In last week's lab, you mapped out the electric fields for different configurations of charged electrodes in a tub of water. While you had the liberty to choose your configurations, we asked you to map the field given by oppositely charged parallel plates, which by now you know is given by $\vec{E} = \frac{V}{d}\hat{d}$, where $V$ is the potential difference between the plates, $d$ is the separation of the plates, and $\hat{d}$ is a unit vector perpendicular to the plates (assuming infinitely large plates). For our experiment, we also used an alternating potential (AC) instead of a constant (DC) potential, the main reason being to avoid the chemical reactions that would occur between the electrodes if we generated a steady current of ions.

This week, however, we want to see what happens when a charged particle suspended in water is subjected to a DC electric field between parallel plates. Indeed, you can imagine that the particle will experience a force that is proportional to how much charge it has as well as to the strength of the field, so that it will move parallel to the field lines. The direction of motion will be determined by the sign of its charge.

The motion of suspended charged particles in an electric field is known as electrophoresis. It has many applications in the characterization of chemical and biological systems, and it is often used to separate mixtures of chemical compounds by their charge-to-mass ratios. For instance, electrophoresis can be used to separate nucleic acids by the number of base pairs they contain, which affects the resultant charge-to-mass ratios that they possess. See gel electrophoresis for an example.

However, to properly characterize and analyze mixtures of complex molecules such as nucleic acids and proteins, a number of complications beyond the scope of this course need to be taken into consideration. Instead, in this experiment we will be focusing on the electrophoresis of a relatively simple colloidal system consisting of pigment particles (like those found in food coloring) that are suspended in water. In doing so, we hope to accomplish the learning objectives that follow.

• Develop a model of the forces acting on pigment particles suspended in water.
  • As it is often the case in science, you will need to make some assumptions about the system under consideration.
• Test the validity of the model through the implementation of an experimental procedure.
  • Which parameters in your model can you manipulate experimentally in order to test whether your model matches up wit the experimental observations?
  • Is your data consistent with the model you develop within experimental uncertainties?
• Apply the principles of electrodynamics and fluid dynamics to study and characterize a chemical system.

Above is a footage of a colloidal system consisting of silica ($\text{SiO}_2$) microspheres of roughly 5 microns in diameter. A colloidal system is a mixture of two phases of matter, wherein one of the phases is insolubly dispersed throughout the other through suspension. In our case, the silica microspheres will be suspended in deionized (DI) water. Due to water content in the atmosphere, the silica microspheres will have $\text{SiOH}$ groups adsorbed onto their surfaces. When the spheres are immersed in water, the $\text{H}^+$ ions separate from the surface of the spheres as they dissolve due to $\text{H}_2\text{O}$'s polar nature, leaving a $\text{SiO}^-$ behind. The spheres are thus left with a net negative charge $q$ on their surfaces.

Using a power supply, we apply a potential difference of 7.5 V across the sample through the two parallel plates, therefore generating a uniform field. As you observe the demo, keep the following questions in mind. After seeing the demo, discuss with your lab mates and sketch out answers in your lab notebook.

• In what direction do the particles move?
• What is the shape of the trajectory?
• What happens at larger voltages? Smaller ones?
• What would you think is the role of ions in the water? Would the particles behave the same in salt water?

For your reference, below we include an schematic of the setup.

Let us first begin by thinking about the forces that a small particle with charge $q$ suspended in water would experience in the presence of an electric field $\vec{E}$. By Newton's second law, we know that the acceleration due to the electric field will be given by

But as the particle moves, there will also be a viscous drag force $\vec{f}$ given by the collisions the particle will experience with the water molecules, as well as the electrodynamic interactions between its charge and ions in water. In the case that the velocity and the particle size are small, this drag force will be linearly proportional to the velocity of the particle and the viscosity of water, so that

Here $\alpha$ is some geometric constant, $\eta$ is the viscosity of water, and $\vec{v}$ is the particle's velocity. Newton's second law thus becomes

If we express the acceleration as the rate of change of the velocity, so that $\vec{a_E} = \frac{\text{d}\vec{v}}{\text{d}t}$ and rearrange, the above becomes the following differential equation:

Instead of solving it analytically, let's think about the limiting cases:

• At $\mathbf{t = 0}$. Just after the constant electric field is turned on while the particle is stationary, the drag force is zero. The particle will then accelerate to some velocity $\vec{v}$ at a constant rate given by Eq. 1: $\vec{a_E} = \frac{q}{m}\vec{E}$.
• At $\mathbf{0<t<\infty}$. As the particles acquires a velocity, it will start decelerating at a rate given by $-\frac{\alpha \eta}{m}\vec{v}$. The motion here will be governed by the time-dependent Eq. 4.
• At $\mathbf{t \to \infty}$. In the long-time limit, the electric force and the drag force will cancel each other, so that $\vec{a_E} = 0$. At this point,$\frac{\text{d}\vec{v}}{\text{d}t}=\frac{q}{m}\vec{E}-\frac{\alpha \eta}{m}{\vec{v}}=0$. We can then solve for the terminal velocity,

In Eq. 5, we find that we have introduced the electrophoretic mobility, $\mu_e \equiv \frac{q}{\alpha \eta}$, so that for spheres suspended in a low Reynolds number environment this becomes $\mu_e = \frac{q}{6\pi r \eta}$.

For our experiment, we will not be working with spheres, but rather pigment molecules, which means that our model for spheres will likely not be accurate. However, if we assume a low Reynold's number environment, our model still predicts that the terminal velocity will be proportional to the applied electric field.

One member of the group should click on the link below to start your group lab notebook. (You may be asked to log into your UChicago Google account if you are not already logged in.) Make sure to share the document with everyone in the group (click the “Share” button in the top right corner of the screen) so each member has access to the notebook after you leave lab.

In this experiment, we want you to test the behavior of the particles predicted by Equation 5. More specifically, you will be performing electrophoresis of food pigment molecules in food coloring, using wet chromatography paper as the medium that provides a frictional drag force. Most food coloring dyes consist of a mixture of a handful of pigment molecules, each of which has a specific charge and molecular mass. This means that when the pigments are subjected to an electrical potential difference, they will experience an electric force that is proportional to the charge they carry. As such, you'll be able to separate the pigments based on the magnitude of their charge given that they will move at different rates. Figure 2 below shows some important parameters of these pigments.

Figure 2a: Food coloring pigments' names and relevant parameters. Taken from American Journal of Physics 83, 1003-1011 (2015)	Figure 2b: Pigments we'll use in this lab.

Figure 3a: Volt meter. We will be using it in the DC setting.	Figure 3b: Toothpicks, flathead, pipettes, petri dishes, and trays.

To perform the experiment, we have provided you with the following equipment, seen in the figures above:

• different pigments and pigment mixtures (when you're ready, ask your TA for a few drops of the pigments on a petri dish)
• 3 cm wide strips of chromatography paper (these will be available from a roll in the front of the room - please only take what you need)
• toothpicks (more will be available in the front)
• electrophoresis tray table with ruler
  • rectangular graphite electrodes
  • petri dishes
• 2 g/L baking soda solution (available in front of the room; you'll only need 4 petri dishes filled half-way with this solution)
• pipette
• high voltage power supply
• banana cables with alligator clips
• flathead screwdriver

Let us first get acquainted with the experimental apparatus and samples we will be working with. To perform the experiment, we have provided you with an electrophoresis tray table, a picture of which you can see in Figure 4. It consists of a flat surface with a rectangular well at either end. In this well we can insert rectangular graphite electrodes that we can charge by connecting the metal pins that keep the electrodes in place to the positive and negative leads of a power supply.

Figure 4: Power supply. You will have two of these already wired up. The negative lead should be connected to the ground pin (the three parallel lines), so that we measure voltages relative to ground (zero).

The electrophoresis procedure will be done on the strips of chromatography paper. Each strip will be placed on the tray so that it lays flat on the surface. Both ends of the strip will go over the well so that, by carefully inserting the graphite in the well, the strip remains pinched and stationary. To prepare this:

• Hold the strip of paper parallel to the surface and over the well, as in the picture below.
• Insert the graphite electrode, letting it pull the paper into the well.
• Pull the paper softly so that its surface is flush with the surface of the tray.
• Holding the paper fixed to the table by pressing on it, place the other electrode in its well; it should pull paper from the other end with it inside the well.
• When both electrodes are placed in the wells and the paper is held taut and flush with the surface of the tray, tighten the screws with the lips pressing on the electrodes. These keep the electrodes in place and in electrical contact with the paper, and also serve as a point of contact with the power supply electrodes.
• Using the pipette, wet the paper with the baking soda solution and place the ends of the strip in a baking soda solution in a petri dish.

Figure 5a: Electrophoresis tray.	Figure 5b: Push down on the graphite electrode while holding the paper taut.	Figure 5c: Insert the second electrode so that the middle of the strip on the tray is flush with the surface.

You will be observing what happens to pigment particles as they are subjected to an electric field using this setup. To do so, start by putting a drop of coloring on a petri dish. Then wet the tip of a toothpick with the dye and touch the wet chromatography paper strip with the tip, so that you end up with a spot of reasonable size.

After doing so, think about the following:

• What do you see about the spot of colorant that may pose a challenge in making measurements?
• How will you determine the position of the dye? How will this affect your measurement uncertainty?
• What is the dye doing? Is it stationary? Moving? If so, in what direction?

Now connect each lead of the power supply to either crimping bolt to generate a voltage potential difference across the strip and set the voltage to a high setting between 120 V to 150 V.

• What do you see when you turn on the voltage? Are the different dyes behaving the same?
• At what timescale is the dye moving? How will you be making time measurements?
• In what direction is the dye moving? What does that tell you given the electric field you are applying?

From the theory section above, the dye particles will reach a terminal velocity due to the frictional forces they experience in the medium. We invite you to do the calculation yourselves for a particle of nanometer length scales, which would lead you to discover that the $1/e$ time for this is in the order of femtoseconds, so that for all practical purposes the particle speed is constant. Therefore, we can determine this speed as the slope of a graph of the average position of the particles versus the time elapsed.

For a few colors, measure the displacement as a function of time when a large potential gradient (above 150 V) is applied between the leads.

• How are you determining the average position?
• How are you ensuring that the successive measured positions of the spots are distinct from the previous measurements?
• For how long will you need to collect? Do you see any unexpected behavior as time goes on? Why may that be?
• Does the uncertainty in position change as time evolves? Is the uncertainty in time significant?

Using the second tray provided, repeat the experiment for a particular color using a few drops of that color. Do they all move at the same rate? What does that tell you about the movement of the particles and the assumptions we have made?

Repeat this last experiment for a specific pigment with at least two voltages. That is, for three different voltages, measure the average displacement for a few drops of one pigment.

Make sure to submit your lab notebook by the end of the period. Download a copy of your notebook in PDF format and upload it to the appropriate spot on Canvas. Only one member of the group needs to submit to Canvas, but make sure everyone's name is on the document!

Answer the questions/prompts below in a new document (not your lab notebook) and submit that as a PDF to the appropriate assignment on Canvas when you are done. You should write the answers to these questions by yourself (not in a group), though you are welcome to talk to your group mates to ask questions or to discuss.

The conclusion is your interpretation and discussion of your data.

• What do your data tell you?
• How do your data match the model (or models) you were comparing against, or to your expectations in general? (Sometimes this means using the $t^{\prime}$ test, but other times it means making qualitative comparisons.)
• Were you able to estimate uncertainties well, or do you see room to make changes or improvements in the technique?
• Do your results lead to new questions?
• Can you think of other ways to extend or improve the experiment?

In about one or two paragraphs, draw conclusions from the data you collected today. Address both the qualitative and quantitative aspects of the experiment and feel free to use plots, tables or anything else from your notebook to support your words. Don't include throw-away statements like “Looks good” or “Agrees pretty well”; instead, try to be precise.

In addition to your conclusion, try to include in your report answers to the following questions. Feel free to discuss among your peers or with the TA for ideas, but make sure the answers are your own. Do not worry too much about a precisely correct answer. Instead, think about what you have learned about electric fields and the questions that would arise when thinking about ions moving in electric fields.

1. For the spheres in water, what do you think is the role of ions in the water? Would the particles behave the same in salt water? What about in more acidic or alkaline water?
2. What do you think will happen next week when we use more acidic water (higher concentration of $\text{H}^+$ ions) by using vinegar instead of baking soda? Think about how the positive hydrogen ions will be attracted to the negatively charged pigment particles. Looking up charge screening may or may not be helpful.
3. You saw that different dyes move at different rates depending both on their geometries and the charge they have. How would you design an experiment to separate a mixture of pigments? Can this be applied to other sorts of molecules such as proteins and nucleic acids?