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Article(id=1246045609900229495, tenantId=1146029695717560320, journalId=1241755870837649424, issueId=1246045605047415288, articleNumber=null, orderNo=null, doi=10.19636/j.cnki.cjsm42-1250/o3.2024.019, pmid=null, cstr=null, oa=null, hot=null, price=null, onlineType=0, articleFormat=0, articleType=null, articleTypeStr=null, receivedDate=1714924800000, receivedDateStr=2024-05-06, revisedDate=null, revisedDateStr=null, acceptedDate=null, acceptedDateStr=null, onlineDate=1775010023807, onlineDateStr=2026-04-01, pubDate=1729785600000, pubDateStr=2024-10-25, doiRegisterDate=null, doiRegisterDateStr=null, onlineIssueDate=1775010023807, onlineIssueDateStr=2026-04-01, onlineJustAcceptDate=null, onlineJustAcceptDateStr=null, onlineFirstDate=null, onlineFirstDateStr=null, sourceXml=null, magXml=null, createTime=1775010023807, creator=13701087609, updateTime=1775010023807, updator=13701087609, issue=Issue{id=1246045605047415288, tenantId=1146029695717560320, journalId=1241755870837649424, year='2024', volume='45', issue='5', pageStart='565', pageEnd='708', issueExtLink='null', onlineDate='null', pubDate='null', beforeIssueId=null, nextIssueId=null, price=null, status=1, issueComplete=1, articleOrder=1, issueType=-1, specialIssue=null, createTime=1775010022651, creator=13701087609, updateTime=1775010094858, updator=13701087609, preIssue=null, nextIssue=null, ext={EN=IssueExt(id=1246045907972636933, tenantId=1146029695717560320, journalId=1241755870837649424, issueId=1246045605047415288, language=EN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=), CN=IssueExt(id=1246045907972636934, tenantId=1146029695717560320, journalId=1241755870837649424, issueId=1246045605047415288, language=CN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=)}, issueFiles=null}, startPage=694, endPage=708, ext={EN=ArticleExt(id=1246045610239968127, articleId=1246045609900229495, tenantId=1146029695717560320, journalId=1241755870837649424, language=EN, title=Study on the Dynamic Behaviors in Contact Resonance Atomic Force Microscopy in Liquid Environments, columnId=1244229834482757770, journalTitle=Chinese Journal of Solid Mechanics, columnName=Research Paper, runingTitle=null, highlight=null, articleAbstract=

Contact resonance atomic force microscopy (CR-AFM) is a powerful technique that enables the measurement of topography and the mechanical properties of various materials at the micro/nanoscale. It can be used in both air and liquid environments. However, when CR-AFM is operated in a liquid environment, the dynamic behaviors of the microcantilever can be significantly different from those in air or vacuum due to the complex fluid-solid coupling of the microcantilever-liquid-sample system and the tip-sample interaction. In this study, we explore the effects of liquid density and viscosity, as well as tip-sample normalized contact stiffness and contact damping, on the dynamics of the AFM microcantilever in liquid environments. We treat the influence of the liquid on the dynamics of the AFM microcantilever as added mass and added damping. Our results show that in free vibration, the natural frequencies of the AFM microcantilever are primarily dominated by the liquid density, while the liquid viscosity plays a dominant role in the quality factor compared to the liquid density. Higher modes exhibit higher sensitivity to changes in liquid viscosity and liquid density. As the normalized tip-sample contact stiffness increases, a higher mode shows increased sensitivity to changes in normalized contact stiffness in a liquid environment. On the other hand, a lower mode is more sensitive to changes in normalized contact damping in a liquid environment. In addition, the dynamic responses of the AFM microcantilever under three different excitation approaches are compared and discussed. Variations in boundary conditions and hydrodynamic loads applied to the microcantilever under these approaches lead to diverse dynamic responses. The findings in this study are essential for the development of micro/nanoscale mechanical property imaging techniques using CR-AFM in liquid environments, as well as the improvement of measurement accuracy and sensitivity.

, correspAuthors=Xilong Zhou, authorNote=null, correspAuthorsNote=null, copyrightStatement=null, copyrightOwner=null, extLink=null, articleAbsUrl=null, sourceXml=null, magXml=null, pdfUrl=null, pdf=null, pdfFileSize=null, pdfExtLink=null, richHtmlUrl=null, mobilePdfUrl=null, reviewReport=null, pdfFirstPage=null, abstractGraph=null, abstractGraphContent=null, abstractVideo=null, citation=null, cebUrl=null, magXmlContent=null, mapNumber=null, authorCompany=null, fund=null, authors=null, authorsList=Changyun Yang, Xilong Zhou, Bangzhi Zhang), CN=ArticleExt(id=1246045621203877997, articleId=1246045609900229495, tenantId=1146029695717560320, journalId=1241755870837649424, language=CN, title=液体环境下接触共振原子力显微术探针动力学行为研究, columnId=1241831201896469478, journalTitle=固体力学学报, columnName=研究论文, runingTitle=null, highlight=null, articleAbstract=

接触共振原子力显微术是材料微纳米尺度力学性能测试的有力工具，可用于大气和液体环境. 接触共振原子力显微术在液体环境下工作时，探针同时受到微悬臂-流体-样品之间的流固耦合作用以及针尖与样品相互作用，使得液体环境下探针微悬臂动力学特性与真空环境下有显著差异. 本文将流体对探针微悬臂的影响等效为附加质量及附加阻尼，研究了液体环境下流体密度和粘度对其动力学特性的影响，以及针尖样品接触刚度和接触阻尼对其动力学特性的影响. 结果表明，与液体粘度相比，液体环境下流体密度对微悬臂固有频率影响占主导地位；与液体密度相比，流体粘度对微悬臂品质因子影响占主导地位；探针高阶模态对于流体粘度和密度变化具有更高的灵敏度；随着液体环境下针尖样品接触刚度的增加，高阶模态对高接触刚度具有更高的灵敏度；低阶模态对于接触阻尼变化的灵敏度更高. 此外，对微悬臂在不同激励方式下的动力学响应进行了对比研究，给出了液体环境不同激励方式下微悬臂动力学响应差异的原因. 论文研究结果对于开展液体环境下接触共振原子力显微术微纳米力学成像，提高测量准确度和灵敏度具有重要指导意义.

, correspAuthors=周锡龙, authorNote=null, correspAuthorsNote=
** E-mail：.
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language=null, rfNumber=[1], rfOrder=0, authorNames=周锡龙, 黄震阳, 李法新, journalName=固体力学学报, refType=null, unstructuredReference=周锡龙, 黄震阳, 李法新. 多模态原子力显微术研究进展[J]. 固体力学学报, 2022, 43(05): 519-540., articleTitle=多模态原子力显微术研究进展, refAbstract=null), Reference(id=1246045629621846233, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246045609900229495, doi=null, pmid=null, pmcid=null, year=2022, volume=43, issue=05, pageStart=519, pageEnd=540, url=null, language=null, rfNumber=[1], rfOrder=1, authorNames=Zhou X L, Huang Z Y, Li F, journalName=Chinese Jou-rnal of Solid Mechanics, refType=null, unstructuredReference=(Zhou X L, Huang Z Y, Li F. 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companyId=1246045622986457209, language=CN, country=null, province=null, city=null, postcode=null, companyName=null, departmentName=null, remark=2武汉理工大学新材料力学理论与应用湖北省重点实验室，武汉，430070)])], figs=[ArticleFig(id=1246045625352044712, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246045609900229495, language=EN, label=Fig.1, caption=The schematic diagram of the microcantilever in a liquid environment, figureFileSmall=H0kEbQ621iKu+UY/VoqCjg==, figureFileBig=2TJFWdgz7ltmtNX/2osUrQ==, tableContent=null), ArticleFig(id=1246045625452708010, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246045609900229495, language=CN, label=图1, caption=液体环境下探针微悬臂示意图, figureFileSmall=H0kEbQ621iKu+UY/VoqCjg==, figureFileBig=2TJFWdgz7ltmtNX/2osUrQ==, tableContent=null), ArticleFig(id=1246045625691783345, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246045609900229495, language=EN, label=Fig.2, caption=(a) The natural frequencies, (b) the ratio of the natural frequencies in liquid to those in vacuum, and (c) the quality factor of each mode of the microcantilever far away from the sample in a liquid environment, figureFileSmall=tvaX4JpHSA2Qb6N9grWqqg==, figureFileBig=Q5lHI6ZIJnPlP2CcciAnCw==, tableContent=null), ArticleFig(id=1246045625767280818, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246045609900229495, language=CN, label=图2, caption=液体环境下微悬臂远离样品时各阶模态的：（a）固有频率，（b）液体与真空环境下的固有频率比值，以及（c）品质因子, figureFileSmall=tvaX4JpHSA2Qb6N9grWqqg==, figureFileBig=Q5lHI6ZIJnPlP2CcciAnCw==, tableContent=null), ArticleFig(id=1246045625842778291, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246045609900229495, language=EN, label=Fig.3, caption=Natural frequencies of the first four modes ((a)-(d)) for different liquid densities and viscosities when the microcantilever is far away from the sample, figureFileSmall=wWSdgcO+qDt4O1oYcrjmJw==, figureFileBig=5kzI3LcIudosiSMcsTvcxQ==, 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tableContent=null), ArticleFig(id=1246045627675689148, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246045609900229495, language=EN, label=Fig.5, caption=(a) Contact resonance frequency of each mode of the microcantilever in liquid environment, (b) ratio of the contact resonance frequency of each mode of the microcantilever in liquid environment to the natural frequency under vacuum and the ratio of the contact resonance frequency to the free resonance frequency. The cantilever-sample separation is 10 μm, with dimensionless contact stiffness of 5 and dimensionless contact damping of 0.05, figureFileSmall=OP81ZGP86af6stIioUGxkQ==, figureFileBig=dgzi9rCoiVY9tvoVr2By5g==, tableContent=null), ArticleFig(id=1246045627759575227, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246045609900229495, language=CN, label=图5, caption=（a）液体环境下微悬臂各阶模态接触共振频率，（b）液体环境下微悬臂各阶模态接触共振频率与真空下固有频率比值以及自由共振频率与接触共振频率的比值. 微悬臂与样品间距为10 μm，无量纲接触刚度为5，无量纲接触阻尼为0.05, figureFileSmall=OP81ZGP86af6stIioUGxkQ==, figureFileBig=dgzi9rCoiVY9tvoVr2By5g==, tableContent=null), ArticleFig(id=1246045627897987261, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246045609900229495, language=EN, label=Fig.6, caption=(a) The contact resonance frequencies for different dimensionless contact stiffness (contact damping is 0) and (b) the contact resonance frequencies for different dimensionless contact damping (contact stiffness is 0), figureFileSmall=bJGskzlLSfYmW5lnqFOE5Q==, figureFileBig=PHLV2FjsNLxTJzJX1Rc4JQ==, tableContent=null), ArticleFig(id=1246045627986067647, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246045609900229495, language=CN, label=图6, caption=（a）不同无量纲接触刚度（接触阻尼为0）时各阶模态接触共振频率，（b）不同无量纲接触阻尼（接触刚度为0）时各阶模态接触共振频率, figureFileSmall=bJGskzlLSfYmW5lnqFOE5Q==, figureFileBig=PHLV2FjsNLxTJzJX1Rc4JQ==, tableContent=null), ArticleFig(id=1246045628065759425, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246045609900229495, language=EN, label=Fig.7, caption=The contact resonance frequencies of the first four modes of the microcantilever for different dimensionless contact stiffness and dimensionless contact damping, figureFileSmall=g30OiHPiCY7YUNXoHlVq5w==, figureFileBig=RGy3IvukIf9ZARkpjHMAEw==, tableContent=null), ArticleFig(id=1246045628166422722, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246045609900229495, language=CN, label=图7, caption=不同无量纲接触刚度和无量纲接触阻尼下微悬臂前四阶模态接触共振频率, figureFileSmall=g30OiHPiCY7YUNXoHlVq5w==, figureFileBig=RGy3IvukIf9ZARkpjHMAEw==, tableContent=null), ArticleFig(id=1246045628262891716, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246045609900229495, language=EN, label=Fig.8, caption=The sensitivity of free resonance frequency of the microcantilever to variations of liquid density with the liquid damping of (a)0.5 mPa·s, (b)1.0 mPa·s and (c)1.5 mPa·s, respectively, figureFileSmall=3c1dfPIBszgo9jAnpKVhBA==, figureFileBig=kI0jbx3L9Hgw9LAo2RSCwg==, tableContent=null), ArticleFig(id=1246045628376137926, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246045609900229495, language=CN, label=图8, caption=微悬臂各阶模态自由共振频率对液体密度变化的灵敏度.（a），（b），（c）的液体粘度分别为0.5 mPa·s、1.0 mPa·s和1.5 mPa·s, figureFileSmall=3c1dfPIBszgo9jAnpKVhBA==, figureFileBig=kI0jbx3L9Hgw9LAo2RSCwg==, tableContent=null), ArticleFig(id=1246045628468412616, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246045609900229495, language=EN, label=Fig.9, caption=The sensitivity of free resonance frequencies of microcantilever to variations of liquid viscosity with the liquid density of (a)500 kg/m3, (b)1000 kg/m3 and (c)1500 kg/m3 respectively, figureFileSmall=IHwJP3X7QJJIFkum4DQthQ==, figureFileBig=rtnFIRUpjQwBleyNqpKAsQ==, tableContent=null), ArticleFig(id=1246045628581658826, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246045609900229495, language=CN, label=图9, caption=微悬臂各阶模态自由共振频率对液体粘度变化的灵敏度.（a），（b），（c）的液体密度分别为500 kg/m3，1000 kg/m3和1500 kg/m3, figureFileSmall=IHwJP3X7QJJIFkum4DQthQ==, figureFileBig=rtnFIRUpjQwBleyNqpKAsQ==, tableContent=null), ArticleFig(id=1246045628686516429, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246045609900229495, language=EN, label=Fig.10, caption=The sensitivity of contact resonance frequencies of the microcantilever to the change of dimensionless contact stiffness for (a) the first mode, (b) the second mode, and (c) the third mode with different dimensionless contact damping. (d) is the sensitivity of contact resonance frequencies to the change of dimensionless contact stiffness with dimensionless contact damping of 0.05, figureFileSmall=Tcaqedj5iiTPnGeyHXrrUw==, figureFileBig=obgAMux5cqkHdbxQdRw5pQ==, tableContent=null), ArticleFig(id=1246045628795568332, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246045609900229495, language=CN, label=图10, caption=不同接触阻尼时微悬臂的：（a）一阶模态，（b）二阶模态，（c）三阶模态的接触共振频率对无量纲接触刚度变化的灵敏度以及（d）无量纲接触阻尼为0.05时前三阶模态的灵敏度, figureFileSmall=Tcaqedj5iiTPnGeyHXrrUw==, figureFileBig=obgAMux5cqkHdbxQdRw5pQ==, tableContent=null), ArticleFig(id=1246045628938174669, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246045609900229495, language=EN, label=Fig.11, caption=The sensitivity of contact resonance frequencies of the microcantilever to the change of dimensionless contact damping for (a) the first mode, (b) the second mode, and (c) the third mode with different dimensionless contact stiffness. (d) is the sensitivity of contact resonance frequencies to the change of dimensionless contact damping with dimensionless contact stiffness of 5, figureFileSmall=pWDrHUm7CdInp2FeKPGCBw==, figureFileBig=1fNse/Jna7us2nloryUoRA==, tableContent=null), ArticleFig(id=1246045629017866447, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246045609900229495, language=CN, label=图11, caption=不同接触刚度时微悬臂的：（a）一阶模态，（b）二阶模态，（c）三阶模态的接触共振频率对无量纲接触阻尼变化的灵敏度以及（d）无量纲接触刚度为5时前三阶模态的灵敏度, figureFileSmall=pWDrHUm7CdInp2FeKPGCBw==, figureFileBig=1fNse/Jna7us2nloryUoRA==, tableContent=null), ArticleFig(id=1246045629114335440, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246045609900229495, language=EN, label=Fig.12, caption=(a), (b), (c) are the contact resonance responses of microcantilever under different excitation approaches in vacuum for the first, the second and the third mode, respectively. (d), (e), (f) are the contact resonance response of microcantilever under different excitation approaches in liquid for the first, the second, and the third mode, respectively. 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液体环境下接触共振原子力显微术探针动力学行为研究
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杨长运 1 , 周锡龙 1, 2, ** , 张帮志 1
固体力学学报 | 研究论文 2024,45(5): 694-708
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固体力学学报 | 研究论文 2024, 45(5): 694-708
液体环境下接触共振原子力显微术探针动力学行为研究
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杨长运1, 周锡龙1, 2, ** , 张帮志1
作者信息
  • 1山东科技大学能源与矿业工程学院，青岛，266590
  • 2武汉理工大学新材料力学理论与应用湖北省重点实验室，武汉，430070

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** E-mail：.
Study on the Dynamic Behaviors in Contact Resonance Atomic Force Microscopy in Liquid Environments
Changyun Yang1, Xilong Zhou1, 2, ** , Bangzhi Zhang1
Affiliations
  • 1College of Energy and Mining Engineering, Shandong University of Science and Technology, Qingdao, 266590
  • 2Hubei Key Laboratory of Theory and Application of Advanced Materials Mechanics, Wuhan University of Technology, Wuhan, 430070
出版时间: 2024-10-25 doi: 10.19636/j.cnki.cjsm42-1250/o3.2024.019
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接触共振原子力显微术是材料微纳米尺度力学性能测试的有力工具，可用于大气和液体环境. 接触共振原子力显微术在液体环境下工作时，探针同时受到微悬臂-流体-样品之间的流固耦合作用以及针尖与样品相互作用，使得液体环境下探针微悬臂动力学特性与真空环境下有显著差异. 本文将流体对探针微悬臂的影响等效为附加质量及附加阻尼，研究了液体环境下流体密度和粘度对其动力学特性的影响，以及针尖样品接触刚度和接触阻尼对其动力学特性的影响. 结果表明，与液体粘度相比，液体环境下流体密度对微悬臂固有频率影响占主导地位；与液体密度相比，流体粘度对微悬臂品质因子影响占主导地位；探针高阶模态对于流体粘度和密度变化具有更高的灵敏度；随着液体环境下针尖样品接触刚度的增加，高阶模态对高接触刚度具有更高的灵敏度；低阶模态对于接触阻尼变化的灵敏度更高. 此外，对微悬臂在不同激励方式下的动力学响应进行了对比研究，给出了液体环境不同激励方式下微悬臂动力学响应差异的原因. 论文研究结果对于开展液体环境下接触共振原子力显微术微纳米力学成像，提高测量准确度和灵敏度具有重要指导意义.

接触共振  /  微悬臂  /  流固耦合  /  复频率  /  动力学分析

Contact resonance atomic force microscopy (CR-AFM) is a powerful technique that enables the measurement of topography and the mechanical properties of various materials at the micro/nanoscale. It can be used in both air and liquid environments. However, when CR-AFM is operated in a liquid environment, the dynamic behaviors of the microcantilever can be significantly different from those in air or vacuum due to the complex fluid-solid coupling of the microcantilever-liquid-sample system and the tip-sample interaction. In this study, we explore the effects of liquid density and viscosity, as well as tip-sample normalized contact stiffness and contact damping, on the dynamics of the AFM microcantilever in liquid environments. We treat the influence of the liquid on the dynamics of the AFM microcantilever as added mass and added damping. Our results show that in free vibration, the natural frequencies of the AFM microcantilever are primarily dominated by the liquid density, while the liquid viscosity plays a dominant role in the quality factor compared to the liquid density. Higher modes exhibit higher sensitivity to changes in liquid viscosity and liquid density. As the normalized tip-sample contact stiffness increases, a higher mode shows increased sensitivity to changes in normalized contact stiffness in a liquid environment. On the other hand, a lower mode is more sensitive to changes in normalized contact damping in a liquid environment. In addition, the dynamic responses of the AFM microcantilever under three different excitation approaches are compared and discussed. Variations in boundary conditions and hydrodynamic loads applied to the microcantilever under these approaches lead to diverse dynamic responses. The findings in this study are essential for the development of micro/nanoscale mechanical property imaging techniques using CR-AFM in liquid environments, as well as the improvement of measurement accuracy and sensitivity.

contact resonance  /  microcantilever  /  fluid-structure coupling  /  complex frequency  /  dynamic analysis
杨长运, 周锡龙, 张帮志. 液体环境下接触共振原子力显微术探针动力学行为研究. 固体力学学报, 2024 , 45 (5) : 694 -708 . DOI: 10.19636/j.cnki.cjsm42-1250/o3.2024.019
Changyun Yang, Xilong Zhou, Bangzhi Zhang. Study on the Dynamic Behaviors in Contact Resonance Atomic Force Microscopy in Liquid Environments[J]. Chinese Journal of Solid Mechanics, 2024 , 45 (5) : 694 -708 . DOI: 10.19636/j.cnki.cjsm42-1250/o3.2024.019
正文
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0　引言
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原子力显微镜广泛应用于样品微纳米尺度形貌测量和力学性能表征，其核心检测部件为末端有微小针尖的微悬臂梁. AFM测试或成像时，通过检测微悬臂静态或动态响应的变化，可实现对样品纳米级分辨率的测试或成像[1-3].
接触共振原子力显微术为AFM的一种动态成像模式，可用于大气或液体环境下样品力学性能测试. Stan等[4]基于接触共振原子力显微术和针尖与层状样品之间的接触相互作用，建立共振频率与接触刚度之间的关系，可用于准确测量样品层间界面位置和材料的力学性能. Kakar等[5]采用接触共振原子力显微术对牛血清白蛋白和牛颌下腺粘蛋白薄膜的力学性能进行了检测. Robins等[6]等提出了一种新型的接触共振成像技术，可通过恒定的接触共振频率获得针尖-样品接触刚度. 该技术通过力与距离测量来改变每个像素的作用力，以实现恒定的接触共振频率. 一些学者[7,8]研究发现使用光热激励可以获得理想的液体环境下的接触共振频谱，以便对液体环境下微悬臂的共振频率和品质因子进行定量化测量. Lei等[9]通过实验研究了镍微悬臂梁的振动，发现微悬臂梁厚度减小会增加其无量纲固有频率和无量纲弯曲刚度. Li等[10,11]基于修正偶应力理论提出了一种确定材料特征尺度参数的标准实验方法，并研究了微悬臂梁非线性振动的尺寸依赖性.
接触共振原子力显微术在液体环境下成像时，探针微悬臂与周围流体会产生复杂的流固耦合效应，使得微悬臂在液体环境下的动力学特性与真空或大气环境下存在显著差异[12-14]. Tung等[15]基于大气和液体环境下微悬臂接触共振频率及品质因子实验结果，拟合获得了施加在微悬臂上的流体动力学载荷表达式，获得了微悬臂的附加质量和附加阻尼. 之后，Shihab和Tung[16]基于数值模拟对构建的流体动力学载荷进行了验证，分析了其适用性. Parlak等[17]类比在大气环境下计算针尖样品接触刚度的方法，通过引入流体阻尼以及流体质量载荷的简化公式，获得了计算液体环境下接触刚度的简单模型. 该模型不考虑压膜阻尼和接触阻尼，仅适用于低雷诺数和探针的低阶模态. Payam和Fathipour[18]研究了针尖质量对液体环境下探针接触共振频率灵敏度的影响. 结果显示，液体环境下针尖质量对微悬臂接触共振频率影响很小，基本可以忽略. Payam[19]分析了微悬臂在液体环境下的共振频率对样品表面刚度变化的灵敏度. 结果表明，低表面刚度下一阶模态灵敏度最高，而高阶模态在高表面刚度下更具优势，且大气中的图像对比度优于液体中的图像对比度.
液体环境下接触共振时压膜阻尼和接触阻尼对微悬臂动力学特性具有显著影响，但是目前尚未有同时考虑液体压膜阻尼和接触阻尼对微悬臂动力学特性影响的研究. 同时，为简单起见，目前研究通常将波数的虚部直接省略，并未考虑环境介质阻尼对波数的影响. 此外，对于液体环境不同激励方式下微悬臂接触共振响应差异对比及其机理研究尚未开展.
本文采用复波数和复频率分析液体环境下接触共振原子力显微术探针动力学特性，研究了流体和接触力学参数对微悬臂动力学特性的影响，计算了微悬臂不同模态共振频率对流体和样品力学性能变化的灵敏度，推导了真空和液体环境下微悬臂受不同激励方式的动力学响应，并对其动力学响应的差异进行了机理分析. 研究结果对于定量化纳米力学测试技术开发以及提高测量准确度和灵敏度具有重要指导意义.
1　液体环境下接触共振原子力显微术探针动力学模型
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1.1　液体环境下探针远离样品时动力学特性
微悬臂振动过程中周围流体会在微悬臂上施加流体动力学载荷. 目前一种有效的处理方法是将流体对微悬臂的作用等效为微悬臂的附加质量和附加阻尼[20]. 假设微悬臂在振动过程中的振幅很小，即微悬臂梁发生小应变小转动，且阻尼力与速度之间呈线性比例关系，则液体环境下探针微悬臂的振动控制方程为：
式中，y为微悬臂瞬时位移；E为微悬臂弹性模量；I为横截面惯性矩；ρb为微悬臂密度；A为横截面面积；L为微悬臂梁长度；ma为单位长度附加质量；ca为单位长度附加阻尼. ma和ca的表达式为[19]：
式中，c∞为流体阻尼系数；ρf为流体密度；b为微悬臂宽度；Γ（ω）为流体动力学函数，表达式为[21]：
式中，a1=1.0553，a2=3.7997，b1=3.8018，b2=2.7364；Re为雷诺数，其表达式为[22]：
需要说明的是，部分学者采用修正偶应力理论考虑材料尺度效应对微悬臂动力学特性影响. 此时，只需将动力学方程中EI替换为EI+GAl2. 其中，G为微悬臂剪切模量，l为材料特征尺度常数[11].
微悬臂振动控制方程的通解可写为：
式中，c1，c2，c3，c4为常数. 将通解表达式代入控制方程，可得频散方程为：
微悬臂远离样品表面振动时的边界条件为：
将通解代入以上边界条件，可得远离样品表面时微悬臂振动特征方程为：
联立频散方程和特征方程，可得液体环境下微悬臂远离样品时的各阶模态固有频率. 微悬臂的品质因子为[23]：
1.2　液体环境下探针微悬臂接触共振动力学特性
液体环境下接触共振原子力显微术对样品进行测试或成像时，需要将针尖与样品之间的作用等效为接触刚度和接触阻尼. 此外，还需要考虑微悬臂在样品表面附近振动时的压膜阻尼效应. 考虑压膜阻尼效应时微悬臂单位长度附加阻尼为[24]：
式中，cs为压膜阻尼系数；D为微悬臂与样品之间的间距.
采用开尔文粘弹性力学模型描述针尖与样品之间的相互作用，此时微悬臂的边界条件为：
式中，k*为接触刚度；η为接触阻尼. 将通解代入以上边界条件，可得特征方程为：
式中，α=k*/kc为无量纲接触刚度，弹性常数kc=3EI/L3；β=ηL/（9EIρbA）1/2为无量纲接触阻尼；ξ=ωL2（ρbA/EI）1/2为无量纲复频率. 联立频散方程和特征方程即可求得液体环境下微悬臂接触振动时各阶模态共振频率.
1.3　测试灵敏度
液体环境下微悬臂不同模态对所在流体介质力学性能变化以及样品力学性能差异的测试灵敏度不同. 对频散方程求全微分，可得微悬臂固有频率对液体密度和粘度变化的灵敏度为：
对特征方程求全微分并联立频散方程，可得共振频率对无量纲接触刚度和接触阻尼的灵敏度表达式：
式中，∂ξ/∂f，∂f/∂k*，∂k*/∂α，∂f/∂η和∂η/∂β的表达式分别为：
式中，f1，f3，f4和f5的表达式如下：
1.4　不同激励方式探针微悬臂动力学响应
接触共振原子力显微术测试时可以采用不同的激励方式对探针进行激励. 液相环境下采用不同方式激励探针时，流体对微悬臂的作用存在差异，从而引起探针微悬臂动力学响应的不同. 本文对液相环境下固定端位移激励、样品位移激励和微悬臂自由端集中力激励三种方式下微悬臂动力学响应进行对比. 微悬臂固定端受正弦位移激励时，其边界条件为：
式中，u0为位移激励幅值. 将通解表达式代入以上边界条件，经过推导可得微悬臂任意位置处的位移响应为：
当x=L时，上式可简化为：
类似地，可得液体环境下底部样品位移激励时探针响应为：
以及微悬臂自由端受正弦集中力激励时位移响应表达式：
式中，F0为激励力幅值. 取k*=0，η=0，D=∞，可简化为微悬臂远离样品表面时的位移响应；取ma=0，ca=0，可简化为真空环境下微悬臂的位移响应.
2　结果与讨论
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2.1　液体环境下探针远离样品动力学分析
微悬臂长度、宽度和厚度分别为125 μm、25 μm和2.1 μm. 材料为单晶硅，密度2330 kg/m3，弹性模量160 GPa，泊松比0.3. 流体为纯净水，密度1000 kg/m3，动力黏度系数1 mPa·s. 液体环境下微悬臂远离样品振动时各阶模态固有频率、与真空环境下固有频率的比值以及品质因子如图2所示. 液体环境下微悬臂一阶模态固有频率为66.74 kHz，与真空下固有频率比值为0.37，品质因子为3.47. 随着模态阶数的增加，微悬臂固有频率近似呈幂函数增长，液体对微悬臂振动的影响不断减小，液体与真空环境下固有频率比值近似呈对数增长，品质因子近似呈线性增加. 微悬臂第十二阶固有频率为28.94 MHz，与真空下固有频率比值增加到0.43，品质因子增长为58.58.
图3为不同液体粘度及密度下微悬臂前四阶模态的固有频率. 液体粘度及密度较小时微悬臂具有较高的固有频率. 微悬臂各阶固有频率随着液体密度的增加而迅速降低，随着液体粘度的增加而缓慢降低. 液体密度对微悬臂固有频率的影响占主导地位. 同时，随着模态阶数的增加，固有频率等高线逐渐趋于与y轴平行，液体密度对固有频率的影响逐渐增强.
不同液体粘度及密度下微悬臂前四阶模态的品质因子如图4所示. 微悬臂在低液体粘度及密度下的品质因子较高. 各阶模态品质因子均随液体粘度的增加而逐渐减小，且当液体密度较小时品质因子减小的速率更高. 与液体密度相比，液体粘度对微悬臂品质因子的影响占主导地位.
2.2　液体环境下探针微悬臂接触共振动力学分析
图5（a）给出了液体环境下微悬臂各阶接触共振频率，图5（b）给出了微悬臂在液体与真空环境下接触共振频率的比值以及微悬臂自由共振频率与接触共振频率的比值. 微悬臂接触共振频率随模态阶数增加近似呈幂函数形式增长. 液体与真空环境下固有频率比值近似呈对数形式增加. 微悬臂振动模态从一阶模态转为二阶模态后，微悬臂二阶模态自由共振频率与接触共振频率比值从一阶模态的0.417增加到二阶模态的0.937，并随模态阶数的增加而趋于1.
不同接触刚度和接触阻尼时微悬臂各阶模态固有频率如图6和图7所示. 其中，微悬臂与样品间距为10 μm. 微悬臂各阶模态固有频率随接触刚度的增加而不断增大. 随着接触阻尼的增加，微悬臂一阶模态固有频率缓慢增加，二阶及高阶模态固有频率则缓慢减小.
2.3　灵敏度分析
图8给出液体粘度分别为0.5 mPa·s、1.0 mPa·s和1.5 mPa·s时微悬臂各阶模态自由共振频率对液体密度变化的灵敏度. 由图可知，高阶模态自由共振频率对液体密度变化的灵敏度更高. 此外，自由共振频率对液体密度变化灵敏度随液体密度的增加而迅速降低；液体密度一定时，自由共振频率对液体密度变化灵敏度随着液体粘度的增加而缓慢增大. 液体密度为500 kg/m3、1000 kg/m3和1500 kg/m3时微悬臂各阶模态自由共振频率对液体粘度变化的灵敏度如图9所示. 微悬臂的高阶模态对液体粘度变化具有较高灵敏度. 液体密度一定时，微悬臂自由共振频率对液体粘度灵敏度随着液体粘度的增加而不断降低；液体粘度一定时，自由共振频率对液体粘度变化灵敏度随着液体密度不断增加而逐渐减小.
图10为微悬臂各阶模态接触共振频率对无量纲接触刚度变化的灵敏度. 由图可知，当接触阻尼一定时，随着无量纲接触刚度的增加，一阶模态接触共振频率对无量纲接触刚度变化的灵敏度不断降低，高阶模态接触共振频率对无量纲接触刚度变化的灵敏度先缓慢增加，随后不断降低. 与低阶模态相比，高阶模态在高无量纲接触刚度范围内的灵敏度更高. 随着无量纲接触阻尼的降低，各阶模态对无量纲接触刚度变化的灵敏度逐渐增加.
各阶模态共振频率对无量纲接触阻尼的灵敏度如图11所示. 由图可知，微悬臂低阶模态共振频率对无量纲接触阻尼变化具有更高的灵敏度. 接触刚度一定时，微悬臂各阶模态接触共振频率对无量纲接触阻尼的灵敏度随着接触阻尼的增加而增大. 无量纲接触阻尼较小时，一阶模态接触共振频率灵敏度远高于高阶模态. 随着无量纲接触阻尼的增加，一阶模态与高阶模态之间的灵敏度差异逐渐减小.
2.4　不同激励方式探针动力学响应分析
真空环境和液体环境不同激励方式下微悬臂前三阶模态接触共振响应如图12所示. 真空环境下微悬臂受样品位移激励和末端集中力激励时的位移响应完全一致，而微悬臂受固定端位移激励时位移响应与其他两种激励方式相比存在较大差异. 这是因为真空环境下微悬臂受样品位移激励与末端集中力激励时边界条件相似且微悬臂振动时无环境阻尼影响，复频率和复波数中虚部为0，微悬臂受样品位移激励时动力学响应可写为y（x）=k*f（x，knL）的形式；微悬臂受力激励时动力学响应可写为y（x）=F0f（x，knL）的形式. 故真空环境两种激励方式下微悬臂动力学响应体现出一致性. 微悬臂受固定端位移激励时边界条件与两者有明显的差异，故其动力学响应与其他激励方式下位移响应存在显著差异. 此外，随着模态阶数的增加，固定端位移激励与其他两种激励方式的微悬臂位移响应差异逐渐减小.
液体环境下微悬臂样品位移激励和受末端集中力激励时一阶模态位移响应近乎重合. 随着模态阶数的增加，集中力激励和样品位移激励下位移响应的差异不断增加. 这是因为液体环境下微悬臂的复频率和复波数中存在虚部，而两种激励方式的位移响应体现出了对虚部和激励频率的不同依赖性. 此外，受流体动力学载荷影响，液体环境中微悬臂受固定端位移激励时位移响应与真空环境下有较大差异，且差异随着模态阶数增加而不断减小. 这是由于随模态阶数增加，液体对微悬臂梁动力学特性的影响不断减小[25].
3　结论
收起
本文对液体环境下接触共振原子力显微术探针动力学行为进行了研究，分析了液体环境下微悬臂自由振动和接触共振时的动力学特性，计算了微悬臂各阶模态共振频率对液体密度、粘度、接触刚度和接触阻尼的灵敏度，并推导和对比了真空和液体环境下微悬臂在不同激励方式下的动力学响应.
研究结果显示，液体环境下微悬臂自由共振频率和接触共振频率随模态阶数增加近似呈幂函数形式增长，品质因子近似呈线性增长，液体和样品力学性能对微悬臂动力学特性的影响不断减小. 微悬臂高阶模态共振频率对液体密度、液体粘度以及高接触刚度的变化具有更高的灵敏度；微悬臂低阶模态固有频率对接触阻尼的变化更为灵敏. 真空环境下，力激励和样品位移激励下微悬臂动力学响应完全一致；液体环境下，微悬臂在以上两种激励方式下的动力学响应存在差异，且差异随着激励频率的增加而增大. 真空与液体环境下微悬臂受固定端位移激励和其他激励方式下动力学响应均存在显著差异. 这是由于边界条件的差异性以及不同激励方式下微悬臂所受流体动力学载荷不同而导致的.
基金
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  • 新材料力学理论与应用湖北省重点实验室开放基金项目(TAM202205)
文献
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2024年第45卷第5期
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doi: 10.19636/j.cnki.cjsm42-1250/o3.2024.019
  • 接收时间：2024-05-06
  • 首发时间：2026-04-01
  • 出版时间：2024-10-25
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  • 收稿日期：2024-05-06
基金
新材料力学理论与应用湖北省重点实验室开放基金项目(TAM202205)
作者信息
    1山东科技大学能源与矿业工程学院，青岛，266590
    2武汉理工大学新材料力学理论与应用湖北省重点实验室，武汉，430070

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2种不同金属材料的力学参数
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鹅膏菌科Amanitaceae 2 11 5.26 鹅膏菌属 Amanita 10 4.78
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