From 3da588370a9bcd37c8ad7afc77852f04a1a44a94 Mon Sep 17 00:00:00 2001 From: "W. Trevor King" Date: Tue, 14 May 2013 11:59:31 -0400 Subject: [PATCH] apparatus/cantilever-calib.tex: Shift math intro to calibcant/overview.tex We don't want to start glazing eyes right out of the gate ;). I try and focus on the text in the first two chapters, leaving the math to the hardy souls who continue into the meat of the thesis. --- src/apparatus/cantilever-calib.tex | 94 +++++++++--------------------- src/calibcant/overview.tex | 71 ++++++++++++++++++---- 2 files changed, 85 insertions(+), 80 deletions(-) diff --git a/src/apparatus/cantilever-calib.tex b/src/apparatus/cantilever-calib.tex index af1d3f5..961630b 100644 --- a/src/apparatus/cantilever-calib.tex +++ b/src/apparatus/cantilever-calib.tex @@ -4,29 +4,28 @@ In order to measure forces accurately with an AFM, it is important to measure the cantilever spring constant $\kappa$\index{$\kappa$}. The force exerted on the cantilever can then be deduced from its -deflection via Hooke's law\index{Hooke's law} +deflection via Hooke's law,\index{Hooke's law} \begin{equation} - F=-\kappa x \;. \label{eq:hooke} + F=-\kappa x \;, \label{eq:hooke} \end{equation} - +where $x$ is the perpendicular displacement of the cantilever tip +($x_c$ in \cref{fig:unfolding-schematic}). +% \nomenclature{$F$}{Force (newtons)} \nomenclature{$\kappa$}{Spring constant (newtons per meter)} \nomenclature{$x$}{Displacement (meters)} -The basic idea is to use the equipartition theorem\citep{hutter93}, +The basic idea is to use the equipartition theorem, which gives the +thermal energy per degree of freedom. For a simple harmonic +oscillator, the only degree of freedom is $x$, so we +have\citep{hutter93} \begin{equation} \frac{1}{2} \kappa \avg{x^2} = \frac{1}{2} k_BT \;, \label{eq:equipart} \end{equation} where $k_B$ is Boltzmann's constant, $T$ is the absolute temperature, -and $\avg{x^2}$ denotes the expectation value of $x^2$ as measured -over a very long interval $t_T$. Solving the equipartition theorem -for $\kappa$ yields -\begin{equation} - \kappa = \frac{k_BT}{\avg{x^2}} \;, \label{eq:equipart_k} -\end{equation} -so we need to measure (or estimate) the temperature $T$ and variance -of the cantilever position $\avg{x^2}$ in order to estimate $\kappa$. - +and $\avg{x^2}$ is the average value of $x^2$ measured +over a long time interval. +% \nomenclature{$k_B$}{Boltzmann's constant, $k_B = 1.380 65\E{-23}\U{J/K}$\cite{codata-boltzmann}} \nomenclature{$T$}{Absolute temperature (Kelvin)} @@ -35,58 +34,17 @@ of the cantilever position $\avg{x^2}$ in order to estimate $\kappa$. \avg{A} \equiv \iLimT{A} \;. \end{equation}} -To find $\avg{x^2}$, the raw photodiode voltages -$V_p(t)$\index{$V_p(t)$} are converted to distances $x(t)$ using the -photodiode sensitivity $\sigma_p$\index{$\sigma_p$} (the slope of the -voltage vs.~distance curve of data taken while the tip is in contact -with the surface) via -\begin{equation} - x(t) = \frac{V_p(t)}{\sigma_p} \;. \label{eq:x-from-Vp} -\end{equation} -By keeping $V_p$ and $\sigma_p$ separate in our calculation of -$\kappa$, we can gauge the relative importance errors in each -parameter and calculate the uncertainty in our estimated $\kappa$ -(\cref{sec:calibcant:discussion:errors}). - -In order to filter out noise in the measured value of $\avg{V_p^2}$ we -fit the measured cantilever deflection to the expected theoretical -power spectral density ($\PSD_f$\index{PSD@\PSD!in frequency space}) -of a damped harmonic oscillator exposed to thermal noise -\begin{equation} - \PSD_f(V_p, f) = \frac{G_{1f}}{(f_0^2-f^2)^2 + \beta_f^2 f^2} \;. - \label{eq:psd-Vp} -\end{equation} -In terms of the fit parameters $G_{1f}$\index{$G_{1f}$}, -$f_0$\index{$f_0$}, and $\beta_f$\index{$\beta_f$}, the expectation -value for $V_p$ is given by -\begin{equation} - \avg{V_p(t)^2} = \frac{\pi G_{1f}}{2\beta_f f_0^2} \;. - \label{eq:Vp-from-freq-fit} -\end{equation} -% -\nomenclature[PSDf]{$\PSD_f$}{Power spectral density in - frequency space - \begin{equation} - \PSD_f(g, f) \equiv \normLimT 2 \magSq{ \Fourf{g(t)}(f) } - \end{equation}} -\nomenclature{$f$}{Frequency (hertz)} -\nomenclature{$f_0$}{Resonant frequency (hertz)} - -Combining \cref{eq:equipart_k,eq:x-from-Vp,eq:Vp-from-freq-fit}, we -have -\begin{align} - \kappa &= \frac{\sigma_p^2 k_BT}{\avg{V_p(t)^2}} - = \frac{2 \beta_f f_0^2 \sigma_p^2 k_BT}{\pi G_{1f}} \;. - \label{eq:kappa} -\end{align} -A calibration run consists of bumping the surface with the cantilever -tip to measure $\sigma_p$ (\cref{sec:calibcant:bump}), measuring the -buffer temperature $T$ with a thermocouple -(\cref{sec:calibcant:temperature}), and measuring thermal vibration -when the tip is far from the surface to extract the fit parameters -$G_{1f}$, $f_0$, and $\beta_f$ (\cref{sec:calibcant:vibration}). - -For a complete derivation of the procedure presented in this section, -see \cref{sec:calibcant:theory}. The rest of \cref{sec:calibcant} -describes my open source implementation for the automatic calibration -procedure. +To calculate the spring constant $\kappa$ using \cref{eq:equipart}, we +need to measure the buffer temperature $T$ and the thermal vibration +variance $\avg{x^2}$. We measure the temperature with a thermocouple +inserted into the AFM fluid cell, and we measure the thermal vibration +by monitoring the cantilever during thermal oscillation when it is far +from the substrate surface. + +The raw cantilever deflection data will have sources of noise that are +not due to the cantilever's thermal vibration (e.g.~electronic noise +in the detector). To avoid biasing $\kappa$, there is a fairly +elaborate theory behind extracting $\avg{x^2}$. For more detail, see +\cref{sec:calibcant}, where I discuss the $\avg{x^2}$ extraction in +detail and present my open source \calibcant\ tool for automated +cantilever calibration. diff --git a/src/calibcant/overview.tex b/src/calibcant/overview.tex index caebb60..a3856a9 100644 --- a/src/calibcant/overview.tex +++ b/src/calibcant/overview.tex @@ -2,23 +2,70 @@ The most common method for calibrating cantilevers for atomic force microscopes is via thermal vibration\citep{florin95}. In this chapter, I'll derive the theory behind this procedure and introduce my \calibcant\ package for performing this calibration automatically. -For a quick overview of the theory, -see \cref{sec:cantilever-calib:intro}. -The basic approach is to treat the cantilever as a simple harmonic -oscillator (\cref{eq:hooke}) and use the equipartition theorem to -connect the cantilever's thermal vibration with the temperature -(\cref{eq:equipart}). The resulting calibration formula for the -cantilever spring constant $\kappa$ is \cref{eq:kappa}, which I'll -reproduce here for easy reference: +We know the energy of the cantilever's thermal vibration from the +equipartion theorem (\cref{eq:equipart,sec:cantilever-calib:intro}). +Solving the equipartition theorem for $\kappa$ yields +\begin{equation} + \kappa = \frac{k_BT}{\avg{x^2}} \;, \label{eq:equipart_k} +\end{equation} +so we need to measure (or estimate) the temperature $T$ and variance +of the cantilever position $\avg{x^2}$ in order to estimate $\kappa$. + +We don't measure $x$ directly, though. We reflect a laser off the +back of the cantilever and measure the position of the deflected beam +with a photodiode (\cref{fig:afm-schematic}). In order to convert the +photodiode signal $V_p$\index{$V_p$} to a tip displacement $x$, we +scale $V_p$ by a linear photodiode sensitivity +$\sigma_p$\index{$\sigma_p$}. +\begin{equation} + x(t) = \frac{V_p(t)}{\sigma_p} \;. \label{eq:x-from-Vp} +\end{equation} +We measure $\sigma_p$ by pushing the tip against the substrate surface +and measuring the slope (deflection volts per piezo meter) of the +resulting contact-deflection trace (\cref{sec:calibcant:bump}). By +keeping $V_p$ and $\sigma_p$ separate in our calculation of $\kappa$, +we can gauge the relative importance errors in each parameter and +calculate the uncertainty in our estimated $\kappa$ +(\cref{sec:calibcant:discussion:errors}). + +In order to filter out noise in the measured value of $\avg{V_p^2}$ we +fit the measured cantilever deflection to the expected theoretical +power spectral density ($\PSD_f$\index{PSD@\PSD!in frequency space}) +of a damped harmonic oscillator exposed to thermal noise +\begin{equation} + \PSD_f(V_p, f) = \frac{G_{1f}}{(f_0^2-f^2)^2 + \beta_f^2 f^2} \;. + \label{eq:psd-Vp} +\end{equation} +In terms of the fit parameters $G_{1f}$\index{$G_{1f}$}, +$f_0$\index{$f_0$}, and $\beta_f$\index{$\beta_f$}, the expectation +value for $V_p$ is given by +\begin{equation} + \avg{V_p(t)^2} = \frac{\pi G_{1f}}{2\beta_f f_0^2} \;. + \label{eq:Vp-from-freq-fit} +\end{equation} +% +\nomenclature[PSDf]{$\PSD_f$}{Power spectral density in + frequency space + \begin{equation} + \PSD_f(g, f) \equiv \normLimT 2 \magSq{ \Fourf{g(t)}(f) } + \end{equation}} +\nomenclature{$f$}{Frequency (hertz)} +\nomenclature{$f_0$}{Resonant frequency (hertz)} + +Combining \cref{eq:equipart_k,eq:x-from-Vp,eq:Vp-from-freq-fit}, we +have \begin{align} \kappa &= \frac{\sigma_p^2 k_BT}{\avg{V_p(t)^2}} = \frac{2 \beta_f f_0^2 \sigma_p^2 k_BT}{\pi G_{1f}} \;. + \label{eq:kappa} \end{align} -where $\sigma_p$ is the photodiode sensitivity, $k_B$ is Boltzmann's -constant, and $T$ is the absolute temperature. The remaining -parameters---$G_{1f}$, $f_0$, and $\beta_f$---come from fitting the -thermal vibration of the cantilever when it is far from the surface. +A calibration run consists of bumping the surface with the cantilever +tip to measure $\sigma_p$ (\cref{sec:calibcant:bump}), measuring the +buffer temperature $T$ with a thermocouple +(\cref{sec:calibcant:temperature}), and measuring thermal vibration +when the tip is far from the surface to extract the fit parameters +$G_{1f}$, $f_0$, and $\beta_f$ (\cref{sec:calibcant:vibration}). \section{Related work} \label{sec:calibcant:survey} -- 2.26.2