Datos comparativos de las vías de mayor peligrosidad

3 The coincidence model. 2.3*2.1 The dislocation model

(20 )

Read and Shockley' J _{assumed the grain boundary to be an array of}

dislocations'and if Tb' is the Burgers vector of the dislocations of mean spacing 'd1, then:-

£ - -J2 — £, (e)

a Sin (9 <9

misorientation between the two grains causes fd* to decrease. At a misorientation of <^10° the Read-Shockly concept breaks down, because the interaction of the closely adjacent dislocations changes the core structure and the boundary is no longer simple. At somewhat higher angles, dislocations would be so closely spaced that they would lose their identity and the dislocation model becomes of little or

t

(

)

no value' .

2.3.2.2 The island model

(22)

This was developed by Mott' ' who suggested that if two crystal planes are in contact, but cannot fit owing to different orientations, then the surface of contact is divided in to islands where the fit is reaso­ nably good separated by areas where the fit is poor. He then set out to characterize the good and the bad fit in terras of the "melting” of the good fit islands, which must occur to allow* boundary migration. To remove an atom from an island requires energy equivalent to the latent heat of fusion. In some respects, this model has similarities with the coincidence model.

2.3.2.3 The coincidence model

The significant feature of the coincidence boundary is that some atoms .occupy sites that belong to both crystals, and there is a periodic

repetition of very small units of structure. Such boundaries have low (23)

energy. Bishop and Chalmers' ' proposed a model to characterize speci­ al orientations which is entirely based on the concept of boundary coincidence. Fig. 7 shows the structure of a symmetrical tilt boundary with a rotation of 23,1° about <1.00)> axis in f.c.c lattice. Any devi­ ation from this orientation because of the disturbances in the ledge sequences in the boundary may lead to an increase in the grain boundary energy.

2.3.3 Mechanism of grain boundary migration

Turnbull by assuming that the atoms are transferred singly across a migrating boundary, derived an expression for the boundary migration rate based on absolute reaction rate.theory. The rate of grain boun­ dary migration 'G1 is:-

G = G e" ^_o (6) where Q is the activation energy, T the absolute temperature and R the gas constant. The value of Gq depends on

(a) -the local boundary movement when an atom is transferred from one grain to the other,

(b) the lattice parameter, (c) the driving force, and (d) the temperature.

Most, although not all, of the experimental results are consistent with (25}

this relationship' \ Since it has been assumed that all the atoms at the grain boundary have the same jump probabilities and always find a free place on the adjoining grain surface, this relationship will give the maximum possible value of the migration rate.

Hofmann and Haessner^^ modified this model in the following way:- (a) The grain boundary is assumed to be a layer a few atoms

thick.

(b) The probability of an atom jumping across the boundary is different in the two jump directions.

(c) An atom can make a jump only when there is a vacant site into which it can move.

The migration rate 'G* was

J -\Jt. Cp. -c~. . expC-AEII /r t)

where a is the lattice parameter; \y. is the vibrational frequency;

<$ is the grain boundary thickness;

2

Cp t Cj. and are the vacant sites/ cm in boundary, grain and matrix respectively; ₂

Zj. is the number of atoms/cm at the surface of the grain, which are capable of jumping;

A E is the difference in free energy between the matrix and the grain;

and Gibbs free energy of activation per moUfi for the jump of one atom across the boundary.

This relationship has been simplified for the following two limiting cases;-

(a) The rate cotrolling process may be the transport of atoms through the boundary and in this case:-

^ ‘* C P, (k C + (J ) 1 which gives II

6 . 0

_.

_{C| . A ® .Zj,}

_/

— --- — exp I —

cf . RT Cj

( - ^ )

(b) The rate controlling process may be the emission of the atoms from the unrecrystallized matrix and their incorporation

in the recrystallized grain. This corresponds to the condition:

f * ce + 1 * sothat a*\ \£ . Cs . A E ZT r AEtT \ G== --- !--- I - exp / (9) Rm / 1. . JL \ CI V RT ' \ °I °II /

(

)

Mott ' assumed that in the "island" grain boundary model, atoms are activated- in groups during their transfer across the boundary and that the basic process involves the "melting” of a group of atoms belonging to one crystal followed by their re-solidification on the other crystal. This gave a similar relationship as that of Turnbull, equation (6). However, in Mott*s theory *G * is related to the number of atoms in the transferring group and is an exponential function of

(27)