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. Whilst 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.^[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. Therefore, there should be three rotational diffusion constants - the eigenvalues of the rotational diffusion tensor - resulting in five rotational time constants.^[3][4] 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.

Relation to translational diffusion

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.

Rotational version of Fick's law

A rotational version of Fick's law of diffusion can be defined. Let each rotating molecule be associated with a unit vector ${\displaystyle {\hat {n}}}$; for example, ${\displaystyle {\hat {n}}}$ 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 ${\displaystyle {\hat {n}}}$ at time t. Here, θ and φ represent the spherical angles, with θ being the polar angle between ${\displaystyle {\hat {n}}}$ and the z-axis and φ being the azimuthal angle of ${\displaystyle {\hat {n}}}$ in the x-y plane.

The rotational version of Fick's law states

${\displaystyle {\frac {1}{D_{\mathrm {rot} }}}{\frac {\partial f}{\partial t}}=\nabla ^{2}f={\frac {1}{\sin \theta }}{\frac {\partial }{\partial \theta }}\left(\sin \theta {\frac {\partial f}{\partial \theta }}\right)+{\frac {1}{\sin ^{2}\theta }}{\frac {\partial ^{2}f}{\partial \phi ^{2}}}}$.

This partial differential equation (PDE) may be solved by expanding f(θ, φ, t) in spherical harmonics ${\displaystyle Y_{l}^{m}}$ for which the mathematical identity holds

${\displaystyle {\frac {1}{\sin \theta }}{\frac {\partial }{\partial \theta }}\left(\sin \theta {\frac {\partial Y_{l}^{m}}{\partial \theta }}\right)+{\frac {1}{\sin ^{2}\theta }}{\frac {\partial ^{2}Y_{l}^{m}}{\partial \phi ^{2}}}=-l(l+1)Y_{l}^{m}(\theta ,\phi )}$.

Thus, the solution of the PDE may be written

${\displaystyle f(\theta ,\phi ,t)=\sum _{l=0}^{\infty }\sum _{m=-l}^{l}C_{lm}Y_{l}^{m}(\theta ,\phi )e^{-t/\tau _{l}}}$,

where C_lm are constants fitted to the initial distribution and the time constants equal

${\displaystyle \tau _{l}={\frac {1}{D_{\mathrm {rot} }l(l+1)}}}$.

Two-dimensional rotational diffusion

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. In this example, a sphere which is fixed on the vertical axis rotates around that axis only, meaning that the particle can have a θ value of 0 through 360 degrees, or 2π Radians, before having a net rotation of 0 again.^[5]

These directions can be placed onto a graph which covers the entirety of the possible positions for the face to be at relative to the starting point, through 2π radians, starting with -π radians through 0 to π radians. 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. Then as more time passes, the chance for the molecule to rotate further from its starting point increases which widens the peak, until enough time has passed that the measurements will be evenly distributed across all possible directions.

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

1. For a single particle with multiple measurements taken over time. A particle which has an area designated as its face pointing in the starting orientation, starting at a time t₀ will begin with an orientation distribution resembling a single line as it is the only measurement. Each successive measurement at time greater than t₀ will widen the peak as the particle will have had more time to rotate away from the starting position.
2. For multiple particles measured once long after the first measurement. The same case can be made with a large number of molecules, all starting at their respective 0th orientation. Assuming enough time has passed to be much greater than t₀, the molecules may have fully rotated if the forces acting on them require, and a single measurement shows they are near-to-evenly distributed.

Basic equations

For rotational diffusion about a single axis, the mean-square angular deviation in time ${\displaystyle t}$ is

${\displaystyle \left\langle \theta ^{2}\right\rangle =2D_{r}t}$,

where ${\displaystyle D_{r}}$ is the rotational diffusion coefficient (in units of radians²/s). The angular drift velocity ${\displaystyle \Omega _{d}=(d\theta /dt)_{\rm {drift}}}$ in response to an external torque ${\displaystyle \Gamma _{\theta }}$ (assuming that the flow stays non-turbulent and that inertial effects can be neglected) is given by

${\displaystyle \Omega _{d}={\frac {\Gamma _{\theta }}{f_{r}}}}$,

where ${\displaystyle f_{r}}$ is the frictional drag coefficient. The relationship between the rotational diffusion coefficient and the rotational frictional drag coefficient is given by the Einstein relation (or Einstein–Smoluchowski relation):

${\displaystyle D_{r}={\frac {k_{\rm {B}}T}{f_{r}}}}$,

where ${\displaystyle k_{\rm {B}}}$ is the Boltzmann constant and ${\displaystyle T}$ is the absolute temperature. These relationships are in complete analogy to translational diffusion.

The rotational frictional drag coefficient for a sphere of radius ${\displaystyle R}$ is

${\displaystyle f_{r,{\textrm {sphere}}}=8\pi \eta R^{3}}$

where ${\displaystyle \mathrm{\eta <!--\; \eta \; -->}}$Zdroj:https://en.wikipedia.org?pojem=Rotational_diffusion
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