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The dot chosen depended on measurement condition such as contamination and phase characteristics of the dot. The data shows the smaller penumbra $\sigma _s$ (\subref {fig:evaporation_measured_penumbra_sigs}) the larger penumbra $\sigma _l$ (\subref {fig:evaporation_measured_penumbra_sigl}), the height of the dot (\subref {fig:evaporation_measured_penumbra_height}) and the diameter of the circle (\subref {fig:evaporation_measured_penumbra_circle_r}).}}{75}{figure.caption.81}\protected@file@percent } \newlabel{fig:evaporation_measured_penumbra}{{5.7}{75}{Data obtained from the previously described method for each of the 5 evaporations, from evaporated dot each from the center of the field, the left, the right, the bottom and the top. The dot chosen depended on measurement condition such as contamination and phase characteristics of the dot. The data shows the smaller penumbra $\sigma _s$ (\subref {fig:evaporation_measured_penumbra_sigs}) the larger penumbra $\sigma _l$ (\subref {fig:evaporation_measured_penumbra_sigl}), the height of the dot (\subref {fig:evaporation_measured_penumbra_height}) and the diameter of the circle (\subref {fig:evaporation_measured_penumbra_circle_r})}{figure.caption.81}{}} -\@writefile{toc}{\contentsline {section}{\numberline {5.4}Tilt}{77}{section.5.4}\protected@file@percent } +\@writefile{toc}{\contentsline {section}{\numberline {5.4}Tilt and deformation}{77}{section.5.4}\protected@file@percent } \newlabel{fig:evaporation_tilts_example}{{5.8a}{77}{\relax }{figure.caption.82}{}} \newlabel{sub@fig:evaporation_tilts_example}{{a}{77}{\relax }{figure.caption.82}{}} \newlabel{fig:evaporation_tilts_all}{{5.8b}{77}{\relax }{figure.caption.82}{}} @@ -91,6 +91,7 @@ \@writefile{lof}{\contentsline {figure}{\numberline {5.15}{\ignorespaces Image of final simulation with parameters given in Figure \ref {fig:evaporation_simulation_first_compare} and an anharmonic oscillation with a power of $20$.}}{84}{figure.caption.89}\protected@file@percent } \newlabel{fig:evaporation_simulation_progression}{{5.15}{84}{Image of final simulation with parameters given in Figure \ref {fig:evaporation_simulation_first_compare} and an anharmonic oscillation with a power of $20$}{figure.caption.89}{}} \@writefile{toc}{\contentsline {paragraph}{Software improvements}{84}{section*.90}\protected@file@percent } +\@writefile{toc}{\contentsline {paragraph}{Conclusion}{85}{section*.91}\protected@file@percent } \@setckpt{chap05}{ \setcounter{page}{86} \setcounter{equation}{1} diff --git a/chap05.tex b/chap05.tex index 694ff01163085182e5eed82db579051ca129ea6f..0da0fc0d3f611936842ac1425544eb047d729ad3 100644 --- a/chap05.tex +++ b/chap05.tex @@ -190,7 +190,7 @@ The diameter of the \ce{Pb} dots would be expected to decrease with subsequent e The larger penumbra data (Figure \ref{fig:evaporation_measured_penumbra_sigl}) indicates no pattern for each field and only possibly a reduction in penumbra for the bottom and center dots. This might be due to different dots being chosen for each analysis, some of which are not completely at the top or bottom (or left and right), but one row below or above. In the following, the penumbra and direction of tilt will be treated in a more thorough manner. \\ -\section{Tilt} +\section{Tilt and deformation} \begin{figure}[H] \centering @@ -271,7 +271,7 @@ At a time $0$ at a distance $L$ from the sample a random point inside the circle Objects in the Godot game engine are moved, rotated and scaled with a $3 \times 4$ matrix called a transform matrix. This matrix performs rotations via their quaternion representation, which is a way to represent $3$-dimensional rotations as a $4$ component complex number. Modifying the transform matrix directly is possible, but would be very unintuitive and cumbersome, so the engine allows modification of the component's displacement and scale via $3$D vectors. The components of the displacement vector will be called x, y and z. The rotation can be modified via Euler angles. Internally the Euler angles are called, based on the axis they rotate around, x, y and z as well. To avoid confusion the angles will be called $\alpha$, $\beta$ and $\gamma$, where $\alpha$ rotates around the x-axis, $\beta$ around the y-axis and $\gamma$ around the z-axis. -In order to simulate vibration effects, the cylinder collider for the mask can be moved and rotated in a periodic manner. The rotation, position and oscillation period are user parameters. And after each time step the collider is moved and in the next iteration the new collider position and rotation is checked against. The position of the current time step is obtained by linear interpolation between the start position and tilt and the end position and tilt. The interpolation parameter is determined with the function $|\sin(\frac{t}{T})|$, where $T$ is the period of the oscillation in time steps and $t$ is the current time step. This allows the simulation of $3$D vibrations in the resulting image. \\ +In order to simulate vibration effects, the cylinder collider for the mask can be moved and rotated in a periodic manner. The rotation, position and oscillation period are user parameters. And after each time step the collider is moved and in the next iteration the new collider position and rotation is checked against. The position of the current time step is obtained by linear interpolation between the start position and rotation and the end position and rotation. The interpolation parameter is determined with the function $|\sin(\frac{t}{T})|$, where $T$ is the period of the oscillation in time steps and $t$ is the current time step. This allows the simulation of $3$D vibrations in the resulting image. It does not take into account possible bending of the mask, since the colliders are stiff rigid bodies, but using rotation, bending can be locally approximated. \\ After a user specified time has passed, the amount of hits on each pixel is saved into a file and the image can then be displayed using a python script. For a more detailed look at the different parameters the script provides, see the Appendix \ref{sec:appendix_raycast}.\\ @@ -294,7 +294,7 @@ After a user specified time has passed, the amount of hits on each pixel is save An image of a simple simulation for an oscillating mask dot with parameters obtained from the AFM measurement can be seen in Figure \ref{fig:evaporation_simulation_first_compare_SIM}. The parameters for the amplitude of the oscillation were extracted from the AFM image shown in Figure \ref{fig:evaporation_simulation_first_compare_AFM}. The values were $0.143$ $\mu$m in x and $-0.358$ $\mu$m in z direction and a tilt of $-41.12^\circ$ in $\alpha$ and $31^\circ$ in $\gamma$. \\ -The mask being deformed by nearly $45^\circ$ at a single hole site locally would induce large strain upon the mask. The visible tilt is most likely an outcome of both an x-y displacement and a bending of the mask. If there was just a displacement due to the vibration the mask would shift between 2 lateral positions with a certain frequency. If there is strong overlap the $2$ extreme positions would have a certain overlap, which is elliptical. If there is now an additional displacement component in the z direction this causes a smaller circle on top of the flat mask position. It is likely that the effect on the edge is an overlap of both a bending of the mask giving the mask some angle and an additional contribution from the displacement in both x-y and z direction. +The mask being deformed by nearly $45^\circ$ at a single hole site locally would induce large strain upon the mask. The visible tilt is most likely an outcome of both an x-y displacement and a bending of the mask. If there was just a displacement due to the vibration the mask would shift between 2 lateral positions with a certain frequency. If there is strong overlap the $2$ extreme positions would have a certain overlap, which is elliptical. If there is now an additional displacement component in the z direction this causes a smaller circle on top of the flat mask position. It is likely that the effect on the edge is an overlap of both a bending of the mask giving the mask some angle and an additional contribution from the displacement in both x-y and z direction. A simulated of this is shown in Figure \ref{fig:evaporation_simulation_overlap} \begin{figure}[H] \centering @@ -303,9 +303,9 @@ The mask being deformed by nearly $45^\circ$ at a single hole site locally would \label{fig:evaporation_simulation_overlap} \end{figure} - The amplitude of displacement in the case in Figure \ref{fig:evaporation_simulation_first_compare_SIM} is $\approx 0.4$ $\mu$m, this is in line with the peak to peak amplitude of an active turbomolecular pump given by $1$ $\mu$m, obtained in the PhD thesis of Priyamvada Bhaskar.~\cite{Bhaskar} Some features of the AFM measurement are mirrored in the simulation, however it does not match the simulated image in a decent number of characteristics. The "half moon" shaped penumbra (\textcolor{tab_red}{red} in Figure \ref{fig:evaporation_simulation_first_compare_AFM}) in the AFM image is very rough, but on average of equal height, while in the simulation the penumbra gradually lowers from the highest part. The lower edge of the elliptical shape visible in the AFM dot below the circle (\textcolor{tab_cyan}{cyan} in Figure \ref{fig:evaporation_simulation_first_compare_AFM}) is so faint as to be invisible in the AFM image, while it is very pronounced in the simulated image and the lower edge is sharp in the AFM image $61 \pm 9$ nm while it is smeared out in the simulated image. \\ +The different roughness from circle and ellipse might suggest different possible reasons. First it could be a chronological effect where the circle is deposited first and the ellipse is deposited second. Another possiblity is that the vibration cause the displacement and bending of the mask in an pattern that is anharmonic, which causes the extreme points of the oscillation to be preferred. In order to investigate possible sources of this effect the simulation was amended. \\ \begin{figure}[H] \centering @@ -328,9 +328,9 @@ The amplitude of displacement in the case in Figure \ref{fig:evaporation_simulat \label{fig:evaporation_simulation_sharpness} \end{figure} -The effect of this can be seen in Figure \ref{fig:evaporation_simulation_sharpness_stick_initial} compared with the simpler model Figure \ref{fig:evaporation_simulation_sharpness_stick_simple}) this gives an image more simple to the AFM measurement. Another possibility is an oscillation, which is not harmonic. For this instead of choosing the oscillation as $\sin(\frac{t}{T} + \phi)$ with $t$ being current time, $T$ the oscillation period and $\phi$ being a phase shift, the oscillation is instead parametrized as $\sin(\frac{t}{T} + \phi)^p$ with $p$ being the oscillation power. The resulting image can be seen in Figure \ref{fig:evaporation_simulation_sharpness_stick_power}. The effect of this is very similar to the initial circular shape. +The effect of this can be seen in Figure \ref{fig:evaporation_simulation_sharpness_stick_initial} compared with the simpler model Figure \ref{fig:evaporation_simulation_sharpness_stick_simple}) this gives an image more simple to the AFM measurement. Another possibility is an oscillation, which is not harmonic. For this instead of choosing the oscillation as $\sin(\frac{t}{T} + \phi)$ with $t$ being current time, $T$ the oscillation period and $\phi$ being a phase shift, the oscillation is instead parametrized as $\sin(\frac{t}{T} + \phi)^p$ with $p$ being the oscillation power. The resulting image can be seen in Figure \ref{fig:evaporation_simulation_sharpness_stick_power}. The effect of this is very similar to the initial circular shape. The vibrations causing the deformation and tilt are unlikely to be very anharmonic, but due to growth of thin films happening near grains the actual growth of \ce{Pb} on the \ce{Si} is concentrated near the extreme positions of the oscillation. -When looking at the measured AFM image, it is very noticeable, that the surface of the "half moon" is rougher than the surface of the inner circle. On average, the roughness is $1.7 \pm 0.4$ times higher. This might be due to the reduced height the outer circle has, or due to being deposited at a different time. \\ +When looking at the measured AFM image, it is very noticeable, that the surface of the "half moon" is rougher than the surface of the inner circle. On average, the roughness is $1.7 \pm 0.4$ times higher. This could be due to the \ce{Pb} preferring already established particle sites to diffuse and grow near another possible reason is a chronology of events where the growth happens first on the outer circle and then on the elliptical shape, as previously looked at.\\ Lead or in general any deposited material deposits more easily, when there is already some of the same material deposited. This nucleation effect can be relatively simply be modeled in the simulation by penalizing deposition for pixels, where no material has been deposited previously. The probability to deposit on an empty surface is a user controlled parameter called "first\_layer\_depo\_prob". It controls the probability with which a particle hitting the sample is deposited, when no material has previously been deposited on the relevant pixel. \\ @@ -366,10 +366,14 @@ The simulation image matches the one given by the AFM measurement pretty well, w \label{fig:evaporation_simulation_progression} \end{figure} -The simulation allows for taking in progress images at specified time intervals, with this a progression of the evaporation can be created. An example for the previously discussed example can be seen in Figure \ref{fig:evaporation_simulation_progression}. With this the chronology of events can be made visible more easily and visualizations could easily be created. +The simulation allows for taking in progress images at specified time intervals, with this a progression of the evaporation can be created. An example for the previously discussed example can be seen in Figure \ref{fig:evaporation_simulation_progression}. With this the chronology of events can be made visible more easily and visualizations could easily be created. \\ \paragraph{Software improvements} The simulation is accurate in geometrical configuration of the Mask Aligner setup, but it assumes each particle hitting the surface either sticks to it or is rejected with a certain probability, which is a reasonable approximation as it follows the linear behavior from the Knudsen equation (Eq. \ref{eq:hertz_knudsen}), but it does not currently take into account grain size and diffusion of particles, which makes the graininess of the image resolution dependent.\\ The current way of implementing the simulation using Godot allowed for very quick implementation and bug fixing, but lacks in performance. Each ray is cast sequentially on the CPU and significant overhead is caused by the game engine computing things necessary for games, but unnecessary for the purposes of this simple simulation. This causes the render time of each image to be in the minute to hour range for images of higher resolutions. \\ In order to improve performance, a dedicated ray tracing engine with support for threading and maybe even parallel deployment on the \textbf{G}raphics \textbf{P}rocessing \textbf{U}nit (GPU) using \textbf{A}pplication \textbf{P}rogramming \textbf{I}nterfaces (APIs) like for example CUDA or OpenCL could give massive performance improvements since rays many thousands of rays could be cast in parallel this way. This would most likely boost generation times by several orders of magnitude. \\ -Since Godot uses its own units for length measurement, which are stored as $32$-bit floating point numbers, this also causes unit conversion from real world units to Godot's units to be time-consuming and can potentially cause float point rounding issues. With a dedicated ray casting engine, real world units could be used and accuracy of the simulation could be improved by using higher precision floating point numbers. \\ \ No newline at end of file +Since Godot uses its own units for length measurement, which are stored as $32$-bit floating point numbers, this also causes unit conversion from real world units to Godot's units to be time-consuming and can potentially cause float point rounding issues. With a dedicated ray casting engine, real world units could be used and accuracy of the simulation could be improved by using higher precision floating point numbers. \\ + +\paragraph{Conclusion} + +The results of the simulation show that a x-y-z vibration with a component of "bending" simulated as a strong tilt can explain the shape of the penumbra obtained in the AFM and that its peak to peak amplitude is within the expected for this system. This also shows that the sharper penumbra edge, which for this evaporation was measured to be $\approx 60$ nm is penumbras that would likely be obtained had there been no vibrational influence on the experiment. 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