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A study of the frequency response of AFM microcantilevers in liquid media contained in a commercial fluid cell is presented. Such systems exhibit complicated dynamics which are often not well described by available theories. Their dynamic behavior has a direct effect on the use of the AFM in dynamic mode while imaging in liquid or while extracting the rheological properties of the fluid. We explore the issues related to the design of the cantilever holder/fluid cell and propose an approach for evaluating, minimizing and recognizing the ultimate limitations of commercial cantilever holders. A technique for estimating the frequency response spectrum of the fluid cell itself from experimental data is presented. This spectrum can then be used to evaluate whether or not the fluid cell is suited for the desired purpose.

Since the invention of the atomic force microscope (AFM) [

There are three main techniques to excite an AFM cantilever: thermally, acoustically and magnetically. In a liquid environment the response of the cantilever strongly depends on the excitation technique. In the case thermal excitation [

Although the thermal and magnetic driving techniques produce smoother cantilever responses, they have some drawbacks which make working with acoustic excitation desirable. Firstly, these techniques require additional hardware such as a signal conditioner, a data acquisition system, special cantilevers, and a magnetic field system making these techniques more complex and costly. Secondly, in the magnetic technique, the fluid is heated by the electromagnetic field and the magnetic coating changes the vibrational properties and bending angle of the cantilever. For these reasons, many studies have been aimed at understanding and removing the redundant peaks in the response of the cantilever to acoustic excitation.

Putman

Although the effects of the various design problems on the cantilever response were previously recognized, the exact relationships were not understood and improvement of the frequency response based on control of these factors has not previously been considered. Instead efforts were focused on other approaches. Tamayo

Beside these practical investigations, a lot of effort has been focused on the evaluation of cantilever response theoretically. Schaffer

Also, because of the particular design of commercial fluid cells, it is impossible to apply ideal acoustic excitation to the cantilever which causes in an even more complicated frequency response. In this work, we apply some simple modifications to a widely used commercial fluid cell from Veeco [

In ideal acoustic excitation, the cantilever response is due to the movement of cantilever base, as shown schematically in

Since the internal friction of the cantilever itself is negligible compared to fluid damping, the governing equation for the deflection of the cantilever can be written in the following form:
_{c}_{h}

To continue, we require a general form for the hydrodynamic force. Because the amplitude of vibration of the cantilever is very small, the hydrodynamic force on each point of the cantilever can be approximated by the hydrodynamic force that would be applied on an infinitely long rigid beam that oscillates transversely with the same amplitude, _{f}

The solution of _{i}(ω)^{th} mode, and _{i}(x)^{th} vibrational mode of the undamped cantilever. The eigenfunctions are normalized in a way that _{i}(L)

After substituting _{i}(x)_{i}^{th} modal wavelength. herefore, given the amplitude of excitation,

It should be noted that the quantity measured by AFM is in fact the inclination of the cantilever. For this case, the theoretical response is simply the spatial derivative of the cantilever deflection:

In our studies we have used four different cantilevers selected according to the requirements of each experiment. Cantilevers 1 and 2 were used in the initial experiments (results in

As is clear from

This fluid cell design has several drawbacks. One problem is the holding clip because first of all, its spring is not strong enough to secure the cantilever base tightly, and secondly it does not necessarily hold the cantilever such that its axis is perpendicular to the clip rod. Since the surface of the cantilever chip is sloped, any configuration other than perpendicular results in only a single point of contact reducing the overall stability of the connection. The other end of the clip, which is above the fluid cell, can easily be moved or rotated during handling and mounting of the fluid cell on the AFM head thus changing the connection between the clip and the cantilever base. Moreover this can result in displacement of the cantilever chip in its groove and consequent misalignment of the laser beam from the AFM head. This is especially important because when the cantilever base moves to another position in its groove it creates a new vibrational system with a different frequency response. Therefore, the clip and spring system does not allow for reproducible experiments as shown in

This problem was solved by removing the clip and gluing the cantilever base to the fluid cell using silicone glue [

The second problem arising from the design of the fluid cell is that it causes an unsteady, free surface flow of the fluid trapped between the cell and scanner (See

This problem can be solved by making a small fluid reservoir from glass and gluing it into o-ring groove of the fluid cell as shown in

The last and most important problem with the fluid cell design is that the measured vibration response is the combination of the cantilever vibration and the fluid cell vibration. The response of the fluid cell itself to the excitation is frequency dependent and not the same as the movement of the piezoelectric actuator. This means that the driving motion experienced by the cantilever is not the ideal constant amplitude sine wave. Therefore the presence of the fluid cell and anything else between the piezoelement and the cantilever base make it impossible to measure the real frequency response of the cantilever.

In order to experimentally verify the above theory, we measured the response of cantilevers 3 and 4 in three different solutions of glycerin and water using the modified fluid cell. Also after filling the reservoir, the inlet and outlet channels were blocked to prevent any evaporation. In this way we can be sure that fluid borne excitation of the cantilever is negligible. Recall that the properties of the cantilever and the surrounding liquids are summarized in

Comparing

Based on the results shown on _{F}(ω)_{C}(x,w)_{exp}(x,ω)_{F}(ω)_{F}(ω)

To further verify the linearity of the model in

_{C}(x,w)

The results presented here prove that with this type of fluid cell the frequency response is dominated by the dynamics of the cell itself rather than the cantilever and that fluid borne excitation is less important than previously thought. This problem can only be solved by placing the peizoelectric actuator directly under the cantilever base as in the regular tip holders. Maali

We have further demonstrated that the important design issues for fluid cells are: (1) the method of attaching the cantilever chip to the cell, (2) the location of the piezoelectric actuator and (3) the occurrence of fluid borne excitation. Of these issues, the first relates to repeatability and the second and third determine the potential for achieving ideal acoustic actuation. If the actuator is placed exactly at the base of the cantilever following the approach of Maali

The frequency response of AFM microcantilevers, in a commercial fluid cell, was investigated while the cantilevers were immersed in different liquids. The dynamic characteristics of the fluid cell were determined by combining the experimental and theoretical results. It was shown that in fluid cells in which the piezoelectric element is removed from the cantilever base, ideal acoustic excitation cannot be achieved. Moreover in this case, the measured frequency response is dominated by the dynamics of the fluid cell potentially leading to significant misinterpretation of data. Contrary to previous reports, fluid borne excitation is shown to be a less significant effect.

Funding was provided by NSERC and CFI.

Cantilever movement in acoustic excitation. _{(x}_{,}_{t)}_{(x}_{,}_{t)}_{(t)}

Schematic of a fluid cell. In this picture (1) is the cantilever, (2) is the clip and spring, (3) is the circular groove for o-ring, (4) are the inlet and outlet channels for exchanging liquids, (5) is the moving support, (6) is the connecting chip, and (7) is the fixed support.

Frequency responses of cantilever 1 with different cantilever base and clip positions (in water). The fundamental resonant frequency of this cantilever is approximately 20 kHz.

Repeatability of the frequency response of cantilever 2 when glued to the fluid cell (in water). The fundamental resonant frequency of this cantilever in water is approximately 35 kHz.

Cross section of the fluid cell defined in

Frequency responses of cantilever 3 in 50% glycerin-water solution before (black line) and after (gray line) installing the reservoir. The cantilever is glued to the fluid cell.

Response of cantilever 4 in three solutions of glycerin and water; a) measured by AFM optics, b) determined theoretically.

Fluid cell frequency responses when containing solutions of glycerin and water, observed with cantilever 4.

Fluid cell frequency responses obtained from the excitation of two different cantilevers in 75% glycerin-water solution. Cantilevers 3 and 4 have lengths of 397 and 197 mm respectively.

Cantilever properties.

29 (mm) | |

2 (mm) | |

2300 (kg/m^{3}) | |

170 (GPa) |

Properties of glycerin-water solutions.

^{3}) |
||
---|---|---|

0% | 997 | 0.8628 |

50% | 1122 | 4.747 |

75% | 1191 | 25.49 |