本研究之目的為使用Mori-Tanaka 微觀力學模型模擬內含物與母材於不同極化方 向之雙相壓電壓磁顆粒複合材料之等效材料性質，尋找出最佳磁電電壓係數之極化方 向配置，並使用COMSOL Multiphysics 有限元素分析軟體驗證微觀力學理論之正確性。
本章將闡述研究結果及相關討論與未來展望。
5-1 結論
1. 驗證單位立方晶格與 Mori-Tanaka 模式之差異性
微觀力學模型Mori-Tanaka 模式無法模擬內含物之排列，在此選用鈮酸鋰 LiNbO3 (3m 對稱性)、鈦酸鋇 BaTiO3 (6mm 對稱性)作為壓電材料並置入鈷鐵氧 CoFeO4 (6mm 對稱性)，利用 COMSOL Multiphysics 有限元素法分析軟體模擬簡 單立方(SC)、體心立方(BCC)與面心立方(FCC)之雙相顆粒複合材料。結果顯示 Mori-Tanaka 模式最適合模擬鈮酸鋰 LiNbO3置入鈷鐵氧CoFeO4之BCC 與 FCC 晶 格結構、鈦酸鋇 BaTiO3置入鈷鐵氧CoFeO4則為FCC 晶格結構。
2. 圓球顆粒內含物之對稱性
圓球顆粒內含物不具有方向性，只要包含圓心之平面皆為顆粒之對稱面，複 合材料極化方向[100]/[100]如同將[001]/[001]對全域座標之 x2旋轉 90°。由第三章 之結果觀察到[001]/[001]之 αE,11* 為[100]/[100]之 αE,33* ；由第四章結果顯示最佳化之 αE,11* 、 αE,22* 與 αE,33* 數值完全相同。
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3. 磁電耦合效應於最佳化前後之比較
LNO[001]/CFO[001]之 αE,11* = -8.0548 V/cmOe，經過旋轉極化方向後提升至 -9.5948 V/cmOe，增加了 1.19 倍；CFO[001]/LNO[001]之 αE,11* = ‐2.1023 V/cmOe，
經過最佳化後提升至-2.6711 V/cmOe，增加了 1.27 倍。
BTO/CFO 與 CFO/BTO 經過旋轉壓電材料之極化方向與壓磁材料之磁軸後，
無法提升磁電電壓係數，兩複合材料於[001]/[001]之極化方向即呈現最佳之磁電 電壓係數。
4. 磁電電壓係數與壓電材料壓電係數之關係
LNO/CFO 與 CFO/LNO 可以藉由旋轉極化方向得到最佳磁電電壓係數，以最 佳磁電電壓係數 αE,11* 而言，兩複合材料之CFO 極化方向維持[001]，LNO 之極化 方向則很有趣地與其壓電係數 e15最佳之極化方向相當接近。LNO 壓電係數 e15與 兩複合材料 αE,11* 在改變LNO 極化方向會出現相同的趨勢，因此可以藉由最佳壓 電係數 e15之極化方向當作參考，尋找出最佳磁電電壓係數對應之極化方向。
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5-2 未來展望
1. 使用非對稱性較高之材料
本次研究之壓電材料LNO 為 3m 對稱結構，旋轉極化方向提升壓電係數 e15相 當有限，而BTO 為 6m 對稱結構，完全不能藉由旋轉極化方向提升壓電係數 e15， 因此壓電材料選擇非對稱性較高之材料，在旋轉極化方向下有可能大幅地增加壓電 係數 e15，進而提升 αE,11* 。
2. 不同形狀內含物對磁電耦合效應之影響
複合材料之內含物不僅限於圓球顆粒，還有層狀與橢球狀顆粒之內含物，其中 橢球內含物於母材中可以旋轉其長軸方向，也可以改變內含物之極化方向，這兩種 組合皆可以影響磁電耦合效應，因此相當值得探討。
3. 非完美交界面對磁電耦合效應之影響
本研究採用完美交界面之假設，而實際上製作出來之複合材料，母材與內含 物之間存在非完美交界面，造成應變、位移、電勢能與磁勢能不連續之現象，進 而影響磁電耦合效應，因此對非完美介面之探討可以讓理論越接近實際材料之現 象。
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附錄 A BaTiO
3置入CoFeO
4之壓電壓磁複合材料
本附錄所使用之材料為鈦酸鋇(BaTiO3,BTO)與鈷鐵氧(CoFeO4,CFO)，BTO 為現在 廣為使用之材料，其與CFO 相同具有 6mm 之晶格對稱結構，且同為陶瓷材料。壓電 材料BTO 與 LNO 之結構對稱性不同，所以研究結果可以與第三張及第四章相互比較。
由材料對稱性可知材料配置[001]/[001]與[001]/[100]之等效材料性質對全域座標 x2順 時針旋轉90°即分別轉為[100]/[100]與[100]/[001]之等效材料性質，因此本章僅討論四 種配置形式(表 A-1)。BTO[001]與 BTO[100]之材料性質 L 如表 A-2，CFO[001]與 CFO[100]之材料性質 L 如表 3-2。
表A-1 BTO-CFO 複合材料配置形式
案例 內含物 極化方向 母材 極化方向
Ⅰ LNO [001]
CFO [001]
Ⅱ [001] [100]
Ⅲ CFO [001]
LNO [001]
Ⅳ [001] [100]
表A-2 BTO 於[001]與[100]之材料性質 L (a) BTO[001]之材料性質 L
1.66E+11 7.7E+10 7.8E+10 0 0 0 0 0 -4.4 0 0 0
7.7E+10 1.66E+11 7.8E+10 0 0 0 0 0 -4.4 0 0 0
7.8E+10 7.8E+10 1.62E+11 0 0 0 0 0 18.6 0 0 0
0 0 0 4.3E+10 0 0 0 11.6 0 0 0 0
0 0 0 0 4.3E+10 0 11.6 0 0 0 0 0
0 0 0 0 0 4.45E+10 0 0 0 0 0 0
0 0 0 0 11.6 0 1.12E-08 0 0 0 0 0
0 0 0 11.6 0 0 0 1.12E-08 0 0 0 0
-4.4 -4.4 18.6 0 0 0 0 0 1.26E-08 0 0 0
0 0 0 0 0 0 0 0 0 5.00E-06 0 0
0 0 0 0 0 0 0 0 0 0 5.00E-06 0
0 0 0 0 0 0 0 0 0 0 0 1.00E-05
(b) BTO[100]之材料性質 L
1.62E+11 7.8E+10 7.8E+10 0 0 0 18.6 0 0 0 0 0
7.8E+10 1.66E+11 7.7E+10 0 0 0 -4.4 0 0 0 0 0
7.8E+10 7.7E+10 1.66E+11 0 0 0 -4.4 0 0 0 0 0
0 0 0 4.45E+10 0 0 0 0 0 0 0 0
0 0 0 0 4.3E+10 0 0 0 11.6 0 0 0
0 0 0 0 0 4.3E+10 0 11.6 0 0 0 0
18.6 -4.4 -4.4 0 0 0 1.26E-08 0 0 0 0 0
0 0 0 0 0 11.6 0 1.12E-08 0 0 0 0
0 0 0 0 11.6 0 0 0 1.12E-08 0 0 0
0 0 0 0 0 0 0 0 0 1.00E-05 0 0
0 0 0 0 0 0 0 0 0 0 5.00E-06 0
0 0 0 0 0 0 0 0 0 0 0 5.00E-06
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A-1 BTO[001]置入 CFO[001]
此模型運用Mori-Tanaka 模式與有限元素法探討 BTO[001]/CFO[001]形式之壓電 壓磁複合材料行為，其中BTO[001]與 CFO[001]之性質為原始之性質，材料性質如表 A-2a 與表 3-2b。
使用Mori-Tanaka 模式可以模擬無內含物存在到母材完全被內含物佔據之狀況。
因此，設定體積比為0 時，複合材料之等效材料性質為 CFO[001]之性質；體積比為 1 時，複合材料之性質為BTO[001]之性質，由此狀況來判斷 Mori-Tanaka 模式與程式碼 為可靠的。
由觀察得知(圖 A-1)，Mori-Tanaka 模式與有限元素之數據於非耦合的等效係數都 相當吻合，驗證Mori-Tanaka 模式之可靠性。等效壓電係數 e*與等效介電常數 κ*於體 積比0.9 之前呈現單調遞增且相當緩慢，體積比 0.9 後，其數據忽然快速增加轉向 BTO 之性質；等效彈性係數 C*、等效壓磁係數 q*與等效磁導率 μ*皆隨體積比增加呈現單 調遞增或遞減之現象。
等效磁電係數 λij*之數據隨體積比呈現類似拋物線的路徑(圖 A-1f)，在體積比 0.8 前和緩地成長，體積比0.8 後迅速地歸零。比較 Mori-Tanaka 模式與有限元素法之數 據，在體積比0.3 之前兩數據相當吻合，而體積比 0.3 之後誤差增加，不過變化之趨勢 一致。此外， λ11* 於體積比0.85 有最佳值3.7641×10-10 Ns/VC；λ33* 於體積比0.85 有最
佳值1.4652×10-10 Ns/VC。以上複合後之等效材料性質，因兩材料皆屬於 6mm 對稱性，
所以複合後也符合6mm 對稱性。
最重要之磁電電壓係數αE,ij* 中(圖 A-2)，其數據隨體積比呈現類似拋物線之路徑，
比較Mori-Tanaka 模式與有限元素分析，其狀況與等效磁電係數 λ*相同，兩數據在體 積比0.3 之後誤差開始增加，不過變化之趨勢一致。此外， E,11* 於體積比0.31 時有最 佳值-1.2652 V/cmOe； E,33* 於體積比0.33 有最佳值-0.4294 V/cmOe。
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Volume Fraction of Inclusion C* (N/m2 )
Volume Fraction of Inclusion
e* (C/m2)
Volume Fraction of Inclusion
* (N/V2 )
Volume Fraction of Inclusion
q* (N/Am)
Volume Fraction of Inclusion
* (Ns2 /C2 )
Volume Fraction of Inclusion
* (Ns/VC)
M-T FEM
121
圖A-2 磁電電壓係數αE*與體積比f 之關係
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -1.4
-1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2
*E,11
*E,22
*E,33
BTO[001]/CFO[001]
Volume Fraction of Inclusion
* E (V/cmOe)
M-T FEM
122
A-2 BTO[001]置入 CFO[100]
本節討論內含物為BTO，其極化方向為[001]，母材為 CFO，其極化方向為[100]
之複合材料，在此使用尤拉角將CFO 之材料性質對局域座標之 x2'軸順時針旋轉90°， 材料性質如表A-2a 與表 3-2d。
使用Mori-Tanaka 模式模擬複合材料中，在體積比為 0 時，得到等效材料性質為 CFO[100]之材料性質；在體積比為 1 時，得到等效材料性質為 BTO[001]之材料性質，
藉此可以判定Mori-Tanaka 模式與程式碼之準確性。
由觀察得知(圖 A-3)，Mori-Tanaka 模式與有限元素分析於非耦合項之數據皆相當 吻合。等效壓電係數與等效介電常數於體積比0.9 之前成長相當緩慢，體積比 0.9 後曲 率相當大地轉向BTO 之性質，等效彈性係數、等效壓磁係數與等較果磁導率皆隨體積 比呈現單調遞增或遞減。由於CFO 極化方向的改變，使得等效介電常數 κ11*、κ22*、κ33* 與
由觀察得知(圖 A-3)，Mori-Tanaka 模式與有限元素分析於非耦合項之數據皆相當 吻合。等效壓電係數與等效介電常數於體積比0.9 之前成長相當緩慢，體積比 0.9 後曲 率相當大地轉向BTO 之性質，等效彈性係數、等效壓磁係數與等較果磁導率皆隨體積 比呈現單調遞增或遞減。由於CFO 極化方向的改變，使得等效介電常數 κ11*、κ22*、κ33* 與