaaIYaaajiaibeaaaKazba4=beaaaaaajaayca GLOaGaayzkaaWcdaahaaqcKfaG=hqabaqcLbmacaaIYaaaaKqzGeGa aiilaaaa@553D@ (10)

In the case of zero pressure,

T s = T S 2 M S 2 ,T S 2 = a AB 2 6 [ ( 115 c B ) 2 U 0A a A 2 + c B 2 U 0B a B 2 +6 c B 2 U 0 A 1 a A 1 2 +8 c B 2 U 0 A 2 a A 2 2 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGub WcdaWgaaqcbasaaKqzadGaam4CaaqcbasabaqcLbsacqGH9aqpjuaG daWcaaGcbaqcLbsacaWGubGaam4uaKqbaoaaBaaajeaibaqcLbmaca aIYaaaleqaaaGcbaqcLbsacaWGnbGaam4uaSWaaSbaaKqaGeaajugW aiaaikdaaKqaGeqaaaaajugibiaacYcacaWGubGaam4uaSWaaSbaaK qaGeaajugWaiaaikdaaKqaGeqaaKqzGeGaeyypa0tcfa4aaSaaaOqa aKqzGeGaamyyaKqbaoaaDaaaleaajugibiaadgeacaWGcbaajeaiba qcLbmacaaIYaaaaaGcbaqcLbsacaaI2aaaaKqbaoaadmaakeaajuaG daqadaGcbaqcLbsacaaIXaGaeyOeI0IaaGymaiaaiwdacaWGJbqcfa 4aaSbaaKqaGeaajugWaiaadkeaaSqabaaakiaawIcacaGLPaaajuaG daWcaaGcbaqcLbsacqGHciITlmaaCaaajeaibeqaaKqzadGaaGOmaa aajugibiaadwfalmaaBaaajeaibaqcLbmacaaIWaGaamyqaaqcbasa baaakeaajugibiabgkGi2kaadggalmaaDaaajeaibaqcLbmacaWGbb aajeaibaqcLbmacaaIYaaaaaaajugibiabgUcaRiaadogalmaaBaaa jeaibaqcLbmacaWGcbaajeaibeaajuaGdaWcaaGcbaqcLbsacqGHci ITlmaaCaaajeaibeqaaKqzadGaaGOmaaaajugibiaadwfalmaaBaaa jeaibaqcLbmacaaIWaGaamOqaaqcbasabaaakeaajugibiabgkGi2k aadggalmaaDaaajeaibaqcLbmacaWGcbaajeaibaqcLbmacaaIYaaa aaaajugibiabgUcaRiaaiAdacaWGJbWcdaWgaaqcbasaaKqzadGaam Oqaaqcbasabaqcfa4aaSaaaOqaaKqzGeGaeyOaIy7cdaahaaqcbasa beaajugWaiaaikdaaaqcLbsacaWGvbqcfa4aaSbaaSqaaKqzadGaaG imaiaadgeajuaGdaWgaaadbaqcLbsacaaIXaaameqaaaWcbeaaaOqa aKqzGeGaeyOaIyRaamyyaSWaa0baaKqaGeaajugWaiaadgealmaaBa aajiaibaqcLbmacaaIXaaajiaibeaaaKqaGeaajugWaiaaikdaaaaa aKqzGeGaey4kaSIaaGioaiaadogalmaaBaaajeaibaqcLbmacaWGcb aajeaibeaajuaGdaWcaaGcbaqcLbsacqGHciITlmaaCaaajeaibeqa aKqzadGaaGOmaaaajugibiaadwfajuaGdaWgaaqcbasaaKqzadGaaG imaiaadgealmaaBaaajiaibaqcLbmacaaIYaaajiaibeaaaSqabaaa keaajugibiabgkGi2kaadggalmaaDaaajeaibaqcLbmacaWGbbWcda WgaaqccasaaKqzadGaaGOmaaqccasabaaajeaibaqcLbmacaaIYaaa aaaaaOGaay5waiaaw2faaKqzGeGaeyOeI0caaa@BFC8@

( 115 c B )[ a AB 2 k A ( k A a A ) 2 a AB 2 k A a A 2 ] ω A a AB 4 k A c B [ a AB 2 k B ( k B a B ) 2 a AB 2 k B a B 2 ] ω B a AB 4 k B         6 c B [ a AB 2 k A 1 ( k A 1 a A 1 ) 2 a AB 2 k A 1 a A 1 2 ] ω A 1 a AB 4 k A 1 9 c B [ a AB 2 k A 2 ( k A 2 a A 2 ) 2 a AB 2 k A 2 a A 2 2 ] ω A 2 a AB 4 k A 2 , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaajugibi abgkHiTKqbaoaabmaakeaajugibiaaigdacqGHsislcaaIXaGaaGyn aiaadogajuaGdaWgaaqcbasaaKqzadGaamOqaaWcbeaaaOGaayjkai aawMcaaKqbaoaadmaakeaajuaGdaWcaaGcbaqcLbsacaWGHbWcdaWg aaqcbasaaKqzadGaamyqaiaadkeaaKqaGeqaaaGcbaqcLbsacaaIYa Gaam4AaSWaaSbaaKqaGeaajugWaiaadgeaaKqaGeqaaaaajuaGdaqa daGcbaqcfa4aaSaaaOqaaKqzGeGaeyOaIyRaam4AaSWaaSbaaKqaGe aajugWaiaadgeaaKqaGeqaaaGcbaqcLbsacqGHciITcaWGHbqcfa4a aSbaaKqaGeaajugWaiaadgeaaSqabaaaaaGccaGLOaGaayzkaaWcda ahaaqcbasabeaajugWaiaaikdaaaqcLbsacqGHsislcaWGHbWcdaWg aaqcbasaaKqzadGaamyqaiaadkeaaKqaGeqaaKqbaoaalaaakeaaju gibiabgkGi2MqbaoaaCaaaleqajeaibaqcLbmacaaIYaaaaKqzGeGa am4AaSWaaSbaaKqaGeaajugWaiaadgeaaKqaGeqaaaGcbaqcLbsacq GHciITcaWGHbWcdaqhaaqcbasaaKqzadGaamyqaaqcbasaaKqzadGa aGOmaaaaaaaakiaawUfacaGLDbaajuaGdaWcaaGcbaqcLbsacqWIpe cAcqaHjpWDlmaaBaaajeaibaqcLbmacaWGbbaajeaibeaajugibiaa dggalmaaBaaajeaibaqcLbmacaWGbbGaamOqaaqcbasabaaakeaaju gibiaaisdacaWGRbWcdaWgaaqcbasaaKqzadGaamyqaaqcbasabaaa aKqzGeGaeyOeI0Iaam4yaSWaaSbaaKqaGeaajugWaiaadkeaaKqaGe qaaKqbaoaadmaakeaajuaGdaWcaaGcbaqcLbsacaWGHbqcfa4aaSba aKqaGeaajugWaiaadgeacaWGcbaaleqaaaGcbaqcLbsacaaIYaGaam 4AaKqbaoaaBaaajeaibaqcLbmacaWGcbaaleqaaaaajuaGdaqadaGc baqcfa4aaSaaaOqaaKqzGeGaeyOaIyRaam4AaSWaaSbaaKqaGeaaju gWaiaadkeaaKqaGeqaaaGcbaqcLbsacqGHciITcaWGHbWcdaWgaaqc basaaKqzadGaamOqaaqcbasabaaaaaGccaGLOaGaayzkaaWcdaahaa qcbasabeaajugWaiaaikdaaaqcLbsacqGHsislcaWGHbWcdaWgaaqc basaaKqzadGaamyqaiaadkeaaKqaGeqaaKqbaoaalaaakeaajugibi abgkGi2UWaaWbaaKqaGeqabaqcLbmacaaIYaaaaKqzGeGaam4AaSWa aSbaaKqaGeaajugWaiaadkeaaKqaGeqaaaGcbaqcLbsacqGHciITca WGHbWcdaqhaaqcbasaaKqzadGaamOqaaqcbasaaKqzadGaaGOmaaaa aaaakiaawUfacaGLDbaajuaGdaWcaaGcbaqcLbsacqWIpecAcqaHjp WDlmaaBaaajeaibaqcLbmacaWGcbaajeaibeaajugibiaadggalmaa BaaajeaibaqcLbmacaWGbbGaamOqaaqcbasabaaakeaajugibiaais dacaWGRbWcdaWgaaqcbasaaKqzadGaamOqaaqcbasabaaaaKqzGeGa eyOeI0cakeaajugibiaabccacaqGGaGaaeiiaiaabccacaqGGaGaae iiaiaabccacaqGGaGaeyOeI0IaaGOnaiaadogalmaaBaaajeaibaqc LbmacaWGcbaajeaibeaajuaGdaWadaGcbaqcfa4aaSaaaOqaaKqzGe GaamyyaSWaaSbaaKqaGeaajugWaiaadgeacaWGcbaajeaibeaaaOqa aKqzGeGaaGOmaiaadUgalmaaBaaajeaibaqcLbmacaWGbbWcdaWgaa qccasaaKqzadGaaGymaaqccasabaaajeaibeaaaaqcfa4aaeWaaOqa aKqbaoaalaaakeaajugibiabgkGi2kaadUgalmaaBaaajeaibaqcLb macaWGbbWcdaWgaaqccasaaKqzadGaaGymaaqccasabaaajeaibeaa aOqaaKqzGeGaeyOaIyRaamyyaSWaaSbaaKqaGeaajugWaiaadgealm aaBaaajiaibaqcLbmacaaIXaaajiaibeaaaKqaGeqaaaaaaOGaayjk aiaawMcaaSWaaWbaaKqaGeqabaqcLbmacaaIYaaaaKqzGeGaeyOeI0 IaamyyaSWaaSbaaKqaGeaajugWaiaadgeacaWGcbaajeaibeaajuaG daWcaaGcbaqcLbsacqGHciITlmaaCaaajeaibeqaaKqzadGaaGOmaa aajugibiaadUgalmaaBaaajeaibaqcLbmacaWGbbWcdaWgaaqccasa aKqzadGaaGymaaqccasabaaajeaibeaaaOqaaKqzGeGaeyOaIyRaam yyaSWaa0baaKqaGeaajugWaiaadgealmaaBaaajiaibaqcLbmacaaI XaaajiaibeaaaKqaGeaajugWaiaaikdaaaaaaaGccaGLBbGaayzxaa qcfa4aaSaaaOqaaKqzGeGaeS4dHGMaeqyYdCxcfa4aaSbaaKqaGeaa jugWaiaadgealmaaBaaajiaibaqcLbmacaaIXaaajiaibeaaaSqaba qcLbsacaWGHbWcdaWgaaqcbasaaKqzadGaamyqaiaadkeaaKqaGeqa aaGcbaqcLbsacaaI0aGaam4AaSWaaSbaaKqaGeaajugWaiaadgealm aaBaaajiaibaqcLbmacaaIXaaajiaibeaaaKqaGeqaaaaajugibiab gkHiTiaaiMdacaWGJbWcdaWgaaqcbasaaKqzadGaamOqaaqcbasaba qcfa4aamWaaOqaaKqbaoaalaaakeaajugibiaadggalmaaBaaajeai baqcLbmacaWGbbGaamOqaaqcbasabaaakeaajugibiaaikdacaWGRb WcdaWgaaqcbasaaKqzadGaamyqaSWaaSbaaKGaGeaajugWaiaaikda aKGaGeqaaaqcbasabaaaaKqbaoaabmaakeaajuaGdaWcaaGcbaqcLb sacqGHciITcaWGRbWcdaWgaaqcbasaaKqzadGaamyqaSWaaSbaaKGa GeaajugWaiaaikdaaKGaGeqaaaqcbasabaaakeaajugibiabgkGi2k aadggalmaaBaaajeaibaqcLbmacaWGbbWcdaWgaaqccasaaKqzadGa aGOmaaqccasabaaajeaibeaaaaaakiaawIcacaGLPaaalmaaCaaaje aibeqaaKqzadGaaGOmaaaajugibiabgkHiTiaadggalmaaBaaajeai baqcLbmacaWGbbGaamOqaaqcbasabaqcfa4aaSaaaOqaaKqzGeGaey OaIy7cdaahaaqcbasabeaajugWaiaaikdaaaqcLbsacaWGRbqcfa4a aSbaaKqaGeaajugWaiaadgealmaaBaaajiaibaqcLbmacaaIYaaaji aibeaaaSqabaaakeaajugibiabgkGi2kaadggalmaaDaaajeaibaqc LbmacaWGbbWcdaWgaaqccasaaKqzadGaaGOmaaqccasabaaajeaiba qcLbmacaaIYaaaaaaaaOGaay5waiaaw2faaKqbaoaalaaakeaajugi biabl+qiOjabeM8a3TWaaSbaaKqaGeaajugWaiaadgealmaaBaaaji aibaqcLbmacaaIYaaajiaibeaaaKqaGeqaaKqzGeGaamyyaSWaaSba aKqaGeaajugWaiaadgeacaWGcbaajeaibeaaaOqaaKqzGeGaaGinai aadUgalmaaBaaajeaibaqcLbmacaWGbbWcdaWgaaqccasaaKqzadGa aGOmaaqccasabaaajeaibeaaaaqcLbsacaGGSaaaaaa@860F@

M S 2 =( 115 c B ) a AB 2 k Bo 4 k A 2 ( k A a A ) 2 + c B a AB 2 k Bo 4 k B 2 ( k B a B ) 2 +6 c B a AB 2 k Bo 4 k A 1 2 ( k A 1 a A 1 ) 2 +8 c B a AB 2 k Bo 4 k A 2 2 ( k A 2 a A 2 ) 2 , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGnb Gaam4uaKqbaoaaBaaajqwaa+FaaKqzadGaaGOmaaqcbawabaqcLbsa cqGH9aqpjuaGdaqadaqcaawaaKqzGeGaaGymaiabgkHiTiaaigdaca aI1aGaam4yaSWaaSbaaKazba4=baqcLbmacaWGcbaajqwaa+Fabaaa jaaycaGLOaGaayzkaaqcfa4aaSaaaKaaGfaajugibiaadggalmaaDa aajqwaa+FaaKqzadGaamyqaiaadkeaaKazba4=baqcLbmacaaIYaaa aKqzGeGaam4AaKqbaoaaBaaajqwaa+FaaKqzadGaamOqaiaad+gaaK qaGfqaaaqcaawaaKqzGeGaaGinaiaadUgalmaaDaaajqwaa+FaaKqz adGaamyqaaqcKfaG=haajugWaiaaikdaaaaaaKqbaoaabmaajaayba qcfa4aaSaaaKaaGfaajugibiabgkGi2kaadUgalmaaBaaajqwaa+Fa aKqzadGaamyqaaqcKfaG=hqaaaqcaawaaKqzGeGaeyOaIyRaamyyaS WaaSbaaKazba4=baqcLbmacaWGbbaajqwaa+FabaaaaaqcaaMaayjk aiaawMcaaSWaaWbaaKazba4=beqaaKqzadGaaGOmaaaajugibiabgU caRiaadogalmaaBaaajqwaa+FaaKqzadGaamOqaaqcKfaG=hqaaKqb aoaalaaajaaybaqcLbsacaWGHbWcdaqhaaqcKfaG=haajugWaiaadg eacaWGcbaajqwaa+FaaKqzadGaaGOmaaaajugibiaadUgalmaaBaaa jqwaa+FaaKqzadGaamOqaiaad+gaaKazba4=beaaaKaaGfaajugibi aaisdacaWGRbWcdaqhaaqcKfaG=haajugWaiaadkeaaKazba4=baqc LbmacaaIYaaaaaaajuaGdaqadaqcaawaaKqbaoaalaaajaaybaqcLb sacqGHciITcaWGRbWcdaWgaaqcKfaG=haajugWaiaadkeaaKazba4= beaaaKaaGfaajugibiabgkGi2kaadggalmaaBaaajqwaa+FaaKqzad GaamOqaaqcKfaG=hqaaaaaaKaaGjaawIcacaGLPaaalmaaCaaajqwa a+FabeaajugWaiaaikdaaaqcLbsacqGHRaWkcaaI2aGaam4yaKqbao aaBaaajqwaa+FaaKqzadGaamOqaaqcbawabaqcfa4aaSaaaKaaGfaa jugibiaadggalmaaDaaajqwaa+FaaKqzadGaamyqaiaadkeaaKazba 4=baqcLbmacaaIYaaaaKqzGeGaam4AaKqbaoaaBaaajeaybaqcLbsa caWGcbGaam4BaaqcbawabaaajaaybaqcLbsacaaI0aGaam4AaSWaa0 baaKazba4=baqcLbmacaWGbbWcdaWgaaqccasaaKqzadGaaGymaaqc casabaaajqwaa+FaaKqzadGaaGOmaaaaaaqcfa4aaeWaaKaaGfaaju aGdaWcaaqcaawaaKqzGeGaeyOaIyRaam4AaSWaaSbaaKazba4=baqc LbmacaWGbbWcdaWgaaqccasaaKqzadGaaGymaaqccasabaaajqwaa+ FabaaajaaybaqcLbsacqGHciITcaWGHbqcfa4aaSbaaKqaGfaajugW aiaadgeajuaGdaWgaaqccawaaKqzGeGaaGymaaqccawabaaajeaybe aaaaaajaaycaGLOaGaayzkaaWcdaahaaqcKfaG=hqabaqcLbmacaaI YaaaaKqzGeGaey4kaSIaaGioaiaadogalmaaBaaajqwaa+FaaKqzad GaamOqaaqcKfaG=hqaaKqbaoaalaaajaaybaqcLbsacaWGHbWcdaqh aaqcKfaG=haajugWaiaadgeacaWGcbaajqwaa+FaaKqzadGaaGOmaa aajugibiaadUgalmaaBaaajqwaa+FaaKqzadGaamOqaiaad+gaaKaz ba4=beaaaKaaGfaajugibiaaisdacaWGRbWcdaqhaaqcKfaG=haaju gWaiaadgealmaaBaaajiaibaqcLbmacaaIYaaajiaibeaaaKazba4= baqcLbmacaaIYaaaaaaajuaGdaqadaqcaawaaKqbaoaalaaajaayba qcLbsacqGHciITcaWGRbWcdaWgaaqcKfaG=haajugWaiaadgealmaa BaaajiaibaqcLbmacaaIYaaajiaibeaaaKazba4=beaaaKaaGfaaju gibiabgkGi2kaadggalmaaBaaajqwaa+FaaKqzadGaamyqaSWaaSba aKGaGeaajugWaiaaikdaaKGaGeqaaaqcKfaG=hqaaaaaaKaaGjaawI cacaGLPaaalmaaCaaajeaibeqaaKqzadGaaGOmaaaajugibiaacYca aaa@4E3A@ (11)

Because the curve of absolute stability limit for crystalline state is not far from the MS of crystal, the Ts temperature usually is large and x X coth x X 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqi=KI8=feuYtb91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamiEaS WaaSbaaKqaGeaajugWaiaadIfaaSqabaqcLbsaciGGJbGaai4Baiaa cshacaGGObGaamiEaSWaaSbaaKqaGeaajugWaiaadIfaaSqabaqcLb sacqGHijYUcaaIXaaaaa@4641@ at T s MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGub WcdaWgaaqcfasaaKqzadGaam4Caaqcfasabaaaaa@3A0C@ . Therefore,

1 ( 115 c B ) a AB 2 k Bo 4 k A 2 ( k A a A ) 2 + c B a AB 2 k Bo 4 k B 2 ( k B a B ) 2 +6 c B a AB 2 k Bo 4 k A 1 2 ( k A 1 a A 1 ) 2 +8 c B a AB 2 k Bo 4 k A 2 2 ( k A 2 a A 2 ) 2 × MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaaO qaaKqzGeGaaGymaaGcbaqcfa4aaeWaaOqaaKqzGeGaaGymaiabgkHi TiaaigdacaaI1aGaam4yaSWaaSbaaKqaGeaajugWaiaadkeaaKqaGe qaaaGccaGLOaGaayzkaaqcfa4aaSaaaOqaaKqzGeGaamyyaSWaa0ba aKqaGeaajugWaiaadgeacaWGcbaajeaibaqcLbmacaaIYaaaaKqzGe Gaam4AaSWaaSbaaKqaGeaajugWaiaadkeacaWGVbaajeaibeaaaOqa aKqzGeGaaGinaiaadUgalmaaDaaajeaibaqcLbmacaWGbbaajeaiba qcLbmacaaIYaaaaaaajuaGdaqadaGcbaqcfa4aaSaaaOqaaKqzGeGa eyOaIyRaam4AaSWaaSbaaKqaGeaajugWaiaadgeaaKqaGeqaaaGcba qcLbsacqGHciITcaWGHbWcdaWgaaqcbasaaKqzadGaamyqaaqcbasa baaaaaGccaGLOaGaayzkaaWcdaahaaqcbasabeaajugWaiaaikdaaa qcLbsacqGHRaWkcaWGJbWcdaWgaaqcbasaaKqzadGaamOqaaqcbasa baqcfa4aaSaaaOqaaKqzGeGaamyyaSWaa0baaKqaGeaajugWaiaadg eacaWGcbaajeaibaqcLbmacaaIYaaaaKqzGeGaam4AaSWaaSbaaKqa GeaajugWaiaadkeacaWGVbaajeaibeaaaOqaaKqzGeGaaGinaiaadU galmaaDaaajeaibaqcLbmacaWGcbaajeaibaqcLbmacaaIYaaaaaaa juaGdaqadaGcbaqcfa4aaSaaaOqaaKqzGeGaeyOaIyRaam4AaSWaaS baaKqaGeaajugWaiaadkeaaKqaGeqaaaGcbaqcLbsacqGHciITcaWG HbWcdaWgaaqcbasaaKqzadGaamOqaaqcbasabaaaaaGccaGLOaGaay zkaaWcdaahaaqcbasabeaajugWaiaaikdaaaqcLbsacqGHRaWkcaaI 2aGaam4yaSWaaSbaaKqaGeaajugWaiaadkeaaKqaGeqaaKqbaoaala aakeaajugibiaadggalmaaDaaajeaibaqcLbmacaWGbbGaamOqaaqc basaaKqzadGaaGOmaaaajugibiaadUgajuaGdaWgaaqcbasaaKqzad GaamOqaiaad+gaaSqabaaakeaajugibiaaisdacaWGRbWcdaqhaaqc basaaKqzadGaamyqaSWaaSbaaKGaGeaajugWaiaaigdaaKGaGeqaaa qcbasaaKqzadGaaGOmaaaaaaqcfa4aaeWaaOqaaKqbaoaalaaakeaa jugibiabgkGi2kaadUgalmaaBaaajeaibaqcLbmacaWGbbWcdaWgaa qccasaaKqzadGaaGymaaqccasabaaajeaibeaaaOqaaKqzGeGaeyOa IyRaamyyaKqbaoaaBaaajeaibaqcLbmacaWGbbWcdaWgaaqccasaaK qzadGaaGymaaqccasabaaaleqaaaaaaOGaayjkaiaawMcaaSWaaWba aKqaGeqabaqcLbmacaaIYaaaaKqzGeGaey4kaSIaaGioaiaadogalm aaBaaajeaibaqcLbmacaWGcbaajeaibeaajuaGdaWcaaGcbaqcLbsa caWGHbWcdaqhaaqcbasaaKqzadGaamyqaiaadkeaaKqaGeaajugWai aaikdaaaqcLbsacaWGRbWcdaWgaaqcbasaaKqzadGaamOqaiaad+ga aKqaGeqaaaGcbaqcLbsacaaI0aGaam4AaSWaa0baaKqaGeaajugWai aadgealmaaBaaajiaibaqcLbmacaaIYaaajiaibeaaaKqaGeaajugW aiaaikdaaaaaaKqbaoaabmaakeaajuaGdaWcaaGcbaqcLbsacqGHci ITcaWGRbWcdaWgaaqcbasaaKqzadGaamyqaSWaaSbaaKGaGeaajugW aiaaikdaaKGaGeqaaaqcbasabaaakeaajugibiabgkGi2kaadggalm aaBaaajeaibaqcLbmacaWGbbWcdaWgaaqccasaaKqzadGaaGOmaaqc casabaaajeaibeaaaaaakiaawIcacaGLPaaalmaaCaaajeaibeqaaK qzadGaaGOmaaaaaaqcLbsacqGHxdaTaaa@ED69@

×{ 2P V AB + a AB 2 6 [ ( 115 c B ) 2 U 0A a A 2 + c B 2 U 0B a B 2 +6 c B 2 U 0 A 1 a A 1 2 +8 c B 2 U 0 A 2 a A 2 2 ] ( 115 c B )[ a AB 2 k A ( k A a A ) 2 a AB 2 k A a A 2 ] ω A a AB 4 k A c B [ a AB 2 k B ( k B a B ) 2 a AB 2 k B a B 2 ] ω B a AB 4 k B MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaajugibi abgEna0MqbaoaaceaakeaajugibiaaikdacaWGqbGaamOvaSWaaSba aKqaGeaajugWaiaadgeacaWGcbaajeaibeaajugibiabgUcaRKqbao aalaaakeaajugibiaadggalmaaDaaajeaibaqcLbmacaWGbbGaamOq aaqcbasaaKqzadGaaGOmaaaaaOqaaKqzGeGaaGOnaaaajuaGdaWada Gcbaqcfa4aaeWaaOqaaKqzGeGaaGymaiabgkHiTiaaigdacaaI1aGa am4yaSWaaSbaaKqaGeaajugWaiaadkeaaKqaGeqaaaGccaGLOaGaay zkaaqcfa4aaSaaaOqaaKqzGeGaeyOaIy7cdaahaaqcbasabeaajugW aiaaikdaaaqcLbsacaWGvbWcdaWgaaqcbasaaKqzadGaaGimaiaadg eaaKqaGeqaaaGcbaqcLbsacqGHciITcaWGHbWcdaqhaaqcbasaaKqz adGaamyqaaqcbasaaKqzadGaaGOmaaaaaaqcLbsacqGHRaWkcaWGJb WcdaWgaaqcbasaaKqzadGaamOqaaqcbasabaqcfa4aaSaaaOqaaKqz GeGaeyOaIyBcfa4aaWbaaSqabKqaGeaajugWaiaaikdaaaqcLbsaca WGvbWcdaWgaaqcbasaaKqzadGaaGimaiaadkeaaKqaGeqaaaGcbaqc LbsacqGHciITcaWGHbWcdaqhaaqcbasaaKqzadGaamOqaaqcbasaaK qzadGaaGOmaaaaaaqcLbsacqGHRaWkcaaI2aGaam4yaSWaaSbaaKqa GeaajugWaiaadkeaaKqaGeqaaKqbaoaalaaakeaajugibiabgkGi2U WaaWbaaKqaGeqabaqcLbmacaaIYaaaaKqzGeGaamyvaSWaaSbaaKqa GeaajugWaiaaicdacaWGbbWcdaWgaaqccasaaKqzadGaaGymaaqcca sabaaajeaibeaaaOqaaKqzGeGaeyOaIyRaamyyaSWaa0baaKqaGeaa jugWaiaadgealmaaBaaajiaqbaqcLbmacaaIXaaajiaqbeaaaKqaGe aajugWaiaaikdaaaaaaKqzGeGaey4kaSIaaGioaiaadogalmaaBaaa jeaibaqcLbmacaWGcbaajeaibeaajuaGdaWcaaGcbaqcLbsacqGHci ITjuaGdaahaaWcbeqcbasaaKqzadGaaGOmaaaajugibiaadwfalmaa BaaajeaibaqcLbmacaaIWaGaamyqaSWaaSbaaKGaGeaajugWaiaaik daaKGaGeqaaaqcbasabaaakeaajugibiabgkGi2kaadggalmaaDaaa jeaibaqcLbmacaWGbbWcdaWgaaqccasaaKqzadGaaGOmaaqccasaba aajeaibaqcLbmacaaIYaaaaaaaaOGaay5waiaaw2faaaGaay5Eaaqc LbsacqGHsislaOqaaKqzGeGaeyOeI0scfa4aaeWaaOqaaKqzGeGaaG ymaiabgkHiTiaaigdacaaI1aGaam4yaSWaaSbaaKqaGeaajugWaiaa dkeaaKqaGeqaaaGccaGLOaGaayzkaaqcfa4aamWaaOqaaKqbaoaala aakeaajugibiaadggalmaaBaaajeaibaqcLbmacaWGbbGaamOqaaqc basabaaakeaajugibiaaikdacaWGRbWcdaWgaaqcbasaaKqzadGaam yqaaqcbasabaaaaKqbaoaabmaakeaajuaGdaWcaaGcbaqcLbsacqGH ciITcaWGRbWcdaWgaaqcbasaaKqzadGaamyqaaqcbasabaaakeaaju gibiabgkGi2kaadggalmaaBaaajeaibaqcLbmacaWGbbaajeaibeaa aaaakiaawIcacaGLPaaalmaaCaaajeaibeqaaKqzadGaaGOmaaaaju gibiabgkHiTiaadggalmaaBaaajeaibaqcLbmacaWGbbGaamOqaaqc basabaqcfa4aaSaaaOqaaKqzGeGaeyOaIy7cdaahaaqcbasabeaaju gWaiaaikdaaaqcLbsacaWGRbWcdaWgaaqcbasaaKqzadGaamyqaaqc basabaaakeaajugibiabgkGi2kaadggalmaaDaaajeaibaqcLbmaca WGbbaajeaibaqcLbmacaaIYaaaaaaaaOGaay5waiaaw2faaKqbaoaa laaakeaajugibiabl+qiOjabeM8a3TWaaSbaaKqaGeaajugWaiaadg eaaKqaGeqaaKqzGeGaamyyaSWaaSbaaKqaGeaajugWaiaadgeacaWG cbaajeaibeaaaOqaaKqzGeGaaGinaiaadUgalmaaBaaajeaibaqcLb macaWGbbaajeaibeaaaaqcLbsacqGHsislcaWGJbWcdaWgaaqcbasa aKqzadGaamOqaaqcbasabaqcfa4aamWaaOqaaKqbaoaalaaakeaaju gibiaadggalmaaBaaajeaibaqcLbmacaWGbbGaamOqaaqcbasabaaa keaajugibiaaikdacaWGRbWcdaWgaaqcbasaaKqzadGaamOqaaqcba sabaaaaKqbaoaabmaakeaajuaGdaWcaaGcbaqcLbsacqGHciITcaWG RbWcdaWgaaqcbasaaKqzadGaamOqaaqcbasabaaakeaajugibiabgk Gi2kaadggalmaaBaaajeaibaqcLbmacaWGcbaajeaibeaaaaaakiaa wIcacaGLPaaalmaaCaaajeaibeqaaKqzadGaaGOmaaaajugibiabgk HiTiaadggalmaaBaaajeaibaqcLbmacaWGbbGaamOqaaqcbasabaqc fa4aaSaaaOqaaKqzGeGaeyOaIy7cdaahaaqcbasabeaajugWaiaaik daaaqcLbsacaWGRbqcfa4aaSbaaKqaGeaajugWaiaadkeaaSqabaaa keaajugibiabgkGi2kaadggalmaaDaaajeaibaqcLbmacaWGcbaaje aibaqcLbmacaaIYaaaaaaaaOGaay5waiaaw2faaKqbaoaalaaakeaa jugibiabl+qiOjabeM8a3TWaaSbaaKqaGeaajugWaiaadkeaaKqaGe qaaKqzGeGaamyyaSWaaSbaaKqaGeaajugWaiaadgeacaWGcbaajeai beaaaOqaaKqzGeGaaGinaiaadUgalmaaBaaajeaibaqcLbmacaWGcb aajeaibeaaaaqcLbsacqGHsislaaaa@4B8E@

6 c B [ a AB 2 k A 1 ( k A 1 a A 1 ) 2 a AB 2 k A 1 a A 1 2 ] ω A 1 a AB 4 k A 1 8 c B [ a AB 2 k A 2 ( k A 2 a A 2 ) 2 a AB 2 k A 2 a A 2 2 ] ω A 2 a AB 4 k A 2 }+ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqGHsi slcaaI2aGaam4yaSWaaSbaaKqaGeaajugWaiaadkeaaKqaGeqaaKqb aoaadmaakeaajuaGdaWcaaGcbaqcLbsacaWGHbWcdaWgaaqcbasaaK qzadGaamyqaiaadkeaaKqaGeqaaaGcbaqcLbsacaaIYaGaam4AaKqb aoaaBaaaleaajugWaiaadgeajuaGdaWgaaadbaqcLbsacaaIXaaame qaaaWcbeaaaaqcfa4aaeWaaOqaaKqbaoaalaaakeaajugibiabgkGi 2kaadUgajuaGdaWgaaqcbasaaKqzadGaamyqaSWaaSbaaKGaGeaaju gWaiaaigdaaKGaGeqaaaWcbeaaaOqaaKqzGeGaeyOaIyRaamyyaSWa aSbaaKqaGeaajugWaiaadgealmaaBaaajiaibaqcLbmacaaIXaaaji aibeaaaKqaGeqaaaaaaOGaayjkaiaawMcaaSWaaWbaaKqaGeqabaqc LbmacaaIYaaaaKqzGeGaeyOeI0IaamyyaSWaaSbaaKqaGeaajugWai aadgeacaWGcbaajeaibeaajuaGdaWcaaGcbaqcLbsacqGHciITlmaa CaaajeaibeqaaKqzadGaaGOmaaaajugibiaadUgalmaaBaaajeaiba qcLbmacaWGbbWcdaWgaaqccasaaKqzadGaaGymaaqccasabaaajeai beaaaOqaaKqzGeGaeyOaIyRaamyyaSWaa0baaKqaGeaajugWaiaadg ealmaaBaaajiaibaqcLbmacaaIXaaajiaibeaaaKqaGeaajugWaiaa ikdaaaaaaaGccaGLBbGaayzxaaqcfa4aaSaaaOqaaKqzGeGaeS4dHG MaeqyYdC3cdaWgaaqcbasaaKqzadGaamyqaSWaaSbaaKGaGeaajugW aiaaigdaaKGaGeqaaaqcbasabaqcLbsacaWGHbWcdaWgaaqcbasaaK qzadGaamyqaiaadkeaaKqaGeqaaaGcbaqcLbsacaaI0aGaam4AaSWa aSbaaKqaGeaajugWaiaadgealmaaBaaajiaibaqcLbmacaaIXaaaji aibeaaaKqaGeqaaaaajugibiabgkHiTiaaiIdacaWGJbWcdaWgaaqc basaaKqzadGaamOqaaqcbasabaqcfa4aaiGaaOqaaKqbaoaadmaake aajuaGdaWcaaGcbaqcLbsacaWGHbWcdaWgaaqcbasaaKqzadGaamyq aiaadkeaaKqaGeqaaaGcbaqcLbsacaaIYaGaam4AaSWaaSbaaKqaGe aajugWaiaadgealmaaBaaajiaibaqcLbmacaaIYaaajiaibeaaaKqa GeqaaaaajuaGdaqadaGcbaqcfa4aaSaaaOqaaKqzGeGaeyOaIyRaam 4AaSWaaSbaaKqaGeaajugWaiaadgealmaaBaaajiaibaqcLbmacaaI YaaajiaibeaaaKqaGeqaaaGcbaqcLbsacqGHciITcaWGHbWcdaWgaa qcbasaaKqzadGaamyqaSWaaSbaaKGaGeaajugWaiaaikdaaKGaGeqa aaqcbasabaaaaaGccaGLOaGaayzkaaWcdaahaaqcbasabeaajugWai aaikdaaaqcLbsacqGHsislcaWGHbWcdaWgaaqcbasaaKqzadGaamyq aiaadkeaaKqaGeqaaKqbaoaalaaakeaajugibiabgkGi2UWaaWbaaK qaGeqabaqcLbmacaaIYaaaaKqzGeGaam4AaSWaaSbaaKqaGeaajugW aiaadgealmaaBaaajiaibaqcLbmacaaIYaaajiaibeaaaKqaGeqaaa GcbaqcLbsacqGHciITcaWGHbWcdaqhaaqcbasaaKqzadGaamyqaSWa aSbaaKGaGeaajugWaiaaikdaaKGaGeqaaaqcbasaaKqzadGaaGOmaa aaaaaakiaawUfacaGLDbaajuaGdaWcaaGcbaqcLbsacqWIpecAcqaH jpWDlmaaBaaajeaibaqcLbmacaWGbbWcdaWgaaqccasaaKqzadGaaG OmaaqccasabaaajeaibeaajugibiaadggalmaaBaaajeaibaqcLbma caWGbbGaamOqaaqcbasabaaakeaajugibiaaisdacaWGRbWcdaWgaa qcbasaaKqzadGaamyqaSWaaSbaaKGaGeaajugWaiaaikdaaKGaGeqa aaqcbasabaaaaaGccaGL9baajugibiabgUcaRaaa@EADC@

+ 2 V AB k Bo a AB [ 1 k A k A a A ( 115 c B )+ 1 k B k B a B c B + 1 k A 1 k A 1 a A 1 6 c B + 1 k A 2 k A 2 a A 2 8 c B ] × MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqGHRa WkjuaGdaWcaaGcbaqcLbsacaaIYaGaamOvaKqbaoaaBaaajeaibaqc LbmacaWGbbGaamOqaaWcbeaaaOqaaKqzGeGaam4AaSWaaSbaaKqaGe aajugWaiaadkeacaWGVbaajeaibeaajugibiaadggalmaaBaaajeai baqcLbmacaWGbbGaamOqaaqcbasabaqcfa4aamqaaOqaaKqbaoaala aakeaajugibiaaigdaaOqaaKqzGeGaam4AaSWaaSbaaKqaGeaajugW aiaadgeaaKqaGeqaaaaaaOGaay5waaqcfa4aaSaaaOqaaKqzGeGaey OaIyRaam4AaSWaaSbaaKqaGeaajugWaiaadgeaaKqaGeqaaaGcbaqc LbsacqGHciITcaWGHbWcdaWgaaqcbasaaKqzadGaamyqaaqcbasaba aaaKqbaoaabmaakeaajugibiaaigdacqGHsislcaaIXaGaaGynaiaa dogalmaaBaaajeaibaqcLbmacaWGcbaajeaibeaaaOGaayjkaiaawM caaKqzGeGaey4kaSscfa4aaSaaaOqaaKqzGeGaaGymaaGcbaqcLbsa caWGRbqcfa4aaSbaaKqaGeaajugWaiaadkeaaSqabaaaaKqbaoaala aakeaajugibiabgkGi2kaadUgalmaaBaaajeaibaqcLbmacaWGcbaa jeaibeaaaOqaaKqzGeGaeyOaIyRaamyyaKqbaoaaBaaajeaibaqcLb macaWGcbaaleqaaaaajugibiaadogalmaaBaaajeaibaqcLbmacaWG cbaajeaibeaajugibiabgUcaRKqbaoaalaaakeaajugibiaaigdaaO qaaKqzGeGaam4AaSWaaSbaaKqaGeaajugWaiaadgealmaaBaaajiai baqcLbmacaaIXaaajiaibeaaaKqaGeqaaaaajuaGdaWcaaGcbaqcLb sacqGHciITcaWGRbWcdaWgaaqcbasaaKqzadGaamyqaSWaaSbaaKGa GeaajugWaiaaigdaaKGaGeqaaaqcbasabaaakeaajugibiabgkGi2k aadggalmaaBaaajeaibaqcLbmacaWGbbWcdaWgaaqccasaaKqzadGa aGymaaqccasabaaajeaibeaaaaqcLbsacaaI2aGaam4yaSWaaSbaaK qaGeaajugWaiaadkeaaKqaGeqaaKqzGeGaey4kaSscfa4aaSaaaOqa aKqzGeGaaGymaaGcbaqcLbsacaWGRbWcdaWgaaqcbasaaKqzadGaam yqaSWaaSbaaKGaGeaajugWaiaaikdaaKGaGeqaaaqcbasabaaaaKqb aoaadiaakeaajuaGdaWcaaGcbaqcLbsacqGHciITcaWGRbWcdaWgaa qcbasaaKqzadGaamyqaSWaaSbaaKGaGeaajugWaiaaikdaaKGaGeqa aaqcbasabaaakeaajugibiabgkGi2kaadggalmaaBaaajeaibaqcLb macaWGbbWcdaWgaaqccasaaKqzadGaaGOmaaqccasabaaajeaibeaa aaqcLbsacaaI4aGaam4yaSWaaSbaaKqaGeaajugWaiaadkeaaKqaGe qaaaGccaGLDbaaaaqcLbsacqGHxdaTaaa@BC9F@

×{ P+ a AB 6 V AB [ ( 115 c B ) U 0A a A + c B U 0B a B +6 c B U 0 A 1 a A 1 +8 c B U 0 A 2 a A 2 ] }=0. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqGHxd aTjuaGdaGadaGcbaqcLbsacaWGqbGaey4kaSscfa4aaSaaaOqaaKqz GeGaamyyaSWaaSbaaKqaGeaajugWaiaadgeacaWGcbaajeaibeaaaO qaaKqzGeGaaGOnaiaadAfalmaaBaaajeaibaqcLbmacaWGbbGaamOq aaqcbasabaaaaKqbaoaadmaakeaajuaGdaqadaGcbaqcLbsacaaIXa GaeyOeI0IaaGymaiaaiwdacaWGJbWcdaWgaaqcbasaaKqzadGaamOq aaqcbasabaaakiaawIcacaGLPaaajuaGdaWcaaGcbaqcLbsacqGHci ITcaWGvbWcdaWgaaqcbasaaKqzadGaaGimaiaadgeaaKqaGeqaaaGc baqcLbsacqGHciITcaWGHbWcdaWgaaqcbasaaKqzadGaamyqaaqcba sabaaaaKqzGeGaey4kaSIaam4yaSWaaSbaaKqaGeaajugWaiaadkea aKqaGeqaaKqbaoaalaaakeaajugibiabgkGi2kaadwfalmaaBaaaje aibaqcLbmacaaIWaGaamOqaaqcbasabaaakeaajugibiabgkGi2kaa dggalmaaBaaajeaibaqcLbmacaWGcbaajeaibeaaaaqcLbsacqGHRa WkcaaI2aGaam4yaSWaaSbaaKqaGeaajugWaiaadkeaaKqaGeqaaKqb aoaalaaakeaajugibiabgkGi2kaadwfalmaaBaaajeaibaqcLbmaca aIWaGaamyqaSWaaSbaaKGaGeaajugWaiaaigdaaKGaGeqaaaqcbasa baaakeaajugibiabgkGi2kaadggajuaGdaWgaaqcbasaaKqzadGaam yqaSWaaSbaaKGaGeaajugWaiaaigdaaKGaGeqaaaWcbeaaaaqcLbsa cqGHRaWkcaaI4aGaam4yaSWaaSbaaKqaGeaajugWaiaadkeaaKqaGe qaaKqbaoaalaaakeaajugibiabgkGi2kaadwfalmaaBaaajeaibaqc LbmacaaIWaGaamyqaSWaaSbaaKGaGeaajugWaiaaikdaaKGaGeqaaa qcbasabaaakeaajugibiabgkGi2kaadggalmaaBaaajeaibaqcLbma caWGbbWcdaWgaaqccasaaKqzadGaaGOmaaqccasabaaajeaibeaaaa aakiaawUfacaGLDbaaaiaawUhacaGL9baajugibiabg2da9iaaicda caGGUaaaaa@A138@  (12)

That is the equation for the curve of absolute stability limit for crystalline state. Therefore, the pressure is a function of the mean nearest neighbor distance

P = P ( a A B ) . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqi=KI8=feuYtb91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamiuai abg2da9iaadcfacaGGOaGaamyyaWWaaSbaaKqaGeaajugWaiaadgea caWGcbaajeaibeaajugibiaacMcacaGGUaaaaa@4191@ (13)

Temperature Ts (0) at zero pressure has the form

T s (0)= a AB 18 γ G T k Bo [ ( 115 c B ) U 0A a A + c B U 0B a B +6 c B U 0 A 1 a A 1 +8 c B U 0 A 2 a A 2 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqi=KI8=feuYtb91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamivaW WaaSbaaKqaGeaajugWaiaadohaaKqaGeqaaKqzGeGaaiikaiaaicda caGGPaGaeyypa0ZcdaWcaaGcbaqcLbsacaWGHbWcdaWgaaqcbasaaK qzadGaamyqaiaadkeaaSqabaaakeaajugibiaaigdacaaI4aGaeq4S dCgddaqhaaqcbasaaKqzadGaam4raaqcbasaaKqzadGaamivaaaaju gibiaadUgammaaBaaajeaibaqcLbmacaWGcbGaam4Baaqcbasabaaa aSWaamWaaOqaaSWaaeWaaOqaaKqzGeGaaGymaiabgkHiTiaaigdaca aI1aGaam4yaWWaaSbaaKqaGeaajugWaiaadkeaaKqaGeqaaaGccaGL OaGaayzkaaWcdaWcaaGcbaqcLbsacqGHciITcaWGvbaddaWgaaqcba saaKqzadGaaGimaiaadgeaaKqaGeqaaaGcbaqcLbsacqGHciITcaWG HbWcdaWgaaqcbasaaKqzadGaamyqaaWcbeaaaaqcLbsacqGHRaWkca WGJbaddaWgaaqcbasaaKqzadGaamOqaaqcbasabaWcdaWcaaGcbaqc LbsacqGHciITcaWGvbWcdaWgaaqcbasaaKqzadGaaGimaiaadkeaaS qabaaakeaajugibiabgkGi2kaadggammaaBaaajeaibaqcLbmacaWG cbaajeaibeaaaaqcLbsacqGHRaWkcaaI2aGaam4yaWWaaSbaaKqaGe aajugWaiaadkeaaKqaGeqaaSWaaSaaaOqaaKqzGeGaeyOaIyRaamyv aWWaaSbaaKqaGeaajugWaiaaicdacaWGbbaddaWgaaqccasaaKqzad GaaGymaaqccasabaaajeaibeaaaOqaaKqzGeGaeyOaIyRaamyyaWWa aSbaaKqaGeaajugWaiaadgeammaaBaaajiaibaqcLbmacaaIXaaaji aibeaaaKqaGeqaaaaajugibiabgUcaRiaaiIdacaWGJbaddaWgaaqc basaaKqzadGaamOqaaqcbasabaWcdaWcaaGcbaqcLbsacqGHciITca WGvbaddaWgaaqcbasaaKqzadGaaGimaiaadgeammaaBaaajiaibaqc LbmacaaIYaaajiaibeaaaKqaGeqaaaGcbaqcLbsacqGHciITcaWGHb addaWgaaqcbasaaKqzadGaamyqaWWaaSbaaKGaGeaajugWaiaaikda aKGaGeqaaaqcbasabaaaaaGccaGLBbGaayzxaaaaaa@A34B@  (14)

where the parameters a AB , U 0X a X , γ G T MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqi=KI8=feuYtb91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamyyaW WaaSbaaKqaGeaajugWaiaadgeacaWGcbaajeaibeaajugibiaacYca caaMc8UaaGPaVlaaykW7caaMc8+cdaWcaaGcbaqcLbsacqGHciITca WGvbWcdaWgaaqcbasaaKqzadGaaGimaiaadIfaaSqabaaakeaajugi biabgkGi2kaadggammaaBaaajeaibaqcLbmacaWGybaajeaibeaaaa qcLbsacaGGSaGaaGPaVlaaykW7cqaHZoWzmmaaDaaajeaibaqcLbma caWGhbaajeaibaqcLbmacaWGubaaaKqzGeGaaGPaVdaa@5BD6@ are determined at T s (0) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqi=KI8=feuYtb91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamivaW WaaSbaaKqaGeaajugWaiaadohaaKqaGeqaaKqzGeGaaiikaiaaicda caGGPaaaaa@3E47@ . Temperature Ts at pressure P has the form

T s T s (0)+ V AB P 3 k Bo ( γ G T ) 2 ( γ G T ) a AB T s . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqi=KI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamivaW WaaSbaaKqaGeaajugWaiaadohaaKqaGeqaaKqzGeGaeyisISRaamiv aSWaaSbaaKqaGeaajugWaiaadohaaSqabaqcLbsacaGGOaGaaGimai aacMcacqGHRaWklmaalaaakeaajugibiaadAfalmaaBaaajeaibaqc LbmacaWGbbGaamOqaaWcbeaajugibiaadcfaaOqaaKqzGeGaaG4mai aadUgammaaBaaajeaibaqcLbmacaWGcbGaam4BaaqcbasabaWcdaqa daGcbaqcLbsacqaHZoWzmmaaDaaajeaibaqcLbmacaWGhbaajeaiba qcLbmacaWGubaaaaGccaGLOaGaayzkaaWcdaahaaqabKqaGeaajugW aiaaikdaaaaaaSWaaeWaaOqaaSWaaSaaaOqaaKqzGeGaeyOaIyRaeq 4SdC2cdaWgaaqcbasaaKqzadGaam4raaWcbeaaaOqaaKqzGeGaeyOa IyRaamivaaaaaOGaayjkaiaawMcaaWWaaSbaaKqaGeaajugWaiaadg gammaaBaaajiaibaqcLbmacaWGbbGaamOqaaqccasabaaajeaibeaa jugibiaadsfammaaBaaajeaibaqcLbmacaWGZbaajeaibeaajugibi aac6caaaa@6FAC@  (15)

Here, k Bo MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGRb WcdaWgaaqcbasaaKqzadGaamOqaiaad+gaaKqaGeqaaaaa@3ADE@ is the Boltzmann constant, V ABC ,  γ G T ,  γ G T /T MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGwb WcdaWgaaqcbasaaKqzadGaamyqaiaadkeacaWGdbaajeaibeaajugi biaacYcacaqGGaGaeq4SdC2cdaqhaaqcbasaaKqzadGaam4raaqcba saaKqzadGaamivaaaajugibiaacYcacaqGGaGaeyOaIyRaeq4SdC2c daqhaaqcbasaaKqzadGaam4raaqcbasaaKqzadGaamivaaaajugibi aac+cacqGHciITcaWGubaaaa@5060@ dare determined at T s MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGub WcdaWgaaqcbasaaKqzadGaam4Caaqcbasabaaaaa@3A04@ . Approximately, T m T s MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGub qcfa4aaSbaaKqaGeaajugWaiaad2gaaSqabaqcLbsacqGHijYUcaWG ubWcdaWgaaqcbasaaKqzadGaam4Caaqcbasabaaaaa@4021@ . In order to solve equation (15), we can use the approximate iteration method. In the first approximate iteration,

T s1 T s (0)+ V AB ( T s (0))P 3 k Bo γ G ( T s (0)) . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqi=KI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamivaS WaaSbaaKqaGeaajugWaiaadohacaaIXaaaleqaaKqzGeGaeyisISRa amivaSWaaSbaaKqaGeaajugWaiaadohaaSqabaqcLbsacaGGOaGaaG imaiaacMcacqGHRaWklmaalaaakeaajugibiaadAfammaaBaaajeai baqcLbmacaWGbbGaamOqaaqcbasabaqcLbsacaGGOaGaamivaWWaaS baaKqaGeaajugWaiaadohaaKqaGeqaaKqzGeGaaiikaiaaicdacaGG PaGaaiykaiaadcfaaOqaaKqzGeGaaG4maiaadUgalmaaBaaajeaiba qcLbmacaWGcbGaam4BaaWcbeaajugibiabeo7aNTWaaSbaaKqaGeaa jugWaiaadEeaaSqabaqcLbsacaGGOaGaamivaWWaaSbaaKqaGeaaju gWaiaadohaaKqaGeqaaKqzGeGaaiikaiaaicdacaGGPaGaaiykaaaa caGGUaaaaa@64AE@  (16)

Here T s (0) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqi=KI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamivaW WaaSbaaKqaGeaajugWaiaadohaaKqaGeqaaKqzGeGaaiikaiaaicda caGGPaaaaa@3D0A@ is the temperature of absolute stability limit for crystalline state at pressure P in the first approximate iteration of equation (15). Substituting Ts1 into equation (15), we obtain the better approximate value Ts2 of Ts at pressure P in the second approximate iteration

T s2 T s (0)+ V AB ( T s1 )P 3 k Bo γ G ( T s1 ) V AB ( T s1 )P 3 k Bo γ G 2 ( T s1 ) ( γ G T ) a AB T s1 . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqi=KI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamivaW WaaSbaaKqaGeaajugWaiaadohacaaIYaaajeaibeaajugibiabgIKi 7kaadsfammaaBaaajeaibaqcLbmacaWGZbaajeaibeaajugibiaacI cacaaIWaGaaiykaiabgUcaRSWaaSaaaOqaaKqzGeGaamOvaSWaaSba aKqaGeaajugWaiaadgeacaWGcbaaleqaaKqzGeGaaiikaiaadsfamm aaBaaajeaibaqcLbmacaWGZbGaaGymaaqcbasabaqcLbsacaGGPaGa amiuaaGcbaqcLbsacaaIZaGaam4AaWWaaSbaaKqaGeaajugWaiaadk eacaWGVbaajeaibeaajugibiabeo7aNXWaaSbaaKqaGeaajugWaiaa dEeaaKqaGeqaaKqzGeGaaiikaiaadsfammaaBaaajeaibaqcLbmaca WGZbGaaGymaaqcbasabaqcLbsacaGGPaaaaiabgkHiTSWaaSaaaOqa aKqzGeGaamOvaWWaaSbaaKqaGeaajugWaiaadgeacaWGcbaajeaibe aajugibiaacIcacaWGubaddaWgaaqcbasaaKqzadGaam4Caiaaigda aKqaGeqaaKqzGeGaaiykaiaadcfaaOqaaKqzGeGaaG4maiaadUgamm aaBaaajeaibaqcLbmacaWGcbGaam4BaaqcbasabaqcLbsacqaHZoWz mmaaDaaajeaibaqcLbmacaWGhbaajeaibaqcLbmacaaIYaaaaKqzGe GaaiikaiaadsfammaaBaaajeaibaqcLbmacaWGZbGaaGymaaqcbasa baqcLbsacaGGPaaaaSWaaeWaaOqaaSWaaSaaaOqaaKqzGeGaeyOaIy Raeq4SdCgddaWgaaqcbasaaKqzadGaam4raaqcbasabaaakeaajugi biabgkGi2kaadsfaaaaakiaawIcacaGLPaaammaaBaaajeaibaqcLb macaWGHbaddaWgaaqccasaaKqzadGaamyqaiaadkeaaKGaGeqaaaqc basabaqcLbsacaWGubaddaWgaaqcbasaaKqzadGaam4CaiaaigdaaK qaGeqaaKqzGeGaaiOlaaaa@9884@ (17)

Analogouslty, we can obtain the better approximate values T s3 , T s4 ,... MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqi=KI8=feuYtb91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamivaW WaaSbaaKqaGeaajugWaiaadohacaaIZaaajeaibeaajugibiaacYca caaMc8UaaGPaVlaadsfammaaBaaajeaibaqcLbmacaWGZbGaaGinaa qcbasabaqcLbsacaGGSaGaaiOlaiaac6cacaGGUaaaaa@484B@  of Ts at pressure P in the thierd, fourth, etc. approximate iterations. These approximations are applyed at low pressures.

In the case of high pressure, the MT of alloy at pressure P is calculated by

T m (P)= T m (0) B 0 1 B 0 G(0) . G(P) ( B 0 + B 0 P) 1 B 0 , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqi=KI8=feuYtb91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamivaW WaaSbaaKqaGeaajugWaiaad2gaaKqaGeqaaKqzGeGaaiikaiaadcfa caGGPaGaeyypa0ZcdaWcaaGcbaqcLbsacaWGubWcdaWgaaqcbasaaK qzadGaamyBaaWcbeaajugibiaacIcacaaIWaGaaiykaiaadkealmaa BaaajeaibaqcLbmacaaIWaaaleqaamaaCaaabeqcbasaaWWaaSaaaK qaGeaajugWaiaaigdaaKqaGeaajugWaiqadkeagaqbaWWaaSbaaKGa GeaajugWaiaaicdaaKGaGeqaaaaaaaaakeaajugibiaadEeacaGGOa GaaGimaiaacMcaaaGaaiOlaSWaaSaaaOqaaKqzGeGaam4raiaacIca caWGqbGaaiykaaGcbaqcLbsacaGGOaGaamOqaSWaaSbaaKqaGeaaju gWaiaaicdaaSqabaqcLbsacqGHRaWkceWGcbGbauaammaaBaaajeai baqcLbmacaaIWaaajeaibeaajugibiaadcfacaGGPaWcdaahaaqabK qaGeaammaalaaajeaibaqcLbmacaaIXaaajeaibaqcLbmaceWGcbGb auaammaaBaaajiaibaqcLbmacaaIWaaajiaibeaaaaaaaaaajugibi aacYcaaaa@6B79@  (18)

where T m (P) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGub WcdaWgaaqcbasaaKqzadGaamyBaaqcbasabaqcLbsacaGGOaGaamiu aiaacMcaaaa@3CBB@ and T m (0) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGub WcdaWgaaqcbasaaKqzadGaamyBaaqcbasabaqcLbsacaGGOaGaaGim aiaacMcaaaa@3CA0@ respectively are the MT at pressure P and zero pressure, G(P) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGhb GaaiikaiaadcfacaGGPaaaaa@397F@ and G(0) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGhb GaaiikaiaaicdacaGGPaaaaa@3964@ respectively are the rigidity bulk modulus at pressure P and zero pressure, B 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGcb WcdaWgaaqcbasaaKqzadGaaGimaaqcbasabaaaaa@39B4@ is the isothermal elastic modulus at zero pressure, B 0 = ( d B T dP ) P=0 ,  B T = B T (P) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqi=KI8=feuYtb91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGabmOqay aafaaddaWgaaqcbasaaKqzadGaaGimaaqcbasabaqcLbsacqGH9aqp lmaabmaakeaalmaalaaakeaajugibiaadsgacaWGcbaddaWgaaqcba saaKqzadGaamivaaqcbasabaaakeaajugibiaadsgacaWGqbaaaaGc caGLOaGaayzkaaaddaWgaaqcbasaaKqzadGaamiuaiabg2da9iaaic daaKqaGeqaaKqzGeGaaiilaiaabccacaWGcbaddaWgaaqcbasaaKqz adGaamivaaqcbasabaqcLbsacqGH9aqpcaWGcbaddaWgaaqcbasaaK qzadGaamivaaqcbasabaqcLbsacaGGOaGaamiuaiaacMcaaaa@56F7@ is the isothermal elastic modulus at pressure.

Numerical results for alloys AuSi and AgSi
For alloys AuSi and AgSi, we use the n–m pair potential

φ(r)= D nm [ m ( r 0 r ) n n ( r 0 r ) m ], MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeqOXdO MaaiikaiaadkhacaGGPaGaeyypa0tcfa4aaSaaaOqaaKqzGeGaamir aaGcbaqcLbsacaWGUbGaeyOeI0IaamyBaaaajuaGdaWadaGcbaqcLb sacaWGTbqcfa4aaeWaaOqaaKqbaoaalaaakeaajugibiaadkhalmaa BaaajeaibaqcLbmacaaIWaaajeaibeaaaOqaaKqzGeGaamOCaaaaaO GaayjkaiaawMcaaSWaaWbaaKqaGeqabaqcLbmacaWGUbaaaKqzGeGa eyOeI0IaamOBaKqbaoaabmaakeaajuaGdaWcaaGcbaqcLbsacaWGYb qcfa4aaSbaaKqaGeaajugWaiaaicdaaSqabaaakeaajugibiaadkha aaaakiaawIcacaGLPaaalmaaCaaajeaibeqaaKqzadGaamyBaaaaaO Gaay5waiaaw2faaKqzGeGaaiilaaaa@5EB0@  (19)

where potential parameters are given in Table 1.8

Material

m

n

D[ 10 16 erg ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGeb qcfa4aamWaaOqaaKqzGeGaaGymaiaaicdajuaGdaahaaWcbeqcbasa aKqzadGaeyOeI0IaaGymaiaaiAdaaaqcLbsacaqGLbGaaeOCaiaabE gaaOGaay5waiaaw2faaaaa@43B7@

width="130"

r 0 [ 10 10 m ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGYb WcdaWgaaqcbasaaKqzadGaaGimaaqcbasabaqcfa4aamWaaOqaaKqz GeGaaGymaiaaicdalmaaCaaajeaibeqaaKqzadGaeyOeI0IaaGymai aaicdaaaqcLbsacaqGTbaakiaawUfacaGLDbaaaaa@43E2@

Au
Ag

5.5
5.5

10.5
11.5

6462.540
4589.328

2.8751
2.8760

Si

6.0

12.0

45128.340

2.2950

Table 1 Potential parameters m,n,D, r 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGTb Gaaiilaiaad6gacaGGSaGaamiraiaacYcacaWGYbWcdaWgaaqcbasa aKqzadGaaGimaaqcbasabaaaaa@3EA2@ of materials

Considering the interaction between atoms Au(Ag) and Si in the above mentioned alloys, we use the potential (19) but calculating approximately D ¯ = D Au(Ag) D Si , r ¯ 0 = r 0Au(Ag) r 0Si MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaceWGeb GbaebacqGH9aqpjuaGdaGcaaGcbaqcLbsacaWGebWcdaWgaaqcbasa aKqzadGaaeyqaiaabwhacaqGOaGaaeyqaiaabEgacaqGPaaajeaibe aajugibiaadsealmaaBaaajeaibaqcLbmacaqGtbGaaeyAaaqcbasa baaaleqaaKqzGeGaaiilaiqadkhagaqeaKqbaoaaBaaajeaibaqcLb macaaIWaaaleqaaKqzGeGaeyypa0tcfa4aaOaaaOqaaKqzGeGaamOC aSWaaSbaaKqaGeaajugWaiaaicdacaqGbbGaaeyDaiaabIcacaqGbb Gaae4zaiaabMcaaKqaGeqaaKqzGeGaamOCaKqbaoaaBaaajeaibaqc LbmacaaIWaGaae4uaiaabMgaaSqabaaabeaaaaa@5BC8@ . Parameters m ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaceWGTb Gbaebaaaa@378F@ and n ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaceWGUb Gbaebaaaa@3790@ are taken empirically.

At 0.1 MPa, Au has a FCC structure with a=4, 0785.10 10 m MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGHb Gaeyypa0JaaGinaiaacYcacaaIWaGaaG4naiaaiIdacaaI1aGaaiOl aiaaigdacaaIWaWcdaahaaqcbasabeaajugWaiabgkHiTiaaigdaca aIWaaaaKqzGeGaaeyBaaaa@4468@ at 300K and the melting point at 1337 K. The melting curve of Au is determined up to 1673 K and 6.5 GPa with the slope dT/dP = 60 K/GPa9,10 and up to 1923 K and 12 Gpa.11

At 0.1 MPa, Ag has a FCC structure with a=4, 0862.10 10 m MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGHb Gaeyypa0JaaGinaiaacYcacaaIWaGaaGioaiaaiAdacaaIYaGaaiOl aiaaigdacaaIWaWcdaahaaqcbasabeaajugWaiabgkHiTiaaigdaca aIWaaaaKqzGeGaaeyBaaaa@4464@ at 300K and the melting point at 1235 K. The melting curve of Au is determined up to 1563 K and 6.5 GPa with the slope dT/dP = 60 K/Gpa.9

Our numerical results are summarized in tables from Table 2 to Table 3 and illustrated in figures from Figure 1 to Figure 2. The concentration of interstitial atoms changes from 0 to 5% and the pressure changes from 0 to 10 GPa for Au and from 0 to 6 GPa for Ag.

Figure 1 Dependence of melting temperature on pressure and concentration of interstitial atoms for alloy AuSi.

Figure 2 Dependence of melting temperature on pressure and concentration of interstitial atoms for alloy AgSi.

x(%)

P(GPa)

0

2

4

6

8

10

0

SMM

1337.33

1397.27

1454.33

1514.75

1568.48

1629.22

 

EXPT [31]

1337.33

1432.90

1530.15

1628,63

1727.10

1824.36

 

EXPT [44]

1340.30

1426.80

1515.77

1599,80

1683.84

1770.34

1

SMM

1382.99

1443.15

1500.10

1560,32

1613.50

1673.78

3

SMM

1505.78

1567.89

1625.75

1686,56

1739.30

1799.29

5

SMM

1701.80

1771.20

1834.58

1900,23

1955.89

2018.94

Table 2 Dependence of melting temperature on pressure and concentration of interstitial atoms for alloy Au–xSi

x(%)

P(GPa)

0

2

4

6

0

SMM

1234.93

1322.75

1408.68

1492.95

 

EXPT[31]

1234.93

1334.42

1438.19

1538.81

1

SMM

1257.82

1349.70

1438.34

1524.38

3

SMM

1319.47

1423.32

1520.45

1612.43

5

SMM

1418.12

1544.47

1659.14

1764.81

Table 3 Dependence of melting temperature on pressure and concentration of interstitial atoms for alloy Ag–xSi

According to our numerical results for alloy AuSi at the same concentration of interstitial atoms Si when pressure increases, the melting temperature increases. For example at c Si =5% MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGJb WcdaWgaaqcbasaaKqzadGaae4uaiaabMgaaKqaGeqaaKqzGeGaeyyp a0JaaGynaiaacwcaaaa@3DDA@ when P increases from 0 to 10 GPa, T melting MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGub WcdaWgaaqcbasaaKqzadGaaeyBaiaabwgacaqGSbGaaeiDaiaabMga caqGUbGaae4zaaqcbasabaaaaa@3F91@ of alloy AuSi increases from 1701.8 K to 2018.94 K. At the same pressure when the concentration of interstitial atoms Si increases, the melting temperature increases. For example at P=10 GPa, when c Si MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGJb WcdaWgaaqcbasaaKqzadGaae4uaiaabMgaaKqaGeqaaaaa@3ADD@ increases from 0 to 5%, T melting MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGub WcdaWgaaqcbasaaKqzadGaaeyBaiaabwgacaqGSbGaaeiDaiaabMga caqGUbGaae4zaaqcbasabaaaaa@3F91@ of alloy AuSi increases from 11629.22 K to 2018.94K.

According to our numerical results for alloy AgSi at the same concentration of interstitial atoms Si when pressure increases, the melting temperature increases. For example at c Si =5% MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGJb WcdaWgaaqcbasaaKqzadGaae4uaiaabMgaaKqaGeqaaKqzGeGaeyyp a0JaaGynaiaacwcaaaa@3DDA@ when P increases from 0 to 6 GPa, T melting MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGub WcdaWgaaqcbasaaKqzadGaaeyBaiaabwgacaqGSbGaaeiDaiaabMga caqGUbGaae4zaaqcbasabaaaaa@3F91@ of alloy AgSi increases from 1418.12 K to 1764.81 K. At the same pressure when the concentration of interstitial atoms Si increases, the melting temperature increases. For example at P=6 GPa, when c Si MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGJb WcdaWgaaqcbasaaKqzadGaae4uaiaabMgaaKqaGeqaaaaa@3ADD@ increases from 0 to 5%, T melting MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGub WcdaWgaaqcbasaaKqzadGaaeyBaiaabwgacaqGSbGaaeiDaiaabMga caqGUbGaae4zaaqcbasabaaaaa@3F91@ of alloy AuSi increases from 1492.95 K to 1764.81K (Table 3).

At zero concentration of interstitial atoms Si, the melting temperature of alloys AuSii and AgSi respectively becomes the melting temperature of metals Au and Ag. The melting temperature of substitution alloys AuCu and AgCu is smaller than the melting temperature of metals Au and Ag, respectively. The dependences of melting temperature on pressure and concentration ò interstirtial atoms Si for alloy AuSi and AgSi are shown in (Figure 1) (Figure 2).

The calculated results for the melting temperature of metals Au and Ag are in goog agreement with the experimental data12 (deviation is about sereral percents) (Table 4) (Table 5).

Metal

Au

Ag

T m (K) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGub qcfa4aaSbaaKqaGeaajugWaiaad2gaaSqabaqcLbsacaGGOaGaae4s aiaacMcaaaa@3D18@
–SMM
–EXPT[12]

 

1400
1336

 

1190
1234

Table 4 The melting temperatures of metals Au and Ag at zero pressure

P(GPa)

Method

1

2

3

4

5

6

Au

SMM
EXPT[12]

1476
1383

1543
1455

1602
1513

1651
1575

1693
1625

1728
1667

Ag

SMM
EXPT[12]

1282
1300

1367
1353

1443
1403

1513
1462

1575
1509

1631
1549

Table 5 The melting temperatures of metals Au and Ag under pressure

Conclusion

In aid of the SMM, we derive the analytic expressions of the temperature of absolute stability limit for crystalline state and the melting temperature together with the melting curve of the binary interstitial alloy depending on pressure and concentration of interstitial atoms. In limit case, we obtain the melting theory of main metal with FCC structure. The theoretical results are numerically applied for alloys AuSi and AgSi.

Acknowledgements

None.

Conflict of interest

Author declares there is no conflict of interest.

References

  1. Belonoshko AB, Simak SI, Kochetov AE, et al. High–pressure melting of molybdenum. Physical Review Letters. 2004;92(19).
  2. Burakovsky L, Preston DL, Silbar RR. Analysis of dislocation mechanism for melting of elements: Pressure dependence. Journal of Applied Physics. 2000;88(11):6294–6331.
  3. Kumari M, Kumari K, Dass N. Onthe melting law at high pressure. Physica Status Solidi (a). 1987;99(1):22–26.
  4. Tang N, Hung VV. Investigation of the thermodynamic properties of anharmonic crystals by the momentum method. I. General results for face–centred cubic crystals. Physica Status Solidi (B). 1998;149(2):511–519.
  5. Hung VV, Masuda–Jindo K. Application of statistical moment method to thermodynamic properties of metals at high pressures. Journal of the Physical Society of Japan. 2000;69.
  6. Hoc NQ, Tinh BD, Tuan LD, et al. Elastic deformation of binary and ternary interstitial alloys with FCC structure and zero pressure: Dependence on temperature, concentration of substitution atoms and concentration of interstitial atoms. Mathematical and Physical Sciences. 2016;61(7):47–57.
  7. Hoc NQ, Vinh DQ, Viet LH. Thermodynamic property of binary interstitial alloy with FCC structure: Dependence on temperature and concentration of interstitial atoms. Vietnam: Report at the 41th National Conference on Theoretical Physics (NCTP–41); 2016.
  8. Magomedov MN. J Fiz Khimic. 1987;61:1003.
  9. Akella J, Kennedy GC. Melting of Au, Ag and Cu–proposal for a new high–pressure calibration scale. Journal of Geophysical Research. 1971;76(20):4969–4977.
  10. Mirwald PW, Kennedy GC. The melting curve of Au, Ag and Cu to 60–kbar pressure: A reinvestigation. Journal of Geophysical Research. 1979;84(B12):6750–6756.
  11. Sumita T, Kato M, Yoneda A. The thermal analysis in an MA–8 type apparatus: the melting of gold at 12 GPa. The Review of High Pressure Science and Technology. 1998;7:254–256.
  12. Tonkov EY, Ponyatovsky EG. Phase transformations of elements under high pressure. USA: CRC Press; 2005.
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