\section{Contour integration}
-As a brief review, some definite integrals from $-\infty$ to $\infty$
-can be evaluated by integrating along the contour \C\
-shown in \cref{fig:UHP-contour}.
+As a brief review, some definite integrals from $-\infty$ to $\infty$%
+\nomenclature{$\infty$}{Infinity} can be evaluated by integrating
+along the contour \C\ shown in \cref{fig:UHP-contour}.
\begin{figure}
\asyfig{figures/contour/contour}
\caption{Integral contour \C\ enclosing the upper half of the
complex plane. If the integrand $f(z)$ goes to zero ``quickly
- enough'' as the radius of \C\ approaches
- infinity\nomenclature{$\infty$}{Infinity}, then the only
+ enough'' as the radius of \C\ approaches infinity, then the only
contribution comes from integration along the real axis (see text
for details).\label{fig:UHP-contour}}
\end{figure}
We can evaluate the integral using the residue theorem\index{residue theorem},
\begin{equation}
- \iC{f(x)} = \sum_{z_p \in \text{poles in \C}} 2\pi i \Res{z_p}{f(z)},
+ \iC{f(x)} = \sum_{z_p \in \text{poles in \C}} 2\pi i \Res{z_p}{f(z)} \;,
\label{eq:res-thm}
\end{equation}
where for simple poles (single roots)
\begin{equation}
- \Res{z_p}{f(z)} = \limZp(z-z_p) f(z), \label{eq:res-simple}
+ \Res{z_p}{f(z)} = \limZp(z-z_p) f(z) \;, \label{eq:res-simple}
\end{equation}
and in general for a pole of order $n$
\begin{equation}
\Res{z_p}{f(z)} = \frac{1}{(n-1)!} \cdot\limZp
- \nderiv{n-1}{z}{}\left[ (z-z_p)^n \cdot f(z) \right]
+ \nderiv{n-1}{z}{}\left[ (z-z_p)^n \cdot f(z) \right] \;.
\label{eq:res-general}
\end{equation}
\subsection{Highly damped integral}
-\begin{align}
- I &= \iOInf{z}{\frac{1}{k^2 + z^2}} \\
- &= \frac{1}{2} \iInfInf{z}{\frac{1}{k^2 + z^2}} \\
- &= \frac{1}{2k} \iInfInf{u}{\frac{1}{u^2+1}} \\
-\end{align}
-where $u \equiv z/k$, $du = dz/k$.
-There are simple poles at $u = \pm i$
-\begin{align}
- I &= \frac{1}{2k} \cdot 2 \pi i \Res{i}{f(u)} \\
- &= \frac{1}{2k} \cdot \frac{2 \pi i}{i+i} \\
- &= \frac{1}{2k} \pi \\
- &= \frac{\pi}{2 k},
-\end{align}
+\begin{equation}
+ I = \iOInf{z}{\frac{1}{k^2 + z^2}}
+ = \frac{1}{2} \iInfInf{z}{\frac{1}{k^2 + z^2}}
+ = \frac{1}{2k} \iInfInf{u}{\frac{1}{u^2+1}} \;,
+\end{equation}
+where $u \equiv z/k$ and $du = dz/k$.
+There are simple poles at $u = \pm i$.
+\begin{equation}
+ I = \frac{1}{2k} \cdot 2 \pi i \Res{i}{f(u)}
+ = \frac{1}{2k} \cdot \frac{2 \pi i}{i+i}
+ = \frac{\pi}{2 k} \;.
+\end{equation}
-\subsection{General case integral}
+\subsection{General case integral}
-We will show that for any $(a,b > 0) \in \Reals$
+We will show that, for any $(a,b > 0) \in \Reals$,%
+\nomenclature[aR]{\Reals}{Real numbers}
\begin{equation}
- I = \iInfInf{z}{\frac{1}{(a^2-z^2) + b^2 z^2}} = \frac{\pi}{b a^2}.
+ I = \iInfInf{z}{\frac{1}{(a^2-z^2) + b^2 z^2}} = \frac{\pi}{b a^2} \;.
\end{equation}
First we note that $|f(z)| \rightarrow 0$ like $|z^{-4}|$ for $|z| \gg 1$,
and that $f(z)$ is even, so
\begin{equation}
- I = \iC{\frac{1}{(a^2-z^2)^2 + b^2 z^2}},
+ I = \iC{\frac{1}{(a^2-z^2)^2 + b^2 z^2}} \;,
\end{equation}
where \C\ is the contour shown in \cref{fig:UHP-contour}.
= (a^2-z^2 \colA{+} ibz)(a^2-z^2 \colA{-} ibz)
\end{equation}
And the roots of $z^2 \colA{\pm} ibz - a^2$
-\begin{align}
+\begin{equation}
z_{r\colB{\pm}}
- &= \colA{\pm}\frac{ib}{2} \left(
+ = \colA{\pm}\frac{ib}{2} \left(
1 \colB{\pm} \sqrt{1-4\frac{-a^2}{(ib)^2}}
- \right) \\
- &= \pm\frac{ib}{2} \left(
+ \right)
+ = \pm\frac{ib}{2} \left(
1 \pm \sqrt{1-4\frac{a^2}{b^2}}
- \right) \\
- &= \pm\frac{ib}{2} \left(
- 1 \pm S
\right)
-\end{align}
-Where $S \equiv \sqrt{1-4\frac{a^2}{b^2}}$.
+ = \pm\frac{ib}{2} \left(
+ 1 \pm S
+ \right) \;,
+\end{equation}
+where $S \equiv \sqrt{1-4\frac{a^2}{b^2}}$.
%critical damping when $\omega_0^2 = \beta'^2$ % TM
%where our $a = \omega_0$ and $b = \beta$,
%and $\beta = \gamma/m = 2 \beta'$
%Critical damping when $a^2 = b^2/4$, so $S = 0$
To determine the nature and locations of the roots, consider the following
-cases (in order of increasing $a$).
+cases
\begin{itemize}
\item $a < b/2$, overdamped.
\item $a = b/2$, critically damped.
\end{itemize}
In the overdamped case $S \in \Reals$ and $S > 0$,
-so $z_{r\pm}$ is purely imaginary, and $z_{r+} != z_{r-}$.
+so $z_{r\pm}$ is purely imaginary, and $z_{r+} \ne z_{r-}$.
For any $a < b/2$, we have $0 < S < 1$, so $\Imag(z_{r\pm}) > 0$.
Thus, there are two single poles in the upper half plane ($z_{r\pm}$),
and two single poles in the lower half plane ($-z_{r\pm}$).
Our factored function $f(z)$ is
\begin{equation}
- f(z) = \frac{1}{(z-z_{r+})(z+z_{r+})(z+z_{r-})(z-z_{r-})}
+ f(z) = \frac{1}{(z-z_{r+})(z+z_{r+})(z+z_{r-})(z-z_{r-})} \;.
\end{equation}
Applying \cref{eq:res-thm,eq:res-simple} we have
&= \frac{\pi i}{\colA{z_{r+}^2-z_{r-}^2}} \left(
\frac{1}{z_{r+}}
\colA{-} \frac{1}{z_{r-}}
- \right) \\
- &= \frac{\pi i}{ \left( \colB{\frac{ib}{2}} (1+S) \right)^2
+ \right)
+ = \frac{\pi i}{ \left( \colB{\frac{ib}{2}} (1+S) \right)^2
- \left( \colB{\frac{ib}{2}} (1-S) \right)^2 }
\cdot \frac{z_{r-}-z_{r+}}{z_{r+}z_{r-}} \\
&= \frac{\colB{-4}\pi i / \colB{b^2}}{ (1+2S+S^2) - (1-2S+S^2) }
\cdot \frac{ \colA{\frac{ib}{2}} [(1-S) - (1+S)] }
- { \left(\frac{ib}{2}\right)^{\colA{2}} (1+S)(1-S) } \\
- &= \frac{-8\pi / b^3}{ 4S }
+ { \left(\frac{ib}{2}\right)^{\colA{2}} (1+S)(1-S) }
+ = \frac{-8\pi / b^3}{ 4S }
\cdot \frac{-2S}
{(1 - S^2)} \\
- &= \frac{ 4\pi }{ b^3 (1 - S^2)} \\
- &= \frac{ 4\pi }{ b^3 [1 - (1-4\frac{a^2}{b^2})]} \\
- &= \frac{ 4\pi }{ b^3 \cdot 4\frac{a^2}{b^2}} \\
- &= \frac{ \pi }{ b a^2 } \label{eq:gen-int-noncrit}
+ &= \frac{ 4\pi }{ b^3 (1 - S^2)}
+ = \frac{ 4\pi }{ b^3 [1 - (1-4\frac{a^2}{b^2})]}
+ = \frac{ 4\pi }{ b^3 \cdot 4\frac{a^2}{b^2}}
+ = \frac{ \pi }{ b a^2 } \;. \label{eq:gen-int-noncrit}
\end{align}
-Hooray!
+
\subsubsection{Critically damped}
Our factored function $f(z)$ is
\begin{equation}
- f(z) = \frac{1}{(z-z_{r+})^2(z-z_{r-})^2}
+ f(z) = \frac{1}{(z-z_{r+})^2(z-z_{r-})^2} \;.
\end{equation}
Applying \cref{eq:res-thm,eq:res-general} we have
\begin{align}
- I &= 2\pi i \Res{z_{r+}}{f(z)} \\
- &= \colA{2}\pi i \left( \colA{\frac{1}{2!}}
+ I &= 2\pi i \Res{z_{r+}}{f(z)}
+ = \colA{2}\pi i \left( \colA{\frac{1}{2!}}
\limZ{z_{r+}}
\deriv{z}{} \frac{1}{(z + z_{r+})^2}
- \right) \\
- &= \pi i \limZ{z_{r+}} -2 \cdot \frac{1}{(z_{r+} + z_{r+})^3} \\
- &= - 2 \pi i \frac{1}{z_{r+}^3} \\
- &= \colA{-} 2 \pi \colA{i} \frac{1}{(\frac{\colA{i}b}{2})^3} \\
- &= \frac{\pi}{b (\frac{b}{2})^2} \\
- &= \frac{\pi}{b a^2}, \label{eq:gen_int_crit}
+ \right)
+ = \pi i \limZ{z_{r+}} -2 \cdot \frac{1}{(z_{r+} + z_{r+})^3} \\
+ &= - 2 \pi i \frac{1}{z_{r+}^3}
+ = \colA{-} 2 \pi \colA{i} \frac{1}{(\frac{\colA{i}b}{2})^3}
+ = \frac{\pi}{b (\frac{b}{2})^2}
+ = \frac{\pi}{b a^2} \;, \label{eq:gen_int_crit}
\end{align}
-which matches \cref{eq:gen-int-noncrit}
+which matches \cref{eq:gen-int-noncrit}.
% Particular to this section.
-@Misc{mathworld_lorentzian,
+@Misc{mathworld-lorentzian,
author = "Eric W.\ Weisstein",
title = "Lorentzian Function",
publisher = "MathWorld--A Wolfram Web Resource",
note = "Defines the standard Lorentzian function."
}
-@Inbook{cos_halfangle,
+@Inbook{cos-halfangle,
crossref = "thornton04",
chapter = "Appendix D",
pages = 609,
note = "See Eq.~12.0.13",
}
-@Misc{four_deriv,
+
+@Misc{four-deriv,
note = "Hmm, it is suprisingly difficult to find an `official' reference for this.
I obviously need to get a spectral analysis book :p.
See Wikipedia's currently excellent page (Feb 15th, 2008) \\
\url{http://en.wikipedia.org/wiki/Fourier_transform#Functional_relationships},\\
or derive it for yourself in about three lines :p.",
+ year = 2008,
}
@Inbook{parseval,
note = "See Eq.~12.0.14",
}
-@Inbook{wiener_khinchin,
+@Inbook{wiener-khinchin,
crossref = "press02",
chapter = 12,
pages = 498,
note = "See Eq.~12.0.12",
}
-@Misc{wikipedia_wiener_khinchin,
+@Misc{wikipedia-wiener-khinchin,
title = "Wiener-Khinchin theorem",
publisher = "Wikipedia",
url = "http://en.wikipedia.org/wiki/Wiener\%E2\%80\%93Khinchin_theorem",
year = "TODO",
}
-@Misc{tweezer_lab_notes,
+@Misc{tweezer-lab-notes,
author = "C.\ Grossman and A.\ Stout",
title = "Optical Tweezers Advanced Lab",
month = "Fall",
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 $F = -kx$.
+\nomenclature{$F$}{Force (newtons)}
+\nomenclature{$k$}{Spring constant (newtons per meter)}
+\nomenclature{$x$}{Displacement (meters)}
-The basic idea is to use the equipartition theorem\cite{hutter93},
+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} k \avg{x^2} = \frac{1}{2} k_BT \;, \label{eq:equipart}
\end{equation}
-where $k_B$ is Boltzmann's constant,
+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 $t_T$,
+\nomenclature{$k_B$}{Boltzmann's constant, $k_B = 1.380 65\E{-23}\U{J/K}$\citep{codata-boltzmann}}
+\nomenclature{$\avg{s(t)}$}{Mean (expectation value) of a time-series $s(t)$}
\begin{equation}
- \avg{A} \equiv \iLimT{A}.
+ \avg{A} \equiv \iLimT{A} \;.
\end{equation}
Solving the equipartition theorem for $k$ yields
\begin{equation}
- k = \frac{k_BT}{\avg{x^2}}, \label{eq:equipart_k}
+ k = \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 $k$.
\subsection{Related papers}
Various corrections taking into acount higher order modes
-\cite{butt95,stark01}, and cantilever tilt \cite{hutter05} have been
-proposed and reviewed \cite{florin95,levy02,ohler07}, but we will
+\citep{butt95,stark01}, and cantilever tilt\citep{hutter05} have been
+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.
Roters and Johannsmann describe a similar approach to deriving the Lorentizian
-power spectral density\cite{roters96}. %,
+power spectral density\citep{roters96}. %,
%as do
% see Gittes 1998 for more thermal noise details
% see Berg-Sorenson for excellent overdamped treament.
\emph{WARNING}: It is popular to refer to the power spectral density
-as a ``Lorentzian''\cite{hutter93,roters96,levy02,florin95} even
+as a ``Lorentzian''\citep{hutter93,roters96,levy02,florin95} even
though \cref{eq:model-psd} differs from the classic
-Lorentzian\cite{mathworld_lorentzian}.
+Lorentzian\citep{mathworld-lorentzian}.
\begin{equation}
L(x) = \frac{1}{\pi}\frac{\frac{1}{2}\Gamma}
- {(x-x_0)^2 + \p({\frac{1}{2}\Gamma})^2}
+ {(x-x_0)^2 + \p({\frac{1}{2}\Gamma})^2} \;,
\end{equation}
+where $x_0$ sets the center and $\Gamma$ sets the width of the curve.
It is unclear whether the references are due to uncertainty about the
definition of the Lorentzian or to the fact that
\cref{eq:model-psd} is also peaked. In order to avoid any
$\sigma_p$ (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}
+ x(t) = \frac{V_p(t)}{\sigma_p} \;.
\end{equation}
Rather than computing the variance of $x(t)$ directly, we attempt to
-filter out noise by fitting the spectral power density (\PSD) of
-$x(t)$ to the theoretically predicted \PSD\ for a damped harmonic
-oscillator (\cref{eq:model-psd})
+filter out noise by fitting the power spectral density (\PSD)%
+\nomenclature[aPSD]{$\PSD$}{Power spectral density in angular
+ frequency space}\index{PSD@\PSD}\nomenclature{$\omega$}{Angular
+ frequency (radians per second)} of $x(t)$ to the theoretically
+predicted \PSD\ for a damped harmonic oscillator (\cref{eq:model-psd})
\begin{align}
\ddt{x} + \beta\dt{x} + \omega_0^2 x &= \frac{F_\text{thermal}}{m} \\
- \PSD(x, \omega) &= \frac{G_1}{(\omega_0^2-\omega^2)^2 + \beta^2\omega^2},
+ \PSD(x, \omega) &= \frac{G_1}{(\omega_0^2-\omega^2)^2 + \beta^2\omega^2} \;,
\end{align}
+\index{Damped harmonic oscillator}
where $G_1\equiv G_0/m^2$, $\omega_0$, and $\beta$ are used as the
fitting parameters (see \cref{eq:model-psd}). The variance of $x(t)$
is then given by \cref{eq:DHO-var}
+\index{$\beta$}
+\index{$\gamma$}
+
\begin{equation}
- \avg{x(t)^2} = \frac{\pi G_1}{2\beta\omega_0^2},
+ \avg{x(t)^2} = \frac{\pi G_1}{2\beta\omega_0^2} \;,
\end{equation}
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}.
+ k = \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
\begin{equation}
- G_1 \equiv G_0/m^2 = \frac{2}{\pi m} k_BT \beta. \label{eq:Gone}
+ G_1 \equiv G_0/m^2 = \frac{2}{\pi m} k_BT \beta \;. \label{eq:Gone}
\end{equation}
{ (\omega_0^2-\omega^2)^2 + \beta^2\omega^2 } \\
\avg{V_p(t)^2} &= \frac{\pi G_{1p}}{2\beta\omega_0^2}
= \frac{\pi \sigma_p^2 G_{1}}{2\beta\omega_0^2}
- = \sigma_p^2 \avg{x(t)^2},
+ = \sigma_p^2 \avg{x(t)^2} \;,
\end{align}
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}}
- = \frac{2 \beta\omega_0^2 \sigma_p^2 k_BT}{\pi G_{1p}}.
+ = \frac{2 \beta\omega_0^2 \sigma_p^2 k_BT}{\pi G_{1p}} \;.
\end{align}
From \cref{eq:Gone}, we find the expected value of $G_{1p}$ to be
\begin{equation}
- G_{1p} \equiv \sigma_p^2 G_1 = \frac{2}{\pi m} \sigma_p^2 k_BT \beta.
+ G_{1p} \equiv \sigma_p^2 G_1 = \frac{2}{\pi m} \sigma_p^2 k_BT \beta \;.
\label{eq:Gone-p}
\end{equation}
&\equiv \frac{1}{\sqrt{2\pi}} \iInfInf{t}{x(t) e^{-i \omega t}} \\
\Fourf{x(t)}(f) &\equiv \iInfInf{t}{x(t) e^{-2\pi i f t}}
= \iInfInf{t}{x(t) e^{-i \omega t}}
- = \sqrt{2\pi}\cdot\Four{x(t)}(\omega=2\pi f),
+ = \sqrt{2\pi}\cdot\Four{x(t)}(\omega=2\pi f) \;,
\end{align}
from which we can translate the \PSD
\begin{align}
\PSD(x, \omega) &\equiv \normLimT 2 \magSq{ \Four{x(t)}(\omega) } \\
+ \begin{split}
\PSD_f(x, f) &\equiv \normLimT 2 \magSq{ \Fourf{x(t)}(f) }
- = 2\pi \cdot \normLimT 2 \magSq{ \Four{x(t)}(\omega=2\pi f) }
- = 2\pi \PSD(x, \omega=2\pi f).
+ = 2\pi \cdot \normLimT 2 \magSq{ \Four{x(t)}(\omega=2\pi f) } \\
+ &= 2\pi \PSD(x, \omega=2\pi f) \;.
+ \end{split}
\end{align}
+\nomenclature[aPSD]{$\PSD_f$}{Power spectral density in frequency space}
+\nomenclature{$f$}{Frequency (hertz)}
+\nomenclature{$t$}{Time (seconds)}
+\index{PSD@\PSD!in frequency space}
The variance of the function $x(t)$ is then given by plugging into
\cref{eq:parseval-var} (our corollary to Parseval's theorem)
\begin{align}
\avg{x(t)^2} &= \iOInf{\omega}{\PSD(x,\omega)}
= \iOInf{f}{\frac{1}{2\pi}\PSD_f(x,f)2\pi\cdot}
- = \iOInf{f}{\PSD_f(x,f)}.
+ = \iOInf{f}{\PSD_f(x,f)} \;.
\end{align}
Therefore
\begin{align}
+ \begin{split}
\PSD_f(V_p, f) &= 2\pi\PSD(V_p,\omega)
= \frac{2\pi G_{1p}}{(4\pi f_0^2-4\pi^2f^2)^2 + \beta^2 4\pi^2f^2}
- = \frac{2\pi G_{1p}}{16\pi^4(f_0^2-f^2)^2 + \beta^2 4\pi^2f^2}
- = \frac{G_{1p}/8\pi^3}{(f_0^2-f^2)^2 + \frac{\beta^2 f^2}{4\pi^2}} \\
- &= \frac{G_{1f}}{(f_0^2-f^2)^2 + \beta_f^2 f^2} \\
- \avg{V_p(t)^2} &= \frac{\pi G_{1f}}{2\beta_f f_0^2}.
+ = \frac{2\pi G_{1p}}{16\pi^4(f_0^2-f^2)^2 + \beta^2 4\pi^2f^2} \\
+ &= \frac{G_{1p}/8\pi^3}{(f_0^2-f^2)^2 + \frac{\beta^2 f^2}{4\pi^2}}
+ = \frac{G_{1f}}{(f_0^2-f^2)^2 + \beta_f^2 f^2}
+ \end{split} \\
+ \avg{V_p(t)^2} &= \frac{\pi G_{1f}}{2\beta_f f_0^2} \;.
% = \frac{\pi G_{1p} / (2\pi)^3}{2\beta/(2\pi) \omega_0^2/(2\pi)^2}
% = \frac{\pi G_{1p}}{2\beta\omega_0^2} = \avg{V_p(t)^2} % check!
\end{align}
$G_{1f}\equiv G_{1p}/8\pi^3$. Finally
\begin{align}
k &= \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}}.
+ = \frac{2 \beta_f f_0^2 \sigma_p^2 k_BT}{\pi G_{1f}} \;.
\end{align}
From \cref{eq:Gone}, we expect $G_{1f}$ to be
G_{1f} = \frac{G_{1p}}{8\pi^3}
= \frac{\sigma_p^2 G_1}{8\pi^3}
= \frac{\frac{2}{\pi m} \sigma_p^2 k_BT \beta}{8\pi^3}
- = \frac{\sigma_p^2 k_BT \beta}{4\pi^4 m}.
+ = \frac{\sigma_p^2 k_BT \beta}{4\pi^4 m} \;.
\label{eq:Gone-f}
\end{equation}
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} + k x = F(t) \;, \label{eq:DHO}
% DHO for Damped Harmonic Oscillator
\end{equation}
where $x$ is the displacement from equilibrium,
$F(t)$ is the external driving force.
During the non-contact phase of calibration,
$F(t)$ comes from random thermal noise.
+\nomenclature{$\beta$}{Damped harmonic oscillator drag-acceleration
+ coefficient $\beta \equiv \gamma/m$}\index{$\beta$}%
+\nomenclature{$\gamma$}{Damped harmonic oscillator drag coefficient
+ $F_\text{drag} = \gamma\dt{x}$}\index{$\gamma$}%
+\index{damped harmonic oscillator}%
+\nomenclature{$\dt{s}$}{First derivative of the time-series $s(t)$
+ with respect to time. $\dt{s} = \deriv{t}{s}$}%
+\nomenclature{$\ddt{s}$}{Second derivative of the time-series $s(t)$
+ with respect to time. $\ddt{s} = \nderiv{2}{t}{s}$}%
In the following analysis, we use the unitary, angular frequency Fourier transform normalization
\begin{equation}
- \Four{x(t)} \equiv \frac{1}{\sqrt{2\pi}} \iInfInf{t}{x(t) e^{-i \omega t}}
+ \Four{x(t)} \equiv \frac{1}{\sqrt{2\pi}} \iInfInf{t}{x(t) e^{-i \omega t}}\;.
\end{equation}
+\nomenclature{\Four{s(t)}}{Fourier transform of the time-series
+ $s(t)$. $s(f) = \Four{s(t)}$}\index{Fourier transform}
We also use the following theorems (proved elsewhere):
\begin{align}
- \cos\left(\frac{\theta}{2}\right) &= \pm\sqrt{\frac{1}{2}[1+\cos(\theta)]}
- &\text{\cite{cos_halfangle},} \label{eq:cos_halfangle} \\
- \Four{\nderiv{n}{t}{x(t)}} &= (i \omega)^n x(\omega)
- &\text{\cite{four-deriv},} \label{eq:four-deriv} \\
-% \Four{x*y} &= x(\omega) y(\omega), \label{eq:four_conv}
+ \cos\left(\frac{\theta}{2}\right) &= \pm\sqrt{\frac{1}{2}[1+\cos(\theta)]}\;,
+ &\text{\citep{cos-halfangle}} \label{eq:cos-halfangle} \\
+ \Four{\nderiv{n}{t}{x(t)}} &= (i \omega)^n x(\omega) \;,
+ &\text{\citep{four-deriv}} \label{eq:four-deriv} \\
+% \Four{x*y} &= x(\omega) y(\omega), \label{eq:four-conv}
% & \text{and} \\
- \iInfInf{t}{\magSq{x(t)}} &= \iInfInf{\omega}{\magSq{x(w)}}
- &\text{(Parseval's)\cite{parseval}.} \label{eq:parseval}
+ \iInfInf{t}{\magSq{x(t)}} &= \iInfInf{\omega}{\magSq{x(w)}} \;.
+ &\text{(Parseval's)\citep{parseval}} \label{eq:parseval}
\end{align}
+\index{cosine half-angle}
+\index{Parseval's theorem}
%where $x*y$ denotes the convolution of $x$ and $y$,
%\begin{equation}
% x*y \equiv \iInfInf{\tau}{x(t-\tau)y(\tau)}.
%\end{equation}
As a corollary to Parseval's theorem, we note that the one sided power spectral density per unit time (\PSD) defined by
\begin{align}
- \PSD(x, \omega) &\equiv \normLimT 2 \left| x(\omega) \right|^2
- &\text{\cite{PSD}} \label{eq:psd-def}
+ \PSD(x, \omega) &\equiv \normLimT 2 \left| x(\omega) \right|^2
+ &\text{\citep{PSD}} \label{eq:psd-def}
\end{align}
+\index{PSD@\PSD}
relates to the variance by
\begin{align}
\avg{x(t)^2}
&= \iLimT{\magSq{x(t)}}
= \normLimT \iInfInf{\omega}{\magSq{x(\omega)}}
- = \iOInf{\omega}{\PSD(x,\omega)}, \label{eq:parseval-var}
+ = \iOInf{\omega}{\PSD(x,\omega)} \;, \label{eq:parseval-var}
\end{align}
where $t_T$ is the total time over which data has been aquired.
-
We also use the Wiener-Khinchin theorem,
which relates the two sided power spectral density $S_{xx}(\omega)$
to the autocorrelation function $r_{xx}(t)$ via
\begin{align}
- S_{xx}(\omega) &= \Four{ r_{xx}(t) }
- &\text{(Wiener-Khinchin)\cite{wiener_khinchin},} \label{eq:wiener_khinchin}
+ S_{xx}(\omega) &= \Four{ r_{xx}(t) } \;,
+ &\text{(Wiener-Khinchin)\citep{wiener-khinchin}} \label{eq:wiener_khinchin}
\end{align}
+\index{Wiener-Khinchin theorem}
where $r_{xx}(t)$ is defined in terms of the expectation value
\begin{align}
- r_{xx}(t) &\equiv \avg{x(\tau)\conj{x}(\tau-t)}
- &\text{\cite{wikipedia_wiener_khinchin}}
+ r_{xx}(t) &\equiv \avg{x(\tau)\conj{x}(\tau-t)} \;,
+ &\text{\citep{wikipedia-wiener-khinchin}}
\end{align}
and $\conj{x}$ represents the complex conjugate of $x$.
+\nomenclature{$\conj{z}$}{Complex conjugate of $z$}
\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},
+ {(\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-acceleration coefficient.
+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{$\omega_0$}
We compute the \PSD\ by plugging \cref{eq:DHO-xmag} into \cref{eq:psd-def}
\begin{equation}
\PSD(x, \omega)
= \normLimT \frac{2 |F(\omega)|^2/m^2}
- {(\omega_0^2-\omega^2)^2 + \beta^2\omega^2}.
- \label{eq:DHO-psd}
+ {(\omega_0^2-\omega^2)^2 + \beta^2\omega^2} \;.
+ \label{eq:DHO-psd-F}
\end{equation}
+\index{PSD@\PSD}
-Plugging \cref{eq:GOdef} into \cref{eq:DHO-psd} we have
+Plugging \cref{eq:GOdef} into \cref{eq:DHO-psd-F} we have
\begin{equation}
- \PSD(x, \omega) = \frac{G_0/m^2}{(\omega_0^2-\omega^2)^2 + \beta^2\omega^2}.
- \label{eq:model-psd}
+ \PSD(x, \omega) = \frac{G_0/m^2}{(\omega_0^2-\omega^2)^2 +\beta^2\omega^2}\;.
\end{equation}
Integrating over positive $\omega$ to find the total power per unit time yields
\begin{align}
\iOInf{\omega}{\PSD(x, \omega)}
&= \frac{G_0}{2m^2}
- \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}
+ \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} \;.
\end{align}
The integration is detailed in \cref{sec:integrals}.
By the corollary to Parseval's theorem (\cref{eq:parseval-var}), we have
\begin{equation}
- \avg{x(t)^2} = \frac{G_0 \pi}{2m^2\beta\omega_0^2} \label{eq:DHO-var}
+ \avg{x(t)^2} = \frac{G_0 \pi}{2m^2\beta\omega_0^2} \;. \label{eq:DHO-var}
\end{equation}
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 \\
- G_0 &= \frac{2}{\pi} k_BT m \beta. \label{eq:GO}
+ G_0 &= \frac{2}{\pi} k_BT m \beta \;. \label{eq:GO}
\end{align}
So we expect $x(t)$ to have a power spectral density per unit time given by
\begin{equation}
\PSD(x, \omega) = \frac{2 k_BT \beta}
- {\pi m \left[
- (\omega_0^2-\omega^2)^2 + \beta^2\omega^2
- \right] }
+ { \pi m \p[{(\omega_0^2-\omega^2)^2 + \beta^2\omega^2}] }\;.
+ \label{eq:DHO-psd}
\end{equation}
+\index{PSD@\PSD}
+
+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 $\omega_0\equiv\sqrt{k/m}$,
+\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}] }
+ = \lim_{m\rightarrow 0} \frac{2 k_BT \gamma}
+ { \pi \p[{(k-m\omega^2)^2 + \gamma^2\omega^2}] } \\
+ &= \frac{2}{\pi}
+ \cdot
+ \frac{\gamma k_BT}{k^2 + \gamma^2\omega^2} \;.
+\end{align}
For highly damped systems, the inertial term becomes insignificant
($m \rightarrow 0$).
-This model is commonly used for optically trapped beads. % \cite{}
-Because it is simpler and solutions are more easily available,
- %cite{grossman05}{}{}{}
+This model is commonly used for optically trapped beads\citep{TODO}.
+Because it is simpler and solutions are more easily available%
+\citep{grossman05,TODO},
it will server to outline the general approach before we dive into the
general case.
% 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}.
+ |x(\omega)|^2 &= \frac{|F(\omega)|^2}{k^2 + \gamma^2 \omega^2} \;.
\label{eq:ODHO-xmag}
\end{align}
+\index{Damped harmonic oscillator!extremely overdamped}
We compute the \PSD\ by plugging \cref{eq:ODHO-xmag} into
\cref{eq:psd-def}
\begin{equation}
\PSD(x, \omega)
- = \normLimT \frac{2\magSq{F(\omega)}}{k^2 + \gamma^2\omega^2}.
- \label{eq:ODHO-psd}
+ = \normLimT \frac{2\magSq{F(\omega)}}{k^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
\begin{equation}
\PSD(F, \omega) = G_0
- = \normLimT 2 \magSq{F(\omega)} \label{eq:GOdef} % label O != zero
+ = \normLimT 2 \magSq{F(\omega)} \;. \label{eq:GOdef} % label O != zero
\end{equation}
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}{k^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
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}{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} \;,
\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 k} \;. \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 \\
- G_0 &= \frac{2 \gamma k_BT}{\pi}.
+ G_0 &= \frac{2 \gamma k_BT}{\pi} \;.
\end{align}
-So we expect $X(t)$ to have a power spectral density per unit time given by
+So we expect $x(t)$ to have a power spectral density per unit time given by
\begin{equation}
\PSD(x, \omega) = \frac{2}{\pi}
\cdot
- \frac{\gamma k_BT}{k^2 + \gamma^2\omega^2}.
+ \frac{\gamma k_BT}{k^2 + \gamma^2\omega^2} \;.
+ \label{eq:ODHO-psd}
\end{equation}
+\index{PSD@\PSD}
I27's unfolding mechanism seems to involve stretching into a metastable
intermediate state followed by Bell-model escape to the unfolded
-state\cite{marszalek99}, although there is not yet a consensus of
-the presense of the proposed intermediate\cite{TODO}.
+state\citep{marszalek99}, although there is not yet a consensus of
+the presense of the proposed intermediate\citep{TODO}.
\begin{figure}
\includegraphics[width=2in]{figures/i27/1TIT}
energy landscape is generally ignored. However, in AFM
biotin-streptavidin unbinding experiments last year, Walton et al.\
demonstrated a surprisingly strong effect on unbinding force due to
-cantilever stiffness\cite{walton08}. The unbinding force
+cantilever stiffness\citep{walton08}. The unbinding force
approximately doubled due to a change from a $35\U{pN/nm}$ cantilever
to a $58\U{pN/nm}$ cantilever. Alarmed by the magnitude of the shift,
we repeated their experiment on octomeric I27 to determine the
and stiffer linkers will increase the mean unfolding force.
Unfolded I27 domains can be well-modeled as wormlike chains (WLCs,
-\cref{sec:tension:wlc})\cite{carrion-vazquez99b}, where $p \approx
+\cref{sec:tension:wlc})\citep{carrion-vazquez99b}, where $p \approx
4\U{\AA}$ is the persistence length, and $L \approx 28\U{nm}$ is the
contour length of the unfolded domain. Obviously effective stiffness
of an unfolded I27 domain is highly dependent on the unfolding force,
% titles would overlap.
\fancyfoot[RE,LO]{}
-\usepackage[super,sort&compress]{natbib} % fancy citation extensions
+\usepackage[super,sort&compress,comma]{natbib} % fancy citation extensions
% super selects citations in superscript mode
-% sort&compress automatically sorts and compresses compound citations (\cite{a,b,...})
+% sort&compress automatically sorts and compresses compound citations (\citep{a,b,...})
+% comma seperates multiple citations with commas rather than the default semicolons.
-%\bibliographystyle{ieeetr} % pick the bibliography style, short and sweet
-%\bibliographystyle{plain} % pick the bibliography style, includes dates
-\bibliographystyle{plainnat}
+\bibliographystyle{unsrtnat} % Number citations in the order referenced.
% Nicer references with \cref, \Cref, etc.
\usepackage[capitalize]{cleveref}
\advance\leftmargin\labelsep
\itemsep\nomitemsep
\let\makelabel\nomlabel}}
-\if@final
- \relax
-\else
+\makeatother
+\iffinal{}{
%\usepackage{showidx} % Print index keys in margins
% for some reason, showidx disables Index generation...
%\usepackage{showkeys} % Print labels in margins
- \relax
-\fi
-\makeatother
+}
% environments for multiline displayed equations, and other enhancements
\usepackage{amsmath}
doi = "10.1073/pnas.1833310100",
URL = "http://www.pnas.org/cgi/content/abstract/100/18/10249",
eprint = "http://www.pnas.org/cgi/reprint/100/18/10249.pdf",
- note = "Derives the major theory behind my thesis. The Kramers rate equation is H{\"a}nggi Eq. 4.56c (page 275)\cite{hanggi90}.",
+ note = "Derives the major theory behind my thesis. The Kramers rate equation is \citet{hanggi90} Eq.~4.56c (page 275).",
project = "Energy Landscape Roughness",
}
doi = "10.1038/sj.embor.7400403",
URL = "http://www.nature.com/embor/journal/v6/n5/abs/7400403.html",
eprint = "http://www.nature.com/embor/journal/v6/n5/pdf/7400403.pdf",
- note = "Applies H\&T\cite{hyeon03} to ligand-receptor
+ note = "Applies \citet{hyeon03} to ligand-receptor
binding.",
project = "Energy Landscape Roughness",
}
URL = "http://www.sciencemag.org/cgi/content/abstract/276/5315/1109",
eprint = "http://www.sciencemag.org/cgi/reprint/276/5315/1109.pdf",
note = "Seminal paper for force spectroscopy on Titin. Cited
- by Dietz '04\cite{dietz04} (ref 9) as an example of how
+ by \citet{dietz04} (ref 9) as an example of how
unfolding large proteins is easily interpreted (vs.\
confusing unfolding in bulk), but Titin is a rather
simple example of that, because of its globular-chain
season = "Fall",
eprint = "http://chirality.swarthmore.edu/PHYS81/OpticalTweezers.pdf",
note = "Fairly complete overdamped PSD derivation in section
- 4.3., cites \cite{tlusty98} and \cite{bechhoefer02} for
+ 4.3., cites \citet{tlusty98} and \citet{bechhoefer02} for
further details. However, Tlusty (listed as reference
8) doesn't contain the thermal response fn.\ derivation
it was cited for. Also, the single sided PSD definition
credited to reference 9 (listed as Bechhoefer) looks
more like Press (listed as reference 10). I imagine
Grossman and Stout mixed up their references, and meant
- to refer to \cite{bechhoefer02} and \cite{press92}
+ to refer to \citet{bechhoefer02} and \citet{press92}
respectively instead.",
project = "Cantilever Calibration",
}
eprint = "http://prola.aps.org/pdf/PRL/v81/i8/p1738_1",
note = "also at
\url{http://nanoscience.bu.edu/papers/p1738_1_Meller.pdf}.
- Cited by \cite{grossman05} for derivation of thermal
+ Cited by \citet{grossman05} for derivation of thermal
response fn. However, I only see a referenced thermal
energy when they list the likelyhood of a small
partical (radius < $R_c$) escaping due to thermal
note = "Good discussion of the effect of correlation time on
calibration. Excellent detail on power spectrum
derivation and thermal noise for extremely overdamped
- oscillators in Appendix A (references \cite{reif65}).
+ oscillators in Appendix A (references \citet{reif65}).
References work on deconvolving thermal noise from
- other noise\cite{cowan98}",
+ other noise\citep{cowan98}",
}
@Book{press02,
note = "The inspiration behind my sawtooth simulation.
Bell model fit to $f_{unfold}(v)$, but
Kramers model fit to unfolding distribution for a given $v$.
- Eqn.~3 in the supplement is Evans-Ritchie 1999's Eqn.~2\cite{evans99}, but it is just ``[dying percent] * [surviving population] = [deaths]'' (TODO, check).
+ Eqn.~3 in the supplement is \citet{evans99} 1999's Eqn.~2, but it is just ``[dying percent] * [surviving population] = [deaths]'' (TODO, check).
$\nu \equiv k$ is the force/time-dependent off rate... (TODO)
- The Kramers' rate equation (second equation in the paper) is H{\"a}nggi Eq.~4.56b (page 275)\cite{hanggi90}.
+ The Kramers' rate equation (second equation in the paper) is \citet{hanggi90} Eq.~4.56b (page 275).
It is important to extract $k_0$ and $\Delta x$ using every
available method.",
}
@Article{hanggi90,
title = {Reaction-rate theory: fifty years after {K}ramers},
author = {H\"anggi, Peter and Talkner, Peter and Borkovec, Michal },
- journal = {Rev. Mod. Phys.},
+ journal = RMP,
volume = {62},
number = {2},
pages = {251--341},
ISSN = "1542-0086",
doi = "10.1529/biophysj.108.141580",
eprint = "http://www.biophysj.org/cgi/reprint/95/6/L42.pdf",
- note = "Cites \cite{dudko03} for Kramers' description of
+ note = "Cites \citet{dudko03} for Kramers' description of
irreversible rupture, and claims it is required to
- explain the deviations in <F> at the same loading
+ explain the deviations in $\avg{F}$ at the same loading
rate. Proposes Moese equation as an example
- potential. Cites \cite{walton08} for experimental
- evidence of <F> increasing with linker stiffness.",
+ potential. Cites \citet{walton08} for experimental
+ evidence of $\avg{F}$ increasing with linker stiffness.",
}
@Article{walton08,
url = "http://www.ncbi.nlm.nih.gov/pmc/articles/PMC2426622/",
eprint = "http://www.ncbi.nlm.nih.gov/pmc/articles/PMC2426622/pdf/426.pdf",
}
+
+@Misc{codata-boltzmann,
+ crossref = "codata06",
+ url = "http://physics.nist.gov/cgi-bin/cuu/Value?k",
+}
+
+@Article{codata06,
+ title = "{CODATA} recommended values of the fundamental physical constants: 2006",
+ author = "Mohr, Peter J. and Taylor, Barry N. and Newell, David B.",
+ journal = RMP,
+ volume = 80,
+ number = 2,
+ pages = {633--730},
+ numpages = 97,
+ year = 2008,
+ month = jun,
+ doi = "10.1103/RevModPhys.80.633",
+ publisher = APS,
+}
]{drexel-thesis}
% See drexel-thesis.pdf for more options.
-%\includeonly{%
+\includeonly{%
% cantilever/main,%
% temperature/main%
-%}
+ cantilever-calib/main
+}
\author{William Trevor King}
\title{Temperature and cantilever dependent protein unfolding}
\appendix
\include{cantilever-calib/main}
\include{viscocity/main}
+
\printnomenclature
+% avoid index's second column overlapping the nomenclature space.
+\iffinal{}{\pagebreak}
\printindex
\begin{vita}
parameters from typical experimental settings. The order of the peaks
in the force curves reflects the temporal sequence of the unfolding
events instead of the positions of the protein molecules in the
-polymer\cite{li00}. As observed experimentally
+polymer\citep{li00}. As observed experimentally
(\cref{fig:expt-sawtooth}), the forces at which identical protein
molecules unfold fluctuate, revealing the stochastic nature of protein
unfolding since no instrumental noise is included in the simulation.
x_u$, and $\alpha\equiv-\rho\ln(N_fk_{u0}\rho/\kappa v)$. The event
probability density for events with an exponentially increasing
likelihood function follows the Gumbel (minimum) probability
-density\cite{NIST:gumbel}, with $\rho$ and $\alpha$ being the scale
+density\citep{NIST:gumbel}, with $\rho$ and $\alpha$ being the scale
and location parameters, respectively
\begin{equation}
\mathcal{P}(F) = \frac{1}{\rho} \exp\p[{\frac{F-\alpha}{\rho}
where $L_{f1}$ is the separation of the two linking points of a folded
domain, and $x_f$ is the end-to-end length of the chain of folded
domains. In this model, any non-zero tension will fully extend these
-folded domains. As discussed in \cref{sec:sawsim:results-folded-tension},
-the contribution of the folded domains to the elastic behavior of the
+folded domains. As discussed in \cref{sec:tension:folded}, the
+contribution of the folded domains to the elastic behavior of the
polymer-cantilever system is relatively insignificant.
% address assumptions & caveats
The relaxation time of the cantilever can be determined by measuring
the cantilever deflection induced by liquid motion and fitting the
time dependence of the deflection to an exponential
-function\cite{jones05}. For a $200\U{$\mu$m}$ rectangular cantilever
+function\citep{jones05}. For a $200\U{$\mu$m}$ rectangular cantilever
with a bending spring constant of $20\U{pN/nm}$, the measured
relaxation time in water is $\sim50\U{$\mu$/s}$ (data not shown.
TODO: show data). This relatively large relaxation time constant
\end{multline*}
We simplify by dropping the 2\nd term
-(``In obtaining Eq.\ \textbf{9}, we have assumed that the second term in Eq.\ \textbf{8} is small.''),
+(``In obtaining Eq.~\textbf{9}, we have assumed that the second term in Eq.~\textbf{8} is small.''),
and defining $\alpha \equiv \kT$,
$\rho \equiv \logp{ \frac{\r \dx}{\kexp \kT} }$, and
$e^{\bt \ep} \equiv \avg{e^{\bt F_1}}$, yielding
\section{Polymer Models}
-\subsection{Worm-like chains}
+\subsection{Wormlike chains}
\label{sec:tension:wlc}
The unfolded forms of many domains can be modeled as Worm-Like Chains
(WLCs)\citep{marko95,bustamante94}
-\index{WLC|textbf}\nomenclature{WLC}{Wormlike Chain}, which treats the
+\index{WLC}\nomenclature{WLC}{Wormlike Chain}, which treats the
unfolded polymer as an elastic rod of persistence length $p$ and
contour length $L$. The relationship between tension $F$ and
extension (end-to-end distance) $x$ is given to within XX\% by
\begin{equation}
F(x, p_u, L_u, N_u) = F_\text{WLC}(x, p_u, N_uL_{u1})
\end{equation}
+
+\subsection{Freely-jointed chains}
+\label{sec:tension:fjc}
projects / thesis.git / commitdiff