# Brownian_Motion_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 1

Read chapter 1 from Howard Berg's book on random walks (link here).

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)

#### week 2

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.

#### week 3

Note that office hours will be held on Tuesday starting at 14:30. We will likely be in the lab on the ground floor. If you don't find us, call John's office number: +41 21 693 02 70

I suggest you look at the scientific literature to help develop your discussion and conclusion. 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 4

Exceptionally, the Tuesday afternoon office hours will not be held this week. If you require additional measurements, we can make arrangements to do this.

Note that instead of meeting in person, John will hold online office hours on Zoom at meeting no.  932 9511 6450. The office hours will be held from 8 until 12 on Wednesday, 22.12.

A second office hour session will be held after the holiday and prior to the 3rd report submission deadline, on 5.1.2022, again from 8 until 12, at the same Zoom room.

### Exercises

#### 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

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