*3.1. Simulator*

The Monte Carlo suite ("simulator") consists of four different classes of algorithms: (1) Standard localized moves that entail the displacement of a single or a sequence of atoms, (2) chain-connectivity-altering moves (CCAMs), (3) cluster-based moves, and (4) identity exchange moves, all being executed at a constant volume. When shrinkage or NPT simulations are conducted, the regular volume fluctuation moves and/or the MRoB algorithm [75] undertake the task of changing the dimensions of the orthogonal simulation cell. This size alteration can be isotropic or anisotropic.

The local moves have been described exhaustively in numerous past publications. For single monomers, the simplest possible move is that of a displacement in a random direction and length within a preset amplitude [0, *ldisp*(*i*)], which again can be different for each dimension, *i*. With respect to chains, the corresponding set consists of: (i) Flip (internal libration), (ii) end-mer rotation, (iii) reptation, (iv) intermolecular reptation, and (v) end-segment re-arrangemen<sup>t</sup> (or CCB as in [76,77]; the reason we use a different notation here is to avoid confusion with the general scheme employed in all moves is explained next). All polymer-related moves can be executed in a configurational bias (CB) pattern (as seen in Figure 1 for the reptation move), with the number of trial configurations per attempted move, *ntrials*, being an input variable in the simulator. Due to the introduction of energetic bias in the forward transition, the reverse transition must be attempted *ntrials*-1 times to guarantee microscopic reversibility. In general, the number of attempts can be different for each local move, *ntrials*(*i*), where index *i* runs over all available polymer-based moves. This is because the individual MC moves are characterized by distinctly different acceptance rates, which are further heavily affected by simulation conditions, chain stiffness and especially by concentration (packing density). As intuitively expected, increasing the number of trial configurations leads to a significant increase in the computational time required per MC move. Setting *ntrials* = 1 enables the conventional execution of the local moves and eliminates the necessity to perform the reverse transition. The selection of *ntrials* is highly system dependent; for example, optimal values for hard-sphere chains in the bulk as a function of the volume fraction from dilute conditions up to the maximally random jammed (MRJ) state can be found in Table 1 of Ref. [59].

**Figure 1.** Schematic of the reptation move implemented through a configurational bias pattern with *ntrials* = 3. Different candidate positions could be picked by the selection of the bond length, bending, and torsion angles used for the re-construction of the monomer.

The set of chain-connectivity-altering moves consists of the simplified end-bridging (sEB), simplified intramolecular end-bridging (sIEB), and simplified double bridging (sDB) [59,75] moves. All constitute simplified versions of the original EB [74,78] and DB [79,80] algorithms, initially developed for the rapid equilibration of atomistically detailed polyethylene chains of high molecular weight. The main difference with respect to the original moves is that none of the simplified versions entails the displacement of atoms; rather they proceed by deleting and forming properly selected bonds in a pair (sEB, sDB) of chains or a single (sIEB) chain. The main advantage of the sDB algorithm is that it can be applied to strictly monodisperse systems and primarily to non-linear molecular architectures. Its main disadvantage is that it requires a bridgeable distance between two different pairs of atoms. For systems of very small bond gaps (*dl* → 0), this condition is very rarely met except very near the jammed state where the contact network is rich as a result of the isostaticity condition [65]. Additionally, all systems to be reported in the continuation are composed of linear chains. Furthermore, it has been found that dispersity in chain lengths has no effect on the universal static scaling laws [66,67] and phase behavior [71,72] of the simulated thermal and athermal polymer packings. Based on the above, sDB is excluded from the mix of moves for all cases studied here.

The third class of MC moves is that of cluster-based ones. The two variations, implemented in Simu-D, are cluster rotation (CluRot) and cluster displacement (CluDis) as first introduced in the home-made cluster code reported in [63]. The execution of the moves proceeds according to the schematic in Figure 2. In the first step, the cluster is identified. Group similarity for cluster detection is conducted first through a Euclidean distance criterion, independently of the identity of the constituent atoms (chain versus single monomers etc.). Further linkage criteria can include additional common elements such as the same crystal similarity (as detected for example by the CCE analysis, see below). Once the clusters are identified with the corresponding members labelled accordingly, one cluster is selected randomly. That cluster, as a whole (i.e., a single object made of the corresponding sites), can be displaced by a random amount in a random direction (CluDis) or be rotated randomly with respect to its center of mass (CluRot). The cluster-related moves can be optionally and automatically de-activated when a single cluster exists in the system.

The cluster detection is a computationally demanding step, so the CluDis and CluRot moves have low attempt probabilities, as also happens with the chain-connectivity-altering ones and the algorithms that alter cell dimensions.

The fifth and final set of moves consists of algorithms that change the identity of atoms and can be applied in the case of blends of monomers and polymers but also of polymers composed of different monomers. Figure 3 presents three such identity exchange (IdEx) moves, involving a single monomer and a single chain or a pair of chains. In the top panel of Figure 3, the execution of IdEx1 is shown once a single monomer (shown in red) is within a bridgeable distance to one of the ends of the chain molecule (shown in blue). The move proceeds by connecting, via a bond, the chain end and the single monomer so that the newly incorporated atom becomes the new chain end. In parallel, the last bond connecting the other end of the chain is deleted and the end is converted to a single monomer. By construction, the move does not entail atom displacement but rather the reconstruction of properly selected bonds. Accordingly, the change in energy entering the Metropolis criterion for acceptance or rejection of the move is due to the bonded term (variation of one bond length, one bending, and one torsion angle), along with any non-bonded change due to the swap of identities. The concept of IdEx2 (middle panel) is very similar. The single

monomer needs to be within a bridgeable distance from the second or penultimate atom of the chain. If the proximity condition is fulfilled, it becomes, through bond formation, the new chain end, and the corresponding chain end is converted into a single atom through bond deletion. Finally, IdEx3 (bottom panel) entails two chains. The difference with respect to the single-chain version is that the new single monomer is created by the deletion of a terminal bond of a randomly selected chain, different than the one that gains the monomer. Clearly, the implementation of IdEx3 requires dispersity in chain lengths.

**Figure 2.** Schematic representation of the cluster displacement (CluDis) and cluster rotation (CluRot) moves in a mixed system of chain (**blue**) and single (**red**) monomers. The initial step of cluster detection is performed based solely on proximity criterion. The identified cluster is shown by the contour line. The cluster, as a whole unified group of monomers, can then be displaced in a random direction and length (ClusDis) or be rotated by a random amount around its center of mass (ClusRot).

**Figure 3.** Schematic representation of (**top**) IdEx1 involving a chain and a single monomer within a bridgeable distance from a chain end, (**middle**) IdEx2 involving a chain and a single monomer within a bridgeable distance from the atom lying in the second of penultimate position in the chain, and (**bottom**) IdEx3, which includes a pair of chains and a single monomer. In IdEx3, the single monomer, lying within a bridgeable distance from an end of the blue chain becomes part of it, while a randomly selected chain (shown here in green) loses a randomly selected end, which becomes a single monomer. None of the moves depicted above include the displacement of atoms, rather only deletion and formation of bonds. IdEx3 requires dispersity in chain lengths in order to be applicable. In all cases, chain monomers are shown in blue (or green) and single monomers in red.

Based on the concept of identity exchange, as presented above, one can envision variations with monomers being incorporated into the inner segments of the polymer chains. However, such an approach would require the double fulfillment of the bridgeable distance and would therefore significantly reduce the pair of sites that could trigger such IdEx moves. For this reason, no further modifications have been incorporated in the present implementation of the simulator.
