Shape-driven solid–solid transitions in colloids Chrisy Xiyu Dua,1, Greg van Andersb,1,2, Richmond S. Newmanb, and Sharon C. Glotzera,b,c,d,3 aDepartment of Physics, University of Michigan, Ann Arbor, MI 48109-1040; bDepartment of Chemical Engineering, University of Michigan, Ann Arbor, MI 48109-2136; cDepartment of Materials Science and Engineering, University of Michigan, Ann Arbor, MI 48109-2136; and dBiointerfaces Institute, University of Michigan, Ann Arbor, MI 48109-2800 Contributed by Sharon C. Glotzer, April 3, 2017 (sent for review December 29, 2016; reviewed by Oleg Gang and David A. Kofke) Solid–solid phase transitions are the most ubiquitous in nature, 29), and vitrification (30). Pioneering realizations of solid–solid and many technologies rely on them. However, studying them in phase transitions in colloids (6, 10, 31–35) have revealed detailed detail is difficult because of the extreme conditions (high pres- information about the transition mechanisms, including show- sure/temperature) under which many such transitions occur and ing experimentally a first-order transition that goes through an the high-resolution equipment needed to capture the interme- intermediate fluid (10, 32), an experimental system that showed a diate states of the transformations. These difficulties mean that variety of different transition pathways (33), and the experimen- basic questions remain unanswered, such as whether so-called tal observation of a pressure-dependent transition in colloidal diffusionless solid–solid transitions, which have only local par- superballs (35). Recently, interaction shifting via DNA program- ticle rearrangement, require thermal activation. Here, we intro- ing has been used to construct colloidal solid–solid transitions duce a family of minimal model systems that exhibits solid–solid (36), including showing that a single solid mother phase can be phase transitions that are driven by changes in the shape of col- reprogrammed to yield multiple daughter phases through diffu- loidal particles. By using particle shape as the control variable, we sionless transitions (37). entropically reshape the coordination polyhedra of the particles Here, we change particle shape in situ to directly control par- in the system, a change that occurs indirectly in atomic solid–solid ticle coordination in colloidal crystals to create minimal models phase transitions via changes in temperature, pressure, or den- of solid–solid phase transitions. Our models mimic the changes sity. We carry out a detailed investigation of the thermodynam- in coordination that occur indirectly in conventional atomic ics of a series of isochoric, diffusionless solid–solid phase transi- solid–solid transitions through changes in temperature, pressure, tions within a single shape family and find both transitions that or density. We find isochoric solid–solid transitions in “shape require thermal activation or are “discontinuous” and transitions space” in a simple two-parameter family (38) of convex colloidal that occur without thermal activation or are “continuous.” In the polyhedra with fixed point group symmetry that exhibits transi- discontinuous case, we find that sufficiently large shape changes tions between crystals with one [simple cubic (SC)], two [body- can drive reconfiguration on timescales comparable with those for centered cubic (BCC)], and four [face-centered cubic (FCC)] self-assembly and without an intermediate fluid phase, and in the particles in a cubic unit cell. We study the thermodynamics of continuous case, solid–solid reconfiguration happens on shorter the transitions using a consistent thermodynamic parametriza- timescales than self-assembly, providing guidance for developing tion of particle shape via the recently proposed approach of “dig- means of generating reconfigurable colloidal materials. ital alchemy” (39) combined with the rare event sampling tech- nique of umbrella sampling (40), which is commonly used to colloids j self-assembly j phase transitions j nanoparticles calculate free energy difference between two different states (41). We investigate solid–solid transitions between BCC and FCC espite wide-ranging implications for metallurgy (1), ceram- crystals and between BCC and SC crystals. We find that both Dics (2), earth sciences (3, 4), reconfigurable materials (5, 6), and colloidal matter (7), fundamental questions remain Significance about basic physical mechanisms of solid–solid phase transi- tions. One major class of solid–solid transitions is diffusion- Despite the fundamental importance of solid–solid transitions less transformations. Although in diffusionless transformations, for metallurgy, ceramics, earth science, reconfigurable materi- particles undergo only local rearrangement, the thermodynamic als, and colloidal matter, the details of how materials trans- nature of diffusionless transitions is unclear (8). This gap in form between two solid structures are poorly understood. our understanding arises from technical details that limit what We introduce a class of simple model systems in which the we can learn about solid–solid transitions from standard labora- direct control of local order, via colloid shape change, induces tory techniques, such as X-ray diffraction or EM (9). The use solid–solid phase transitions and characterize how the transi- of a broader array of experimental, theoretical, and computa- tions happen thermodynamically. We find that, within a single tional techniques could provide better understanding of solid– shape family, there are solid–solid transitions that can occur solid transitions if an amenable class of models could be devel- with or without a thermal activation barrier. Our results pro- oped (10). To develop minimal models, it is important to note vide means for the study of solid–solid phase transitions and that solid–solid transitions are accompanied by a change in shape have implications for designing reconfigurable materials. of the coordination polyhedra in the structure (11). Coordina- tion polyhedra reflect the bonding of atoms in a crystal, which Author contributions: C.X.D., G.v.A., and S.C.G. designed research; C.X.D., G.v.A., and suggests that minimal models of solid–solid transitions could be S.C.G. performed research; C.X.D., G.v.A., and R.S.N. contributed new reagents/analytic provided by systems in which the shape of coordination poly- tools; C.X.D., G.v.A., and S.C.G. analyzed data; and C.X.D., G.v.A., and S.C.G. wrote the hedra is directly manipulated. Direct manipulation of coordi- paper. nation polyhedra may be achieved in systems of anisotropically Reviewers: O.G., Brookhaven National Laboratory; and D.A.K., University at Buffalo, shaped colloids (12, 13). Anisotropic colloids manifest an emer- State University of New York. gent, shape-dependent entropic valence (12, 13) that is respon- The authors declare no conflict of interest. sible for the stabilization of a wide variety of structures, even 1C.X.D. and G.v.A. contributed equally to this work. in the absence of direct interparticle forces (12, 14–23). More- 2Present address: Department of Physics, University of Michigan, Ann Arbor, MI 48109. over, colloids are amenable to a wide range of observational and 3To whom correspondence should be addressed. Email: [email protected]. experimental techniques, and colloids have been widely used to This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10. investigate melting (24, 25), sublimation (26), crystallization (27– 1073/pnas.1621348114/-/DCSupplemental. E3892–E3899 j PNAS j Published online May 1, 2017 www.pnas.org/cgi/doi/10.1073/pnas.1621348114 Downloaded by guest on September 27, 2021 transitions are diffusionless transformations between lattices that have rich self-assembly behavior (Fig. 2 shows three examples) with both PNAS PLUS are continuously related by linear mathematical transforma- wide and narrow regions of thermodynamic stability for a number of dif- tions. We study four cases of the BCC$FCC transition and ferent bulk structures (22). We denote shapes according to the conven- find that, in all cases, the transition is thermodynamically dis- tions (αa, αc), where 0 ≤ αa, c ≤ 1 defines the boundaries of shape space continuous (i.e., first order). We also study two cases of the in this shape family. With these conventions, (0; 0) is an octahedron, (0, 1) $ and (1, 0) are tetrahedra (which is self-dual), and (1, 1) is a cube (dual BCC SC transition and find, in contrast, that, in both cases, to the octahedron). We use conventions in which all particles have unit the transition is thermodynamically continuous (i.e., second or volume. higher order). We study the dynamics of the solid–solid transi- We investigate shape change-induced solid–solid transitions in ∆323 in tions and find no evidence of intermediate fluid states, regard- the regions indicated in Fig. 1, focusing on BCC, FCC, and SC structures. FCC less of whether the transition is discontinuous or continuous. and BCC as well as BCC and SC can be found in neighboring regions of ∆323. Our results show that diffusionless solid–solid phase transitions We study FCC$BCC and BCC$SC transitions, and the regions of investiga- can be thermodynamically discontinuous or continuous, even in tion indicated in Fig. 1 are all known boundaries between the phases of cases where transitions are induced by symmetry-invariant shape interest in ∆323. change driven solely by entropy maximization and where transi- We study the thermodynamics of solid–solid transitions using both the Ehrenfest and