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When microcantilevers are used in the dynamic mode, the resonance shift upon material adsorption depends on the position of the adsorbate along the microcantilever. We have previously described that the adsorbate stiffness needs to be considered in addition to its mass in order to correctly interpret the resonance shift. Here we describe a method that allows obtaining the Young's modulus of the adsorbed bacteria derived from the measurement of the frequency shift when adsorbates are placed close to the clamping region. As a model system we have used E. Coli bacteria deposited on the cantilever surface by the ink-jet technique. We demonstrate that the correct information about adsorbed mass can be extracted by recording the cantilever profile and its resonance response. Also, the position and extent of adsorbates is determined by recording the microcantilever profile. We use a theoretical model based on the Euler – Bernouilli equation for a beam with both mass and flexural rigidity local increase due to the deposited material.

Microcantilevers have demonstrated their potential to be used as highly sensitive sensors. They have been applied in diverse fields, such as nanocalorimetry [

The shrinking of cantilever devices raises new questions when adsorbed masses are to be detected and measured with them. It is known that when the adsorbate is not uniformly distributed over the cantilever, the position of the adsorbate affects the dynamic response in different ways depending on the mass position with respect to the cantilever end [

Here we describe a method that allows fast and reliable determination of the mass position through the recording of the cantilever profile. This information, together with the read-out of the vibration response, allows the correct determination of the adsorbed mass. The model used here accounts for both the effect of the adsorbate stiffness and the effect of the added mass [

In this work, the resonance frequencies of the cantilevers are measured by acquiring the spectrum of the Brownian motion of the cantilevers vibrating in air. The relative frequency shift of the first three resonant modes is compared in order to choose the mode that provides the major sensitivity for the mass measurements. Here, we present experiments showing how the resonance is shifted to lower frequencies when the bacteria cells are placed at the free end of the cantilever and to higher frequencies when the same added mass is placed close to the clamped end. We show that the stiffness can be experimentally measured if the adsorbates are placed near the clamping of the microcantilever.

The E. Coli suspension was obtained by centrifugation of the Luria Bertani (LB) broth in which the E. Coli bacteria were grown. The resulting pellet was resuspended in ultrapure water to avoid the effects of the salt crystals when buffer solutions are used. By using the ink–jet technique, 0.5 nanoliter droplets of bacteria suspension were achieved. The micropipette (Microdrop Gmbh) was mounted on a microposition XY stage in order to control the position of the deposited drop along the cantilever with ±10 μm accuracy. Commercial rectangular with triangular free end tipless silicon cantilevers (NanoWorld) were used for the experiments. Their length, width and thickness were 500 μm, 100 μm and 1 μm respectively. In order to simplify the theoretical calculations, the triangular free end was substituted by an effective rectangular cantilever, taking the barycentre of the triangle as the end of the effective rectangular shape.

Since the quality factor of the oscillation of cantilevers immersed in liquids is very low, the resonance frequency was measured in air, comparing the effects of the adsorption on the resonance before and after the deposition of the cells. The short time delay between the measurements (about 15 min) assures that the temperature and pressure of the surrounding air in the laboratory conditions do not sufficiently change to affect the resonance of the cantilevers.

The out of plane vibrational resonant modes of the cantilever used in this work were measured using the optical beam deflection method. A laser diode (3 mW, 635 nm, Edmund optics Ltd.) was focused on the free end of the cantilever and the reflected spot was collected by a double split photodiode used as a position sensitive detector (Hamamatsu). The photocurrents are preamplified by current to voltage converters (HMS). The signals of the different quadrants of the photodiode are then processed by a summing amplifier (Stanford Research Systems, SIM 980). The amplified signals are connected to the PC via digital-to-analog converters (National Instruments) which separate the AC and the DC coupling of the signal. To measure the resonant peaks, fast frequency sweeps were performed in the range of 1 ms. Then a Fast Fourier Transform of the vibration is performed to obtain the resonant peaks.

The profile of the cantilever before and after bacteria deposition is recorded by a scanning laser set-up. This readout technique combines the optical beam deflection method and the automated two-dimensional scanning of a single laser beam by voice-coil actuators.

In order to describe the response of the microcantilevers upon adsorption of an inhomogeneous layer, we propose a model that is based on the Rayleigh's method. This method consists of an energy – work balance during a vibration cycle. Assuming a harmonic vibration _{n}_{n}t_{n}_{n}

Where x is the spatial coordinate along the cantilever, ρ_{c} and T_{c} are the cantilever mass density and thickness, respectively; ρ_{a} is the mass density of the material adsorbed on the cantilever, T_{a} is the adsorbate thickness that depends on x,

where E_{c} and E_{a} are the Young's modulus of the cantilever and adsorbed material, respectively. The flexural rigidity is a measure of the stiffness of the cantilever. It can be obtained as the product of the modulus of elasticity and the moment of inertia per unit length.

The flexural vibration shape of the unloaded cantilever is given by [_{n} satisfy the boundary conditions and are given by k_{n} =1.8751, 4.6941, 7.8548,…

We have applied the above equations to estimate the frequency shift of the first three resonant modes due to the adsorption of a finite structure on the cantilever. In order to simplify the theoretical calculation and compare it to the experiments, the adsorbates are modelled as a solid disc. The diameter of this disc is 120 μm, approximately the diameter of the droplet obtained by the ink-jet technique. By equalling the volume of the all bacterial cells to the volume of a pad, we deduce an effective thickness of 800 nm, smaller than the bacteria thickness, 1 μm. The estimated adsorbed mass is 3.3 ng which is equivalent to 4200 cells, and the Young's modulus is 1GPa, which is in the range of values obtained by atomic force microscopy measurements in dried bacteria [

We observe that the mass and the flexural rigidity have opposite effects on the resonance. The added mass always produces a decrease of the resonance, whereas the increase of the rigidity produces a positive shift of the frequency. The different maxima and minima of the frequency shift are related with the shape of the eigenmodes. As can be deduced from

Therefore, from the theoretical analysis it can be derived a frequency shift that depends on the position of the adsorbate along the cantilever length and on its mechanical properties in addition to the added mass. It is demonstrated also that the relative frequency shift is higher for the first resonant mode, although its resonant frequency is smaller than the rest of the modes. This is due to the presence of the nodes in the higher modes where the stiffness contribution is as sufficient as to practically cancelling the mass effect.

In

We have also measured the spectra of the Brownian motion of this cantilever before and after bacteria deposition,

By using

On repeated measurements, we observed that the third mode produces a value with the larger dispersion. This can be explained as the bacteria are adsorbed close to a node of the vibration so the error in the mass determination is maximal. The main source of experimental error in this measurements is the determination of the size of the drop. The above calculated errors take into account an indetermination of 20% in this parameter and 50% in the Young's modulus of the bacteria, the later considering the different values reported for the bacteria in the literature [

If the harmonic oscillator approximation Δm/m=-2Δω/ω is employed to calculate the mass, the mass calculated from the measured resonance shift corresponding to the first flexural mode is 4.4 ng, the second resonance mode provides a value of 2.3 ng and the shift for the third mode gives 2.2 ng. In this case the mass calculated using the first flexural mode overestimates the adsorbed mass by a factor of four because this model neglects the contribution of the flexural rigidity to the resonance shift.

From the

where D_{0} is the flexural rigidity of the unloaded cantilever and a and b are the limits over the cantilever in which the mass is spread out expressed as spatial coordinates.

For the cantilever shown in

Note that this value does not correspond to what was used in the previous section to simulate the biological layer. In

We have presented a method to accurately determine the adsorbed mass on a cantilever when adsorbates are inhomegeneously distributed over the cantilever surface. The measurement of the cantilever profile is needed to determine the position and extension of the adsorbed material. Then, these data can be used, together with the measured resonance shift, to derive the adsorbed mass by using the presented theoretical model. By recording the profile, the local surface stress induced by the adsorbates can also be measured.

In addition, we have theoretically demonstrated and experimentally tested that the use of higher vibration modes gives a relative frequency shift smaller than the first mode. This is due to the presence of vibration nodes. At these positions the amplitude of oscillation is zero, hence the added mass has no effect in the dynamic response. Therefore, when the added mass has a size comparable to the node – antinode separation, the use of higher flexural modes overestimates the measured mass values. Thus, to improve the sensitivity by working at high frequencies it is necessary to shrink the cantilevers.

As a model system, we have chosen the detection of E.Coli bacteria. We have also shown that by placing the bacteria close to the clamping region, we can measure its rigidity by recording the resonance frequency and the cantilever profile. From the results shown in this work, we conclude that the design of highly sensitive cantilevers to be used for pathogen detection must take into account the inhomogeneous nature of the adsorbed layers and also the important effect of the mechanical properties of the adsorbates in the dynamic response. Designs of microcantilevers that promote adsorption near the clamping region will take advantage of the large positive resonance shifts produced due to the local increase of stiffness given by pathogen adsorption. These new designs can produce highly sensitive biosensors that rely on the response due to the mechanical properties of the adsorbed pathogens rather than on the change of mass.

The authors acknowledge financial support from Ministerio de Educación y Ciencia (TEC2006-10316/MIC), CSIC and Comunidad de Madrid (200550M056). D. Ramos acknowledges the Comunidad de Madrid fellowship.

Schematic drawing of the set-up for the read-out of the cantilever profiles in arrays. The laser is mounted in a 2-D voice-coil scanner that allows obtaining the profile of an arbitrary number of cantilevers in the array. Measurements can also be performed in real-time.

Experimental measurements of the resonance frequency shift as a function of the longitudinal position of the adsorbed bacteria with respect to the clamping for the first (open squares), second (open circles) and third (open triangles) flexural modes. The lines represent the theoretical calculations for the first (solid), second (dash) and third (dotted) flexural modes.

Optical micrographs of the cantilevers after deposition of droplets containing E. Coli bacteria. Above each picture the cantilever profile measured before (dotted line) and after (continuous line) bacteria deposition are presented. The cantilever profile is modified at the position of the adsorbed bacteria due to the induced surface stress by the adsorbed inhomogeneous layer of bacteria.

Curvature of the cantilever recorded by the scanning laser set-up. The local change of curvature indicates the position and extension of the adsorbed droplet on the cantilever.

Spectra of the Brownian motion of the cantilever before (solid line) and after (dashed line) bacteria adsorption.

Theoretical calculation of the frequency shift as a function of the adsorbate thickness for three different values of the Young's Modulus: 1Gpa (solid line), 100 Mpa (dashed line) and 500 Mpa (dotted line).