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)}
\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.
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}
projects / thesis.git / commitdiff