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TECHNICAL PAPERS: Other

Thermo-Mechanical Responses of a Surface-Coupled AFM Cantilever

[+] Author and Article Information
Jianhua Wu, Ying Fang1

George W. Woodruff School of Mechanical Engineering,  Georgia Institute of Technology, Atlanta, GA 30332

Dong Yang

George W. Woodruff School of Mechanical Engineering,  Georgia Institute of Technology, Atlanta, GA 30332

Cheng Zhu2

George W. Woodruff School of Mechanical Engineering and Wallace H. Coulter Department of Biomedical Engineering,  Georgia Institute of Technology, Atlanta, GA 30332cheng.zhu@me.gatech.edu

1

Present address: School of Life Sciences, Sun Yat-Sen University, Guangzhou 510275, China.

2

Corresponding author.

J Biomech Eng 127(7), 1208-1215 (Aug 15, 2005) (8 pages) doi:10.1115/1.2073647 History: Received February 08, 2005; Revised August 05, 2005; Accepted August 15, 2005

Atomic force microscopy (AFM) has been widely used for measuring mechanical properties of biological specimens such as cells, DNA, and proteins. This is usually done by monitoring deformations in response to controlled applied forces, which have to be at ultralow levels due to the extreme softness of the specimens. Consequently, such experiments may be susceptible to thermal excitations, manifested as force and displacement fluctuations that could reduce the measurement accuracy. To take advantage of, rather than to be limited by, such fluctuations, we have characterized the thermomechanical responses of an arbitrarily shaped AFM cantilever with the tip coupled to an elastic spring. Our analysis shows that the cantilever and the specimen behave as springs in parallel. This provides a method for determining the elasticity of the specimen by measuring the change in the tip fluctuations in the presence and absence of coupling. For rectangular and V-shaped cantilevers, we have derived a relationship between the mean-square deflection and the mean-square inclination and an approximate expression for the specimen spring constant in terms of contributions to the mean-square inclination from the first few vibration modes.

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Copyright © 2005 by American Society of Mechanical Engineers
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Figures

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Figure 1

Schematics of a surface-coupled V-shaped cantilever loaded by thermal excitations ρf, which are highly random in both magnitude and direction. The coupling of the cantilever tip and the surface is modeled by a string (spring constant k). The values for length L, width W, and thickness h for commercial cantilevers are listed in Table 1. A Side view. B Top view.

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Figure 2

(Color) Plots of (A) static deflection, z, versus cantilever length times static inclination, L(dz∕dx), and (B) ratio of mean-square virtual deflection to mean-square real deflection at the cantilever tip, ⟨z*2⟩∕⟨z2⟩, minus a constant, b, versus the ratio of specimen spring constant to the cantilever spring constant, k∕kc, for the indicated Veeco cantilevers. The black lines represent the theoretical results for rectangular cantilever as indicated. The numerical results were fit by the same equations in each panel. The best-fit proportionality constants α in A and the best-fit slopes a and y-axis intercepts b in B are summarized in Table 1 for the different Veeco cantilevers.

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Figure 3

(Color) Comparison of the relative contributions to the mean-square real deflections from the the first (A), second (B), third (C), and fourth (D) vibration modes for a rectangular cantilever solved analytically and the indicated Veeco cantilevers calculated numerically

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Figure 4

(Color) Comparison of the relative contributions to the mean-square virtual deflections from the first (A), second (B), third (C), and fourth (D) vibration modes for a rectangular cantilever solved analytically and the indicated Veeco cantilevers calculated numerically

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Figure 5

(Color) Relative errors of the specimen spring constant resulted from bandwidth limitation. k is the exact specimen spring constant determined from a complete signal from all vibration modes. kN is the approximate specimen spring constants when only the first (A), the first two (B), and first three (C) vibration modes contribute to the mean-square virtual deflection. It is calculated using either Eq. 28 (analytical without correction) or Eq. 31 (with correction) analytically for the rectangular cantilever and numerically for the indicated V-shaped Veeco cantilevers.

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