Figure 1

Images of a STM tip (left) and a blunt STM tip (right) by Olfert [7].Reuse & Permissions

Figure 2

Image of blunting of conical tungsten tips for (left) cone angle $\alpha =1^{\circ }$ and (right) cone angle $\alpha ={11}^{\circ }$ [12].Reuse & Permissions

Figure 3

Change of variables used near the tip.Reuse & Permissions

Figure 4

Numerical solutions of $H\left(\xi \right)$ for ${\xi }_{\mathrm{tip}}=0.05,0.6,1,1.5,3,4,6,9$, and 12. The cone angles are $\alpha \cong {87}^{\circ },{63}^{\circ },{50}^{\circ },{37}^{\circ },{17}^{\circ },{12}^{\circ },5.9^{\circ },2.6^{\circ }$, and $1.4^{\circ }$, respectively.Reuse & Permissions

Figure 5

Comparison of numerical simulations with experimental photographs [12] of blunting conical tungsten tips. Theory (dark gray lines) has been superimposed onto the photographs, whose contrast has been adjusted for clarity.Reuse & Permissions

Figure 6

Two sections of the profile $H\left(\xi \right)$ for ${\xi }_{\mathrm{tip}}=100$, with cone angle $\alpha =0.{019}^{\circ }$. The amplitude is decreasing slowly, while the wavelength is increasing.Reuse & Permissions

Figure 7

Experimental images of the spheroidization of a conical tungsten tip of cone angle $\alpha =1^{\circ }$ [12]. Below each frame, the time in minutes from the beginning of the experiment.Reuse & Permissions

Figure 8

Plot of $\ln \left({\xi }_{\mathrm{tip}}\right)$ against $\ln \left[\tan \right(\alpha \left)\right]$ from numerical solutions.Reuse & Permissions

Figure 9

Asymptotic and numerical solution for $\delta \left(\xi \right)$ for ${\xi }_{\mathrm{tip}}=12,\alpha =0.0243$. The full lines are numerical solutions of the similarity Eq. (5), with the base solution $\overline{H}\left(\xi \right)$ subtracted. The dashed lines are the results of our asymptotic calculations. We find two asymptotic regions, separated by a crossover value ${\xi }_{\mathrm{cr}}=22.9$, marked by the dotted line. To the right of ${\xi }_{\mathrm{cr}}$, we find a strongly damped far-field region, to the left of ${\xi }_{\mathrm{cr}}$, an oscillating region near the tip. To show some of the oscillations in the far-field region, we also include a magnified view. The free constants in Eqs. (35) and (36) that best fit the numerical solution have been found to be $A_1=0.21$, ${\varphi }_1=\pi +0.7$ and $B_1=0.0025$, for $\xi <{\xi }_{\mathrm{cr}}$ and $A_2=1.33\times {10}^7$, ${\varphi }_2=0.6$ and $B_2=0.0025$ for $\xi >{\xi }_{\mathrm{cr}}.$Reuse & Permissions