This document discusses phase transformation and freezing and stable interface freezing. It first discusses assumptions about nucleation rate and freezing of pure metals. It describes that nucleation rate depends on the average number of critical clusters and diffusion of molecules to the cluster. It then discusses stable interface freezing, noting that if the temperature gradient is linear and perpendicular to the interface, the interface will maintain a stable planar shape as it moves forward. However, if facets have different accommodation factors, the interface will develop a series of closed packed planar steps with a curved shape due to uneven growth rates across each facet.

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3. Rafaqat Ali
(MME-10M-)
Muhammad Akbar
(MME-10M-18)
Muhammad Rameez
(MME-10M-07)
Muhammad Shafqat
(MME-10M-11)

4. Freezing:
Consider the freezing of pure metal. Lets consider
following assumptions…
1st assumption is rate of nucleation, does not
influence the attachment of atoms to the surface of
solid.
2nd,we assume that solid embryos have
simple spherical shape so that atoms attached to
nucleus from all possible directions.

5. Nucleation rate
Rate of nucleation
The nucleation rate, I, depends on the average number of
critical clusters, n* and the diffusion of molecules to the
cluster, .
where the average population of critical nuclei is

6. where:
ΔG* is critical free energy needed corresponding to
that of the critical radius.
N is the number of potential nucleation sites per unit
volume
kB is the Boltzmann constant

8. Once the critical mass of nuclei is reached, the system
nucleates at point C in the figure and releases its latent
heat faster than heat is being removed from the system.
In pure water, the time line from D to E . Fast freezing
rates promote the formation of many small ice crystals
during this period. The partially frozen mixture will not
cool until all of the "freezable" water has crystallized;
hence, the line DE for pure water occurs at constant
temperature. The freezing time is usually defined as the
time from the onset of nucleation to the end of the
crystal growth phase. After crystallization is completed,
the temperature drops from E to F as sensible heat of
ice is removed.

9. Stable Interface Freezing:
(temp. gradient advanced of interface increases)
One factor that has a strong effect on the shape of
interface on a macroscopic scale is removal of heat of
fusion from interface.
Let it be assumed that temp. gradient is linear &
perpendicular to interface. Under these conditions,
interface is believed to maintain a stable planer shape &
move forward as a unit.
If by chance a close-packed in exactly right(90*)
position is very small, also consider a case where
crystallographic plane of high atomic density parallel to
interface.

10. Heat flow direction
Area of lower
accommodation factor
Area of higher
accommodation factor
solid liquid

11. Stable Interface Freezing:
(temp. gradient advanced of interface increases)
Fig. shows the interface develop a series of closed
packed planar steps.
Each facet has an inclination in the heat flow
direction,temp. cannot uniform over its entire area.
Because of the rising of temp. is advance of interface,
those points of each facet which are most advanced
will be in contact with hotter liquid than those
positions to the rear.
As rate of Growth is function of degree of super
cooling, it is not possible for facet to maintain a
strictly crystallographic surface & grow with constant
velocity.

12. Stable Interface Freezing:
(temp. gradient advanced of interface increases)
As a consequence curved shape interface is
observed.Also most hotter/advanced position of
facet has higher accommodation factor while most
retard or cooled portion correspond to slow
growing/low accommodation factor.