Role of local assembly in the hierarchical crystallization of associating colloidal hard hemispheres
aa r X i v : . [ c ond - m a t . s o f t ] S e p Role of local assembly in the hierarchical crystallization of associating colloidal hardhemispheres
Qun-li Lei, Kunn Hadinoto, and Ran Ni ∗ School of Chemical and Biomedical Engineering,Nanyang Technological University, 637459, Singapore
Hierarchical self-assembly consisting of local associations of simple building-blocks for the forma-tion of complex structures widely exists in nature, while the essential role of local assembly remainsunknown. In this work, by using computer simulations, we study a simple model system consist-ing of associating colloidal hemispheres crystallizing into face-centered-cubic crystals comprised ofspherical dimers of hemispheres, focusing on the effect of dimer formation on the hierarchical crystal-lization. We found that besides assisting the crystal nucleation because of increasing the symmetryof building-blocks, the association between hemispheres can also induce both re-entrant melting andre-entrant crystallization depending on the range of interaction. Especially when the interaction ishighly sticky, we observe a novel re-entrant crystallization of identical crystals, which melt only incertain temperature range. This offers a new axis in fabricating responsive crystalline materials bytuning the fluctuation of local association.
PACS numbers: 82.70.Dd, 64.75.Xc, 64.60.Q-,68.35.Rh
Hierarchical self-assembly, where the products fromthe lower-level assembly act as building-blocks for thehigher-level self-assembly, is first used by nature to ac-curately build complex micro-structures [1–3]. The pro-cesses are usually accompanied with the formation of lo-cal assemblies, e.g. dimerization [4, 5], with which higherlevel complex structures can be built with ease [6–10].For example, in the self-assembly of icosahedral viruscapsids, anisotropic protein monomers first form dimersto gain centrosymmetry, then the dimers assemble intopentamer blocks, which crystallize into “spherical crys-tals” [11, 12]. Accordingly, a new racemic protein crys-tallography [13] method was recently proposed, wheresynthesized enantiomers or enontiomorphs are used toco-crystallize some natural chiral proteins, whose crys-tals are difficult to obtain using traditional crystallogra-phy. [14].In colloidal self-assembly, one of the major tasks is todesign anisotropic particles to fabricate crystalline mate-rials with desired properties [15–18]. It was recently sug-gested that for self-assembly of complex colloidal crystals,one can pre-assemble the local structures to help the hier-archical crystallization [19–21]. However, the role of thelocal assembly for the hierarchical crystallization remainsunclear. Here we investigate the hierarchical crystalliza-tion of a simple yet representative system consisting of as-sociating colloidal hemispheres without centrosymmetry,which at high density self-assemble into a face-centered-cubic (FCC) crystal of spherical dimers of hemispheres,i.e. FCC crystal. We found that besides assisting thehierarchical nucleation of FCC crystal of colloidal hemi-spheres, the formation of local assemblies can induce, de-pending on the interaction range of association, both re-entrant melting and re-entrant crystallization of FCC crystals within certain density range. This suggests anew way of fabricating responsive photonic materials by controlling local structural fluctuations.We consider a system of N colloidal hard hemispheres,which at high density can crystallize into an FCC crys-tal [22–24]. To control the formation of local structures,i.e. spherical dimers, we introduce an attraction betweenhemispheres. The total energy of the system is given by U = X i
0 0.02 0.04 0.06 0.08 〈 ( ∆ U) 〉 /(N ε ) liquidsolid
0 20 40 60 pressure (k B T/ σ )P co ε / ( k B T ) 〈 U 〉 /(N ε )liquidsolid φ liquidsolid
0 0.02 0.04 0.06 0.08 〈 ( ∆ U) 〉 /(N ε ) liquidsolid
0 20 40 60 pressure (k B T/ σ )P co ε / ( k B T ) 〈 U 〉 /(N ε )liquidsolid φ liquidsolid
0 0.02 0.04 0.06 0.08 〈 ( ∆ U) 〉 /(N ε ) liquidsolid
0 20 40 60 pressure (k B T/ σ )P co FIG. 2: (Color online) Phase diagrams of associating colloidal hemispheres for various interaction ranges, i.e. r c =0 . σ (b), 0 . σ (f) and0 . σ (j), in the representation of volume fraction vs. inverse temperature ǫ/k B T . The average energy per particle h U i /Nǫ , normalizedenergy fluctuation h ∆ U i /Nǫ , and the co-existing pressure P co are shown in column one (a,e,i), three (c,g,k) and four (d,h,l),respectively. hemisphere. The reduced temperature T ∗ = k B T /ǫ con-trols the associating degree, or the dimer fraction θ , with k B and T the Boltzmann constant and temperature of thesystem, respectively. Here a spherical dimer is defined asa collection of two hemispheres, whose center-to-centerdistance is smaller than r c . In the limit of T ∗ →
0, allhemispheres form spherical dimers in the fluid recoveringa system of hard spheres. [25].We first calculate the phase diagram for the system ofhard hemispheres, i.e. ǫ/k B T = 0, by using the Einsteinintegration, where all particles are modelled as penetra-ble repulsive hemispheres and each particle is attachedto a crystalline lattice site via a spring. By increasingthe strength of the spring and decreasing the strength ofrepulsion, the system recovers a non-interacting Einsteinplastic crystal [26]. However, different from conventional plastic crystals, in the FCC crystal of hemispheres, thetwo particles on the same lattice site are exchangeablecontributing a free energy of ln 2 k B T per particle, andthe resulting free energy of the FCC Einstein crystal is F Einst k B T = − N − (cid:18) πk B Tλ max (cid:19) + ln (cid:18) σ V N / (cid:19) + N , (3)where λ max is the strength of spring with V the vol-ume of the system. This extra free energy contributionfrom indistinguishness generally exists in all hierarchicalplastic crystals, whose building-blocks are local assem-blies of smaller particles. By using this Einstein crystalcombined with thermodynamic integrations [27], we ob-tain [ φ f , φ F CC ] = [0 . , . φ f and φ F CC theco-existing packing fraction of the fluid and FCC crys- -5 0 5 10 15 20 25-0.5 -0.25 0 g / ( k B T ) 〈 U 〉 /(N ε ) (a) (b) (c) (d) r c =0 σ FCC fluid liquidsolid φ liquidsolid
0 0.1 0.2 0.3 〈 ( ∆ U) 〉 /(N ε ) liquidsolid
0 10 20 30 40 50 60 pressure (k B T/ σ ) P co FIG. 3: (Color online) Phase diagrams of sticky colloidal hemispheres, i.e. r c =0 (b), in the representation of volume fraction vs.association strength g/k B T . Corresponding average energy per particle h U i /Nǫ , normalized energy fluctuation h ∆ U i /Nǫ , and theco-existing pressure P co are shown in (a), (c) and (d), respectively. tal, respectively. These are substantially higher than thevalues obtained in Ref. [24], and the reason is the lastterm of Eq. 3 missing in the previous works [28], as byreducing the free energy of our crystal phase by ln 2 k B T per particle, we obtain the same phase boundaries as inRef. [24].Next we trace the change of phase boundaries as afunction of ǫ/k B T by using the Gibbs-Duhem integration (cid:20) d ln P d( ǫ/k B T ) (cid:21) coex = − ∆ hP ∆ vǫ/k B T , (4)where ∆ h and ∆ v are the difference of enthalpy and spe-cific volume between two coexisting phases, respectively.We perform isobaric-isothermal Monte Carlo simulationswith N = 1 ,
000 hemispheres to solve Eq. 4 starting fromthe system of hard hemispheres, i.e. ǫ/k B T = 0, and theresulting phase diagrams for various attraction rangesare shown in the second column of Fig. 2. For the caseof relatively long range attraction, i.e., r c = 0 . σ , one cansee that with increasing ǫ/k B T from 0, both the phaseboundaries of fluid and FCC phases first decrease andthen increase approaching the limit of hard-sphere sys-tems. They reach [ φ f , φ F CC ] = [0 . , .
52] at an interme-diate association ǫ/k B T ≃
17, which are even lower thanthose of hard-sphere systems. This non-monotonic be-haviour of the crystallization packing fraction implies aninteresting re-entrant melting at certain packing fractionrange with increasing the attraction. Moreover, from thehard-sphere limit, with decreasing the attraction, thephase boundaries shift to lower values, which suggeststhat at fixed packing faction, the crystal nucleation ratein the fluid increases when the spherical dimers have cer-tain shape fluctuations. With decreasing the attractionrange r c , the re-entrant melting becomes weaker, and italmost disappears at r c = 0 . . σ . Surprisingly,when the r c is very small, i.e. 0 . σ , the melting packingfraction of the FCC crystal changes non-monotonicallywhen approaching the system of hard hemispheres, and itreaches the maximal value of φ F CC ≃ .
64 at ǫ/k B T = 10. With further decreasing the attraction, the melting lineof FCC crystal moves down to φ F CC ≃ .
62 at thehard-hemisphere limit. This non-monotonic behaviourof φ F CC suggests that at certain fixed packing frac-tion between 0.62 and 0.64, by increasing the strengthof short range attraction, the system undergoes a novelre-entrant crystallization by forming identical crystals atboth strong and weak attraction limits which melt atcertain intermediate attraction. However, although there-entrant melting and re-entrant crystallization both ex-ist in the system of associating colloidal hemispheres de-pending on the associating range, the co-existing pressurealways monotonically decreases with increasing ǫ/k B T (Fig. 2 right column). Additionally, by using the Gibbs-Duhem integration from hard-hemisphere systems withincreasing attraction, we reproduce the phase boundaryof hard sphere systems at ǫ/k B T → ∞ , which verifies ourfree energy calculation of hard hemisphere systems. Herewe focus on the phase transition between fluid and theFCC crystal, and full phase diagrams can be found inRef [28].To understand the physics behind these re-entrant be-haviours, we plot average energy per particle h U i /N ǫ and the energy fluctuation h ∆ U i /N ǫ on the fluid-FCC phase boundaries jointly with the phase digram in thefirst and third column of Fig. 2. In the systems ofshort range attractive hard hemispheres, the change of h U i /N ǫ is very similar to that of θ [28]. As shown inFig. 2a, e, and i, for r c = 0 . , . . σ , h U i /N ǫ of co-existing phases matches with each other at high attrac-tion strength, where all hemispheres form dimers. De-creasing ǫ/k B T increases h U i /N ǫ of co-existing phasesgradually, which implies that the average distance be-tween two hemispheres in spherical dimers increases.This change has little influence on the phase bound-ary when r c is small, i.e. 0 . σ . However, in the sys-tem of relatively longer range attraction, i.e. r c = 0 . σ ,this effectively increases the “size” of the spheres mov-ing the phase boundary to lower packing fractions. Fur- ∆ G ( n ) / ( k B T ) n g = 6.7 k B Tg = 1.7 k B Tg = -3.3 k B T
25 50 75 100 0 0.1 0.2 0.3 ∆ G * / ( k B T ) θ r c =0.3 σ r c =0.1 σ r c =0.05 σ r c =0 σ FIG. 4: (Color online) Nucleation barrier of FCC crystals∆ G ( n ) /k B T as a function of nucleus size n in systems of stickycolloidal hemispheres, i.e. r c =0, with various association strength g at the supersatruation of 0 . k B T per spherical dimer. Inset:the heights of nucleation barrier as a function of the freemonomer fraction 1 − θ for various attraction strength andinteraction range at the supersatruation of 0 . k B T per sphericaldimer, where the dash line is to guide the eye. ther decreasing the attraction induces deviation between h U i /N ǫ in the two co-existing phases, and the energy offluid increases faster than solid indicating that the disso-ciation of spherical dimers occurs first in the fluid. Thisimbalance implies that the fluid gains more entropy fromthe dissociation than solid. Then the co-existing pack-ing fractions shift to high values to equalize the chemicalpotentials of co-existing phases. This effect, along withthe increased number of free hemispheres, explains theincrease of co-existing pressure as well as the re-entrantmelting. Moreover, as shown in Fig. 2c, g, and k, theenergy fluctuations on the coexisting phases, especiallyin the coexisting FCC crystal, changes differently withdecreasing attraction for different r c . When the attrac-tion range is relatively long, i.e. r c = 0 . σ , the energyfluctuation h ∆ U i /N ǫ increases monotonically when de-creasing the attraction strength, while at short range at-tractions, it develops a maxima when approaching thehard hemisphere limit. Interestingly, the location of theenergy fluctuation maxima is very close to the maximalmelting packing fraction of FCC crystals leading to there-entrant crystallization of identical FCC crystals withincreasing attraction.To further explore the nature of this intriguing re-entrant crystallization, we simulate a system of associ-ating hard hemispheres with r c →
0. In this limit, tobind two hemispheres forming a spherical dimer, ǫ/k B T needs to approach infinity, and the dimerization fraction θ = − h U i /N ǫ . Therefore, instead of ǫ/k B T ,we definea dimerization free energy g to describe the associationstrength between hemispheres as g = − k B T ln Z b = − k B T ln (cid:26)Z exp [ − βU b ( r )] d s (cid:27) , (5) where Z b can be seen as the internal partition functionof a spherical dimer with s the internal degrees of free-dom of two hemispheres. Since the entropic barrier fordimerization increasing dramatically when r c →
0, we de-vise a modified aggregation-volume-bias Monte Carlo al-gorithm [29] to accelerate the simulation [28], and theresults are shown in Fig. 3. Compared with r c = 0 . σ , amore pronounced re-entrant crystallization is observed insystems with r c → crystalstill remain dimerized. A small further increase of tem-perature induces a sharp change of energy in the FCC crystal, and a pronounced energy fluctuation peak ap-pears suggesting a collective dissociation in the crystal,which is stronger at smaller r c . This collective behaviourcan be seen as a kind of weak solid-solid transition fromhigh density to low density similar to the solid-solid tran-sition in systems of sticky hard spheres [30]. However,in our systems of sticky hard hemispheres, the natureof dissociation of spherical dimers is continuous, and notstrong enough to drive a first order phase separation, butproduces a new re-entrant crystallization in the systemto form identical crystals with changing temperature.Furthermore, we study the nucleation of FCC crys-tals from the fluids of colloidal hemispheres. We performumbrella sampling Monte Carlo simulations [31, 32] tocalculate the free energy barrier ∆ G ( n ) /k B T = − ln P ( n )with P ( n ) the probability of finding a nucleus contain-ing n solid crystal-like dimers, which is determined byusing the bond orientation order parameter [28, 33]. Theobtained nucleation barriers for systems at the supersat-uration of | ∆ µ | = | µ F CC − µ fluid | = 0 . k B T per spheri-cal dimer with various association strength at r c = 0 areshown in Fig. 4. One can see that with decreasing theassociation strength g/k B T , at the same supersatura-tion, the nucleation barrier dramatically increases. Asshown in the inset of Fig. 4, nucleation barrier heightsof systems with different interaction ranges change verysimilarly with decreasing the fraction of spherical dimersin the supersaturated fluids. This suggests that the de-termining factor for the nucleation rate of FCC crys-tal is the fraction of spherical dimers in the fluid, whilethe exact form of interaction is less important. More-over, as our simulations are performed at the constantsupersaturation, the higher nucleation rate in strongertwo-step hierarchical self-assembling systems cannot beexplained by the increase of driving force. Instead, ourresults demonstrate that for particles of low-symmetry,like hemispheres, locally self-assembling into secondarybuilding-blocks of high-symmetry can dramatically in-crease the self-assembly efficiency [34, 35]. This gives ageneric explanation on why dimerization or local struc-tural formation is usually the first step in the proteinself-assembly, and why racemic protein crystallographyworks better by introducing local association of enan-tiomers [13].In conclusion, by performing computer simulations fora simple yet representative model system of colloidalhemispheres, we investigate the role of local assemblyin hierarchical crystallization. We found that dependingon the range of attraction driving the formation of lo-cal structures, i.e. spherical dimers, the system possesnovel re-entrant melting and re-entrant crystallization atcertain densities. Especially in the system of the stickycolloidal hemispheres, i.e. r c →
0, where the exact formof attraction is not important, increasing the strengthof attraction can induce a new re-entrant crystallizationby forming identical FCC crystals at both weak andstrong attraction limits which melts at intermediate at-traction strength. This is due to the collective disso-ciation of spherical dimers. We argue that this stickyassociation induced new re-entrant crystallization gen-erally should exist in many hierarchical self-assemblingsystems, and more subunits in each local assembly canproduce stronger re-entrant crystallization, which couldbe interesting for future investigations. In experiments,such sticky attraction, for example, can be realized byusing hydrophobic coatings on the flat surface of col-loidal hemispheres [36–38], which may open up a newway of making novel responsive photonic materials [39].Moreover, we also studied the nucleation of FCC crys-tal from supersaturated fluids, and we demonstrated thatat the same supersaturation, the increase of the frac-tion of spherical dimers in fluids significantly lowers thenucleation barrier suggesting that the existence of pre-assembled local structures is of primary importance forthe hierarchical crystallization, which is relevant for de-signing the self-assembly of anisotropic colloids [21] andprotein crystallization [13]. Our results lay the first stonein understanding the role of local structural formation inthe multi-scale hierarchical assembly, and a number ofinteresting questions can be further explored in this di-rection, e.g. the effect of local structural fluctuations onhierarchical glass transitions [40].This work is supported by Nanyang TechnologicalUniversity Start-Up Grant (NTU-SUG: M4081781.120),Academic Research Fund Tier 1 from Singapore Min-istry of Education (M4011616.120), the Advanced Man-ufacturing and Engineering Young Individual ResearchGrant (M4070267.120) by the Science & Engineering Re-search Council of Agency for Science, Technology andResearch Singapore, and Green and Sustainable Manu-facturing Trust Fund 2013 by GlaxoSmithKline (Singa-pore). We are grateful to the National SupercomputingCentre (NSCC) of Singapore for supporting the numeri-cal calculations. ∗ [email protected][1] J. A. Elemans, A. E. Rowan, and R. J. Nolte, Journal ofMaterials Chemistry , 2661 (2003).[2] B. J. Pieters, M. B. van Eldijk, R. J. Nolte, and J. Me-cinovi´c, Chemical Society Reviews , 24 (2016).[3] B. Alberts et al. , Molecular Biology of the Cell (Garlandscience, New York, 2002).[4] N. J. Marianayagam, M. Sunde, and J. M. Matthews,Trends in biochemical sciences , 618 (2004).[5] J. M. Matthews, editor, Protein dimerization andoligomerization in biology (Springer Science& BusinessMedia, 2012).[6] S. E. Ahnert, J. A. Marsh, H. Hern´andez, C. V. Robin-son, and S. A. Teichmann, Science , aaa2245 (2015).[7] N. P. King and Y.-T. Lai, Current opinion in structuralbiology , 632 (2013).[8] D. S. Goodsell and A. J. Olson, Annual review of bio-physics and biomolecular structure , 105 (2000).[9] J. Zhang, F. Zheng, and G. Grigoryan, Current opinionin structural biology , 79 (2014).[10] J. B. Bale et al. , Science , 389 (2016).[11] V. Krishnamani, C. Globisch, C. Peter, and M. Deserno,The European Physical Journal Special Topics , 1757(2016).[12] J. E. Baschek, H. C. Klein, and U. S. Schwarz, BMCbiophysics , 22 (2012).[13] T. O. Yeates and S. B. Kent, Annual review of biophysics , 41 (2012).[14] A. Laganowsky et al. , Protein Science , 1876 (2011).[15] S. C. Glotzer and M. J. Solomon, Nature Materials ,557 (2007).[16] B. Li, D. Zhou, and Y. Han, Nature Reviews Materials , 15011 (2016).[17] P. Damasceno, M. Engel, and S. Glotzer, Science ,453 (2012).[18] U. Agarwal and F. Escobedo, Nature Materials , 230(2011).[19] A.-P. Hynninen, J. H. Thijssen, E. C. Vermolen, M. Di-jkstra, and A. Van Blaaderen, Nature materials , 202(2007).[20] G. Avvisati, T. Dasgupta, and M. Dijkstra,arXiv:1603.07591 (2016).[21] ´E. Ducrot, M. He, G.-R. Yi, and D. J. Pine, NatureMaterials , doi:10.1038/nmat4869 (2017).[22] M. Marechal, R. J. Kortschot, A. F. Demirors, A. Imhof,and M. Dijkstra, Nano letters , 1907 (2010).[23] M. Marechal and M. Dijkstra, Physical Review E ,031405 (2010).[24] J. M. McBride and C. Avenda˜no, Soft Matter , 2085(2017).[25] W. Hoover and F. Ree, J. Chem. Phys. , 3609 (1968).[26] A. Fortini, M. Dijkstra, M. Schmidt, and P. Wessels,Phys. Rev. E , 051403 (2005).[27] D. Frenkel and B. Smit, Understanding Molecular Sim-ulations: From Algorithms to Applications (AcademicPress, 2002).[28] Supplementary information .[29] B. Chen and J. Siepmann, J. Phys. Chem. B , 11275(2001).[30] P. Bolhuis, M. Hagen, and D. Frenkel, Physical ReviewE , 4880 (1994). [31] L. Filion, M. Hermes, R. Ni, and M. Dijkstra, J. Chem.Phys. , 244115 (2010).[32] R. Ni and M. Dijkstra, J. Chem. Phys. , 034501(2011).[33] P. Steinhardt, D. Nelson, and M. Ronchetti, Phys. Rev.B , 784 (1984).[34] J. Glaser, A. Karas, and S. Glotzer, J. Chem. Phys. ,184110 (2015).[35] B. Schultz, P. Damasceno, M. Engel, and S. Glotzer, ACSNano , 2336 (2015).[36] Q. Chen, S. Bae, and S. Granick, Nature , 381 (2011).[37] Q. Chen et al. , Science , 199 (2011).[38] K. Chaudhary, Q. Chen, J. Juarez, S. Granick, andJ. Lewis, J. Am. Chem. Soc , 12901 (2012).[39] J. Ge and Y. Yin, Angew. Chem. Int. Ed. , 1492(2011).[40] T. Speck, A. Malins, and C. Royall, Phys. Rev. Lett.109