Lecture 17: Diffusion

PHYS 430/603 materialLaszlo Takacs

UMBC Department of Physics

Diffusion in gases and liquids

Types of diffusion

• Interstitial diffusion - C in Fe, H in Ti, etc.• Vacancy diffusion• Self diffusion, tracer diffusion - healing of defects, time-

dependent mechanical phenomena• Chemical diffusion - diffusion of one element through another,

solid state chemical reactions

• Bulk diffusion• Diffusion in grain boundaries, lattice defects• Surface diffusion

• Diffusion is a statistical phenomenon, driven by the random motion of atoms.

Phenomenological description

Fick’s law:

Continuity:

Combined:

If D = const.

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jD = −∇c jD in cm2/s; c in particles/cm3

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∂c∂t

+∇jD = 0

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∂c∂t

=∇ D∇c( )

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∂c∂t

= D∇ 2cTo be solved for c(x,t) or c(r,t)

Typical solution depends on x/(Dt)1/2.

One can define a diffusion front that travels as x2 ~ 6Dt

If defined that way, about 99% of the diffusing element is still behind the diffusion front.

Evaluating a diffusion couple

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∂c∂t

= D∂2c∂x 2 , c = c(x,t)

c(0,t) = (c1 + c2) /2;c(x → −∞,t) = c1; c(x → ∞,t) = c2

Assume constant D and symmetrical solution to x = 0:

Introduce new variables:

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η =xt

∂c∂x

= dcdη∂η∂x

= dcdη

1t

; ∂ 2c∂x 2 = d

2cdη 2

1t

∂c∂t

= dcdη∂η∂t

= dcdη

− x2t t

⎛ ⎝ ⎜

⎞ ⎠ ⎟= − η

2tdcdη

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D d2cdη 2 + η

2dcdη

= 0

with g = dc /dη D dgdη

+ η2g = 0

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D dgdη

+ η2g = 0 ⇒ g = A exp(−η 2 /4D)

c(η ) = B exp(−ξ 2)dξ with ξ =η /2 D∫c(η ) =G erf η

2 D ⎛ ⎝ ⎜

⎞ ⎠ ⎟+ H =G erf x

2 Dt ⎛ ⎝ ⎜

⎞ ⎠ ⎟+ H

We can guess the solution

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erf (x) = 2π

exp(−y 2)0

x

∫ dy

Adjust G and H to the boundary conditions

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c(x, t) = c1 + c2

2+ c2 − c1

2erf x

2 Dt ⎛ ⎝ ⎜

⎞ ⎠ ⎟

Diffusion is a thermally activated process, resulting in an exponential temperature dependence:

D = D0 exp(-Q/kT)

It can also depend on orientation relative to the crystal axes.

Interstitials diffuse much faster than substitutional impurities and matrix atoms.

Rough correlations

• D0 for self diffusion in metals ~ 1 cm2/s (0.1 to 10)

• ∆HD / Tm ~ 1.5*10-3 eV/K

• ∆HD / Lm ~ 15

These relationships are useful if no measured data are available. Typically estimates of diffusion are quite rough, there is a huge variation, high accuracy is not needed. Also, defects and impurities influence diffusion very much, it is difficult to control it.

Examples

Temperature is uniformly T = 912°C, the alpha-gamma transition of iron.Diffusion time t = 1 h =3600 s, distance x2 = 6Dt

Self diffusion in alpha Fe: D0 = 2.0 cm2/s, ∆HD = 2.5 eV

D = 2.0 exp(-2.5/8.6*10-5*11850) = 4.4*10-11 cm2/s, x ~ 10 µmSelf diffusion in gamma Fe: D0 = 0.4 cm2/s, ∆HD = 2.8 eV

D = 4.6*10-13 cm2/s, x ~ 1 µm60Co in alpha Fe: D0 = 0.2 cm2/s, ∆HD = 2.4 eV

D = 1.2*10-11 cm2/s, x ~ 5 µmC in alpha Fe: D0 = 0.004 cm2/s, ∆HD = 0.83 eV

D = 1.2*10-6 cm2/s, x ~ 1.6 mmC in gamma Fe: D0 = 0.67 cm2/s, ∆HD = 1.6 eV

D = 1.0*10-7 cm2/s, x ~ 0.5 mm

Notice the very large differences.

Vacancy diffusion in a solid anddislocation climb due to capturing vacancies

A simple atomistic view of diffusion

Let’s consider two neighboring lattice planes of area A (left and right) at a distance d, perpendicular to the x direction.Nl = Ad c and Nr = Ad [c + (dc/dx)*d]

If is the jump frequency and jump in 6 mutually perpendicular directions has the same probability:jr = /6 (Adc) / A and jl = /6 [Ad [c + (dc/dx) d] / A

j = jr - jl = - d2/6 dc/dx = - D dc/dx,

thusD = d2/6

Problems

• Geometry is more complicated, even for the simplest case of interstitial diffusion. 1/6 assumption is good for simple cubic, bcc gives the same result. An anisotropic hcp lattice is certainly different.

• At higher concentration, the jump probability depends on the occupation of the potential target sites.

• For substitutional atoms (self diffusion included) jump can only occur, if the target site is vacant. Estimate for D must be multiplied by the vacancy concentration, cv = exp( - Gf/kT ).This factor is not needed for vacancy diffusion, it is much faster.

• Probability of jump depends on previous jump - correlation effects.• Estimating jump frequency is very difficult - but needed for quantitative

estimates: = exp( - Gm /kT )Gm migration free energy, Debye frequency.

• Diffusion current should be related to the Gibbs free energy, rather than concentration.

More than one diffusing species

Diffusion is usually asymmetric, the boundary between regions is shifting, often difficult to define.

Very importantly, solid state reactions always take place at interfaces via diffusion. The products form intermediate layers that often act as diffusion barriers. Decreasing the particle size - e.g. by ball milling - accelerates the reactions.

Diffusion relates to volume changes, formation of voids, etc.

According to the experts, diffusion is much more complicated than it looks for a superficial first look.Rather than focusing on the difficult delicacies of the theoretical description, let’s consider a few examples.

The Kirkendall experiment

Zn diffuses faster than Cu in this system. As a result, the Cu - brass interface shifts, the Mo wires marking the interface drift toward each other.

After 15 h at 500°C layered structure developed that exhibits every compound in the phase diagram in order of composition.The layer thickness depends on the diffusion coefficients. Concentration gradient exists within the layer, if the phase diagram allows.

Each layer is single-phase, two phase regions do not form.

Surface and interface diffusion

• More open, diffusion can take place faster. In fact, dislocations provide routes for fast diffusion also. Diffusion in grain boundaries makes predicting the behavior of polycrystals difficult.

• Activation energy is lowest for surface diffusion, larger for grain boundary diffusion, and the highest for bulk diffusion. Therefore, bulk diffusion dominates at high temperature (larger number of possible sites) but grain boundary and surface diffusion takes over at lower temperature.

• Technological importance: Thin film deposition, weakening of grain boundaries by impurities, shape changes of particles, etc.

• Diffusion can be accelerated by introducing grain boundaries and other lattice defects. This is an important benefit from MA (mechanical alloying).