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\begin{document}
\title{Density Of Ice}
\author{Ashley Ong$^\ast$}
\affiliation {\it Physics Department, University of California, Santa Barbara, CA 93106}
\date{submitted June 4, 2016}
\begin{abstract}
%First sentence (or two) says something about the importance of the parameter being measured.
The density of solid water, unlike most molecules, is less than that of its liquid form.
Its precise value is of use in many applications, including $\ldots$.
%The next sentence (or two) summarizes the measurement approach used.
Freezing a spherical droplet of water and analyzing the changed shape from a sphere to a sphere with a slight peak in order to find the density of ice.
%The final sentence (or two) state the result (measurement with its 68% confidence interval) and insight(s) concerning the measurement approach, if any.
We find the density of ice to be at $0.90 \pm 1.66 \cdot 10^6$ g/mL.
The precision of our measurement was limited by uncertainty in the angle measurements of the peak of the droplet.
\end{abstract}
%For fun, see PACS list at https://www.aip.org/publishing/pacs/pacs-2010-regular-edition
\pacs{06.30.-k,42.25.Bs}
\maketitle
%P1: Intro: scientific motivation
Ice floating on water is such a common occurrence in everyday life. Typically, substances have liquid forms that are less dense than their solid form.
The fascinating, counter-intuitive consequences of ice's lighter density has biological applications, such as forming ice covers to protect aquatic life in frozen lakes over the winter \cite{Watz16}.
In addition to satisfying curiosity, precise knowledge of the density of ice is necessary in determining whether or not a frozen lake can sustain life beneath it, or keep life afloat above it... \cite{Watz16}.
%P2: More intro: approach, definitions, history, context, rationale
Water droplets are held in a spherical shape by its property of cohesion in the hydrogen bonds of water. As it freezes, the water becomes less dense, and expands. Due to the force of gravity, the spherical droplet takes on a slight peak once frozen solid \cite{Sanz86}
The value obtained, $0.90 \pm 1.66 \cdot 10^6$ g/mL, was limited in accuracy by the pictures taken of the droplet and the accuracy of the image processing program, ImageJ \cite{Watz16}.
The expected value of the density of ice was 0.917 g/mL \cite{Watz16}.
%P3: What you did, what you found
Here we report a simple approach to measuring the density of ice using volume ratio analysis of water droplets and ice peaks $0.90 \pm 1.66 \cdot 10^6$ g/mL.
The precision of this measurement was limited by uncertainty in the angle measurements of the peak of the droplets.
%P4: Approach 1 - Setup (The nonsense below aims to give you a sense for style. It is purely imaginary.)
Our approach was based on the one described by J. Watz {\it et al.} \cite{Watz16}. We placed water droplets using a hospital-grade micro pipette on a metal sheet cooled by dry ice to a temperature of -78.5'C. The droplet would freeze from the bottom upwards, forming the distinct peak-like shape. It would take around ten to thirty seconds to freeze.
Once completely frozen, we immediately took a picture. If the picture was not captured immediately, sometimes the peak would collapse on itself, rendering that trial invalid for analysis.
%P5: Approach 2 - Methods (It may take several paragraphs to describe how you made your measurement. Look at the paper from which you drew inspiration to get a sense of what to write.)
To find the density of ice, we followed the method outlined by Schetnikov, Matiunin, and Chernov \cite{Schetnikov10}. They found that the density ratio of ice to water, $\rho$ was only dependent on the vertex angle $\alpha$.They found the mathematical relationship by removing the top of the water droplet before it froze entirely, leaving a dip (Fig.~\ref{figure1}). By comparing that to the droplet allowed to freeze entirely, they found the following relationships to be true:
$$V_1 = \frac{{\pi}r^3}{3} \frac{(1-cos\alpha)^2(2+cos\alpha)}{sin^3\alpha}$$
$$V_2 = \frac{{\pi}r^3}{3} \frac{(1-sin \alpha)^2(2+sin\alpha)}{cos^3\alpha}$$
$$V_c = \frac{{\pi}r^3}{3} \frac{cos\alpha}{sin\alpha}$$
$$\rho = \frac{V_1(\alpha) + V_2(\alpha)}{V_1(\alpha) + V_c(\alpha)}$$
Notice that the density ratio only depends on $\alpha$. As a result, we used a USB microscope camera mounted at equal elevation to the water droplets in order to find the angle $\alpha$ (Fig.~\ref{figure2}). These pictures were analyzed in ImageJ to find $\alpha$. Since the density of water is 1 g/mL, we could take $\rho$ to be equal to the density of ice in g/mL.
The measurement was repeated seventeen times. The first three trials and the last ten trials were taken in an uncovered apparatus. We initially thought fluctuations in temperature would affect the freezing of the droplet, and attempted to cover the apparatus for trials four to nine. Instead, we found that carbon dioxide gas released by the dry ice was dissolving in the droplet as it froze, causing the measured density to be lighter than expected. This problem was less noticeable when we froze the droplets in an uncovered apparatus, but may have affected our results.
%P6: Approach 3 - Methods (There is no hard and fast rule about how many paragraphs to devote to describing your approach. Use as many as you need in order to convey whatever information necessary for someone else to get the same result you got using your approach.)
%%%%%%%%%%%%%%%%%%%%Figure 1 begin %%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}
%
\includegraphics[width=3.0in]{figure1.png}
%
\caption{\label{figure1}
%Start with a clause summarizing what is presented in the figure.
Mathmatical visual of the top of the droplet and peak.
%Then add a sentence or two pointing out key features.
On the left is the top of the spherical, unfrozen water droplet. $V_2$ is the volume of water taken from the drop before it froze.
On the right is the peak of the frozen droplet. Notice the vertex angle, $\alpha$ that the density ratio $\rho$ depends on.}
%
\end{figure}
%%%%%%%%%%%%%%%%%%%%Figure 1 end %%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%Figure 2 begin %%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[t]
%
\includegraphics[width=3.0in]{densityoficesetup.jpg}
%
\caption{\label{figure2} Apparatus used to freeze water
A properly formatted figure would have axis labels and error bars, of course, and a proper functional fit, instead of connected dots, if appropriate.
}
%
\end{figure}
%%%%%%%%%%%%%%%%%%%% Figure 2 end %%%%%%%%%%%%%%%%%%%%%%%
%P7: Results 1 - Start by describing the raw data. Raw data is what you directly measured, with no calculations or other processing.
We measured the vertex angle $\alpha$ for each of these trials. For the first few trials, we also measured the radius $r$, and the total volume of the droplet $V_{tot}$.
Upon further investigation, we realized that the volume ratio $\rho$ was only dependent on $\alpha$
Uncertainty in the vertex angle was due to the display of ImageJ, which only showed three decimal places. Uncertainty was then calculated to be ${\delta}\alpha = \frac{0.001}{\sqrt{12}}$.
%P8: Results 2 - If you have two types of raw data, they may each require their own paragraph. Once you are done describing the raw data, then describe the measurement itself, making it clear what, if any, calculations were done.
Using the equations given by Schetnikov, Matiunin, and Chernov; we calculated $\rho$ as given by each $\alpha$. This data was used to calculate the uncertainty in $\rho$. Since $\rho$ is a function of $\alpha$,
$$\delta \rho = \frac{d\rho}{d\alpha} \delta \alpha$$
A linear fit to (Fig.~\ref{processedData}) shows that $\frac{d\rho}{d\alpha}$ is the slope, 0.0057575 g/mL*degree.
%%%%%%%%%%%%%%%%%%%% Figure 3 begin %%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[!h]
%
\includegraphics[width=3.38in]{densityvsangle.jpg}
%
\caption{\label{processedData} The graph of the density $\rho$ and vertex angle, $\alpha$.
Ideally, the slope of this graph would be zero, since there is no mathematical dependence of the density on angle. However, due to slight variations in droplets, the slope of this graph is part of the calculation to find the uncertainty in $\rho$.}
%
\end{figure}
%%%%%%%%%%%%%%%%%%%% Figure 3 end %%%%%%%%%%%%%%%%%%%%%%%
%P9: Discussion - After presenting your result you may have ideas about it you want to share. You may want to say something about how your result compares with the results of others (so-called ``accepted values'', published elsewhere).
This result is consistent with measurements reported by Schetnikov {\it et al.} \cite{Schetnikov10}, who froze water droplets inside of a cooling chamber at -15'C, likely similar to a refrigerator. It is also consistant with the accepted value of the density of ice, 0.9167 g/mL. The approach used here is less accurate than Schetnikov {\it et al.}, due to the issues with sublimation as mentioned previously.
%P10: Discussion - another good subject for discussion is your ideas for how the measurement approach could be improved.
Air bubbles within the water droplet was likely the cause of discrepancy between our measurement and the accepted value. We used water degassed by boiling, but perhaps future measurements could use water degassed by a vacuum. Furthermore, we believe that carbon dioxide gas from the dry ice was dissolving into the droplets before or as they froze. A full cooling chamber, such as the one used by Schetnikov {\it et al.}, would give a more accurate measurement.
%P11: Discussion - use as many paragrahs as you like for the discussion. It is fine to include matters of opinion, but be sure to support each assertion with clear reasoning.
%P12: Summary & Vision
In summary, we have measured the density of ice to be $0.90 \pm 1.66 \cdot 10^6$ g/mL.
Our measurement suffered systematic errors due to the method of cooling water droplets.
The most marked improvements to this measurement approach would come from purifying the water further. A better method of degassing the water and of cooling the droplet into ice would give a more accurate measurement.
%P21: Acknowledgements
I thank my lab partner, Dylan Karr, for insightful conversations and company while watching water freeze. I am grateful to Professor Fygenson and Ryan DeCrescent for patient explanations and thoughtful encouragement.
This work was supported by Physics 25L lab fees.
\begin{thebibliography}{10}
\bibitem{Watz16} J. Watz {\it et al.}, ``Ice cover affects the growth of stream-dwelling fish'', {\it Oecologica} {\bf 181}, 299–311 (2016).
\bibitem{Sanz86} A. Sanz {\it et al.}, ``The influence of gravity on the solidification of a drop'', {\it Journal of Crystal Growth} {\bf 82}, 81-88 (1986).
\bibitem{Schetnikov10} A. Schetnikov {\it et al.}, ``Conical shape of frozen water droplets'', {\it American Journal of Physics} {\bf 83}, 36 (2010).
\end{thebibliography}
\end{document}
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