Percolation in binary mixtures of linkers and particles: Chaining vs branching

The theoretical and simulation studies of patchy particle models^1–4 have opened the way to the concept of equilibrium gels, a thermodynamically stable phase formed by a percolated network of particles, not related to phase separation arrest. These works have shown that the formation of bonds between particles with low functionality leads to the emergence of a percolated fluid at low temperatures and low densities, which does not phase separate with a vapor. Experimental evidence of the existence of equilibrium gels was first found in single component systems, like laponite⁵ (a colloidal clay), DNA nanostars,⁶ and a solution of Fmoc-diphenylalanine molecules in dimethyl sulfoxide.⁷ 

In recent years, significant attention has been given to the study of aggregation in binary mixtures where the self-assembly of one of the components is mediated and controlled by the other. Examples span a large variety of soft matter and biological systems, such as, among many others, protein–protein aggregation,⁸ cross-linking of actin filaments,⁹ colloidal dispersions and protein fibrils,¹⁰ nanoparticles and globular proteins,¹¹ microparticles and small soft microgels,¹² and cell-mediated colloidal scaffolds.^13–15 Following a similar line of reasoning, the search for realizing equilibrium gels with an extra degree of control has led to the study of model systems where bonds between particles are mediated by another component, linkers.^16–20 In these works, the linkers are bifunctional (i.e., they bond two particles) and can be polymers of different lengths,¹⁷ DNA strands,^18,19 or patchy particles.^16,20 In any case, their properties (size, shape, and concentration) can be used to control the aggregation process and the thermodynamics, and they can be adjusted to obtain the proper conditions for the formation of equilibrium gels. These works establish some interesting results for linker–particle aggregating systems: A percolated network appears at low temperatures in a finite range of linker concentrations^16,20 that depends on the functionality of particles; the connectivity properties of this network can be controlled by the amount of linkers;^16,17,20 it is possible to find single phase percolated fluids at low densities and temperatures,^16,17,20 and these densities can be further reduced using longer linkers.¹⁷ An experimental system²¹ formed by nanoparticles and telechelic polymer chains (the linkers) confirms some of these predictions since it is found that the gelation of particles is controlled by the relative concentration of polymeric linkers.

The aim of the present work is to thoroughly investigate the percolation thresholds of linker–particle patchy models in which linkers can also assemble into linear chains, using theory and simulation. This is accomplished by introducing bonds between linkers. While linker–linker bonds promote chaining, the usual particle–linker bonds lead to branching of those chains and eventually to the formation of percolated networks. Two types of interplay between chaining and branching are addressed (in two variants of this model): (i) A competition between chaining and branching is set by letting the two patches of the linkers bond either to another linker or to a particle (model A). (ii) Chaining and branching are set independently by letting the linkers have two patches that bond only to linkers and other patches that bond only to particles (model B). It will be shown that the chaining of linkers strongly affects the conditions at which percolation occurs and that these changes are different in models A and B. It should be emphasized that percolation is only a necessary condition to obtain equilibrium gels. The investigation of the phase diagrams of the percolated phases of these models is left for future work. Still, it is important to stress that in almost all models of binary mixtures of patchy particles with low valence,^{16,20,22–24} the phase diagrams (calculated using Wertheim’s theory) always exhibit percolated single phases at low temperatures for a range of low densities.

The paper is organized as follows: In Sec. II, the patchy particle models are introduced in detail, and the simulation and theoretical methods employed to study the percolation properties of the models are described. In Sec. III, the percolation thresholds obtained for the two models are described and analyzed. Finally, in Sec. IV, the results are discussed and conclusions are drawn.

A binary mixture of N_P particles and N_L linkers in a volume V is considered. Particles and linkers are hard spheres (HSs) of diameter σ. The (reduced) density of the system is ρ = (N_L + N_P)σ³/V and the composition x is the fraction of particles, x = N_P/(N_L + N_P). Particles are decorated with f_C ≥ 3 patches of type C on its surface, while linkers are decorated with 2 patches of type A and up to f_B patches of type B (see Fig. 1).

The interaction potential defined in (1) depends on the number of patches of different types on each species (f_C and f_B) and on the energy scales ɛ_αβ between all pairs of types of patches. The choice of f_C, f_B and of the nonzero ɛ_αβ (i.e., of which types of bonds can be formed) will define a particular model of a binary mixture of patchy particles. The types of bonds allowed will then determine the types of self-assembled structures that emerge on the model. This relation is illustrated in Fig. 2. If the only nonzero energy is ɛ_AA, then only bonds between linkers can be formed, and chains of linkers are the only self-assembled structures expected—Fig. 2(a). The formation of clusters where a single linker mediates the bonds between two (and only two) particles^16,20 is obtained for a model where only ɛ_AC ≠ 0 (or only ɛ_AB ≠ 0 and f_B = 2)—Fig. 2(b).

The goal of this work is to study the percolation thresholds in systems where the linkers, besides mediating the bonding of particles, can also self-assemble into chains. This will be accomplished by investigating the following models:

• Model A: Only AA and AC bonds can form, i.e., only ɛ_AC and ɛ_AA are nonzero; in simulations (and numerical calculations), f_C = 4, and several values of the energy ratio ɛ_AA/ɛ_AC are considered.

• Model B: Only AA and BC bonds can form, i.e., only ɛ_BC and ɛ_AA are nonzero; in simulations (and numerical calculations), f_C = 4, f_B = 2, and several values of the energy ratio ɛ_AA/ɛ_BC are considered.

In both models, particles and linkers self-assemble into structures formed by chains of linkers (sequences of AA bonds) that may branch when bonded to particles [through AC bonds in model A or BC bonds in model B—see Figs. 2(c) and 2(d)]. However, the interplay between chaining and branching is different: In model A, a patch A can bond to patches A or to patches C, setting a competition between chaining and branching [see Fig. 2(c)]; in model B, each type of patch is only engaged in one type of bond and this competition is absent [see Fig. 2(d)].

It is worth noticing that model A and model B (with f_B = 2) become the thoroughly studied linker–particle model of Refs. 16 and 20 when ɛ_AA = 0. Therefore, the present work can be seen as an extension of the study of linker–particle models to a case where the self-assembly of linkers into chains is present.

Structural properties of the models were obtained as a function of temperature, density, and composition with a classic Metropolis Monte Carlo (MC) simulation in the canonical ensemble using Ref. 26 as reference. A mixture of 1200 particles and linkers in a cubic box with periodic boundary conditions was set up. Particles and linkers were randomly placed in the box with random orientations and then moved around until equilibrium was reached. A move was defined as a simultaneous displacement between ±0.05σ in each direction and a rotation between ±0.1 rad around a random axis, with all quantities being drawn from uniform distributions. Simulations ran for a minimum of 10⁵ MC steps (each step is defined as 50 000 attempts to move a particle or a linker). Steadiness of the bonding probabilities (i.e., of the number of bonds formed in the system) and the fraction of particles belonging to the largest cluster were used to assess equilibrium. Cluster size distributions were obtained using the Hoshen–Kopelman algorithm.²⁷ 

The particles were decorated with f_C = 4 patches of type C, placed on their surface as vertices of a tetrahedron. The linkers were decorated with 2 patches of type A as (opposite) poles, and with f_B = 2 patches of type B, placed at the equator and as opposite poles. The radius of the patches (or the range of the patch–patch potential) was set to $δ=(5−23−1)σ/2≈0.119σ$⁠, the maximum value that guarantees that each patch is engaged in a single bond.

In what follows, we present the specific expressions of matrix $T̃$ and of Eqs. (3) and (7) for models A and B.

The percolation diagram of model A when linkers do not form chains (i.e., when ɛ_AA = 0) has already been obtained^16,20 and is represented in Fig. 3 (solid line in all panels). Percolation is present only for a limited range of compositions, 1/7 ≤ x ≤ 0.6, at low temperatures—Fig. 3(a). These two limiting compositions can be determined from (9) and (12) with p_A→A = 0 (and f_C = 4): when p_C→A → 1 (i.e., all patches C are bonded), x → 1/7; when p_A→C → 1 (i.e., all patches A are bonded), x → 0.6. This has a transparent physical meaning: When chains of linkers are absent, the system needs both a minimum amount of particles (1/7) and a minimum amount of linkers (0.4) to form large clusters. Below x = 1/7, there are too many linkers: Part of them bond to patches C, almost fully covering the particles and preventing significant bonding between two particles (patches C are “blocked”); the other linkers are free and do not contribute to clustering. Above x = 0.6, the linkers can promote the bonding of particles, but their number is not enough to form large clusters. The introduction of chains of linkers changes qualitatively the temperature–composition percolation diagram for x ≤ 0.6—see Fig. 3(a). In particular, the limit x = 1/7 disappears and percolation can be obtained for all compositions up to x = 0.6 at low temperatures. The cause of this change is energy minimization. A bond AC decreases the energy by ɛ_AC; the formation of a bond AA from bonds AC requires the breaking of two bonds AC (to free two patches A), and the energy variation resulting from this process is −ɛ_AA + 2ɛ_AC. As a consequence, at low temperatures and when ɛ_AA < 2ɛ_AC, patches A will bond preferentially to patches C. At low composition x, bonds AC will saturate patches C, but all the remaining patches A can now bond to form AA bonds. Therefore, the energy is minimized by forming chains of linkers (AA bonds) that branch when connected to particles (AC bonds). This structure is percolated and will always form for 0 < ɛ_AA/ɛ_AC < 2 and x < 0.6 at sufficiently low temperatures. Therefore, the self-assembly of linkers into chains opens the possibility of percolation with fewer particles (for the same amount of linkers).

These theoretical predictions are confirmed with simulations, as shown in Fig. 4. For a series of equilibrated simulations at fixed (x, ρ, T), the cluster size distribution was recorded at each 10³ MC steps, the fraction of particles and linkers that belong to the largest cluster was determined, and its mean value (the points in Figs. 4 and 8) and standard deviation (the error bars in Figs. 4 and 8) were calculated. The fraction of particles and linkers that belong to the largest cluster, p_∞, is displayed, for ɛ_AA/ɛ_AC = 0 and ɛ_AA/ɛ_AC = 1 at a fixed low density as a function of x for a given low temperature in Fig. 4(a) and as a function of temperature for a given low x in Fig. 4(b). p_∞ is the order parameter for the percolation transitions, going from 0 (in the thermodynamic limit) below the percolation threshold to a nonzero value above the percolation threshold. For the finite systems employed, the sudden increase of p_∞ signals the onset of percolation. The simulation results of Fig. 4(a) confirm that the formation of chains of linkers decreases the fraction of particles needed to obtain percolation: A strong variation of p_∞ is observed at lower x for ɛ_AA/ɛ_AC = 1. On the other hand, the results depicted in Fig. 4(b) show that for a fixed, low x, percolation is only obtained in the case ɛ_AA/ɛ_AC = 1.

The temperature at which, for low x, percolation disappears, has a non-monotonic dependence on ɛ_AA/ɛ_AC, as can be seen in Figs. 3(a) and 3(c). This can be understood by recognizing that there are three ways of disrupting the fully bonded percolated network that is formed at low temperatures: (a) breaking the chains (or AA bonds) that connect the particles; (b) breaking the bonds between the chains and the particles (or AC bonds); (c) a combination of (a) and (b). These three regimes can be identified by determining the fraction of bonds AA and AC that do not break when the system is heated from T = 0 to the percolation temperature. These quantities [calculated at (ρ, x) = (0.1, 0.1)] are represented, as a function of ɛ_AA/ɛ_AC, in Fig. 5:

• For low values of ɛ_AA/ɛ_AC (up to 0.5), percolation disappears due to the breaking of chains: Around 60% of bonds AA are broken, while AC bonds remain unchanged. In this case, the energy cost of breaking AA bonds is low, and so the breaking of chains to an extent that leads to the disappearance of percolation may happen at a low temperature.

• For the larger values of ɛ_AA/ɛ_AC (in the range $≈1.5$ to 2), the network is disrupted due to the breaking of bonds between chains and particles: 2/3 of the bonds AC break, while new bonds AA are formed. In this regime, the energy cost of replacing two AC bonds by one AA bond and two free patches C is low, and therefore the percolation threshold occurs at low temperatures.

• For intermediate values of ɛ_AA/ɛ_AC, percolation disappears through both mechanisms: When ɛ_AA = ɛ_AC, around 25% of bonds AA and bonds AC are broken at the percolation threshold. In this regime, the energy cost of breaking chains is larger than in (a) and the energy cost of replacing bonds AC by bonds AA and two free C patches is larger than in (c). As a consequence, the temperature at which percolation disappears is larger than in those regimes.

The dependence of the percolation threshold on density is illustrated in Figs. 3(b)–3(d). For a fixed intermediate temperature [k_BT/ɛ_AC = 0.075 in Fig. 3(b)], percolation is observed above a minimum density, for a range of compositions x. For values of ɛ_AA/ɛ_AC up to 1, this minimum density is low and does not change; the lower limit of the range of compositions, on the other hand, decreases significantly when ɛ_AA/ɛ_AC ≈ 1. This effect is evident in Fig. 3(d): The minimum number of particles needed to obtain percolation for a fixed number of linkers decreases significantly for those values of ɛ_AA/ɛ_AC. On the contrary, the maximum number of particles for which, for a given number of linkers, percolation still exists is unaffected by ɛ_AA/ɛ_AC. This behavior is totally different for ɛ_AA/ɛ_AC = 1.5: At k_BT/ɛ_AC = 0.075, percolation is obtained only at high densities [Figs. 3(b) and 3(d)]; percolation takes place at low densities only if the temperature is significantly lowered—see Fig. 3(c).

The structure of the ground state of model A changes when ɛ_AA > 2ɛ_AC. It becomes energetically more favorable to form bonds AA than bonds AC, and all linkers tend to assemble into long chains without connecting to the particles. Only chains with no branching are formed at low temperatures and so percolation is not energetically favorable. We found no numerical solutions to Eqs. (9)–(14) using δ_AC = δ_AA = δ = 0.119σ and ɛ_AA > 2ɛ_AC.

However, it is possible to envisage a mechanism that promotes percolation when it is not energetically favorable to form branched structures. As temperature is raised, bonds AA start to break and to free A patches that can bond to C patches. The resulting structure will be determined by the competition between these two types of entropic defects:^3,31 free patches A and bonds AC. Favoring bonds AC promotes branching and the possibility of percolation. These bonds can be made entropically more favorable by increasing their volume v_AC. Figure 6 shows the percolation thresholds for ɛ_AA/ɛ_AC = 2.1 and values of δ_AC/σ = 0.45, 0.5, 0.6 (i.e., larger values of the bonding volume v_AC). δ_AA is kept equal to 0.119σ. The temperature–composition diagram of Fig. 6(a) exhibits a percolation loop that appears only when δ_AC ≳ 0.4 and that increases in size when δ_AC is increased. This result shows that, at fixed low ρ and intermediate x, the temperature increase favors, at first, “entropic” branching to an extent that makes percolation possible. Further increase in the temperature promotes breaking of both types of bonds and, at some point, the vanishing of percolation. Therefore, the nonintuitive phenomena of percolation (or gelation) by heating³² is obtained as a result of an entropic competition in bond formation. The density of the percolation threshold is represented in Fig. 6(b) as a function of temperature, at x = 0.1: There is a density below which percolation is not possible; for larger densities, percolation occurs within a range of temperatures.

The use of δ_AC > 0.119σ in the theoretical framework developed in Sec. II B is, strictly speaking, inconsistent. Both the approximation for Δ_AC in (4) and the tree-like cluster description employed for the structure assume that each patch can only be engaged in one bond (single bond per patch condition). This hypothesis is violated, for the interacting potential given by (1), when δ_αβ > 0.119σ. However, the results represented in Fig. 6 are still physically meaningful. In fact, flexible linkers (or particles) and mobile patches on particles could give origin to larger bonding volumes without compromising the single bond per patch condition. Therefore, the results of Fig. 6 suggest that in systems where linear linker–linker and low valence linker–particle self-assembly occur, entropic effects related to the flexibility and mobility of linkers can give rise to unexpected features like percolation (or gelation) by heating and closed loops in percolation diagrams.

We conclude that model A exhibits two distinct regimes for percolation that are set solely by the relation between the energy scales ɛ_AA and ɛ_AC, irrespective of the values of δ_AC and δ_AA: When ɛ_AC > ɛ_AA/2, percolated sates are always obtained when T → 0 and x < 2(f_C − 1)/(3f_C − 2) (= 0.6 for f_C = 4); when ɛ_AC < ɛ_AA/2, percolated states are absent when T → 0.

Model B is characterized by the self-assembly of linkers into chains through AA bonds and by the branching of these chains when they connect to the particles through BC bonds. Since each type of patch is only involved in one type of bond, the formation of bonds AA does not constrain the formation of BC bonds (and vice versa). As a consequence, the structure of the ground state is independent of ɛ_AA/ɛ_BC and corresponds to the maximization of the number of both types of bonds.

The percolation diagrams of Fig. 7 (for f_B = 2) demonstrate that the introduction of chain assembly in model B leads to substantial changes. The temperature–composition diagram of Fig. 7(a) shows that percolation is obtained for all compositions x at sufficiently low temperatures. This is a drastic change from the case ɛ_AA = 0 and f_C = 4, where percolation could only be found for 1/7 < x < 0.6, and from model A, where percolation was never found for x > 0.6. The presence of percolation at extremely low x and at low temperatures means that a vanishingly small amount of particles is enough to connect, through BC bonds, the abundant and long chains of linkers formed by AA bonds, to an extent that leads to the formation of percolating clusters. On the other hand, the emergence of percolation at x close to 1 and low temperatures means that a vanishingly small amount of linkers is enough to connect the particles in a percolated cluster, through the combination of a few bonds BC that connect linkers to particles and of bonds AA that connect linkers to linkers and, indirectly, particles to particles.

The density dependence of percolation is shown in Figs. 7(b)–7(d). At a given temperature [k_BT/ɛ_BC = 0.075 in Fig. 7(b)], percolation only exists above a certain density, and this minimum density decreases with ɛ_AA/ɛ_BC. Above this density, percolation occurs for a range of compositions that increases with increasing ɛ_AA/ɛ_BC. In practice, when ɛ_AA/ɛ_BC is high, percolation occurs for every composition. At fixed composition [x = 0.1 in Fig. 7(c)], the temperature at which percolation occurs for a given density increases with ɛ_AA/ɛ_BC. At a fixed temperature, the density of particles (linkers) needed to obtain percolation for a given density of linkers (particles) decreases significantly with increasing ɛ_AA/ɛ_BC—see Fig. 7(d). The results of Fig. 7 show that, in model B, when chaining of linkers is favored, percolation is controlled mainly by temperature (see the results for ɛ_AA/ɛ_BC = 1.5): Below a threshold temperature (almost constant), percolation occurs for all compositions and all densities above a very small minimum density.

Some predictions of the theory for model B were tested (and confirmed) by simulations. Their results are shown in Fig. 8. The introduction of chains of linkers reduces the fraction of particles at which, at fixed low density and low temperature, percolation occurs, as shown by the comparison between the results for ɛ_AA/ɛ_BC = 0.8 and for ɛ_AA/ɛ_BC = 0 in Fig. 8(a). The results shown in Fig. 8(b) confirm the increase of the temperature at which percolation occurs (for fixed low density and composition) when chains of linkers are present.

The theory presented in Sec. II B can be developed to obtain some properties of the clusters, within its assumption that clusters are tree-like. In particular, the percolation threshold Eqs. (9) and (16) can be expressed in terms of two structural properties: ⟨n_C⟩, the mean number of bonds formed by one particle, which quantifies the number of chains that branch in one particle; and ⟨ℓ_L⟩, the mean length of the chains formed by linkers, which quantifies the extent of chaining.

The percolation thresholds expressed by Eqs. (24) and (25) can be used to obtain (⟨n_c⟩, ⟨ℓ_L⟩) percolation diagrams. Figure 9 depicts the four representative diagrams, each obtained at a fixed composition x. The thick lines represent the physical limits imposed by the restrictions: (i) 0 ≤ p_A→A ≤ 1, which translates into ⟨ℓ_L⟩≥ 1; (ii) 0 ≤ p_C ≤ 1, which is equivalent to 0 ≤ ⟨n_C⟩ ≤ f_C; (iii) for model A, p_A→A + p_A→C ≤ 1, which is equivalent to ⟨n_C⟩⟨ℓ_L⟩ ≤ 2(1 − x)/x, and, for model B, p_B→C ≤ 1, which is equivalent to ⟨n_C⟩ < f_B(1 − x)/x. In model A [see Figs. 9(a) and 9(b)], percolation only occurs for a limited range of values (⟨n_c⟩, ⟨ℓ_L⟩): ⟨n_C⟩ ≥ f_C/(f_C − 1) and $⟨ℓL⟩≤⟨ℓL⟩max$⁠, with $⟨ℓL⟩max=2(fC−1)(1−x)/(fCx)$⁠. Therefore, larger chains of linkers in a percolated network can only be obtained if composition x is lowered. The percolation diagrams for model A display another dependence on composition: For 2/(f_C + 2) ≤ x ≤ 2(f_C – 1)/(3f_C – 2), the percolated states exhibit limited branching since ⟨n_C⟩ < f_C [see Fig. 9(b)]; for x > 2(f_C − 1)/(3f_C − 2), percolation disappears, as already seen in Secs. III A and III B. The percolation behavior of model B is different—see Figs. 9(c) and 9(d): For all compositions, percolated networks with all possible lengths of chains of linkers can be obtained. The composition only restricts branching in percolated states: If x > f_B/(f_B + f_C), then ⟨n_C⟩ < f_B(1 − x)/x [see Fig. 9(d)].

The present work shows that in linker-mediated aggregation of particles, the introduction of self-assembled chains of linkers gives an extra control over percolation, compared to the case where chains of linkers are absent. The choice of the specific interactions between linkers (model A or model B), the strength of those interactions (value of ɛ_AA/ɛ_AC or ɛ_AA/ɛ_BC), and the bonding volumes can be used to tune the composition, temperature, and density at which percolation occurs and to change the structure of the percolated clusters.

In model B, where chaining of linkers and branching through bonds between chains and particles are independent, it is possible to reach percolation at both low and high concentrations of particles: A vanishingly small number of particles (linkers) is enough for percolation to occur in systems with a large fraction of linkers (particles). Model A—where competition between chaining of linkers and branching of chains (through their bonds to particles) is present—exhibits a more complex behavior, which depends on the ground state of the model. For ɛ_AA/ɛ_AC < 2, branching is energetically favored, and percolation is extended to low particle composition. The temperature at which, at low compositions, percolation disappears depends non-monotonically on ɛ_AA/ɛ_AC, being maximum for ɛ_AA ≈ ɛ_AC. This non-monotonic behavior is related to the energy costs of disrupting the percolated network, either by breaking the chains or by breaking the bonds between chains and particles. Percolation occurs in model A with ɛ_AA > 2ɛ_AC only when the entropic gain of branching is increased. In this case, percolation by heating and closed loops of percolated phases in temperature–composition diagrams are obtained. The number of branching bonds per particle is related to the mean length of chains of linkers through a function that depends only on composition. This means that composition controls the structure of the percolated clusters: It is possible to obtain clusters with larger (smaller) chains at the cost of decreasing (increasing) branching. Table I shows a summary of the main results.

The simulations performed for model A (for ɛ_AA < 2ɛ_AC) and for model B have confirmed the qualitative differences predicted by the theory for percolation diagrams obtained when chains of linkers are present. Figures 4 and 8(a) show, through the size of the error bars, that the cluster size distribution displays much larger fluctuations in a broader region of composition or temperature when linkers are dominant (x ≲ 0.2) and can bond to each other (ɛ_AA ≠ 0). Therefore, the introduction of self-assembled chains of linkers will require larger systems and longer simulations if more precise measurements of percolation thresholds are desired. Moreover, the quantitative comparison between simulation and theoretical results for model B would have to take into account a fairly strong tendency for loop formation, which we have checked in a more detailed analysis of clusters in the simulations of this model.

Percolation is only a necessary condition to obtain an equilibrium gel. The phase diagrams of the models of this work are needed to know the thermodynamic conditions at which percolation occurs in single phase regions. These calculations are left for future work, but some hints can be found in previous studies.^20,22,23

The case of model A (or model B with f_B = 2) with ɛ_AA = 0 (particle–linker mixture with no chains) and f_C = 3 was investigated in Ref. 20. Percolated liquids, not affected by phase separation, were found at intermediate compositions, low densities, and low temperatures (see Figs. 4–6 in Ref. 20). Moreover, in Ref. 23, the phase diagram of model A with ɛ_AA = ɛ_AC and f_C = 3 shows that at high linker concentrations, the percolated liquids have less tendency to phase separate with a vapor (see Fig. 6 in Ref. 23). Even in a mixture of patchy particles with 2 and 4 patches, all equal and all forming bonds, large regions of a single phase percolated liquid are found.²² Therefore, it is much likely that the models studied in this work will exhibit regions of phase diagrams with a single percolated liquid phase.

Finally, it should be mentioned that, to complete the study of equilibrium gels, the onset of mechanical stability in the percolated network has to be addressed.^33,34 The emergence of rigidity in equilibrium self-assembling thermodynamic systems is poorly understood.³⁵ However, recent developments of a theory that describes the mechanical properties of ideal reversible polymer networks at thermodynamic equilibrium may apply to linker–particle systems,³⁶ if properly generalized to the case of a mixture.

We acknowledge financial support from the Portuguese Foundation for Science and Technology (FCT) under Contract Nos. PTDC/FIS-MAC/28146/2017, LISBOA-01-0145-FEDER-028146, PTDC/FIS-MAC/5689/2020, CEECIND/00586/2017, UIDB/00618/2020, and UIDP/00618/2020.

The authors have no conflicts to disclose.

M. Gouveia: Investigation (equal); Software (equal); Visualization (equal); Writing – review & editing (equal). C. S. Dias: Conceptualization (equal); Investigation (equal); Supervision (equal); Writing – review & editing (equal). J. M. Tavares: Conceptualization (equal); Investigation (equal); Supervision (lead); Writing – original draft (lead).

The data that support the findings of this study are available from the corresponding author upon reasonable request.

Let us consider the levels of a tree-like cluster (with no loops) formed by linkers and particles connected through the bonding of its patches as described in Sec. II B. b_i,γ is the number of patches of type γ that belong to particles or linkers in level i and that are bonded to the previous level. Matrix $T̃$ in (8) and (15) is obtained from the relation between the number of patches b_i,γ of different levels. Figure 10 represents, for models A and B, all the possible ways of obtaining a patch of level i + 1 that is bonded to level i, from the patches of level i that are bonded to level i − 1.

In model B, bonds AA and BC are allowed and 3 different types of patches are involved in bonds.

We conclude that the general definition for the percolation threshold, that is, the thermodynamic conditions for which the maximum absolute value of all eigenvalue of matrix $T̃$ is 1, reduces, for model B, to the conditions at which one of the eigenvalues is equal to 1.