Diffusion
Diffusion can be defined as the mass flow process in
which atoms change their positions relative to neighbors in a
given phase under the influence of thermal agitation and a
gradient. The gradient can be a compositional gradient, an
electric or magnetic gradient, or stress gradient.
Steady-State Diffusion: Fick’s first law
Fick’s first law: The rate of diffusion is characterized by
describing atomic fluxes at particular locations in the material
J = atomic flux (atoms/m2-s, kg/ m2-s)(the number of atoms
passing through a plane of unit area per unit time)
(dc/dx) = concentration gradient(atoms/m4)
D = diffusion coefficient (m2/s)
Concentration Gradient:The concentration gradient shows the
composition of the material varies with distance;
is the
difference in concentration over the distance
. The flux is
initially high when the concentration gradient is high and
gradually decreases as the gradient is reduced.
Diffusion – Temperature Dependence:
D0 – temperature-independent preexponential (m2/s)
Qd – the activation energy for diffusion (J/mol R – the
gas constant (8.31 J/mol-K)
T – absolute temperature (K)
The above equation can be rewritten as
If the temperature of a material increases the diffusion
coefficient and the flux of atoms increase as well. At
higher temperatures the thermal energy that supplied to
the diffusing atoms permits the atoms to overcome the
activation energy barrier and more easily move to a new
lattice sites.
Nonsteady-State Diffusion: Fick’s second law
most real situations the concentration profile and the
concentration gradient are changing with time. The
changes of the concentration profile is given in this case
by a differential equation, Fick’s second law.
The solution to this differential equation with the
given boundary condition is:
CS = constant concentration of the diffusing atoms at the
surface of the material.
CO = initial uniform concentration of the diffusing atoms
in the material.
CX = the concentration of the diffusing atom at location
X below the surface after time t.
erf: is error function.
The solution of second Fick's law permits us to calculate
the concentration of one diffusing species near the
surface of the material as a function of time and distance
provided that the diffusion coefficient D remains
constant and the concentrations of the diffusing atom at
the surface CS and within the material Co remain
unchanged.