Premelting

Boundary between phase (1) and phase (2) without and with intermediate phase (3)

In the following thermodynamical equilibrium is assumed, as well as for simplicity (2) should be a vaporous phase.

The first (1) and the second (2) phase are always divided by some form of interface, what results in an interfacial energy ${\displaystyle \gamma _{1-2}}$. One can now ask whether this energy can be lowered by inserting a third phase (l) in between (1) and (2). Written in interfacial energies this would mean:

${\displaystyle \gamma _{1-2}>\gamma _{1-l}+\gamma _{l-2}}$

If this is the case then it is more efficient for the system to form a separating phase (3). The only possibility for the system to form such a layer is to take material of the solid and "melt" it to a quasi-liquid. In further notation there will be no distinction between quasi-liquid and liquid but one should always keep in mind that there is a difference. This difference to a real liquid becomes clear when looking at a very thin layer (l). As, due to the long range forces of the molecules of the solid material the liquid very near the solid still "feels" the order of crystalline solid and hence itself is in a state providing a not liquid like amount of order. As considering a very thin layer at the moment it is clear that the whole separating layer (l) is too well ordered for a liquid. Further comments on ordering can be found in the paragraph on Landau theory.

Now, looking closer at the thermodynamics of the newly introduced phase (l), its Gibbs energy can be written as:

${\displaystyle G\left(T,P,d\right)=\left(n_{l}\mu _{l}\left(T,P\right)\right)d+\gamma _{total}\left(d\right)}$,

where ${\displaystyle T}$ is the temperature, ${\displaystyle P}$ the pressure, ${\displaystyle d}$ the thickness of (l) corresponding to the number or particles ${\displaystyle N}$ in this case. ${\displaystyle n_{l}}$ and ${\displaystyle \mu _{l}}$ are the atomic density and the chemical potential in (l) and ${\displaystyle \gamma _{total}=\gamma _{1-l}+\gamma _{l-2}}$. Note that one has to consider that the interfacial energies can just be added to the Gibbs energy in this case. As noted before ${\displaystyle d}$ corresponds ${\displaystyle N}$ so the derivation to ${\displaystyle d}$ results in:

${\displaystyle \mu \left(T,P,d\right)=\mu _{l}\left(T,P\right)+{\frac {\Delta \gamma }{n_{l}}}{\frac {\partial f\left(d\right)}{\partial d}}=\mu _{1}}$

Where ${\displaystyle \gamma _{total}=\Delta \gamma _{1-l}\cdot f\left(d\right)+\gamma _{1-2}}$. Hence ${\displaystyle \mu _{1}}$ and ${\displaystyle \mu _{l}}$ differ and ${\displaystyle \Delta \mu =\mu _{1}-\mu _{l}}$ can be defined. Assuming that a Taylor expansion around the melting point ${\displaystyle \left(T_{m},P_{m}\right)}$ is possible and using the Clausius–Clapeyron equation one can get the following results:

• For a long range potential assuming ${\displaystyle f\left(d\right)=1-\sigma ^{2}/d^{2}}$ and ${\displaystyle d>>\sigma }$:

${\displaystyle \left(d=-{\frac {2\sigma ^{2}\Delta \gamma }{n_{l}q_{m}t}}\right)^{1/3}}$

• For short range potential of the form ${\displaystyle {\frac {\partial f}{\partial d}}~e^{-d/d_{0}}}$:

${\displaystyle d\propto \left|ln\left|t\right|\right|}$

Where ${\displaystyle \sigma }$ is in the order of molecular dimensions ${\displaystyle q_{m}}$ the specific melting heat and ${\displaystyle t={\frac {T_{m}-T}{T_{m}}}}$

These formulas also show that the more the temperature increases, the more increases the thickness of the premelt as this is energetically advantageous. This is the explanation why no overheating exists for this type of phase transition.[5]

With the help of the Lifshitz Theory on Casimir, respectively van der Waals, interactions of macroscopic bodies premelting can be viewed from an electrodynamical perspective. A good example for determining the difference between complete and incomplete premelting is ice. From vacuum ultraviolet (VUV) frequencies upwards the polarizability of ice is greater than that of water, at lower frequencies this is reversed. Assuming there is already a film of thickness ${\displaystyle d}$ on the solid it is easy for any components of electromagnetic waves to travel through the film in the direction perpendicular to the solid surface as long ${\displaystyle d}$ is small. Hence as long as the film is thin compared to the frequency interaction from the solid to the whole film is possible. But when ${\displaystyle d}$ gets large compared to typical VUV frequencies the electronic structure of the film will be too slow to pass the high frequencies to the other end of the liquid phase. Thus this end of the liquid phase feels only a retarded van der Waals interaction with the solid phase. Hence the attraction between the liquid molecules themselves will predominate and they will start forming droplets instead of thickening the film further. So the speed of light limits complete premelting. This makes it a question of solid and surface free energies whether complete premelting occurs. Complete surface melting will occur when ${\displaystyle \gamma _{total}\left(d\right)}$ is monotonically decreasing. If ${\displaystyle \gamma _{total}\left(d\right)}$ instead shows a global minimum at finite ${\displaystyle d}$ than the premelting will be incomplete. This implies: When the long range interactions in the system are attractive than there will be incomplete premelting — assuming the film thickness is larger than any repulsive interactions. Is the film thickness small compared to the range of the repulsive interactions present and the repulsive interactions are stronger than the attractive ones than complete premelting can occur. For van der Waals interactions Lifshitz theory can now calculate which type of premelting should occur for a special system. In fact small differences in systems can affect the type of premelting. For example, ice in an atmosphere of water vapour shows incomplete premelting, whereas the premelting of ice in air is complete.

For solid–solid interfaces it cannot be predicted in general whether the premelting is complete or incomplete when only considering van der Waals interactions. Here other types of interactions become very important. This also accounts for grain boundaries.[5]

A qualitative picture of the order parameter of a premelting solid for temperatures below the melting point. One can see that there is still a high amount of order in the liquid, which decreases with rising Temperature

Most insight in the problem probably emerges when approaching the effect form Landau Theory. Which is a little bit problematic as the melting of a bulk in general has to be considered as a first order phase transition, meaning the order parameter ${\displaystyle \eta }$ jumps at ${\displaystyle t=0}$. The derivation of Lipowski (basic geometry shown in fig.2) leads to the following results when ${\displaystyle T\leq T_{m}}$: ${\displaystyle \eta _{0}\propto {\begin{cases}const.&{\text{, }}a<{\sqrt {a_{m}}}\\\left|t\right|^{1/4}&{\text{, }}a={\sqrt {a_{m}}}\\\left|t\right|^{1/2}&{\text{, }}a>{\sqrt {a_{m}}}\end{cases}}}$

Where ${\displaystyle \eta _{0}}$ is the order parameter at the border between (2) and (l), ${\displaystyle 1/a}$ the so-called extrapolation length and ${\displaystyle a_{m}}$ a constant that enters the model and has to be determined using experiment and other models. Hence one can see that the order parameter in the liquid film can undergo a continuous phase transition for large enough extrapolation length. A further result is that ${\displaystyle d\propto \left|ln\left|t\right|\right|}$ what corresponds to the result of the thermodynamical model in the case of short range interactions. Landau Theory does not consider fluctuations like capillary waves, this could change the results qualitatively.[6]