Rotational diffusion is the rotational movement which acts upon any object such as particles, molecules, atoms when present in a fluid, by random changes in their orientations. Although the directions and intensities of these changes are statistically random, they do not arise randomly and are instead the result of interactions between particles. One example occurs in colloids, where relatively large insoluble particles are suspended in a greater amount of fluid. The changes in orientation occur from collisions between the particle and the many molecules forming the fluid surrounding the particle, which each transfer kinetic energy to the particle, and as such can be considered random due to the varied speeds and amounts of fluid molecules incident on each individual particle at any given time. The analogue to translational diffusion which determines the particle's position in space, rotational diffusion randomises the orientation of any particle it acts on. Anything in a solution will experience rotational diffusion, from the microscopic scale where individual atoms may have an effect on each other, to the macroscopic scale.
Applications 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. In order to communicate, the proteins must both come into contact with each other and be facing the appropriate way to interact with each other's binding site, which relies on the proteins ability to rotate. 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. 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. Therefore, there should be three rotational diffusion constants - the eigenvalues of the rotational diffusion tensor - resulting in five rotational time constants. If two eigenvalues of the diffusion tensor are equal, the particle diffuses as a spheroid with two unique diffusion rates and three time constants. And if all eigenvalues are the same, the particle diffuses as a sphere with one time constant. The diffusion tensor may be determined from the Perrin friction factors, in analogy with the Einstein relation of translational diffusion, but often is inaccurate and direct measurement is required. 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. In some techniques such as fluorescence it may be very difficult to characterize the full diffusion tensor, for example measuring two diffusion rates can sometimes be possible when there is a great difference between them, e.g., for very long, thin ellipsoids such as certain viruses. 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.
The diffusion equation and the rotational diffusion constant To model the diffusion process, consider a large number of identical rotating particles. The orientation of each particle is described by a unit vector n ^ {\displaystyle {\hat {n}}} ; for example, n ^ {\displaystyle {\hat {n}}} might represent the orientation of an electric or magnetic dipole moment. Let f(θ, φ, t) represent the probability density distribution for the orientation of n ^ {\displaystyle {\hat {n}}} at time t. Here, θ and φ represent the spherical angles, with θ being the polar angle between n ^ {\displaystyle {\hat {n}}} and the z-axis and φ being the azimuthal angle of n ^ {\displaystyle {\hat {n}}} in the x-y plane. Fick's second law of diffusion, applied to angular diffusion, states that in the absence of an external torque on the particles, the evolution of f(θ, φ, t) obeys
1 D r o t ∂ f ∂ t = ∇ θ ϕ 2 f . {\displaystyle {\frac {1}{D_{\mathrm {rot} }}}{\frac {\partial f}{\partial t}}=\nabla _{\theta \phi }^{2}f.}
Here D r o t {\displaystyle D_{\mathrm {rot} }} is the angular diffusion coefficient, whose units are rad2/s.
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