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Chapter 6. Diffusion6.1. Introduction6.2. Diffusion mechanisms6.3. Steady-state diffusion

6.4. Nonsteady-state diffusion6.5. Factors that influence diffusion6.6. Other diffusion paths6.7. Materials processing and diffusion

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6.1. Introduction

Diffusion is a phenomenon of material transport by atomic motion.

Inhomogeneous materials can become homogeneous by diffusion.For an active diffusion to occur, the temperature should be highenough to overcome energy barriers to atomic motion.

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Interdiffusion or impurity diffusion occurs in response to a concentration

gradient.

Heat

Before After

Self diffusion is diffusion in one-component material, when all atomsthat exchange positions are of the same type.

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6.2. Diffusion mechanisms

Vacancy diffusion

To jump from lattice site to lattice site, atoms need energy tobreak bonds with neighbors, and to cause the necessary

lattice distortions during jump. This energy comes from thethermal energy of atomic vibrations (Eav ~ kT).Materials flow (the atom) is opposite the vacancy flowdirection.

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Interstitial diffusion

Interstitial diffusion is generally faster than vacancy diffusionbecause bonding of interstitials to the surrounding atoms isnormally weaker and there are many more interstitial sites thanvacancy sites to jump to.Requires small impurity atoms (e.g. C, H, O) to fit into intersticesin host.

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6.3. Steady-state diffusion

Diffusion is a process of a quantity of an element that is transportedwithin another as a function of time or the rate of mass transfer.Diffusion flux (J) is defined as the mass (or the number of atoms) Mdiffusing through and perpendicular to a unit cross-sectional areaof solid, A, per unit of time, t.

At

MJ =

dt

dM

A

1J =or

with the unit of J is kg/m2s or atoms/m2s.

Steady-state diffusion occurs if the diffusion flux does not change

with time.Example: diffusion of atoms of a gas through a plate of metal forwhich concentrations (pressures) of the diffusing species on bothsurfaces of the plate are held constant.

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Concentration profile: concentration of atoms/molecules ofinterest as function of position in the sample.

Concentration gradient: dC/dx (Kg.m-3): the slopeat a particular point on concentration profile.

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Ficks first law

dx

dCgradientionconcentrat =

x

Cgradientionconcentrat

=

The slope of a particular point on the concentration profile:

or

dx

dCDJ =

where: D is the diffusion coefficient (m2/s).The negative sign indicates that the direction of diffusion is

down the concentration gradient, from high to low concentration.

Example: purification of hydrogen gas

Sometimes the concentration gradient is called the driving force.

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6.4. Nonsteady-state diffusion

Most of diffusion cases are nonsteady-state ones, which means thatthe concentration gradient at some particular point vary with time.

Ficks second law 2

2

xCD

tC

=

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Some assumptions are made:

1. Before diffusion, any of the diffusing solute atoms in the solid areuniformly distributed with concentration of C0.

2. The value of a at the surface is zero and increases with distance intothe solid

3. The time is taken to be zero the instant before the diffusion processbegins.

Boundary conditions of Ficks second law:

=

Dt2

xerf1

CC

CC

0s

0x

where: Cx is the concentration at depth x after time t,C0 is the concentration at time =0,Cs is the concentration at x = 0 (surface concentration),

erf(x/2DT) is the Gaussian error function (see Table 6.1).

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When a specific concentration of solute is achieved, C1, the left-handside of Ficks second law equation becomes:

constant

Dt

x 2=

constantCC

CC

0s

01 =

and the right-hand side of Ficks second law equation becomes:

constantDt2

x=

or

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6.5. Factors that influence diffusion

Both diffusing species and host material influence the diffusioncoefficient.

Smaller atoms diffuse more readily than big ones, and diffusion isfaster in open lattices or in open directions.

6.5.1. Diffusing species

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6.5. Factors that influence diffusion

6.5.2. Temperature

Temperature has a big influence on the coefficients and diffusionrates Arrhenius dependence. See examples in Table 5.2.

where: D0 is a temperature-independent pre exponential (m2/s),

Qd is the activation energy for diffusion (J/mol),R is the universal gas constant (= 8.31 J/mol K),T is the absolute temperature (K).

The activation energy is an energy required to produce the diffusivemotion of one mole of atoms.Large Qd small diffusion coefficient

=RT

QexpDD d0

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

intercept of ln Do

In logarithm function:

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Arrhenius plot of diffusivity data for some metallic systems

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6.6. Other diffusion paths

Short-circuit diffusion paths may also occur along dislocations,grain boundaries, and external surfaces.

Even the rates are faster than the bulk diffusion, compared theoverall diffusion flux, the short-circuit diffusion paths areinsignificant.

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6.7. Materials processing and diffusion

Heat treatment is a process that involves atomic diffusion and occursover reasonable time periods at elevated temperatures. The diffusionrate is usually relatively rapid and a common practice for all most

materials (metals, ceramics, and polymers).Example: the strength of some steel changes significantly after heattreatment.