Reading an Isomorphous Diagram and the Lever Rule

From vocabulary to a working map

In guide 1 you learned the words. A component is a pure ingredient of the alloy; a phase is a region that is uniform in both structure and composition; and the solubility limit is how much of one thing will dissolve into another before a second phase must appear. Now we put those words onto a real map — temperature up the side, composition across the bottom — and learn to read not just which phases exist, but exactly how much of each. The gentlest map to start on is the binary isomorphous system.

"Isomorphous" means there is a single solid phase across the entire composition range: the two metals dissolve in each other in all proportions, and no solubility limit is ever reached. The classic example is copper-nickel. Copper and nickel are both face-centred cubic, their atomic radii differ by only about 2.5 percent, and they sit side by side in the periodic table — so they satisfy the Hume-Rothery rules beautifully and form a single substitutional solid solution (call it alpha) at every ratio from pure Cu to pure Ni. That complete mixing is exactly what makes the diagram so simple.

Three regions, two lines

The isomorphous diagram has two curves that split the space into three fields. The upper curve is the liquidus: above it, everything is molten liquid, written L. The lower curve is the solidus: below it, everything has frozen into the solid solution alpha. Between the two curves is a lens-shaped lagoon where liquid and solid coexist — the two-phase region, L + alpha.

To find WHICH phases are present, you just locate your point (its composition and temperature) and see which field it lands in. Land in L, you have one phase, all liquid. Land in alpha, you have one phase, all solid. Land in the lens, you have two. Notice a consequence: unlike a pure metal, which melts at a single sharp temperature, an alloy melts over a RANGE — from the solidus up to the liquidus — which is why a freezing alloy passes through a soft, mushy stage.

 T |          L  (all liquid)
   |     ..-----------------.. liquidus
   |    /                       
   |   /       L + alpha         
 T1|--+--------+--------+---   <-- tie line
   |  CL       C0       Ca       
   |  |<--a--->|<--b--->|        
   |   \                        
   |    ``-----------------`` solidus
   |           alpha (all solid)
   +--------------------------------> wt% B
      20       40       60

  fraction of alpha (solid) = a/(a+b) = (C0 - CL)/(Ca - CL)
  fraction of liquid        = b/(a+b) = (Ca - C0)/(Ca - CL)
  worked: C0=40, CL=20, Ca=60  ->  a=20, b=20  ->  50% solid, 50% liquid

The tie line: what each phase is made of

Here is the point beginners miss. Inside the two-phase lens, the overall composition you started with does NOT tell you the composition of either phase. The liquid and the solid have different compositions — from each other, and from the average. To read them off, draw the tie line: a horizontal line at your temperature, stretching across the lens from the liquidus to the solidus. Where it touches the liquidus, drop straight down to the axis — that is the liquid's composition, CL. Where it touches the solidus, drop down — that is the solid's composition, Ca.

Why does fixing the temperature pin those two compositions to single, definite values? Because of the Gibbs phase rule you met in guide 1. In a binary alloy at fixed pressure the reduced rule gives degrees of freedom F = C - P + 1; inside the two-phase region C = 2 and P = 2, so F = 1. One degree of freedom means once you choose the temperature, nothing else is free — the compositions of both phases are locked. That is exactly why the tie-line endpoints are not a matter of choice.

A concrete read. Take an A-B alloy whose overall composition is 40 wt% B, at a temperature where the tie line runs from 20 wt% B on the liquidus to 60 wt% B on the solidus. Then the liquid is 20 percent B and the solid is 60 percent B — one leaner than the 40 average, one richer. In general the solid is richer in the higher-melting component, because that is what wants to freeze first. Good — we now know WHAT the two phases are. But in what amounts?

The lever rule: how much of each

The tie line told you WHAT; the lever rule tells you HOW MUCH. Picture the tie line as a see-saw with the fulcrum sitting at the overall composition C0, and a phase hanging off each end. For the see-saw to balance, each phase's amount times its arm length must match the other's — so the phase whose end is CLOSER to the fulcrum is the one you have MORE of, just as a heavier child must sit nearer the pivot. That single picture is the whole rule.

In symbols, fraction of solid = (C0 - CL)/(Ca - CL) and fraction of liquid = (Ca - C0)/(Ca - CL). The subtlety that trips everyone: to get a phase's amount you use the arm on the OPPOSITE side of C0. Work the earlier numbers — C0 = 40 between CL = 20 and Ca = 60. Fraction solid = (40 - 20)/(60 - 20) = 20/40 = 0.5; fraction liquid = (60 - 40)/40 = 0.5. Half and half, because C0 sits dead centre. Now slide C0 to 30, nearer the liquid end: fraction solid = (30 - 20)/40 = 0.25, fraction liquid = 0.75 — nearer an end means more of the phase that lives at that end, exactly as the see-saw promised.

1. Check you are in a two-phase region. If your point sits in a single-phase field (L or alpha), that phase is simply 100 percent and there is no tie line to draw.
2. Draw the horizontal tie line across the region at your temperature, and read the two endpoint compositions CL and Ca off the bounding curves.
3. Mark your overall composition C0 on that same line — it must fall between CL and Ca.
4. Fraction of a phase = length of the OPPOSITE arm divided by the total tie-line length. The solid fraction uses the arm reaching toward the liquid end, and vice versa.
5. Multiply by the total mass for actual amounts. Because the axis is weight percent, these are WEIGHT fractions — convert with densities if you need volume fractions.

Cooling for real: coring, and why the map is equilibrium

Everything above assumes equilibrium — cooling so slow that diffusion keeps every grain uniform all the way down. Real castings cool in seconds. The first solid to appear is rich in the high-melting component (nickel, in Cu-Ni); as the temperature falls the solidus says the solid SHOULD become more copper-rich, but atoms buried inside a growing grain cannot diffuse fast enough to even things out. So each grain freezes with a layered gradient — a nickel-rich core and a copper-rich rim. This is called a cored structure, and it usually grows in tree-like dendrites.

Coring matters. The copper-rich rims melt at a lower temperature than the equilibrium solidus predicts, so a cored casting can begin to melt locally well below the temperature the diagram promises — a genuine hazard during service or a later heat treatment. The cure is a homogenization anneal: hold the part hot for long enough that diffusion finally levels the composition out. This is your first hard taste of the honesty rule for every phase diagram: it maps EQUILIBRIUM only, and it says nothing about how fast you actually cooled.

The isomorphous system is the gentle case — one solid phase everywhere, no reactions to memorise. Most alloy pairs are not so accommodating: cross the solubility limit and a second solid appears, giving the far richer eutectic diagrams and the striking layered microstructures of guide 3. But the two tools you built here — the tie line for compositions, the lever rule for amounts — work on EVERY two-phase region of EVERY diagram, right through to the iron-carbon diagram at the end of this rung. Learn them once; use them forever.