Kinetics and Diffusion Basic concepts in...

MSE 3050, Phase Diagrams and Kinetics, Leonid Zhigilei

Kinetics and Diffusion

Basic concepts in kineticsKinetics of phase transformations Activation free energy barrier Arrhenius rate equation

Diffusion in Solids - Phenomenological description

Flux, steady-state diffusion, Fick’s first law

Nonsteady-state diffusion, Fick’s second law

Atomic mechanisms of diffusionHow do atoms move through solids?

● Substitutional diffusion

● Interstitial diffusion

● High diffusivity paths, diffusion along grain

boundaries, free surfaces, dislocations

Factors that influence diffusion

● Diffusing species and host solid (size, bonding)

● Temperature

● Microstructure


Kinetics: basic concepts

Thermodynamics can be used to predict what is theequilibrium state for a system and to calculate the drivingforce (ΔG) for a transformation from a metastable state toa stable equilibrium state.

How fast the transformation occurs is the questionaddressed by kinetics.

Let’s consider transition from a metastable to theequilibrium state. The transformation between the initialand final states involves rearrangement of atoms – thesystem should go through a transformation (or reaction)path. Since the initial and final states are metastable orstable ones, the energy of the system increases along anytransformation path between them


Kinetics: basic concepts

G1 and G2 are the Gibbs free energies of the initial and finalstates of the system

G = G2 - G1 is the driving force for the transformation.

Ga is the activation free energy barrier for the transition - themaximum energy along the transformation path relative to theenergy of the initial state.

Activated stateFinal state

(equilibrium or another metastable)

Transition pathG

G1G

2G

aG

Initial state (metastable)


In order for a system to proceed through the transition path to theequilibrium state, it has to obtain the energy that is sufficient toovercome the activation barrier.

The energy can be obtained from thermal fluctuation (when thethermal energy is “pooled together” in a small volume).Statistical mechanics can be used to predict the probability that asystem gets an energy that exceeds the activation energy. Thisprocess is called thermal activation.

The probability of such thermal fluctuation or the rate at which atransformation occurs, depends exponentially on temperature andcan be described by equation that is attributed to Swedishchemist Svante Arrhenius*:

* Arrhenius equation was first formulated by J. J. Hood on the basis of his studiesof the variation of rate constants of some reactions with temperature. Arrheniusdemonstrated that it can be applied to any thermally activated process.

Ga = Ha -TSa

Arrhenius equation can be applied to a wide range of thermallyactivated processes, including diffusion that we consider next.

The concept of thermal activation

Tk

HTk

Hk

S~TkGrate~

B

a

B

a

B

a

B

a Δexp~ΔexpΔexpΔexp


What is diffusion?

Most kinetic processes in materials involve diffusion.Inhomogeneous materials can become homogeneous bydiffusion, compositions of phases can change by diffusion, etc.

For an active diffusion to occur, the temperature should be highenough to overcome energy barriers to atomic motion.

“Diffusion” is transport through “random walk” - atoms,molecules, electrons, phonons, etc. are moving around randomlyin a crystal. This random motion can lead to mass, heat, orcharge transport. We will consider atomic diffusion that isinvolved in most phase transformations.


The flux of diffusing atoms, J, is used to quantify how fastdiffusion occurs. The flux is defined as either number of atomsdiffusing through unit area and per unit time (e.g., atoms/m2-second) or in terms of the mass flux - mass of atoms diffusingthrough unit area per unit time, (e.g., kg/m2-second).

Phenomenological description of diffusion: diffusion flux

AJ

Let’s consider steady state diffusion - the diffusion flux does notchange with time. Example: diffusion of gas molecules througha thin metal plate.

gas at pressure P1

gas at pressure P1 < P2

concentration profile

C2

C1


Fick’s first law: the diffusion flux along direction x isproportional to the concentration gradient

Steady-State Diffusion: Fick’s first law

where D is the diffusion coefficientdx

dCDJ

The minus sign in the equation means that diffusion is down theconcentration gradient.

Fick’s first law describes steady state flux in a uniformconcentration gradient.

Concentration gradient: dC/dx (Kg m-4) is the slope at aparticular point on concentration profile.

A J

C

x

dx

dC


dxxx JJdx

t

tx,C

In most practical cases steady-state conditions are notestablished, i.e. concentration gradient is not uniform and varieswith both distance and time. Let’s derive the equation thatdescribes nonsteady-state diffusion along the direction x.

Consider an element of material with dimensions dx, dy, and dz

x

tx,CDJ x

dxx

JJJ x

xdxx

From the balance of incoming and outgoing particles:

x x+dx

dVJx Jx+dxdV =dx dy dz

dAx = dy dz

Nonsteady-State Diffusion: Fick’s second law (I)

The number of particles that diffuse into the volume dV duringtime dt is JxdAxdt from the left and -Jx+dxdAxdt from the right.

dVtx,dCdtdAJ-J xdxxx

x

tx,CD

t

tx,C

x

and using expressions for Jx and Jx+dx we have


Fick’s second law relates the rate of change of composition withtime to the curvature of the concentration profile:

Nonsteady-State Diffusion: Fick’s second law (II)

[m-3t-1] = [m2t-1][m-3m-2] 2

2

x

tx,CD

t

tx,C

x

tx,CD

t

tx,C

x

If dependence of D on x (and C !) is ignored,

Concentration increases with time in those parts of the systemwhere concentration profile has a positive curvature. Anddecreases where curvature is negative.

C

x

C

x

C

x


Nonsteady-State Diffusion: Fick’s second law (III)

C

x

Can we apply this equation tosteady state diffusion?

2

2

x

tx,CD

t

tx,C


Microscopic picture of diffusion (I)

N

iiii

))(r(t)r(N

(t)rsMSD1

222 01

Δ

At the microscopic (atomic) level, diffusion is defined by randommovement (“random walk”) of the diffusing species (atoms,molecules, Brownian particles). The mobility of the diffusingspecies can be described by their mean square displacement:

(t)ri(0)ri

How to relate this microscopic characteristic (the mean square distance of atomic migration) to the macroscopic transport coefficient D used in the phenomenological Fick’s laws?

s

Let’s first consider an intuitive (not rigorous) “derivation” of therelationship between s and D.

Suppose that in time t, the average particle moved a distance sx along the direction in which diffusion is occurring.

X0

CRCL

JL

JR

sx sx

Assuming that the movement of particles is random, half of theparticles moved to the left, half to the right. The total flux ofparticles from left to right is JLt.

If CL is the average concentration of diffusing particles in the leftzone, than the total flux per unit area is JLt = (sxCL)/2 - half ofthe particles will cross the plane X0 from left to right.

The same for the flux from right to left, JRt = (sxCR)/2


Microscopic picture of diffusion (II)

Since JLt = (sxCL)/2 and JRt = (sxCR)/2, the net flow across X0

is J = JL – JR = sx(CL – CR)/2t

We can express (CL–CR) in terms of concentration gradient dc/dx:

(CL – CR)/sx = -(CR – CL)/sx = - dc/dxTherefore,

J = sx(CL – CR)/2t = -sx2/2t dc/dx

From the Fick’s law we

also have J = -D dc/dx

Thus, D = sx2/2t or sx

2 = 2tD for 1D diffusion

For 3D diffusion s2 = sx2 + sy

2 +sz2 = 3sx

2, and D = s2/6t

X0

CRCL

JL

JR

sx sx

In general, D = s2/2dt

This expression is called Einstein relation since it was first derived byAlbert Einstein in his Ph.D. thesis in 1905. It relates macroscopictransport coefficient D with microscopic information on the mean squaredistance of molecular migration.

where d is the dimensionality of the system


Atomic mechanisms of diffusion

Two main mechanisms of atomic diffusion in crystals:

Atoms located at the crystal lattice sites, usually diffuse by avacancy mechanism.

Interstitial atoms diffuse by jumping from one interstitial site toanother interstitial site without permanently displacing any of thematrix/solvent atoms:

interstitial mechanism

The phenomenological description in terms of 1st and 2nd Fick’slaws is valid for any atomic mechanism of diffusion.Understanding of the atomic mechanisms is important, however,for predicting the dependence of the atomic mobility (and,therefore, diffusion coefficient) on the type of interatomicbonding, temperature, and microstructure.

In both cases the moving atom must pass through a state of highenergy – this creates energy barrier for atomic motion.

Substitutional impurities

Substitutional self-diffusion – can be studied by depositing ofa small amount of radioactive isotope of the element (tracerdiffusion)


Diffusion Mechanisms: Vacancy diffusion (I)

To jump from lattice site to lattice site, atoms need energy tobreak bonds with neighbors, and to cause the necessary latticedistortions during jump. This energy necessary for the jump,ΔGm

v, is called the activation free energy for vacancy motion. Itcomes from the thermal energy of atomic vibrations (thermalenergy of an atom in a solid <Uatom> ≈ 3kT).

AtomVacancy

Distance

G

vmGΔ

The average thermal energy of an atom (3kBT ≈ 0.08 eV at roomtemperature) is usually much smaller that the activation freeenergy ΔGm

v (~ 1 eV/vacancy) and a large thermal fluctuation isneeded for a jump.


Diffusion Mechanisms: Vacancy diffusion (II)

For a simple one-dimensional case, the probability of suchfluctuation or frequency of jumps, Rj, can be described by theArrhenius equation:

where ν0 is an attempt frequency related to the frequency ofatomic vibrations. The value of ν0 is of the order of the meanvibrational frequency of an atom about its equilibrium site(usually taken to be equal to the Debye frequency).

Tk

GRB

vm

jΔexp0

frequency of atom vibrationsin the diffusion direction ν0

probability that a givenoscillation will move theatom to an adjacent site

jR

To relate this to the diffusion of atoms we have to consider thejump frequency of a given atom in a 3D crystal.

Moreover, for an atom to jump, there must be a vacancy next to it


Diffusion Mechanisms: Vacancy diffusion (III)

The probability for any atom in a solid to move is the product of

The rate at which atom jumps from place to place in the crystal istherefore

Tk

Gzz

N

n

B

vfeq exp

Tk

GTk

Gz

B

vm

B

vf expexp

τ

1R 0

j

atomj

the probability of finding a vacancy inan adjacent lattice site (fraction of atomsthat have a vacancy as a neighbor):

the rate of jumps of a vacancy (definedby a probability of a thermal fluctuationneeded to overcome the energy barrierfor vacancy motion):

where j is the average time between jumps for atoms.

Tk

GRB

vm

j exp0

If the distance atoms cover in each jump is a, the Einstein

relation can be used to estimate the diffusion

coefficient from the average time between jumps:

Tk

GG

za

τ

aD

B

vm

vf

j

exp66

022

where D0 is a parameter of material (both matrix and diffusingspecies) and is independent of temperature, Ed is activationenergy for diffusion:

Dt(t)rΔii 62

vm

vfd hhE

Tk

ED

Tk

hh

k

ss

za

B

d

B

vm

vf

B

vm

vf expexpexp

6 00

2


Diffusion Mechanisms: Vacancy diffusion (IV)

Let’s perform an order of magnitude estimate of the average timebetween jumps and the diffusion coefficient for self-diffusion inaluminum

eV 0.72vf h eV 0.68v

m h -1130 s10

21z m103a 10

j 6 1011s(less than one jump in 20000 years)

D 310-32 m2/s

17 orders of magnitude difference!

Tk

hh

k

ss

zτ

B

vm

vf

B

vm

vf

j expexp1

0

1~exp

B

vm

vf

k

ss- e.g., Shewmon, Diffusion in solids, Ch. 2.4-2.7

Tk

hh

k

ss

zaD

B

vm

vf

B

vm

vf expexp

60

2

Tk

ED

Tk

hh

k

ss

za

τ

aD

B

d

B

vm

vf

B

vm

vf

j

expexpexp66 0

022

Al:

j 4 10-7s(2.5 million jumps per second)

T = 0ºC T = 650ºC

D 410-14 m2/s


Substitutional self-diffusion

For a given crystal structure and bonding type Ed/RTm is roughlyconstant → D(T/Tm) const

Tm

K

D0

10-6 m2/s

Ed

kJ/mol

Ed

eVEd/RTm D(Tm)

10-12 m2/s

Al 933 170 142 1.47 18.3 1.9

Cu 1356 31 200.3 2.08 17.8 0.59

Ni 1726 190 279.7 2.90 19.5 0.65

-Fe 1805 49 284.1 2.94 18.9 0.29

Cr 2130 20 308.6 3.20 17.4 0.54

V 2163 28.8 309.2 3.20 17.2 0.97

Nb 2741 1240 439.6 4.56 19.3 5.2

K 337 31 40.8 0.42 14.6 15

Na 371 24.2 43.8 0.45 14.2 16

Li 454 23 55.3 0.57 14.7 9.9

Ge 1211 440 324.5 3.36 32.3 4.410-5

Si 1683 900000 496.0 5.14 35.5 3.610-4

from Porter and Easterling textbook

fcc metals

bcc transition metals

bcc alkali metals

diamondcubic semicond.

Not surprisingly, we see a correlation between Tm and Ed → stronger interatomic bonds make it more difficult to melt material and increase both v

mvf hh and


Diffusion of interstitial atoms

Interstitial diffusion also involve transition through the energybarrier and can be discussed in a manner similar to the vacancydiffusion mechanism.

The difference is that there are always sites for an interstitialatom to jump to.

Small interstitial atoms of a foreign (extrinsic) type, e.g., C in Feor O in Si may diffuse directly through the lattice (i.e., withoutthe help of vacancies) and play an important role in definingproperties of materials.

Tk

G

pa

τ

aD

B

im

j

exp66

022

where p is number of neighbor interstitial sites andimd hE

Tk

ED

Tk

h

k

s

pa

B

d

B

im

B

im expexpexp

6 00

2


Diffusion of interstitial atoms

Impurity D0, mm2/s-1 Ed, kJ/mol

C in FCC Fe 23.4 148

C in BCC Fe 2 84.1

N in FCC Fe 91 168.6

N in BCC Fe 0.3 76.1

H in FCC Fe 0.63 43

H in BCC Fe 0.1 13.4

D0, mm2/s-1 Ed, kJ/mol

Fe in γ-Fe 49 284

Fe in α-Fe 276 250.6

Fe in -Fe 201 240.7

Fe in Cr 47 332

Au in Ag 85 202.1

Si in Si 146000 484.4

from Porter and Easterling textbook & Smithells Metals Reference Book

vacancymechanism →

← interstitial impurities

• Smaller atoms cause less distortion of the lattice duringmigration and diffuse more readily than big ones (the atomicdiameters decrease from C to N to H).

• Diffusion is faster in more open lattices

Diffusion of interstitials is typically faster as compared to thevacancy diffusion mechanism (self-diffusion or diffusion ofsubstitutional atoms).

Tk

EDD

B

dexp0


Diffusion of self-interstitialsIntrinsic interstitials, also called “self-interstitials” are interstitialsatoms of the same kind as the atoms of the crystal.

Self-interstitials in most materials introduce strong deformationsinto the lattice and have very high formation energy, Δhf

i 3Δhfv

for metals. The number of equilibrium interstitials can beestimated by an equation similar to the one derived for vacancies:

Tk

ΔhNn

B

fii

eq expWe can estimate that at roomtemperature in copper there is lessthan one interstitial per cm3, whereas

In Si, however, intrinsic interstitials play an important role indiffusion and formation of defect structures.

Non-equilibrium self-interstitials in most materials are verymobile, e.g., Δhm

i 0.5Δhmv for metals and they quickly diffuse

out of the bulk of the crystal after being formed.

Self-interstitials can move through formation of intermediatelow-energy configurations, e.g., dumbbells (two atoms share thespace of one), and crowdions.

just below the melting point there is one interstitial for every 1012

atoms – there are virtually no “equilibrium” interstitials in metalsand most other elemental crystals.


Diffusion of clusters of interstitials One-dimensional motion of an almost isolated ½[111] loop at 575 K. A loop continuously moves in a direction parallel to its Burgers vector

Arakawa et al., Science 318, 956, 2007

Schematic view of the observation of the 1D glide motion of a interstitial-type prismatic perfect dislocation loop by TEM.

175 ps 200 ps 400 ps 450 ps

vacancy interstitial <110>-dumbbell

cluster of 4 interstitials

<111>-crowdion

Point defects in atomistic simulations [Lin et al., Phys. Rev. B 77, 214108, 2008]


Diffusion – Temperature Dependence

The activation energy Ed and pre-exponential D0, therefore, canbe estimated by plotting lnD vs. 1/T or logD vs. 1/T. Such plotsare called Arrhenius plots.

D0 – T-independent pre-exponential (m2/s)Ed – activation energy for diffusion (J/mol)R – the gas constant (8.31 J/mol-K)T – absolute temperature (K)

TR

EDD d 1

lnln 0

TR.

EDD d 1

32loglog 0

or

The above equation can be rewritten as

RT

EDD dexp0

baxy

0logDb

R.

Ea d

32

/Tx 1

Graph of log D vs 1/Thas slop of –Ed/2.3R, intercept of ln Do

2121 11loglog32 TTDDR.Ed


Diffusion of interstitials is typically faster as compared to thevacancy diffusion mechanism (self-diffusion or diffusion ofsubstitutional atoms).

TR.

EDD d 1

32loglog 0

Diffusion of interstitial and substitutional impurities


The plots are from the computer simulation by T. Kwok, P. S.Ho, and S. Yip. Initial atomic positions are shown by the circles,trajectories of atoms are shown by lines. We can see thedifference between atomic mobility in the bulk crystal and in thegrain boundary region.

Mean-square displacement of all

atoms in the system (B), atoms in

the grain boundary region (C), and

bulk region of the system (A).

More open atomic structure at defects (grain boundaries,dislocations) can result in a significantly higher atomic mobilityalong the defects.

Fast diffusion paths (I)

Dt(t)rΔMSDii 62


Fast diffusion paths (II)

Diffusion coefficient along a defect (e.g. grain boundary) can bealso described by an Arrhenius equation, with the activationenergy for grain boundary diffusion significantly lower than theone for the bulk.

Self-diffusion coefficients forAg. The diffusivity if greaterthrough less restrictivestructural regions – grainboundaries, dislocation cores,external surfaces.

Tk

EDD

B

defddefdef exp0

However, the effective cross-sectional area of the defects is onlya small fraction of the total area of the bulk (e.g., an effectivethickness of a grain boundary is ~0.5 nm). The diffusion alongdefects is less sensitive to the temperature change → becomesimportant at low T.


Arrhenius plots for 59Fe diffusivities in nanocrystalline Fe and other alloys compared to the crystalline Fe (ferrite).

[Wurschum et al. Adv. Eng. Mat. 5, 365, 2003]

image by Zhibin Lin et al.J. Phys. Chem. C 114, 5686, 2010

Fast diffusion paths: Diffusion in nanocrystalline materials


High-resolution electron micrograph (left, [Acta Mater. 56, 5857,2008]) and computed atomic structure (right, [Acta Mater. 45,987, 1997]) of nanocrystalline Si.

Wurschum et al., Defect Diffus. Forum 143-147, 1463, 1997]

volume fraction of grain boundary regions: ~50%.

diffusivity is enhanced by ~30 orders of magnitude.

Fast diffusion paths: Diffusion in nanocrystalline materials


Factors that Influence Diffusion:

Temperature - diffusion rate increases very rapidly withincreasing temperature (Arrhenius dependence)

Diffusion mechanism - interstitial is usually faster thanvacancy

Diffusing and host species - Do, Ed is different for everysolute - solvent pair

Microstructure – low-temperature diffusion is faster inpolycrystalline vs. single crystal materials because of theaccelerated diffusion along grain boundaries and dislocationcores.

Summary on the Diffusion Mechanisms

Make sure you understand language and concepts:

Mobility of atoms and diffusion Activation energy High diffusivity path Arrhenius equation Interstitial diffusion Self-diffusion Vacancy diffusion

Connection between the microscopic picture of diffusion (meansquare displacement of atoms) and the diffusion coefficient

Date post:	03-May-2018
Category:	Documents
Upload:	phamnhan
View:	253 times
Download:	4 times

Kinetics and Diffusion Basic concepts in...

Documents