In order to measure forces accurately with an AFM, it is important to
measure the cantilever spring constant. The force exerted on the
cantilever can then be deduced from it's deflection via Hooke's law
-$F=-kx$.
+$F=-\kappa x$.
\nomenclature{$F$}{Force (newtons)}
-\nomenclature{$k$}{Spring constant (newtons per meter)}
+\nomenclature{$\kappa$}{Spring constant (newtons per meter)}
\nomenclature{$x$}{Displacement (meters)}
The basic idea is to use the equipartition theorem\citep{hutter93},
\begin{equation}
- \frac{1}{2} k \avg{x^2} = \frac{1}{2} k_BT \;, \label{eq:equipart}
+ \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
\begin{equation}
\avg{A} \equiv \iLimT{A} \;.
\end{equation}}
-Solving the equipartition theorem for $k$ yields
+Solving the equipartition theorem for $\kappa$ yields
\begin{equation}
- k = \frac{k_BT}{\avg{x^2}} \;, \label{eq:equipart_k}
+ \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 $k$.
+of the cantilever position $\avg{x^2}$ in order to estimate $\kappa$.
To find $\avg{x^2}$, the raw photodiode voltages $V_p(t)$ are
converted to distances $x(t)$ using the photodiode sensitivity
\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 $k$, we
-can gauge the relative importance errors in each parameter and
-calculate the uncertainty in our estimated $k$.
+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 $\kappa$.
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
Combining \cref{eq:equipart_k,eq:x-from-Vp,eq:Vp-from-freq-fit}, we
have
\begin{align}
- k &= \frac{\sigma_p^2 k_BT}{\avg{V_p(t)^2}}
+ \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}} \;.
\end{align}
I didn't have a good understanding of the theory behind thermally
calibrating an AFM cantilever, so I work it out here with all the
-gorey details :p.
+gory details :p.
The testq subdirectory contains some python scripts I used to test my
algebra and get a better feel for what was going on. The dot
proposed and reviewed\citep{florin95,levy02,ohler07}, but we will
focus here on the derivation of Lorentzian noise in damped simple
harmonic oscillators that underlies all frequency-space methods for
-improving the basic $k\avg{x^2} = k_BT$ method.
+improving the basic $\kappa\avg{x^2} = k_BT$ method.
Roters and Johannsmann describe a similar approach to deriving the Lorentizian
power spectral density\citep{roters96}. %,
which we can plug into the equipartition theorem
(\cref{eq:equipart}) yielding
\begin{align}
- k = \frac{2 \beta \omega_0^2 k_BT}{\pi G_1} \;.
+ \kappa = \frac{2 \beta \omega_0^2 k_BT}{\pi G_1} \;.
\end{align}
From \cref{eq:GO}, we find the expected value of $G_1$ to be
where $m_p\equiv m/\sigma_p$, $G_{1p}\equiv G_0/m_p^2=\sigma_p^2 G_1$.
Plugging into the equipartition theorem yeilds
\begin{align}
- k &= \frac{\sigma_p^2 k_BT}{\avg{V_p(t)^2}}
+ \kappa &= \frac{\sigma_p^2 k_BT}{\avg{V_p(t)^2}}
= \frac{2 \beta\omega_0^2 \sigma_p^2 k_BT}{\pi G_{1p}} \;.
\end{align}
where $f_0\equiv\omega_0/2\pi$, $\beta_f\equiv\beta/2\pi$, and
$G_{1f}\equiv G_{1p}/8\pi^3$. Finally
\begin{align}
- k &= \frac{\sigma_p^2 k_BT}{\avg{V_p(t)^2}}
+ \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}} \;.
\end{align}
Our cantilever can be approximated as a damped harmonic oscillator
\begin{equation}
- m\ddt{x} + \gamma \dt{x} + k x = F(t) \;, \label{eq:DHO}
+ m\ddt{x} + \gamma \dt{x} + \kappa x = F(t) \;, \label{eq:DHO}
% DHO for Damped Harmonic Oscillator
\end{equation}
where $x$ is the displacement from equilibrium,
$m$ is the effective mass,
$\gamma$ is the effective drag coefficient,
- $k$ is the spring constant, and
+ $\kappa$ is the spring constant, and
$F(t)$ is the external driving force.
During the non-contact phase of calibration,
$F(t)$ comes from random thermal noise.
Fourier transforming \cref{eq:DHO} and applying \cref{eq:four-deriv} we have
\begin{align}
- (-m\omega^2 + i \gamma \omega + k) x(\omega) &= F(\omega)
+ (-m\omega^2 + i \gamma \omega + \kappa) x(\omega) &= F(\omega)
\label{eq:DHO-freq} \\
(\omega_0^2-\omega^2 + i \beta \omega) x(\omega) &= \frac{F(\omega)}{m} \\
|x(\omega)|^2 &= \frac{|F(\omega)|^2/m^2}
{(\omega_0^2-\omega^2)^2 + \beta^2\omega^2} \;,
\label{eq:DHO-xmag}
\end{align}
-where $\omega_0 \equiv \sqrt{k/m}$ is the resonant angular frequency
-and $\beta \equiv \gamma / m$ is the drag-aceleration coefficient.
+where $\omega_0 \equiv \sqrt{\kappa/m}$ is the resonant angular
+frequency and $\beta \equiv \gamma / m$ is the drag-aceleration
+coefficient.
\index{Damped harmonic oscillator}\index{beta}\index{gamma}
\nomenclature{$\omega_0$}{Resonant angular frequency (radians per second)}
\index{$\omega_0$}
\iInfInf{\omega}{\frac{1}{(\omega_0^2-\omega^2)^2 + \beta^2\omega^2}}
= \frac{G_0}{2m^2} \cdot \frac{\pi}{\beta\omega_0^2}
= \frac{G_0 \pi}{2m^2\beta\omega_0^2}
- = \frac{G_0 \pi}{2m^2\beta \frac{k}{m}} \\
- &= \frac{G_0 \pi}{2m \beta k} \;.
+ = \frac{G_0 \pi}{2m^2\beta \frac{\kappa}{m}} \\
+ &= \frac{G_0 \pi}{2m \beta \kappa} \;.
\end{align}
The integration is detailed in \cref{sec:integrals}.
By the corollary to Parseval's theorem (\cref{eq:parseval-var}), we have
Plugging \cref{eq:DHO-var} into the equipartition theorem
(\cref{eq:equipart}) we have
\begin{align}
- k \frac{G_0 \pi}{2m \beta k} &= k_BT \\
+ \kappa \frac{G_0 \pi}{2m \beta \kappa} &= k_BT \\
G_0 &= \frac{2}{\pi} k_BT m \beta \;. \label{eq:GO}
\end{align}
As expected, the general form \cref{eq:DHO-psd} reduces to the
extremely overdamped form \cref{eq:ODHO-psd}. Plugging in for
-$\beta\equiv\gamma/m$ and $\omega_0\equiv\sqrt{k/m}$,
+$\beta\equiv\gamma/m$ and $\omega_0\equiv\sqrt{\kappa/m}$,
\begin{align}
\lim_{m\rightarrow 0} \PSD(x, \omega)
&= \lim_{m\rightarrow 0} \frac{2 k_BT \gamma}
- { \pi m^2 \p[{(k/m-\omega^2)^2 + \gamma^2/m^2\omega^2}] }
+ { \pi m^2 \p[{(\kappa/m-\omega^2)^2 + \gamma^2/m^2\omega^2}] }
= \lim_{m\rightarrow 0} \frac{2 k_BT \gamma}
- { \pi \p[{(k-m\omega^2)^2 + \gamma^2\omega^2}] } \\
+ { \pi \p[{(\kappa-m\omega^2)^2 + \gamma^2\omega^2}] } \\
&= \frac{2}{\pi}
\cdot
- \frac{\gamma k_BT}{k^2 + \gamma^2\omega^2} \;.
+ \frac{\gamma k_BT}{\kappa^2 + \gamma^2\omega^2} \;.
\end{align}
\cref{eq:four-deriv} we have
% ODHO stands for very Over Damped Harmonic oscillator
\begin{align}
- (i \gamma \omega + k) x(\omega) &= F(\omega) \label{eq:ODHO-freq} \\
- |x(\omega)|^2 &= \frac{|F(\omega)|^2}{k^2 + \gamma^2 \omega^2} \;.
+ (i \gamma \omega + \kappa) x(\omega) &= F(\omega) \label{eq:ODHO-freq} \\
+ |x(\omega)|^2 &= \frac{|F(\omega)|^2}{\kappa^2 + \gamma^2 \omega^2} \;.
\label{eq:ODHO-xmag}
\end{align}
\index{Damped harmonic oscillator!extremely overdamped}
\cref{eq:psd-def}
\begin{equation}
\PSD(x, \omega)
- = \normLimT \frac{2\magSq{F(\omega)}}{k^2 + \gamma^2\omega^2} \;.
+ = \normLimT \frac{2\magSq{F(\omega)}}{\kappa^2 + \gamma^2\omega^2} \;.
\label{eq:ODHO-psd-F}
\end{equation}
\index{PSD@\PSD}
-Because thermal noise is white (not autocorrelated + Wiener-Khinchin Theorem),
-we can denote the one sided thermal power spectral density per unit time by
+Because thermal noise is white (not autocorrelated + Wiener-Khinchin
+Theorem), we can denote the one sided thermal power spectral density
+per unit time by
\begin{equation}
\PSD(F, \omega) = G_0
= \normLimT 2 \magSq{F(\omega)} \;. \label{eq:GOdef} % label O != zero
Plugging \cref{eq:GOdef} into \cref{eq:ODHO-psd} we have
\begin{equation}
- \PSD(x, \omega) = \frac{G_0}{k^2 + \gamma^2\omega^2} \;.
+ \PSD(x, \omega) = \frac{G_0}{\kappa^2 + \gamma^2\omega^2} \;.
\end{equation}
-This is the formula we would use to fit our measured \PSD, but let us go a
-bit farther to find the expected \PSD\ and thermal noise
- given $m$, $\gamma$ and $k$.
+This is the formula we would use to fit our measured \PSD, but let us
+go a bit farther to find the expected \PSD\ and thermal noise given
+$m$, $\gamma$ and $\kappa$.
-Integrating over positive $\omega$ to find the total power per unit time yields
+Integrating over positive $\omega$ to find the total power per unit
+time yields
\begin{align}
\iOInf{\omega}{\PSD(x, \omega)}
- = \iOInf{\omega}{\frac{G_0}{k^2 + \gamma^2\omega^2}}
- = \frac{G_0}{\gamma}\iOInf{z}{\frac{1}{k^2 + z^2}}
- = \frac{G_0 \pi}{2 \gamma k} \;,
+ = \iOInf{\omega}{\frac{G_0}{\kappa^2 + \gamma^2\omega^2}}
+ = \frac{G_0}{\gamma}\iOInf{z}{\frac{1}{\kappa^2 + z^2}}
+ = \frac{G_0 \pi}{2 \gamma \kappa} \;,
\end{align}
where the integral is solved in \cref{sec:integrals}.
Plugging into our corollary to Parseval's theorem (\cref{eq:parseval-var}),
\begin{equation}
- \avg{x(t)^2} = \frac{G_0 \pi}{2 \gamma k} \;. \label{eq:ODHO-var}
+ \avg{x(t)^2} = \frac{G_0 \pi}{2 \gamma \kappa} \;. \label{eq:ODHO-var}
\end{equation}
Plugging \cref{eq:ODHO-var} into \cref{eq:equipart} we have
\begin{align}
- k \frac{G_0 \pi}{2 \gamma k} &= k_BT \\
+ \kappa \frac{G_0 \pi}{2 \gamma \kappa} &= k_BT \\
G_0 &= \frac{2 \gamma k_BT}{\pi} \;.
\end{align}
\begin{equation}
\PSD(x, \omega) = \frac{2}{\pi}
\cdot
- \frac{\gamma k_BT}{k^2 + \gamma^2\omega^2} \;.
+ \frac{\gamma k_BT}{\kappa^2 + \gamma^2\omega^2} \;.
\label{eq:ODHO-psd}
\end{equation}
\index{PSD@\PSD}
projects / thesis.git / commitdiff