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# Shape-driven solid–solid transitions in colloids

Contributed by Sharon C. Glotzer, April 3, 2017 (sent for review December 29, 2016; reviewed by Oleg Gang and David A. Kofke)

## Significance

Despite the fundamental importance of solid–solid transitions for metallurgy, ceramics, earth science, reconfigurable materials, and colloidal matter, the details of how materials transform between two solid structures are poorly understood. We introduce a class of simple model systems in which the direct control of local order, via colloid shape change, induces solid–solid phase transitions and characterize how the transitions happen thermodynamically. We find that, within a single shape family, there are solid–solid transitions that can occur with or without a thermal activation barrier. Our results provide means for the study of solid–solid phase transitions and have implications for designing reconfigurable materials.

## Abstract

Solid–solid phase transitions are the most ubiquitous in nature, and many technologies rely on them. However, studying them in detail is difficult because of the extreme conditions (high pressure/temperature) under which many such transitions occur and the high-resolution equipment needed to capture the intermediate states of the transformations. These difficulties mean that basic questions remain unanswered, such as whether so-called diffusionless solid–solid transitions, which have only local particle rearrangement, require thermal activation. Here, we introduce a family of minimal model systems that exhibits solid–solid phase transitions that are driven by changes in the shape of colloidal particles. By using particle shape as the control variable, we entropically reshape the coordination polyhedra of the particles in the system, a change that occurs indirectly in atomic solid–solid phase transitions via changes in temperature, pressure, or density. We carry out a detailed investigation of the thermodynamics of a series of isochoric, diffusionless solid–solid phase transitions within a single shape family and find both transitions that require thermal activation or are “discontinuous” and transitions that occur without thermal activation or are “continuous.” In the discontinuous case, we find that sufficiently large shape changes can drive reconfiguration on timescales comparable with those for self-assembly and without an intermediate fluid phase, and in the continuous case, solid–solid reconfiguration happens on shorter timescales than self-assembly, providing guidance for developing means of generating reconfigurable colloidal materials.

Despite wide-ranging implications for metallurgy (1), ceramics (2), earth sciences (3, 4), reconfigurable materials (5, 6), and colloidal matter (7), fundamental questions remain about basic physical mechanisms of solid–solid phase transitions. One major class of solid–solid transitions is diffusionless transformations. Although in diffusionless transformations, particles undergo only local rearrangement, the thermodynamic nature of diffusionless transitions is unclear (8). This gap in our understanding arises from technical details that limit what we can learn about solid–solid transitions from standard laboratory techniques, such as X-ray diffraction or EM (9). The use of a broader array of experimental, theoretical, and computational techniques could provide better understanding of solid–solid transitions if an amenable class of models could be developed (10). To develop minimal models, it is important to note that solid–solid transitions are accompanied by a change in shape of the coordination polyhedra in the structure (11). Coordination polyhedra reflect the bonding of atoms in a crystal, which suggests that minimal models of solid–solid transitions could be provided by systems in which the shape of coordination polyhedra is directly manipulated. Direct manipulation of coordination polyhedra may be achieved in systems of anisotropically shaped colloids (12, 13). Anisotropic colloids manifest an emergent, shape-dependent entropic valence (12, 13) that is responsible for the stabilization of a wide variety of structures, even in the absence of direct interparticle forces (12, 14⇓⇓⇓⇓⇓⇓⇓⇓–23). Moreover, colloids are amenable to a wide range of observational and experimental techniques, and colloids have been widely used to investigate melting (24, 25), sublimation (26), crystallization (27⇓–29), and vitrification (30). Pioneering realizations of solid–solid phase transitions in colloids (6, 10, 31⇓⇓⇓–35) have revealed detailed information about the transition mechanisms, including showing experimentally a first-order transition that goes through an intermediate fluid (10, 32), an experimental system that showed a variety of different transition pathways (33), and the experimental observation of a pressure-dependent transition in colloidal superballs (35). Recently, interaction shifting via DNA programing has been used to construct colloidal solid–solid transitions (36), including showing that a single solid mother phase can be reprogrammed to yield multiple daughter phases through diffusionless transitions (37).

Here, we change particle shape in situ to directly control particle coordination in colloidal crystals to create minimal models of solid–solid phase transitions. Our models mimic the changes in coordination that occur indirectly in conventional atomic solid–solid transitions through changes in temperature, pressure, or density. We find isochoric solid–solid transitions in “shape space” in a simple two-parameter family (38) of convex colloidal polyhedra with fixed point group symmetry that exhibits transitions between crystals with one [simple cubic (SC)], two [body-centered cubic (BCC)], and four [face-centered cubic (FCC)] particles in a cubic unit cell. We study the thermodynamics of the transitions using a consistent thermodynamic parametrization of particle shape via the recently proposed approach of “digital alchemy” (39) combined with the rare event sampling technique of umbrella sampling (40), which is commonly used to calculate free energy difference between two different states (41). We investigate solid–solid transitions between BCC and FCC crystals and between BCC and SC crystals. We find that both transitions are diffusionless transformations between lattices that are continuously related by linear mathematical transformations. We study four cases of the BCC

## Model and Methods

A goal of this study is to determine if shape-driven solid–solid transitions are viable for developing reconfigurable materials. Our study considers scenarios in which structural reconfiguration happens on much longer timescales than particle shape change. Moreover, because particle shape change can occur in systems of colloidal particles that are on the scale of hundreds of nanometers (42), shape variability is typically small at those scales, and therefore, we will model all particles as having the same shape.

A family of shapes that have the same point group symmetry and self-assemble crystals with small unit cells (1-SC, 2-BCC, and 4-FCC) in adjacent regions of shape space is found in the spheric triangle invariant 323 family (

We investigate shape change-induced solid–solid transitions in

We study the thermodynamics of solid–solid transitions using both the Ehrenfest and Landau approaches (44). All Monte Carlo (MC) simulations and computations were done at fixed packing fraction

First, to estimate the location of phase boundaries, we use the notion of generalized “alchemical” structure–property relationships (39). Alchemical structural–property relationships are derived from a consistent statistical mechanics treatment of particle attributes (here shape) as thermodynamic quantities. These thermodynamic quantities enter extended *A* and *C*) in perfect BCC (*B* and *D*) after

Second, having located discontinuities in *B* and 5*B*. To confirm the validity of *B*. Data indicate that BCC crystals have a peak near *A*). For BCC*A*). In all cases, five independent replicates were used to generate umbrella samples. Umbrella samples were used to reconstruct free energy curves using the weighted histogram analysis method (50), and errors were estimated using jackknife resampling (51). Additional methodological details are in *SI Text*. Data, documentation, simulation code, and analysis code are available on request from S.C.G.

## SI Text

### Averaged Bond Order Parameter.

The averaged bond order parameter (49) associates a set of spherical harmonics with every fictitious bond connecting one particle to its near neighbors. Here, a “bond” refers to a vector from a particle to a neighboring particle. The mathematical formalism of this parameter is as follows.

Following ref. 51, we denote

Lechner and Dellago (48) have introduced a second neighbor-averaged

After comparing different

### Umbrella Sampling.

The basic idea of umbrella sampling (40, 41) is to divide the order parameter space (here,

In the following derivation, superscript

To obtain the unbiased free energy **S7** and **S9**, we get

### Pressure 𝑷 ( 𝜶 𝒂 , 𝜶 𝒄 ) Plots.

Six different groups of polyhedra were used in umbrella sampling and chosen from three different slices of the 323 family (38). The pressure

### Free Energy Plots.

Similar to Figs. 2 and 3, which show the free energy calculation for two groups of polyhedra, Figs. S2–S5 show the free energy calculation of the remaining four groups. Figs. S2–S5 follow the same style, with Figs. S2*A*, S3*A*, S4*A*, and S5*A* indicating the polyhedra used, Figs. S2*B*, S3*B*, S4*B*, and S5*B* showing the pressure–shape constitutive relationship, and Figs. S2 *C* and *D*, S3 *C* and *D*, S4 *C* and *D*, and S5*C* showing the free energy calculation.

### Error Analysis.

All of the error bars in the free energy plots are generated using jackknife resampling. We selected 10 different subsets of all of the data from the replica runs and stitched together the resulting free energy curves from the subsets. The error from the weighted histogram analysis method (50) is negligible. However, jackknife resampling can only compute the statistical error; because of the large sample size (50,000 × 5 = 250,000), the statistical error is still small. The largest and most difficult error to calculate is the systematic error of umbrella sampling. This systematic error can come from multiple sources. First, because of the equilibration routine of the umbrella sampling simulation, for each individual window, the final distribution can be shifted slightly to the left or right of the target

## Results

We first present thermodynamic findings followed by dynamics results for the FCC

### FCC↔ BCC Transition.

We investigated the thermodynamics of shape change-driven FCC*B*), indicating a phase transition that is either first or second order in the Ehrenfest classification (44). Additional investigation via umbrella sampling yields the Landau free energy near the putative solid–solid transition for six different shapes depicted in Fig. 4*A*. Note that the similarity in particle shapes makes them difficult to distinguish by eye but is most clearly indicated by the relative size of the square face. Particles are colored from blue (BCC) to red (FCC) according to the structures that they spontaneously self-assemble. More blue (more red) shapes are more likely to form BCC (FCC). Shapes colored purple exhibit an almost equal probability to form either BCC or FCC. We computed the Landau free energy using the order parameter *C* and *D*, we plot Landau free energies obtained from umbrella sampling after averaging from five independent replica runs on both sides of the solid–solid phase transition. Calculations at *C*) show that, sufficiently far into the BCC phase, there is no metastable FCC free energy basin; however, as *D*), umbrella sampling calculations show that FCC becomes the stable free energy basin and that the BCC basin becomes metastable. Well below the transition, the BCC basin disappears, but a metastable basin develops that corresponds to mixed FCC and HCP stacking. Corresponding plots that lead to the same conclusions for the other regions of shape space are shown in *SI Text*. Together, our results show that shape change-driven FCC

We investigated the dynamics of the FCC

### BCC↔ SC Transition.

We investigated the thermodynamics of BCC*D*, we plot the *B*. Fig. 6*A* shows *A*. Particles are colored from blue (BCC) to green (SC) according to the value of the order parameter *B*) of the order parameter *C* and show no evidence of secondary local minima that would indicate a discontinuous (i.e., first-order) phase transition. Umbrella sampling computations were performed at a higher resolution of shape space below the putative transition (*D*) and are consistent with the self-assembled *SI Text*.

As in the FCC

## Discussion

Motivated by the need for minimal models to study solid–solid transitions (10), the observation that, in these transitions, coordination polyhedra change shape (11), the connection between anisotropic colloid shape and valence (12, 13), the large body of work on entropy-driven ordering in systems of colloids with anisotropic shape (12⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓–23), and recently developed techniques for treating particle shape thermodynamically (39), we studied a class of minimal model systems exhibiting solid–solid phase transitions driven by changes in particle shape. We showed that particle shape change gives rise to several distinct solid–solid transitions in a single family of shapes and via MC simulation and umbrella sampling techniques. We investigated FCC

The physics of FCC

Constructing shape-driven solid–solid transitions furthers the aim of developing minimal models of these transitions because it allows the direct manipulation of coordination polyhedra. As we noted above, coordination polyhedra also reconfigure in solid–solid transitions in metallurgy with changes in pressure, density, or temperature. An additional complicating factor in those transitions is that both enthalpy and entropy play a role, and decoupling their effects is difficult (8). A side benefit of this approach is that, in the hard particle systems that we present here, the behavior is entirely driven by entropy. Future studies of systems with controllable shape and enthalpic interactions (12) could allow enthalpic and entropic contributions to be disentangled. An important question for additional investigation is whether the physics of solid–solid transitions is determined by the structures, the particle shapes, or an interplay between the two.

Another fundamental question that calls for additional investigation is the study of the kinetics of colloidal solid–solid phase transformations through nonclassical nucleation and growth. It is expected that the nucleation and growth of solid–solid transitions will be rich because crystals break the rotational symmetry required by classical nucleation theory, and recent experimental evidence (10, 56) shows evidence for two-step nucleation in quasi-2D systems. Minimal colloidal models of the type constructed here provide an avenue for the study of full 3D transformations.

Our results can also help to guide the synthesis of reconfigurable colloidal material (Fig. 7). Experiments have shown systems with changeable building block shape either directly (5, 52, 53, 57, 58) or effectively via depletion (34). Here, we show that, for colloidal particles that can be synthesized in the laboratory (18, 20), changing particle shape can be used to induce transformations between FCC

## Acknowledgments

We thank K. Ahmed, J. Anderson, J. Dshemuchadse, O. Gang, E. Irrgang, and D. Klotsa for helpful discussions and encouragement; and D. Klotsa for sharing with us an early version of the follow-up work of ref. 39. This material is based on work supported in part by the US Army Research Office under Grant Award W911NF-10-1-0518 and by a Simons Investigator award from the Simons Foundation to S.C.G. This work used the Extreme Science and Engineering Discovery Environment (XSEDE) (61), which is supported by National Science Foundation Grant ACI-1053575 (XSEDE Award DMR 140129). Additional computational resources and services were supported by Advanced Research Computing at the University of Michigan.

## Footnotes

↵

^{1}C.X.D. and G.v.A. contributed equally to this work.↵

^{2}Present address: Department of Physics, University of Michigan, Ann Arbor, MI 48109.- ↵
^{3}To whom correspondence should be addressed. Email: sglotzer{at}umich.edu.

Author contributions: C.X.D., G.v.A., and S.C.G. designed research; C.X.D., G.v.A., and S.C.G. performed research; C.X.D., G.v.A., and R.S.N. contributed new reagents/analytic 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 paper.

Reviewers: O.G., Brookhaven National Laboratory; and D.A.K., University at Buffalo, State University of New York.

The authors declare no conflict of interest.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1621348114/-/DCSupplemental.

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