Strategies for the AFM-based manipulation of silver nanowires on a flat surface

An NW manipulated by an AFM probe is often considered to be a beam lever under concentrated and distributed force loads [21–24], where the concentrated load represents the pushing force from the AFM probe, and the distributed load represents the friction on the substrate. Nonrigid 1D materials, like Ag NWs, are usually partly bent by a single push manipulation. Assuming that the pushing force is perpendicular to the NW and the displacement is short, then at each moment during the pushing process the bending part of the NW can be modelled as a fixed–fixed beam, as shown in figures 1(a)–(b). If the ends of the NW are not moved in this process, the bending part is symmetric about the force application point. The frictional force acting on this part is dynamic friction, which is supposed to be equivalent everywhere along the curved NW. The other parts of the NW are fixed by unevenly distributed static friction. According to the elastic mechanics, the curved segment of the NW can be described by the following deflection equations of Euler's beam:

$$\begin{eqnarray}\scriptsize{{y}=\left\{\begin{array}{ll}\displaystyle \frac{1}{{EI}}\left[-\displaystyle \frac{{q}}{24}{\left({x}+\tfrac{{l}}{2}\right)}^{4}-\displaystyle \frac{{F}-{ql}}{12}{\left({x}+\tfrac{{l}}{2}\right)}^{3}+\displaystyle \frac{3{Fl}-2{q}{{l}}^{2}}{48}{\left({x}+\tfrac{{l}}{2}\right)}^{2}\right], & -\displaystyle \frac{{l}}{2}\leqslant {x}\leqslant 0\\ \displaystyle \frac{1}{{EI}}\left[-\displaystyle \frac{{q}}{24}{\left({x}+\tfrac{{l}}{2}\right)}^{4}+\displaystyle \frac{{F}+{ql}}{12}{\left({x}+\tfrac{{l}}{2}\right)}^{3}-\displaystyle \frac{9{Fl}+2{q}{{l}}^{2}}{48}{\left({x}+\tfrac{{l}}{2}\right)}^{2}+\displaystyle \frac{{F}{{l}}^{2}}{8}\left({x}+\tfrac{{l}}{2}\right)-\displaystyle \frac{{F}{{l}}^{3}}{48}\right], & 0\lt {x}\leqslant \displaystyle \frac{{l}}{2}\end{array}\right.}\end{eqnarray} \tag{ 1 }$$

where E and I are the elastic modulus and sectional area moment of inertia of the NW, l is the initial length of the curved segment, q is the dynamic frictional force per unit length, F is the pushing force and (x, y) are the coordinates of each point on the curved segment in the coordinate system marked in the graph. Because the shear forces at the two ends of the curved segment are always small in this situation, the pushing force F is approximately balanced by the dynamic friction, namely F ≈ ql. Thus the deflection of the NW at the force application point can be given by:

$$\begin{eqnarray}&&{y}_{0}=\displaystyle \frac{q{l}^{4}}{384EI}\end{eqnarray} \tag{ 2 }$$

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In most AFM-based manipulation, the actively controlled argument is the displacement rather than the pushing force. So the length of the curved NW is actually the function of the deflection at the force application point as well as the mechanical properties of the NW and the friction. To clear this relationship, equation (2) is converted into the following form:

$$\begin{eqnarray}&&l={\left(\displaystyle \frac{384EI{y}_{0}}{q}\right)}^{\displaystyle \frac{1}{4}}.\end{eqnarray} \tag{ 3 }$$

By substituting equation (3) into equation (1), the whole profile of the curved NW can be determined. It is known from the above equation that a longer pushing distance will bend a larger segment of the NW. Meanwhile, the higher elastic modulus and the area moment of inertia will also result in deflection in a wider part of the NW. In contrast, the friction always hampers the movement of the NW and impedes the extension of the bending part. Figure 1(c) depicts how these parameters respectively influence the shape of the NW.

When the pushing force is retracted, the NW may remain at the new position or rebound back depending on whether the static friction is strong enough to hold the NW. Let q_s be the maximum static frictional force per unit length of the NW, and φ and s be the local slope angle and the arc length of the NW at the last moment before the probe is withdrawn; then the following formula can be used as a judgement:

$$\begin{eqnarray}&&{q}_{s}\geqslant EI\displaystyle \frac{{{\rm{d}}}^{3}\varphi }{{\rm{d}}{s}^{3}}.\end{eqnarray} \tag{ 4 }$$

If the above formula is true at any point on the curved NW, its shape can be well maintained; otherwise rebounding will happen.

The above analysis indicates that the nonrigid 1D material will always deflect when it is moved by only one push action. Therefore, we suppose that the Ag NW should be re-straightened after multiple push actions despite being bent during the process. A possible strategy for this is using a group of parallel pushing vectors (PPVs) to move the NW section by section. The effect of the PPVs on the shape of the NW may be understood simply looking at figure 2, where two pushing vectors of the same length and direction are applied to the NW in sequence. The vector executed first P₁ bends segment A₁B₁, and the following vector P₂ causes the deflection of segment A₂B₂. Supposing formula (4) is satisfied, the curved NW is well held by the substrate. If the distance between P₁ and P₂ is shorter than the length of the curved segment, A₁B₁ and A₂B₂ partly overlap. The deflection of each point in the overlap region mainly depends on the larger value of the two deflections respectively induced by P₁ and P₂. If these two vectors are close enough, the NW between them will approach the straight line connecting the ends of P₁ and P₂. If a number of PPVs are applied in this way, all the NW segments between adjacent vectors will be approximately on the same line.

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Based on the above discussion, a translation strategy is proposed in figure 3(a). Assuming the length of a straight Ag NW is L, and the coordinates of its two ends are A (0, 0) and B (L, 0), for the nth pushing vector the coordinates of the start point (S_n[X], S_n[Y]) and the end point (E_n[X], E_n[Y]) can be defined as:

$$\begin{eqnarray}\left\{\begin{array}{lll}{S}_{n}[X] & = & (n-1){\rm{\Delta }}x\\ {S}_{n}[Y] & = & -(R+r+t)\\ {E}_{n}[X] & = & {S}_{n}[X]\\ {E}_{n}[Y] & = & {S}_{n}[Y]+t+{\rm{\Delta }}y\end{array}\right..\end{eqnarray} \tag{ 5 }$$

Here, Δx is the interval between adjacent pushing vectors, Δy is the effective length of each vector, namely the displacement (deflection) of the sample at the force application point; n is the vector number, which must be an integer ranging from 1 to $1+\lfloor L/{\rm{\Delta }}x\rfloor ,$ R and r are the radii of the Ag NW and the probe tip, respectively and t is the recession distance (usually a few tens of nanometres). With an appropriate Δx and Δy, each vector will move a small part of the Ag NW around the force application point to the target, and the whole sample will reach the target straight line when all the PPVs have been processed. By shifting forward and re-executing the PPVs, the Ag NW can be translated further in steps.

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The rotation strategy is quite similar to the translation. The difference is that the lengths of these PPVs are not identical. In a former paper, we demonstrated that a single pushing vector rotates a rigid nanotube around a pivot point [20]. Here we also suppose that a pivot point exists on the flexible NW. The length of each vector is proportional to the distance between the force application point and the pivot point, and the direction of the vector points from the application point to the corresponding point on the target straight line.

Figure 3(b) schematically illustrates the PPVs for rotation, in which the pivot is located at the mid-point of the Ag NW. The pushing vectors applied to the two sides of the Ag NW are symmetrical about the pivot point. In the case of a small angle rotation, the coordinates of the start point and the end point of the nth vector are determined by the following equations:

$$\begin{eqnarray}{\rm{If}}\,1\leqslant {\rm{n}}\leqslant 1+\lfloor 0.5L/{\rm{\Delta }}x\rfloor ,\left\{\begin{array}{lll}{S}_{n}[X] & = & L-(n-1)\cdot {\rm{\Delta }}x+(R+r+t)\cdot \,\tan \,\displaystyle \frac{{\rm{\Delta }}\theta }{2}\\ {S}_{n}[Y] & = & -(R+r+t)\\ {E}_{n}[X] & = & 0.5L+[0.5L-(n-1)\cdot {\rm{\Delta }}x]\cdot \,\cos \,{\rm{\Delta }}\theta +(R+r)\cdot \,\sin \,\displaystyle \frac{{\rm{\Delta }}\theta }{2}\\ {E}_{n}[Y] & = & [0.5L-(n-1)\cdot {\rm{\Delta }}x]\cdot \,\sin \,{\rm{\Delta }}\theta -(R+r)\cdot \,\cos \,\displaystyle \frac{{\rm{\Delta }}\theta }{2}\end{array}\right.\end{eqnarray} \tag{ 6a }$$

$$\begin{eqnarray}&&{\rm{If}}\,1+\lfloor 0.5L/{\rm{\Delta }}x\rfloor \lt {\rm{n}}\leqslant 1+\lfloor L/{\rm{\Delta }}x\rfloor ,\\ \,\left\{\begin{array}{lll}{S}_{n}[X] & = & L-{S}_{n-1-\lfloor 0.5L/{\rm{\Delta }}x\rfloor }[X]\\ {S}_{n}[Y] & = & R+r+t\\ {E}_{n}[X] & = & L-{E}_{n-1-\lfloor 0.5L/{\rm{\Delta }}x\rfloor }[X]\\ {E}_{n}[Y] & = & -{E}_{n-1-\lfloor 0.5L/{\rm{\Delta }}x\rfloor }[Y]\end{array}\right..\end{eqnarray} \tag{ 6b }$$

In the above equations, Δθ is the rotation angle, and the other parameters have the same meaning as in equation (5). For either half of the Ag NW, the pushing vectors located further from the pivot are executed with higher priority. This is because the NW is easier to rotate from the ends. Like the process of translation, the whole NW is expected to achieve the target direction and be re-straightened when all the vectors are completed. By accumulating small angle rotation the larger angle rotation can be realized.

Though in figure 3(b) the Ag NW is supposed to rotate around its mid-point, the pivot may actually be in another position on a flexible sample, especially when it is very soft. We prefer to use the mid-point because in this case the total length of the PPVs reaches a minimum and hence the rotation process is more efficient. But if the NW has a large amount of bending stiffness, the mid-point is perhaps not a reasonable choice. Let us consider the extreme condition, in which the NW is rigid, and the pivot point is then uniquely determined by the self-balance equations [20]. If the pushing force is applied to B(L, 0), the pivot will be located around the point R(0.3L, 0). The R point is also a possible pivot for those NWs that are stiff but still flexible, and in this case the PPVs are determined as follows:

$$\begin{eqnarray}\scriptsize{{\rm{If}}\,1\leqslant {\rm{n}}\leqslant 1+\lfloor 0.7L/{\rm{\Delta }}x\rfloor ,\left\{\begin{array}{lll}{S}_{n}[X] & = & L-(n-1)\cdot {\rm{\Delta }}x+(R+r+t)\cdot \,\tan \,\displaystyle \frac{{\rm{\Delta }}\theta }{2}\\ {S}_{n}[Y] & = & -(R+r+t)\\ {E}_{n}[X] & = & 0.3L+[0.7L-(n-1)\cdot {\rm{\Delta }}x]\cdot \,\cos \,{\rm{\Delta }}\theta +(R+r)\cdot \,\sin \,\displaystyle \frac{{\rm{\Delta }}\theta }{2}\\ {E}_{n}[Y] & = & [0.7L-(n-1)\cdot {\rm{\Delta }}x]\cdot \,\sin \,{\rm{\Delta }}\theta -(R+r)\cdot \,\cos \,\displaystyle \frac{{\rm{\Delta }}\theta }{2}\end{array}\right.}\end{eqnarray} \tag{ 7a }$$

$$\begin{eqnarray}\scriptsize{{\rm{If}}\,1+\lfloor 0.7L/{\rm{\Delta }}x\rfloor \lt {\rm{n}}\leqslant 1+\lfloor L/{\rm{\Delta }}x\rfloor ,\left\{\begin{array}{rcl}{S}_{n}[X] & = & \left(n-1-\lfloor \displaystyle \frac{0.7L}{{\rm{\Delta }}x}\rfloor \right)\cdot {\rm{\Delta }}x-(R+r+t)\cdot \,\tan \,\displaystyle \frac{{\rm{\Delta }}\theta }{2}\\ {S}_{n}[Y] & = & R+r+t\\ {E}_{n}[X] & = & 0.3L-\left[0.3L-\left(n-1-\lfloor \displaystyle \frac{0.7L}{{\rm{\Delta }}x}\rfloor \right)\cdot {\rm{\Delta }}x\right]\cdot \,\cos \,{\rm{\Delta }}\theta -(R+r)\cdot \,\sin \,\displaystyle \frac{{\rm{\Delta }}\theta }{2}\\ {E}_{n}[Y] & = & -\left[0.3L-\left(n-1-\lfloor \displaystyle \frac{0.7L}{{\rm{\Delta }}x}\rfloor \right)\cdot {\rm{\Delta }}x\right]\cdot \,\sin \,{\rm{\Delta }}\theta +(R+r)\cdot \,\cos \,\displaystyle \frac{{\rm{\Delta }}\theta }{2}\end{array}\right.}\end{eqnarray} \tag{ 7b }$$

It should be noted that there is not a clearly defined border between the stiff and soft NWs. In principle, an NW with large thickness and elastic modulus has high bending stiffness, but it may still behave like a soft wire if it is long enough or the friction is extremely strong. In most cases, the modulus and the friction are not precisely measured, so the selection of the pivot is somewhat empirical. A simple push test would help to find the proper model.

Although the behaviour of the Ag NW under a single push manipulation can be described using equations (1)–(4), we found it was difficult to give a precise analytical expression of the final state of the NW after multiple pushing. Instead of a formulation, the simulation based on a finite element method (FEM) was used to investigate the effects of the present strategies and to guide the parameter selection for the PPVs. The simulation was performed using ANSYS. The Ag NW model in the simulation was a 2 μm long prism with a pentagon cross section. It was laid on a flat hard substrate with one of its faces in contact with the substrate surface. The height of the NW, i.e. the distance from the bottom face to the upper ridge, was 50 nm. The density and the Poisson ratio were 10.5 ×10³ kg m⁻³ and 0.38. The elastic modulus was 80 ∼ 120 GPa according to the early literature [25]. A pressure of 0.5 ×10⁸ Pa was added to the NW substrate contact area in order to generate friction. The dynamic friction per unit length was given by q = 0.65 P · h · μ_d, where μ_d represents the dynamic friction coefficient, which is half the static friction coefficient. In this simulation, μ_d varied between 0.05 ∼ 0.2, so the dynamic friction and the maximum static friction were respectively in the range of 0.12 ∼ 0.5 nN nm⁻¹ and 0.2 ∼ 1.0 nN nm⁻¹, which is consistent with our direct measurement result (see supporting data is available online at stacks.iop.org/NANO/28/365301/mmedia). The AFM tip was modelled as a silicon hemisphere with a radius of 35 nm. The pushing vectors were sequentially applied to a series of evenly spaced positions along the Ag NW through the displacement of the tip. The interval of these vectors, i.e. Δx, was 150 ∼ 600 nm. The displacement of each step in the translation, i.e. Δy, was 50 ∼ 200 nm. The rotation angle Δθ was 3 ∼ 5°.

Figure 4 displays a typical translation simulation result, in which E = 80 GPa, μ_d = 0.2 (i.e. q = 0.5 nN nm⁻¹), Δx = 321 nm and Δy = 100 nm. As seen in the figure, the parallel pushing strategy resulted in the wriggling behaviour of the Ag NW. In each step, a small part of the NW around the force application point was moved 100 nm forward. When all seven pushing steps were completed, the whole body of the NW was on the target line. Although this NW still appeared to be slightly curved after the last pushing step, the straightness was apparently improved compared to the intermediate steps. The maximum stress introduced by this manipulation was about 1.8 GPa, occurring at the most curved point in the sixth step. After the last pushing step, the residual stress significantly decreased. In most regions, the residual stress was below 0.3 GPa. The residual stress was due to local deformation of the NW during movement. If the inner tensile force is not able to overcome the static friction, the deformation cannot be released. A smaller Δx and Δy would help to reduce the residual stress.

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Figure 5 displays two typical simulation results of rotation around the mid-point and the R-point. In this simulation, Δθ = 5°, Δx = 400 nm and E and μ_d are the same as in figure 4. The maximum bending stress introduced by rotation always occurred in the first step. The peak value was about 1.3 GPa when the NW was rotated around the mid-point. When the pivot shifted to R, the lever arm from the first pushing vector to the pivot was extended. Thus, the deflection of the NW in the first step grew larger, which increased the maximum bending stress to 1.6 GPa. After the last manipulation step in both the upper and lower images, the Ag NW was successfully turned in the expected direction with an almost straight shape. The residual stresses were 0 ∼ 0.8 GPa and 0 ∼ 0.6 GPa, respectively.

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From a series of simulations with different E, μ_d, Δx, Δy and Δθ, we learnt that all these parameters had a significant influence on the final state of the NW. With a higher modulus and lower friction, the NW will have a smoother shape after manipulation, because in this case the same deflection drives a longer segment of the NW to be bent. This is also in agreement with the analysis for a single push, as illustrated in figure 1. However, the final straightness could be worse due to the increased possibility of rebounding. Rebounding can occur in a wide region around the force application point, and the rebounding parts can stop in unpredictable positions. In comparison, soft NWs are more controllable, as each part of the NW can be directly moved to the target position and held stably by the static friction. High amounts of friction on the substrate help to maintain the shape and position of the NW, but on the other hand, it also makes movement difficult. The NWs are more likely to be broken, and the higher residual stress will cause them to become unstable.

For a given modulus and friction properties, shorter vectors and smaller intervals ensure lower amounts of stress as well as better straightness. Nevertheless, it requires more steps to transfer the NW over the same distance, and hence the time consumption increases. Therefore, a trade-off must be made between the manipulation efficiency and the final straightness of the sample. Two principles for selecting Δx, Δy and Δθ are as follows:

• 1.  
  The maximum stress caused by Δy or Δθ must be lower than the yield strength of the material;
• 2.  
  Δx must be no more than half the length of the curved segment caused by a single pushing vector.

In the simulations shown in figures 4 and 5, the maximum stresses are 1 ∼ 2 GPa, which are close to the reported yield strength of Ag NWs [26–28]. Thus 100 nm and 5° can be taken as the upper limits of Δy and Δθ under the related conditions. Nevertheless, because the maximum friction coefficient measurement is used in this simulation, Δy and Δθ are allowed to be a few times larger in the experiment. According to equation (3), the length of the curved segment is about 10 times that of Δy; thus Δx should be smaller than 5Δy.

In our simulation, no rolling behaviour was observed. Although there was a twisting trend when the first pushing vector was applied to the end of the NW, the torque induced by the AFM probe was insufficient to encounter adhesion on the substrate surface. Sliding is the only form of motion that should be considered in this strategy.