UChicago Instructional Physics Laboratories

In this experiment, we will be studying electrophoresis, the motion of charged particles suspended in a medium when they are subjected to an electric field. The dynamics of charged objects in aqueous media is particularly relevant in the study of living systems. Charged particles, such as proteins and ions in cells, generate electric fields that that act on other charged particles, forming the basis for many transport phenomena in living systems. How suspended particles behave in the presence of an electric field has significant applications in analytical methods for studying biomolecules, such as DNA gel electrophoresis for separating nucleic acid chains based on their charge to mass ratios. As such, understanding the dynamics of charged particles suspended in aqueous media if of great interest in the study of living systems. This lab will consider a relatively simple system to shed light on this phenomenon.

By working through this experiment, you will:

1. develop a model of the forces acting on a system in aqueous solution and test the validity of that model by manipulating observable parameters;
2. think about the regimes in which a model is valid and where it breaks down or is no longer applicable;*
3. apply knowledge of electrodynamics and fluid dynamics to study chemical and biological systems.

For this experiment, you will be working with a colloidal system consisting of silica ($\text{SiO}_2$) microspheres. 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.

In addition to this surface charge, there are freely flowing $\text{H}^+$ and $\text{OH}^-$ ions in water that are affected by the resulting potentials from the negatively charged spheres. In fact, positively charged ions are attracted to the negative surface charge, creating a layer of oppositely charged freely flowing ions around the surface charge of the spheres, resulting in two layers of charge, otherwise known as an electric double layer. This added layer of charge essentially “screens” the surface charge of the spheres, resulting in a smaller effective surface charge on which external electric fields may act. What would happen if, in addition of the naturally arising ions in DI water, we added ions by dissolving a salt, such as $\text{NaCl}$, in the medium? One can imagine the extra $\text{Na}^+$ and $\text{Cl}^-$ ions interacting with this double layer, further screening the charge on the sphere. It therefore becomes important to take into account the ionic charge distribution in the medium in addition to the adsorbed charge on the particles when analyzing colloidal systems under electric potential gradients.

In this experiment, we would like you to investigate the motion of the microspheres in suspension when a uniform electric field is applied to them. More specifically, using your knowledge of electrodynamics and fluid dynamics, develop a model to determine the effective charge of the spheres. What are the forces acting on a sphere as it moves in the fluid due to an external electric field $\textbf{E}$? Is the sphere's speed constant? Recall the Stoke's drag for small spheres that we investigated last quarter.$^1$

For the first part of this experiment, we would like for you to develop and test a model for the charge that the spheres acquire when immersed in DI water. Think about the forces that a charge feels in the presence of an electric field, as well as the force that a small sphere feels when moving through a fluid. What happens when the sphere is moving at constant speed? Discuss with your lab partners and in your lab notebook sketch a diagram of the system, including the forces, the field, and any other aspect you think is important to understand and model this system. Then try to come up with an experiment or measurement you can perform in order to test your model. The following description of the experimental tools you have at your disposal may be helpful.

To test your model, we have provided you with a microscope equipped with a CCD camera along with a chambered microscope slide with two copper electrodes separated by some distance $l$ (you can measure this distance using the calipers provided). You will be able to apply a voltage $V$ across these electrodes using the power supply provided. We have also provided a suspension of silica microspheres of diameter $d = 4.89 \text{ }\mu \text{m} \text{ }\pm \text{ } 10\%$ in DI water, some of which you can put in the chamber to observe under the microscope.

• What is the field between the electrodes when you apply a voltage across them? What assumptions are you making?
• If you were to put an electron of charge $e$ between the electrodes while there is a voltage applied across them, what would be the force it would experience?
• What would be the drag force that a small sphere would experience in water as it is moving at constant speed? What assumptions are you making?
• Based on what you've learned thus far, how would you determine the charge of the spheres in suspension with the tools you're provided? How does the speed of the particles relate to the field and the charges they acquire? Describe the experiment that you will be performing to determine this charge.

Your TA will be coming to you to discuss what you have discovered and aid you with any question. Once you feel confident with your model and your experiment, you can start calibrating the microscope.

To see the microspheres, you will be using the microscope with a CCD camera connected to a computer. Using the camera software, this will allow you to see the microspheres and record a video of their movement in suspension. In order to make distance measurements of the spheres you record, you will have to calibrate the microscope, converting pixels measured to distances on the slides. We have provided you with a microscope slide that has markings at known distances of ($0.01 \text{mm}$). Using the software, you can then measure the distance in pixels between these markings and relate it to the distance in meters. Note that this calibration will be specific to the objective you're using, so you may want to look at the suspensions under the microscope first to determine which objective may suit your needs. Make sure that you record these calibration parameters, as you will be using them throughout this lab.

Now that you have calibrated the microscope and are able to see the microspheres in suspension, what happens when you apply an electric field to the spheres? Connect the electrodes from the chambered slide to the power supply and apply a voltage across the electrodes. Do they seem to be moving at constant speed? In what direction do the spheres move? Are they getting closer to the positive or negative electrode? What does this imply for their charge? Make note of your observations.

Now record a video of them moving as you apply an electric field. To do this, hit the record button on the software and select where you want to store your video; you may rename it if you want to include any relevant information in its title. After you have collected enough footage, hit the stop button.

Open the Tracker software on your computer and browse for the video you intend on analyzing. Open it on the software. Using the calibration tool, calibrate the pixels on the video using the Calibrate tool and converting the width of the video in pixels to the width of the sensor you measured in mm when you calibrated the microscope camera. Hide the calibration tool once you're done.

Next, you need to determine the time interval where you'll be making measurements. By default, the software measures time by frame, but we want to measure time in seconds. On Clip Settings, configure the program to measure time in seconds, setting the 0.0 s point at the beginning of the video. Select the interval by tuning the left and right black triangles on the time bar at the bottom. When you play the video on the software, you may see that they are moving rather slowly and that they don't change position too much from one frame to the next. To make your life easier, you may want to skip some frames using the Frame Skip tool on the bottom right. Try different values and see which ones allow you to not lose the sphere when you're tracking it while also saving you some time in measuring its position (Do you really need 20 measurements of position vs time in your time interval to determine average velocity?). Once you have done this, you are ready to track the spheres.

Click on the Track tool and select “Point Mass”. Now find a sphere that has a clear trajectory in the interval you're considering and, starting on the first frame of that interval, click on it while pressing the Shift key to record its position. The program will jump to the next frame. You will see the position and time on the table at the right of the screen. Continue recording its position until you reach the last frame. Now you have tracked one sphere, the velocity of which is given by the total change in displacement over the total change in time. Continue tracking the velocity of other spheres.

Once you have an adequate number of measurements of the speed of the particles, using the average speed, estimate the effective surface charge that the spheres acquire in DI water. Is this reasonable? (Recall that this charge is due to the accumulation of $\text{SiO}^-$ on the surface of the sphere, so each molecule will contribute one unit of charge $e = 1.6 \times 10^{-19} \text{ C}$.) Record this value along with pertinent uncertainties in your notebook.

There will not be a report due this week. Instead, we want you to turn in your notebook and think about what you may see next week when you investigate the spheres in saline solution of different $\text{NaCl}$ concentrations. For next week, make sure to include in your notebook answers to the questions in this wiki. Additionally:

1. What happens to the effective charge that the spheres acquire in suspension when extra ions are added to the water? What would this mean for the observed velocity of the spheres when you apply a constant electric field across the suspension?
2. Using the ideas you have explored so far, do you think you could devise a method for separating particles by size? How would this look like in the microscope?

Last week we studied the electrophoretic motion of silica microspheres suspended in DI water in order to estimate the charge that they acquire in suspension. After discussing the system with your lab partners and TA, you likely arrived at the following expression for the effective charge on the spheres based on an electro- and fluid dynamic analysis:

$q_{\text{eff}} = \dfrac{6\pi\eta a v}{E}$

where $\eta$ is the viscosity of water, $a$ is the sphere radius, and $\frac{v}{E}$ is the measured velocity per unit field.

This week, we will consider the same system, but instead of DI water, we will be using saline water of varying concentrations. We have provided you with 2-4 suspensions of the same $4.89 \text{ }\mu \text{m}$ microspheres in solutions ranging from 0.5 mg/L to 50 mg/L $\text{NaCl}$. For each of these solutions, repeat the experiment from last week and determine the effective charge of the spheres in the different suspensions. Try to keep the other parameters as constant as possible (i.e., same potential, same distance, etc.) and keep track of all the assumptions that you are making. Once you have a measurement based on the video analysis (remember to calibrate the video grid as last week), plot your results for charge as a function of concentration using the provided script.

• What does your data tell you? Is this consistent with the screening description we provided in the theory section?

In this experiment, we have asked you to develop your own model to determine the charge that silica microspheres acquire when suspended in water with different ionic concentrations. For the first part of your report, describe the model and experiment you devised, along with the physical reasoning and assumptions that went into it. Include any diagram or sketch that may aid your explanation (sometimes a clear figure with a brief description is more helpful for the reader than a wall of text). It will be helpful to look at your notebook from the first week.

For the second part of your report, discuss and analyze the data that you have gathered. The following questions may be helpful when crafting your text:

• 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?

It actually turns out that the model we have studied here does not generally describe the dynamics of the system. In fact, using an analysis vastly beyond the scope of this course, it turns out that the velocity per unit field of the spheres ($\frac{v}{E}$), otherwise known as the electrophoretic mobility $\mu_e \equiv \frac{v}{E}$, is independent of the sphere radius (more generally, independent of the suspended particle's geometry). Instead, it depends only on the viscosity of the medium, $\eta$, its dielectric constant $\epsilon$, and the ion concentration $\nu$ in the medium. To arrive at this, the presence of charges due to freely flowing ions around the spheres needs to be taken into consideration, which is not something we did in this experiment.

If this is the case, then how can electrophoresis be used to analyze the charge to mass ratios of biomolecules? One method that is commonly used in the analysis of nucleic acids is gel electrophoresis. Instead of having particles freely suspended in water, they are suspended in a gel matrix that provides some frictional drag to the particles' motion. This friction is dependent on their masses, which allows the particles of similar charges to be separated by their masses.