\chapter{Tetrahedral Colloidal Clusters from Random Parking of Bidisperse Spheres}
\label{ch:random_parking}
\graphicspath{{figures/ch3/}} 

%%%%%%%%%%%%%%%%%%%%%%%%%%

Reprinted figures and text with permission from \href{http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.110.148303}{Schade \emph{et al.}, \emph{Phys. Rev. Lett.}, \textbf{110}, 148303 (2013)}. Copyright 2013 by the American Physical Society.

\section{Introduction}

Understanding the geometry of clusters formed from small particles is 
a fundamental problem in condensed matter physics, with implications 
for phenomena ranging from nucleation \cite{Kelton2003} to self-assembly 
\cite{Manoharan2003}. Colloidal particles are a useful experimental 
system for studying cluster geometry and its relation to phase 
behavior \cite{Gasser2001} for several reasons: they are large enough to 
be directly observed using optical microscopy; their assembly can be 
understood in terms of geometry \cite{Arkus2009, Hoy2010}; and they can be 
driven to cluster by a variety of controllable interactions, including 
capillary forces \cite{Manoharan2003}, depletion \cite{Meng2010}, 
fluctuation-induced forces \cite{Yunker2011}, or DNA-mediated attraction 
\cite{Soto2002}. Colloidal clusters are also useful materials in their 
own right. They can be used, for example, as building blocks for 
isotropic optical metamaterials known as metafluids \cite{Urzhumov2007,
  Alu2009, Fan2010_Science}. Tetrahedral clusters are of particular interest 
for metafluids since the tetrahedron is the simplest cluster with 
isotropic dipolar symmetry \cite{Urzhumov2007}. An unsolved challenge 
for this application is to determine the interactions and conditions 
that enable assembly of bulk quantities of highly symmetric, uniform 
clusters such as tetrahedra.

With this motivation in mind, we study experimentally the geometry and 
size distribution of binary clusters formed when small colloidal 
spheres are mixed with an excess of large spheres that stick 
\emph{irreversibly and randomly} to their surfaces 
(Figure~\ref{fig:binaryMixture}). 
\begin{figure}[b]
\begin{center}
\includegraphics[width=10.0cm]{binaryMixture}
\caption{\label{fig:binaryMixture} Two colloidal sphere 
  species are mixed together to form clusters.
}
\end{center}
\end{figure}
An obvious way to control 
the cluster geometry in such binary systems is to vary the size 
ratio. One might expect that at certain ratios the particles could 
arrange into dense clusters or ``spherical packings'' -- arrangements 
of spheres around a central sphere that maximize surface density 
\cite{Melnyk1977, Conway1993, Phillips2012}. Such packings have long been 
used in modeling the microstructure of dense, disordered atomic 
systems \cite{Egami1997, Miracle2004}. But unlike atoms, colloidal 
particles can stick irreversibly, such that two particles bound to a 
third show no motion relative to one another. This type of binding 
occurs frequently in strongly interacting, monodisperse colloidal 
suspensions, which consequently form fractal aggregates instead of 
dense glasses \cite{Weitz1984, Lin1989}. Similarly, in the binary systems 
we study, the irreversible and stochastic process of sticking 
precludes the formation of dense or symmetric packings. The large 
spheres \emph{park}, rather than pack, on the surfaces of the small 
spheres.

\section{Results and Discussion}

\subsection{Experimental Results}

Surprisingly, this random and nonequilibrium process can produce 
clusters of uniform size. Our experiments show that at a size ratio 
$\alpha = R_{\mathrm{large}}/R_{\mathrm{small}} = 2.45$, where 
$R_{\mathrm{large}}$ and $R_{\mathrm{small}}$ are the sphere radii, 
nearly all of the clusters contain four large spheres stuck to a 
smaller sphere (Table~\ref{table:histograms}). In these experiments we 
use a 100:1 stoichiometric ratio of the two sphere species, 
statistically ensuring that each cluster contains only one small 
sphere surrounded by two or more larger spheres. After waiting 
several days for the average cluster size to saturate, we measure the 
distribution of $N$, the number of large spheres bound to each small 
sphere. (See Section~\ref{sec:cluster_size_dist} for details.) We do not include single large spheres, non-specifically 
aggregated clusters of large spheres, or clusters with multiple small 
spheres. While there are many isolated large spheres due to the high 
stoichiometric ratio, the latter two types of cluster are rare.

\begin{table}[t]
  \caption{\label{table:histograms} Experimentally observed cluster size 
    distributions for charged colloids. Percentages of total are 
   listed. The distribution for $\alpha=2.45$ (bold) is sharply peaked 
   at $N=4$.} 
\begin{tabular}{c|r r r r}
Size ratio $\alpha$ & 1.94 & \textbf{2.45} & 3.06 & 4.29 \\ [0.5ex]
\hline
$N=6$ &  6.3 & \textbf{ 0.0} &  0.0 &  0.0 \\
$N=5$ & 39.2 & \textbf{ 0.8} &  0.0 &  0.0 \\
$N=4$ & 54.4 & \textbf{90.2} & 18.6 &  0.7 \\
$N=3$ &  0.0 & \textbf{ 6.6} & 69.9 & 35.9 \\
$N=2$ &  0.0 & \textbf{ 0.8} & 10.9 & 51.0 \\
$N=1$ &  0.0 & \textbf{ 0.8} &  0.6 & 11.1 \\
$N=0$ &  0.0 & \textbf{ 0.8} &  0.0 &  1.3 \\ [1ex]
\end{tabular}
\end{table}

\begin{figure}[ht]
\begin{center}
\includegraphics[width=8.6cm]{experimentalSystem}
\caption{\label{fig:experimentalSystem} (a) Oppositely charged 
  polystyrene spheres cluster due to electrostatic attraction. Optical 
  micrograph shows a tetramer ($N=4$). (b) Po\-ly\-sty\-rene spheres labeled 
  with complementary DNA strands (not to scale) cluster due to DNA 
  hybridization. Optical micrograph shows a trimer ($N=3$); the small, 
  central sphere is fluorescent.  
}
\end{center}
\end{figure}

The $N=4$ tetramers that we observe are not dense packings or, in 
general, symmetric arrangements. As can be seen from the images in 
Figure~\ref{fig:experimentalSystem}, there is space between the large 
particles, and the resulting tetrahedra are irregular. Moreover, the 
ratio $\alpha=2.45$ is well below the value $\alpha = 4.44$ found by 
Miracle \textit{et al.} \cite{Miracle2003} for efficient tetrahedral 
packing in binary atomic clusters. In fact, at $\alpha=4.29$, closer 
to this bound, we see much smaller clusters and few tetrahedra. The 
sparsity of large spheres in the clusters is a result of the 
irreversible, nonequilibrium, random binding: once the big particles 
stick to the smaller ones, we do not see them detach or move relative 
to one another. We expected such a stochastic process to lead to a 
much broader distribution of clusters. At other values of $\alpha$ it 
does (Table~\ref{table:histograms}), but at $\alpha=2.45$ we obtain 
90\% tetramers.

The high yield of tetramers occurs in two experimental systems with 
different types of interactions. In both systems the interactions are 
specific, strong, and short-ranged, and the particles do not rearrange 
once bound. In the first system the clustering is driven by 
electrostatic interactions. We mix large, positively-charged 
particles with small, negatively-charged particles, as shown in 
Figure~\ref{fig:experimentalSystem}(a). To adjust $\alpha$ we use 
several different particle sizes (Section~\ref{sec:electrostatic}). We add salt to 
reduce the Debye length to approximately \unit{3}{\nano \meter}, small 
enough to ensure that the interaction range does not significantly 
influence the effective particle size. In the second system the 
clustering is driven by hybridization of grafted DNA strands (Section~\ref{sec:DNA-colloid}). As shown in Figure~\ref{fig:experimentalSystem}(b), we 
mix small and large spheres labeled with complementary DNA 
oligonucleotides \cite{Dreyfus2009}. We work well below the DNA melting 
temperature so that the attractive interaction is many times the 
thermal energy \cite{Dreyfus2010}.

\subsection{Random Parking Theory}

To better understand why the distribution is sharply peaked at $N=4$ 
for $\alpha = 2.45$, we use simulations and analytical techniques that 
account for the irreversibility of the aggregation process.

\subsubsection{Simulation Results}

Our simulations use a ``random parking'' algorithm \cite{Mansfield1996,
  Rosen1986, Wouterse2005, Talbot2000} to model the formation of clusters. 
The algorithm involves attaching large spheres to randomly selected 
positions on the surface of a small sphere, subject to a no-overlap 
constraint (Section~\ref{sec:parking_calc}). We do not model the finite range of the 
interactions, which in both experimental systems is small compared to 
the particle size, or the diffusion of the particles prior to binding. 
In accord with experimental observations, the particles are not 
allowed to rearrange once bound. We repeat the process numerically to 
obtain distributions of cluster sizes as a function of a single 
parameter, $\alpha$.

The simulations find a 100\% yield of tetramers at the size ratio 
$\alpha \approx 2.41$. As in the experiments, the large particles in 
these tetramers are not densely packed, and the clusters are therefore 
distorted tetrahedra. We also find that while the yield of any 
particular cluster can be maximized by varying $\alpha$ 
[Figure~\ref{fig:histograms}(a)], the yield approaches 100\% only for 
dimers ($N=2$) and tetramers ($N=4$). Interestingly, the yield curve 
for tetramers has a cusp at its peak, showing that the size ratio 
$\alpha_c$ at the maximum is a mathematical critical point.

\begin{figure}
\begin{center}
% \includegraphics[totalheight=0.6\textheight]{histograms}
\includegraphics[width=12.0cm]{histograms}
\caption{\label{fig:histograms} (a) Yield curves, as determined by 
  simulations, for $N$-particle clusters, $2 \le N \le 8$, where the 
  critical size ratio $\alpha_c$ is marked with a black line. 
  Below are histograms for (b) DNA-labeled particles (left) at $\alpha
  = 1.90$ and (c) charged particles (right) at $\alpha = 2.45$, as 
  observed in experiments (colored bars) and as predicted from 
  simulations (gray bars). Error bars are 95\% confidence intervals 
  (Wilson score interval method).}
\end{center}
\end{figure}

The simulated distributions agree well with those found experimentally 
[Figs.~\ref{fig:histograms}(b) and ~\ref{fig:histograms}(c)] for both electrostatic and DNA-mediated interactions. For instance, at $\alpha=2.45$ with 
electrostatic interactions, we find a sharply peaked distribution 
consisting almost entirely of tetramers. This value of $\alpha$ is 
close to but not precisely at the critical value, so a small yield of 
trimers is predicted and observed experimentally. In contrast, at 
$\alpha=1.90$ we find a mixture of mostly $N=4$ and $N=5$ clusters in 
both the DNA system and simulations. Some discrepancy arises between 
the simulated and experimental histograms because the yield curves in 
Figure~\ref{fig:histograms}(a) are steep; a slight error in the 
effective size ratio can shift the cluster distribution. Nevertheless, 
the random sphere parking model successfully reproduces both the large 
yield of tetrahedra near $\alpha_c$ and the details of the measured 
histograms at various other $\alpha$.

\subsubsection{Origin of Geometrical Critical Point}

That we can reproduce the same phenomenon in two different 
experimental systems and in a one-parameter model suggests that the 
critical size ratio $\alpha_c$ has a universal, geometrical origin. 
Intuitively, one might expect that it is related to packing 
constraints on the large spheres. Other theoretical studies of random 
sphere parking \cite{Rosen1986, Mansfield1996} have calculated the maximum 
number of large spheres $N_{\max}$ that can fit around a small sphere 
at a given $\alpha$. However, this bound cannot by itself explain why 
the yield of tetramers can reach 100\% while that of other clusters, 
such as trimers or hexamers, cannot. At a given $\alpha$, it tells us 
only why no clusters larger than $N_{\max}(\alpha)$ can form, but it 
says nothing about the probability of forming smaller clusters with 
different arrangements.

Therefore we also examine a different bound, one not previously 
discussed in the context of random sphere parking: the ``minimum 
parking'' curve $N_{\min}(\alpha)$.  $N_{\min}$ is the smallest number 
of hard spheres that can be positioned on a smaller sphere such that 
another sphere cannot fit. To understand this bound, consider a 
simple, one-dimensional analogy to car parking on a busy city street, 
where if a space opens up that is large enough to fit a car, it is 
filled. The minimum parking number occurs when all drivers have been 
equally inconsiderate, leaving spaces between their parked cars that 
are all slightly too small for another car to fit. This lower bound is 
meaningful only at long times, when all available parking spaces have 
been filled. The long-time limit holds also for our experiments and 
simulations, which we carry out until the average cluster size has 
saturated.

Whereas the upper bound $N_{\max}(\alpha)$ is straightforwardly related 
to solutions of the well-known spherical packing problem 
\cite{Conway1993, Sloane2011}, the calculation of the lower bound 
$N_{\min}(\alpha)$ requires a different approach. In our clusters,
the distance between the centers of any two big spheres must be at 
least $2R_{\mathrm{large}}$. Consider then a sphere of radius 
$(R_{\mathrm{small}}+R_{\mathrm{large}})$ that circumscribes the centers 
of the parked spheres. If this sphere is completely covered with $N$ 
circles of radius $2R_{\mathrm{large}}$, it will be impossible to add an 
$(N+1)^{\mathrm{th}}$ large sphere. We are led naturally to the 
\emph{spherical covering} problem, a problem with a rich history in 
mathematics. Like spherical packings, spherical coverings are 
solutions to an extremum problem: they are arrangements of $N$ points 
on a sphere that minimize the largest distance between any location on 
the sphere surface and the closest point \cite{Conway1993}. But unlike 
spherical packings, spherical coverings need not correspond to 
arrangements of nonoverlapping spheres. We therefore solve for the 
minimum parking curve by examining the solutions to the spherical 
covering problem \cite{Sloane2011} at each $N$ and manually 
verifying that they correspond to nonoverlapping 
configurations.\footnote{See Appendix~\ref{ch:appendix_A} for additional details.}

Our analytical results for the bounds reveal why $\alpha_c$ is a 
special point: it is the only nontrivial point where the calculated 
maximum and minimum parking curves come together 
(Figure~\ref{fig:expAndTheory}). 
\begin{figure}[t]
\begin{center}
\includegraphics[width=12.0cm]{expAndTheory}
\caption{\label{fig:expAndTheory} $N_{\max}$ (solid gray) and 
  $N_{\min}$ (black) as functions of $\alpha$. Cluster images show 
  sphere configurations at discontinuities of these curves. Average 
  cluster sizes from simulations (dashed gray line) and experiments 
  (blue and red data points) are shown. We characterize the 
  statistical dispersion in each distribution by the average absolute 
  deviation from the median, indicated by dotted light gray lines for 
  simulations and vertical bars for experiments.}
\end{center}
\end{figure}
Analytically we find the location 
of the critical value to be $\alpha_c=(1+\sqrt{2})\approx 2.41$, very 
close to the values where the experimental distributions are 
peaked. At $\alpha$ slightly larger than this value, the minimum 
parking configuration corresponds to two spheres placed at opposite 
poles ($N_{\min}=2$), and the maximum $N$ is obtained by first parking 
three large spheres next to one another, so that there is room for one 
more sphere to park ($N_{\max}=4$). At $\alpha$ slightly smaller than 
$\alpha_c$ , the big spheres can park along orthogonal axes about the 
small sphere to make an octahedron ($N_{\max}=6$). The minimum $N$ is 
obtained by placing four spheres as far from each other as possible, 
so as to make the addition of a fifth impossible ($N_{\min}=4$). Thus, 
as we increase $\alpha$ through $\alpha_c$, $N_{\max}$ goes from $6$
to $4$ and $N_{\min}$ from $4$ to $2$, and the two curves become 
infinitesimally close.

The parking process is therefore geometrically constrained to yield 
clusters with exactly $N=4$ particles in the limit 
$\alpha\rightarrow\alpha_c$. A simple geometric argument sheds some 
light on this result. At $\alpha_c$ there is always room for four 
large spheres to park. Parking more spheres requires that at least 
three park precisely along a great circle of the smaller particle, but 
the probability of this happening randomly is zero. Thus irreversible 
binary aggregation, a stochastic process, has a deterministic feature 
at the critical size ratio: although the space between the large 
spheres can vary, all clusters must be tetramers. Our numerical 
approach confirms that the statistical dispersion in the cluster size 
distribution vanishes at $\alpha_c$, as shown in 
Figure~\ref{fig:expAndTheory}.

The experimental and simulated distributions differ slightly due to 
two effects. First, the measured sizes tend to be smaller than the 
simulated ones because a few parking spaces remain unfilled even at 
long times. This effect is more pronounced for larger spheres, which 
diffuse more slowly and encounter the small spheres less frequently. 
The systems most affected are the electrostatic ones at $\alpha =
3.06$ and $4.29$. Second, the experimental size ratios can vary by 
5\% due to polydispersity. Both of these factors increase the width in 
the experimental distributions and diminish the achievable yield of 
tetramers near $\alpha_c$. The random parking model also assumes the 
interactions are infinitesimally short-ranged and isotropic. It would 
not be valid if, for example, there were surface inhomogeneities on 
length scales comparable to the particle radii. Nevertheless, the 
experimental data indicate that near $\alpha_c$ a tetramer yield of at 
least 90\% is possible, and the model is useful for predicting cluster 
size distributions in two very different colloidal systems.

\section{Conclusion}

These results have both fundamental and practical consequences. On 
the fundamental side, the particle size ratio could affect the jamming 
threshold in bulk packings of bidisperse spheres. Previous 
simulations of these systems have shown that the distribution of 
coordination numbers also depends on the size ratio \cite{He1999} and 
may be modeled using random parking \cite{Wouterse2005}. This contrasts 
with dense atomic systems like metallic glasses \cite{Egami1997,
  Miracle2004} in which the atoms have some freedom to rearrange 
locally. In these systems packing constraints may explain structure 
and coordination better than parking arguments.

On the practical side, this random aggregation process is a simple way 
to mass produce tetrahedral clusters in theoretically 100\% yield. 
Although the tetrahedra we produce are irregular in that the distance 
between the large spheres can vary, it may well be possible to form 
large quantities of symmetric tetrahedra simply by shrinking the small 
spheres after the tetramers have formed \cite{Lyon2004}. An additional 
step, such as density gradient centrifugation \cite{Manoharan2003}, will 
also be required to separate the assembled clusters from the many 
unbound large particles. Furthermore, although the yield will 
approach 100\% only for dimers and tetramers, the yield of any 
$N$-particle cluster can be maximized by choosing the appropriate size 
ratio. For instance, the yield of octahedral clusters, also promising 
candidates for building metamaterials \cite{Alu2009}, may surpass 70\% 
at $\alpha = 1.42$.

The size ratio in binary colloidal systems thus emerges as a valuable 
control parameter for directed self-assembly. Moreover, because it 
does not require precise control over the interactions, random parking 
offers a robust and simple way to make colloidal clusters that are 
more monodisperse than those prepared through other methods 
\cite{Manoharan2003}.

\section{Materials and Methods}

\subsection{Electrostatic Interactions}
\label{sec:electrostatic}

Charged colloidal polystyrene spheres were purchased from Invitrogen
as ``IDC surfactant-free latex'' in batches as listed in
Table~\ref{table:stocks}.

\begin{table}[h]
  \caption{Charged particles used in
    electrostatic system experiments. Values for surface charge from
    data sheets provided by manufacturers.} \label{table:stocks}
\begin{tabular}{c c c r}
Mean diameter & Surface functionality & Fluorescent? & Surface charge \\ [0.5ex]
\hline
\unit{0.49}{\micro \meter} & carboxylate-modified latex (CML) & yes & \unit{-262}{\micro \coulomb / \centi \meter ^2} \\
\unit{0.95}{\micro \meter} & amidine & no & \unit{+23.7}{\micro \coulomb / \centi \meter ^2} \\
\unit{1.2}{\micro \meter} & amidine & no & \unit{+18.2}{\micro \coulomb / \centi \meter ^2} \\
\unit{1.5}{\micro \meter} & aldehyde-amidine & no & \unit{+18.2}{\micro \coulomb / \centi \meter ^2} \\
\unit{2.1}{\micro \meter} & amidine & no & \unit{+30.2}{\micro \coulomb / \centi \meter ^2} \\ [0.5ex]
\end{tabular}
\end{table}

A \unit{100}{\micro \liter} sample of each colloid was diluted to 1\%
weight by volume.  This was then vortexed for a few seconds and
bath-sonicated for 10 seconds.  We cleaned the particles by
centrifuging and redispersing them several times in deionized (DI)
water, using the following wash procedure.

\begin{enumerate}
\item Colloids were centrifuged for 5 minutes at $6\,600g$.
\item Supernatant was removed and \unit{190}{\micro \liter} of DI
  water was added to each sample.
\item Samples were vortexed for 5 seconds each and then bath-sonicated
  for 10 seconds.
\end{enumerate}

Because the washing procedure involves centrifugation at high
acceleration, it can cause the particles to aggregate.  The vortexing
and sonication steps break apart the majority of these aggregates, but
some small aggregates remain and can later appear as clusters showing
non-specific aggregation.  We do not count these non-specific
aggregates when we characterize the cluster size distribution.

We performed six wash cycles.  After the last centrifugation, the
supernatant was replaced with \unit{40}{\micro \liter} of DI water,
rather than \unit{190}{\micro \liter} as before.  Then
\unit{50}{\micro \liter} of 20 mM NaCl was added to each sample to
achieve an overall salt concentration of 10 mM.  This screens the
repulsion between like particles before mixing.

We prepared mixtures of the positively and negatively charged
particles such that each mixture contained one batch of positively
charged particles at 1\% w/v.  In each mixture, the number ratio of
the large (positively charged) to small (negatively charged) spheres
was $100:1$.  The salt concentration in each mixture was 10 mM
NaCl.  Each mixture consisted of large and small particles with a
different size ratio $\alpha = R_{\mathrm{large}} / R_{\mathrm{small}}$, 
as listed in Table~\ref{table:mixtures}.

\begin{table}[h]
  \caption{\label{table:mixtures} Size ratios and components of
    binary mixtures of charged colloids.}
\begin{tabular}{c l l}
$\alpha$ & Large particles & Small particles \\ [0.5ex]
\hline
$1.94$ & \unit{0.95}{\micro \meter} amidine (+) & \unit{0.49}{\micro
  \meter} CML (-) \\
$2.45$ & \unit{1.2}{\micro \meter} amidine (+) & \unit{0.49}{\micro
  \meter} CML (-) \\
$3.06$ & \unit{1.5}{\micro \meter} aldehyde-amidine (+) &
\unit{0.49}{\micro \meter} CML (-) \\
$4.29$ & \unit{2.1}{\micro \meter} amidine (+) & \unit{0.49}{\micro
  \meter} CML (-) \\ [0.5ex]
\end{tabular}
\end{table}

Each mixture was stored in a micro-centrifuge tube and vortexed at
$3\,000$ rpm, bath-sonicated for 20 seconds, and then mounted on a
Glas-Col Rugged Rotator to tumble slowly at 4 $^\circ$C.  Each mixture
tumbled for at least three days before observation to reduce the
effects of sedimentation on local particle concentrations throughout
the mixture. To make it easier to identify and characterize one
cluster at a time, we diluted samples of the mixtures to 0.1\% w/v
just prior to observing the distribution of cluster sizes.

\subsubsection{Electrostatics Control Experiment}

In the experiments outlined above, each mixture contained particles
with surface charges of opposite sign.  In a separate control
experiment, we mixed particles of two different sizes but with surface
charges of the same sign.  Both components in our control mixture were
carboxylate-modified latex (CML) colloids with a size ratio $\alpha =
2.24$, as listed in Table~\ref{table:elec_control}.

\begin{table}[h]
  \caption{\label{table:elec_control} Colloids used in electrostatic
    system control experiment.}
\begin{tabular}{c c c r}
Mean diameter & Surface functionality & Fluorescent? & Surface charge \\ [0.5ex]
\hline
\unit{0.49}{\micro \meter} & CML & yes & \unit{-262}{\micro \coulomb / \centi \meter ^2} \\
\unit{1.1}{\micro \meter} & CML & no & \unit{-31.5}{\micro \coulomb / \centi \meter ^2} \\ [0.5ex]
\end{tabular}
\end{table}

\begin{figure}[h]
\begin{center}
\includegraphics[width=10.0cm]{electrostaticControl}
\caption{\label{fig:electrostaticControl} Cluster size distribution
  in a mixture of \unit{1.1}{\micro \meter} non-fluorescent CML
  particles and \unit{0.49}{\micro \meter} fluorescent CML particles
  in a $100:1$ number ratio, showing that particles with surface
  charge of the same sign rarely form clusters.}
\end{center}
\end{figure}

We washed these colloids using the procedure outlined above and then
mixed them in a $100:1$ number ratio.  After letting them tumble at 4 $^\circ$C 
for several days, we measured the cluster size distribution. As shown in
Figure~\ref{fig:electrostaticControl}, fewer than 1\% of the small
particles bind to large particles when they have surface charges of
the same sign.  Non-specific aggregation is therefore rare in the
charged colloidal systems.

\subsubsection{Experiments Without Salt}

In another set of experiments, mixtures of charged colloidal particles
were prepared as described above, but without salt.  The number ratio
of positively charged spheres to negatively charged spheres was again
$100:1$, and several size ratios $\alpha$ were investigated, as
listed in Table~\ref{table:no_salt_mixtures}.  As in the experiments
with 10 mM NaCl, we tumbled the mixtures for several days before
measuring the distribution of clusters.

\begin{table}[h]
  \caption{\label{table:no_salt_mixtures} Size ratios and components of
    binary mixtures without salt.} 
\begin{tabular}{c l l}
$\alpha$ & Large particles & Small particles \\ [0.5ex]
\hline
$1.09$ & \unit{1.2}{\micro \meter} amidine (+) & \unit{1.1}{\micro
  \meter} CML (-) \\
$1.36$ & \unit{1.5}{\micro \meter} aldehyde-amidine (+) &
\unit{1.1}{\micro \meter} CML (-) \\
$1.94$ & \unit{0.95}{\micro \meter} amidine (+) & \unit{0.49}{\micro
  \meter} CML (-) \\
$2.45$ & \unit{1.2}{\micro \meter} amidine (+) & \unit{0.49}{\micro
  \meter} CML (-) \\
$3.06$ & \unit{1.5}{\micro \meter} aldehyde-amidine (+) &
\unit{0.49}{\micro \meter} CML (-) \\
$4.29$ & \unit{2.1}{\micro \meter} amidine (+) & \unit{0.49}{\micro
  \meter} CML (-) \\ [0.5ex]
\end{tabular}
\end{table}

Figure~\ref{fig:electrostaticSaltComp} shows that the average cluster
sizes in these mixtures were smaller than the average sizes predicted
from simulation and those observed in mixtures containing 10 mM NaCl.
For instance, at $\alpha=2.45$ with 10 mM NaCl the average cluster size
is $N = 3.9$, but when there is no salt in the system the average
cluster size is $N = 2.7$.  For the four size ratios for
which there is data both without salt and with 10 mM NaCl, we found
that clusters are 20\% to 35\% smaller when no salt is added.

\begin{figure}[h]
\begin{center}
\includegraphics[width=8.6cm]{electrostaticSaltComp}
\caption{\label{fig:electrostaticSaltComp} Average cluster sizes from
  simulations (dashed dark gray) and electrostatic experiments  with
  (red data points) and without (cyan) salt.  Widths of the cluster
  size distributions are indicated by dotted light gray lines for
  simulations and vertical error bars for experiments.}
\end{center}
\end{figure}

These experiments show that electrostatic repulsion affects the
cluster assembly.  This observation is consistent with other recent
experiments \cite{Wang2010} showing that the cluster size distribution
in binary mixtures depends on ionic strength when the salt
concentration is less than 10 mM.  In our mixtures with 10 mM NaCl,
the Debye length is approximately \unit{3}{\nano \meter}, very small
compared to the particle sizes, so the large spheres do not interact
with each other except at small distances.  The random parking model
should be more appropriate for systems like these where the
interaction range is much smaller than the particle size.  This is
because the random parking model assumes no interactions between the
particles except for a hard-core repulsion and irreversible binding on
contact between spheres of two different types.

\subsection{DNA-Colloid Interactions}
\label{sec:DNA-colloid}

In another set of experiments, we mixed small and large spheres
labeled with complementary 65-base ssDNA oligonucleotides purchased
from Integrated DNA Technologies:
\begin{itemize}
\item Sequence A: 5'-biotin-51xT-TGTTGTTAGGTTTA-3'
\item Sequence B: 5'-biotin-51xT-TAAACCTAACAACA-3'
\end{itemize}

The oligonucleotides terminate with a biotin group, which allows us to
graft them to streptavidin-coated polystyrene particles using a
protocol from Dreyfus \textit{et al.} \cite{Dreyfus2009}.  We purchased the
streptavidin-coated polystyrene particles from Bangs
Laboratories, Inc. with the following diameters:
\begin{itemize}
\item \unit{0.21}{\micro \meter}, fluorescent, to be coated with sequence B
\item \unit{0.39}{\micro \meter}, fluorescent, to be coated with sequence B
\item \unit{0.51}{\micro \meter}, fluorescent, to be coated with sequence B
\item \unit{0.97}{\micro \meter}, non-fluorescent, to be coated with sequence A
\end{itemize}

We dissolved the DNA strands in water in \unit{20}{\micro}M
concentration.  We mixed \unit{20}{\micro \liter} of this DNA 
solution with \unit{10}{\micro \liter} of 1 wt \% streptavidin-coated
polystyrene particles with \unit{120}{\micro \liter} of phosphate
buffer in a 1.7 mL propylene micro-centrifuge tube.  Each 50 mL batch
of buffer contained 0.0128 g KH$_2$PO$_4$, 0.0707 g K$_2$HPO$_4$,
0.1467 g NaCl, and 0.250 g F108 surfactant in 50 mL of deionized
water.  We filtered the buffer through a \unit{0.2}{\micro \meter}
membrane before use.  We prepared a separate batch containing no 
salt so that the salt concentration could be adjusted by combining
the two.  We vortexed the mixtures at $3\,000$ rpm for 5 seconds and
bath-sonicated them for 10 seconds. We incubated them at room temperature
for 30 minutes to allow the ssDNA to graft to the surface of the
particles.  Then we washed the colloids using the following procedure:

\begin{enumerate}
\item We centrifuged the colloids for 3 minutes at $12\,000g$.
\item We removed the supernatant and added \unit{100}{\micro \liter} of 50 mM
  NaCl buffer to each sample.
\item We vortexed the samples for 5 seconds at $3\,000$ rpm and then
  bath-sonicated them for 10 seconds.
\end{enumerate}

As in the charged particle system, the centrifugation step can create
aggregates, some of which survive the vortexing and sonication steps
and appear as non-specifically aggregated clusters.  We washed each
colloid three times to remove excess DNA from the system, and then
incubated each at 55 $^\circ$C for 30 minutes.  We then washed three
more times, incubated at 55 $^\circ$C for another 30 minutes, and
washed three times again.  At this point the salt concentration in the
buffer was adjusted to 20 mM.  The A- and B-labeled particles were
mixed in a $100:1$ (large : small) number ratio such that the larger
particles were at a volume fraction of about 0.1\%.  Three separate
mixtures were prepared, as listed in Table~\ref{table:DNAmixtures}.

\begin{table}[h]
  \caption{\label{table:DNAmixtures} Size ratios and components of
    mixtures with DNA-driven interactions.} 
\begin{tabular}{c l l}
$\alpha$ & Large particles & Small particles \\ [0.5ex]
\hline
$1.90$ & \unit{0.97}{\micro \meter}, sequence A & \unit{0.51}{\micro
  \meter}, sequence B \\
$2.49$ & \unit{0.97}{\micro \meter}, sequence A & \unit{0.39}{\micro
  \meter}, sequence B \\
$4.62$ & \unit{0.97}{\micro \meter}, sequence A & \unit{0.21}{\micro
  \meter}, sequence B \\ [0.5ex]
\end{tabular}
\end{table}

After the mixtures had been prepared, we followed the same procedure
that we used for the charged colloid system.

\subsubsection{DNA-Colloid Control Experiment}

In the experiments outlined above, each mixture contained particles
labeled with complementary DNA strands.  In a separate control
experiment, we mixed particles of two different sizes but labeled with
identical ssDNA.  

\begin{figure}[h]
\begin{center}
\includegraphics[width=10.0cm]{DNAcolloidControl}
\caption{\label{fig:DNAcolloidControl} Cluster size distribution in a mixture of \unit{0.97}{\micro \meter} non-fluorescent particles and \unit{0.51}{\micro \meter} fluorescent particles in a $100:1$ number ratio, coated with the same non-self-complementary DNA sequence (A).}
\end{center}
\end{figure}

Both components in our control mixture were streptavidin-coated
polystyrene colloids labeled with sequence A.  We used
\unit{0.97}{\micro \meter} (non-fluorescent) and \unit{0.51}{\micro
  \meter} (fluorescent) particles, for a size ratio $\alpha =
1.90$.  We functionalized the particles with DNA using the same
procedure described above and then mixed them in a $100:1$ number ratio.
After tumbling them at 4 $^\circ$C for several days, we measured the 
cluster size distribution. Figure~\ref{fig:DNAcolloidControl} shows
that fewer than 2\% of the small spheres bind to large spheres when
they are coated with the same DNA sequence.  The low amount of
non-specific aggregation is expected, since we designed sequence A to
have a negligible amount of self-hybridization even at 0 $^\circ$C.

\subsection{Measurement of Cluster Size Distribution}
\label{sec:cluster_size_dist}

To measure the distribution of cluster sizes in a particular mixture,
we placed a \unit{5}{\micro \liter}, 0.1\% w/v sample between two
cover slips and then sealed it at the edges with UV-curable epoxy
(Norland Optical Adhesive 61).  We observed each sample under
differential interference contrast with a 100X oil-immersion objective
on a Nikon Eclipse TE2000-E inverted microscope.  We used fluorescence
to identify the small spheres.

We obtained the distribution by counting the number of clusters at
each $N$.  $N=0$ is a small sphere without any large spheres adsorbed,
$N=1$ a small sphere attached to a single large sphere, etc.  We also
counted clusters containing multiple small spheres and clusters formed
through non-specific aggregation.  The number of such clusters is
small compared to the total number of clusters counted.  Thus, we do
not include these in the histograms. 

Our counting procedure is designed to avoid errors from double
counting.  We begin by counting clusters in the field of view (FOV) on
the microscope, scanning through $z$ to find clusters that may
initially be out of focus.  The Brownian motion of each cluster
eventually brings all particles into view.  We can therefore determine
the number of large and small spheres in each cluster through direct
observation.  Once we have recorded all clusters in the FOV, we
translate the FOV to another part of the sample located more than one
FOV away.  We repeat this process to build up the histogram, rastering
the FOV over the sample.  We count more than 60 clusters at each size
ratio and more than 120 clusters at size ratios greater than $2$.  The
entire histogram for a given mixture is recorded in one session on the
microscope.

\subsection{Random Sphere Parking Calculations}
\label{sec:parking_calc}

\subsubsection{Monte Carlo Simulations}

We simulate the cluster size distribution for spheres undergoing a 
random sequential adsorption or ``random parking'' process.  We use 
an algorithm based on Monte Carlo trials.  These consist of two 
stages:

\paragraph{Coarse stage}  
We repeatedly try to insert a disc of radius $r = R_{\mathrm{large}} /
(R_{\mathrm{large}}+R_{\mathrm{small}})$ on the surface of the unit
sphere. The center of the disc is randomly and uniformly distributed
on the sphere.  If the disc overlaps any discs that are already
``parked'', we reject it; otherwise we add it to the list of parked
discs. After a fixed number of consecutive rejections, (typically
$N_{\text{coarse}} = 10^4$), we switch to the fine stage.

\paragraph{Fine stage}
We compute the remaining regions of possible insertion and then try
to insert an additional disc.  We choose the center of the disc
randomly and uniformly from these regions.

To find the regions of possible insertion, we first compute the arcs
that form their boundaries: Given a central disc of radius $r$,
neighboring disc centers must lie on or outside a concentric circle of
radius $2r$. We erase arcs that lie in the interior of the concentric
circles of radius $2r$ about each neighboring disc. If the central
circle contains any unerased arcs, these are added to our roster of
arcs.

We do this for each parked disc center, and then stitch together the
remaining arcs to form the regions to which another disc center could
be added.  To find the area of each region, we inscribe the region in
a circumcircle and use a Monte-Carlo integration method to determine
the ratio of the area of the region to the area of its circumcircle.

The algorithm terminates when the list of remaining arcs in the fine
stage is empty, at which point the final number of inserted discs is
recorded.

\subsubsection{Bounds on Cluster Size Distribution}

The upper and lower bounds $N_{\max}$ and $N_{\min}$ on the cluster size
distribution can be calculated from \textit{spherical codes}
\cite{Sloane2011} corresponding to known solutions to the problems of
spherical packings ($N_{\max}$) and spherical coverings ($N_{\min}$).
Spherical packings are arrangements of $N$ points on a unit sphere
that maximize the smallest distance between any two of them
\cite{Conway1993, Sloane2011}.  For a given $\alpha$ we determine
$N_{\max}(\alpha)$ by looking up the spherical packing \cite{Sloane2011}
with the largest $N$ for which the minimal distance between points is
at least $2\alpha / (1 + \alpha)$. Our calculation of
$N_{\min}(\alpha)$ is more involved, as explained in 
Appendix~\ref{ch:appendix_A}.
