Calculation of the effect of tip geometry on noncontact atomic force microscopy using a qPlus sensor

Summary In qPlus atomic force microscopy the tip length can in principle approach the length of the cantilever. We present a detailed mathematical model of the effects this has on the dynamic properties of the qPlus sensor. The resulting, experimentally confirmed motion of the tip apex is shown to have a large lateral component, raising interesting questions for both calibration and force-spectroscopy measurements.

where γ i is a dimensionless parameter defined as Writing Equations 36, 37, 38, and 39 as a matrix equation in the form it becomes clear that if det(D) = 0, then the solution would trivially be b 1..4 = 0, a stationary beam.
Hence, det(D) = 0, giving the following condition for β i : from Equations 5 and 40, where m * is the ratio of the tip mass to the mass of the tine. Equation 42 can be numerically solved simply and quickly by using the Newton-Raphson method.

Appendix B
Equation 12 can be expanded to give Butt and Jaschke [1] use (which is clearly still true for our boundary conditions from the general form of Φ given in Equation 4) to show that the integral with mixed (Φ i (x) and Φ k (x)) second derivatives can be written

S2
If we combine Equations 40 and 43 to give then we can rewrite boundary conditions Equations 8 and 9 as From these conditions it becomes clear that the first and third terms in the square brackets of Equation 46 cancel, as do the second and fourth. Thus, and

Appendix C
To solve Λ i we first integrate by parts twice to give The square brackets can be evaluated using boundary conditions from Equations 6, 7, 48, and 49.
Furthermore, with Equation 45 the integral can be written in terms of Φ i (x) only:

S3
The integral can be solved by substituting in ζ = x/L, and writing β i L as α i , Returning back to notation without α i or γ i and using Equation 42 to replace (1 + cos(β i L) cosh(β i L)) in the second term, after some manipulation gives

S5
Using Equation 42 again, this time to replace β 3 i Jm * L(cosh(β i L) sin(β i L) + cos(β i L) sinh(β i L)) in the final term, and rearranging will give It can be shown simply that and that

S6
Note that this form of Φ 2 i (L) appears in Equation 57, allowing us to reduce it to Combining Equations 58, 59, and 60 after some manipulation gives Which for simplicity, can be written as where f (m * , J) =3 sin 2 (β i L) sinh 2 (β i L)

By substituting Equation 42 into Equation 59 and rearranging we get
Initially this form appears to be more complicated. However, as will be demonstrated, because it contains the boundary conditions it produces a final result that is more physically understandable.
For simplicity it can be written as where Thus, finally we can write It is clear that in the case of no tip, m * = 0, this reduces to just Hooke's law in terms of the static spring constant: and inserting Equation 15 we get S9 Using the same model qPlus sensor as described above, this has been plotted in Figure 1, for i = 1..8 showing excellent agreement with theory, and faster convergence for larger tips.

Figure S 1: Agreement with lim
= 3, plotted for N = 8 roots to show that the equation is consistent with equipartition theorem. Where H is the length and D tip is the diameter of a cylindrical tip.