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1 Jurnal f the Krean Ceramic Sciety Vl. 47, N. 6, pp. 491~497, DOI: /KCERS Temperature Dependence n Elastic Cnstant f SiC Ceramics Jng In Im, Byung-W Park, H-Yng Shin, and Jng-H Kim Simulatin Center, Business Supprt Divisin, KICET, Seul , Krea (Received September 24, 2010; Revised Octber 19, 2010; Accepted Octber 25, 2010) SiC k Á Á y Á½ y w»» l ( ; ; ) ABSTRACT In this paper, we emplyed the classical mlecular dynamics simulatins using Tersff s ptential t calculate the elastic cnstants f the silicn carbide (SiC) crystal at high temperature. The elastic cnstants f the SiC crystal were calculated based n the stressstrain characteristics, which were drawn by the simulatin using LAMMPS sftware. At the same time, the elastic cnstants f the SiC ceramics were measured at different temperatures by impulse excitatin testing (IET) methd. Based n the simulated stress-strain results, the SiC crystal shwed the elastic defrmatin characteristics at the lw temperature regin, while a slight plastic defrmatin behavir was bserved at high strain ver 1,000 C temperature. The elastic cnstants f the SiC crystal were changed frm abut 475 GPa t 425 GPa by increasing the temperature frm RT t 1,250 C. When cmpared t the experimental values f the SiC ceramics, the simulatin results, which are unable t btain by experiments, are fund t be very useful t predict the stress-strain behavirs and the elastic cnstant f the ceramics at high temperature. Key wrds : Silicn carbide (SiC), Mlecular dynamics simulatins, Impulse excitatin test, Stress-strain characteristics, Elastic cnstants 1. w ƒ š w š w p š š, ü ü w œ, ü š t, ƒ l t t w š. 1) t Al 2 O 3, ZrO 2 y SiC, Si 3 N 4, B 4 C, AlN, TiC y w ƒ w. š t w» w,, q wì w y, k, - x p» v w. k d w x(tensile test) 3 y 4 š x(bending test). x w, r Crrespnding authr : Jng-In IM jngin@kicet.re.kr Tel : Fax : š x w. š xw, w q d w - x š» k d w». š x k d ƒ w š w w k d w» w r v w. š k q y d w d w wš. w q w d q k y w k p sƒw» w ƒ. ful š y y w, w w w ƒ y š š, k w w w» w š. w 2-5)» dw, š t 491

2 492 Á Á y Á½ y Table 1. Tersff s Ptential Parameter f the SiC Crystal (ref.9) C Si A(eV) B(eV) λ(å -1 ) µ(å -1 ) β n c d h X c si = Fig. 1. Unit cell structure f the SiC crystal. q. w w w š SiC - k wš w. w Tersff ptential w SiC wš, š w v LAMMPS S/W w l 1,250 C¾ - p w w k y wš, SiC k d e w wš w. 2. š 2.1. SiC w sl SiC (unit cell) Fig. 1 ùkü.» Si ü ƒ ewš š, C Si» ewš. w SiC ü C-C- C, C-C-Si, C-Si-Si, Si-Si-Si, C-Si-C, Si-C-Si xk š, w w s l ƒ v w. Tersff š sl w SiC w w sl w. 6-9) Tersff sl k, g, 4 œ w w w w» w sl w œ y. Tersff y x w (empirical bnd-rder) w sl w» w. (E) (1)». 1 E = i E i = 2 -- V i j ij (1) R C C S C C R Si Si S Si Si R C Si S C Si » V ij V ij = f c ( r ij )[ a ij f R ( r ij ) + b ij f A ( r ij )], ƒƒ. f R f A f c ( r) = Aexp( λ 1 r) ( r) = Aexp( λ 2 r) π ( r) = ( r R) sin R D< D 0 r< R D r< R+ D r> R+ D w b ij w- (bnd-frmatin energy) ùkü (many-bdy) w. b ij 1 β n n = ( + ξ ij ) 1 2n ξ ij = ( ) m f ( )g( θ ijk ) exp[ λ 3 ( rij r ik ) m ] r k ( i, j) c ij c 2 g( θ) = c d 2 d 2 + ( h csθ) 2» ξ ij t (crdinatin number) ùkü, g( θ) ƒ w. Table 1 SiC w Tersff sl ù kü. SiC w y w Tang w 9) w r ƒ w k d SiC k x š x d š, q (2) (3) (4) (5) (6) (7) w wz

3 SiC k 493 T 1 = ( v v 2 ) t t -- 4 L L ( v v 2 t ) 4 L ( v v 2 t ) 2 + L (9) ùkü. k (shear mdulus: G) Fig. 2(b) p (trsinal vibratin) œ q ( f t ) l 4Lmf 2 t B G = ( 1+ A) b t (10) w.» A B x ƒƒ Fig. 2. Simple gemetry f the elastic cnstant measurement using the IET: (a) flexural vibratin mde and (b) trsinal vibratin mde. d ƒ w. k š w» œ q ƒ š š, œ q r x, k w. š x r w œ q l k w ƒ w. w r ƒ (Impulse Excitatin Testing methd: IET) Fig. 2 ùkü š x r ƒwš, j s œ q d w k w w p w ƒœ y w k d w 10-12). k (E) Fig. 2(a) { (flexural vibratin) œ q ( f f ) f 2 m f E = L T b 1 t 3 tx.» L, b, t ƒƒ r ¼, s, Ì wš, m r w. w x (T-) s (Pissn s rati: v) r x ƒ š, (8) b b b -- 3 A = b b B = b t t b t t t b b b (11) (12) ùkü. w k (E) (G), s (v) G = E/[2 (1 +v)] (13) ƒ r { p w œ q d w k d ƒ w. 3. x SiC k w w» w SiC w w w. SiC Fig. 1 ùk ü w x, y, z w ƒƒ 5 w. SiC w Fig. 3 1,000 š, w ¼ ƒ Å x. SiC ü C-C-C, C-C-Si, C-Si-Si, Si- Si-Si, C-Si-C, Si-C-Si xk š, w w sl Table 1 ùkü Tersff sl w w. 47«6y(2010)

4 임종인 박병우 신호용 김종호 494 Fig. 3. Simulatin mdels f the SiC crystal. Fig. 5. Fig. 4. The MD simulatin prcedures fr the elastic cnstant f the SiC crystal. 또한 SiC 결정이 무한이 반복되는 상황을 가정하여 x, y, z축과 수직한 단면에 주기적인 경계조건을 적용하였다. Fig. 4는 SiC 결정의 탄성계수 분석을 위한 분자동역학 한국세라믹학회지 Measured acustic signals f the SiC ceramics at RT: (a) time dmain and (b) frequency dmain. 시뮬레이션 절차를 나타낸 것이다. 여기에서 알 수 있듯 이, SiC 결정에 온도를 인가하여 안정화 시킨 후, 변위를 인가하여 포텐셜 에너지 및 응력을 계산한다. 다양한 변 위에 대해 반복 계산한 응력 값으로부터 응력-변형 특성 을 구하고, 다양한 온도구간에서 SiC 결정의 탄성률을 분 석할 수 있다. SiC 결정의 탄성계수를 계산하기 위해서는 결정에 걸 리는 응력 및 변위를 계산하여야 하고, 이는 온도에 많은 영향을 받는다. 이를 위하여 SiC 결정 내부의 원자에 온 도를 인가하고, 원자들의 열적 진동(thermal vibratin)을 안정화 시킨 후, 원자들을 강제적으로 변형시켜 내부 원 자의 포텐셜 에너지 및 응력을 계산하였다. SiC 결정의 온도는 상온부터 1,250 C까지 변화시켰고, 각각의 온도구 간에서 인가되는 변위는 0.02 간격으로 증가시키면서 SiC 결정 내부의 각 원자에 걸리는 응력을 해석하였다. 그리 고 SiC 결정에 인가된 변위의 높은 에너지로 인해 외각 의 특정 원자가 단위 격자 위치에서 많이 벗어날 경우,

5 SiC 세라믹스 탄성률의 온도 의존성 Fig Defrmed shapes f the SiC crystal simulated with MD at 1000 C: (a) befre and (b) after applied 0.15 strain. Fig. 8. The simulated elastic cnstant f the SiC crystal with the temperature. SiC 결정이 인가된 변위에 의해 1,000 C에서 변형된 모 양을 Fig. 6에 나타낸 것이다. 여기서 알 수 있듯이, 고온 에서 원자의 열적 진동에 의해 SiC 결정이 안정화 되었 다 하더라도 인가된 변위 높은 에너지로 인해 최외각 특 정 원자는 단위 격자 위치에서 상당히 이탈한 것을 확인 할 수 있다. 특히 SiC 결정의 온도가 1,000 C 이상 고온 으로 상승할 경우, 원자의 열적 진동과 인가된 변위 에너 지가 합해져 이러한 원자 이탈이 발생했다고 판단된다. 그러므로 해석의 정밀도 향상을 위해 단위 격자 위치에 서 이탈한 원자를 제외하고, 구성 원자들 만으로 내부 단 위 격자의 평균 응력을 계산하여 SiC 결정의 응력-변위 특성 및 탄성률을 분석하였다. 분자동역학을 이용하여 상온부터 1,250 C 온도까지 SiC 결정의 응력-변위 특성을 분석한 결과를 Fig. 7에 나타내 었다. 상온의 SiC 결정의 경우, 압축 변위에 의해 원자간 간격이 가까워지면서 전체 에너지가 증가함으로 선형적 으로 응력이 증가한다. 그리고 0.2 이상의 압축 변위가 인 가되면, 전체 응력은 일정한 값에 수렴하다 파괴되는 특 성을 나타내었다. 또한 결정의 온도가 500 C까지 증가하 여도 SiC 결정은 상온과 유사한 응력-변위 특성을 보이 고 있다. 그러나 결정의 온도가 1,000 C 이상 증가하게 되 면, SiC 결정은 0.1 정도의 변위에서 탄성-소성 한계인 항 복 응력 (yield stress)과 유사한 변곡점을 나타난 것을 확 인할 수 있다. 그리고 변위가 계속 증가하여 0.2 변위에 서 도달하게 되면, 탄성변형 (elastic defrmatin)이 아닌 소성변형 (plastic defrmatin) 변화되는 응력-변위 특성을 보이고 있다. 이 결과는 지금까지 알려진 SiC 등 세라믹 스는 탄성변형 후 파괴된다는 결과와 전혀 다르지만, SiC 결정도 고온에서 소성변형 할 가능성도 있음을 암시하는 것으로 추가적인 분석이 필요하다고 판단된다. 분자동역학을 이용해 해석한 응력-변위 특성으로부터 온도 증가에 따른 SiC 결정의 탄성률 변화를 Fig. 8에 나 Fig. 7. Stress-strain characteristics f the SiC crystal at varius temperatures. 해석 정밀도 향상을 위해 이 원자를 제외한 구성 원자만 을 이용해 응력-변위 특성 및 탄성률을 해석하였다. SiC 세라믹스의 탄성률은 길이 50 mm, 폭 12 mm, 두께 5 mm 크기의 직육면체 시편으로 동적 탄성률 측정기 (RFDA HTVP 1600, IMCE사)를 이용하여 측정하였다. Fig. 5는 SiC세라믹스에 임펄스를 인가하여 측정한 진동 신호를 나타낸 것이다. 이렇게 측정한 시편의 공진주파수 와 식(8)부터 식(13)의 관계식을 이용하여 SiC 세라믹스 의 탄성률을 계산할 수 있다. 그리고 시편을 상온부터 1,500 C 까지 3 C/min의 속도로 승온 및 냉각하면서 반복 시험을 실시하고, 온도 변화에 따른 SiC 세라믹스의 탄성 률을 측정하였다. 4. 결과 및 고찰 SiC 결정의 온도를 상온부터 1,500 C까지 변화시키면서 응력-변위(stress-strain) 특성 및 탄성률을 분석하고, 그 결 과를 Fig. 6부터 Fig. 8까지 나타내었다. 제 47 권 제6호(2010)

6 496 Á Á y Á½ y Fig. 9. The measured elastic cnstants f the SiC ceramics with the temperature during the heating and the cling stage. Fig. 10. Cmparisn the simulated results with the measured elastic cnstant f the SiC material. kü.», SiC k 475 GPa š, ƒ 1,250 C¾ ƒ w 425 GPa ¾ w ùkû. IET y SiC k d w w SiC k w ƒƒ Fig. 9 Fig. 10 ùkü. Fig. 9 SiC 440 GPa k wš š, ƒ 1,500 C ¾ ƒw 410 GPa ¾ x w p š. SiC þƒw w, 700~800 C k w ƒw š w y w. w Fig. 10, l 750 C SiC ƒ 40 GPa ƒ š, w xk w š. 750 C ƒ ƒw SiC k w w 1,250 C SiC w k q. l SiC k IET d w SiC w 40 GPa sƒ. ƒ w SiC w (grain bundary)»œ, ƒ,, š w. ù w ƒ v w. mw SiC l 1,250 C¾ ƒ ƒw k 475 GPa l 425 GPa ¾ yw y w š, SiC k d w y w. w š x y w» ƒ w - p k p w mw dw. 5. SiC, w š, š w š š w. w w y SiC k w. w v LAMMPS S/W w l 1,250 C¾ SiC - (stress-strain)p k y w w š, SiC k d e w., SiC k 1,250 C ƒw 475 GPa l 425 GPa yw y w š, SiC k d w y w. š š x y w» ƒ w - p k p w mw dw. REFERENCES 1. Z.H. Huang, D.C. Jia, Y. Zhu, and Y.J. Wang, Effect f a New Additive n Mechanical Prperties f Ht-Pressed Silicn Carbide Ceramics, Mat. Res. Bull., (2002). 2. H. Ya, L. Ouyang, and W. Y. Ching, Ab Initi Calculatin f Elastic Cnstants f Ceramic Crystal, J. Am. Ceram. Sc., 90 [10] (2007). 3. S. H. Garfalini and W. Lu, Mlecular Dynamics Simulatins f Calcium Silicate Intergranular Films between Silicn Nitride Crystals, J. Am. Ceram. Sc., 86 [10] (2003). w wz

7 SiC k D. A. Littn and S. H. Garfalini, Mlecular Dynamics Simulatins f Calcium Aluminsilicate Intergranular Films n (0001) Al 2 O 3 Facets, J. Am. Ceram. Sc., 83 [9] (2000). 5. P. K. Schelling and S. R. Phillpt, Mechanism f Thermal Transprt in Zircnia and Yttria-Stabilized Zircnia by Mlecular-Dynamics Simulatin, J. Am. Ceram. Sc., 84 [12] (2001). 6. J. Tersff, Mdelling Slid-state Chemistry: Inter-Atmic Ptentials fr Multi-Cmpnent Systems, Phys. Rev. B., (1989). 7. J. Tersff, Empirical Interatmic Ptential fr Carbn, with Applicatins t Amrphus Carbn, Phys. Rev. Lett., (1988). 8. J. Tersff, New Empirical Mdel fr the Structural Prperties f Silicn, Phys. Rev. Lett., (1986). 9. M. Tang and S. Yip, Atmistic Simulatin f Thermmechanical Pperties f β-sic, Phys. Rev. B, (1995). 10. ASTM Standard C 1259, Test Methd fr Yung s Mdulus, Shear Mdulus, and Pissn s Rati fr Advanced Ceramics by Impulse Excitatin Of Vibratin, Annual Bk f Astm Standards, 15 [01] (1999). 11. J.D. Lrd and R. Mrrell, Measurement Gd Practice Guide N. 98: Elastic Mdulus Measurement, pp , Natinal Physical Lab. Reprt., G. Rebben, et al., High-temperature Stiffness and Damping Measurements t Mnitr the Glassy Intergranular Phase in Liquid-Phase-Sintered Silicn Carbide, J. Am. Ceram. Sc., 88 [8] (2005). 47«6y(2010)

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