微纳通道谐振器检测与表征中的动力学问题
闫寒, 张文明
上海交通大学机械系统与振动国家重点实验室, 上海 200240
Dynamics problems of micro/nano channel resonators for detection and characterization
YAN Han, ZHANG Wenming
中图分类号: O326
文献标识码: A
通讯作者:
收稿日期: 2018-04-22
接受日期: 2018-06-19
网络出版日期: 2019-01-15
版权声明: 2019 中国力学学会 This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
基金资助:
作者简介:
张文明, 上海交通大学特聘教授, 博士生导师.国家杰出青年科学基金获得者、国家优秀青年科学基金获得者、国家“万人计划”中组部青年拔尖人才、教育部霍英东青年基金获得者、上海市曙光学者、上海市青年科技启明星、上海市青年五四奖章标兵.2010年在日本京都大学做JSPS特别研究员.主要从事机械动力学设计理论与方法及振动控制等方面的教学与科研工作.在国际权威期刊发表SCI论文110余篇, SCI他引1000余次;公开/授权国家发明专利30余项, 软件著作权3项; 出版学术专著1部,参编中英文专著3部.中国微米纳米技术学会微纳执行器与微系统分会副理事长、中国力学学会青年工作委员会委员、中国振动工程学会非线性振动专业委员会委员、转子动力学专业委员会常务理事;担任多个国内外期刊编委;获教育部自然科学奖一等奖、中国振动工程学会青年科技奖等多项奖励.
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摘要
微纳通道机械谐振器在液体环境中具有超高的谐振频率、品质因子和灵敏度,常用于液体环境中的高精度检测与表征,在生物、医药、化工等领域有着广阔的应用前景.微纳通道机械谐振器的检测与表征功能高度依赖其动力学特性,而此类器件是由谐振结构、内部流体、被检测物和外部激励等多因素组成的耦合系统,涉及的动力学问题较为复杂,已成为谐振器件研究中的前沿热点和瓶颈问题.本文综述了微纳通道机械谐振器的研究进展,总结了谐振器件实现高精度检测与表征功能时的动力学设计原理,详细讨论了谐振器件的稳定性、频响特性、能量耗散、频率波动等动态特性,阐明了不同动力学问题的物理机制及其对谐振器性能的影响规律,可为深入厘清微纳通道机械谐振器的动力学设计问题,提高器件动态性能提供理论参考和技术支撑,对超高频、超高灵敏度谐振器的设计、制造及应用发展具有重要意义.
关键词：
Abstract
Micro/nano-channel mechanical resonators have ultra-high resonance frequency, quality factor, and sensitivity in liquid environment. Hence they are usually used for high-precision detection and characterization in liquid environments. These resonators have broad application prospects in the fields of biology, medicine, and chemical industry. The detection and characterization functions of micro/nano-channel mechanical are highly dependent on their dynamic characteristics. Such devices are coupled systems composed of multiple components, including resonant structure, internal fluid, detected object, external excitation and so on. As a result, the involved dynamic problems are much complicated, and they have become a hotspot and bottleneck in the research of resonant devices. In this paper, the research progress of micro/ nano-channel mechanical resonators is reviewed. The dynamic design principles for high-precision detection and characterization are summarized. The dynamic characteristics, including stability, frequency response characteristics, energy dissipation, frequency fluctuations and so on, are discussed in detail. The physical mechanism of different dynamics and its influence on the performance of the resonator are expounded. It can provide theoretical reference and technical support for deep understanding of the dynamic design problem of micro/nano-channel mechanical resonators and improve the dynamic performance of the devices. And it is of great significance for the design, manufacture, and application of ultra-high frequency and ultra-high sensitivity devices.
Keywords：
1 引 言
微纳机械谐振器作为微纳机电系统(micro/nano-electro-mechanical systems)中实现电能与 机械能转换的重要传感与执行器件,在信息技术、生物医学、机电工业等领域有着广泛应用, 已成为国内外研究的热点和前沿问题(Cermak et al. 2016, Jensen et al.2008a, Lei et al. 2013). 在真空环境中,微纳机械谐振器具有超高的谐振频率$(10^3\sim 10^6$Hz)和品质因子$(10^3\sim 10^4)$, 可用于超高精度的质量传感器, 其质量分辨率可达原子量级(Jensen et al.2008b). 而在生物、医药、化工 等诸多领域,微纳机械谐振器的谐振结构(通常为微梁或微板)通常需要浸入液体环境中进行检测与表征,液体会对谐振结构产生附加质量和附加阻尼效应(Sader 1998),使得谐振器件的频率大幅下降, 最高可达50%, 品质因子也会下降$3\sim4$个数量级(Beardslee et al. 2010). 谐振频率和品质因子的大幅降低,严重限制了微纳机械谐振器在液体环境中的检测精度(Sader et al.2010a). 针对该问题,许多学者提出了优化谐振器几何结构、利用高阶振动模态等方法(Ghatkesar et al. 2008, Green & Sader 1998, Johnson & Mutharasan, 2011, Yang et al. 2013), 有效降低了流体阻尼,使品质因子也提高了大约一个数量级(Beardslee et al. 2010),但仍不足以满足高精度检测的需求.
为了解决液体环境中的高精度检测与表征难题, Burg等(Burg et al. 2007,Burg & Manalis 2003, Burg et al.2006)率先研发了微纳通道机械谐振器件, 通过在悬臂梁中嵌入微通道,将液体从谐振结构的外部转移到内部,从而改变了弹性结构与液体之间的流固耦合作用,显著降低了流体的附加质量和附加阻尼,使得谐振器件频率和品质因子得到大幅提升. 随着微纳加工技术的发展,谐振器件设计性能得到大幅度提升, 谐振频率从千赫兹 (Son et al.2008)逐步提高至兆赫兹 (Barton et al. 2010); 同时,谐振器件的结构设计也趋于多样化, 在早期悬臂梁式谐振器(Burg et al. 2007)基础上, 学者们设计研发了固支梁式(Barton et al. 2010)、微板式(Agache et al. 2011)、微管式(Kim et al.2016)等多种谐振器件, 如图1所示. 与悬臂梁式相比,固支梁式和微板式谐振器件的频率较高, 有利于提高分辨率;而且微通道易加工, 成本较低(Kim et al. 2016).
微纳通道谐振器可用于表征液体中的微纳米颗粒,包括细菌、病毒、蛋白质分子、纳米颗粒等,质量分辨率可达$10^{-18}$g量级,常用于质量、密度、体积等参数的检测. 此外,器件还可以用于表征流体密度(Khan et al. 2013)、流体黏度(Lee et al.2012)、流体相变(Minhyuk et al. 2014)、颗粒位置(Olcum et al.2015))等.微纳通道机械谐振器的检测与表征功能高度依赖其动力学特性(Zhang et al. 2016),而该谐振器是包含谐振结构、内部流体、被检测物、激励源等多种因素的复杂系统,在检测与表征的过程中受到流--固--电多场耦合效应以及颗粒吸附、颗粒运动等因素的影响,表现出刚度硬化或软化、颤振、吸合等非线性现象,因此微纳通道机械谐振器的动力学问题引起了研究人员的广泛关注.
本文围绕微纳通道谐振器检测与表征中的动力学问题,在第2节介绍谐振器检测与表征的动力学设计原理以及最新研究进展,第3节综述谐振系统中的动力学问题,重点讨论谐振器件的稳定性、频率响应、能量耗散、频率波动等动力学特性及其对谐振器性能的影响规律,第4节对全文进行总结, 并对未来的研究进行力所能及的展望.
2 设计原理与表征
2.1 悬浮质量检测
2.1.1 动力学原理
微纳通道机械谐振器件最初用于液体环境中颗粒检测与表征.被检测颗粒的悬浮质量与谐振器频率之间的关系为(Sarid 1994)
$$ f = \dfrac{1}{2\pi }\sqrt {\dfrac{k}{m_0 + \alpha\cdot \Delta m}} (1) $$
其中, $f$为谐振频率,$k$为谐振器刚度, $m_{0}$为谐振器等效质量, $\alpha$为颗粒位置有关的参数, 当颗粒位于悬臂梁自由端时, $\alpha = 1$,$\Delta m$为颗粒的悬浮质量, 定义为
$$ \Delta m = \left( {\rho _{\rm P} - \rho _{\rm f} } \right)V_{\rm P} (2) $$
其中, $\rho _{\rm P} $为颗粒密度, $\rho _{\rm f}$为流体密度, $V_{\rm P}$为颗粒的体积.由于颗粒的悬浮质量远小于谐振器的有效质量, 即$\Delta m \ll m_0 $,因此 式(1)可以写作
$$ f_0 + \Delta f \approx \dfrac{1}{2\pi }\sqrt k\left( {m_0 - \dfrac{1}{2}\alpha \dfrac{\Delta m}{m_0 }} \right) (3) $$
其中, $\Delta f$表示由颗粒引起的频率漂移,$f_{0}$表示无颗粒时的谐振频率. 当颗粒位于悬臂梁自由端时,由上式可得 (Zhang 2014)
$$ \dfrac{\Delta f}{f_0 } \approx -\dfrac{1}{2}\dfrac{\Delta m}{m_0 } (4) $$
由式(4)可知, 颗粒的悬浮质量与谐振器的频率漂移呈线性关系,当已知谐振频率与谐振器有效质量时,可通过测量频率漂移直接确定颗粒的悬浮质量.式(4)是微纳通道谐振器检测颗粒悬浮质量的动力学原理.
2.1.2 检测方法
目前, 微纳通道机械谐振器主要采用吸附式(Burg et al.2007)和流动式(Olcum et al. 2015)两种检测方法.表1列出了两种不同方法的优、缺点和误差来源.两种检测方法的原理如图2所示,吸附式检测通过对谐振器的内壁进行表面修饰(surfacefunctionalization), 使内壁可以吸附特定的颗粒,当不同的颗粒流经内部通道时, 由于修饰后的表面具有选择性,目标颗粒吸附在表面上, 而其余颗粒从通道的另一端流出.由于谐振器的比表面积很大, 吸附颗粒的数量远大于通道内的自由颗粒,因此忽略自由颗粒对谐振频率的影响. 测量吸附前后谐振器的频率变化,可以利用式(1)确定吸附颗粒的质量, 需要注意的是,当颗粒均匀吸附在内部通道的壁面时, 位置参数$\alpha $取0.24.吸附式检测具有选择性, 并且由于谐振器表面积--体积比很大(10$^{4}$cm$^{ - 1})$, 可以同时检测大量颗粒, 检测效率高. 然而,颗粒的吸附效应不仅使谐振器质量变化, 也会改变谐振器的刚度 (Tamayo et al. 2006, Zhang 2013, Zhang & Zhao 2015),因此谐振系统的频率漂移可以写作$\Delta f = \Delta f_{{\rm mass}} +\Delta f_{{\rm stiffness}} $, 其中$\Delta f_{{\rm mass}}$和$\Delta f_{{\rm stiffness}}$分别表示质量变化和刚度变化引起的频率漂移. 当$\Delta f_{{\rm stiffness}} $不为0时, 基于式(4)的质量检测存在检测误差, 而当$\Delta f_{{\rm stiffness}} $过大时, 谐振器甚至无法正常检测.
表1 吸附式检测与流动式检测的对比
| 检测方式 | 优点 | 缺点 | 误差源 |
|---|---|---|---|
| 吸附式 | 具有可选择性; 可检测大量颗粒 | 需要表面修饰 | 吸附效应 |
| 流动式 | 无需表面修饰,使用 方便;可以检测单个颗粒 | 通量较低 | 颗粒流动路径的不确 定性;颗粒的流固耦合运动 |
流动式检测的原理如图2(b)所示, 颗粒在通道中流动,谐振器的频率随着颗粒的位置而变化, 当颗粒位于通道的末端时,频率的变化最大. 与吸附式检测相比,流动式检测不需要对谐振器进行化学修饰,也不需要在检测之后采用化学方法处理通道内壁,并且可以控制单个颗粒流经微通道, 从而实现单颗粒的表征. 然而,受到溶液浓度(Burg et al. 2007)以及流动速度的限制,在一定时间内检测的颗粒数量较少, 检测通量较低. 此外, 根据式(1)可知,颗粒位置影响谐振器的频率,因此颗粒以不同路径通过微通道会引起不同的频率变化,从而产生检测误差. 研究发现,由流动路径引入的测量误差可达5%$\sim $11%(Lee et al.2011). 并且, 由于颗粒是悬浮在液体中而不是与谐振器刚性连接,颗粒在液体中的流固耦合运动也会引起频率的改变, 从而产生检测误差(Yan et al. 2017b).
学者们还设计研发了捕捉式检测方法(Weng et al. 2011b),该方法是通过微流控技术、结构设计等来控制颗粒在微通道中的位置,从而实现特定的检测目的.Godin等(2010)设计了微流控系统用于调节通道内压力, 改变流体流动方向,使被检测颗粒在悬臂梁末端的通道内往复流动, 可实现颗粒的动态捕捉,如图3(a)所示. 利用该方法,可以将细胞长时间控制在通道末端,通过测量细胞悬浮质量随时间的变化情况, 获得细胞的生长率特性.Lee等(2010)提出了一种基于离心力的捕捉方法,如图3(b)所示,悬臂梁振动使颗粒受到沿梁长度方向的离心力作用, 当流动速度较低时,颗粒受到的流体力小于离心力, 颗粒在离心力作用下被限制在通道末端;而随着流动速度提高, 流体作用力增大, 颗粒被释放. 通过这种方法,可以消除颗粒位置的不确定性, 提高检测精度.
2.1.3 质量分辨率与谐振器设计
质量分辨率是表征谐振器性能的关键参数,超高的分辨率是实现微纳米颗粒质量检测的重要保障.为了提高质量分辨率, 学者们不断设计、开发新的微纳通道谐振器.根据式(4)可知: ${\Delta f} /{\Delta m} = - {f_0 } / {2m_0 }$,因此, 提高$\Delta f$的检测精度、降低等效质量$m_{0}$、提高谐振频率$f_{0}$都有助于质量分辨率的提升.微纳通道机械谐振器将流体限制在内部通道中,避免了附加阻尼效应的影响, 品质因子可达$10^{3}\sim 10^{5}$,保证了$\Delta f$的测量精度. 此外,等效质量与特征尺寸的三次方成正比$m_0 \propto L_{\rm s}^3 $,而谐振频率与梁长度的平方成反比$f_0 \propto 1 / {L^2}$, 因此,减小谐振结构的尺寸可以显著减少谐振器质量、提高谐振频率,从而提升器件的质量分辨率, 如图4所示,图中标识含义见表2.
图4 典型微纳通道机械谐振器质量分辨率与谐振器结构参数的关系.蓝色表示质量分辨率与谐振器长度的关系,红色表示谐振频率与谐振器长度的关系,黑色表示质量分辨率与谐振频率的关系
微纳通道机械谐振器问世以来 (Burg & Manalis, 2003),其质量分辨率已从742fg (10$^{ - 15}$g) (Burg & Manalis,2003)提升到0.3ag (Modena et al. 2014), 提升幅度达6个数量级.2003年, Burg等首创研发了微通道机械谐振器件,采用多晶硅镶嵌工艺和牺牲层刻蚀技术, 加工了长度为300$\mu$m、通道高度为1.2$\mu $m的悬臂梁式谐振器, 频率为40kHz,质量分辨率约为742fg. 该谐振器受到空气阻尼的影响,品质因子较低(约90), 限制了质量分辨率的提高, 约为742fg. 2007年,Burg等(2007)改进了谐振器的结构设计并利用真空腔抑制空气阻尼,有效提高了谐振频率和品质因子, 器件的质量分辨率达到1fg,可以检测液体环境中单个微纳米颗粒的质量. 2010年,Lee等(2010))进一步优化了谐振器的结构,并在梁结构中嵌入纳尺度通道(3.0$\mu $m$\times $0.7$\mu $m),使质量分辨率提高到27ag.
2014年, 在Lee等(2010)工作的基础上,Olcum等(2014)通过减小谐振器尺寸、改进激励形式等方法,进一步提高了质量分辨率. Olcum等开发了4种尺寸的微纳通道机械谐振器,尺寸最小的谐振器采用22.5$\mu $m$\times $7.5$\mu$m$\times $1.0$\mu $m的微悬臂梁, 梁中嵌入1.0$\mu$m$\times $0.4$\mu $m的纳通道, 谐振频率高达2.87MHz.此外, 采用压电激励方式代替静电激励, 增大了谐振器的振幅,有效降低了频率噪声. 通过以上方法,将谐振器的质量分辨率提高到了0.85ag.Modena等(2014)采用扭转振动代替弯曲振动,并在测量中引入相关分析方法, 使谐振器的动态范围提高5个量级,质量分辨率高达0.3ag, 但是该方法只适用于检测多个颗粒的平均质量,不能用于单个颗粒的测量.
图4还总结了微纳通道机械谐振器采用的激励和检测方式,静电激励和压电激励是最常用的两种激励方式. 其中,静电驱动是一种将电能转换成机械能的方法,具有效率高、损耗低、结构简单、响应迅速等优点,广泛应用于微机电系统中(Zhang et al. 2014).但是静电激励具有固有的不稳定性, 当电压过大时, 谐振器发生吸合失稳.压电激励是利用压电材料的机电耦合特性来实现的,具有高应力、高带宽和高能量密度等优点.光学式检测在微纳通道机械谐振器有广泛的应用, 然而,光学测量设备价格高昂、体积较大、使用不便,不利于谐振器的集成化$\mu $s (Agache et al. 2011, Lee et al.2011). 为此, Lee等(2011)开发了基于压阻单元的微纳通道机械谐振器,Agache等(2011)设计了基于静电激励和静电测量的板式微通道机械谐振器.由于不需要复杂的光学测量设备, 采用电学方法检测的谐振器体积很小,方便使用和集成, 如图5所示.
图5 (a) 采用静电测量的板式微纳通道机械谐振器(
2.1.4 颗粒位置表征
式(1)和式(4)描述了悬浮质量与频率漂移的关系,但是难以表征颗粒位置对谐振频率的影响,因此无法用于设计检测颗粒位置的谐振器件.Dohn等(2007)考虑颗粒附加质量、颗粒位置的影响,利用能量法和Rayleigh-Ritz理论,推导得到了谐振器不同弯曲模态下的谐振频率与颗粒附加质量和位置之间的解析关系式
$$ \dfrac{f_{\Delta m,n} }{f_n } = \dfrac{1}{\sqrt {1 +\dfrac{\Delta m}{m_0 }U_n^2 \left( {x_0 } \right)} } (5) $$
其中, $f_n $为无颗粒时谐振器的第$n$阶频率, $f_{\Delta m,n}$为附加质量$\Delta m$作用下第$n$阶频率$f_n $的变化量, $U_n$为第$n$阶模态振型, $x_{0}$为颗粒在悬臂梁长度方向上的位置.由式(5)可知, 测量不同模态下谐振器的频率漂移,可以确定颗粒的附加质量和位置.
Olcum等(2015)开发了一种通用平台,能够同时激励微纳通道谐振器的多阶模态,并测量颗粒流经微通道时不同模态下谐振器的频率变化,如图6所示. 根据器件在各阶模态下的频率漂移,利用式(5)即可以确定通道内流动颗粒的悬浮质量和位置. 研究结果表明,采用前四阶模态, 可以精确测量颗粒的悬浮质量和位置, 其中,质量精度达40ag, 位置精度达到150nm.
图6 (a)谐振器前四阶模态振型以及颗粒在内部通道匀速流动时谐振频率的变化情况,(b) 直径150nm和100nm黄金颗粒流经通道时实验测得的前四阶模态频率漂移
2.2 生物颗粒表征
2.2.1 密度表征
微纳通道机械谐振器在液体环境中具有高谐振频率、高品质因子、高质量分辨率等优越性能,因此广泛应用于生物颗粒的表征, 可以检测蛋白质密度(Folzer et al.2015)、单个细胞的密度和体积 (William et al.2011)、细胞生长率(Cermak et al. 2016)等. 由式(2)和式(4)可知,根据频率漂移所确定的质量为颗粒的悬浮质量$\Delta m = \left( {\rho_{\rm P} - \rho _{\rm f} } \right)V_{\rm P} $,由于颗粒的体积是未知的, 因此无法通过悬浮质量直接确定颗粒密度,而需要检测颗粒在不同密度液体中的悬浮质量, 利用阿基米德原理(William et al. 2011)得到颗粒密度, 如图7所示:测量颗粒在两种(或以上)不同密度流体中的悬浮质量,采用线性插值或最小二乘方法拟合悬浮质量与流体密度之间的线性关系,直线与$y$轴的交点为颗粒的总质量, 与$x$轴的交点为颗粒密度,直线斜率则代表颗粒体积. 因此,确定颗粒密度的关键在于测量颗粒在不同密度液体中的悬浮质量.
Godin等(2007)最早使用微纳通道机械谐振器测量了大肠杆菌、红细胞等生物颗粒的密度,分辨率为10$^{ - 4}$g/cm$^{3}$. 在实验中,通过测量颗粒在纯水、水--重水混合物、纯重水3种不同密度液体中的悬浮质量,得到了颗粒密度. Folzer等(2015)首次测量了尺寸为0.2$\mu$m$\sim $5.0$\mu $m的蛋白质分子密度, 结果表明,蛋白质密度在$1.28 \sim 1.33$g/cm$^{3}$之间, 低于理论预测值.实验中, 检测蛋白质分子在多种不同密度液体中的悬浮质量,然后采用最小二乘法确定悬浮质量--溶液密度之间的线性关系,从而根据阿基米德原理获得蛋白质密度. 需要注意的是,Godin等(2007)和Folzer等(2015)测量的是多个颗粒的平均密度,而没有实现单个细胞的密度测量. 此外,测量颗粒在不同液体中的悬浮质量时, 需要更换液体重新测量,因此操作较为复杂.
William等(2011)采用动态捕捉方法, 实现了对单个细胞密度的检测,测量过程如图8所示. 第1步, 在通道中充满液体1,液体密度小于细胞密度; 第2步, 被检测细胞从左侧支路管道流入悬臂梁内,并进入右侧的支路管道, 通过测量细胞流经内部通道时谐振器的频率漂移,可以确定细胞在液体1中的悬浮质量; 第3步, 在悬臂梁通道中充满液体2,液体2的密度大于细胞密度; 第4步, 细胞流经充满液体2的内部通道,确定细胞的悬浮质量.根据细胞在两种不同液体中的悬浮质量进行线性插值,可以得到单个细胞的密度. 由于颗粒两次流经微通道,并且两种液体的混合也需要时间,因此检测一个细胞所需的时间大约为10s, 通量较低,并且由于检测方式的限制, 通量难以获得提升.
为了降低细胞检测时间, 提高检测通量,Bryan等(2014)设计了串联式谐振器, 如图9所示.谐振器1与谐振器2之间由通道相连, 谐振器1之中充满低密度流体,谐振器2中则是高密度液体, 两种液体的混合在连接通道之中完成.根据细胞在谐振器1和谐振器2中的悬浮质量, 可以计算细胞密度.采用这种方法, 细胞无需两次流经同一个谐振器,也不需要等待流体在谐振器中的混合, 因此可以降低细胞的检测时间,提高通量. 理想情况下,使用该谐振器检测一个细胞所需的时间约为0.5s. 然而,由于流体混合时引起的不稳定性以及通道内流动阻力很高,检测单个细胞所需的实际时间达到10s.
William等(2011)和Bryan等(2014)开发的谐振器适合测量单个细胞的密度,但难以检测细胞在药物或化学品作用下密度的变化情况.为了解决这一问题,Weng等(2011a)开发了三通道式和圆柱式微纳通道机械谐振器.图10(a)所示为三通道式谐振器, 悬臂梁末端有细胞捕捉区域,中部的通道用于控制捕捉区域中流体的流入和流出. 检测细胞密度时,通过控制通道内的压力, 将被测细胞限制在梁末端的捕捉区域,之后更换细胞周围的液体, 测量细胞在不同液体中的悬浮质量,从而得到细胞密度. 由于不需要等待两种液体的混合,采用这种方法检测所需的时间大约为3$\sim $5s.图10(b)所示为圆柱式微纳通道机械谐振器,在通道末端有均匀分布的圆柱, 实现对细胞的机械式捕捉.该谐振器可用于检测细胞在药物作用前后的参数变化情况.采用图10所示微机械谐振器还可以消除颗粒位置不确定性所引起的误差,提高检测精度. 利用这两种微纳通道机械谐振器,Weng等(2011a)测量了单个酵母菌细胞在培养基中的质量、体积、密度,以及小鼠淋巴细胞在药物或其他刺激物影响下质量的动态变化情况.表3总结了基于微纳通道机械谐振器的颗粒密度表征方法,对比了不同方法所采用的谐振器形式、检测对象、检测时间以及优缺点.
2.2.2 生长率表征
细胞生长率是表征细胞特性的重要参数,检测细胞生长率不需要测量细胞在不同液体中的悬浮质量,但为了实现实时检测, 需要使细胞长时间处于通道中.Godin等(2010)提出了动态捕捉方法, 如图11所示,通过反馈控制算法, 在探测到细胞从悬臂梁中流出时,立刻调整通道内流体流动的方向, 使细胞重新流入悬臂梁. 利用这种方法,可长时间得将颗粒控制在悬臂梁的内部通道中. 图11(b)显示了随着时间推移谐振器频率的变化情况, 可以发现, 随着细胞的生长,细胞质量增大, 频率漂移量提高. 考虑到在细胞的生长周期内,谐振器只能检测一个细胞的质量变化, 因此检测通量很低.
为了提高细胞生长率的检测通量,Cermak等(2016)开发了串联的阵列式微纳通道机械谐振器,如图12所示.一系列的微纳通道机械谐振器通过延迟通道(delay channel)串联起来,而细胞可以在延迟通道内生长.当单个的细胞在某一个谐振器中检测完毕并进入延迟通道后,其余的细胞可以流入该谐振器进行质量检测. 因此,使用该谐振器阵列能够同时检测多个细胞, 从而实现高通量检测.结果表明, 利用该谐振器阵列,可以在1h内检测60个哺乳动物细胞或者150个细菌的生长率.
2.3 流体性质表征 2.3.1 密度检测
假设悬臂梁的质量、刚度以及通道内的流体质量均匀分布,微纳通道机械谐振器可以近似为线性的弹簧振子模型. 当阻尼较低时,可以认为测量得到的谐振频率$\omega _{\rm r}$与系统的无阻尼固有频率$\omega _0 $相等
$$ 2\pi f_{\rm r} = \omega _{\rm r} \approx \omega _0 = \sqrt{\dfrac{k}{m}} (6) $$
其中, $m$表示谐振器的总质量,等于悬臂梁质量$m_{\rm c}$与流体质量$m_{\rm f}$之和
$$ m = m_{\rm c} + m_{\rm f} = \rho _{\rm c} V_{\rm c} + \rho _{\rm f}V_{\rm f} (7) $$
其中, $\rho _{\rm c} $和$\rho _{\rm f}$分别代表悬臂梁和流体的密度, $V_{\rm c} $和$V_{\rm f}$分别表示悬臂梁和流体的体积. 考虑到通道内的流体不会改变谐振器刚度,式(6)可以表示为
$$ f_{\rm r} = \dfrac{1}{2\pi }\sqrt {\dfrac{k/{V_{\rm f} }}{{m_{\rm c} }/{V_{\rm f} } + \rho _{\rm f} }} =\dfrac{1}{2\pi }\sqrt {\dfrac{A}{B + \rho _{\rm f} }} (8)$$
$$其中$A$和$B$表示与谐振器结构、材料有关的常数,可以通过测量两种已知密度的流体来确定. 根据式(7),通道内流体的密度可以表示为 (Khan et al. 2013)
\rho _{\rm f} = \left( {\dfrac{A}{2\pi f_{\rm r} }}\right)^2 - B (9) $$
根据以上原理, Khan等(2013)利用微纳通道机械谐振器测量了不同流体的密度,谐振频率与密度的关系如图13所示.图中的直线是由式(9)得到的理论结果,其中的常数$A$与$B$通过测量通道中充满空气和水时的谐振频率来确定.可以发现, 理论结果与实验结果吻合得很好.采用微纳通道机械谐振器检测流体密度, 检测原理清晰, 测量范围广,并且所需样本量少(10$^{ - 12}$L)、质量分辨率高(0.001kg/m$^{3})$(Son et al. 2008), 因此具有广阔的应用前景.
2.3.2 黏度检测
流体的黏性影响系统的阻尼, 从而影响谐振器的品质因子以及振动幅值.因此, 可以通过测量品质因子或振动幅值的变化检测流体黏度 (Khan et al. 2013, Lee et al. 2012).品质因子定义为一个振动周期内谐振器储存能量$W$与损耗能量$\Delta W$之比
$$ Q = 2\pi \dfrac{W}{\Delta W} = \dfrac{m\omega _{\rm r}}{c} (10) $$
其中$c$表示黏性阻尼系数.系统总的阻尼包括悬臂梁的阻尼$c_{\rm c}$以及通道内的流体阻尼$c_{\rm f}$
$$ c = c_{\rm c} + c_{\rm f} (11) $$
根据式(10)和式(11)可得
$$c = \dfrac{m\omega _{\rm r} }{Q} \approx \dfrac{k}{\omega _{\rm r}Q} (12) $$
谐振器的刚度$k$不受内部流体的影响, 并假设通道内充满气体时,气体引起的阻尼$c_{\rm f}$很小, 则$k$可以表示为
$$ k = c_{\rm c} \omega _{\rm r,air} Q_{\rm air} (13) $$
根据以上3个公式, 被测流体的阻尼系数与悬臂梁阻尼系数之比可以表示为
$$ \dfrac{c_{\rm f} }{c_{\rm c} } = \dfrac{\omega _{\rm r,air}Q_{\rm air} }{\omega _{\rm f} Q_{\rm f} } - 1 (14) $$
式(14)表征了谐振频率、品质因子与阻尼系数之间的关系,但没有给出流体黏性与品质因子的解析关系,因此不能直接用于流体黏度检测. 为此, Khan等(2013)测量了通道内充满不同黏度液体时谐振器的阻尼系数比${c_{\rm f}} / {c_{\rm c} }$, 如图14所示. 结果表明, 在$1 \sim 10{\rm mPa} \cdot {\rm s}$的黏度范围内, ${c_{\rm f} } /{c_{\rm c}}$与流体黏性表现出线性关系. 利用拟合得到的${c_{\rm f} }/{c_{\rm c}}$与黏度关系曲线, 测量了不同浓度乙醇--水混合溶液的黏度,检测精度约为0.025${\rm mPa} \cdot {\rm s}$.
Lee等(2012)提出了两种流体黏度检测方法, 一种基于谐振器的品质因子,另一种基于谐振器振幅, 其原理如图15(a)所示.图15(b)显示了不同黏度液体作用下,谐振器品质因子和振幅的变化情况. 可以发现,品质因子和振幅随黏度的变化趋势是一致的,因此二者都可以用于检测流体黏度.基于品质因子的检测方法具有较高的分辨率, 约为0.035${\rm mPa}\cdot {\rm s}$, 但检测时需要进行扫频分析, 因此检测时间较长,大约需要30s. 而基于振动幅值的检测方法只需测量谐振器的时域信号,因此检测速度很快, 所需时间约为0.1$\sim $1.0ms. 然而,这种方法的分辨率较低, 为0.096${\rm mPa} \cdot {\rm s}$.
随着通道内液体黏度的增大,谐振器的品质因子呈现非单调的变化趋势(Burg et al. 2009). 因此,利用微纳通道机械谐振器只能测量较小范围内的液体黏度. 并且,由于品质因子与谐振器结构参数有关(Burg et al. 2009),对于不同的谐振器, 品质因子随液体黏度的变化规律也不同.品质因子与内部液体黏度之间的复杂关系,限制了其作为黏度传感器的应用.
3 动力学问题
3.1 稳定性
微纳通道机械谐振器是一个包含微梁、内部流体、激励源的复杂耦合系统,当流动速度过大时, 系统会发生失稳(Zhang et al. 2016), 并且,谐振器的激励方式也会影响系统稳定性 (Abbasnejad et al. 2015, Yan et al. 2017a). 由于稳定性是谐振器正常工作的前提,因此该问题引起了学者的广泛关注.
梁在内部流体作用下的稳定性是一类典型的流固耦合动力学问题,长期以来都是国内外研究的热点问题 (Dai et al. 2014, 2017; Ghayesh& Farokhi, 2018; He et al. 2017; Paidoussis, 1998; Wang et al.2013, 2017a, 2017b, 2018; Zhou et al. 2018). Paidoussis,(1998)提出了梁在内部流体作用下的经典流固耦合动力学模型,系统分析了内流体对梁动力学特性及稳定性的影响规律. 结果表明,当内部流体的速度超过某一临界值时, 系统发生失稳,该临界值称为临界速度. 对于悬臂梁, 失稳形式通常是颤振,而对于两端固支梁, 失稳形式为屈曲. Rinaldi等(2010)采用经典模型(Paidoussis 1998)分析内部流体作用下微尺度梁的振动特性,研究了尺度效应、材料性质以及流体速度对微梁频率响应和稳定性的影响.Wang(2010)基于修正偶应力理论,提出了一种新的描述输流微梁动力学行为的理论模型. 研究发现,基于修正偶应力理论得到的临界速度高于经典梁理论给出的值,这说明输流微梁的稳定性高于宏观尺度的梁.Wang等(2013)考虑结构和流体的尺度效应, 修正了Paidoussis(1998)提出的经典模型, 并基于修正模型研究了输流微梁的稳定性.结果表明, 微梁的材料长度尺寸参数增大了梁的刚度,从而使临界速度增大, 稳定性提高;而表征微流体速度轮廓的参数会使临界速度减小, 降低系统稳定性.Setoodeh和Afrahim(2014)基于应变梯度理论,研究了输流微梁的非线性动力学行为,发现材料尺度参数对梁的基础频率和临界速度都有重要影响.Zhang和Meguid(2016)考虑表面弹性的影响,提出了内部流体作用下纳米梁的修正连续模型, 发现对于铝材料纳米梁,表面能使梁的临界速度升高, 而对于硅材料纳米梁,表面能的作用恰好相反.Zhou等(2018)研究了具有功能梯度材料的悬臂梁在内部流体作用下的稳定性,分析了沿轴向的弹性模量梯度和密度梯度对临界速度的影响,研究结果表明, 与均质梁相比, 当梁的密度沿长度方向减小时,临界速度更高, 系统更加稳定,而当质量比较小并且弹性模量沿长度方向逐渐降低时, 系统更容易失稳.Ghayesh和Farokhi(2018)分析了输流微梁的黏弹性动力学特性, 研究发现,包含内部流体的黏弹性输流管表现出刚度硬化现象,并且对称模态和非对称模态之间存在相互作用.
以上研究工作充分阐释了内部流体流动对输流微梁稳定性的影响规律,揭示了微尺度梁的流致失稳机理. 然而,这些工作主要针对包含平直通道的微梁,而微纳通道机械谐振器根据加工方法的不同,其内部通道可以是平直通道或U形通道, 并且,包含U形通道的谐振器在实际中的应用更加广泛 (Hashem et al. 2016,Nejadnik & Jiskoot 2015, Wang et al. 2015). 对于平直通道,流体从梁的一端流入, 从另一端流出, 流动为单向; 对于U形通道,流体从梁的固支端流入, 到达自由端之后返回固支端, 流动为双向. 因此,两种通道内的流体与微梁之间的流固耦合行为存在差异.Zhang等(2016)考虑谐振器内的双向流体流动特性、流体黏性以及速度轮廓,建立了包含U形通道的谐振器动力学模型,发现内部流体的双向流动使科氏力相互抵消. 因此,与经典的流固耦合动力学模型相比, 新模型中不包含科氏力项.图16显示了由两种模型得到的Argand图, 可以发现,两种系统的动力学特性有显著区别(Zhang et al. 2016). 此外,Vakilzadeh等(2017)基于应力梯度理论,研究了尺寸效应对微纳通道机械谐振器的影响,发现当梁的厚度接近材料的尺度参数时,尺寸效应使谐振器的临界速度提高.Belardinelli等(2017)考虑内部流体和颗粒悬浮质量的影响,利用Hamilton原理建立了微纳通道机械谐振器的非线性动力学模型,分析了激励频率和颗粒运动对谐振器时域特性、频响特性和频率漂移的影响规律.研究结果表明,谐振器的非线性动力学特性有助于提高时域和频域的分辨率.
图16 (a) 包含U形通道微机械谐振器的Argand图 (
需要注意的是, 在输流微梁的稳定性研究中, 为了分析的方便,通常对动力学模型进行无量纲化, 所得到的临界速度为无量纲速度.无量纲速度与实际流动速度、梁的长度和弹性性质、单位长度的流体质量等参数有关(Wang et al. 2013).Rinaldi等(2010)分析了材料弹性性质和梁的截面形状对输流微梁实际临界速度的影响,发现随着微梁长度的提高, 实际临界速度降低, 而随着弹性模量的增大,实际临界速度升高. 对于微通道机械谐振器, 材料通常为硅 (Burg et al.2007, Kim et al. 2016), 弹性模量达190GPa,实际失稳速度的量级约100m/s, 远远高于通道内流体的流动速度. 然而,为了降低检测时间、提高检测通量, 需要增大通道内流体的流动速度;并且, 随着设计手段和加工工艺的进步,研究人员开发出了基于其他材料的微通道机械谐振器, 例如Marzban等(2017)开发了基于高分子聚合物的谐振器, 其弹性模量为700kPa,实际失稳速度的量级为0.1m/s. 因此,有必要研究微通道机械谐振器的流致稳定性.
微纳通道机械谐振器的激励方式也是影响系统稳定性的重要因素.压电激励和静电激励是最常用的激励方式, 其中,压电激励会影响谐振器的流致稳定性.Abbasnejad等(2015)考虑梁上下表面压电层的影响,导出了输流微梁的流固耦合动力学模型, 结果表明,压电层之间的电压差显著降低了流动速度对振动频率的作用,从而提高了系统的稳定范围. 而对于静电激励, 当外加电压过大时,梁的弹性回复力无法平衡静电力, 引起谐振器的吸合失稳, 因此,静电激励下的微纳通道机械谐振器存在两种失稳形式:吸合失稳和流致失稳. Yan等(2017a)分析了谐振器的两种失稳形式,发现增大流速可以提高吸合电压, 有助于增强系统的吸合稳定性,而增大电压对临界速度没有明显影响. 根据分析结果,得到了系统的静态和动态稳定范围, 如图17所示.
在稳定性研究方面, 国内外学者已经取得了一系列的成果. 然而,现有的工作多集中于理论分析, 实验研究较少. 因此,为了更深入地理解微纳通道机械谐振器的失稳机理,建立更加精确的动力学模型, 需要加强实验方面的研究.
3.2 品质因子
品质因子是衡量谐振器动力学性能的重要指标, 与谐振器的能量耗散有关,定义为
$$ Q = 2\pi \dfrac{E_{{\rm stored}} }{E_{{\rm diss}} } (15) $$
其中, $E_{\rm stored}$表示一个周期内谐振器储存的能量, $E_{\rm diss}$表示一个周期内耗散的能量. 由上式可知, 损耗的能量越少,系统品质因子越高. 对于理想的谐振器件, 能量耗散为0,品质因子为无穷大. 但微机械谐振器在运行过程中总会产生能量耗散,并且有多种能量耗散机制, 限制了谐振器件的品质因子(张文明等 2017).目前对各种能量耗散机理的认识还不够清楚,未能建立起描述各种耗散机制的精确模型,因此准确获得谐振器的品质因子是非常困难的. 通常情况下,在研究谐振系统的品质因子时, 先针对性地分析每一种能量耗散机制,再利用总的品质因子模型来描述
$$ \dfrac{1}{Q_{{\rm total}} } = 2\pi \dfrac{E_{{\rm anchor}} + E_{{\rm fluid}} + E_{{\rm surface}} + E_{{\rm TED}} +E_{{\rm phonon}} + E_{{\rm other}} }{E_{{\rm stored}} } (16) $$
其中, $Q_{\rm total}$表示总品质因子, $E_{\rm anchor}$代表锚点耗散,$E_{\rm fluid}$代表流体引起的能量耗散, $E_{\rm surface}$表示界面耗散, $E_{\rm TED}$代表热弹性阻尼, $E_{\rm phonon}$代表声子耗散, $E_{\rm other}$为其他机制造成的耗散.张文明等(2017)综述了微纳机械谐振器中的能量耗散机理与非线性阻尼效应的研究进展,阐明了不同能量耗散的产生机理及影响规律. 对于微纳通道机械谐振器,内部液体引起的能量耗散是限制品质因子的主要因素,引起了研究人员的广泛关注 (Burg et al. 2009; Sader et al. 2011, 2010a, 2010b).
Burg等(2009)测试了通道高度分别为3$\mu $m和8$\mu$m的微纳通道机械谐振器的品质因子, 结果如图18所示.从图中可以看出, 对于通道高度为8$\mu $m的谐振器,当通道内充满氮气 (0.017mPa$ \cdot $s)时, 其品质因子为9172;通道内充满纯水 (1mPa$ \cdot $s), 品质因子降为5653;将纯水替换为浓度为24%的甘油 (2mPa$ \cdot $s),品质因子仅有微弱的减小; 而当液体黏度在2$ \sim $5mPa$ \cdot$s之间时, 品质因子降至极小值点后升高, 直至黏度达到385mPa$ \cdot$s (浓度92%的甘油)时, 品质因子开始下降.这种趋势同样表现在高度为3$\mu $m的谐振器中.而如果将谐振器直接浸入相应的黏性液体中, 则如图中的直线所示,谐振器品质因子小于15, 并且随着液体黏度增大而减小.
理论研究发现 (Burg et al. 2009,
$$ Q = F\left( \beta \right)\left( {\dfrac{\rho _{\rm c}}{\rho _{\rm f} }} \right)\left( {\dfrac{h_{\rm c} }{h_{\rm f} }}\right)\left( {\dfrac{b_{\rm c} }{b_{\rm f} }} \right)\left({\dfrac{L}{h_{\rm f} }} \right)^2 (17) $$
其中, $\rho _{\rm c} $和$\rho _{\rm f} $分别表示悬臂梁、流体的密度, $h_{\rm c}$和$h_{\rm f} $分别为悬臂梁的高度和内部通道的高度, $b_{\rm c}$和$b_{\rm f} $分别表示悬臂梁宽度和通道宽度, $L$为悬臂梁的长度,$\beta $为雷诺数, 定义为$\beta = {\rho _{\rm f} \omega h_{\rm f}^2} /\mu $, $F\left( \beta \right)$为标准化的品质因子, 可以表示为
$$ F\left( \beta \right) = \dfrac{\beta }{16\int_{ - \bar{L}_{\rm c} }^1 {\int_{ - 1/ 2}^{1/ 2} {\left| {G\left( {X,Z}\right)} \right|^2{\rm d}X {\rm d}Z} } } (18)
$$其中, $\bar {L}_{\rm c} = {L_{\rm c} } / L$, $L_{\rm c}$为基底内的通道长度, $G\left( {X,Z} \right)$表示为
$$ G\left( {X,Z} \right) = \left[ {1 - \dfrac{1 - {\rm i}}{2}\sqrt{\dfrac{\beta }{2}} \dfrac{\cosh \left( {\left( {1 - {\rm i}}\right)\sqrt {\dfrac{\beta }{2}} Z} \right)}{\sinh \left( {\left({1 - {\rm i}} \right)\sqrt {\dfrac{\beta }{2}} } \right)}}\right]\dfrac{{\rm d}\bar {W}}{{\rm d}X} + \dfrac{{\rm i}\beta\bar {Z}_0 }{2}\cdot $$
$$ \left[ {\dfrac{\sinh \left( {\left( {1 - {\rm i}} \right)\sqrt{\dfrac{\beta }{2}} Z} \right)}{\left( {1 - {\rm i}} \right)\sqrt{\dfrac{\beta }{2}} \cosh \left( {\left( {1 - {\rm i}}\right)\sqrt {\dfrac{\beta }{2}} } \right) - 2\sinh \left({\dfrac{1 - {\rm i}}{2}\sqrt {\dfrac{\beta }{2}} } \right)}}\right]\left[ {S\left( \bar {x} \right) - h\left( \bar {x}\right)} \right]\dfrac{{\rm d}\bar {W}}{{\rm d}X}\bigg|_{X = 1} (19) $$
其中, $X = x /L$, $Z = {z - z_0 }/{h_{\rm f} }$,$\bar {Z}_0 = {z_0 }/{h_{\rm f} }$, $\bar{W}$为标准化后的悬臂梁振型函数, 满足$\bar {W}\left( 1 \right) =1$.
根据以上理论模型计算谐振器的品质因子, 将其与实验结果进行对比,如图19所示. 可以发现,理论模型能够准确预测品质因子随雷诺数的变化趋势. 当流体黏度较低时,雷诺数较大, 流体的惯性效应显著, 剪切能量耗散机制占据主导作用,品质因子随黏度的增大而降低; 当黏度增大到一定值时,流体黏性边界层发生重叠, 使品质因子随着黏度的增大而升高;当黏度足够高时, 抽吸式能量耗散起主导作用,流体的可压缩性决定品质因子的变化情况. 从图中还可以看出,对于通道高度为3$\mu $m的谐振器, 理论结果与实验数据吻合得很好,而对于高度为8$\mu $m的谐振器, 理论值与实验结果有较大偏差.这是因为理论模型的前提之一是通道高度远小于通道宽度,而通道高度为3$\mu $m的谐振器显然更接近这一假设. Sader等(2011,2010b)还研究了模态阶数以及材料泊松比对品质因子的影响规律.研究发现, 随着模态阶数的升高, 微纳通道机械谐振器的品质因子降低.并且在高阶模态下, 流体可压缩性的影响更为显著.悬臂梁材料的泊松比对谐振器的品质因子也有重要影响:随着泊松比的增大, 能量耗散降低, 品质因子升高,而对于理想的不可压缩材料,由通道偏心距引起的抽吸式能量耗散可以忽略.
图19 微纳通道机械谐振器品质因子的理论分析与实验结果对比. (a)通道高度8$\mu $m, $\bar {Z}_0 = 0.06$; (b) 通道高度3$\mu$m, $\bar {Z}_0= 0.14$
Burg等(2009)、Sader等(2010a)的研究工作阐释了微纳通道机械谐振器的能量耗散机理,揭示了品质因子与流体黏度、流体可压缩性、振动模态、材料特性、通道结构等多种因素之间的关系.在理论模型的建立过程中, 为了使复杂的能量耗散问题得到简化,Burg等(2009)、Sader等(2010a)引入了多种假设,包括细长梁假设、小振幅假设、矩形通道假设等,并且没有考虑通道宽度、流动速度、支承形式等因素的影响.
在实际应用中, Sader等(2010a)发现当通道宽度接近通道高度时,理论预测的品质因子显著高于谐振器实际的品质因子;Barton等(2010)开发了两端固支的微通道谐振器,研究表明该谐振器的品质因子比理论值低了2$\sim $3个数量级,这种现象可能是由固支梁的中平面拉伸效应引起的,但具体的能量耗散机制尚不清晰. 因此,现有的品质因子模型通用性不强、精度较低,需要研究通道截面形状、流动速度、支承方式等对微纳通道机械谐振器能量耗散的影响规律,建立适用范围更广、精度更高的品质因子模型,为谐振器的设计和使用提供理论指导.
3.3 谐振频率
谐振频率是表征颗粒质量、流体密度等被检测物性质的关键参数,谐振器件的分辨率会随着谐振频率增大而提高. 因此,谐振频率的变化不仅会影响谐振器的分辨率, 也会引入检测误差,降低谐振器的检测精度.
影响谐振频率的主要因素有载荷和颗粒, 其中,载荷包括流体载荷以及激励载荷.Yan等(2017a)研究了流体流动对谐振频率的影响, 随着流动速度的增大,微纳通道谐振器的一阶频率升高, 而二阶频率降低.图20给出了流体载荷作用下谐振频率的变化, 对于一阶模态,流体流动产生的向心力总是指向谐振器的平衡位置, 增强了系统的回复力,使有效刚度增大, 谐振频率提高; 对于二阶模态,向心力的方向背离平衡位置, 削弱了系统的回复力, 使有效刚度减小,谐振频率降低. 当谐振器受到外加静电力载荷时,谐振频率会受到显著的影响,许多学者研究了静电力对谐振器动力学特性的影响 (De & Aluru 2004,Mojahedi et al. 2010, Nayfeh et al. 2005, Nayfeh et al. 2007,Rhoads et al. 2006), 研究表明, 静电力会使谐振器频率降低,并且当外加电压达到吸合电压时, 频率下降为0. 值得注意的是,谐振频率会随着流动速度和外加电压变化显著增大或降低:当流速增大到临界速度时, 一阶频率大约提高180%,而当电压增大至吸合电压时, 一阶频率下降为0. 因此,可以通过改变载荷来调节谐振频率, 从而调节谐振器的灵敏度. 此外,在实际检测过程中,流体、激励载荷的影响可以用谐振器件的等效刚度来表征,等效刚度可以通过实际测量得到. 因此,载荷作用下谐振频率的变化不会引入额外的检测误差.
颗粒的附加质量会使谐振器的频率发生改变, 如式(1)和式(4)所示, 然而,式(1)和式(4)成立的内在假设是颗粒与谐振器刚性连接,颗粒只改变谐振器的质量, 而不会改变刚度和阻尼, 在这种假设下,颗粒与谐振器的位移完全相同, 如图21(a)所示.但是对于吸附式检测, 吸附效应不仅会增大谐振器质量,还会改变谐振器的刚度和阻尼 (Yin 2014), 从而引起谐振器的频率变化,如图21(b)所示. 对于流动式检测,由于颗粒没有与谐振器刚性连接, 而是悬浮在液体中,颗粒与液体的流固耦合运动会影响谐振频率, 如图21(c)所示.
对于吸附式检测来说, 如果不考虑吸附效应引起的刚度变化,颗粒的吸附使谐振器件质量增加, 谐振频率降低. 然而,Ramos等(2006)通过实验研究了大肠杆菌的吸附,发现当大肠杆菌吸附在谐振器上时, 谐振器件的频率升高;Tamayo等(2006)也在实验中发现有机分子的吸附会使悬臂梁结构的谐振频率升高.频率升高说明吸附引起的刚度增加占据主导作用, 在这种情况下,基于式(4)设计的谐振器将无法正常进行检测. 对于谐振频率降低的情况,如果不考虑吸附效应引起的刚度变化,通过频率漂移确定的附加质量可能小于颗粒的实际质量(Lee et al. 2009).因此,研究颗粒的吸附效应对指导谐振器设计、提高谐振器性能有着重要意义.
吸附效应对谐振器件刚度的影响主要表现在以下两个方面:吸附物刚度(Tamayo et al. 2006, Yin 2014)和吸附诱导的表面应力(Cherian & Thundat 2002, Dareing & Thundat 2005, Zhang et al. 2004). 其中, 吸附物刚度引起谐振器刚度变化的原理是:被检测颗粒与谐振器的表面层形成具有一定厚度的吸附层,谐振器与吸附层构成复合材料层合结构,层合结构的等效抗弯刚度可表示为(Ramos et al. 2006)
$$ D_{\rm e} = \dfrac{b_{\rm c} }{12}\dfrac{E_{\rm c}^2 t_{\rm c}^4 + E_{\rm a}^2 t_{\rm a}^4 + 2E_{\rm c} E_{\rm a} \left({2t_{\rm c}^2 + 3t_{\rm c} t_{\rm a} + 2t_{\rm a}^2 }\right)}{E_{\rm c} t_{\rm c} + E_{\rm a} t_{\rm a} } (20) $$
其中, $D_{\rm e} $为等效弯曲刚度, $b_{\rm c} $为梁的宽度, $E_{\rm c} $和$E_{\rm a} $分别为梁和吸附层的弹性模量, $t_{\rm c}$和$t_{\rm a} $分别为梁和吸附层的厚度. 根据公式可知,层合结构的刚度依赖于吸附层的弹性模量和厚度,并且吸附物总是增大谐振器刚度. 因此,谐振器的频率不仅与被检测物的质量有关, 也与吸附层的刚度和厚度有关.Tamayo等(2006)建立了考虑吸附层的谐振器动力学模型,导出了相对频率漂移与吸附层性质之间的解析关系式
$$ \dfrac{\omega _n - \omega _{0n} }{\omega _{0n} } =\alpha _1 \left( {\dfrac{t_{\rm a} }{t_{\rm c} }} \right) + \alpha_2 \left( {\dfrac{t_{\rm a} }{t_{\rm c} }} \right)^2 (21) $$
式中, $\omega _{0n} $表示没有发生吸附时谐振器的第$n$阶频率,$\omega _n $为吸附后的第$n$阶谐振频率, $\alpha _1 $和$\alpha _2$是与吸附层性质有关的系数, 可表示为
$$ \alpha _1 = \dfrac{1}{2}\left( {3\dfrac{E_{\rm a}}{E_{\rm c} } - \dfrac{\rho _{\rm a} }{\rho _{\rm c} }}\right),\qquad \alpha _2 = \dfrac{3}{8}\left[ {\left( {\dfrac{\rho_{\rm a} }{\rho _{\rm c} }} \right)^2 + 2\dfrac{E_{\rm a} }{E_{\rm c} }\left( {4 - \dfrac{\rho _{\rm a} }{\rho _{\rm c} }} \right) -7\left( {\dfrac{E_{\rm a} }{E_{\rm c} }} \right)^2} \right] (22) $$
式中, $\rho _{\rm a} $和$\rho _{\rm c} $分别为吸附层和梁的密度.吸附层对谐振频率的影响如图22所示,图中SI表示梁的材料为硅$(\rho _{\rm c} = 2330{\rm kg / m}^3$,$E_{\rm c} = 169{\rm GPa})$, Protein表示肌球蛋白$(\rho _{\rm a}= 183{\rm kg / m}^3$, $E_{\rm a} = 0.7{\rm GPa})$,SU-8表示梁的材料为SU-8光刻胶$(\rho _{\rm c} = 1190{\rm kg /m}^3$, $E_{\rm c} = 4.0{\rm GPa})$,SAM表示由烷烃硫醇通过自组装形成的吸附层$(\rho _{\rm a} = 675{\rm kg / m}^3$, $E_{\rm a} = 12.9{\rm GPa})$.从图22中可以发现, 吸附层刚度对谐振系统的频率有显著影响,并且随着吸附层与谐振器的刚度比增大, 系统的谐振频率升高.
当被检测物吸附在谐振器件表面时,吸附物之间、吸附物与谐振器表面分子之间的静电力(Cherian &Thundat 2002)或Lennard--Jones势 (Dareing & Thundat 2005)的相互作用均会产生表面应力,表面应力会导致谐振器件的刚度发生变化 (Zhang et al. 2004).对于固支梁来说, 表面应力会引起轴向力, 轴向力会改变谐振器的刚度(Karabalin et al. 2012, Villanueva et al. 2011). 此外,由于轴向力可能是拉伸力或压缩力 (Berger et al. 1997, Fritz et al.2000), 使得谐振器件的刚度升高或降低 (Cherian & Thundat 2002,Zhang et al. 2013). 对于悬臂梁结构来讲,表面应力对刚度的影响机理尚未厘清,Lagowski等(1975)提出了表面应力作用下悬臂梁刚度的轴向力模型,通过模型分析得知: 表面应力会使悬臂梁内部产生轴向力,轴向力改变了梁的刚度, 所得结果与固支梁模型相似, 并且得到了实验验证(Hwang et al. 2006, McFarland et al. 2005). 然而, 研究人员也发现,悬臂梁结构的轴向力模型无法满足力平衡这一基本物理规律,与牛顿第三定律相矛盾 (Lachut & Sader 2007, Lu et al. 2005).针对这一问题, Lachut和Sader(2007)建立三维模型并分析了表面应力对悬臂梁刚度的影响, 研究表明,在表面应力作用下, 悬臂梁靠近固支端位置会产生面内应力,从而改变了悬臂梁的刚度, 推导所得频率漂移与表面应力之间的关系如下
$$ \dfrac{\Delta \omega }{\omega _0 } \approx -0.042\dfrac{\nu \left( {1 - \nu } \right)\sigma _{\rm s}^{\rm T}}{Eh}\left( {\dfrac{b}{L}} \right)\left( {\dfrac{b}{h}} \right)^2 (23) $$
其中, $\Delta \omega $为频率漂移量, $\omega _0$为无表面应力时的谐振频率, $\nu $为泊松比, $E$为弹性模量, $\sigma_{\rm s}^{\rm T} $为表面应力, $L$, $b$,$h$分别为悬臂梁的长度、宽度和厚度. Karabalin等(2012)在上述理论基础上研究发现, 除了面内应力以外,表面应力还会使悬臂梁的几何结构发生改变, 从而引起刚度的变化.并且采用可靠性高、重复性好的测试方法,实验验证了该理论模型的正确性.
在流动式检测过程中, 颗粒不会发生吸附, 可以避免复杂的吸附效应.但是在液体工作环境中, 颗粒的流固耦合运动会影响谐振频率,从而导致检测误差. 为了研究颗粒运动对器件的影响,Yan等(2017b)建立了考虑颗粒运动的谐振器件流固耦合动力学模型, 即
$$ EI\dfrac{\partial ^4w}{\partial x^4} + \left( {m_{\rm c} + M_{\rm f} } \right)\dfrac{\partial ^2w}{\partial t^2} +\left( {\varepsilon \cdot \Delta m} \right)\delta \left( {x - x_0} \right)\dfrac{\partial ^2w}{\partial t^2} = 0 (24) $$
其中, $\varepsilon $为幅值比,即颗粒运动幅值与谐振器振动幅值之间的比值. 从模型中可以看出,实际影响谐振频率的不是悬浮质量$\Delta m$, 而是$\varepsilon \cdot\Delta m$, 只有当幅值比等于1时,由频率漂移推算出的质量才是实际的悬浮颗粒质量. 因此,幅值比直接影响着谐振器件的谐振频率和质量检测精度. 研究表明,密度比和雷诺数是影响检测误差的主要因素,其中密度比表示颗粒密度与流体密度的比值, 雷诺数$Re = {\rho _{\rm f}\omega R^2} / \mu $, 式中$\rho _{\rm f} $为流体密度, $\omega$为谐振频率, $R$是颗粒半径,$\mu $为流体黏度.图23描述了密度比和雷诺数对器件检测误差的影响规律,误差会随着雷诺数和密度比的增大而增大, 并且当雷诺数低于临界值时,误差为0. 因此, 为了保证颗粒质量的检测精度,需要使测量时的雷诺数低于临界雷诺数.
3.4 频率稳定性
频率稳定性是反映谐振器件在一定时间范围内产生相同频率的综合能力,是表征微纳米机械谐振器性能的重要参数(Sansa et al. 2016),直接决定谐振器件的检测极限(limit of detection). 因此,谐振器件的频率稳定性问题得到了学者们的普遍关注 (Arlett &Roukes, 2010, Cleland & Roukes 2002, Ekinci et al. 2004, Vig& Kim 1999).
频率波动$\delta \omega $定义为
$$ \delta \omega = \dfrac{1}{N}\sqrt {\sum\nolimits_{i =1}^N {\left( {\omega _i - \omega _0 } \right)^2} } ,\quad { SNR} =1 (25) $$
其中$SNR$表示信噪比. $\delta \omega$表征谐振器件的频率稳定性, $\delta \omega $越小, 频率稳定性越好.频率稳定性主要受噪声影响, 根据噪声产生机制不同,包括热机械噪声、温度波动噪声、吸附--脱附噪声、动量交换噪声等(Ekinci et al. 2004). 在噪声源影响下, 谐振器件会产生随机运动,从而引起频率的波动. 频率波动的有效谱密度记为$S_\omega \left(\omega \right)$, $\delta \omega $与$S_\omega \left( \omega\right)$之间的关系为
$$ \delta \omega = \left[ {\int_{\omega _0 - \pi \Delta f}^{\omega _0 + \pi \Delta f} {S_\omega \left( \omega\right){\rm d}\omega } } \right]^{1/ 2} (26) $$
式中,$\Delta f$为测量带宽, $\Delta f = 1 /2\pi \tau $, 其中$\tau$是测量的平均时间.
分子热运动会使微纳机械谐振器产生随机运动, 引起热机械噪声;由于谐振器件的热容很小, 环境变化很容易引起器件的温度波动,而谐振器件的尺寸和材料参数都与温度有关,因而温度波动也会导致频率波动; 吸附在谐振器件表面的气体分子,会改变谐振器件的质量, 从而影响谐振频率,分子的随机吸附和脱附会使谐振器件的频率降低或升高,引起吸附--脱附噪声;气体分子与谐振器之间的动量交换也会引起器件的随机运动,从而产生动量交换噪声. 不同噪声类型均会引起频率波动$\delta \omega$, 如表4所示, 其中$\omega _0 $为谐振频率,$Q$表示谐振器品质因子, $k_{\rm B} $是Boltzmann常数,$T$是热力学温度, $E_{\rm C} $为谐振器的最大驱动能力, $\omega$为振动频率, $c_{\rm s}$表示谐振器材料的声速, $\alpha _{\rm T}$为材料的热膨胀系数, $g$表示热导, $\tau _{\rm T}$为结构的热时间常数, $N_{\rm a} $表示谐振器表面吸附位点的数量,$\sigma _{{\rm occ}}^2 $表示每个位点占用概率的方差, $\tau _{\rm r}$为一个吸附--脱附周期的相关时间, $m_{{\rm mole}}$表示单个气体分子的质量, $Q_{\rm gas}$表示谐振器在气体耗散影响下的品质因子.
在这些噪声中,温度波动噪声、吸附脱附噪声以及动量交换噪声可以通过调节谐振器的工作环境来减弱甚至消除,而热机械噪声是由分子热运动引起, 只有当温度达到绝对零度时才会降为0.因此, 热机械噪声在微纳机械谐振器中普遍存在,并且由热机械噪声引起的$\delta \omega$是理论上能够测量的最小频率漂移,决定了谐振器频率稳定性的极限(Sansa et al. 2016).
频率稳定性对微纳通道机械谐振器的影响主要有两个方面:一方面影响谐振器的质量分辨率,另一方面影响被检测颗粒的最小停留时间.根据谐振器的质量检测原理可知, 噪声影响下的最小可检测质量$\delta m_{\min } $可表示为$\delta m_{\min } = -2m_0 \cdot {\delta \omega} / {\omega _0 }$, $\delta \omega $越小, 分辨率越高.当采用流动式检测时, 颗粒必须在通道中停留足够的时间,从而使可检测信号高于本底噪声. 最小停留时间可写成
$$\tau _{\rm acq} = \dfrac{S_x }{Q^2\Delta x_{\max }^2 }\dfrac{m_0^2}{\Delta m_{\rm p}^2 }\left( {{\rm SNR}} \right)^2 (27) $$
其中, $\Delta x_{\max } $为谐振器在线性范围内的最大振幅, $S_x$为噪声引起的位移功率谱密度, 与谐振器的频率稳定性有关, $\Delta m_{\rm p} $为被检测颗粒的悬浮质量. 根据上式可知,被检测物的悬浮质量越大, 颗粒的最小停留时间越小. 对于长20$\mu$m、宽4$\mu $m、厚0.7$\mu $m的微通道谐振器,检测悬浮质量为80ag的颗粒时, 最小停留时间为0.3$\mu $s,而检测悬浮质量为4ag的颗粒时, 最小停留时间达到0.13ms (Arlett & Roukes 2010). 最小停留时间影响检测通量,停留时间越短, 检测通量越高. 因此, 检测微小颗粒时, 检测通量较低.
最近, Sansa等(2016)研究发现,现有微纳机械谐振器的频率稳定性远没有达到由热机械噪声引起的稳定性极限(见图24),而是大约高2.1个数量级. 为了解释这一现象,他们通过实验研究了室温下单晶硅谐振器的频率特性, 结果表明,除了仪器噪声、热机械噪声、温度波动噪声等已知噪声源之外,还存在引起谐振器频率波动的未知噪声源,这可能是谐振器件无法达到稳定性极限的关键原因. 因此,迫切需要深入研究谐振器件频率波动的物理机理,为提高谐振器的频率稳定性、开发更高性能的谐振器奠定理论基础.
4 总结与展望
本文综述了微纳通道机械谐振器的研究进展,总结了谐振器的检测与表征功能及其相应的动力学设计原理,阐明了稳定性、频响特性、能量耗散、频率波动等动力学特性的内在作用与影响机理,分析了这些性质对谐振器性能的影响规律. 目前,许多学者在微纳通道谐振器设计及应用方面做了大量的研究工作,并取得了一些重要成果, 但是仍有许多问题需要进一步深入研究和探索,主要体现在以下几个方面:
(1) 微纳通道机械谐振器结构的局限性. 尺寸较大的谐振器,通道的截面较大, 可以检测多种直径的颗粒, 检测范围广,并且大尺寸通道的流动阻力较小, 有利于检测通量的提高;小尺寸谐振器的谐振频率、灵敏度和质量分辨率高于大尺寸谐振器,但是通道截面很小, 限制了检测范围, 此外,由于通道尺寸很小甚至达到了纳尺度, 因此通道内流动阻力很大,检测通量难以提升. 由于微纳通道机械谐振器结构的局限性,质量分辨率、检测范围、检测通量之间的矛盾十分突出,难以获得质量分辨率高、检测范围广、检测通量大的谐振器件.
(2) 谐振器件流致稳定性方面的实验研究较为缺乏. 目前,国内外学者针对微尺度输流管的稳定性问题提出了许多理论模型,采用经典输流管理论、非局部理论、应变梯度理论等分析了流动速度对稳定性的影响规律,但是这方面的实验研究较少, 理论模型缺乏有效的实验验证. 因此,需要开展微纳谐振器流致稳定性方面的实验研究,为更加深入认识稳定性的内在机理、建立精确有效的理论模型提供实验依据.
(3) 通用性强、精度高的品质因子理论模型有待建立.由于能量耗散机制的多样性、耦合性, 现有的模型基于线性动力学框架,并且没有考虑流动速度、支承方式等多种因素的影响,限制了模型的应用范围.当通道宽高比较小、支承方式改变、非线性效应显著时,由理论模型得到的预测值和实际值有较大差距, 甚至相差2$\sim$3个数量级. 因此,需要充分考虑小宽高比、大流动速度、不同支承方式以及大振幅、中平面拉伸等非线性效应对能量耗散机制的影响,建立通用性更强、精度更高的品质因子理论模型.
(4)加工误差对微通道机械谐振器动力学特性和测量精度的影响有待进一步研究.由于微加工工艺的局限性,在制造微通道机械谐振器的过程中必然存在加工误差, 包括:微通道截面关于梁的中性面不对称、通道截面不是标准的矩形或圆形、谐振器内表面和外表面存在表面粗糙度等.这些加工误差可能对微通道谐振器的动态性能和测量精度产生影响, 比如,微通道截面的不对称会引起谐振器的抽吸式能量耗散,制约谐振器品质因子的提高(Sader et al. 2010a);微通道的截面形状影响速度轮廓, 而根据Wang等(2013)的研究工作可知,谐振器的频响特性和稳定性与速度轮廓有关;谐振器外表面的表面粗糙度会影响谐振器与周围气体分子的相互作用,从而影响谐振器的吸附-脱附噪声以及动量交换噪声,而谐振器的噪声制约测量精度的提高;谐振器内表面的表面粗糙度影响内部流体的流动,进而影响谐振器的动力学特性. 然而, 针对以上问题的工作很少,有待进一步的研究与分析.
致谢
The authors have declared that no competing interests exist.
参考文献
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微纳机械谐振器能量耗散机理研究进展 .
微纳机械谐振器因其具有超高的谐振频率、品质因子和灵敏度等优越特性，在物理传感、生物与化学检测、射频通信、能量收集等方面表现出了卓越的性能而倍受关注，已成为当前微/纳机电系统(MEMS/NEMS)领域的研究重点和热点之一。能量耗散一直以来都是制约微纳机械谐振器性能提升与应用发展的瓶颈问题，且耗散机制具有多样性、不确定性和尺度相关性。本文综述了微纳机械谐振器中的能量耗散机理与非线性阻尼效应的研究进展，主要针对热弹性阻尼、声子相互作用、粘性阻尼、支撑损耗、表面与界面损耗等内禀和外部耗散机制进行了综述，阐明了不同能量耗散的产生机理及影响规律，可为降低能量损耗和结构优化设计、提高谐振器件的品质因子和动态性能提供参考，对微纳机械谐器的设计、制造及应用发展具有重要意义。
Research progress on energy dissipation mechanisms in micro- and nano-mechanical resonators .
微纳机械谐振器因其具有超高的谐振频率、品质因子和灵敏度等优越特性，在物理传感、生物与化学检测、射频通信、能量收集等方面表现出了卓越的性能而倍受关注，已成为当前微/纳机电系统(MEMS/NEMS)领域的研究重点和热点之一。能量耗散一直以来都是制约微纳机械谐振器性能提升与应用发展的瓶颈问题，且耗散机制具有多样性、不确定性和尺度相关性。本文综述了微纳机械谐振器中的能量耗散机理与非线性阻尼效应的研究进展，主要针对热弹性阻尼、声子相互作用、粘性阻尼、支撑损耗、表面与界面损耗等内禀和外部耗散机制进行了综述，阐明了不同能量耗散的产生机理及影响规律，可为降低能量损耗和结构优化设计、提高谐振器件的品质因子和动态性能提供参考，对微纳机械谐器的设计、制造及应用发展具有重要意义。
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Stability analysis of a piezoelectrically actuated micro-pipe conveying fluid .
This paper presents the stability analysis of a fluid-conveying micro-pipe axially loaded with a pair of piezoelectric layers located at its top and bottom surfaces. Based on Euler–Bernoulli beam theory, the governing equations of the system are derived by applying Hamilton’s variational principle. Galerkin projection technique is used to extract the frequency equations. Taking into account clamped-free boundary conditions with and without intermediate support, stability of the system is investigated to demonstrate the influence of flow velocity as well as the voltage of the piezoelectric layers on the flow-induced flutter instability. It is shown that imposing voltage difference to piezoelectric layers can significantly suppress the effect of fluid flow on vibrational frequencies and thus extend the stable margins. Moreover, effects of the intermediate support on the stability of the system are examined and it is shown that for some particular range of system configuration, the instability type may change from flutter to divergence.
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An embedded microchannel in a MEMS plate resonator for ultrasensitive mass sensing in liquid .
A mass sensor innovative concept is presented here, based on a hollow plate Micro Electro Mechanical System (MEMS) resonator. This approach consists in running a solution through an embedded microchannel, while the plate resonator is actuated according to a Lam茅-mode by electrostatic coupling in dry environment. The experimental results have shown a clear relationship between the measured shift of the resonance frequency and the sample solution density. Additionally, depending on the channel design and the solution properties, the quality factor (Q-factor) was noticed maintaining its level and even substantial improvement in particular cases. Resonators demonstrate resonance frequencies close to 78 MHz andQ-factor of a few thousands for liquid phase detection operating at ambient temperature and atmospheric pressure. Frequency fluctuations study revealed a 13 Hz instability level, equivalent to 1.5 fg in mass. Using a fully electronic readout configuration, a mass responsivity ofca.850 fg kHz 1was monitored.
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Ultimate and practical limits of fluid-based mass detection with suspended microchannel resonators .
Suspended microchannel resonators (SMRs) are an innovative approach to fluid-based microelectromechanical mass sensing that circumvents complete immersion of the sensor. By embedding the fluidics within the device itself, vacuum-based operation of the resonator becomes possible. This enables frequency shift-based mass detection with high quality factors, and hence sensitivity comparable to vacuum-based micromechanical resonators. Here we present a detailed analysis of the sensitivity of these devices, including consideration of fundamental and practical noise limits, and the important role of binding kinetics in sensing. We demonstrate that these devices show significant promise for protein detection. For larger, biologically-important targets such as rare whole virions, the required analysis time to flow sufficient sample through the sensor can become prohibitively long unless large parallel arrays of sensors or preconcentrators are employed.
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| [5] |
Fabrication of a Nanomechanical Mass Sensor Containing a Nanofluidic Channel .
Nanomechanical resonators operating in vacuum are capable of detecting and weighing single biomolecules, but their application to the life sciences has been limited by viscous forces that impede their motion in liquid environments. A promising approach to avoid this problem, encapsulating the fluid within a mechanical resonator surrounded by vacuum, has not yet been tried with resonant sensors of mass less than approximately 100 ng, despite predictions that devices with smaller effective mass will have proportionally finer mass resolution. Here, we fabricate and evaluate the performance of doubly clamped beam resonators that contain filled nanofluidic channels and have masses of less than 100 pg. These nanochannel resonators operate at frequencies on the order of 25 MHz and when filled with fluid have quality factors as high as 800, 2 orders of magnitude higher than that of resonators of comparable size and frequency operating in fluid. Fluid density measurements reveal a mass responsivity of 100 Hz/fg and a noise equivalent mass of 2 fg. Our analysis suggests that realistic improvements in the quality factor and frequency stability of nanochannel resonators would render these devices capable of sensing attogram masses from liquid.
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| [6] |
Thermal excitation and piezoresistive detection of cantilever in-plane resonance modes for sensing applications .
Thermally excited and piezoresistively detected bulk-micromachined cantilevers vibrating in their in-plane flexural resonance mode are presented. By shearing the surrounding fluid rather than exerting normal stress on it, the in-plane mode cantilevers exhibit reduced added fluid mass effects and improved quality factors in a fluid environment. In this letter, different cantilever geometries with in-plane resonance frequencies from 50 kHz to 2.2 MHz have been tested, with quality factors as high as 4200 in air and 67 in water.
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| [7] |
Linear and non-linear vibrations of fluid-filled hollow microcantilevers interacting with small particles .
Linear and non-linear vibrations of a U-shaped hollow microcantilever beam filled with fluid and interacting with a small particle are investigated. The microfluidic device is assumed to be subjected to internal flowing fluid carrying a buoyant mass. The equations of motion are derived via extended Hamilton's principle and by using Euler-Bernoulli beam theory retaining geometric and inertial non-linearities. A reduced-order model is obtained applying Galerkin's method and solved by using a pseudo arc-length continuation and collocation scheme to perform bifurcation analysis and obtain frequency response curves. Direct time integration of the equations of motion has also been performed by using Adams-Moulton method to obtain time histories and analyze transient cantilever-particle interactions in depth. It is shown that exploiting near resonant non-linear behavior of the microcantilever could potentially yield enhanced sensor metrics. This is found to be due to the transitions that occur as a matter of particle movement near the saddle-node bifurcation points of the coupled system that lead to jumps between coexisting stable attractors.
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| [8] |
Surface stress in the self-assembly of alkanethiols on gold .
Surface stress changes and kinetics were measured in situ during the self-assembly of alkanethiols on gold by means of a micromechanical sensor. Self-assembly caused compressive surface stress that closely followed Langmuir-type adsorption kinetics up to monolayer coverage. The surface stress at monolayer coverage increased linearly with the length of the alkyl chain of the molecule. These observations were interpreted in terms of differences in surface potential. This highly sensitive sensor technique has a broad range of applicability to specific chemical and biological interactions.
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| [9] |
Measuring single cell mass, volume, and density with dual suspended microchannel resonators .
Cell size, measured as either volume or mass, is a fundamental indicator of cell state. Far more tightly regulated than size is density, the ratio between mass and volume, which can be used to distinguish between cell populations even when volume and mass appear to remain constant. Here we expand upon a previous method for measuring cell density involving a suspended microchannel resonator (SMR). We introduce a new device, the dual SMR, as a high-precision instrument for measuring single-cell mass, volume, and density using two resonators connected by a serpentine fluidic channel. The dual SMR designs considered herein demonstrate the critical role of channel geometry in ensuring proper mixing and damping of pressure fluctuations in microfluidic systems designed for precision measurement. We use the dual SMR to compare the physical properties of two well-known cancer cell lines: human lung cancer cell H1650 and mouse lymphoblastic leukemia cell line L1210.
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Weighing of biomolecules, single cells and single nanoparticles in fluid .
Nanomechanical resonators enable the measurement of mass with extraordinary sensitivity. Previously, samples as light as 7 zeptograms (1 zg = 10(-21) g) have been weighed in vacuum, and proton-level resolution seems to be within reach. Resolving small mass changes requires the resonator to be light and to ring at a very pure tone-that is, with a high quality factor. In solution, viscosity severely degrades both of these characteristics, thus preventing many applications in nanotechnology and the life sciences where fluid is required. Although the resonant structure can be designed to minimize viscous loss, resolution is still substantially degraded when compared to measurements made in air or vacuum. An entirely different approach eliminates viscous damping by placing the solution inside a hollow resonator that is surrounded by vacuum. Here we demonstrate that suspended microchannel resonators can weigh single nanoparticles, single bacterial cells and sub-monolayers of adsorbed proteins in water with sub-femtogram resolution (1 Hz bandwidth). Central to these results is our observation that viscous loss due to the fluid is negligible compared to the intrinsic damping of our silicon crystal resonator. The combination of the low resonator mass (100 ng) and high quality factor (15,000) enables an improvement in mass resolution of six orders of magnitude over a high-end commercial quartz crystal microbalance. This gives access to intriguing applications, such as mass-based flow cytometry, the direct detection of pathogens, or the non-optical sizing and mass density measurement of colloidal particles.
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| [11] |
Suspended microchannel resonators for biomolecular detection .
We present a resonant mass sensor for specific biomolecular detection in a subnanoliter fluid volume. The sensing principle is based on measuring shifts in resonance frequency of a suspended microfluidic channel upon accumulation of molecules on the inside walls of the device. Confining the fluid to the inside of a hollow cantilever enables direct integration with conventional microfluidic systems, significantly increases sensitivity by eliminating high damping and viscous drag, and allows the resonator to be actuated by electrostatic forces. Fluid density measurements reveal a mass resolution of106117g/μm2in a 4 mHz–4 Hz bandwidth. To demonstrate biomolecular detection, we present real-time measurements of the specific binding between avidin and biotinylated bovine serum albumin. Based on these measurements, we expect that changes in surface mass loading on the order of106119g/μm2can be detected in an optimized system.
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| [12] |
Vacuum-packaged suspended microchannel resonant mass sensor for biomolecular detection .
There is a great need in experimental biology for tools to study interactions between biological molecules and to profile expression levels of large numbers of proteins. This paper describes the fabrication, packaging and testing of a resonant mass sensor for the detection of biomolecules in a microfluidic format. The transducer employs a suspended microchannel as the resonating element, thereby avoiding the problems of damping and viscous drag that normally degrade the sensitivity of resonant sensors in liquid. Our device differs from a vibrating tube densitometer in that the channel is very thin, which enables the detection of molecules that bind to the channel walls; this provides a path to specificity via molecular recognition by immobilized receptors. The fabrication is based on a sacrificial polysilicon process with low-stress low-pressure chemical-vapor deposited (LPCVD) silicon nitride as the structural material, and the resonator is vacuum packaged on the wafer scale using glass frit bonding. Packaged resonators exhibit a sensitivity of 0.8 ppm/(ngmiddotcm2) and a mechanical quality factor of up to 700. To the best of our knowledge, this quality factor is among the highest so far reported for resonant sensors with comparable surface mass sensitivity in liquid
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| [13] |
Nonmonotonic energy dissipation in microfluidic resonators . |
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High-throughput measurement of single-cell growth rates using serial microfluidic mass sensor arrays .
Methods to rapidly assess cell growth would be useful for many applications, including drug susceptibility testing, but current technologies have limited sensitivity or throughput. Here we present an approach to precisely and rapidly measure growth rates of many individual cells simultaneously. We flow cells in suspension through a microfluidic channel with 10–12 resonant mass sensors distributed along its length, weighing each cell repeatedly over the 4–20 min it spends in the channel. Because multiple cells traverse the channel at the same time, we obtain growth rates for >60 cells/h with a resolution of 0.2 pg/h for mammalian cells and 0.02 pg/h for bacteria. We measure the growth of single lymphocytic cells, mouse and human T cells, primary human leukemia cells, yeast,Escherichia coliandEnterococcus faecalis. Our system reveals subpopulations of cells with divergent growth kinetics and enables assessment of cellular responses to antibiotics and antimicrobial peptides within minutes.
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| [15] |
Determination of adsorption-induced variation in the spring constant of a microcantilever .
Although resonating microcantilevers are demonstrated to be excellent mass sensors, adsorption-induced changes in the spring constant result in errors in the calculation of adsorbed mass from shifts in resonance frequencies. However, simultaneous measurement of resonance frequency and adsorption-induced cantilever bending can be used to determine the variation in spring constant. Plotting the change in surface stress as a function of analyte concentration, the surface excess of adsorbed molecules and, therefore, the mass adsorbed can be determined. Here, we demonstrate this concept for adsorption of Na+ ions on microcantilevers in NaCl solutions where a change in the spring constant was found to increase from 9.5x10(-4) to 7.5x10(-3) N/m as the NaCl concentration increased from 0.05 to 0.8 M. (C) 2002 American Institute of Physics.
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| [16] |
Noise processes in nanomechanical resonators .
Nanomechanical resonators can be fabricated to achieve high natural resonance frequencies, approaching 1 GHz, with quality factors in excess of104.These resonators are candidates for use as highly selective rf filters and as precision on-chip clocks. Some fundamental and some nonfundamental noise processes will present limits to the performance of such resonators. These include thermomechanical noise, Nyquist–Johnson noise, and adsorption–desorption noise; other important noise sources include those due to thermal fluctuations and defect motion-induced noise. In this article, we develop a self-contained formalism for treating these noise sources, and use it to estimate the impact that these noise processes will have on the noise of a model nanoscale resonator, consisting of a doubly clamped beam of single-crystal Si with a natural resonance frequency of 1 GHz.
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| [17] |
Modeling and nonlinear dynamics of fluid-conveying risers under hybrid excitations .
The nonlinear dynamical responses of a vertical riser concurrently subjected to hybrid excitations, namely, vortex-induced vibrations (VIVs) and base excitations are investigated. The riser conveying fluid is placed in a uniform cross-flow and subjected to direct harmonic excitations. A van der Pol wake oscillator is used to model the fluctuating lift coefficient. The extended Hamilton principle and the Galerkin procedure are used to derive a nonlinear distributed-parameter model for a vertical riser under a combination of vibratory base excitations and vortex-induced vibrations. Linear and nonlinear analyses are performed to investigate the effects of internal fluid velocity, cross-flow speed, and base acceleration on the coupled frequency, onset speed of synchronization, and vibration amplitudes of the riser. The results show that when the cross-flow speed becomes in the synchronization region, vibration behaviors of the riser change from aperiodic to periodic motions, with a jumping phenomenon between these two kinds of motions. It is also demonstrated that the amplitude of the riser can be increased or decreased under combined effects of vortex-induced vibrations and base excitations compared to the separate effect of vortex-induced vibrations or base excitations. The results also show that an increase of the base acceleration results in a wider synchronization region and a significant effect associated with the quenching phenomenon.
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| [18] |
Nonlinear dynamic responses of electrostatically actuated microcantilevers containing internal fluid flow .
A nonlinear theoretical model for electrostatically actuated microcantilevers containing internal fluid flow is developed in the present study, which takes into account the geometric and electrostatic
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| [19] |
Simulation of adsorption-induced stress of a microcantilever sensor .
Molecular adsorption on a microcantilever surface causes the cantilever to bend as a result of adsorption-induced stresses. A model for adsorption-induced surface stress based on atomic or molecular interaction is proposed. The model was tested with mercury adsorption on gold-coated cantilevers. Calculated deflections based on the proposed simulation model agree well with measured deflection data. This model gives insight into the interatomic forces that play a significant role in creating adsorption-induced surface stresses and resultant mechanical bending of microcantilevers.
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| [20] |
Full-Lagrangian schemes for dynamic analysis of electrostatic MEMS .
Dynamic analysis of microelectromechanical systems (MEMS) is characterized by the nonlinear coupling of electrical and mechanical domains. The nonlinear coupling between the two domains gives rise to several interesting dynamic phenomena besides the well established pull-in phenomenon in electrostatic MEMS. For proper understanding and detailed exploration of MEMS dynamics, it is important to have a reliable and efficient physical level simulation method. In this paper, we develop relaxation and Newton schemes based on a Lagrangian description of both the mechanical and the electrical domains for the analysis of MEMS dynamics. The application of a Lagrangian description for both mechanical and electrostatic analysis makes this method far more efficient than standard semi-Lagrangian scheme-based analysis of MEMS dynamics. A major advantage of the full-Lagrangian scheme is in the accurate computation of the interdomain coupling term (mechanical to electrical) in the Jacobian matrix of the Newton scheme which is not possible with a semi-Lagrangian scheme. The full-Lagrangian based relaxation and Newton schemes have been validated by comparing simulation results with published data for cantilever and fixed-fixed MEM beams. The Newton scheme has been used for the dynamic analysis of two classes of comb-drives widely used in MEMS, namely, transverse and lateral comb-drives. Several interesting MEM dynamic phenomena and their possible applications have been presented. Spring-hardening and softening of MEM devices has been shown. The existence of multiple resonant peaks in MEM devices has been analyzed under different electrical signals and their possible applications in multiband/passband MEM filters/oscillators is discussed. Switching speed is a serious constraint for capacitive based RF-MEM switches. We have shown that a DC bias along with an ac bias at the resonant frequency can give very fast switching at a considerably less peak power requirement.
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| [21] |
Mass and position determination of attached particles on cantilever based mass sensors .
An analytical expression relating mass and position of a particle attached on a cantilever to the resulting change in cantilever resonant frequency is derived. Theoretically, the position and mass of the attached particle can be deduced by combining measured resonant frequencies of several bending modes. This finding is verified experimentally using a microscale cantilever with and without an attached gold bead. The resonant frequencies of several bending modes are measured as a function of the bead position. The bead mass and position calculated from the measured resonant frequencies are in good agreement with the expected mass and the position measured.
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| [22] |
Ultimate limits to inertial mass sensing based upon nanoelectromechanical systems .
Nanomechanical resonators can now be realized that achieve fundamental resonance frequencies exceeding 1 GHz, with quality factors (Q) in the range10381Q81105.The minuscule active masses of these devices, in conjunction with their highQs, translate into unprecedented inertial mass sensitivities. This makes them natural candidates for a variety of mass sensing applications. Here we evaluate the ultimate mass sensitivity limits for nanomechanical resonators operatingin vacuothat are imposed by a number of fundamental physical noise processes. Our analyses indicate that nanomechanical resonators offer immense potential for mass sensing—ultimately with resolution at the level of individual molecules.
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| [23] |
Determination of the Density of Protein Particles Using a Suspended Microchannel Resonator .
One of the analytical tools for characterization of subvisible particles, which gained popularity over the last years because of its unique capabilities, is the resonance mass measurement technique. However, a challenge that this technique presents is the need to know the exact density of the measured particles in order to obtain accurate size calculations. The density of proteinaceous subvisible particles has not been measured experimentally yet and to date researchers have been using estimated density values. In this paper, we report for a first-time experimental measurements of the density of protein particles (0.2–5 μm in size) using particles created by stressing three different proteins using four different types of stress conditions. Interestingly, the particle density values that were measured varied between 1.28 and 1.33 g/cm3 and were lower than previous estimates. Furthermore, it was found that although the density of proteinaceous particles was affected to a very low degree by the stress conditions used to generate them, there is relatively larger difference between particles originating from different classes of proteins (e.g., monoclonal antibody vs. bovine serum albumin). 08 2015 Wiley Periodicals, Inc. and the American Pharmacists Association J Pharm Sci 104:4034–4040, 2015
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| [24] |
Translating biomolecular recognition into nanomechanics .
We report the specific transduction, via surface stress changes, of DNA hybridization and receptor-ligand binding into a direct nanomechanical response of microfabricated cantilevers. Cantilevers in an array were functionalized with a selection of biomolecules. The differential deflection of the cantilevers was found to provide a true molecular recognition signal despite large nonspecific responses of individual cantilevers. Hybridization of complementary oligonucleotides shows that a single base mismatch between two 12-mer oligonucleotides is clearly detectable. Similar experiments on protein A-immunoglobulin interactions demonstrate the wide-ranging applicability of nanomechanical transduction to detect biomolecular recognition.
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| [25] |
Resonating modes of vibrating microcantilevers in liquid .
A study of nanomechanical cantilevers vibrating at various resonating modes in liquid is presented. Resonant frequency spectrum with 16 well resolved flexural modes is obtained. The quality factor increased from 1 at mode 1 to 30 at mode 16. The theoretical estimate of eigenfrequency using the Elmer–Dreier model [F.-J. Elmer and M. Dreier,J. Appl. Phys.81,12(1997)] and Sader’s extended viscous model [C. A. Van Eysden and J. E. Sader,J. Appl. Phys.101,044908(2007)] matched well with the experimental data. The apparent mass of the liquid comoved by the oscillating cantilevers decreased asymptotically with mode number.
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| [26] |
On the viscoelastic dynamics of fluid-conveying microtubes .
This paper is the first to analyse the coupled fluid-structure viscoelastic dynamical characteristics of a fluid-conveying viscoelastic microtube resting on a nonlinear elastic bed subject to large rotations. None of the axial and transverse motions/accelerations is neglected in the modelling and simulations. The dissipation is modelled using the Kelvin oigt scheme for the deviatoric segment of the symmetric couple stress tensor and the stress tensor. Based on the Euler ernoulli theory, in which the microtube cross-section remains perpendicular to the centreline, and the modified couple stress theory (MCST), the energies and the work of external load and damping are formulated. Through use of Hamilton's principle, the coupled transverse-longitudinal equations governing the motion of the fluid-conveying viscoelastic microtube are developed. A weighted-residual-based discretisation method is applied to the continuous vibration model and the resultant reduced model is simulated via a continuation technique. The coupled fluid-structure dynamical characteristics of the fluid-conveying viscoelastic microtube are analysed by constructing the frequency-amplitude diagrams. It is shown that slight changes in the flow speed significantly affects the resonant response and modal interactions.
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| [27] |
Measuring the mass, density, and size of particles and cells using a suspended microchannel resonator .
We demonstrate the measurement of mass, density, and size of cells and nanoparticles using suspended microchannel resonators. The masses of individual particles are quantified as transient frequency shifts, while the particles transit a microfluidic channel embedded in the resonating cantilever. Mass histograms resulting from these data reveal the distribution of a population of heterogeneously sized particles. Particle density is inferred from measurements made in different carrier fluids since the frequency shift for a particle is proportional to the mass difference relative to the displaced solution. We have characterized the density of polystyrene particles,Escherichia coli, and human red blood cells with a resolution down to10614g∕cm3.
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| [28] |
Using buoyant mass to measure the growth of single cells .
We used a suspended microchannel resonator (SMR) combined with picoliter-scale microfluidic control to measure buoyant mass and determine the 'instantaneous' growth rates of individual cells. The SMR measures mass with femtogram precision, allowing rapid determination of the growth rate in a fraction of a complete cell cycle. We found that for individual cells of Bacillus subtilis, Escherichia coli, Saccharomyces cerevisiae and mouse lymphoblasts, heavier cells grew faster than lighter cells.
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| [29] |
Torsional frequency response of cantilever beams immersed in viscous fluids with applications to the atomic force microscope .
The vibrational characteristics of a cantilever beam are well known to strongly depend on the fluid in which the beam is immersed. In this paper, we present a detailed theoretical analysis of the frequency response of a cantilever beam, that is immersed in a viscous fluid and excited by an arbitrary driving force. Due to its practical importance in application to the atomic force microscope (AFM), we consider in detail the special case of a cantilever beam that is excited by a thermal driving force. This will incorporate the presentation of explicit analytical formulae and numerical results, which will be of value to the users and designers of AFM cantilever beams.
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| [30] |
Microfluidic cantilever detects bacteria and measures their susceptibility to antibiotics in small confined volumes .
In the fight against drug-resistant bacteria, accurate and high-throughput detection is essential. Here, a bimaterial microcantilever with an embedded microfluidic channel with internal surfaces chemically or physically functionalized with receptors selectively captures the bacteria passing through the channel. Bacterial adsorption inside the cantilever results in changes in the resonance frequency (mass) and cantilever deflection (adsorption stress). The excitation of trapped bacteria using infrared radiation (IR) causes the cantilever to deflect in proportion to the infrared absorption of the bacteria, providing a nanomechanical infrared spectrum for selective identification. We demonstrate thein situdetection and discrimination ofListeria monocytogenesat a concentration of single cell per l. TrappedEscherichia coliin the microchannel shows a distinct nanomechanical response when exposed to antibiotics. This approach, which combines enrichment with three different modes of detection, can serve as a platform for the development of a portable, high-throughput device for use in the real-time detection of bacteria and their response to antibiotics. Analysis of bacteria and their response to antibiotics in real time is challenging. Here the authors report a microcantilever based system that can detect and discriminate between bacteria species and, due to the ability to discriminate between alive and dead samples, measure response to antibiotics.
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| [31] |
Nonlinear dynamics of a fluid-conveying pipe under the combined action of cross-flow and top-end excitations .
We report a theoretical investigation of an elastic and slender fluid-conveying pipe with a top-end excitation subjected to uniform cross flows. Considering the mean drag force and the time varying vortex-induced lift force which is modeled using a nonlinear van der Pol oscillator, the nonlinear partial differential equations of the motion of coupled fluid-structure system are constructed and simplified to a reduced-order model through the Galerkin-type discretization. By virtue of quasi-static displacement conditions, the characteristics of vortex-induced vibration of the pipe are evaluated for the first two lock-in modes. The results show that the top-end excitation can increase the vibration amplitude of the pipe when the cross-flow speed is out of the lock-in regions. When the cross-flow speed is within the lock-in region, however, the top-end oscillation causes a transition between quasi-periodic and periodic in the responses of the pipe, significantly reducing or increasing the vibration amplitudes depending on the excitation acceleration and frequency. This finding has an important guidance in suppressing vortex-induced vibrations by balancing the internal fluid velocity and the top-end excitation.
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| [32] |
Dominant surface stress driven by biomolecular interactions in the dynamical response of nanomechanical microcantilevers .
Nanomechanical microcantilevers have played a vital role in detecting biomolecular interactions. The ability of microcantilevers to detect biomolecular interactions is ascribed to the principle that the surface stress, caused by biomolecular interactions, dominates the dynamical response of the microcantilever. Here we have experimentally studied the correlation between biomolecular interactions and the dynamical response of microcantilevers. Moreover, the authors employed a mechanical beam model to calculate the surface stress, representing the biomolecular interactions, through measuring the resonant frequency shift. The quantitative analysis of surface stress, driven by the specific protein-protein interactions, demonstrated that microcantilevers enable the quantitative study of biomolecular interactions.
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| [33] |
An atomic-resolution nanomechanical mass sensor .
Abstract Mechanical resonators are widely used as inertial balances to detect small quantities of adsorbed mass through shifts in oscillation frequency. Advances in lithography and materials synthesis have enabled the fabrication of nanoscale mechanical resonators, which have been operated as precision force, position and mass sensors. Here we demonstrate a room-temperature, carbon-nanotube-based nanomechanical resonator with atomic mass resolution. This device is essentially a mass spectrometer with a mass sensitivity of 1.3 x 10(-25) kg Hz(-1/2) or, equivalently, 0.40 gold atoms Hz(-1/2). Using this extreme mass sensitivity, we observe atomic mass shot noise, which is analogous to the electronic shot noise measured in many semiconductor experiments. Unlike traditional mass spectrometers, nanomechanical mass spectrometers do not require the potentially destructive ionization of the test sample, are more sensitive to large molecules, and could eventually be incorporated on a chip.
|
| [34] |
An atomic-resolution nanomechanical mass sensor .
Abstract Mechanical resonators are widely used as inertial balances to detect small quantities of adsorbed mass through shifts in oscillation frequency. Advances in lithography and materials synthesis have enabled the fabrication of nanoscale mechanical resonators, which have been operated as precision force, position and mass sensors. Here we demonstrate a room-temperature, carbon-nanotube-based nanomechanical resonator with atomic mass resolution. This device is essentially a mass spectrometer with a mass sensitivity of 1.3 x 10(-25) kg Hz(-1/2) or, equivalently, 0.40 gold atoms Hz(-1/2). Using this extreme mass sensitivity, we observe atomic mass shot noise, which is analogous to the electronic shot noise measured in many semiconductor experiments. Unlike traditional mass spectrometers, nanomechanical mass spectrometers do not require the potentially destructive ionization of the test sample, are more sensitive to large molecules, and could eventually be incorporated on a chip.
|
| [35] |
Persistence of bending and torsional modes in piezoelectric-excited millimeter-sized cantilever (PEMC) sensors in viscous liquids - 1 to 10 3 cP .
Cantilever sensors consisting of only a piezoelectric layer express both bending and torsional modes near 6525 kHz that persist with reasonable Q-values (6515) in liquids of high viscosity (>100 cP). Responses of both bending and torsional modes in liquids (1611019 cP) were measured simultaneously. The bending mode response was more sensitive to mass-change effects than the torsional mode, and the response in liquids <70 cP matched theoretical values within 0.9%. At 1019 cP the bending mode response was within 10.4% of theory. The bending and torsional modes in PEMC can potentially be used simultaneously for bio-chemical sensing in very viscous samples.
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| [36] |
Stress-induced variations in the stiffness of micro- and nanocantilever beams .
The effect of surface stress on the stiffness of cantilever beams remains an outstanding problem in the physical sciences. While numerous experimental studies report significant stiffness change due to surface stress, theoretical predictions are unable to rigorously and quantitatively reconcile these observations. In this Letter, we present the first controlled measurements of stress-induced change in cantilever stiffness with commensurate theoretical quantification. Simultaneous measurements are also performed on equivalent clamped-clamped beams. All experimental results are quantitatively and accurately predicted using elasticity theory. We also present conclusive experimental evidence for invalidity of the long-standing and unphysical axial force model, which has been widely applied to interpret measurements using cantilever beams. Our findings will be of value in the development of micro- and nanoscale resonant mechanical sensors.
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| [37] |
Online measurement of mass density and viscosity of pL fluid samples with suspended microchannel resonator .
Physical characterization of viscous samples is crucial in chemical, pharma and petroleum industry. For example, in the refining industry of petroleum, water percentage is verified by measuring the density of a sample. In this article we present a suspended microchannel resonator (SMR) which uses 5 pL of a fluid sample and measures its density with a resolution of 0.01 kg/m(3) and a sensitivity of 16 Hz/kg/m(3). The resonator can also simultaneously measure viscosity of the solutions with an accuracy of 0.025 mPa s. The SMR is part of a system which contains packaging and tubing to deliver samples to the resonator. The system can easily handle multiple viscous fluids to measure their densities and viscosities. The SMR is transparent, facilitating visual inspection of the microchannel content. (C) 2013 Elsevier B.V. All rights reserved.
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| [38] |
Hollow microtube resonators via silicon self-assembly toward subattogram mass sensing applications .
Fluidic resonators with integrated microchannels (hollow resonators) are attractive for mass, density, and volume measurements of single micro/nanoparticles and cells, yet their widespread use is limited by the complexity of their fabrication. Here we report a simple and cost-effective approach for fabricating hollow microtube resonators. A prestructured silicon wafer is annealed at high temperature under a controlled atmosphere to form self-assembled buried cavities. The interiors of these cavities are oxidized to produce thin oxide tubes, following which the surrounding silicon material is selectively etched away to suspend the oxide tubes. This simple three-step process easily produces hollow microtube resonators. We report another innovation in the capping glass wafer where we integrate fluidic access channels and getter materials along with residual gas suction channels. Combined together, only five photolithographic steps and one bonding step are required to fabricate vacuum-packaged hollow microtub...
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| [39] |
Effect of surface stress on the stiffness of cantilever plates . |
| [40] |
Surface stress and the normal mode of vibration of thin crystals: GaAs. Photopiezoelectric effect .
The normal mode of vibration of (111) GaAs wafers with a thickness below about 15 μm was found to depend strongly on the surface preparation and on the ambient atmosphere. This dependence was attributed to effects directly related to the surface stress σs. It was shown that σscan be evaluated from the natural frequency of vibration. The values of σs, in the 〈110〉 direction, for etched and unetched (111) GaAs wafers in room atmosphere were found to be 325 and 570 dyn/cm, respectively. It was further demonstrated that surface stress transients due to the adsorption processes (adsorption transients) can be determined by corresponding changes in the natural frequency of vibration.
|
| [41] |
Effects of gold patterning on the bending profile and frequency response of a microcantilever .
We have systematically investigated the effect of various gold patterns on the bending profile and frequency response of a microcantilever. The gold patterns were deposited on the cantilever arrays using four types of shadow mask. The local bending profile, slope, and curvature varied depending on the area and position of the gold pattern. Also, the variations in the first three modes of the flexural resonance frequencies of the gold patterned cantilevers were measured to understand the opposing effects of mass loading and flexural rigidity; both of these parameters are dependent on the position and area of the gold pattern. The experimental results validated the theoretical one-dimensional model introduced byTamayoet al.[Appl. Phys. Lett.89,224104(2006)]and our calculations using the finite element method. The gold patterns giving the maximum response of the mass loading and flexural rigidity change were determined by examining how the relative resonance frequency shifts as a function of the distance of the gold coating from the free end or clamping region. The results of this study can potentially be applied in the design of a microcantilever sensor in which pattern analysis is utilized to determine the presence of adsorbed biological and chemical molecules.
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| [42] |
Note: precision viscosity measurement using suspended microchannel resonators .
We report the characterization of a suspended microchannel resonator (SMR) for viscosity measurements in a low viscosity regime (<10 mPa65s) using two measurement schemes. First, the quality factor (Q-factor) of the SMR was characterized with glycerol-water mixtures. The measured Q-factor at 2065°C exhibits a bilinear behavior with the sensitivity of 1281 (mPa65s)611for a lower (1-4 mPa65s) and 355 (mPa65s)611for a higher viscosity range (4-8 mPa65s), respectively. The second scheme is the vibration amplitude monitoring of the SMR running in a closed loop feedback. When compared in terms of the measurement time, the amplitude-based measurement takes only 0.1 65 1 ms while the Q-factor-based measurement takes 6530 s. However, the viscosity resolution of the Q-factor-based measurement is at least three times better than the amplitude-based measurement. By comparing the Q-factors of heavy water and 9.65 wt.% glycerol-water mixture that have very similar viscosities but different densities, we confirmed that the SMR can measure the dynamic viscosity without the density correction. The obtained results demonstrate that the SMR can measure the fluid viscosity with high precision and even real-time monitoring of the viscosity change is possible with the amplitude-based measurement scheme.
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| [43] |
High precision particle mass sensing using microchannel resonators in the second vibration mode .
An intrinsic uncertainty in particle mass sensing with the suspended microchannel resonator results from variation in a particle's position near the free end of the resonator. To circumvent this error we employ the second flexural bending mode. This mode exhibits additional frequency peaks while particles pass over the antinode, a point where the frequency shift is insensitive to the lateral position of the particle. We measure polystyrene beads with the first and second modes and confirm that the second mode sensing provides a narrower mass histogram. For 3 m diameter beads, second mode sensing at the antinode improves the coefficient of variation in buoyant mass from 1.76% to 1.05% for population measurements and from 1.40% to 0.53% for a single trapped particle.
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| [44] |
Suspended microchannel resonators with piezoresistive sensors .
Precision frequency detection has enabled the suspended microchannel resonator (SMR) to weigh single living cells, single nanoparticles, and adsorbed protein layers in fluid. To date, the SMR resonance frequency has been determined optically, which requires the use of an external laser and photodiode and cannot be easily arrayed for multiplexed measurements. Here we demonstrate the first electronic detection of SMR resonance frequency by fabricating piezoresistive sensors using ion implantation into single crystal silicon resonators. To validate the piezoresistive SMR, buoyant mass histograms of budding yeast cells and a mixture of 1.6, 2.0, 2.5, and 3.0 m diameter polystyrene beads are measured. For SMRs designed to weigh micron-sized particles and cells, the mass resolution achieved with piezoresistive detection ( 3.4 fg in a 1 kHz bandwidth) is comparable to what can be achieved by the conventional optical-lever detector. Eliminating the need for expensive and delicate optical components will enable new uses for the SMR in both multiplexed and field deployable applications.
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| [45] |
Toward attogram mass measurements in solution with suspended nanochannel resonators .
Abstract Using suspended nanochannel resonators (SNRs), we demonstrate measurements of mass in solution with a resolution of 27 ag in a 1 kHz bandwidth, which represents a 100-fold improvement over existing suspended microchannel resonators and, to our knowledge, is the most precise mass measurement in liquid today. The SNR consists of a cantilever that is 50 microm long, 10 microm wide, and 1.3 microm thick, with an embedded nanochannel that is 2 microm wide and 700 nm tall. The SNR has a resonance frequency near 630 kHz and exhibits a quality factor of approximately 8000 when dry and when filled with water. In addition, we introduce a new method that uses centrifugal force caused by vibration of the cantilever to trap particles at the free end. This approach eliminates the intrinsic position dependent error of the SNR and also improves the mass resolution by increasing the averaging time for each particle.
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| [46] |
An atomic-resolution nanomechanical mass sensor based on circular monolayer graphene sheet: Theoretical analysis of vibrational properties .
Magnetron sputtered permalloy films are treated by wet chemical etchings with acid etchant as well as fluorine based reactive ion etch (RIE). Upon these treatments, the resistivity and coercivity of the permalloy film increase is within 10 %. No significant increase observed with prolonged etching time. The effective magnetization change of the permalloy films are within 5 % post the treatments. Atomic force microscope (AFM) and transmission electron microscope (TEM) are used to study the surface and interface evolution of permalloy film upon etching. The small impact on the electrical and magnetic properties of permalloy films can be correlated with the surface oxide protecting layer formation during the etch. Consequently, sputtered NiFe is a safe material to expose to these etching processes for write pole shield application.
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| [47] |
Surface stress effects on the resonance properties of cantilever sensors . |
| [48] |
3D suspended polymeric microfluidics (SPMF3) with flow orthogonal to bending (FOB) for fluid analysis through kinematic viscosity . |
| [49] |
Influence of surface stress on the resonance behavior of microcantilevers .
This work presents a model to predict the effect of surface stresses on theith-mode bending resonant frequency of microcantilevers and its experimental validation. With this model, one can calculate the surface stress acting upon the microcantilever solely by measuring resonant frequencies whereas previously one needed to measure the deflection. Resonant frequency measurement has distinct advantages in terms of ease and accuracy of measurement.
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| [50] |
Facile phase transition measurements for nanogram level liquid samples using suspended microchannel resonators .
We investigated phase transitions of a PEO-PPO-PEO triblock copolymer and n-heptadecane using a suspended microchannel resonator (SMR). After filling the microchannel of the SMR with each sample in liquid state, changes in the resonance frequency of the SMR were measured as a function of temperature, and then converted into changes in the density of each sample. As temperature increases, PEO-PPO-PEO unimers aggregate and form micelles (unimer-micelle transition), so the density of the polymer decreases and the resonance frequency of the SMR increases. As temperature decreases, n-heptadecane undergoes liquid to rotator phase (liquid-rotator transition), which increases the sample density and decreases the resonance frequency of the SMR. In addition, the liquid-rotator transition of n-heptadecane exhibits a sudden change in the quality factor of the SMR.
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| [51] |
Resolution enhancement of suspended microchannel resonators for weighing of biomolecular complexes in solution .
We introduce the use of correlation analysis to extend the dynamic range of suspended micro- and nanochannel resonator (SMR/SNR) mass sensors by over five orders of magnitude. This method can analyze populations of particles flowing through an embedded channel micromechanical resonator, even when the individual particle masses are far below the noise floor. To characterize the method, we measured the mass of polystyrene nanoparticles with 300 zg resolution. As an application, we monitored the time course of insulin amyloid formation from pre-fibrillar aggregates to mature fibrils of 15 MDa average mass. Results were compared with thioflavin-T (ThT) assays and electron microscopy (EM). Mass measurements offer additional information over ThT during the fluorescent inaccessible lag period, and the average fibril dimensions calculated from the mass signal are in good accordance with EM. In the future, we envision that more detailed modeling will allow the computational deconvolution of multicomponent samples, enabling the mass spectrometric characterization of a variety of biomolecular complexes, small organelles, and nanoparticles in solution.
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| [52] |
Static pull-in analysis of electrostatically actuated microbeams using homotopy perturbation method .
In this study, static pull-in instability of electrostatically-actuated microbridges and microcantilevers is investigated considering different nonlinear effects. Galerkin’s decomposition method is utilized to convert the nonlinear differential equations of motion to nonlinear integro-algebraic equations. Afterward, analytic solutions to static deflections of the microbeams are obtained using the homotopy perturbation method. Results are in excellent agreement with those presented in the literature.
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| [53] |
Reduced-order models for MEMS applications .
We review the development of reduced-order models for MEMS devices. Based on their implementation procedures, we classify these reduced-order models into two broad categories: node and domain methods. Node methods use lower-order approximations of the system matrices found by evaluating the system equations at each node in the discretization mesh. Domain-based methods rely on modal analysis and the Galerkin method to rewrite the system equations in terms of domain-wide modes (eigenfunctions). We summarize the major contributions in the field and discuss the advantages and disadvantages of each implementation. We then present reduced-order models for microbeams and rectangular and circular microplates. Finally, we present reduced-order approaches to model squeeze-film and thermoelastic damping in MEMS and present analytical expressions for the damping coefficients. We validate these models by comparing their results with available theoretical and experimental results.
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| [54] |
Dynamic pull-in phenomenon in MEMS resonators .
We study the pull-in instability in microelectromechanical (MEMS) resonators and find that characteristics of the pull-in phenomenon in the presence of AC loads differ from those under purely DC loads. We analyze this phenomenon, dubbed dynamic pull-in, and formulate safety criteria for the design of MEMS resonant sensors and filters excited near one of their natural frequencies. We also utilize this phenomenon to design a low-voltage MEMS RF switch actuated with a combined DC and AC loading. The new switch uses a voltage much lower than the traditionally used DC voltage. Either the frequency or the amplitude of the AC loading can be adjusted to reduce the driving voltage and switching time. The new actuation method has the potential of solving the problem of high driving voltages of RF MEMS switches.
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| [55] |
Measurement of the average mass of proteins adsorbed to a nanoparticle by using a suspended microchannel resonator .
We assessed the potential of a suspended microchannel resonator (SMR) to measure the adsorption of proteins to nanoparticles. Standard polystyrene beads suspended in buffer were weighed by a SMR system. Particle suspensions were mixed with solutions of bovine serum albumin (BSA) or monoclonal human antibody (IgG), incubated at room temperature for 3h and weighed again with SMR. The difference in buoyant mass of the bare and protein-coated polystyrene beads was calculated into real mass of adsorbed proteins. The average surface area occupied per protein molecule was calculated, assuming a monolayer of adsorbed protein. In parallel, dynamic light scattering (DLS), nanoparticle tracking analysis (NTA), and zeta potential measurements were performed. SMR revealed a statistically significant increase in the mass of beads because of adsorption of proteins (for BSA and IgG), whereas DLS and NTA did not show a difference between the size of bare and protein-coated beads. The change in the zeta potential of the beads was also measurable. The surface area occupied per protein molecule was in line with their known size. Presented results show that SMR can be used to measure the mass of adsorbed protein to nanoparticles with a high precision in the presence of free protein. 2014 Wiley Periodicals, Inc. and the American Pharmacists Association J Pharm Sci 104:698-704, 2015
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| [56] |
Weighing nanoparticles in solution at the attogram scale .
Physical characterization of nanoparticles is required for a wide range of applications. Nanomechanical resonators can quantify the mass of individual particles with detection limits down to a single atom in vacuum. However, applications are limited because performance is severely degraded in solution. Suspended micro- and nanochannel resonators have opened up the possibility of achieving vacuum-level precision for samples in the aqueous environment and a noise equivalent mass resolution of 27 attograms in 1-kHz bandwidth was previously achieved by Lee et al. [(2010) Nano Lett 10(7):2537-2542]. Here, we report on a series of advancements that have improved the resolution by more than 30-fold, to 0.85 attograms in the same bandwidth, approaching the thermomechanical noise limit and enabling precise quantification of particles down to 10 nm with a throughput of more than 18,000 particles per hour. We demonstrate the potential of this capability by comparing the mass distributions of exosomes produced by different cell types and by characterizing the yield of self-assembled DNA nanoparticle structures.
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| [57] |
High-speed multiple-mode mass-sensing resolves dynamic nanoscale mass distributions .
Simultaneously measuring multiple eigenmode frequencies of nanomechanical resonators can determine the position and mass of surface-adsorbed proteins, and could ultimately reveal the mass tomography of nanoscale analytes. However, existing measurement techniques are slow (<165Hz bandwidth), limiting throughput and preventing use with resonators generating fast transient signals. Here we develop a general platform for independently and simultaneously oscillating multiple modes of mechanical resonators, enabling frequency measurements that can precisely track fast transient signals within a user-defined bandwidth that exceeds 50065Hz. We use this enhanced bandwidth to resolve signals from multiple nanoparticles flowing simultaneously through a suspended nanochannel resonator and show that four resonant modes are sufficient for determining their individual position and mass with an accuracy near 15065nm and 40 attograms throughout their 150-ms transit. We envision that our method can be readily extended to other systems to increase bandwidth, number of modes, or number of resonators. Nanomechanical resonators are sensitive to tiny changes in their mass. Here, the authors demonstrate a method for quickly measuring many resonator modes and use it to analyse the mass and position of multiple nanoparticles flowing in a fluid channel with a precision of 40 attograms and 15065nm, respectively.
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| [58] |
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| [59] |
Origin of the response of nanomechanical resonators to bacteria adsorption .
Resonant microcantilevers are being actively investigated as sensitive mass sensors for biological detection. By performing experiments of adsorption of the bacteriaEscherichia colion singly clamped microcantilevers, we demonstrate that the effect of the added mass is not the only and may not be the main origin of the response of these sensors. The experiments show that the magnitude and sign of resonance frequency shift both depend critically on the distribution of the adsorbed bacterial cells on the cantilever. We relate this behavior to the added mass that shifts the resonance to lower frequencies and the higher effective flexural rigidity of the cantilever due to the bacteria stiffness that shifts the resonance to higher frequencies. Both effects can be uncoupled by positioning the cells where each effect dominates, near the free cantilever end for measuring the added mass or near the clamping for measuring the increase of flexural rigidity.
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| [60] |
The nonlinear response of resonant microbeam systems with purely-parametric electrostatic actuation .
Electrostatically-actuated resonant microbeam devices have garnered significant attention due to their geometric simplicity and broad applicability. Recently, some of this focus has turned to comb-driven microresonators with purely-parametric excitation, as such systems not only exhibit the inherent benefits of MEMS devices, but also a general improvement in sensitivity, stopband attenuation and noise rejection. This work attempts to combine each of these areas by proposing a microbeam device which couples the inherent benefits of a resonator with purely-parametric excitation with the simple geometry of a microbeam. Theoretical analysis reveals that the proposed device exhibits desirable response characteristics, but also quite complex dynamics. Of particular note is the fact that the device's nonlinear frequency response is found to be qualitatively dependent on the system's ac excitation amplitude. While this flexibility can be desirable in certain contexts, it introduces additional design and operating limitations. While the principal focus of this work is the proposed system's nonlinear response, the work also contains details pertaining to model development and device design.
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| [61] |
Dynamics of microscale pipes containing internal fluid flow: Damping, frequency shift, and stability .
This paper initiates the theoretical analysis of microscale resonators containing internal flow, modelled here as microfabricated pipes conveying fluid, and investigates the effects of flow velocity on damping, stability, and frequency shift. The analysis is conducted within the context of classical continuum mechanics, and the effects of structural dissipation (including thermoelastic damping in hollow beams), boundary conditions, geometry, and flow velocity on vibrations are discussed. A scaling analysis suggests that slender elastomeric micropipes are susceptible to instability by divergence (buckling) and flutter at relatively low flow velocities of 10 m/s.
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| [62] |
Frequency response of cantilever beams immersed in viscous fluids with applications to the atomic force microscope .
The vibrational characteristics of a cantilever beam are well known to strongly depend on the fluid in which the beam is immersed. In this paper, we present a detailed theoretical analysis of the frequency response of a cantilever beam, that is immersed in a viscous fluid and excited by an arbitrary driving force. Due to its practical importance in application to the atomic force microscope (AFM), we consider in detail the special case of a cantilever beam that is excited by a thermal driving force. This will incorporate the presentation of explicit analytical formulae and numerical results, which will be of value to the users and designers of AFM cantilever beams.
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| [63] |
Energy dissipation in microfluidic beam resonators: Effect of Poisson's ratio .
Dissipation of mechanical energy underlies the sensitivity of many nanomechanical devices, with environmental effects often having a significant effect. One case of practical relevance is the interaction of elastic beam resonators with fluid, which is known to dramatically increase energy dissipation. Recently, we investigated energy dissipation in a different class of elastic beam resonator that embeds a microfluidic channel in its interior. In this paper, we examine the effect of the beam material Poisson ratio on these devices and discover that it can strongly affect energy dissipation--this is in direct contrast to conventional cantilever beams immersed in fluid. Increasing the Poisson ratio in these microfluidic devices is found to decrease energy dissipation, with the incompressible material limit providing minimum energy dissipation. Our paper establishes that, in this limit, placement of the fluid channel away from the beam neutral axis has negligible effect on energy dissipation in many cases of practical interest. The physical implications of these findings are discussed, and a detailed comparison with available experimental results is provided.
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| [64] |
Energy dissipation in microfluidic beam resonators .
Energy dissipation experienced by vibrating microcantilever beams immersed in fluid is strongly dependent on the mode of vibration, with quality factors typically increasing with mode number. Recently, we examined energy dissipation in a new class of cantilever device that embeds a microfluidic channel in its interior he fundamental mode of vibration only was considered. Due to its importance in practice, we examine the effect of mode number on energy dissipation in these microfluidic beam resonators. Interestingly, and in contrast to other cantilever devices, we find that the quality factor typically decreases with increasing mode number. We explore the underlying physical mechanisms leading to this counterintuitive behavior, and provide a detailed comparison to experimental measurements for which good agreement is found.
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| [65] |
Energy dissipation in microfluidic beam resonators: Dependence on mode number . |
| [66] |
Frequency fluctuations in silicon nanoresonators .
Frequency stability is key to performance of nanoresonators. This stability is thought to reach a limit with the resonator ability to resolve thermally-induced vibrations. Although measurements and predictions of resonator stability usually disregard fluctuations in the mechanical frequency response, these fluctuations have recently attracted considerable theoretical interest. However, their existence is very difficult to demonstrate experimentally. Here, through a literature review, we show that all studies of frequency stability report values several orders of magnitude larger than the limit imposed by thermomechanical noise. We studied a monocrystalline silicon nanoresonator at room temperature, and found a similar discrepancy. We propose a new method to show this was due to the presence of frequency fluctuations, of unexpected level. The fluctuations were not due to the instrumentation system, or to any other of the known sources investigated. These results challenge our current understanding of frequency fluctuations and call for a change in practices.
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| [67] |
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| [68] |
Nonlinear dynamic analysis of FG micro-pipes conveying fluid based on strain gradient theory .
In this article, an analytical solution is presented for the size dependent nonlinear vibration behavior of micro-pipes conveying fluid made of functionally graded materials (FGMs). On the basis of the Euler-Bernoulli beam model, the strain gradient theory and von Karman geometric nonlinearity, the mathematical formulations are developed in terms of three length scale parameters. The material properties of the functionally graded (FG) micro-pipes vary continuously across the thickness according to the power law distribution. The Hamilton's principle is employed to obtain the differential equation of motion and the corresponding boundary conditions. Without loss of generality, simply supported pipes are considered. The governing equation is written in the form of duffing equation by using Galerkin method. Subsequently, a powerful analytical technique called the homotopy analysis method (HAM) is employed to determine the explicit expressions for nonlinear fundamental frequency for different fluid velocities and power law gradient indices. Comprehensive comparison studies between linear and nonlinear theories using the strain gradient, the couple stress and classical theories are conducted. The results show that the length scale parameter and the FG power law index have significant effect on the fundamental frequency of the FG micro-pipes and the fluid critical velocity. (C) 2014 Elsevier Ltd. All rights reserved.
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| [69] |
Suspended microchannel resonators for ultralow volume universal detection .
Universal detectors that maintain high sensitivity as the detection volume is reduced to the subnanoliter scale can enhance the utility of miniaturized total analysis systems (mu-TAS). Here the unique scaling properties of the suspended microchannel resonator (SMR) are exploited to show universal detection in a 10 pL analysis volume with a density detection limit of approximately 1 microg/cm (3) (10 Hz bandwidth) and a dynamic range of 6 decades. Analytes with low UV extinction coefficients such as polyethylene glycol (PEG) 8 kDa, glucose, and glycine are measured with molar detection limits of 0.66, 13.5, and 31.6 microM, respectively. To demonstrate the potential for real-time monitoring, gel filtration chromatography was used to separate different molecular weights of PEG as the SMR acquired a chromatogram by measuring the eluate density. This work suggests that the SMR could offer a simple and sensitive universal detector for various separation systems from liquid chromatography to capillary electrophoresis. Moreover, since the SMR is itself a microfluidic channel, it can be directly integrated into mu-TAS without compromising overall performance.
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| [70] |
Effect of the adsorbate stiffness on the resonance response of microcantilever sensors .
The authors present a theoretical model to predict the resonance frequency shift due to molecule adsorption on micro- and nanocantilevers. They calculate the frequency shift experienced by cantilevers made of either silicon or the polymer SU-8, when two adsorbates, myosin protein and an alkanethiol, are attached to the cantilever surface. They demonstrate that the effect of the adsorbate stiffness can be comparable or even larger than the mass effect, producing positive frequency shifts. The results provide methods for decoupling both opposite effects and routes for the design of resonators with high sensitivity to molecule adsorption based on either stiffness or mass effects.
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| [71] |
Dynamics and vibration analysis of suspended microchannel resonators based on strain gradient theory .
In this paper, dynamics of a suspended microchannel resonator (SMR) will be derived using the strain gradient theory. Accordingly, the size dependent governing equation and corresponding boundary...
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| [72] |
Noise in microelectromechanical system resonators .
Microelectromechanical system (MEMS) and nanoelectromechanical system (NEMS) based resonators and filters, ranging in frequencies from kHz to GHz, have been proposed. The question of how the stabilities of such resonators scale with dimensions is examined in this paper, with emphasis on the noise characteristics. When the dimensions of a resonator become small, instabilities that are negligible in macro-scale devices become prominent. The effects of fluctuations in temperature, adsorbing/desorbing molecules, outgassing, Brownian motion, Johnson noise, drive power and self-heating, and random vibration are explored. When the device is small, the effects of fluctuations in the numbers of photons, phonons, electrons and adsorbed molecules can all affect the noise characteristics. For all but the random vibration-induced noise, reducing the dimensions increases the noise. At submicron dimensions, especially, the frequency noise due to temperature fluctuations, Johnson noise, and adsorption/desorption are likely to limit the applications of ultra-small resonators.
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| [73] |
A Nanoscale Parametric Feedback Oscillator .
We describe and demonstrate a new oscillator topology, the parametric feedback oscillator (PFO). The PFO paradigm is applicable to a wide variety of nanoscale devices and opens the possibility of new classes of oscillators employing innovative frequency-determining elements, such as nanoelectromechanical systems (NEMS), facilitating integration with circuitry and system-size reduction. We show that the PFO topology can also improve nanoscale oscillator performance by circumventing detrimental effects that are otherwise imposed by the strong device nonlinearity in this size regime.
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| [74] |
Size-dependent vibration characteristics of fluid-conveying microtubes .
In this paper, a new theoretical model is developed, based on the modified couple stress theory, for the vibration analysis of fluid-conveying microtubes by introducing one internal material length scale parameter. Using Hamilton's principle, the equations of motion of fluid-conveying microtubes are derived. After discretization via the Differential Quadrature Method (DQM), the analytical model exhibits some essential vibration characteristics. For a microtube in which both ends are supported, it is found that the natural frequencies decrease with increasing internal flow velocities. It is also shown that the microtube will become unstable by divergence at a critical flow velocity. More significantly, when the outside diameter of the microtube is comparable to the material length scale parameter, the natural frequencies obtained using the modified couple stress theory are much larger than those obtained using the classical beam theory. It is not surprising, therefore, that the critical flow velocities predicted by the modified couple stress theory are generally higher than those predicted by the classical beam theory.
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| [75] |
Flexural vibrations of microscale pipes conveying fluid by considering the size effects of micro-flow and micro-structure .
In this paper, the in-plane and out-of-plane flexural vibrations of microscale pipes conveying fluid with clamped-clamped ends are examined theoretically, by modifying the classical equations of motion with consideration of the size effects of micro-flow and micro-structure. The theoretical model including an additional parameter associated with the flow velocity profile and another material length scale parameter representing the size effect of micro-structure is valid for both straight and curved pipe systems. The results of calculating the evolution of natural frequencies for straight pipes with increasing mean flow velocity show that the material length scale parameter tends to stiffer the microscale pipe and hence increases the critical flow velocity while the parameter describing the effect of flow velocity profile would decrease the critical flow velocity. For curved pipes, however, the modified inextensible theory predicts that the effect of micro-flow on the in-plane and out-of-plane natural frequencies may be not visible. [All rights reserved Elsevier].
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| [76] |
Label-free measurement of amyloid elongation by suspended microchannel resonators .
Protein aggregation is a widely studied phenomenon that is associated with many human diseases and with the degradation of biotechnological products. Here, we establish a new label-free method for characterizing the aggregation kinetics of proteins into amyloid fibrils by suspended microchannel resonators (SMR). SMR devices are unique in their ability to provide mass-based measurements under reaction-limited conditions in a 10 pL volume. To demonstrate the method, insulin seed fibrils of defined length, characterized by atomic force microscopy (AFM) and transmission electron microscopy (TEM), were covalently immobilized inside microchannels embedded within a micromechanical resonator, and the elongation of these fibrils under a continuous flow of monomer solution (rate 651 nL/s) was measured by monitoring the resonance frequency shift. The kinetics for concentrations below 650.602mg/mL fits well with an irreversible bimolecular binding model with the rate constant kon = (1.2 ± 0.1) × 10(3) M(-1) s(-1). Rate saturation occurred at higher concentrations. The nonlinear on-rate for monomer concentrations from 0 to 6 mg/mL and for temperatures from 20 to 42 °C fit well globally with an energy landscape model characterized by a single activation barrier. Finally, elongation rates were studied under different solution conditions and in the presence of a small molecule inhibitor of amyloid growth. Due to the low volume requirements, high precision, and speed of SMR measurements, the method may become a valuable new tool in the screening for inhibitors and the study of fundamental biophysical mechanisms of protein aggregation processes.
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| [77] |
Nonlinear impacting oscillations of pipe conveying pulsating fluid subjected to distributed motion constraints . |
| [78] |
Non-planar responses of cantilevered pipes conveying fluid with intermediate motion constraints .
In this paper, the nonlinear responses of a loosely constrained cantilevered pipe conveying fluid in the context of three-dimensional (3-D) dynamics are investigated. The pipe is allowed to oscillate...
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| [79] |
Three-dimensional nonlinear dynamics of a cantilevered pipe conveying fluid subjected to loose constraints .
The dynamics of pipes conveying fluid has become a hot topic in the research field of fluid-structure interactions.Perhaps one of the main reasons why the dynamics of pipes conveying fluid has remained of intense interest to dynamicists is the fact that it displays interesting and sometimes unexpected nonlinear dynamical behavior and it has become a handy tool in developing or testing modern dynamics theory.In physical terms,the dynamical system of a loosely supported pipe conveying fluid is an example of large class of problems involving self-excited oscillations and interactions with loose constraints.Hence,understanding and modeling the dynamics of such systems is of both fundamental and practical interests.The two-dimensional(2-D) vibration of a cantilevered pipe conveying fluid has been studied for a long time,but for the moment,only few studies discussed the three-dimensional(3-D) planar and non-planar dynamics of cantilevered pipes subjected to nonlinear loose constraints.This paper establishes the 3-D governing equations and explores the 3-D dynamics of a cantilevered pipe with loose constraints somewhere along its length,with consideration of the friction effect during impacting between the pipe and the loose constraints.The loose constraints consist of two parallel bars(TPBs) on both sides of the pipe in one fixed lateral direction,with free gaps between the pipe and the restraining bars.The impacting force between the pipe and loose constraints is depicted by a smoothened-trilinear spring.In the theoretical analysis,the two governing equations were discretized via Galerkin's approach and solved using a fourth-order Runge-Kutta method.As the flow velocity becomes sufficiently high,flutter instability of the pipe occurs and limit cycle motions would be generated.When the lateral displacement of the pipe exceeds the free gap,effective impacting occurs.During impacting between the pipe and loose constraints,either static or dynamic friction forces would occur along the other lateral direction.The nonlinear behavior in two perpendicular planes and the influence of friction coefficient were analyzed with special attention.The dynamic responses of the pipe system for various internal flow velocities are exhibited in the form of bifurcation diagrams,time traces and phase plots.Comparisons of the planar and non-planar motions of the pipe with or without loose constraints are conducted for various internal flow velocity ranges.Results show that the constrained pipe is capable of displaying interesting dynamics in the presence of nonlinear impacting force induced by the loose constraints.Both 3-D periodic and quasi-periodic oscillations are observed in a wide range of internal flow velocities.It is found that the introducing of nonlinear impacting constraints slightly enlarges the internal flow velocity range for non-planar motions.It is also shown that the orientation of 2-D planar vibrations of the pipe may be changed with the increase of the friction coefficient.The results obtained in this work may be useful for further understanding other problems in fluid-structure interactions involving slender structures and axial flows,such as the dynamics of slender cylinders in axial flow and deep water risers concurrently subjected to axial and cross flows.
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| [80] |
Mass sensors with mechanical traps for weighing single cells in different fluids .
We present two methods by which single cells can be mechanically trapped and continuously monitored within the suspended microchannel resonator (SMR) mass sensor. Since the fluid surrounding the trapped cell can be quickly and completely replaced on demand, our methods are well suited for measuring changes in cell size and growth in response to drugs or other chemical stimuli. We validate our methods by measuring the density of single polystyrene beads andSaccharomyces cerevisiaeyeast cells with a precision of approximately 10613g cm613, and by monitoring the growth of single mouse lymphoblast cells before and after drug treatment.
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| [81] |
Mass sensors with mechanical traps for weighing single cells in different fluids .
We present two methods by which single cells can be mechanically trapped and continuously monitored within the suspended microchannel resonator (SMR) mass sensor. Since the fluid surrounding the trapped cell can be quickly and completely replaced on demand, our methods are well suited for measuring changes in cell size and growth in response to drugs or other chemical stimuli. We validate our methods by measuring the density of single polystyrene beads andSaccharomyces cerevisiaeyeast cells with a precision of approximately 10613g cm613, and by monitoring the growth of single mouse lymphoblast cells before and after drug treatment.
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| [82] |
Measuring single-cell density . |
| [83] |
Pull-in effect of suspended microchannel resonator sensor subjected to electrostatic. Actuation .
In this article, the pull-in instability and dynamic characteristics of electrostatically actuated suspended microchannel resonators are studied. A theoretical model is presented to describe the pull-in effect of suspended microchannel resonators by considering the electrostatic field and the internal fluid. The results indicate that the system is subjected to both the pull-in instability and the flutter. The former is induced by the applied voltage which exceeds the pull-in value while the latter occurs as the velocity of steady flow get closer to the critical velocity. The statically and dynamically stable regions are presented by thoroughly studying the two forms of instability. It is demonstrated that the steady flow can remarkably extend the dynamic stable range of pull-in while the applied voltage slightly decreases the critical velocity. It is also shown that the dc voltage and the steady flow can adjust the resonant frequency while the ac voltage can modulate the vibrational amplitude of the resonator.
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| [84] |
A measurement criterion for accurate mass detection using vibrating suspended microchannel resonators .
In this paper, a measurement criterion is presented to ensure the accuracy of mass detection using suspended microchannel resonators. The dynamic characteristics of the microchannel resonator with an added suspended particle are investigated to analyze and clarify the measurement principle for mass detection. It indicates that the vibration properties of the suspended particle have a significant effect on the measurement accuracy and the generation of systematic error. The fluid-structure interaction vibration of the particle driven by the resonator is solved by the lattice Boltzmann method integrated with the immersed boundary method. The effects of several important factors, including the particle density, fluid viscosity, and oscillating frequency, are studied and discussed. The results demonstrate that the Reynolds number and the density ratio are the two crucial parameters which affect the measurement accuracy and error generation. As the Reynolds number or the density ratio increases, the measurement precision degrades. Moreover, once the Reynolds number decreases to a critical value, the systematic error of mass detection reduces to zero. The measurement criterion can be taken as the guide to accurately detect the mass of suspended particle in the resonator.
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| [85] |
Dynamical characteristics of fluid-conveying microbeams actuated by electrostatic force .
In this paper, the dynamic characteristics and pull-in instability of electrostatically actuated microbeams which convey internal fluids are investigated. A theoretical model is developed by consideri
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| [86] |
Torsional resonance mode atomic force microscopy in liquid with Lorentz force actuation .
In this work, we present a design based on Lorentz force induction to excite pure torsional resonances of different types of cantilevers in air as well as in water. To demonstrate the atomic force microscopy imaging capability, the phase-modulation torsional resonance mode is employed to resolve fine features of purple membranes in a buffer solution. Most importantly, force-versus-distance curves using a relatively stiff cantilever can clearly detect the characteristic oscillatory profiles of hydration layers at a water-mica interface, indicating the high force sensitivity of the torsional mode. The high resonance frequencies and high quality-factors for the torsional mode may be of great potential for high-speed and high-sensitivity imaging in aqueous environment.
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| [87] |
Detecting the stiffness and mass of biochemical adsorbates by a resonator sensor .
The biochemical adsorption on a resonator sensor can result in the changes of both stiffness and mass. If the effect of stiffness is not considered, the resonator response will be wrongly interpreted. Determining the adsorbate stiffness and mass by the shifts of resonant frequency formulates an inverse problem. The inverse problem is solved by varying the adsorbate thickness and measuring the corresponding shifts of resonant frequencies. With the technique of solving the inverse problem, a micro/nanomechanical resonator can be used to identify what kind of material an adsorbate is, which is more than a mass resonator sensor.
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| [88] |
Effect of surface energy on the dynamic response and instability of fluid-conveying nanobeams. European Journal of Mechanics-A/
61A modified Timoshenko nanoscale beam conveying fluid is presented.61Surface effects on the natural frequencies and the critical flow velocities are discussed.61The influences of shear deformation and rotary inertia on the natural frequencies and the critical flow velocities are discussed.61Surface energies affect the results through the presence of some intrinsic length scales.
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| [89] |
Electrostatic pull-in instability in MEMS/NEMS: A review .
Pull-in instability as an inherently nonlinear and crucial effect continues to become increasingly important for the design of electrostatic MEMS and NEMS devices and ever more interesting scientifically. This review reports not only the overview of the pull-in phenomenon in electrostatically actuated MEMS and NEMS devices, but also the physical principles that have enabled fundamental insights into the pull-in instability as well as pull-in induced failures. Pull-in governing equations and conditions to characterize and predict the static, dynamic and resonant pull-in behaviors are summarized. Specifically, we have described and discussed on various state-of-the-art approaches for extending the travel range, controlling the pull-in instability and further enhancing the performance of MEMS and NEMS devices with electrostatic actuation and sensing. A number of recent activities and achievements methods for control of torsional electrostatic micromirrors are introduced. The on-going development in pull-in applications that are being used to develop a fundamental understanding of pull-in instability from negative to positive influences is included and highlighted. Future research trends and challenges are further outlined. (C) 2014 Elsevier B.V. All rights reserved.
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| [90] |
Dynamics of suspended microchannel resonators conveying opposite internal fluid flow: Stability, frequency shift and energy dissipation .
In this paper, the dynamics of suspended microchannel resonators which convey internal flows with opposite directions are investigated. The fluid tructure interactions between the laminar fluid flow and oscillating cantilever are analyzed by comprehensively considering the effects of velocity profile, flow viscosity and added flowing particle. A new model is developed to characterize the dynamic behavior of suspended microchannel resonators with the fluid tructure interactions. The stability, frequency shift and energy dissipation of suspended microchannel resonators are analyzed and discussed. The results demonstrate that the frequency shifts induced by the added flowing particle which are obtained from the new model have a good agreement with the experimental data. The steady mean flow can cause the frequency shift and influence the stability of the dynamic system. As the flow velocity reaches the critical value, the coupled-mode flutter occurs via a Hamiltonian Hopf bifurcation. The perturbation flow resulted from the vibration of the microcantilever leads to energy dissipation, while the steady flow does not directly cause the damping which increases with the increasing of the flow velocity predicted by the classical model. It can also be found that the steady flow firstly changes the mode shape of the cantilever and consequently affects the energy dissipation.
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| [91] |
Determining the adsorption-induced surface stress and mass by measuring the shifts of resonant frequencies. Sensors and Actuators a
The adsorption-induced surface stress and mass can cause the resonant frequency shifts of a microcantilever, which is used as the sensing mechanism for a resonator sensor. Determining the adsorption-induced surface stress and mass from the experimentally measured data of resonant frequencies forms an inverse problem. Because there are infinite combinations of surface stress and mass which can result in the same change of one resonant frequency, the previous studies usually measure surface stress or mass by one measurement method and then find the other by another different measurement method. This study shows that surface stress and mass have different impacts on the resonant frequencies of a microcantilever. Two resonant frequencies are used to uniquely determine the adsorption-induced surface stress and mass. Mathematically, the new method presented in this study provides an efficient and straightforward solution to the inverse problem and its accuracy is also demonstrated. Physically, the new method only requires the dynamic mode to measure the resonant frequencies, which should be of a great help to various sensor applications.
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| [92] |
Detecting the stiffness and mass of biochemical adsorbates by a resonator sensor .
The biochemical adsorption on a resonator sensor can result in the changes of both stiffness and mass. If the effect of stiffness is not considered, the resonator response will be wrongly interpreted. Determining the adsorbate stiffness and mass by the shifts of resonant frequency formulates an inverse problem. The inverse problem is solved by varying the adsorbate thickness and measuring the corresponding shifts of resonant frequencies. With the technique of solving the inverse problem, a micro/nanomechanical resonator can be used to identify what kind of material an adsorbate is, which is more than a mass resonator sensor.
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| [93] |
Modelling analysis of surface stress on a rectangular cantilever beam .
Three models of surface stress on a rectangular cantilever beam are presented. The surface stress is modelled as a corresponding concentrated moment at the beam free end, a corresponding concentrated moment plus a corresponding concentrated axial load at the beam free end, and a corresponding uniformly distributed axial stress plus bending moment per unit length along the beam span, respectively. The results of the three models are compared under three different loading scenarios. We also present an analysis of the error source, when using Stoney formula to predict the surface stress, by comparing the kinematic and loading assumptions of the three models. The surface stress effects on structure deflection are usually modelled as bending moments applied at structure free edge(s)/end(s). Modelling the surface stress effect along the beam neutral axis is presented and compared with modelling its effect at free edge(s)/end(s). The stiffening effect of tensile surface stress is also studied.
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| [94] |
Mass and force sensing of an adsorbate on a beam resonator sensor .
The mass sensing superiority of a micro-/nano-mechanical resonator sensor over conventional mass spectrometry has been, or at least is being firmly established. Because the sensing mechanism of a mechanical resonator sensor is the shifts of resonant frequencies, how to link the shifts of resonant frequencies with the material properties of an analyte formulates an inverse problem. Besides the analyte/adsorbate mass, many other factors, such as position and axial force, can also cause the shifts of resonant frequencies. The in situ measurement of the adsorbate position and axial force is extremely difficult if not impossible, especially when an adsorbate is as small as a molecule or an atom. Extra instruments are also required. In this study, an inverse problem of using three resonant frequencies to determine the mass, position and axial force is formulated and solved. The accuracy of the inverse problem solving method is demonstrated, and how the method can be used in the real application of a nanomechanical resonator is also discussed. Solving the inverse problem is helpful to the development and application of a mechanical resonator sensor for two reasons: reducing extra experimental equipment and achieving better mass sensing by considering more factors.
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| [95] |
Determining the effects of surface elasticity and surface stress by measuring the shifts of resonant frequencies .
Both surface elasticity and surface stress can result in changes of resonant frequencies of a micro/nanostructure. There are infinite combinations of surface elasticity and surface stress that can cause the same variation for one resonant frequency. However, as shown in this study, there is only one combination resulting in the same variations for two resonant frequencies, which thus provides an efficient and practical method of determining the effects of both surface elasticity and surface stress other than an atomistic simulation. The errors caused by the different models of surface stress and mode shape change due to axial loading are also discussed.
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| [96] |
Dynamics of axially functionally graded cantilevered pipes conveying fluid .
The linear dynamics of axially functionally graded (AFG) cantilevered pipes conveying fluid is studied, aiming at enhancing the dynamic stability of such fluid-interaction systems. Either the elastic modulus or the density of the AFG cantilevered pipe is assumed to be varied from the clamped to the free ends. The governing equation of the AFG pipe is derived first and then discretized by using the differential quadrature method (DQM). The effects of elastic modulus gradient and density gradient on the critical flow velocity for flutter instability are analyzed. It is found that, compared with uniform pipes, the decrease of density along the pipe length leads to a more stable system, while the opposite result may be obtained for the decrease of elastic modulus for small values of mass ratio. From the boundary curves of critical flow velocity versus density gradient and of critical flow velocity versus elastic modulus gradient, it is shown that the occurrence of Z-shape segments is possible when the mass ratio becomes large. Furthermore, the phenomenon of mode exchange has also been detected with increasing density gradient or elastic modulus gradient within a certain range of mass ratio.
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