Entropy-driven phase behavior of associative polymer networks

Dear Editor,

Thank you for providing us with two constructive reports. We believe that taking into considerations the points raised by both Reviewers helped us to improve the clarity of the manuscript. The changes we carried out comprise adding a table and an inset to figure 2, as well as several new paragraphs of discussions.

We hope the attached reply and the changes reported therein will make the manuscript worth of publication in SciPost Physics.

Best regards,

Lorenzo Rovigatti and Francesco Sciortino

Changes made in response to Reviewer's 1 remarks:

- Added the following paragraph to discuss the new inset of figure 2:

"In order to gain more insight on the differences between the $A_{24}$ and $(AB)_{12}$ systems, we have also computed the number of inter-molecular bonds formed by each chain, $N_{\rm inter}$, for the same two systems of Figure~1. The inset of Figure~2(a) shows $N_{\rm inter}$ for the $A_{24}$ and $(AB)_{12}$ systems at $\rho\sigma^3 = 0.00049$ (the highest investigated density), as a function of the attraction strength $\beta epsilon$. At this high density no signs of phase separation have been detected in either systems. At the lowest value of $\beta \epsilon$ $N_{\rm inter}$ is comparable between the two systems, although in the $A_{24}$ more inter-molecular bonds are present in virtue of the higher bonding probability (see insets of Fig.~1). However, as the attraction reaches $\beta \epsilon = 10$, the number of inter-molecular bonds per chain in the $(AB)_{12}$ system, for which direct coexistence results demonstrate phase separation at lower density, experiences a steep increases that is not present in the $A_{24}$ system and that drives the transition. We also find that the $(AB)_{12}$ system with $\beta \epsilon = 10$ and $\rho\sigma^3 = 0.00025$, which, according to direct coexistence results, is near-critical and close to the density of the liquid phase, has a $N_{\rm inter} \approx 3$, making it a rather low-valence and sparsely-connected system."

- added the following sentence to the conclusions:

"As a further step forward, we plan to investigate how the size of the polymers and the number of reactive sites affect the material thermodynamics in future work."

- added the following paragraphs to discuss the new Table I:

"We start by considering the effect that the polymer architecture has on the average loop length, \textit{i.e.} the average chemical distance between two reactive monomers involved in an intra-molecular bond. Table I shows this quantity for ordered and (selected) disordered chains, as well as for rings, for the highest probed density ($\rho \sigma^3 = 0.00049$) and strongest attraction strength ($\beta \epsilon = 20$). As we will discuss and show below, under these conditions all the systems are homogeneous (no signs of phase separation) and essentially fully bonded, so that the average loop length can be considered a proxy for the entropic cost of forming an intra-molecular bond. Focussing on the chain systems (top four rows), it is clear that $(AB)$ chains tend to form larger loops compared to $A$ chains, which causes an increased polymer-polymer effective attraction that can drive phase separation [21]. However, in the presence of disordered arrangements of the reactive monomers the average loop length decreases. In the case of maximum disorder ($S = 1$), the average loop length of the $(AB)_{12}$ system is comparable to the value found for the ordered $A_{12}$ chains and, in fact, the consequential reduced cost of intra-molecular bonds is enough to suppress phase separation in the $(AB)_{12}$, $S = 1$ system (see below).

Table I also shows results for ring systems, for which going from one to two types of reactive monomers has the same qualitative effect observed for chains: the entropic cost of forming an intra-molecular bond goes up, and therefore the average loop length increases, causing a larger ring-ring effective attraction (see below for the consequences on the thermodynamics). However, the values in these case are larger than what observed in chains, owing to the smaller radius of gyration of (and therefore of the increased probability of intra-molecular contacts in) rings compared to chains of the same chemical size [32]."

- added the following sentence to clarify how the non-ordered arrangements are randomly generated:

"In the case of disordered arrangements, the position of the reactive monomers are generated as follows: (i) a non-reactive monomer $i$ is selected at random; (ii) if $i$ is separated by more than $S$ inert monomers from any other reactive monomer then $i$ is flagged as a reactive monomer; (iii) if the number of reactive monomers is equal to the target $N_r$ value the procedure is stopped, otherwise steps (i)-(iii) are repeated until such a condition is fulfilled."

Changes made in response to Reviewer's 2 remarks:

- added the following paragraph to the methods section:

"We study linear polymers (\textit{i.e.} chains) made of $N_m = 243$ monomers, with both ordered and disordered arrangements of the reactive monomers. The ordered chains start and end with $6$ inert monomers, and then the $N_r$ reactive monomers are placed equispatially, with $9$ inert monomers separating each pair of neighbouring reactive monomers. Note that we choose a slightly different system than the one in Ref.~21 to show that small differences in the number of inert monomers separating neighbouring reactive monomers (9 \textit{vs.} 10) and the presence of short inert segments at the beginning and at the end of the chain, absent in Ref.~21, do not alter the phenomenon we observe. In the case of disordered arrangements, the position of the reactive monomers are generated as follows: (i) a non-reactive monomer $i$ is selected at random; (ii) if $i$ is separated by more than $S$ inert monomers from any other reactive monomer then $i$ is flagged as a reactive monomer; (iii) if the number of reactive monomers is equal to the target $N_r$ value the procedure is stopped, otherwise steps (i)-(iii) are repeated until such a condition is fulfilled. By contrast, polymer rings are made by $N_m = 264$ monomers and $N_r = 24$ reactive monomers, with pairs of neighbouring reactive monomers being separated by $10$ inert monomers."

- added the following two sentences to compare results discussed here with those in Ref. 21:

"In the following we will set $\beta \epsilon = 20$ as in Ref.~21 to obtain systems that are essentially fully bonded, or smaller values to probe thermal effects."

"Under these conditions, for which we find results compatible with those of Ref.~21, the effective attraction [...]"