Application of Multirank Lamination Theory to the Modeling of Ferroelectric and Martensitic Materials

「鐵電材料」與「形狀記憶合金」可說是當今智能材料中最主流的兩大類。這些材料由於具有能量轉換的性質，以及在外界刺激下可引發巨大非線性、可回復之物理反應等特點，成為許多先進設備與儀器的基本元件。然而，這些獨特的性質，實際上源自於這些材料內部秩序性的微結構排列與演化所導致的宏觀反應。因此，瞭解這些材料之微觀結構，以及其導致宏觀行為的介觀機制，是有效利用這些材料的基本工作。本論文即是從材料的能量描述出發，建立鐵電材料和形狀記憶合金之介觀與微觀材料模型。文發展的第一個模型架構，是一套用來探討鐵電材料晶域轉換導致宏觀致動應變的介觀力電耦合模型。此模型首創之特點，在於引入「多階層狀結構」的觀念來描述鐵電材料晶域之排列，這使得所有晶域彼此間滿足力電諧和條件，也因此提供了極化向量轉換之最低能量路徑。另一特點為藉由此多階層狀結構對晶域壁移動之描述，可清楚區分不同極化向量翻轉所對應之頑強電場，如 90° 與 180° 頑強電場。利用此模型架構，我們模擬鐵電材料在力電耦合作用下引發致動應變的行為。模擬結果得到之致動應變量與實驗量測有很好的一致性。此外，透過此模型，我們提出一個新的觀點：「消極化能」對於力電耦合作用引發的致動應變量，有不可忽略的影響。著，我們建立可模擬鐵電材料微結構排列與演化的「新式相場模型」。藉由能量描述，穩態之鐵電微結構乃決定於系統總能量的最低狀態。系統在降低個別能量項的過程中，會引發使微結構聚結化、細微化、選擇與校列等相互抗衡的驅動力，因此最終穩定狀態為能量間競爭與妥協的自然結果。啟發於上述之多階層 結構，本文建立的新式相場法引用一組新的場變數來表示鐵電兄弟晶。利用這組新的場變數，系統的能量基態結構便可以用解析的數學式描寫，且其數學形式可適用於所有的晶體對稱性。我們用這套新式相場法模擬鐵電材料在四方晶系與菱形晶系兩種常見固態相中所構成的穩定微結構。模擬結果呈現各種滿足宏觀邊界條件的自主性調適之微結構圖樣，以及一個在外加電場操控下產生之晶域組態。這些模擬所得到的微結構，與諸多實驗的觀察相當符合。文最後一部份的研究工作，則是在建立適用於麻田散鐵材料微結構模擬的新式相場法。此相場法與上述鐵電材料之架構相同，唯略去電學的因素。我們應用此模型進行三方晶系之「薄膜」麻田散鐵材料微結構模擬。模擬結果呈現許多與實驗觀察相符的微結構圖樣。此外，針對常見之圓頂狀和隧道狀致動器，我們也探討薄膜之晶格方向以及微結構圖樣，對這些設計所能獲得之應變量的影響。最後，觀察微結構演化的過程，我們發現晶域間始終滿足應變諧和條件，這個現象除了證實滿足應變諧和條件之介面移動是一條最低能量的演化路徑外，也提供了大應變致動器設計上的一個參考準則。Ferroelectrics and shape-memory alloys are two major families of smart materials, and are both key units for several advanced devices. Their unique features and nonlinear behaviors, however, originate from the arrangements and evolution of the underlying microstructures. Therefore, to take full advantage of these materials, it is essential to study the microscopic and mesoscopic physics of them. This thesis addresses these topics via developing novel models based on energy arguments and verifying them. mesoscopic electromechanical switching model is developed to investigate the switching behavior of ferroelectric single crystals. The theoretical model makes an assumption that switching follows the evolution of a particular domain pattern. The construction of this pattern is achieved using multirank laminates, which guarantee that domains remain compatible during evolution. This in turn provides a low-energy path for the overall switching. It offers an advantage of specifying different types of domain wall movements, leading to a distinction for the switching types.he required input parameters, 180◦ and 90◦ coercive fields, are taken from measured data. Simulation results show good agreement with the measured strains in experiments. It is found that depolarization has a non-trivial influence on attainable actuation strains.ext, to investigate the formation and evolution of microscopic domain patterns in ferroelectrics, a non-conventional phase-field model is developed through competing energetics to describe the coarsening, refinement, selection, and alignment of microstructure. It employs a set of field variables motivated by multirank laminates to represent energy-minimizing domain configurations. As a result, the energy-well structure can be expressed explicitly in a unified fashion. The framework is applied to domain simulation in both the tetragonal and rhombohedral phases assuming that polarization is close to one of their ground states. Several electromechanical self-accommodation patterns and an engineered domain configuration are predicted and found in good agreement with those observed in experiment.inally, we build a phase-field model for martensitic materials. The main feature of this novel model is also built on the ideas of multirank lamination. The model is applied to the investigation of pattern formation in martensitic thin films with trigonal symmetry. Various intriguing and fascinating patterns are predicted and are found in good agreement with those observed in experiments. In addition, the film orientations and patterns to achieve large actuation strains are suggested for dome-shaped and tunnel-shaped microactuators. It is found that the resultingorphologies evolve with coherent interfaces under various loading conditions. This suggests that compatible walls provide a low-energy path during evolution, and thenderstanding of them leads to novel strategies of large strain actuation.Acknowledgements (Chinese) ibstract (Chinese) iiibstract vontents viiist of Figures ixist of Tables xv Introduction 1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Ferroelectrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Electromechanical Switching in Ferroelectric Crystals . . . . . 5.2.2 Phase-Field Modeling of Ferroelectric Domain Patterns . . . . 8.3 Martensites and Martensitic Thin Films . . . . . . . . . . . . . . . . 10.3.1 Phase-Field Modeling of Martensitic Microstructures . . . . . 13.4 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Framework 17.1 Kinematics and Ferroelectric Phase Transformations . . . . . . . . . . 17.2 Electroelastic Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.3 Multirank Laminated Domain Pattern . . . . . . . . . . . . . . . . . 21.4 Electromechanical Switching Model . . . . . . . . . . . . . . . . . . . 25.4.1 Domain Switching . . . . . . . . . . . . . . . . . . . . . . . . 26.4.2 Flat-Plate Configuration . . . . . . . . . . . . . . . . . . . . . 29.4.3 Other Configurations . . . . . . . . . . . . . . . . . . . . . . . 32.5 Phase-Field Model for Ferroelectrics . . . . . . . . . . . . . . . . . . . 35.5.1 Modified Free Energy Functional . . . . . . . . . . . . . . . . 36.5.2 Thermodynamic Driving Force and Evolution of Ferroelectricomains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39ii.5.3 Constrained Model . . . . . . . . . . . . . . . . . . . . . . . . 44.6 Phase-Field Model for Martensites . . . . . . . . . . . . . . . . . . . 48.6.1 Transformation Strain, Free Energy and Microstructure Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48.6.2 Thin-Film Limit . . . . . . . . . . . . . . . . . . . . . . . . . 51 Numerical Implementations 57.1 Implementation for Electromechanical Switching . . . . . . . . . . . . 57.2 Implementation for Phase-Field Model . . . . . . . . . . . . . . . . . 62 Results and Discussions 67.1 Electromechanical Switching in Flat-Plate Configuration . . . . . . . 67.1.1 Experiment under Flat-Plate Configuration . . . . . . . . . . 67.1.2 Depolarization Effect . . . . . . . . . . . . . . . . . . . . . . . 69.1.3 Comparison with Experiments . . . . . . . . . . . . . . . . . . 75.2 Phase-Field Simulation of Ferroelectrics . . . . . . . . . . . . . . . . . 77.2.1 Ferroelectric Single Crystal in the Tetragonal Phase . . . . . . 77.2.2 Ferroelectric Single Crystal in the Rhombohedral Phase . . . . 82.3 Phase-Field Simulation of Martensitic Thin Films . . . . . . . . . . . 92.3.1 (001) Film . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.3.2 (110) Film . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95.3.3 (111) Film . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97.3.4 Design of Large Strain Microactuators . . . . . . . . . . . . . 98 Conclusions and Future Works 107.1 Electromechanical Switching Model . . . . . . . . . . . . . . . . . . . 107.2 Phase-Field Simulation of Ferroelectrics . . . . . . . . . . . . . . . . . 109.3 Phase-Field Simulation of Martensitic Thin-Films . . . . . . . . . . . 112ibliography 115 Depolarization of a Uniformly Polarized Body 129 Depolarizing Factor of a Rectangle 133 Depolarizing Factor of a Cylinder 135 Solution of the Stress Field 139 Solution of the Depolarization Field 143urriculum Vitae 14