Why Is Gyroid More Difficult to Nucleate from Disordered Liquids than

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Why Is Gyroid More Difficult to Nucleate from Disordered Liquids Than Lamellar and Hexagonal Mesophases? Abhinaw Kumar, and Valeria Molinero J. Phys. Chem. B, Just Accepted Manuscript • Publication Date (Web): 05 Apr 2018 Downloaded from http://pubs.acs.org on April 5, 2018

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The Journal of Physical Chemistry

Why Is Gyroid More Difficult to Nucleate from Disordered Liquids than Lamellar and Hexagonal Mesophases? Abhinaw Kumar and Valeria Molinero* Department of Chemistry, The University of Utah 315 South 1400 East, Salt Lake City, Utah 84112-0850

Abstract Block copolymers, surfactants and biomolecules form lamellar, hexagonal and gyroid mesophases. Across these systems, the nucleation of lamellar from the disordered liquid is the easiest, and the nucleation of gyroid the most challenging. This poses the question of what are the factors that determine the rates of nucleation of the mesophases and whether they are controlled by the complexity of the structures or the thermodynamics of nucleation. Here we use molecular simulations to investigate the nucleation and thermodynamics of lamellar, hexagonal and gyroid in a binary mixture of particles that produces the same mesophases as surfactants and block copolymers. We demonstrate that a combination of averaged bond-order parameters 𝑞! and 𝑞! identifies and distinguishes the three mesophases. We use these parameters to track the microscopic process of nucleation of each mesophase and investigate the existence of heterogeneous nucleation (cross-nucleation) between mesophases. We estimate the surface tensions of the liquid/mesophase interfaces from nucleation rates using Classical Nucleation Theory, and find that they are comparable for the three mesophases with values that are about a third of those expected for liquid-crystal interfaces. The driving forces for nucleation, on the other hand, are quite different and increase in the order gyroid < hexagonal < lamellar at any temperature. We find that the nucleation rates of the mesophases follow the order of their driving forces. We conclude that the difficulty to nucleate the gyroid originates in its lower entropy and temperature of melting compared to the hexagonal and lamellar mesophases.

* corresponding author, email: [email protected]

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1. Introduction Lamellar, hexagonal and gyroid mesophases are common in block copolymers, surfactants, and biomolecules.1-3 Lamellar order is common in membranes and other biological systems.4-5 Layered architectures based on lamellar mesophases have applications in lubricants, data storage devices, and photoconductors.6-9 Hexagonal mesophases have been used for drug delivery and to template the assembly of mesoporous silicas.10-11 Double gyroid is particularly interesting because the minor component forms two interpenetrated chiral three-dimensional networks,12 which makes it valuable in photonics, and to produce negative refractive index materials that have applications in super-lensing and cloaking.13-14 Gyroidal order is also responsible for the iridescence of the wing-scales of some butterflies15-16 and has been used as components for solar cells.17-18 Voiding of the minor component of the gyroid has been used to produce porous materials with high contact surface.19 There is a growing interest in controlling the formation of mesophases at nano- and micro-scales. This requires an understanding of the microscopic process of nucleation of the mesophases, what factors determine their formation rate, and whether they can be cross-nucleated20-25 from other mesophases. It is generally more challenging to produce gyroid than lamellar and hexagonal mesophases.26-31 We are not aware of any experimental report of direct nucleation of gyroid from the disordered liquid phase. Instead, metastable perforated lamellar nucleates and grows from the disordered phase, and then transitions to gyroid.26,

32-34

To date, it is not understood what

precludes or slows down the direct formation of the gyroid from the liquid phase. Classical Nucleation Theory (CNT)35 has been used to analyze the nucleation of lamellar and bodycentered-cubic (bcc) mesophases from liquid,36-37 hexagonal from lamellar38, and lamellar from hexagonal.39 It was found that the nuclei of lamellar and hexagonal are anisotropic in shape and


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in their interfacial free energies, while the nuclei of BCC is spherical. According to CNT, the barrier for nucleation increases with the interfacial tension between the new and parent phases and decreases with the free energy difference between these phases. To our knowledge, the role of these two factors in the distinct rates of nucleation of gyroid, hexagonal and lamellar from the disordered phase have not yet been investigated. Mesophases have been obtained in molecular simulations with several models that encode two length-scales or frustration between attraction and repulsion in the interaction potentials. These include not only models of surfactants40-45 and block copolymers,26, 29, 46-56 but also short-range attraction and long-range repulsive potentials (SALR), hard-core soft-shoulder (HCSS) potentials,57-64 and mesogenic mixtures of nanoparticles.65 Nucleation of gyroid from liquid is challenging in molecular simulations, but has been achieved in Monte Carlo simulations of block copolymers, dumbbells, Brownian dynamics simulation of tethered nanoparticles, and dissipative particle dynamics of block copolymers.27-28, 48, 66-74 We have recently shown that binary mixtures of soft spherical particles in which unlike particles interact stronger than like particles, but cannot get as close, form the same mesophases as block copolymers and surfactants.65 Molecular dynamics simulations of these mesogenic binary mixtures easily produce lamellar, gyroid, hexagonal, body-centered cubic, body-centered tetragonal, and face-centered cubic mesophases at 1/2, 1/3, 1/4, 1/7, 1/9, and 1/12 fraction of the minority components, respectively.65 Metastable perforated lamellar, common in block copolymers,75 is also produced by the mesogenic mixtures.65 All the order-disorder transitions in the mesogenic binary mixture are first order.65 The phase diagram of the binary mixture has regions of coexistence between two phases,65 different from those of one component block copolymers,76 but similar to the phase diagrams of surfactants-water mixtures2 or diblock copolymer-homopolymer blends.77


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In this study, we use the simple and computationally efficient model of binary mesogenic particles of ref.

65

to investigate how do gyroid, hexagonal and lamellar mesophases nucleate

from the liquid phase, unravel the thermodynamic and structural factors that control the relative rates of formation of these mesophases, and investigate whether in two-component systems, such as lyotropic mesogens, the gyroid can be heterogeneously nucleated (cross-nucleated) by hexagonal/liquid or lamellar/liquid interfaces.

2. Methods A. Model. We use the model of binary mixtures of soft spherical particles A and B of ref. 65. In that model, all interactions are modeled with the two-body term of the Stillinger-Weber (SW) potential:78

𝐸(𝑟!" ) = 7.049556277 𝜀 0.6022245584

! !!"

!

− 1 exp (

! !!" !! !

),

(1)

where ε and σ represent the strength of attraction and the size of particles and rij is the distance between particles i and j. The cutoff value a = 1.8 makes the potential vanish at 1.8 σ. We report all the thermodynamic quantities in reduced units, except for the temperature and depth of the potential well. Throughout all the simulations we keep σAA = σBB = 1, σAB = 1.15, εAA = εBB = 1.0 kcal mol-1 and εAB = 1.8 kcal mol-1. These parameters produce stable lamellar, hexagonal and gyroid mesophases for appropriate compositions and temperatures.65

B. Simulation settings.


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We perform all molecular dynamics simulations using LAMMPS.79 We prepare simulation boxes with a total of 64000 A and B particles. The simulation boxes are periodic in the three directions and have initial dimensions 65.6 ×17.8 × 30. We start each simulation of a liquid from a random configuration with the selected composition. The system is evolved with molecular dynamics simulations in the isobaric-isothermal (NpT) ensemble at T = 300 K and p = 0. All isobaric simulations are performed at p = 0. The equations of motion are integrated using the velocity-Verlet algorithm with a time step of ∆t* = 0.00535. Temperature and pressure are controlled with the Nose-Hover thermostat and barostat with time constants 12.5 and 62.5, respectively. For simulations performed at constant cooling rates, the target temperature of the thermostat is linearly adjusted on each step at the rate indicated. C. Nucleation. To identify the pathway of nucleation for the lamellar, gyroid, and hexagonal mesophases we perform simulations at the heart of the regions of stability of each of these mesophases: XA= 0.5, 0.68 and 0.76, respectively (the phase diagram is symmetric in composition). We perform at least five simulations of nucleation with different random initial positions for each composition at each selected temperature. From these simulations we compute the average time of nucleation,

τN. The simulations of nucleation are evolved until either the stable mesophase at that composition nucleates or 100 million steps have been integrated. We determine the nucleation of each phase using the using order parameters described in section 2E. To determine whether the hexagonal/liquid interface can promote the nucleation of gyroid, we prepare simulation boxes in which the hexagonal phase (XA = 0.752) is in coexistence with a metastable liquid mixture with XA = 0.695. The hexagonal mesophase occupies only ~20%


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of the simulation box. We pre-equilibrate the liquid mixture at 300 K and p = 0 for 2×105 time steps, without integrating the equations of motion of the particles in the hexagonal region. To find whether lamellar can promote nucleation of the gyroid, we prepare a system where the lamellar phase of composition XA = 0.5 interfaces with a metastable liquid mixture with XA = 0.68. The liquid is pre-equilibrated at 300 K and p = 0 for 2×105 time steps, without integrating the equations of motion of the particles in the lamellar region. We determine the temperature of maximum formation rate of gyroid from liquid with composition XA = 0.68 using the procedure of refs.80-81, cooling the liquid at constant rates ranging from -0.025 K every 105 time steps to -0.1 K every 105 time steps. The temperature of formation of the mesophase at the fastest cooling rate that still results in nucleation is the temperature of maximum formation rate, the tip of the time-temperature-transformation diagram.80 D. Thermodynamic properties. We compute the equilibrium melting temperature Tm of each mesophase to liquid using the phase-coexistence method, following ref.

82

. We start in each case with the mesophase in

coexistence with a liquid of the same composition, which we obtain by heating half of simulation cell containing only the mesophase. If the mesophase grows (melts), the temperature is below (above) the melting point. We scan temperatures until one is found for which the system does not evolve in any direction. This procedure results in melting points with ±1 K accuracy for the lamellar, hexagonal and gyroid phases. To calculate the enthalpy of melting ∆Hm of each phase, we perform separate one-phase simulations of each mesophase and liquid of the same composition at the corresponding


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equilibrium melting point. The one-phase systems contain 13824 particles; and their enthalpies are averaged over 2×106 steps long simulations. We find identical molar enthalpies for lamellar using 13824 and 27648 particles, from what we conclude that finite size effects do not impact our results. The entropy of melting at the melting point is computed as ∆Sm = ∆Hm / Tm. The excess molar free energy of the liquid with respect to the mesophase of the same composition is computed as83 ΔGm(T) = (Tm-T) ΔSm(Tm)+(ΔCp/2Tm) ΔT2,

(2)

where ΔCp is the difference in the heat capacities of the liquid and mesophase. The heat capacities are computed from the slopes of the enthalpy vs temperature for each mesophase and the disordered liquid with the same composition at temperatures that range from the melting temperature of the mesophase to the temperature at which it spontaneous nucleates from the supercooled disordered liquid when the system is cooled at a constant rate of 5×10-6 K/step. This cooling rate is sufficiently slow to allow for equilibration of the simulation boxes in the whole temperature range. To interpret the differences in nucleation rates between the mesophases, we need to compute the surface free energies between disordered liquids and mesophases of the same composition. The test area method84 is usually employed for the calculation of fluid/fluid interfacial tensions. This method determines the surface tension from the anisotropy in the components of the pressure tensor perpendicular (pxx) and parallel (pyy and pzz) to the fluid/fluid interface:


7


𝛾=

!! !

< 𝑝!! > −
,

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(3)

where Lx is the length of the simulation cell in the direction perpendicular to the surface and the brackets indicate a time average over the length of the simulation. We test the test area method for the calculation of the lamellar/liquid surface tension through two sets of 3×107 time steps long NVT simulations of two-phase lamellar/liquid systems with composition XA = 0.5 and 10600 particles at T = 655.5 K. The only difference between the simulations is the aspect ratio of the cells: 43.5×13.44×16.46 in the first simulation and 43.28×13.33×16.69 in the second. The density and temperature of the cells are the same, and should result in identical surface tensions if the phases involved were fluid. However, the surface tensions obtained by the test area method are widely different: −6.54 ± 0.52 mJ m-2 in the first case and 41.68 ± 3.86 mJ m-2 in the second. The large variability and negative value of the surface tension indicates that the system cannot relax the imposed stress, invalidating the use of this methodology. Instead, we use Classical Nucleation Theory35 to estimate orientiationally-averaged interfacial tensions of the lamellar/liquid, hexagonal/liquid and gyroid/liquid interfaces from the ratio between the homogeneous nucleation rate J of a mesophase from the liquid at two temperatures, T1 and T2: J(T2)/J(T1) = A(T2) / A(T1) (exp(ΔG*(T1) / KBT1 − ΔG*(T2) / KBT2),

(4)

where A(T) is a pre-exponent that mostly depends on the diffusion coefficient, the free energy barrier for nucleation, ΔG*=16πγ3/(3ρ2ΔGm2), depends on the surface tension γ of the mesophase/liquid interface, the driving force ΔGm, and the number density ρ of the mesophase. On using this expression for the barrier to determine γ we are approximating that the nuclei of the mesophases are spherical and that their surface tension is isotropic. However, surface tensions of lamellar and hexagonal are known to depend on the orientation,36, 38-39 resulting in


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non-spherical nuclei (in agreement with our findings for the binary mesogenic mixture, see Results). The values of γ estimated from the rates should be considered an average weighted over all the area of the mesophase in the critical nucleus. While not as rich in information as the orientational dependent surface tensions, the average values are useful to assess whether differences in the surface free energy cost or the bulk free energy gain are responsible for the distinct nucleation rates of the mesophases. As the volumes of the cells do not change significantly with temperature, we use the ratio of the average nucleation times in lieu of the ratio of the rates, J(T2)/J(T1) = τN(T1)/τN(T2). We calculate nucleation times of lamellar at 610 and 620 K, of hexagonal at 335 K with 348 K, and gyroid at 300 K and 304 K. We further assume that γ and the pre-exponent A do not change with temperature and obtain γ from the slope of log(τ2/τ1) vs 1/(ρ2 ΔGm2RT) for each of the mesophases. In all cases we strive to find temperatures for which the nucleation rates change between one to two orders of magnitude and for which the formation of the mesophase is limited by nucleation and not growth. E. Order parameters. The identification of mesophases through order parameters is challenging because of the variability in local order in these phases. The average hexatic order parameter85-86 has been used to quantify the phase transition from lamellar to liquid, however we find that it does not suffice to distinguish the small nucleus of the lamellar mesophase from liquid. Here we identify the mesophases using bond-orientation order parameters of the minority component (or any of them, in the case of lamellar). We compute the per-atom average bond orientation order parameter following ref. 87:


9


𝑞! (i) =

!! !!!!

! !!!!

𝑞!" (𝑖)

!

,

Page 10 of 38

(5)

where l is the rank of the spherical harmonics, and m are the integers from –l to l and 𝑞!" (i) = ! !! !

!! ! !!!

𝑞!" k , where 𝑞!" (k) =

! !! !

!! ! !!!

𝑌!" 𝑟!" and Nb(i) is the number of nearest

neighbors of atom i within a specified cutoff distance rcut, and 𝑌!" 𝑟!" are spherical harmonics. We evaluate 𝑞! with l = 2 to 12 for lamellar, gyroid, and hexagonal and the liquid phase at the corresponding compositions with various cutoff values to identify what is the optimum l and rcut to identify each mesophase. In section 3B we show that the optimal identification of the mesophases is attained through a combination of 𝑞! and 𝑞! .

3. Results and Discussion A. Thermodynamic driving forces for nucleation of the mesophases. All order-disorder transitions in the binary mixture of mesogenic particles are first order.65 The shaded areas in Figure 1 show the regions of thermodynamic stability of lamellar, gyroid, and hexagonal mesophases as a function of the fraction of A particles, XA, and the strength of the AB attraction, εAB, for particles with σAA = σBB = 1.0, σAB = 1.15, εAA = εBB = 1.0, at T = 300 K and p = 0. The Roman numerals in the phase diagram indicate the five different compositions with εAB = 1.8 kcal mol-1 for which we investigate the nucleation of the mesophases, always starting from a metastable liquid of the same composition. The interaction strength εAB = 1.8 kcal mol-1 is chosen because it allows the three mesophases to be stable, at different compositions, and the attractions are not so strong as to hamper the mobility of the particles.


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AB attraction, εAB (kcal/mol)

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2

LXs I

1.6

Gyroid (PL)

II Lam + Gyr

III

Hexagonal IV

Gyr + Hex

V

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 1. Phase diagram of the binary mixture of mesogenic particles at T = 300 K and p = 0. Regions of stability of the lamellar (red area), lamellar crystal LXs (orange), gyroid (green area), hexagonal (cyan area), liquid (gray), perforated lamellar (PL, red empty circles), and phase-segregated region as a function of the attraction between A and B particles, εAB, and the fraction of A, XA. We choose εAA = εBB = 1.0 kcal mol-1, σAA = σBB = 1, σAB = 1.15. The phase diagram for XA 0 to 0.5 is a mirror image of the one presented here. Adapted from ref. 65.

The rate of homogeneous nucleation strongly depends on the thermodynamic driving force, i.e. the excess molar free energy ΔGm of the mesophase with respect to the liquid, which we determine from the degree of supercooling and entropy change of the phase transition using eq. 2. Table 1 shows the melting temperatures, and the difference in enthalpies, entropies and heat capacities of lamellar (XA = 0.5), gyroid (XA = 0.68), and hexagonal (XA = 0.76) mesophases to the liquids of the corresponding compositions for εAB = 1.8 kcal mol-1 and 0 bar. The magnitude of the enthalpies and entropies of the 1st order-disorder transitions (ODT) indicate that these are all very weak, in agreement with experimental results88 and theoretical predictions for ODT in block copolymers89 and other systems that belong to the Brazovskii universality class.9091

The melting temperatures follow the order lamellar > hexagonal > gyroid (Table 1). The same order is found in experiments with block copolymers and surfactants.2,

32

The entropies of

melting of the mesophases follow the same order as the melting temperatures (Table 1); i.e. the


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transition is the strongest for lamellar, and the weakest for gyroid. While experimental entropies or enthalpies of phase transitions of mesophases are rare in the literature, a recent calorimetry study of a series of block copolymers indicates that the entropy of melting of lamellar is about 2.4 times larger than for hexagonal,88, 92-93 a ratio close to the 2.1 we find for the binary mixture of particles in this study (Table 1). This implies that both in experiments of block copolymers and surfactants and in the simulations with the mesogenic mixture of this work, the driving force for nucleation at any temperature follows the order lamellar > hexagonal > gyroid. For example, at 300 K the driving force for the nucleation of lamellar in the mixture is 6 times larger than for hexagonal and 12 times larger than for gyroid (Table 1). In sections C-E below we show that the nucleation rates of lamellar, hexagonal and gyroid follow the order of their driving forces. Table 1: Thermodynamics of the phase transition between mesophases and liquid mixtures of same composition. Tm

∆Hm at Tm

Cp,meso

ΔCp

∆Sm at Tm

∆Gm at 300K

γmeso/liqa

(K)

(kcal mol-1)

(cal K-1 mol-1)

(cal K-1 mol-1)

(cal K-1 mol-1)

(kcal mol-1)

(mJ m-2)

0.5

655 ± 1

1.41

8.27

0.29

2.15

0.73

11 ± 2

hexagonal

0.76

417 ± 1

0.43

7.10

0.89

1.03

0.11

9±2

gyroid

0.68

399 ± 1

0.25

7.78

0.35

0.63

0.06

9±2

Mesophase

XA

lamellar

a)

Surface tensions estimated through CNT correspond to the temperatures 610 K for lamellar, 335 K for hexagonal, and 300 K for gyroid. We report surface tensions in units that map the density and enthalpy of vaporization to those of liquid water (see text).

B. Order parameters to identify the mesophases. To follow the nucleation and growth of the mesophases, it is key to have order parameters able to identify lamellar, hexagonal, and gyroid when they are in contact with each


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other and in the presence of liquid. To this end, we evaluate the bond orientation order parameters 𝑞! , with l = 2 to 12 for the particles of the minor component, B, of each of the mesophases and the liquid phase of the corresponding composition. We show below that l = 2 counting BB neighbors up to distances rcut* = 1.91 and l = 8 counting BB neighbors up to rcut* = 3.39 suffice to distinguish lamellar, gyroid and hexagonal from each other and from the liquids. We need to account for neighbors beyond the first shell to identify the mesophases because the local, first shell order in the liquid and mesophases of the corresponding composition are quite similar (Figure 2).

gBB(r)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


7 6 5 4 3 2 1 0 0

Figure 2. Radial distribution functions of the minority component, B, in the mesophases and liquids of the same composition. Distributions for mesophases are shown with full lines: lamellar at 610 K (red), hexagonal at 348 K (blue) and gyroid at 300 K (green). Distributions for the liquids with identical compositions XB and temperatures are shown with same colors using dashed lines.

XB = 0.50 XB = 0.24 XB = 0.32

1

2 3 r/σΒΒ

4

Figure 3 displays the structures of the lamellar, hexagonal, and gyroid mesophases and the order parameters that distinguish them. The lamellar phase of the mixture of particles has alternating disordered layers of components A and B (Figure 3a), where each layer is oneparticle thick and lacks long-range order (Figure 3d), similar to the layered structures formed by surfactants and block copolymers.1-2 We find that 𝑞! distinguishes very well lamellar from the


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liquid (Figure 3g), gyroid (Figure 3h) and hexagonal (Figure 3i). We identify the particles that have 0.42 < 𝑞! < 0.52 as part of the lamellar mesophase. The minority component in the hexagonal mesophase forms two-coordinated wires, arranged in a triangular lattice (Figure 3f). The majority component is amorphous and surrounds the wire-like structure (Figure 3c). We find that by selecting 𝑞! > 0.7, the hexagonal phase is distinguished from the liquid and the other mesophases (Figure 3i). The gyroid is arguably the most challenging mesophase to identify. The minority component of the gyroid mesophase has two interpenetrated networks of three-coordinated particles that do not intersect (Figure 3e). The majority component surrounds the minority component and has an amorphous structure. The distributions of 𝑞! for gyroid and liquid with composition XA = 2/3 are strongly overlapped (Figure 3h). We find that 𝑞! of the minor component counting up to third neighbors distinguishes gyroid from liquid and the other mesophases (Figure 3h). We identify the minor component particles with 0.045 < 𝑞! 4; gyroid has nB < 4.

C. Nucleation of the lamellar mesophase. Lamellar can be nucleated easily in both experiments and simulations.47,

94

We first

investigate the nucleation of lamellar mesophase from liquid at 300 K and XA = 0.5 (I in figure 1), 355 K below its equilibrium melting point (Table 1). Under these extremely undercooled conditions, lamellar forms through what seems to be spinodal decomposition, resulting in multiple domains of lamellar mesophase that slowly coarsen to a single, straight domain (Figure 4). Nucleation of lamellar seems to be barrierless at temperatures up to 8% below its melting point. This is partly due to the large driving forces for nucleation of lamellar from liquid, arising from its melting temperature and entropy of melting relative to other mesophases (Table 1). We note that lamellar has also the highest melting temperature among mesophases in block copolymers32 and –consistently– spinodal decomposition has been reported for its formation in these systems.1, 95-96


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The formation of lamellar occurs through a nucleation and growth mechanism in experiments only at temperatures close to its melting point.94 Likewise, lamellar forms through a mechanism involving nucleation and growth in simulations of the binary mixture at supercooling less than 56 K. The growth of the lamellar mesophase is anisotropic with lens-like shape, piling new layers at a rate faster than it widens them (Figure 5), in agreement with previous reports for block copolymers.36, 97 The nucleation of lamellar takes about t* = 1700 (~3.2 ×105 steps) at 45 K below its melting point, and about t* = 17000 (~3.2 ×106 steps) when the supercooling is 35 K. The slow change with temperature suggests a relatively low value of surface tension for the lamellar/liquid interface. It has been proposed that the value of the surface tension between two phases correlates with their structural and enthalpy gap.98 We obtain the lamellar/liquid surface tension from the ratio of two nucleation times using Classical Nucleation Theory.35 We find that the surface tension of the lamellar-liquid interface is small, γlamellar/liquid* = 0.636. For comparison, we note that this value corresponds to 11 ± 2 mJ m-2 in units that map the density and enthalpy of vaporization to those of liquid water. This is about a third of the surface tension of the liquid/ice interface, 32 ± 3 mJ m-2,99-102 and comparable to the ratio in the change in enthalpy from liquid to lamellar to total enthalpy change from liquid to layered crystal for the mesogenic particles.65 We conclude that the nucleation of lamellar mesophase is generally favored by its high melting temperature and its relatively low interfacial tension.


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t* = 0

Figure 4. Spinodal-like decomposition to form lamellar mesophase from liquid at 300 K and XA = 0.50. The two components of the binary mixture are shown with red and blue spheres. The simulation cell is periodic in the three directions. The time sequence shows the immediate nucleation throughout the cell and coarsening of the lamellar.

t* = 50

t* = 250

t*= 10000

A) t* = 0

C) t* = 700

B) t* = 625

D) t* = 1000

Figure 5. Nucleation and growth of lamellar mesophase from liquid at 610 K and XA = 0.50. Only the minority component, B, is shown. The particles are colored blue if they have the structure of lamellar and grey if liquid. The simulation cell is periodic in the three directions. The lens-like shape of the lamellar nucleus resembles the one reported for block copolymers.97, 103


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D. Nucleation of the hexagonal mesophase. We investigate the formation of the hexagonal mesophase from the liquid with composition XA = 0.76 at temperatures from 300 to 350 K, i.e. with supercooling ranging from 117 to 67 K. Starting from a randomly mixed liquid at 300 K, we find that the order parameter that identifies hexagonal grows steadily without a noticeable induction time. Initially, the minority component, B, aggregates to form throughout the whole system entangled hair-like clusters in which the B particles are two-coordinated. The lack of induction time and existence of multiple nucleation sites indicate that at 300 K the rate of formation of hexagonal is already limited by growth. The entangled hair-like structures straighten and order into the characteristic triangular lattices of the hexagonal phase within Δt* = 1000 at 300 K. Dissipative particle dynamics simulations of block copolymers also display intermediate entangled hair-like structures (also referred to as gyroid-like47) in the pathway to the hexagonal mesophase.47 The formation of the hexagonal mesophase at 348 K proceeds through a nucleation and growth mechanism, with a clear induction time and propagation from a single nucleus (Figure 6). The nucleus of hexagonal has a slightly anisotropic shape, consistent with previous reports for hexagonal nuclei in lamellar for block copolymers.38 The temperature dependence of the rate of formation of the hexagonal mesophase is comparable to that of lamellar, suggesting that the surface tension of the liquid to hexagonal transition is equally small. Indeed, we find that γhexagonal/liquid is, within the error bar, identical to γlamellar/liquid (Table 1). We do not find a correlation between the enthalpy of melting of the mesophases and the value of the surface tension of the mesophase/liquid interface (Table 1). We attribute the low surface tensions of the liquid/mesophase interfaces to the similarity in local order in the mesophases and the supercooled liquids with the same composition (Figure 2).


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t* = 48000

Figure 6. Nucleation and growth of the hexagonal mesophase from liquid at 348 K and XA = 0.76. Only the minor component, B, is shown. The particles are colored blue if they have the structure of hexagonal and grey if liquid. The simulation cell is periodic in the three directions. The nucleus of hexagonal displays a small anisotropy, being slightly longer in a direction normal to the hexagonal axis.

t* = 49000

t* = 50000

t* = 51000

t* = 55000

E. Nucleation of the gyroid mesophase. Gyroid has lower melting temperature than lamellar and hexagonal. This is the case not only for the mixture of mesogenic particles of this study (Figure 1 and Table 1),65 but also for simulations with other models and experiments with block copolymers and surfactants.1, 104-105 We find that the temperature of maximum formation rate of gyroid from the liquid mixture with XA = 0.68 is around 298 K. Below that temperature, the gyroid is still stable but its rate of formation decreases, limited by the kinetics of growth. Combined with the relatively low melting


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point of the gyroid, this results in a relatively narrow range of temperatures for which the rate of formation of gyroid increases with supercooling. The formation of gyroid from liquid at 300 K and XA = 0.68 (point III of Figure 1), in the middle of the region of stability of the mesophase, occurs through a nucleation and growth mechanism. The nucleation of gyroid at 300 K takes an average of 1.8 × 107 time steps (t* ≈ 100,000), and is preceded by fast nucleation of the lamellar mesophase from the liquid mixture (Figure 7). The initial fast growth of lamellar, which has XA ≈ 0.5, leads to enrichment in component A of the liquid at its boundary and limits further growth of lamellar. We find that the nucleation of gyroid always occurs in the liquid region, away from lamellar. The global change in XA of the liquid due to the formation of lamellar is small, from 0.68 to 0.695, and within the region of stability of the gyroid phase (Figure 1). The gyroid overgrows the lamellar phase, and is the only phase present at the end of the simulations. We find the same results in multiple simulations of nucleation of gyroid starting from liquids with XA = 0.68. Likewise, liquids with XA = 0.65 (point II in the phase diagram of Figure 1), nucleate first lamellar and then gyroid; the latter always nucleates homogeneously from the liquid phase. The preference for homogeneous nucleation of the gyroid may be due to the existence of gyroid-like local motifs in the liquids with XA = 0.68. The commonality of local motifs is the reason we need to account up to 3rd neighbors of the minor component to identify the gyroid with the bond order parameter. The similarity is also evident in the radial distribution functions of the metastable liquid and gyroid, which have peaks at the same positions and same number (three) of B neighbors around B particles. The difference between the metastable liquid and the gyroid is that in the latter the minor component makes long-range pores in the A matrix.


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The fast growth of metastable lamellar in the direction perpendicular to the layers disfavors configurations in which lamellar exposes the plane of the layers to the liquid. That orientation could allow for insertion of A particles into the layers, resulting in perforated layers from which gyroid could epitaxially (i.e. heterogeneously) nucleate. To test whether lamellar exposing the surface of the layers to the liquid could assist in the nucleation of gyroid, we prepare two-phase simulation cells in which the lamellar/liquid interface is normal to the layers and the liquid has the composition of the gyroid, XA = 0.68. We find those configurations to be unstable: the layers in lamellar reorient to expose the edges to the liquid phase in a time shorter than the nucleation of the gyroid that, again, occurs homogeneously within the liquid phase. The resulting system has lamellar and gyroid in coexistence, and the two mesophases epitaxially aligned. We conclude that lamellar does not assist in the nucleation of the gyroid mesophase from the mesogenic mixture, and that epitaxial alignment does not necessarily imply crossnucleation between the mesophases. Perforated lamellar plays an important role as intermediate in the transformation between lamellar and gyroid in block copolymers.26,

32-34, 75, 106

Although perforated lamellar is a

metastable phase in the mesogenic mixture of this study (Figure 1), it is not involved in the nucleation pathway of gyroid from liquid. The lack of nucleation of gyroid by lamellar may seem at odds with the results for block copolymers and surfactants.107-114 However, a fair comparison cannot be made because a liquid phase does not seem to be involved in the transition from lamellar to gyroid in these experiments. This difference with block copolymers –which can nucleate gyroid from either lamellar or perforated lamellar mesophases110- is the result of the two-component nature of the mixture of mesogenic particles that results in a gap in composition between lamellar and gyroid (Figure 1) which prevents the direct transformation between these


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phases. We conjecture that the lamellar/liquid interface does not promote the nucleation of gyroid because lamellar exposes to the liquid only the “edge” of the layers, preventing the insertion of particles of the major component that would be needed to form metastable perforated lamellar and transition from there to the gyroid mesophase.

A) t* = 0

Figure 7. Nucleation and growth of gyroid from liquid at 300 K and XA = 0.68 is preceded by the formation of lamellar. Minor component B is shown in gray in the liquid, green in the gyroid, and red in lamellar. Particles are connected with bonds if they are within 1.22σ . The largest clusters of gyroid in the induction period contain around 150 B particles. This sets a lower limit for the size of the critical gyroid nucleus for XA = 0.68 at 300 K. Lamellar nucleates and grows first (B), but does not play a role in the nucleation of the gyroid. Major component is not shown, for clarity. The green halo around lamellar does not correspond to gyroid but small clusters of particles that have order parameter 𝑞! intermediate between lamellar and liquid. Gyroid nucleates homogeneously within the liquid (C), and grows over lamellar (D). The final state for this composition corresponds to only the gyroid mesophase (E). The lower panel shows the fraction of the B particles involved in the lamellar and gyroid domains along the simulation (letters indicate the corresponding snapshots).

B) t* = 51000

ΒΒ

C) t* = 70000

D) t* = 120000

E) t* = 380000

E

0.8 D

Fraction

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


0.6 0.4 0.2

C A

0 0

Gyroid

B

Lamellar 2 3 1 5 Time (t*/10 )

4


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The gyroid mesophase is generally difficult to nucleate in experiments and in simulations.28, 115 From the analysis of the rates of nucleation of gyroid, we find that the average surface tension for the gyroid/liquid interface is identical to that of the other mesophases (Table 1). To assess whether the slow rate of nucleation of gyroid is due to its complex structure or to its smaller range of thermodynamic stability, we compare the rate of homogeneous nucleation of hexagonal and gyroid mesophases at their canonical compositions under identical driving forces, which we tune through control of the nucleation temperature T (eq. 2). The driving force for nucleation of the hexagonal phase at 356 K is the same than for the gyroid phase at 300 K. Under those conditions, we find comparable times for the formation of these two mesophases. We conclude that the gyroid is not intrinsically more difficult to nucleate from liquid than the other mesophases: the low rates of nucleation of gyroid arise from its lower melting temperature and extremely low entropy of melting (Table 1), which result in lower driving forces for nucleation at any given temperature. To determine whether hexagonal can promote the nucleation of gyroid, we first prepare a liquid with XA= 0.71 at 300 K (labeled with IV in Figure 1), for which the stable state corresponds to phase coexistence of gyroid and hexagonal. That liquid, however, does not nucleate any mesophase in any of five 108 steps long simulations. To test whether hexagonal assists in the nucleation of gyroid, we prepare two-phase simulation cells in which hexagonal coexists with liquid that has composition XA = 0.695, the composition of gyroid in coexistence with hexagonal at that temperature (Figure 1). The gyroid nucleates rapidly (within t* = 10000) at 300 K, and the critical nucleus of the gyroid consistently forms at the hexagonal/liquid interface (Figure 8). We conclude that the hexagonal/liquid interface promotes the heterogeneous nucleation of the gyroid.


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A) t* = 0

B) t* = 10000

Figure 8. Heterogeneous nucleation and growth of the gyroid mesophase at the hexagonal/liquid interface at 300 K and XA = 0.695. The minority component is shown in gray when part of the liquid, blue when hexagonal, and green when gyroid. Particles are connected if the distance between them is less than 1.22*σ . The majority component is not shown, for clarity. In this pictures, the upper value of 𝑞! identifies gyroid is expanded from 0.065 (Table 2) to 0.1 to capture the interface between hexagonal and gyroid mesophase. The starting configuration has hexagonal in coexistence with metastable liquid with XA = 0.695. Gyroid epitaxially nucleates at the hexagonal/liquid interface. The details of that process are shown in Figure 9. ΒΒ

Time Evolution

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


C) t* = 25000

D) t* = 30000

E) t* = 40000

The mechanism of epitaxial heterogeneous nucleation and growth of gyroid at the hexagonal/liquid interface is illustrated in Figure 9. The hexagonal phase has two-coordinated strings of the minority component. These transform into the three-coordinated structures characteristic of the gyroid through addition of extra B particles. A rotation of the triangles, to form a dihedral angle of 70.52º, results in the formation of the gyroid. The mechanism is akin to that previously reported for the transformation of hexagonal to gyroid in block copolymers.110


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However, in the binary mixture the transformation can only happen at the interface of hexagonal and the B-richer liquid phase because the composition of the two phases is quite different.

A

B

D

C

Figure 9. Mechanisms of epitaxial nucleation of gyroid at the hexagonal/liquid interface. Upper panel shows a sketch of the transformation of the minority network of hexagonal (blue) into gyroid (green) through addition of particles of the minority component. The intermediate state has both two-, and three-coordinated particles. Snapshots A-D in the lower panels illustrate the development of gyroid order at the hexagonal/isotropic interface. The hexagonal /liquid interface is parallel to the page.

4. Conclusions. In this study we use molecular dynamics simulations to investigate the nucleation of lamellar, hexagonal, and gyroid mesophases in a model that produces the same mesophases as surfactants and block copolymers.65 We find that a combination of neighbor-averaged bondorder parameters 𝑞! and 𝑞! suffices to identify and distinguish these three mesophases and the liquid mixture. With the help of these order parameters, we investigate the mechanism of nucleation of lamellar, gyroid and hexagonal from isotropic liquids and assess the existence of cross-nucleation (heterogeneous nucleation) between the mesophases. We find that the surface tensions of all mesophase/liquid interfaces are similar, and about 1/3 of those expected for crystal/liquid interfaces. These results are in agreement with previous experimental reports for polymers.116 The low surface tensions favor the nucleation of the


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mesophases at relatively low driving forces compared to those required for the nucleation of crystals from liquids. This suggests that easily formed mesophases can be used as steppingstones for the nucleation of complex crystal phases, reducing the overall nucleation barriers.117 The liquid-mesophase transitions of the mesogenic model are first order. We find that all these order-disorder transitions are quite weak, and that the entropy of melting of lamellar is about twice larger than for hexagonal and over three times larger than for gyroid. This order agrees with the thermodynamic data on phase transitions of mesophases available in the literature.88, 92 Moreover, the model reproduces the experimental ratio of the enthalpy of melting of lamellar vs hexagonal in the only report to date for a block copolymer.88 The equilibrium melting temperatures also follow the order lamellar > hexagonal > gyroid in the simulations and experiments.1, 32 This implies that, at any given temperature, the driving force for nucleation is always the largest for lamellar and the smallest for gyroid. As the values of surface tensions are comparable for the three mesophases, the rate of homogeneous nucleation of the mesophases from liquids follows the order of their driving forces. Lamellar is the easiest mesophase to nucleate from liquid because it has the highest melting point and largest entropy of melting, which together result in the highest driving force at any temperature. The driving force can become so large for lamellar, that it forms through spinodal decomposition even at high temperatures at which the other phases only display nucleation and growth or are difficult or impossible to nucleate. It has been noted that in block copolymers the lamellar phase is generally obtained via spinodal decomposition, whereas the hexagonal phase forms via nucleation and growth.26 We find that lamellar and hexagonal can be obtained by both mechanisms, depending on the driving force. The pervasiveness of a spinodal


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mechanism of formation of lamellar reflects the higher stability and entropy of melting of that mesophase. Our analysis indicates that gyroid is not intrinsically more difficult to nucleate from the liquid than the other mesophases, its slow rate of nucleation arises from the extreme weakness of its first order transition from the liquid phase. Likewise, Bates and coworkers measured the experimental enthalpy of melting of lamellar, hexagonal and BCC for a block copolymer and found that the order-disorder transition for BCC was the weakest and the nucleation rates of BCC from liquid the lowest.88 Our simulations indicate that when nucleation rates are measured at the same driving force, the rates of nucleation of gyroid and hexagonal are essentially the same. We conclude that the difficulty to nucleate gyroid arises from it having the narrowest temperature range of stability and its very weak order-disordered transition, which results in a slower increase of the driving force with supercooling compared to the other mesophases in both simulations and experiments.1, 29, 31-32 Different from single component mesogenic systems, such as block copolymers, the twocomponent nature of the mixture of this study results in gaps in composition between lamellar, gyroid and hexagonal mesophases that prevents them from experiencing direct order-order transitions. It would still be possible, however, for one mesophase to cross-nucleate to other mesophase at the interface with the liquid phase. We investigate the possibility of heterogeneous nucleation of gyroid in the binary mixture by the lamellar/liquid and gyroid/liquid interfaces. We find that gyroid nucleates heterogeneously at the hexagonal/liquid interface but not at the lamellar/liquid interface. The nucleation of gyroid at the hexagonal/liquid interface proceeds through the addition of particles of the minority component to the network of the hexagonal and their reconnection into the motif of the gyroid. This process is similar to what has been reported


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in SCF simulations for nucleation of block copolymers mesophases.96, 108, 110, 118 Our analysis suggests that matching of local orders is key for the cross-nucleation between mesophases. We interpret that lamellar does not nucleate gyroid because the surface that exposes the plane of the layers, where the epitaxial transformation to gyroid would be geometrically possible, is not a stable interface in contact with the disordered liquid. We note that although gyroid nucleates homogeneously in the liquid phase in two-phase lamellar/liquid simulations, gyroid and lamellar are epitaxially aligned in the final state. These results demonstrate that epitaxial alignments between coexisting mesophases can result from the minimization of interfacial free energies during the growth process and are not proof of a heterogeneous nucleation mechanism.

Acknowledgements. This work was partially supported by the Camille and Henry Dreyfus Foundation through a Camille Dreyfus Teacher-Scholar Award and by the United States Army Research Laboratory under Cooperative Agreement Number W911NF-12-2-0023. We thank the Center for High Performance Computing at the University of Utah for technical support and a grant of computer time.

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Why Is Gyroid More Difficult to Nucleate from Disordered Liquids than

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