Chapter 3: Polycrystalline Microstructures

Chapter 3: Polycrystalline Microstructures

3.1 Interfacial Tension and Microstructure

Microstructure after sintering is determined by the interfacial tension.

_ij = interfacial tension force per unit length between phases i and j.

If we consider an equilibrium between three interfacial tensions:

sine law

a/sin A=b/sin B=c/sinC ¹

3.1.1 Wetting Angle

Assume interfacial tension = interfacial energy 

 = wetting angle



= solid surface energy



= liquid surface energy



= solid/liquid interfacial energy

Wetting angle: the angle between

solid/liquid and liquid/vapour interfaces when a liquid drop is placed on a solid substrate

 > 90: nonwetting

 < 90: wetting

 = 0: spreading

 > 90  < 90  = 0

N.b. force balance only satisfied in x

direction  real equilibrium not maintained The smaller  (higher wettability)

 enhanced densification in liquid phase sintering

 improved bonding in soldering/brazing

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3.1.2 Dihedral Angle

Dihedral angle: the angle at the junction between two grains

_b = grain boundary energy of  phase

_ = interfacial energy between  and  phases

 = dihedral angle

• Solid/solid, solid/liquid and solid/vapour

systems also reach an equilibrium defined by the surface energies.

• If a polycrystalline solid is immersed in a liquid or vapour, grooves form where grain boundaries meet the surface.

•  is determined only by  i.e. is independent of pressures in the phases.

• If  is a vapour,  is > 120, because _s > _b

• Usually _s  2 - 3  _b and   150

Dihedral angles also exist at internal inclusions.

Solid grain

Solid grain Solid grain

Distribution of dihedral angles where grain boundaries meet the surface

A range of dihedral angles will exist:

_b (_ss) varies with grain misorientation

_sv and _sl vary with crystal orientation

For ceramics,  can be < 120 i.e. _s (_sv)< _b May be due to:

bond distortion at grain boundaries;

differences in impurity adsorption.

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3.2 Single-Phase Microstructures

Microstructure is determined by grain boundary energy _b.

For an equilibrium grain shape (i.e. all grains have same shape, size and _b is constant), the microstructure must:

1. Minimize total interfacial area under interfacial tension equilibrium 2. Completely fill space without voids (holes)

For constant _b,  = 120.

In 2D, equilibrium grain shape is a hexagon.

In 3D, no equilibrium grain shape exists.

Closest shape is a tetrakaidecahedron (truncated octahedron)

When packed in bcc lattice, space is completely filled but interfacial tension equilibrium is not met.

d

d

Grain Boundary as Dislocation Array:

Small (Low)-angle grain boundary

sin (θ/2) = b/(2d)

θ/2  b/2d θ  b/d

sin x  x for small x e.g. Tilt (: misorientation) boundary with edge dislocations

at intervals of d

_b changes with crystallographic orientation between grains.

 _b is not constant and microstructure can vary locally.

However, in reality:

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e.g.) Tetrahedron C = 4

E = 6 P = 4 B = 1

4 - 6 + 4 - 1 = 1

All microstructures satisfy Euler's law

C - E + P - B = 1

C = corner E = edge P = polygon

B = body (i.e. 3D shape)

For a 2D figure, C - E + P = 1

e.g.) Hexagon C = 6

E = 6 P = 1

6 - 6 + 1 = 1

Schematic 2D microstructure 3 polygons meet at each point:

 ^

 nP

E

C

3

P_n = no. of polygons with n edges

E_C = no. of partially shared corners at edge of figure

2 polygons meet at each edge:

 ^

 nP

E

E

2

E_B = no. of unshared edges at edge of figure

If we put these equations into Euler's law:

 ^{( 6  n ) P}

^{ E}

^{ 6}

If P = total no. of polygons and p_n = fraction of n-sided polygons, then:

 ( 6  n ) p

 ( 6  E

) / P

If P is large, then:

 ^{( 6  n ) p}

^{ 0}

The average grain is a hexagon and polygons are distributed to satisfy this e.g. if a 3-sided grain is present, there must also be a 9-sided grain or two grains with 7 and 8 sides. ₈

Polygons with > 6 sides have concave boundaries Polygons with 6 sides have flat boundaries

Polygons with < 6 sides have convex boundaries

Concave boundaries migrate outwards  grain grows

Convex boundaries migrate inwards  grain shrinks

3.3 Multiphase Microstructures

 = 150?  = 90? = 60?

 = 30?  = 0?

More than one phase e.g. solid + liquid, solid + gas (pore), solid + liquid + gas Microstructure determined by

equilibrium among interfacial tensions.

Shape of second phase at 3-grain junction (triple junction) depends on dihedral angle  .

Distribution of 2nd phase in a 2D microstructure

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In a 3D microstructure:

 > 60: isolated second phase.

  60: continuous network along

grain edges.  > 60   60

As  , grain boundary (i.e. solid/solid) area 

W-Ni-Fe heavy alloy

W(Ni, Fe) grains

Ni-Fe-W 2nd phase

Matrix network after sintering

If  = 0:

liquid phase at all grain boundaries:

all grains are separated.

Some (low energy) grain boundaries remain.

Wet and dry boundaries can coexist.

anisotropy in _gb & _sl

W-Ni-Fe heavy alloy

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Thickness of liquid films  1-2 nm.

Liquid films can form at dry grain boundaries and thicken during grain growth.

1350C, 5 hrs

1350C, 20 hrs

1350C, 50 hrs

BaTiO

Causes: accumulation of segregated impurities penetration of liquid phase at triple junctions into grain boundaries.

BaTiO₃

For given values of  and 2nd phase volume fraction, the grains change shape to obtain minimum interfacial energy.

E = total interfacial energy per grain A_g = grain boundary area

A_m = grain-matrix interface area

_g = grain boundary energy

_m = grain/matrix interfacial energy

m m g

g

A A

E     2

1

Park and Yoon* calculated E as a function of matrix volume fraction V_m and .

V_m For  > 90, _m >> _g and E  as V_m 

For  = 0, _m << _g and E  as V_m 

For 0 < < 90, E vs. V_m curve has a minimum.

As E , grain shape becomes more rounded.

*H.H. Park and D. Y. Yoon, Metall. Trans. A, 16, 923-28 (1985)

Minimum

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If  is > 90, the 2nd phase will exude from (come out of) the sample. All boundaries will be solid/solid grain boundaries.

For 0 < < 90, the sample will contain solid/solid grain boundaries and solid/matrix interfaces. Any excess liquid (beyond the minimum point of the E vs.

V_m curve) will remain outside the sample during and after sintering.

For  = 0, all boundaries will be solid/matrix interfaces.

Exuded liquid phase after liquid phase sintering a system with

high 

N.b. matrix phase can be liquid or vapour

V_m

If E vs. V

curve isn't at minimum point:

• there will be a driving force to reach the minimum point

• the grains will try to change shape to reduce E.

 

Vg

m g

m

e

V

E V

P V 



 









 



1

P

= effective pressure

V

= volume fraction of matrix V

= volume of grain

E = total interfacial energy of grain For const. V

and variable V

:

P

pushes grains to achieve minimum E by changing shape.

If  is small, there is a limiting porosity in solid state systems.

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