Tie Line
A binary phase diagram between
two elements A and B. When an
alloy is present in a two phase
region, a tie line at the
temperature of interest fixes the
composition of the two phases.
This is a consequence of the
Gibbs phase rule, which provides
for only one degree of freedom.
Lever Rule
The Lever Rule is used to calculate the weight % of the phase in any two-phase
region of the Phase diagram (and only the two phase region!)
In general:
• Phase percent = opposite arm of lever
x 100
total length of the tie line
For example,
weight fraction, %X s =
wo − wl
x 100
ws − wl
(of solid phase)
ws − wo
weight fraction, %X l =
x 100
ws − wl
(of liquid phase)
Lever Rule
When a material solidifies it
does not have a constant
concentration throughout the
material but there will be
concentration gradients,
which will significantly alter
the properties of the material.
This is an important concept.
In the example of Cu and Ni, the
concentration of Ni that freezes
at 1270 oC is 50 wt%, at 1250 oC
is 45 wt% and 1200 oC is 40
wt%.
Lever Rule
Calculate the amount of α phase and L phase present in a Cu 40% Ni alloy at 1250 C
In general:
• Percent α phase = (% Ni in alloy) – (% Ni in L)
% Ni in L - % Ni in α
weight fraction, %X s =
(of solid α phase)
40 − 32
x 100 = 62 %
45 − 32
weight fraction, % L = 38 %
(of liquid phase)
x 100
Solidification of a Solid-Solution Alloy
The change in structure
and composition of a Cu40% Ni alloy during
equilibrium solidification
showing that the liquid
contains 40% Ni and the
first solid contains Cu-52%
Ni. At 1250 C,
solidification has advanced
and the phase diagram tells
us that the liquid contains
32% Ni and the solid
contains 45% Ni, which
continues until just below
the solidus, all of the solid
contains 40% Ni, which is
achieved through diffusion.
Nonequilibrium Solidification and Segregation
When cooling is too
fast for atoms to
diffuse and produce
equilibrium
conditions,
nonequilibrium
concentrations are
produced. The first
solid formed contains
52% Ni and the last
solid only 25% Ni
with the last liquid
containing only 17%
Ni. The average
composition of Ni is
40% but it is not
uniform.