**Micelle ordering resolved by s****mall angle X-ray scattering (SAXS)****. **Figure 3(a) shows the SAXS profiles collected by heating the solvent-cast P2VP-*b*-PDMS. At the onset temperature (i.e., 30 oC), the SAXS curve showed a broad peak centering at 0.47 nm-1 along with a shoulder at 0.75 nm-1. This scattering profile was fitted well by the Percus-Yevick model of polydisperse spherical particles (see Figure S2 of the Supplementary Information), indicating that the micellar entity of the bcp still retained, but the micelles exhibited only short-range order. That is, the bcp formed the micellar liquid phase. The broad hump marked by “*i=1*” was the first-order form factor maximum of P2VP core. The fact that this peak was broad and there was no discernible higher-order peaks attested that the distribution of the core size was quite broad, as manifested by the relatively large polydispersity index (=0.154) given by the ratio of the standard deviation to the mean value (= 4.77 nm) of the sphere radius assuming Schultz size distribution.

As the temperature was raised to 80 oC, which situated 15 oC above the Tg of P2VP core (Tgcore), the micelles organized into BCC lattice with the unit cell dimension of *a *= 22.0 nm, as evidenced by the emergence of sharp peaks with the position ratio of 1: 21/2: 31/2. The SAXS results suggested that the micelles developed during the solvent evaporation and subsequent drying processes were unable to undergo a fast ordering and were trapped into a metastable micellar liquid phase at 30 oC. Upon heating above Tgcore, the micelles gained sufficient mobility to proceed with the ordering into the stable BCC phase within the experimental time scale. An order-disorder transition (ODT) occurred upon further heating to T > 160 oC, where the BCC phase turned into a micellar liquid phase exhibiting broader interaction and form factor peaks in the SAXS curve.

The micellar liquid phase attained at high temperature persisted in the subsequent cooling cycle, as demonstrated in Figure 3(b). The copolymer sample thus cooled was then stored at 30 oC for prolonged annealing. Interestingly, the sample having been annealed for 60 days was found to exhibit a large number of diffraction peaks in the SAXS curve, as shown in Figure 3(b). The diffraction peaks were indexed well according to the P63/mmc space group of hexagonal unit cell (see Table S2 and Figure S3 of the Supplementary Information) and the entire diffraction pattern was consistent with that of Laves C14 phase of other bcp systems reported previously.14–17 The dimensions of the large hexagonal unit cell deduced from the peak positions were *a* = 37.59 nm and *c* = 61.39 nm, yielding the ratio of *c/a* = 1.633, in accord with that associated with an ideal hexagonal cell. The unit cell of the C14 phase composes of 12 particles and is filled by three types of Voronoi cells, i.e., two types of Z12 and one Z16 cells,14 as schematically illustrated in Figure 1(d).

The present study revealed that P2VP-*b*-PDMS was another bcp system capable of forming FK phase, where the micelles in the supercooled micellar liquid phase at 30 oC underwent a slow organization to form Laves C14 phase. According to the conventional Voronoi tessellation, the 12 particles in the unit cell of C14 phase have three different volumes, with the ratio of the largest cell volume to the smallest one being 1.23. On the other hand, the micelles in the micellar liquid phase, from which the C14 phase developed, displayed unimodal size distribution. A redistribution of the association numbers of the micelles should in principle occur during the phase transition, transforming the unimodal distribution in the micellar liquid into the multimodal distribution in C14 phase.24 However, such a symmetry breaking process was not accessible here, since the structural organization occurred at 35 oC below Tgcore, thereby prohibiting the mass transport required for redistributing the association number. As a matter of fact, the SAXS profiles at q > 0.7 nm-1, which were dominated by the form factor scattering of the P2VP core, associated with micellar liquid and C14 phase were superimposable (see Figure 4), confirming that the micelle size distribution was preserved upon the phase transformation.

Strictly speaking, the unimodal distribution of micelle size in micellar liquid phase did not fit the multiplicity of the cell volume in C14 phase; nevertheless, comparing to the scenario of monodisperse particle size, the relatively high polydispersity of micelle size in the present system could be advantageous for accommodating the volume asymmetry underlying the C14 phase.36 Moreover, the lattice formed by bcp micelles is usually distorted, where the centroids of the micelles deviate from the ideal positions due to size distribution and thermal fluctuations. There is an allowable range of distortion within which the scattering pattern still contains sufficient number of diffraction peaks (but with broadening in peak breadth) for assigning the packing structure. In the case where mass transport is forbidden, the requirement of volume asymmetry for FK phase formation can be alleviated by lattice distortion.

The C14 phase dissipated almost completely upon heating to 90 oC, as demonstrated in the temperature-dependent SAXS profiles of the C14-forming sample collected in a heating cycle shown in Figure 5. The TODT of C14 phase was ca. 25 oC higher than Tgcore and was much lower than the TODT of the BCC phase observed in Figure 3(a). The result suggests that the C14 phase developed here was metastable relative to BCC phase. This is understandable in that the glassy state of polymer is nonequilibrium in nature, such that the micelle was indeed a metastable entity below Tgcore. If the vitrification of the core did not occur in the cooling process, the micelles would have been able to adjust their association numbers (and hence sizes) in response to the change of segregation strength; in this case, BCC should have been the thermodynamically stable ordered structure along the equilibrium free energy path representing the temperature change of the structure for the micelles with fluid core and corona. On the other hand, once the micelle size was frozen in by the vitrification of the core, the system would go through another free energy path representing the change of the structure for micelles composed of a glassy core and fluid corona. C14 phase then became the favored packing structure under this metastable condition.

**Thermodynamic driving force leading to the formation of C14 phase. ** The key issue remained is the thermodynamic driving force leading to the formation of C14 phase at temperature Tgcorona < T < Tgcore. On basis of the DFM, Reddy et al. have calculated the free energy of the micelle confined within the Voronoi cells associated with various lattice structures in the polyhedral interface limit (PIL), and demonstrated that C14 phase was unstable relative to BCC, A15 and s phases.27 In this model, the interfacial free energy governed by the surface area per unit volume of the core is coupled with the geometry of the Voronoi cell, as the core is assumed to be the affinely shrunk copy of the cell. Therefore, the interfacial free energy is directly determined by the lattice structure chosen to calculate the total free energy of the micelle. When the micelles are brought below Tgcore, the core geometry is arrested upon vitrification; in this case, the PIL *ansatz *is no longer applicable in that the interfacial free energy becomes a constant and does not vary with the lattice structure chosen for calculating the total free energy of the micelles in the Voronoi cell below Tgcore. Now the conformational free energy of the coronal block becomes the sole variable in DFM, and BCC will be the favored packing lattice for minimizing the entropic penalty arising from stretching of the coronal blocks in the Voronoi cell, provided that the micelle core arrested (e.g. from the micellar liquid phase) adopts sphere geometry.27 Nevertheless, the micelles of P2VP-*b*-PDMS were found to organize spontaneously into C14 phase below Tg core, implying that there exists a free energy component not considered explicitly in DFM.

At Tgcoronna < T < Tgcore, the micelle approaches the so-called “fuzzy colloid” defined by Ziherl and Kamien to describe the particle composed of a hard core and a thin soft corona such as dendrimer.28 In the treatment of the packing problem of the fuzzy colloid, Ziherl-Kamien (Z-K) model postulated that the packing lattice is governed by the balance between two free energy components, i.e., the bulk free energy arising from the interaction between the cores which were treated as hard spheres and the surface free energy arising from the loss of orientational entropy of the chain segments constituting the corona upon overlapping with the segments associated with the neighboring particles. This model predicted that, if the corona is thin compared to the core, the bulk free energy dominates and close-packed lattice such as FCC is favored. But when the corona is sufficiently thick, A15 lattice becomes the stable packing symmetry, in that it minimizes the contact area and hence the surface free energy of the Voronoi cells with fixed volume.28 The micelles of P2VP-*b*-PDMS however formed neither close-packed lattice nor A15 phase predicted by Z-K model.

Z-K model was originally developed to predict the packing of fuzzy colloids composed of a thin corona formed by short alkyl chains, so the conformational free energy of these short chains was not taken into account. Recently, Pansu and Sadoc extended Z-K model to include the conformational free energy change of the coronal chains attached with hard spherical particles packed in the lattice by treating the chain as an entropic spring.29 Moreover, the hard sphere interaction assumed in Z-K model was replaced by the van der Waals attraction in formulating the bulk free energy. As expected, the theory predicts BCC as the packing lattice that minimizes the conformational free energy. Most intriguingly, the distribution of the interparticle distance in the lattice of C14 phase was found to minimize the van der Waals interaction energy of the particles. Consequently, the fuzzy colloids prefer to organize in C14 phase once they experience strong van der Waals attraction. This theoretical prediction was consistent with the experimental finding of C14 phase in gold nanoparticles coated with hydrophobic ligands.37

The energy of interaction between the cores was normally neglected in treating the packing problem of bcp micelles. This is a good assumption for weaker inter-core interaction; under this condition, the calculation of the intramicellar free energy associated with the Voronoi cells is sufficient to evaluate the stability of the corresponding lattice. Once the bcp micelles fall into the fuzzy colloid regime, the interfacial free energy of the micelle becomes independent of the lattice structure; the inter-core interaction energy determined by the positions of the micelles in the lattice may then emerge as an important variable in the total free energy. Because the particles showing poor affinity to the matrix phase tend to aggregate, the van der Waals attraction between the cores of the micelles is expected to be stronger in the bcp displaying larger Flory-Huggins interaction parameter c. In other words, the contribution of the inter-core interaction energy will be particularly important in high-c bcps, where Pansu-Sadoc (P-S) model will serve as the appropriate tool for analyzing the stabilities of the packing lattices. The solubility parameters of P2VP and PDMS are 20.6 and 15.5 MPa1/2, respectively; the large difference in their solubility parameters prescribes a large c for P2VP-*b*-PDMS. Therefore, we believe that the formation of C14 phase in P2VP-*b*-PDMS was driven mainly by the strong attractive force between P2VP cores, according to the P-S model.29,38

In summary, we have disclosed a new approach for generating the Laves C14 phase of bcp through accessing the fuzzy colloid regime at Tgcoronna < T < Tgcore. This approach is particularly plausible for bcp displaying large c, as the strong van der Waals attraction between the cores could outweigh the conformational free energy of the coronal blocks to drive the organization of the micelles into C14 phase that minimizes the interaction energy under the metastable condition. The FK phases having been disclosed for bcp thus far include σ, A15 and Laves C14 and C15 phases, with σ phase being the most common. Bates and coworkers reported the first discovery of C14 and C15 phases via thermal path dependent processes in compositionally asymmetric polyisoprene-*block*-polylactide.14 Their work highlighted the strong effect of thermal pathway on the final structure formed. Ryu et al. investigated the ordered structure in poly(dimethylsiloxane)-*block*-poly(2,2,2- triflouroethyl acrylate) with different block ratios and detected C14 phase at the volume fraction fPDMS = 0.85.17 For all the systems showing C14 phase reported thus far, the quasicrystal approximant order was established via thermal treatment above the Tgs of the constituent blocks, such that the mass transport process for redistributing the association number of the micelles was accessible. The development of the Laves C14 phase in the fuzzy colloid regime unveiled here however did not involve mass transport and would hence represent a new regime for the FK phase formation in bcp.