Cantilever Dynamics: Theoretical Modeling

Abstract

To provide a measure of flexibility and symmetry regarding the description of tip-sample interactions, a dynamical model is presented for which the cantilever tip and the sample surface are treated as independently damped simple harmonic oscillators passively coupled via the nonlinear tip-sample interaction forces. The sample oscillations are assumed to occur in the coupling from a small element of surface mass (active mass) attached to the remainder of the sample for which the spring constant is the sample stiffness constant. The analytical model reduces to a pair of coupled nonlinear differential equations, the general solutions of which are obtained using a matrix iteration procedure. The general solutions are applied to the quantitative assessment of signal generation and contrast mechanisms in atomic force acoustic microscopy (AFAM), force modulation microscopy (FMM), ultrasonic force microscopy (UFM), ultrasonic atomic force microscopy (UAFM), amplitude modulation atomic force microscopy (AM-AFM), and scanning near-field ultrasonic holography (SNFUH) including the related heterodyne force atomic force microscopy (HF-AFM) and resonant difference-frequency atomic force ultrasonic microscopy (RDF-AFUM). In addition to obtaining quantitative expressions for surface contrast mechanisms, contrast mechanisms from subsurface features are accounted in the model for AFAM, FMM, UFM, and SNFUH.

Appendices

Appendix A: Amplitude Bifurcations

A general assessment of the effects of the various expansion coefficients $F_{0}$, $F_{1 }$, $F_{2}$, etc., on the frequency and amplitude in the solution to Eq. (3.21) can be achieved most expediently by taking $\eta _{cn} \rightarrow \eta _c , \gamma _{cn} \rightarrow \gamma _c$, and $k_{cn} \rightarrow k_c$ in Eq. (3.21), and assuming that in first-order approximation $\eta _{c} = c + \eta _{c0}\cos \omega _{c}$ t where c is a constant. Substituting the expression in Eq. (3.21), neglecting harmonic frequency terms in the resulting expression, and solving the resulting equation, we obtain

$$\begin{aligned} \omega _{c} \approx \left\{ {\left[{\left({\omega _{0}^{2}-\frac{F_{1} }{m_{c} }} \right)^{2}-\frac{4F_{0} F_{2} }{m_{c}^{2} }-\frac{2F_{2}^{2} }{m_{c}^{2} }\eta _{c0}^{2} } \right]^{1/2}-\frac{F_{c} }{m_{c} \eta _{c0} }} \right\} ^{1/2} \end{aligned}$$

(A.1)

where $\omega _{0}$ is the free-space resonance frequency of the cantilever. It is seen directly from Eq. (A.1) that $F_{1}$ leads to a shift in $\omega _{0}$ that is dependent via Eqs. (3.14) and (3.16) on the amplitude of cantilever oscillation, the quiescent separation distance z $_{0}$, and the sign of $F_{1}$.

The product term $F_{0}$ $F_{2}$ in Eq. (A.1), where $F_{0}$ is the static term and $F_{2}$ is the first nonlinear term in the polynomial expansion of the interaction force, also leads to a shift in $\omega _{0}$, the shift direction being dependent on the signs of $F_{0}$ and $F_{2}$. However, the $F_{2}$ term alone in Eq. (A.1) leads to a bifurcation in the cantilever oscillation amplitude that produces upper and lower amplitude versus frequency curves over specific ranges of drive frequency. The bifurcation results from the left-oriented fold shown in Fig. 3.4 where from Eq. (A.1) the absolute oscillation amplitude ${\vert } \eta _{c} {\vert }$ is plotted as a function of the drive frequency $\omega _{c}$. We point out that since Eq. (A.1) does not include the effects of damping, the resonance “peak” in the figure is not closed. The factor $({2F_2^2 \eta _{c0}^2 /m_c^2 })$ in Eq. (A.1) is responsible for the frequency shift in the “peak” of the resonance curve that increases with an increase in the oscillation amplitude $\eta _{c0}$. The increasing shift in the resonance peak with increasing amplitude produces the fold shown in Fig. 3.4.

Fig. 3.4

Amplitude bifurcation resulting from the creation of a fold in the amplitude versus frequency curve. The left-oriented fold for the case depicted in the figure gives rise to stable upper ($\eta _{c} <$ 0) and lower ($\eta _{c} <$ 0) oscillation amplitudes (amplitude bi-stability) represented by the solid portion of the curves for drive frequencies below the free-space resonance frequency $\omega _{0}$. The dashed portion of the lower curve is an unstable region of the curve that produces a third oscillation amplitude not observed experimentally

Since $F_{2}$ is squared in Eq. (A.1), the frequency of the resonance “peak” decreases with increasing amplitude irrespective of the sign of $F_{2}$, and results in the left-oriented fold shown in the figure. It is apparent from the figure that three possible oscillation amplitudes $\eta _{c0}$ are possible when the drive frequency $\omega _{c}$ lies within the fold. Only the upper and lower amplitudes are stable and lead to the experimentally observed amplitude bi-stability; the middle amplitude is unstable and not seen in experiments. It is emphasized that the curve folding (peak frequency shifting) from $F_{2}$ is quite independent of the frequency shift produced by $F_{1}$ and the product $F_{0}$ $F_{2}$. $F_{1}$ and $F_{0}$ $F_{2 }$ shift the entire resonance curve to larger or smaller frequencies – not a portion of the curve leading to folding, as is the case for $F_{2}$.

To more clearly delineate the role of $F_{3}$ in the frequency-amplitude relation, we repeat the derivation leading to Eq. (A.1) using only the terms $F_{1}$ and $F_{3}$ in the polynomial expansion. We now assume that in first-order approximation $\eta _{c}$ = $\eta _{c0}$cos$\omega _{c}$ t and substitute the expression in Eq. (3.21). Again neglecting harmonic frequency terms in solving the resulting expression, we obtain

$$\begin{aligned} \omega _{c}=\left({\omega _{0}^{2}-\frac{F_{1} }{m_{c} }-\frac{3}{4}\frac{F_{3} }{m_{c} }\eta _{c0}^{2}-\frac{F_{c} }{m_{c} \eta _{c0}}} \right)^{1/2}. \end{aligned}$$

(A.2)

Equation (A.2) shows that $F_{3}$ plays a role similar to that of $F_{2}$ in the frequency-amplitude relationship. However, unlike $F_{2}$ in Eq. (A.1), $F_{3}$ appears to the first power in Eq. (A.2). This means that the frequency of the resonance “peak” increases or decreases in accordance with the sign of $F_{3}$, producing either a right-oriented or left-oriented fold, respectively.

Amplitude bifurcation is well-documented in the literature [34, 45, 46] and is often attributed to the occurrence of attractive and repulsive components in the force-separation curve. Equations (A.1) and (A.2) show, however, that amplitude bi-stability is a consequence only of the nonlinearity in F(z) (represented in Eq. (A.1) by the tem $F_{2}$ and in Eq. (A.2) by ${ F}_{3}$). The attractive and repulsive components in the interaction force affect only the magnitude and sign of the expansion coefficients. The upper amplitude curve in Fig. 3.4 corresponds to the case where the cantilever oscillations are 180 degrees out of phase with the drive force ($F_{c}/ \eta _{c0} <$ 0), while the lower amplitude curve corresponds to the case where the cantilever oscillations and drive force are in phase ($F_{c} / \eta _{c0} >$ 0).

Appendix B: Subharmonic Generation and Routes to Chaos

We show in Sect. 3.3.2 that nonlinearity in the interaction force leads to a shift in the resonance frequency with increasing drive amplitude. In Sect. 3.3.4 we find that the nonlinearity also generates harmonics of the driving force frequency. For an appropriate range of values of the dynamical parameters, including drive frequency, the interaction force nonlinearities that lead to the generation of harmonics is known to stimulate the generation of subharmonics [51]. This is seen, for example, from a consideration of the forced Duffing equation$\ddot{x}+\alpha x+\varepsilon x^{3}=\Gamma \cos \omega t$.

Let $\tau =\omega t$ and re-write the Duffing equation as [51]

$$\begin{aligned} \omega ^{2}{{x}^{\prime \prime }}+\alpha x+\varepsilon x^{3}=\Gamma \cos \tau \end{aligned}$$

(B.1)

where the prime symbol in Eq. (B.1) denotes the operator d/d$\tau$. We look for solutions of Eq. (B.1) having period 6$\pi$ (corresponding to frequency $\omega$/3). Letting $\omega$ = $\omega _{0}$ + $\varepsilon _{s}$ $\omega _{1}$, substituting in Eq. (B.1), and using the perturbation procedure given Sect. 3.3.4 where $\varepsilon = \varepsilon _{s}$, we obtain the zeroth- and first-order perturbation equations, respectively, as

$$\begin{aligned} \omega _{0}^{2} {{x}^{\prime \prime }}_{0}+\alpha x_{0}=\Gamma \cos \tau \end{aligned}$$

(B.2)

and

$$\begin{aligned} \omega _{0}^{2} {{x}^{\prime \prime }}_{1}+\alpha x_{1}=-2\omega _{0} \omega _{1} {{x}^{\prime }}^{\prime }_{0}-x_{0}^{3}. \end{aligned}$$

(B.3)

The periodicity condition, 6$\pi$, applied to Eq. (B.2) means that $\omega _0 =3\sqrt{\alpha }$. The solution to Eq. (B.2) [zeroth-order solution to Eq. (B.1)] is thus

$$\begin{aligned} x_{0} (\tau )=a_{1/3} \cos \frac{1}{3}\tau +b_{1/3} \sin \frac{1}{3}\tau -\frac{\Gamma }{8\alpha }\cos \tau . \end{aligned}$$

(B.4)

Although Eq. (B.4) suggests the possibility of a period-3 solution, it does not guarantee the solution. The second-order perturbation equation, Eq. (B.3), is used to determine the conditions on the coefficients $a_{1/3}$ and $b_{1/3}$ that allows the generation of a subharmonic. Such a determination is not straightforward and shall not be presented here (for details see Jordan and Smith [51]). We simply point out that the conditions allow the generation of the 1/3 subharmonic for Eq. (B.1), but a similar solution for the 1/2 subharmonic is not allowed when $\varepsilon$ is small. Even when the conditions permit a given subharmonic, the subharmonic may not be observed experimentally. Whether a given subharmonic is observed experimentally depends on the stability of the subharmonic.

We now consider the cantilever dynamical equation. Subharmonic stability for the cantilever dynamical equation (3.21) is more conveniently addressed by re-writing Eq. (3.21) in dimensionless form as

$$\begin{aligned} {x}^{\prime \prime }+\Gamma _{c} {x}^{\prime }+\Omega ^{2}x=f_{c} \cos \tau +f_{0}+f_{2} x^{2}+{\cdots } \end{aligned}$$

(B.5)

where

$$\begin{aligned} \begin{array}{l} x=\eta _{cn}, \quad \tau =\omega t, \quad \Gamma _{c}=\dfrac{\gamma _\mathrm{eff} }{m_{c} \omega ^{2}},\quad \Omega ^{2}=\dfrac{k_\mathrm{eff} }{m_{c} \omega ^{2}}=\dfrac{\omega _{0}^{2} }{\omega ^{2}},\\ f_{c} =\dfrac{F_{c} }{m_{c} \omega ^{2}}, \qquad f_{0}=\dfrac{F_{0} }{m_{c} \omega ^{2}}, \qquad f_{2}=\dfrac{F_{2} }{m_{c} \omega ^{2}}. \end{array} \end{aligned}$$

(B.6)

Equation (B.5) is an example of a second-order nonlinear ordinary differential equation of general form

$$\begin{aligned} {x}^{\prime \prime }=f(x,{x}^{\prime },\tau ). \end{aligned}$$

(B.7)

The stability of any solution to Eq. (B.7) can be assessed by first reducing Eq. (B.7) to a system of first-order equations as

$$\begin{aligned} x^{\prime }=g_{1} (x,y,\tau ), \quad {y}^{\prime }=g_{2} (x,y,\tau ) \end{aligned}$$

(B.8)

or in matrix form as

$$\begin{aligned} {X}^{\prime }=G\left({X,\tau } \right) \end{aligned}$$

(B.9)

where

$$\begin{aligned} X=\left({{\begin{array}{l} x \\ y \\ \end{array} }} \right), \qquad {X}^{\prime }=\left({{\begin{array}{l} {{x}^{\prime }} \\ {{y}^{\prime }} \\ \end{array} }} \right), \qquad G\left({X,\tau } \right)=\left({{\begin{array}{l} {g_1 (x,y,\tau )} \\ {g_2 (x,y,\tau )} \\ \end{array} }} \right). \end{aligned}$$

(B.10)

Let ${\varvec{X}}_{s}$ be a solution to Eq. (B.9) and $\Xi$ be a small perturbation to the solution such that

$$\begin{aligned} X=X_{s}+\Xi \end{aligned}$$

(B.11)

where

$$\begin{aligned} X_{s}=\left({{\begin{array}{l} {x_s } \\ {y_s } \\ \end{array} }} \right), \quad \Xi =\left({{\begin{array}{l} \xi \\ \eta \\ \end{array} }} \right). \end{aligned}$$

(B.12)

Substituting Eq. (B.11) in Eq. (B.9), we obtain

$$\begin{aligned} {X}^{\prime }_{s}+{\Xi }^{\prime }=G\left({X_{s}+\Xi ,\tau } \right). \end{aligned}$$

(B.13)

If $\Xi$ is sufficiently small, we may reduce Eq. (B.13) to the linear matrix equation

$$\begin{aligned} {\Xi }^{\prime }=G\left({X_{s}+\Xi ,\tau } \right)-G\left({X_{s} ,\tau } \right)\approx A(\tau )\Xi \end{aligned}$$

(B.14)

where A is a 2 x 2 linear matrix. Equation (B.13), known as the first variational equation, implies that the stability of the solution x $_{s}$ to Eq. (B.5) is the same as the stability of solutions $\Xi$ to Eq. (B.9) and, hence, to Eq. (B.7) [51].

We apply the above results to Eq. (B.5) by re-writing Eq. (B.5) as a pair of first order differential equations given in matrix form by

$$\begin{aligned} {X}^{\prime }=\left({{\begin{array}{l} {{x}^{\prime }} \\ {{y}^{\prime }} \\ \end{array} }} \right)=\left({{\begin{array}{c} y \\ {-\Gamma _c y-\Omega ^{2}x+f_0 +f_2 x^{2}+f_c \cos \tau } \\ \end{array} }} \right)=\left({{\begin{array}{l} {g_1 (x,y,\tau )} \\ {g_2 (x,y,\tau )} \\ \end{array} }} \right)=G\left({X,\tau } \right). \end{aligned}$$

(B.15)

It is straightforward to show that Eq. (B.15) is identical to Eq. (B.5) by differentiating the top element on both sides of the second equality and substituting the bottom elements in the resulting expression. We now write

$$\begin{aligned} X=\left({{\begin{array}{l} x \\ y \\ \end{array} }} \right)=\left({{\begin{array}{l} {x_s +\xi } \\ {y_s +\eta } \\ \end{array} }} \right)=X_s +\Xi \end{aligned}$$

(B.16)

and substitute Eq. (B.16) in (B.14) to get

$$\begin{aligned} \left({{\begin{array}{c} {{x}^{\prime }} \\ {{y}^{\prime }} \\ \end{array} }} \right)=\left({{\begin{array}{c} {{x}^{\prime }_s +{\xi }^{\prime }} \\ {{y}^{\prime }_s +{\eta }^{\prime }} \\ \end{array} }} \right)=\left({{\begin{array}{c} {y_s +\eta } \\ {-\Gamma _c (y_s +\eta )-\Omega ^{2}(x_s +\xi )+f_0 +f_2 (x_s +\xi )^{2}+f_c \cos \tau } \\ \end{array} }} \right). \end{aligned}$$

(B.17)

Expanding the quadratic term in Eq. (B.17), retaining only the first power of $\xi$, and subtracting ${X}^{\prime }_s =G\left({X_s ,\tau } \right)$ from the resulting expression, we obtain

$$\begin{aligned} \left({{\begin{array}{c} {{\xi }^{\prime }} \\ {{\eta }^{\prime }} \\ \end{array} }} \right)=\left({{\begin{array}{c} \eta \\ {-\Gamma _c \eta -\Omega ^{2}\xi +2f_2 x_s \xi } \\ \end{array} }} \right). \end{aligned}$$

(B.18)

Differentiating the top elements on both sides of the equality in Eq. (B.18) and substituting the bottom elements in the resulting expression, we obtain the second order linear differential equation

$$\begin{aligned} {\xi }^{\prime \prime }+\Gamma _{c} {\xi }^{\prime }+(\Omega _{c}^{2}-2f_{2} x_{s} )\xi =0. \end{aligned}$$

(B.19)

The factor $x_{s}$ in Eq. (B.19) is a solution to Eq. (B.5) which, from Sect. 3.3.4, we approximate to first order as $x_s \approx (f_0 /\Omega ^{2})+A\cos \tau$. Writing

$$\begin{aligned} \xi (\tau )=e^{-(1/2)\Gamma _{c} \tau }\zeta (\tau ) \end{aligned}$$

(B.20)

and substituting both x $_{s}$ and Eq. (B.20) in (B.19), we obtain Mathieu’s equation (in standard form) [55]

$$\begin{aligned} \zeta ^{\prime \prime }+(\alpha +\beta \cos \tau )\zeta =0 \end{aligned}$$

(B.21)

where in the present case

$$\begin{aligned} \alpha =\Omega _{c}^{2}-\frac{1}{4}\Gamma _{c}^{2}-2\frac{f_{0} f_{2} }{\Omega _{c}^{2} }=\frac{\omega _{0}^{2} }{\omega }-\frac{1}{4}\left({\frac{\gamma _\mathrm{eff} }{m_{c} \omega ^{2}}} \right)^{2}-2\frac{F_{0} F_{2} \omega ^{2}}{\omega _0^2}, \end{aligned}$$

(B.22)

and

$$\begin{aligned} \beta =-2f_{2} A=-2\frac{F_{2} A}{m_{c} \omega ^{2}}. \end{aligned}$$

(B.23)

The solutions to Mathieu’s equation for a given value of $\alpha$ occur in alternating regions or bands of stability and instability as the magnitude of $\beta$ increases [note from Eq. (B.23) that $\beta$ increases with oscillation amplitude A]. The Mathieu equation belongs to the class of linear differential equations with periodic coefficients. The general solutions to such equations, known as Floquet solutions, are of the form [51]

$$\begin{aligned} \zeta (\tau )=ce^{\rho \tau }P(\tau ) \end{aligned}$$

(B.24)

where P($\tau$) is a periodic function with minimum period T, $\rho$ is the Floquet index, and c is a constant. The Floquet indices for Eq. (B.21) are obtained from solutions to the expression

$$\begin{aligned} e^{2\rho T}-\phi (\alpha ,\beta )e^{\rho T}+1=0. \end{aligned}$$

(B.25)

The factor $\phi$($\alpha$,$\beta$) is not known explicitly. However, writing $\mu$ = $\exp$($\rho$ T) and substituting in Eq. (B.25) yields a quadratic equation that can be solved for $\mu$, hence for $\rho$, as a function of $\phi$($\alpha$,$\beta$). Thus, the solution $\rho$ to Eq. (B.25) in terms of $\phi$($\alpha$,$\beta$) determines the stability or instability of the solution to the Mathieu equation, Eq. (B.21), for a range of values of $\phi (\alpha ,\beta )$. The ranges of values of $\phi$($\alpha$,$\beta$) lead to a plot of $\alpha$ versus $\beta$ showing alternating regions of stability and instability.

The solutions to Eq. (B.25) corresponding to regions of stability lead to solutions of the Mathieu equation of the form [56]

$$\begin{aligned} \zeta (\tau )=e^{i\nu \tau }\sum _{n=-\infty }^\infty {c_{n}} e^{\mathrm{in}\tau }=e^{i\nu \omega t}\sum _{n=-\infty }^\infty {c_{n}} e^{\mathrm{in} \omega t} \end{aligned}$$

(B.26)

where the Floquet index takes the value $\rho =$ i $\nu$ ($\nu$ real) and c $_{n}$ are constants. A fractional value of $\nu$ leads to fractional harmonics (including subharmonics). In the unstable regions of the $\alpha$-$\beta$ plot the solutions are given as [56]

$$\begin{aligned} \zeta (\tau )=c_{1} e^{\sigma \tau }P_{1} (\tau )+c_{2} e^{-\sigma \tau }P_{2} (\tau ) \end{aligned}$$

(B.27)

where $\rho =\sigma$ ($\sigma$ real), c $_{1}$ and c $_{2}$ are constants, and P $_{1}$($\tau$) and P $_{2}$($\tau$) are periodic functions. It is clear from Eq. (B.25) that in the regions of instability at least one solution is unbounded as the result of exponential growth and is, in fact, the origin of the instability.

Numerical solutions of the Mathieu equation reveal that not all solutions in the regions of instability are unbounded. Kim and Hu [57] show from numerical calculations that upon entering regions of instability from a region of stability the fundamental oscillation undergoes a cascade of period-doubling or pitchfork bifurcations that culminates in the establishment of bounded, chaotic motion. They also found that upon encountering the region of stability from a region of instability, the instability becomes stable through a reverse pitchfork or period-doubling bifurcation. However, the occurrence of a stable oscillation does not necessarily mean re-establishment of the fundamental oscillation frequency. For example, numerical solutions of the damped Duffing equation with $\alpha =-1$ and $\varepsilon =1$ reveal the appearance of a period-five stable solution upon entering the second stable region from the preceding unstable region [51]. Such a solution is allowed by Eq. (B.26). Experimental AM-AFM measurements show that the stable oscillation frequency upon entering a second stability region is highly dependent on the initial conditions [58].

It is important to note from Eqs. (B.22) and (B.23) that both $\alpha$ and $\beta$ are dependent on $F_{2}$ and $\omega$, while $\alpha$ depends additionally on $\gamma _\mathrm{eff}$ and $\omega _{0}$. From Eqs. (3.19) and (3.20) $F_{2}$ and $\gamma _\mathrm{eff} = \gamma _{c} - S_{1}$ vary with the amplitude of oscillation. When the amplitude of oscillation is such that the tip-sample separation distance enters the strong force region (near the sample surface) of the force-separation curve, the variation in $F_{2}$ and S $_{1}$ can be substantial. Such changes affect $\alpha$ and $\beta$ and thus the solutions of the Mathieu equation with an increase in drive amplitude.

Appendix C: Renormalization Methods

As indicated in Sect. 3.2 the general cantilever dynamics is quite properly described in terms of an infinite set of superimposed, damped, harmonic oscillators (modes), each with an associated free-space resonance frequency. Typically, the mode with the largest contribution to the cantilever displacement amplitude is chosen for consideration, the others ignored, and the cantilever modeled as a set of decoupled oscillators as given by Eqs. (3.21). However, the nonlinearity of the tip-sample interaction force leads to the possibility of interactions among the modes that produce significant effects in the cantilever dynamics. Such possibilities are affirmed in amplitude versus frequency spectra taken from AFAM experiments [52, 53]. The spectra very often reveal the bifurcation of a single free-space resonance into multiple resonances upon cantilever-sample contact.

We show that resonance bifurcation is analytically predictable and occurs as the result of nonlinear modal interactions [44]. Our analytical approach is quite similar to that of group renormalization used in quantum field theory and in the description of critical phase transitions in materials. In the present application of renormalization, deviations of the cantilever displacements $\eta _{cn}$ from that expected for the spring model at frequencies well away from some initially chosen renormalization reference frequency are formally absorbed into the model by allowing the parameters in the model to vary with frequency. Since the mathematical machinery is analogous to that of the renormalization group, we adopt the language used in the mathematical formalism for renormalization: in the present model the chosen reference frequency is the ‘renormalization scale,’ the model parameters are said to ‘run’ with the scale, and the theory is said to be ‘renormalized.’

To obtain the appropriate equations of motion that couple the separate modes, we take $\eta _{cn} \rightarrow \eta _n$, $\gamma _{cn} \rightarrow \gamma _n$, and $k_{cn} \rightarrow k_n$ on the left-hand side of Eq. (3.21), and $\omega _c \rightarrow \omega$ and $\eta _{cn} \rightarrow \sum \limits _{m\in Z^+} {\eta _m }$ (where $Z^{+} = \{1 , 2, \ldots$}) on the right-hand side of Eq. (3.21). For present purposes it is expedient to ignore the dissipative terms in the expansion of Eq. (3.12) involving the coefficients S $_{n}$. The implications of nonzero S $_{n}$ will be considered following the renormalization. We thus obtain the dynamical equations that account for mode coupling as [44]

$$\begin{aligned} L_{n} \eta _{n}=F_{c} \cos \omega t+F_{00}+F_{10} \sum _{m\in Z^{+}} {\eta _m } +F_{20} \left({\sum _{m\in Z^{+}} {\eta _m } } \right)^{2}+\cdots \end{aligned}$$

(C.1)

where the operator $L_{n}=\left(m_{c} \frac{\mathrm{d}^{2}}{\mathrm{d}t^{2}}+\gamma _{n} \frac{\mathrm{d}}{\mathrm{d}t}+k_{n} \right)$ . The greater difficulty in accounting for mode coupling is apparent: a single, nonlinear differential equation, Eq. (3.21), has been traded for an infinite number of coupled ones, Eq. (C.1). The renormalization method is quite appropriate to handle such a situation.

${L}_{n}$ can be inverted outside its nullspace to obtain the particular solution [44]

$$\begin{aligned} \eta _{n}&=2\pi \tilde{G}_{n}^{0} (0)F_{00}+\frac{1}{2}F_{c} \left({2\pi \tilde{G}_{n}^{0} (\omega )e^{i\omega t}+2\pi \tilde{G}_{n}^{0} (-\omega )e^{-i\omega t}} \right)\\&\quad \; +F_{10} \int d{t}^{\prime } G_{n}^{0} (t-{t}^{\prime })\left(\sum _{m\in Z^{+}} \eta _{m} ({t}^{\prime }) \right)\\&\quad \; +F_{20} \int dt^{\prime } G_{n}^{0} (t-{t}^{\prime })\left(\sum _{m\in Z^{+}} \eta _{m} ({t}^{\prime }) \right)^{2}+\cdots \end{aligned}$$

(C.2)

(Note: the 0 superscript is a label, not an exponent) where $G_{n}^{0} (t-{t}^{\prime })$ is the Green function for L$_{n}$, $L_{n} G_{n}^{0} =\delta (t-{t}^{\prime })$:

$$\begin{aligned} G_{n}^{0} (t-{t}^{\prime })=\int {d{\omega }^{\prime }} \tilde{G}_{n}^{0} ({\omega }^{\prime })e^{i{\omega }^{\prime }(t-{t}^{\prime })}, \end{aligned}$$

(C.3)

$$\begin{aligned} \tilde{G}_{n}^{0} ({\omega }^{\prime })=\frac{-1}{2\pi m_{c} \left({{\omega }^{\prime 2}-\dfrac{k_{n} }{m_{c} }-i\dfrac{\gamma _{n} }{m_{c} }{\omega }^{\prime }} \right)}. \end{aligned}$$

(C.4)

We sum over all modes to obtain the total cantilever deflection $\eta _{c}$ and define the function

$$\begin{aligned} G^{0}\equiv \sum _{n\in Z^{+}} {G_n^0 } \end{aligned}$$

(C.5)

along with the corresponding Fourier transform $\tilde{G}^{0}\equiv \sum \limits _{n\in Z^{+}} {\tilde{G}_n^0 }$. We obtain from Eq. (C.2) an integral equation for the total cantilever deflection $\eta _{c}$ (the exact parameter of interest) as [44]

$$\begin{aligned} \eta _{c} (t)=&\,\sum _{n\in Z^{+}} {\eta _{n}=} 2\pi \tilde{G}^{0}(0)F_{00} +\frac{1}{2}F_{c} \left({2\pi \tilde{G}^{0}(\omega )e^{i\omega t}+2\pi \tilde{G}^{0}(-\omega )e^{-i\omega t}}\right)\\&+F_{10} \int {d{t}^{\prime }} G^{0}(t-{t}^{\prime })[\eta _c ({t}^{\prime })]+F_{20} \int {d{t}^{\prime }} G^{0}(t-{t}^{\prime })[\eta _c ({t}^{\prime })]^{2}+\cdots . \end{aligned}$$

(C.6)

For a given driving frequency $\omega$, the nonlinearity of the tip-sample interaction force is known to generate harmonic terms I$\omega$ ($I\in Z^{+})$ as shown in Sect. 3.3.4. In solving Eq. (C.6) the generation of harmonics leads us to consider the ansatz [44]

$$\begin{aligned} \eta _{c} (t)=\sum _{I\in Z} {a_I } e^{iI\omega t}\Rightarrow \end{aligned}$$

(C.7)

$$\begin{aligned} \sum _{I\in Z} {a_I } e^{iI\omega t}=&\,2\pi \tilde{G}^{0}(0)F_{00} +\frac{1}{2}F_c \left({2\pi \tilde{G}^{0}(\omega )e^{i\omega t}+2\pi \tilde{G}^{0}(-\omega )e^{-i\omega t}} \right)\\&+F_{10} \sum _{J\in Z} {\tilde{G}^{0}(J\omega )} a_J e^{iJ\omega t}\\&+F_{20} \sum _{(J,K)\in Z^{2}} {\tilde{G}^{0}[(J+K)\omega ]} a_J a_K e^{i(J+K)\omega t}+\cdots \end{aligned}$$

(C.8)

where Z is the set of integers. The Fourier coefficients $a_{\pm I}$ determine the cantilever amplitude that is experimentally observed at the frequency ${\vert }$I${\vert }$$\omega$. Explicitly, the amplitude is given as $A(I\omega )=2\left| {a_I } \right|$, which follows from the reality condition $\eta _c =\eta _c^{*}$, where the star denotes complex conjugation. The non-triviality of the coefficients $a_I$${\vert }$I${\vert }$$>$ 1) reflects the nonlinearity of the interaction force.

Using the orthogonality of the Fourier basis, we obtain from Eq. (C.8) the recursion relation [44]

$$\begin{aligned} a_{I}=2\pi \tilde{G}^{0}(I\omega )\left[{\frac{1}{2}F_{c} (\delta _{I,1} +\delta _{I,-1})+F_{00} \delta _{I,0}+F_{10} a_{I}+F_{20} \sum _{J\in Z} {a_{J} a_{I-J} +\cdots } } \right] \end{aligned}$$

(C.9)

We note that if $\tilde{G}^{0}(I\omega )$ were the Fourier-space Green function$\tilde{G}_n^0 (I\omega )$ for a single harmonic oscillator mode n, Eq. (C.9) would have the same form as that for the typical solution found for a decoupled mode n. In such case the exact solution for the total cantilever displacement would look identical to that of a linear spring subjected to nonlinear forces—a more immediately tractable problem mathematically than the one at hand. This suggests that, in analogy to Eq. (C.4) for $\tilde{G}_n^0 (I\omega )$, $\tilde{G}^{0}$ should be expressed as

$$\begin{aligned} \tilde{G}^{0}(\omega )=\frac{-1}{2\pi m_{c} \left({\omega ^{2}-\dfrac{K(\omega )}{m_{c} }-i\dfrac{G(\omega )}{m_{c} }\omega } \right)} \end{aligned}$$

(C.10)

where K and G are the renormalized cantilever spring and damping coefficients, respectively.

When transforming Eq. (C.10) to ‘time’-space, the Green function is completely characterized for purposes of integration by its poles and residues. Poles must necessarily occur in Eq. (C.10) at

$$\begin{aligned} \Omega _{n}=i\frac{\gamma _{n} }{2m_{c} }+\sqrt{\frac{4k_{n} m_{c} -\gamma _{n}^{2} }{4m_{c}^{2}}} \end{aligned}$$

(C.11)

where the real part of $\Omega _{n}$ is the free-space resonance of the cantilever corresponding to the nth mode. $K$ and $G$ must be real-valued functions that, due to constraints on the location of the poles, must satisfy

$$\begin{aligned} K(\Omega _{n})=k_{n} \end{aligned}$$

(C.12)

$$\begin{aligned} G(\Omega _{n})=\gamma _{n}. \end{aligned}$$

(C.13)

With these constraints the poles of $\tilde{G}^{0}$ are {$\Omega _{n}$} and $\mathrm{Re}\,s\left[{\tilde{G}^{0}(\Omega _n )} \right]=\mathrm{Re}\,s\left[{\tilde{G}_n^0 (\Omega _n )} \right]=-\left({2\pi \sqrt{4m_c k_n -\gamma _n^2 }} \right)^{-1}$. Equations (C.10)–(C.13) lead to the explicit functional forms

$$\begin{aligned} K(\omega )=\frac{\mathrm{Im}(\omega )}{\mathrm{Re}(\omega )}\mathrm{Im}\left[{m_{c} \omega ^{2}+\frac{1}{2\pi \tilde{G}^{0}(\omega )}} \right]+\mathrm{Re}\left[{m_{c} \omega ^{2}+\frac{1}{2\pi \tilde{G}^{0}(\omega )}} \right] \end{aligned}$$

(C.14)

$$\begin{aligned} G(\omega )=\frac{1}{\mathrm{Re}(\omega )}\mathrm{Im}\left[{m_{c} \omega ^{2}+\frac{1}{2\pi \tilde{G}^{0}(\omega )}} \right] \end{aligned}$$

(C.15)

where $\tilde{G}^{0}$ in Eqs. (C.14) and (C.15) is calculated from the infinite sum given by Eqs. (C.4) and (C.5).

From Eqs. (C.9) and (C.10) we re-write the recursive solution as

$$\begin{aligned} a_{I}=2\pi \tilde{G}(I\omega )\left[{F_{00} \delta _{I,0} +\frac{1}{2}F_{c} (\delta _{I,1}+\delta _{I,-1})+F_{20} \sum _{J\in Z\backslash \{0,I\}} {a_{J} a_{I-J} +\cdots } } \right] \end{aligned}$$

(C.16)

where the interaction force-modified Green function $\tilde{G}$ is given by

$$\begin{aligned} \begin{array}{l} \tilde{G}(\omega )\equiv \left[{1-2\pi \sigma \tilde{G}^{0}(\omega )} \right]^{-1}\tilde{G}^{0}(\omega )\\ [8pt] =\frac{-1}{2\pi m_c \left({\omega ^{2}-\dfrac{(K(\omega )+\mathrm{Re}(\sigma ))}{m_c }-i\dfrac{(\omega G(\omega )+\mathrm{Im}(\sigma ))}{m_c }} \right)} \end{array} \end{aligned}$$

(C.17)

$$\begin{aligned} \sigma =(F_{10}+2F_{20} a_{0} +\cdots ). \end{aligned}$$

(C.18)

We note that for nonzero dissipation terms S $_{n}$ and R $_{mn }$ the expansion in Eq. (C.1) would pick up S $_{n}$ and R $_{mn }$ terms that lead to additional sigma-like terms, similar to that of Eq. (C.18), in the denominator of Eq. (C.17).

We now consider a particular application of the renormalized model that demonstrates resonance bifurcation. Since the running of the renormalized parameters with frequency necessarily results in the appropriate multi-peak, free-space resonance spectrum, it is reasonable to suspect that the nonlinear interactions, smoothly introduced to the cantilever system as the cantilever engages the sample surface, would mix the peaks and bifurcate the resonances. For example, the running of the parameters in the vicinity of a given resonance peak from one parameter value to a second value should result in two contact resonance peaks in place of the given free-space resonance peak. To demonstrate resonance bifurcation, we wish to plot the cantilever amplitude as a function of its driving frequency and observe a splitting in the local maxima of the curve.

We begin by assuming that the cantilever output signal is passed through a lock-in amplifier such that all frequencies except the drive frequency are filtered out. Thus, only the amplitudes corresponding to I = $\pm$ 1 are of interest and the amplitude is $A=2 {\vert }a_{I}{\vert }$. In such case the nonlinear components of the interaction force vanish from the recursion relation given by Eq. (C.16), if Eq. (C.16) is solved iteratively for $a_{\pm 1}$ by recursively substituting the relation in Eq. (C.16) into the a $_{I}$ on the right-hand side. The resulting expression is

$$\begin{aligned} a_{\pm 1}=\pi \tilde{G}(\pm \omega )F_{c}. \end{aligned}$$

(C.19)

It is emphasized that for an infinitely stiff sample surface Eq. (C.19) is exact and is numerically equivalent to results obtained by starting with the beam equation and applying a nonlinear interaction force at the tip-sample boundary. In most cases the sums defining $\tilde{G}^{0}$ and $\sigma$ converge quite rapidly, so practicably the sums can be truncated with accurate results. The extension of the model to include the more realistic case of an elastic sample surface is more complicated and is the subject of current research. The present model, nonetheless, predicts the resonance bifurcation observed in AFAM experiments quite well.

To calculate the right-hand side of Eq. (C.19), we must determine $\tilde{G}^{0}(\pm \omega )$ and $\sigma$, and use the relationship given by Eq. (C.17) to obtain $\tilde{G}(\pm \omega )$. To determine $\tilde{G}^{0}(\pm \omega )$, we use the rough approximations $k_n \approx k_1 n^{4}$ and $\gamma _n \approx \gamma _1 n^{2}$ in Eqs. (3.14) and (3.15) for the cantilever modal stiffness constants and damping coefficients, respectively [44]. The exact relationships depend, of course, on the cantilever shape and the experimental environment. If the driving frequency is close to the free-space cantilever resonance frequency, then large n terms contribute minimally to the calculation of $\tilde{G}^{0}$. $a_0$, used in Eq. (C.18) to obtain $\sigma$, is calculated recursively from Eq. (C.16) to order ($F_{c})^{2}$ $F_{2}$ as

$$\begin{aligned} a_{0} \approx 2\pi \tilde{G}^{0}(0)\left[{F_{00}+(2\pi \tilde{G}^{0}(\omega ))^{2}(F_{c} )^{2}F_{20} } \right]. \end{aligned}$$

(C.20)

We assume a damping coefficient $\gamma _{1} =10^{-6}$ N s m$^{-1}$, cantilever mass m $_{c} = 10^{-9}$ kg, fundamental resonance frequency $\omega _{1} = 22$ kHz, and cantilever stiffness constant $k_{1} = 0.484$ N m$^{-1}$. We assume typical force parameters $F_{c} = 10^{-7}$ N, $F_{00} = -10^{-6}$ N, $F_{10} = -1$ N m$^{-1}$, and $F_{20} = 10^{6}$ N m$^{-2}$. We point out that not all values of the force and damping parameters are found experimentally to give rise to resonance bifurcations. The same is true for the present renormalization model. The values chosen above are all within the range of parameter values typically found for AFAM operation. The specific values given above are found to generate the triple bifurcation resonance bifurcation given in Fig. 3.5, while many other parameters values do not generate resonance bifurcations in the model at all.

Fig. 3.5

Cantilever displacement amplitude plotted as a function of drive frequency for free-space cantilever oscillations (dashed curve) and for cantilever engagement with the sample surface (solid curve) [44]

The calculated amplitude of the cantilever as a function of the driving frequency $\omega$ is plotted in Fig 3.5 for both the free-space and surface-engaged cantilever using the renormalized model. The multiple free-space resonance modal peaks (dashed line) are clearly shown. Resonance bifurcation and frequency shifting is apparent in the curve for the engaged cantilever (solid line). The free-space resonance at angular frequency 22 kHz is shown to bifurcate into three resonances: at angular frequencies 19, 22, and 51 kHz. We point out that the number of bifurcation resonances predicted in the present model is quite sensitive to the values of the cantilever parameters used in the calculation. This is in agreement with the findings of Arnold et al. [52] who report both double and triple bifurcation resonances in the frequency spectra for various materials and cantilevers. Zhao et al. [53] report experimental data showing the bifurcation of a 22 kHz free-space resonance ($\omega$/2$\pi )$ into resonances at roughly 19, 31, and 60 kHz.

Renormalization can also be used to address the concern that conventional spring models (for which the cantilever has fixed cantilever spring and damping constants) fail to describe cantilever dynamics adequately, particularly at drive frequencies much larger than the fundamental cantilever resonance frequency. We begin by noting that although the form of the Fourier-space Green function in Eq. (C.10) is similar to that of the harmonic oscillator, the differential operator associated with the Green function $G^{0}(t-{t}^{\prime })=-\int {[2\pi (m_c \omega ^{2}-K(\omega )} -iG(\omega )\omega ]^{-1}\exp [i\omega (t-{t}^{\prime })]\mathrm{d}\omega$ does not actually correspond to that of a conventional harmonic oscillator due to the running of the renormalized stiffness and damping parameters with frequency. Rather, it corresponds, as shown above, to that of a superposition of harmonic oscillators due to the structure of the poles. However, Eq. (C.9) reveals that the amplitude of the cantilever at the excited frequencies ($\omega$ and its harmonics) depends solely on the Fourier-space Green function at those frequencies. Thus, if the renormalization scale $\omega _{0}$ is chosen to be sufficiently close to an integral multiple of the driving frequency of interest, we can use the renormalized values $K(\omega _0 )$ and $G(\omega _0 )$ in $\tilde{G}^{0}$ to recover approximately harmonic behavior in a neighborhood of frequencies around the renormalization scale.

To better illustrate why this is so, consider a function f(x) in a neighborhood of some point x $_{0}$. One can obtain a good approximation to f(x) in a given neighborhood by expanding f(x) about x $_{0}$ in a Taylor series and keeping only the zeroth order term provided the neighborhood is sufficiently small. As one moves the neighborhood by changing x $_{0}$, the parameter f(x $_{0})$ also changes. In the case of the renormalized cantilever parameters, if one intends to measure cantilever behavior at a frequency I $\omega$, it is necessary to determine experimentally $K(\omega _0 )$ and $G(\omega _0 )$ for a renormalization scale $\omega _{o}$ near I $\omega$ before the theoretical spring model gains predictive power. Once the parameters at the renormalization scale are determined, $\omega$ can be changed slightly without necessarily needing to determine new values.

There may be situations where the frequency at which cantilever dynamics of interest are probed is not an integral multiple of the driving frequency. In such cases, the procedure generalizes naturally to choosing a renormalization scale in the neighborhood of this probed frequency. In application, this is similar to the previously established practice of throwing away the least excited modes in the modal sum that determines the total cantilever deflection amplitude and keeping only the mode with resonance closest to the probed frequency. However, in this former practice, it would become awkward to choose a ‘most excited’ mode if the probed frequency were between two resonance frequencies. Moreover, throwing away an infinite number of modes could underestimate the total cantilever deflection even if the mode contributions are individually small.

Quantitatively, the above amounts to the following. Eq. (C.6) demonstrates that the cantilever deflection can be expressed as

$$\begin{aligned} \eta _{c} (t)=\int {\mathrm{d}t^{\prime }G^{0}\left({t-t^{\prime }} \right)} N[\eta _{c} \left({t^{\prime }} \right),t^{\prime }] \end{aligned}$$

(C.21)

where N is some function that characterizes the interactions governing cantilever dynamics. Writing $N[\eta _c (t),t]=\frac{1}{2\pi }\int {\mathrm{d}\omega \tilde{N}(\omega )e^{i\omega t}}$ and using Eqs. (C.3), (C.5), and (C.10), we obtain

$$\begin{aligned} \eta _{c} (t)=\int {\mathrm{d}\omega \tilde{G}^{0}(\omega )\tilde{N}(\omega )e^{i\omega t}=\int {\mathrm{d}\omega \frac{-1}{2\pi m_{c} \left({\omega ^{2}-\frac{K(\omega )}{m_{c} }-i\frac{G(\omega )}{m_{c} }\omega } \right)}\tilde{N}(\omega )e^{i\omega t}} }. \end{aligned}$$

(C.22)

If $\tilde{N}(\omega )$ is sharply peaked in the range $\left[{\omega _1 ,\omega _2 } \right]$ or if the signal is probed in a frequency range $\left[{\omega _1 ,\omega _2 } \right]$, then we have

$$\begin{aligned} \eta _{c} (t)&=\int \limits _{\omega _{1} }^{\omega _{2} } {\mathrm{d}\omega \frac{-1}{2\pi m_{c} \left({\omega ^{2}-\frac{K(\omega )}{m_{c} }-i\frac{G(\omega )}{m_{c} }\omega } \right)}\tilde{N}(\omega )e^{i\omega t}}\\&\approx \int \limits _{\omega _{1} }^{\omega _{2}} {\mathrm{d}\omega \frac{-1}{2\pi m_{c} \left({\omega ^{2}-\frac{K(\omega _{0} )}{m_{c} }-i\frac{G(\omega _{0} )}{m_{c} }\omega } \right)}\tilde{N}(\omega )e^{i\omega t}} \end{aligned}$$

(C.23)

for some $\omega _0 \in \left[{\omega _1 -\varepsilon _1 ,\omega _2 +\varepsilon _2 } \right]$, $0<\varepsilon _i \ll \omega _i$.

Remarkably, Eq. (C.23) shows that the (observed) cantilever behavior is very nearly identical to that of a conventional spring. We note that in most cases, the integral bounds should symmetrically include $\left[{-\omega _2 ,-\omega _1 } \right]$, but, due to the reality conditions imposed on the integrand, the bounds $\left[{-\omega _2 ,-\omega _1 } \right]$ can be accounted by keeping the original bounds and adding a term in the integrand that differs trivially from the integrand already considered. Consequently, the above result is quite general.

If the probed frequency deviates significantly from $\omega _{0}$ (that is, if in the above $\omega _0 \notin \left[{\omega _1 -\varepsilon _1 ,\omega _2 +\varepsilon _2 } \right])$, it becomes necessary to calculate $K(\omega )$ and $G(\omega )$, or $\tilde{G}^{0}$, explicitly, or to measure a new set of renormalized parameters at a new renormalization scale. Exact precision requires calculating $\tilde{G}^{0}$ using an infinite sum of terms, each term being given by Eq. (C.4). However, calculations to any desired accuracy can be obtained by truncating the sum and measuring a finite number of parameters $k_{n}$ and $\gamma _{n}$ corresponding to modes {n} closest to $\omega _{0}$ and $\omega$.

Although renormalization methods are initially applied here as a means to explain resonance bifurcation, the utility of renormalization in AFM modeling cannot be over-stated. Since d-AFM modalities are controlled completely by the cantilever driving frequency, the application of the renormalization method allows for the accurate interpretation and modeling of cantilever dynamics as conventional spring and point-mass dynamics with fixed cantilever parameters for driving frequencies sufficiently close to the renormalization scale regardless of the value of scale. This suggests that although quantitative, conventional spring models of cantilever dynamics are insufficient over a large range of frequencies, they can be ‘tuned’ to any frequency such that over a given, sufficiently smaller range they, indeed, yield accurate predictions. Over wider ranges, cantilever dynamics can be understood qualitatively as spring dynamics with frequency-dependent stiffness and damping parameters.