Crystal Interfaces - Microstructure

Basically three different types of interface are important in metallic systems:1. The free surfaces of a crystal (solid/vapour interface)2. Grain boundaries (- interfaces)3. Interphase interfaces ( /ß interfaces).

• All crystals possess the first type of interface.

• The second type separates crystals with essentially the same composition and crystal structure, but a different orientation in space.

• The third interface separates two different phases that can have different crystal structures and/or compositions and therefore also includes solid/liquid interfaces.

Ref: Phase transformations in metals and alloys by Porter and Easterling

Crystal Interfaces - Microstructure

• The great majority of phase transformations in metals occur by the growth of a newphase () from a few nucleation sites within the parent phase ()- a nucleation andgrowth process.

• The a/ interface plays an important role in determining the kinetics of phasetransformations and is the most important.

• It is, however, also the most complex and least understood

• The solid/vapour interface is of course itself important in vaporization andcondensation transformations, while grain boundaries are important inrecrystallization, i.e. the transformation of a highly deformed grain structure intonew undeformed grains.

• Although no new phase is involved in recrystallization it does have many features incommon with phase transformations.

Interfacial Free Energy

The free energy of a system containing an interface of area A and free energy per unit area is given by

G = G0 + A

where Go is the free energy of the system assuming that all material in the system has the properties of the bulk. is the excess free energy arising from the fact that some material lies in or close to the interface. It is also the work that must be done at constant T and P to create unit area of interface.

Consider for simplicity a wire frame suspending a liquid film If one bar of the frame is movable it is found that a force F per unit length must be applied to maintain the bar in position.

If this force moves a small distance so that the total area of the film is increased by dA the work done by the force is FdA. This work is used to increase the free energy of the system by dG.

Interfacial Free Energy

dG = dA + Ad = FdA

Equating to FdA gives F = + A (d/dA)

In the case of a liquid film the surface energy is independent of the area of the interface and d/dA = 0. This leads to the well-known result F = , a surface with a free energy , J m-2 exerts a surface tension of , N m-1

In solids, d/dA 0, surface free energy and surface tension will not be identical

Solid / Vapour Interfaces

To a first approximation the structure of solid surfaces can be discussed in terms of a hard sphere model.

The origin of the surface free energy is that atoms in the layers nearest the surface are without some of their neighbors.

For example: the atoms on a {111} surface (FCC structure) are deprived of three of their twelve neighbors.

• If the bond strength of the metal is , each bond can be considered as lowering theinternal energy of each atom by /2. Therefore every surface atom with three‘broken bonds’ has an excess internal energy of 3/2 over that of the atoms in thebulk.

• For a pure metal, can be estimated from the heat of sublimation LS. The latentheat of sublimation is equal to the sum of the latent heat of melting (or fusion)and the latent heat of vaporization.

• If 1 mol of solid is vaporized, 12Na broken bonds are formed. (Na Avagadronumber) (number of atoms per mole i.e., 6.022 1023)

Solid / Vapour Interfaces

𝐿𝑠 = 12𝑁𝑎𝜖

2= 6𝑁𝑎𝜖

𝜖 =𝐿𝑠6𝑁𝑎

Assume that the strengths of the remaining bonds in the surface are

unchanged from the bulk values.

The surface energy, Esv, of a {111} surface (FCC structure) should be given by

Solid / Vapour Interfaces

Different planes and different crystals could have different values for depending on the number of broken bonds

In FCC, the number of broken bonds at the surface will increase through the series {111} {200} {220}

J per surface atom𝑬𝒔𝒗 = 𝟑𝝐

𝟐= 𝟑 ⋅

𝟏

𝟐⋅𝑳𝒔𝟔𝑵𝒂

= 𝟎. 𝟐𝟓𝑳𝒔𝑵𝒂

From the definition of Gibbs free energy the surface free energy will be given by

Solid / Vapour Interfaces

Y = E + PV – TS

Even if the 'PV' term is ignored surface entropy effects must be taken into account. Itmight be expected that the surface atoms will have more freedom of movement andtherefore a higher thermal entropy compared atoms in the bulk.

Extra configurational entropy can also be introduced into the surface by the formationof surface vacancies for example. The surface of a crystal should therefore be associatedwith a positive excess entropy which will partly compensate for the high internalenergy.

Experimental determination of 'Ysv is difficult but the measured values for pure metals indicate that near the melting temperature the surface free energy averaged over many surface planes is given by Ysv = 0.15 Ls / Na (J/surface atom)

Equilibrium shapes of crystals

Wulf plot represents the dependence of the interfacial energy on the surface plane orientation. Inthis method, the lattice planes are identified by their normals. Given a normal <hkl> and the surfaceenergy hkl associated with the surface plane with <hkl> as the normal, from an origin, we plot apoint in the <hkl> direction at a distance which is equal to hkl.

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• Given a Wulf plot, it is possible to identify the equilibrium shape

• The equilibrium shape is that shape which minimizes Aii where Ai is the area of the plane with an interfacial energy of i

• From the Wulf plot, it is clear that planes whose energies lie at the cusps of the Wulf plot have lower energies

• At every point on the Wulf plot, we draw a tangent (which is perpendicular to the radial line). The inner envelope of such tangents gives us the equilibrium shape

• If there are cusps in the Wulf plot, this construction give facetted equilibrium shapes

Equilibrium shapes of crystals

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Equilibrium shapes of crystals

Equilibrium shapes of crystals

Wulf plot Equilibrium shape

Equilibrium shapes of crystals

Wulf plot Equilibrium shape

Surface Engineering Volume 34(6):485-492

Equilibrium shapes of crystals

• Crystalline solids (most materials) generally consist of millions of individualgrains separated by boundaries.

• Each grain (or subgrain) is a single crystal.

• Within each individual grain there is a systematic packing of atoms.Therefore each grain has different orientation and is separated from theneighboring grain by grain boundary.

• When the misorientation between two grains is small, the grain boundarycan be described by a relatively simple configuration of dislocations (e.g., anedge dislocation wall) and is, fittingly, called a low-angle boundary.

Grain Boundaries

Grains in a metal or ceramic; the cube depicted in each grain indicates the crystallographic orientation of the grain in schematic fashion with respect to an external reference frame

Grain Boundaries

At the grain boundary, there is a disturbance in the atomic packing.

Grain Boundaries

• When the misorientation is large (high-angle grain boundary), more complicated structures are involved (as in a configuration of soap bubbles simulating the atomic planes in crystal lattices).

• The grain boundaries are therefore:• where grains meet in a solid.

• transition regions between the neighboring crystals.

• Where there is a disturbance in the atomic packing

• These transition regions (grain boundaries) may consist of various kinds of dislocation arrangements.

Grain Boundaries

Role of Grain Boundaries

• Grain boundaries have very important role in plastic deformation ofpolycrystalline materials.

• At low temperature (T<0.5Tm, where Tm is the melting point in K), thegrain boundaries act as strong obstacles to dislocation motion.

• Mobile dislocations can pile up against the grain boundaries and thusgive rise to stress concentrations that can be relaxed by initiatinglocally multiple slip.

• There exists a condition of compatibility among the neighboring grainsduring the deformation of polycrystals; that is, if the development ofvoids or cracks is not permitted, the deformation in each grain must beaccommodated by its neighbors.

• This accommodation is realized by multiple slip in the vicinity of theboundaries which leads to a high strain hardening rate.

• It can be shown, following von Mises, that for each grain to stay incontiguity with others during deformation, there must be operating atleast five independent slip systems - Taylors Theorem.

• This condition of strain compatibility leads a polycrystalline sample to have multiple slip in the vicinity of grain boundaries.

• The smaller the grain size, the larger will be the total boundary surface area per unit volume.

• In other words, for a given deformation in the beginning of the stress-strain curve, the total volume occupied by the work-hardened materialincreases with the decreasing grain size.

• This implies a greater hardening due to dislocation interactions inducedby multiple slip.

Role of Grain Boundaries

• At high temperatures the grain boundaries function as sites of weakness.

• Grain boundary sliding may occur, leading to plastic flow and/or openingup of voids along the boundaries.

• Grain boundaries can act as sources and sinks for vacancies at hightemperatures, leading to diffusion currents as, for example, in the NabarroHerring creep mechanism.

• In polycrystalline materials, the individual grains usually have a randomorientation with respect to one another.

Role of Grain Boundaries

• The term polycrystalline refers to any material which is composed of many individual grains.

• However, some materials are actually used in their single crystal state: silicon for integrated circuits and nickel alloys for aircraft engine turbine blades are two examples.

• The sizes of individual grains vary from submicrometer (for nanocrystallinestructures) to millimeters and even centimeters (for materials especially processed for high-temperature creep resistance).

Role of Grain Boundaries

Micrographs showing polycrystalline Tantalum

Role of Grain Boundaries

A polycrystalline zirconium specimen photographed with polarized light. In this photograph, individual crystals can be distinguished by a difference in shading, as well as by the thin dark lines representing grain boundaries. 350 X. Note that most of the triple junctions form 120° angles.

Role of Grain Boundaries

• One example of a material property that is dependent on grain size is the strength of a material; as grain size is increased the material becomes softer .

• strength is expressed in units of stress (MN/m2)• grain size of a material can be altered (increased) by annealing

• Hardness measurement (e.g., by Vickers indenter) can provide a measure of the strength of the material.

Role of grain Boundaries

The dependence of strength on grain size for a number of metals and alloys.

Role of grain Boundaries

Role of Grain Boundaries

Transgranular fracture

Role of Grain Boundaries

Intergranular fracture

Grain Size Measurements

Grain structure is usually specified by giving the average diameter.

Grain size can be measured by two methods.

(a) Lineal Intercept Technique: This is very easy and may be the preferred method for measuring grain size.

(b) ASTM Procedure: This method of measuring grain size is common in engineering applications

In this technique, lines are drawn in the photomicrograph, and thenumber of grain-boundary intercepts, Nl along a line is counted.

• The mean lineal intercept is then given as:

where L is the length of the line and M is the magnification in thephotomicrograph of the material.

MN

Ll

l

=−

Lineal Intercept Technique

Micrographs showing polycrystalline TiC

Lineal Intercept Technique

• In Figure a line is drawn for purposes of illustration.

• The length of the line is 100 mm. The number of intersections, Nl, is equal to 7, and the

magnification M = 1,300. Thus,

• Several lines should be drawn to obtain a statistically significant result.

mX

Xl 11

13007

10100 3

==−

Lineal Intercept Technique

• The mean lineal intercept l does not really provide the grain size, but is related to a fundamental size parameter, the grain-boundary area per unit volume, Sv by the equation

• The most correct way to express the grain size (D) from lineal intercept measurements is:

• Therefore, the grain size (D) of the material of Figure is:

vSl

2=

−

lD−

=2

3

5.16112

3== XD

ASTM Procedure

• With the ASTM method, the grain size is specified by the number n in the expression N = 2 n-1, where N is the number of grains per square inch (in an area of 1 in2), when the sample is examined at 100 power micrograph.

Example

In a grain size measurement of an aluminum sample, it was found that there were 56 full grains in the area, and 48 grains were cut by the circumference of the circle of area 1 in2. Calculate ASTM grain size number n for this sample.

Solution

The grains cut by the circumference of the circle are taken as one-half the number. Therefore,

( ) 35.7169.0

38.4

12ln

80ln

12ln

ln

2802456

24856

1

=+=

+

=

+

=

==+=

+=

−

n

NnBut

N

n