Coarsening Kinetics of Complex Macromolecular Architectures in Bad Solvent

This study reports a general scenario for the out-of-equilibrium features of collapsing polymeric architectures. We use molecular dynamics simulations to characterize the coarsening kinetics, in bad solvent, for several macromolecular systems with an increasing degree of structural complexity. In particular, we focus on: flexible and semiflexible polymer chains, star polymers with 3 and 12 arms, and microgels with both ordered and disordered networks. Starting from a powerful analogy with critical phenomena, we construct a density field representation that removes fast fluctuations and provides a consistent characterization of the domain growth. Our results indicate that the coarsening kinetics presents a scaling behaviour that is independent of the solvent quality parameter, in analogy to time-temperature superposition principle. Interestingly, the domain growth in time follows a power-law behaviour that is approximately independent of the architecture for all the flexible systems; while it is steeper for the semiflexible chains. Nevertheless, the fractal nature of the dense regions emerging during the collapse exhibits the same scaling behaviour for all the macromolecules. This, suggests that the faster growing length scale in the semiflexible chains originates just from a faster mass diffusion along the chain contour, induced by the local stiffness. The decay of the dynamic correlations displays scaling behavior with the growing length scale of the system, which is a characteristic signature in coarsening phenomena.

Recent computational works have proposed analogies between the collapse of linear polymer chains in bad solvent [3,15,16] and phase-ordering phenomena or coarsening systems (as foams [17] or polymer blends [18]), where coarsening refers to any out-of-equilibrium relaxation process involving the growth of two separated phases from an initially homogeneous mixture. In a coarsening system the characteristic length scale grows over time and exhibits key universal features like dynamic scaling and self-similarity [19]. In the context of macromolecular collapse in bad solvent, the coarsening process initiates with the formation of small clusters of monomers along the polymeric strands. Afterwards, these clusters become stable and start to grow by withdrawing monomers from the bridges connecting them or by coalescence with other clusters. This moves forward until all the monomers pile up to a single cluster. Finally, the single cluster collapses to the ultimate equilibrium state, characterized by a compact object of melt-like density [3]. The characteristic length scale of the coarsening process, associated to the collapse from the initial self-avoiding conformations of the polymer strands to the late fully-collapsed state, is given by the size of the growing aggregates of monomers. Some theoretical approaches have been developed for describing the different scaling regimes associated to the coarsening kinetics. In binary liquids, where the chemical potential gradient acts as the driving force [20], the domains grow with time as ∼ t 1/3 . For fluids and polymers, where hydrodynamic contributions are relevant, linear growth is predicted at late times in the viscous regime [21] . An apparent sublinear regime for the domain growth, ∼ t 1/2 , has been observed at intermediate times in simulations of a gas-liquid separating system [22], suggesting an effective interpolation between the former early and late regimes. In a previous work by some of us [23] a scaling exponent of ∼ 0.7 has been found at intermediate times for the growth of the domains during the deswelling of microgels in bad solvent. This accelaration (higher exponent) with respect to the system of Ref. [22] is tentatively related to the polymer connectivity of the microgels, which facilitates the merging of close clusters.
In this contribution we investigate whether the scaling behaviour found for the microgels of Ref. [23] is a general result for other macromolecules or if it depends significantly on the architecture of the system. In order to quantify the growing length scale we introduce a density field representation of the macromolecules that removes artifacts arising from the local, fast density fluctuations in the coarsening structure [22]. We establish a set of scaling laws for the time dependence of the growing domain size during the coarsening, which are independent of the solvent quality parameter in analogy to time-temperature superposition principle. Domain growth in mi-crogels shows a power law, though an overshoot is found in the late stage of the collapse for the case of diamond-like networks. This unusual behavior is related to the fast late merging of the regularly distributed nucleating centers. Power-laws are also observed for the coarsening dynamics of collapsing flexible linear chains and star polymers, though with slightly smaller exponents imputable to a lower number of nucleation centers in the absence of a network structure. Semiflexible chains present a significantly higher exponent for the domain growth. We also analyze the fractal structure of the clusters of different sizes that are formed during the coarsening process, and find no differences between the flexible and semiflexible systems, concluding that the higher exponent for the growing length scale originates from a faster mass diffusion in the semiflexible chains induced by the stiffness. Finally we characterize the dynamic density correlations during the collapse and relate them with the growing length scale. A common power-law is found for this relation, irrespective of the solvent quality and macromolecular architecture, which is a typical signature of the critical nature of the process.
The paper is organised as follows. In Section 2 the model and the simulation details are presented. In Section 3 a deep analysis of the kinetics of the collapse for all the investigated systems is reported. In particular, the density field construction is introduced, and data for the domain growth and the scaling of the dynamic correlations are discussed. Section 4 summarizes our conclusions.

II. MODEL AND SIMULATION DETAILS
We performed NVT molecular dynamics simulations of star polymers with 3 and 12 arms, semiflexible and flexible linear chains, and migrogels with a disordered and a diamond-like network structure. We used the bead-spring model of Kremer and Grest [24] for the interactions. All the non-bonded interactions were implemented by a, purely repulsive, Weeks-Chandler-Andersen (WCA) potential [25] that was modified in order to tune the quality of the solvent. This was achieved by introducing an attractive tail, regulated by a solvent quality parameter φ that sets the solvophobicity of the monomers [23,[26][27][28]. Therefore this parameter plays the role of an effective temperature in the model. The non-bonded interactions were given by: The values for the parameters α = π(2.25 − 2 1/3 ) −1 and β = 2π − 2.25α are chosen in order to satisfy the condition that the non-bonded potential and its first derivative are continuous both at r = 2 1/6 σ and at the cutoff r c = 1.5σ [26]. For the case φ = 0 the purely repulsive WCA potential is recovered and the system is in good solvent conditions. The quality of the solvent is worsened by choosing φ > 0. Bad solvent conditions are reached when φ is higher than some critical value and the system collapses. In all the studied systems the collapse transition (θ-point) occurred at φ ∼ 0.6 (this was estimated from the maximum of the derivative of the radius o gyration vs. φ).
In addition, bonded monomers interact via a finite extensible nonlinear elastic (FENE) potential, which implements the molecular connectivity. The FENE potential reads [24]: where K F = 30 is the spring constant and R 0 = 1.5 is the maximum elongation. The sum of the FENE and the non-bonded potential provides a total interaction between two connected monomers showing a deep minimum at r ∼ 0.95, which guarantees uncrossability and prevents violation of the topological constraints. In the case of the semiflexible linear chains the bending stiffness was implemented through the worm-like model [29,30]. Thus, the interaction for the polymer bending has the form: where θ is the angle between two successive bonds and K s is the strength of the bending. We used a value K s = 5. The units of energy, length, mass and time are respectively , σ, m and τ = (σ 2 m/ ) 1/2 where m is the mass of a monomer. In the rest of the paper all the numerical values will be given in reduced units = σ = m = τ = 1.
We performed MD simulations at temperature T = /k B = 1.0 (with k B the Boltzmann constant) using a Langevin thermostat [31,32]. Thus, the force experienced by the monomers is: where r i is the position vector and V (r i ) is the total interaction potential for the monomer of index i.
The second term of the right side of Equation (4) represents viscous damping, with γ the friction coefficient. The last term is a random uncorrelated force, ζ α i (t)ζ β j (t ) = δ i,j δ α,β δ(t − t ) (with α, β the Cartesian components), representing the collisions with solvent particles. The Langevin thermostat acts therefore as an implicit solvent, in which every particle interacts independently with the solvent, but hydrodynamic interactions between solute particles are not considered. Their inclusion would require the use of e.g., Lattice Boltzmann [33], multi-particle collision dynamics (MPCD) [34] or dissipative particle dynamics [35] methods, involving a huge computational cost due to the size of the investigated systems and the big boxes needed to avoid finite size effects. For example, in the widely used MPCD method, radii of gyration R g ∼ 50 would require using boxes of side L box ∼ 200 and at least 5L 3 box = 4 × 10 7 solvent particles for both correctly implementing the hydrodynamic interactions and avoiding significant finite-size effects [36]. Still, based on previous evidence [37][38][39][40] we do not expect that hydrodynamics will lead to qualitatively different results from those presented here. We used a time step δt = 0.005, and a friction γ = 0.05, which is high enough for good thermalization and low enough to prevent strong damping that would slow down the dynamics to time scales requiring a huge computational cost.
After generating them, equilibration of all the investigated systems (flexible and semiflexible chains, stars, disordered and diamond microgels) was performed in the limit of good solvent (φ = 0).
To investigate the coarsening kinetics, the systems equilibrated at φ = 0 were quenched at infinite rate to different values of the solvent parameter φ well below the θ-point, and coarsening was analyzed from the quenching instant (t = 0) until the macromolecule reached its equilibrium fully collapsed state.
The linear chains were constructed as simple strings of beads and springs (adding the bending interaction in the case of the semiflexible chains). We used N = 1600 and 400 beads for the flexible and semiflexible chain, respectively. The star polymers were constructed by linking flexible linear arms of 800 beads to a central bead. Several protocols have been recently proposed to generate realistic models of microgels [23,28,[41][42][43][44] beyond the regular networks usually employed in the literature. In our method we take inspiration from the synthesis of microgels in microfluidic cavities. This experimental route takes place via confinement in a droplet of pre-existing linear polymer chains, and further inter-and intra-molecular irreversible association [45]. In our model each confined single chain consists of N beads, and in each chain N r of these beads are reactive (cross-linkable) groups, randomly distributed along the polymer backbone and with at least one inert group between consecutive reactive groups to prevent trivial cross-links. In order to implement the cavity, a confining spherical, purely repulsive, LJ potential is applied to each monomer: where r w represents the shortest distance from the monomer to the spherical wall.
We performed the cross-linking of N cha = 36 chains in a cavity of radius R cav = 55. Each chain had N = 600 beads, so that the finally generated microgel had 21600 beads. The number of reactive beads in each chain was N r = 12, i.e., the fraction of cross-linked monomers in the microgel was f = N r /N = 0.02. The number density used in the synthesis was 3N cha N/(4πR 3 cav ) ≈ 0.03, which qualitatively corresponds to experimental concentrations of about 30 mg/mL [24,46,47]. After equilibration of the chains inside the cavity the cross-linking of the reactive groups was activated.
A permanent bond (modeled by the FENE potential) between two reactive groups was formed if: (i) none of them was already bonded to another reactive group, and (ii) they were at a mutual capture distance r < 1.3σ. A random choice was made in case of multiple candidates within the capture distance. To speed up the late stage of the cross-linking process (≤ 6 remaining unbonded reactive groups), a random pair was chosen from the unreacted groups and an attractive harmonic interaction between the constituents of the pair was implemented, in order to approach them to the capture distance and form the bond. After forming the bond the microgel was equilibrated and the procedure was repeated until full completion of the cross-linking. Then the cavity was removed to allow for swelling and equilibration of the obtained microgel. Cross-linking of 50 initial realizations, with the same former values of N , N cha , N r and R cav was perfomed, leading to microgels with the same number of monomers and cross-links but topologically polydisperse [23]. It was found that a large number of the cross-links (about 65 %) occurred between reactive groups belonging to the same polymer chain, forming loops. These kind of cross-links do not contribute to the connectivity of the network and are elastically inactive. On the other hand in the diamond-like microgels no loops are present and all the cross-links are elastically active. Therefore, for a fair comparison with the disordered ones, the diamond-like microgels were constructed with the same fraction of cross-links (nodes) as the average number of intermolecular bonds in the disordered microgels, tuning the number of nodes so that the total mass of both kinds of microgels was essentially the same [23]. Thus, we simulated a diamond-like microgel of N = 21615 beads containing 78 nodes.
The diamond-microgel was generated by placing the cross-links in the nodes of a regular diamond network, and by connecting every pair of nearest-neighbour nodes through linear bead-spring rods [48][49][50][51][52]. All the beads out of a sphere containing the selected N beads were removed.
We computed the time-averaged asphericity a [53] of each disordered microgel in the swollen state (φ = 0) and obtained the corresponding distribution P (a). For the analysis of the coarsening kinetics we selected three disordered microgels, at the center and at the two extremes of the distribution of asphericities. The corresponding radius of gyration, at φ = 0, of the selected disordered microgels is R g = 49.3, 52.4 and 64.0 for a = 0.02, 0.06 and 0.14, respectively. In the rest of the paper these disordered microgels will be denoted in the figure legends as I-II-III from lower to higher asphericity. The size of the other investigated systems at φ = 0 is R g = 51.8 (diamond-like microgel), 45.3 (3-arm star), 48.9 (12-arm star), 39.9 (flexible chain) and 26.4 (semiflexible chain).

III. COARSENING KINETICS
The coarsening kinetics is characterized by a growing length scale. The quantitative characterization of such a length scale can be easily affected by artifacts originating from the 'structural noise' emerging during the macromolecular collapse (bridges that connect the clusters of monomers, small halls, protrusions, etc). To avoid these artifacts we introduce a smooth representation of the macromolecules through a coarse-grained density field. This method is based, originally, on the characterization of the growing length scales in a coarsening binary Ising system [54] and was later applied in the continuous space to a liquid-gas phase separating system [22,55]. In the first case the thermal noise effects on the coarsened structure were removed by using a majority spin rule, i.e. by replacing each spin by the majority spin of its nearest neighbours. In the second case, the real particles were substituted by their local densities averaged over their nearest environment.
These averaging procedures smooth the interface corrugations, fill the smallest holes, and delete the smallest clusters in the coarsening structure. In this way the smooth density field avoids, in the calculation of the growing length scale, the effect of non-relevant minimal paths or artificial interruptions of long paths within the dense domains. In our systems we construct our coarse-grained density field as follows: i) we divide the space into cubic cells of side δ; ii) we define the local density (for each cell) by the number of monomers in a sphere: where r c is the cutoff radius of the sphere and n(r) is the number of monomers at a distance d ≤ r c from the position r of the cell center; iii) we fix the value of the coarse-grained density at r as a weighted average of the local density over the surrounding cells:ρ where the sum is performed over the directions k ∈ {(±1, 0, 0), (0, ±1, 0), (0, 0, ±1)}. The grid size δ and the cutoff radius r c are chosen in order to get a smooth density field while keeping sufficient spatial resolution in the representation of the real system. This is achieved by using values δ ∼ 0.5 and r c ∼ 1. In what follows we will present results for δ = 0.5 and for two specific values r c = 1.0 and r c = 1.2. By using a threshold value ρ min in the density field construction the macromolecules can be seen as a coarsening biphasic system composed by 'empty' and 'filled' domains. These domains are identified according to their low (ρ(r) ≤ ρ min ) or high (ρ(r) > ρ min ) local density, respectively. The construction of the coarse-grained density field allows to measure the distribution of the domain size from the obtained smooth biphasic structure. We define an 'interfacial cell' [23] in our system as a filled cell with at least one adjacent empty cell. In order to compute the characteristic The chord lengths are calculated for a density field with r c = 1.0 and ρ min = 0.6. the filled cells. To improve statistics 5 random rotations of the former configuration were taken, and the whole procedure was repeated over 5 independent realizations of the same macromolecule. Figure 2 shows the normalized distributions of chord lengths, P (L), at φ = 1.2 and different times during the collapse of the disordered microgel of middle asphericity (Figure 2a), the semiflexible chain (2b), 12-arm star polymer (2c) and 3-arm star polymer (2d). P (L) shows the characteristic length distribution observed in coarsening systems [3,22,54,[56][57][58]. In particular, for earlier times P (L) shows an exponential decay and extends over longer distances as time increases, which is a consequence of the growth of the filled domains during the coarsening. As expected, the exponential behavior saturates at long times when the fully collapsed state is reached.
The flat plateau originates from the equiprobable different straight paths that connect two points, at both sides of the outer interface, in the fully collapsed state (where empty cells are absent). The drop from the plateau obviously reflects the finite size of the collapsed object . Interestingly, the P (L) of the semiflexible chain shows an approximate self-similar behavior over time, i.e., the decay is shifted to longer times but shows roughly the same shape, in contrast with the other systems for which the slope of the exponential is strongly time-dependent. This is consistent with the growth and transport process of the nucleation centers anticipated in Figure 1c. In fact, as shown there, the clusters are better defined and further apart, and they grow considerably before coalescing in a single cluster. Instead, in the flexible systems (panels (a,b) of Figure 1) the clusters grow and merge in a much more gradual way until the single globule is formed at late times.
We use the information on the former distributions to quantify the growing length scale characteristic of the coarsening process. The mean domain length for a given time t is obtained from the first momentum of the distribution P at time t , i.e., L(t) = L P (L ; t)dL . We find the same qualitative behavior of P (L) for several choices of r c and ρ min , whereas quantitatively L(t) depends on the specific parameters used to construct the density field. Indeed, choosing, e.g., a lower value for ρ min implies having a higher number of filled cells in the density field, and consequently a larger value of L. For this reason we quantify the domain growth rate in terms of the relative domain size, defined as: In this way C(t) represents a normalized mean chord length growing from zero at t = 0 to 1 at late times in the collapsed state. Figure 3 shows the domain growth for three different topologies of the disordered microgels (corresponding to a low, middle, and large value in the distribution of the asphericity parameter), in comparison with the diamond-like network. For each system the panel includes data of C(t) for several selections of the parameters r c and ρ min . The absolute times have been rescaled by the time τ 0.5 , defined as C(τ 0.5 ) = 0.5. As can be seen for all systems, after this time rescaling the different data sets of C(t) nicely overlap. This demonstrates that the density field approach is consistent, as it provides a time dependence of C(t) that is independent of any reasonable choice of the parameters defining the density field. Furthermore a good overlap is observed also for different values of the solvent quality parameter (φ = 1.2 and φ = 1.5), indicating that the coarsening kinetics follows an effective time-temperature superposition principle.
As can be seen in Figure 3, the function C(t) reveals that the coarsening length scale grows by following a sublinear power law ∼ t x . The exponents are x 0.6 for all the microgels, irrespective of the microstructural degree of disorder. The exponents obtained here for f = 0.02 are slightly smaller than those found in our previous work [23] (x ∼ 0.7) for a much higher degree of crosslinking (f = 0.1). We believe that the slightly higher value of the exponent for f = 0.1 is just an artifact originating from the proximity of the cross-links in such a system. Namely, since the Our previous work in microgels with higher f suggested that the diamond network exhibited the same power-law in C(t) that the disordered networks. However, for the diamond network this behaviour was developed only in a narrow time window and the results were not conclusive [23]. The results in Figure 3d for f = 0.02 clearly confirm the power law regime over 3 time decades. These data also confirm the acceleration of the domain growth in the late stage of  the collapse of the diamond network [23], which is at most a marginal effect in the disordered networks, where the power-law continues and is a good approximation until the collapsed state is reached and the ultimate plateau emerges. The strong late acceleration in the diamond network is tentatively related to the regular spatial distribution of the cross-links acting as preferential nucleating centers, which leads to a more homogeneous collapse until all holes in the structure vanish in a short time window (compare snapshots in panels (b) and (d)), and the filled paths in the density field experience a late sudden growth.
The results presented above confirm the common scaling law C(t) ∼ t 0.6 for the domain growth in collapsing microgels, irrespective of their topology (disordered or regular networks). In order to search for a more general scenario, we investigate the scaling properties of the coarsening kinetics in other very different architectures, as stars with different number of arms, and flexible and semiflexible linear chains. We follow the same procedure as for the microgels, constructing the density field and analyzing the domain growth in this representation of the macromolecule.  shows the corresponding results of C(t) for the stars and linear chains. We find that the systems without bending stiffness (the stars and the flexible linear chains) essentially show the same powerlaw C(t) ∼ t 0.56 , with an exponent that is just slightly smaller than those found for the microgels.  N clus (0))/(N clus (∞) − N clus (0)) and r(t) = (R clus (t) − R clus (0))/(R clus (∞) − R clus (0)), respectively, so that both n(t) and r(t) grow from 0 to 1. The arrows in all panels of Figures 6 and 7 indicate the time scale for which N clus = N * . As shown in Figure 5, the clusters in all systems have the same fractal behaviour. However, their evolution in time (Figures 6 and 7) can strongly depend on the system, namely there are clear differences between the flexible and semiflexible systems. In other words, coarsening leads to mass aggregation into the same kind of clusters for all the systems, but the rate of mass aggregation can be significantly affected by bending stiffness. For the time scales where the mean population of the clusters is smaller than N * (clusters are still relatively small and approximately Gaussian), we find a power-law n(t) ∼ t β , with exponent β ≈ 1.2 for the flexible systems, and a much higher value β = 1.6 for the semiflexible chains. Not surprisingly, the This indeed reflects the scaling R clus ∼ N 0.5 clus found in Figure 5. In the case of the flexible and semiflexible chains, and in the stars (not shown), no apparent or significant change is observed in the power-law behavior of n(t) and r(t) at later times when larger clusters are formed, until the former quantities finally saturate when all clusters have merged into a single one and the systems approach the equilibrium fully collapsed state. An overshoot in n(t) and r(t) before saturation is however found in the microgels. This effect is specially pronounced in the diamond network, and not surprisingly occurs roughly in the same time window as for the normalized domain length C(t) (Figure 3d). Since r ∼ n ν with ν ≤ 0.5, the overshoot is much less pronounced in the growing size of the clusters than in the mass aggregation.
In summary, the analysis of clusters reveals that there are no significant differences in the conformations of the dense regions that are formed during the coarsening of the different macromolecular architectures, but on the rate at which they are formed, which is essentially the same in flexible systems but can be accelerated through bending stiffness in semiflexible systems.

C. Dynamic correlations
A well-known feature of phase separating and coarsening systems is the scaling of the dynamic correlations with the growing length scale. To test this possibility we first define a 'spin' selfcorrelation function P s (t) for the density-field representation. The function is defined as [59]: where the variable S is computed for each cell of the density field, and it is assigned a value 0 or 1 if the cell is empty (ρ ≤ ρ min ) or filled (ρ > ρ min ), respectively.
In Figure 8 we show the correlation function P s (t) for the three selected disordered microgels (low, middle and high asphericity) and for the diamond-like microgel, at solvent parameter Φ = 1.2.
Data are shown for two different couples of the parameters (r c , ρ min ) used to define the density field, and the times are rescaled by τ 0.5 as defined above. Again, the good overlap of the different data sets confirms the consistency of the density field representation for characterizing the coarsening.
The underlying complex dynamics associated to the coarsening process is reflected by the nonexponential decay of the dynamic correlations, which indeed follows a power-law, P s (t) ∼ t −y .
Results for all the microgels are consistent with an exponent y 0.5, irrespective of the network topology. This is analogous to our results for the domain growth (C(t), Figure 3) in the microgels, for which the specific network topology has no significant effect on the observed power-law.
Moreover, from the computation of P s (t) and the average domain size L(t), we can establish a direct relation between the dynamic self-correlations and the growing characteristic length scale, just by taking at each time t the corresponding values of P s and L. Figure 9 shows the results for this relation in the diamond-like microgel and in the three selected disordered microgels (panel (a)) and in the 3-arm and 12-arm stars (panel (b)). The data are computed for several values of the solvent parameter φ and the parameters (r c , ρ min ) used to define the density field. Timetemperature superposition is also confirmed for this scaling relation, and data for all flexible systems are consistent with a common power law P s ∼ L −1.1 . Unfortunately we could not confirm this observation for the semiflexible case, for which the power-law regime in P s (L) developed over a short range -note that the size of the investigated semiflexible chains is much smaller than for the flexible systems -and had very poor statistics. Confirmation would require to simulate much longer semiflexible chains, which would highly complicate the analysis. Tests for much longer chains showed that they tend to collapse into rods and in an extremely heterogeneous fashion (visiting several long-living intermediate conformations prior to the equilibrium state).

IV. CONCLUSIONS
By means of simulations, we have investigated the coarsening kinetics emerging during the collapse of several macromolecular architectures in bad solvent: microgels with realistic (disordered) and with ideal regular (diamond) networks, star polymers of 3 and 12 arms, and linear chains. We have also investigated the effect of bending stiffness on the coarsening kinetics by simulating the collapse of semiflexible linear chains. In order to remove fast fluctuations that can lead to artifacts in the characterization of the growing length scale, we have made use of a smooth density field representation of the macromolecules. The domain growth during the coarsening follows a powerlaw behaviour that is independent of any reasonable selection of the parameters used to construct the density field (grid size and threshold for defining dense regions). The scaling behaviour is independent of the solvent quality parameter, in analogy to time-temperature superposition. All and not from a distinct structural feature of the aggregates formed during the coarsening process.
We have analyzed dynamic correlations, and in analogy with critical phenomena, investigated their dependence on the growing length scale L. We find an apparent common power-law dependence of the correlations (∼ L −1.1 ) for all the flexible systems.
This work reports, to the best of our knowledge, the first comparison of the coarsening kinetics in bad solvent for a broad range of macromolecular architectures, and explores the role of chain stiffness. As such, it provides a general physical scenario, and a valuable set of results for future theoretical developments in this, still scarcely studied, fundamental problem with potential applications to e.g., protein folding.