
Rotational diffusion is the rotational movement which acts upon any object such as
particle
In the physical sciences, a particle (or corpuscle in older texts) is a small localized object which can be described by several physical or chemical properties, such as volume, density, or mass.
They vary greatly in size or quantity, from s ...
s,
molecule
A molecule is a group of two or more atoms that are held together by Force, attractive forces known as chemical bonds; depending on context, the term may or may not include ions that satisfy this criterion. In quantum physics, organic chemi ...
s,
atom
Atoms are the basic particles of the chemical elements. An atom consists of a atomic nucleus, nucleus of protons and generally neutrons, surrounded by an electromagnetically bound swarm of electrons. The chemical elements are distinguished fr ...
s when present in a
fluid
In physics, a fluid is a liquid, gas, or other material that may continuously motion, move and Deformation (physics), deform (''flow'') under an applied shear stress, or external force. They have zero shear modulus, or, in simpler terms, are M ...
, by random changes in their
orientations
''Orientations'' is a bimonthly print magazine published in Hong Kong and distributed worldwide since 1969.
History
''Orientations'' was launched in 1969 by Adrian Zecha (who was later the founder of Aman Resorts) to showcase Asian art and cu ...
.
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
colloid
A colloid is a mixture in which one substance consisting of microscopically dispersed insoluble particles is suspended throughout another substance. Some definitions specify that the particles must be dispersed in a liquid, while others exte ...
s, where relatively large
insoluble
In chemistry, solubility is the ability of a substance, the solute, to form a solution with another substance, the solvent. Insolubility is the opposite property, the inability of the solute to form such a solution.
The extent of the solub ...
particles are suspended in a greater amount of fluid. The changes in orientation occur from
collision
In physics, a collision is any event in which two or more bodies exert forces on each other in a relatively short time. Although the most common use of the word ''collision'' refers to incidents in which two or more objects collide with great for ...
s between the particle and the many molecules forming the fluid surrounding the particle, which each transfer
kinetic energy
In physics, the kinetic energy of an object is the form of energy that it possesses due to its motion.
In classical mechanics, the kinetic energy of a non-rotating object of mass ''m'' traveling at a speed ''v'' is \fracmv^2.Resnick, Rober ...
to the particle, and as such can be considered
random
In common usage, randomness is the apparent or actual lack of definite pattern or predictability in information. A random sequence of events, symbols or steps often has no order and does not follow an intelligible pattern or combination. ...
due to the varied
speed
In kinematics, the speed (commonly referred to as ''v'') of an object is the magnitude of the change of its position over time or the magnitude of the change of its position per unit of time; it is thus a non-negative scalar quantity. Intro ...
s 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
Space is a three-dimensional continuum containing positions and directions. In classical physics, physical space is often conceived in three linear dimensions. Modern physicists usually consider it, with time, to be part of a boundless ...
, rotational diffusion randomises the orientation of any
particle
In the physical sciences, a particle (or corpuscle in older texts) is a small localized object which can be described by several physical or chemical properties, such as volume, density, or mass.
They vary greatly in size or quantity, from s ...
it acts on.
Anything in a solution will experience rotational diffusion, from the
microscopic scale
The microscopic scale () is the scale of objects and events smaller than those that can easily be seen by the naked eye, requiring a lens (optics), lens or microscope to see them clearly. In physics, the microscopic scale is sometimes regarded as ...
where individual atoms may have an effect on each other, to the
macroscopic scale
The macroscopic scale is the length scale on which objects or phenomena are large enough to be visible with the naked eye, without magnifying optical instruments. It is the opposite of microscopic.
Overview
When applied to physical phenom ...
.
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
In biochemistry and molecular biology, a binding site is a region on a macromolecule such as a protein that binds to another molecule with specificity. The binding partner of the macromolecule is often referred to as a ligand. Ligands may includ ...
, 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 star
A binary star or binary star system is a system of two stars that are gravitationally bound to and in orbit around each other. Binary stars in the night sky that are seen as a single object to the naked eye are often resolved as separate stars us ...
s, as well as the seemingly random spin axes of
supermassive black hole
A supermassive black hole (SMBH or sometimes SBH) is the largest type of black hole, with its mass being on the order of hundreds of thousands, or millions to billions, of times the mass of the Sun (). Black holes are a class of astronomical ...
s.
[ Merritt, D. (2002)]
Rotational Brownian Motion of a Massive Binary
''The Astrophysical Journal'', 568, 998-1003. Retrieved 28 March 2022
The random re-orientation of molecules (or larger systems) is an important process for many
biophysical probes. Due to the
equipartition theorem
In classical physics, classical statistical mechanics, the equipartition theorem relates the temperature of a system to its average energy, energies. The equipartition theorem is also known as the law of equipartition, equipartition of energy, ...
, 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
In physics, angular velocity (symbol or \vec, the lowercase Greek letter omega), also known as the angular frequency vector,(UP1) is a pseudovector representation of how the angular position or orientation of an object changes with time, i ...
about each of an object's
principal axes is inversely proportional to its
moment of inertia
The moment of inertia, otherwise known as the mass moment of inertia, angular/rotational mass, second moment of mass, or most accurately, rotational inertia, of a rigid body is defined relatively to a rotational axis. It is the ratio between ...
about that axis. Therefore, there should be three rotational diffusion constants - the eigenvalues of the rotational diffusion tensor - resulting in five rotational
time constant
In physics and engineering, the time constant, usually denoted by the Greek language, Greek letter (tau), is the parameter characterizing the response to a step input of a first-order, LTI system theory, linear time-invariant (LTI) system.Concre ...
s.
If two eigenvalues of the diffusion tensor are equal, the particle diffuses as a
spheroid
A spheroid, also known as an ellipsoid of revolution or rotational ellipsoid, is a quadric surface (mathematics), surface obtained by Surface of revolution, rotating an ellipse about one of its principal axes; in other words, an ellipsoid with t ...
with two unique diffusion rates and three time constants. And if all eigenvalues are the same, the particle diffuses as a
sphere
A sphere (from Ancient Greek, Greek , ) is a surface (mathematics), surface analogous to the circle, a curve. In solid geometry, a sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
with one time constant. The diffusion tensor may be determined from the
Perrin friction factors In hydrodynamics, the Perrin friction factors are multiplicative adjustments to the translational and rotational friction of a rigid spheroid, relative to the corresponding frictions in spheres of the same volume. These friction factors were first ...
, 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
Fluorescence anisotropy or fluorescence polarization is the phenomenon where the light emitted by a fluorophore has unequal intensities along different axes of polarization. Early pioneers in the field include Aleksander Jablonski, Gregorio Webe ...
,
flow birefringence
In biochemistry, flow birefringence is a hydrodynamic technique for measuring the rotational diffusion constants (or, equivalently, the rotational drag coefficients). The birefringence
Birefringence, also called double refraction, is the opt ...
,
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
virus
A virus is a submicroscopic infectious agent that replicates only inside the living Cell (biology), cells of an organism. Viruses infect all life forms, from animals and plants to microorganisms, including bacteria and archaea. Viruses are ...
es. 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
Fluorescence correlation spectroscopy (FCS) is a statistical analysis, via time correlation, of stationary fluctuations of the fluorescence intensity. Its theoretical underpinning originated from L. Onsager's regression hypothesis. The analysis ...
.
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
In mathematics, a unit vector in a normed vector space is a Vector (mathematics and physics), vector (often a vector (geometry), spatial vector) of Norm (mathematics), length 1. A unit vector is often denoted by a lowercase letter with a circumfle ...
; for example,
might represent the orientation of an
electric
Electricity is the set of physical phenomena associated with the presence and motion of matter possessing an electric charge. Electricity is related to magnetism, both being part of the phenomenon of electromagnetism, as described by Maxwel ...
or
magnetic dipole moment
In electromagnetism, the magnetic moment or magnetic dipole moment is the combination of strength and orientation of a magnet or other object or system that exerts a magnetic field. The magnetic dipole moment of an object determines the magnitude ...
. Let ''f''(''θ, φ, t'') represent the
probability density distribution for the orientation of
at time ''t''. Here, ''θ'' and ''φ'' represent the
spherical angles, with ''θ'' being the polar angle between
and the ''z''-axis and ''φ'' being the
azimuthal angle
An azimuth (; from ) is the horizontal angle from a cardinal direction, most commonly north, in a local or observer-centric spherical coordinate system.
Mathematically, the relative position vector from an observer ( origin) to a point of in ...
of
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
:
Here
is the angular diffusion coefficient, whose units are rad
2/s.
This equation contains the
angular Laplace operator , which can be written
:
Solution of the diffusion equation
This
partial differential equation
In mathematics, a partial differential equation (PDE) is an equation which involves a multivariable function and one or more of its partial derivatives.
The function is often thought of as an "unknown" that solves the equation, similar to ho ...
may be solved using the method of
separation of variables
In mathematics, separation of variables (also known as the Fourier method) is any of several methods for solving ordinary differential equation, ordinary and partial differential equations, in which algebra allows one to rewrite an equation so tha ...
by expanding
in
spherical harmonics
In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations in many scientific fields. The table of spherical harmonics co ...
Since spherical harmonics satisfy the identity
:
.
the solution may be written
:
,
where ''C
lm'' are constants (which depend on the initial distribution
) and the time constants are
:
.
Two-dimensional rotational diffusion

A sphere rotating around a fixed axis will rotate in two
dimension
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coo ...
s 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.
These directions can be placed onto a graph which covers the entirety of the possible positions for the
face
The face is the front of the head that features the eyes, nose and mouth, and through which animals express many of their emotions. The face is crucial for human identity, and damage such as scarring or developmental deformities may affect th ...
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
In mathematical analysis, the Dirac delta function (or distribution), also known as the unit impulse, is a generalized function on the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire real lin ...
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
A uniform is a variety of costume worn by members of an organization while usually participating in that organization's activity. Modern uniforms are most often worn by armed forces and paramilitary organizations such as police, emergency serv ...
as they all randomly
disperse to be nearly equal in all directions. This can be visualized in two ways.
# 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 t
0 will begin with an orientation distribution resembling a single line as it is the only measurement. Each successive measurement at time greater than t
0 will widen the peak as the particle will have had more time to rotate away from the starting position.
# 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 t
0, 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
is
:
,
where
is the rotational diffusion coefficient (whose units are radians
2/s).
The angular drift velocity
in response to an external torque
(assuming that the flow stays non-
turbulent
In fluid dynamics, turbulence or turbulent flow is fluid motion characterized by chaotic changes in pressure and flow velocity. It is in contrast to laminar flow, which occurs when a fluid flows in parallel layers with no disruption between ...
and that inertial effects can be neglected) is given by
:
,
where
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):
:
,
where
is the
Boltzmann constant
The Boltzmann constant ( or ) is the proportionality factor that relates the average relative thermal energy of particles in a ideal gas, gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin (K) and the ...
and
is the absolute temperature. These relationships are in complete analogy to translational diffusion.
The rotational frictional drag coefficient for a sphere of radius
is
:
where
is the
dynamic (or shear) viscosity.
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. This deviation can be explained by the formation of a depletion layer around the nanoparticle.
Langevin dynamics
Collisions with the surrounding fluid molecules will create a fluctuating torque on the sphere due to the varied speeds, numbers, and directions of impact. When trying to rotate a sphere via an externally applied torque, there will be a systematic drag resistance to rotation. With these two facts combined, it is possible to write the
Langevin-like equation:
Where:
*''L'' is the angular momentum.
*
is
torque
In physics and mechanics, torque is the rotational analogue of linear force. It is also referred to as the moment of force (also abbreviated to moment). The symbol for torque is typically \boldsymbol\tau, the lowercase Greek letter ''tau''. Wh ...
.
*''I'' is the moment of inertia about the rotation axis.
*''t'' is the time.
*''t''
0 is the start time.
*''θ'' is the angle between the orientation at ''t''
0 and any time after, ''t''.
*''ζ''
r is the rotational friction coefficient.
*''TB(t)'' is the fluctuating Brownian torque at time ''t''.
The overall Torque on the particle will be the difference between:
and
.
This equation is the rotational version of
Newtons second equation of motion. For example, in standard translational terms, a
rocket
A rocket (from , and so named for its shape) is a vehicle that uses jet propulsion to accelerate without using any surrounding air. A rocket engine produces thrust by reaction to exhaust expelled at high speed. Rocket engines work entirely ...
will experience a boosting force from the engine while simultaneously experiencing a
resistive force from the air it is travelling through. The same can be said for an object which is rotating.
Due to the random nature of rotation of the particle, the ''average'' Brownian torque is equal in both directions of rotation. symbolised as:
This means the equation can be averaged to get:
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.
As
is the rate of change of angular momentum over time, and is equal to a negative value of a coefficient multiplied by
, this shows that the angular momentum is decreasing over time, or decaying with a decay time of:
.
For a sphere of mass ''m'', uniform density ''ρ'' and radius ''a'', the moment of inertia is:
.
As mentioned above, the rotational drag is given by the
Stokes friction for rotation:
Combining all of the equations and formula from above, we get:
where:
*
is the momentum relaxation time
*''η'' is the
viscosity
Viscosity is a measure of a fluid's rate-dependent drag (physics), resistance to a change in shape or to movement of its neighboring portions relative to one another. For liquids, it corresponds to the informal concept of ''thickness''; for e ...
of the Liquid the sphere is in.
Example: Spherical particle in water

Let's say there is a virus which can be modelled as a perfect sphere with the following conditions:
*
Radius (a) of 100 nanometres, ''a'' = 10
−7m.
* Density: ''ρ'' = 1500 kg m
−3
* Orientation originally facing in a direction denoted by ''π''.
* Suspended in water.
* Water has a viscosity of ''η'' = 8.9 × 10
−4 Pa·s at 25 °C
*Assume uniform mass and density throughout the particle
First, the mass of the virus particle can be calculated:
From this, we now know all the variables to calculate moment of inertia:
Simultaneous to this, we can also calculate the rotational drag:
Combining these equations we get:
As the
SI units
The International System of Units, internationally known by the abbreviation SI (from French ), is the modern form of the metric system and the world's most widely used system of measurement. It is the only system of measurement with official st ...
for
Pascal are kg⋅m
−1⋅s
−2 the units in the answer can be reduced to read:
For this example, the decay time of the virus is in the order of nanoseconds.
Smoluchowski description of rotation
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:
where the current can be written as:
Which can be combined to give the rotational diffusion equation:
We can express the current in terms of an angular velocity which is a result of Brownian torque T
B through a rotational mobility with the equation:
Where:
*
*
*
The only difference between rotational and translational diffusion in this case is that in the rotational diffusion, we have periodicity in the angle θ. As the particle is modelled as a sphere rotating in two dimensions, the space the particle can take is compact and finite, as the particle can rotate a distance of 2π before returning to its original position
We can create a conditional probability density, which is the probability of finding the vector u at the angle θ and time t given that it was at angle θ
0 at time t=0 This is written as such:
The solution to this equation can be found through a Fourier series:
Where
is the Jacobian theta function of the third kind.
By using the equation
[Whittaker, E.T., Watson, G.N. ''A course of modern analysis'', (1965)]
The conditional probability density function can be written as :
For short times after the starting point where t ≈ t
0 and θ ≈ θ
0, the formula becomes:
The terms included in these are exponentially small and make little enough difference to not be included here. This means that at short times the conditional probability looks similar to translational diffusion, as both show extremely small perturbations near t
0. However at long times, t » t
0 , the behaviour of rotational diffusion is different to translational diffusion:
The main difference between rotational diffusion and translational diffusion is that rotational diffusion has a periodicity of
, meaning that these two angles are identical. 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
. This is opposed to translational diffusion, which has no such periodicity.
The conditional probability of having the angle be θ is approximately
.
This is because over long periods of time, the particle has had time rotate throughout the entire range of angles possible and as such, the angle θ could be any amount between θ
0 and θ
0 + 2 π. The probability is near-evenly distributed through each angle as at large enough times.
This can be proven through summing the probability of all possible angles. As there are 2π possible angles, each with the probability of
, the total probability sums to 1, which means there is a certainty of finding the angle at some point on the circle.
See also
*
Diffusion equation
The diffusion equation is a parabolic partial differential equation. In physics, it describes the macroscopic behavior of many micro-particles in Brownian motion, resulting from the random movements and collisions of the particles (see Fick's l ...
*
Perrin friction factors In hydrodynamics, the Perrin friction factors are multiplicative adjustments to the translational and rotational friction of a rigid spheroid, relative to the corresponding frictions in spheres of the same volume. These friction factors were first ...
*
Rotational correlation time
Rotational correlation time (\tau_c) is the average time it takes for a molecule to rotate one radian.
In solution, rotational correlation times are in the order of picoseconds. For example, the \tau_c = 1.7 ps for water, and 100 ps for a pyrrol ...
*
False diffusion
False diffusion is a type of error observed when the upwind scheme is used to approximate the convection term in convection–diffusion equations. The more accurate Finite difference#central difference, central difference scheme can be used for th ...
Notes
References
Further reading
*
*
{{Authority control
Diffusion
Rotation