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Self-Assembly of Mesophases from Nanoparticles Abhinaw Kumar, and Valeria Molinero J. Phys. Chem. Lett., Just Accepted Manuscript • DOI: 10.1021/acs.jpclett.7b02237 • Publication Date (Web): 29 Sep 2017 Downloaded from http://pubs.acs.org on September 30, 2017

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The Journal of Physical Chemistry Letters

Self-assembly of Mesophases from Nanoparticles Abhinaw Kumar and Valeria Molinero* Department of Chemistry, The University of Utah, 315 South 1400 East, Salt Lake City, Utah 84112-0850 ABSTRACT: A growing number of crystalline and quasi-crystalline structures have been formed by coating nanoparticles with ligands, polymers and DNA. The design of nanoparticles that assemble into mesophases such as those formed by block copolymers would combine the order, mobility and stimuli responsive properties of mesophases with the electronic, magnetic and optical properties of nanoparticles. Here we use molecular simulations to demonstrate that binary mixtures of unbound particles with simple short-ranged pair interactions produce the same mesophases as block copolymers and surfactants, including lamellar, hexagonal, gyroid, bcc, fcc, perforated lamellar and semicrystalline phases. The key to forming the mesophases is the frustrated attraction between particles of different types, achieved through control over interparticle size, strength and softness of the interaction. Experimental design of nanoparticles with effective interactions described by the potentials of this work would provide a distinct, robust route to produce ordered tunable liquid crystalline mesophases from nanoparticles.

AA BB

AB

KEYWORDS: liquid crystals, block copolymers, gyroid, lamellar, microphase separation

Molecules that have two moieties that “repel” each other yet are chemically bound, such as block copolymers and surfactants, form a wealth of mesophases, including lamellar, gyroid and hexagonal.1-5 These mesophases result from the repulsion between incompatible groups that cannot separate macroscopically because they are chemically bound. Stillinger proposed that the physics of microphase separation in surfactants is equivalent to that of oppositely charged particles subject to geometric constraints.6 Wu et al. showed that this physics could be captured with a charge-frustrated model with competing short range (spin) and long-range (electrostatic) interactions.7-9 The same frustration can be built into single component systems with two interaction length scales: simulations of particles with short-range attraction and long-range repulsion10-13 (SALR) produce the same mesophases as block copolymers.14-16 Likewise, simulations of two-dimensional hardcore soft-shoulder (HCSS) potentials with two length scales can stabilize stripe phases, the two-dimensional analog of the lamellar mesophase.17-19 The ratio of shoulder width to core diameter is key for the formation of the mesophases in HCSS models: potentials with a ratio larger than 2 produce stripes and other mesophases, while shorter

ratios of shoulder to core produce only liquid, crystal and quasicrystal phases.19-25 A wealth of structures, including porous materials, can be designed by proper selection of the effective interaction potential.26-41 However, to date it has not been possible to realize particles interacting with SALR and HCSS potentials that produce the mesophases in experiments.16 This poses the question of whether robust, simpler interactions potentials between nanoparticles can result in the assembly of mesophases and open an avenue for their realization in experiments. Here we show that simulations of binary mixtures of two types of particles, A and B, that interact exclusively through simple, short-range isotropic potentials produce the same mesophases as block copolymers. For simplicity, we start by assuming that A and B particles are identical and interact through a pair potential with a repulsion that scales with r4 and an attraction with length-scale comparable to the size of the repulsive core of the particle (Figure 1). The frustration needed to produce microphase separation is implemented through the condition that while unlike particle cannot come as close as particles of same type (σAB/σAA > 1), they do nevertheless experience more total attraction than like particles. The ratio between the AB and

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AA attraction is controlled by the depth of the interaction potentials ε /ε and the ratio of the total number of AB and AA neighbors; the latter increases with σAB/σAA. Particles of the same type are pushed together to maximize the strong effective AB attraction, introducing two characteristic length scales in the interactions of the binary mixture. The result is an effective interaction with two-length scales that leads to the formation of the mesophases. ΑΑ

6 σAA

a)

σBB

A

AB attraction,"εAB/εAA

4

σAB > σAA = σBB

-2 0.9

LXs

LXa

Lamellar Mesophase

1 Isotropic Thick Lamellar

0.5

AB

0.6

LXa

1.5

B

2 BB 0

AG

2

1.2

1.5

0 1

1.8

Interparticle Distance (r/σ) Figure 1. Interaction potential between the nanoparticles. Particles interact through two-body part of the StillingerWeber interaction potential,42 shown equation s1 of Supporting Information A. Interaction potentials between pairs of A particles (red curve) and pairs of B particles (blue curve) are identical (σAA= σBB=1, εAA= εBB=1 kcal mol-1). Interaction between A and B shown here with the black curve has σAB/σAA = 1.045 and εAB = 2.1 kcalmol-1. The potentials have minima at 1.112*σ and vanish at 1.8*σ. Supporting Information C shows that the mesophases can be stabilized also when the AA or BB interaction is purely repulsive. The softness of the AA and BB repulsion, however, is needed to produce the mesophases. Figure 2a shows the phase diagram for the equimolar mixture of identical A and B particles as a function of the strength of the AB attraction εAB and the size repulsion σAB/σAA. The phase diagram is computed keeping εAA = 1 kcal mol-1, T = 300 K, and p = 0 atm. There is a wide region of frustrated attraction where the equimolar binary mixture of A and B particles produces lamellar mesophases. The lamellar mesophases of the binary mixture can be thin or thick; these structures shown in Figure 3. Thin lamellar (also called lamellar in what follows) has stacks of alternated disordered single layers of A and B particles (Figure 3 and Supporting Information B) and is largely stabilized by AB attraction (Supporting Information C). Figure 2a shows that the stabilization of the lamellar mesophase at T = 300 K and p = 0 requires a minimum size repulsion σAB/σAA = 1.10.

Phase Segregated A and B

1.1 1.2 1.3 Size repulsion,"σAB/σAA

b) 450 400

350

1.4

c) Isotropic

Lamellar

LXa

300 1.4 1.5 1.6 1.7 1.8 AB attraction,"εAB (kcal/mol)

Temperature (K)

AA

However, we find that at higher temperatures the lamellar mesophase is thermodynamically stable at p = 0 for mixtures with size repulsion σAB/σAA as low as 1.045 (Figure 2b & 2c). Below that ratio, e.g. at σAB/σAA = 1.04, we do not find any combination of T and εAB for which lamellar is the most stable phase, although it can remain as a metastable phase. We conclude that a small repulsion suffices to produce a stable lamellar mesophase.

Temperature (K)

ΑΒ

Interaction Energy (kcal/mol)

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450 400

Isotropic Lam

350 LXa

AG

300 1.4 1.6 1.8 2 2.2 AB attraction,"εAB (kcal/mol)

Figure 2. Phase diagrams of the equimolar mixture of A and B particles. a) Regions of thermodynamic stability of the phases of the equimolar mixture as a function of the relative strength of AB attraction and size repulsion at T = 300 K and p = 0 with σAA = σBB = 1, and εAA = εBB=1 kcal mol-1. The equimolar mixture stabilizes two mesophases: a thin lamellar mesophase (red area) and thick lamellar mesophase (maroon area). The larger the size repulsion, the thicker is each layer of the lamellar phase. The equimolar mixture also stabilizes three crystals aligned layered crystal (LXa, orange area), shifted layered crystal (LXs, orange area), and alternating gyroid crystal (AG, brown area)- and two liquids-isotropic liquid (gray area) and phase separated liquid (white area). b) Phase diagram of the equimolar mixtures shown in panel a), fixing σAB/σAA = 1.08 and varying temperature and strength of the AB attraction. c) Same as in (b) but with σAB/σAA = 1.045. Comparison of panels b) and c) shows that the region of stability of the lamellar mesophase narrows on decreasing the size repulsion σAB/σAA. The region of stability of the layered crystal also decreases, outcompeted by the alternate gyroid (AG) crystal.

Heating the lamellar mesophase at constant pressure results in a transition to an isotropic liquid, while isobaric cooling of lamellar produces a transition to a layered crystal


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(Supporting Information D). These transitions are first order, reversible and with hysteresis, as seen in Figure 4a. The diffusivity of the lamellar phase is strongly anisotropic: near the isotropic-lamellar melting point the diffusion coefficient along the layer is ~130 times higher than in the perpendicular direction; the ratio increases on approaching

Lamellar

Thick Lam.

Semicr. Lam

LXa

the lamellar-crystal equilibrium line (Supporting Information B). Strongly anisotropic mobility is also a characteristic of smectic liquid crystals and lamellar mesophases in block copolymers and surfactants.43-44

Alt. Gyroid

Gyroid

Hexagonal

!!!!!!!!!!!!!!!!!!!!!!!!!!1A:1B!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!2A:1B!!!!!!!!!!!!!3A:1B! !

Figure 3. Mesophases and crystals formed by the binary mixture of nanoparticles. Details of their local structures are shown in the upper panel and representative simulation cells in the lower panel. Mixtures with ratio molar fraction of A XA = 0.5 produce lamellar mesophase, thick lamellar mesophase, semicrystalline lamellar, aligned layered crystal, shifted layered crystal (not shown), and alternating gyroid crystal. Mixtures with XA = 0.66 stabilize the double gyroid mesophase, and those with XA = 0.75 stabilize the hexagonal mesophase. Body centered cubic (BCC) at XA = 0.85, body centered tetragonal (BCT) XA = 0.9, face centered cubic FCC phases XA = 0.92 are also stabilized by the mixture, but not shown here.

Total Energy (kcal/mol)

a

b

Total Energy (kcal/mol)

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Lamellar Mesophase

Crystals

Coo

ling

ting Hea

Tm

Isotropic

Tm

Temperature (K)

Co

oli

ng

a He

g tin

Lamellar

Thick lamellar

Temperature (K) Figure 4. Phase transformations of the lamellar mesophase. a) First order transitions of the equimolar mixture with σAB/σAA

=1.15, εAB/εAA = 1.2, σAA = σBB, and εAA = εBB = 1 kcal mol-1 at p = 0. The layered crystal transforms to lamellar and then to isotropic mixture on heating. The equilibrium melting temperature for lamellar-layered crystal and lamellar-isotropic equilibrium temperatures computed from two-phase coexistence simulations are 276 ± 2 K and 384 ± 2 K, respectively. Between these temperatures, lamellar is the stable phase (red area). The hysteresis is a signature of 1st order transitions, which occur via nucleation and growth. The enthalpies of the layered crystal to lamellar and lamellar to isotropic transitions are 0.23 and 0.75 kcal mol-1, respectively; the corresponding entropy changes are 0.83 and 2.16 cal K-1mol-1. b) Transition between lamellar and thick lamellar mesophases for an equimolar mixture with σAB/σAA = 1.3 and εAB/εAA = 0.8, σAA = σBB and εAA = εBB.. The transition between the mesophases is continuous; second order. The temperature at which the heat capacity is maximum, 345 K, is taken to be the one for the transformation. Supporting Information D shows the continuous reversible change in volume on cooling and heating.

Thick lamellar mesophases, with structures intermediate between lamellar and phase-segregated A and B liquids, nucleate and grow from the isotropic mixture at 300 K for large values of size frustration, σAB/σAA ≥ 1.28 (Figure 2a). The larger the size frustration σAB/σAA, the thicker is each stack of the mesophase. We compute the contribution of interactions between like and unlike particles energy to the total energy of the lamellar mesophase (Supporting Information C) and find that while the AB attraction dominates


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the stability of thin lamellar, thick lamellar is stabilized equally by AA, BB and AB interactions. This explains the transition from thick to thin lamellar on increasing εAB (Figure 2a). Figure 4b and Supporting Information D show that cooling thick lamellar results in a reversible transition to the thin lamellar mesophase. Interestingly, the transition between thick and thin lamellar does not have hysteresis; it is a second order phase transition. The softness of the repulsive part of the AA and BB interactions is key to stabilize the mesophases. Equimolar binary mixtures of Lennard-Jones (LJ) particles, with r12 repulsive potential, can produce layered crystals,45-49 but do not stabilize a lamellar mesophase. Supporting Information E shows that to stabilize a mesophase, the AB interaction can be soft or hard; however the AA and BB interaction must be soft to allow for the large positional disorder within each layer required to make the layers fluid. Semicrystalline phases have been observed in block copolymers in which unlike segments have quite different crystallization temperatures.50 A large difference in the interaction strength between AA and BB interactions in the mesogenic mixture results in semicrystalline phases in which the weakly interacting component is disordered, while the strongly interacting one is crystalline. In those cases, there are two melting transitions between the crystal and the liquid crystal mesophase (Supporting Information F). Semicrystalline materials made of nanoparticles could be used in applications that require the mechanical properties of a solid and the high mobility of a liquid phase. We find that the disorder and thickness of the stack of each component in lamellar mesophases can be independently controlled: the strength of the size repulsion, σAB/σAA, controls the stability and thickness of the lamellar mesophases, whereas the attractions εAB/εAA and εAB/εBB controls the transition temperatures and determines which component in the layered structure is crystalline or disordered. Suppporting Information F presents guidelines to design a mixture of nanoparticles that stabilizes a semicrystalline lamellar with alternating m-layer thick liquid layers of A and nlayer thick crystalline layers of B. Figure 5 illustrates the structures and transitions of a system that alternates layers of thin B and thick A layers and its melting through a semicrystalline phase. The versatility of the mesogenic mixtures to produce modulated phases with controllable width and degree of order could make them an alternative to bottomup assembly of nanoparticles for the formation of materials with nano-scale patterns.

b

a

Total energy (kcal/mol)

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Mesophase

-3 -3.3

Semicrystalline

-3.6 -3.9

c

Crystalline

125 150 175 200 Temperature (K)

Mesophase

d

Crystalline

Semicrystalline

Figure 5: The mesogenic mixture can be designed to display a semicrystalline phase. Panel a shows the energy per particle along the transitions from layered crystal to semicrystalline lamellar to lamellar for a system that alternates thick layers of A (red balls in panels B, C and D) and thin layers of B (blue balls in panels B, C and D). The mixture has interactions give by εAB = 0.5 kcal mol-1, εAA = 0.2 kcal mol-1, εBB = 1.0 kcal mol-1, σAB/σAA = 1.35, and σAA = 1, σBB = 1, a molar fraction of A, XA= 2/3 and is evolved at p = 0.

Common block copolymer mesophases include lamellar, hexagonal, gyroid, and body-centered cubic (bcc).1 The volume fractions of the majority component in the chain of the block copolymer determine which mesophase is stable: around 50% for lamellar, 67% for gyroid, 75% for hexagonal, and 85% for bcc. The binary mixtures of this work stabilize these and other mesophases. Figure 5 shows the region of thermodynamic stability of lamellar, gyroid and hexagonal mesophases as a function of the strength of attraction εAB and molar fraction XA at T= 300 K and p = 0. These mesophases spontaneously nucleate and grow from the isotropic mixture with the corresponding composition. Thick versions of each mesophase can be produced on increasing the size repulsion σAB/σAA. For example, a thick hexagonal phase forms at 1/3 volume fraction of the minority component with σAB/σAA = 1.35. Distinct εAA and εBB can be used to make semicrystalline variants of gyroid and hexagonal mesophases. Same as in block copolymers, metastable perforated lamellar occurs in the mesogenic mixture (Figure 6). Face centered cubic (fcc), body centered tetragonal (bct), and bcc phases are stabilized for molar fraction of minority component 1/12, 1/9 and 1/7 at high AB attraction (Supplementary Information B). We expect that all the stable and metastable phases of block copolymers could be formed using spherical nanoparticles.


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AB attraction, εAB (kcal/mol)

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The Journal of Physical Chemistry Letters 2

LXs

Gyroid

Hexagonal

(PL) 1.6

Lamellar

1.2

Isotropic Liquid 0.8

Phase Segregated A and B 0.5

0.6

0.7

0.8

Fraction of A, XA

Figure 6. Phase diagram of the binary mixture as a function of composition and strength of the AB attraction. Phase diagram is computed at T = 300 K and p = 0, with εAA = εBB = 1 kcal mol-1, σAA = σBB, and σAB/σAA = 1.15. As particles A and B are identical the phase diagram for XA= 0 to 0.5 is the mirror image of the one shown. The regions of stability of a single phase are shown with different colors: lamellar in red, gyroid in green, hexagonal in cyan, isotropic mixture in gray, LXs layered crystal in orange. Metastable perforated lamellar phase is shown with empty red circles. The white regions of the phase diagram correspond to coexistence of two phases, with proportions established by the lever rule.

Different from block copolymers and surfactants, the mesogenic binary mixture can fully phase segregate for low AB attraction strengths. Other difference is that block copolymers microphase-segregate into mesophases to minimize AB interactions, whereas this interaction is maximized in the mesogenic binary mixtures. This results in opposite effect of the strength of AB attraction on the stability of the mesophases: increase in the AB attraction favors microphase segregation in the mesogenic mixtures (Figure 6), while it favors mixing into an isotropic phase in block copolymers. However, the temperature dependence of the transitions between crystals, mesophases and isotropic liquids depends only on their relative entropy, and is the same for the mesogenic mixture of nanoparticles and block copolymers and surfactants.

This study provides a novel avenue to assemble mesophases from mixtures of non-mesogenic nanoparticles. Control over particle sizes and distances of approach, strength of attraction, and softness are used here to generate the same mesophases as block copolymers or surfactants. Nanoparticles coated with rod or disc-shaped molecules have been shown to form nematic, smectic and discotic liquid crystals.51-55 The shape-anisotropy of the particles, imparted by the mesogenic character of the coating molecules, is key for the formation of liquid crystals by these particles.51 The mesogenic mixture of spherical particles of this work, on the other hand, relies on frustrated attraction of different types of particles, and not on shape anisotropy, to produce modulated structures. Nanoparticles coated with functionalized organics can have a length of attraction comparable to the size of their cores,56 making them candidates to assemble into mesophases. To that end, the strength of all interparticle attractions should be comparable to the thermal energy, and particles of different type must experience a stronger attraction than those of same type, while also being unable to approach as closely. The two length-scales that results from these competing attractive and repulsive forces is the key to stabilize the mesophases. Coating nanoparticles with ligands, polymers, or complementary DNA strands could be used to keep the repulsion between like-particles soft and control the strength and distance of approach of the particles. The strength and cooperativity of the DNA pairing can be decreased through DNA strand displacement.57 However, only crystals or quasicrystals have been produced to date from nanoparticles covered with DNA, ligands, or polymers.58-63 The crystals can be generated in simulations with soft repulsive potentials64-67 using additive combination rules for the sizes, σAB = (σAA+σAA)/2. Our work suggests that efforts to design and produce mesogenic nanoparticles should focus on achieving the competing demands of strong AB attraction combined with longer AB distances of approach. Achieving these interactions with nanoparticles would give access to a new range of materials with potential applications in stimuli responsive devices and energy materials.

ASSOCIATED CONTENT

Acknowledgements

Supporting Information. Supporting Information is available free of charge on the ACS Publications website. It contains the models and methods, and supplementary data and discussion on the structure, mobility and stability of the mesophases.

We gratefully acknowledge discussions with Fernando Escobedo and Michael Grünwald. This work was supported by the Camille and Henry Dreyfus Foundation through a Camille Dreyfus Teacher-Scholar Award. We thank the Center for High Performance Computing at the University of Utah for technical support and a grant of computer time.

AUTHOR INFORMATION Corresponding Author

References

[email protected]

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Notes The authors declare no competing financial interest.


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