**Web-based App to Implement Theories**

**Here we provide the theoretical tools we developed to help you calculate the topology and elasticity of your own gels.**

**Modified Rate Theory for Polymer Network Topology**

To describe both the topology of polymer networks and the kinetics of network formation, we develop a modified rate theory based on the work of Stepto, which uses a set of finite number of subgraphs to represent the unmanageably large network. The synthesis of the subgraphs is tracked through rate equations. Subgraphs are restricted to a critical size; cyclic topologies formed within the critical size are recorded. Beyond the critical size, junctions are assumed to be uncorrelated, which can randomly reacted with each other.

**Monte Carlo Simulation for Counting Loops**

Monte Carlo simulations were performed following the general algorithm developed by Stepto. According to the desired concentration, A and B groups are initially put in the system. Networks are constructed by sequentially selecting a pair of unreacted A-B groups at each step with a probability reflecting the connectivity between these two groups. The topology of the entire network is updated after the selected A-B pair reacts: the distance between all groups that are connected to these reacted A and B groups are recalculated. The process repeated until the full conversion of reactive groups is reached. Fractions of different orders of loops are then recorded. For example, primary loop is counted if two A groups belonging to the same polymer chain connect to the same junction (two adjacent B groups).

**Modified Phantom Network Theory for Gel Elasticity**

A phantom network containing loops of different orders are considered. Different cyclic defects are assumed to be independent in the contribution of the gel modulus. The effective chain length of the polymer strands in the loop is calculated by using the property of Gaussian chain.