thermodynamics of diffusion - university of...

MSE 3050, Phase Diagrams and Kinetics, Leonid Zhigilei

Thermodynamics of diffusion

(extracurricular material - not tested)

Driving force for diffusion

Diffusion in ideal and real solutions

Thermodynamic factor

Diffusion against the concentration gradient

Spinodal decomposition vs. nucleation and growth


In general, it is common for atoms to diffuse from regionsof high concentration towards the regions of lowconcentration. Thus, the phenomenological Fick’s lawsdescribe the diffusion in terms of the relationships betweenthe diffusion flux and concentration gradient.E.g., consider ideal solution:

Driving force for diffusion

Atoms here jumprandomly both rightand left

But there are notmany atoms here tojump to the left

As a result there is a net flux of atoms from left to right.

The thermodynamic properties of solid solutions, however,play an important role in diffusion and, under certainconditions, may even induce the diffusion against theconcentration gradient (D<0)!


The empirical Fick’s first law assumes proportionalitybetween the diffusion flux and the concentration gradient. Butthermodynamics tells us that any spontaneous process shouldgo in the direction of minimization of the free energy.

As we can see from the examples below, atoms can diffusefrom regions of high concentration towards the regions of lowconcentration – down the concentration gradient (left) as wellas from the regions of low concentration towards the regionsof high concentration – up the concentration gradient? (right)

Driving force for diffusion (I)

BX

G

10BX

G

10

A-rich B-rich

A-rich B-rich

1α 2α


Driving force for diffusion (II)

BX

G

10BX

G

10

B

Diffusion occur so that the free energy is minimized and istherefore driven by the gradient of free energy.

The chemical potential of atoms of type A can be definedas the free energy per mole of A atoms.

2121 αA

αA

αB

αB μμ and μμ

B1α 2α

BBAA XμXμG

x

μCMJ A

AAx

Therefore, the free energy gradient can be expressedthrough the chemical potential gradient:

In both cases the A and B atoms are diffusing from theregions where chemical potential is high to the regionswhere chemical potential is lower. The driving force fordiffusion is gradient of chemical potential.

Atoms migrate so as to remove differences in chemicalpotential. Diffusion ceases at equilibrium, when

where MA is the atomic mobilityof A atoms.


Chemical potential gradient is the driving force fordiffusion:

Driving force for diffusion (III)

where MB is mobility of B atomsx

μCMJ B

BBx

B

BBB

B

BBBB X

μXM

C

μCMD

x

CDJ B

Bx

- gradient of chemical potentialis in the same direction as theconcentration gradient.

0D then 0,X

μ if B

B

B

- diffusion occurs against theconcentration gradient!

0D then 0,X

μ if B

B

B

For example, we can identify regions with negative /XB ina system with miscibility gap:

BX

G

10

1α 2α

B

BX

0X

μ

B

B

We will discuss thebehavior of homogeneoussolution cooled within themiscibility gap later, afterderiving equations fordiffusion flux in ideal andregular solutions.


Lets consider diffusion driven by the chemical potentialgradient for ideal and regular solutions.

Driving force for diffusion (IV)

For an ideal solution:

The factor in brackets is termed the thermodynamic factor F.It defines how inter-atomic interaction affects the diffusion ofthe atoms in the presence of concentration gradient.

x

μCMJ B

BBx

BBB RTlnXGμ

x

C

C

RT

x

X

X

RT

x

X

X

μ

x

μ B

B

B

B

B

B

BB

x

CD

x

CRTMJ B

BB

BB

For a regular solution:

x

X

RT

XX12Ω1

X

RT

x

X

X

μ

x

μ BBB

B

B

B

BB

B2

BBB RTlnXX1ΩGμ

x

C

RT

XX 2Ω1

C

RT BBA

B

RTMD BB

RTFMD BB

F


As shown below, the thermodynamic factor is the same forboth species A and B at a given composition and is relatedto the curvature of the free energy curve.

Driving force for diffusion (V)

x

CF

C

RT

x

C

RT

XX 2Ω1

C

RT

x

μ B

B

BBA

B

B

BBBBBBBBAB lnXXX-1lnX-1RTXX-1GXGX-1

B

BB

B

BBBBA

B X

XlnX

X-1

X-1X-1ln-RTX 2ΩG-G

X

G

BBBBA lnXX-1ln-RTX 2ΩG-G

BBAABABBAAreg lnXXlnXXRTXΩXGXGXG

B

BBAB X-1

XRTln2X1G-G

BABB2

B

2

XX

RT 2Ω

X

1

X-1

1RT 2Ω

X

G

2B

2BABA

X

G

RT

XX

RT

XX 2Ω1F


The presence of a strain energy gradient, an electric field,or a temperature gradient can also affect the diffusion and,in particular, can induce diffusion of atoms against theconcentration gradient.

For example, in the presence of a strain energy gradient theequation for the chemical potential will include an elasticstrain energy term E(x). For a regular solution we have

Driving force for diffusion (VI)

x

E

x

CF

C

RT

x

μ B

B

B

ERTlnXX1ΩGμ B2

BBB

x

E

RTF

CD

x

CDJ BBB

BB

x

ECM

x

CRTFM

x

μCMJ BB

BB

BBBB

RTFMD BB


When the free energy curvature is negative, thethermodynamic factor F is negative, and the diffusion isdirected against the concentration gradient:

Diffusion against the concentration gradient:Spinodal Decomposition

x

CDJ B

BB

0RTFMD BB

0X

G2

B

2

0F

BX

G

101α 2α

0X

G2B

2

BX

T

10

1T

Chemical spinodal

Miscibility gap

21 αα 2α

1α


Homogeneous solution cooled into the miscibility gap willdecompose into 1 and 2 so that the total free energy of thesystem decreases.

The mechanism of decomposition into 1 and 2 is differentwithin the chemical spinodal region and outside (in thenucleation regions). Let’s consider small fluctuations aroundthe average composition XB

0:

Spinodal Decomposition

BX

G

BX

G

-BX

BX 0BX -

BX BX 0

BX

0X

G2B

2

0X

G2B

2

Free energy decreases as aresult of an arbitraryinfinitesimal fluctuation incomposition – the system isunstable

Free energy increases as aresult of an infinitesimalfluctuation in composition– the system is stable withrespect to small fluctuations

-BX

BX

0BX

coordinate spatial

2

B αX

1

B αX

0 B

B

B XXX

(fluctuations are small)


Although the system within the miscibility gap but outside thespinodal region is stable (metastable) with respect to smallfluctuations, it is unstable to the separation into 1 and 2

determined by the common tangent construction. There islarge difference in composition between 1 and 2 and largecomposition fluctuations are required in order to decrease thefree energy. A process of formation of a large compositionfluctuation is called nucleation. The phase separation isoccurring in this case by nucleation and growth (will bediscussed later).

Spinodal Decomposition

Nucleation and growthSpinodal decomposition

αB

1X

αB

2X

0BX

coordinate spatial

αB

1X

αB

2X

0BX

αB

1X

αB

2X

0BX

atoms B

atoms B

αB

1X

αB

2X

0BX

coordinate spatial

αB

1X

αB

2X

0BX

αB

1X

αB

2X

0BX

atoms B

atoms B


Region of spinodal decomposition on a phase diagramwith a miscibility gap

BX

T

10

1T

Nucleationand growth

Spinodaldecomposition

1α 2α

21 αα

0X

G2B

2

0X

G2B

2

0X

G2B

2

BX

G

101α 2α

0X

G2B

2

1TT

21 αα 21 αα


Computer simulation of spinodal decomposition in a binary alloy

http://math.gmu.edu/~sander/movies/spinum.html


Computer simulation of laser overheating & explosive boiling

Short pulse laser irradiation

leads to strong superheating

and rapid decomposition of a

surface region of the target

into a mixture of gas phase

atoms and liquid droplets

http://www.faculty.virginia.edu/CompMat/ablation/animations/

thermodynamics of diffusion - university of...

Documents