Kanes Sumetpipata,
Duangkamon Baowan*a,
Barry J. Coxb and
James M. Hillc
aDepartment of Mathematics, Faculty of Science, Mahidol University, Centre of Excellence in Mathematics, CHE, Rama VI Rd., Si Ayutthaya Rd, Bangkok 10400, Thailand. E-mail: duangkamon.bao@mahidol.ac.th
bSchool of Mathematical Sciences, The University of Adelaide, Adelaide, SA 5005, Australia
cSchool of Information Technology and Mathematical Sciences, University of South Australia, Mawson Lakes, South Australia 5095, Australia
First published on 5th May 2016
The ever increasing demand to analyse substrates means that an improved theoretical understanding is necessary for atomic force microscope cantilevers. In this study, we utilize fundamental mathematical modelling, comprising the Lennard-Jones potential and techniques involving the calculus of variations, to obtain the energy equations arising from the probe and the substrate, leading to the deflection equations of the cantilever. Here we assume a silicon tip and the substrate surface is assumed to be a graphene sheet. Based on an energy calculation, the most stable system occurs where the probe is 0.206 nm away from the substrate, and this value exists independently of the size and tilt angle of the probe. For the deflection of the cantilever, we apply the calculus of variations to the separate domains, considering derivatives up to third order at the connection point. The deflection behaviour of a V shaped plate depends primarily on its length, and the spring constants of various cantilevers are calibrated from the deflection equations. In comparison to the zeroth order method of previous studies, our method predicts a 30–50% difference in the value of their spring constants. Moreover, we observe the bending behaviour of cantilever systems by considering the energy between the probe and the substrate together with the bending energy in the cantilever, and we find that the maximum bending distance at the tip is in the range between 0.09 nm and 0.2 nm.
There have been numerous investigations on AFM cantilever properties.3 The most important aspect of any AFM cantilever is the accuracy of the measurements involving elasticity, geometric shape, temperature, spring constant including adhesive force, type of substrate and the loading position. To obtain this data and minimize any errors in measurement, many interesting techniques have been proposed, such as uncertainty analysis, focused curve analysis, focused ion beam with spatial marker, where the shape of the cantilever and the tip, and the spring calibration constant are the two main aspects of the investigations.4–7 Sader et al.8 propose a theoretical model to determine the spring constant for isosceles trapezoidal and V-shaped thin plates. Based on the well-known elastic strain equations for an arbitrary thin plate,9 these authors derive the deflection equations for such cantilevers. The deflection equations are solved using a zeroth order method and a second order method where the former method assumes a solution in one variable. Their results show that both methods give similar results in determining the spring constant and they are in a good agreement as compared to a finite element calculation. Moreover, their second order method can also be used for more general cantilever geometries.
Cleveland et al.10 also derive the spring constant for AFM cantilevers, and measure the resonant frequency of tungsten spheres with different masses with a force loading at the end. The spring constants obtained in their study are in agreement with the theoretical result shown in.11 Sader et al.12 analyze the effect of air damping and the cantilever position where the force is applied, and the spring constant obtained after considering the air damping term shows excellent agreement with Cleveland et al.10 Sader et al.12 comment that the cantilever thickness and the gold coating also affect the observed frequency.
Besides direct calculation of the spring constant, a number of researchers consider torsional spring constants, because AFM cantilevers might be acted upon by lateral forces during their operation. Green et al.13 extend the solutions of Sader et al.14 and Cleveland et al.10 to find such torsional spring constants. Hoogenboom et al.15 employ the Fabry–Perot interferometer together with the cantilever to scan the surface of aqueous or liquid environments by considering thermal noise and the frequency of the silicon cantilever. Fukuma et al.16 develop a cantilever system that is more sensitive to the noise so that it can be applied in diverse environments.
Cai et al.17 design the shape of cantilevers to achieve a wide range of frequencies so that small changes in the cantilever probe can be detected. Stevens et al.18 utilize the nanotube as a probe and report that the image resolution of the general silicon tip and nanotube tip are similar. Further, the nanotube can give better accuracy of the image height than the general silicon tip. There are also other studies that calibrate the spring constant and propose new techniques for cantilever use and we direct the interested reader to the following studies.19–28
Even though the literature contains many results from experiments on AFM cantilevers, further analytical and theoretical methods are required to better explain the function of the cantilevers. In this work, we study the cantilever system by employing classical mathematical modelling techniques to calculate forces and energies of the system, and involving two procedures. Firstly, we discuss the energy between the probe and the surface that is being scanned. Assuming only van der Waals forces, we utilize the Lennard-Jones potential to determine the energy of the system, where the probe is assumed to be a conical shape, and the surface is modelled as an infinite flat plane. Secondly, we discuss the bending of the cantilever to calculate the potential energy including its spring constant. The cantilever geometry considered in this work is the isosceles trapezoidal shape both with and without an isosceles trapezoidal hole. We then combine the results of these two calculations to determine the total energy of the system.
The calculation of the interaction between the probe and the surface and the calculation of the deflecting cantilever are presented in Section 2. Additionally, the basic formulae for the molecular interactions and the calculus of variations formulae are also detailed in this section. Results for the mechanical system for the AFM cantilever are discussed in Section 3, and finally, we give some concluding remarks in Section 4.
We make the substitution to obtain
(1) |
Assume also that is the plane is located below the vertex of the cone at a distance ε, so that the distance between any point of the cone located on the z-axis and the plane is ω = z + ε. The cone base area that is parallel to the plane at a height ω = z + ε has the area Az = πk2z2, and the cone angle is denoted by α, so tan(α/2) = k. Let the integral Jn be defined by
Ecp = ηpηc(−AJ3 + BJ6). |
For the right cone, we may have
The tilted cone system can be established by rotating the right cone through an angle θ as shown in Fig. 3, so that we have
xN = x, yN = ycosθ − zsinθ, zN = ysinθ + zcosθ. |
From the equation for a right cone, we substitute the relations for x, y and z to obtain
(x/k)2 + [(ycosθ − zsinθ)/k]2 = (ysinθ + zcosθ)2, |
(x/k)2 + Gy2 − Hy = F, |
The equation for the tilted cone may be written in the elliptic form given by
Similarly for the case of the right cone, the integral Jntilt becomes
y(x0) = y0, y′(x0) = y′0,…,y(n−1)(x0) = y0(n−1), |
(2) |
From the Euler–Lagrange equation of second order
w(x) = T3x3 + T2x2 + T1x + T0, | (3) |
w(4)(x) − 2w′′′(x)/(ξ − x) = 0, | (4) |
w(x) = λ(ξ − x)(log|ξ − x| − 1) + K2x2 + K1x + K0, | (5) |
Next, we use the boundary conditions to find the unknown constants. Considering w(0) = 0 and w′(0) = 0 in (3) shows that T0 = T1 = 0, and the solution becomes
w(w) = T3x3 + T2x2. | (6) |
The other two conditions, w(L) = δ and w′′(L) = 0, are now applied to (5) to obtain
δ = λ(ξ − L)(log|ξ − L| − 1) + K2L2 + K1L + K0, | (7) |
K2 = −λ/[2(ξ − L)]. | (8) |
We now need to find the constants K0, K1, T2 and T3 assuming that the function w(x) at x = is continuous and smooth up to the third derivative; which means, w(), w′(), w′′() and w′′′() from (5) and (6) must coincide, and the four equations are
T33 + T22 = λ(ξ − )(log|ξ − | − 1) + K22 + K1 + K0, 3T32 + 2T2 = −λlog|ξ − | + 2K2 + K1, 6T3 + 2T2 = λ/(ξ − ) + 2K2, T3 = λ/[6(ξ − )2]. | (9) |
From (7)–(9), both solutions from (5) and (6) may be given by
(10) |
(11) |
From Fig. 5, the energy of the system becomes positive when the vertex of the cone becomes close to the graphene plane. The systems are unstable when ε is less than 0.149 nm, and the equilibrium distance is obtained as εmin = 0.206 nm. Our results show that the cone vertex experiences a strong repulsive force, whereas the bulk of the cone makes a much smaller contribution to the energy. Fig. 6 shows that the larger the cone angle α, the lower the energy. In other words, more atoms on the cone give rise to a larger interaction force between the cone and the plane. The smallest cone (α = π/16 or 11.25°) gives rise to an energy of −1.65 × 10−21 J while the largest one (α = π/4 or 45°) gives rise to an energy of −29.19 × 10−21 J.
Fig. 6 Relation between energy of right cone and cone angle α at equilibrium distance εmin = 0.206 nm. |
The tilted cone system behaves similarly in terms of the ε value. However, in Fig. 7, the system which has a larger rotational angle θ will have a lower energy. Moreover, the value of εmin for the tilted cone system is the same as that for the right cone case, that is, 0.206 nm because θ + α/2 is not much greater than π/2. This means that the closest point between the cone and the plane is at the tip, and therefore the most important parameter is the interspacing ε. We note that the energy of the right cone system can also be obtained from the tilted cone case by using Jn(θ→0)tilt. Fig. 8 shows the possible lowest energy of the tilted cone for any θ. From the graph, the cone with α = π/16 has a difference in energy around 10 × 10−21 J on rotating θ by 1 rad, whereas the cone with α = π/4 has an energy difference of around 30 × 10−21 J corresponding to rotating θ through 0.6 rad.
L (μm) | (μm) | C (μm) | E (μm) | m | |
---|---|---|---|---|---|
Type 1 | 182 | 107.2 | 91 | 53.6 | 0.5 |
Type 2 | 176 | 139.46 | 79 | 62.6 | 0.4489 |
Type 3 | 89.2 | 61.34 | 45.15 | 31.05 | 0.5062 |
Type 4 | 89.4 | 41.54 | 45.2 | 21 | 0.5056 |
The bending distance of Type 4 cantilever, δType 4, is calculated from the bending distance of Type 1 cantilever, δType 1, with the conditions that the deflection distance w(x) is equal (see Fig. 9) and the energy is equal (see Fig. 10). Further, we examine two values of δType 1 which are 0.4 nm and 0.5 nm. We note that these two values of the bending distances are very small compared with the length of cantilever L, and the bending distance δ measured vertically in the x-direction.
From (10) and (11), our bending behaviour does not depend on D. So, even for different materials, the same bending profiles (Fig. 9 and 10) will be obtained converging to the parabolic profile δx2/L2. Furthermore, our solutions approach δx2/L2 indicating that the bending behaviour is the same when C and E are different. Moreover, does not appear in the parabolic solution, that is, the equivalent V-shaped cantilevers with and without holes give the same bending behaviour. This is because our approach ignores the variable y, which is acceptable if twisting or distortions do not occur in the y-direction.8 The major factor on the bending behaviour is the length of the cantilever L. From Fig. 9, the bending behaviour of Type 1 and Type 4 cantilevers are similar but the parabolae have different deflection magnitudes. In Fig. 10 (left), there is more interesting behaviour showing that Type 1 cantilevers bend more than Type 4 cantilevers with the same energy in the two systems. The ratio of the bending distance between Type 1 and Type 4 cantilevers δType 4/δType 1 is 0.4458 at the equivalent energy condition. We may infer that a longer cantilever bends more than a shorter one, as expected. Additionally, when we compare the gradient between Type 1 and Type 4 cantilevers as shown in Fig. 10 (right), Type 1 cantilever has the larger gradient as compared to that of Type 4, and therefore, Type 1 would require a larger bending angle θ.
Next, we consider the spring constants, k, presented in Table 2, where kclv is obtained by Cleveland et al.,10 ksd is taken from Sader et al.,12 while kz is obtained by the zeroth order method given in (ref. 8) and kw is that obtained in this study. We find that kz is greater than kclv and ksd, while kw is less than kclv and ksd for all four types of cantilevers. The differences are due to the different calculations, and as shown in (ref. 12) who discuss the effect of air damping and loading position which also affect the spring constant. We comment that from Table 2, the value of kz is larger than the value of kw by approximately 30–50%. However, Sader et al. state that their spring constant kz differs by 13%,8,14 and when we decrease our kw values by 13%, kw differs from kclv and ksd by approximately 25–36%. This predicted difference is still large for a nano-scaled system where an angstrom change will make a major difference to the overall system.
kclv | ksd | kz | kw | |
---|---|---|---|---|
Type 1 | 0.091 | 0.092 | 0.102 | 0.06 |
Type 2 | 0.044 | 0.044 | 0.05 | 0.033 |
Type 3 | 0.28 | 0.29 | 0.333 | 0.203 |
Type 4 | 0.46 | 0.47 | 0.536 | 0.301 |
In Fig. 11, we use A′ = 2C/L to indicate the different types of cantilevers and r = (C − E)/2C to represent the distance in the x-axis, in order to mimic the situation shown in Fig. 7 of Sader and White.8 Fig. 11 shows that kw is 30–50% different from kz as obtained in,8 but for small values of r, kw and kz are comparable. However, the value of krel = kw/kz is as much as 0.5 for a larger value of r, and it gives a difference of approximately 50%. Moreover, the relation between krel and r does not depend on A′. We believe that the differences between the two methods occurs because the potential energy equation is assumed to have only a force applying at the end of the tip and does not have any boundary edge forces. Our method also predicts bending of the plate smoothly up to the third order derivatives at x = l. So, the curvature the cantilever does not arise and is assumed to be small in the zeroth order method. For the case of Poisson's ratio ν = 0.25, the differences are also approximately 30–50%. Moreover, krel is around 0.53 when r is very large. In comparison with Fig. 7 presented in,8 both Fig. 11 (left) and Fig. 11 (right) are completely different, since Sader's graphs8 evidently show different behaviour for each A′. Additionally, when r tends to 0.5, their krel approaches unity whereas our krel approaches 0.5. We comment that when r is zero, there is a singular point, and that from a reduction of 13% difference in kz, our kw shows a 25–43% difference.
Fig. 12 (a) Geometry of cantilever, (b) relation between θ and ω′(L) where θ = arctan (w′(L)), and (c) geometry of cone with height p and base q. |
Fig. 13 Energy comparison, Ecp for cone and plane, and Ebend is potential energy in cantilever, (left) Type 1 and (right) Type 4 cantilevers. |
In the first procedure, we obtain the energy equations of the right cone and that of the tilted cone. In the case of the tilted cone, when the tilted angle θ is zero, the energy equation gives rise to the energy of the right cone of the same cone angle α. Both cases have an equilibrium distance εmin at [B/(30A)]1/6. Moreover, the cone angle and the tilted angle do not effect the value of εmin where it is obtained as 0.206 nm. Both cases are unstable if ε is less than 0.149 nm, and the different values of εmin occur if and only if we use different materials for the tip and the plane.
In the second procedure, the bending profile tends to a parabola (δ/L2)x2. For any given bending distance δ, L will be a dominant parameter that affects the bending profile. On comparison with the spring constant given in (ref. 12), our kw gives a lower value while kz given in (ref. 8) shows larger values for all types of cantilevers. By comparing between kz and kw, the results show 30–50% difference and when r increases, this difference increases. Additionally, the 13% difference for kz mentioned in (ref. 8) is reduced, but still gives 25–36% difference for our spring constant.
Finally, on combining the first and the second procedures to observe the energy relation, we choose a cone radius α = 0.234 rad and use the parameter values given in Table 1. We predict a maximum bending distance δ between 0.2 nm and 0.09 nm corresponding to tilted angles θ = 2.18 × 10−6 rad and θ = 9.22 × 10−6 rad for Type 1 and Type 4, respectively, and the energies arising from both procedures are −2.35 × 10−21 J.
Our method gives an alternative approach to determine the bending behaviour of the cantilever and the tip response to the surface. This analytical method is relevant in the determination of the distance, energy, and force that are the main considerations in any study of the mechanics of the system. Our approach gives a better understanding of the relations between the substrate surface, the bending distance and the properties of the surface. Moreover, this approach can be applied to any shape of the cantilever and any surface. For a given substrate, we can calculate the appropriate distance between the tip and the sample so as to fix the position of the cantilever to accommodate the bending angle. Further, if we know the surface level from the monitor and we know the bending distance or angle, we can then predict the molecules on the substrate. In the theoretical analysis, our calculation utilizes the calculus of variation, and coordinate transformation of the atoms to model the cantilever system. The approach may involve some approximations but it gives a numerical solution that is faster than that obtained by computer simulation. Hence, we believe that the approach adopted here might be used in many future studies, not only for the cantilever system but also for any mechanical systems involving a scanning step, so that we might quickly determine a reliable solution by means of a simple mathematical formula.
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