On the Nonlinear Stochastic Dynamics of an Atomic Force Microscope Cantilever

Kumar Singh, Aman; Ramakrishnan, Subramanian

doi:10.1115/1.4063601

Abstract

Atomic force microscopy (AFM) serves characterization and actuation in nanoscale applications. We study the stochastic dynamics of an AFM cantilever under tip-sample interactions represented by the Lennard–Jones and Morse potential energy functions. In both cases, we also study the contrasting dynamic effects of additive (external) and multiplicative (internal) noise. Moreover, for multiplicative noise, we study the two sub-cases arising from the Itô and Stratonovich interpretations of stochastic integrals. In each case, we also investigate the stochastic stability of the system by tracing the time evolution of the maximal Lyapunov exponent. Additionally, we obtain stationary probability densities for the unforced dynamics using stochastic averaging.

1 Introduction

Since its invention in the mid-1980s, the atomic force microscopy (AFM) has become the preeminent tool for imaging, measuring, and manipulating matter in the micro and nano scales [1–3]. An AFM comprises a cantilever with a sharp tip at the free end which interacts with the material specimen of interest. An AFM can operate in contact and non-contact modes. Nonlinearity and randomness are two fundamental aspects of AFM dynamics. While nonlinearity characterizes tip-sample interactions, randomness becomes important in the dynamical scale of AFM operation. Furthermore, phenomena such as stochastic resonance that arise when noise interacts with nonlinearity [4], can impact AFM design and operation. Noting that some of these aspects are recognized [5–7], we now turn to the open questions that motivate this work.

A fundamental point in stochastic dynamics is the contrasting influence of distinct types of noise on the system response. This distinction is brought into sharp relief when considering the effects of additive versus multiplicative noise. We note that while additive noise can represent the random effects of the environment (i.e., external noise), multiplicative noise—also known as state-dependent noise—better represents the random effects arising from within the system (i.e., internal noise). While these two types of noise can engender vastly different dynamics, their contrasting effects on AFM dynamics, to the best of our knowledge, remain yet unexplored.

Allied with the distinction between additive and multiplicative noise is the second fundamental point that unlike a deterministic integral a stochastic integral can be interpreted in two distinct ways—the Itô and Stratonovich interpretations. In the case of multiplicative noise, the two interpretations yield different values of the same stochastic integral. Understanding the consequences of this contrast for AFM dynamics is also an open question that motivates this paper.

Turning now to the nonlinear tip-sample interaction, typical representations include the potential corresponding to the Derjaguin–Muller–Toporov contact force as well as the Lennard–Jones (L–J) potential [8,9]. However, the Morse potential provides an analytically smoother alternative to the L–J potential, while preserving the key features of the latter. However, the role of the Morse potential in AFM dynamics is yet to be thoroughly studied. Hence it is of interest to compare the L–J and Morse potentials, for both the additive versus multiplicative contrast, as well as in the Itô versus Stratonovich interpretations.

We now focus on the specific questions studied. On the analytical front, we investigate AFM dynamics driven by multiplicative noise by deriving closed-form, stationary solutions to the F–P equation for both the L–J and Morse potentials. Stochastic averaging is used to accomplish this. On the computational front, we investigate the stochastic stability of AFM dynamics by computing the maximal Lyapunov exponent from numerical solutions of stochastic differential equations corresponding to each case. The contribution of the paper is two-fold. From the AFM viewpoint, the currently unexplored dynamics of the AFM cantilever under combined external (additive) and internal (parametric) stochastic excitations is an important aspect due to the dynamic scale and the nature of the forces involved. To our knowledge, this paper presents the first results in the AFM literature related to the contrast between the Itô and Stratonovich interpretations of a stochastic integral, the comparison between the Lennard–Jones and Morse potentials for tip-sample interactions, and related aspects. Taken together, our results are expected to motivate and serve novel AFM experiments and contribute to the design of the AFM cantilever for applications where noise is an inevitable dynamic consideration. From the viewpoint of stochastic dynamic systems, the choice between the Itô and Stratonovich interpretations for analysis of multiplicative noise remains a modeling choice and our results suggest yet another example of an important physical system where the Stratonovich interpretation is the more appropriate choice.

The rest of the paper is set as follows. In Sec. 2, we present the requisite analytic framework. Specifically, Sec. 2.1 presents the L–J and Morse potentials, as well as the stochastic differential equations representing a reduced-order model of the AFM cantilever dynamics. Section 2.2 provides a discussion of the Itô and Stratonovich interpretations of a stochastic integral, followed by explicit stochastic differential equations of the AFM cantilever motion under both interpretations. Section 2.3 presents a brief discussion of the Lyapunov exponent. Closed-form, analytical solutions of the Fokker-Planck (F–P) equations for the distinct cases are presented in Sec. 3. Numerical solutions yielding the system response and also the maximal Lyapunov exponent are presented in Sec. 4. A brief discussion and concluding remarks are presented in Sec. 5.

2 Analytic Framework

2.1 Atomic Force Microscope Cantilever Dynamics: Reduced-Order Stochastic Model.

The dynamics of an AFM cantilever can be represented by a reduced-order model involving a forced Duffing equation with hardening stiffness term as

\ddot{x} + γ \dot{x} + x + β x^{3} = f \cos (ω t) + f_{1} (\dot{x}) + f_{2} (x)

(1)

where γ is the damping coefficient, β is the coefficient of nonlinear restoring force, f, and ω are amplitude and frequency of the external periodic force, respectively,

f_{1} (\dot{x}) = - Δ \dot{x}

represents the squeeze—film damping on the cantilever, and f₂(x) models the interaction between the cantilever tip and the surface of the sample (see, for instance, Refs. [1,5,9,10]). In this work, we consider two variants for the interaction force f₂(x), arising from the L–J (f_LJ) and Morse potentials (f_M). These forces corresponding to the two potentials are defined as [8,11,12]

f_{L J} (x) = - \frac{d}{(α + x)^{2}} + \frac{δ^{6} d}{30 (α + x)^{8}}

(2a)

f_{M} (x) = - k_{0} \exp [k_{1} (x - x_{e})] [1 - \exp (k_{1} (x - x_{e}))]

(2b)

where

d = \frac{4}{16}

⁠, α is the ratio of initial equilibrium distance to static distance between tip and sample, δ is related to the ratio of Hamaker constants of the attractive and repulsive potentials, k₀, and k₁ are is the depth and width of Morse potential, respectively [10,13]. If one then considers (random) fluctuations in the damping coefficient (γ), Eq. may be rewritten as

\ddot{x} + (γ + σ ξ (t)) \dot{x} + x + β x^{3} = f \cos (ω t) + f_{1} (\dot{x}) + f_{2} (x)

(3)

where σ is the strength of the fluctuations. The random variable ξ(t) is a delta-correlated Gaussian white noise with zero-mean [14]. We note that Eq. may be rewritten as a pair of coupled first-order differential equations as

\dot{x} = y

(4a)

\dot{y} = - γ y - x - β x^{3} + f \cos (ω t) + f_{1} (\dot{x}) + f_{2} (x) - σ ξ (t) y

(4b)

Following standard practice in stochastic analysis [14], we can rewrite Eq. as stochastic differential equations in the following form:

d x = y d t

(5a)

d y = a_{s} (t, x, y) d t + b_{s} (y) d W (t)

(5b)

where

ξ (t) = \frac{d W (t)}{d t}

is a Wiener process satisfying 〈dW²〉 = dt, and

\begin{aligned} a_{s} (x, y, t) & = - γ y - x - β x^{3} \\ + f \cos (ω t) + f_{1} (\dot{x}) + f_{2} (x) \end{aligned}

(6a)

b_{s} (y) = - σ y

(6b)

The functions a_s and b_s are called the drift and diffusion coefficients, respectively. Thus, the two distinct forms of the force f₂ (corresponding to the L–J and Morse potentials) would yield two distinct stochastic differential equations. The non-dimensional parameters of the system and potentials are taken as [10]: σ = 0.5, γ = 0.6, β = 0.2, f = 0.2, ω = 5, d = 0.1, δ = 1, and Δ = 0.1, k₀ = 0.2, k₁ = 0.1, x_e = 0.01. These values shall be used to generate the results.

2.2 Stochastic Differential Equations: Itô and Stratonovich Interpretations.

We refer to the standard literature on stochastic analysis for further details of the ideas summarized in this section [14,15]. Consider a stochastic differential equation (SDE) given by

d Y = f (Y, t) d t + b (Y, t) d W_{t}

(7)

where the first term on the right-hand side (RHS) is the deterministic part and the second term represents the randomness wherein W_t denotes a Wiener process. Moreover, the Wiener process is related to the white noise process ξ(t) by the relation dW_t = ξ(t)dt. The function f(Y, t) is called the drift coefficient, and b(Y, t) is related to the diffusion coefficient by D = b²(Y, t)/2. Solving for Y, one integrates Eq. to formally write

Y_{t} = Y_{0} + \int_{0}^{t} f (Y_{s}, s) d s + \int_{0}^{t} b (Y_{s}, s) d W_{s}

(8)

Here the first integral term on the RHS can be solved using familiar integration rules of standard calculus. However, the evaluation of the second integral term is less straightforward due to stochastic term [14]. There are two distinct ways of computing the second integral based on the Stratonovich and Itô interpretations, denoted respectively by

\int_{0}^{t} b (Y_{s}, s) d W_{s} ⟹ \int_{0}^{t} b (Y_{s}, s) \circ d W_{s}

(9a)

\int_{0}^{t} b (Y_{s}, s) d W_{s} ⟹ \int_{0}^{t} b (Y_{s}, s) d W_{s}

(9b)

Consider the SDE in the Stratonovich interpretation as given by Eq. with time-independent coefficients f(Y) and b(Y),i.e.,

d Y = f (Y) d t + b (Y) d W_{t}

(10)

We note that the Stratonovich interpretation follows the rules of standard calculus. However, consistent rules of the Itô calculus provide passage between the Stratonovich and the Itô versions of an SDE. Thus the Stratonovich SDE Eq. may be mapped to its Itô counterpart by the transformation rule

f (Y) \to f (Y) - \frac{1}{2} b (Y)

\frac{\partial}{\partial Y} b (Y)

⁠. In the other direction, an Itô SDE in the Stratonovich interpretation is given by

d Y = [f (Y) - \frac{1}{2} b (Y) \frac{\partial}{\partial Y} b (Y)] d t + b (Y) d W_{t}

(11)

and its solution is same as the solution of Eq. [14,15]. This differs from Eq. only in terms of the drift coefficient. Interestingly, if the noise is additive (in which case the diffusion coefficient b is a constant) it follows from the transformation rule that the drift coefficient will be identical in both interpretations, i.e., for both Stratonovich and Itô SDEs.

Finally, we note that the SDE given by Eq. is in the Stratonovich interpretation. We obtain the corresponding Itô SDE as

d x = y d t

(12a)

d y = a_{I} (t, x, y) d t + b_{I} (y) d W (t)

(12b)

where the drift coefficient a_I can be obtained using Eq. as

b_{I} = b_{s}

(13a)

a_{I} = a_{s} - \frac{σ^{2}}{2} y

(13b)

We reiterate here that the interaction force f₂(x) for the L–J and Morse potentials are defined in Eq. .

2.3 Lyapunov Exponents.

The Lyapunov exponent is a quantitative measure of how rapidly an infinitesimally small distance between two initially close states grows over time [10,16]. For instance, consider a dynamical system given by

\frac{d x}{d t} = F (x, t)

that yields two nearby trajectories corresponding to initial states x₀ and

x_{0} + ε

⁠. The distance between these two trajectories grows as [17]

F (x_{0} + ε) - F (x_{0}) = ε e^{λ t}

(14)

The left-hand side is the distance between two initially close states after t steps, and the RHS is based on the assumption that the distance grows exponentially over time. The exponent λ measured for a long period of time (ideally t → ∞) is the Lyapunov exponent. If λ > 0, small distances grow indefinitely over time, which also implies that a stretching mechanism is in effect. On the other hand, if λ < 0, small distances do not grow indefinitely, i.e., the system settles down into a periodic trajectory eventually. In this paper, we have used the algorithm described in Ref. [17] in order to numerically compute the largest Lyapunov exponent.

2.4 Stochastic Stability.

Stochastic stability refers to the stability properties of solutions to stochastic differential equations. Stochastic stability can be characterized in multiple ways, including stability in the mean and stability in probability (see, for instance, Ref. [18]). Here we consider stability in the mean. Specifically, a solution X(t) to a stochastic differential equation is stable in the mean if [18]

lim_{t \to \infty} ⟨ | X (t) | ⟩ < α

where α is an arbitrary constant. Notably, stability in the mean can be evaluated by computing the largest Lyapunov exponent for the statistically averaged response [19]. In particular, for a stochastic dynamic system the largest Lyapunov exponent can be related to the probability density function (PDF) that characterizes the solution trajectories. Specifically, if the probability density describing the solution of the SDE Eq. is ρ(t), the corresponding Lyapunov exponent may be derived using the Khasminskii procedure, as

λ = \frac{ρ (t)}{t}

for t → ∞ (see Ref. [19]).

Comparing the relationship of the largest Lyapunov exponent to stability of solutions between deterministic and stochastic systems, we first note that for a deterministic system in stable limit cycle oscillations, the largest Lyapunov exponent converges to zero in time. For a stochastic system in stable limit cycle oscillations, the largest Lyapunov exponent similarly tends to zero in time, but now will be described as tending to zero with probability one. Stochastic systems are essentially described in a probabilistic sense and hence such limits are always described as being approached with probability one or almost surely [19–21].

3 Steady-State Behavior: Analytic Stationary Solutions to F–P Equations

An alternative to directly solving SDE is to solve for probability density functions that equivalently characterize the dynamics. For AFM dynamics (represented by the stochastic process known as the standard diffusion), it can be shown in Ref. [14] that the corresponding probability density function satisfies a (deterministic) partial differential equation known as the F–P equation. While F–P equations can be challenging for nonlinear systems (particularly those driven by multiplicative noise), their closed-form solutions are desirable. Stationary solutions to the F–P equation can illuminate a variety of phenomena including stochastic bifurcations.

3.1 F–P Equation in L–J Potential: Multiplicative Noise.

Considering the unforced system, we obtain the stationary probability density using stochastic averaging. For this, we use the binomial expansion in Eq. up to the third order, assuming that

\frac{| x |}{α} < 1

⁠. Thus, we have

\begin{aligned} f_{L J} (x) & = (\frac{δ^{6} d - d}{α}) + (\frac{2 d - 8 δ^{6} d}{α^{2}}) x \\ + (\frac{- 3 d + 36 δ^{6} d}{α^{3}}) x^{2} + (\frac{4 d - 120 δ^{6} d}{α^{4}}) x^{3} \end{aligned}

(15)

We write Eq. for f_LJ given in Eq. as

\ddot{x} + μ \dot{x} + Ω^{2} x + β_{1} x^{2} + β_{2} x^{3} + σ ξ (t) \dot{x} = (\frac{δ^{6} d - d}{α})

(16)

where the new parameters are

μ = γ + Δ, $Ω^{2} = 1 - (\frac{2 d - 8 δ^{6} d}{α^{2}})$ ⁠, $β_{1} = (\frac{3 d - 36 δ^{6} d}{α^{3}}),$ $β_{2} = β - (\frac{4 d - 120 δ^{6} d}{α^{4}})$

For stochastic averaging in Eq. , we substitute [22], x = a cos θ,

\dot{x} = - a Ω \sin θ

⁠, where θ = Ωt + ϕ(t). We obtain the evolution for amplitude as

\begin{aligned} \dot{a} & = - \frac{\sin θ}{Ω} (\frac{δ^{6} d - d}{α}) - \frac{1}{Ω} a \sin^{2} θ σ ξ (t) \\ - \frac{\sin θ}{Ω} (a μ Ω \sin θ - β_{1} a^{2} \cos^{2} θ - β_{2} a^{3} \cos^{3} θ) \end{aligned}

(17)

Since our interest is restricted to amplitude dynamics, we apply the stochastic averaging method to Eq. . Thus, we have

d a = - \frac{1}{2} μ a d t + \frac{\sqrt{3} σ a}{2 Ω} d W

(18)

This is the Stratonovich SDE. We can also find the Itô SDE in the same interpretation as

d a = - (\frac{1}{2} μ a + \frac{3}{8 Ω^{2}} σ^{2} a) d t + \frac{\sqrt{3} σ a}{2 Ω} d W

(19)

The F–P equation for Stratonovich SDE in Eq. is given by

\frac{\partial P^{S}}{\partial t} = \frac{\partial}{\partial a} (\frac{1}{2} μ a) P^{S} + \frac{3 σ^{2} a^{2}}{8 Ω^{2}} \frac{\partial^{2} P^{S}}{\partial a^{2}}

(20)

The stationary solution is obtained as

P_{s}^{S} = (\frac{4 μ Ω^{2}}{3 σ^{2}} - 1) a^{(\frac{4 μ Ω^{2}}{3 σ^{2}} - 2)}

(21)

where the prefactor

(\frac{4 μ Ω^{2}}{3 σ^{2}} - 1)

is due to the normalization. Similarly, we obtain the stationary solution for the probability density of Itô SDE as

P_{s}^{I} = (\frac{4 μ Ω^{2}}{3 σ^{2}}) a^{(\frac{4 μ Ω^{2}}{3 σ^{2}} - 1)}

(22)

3.2 F–P Equation and Probability Density in Morse Potential: Multiplicative Noise.

Following our previous discussion, for smaller x in Eq. for the Morse potential, we can expand the exponential term and obtain

f_{M} (x) = p_{0} + p_{1} x + p_{2} x^{2} + p_{3} x^{3}

(23)

where we have

\begin{aligned} p_{0} & = - k_{0} e^{- k_{1} x_{e}} + k_{0} e^{- 2 k_{1} x_{e}} \\ p_{1} & = - k_{0} k_{1} e^{- k_{1} x_{e}} + 2 k_{0} k_{1} e^{- 2 k_{1} x_{e}} \\ p_{2} & = - \frac{k_{0} k_{1}^{2}}{2} e^{- k_{1} x_{e}} + 2 k_{0} k_{1}^{2} e^{- 2 k_{1} x_{e}} \\ p_{3} & = - \frac{k_{0} k_{1}^{3}}{6} e^{- k_{1} x_{e}} + \frac{8 k_{0} k_{1}^{3}}{6} e^{- 2 k_{1} x_{e}} \end{aligned}

Proceeding with similar steps as for L–J potential, we obtain the stationary probability density for Stratonovich case as

ρ_{s}^{S} = (\frac{4 μ Ω_{M}^{2}}{3 σ^{2}} - 1) a_{M}^{(\frac{4 μ Ω_{M}^{2}}{3 σ^{2}} - 2)}

(24)

where

Ω_{M}^{2} = 1 - p_{1}

⁠. Similarly, we obtain the stationary solution for the probability density of Itô SDE as

ρ_{s}^{I} = (\frac{4 μ Ω^{2}}{3 σ^{2}}) a^{(\frac{4 μ Ω_{M}^{2}}{3 σ^{2}} - 1)}

(25)

where a_M represents amplitude in Morse potential.

3.3 F–P Equation and Probability Density: Additive Noise.

For L–J potential, we can write the evolution of amplitude in a similar form as that of Eq. , i.e.,

\begin{aligned} \dot{a} & = - \frac{\sin θ}{Ω} (\frac{δ^{6} d - d}{α}) - \frac{1}{Ω} \sin θ σ ξ (t) \\ - \frac{\sin θ}{Ω} (a μ Ω \sin θ - β_{1} a^{2} \cos^{2} θ - β_{2} a^{3} \cos^{3} θ) \end{aligned}

(26)

The stochastic averaging gives

d a = (- \frac{1}{2} μ a + \frac{σ}{2 a Ω}) d t + \frac{σ}{Ω} d W

(27)

The stationary probability density is obtained as

ρ = \frac{N_{a} Ω^{2}}{σ^{2}} a^{- \frac{Ω}{2 σ}} \exp (\frac{μ Ω^{2} a^{2}}{2 σ^{2}})

(28)

where N_a is the normalization constant. For Morse potential, we have

d a = (- \frac{1}{2} μ a + \frac{σ}{2 a Ω_{M}}) d t + \frac{σ}{Ω_{M}} d W

(29)

The stationary probability density is obtained as

ρ_{M} = \frac{N_{b} Ω_{M}^{2}}{σ^{2}} a^{- \frac{Ω_{M}}{2 σ}} \exp (\frac{μ Ω_{M}^{2} a^{2}}{2 σ^{2}})

(30)

where N_b is the normalization constant. We note that in the additive noise case, the Itô and Stratonovich SDEs are indeed identical. The probability densities for various cases are provided in Figs. 1 and 2.

Fig. 1

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The stationary probability densities for amplitude of AFM under multiplicative noise corresponding to L–J and Morse potentials, respectively

Fig. 2

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The stationary probability densities for the amplitude of AFM response under additive noise corresponding to L–J and Morse potentials, respectively

4 System Response and Stability: Numerical Solutions of Stochastic Differential Equation

We now study both the system response and stochastic stability by directly numerically solving the corresponding SDE in each case. The algorithm used is Heun’s method [23].

For the L–J potential, numerically obtain the average response (output) from the SDE under additive and multiplicative noise in Fig. 3. The displacement of the AFM cantilever in both Stratonovich and Itô cases is approximately identical. However, the displacement is higher in the additive noise case for the L–J potential. In contrast, for Morse potential, one finds that displacement is identical in additive and multiplicative noises in Fig. 4. The amplitude level of AFM in L–J potential is also higher than that of Morse potential. For clarity of periodic motions, 3D plots for the displacement, velocity, and time are given in Figs. 5–7. We can see that the system, while evolving from the same initial states, approaches different attractors for the two potentials. The nature of the nonlinearity in these potentials being different, this behavior may be expected. Also, if we restrict the analysis to weak noise and periodic force of small magnitude, these limit cycle oscillations are stable. However, if the periodic forcing amplitude is increased, these periodic orbits can become unstable, leading to the chaotic regime. Moreover, increasing the noise intensity can lead as well to chaotic dynamics [24].

Fig. 3

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The time series with L–J potential

Fig. 4

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The time series with Morse potential

Fig. 5

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Phase trajectories of AFM under multiplicative noise corresponding to the time series for L–J potential for Stratonovich and Itô cases, respectively

Fig. 6

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Phase trajectories of AFM under multiplicative noise corresponding to the time series for Morse potential for Stratonovich and Itô cases, respectively

Fig. 7

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Phase trajectories of AFM under additive noise corresponding to L–J and Morse potentials, respectively

Turning now to stability, we present the time evolution of the maximal Lyapunov exponents (LLE) for L–J and Morse potentials, respectively in Figs. 8 and 9. We see that the asymptotic values of LLEs are zero, hence, we have stable limit cycle oscillations in Figs. 3 and 4, respectively. It is also possible to get noise-induced chaotic motion in AFM but this lies outside the scope of this paper [1].

Fig. 8

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The largest Lyapunov exponent with L–J potential

Fig. 9

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The largest Lyapunov exponent with Morse potential

Next, we turn to the stationary PDF for multiplicative noise, presented in Fig. 1, and for additive noise in Fig. 2. In the Stratonovich interpretation, the stationary PDF for L–J potential has a smaller value than the stationary PDF in Morse potential. However, in Itô interpretation, this behavior is the opposite, i.e., stationary PDF in L–J potential is greater in value than the stationary PDF in Morse potential (see Fig. 1). Interestingly, for each type of potential, the stationary probability densities have contrasting behaviors for the Stratonovich and Itô interpretations. In low amplitude region, the Stratonovich interpretation has higher PDF values than Itô. In larger amplitude region, the Itô interpretation yields a PDF larger in value than in the Stratonovich case.

Also, a damped deterministic system, in the absence of external forcing must converge to stable fixed point states (oscillation amplitude zero) due to the inevitable loss of energy. Correspondingly, for a stochastic system, the most probable values of amplitude must be zero as there are no oscillations in the fixed point states. For additive noise, the plots in Fig. 2 affirm this conclusion. However, it is observed from the plots in Fig. 1 that, for the multiplicative noise case (for both L–J and Morse potentials), this conclusion is supported only by the stationary densities obtained from the Stratonovich interpretation. This suggests that the latter interpretation might be the more appropriate model for AFM dynamics driven by multiplicative noise.

Under additive noise, the Stratonovich and Itô interpretation have no distinction. In this case, the PDF in L–J potential is higher in value than for the Morse potential. However, this difference in the stationary PDF is quite small (see Fig. 2).

5 Discussion and Conclusion

Comparing the effects of additive versus multiplicative noise on AFM dynamics, the results show that in the L–J potential case, additive noise yields a system response of higher magnitude; little difference exists between the two types of noise under the Morse potential. By way of the Itô versus Stratonovich contrast, only the stationary densities in the Stratonovich version indicate zero-amplitude oscillations with probability one, in the fixed state. This suggests that the Stratonovich interpretation might be the more appropriate choice for modeling AFM dynamics. All cases yielded stable, limit cycle oscillations. However, in the case of additive noise and the L–J potential the LLE shifted from positive values in the transient phase to ultimately converge to values indicating stability. Overall, the results advance our understanding of stochastic AFM dynamics in the presence of multiplicative noise. They clarify that environmental and internal noise result in distinct dynamic behaviors, also depending on the specifics of the tip-surface interaction. From the stochastic analysis viewpoint, the results indicate the AFM to be yet another physical system where the Stratonovich interpretation yields physically meaningful results. Finally, the need for experiments to validate the reported results cannot be overstated. Noise is an important aspect of AFM dynamics and ingeniously devised experiments are essential to validate and build upon theoretical results. We trust that the reported results can serve to motivate such experiments and conclude with the hope that the paper contributes to future AFM research.

Footnote

1

Paper presented at the 2023 Modeling, Estimation, and Control Conference (MECC 2023), Lake Tahoe, NV, Oct. 2–5. Paper No. MECC2023-106.

Acknowledgment

The authors would like to acknowledge critical and helpful comments from the anonymous referees.

Funding Data

This work was partially supported by the U.S. National Science Foundation (NSF Award: CMMI-2140405).

Conflict of Interest

There are no conflicts of interest.

Data Availability Statement

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.

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On the Nonlinear Stochastic Dynamics of an Atomic Force Microscope Cantilever^¹

Abstract

1 Introduction

2 Analytic Framework

2.1 Atomic Force Microscope Cantilever Dynamics: Reduced-Order Stochastic Model.

2.2 Stochastic Differential Equations: Itô and Stratonovich Interpretations.

2.3 Lyapunov Exponents.

2.4 Stochastic Stability.

3 Steady-State Behavior: Analytic Stationary Solutions to F–P Equations

3.1 F–P Equation in L–J Potential: Multiplicative Noise.

3.2 F–P Equation and Probability Density in Morse Potential: Multiplicative Noise.

3.3 F–P Equation and Probability Density: Additive Noise.

4 System Response and Stability: Numerical Solutions of Stochastic Differential Equation

5 Discussion and Conclusion

Footnote

Acknowledgment

Funding Data

Conflict of Interest

Data Availability Statement

References

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On the Nonlinear Stochastic Dynamics of an Atomic Force Microscope Cantilever1

Abstract

1 Introduction

2 Analytic Framework

2.1 Atomic Force Microscope Cantilever Dynamics: Reduced-Order Stochastic Model.

2.2 Stochastic Differential Equations: Itô and Stratonovich Interpretations.

2.3 Lyapunov Exponents.

2.4 Stochastic Stability.

3 Steady-State Behavior: Analytic Stationary Solutions to F–P Equations

3.1 F–P Equation in L–J Potential: Multiplicative Noise.

3.2 F–P Equation and Probability Density in Morse Potential: Multiplicative Noise.

3.3 F–P Equation and Probability Density: Additive Noise.

4 System Response and Stability: Numerical Solutions of Stochastic Differential Equation

5 Discussion and Conclusion

Footnote

Acknowledgment

Funding Data

Conflict of Interest

Data Availability Statement

References

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Cited By

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On the Nonlinear Stochastic Dynamics of an Atomic Force Microscope Cantilever^¹