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Shape-driven –solid transitions in

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 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/) 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 -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 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 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 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 | self-assembly | phase transitions | nanoparticles calculate free energy difference between two different states (41). We investigate solid–solid transitions between BCC and FCC espite wide-ranging implications for (1), ceram- crystals and between BCC and SC crystals. We find that both Dics (2), sciences (3, 4), reconfigurable materials (5, 6), and colloidal (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, , 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 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 , 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 . 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 (24, 25), sublimation (26), (27– 1073/pnas.1621348114/-/DCSupplemental.

E3892–E3899 | PNAS | Published online May 1, 2017 www.pnas.org/cgi/doi/10.1073/pnas.1621348114 Downloaded by guest on September 27, 2021 Downloaded by guest on September 27, 2021 ersnainfo h culsl-sebe hss hssi h ht einaento neeti hsppr eut o rniin n r given are 2 and 1 transitions approximated for an Results paper. are this indicated in phases in interest self-assembled are of The 3–6 not green. transitions are is for region SC Results white and respectively. the blue, 5, in and is 4 Phases BCC Figs. phases. red, of in self-assembled is density actual FCC packing the where a from structures, representation at self-assembled the structure indicate equilibrium colors the The in change a is ue al. et Du [(α octahedron the by 1. Fig. fam- 323 invariant triangle spheric adjacent the in in 4-FCC) found (∆ and is ily 2-BCC, space (1-SC, shape cells of unit regions small with hundreds crystals and shape. assemble of same scales, the those scale having at as the particles small on all typically model are is will we that variability therefore, shape particles (42), can colloidal nanometers change of of shape systems particle because in timescales Moreover, longer occur change. much shape on particle happens reconfiguration than scenar- structural considers which study in Our are ios transitions materials. solid–solid reconfigurable shape-driven developing if determine for to viable is study this of goal A Methods and Model colloidal reconfigurable designing materials. time- suggest- rationally self-assembly, on for for approaches occur those ing transitions with comparable solid–solid Finally, are which that polyhedra. scales in coordination systems particle find using manipulate we by directly array transitions wide to solid–solid a in up shape studying particles opens for fewer approach systems our or generally, of four More with cell. crystals unit transi- a cubic where between and are maximization tions by solely shape driven symmetry-invariant by change in induced are even transitions continuous, where continuous. or cases transitions discontinuous or phase thermodynamically solid–solid discontinuous be diffusionless can is that regard- transition show states, results the fluid Our intermediate whether transi- of solid–solid of evidence the less no of find dynamics cases, and or the the tions second both study of (i.e., We in cases continuous order). that, higher thermodynamically two contrast, is study in transition dis- also find, the thermodynamically and and We is transition transition order). transition BCC↔SC BCC↔FCC first the the cases, (i.e., of all continuous transforma- in cases mathematical that, four find linear study by We tions. related continuously that are lattices between transformations diffusionless are transitions 3.Smltoso adclodlplhdain polyhedra colloidal hard of intermedi- Simulations as 43). well as in shapes shapes These includes ate octahedron. and and truncations tetrahedron, tetrahedral of cube, sets the distinct two to cubes subjecting AB 0.25 0.75 aiyo hpsta aetesm on ru ymtyadself- and symmetry group point same the have that shapes of family A 0.5 0 1 323 fhr oyer 3)(i.1sosshapes); shows 1 (Fig. (38) polyhedra hard of ) pei rageivrat(∆ invariant triangle Spheric (A) .50507 1 0.75 0.5 0.25 0 ∆ 323 aebe yteie tteclodlsae(,1,20, 18, (6, scale colloidal the at synthesized been have a , α c ) eso i ie niaigrgoso hp pc nwihthere which in space shape of regions indicating lines six show We (B) (1,1). cube and (1,0)], and [(0,1) tetrahedron =(0,0)], 323 oyer omacniuu w-aaee (α two-parameter continuous a form polyhedra ) ∆ 323 aesonte to them shown have ∆ 323 IText SI sfre by formed is η = . 5 h ie r noae ihterlvn tutrltasto n direction. and transition structural relevant the with annotated are lines The 0.55. 0.25 0.75 0.5 0 1 inidctdi i.1aealkonbudre ewe h hssof phases the between boundaries known all are investiga- 1 in of interest Fig. regions of the in regions and neighboring indicated transitions, in tion found BCC↔SC FCC be and structures. can FCC↔BCC SC SC study and and We FCC, BCC as BCC, well on as focusing (dual BCC 1, and unit Fig. cube in have a indicated particles regions is the all 1) which (1, in and conventions self-dual), use is We volume. (which octahedron). tetrahedra the are to (0, conventions, 0) (1, these With and family. shape this in ia ehnc ramn fpril trbts(eesae sthermody- for as that, extended ensembles shape) enter (here quantities statis- thermodynamic attributes These consistent quantities. particle namic a of from treatment derived mechanics tical are relationships Alchem- (39). structural–property relationships ical structure–property “alchemical” generalized of use we and (46), were infi- which HPMC-plugin simulations in complicated All the units 38). using the (22, (45) avoid shapes HOOMD-Blue to of with family as performed this tar- so of the dilute behavior of pressure sufficiently each nite and of (22) assembly phases spontaneous fraction the get simulations packing observe (MC) fixed to Carlo at dense sufficiently Monte done were All computations (44). and approaches Landau and Ehrenfest conven- the to according shapes dif- denote of number We (α a both (22). tions for with structures stability examples) thermodynamic bulk three of ferent shows regions 2 narrow and (Fig. wide behavior self-assembly rich have h hmcleeet.I scnein omk eedetransforma- Legendre a make in energy to system transmute free convenient the to the is in alchemists of It ancient tion particles the elements. the of chemical of attempts the (failed) shape) the (here with attributes analogy alchemical basic term that the the fact of ifying the use from The (39). comes potentials” here “alchemical as where to and (trans- referred particles ordinary the the of as well space and as phase space rotational) shape and over lational taken is integral the where Z (N eivsiaesaecag-nue oi–oi rniin in transitions solid–solid change-induced shape investigate We is,t siaetelcto fpaebudre,w s h notion the use we boundaries, phase of location the estimate to First, the both using transitions solid–solid of thermodynamics the study We µ , V a, , c a .50507 1 0.75 0.5 0.25 0 T , r hroyaial ojgt otesaevariables shape the to conjugate thermodynamically are , α ∆ µ c 323 ,wee0 where ), a , k a µ . B , c T ) α = ≡ c BCC aiyo ymti ovxsae htaebounded are that shapes convex symmetric of family ) ∆ 4 1. e 323

FCC −βφ ≤ φ aeteform the have , 1 3 α rmthe from = a, PNAS c Z

BCC FCC FCC BCC ≤ 6 µ

FCC 5 d enstebudre fsaespace shape of boundaries the defines 1 a, α

BCC c | a NVT noe h reeeg oto mod- of cost energy free the encodes BCC SC d ulse nieMy1 2017 1, May online Published α c µ 2 [

BCC SC dp][dq]e a µ c nebet h reenergy free the to ensemble )i nothdo,(,1) (0, octahedron, an is 0) −β (H −µ η a N = α a 5 hc is which 0.55, −µ β c N NVT BCC FCC α = α | SC ∆ a, c 1/k ) E3893 323 , c ∆ µ and a 323 [1] B µ in T c .

APPLIED PHYSICAL PNAS PLUS SCIENCES Fig. 2. Sample self-assembled colloidal crystals formed by shapes in the ∆323 triangle-invariant family of hard polyhedra, with images showing par- ticle shape and bond order diagram. (A) An FCC crystal self-assembled from shape (αa, αc) = (0.4, 0.525). (B) A BCC crystal self-assembled from shape (αa, αc) = (0.4, 0.59). (C) An SC crystal self-assembled from shape (αa, αc) = (0.76, 0.76). Note the similarity of shapes in A and B; even small shape differ- ences can affect the bulk self-assembly of hard polyhedra. Shapes in A and B are both on line 1 in Fig. 1, and the shape in C is on line 2 in Fig. 1.

F = φ + µaNαa + µcNαc of the NVTαaαc ensemble, from which we can shapes near the solid–solid transition; all systems contained N = 500 particles extract the constitutive relation (Fig. 4A). For BCC↔SC transitions, we studied two distinct regions of shape  ∂F  space, in both cases using different polyhedra in systems of N = 432 parti- P(αa, αc) = − . [2] cles (Fig. 5A). In all cases, five independent replicates were used to generate ∂V N, T, α , α a c umbrella samples. Umbrella samples were used to reconstruct free energy A thermodynamic , by the standard approach of Ehrenfest curves using the weighted histogram analysis method (50), and errors were (44), is indicated if a thermodynamic quantity [e.g., P(αa, αc)] or any of its estimated using jackknife resampling (51). Additional methodological details derivatives is discontinuous. A discontinuity of P(αa, αc) signals a thermody- are in SI Text. Data, documentation, simulation code, and analysis code are namic phase transition in shape space because of the explicit shape depen- available on request from S.C.G. dence in this relation. Accordingly, we searched for discontinuities by initial- izing systems with different building blocks (examples are in Fig. 3 A and C) Results in perfect BCC (N = 2,000), FCC (N = 2,048), or SC (N = 2,197) structures and 7 We first present thermodynamic findings followed by dynamics computed P(αa, αc) (Fig. 3 B and D) after 1.5 × 10 MC steps to ensure that systems reach equilibrium or metastable equilibrium using standard tech- results for the FCC↔BCC transition. Next, we present the same niques (47). for the BCC↔SC transition.

Second, having located discontinuities in P(αa, αc) and its derivatives, we computed the free energy as a function of order parameter (i.e., the Landau FCC↔BCC Transition. We investigated the thermodynamics of free energy) (44) for a series of fixed particle shapes near the solid–solid tran- shape change-driven FCC↔BCC solid–solid phase transitions in sition using umbrella sampling (40). To quantify the system crystal structure, four distinct regions of shape space (indicated by lines 1 and 3–5 we used a neighbor-averaged (48) version of the standard spherical harmonic in Fig. 1). In each region, at the FCC↔BCC cross-over, we find bond order parameters (49). To achieve good order parametric separation of that the P(αa , αc ) constitutive relation exhibits a discontinu- our crystal phases of interest, we used the second neighbor-averaged l = 4 ous first derivative (Fig. 3B), indicating a phase transition that

parameter Q4, which distinguishes BCC from FCC, SC, and hexagonally close- is either first or second order in the Ehrenfest classification (44). packed (HCP) phases in our systems as shown in Figs. 4B and 5B. To confirm Additional investigation via umbrella sampling yields the Landau

the validity of Q4 as an order parameter for monitoring the FCC↔BCC transi- free energy near the putative solid–solid transition for six dif-

tions, we plot thermal averages of Q4 computed in BCC, HCP, and FCC in Fig. ferent shapes depicted in Fig. 4A. Note that the similarity in

4B. Data indicate that BCC crystals have a peak near Q4 = 0.06, that HCP has a particle shapes makes them difficult to distinguish by eye but

peak near Q4 = 0.09 and a second smaller peak around Q4 = 0.13 because of is most clearly indicated by the relative size of the square face.

mixed HCP–FCC stacking, and that FCC has a peak near Q4 = 0.17. The peaks Particles are colored from blue (BCC) to red (FCC) according

are well-separated, and therefore, Q4 is a good distinguishing measure of to the structures that they spontaneously self-assemble. More a crystal phase. Umbrella sampling calculations used 5 × 104 samples in 32 blue (more red) shapes are more likely to form BCC (FCC).

equally spaced windows in Q4 across each transition with a harmonic con- Shapes colored purple exhibit an almost equal probability to straint of spring constant k = 3.5 × 104.[k is parametrically large because it form either BCC or FCC. We computed the Landau free energy

scales like the inverse square of the resolution of the order parameter, δQ4. using the order parameter Q 4 defined above. In Fig. 4 C and For our crystals of interest, Q4 falls in the range of 0.05–0.2, so that we need D, we plot Landau free energies obtained from umbrella sam-

to be able to resolve order parameter intervals of δQ4 ≈ 0.005. The value pling after averaging from five independent replica runs on both 4 ∗ k = 3.5 × 10 that we found to be consistent with efficient sampling is con- sides of the solid–solid phase transition. Calculations at αc > αc −2 sistent with a naive estimate k ≈ (δQ4) .] We study FCC↔BCC transitions (i.e., above the FCC↔BCC transition) (Fig. 4C) show that, suf- in four distinct regions of shape space, in each case using six polyhedra with ficiently far into the BCC phase, there is no metastable FCC free

E3894 | www.pnas.org/cgi/doi/10.1073/pnas.1621348114 Du et al. Downloaded by guest on September 27, 2021 Downloaded by guest on September 27, 2021 in akrclr niaetevleo h re parameter order initializa- the BCC of value indicate the triangles indicate colors initialization; Marker system tion. SC indicate Squares bu)t C(re) ( (green). SC to (blue) parameter indicate order colors the Marker of initialization. value system the BCC Pressure–shape indicate (B) triangles (blue). and BCC tion, to fixed back at then and relation (red) constitutive FCC to (blue) BCC into orsodn olns1ad3i i.1 rosaesalrta akrsize. marker than smaller are Errors 1. Fig. boundaries in ↔FCC BCC 3 (C and the 1 show lines regions to corresponding Boxed equilibration. after system in evolution Shape (A) xs o yaial leigclodsae(,4,5,5) Here, 53). 52, 42, (5, techniques shape experimental colloid altering Several sud- dynamically shape. a for to particle exist responds in system change the how den modeling by transition phase solid–solid transitions. FCC↔BCC in in change-driven transitions shown other shape phase the are that for show space conclusions results same shape Corre- the stacking. of to HCP regions lead and that devel- FCC plots basin mixed tran- metastable sponding to a the corresponds but below that disappears, and Well ops basin basin metastable. BCC energy becomes the free basin sition, stable BCC the the becomes that FCC that show tions At appears. basin as however, basin; energy not did in that systems 5 are and Outliers 4, size. 2 marker 2, in than equilibrate lines smaller are to show Errors regions corresponding 1. Boxed Fig. boundaries equilibration. after BCC↔SC system and the BCC↔FCC of structure final the in 3. Fig. ue al. et Du D C A B hp vlto for evolution Shape ) C solid–solid FCC↔BCC the of dynamics the investigated We rsuesaecntttv eainfor relation constitutive Pressure–shape BCC-Like BCC-Like Line 2 × 10 7 α Csteps. MC rsuesaecntttv eainfor relation constitutive Pressure–shape D) α c ∆ c α α < Line 3 tfixed at a 323 = FCC-Like α c ∗ α a α r rtodrtemdnmcphase thermodynamic first-order are ,ubel apigcalcula- sampling umbrella 4D), (Fig. c = hpsvr rmsl-sebeit BCC into self-assemble from vary Shapes . c Q .Crlsidct C ytminitializa- system FCC indicate Circles 0.4. α approaches 4 a BCC FCC optdi h nlsrcueo the of structure final the in computed = .Sae ayfo self-assemble from vary Shapes 0.4. oehr our Together, Text. SI α c ∗ SC-Like α eatbeFCC metastable a , BCC-Like a = Line 1 BCC SC and 0.4 Q 4 computed α α a a = = BCC FCC BCC SC α α c c . . nemdaefli hs.Teeitneo eatblt nshape in of discernible existence no which The with phase. in fluid reconfiguration region, intermediate structural metastable observed did the we we beyond case However, were shapes 19). systems 17, with when 15, transition initialized solid–solid (12, spontaneous type a tran- this observe of phase systems are fluid–solid observe in that first-order to sitions timescales observe barrier to on energy sufficient transition is typically free phase reconfiguration solid–solid the first-order structural overcome space a for to shape force of sufficient driving regions not the metastable in that simulations suggest as event, MC rare any indicated our a for and inherently computation is transformation sampling Homogeneous umbrella structural metastable. our observe that not systems did metastable longer the we outside much transition region, and solid–solid the times for relaxation needed than structural typical with pared S1 density Movie packing vice and shape (and particle for lattices relaxation fixed reconfiguration at BCC structural simulating shape in than and versa) particles particle timescale FCC-forming shorter which initializing much by in a process on occurs a model we C reeeg ai eeosnear develops basin (α energy free trian- spheric FCC (C of systems polyhedra. thermal in invariant phases gle crystal HCP and Sec- FCC, BCC, ( B) guishes calculations. sampling umbrella neighbor-averaged in ond used Shapes invariant triangle (A) spheric polyhedra. in hard transition phase thermodynamic first-order a by 4. Fig. ntbe n eodmtsal C ai appears. basin HCP metastable second a (α and unstable, transition the above well D C B A c α < 1.5 c ∗ × C eofiuaini accompanied is reconfiguration FCC↔BCC structural Shape-induced ,teFCfe nrybsnbcmsdmnn ( dominant becomes basin energy free FCC the ), 10 .O hs iuaintmsae,wihaeln com- long are which timescales, simulation these On ). 7 Csep eapesmlto eut r hw in shown are results simulation (example sweeps MC (0.4, 0.56) (0.4, 0.57) (0.4, 0.59) l BCC = bv h rniin(α transition the Above ) peia amncodrparameter order harmonic spherical 4 c PNAS = 4,teBCfe nrybsnbecomes basin energy free BCC the 0.54), HCP | ulse nieMy1 2017 1, May online Published α c = eo h transition the Below (D) 0.58. c α > c ∗ (0.4, 0.525) (0.4, 0.540) (0.4, 0.550) α ,ametastable a ), FCC c = 5,and 0.55), Q | 4 distin- E3895

APPLIED PHYSICAL PNAS PLUS SCIENCES gies computed via umbrella sampling are plotted in Fig. 5C and A show no evidence of secondary local minima that would indi- cate a discontinuous (i.e., first-order) phase transition. Umbrella sampling computations were performed at a higher resolution of 0.575 0.6 0.625 0.65 0.675 0.7 0.725 shape space below the putative transition (α . 0.6) to extract the B Q D expected value of the order parameter 4 (Fig. 5D) and are con- sistent with the self-assembled Q 4 measurements (Fig. 6), sug- gesting that Q 4 has a discontinuous derivative at the transition. Together, the P(αa , αc ) constitutive relation, the direct evalu-

ation of the order parameter Q 4, and the umbrella sampling results all indicate that the BCC↔SC solid–solid phase transition is a continuous (i.e., second- or higher-order) thermodynamic phase transition in ∆323. Evidence that the BCC↔SC solid–solid phase transition is also continuous in shape space region 6 (Fig. 1) is given in SI Text. C 0.575 As in the FCC↔BCC case, we investigated the dynamics of 0.58 the BCC↔SC solid–solid phase transformation by modeling how 0.585 the system responds to a sudden change in particle shape. We 0.59 model a process in which particle shape reconfiguration occurs 0.595 0.6 on a much shorter timescale than the structural relaxation by ini- 0.625 tializing BCC-forming particles in SC lattices (and vice versa) 0.65 and simulating at fixed particle shape and packing density for 0.675 1.5 × 107 MC sweeps (example simulation results are shown 0.7 0.725 in Movie S2). On these simulation timescales, in all cases, we observed dynamic solid–solid phase transformations via a tran- sition pathway through intermediate structures that follow the order parameter that we used for umbrella sampling. Moreover, Fig. 5. Shape-induced structural BCC↔SC reconfiguration occurs continu- we observe that, for α above the transition, any shape perturba- ously in spheric triangle invariant hard polyhedra. (A) Sample shapes used tion induces a structural change with no evidence of metastabil- in umbrella sampling calculation from the start to end in equal space. C ity. We also observe that the dynamics of the solid–solid trans- shows all shapes. (B) Second neighbor-averaged l = 4 spherical harmonic formation occurs on typical timescales of 2 × 106 MC sweeps. order parameter Q4 shows a series of structures between BCC and SC. Taken together, these results provide additional corroboration (C) Umbrella sampling shows a continuous phase transition. (D) Location of free energy minima extracted from umbrella sampling simulations as a of our observation that BCC↔SC is a continuous thermody- namic phase transition in ∆323 at fixed packing fraction η = 0.55, function of αa, c. and the timescale under which the solid–solid transition occurs dynamically is shorter by nearly an order of magnitude than in space provides an additional confirmation that the solid–solid the case of FCC↔BCC. transition is first order. Moreover, by measuring the order parameter evolution in MC simulation, our results indicate Discussion that the transition pathway in MC simulation follows the order Motivated by the need for minimal models to study solid– parameter that we chose in umbrella sampling, providing addi- solid transitions (10), the observation that, in these transitions, tional confirmation that it appropriately parametrizes the FCC↔ BCC solid–solid transition. BCC↔SC Transition. We investigated the thermodynamics of A BCC↔SC solid–solid phase transitions in two distinct regions SC of shape space (lines 2 and 6 in Fig. 1). In Fig. 3D, we plot BCC SC the P(αa , αc ) constitutive relation with αa = αc ≡ α for region 2, which shows a discontinuity in pressure near α ≈ 0.6 consis- tent with a phase transition that is, at most, second order in the Ehrenfest classification. A close-up view of these data is pre-

sented in Fig. 6B. Fig. 6A shows Q 4 order parameter measure- ments that suggest a discontinuous first derivative with respect to α, which is also consistent with a continuous (i.e., second or B SC higher order) thermodynamic phase transition. Corroborating BCC evidence is provided by computing the Landau free energy as a

function of the order parameter Q 4 near the putative solid–solid transition via umbrella sampling for a range of shapes indicated BCC in Fig. 5A. Particles are colored from blue (BCC) to green (SC)

according to the value of the order parameter Q 4 of the struc- tures into which they self-assemble. Computed thermal averages

(Fig. 5B) of the order parameter Q 4 in BCC and SC crystals show that BCC crystals have a peak near 0.08 and that SC has a peak near 0.17; however, our results also suggest the exis- Fig. 6. BCC↔SC solid–solid phase transition is a continuous (i.e., second- or higher-order) thermodynamic phase transition. (A) Order parameter Q4 vs. tence of structures with intermediate Q 4 for intermediate parti- shape suggests the derivative of the order parameter changes discontinu- cle shapes, where their self-assembled structures are in between ously near αa, c = 0.6. (B) P(αa, αc) also indicates a discontinuous derivative BCC and SC as shown in their Q 4 distribution. Landau free ener- near αa, c also consistent with a continuous phase transition.

E3896 | www.pnas.org/cgi/doi/10.1073/pnas.1621348114 Du et al. Downloaded by guest on September 27, 2021 Downloaded by guest on September 27, 2021 oi–oi hs rniin cu yaial nM iuain ntmsae (τ timescales on simulations MC in dynamically occur transitions phase solid–solid 7. Fig. ue al. et Du solid–solid on was here be focus could our future in there although question Additionally, shifting, this address work. shape will of we activation; that, paths thermal possible less/no other is with It particles FCC discontinuous. for the is space, transition shape phase of solid–solid regions distinct several for that, between so-called transition the the in [e.g., metallurgy in interest addi- longstanding for directions several suggest investigation. results tional Our or second order). (i.e., and continuous higher FCC order) thermodynamically the first is BCC↔SC that (i.e., that find discontinuous thermodynamically we is shape, sition particle the sym- group of point both common metry transi- the despite related and Surprisingly, Both are transitions diffusionless. (55). diffusionless SC be being cell to unit and expected the BCC are within tions (54); transformation shear direction a related one by are in FCC and elongation BCC centers. by particle transformations the of linear positions by the related of are are BCC↔SC that simulation and transitions MC ↔BCC FCC solid–solid Both via transitions. and FCC BCC↔SC investigated shapes and We of techniques. sampling family umbrella single and a solid–solid distinct in several to transitions rise gives showed change We solid–solid shape shape. particle exhibiting particle that in systems changes we by model driven (39), minimal transitions phase tech- of thermodynamically class developed shape a particle recently studied treating colloids and of for (12–23), systems niques shape in connection ordering anisotropic entropy-driven the with large on the work 13), (11), of (12, valence body and shape shape colloid anisotropic change between polyhedra coordination on simulations (τ MC in timescales dynamically on occur occur transitions not phase does solid–solid BCC↔SC reconfiguration continuous solid–solid Thermodynamically (τ region, times. timescales metastable self-assembly typical the than In longer region. much metastable the beyond sweeps) C oi–oi hs rniin sof is transitions phase solid–solid FCC↔BCC of physics The Timescale Timescale Timescale Initial Phase Initial Phase Initial Phase Final Phase Final Phase Final Phase hp-rvnsldsldrcngrto n efasml ieclsfrBC C,adS tutrs hroyaial icniuu FCC↔BCC discontinuous Thermodynamically structures. SC and FCC, BCC, for timescales self-assembly and reconfiguration solid–solid Shape-driven . γ 10 FC and (FCC) 6 Csep)ta r oprbewt rls hntpclsl-sebytimes. self-assembly typical than less or with comparable are that sweeps) MC α

BC om fio 5).W found We (54)]. iron of forms (BCC) Continuous Discontinuous Discontinuous Unstable Unstable Metastable rniinfo oi AssemblyfromFluid Transition fromSolid BCC FCC BCC FCC BCC C tran- ↔BCC SC ↔BCC ↔BCC r eurdb lsia ulainter,adrcn experi- nucleation recent two-step for and evidence symme- theory, shows 56) rotational nucleation (10, the evidence classical mental break by crystals required transi- because solid–solid try rich of be growth will and nucleation tions It the growth. that phase expected and solid–solid is colloidal nucleation of nonclassical kinetics through the transformations of study the is between tigation interplay an or determined shapes, is particle transitions two. the the solid–solid structures, of investigation the physics by additional the for dis- whether be question is to important contributions stud- An entropic Future and entangled. interactions enthalpic entropy. present enthalpic allow and by we could shape this (12) driven controllable that with of systems entirely systems benefit of particle is ies side hard and behavior the A role, the in a (8). here, that, play difficult is entropy is approach and effects enthalpy their den- those both decoupling in pressure, that factor in is complicating additional changes transitions An with temperature. or metallurgy solid– sity, in in reconfigure transitions also solid polyhedra we coordination As polyhedra. above, coordination noted of it manipulation because direct transitions these the of allows models minimal developing of aim wide of a structures. study relevant between technologically the transitions of phase for array solid–solid setting of powerful physics have basic a we the provide that approach will the here that generaliza- expect developed We straightforward approach. more only our with of require tions crystals or cells crystals unit noncubic complicated of per studies particles cell, fewer unit or cubic four with crystals cubic between transitions . nte udmna usinta al o diinlinves- additional for calls that question fundamental Another the furthers transitions solid–solid shape-driven Constructing 10 7 FCC BCC FCC BCC BCC Csep)ta r iia osl-sebytmsae (τ timescales self-assembly to similar are that sweeps) MC SC PNAS BCC Fluid BCC Fluid BCC Fluid | ulse nieMy1 2017 1, May online Published  10 7 Csep)ta are that sweeps) MC FCC Fluid FCC Fluid Fluid SC ≈ | 10 E3897 6 MC

APPLIED PHYSICAL PNAS PLUS SCIENCES in quasi-2D systems. Minimal colloidal models of the type observed spontaneous FCC↔BCC reconfiguration on time- constructed here provide an avenue for the study of full 3D scales of τ ∼ 106 MC sweeps. The relatively short timescales transformations. observed for reconfiguration suggests that, for sufficiently large Our results can also help to guide the synthesis of reconfig- shape deformations, although the phase transition is first order, urable colloidal material (Fig. 7). Experiments have shown sys- direct solid–solid reconfiguration without an intermediate fluid tems with changeable building block shape either directly (5, 52, can occur on comparable physical timescales to self-assembly 53, 57, 58) or effectively via depletion (34). Here, we show that, and therefore, is a viable means of designing reconfigurable col- for colloidal particles that can be synthesized in the laboratory loidal materials. In the BCC↔SC case, the continuous nature (18, 20), changing particle shape can be used to induce trans- of the transition implies that there is no nucleation barrier, and formations between FCC↔BCC and BCC↔SC. What implica- indeed, we observed structural reconfiguration in MC simula- tions are there for the rational design of reconfigurable colloidal tions of N ∼ 2 × 103 particles on typical timescales of τ ∼ 106, materials? To answer this question, it is important to understand which are less than what is typically observed for self-assembly of how structural reconfiguration compares with self-assembly in comparably sized systems of hard anisotropic colloids. The rela- terms of typical timescales. We obtain “timescales” via MC sim- tively fast speed at which structural reconfiguration occurs in this ulations involving local translations and rotations of individual case of a continuous solid–solid transition suggests that a broader particles to approximate the Brownian dynamics (59) of physi- search for other systems of anisotropic colloids that exhibit con- cal colloids (60). In the case of FCC↔BCC, for shapes near the tinuous solid–solid phase transitions could yield candidate sys- ∗ discontinuous transition (|α − α | . 0.05), we did not observe tems for developing rapidly switchable reconfigurable colloidal spontaneous structural reconfiguration in systems of N ∼ 2,000 materials. 7 particles on timescales of τ . 10 MC sweeps. This timescale is much longer than the typical time that it takes to observe ACKNOWLEDGMENTS. We thank K. Ahmed, J. Anderson, J. Dshemuchadse, spontaneous crystallization or melting in MC simulations of the O. Gang, E. Irrgang, and D. Klotsa for helpful discussions and encourage- N ∼ 2 × 103 τ ∼ 107 ment; and D. Klotsa for sharing with us an early version of the follow-up self-assembly of particles, for which . work of ref. 39. This material is based on work supported in part by the The contrasting timescales for self-assembly vs. solid–solid US Army Research Office under Grant Award W911NF-10-1-0518 and by a reconfiguration suggest that, for small shape deformations of Simons Investigator award from the Simons Foundation to S.C.G. This work ∗ |α − α | . 0.05, spontaneous, shape change-driven, dynamic used the Extreme Science and Engineering Discovery Environment (XSEDE) ↔ ∆ (61), which is supported by National Science Foundation Grant ACI-1053575 FCC BCC reconfiguration in 323 can be achieved on shorter (XSEDE Award DMR 140129). Additional computational resources and ser- timescales by completely melting and then recrystallizing the vices were supported by Advanced Research Computing at the University of ∗ system. However, for larger shape changes |α − α | & 0.05, we Michigan.

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