UPDATED

Something I learned that I have found invaluable in describing and understanding microstructural materials phenomena, especially in metallurgical systems, is that the answer is almost always that the system is lowering the Gibb's free energy.

In this example, what I suspect is a combination of things. You have to atomic ratios, 1:3 and 2:3, and both compositions are attributed to a unique crystal structure, such that the atomic ratio within the unit cell is 1:3 or 2:3 respectively. The resulting decrease in entropy of forming these phases, and therefore the decrease in Gibbs free energy, is maximized when some linear combination of these two are formed. At a specific composition, you have a split, determined by lever rule, that will tell you how much of each you form.

Now, you mentioned quenching in particular. Keep in mind that with quenching, you have the issue of rate of quench. The phase diagram is simply a series of points of data plotted together at equilibrium. It's all thermodynamics, no kinetics. As soon as you change the rate of your cooling, you are introducing a parameter of kinetics that will not be captured by the model of the phase diagram. Take care when using phase diagrams to make conclusions for quenching experiments. Instead, perhaps use a TTT diagram.

UPDATED

Something I learned that I have found invaluable in describing and understanding microstructural materials phenomena, especially in metallurgical systems, is that the answer is almost always that the system is lowering the Gibb's free energy.

In this example, what I suspect is a combination of things. You have to atomic ratios, 1:3 and 2:3, and both compositions are attributed to a unique crystal structure, such that the atomic ratio within the unit cell is 1:3 or 2:3 respectively. The resulting decrease in entropy of forming these phases, and therefore the decrease in Gibbs free energy, is maximized when some linear combination of these two are formed. At a specific composition, you have a split, determined by lever rule, that will tell you how much of each you form.

Now, you mentioned quenching in particular. Keep in mind that with quenching, you have the issue of rate of quench. The phase diagram is simply a series of points of data plotted together at equilibrium. It's all thermodynamics, no kinetics. As soon as you change the rate of your cooling, you are introducing a parameter of kinetics that will not be captured by the model of the phase diagram. Take care when using phase diagrams to make conclusions for quenching experiments. Instead, perhaps use a TTT diagram.

@ V. Mambakkam: could you please formulate your contribution without mixing the term LATTICE which describes a mathematical description of a translation symmetry only and the term STRUCTURE which explains the atomic positions?

perhaps therefore "..such that the lattice has the atomic ratio 1:3 or 2:3..." sounds a bit weird.

I know that even in several books the term lattice and structure is mixed but this doesn't mean it is acceptable. It is hard to follow if terms are mixed up, and many things cannot be explained anymore because you always have to express which meaning of lattice you are using.  

@Gert Nolze My apologies. I am still a student and you caught me on a rookie mistake. I will reword my previous contribution to reflect the error.

Hi,

not sure I understand the question, but here is the answer to what I guess the question is: just above the peritectic line your alloy of overall composition of 43%Ni and 57% Al is made of two phases: a liquid with a composition of 70% Al 30% Ni and a solid Ni2Al3 with a composition of 48% Ni and 52 % Al. (values in weight percent!). At the peritectic reaction the liquid and the solid react to form a new solid Al3Ni with a composition of 42% Ni and 58% Al. since you are with the global composition to the right of the Al3Ni line !at equilibrium! not all solid Ni2Al3 will be consumed before there is no liquid left to react. Hence you will end up with a mixture of Al3Ni and Ni2Al3.

You seem puzzled by the fact that a liquid that contains 70%Al and 30%Ni should form two solids that have lower Al content than the liquid. The key to your question is hence that it is not just the liquid that transforms but also a large part of the solid rich in Ni that enters the reaction. The relative quantities of liquid and solid (before the reaction) and the two solids (after the reaction) can be determine by the lever rule.

A further issue is that at technical cooling rates you hardly ever reach equilibrium. Hence you may end up with some (divorced) eutectic (Al+Al3Ni) that forms in the spaces between areas that are formed by a Ni2Al3 core and a Al3Ni shell. The reason for this is that as you keep cooling down the liquid gets more and more enriched in Al (since the solid state diffusion necessary to convert the Ni2Al3 core is slow) and eventually solidifies at teh eutectic point (far to the left). If the cooling speed is such that the interfiber distance in the eutectic is of the same order of magnitude as the distances of the core shell structures you will end up with a divorced eutectic i.e. the Al3Ni phase will grow on the existing Al3Ni. and the aluminum will form in the spaces in between.

Hope that helps

@Ludger Weber

Thanks for exploring my question, it may help other to understand, it's right what i'm asking. I get some solution from your answer. 

For my curiosity, I want to know what is that magical peritectic temperature(854 C) at [[roughly numerical value I'm giving you to understand what I'm saying]] which liquid (20%) will diffuse and join with some solid Ni2Al3 (30%) and make new intermettalic NiAl3 (50%) which is richer in Al than liquid.

1) Above this temperature things happen like reduce liquid portion (become richer in Al side) and increase Ni2Al3 alloy ,but after peritectic point it suddenly starts to  form NiAl3, So why at this magical temperature?

2) What properties change at that composition of Ni and Al in liquid which leads to form intermettalic alloy?

The electron-electron interactions and atomic interactions at nano scale decides what to form actually in any systems. Moreover, such phenomena are predominant in solid system, whereas in liquid you cannot expect it. such interactions are closely controlled by compositions, temperature and pressure. I suggest you to go through solution thermodynamics of materials, you may get clue or idea about compound formation and solution formation and their respective stability. It is always better to have clear idea about solution thermodynamics before going to understand any system's phase diagram.

Sorry for the late reply: the "magical" temperature comes from a global minimum Gibbs free energy analysis. This is not really an answer for the "why" because there is not a simple link between structure and atomic arrangement (Density functional theory might give some clues but often energy differences are very small and predicted transformation temperatures are easily 100K off.) Hence if you are fine with the observation that every phase can be associated with a Gibbs free energy function the answer is there.

I recommend you to read this paper:

http://ijens.org/Vol_18_I_05/180505-3737-IJMME-IJENS.pdf

Best regards