You might take a look at this video:

(not mine but I think it illustrates the idea quite well)

https://www.youtube.com/watch?v=n7DPiuHllZE

Dear sir Karl Michael Kraemer ,

Thanks for the kind reply and reference of the video. The video talks about "interpretation of a tie line in ternary diagram" and drawing of a tie line based on "conjugate line".

My requirement is to draw the tie line itself in the condition when conjugate curve is not provided already.

Dear all,

If you want to draw the direction of the tielines in a ternary phase diagram (isothermal section), you have to perform some appropriated post-mortem analyses of the samples to determine the composition of the phases at equilibrium. These composition will determine the extremity of the line.

They can also be determined experimentally from isopiestic measurements. Of course, these tielines only have meaning in two-phases region were chemical potentials of both phases are fixed along the line.

In the case of a three phase equilibrium (tie-triangle), the compositions are determined by the apex of the triangle. All the chemical potentials are fixed inside the triangle due to the Gibbs phase rule.

If you can perform thermodynamic calculations using the Calphad method, the tielines are automatically calculated.

So are you in a 2 or 3 phase equilibrium ?

Regards

Dear sir Gossé Stéphane,

Thanks for the nice reply. Insights provided by you are of much utility and are much appreciated.

I wish to calculate tie lines for a 3 phase region.

Thanking you.

In a two phase equilibrium, a tieline represents an iso-chemical potential line at fixed (T,p). The variance of the ternary system is equal to 1.

Since all the variables are fixed inside a three phase equilibrium, the variance of the ternary system is equal to zero. There is no degree of liberty inside the triangle. So all the chemical potentials are equal: there are no tielines inside a three phase equilibria.

If you want to determine the amount of each phase for a specific composition inside the triangle (different from the chemical potential), please see: Chemical thermodynamics of materials, C. H. P. Lupis, North-Holland, 1983.

Regards

Dear Adhidesh,

Dr. Stephane has provided a very nice explanation of tie lines in context of degrees of freedom. A tie line is specifically for a two-phase region, regardless of the dimension of the phase diagram (binary/ternary/quaternary). So in a 2-phase region, simply proceed as in the binary case.

Coming to 3-phase region, as mentioned in the previous answer, there is no tie line. But you will notice that it is triangular, has single phase regions at the corners and the sides are straight (why?- an interesting question).Now, since you have a 3-phase triangle, say ABC, and the phases at each corner are known, you can proceed geometrically. Suppose the point of interest is D. Draw lines from points the corners (A,B,C) through D to the opposite side. Say they intersect the sides at E,F,G respectively. Now you can use a lever-rule type formulation to get:

%A at D = 100 * DE/AE, %B at D =100* DF/BF, %C at D= 100*DG/CG

This way you have the phase composition at D in terms of A,B,C.

Do note though, that there is nothing thermodynamic about this, this is just geometrical construction; you can use it for any similar diagram. But do take care that this calculation is performed on an isothermal section only, not on an isopleth.

Dear Aashranth ,

The answer you have provided gives us the information regarding what composition (overall) we have at any given point inside the triangular composition diagram of three components, but what I am asking is about drawing tie lines inside that triangular diagram on the basis on various phases present.

Thanks for the reply.

No, I'm talking about the composition of different phases; A, B, C represent phases in the above answer, not the components. I should have said 'phase A', 'phase B', 'phase C' for clarity. Alternately you could think of them as A-water, B-benzene, C-ether (arbitrary choice) if that helps. Or ferrite-austenite-liquid if you're looking at metals.

Getting the overall chemical composition in terms of components is quite direct and independent of whether its a 2-phase or 3-phase region. That question wouldn't warrant this much discussion. :)

To re-iterate, the triangle ABC (=water-benzene-ether, or ferrite-austenite-liquid) referred to is a specific 3-phase region WITHIN a triangular (isothermal) section, not the entire isothermal section. There are no tie-lines in this 3-phase region, for the reason mentioned by Dr. Stephane. The lines AD, BE, CF are similar to tie-lines geometrically, but they are not actually tie-lines in the thermodynamic sense.

Sorry for the ambiguity in the previous answer.

Dear Adhidesh,

Unfortunately, you can not predict those tie lines that you see in ternary phase diagrams, they are experimentally determined by different techniques depending on what kind of sample you have (NMR spectroscopy, low-angle x-ray scattering and etc...). Usually, when the two phases region is determined, the tie lines are determined as well. I think your question was: can i draw the tie limes (by geometry or any other mathematical means) if i know the limits of the two phases region? The answer is no, those lines are experimentally determined by analyzing the composition of the two phases.

For example, if you know the limits of the two phases, what you do is take many samples inside the two phases region and determine analytically the composition of each phase: if you have one phase with 25% of A, 50% of B and 25% of C and the other phase with 60% of A, 30% of B and 10% of C, you now have two points that can be connected to form one tie line.

Once you have sufficiently tie lines inside your two phases region, other tie lines can be predicted (conjugate lines).

Best,

Thiago.