At σ _S = 1.0 (q _L = 1.0), the self-assembly occurs at all five temperatures (k _B T = 0.1, 0.2, 0.3, 0.4, 0.5) studied in this work. The RDFs shown in Figure 3A reveal that the self-assembled structures formed at σ _S = 1.0 are mainly random aggregates. At lower temperatures, the lobes are sticky due to stronger interactions between them. As a result, when the smaller aggregates generated at the initial stages in simulations merge to form larger aggregates, the lobed particles cannot rearrange their positions inside these aggregates, thereby leading to different shapes. The aggregates formed at lower temperatures are composed of both snowman and dumbbell shaped particles (Figure 3B). However, at higher temperatures, the lobes are not so sticky as the interactions between them are moderate, and the lobed particles can rearrange their positions to reduce the interfacial tension and finally appear in a spherical shape (Figure 2 and Figures 3D,F). The spherical aggregates formed at higher temperatures are homogeneous, as revealed by the parameter Φ at different temperatures (Figure 4A; black curve). The cross sectional views of the random aggregates shown in Figures 3C,E reveal that the cores of the random aggregates are formed mainly by the dumbbell shaped particles and the outer most shell is covered by a single layer of snowman shaped particles. This is partly driven by the excluded volume effect and partly driven by the energy. As the volume of the dumbbell shaped particles are higher than the snowman shaped particles, the formation of the central cores of the self-assembled structure only by the dumbbell shaped particles reduces more excluded volume, which makes it favorable to form these morphologies. In addition, the dumbbell shaped particles, when they are at the cores of the self-assembled structures, utilize their two lobes to interact with the other particles, and lower the energy of the system. And the snowman shaped particles due to having one lobe interact with the other particles from one direction only, and form the outer surfaces of the self-assembled structures, making the process partly energy driven. We suggest that the self-assembled monolayers formed at lower temperatures by the snowman shaped particles can stabilize, and control several properties like wetting, adhesion, and chemical resistance of the inner self-assembled structures formed by the dumbbell shaped particles [43, 44].

We also estimated the average number of particles present in the first coordination shells of dumbbell shaped ( N ̄ D ) and snowman shaped ( N ̄ S ) particles at different seed diameters and temperatures. The changes in N ̄ D and N ̄ S with temperature for all seed diameters are shown in Figures 4B,C, respectively. At σ _S = 1.0, the values of N ̄ D (Figure 4B, black curve) remain nearly constant with an increase in temperature, revealing that the dumbbell shaped particles participate in the self-assembly for the entire range of temperatures. But, the values of N ̄ S , at σ _S = 1.0 (Figure 4C, black curve) reveal that the snowman shaped particles participate in the self-assembly until k _B T = 0.3, and then they begin to disassemble and at k _B T = 0.5, most of the snowman shaped particles remain in a dissociated state. At k _B T = 0.5, the aggregates are homogeneous, formed predominantly by the dumbbell shaped particles.

At σ _S = 1.2 (q _L = 0.83), we observed a very similar kind of self-assembly phenomenon that we observed at σ _S = 1.0 (Supplementary Figure S4). At lower temperatures (k _B T = 0.1, 0.2), random aggregates of different shapes are formed (Supplementary Figures S4A,D), and at higher temperatures (k _B T = 0.3, 0.4, 0.5), these aggregates become spherical (Supplementary Figures. S4B,C, E–G). With an increase in temperature, the aggregates become homogeneous (Figure 4A; red curve), formed mainly by the dumbbell shaped particles. The variations in N ̄ D and N ̄ S with temperature reveal that the dumbbell shaped particles self-assemble at a wider range of temperatures (Figure 4B; red curve), but the snowman shaped particles start to disassemble from k _B T = 0.3 and remain in a dissociated state (Figure 4C; red curve).

At σ _S = 1.4 (q _L = 0.71) and σ _S = 1.6 (q _L = 0.63), we observed a transition from random aggregates to a liquid droplet state with an increase in temperature. At σ _S = 1.4, the self-assembly occurs at k _B T = 0.1 to 0.4. At k _B T = 0.1 and 0.2, heterogeneous random aggregates form (Supplementary Figures S5A,C,E, and F). At higher temperatures (k _B T > 0.2), the self-assembled structures become homogeneous, formed predominantly by the dumbbell shaped particles (Figure 4A; green curve). At higher temperatures, the self-assembled structures formed by the dumbbell shaped particles attain a liquid droplet state (Supplementary Figures S5B,D,G), and the snowman shaped particles remain in a dissociated state.

At σ _S = 1.6, the self-assembly occurs from k _B T = 0.1 to 0.3 (Figure 5 and Supplementary Figure S6). At a lower temperature, k _B T = 0.1, we observed wire-like structures, that are amorphous and heterogeneous in nature, formed by both types (dumbbell and snowman shaped) of particles (Figure 5B and Supplementary Figure S6B). At k _B T = 0.2, most of the snowman shaped particles disassemble (Figure 4C, blue curve) and the dumbbell shaped particles self-assemble into amorphous three-dimensional aggregates (Figure 5C and Supplementary Figure S6C). At both k _B T = 0.1 and 0.2, the dumbbell shaped particles form the cores of the self-assembled structures and the snowman shaped particles are found in the outer shell (Figures 5B,C). At both temperatures, the average number of particles present in the first coordination shells of the dumbbell shaped particles ( N ̄ D ) is 10 (Figure 4B; blue curve), where each of the two lobes are attached with five other particles (Figures 5E,F). From the top-view, the first coordination shell resembles a starfish (Figure 5E). At a higher temperature (k _B T = 0.3), mainly the dumbbell shaped particles participate in the self-assembly, and form a liquid droplet state (Figure 5D and Supplementary Figure S6D).

For σ _S ≥ 1.6, we observed a rapid phase separation with an increase in temperature. For an increase in temperature (k _B T) from 0.1 to 0.2 at σ _S = 1.6, the values of Φ were observed to significantly increase from 0.14 to 0.61 (Figure 4A; blue curve). The values of N ̄ S (i.e., the average number of particles in the first coordination shell of the snowman shaped particles) were also observed to undergo a significant decrease from 6.3 to 1.5 with an increase in temperature from 0.1 to 0.2 (Figure 4C; blue curve), while the values of N ̄ D remain constant at around 8.5 for the entire range of temperature (Figure 4B; blue curve). These data establish the fact that the snowman shaped particles get separated from the aggregates rapidly at higher seed diameters than they do at lower seed diameters. A similar phase separation phenomenon was observed at σ _S = 1.8.

At σ _S = 1.8, the self-assembly occurs only at lower temperatures (k _B T = 0.1 and 0.2), and a transition from a wire-like structure (Figure 6B and Supplementary Figure S7B) to a crystalline structure (Figure 6D and Supplementary Figure S7C) is observed with an increase in temperature. At k _B T = 0.1, the heterogeneous wire-like structures (Figure 6B and Supplementary Figure S7B) are formed by both types (snowman and dumbbell shaped) of particles. The lobed particles were observed to self-assemble into tetrahedral or octahedral clusters (Figure 6C) before forming an extended wire-like structure. The dumbbell shaped particles contribute to form the central cores of the wires, thereby helping in lengthening the wires via two lobes at the opposite poles. The snowman shaped particles are mainly found at the surfaces of wires, which prevents the growth of wires orthogonal to the axes of the wires. At higher seed diameters, the volumes of the snowman and the dumbbell shaped particles are almost equal. Therefore, the excluded volume effect almost diminishes at this condition. So, the formation of the central cores of the self-assembled structures by the dumbbell shaped particles and the outer surface by the snowman shaped particles is mainly energy driven.

At higher temperatures, the snowman shaped particles separate from the assembly leaving only homogeneous self-assembled structures formed by the dumbbell shaped particles. We observed a large increase in the Φ value from 0.21 to 0.82 (Figure 4A; the magenta curve) with a small increment in the temperature from 0.1 to 0.2, revealing a rapid transition from the heterogeneous to homogeneous structures. The variations in N ̄ D (Figure 4B; the magenta curve) and N ̄ S (Figure 4C; the magenta curve) values with temperature also support the observation of a rapid phase separation. The self-assembled structures formed at k _B T = 0.2 are crystalline where the unit cell is a face-centered cubic lattice (Figure 6F and Supplementary Figure S7D). In Figures 6D,E, we show a self-assembled cluster and its cross sectional view formed at k _B T = 0.2. It reveals that the bulk of the crystalline structures is constituted by the dumbbell shaped particles only, and few lattice points at the surfaces of the crystalline structures are occupied by the snowman shaped particles (the red spheres).

At σ _S = 2.0, the self-assembly occurs only at k _B T = 0.1, predominantly by the dumbbell shaped particles (Figure 6G). The self-assembled structures are amorphous and made up of ring like structures (Figure 6H). We did not observe any long-range order between the particles (Figure 6A; the green curve) but a short-range order is observed where the first coordination shell possesses a trigonal prismatic geometry. The lobed particles were observed to self-assemble into a tetrahedral geometry before forming a ring like structure (Figure 6G). The dumbbell shaped particles form the bulk of the clusters, and the snowman shaped particles are observed only at the surfaces of the clusters (Figure 6G).

The self-assembled morphologies, that we have described so far, were obtained at the SSLJ parameter σ = 1.0. Now we will describe the self-assembled morphologies obtained at higher σ. For the seed diameters (σ _S) equal to 1.2, 1.4, 1.6, 1.8, and 2.0, we carried out a second set of simulations where the values of σ were set equal to their respective σ _S values. At these higher σ values, we observed the self-assembly only at σ _S = 1.2 and 1.4. At σ _S = 1.2, we observed self-assembly at four different temperatures (k _B T = 0.1, 0.2, 0.3, and 0.4). At lower temperatures, heterogeneous random aggregates are formed, and at higher temperatures, homogeneous liquid droplets primarily formed by the dumbbell shaped particles are observed (Supplementary Figure S8). At σ _S = 1.4, we observed the self-assembly at two different temperatures (k _B T = 0.1 and 0.2). At k _B T = 0.1, the heterogeneous wire-like structures (Figure 7A), that are amorphous in nature, are formed. The lobed particles self-assemble into octahedral clusters (Figures 7B,C) before forming the extended wire-like structures. At k _B T = 0.2, homogeneous liquid droplets are formed mainly by the dumbbell shaped particles, and the snowman shaped particles are found in a dissociated state (Figure 7D).

At any seed diameter and temperature, the self-assembled morphologies remain same at different number ratios (r _N ) of the lobed particles. At any combination of the seed diameter (σ _S), temperature (k _B T), and the SSLJ parameter, σ, the morphologies of the self-assembled structures formed at two other number ratios (r _N = 0.5 and 2.0) of the lobed particles are similar to the structures formed at r _N = 1.0 (Supplementary Figures S9, S10). At any seed diameters, homogeneous structures are formed at lower temperatures, and heterogeneous structures are formed at higher temperatures (Supplementary Figure S11). The variations in N ̄ D and N ̄ S (Supplementary Figure S12) with an increase in temperature are also similar to the variations that we observed in the case of r _N = 1.0 (Figure 4). The values of N ̄ D (Supplementary Figures S12A,B) remain almost same at the entire range of temperature. But, the values of N ̄ S (Supplementary Figures S12C,D) start to decrease after a certain temperature at any seed diameter as the snowman shaped particles remain in a dissociated state at higher temperatures. At r _N = 2.0, as the number of the snowman shaped particles is twice the number of the dumbbell shaped particles, the self-assembled clusters formed at r _N = 2.0 are fully wrapped by the snowman shaped particles. In contrast, at r _N = 0.5, the clusters are partially wrapped due to the less number of snowman shaped particles in the mixture.

At the SSLJ patameter σ = 1.0, the largest pores are found at σ _S = 2.0 and k _B T = 0.1. We estimated pore diameters for those structures that are three dimensional, and not extended wire-like. At any seed diameter, the three-dimensional structures usually form at higher temperatures, and the extended wire-like structures or narrow random aggregates form at lower temperatures. Therefore we estimated the pore size distributions, which are shown in Figure 8, Supplementary Figures S13, S14, for the higher temperature structures only. For three-dimensional self-assembled structures, we extracted large cuboids, and then used the Zeo++ software [45, 46] to estimate the pore size distributions, as carried out in our previous work [30, 31]. For each seed diameter, the pore diameters were observed to increase with an increase in the temperature. The lobed particles having larger seeds or smaller lobes were observed to produce bigger pores than the particles having smaller seeds. At σ = 1.0, the largest pores were found for σ _S = 2.0 at k _B T = 0.1 (Supplementary Figure S13F). We obtained a broader pore size distribution because of the amorphous ring structures formed at this condition. The pore diameters were found to vary between ∼ 1.5σ _B and ∼ 3.5σ _B .

The self-assembled structures formed at higher σ values are observed to form larger pores than the corresponding structures formed at lower σ values (Figure 8). Among the self-assembled structures formed at higher σ values, the structures self-assembled at σ _S = 1.4 and σ = 1.4 were found to possess the largest pores at k _B T = 0.2 (Figure 8B, Supplementary Figure S14B). At this condition, the pore diameter varies between ∼ 1.0σ _B and ∼ 2.5σ _B. The structures self-assembled at the same seed diameter and temperature (σ _S = 1.4 and k _B T = 0.2) but at a lower σ value (σ = 1.0) produce very small pores (∼ 0.2σ _B to ∼ 0.7σ _B), revealing the effect of the Lennard-Jones parameters σ on the porosities of the self-assembled structures. The central seeds, at a higher σ value, become more repulsive and therefore remain further from each other compared to a lower σ, which is why the pore diameters at a higher σ are much larger than at a lower σ.

As all the simulation domains in our work contain a mixture of snowman and dumbbell shaped particles, the self-assembly phenomenon is observed mainly at lower temperatures. For example, in our earlier study [31], we observed that the highest temperature at which the dumbbell shaped particles of σ _S = 1.6 self-assemble is 0.4, and at that condition the pore diameters vary between 0.1σ _B and 2.3σ _B [31]. But in the presence of snowman shaped particles, the highest temperature at which the self-assembly occurs at σ _S = 1.6 is 0.3. Although the self-assembled structures formed at k _B T = 0.3 in the present work are almost similar to the structures formed at k _B T = 0.4 in our previous study (in both cases liquid droplets are formed by the dumbbell shaped particles), the pore diameters found in the present work (Supplementary Figure S13D, red curve) are relatively smaller (0.1σ _B to 1.2σ _B ) than the previous work due to the difference in the temperature. As pore diameters increase with the temperature, and the presence of the snowman shaped particles inhibits the self-assembly at higher temperatures, we recommend the usage of only dumbbell shaped particles for better porous self-assembled structures. Moreover, as the snowman shaped particles remain only at surfaces of the self-assembled structures and the bulk of the self-assembled structures are formed mainly by the dumbbell shaped particles at any seed diameter and temperature, the snowman shaped particles do not have a significant impact on the porosities of the self-assembled structures.