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Statistical and Systematic Errors of the Surface Forces Apparatus M. Za¨ch and M. Heuberger* Department of Materials, ETH Zentrum, 8092 Zu¨ rich, Switzerland Received March 7, 2000. In Final Form: May 23, 2000 Spurred by the observation that force-distance profiles measured with the interferometric surface forces apparatus are often presented without error bars, we have considered the most important sources of statistical and systematic errors. We have also performed a calculation that yields quantitative error bars and insights into the most important parameters. In particular, we found instrumental drift as a subtle source of systematic error of unexpected magnitudesimplying the exigency for clear error and drift declaration together with experimental data. Our results indicate that the spring constant of the forcemeasuring spring plays a pivotal role. Namely, for spring constants above some 103 N/m, systematic errors may result in misinterpretation of results. In typical instrumental designs, the spring constant is adjustable in a range from 101 N/m to some 105 N/m.
Introduction The surface forces apparatus (SFA) has played a pivotal role in the discovery and experimental assessment of a variety of surface forces since the early seventies. Illustrious and well-known examples are dispersion forces,1-3 double-layer forces,3-5 and structural and short-range forces,6-8 as well as a variety of biological interactions.9-11 To measure intermolecular potentials, the interferometric SFA makes use of an optical distance measurement technique, known as multiple beam interferometry (MBI). This approach allows both the contact geometry as well as the separation between the surfaces to be determined at subnanometer resolution. The two surfaces under investigation normally consist of two equally thick sheets of atomically smooth mica, which are brought together in crossed-cylinder geometry. One of the surfaces is mounted on a force-measuring spring, which deflects as soon as a surface force is present. The force (i.e. spring deflection) is not directly measuredsrather is it calculated using the simple expression
F(D) ) k(D - EC) ) k(D - M)
two surfaces together from a sufficiently large separation, such that the spring is initially not deflected by surface forces. Additional motion of the approach mechanism brings the surfaces closer together until the spring is eventually deflected, when the surfaces are close enough to interact. Due to spring deflection, the optically measured distance, D, between the surfaces is no longer a linear function of the actuator reading, E. If one knows the spring constant, k, the surface force, F, can now be calculated from the difference D - EC using eq 1. In the case of an attractive potential, this scheme can be continued until the (attractive) interaction force gradient equals the spring constant. At this point the apparatus displays a mechanical instability and the surfaces snap into contact. The actuator calibration is readily obtainable at large surface separations, i.e., outside the range of surface forces, where C ) ∆D/∆E. A disadvantage of this method is that the unavoidable instrumental drift is not explicitly taken into account. The associated systematic errors have, to our knowledge, never been estimated or calculated in detail.
(1)
where k is the previously calibrated spring constant, D the distance measured between the surfaces, E the reading of the approach mechanism, and C its linear calibration, such that M ) EC is the real-space displacement of the approach mechanism. The approach actuator may comprise a dc motor or piezo crystal. The well-established method used to measure an intermolecular potential with an SFA is to start bringing (1) Israelachvili, J. N.; Tabor, D. Proc. R. Soc. London 1972, A331, 19-38. (2) Israelachvili, J. N.; Tabor, D. Nat. Phys. Sci. 1972, 236, 106-108. (3) Israelachvili, J. N.; Adams, G. E. J. Chem. Soc., Faraday Trans. 1978, I74, 975-1001. (4) Israelachvili, J. N.; Adams, G. E. Nature 1976, 262, 774. (5) Xu, Z. H.; Ducker, W.; Israelachvili, J. Langmuir 1996, 12, 22632270. (6) Israelachvili, J. N.; Pashley, R. M. Nature 1983, 306, 249-250. (7) Kumacheva, E. Klein, J. J. Chem. Phys. 1998, 108, 7010-7021. (8) Richetti, P.; Moreau, L.; Barois, P.; Kekicheff, P. Phys. Rev. E 1996, 54, 1749-1762. (9) Leckband, D. Nature 1995, 376, 617-618. (10) Helm, C. A.; Israelachvili, J. N.; McGuiggan, P. M. Biochemistry 1992, 31, 1794-1805. (11) Wong, J. Y.; Kuhl, T. L.; Israelachvili, J. N.; Mullah, N.; Zalipsky, S. Science 1997, 275, 820-822.
Statistical Errors Force-versus-distance curves, F(D), measured with an SFA naturally exhibit statistical errors that have a finite magnitude in both dimensions, F and D. Here, we give a quantitative estimation of the statistical error bars that are relevant for the measurement of intermolecular potentials. The primary quantity that is measured in the SFA is the distance, D, between the surfaces. The surface force, F, is then calculated using eq 1. We will thus apply the laws of error propagation to estimate the statistical error bars on the force axis. The theoretical resolution of a modern half-meter spectrograph with a 1200 lines/mm grating is in the order of γ ) (0.05 nm. This signifies the half-width of a monochromatic line. Using fitting methods (manual or mathematical), it is possible to determine the position of a line or fringe about twice as precisely, so σλ ≈ 0.025 nm. The chromatic order, N, associated with the lead-fringe used in a high-resolution setup is N ≈ 15 and, being an integer number, is error free. The refractive index, µ, is assumed to have a realistic standard deviation, σµ ) 10-3.
10.1021/la0003387 CCC: $19.00 © 2000 American Chemical Society Published on Web 08/12/2000
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From the following well-known expression12
D=
N(λ - λ0) 2µ
(2a)
we can obtain an estimation of the statistical error associated with odd-order fringes.
D )
x( ) 2
( )
N 2 2 N∆λ 2 2 σλ + σµ 2µ 2µ2
(2b)
The prefactor of 2 in the first term under the square root reflects our assumption that the measurements of λ0 and λ exhibit the same standard deviations. For a gap of up to a few nanometers we therefore estimate D ≈ 0.10.2 nm. Note that this error estimation is based on an optimal acquisition of one single fringe wavelength. Since the first term in eq 2b is predominant, statistical errors can further be reduced by repeated acquisition or, alternatively, by simultaneously acquiring fringes of different chromatic order. We have utilized the latter approach (16 fringes) to precisely measure the standard deviation of both piezo and dc-motor motion in our SFA (i.e. positioning precision). We found a value of σM ) 0.2 nm for the piezo when moving not more than 30 nm and a value of σM ) 0.5 nm for the dc motor over all tested ranges of motion. For motions up to 300 nm, the performance of the piezo is inferior (σM ) 1.7 nm). For simplicity, we assume that all variables are statistically independent and that the standard deviation of D is equal to the associated error calculated in eq 2b (D ∼ σD). We find an expression for the statistical error of F/R, based on eq 1:
F/R )
x( )
( )
( )
k 2 2 F 2 2 F 2 (σD + σM2) + σ + 2 σR2 R kR k R
(3)
If we insert realistic values, e.g. k ) 103 N/m, R ) 10-2m, and F ) 10 µN, we readily see that the standard deviation of the spring constant, k, is negligible. Nonetheless, the actual value of k plays a crucial role as can be seen from comparison between Figure 1, parts a and b. Below some 500 N/m, the third term in eq 3 dominates. Statistical errors then scale with the absolute value of the force, F (Figure 1a). Typical relative errors for the determination of R are in the order of 10%-20%.13 These variations are, however, not visible as scatter in the data points, since one never determines R for each point separately. Therefore, this component of statistical error is in practice effectively transmuted into a hidden systematic error. Likewise, the statistical error associated with σM is partially transformed into an apparent systematic error, since the approach actuator calibration, C, is taken to be the same for all points of the force-distance profile. For a spring constant greater than about 103 N/m, the first term dominates and the size of the error bars predominantly scales with k (Figure 1b). Systematic Errors Systematic errors are often difficult to detect, since repeated measurements do not necessarily reveal thems they may be highly reproducible. Fortunately however, systematic errors are deterministic, which allows their exact calculation, once the source of the error is recognized. (12) Israelachvili, J. N. J. Colloid Interface Sci. 1973, 44, 259-271. (13) Stewart, A. M. J. Colloid Interface Sci. 1995, 170, 287-289.
Figure 1. Statistical error bars calculated for a Lennard-Jonestype potential (eq 4 and 5) and for two different spring constants. At k ) 100 N/m the third term in eq 3 dominates, whereas at k ) 5000 N/m the first term clearly dominates. The snap-in distances for direct force measurement are Dk)100 ) 9 nm and Dk)5000 ) 2.4 nm.
To this end, we demonstrate that instrumental drift may cause systematic errors in force-distance profiles in conjunction with excessively high spring constants. Unfortunately, the embedding of drift-induced force deviations is rather subtle. To illustrate this problem, we have chosen an iterative approach. The following procedure involves assuming a realistic interaction potential as a starting point and then calculating the outcome of a driftaffected SFA measurement of this realistic interaction potential. As above, the well-known Lennard-Jones interaction pair potential serves as a basis:
A B w(r) ) - 6 + 12 r r
(4)
Here A and B are constants. Integrating over the macroscopic geometry of two crossed cylinders with radius, R using the Derjaguin approximation14,15 leads to an expression for the macroscopically measured force in an SFA:
{
}
A B F(D) ) π2F2R - 2 + 6D 180D8
(5)
Here D is the surface separation and F is the number density of atoms. For mica, F is 4 × 1027 m-3.16 The (14) Derjaguin, B. V. Kolloid-Z. 1934, 69, 155-164. (15) Israelachvili, J. Intermolecular & Surface Forces, 2nd ed.; Academic Press: London, 1991.
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Figure 2. Schematic representation of the geometrical quantities used to describe a drift-affected SFA measurement. S is the frame distance, or the delimiter of the mechanical loop, which consists of the surface separation, D, the spring deflection, Z, the motion of SFA approach actuator, M, and drift displacement, M′.
constants A ) 13.9 × 10-77 J m6 and B ) 1 × 10-134 J m12 were chosen such that the resulting potential corresponds exactly to the established dispersion force measurement between two sheets of mica by Israelachvili and Adams3 (i.e. Hamaker constant H ) π2F2A ) 2.2 × 10-20 J). To simplify the following calculation, we define an ancillary function S(D). As pictured in Figure 2, S represents the distance in the SFA frame. In addition to the approach displacement, M, we can now incorporate a variable instrumental drift into our model in the form of a drift displacement, M′. The real-world user has control over M exclusively, but our calculation will allow us varying M′ as well. While both displacement’s positional readings E, E′ are in hardware units (V, encoder pulse, or kΩ), the real-space actuator positions are denoted with M and M′, respectively. This is an important distinction since an SFA operator has direct access only to the positional readings in hardware units. From Figure 2, one can readily see that
S ) Z + D ) M + M′
(6a)
Figure 3. Ancillary function S(D, k ) 5000 N/m) for A ) 1 × 10-77 J m6. Upon approach, the system has a mechanical instability at the local minimum (1) and the surfaces snap into contact 1 f 2. A similar mechanical instability occurs when the surfaces are separated 3 f 4. To invert the function S(D) we thus consider two partial functions Sin(D) and Sout(D) as illustrated with the graphical insets.
∂D ) k. In terms of Sk(D), this means that ∂S(D)/∂D ) -1/k[∂F(D)/∂D] + 1 ) 0. The corresponding snap-in and jump-out conditions are hence given as the local minimum and maximum of the Sk(D) function. The function S can be rendered reversible, if we define two subfunctions Skin(D) for loading and Skout(D) for unloading (Figure 3) of this nonadhesive contact. The outcome of a drift-affected SFA measurement is finally calculated with an iterative algorithm, following the same scheme as in a realistic experiment. Thereby, S is successively decreased in steps, starting from an initial value S0. The exact iterative form is therefore
Sj+1 ) Sj + ∆Sj+1
In the case of an equilibrium force measurement, we can thus write
-Zk ) Fspring ) Finteraction ) F(D)
(6b)
Afterward, we calculate the new equilibrium separation Dj+1(Sj+1) by inversion of the appropriate function, Sin(D,k) or Sout(D,k), such that
Thus, we obtain an expression for S, as a function of D and k:
S(D, k) )
-F(D) +D k
(7)
An example of a function S(D,k) is plotted in Figure 3. For positive values of D, the function exhibits exactly one local maximum and one local minimum. Furthermore, Sk(D . 0) ≈ D, since surface forces are insignificant at large distances. In a realistic measurement with a spring apparatus, one always observes an intrinsic instability in the system. Mathematically, the instability occurs when the interaction force gradient equals the springs constant, i.e., ∂F(D)/ (16) Ondik, H. M.; Wolten, G. M. Crystal Data, Determinative Tables, Vol. II: Inorganic Compounds; National Standard Reference Data System, 3rd ed.; U.S. Department of Commerce, National Bureau of Standards, and the Joint Committee on Powder Diffraction Standards, 1973.
(8a)
Dj+1 ) D(Sj+1)
(8b)
The time, ∆t, used to perform one step, including the data acquisition time of a distance measurement, τ, is given by
∆tj+1 )
∆Sj+1 - v′τ ∆Sj+1 - v′τ + τ with g 0 (9a) v + v′ v + v′ tj+1 ) tj + ∆tj+1
(9b)
where v is the speed of the approach actuator and v′ signifies the drift rate. We note that the first term in eq 9a corresponds to the approach actuator moving time and the second to the approach actuator rest time. The signs of v and v′ are chosen such that a negative value means inward motion (approach). Finally, we calculate the drift-induced deviation in the reading of the approach actuator position, E. After one
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Figure 4. Drift-induced systematic errors obtained when measuring a Lennard-Jones-type potential (plain line). The following parameters were held constant: v ) 10-8 m/min; τ ) 10 s; R ) 10-2 m. (a) If all quantities in the drift factor of eq 13 are constant throughout a measurement, the real potential is reproduced by the measurement (open squares). The small remaining deviation is realistic and stems from the actuator calibration, C, which is still slightly affected by the potential at large separations. A 3-fold reduction of the step size ∆S at distances below 10 nm may significantly change the outcome of the measurement (plain circles). (b) Similar situation, but with a gradually decreasing step size to produce a negatively curved artifact. (c) Alternating variation of step size can induce apparent oscillations of significant amplitude. (d) Small variation of drift rate.
different from the real calibration, C0. Inserting eqs 9a and 10a into eq 12 leads to the desired expression
approach step we thus have
∆Ej+1 )
∆Sj+1 - v′∆tj+1 C0
Ej+1 ) Ej + ∆Ej+1
(10a) (10b)
where C0 is the real motor calibration (i.e. without drift). Upon iteration, we will obtain a data set Dj(E,t) that corresponds to a drift-affected SFA measurement. These data are subsequently analyzed following the scheme mentioned above. The effective motor calibration, C, is resolved by a linear fit to the data Dj (Ej) at large distances, where F(D . 0) ≈ 0. Finally, the force-distance profile, F(D), is determined, using the following equation for each point of the data set:
F(D) ) k[D(E) - EC] - offset
(11)
The offset is introduced to account for the motor offset at the start position and effectively shifts the absolute zero of the force axis so that F(D . 0) ) 0. As mentioned above, the effective calibration, C, is obtained at large separations, where surface forces are negligible, and we can thus write
C)
∆D ∆S = ∆E ∆E
(12)
In the presence of drift, this effective calibration, C, is
[ ]
v′ v C ) C0 v′τ 1∆S 1+
(13)
which states that the real calibration, C0, multiplied by a drift factor equals the effective calibration, C. As expected, the drift factor is unity if there is no drift (i.e. v′ ) 0). Apart from the drift rate v′, the drift factor also depends on three additional parameters, namely v, τ, and ∆S. If all four parameters remain constant throughout a force run, the measured calibration will be generally different from the real calibration, C0, but valid for all points of the data set. In thissand only thisssituation, drift is implicitly accounted for. The measured points will hence reproduce the real potential, as illustrated in Figure 4a. If one or more parameters change during a force run, the measured calibration is not valid for all points of the data set and we produce a systematic error. A possible manifestation of such a systematic error could be due to an increase of point density at smaller separations, i.e., the measurement of more points in the interesting regions. This practice can change ∆S fairly reproducibly. Figure 4a shows how a three-times-higher point density below a certain distance can change the apparent shape of a measured potential. For high spring
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constants, Figure 4b,c shows selected examples of artifacts produced by gradually increasing or oscillating point density, respectively. Contrary to intuition, choosing a higher approach actuator velocity, v, does not reduce the extent of this type of systematic error. This can readily be seen from the drift factor in eq 13, since v and ∆S appear in independent factors (i.e. denominator/numerator). In the case of a small outward drift, an increased point density always leads to a shift of the apparent force toward repulsion. Analogously, the opposite is true for inward drifts. To extend the picture, Figure 4d illustrates a selected example, where the drift rate itself varies in the order of 1 pm/min per minutesall other parameters are held constant. Since this other type of systematic error is based on the variation of the drift rate, active drift compensation (e.g. a ramp function) cannot simply account for it. Another parameter that might change the drift factor is the data acquisition time, τ. Especially for manual measurements, this value is strongly affected by individual circumstances, so we do not illustrate the corresponding systematic error here. However, it is also clear from eq 13 that the extent of this latter type of systematic error is also independent of the approach actuator velocity, v. A possible improvement could be automated data acquisition, where τ is constant throughout the measurement. A variation of the actuator speed, v, during the measurement would potentially produce a qualitatively similar effect to the variation of ∆S illustrated above. In many cases however, it seems justified to assume that the SFA approach actuator moves at a constant velocity. From eq 1, we notice that a systematic deviation of the calibration, ∆C, leads to an apparent force variation
∆F ∝
k ∆C R
(14)
Therefore, all drift-induced systematic errors scale with k/R. We note that error bars are hence amplified with small radii of surface curvature, R. The points presented in Figure 4 have been calculated by following the iterative process described in eqs 8-10. A corresponding program has been implemented in LabView 5.0, which allows all relevant parameters to be changed in real time. It can be freely downloaded and is compatible with both Macintosh and PC platforms.17 Discussion and Conclusions We have shown that all important errors scale with k/R. However, we have seen that it is solely the error of the distance measurement σD and partially of σM that becomes evident as scatter of the data points. All other errors are hidden systematic errors, unless, of course, error bars are explicitly marked in the graphs. For convenience, we have summarized the extent of drift-induced systematic errors in Figure 5, where the relative error bars at snap-in separation are illustrated as a function of drift rate and spring constant for the realistic example of a known dispersion potential between mica. The illustrated errors were calculated by averaging over the errors produced by drift in both directions. We know from our experience that a drift rate between 0.1 nm/min and 1 nm/min is routinely achievable in our SFA after an equilibration time of several hours. Comparing these values with Figure 5 reveals that a spring constant (17) Web address: http://www.surface.mat.ethz.ch/users/manfred/ SFA_DRIFT.
Figure 5. Summarizing relative error bars due to drift-induced systematic error at 3-fold reduction of step size, ∆S from 1 to 0.33 nm below S ) 10 nm, for drift rates, v′, between 0.01 and 1 nm/min and spring constants, k, between 5 × 102 and 5 × 104 N/m. All other parameters are constant and the same as in Figure 4.
softer than some 103 N/m is needed to securely reduce the relative error below the 10% mark. We have also conducted a literature survey to obtain information on recommended spring constants. We have selected 50 relevant publications across three decades of SFA work covering various scientific areas, laboratories, and instrumental designs. The actual value of the spring constant is declared in 14% of the publicationssroughly half of them indicate values below 103 N/m. Another 16% cite the entire accessible range (e.g. 101-106 N/m) of available spring constants in the instruments used but without giving the actual values. The remaining 70% give no value for the spring constant. We also note that information concerning instrumental drift rates was lacking in 100% of the considered publications and less than 10% indicated error bars in the graphics. It hence appears that spring constant, instrumental drift, and error bars are not frequently discussed topics in the SFA literature. The question thus remains: What is the optimal spring constant? Obviously, there are limits to the useful range of spring constants. The upper limit, which is given by the above presented error calculation, seems to be around 1000 N/m. The lower limit is around 100 N/m and often determined by the total range of the approach mechanism and the required maximum forces. Consequently, the useful range is, say, 100-1000 N/m and therefore quite narrow. To measure a purely repulsive potential, a spring constant of the order of 100 N/m seems adequate, since it is small enough to reduce errors and guarantee a good force sensitivity. However, if we want to measure an attractive potential with the SFA directly, the situation becomes more complicated. To avoid early snap-in, one would eventually be tempted to choose a spring considerably stiffer than 1000 N/m, yet we know that this may lead to unwanted errors. Fortunately, there are ways out of this dilemma. One example is to measure an attractive potential with an even stronger but well-known repulsive potential superimposed to it, such that the overall force is always repulsive. This would allow measuring the potential using a smaller spring constant without the occurrence of the mechanical instability. The extra repulsive force could be (18) Note: scanning probe microscopes operate on different time scales and k/R ratios. The presence of an active feedback-loop and the fact that the spring deflection is directly measured presents a situation different from that in the SFA.
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in form of an intrinsic repulsion, such as double-layer repulsion in aqueous salt solutions. We note that this procedure has been successfully practiced previously to measure the dispersion force between mica,3 assuming the force superposition principle and exact knowledge of the double-layer repulsion. We have shown that the issue of statistical and systematic errors is rather complex, and we were able to illustrate only a few special cases here. The embedding of systematic errors can be very subtle and still produce considerable effects. It takes careful analysis of the exact experimental conditionssfrom case to casesto calculate the magnitude of such errors. To measure force-distance profiles with the SFA, it is strongly recommended to use spring constants below 103 N/m and declare error bars. A central element of this communication is certainly the affirmation that many SFAs, by design, measure
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distance and not forcesan important nuance that imposes narrow limitations on the reasonable range of spring constants. Some aspects of the here presented results might be relevant to scanning probe microscopes but need specific adaptation to the experimental setup or mode of operation.18 Acknowledgment. This work was performed at the Laboratory for Surface Science and Technology, ETH Zu¨rich, and the authors thank Prof. N. D. Spencer for his generous support. We are also grateful for the financial support provided by the Swiss National Science Foundation (M.Z.). LA0003387
