Heat-Conduction Equations - Heat Conduction: Mathematical Models and Analytical Solutions - page 138

3.6.1 Fick's Law of Diffusion and Diffusion Equation

The diffusion of material is widely encountered in many diverse fields of science
and technology. In manufacturing silicon transistors, for example, we use diffusion
to mix silicon with other materials. The silicon plates are surrounded by steam con-
taining the mixed material in a high-temperature oven. The mixed material diffuses
into the plates like the diffusion of red ink in water. Using planographic technol-
ogy, the diffusion of the mixed material is along one direction only. Such kinds of
diffusion are termed as unidirectional diffusion. We refer to this kind of diffusion
hereinafter in this section unless otherwise stated.
Diffusion always flows from a place with high density to a place of low density.
The density of the mixed material is defined as the particle number per unit volume
and is denoted by N . N is normally a function of position and time. In unidirectional

. ∂

N
∂

diffusion, N

x is called the density gradient and plays the same role in
material diffusion as the temperature gradient in heat conduction. The flux density
is the particle number passing through the unit area in a unit time period and is
denoted by j .
Fick's first law of diffusion . The flux density j is proportional to the density gra-
dient. Therefore,

= N ( x , t )

D ∂

N
∂

j = −

x ,

(3.55)

where D is called the diffusion coefficient . The negative sign accounts for the fact
that the diffusion is from an area with high density to another of low density. The
unit of the diffusion coefficient is m 2
s

. The unit of Eq. (3.55) is thus

m 2
s ·

Part icl e Number
m 2

Part icl e Number
m 4

= .

· s

Fick's law of diffusion is similar to the Fourier law of heat conduction. By the con-
servation of mass and Eq. (3.55), we can readily obtain the one-dimensional diffu-
sion equation

2 N

∂ N

∂

D ∂

t =

or N t = DN xx . (3.56)

∂ x 2

Similarly, the two- and the three-dimensional diffusion equation is

∂ N

∂

t =

D Δ N ,

where

2 2 2 2 2

Δ = ∂

x 2 + ∂

Δ = ∂

x 2 + ∂ y 2 + ∂

or

z 2 . ∂ ∂ y 2 ∂ ∂ ∂