Rotational diffusion

One example occurs in colloids, where relatively large insoluble particles are suspended in a greater amount of fluid.

Rotational diffusion has multiple applications in chemistry and physics, and is heavily involved in many biology based fields.

For example, protein-protein interaction is a vital step in the communication of biological signals.

[1] As an example concerning physics, rotational Brownian motion in astronomy can be used to explain the orientations of the orbital planes of binary stars, as well as the seemingly random spin axes of supermassive black holes.

[2] The random re-orientation of molecules (or larger systems) is an important process for many biophysical probes.

Due to the equipartition theorem, larger molecules re-orient more slowly than do smaller objects and, hence, measurements of the rotational diffusion constants can give insight into the overall mass and its distribution within an object.

Quantitatively, the mean square of the angular velocity about each of an object's principal axes is inversely proportional to its moment of inertia about that axis.

The rotational diffusion tensor may be determined experimentally through fluorescence anisotropy, flow birefringence, dielectric spectroscopy, NMR relaxation and other biophysical methods sensitive to picosecond or slower rotational processes.

This is however not the case of the extremely sensitive, atomic resolution technique of NMR relaxation that can be used to fully determine the rotational diffusion tensor to very high precision.

Rotational diffusion of macromolecules in complex biological fluids (i.e., cytoplasm) is slow enough to be measurable by techniques with microsecond time resolution, i.e. fluorescence correlation spectroscopy.

[5] Much like translational diffusion in which particles in one area of high concentration slowly spread position through random walks until they are near-equally distributed over the entire space, in rotational diffusion, over long periods of time the directions which these particles face will spread until they follow a completely random distribution with a near-equal amount facing in all directions.

As impacts from surrounding particles rarely, if ever, occur directly in the centre of mass of a 'target' particle, each impact will occur off-centre and as such it is important to note that the same collisions that cause translational diffusion cause rotational diffusion as some of the impact energy is transferred to translational kinetic energy and some is transferred into torque.

Let f(θ, φ, t) represent the probability density distribution for the orientation of

The rotational version of Fick's law states This partial differential equation (PDE) may be solved by expanding f(θ, φ, t) in spherical harmonics

for which the mathematical identity holds Thus, the solution of the PDE may be written where Clm are constants fitted to the initial distribution and the time constants equal A sphere rotating around a fixed axis will rotate in two dimensions only and can be viewed from above the fixed axis as a circle.

Assuming all particles begin with single orientation of 0, the first measurement of directions taken will resemble a delta function at 0 as all particles will be at their starting, or 0th, position and therefore create an infinitely steep single line.

Over time, the increasing amount of measurements taken will cause a spread in results; the initial measurements will see a thin peak form on the graph as the particle can only move slightly in a short time.

The distribution of orientations will reach a point where they become uniform as they all randomly disperse to be nearly equal in all directions.

For rotational diffusion about a single axis, the mean-square angular deviation in time

(assuming that the flow stays non-turbulent and that inertial effects can be neglected) is given by where

[7] The rotational diffusion of spheres, such as nanoparticles, may deviate from what is expected when in complex environments, such as in polymer solutions or gels.

[8] Collisions with the surrounding fluid molecules will create a fluctuating torque on the sphere due to the varied speeds, numbers, and directions of impact.

Which is to say that the first derivative with respect to time of the average Angular momentum is equal to the negative of the Rotational friction coefficient divided by the moment of inertia, all multiplied by the average of the angular momentum.

is the rate of change of angular momentum over time, and is equal to a negative value of a coefficient multiplied by

For a sphere of mass m, uniform density ρ and radius a, the moment of inertia is:

To write the Smoluchowski equation for a particle rotating in two dimensions, we introduce a probability density P(θ, t) to find the vector u at an angle θ and time t. This can be done by writing a continuity equation:

We can express the current in terms of an angular velocity which is a result of Brownian torque TB through a rotational mobility with the equation:

This means that at short times the conditional probability looks similar to translational diffusion, as both show extremely small perturbations near t0.

This is because a circle can rotate entirely once before being at the same angle as it was in the beginning, meaning that all the possible orientations can be mapped within the space of

, the total probability sums to 1, which means there is a certainty of finding the angle at some point on the circle.

A molecule with a red cross on its front undergoing 3 dimensional rotational diffusion. The red cross moves erratically as the sphere is made to randomly rotate by collisions with surrounding molecules.
The standard translational model of Brownian motion
A sphere rotating around a fixed central axis can be modelled as a circle rotating in 2-dimensions when viewed from the axis of rotation. Here A 0 is the starting position at t 0 and A is the position at time t when the circle has rotated by θ.
Water particles (blue) and larger virus particle (red). The impact between the virus and water molecules will cause translational and rotational movement with varying speeds depending on the angle and speed of impact.