Yoni Mehlman

Scanning Probe Microscopy Force Reconstruction from Non-Equilibrium Dynamics Thesis Submitted in Partial Fulfillment of the Requirements of the Jay and Jeanie Schottenstein Honors Program

Yeshiva College Yeshiva University May 2014

Yoni Mehlman Mentor: Dr. Fredy Zypman, Physics

Abstract Atomic Force Spectroscopy generates a voltage time trace which contains physical information of the sample under study. This information is hidden in the trace and a central challenge is the recovery of a force vs. separation curve, which characterizes the physical and chemical properties of the sample. Often theoretical approaches approximate the motion of the AFM cantilever as a mass-spring system. These models assume that the motion of the cantilever is either quasi-static or dominated by a single mode. However, under relevant realistic measuring conditions, the cantilever is likely to accelerate appreciably and its motion may become a sum of many modes. Furthermore, these models (and others that go beyond a single mode) require that the voltage be related to deflection when, in reality, the voltage relates to the slope of the cantilever end. In this paper we explore beyond these constrains by considering the dynamics of a flexible cantilever satisfying the Euler-Bernoulli equation including an appropriate boundary condition that interprets the voltage as a slope. With this explicit boundary condition in conjunction with standard boundary conditions we are able to calculate the force in the snap-tocontact region. The snap-to-contact approach may contain high velocities and acceleration events. To the best of our understanding, the model and solutions we propose here are based on a physically sound basis. A central result of this thesis is the assessment of the accuracy of previous and current models. We show that the accuracy is related to a single constant α which characterizes the curvature of the snap-to-contact in relation to the frequency of the slowest mode of oscillations.

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Table of Contents §1 Introduction ................................................................................................................................ 3 §2 Numerical Euler-Bernoulli (NEB) Method of Force Curve Construction............................... 12 §2.1 The First Three Boundary Conditions .............................................................................. 15 §2.2 The Fourth Boundary Condition ....................................................................................... 16 §2.3 Numerical Technique to Solve Euler-Bernoulli Equation ................................................ 20 §2.4 Dimensionless Numerical Solution................................................................................... 28 §3 Theoretical Limits of the Spring Models of the AFM Cantilever ........................................... 30 §3.1 Force Characteristics of Proportional Voltage Outputs .................................................... 31 §3.2 Proportionality Conditions of Mass-Spring Model........................................................... 35 §3.3 Limits and Breakdown of DSM ........................................................................................ 44 §4 Conclusion ............................................................................................................................... 47 References ..................................................................................................................................... 48 Appendix A: Implementation of Numerical Algorithm................................................................ 49

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§1 Introduction Of the many imaging techniques available today, the Atomic Force Microscope (AFM) stands out as the most commonly used Scanning Probe Microscope in science research and technological applications. An AFM is typically used to produce topographical images with nano-metric resolution. This feature is a big advantage over many other microscopes (including ultra-high resolution microscopes) which cannot detect topography. An AFM typically produces 1,000,000X magnification, outshining optical microscopes which are limited at about 1,000X magnification and Scanning Electron Microscopes, which normally produce 100,000X magnification. [1] Furthermore, an AFM can probe nearly any type of material and can be used in ambient as well as liquid conditions.[2] With the growth of the nanotechnology industry and the constant push to scale down the size of a computer, imaging at the nanoscale is becoming increasingly important. Most relevant to our work is a slightly different use of the AFM known as AFM spectroscopy. Not only is the AFM capable of imaging, but it is also used to measure interatomic forces. While the goal in generic microscopy is to produce an image, the goal when probing forces is to produce a force vs. separation curve. This curve displays the force which the sample exerts on the AFM probe as a function of separation between the probe and the sample, as seen in Figure 2. Typical forces are on the scale of pico-Newtons and the separation in nanometers.[3,4] In essence, a force curve displays the close range forces the sample will exert on its surroundings. However, the AFM output relates to the geometric properties of the probe rather than the force. Therefore, a key challenge in force curve construction is inferring the force from the geometry. This thesis will attempt to assess the limitations of the current methods used for this conversion and will present a method which can go beyond the limits of these techniques. 3

Head

Platform

Figure 1. Atomic Force Microscope (Yeshiva University, Belfer Hall C08)

Figure 2. Sample force vs. separation curve.

To appreciate this problem it is important to understand how an AFM works. Inside of an AFM is a bendable cantilever, with a very fine tip at the end, as shown in Figure 3. Typically, the length of the cantilever is ~100 μm and the tip has a radius of curvature of a few nanometers at its apex. One end of the cantilever is fixed to the body of the AFM structure whereas the end

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with the tip is free-hanging. The tip interacts with the sample causing the cantilever to bend. Because the tip is so fine, it is able to capture detail at the nanoscale.

Figure 3. AFM cantilever and cantilever tip.[5]

To amplify the bending of the cantilever, a laser which is very far away shines a beam on the end of the cantilever, as shown in Figure 4. This reflects off of the cantilever and hits a photodetector system which converts the position of the beam upon the photodetectors to a voltage. This voltage is sampled at a very high frequency and is sent to the user’s computer. As the cantilever deflects, the slope at the end of the cantilever changes, slightly altering the angle of incidence of the laser beam. Because the photodetector system is placed very far away from the cantilever, any small change in angle will produce a large change in position and, by extension, a measurable change in voltage.[3,6,7,8]

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Photodetector Laser Tip (Probe)

Cantilever

z

Piezoelectric stage

x y

Figure 4. Diagram of the core components of an AFM. The piezoelectric stage moves the sample as the cantilever bends in response. The laser reflects off of the cantilever and the change in geometry of the cantilever is output as a voltage by the photodetector system.[9]

This design can be harnessed for both imaging and spectroscopy. For imaging, the simplest technique, known as contact mode, is when the sample is moved close enough to the cantilever tip that the tip touches the sample. The tip is then dragged along the sample traversing, as it were, the hills and valleys associated with the sample. The differences in topography bend the cantilever up or down, which is ultimately recorded as a voltage. The sample is moved around slowly in the x-y plane using a voltage-controlled piezoelectric base, raster scanning the surface. The voltage is recorded at a high frequency to produce a high resolution image. After probing the sample, the voltage is converted to a height, and each data point in the voltage is mapped to the x-y position at which it was recorded.[10] 6

On the other hand, in force curve construction, the user analyzes the voltage output before it makes contact. Once contact is made, adhesive forces dominate and the force characteristics are no longer of interest. Prior to contact, the cantilever is either attracted to or repelled by the sample, providing information about the force characteristics of the sample. To measure this, a voltage is applied to the piezoelectric stand upon which the sample rests. This applied voltage causes the piezoelectric to move the sample up or down in very small amounts. Therefore, the user can carefully and accurately control the vertical movement and position of the sample. By very slowly moving the sample closer to the cantilever tip, the separation is changed and the force is altered accordingly. The change in force causes a change in the bending of the cantilever and is recorded by the voltage output of the photodetector system, as shown in Figure 5. The sample is moved closer and closer to the tip until eventually contact is made and adhesive forces take over.[3,6,10,11] As stated above, one major challenge in force curve construction is converting the raw voltage output to force. Since the voltage reflects the geometry of the cantilever, the topography can be easily inferred from the voltage. In other words, when imaging with an AFM, the conversion from voltage to topography is very natural since both are rooted in geometrical properties. Although force curves are a natural extension to AFM imaging since the AFM operates through the forces the sample exerts on the tip, the final voltage output doesn’t immediately lend itself to being interpreted as a force.

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Figure 5. Typical voltage data collected during a force measurement.

To address this challenge, there are three different techniques which are used to convert voltage to force. Because the dynamics of a bending cantilever are quite complex, the first two techniques calculate the force by resorting to simpler models of the cantilever’s motion. The third technique uses an accurate model of cantilever dynamics. However, it resorts to questionable math when using the model to calculate force. The three methods are: 1) Static-Spring Model (SSM). A very simple and intuitive model is to view the cantilever as a simple spring. For example, imagine a diver jumping on a diving board. The board acts as a sort of spring which resists changes from equilibrium. While the spring may not be massless, it is easiest to treat the cantilever as a massless spring which will correspond to a quasi-static mass-spring system. In this case any inertial properties and acceleration of the spring can be disregarded, and the force the sample exerts on the tip is given by, (1)

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where

is the deflection of the tip of the cantilever from its equilibrium position

and k is the effective spring constant of the cantilever.[6] This equation is simply the equation of an external force on a spring in equilibrium. This is the simplest and most common technique employed. We refer to this model as the static-spring model (SSM). 2) Dynamic-spring Model (DSM). The second most common technique treats the cantilever as a mass-spring system, moving beyond the static approximation by incorporating the mass and motion of the cantilever. Since the mass-spring system is no longer assumed to be static the external force is given by, (2) where m is the effective mass of the cantilever in a mass-spring system. Since this technique incorporates the motion of the cantilever, we call this the Dynamic-Spring Model (DSM).[6,12,13] The DSM and SSM methods, by simplifying the underlying physics of cantilever motion, enable the AFM user to very easily and quickly calculate the force from the voltage output. The user first converts the voltage to deflection (u) by assuming that they are proportional and then simply substitutes that result into Equations (1) or (2). 3) Euler-Bernoulli Model. The third technique is to use a partial differential equation, known as the Euler-Bernoulli equation, which describes the vibrations and shape of a rigid bar to calculate the force. The advantage of this approach is that it correctly describes the dynamics of the cantilever. However, unlike the previous techniques, there is no known analytical expression which can calculate the force from the voltage given the boundary conditions associated with AFM spectroscopy. Generally, 9

an approximation is made using a spectral approach.[6,14,15] However, this method is highly suspect. Due to its questionability and complexity, this method is the least commonly used. In §2, we will introduce a new method to convert the voltage to force. Using the Euler-Bernoulli model to accurately capture the dynamics of the cantilevers motion, we will solve the differential equation using a direct numerical method, avoiding the questionable assumptions of the spectral method. Furthermore, the use of this method will allow us to introduce appropriate boundary conditions for the hanging end of the cantilever, something which could not be captured by the spectral methods. We call our technique the Numerical Euler-Bernoulli (NEB) method. This technique will provide a straightforward and physically accurate method of calculating the force from the voltage. The SSM and DSM methods make one critical assumption: the cantilever can be modeled as a simple harmonic oscillator. While this is true within certain limitations, it is not correct in general. In particular, these methods are likely to fail in a region of the voltage output known as the snap-to-contact.[6,12] Typical output can be divided into three regions. In the first region, the voltage is nearly constant. The constant voltage implies that the shape of the cantilever is unchanging and the cantilever is static. In this region, the SSM can be assumed to correctly characterize the physics of the cantilever tip. The second region is known as the “snap-tocontact.” It is given this name because it is generally assumed that it is within this region that the cantilever passes a point of no return and accelerates toward the sample ending in contact. The third region is when the cantilever begins to bend upwards, and is typically assumed to be in contact throughout. Since we are interested in studying the force before contact is made and we

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can only assume that the tip and sample are not in contact prior to Region III, we will focus only on the first two regions.

Region I

Region III Region II

Figure 6. Plot of typical AFM voltage output, highlighting the three key regions.

The “snap-to-contact” is a potentially problematic region when using either type of spring model. Since it begins to accelerate quickly, even if one were to include the acceleration by using the DSM, the cantilever may no longer behave like a simple mass-spring system. In this region, it is likely that only a complete physical description of the cantilever as given by the Euler-Bernoulli equation will accurately capture its motion. Furthermore, calculating the force during the snapto-contact is of great interest because it can provide the user with information about sample forces at the closest range. Therefore, a critical challenge in force curve construction is the accurate calculation of force during the snap-to-contact. Therefore, in §3 we develop a theory for predicting when the behavior of the cantilever during the snap-to-contact can no longer be modeled by either SSM or DSM. We show that instead, the NEB method can and must be used.

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§2 Numerical Euler-Bernoulli (NEB) Method of Force Curve Construction The motion of a cantilever is governed by,[14,15,16] (3) where u(x,t) is the vertical displacement of the cantilever at (horizontal) position x and time t, as shown in Figure 7. c is a constant determined by the internal properties of the cantilever equal to, (4) where E is Young’s modulus, ρ is the density, and R2 is the radius of gyration of the cantilever.1 Eq. (3) is known as the Euler-Bernoulli equation. It is correct for all vibrating bars with small deflections, and, therefore, accurately models the motion of an AFM cantilever. Typical cantilever lengths are on the order of 100 μm whereas even extreme deflections are near 10 nm. Thus, the deflections are normally less than 1/10,000 of the cantilever’s length, justifying the assumptions of the Euler-Bernoulli equation. Hence, Eq. (3) is the best model for calculating the external force on the cantilever tip.

u x Cantilever Figure 7. Snapshot of a cantilever bending in x-u plane at a given instant t.

Solving this differential equation given certain boundary conditions describes the time-dependent shape of the cantilever, u(x,t). Once this is known, the external force at the tip is given by:[17]

1

For a clear treatment of calculating the radius of gyration see Engineering Mathematics by J. Bird pages 505-512 (Oxford: Newnes, 2010).

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|

(5)

where A is the cross sectional area and L the length of the cantilever. Since our goal is to determine the force as a function of separation, and the signal output is given in terms of time, we must convert Fext(t) to Fext(s), where s is the separation (see Figure 8). The separation can be calculated by the following equation: (6) where

is the speed of the piezoelectric and y0 is the initial position of the sample (see Figure

8).2 We use the vertical displacement at the end of the cantilever to calculate the separation because the tip of the cantilever interacts with the sample. Since all constants are known, if we solve for u(x,t), then we can find Fext(t) and s(t), giving a force vs. separation curve. Hence, the bottleneck of constructing a force curve reduces to solving Eq. (3). Because finding a solution to this equation is difficult for the given boundary conditions, users resort to the SSM or DSM to calculate the force. It can be readily seen that these models are based upon certain conditions which reduce the Euler-Bernoulli equation to that of a simple harmonic oscillator, justifying the use of the SSM or DSM when and only when these conditions are met. The most common model employed, SSM, assumes that the cantilever is quasi-static.[19] Under this condition, the acceleration is zero so Eq. (3) simplifies to, (7)

2

Determining the initial position of the piezoelectric and consequently the separation is a well-known problem in AFM measurement science.[6] While an arbitrary initial separation can be used to investigate how the force changes with separation, to find the actual separation this value must be known. We will propose a method to estimate this value in our implementation of the NEB method. Refer to Appendix A.

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y

u(L)

x

Fexternal

s

v

vt y0 z Figure 8. Diagram of interaction between moving sample and deflecting AFM cantilever.[18]

Since the tip of the cantilever (x=L) is under the influence of a time varying external force F(t), the solution u(x,t) which satisfies both Equations (5) and (7) is, (8) From Eq. (8) the deflection at the tip equals, (9) Since the force is the unknown, using Eq. (9) to solve for F(t) yields a very simple formula for the external force. We shall write this equation in a telling form: (10) where k, the effective spring constant, is, (11) Notice that Eq. (10) is the equation of an external force acting on a static spring. Therefore, when the motion of the tip is quasi-static, modeling the AFM cantilever as a simple spring is justified. 14

However, in the snap-to-contact region, where the cantilever begins to rapidly deflect, the assumptions of the spring-model may no longer hold true.[6,12] First, the possible large acceleration may imply that the cantilever is no longer in instantaneous equilibrium and can no longer be viewed as static. Second, multiple modes of oscillations may begin to dominate, invalidating even the DSM, which captures only a single mode of oscillation (since it models the cantilever as a single spring with a single mass). Therefore, to calculate the external force when the spring-model conditions are violated or suspect, we must use the general non-static EulerBernoulli equation as given by Eq. (3). Once u(x,t) is known, the force can be easily computed using Eq. (5).

§2.1 The First Three Boundary Conditions To solve Eq. (3), we need 4 boundary conditions and 2 initial conditions. The first 3 boundary conditions are standard for any cantilever which is fixed at one end and hanging at the other.[12] The first two boundary conditions are: (12) and |

(13)

which result from the fact that one end (which we place at x=0) is fixed and the cantilever cannot be infinitely bent. The third boundary condition, at the hanging end of the cantilever, is: |

(14)

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Since the vertical component of the external force dominates any horizontal component of force, the torque at the tip of the cantilever is effectively zero. The torque is related to the second spatial derivative, which is set to 0 at x=L, as seen in Eq. (14).

§2.2 The Fourth Boundary Condition We must define one more boundary condition at the hanging end to find a unique solution to Eq. (3). This boundary condition is related to the voltage output and is ultimately how the output will determine the force. Typically, the voltage is assumed to be proportional to the deflection of the cantilever tip. While this is true under the conditions of the SSM (and, by extension, the DSM), this is not true generally.[6] In fact, the voltage may not even have a one-to-one relationship with the deflection. As seen in Figure 9, two cantilever shapes which will result in the same voltage have different deflections. Therefore, as we move beyond quasi-static conditions, the deflection is no longer a function of voltage. (nm) 2

1

0 0

25

50

75

100

(μm)

Figure 9. Two possible deflections for an AFM cantilever. Different voltage data can results in the same slope at the end of the cantilever but drastically different deflections. The laser will reflect at same angle, inducing the same voltage on the photodetector system. The vertical deflections are extremely exaggerated.

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In reality, the voltage is related to the slope of the end of the cantilever. As seen in Figure 4, the laser beam reflects off of the end of the cantilever. The slope of the cantilever changes the angle of reflection of the laser, causing a shift in where the light beam hits the photodetector system. The shift of the light spot on the photodetectors is then converted to a voltage, giving the output. On the other hand, the deflection’s effect on the voltage is negligible for two reasons. First, since the deflections of the cantilever are very small and the laser beam is wide, we can assume that the shift in the position of incidence due to the deflection is negligible. Additionally, since the distance of the photodetector from the cantilever is much greater than the deflection of the cantilever, the vertical distance the light beam must travel can be treated as a constant. Consequently, deflection alone does not shift the location at which the beam hits the photodetector. Therefore, we only consider the large effect of the slope on the voltage output. The fourth boundary condition is given by, |

(15)

where f(V) is some function which maps voltage to slope. To use this boundary condition, one must find this function. This could be done by compiling a table from experimentation. In practice, however, this is not necessary. Since the change in slope is small, the change in position of where the laser beam hits the photodetector can be approximated by a linear relationship to the change in slope. By extension, the voltage is proportional to the slope. This can be easily seen by considering the diagram in

Figure 10 of the laser beam reflecting off of the cantilever.

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Laser

Photodetector p

α

h

Cantilever Figure 10. Diagram of laser beam reflecting off of AFM cantilever tip and hitting the photodetector.

The variable p is the position at which the laser beam hits the photodetector. This is given by, (

)

(16)

where h is the distance between the undisturbed cantilever and the photodetector, utip is the deflection of the tip (up is the positive direction), and α is the angle of incidence of the beam on the photodetector, which is linearly related to the slope of the tip of the cantilever. Since , Eq. (16) can be approximated as, (17) This form of the equation justifies the argument that a change in deflection will not have an effect on the position at which the beam hits the photodetector. Instead, the voltage output is completely captured by the angle of incidence α. Clearly Eq. (17) is a non-linear relationship between p and α. However, since changes in the slope of the cantilever tip are small, changes in α are small, and a linear approximation of Eq. (17) is justified. Let the angle of incidence and position of the undisturbed cantilever be defined 18

as α0 and p0. α is given by

, where

where

. Since

is a small change in angle, and is very small, this can be written as, |

Since

|

is a constant,

is proportional to

(18) . Ordinarily, the photodetector is

calibrated that the initial voltage is equal to zero. With this as the reference point, p 0 corresponds to the origin, resulting in the following linear approximation for p, |

(19)

This equation fully justifies the assumption that the relationship between p and α is linear. The position is then converted linearly to voltage. Consequently, for small changes in the angle of incidence, the voltage will be directly proportional to the slope of the cantilever end. Because of this proportionality, the manufacturer provided ratio between voltage and deflection can be exploited to quickly find the ratio between voltage and slope, which reflects the true physics of the cantilever system. The proportionality constant between deflection and voltage, r, is based upon quasi-static conditions, when a relationship between voltage and deflection is justified.[20,21] This is a consequence of the fact that the deflection of a static cantilever is uniquely related to its slope. This can be easily seen by differentiating Eq. (8) with respect to x at L and comparing to the deflection at x = L (Eq. (9)): |

(20)

Since the deflection is proportional to slope and the slope to voltage, the manufacturer is justified in assuming the static deflection is proportional to the voltage. Essentially, the proportionality constant the manufacturer provides (r) is a product of the proportionality factor between voltage 19

and slope, which is generally true, and the proportionality factor between slope and deflection, which is only true under static conditions. Mathematically, (

)

(21)

where γ is the general proportionality factor between voltage and slope. From this equation it is readily seen that

. However, once the cantilever is no longer assumed to be static, the

deflection is no longer a function of slope: Eq. (20) no longer holds. Accordingly, the deflection is no longer related to the voltage, as shown in Figure 9. Nonetheless, because the user knows r, γ can be easily derived: (22) This result allows us to quickly adapt the information provided by the manufacturer to input the fourth boundary condition and ultimately solve Eq. (3) without any additional experimentation. Therefore, in practice, the fourth boundary condition is, |

(23)

This is used because it captures the general relationship between the voltage and the cantilever geometry with a simple and easily derived linear expression for f(V).

§2.3 Numerical Technique to Solve Euler-Bernoulli Equation There is no known exact analytical solution to Eq. (3) given the boundary condition in Eq. (23). One technique which is sometimes used is the spectral method, which expresses the solution as a sum of normal modes for a cantilever with, for example, one fixed and one free end. [14,15] The problem with this method is that the cantilever’s free end is, in fact, under an external force. Therefore, there is no assurance that a sum of normal modes to a different problem can model the 20

motion of the AFM cantilever. Instead, we propose using a more direct numerical technique for solving the Euler-Bernoulli equation. Of the common techniques available, we use the method of finite-differences as it is the most straightforward method given our problem. This technique is a fairly common numerical method for solving differential equations, but we shall review the details here, particularly as they relate to our problem.[22] The implementation of the finite difference method for our problem can be broken down into six steps as shown in Figure 11. We begin by discretizing space and time. We divide time into M sections (M+1 total points). Δt will be given by the (N+1 total points) so

. Similarly, we divide the cantilever along its length into N sections . We must choose sufficiently large values of N and M to fully

capture all the detail of the cantilever’s shape and to accurately approximate the derivatives. M and N will ultimately depend on the total time and cantilever length. Let us describe this discretization by

where

and

with n and m going

from 0 to N and 0 to M respectively. Once x and t have been discretized, we can approximate the differentials using finite difference approximations. This must be done for Eq. (3) and for Equations (13)-(14). One must be careful when selecting finite difference formulas to use forward and backward finite differences at the boundaries and central differences at the midpoints.

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Discretize u(x,t) in space and time -> u(xn,tm) -> unm

Express differential equation and boundary conditions as finite difference approximations over the unm grid

Express differential equation as a system of linear equations, one for each xn

Include boundary conditions amongst system of equations

Solve this system of equations at time tm using information from previous times

Repeat process for time tm+1

Figure 11. Diagram of steps for solving a partial differential equation using the finite-differences method

For the second-order time derivative (the left-hand-side of Eq. (3)), we will always use a backwards finite difference formula, inferring the current time’s solution from past times’ solutions. A second-order differential requires at least 3 terms in a finite difference formula. Therefore, we need at least two known initial states to begin the recursion. We set the initial conditions using the static solution of a cantilever’s shape under an external force applied to its 22

hanging end, Eq. (8). This assumption is justified because at the start of most experiments the force between the sample and the tip is negligible and the cantilever is certainly quasi-static. To proceed, we must find

(see Eq. (8)). This is easily computed using the voltage output V.

Since the cantilever is assumed to be quasi-static,

, and (24)

Using Eq. (24) in Eq. (8) gives us the following expression for the quasi-static shape of the cantilever in terms of discretized position and time variables: (25) Since we need two initial conditions, we use

as the solution at t0 and t1. In

theory, one can start with w total initial states, which will then permit a higher order finitedifference formula to approximate the second-derivative (w+1 terms). Therefore, we will express our algorithm generally, using w initial states. In practice, accuracy can be increased by decreasing

, allowing one to use a first-order finite difference formula (w=2) for any desired

accuracy. Furthermore, Eq. (25) is more accurate for smaller

rather than higher-order finite

differences. For the approximation of the fourth-order spatial derivative (right-hand-side of Eq. (3)), we are limited to using finite difference formulas which contain no more terms than the number of points which we are using to estimate the shape of the cantilever (N+1) and must use a minimum of five terms. For typical experiments, we found that using the minimal number of points sufficed to estimate the derivatives. Going beyond this number would be an attempt to achieve accuracy well beyond the uncertainties associated with the measurement of voltage and the cantilever constants. 23

The discretized approximation of Eq. (3) for a given unm is,

∑ where

and

∑

(26)

are the coefficients in the finite difference formulas, w determines the number

of terms used in the finite difference formula for time, and Qh determines the number of terms used for the finite difference in space. We include a variable h which determines whether the finite difference is forward, backward, or central (or somewhere in between). Q and α are indexed by h since the coefficients and number of points used may vary based on the type of finite difference formula used. For example, when using first-order finite difference formulas, h=2 corresponds to the central finite difference. α2,0 will be the coefficient for un-2,m. Consider, as an example, an approximation using all first-order finite differences. Suppose N = 9. Choose n = 4 and m = 3. A fourth derivative is approximated by five terms, so the right hand side of Eq. (26) for u4,3 is a central difference formula. Accordingly, h = 2, Qh = 5, and the coefficients are given by:

. The second derivative in time is always

modeled using a backwards finite difference formula. For this example, a first order approximation has 3 terms (w=3) and (

(26) reads:

and

. Putting this together, Eq.

)

. Using Eq. (26) we can write an equation for each unm. If we group all points at time tm on one side of the equation, we will have all of our unknowns on one side of the equation and a constant on the other. Rearranging Eq. (26) we get:

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(

)

∑

∑

(27)

Notice that the right hand side begins at i=1 as opposed to 0, since the i=0 term corresponds to unm and has been moved to the left hand side. Since the right hand side only includes previous times where the solution is known, the entire side is known. All unknowns remain on the left hand side. Eq. (27) for the example above is: ( )

(

)

(

).

Doing this for all n’s at a given time tm will give us a system of linear equations describing the vertical position of the cantilever at any position xn at a set time tm. These can be written as a matrix and quickly solved using a computer. Afterwards, we can let m → m+1 and repeat the process. Thereby, we sequentially solve this system of equations, iterating in time, up to tM. The entries of the matrix (call it E) which represents the left hand side of Eq. (27) can be written as follows: (28)

Each row, n, is derived from the equations for unm (Equations (26) and (27)). p, which indexes the column, indicates what coefficient to place at position Enp. Both n and p go from 0 to N.3 is the Kronecker delta. Therefore, only when n=p, i.e. when we are considering the coefficient in front of unm in Eq. (27), does an additional time term appear. For all

3

To simplify the notation, we will call the first row and column (0,0).

25

, the coefficient is

either zero (when the position is too far away from unm that the term upm will not appear in the finite difference formula) or is some value determined by the coefficient in the spatial finite difference formula. The spatial value is determined by . Otherwise,

. If

or

,

, where h is determined by the type of finite-difference

used in row n. For the example above, again consider n=4. If p < 2 or p > 6, the entry will be zero since this term is not represented in the equation for u4,3. The rest of the entries will be: ,

,

,

, and

.4

The matrix E multiplied by the variable vector U = [un] is equal to the solution vector. The solution vector is simply defined by the right hand side of Eq. (27) for all n’s from 0 to N.

∑

(29)

In our example this will be:

(

4

)

A sample 10x10 matrix E:

(

)

26

All of these values are known from the solutions at t2 and t1 which will have already been calculated in an actual implementation of this algorithm. Equations (28) and (29), the system of equations, written in matrix form, is: (30) To solve the equation for the AFM system, the boundary conditions must be included in E. Since we have four boundary conditions, two at each boundary, we must replace the first and last two rows, corresponding to the solutions at the two ends of the cantilever, with equations describing the boundary conditions. The first boundary condition (Eq. (12)) replaces row zero with a 1 followed by all 0s. The first entry in S must also be replaced by a 0. The second boundary condition (Eq. (13)) replaces the second row in E with coefficients of a forward finite difference formula which approximates the first derivative of u at x=0 (or in our discretized version x0). The second row in S is replaced by a 0. The third boundary condition (Eq. (23)) replaces the second to last row in E with the coefficients of a backward finite difference formula for the first derivative of u at x=L (xN). The second to last row of S is replaced by

. Finally, the last

boundary condition (Eq. (14)) replaces the last row in E with the coefficients of a backward finite difference formula for the second derivative of u at xN. The last row of S equals 0. With these replacements made, solving Eq. (30) for U gives an approximate numerical solution for u at all xn’s of a given time tm. Using these results to calculate the external force given by Eq. (5) and the separation given by Eq. (6) at every time tm and plotting parametrically gives the desired force vs. separation curve.

27

§2.4 Dimensionless Numerical Solution In practical implementations of this algorithm, greater numerical stability is attained by redefining the parameters to be dimensionless. The quantities take values close to unity avoiding the multiplication of very large and very small numbers. We begin by defining the vertical displacement u(x,t) relative to the length of the cantilever, L, by the new variable, (31) Similarly, x is defined relative to L: ̂

(32)

We also define dimensionless time in units of the period of the of the slowest mode of oscillation of the cantilever, T; thus, (33) On substitution of Equations (31) - (33) into Eq. (3) we get:

̂

(34)

is a dimensionless quantity. Therefore, Eq. (34) is a dimensionless form of Eq. (3). When solving Eq. (3) with the boundary conditions for one fixed and one free end, the first excited frequency is given by, (35) Because

, rearranging the above equation in terms of T2 gives, (36)

Substituting Eq. (36) into the dimensionless constant in Eq. (34) gives, 28

(37) Consequently, regardless of L, c, and T, the dimensionless constant will always be 3.16. All of the boundary conditions are already dimensionless. Therefore, simply replace x with ̂, with ̂

, t with , and u with ξ in Equations (12) - (14). To make the initial conditions

dimensionless substitute Equations (31) - (33) into Eq. (25). ̂ Note that

̂

(38)

̂

L2 is also a dimensionless quantity.

Finally, we must redefine the differentials according to the following relations: (39) and ̂ Using the newly defined variables, replace

̂ with

̂ ̂,

(40) with

,

with , and

with 3.16 in

equations (27) - (29). Solving Eq. (30) with these substitutions gives the solution to the dimensionless Euler-Bernoulli equation. Afterwards, by multiplying every in standard units is recovered.

29

by L, the matrix U

§3 Theoretical Limits of the Spring Models of the AFM Cantilever Given its extensive use in the AFM community, it is pertinent to enquire into the accuracy of the simple mass-spring system to model the cantilever dynamics (either SSM or DSM). To that effect, we develop a method to quickly estimate whether or not a mass-spring model can be used for a particular voltage output or if the numerical Euler-Bernoulli equation must be used. The potential breakdown of the mass-spring model is particularly relevant in the snap-to-contact region. Therefore, we will focus on the properties of that region. It is useful to identify three situations, which we wish to infer from the voltage output. 1) The force and voltage are proportional. In this case the cantilever is in constant instantaneous equilibrium and can be modeled by the SSM. Voltage data with this property can be easily converted to force with the factory provided or user derived spring constant. In practice, this is the most common approach, but is highly questionable, mostly in the snap-to-contact region. When the force is proportional, snap-to-contact could be a misnomer since there is no assurance that contact was made. For example, if repulsive forces act at a greater distance than adhesive forces, the cantilever tip will reverse direction prior to contact. This is very likely to happen if the cantilever is quasistatic and never breaks free of equilibrium. Therefore, determining if the force is proportional will also tell the user whether or not contact was definitely achieved. 2) The force is given by an accelerating mass-spring system (DSM). Unlike the first category, voltage data of this type is not proportional to force. One must include an acceleration term when calculating the force, disrupting the proportionality when acceleration is large. In theory, this method is very simple to implement. However, in

30

practice it is often impractical since it requires very smooth data. More importantly, we shall demonstrate that there is only a small range of data for which this model is reliable. 3) The force can only be calculated using the Euler-Bernoulli equation (NEB). While this method is the most involved in terms of computation and determining various properties of the cantilever, it is the most reliable. It gives correct results even in noisy environments, allowing the AFM user to calculate the force without excessive data smoothing (standard oscillations arising from noise need not be filtered). As it turns out, this technique is also very fast. Our implementation of the algorithm (see Appendix A) completes the calculation of force vs. separation in under 30 seconds on a low-end laptop (Intel Core i3, 2.5 GHz processor). Since the AFM user starts with voltage output, in this section we present a quick method for determining from the output whether the SSM, DSM, or NEB method must be used to calculate the force. In §3.1, to gain insight into the system dynamics, we develop the physical basis for when a force will result in a proportional voltage signal. In §3.2, we model the experimental data using a family of functions parameterized by their curvature. Adjusting the curvature of this analytical model, we determine what type of voltage output can be assumed to be proportional to the force. Finally, in §3.3, we perform a similar calculation to determine the limits of the DSM.

§3.1 Force Characteristics of Proportional Voltage Outputs While the SSM and DSM are not generally correct, they can provide intuition regarding the dynamics of the interaction between the sample and the cantilever tip. In particular, a massspring model of the cantilever tip helps explain when the force characteristics of the sample will induce quasi-static motion and when equilibrium is disturbed. Although the AFM user converts

31

voltage to force, this physical intuition is based on looking at the problem from the opposite perspective: knowing the force vs. separation, will the voltage output produced be proportional? The sample is moved up at a very small velocity, allowing for the possibility that equilibrium will be maintained throughout the probing. This will depend on the relationship between the external force on the tip and the internal force of the cantilever. In a mass-spring system, the internal restoring force the spring exerts on the mass is simply, (41) where u is the deflection of the tip from equilibrium. Therefore, the total force on the tip is, (42) where s is the separation and Fext is the external force the sample exerts on the tip. So long as the total force remains small, the acceleration is negligible and the system is in approximate equilibrium. In this case, the total force is about zero so, (43) as the SSM predicts, and the deflection will be proportional to the force. Since the sample is moving slowly, in order for the external force to cause the tip to break free of equilibrium so that at a given moment

, the degree to which the external force changes with

separation must be greater than k. Consider a case where s0, u0 correspond to a situation in which the cantilever tip is in equilibrium. Suppose the sample is now moved up a small amount to a new separation of s1. This is analogous to the slow movement of the sample. Now that the separation has changed, the essential question becomes whether or not equilibrium can be reached starting from u0 and s1. To simplify the analysis, we derive whether or not equilibrium can be reached when the new position of the sample is kept constant. Therefore, the change in separation, change in cantilever deflection,

, is equal to the

. In reality, since the sample is moving, the ability to reach 32

equilibrium can be affected due to the changes in separation arising from the velocity of the sample. This can either assist or harm the system from reaching equilibrium. Nonetheless, because the velocity is very small, analysis of a system with a static sample can provide sufficient physical intuition and a decent estimate for the dynamics at play. Since we are considering only small changes from s1, the external force can be written in a linear form:

where

|

(44)

|

(45)

is a small change in separation.

Expressing the total force with Eq. (44) gives,

where

is the change in deflection from u0. This equation highlights the essential property for

determining whether or not the system will remain in equilibrium. It expresses that starting from a state of slightly perturbed equilibrium corresponding to separation s1 and deflection u0, the ability to find a new equilibrium depends upon the relationship between

and

|

. To

highlight this, consider Eq. (45) written in the following form, (46) where

|

. This equation demonstrates that the external force near s1 can be

viewed as a force proportional to deflection, similar to a spring, with a “spring constant” given by the slope of the force at that point. If Therefore, if negative initial force,

, then . However, if

is negative, then initially

will be negative.

and will eventually cancel out the , then

and the

magnitude of the total force will only increase. There will be no ability to recover equilibrium. 33

Similarly, if

is positive, then

will initially be positive. Only if

will

equilibrium be recovered. Therefore, we can state simply that equilibrium can only be maintained throughout the probing of the sample if

for all separations s.

Consider, as an example, an external Lennard-Jones force, as shown in Figure 12. This force can be divided into two regions. Region I is where the force increases with separation, and Region II is where the force decreases with separation. Force 0.2

II

I

0.1

1.2

1.4

1.6

1.8

2.0

Separation

0.1

0.2

Figure 12. Sample plot of force vs. separation obeying a Lennard-Jones potential. The units are arbitrary.

Now consider a cantilever tip in the presence of this force. Initially, the separation is large, corresponding to the right end of Figure 12. The sample is moved closer to the cantilever (to the left on the x-axis). In Region I, kext(s) is strictly positive. Since the overall behavior is determined by

and k is always positive, whichever spring constant is greater will win out. Suppose

that kext is less than k throughout all of Region I. Then, the system will be in equilibrium at the start of Region II. Since kext will be negative throughout this region, it will always be less than k and equilibrium will be maintained.

34

It is generally assumed that the snap-to-contact region will correspond to a region of the external force which has the potential to knock the cantilever out of equilibrium. Only once contact is made and adhesive forces dominate will the cantilever reverse direction. However, with this physical background it is clear that this is not necessarily the case. Not every external force will necessarily produce non-equilibrium motion. If this is the case, the entire voltage output, including the snap-to-contact, may correspond to quasi-static motion of the cantilever. The reversal of direction of motion of the cantilever need not result from adhesive forces. Instead, it could correspond to a Region II like portion of the external force. Consequently, snap-to-contact may be a misnomer. Furthermore, the SSM solution for force will accurately describe even the “snap-to-contact” region of the voltage output. However, if equilibrium is not maintained, then either the DSM or NEB must be used to calculate the force from the voltage. Since in practice the force is the unknown, in the coming sections we develop a method to determine from the voltage output if it is proportional to the force and which model of the cantilever should be used to calculate the force.

§3.2 Proportionality Conditions of Mass-Spring Model Having demonstrated that certain external forces can induce quasi-static responses throughout the entire voltage output, we will now provide a way in which one can determine from the output whether or not it corresponds to such a force. Outputs of quasi-static motion correspond to the SSM solution, where the force and voltage are proportional. Therefore, we wish to determine which voltage outputs can be analyzed using the SSM. Since the DSM method works for a wider range of data, the breakdown of the SSM method can be determined by comparison to DSM results. Furthermore, an analytical equation giving the accuracy of SSM results can be found. 35

To model typical voltage outputs we propose the use of the following empirical voltage equation (EVE): (47) This model captures the essence of typical voltage output before it reaches its maximal downward slope at the voltage minimum as shown in Figure 13. The right tail of Eq. (47) obviously fails to correctly characterize the voltage output since the voltage output continues to rise with a constant slope while the EVE has a horizontal asymptote. However, we can ignore this region with no penalty since our region of interest is to the left of the snap-to-contact.

-5

Voltage (mV)

-10 A

-15

EVE Experimental Output

-20

C

-25

B

-30 -35 Time (ms) .2

.4

.6

.8

1

Figure 13. Comparison of EVE model of voltage output to sample AFM voltage output.

Typically, the snap-to-contact contains three parts. 

First is the start of the descent (region A in Figure 13). In this part, the cantilever begins to rapidly accelerate. Care must be taken when characterizing experimental data by the EVE to fit to this region. Underestimation or overestimation of the acceleration must be carefully considered. 36



The second part is usually a fairly steady decrease with a nearly constant slope (region B). This is modeled well by the EVE, as seen in Figure 13.



Third is the sudden reversal of motion (region C). In situations where contact is reached, this change in direction is very sharp and happens almost instantaneously. While the EVE may not be able to capture the sharpness of this reversal, the effect of this feature for our purposes is negligible. Therefore, the EVE can be used for this analysis without concern for Region C

Eq. (47) contains a number of adjustable parameters. V0 sets the vertical shift of the voltage. It essentially represents an initial deflection. Since the initial deflection is typically very small, we set it to zero. A determines the amplitude of the voltage. This parameter can be viewed as a typical magnitude setting the units of voltage being used. Therefore, it has no impact on proportionality. t1 sets the time at which the minimum is reached. Since one can choose an arbitrary time as a reference, t1 does not impact proportionality. Finally, α determines the steepness and concavity of the EVE snap-to-contact, as shown in Figure 14. This parameter will impact proportionality, as we shall demonstrate shortly, since it reflects the acceleration of the cantilever tip. Therefore, by adjusting α alone the proportionality of the voltage to the force will be determined for a given cantilever.

37

0

Voltage mV

5

9 10

10

15

0

20

40

60 Time s

80

100

120

Figure 14. Comparison of EVE for two different values of α

Using the EVE in conjunction with the DSM provides insight into the limitations of the SSM method. Under the DSM, the external force is given by:

(48) where k is the effective spring constant of the cantilever (Eq. (11)), m is the effective mass of the cantilever, and y(t) is the vertical displacement of the cantilever as a function of time. The mass in this model is not equal to the actual mass of the cantilever. Rather, the effective mass can be estimated from the frequency of the first normal mode of a cantilever with one fixed end and one free end. Under these conditions, the frequency is given by:

√

√

(49)

Rearranging this equation, the mass is (50) 38

The external force as given by the SSM is, (51) This equation highlights two essential characteristics of the SSM. 1) The force is proportional to the displacement. Therefore, if the force is proportional to voltage, the SSM sufficiently describes the deflection of the cantilever tip and can be used to construct a force curve. 2) The force as calculated by the SSM can be viewed as a particular case within the DSM where the acceleration term can be neglected. Hence, the SSM is a static spring model. Therefore, when the DSM is no longer approximated well by the SSM, we conclude that the SSM is no longer an appropriate model for the AFM cantilever. Accordingly, the force will no longer be proportional to voltage. In any spring model, the displacement y(t) is considered to be proportional to the voltage V(t), by the constant r (see §2.2). Therefore, we can substitute V(t) into Eq. (48), resulting in a formula to calculate the force directly from the voltage. (52) This equation provides insight about how to view the voltage. Since the voltage is proportional to displacement, it can be viewed directly as displacement with non-standard units. Therefore, the right hand side of Eq. (52) is identical to Eq. (48), but with V(t) instead of y(t). The left hand side is akin to a force calculated in units adjusted by a factor of 1/r. By inspection of Eq. (52), it is apparent that when the kV(t) term dominates (

, the

force will be proportional. Otherwise, the force will not be proportional. In other words, this condition is necessary for the validity of the SSM, which neglects the

term. Therefore, the

proportionality will depend on both the relationship between k and m and on the relationship 39

between the displacement and the acceleration. In essence, these four quantities capture both the properties of the cantilever (k and m) and the characteristics of the cantilever’s response to an external force (V(t) and V’’(t)). Substituting Eq. (47) into Eq. (52) to explicitly solve for the force given by the EVE yields, (

) (53)

As before, Eq. (53) demonstrates that proportionality is a competition between the first and second terms. When the following condition is met for all t, there will be proportionality: (54) To measure the degree of overall proportionality between force and voltage, we define a proportionality metric ϕ as,

√

∫

(55) ∫

To calculate the force using the SSM (i.e. the proportional solution) the force simply equals Therefore, the difference between force and

.

in Eq. (55) measures the difference between the

DSM force and the proportional SSM force. This is squared so that the differences are always positive and calculated over all time. Essentially, the numerator calculates the area between a proportional force curve and the force given by Eq. (53). This is divided by the squared-area under the SSM solution. We take the square root to return to non-squared units, giving us the relative difference of the SSM and DSM solutions. Solving this equation using Equations (53) and (47) yields the following simple expression for ϕ:

40

√

(56)

This confirms our intuition that for small initial deflections, the impact of the voltage output on proportionality depends entirely on α. In particular, this equation shows that with a spring model of the cantilever, the overall degree of proportionality between the force and voltage is linearly related to α. Given a particular cantilever (which set m and k), the degree of proportionality can be quickly computed using an estimate for α. Since we wish to give general recommendations regarding the overall proportionality of the force and voltage, it is useful to define a dimensionless quantity which indicates this proportionality. Since units s-2,

has units of s2 and α has

is a dimensionless quantity which captures all of the information necessary to

determine the overall agreement between the SSM solution and the DSM solution. Therefore, we define (57) Another way of looking at this quantity is to first realize that

is equal to

, where ω is the

frequency of the slowest mode oscillation. Since α also has units of s-2, it can be viewed as a frequency squared, measuring the frequency, as it were, of the snap-to-contact of the voltage output. Therefore, β is a measure of the curvature of the voltage output in units of the cantilever frequency. Substituting β into Eq. (56) gives, √

(58)

After calculating the β of the system, the user can quickly determine with this equation whether or not the proportional SSM solution is within some desired accuracy. The AFM we used for our experiments has an effective spring constant effective mass

and an

. The tiny mass in comparison to the spring constant indicates 41

that a very large acceleration (or α) is necessary to break proportionality. At α = 1010 s-2, and

. Qualitatively, only the bottom of the snap to contact deviates

slightly from the proportional solution. For most applications this force can be treated as proportional. However, at α = 1011 s-2,

and

implying a large deviation from

the correct solution. Qualitatively, the bulk of the force is non-proportional to the voltage. Yet, the overall shape of the curves are similar. At α = 1012 s-2, a qualitative change the behavior of the force occurs: the force becomes positive before the minimum voltage. This feature is important because it is reasonable to expect the force to be repulsive before contact is made. To ensure accuracy within 10% and qualitatively correct behavior, the SSM can only be used for , which in our system corresponds to

.

42

α = 109 s-2

α = 1011 s-2

Proportional SSM Solution DSM Solution

α = 1010 s-2

α = 1012 s-2

Figure 15. Plots of force vs. time comparing the SSM solution to the DSM solution. Plots are shown for four values of α (109, 1010, 1011, and 1012 s-2) demonstrating progression of loss of proportionality between the force and the voltage.

43

§3.3 Limits and Breakdown of DSM Using the DSM to calculate the force the sample exerts on the AFM tip can theoretically be used beyond the limits of the SSM since it takes into account the tip’s acceleration. However, because it only takes into consideration a single mode of oscillation, it is likely to fail when acceleration is high and the motion of the cantilever is governed by a sum of many modes. In this case, only the most general description of cantilever motion, namely the Euler-Bernoulli model, can accurately capture the dynamics. Therefore, to assess the range of voltage output which allows for the DSM, we will assess the accuracy of the DSM solution using the NEB solution. As before, varying α in Eq. (47) for a given cantilever will determine the accuracy of the DSM solution. Because the NEB gives a numerical solution for the force as opposed to an analytical expression, a discrete numerical version of ϕ is used to compare the two solutions. Hence, ϕ is defined as, ∑ √

(59)

∑

where FDSM corresponds to the force as given by the DSM, FNEB corresponds to the NEB solution, i indexes time, and G corresponds to the instant at which the voltage reaches a minimum (the end of the snap-to-contact). This equation will compute the relative overall deviation of the DSM solution from the NEB solution. Since ϕDSM(β) cannot be expressed analytically, the results are shown in Figure 16. At about , the difference between the solutions begins to slowly increase. When

, there is

a deviation of about 1%. For 10% accuracy and a qualitatively reliable result, the DSM can be used for

, corresponding to

for our experimental system.

44

Figure 16. Plot of ϕDSM vs. β. The x-axis is log scale.

Figure 17 displays the evolution of the agreement between the two solutions as β increases. Whereas the SSM solution deviated by 10% when β = .06 the DSM solution is in very close agreement with the NEB solution, as shown in Figure 17. This permits the DSM to be used beyond the limitations of the SSM. When

, the deviation between the NEB and DSM

solutions becomes more significant although still small. At this point, the qualitative shape begins to change. One can see that the very end of the NEB solution predicts a slight increase in the force, whereas the DSM does not capture this feature. While this is only slight for when

,

the difference between the two solutions differ significantly in a qualitative as well

as quantitative sense. In particular, there is a disagreement as to when the force begins to increase and when it becomes positive, two very important features in a force curve.

45

β = .06

β = .15

β = .6

Figure 17. Comparison of force curves with DSM (blue) and NEB (purple) solutions. Progression is shown for changing β.

46

§4 Conclusion We successfully solved the Euler-Bernoulli equation interpreting the voltage as the slope of the end of the cantilever, as opposed to the ordinary practice of viewing it as a deflection. This was reflected by the fourth boundary condition used to solve the Euler-Bernoulli equation. With this new technique one can calculate the force in general and in particular in the relevant snap-tocontact region, without concern for a loss of instantaneous equilibrium. With the NEB, the AFM community can reliably calculate the force at even closer ranges, allowing more detailed research into the interactions between molecules, atoms, and surfaces. We also used this new approach to assess the accuracy of previous models and our own. While we were able to determine the accuracy of the SSM with the DSM, use of the NEB model enabled us to characterize the limits of the DSM model. We summarize these results with the following recommendations: 

If β < .06, the SSM can be used with a target accuracy of 10%.



If

, the DSM can be used with a target accuracy of 10%.. However,

because the inaccuracy for larger β is primarily in the snap to contact region, it may be better to use the DSM only for 

, depending on the desired accuracy.

For β > .27, the Euler-Bernoulli equation should be used to find the force.

47

References [1] L. Zang, "Lecture 10: Basics of Atomic Force Microscope," http://www.eng.utah.edu/~lzang/images/Lecture_10_AFM.pdf. [2] M. Rouse, "Atomic Force Microscopy," 2014, (2011), http://whatis.techtarget.com/definition/atomic-force-microscopy-AFM. [3] V. S. J. Craig, "A Historical Review of Surface Force Measurement Techniques," Colloids and Surfaces A: Physiochem. Eng. Aspects 129-130, 75 (1997). [4] V.S.J. Craig, A.M. Hyde, R.M. Pashley, Langmuir 12 3557 (1996). [5] Image Source: "Atomic Force Microscopy.” http://en.wikipedia.org/wiki/Atomic_force_microscopy. [6] H. J. Butt, B. Cappella and M. Kappl, "Force Measurements with the Atomic Force Microscope: Technique, Interpretation, and Applications," Surface Science Reports 59, 1-152 (2005). [7] G. Meyer, N.M. Amer, Appl. Phys. Lett. 53 1045 (1988). [8] S. Alexander, L. HeUemans, O. Marti, J. Schneir, V. Elings, P.K. Hansma, M. Longmire, J. Gurley, J. Appl. Phys. 65 164 (1989). [9] Image Source: http://www.farmfak.uu.se/farm/farmfyskem/instrumentation/afm.html [10] "Fundamentals of Contact Mode and Tapping Mode Atomic Force Microscopy," (2014), http://www.azonano.com/article.aspx?ArticleID=3010. [11] W.A. Ducker, T.J. Senden, R.M. Pashley, Nature 353 239 (1991). [12] B. Todd, S. Eppell and F. Zypman, "Improved Analysis of the Time Domain Response of Scanning Force Microscope Cantilevers," J. Appl. Phys. 88 (12), 7321 (2000). [13] H.-J. Butt, J. Colloid Interf. Sci. 166 109 (1994). [14] Todd and S. Eppell, "Inverse Problem of Scanning Force Microscope Force Measurements," J. Appl. Phys. 94 (5), 3563 (2003). [15] Todd, S. Eppell and F. Zypman, "Squeezing Out Hidden Information from Scanning Force Misroscopes," Appl. Phys. Lett. 79 (12), 1888 (2001). [16] S. M. Han, H. Benaroya and T. Wei, "Dynamics of Transversely Vibrating Beams Using Four Engineering Theories," J. Sound and Vibration 225 (5), 935 (1999). [17] F. Zypman, Mathematical Physics, Lecture Series, Yeshiva University (2012) [18] Image Source: http://www.azonano.com/images/Article_Images/ImageForArticle_3010%283%29.jpg (2014). [19] J. Ralston et al., "Atomic Force Microscopy and Direct Surface Force Measurments," Pure Appl. Chem. 77 (12), 2149 (2005) [20] T.R. Albrecht, S. Akamine, T.E. Carver, C. F. Quate, J. Vac. Sci. Technol. A 8 3386 (1990). [21] T.A. Senden, W.A. Ducker, Langmuir 10 1003 (1994). [22] Y. Pinchover, J. Rubinstein, “An Introduction to Partial Differential Equations,” Cambridge University Press, Cambridge, (2005).

48

Appendix A: Implementation of Numerical Algorithm We present the computer code (in Mathematica) we used to implement the numerical algorithm to solve the Euler-Bernoulli equation. We begin by reading in the data from the AFM voltage output. 1 2 3 4 5 Line 1 reads in a file “Raw Data.csv” which contains the data from the AFM output (voltage vs. time) in table form. It assigns this to a variable named Vvt (corresponding to voltage vs. time). In Line 2, the frequency of the sampling rate of the AFM is defined. In our example, the frequency is 10-7 Hz. This is used in line 5 to convert the time variable in our data to seconds (which was originally just an index). We then define (line 3) a starting time. Since this choice can be anything, we choose tmin=0. Therefore, our data begins at t=0. In line 4, we define the maximum time of our data, tmax. This is done by multiplying the total number of data points by the frequency and shifting by tmin. In line 5, we define an interpolating function V. This function is equal to the voltage vs. time in seconds. A sample plot of V(t) is shown in Figure 18.

49

Figure 18. Sample plot of interpolating function of experimental data.

6 7 8 9 10 11 12 13 14 15 In this block of code, we define the following constants: 

Nx - the total number of x divisions in the discretization



Nt - the total number of time divisions in the discretization



r – the ratio between voltage and displacement for quasi-static cantilever motion



cantileverLength – the length of the cantilever (100 μm)



γ – the ratio between slope and voltage



T – the period of first-mode oscillations



tot – the total elapsed time (in seconds) 50



̂ – the dimensionless position differential



Δτ – the dimensionless time differential



c – the constant in the Euler-Bernoulli equation

For Nx and Nt we use 499 and 199 respectively. While a qualitatively accurate result can be achieved with many fewer points in space (at around Nx=9), our experience shows that a numerically convergent solution is not achieved until an Nx of about 500 (at Nx=499 the solution is within 1% of the correct solution). As Δτ decreases, the stability of the algorithm increases. However, we did not observe any errors in our solution even for large values of Δτ. We choose Nt = 199 because this provides sufficient resolution to properly sample the data in Figure 18. When defining the period in Line 11, 5*10-6 m4/s2 comes from the constant c in Eq. (3). Our constant c is equal to

in Eq. (34) multiplied by Δτ2 and divided by ̂ , since these

values are constant for our finite difference approximation. This is done to make the code neater later on. Next, we construct a discretized version of the initial conditions based upon a static cantilever as described in Equations (25) and (38). 16 17 In line 16 we define a function u[k,j] which describes the shape of the cantilever at any time τk and position ̂ (note that u(k,j) refers to the dimensionless displacement, defined as ξ in Eq. (31)). This function is defined so that discretized entries of the static solution can be quickly generated. Accordingly, in Line 17, we define a matrix solution to store the solutions to Eq. (30) at every instant. Each row corresponds to the solution at a given instant, and each column a point in space. We immediately define the first two rows of solution by assuming the cantilever is 51

static, using u[k,j] to generate the values. These two rows correspond to τ = 0 and Δτ, and can be viewed as the equivalents of an initial position and initial velocity. We only do this for the first two instants in time since we use three terms for our approximation of the second derivative with respect to time. It turns out that for all the derivatives, the lowest-order finite difference formula is sufficiently accurate. Next, we construct the matrix E of Eq. (30). This matrix depends only on the differential equation and is therefore invariant in time. 18 19 20 21 22 23 24 25 26 27 28 29 30

In line 18 a matrix coefMatrix is defined. This is used to store matrix E. In lines 19-26, the left hand side of the boundary conditions are input. Later, we define the matrix S (ref. Eq. (30)) into which we input the right hand side of those boundary conditions into corresponding rows. In line 19, the boundary condition u(0,t)=0 is input. Therefore, a row consisting of a 1 followed by all 0s is defined. In line 20, this is appended to coefMatrix. In line 21 a row corresponding to the boundary condition in Eq. (13) is defined. Here the row consists of a finite difference formula for the first derivative at x=0, followed by all 0s. As before, this is appended to coefMatrix in line 22. The same process is repeated in lines 23-26 for the two boundary conditions at the hanging end of the cantilever. Lines 23-24 store

|

into coefMatrix. Again a first-order finite

52

difference formula is used. Finally, lines 25-26 store

|

. These boundary conditions

ultimately define the solutions at the fixed end (u1, u2) and hanging end (uNx+1, uNx) of the cantilever. To solve for the remaining ui’s, we populate coefMatrix with rows corresponding to the Euler-Bernoulli equation at ui. These rows are given by Eq. (28), resulting from the equation of cantilever motion, as demonstrated in §2.3. Lines 27-30 define these rows and store them in coefMatrix. The rows are all defined by the identical finite difference formula as given in line 28. The position of this formula is merely shifted to correspond to the correct ui. In the next block of code, we generate the matrix S and solve Eq. (30). 31 32

33 34 In line 31, we open a do loop which iterates in time, solving for the U-vector at every point in time. It begins with the 3rd time step since the first two have been defined based on the static solution. First, the variable solMatrix is defined to store the S matrix for that instant. The first four terms are the right hand sides of the boundary conditions. The rest of the terms are given by a list of Eq. (29) for each ui. Notice that the third boundary condition and the table depend on the voltage at the given time and the previous solutions. In line 33, the solution to Eq. (30) is found and stored in the solution variable. The program to this point solves the Euler-Bernoulli equation, thus providing the shape of the cantilever for all times. Next we proceed to calculate the force vs. separation.

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35 36 37 38

39 40 41

42

In line 35, we define v which is equal to the constant speed of the platform which raises the sample towards the AFM cantilever. In line 36, we define the constant FC which converts the third spatial derivative of the cantilever deflection into force, as given by Eq. (5). This equals . Next, the force is calculated at each time step, denoted in line 37 by the index i. This is done by using a finite difference approximation for the third derivative at the very end of the cantilever (dented by index Nx+1) and multiplying by FC. This result is divided by the square of cantileverLength to recover dimensionality. Finally, this is multiplied by 1012 pN/N so the results will be in pN. The variable force is a list of the force at each time step. Lines 38-40 solve for the separation of the cantilever tip and the sample. This is done using the velocity of the sample and the deflection of the tip of the cantilever. However, since the initial position of the sample is unknown, there is an offset. To correct for this, we approximate the zero-separation as the separation when the voltage minimum is reached in the experimental data, Vvt. Therefore, in line 38, we find the time-index at which the voltage is a minimum. Ordering[Vvt[[All, 2]], 1] finds the index in Vvt at which the voltage is a minimum. Vvt[[Ordering[Vvt[[All, 2]], 1][[1]], 1]] therefore gives the time at that index. The total elapsed time until that point is found by subtracting the initial time Vvt[[1,1]]. This is converted to seconds by multiplying by the 54

frequency and divided by the total elapsed time. This quantity provides the percentage of the total time elapsed when the minimum voltage is reached. This is multiplied by Nt. After rounding the result and adding 1, we have the index of the solution which corresponds to the minimum voltage (somewhere between 1 and Nt+1). Finally, offset is found by calculating the separation at this time step assuming the separation is 0. In line 40, we calculate the separation using negative offset as the initial position of the sample. Notice that the results are multiplied by T to recover dimensionality and then by 109 nm/m to convert to nm. In line 41, force and separation, which are both lists, are joined together to form a table of force vs. separation, the ultimate goal of AFM spectroscopy. This is assigned to variable FvS. Figure 19 shows a sample plot of FvS.

Force pN 60 40 20 10

20

30

20 40 60 Figure 19. Force vs. separation curve for data in Figure 18

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Separation nm