Symmetry of solids/Topic 15

From WolfWikis

Phase diagrams with continuous transitions

For general background on phase diagrams see here.

Phase gaps and first order transitions

A simple binary diagram with three points of contact

In a typical pseudo-binary temperature-composition (T-X) phase diagram two phases related by a first order transition can only be at equilibrium in a single point. I.e. there is only one composition for which phase 1 will transform into phase 2 without a third phase present when e.g. heating up. Another way of saying this is that homogeneity ranges touch each other in single points only. In the first figure there are three such points: the melting points of the three 'pure' phases, where A, B and C touch the broad liquid homogeneity range. The homogeneity ranges of the three phases are shown as vertical lines, assuming that they are very narrow.

Otherwise all homogeneity ranges are separated by two-phase regions, aka phase gaps. This fact is a direct consequence of the phase rule.

In the case of a peritectic there is even less contact as shown in the peritectic diagram on the left: Only the melting points of the pure compounds at the two extremes of the diagram now touch the homogeneity region of the melt. The compound's homogeneity range is entirely separated from other homogeneity ranges (shown in color) by two phase regions ('phase gaps', shown in white). (In this diagram the width of the ranges for the three low temperature solids was assumed to be a bit broader)

A peritectic diagram

Transition lines and the entropy issue

If the transition is second order or continuous these limitations no longer hold. A continuous transition is in a sense 'internal', it takes place as a distortion inside a phase rather than between two phases with potentially different compositions. There is never more than one phase present at the time. Thus, the set of points for which the symmetry changes may well form a line in the T-X diagram. We could think of such a line as a set of critical points as a function of composition: T_c(X).

Usually the higher symmetry 'parent' phase is to be found at the high temperature end of the line. There is a sound thermodynamic reason for that. Symmetry elements typically take one atom into an identical atom. This means that if we swap the atoms we get the same object back and in the language of statistical thermodynamics that means that the system has two equivalent indistinguishable realizations, i.e. W=2. This adds to the entropy S of the system because S=k.ln(W). Thus higher symmetry (more rotations, more translations, the latter being equivalent to smaller unit cells) tend to have more entropy and therefore be favored at higher temperatures.

Critical points

Hypothetical phase diagram with continuous transitions and two kinds of critical points

Tricritical points

If a transition is Landau-allowed, i.e. it if fulfills all the requirements shown above a continuous transition may occur but it must not, depending on the sign of the fourth order term. That in turn means that in one part of the diagram the transition may be continuous but in an other part it may become first order. In the latter case we must have phase gaps (apart from single points touching) and so at the transition from continuous to discontinuous one can find a critical point where the phase gap opens up. This is called a tricritical point. This is shown as T_c¹ in the diagram to the right. Notice that strictly speaking we cannot apply Landau theory to the left of this point because the transition is not continuous. However, we can still think of the two phase as a parent-daughter pair in the Landau sense of the word and the discontinuity is likely to be rather 'weak'.

'Sibling' points

There is a second kind of critical point^[1] possible if the system has two daughter solutions that transform as the same irrep. The transition between the two daughters is usually not Landau-allowed because they are not related by a sub-group. (They are sisters, not parent-child). Thus there must be a phase gap between them. The figure on the right shows a hypothetical diagram of a system that has both kinds of critical points.^[2] On the left edge parent and daughter II undergo a regular ('weak') first order transition for the pure compound. This is why the two colored regions touch in a single point at the edge. When changing the composition towards the right, a phase gap P+II is shown as demanded by the first order nature of the transition. However, it closes at the tricritical point T¹_c, where the transition becomes second order. Further to the right the transition towards the parent phase is a continuous one and remains so to the other end of the diagram but at T²_c (a 'sibling' point) a different daughter symmetry becomes stable at lower temperatures. This opens up a phase gap between the two daughters.

How to study continuous transitions

For solid-solid transitions a good way to study a continuous transition is to measure the XRD pattern as a function of temperature. If the transition is continuous but the symmetry does change often it is seen that lines in the pattern split up smoothly when lowering the temperature (or merge while going up). Of course it is impossible to 'prove' that the split occurs without any discontinuous 'jump'. In other words you can prove something is first order by demonstrating a finite jump, but you cannot prove it is continuous, because someone else may always come along with a better measurement that shows that you overlooked a teeny weeny little jump.

In thermal analysis the usual DSC-peak due to the discontinuity in the enthalpy is missing so that the signal is more more subtle. In principle the heat capacity is discontinuous which ought to give a more or less sudden baseline shift. In practice the baseline often shows a sort of lambda shape as observed for the famous transition from superfluid He to ordinary liquid Helium. Much could be said about that that would be far beyond the scope of this short course.

Ag-Au-S

An example of a second order phase transition line can be found in the pseudo-binary system Ag₂S-Au₂S. This system is only stable up to AgAuS and at low temperature it has three discrete solid phases with relatively narrow homogeneity ranges. We can conveniently call them Ag₂S and Ag₃AuS₂ and AgAuS. At relatively low temperatures (the lowest one being the eutectic temperature of 113^oC) the silver atoms become highly mobile in the lattice and we find a continuous series of solid solutions that spans the whole range. However they have a different symmetry on the left than on the right and there is a second order phase transition line between them that leans a little to the right. The one on the left is the parent Im3m symmetry the one on the right the Pn3m daughter. The higher symmetry phase Im3m becomes stabilized at higher temperatures as one would expect from the symmetry-gives-entropy argument made earlier, so the line leans the right way!

In this particular example the second order transition line impinges upon one of the phase gaps at lower temperatures. It can be shown theoretically that this should induce a little kink in the line that forms the lower temperature limit of the cubic homogeneity ranges.

The Ag-Au-S pseudo-binary ^[3]The temperature range is from 0 - ca. 300 C. The materials melt at much higher temperatures (>800). Notice the lack of a phase gap between the two cubic solid solution ranges

Other types of systems

So far we have considered the phases to be true solids with full 3D translation symmetry but it should be noted that Landau theory does not require the symmetry to be translational or the material to be solid. The theory can also be applied to e.g. transitions between liquid crystal phases. ^[4]. What holds for crystalline solids also holds for liquid crystals: Some transitions are Landau-allowed others are not. For example, the transition from simple nematic liquid crystal to isotropic liquid is typically weakly first order at least in the bulk liquid crystal without interference from external fields (such as electrical or magnetic fields or even the van der Waals interaction with a container wall). That means that in a T-X phase diagram we should expect phase gaps except at special points.

From an experimental point of view however it is far more difficult to determine whether a sample is single phase or a two-phase mixture for such systems so that the presence of phase gaps is much more difficult to demonstrate for liquid crystals. Other experimental methods e.g. DSC are also more difficult to interpret because heat effects are much more subtle for liquid crystals. This is why phase gaps are usually not shown in diagrams involving liquid crystals. This does not necessarily imply that they are absent, but just that they have not been identified experimentally.

References and notes

1. ↑ To my knowledge it has never been given a special name: if it has, I gladly retract the 'sibling' in the title
2. ↑ To my knowledge no such system has ever been identified in a single diagram, but then most people solving phase diagrams are not very aware of the possibility of continuous transitions and what could do to a phase diagram. Parent structures with more than one daughter have been shown to exist but often one daughter is found for one compound and the other for another. Or they are found in the same diagram but without continuous transitions. See e.g.: :Group-theoretical treatment of the sphalerite-chalcopyrite order-disorder transition in copper indium selenide (CuInSe2).

   Folmer, J. C. W.; Franzen, Hugo F. Sol. Energy Res. Inst., Golden, CO, USA.

   Physical Review B: Condensed Matter and Materials Physics (1984), 29(11),

3. ↑ Order-disorder transitions in the system Ag2−xAuxS (0 x 1)

   J. C. W. Folmer, P. Hofman and G. A. Wiegers

   Journal of the Less Common Metals

   Volume 48, Issue 2, August 1976, Pages 251-268

4. ↑ An interesting study of a liquid crystal system that shows a tricritical point is :http://www.informaworld.com/smpp/content~content=a757041480~db=all~order=page

   Landau second order to tricritical crossover behaviour for smectic A-smectic C* transitions

   Authors: T. Chan a; Ch. Bahr b; G. Heppke b; C. W. Garland a

   Affiliations: a Center for Materials Science and Engineering and Department of Chemistry, Massachusetts Institute of Technology, Cambridge, Massachusetts, U.S.A.

   Published in: journal: Liquid Crystals, Volume 13, Issue 5 May 1993 , pages 667 - 675