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ENS 205Materials Science I

Chapter 5: Diffusion

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What is Diffusion? • Many reactions and processes rely on the transfer of mass accomplished by

diffusion – the phenomenon of material transport by atomic motion

– Helium atoms inside a balloon can diffuse through the wall of the balloon and escape, resulting in the balloon slowly deflating

– Consider a drop of ink in a glass of water, drop spreads out until evenly colored water- the movement of molecules from an area of higher concentration to an area of lower concentration diffusion example

– Difficult to visualize in solids – lower rate of diffusion

• Understanding the movements of the atoms within the materials can be critically important both in producing the material and in applying it within an engineering design. The chemical composition of engineering materials changes as a result of the movements of atoms (solid-state diffusion).

• Atoms are re-distributed within the microstructure• Atoms are added from the materials environment • Atoms from the material might be discharged to the environment

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Diffusion of copper atoms into nickel

• Eventually, the copper atoms are randomly distributed throughout the nickel. Concentration gradient is a driving force.

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

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Arrhenius Equation• Diffusion is the movement of atoms in materials• The ability of an atom or an atomic imperfection to diffuse increases with

temperature as expected since its thermal energy increases.• The rate of diffusion is a thermally activated process governed by the

Arrhenius equation.RTQCerate /

RT

QCrate ln)ln(

where C is a constant, R is the gas constant (1.987 cal/(mole K), T is the absolute temperature (K) and Q is the activation energy (cal/mol) to cause an Avogadro’s number of atoms or ions to move.

We can rewrite this equation by taking the natural logarithms on both sides:

If we plot ln(rate) of some reaction versus 1/T, the slope of the curve will be –Q/R, Thus Q can be measured. The constant, C, corresponds to the intercept at ln C when 1/T is zero.

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An Arrhenius plot of ln (rate) versus 1/T can be used to determine the activation energy required for a reaction. This has many applications to science and engineering.

Arrhenius Equation

RT

QCrate ln)ln(

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Activation Energy for Diffusion

A high energy is required to squeeze atoms past each other during diffusion, which is known as the activation energy. The higher the activation energy, the harder it is for diffusion to occur. In general more energy is required for a substitutional atom, Qv, than for an interstitial atom, Qi.

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Activation Energy

• Energy barrier that must be overcome by thermal activation in order for a reaction to occur

• Q energy per mole

• q = Q/Nav energy per atomic scale unit (atom, electron and ion)

• k = R /Nav = 13.8 x 10-24 J/K Boltzmann’ s constant

– As T , a larger number of atomic scale unit is available to overcome a given energy barrier, q

kTqCerate /

q

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Thermal Production of Point Defects• Energy requirements to squeeze atoms through perfect crystal structures

are too high, points defects are generally required• Points defects occur due to thermal vibration. T↑, intensity of vibration

increases, thereby likelihood of structural disruption and the development of point defects increase

• Concentration of point defects increases exponentially

)/( kTE

sites

defects defectCen

n

Edefect energy needed to create a single-point defect in the crystal structure, nsites=ideal crystal-lattice sites

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Tk

EC

n

n v

sites

v 1lnln

The slight difference between the thermal expansion measured by overall sample dimensions ΔL/L and by x-ray diffraction Δa/a is the result of vacancies

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Atomic Movement in Materials

For an atom to make a move:

There must be an empty adjacent site

The atom must have enough energy to break bonds with the neighbor atoms and then cause some lattice distortion during the displacement. With T increased, fraction of atoms capable of diffusive motion due to their vibrational energies increases (thermal energy of atomic vibrations, Eav ~ kT)

• There are several atomic diffusion mechanisms in materials, which include:

– Vacancy

– Interstitial

– Interstitialcy

– Exchange and Ring.

• Note: We will see other types of diffusion including volume, surface, grain boundaries and dislocations.

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Atomic Diffusion Mechanisms• Vacancy Diffusion – In self-diffusion and diffusion involving substitutional atoms,

an atom leaves its lattice site to fill a nearby vacancy (thus creating a new vacancy at the original lattice site). As the diffusion continues, we have a countercurrent flow of atoms and vacancies.

• Interstitial Diffusion – When a small interstitial atom is present in the crystal structure, the atom moves from one interstitial site to another. No vacancies are required. Interstitial diffusion is generally faster than vacancy diffusion because bonding of interstitials to the surrounding atoms is normally weaker and there are many more interstitial sites than vacancy sites to jump to. Requires small impurity atoms (e.g. C, H, O) to fit into interstices in host.

• Interstitialcy Diffusion – When a “normal” atom occupies an interstitial site in the crystal, it moves from one interstitial site to another. Normally this is not a common method of diffusion except when irradiation damage occurs, which happens when semiconductor materials are doped with foreign atoms and in nuclear reactors when high-energy neutrons pass through materials.

• Exchange and Ring Diffusion – Atoms are always vibrating and can exchange position with their adjacent neighbor or many atoms can rotate in unison around in a ring fashion.

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Interstitialcy diffusion (This is energetically unfavorable so is very rare)

Ring diffusion

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Fick’s First Law (Rate of Diffusion)

• The rate at which atoms, ions, particles or other species diffuse in a material can be measured by the flux, J. The flux J is defined as the number of atoms passing through a plane of unit area per unit time, given by:

• Where D is the diffusivity or diffusion coefficient (cm2/s) and dc/dx is the concentration gradient

• The negative sign tells us that the flux of diffusing species is from higher to lower concentration making the dc/dx term negative and hence J positive.

• The thermodynamic driving force for diffusion is the concentration gradient.

• A net or observable flux is created depending upon temperature and concentration gradient.

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12

12

xx

ccD

xx

ccD

x

cD

dx

dcDJ

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Illustration of a concentration gradient. Notice that it is negative. The concentration gradient is the driving force for diffusion.

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Factors Affecting Diffusion• Several factors affect diffusion and these include:

– Temperature– Type of diffusion– Time– Crystal structure and type of bonding

• We will look at some of these in detail.

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Temperature and Diffusion• The diffusion coefficient is related to temperature by an Arrhenius-type

equation given as:

• Where Q is the activation energy for diffusion (cal/mol) and Do is the pre-exponential constant, Covalently bonded materials such as Si and C have unusually high activation energies owing to their high-strength atomic bonds.

• Typical values for D are given in the next two tables for some metals ceramics and semiconductors.

RT

QDD o exp

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Diffusion coefficients as a function of reciprocal temperature for some metals and ceramics. A steep slope represents a high activation energy.

Diffusion coefficients for dopants in Silicon. Why does self-diffusion in Si have such a high activation energy and corresponding low diffusion?

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Types of Diffusion

• In volume diffusion or bulk diffusion, the atoms move through the crystal from one regular or interstitial site to another. Because of the surrounding atoms, the activation energy is large and the rate of diffusion is relatively slow.

• Atoms move easily along grain boundaries because the atom packing is poor resulting in a low relative activation energy.

• Surface diffusion is easier still because there is even less constraint on the diffusing atoms at the surface. surfaceboundarygrainvolume

surfaceboundarygrainvolume

DDD

QQQ

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Bonding and Crystal Structure

Activation energies are usually lower for atoms diffusing through open crystal structures than for closed or close-packed crystal structures. Because the activation Because the activation energy depends on the energy depends on the strengthstrength of of atomic atomic bondingbonding, it is higher for , it is higher for diffusion of atoms in materials diffusion of atoms in materials with a high with a high melting temperaturemelting temperature..

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Carbon can diffuse through BCC Fe more readily than through FCC Fe because of the greater openness of the BCC structure. Similarly, the self-diffusion of Fe by a vacancy mechanism is greater in BCC Fe than in FCC. In many compounds (Al2O3), the smaller ionic species (Al+3) diffuse much more readily through the system. Smaller atoms diffuse more readily than big ones, and diffusion is faster in open lattices or in open directions

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Fick’s Second Law• Fick’s second law describes the dynamic, or non-steady state, diffusion

of atoms in materials, given by:

• If we assume that the diffusion coefficient D is not a function of location, x, and the concentration, c, of the diffusing species then Fick’s second law can be simplified to:

x

cD

xdt

dc

2

2

x

cD

dt

dc

• The solution to this equation depends on the boundary conditions for a particular situation where one common solution is given by the error function (erf) and expressed as:

Dt

xerf

cc

cc

s

x

21

0

0

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Fick’s Second LawDiffusion of atoms into the surface of a material.

Dt

xerf

cc

cc

s

x

21

0

0

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Master plot generated using all of the time-dependent concentration profiles. This plot permits rapid calculation of the time necessary for relative saturation of the solid.

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Diffusion and Materials Processing• Diffusion is involved in many types of materials processing, some of which we

will discuss during this course.

• Some examples include solidification, phase transformation, heat treatment, sintering, grain growth etc.

During sintering, atoms diffuse to points of contact, creating bridges and reducing pore size.

Microstructure of barium magnesium tantalate, a ceramic, obtained by compaction and sintering of powders.

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Diffusion and Materials Processing• Polycrystalline materials contain a large number of grain boundaries, which

represent a high-energy area because of the inefficient packing of atoms.

• A lower energy is obtained in the material if the amount of A lower energy is obtained in the material if the amount of grain boundary grain boundary areaarea is is reducedreduced by grain growth. by grain growth.

• Grain growth involves the movement of grain boundaries, which results in larger grains growing at the expense of smaller grains.

• An analogy is big fish getting bigger by eating small fish.

Grain growth results from atoms diffusing across the grain boundary from one grain to another.

Grain growth in alumina ceramic sintered at 1350 C for 15 hours (left) and 300 hours (right).

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Steady State Diffusion

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Diffusion – Temperature Dependence (I)

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Factors that Influence Diffusion: Summary• Temperature - diffusion rate increases very rapidly

with increasing temperature

• " Diffusion mechanism - interstitial is usually faster than vacancy

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