introduction to materials science, chapter 5, diffusion university of virginia, dept. of materials...

Introduction To Materials Science, Chapter 5, Diffusion

University of Virginia, Dept. of Materials Science and Engineering 1

Diffusion how atoms move in solids

Diffusion mechanisms Vacancy diffusion Interstitial diffusion Impurities

Mathematics of diffusion Steady-state diffusion (Fick’s first law) Nonsteady-State Diffusion (Fick’s second law)

Factors that influence diffusion Diffusing species Host solid Temperature Microstructure5.4 Nonsteady-State Diffusion – Not Covered / Not Tested

Chapter 5 Outline



Diffusion transport by atomic motion.

Inhomogeneous material can become homogeneous by diffusion. Temperature should be high enough to overcome energy barrier.

What is diffusion?



Concentration Gradient Interdiffusion (or Impurity Diffusion).

Self-diffusion: one-component material, atoms are of same type.

Inter-diffusion vs. Self-diffusion

Before After

(Heat)



Vacancy diffusion

To jump from lattice site to lattice site, atoms need energy to break bonds with neighbors, and to cause the necessary lattice distortions during jump.Therefore, there is an energy barrier.Energy comes from thermal energy of atomic vibrations (Eav ~ kT)

Atom flow is opposite to vacancy flow direction.

Diffusion Mechanisms (I)

Atom migration Vacancy migration

AfterBefore



Interstitial diffusion

Diffusion Mechanisms (II)

Interstitial atom before diffusion

Interstitial atom after diffusion

Generally faster than vacancy diffusion because bonding of interstitials to surrounding atoms is normally weaker and there are more interstitial sites than vacancy sites to jump to.Smaller energy barrier Only small impurity atoms (e.g. C, H, O) fit into interstitial sites.



Flux of diffusing atoms, J.

Number of atoms diffusing through unit area per unit time [atoms/(m2s)]

or

Mass of atoms diffusing through unit area per unit time [kg/(m2 s)]

Mass: J = M / (A t) (1/A) (dM/dt)

AJ



Diffusion flux does not change with time

Concentration profile: Concentration (kg/m3) vs. position

Concentration gradient: dC/dx (kg / m4)

Steady-State Diffusion

BA

BA

xx

CC

x

C

dx

dC



Fick’s first law: J proportion to dC/dx

Steady-State Diffusion

D=diffusion coefficientdC

J Ddx

Concentration gradient is ‘driving force’

Minus sign means diffusion is ‘downhill’: toward lower concentrations



Concentration changing with time

Fick’s second law

Nonsteady-State Diffusion(not tested)

Find C(x,t)

2

2

C J

t x

CD

x



Atom needs enough thermal energy to break bonds and squeeze through its neighbors.

Energy needed energy barrier Called the activation energy Em (like Q)

Diagram for Vacancy Diffusion

Diffusion Thermally Activated Process

AtomVacancyEm

Distance

Ene

rgy



Room temperature (kBT = 0.026 eV) Typical activation energy Em (~ 1 eV/atom) (like Qv)

Therefore, a large fluctuation in energy is needed for a jump.

Probability of a fluctuation or frequency of jump, Rj

Diffusion – Thermally Activated Process

R0 = attempt frequency proportional to vibration frequency

Swedish chemist Arrhenius

0 exp mj

B

ER R k T



1. Probability of finding a vacancy in an adjacent lattice site (Chap. 4):

times2. Probability of thermal fluctuation

Calculate Activated Diffusion

The diffusion coefficient = Multiply

0 exp mj

B

ER R k T

.exp v

B

QP Const k T

.exp expm V

B B

E QD Const k T k T

0 exp d

B

QD D k T

Arrhenius dependence.



Diffusion coefficient is the measure of mobility of diffusing species.

Diffusion – Temperature Dependence

Arrhenius Plots (lnD) vs. (1/T) or (logD) vs. (1/T)

0 0exp expd dQ QD D DRT kT

dx

dCDJ

D0 – temperature-independent (m2/s)Qd – the activation energy (J/mol or eV/atom)R – the gas constant (8.31 J/mol-K) orkB - Boltzman constant ( 8.6210-5 eV/atom-K)T – absolute temperature (K)

0

1ln ln dQ

D DR T

0

1log log

2.3dQ

D DR T

or



Diffusion – Temperature Dependence (II)

Graph of log D vs. 1/T has slop of –Qd/2.3R,

intercept of ln Do

21

21d T1T1

DlogDlogR3.2Q

T

1

R3.2

QDlogDlog d

0



Diffusion – Temperature Dependence (III)

Arrhenius plot:Diffusivity for metallic systems



Diffusion of different species

Smaller atoms diffuse more readily

Diffusion faster in open lattices or in open directions



Diffusion: Role of the microstructure (I)

Self-diffusion coefficients for Ag

Depends on diffusion path

Grain boundaries and surfaces less restrictive



Diffusion: Role of the microstructure (II)

Plots are from computer simulations

Initial positions are shown by the circles, paths are shown by lines. See difference between mobility in the bulk and in the grain boundary.



Factors that Influence Diffusion

Temperature - diffusion rate increases very rapidly with increasing temperature

Diffusion mechanism - interstitial is usually faster than vacancy

Diffusing and host species - Do, Qd is different for every solute, solvent pair

Microstructure - diffusion faster in polycrystalline vs. single crystal materials because of the rapid diffusion along grain boundaries and dislocation cores.



Summary

Activation energy Concentration gradient Diffusion Diffusion coefficient Diffusion flux Driving force Fick’s first and second

laws Interdiffusion Interstitial diffusion Self-diffusion Steady-state diffusion Vacancy diffusion

Make sure you understand

introduction to materials science, chapter 5, diffusion university of virginia, dept. of materials...

Documents