Page 334 - Materials Science and Engineering An Introduction

306 • Chapter 9 / Phase Diagrams

If the composition and temperature position is located within a two-phase region,
things are more complex. The tie line must be used in conjunction with a procedure that
lever rule is often called the lever rule (or the inverse lever rule), which is applied as follows:
1. The tie line is constructed across the two-phase region at the temperature of the alloy.
2. The overall alloy composition is located on the tie line.
3. The fraction of one phase is computed by taking the length of tie line from the
overall alloy composition to the phase boundary for the other phase and dividing
Tutorial Video: by the total tie line length.
Phase Diagram
Calculations and 4. The fraction of the other phase is determined in the same manner.
Lever Rule 5. If phase percentages are desired, each phase fraction is multiplied by 100. When
The Lever Rule the composition axis is scaled in weight percent, the phase fractions computed
using the lever rule are mass fractions—the mass (or weight) of a specific phase
divided by the total alloy mass (or weight). The mass of each phase is computed
from the product of each phase fraction and the total alloy mass.

In the use of the lever rule, tie line segment lengths may be determined either by
direct measurement from the phase diagram using a linear scale, preferably graduated in
millimeters, or by subtracting compositions as taken from the composition axis.
Consider again the example shown in Figure 9.3b, in which at 1250 C both a and
liquid phases are present for a 35 wt% Ni–65 wt% Cu alloy. The problem is to com-
pute the fraction of each of the a and liquid phases. The tie line is constructed that
was used for the determination of a and L phase compositions. Let the overall alloy
composition be located along the tie line and denoted as C 0 , and let the mass fractions
be represented by W L and W a for the respective phases. From the lever rule, W L may
be computed according to

S
W L = (9.1a)
R + S
or, by subtracting compositions,

Lever rule expression
for computation of W L = C a - C 0 (9.1b)
liquid mass fraction C a - C L
(per Figure 9.3b)
Composition need be specified in terms of only one of the constituents for a binary al-
loy; for the preceding computation, weight percent nickel is used (i.e., C 0 = 35 wt% Ni,
C a = 42.5 wt% Ni, and C L = 31.5 wt% Ni), and

42.5 - 35
W L = = 0.68
42.5 - 31.5
Similarly, for the a phase,

R
W a = (9.2a)
Lever rule expression R + S
for computation of
a-phase mass fraction
(per Figure 9.3b) = C 0 - C L (9.2b)
C a - C L
35 - 31.5
= = 0.32
42.5 - 31.5