# What is the topology of a phase diagram?

This question is cross-posted from phys

Looking at various two-variable phase diagrams I was struck by that on every one I have seen so far all the phases formed simple connected regions; see, for example the phase diagrams of $$\ce{H2O}$$ or of $$\ce{Fe-Fe_3C}$$ in 1. Every phase is a connected region, no two disjoint pieces correspond to the same phase. Is this really true in general and if so why?

Going further, let us consider a three-variable phase diagram, say $$p, T, y$$, where $$y$$ maybe electric or magnetic field intensity, or some other controllable extensive/intensive variable. Is it possible that now a 3d region occupied by some phase is a kind of tube that makes a half turn so that if we fix, say, $$y$$ then the resulting 2d section would have two disjoint $$p,T$$ regions of the same phase? Can such phase "tube" exist or by some thermodynamic-geometric rule is it excluded?

A possibly more difficult question if a phase can form an annular region, ie., homeomorphic to an annulus while surrounding another phase homeomorphic to a disk?

In short, what is the topology of the phase diagram and its sections?

• There are any number of binary phase diagrams where the same crystal structure is observed in several discrete regions. For some of them, it is known that those regions can be connected using, e.g., a third element. Fe is a common case, since the fcc $\gamma$ phase sits between regions $\alpha$ and $\delta$ bcc. So, with, say, Fe-Si the bcc phase exists continually around the fcc phase, while in Fe-Ni the fcc phase extends across the entire phase diagram. – Jon Custer Oct 15 '18 at 13:22
• @Jon_Custer Thank you and very interesting but is it fair to say that $\alpha-Fe$ and $\delta-Fe$ are the same phase when the former is ferromagnetic and hence has domains while the latter is above the Curie temperature and is not magnetic at all? – hyportnex Oct 15 '18 at 13:42
• As mentioned, in many binary diagrams the $\alpha$ and $\delta$ regions are connected, and the only question is where the Curie temperature occurs as a function of composition. But, above or below the Curie temperature the phase is considered the same. – Jon Custer Oct 15 '18 at 13:48
• Cross-posting the same question on two sites is not recommended. – Jan Oct 16 '18 at 15:37

Particularly since Fe was mentioned, I'll focus there. For pure iron, it is well known that $$\alpha$$-Fe, the stable phase at room temperature, is bcc. This is stable up to 1184K where it transitions to fcc $$\gamma$$-Fe. In turn, this transforms back to bcc $$\delta$$-Fe at 1667K before melting at 1810K. Yet, both $$\alpha$$ and $$\delta$$ have a bcc crystal structure.
If one looks at the Fe-Zn phase diagram: one sees that the region of bcc phase, enclosed in dark blue, fully encompasses the green sliver of fcc $$\gamma$$. Using the Zn concentration as a free parameter, one can go from $$\alpha$$ to $$\delta$$ without changing phase.
In contrast, in the Fe-Pd phase diagram: one sees a different picture. Here, Fe and Pd show complete miscibility in the fcc phase across the entire diagram. This means you can't go from $$\alpha$$ to $$\delta$$ bcc phase without going through the fcc phase. Yet, one can see that, by trading off Zn for Pd, one could manage to get around the fcc phase by some path in composition space.
• You seem to be saying that the phases $\alpha$ and $\delta$ are identical because they have the same bcc crystal structure. If they are really the same phase then you have answered part of my question with a valid counter example but then why would we even bother distinguishing them? This is not just a matter of verbiage, are they really the same phase? – hyportnex Oct 15 '18 at 14:26
• The $\alpha$/$\delta$ labeling is more historic than anything else, dating to early x-ray measurements of elements - as the temperature was raised the transition from bcc to fcc was seen, then from fcc to bcc and each was given their own greek letter (not quite sure what $\beta$ was mind you!). Many modern phase diagrams just call it all bcc-Fe and be done with it. If it is Fe atoms in a bcc crystal structure, it is bcc-Fe. – Jon Custer Oct 15 '18 at 14:34
• Is the behavior in an external magnetic field the only physical difference between the $\alpha-Fe, \beta-Fe$ and $\delta-Fe$? According to en.wikipedia.org/wiki/Allotropy $\beta-Fe$ is the paramagnetic bcc-Fe between 770 and 912 °C and $\alpha-Fe$ is the ferromagnetic bcc-Fe below 770°C. – hyportnex Oct 15 '18 at 15:11
• Looking at modern references such as W. Xiong et al., CALPHAD 33 433-440 (2009), they call the whole bcc region $\alpha$Fe, but indicate the magnetic transition temperature across the phase. IT is just an evolution of how it is described. – Jon Custer Oct 15 '18 at 15:31