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\begin{document}

\title{Dynamic light scattering from the water-air interface}


\author{Jacinto Rosas}
\affil{The German University in Cairo}



\date{\today}

\maketitle




\begin{Abstract}
Dynamic light scattering is a useful technique for studying the dynamics of colloids in bulk fluids. It has been also used to study near wall dynamics of colloids, and colloidal dynamics at the water-oil interface. Here, we report the implementation of dynamic light scattering for studying dynamic of colloids at the water-air interface. Further insight into the sample preparation and the analysis of the data is revealed.
\end{Abstract}

\section{Introduction}
Dynamic light scattering is a well established experimental technique for measuring dynamics of colloids at bulk fluid. It have been also applied for evaluating dynamics of colloids at the solid-fluid interface, and at the water-oil interface \cite{Liu_2015, Stocco_2009}. Each application requiring an specific instrumental configuration. When applied at bulk fluids DLS enables size characterization of colloidal particles, or rheological characterization of the evaluated solvent. At solid-fluid interface, DLS enables studying the near-wall effect and its relevance in the hydrodynamics interactions of the colloidal system. Moreover, at the water-oil interface it enables evaluating the effect that a fluid-fluid interface has on the dynamics of colloids, and could be also employed for rheological characterization of interfacial films or the interface itself \cite{Samaniuk_2014, FISCHER_2006, Danov_2000, Prasad_2006}. Here, we present the implementation of DLS for evaluating the diffusion of polymer colloids at the water-air interface, and we present further insight into the interfacial dynamics of the colloids.
The physical dynamics of a particle immersed in a bulk solvent, and describing a Brownian trajectory is ruled by the following Newton's equation of motion \cite{1996}:

\begin{equation}
\label{eq:newtoneq}
-\gamma{p \over M} + f(t) = {dp \over dt}
\end{equation}

Where the term $\gamma{p/M}$ accounts for the friction force the particle experiences due to systematic collisions with the solvent molecules, and the term $f(t)$ is a fluctuating force the particle experiences owing to random collisions with the solvent molecules. Moreover, the position coordinate of the particle is by definition:

\begin{equation}
\label{eq:position}
{dr \over dt} = {p \over M}
\end{equation}

The solution to equations \ref{eq:newtoneq} and \ref{eq:position} is given by the conditional probability density function, $P$, for the momentum and position coordinates of the Brownian particle:

\begin{equation}
\label{eq:pdf}
P(\textbf{p}, \textbf{r}, t| \textbf{p}(0), \textbf{r}(0), t=0) = {1 \over (2\pi)^{n/2}\sqrt{det\textbf{D}}}exp\left\{-{1 \over 2}\textbf{X}\cdot \textbf{D}^{-1}\cdot \textbf{X}\right\}
\end{equation}

Where $\textbf{X}$ is a Gaussian variable defined by:

\begin{equation}
\label{eq:xonly}
\textbf{X} \equiv (x_{1}, x_{2})
\end{equation}

Where $x_{1}$ and $x_{2}$ are also Gaussian variables defined as:

\begin{equation}
\label{eq:x1}
x_{1} \equiv p(t) - p(0) exp\left\{-{\gamma \over M}t\right\}
\end{equation}

\begin{equation}
\label{eq:x2}
x_{2} \equiv r(t) - r(0) - {p(0) \over \gamma} \left[1- exp\left\{-{\gamma \over M}t\right\}\right]
\end{equation}

$\textbf{X}$ can be also written as:

\begin{equation}
\label{eq:x}
\scriptsize
\textbf{X} \equiv (x_{1}, x_{2}) = \left(\int^{t}_{0}{dt^{\prime} f(t^{\prime})}exp\left\{-{\gamma \over M}(t-t^{\prime}) \right\}\\,{1\over \gamma}\int^{t}_{0}{dt^{\prime} f(t^{\prime})}\left[1-exp\left\{-{\gamma \over M}(t-t^{\prime}) \right\}\right]\right)
\end{equation}

and

\begin{equation}
\label{eq:d}
\textbf{D} = \left\langle \textbf{X} \textbf{X} \right\rangle \equiv \left( \begin{array}{ccc}
<x_{1}x_{1}> & <x_{1}x_{2}>\\
<x_{2}x_{1}> & <x_{2}x_{2}>\\
\end{array} \right)
\end{equation}

Where the ensemble averages $<x_{i}x_{j}>$ are given by:

\begin{equation}
\label{eq:x1x1}
<x_{1}x_{1}>={MG \over 2\gamma}\left[1-exp\left\{-{2\gamma \over M}t \right\}\right]
\end{equation}

\begin{equation}
\label{eq:x1x2}
<x_{1}x_{2}>=<x_{2}x_{1}>={MG \over {\gamma}^2}\left[1-exp\left\{-{\gamma \over M}t \right\}\right]^2
\end{equation}

\begin{equation}
\label{eq:x2x2}
<x_{2}x_{2}>={MG \over \gamma^3}\left({\gamma \over M}t-{1\over 2}\left[exp\left\{-{2\gamma \over M}t \right\}-1\right]-2\left[1 - exp\left\{-{\gamma \over M}t \right\}\right]\right)
\end{equation}

Where fluctuation strength, $G$, is given by the fluctuation dissipation theorem:
\begin{equation}
\label{eq:G}
\textbf{G} = \hat{\textbf{I}}{2\gamma \over \beta}
\end{equation}

What is experimentally recorded in a dynamic light scattering measurement is the intensity autocorrelation function, $g_{I}(\textbf{k},t)$, which is related to the field autocorrelation function, $g_{E}(\textbf{k},t)$, by way of the Siegert relation, namely:

\begin{equation}
\label{eq:siegert}
\hat{g}_{I}(\textbf{k},t)= 1 + \left|\hat{g}_{E}(\textbf{k},t)\right|^{2}
\end{equation}

Where $\hat{g}_{I}(\textbf{k},t)$ and $\hat{g}_{E}(\textbf{k},t)$ are intensity normalized functions:

\begin{equation}
\label{eq:iacfnor}
\hat{g}_{I}(\textbf{k},t) \equiv {g_{I}(\textbf{k},t) \over I^{2}}
\end{equation}

\begin{equation}
\label{eq:eacfnor}
\hat{g}_{E}(\textbf{k},t) \equiv {g_{E}(\textbf{k},t) \over I}
\end{equation}

and by definition,

\begin{equation}
\label{eq:iacfnordef}
\hat{g}_{I}(\textbf{k},t=0) = 2 
\end{equation}

\begin{equation}
\label{eq:eacfnordef}
\hat{g}_{E}(\textbf{k},t=0) = 1 
\end{equation}



The electric field auto-correlation function, $\hat{g}_{E}(\textbf{k},t)$, is given by the Fourier transform of the probability density function describing the dynamics of the Brownian particle:

\begin{equation}
\label{eq:EACF}
\hat{g}_{E}(\textbf{k},t)= \int{d{r^{\prime}}P({r^{\prime}},t)exp\left\{ik\cdot {r^{\prime}}\right\} }
\end{equation}

Equation \ref{eq:EACF} can be also written as:

\begin{equation}
\label{eq:EACF2}
\hat{g}_{E}(\textbf{k},t)= exp\left\{ -D_{0}k^{2}t\right\}=exp\left\{ -t/\tau\right\}
\end{equation}

With

\begin{equation}
\label{eq:tau}
\tau^{-1}=D_{0}k^{2}
\end{equation}

The dependency of the decay constant of the electric field auto-correlation function with the scattering wave vector sets a methodology for determining experimentally the diffusion coefficient, $D_{0}$. This methodology is usually referred as multi-angle dynamic light scattering (Multi-angle DLS).

\section{Material and methods}
\subsection{Sample preparation}
We employ polystyrene colloids with a diameter of $200nm$. The colloids are stabilized by sulphate groups. A suspension of these colloids is obtained from \textit{Molecular Probes}. The spreading solution is prepared by diluting a small volume of the stock solution in ultra-pure water, and mixing with up to $50\%$ of $1-propanol$. An small volume of the spreading solution is added at the liquid surface. The spreading solution of interfacial material, namely, polymers, phospholipids or surfactants is prepared by dissolving the material in a proper solvent. An small volume of the interfacial material solution is added to the interface, and sometime is left for solvent evaporation before starting measurements. The sub-phase is prepared with water at degrees of processing, namely, molecular biology reagent, deionized, and HPLC reagent. All from Sigma-Aldrich. The glass cell containing the sample is previously cleaned with a diluted solution of \textit{Hellmanex} and rinsed with ultra-pure water. Furthermore, an important step in the sample preparation is checking that a flat auto-correlation function is measured for the clean surface of water, which indicates there are not impurities at the interface being able to generate scattering. The temperature in the cell is regulated by a MINISTAT temperature control system, and sometime is left before starting measurements for allowing the sample to reach the thermal equilibrium.
\subsection{Light scattering set-up}
The illumination source for the light scattering experiments is a laser spot created at the surface of the water by setting the laser beam in a total reflection geometry. The instrument consist of a rotating goniometer containing the laser and detector arms, and in the center the cell holder. The goniometer can be tilted at any angle, which enables to detection in the three dimensional space. The scattered light is collected at a single-mode optical fiber, which is connected to a photon counting and a avalanche photo-diode. The auto-correlation function is build up by using an ALV correlator, and visualized with the ALV-5000 software. The whole set-up lies on a anti-vibration table which isolate the liquid interface from external mechanical perturbations.
\subsection{Data analysis}
The auto-correlation function is fitted using the ALV-5000 software. The multi-angle plot is build up from the fitting of the auto-correlation function at several scattering vectors. The data can be also analysed employing the polydisperse multi-angle plot as will be show later.


\section{Results and discussion}
Dynamic light scattering is a well established experimental technique for evaluating the diffusion of colloids at bulk fluids. Its implementation for accessing the dynamics of colloids at the water-air interface requires modifying the way the laser beam hits the sample, as well as, the way the intensity of the scattered light is detected. However, the whole theoretical framework remains essentially the same. For implementing DLS at the water-air interface, the scattered light is detected on a tilted plane respect to the interfacial plane. This configuration allows separating the colloidal diffusion into its components normal and parallel to the plane of the interface, as well as, the independent variation of these components. The detected scattered light is the one coming from a spot on the fluid's surface which is generated by positioning the laser beam on a total internal reflection configuration from the water sub-phase. The average intensity of the scattered light as a function the scattering angle for the pure solvent is shown in figure \ref{fig:aveint}. The results show no major variation of the light scattered by the solvent at the different angles evaluated. The detected intensity auto-correlation function is shown in figure \ref{fig:acf1} for three different experimental condition, namely, a clean water surface, the surface of water containing a film of interfacial material, and the surface of water containing a dilute colloidal solution. The results show that only the case when there are present colloids at the fluid interface enables the detection of an auto-correlation function. The intensity autocorrelation function of the light scattered by the colloids is presented in figure \ref{fig:acfs1} for different detection angles. The fitting of the auto-correlation function for each detection angle enables the construction of the DLS multi-angle plot, which correspond to the inverse of the time constant versus the squared modulus of the scattering vector component parallel to the interface plane, figure \ref{fig:waterquality}. The diffusion constant of the colloids is determined from the lineal fitting of the multi-angle plot.

Moreover, it is worth mentioning several difficulties that were needed to identify and overcome in order to come to the presented results. The most important has been the degree of processing of the water phase employed for the preparation of the interface. Experimental results show how the DLS measurements are highly influenced by the degree of processing of the water phase, and how a diligent testing of the purity of the employed reagents is needed for arriving to replicable and meaningful experimental results. The DLS multi-angle plot for polymer colloids at the surface of water is presented in figure \ref{fig:waterquality} . Three degree of processing of the water phase are tested, namely, one coming from a Milli-Q system and two obtained from Sigma-Aldrich, which correspond to deionized water, water reagent for molecular biology, the latest having the higher degree of processing. Figure 6a shows a DLS multi-angle plot with two clearly different slopes which could in principle be mistaken as a colloidal system containing particles of two different sizes. However, this effect vanish when water with a better degree of processing is employed. A more complex situation is experienced when the employed water reagent is simply double distilled water. In this case, even no correlation function can be measure for scattering vectors larger than zero, which is shown in figure \ref{fig:doubledistilledwater}. In this case no multi-angle plot can be built from the experimental data.

The effect of the polydispersity of the colloidal system can be observed in figure \ref{fig:oscillationd4} and \ref{fig:oscillationd5}, which shows the polydisperse diffusion coefficient as a function of the squared modulus of the scattering vector component parallel to the interface plane. In this case, the diffusion coefficient exhibit an 'oscillatory' behavior which is typical for polydisperse colloidal systems.

\section{Conclusions and outlook}
Dynamic light scattering represents an interesting and alternative method for experimentally measuring the diffusion of colloids at the water-air interface. Despite the difficulties concerning the sample preparation, it involves a simple and fast data analysis process while keeping the advantages of light scattering methods.\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/int-check1/int-check1}
\caption{{\label{fig:aveint} The average intensity of the scattered light as a function the scattering angle for the pure solvent%
}}
\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/acfs1/acfs1}
\caption{{\label{fig:acfs1} The intensity autocorrelation function of the light scattered by the colloids for different detection angles.%
}}
\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/ACF1/ACF1}
\caption{{\label{fig:acf1}Dynamic light scattering measurements at the water-air interface. Figure $a$ represents the intensity auto-correlation function measured for the clean surface of water. Figure $b$ represents the intensity auto-correlation function measured for the surface of water containing a film of hydrophobic polymer of surface concentration $3.00 mg/m^{2}$. Figure $c$ represents the intensity auto-correlation function measured for polymer colloids at the surface of water containing the film of hydrophobic polymer.The measurement was performed at a scattering vector of $38.93{\mu}m^{-2}$, and controlled temperature of $305.15K$.%
}}
\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/waterQuality/waterQuality}
\caption{{\label{fig:waterquality} Multi-angle dynamic light scattering.%
}}
\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/doubledistilledwater1/doubledistilledwater1}
\caption{{\label{fig:doubledistilledwater}Dynamic light scattering measurements at the water-air interface. The picture shows the intensity autocorrelation function for the light scattered by polystyrene colloids attached to the surface of water at given scattering vectors: $a=0.00302 {\mu}m^{-2}$, $b=0.74029 {\mu}m^{-2}$, $c=3.15306 {\mu}m^{-2}$. The degree of processing of the water is double distilled. %data 21/04/2015%
}}
\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/oscillation-D2/oscillation-D2}
\caption{{\label{fig:oscillationd4}The polydisperse diffusion coefficient versus the square modulus of the scattering wave vector. Milli-Q water (a). Deionized water Sigma-Aldrich (b). Molecular biology water Sigma-Aldrich (c).%
}}
\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/oscillation-D3/oscillation-D3}
\caption{{\label{fig:oscillationd5}The polydisperse diffusion coefficient, relative to the monodisperse diffusion coeficient versus the square modulus of the scattering wave vector. Milli-Q water (a). Deionized water Sigma-Aldrich (b). Molecular biology water Sigma-Aldrich (c).%
}}
\end{center}
\end{figure}

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