Chapter 4
The Smoluchowski Model
4.1 Diffusion
If a given sample consists of particles or molecules in random motion, then one
would expect a net flux from a region of higher concentration to one of lower
concentration. The question has been approached from the empirical side leading
to a relation known as Fick’s first law. Which states that the flow per unit area J
[#/(nm 2 .s)], across a surface yz plane is given by 1
J = −D
∂C( r, t)
∂x
(4.1)
where r = ˆ
ix + ˆ
jy + ˆ
kz, the quantity
∂C( r,t)
∂x
is the concentration gradient in the x
direction and the flow is in fact a vector quantity. Thus Fick’s law when extended to
three dimensions becomes
J = −D
ˆ
i
∂C( r, t)
∂x
+ ˆ
j
∂C( r, t)
∂y
+ ˆ
k
∂C( r, t)
∂z
(4.2)
where ˆ
i, ˆ
j , and ˆ
k are unit vector along the x, y, and z axes. The constant D is known
as the diffusion coefficient. The minus sign expresses the fact that flow is from a
higher to lower concentration. From now on we consider flow in one dimension, so
we omit the vectorial notation. We will use Eq. 4.3 another form of the diffusion
equation more useful for solving certain problems. Consider a slab of thickness
dx perpendicular to the x axis (see Fig. 4.1). Consider an area A of this slab. The
diffusion flow into the slab is given from Eq. 4.3 as −DA
∂C
∂x . The diffusion flow
across the corresponding face on the other side of the slab is given by
1 Where # means the number of particles.
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
J. A. Fornés, Principles of Brownian and Molecular Motors, Springer Series in
Biophysics 21, https://doi.org/10.1007/978-3-030-64957-9_4
49
The Smoluchowski Model
4.1 Diffusion
If a given sample consists of particles or molecules in random motion, then one
would expect a net flux from a region of higher concentration to one of lower
concentration. The question has been approached from the empirical side leading
to a relation known as Fick’s first law. Which states that the flow per unit area J
[#/(nm 2 .s)], across a surface yz plane is given by 1
J = −D
∂C( r, t)
∂x
(4.1)
where r = ˆ
ix + ˆ
jy + ˆ
kz, the quantity
∂C( r,t)
∂x
is the concentration gradient in the x
direction and the flow is in fact a vector quantity. Thus Fick’s law when extended to
three dimensions becomes
J = −D
ˆ
i
∂C( r, t)
∂x
+ ˆ
j
∂C( r, t)
∂y
+ ˆ
k
∂C( r, t)
∂z
(4.2)
where ˆ
i, ˆ
j , and ˆ
k are unit vector along the x, y, and z axes. The constant D is known
as the diffusion coefficient. The minus sign expresses the fact that flow is from a
higher to lower concentration. From now on we consider flow in one dimension, so
we omit the vectorial notation. We will use Eq. 4.3 another form of the diffusion
equation more useful for solving certain problems. Consider a slab of thickness
dx perpendicular to the x axis (see Fig. 4.1). Consider an area A of this slab. The
diffusion flow into the slab is given from Eq. 4.3 as −DA
∂C
∂x . The diffusion flow
across the corresponding face on the other side of the slab is given by
1 Where # means the number of particles.
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
J. A. Fornés, Principles of Brownian and Molecular Motors, Springer Series in
Biophysics 21, https://doi.org/10.1007/978-3-030-64957-9_4
49