Entropy compartmentalization stabilizes open host–guest colloidal clathrates

Particle design for host–guest hard truncated TBP clathrates

In clathrate hydrates2, the size ratio of guest to cage determines the clathrate structure. Inspired by this knowledge, we engineered the shape of truncated TBP particles to tune guest/cage size ratios. Previous work8 has shown that, for regular hard TBPs, the cage interiors are almost fully filled with the body of the host particles such that there is no room for a guest particle (Fig. 1b). However, a cavity can be made by truncating the equatorial tips of the TBP (Fig. 1a,b and Extended Data Fig. 1). The truncation parameter S (0 ≤ S ≤ 1) is defined by the distance between the tip of the regular TBP (S = 0) and the centre of the truncated face (Extended Data Fig. 1). We measured the average radius of the cavity (rcavity) in the four types of cage possible for a clathrate crystal (A, 512; B, 51262; C, 51263; D, 51264, following the usual clathrate nomenclature8) from ideally constructed clathrates thermalized in hard-particle Monte Carlo (HPMC20) simulations at equilibrium (Extended Data Fig. 2 and Monte Carlo simulation section in the Methods). HPMC attempts small local particle moves (both translational and rotational) while avoiding particle overlap to mimic the Brownian motion expected for colloidal particles in water or other solvent (see the Monte Carlo simulation section in the Methods and Supplementary Table 1 for simulation details). The range of S over which a cage has a cavity large enough to contain a guest can be estimated from the size ratio α between guest and cavity (α ≡ rguest/rcavity). The size ratio α decreases monotonically with increasing S for all four cages (Fig. 1c) over the entire S range, with relative cavity sizes D > C > B > A for any given value of S. This observation indicates that the type of cage that can house a guest particle varies with S. Because the stability of any clathrate structure requires a stable combination of cage types, we hypothesized that the clathrate type obtained in self-assembly simulations would vary with S. For example, if S is large enough for the D-cage—but not other cages—to accommodate a guest, we might expect the self-assembly of a clathrate type II structure (Clath II–A0D1) with guest-free A-cages and single-guest D-cages (Fig. 1d). The A-cages are empty because the size of the empty pore is smaller than the host particle. We show in a later section that these empty cages can be filled with particles of arbitrary shape during self-assembly, provided that the size is small enough to fit inside the pore.

Fig. 1: Particle design for host–guest colloidal clathrates.figure 1

a, TBP with truncation parameter, from left to right, of S = 0, 0.3, 0.5, 0.7 and 1.0. b, Thirty truncated TBPs form a cage-type clathrate cluster. The truncation creates a cavity (red dotted circle) at the centre of the cluster. The size of the cavity increases as S increases. c, The change of size ratio α = rguest/rcavity as a function of S for the four clathrate cages measured at a constant volume fraction, ϕ = 0.65 (Extended Data Fig. 2 and Guest-to-cavity size ratio section in the Methods). Data points are obtained from HPMC simulations; solid lines are guides to the eye. The location where the red dotted line meets each curve indicates the minimum S for the cage to have a single truncated TBP guest. d, A model structure of Clath II–A0D1 (S = 0.42), composed of guest-free A-cages and single-guest D-cages.

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Self-assembly simulations

To test our hypothesis, we conducted self-assembly HPMC simulations of hard truncated TBPs with varying truncation S. For 0 ≤ S < 0.40, a guest-free Clath I–A0B0 formed via self-assembly from a disordered fluid phase; this is the same type of clathrate found in the hard regular TBP system8, but now with an empty cavity inside the cages that increases in size with increasing S while remaining smaller than the host particle (Fig. 1c and Supplementary Fig. 1a). Above S = 0.40, we observed only host–guest clathrates. For 0.40 ≤ S ≤ 0.71, five different types of host–guest colloidal clathrate crystal self-assembled (Fig. 2a–e). The type of clathrate was confirmed by the cage identification method and the arrangement of the cages (Clathrate identification section in the Methods). For 0.40 ≤ S < 0.46, we obtained Clath II–A0D1 with empty (that is, guest-free) A-cages and single-guest D-cages (Fig. 2a). In this structure, the A-cages arrange into rhombs when viewed along the two-fold axis and, in between, the D-cages are tetrahedrally arranged. For 0.46 ≤ S < 0.49, we obtained Clath III–A0B1C1 with empty, guest-free A-cages, single-guest B-cages and single-guest C-cages (Fig. 2b). In this structure, the B-cages arrange in square and triangular tiles, which can be mapped to the Frank–Kasper σ phase21. For 0.49 ≤ S < 0.55, we observed the assembly of Clath I–A0B1 with empty, guest-free A-cages and single-guest B-cages (Fig. 2c, Supplementary Fig. 1b and Supplementary Video 1). The B-cages are arranged in square tiles, which is a unique feature of the Clath I structure. For 0.55 ≤ S < 0.57, Clath I–A1B1 was obtained, with both A-cages and B-cages occupied by single guests. Finally, for 0.57 ≤ S < 0.71, we obtained a clathrate with single-guest A-cages and tetramer-guest D-cages (Fig. 2d). Each tetramer is composed of four truncated TBPs that form a cluster with tetrahedral symmetry, reminiscent of the tetrahedral cluster of hydrogen comprising the guest in the D-cage of a clathrate hydrate system9. Here, the A-cages arrange into a rhombus, corresponding to Clath II–A1D4. For S ≥ 0.72, crystallization was not observed during the simulation time. Our findings are summarized in the phase diagrams in Fig. 2e and Supplementary Fig. 2.

Fig. 2: Self-assembly simulation results.figure 2

ae, MC simulation results for Clath II–A0D1 at S = 0.42 (a), Clath III–A0B1C1 at S = 0.48 (b), Clath I–A0B1 at S = 0.52 (c) and Clath II–A1D4 at S = 0.64 (d). Red and green truncated TBPs are guest particles, and blue truncated TBPs are host particles. Each clathrate can be described along an axis by a unique tiling of polygons (orange lines and green/yellow dashed circles) and a bond orientational order diagram (BOOD; insets). The BOODs are a histogram of the directions of bonds connecting nearest neighbours. Bottom left: network representation of each clathrate (Network representation of clathrate section in the Methods) for clear visualization of host particle cages. Bottom right: number ratio of cages identified in each clathrate (Clathrate identification section in the Methods). e, Phase diagram as a function of S obtained from self-assembly simulation results. Coloured dots are data points, and the four systems shown in ad are indicated by red arrows. f,g, Self-assembly simulation results of truncated TBP clathrate with other polyhedron guests. f, A binary mixture of truncated TBPs (S = 0.52) and tetrahedra (αA ≈ 0.89) forms Clath 1–A1B1. The A-cage has a tetrahedron guest and the B-cage has a truncated TBP guest. g, A binary mixture of truncated TBPs (S = 0.52) and dodecahedra (αA = 0.97) forms Clath I–A1B1. The A-cage has a dodecahedron guest and the D-cage has a tetramer truncated TBP guest.

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From the self-assembly results, we find a critical value of α below which a truncated TBP guest enters the cavities (αc ≈ 1.2, the red dotted line in Fig. 1c), corresponding to Sc = 0.40, 0.46, 0.48 and 0.58 for cage types A, B, C and D, respectively. Thus, as hypothesized, specifying the value of S determines the type of cage that can house a guest particle, which in turn determines the type of clathrate structure that self-assembles. Because there is no potential energy in any of these systems, the clathrate structures that minimize free energy maximize entropy. Although the diversity and complexity of entropic colloidal crystals is well established16,17, the way in which these clathrates maximize entropy, which we discuss next, is not.

Entropy compartmentalization

In an isotropic fluid phase of a single-component system of hard particles, the free volume is distributed uniformly (Fig. 3a). In the clathrate structures, the free volume is not uniformly distributed among hosts and guests. For example, in the Clath I–A0B1 crystal (S = 0.56), both host and guest particles in equilibrium exhibit similar translational displacement (quantified by the mean-squared displacement \(}}} - }}}_0^2}\), where r is the centre-of-mass position) with comparable fluctuations about the mean, over 1 × 107 MC sweeps (Fig. 3b, upper panel), indicating a similar free volume in the two sublattices in which translational motion can occur. In contrast, guest particles exhibit much larger rotational displacements (\(}}} - }}}_0^2}\), where q is a unit quaternion describing orientation) than host particles (Fig. 3b, lower panel and Supplementary Video 2), indicating unequal free volume partitioning. We observe the same behaviour in all seven colloidal clathrates. Thus, the many microstates available to these self-assembled crystal phases arise from the many rotational degrees of freedom available to the guest particles. In this way, we say that the entropy is compartmentalized, and can be almost entirely attributed to the sublattice(s) of guest particles. When multiple cage types contain guests, the rotational degrees of freedom of the guests in all cages contribute to the system entropy.

Fig. 3: Entropy compartmentalization and free energy calculation.figure 3

a, A schematic of entropy compartmentalization in a single-component system. b, Translational (upper) and rotational (bottom) displacement of the host and the guest truncated TBPs of Clath I–A0B1 (S = 0.56) at P* = 13 for 1 × 107 MC sweeps. c, Left: Helmholtz free energy of the S = 0.56 system at ϕ = 0.60 in fluid and crystal (Clath I–A0B1) phases. Right: Helmholtz free energy of host and guest sublattices in the crystal. d, Gibbs free energy of clathrates at P* = 10.0 in 0.35 ≤ S ≤ 0.75 systems. e, Phase diagram of truncated TBP clathrates (P* versus S). The melting pressure is obtained from where the free energy of the clathrate crystal crosses the free energy of the fluid as the pressure increases (Extended Data Fig. 5). The colour key is the same as in d.

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Entropy compartmentalization is confirmed in these clathrate structures by free energy calculations. Based on the Frenkel–Ladd method22,23, we developed a way to calculate the free energy of the two subsystems of host and guest particles independently by controlling each subsystem with different paths (Extended Data Figs. 3 and 4 and Free-energy calculation for the host and guest sublattices section in the Methods). For example, in the Clath I–A0B1 system (ϕ = 0.60 and S = 0.56), the per-particle Helmholtz free energy F of the guest subsystem (Fguest/Nguest ≈ 0.72kBT) has only 1/8 the per-particle free energy of the host subsystem (Fhost/Nhost ≈ 5.65kBT). Because there are far fewer guest particles than host particles and no potential energy, the guest subsystem has substantially lower free energy and thus much higher entropy than the host subsystem. The Clath I–A0B1 system has 92 host particles and 6 guest particles per unit cell (Supplementary Table 1), and thus Fguest ≈ 4.32kBT and Fhost ≈ 519.8kBT, a difference of over two orders of magnitude.

We compared the bulk free energies of the different clathrate phases to confirm the relative thermodynamic stability among the various clathrate types at a given S. We calculated the Gibbs free energy G of seven types of clathrate (six of which self-assembled and one, Clath IV–A0B1C1, did not and is metastable in all S ranges) for 0.35 ≤ S ≤ 0.71 at constant pressure (P* = 11.0) (Fig. 3d and Free-energy calculation of bulk phases section in the Methods). The calculation accurately predicts the range of S over which each of the six self-assembled clathrate crystals is stable (Fig. 3d). Melting pressure (\(}^ \ast}\)) is another indicator to estimate the stability of a crystal in hard particle systems. \(}^ \ast}\) can be determined from where the free energy of the fluid and crystal cross (Extended Data Fig. 5a), and the free energy of the fluid can be calculated using thermodynamic integration (Free-energy calculation of bulk phases section in the Methods). We confirmed that the values of \(}^ \ast}\) obtained from the free-energy calculations are consistent with those we find in the simulations (Extended Data Fig. 5b). Based on the \(}^ \ast}\) of each clathrate phase, we constructed a phase diagram (P* versus S) for 0.35 ≤ S ≤ 0.71 (Fig. 3e). We also tested the importance of the guest for the stability of the host clathrate framework by removing the guest particles from a previously self-assembled Clath I–A0B1 (S = 0.52). Upon further equilibration, the host network reorganized to sacrifice the required number of host particles to fill the empty cages, forming a new, complete host–guest structure (Extended Data Fig. 6). This indicates that the presence of the guest particle is essential to stabilize the host framework.

Three types of rotational dynamics of guest particles

The rotational dynamics of the guest particles varies depending on the relationship between the shape and symmetry of the guest particle and cavity. To investigate the rotational dynamics of the guests in each of the colloidal clathrates, we tracked the motion of the guests in equilibrium. We assigned a direction vector of unit length to the guest particle (Fig. 4a,f), tracked its position for 3 × 107 MC sweeps, and mapped the accumulated positions of the direction vector onto a unit sphere. We identified three distinct types of rotational motion of the guest particles: (1) free rotation (Fig. 4b), (2) rotation around an axis (Fig. 4d) and (3) discrete rotation (Fig. 4g). For the single truncated TBP guest of the D-cage in the Clath II–A0D1 phase (S = 0.42), we assigned a direction vector from the centre of the particle to a tip (Fig. 4a) and observed that the accumulated positions of the direction vector show a random distribution (Fig. 4b), indicating the free rotation of the guest (Supplementary Video 3). For the single truncated TBP guest of the B-cage in the Clath I–A0B1 phase (S = 0.52), the accumulated positions of the direction vector distribute along a continuous band (Fig. 4d), indicating that the guest is rotating around an axis (Supplementary Video 2). For the tetramer truncated TBP guests of the D-cage in the Clath II–A1D4 phase (S = 0.64), we assigned direction vectors from the tetramer guest centre to the centre of each truncated TBP (Fig. 4f) and observed that the accumulated positions form several clusters (Fig. 4g), indicating that the four truncated TBPs comprising the tetramer move as one, and with preferred orientations. Each cluster has a mixture of the four direction vectors (Fig. 4g, right), indicating that the tetramer guest rotates through all four discrete directions (Supplementary Video 4). We refer to this type of rotational motion as discrete rotation.

Fig. 4: Rotational motion of guests in clathrate cages.figure 4

a, For a single truncated TBP guest, a direction vector v with unit length σ from the particle centre to the centre of a truncated face quantifies the rotational motion of the guest. b,c, Accumulated points of v of a guest particle in a D-cage of a Clath II–A0D1 crystal phase (S = 0.42) at constant pressure (P* = 10.0) from MC simulation (b) and EBT calculation (c), represented on a unit sphere (left) and spherical coordinates (r = 1.0σ) (right). d,e, Accumulated points of v of a guest particle in a B-cage of a Clath I–A0B1 crystal phase (S = 0.52) at P* = 12.5 from MC simulation (d) and the lower energy distribution as computed from EBT (e), in each case represented on a unit sphere (left) and spherical coordinates (r = 1.0σ) (right). f, For a truncated TBP tetramer guest, four direction vectors (v1, v2, v3 and v4) with unit length σ were assigned from the tetramer centre to the centre of each truncated TBP. g, Accumulated points of the direction vectors of a tetramer guest in a D-cage of a Clath II–A1D4 crystal phase (S = 0.64) at constant pressure P* = 15.0 on a unit sphere (left) and spherical coordinates (r = 1.0σ) (right). h, Distribution of prolateness Pr of the cavity of the B-cage of Clath I–A0B1 (the same system as in d) and the D-cage of Clath II–A0D1 (the same system as in b). The negative Pr of the B-cage indicates an oblate shape of the cavity, and the Pr ≈ 0 of the D-cage indicates a spherical shape of the cavity. i, Both the D-cage and the tetramer guest in Clath II–A1D4 (the same system as in g) have tetrahedral symmetry, inducing the discrete rotation of the guest. jl, Schematic representation (left) and rotational motion (right) of the tetrahedron in cage A (free rotation) (j), the hexagonal prism in cage B (rotation around an axis) (k) and the icosahedron in cage A (discrete rotation) (l). The three cages are composed of S = 0.52 truncated TBPs.

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The shape of a guest relative to that of the cavity that contains it correlates with the three types of rotational motion. We characterized cavity shape using the prolateness Pr (Dynamics of guest particles section in Methods) and confirmed that the cavity of the D-cage is almost spherical (Pr ≈ 0), while the cavity of the B-cage is oblate (Pr ≈ −0.02) (Fig. 4h). Combined with the above findings, the spherical cavity of the D-cage does not hinder the single truncated TBP guest (~0.4 aspect ratio), resulting in free rotation (Fig. 4h, right inset). In contrast, continuous rotation around an axis and discrete rotation occur when the guest shares symmetry elements with the cavity. For example, the oblate cavity of the B-cage (symmetry group D6d) allows the oblate guest (symmetry group D3h) to align along, and rotate around, the same symmetry axis (Fig. 4h, left inset). In the case of discrete rotation, both the truncated TBP tetramer guest and the D-cage cavity have tetrahedral (T) symmetry (Fig. 4i), producing four preferred, discrete orientations within the cavity. Similar types of rotational behaviour were reported for two-dimensional (2D) colloidal crystals of polygons when the particle symmetry matches the symmetry of its local environment24.

To elucidate the correlation between guest and cage shape/symmetry, we used EBT25 to calculate the ground-state free energy for each individual orientation of a guest particle (Entropic bond calculations section in the Methods). We first validated the applicability of EBT to the host–guest clathrate systems by constructing a phase diagram for the simulated structures, finding good agreement with both simulations and Frenkel–Ladd free energy calculations (Extended Data Fig. 7). We then used EBT to compute the free energy of a guest-occupied D-cage in Clath II–A0D1 and B-cage in Clath I–A0B1 at S = 0.42 and 0.52, respectively, for all possible guest orientations. Histograms of the computed free energies from EBT show two distinct distributions for the B-cage (Extended Data Fig. 8b), but only one continuous distribution for the D-cage (Extended Data Fig. 8a). Employing the same definition of directional vector in Fig. 4a, mapping the lower free energy distribution for the B-cage to a unit sphere predicts the same rotational pattern about an axis observed in the simulations (Fig. 4e). EBT predicts free rotation for the case of the D-cage (Fig. 4c), in agreement with simulation. The excellent agreement between EBT predictions, the observed rotational dynamics and the corresponding clathrate structures reveals that thermodynamics not only drives the formation of the clathrate structures but also controls the allowed/disallowed guest dynamics for each respective clathrate cage.

The above results suggest that one can a priori select the shape of the guest particles relative to the cage shape to target specific rotational patterns, which is potentially useful for the design of stimuli-responsive sensors or signal-amplifying/propagating mesostructures. To test this idea, we replaced the truncated TBP guests with tetrahedral, hexagonal prism and icosahedral guests in S = 0.52 clathrate cages (Fig. 4j–l) and observed their rotational motion using HPMC simulation. As hypothesized, the tetrahedron guest (αA ≈ 0.89) freely rotates in the A-cage due to the symmetry mismatch between the guest (tetrahedral symmetry) and the cage (icosahedral symmetry); the hexagonal prism (αB ≈ 0.95) rotates continuously around an axis in the B-cage because both guest and cage have an oblate shape; and the icosahedron guest (αA ≈ 1.01) rotates among several discrete orientations in the A-cage due to the symmetry match between the guest (icosahedral symmetry) and the cage (icosahedral symmetry). When the guest particles are sufficiently small relative to the cage size, the shape and symmetry of cage versus particle is irrelevant and all guests rotate freely. This shape control over the rotational dynamics of the guest particles could be especially powerful in the self-assembly of host–guest clathrates from binary mixtures of shapes, as discussed next.

Self-assembly of binary host–guest colloidal clathrates

The cavities of the clathrate cages can be filled by particles different from that of the host during self-assembly, as confirmed by HPMC simulations of two binary hard-particle systems: (1) truncated TBPs with tetrahedra and (2) truncated TBPs with dodecahedra (Fig. 2f,g, Supplementary Fig. 3 and Supplementary Table 2). We chose the size of the second particle type to fit into the cavity of the A-cage in Clath I in both cases, but selected different symmetries to induce different rotational dynamics. For the first system, a fluid mixture of truncated TBPs (S = 0.52) and tetrahedra (αA ≈ 0.89) self-assembled into Clath I–A1B1 at ϕ = 0.55, with the A-cages now occupied by a single tetrahedron guest (rather than remaining empty or being filled by a host particle) and the B-cages occupied by a single truncated TBP guest (Fig. 2f). Similarly, the second system (S = 0.52) formed Clath I–A1B1 at ϕ = 0.55, with the A-cages containing a single dodecahedron guest (αA = 0.97) and the B-cages containing a single truncated TBP guest (Fig. 2g). As expected, the guest and host particles again show different rotational dynamics and entropy compartmentalization, and the second particle types with different symmetries show different rotational dynamics. Tetrahedron guests in the truncated TBP–tetrahedron system rotate freely in the A-cages (Extended Data Fig. 10a) due to the symmetry mismatch with the cage, whereas the dodecahedron guests in the truncated TBP-dodecahedron system rotate among several discrete orientations in the A-cages (Extended Data Fig. 10d) due to the symmetry match with the cage. For both systems, the single truncated TBP guest in the B-cages rotates continuously around an axis (both the truncated TBP and B-cage have an oblate shape; Extended Data Fig. 10b,d); at the same time, the host truncated TBP particles exhibit only local vibration (Extended Data Fig. 10c), as in all of the one-component self-assembled colloidal clathrates. The multiple types of rotational motion identified for the different species of guest particles in the different cages of the binary clathrates indicate that the entropy of the guest subsystem may be even further compartmentalized among different guest sublattices, by design.

Host–guest colloidal clathrate with attractive polyhedra

We anticipate that these and other open-caged colloidal clathrate crystals can be realized in the laboratory in several ways, even when purely hard particles are not available or desired. EBT calculations indicate that the key ingredient is a net attraction of at least 3 kT between the host particles to ‘freeze’ them into the cage network (Extended Data Fig. 9). In the hard-particle systems studied here, this attraction is emergent and implicit—a statistical, effective force of attraction driven by entropy maximization. However, an attractive interaction energy between particles can be explicitly encoded through the use of attractive ligands26. One way to achieve this would be to use DNA-mediated interactions to explicitly program a minimum 3-kT attraction between the facets of the host truncated TBPs. For example, mixing a batch of non-attractive, truncated TBPs (or other suitable shapes) with truncated TBPs functionalized with self-complementary DNA should produce a relatively rigid host clathrate network composed of DNA-linked particles, and a guest sublattice of unlinked particles that rotate in the cages of the network (to make them non-attractive, the guest particles could be functionalized with non-complementary DNA). Thermal annealing about the DNA melting temperature should produce net DNA base-pairing interactions between truncated TBP faces on the order of kT. Once this thermal energy scale is achieved, entropy maximization will dominate and drive the particles functionalized with non-complementary DNA to occupy the interior of the clathrate cages. A similar scheme should work for other types of patchy particle with attractive interactions between host particles and no attraction between guests or between guests and hosts.

We tested these possibilities using two different molecular dynamics (MD) simulation models (Methods): a mean-field, pairwise interaction potential using the anisotropic Lennard–Jones potential (ALJ)27 and a patchy particle model of DNA-functionalized colloidal particles7,28. Both models have successfully predicted nanoparticle assemblies7,28,29. In the ALJ model, we checked that a B-cage (S = 0.52) of attractive truncated TBPs with a purely repulsive guest TBP self-assembles at ~5-kT attraction between host particles (Supplementary Video 5). Here, the choice of 5 kT as opposed to 3 kT was chosen because EBT does not consider thermal fluctuations present in the simulations. That is, 3 kT is the minimum attractive strength in the absence of fluctuations. By selecting 5 kT we ensure that the attraction is sufficiently strong to overcome thermal fluctuations. For the patchy particle model, we constructed a Clath I–A0B1 (N = 2,315, S = 0.52) system with attractive host truncated TBPs and repulsive guest truncated TBPs. We confirmed that the clathrate is stable at \(}^ \ast \approx 0.90}\) as shown in the potential energy of the system over time (Supplementary Fig. 4). The fact that the clathrate structure can be stabilized with attractive host particles suggests that the design approach used here to select for different clathrate structures using particle shape should be useful in guiding experiments, regardless of the source of interaction (and whether explicit or implicit) between particles.

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