个人简历

课题组新闻动态
2020/06, 热烈祝贺乔灵霞、郭晓露、韩雨岑、赵伟四位同学顺利通过博士论文答辩！他们将分别在UCSD，UCLA，U. of Strathclyde, UCI做博士后。
2020/04, 祝贺陶宽的文章“Tuning cell motility via cell tension with a mechanochemical cell migration model”被Biophysical Journal接受。此项工作与北大定量生物学中心刘峰课题组合作完成。
2020/04, 祝贺李若泰的文章“Variational Phase Field Formulations of Polarization and Phase Transition in Ferroelectric Thin Films”被SIAM Journal on Applied Mathematics接受。此项工作与哥伦比亚大学杜强教授合作完成。
2020/04, 祝贺韩雨岑的文章“A Reduced Study for Nematic Equilibria on Two-Dimensional Polygons”被SIAM Journal on Applied Mathematics接受。此项工作与英国Strathclyde大学Apala Majumdar教授合作完成。
2020/03, 祝贺俞炳的文章“Global optimization-based dimer method for finding saddle points”被Discrete Contin. Dyn. Syst. Ser. B接受。
2020/03, 祝贺殷鉴远在《Phys. Rev. Lett.》发表论文“Construction of a pathway map on a complicated energy landscape”。该论文发展了一种在复杂能量景观上构建“路径图”的计算方法，为解决多解问题提供了一个高效的工具。此项工作与张平文院士、加拿大滑铁卢大学陈征宇教授合作完成。《北大新闻网》。
2019/12, 祝贺韩雨岑和徐子睿同学合作在《Soft Matter》发表论文“Pathways connecting two opposed bilayers with a fusion pore: a molecularly-informed phase field approach”。该论文利用分子信息构建的相册模型揭示了完整的生物双层膜的融合过程，此项工作与加拿大McMaster大学史安昌教授合作完成。
2019/11, 祝贺殷鉴远在《SIAM Journal on Scientific Computing》发表论文“High-index Optimization-based Shrinking Dimer Method for Finding High-Index Saddle Points”。该论文发展了一种计算高阶鞍点的高效算法，在此算法基础上，我们开展了一系列后续工作。此项工作与张平文院士合作完成。
2019/09, 欢迎张雨泽博士 (香港理工大学) 加入我们课题组！
2019/09, 我们课题组在《Cell》子刊《Cell Systems》发表题为“Network topologies that can achieve dual function of adaptation and noise attenuation”的研究论文。该工作通过数学与生命科学的交叉研究，利用理论分析与大规模科学计算，揭示了生物网络实现适应-抗噪双功能的设计原理。祝贺乔灵霞和赵伟，他们为论文的共同第一作者。《北大新闻网》。
2019/07, 欢迎曹露博士（悉尼大学）、徐真博士（北京师范大学）加入我们课题组！
2019/06, 祝贺韩雨岑同学在《Journal of Computational Physics》发表论文“Transition Pathways between Defect Patterns in Confined Nematic Liquid Crystals”。该论文利用Landau-de Gennes模型研究了受限在三维圆柱内向列相液晶缺陷构型的过渡态和转移路径，此项工作与张平文院士和清华大学胡煜成助理教授共同合作完成。
2019/06, 热烈祝贺陶宽同学通过博士论文答辩！他将入职北京体育大学体育工程学院。
2019/05, 我们在北京大学北京国际数学研究中心成功举办了第四届中日韩数学生命科学研讨会（A3 Workshop on Mathematical Life Science）。
2018/03, 张磊获得中英人才项目“牛顿高级学者基金”（Newton Advanced Fellowship）资助，项目名称为《Solution Landscapes in the Landau-de Gennes theory for Nematic Liquid Crystals》。
2018/02, 我们课题组在《BMC Systems Biology》发表文章利用基因调控模型揭示了一种关键的多能性基因Nanog在干细胞分化和重编程过程中的双重作用以及细胞状态转换过程中中间状态的重要性，提供了一种分析和控制细胞分化和重编程的关键调控因子的通用方法。我们课题组轮转研究生于沛加（定量生物学中心）为论文第一作者，张磊、聂青和汤超为共同通讯作者。
2018/01, 我们与中科院遗传发育所焦雨铃课题组合作在《Developmental Cell》上通过学科交叉研究发现了生长素长距离运输在侧生器官对茎尖干细胞的反馈中扮演了重要角色，揭示了植物干细胞稳态的维持与动物干细胞的稳态维持在分子水平采取了不同的策略。焦雨铃研究组博士研究生石碧海和我们课题组博士研究生郭晓露为该论文共同第一作者，张磊和焦雨铃为共同通讯作者。
招收优秀博士后、博士研究生、硕士研究生
博士、硕士：我们课题组每年招收1-2名博士生和硕士生，数学专业或系统生物学背景的同学均欢迎加入！
博士后：我们课题组目前需要计算与应用数学专业或系统生物学背景的优秀博士后。我们将提供18+万年薪（税前）的工资及北大的住房与医疗等待遇。
本科生科研：每年级招2-3名本科生科研，无先修课程要求，数学基础扎实，科研要求积极主动，每周可以投入固定时间做课题，参加讨论班并积极参与讨论。
感兴趣的同学欢迎email联系本人，请附个人简历。
科研
科学计算，稀有事件及其鞍点问题的算法和应用
稀有事件及相关鞍点问题在物理、化学、生物等很多领域都有着重要的应用，例如材料中的成核问题，化学反应的随机动力学，蛋白质的折叠构象等[1][1][4]。因为鞍点是代表过渡态的不稳定点，从数学上来讲比求解一般的稳定点更加困难，特别是对高维问题，其能量曲面非常复杂，这些对鞍点的计算方法提出了更高的要求。为此我们发展了一系列高效而稳定的数值方法，例如约束弦方法[2]、优化二聚体方法[3][5]等,算法具有很好的稳定性和适应性，可以有效解决各类鞍点问题。
在偏微分方程数值解中，对于高阶导数和刚性反应经常要求数值方法的时间步长引入严格的时间稳定性约束。隐式积分法通过解耦扩散和反应，为算法提供了优良的稳定性和高效率。 为了求解高维系统，我们利用阵列表示来有效处理从一般线性微分算子得到的指数矩阵，些矩阵只取决于导数的阶数和离散点的数量，而与空间维度的大小无关。这种基于阵列表示的方法可以模拟涉及高维数据的复杂系统等广泛应用[6]。
[1] Lei Zhang, Long-qing Chen, Qiang Du, Mathematical and Numerical Aspects of Phase-field Approach to Critical Morphology in Solids, Journal of Scientific Computing, 37, 89-102, 2008.
[2] Qiang Du, Lei Zhang, A constrained string method and its numerical analysis, Commun. Math. Sci., 7, 1039-1051, 2009.
[3] Lei Zhang, Jingyan Zhang, Qiang Du, Finding Critical Nuclei in Phase Transformations by Shrinking Dimer Dynamics and its Variants, Commun. Comput. Phys., 16, pp. 781-798, 2014.
[4] Lei Zhang, Weiqing Ren, Amit Samanta, Qiang Du,Recent Developments in Computational Modeling of Nucleation in Phase Transformations, npj Computational Materials (2016) 2, 16003.
[5] Lei Zhang, Qiang Du, Zhenzhen Zheng, Optimization-based Shrinking Dimer Method for Finding Transition States, SIAM Journal on Scientific Computing 38:1, A528-A544, 2016.
[6] Dongyong Wang, Lei Zhang, Qing Nie, Array-representation Integration Factor Method for High-dimensional Systems, J. Comput. Phys., 258, 585-600, 2014.

数学与生命科学的交叉，计算系统生物学
近年来随着生物海量数据的产生和计算机能力的显著提高，对复杂生物系统进行建模和计算模拟已成为生命科学领域作为继实验和理论方法外的又一重要科学研究手段。数学与生命科学的交叉研究可以探求生命机制的内在特征、定量规律及普适原理，正广泛应用于生命科学的各个领域。我们主要利用计算系统生物学手段，紧密结合生物实验，构建基因-细胞-组织的多层次模型，开展理论分析和科学计算，解决生命科学中一些重要的共性问题[1-5]。
近年来我们主要针对发育细胞、干细胞等为主要的研究领域，开展了一系列工作：
1. 提出了以噪声诱导的斑马鱼后脑的边界锐化机制，为研究分子噪声对细胞命运和基因不确定性提供了重要的理论依据和方法[1]。
2. 针对肠道隐窝的形成和再生，发展了基于反应扩散机制的数学框架[2]。
3. 我们与香港科技大学阎言课题组合作建立相场模型，利用稀有事件的机制揭示了果蝇胚胎成神经细胞分层过程中顶端肌球蛋白的重要作用[3]。
4. 我们与刘峰课题组构建力-化耦合的细胞极化模型，通过单细胞实验和理论模型阐明细胞膜表面张力对细胞极化有抑制作用，较高的表面张力能维持极化的稳定性[4]。
5. 干细胞是器官的来源，但器官能否反馈调控干细胞？分化细胞对干细胞的反馈有助于动态维持干细胞的稳态，使干细胞的数量和活性适应内源变化。在动物中，分化细胞通过分泌可扩散的信号分子反馈于干细胞。植物中是否存在反馈调控，在分子水平如何实现反馈调控？我们通过学科交叉研究发现了生长素长距离运输在侧生器官对茎尖干细胞的反馈中扮演了重要角色。研究发现，茎尖干细胞区域的生长素含量负调控干细胞平衡，高生长素含量导致干细胞分化加速。计算模拟表明，侧生器官原基外运的生长素能够抑制茎尖的生长素外运，从而维持茎尖生长素浓度的稳态，进而维持干细胞的稳态。遗传分析和显微手术实验验证了模型结论，发现在叶片、花原基发育不良的突变体中或在切除原基后，茎尖干细胞区域生长素外运加强，生长素浓度降低，茎尖干细胞增多。增大的干细胞团进而产生更多侧生器官。该模型解释了侧生器官突变体中茎尖干细胞区域增大，特别是周期性变化的表型。本研究揭示了植物干细胞稳态的维持与动物干细胞的稳态维持在分子水平采取了不同的策略。
[1] Lei Zhang#, Kelly Radtke#, Likun Zheng, Anna Cai, Tom Schilling, Qing Nie, Noise Drives Sharpening of Gene Expression Boundaries in the Zebrafish Hindbrain, Molecular Systems Biology 8:613, 2012.
[2] Lei Zhang, Arthur Lander, Qing Nie, A Reaction-Diffusion Mechanism Influences Cell Lineage Progression as a Basis for Formation, Regeneration, and Stability of Intestinal Crypts, BMC Systems Biology 6:93, 2012.
[3] Yanru An, Guosheng Xue, Xiaowei Zhou, Weichuan Yu, Toyotaka Ishibashi, Lei Zhang*, Yan Yan*, Apical constriction is driven by a pulsatile apical myosin network in delaminating Drosophila neuroblasts, Development (2017): dev-150763.
[4] Weikang Wang#, Kuan Tao#, Jin Wang, Gen Yang, Qi Ouyang, Yugang Wang, Lei Zhang*, Feng Liu*, Exploring the inhibition effect of membrane tension on cell polarization, PLoS Comput Biol 13(1): e1005354. doi:10.1371/ journal.pcbi.1005354, 2017.
[5] Bihai Shi#, Xiaolu Guo#, Ying Wang, Ken-ichiro Hayashi, Jinzhi Lei, Lei Zhang*, Yuling Jiao*, Feedback from Lateral Organs Controls Shoot Apical Meristem Growth by Modulating Auxin Transport, Developmental Cell, 44, 204-216, 2018.

计算材料科学
设计材料的性能最有效的方法是控制其相变和结构演变，它始于初期阶段纳米核的核变产生新的粒子，然后经历粒子增长，撞击或分裂的过程。在很多时候，核过程最终决定了材料的微观结构。由于各项异性的表面能和非局部弹性能的存在，人们普遍认为固体成核的建模和预测是迄今为止最困难的物理过程之一。我们主要研究相变中的各类成核问题，利用稳定的数值算法寻找不稳定的临界核和转变路径 [1-7]，其中包括固态相变、液晶、超导材料等。
我们对固态成核问题建立了相场模型，结合稀有事件的数值算法准确地找到了固态中的临界核 [1]，系统研究了固态成核及其动力学[2-7]，解决了固态相变中长期存在的一个核心问题。
[1] Lei Zhang, Long-qing Chen, Qiang Du, Morphology of critical nuclei in solid state phase transformations, Physical Review Letters, 98, No.25, 265703, 2007.
[2] Lei Zhang, Long-qing Chen, Qiang Du, Diffuse-Interface Description of Strain-Dominated Morphology of Critical Nuclei in Phase Transformations, Acta Materialia, 56, 3568-3576, 2008
[3] Lei Zhang, Long-qing Chen, Qiang Du, Diffuse-Interface Approach to Predicting Morphologies of Critical Nucleus and Equilibrium Structure for Cubic to Tetragonal Transformations, J. Comput. Phys., 229, 6574-6584, 2010.
[4] Tae Wook Heo, Lei Zhang, Qiang Du, and Long-Qing Chen, Incorporating Non-classical Nuclei in Phase-field Simulations, Scripta Materialia, 63, 8-11, 2010
[5] Lei Zhang, Long-qing Chen, Qiang Du, Simultaneous Prediction of Morphologies of a Critical Nucleus and an Equilibrium Precipitate in Solids, Commun. Comput. Phys., 7, 674-682, 2010.
[6] Yulan Li, Shenyang Hu, Lei Zhang, Xin Sun, Non-classical nuclei and growth kinetics of Cr precipitates in FeCr alloys during aging, Modelling Simul. Mater. Sci. Eng., 22, 025002, 2014.
[7] Boyan Li, Lei Zhang*, Chengliang Li, Shenyang Hu, Qiulin Li, Jun Chen, Guogang Shu, Yuqing Weng, Ben Xu*, Wei Liu*, Finding Non-Classical Critical Nuclei and Minimum Energy Path of Cu Precipitates in Fe-Cu Alloys, Modelling Simul. Mater. Sci. Eng. 25 085006, 2017.
# equal-contribution, * Co-corresponding authors.
Teaching at Peking University
Spring 2018, 计算系统生物学 (Computational Systems Biology)
Fall 2017, 数学与物理的整合 I (Integrating Sciences - Mathematics & Physics I)
Fall 2017, 高等数学 (B) (Advanced Mathematics B)
Spring 2017, 计算系统生物学 (Computational Systems Biology)
Fall 2016, 数学与物理的整合 I (Integrating Sciences - Mathematics & Physics I)
Fall 2016, 数值线性代数 (Numerical Linear Algebra)
Spring 2016, 毕业论文 (Undergraduate Thesis)
Fall 2015, 数学与物理的整合 I (Integrating Sciences - Mathematics & Physics I)
Spring 2015, 毕业论文 (Undergraduate Thesis)
Fall 2014, 数学与物理的整合 I (Integrating Sciences - Mathematics & Physics I)

Teaching at City University of Hong Kong
Spring 2013, MA2506: Probability and Statistics
Fall 2012, MA5601: Applied Partial Differential Equations

Teaching at University of California, Irvine
Spring 2012, Math 2D: MULTIVARIABLE CALCULUS
Winter 2012, Math 105B: NUMERICAL ANALYSIS
Fall 2011, Math 105A: NUMERICAL ANALYSIS
Spring 2011, Math 6G: Linear Algebra
Winter 2011, Math 3D: ELEM DIFF EQUATIONS
Fall 2010, Math 6G - Linear Algebra
Spring 2010, Math 112C - INT PARTIAL DIF EQU
Winter 2010, Math 121B - Linear Algebra
Fall 2009, Math 121A - Linear Algebra

Teaching at Penn State University
Fall 2007, Math 21 - College Algebra I
Spring 2007. Math 110 - Techniques of Calculus I
Fall 2006, Math 110 - Techniques of Calculus I
Spring 2006, Math 110 - Techniques of Calculus I
Fall 2005, Math 22 - College Algebra II

研究领域

研究方向
科学计算，偏微分方程数值解，稀有事件及其鞍点问题的算法；
数学与生命科学的交叉，计算系统生物学；
计算材料科学，各类相变中的形核问题。

近期论文

Journal Publications
Yucen Han, Jianyuan Yin, Pingwen Zhang, Apala Majumdar, Lei Zhang. Solution landscape of a reduced Landau--de Gennes model on a hexagon, arXiv:2003.07643. [Link]
Ruotai Li, Qiang Du, Lei Zhang. Efficient Variational Phase Field Method for Phase Transitions in Ferroelectric Thin Films, arXiv:2003.00416. [Link]
Jianyuan Yin, Bing Yu, Lei Zhang. Searching the solution landscape by generalized high-index saddle dynamics, arXiv:2002.10690. [Link]
Xiaolu Guo, Tao Tang, Minxuan Duan, Lei Zhang, Hao Ge. The Nonequilibrium Mechanism of Noise Enhancer synergizing with Activator in HIV Latency Reactivation, bioRxiv 2020.01.14.905653. [Link]
Kuan Tao, Jin Wang, Xiangyu Kuang, Weikang Wang, Feng Liu, Lei Zhang. Tuning cell motility via cell tension with a mechanochemical cell migration model, Biophysical Journal, accepted. [Link]
Yucen Han, Apala Majumdar, Lei Zhang. A Reduced Study for Nematic Equilibria on Two-Dimensional Polygons, SIAM Journal on Applied Mathematics, accepted. [Link]
Qiang Du, Ruotai Li, Lei Zhang. Variational Phase Field Formulations of Polarization and Phase Transition in Ferroelectric Thin Films, SIAM Journal on Applied Mathematics, accepted. [Link]
Bing Yu, Lei Zhang. Global optimization-based dimer method for finding saddle points, Discrete Contin. Dyn. Syst. Ser. B (2020), doi: 10.3934/dcdsb.2020139. [Link]
Jianyuan Yin, Yiwei Wang, Jeff Z.Y. Chen, Pingwen Zhang, Lei Zhang. Construction of a pathway map on a complicated energy landscape, Physical Review Letters 124, 090601 (2020). [Link]
Qing Nie, Lingxia Qiao, Yuchi Qiu, Lei Zhang, Wei Zhao. Noise control and utility: from gene regulatory network to spatial patterning, Sci. China Math 63, 425–440 (2020). [Link]
Yucen Han, Zirui Xu, An-Chang Shi,Lei Zhang. Pathways connecting two opposed bilayers with a fusion pore: A molecularly-informed phase field approach, Soft Matter, 2020, 16, 366-374. [Link]
Jianyuan Yin, Lei Zhang. Pingwen Zhang, High-index Optimization-based Shrinking Dimer Method for Finding High-Index Saddle Points, SIAM Journal on Scientific Computing, 41 (6), A3576-A3595. 2019.[Link]
Lingxia Qiao, Wei Zhao, Chao Tang, Qing Nie, Lei Zhang. Network Topologies That Can Achieve Dual Function of Adaptation and Noise Attenuation, Cell Systems 9, 1–15, 2019.[Link]
Yucen Han, Yucheng Hu, Pingwen Zhang, Lei Zhang. Transition Pathways between Defect Patterns in Confined Nematic Liquid Crystals, J. Comput. Phys. 396 (2019) 1–11.[Link]
Chunhe Li, Lei Zhang,Qing Nie, Landscape reveals the underlying mechanisms for sharpening the gene expression boundaries, BMC Systems Biology 12:67, 2018. [Link]
Lina Meinecke, Praveer Sharma, Huijing Du, Lei Zhang, Qing Nie, Thomas Schilling. Modeling craniofacial development reveals spatiotemporal constraints on robust patterning of the mandibular arch, PLoS Comput Biol 14(11): e1006569, 2018. [Link]
Boyan Li, Lei Zhang, Chengliang Li, Qiulin Li, Jun Chen, Guogang Shu, Yuqing Weng, Ben Xu, Shenyang Hu, Wei Liu. Phase Field Simulation of Critical Nuclei and Minimum Energy Paths of Cu/Mn/Ni Precipitates in Fe-Cu-Mn (Ni) Alloys, Journal of Nuclear Materials 507, 59-67, 2018. [Link]
Peijia Yu, Qing Nie, Chao Tang, Lei Zhang. Nanog induced intermediate state in regulating stem cell differentiation and reprogramming, BMC Systems Biology 12:22, 2018. [Link]
Bihai Shi, Xiaolu Guo, Ying Wang, Ken-ichiro Hayashi, Jinzhi Lei, Lei Zhang, Yuling Jiao. Feedback from Lateral Organs Controls Shoot Apical Meristem Growth by Modulating Auxin Transport, Developmental Cell, 44, 204-216, 2018. [Link]
Boyan Li, Lei Zhang, Chengliang Li, Shenyang Hu, Qiulin Li, Jun Chen, Guogang Shu, Yuqing Weng, Ben Xu, Wei Liu. Finding Non-Classical Critical Nuclei and Minimum Energy Path of Cu Precipitates in Fe-Cu Alloys, Modelling Simul. Mater. Sci. Eng. 25 085006, 2017.
Yanru An, Guosheng Xue, Xiaowei Zhou, Weichuan Yu, Toyotaka Ishibashi, Lei Zhang, Yan Yan. Apical constriction is driven by a pulsatile apical myosin network in delaminating Drosophila neuroblasts, Development, 144, 2153-2164, 2017. [Link]
Weikang Wang, Kuan Tao, Jin Wang, Gen Yang, Qi Ouyang, Yugang Wang, Lei Zhang, Feng Liu. Exploring the inhibition effect of membrane tension on cell polarization, PLoS Comput Biol 13(1): e1005354. doi:10.1371/ journal.pcbi.1005354, 2017. [Link]
Lei Zhang, Weiqing Ren, Amit Samanta, Qiang Du. Recent Developments in Computational Modeling of Nucleation in Phase Transformations, npj Computational Materials (2016) 2, 16003.
Lei Zhang, Qiang Du, Zhenzhen Zheng. Optimization-based Shrinking Dimer Method for Finding Transition States, SIAM Journal on Scientific Computing 38:1, A528-A544, 2016.
Amanda Cinquin, Likun Zheng, Harry P. Taylor, Adrian Paz, Lei Zhang, Michael Chiang, Josh Snow, Qing Nie, Olivier Cinquin. Semi-permeable diffusion barriers provide patterning robustness against spatial noise in morphogen release, Developmental Cell 35, 405-417, 2015.
Tian Hong, Ernest S. Fung, Lei Zhang, Edwin S. Monuki, Qing Nie. Semi-adaptive response and noise attenuation in bone morphogenetic protein signalling, J. R. Soc. Interface 12: 20150258, 2015.
Lei Zhang, Jingyan Zhang, Qiang Du. Finding Critical Nuclei in Phase Transformations by Shrinking Dimer Dynamics and its Variants, Commun. Comput. Phys., 16, pp. 781-798, 2014.
Yulan Li, Shenyang Hu, Lei Zhang, Xin Sun. Non-classical nuclei and growth kinetics of Cr precipitates in FeCr alloys during aging, Modelling Simul. Mater. Sci. Eng., 22, 025002, 2014.
Dongyong Wang, Lei Zhang, Qing Nie. Array-representation Integration Factor Method for High-dimensional Systems, J. Comput. Phys., 258, 585-600, 2014.
Lei Zhang, Kelly Radtke, Likun Zheng, Anna Cai, Tom Schilling, Qing Nie. Noise Drives Sharpening of Gene Expression Boundaries in the Zebrafish Hindbrain, Molecular Systems Biology 8:613, 2012.
Lei Zhang, Arthur Lander, Qing Nie. A Reaction-Diffusion Mechanism Influences Cell Lineage Progression as a Basis for Formation, Regeneration, and Stability of Intestinal Crypts, BMC Systems Biology 6:93, 2012.
Lei Zhang, Long-qing Chen, Qiang Du. Simultaneous Prediction of Morphologies of a Critical Nucleus and an Equilibrium Precipitate in Solids, Commun. Comput. Phys., 7, 674-682, 2010.
Tae Wook Heo, Lei Zhang, Qiang Du, and Long-Qing Chen. Incorporating Non-classical Nuclei in Phase-field Simulations, Scripta Materialia, 63, 8-11, 2010.
Lei Zhang, Long-qing Chen, Qiang Du. Diffuse-Interface Approach to Predicting Morphologies of Critical Nucleus and Equilibrium Structure for Cubic to Tetragonal Transformations, J. Comput. Phys., 229, 6574-6584, 2010.
Qiang Du, Lei Zhang. A constrained string method and its numerical analysis, Commun. Math. Sci., 7, 1039-1051, 2009.
Lei Zhang, Long-qing Chen, Qiang Du. Mathematical and Numerical Aspects of Phase-field Approach to Critical Morphology in Solids, Journal of Scientific Computing, 37, 89-102, 2008.
Lei Zhang, Long-qing Chen, Qiang Du. Diffuse-Interface Description of Strain-Dominated Morphology of Critical Nuclei in Phase Transformations, Acta Materialia, 56, 3568-3576, 2008.
Lei Zhang, Long-qing Chen, Qiang Du. Morphology of critical nuclei in solid state phase transformations, Physical Review Letters, 98, No.25, 265703, 2007.