ms516 kinetic processes in materials lecture note 2....

MS516 Kinetic Processes in MaterialsLecture Note

2. Macroscopic Diffusion

Byungha ShinDept. of MSE, KAIST

1

2016 Spring Semester

Course InformationSyllabus

1. Atomistic mechanisms of diffusion (3 classes)

2. Macroscopic diffusion2.1. Diffusion under chemical driving force (2 classes)2.2. Other driving forces for diffusion (2 classes)2.3. Solving diffusion equations (2 classes)

3. Diffusion (flow) in glassy states (2 classes)

4. Kinetics of surfaces and interfaces4.1. Thermodynamics of surfaces and interfaces (4 classes)4.2. Capillary-induced morphology evolution (2 classes)

4.2.1. Surface evolution4.2.2. Coarsening

5. Phase transformation5.1. Phenomenological theory (1 class)5.2. Continuous phase transformation (3 classes)

5.2.1. Spinodal decomposition5.2.2. Order-disorder transformation

5.3. Nucleation and growth (Solidification) (3 classes)

Course InformationUphill diffusion

Phase separation in oil & water mixture

Mo

le f

ract

ion

of

oil

Upon vigorous stirring of oil and water mixture

z (height direction)

zMo

le f

ract

ion

of

oil

Inconsistent with Fick’s law, 𝐽 = −𝐷𝛻𝐶 ?

Tracer diffusivity• Self-diffusion in a pure (i.e., chemically homogeneous) material

𝐽 = −𝐷𝛻𝐶𝐴∗

A

A*: radioactive isotope of A

CA* (conc. of A*)

𝐷 = 𝑓𝑎02𝑥𝑉

0𝜈 (in case of diffusion via vacancy mechanism)

Self-diffusivity (tracer diffusivity), D

= 𝑓𝑎02 exp

−Δ𝐺𝑓𝑜𝑟𝑚𝑎𝑡𝑖𝑜𝑛0

𝑘𝐵𝑇𝜈0 exp

−Δ𝐺𝑚𝑖𝑔𝑟𝑎𝑡𝑖𝑜𝑛

𝑘𝐵𝑇

• Diffusion of solute in a dilute (~chemically homogeneous) solution

𝐷𝐼 = 𝑓𝑎02𝑥𝐼𝑉

0 𝜈(in case of diffusion of impurities via vacancy mechanism)

≈ 𝑓𝑎02𝑥𝐼 𝜈

= 𝑓𝑎02𝑥𝐼 𝜈0 exp

−Δ𝐺𝑚𝑖𝑔𝑟𝑎𝑡𝑖𝑜𝑛

𝑘𝐵𝑇

• In both cases, diffusion occurs via random walk (except for f)

(if impurity-vacancy pair more stable than impurity alone; every impurity atom has a vacancy associated with it)Zn in Cu

Equation for flux

𝐽𝑖 = 𝐶𝑖𝑀𝑖𝐹′Ci : concentration of the species i [#/cm3]Mi : mobility ≡ vel. / force [cm/sec /force]𝐹′ : driving force acting on the species i= 𝐶𝑖𝑀𝑖[−𝛻 potential ]

= 𝐶𝑖𝑀𝑖[−𝛻𝜇𝑖] mi : chemical potential of the species i

𝜇𝑖 = 𝜇𝑖0(𝑇, 𝑃) + 𝑘𝐵𝑇 ln 𝑎𝑖

= 𝜇𝑖0 + 𝑘𝐵𝑇(ln𝑋𝑖 + ln 𝛾𝑖)

ai: activity (= gi Xi)Xi: mole fraction (= Ci / C)gi: activity coefficient

- ideal: non-interacting, gi = 1- dilute: gi =gi

0 (constant), when gi >1 and when gi <1?

Ideal or dilute solution case:

𝐽𝑖 = −𝐶𝑖𝑀𝑖𝑘𝐵𝑇𝛻 ln𝑋𝑖

= −𝐶𝑖𝑀𝑖𝑘𝐵𝑇𝛻𝑋𝑖

𝑋𝑖

= −𝑀𝑖𝑘𝐵𝑇𝛻𝐶𝑖

= −𝐷𝑖𝛻𝐶𝑖

𝐷𝑖 = 𝑀𝑖𝑘𝐵𝑇

Einstein relation𝐽𝑖 = −

𝐷𝑖𝐶𝑖

𝑘𝐵𝑇𝛻𝜇𝑖

[#/cm2∙sec]

(Di : tracer diffusivity)

General equation for flux

𝜇𝑖 = 𝜇𝑖0 + 𝑘𝐵𝑇(ln𝑋𝑖 + ln 𝛾𝑖)

𝐽𝑖 = −𝐷𝑖𝐶𝑖

𝑘𝐵𝑇𝛻𝜇𝑖 =

𝐷𝑖𝐶𝑖

𝑘𝐵𝑇𝐹′

= −𝐷𝑖𝛻𝐶𝑖 −𝐷𝑖𝐶𝑖

𝑘𝐵𝑇𝛻(𝑘𝐵𝑇 ln 𝛾𝑖)

chemical driving force (other than concentration gradient)

= −𝐷𝑖

𝜕𝐶𝑖

𝜕𝑥− 𝐷𝑖𝐶𝑖

𝜕 ln 𝛾𝑖

𝜕𝑥= −𝐷𝑖

𝜕𝐶𝑖

𝜕𝑥− 𝐷𝑖𝐶𝑖

𝜕 ln 𝛾𝑖

𝜕 ln 𝑋𝑖

𝜕 ln𝑋𝑖

𝜕𝑥

= −𝐷𝑖

𝜕𝐶𝑖

𝜕𝑥− 𝐷𝑖

𝜕 ln 𝛾𝑖

𝜕 ln𝑋𝑖

𝜕𝐶𝑖

𝜕𝑥= −𝐷𝑖 1 +

𝜕 ln 𝛾𝑖

𝜕 ln 𝑋𝑖

𝜕𝐶𝑖

𝜕𝑥= − 𝐷𝑖

𝜕𝐶𝑖

𝜕𝑥

𝐷𝑖 = 𝐷𝑖 1 +𝜕 ln 𝛾𝑖

𝜕 ln 𝑋𝑖

Intrinsic diffusivity, is related to tracer diffusivity, Di : 𝐷𝑖

𝜕 ln 𝛾𝑖

𝜕 ln 𝑋𝑖< -1. 𝐷𝑖 can be negative when

(the species i does not want to mix with the host atoms; but this is not a sufficient condition for a negative 𝐷𝑖 , only necessary)

Miscibility gap

Cu-Ni Phase Diagram 1 +𝜕 ln 𝛾𝑖

𝜕 ln 𝑋𝑖=

1

𝑘𝐵𝑇

1

𝑋𝑖(1 − 𝑋𝑖)

𝑑2𝐺

𝑑𝑋𝑖2

Immiscible when d2G/dXi2 < 0

Questions:• Plots of G vs XNi at 500K and 1000K ?• Enthalpy of mixing (DHmix)of Cu-Ni

positive or negative?• At 300K, why solubility of Ni in Cu

(or Cu in Ni) is not zero?• Why Cu-Ni completely miscible at a

high temperature when it is immiscible at T < 698K?

Kinetic driving force

DGm DGm21

DGm12

F: driving force other than concentration gradient = 1 2 Γ12 + Γ21 𝑛1 − 𝑛2

+ 1 2 𝑛1 + 𝑛2 Γ12 − Γ21

𝐽 = 𝑛1Γ12 − 𝑛2Γ21

= −𝐷𝑖 ( 𝜕𝐶𝑖 𝜕𝑥) + 𝐶𝑖𝜆 Γ12 − Γ21

For small driving force, F

Γ12 = Γ0 exp −∆𝐺𝑚12

𝑘𝐵𝑇= Γ0 exp −

∆𝐺𝑚 −12

𝜆𝐹

𝑘𝐵𝑇= Γ𝑖 exp

12

𝜆𝐹

𝑘𝐵𝑇≈ Γ𝑖 1 +

𝜆𝐹

2𝑘𝐵𝑇

Γ21 = Γ0 exp −∆𝐺𝑚21

𝑘𝐵𝑇= Γ0 exp −

∆𝐺𝑚 +12

𝜆𝐹

𝑘𝐵𝑇= Γ𝑖 exp

12

𝜆𝐹

𝑘𝐵𝑇≈ Γ𝑖 1 −

𝜆𝐹

2𝑘𝐵𝑇

= −𝐷𝑖 ( 𝜕𝐶𝑖 𝜕𝑥) + 𝐶𝑖Γ𝑖𝜆2

𝐹

𝑘𝐵𝑇

= −𝐷𝑖𝛻𝐶𝑖 +𝐷𝑖𝐶𝑖

𝑘𝐵𝑇𝐹

(compare this with the eq. in slide 6)

all driving forces including chemical one (except conc. gradient)

Course InformationHollow nanostructure based on Kirkendall effect

Geosele et al. “Monocrystalline spinel nanotube fabrication based on the Kirkendall effect” Nature Materials 5, 627 (2006).

Diffusion couple

“Marker”

A-rich B-rich

A-B alloy

Consider a single phase binary diffusion couple (with markers embedded at the boundary) where both components are mobile and utilize the vacancy diffusion mechanism, but 𝐷𝐴 > 𝐷𝐵

• Diffusion via vacancy mechanism: same flux of vacancies opposite direction to flux of each element

𝐽𝐴 + 𝐽𝐵 + 𝐽𝑉 = 0• ( ) of vacancies on the A-side

and ( ) of vacancies on the B-side

• Assuming equilibrium vacancy conc. is maintained throughout the specimen:A-side: ( ) of vacancies via dislocation ( ), grain boundaries, or surface ( ) in length on the left side of the markerB-side: ( ) of vacancies via dislocation ( ), grain boundaries, or surface ( ) in length on the right side of the marker

flux of A, 𝐽𝐴flux of B, 𝐽𝐵net flux of vacancies, 𝐽𝑉

Diffusion couple

XV0 = 6/5*12 = 0.1

jA

jBjV

6

1

A B

A-side: increase (+5) of NV

B-side: decrease (+5) of NV

“marker”

5

A-side: destruction of NV

shrink in lengthB-side: creation of NV

elongation in length

To maintain XV0 (and if

vacancy kinetics is fast enough),

Diffusion coupleA B“marker”

x’

Reference of frame fixed at the marker (the moving frame),

𝑗𝐴 = − 𝐷𝐴

𝜕𝐶𝐴

𝜕𝑥′

𝑗𝐵 = − 𝐷𝐵

𝜕𝐶𝐵

𝜕𝑥′

jA = jB + jV

xOriginal interface (Matano interface), across which the same number of A and B passed

Reference of frame fixed at the laboratory wall (the fixed frame):

JA = JB

3 3

“marker” moves to the left

Diffusion couple

“Marker”

A-rich B-rich

A-B alloy

A-rich B-rich

x

The laboratory frame of reference:

v

𝐽𝐴 = − 𝐷𝐴

𝜕𝐶𝐴

𝜕𝑥+ 𝑣𝐶𝐴

Flux due to diffusive motion of atoms relative to the moving atomic planes

Flux due to the motion of the atomic planes

velocity of the marker, 𝑣 ?

Matano plane

Velocity of the marker

𝐶(total # atoms/vol) = 𝐶𝐴 + 𝐶𝐵 = constant (for any x between 0 and L)

Assuming the molar volume is independent of composition:

𝜕𝐶

𝜕𝑡=

𝜕𝐶𝐴

𝜕𝑡+

𝜕𝐶𝐵

𝜕𝑡= 0

𝜕𝐶𝐴

𝜕𝑡= −

𝜕𝐽𝐴𝜕𝑥

;𝜕𝐶𝐵

𝜕𝑡= −

𝜕𝐽𝐵𝜕𝑥

𝜕𝐶

𝜕𝑡=

𝜕

𝜕𝑥 𝐷𝐴

𝜕𝐶𝐴

𝜕𝑥− 𝑣𝐶𝐴 + 𝐷𝐵

𝜕𝐶𝐵

𝜕𝑥− 𝑣𝐶𝐵 =

𝜕

𝜕𝑥 𝐷𝐴

𝜕𝐶𝐴

𝜕𝑥+ 𝐷𝐵

𝜕𝐶𝐵

𝜕𝑥− 𝑣𝐶 = 0

𝐷𝐴

𝜕𝐶𝐴

𝜕𝑥+ 𝐷𝐵

𝜕𝐶𝐵

𝜕𝑥− 𝑣𝐶 = constant (for any 𝑥 between 0 and 𝐿)

In a region far away from the interface, no diffusion occurs:

𝜕𝐶𝐴

𝜕𝑥=

𝜕𝐶𝐵

𝜕𝑥= 0 and 𝑣 = 0

𝑣 =1

𝐶 𝐷𝐴

𝜕𝐶𝐴

𝜕𝑥+ 𝐷𝐵

𝜕𝐶𝐵

𝜕𝑥= 𝐷𝐴 − 𝐷𝐵

1

𝐶

𝜕𝐶𝐴

𝜕𝑥= 𝐷𝐵 − 𝐷𝐴

1

𝐶

𝜕𝐶𝐵

𝜕𝑥

Darken’s 1st equation

Interdiffusivity

𝐽𝐴 = − 𝐷𝐴

𝜕𝐶𝐴

𝜕𝑥+ 𝑣𝐶𝐴 = − 𝐷𝐴

𝜕𝐶𝐴

𝜕𝑥+ 𝐷𝐴 − 𝐷𝐵

1

𝐶

𝜕𝐶𝐴

𝜕𝑥𝐶𝐴

=−𝐶𝐴

𝐷𝐴 − 𝐶𝐵 𝐷𝐴 + 𝐶𝐴

𝐷𝐴 − 𝐶𝐴 𝐷𝐵

𝐶

𝜕𝐶𝐴

𝜕𝑥= −

𝐶𝐵 𝐷𝐴 + 𝐶𝐴

𝐷𝐵

𝐶

𝜕𝐶𝐴

𝜕𝑥

𝐽𝐴 = − 𝐷𝜕𝐶𝐴

𝜕𝑥, where 𝐷 = 𝑋𝐴

𝐷𝐵 + 𝑋𝐵 𝐷𝐴

Flux of each component is now expressed by a single diffusivity, 𝐷(𝐶)and its concentration gradient.

Darken’s 2nd equation

• By knowing 𝐷 and v, 𝐷𝐴 and 𝐷𝐵 can be determined.

𝐽𝐵 = − 𝐷𝜕𝐶𝐵

𝜕𝑥, where 𝐷 = 𝑋𝐴

𝐷𝐵 + 𝑋𝐵 𝐷𝐴

Similarly,

Boltzmann-Matano Analysis

Continuity,

𝜕𝐶

𝜕𝑡=

𝜕

𝜕𝑥 𝐷

𝜕𝐶

𝜕𝑥= 𝐷 𝐶

𝜕2𝐶

𝜕𝑥2+

𝑑 𝐷(𝐶)

𝑑𝐶

𝜕𝐶

𝜕𝑥

2

𝜂 ≡𝑥

4𝑡;

Consider a diffusion couple with:

(IC’s) C (x, t=0) = C0 -∞ < x < 00 0 < x < ∞

(BC’s) C (-∞, t) = C0 ; C(-∞, t) = 0

𝜕𝐶

𝜕𝑡=

𝜕

𝜕𝑥 𝐷

𝜕𝐶

𝜕𝑥−2𝜂

𝑑𝐶

𝑑𝜂=

𝑑

𝑑𝜂 𝐷(𝐶)

𝑑𝐶

𝑑𝜂

(BC’s) C (h=-∞) = C0 ; C (h=∞) = 0

Boltzmann-Matano Analysis: determination of D(C) from measured C(x)

−2𝜂𝑑𝐶

𝑑𝜂=

𝑑

𝑑𝜂 𝐷(𝐶)

𝑑𝑐1

𝑑𝜂(BC’s) C (h=-∞) = C0 ; C (h=∞) = 0

integration

At time

This relates with C(x), which can be determined experimentally.

𝐷

0

𝐶0

𝑥𝑑𝐶 = 0 , which determines the original interface (x=0) or Matano interface


𝑡 = 𝜏, 𝑥 = 𝜂 4𝜏,

−2 𝜂=∞

𝜂=𝑢

𝜂𝑑𝐶

𝑑𝜂𝑑𝜂 = 𝐷(𝐶)

𝑑𝐶

𝑑𝜂𝜂=∞

𝜂=𝑢

−2 0

𝐶(𝑢)

𝜂𝑑𝐶 = 𝐷(𝐶)𝑑𝐶

𝑑𝜂0

𝐶(𝑢)

0

𝐶(𝑥)

𝑥(𝐶)𝑑𝐶 = −2𝜏 𝐷(𝐶) 𝑑𝐶

𝑑𝑥𝐶(𝑥)

Graphical interpretation of Matano Analysis


Matano interface,which sets x=0

𝐶0

𝐶

𝐶(𝑥)

0

𝑑𝐶

𝑑𝑥𝐶(𝑥)

𝑥

0

𝐶(𝑥)

𝑥(𝐶)𝑑𝐶

0

𝐶(𝑥)

𝑥(𝐶)𝑑𝐶 = −2𝜏 𝐷(𝐶) 𝑑𝐶

𝑑𝑥𝐶(𝑥)

𝑑𝐶

𝑑𝑥𝐶(𝑥)

𝐶0

𝐶𝐶(𝑥)

• Cu0.7Zn0.3 bar wound with thin W marker wire then electrodeposition of Cu• During annealing, the marker (staying at 22.5% Zn) moved inwards• Boltzmann-Matano analysis interdiffusivity 𝐷 measured• 𝐷 = 𝑋𝐶𝑢

𝐷𝑍𝑛 + 𝑋𝑍𝑛 𝐷𝐶𝑢 at 1058 K, 𝐷𝑍𝑛/ 𝐷𝐶𝑢(@ 22.5% Zn) = 2.3

• Indirect evidence for diffusion of Cu and Zn via vacancy mechanism

Kirkendall Experiment

Kirkendall voids:If kinetics of vacancy destruction is not fast enough to maintain the equilibrium vacancy concentration agglomeration of vacancies leading to a hollow structure

Kirkendall void

JSe

JCo

Co

CoSe CoSe

hollow

Jco > JSe

Yin, Rioux, Erdonmez, Huges, Somorjai, Alivisatos, “Formation of Hollow Nanocrystals Through the Nanoscale Kirkendall Effect”, Nature 304, p. 711 (2004)

Summary of diffusivity

Tracer Diffusivity, Di • Self-diffusivity in a chemically homogeneous material composed of only one species

• Diffusivity of component i in a chemically homogeneous (ideal or dilute) solution

Intrinsic Diffusivity, 𝐷𝑖 • Composition-dependent intrinsic diffusivityof component i in a non-ideal solution

Interdiffusivity, 𝐷 • Composition-dependent interdiffusivity in a chemically inhomogeneous system

• In a binary system, it relates the flux of either component A or B to its corresponding concentration gradient via Fick’s law in a reference frame fixed at a wall of the laboratory

𝐷𝑖 = 𝐷𝑖 1 +𝜕 ln 𝛾𝑖

𝜕 ln𝑋𝑖

𝐷 = 𝑋𝐴 𝐷𝐵 + 𝑋𝐵

𝐷𝐴

= 𝑋𝐴𝐷𝐵 1 +𝜕 ln 𝛾𝐵

𝜕 ln𝑋𝐵+

𝑋𝐵𝐷𝐴 1 +𝜕 ln 𝛾𝐴

𝜕 ln𝑋𝐴

= (𝑋𝐴𝐷𝐵 + 𝑋𝐴𝐷𝐵) 1 +𝜕 ln 𝛾𝐴

𝜕 ln𝑋𝐴

Course InformationDiffusional creep

Simple creep test apparatus

www.mutiaranata.com

• Creep: time-dependent deformation under a constant load (stress); distinguished from instantaneous elastic and plastic strain

• Dislocation creep and diffusional creep (stress is driving force for diffusion)• Creep occurs in nearly all materials (metals, ceramics, polymers, cements,…)

Course InformationKinetic driving forces for diffusion

Driving force F includes:• Activity coefficient gradient in a non-ideal solid solution (where

energy of mixing is non-zero), • Electric potential gradient (electric field)• Stress gradient• Pressure gradient• Temperature gradient• Electron wind effects from an electric current (electromigration)• And etc…..

−𝛻(𝑘𝐵𝑇 ln 𝛾𝑖)

due to entropy of mixing, 𝛻(𝑘𝐵𝑇 ln𝑋𝑖)

𝐽𝑖 = −𝐶𝑖𝐷𝑖

𝑘𝐵𝑇𝛻𝜇𝑖 = −𝐷𝑖𝛻𝐶𝑖 +

𝐶𝑖𝐷𝑖𝐹

𝑘𝐵𝑇

Course InformationDiffusion in an electric potential gradient

Flux of charged particles, 𝐽𝑖 = −𝐷𝑖𝛻𝐶𝑖 +𝐶𝑖𝐷𝑖𝐹

𝑘𝐵𝑇=

𝐶𝑖𝐷𝑖(−𝑞𝑖𝛻𝜙)

𝑘𝐵𝑇=

𝐶𝑖𝐷𝑖𝑞𝑖𝐸

𝑘𝐵𝑇

Flux of charge, 𝐽𝑞 = 𝑞𝑖𝐽𝑖 =𝐶𝑖𝐷𝑖𝑞𝑖

2𝐸

𝑘𝐵𝑇= 𝜎𝐸

Charged ions in ionic conductors (in the absence of concentration gradient)

Ohm’s law (V = IR)

𝜎 =𝐶𝑖𝐷𝑖𝑞𝑖

2

𝑘𝐵𝑇= 𝐶𝑖𝑞𝑖𝑀𝑖

𝑀𝑖 =𝑞𝑖𝐷𝑖

𝑘𝐵𝑇Einstein relation

Course InformationDiffusion in an electric potential gradientElectrons in p-n junction (both concentration gradient & electric potential gradient present)

EF

n-type side

p-type side

p-n junction under equilibrium

EV

EC

𝐽𝑛 = −𝐷𝑛

𝜕𝑛

𝜕𝑥+

𝑛𝐷𝑛

𝑘𝐵𝑇

[− −𝑞 𝜕𝜙]

𝜕𝑥

𝑛 ≈ 𝑁𝐶 exp −𝐸𝐶 − 𝐸𝐹

𝑘𝐵𝑇,

𝜕𝑛

𝜕𝑥≈ −

𝑛

𝑘𝐵𝑇

𝜕𝐸𝐶

𝜕𝑥

“diffusion current”

“drift current”

Increa-sing f

−𝑞𝐽𝑛 = −𝑞𝑛𝐷𝑛

𝑘𝐵𝑇

𝜕𝐸𝐶

𝜕𝑥+

𝑞𝑛𝐷𝑛

𝑘𝐵𝑇

𝜕𝐸𝐶

𝜕𝑥= 0 =

𝑞𝑛𝐷𝑛

𝑘𝐵𝑇

𝜕𝜇𝑛

𝜕𝑥=

𝑞𝑛𝐷𝑛

𝑘𝐵𝑇

𝜕𝐸𝐹

𝜕𝑥

mn: Electrochemical potential of an electron ≡ EF No current flow if EF is constant.

electron flux

electron current density

−𝑞𝐽𝑛 = 𝑞𝐷𝑛

𝜕𝑛

𝜕𝑥−

𝑞2𝑛𝐷𝑛

𝑘𝐵𝑇

𝜕𝜙

𝜕𝑥

𝑞𝜕𝜙

𝜕𝑥= −

𝜕𝐸𝐶

𝜕𝑥

f in V and EC in eV (qV)Increasing EC decreasing f

Course InformationDiffusion in an electric potential gradientElectromigration

𝐽𝑖 = −𝐷𝑖𝛻𝐶𝑖 +𝐷𝑖𝐶𝑖

𝑘𝑇𝐹 = −𝐷𝑖𝛻𝐶𝑖 +

𝐷𝑖𝐶𝑖

𝑘𝑇𝛽𝐸

Proposed physical mechanism behind electromigration: Change in the electronic charge distribution surrounding an interstitial or a vacancy formation of a dipole (between the defect and current-carrying electrons) exertion of electrostatic force on the interstitial or vacancy

Flux of atoms,

Average driving force F = b E

Hu et al. “Microstructure, impurity and metal cap effects on Cu electromigration”, AIP Conference Proceedings 1061, 67 (2014)

Course InformationDiffusion in a pressure gradient

𝐽𝑖 = −𝐶𝑖𝐷𝑖

𝑘𝐵𝑇𝛻𝜇𝑖 = −𝐷𝑖𝛻𝐶𝑖 −

𝐶𝑖𝐷𝑖

𝑘𝐵𝑇𝛻(ΔΩ𝑃)

Formation of solute-atom atmosphere around dislocations

𝜇𝑖 = 𝑘𝐵𝑇 ln 𝑋𝑖 + 𝑃ΔΩ

Increase in chemical potential (PV work)

• Dilation caused by an interstitial solute atom, DW

• Additional pressure experienced by the interstitial, P (hydrostatic pressure)

• Chemical potential of the interstitial

• Interstitial atoms “wish” to migrate to a place of lower pressure such as around an edge dislocation

𝐽𝑖

= −𝐷𝑖𝛻𝐶𝑖 −𝐶𝑖𝐷𝑖ΔΩ

𝑘𝐵𝑇𝛻𝑃

Course InformationDiffusional creep (diffusion in a stress gradient)

Nabaro-Herring creep (bulk diffusion)

before

afterd

- s

- s

d

one quadrant

Chemical potential at the side wallm = m0 + s W

ss

Chemical potential at the vertical boundariesm = m0 – s W 𝐽 = −

𝐶𝐷𝑏𝑢𝑙𝑘

𝑘𝐵𝑇𝛻𝜇 = −

𝐷𝑏𝑢𝑙𝑘

𝑘𝐵𝑇

1

Ω𝛻𝜇

(W: atomic volume)

𝛻𝜇 ≈ −2𝜎Ω

2𝑑2

= −2 2𝜎Ω

𝑑

=2 2

𝑑

𝜎𝐷𝑏𝑢𝑙𝑘

𝑘𝐵𝑇

# of atoms moved per time

= # of quadrants flux area

= 4 𝐽 𝑑 ∙ 𝑑/ 2

Volume moved per time = 4 𝐽 𝑑 ∙ 𝑑/ 2 Ω

Extension per time = volume moved per time / d2

Strain rate, = Dd per time / d 휀

(Dd)

=4𝐽Ω

2𝑑=

8𝐷𝑏𝑢𝑙𝑘Ω

𝑘𝐵𝑇𝑑2 𝜎

Course InformationDiffusional creep (diffusion in a stress gradient)

Coble creep (surface/grain boundary diffusion)

Chemical potential at the side wallm = m0 + s W

s

Chemical potential at the vertical boundariesm = m0 – s W

𝐽 = −𝐷𝑠𝑢𝑟𝑓𝑎𝑐𝑒

𝑘𝐵𝑇

1

Ω𝛻𝜇 =

2

𝑑

𝜎𝐷𝑠𝑢𝑟𝑓𝑎𝑐𝑒

𝑘𝐵𝑇

𝛻𝜇 ≈−2𝜎Ω

𝑑= −

2𝜎Ω

𝑑

# of atoms moved per time

= # of quadrants flux area

= 4 𝐽 𝑑 ∙ 𝛿

Volume moved per time = 4 𝐽 𝑑 ∙ 𝛿 Ω

Extension per time = volume moved per time / d2

Strain rate, = Dd per time / d 휀

(Dd)

=4𝐽𝛿Ω

𝑑2 =8𝐷𝑠𝑢𝑟𝑓𝑎𝑐𝑒𝛿Ω

𝑘𝐵𝑇𝑑2 𝜎

- s

- s

s

d

Course InformationSurface transport

Activation energy:surface diffusion < bulk diffusion

0

Course InformationSurface transport

Surface diffusion dominates at low temperatures (or small grains), while bulk diffusion dominates at high temperatures (or large grains).

high T Low T

Course InformationDeformation mapPure Ni (d = 1 mm) Pure Ni (d = 0.1 mm)

Dislocation creep

Dislocation creep

Coble N-HCoble N-H

Transition between Coble and N-H occurs at a higher temperature in Ni with smaller grain structure

Course InformationReciprocal relations in a reversible process

In many reversible processes, there is a liner response matric that is symmetric:

𝑃 = 𝜒𝐸Dielectric polarization

Electric field

Susceptibility

𝑃𝑥 = 𝜒11𝐸𝑥 + 𝜒12𝐸𝑦 + 𝜒13𝐸𝑧

𝑃𝑦 = 𝜒21𝐸𝑥 + 𝜒22𝐸𝑦 + 𝜒23𝐸𝑧

𝑃𝑧 = 𝜒31𝐸𝑥 + 𝜒32𝐸𝑦 + 𝜒33𝐸𝑧

𝑃𝑖 = 𝜒𝑖𝑗𝐸𝑗

𝑑𝐹 = 𝐏 ∙ 𝑑𝐄 − 𝑆𝑑𝑇;

𝜒𝑖𝑗 =𝜕𝑃𝑖

𝜕𝐸𝑗 𝑇

=𝜕2𝐹

𝜕𝐸𝑗𝜕𝐸𝑖 𝑇

=𝜕2𝐹

𝜕𝐸𝑖𝜕𝐸𝑗 𝑇

= 𝜒𝑗𝑖

𝑃𝑖 =𝜕𝐹

𝜕𝐸𝑖 𝑇

;Helmholtz free energy

Course InformationThermodynamics of irreversible processesIrreversible processes not covered by equilibrium Thermodynamics. There are often phenomenological relationships for such processes:

𝐽𝑒 = 𝜎𝐸 = −𝜎𝜕𝜑

𝜕𝑥Ohm’s Law

Flux (or current density) = constant * driving force = constant * (- gradient of potential)

𝐽𝑄 = −𝐾𝜕𝑇

𝜕𝑥Fourier’s Law 𝐽𝑛 = −𝐷

𝜕𝑐

𝜕𝑥Fick’s “Law”

𝐽𝑒,𝑥

𝐽𝑒,𝑦

𝐽𝑒,𝑧

𝐽𝑄,𝑥

𝐽𝑄,𝑦

𝐽𝑄,𝑧

𝐽𝑛,𝑥

𝐽𝑛,𝑦

𝐽𝑛,𝑧

⋮

=

⋯

𝐿𝑖𝑗

⋯

𝐸𝑥

𝐸𝑦

𝐸𝑧

𝜕𝑇/𝜕𝑥𝜕𝑇/𝜕𝑦𝜕𝑇/𝜕𝑧𝜕𝑐/𝜕𝑥𝜕𝑐/𝜕𝑦𝜕𝑐/𝜕𝑧

⋮

“Phenomenological Equations”

or “Thermodynamic

Equations of Motion”

𝐽𝑖 = 𝐿𝑖𝑗𝐹𝑖

Lne: electromigrationLeQ (or LQe): thermoelectric

Course InformationOnsager reciprocal relations

Lij = Lji in irreversible processes?

𝐽𝑖 = 𝐿𝑖𝑗𝐹𝑖

Onsager’s theorem says, “yes”, for certain specific choices of the Ji and Fi

The Nobel Prize in Chemistry 1968 was awarded to Lars Onsager "for the discovery of the reciprocal relations bearing his name, which are fundamental for the thermodynamics of irreversible processes".

Course InformationOnsager reciprocal relations

Lij = Lji in irreversible processes?𝐽𝑖 = 𝐿𝑖𝑗𝐹𝑖

A BReservoirs connected by a homogeneous conducting wireT, j T + DT,

j + Dj

electric current I = L’11 Dj + L’12 DTthermal current W = L’21 Dj + L’22 DT

L’12 = L’21 because fluxes (I, W) and “forces” (Dj, DT) not chosen in the special way required by the Onsager Theorem

𝐼 = 𝐿11 −Δ𝑇

𝑞

𝜕

𝜕𝑇

𝜇

𝑇−

Δ𝜑

𝑇+ 𝐿12 −

Δ𝑇

𝑇2

𝑊 = 𝐿21 −Δ𝑇

𝑞

𝜕

𝜕𝑇

𝜇

𝑇−

Δ𝜑

𝑇+ 𝐿22 −

Δ𝑇

𝑇2

𝐿12 = 𝐿21

Course InformationDiffusion equation: constant diffusivity

Fick’s 2nd “Law”Continuity

If there is no creation or destruction of particles, 𝑛=0

𝜕𝑐

𝜕𝑡= 𝑛 − 𝛻 ∙ 𝐽 = 𝛻 ∙ 𝐷𝛻𝑐 = 𝐷𝛻2𝑐

For constant diffusivity

𝜂 ≡𝑥

4𝐷𝑡;

𝜕𝑐

𝜕𝑡= 𝐷

𝜕2𝑐

𝜕𝑥2 −2𝜂𝑑𝑐

𝑑𝜂=

𝑑2𝑐

𝑑𝜂2

𝑐 𝜂 − 𝑐 𝜂 = 𝜂0 = 𝑎1 𝜂0

𝜂

𝑒−𝜉2𝑑𝜉

applying IC:

𝑐 𝜂 = 𝑐𝐿 + 𝑎2

2

𝜋 −∞

0

𝑒−𝜉2𝑑𝜉 +

2

𝜋 0

𝜂


𝑑𝑐

𝑑𝜂= 𝑒−2𝜂2

𝑐 −∞ = 𝑐𝐿

Course InformationDiffusion equation: constant diffusivity

𝑐 𝜂 = 𝑐𝑥

4𝐷𝑡= 𝑐𝐿 + 𝑎2

2

𝜋 −∞

0

𝑒−𝜉2𝑑𝜉 +

2

𝜋 0

𝜂


applying BC’s

𝑐 𝑥, 𝑡 =𝑐𝑅 + 𝑐𝐿

2+

𝑐𝑅 − 𝑐𝐿

2

2

𝜋 0

𝑥/ 4𝐷𝑡


=𝑐𝑅 + 𝑐𝐿

2+

𝑐𝑅 − 𝑐𝐿

2erf

𝑥

4𝐷𝑡

−∞

∞

𝑒−𝜉2𝑑𝜉 = 𝜋

−∞

0

𝑒−𝜉2𝑑𝜉 =

𝜋

2

𝑐 −∞ = 𝑐𝐿

𝑐 ∞ = 𝑐𝑅

erf(−𝑥) = − erf(𝑥)

erf(∞) = 1

erf(−∞) = − erf ∞ = −1

erf 0 = 0

Course InformationPoint source (Thin film solution)

Diffusion equation is a linear second-order differential equation the superposition of two solutions also solves the diffusion equation with superposed initial and boundary conditions.

𝑐 𝑥, 𝑡 =𝑐0

2+

𝑐0

2

2

𝜋 0

𝑥/ 4𝐷𝑡

𝑒−𝜉2𝑑𝜉 + +

2

𝜋 0


=𝑐0

𝜋 (𝑥−∆𝑥)/ 4𝐷𝑡

𝑥/ 4𝐷𝑡

𝑒−𝜉2𝑑𝜉 =

𝑐0∆𝑥

4𝜋𝐷𝑡𝑒−𝑥2/(4𝐷𝑡) =

𝑛𝑑

4𝜋𝐷𝑡𝑒−𝑥2/(4𝐷𝑡)

, where source strength, 𝑛𝑑 = −∞

∞

𝑐(𝑥)𝑑𝑥

for a very small Dx

• This applies to point source in 1D, line source in 2D, plane source (thin film) in 3D• Solution for 2D, 3D point source, ? 𝑐 𝑟, 𝑡

Course InformationSteady-state solution to diffusion eq’n

One dimension

𝑐 𝑥 = 𝑐0 − 𝑐0 − 𝑐𝐿𝑥

𝐿

𝐽 = −𝐷𝑐0 − 𝑐𝐿

𝐿

Stead-state, 𝛻2𝑐 = 0

𝑑2𝑐

𝑑𝑥2= 0

Cylindrical shell

Boundary conditions

𝑑

𝑑𝑟𝑟

𝑑𝑐

𝑑𝑟= 0

,

Course InformationSteady-state solution to diffusion eq’n

𝑐 𝑟 = 𝑎1 ln 𝑟 + 𝑎2 = 𝑐in −𝑐in − 𝑐out

ln(𝑟out /𝑟in)ln

𝑟

𝑟in

𝐽 𝑟 = −𝐷𝑑𝑐

𝑑𝑟= 𝐷

𝑐in − 𝑐out

ln(𝑟out /𝑟in)

1

𝑟

Cylindrical shell

r1r2

J(r1)J(r2)

Total current across the wall at r1 = 2pr1d J(r1)Total current across the wall at r2 = 2pr2d J(r2) 2prd J(r) = const. J(r) ∝ 1/r c(r) ∝ ln r

Spherical shell

Laplace operator in spherical coordination:

𝑑

𝑑𝑟𝑟2

𝑑𝑐

𝑑𝑟= 0

Total current across the wall at r1 = 4pr12 J(r1)

Total current across the wall at r2 = 4pr22 J(r2)

4pr2 J(r) = const. J(r) ∝ 1/r2 c(r) ∝ 1/r

(BC’s)

Course InformationTime-dependent solution to diffusion eq’nInstantaneous Localized Sources

Time-dependent solution to the case of a semi-infinite source can be obtained by a infinite series of localized sources separated by an infinitesimally small distance, dx

Method of Separation of Variables (best used for describing a system at long times)

𝜕𝑐

𝜕𝑡= 𝐷

𝜕2𝑐

𝜕𝑥2

1

𝐷𝑇

𝑑𝑇

𝑑𝑡=

1

𝑋

𝜕2𝑋

𝜕𝑥2= −𝜆

IC: c(x,0) = c0 (0 < x < L); BC’s: c(0,t) = 0, c(L,t) = 0

𝑐 𝑥, 𝑡 = 𝑋 𝑥 𝑇(𝑡)

𝑋 𝑥 =

Applying BC’s,

𝜆 > 0 : 𝜆𝑛 = 𝑛2𝜋2

𝐿2, 𝑋𝑛 𝑥 = 𝑎𝑛 sin 𝑛𝜋

𝑥

𝐿

Time-dependent solution to diffusion eq’n

(𝜆 = 0)

(𝜆 < 0)

(𝜆 > 0)

𝐴′′𝑥 + 𝐵′′

𝐴′𝑒 −𝜆𝑥 + 𝐵′𝑒− −𝜆𝑥

𝐴 sin( 𝜆𝑥) + 𝐵 cos( 𝜆𝑥)

1D diffusion problem

0 L

c0

0

c

x

Course Information

Method of Separation of Variables

𝜕𝑐

𝜕𝑡= 𝐷

𝜕2𝑐

𝜕𝑥2

1

𝐷𝑇

𝑑𝑇

𝑑𝑡=

1

𝑋

𝜕2𝑋

𝜕𝑥2= −𝜆

𝑐 𝑥, 𝑡 = 𝑋 𝑥 𝑇(𝑡)


𝑇 𝑡 = 𝑇0𝑒−𝜆𝐷𝑡

𝑇𝑛 𝑡 = 𝑇𝑛0𝑒−𝑛2𝜋2𝐷𝑡/𝐿2

𝑐 𝑥, 𝑡 =

𝑛=1

∞

𝑋𝑛(𝑥)𝑇𝑛(𝑡) =

𝑛=1

∞

𝐴𝑛 sin 𝑛𝜋𝑥

𝐿𝑒−𝑛2𝜋2𝐷𝑡/𝐿2

𝐴𝑛?

𝜆𝑛 = 𝑛2𝜋2

𝐿2 (from the previous slide)

From the boundary condition, c(x,0) = c0: 𝑐0 =

𝑛=1

∞


𝐿

IC: c(x,0) = c0 (0 < x < L); BC’s: c(0,t) = 0, c(L,t) = 0

0 L

c0

0

c

x

Course InformationFourier series

Individual Fourier components (first twelve components)

Sum of the Fourier components shown above

Course InformationFourier series

𝑢 𝑥 =𝑏0

2+

𝑛=1

∞

𝑎𝑛 sin 𝑛𝜋𝑥

𝐿+ 𝑏𝑛 cos 𝑛𝜋

𝑥

𝐿

𝑎𝑛 =1

𝐿 −𝐿

𝐿

𝑢(𝑥) sin 𝑛𝜋𝑥

𝐿𝑑𝑥 , 𝑏𝑛 =

1

𝐿 −𝐿

𝐿

𝑢(𝑥) cos 𝑛𝜋𝑥

𝐿𝑑𝑥

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

𝑢 𝑥 =

𝑛=1

∞

𝑎𝑛 sin 𝑛𝜋𝑥

𝐿, 𝑎𝑛 =

2

𝐿 0

𝐿

𝑢(𝑥) sin 𝑛𝜋𝑥

𝐿𝑑𝑥 ,

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

𝑢 𝑥 =𝑏0

2+

𝑛=1

∞

𝑏𝑛 cos 𝑛𝜋𝑥

𝐿, 𝑏𝑛 =

2

𝐿 0

𝐿

𝑢(𝑥) cos 𝑛𝜋𝑥

𝐿𝑑𝑥 ,

For u(x) that exists –L < x < L,

Course InformationTime-dependent solution to diffusion eq’n

Method of Separation of Variables

𝑐0 =

𝑛=1

∞


𝐿

𝐴𝑛 =2𝑐0

𝐿 0

𝐿

sin 𝑛𝜋𝑥

𝐿𝑑𝑥 = −

2𝑐0

𝑛𝜋cos 𝑛𝜋 − 1 =

𝑐 𝑥, 𝑡 =4𝑐0

𝜋

𝑗=0

∞1

2𝑗 + 1sin 2𝑗 + 1 𝜋

𝑥

𝐿𝑒− 2𝑗+1 𝜋/𝐿 2𝐷𝑡

≈4𝑐0

𝜋sin 𝜋

𝑥

𝐿𝑒−𝜋2𝐷𝑡/𝐿2

𝑐 𝑡 =1

𝐿 0

𝐿

𝑐 𝑥, 𝑡 𝑑𝑥 ≈8𝑐0

𝜋2𝑒−𝜋2𝐷𝑡/𝐿2

(only considering the first order term b/c higher order terms, i.e. shorter wavelengths, decay faster)

c0 is an odd function

4𝑐0

𝑛𝜋

0

(n odd)

(n even)

(average composition)

Course InformationTime-dependent solution to diffusion eq’n

Decay time of a Fourier component proportional to

𝑒− 𝑛𝜋/𝐿 2𝐷𝑡, where n is odd (shorter wavelengths faster decay due to shorter diffusion distance)

n=1n=3

Time is in units of L2/D

Laplace Transform

ℒ 𝑓(𝑥, 𝑡) = 𝑓 𝑥, 𝑝 = 0

∞

𝑒−𝑝𝑡𝑓 𝑥, 𝑡 𝑑𝑡

ℒ𝜕𝑓

𝜕𝑡=

0

∞

𝑒−𝑝𝑡𝜕𝑓(𝑥, 𝑡)

𝜕𝑡𝑑𝑡

0

∞


𝜕𝑡𝑑𝑡 = 𝑓 𝑥, 𝑡 𝑒−𝑝𝑡

0

∞+ 𝑝

0

∞

𝑓(𝑥, 𝑡)𝑒−𝑝𝑡𝑑𝑡

ℒ𝜕𝑓

𝜕𝑡= 𝑝ℒ 𝑓 − 𝑓(𝑥, 𝑡 = 0)

ℒ𝜕𝑛𝑓

𝜕𝑥𝑛=

𝜕𝑛 𝑓 𝑥, 𝑝

𝜕𝑥𝑛

(integration by parts:

𝑓′𝑔 = 𝑓𝑔 − 𝑓𝑔′)

0

∞


𝜕𝑥𝑑𝑡 =

𝜕 0

∞𝑒−𝑝𝑡𝑓(𝑥, 𝑡)𝑑𝑡

𝜕𝑥

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

𝑝 𝑐 𝑥, 𝑝 − 𝑐(𝑥, 𝑡 = 0) = 𝐷𝜕2 𝑐

𝜕𝑥2

𝑐 𝑥, 𝑝 = 𝑎1𝑒𝑝/𝐷𝑥 + 𝑎2𝑒

− 𝑝/𝐷𝑥

BCs: 𝑐 𝑥 = 0, 𝑝 = 𝑐0 0

∞

𝑒−𝑝𝑡𝑑𝑡 =𝑐0

𝑝;

𝜕 𝑐

𝜕𝑥𝑥 = ∞, 𝑝 = 0

𝑐 𝑥, 𝑝 =𝑐0

𝑝𝑒− 𝑝/𝐷𝑥


𝜕2 𝑐

𝜕𝑥2=

𝑝

𝐷 𝑐

c0

c

x

increasing time



𝑝𝑒− 𝑝/𝐷𝑥


𝑐 𝑥, 𝑝 𝑐 𝑥, 𝑡

𝑐 𝑥, 𝑡 =

0

2𝑐0

𝑐0

𝑥 = 0

𝑐 𝑥, 𝑡 = 𝑐0 − 𝑐0 erf𝑥

4𝐷𝑡

= 𝑐0 erfc𝑥

4𝐷𝑡


A constant flux J0 imposed on the surface of a semi-infinite sample(for example, doping of Si wafers by flowing dopant-containing gas)

BC’s:

𝜕𝑐

𝜕𝑥𝑥 = 0, 𝑡 = −

𝐽0𝐷

= constant

(for 0 ≤ x < ∞)

𝑐 𝑥 = ∞, 𝑡 = 𝑐0

𝑐 𝑥, 𝑡 = 0 = 𝑐0

ℒ𝜕𝑐

𝜕𝑡= 𝐷ℒ

𝜕2𝑐

𝜕𝑥2

𝑝 𝑐 − 𝑐(𝑥, 𝑡 = 0) = 𝐷𝜕2 𝑐

𝜕𝑥2


𝑝+𝑎1 𝑒− 𝑝/𝐷𝑥 +𝑎2 𝑒− 𝑝/𝐷𝑥


(𝑐0)


BC’s:

𝜕𝑐

𝜕𝑥𝑥 = 0, 𝑡 = −

𝐽0𝐷

= constant

(for 0 ≤ x < ∞)

𝑐 𝑥 = ∞, 𝑡 = 𝑐0

𝑐 𝑥, 𝑡 = 0 = 𝑐0


𝑝+𝑎1 𝑒− 𝑝/𝐷𝑥 +𝑎2 𝑒− 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


Laplace transform

ms516 kinetic processes in materials lecture note 2....

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