This document discusses diffusion in solids. It defines diffusion as the process of molecules moving from one area of a solid to another due to concentration differences. There are three main types of diffusion: self-diffusion, inter-diffusion/impurity diffusion, and surface diffusion. For an atom to diffuse, it must have an empty adjacent site available and enough energy to move into that site. Diffusion occurs via vacancies or interstitial spaces in the crystal lattice. Fick's laws provide a way to quantify the rate of diffusion based on flux, concentration gradients, and diffusion coefficients.

1. Diffusions in
Solids

2. Group 2
By: Belamide, Rannelle O.
Dela Cruz, Mary Adela L.

3. SOLID RECORDS
What is
Diffusion in
Solids?
• The process of molecules moving
from one area of a solid to
another due to differences in
concentration.
• It is a type of transport
phenomenon that is driven by a
concentration gradient and is
affected by temperature and
pressure.
• It is used to explain the
behavior of materials under
certain conditions, such as the
movement of heat, mass, and
electric charge.

4. SOLID RECORDS
Types of
Diffusions
• Three main types of diffusion in solids
• Self-diffusion - occurs for pure metals, but
all atoms exchanging positions are of the
same type
• Inter-diffusion/ Impurity Diffusion - The
process by which atoms of one metal diffuse
into another.
• Surface diffusion - the movement of atoms
or molecules along the surface of a solid due
to differences in concentration. It is driven
by an energy gradient and is affected by
temperature and pressure.

6. SOLID RECORDS
Diffusion
Mechanism
• For an atom to move, it must meet two
conditions:
• an empty adjacent site must be
available,
• the atom must have enough energy to
disrupt its neighboring atoms and
cause lattice distortion during
displacement.

7. SOLID RECORDS
Vacancy Diffusion- the process by which
atoms or molecules move within a material through
vacancies or empty spaces in its crystal lattice.
• occurs when atoms or molecules leave or enter
these vacancies through thermal motion or heat
energy.
• Interstitial Diffusion-refers to the
process wherein atoms or molecules move through
the spaces between the atoms or molecules of a
crystalline material
• occurs when atoms or molecules move into these
interstitial spaces.

9. Fick’s Law

10. How do we quantify the rate of
Diffusion?
• J = Flux =
𝑚𝑜𝑙𝑒𝑠 𝑚𝑎𝑠𝑠 𝑑𝑖𝑓𝑓𝑢𝑠𝑖𝑛𝑔
(𝑎𝑟𝑒𝑎)(𝑡𝑖𝑚𝑒(𝑡𝑖𝑚𝑒)
=
𝑚𝑜𝑙
𝑐𝑚2𝑠
𝑜𝑟
𝑘𝑔
𝑚2𝑠
• Measured empirically
• Make thin film (membrane) of known cross-sectional area
• Impose concentration gradien
• Measure how fast atoms or molecules diffuse through the membrane

11. Fick’s First Law / Steady State Diffusion
• Diffusion condition for which the flux is independent of time
• Flux proportional to concentration gradient =
𝑑𝑐
𝑑𝑥
• Fick’s First Law of Diffusion
J= -D
𝒅𝒄
𝒅𝒙
D = diffusion coefficient

12. Sample Problem

13. A glass beaker contains a solution of salt in water. The
concentration of salt in the beaker is 0.02 g/cm^3. The
beaker is left uncovered, and over time, salt diffuses into
the air above the solution. After 5 hours, the
concentration of salt in the air above the solution is
measured to be 0.001 g/cm^3. The diffusion coefficient
of salt in air is 0.1 cm^2/hour. Calculate the rate of mass
transfer (flux) of salt from the solution into the air using
Fick's First Law.

14. Answer
C1 = 0.02 g/cm^3 C2 = 0.001 g/cm^3
t = 5 hours
𝑑𝐶
𝑑𝑥
=
0.001𝑔/𝑐𝑚^3− 0.02𝑔/𝑐𝑚^3
5 ℎ𝑟
𝑑𝐶
𝑑𝑥
= −0.0038
𝑔
𝑐𝑚3∗ ℎ𝑟
J = −𝐷 ∗
𝑑𝐶
𝑑𝑥
J = -0.1
𝑐𝑚^2
ℎ𝑟
∗ (−0.0038
𝑔
𝑐𝑚3∗ ℎ𝑟
)
J = 0.00038
𝑔
𝑐𝑚2∗ ℎ𝑟

15. Fick’s Law

16. Fick’s Second
Law
• The concentration of diffusing species is a function of both time and position C = C(x,t)
• In this case Fick’s Second Law is used
• Fick’s Second Law of Diffusion
∂𝑪
∂𝒕
=
∂𝟐
𝑪
∂𝒙𝟐
• This is a partial differential equation and it has been assumed here that the diffusivity
is independent of composition

17. To Solve a PDE, we need boundary and initial
conditions
• Boundary conditions
At t=0, C=𝑪𝒐 for 0 ≤ x ≤ ∞ (initial conditions)
At t › 0, C= 𝑪𝒔 for x = 0 (constant surface conc.)
C = 𝑪𝒐 for x = ∞ (con. At infinity)

18. With increasing time
concentration increases
further into the Al bar
Semi-infinite solid diffusion: Cu diffusing into
aluminum bar
Pre-existing
conc., 𝐂𝐨 𝐨𝐟 𝐜𝐨𝐩𝐩𝐞𝐫

19. Solution to PDE for semi-infinite solid diffusion

20. As a result, the right side is constant and
also the term inside the error function
For Specific concentration, left side of equation
is constant
For a constant D and C, increasing depth
will require a longer time

21. Effective penetration depth, 𝑥𝑒𝑓𝑓 is where C is in
the middle

22. erf(): Gaussian error function and is normally
tabulated
Note that there are 3 columns of z-values. Make sure to match the correct value of z with the
appropriate value of erf(z)

23. Sample Problem

24. A thin sheet of paper with a thickness of 0.2 cm is
initially coated with a dye on one side. The
concentration of the dye on the coated side is 1.5
g/cm^3, and the concentration on the uncoated side is
0.2 g/cm^3. The diffusion coefficient of the dye in paper
is 0.01 cm^2/hour. After 3 hours, the concentration of
the dye at a distance of 0.1 cm from the coated side is
measured to be 0.8 g/cm^3. Calculate the rate of
change of concentration with respect to time (dc/dt) at
that location using Fick's Second Law.

25. Answer
C = 0.8 𝑔/𝑐𝑚^3 D = 0.01 cm^2/hour
C2 = 0.2 g/cm^3
𝑑𝑐
𝑑𝑥
= 6 g/(cm^3 * cm)
x = 0.1 cm
𝑑𝑐
𝑑𝑡
= 0.01 cm^2/hour * (6 g/(cm^3 * cm))
𝑑𝑐
𝑑𝑥
= (0.8 - 0.2) / 0.1
𝑑𝑐
𝑑𝑡
= 0.06 g/(cm^3 * hour)
𝑑𝑐
𝑑𝑥
= 6 g/(cm^3 * cm)

26. Ready For the Quiz!!

27. THANK YOU