Fig 1.
Lattice model for phase separation by polymers with one-to-one interacting motifs.
(a) Each polymer is defined by its sequence of motifs, which come in types “A” (red) and “B” (blue). The class of sequences shown consists of repeated blocks of As and Bs, labeled by their block size ℓ. (b) In lattice simulations, an A and a B motif on the same lattice site form a specific, saturating bond (green) with binding energy ϵ. Monomers of any type on adjacent lattice sites have an attractive nonspecific interaction energy J = 0.05ϵ. A-A and B-B overlaps are forbidden. (c) Polymer number distribution P(N) at the phase boundary of the ℓ = 3 sequence (βϵ = 0.9287, μ = −9.9225ϵ). At fixed μ the system fluctuates between two phases. Inset: Snapshots of the GCE (fixed μ) simulation at ϕdilute and ϕdense.
Fig 2.
The sequence of binding motifs strongly affects a polymer’s ability to phase separate.
(a) Binodal curves defining the two-phase region for the six sequences of length L = 24 shown in Fig 1a. Stars indicate the critical points and the solid curves are fits to scaling relations for the 3D Ising universality class. Mean ± SD for three replicates. (Uncertainties are too small to see for most points.) Color key applies to all panels. (b) When rescaled by the critical temperature Tc and critical density ϕc, the phase boundaries in (a) collapse, even far from the critical point. (c) The tendency to phase separate is inversely related to the density of states g(s), i.e. the number of ways a given sequence can form s bonds with itself. Inset: Snapshots of ℓ = 3 polymer with s = 5 (top) and s = 10 (bottom). Black lines show the polymer backbone. (d) Phase boundaries from mean-field theory using g(s) (Eq 1).
Fig 3.
Ability to phase separate is determined by the sequence of binding motifs for polymers of different lengths, patterns, and motif stoichiometries.
(a) Tc and ϕc for L = 24 polymers with scrambled sequences and block sequences of various lengths. Mean ± SD over three replicates. (Temperature uncertainties are too small to see in (a) and (c).) (b) Tc as a function of motif stoichiometry a/L. The solid curve corresponds to ℓ = 3 sequences where a number of B motifs are randomly mutated to A motifs, and the dashed curve shows scrambled sequences. Mean ± SD over four different sequences. (c) Tc from Monte Carlo simulations versus mean-field theory (blue) and condensation parameter (orange) for block sequences, scrambled sequences, and sequences with unequal motif stoichiometry, all L = 24. Mean ± SD over three replicates for simulation Tc. (d) Distribution of Tc values for 20, 000 random sequences of length L = 24 with a = b, calculated from Ψ values and the linear Tc versus Ψ relation for block sequences. Block sequence Tc values are marked.
Fig 4.
The structure of the dense phase depends on the motif sequence.
(a) Number of self-bonds s in the dense phase as a function of reduced temperature for block sequences (symbols as in (c)). Each point shows s (mean ± SD) over all configurations with |ϕ − ϕdense| ≤ 0.01. Color bar: droplet density. (b) Number of trans-bonds t (bonds with other polymers) versus temperature as in (a). (c) “Viscosity” (Eq 5) of the dense phase, shown as in (a). Symbol key applies to all panels. (d) Radius of gyration Rg of polymers in the dense phase (shown as in (a)) and in the dilute phase. Dilute-phase points show Rg (mean ± SD) over all configurations with |ϕ − ϕdilute| ≤ 0.01. They share reduced temperatures with the dense phase points but are shifted for clarity. Color bar: dilute phase density.
Fig 5.
The polymer moves used to update Monte Carlo simulations at each step.
We also allow translation of connected clusters of polymers and insertion/deletion of polymers. (a) End move. (b) Corner move. (c) Reptation. (d) Contraction. (e) Expansion.
Fig 6.
Multicanonical sampling makes it possible to determine the phase boundary at temperatures substantially below Tc.
(a) The polymer number distribution produced in a multicanonical simulation with
. Block sequence with ℓ = 2, βϵ ≈ 0.94, J = 0.05ϵ. (b) The true distribution P(N), obtained by reweighting
from (a) to remove h(N). (c) The distribution at the phase boundary, obtained by reweighting (b) to the chemical potential μ* at which both peaks have equal weight.