Interplay between the Formation of Colloidal Clathrate and Cubic Diamond Crystals

Controlling the valency of directional interactions of patchy particles is insufficient for the selective formation of target crystalline structures due to the competition between phases of similar free energy. Examples of such are stacking hybrids of interwoven hexagonal and cubic diamonds with (i) its liquid phase, (ii) arrested glasses, or (iii) clathrates, all depending on the relative patch size, despite being within the one-bond-per-patch regime. Herein, using molecular dynamics simulations, we demonstrate that although tetrahedral patchy particles with narrow patches can assemble into clathrates or stacking hybrids in the bulk, this behavior can be suppressed by the application of external surface potential. Depending on its strength, the selective growth of either cubic diamond crystals or empty sII clathrate cages can be achieved. The formation of a given ordered network depends on the structure of the first adlayer, which is commensurate with the emerging network.


Model details
In our previous work, 1 we have demonstrated that the set of parameters shown in Methods section in the main article, is ecient to describe relatively strong but reversible interactions.
The parameter l (Figure 1 in the main text), depicting to what degree the active sites are embedded into the core's surface, allows for the manipulation of the patch's valency.This is because the spherical Lennard-Jones potential is screened for certain orientations of patchy particles, allowing to eectively control the extent of directionality of the interparticle interactions by the parameter l.As shown previously, 1 values larger than l = 0.36σ allow for the creation of multiple bonds by a single active site which is an undesired eect for a current study.
In Figure S1 we demonstrate the distribution of a number of nearest neighbors (N nn ) around a patchy particle, calculated for the distances up to the rst minimum in the radial distribution function, r min = 1.2σ.The thermodynamic states were T = 0.185 and T = 0.3 for l = 0.34σ and l = 0.36σ, respectively, for both cases in the bulk density equal to ρ = 0.4.
It can be seen that in both cases of l = 0.34σ and l = 0.36σ, are within the single-bondper-patch regime.However, as reported in the previous article, 2 and demonstrated in the main document, narrowing the patches signicantly changes the structural behaviour of tetrahedral patchy particles.

The bulk phase behavior
The bulk phase diagram in the ρ − T plane has been evaluated using the block's analysis method proposed by Binder. 3The dierence between simulations from the N V T and N pT ensembles can be attributed to dierent approaches used.In the latter, the starting point for every pressure studied was a disordered and low-density conguration which after equilibration time reached the density as shown in the isotherm in Figure 1-b in the main text.
On the other hand, in the N V T ensemble two protocols have been used, depending on the system's density.For the range of ρ = 0.4 − 0.8 we have started from a disordered conguration at high temperature and gradually cooled down to the target temperature.This causes that some replicas crystallized into either DC/DH hybrids, sII clathrates, or their mixture.To estimate higher density branches of the phase diagram, we have selected congurations of either sII clathrates or DC/DH hybrids and compressed them to higher densities, eventually reaching ρ = 1.2.After compression, we have xed the density again and performed simulations in the N V T ensemble, gathering block density distributions.We have found that DC/DH hybrids remain stable at densities higher than ρ ≈ 0.72, and the contribution of the BCC phase increases with density.The latter was expected since the BCC network is composed of two interwoven diamond phases.On the other hand, the same protocol used for sII clathrate conguration leads to the formation of a dense amorphous uid, as in the case of simulations in the N pT ensemble.However, it is worth noticing that in the case of the N pT ensemble, the formation of the BCC network has also been observed at the density ρ = 1.2.
On top of that, Figure S2 demonstrated the comparison between the phase diagrams evaluated for narrow (l = 0.34σ) and wider (l = 0.36σ) patches.It is evident that the topology of the phase diagram is similar.However, the most pronounced dierences are (i) the emergence of the sII clathrates and (ii) the lack of a metastable liquid region for narrow patches.Moreover, the temperature scale is almost two times smaller for the current case.

Distinction between the clathrate networks
The CHILL+ parameter 4 allows to discriminate between dierent local crystalline environments including diamond and clathrate networks, based on the number of staggered or eclipsed bonds that a given particle forms.Unfortunately, using this parameter, it is not possible to dierentiate between the sI and sII clathrates that are comprised of a dierent number of molecules which form dierent types of cages.In order to avoid lengthy computations that would identify vertices and cages of clathrates, 5 we have demonstrated an alternative way based on behavior of the c 3 (i, j) and c 6 (i, j) correlation functions which enabled the distinction between sI and sII clathrate networks.
The ideal sI and sII clathrate networks were generated using GenIce2 package 6 and for each of them, we have calculated the distribution of the c 3 (i, j) and c 6 (i, j) correlation functions and compared them with the simulation results, given in Figure S3.One can see that as dened in CHILL+, the distribution functions of c 3 (i, j) correlation functions are almost identical for both clathrate networks and overlap with the simulation results (Figure S3 c, d).However, this is not the case for c 6 (i, j).The functions despite being similar for sI and sII clathrates, exhibit a systematic shift in the distributions to higher values for the sII network allowing for the discrimination.To prove that the structure formed in current simulations is sII clathrate, we have compared the distribution functions obtained from simulations with the ideal cases (Figure S3 c, d).

The formation of sII clathrates
In the main article, we have mentioned the formation of sII clathrate networks for two external potential eld strengths that is ξ = 0.1 and ξ = 0.4.However, the density proles shown in Figure S5 (a), demonstrate that these ordered clusters dier.For ξ = 0.1 we can see some intermediate peaks with lower intensity which are absent for ξ = 0.4.We have found that this is due to the dierent orientation of these networks with respect to the wall (cf.parts (b) and (d) of Figure S5).If one rotates the network obtained for in the system with ξ = 0.1 by 35 • around the (0,1,0) vector, the structures appear to be identical as demonstrated in parts (c) and (d) of Figure S5).Such dierence can be attributed to the formation of a primary adsorption layer for ξ = 0.4, which is absent for ξ = 0.1.Pentagonal rings which are present for the latter case promote the growth of clathrate cages in this particular direction, commensurate with the formed structure of the primary layer.

Figure S1 :
Figure S1: Distribution of the nearest neighbours (N nn ) for systems with embedding distance l = 0.34σ and l = 0.36σ.

Figure S2 :
Figure S2: The comparison between the phase diagrams obtained for narrow (part a, l = 0.34σ) and wider (part b, l = 0.36σ) patches.The latter diagram is taken from Ref. 1 with permission from the Royal Society of Chemistry.