3.3 Diffusion pictures

two concentration regions, then the concentration in B increases as much as that in A decreases because mass is neither created nor annihilated inside the container. This spontaneous mixing continues until both concentrations become equal and, hence, reach a thermodynamic equilibrium consisting of a half-seawater/half-fresh water concentration throughout the entire box. Diffusion occurs wherever and whenever the concentration gradient exists, and diffusive solute flux is represented

Js ¼ �<sup>D</sup> <sup>d</sup><sup>c</sup>

A length scale of diffusion can be estimated by ffiffiffiffiffiffiffiffi

distance between two consecutive collisions).

3.2 Stokes-Einstein diffusivity

1D, 2D, and 3D spaces).

potential is denoted as μ.

84

dx

where D is diffusion coefficient (also often called diffusivity) of a unit of m<sup>2</sup>=s.

time interval. In molecular motion, δt can be interpreted as a time duration required for a molecule to move as much as a mean free path (i.e., a statistical averaged

When the solute concentration is low so that interactions between solutes are negligible, the diffusion coefficient, known as the Stokes-Einstein diffusivity, may

> <sup>D</sup><sup>0</sup> <sup>¼</sup> kBT 6πηa

(hydrodynamic) radius of solute particles. Stokes derived hydrodynamic force that a stationary sphere experiences when it is positioned in an ambient flow [5]:

where v represents a uniform fluid velocity, which can be interpreted as the velocity of a particle relative to that of an ambient fluid. FH is linearly proportional to v, and its proportionality 6πηa is the denominator of the right-hand side of Eq. (16). Einstein used the transition probability of molecules from one site to the another, and Langevin considered the molecular collisions as random forces acting on a solute (see Section 3.3 for details). Einstein and Langevin independently derived the same equation as (16) of which the general form can be rewritten as

> <sup>D</sup><sup>0</sup> <sup>¼</sup> kBT ð Þ 2d πη

<sup>1</sup> Greek symbol μ is also often used for viscosity in fluid mechanics literature. In this book, chemical

where d is the spatial dimension of the diffusive system (i.e., d ¼ 1, 2, and3 for

where kB is the Boltzmann constant, η is the solvent viscosity<sup>1</sup>

in 1 � D (14)

<sup>D</sup>δ<sup>t</sup> <sup>p</sup> where <sup>δ</sup><sup>t</sup> is a representative

(16)

(18)

, and a is the

J<sup>s</sup> ¼ �D∇c in 3 � D (15)

FH ¼ 6πηav (17)

using Fick's law as follows [3, 4]:

Non-Equilibrium Particle Dynamics

or

be given by

Several pictures of diffusion phenomena are discussed in the following section, which give probabilistic and deterministic viewpoints. If one considers an ideal situation where there exists only one salt molecule in a box filled with solvent (e.g., water) of finite T, P, and V. Since the sole molecule exists, there is no concentration gradient. Mathematically, the concentration is infinite at the location of the molecule and absolutely zero anywhere else: <sup>c</sup> <sup>¼</sup> <sup>V</sup>�<sup>1</sup> δð Þ r � r<sup>0</sup> where r<sup>0</sup> is an initial position of the solute and r is an arbitrary location within the volume. However, the following question arises. Why does a single molecule diffuse without experiencing any collisions in the absence of other molecules? The answer is that the solvent medium consists of a number of (water) molecules having a size of an order of O 10�<sup>10</sup> m. The salt molecule will suffer a tremendous number of collisions with solvent molecules of a certain kinetic energy at temperature T. Since each of these collisions can be thought of as producing a jump of the molecule, the molecule must be found at a distance from its initial position where the diffusion started. In this case, the molecule undergoes Brownian motion. Note that the single molecule collides only with solvent molecules while diffusing, which exists as a type of diffusion called self-diffusion.
