Nemrg

Error mode images taken 5 minutes apart of an indented cell undergoing a slow recovery

A clear understanding of cell structure, requires accurate measurements of its structural properties. Because of the complex nature of the cell's interior with its polymer networks and cytoplasm within a fluid membrane, it can be characterized as a viscoelastic medium. Thus, the cells structural properties are governed by both a time independent elastic response and a time-dependent viscous response. An AFM measurement of the viscoelastic properties requires alterations in both the standard AFM set-up, and the traditional data analysis.

Experimental

One key component to a quantitative assessment of the viscoelastic properties of a material through indentation, is the indenter itself. This must be well-defined or characterized such that the surface area is known throughout the indentation. One way to accomplish this is with the addition of a spherical bead to the cantilever to provide a symmetric spherical indenter. This is shown above schematically and an inverse AFM image of such a probe is also shown( white bar = 1 mm). The radius of the probe can vary and will define the stress range obtainable with this probe. In our experiments it generally varies from 3-10 microns.

In addition to a well-defined surface area, a the spherical probes also provide a larger contact surface area with a correspondingly lower stress for the same total force. These lower stresses in turn give lower indentations. For thinner samples this is a great advantage because for larger indentations the sample substrate plays a role in the measured force responses and simple models for a supposed infinite sample could no longer be used.

Hertz Model

The Hertz model can be used for approximately infinite samples when indented with a spherical or paraboloid probe. It describes the relation between the Young's modulus(E) and the force and indentation with factors that reflect the tips geometry. Experimentally, the force and indentation are detyermined by obtaining a force curve at a point on the sample, which is simply a measurement of cantilever response as the tip is pushed into the sample.

Using this equation and the force vs. distance data obtain with the AFM, we can calculate the Young's modulus. This is the zero-frequency result. There is no time dependence in this equation and there is no viscous contribution assumed.

The Extended Hertz Model

To obtain a viscous response, a higher frequency signal (20-400 Hz) is added to the existing force curve. When this signal is significantly faster than the rate of the force curve, then the rate of the force curve can be ignored. The additional signal is small in amplitude (2-20 nm) and sinusoidal in nature.

The response of the cantilever is also sinusoidal with properties that reflect the material under the probe. The signal is phase shifted due to the property's viscous response and the amplitude decreases due to the elastic properties. The cantilever component thus has a real (in-phase) component and an imaginary out of phase component.

For data analysis, the Hertz model is extended to first order in the indentation. There now appears a second term that conains the oscillating signal and is proportional to a complex viscoelastic constant E*. A lock-in amplifier is used to isolate the second term.

The measured force is a combination of the force of the probe on the sample and the force of the cantilever dragging through the liquid. This second term must be subtracted to obtain solely the force of the probe on the sample.

Having successfully subtracted the drag force, the oscillating portion of the force and the indentation, both total and oscillating, can be used to determine viscoelastic constants for cells and polymer gels.

Data and Results

This data were all taken at 50Hz. The polyacrylamide gel shows a strong elastic component and a weak viscous component while the cells show strong viscous components. Also the value of E* remains constant only for low indentation depths in the fibroblast cells. This is probably due to the substrate effect. Values of E* can be determined for different frequency and the frequency dependance of the viscoelastic constants can be determined.


	Nano Electronic Materials Research Group		July 16, 2007
	Nano Electronic Materials Research Group		Bio


Home Research Members Publications Links	Error mode images taken 5 minutes apart of an indented cell undergoing a slow recovery A clear understanding of cell structure, requires accurate measurements of its structural properties. Because of the complex nature of the cell's interior with its polymer networks and cytoplasm within a fluid membrane, it can be characterized as a viscoelastic medium. Thus, the cells structural properties are governed by both a time independent elastic response and a time-dependent viscous response. An AFM measurement of the viscoelastic properties requires alterations in both the standard AFM set-up, and the traditional data analysis. Experimental One key component to a quantitative assessment of the viscoelastic properties of a material through indentation, is the indenter itself. This must be well-defined or characterized such that the surface area is known throughout the indentation. One way to accomplish this is with the addition of a spherical bead to the cantilever to provide a symmetric spherical indenter. This is shown above schematically and an inverse AFM image of such a probe is also shown( white bar = 1 mm). The radius of the probe can vary and will define the stress range obtainable with this probe. In our experiments it generally varies from 3-10 microns. In addition to a well-defined surface area, a the spherical probes also provide a larger contact surface area with a correspondingly lower stress for the same total force. These lower stresses in turn give lower indentations. For thinner samples this is a great advantage because for larger indentations the sample substrate plays a role in the measured force responses and simple models for a supposed infinite sample could no longer be used. Hertz Model The Hertz model can be used for approximately infinite samples when indented with a spherical or paraboloid probe. It describes the relation between the Young's modulus(E) and the force and indentation with factors that reflect the tips geometry. Experimentally, the force and indentation are detyermined by obtaining a force curve at a point on the sample, which is simply a measurement of cantilever response as the tip is pushed into the sample. Using this equation and the force vs. distance data obtain with the AFM, we can calculate the Young's modulus. This is the zero-frequency result. There is no time dependence in this equation and there is no viscous contribution assumed. The Extended Hertz Model To obtain a viscous response, a higher frequency signal (20-400 Hz) is added to the existing force curve. When this signal is significantly faster than the rate of the force curve, then the rate of the force curve can be ignored. The additional signal is small in amplitude (2-20 nm) and sinusoidal in nature. The response of the cantilever is also sinusoidal with properties that reflect the material under the probe. The signal is phase shifted due to the property's viscous response and the amplitude decreases due to the elastic properties. The cantilever component thus has a real (in-phase) component and an imaginary out of phase component. For data analysis, the Hertz model is extended to first order in the indentation. There now appears a second term that conains the oscillating signal and is proportional to a complex viscoelastic constant E. A lock-in amplifier is used to isolate the second term. The measured force is a combination of the force of the probe on the sample and the force of the cantilever dragging through the liquid. This second term must be subtracted to obtain solely the force of the probe on the sample. Having successfully subtracted the drag force, the oscillating portion of the force and the indentation, both total and oscillating, can be used to determine viscoelastic constants for cells and polymer gels. Data and Results This data were all taken at 50Hz. The polyacrylamide gel shows a strong elastic component and a weak viscous component while the cells show strong viscous components. Also the value of E remains constant only for low indentation depths in the fibroblast cells. This is probably due to the substrate effect. Values of E* can be determined for different frequency and the frequency dependance of the viscoelastic constants can be determined.


		Nano Electronic Material Research Group The University of Texas at Austin