Technical Briefs

Measuring Viscoelasticity of Soft Samples Using Atomic Force Microscopy

[+] Author and Article Information
S. Tripathy1

Department of Mechanical and Aerospace Engineering, University of Virginia, Charlottesville, VA 22904sakya@virginia.edu

E. J. Berger

Department of Mechanical and Aerospace Engineering, University of Virginia, Charlottesville, VA 22904berger@virginia.edu


Corresponding author.

J Biomech Eng 131(9), 094507 (Aug 28, 2009) (6 pages) doi:10.1115/1.3194752 History: Received July 11, 2008; Revised June 02, 2009; Published August 28, 2009

Relaxation indentation experiments using atomic force microscopy (AFM) are used to obtain viscoelastic material properties of soft samples. The quasilinear viscoelastic (QLV) model formulated by Fung (1972, “Stress Strain History Relations of Soft Tissues in Simple Elongation,” in Biomechanics, Its Foundation and Objectives, Prentice-Hall, Englewood Cliffs, NJ, pp. 181–207) for uniaxial compression data was modified for the indentation test data in this study. Hertz contact mechanics was used for the instantaneous deformation, and a reduced relaxation function based on continuous spectrum is used for the time-dependent part in the model. The modified QLV indentation model presents a novel method to obtain viscoelastic properties from indentation data independent of relaxation times of the test. The major objective of the present study is to develop the QLV indentation model and implement the model on AFM indentation data for 1% agarose gel and a viscoelastic polymer using spherical indenter.

Copyright © 2009 by American Society of Mechanical Engineers
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Grahic Jump Location
Figure 1

Hertz contact theory for spherical indenter (34). For incompressible materials, the Poisson ratio is ν=0.5. This reduces the relation to Eq. 3. To preclude any substrate effects the sample thickness, h is large with respect to the indenter size and the maximum penetration depth. Furthermore, since the modulus of the indenter is considerably larger than the modulus of the material, it is considered to be rigid with respect to the sample material.

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Figure 2

The indentation depth as a function of time for the AFM experiment. Due to the relaxation of the material, we observe an increase in the indentation depth. This study presents two approximations for the indentation depth: an average of the entire hold region (LA) and an exponential fit (p1ep2t+p3ep4t) (EA). (a) presents the data for 1% agarose while (b) presents the data for the polymer. Although the experiments were performed for longer times for agarose to reduce the number of data points and clarity, only until t=3 s is shown.

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Figure 3

The force response of the AFM indentations along with curve fits. The indentation depths of each of these plots are given in Fig. 2. Fits using both the approximations, linear average and exponential approximation, are plotted for comparison for both (a) 1% agarose and (b) polymer. For agarose, we only used the data until t=3 s to reduce the data points since our model is not dependent on the hold time. Moreover, 99% of the relaxation occurs within 3 s.

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Figure 4

The relaxation ratio at t=10 s is compared for difference ramp speeds. As the approach velocity is not equal to the ramp speed because of the material relaxation, there was an error associated with it, too. The x-axis is not in the scale. The data are presented as mean±SD.



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