
Global thermal equilibrium at arbitrary
temperatures, i.e. the absolute minimum of
the free enthalpy, can only be achieved if there are mechanisms for the
generation and total
annihilation of point
defects. 


This means there must be
sources and
sinks for vacancies and
(intrinsic) interstitials that operate with small activation energies 
otherwise it will take a long time before global equilibrium will be achieved.


At this point it is essential to
appreciate that an ideal perfect (=
infinitely large) crystal has no sources
and sinks  it can never be in thermal
equilibrium. 


An atom, to be sure, cannot simply disappear
leaving a vacancy behind. Even if the crystal is finite, it cannot simply
disappear leaving a vacancy behind and then miraculously appear at the surface,
as we assumed in equilibrium thermodynamics, where it does not matter how a state is reached. 


On the other hand, infinitely large perfect
crystals do not exist  but semiconductorgrade dislocationfree single
Si crystals with diameters of 300 mm and beyond, and lengths of
up to 1 m are coming reasonably close. These crystals form a special
case as far as point defects are concerned but nevertheless incorporate point
defects in equilibirum. 

In real life we need other defects  surfaces, crystalmelt interfaces,
grain boundaries, dislocations, precipitates, and so on, as sources and sinks
for point defects. In regular metals or ceramics and so on, we have almost
always plenty of those defects. 

How a
grain boundary
can act as source or sink for vacancies is schematically shown in the pictures
below. 


It is clear from these drawings that the
activation energy (which is not the
formation energy of a vacancy!!) needed to emit (not to form from scratch!) a vacancy from a
grain boundary is small. 




Grain boundary absorbs 1 vacancy, i.e. acts as
sink after one more jump of the proper
atom. 
Grain boundary emits 3 vacancies, i.e. acts as
source after one more jump of the 3
proper atoms. 
The red arrows indicate the jumps of individual
atoms. The flux of the vacancies is always opposite to the flux of diffusing
atoms. 


We thus may expect that at
sufficiently high temperatures (meaning temperatures large enough to allow
diffusion), we will be able to establish global point
defect equilibrium in a real (= nonideal) crystal, but not really
global crystal equilibrium, because a
crystal with dislocations and grain boundaries is never at global
equilibrium. 

Sources and sinks are a thus a
necessary, but not a sufficient ingredient for point defect
equilibrium. We also must require that the point defects are able to move,
there must be some diffusion  or you must resign yourself to waiting for a
long time. In other words, we must look at the temperature now. 


At low
temperatures, when all diffusion effectively stops, nothing goes
anymore. Equilibrium is unreachable. For many practical cases however, this is
of no consequence. At temperatures where diffusion gets
sluggish, the equilibrium
concentration c_{eq} is so low, that you cannot measure
it. For all practical purposes it surely doesn't matter if you really achieve,
for example, c_{eq} = 10^{–14}, or if you
have nonequilibrium with the actual concentration c a thousand
times larger than c_{eq} (i.e. c =
10^{–11}). For all practical purposes we have simply
c = 0. 


At high
temperatures, when diffusion is fast, point defect equilibrium will
be established very quickly in all real crystals with enough sources and
sinks. 

The intermediate temperatures thus are of interest. The
mobility is not high enough to allow many point defects to reach convenient
sinks, but not yet too small to find other point defects. 


In other words, the average diffusion length or mean distance covered by a
randomly diffusing point defect in the time interval considered, is smaller
than the average distance between sinks, but larger than the average distance
between point defects. 


This is important, so let's say it once more in
yet other words: In the intermediate temperature range we are considering here,
a given vacany will still be able to move around sufficiently to encounter
another vacancy, but not a dislocation, precipitate or grain boundary. 

Global point defect equilibrium as
the best state of being is thus unattainable at medium
temperatures. Local
equilibrium is now the second best choice and far preferable to a huge
supersaturation of single point defects slowly moving through the crystal in
search of sinks. 


Local equilibrium then simply refers to the state
with the smallest free enthalpy taking into account
the restraints of the system. The most simple restraint is that the
total number of vacancies in vacancy clusters of all sizes (from a single
vacancy to large "voids") is constant. This acknowledges that
vacancies cannot be annihilated at sinks under these conditions, but still are
able to cluster. 

Let us illustrate this with a
relevant example. Consider vacancies in a metal crystal that is cooled down
after it has been formed by casting. 


As the temperature decreases, global equilibrium
demands that the vacancy concentration decreases exponentially. As long as the
vacancies are very mobile, this is possible by annihilation at internal
sinks. 


However, at somewhat lower temperatures, the
vacancies are less mobile and have not enough time to reach sinks like grain
boundaries, but can still cover distances much larger than their average
separation. This means that divacancies, trivacancies and so on can still form
 up to large clusters of vacancies, either in the shape of a small hole or
void, or, in a twodimensional form, as small dislocation loops. Until they
become completely immobile, the vacancies will be able to cover a distance
given by the diffusion length L (which depends, of course, on how
quickly we cool down). 


In other words, at intermediate temperatures
small vacancy clusters or agglomerates can be formed. Their maximum size is
given by the number of vacancies within a volume that is more or less given by
L^{3}  more vacancies are simply not available for any
one cluster. 


Obviously, what we will get depends very much on
the cooling rate and the mobility or diffusivity of the vacancies. We will
encounter that again; here is a
link looking a bit
ahead to the situation where we cool down as fast as we can. 

It remains to find out which mix of
single vacancies and vacancy clusters will have the smallest free enthalpy,
assuming that the total number of vacancies  either single or in clusters 
stays constant. This minimum enthalpy for the specific restraint (number of
vacancies = const.) and a given temperature then would be the local
equilibrium configuration of the system. 

How do we calculate this? The
simplest answer, once more, comes from using the the massaction law. We
already used it for deriving the
equilibrium concentration of the divacancies. And we did not assume that the vacancy concentration was in
global thermal equilibrium! The mass action law is valid for any starting concentrations of the ingredients  it
simply describes the equilibrium concentrations for the set of reacting
particles present. This corresponds to what we called local equilibrium
here. 


The reaction equation from subchapter
2.2.1 was 1V + 1V
Û V_{2} and in this case this is a
valid equation for using the mass action law. The result obtained for the
concentration of divacancies with the single vacancy concentration in global thermal equilibrium was 


c_{2V} 
= (c_{1V})^{2} · 
z
2 
· exp 
DS_{2V}
k 
· exp 
B_{2V}
kT 




Don't forget that concentrations here are defined
as n/N, i.e. in relative units (e.g. c = 3,5 ·
10^{–5}) and not in absolute units, e.g. c = 3,5
· 10^{15} cm^{–3}. 


For arbitrary clusters with
n vacancies (1V + 1 V + ... + 1V Û V_{n}) we obtain in an analogous way
for the concentration c_{nV} of clusters with
n vacancies 


c_{nV} 
= (c_{1V})^{n} · 
a 
· exp 
DS_{nV}
k 
· exp 
B_{nV}
kT 




with B_{nV} = average
binding energy between vacancies in an ncluster,
c_{1V} = const. concentration of the vacancies (and no longer the thermal equilibrium concentration
!), and a = number of possible
"orientations" of the ncluster divided by the
indistinguishable permutations. The value of a will depend to some extent how we arrange
n vacanies: in a row, on a plane or threedimensionally  but we won't
worry about that because the other factors are far more important. 

The essential point now is to realize
that these equations still work for local equilibrium! They now describe the
local equilibrium of vacancy clusters if a
fixed concentration of vacancies is given.
The situation now is totally different from
global equilibrium. If we consider divacancies for example, we have: 

