Entropy-driven phase behavior of associative polymer networks

The manuscript addresses a very general bonding mechanism leading to transient polymer networks that occurs in diverse polymeric materials through a minimal modelling allowing the detection of the essential physical ingredients involved.

Within this framework, polymers capable of forming transient networks are flexible Kremer-Grest chains decorated with monomers able to form reversible bonds.
A previous paper from the same authors found that if monomers are of two different species arranged alternately on the chain, an entropy-driven first order phase transition appears between a regime dominated by intra-molecular towards one in which inter-molecular bonds are prevalent.
This finding is at odd with the case in which attractive monomers are all of the same species and the network formation takes place continuously.

The current paper reinforces and extends the findings of Ref.21 showing that previous results are robust against thermal fluctuations, against disorder, and changing the model topology, i.e. if rings are considered instead of chains. The paper is interesting for a large public , unraveling a fundamental mechanics, and the authors' conclusions are supported from the solid, rigorous and nicely organized results.
I therefore recommend its publication in SciPost Physics after addressing some minor comments I list hereafter.

-The authors employ chains of length $N_m=243$, differently from Ref.21 where they used $N_m=254$. This last accounts for 23 trunks made of 10 inert monomers and 24 active monomers. To me, it is not clear the length of the inert truck with the current choice.

-How does the phenomenology of Fig.1 compares quantitatively with the previous paper? At which $\beta\epsilon$ was previously investigated the system? A comment on this would help the comparison

-Please add legends (a,b) on figs.1,3. I believe legend of fig.3 incorrectly refers to the $S=1$ case for the inset

-As follows from the first point, how many inert monomers separate attractive monomers in the rings? Is this a constant number?

The manuscript contributes to broadening our understanding of the phase behavior of an important class of materials.

The analysis of the results could be improved

The manuscript studies the phase behavior of polymers carrying reactive monomers that dimerize forming intra- and intermolecular, reversible bonds. Reactive chains are currently used to study multiple soft and biological systems including chromatin organization, phase separation of disordered proteins, or folding of single-chain polymeric nanoparticles.

The current submission is a follow-up of a previous contribution by the same group of authors (Phys. Rev. Lett. 2022 or Ref. 21). While increasing the chemical potential, systems of self-avoiding chains functionalized by a single type of (self-complementary) reactive monomers continuously transit from a diluted to a percolated phase without a discontinuity in density. In Ref. 21, the authors designed a system featuring a phase transition by decorating chains with two sets of self-complementary reactive monomers in which different types of monomers do not interact. Due to configurational terms, such a design penalizes intramolecular bonds in favor of intermolecular contacts. The current submission assesses the robustness of this finding by studying how changes in the original design impact the gas-liquid transition. In particular, chains with different topologies (ring vs linear) and organizations of the reactive monomers are considered.

The manuscript is certainly interesting in the way it contributes to broadening our understanding of the phase behavior of an important class of materials. The results are sound but it seems that their analysis could be improved as discussed below.

-The authors correlate the tendency of the system to phase separate with the fraction of bonds, pb. It seems that studying the fraction of intermolecular bonds would have been more insightful given that, because of intra-molecular bonds, pb seems to be maximized in the monofunctional system (which does not phase separate).

-It would be insightful to discuss how the critical point is affected by the number of reactive monomers and the chain’s length.

-It is not obvious to me that for disordered chains (low values of S) the configurational cost of forming loops is smaller; some loops will be smaller but others longer. Disordered chains may also find it harder to maximize the number of loops resulting in behaving as in the ‘defective’ arrangement of Fig. 3a.

-The ring topology has a major impact on the equation of state with the pressure of the A24 system at high density twice as big as the one in the linear topology. Beyond excluded volume considerations, I wonder if the ring topology also increases the propensity to form intramolecular bonds (e.g. with the appearance of zipped configurations).

-Following on from the previous point, the analysis of Fig. 3 and 4 could be improved by the study of the corresponding pb (ideally, disentangling intramolecular from intermolecular bonds). Similarly, the loop length distributions may help corroborate some of the claims made in the manuscript.

-I am wondering if the morphological and structural properties of the percolated phase (in particular the valency) are the same in all cases irrespective of the presence or not of the gas-liquid transition.

Minor

-Specify the protocol used to distribute the reactive monomers over the chain in the non-ordered system (multiple reactive monomers are allowed).