Introduction

Daniel FitzGreen

Diffusion by Brownian Motion

Introduction

2.1 Diffusion

Diffusion is the movement of a substance from an area of high concentration to an area of low concentration. Diffusion dictates the spatial variation of neutron density in a nuclear reactor [2], coordinates DNA replication [3], and many other processes in a wide variety of systems.

Fick’s law relates the diffusive flux (the rate of flow per unit area) to concentration:

(1) $\begin{equation*} J = -D\cfrac{\dd\phi}{\dd x},\end{equation*}$

where $J$ is the diffusion flux, $D$ is the diffusion coefficient (typically expressed in units of m²/s), and $\phi$ is the concentration. As a direct result of this law, one can predict how the concentration of a substance changes with time:

(2) $\begin{equation*} \cfrac{\partial\phi}{\partial t}=D\cfrac{\partial^2\phi}{\partial x^2}.\end{equation*}$

In the reference paper, this equation is expressed as

(3) $\begin{equation*} \cfrac{\partial\rho(r,t)}{\partial t}=D\nabla^2_r \rho(r,t),\end{equation*}$

which is pretty much the same equation with a minor variable change, and generalized to more than one dimension.

It turns out that Einstein was very active in many fields of physics, not just relativity. The Stokes-Einstein equation (from Einstein’s PhD thesis) for a perfectly spherical particle is

(4) $\begin{equation*} D = \cfrac{k_BT}{6\pi R\eta},\end{equation*}$

where $k_B$ is the Boltzmann constant, $T$ is the temperature, $\eta$ is the viscosity, and $R$ is the radius of the particle.

One can solve Equation (3) (in one dimension) to determine the concentration of the diffusing substance for $t>0$ :

(5) $\begin{equation*} \rho (r,t) = \cfrac{\rho_0}{(4\pi Dt)^{1/2}}\exp[-\cfrac{(r-r_0)^2}{4D(t-t_0)}].\end{equation*}$

You should recognize the form of that equation: It’s a Gaussian! But it isn’t normalized since we’re not talking about probability distributions. As the particles diffuse from areas of high concentration to low concentration, the Gaussian curve will become wider. A standard measure of the width of a Gaussian curve is its full width at half-maximum (FWHM), which is

(6) $\begin{equation*} W = 4\sqrt{\ln(2)Dt}.\end{equation*}$

It should be quite obvious how the width of this curve changes in time.

2.2 Superparamagnetic colloids

Paramagnetism is a property of some materials whereby they are only weakly attracted by external magnetic fields, but align themselves in the direction of that external field, and generate a magnetic field of their own.

A colloid is a homogeneous, non-crystalline substance consisting of large molecules or ultramicroscopic particles of one substance dispersed through a second substance. In this experiment, the colloids are comprised of iron-oxide crystals dispersed in a microscopic polymer microsphere. The iron oxide crystals have a strong magnetic moment which, if randomly aligned, creates a microsphere with no net magnetic moment. When a magnetic field is applied, the magnetic moments align, creating a strong net magnetic moment.

The reference paper goes through a lot of math to explain that, when exposed to a magnetic field, the superparamagnetic colloids are attractive in the radial direction, and repulsive in the axial direction. Due to this property, the colloids will form chains when exposed to a uniform magnetic field. Don’t worry too much about all this theory unless you are very interested in electromagnetism. It’s not super relevant to the data analysis or understanding the concept of diffusion. Just know that we are choosing these colloids because we can easily compress them into chains, providing a suitable initial condition for this diffusion experiment, and by removing the magnetic field, the colloids become a collection of non-interacting, free particles undergoing Brownian motion.

Introduction

2.1 Diffusion

2.2 Superparamagnetic colloids

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