Summary of faceting paper

Here is my rough outline, with figures, of what I think should go into a paper on our faceting project. In most respects it follows the summary page Yanting wrote. All the figures below are taken from Yanting's website.

First order melting from energy curve

Caloric curve of potential energy - shows melting transition to be about 1100K

Ichosohedral order from bulk bond orientation

Bond orientation order parameter of bulk atoms (excludes surface and first 4 sublayers) - shows melting again at 1100K, but also shows that bulk order is ichosohedral.

We should add to this graph results from new simulations at 850K, 950K, 1050K and 1075K that Yanting is doing.

Surface stays ordered up to melting, from surface bond orientation

Bond orientation order parameter of surface atoms - seems to show that the surface atoms remain ordered up until bulk melting at 1100K. Note, in computing this surface bond order parameter, only bonds between atoms on the surface are included (no bonds to atoms in first sublayer included). This compares with bulk calculation in which bonds to all neighbors in the 3d local environment are included. This is presumably the reason for the big difference between W6 (blue curve) in the two figures.

We should add to this graph results from new simulations at 850K, 950K, 1050K and 1075K that Yanting is doing.

Diffusion of atoms, layer by layer

At the start of each temperature, we label each atom with a layer number n=0, 1, 2, 3, 4, 5 according to whether it lies on the surface, or in 1st sublayer, 2nd sublayer, etc., 5=bulk atom. We then compute the mean square distance (MSD) traveled by the atoms which start with a given layer number n, as function of time. For diffusion, we expect this curve to be linear in time; but if one diffuses too far, this curve will saturate at a distance of order the cluster size. Below are plots at various temperatures.

Observations: For T=1100K above melting, we see saturation of diffusion for all layers - everyone moves around a lot! For T less than melting, we see primarily diffusion of surface and 1st sublayer. At the higher T, there is some diffusion of the lower sublayers. But the diffusion of sublayers is most likely due to the following effect: atoms only diffuse on surface; sublayer atoms diffuse only because they can get excited up to the surface and start diffusing from there (see figure in next section) - recall, we give each atom its layer number at the start of the simulation at each T, and do not relabel even if atoms move from one layer to another.

It is still odd to me that the 1st sublayer is diffusing more than the surface layer. Any idea why this is so? Is it possibly a misslabeling of the graphs?

Perhaps we should include here curves from the new simulations at T=950, 1050, 1075, depending on how they look.

Should we be concerned about the "glitches" in the data near 800ps? I recall we talked about his earlier, but I forget the reason for the glitch.

Do I recall correctly, that the average spacing between atoms is about 3.6 angstrom? So to set the scale of these graphs, a MSD of about 13 corresponds to diffusing one inter-atomic spacing?

Diffusion constant from above MSD vs. t graphs

Fitting the MSD vs. t curves above (for T<1100K) to MSD = Dt (i.e. curves pass through origin; Yanting, do I get the coefficient you used correct?) we get the average diffusion constant of atoms in each layer. Diffusion constants here are quite small compared to what they are in the liquid - Yanting, can you use the short time part of the 1100K MSD vs t curve to find D in the liquid just above melting? We will argue below that, below melting, it is not all the atoms that diffuse with the diffusion constant shown below, but rather only a small subset of atoms (those at vertices and edges) that diffuse. Hence, although there is diffusion as low as 600K, until around 900K, most of the atoms on the surface are NOT diffusing, hence there is no surface melting (but this is less clear above 900K).

Interlayer diffusion of atoms

At the start of each temperature, we label each atom with a layer number n=0, 1, 2, 3, 4, 5 according to whether it lies on the surface, or in 1st sublayer, 2nd sublayer, etc., 5=bulk atom. Then we run the simulation at the given temperature, and look at the ending configuration to see where each of the initial population of atoms ends up. We give each atom a new layer number N=0, 1,...,5 according to whether it has ended on the surface, in 1st sublayer, etc. For the set of atoms which started with an initial layer number n, we then compute the average of their final layer number N. Plotted below is this average N, for each of the initial populations n. We see that surface and 1st sublayer have fair amount of mixing above 900K. The 2nd sublayer also mixes in around 1000K. The other layers do not mix noticeably until melting at 1100K, when all the layers equally mix together.

We should add to this graph results from new simulations at 850K, 950K, 1050K and 1075K that Yanting is doing.

Faceting

To investigate faceting, we wish to find the equilibrium shape of the crystal - that is the average shape after averaging over thermal fluctuations. For an atomistic model (as opposed for example to a model of a fluctuating interface) there is no (not that we could find!) simple way to do this. We therefore take the following approach. We consider each atom in the cluster, and compute its average position and its root mean square deviation from this average position, over the course of the simulation at a fixed temperature. We then assign each atom to its average position in order to construct the average cluster shape. This is quite reasonable for an atom which is NOT diffusing around the cluster, but is only making finite distance thermal vibrations about its average position. It is less clear how reasonable this is for an atom which is diffusing, and hence making, as time increases, ever larger excursions from its average position. However, even for a diffusion atom, its average position (regarded now as an average over many repeated simulations) is in principle not moving. Hence we use our procedure even in this case, and regard fluctuations away from the average position (which increase in time if atom diffuses) as representing the smearing of the shape due to diffusing atoms. Does this make any sense?

Using the above procedure to define the position of the atoms in the average shape, we then compute the two principle local curvatures of each atom on the surface using Yanting's algorithm The equilibrium shapes, with each atom colored according the the value of its largest of the two principle curvatures, are shown below - scale is blue to red corresponds to small to large curvature.

Observations: at low T we have a very nice ichosohedral fully faceted crystal. It remains like this up to about 600K. Vertices have largest curvature (red), followed by edges (orange or yellow), with facets the smallest curvature (light blue). At 700K we start to see some rounding of the vertices (note: the orientation at 700K is not necessarily the same orientation as shown at 600K, if I remember correctly?). At 900K we see rounded vertices and edges. At 1000K things are now quite rounded, with maybe very small facets still remaining. It will be good to see similar picutres for the new temperatutres 950, 1050, 1075, to see if facets totally dissappear before melting or not.

Yanting, for publication, we should probably take the numbers off the atoms.

400K

600K

700K

800K

900K

1000K

Diffusion at vertices and edges

As mentioned in the section on diffusion, most of the diffusion below melting comes from the atoms at the vertices and at the edges. This seems reasonable as these are the atoms with fewest neighbor bonds, hence it is easiest to excite them away from their bound positions. As T increases, one expects first vertex atoms to diffuse, then edge atoms to diffuse, and finally all but the few facet atoms will diffuse.

To see this we plot pictures similar to above, ie each atom is located at its average position, however we now adjust the size of each sphere representing a given atom to equal the root mean square distance moved by the atom about its average position, throughout the course of our simulation. Below, for each temperature shown, we plot two such pictures: one averaged over the first 25 stored configurations of the simulation, and the second averaged over 100 stored configurations (Yanting, what simulation time does this correspond to in ps?). The point here is that if a particle is NOT diffusing, the size of its sphere will be the same in both pictures, but if a particle is diffusing, the size of its sphere will roughly double from the first picture to the second (since root mean square distance will grow as square root of time).

Pictures are on Yanting's page, here.

I think these look really nice. It is quite clear that at the lower T<900K, it is only vertices and edges that are diffusing. And even at higher T=900 and 950 one can see facet atoms which are not diffusing.

For a paper, I suggest we show the same temperatures here as we do for the curvature colors pictures in previous section.

Things still to so

It would be good, for the pictures of previous section, to draw ellispoids, rather than spheres, where the length of each axis of the ellipsoid corresponds to the root mean square distance traveled in that direction. This will tell the direction, as well as magnitude, of the diffusion. If, for a given atom, one computes the matrix
Cij=<(ri-ri ave)(rj-rj ave)>
where ri denotes the ith component of the position vector r, rave=<r> is the average position, and by average we mean the average over configurations, then the ellispoid surface is given by the equation:
(ri-ri ave)C-1ij(rj-rj ave)=1
where C-1ij is the inverse matrix of Cij.

Try to make similar pictures for the nanorod, to see if we can correlate what happens here to what happens there. Can one say that the onset of shape change in the nanorod is due to onset of diffusion at vertices and edges?

It the end, it is still unclear how our results are related to the faceting transition. The equilibrium shape in the "Wolf" plot way of thinking has to do with the dependence of surface tension, for in principle an infinite flat plane, on surface orientation. Here, things seem to be goverened by curvature (not included in Wolf construction method) ie by the atoms that have fewest nearest neighbor bonds. On the other hand, even in the Wolf construction point of view, the crystal will round out first at its edges. So I am not sure if there is a connection or if there is not. Also, our cluster is so small that it might not be realistic to use the "infinite size" assumption behind the Wolf construction. Clearly the bulk crystal structure (ichosohedral) we have here is different from that (fcc/hcp) of the rod, so size is obviously important. I will try to think more about this. Any thoughts from others are welcome!