# Module 3: Brownian motion with colloids

## 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 10 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

#### Groups for module 3

 G1 Colleen Algisi Joseph Bernaert Andrea Leon De la Torre Karim Azzouzi G2 Cassandre Pitz Cameron Bush Syrielle Wicki Alexandre Mehdi Clement G3 Joao Barini Ramos Dogukan Ustun Filippo Brignolo Abhijeet Singh G4 Benjamin Colety Nathan Soury-Lavergne Julien Jeandroz Axel Guede G5 Aymeric Roy Nicolas Avellan Marin Pierre Garrabos Victor Solelhac G6 Joseph Bejjani Mi-Lane Bouchez Leo Assier de Pompignan James Ziadeh G7 Xinyue Wei Lebo Molefe Renata Osypova Tian Yang

#### Group Submissions

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'.

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