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Strain effects of periodic thickness-gradient films on flexible substrates

Li Hui-Hua Zhang Jia-Hui Yu Sen-Jiang Lu Chen-Xi Li Ling-Wei

Strain effects of periodic thickness-gradient films on flexible substrates

Li Hui-Hua, Zhang Jia-Hui, Yu Sen-Jiang, Lu Chen-Xi, Li Ling-Wei
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  • Controlled surface structures have important applications in the fields of flexible electronics, bionic devices, smart materials and surface engineering. Although various instability modes induced by stresses in homogeneous and heterogeneous film systems have been investigated extensively in the past decades, understanding the stress relief mechanisms of gradient film systems is still a challenge. In this work, periodic thickness-gradient metal silver films on flexible polydimethylsiloxane (PDMS) substrates are prepared by using magnetron sputtering technique and by masking weaved copper grid during deposition. The morphological evolutions and structural characteristics of the gradient films in the uniaxial mechanical compression/tension process are detected by using optical microscopy and atomic force microscopy. It is found that the thickness-gradient film spontaneously forms on the PDMS surface due to the specific three-dimensional structure of the weaved copper grid. The maximum film thickness in the mesh center is about twice larger than the minimum one in the region covered by the copper wire. Under the uniaxial mechanical loading, the silver film surface will form stripe wrinkles and straight cracks, which are aligned perpendicular to each other. The variation of film thickness can well modulate the in-plane strain distribution of the film and thus the stress relief patterns. As the compressive strain increases, the wrinkles first form in the region with smaller film thickness, and then extend into the region with larger thickness. The wrinkle wavelength decreases with compressive strain increasing, which is in consistent with the theoretical prediction of accordion model. The wavelength and amplitude of the wrinkles exhibit periodically oscillatory behaviors across the film surface. As the tensile strain increases, the cracks start to propagate in the region with smaller thickness, and then the crack width and number both increase gradually, leading the multiple crack modes to form finally. The crack width and average spacing both increase with film thickness increasing. The morphologies and evolutions of such wrinkles and cracks in the thickness-gradient films are analyzed in depth based on the stress theory. The wrinkle patterns are further simulated by the finite element method, and the simulation results are in good agreement with the experimental results. This study could promote a better understanding of the strain effects of heterogeneous film systems and find the applications in the field of flexible electronics by the structural design of thickness-gradient films.
      Corresponding author: Yu Sen-Jiang, sjyu@hdu.edu.cn ; Li Ling-Wei, lingwei@hdu.edu.cn
    [1]

    Li B, Cao Y P, Feng X Q, Gao H J 2012 Soft Matter 8 5728

    [2]

    许宏, 苑争一, 黄彤飞, 王啸, 陈正先, 韦进, 张翔, 黄元 2020 物理学报 69 026101

    Xu H, Yuan Z Y, Huang T F, Wang X, Chen Z X, Wei J, Zhang X, Huang Y 2020 Acta Phys. Sin. 69 026101

    [3]

    倪勇, 刘佩琳, 马龙, 李世琛, 何陵辉 2018 固体力学学报 39 113

    Ni Y, Liu P L, Ma L, Li S C, He L H 2018 Chin. J. Solid Mech. 39 113

    [4]

    Lee W K, Odom T W 2019 ACS Nano 13 6170

    [5]

    Ma L, He L, Ni Y 2020 J. Appl. Phys. 127 111101

    [6]

    冯雪, 陆炳卫, 吴坚, 林媛, 宋吉舟, 宋国锋, 黄永刚 2014 物理学报 63 014201

    Feng X, Lu B W, Wu J, Lin Y, Song J Z, Song G F, Huang Y G 2014 Acta Phys. Sin. 63 014201

    [7]

    Kim H N, Ge D, Lee E, Yang S 2018 Adv. Mater. 30 1803847

    [8]

    Gao N, Zhang X, Liao S, Jia H, Wang Y 2016 ACS Macro Lett. 5 823

    [9]

    Lee W K, Jung W B, Rhee D, Hu J, Lucy Lee Y A, Jacobson C, Jung H T, Odom T W 2018 Adv. Mater. 30 1706657

    [10]

    王月桐, 商珞然, 赵远锦 2020 物理学报 69 084701

    Wang Y T, Shang L R, Zhao Y J 2020 Acta Phys. Sin. 69 084701

    [11]

    Yu S, Liu X, Sun Y, Zhou H, Cai P 2019 Thin Solid Films 669 355

    [12]

    Yin J, Yagüe J L, Eggenspieler D, Gleason K K, Boyce M C 2012 Adv. Mater. 24 5441

    [13]

    Bowden N, Brittain S, Evans A G, Hutchinson J W, Whitesides G M 1998 Nature 393 146

    [14]

    Wu D, Yin Y J, Xie H M, Shang Y F, Li C W, Wu L F, Dai X L 2014 Sci. China-Phys. Mech. Astron. 57 637

    [15]

    Um D S, Lim S, Lee Y, Lee H, Kim H, Yen W C, Chueh Y L, Ko H 2014 ACS Nano 8 3080

    [16]

    Wang D, Cheewaruangroj N, Li Y, McHale G, Jiang Y, Wood D, Biggins J S, Xu B B 2018 Adv. Funct. Mater. 28 1704228

    [17]

    Chan E P, Crosby A J 2006 Soft Matter 2 324

    [18]

    Ouchi T, Yang J, Suo Z, Hayward R C 2018 ACS Appl. Mater. Interfaces 10 23406

    [19]

    Ding W, Yang Y, Zhao Y, Jiang S, Cao Y, Lu C 2013 Soft Matter 9 3720

    [20]

    Gurmessa B J, Croll A B 2017 Soft Matter 13 1764

    [21]

    Nogales A, Del Campo A, Ezquerra T A, Rodriguez-Hernandez J 2017 ACS Appl. Mater. Interfaces 9 20188

    [22]

    Li B, Zeng C Q, Yin S F, Feng X Q 2019 Sci. China-Technol. Sci. 62 747

    [23]

    Ni Y, Yang D, He L 2012 Phys. Rev. E 86 031604

    [24]

    Yu S, Ni Y, He L, Ye Q L 2015 ACS Appl. Mater. Interfaces 7 5160

    [25]

    Yu S, Ma L, Sun Y, Lu C, Zhou H, Ni Y 2019 Langmuir 35 7146

    [26]

    Zhang Q, Tang Y, Hajfathalian M, Chen C, Turner K T, Dikin D A, Lin G, Yin J 2017 ACS Appl. Mater. Interfaces 9 44938

    [27]

    Cerda E, Mahadevan L 2003 Phys. Rev. Lett. 90 074302

    [28]

    Sun J Y, Xia S, Moon M W, Oh K H, Kim K S 2012 Proc. R. Soc. A 468 932

    [29]

    Marthelot J, Roman B, Bico J, Teisseire J, Dalmas D, Melo F 2014 Phys. Rev. Lett. 113 085502

    [30]

    Kappert E J, Pavlenko D, Malzbender J, Nijmeijer A, Benes N E, Tsai P A 2015 Soft Matter 11 882

    [31]

    Volynskii A L, Bazhenov S, Lebedeva O V, Ozerin A N, Bakeev N F 1999 J. Appl. Polym. Sci. 72 1267

    [32]

    Chen X, Hutchinson J W 2004 J. Appl. Mech. 71 597

    [33]

    Jiang H, Khang D Y, Song J, Sun Y, Huang Y, Rogers J A 2007 Proc. Natl. Acad. Sci. U. S. A. 104 15607

    [34]

    Huang Z Y, Hong W, Suo Z 2005 J. Mech. Phys. Solids 53 2101

    [35]

    Pan K, Ni Y, He L, Huang R 2014 Int. J. Solids Struct. 51 3715

    [36]

    Cao G, Chen X, Li C, Ji A, Cao Z 2008 Phys. Rev. Lett. 100 036102

    [37]

    Zhang X, Mather P T, Bowick M J, Zhang T 2019 Soft Matter 15 5204

    [38]

    Hibbett Karlsson Sorensen Inc. 2001 ABAQUS/Standard user's manual: version 6.2

    [39]

    Riks E 1979 Int. J. Solids Struct. 15 529

  • 图 1  PDMS基底上周期性厚度梯度薄膜的制备流程及应变施加过程 (a) PDMS基底预拉伸至20%的预应变; (b)编织铜网覆盖到预拉伸的PDMS表面; (c)编织铜网的光学显微镜照片; (d)银薄膜沉积之后, 移除铜网, 并释放预应变至PDMS的原长; (e)进一步拉伸PDMS至任意长度L

    Figure 1.  Schematic illustration of sample preparation and loading sequence of periodic thickness-gradient films on PDMS substrates: (a) The PDMS substrate was stretched to 20% prestrain; (b) a weaved copper grid was placed on the pre-stretched PDMS surface; (c) optical micrograph of the weaved copper grid; (d) after deposition of silver film, the copper grid was removed and the prestrain was released to the original length of PDMS; (e) the PDMS substrate was further stretched to a random length, namely L.

    图 2  PDMS基底上周期性厚度梯度Ag薄膜受单轴压缩的形貌演化及结构特征 (a)光学显微镜放大50倍的图片, 尺寸均为1042 μm × 1390 μm, 箭头指向代表压缩方向; (b)光学显微镜放大200倍的图片, 尺寸均为261 μm × 348 μm; (c)单个网孔内薄膜表面形貌的原子力显微镜图片; (d)不同位置的剖面线结构

    Figure 2.  Morphological evolutions and structural characteristics of periodic thickness-gradient Ag films on PDMS substrates under uniaxial compression: (a) Optical micrographs with 50× magnification. All the images have the size of 1042 μm × 1390 μm. The arrows represent the direction of compression. (b) Optical micrographs with 200× magnification. All the images have the size of 261 μm × 348 μm. (c) Atomic force microscopy (AFM) image of the film surface within a grid period. (d) Cross-sectional profiles of the film surface at different positions.

    图 3  (a)一个周期内褶皱波长沿着中心线的演化规律; (b)中心区域褶皱的波长与机械压缩应变的关系, 实线为手风琴模型的理论结果; (c)归一化的裂纹长度(裂纹长度与网孔宽度之比)与机械应变的关系, 实线为实验数据的线性拟合; (d)裂纹宽度与机械应变的关系, 插图为不同应变下的裂纹形貌, 尺寸均为70 μm × 16 μm

    Figure 3.  (a) Evolution of wrinkle wavelength along the center line within a grid period. (b) dependence of the wrinkle wavelength at the mesh center on the mechanical compressive strain. The solid line represents theoretical prediction of accordion model. (c) dependence of normalized crack length (the ratio of crack length to mesh width) on the mechanical strain. The solid line is a linear fit to the experimental data. (d) dependence of the crack width on the mechanical strain. The insets (both 70 μm × 16 μm) show the crack morphologies under different strains.

    图 4  PDMS基底上周期性厚度梯度Ag薄膜受单轴拉伸的形貌演化及结构特征: (a)光学显微镜放大50倍的图片, 尺寸均为1042 μm × 1390 μm, 箭头指向代表拉伸方向; (b)光学显微镜放大200倍的图片, 尺寸均为261 μm × 348 μm; (c)光学显微镜放大500倍的图片, 尺寸均为104 μm × 139 μm; (d)中心区域的褶皱波长与机械拉伸应变的关系, 实线为手风琴模型的理论结果; (e)沿中心线的裂纹宽度和裂纹数目随应变的演化规律; (f)总的裂纹宽度与机械应变的关系, 实线为实验数据的线性拟合.

    Figure 4.  Morphological evolutions and structural characteristics of periodic thickness-gradient Ag films on PDMS substrates under uniaxial tension: (a) Optical micrographs with 50× magnification. All the images have the size of 1042 μm × 1390 μm. The arrows represent the direction of tension. (b) Optical micrographs with 200× magnification. All the images have the size of 261 μm × 348 μm. (c) Optical micrographs with 500× magnification. All the images have the size of 104 μm × 139 μm. (d) Dependence of the wrinkle wavelength at the mesh center on the mechanical tensile strain. The solid line represents theoretical prediction of accordion model. (e) Evolutions of the crack width and crack number along the center line with the tensile strain. (f) Dependence of the total crack width on the tensile strain. The solid line is a linear fit to the experimental data.

    图 5  (a)周期性厚度梯度Ag薄膜的表面形貌, $\varepsilon = 0$; (b)沿中心线薄膜厚度分布的示意图; (c)膜厚最小区域和最大区域的薄膜横截面示意图. 注意: 图中薄膜厚度h与网孔周期w并不按比例绘制, 膜厚被极大地放大了, 事实上膜厚在纳米量级(最大约90 nm), 而网孔周期在微米量级(约125 μm), 它们之间相差3个数量级

    Figure 5.  (a) Surface morphology of a periodic thickness-gradient Ag film when $\varepsilon = 0$; (b) schematic diagram of the periodic thickness-gradient film along the center line. Note that the film thickness is greatly exaggerated. In fact, the grid period (w ~ 125 μm) is three orders of magnitude larger than the film thickness (hmax ~ 90 nm).

    图 6  (a)$\varepsilon = 20\% $时单个周期的典型结构特征; (b)在xy方向上, 褶皱波长随距离的演化规律; (c)裂纹数目和裂纹平均间距随距离y的变化规律; (d)在y方向上, 裂纹平均间距和褶皱波长的依赖关系, 实线为实验数据的线性拟合

    Figure 6.  (a) Typical surface morphology of Ag film within a grid period under $\varepsilon = 20\% $; (b) evolutions of the wrinkle wavelength with the distance both in x and y directions; (c) evolutions of the crack number and average crack spacing with the distance y; (d) dependence of the average crack spacing on the wrinkle wavelength in the y direction and the solid line is a linear fit to the experimental data.

    图 7  厚度梯度薄膜-基底体系的截面示意图

    Figure 7.  Schematic diagram of the thickness-gradient film-substrate system.

    图 8  有限元模拟结果 (a)一个周期内薄膜面内压应变的分布图; (b)褶皱幅值随薄膜面内压应变的演化规律; (c)一个周期内褶皱形貌的模拟图; (d)褶皱波长在一个周期的变化规律

    Figure 8.  Finite element simulation results: (a) Distribution of the in-plane compressive strain within a period; (b) evolution of the wrinkle amplitude with the in-plane compressive strain; (c) simulated wrinkle morphology within a period; (d) distribution of the wrinkle wavelength within a period.

  • [1]

    Li B, Cao Y P, Feng X Q, Gao H J 2012 Soft Matter 8 5728

    [2]

    许宏, 苑争一, 黄彤飞, 王啸, 陈正先, 韦进, 张翔, 黄元 2020 物理学报 69 026101

    Xu H, Yuan Z Y, Huang T F, Wang X, Chen Z X, Wei J, Zhang X, Huang Y 2020 Acta Phys. Sin. 69 026101

    [3]

    倪勇, 刘佩琳, 马龙, 李世琛, 何陵辉 2018 固体力学学报 39 113

    Ni Y, Liu P L, Ma L, Li S C, He L H 2018 Chin. J. Solid Mech. 39 113

    [4]

    Lee W K, Odom T W 2019 ACS Nano 13 6170

    [5]

    Ma L, He L, Ni Y 2020 J. Appl. Phys. 127 111101

    [6]

    冯雪, 陆炳卫, 吴坚, 林媛, 宋吉舟, 宋国锋, 黄永刚 2014 物理学报 63 014201

    Feng X, Lu B W, Wu J, Lin Y, Song J Z, Song G F, Huang Y G 2014 Acta Phys. Sin. 63 014201

    [7]

    Kim H N, Ge D, Lee E, Yang S 2018 Adv. Mater. 30 1803847

    [8]

    Gao N, Zhang X, Liao S, Jia H, Wang Y 2016 ACS Macro Lett. 5 823

    [9]

    Lee W K, Jung W B, Rhee D, Hu J, Lucy Lee Y A, Jacobson C, Jung H T, Odom T W 2018 Adv. Mater. 30 1706657

    [10]

    王月桐, 商珞然, 赵远锦 2020 物理学报 69 084701

    Wang Y T, Shang L R, Zhao Y J 2020 Acta Phys. Sin. 69 084701

    [11]

    Yu S, Liu X, Sun Y, Zhou H, Cai P 2019 Thin Solid Films 669 355

    [12]

    Yin J, Yagüe J L, Eggenspieler D, Gleason K K, Boyce M C 2012 Adv. Mater. 24 5441

    [13]

    Bowden N, Brittain S, Evans A G, Hutchinson J W, Whitesides G M 1998 Nature 393 146

    [14]

    Wu D, Yin Y J, Xie H M, Shang Y F, Li C W, Wu L F, Dai X L 2014 Sci. China-Phys. Mech. Astron. 57 637

    [15]

    Um D S, Lim S, Lee Y, Lee H, Kim H, Yen W C, Chueh Y L, Ko H 2014 ACS Nano 8 3080

    [16]

    Wang D, Cheewaruangroj N, Li Y, McHale G, Jiang Y, Wood D, Biggins J S, Xu B B 2018 Adv. Funct. Mater. 28 1704228

    [17]

    Chan E P, Crosby A J 2006 Soft Matter 2 324

    [18]

    Ouchi T, Yang J, Suo Z, Hayward R C 2018 ACS Appl. Mater. Interfaces 10 23406

    [19]

    Ding W, Yang Y, Zhao Y, Jiang S, Cao Y, Lu C 2013 Soft Matter 9 3720

    [20]

    Gurmessa B J, Croll A B 2017 Soft Matter 13 1764

    [21]

    Nogales A, Del Campo A, Ezquerra T A, Rodriguez-Hernandez J 2017 ACS Appl. Mater. Interfaces 9 20188

    [22]

    Li B, Zeng C Q, Yin S F, Feng X Q 2019 Sci. China-Technol. Sci. 62 747

    [23]

    Ni Y, Yang D, He L 2012 Phys. Rev. E 86 031604

    [24]

    Yu S, Ni Y, He L, Ye Q L 2015 ACS Appl. Mater. Interfaces 7 5160

    [25]

    Yu S, Ma L, Sun Y, Lu C, Zhou H, Ni Y 2019 Langmuir 35 7146

    [26]

    Zhang Q, Tang Y, Hajfathalian M, Chen C, Turner K T, Dikin D A, Lin G, Yin J 2017 ACS Appl. Mater. Interfaces 9 44938

    [27]

    Cerda E, Mahadevan L 2003 Phys. Rev. Lett. 90 074302

    [28]

    Sun J Y, Xia S, Moon M W, Oh K H, Kim K S 2012 Proc. R. Soc. A 468 932

    [29]

    Marthelot J, Roman B, Bico J, Teisseire J, Dalmas D, Melo F 2014 Phys. Rev. Lett. 113 085502

    [30]

    Kappert E J, Pavlenko D, Malzbender J, Nijmeijer A, Benes N E, Tsai P A 2015 Soft Matter 11 882

    [31]

    Volynskii A L, Bazhenov S, Lebedeva O V, Ozerin A N, Bakeev N F 1999 J. Appl. Polym. Sci. 72 1267

    [32]

    Chen X, Hutchinson J W 2004 J. Appl. Mech. 71 597

    [33]

    Jiang H, Khang D Y, Song J, Sun Y, Huang Y, Rogers J A 2007 Proc. Natl. Acad. Sci. U. S. A. 104 15607

    [34]

    Huang Z Y, Hong W, Suo Z 2005 J. Mech. Phys. Solids 53 2101

    [35]

    Pan K, Ni Y, He L, Huang R 2014 Int. J. Solids Struct. 51 3715

    [36]

    Cao G, Chen X, Li C, Ji A, Cao Z 2008 Phys. Rev. Lett. 100 036102

    [37]

    Zhang X, Mather P T, Bowick M J, Zhang T 2019 Soft Matter 15 5204

    [38]

    Hibbett Karlsson Sorensen Inc. 2001 ABAQUS/Standard user's manual: version 6.2

    [39]

    Riks E 1979 Int. J. Solids Struct. 15 529

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Publishing process
  • Received Date:  28 June 2020
  • Accepted Date:  02 September 2020
  • Available Online:  12 January 2021
  • Published Online:  05 January 2021

Strain effects of periodic thickness-gradient films on flexible substrates

    Corresponding author: Yu Sen-Jiang, sjyu@hdu.edu.cn
    Corresponding author: Li Ling-Wei, lingwei@hdu.edu.cn
  • 1. Key Laboratory of Novel Materials for Sensor of Zhejiang Province, College of Materials and Environmental Engineering, Hangzhou Dianzi University, Hangzhou 310018, China
  • 2. School of Mechanical Engineering, Hangzhou Dianzi University, Hangzhou 310018, China
  • 3. CAS Key Laboratory of Mechanical Behavior and Design of Materials, Department of Modern Mechanics, University of Science and Technology of China, Hefei 230026, China

Abstract: Controlled surface structures have important applications in the fields of flexible electronics, bionic devices, smart materials and surface engineering. Although various instability modes induced by stresses in homogeneous and heterogeneous film systems have been investigated extensively in the past decades, understanding the stress relief mechanisms of gradient film systems is still a challenge. In this work, periodic thickness-gradient metal silver films on flexible polydimethylsiloxane (PDMS) substrates are prepared by using magnetron sputtering technique and by masking weaved copper grid during deposition. The morphological evolutions and structural characteristics of the gradient films in the uniaxial mechanical compression/tension process are detected by using optical microscopy and atomic force microscopy. It is found that the thickness-gradient film spontaneously forms on the PDMS surface due to the specific three-dimensional structure of the weaved copper grid. The maximum film thickness in the mesh center is about twice larger than the minimum one in the region covered by the copper wire. Under the uniaxial mechanical loading, the silver film surface will form stripe wrinkles and straight cracks, which are aligned perpendicular to each other. The variation of film thickness can well modulate the in-plane strain distribution of the film and thus the stress relief patterns. As the compressive strain increases, the wrinkles first form in the region with smaller film thickness, and then extend into the region with larger thickness. The wrinkle wavelength decreases with compressive strain increasing, which is in consistent with the theoretical prediction of accordion model. The wavelength and amplitude of the wrinkles exhibit periodically oscillatory behaviors across the film surface. As the tensile strain increases, the cracks start to propagate in the region with smaller thickness, and then the crack width and number both increase gradually, leading the multiple crack modes to form finally. The crack width and average spacing both increase with film thickness increasing. The morphologies and evolutions of such wrinkles and cracks in the thickness-gradient films are analyzed in depth based on the stress theory. The wrinkle patterns are further simulated by the finite element method, and the simulation results are in good agreement with the experimental results. This study could promote a better understanding of the strain effects of heterogeneous film systems and find the applications in the field of flexible electronics by the structural design of thickness-gradient films.

    • 薄膜-基底双层(或多层)结构体系在自然界中普遍存在, 如地质板块、动物皮肤、植物果实和生物组织[1,2]. 薄膜材料当前也被广泛应用于各类高科技的工程领域中, 如热障涂层、磁存储介质、光学器件、微机电系统等. 由于受到基底的界面约束, 薄膜中往往存在较大的残余内应力, 促使薄膜表面产生褶皱、脱层和开裂等失稳模态[3-5]. 这些模态通常被视为薄膜的力学失效, 严重影响器件的功能质量和使用寿命. 然而, 最近的研究发现, 可控的表面微结构(尤其是褶皱模态)在柔性电子[6]、弹性光学[7]、压力传感[8]、表面润湿[9]、生物界面[10]等领域都具有重要的应用价值, 因而受到了研究人员的极大关注. 人们通过对刚性薄膜-弹性基底体系施加单轴或双轴压缩应力来构筑各类褶皱结构. 对于均匀的薄膜体系, 单轴应力产生周期分布的条纹型褶皱[11], 而双轴应力将诱导出无序的迷宫型褶皱或有序的人字型结构[12].

      最近, 非均匀薄膜体系中的褶皱现象也引起了人们的极大兴趣. 事实上, 不管是自然界中的地质板块和生物组织, 还是工程技术中的人工微结构, 薄膜体系都是不均匀分布的. 因而, 研究非均匀体系的褶皱现象及其对外界的响应具有更加重要的意义. 构筑非均匀薄膜体系最常用的方法是引入周期性的台阶或缺陷等结构. Bowden等[13]利用边界诱导的应力各向异性调控出有序的褶皱斑图; Wu等[14]利用预制裂纹的曲率诱导或抑制薄膜的褶皱; Um等[15]利用聚二甲基硅氧烷(PDMS)的微孔结构调控出有序褶皱阵列; Wang等[16]探索了单轴载荷下微孔型薄膜体系的褶皱形貌演化. 利用金属板或光刻胶掩模是构筑非均匀薄膜体系的另一类方法, 可制备各种类型的薄膜体系. Chan和Crosby[17]对PDMS表面进行选择性氧化处理, 调控出取向生长的褶皱斑图; Ouchi等[18]报道了PDMS基底上周期性光刻胶阵列中由单轴压缩诱导的多重褶皱模态; Ding等[19]通过铜网掩模在PDMS表面形成正交的褶皱斑图; Gurmessa和Croll[20], Nogales等[21]研究了铜网掩模构筑的薄膜体系在单轴载荷作用时的力学响应; Li等[22]则进一步探索了铜网构筑的异质薄膜体系在双轴载荷下的褶皱结构.

      上述提到的非均匀薄膜体系通常在边界处具有不连续的特性, 比如薄膜厚度、基底高度、薄膜或基底的硬度等具有突变性. 最近, 人们对柔性基底上梯度薄膜的褶皱现象也进行了探索. 研究发现, 单轴压缩通常导致条纹褶皱的自发分叉[23], 而双轴压缩与厚度梯度的竞争将导致条纹型、人字型和迷宫型褶皱的共存[24]. 在此基础上, 本研究组利用编织铜网构筑了周期性的厚度梯度薄膜, 研究了厚度梯度对自发的褶皱形貌的调控效应[25]. 然而到目前为止, 周期性厚度梯度薄膜在外加载荷作用下的响应行为和力学机制, 尚不清楚. 本文采用光学显微镜和原子力显微镜研究柔性PDMS基底上周期性厚度梯度银薄膜在单轴压缩/拉伸过程中的形貌演化, 重点分析褶皱和裂纹的分布特征和演化行为, 并采用应力理论和有限元模拟对实验现象给出了合理解释.

    2.   实验方法
    • 实验用的柔性基底为聚二甲基硅氧烷(PDMS, Sylgard 184, Dow Corning), 按主剂和固化剂10∶1的重量比混合, 用玻璃棒搅拌均匀后水平静置30 min. 待气泡完全消除后, 将混合溶液倾倒在平整的器皿内, 随后在温度为70 °C的加热台上固化5 h, 使PDMS完全交联成为弹性聚合物材料. 固化后的PDMS厚度约为1.2 mm, 用刀具将其切割成宽为12 mm的矩形片. 使用自制的拉伸装置将PDMS沿长度方向固定, 夹具内的PDMS长度为25 mm, 然后将PDMS预拉伸到30 mm, 即预应变为20%(图1(a)). 在预拉伸的PDMS上覆盖编织铜网, 表面贴紧(图1(b)). 铜网由直径约为52 μm的铜线正交编织而成, 网孔的宽度(即周期)w约为125 μm(图1(c)). 应当指出, 当网孔周期与褶皱波长相当时, 它们之间会存在复杂的竞争[18,21,22]. 而本文采用的网孔周期比褶皱波长大很多, 不需要考虑这种竞争关系. 本文主要通过薄膜的厚度梯度诱导应力的重排, 从而调控褶皱和裂纹的形成与分布.

      Figure 1.  Schematic illustration of sample preparation and loading sequence of periodic thickness-gradient films on PDMS substrates: (a) The PDMS substrate was stretched to 20% prestrain; (b) a weaved copper grid was placed on the pre-stretched PDMS surface; (c) optical micrograph of the weaved copper grid; (d) after deposition of silver film, the copper grid was removed and the prestrain was released to the original length of PDMS; (e) the PDMS substrate was further stretched to a random length, namely L.

    • 采用直流磁控溅射技术在室温条件下沉积金属银薄膜, 真空腔的本底真空优于2 × 10–4 Pa, 溅射时氩气压固定在0.5 Pa, 靶基距约为80 mm, 溅射功率为76 W (电流为0.2 A, 电压为380 V). 在此功率下银薄膜的沉积速率约为30 nm/min, 溅射时间为3 min, 即平整处薄膜厚度约为90 nm. 本实验采用铜网掩模的方法, 因而薄膜厚度并不均匀分布. 由于编织铜网特定的空间三维结构, 铜线与PDMS表面并不都紧密接触, 而是大部分呈悬空状态. 在薄膜沉积过程中, 金属原子受到氩原子等粒子的频繁碰撞而改变方向, 部分金属原子能沉积到铜线覆盖处的PDMS表面, 因而在该处自然形成厚度呈连续变化的梯度薄膜[24,25]. 由于铜网为周期性排布的阵列结构, 因而最终形成的薄膜呈周期性的厚度梯度变化.

    • 沉积完成后, 将编织铜网小心移除, 缓慢释放预应变至PDMS的原长(图1(d)), 然后再次回到20%的预应变状态, 并进一步拉伸PDMS至任意长度(图1(e)). 实验上, 也可以先进行拉伸再逐渐释放应变, 甚至改变预应变的值. 结果表明, 应变的施加方式会改变褶皱和裂纹的形成顺序及朝向, 但不会改变褶皱和裂纹的形貌特征和演化规律. 设薄膜的原长为L0, 释放/拉伸过程中的任意长度为L, 则薄膜的机械应变可表示为$\varepsilon = ({{L - {L_0}}})/{{{L_0}}}$. 在预应变释放过程中, L小于L0, 机械应变为负值, 表示薄膜在加载方向上受压缩应力; 而在进一步拉伸过程中, L大于L0, 机械应变为正值, 表示薄膜在加载方向上受拉伸应力. 薄膜的表面形貌由光学显微镜(Olympus BX41)和原子力显微镜(AFM, JPKSPM)进行观察和拍摄. 梯度薄膜的褶皱行为采用有限元方法进行模拟计算.

    3.   实验结果与讨论
    • 图2(a)(b)为周期性厚度梯度薄膜在预应变释放过程中的原位形貌演化规律. 由图2可知, 编织铜网的存在使得薄膜表面形成与铜网结构一致的周期性阵列, 样品表面深色部分为铜线覆盖区域, 膜厚较小, 浅色部分为网孔中心区域, 膜厚较大. 由于薄膜的残余应力作用, 制备态的样品($\varepsilon = 0$)自发形成了褶皱结构, 其主要分布在膜厚较小区域. 随着预应变的释放, 膜厚较小处开始形成新的褶皱结构, 并逐渐扩展至网孔中心处. 由于泊松效应, PDMS在载荷方向上的收缩将使其在垂直方向上产生拉伸[11,26]. PDMS的泊松比约为0.48, 因而垂直方向上产生的应变约为机械应变的一半. 在$\varepsilon \approx - 3.3\% $时, 受横向拉伸应变的作用, 薄膜开始在垂直于褶皱的方向上出现裂纹[11,26]. 裂纹的长度和数量随着机械应变的增加而增加, 但都限定在膜厚较小的区域.

      Figure 2.  Morphological evolutions and structural characteristics of periodic thickness-gradient Ag films on PDMS substrates under uniaxial compression: (a) Optical micrographs with 50× magnification. All the images have the size of 1042 μm × 1390 μm. The arrows represent the direction of compression. (b) Optical micrographs with 200× magnification. All the images have the size of 261 μm × 348 μm. (c) Atomic force microscopy (AFM) image of the film surface within a grid period. (d) Cross-sectional profiles of the film surface at different positions.

      图2(c)显示了单个网孔内薄膜表面形貌的原子力显微镜图像, 图2(d)为不同位置的剖面线结构. 由图2可知, 褶皱均沿着垂直于机械载荷的方向排列, 呈条纹型结构, 这是由单轴机械压缩引起的[11,26]. 沿着网孔中心线的褶皱(剖面线1)具有良好的周期性变化规律, 其波长和振幅均呈现振荡行为, 在网孔中心处, 波长和振幅达到最大值, 随后逐渐衰减, 至铜网覆盖处达到最小. 偏离网孔中心线的褶皱(剖面线2)仍显示较好的振荡规律, 但波长和振幅均相应变小. 铜线覆盖处(剖面线3)褶皱的波长和振幅都显著变小, 但仍显示一定的振荡规律. 在原子力显微图的右下角(剖面线3的右侧), 褶皱结构消失, 这是因为该处正好是铜线与PDMS表面的接触点, 薄膜厚度为零. 采用编织铜网作掩模来构筑周期性厚度梯度薄膜, 简单方便, 但接触点导致的薄膜不均匀性(或缺陷)没法避免. 可以采用TEM用的平板铜网, 在沉积过程中抬起一定的高度来解决这一问题, 这也是后续要研究的课题.

      为了进一步理解预应变释放过程中薄膜结构的演化规律, 系统测量了褶皱波长、裂纹长度和宽度等信息(图3). 图3(a)为预应变释放完毕后褶皱波长在一个周期范围内的分布规律, 可以发现, 中心区域的褶皱波长最大, 而铜网覆盖处的波长最小, 最大波长和最小波长之比接近于2, 与之前的实验结果一致[25]. 图3(b)为中心区域的褶皱波长(即最大波长)随机械应变的演化规律, 可以发现褶皱波长随着压缩应变的增加而持续减小.

      Figure 3.  (a) Evolution of wrinkle wavelength along the center line within a grid period. (b) dependence of the wrinkle wavelength at the mesh center on the mechanical compressive strain. The solid line represents theoretical prediction of accordion model. (c) dependence of normalized crack length (the ratio of crack length to mesh width) on the mechanical strain. The solid line is a linear fit to the experimental data. (d) dependence of the crack width on the mechanical strain. The insets (both 70 μm × 16 μm) show the crack morphologies under different strains.

      根据连续介质弹性理论, 当薄膜-基底体系受到的压缩应力超过某个临界值时, 薄膜将产生具有面外变形的褶皱结构. 该临界应变可表示为[13,27]

      其中, E为杨氏模量, $\nu $为泊松比, 下标fs分别代表薄膜和基底. (1)式显示褶皱的临界应变只与薄膜和基底的杨氏模量和泊松比有关, 与薄膜厚度等参数无关. 对于Ag/PDMS体系, 银薄膜厚度较小, 直接测定其杨氏模量比较困难, 因而采用银的材料参数(杨氏模量约为70 GPa); 而PDMS的杨氏模量通过材料拉伸机测定, 约为2 MPa. 因而该体系的临界应变非常小(约0.05%), 说明在金属薄膜/PDMS体系中褶皱很容易形成, 这已被众多实验证实[13,24,25]. 需要指出的是, 褶皱形成的临界应变不仅与薄膜厚度无关, 也不受薄膜厚度梯度的限制. 本实验中观察到膜厚较小区域先形成褶皱, 是由于该区域的应变较大, 率先超过褶皱的临界应变所致(详见下文阐述).

      膜基体系的能量最小化(薄膜的弯曲能和基底的变形能)可导致稳定的褶皱波长, 表示为[13,27]

      其中h为薄膜厚度. 需要指出的是(2)式仅适用于非常小的应变条件(接近于褶皱的临界应变). 在大应变条件下, 褶皱的波长可根据手风琴模型简单修正为[28]

      其中, $\varepsilon $为压缩应变, ${\varepsilon _w}$为褶皱形成的临界应变. 由图3(b)可知, (3)式的理论预言(红色实线)与实验结果(空心方块)一致.

      另一方面, 在实验中, 褶皱的临界应变、薄膜和基底的杨氏模量和泊松比均为常数, 当预应变释放完毕后, 薄膜的压缩应变也为常数(约16.7%). 因而, 根据(2)式和(3)式, 褶皱的波长只与薄膜厚度有关, 可表示为

      褶皱波长与膜厚的正比关系已在先前的研究中得到充分证实[11,24-26]. 本实验中, 薄膜厚度的空间分布很难直接测定, 可以根据褶皱的波长反推薄膜的厚度. 图3(a)事实上也反映了薄膜厚度的周期性变化规律, 最大膜厚与最小膜厚的比值约为2∶1. 图2(c)的原子力显微图则更加直观地展示了薄膜厚度的空间分布规律, 在网孔中心处膜厚具有最大值, 并沿着四周逐渐衰减, 可理想化成球冠或圆锥形结构. 为了进一步验证膜厚的变化规律, 在硅片上沉积梯度薄膜, 分别采用台阶仪和扫描电镜观测薄膜的表面轮廓和剖面结构, 证实沿着中心线的最大膜厚约为最小膜厚的2倍[25].

      此外, 在预应变的释放过程中, 泊松效应引起的拉伸应变会导致薄膜产生垂直于褶皱的裂纹(图2(b)). 所有裂纹都限定在膜厚较小的区域, 开始为单条裂纹的逐步扩展, 随后呈现多重裂纹的平行排布. 图3(c)显示了裂纹长度与机械应变的定量关系, 对裂纹长度进行归一化处理, 即采用裂纹长度与网孔周期的比值Lcrack/w. 由图3(c)可知, 随着机械应变的增加, Lcrack/w呈线性增加. 当Lcrack/w < 1时, 薄膜的机械应变较小, 表现为单条裂纹的逐步扩展. 当Lcrack/w = 1时, 裂纹正好贯穿整个网孔, 表现为周期性阵列内单条裂纹相互连接, 图3(c)机械应变约为6.7%. 当Lcrack/w > 1时, 膜厚较小区域不再仅限于单条裂纹, 即裂纹的数目随着机械应变的增加而增加. 图3(d)显示了裂纹的宽度随机械应变的演化规律, 随着机械应变的增加, 裂纹的宽度先快速增加, 随后逐渐达到饱和值(约为4.5 μm). 当裂纹的宽度和长度达到饱和之后, 薄膜将产生新的裂纹来释放增加的拉伸应变, 因而产生多重裂纹模态.

    • 在薄膜器件(特别是柔性电子器件)的使用过程中, 其往往受到复杂的载荷作用, 而拉伸应变对器件的力学稳定性和使用寿命影响更大. 因而, 在机械压缩的基础上, 进一步考察了Ag/PDMS体系在单轴拉伸载荷下的结构演化. 图4(a)(b)为周期性厚度梯度薄膜在拉伸过程中的原位形貌演化规律. 可以发现, 在平行于加载方向上出现褶皱, 而在垂直方向上则出现裂纹, 该结果正好与单轴压缩时相反, 这是由应力的性质(拉伸或压缩)决定的. 与单轴压缩情况相似的是, 褶皱最先在膜厚较小区域产生, 随后逐渐扩展到膜厚较大区域. 图4(d)显示中心处褶皱的最大波长随着机械应变的增加而逐渐减小, 与图3(b)的结果一致. 同样, 褶皱的波长也符合手风琴模型的理论预测(图4(d)中红色实线).

      Figure 4.  Morphological evolutions and structural characteristics of periodic thickness-gradient Ag films on PDMS substrates under uniaxial tension: (a) Optical micrographs with 50× magnification. All the images have the size of 1042 μm × 1390 μm. The arrows represent the direction of tension. (b) Optical micrographs with 200× magnification. All the images have the size of 261 μm × 348 μm. (c) Optical micrographs with 500× magnification. All the images have the size of 104 μm × 139 μm. (d) Dependence of the wrinkle wavelength at the mesh center on the mechanical tensile strain. The solid line represents theoretical prediction of accordion model. (e) Evolutions of the crack width and crack number along the center line with the tensile strain. (f) Dependence of the total crack width on the tensile strain. The solid line is a linear fit to the experimental data.

      图4(b)可知, 尽管裂纹也最先在膜厚较小区域产生, 并基本限定在这一区域, 但其结构和数目与单轴压缩时具有显著差异. 为此, 特别考察了单轴拉伸时薄膜裂纹的原位演化行为(图4(c)). 当机械应变$\varepsilon = 0$时, 膜厚较小区域没有出现裂纹, 仅存在由残余应力引起的轻微的薄膜褶皱. 随着机械应变的增加, 当$\varepsilon \approx 1.7\% $时, 裂纹开始出现, 并直接贯穿整个网孔. 随后裂纹的宽度和数目逐渐增加, 并最终形成平行的多重裂纹模态. 本实验施加的最大拉伸应变为$\varepsilon \approx 26.7\% $, 此时, 裂纹已经扩展到膜厚较大的区域, 甚至网孔中心处也开始形成裂纹(图4(b)).

      图4(e)显示了沿着中心线(图4(c)中的红色虚线)的裂纹宽度和裂纹数目随应变的演化规律. $\varepsilon \leqslant 6.7\% $时, 中心处只出现单条裂纹, 其宽度快速增加并逐渐达到饱和(约为4.5 μm), 与图3(d)的结果类似. 可以认为这一过程中拉伸应变是以裂纹宽度的增加来承担的. 当$\varepsilon = 10.0\% $$13.3\% $时, 裂纹数目快速增加, 而单条裂纹的宽度则明显减小, 表明多重裂纹的出现有效分担了拉伸应变. 随着应变的继续增加, 裂纹数目呈缓慢增加状态, 而裂纹宽度基本保持在4 μm左右. 显然, 裂纹数目和裂纹宽度之间存在相互竞争, 它们共同承担了薄膜的拉伸应变能. 图4(f)显示了总的裂纹宽度(沿中心线裂纹宽度之和)随机械应变的演化规律, 可以发现它们之间存在良好的线性关系.

      根据材料的断裂理论[29,30], 当薄膜中储存的弹性应变能超过材料的断裂能时, 薄膜将产生开裂, 即$2\gamma he \geqslant {G_{\rm{c}}}h$. 其中$e = {{h{\sigma ^2}(1 - {\nu _f})} / {{E_f}}}$为薄膜的弹性能量密度; ${G_{\rm{c}}}$为薄膜的断裂能密度, 即断裂韧性; $\gamma $为一个重要的无量纲参数, 其值取决于断裂模式和膜基的弹性模量失配度; $\sigma $为拉伸应力, 在各向同性材料中, 应力与应变之间具有简单的正比关系, 即$\sigma = E\varepsilon $. 因而上式可整理成

      其中G为薄膜的能量释放率. 本实验中, 薄膜厚度呈周期性变化, 导致应力(或应变)也并不均匀分布在膜面上. 周期性厚度梯度薄膜的表面形貌($\varepsilon = 0$)和膜厚分布如图5(a)图5(b)所示. 考虑单个周期的情况, 应力可表示成$\sigma = F/S$, 其中F为薄膜两侧的拉力(或压力), $S = wh$为薄膜的横截面. 力具有传递效应, 因而在不同位置力是相等的, 但薄膜横截面与膜厚呈正比, 也呈周期性变化(图5(c)). 因而应力与薄膜厚度之间存在简单的反比关系, 即$\sigma \propto {h^{ - 1}}$. 显然, 在膜厚较小区域, 应力(或应变)具有较大值, 或者说应力倾向于集中在膜厚较小区域, 并率先超过褶皱的临界应力. 因而, 褶皱总是在铜网覆盖区域率先形成, 并逐渐扩展到网孔中心区域(图2图4).

      Figure 5.  (a) Surface morphology of a periodic thickness-gradient Ag film when $\varepsilon = 0$; (b) schematic diagram of the periodic thickness-gradient film along the center line. Note that the film thickness is greatly exaggerated. In fact, the grid period (w ~ 125 μm) is three orders of magnitude larger than the film thickness (hmax ~ 90 nm).

      $\sigma \propto {h^{ - 1}}$代入公式(5), 并考虑到薄膜的杨氏模量、泊松比和$\gamma $值均为不变量, 可得

      即能量释放率与薄膜厚度呈反比. 因而膜厚较小区域具有较大的能量释放率, 同时薄膜的断裂韧性为一常量. 随着拉伸应变的增加, 膜厚较小区域的能量释放率将率先超过薄膜的断裂韧性, 裂纹开始萌生. 该裂纹将有效释放附近的拉伸应力, 使周围薄膜的能量释放率低于临界值, 从而抑制了其它裂纹的萌生. 此时拉伸应变主要通过单条裂纹的长度和宽度的增加得以释放. 当该裂纹的宽度达到饱和之后, 将不能再有效地释放应变能, 此后周围薄膜的能量释放率快速增加, 并超过薄膜的断裂韧性, 从而形成多重裂纹模态. 实验显示, 多重裂纹基本都限定在膜厚较小的区域, 至少在20%的拉伸应变作用下, 网孔中心区域的薄膜仍保持完好(图4(b)).

      为了进一步研究拉伸应变作用下厚度梯度薄膜的形态分布, 考察$\varepsilon = 20\% $时单个周期的结构特征(图6(a)). 方便起见, 选取如图所示的坐标系, 膜厚最小处为原点, 加载方向为x轴, 垂直方向为y轴. 样品在xy两个方向上都具有厚度梯度, 因而褶皱波长随着x的增加而增加, 随着y的增加而减小, 如图6(b)所示, 这一结果与压缩应变诱导的褶皱类似. 另一方面, 在y方向上, 裂纹的形态和数目有显著变化. 随着y的增加, 裂纹的数目增加, 宽度变小, 同时裂纹的平均间距也变小(图6(c)). 根据材料的断裂理论, 薄膜裂纹的平均间距可以表示为[26,31]

      Figure 6.  (a) Typical surface morphology of Ag film within a grid period under $\varepsilon = 20\% $; (b) evolutions of the wrinkle wavelength with the distance both in x and y directions; (c) evolutions of the crack number and average crack spacing with the distance y; (d) dependence of the average crack spacing on the wrinkle wavelength in the y direction and the solid line is a linear fit to the experimental data.

      其中, $\varepsilon $为拉伸应变, ${\varepsilon _{\rm{c}}}$为裂纹萌生的临界应变(本实验中其值约为1.7%).

      对于厚度均匀的薄膜样品, 裂纹的萌生位置是随机的, 一般由薄膜缺陷诱导应力集中, 并进一步引起裂纹扩展. 裂纹形成之后, 能够有效释放周边薄膜的应力能. 在开裂后的薄膜片中, 应力在膜片中间区域具有最大值, 越靠近裂纹边界, 应力值越小. 因而, 新产生的裂纹容易在膜片中间区域萌生, 这一过程将随着应变的增加不断持续下去, 直到形成稳定的裂纹斑图. 由此形成的裂纹通常具有相仿的间距, 并且间距随应变的增加呈指数下降规律[26,31]. 而本实验研究的厚度梯度薄膜样品, 裂纹总是从膜厚较小区域逐渐扩展到膜厚较大区域(图4). 在扩展过程中, 它们的间距保持不变. 由于基底模量、拉伸应变、临界断裂应变和薄膜断裂韧性均为常数, 裂纹的平均间距与薄膜厚度成正比, 而由(4)式可知薄膜厚度与褶皱波长成正比. 因而, (7)式可以写成

      图6(d)显示了在y方向上, 裂纹的平均间距和褶皱的波长之间具有良好的线性依赖关系, 与公式(8)的理论预言吻合. 与均匀薄膜样品相比, 周期性厚度梯度薄膜具有截然不同的裂纹扩展行为, 裂纹在膜厚较小区域的限定效应可为柔性电子器件的结构设计提供有益指导. 只要把柔性电子器件设计成厚度梯度的结构, 并将需要特别保护的单元放置在膜厚较大区域, 就能够承载很大的机械应变而不发生断裂失效.

    • 为了理解薄膜体系的失稳行为及其后屈曲演化, 表面屈曲失稳的稳定性分析成为当前研究的热点. Cerda和Mahadevan[27]通过标度律法研究了薄膜拉伸产生的横向表面褶皱与材料性质的关系; Chen等[32]通过冯·卡门板理论, 得到薄膜-基底体系在均匀单向压应力下的面外正弦型褶皱波长与临界应力的理论解; Jiang等[33]将前者适用于小变形的理论进行了修正, 得到考虑有限变形影响的薄膜褶皱波长和波幅的理论解. 而对于二维和三维复杂系统, 稳定性的后屈曲分析由于涉及到失稳分岔路径的选择和非线性问题求解, 很难得到失稳理论解, 因此数值求解成为常用的分析方法. Huang等[34]通过数值频谱方法得到基底上软膜在不同的双轴加载应力下面外褶皱形貌的选择和转变; Pan等[35]通过相场法研究了薄膜-基底体系的面外褶皱和脱粘翘曲行为; Cao等[36]通过有限元法模拟了球型核壳结构在热应力作用下的表面屈曲形貌和对应的后屈曲演化; Zhang等[37]利用有限元法模拟了甜甜圈型核壳结构在热应力作用下产生的多种失稳褶皱形貌共存的现象, 并研究了曲率不均匀对褶皱形貌产生的影响.

      本文通过有限元方法模拟厚度梯度薄膜单轴受压时的屈曲褶皱行为, 利用有限元计算软件ABAQUS[38]来模拟厚度梯度薄膜体系在均匀单向压应力下的屈曲行为, 并追踪其褶皱的形成和演化过程. 首先对体系进行线性屈曲分析, 此步骤为特征值求解问题, 可以得到体系的屈曲模态和对应的特征值, 以此确定薄膜褶皱的临界应变以及褶皱发生的起始位置; 再将体系线性屈曲的初始模态作为结构的初始几何缺陷引入体系, 通过弧长法[39]来计算褶皱的后屈曲演化, 所引入的微小几何缺陷大小是薄膜最小厚度的3%, 来保证后屈曲模拟的稳定性和收敛性. 为简化计算, 只对薄膜的一个厚度梯度周期进行模拟, 且只考虑薄膜沿一个方向的厚度梯度, 几何截面如图7所示. 按照实验数据, 铜网的周期设定为w = 125 μm, 薄膜的最大膜厚设为h0 = 90 nm, 位于薄膜中间, 最小膜厚设为最大膜厚的一半, 位于左右两边, 将膜厚随位置的变化简化为线性. 薄膜材料为Ag, 杨氏模量${E_f} = 70$ GPa, 泊松比${\nu _f} = 0.38$; PDMS基底的杨氏模量${E_s} = 2$ MPa, 泊松比${\nu _s} = 0.48$. 薄膜和基底均采用八节点缩减积分单元C3D8R[38], 单元尺寸的选择保证长度方向上每个波长存在20个单元, 薄膜沿厚度方向存在4个单元, 来保证模拟的准确性. 薄膜和基底之间采用理想粘结. 基底下表面限制法向位移和横向剪力, 体系左右两端施加相向的位移载荷, 按照实验条件, 施加的压应变为16%. 对体系先进行线性屈曲分析, 再进行非线性后屈曲分析, 得到其屈曲褶皱的演化过程.

      Figure 7.  Schematic diagram of the thickness-gradient film-substrate system.

      由于薄膜厚度梯度的存在, 当面内压应力未达到临界应力时, 面内应变的分布不均匀, 且与膜厚成反比, 薄膜两端厚度最小处的压应变最大, 而薄膜中间处的压应变最小(图8(a)). 最大应变与最小应变之比约为2∶1, 与上文通过力的传递效应推导的结果一致. 这就导致膜厚最小处率先达到临界应变, 首先发生屈曲, 通过面外褶皱释放应变, 并逐步向膜厚处扩展. 薄膜褶皱幅值与面内压应变关系曲线如图8(b)所示, 在薄膜屈曲褶皱的演化过程中, 模拟所得的临界屈曲应变为0.047%, 与通过(1)式所得理论值0.052%接近, 说明模拟结果可信.

      Figure 8.  Finite element simulation results: (a) Distribution of the in-plane compressive strain within a period; (b) evolution of the wrinkle amplitude with the in-plane compressive strain; (c) simulated wrinkle morphology within a period; (d) distribution of the wrinkle wavelength within a period.

      当薄膜全部屈曲后, 其褶皱形貌如图8(c)所示,沿长度方向, 褶皱波长与膜厚成正比. 图8(d)显示了褶皱波长沿一个周期的变化规律, 波长分布情况与薄膜厚度分布一致, 可从波长分布反推得到薄膜的厚度梯度, 波长最大值和最小值之比约为2∶1, 这与(2)式的理论结果及实验中光学显微镜下的褶皱形貌一致.

    4.   结 论
    • 本文采用编织铜网作为掩模板, 利用磁控溅射技术在柔性PDMS基底上制备了周期性厚度梯度的金属银薄膜, 采用光学显微镜和原子力显微镜研究了薄膜在单轴压缩/拉伸过程中的形貌演化规律, 所得的主要结论如下:

      1)由于编织铜网特殊的空间三维结构, 铜线覆盖处将自发形成厚度具有梯度的银薄膜, 最大膜厚与最小膜厚之比约为2∶1.

      2)随着压缩应变的增加, 褶皱在膜厚较小处率先形成, 并逐渐扩展到膜厚较大区域, 褶皱波长和振幅在空间呈周期性振荡行为, 波长与膜厚成正比关系; 褶皱波长随应变的增加而减小, 服从手风琴模型的理论预言.

      3)随着拉伸应变的增加, 裂纹开始在膜厚较小区域萌生扩展, 宽度增加, 最后形成多重裂纹模态; 裂纹的宽度和平均间距随薄膜厚度的增加而增加, 数目随膜厚的增加而减小.

      4)采用应力理论对周期性厚度梯度薄膜的褶皱和裂纹的形貌特征和演化规律进行分析讨论, 采用有限元方法对薄膜的屈曲褶皱进行模拟计算, 完美解释了实验结果.

      5)研究结果有助于深入理解周期性厚度梯度薄膜的应变效应, 并有望通过设计梯度薄膜阵列在柔性电子等领域获得应用.

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