# Brownian Motion and viscometry

## Introduction and motivation: measuring viscosity with colloids

One of the most significant scientific advances in the 20th century was the complete description of Brownian motion. When Einstein's theory for Brownian motion, and Stokes's solution for drag on a sphere in a viscous liquid were merged to form the Stokes-Einstein relation, a whole new world for scientific development opened up. Indeed, this simple equation:

$D = \frac{k_B T}{6 \pi \eta r}$,

where $D$ represents the diffusion coefficient, $k_B$ is Boltzmann's constant, $T$ is the temperature, $\eta$ the liquid viscosity and $r$ the particle radius, has been implicated in no fewer than three Nobel prizes!

In this module, we will derive this expression, and use it to measure the viscosity of a water-glycerol mixture. This will be achieved by embedding a low numerical concentration of 1.1 micron-diameter colloids. To do this, we will track the motion of these particles, and measure their mean-squared displacement, which grows linearly with the lag-time. The constant of proportionality is related to $D$.

#### Week 2

Helpful alternative references for this week's lecture:

Review these notes on Einstein's analysis (link here)

Review these notes on Stokes drag (link here)

Review the particle-handling information sheet (link here), and review our particle's properties (link here)

Carefully read the 1st chapter from Howard Berg's book on random walks (from last week). You should be able to relate the diffusion coefficient to the mean-square displacment to D for a given diffusive trajectory in 2-D. Read chapter 1 from Howard Berg's book on random walks (link here)

#### Week 3

I suggest you look at the scientific literature to help develop your discussion and conclusion. Be sure to cite any references you use. Can you identify an open question? How about a question that was recently addressed using colloids that diffuse? For a point of reference, there are some cool questions related to far-from equilibrium thermodynamics that can be approached with colloids: Spatial Crossover Between Far-From-Equilibrium and Near-Equilibrium Dynamics in Locally Driven Suspensions (aps.org)

#### Week 1

1. Calculate the volume proportions of glycerol and water for the viscosity you will prepare for your assigned viscosity. The assigned viscosity is the number of your group x 3/2 cSt. The total target volume of water + glycerol + particles is 1 ml. Here is a link for an online calculator

2. calculate a `reasonable' concentration of particles for your measurement: approximately 50 particles per 1 mm x 1mm x 10 microns (!)

3. write a protocol for steps 1 & 2 to carry out next week to include as an appendix to your lab reports
4. Download the m-files from here or, if you prefer python, download the trackpy package, located here. There are sample data sets included with either software package; familiarize your selves with the tracking processing workflow so you are prepared to collect and analyze data next week.

#### Week 2

A tiff stack can be found here for typical particle tracking data

Groups for module 3

1. Vianney, Nasiri, Delespaul, Buehler

2. Froessel, Karam, Marciniak

3. Fontbonne, Aebi, Scherrer

4. Bagnoli, Bernard, Sabaghi

6. PhD group

#### Week 3

The goal of today's exercises is to acquire a complete data set or two - comprised of mean-squared displacement as a function of lag time.

With these key data in hand, you are ready to consider hyptotheses. In this process, bear in mind a few points:

• Does the measured $D$ agree with the prediction from the Stokes-Einstein relation?
• If not, why might it not agree?
• What are the sources of error in the experiment? Can the difference, if any, between your measured value and the S-E prediction be accounted for by error?

In light of the readings for this week, what questions emerge that can help complete the scientific context for you? Any points that are unclear, or that you might do differently?

We also have the possibility to measure the viscosity directly using the rheometer for comparison, if that can help your scientific investigation.

#### Group Submissions (2023)

Group 5 - Matlab code to simulate the diffusion of n particules for x steps in 2D.

Make sure to select the number of particules, steps and the length of the displacement you wish to simulate.

#### Group Submissions (2022 and prior years)

Group 4 (2022): Concentration of glycerol in the glycerol bottle.

Group 6 (2022): How to Use a Micropipette.pdf

Group 5 - Matlab code to get the required volumes for a given viscosity and temperature.

Make sure to write the good path in the Matlab code 'visosity2volume.m'.

• This wiki