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Drag forces

The piezostage moves following a sine wave X(t) = X0 sin(wt) with frequency f (w = 2pi f) and amplitude X0. The advantage of moving the stage using a sine instead of a triangular function is that the low-pass filter of the piezo electronics does not modify the shape of the signal sent to the microchamber. It will just decrease the amplitude of the sine wave but not the shape, which is quite important. In order to determine accurately the value of the amplitude and the frequency we suggest the use of an oscilloscope connected to the piezo to read the exiting signal. The main disadvantage of this method in front of the most common one with the triangular function is that there is a restriction in the maximum value of the frequency f, as we will show later.

The movement of the microchamber leads to a displacement of the bead, which is trapped through the focused laser beam. The motion equation is given by

where gamma is the drag coefficient. In the particular case of a sphere, one can use an analytical expression derived from Stokes' formula and Faxen's correction

In the previous equation r indicates the radius of the microsphere. During the experiments, one must consider the real value of r and not the nominal one. On the other hand, l = z - z_0 is the distance between the axial position z of the particle and the coverslip z0. This last value can be estimated from the reflection observed at the coverslip of the microchamber. The dynamic viscosity arises from the product of the density of the aqueous medium and its kinematic viscosity.

The value of the water dynamic viscosity is eta = 1.0020 cP (cP, centipoise) at T = 20ºC, where 1P = 0.1 Pa· s = 0.1 Kg/s·m. This value does not drastically change when adding Sodium Dodecyl Sulfate SDS to the sample. The concentration of SDS that we use is 2.35\, g/L and it has a molar mass of 288.38 g/mol = 288.38 (molecular weight) · Mu (molar mass constant = 1 g/mol). This leads to a molar concentration of 0.00815 M for which the viscosity is close to that of water eta_{SDS} = 1.01 eta_{H_20} (see the link).

Equation (1) can be rewritten in a more convenient way

where fc is the so-called corner frequency. The equation has a general solution given by

Thus, the force applied by the laser on the trapped microbead is