Molecular simulations of phase separation in elastic polymer networks

TL;DR

This study uses coarse-grained molecular dynamics to reveal how elastic polymer networks arrest phase separation and set finite-domain sizes. By systematically varying chain contour length $L_c$, persistence length $L_p$, and network topology (regular vs. entangled), they identify two mechanisms for microphase formation: (i) local bending rigidity when $L_p$ becomes comparable to cross-link spacing $a$, and (ii) topological constraints that generate an emergent length scale $L_e$ from multi-chain entanglements. Domain sizes, quantified by the microphase length $\\lambda$, scale with $L_p/L_c$ in regular networks and with $L_e$ in entangled networks, while the bulk modulus $B$ alone is not predictive of phase behavior. These results bridge molecular architecture and continuum descriptions, offering design principles for synthetic gels and insights into condensate organization in cells, and point to extensions involving activity and network-wetting effects. Overall, the work clarifies that microscopic, architecture-dependent length scales—not bulk elasticity—govern arrested phase separation in elastic networks.

Abstract

Phase separation within polymer networks plays a central role in shaping the structure and mechanics of both synthetic materials and living cells, including the formation of biomolecular condensates within cytoskeletal networks. Previous experiments and theoretical studies indicate that network elasticity can regulate demixing and stabilize finite-sized domains, yet the microscopic origin of this size selection remains elusive. Here, we use coarse-grained molecular dynamics simulations with implicit solvent to investigate how network architecture controls phase separation and limits domain growth. By systematically varying chain contour length, chain rigidity, and network topology, we uncover that finite domains emerge when intrinsic chain- or network-level length scales, such as persistence length or entanglement length, impose local constraints on coarsening. Further, the size of these finite domains is highly correlated with these microscopic network properties, but depends surprisingly little on bulk elasticity. Taken together, our findings establish a molecular basis for understanding droplet formation in polymer networks, and provide guiding principles for engineering materials and interpreting condensate behavior in cells.

Figures (8)

• Figure 1: (a) Preparation of regular networks. (a-1) Conformation of an individual network strand, where blue and orange particles represent monomers and cross-linkers, respectively. Two-dimensional schematics of the regular networks under the condition of (a-2) $L_{\text{p}} \ll a \ll L_{\text{c}}$, and (a-3) $a \lesssim L_{\text{p}} < L_{\text{c}}$. (b) Preparation of entangled networks. (b-1) Molecular structures of a hexa-functional cross-linker and a linear chain. (b-2) Two-dimensional schematic of an entangled network after the cross-linking procedure. (c) Schematic of networks in different solvents: homogeneous phase in good solvent (left panel), macro- or microphase separation in poor solvent (right panel).
• Figure 2: (a) Representative snapshots of three different phase separation behaviors for (i) regular network with flexible chains ($\kappa/k_{\text{B}}T=0$), (ii) regular network with semi-flexible chains ($\kappa/k_{\text{B}}T=10$), and (iii) entangled network with flexible chains ($\kappa/k_{\text{B}}T=0$), under the same cross-linking fraction $\phi_{\text{c}}=0.1034$. (b) Average pore size $\left\langle d_{\text{pore}} \right\rangle$ as a function of system volume $V$ for regular networks with flexible (dotted lines and open symbols) and semi-flexible strands (solid lines and filled symbols). The light red line shows the results from simulations with weaker interaction strength $\varepsilon_{\text{LJ}}=k_{\text{B}}T/2$. (c) $\langle d_{\text{pore}} \rangle$ as a function of $V$ for regular (dotted lines and open symbols) and entangled networks (solid lines and filled symbols). The gray line indicates the scaling of $\langle d_{\text{pore}} \rangle \propto V^{1/3}$ observed for regular networks with flexible chains.
• Figure 3: (a) Average bending potential energy per triplet $\langle U_{\text{bend}} \rangle$ (Eq. \ref{['eq:UBEND']}) as a function of bending stiffness $\kappa$. Inset: Average monomer-monomer interaction energy per monomer $\langle U_{\text{WCA}} \rangle$ (Eq. \ref{['eq:UWCA']}). (b) Bulk modulus $B$ of networks as a function of relative persistence length $L_\text{p}/L_\text{c}$. All data shown for regular networks in a good solvent.
• Figure 4: Probability distribution of radius of gyrations $P(R_{\text{g}})$ of network strands for various bending stiffnesses $\kappa$, with chain lengths (a) $N=20$ and (b) $N=40$. Black dotted vertical lines indicate the computed $R_{\text{g}}$ values, corresponding to different chain conformations characterized by their Flory exponents $\nu$: a collapsed globule in a poor solvent ($\nu=1/3$), an ideal random coil in a theta solvent ($\nu=1/2$), a self-avoiding random walk in a good solvent ($\nu=3/5$), and a fully stretched rigid rod ($\nu=1$). The radius of gyration of a rod $R^{\text{rod}}_{\text{g}}$ can be calculated analytically as $R^{\text{rod}}_{\text{g}} = L_{\text{b}} \sqrt{(N^{2} - 1)/12}$. All data shown for regular networks in poor solvent.
• Figure 5: Relationships between the characteristic length scale $\lambda$ of microphase separation an relative chain rigidity $L_{\text{p}}/L_{\text{c}}$ in regular networks for various chain lengths $N$. Inset: $\lambda$ as a function of stiffness parameter $\kappa$. Error bars indicate the standard deviation of $\lambda$ within one system.