Gel Formation in Reversibly Cross-Linking Polymers

By means of Langevin dynamics simulations, we investigate the gel formation of randomly functionalized polymers in solution, with the ability to form both intra- and intermolecular reversible bonds. Under highly dilute conditions, these polymers form soft nano-objects (so-called single-chain nanoparticles, SCNPs), resulting from the purely intramolecular cross-linking of the reactive functional groups. Here we show that the competition between intra- and intermolecular bonds at finite concentration is governed by a delicate balance of various entropic contributions and leads to a density dependent effective valence. System-spanning networks are formed at relatively low monomer densities and their stability is mediated by just a small number of intermolecular connections per chain. The formation of intermolecular bonds furthermore can induce a non-monotonic dependence of the polymer size on the density for long bond lifetimes. Concomitantly, the polymers in the percolating cluster adopt an intramolecular structure characteristic for self-avoiding chains, which constitutes a strong contrast to the fractal globular behavior of irreversible SCNPs in crowded solutions with purely topological interactions (no intermolecular bonds). Finally, we study the dynamics of the system, which displays signatures expected for reversible gel-forming systems. An interesting behavior emerges in the reorganization dynamics of the percolating cluster. The relaxation is mostly mediated by the diffusion over long distances, through breaking and formation of bonds, of chains that do not leave the percolating cluster. Regarding the few chains that are transiently free, the time they spend until they reattach to the cluster is solely governed by the bond strength.


I. Introduction
Single-chain nanoparticles (SCNPs) are soft nano-objects synthesized from a linear polymer precursor, which is functionalized with reactive groups capable of forming intramolecular bonds. They are typically a few nanometers (≤ 20 nm) in size and possess a large surface-to-volume ratio. The necessary technological ingredients for their synthesis include controlled polymerization, monomer functionalization and cross-linking protocols that ensure their purely intramolecular collapse. Design and function of SCNPs is a rapidly growing area of research due to their promising applications as catalysts, drug delivery vehicles, biosensors or rheology modifying agents. [1][2][3][4][5][6][7][8][9][10][11][12][13][14] Since its beginnings at the start of the 21st century, the synthesis of SCNPs has been dominated by polymer chemistries involving irreversibly crosslinking monomers. In the past years, however, the possibility of exploiting reversible interac-tions to produce stimuli-responsive SCNPs has gained increased interest. 1,2,4,5,[15][16][17][18] Separate classes of reversible interactions with distinct advantages have emerged in the field of singlechain technology: non-covalent and so-called dynamic covalent bonds. Non-covalent bonds are characterized by their relatively low energy (typically a few k B T ), which is modulated smoothly by external variables such as temperature, pH and solvent. Prominent examples of non-covalent interactions used in SCNPs are hydrogen bonds, 19,20 helical 21,22 or π − π stacking, 23 host-guest interactions, 24,25 ionic attraction 26 and metal complex formation. 27,28 In contrast to non-covalent bonds, dynamic covalent bonds are very robust and their formation, breaking or exchange can be induced rapidly by very specific external stimuli. These can be pH, photons, redox potentials or a catalyst. The classical example of a dynamic covalent bond is the disulfide bridge, which plays a prominent role in the stabilization of the native folded state of proteins. It served as inspiration for including disulfide bonds, 29 but also hydrazone, 30 enamine, 31 coumarine 32 and anthrazene 33 bonds in the SCNP chemistry toolbox.
The advantage of dynamic covalent bonds is that the need for an external stimulus to catalyze their formation and breakage opens up the possibility of kinetically trapping the system. Furthermore, reversibility means that synthesis is never 'complete' and individual SCNPs of this kind can form intermolecular bonds in addition to their intramolecular bonds if their concentration is increased above the limit of high dilution. Such intermolecular bonds could potentially lead to aggregation or phase separation but also to the formation of a physical gel. The interplay between intra-and intermolecular bond formation has been exploited recently by Fulton et al. in thermoresponsive polymers to produce a system that reversibly crosses between a SCNP solution and an hydrogel. 34 The thermoresponsive nature of the oligoethyleneglycol methyl ether branches causes the polymers to aggregate upon a rise in temperature, while a mildly acidic pH allows the acylhydrazone bonds to undergo component exchange processes. The combination of these two orthogonal triggers leads to the reversible reorganization of intramolecularly folded SCNPs into a robustly cross-linked hydrogel. The response of this material to multiple external stimuli could be exploited in situations where the behavior of the material should depend on the specific makeup of the environment, for example the specific release of drugs in target tissues.
While the advances in synthesis of such reversible gels made from dynamic covalent SC-NPs are promising, investigations of the structure of such materials has been lacking until now. Moreover, the theoretical description of physical gels in general (as opposed to permanently cross-linked chemical gels), has come into focus in the soft matter community in relatively recent years. This is due in part to the difficulty of precisely defining the meaning of 'gel' (systems exhibiting both dynamical arrest and network formation are generally considered as gels). A common working definition of a gel is a low density disordered state with solid-like properties such as a yield stress. It combines properties of a liquid (through its disordered structure) and a solid (it does not flow). What distinguishes them from glasses is not only their typical low volume fraction but also their retention of quasi-ergodicity on all but the largest length scales dictated by the infinite percolating network. [35][36][37][38][39] A second obstacle for the establishment of a unifying theoretical framework of gel formation is the lack of an ideal model system that incorporates the minimal, necessary ingredients to reproduce the universal features of a gelling system.
In Hill's formalism of liquid condensation in terms of physical clusters, phase separation induced by strong attractive interactions can be avoided by either complementing the attraction by a long-range repulsion [40][41][42] or by modifying the attraction by limiting the valence of the interacting molecules. 43 The former can be induced by excessive surface charges on colloids, 44 while the latter can be achieved, e.g., by decorating colloidal particles with a small number of well defined attractive patches 45,46 or the engineering of specific DNA sequences designed to form star-shaped architectures with sticky ends. 47,48 An advantage of such limited-valence particles lies in the possibility of theoretically calculating their free energy within the formalism of Wertheim theory, 49,50 which allows one to determine the phase diagram of the system. 51 Furthermore, the increased experimental control over such patchy particles achieved in the past decade has paved the way for their use as highly tunable building blocks for the design of self-assembled materials. 52,53 Single-chain nanoparticles with reversible bonds may display characteristics of both microphase separating colloids and patchy particles due to the competition between intra-and intermolecular bonds. At very high dilution, intermolecular bonds should be disfavoured by the long intermolecular distances. Upon increasing the volume fraction, some of the intramolecular bonds will be exchanged for connections with other chains for entropic reasons, possibly forming a system-spanning network for the right combination of system parameters. We expect phase separation of the system to be confined to very small densities through the combination of excluded volume interactions and the inherently limited 'valence' of the polymers that originates from the locally small number of (monovalent) monomers capable of forming bonds. With these ideas in mind, in this article we present Langevin dynamics simulations of solutions of a bead-spring model for SCNPs with reversible bonds, exploring concentrations from high dilution to far beyond the overlap density. We characterize the structural and dynamic changes produced by the competition between intra-and intermolecular bonding as the concentration increases. We find that the number of intramolecular bonds is not constant but decreases with the concentration, which complicates the possibility of considering the SCNPs as objects with a characteristic valence, and thus of straightforwardly applying Wertheim's formalism. A system-spanning cluster is formed above the overlap concentration. The scaling of the cluster size distribution is consistent with nonmean field critical percolation. The connectivity of the percolating cluster is mediated by a few intermolecular connections per chain. In contrast with the fractal ('crumpled') globular conformations adopted by irreversible SCNPs (no intermolecular bonds) under purely steric crowding and topological constraints, 54 intermolecular bonding in the dense solutions just leads to weak perturbations with respect to the limit of high dilution. The analysis of the dynamics reveals that, in analogy to the case of colloids with limited valence, the system forms a reversible gel with no apparent signatures of phase separation in the broad range of investigated densities. Relaxation of the dynamic network is mostly mediated by diffusion, through breaking and formation of bonds, of chains that, without detaching from the percolating cluster, are able to explore distances of several times their size. The article is organized as follows. Section II presents the model and simulation details. Section III shows and discusses structure and dynamics in the scenario emerging as the concentration increases and the system changes from a dilute solution of SCNPs to a reversible gel. Conclusions are given in Section IV.

II. Simulation Details
The polymers are simulated as chains of beads and springs according to the model of Kremer and Grest. 55 As such, they represent uncrossable flexible chains with excluded volume interactions in implicit good solvent conditions, and the bead size σ qualitatively corresponds to a Kuhn length 56 (σ 1 nm). The chains consist of N = 200 bead ('monomers'), consecutively linked together by irreversible backbone bonds modeled via the FENE potential: 55 with K F = 15 and R 0 = 1.5. Monomer excluded volume interactions are given by a purely repulsive Lennard-Jones potential (Weeks-Chandler-Andersen potential): with a cutoff distance r LJ = 2 1/6 σ, at which both the potential and the corresponding forces are continuous. The sum of both potentials leads to a deep minimum at r min σ for the total interaction between two bonded beads, which limits the bond fluctuations and guarantees uncrossability. The units of energy, length, mass and time are, respectively, , σ, m and τ = σ 2 m/ . In the rest of the article all numerical values will be given in reduced units with adjustable parameters K and r 0 . In the case of the irreversible cross-linking, if two reactive monomers form a bond they interact for the rest of the simulation via the same FENE potential (Eq. 1) as the permanent backbone bonds.
Reversible bonds can be broken again if, at any given time step, the participating monomers are separated by a distance r > r c , upon which their interaction via the Morse potential terminates. The parameter K governs the bond strength through modulating the energy barrier that has to be overcome in order to break the bond. Since r c > r LJ and hence U LJ (r c ) = 0, the barrier is given by the energy difference U rev (r c ) − U min , where U min is the minimum of U LJ + U rev . As such, bond formation is independent of K, while bond breakage depends on K. Thus, varying K does not only change the average bond lifetime, but also the average probability of any reactive monomer being bonded at equilibrium. The remaining free parameter r 0 is chosen such that the minimum U min of the sum of the non-bonded and bonded interactions is, with respect to the irreversible case, the same in the energy and as close as possible in the distance (just a slight variation in the range 0.93 < r min < 0.98 is obtained, see Figure 1). Contrary to patchy particle models, in which the monofunctionality of the bonds is encoded in the geometry of the interaction, 48  We perform Langevin dynamics simulations at a fixed temperature T = /k B = 1 (with k B the Boltzmann constant). We use a time-step ∆t = 0.01 and a friction coefficient γ = 0.05.
The equations of motion are integrated following the scheme of Ref. 63. Before simulating reversibly cross-linking chains at various densities, we perform exploratory simulations of isolated single chains (mimicking the limit of high dilution), for various values of f (fraction of reactive groups) and K (bond strength). We calculate, for such isolated chains, the radius of gyration R g0 and the bond probability p B0 , which is defined as the ratio between the average number of formed bonds and the maximum number of bonds (corresponding to the fully cross-linked SCNP). Our results are summarized in Table 1. As expected, the probability of any reactive monomer to be bonded at equilibrium depends more strongly on K than on f .
Since we are interested in exploring the possibility of forming gels from these reversibly cross-linking chains, the bonding probability needs to be high enough for a system-spanning cluster to form and give the material the ability to propagate stress throughout  Table 1. For the remainder of the article, we will use the term 'solutions of SCNPs' only to refer to the solutions of irreversible SCNPs without intermolecular bonds. The reversible systems will be denoted by their values of K and f . The density of the system will be given in reduced with R g0 the radius of gyration of the isolated polymer (ρ → 0, see Table 1). We expect intermolecular cross-links to begin forming significantly around the overlap concentration, when monomers of different chains start to enter the same space. We explore densities in the range 0 ≤ ρ/ρ ≤ 4.4. This corresponds to monomer densities up to ρ ≈ 0.14, which lies below the entanglement density for linear chains of the same polymerization degree, found. 65 For each combination (K, f ) and each density 8 independent simulation runs were carried out, each consisting of 1 × 10 7 equilibration steps and 4 × 10 7 production steps.

Competition between intra-and intermolecular bonds
Ideally, one would wish to derive a thermodynamic description of the system, for example according to Wertheim theory, 49,50 by using inputs from computer simulations, which would allow us to evaluate the complete K − ρ phase diagram of these reversibly cross-linking polymers, and to find the regions in which gel formation is possible. Wertheim thermodynamic perturbation theory (TPT) was originally developed for associating liquids, but has also been successfully employed to elucidate the phase behavior of gel forming systems of limited valence, qualitatively and sometimes even quantitatively reproducing numerical results. 51,66,67 A few fundamental assumptions of TPT have to be satisfied in order to be able to to patchy particles, but with the distinction that the 'patches' are not located at specific points on their surface, but randomly distributed, and fluctuate due to the inherent softness of the polymer and the rearrangement of the intramolecular bonds along the chain backbone.
In this view, the intramolecular bonds solely affect the reference free energy. and f = 0.3, the system with the highest bonding probability p B0 at high dilution. This can be understood in terms of the total (intra-and intermolecular) bonding probability p B , which is displayed in Figure 2b as a function of the density. We notice that at K = 33.7 and f = 0.3 the bonding probability is already saturated at low density and stays almost constant upon crowding the system. On the other hand, the parameter combination with the smallest bonding probability at high dilution, K = 29.6 and f = 0.1, exhibits the strongest increase in p B as a function of density. Here, the loss in entropy stemming from crowding of the surrounding molecules can be largely compensated by the enthalpic gain of forming new additional intermolecular bonds. Since Figure 2a shows that the number of intramolecular bonds is not constant but strongly decays as the density grows, we conclude that a straightforward application of Wertheim's theory is not possible in the present case. The competition between intra-and intermolecular bonds when the system is highly bonded has to be understood in terms of a delicate interplay between various entropic and energetic contributions to the free energy of the system. One might expect that the formation of intramolecular bonds is favourable over the formation of connections with other molecules, as the latter reduces the translational entropy of both molecules without a compensating energetic gain (the bonds are energetically equivalent). On the other hand, depending on the contour length separating the monomers whose intramolecular bond is exchanged for an intermolecular bond, it might be that the breakage of the former could potentially increase the conformational entropy of the molecule that loses the intramolecular link.

Intermolecular Bonding and Percolation
A necessary, albeit not sufficient, prerequisite for gelation is the emergence of a fully connecting network, spanning We explore the formation of a spanning cluster in our system. Figure 4  Since the 'effective valence' of these reversibly cross-linking polymers is dependent on the density, reactive monomer fraction and bond strength, we are interested in how many intermolecular bonds a chain forms on average as the system starts to percolate. One has to keep in mind, however, that multiple bonds are possible between two specific polymers, and every additional bond shared between two given chains does not add to the overall connectivity of the network. In Figure 5 we thus display both the average number of intermolecular bonds per chain as well as the average connectivity C (i.e. to how many different chains is a given chain connected through inter-chain bonds, irrespective of the number of bonds mediating the connection between each pair of chains). The values at which a pronounced peak at large cluster sizes first appears in P (N cluster ) (see Figure 4) are marked by ellipses The values at which a pronounced peak at large cluster sizes first appears in P (N cluster ) (see Figure 4) are marked by ellipses.
in Figure 5. These values provide an estimation of the percolation line and, as can be seen, this occurs at densities above and even far above the overlap concentration. The general trend is that the percolation transition approaches the overlap concentration from above by increasing the fraction of reactive groups and the bond strength. To gain a deeper understanding of the formation of intermolecular bonds, we next investigate the distribution of intermolecular bonds at specific densities.

Structural Properties
In the previous section, we have shown that the intramolecular bonds are not unaffected by the presence of other molecules, but rather that intra-and intermolecular bonds compete with each other, the outcome of which depends on a delicate interplay of various entropic contributions. We expect this exchange of intra-for intermolecular bonds to be accompanied by structural changes in the polymers. The partial unfolding induced by the opening of intramolecular loops might to some degree counteract the collapse found above the overlap concentration in the case of the irreversible SCNPs. A first measure of structural changes upon increasing the density is the size of the polymers, given by the radius of gyration depicted in Figure 8. We find that the competition between steric repulsion and partial unfolding leads to qualitatively different density dependences of the molecular size. For K = 29.6, shrinking due to macromolecular crowding dominates, whereas for the strong association energy K = 33.7, the polymers swell slightly with respect to their conformations at high dilution. At high densities, however, a re-entrance of R g for K = 33.7 can be observed.
For comparison we include the corresponding data for linear chains without reactive groups and for the irreversible SCNPs (no intermolecular bonds). In both cases shrinking is stronger than in the reversible systems. The effect is especially pronounced in the irreversible SCNPs, resembling the observation of ring polymers collapsing to crumpled globular structures. 54,64,77 A more detailed description of the intramolecular structure of the polymers is given by their isotropic form factors, where r jk is the distance between monomers j and k and the sum only includes pairs of monomers belonging to the same molecule. Being fractal objects, the form factors of polymers typically follow a scaling law w(q) ∼ q −1/ν for wave vectors corresponding to length scales bigger than the bond length b, but smaller than the radius of gyration, i.e.
1/R g q 1/b. the macromolecular centers-of-mass through the structure factor where r CM,j denotes the position of the center-of-mass of the macromolecule j. Figure 10 shows S CM (q) for reversible chains with K = 33.7 and f = 0.3 at various monomer densities. The polymers with other values of (K, f ) exhibit qualitatively the same behavior (not shown). For comparison we include the corresponding data for irreversible SCNPs at the same monomer densities. In the limit of q → 0 the structure factor of the macromolecular centers-of-mass is proportional to the compressibility χ of the material At the same monomer density the low-q limit of S CM (q) is significantly and consistently lower for the solution formed by irreversible SCNPs than for the network made up of reversibly bonded chains. The permanent cross-links in the irreversible SCNPs prevent concatenation of loops and lead to less interpenetration of different SCNPs, rendering the system less compressible (lower S(q = 0) and χ) than the reversible network at the same density. Furthermore, in the investigated range of concentrations 0.1 < ρ/ρ * < 4.4, the structure factor shows no sign of phase separation or growing inhomogeneities (which would be manifested by a diverging or strongly growing S(q) as q → 0). To assure that no phase separation is indeed intervening with gelation, we investigate density fluctuations in the system. To this end, the box is divided into sub-boxes of side L s and the 'local' monomer density φ is calculated within each of them. Density fluctuations are then defined by the parameter δφ 2 = (φ − φ ) 2 , where the average is both taken over all sub-boxes and over various realizations (therefore the mean value of φ is identical to the macroscopic monomer density, φ = ρ). In the absence of correlations, fluctuations just behave as 79 δφ 2 ∼ ρ. Figure 11 shows the normalized quantity δφ 2 /ρ as a function of the concentration, so that the effect of the correlations is reflected by deviations from a plateau.
The data in the main panel are shown as a function of the absolute monomer density ρ. In order to higlight the different trends found below and above the overlap concentration, the inset shows the data as a function of ρ/ρ . The choice of the sub-box size L s influences the values of δφ 2 /ρ. If L s is too small, fluctuations are inherently limited, while if it is too big the phase occupying less volume might not be properly sampled. We varied L s in a reasonable range 3 ≤ L s ≤ 20 and the fluctation size showed the same dependence on the concentration irrespective of the specific L s , the numerical values just changing by a scaling factor. Data in Figure 11 were obtained by using L s = 5.
At high dilution, ρ/ρ * 1, δφ 2 /ρ for the linear chains and the irreversible SCNPs shows a plateau. As aforementioned this result reflects vanishing correlations and is fully expected in such systems, due to both high dilution and the absence of bonding and aggregation.
Above the overlap concentration (ρ/ρ > 1), in semidilute conditions, the low-q limit of the structure factor of linear chains is expected to follow the relation 56 where ξ is the correlation length. The portion of the linear chain within this length scale keeps its high-dilution scaling, i.e., ξ ∼ g ν F , with g the number of monomers of the chain in the correlation volume. Combining the former relations we obtain: and the concentration dependence of the fluctuations follows the power-law: Since ν F = 0.588, the fluctations in the solutions of linear chains are expected to decrease with the concentration as δφ 2 /ρ ∼ ρ −1.54 . This is confirmed in Figure 11. Similar results are found for the irreversible SCNPs.
Remarkably, the solutions of reversibly cross-linking polymers show the same qualitative trends as the solutions of non-aggregating linear chains or irreversible SCNPs, in particular the plateau at low concentrations. Quantitative differences are found in the relative amplitudes of the fluctuations, which can be easily rationalized. For the same values of L s and ρ the amplitude of δφ 2 /ρ is systematically higher as the molecular size decreases (see values of R g0 in Table 1). Since all systems have the same molecular weight, smaller macromolecular sizes correspond to more compact objects, which at the same macroscopic density ρ lead to higher values of δφ 2 when they are distributed over the simulation box. In summary, data in Figure 11 are consistent with the results for the structure factors of Figure 10 and no signs of phase separation are found (otherwise consistently growing values of δφ 2 /ρ with decreasing ρ would be found instead of the plateau). Still, phase separation cannot be fully discarded -one should have in mind that fluctuations are limited by the simulation box.
Unfortunately the evaluation of the phase diagram for small densities is quite challenging and we cannot give a fully conclusive answer to this issue.

Dynamic Properties
After having discussed in detail the structural properties of the networks formed through the reversible intermolecular bonds, we now shift our focus to the dynamics of the system. In all the simulated systems monomers reach the diffusive regime at long time scales,  Table 1, and the range ρ/ρ * ∼ 2 − 4 where we observe the crossover to the mild regime in the diffusivity, the former distance is d inter ∼ 13 − 15. On the other hand there are in average N p /3 monomers between consecutive reactive monomers in a same chain (since f = 0.3), which corresponds to a distance in the real space of d intra ∼ (C ∞ N p /3) ν . For the used bead-spring model the characteristic ratio 56 is C ∞ ≈ 1.7, and at high concentrations ν = 0.58 (Figure 9b). Therefore d intra ∼ 16, which is comparable or larger than d inter .
Finally, we investigate the bond dynamics and the reorganization of the percolating cluster once it is formed. We first analyze whether the lifetime of intermolecular bonds is affected by gelation and whether cooperativity plays a role in the formation of intermolecular bonds. To this end, we calculate the intermolecular bond self-correlation where B ij (t) = 1 if the reactive monomers i and j (belonging to different chains) form a bond at time t and this has never been broken since t = 0. If the bond has been broken at least once, then B ij (t) = 0. Furthermore, we calculate the self-correlation of the cluster where M i (t) = 1 if polymer i has been a member of the percolating cluster at all times 0 ≤ t ≤ t and M i (t) = 0 otherwise. As such, the relaxation of S cluster (t) provides a measure for the average dissociation time from the cluster. Figure 14 displays the intermolecular bond selfcorrelation as well as the cluster self-correlation for f = 0.3 and for the two investigated bond strengths. In both cases the intermolecular bond self-correlation is not affected by changes in the density and can be well described by an exponential decay, S inter (t) ∼ e −t/τ . This exponential relaxation is also found for the intramolecular bonds (not shown), demonstrating that bond breaking is purely governed by temperature (as the bond strength is akin to an inverse temperature) and bonding is not cooperative (stretched exponentials would be otherwise observed).
On the other hand, the cluster self-correlation (weakly non-exponential) is strongly af- fected by density. For K = 33.7, at high densities the percolating cluster becomes so stable that almost no polymers leave it within the simulation time window. This, however, does not mean that the cluster does not rearrange. Indeed, within the simulation time scale, the polymers are able to diffuse distances of more than 3R g0 ≈ 26 even at ρ/ρ * = 4.0 (see Figure 12).
An average of 6 intermolecular connections per chain at this concentration (see Figure 5) allows the polymers to move through the cluster, breaking and reforming bonds, without ever detaching from it. This stability of the cluster along with its potential to rearrange could lead to an interesting behavior under external stresses, which should be investigated in future work. Furthermore, it is not clear whether intermolecular or intramolecular bonds would break first under shear, potentially leading to an interesting viscoelastic response.
At intermediate densities, a significant number of chains (see Figure 14b) detaches from the infinite cluster from time to time and moves through the system before getting reabsorbed in the cluster again. We monitor each polymer that leaves the percolating cluster at some time during the simulation, and compute the time it spends since it leaves it until it is reattached. The results of this analysis are shown in Figure 15. Surprisingly, the distributions of times spent outside the percolating cluster are essentially independent of the number of reactive monomers and of the density of the solution. We find a universal distribution solely dependent on the bond strength K that follows a stretched exponential p(t c ) ∼ e −(t/τc) β , with β = 0.55 for K = 29.6 and β = 0.40 for K = 33.7. As shown in Figure 13 the diffusivity strongly depends on the density, e.g., for the case K = 33.7 and f = 0.1 the diffusivity decays by an order of magnitude in the range 1.5 ≤ ρ/ρ * ≤ 3.2, while in that same range p(t c ) is at most marginally dependent on the concentration with no systematic trend.
As can be seen in Figure 15 for K = 33.7 most of the free motions between detachment and reattachment occur within time scales t c < 10 4 (p(t c ) has decayed 2 orders of magnitude at that time), which as seen in the analysis of the MSD correspond to displacements of at most the polymer size. At the relaxation time τ c = 150 the MSD is still in the soft plateau regime, i.e., in that time scale the typical free chain is still localized in the region of its last detachment. In other words, free chains do not perform long flights until they find an unbonded reactive group in the percolating cluster and are reattached to it. This result suggests that the reattachment time is not controlled by the characteristic diffusivity over large scales (which indeed is strongly affected by crowding), but rather by the time it takes to break a bond (essentially controlled by the energy scale K ) so that the new unbonded monomers in the free chain can explore their counterparts in the percolating cluster through conformational fluctuations.

IV. Conclusions
We have investigated the gel formation of linear polymer chains decorated with functional groups with the ability to form reversible bonds. For this purpose, we have employed Langevin dynamics simulations and a bond potential that mimics reversible covalent bonds for the synthesis of reversible single-chain nanoparticles. We have studied in detail the competition between intra-and intermolecular bonding as a function of the concentration, finding that the replacement of intramolecular links by intermolecular ones prohibits a straightforward treatment of the system via Wertheim's theory and its use for elucidating the complete phase diagram. The formation of a system-spanning cluster takes place above the overlap concentration and seems to converge to it by increasing the bond strength and fraction of reactive groups.
Surprisingly, the introduction of intermolecular bonds induces a non-monotonic dependence of the radius of gyration on the density for high bond strengths, while the polymers shrink monotonically for lower bond strengths. In all the cases the variation of the polymer size with respect to high dilution conditions is at most 5%, even at 4 times the overlap con- Finally, we have demonstrated that the dynamics of the system display the soft caging regime expected for gel-forming materials. The reorganization dynamics of the percolating cluster is mostly mediated by the diffusion, through breaking and formation of bonds, of chains that, without detaching from the percolating cluster, are able to explore distances of several times their size. A remarkable universal behavior is found: the time a polymer spends out of the percolating cluster is independent of the density and solely depends on the bond lifetime, which suggests that reattachment to the percolating cluster is limited by the availability of free reactive groups and not by the diffusion of the free chain.
In conclusion, our results present valuable preliminary insights into the gelling process of reversibly cross-linking polymers with randomly distributed functional groups. The competition between intra-and intermolecular bonds leads to complex structural rearrangements purely governed by entropical contributions. Systems in which such a competition is present have not been studied extensively in the literature to this date. We hope that our results will motivate further research efforts in this direction. On the other hand, the possibility of combining advanced functions (catalysis, luminescence, etc), polymer flexibility, dynamic bonds and assembly into equilibrium gels make reversible SCNPs potentially attractive systems as building blocks for the design of smart materials with self-healing properties.
Graphical TOC Entry intramolecular intermolecular unreacted