Development of a mathematical model and analytical solution of a coupled two-beam array with nonlinear tip forces for application to AFM

Abstract

Utilising arrays of cantilevers has been identified as a method of increasing AFM sensitivity and throughput beyond the limitations of current non-contact AFM. In order to develop the technology, we investigate the change in eigenstates of arrays due to interactions that occur between cantilever tip and sample using a model of a base-coupled array of cantilevers with individual mass and stiffness properties influenced by a quadratic nonlinear force to represent tip–sample interactions. An analytical expression is developed for a coupled two-beam array utilising Multiple Scale Lindstedt–Poincare perturbation theory to assess how a selected parameter space alters the array dynamics. Experiments are carried out using a macroscale set-up to validate the developed model. We show that the model captures the eigenstates of the system with good qualitative accuracy and that the perturbation expansion performs well in both the weakly (far from surface) and strongly (near the pull-in point) nonlinear regimes. The results demonstrate that utilising changes in eigenstates due to force interactions could have significant benefits to non-contact AFM with regard to measurement sensitivity and speed.

Appendices

Appendix 1

Expansion of the two-beam Eq. (14) prior to applying perturbation expansions.

$$\begin{aligned}&\ddot{\hat{\varPhi }}_n + W_{cn} \dot{\hat{\varPhi }}_n + W_{kn} (\hat{\varPhi }_n + \bar{\varPhi }_n) + (W_{F_{n1}} AC_1 + W_{F_{n2}} AC_2) \nonumber \\&\qquad \times \cos (\varOmega \tau ) - \epsilon \nu \bigg ( \dfrac{2}{d_{01} \hat{d}_{02}} \bigg ) \bigg [ (d_{01} \overline{W}_{21} + \hat{d}_{02} \overline{W}_{11}) (\hat{\varPhi }_1 + \bar{\varPhi }_1) \nonumber \\&\qquad + (d_{01} \overline{W}_{22} + \hat{d}_{02} \overline{W}_{12}) (\hat{\varPhi }_2 + \bar{\varPhi }_2) \bigg ] (\ddot{\hat{\varPhi }}_n + W_{kn} (\hat{\varPhi }_n + \bar{\varPhi }_n)) \nonumber \\&\qquad + \epsilon ^2 \nu ^2 \bigg [ \dfrac{1}{\hat{d}^2_{02}} (\overline{W}_{21} (\hat{\varPhi }_1 + \bar{\varPhi }_1) + \overline{W}_{22} (\hat{\varPhi }_2 + \bar{\varPhi }_2))^2 \nonumber \\&\qquad +\dfrac{1}{\hat{d}^2_{01}} \left( \overline{W}_{11} (\hat{\varPhi }_1 + \bar{\varPhi }_1) + \overline{W}_{12} (\hat{\varPhi }_2 + \bar{\varPhi }_2)\right) ^2 + 4 \dfrac{1}{d_{01} \hat{d}_{02}} \nonumber \\&\qquad \times (\overline{W}_{11} (\hat{\varPhi }_1 + \bar{\varPhi }_1) + \overline{W}_{12} (\hat{\varPhi }_2 + \bar{\varPhi }_2)) (\overline{W}_{21} (\hat{\varPhi }_1 + \bar{\varPhi }_1) \nonumber \\&\qquad + \overline{W}_{22} (\hat{\varPhi }_2 + \bar{\varPhi }_2)) \bigg ] (\ddot{\hat{\varPhi }}_n + W_{kn} (\hat{\varPhi }_n + \bar{\varPhi }_n)) \nonumber \\&\qquad + \dfrac{1}{\hat{d}^2_{01}} W_{NL_{n1}} \tau _m - 2 \dfrac{1}{\hat{d}^2_{01} \hat{d}_{02}} \epsilon \nu \overline{W}_{22} \hat{\varPhi }_2 W_{NL_{n1}} \tau _m\nonumber \\&\qquad - B_{2n-1} \hat{\varPhi }_n + \epsilon ^2 \nu ^2 \dfrac{1}{\hat{d}^2_{01} \hat{d}^2_{02}} (\overline{W}_{21} \hat{\varPhi }_1 + \overline{W}_{22} \hat{\varPhi }_2)^2 W_{NL_{n1}} \tau _m \nonumber \\&\qquad + \dfrac{1}{\hat{d}^2_{02}} W_{NL_{n2}} \tau _m - 2 \dfrac{1}{\hat{d}_{01} \hat{d}^2_{02}} \epsilon \nu \overline{W}_{11} \hat{\varPhi }_1 W_{NL_{n2}} \tau _m \nonumber \\&\qquad - B_{2n} \hat{\varPhi }_N + \epsilon ^2 \nu ^2 \dfrac{1}{\hat{d}^2_{01} \hat{d}^2_{02}} (\overline{W}_{11} \hat{\varPhi }_1 + \overline{W}_{12} \hat{\varPhi }_2)^2 W_{NL_{n2}} \tau _m\nonumber \\&\quad = 0 \end{aligned}$$

(43)

with the following definitions

$$\begin{aligned}&\hat{d}_{01}=d_{01}-\overline{W}_{11}\bar{\varPhi }_1- \overline{W}_{12}\bar{\varPhi }_2\\&\hat{d}_{02}=d_{02}-\overline{W}_{21}\bar{\varPhi }_1- \overline{W}_{22}\bar{\varPhi }_2\\&B_1=\dfrac{2}{\hat{d}^2_{01}\hat{d}_{02}}W_{NL_{11}}\tau _m \overline{W}_{21}\\&B_2=\dfrac{2}{\hat{d}_{01}\hat{d}^2_{02}}W_{NL_{22}}\tau _m \overline{W}_{12}\\&B_3=\dfrac{2}{\hat{d}_{01}\hat{d}^2_{02}}W_{NL_{12}}\tau _m \overline{W}_{12}\\&B_4=\dfrac{2}{\hat{d}^2_{01}\hat{d}_{02}}W_{NL_{21}}\tau _m \overline{W}_{21} \end{aligned}$$

Appendix 2

Eigenvalues and eigenvectors for the first-order perturbation solution (20).

$$\begin{aligned}&r_1 = \sqrt{\omega ^2_1/2 - \alpha /2 + \omega ^2_2/2} \end{aligned}$$

(44a)

$$\begin{aligned}&r_2 = \sqrt{\alpha /2 + \omega ^2_1/2 + \omega ^2_2/2} \end{aligned}$$

(44b)

$$\begin{aligned}&\mathbf {\mathbf {v}_1} = \begin{bmatrix} v_{11} \\ v_{21} \end{bmatrix} = \begin{bmatrix} 1 \\ (\alpha + \omega ^2_1 - \omega ^2_2)/(2 \overline{\omega }_1) \end{bmatrix} \end{aligned}$$

(44c)

$$\begin{aligned}&\mathbf {\mathbf {v}_2} = \begin{bmatrix} v_{12} \\ v_{22} \end{bmatrix} = \begin{bmatrix} -(\alpha + \omega ^2_1 - \omega ^2_2)/(2 \overline{\omega }_2) \\ 1 \end{bmatrix} \end{aligned}$$

(44d)

$$\begin{aligned}&\alpha = \sqrt{\omega ^4_1 - 2 \omega ^2_1 \omega ^2_2 + \omega ^4_2 + 4 \overline{\omega }_1 \overline{\omega }_2} \end{aligned}$$

(44e)

$$\begin{aligned}&A_1 = \dfrac{1}{2} a_1 e^{i b_1} \end{aligned}$$

(44f)

$$\begin{aligned}&A_2 = \dfrac{1}{2} a_2 e^{i b_2} \end{aligned}$$

(44g)

Appendix 3

Secular terms of the two-beam perturbation model (31) and (32).

$$\begin{aligned} T_1= & {} (\mu ^2 W_{c11} z_{11}+ i r_1 A_1)/(2 v_{11} r_1) \end{aligned}$$

(45)

$$\begin{aligned} V_1= & {} (\mu ^2 (W_{F_{11}} AC_1 + W_{F_{12}} AC_2) e^{i \sigma _1 \tau _1})/(2 v_{11} r_1) \end{aligned}$$

(46)

$$\begin{aligned} T_2= & {} (\mu ^2 W_{c22} z_{21}+ i r_1 A_1)/(2 v_{21} r_1) \end{aligned}$$

(47)

$$\begin{aligned} V_2= & {} (\mu ^2 (W_{F_{21}} AC_1 + W_{F_{22}} AC_2) e^{i \sigma _1 \tau _1})/(2 v_{21} r_1) \end{aligned}$$

(48)

$$\begin{aligned} T_3= & {} (\mu ^2 W_{c11} z_{12} + i r_2 A_2)/(2 v_{12} r_2) \end{aligned}$$

(49)

$$\begin{aligned} V_3= & {} (\mu ^2 (W_{F_{11}} AC_1 + W_{F_{12}} AC_2) e^{i \sigma _2 \tau _1})/(2 v_{12} r_2) \end{aligned}$$

(50)

$$\begin{aligned} T_4= & {} (\mu ^2 W_{c22} z_{22}+ i r_2 A_2)/(2 v_{22} r_2) \end{aligned}$$

(51)

$$\begin{aligned} V_4= & {} (\mu ^2 (W_{F_{21}} AC_1 + W_{F_{22}} AC_2) e^{i \sigma _2 \tau _1})/(2 v_{22} r_2) \end{aligned}$$

(52)

Appendix 4

The following values are used for the single-beam and two-beam experiments.

See Tables 1 and 2.

Table 1 Values used for the single-beam model

Table 2 Values used for the two-beam model