Drift-Induced Systematic Errors in the Surface Forces Apparatus

The Surface Forces Apparatus (SFA) can measure intermolecular potentials by direct force measurement. A central design feature of the SFA is a mechanical loop as depicted in figure 1. Two thin sheets of atomically smooth mica are approached in crossed cylinder geometry. The cylinder curvature of each surface is achieved by gluing the mica sheets onto an optically polished cylinder lens. One of the surfaces is connected to the SFA frame via a force measuring spring with the spring constant (stiffness), k. When there is a surface force, F, between the sample surfaces, the spring will deflect by the amount F/k. A specialty of the SFA is that the deflection of the force measuring spring is not measured directly. Instead, one makes use of a highly precise interferometric distance measurement, called Multiple Beam Interferometry (MBI), to measure the separation between the surfaces, D, while the approach mechanism, M, is varying the total size, S, of the mechanical loop. The reading, E, is then positional reading of the actuator during the measurement. This reading, E, corresponds to hardware units and does not necessarily have to be in distance units (meter).

Enlarged view: Figure 1: Mechanical loop of the Surface Forces Apparatus, to control and measure the distance between two cylindrically curved sheets of mica (red).
Figure 1: Mechanical loop of the Surface Forces Apparatus, to control and measure the distance between two cylindrically curved sheets of mica (red).

Rough data of the SFA thus consist of a set of points Dj(Ej), E being the reading of the actuator. To evaluate this data set, one has to obtain the actuator calibration, C. This calibration relates the actuator positional reading, E, (encoder pulse, kW...) to the effective displacement of the actuator, M. It is common practice to deduce the actuator calibration from points measured at large distances. This is justified by the fact that surface forces have a limited range, so the force is approximatively zero at large distances. We thus have

C = DM/DE = DD/DE

Once the actuator calibration is known, it is possible to calculate the force at smaller distances, since the spring deflection is given by the difference (D - M) = (D - C*E). So, we can write:

Fj(Dj) = k*(Dj- C*Ej)

This scheme does not explicitely take instrumental drift into account. Generally, instrumental drift has the effect to change the SFA frame distance, S, without the action of the actuator. If the SFA exhibits instrumental drift, the measured actuator calibration, C, if determined as described above, will not equal the real (drift-free) actuator calibration, C0. This deviation can be expressed with a drift factor, g, as follows:

C = C0 * g

and g = (1+v'/v)/(1-(v'*t/DS))

The drift factor depends on four parameters, namely the drift rate v', the approach actuator speed v, the time, t, needed to measure/acquire the distance, D, whilst the actuator stands still, and, finally DS, which is the stepsize of the SFA frame from point to point (incl. drift motion).
We see that the drift factor equals unity (g=1), in the abscence of instrumental drift (v'=0). In the presence of instrumental drift, all depends whether the four parameters of the drift factor remain constant throughout the measurement or not. Because, if they are constant, the calculated force is still accurate although the measured calibration is different from the real calibration, This is because the measured calibration then implicitely accounts for drift effects. This situation is illustrated in figure 2 for a drift factor g=0.68.

Enlarged view: Figure 2: Simulated SFA measurement. The brown line shows the force-distance profile measured with an ideal instrument. The yellow dots correspond to measured points with a drift-affected SFA, but withthe parameters of the drift factor are held constant. The parameters of the drift factor were: v'=1nm/min, v=-10nm/min, t=10sec, DS=-0.5nm. The attentive viewer will remark a minute deviation of the measured points from the real potential. This is not due to drift. Rather is it a result of the non-zero potential atseparations where the calibration is taken. This effect is real and occurs also in realistic SFA measurements.
Figure 2: Simulated SFA measurement. The brown line shows the force-distance profile measured with an ideal instrument. The yellow dots correspond to measured points with a drift-affected SFA, but withthe parameters of the drift factor are held constant. The parameters of the drift factor were: v'=1nm/min, v=-10nm/min, t=10sec, DS=-0.5nm. The attentive viewer will remark a minute deviation of the measured points from the real potential. This is not due to drift. Rather is it a result of the non-zero potential atseparations where the calibration is taken. This effect is real and occurs also in realistic SFA measurements.

An often useful strategy is to measure more points at small separations. This is beacuse at large separations, surface forces are weak and rather uniform and thus less interesting. Of course, we still measure some points at large separations to obtain the actuator calibration. In the presence of instrumental drift, this practice results in a nun-uniform drift factor along the force-distance profile. To illustrate the resulting systematic error, we have performed the calculations shown in figures 3 and 4. For these calculations have we assumed a threefold point-density below a frame disnce of 8nm. To better illustrate the effects, we have animated figures 3 and 4. Figure 3 shows how the systematic error varies with drift rate and figure 4 shows how it varies with the SFA spring constant, k.

Figure 3: Simulated SFA measurement. The brown line shows the force-distance profile measured with an ideal instrument. The yellow dots correspond to measured points with a drift-affected SFA. The systematic error shown is based on a threefold point density below 8nm. The parameters of the drift factor were: v'=-1nm/min to +1nm/min (animation), v=-10nm/min, t=10sec, DS=-1nm (above D=8nm) and DS=-0.3333nm (below D=8nm). Drift rate and spring constant are displayed in the animation.
Figure 3: Simulated SFA measurement. The brown line shows the force-distance profile measured with an ideal instrument. The yellow dots correspond to measured points with a drift-affected SFA. The systematic error shown is based on a threefold point density below 8nm. The parameters of the drift factor were: v'=-1nm/min to +1nm/min (animation), v=-10nm/min, t=10sec, DS=-1nm (above D=8nm) and DS=-0.3333nm (below D=8nm). Drift rate and spring constant are displayed in the animation.
Figure 4: Simulated SFA measurement. The brown line shows the force-distance profile measured with an ideal instrument. The yellow dots correspond to measured points with a drift-affected SFA. The systematic error shown is based on a threefold point density below 8nm. The parameters of the drift factor were: v'=1nm/min, v=-10nm/min, t=10sec, DS=-1nm (above D=8nm) and DS=-0.3333nm (below D=8nm). The spring constant is varied from k=500N/m to k=100,000N/m (animation).Drift rate and spring constant are displayed in the animation.
Figure 4: Simulated SFA measurement. The brown line shows the force-distance profile measured with an ideal instrument. The yellow dots correspond to measured points with a drift-affected SFA. The systematic error shown is based on a threefold point density below 8nm. The parameters of the drift factor were: v'=1nm/min, v=-10nm/min, t=10sec, DS=-1nm (above D=8nm) and DS=-0.3333nm (below D=8nm). The spring constant is varied from k=500N/m to k=100,000N/m (animation).Drift rate and spring constant are displayed in the animation.

The following figures 5-8 illustrate a selection of systematic errors with different shapes. It demonstrates the different faces of this type of error and is not further commented.

NOTE: All force-distance profiles presented on this page were obtained with an iterative algorythm, which is implemented in LabView 5.0. We offer this program for free on our download page for Macintosh and PC compatibles.

Enlarged view: Figure 5: Simulated SFA measurement. Continous increase of point density.
Figure 5: Simulated SFA measurement. Continous increase of point density.
Enlarged view: Figure 6: Simulated SFA measurement. Oscillatory point density.
Figure 6: Simulated SFA measurement. Oscillatory point density.
Enlarged view: Figure 7: Simulated SFA measurement. Oscillatory point density.
Figure 7: Simulated SFA measurement. Oscillatory point density.
Enlarged view: Figure 8: Simulated SFA measurement. Continously increasing point density.
Figure 8: Simulated SFA measurement. Continously increasing point density.
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