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## Brownian motion

"...bodies of microscopically-visible size suspended in a liquid will perform movements of such magnitude that they can be easily observed..."

Albert Einstein, 1903

Brownian motion (see right) is the irregular (apparently random) motion of small particles suspended in water (or other fluids) as a result of thermal molecular motions. This irregular movement is observable down to extremely small timescales. It has been known at least as far back as Lucretius in about. 60 BC. It is named after the botanist Robert Brown who described the phenomenon as inanimate in 1829 in spite of their vigorous movement that had previously been thought due to 'living' particles. Early in the 20th century, the following conclusions were drawn,

• The movement never ceases. c
• The movement is very irregular and made up of both translations and rotations.
• The apparent mean velocity varies significantly in magnitude and direction. b
• The apparent velocity does not tend to a limit as the time taken for an observation decreases. b
• The actual velocity cannot be measured. b
• The movement is greater when the particles are smaller.
• The movement is greater if the viscosity is less.
• The movement is greater if the temperature is greater. Viscosity changes partially explain this.
• The composition and density of the particles seem to have no effect.
• The particles move independently even if they closely approach.

This motion can be regarded as that of a particle that moves with velocity v = x/t (m ˣ s-1), that collides every t (s), and comes out of the collision by going forward or backward with equal probability. The mean displacement is zero; <Δx> = <(xt− x0)> = 0, where <> indicates' the mean value of'. The mean squared displacement (<Δx2> = <(xt− x0)2>) is proportional to the time elapsed (t) and the diffusivity (D; m2 ˣ s-1; the diffusion coefficient); the averaged net displacement (Δx) being proportional to the square root of the elapsed time;

the Einstein-Smoluchowski equation) a;

<Δx2> = 2Dt

<Δy2> = 2Dt

<Δz2> = 2Dt

However, the mean velocity (v, m ˣ s-1) diverges as the time (t) approaches zero.

v ≡ (<Δx2>) /t = (2D/t)½ =

with D described by the Stokes-Einstein equation for translational diffusion [806],

where D is the diffusivity (m2 ˣ s-1 ), R is the gas constant (J ˣ mol-1 ˣ K-1; kg ˣ m2 ˣ s-2 ˣ K-1 ˣ mol-1), T is the temperature (K), N is the Avogadro constant, η is the dynamic viscosity (Pa ˣ s; kg ˣ m-1 ˣ s-1), r is the averaged particle radius (m), and t is the time passed (s). The random velocity decreases with increasing particle size. The mean velocity at 25 °C for 10 nm, 100 nm, and 1 µm diameter particles are 9 µm ˣ s-1, 3 µm ˣ s-1, and 0.9 µm ˣ s-1 respectively.

If correct down to very short times (clearly not so), the apparent velocity ((<Δx2>) /t) is inversely proportional to √t and therefore is incorrect and grows without limit when this time interval becomes shorter. Einstein realized that his equation could not be used to determine velocities as his theory only applies at long time-scales. At short time scales, Brownian motion is not entirely random, due to the inertia of the particle and the surrounding fluid. Einstein stated that the kinetic energy of a particle is independent of the size and nature of the particle and independent of the nature of its environment. This kinetic energy (½mv2) is equal to 1.5kBT for all particles where kB is the Boltzmann constant and T is the temperature (K).

½mv2 = 1.5kBT

The motion of a particle through a liquid causes long-lived vortices (a memory effect). These affect the particles' dynamics. The effective mass of a particle in an incompressible liquid is the sum of the mass of the microsphere plus half of the mass of the displaced liquid [3323]. Brownian motion may also produce (thermal) noise in the structure of biomolecules such as proteins and nucleic acids and in water clustering.

Einstein [2480] also derived the equation,

That is, the spread of particles after a time period (t) forms a normal distribution with the mean distance from the origin of μ = 0 with a variance of

σ2 = 2 ˣ D ˣ t

These phenomena can be (at least plausibly) explained by collisions in the gas phase but lead to difficulty in ascribing Brownian motion to that resulting from their collision with the water molecules or clusters in a liquid [2480]. Difficulties in the explanation involve the conservation of energy in the collisions and the distances over which the consequent movement occurs. Einstein seems to have mostly avoided these questions, but he did involve osmotic pressure in the movement of the particles [2480]. The osmotic pressure of the particles certainly seems more likely as the cause of the macroscopic movement of the water and consequent motion of the particles. (see the discussion of the osmotic pressure at surfaces). Also, if the Brownian motion is prevented (for example at a surface), you get osmotic pressure [2480].

The entropy of a particle depends on the number of configurations associated with it being a distance (d) away from a fixed reference point. The entropic force <f> (resulting from the thermodynamical tendency to increase its entropy) acting on the particle is given as:

<f>= T(dS/dd) = 2kBT/d

<f> is the repulsive force that acts so as to drive the particle away from the reference point [3323e].

It appears that Brownian diffusion of a particle at an air/liquid interface is subject to an elastic response at the surface [3020]. However, there is a dispute over whether the response is viscous rather than elastic [3020]. Other work has described the diffusion coefficients in the directions parallel and perpendicular to an air/liquid interface. It was found experimentally that there is enhanced Brownian motion in the direction parallel to the surface [3628].

## Footnotes

a The full relationship is

where D is the diffusivity (m2 ˣ s-1 ), m is the particle mass (kg), η is the dynamic viscosity (Pa ˣ s; kg ˣ m-1 ˣ s-1), and r is the averaged particle radius (m). This reduces to      <Δx2> = 2Dt     at longer time intervals (t ≫ m/6πηr) [2480]. [Back]

b The instantaneous velocity of a Brownian particle in air (3-µm diameter silica bead, νrms = 0.422 mm ˣ s-1 at 99.8 kPa, 297 K; cf. νrms of N2 = 475 m ˣ s-1 ) or acetone (3.7-µm diameter barium titanate glass bead; νrms = 0.174 mm ˣ s-1, 291 K, there was too much noise to determine it in water), has been measured more recently using an optical tweezer [3323]. In water, you need ten picometer resolution within five nanoseconds. It is concluded that, at this limit, the paths of the particles build from short bursts of constant velocity; with this velocity depending on the mass (that includes the inertial mass of the displaced fluid) and temperature. [Back]

c This has caused much discussion over whether it is or is not a perpetual motion machine. [Back]

This page was established in 2016 and last updated by Martin Chaplin on 21 June, 2019