MS 482. Kinetic Processes in Materials

MS 482. Lecture Notes 11-12

Byungha Shin Dept. of MSE, KAIST

1

MS 482. Kinetic Processes in Materials 2

Syllabus 1. Atomistic diffusion

1.1 Basic rate theory (3 classes)

1.2 Atomistic diffusion mechanisms (1 class)

2. Macroscopic diffusion

2.1. Thermodynamics of irreversible processes (2 classes)

2.2. Diffusion under a concentration gradient (2 classes)

2.3. Other driving forces for diffusion (1.5 classes)

2.4. Solving diffusion equations (1.5 classes)

3. Kinetics of interfaces

3.1. Thermodynamics of interfaces (3 classes)

3.2. Capillary-induced morphology evolution

3.2.1. Surface evolution (2 classes)

3.2.2. Coarsening (2 classes)

3.3. Interface motion (2 classes)

4. Phase transformation

4.1. Phenomenological theory (1 class)

4.2. Continuous phase transformation

4.2.1. Spinodal decomposition (2 classes)

4.2.2. Order-disorder transformation (2 classes)

4.3. Nucleation and growth (3 classes)

MS 482. Kinetic Processes in Materials 3

Time-dependent diffusion

Instantaneous Localized Sources

Semi-infinite case?

MS 482. Kinetic Processes in Materials 4

Time-dependent diffusion

Method of Separation of Variables

𝜕𝑐

𝜕𝑡= 𝐷

𝜕2𝑐

𝜕𝑥2

1

𝐷𝑇

𝑑𝑇

𝑑𝑡=

1

𝑋

𝜕2𝑋

𝜕𝑥2=

BC’s: c(x,0)=c0, c(0,t)=0, c(L,t)=0

𝑐 𝑥, 𝑡 = 𝑋 𝑥 𝑇(𝑡)

𝑋 𝑥 =

Applying BCs,

𝜆𝑛 = 𝑛2𝜋2

𝐿2, 𝑋𝑛 𝑥 = 𝑎𝑛 sin 𝑛𝜋

𝑥

𝐿

MS 482. Kinetic Processes in Materials 5

Time-dependent diffusion

Method of Separation of Variables

𝜕𝑐

𝜕𝑡= 𝐷

𝜕2𝑐

𝜕𝑥2

1

𝐷𝑇

𝑑𝑇

𝑑𝑡=

1

𝑋

𝜕2𝑋

𝜕𝑥2= −𝜆

BC’s: c(x,0)=c0, c(0,t)=0, c(L,t)=0

𝑐 𝑥, 𝑡 = 𝑋 𝑥 𝑇(𝑡)

𝑇 𝑡 = 𝑇0𝑒−𝜆𝐷𝑡

𝑇𝑛 𝑡 = 𝑇𝑛0𝑒−𝑛2𝜋2𝐷𝑡/𝐿2

𝑐 𝑥, 𝑡 = 𝑋𝑛(𝑥)𝑇𝑛(𝑡)

∞

𝑛=1

= 𝐴𝑛 sin 𝑛𝜋𝑥

𝐿𝑒−𝑛2𝜋2𝐷𝑡/𝐿2

∞

𝑛=1

𝑐0 = 𝐴𝑛 sin 𝑛𝜋𝑥

𝐿

∞

𝑛=1

MS 482. Kinetic Processes in Materials 6

Fourier Series

𝑢 𝑥 =𝑏0

2+ 𝑎𝑛 sin 𝑛𝜋

𝑥

𝐿+ 𝑏𝑛 cos 𝑛𝜋

𝑥

𝐿

∞

𝑛=1

𝑎𝑛 =1

𝐿 𝑢(𝑥) sin 𝑛𝜋

𝑥

𝐿𝑑𝑥

𝐿

−𝐿

, 𝑏𝑛 =1

𝐿 𝑢(𝑥) cos 𝑛𝜋

𝑥

𝐿𝑑𝑥

𝐿

−𝐿

If u(x) is an odd function [u(x) = -u(-x)],

𝑢 𝑥 = 𝑎𝑛 sin 𝑛𝜋𝑥

𝐿,

∞

𝑛=1

𝑎𝑛 =2

𝐿 𝑢(𝑥) sin 𝑛𝜋

𝑥

𝐿𝑑𝑥

𝐿

0

,

If u(x) is a even function [u(x) = u(-x)],

𝑢 𝑥 =𝑏0

2+ 𝑏𝑛 cos 𝑛𝜋

𝑥

𝐿,

∞

𝑛=1

𝑏𝑛 =2

𝐿 𝑢(𝑥) cos 𝑛𝜋

𝑥

𝐿𝑑𝑥

𝐿

0

,

MS 482. Kinetic Processes in Materials 7

Time-dependent diffusion

Method of Separation of Variables

𝑐0 = 𝐴𝑛 sin 𝑛𝜋𝑥

𝐿

∞

𝑛=1

𝐴𝑛 =

𝑐 𝑥, 𝑡 =4𝑐0

𝜋

1

2𝑗 + 1sin 2𝑗 + 1 𝜋

𝑥

𝐿𝑒− 2𝑗+1 𝜋/𝐿 2𝐷𝑡

∞

𝑗=0

≈4𝑐0

𝜋sin 𝜋

𝑥

𝐿𝑒−𝜋2𝐷𝑡/𝐿2

𝑐 (𝑡) ≈8𝑐0

𝜋2𝑒−𝜋2𝐷𝑡/𝐿2

MS 482. Kinetic Processes in Materials 8

Laplace Transform

ℒ 𝑓(𝑥, 𝑡) = 𝑓 𝑥, 𝑝 = 𝑒−𝑝𝑡𝑓 𝑥, 𝑡 𝑑𝑡∞

0

ℒ𝜕𝑓

𝜕𝑡= 𝑒−𝑝𝑡

𝜕𝑓(𝑥, 𝑡)

𝜕𝑡𝑑𝑡

∞

0

𝑒−𝑝𝑡𝜕𝑓(𝑥, 𝑡)

𝜕𝑡𝑑𝑡

∞

0

=

ℒ𝜕𝑓

𝜕𝑡=

ℒ𝜕𝑛𝑓

𝜕𝑥𝑛=

𝜕𝑛𝑓 𝑥, 𝑝

𝜕𝑥𝑛

MS 482. Kinetic Processes in Materials 9

Time-dependent diffusion

Lapalace Transform: Example 1

Diffusion of constant surface concentration into a semi-infinite body

BCs: 𝑐 𝑥 = 0, 𝑡 = 𝑐0; 𝜕𝑐

𝜕𝑥𝑥 = ∞, 𝑡 = 0; 𝑐 𝑥, 𝑡 = 0 = 0 (for 0 ≤ x < ∞)

ℒ𝜕𝑐

𝜕𝑡= 𝐷ℒ

𝜕2𝑐

𝜕𝑥2

= 𝐷𝜕2𝑐

𝜕𝑥2

𝑐 𝑥, 𝑝 = 𝑎1𝑒 𝑝/𝐷𝑥 + 𝑎2𝑒− 𝑝/𝐷𝑥

BCs: 𝑐 𝑥 = 0, 𝑝 = 𝑐0 𝑒−𝑝𝑡𝑑𝑡∞

0

=𝑐0

𝑝;

𝜕𝑐

𝜕𝑥𝑥 = ∞, 𝑝 = 0

𝑐 𝑥, 𝑝 =𝑐0

𝑝𝑒− 𝑝/𝐷𝑥

MS 482. Kinetic Processes in Materials 10

Time-dependent diffusion

Lapalace Transform: Example 1

𝑐 𝑥, 𝑝 =𝑐0

𝑝𝑒− 𝑝/𝐷𝑥

MS 482. Kinetic Processes in Materials 11

Time-dependent diffusion

Lapalace Transform: Example 2

A constant flux J0 imposed on the surface of a semi-infinite sample

BCs:

𝜕𝑐

𝜕𝑥𝑥 = 0, 𝑡 = −

𝐽0𝐷

= constant

(for 0 ≤ x < ∞)

𝑐 𝑥 = ∞, 𝑡 = 𝑐0

𝑐 𝑥, 𝑡 = 0 = 𝑐0

ℒ𝜕𝑐

𝜕𝑡= 𝐷ℒ

𝜕2𝑐

𝜕𝑥2

= 𝐷𝜕2𝑐

𝜕𝑥2

𝑐 𝑥, 𝑝 =𝑐0

𝑝+𝑎1𝑒− 𝑝/𝐷𝑥 +𝑎2𝑒− 𝑝/𝐷𝑥

MS 482. Kinetic Processes in Materials 12

Time-dependent diffusion

Lapalace Transform: Example 2

BCs:

𝜕𝑐

𝜕𝑥𝑥 = 0, 𝑡 = −

𝐽0𝐷

= constant

(for 0 ≤ x < ∞)

𝑐 𝑥 = ∞, 𝑡 = 𝑐0

𝑐 𝑥, 𝑡 = 0 = 𝑐0

𝑐 𝑥, 𝑝 =𝑐0

𝑝+𝑎1𝑒− 𝑝/𝐷𝑥 +𝑎2𝑒− 𝑝/𝐷𝑥 =

𝑐0

𝑝+

𝐽0𝑝3/2𝐷1/2

𝑒− 𝑝/𝐷𝑥

𝜕𝑐 (𝑥 = 0, 𝑝)

𝜕𝑥= −

𝐽0𝑝𝐷

𝜕𝑐 (𝑥 = ∞, 𝑝) =𝑐0

𝑝

𝑐 𝑥, 𝑡 = 𝑐0 +𝐽0𝐷

4𝐷𝑡

𝜋𝑒−𝑥2/(4𝐷𝑡) − 𝑥 erfc

𝑥

4𝐷𝑡