Alignment in the Wigner glass

In the first part of the discussion, we describe a hypothetical crystallization pathway, which is similar to mesocrystallization. This pathway proceeds via alignment of nanocrystals in a low density arrested state of matter called the Wigner glass and subsequent oriented attachment realized by compressing the Wigner glass. We use the results of this thesis to suggest how to make such a pathway realistic in a simulation and possibly also in an experiment. Briefly, the pathway is initiated by a moderately deep quench of a low density fluid which leads to a phase separation of the system into monodisperse spherical crystalline aggregates. The crystals do not merge and are assumed to have enough time to crystallographically align. The resulting low density system of aligned clusters is slowly compressed to form a single crystalline block.

6.1.1 Formation of monodisperse aggregates

Phase separation of the system into isolated crystals can be achieved at low packing fractions and by moderately deep quenches into a suitable temperature, obtained either from the analysis of fluctuation-dissipation ratios [Jack et al., 2007; Klotsa and Jack, 2011] or from the estimate of the intersection of the glass line and the spinodal [Zaccarelli, 2007]. The temperature must be low enough such that the spinodal

decomposition drives the phase separation, but high enough such that bond breaking and bond making can regularly happen [Grant et al., 2011; Grant and Jack, 2012], and crystallinity within the dense regions can evolve. A simple quench only leads to low quality crystals or polycrystals. A controlled quench [Royall and Malins, 2012] or variation of interactions according to some kind of real time feedback [Klotsa and Jack, 2013] can improve the crystal yield. The size of the system must be reasonably small such that the timescales on which the non-crystalline partitions (liquid drops) dissolve can be achieved by the computer simulation. On the other hand, the system size must be large enough such that a sufficient number of crystals is formed, and further self-assembly into larger structures can be studied. The number of crystals (or equivalently their size) can be controlled by other parameters of the system such as the packing fraction or the shape of the potential.

A common and necessary prerequisite for hierarchical self-assembly or for mesocrystallization is to require that the self-assembled aggregates have low polydis- persity, to provide new building blocks for further self-assembly of larger aggregates. We observed that monodispersity may be achieved in different ways.

First, polydispersity may simply be a consequence of the simulation param- eters such as the size of the maximum displacementδ. Ifδ is small enough, and the quench starts from an equilibrium fluid, the system is expected to condense into a large number of liquid drops. These drops exist for a long time even in the absence of long-range repulsions. If δ is large, the polydispersity of the aggregates is seen to be increased. We thus expect that it is preferable to perform simulations with a low δ, at least at those stages of phase separation where most particles form parts of larger partitions.

Second, polydispersity can also be reduced by a suitable renormalization of the long-range repulsion. In Chap. 5, the amplitude of the pairwise potential was renormalized by the size of the parent partitions, which leads to a great reduction of polydispersity of the aggregates developed after the quench. The renormalization, however, is so approximate that it may be considered non-physical orad hoc with the aim to achieve monodispersity. Other, more physical partition-based renormal- ization models may be derived from integration of long-range repulsion [Sciortino et al., 2004].

Third, if the aim is only to obtain partitions with a low polydispersity, the original non-normalized model defined in Eq. (3.1) may be sufficient, provided the simulation starts from a uniformly distributed set of particles in the fluid phase, and provided a small maximum displacementδ is used at the beginning of the simula- tion. Controlling the nucleation at early times is, indeed, a way of experimentally

achieving low polydispersity (Sec. 1.6). The two previously described renormal- ization models are based on a dynamical biconnectivity routine [Henzinger, 1995], which may be computationally expensive or hard to implement. The advantage of the non-normalized model in Eq. (3.1) is that biconnectivity is not needed, and so the model can be easily implemented. In fact, the computational cost of evaluating the long-range repulsion is comparable to the cost of biconnectivity. Nevertheless, as discussed later, if the monodisperse partitions are to be compressed into a clus- ter crystal or a mesocrystal, a renormalization of the repulsion representing the polymers accumulated on the partitions is needed, to avoid reorganization of the partitions into larger domains during the compression.

Fourth, as recently emphasized by Sweatman et al. [2014] or by Zhang et al. [2012], the polydispersity in SALR system, where the range of attraction is large, may only be a kinetic phenomenon, and a systematic dispersion of smaller partitions may accelerate the simulation with the aim to reach the equilibrium state character- ized by partitions with low polydispersity. It is emphasized here that this may only be valid for longer-range attractions, as the simulation of the short-range attrac- tive model of Charbonneau and Reichman [2007b], which uses non-kinetic moves equilibrating the system, still shows clusters with a high polydispersity.

Let us stress that to achieve timescales on which most drops crystallize and liquid drops dissolve may require a long simulation. Our analysis also suggests that an acceleration might also be achieved, if the expensive simulation of SALR systems by VMMC is approximated by a SPMC simulation of the same system but without the long-range repulsion. A similar approximation may be possible for the renormalized potentials. The collective moves would then be used only sparingly, in order to equilibrate the partition-partition distances. This approximation is made possible by facts that the structure evolution happens mainly via single-particle movements and exchange, and that the contribution of collective modes of motion to long-time structural changes can be neglected (integrated out).

We also point out that if the long timescale cannot be achieved, we can con- struct the state of isolated partitions artificially. Basin hopping techniques [Wales, 2003] allow us to predict the parameters of SALR potentials and a suitable tem- perature at which the spherical partitions are stable [Mossa et al., 2004]. These crystalline partitions can then be periodically replicated in a simulation box such that nearest-neighbour inter-partition distances are roughly equal, and the partitions are rotated such that their relative lattices have random orientations. Although the long-range nature of the potential may make this state of isolated partitions un- stable [Toledano et al., 2009], and particles within the partitions may reorganize to

form a more stable state [Malins et al., 2011], we assume that this effect is negligible at low densities. If the effect is large, we may attempt to equilibrate the system, or chose other properties and parameters of the system.

Generation of monodisperse partitions, both artificially and via quenches, is a preparation protocol, which results in a fluid of clusters or in a non-equilibrium (stationary) system, which has the properties of a cluster fluid for a long time. The protocol can be such that the Wigner glass instead of a fluid of clusters is produced. This can be achieved if the quench is performed slowly, and at suitable packing fractions. The assumption about the possibility of reaching a Wigner glass composed of monodisperse partitions will be discussed in what follows.

6.1.2 Alignment of partitions

To illustrate a possible way of aligning the crystal lattices of the isolated partitions, we will assume that the system is in a kinetically slowed down (arrested) state with properties similar or identical to the properties of a Wigner glass. In particular, we will make use of the numerical evidence that the isolated partitions within a Wigner glass do not move via translational motion, and that the main degrees of freedom are collective rotations [Toledano et al., 2009]. This is indirectly confirmed by our VMMC simulations of fluids of partitions showing that as long as the partitions are spherical, large rotations are accepted more often than large translations. We may thus expect that the clusters within the Wigner glass have enough time to rotate until they align.

In such a crystal alignment process, the rotational entropy contributing to the free energy would compete with the potential energy of the system, and the transition between the aligned and rotationally disordered Wigner glass might be a first order transition with a distinct free energy barrier between the two states. The waiting time needed for the system to cross such barrier can be related to the timescale on which the properties of the Wigner glass persist. In the following, we will compare those timescales to the timescales of an Ostwald ripening process. This allows us to characterize the Wigner glass, and to tell whether it is stable enough for the alignment to occur.

In particular, the timescale of an Ostwald ripening process is given by the time needed for the detachment of a particle from its partition, and the average time needed to diffuse to another partition. We characterize the timescales of the stability of the Wigner glass as the timescale needed for two partitions to merge. These two timescales can be seen as the stability timescales, and we can compare them with the average time needed for all partitions to align. If any of the two timescales needed

for stability is shorter than the timescale needed for alignment, the partitions grow or merge before they align. (Unless the collisions are effective as will be discussed later). Hence, a necessary condition for a system of isolated monodisperse crystalline partitions to align is that the timescale for stability against aggregation or against growth via Ostwald ripening must be longer than the timescale needed for alignment. We note that in a system where the phase of uniformly sized clusters is ther- modynamically stable, a steady exchange of single particles between the partitions as well as continuous splitting and merging of the partitions might occur. The re- versible nature of these changes, and the fact that the partitions are stable, suggests that the earlier defined timescale for stability of a cluster phase can be arbitrary with respect to the timescales needed for alignment.

6.1.3 Compression into a mesocrystal

Wigner crystals are solids whose density is far too low to form a mesocrystal. A higher density may be achieved if the Wigner crystal or the cluster fluid are com- pressed or crunched [Mossa et al., 2002]. Crunching can be an alternative to quench- ing: both can reach a certain state point of the phase diagram [Royall and Malins, 2012]. Here, it is assumed that, depending on whether the crystals within the initial Wigner crystal are aligned or not, the compression leads either to a low density cluster crystal [Sciortino et al., 2004; Toledano et al., 2009] or to a mesocrystal. A similar compression of a very low density cluster fluid was used by Xia et al. [2011] to obtain a high density arrested state of uniformly sized CdSe nanoparticles, which were reminiscent of the cluster crystal or a glass of clusters.

Now the compression of the system may not be straightforward. It is known that the size of the wave vector corresponding to the first peak in the structure factor has been reported to be invariant with respect to the packing fraction [Stradner et al., 2004; Cardinaux et al., 2007]. This means that the position of the peak should remain constant during a sufficiently slow compression, and that slow crunches of the SALR system into lower densities may have a fundamental significance. However, it is known that partitions reorganize [Malins et al., 2011] and their size grows with increasing packing fraction [Sciortino et al., 2004]. Also, higher densities may be accompanied with transitions into modulated phases [Archer and Wilding, 2007]. Moreover, it is expected that the growth is not only given by the partition-partition interactions or translational entropy, but also by the temperature dependence of the interfacial tension [Stradner et al., 2004]. These results suggest that a fast compression or a non-trivial renormalization of the potential similar to that in Sec. 5 or in Xia et al. [2011] is necessary to produce a low density cluster crystal or a

mesocrystal composed of monodisperse building blocks.

We also point out that the re-entrant percolation in the non-normalized SALR system [Sciortino et al., 2005] may indicate another way of reaching higher packing fractions by compressions in systems where the potential has another shape.